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 Permanent Link:
 http://ufdc.ufl.edu/AA00029876/00001
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 Title:
 Phenomenological aspects of the standard model high energy QCD, renormalization, and a supersymmetric extension
 Creator:
 Mikaelian, Samuel
 Publication Date:
 1996
 Language:
 English
 Physical Description:
 vi, 111 leaves : ill. ; 29 cm.
Subjects
 Subjects / Keywords:
 Approximation ( jstor )
Electric fields ( jstor ) Energy ( jstor ) Mass ( jstor ) Physics ( jstor ) Quarks ( jstor ) Scalars ( jstor ) Scattering amplitude ( jstor ) Solitons ( jstor ) Trajectories ( jstor ) Dissertations, Academic  Physics  UF Nuclear physics ( lcsh ) Physics thesis, Ph. D
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 bibliography ( marcgt )
nonfiction ( marcgt )
Notes
 Thesis:
 Thesis (Ph. D.)University of Florida, 1996.
 Bibliography:
 Includes bibliographical references (leaves 109110).
 General Note:
 Typescript.
 General Note:
 Vita.
 Statement of Responsibility:
 by Samuel Mikaelian.
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 University of Florida
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 Copyright [name of dissertation author]. Permission granted to the University of Florida to digitize, archive and distribute this item for nonprofit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
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Full Text 
PHENOMENOLOGICAL ASPECTS OF
THE STANDARD MODEL: HIGH ENERGY
QCD, RENORMALIZATION, AND A SUPERSYMMETRIC EXTENSION
By
SAMUEL MIKAELIAN
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
1996
ACKNOWLEDGEMENTS
As in any process of learning one is often influenced by many more sources than one can enumerate, the list below, regarding my graduate work, can only be a partial one.
I am thankful to my adviser, Charles Thorn, for suggesting and guiding me through the main topic of this thesis, regarding pQCD Regge trajectories, and, in general, my learning experience in high energy theory. His standards of scholarship and scientific integrity have set ideals for me to strive toward. I also thank Pierre Ramond for suggesting the topic for the latter part of this work, renormalizationgroup study of the standard model. His teachings have also been a guiding influence in my education. The justmentioned part of this thesis would not have been possible without the collaboration of my fellow exstudents Haukur Arason, Diego Castano, Eric Piard and Brian Wright. Thanks also go to Jorge Rodriguez for introducing me to various software packages that facilitated my work. I thank my other committee members for their efforts and indulgence, especially Rick Field for current assistance with completing some aspects of this work.
I also acknowledge the patience and support of my parents (d.) at crucial times in the past.
ii
TABLE OF CONTENTS
Page
ACKNOWLEDGEMENTS ................... ii
ABSTRACT . . . . . . . . . . . . . v
CHAPTERS
1 INTRODUCTION ..................... 1
2 QCD REGGE TRAJECTORIES ............. 3
3 THE POTENTIAL ANALOGY .............. 24
3.1 The Trajectories . . . . . . . . .. 26
3.1.1 The Model Incorporating Asymptotic Freedom 26 3.1.2 The Model with Fixed Coupling ......... 35
3.2 The Scattering Amplitude: The FixedCoupling Case . 37
3.2.1 MandelstamSommerfeldWatson Transform and the Regge
Lim it . . . . . . . . . . 37
3.2.2 Analytical Estimates ......... . 42
3.2.3 Numerics . . . . . .. . . . 47
3.3 The Scattering Amplitude: The Asymptotically Free Case 52 3.4 Summation .. ................... 57
4 RENORMALIZATION GROUP STUDY OF THE STANDARD MODEL
AND AN APPLICATION . . . . . . . 58
4.1 The Initial Values . . . . . . . . . 59
4.2 Results .......... ..... ... 68
4.3 Top Quark and Higgs Boson Masses ........ . 74
CONCLUSIONS 85 5 CONCLUSIONS .....................85
APPENDICES
A 0 FUNCTIONS FOR THE STANDARD MODEL . . . 87 B THE EXPRESSION FOR 6(p) .............. 94
C SELFDUAL SOLITONS IN A CHERNSIMONS MODEL WITHOUT A BROKEN VACUUM ................ 95
C.1 The Model . . . . . . . . .. . 96
C.2 Specific Solutions . . . . . . . . .. 100
C.3 A Note .. .. .................... 106
C.4 Comments .. ........ ........... ... 107
REFERENCES ... .................... 109
BIOGRAPHICAL SKETCH .................. 111
iv
Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy
PHENOMENOLOGICAL ASPECTS OF
THE STANDARD MODEL: HIGH ENERGY QCD, RENORMALIZATION, AND A SUPERSYMMETRIC EXTENSION By
Samuel Mikaelian
May 1996
Chairman: Charles B. Thorn
Major Department: Physics
Two aspects of the standard model are the main focus of this work. One is understanding the high energy behavior of QCD in certain experiments, and the other is a study of the scale dependence of the standard model and its minimal supersymmetric extension.
Available data on exclusive and inclusive pionproton reactions at high energies and large momentum transfers are analyzed for signs of a subleading effect strongly affecting the effective p Regge trajectories. It is found that the data are consistent with a strongly coupled subleading trajectory at around a = 1. Based on this, an estimate for the beam energy of around 103 GeV or more is made in order to detect the onset of the true Regge limit. To shed light on the physics of the Regge limit of QCD, we study a nonrelativistic potential model incorporating asymptotic freedom. The large negative energy limit of the trajectories is shown to simulate those predicted by QCD. It is also found that for a choice of parameters corresponding to those of QCD, the scattering amplitude exhibits a slow approach to the Regge regime, consistent with our data analysis.
v
The scale dependence of the standard model and its minimal supersymmetric extension are determined to two loop order, taking full account of the Yukawa sector. We employ the best available data to extract the quark masses to which the results are sensitive. Demanding the equality of the bottom and r Yukawa couplings at the scale of unification, we place bounds on the top quark and Higgs boson masses. For MSUSY = 1 TeV and Mb = 4.6 GeV, they are 139 < Mt < 194 GeV and 44 < MH < 120 GeV. Subsequently, Fermilab has measured Mt to be 174 10 GeV, consistent with our bounds.
vi
CHAPTER 1
INTRODUCTION
Since QCD has resisted a definitive solution for more than twenty years, any aspect of it that can be confronted with experiment deserves particular attention. Regge trajectories in the large negative momentumtransfer limit of QCD provide an instance. Recent calculations of such meson trajectories are not born out by the available data. In chapter 2, after a brief introduction to the topic, we reanalyze the exclusive and inclusive data for signs of a strong subleading effect in the scattering amplitude taking hold close to the point where the trajectory turns negative. We discuss the similarities and differences between our fits and those already available in the experimental literature. An estimate is made of the beam energy needed in an inclusive reaction to probe the Regge region.
In chapter 3, a potential model is introduced in order to examine the behavior attributed to QCD in a solubleyetapt analogy. The resulting leading trajectories are examined both numerically and analytically and compared to the QCD ones. The scattering amplitude and its implications for the effective trajectories, are studied analytically and numerically for a model corresponding to fixedcoupling behavior. This is then used to infer the corresponding behavior of the asymptotically free model.
The renormalization group is an indispensable tool in extracting lowenergy predictions from proposed extensions of the standard model. In chapter 4 we
1
2
introduce a rather comprehensive numerical treatment of this and examine the relative merits of various corrections to the naive treatment hitherto performed. We apply this machinery to a minimal supersymmetric extension of the standard model subject to grandunified constraints, to place bounds on the top quark and Higgs boson masses.
Chapter 5 summarizes our results. Appendix A contains the 3 functions of the standard model and appendix B regards the radiative correction to the Higgs mass. Appendix C presents a scalar chernsimons model in 2+1 dimensions with novel soliton solutions.
A clarification of notation is in order here. In chapters 2 and 3, which involve Regge theory, a refers to an angular momentum trajectory, unless it is aQCD and as defined in chapter 2 or OQM defined in the chapter 3. In chapter 4 and thereafter, on the other hand, a will exclusively refer to gauge couplings.
In retrospect, a common theme of this thesis could be aspects of mass spectra in particle physics.
CHAPTER 2
QCD REGGE TRAJECTORIES
Quantum Chromodynamics (QCD), an SU(Nc) gauge theory with Nc = 3 describing the interactions among quarks and gluons, is widely held as the correct theory of the strong interactions mainly due to its successes in a particular set of processes. These are large momentumtransfer (hard) processes, such as certain aspects of deep inelastic electronnucleon scattering and jet physics, and heavy quarkonia mass spectra where due to the property of asymptotic freedom weakcoupling calculations can yield quantitatively reliable predictions. Given SU(Nc) as the gauge group of the color force, asymptotic freedom is a consequence of the quark content (number and representations) of the standard model, since the QCD coupling constant has the behavior,
Ncas(t) 4Nc
aQCD ~ = (2.1) 7r (Cadj 3CfN ) ln()
with the Dynkin indices Cadj = Nc, Cf = for a Dirac quark in the fundamental representation, Nf is the number of quark flavors, t is the momentum transfer and A is a constant characteristic of the theory.
On the other hand, much of the low energy data on the strong interactions, such as the mass spectra of hadrons containing light quarks, involve the nonperturbative aspects of the theory such as quark confinement and chiral symmetry breaking, which are not yet subject to a fundamental and quantitative understanding. There is, however, an extrapolation of the spectrum
3
4
problem that is amenable to perturbative calculations. This is the large t limit of the QCD Regge trajectories.
The Regge formalism, introduced in the preQCD era, provides a means of classifying the spectrum of a theory and predicting the high energy behavior of its scattering amplitudes. Here we give a brief but illustrative account of this [1]. (For our future purposes the complications of including spin are not necessary. Hence, for clarity, we limit ourselves to spinless particles.) A 2 + 2 scattering can proceed in the three channels a b  c d schannel
a + b d tchannel
a d  b c uchannel
where
S = (Pa + Pb)2 t = (pa Pc)2 u = (pa Pd)2
are the Mandelstam variables, of which only two are independent. By crossing and the PCT theorem the amplitudes of the above three reactions are the same but correspond to different kinematic ranges of the Mandelstam variables. We can, therefore, write a partialwave expansion of the tchannel amplitude as
00
A(s, t) = 16 E (21+ 1)AI(t)P(zt(s, t)) (2.2) 1=0
where for the case of interest, ma = mc = 0, t2 + 2st m2 2 m
zt = cos(t) = (t )(t m)
Ot being the tchannel scattering angle in the centerofmomentum (c.m.) frame.
5
Partialwave amplitudes of definite signature, A', need to be defined in order to have the appropriate asymptotic behavior for the application of the SommerfeldWatson transform,
A+ (t) = Al(t) for 1= 0,2,4,...
(2.4)
Al(t) = A,(t) for 1= 1,3,5,..., and the physical (unphysical) integer values of I are the right (wrong) signature points. The physical amplitude is related to these by
A(s, t) = (A+(zt, t) + A+(zt, t) + A(zt, t) A(zt, t))
If there is no lefthand cut in the ztplane At has the correct convergence behavior so that we have the socalled exchange degeneracy, i.e., A+ = A = Al. This implies that the corresponding uchannel reaction cannot occur through the production of a single resonance.
The application of the SommerfeldWatson transform to the partialwave expansion of the signatured amplitudes A'(s, t) and the displacement of the contour to enclose the Iplane singularities of the Af(t) follow a familiar line [1], which we only expound upon in section (2.2.1) for the case of potential scattering. The relevant contribution for our discussion here is that of the 1plane poles of Af(t) to the scattering amplitude, which for a given pole has the form
Ap(s, t) 167r2(2a(t) + 1) (t P (zt) (2.5) sin 7ra(t) a(t)
where 3 is the residue of A'(t) at the Regge pole a(t). The physical scattering amplitude then has the form
Ap(s, t) 167r2(2a(t) + 1) (t) (Pa(t)(zt) + oPa(t)(zt)) (2.6) sin 7ra(t)
6
Its poles are at the physical integer values of a as expected. The largezt (larges) limit of eq. (2.6) using eq. (2.3) results in the cross section
dr + eiia(t)
d(s t) = f(t)2(t) 82a(t)2 (t) r(t) (2.7)
dt sin 7ra(t)
( is known as the signature factor and ensures that only right signature poles along a trajectory are present.
The above formalism can be applied to the charge exchange reaction
7 p + 70(q) n (2.8) as in fig. 2.1(1). As the exchanged trajectory has the quantum numbers of an isotriplet nonexotic meson it corresponds to the p (A2) trajectory. These are (in the largeNc limit) exchangedegenerate trajectories, ap = 1 = aA2, with normal parity, P = (1)J, and G = (1)J+1 C = (1)J for the Gparity and charge conjugation at any positive integer J.
Experiments are often performed on inclusive processes, where certain final states are left undetected, since certain particles, such as the neutral ones, may be hard to detect. The Regge formalism has been applied to the following subset of such reactions
ab + c X
where X designates the unidentified flying objects (UFOs) out of the collider. The process is now described by three invariants, which may be chosen as s, t and M2, the invariant masssquared of the UFOs. A parameter used commonly in such processes is
/ PcL X M + 2st (pcL)max c.m. s m2 ( m2)2 (s m )2
7
R 0 (T)
P (A2 )
p n
(1)
2
as c c c a C C
a a X bX )
X b b t=O
(2) (3)
a a
(4)
t=o
b b
Figure 2.1 Figure (1) is the singleRegge limit of an exclusive process and figs. (2) to (4) depict the tripleReggeizaton of an inclusive process.
8
where PL denotes the longitudinal component of the momentum and the last equality is for the case ma = mc = 0. For our case of interest, b is a fixed target in the lab frame and < 1, so that in terms of the lab quantities,
Ea 2mbEa s
The process by which the cross section in the triple Regge limit is obtained as depicted in figs. 2.1(2) through 2.1(4) [1]. The amplitude, represented by the diagram under the sum in fig. 2.1(2), is described in terms of a single Reggeon exchange in the limit > 1 with M and t fixed. The application of optical M2
theorem is represented in fig. 2.1(3). Now taking 74 >> 1 introduces another Reggeon exchange as in fig. 2.1(4). In the appropriate limit s > M2 > t (2.9) the differential cross section is given by
dtd = E Gpp,p(t)(1 xF)aP,()a(t)a () ) (2.10)
ppP
where
GP,4p(t) = 7pp'P(t) 7bbP(O) Tacp(t) p(t)Yacp'(t)p,(t) (2.11) with 7 denoting a vertex coupling and ( the signature factor introduced in eq. (2.7). In the case of the reaction 7r p r0 X (2.12) in the triple Regge limit the exchanged reggeons are the p and the pomeron, P, which is the leading reggeon with the quantum numbers of the vacuum. From the flatness of total elastic cross sections one infers ap(0) 1 (2.13)
9
In the tripleRegge limit of reaction (2.12), expression (2.11) is valid as only the spin nonflip amplitude for the proton contributes.
For large t, where perturbative QCD (pQCD) is applicable, the scattering amplitude can be obtained by a renormalizationgroupimproved sum over the leading double logarithms [2,3]. In the largeNc limit the diagrams are only of the ladder type and for massless quarks the associated BetheSalpeter equation yields the following for the meson trajectories [4]
an(t) t>>Acc 2 (n + 3/4)7r] 2/3)
(2.14)
( 12/11
where aQCD(t) = n(_ t/A2
In contrast, for a fixed coupling constant the leading singularity would be a branch point at a = VQCD. The running of the coupling causes an accumulation of trajectories above zero to the left of what would have been a branch point. The set of trajectories in eq. (2.14) will be referred to as a bundle.
The cross sections for the reaction given in expression (2.12) fitted to a pure Regge form in ref. 5 implied trajectories which flattened off around t = 2 GeV2 to an apparent asymptotic value between .5 and 1. The question then arises as to whether these empirically extracted trajectories essentially represent the leading bundle, or whether they should be viewed as an effective trajectory strongly influenced by the softer physics of the subleading ones [6]. Assuming the applicability of pQCD calculations for t >> A2CD, the first scenario would be problematic. It would imply an eventual rise of the leading trajectory to above zero at large t (in quantum mechanics for I > .5 the trajectories are real and monotonically increasing functions of the energy). This leads eventually to a large increase in the cross section as t increases, for very
10
0
0.1
0.2
0.3
0.4
0.5
0.6 I
1.3 1.2 1.1 1 0.9 0.8 0.7 0.6
t(GeV2)
Figure 2.2 The diamonds are ours and the others, shifted to the right for clarity, are ref. 7's effective trajectories for the exclusive process.
large values of s. We, therefore, consider the second scenario as physically more plausible, and first examine the data for any such indications.
The exclusive interactions of eq. (2.8) were studied in ref. 7. The experiment was conducted at beam energies of 20.8 40.8 64.4 100.7 150.2 and 199.3 GeV, and momentatransfers of up to t = 1.3 GeV2. The trajectories go through zero at around t x .70. Figure 2.2 shows a comparison of our fits for higher values of t to those of the experimentalists. The values of and errors associated with the two fits are practically the same. These trajectories were extracted by fitting the cross sections for all six of the data points at a given
0.3
0.2
0.1
0  0.1
,.I
0.2
0.3
0.4
0.5
0.6 I I I
1.3 1.2 1.1 1 0.9 0.8 0.7 0.6
t(GeV 2)
Figure 2.3 Our effective trajectories for the exclusive process: the diamonds are for the 20.8 < E7r < 199.3 GeV, the squares for 64.4 < Er < 199.3 GeV and the others for 40.8 < E, < 199.3 GeV, where the last two sets of data are shifted to the right.
t to the Regge form
d(s, t) = (t)(Er)2a(t)2 (2.15) dt
We next made fits by excluding first one and then two of the lowest beam energies from the fits. Assuming that the above beam energies, especially the lower ones, are not yet probing the Regge limit, restricting the range of these fits to higher energies should bring a(t) in formula (2.15) closer to the true leading trajectory. The results of these fits are presented in fig. 2.3. There seems to be a tendency for the trajectory to rise as the range of energies is moved towards higher values. This is much less clear for the largest two t values which have
12
big error bars attached to them. The rise for the rightmost (on the graph) values of t is not inconsistent with the assumption that in the region where the leading trajectory is expected to flatten toward zero, the subleading trajectory develops a stronger coupling to the scattering amplitude and hence lowers the effective trajectory.
We next examine the inclusive reactions of eq. (2.12) subject to eqs. (2.10) and (2.13) in the triple Regge limit, i.e.,
r"= G(t) y12a(t) y (1 F)
These were tested for (i) s = 200 GeV2 (E,_ = 100 GeV) in ref. 8 and for (ii) s = 400 GeV2 (E_ = 200 GeV) in ref. 5. Figure 2.4 shows the trajectories extracted in the last two references. Clearly there is a rise in the trajectory as the fitted range of XF is brought closer to one, i.e., the range of E.0 is brought closer to E,_ from the bottom to the top graph. For the sake of clarity, we have plotted case (i) separately as well in fig. 2.5.
Although the above effect is heartening in our interpretation of the extracted trajectories not being the leading one, we cast doubt on the quantitative extent of this trajectory lifting. The problem is our inability to reproduce the experimentalists' fits for either of the s values within the associated tolerance, as shown in fig. 2.6 for the abovementioned two ranges in case (i). In fact our fits for the two respective ranges of XF in fig. 2.7 do not show any indication of the rise in the trajectory that could have signaled the approach to the Regge limit.
Case (ii) is similar although less inconclusive. Here the data available to us were in rather large xF bins so that for a fixed t only five data points were
13
.3 + 200 GeV (E > 160 GeV)
100 GeV (E > 80 GeV)
.7
.. ........... ..... .....
.9
(a)
20 .. .I I .. .It(Ge\? )
.3 + 200 GeV (E > 140 GeV) 100 GeV (E > 70 GeV)
.7
5
Figure 2.4 Reference 5's fitted trajectories for the two indicated values of E y: y :.05 .2 for the top graph and y : .05 .3 for the bottom one. available. The top half of fig. 2.8 shows our fits to the two and three data points
14
0.35
0.4 s=200 GeV2
0.45
0.5
0.55
0.6
0.65
0.7
0.75
0.8
0.85 I I I I
2.5 2.4 2.3 2.2 2.1 2 1.9
t(GeV2)
Figure 2.5 Reference 8's fitted trajectories: y : .05 + .3 for the diamonds and y : .05 + .2 for the others.
that correspond to the ranges of Kennett's fits at each t. Comparing this figure to fig. 2.4 there is the same discrepancy of the results of our fits being lower than theirs. For the smaller t values there is some indication of a rise in the effective trajectory but not as unequivocal as in their fits.
We discount the possibility of some deficiency in our fitting routine, which utilizes Marquardt's method of minimizing x2, since it reproduced the fits of ref. 7 as indicated above and it is a routine in common use by highenergy experimentalists. A more likely situation is that our fits are simply too "raw." The subleading effects for which the experimentalists make subtractions from the data are the ppf (f has the same quantum numbers as the pomeron but a
15
0.65
s=200 GeV2
0.7
0.75
0.8
0.85
0.9
0.95
1 ili
2.5 2.4 2.3 2.2 2.1 2 1.9
t(GeV2)
0.3
s=200 GeV2
0.4
0.5
0.6
S0.7
0.8
0.9
1
1.1 I I i I
2.5 2.4 2.3 2.2 2.1 2 1.9
t(GeV2) Figure 2.6 Reference 8's and our (diamonds) fitted trajectories: y : .05 + .3 for the top and y : .05 + .2 for the bottom figures.
16
0.6
0.65 s =200 GeV2
0.7
0.75
0.8
0.85
0.9
0.95
1
1.05
1.1
2.5 2.4 2.3 2.2 2.1 2 1.9
t(GeV2)
Figure 2.7 Our fitted trajectories at s = 200 GeV2: y : .05 + .3 for the diamonds and y : .05 + .2 for the others, shifted to the right for clarity.
Regge intercept of around .5) effects, low mass resonances and a 27rexchange cut. However, for the t values discussed above these effects were expected or assumed to be small and therefore ignored by them. There are also experimental subtleties discussed for which corrections need to be made. These include smearing the data with beam momentum distribution and energy resolution, and integrating the data over each bin. We included such corrections the best we understood with no substantial effect on the results. In fact in ref. 8 it is indicated that energy resolution and beam momentum distribution did not affect the results appreciably.
17
0.2 I I
s=400 GeV2
0.4
0.6
0.8
1
1.2
1.4
1.6 I
6.5 6 5.5 5 4.5 4 3.5 3 2.5 2
t(GeV
0.5 ,
0.6 s=400 GeV2
0.7
0.8
0.9 1
1.1 j
1.2
1.3 1
1.4
1.5 iI i l ii
6.5 6 5.5 5 4.5 4 3.5 3 2.5 2
t(GeV2) Figure 2.8 Our fitted trajectories at s = 400 GeV2 for the range y: .05 yl, where y1 is .25 and .45 for the diamonds in the top and bottom figures, respectively, and .15 and .35 for the top and bottom fig. nondiamonds, which are shifted to the right, respectively.
18
1.8<t<2.0
00.1
0)
0.01
0.1
lo 0(Y)
1
1.8<t<2.0
00.1
0.01
0.1
log (y) 10 Figure 2.9 Differential cross sections at s = 200 GeV2: ref 8's "raw" (dashed) and our (solid) predictions are plotted for y :.05  .3 in the top figure and for y : .05  .2 in the bottom one.
19
2.0<t<2.2
0.1
M0
0o
0.01
0.001
0.01 o 0W0.1
2.0<t<2.2
0.1
O 0
CD
0.01
0.001
0.01 0.1 log (Y) Figure 2.10 Differential cross sections at s = 200 GeV2: ref. 8's "raw" (dashed) and our (solid) predictions are plotted for y :.05 * .3 in the top figure and for y : .05 + .2 in the bottom one.
20
2.2<t<2.6
0.1
01W
O
o
0.01 ,
0.001
0.01 log () 0.1
2.2<t<2.6
0.1
0.01
0.001
0.01 0.1 logl0(Y) Figure 2.11 Differential cross sections at s = 200 GeV2: ref. 8's "raw" (dashed) and our (solid) predictions are plotted for y :.05 + .3 in the top figure and for y : .05 + .2 in the bottom one.
21
As a means of contrast we give several plots of the experimental cross section and the predictions of their and our fits for various t bins in figs. 2.9 to 2.11. In most of these their "raw" prediction is not close. In the bottom half of fig. 2.8 we present our fits for two different ranges of XF which are not as restricted as the previous two ranges, for the sake of having more data points to fit to. There is again a slight trend for the trajectory to rise although it is marred by big error bars at larger t values.
We proceed to make a simplistic two power fit of the inclusive data. Assuming that the behavior of the fitted trajectory is due to a stronglycoupled trajectory subleading to the p, we would have
a" = G00(t)(1 xF)12ao + 2Gol(t)(1 F)1alao + G11(1 xF)12al
where 0 and 1 refer to the p and the proposed subleading trajectory, respectively, and the G's are as in eq. (2.10). In the next chapter it will be shown that in the context of potential scattering, a squareroot branch cut behaves like a fixed pole at the center of the cut for smaller values of the cut. Since the QCD bundle of trajectories simulates a cut, we assume the bundle can be represented by a pole at a = 0. Thus we set
a =0 .
Indeed, the t dependence of a0(t) is not readily available as eq. (2.14) is valid only for much larger values of t where the perturbative approximation is valid. The G's are taken to be real and, as a further simplification, we assume the factorization Got = p/Go00Gi. Thus we fit to the form
a" = (1 XF)(G + Gi(1 XF)a' )2 (2.16)
22
Table 2.1
Results of a TwoPower Fitting
t al G1/Go X2/d.o.f. s(r = 10)
2.3 1.14 .09 21.6 + 5.7 3.4 5000 2.9 1.64 .1 26.2 + 4.1 .75 1000 3.3 1.17 .19 24.5 9.9 1.8 5000 3.7 1.33 + .16 61.6 + 35. 5.3 5000 4.1 1.87 + .10 33.4 5.0 2.6 500.
4.8 1.71 .20 39.8 13. .15 1000 5.2 1.32 .30 29.6 18. 0.0 2500 5.7 1.45 .29 47.1 33. 2.5 2000 6.3 2.02 .32 41.3 18. 1.8 500.
Note: Above table for s = 400GeV2; for the first four columns y :.05 .45 and for the last M2 = 20
t al G1/Go X2/d.o.f. s(r = 10)
2.3 .992 t .16 29.2 + 20. 3.5 13000 4.1 1.52 .28 25.7 + 10. 3.6 1000 4.8 1.71 .36 39.8 21. 0.3 1000 5.2 1.31 .33 29.6 + 19. 0.0 2500 6.3 1.87 .36 34.7 18. 3.0 500.
Note: Above table for s = 400GeV2; for the first four columns y : .05 * .35 and for the last M2 = 20
The results are presented in the first four columns of tables 2.1 and 2.2 Within the error bars, the subleading power a1 does not show a rise as the range of y is brought closer to the Regge limit. One can estimate the s at which
23
Table 2.2
Results of a TwoPower Fitting
t C1 G1/Go X2/d.o.f. s(r = 10)
2.1 1.36 .18 25.4 9.0 .70 1000 2.4 1.04 .16 56.8 74. 1.5 8500
Note: Above table for s = 200GeV2; for the first four columns y : .05 4 .45 and for the last M2 = 10
t al G1/Go X2/d.o.f. s(r = 10)
2.1 1.18 + .33 27.5 23. .94 2000 2.4 1.06 .22 58.8 93. 1.5 8000
Note: Above table for s = 200GeV2; for the first four columns y : .05 * .29 and for the last M2 = 10
the nexttoleading term in the cross section is r times smaller than the leading one by
(2rG 1/01
In the last column of the above two tables some representative estimates of s are tabulated. The chosen M2 corresponds to xF = .05 in cases (i) and (ii). Three orders of magnitude for s is an estimate compatible with entries that have the lower X2 per degree of freedom.
We point out that more sophisticated analysis of the available data is being considered. However, cleaner experiments probing the Regge limit could be more decisive in settling the discrepancy between data and pQCD.
CHAPTER 3
THE POTENTIAL ANALOGY
The analogy here is between a relativistic physical process and a nonrelativistic unphysical one. In a relativistic 2  2 scattering the physical Regge limit corresponds to extreme forward scattering in the schannel due to reggeon exchange in the tchannel. As only one channel is available in nonrelativistic scattering, the Regge limit we examine corresponds to unphysical elastic scattering parametrized by the limit
2E
and E < 0 is the domain of interest, i.e., t +* E and zt + z is the correspondence between the exclusiverelativistic and nonrelativistic cases. The potential is QCD inspired in its shortdistance behavior, where the running of aQCD is simulated by a shortdistance logarithmic violation of canonical scaling in the Schrodinger equation. We shall see below that this choice for the hard physics of the model leads to an accumulation of the leading trajectories, analogous to eq. (2.14), above a = .5 at large negative energies. Thus we consider the following radial Schr5dinger equation for our spherically symmetric potential (h = c = 1 = 2m)
( 2 r2 +1/V(r))(r)= E(r) with A = + 1/2 ,
c cA 1 (3.2)
V(r) =E + 
r2 In (1 + ) r
24
25
The coefficient of the extra Coulombic term in eq. (3.2) is fixed by requiring the large distance behavior of V to be that of an attractive Coulomb potential of unit strength; that exhausts our freedom to scale the Schrodinger equation. This provides some degree of solubility to the model.
We will also study the scattering amplitude due to the potential V in the above Schridinger equation, where
12 1
Vo 2 r (3.3)
2 r
simulates the short distance behavior of a fixed coupling constant. As we will discuss later the scattering amplitude of the latter model has a fixed cut in the angular momentum plane centered at a = .5 and subleading Regge poles that are Coulombic in character for E > 1. For c, A < 1, V may be considered a coarse graining for the short distance behavior of V, whereby a fixed cut is simulated by the accumulation of the Regge poles at large negative energies. Hence, we would expect subleading trajectories in the model with the potential V that approach the negative integers with large enough E. This is qualitatively in accord with the larget behavior of the effective trajectories in fig. 2.4. As indicated in the previous chapter, simple twopower fittings of the data on the inclusive reaction indicate a subleading power between 1 and 1.5. Therefore, such a choice for the subleading term in our potential, although not based on any fundamental property of QCD, is indicative of the data.
A potential that at large distances behaves like a harmonic oscillator would render a more faithful representation of QCD. It would imply linear trajectories for large E analogous to trajectories that interpolate between qq bound
26
states in the confining limit of QCD. Such asymptotics, however, would create unnecessary complications in defining the scattering amplitude in our model; the very large distance behavior of the theory is not expected to contribute significantly to the scattering amplitude at energies that probe the short distance physics, as we shall substantiate in the next section. Therefore, we consider the potential given in eq. (3.2) sufficient for the purpose of testing the basic premise of this paper.
3.1 The Trajectories
3.1.1 The Model Incorporating Asymptotic Freedom
The trajectories of our proposed models can be obtained by numerically solving the Schrodinger equation. Adding a harmonic oscillator to eq. (3.2), V(r) + V(r) + w2r2 (3.4) linearizes the largepositiveenergy limit of the trajectories as in fig. 3.1.
The influence of the soft physics of the resulting model in the short distance limit can be inferred by comparing the position of the trajectories at moderate
E for representative values of w2, as given in table 3.1 For the first set of values, A = 102, c = .125, the parameter is varied from where there is no harmonic oscillator to an extreme value of w2 = 16, and three moderate negative energies are picked. We see that the locations of the trajectories are indifferent to w2 to three significant figures. The value w2 = 16 is extreme because the trajectories it induces at moderate positive energies are practically flat slightly above the I = .5 line. On the other hand, w2 m .2 induces positiveE trajectories that have a slope of around one; a closer analogy with the strong
27
2.5
2
1.5
I(E) 1
0.5
0   0.5
8 6 4 2 0 2 4 6
E
Figure 3.1 Regge trajectories exhibiting asymptotic freedom and confinement: the first two nodes for A = .5, c = .125 and w2 = .2 are plotted.
interactions. A similar comparison is shown for A = .5 and c = .125, values to be of interest later, for two values of w, and another comparison for an arbitrary value of parameters.
One would expect the same lack of sensitivity to the verylongdistance (confining) physics for the nexttoleading (Coulomblike here) trajectory, except perhaps at eversoslightly more negative energies. If at negative energies the location of the Smatrix singularities are unaffected by the variation of a parameter, it is reasonable to expect the partialwave amplitude to behave similarly. Thus, as advertised earlier, we shall ignore simulating confining trajectories for the main purpose of this work.
28
Table 3.1
Effect of a Confining Potential on the Asymptotically Free Component
w2 1(5) 1(6) 1(7) A c n 0 .4500 .4507 .4511 102 .125 1 1 .4502 .4507 .4511 102 .125 1 9 .4507 .4510 .4513 102 .125 1 16 .4509 .4512 .4515 102 .125 1 1 .3580 .3587 .3592 101' 1 1 9 .3586 .3591 .3595 1010 1 1 0 .4690 .4693 .4696 .5 .125 2 .2 .4690 .4694 .4696 .5 .125 2
The numerical trajectories for the model of eq. (3.2) are plotted in fig. 3.2. The potential becomes more negative with increasing A or c, with the effect of increasing the binding energy and thus raising the trajectories, as in the figure. As evident though, there is more sensitivity to a change in c than A, since the latter only incurs a logarithmic change in the potential.
Let us obtain an analytic expression for the trajectories in the region of interest. The following transformations are helpful. Defining
R = ln(1/Ar) and = eR/2f (3.5) eq. (3.2) can be transformed into the following form
( + V(R))f = A2f, with
c (cA 1) _R E 2R Roo c (3.6) V(R) 2e
ln(1 + eR) + A A2 R
In the above Schrodingertype equation, where A2 plays the role of boundstate energy, the Coulombic tail of the "potential" V(R) implies a logarithmic
29
0.3 0.2
0.1
o     
I 0.1
0.2
0.3
0.4
               
0.5 I I I
8 7 6 5 4 3 2 1 0
E
Figure 3.2 Regge trajectories for the potential V(r): the first node for A = .5, c = .5, the first three nodes for A = .5, c = .125, and the first two nodes for A = .01, c = .125 are plotted as dotted, solid and dashed curves, respectively.
modification of the usual asymptotics (r) 4 r+1 near the origin in eq. (3.2), i.e. for A > 0
f(R) R/ = (r) (3.7)
Hence, there is a zeroA singularity in the asymptotic behavior of the wave function of an asymptotically free potential, reminiscent of the threshold singularity in the wave function asymptotics of a Coulombtailed potential [2]. Given this correspondence, we would expect an accumulation of trajectories
30
above A = 0 for an asymptotically free potential, analogous to such accumulation near threshold of a Coulombtailed potential, i.e., c E
> 1 as a (3.8) There is, of course, the actual Coulombic behavior far from the origin in eq. (3.2) which induces the accumulation of the trajectories just below E = 0. Invoking the theorem on the monotonicity of the trajectories above A = 0, together with the E + 0 and A + 0+ behaviors delineated above, gives us the behavior exhibited in fig. 3.2.
WKB approximation is applicable to a Coulombic potential near threshold and so is justified in the A s 0 region of interest. The eigenvalues An of eq. (3.6) are obtained as solutions to
J dR V(R) A,2 = (n + 7)7r, (3.9) R
where R are zeros of the integrand and qj will be determined by examining the boundary condition across the turning points. In the limit of eq. (3.8) the wave function is concentrated near the origin, R > 1, so that henceforth we may approximate
R+ A n. (3.10)
I dR/W(R) for Wn(R) E + e 2R 2 (3.10)
We determine the constant 9 in eq. (3.9) by examining the exact solution to the SchrSdinger equation. Near RV(R) + A2) e2x where x R R, and the corresponding Schr6dinger equation, eq. (3.6), becomes
( 2 v2e2x)f(x) = v2f(x) for v i A2 (3.11)
31
The inverse of the transformation of the type in eq. (3.5) yields
y 2 214() = (Av)24(y) if V) = ey/2f, e/A. (3.12) f, then, has the Bessel functions as its solutions, out of which we seek the one that is exponentially damped to the left of the turning point R_ (x  oo). The corresponding solution for f(x) is Macdonald's function, Kv(ivex), with the following asymptotics
7ez ivez
rex ei e for x + oo f(x) = Kv(ivex) 2iv (3.13) sin(ivx) for v<1,x+0o.
iv
The integrals relevant to the WKB solution are
1 iv e2' ldx' = iv( e2 1 + tan1 ex 1) 12 ivij 1 e2xd' = iv(  e2x + tanh1 1 e2x )
x00 x00oo with iex 2  iv .
(3.14)
with the solution in the neighborhood of R having the forms
(ivve2x 1) Cexp(I1) if x < 0
WKB (x = (3.15)
(iv1 e2x ) (Cei2 + C+ei2) if x > 0. Matching eq. (3.15) to eq. (3.13) subject to eq. (3.14) gives
C = Cef1/ 2ivr (3.16) Thus, with p as the momentum, the WKB solution to the right of R_ has the form
f(R > R) = cos p dx (3.17)
32
In the vicinity of R+ we have cc
W(R) C C
R R+
and a conventional analysis of the boundary conditions at that turning point, by linearizing the potential there, yields f(R < R+) = cos( pdx (3.18) Comparison of eqs. (3.17) and (3.18) completes eq. (3.9) with r = 3/4, (3.19) the value indicative of a barrier at R consistent with the faster than exponential decay in eq. (3.13) on the classically forbidden side. (The corrections to eq. (3.19) are higher order than the nextorder corrections to An below and hence will be ignored.)
To evaluate the eigenvalues An notice that in the limits given by eq. (3.8), the classically allowed region is small compared to the distances characterizing the turning points and the exponential decay in eq. (3.10) implies that the maximum of Wn occurs very close to R_, as in fig. 3.3. Hence, we may approximate eqs. (3.9) and (3.10) by
ivj X1 e2x' dx' + +x dR = (n + 3/4)7r ,
x R (3.20) with 1 <1. To first order in the above small quantities we can write
(1 R_ /2 RI+ 2  R 12dR (n + 3/4)r R_ + In )E/A2 R+ 2 c/A.
(3.21)
33
1 I I
0.8 0.6
0.4 0.2
0       0.2
0.4 I
9 10 11 12 13 14
R
Figure 3.3 V(R) is plotted for c = 1 A = 104 E = 1. The x dependence drops out so that the trajectories are given by
3c [1 Ini( E/A2 ) 3/= (n + 3/4)r (3.22) This is the same result that would have been obtained had the expression for W(R) in eq. (3.10) been approximated by a purely Coulombic one c A2 and a barrier introduced at R_. Upon inverting eq. (3.22), QMn 1 (n + 3(3.23) for aqM = 1/l n (eq. (3.23) is the quantum mechanical analog of expression (2.14) in QCD. This accumulation of trajectories to the left of A' = 2cV M takes the place of the
34
Table 3.2
Asymptotically Free Trajectories
E A c n AWKB A AA/A
104 1060 10 0 1.56(.897) = .2374 .2389 6.3 x 103 104 4 x 1010 10 0 .617(.819) = .5053 .5170 2.3 x 102 104 1060 10 1 .265(.819) = .2166 .2214 2.2 x 102 104 1060 5 1 .187(.771) = .1443 .1496 3.5 x 102 104 1060 10 3 .265(.698) = .1848 .1974 6.4 x 102 103 1030 5 3 .263(.524) = .1370 .1668 1.8 x 101 500 1012 5 2 .403(.484) = .1953 .2476 2.1 x 101 102 109 1 0 .208(.592) = .1233 .1408 1.2 x 101 102 106 5 0 .557(.731) = .4071 .4312 5.6 x 102
branch cut singularity in the fixedcoupling case (to be discussed in the next section) and in that sense simulates the cut.
We can compare the last equation with some numerically obtained results. First we specify the range of validity of that equation. The WKB approximation is in general more valid for higher node solutions. However, eq. (3.23) is obtained by an expansion in aQM so that the second term in the curly brackets in eq. (3.23) should be small compared to one. Also, in the particular case of a Coulombtailed potential, c/R, the important region of motion is where Ipotentiall lenergyl, or in our notation, R _. The semiclassical approximation applies if this region is large compared to the wavelength of the particle, 1// energy = 1/A, i.e., c > A and from eq. (3.7) the wavefunction is enhanced for large R. Therefore, eq. (3.23) is improved for larger values of C.
35
Table 3.2 compares the numerical values of A from solving the corresponding Schr6dinger equation to those obtained from eq. (3.23) for different choice of the parameters in the range discussed above. In all entries the agreement is better than around 20%. Compaing entries 1 and 2, we see the sensitivity to the value of aQM. Comparing the second and third entries indicates the dependence on the number of nodes. Comparing the first, third and fifth entries shows that the sensitivity is stronger to the perturbative expansion than to the number of nodes (the closer the number in paranthesis under the WKB column is to one, the more justified the perturbative approximation is). Comparing the third and fourth entries indicates the cdependence. In general, the different entries give an indication of where eq. (3.23) is more valid. It should be noted that the large orders of magnitude chosen for values are necessary to satisfy the perturbativity requirement of our expression.
3.1.2 The Model with Fixed Coupling
For the potential V of eq. (3.3) the partialwave scattering amplitude, A/2
and hence its singularities, can be written in a closed form. The part of eq. (3.3) can be combined with the centrifugal term to yield the effective potential, A2/r2, for
Ac vA2 2 Ic Ac (3.24) The solution to the corresponding Schrodinger equation is then that for a Coulomb potential with the A + Ac replacement. The asymptotic form of the wavefunction is then
r+ o 1 r
(r)2 sin kr  In 2kr 2Ic+Coul(lc, k) k= vE ,
i12k l6C1k"' ,) k 1f,
36
0.5
0   0 .5 . ..  :    :: : .           
0   1
1.5
6 5 4 3 2 1 0
E
Figure 3.4 Regge trajectories for the potential Vo(r): the first two nodes and the branch cut (series of flat lines) for A' = .1 are shown.
from which the following partialwave Smatrix can be extracted
S1(k) = e2ibCoul.(lc'k)ei(Ic) F(Ac + j) eix(AAc) (3.25) r(Ac + 2 + A)
The above Smatrix satisfies the generalized unitarity condition
S*.(k*) Sl(k) = 1,
and has a fixed squareroot branch cut whose discontinuity does not vanish for large magnitudes of the energy; the asymptotic strength of the cut singularity is given by the exponential factor. That factor is the Smatrix for a y potential; the energyindependence a reflection of the potential's scale invariance.
37
Nx x
L
Figure 3.5 The Mandelstam contour and the Smatrix singularities it encloses.
The expression in eq. (3.25) also has simple Regge poles at
An = V n k+ + A'2, (3.26) as in fig. 3.4, with the corresponding residues
(1)n 1  n)
n(k) n!A F(i n) exp ir(An A A/2) (3.27) where the +() branch corresponds to n + being positive (negative).
3.2 The Scattering Amplitude: The FixedCoupling Case
3.2.1 MandelstamSommerfeldWatson Transform and the Regge Limit
We first give a brief heuristic derivation of the ReggeWatson continuation, in the zplane, of the partialwave expansion using the Mandelstam representation [9], mainly for application to potential V given in eq. (3.3). For this
38
purpose, the unique analytic continuation of the S, for the potentials under consideration is the one obtained by simply taking I in the radial Schr6dinger equation to be a complex number. Consider, then, the following integral in the Aplane
A L+ioo C dA S_.5(k) 1QA.5(z) +
27rk sin 7A JLioo semi circ (3.28)
where the Q's are the Legendre functions of the second kind, S1 are the Smatrix elements as before and the righthand side of eq. (3.28) defines the contour to be as in fig. 3.5.
The singularities of the above integrand are at the poles of Ql for negative integer 1, possible poles and branch cuts of the Smatrix, and the zeros of sin(7rl). Therefore, if 3 denotes the residue of a pole of S and N the nearest integer to the right of L, rearranging terms results in
00 i An
2 (21+ 1)(S(k) 1)P(z) = n 3A.5(k) QA.5(z)
l=0 poles(n)
cuts N(1)l [S_.5(k) S.5(k)]Ql.5(z)
l=0
7 0 L+ioo
k (1)1(S .5(k) 1)Q 5(z)+ sem circ
I=N+1 Lioo emi cir
(3.29)
The left hand side of the last equation is the partialwave expansion of the scattering amplitude with a finite domain of absolute convergence in the zplane, e.g., within the Lehmann ellipse for a superposition of Yukawa potentials. The domain of convergence can be extended to the whole complex zplane by discarding the part of the contour along the semicircle at infinity.
39
The expression most suited for our purpose, given the asymptotic behavior of the Q's below, is obtained by pushing the background line to infinity, L + oo, which makes its line integral vanish in the Regge limit, with all singularities in the Aplane now contributing. Thus we may ignore the background terms to write for the scattering amplitude
A(k,z) = sin rAn ,.5(k) Q,.5(z)
all poles
2rk cutsdA .5(k) 1) Q _.(z) (3.30)
ir (1)'l[Sl_5(k) S_5(k)] Ql_5(z) 1=0
Because of the branch cut in expression (3.25) the S, do not possess the Mandelstam symmetry necessary for the last term in eq. (3.30) to vanish. That term cancels the poles at integer values of A of the first term on the righthand side of eq. (3.30). Thus the poles of A coincide with those of Q_ .5 which are at the physical (nonnegative integer) values of I.
We would like to specialize expression (3.30) to Vo of eq. (3.3). Substitution of eqs. (3.25) (3.27) in (3.30) and repeated use of wellknown Gammafunction identities yields
ex/2k 00 p eidrA
AO(k, z) k sin (1l)n n! sin(rin) F(n 2p) Q.5(z)
k n=Osn(rA)
exr/2k A' ,eiAr
+ 22k sin rA sinh(2rA'2 A2) I(iA2 2 p) QA.5(z) dA
er/2k 00
1 l sin(27rx2 A'2) F(I2 12 Ap) F ( A2 p) Q1.5(z) l=0
= Ap + Ac + Am ,
(3.31)
for p An (p n)2 + A/2
forP 2k 2'
40
The last line of eq. (3.31) labels the contributions from Regge poles, the fixed cut and the Mandelstam terms, respectively.
We need impose the further restriction
0 < A' < .5 (3.32) on the above formula. That there is trouble above that value can be seen from the partialwave expansion of the scattering amplitude, given on the lefthand side of eq. (3.29) together with eqs. (3.24) and (3.25). Terms for which A'2 > A2 will be illdefined as they correspond to points that are on the branch cut. This is a reflection of a wellknown quantum mechanical result: if a potential behaves as y near the origin, for A'2 > A2 the ground state of the particle is at E = oo, i.e., it tends to fall towards the origin.
The approach to the Regge limit, Izi + oo, is determined by the asymptotics of the Q's and the relative contribution from each of the terms in eq. (3.31). As the series expansion for the Legendre functions of the second kind is given by
r(v + 1)vW (5 + 1( + k 2k Qz)= z 2 (3.33) F(v + )(2z)v+1 k=0 k! (v 3 where (a)k a(a+1)...(a + k1), (a) 1,
in the extreme Regge limit the righthand most singularity in the Aplane would dominate. In general, the presence of a leading branch point (or a pole if 6 is a positive integer) at A', i.e.,
(A(E) A')1(E)
results in the Regge behavior of the scattering amplitude jzj A (_o
A(k, z) + (k)(z) n(E)(z) (3.34)
41
0.55 0.5 0.45 0.4 0.35
0 0.3
0.25 0.2 0.15 0.1
0.05 I I I I ,
2 4 6 8 10 12 14 16
z
Figure 3.6 The solid line is Qo(z) and the dashed its asymptotic expression.
In the model at hand, the particular hierarchy of singularities that would be most relevant to us occurs for E > 1. There the leading singularity of the Smatrix is the fixed cut with a tip below A = 1 and the subleading trajectories flatten off not far below the cut in a Coulomblike manner. Then we could examine the interplay between the hard physics of the cut to the softer one of the poles for signs of the effect we conjectured in the QCD experiments.
The asymptotics of the Q's can give us an indication of the scale relative to which the onset of the Regge limit is to be measured. Given the series expansion in eq. (3.33), the main contributions to the scattering amplitude at moderate values of z will come from Q,'s of smaller orders, approximately 1 < v < 2.5 for the case at hand, for which at z 5 the asymptotic value is not off
42
2.4 2.2
1.8
1.6 1.4 1.2
1
0.8 I
2 4 6 8 10 12 14 16
z
Figure 3.7 The solid line is Q_.6(z) and the dashed its asymptotic expression.
by more than 5% from the exact one. The approach to asymptopia for the Legendre functions is plotted in figs. 3.6 and 3.7. Thus, an order of magnitude would be a conservative scale for z against which to measure the onset of the Regge regime.
3.2.2 Analytical Estimates
Before giving a numerical evaluation of the scattering amplitude, let us give an approximation to the various terms in eq. (3.31) for a range of parameters that will be of interest later on. This is also useful as such approximation may be evaluated for parameter values that would boggle the computer. As A' increases towards the contribution of the cut becomes more and more
43
pronounced relative to that of the poles in approaching the Regge limit due to two effects. On the one hand, the separation between the tip of the cut and the location of the poles in the Aplane increases (i.e., higher power of z from the cut than the pole), and on the other, the strength (width) of the cut increases relative to the residue of the pole. The dependence on A' is rather dramatic as we shall see below.
Here we restrict ourselves to smaller values of A', say A' < .15, a range that will be argued to be of most relevance to us, later. To expand eq. (3.26) in powers of A' we require values of the energy not too close to 1 (where the leading trajectory merges into the cut) say E > 10. To first approximation then the trajectories are those of a pure Coulomb potential, and the sum over poles in eq. (3.31) is the scattering amplitude thereof. But that amplitude is known in closed form, namely the Rutherford amplitude. Hence,
Ap = 4 (1 ) 1+ O(A2) (3.35) where the nextorder correction is not needed below. It is a rather fortunate fact about the Coulomb potential that the exact amplitude has a Regge form in terms of the momentum transfer A2, i.e., the power of (1 z) in the last equation is just the leading trajectory, so that the Regge limit is attained within an order of magnitude for z. This is certainly commensurate with our expectation based on the asymptotics of the Legendre functions. We have yet to examine the other terms.
We now restrict ourselves to the asymptotic limit of the Q's to see how far away from this the Regge asymptopia is situated. So, for now, we work in the range
A'< .15 and z> 10 (3.36)
44
In that case, the contribution of the cut integral, Ac, in eq. (3.31) can be approximated as
e 2Ti 1 '
Ac F Ir( 2 2 2(z) A'/2 A2 exp (iA[7r iln(8z)]) dA
Ae2 I(1 i 1 JI(A'[7 i In(8z)])
=F 2 (z) 2
vF2k 2 2k 2 A'[w iln(8z)]
(3.37)
Taking the largeI z limit of the last expression, Ixl>l V 7r
J1(x) W sin(x ) (3.38) results in
Ac(k,z) ~ e2k A'(ln8+) )2( _z)A' 2 ln(z) (3.39)
2k V2 2k
The logarithmic correction to the tipofthecut power behavior of the scattering amplitude is characteristic of branch point singularities, as indicated in eq. (3.34). The relation that characterizes the Regge limit by examining the nexttoleading correction to eq. (3.38) is
A'ln(z) > .375 (3.40) This is an energyindependent criterion and demonstrates a sharp dependence of the Regge asymptopia on the location of the cut. For a value of interest,
A' = .1 = (z) 1015 to 1030 (3.41) if we want the subleading term in the cross section between 5 to 10 times, respectively, smaller than the Regge term. This is rather astronomical compared to our oneorderofmagnitude scale for the Regge limit of the poles. Although the perturbative approximation would not be valid there, as a rough estimate
45
for A' s .5 eq. (3.40) yields an order of magnitude for the Regge limit. This is not unexpected as when approaching our upper limit on A' we are approaching a pole in the Legendre function in the integrand in eq. (3.31), and we already expect a more rapid approach to Reggeland for pole singularities.
The contribution of the third term on the right hand side of eq. (3.31), Am, is negligible in the region given by eq. (3.36) provided the energy is not too close to 1, say conservatively E > 10,
Am 0 ,
which has been verified numerically. As stated previously the contribution of that term is only significant near values of energy corresponding to integervalued Regge poles An. It drops sharply away from such points as it contains Legendre functions of comparatively large orders. So the approach of the scattering amplitude to the Regge regime is mainly dominated by the two contributions discussed above. For values of z that are not extremely large the zerothorder A'expansion of the integral in eq. (3.31) would suffice:
A, + 2 ) 2Q.5(z) (3.42) the region of validity for which can be deduced from the argument of the Bessel function in eq. (3.37)
A'in(8z) < 1
The order of the Legendre function in eq. (3.42) implies that for smaller values of A', the "Regge" behavior of the branch cut is approximated by a simple pole located at the center of the cut, A = 0, up to sizeable values of z, e.g., up to z 103 for A' = .1. This aspect of the weakcoupling limit of the
46
potential may also be understood by expanding the Smatrix in eq. (3.25) in powers of A' to obtain, for p defined in eq. (3.31),
F(A p) At2
Sg(k) + 1F(A p) / [(A p) V(A + p + 1) ir]). (3.43)
F(A + 1 + p) 1 2A /
This expression has a fixed simple pole at A = 0 and no branch cuts. Therefore, in eq. (3.30) we do away with the integral but now add the fixed pole to the first term on the righthand side of it. The residue can be read off from the last equation and the resulting expression is eq. (3.42).
This, of course, is suggestive of the Born approximation for a V(r) =
A2/r2 potential, with eq. (3.1),
1 f0 7rA2
Aborn 47r e V(r)dr 2 k (1 z) (3.44) The largelkI and jzj limit of eqs. (3.44) and (3.42) coincide. By "largelkl" we mean E s 100 and "largejz means an order of magnitude as usual. Although, in the extreme Regge limit the Born approximation and eq. (3.42) break down, this analysis indicates that they are valid for at least a couple of orders of magnitude in z This lends credence to the calculations of ref. 10 where an improved Born calculation was used to estimate the hard physics at large XF, as long as such calculation is valid in the applied range of t. Again, the above result has a sharp dependence on A'; for A' close to .5 it is the rightmost branch point that rapidly dominates the integral. Also note that from eq. (3.42) (and of course eq. (3.44)) the weakcoupling amplitude is proportional to the coupling A'2, whereas in the Regge limit, eq. (3.39), that factor is much enhanced.
From eqs. (3.35) and (3.42) we can get a rough estimate of the z at which the cut dominates over the pole. This estimate is strongly energy dependent,
47
since the Regge pole term on the right side of eq. (3.31) has an extra factor of 1/k suppression relative to the cut integral, in the largelkI limit. To have the hard physics say around 5 to 10 times larger than the softer contribution, an orderofmagnitude estimate would be
A' = .1 E =10=> z N 105 (3.45) For more negative energies,
l E 10 for E> 100, (3.46)
(z)k _ IEI 4 for46
is a similar estimate for A' < .1.
In short, in the presence and for a judicious choice of a branch point singularity, there is a pronounced hierarchy of scales. The scale at which the leading effect (the center of the cut) is dominant at moderate binding energies is orders of magnitude separated from the analogous scale for the nexttotheleading effects (the poles); and the scale at which the leading branch point dominates is even farther removed.
3.2.3 Numerics
We can now numerically study certain behavior of the scattering amplitude without the approximations of the previous section. The ratio of the cut amplitude, Ac to that of the pole as a function of z is plotted in fig. 3.8. The solid curves validate the estimations of eqs. (3.45) and (3.46). Even for the relatively large value of A' = .25 it takes z 100 for the hard physics to contribute five times more to the cross section (ten times more to the scattering amplitude) than the soft.
48
20
18
16 E=10
14
12 E=5 ,' E=5
8 A 10
AP
8
6
4 E=10
2 E=10
0 1 2 3 4 5 6 log1(z)
Figure 3.8 A for A'=.25 (dashed), .1 (solid) and .05 (dotted).
IApI
A more accurate indication of the relative contribution of the hard physics is given by fig. 3.9, where the ratio of the amplitude of the cut to the total amplitude is plotted as a function of energy for two different values of z. The rise with energy of this ratio due to the 4 suppression of the pole contribution relative to that of the cut is more dramatic for the larger values of the coupling.
In ref. 10 an improved Born approximation of the hard contribution to the process of eq. (2.12) at t = 4 GeV2 yielded the estimate of d2c
ddEo 2.5 x 103 ya (2/y) (3.47) dtdEo
for y (1 XF). For the values of y = 20 and 7 with cas .2 at the relevant scale, the estimate is for the hard part to be 2% and .5% of the experimental
49
1.2
'=.2
0.8
A 0.6
0.4
0.2 55 50 45 40 35 30 25 20 15 10 5 0
E
Figure 3.9 I 12 for z = 10 (solid) and z = 5 (dashed).
cross section, respectively. Although there is no isomorphism of parameters between the triple Regge limit of the inclusive interactions and the Regge limit of our quantum mechanical model, a rough correspondence between y1 and
z, seems to indicate, from fig. 3.9, that the toy model with analogous behavior corresponds to a value of A' < .1, actually closer to A' s .05. The sharp rise with energy of this ratio in this toy model is probably not a characteristic of QCD. This difference may be improved when we consider the model with asymptotic freedom, as we shall discuss below.
The E and zrange of validity of the Born approximation to the cut was discussed in the previous section. Figure 3.10 is a plot of the ratio of the exact amplitude to the Born one for the full potential Vo(r) for different values of z
50
2.8 2.6 2.4 2.2
o
1.6
z=50
z=10{
1.4
55 50 45 40 35 30 25 20 15 10 5
E
Figure 3.10 The lexact/Bornl ratioofamplitudes for the model of Vo(r) for A' = .1 (solid) and A' = .05 (dashed) at the indicated z values.
and A'. It exhibits a stronger sensitivity to variations in energy than z. Also the dependence on z is mainly due to the rapid approach of the pole terms in the scattering amplitude to their Regge asymptotics.
For a comparison of the model with the empirical data we have fitted the differential cross section to an effective power, i.e., do
d G(E)y2aeff with y1 (3.48) dO 2
in fig. 3.11 for two different ranges of y. For A' = .15 the effective trajectory is practically no lower than the bottom of the cut at 1 = .65. The corresponding behavior for A' = .1 is distinctly different. Although it also rises with energy,
51
0.45
0.5
0.55
'=.15.6
0.6
0.65
0.75
0.8 0.85
0.9 '=.1 Coul.
      0.95 I I I I I
50 45 40 35 30 25 20 15 10 5 0
E
Figure 3.11 Effective and Coulombic Regge trajectories: the solid curves are for the range y : 8 + 15, the dashed one at the top is for y1 : 4 + 10.5 and the dashed one at the bottom is the n = 0 Coulomb trajectory for A' = .1.
at moderate energy values it is closer to the subleading trajectory, the plotted Coulomb trajectory, than even the bottom of its associated cut at I = .6. It is quite suggestive of the empirical effective trajectories in fig. 1.A. We note, however, that although there is a rise in the fitted trajectory as the range of y is shifted up, it does not appear as dramatic as in fig. 1.A and might well have been hidden had we associated errors with our fits. This is more in tune with our fittings of the cross sections discussed in the last chapter. We see again that values of A' not higher than .1 are more commensurate with the numbers extracted from the strong interactions (empirically or theoretically).
52
3.3 The Scattering Amplitude: The Asymptotically Free Case
We need to select values for the parameters of the potential V(r). We discussed earlier the indifference of the left end of the Regge trajectories to the parameters that characterize the right end. Had we linearized our trajectories at large positive energies, a natural choice would then have been for A .3 That is we take the dimensionless quantity (trajectory slope/A) to correspond our model to QCD. We are thus contradicting our earlier arguments for the irrelevance of the confining aspect to the physics of asymptotic freedom, only to the extent that an external mechanism is invoked to relate two disparate scales of our model. The mechanism, of course, has to do with chiral symmetry breaking in QCD. We choose
A = .5
as a close enough value since we already noted the insensitivity of the trajectories in fig. 3.2 to an order of magnitude variation in A.
Had we been motivated by economizing the number of parameters in our model, we might have picked c = 7. This would have summarized the whole potential in one term. However, such a choice does not lead to the behavior observed in QCD, since for A of order one, the scattering amplitude at moderate E is domianted by the leading bundle (as can be seen from a born approximation). Hence we proceed differently. To fix a value for c we compare eq. (3.23) to (2.14). Taking 12/11 and numbers of similar order as 1 in the latter equation, for c = the two expressions would coincide. This is because at that value the shortdistance limit of the potential is
V(r) + 1/r2 ln(A2r2) (3.49)
53
similar to the asymptotically free limit of QCD VQCD(r) + 1/r In(AnCDr2). However, we require a further correspondence.
The analog of relation (3.32) in the relativistic case would be a < 1 (for a a/r potential), beyond which the system is unstable with respect to pair production. In QCD at the point where perturbation theory is taken to break down, i.e., aQCD = 1, the potential mimics a fixedcoupling Coulombic one at the threshold of stability. Taking this as a proper guide, the coefficient of V in eq. (3.49) should be adjusted from 1 to :. In that case, in eq. (3.2) we set
c= 1/8.
We point out that since the short distance behavior of V(r) is unaffected if c and the power of Ar are changed in a correlated manner, we also examined the trajectories generated by the potential
.25 1
r2 ln(1 + r4) r
This uplifted the trajectories in the "flat" domain only slightly so that the general results presented below are unaffected.
To apply eq. (3.30) to V(r) requires the trajectories and residues for A' < 0. These correspond to irregular solutions of the Schrodinger equation which can only be obtained by an analytic continuation of the regular ones. As these solutions are not available to us in closed form we do not pursue this approach. Rather we use our original paradigm of the accumulation of trajectories simulating a squareroot branch cut to approximate the model for V by the model for V0. This we argue is a valid approximation in the region of interest because the nexttoleading term in the potential near the origin is CAl and, since
54
Table 3.3
"Cut Profile" for A = .5 c = .125
E A'
4.1 .1051
5.1 .1012
6.1 .09855
7.1 .09687
8.1 .09552
9.1 .09450
10.1 .09348
20.0 .08910
cA < 1, the subleading trajectories induced by V are essentially the Coulombic ones generated by V for large negative energies. Around E = 1 this approximation is not valid so we restrict ourselves to E > 5.
An estimate for the effective cut associated with the bundle of trajectories can be obtained by the ratio of level separations predicted by eq. (3.23). We call this a "cut profile" for the bundle of trajectories and present it in table 3.3 from which the reason for our earlier focus on A' .1 should be clear. As the effective cut decreases with energy we expect a more subdued dependence on energy than exhibited in fig. 3.9.
Figure 3.12 shows the fraction of the total amplitude that is contributed by the bundle of trajectories around A = 0 given the above cut profile, at
z = 10. (The curves above and below it are for the same quantity had the cut remained fixed at its corresponding values at the two ends of the energy range.) The ratio is essentially flat slightly below .1 in the plotted range. The
55
0.18
z=10
0.16 0.14 0.12
2
A 0.1
0.08 0.06 0.04
22 20 18 16 14 12 10 8 6 4
E
Figure 3.12 I i2: the solid curve is for V(r) with A = .5 and c = .125, the dotted and dashed curves are for Vo(r) with A' = .1015 and .0891, respectively.
corresponding effective trajectory would, therefore, be even flatter (and lower) than that in fig. 3.11 for A' = .1, requiring higher values of energy (for a given range of z) to display an upward turnabout. This, we suspect, is a closer picture of what takes place in QCD.
The behavior of the scattering amplitude in the neighborhood of E = 1 where the relevant trajectory of fig. 3.2 is in transition from a Coulombic mode to the asymptotically free limit, is an indicator of how the hard physics departs from its softer progenitor. Taking the largez limit of the Q's in eq. (3.30) for the parialwave amplitude, An, SiAn 13, F(A +.5)2 (3.50) A2 (z)l (3.50) ksinrAn (An+1)'
56
50o 45 40 35
30 1 25
0
: 20 15 10
5
0 I I I I
5.5 5 4.5 4 3.5 3 2.5 2 1.5 1 0.5
E
Figure 3.13 The ICoulomb/exactl2ratio for the "residue function" in the partialwave scattering amplitude: the plot is for the first node and "exact" refers to the model of V(r) with A = .5 c = .125.
we call the coefficient of (z)1n in the above expression the "residue function" and examine the departure of that from a Coulombic behavior around and beyond the transition region. The 3's above can be obtained from the formula [11] n(k) =Ak fo drf2(r)/r2 where fn(r) are the Jost solutions of the Schridinger equation defined by the boundary condition f(r + ei(kr+ ln2kr)
57
The ratiosquared of the residue function of a Coulomb potential to that in eq. (3.50) is plotted in fig. 3.13 for the first node. The transition region is around E .7 where the plotted ratio is below one and shows a rapid rise with energy.
3.4 Summation
To the extent that our choice for the parameters of the model incorporating asymptotic freedom was justified and that its bundle of trajectories simulates the squareroot branch singularity of the fixedcoupling model, the analytic and numerical results are highly supportive of our conjecture regarding the behavior of the QCD trajectories. Although this substantiation can not be made exact, as inclusive strong interactions are described by a richer set of parameters than our model, the various plots support the correspondence of QCD with the quantum mechanical model of effective coupling less than .1. These results also support calculations of ref. 10 both in their range of applicability and the values obtained.
CHAPTER 4
RENORMALIZATION GROUP STUDY OF THE
STANDARD MODEL AND AN APPLICATION
The renormalization group (RG) equations describe the evolution with scale of renormalized parameters of a theory. The standard model parameters can thus be extrapolated from scales at which their values are known to scales where they are subjected to constraints of a hypothetical extension of the standard model [12]. These techniques have helped rule out some simpler extensions of the standard model. An example is the ruling out of simple grand unified extensions of the standard model based on the absence of the gauge coupling unification at grandunified scales [13]. No doubt such conclusions based on evolution over widely separated scales can be sensitive to the initial values of certain parameters. Below we list our choice for the initial values [14]. We employ the modifiedminimalsubtraction (MS) scheme where the running couplings are unphysical, and we collect the 0 functions of the standard model in Appendix A.
From the decoupling theorem we expect the physics at energies below a given mass scale to be independent of the particles with masses higher than this threshold. Therefore for a correct interpretation of these running couplings we must take into account the thresholds [15,16,17]. For the electroweak threshold we use one loop matching functions[17] with the two loop beta functions valid in the standard model regime. These matching functions are obtained in MS renormalization by integrating out the heavy gauge fields in such a way that 58
59
the remaining effective action is invariant under the residual gauge group [16]. At the electroweak threshold, near Mw, we integrate out the heavy gauge fields and the top quark. Below this threshold there is an effective SU(3)c x U(1)EM theory. Thresholds in this region are obtained by integrating out each quark to one loop at a scale equal to its physical mass. At these scales the one loop matching functions in the gauge couplings vanish and the threshold dependence appears through steps in the number of quark flavorq18] as the renormalization group scale passes each physical quark mass.
4.1 The Initial Values
The values of al(MZ) and a2(M), the U(1) and SU(2) running coupling constants at MZ, are obtained from measurements of the QED coupling constant and the Weinberg angle from a fit to all neutralcurrent data. They are aI (MZ) = .01698 0.00009
(4.1)
a2(MZ) = .03364 0.0002 .
The value of as has been more difficult to determine. Due to the larger values of the coupling at lower energies, renormalization scheme dependences of the truncated perturbation series can be nonnegligible. We use the Gaussian weighted average of the as determined from e+ e annihilation to hadrons, T decay, deepinelastic scattering and jet production in e+ e collision, to obtain
as(Mz) = .113 .004 .
The Yukawa matrices were diagonalized at each step of the numerical procedure in order to relate the matrix elements to the known parameters such as
60
the quark masses and the CKM mixing angles both as input and output. We choose the following parametrization of the unitary CKM matrix
( cl 1cl3 s1S3
V slc2 Clc2c3 s2s3ei 1c23 + s2c3ei (4.2)
s8s2 clS2c3 + c2s3ei cls2s3 c2c3eid with si = sin Oi and ci = cos 0i, i = 1, 2, 3. The experimental ranges of these parameters are
V = 0.218 0.224 0.97340.9752 0.0300.058 (4.3)
0.0030.019 0.0290.058 0.99830.9996)
From the above two equations we arrive at the following bounds on the mixing angles
0.2188 < sin 01 < 0.2235 ,
0.0216 < sin 02 < 0.0543 (4.4)
0.0045 < sin 03 < 0.0290 .
However the accuracy with which IVI is known does not constrain sin 6. A set of angles {01, 02, 03, 6} was chosen that falls within the ranges quoted above. It is also not clear at what scale should the above initial values be considered known. However, since for the whole range of initial values the running of the mixing angles is quite flat, in accordance with their being related to ratios of quark masses, the choice for that scale is not crucial.
The physical values of the lepton masses are known to a very good precision, resulting in the running values me(1 GeV) = 0.496 MeV ,
mp(1 GeV) = 104.57 MeV (4.5) mr(1 GeV) = 1.7835 GeV The running top quark mass is chosen arbitrarily, although consistent with the (thenavailable) bound on the physical mass Mt = 122+41GeV. For the
 \V YUI~lV1 YIU LIIU~U L IIC(J(~l IL32
61
physical Higgs boson mass the bound 91 < MH 1TeV is used consistent with observations and perturbativity. The running and physical masses of the top and the Higgs are related thru Mt 4 as(Mt)
=1 + Mt)+ 16.11
mt(Mt) 3 7r
5 (as(MO))2 (4.6)
1.04 (1 )] s(Mt) 2 i=l
and
A(p) = M21 + 6()] (4.7) where Mi, i = 1,..., 5 represent the masses of the five lighter quarks and 6(p) contains the radiative corrections as presented in Appendix B. The masses of the five lighter quarks are discussed separately.
For the vacuum expectation value of the scalar field the following wellknown value suffices:
v = (/ G,) = 246.22 GeV The quark masses are a special class of low energy standard model parameters in that the renormalized quantity which appears in the Lagrangian does not have a direct physical analog. Since quarks are not observed as physical states, their masses do not correspond to poles of a physical propagator. interest. In the past decade a variety of techniques have been developed and utilized to extract quark masses from the observed hadronic spectrum. Below, we shall briefly recount some such techniques. Furthermore, we shall present new values for the heavy quark masses based on the application of our numerical technique to three loops.
62
The light quark masses are the ones least accurately known. They are determined by a combination of chiral perturbation techniques and QCD spectral sum rules (QSSR). In the former case the light quark masses are directly expressible in terms of the parameters of the explicit SU(2) and SU(3) chiral symmetry breakings. One then considers an expansion of the form [19]
Mbaryon = a + b mlight + ... (4.8) for the mass of a baryon from the octet and one of the form
Mmeson = Bmlight + ... (4.9) for a typical member of the pseudoscalar octet. A parameter measuring the strength of the breaking of the more exact SU(2) chiral symmetry in comparison with the SU(3) one is the ratio R ms m
R (4.10) rnd mu
where
1
m' (mu + md) (4.11) To lowest order in isospin splittings, this translates in the meson sector into M2 M2
R= K (4.12) M M2
KO K+
and in the baryon sector into three different determinations of R, S (M MN) (ME MA) Mn Mp
S '=(M M ) + l / A) (4.13) M Mo'
M MN
R
Mr Mr+
63
To make R compatible with all the above mass splittings one has to consider higher order corrections in Eqs. (4.8) and (4.9). Here infrared divergences emerge as one is expanding about a ground state containing NambuGoldstone bosons. Once such singularities are removed within the context of an effective chiral Lagrangian, one finds the following as the optimum value of R R = 43.5 + 2.2 (4.14) Together with the ratic20]
= 25.7 + 2.6 (4.15) also determined by applying eq. (4.9) to the physical masses of 7r, 77 and K, they imply the following renormalization group invariant mass ratio md  mu
= 0.28 0.03. (4.16) 2m'
QCD spectral sum rules are obtained in an attempt to relate the observed low energy spectrum to the parameters describing the high energy domain where perturbation theory becomes applicable to the quarkgluon picture [21] One starts by considering the twopoint correlation functions for the vector ,j = jyli and axial vector A1' = 5'0j y1'Y5i quark currents i d4xe iqx < 01T(V? (x)Vit (0))0 >
(q41qV g qy),Q) (q2) + q1'qvH>) (q2) ( (4.17)
i Id4xeiq.x <01T(A~(x)A1 (0)) 0 >
(qqV g/Vq2)H}l)A(q2) + q j1,All (q 2) ( 9 9 2 A 2 ij,A where i, j = u, d, s are the quark flavors. The current divergences satisfy
Apvi(x) = i(mi mj) :i(x)'(x): (4.18) aPA j(x) = i(m i + rj) : ;i(x) ,50j(x):( .
tj~ \4 "
64
The spectral functions Im{H(q2)} obey certain sum rules based on how their analyticity properties are formulated. Among the QSSRs in vogue are the usual dispersion relations based on a Hilbert transform
II(q2) = 1_ dtImI2 (t)} (4.19) 7rJ0 t + q2
The Laplace transform sum rule is obtained by applying the inverse Laplace operator L to the last expression, 00
LII = dtetIm{II(t)} (4.20)
for r a constant. The longknown finite energy sum rule (FESR) is obtained by applying the Cauchy theorem to II(z)
qdtnIm{Htheor. (t)}m { dt tnIm{Hexp.(t)} (4.21)
7rgo 7rJO
where n is any integer, and the moment sum rules are obtained by taking the nth derivative of eq. (4.19)
.M(n) (_1)n dn (q2
Sn! (dq2)n(
1 1 dt (4.22) Jo (t + q2)n+l
The left hand sides of Eqs. (4.19) through (4.22) follow from the high energy calculations to which various perturbative and nonperturbative corrections have been found, while the right hand sides represent the low energy aspect, such as the hadronic vacuum polarization measured in e+e 4 hadrons. Applied to the light quarks these sum rules imply [20]
rnu + rd = 24.0 + 2.5 MeV (4.23)
65
Together with eq. (4.16) they reduce to meu = 8.7 0.8 MeV ,
(4.24)
rhd = 15.4 0.8 MeV The parameter r~ is a renormalization group invariant which to three loops is related to the MS running mass parameter m(p) via [22] as 2Y1 _2 a
(m() m(;1 1/{1 31 2 s()
1 1 2
1 2 y1 2)2 322 (1 Y2 + [( ) ( (4.25) 02 01 02 12 1 2 + 3(3'1 _3)] (Is (/))2}
3 31 73 a
where the Oi and the yi are the coefficients of the beta functions for as and m given in appendix A. From eq. (4.24) and eq. (4.25) to two loops, one may infer the following values mu(1 GeV) = 5.2 0.5 MeV (4.26)
md(1 GeV) = 9.2 0.5 MeV In applying expression (4.24) it should be kept in mind that the continuity of m(p) across a quark mass threshold requires n to depend on the effective number of flavors at the relevant scale. The strange quark mass is determined, averaging the value extracted from Eqs. (4.23) and (4.15) with those derived using eq. (4.24) and the various QSSR values for mnu + mns to be[23] ^s = 266 29 MeV (4.27) corresponding to the running value ms(1 GeV) = 194 4 MeV (4.28) For the heavier quarks, charm and bottom, one can make a more precise prediction. Here the nonrelativistic boundstate approximation may be applied. The physical mass M(q2 = M2) appearing in the Balmer series may
66
be identified with the gauge and renormalization scheme invariant pole of the quark propagator
S(q) = z(q)[y q M(q2)]1
Corresponding to the above pole mass is its Euclidean version, m(q2), which although renormalization group invariant, is not gauge invariant and therefore, not physical. The Euclidean mass parameter is the one often employed in the J/0 and T sum rules, as it minimizes the radiative corrections in such sum rules. In the Landau gauge the two are related to two loops according to [24]
(M2)= M(M2) [1 ln4 (4.29)
Once the pole mass is determined from the Euclidean one, the running mass at the pole mass is obtained to three loops via
m(q2 = M2) M(q2 (4.30)
1 + a + K(Q' )2
where K = 13.3 for the charm and K = 12.4 for the bottom quarks[25].
From the J/IJ and T sum rules the following values have been extracte420] mc(q2 = M2) = 1.26 + 0.02 GeV (4.31)
rb (92 M2) = 4.23 + 0.05 GeV To obtain an accurate value for the corresponding pole masses, we applied our solution routine to eq. (4.29), with the above values inserted and the three loop beta function for as given in appendix A to selfconsistently obtain the following pole masses
Mc(q2 = M2) = 1.46 0.05 GeV, (4.32)
Mb(q2 = M) 4.58 0.10 GeV.
67
Recently[26], new values for the charm and bottom pole masses have been extracted from CUSB and CLEO II by analysis of the heavylight, B and B*, D and D* meson masses, and the semileptonic B and D decays with the results Mc(q2 = M2) = 1.60 0.05 GeV, (4.33)
Mb(q2 = M2) 4.95 0.05 GeV A weighted average of the values in Eqs. (4.32) and (4.33)yields Mc(q2 = M2) = 1.53 0.04 GeV, (4.34)
Mb(q2 = M2) = 4.89 0.04 GeV The running masses at the corresponding pole masses then follow from eq. (4.30) mc(Mc) = 1.22 0.06 GeV ,
(4.35)
mb(Mb) = 4.32 0.06 GeV .
With these taken as initial data along with the value of the strong coupling at MZ quoted earlier, we run (to three loops) the masses and as to obtain the following values at the conventionally preferred scale of 1 GeV mc(1 GeV) = 1.41 0.06 GeV ,
(4.36)
mb(1 GeV) = 6.33 0.06 GeV.
Our numerical approach does not make any more approximations than the ones assumed in the beta functions and the mass equations used, and is therefore more commensurate with our program than using the "perturbatively integrated" 0 functions. Thus we shall adopt the above values. It should be stressed that at the low scales under consideration the three loop as corrections we have included in our mass and strong coupling beta functions are often comparable to the two loop ones and hence affect the accuracy of our final values noticeably. Nevertheless, it should be noted that the above expressions relating
68
the various mass parameters are not fully loop consistent as to our knowledge eq. (4.29) has only been computed to two loops.
It should be pointed out that although we opted for the QSSR extraction of masses, there are rival models, such as the nonperturbative potential models, which predict appreciably higher values of the heavy quark masses than the ones quoted here. These models, however, are not as fundamental as the approach considered here, and their connection to field theory is rather problematic.
4.2 Results
The results of numerically integrating the / functions of the standard model parameters from 1 GeV to the Planck mass are displayed in the following nine figures. Unless otherwise stated, Mt = MH = 100GeV for these plots. In fig. 4.1 the evolution of the inverse of each of the three gauge couplings are shown, assuming the standard model as an effective theory in the desert up to the Planck scale. The "GUT triangle" signifying the absence of grand unification is evident. Here, the differences between one and two loop evolution appear in the high energy regime, and, for the strong coupling constant, at low energies, where it is stronger, as well. In fig. 4.2 these same inverse couplings are displayed, this time including the associated uncertainties in their values. We note, as is well known, that the uncertainties do not fill in the "GUT triangle".
Figures 4.3 thru 4.5 display the one and twoloop evolution of the light mass fermions (me, mu, and md), the intermediate mass fermions (m/ and ms), and the heavy mass fermions (mr, mc, and mb), respectively. Evidently, the largest differences between one loop vs. two loop evolution occur in the bottom, charm, and strange quark masses in these cases. Figure 4.6 is the plot
69
60
  1 Loop al
50 a 2 Loop
I
40
C 30
1 GUT TRIANGLE a21 50 1., .... .
48
I r(20. 46 44
42
0
40
1 38
12 13 14 15 16 17
0 5 10 15 20 log 0o(#)
Figure 4.1 Running of the inverse gauge couplings with their central value used as initial data.
for the self quartic coupling A and the top Yukawa coupling Yt for (Mt = 100 GeV, MH = 100 GeV) and for (Mt = 200 GeV, MH = 195 GeV). We have studied the effects of changing Mt and MH values in our analyses of the running of the other parameters. For any Mt between 100 GeV and 200 GeV, varying MH, while maintaining perturbativity and vacuum stability, did not affect appreciably the evolution of any of the other parameters. However, varying Mt itself showed a significant difference in the running of the heavier quarks.
This is illustrated in figs. 4.7 and 4.8 which are similar to figs. 4.4 and 4.5 except that Mt = 200 GeV and MH = 195 GeV. Note that in fig. 4.8 the intersection point between the bottom quark and the 7 lepton moves down to a lower scale for this case of a larger top quark mass. This is expected since
70
1
50
40
I a 30
20
40
o 1 12 13 14 115 10 17
0 5 10 15 20 logo( )
Figure 4.2 Running of the inverse gauge couplings at two loops: the pair of dotted lines denote the experimental errors for the central value (solid line).
from eq. A.10 one can see that the bottom type Yukawas decrease with an increasing top Yukawa. In contrast, in the SUSY GUT case the bottom type Yukawa 0 function is such that this crossing point is shifted toward a higher scale with increasing top mass. In an SU(5) SUSY GUT model, the equality of the bottom and 7 Yukawas at the scale of unification will be used in the next chapter to get bounds on the top and Higgs masses.
Figure 4.9 displays the running of the CKM angles using the initial data sin91 = 0.2206, sin92 = 0.0298, and sin93 = 0.0106. We have also taken 6 = 900 which corresponds to the case of maximal CP violation. As mentioned previously, the evolution curves for these angles are effectively flat.
71
0.0 I I I I I
....... 1 oop
2 Loop
0.015 M= 100 GeV
M9 100 GeV
0.010
d
0.005
0.000
m*
0.005
0 5 10 15 20 log Io()
Figure 4.3 Light quark and lepton massses.
In the present case of the standard model, we find that two loop running of the parameters does at times improve on the one loop running. We have tabulated the differences of several parameters in their one versus two loop values at various scales, for the cases (Mt = 100 GeV, MH = 100 GeV) and (Mt = 200 GeV, MH = 195 GeV). Table 4.1 illustrates the difference one loop vs. two loop running make in the ratio mb/mr, for the three scales 102 GeV, 104 GeV, and 1016 GeV. As expected the difference between one and two loop results is more pronounced at higher scales. Over all these scales the difference is never greater than 10%. We note that the ratio becomes equal to one well below the scale of grand unification as noted in the discussion of figs. 4.5 and 4.8. Table 4.2 presents a similar comparison for the top Yukawa. Here, two loops represent a smaller correction with the difference at all scales
72
0.25.. Loop 2 Loop
.2 Mt= 100 GeV
MH=100 GeV
m 0, 1 E 0o.
m
0.10
0.05
0.00
0 5 10 15 20
logloW)
Figure 4.4 Intermediate quark and lepton massses.
always being less than 5%. Lastly, Table 4.3 displays the same analysis for as for the case Mt = MH = 100 GeV. We observe no appreciable deviation from the tabulated values for any Mt < 200 GeV (except in the low energy regime where the difference is at most 4%).
At scales < MZ, the inclusion of two loops is important in the evolution of the strong coupling (and of the quark masses). Indeed, we find that the pure QCD three loop contribution is also significant and therefore include it in the running of the strong coupling and of the quark masses in the low energy region. As seen in this table, the combined two and three loops in the low energy regime account for a 17% difference at 1 GeV in ..
73
.... 1 Loop
6 2 Loop M= 100 GeV
MH=100 GeV
4
mb
o t, ,, , ,
0 5 10 15 20 log20(gi)
Figure 4.5 Heavy quark and lepton massses.
Although in the cases considered in these last two tables there does not appear to be a significant difference in two loop over one loop evolution at scales above MZ, the first table does show a 10% difference at the scale, 1016 GeV. We expect two loop effects to be more important when the theory is extended, e.g., to include supersymmetry and/or grand unification.
The effects of using a naive step approximation vs. a proper treatment of thresholds are numerically unimportant for the cases discussed above. Indeed they are less important than the two loop effects. We note however, that the inclusion of nonnaive thresholds effects is significant in the numerical analysis of extensions of the standard model.
74
Yt  Mt= 100 GeV, MH= 100 GeV 1.00oo M=200 GeV, My=195 GeV
0.75
0.50
0.25
0.00 I
0 5 10 15 20 logoo()
Figure 4.6 Top Yukawa and scalar quartic couplings at two loops.
4.3 Top Quark and Higgs Boson Masses
The renormalizationgroup technique is now applied to extract bounds on the mass of the top quark and the Higgs boson in a minimal supersymmetric extension of the standard model (MSSM) with minimal Higgs structure in the context of a grand unified theory (GUT) [27]. The MS renormalization group equations for the standard model and the MSSM [28] are numerically integrated to evolve the parameters of the model to the Planck scale. As before full account of the Yukawa sector is taken by diagonalizing the Yukawa matrices at every step of the numerical routine.
If the standard model is the low energy manifestation of some yet unknown GUT or of a possible supersymmetric (SUSY) extension thereof, the three
75
025 '.
0 M=200 GeV MH=195 GeV a 0.15
4 m
E oo
0.05 m
0 5 10 15 20 logo(L)
Figure 4.7 Intermediate quark and lepton massses.
6
Mt=200 GeV MH=195 GeV
mb
S 2
0 me
0 5 10 15 20
log massses.o() Figure 4.8 Heavy quark and lepton masses.
76
0.3
sinO1
M 1 0.2
SC Mt=100 GeV
0..1 MH=lOO GeV
SsinU2
0.0 sinO3
0.11 1 1 I I I
0 5 10 15 20 log,10o(/l1GeV) Figure 4.9 CKM mixing angles.
Table 4.1
mb/m,
102GeV 104GeV 1016GeV one loop 1.938 1.499 .8326 two loop 1.937 1.463 .7962 Note: For the above Mt = 100GeV
102GeV 104GeV 1016GeV one loop 1.868 1.392 .6647 two loop 1.769 1.285 .6047 Note: For the above Mt = 200GeV
77
Table 4.2
Yt
102GeV 104GeV 1016GeV one loop .7879 .6076 .2830 two loop .7873 .5940 .2701 Note: For the above Mt = 100GeV
102GeV 104GeV 1016GeV one loop 1.133 .9780 .7145 two loop 1.143 .9700 .6816 Note: For the above Mt = 200GeV
Table 4.3
1GeV 102GeV 104GeV 1016GeV one loop .3128 .1118 .07103 .02229 two loop .3788 .1117 .07039 .02208
couplings g3, g2, and gl corresponding to the standard model gauge groups, SU(3)c x SU(2)w x U(1)Y, should meet at some large grand unification scale. Using the accepted values and associated errors of these couplings we observe unification in the SUSYGUT case but not in the pure GUT case, as noted by several groups[13,29] (see fig. 4.10). However this should not be viewed as proof of supersymmetry since given the values of al, a2, a3 at some scale, and three unknowns (the value of a at the unification scale, the unification scale, and an
78
Msusy = 1 TeV 150
1
as em 28
125
27 .. ..
75  25
a1
,. 15.6 15.8 16 16.2 16.4 I 50
U 25 10 5 10 15 20 logo(ju)
Figure 4.10 Running of the inverse couplings: dotted pair of lines are the experimental errors for the central value (solid line).
extra scale such as the SUSY scale) there is always a solution. The surprising aspect of the analysis of ref. 13 is the numerical output, namely a low SUSY scale, MSUSY, and a perturbative solution below the Planck scale which does not violate proton decay bounds.
Furthermore, in the context of a minimal GUT there are constraints on the Yukawa couplings at the scale of unification. First we restrict ourselves to an SU(5) SUSYGUT where Yb and y,, the bottom and Yukawa couplings, are equal at unification. The crossing of these renormalization group flow lines is sensitive to the physical top quark mass, Mt. This can be seen in the downtype
79
Yukawa renormalization group equation (above MsUSY, for example), from which we extract the evolution of Yb, since the top contribution is large and appears already at the one loop order through the uptype Yukawa dependence:
dYd 1
dtY 1Yd[ 3YdtYd + YutYu + Tr{3YdtYd+ YetYe}
7 2 16 (4.37)
15g + 3 3
where Yu,d,e are matrices of Yukawa couplings. Demanding that their crossing point be within the unification region determined by the gauge couplings allows one to constrain Mt. This yields an upper and lower bound for Mt which is rather restrictive. Our treatment of the thresholds below the SUSY scale was discussed towards the beginning of the chapter. For the threshold at MSUSy, the matching condition is the naive one of simple continuity, due to the lack of knowledge about the superparticle spectrum. We take this scale to be variable to account for this ignorance.
We consider the simplest implementation of supersymmetry and run the couplings above MSUSY to one loop. The superpotential for the supersymmetric theory is:
W = 4)uYufC + dYddc + 4diYec + ,Vd$ (4.38) where the hat denotes a chiral supermultiplet. We assume the MSSM above MsUSY, and a model with a single light Higgs scalar below it. This is done by integrating out one linear combination of the two doublets at MSUSy, thereby leaving the orthogonal combination in the standard model regime as the "Higgs doublet":
AD(sM) = 4IdcOsS + (usinO (4.39)
80
where A = iTr2*, and where tanG is also the ratio of the two vacuum expectation values (vu/vd) in the limit under consideration. This sets boundary conditions on the Yukawa couplings at MSUSy. Furthermore, in this approximation the quartic self coupling of the surviving Higgs at the SUSY scale is given by:
A(Mssy) = (g2 g2)cos2(2) (4.40) This correlates the mixing angle with the quartic coupling and thereby gives a value for the physical Higgs mass, MHiggs. Using the experimental limits on the MHiggs further constrains some of the results. By using the renormalization group we take into account radiative corrections to the light Higgs mass and hence relax the tree level upper bound, MHiggs MZ.
We determine the bounds on Mt and MHiggs by probing their dependence on 0. In SUSYSU(5), tan 0 is constrained to be larger than one in the one light Higgs limit. It seems natural to us to require that Yt > Yb up to the unification scale, thereby yielding an upper bound on tan 3. The initial values at MZ for the gauge couplings are taken to be:
a = 0.016887 0.000040 ,
a2 = 0.03322 0.00025 (4.41)
3 0.109n(+0.004
a3 0.005
where GUT normalization for al is used. We use the set of four quark running masses defined at 1 GeV by the Particle Data book: mu = 5.6 MeV, md = 9.9 MeV, ms = 199 MeV, and mc = 1.35 GeV. For the bottom mass we use the Gasser and Leutwyler bottom mass value of 5.3 GeV at 1 GeV which translates into a physical mass of Mb = 4.6 GeV [19]. To probe the dependence of our results on Mb we also study the case of Mb = 5 GeV, the typical value
81
obtained from potential model fits for bottom quark bound states [30]. We also investigate the effect of varying MSUSY. Given the values of the gauge couplings, we find unification up to a SUSY scale of 8.9 TeV, and as low as MW, below which we did not investigate for empirical reasons.
From fig. 4.10 we determine that the lower end scale, MGL of the unification region corresponds to an a3 value of 0.104 at MZ, while the higher end scale, MGHUT, corresponds to a value of 0.108 at M for a3. We find that the unification region is insensitive to the range of top, bottom, and Higgs masses considered. In our analysis of the bounds for Mt, the values for a, and o2 are chosen to be the central values since their associated experimental uncertainties are less significant than for a3. Demanding that yb and y, cross at MLUT and taking a3 = 0.104 then sets a lower bound on Mt. Correspondingly, demanding that yb and y, cross at MHUT and taking a3 = 0.108 yields an upper bound on Mt. These bounds are found for each possible value of 3.
Figure 4.11 shows the upper and lower bound curves for both Mt and MHiggs as a function of 3 and for MSUSY = 1 TeV and Mb = 4.6 GeV. When applicable we use the current experimental limit of 38 GeV on the light supersymmetric neutral Higgs mass, to determine the lowest possible Mt value consistent with the model. We find 139 < Mt < 194 GeV and 44 < MHiggs < 120 GeV. We investigated the sensitivity of these results on MSUSY in the range, 1.0 + 0.5 TeV. It is found that the bounds on Mt are not modified, but the upper bound on the Higgs is changed to 125 GeV, and the lower bound drops below the experimental lower bound.
For Mb = 5.0 GeV, we see an overall decrease in the top and Higgs mass bounds: 116 < Mt < 181 GeV, MHiggs < 111 GeV. Varying MSUSY as above
82
Msus = 1 TeV, Mb = 4.6 GeV
200
F top
MHiggs
150
00 F "
40 50 60 70 80 90
p (deg.)
Figure 4.11 Mt and MH as a function of the mixing angle 3: the highest (lowest) curves are for highest (lowest) value of a3 consistent with unification as per fig. 4.10.
modifies the upper bound on MHiggs to 115 GeV. We display the results of our analysis for the extreme case, MSUSY = 8.9 TeV, in fig. 4.12, with Mb = 4.6 GeV. This only significantly changes the upper bound on MHiggs to 144 GeV compared to the MSUSY = 1 TeV case.
We have also run Yt up to the unification region and compared it with yb and Yr to see what the angle / must be for these three couplings to meet [31], as in an SO(10) or E6 model with a minimal Higgs structure. It is clear that this angle is precisely our upper bound on / as described earlier. In fig. 4.13 we display Yt/Yb at the GUT scale as a function of tan / for MSusy = 1 TeV and
83
Msus = 8.9 TeV, Mb = 4.6 GeV
200
M.op
0 150
100 MHiggs
0 50
0 "I I I 111 l I
40 50 60 70 80 90
p (deg.)
Figure 4.12 Mt and MH as a function of the mixing angle / for the upper bound on MsUsY.
for the two bottom masses we have considered. If we demand that the ratio be one we can determine the mixing angles for the low and high ends of the unification region. Then going back to fig. 4.11 we find, as expected, a much tighter bound on the masses of the top and of the Higgs. Indeed, for Mb = 4.6 GeV, we have 49.40 < tan/ 5 54.98, which yields 162 < Mt < 176 GeV and 106 < MHiggs < 111 GeV. When Mb = 5.0 GeV, we obtain 31.23 < tan/3 < 41.18, which gives 116 < Mt < 147 GeV and 93 < MHiggs < 101 GeV.
We point out several simplifications made. We have not implemented the supersymmetric two loop beta functions and the corresponding thresholds. The effects of soft SUSY breaking terms were not investigated. Also, we have
84
Msusy= 1 TeV
102
0 MMb=5. GeVGeV
0 10 20 30 40 50 60
tan#f
Figure 4.13 Y vs tan 3 for two distinct bottom masses (solid and dashed
Yb
curves), and highest and lowest values of c3 (high and low curves) consistent with unification as per fig. 4.10.
integrated out all the supersymmetric particles at the same scale. Given the relative crudeness of the approximations in this paper, it is remarkable that the experimental bounds on the pparameter were satisfied. The top quark has since been observed at Fermilab with a mass of 174 f 10GeV.
CHAPTER 5
CONCLUSIONS
We have studied mainly two aspects of standard model phenomenology. The discrepancy between a recent calculation of the leading QCD meson trajectory and the relatively old experimental data on such trajectory is conjectured to be due to strong subleading effects in the scattering amplitude at the explored energies. The extraction of trajectories is subjected to a new analysis for both exclusive and inclusive pexchange processes. In both the exclusive and s = 400 GeV2 inclusive cases restricting the fit range to be closer to the Regge limit induces a slight rise in the trajectory around, but above, the t value where it crosses zero. Although the quantitative extent of this rise is noticeably less than that displayed by the experimentalists, it is consistent with our conjecture.
To estimate the energies needed to reach the Regge limit, a fairly simpleminded twopower fit of the scattering amplitude is made, with the leading power fixed at the predicted effective value of zero. The subleading term shows a strong coupling to the amplitude and the resulting estimate is for a beam energy of three orders of magnitude to probe the Regge limit.
To have a soluble model to directly test the behavior of the trajectories and the scattering amplitude a potential model that incorporates asymptotic freedom is studied. The resulting accumulation of the leading trajectories mimics that of pQCD. The model is then approximated by one where the accumulation
85
86
of trajectories is replaced by a fixed squareroot branch cut at every energy. For parameter values that arguably correspond to QCD, the resulting scattering amplitude is strongly influenced by the subleading effects at moderate energies and the effective trajectory it implies is well below the leading singularity.
The other aspect of our work is a twoloop order renormalization group study of the standard model subject to a full accounting of the Yukawa sector and a more sophisticated treatment of threshold effects. The inclusion of twoloop corrections was shown to be more significant than that of the nonnaive threshold effects. Assuming a desert between the electroweak scale and typical GUT scales, the three gaugecouplings failed to unify. However, extending the RG equations of the standard model by minimal supersymmetry, induced coupling unification at moderate values of the SUSY scale. Demanding the equality of the tau and bottom masses at the resulting unification scale in the extended model yielded top and Higgs mass predictions, the former of which has subsequently been verified at Fermilab.
APPENDIX A
0 FUNCTIONS FOR THE STANDARD MODEL
In this appendix we compile the renormalization group 3 functions of the Standard Model. These have appeared in one form or another in various sources. We have endeavored to confirm their validity through a comparative analysis of the literature. Our main source is ref. 28. Following their conventions,
L =QLJYUtUR + QLYdtd + iLYeten + h.c.
1cb D)2 (A.1) where flavor indices have been suppressed, and where QL and L are the quark and lepton SU(2) doublets, respectively,
Q= (UL) tL = (VL) (A.2)
I and # are the Higgs scalar doublet and its SU(2) conjugate:
= o = il2A. (A.3) UR, dR, and eR are the quark and lepton SU(2) singlets, and Yu,d,e are the matrices of the uptype, downtype, and leptontype Yukawa couplings.
The 3 functions for the gauge couplings are
d 3 2 g3 13
xTr{C1uYtYu + CldYdtYd + CletYe) ,
87
88
where t = In p and 1 = 1,2,3, corresponding to the gauge group SU(3)C x SU(2)L x U(1)y of the Standard Model. The various coefficients are defined to be
4 1
bl = n 10 22 4 1 b2 = 3 ng (A.5)
4
b3 11  ng
3 /
19 1 11 9 3 0 (bkl) = 0 0 n9 5 2 T 0 (A.6)
= 0 102 4 7 0 0 0
and
(Cif)= 3 2 with f = u, d, e, (A.7)
with ng = nft.
In the Yukawa sector the 0 functions are
dYu d,e 1 (1) 1 (2) (A.8)
dt = (162 u,d,e (167r2)2 u,d,e) ude where the one loop contributions are given by
P(1) (YYu YdtYd) + Y2(S) 17 92 17g + +g2 + 8g2) 20 2 4 1) = (YdYd YutYu) + Y2(S) (A.9)
12 + + sg ),
4 4
) YetYe + Y2(S) (g + g2)
2 4 with
Y2(S) = Tr{3YutYu + 3YdtYd + YetYe} (A.10)
89
and the two loop contributions are given by
(2) 3 11(y y 2 4 4
+ Y2(S)( YdtYd YUtYu)
4 4
S4(S) + 3 2 2A(3YuYu + YdtYd) + (1223 + + 16g)Y Y
43 2 9 2 5 9 29
g(9 92 + 16g3 +Y4(S) + (2 + gn 4 80 16 2200 45ng)g
9 2 2 19 2 2 35 2 2 404 80
g 92 g1g3 ng)g + 992g3 ( 3 9 ng)g
20 15 4 3 9
() 3(YdY d)2 YdtdYutYu YUtYuYdtYd + l(YtY)2
+ Y2(S)( Yty ydtyd)
4 4
x4(S) + 2 2A(3YdtYd + YutYu) + ( g + g2 + 16g )Yd 3g80 16 79 9 2 5 29 1
(91 + 16g3 )YutY + Y4(S) ( n)g
80 16 2 200 45
272 31 2 2 35 4 22 404 80 4
9M +9193  tg)g92 + 992g3
20 15 4 3 9
(2) ( )2 92(S)YetY, (S) + 3 e2 (ASeYe
2 4 2
387 2 1352y
+(91 + g
80 15
5 51 11 4 2722 35 A
+ 2Y4(S) + (200 5 ng)g + 20912 ng)g
(A.11)
with
Y4(S) =(g1 + 92 + 8g 12 9
+( 4g + g +8g )Tr{YtY} (A.12)
4 91 2
and
X4(S) 9Tr{3(Yu Yu)2 + 3(Ydtd)2 + (YetYe)2
2 (A.13)
YutyuydtYd
In the Higgs sector we present 0 functions for the quartic coupling and the
vacuum expectation value of the scalar field. Here we correct a discrepancy in
90
the one loop contribution to the self quartic coupling of ref. 28 dA 1 (1) 1 2) (A.14) dt 16r2 A (16r2)2 where the one loop contribution is given by
1) =12A2 2 + 9g)A + (g2 + g4)(A.15)
P 5 4 25 5 (A.15) + 4Y2(S)A 4H(S) ,
with
H(S) = Tr{3(Yu Yu)2 + 3(YdtYd)2 + (YetYe)2} (A.16) and the two loop contribution is given by =2) 78A3 + 18( 3g2 + 3g9)A2
313 4 117 2 9 229
[( 10ng)g + 2012 +25 4 + 2ng)g4 ]A
497 3 97 8 9 239 40
( 8ng)g ( ng)g2 24 ng)g g2
8 59 24 3402524
27 59 + n40 )gl 64gTr{(YutYV)2 + (YdtYd)2} 125 24 + d )2
g2Tr2(YuYu)2 (YdtYd)2 + 3(Ye tYe)2} 4(s)
10A[(7 g 9g 2 8g)Tr{YetY ( + 8g )Tr{YdtY
+ g[ ( g 91 + 21g)Tr{Y Yu} + ( +9g2 + 893)Tr{Y t
20 4 4 41
+( g + 11g)Tr{YetYe}]
24A2y2() AH(S) 6ATr{Yu YuYdt}
12Tr{Yutyu(ytyu + ydtyd) dtYd (
(A.17)
The 3 function for the vacuum expectation value of the scalar field is
dnv (1) 1 (2) (A.18) dt 167r2 (167r2)2
91
where the one loop contribution is given by
(1) = ( + g2) Y2(S) (A.19)
4 5
and the two loop contribution is given by
7(2) =X4(S) 12 23 4 221 4 27 2 2 17 9)  92 5[(172 9 22 16g2)Tr{YtYu} 10 2(A.20)
2 2
+ (g2 + g2 + 16g)Tr{YdtY d}
+ (g + g)Tr{YetYe}
These expressions were arrived at using the general formulas provided in ref.
28 for the anomalous dimension of the scalar field, choosing the Landau gauge.
In the low energy regime the effective theory is SU(3)C x U(1)EM. We
employ the general formula of ref. 32 to arrive at the 0 functions for the
respective gauge couplings:
dg3 2 3
dt=[(nu +nd) 11]( 93)2 dt 3 (47)2
38 g + [ 3(nu + nd) 102] 9
3 (4r)4 (A.21)
8 2 g3e2 5033 + [8nu + nd] (4 + [51 (nu + nd)
325 2857 g7 S(nu + nd)2 7 54 2 (47)6
and
de 16 4 4 e3
[ n + nd + nl 4)2 dt 9 9 3 (4] 4)2 64 4 e5 + [ nu + nd + 4nl e (A.22) 27 27 (47)4 64 16 e3g2 + [ nu + nd] (4)4 9 9 (47r)4
92
where nu, nd, and n are the number of uptype quarks, downtype quark, and leptons, respectively. In eq. (A.21) we have also included the three loop pure QCD contribution to the 3 function of g3 [33].
For the evolution of the fermion masses we used ref. 34 It is known that there is an error in their printed formula [35]. Using the corrected expression, we compute the following mass anomalous dimension. The fermion masses in the low energy theory then evolve as follows: dm
dt = 7(l,q)m (A.23) where the I and q refer to a particular lepton or quark, and where
1 e2 3 93 Y(l) (1,q ) ( (4, )2 '(1,q) (4r )2
1 [ 11 4 33 4 13 2 2 (A.24) + (4(7r 4 1,q)g3 + 2Y(,q)e23] (A.24) S333 93
(q) (47r)6
The superscripts 1 and 3 superscripts refer to the U(1)EM and SU(3)C contributions, respectively. Explicitly, the above coefficients are given by
1 6Q2jq)
(1,q)
3
() = 8
7 13 33
(l) Y(I)
11 34 8, 0 + 2 0 2 (A.25)
(1,q) (1,q) + 9 9 d I Q(l,q)
13
7(q) = 4Qq)
33 404 40
(q) 3 + (nu + nd)
333 = 2140 2 2216
(q) (nu + nd) (16(3) + )(nu + nd) 3747],
93
where Q(l,q) is the electric charge of a given lepton or quark, and ((3)
1.2020569... is the Riemann zeta function evaluated at three. In the mass anomalous dimension for the quarks above, we have also included the three loop pure QCD contribution y333 [33].
(q)
APPENDIX B
THE EXPRESSION FOR 6(p)
In ref. 36 the radiative corrections term 6(p) from eq. (4.7) is derived. In
this appendix, we present its explicit form as it appears in this reference except
for some minor notational changes. In the following, s and c refer to sin 0W
and cos OW, respectively. Also, ( is defined to be the ratio Mj/M.
G M2
6(g) = M { fi((, /) + fo((, P) + C1f1(, )} (B.1)
V2_ 87r2
where the various functions are defined as follows:
p2 3 1 c2 9 25 7r fl (, p) =61n + In l Z( )Z()lnc2 ( M2 2 2 2 9
p 2 2 2 2 p2 3c2 In c2
2fo(2, ) n2 [1+2 2 + In 2 2Z( ) + 4c2Z( +
M2 Mt c2 c2 822
12 lnc2 15 M2 M2 M2
2c2nc (1 + 2c2) 3 t[2Z( Mt ) + 41n t 5]
2 M2 M( M2
fl_(,p) =6 In [1 + 2c4 4 ] 6Z() 12c4Z( ) 12c4 In c2 + 8(1 + 2c4)
M4 2 M2
S24 t [ln t 2+Z( Mt )
M4 M2 M( '
(B.2)
with
2Atan (1/A) for(z > 1) Z(z) = 4 (B.3) A ln[(1 + A)/(1 A)] for(z < 1)
4
A 14z1.
94

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PHENOMENOLOGICAL ASPECTS OF
THE STANDARD MODEL: HIGH ENERGY
QCD, RENORMALIZATION, AND A
SUPERSYMMETRIC EXTENSION
By
SAMUEL MIKAELIAN
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
1996
ACKNOWLEDGEMENTS
As in any process of learning one is often influenced by many more sources
than one can enumerate, the list below, regarding my graduate work, can only
be a partial one.
I am thankful to my adviser, Charles Thorn, for suggesting and guiding
me through the main topic of this thesis, regarding pQCD Regge trajectories,
and, in general, my learning experience in high energy theory. His standards
of scholarship and scientific integrity have set ideals for me to strive toward. I
also thank Pierre Ramond for suggesting the topic for the latter part of this
work, renormalizationgroup study of the standard model. His teachings have
also been a guiding influence in my education. The justmentioned part of this
thesis would not have been possible without the collaboration of my fellow exÂ¬
students Haukur Arason, Diego CastaÃ±o, Eric Piard and Brian Wright. Thanks
also go to Jorge Rodriguez for introducing me to various software packages that
facilitated my work. I thank my other committee members for their efforts and
indulgence, especially Rick Field for current assistance with completing some
aspects of this work.
I also acknowledge the patience and support of my parents (d.) at crucial
times in the past.
n
TABLE OF CONTENTS
Page
ACKNOWLEDGEMENTS ii
ABSTRACT v
CHAPTERS
1 INTRODUCTION 1
2 QCD REGGE TRAJECTORIES 3
3 THE POTENTIAL ANALOGY 24
3.1 The Trajectories 26
3.1.1 The Model Incorporating Asymptotic Freedom ... 26
3.1.2 The Model with Fixed Coupling 35
3.2 The Scattering Amplitude: The FixedCoupling Case ... 37
3.2.1 MandelstamSommerfeldWatson Transform and the Regge
Limit 37
3.2.2 Analytical Estimates 42
3.2.3 Numerics 47
3.3 The Scattering Amplitude: The Asymptotically Free Case . 52
3.4 Summation 57
4 RENORMALIZATION GROUP STUDY OF THE STANDARD MODEL
AND AN APPLICATION 58
4.1 The Initial Values 59
4.2 Results 68
4.3 Top Quark and Higgs Boson Masses 74
5 CONCLUSIONS 85
iii
APPENDICES
A (3 FUNCTIONS FOR THE STANDARD MODEL 87
B THE EXPRESSION FOR
C SELFDUAL SOLITONS IN A CHERNSIMONS MODEL WITHÂ¬
OUT A BROKEN VACUUM 95
C.l The Model 96
C.2 Specific Solutions 100
C.3 A Note 106
C.4 Comments 107
REFERENCES 109
BIOGRAPHICAL SKETCH Ill
IV
Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy
PHENOMENOLOGICAL ASPECTS OF
THE STANDARD MODEL: HIGH ENERGY
QCD, RENORMALIZATION, AND A
SUPERSYMMETRIC EXTENSION
By
Samuel Mikaelian
May 1996
Chairman: Charles B. Thorn
Major Department: Physics
Two aspects of the standard model are the main focus of this work. One is
understanding the high energy behavior of QCD in certain experiments, and
the other is a study of the scale dependence of the standard model and its
minimal supersymmetric extension.
Available data on exclusive and inclusive pionproton reactions at high
energies and large momentum transfers are analyzed for signs of a subleading
effect strongly affecting the effective p Regge trajectories. It is found that the
data are consistent with a strongly coupled subleading trajectory at around
a = â€” 1. Based on this, an estimate for the beam energy of around 103 GeV or
more is made in order to detect the onset of the true Regge limit. To shed light
on the physics of the Regge limit of QCD, we study a nonrelativistic potential
model incorporating asymptotic freedom. The large negative energy limit of
the trajectories is shown to simulate those predicted by QCD. It is also found
that for a choice of parameters corresponding to those of QCD, the scattering
amplitude exhibits a slow approach to the Regge regime, consistent with our
data analysis.
v
The scale dependence of the standard model and its minimal supersymÂ¬
metric extension are determined to two loop order, taking full account of the
Yukawa sector. We employ the best available data to extract the quark masses
to which the results are sensitive. Demanding the equality of the bottom and
r Yukawa couplings at the scale of unification, we place bounds on the top
quark and Higgs boson masses. For MguSY = 1 TeV and Mj, = 4.6 GeV, they
are 139 < < 194 GeV and 44 < <120 GeV. Subsequently, Fermilab
has measured M* to be 174 Â± 10 GeV, consistent with our bounds.
vi
CHAPTER 1
INTRODUCTION
Since QCD has resisted a definitive solution for more than twenty years,
any aspect of it that can be confronted with experiment deserves particular
attention. Regge trajectories in the large negative momentumtransfer limit of
QCD provide an instance. Recent calculations of such meson trajectories are
not born out by the available data. In chapter 2, after a brief introduction to
the topic, we reanalyze the exclusive and inclusive data for signs of a strong
subleading effect in the scattering amplitude taking hold close to the point
where the trajectory turns negative. We discuss the similarities and differences
between our fits and those already available in the experimental literature. An
estimate is made of the beam energy needed in an inclusive reaction to probe
the Regge region.
In chapter 3, a potential model is introduced in order to examine the beÂ¬
havior attributed to QCD in a solubleyetapt analogy. The resulting leading
trajectories are examined both numerically and analytically and compared to
the QCD ones. The scattering amplitude and its implications for the effective
trajectories, are studied analytically and numerically for a model correspondÂ¬
ing to fixedcoupling behavior. This is then used to infer the corresponding
behavior of the asymptotically free model.
The renormalization group is an indispensable tool in extracting lowenergy
predictions from proposed extensions of the standard model. In chapter 4 we
1
2
introduce a rather comprehensive numerical treatment of this and examine
the relative merits of various corrections to the naive treatment hitherto perÂ¬
formed. We apply this machinery to a minimal supersymmetric extension of
the standard model subject to grandunified constraints, to place bounds on
the top quark and Higgs boson masses.
Chapter 5 summarizes our results. Appendix A contains the (3 functions
of the standard model and appendix B regards the radiative correction to
the Higgs mass. Appendix C presents a scalar chernsimons model in 2+1
dimensions with novel soliton solutions.
A clarification of notation is in order here. In chapters 2 and 3, which
involve Regge theory, a refers to an angular momentum trajectory, unless it is
aQCD and as defined in chapter 2 or oÂ¿qM defined in the chapter 3. In chapter
4 and thereafter, on the other hand, a will exclusively refer to gauge couplings.
In retrospect, a common theme of this thesis could be aspects of mass
spectra in particle physics.
CHAPTER 2
QCD REGGE TRAJECTORIES
Quantum Chromodynamics (QCD), an SU(NC) gauge theory with Nc = 3
describing the interactions among quarks and gluons, is widely held as the corÂ¬
rect theory of the strong interactions mainly due to its successes in a particular
set of processes. These are large momentumtransfer (hard) processes, such as
certain aspects of deep inelastic electronnucleon scattering and jet physics, and
heavy quarkonia mass spectra where due to the property of asymptotic freeÂ¬
dom weakcoupling calculations can yield quantitatively reliable predictions.
Given SU(NC) as the gauge group of the color force, asymptotic freedom is a
consequence of the quark content (number and representations) of the standard
model, since the QCD coupling constant has the behavior,
aQCD
Ncas(t)
7T
4Nc
(TCadj â€œ
(2.1)
with the Dynkin indices Ca(jj = Nc, Cf = 5 for a Dirac quark in the fundaÂ¬
mental representation, Nf is the number of quark flavors, t is the momentum
transfer and A is a constant characteristic of the theory.
On the other hand, much of the low energy data on the strong interacÂ¬
tions, such as the mass spectra of hadrons containing light quarks, involve the
nonperturbative aspects of the theory such as quark confinement and chiral
symmetry breaking, which are not yet subject to a fundamental and quanÂ¬
titative understanding. There is, however, an extrapolation of the spectrum
3
4
problem that is amenable to perturbative calculations. This is the large â€”t
limit of the QCD Regge trajectories.
The Regge formalism, introduced in the preQCD era, provides a means of
classifying the spectrum of a theory and predicting the high energy behavior
of its scattering amplitudes. Here we give a brief but illustrative account of
this [1]. (For our future purposes the complications of including spin are not
necessary. Hence, for clarity, we limit ourselves to spinless particles.) A2>2
scattering can proceed in the three channels
a b â€”> c d schannel
a c^yb d Â¿channel
a d â€”Â»â– b c wchannel
where
S = (Pa+ Pbf , t=(Pa~ Pc)2 , U = (Pa~ Pdf
are the Mandelstam variables, of which only two are independent. By crossing
and the PCT theorem the amplitudes of the above three reactions are the same
but correspond to different kinematic ranges of the Mandelstam variables. We
can, therefore, write a partialwave expansion of the Â¿channel amplitude as
00
A(s,t) = lfcrÂ£(21 + l)AÂ¡(t)P,(zt(s,t)) ,
(2.2)
/=0
where for the case of interest, ma â€” mc = 0,
(2.3)
9t being the /channel scattering angle in the centerofmomentum (c.m.) frame.
5
Partialwave amplitudes of definite signature, Af, need to be defined in
order to have the appropriate asymptotic behavior for the application of the
SommerfeldWatson transform,
Af(t) = Al(t) for / = 0,2,4,...
(2.4)
AÂ¡ (t) = AÂ¡(t) for / = 1,3,5,... ,
and the physical (unphysical) integer values of / are the right (wrong) signature
points. The physical amplitude is related to these by
A{s,t) = ^(A+(zt,t) + A+(zt,t) + AT(zt,t)  A~(â€”zt, t)) .
If there is no lefthand cut in the Â¿Â¿plane AÂ¿ has the correct convergence beÂ¬
havior so that we have the socalled exchange degeneracy, i.e., A+ = AJ = AÂ¡.
This implies that the corresponding Â«channel reaction cannot occur through
the production of a single resonance.
The application of the SommerfeldWatson transform to the partialwave
expansion of the signatured amplitudes Aa(s,t) and the displacement of the
contour to enclose the /plane singularities of the Af(t) follow a familiar line
[1], which we only expound upon in section (2.2.1) for the case of potential
scattering. The relevant contribution for our discussion here is that of the
/plane poles of Af (/) to the scattering amplitude, which for a given pole has
the form
AÂ°p(s,t) = â€”16*2(2 c(t) + (2.5)
where (3 is the residue of Af (t) at the Regge pole a(t). The physical scattering
amplitude then has the form
Ap(s,t) = 16^(2a(t) + l)^M(r){Pa{t)(zt)+^Pam(zl)) . (2.6)
6
Its poles are at the physical integer values of a as expected. The large2^
(larges) limit of eq. (2.6) using eq. (2.3) results in the cross section
_ 1 â€žâ€”Ã7TCÃ(t)
= Â«oâ€”^
(2.7)
Â£ is known as the signature factor and ensures that only right signature poles
along a trajectory are present.
The above formalism can be applied to the charge exchange reaction
7T
p â€”> 7t0(t7) n ,
(2.8)
as in fig. 2.1(1). As the exchanged trajectory has the quantum numbers of an
isotriplet nonexotic meson it corresponds to the p (A2) trajectory. These are (in
the largeA^c limit) exchangedegenerate trajectories, crp â€” â€”1 â€” â€” 0 ^ , with
normal parity, P = (â€”1)^, and G â€” (â€”l)'7'1'1 , C = (â€”1)^ for the Gparity
and charge conjugation at any positive integer J.
Experiments are often performed on inclusive processes, where certain final
states are left undetected, since certain particles, such as the neutral ones, may
be hard to detect. The Regge formalism has been applied to the folllowing
subset of such reactions
a b c X
where X designates the unidentified flying objects (UFOs) out of the collider.
The process is now described by three invariants, which may be chosen as
s, t and M, the invariant masssquared of the UFOs. A parameter used
commonly in such processes is
_ / PcL \ _ S  M\ 2St
F (PcI^max'cm s ~ mb + (s  m%)2
7
It0 (TI)
Figure 2.1 Figure (1) is the singleRegge limit of an exclusive process and
figs. (2) to (4) depict the tripleReggeizaton of an inclusive process.
8
where pL denotes the longitudinal component of the momentum and the last
equality is for the case ma = mc â€” 0. For our case of interest, b is a fixed
m2
target in the lab frame and Â¿k
X jp
(
Ec t
Ea + 2mbEa
The process by which the cross section in the triple Regge limit is obtained
as depicted in figs. 2.1(2) through 2.1(4) [1]. The amplitude, represented by the
diagram under the sum in fig. 2.1(2), is described in terms of a single Reggeon
exchange in the limit Â» 1 with M and t fixed. The application of optical
Ml
1*1
theorem is represented in fig. 2.1(3). Now taking >â– 1 introduces another
Reggeon exchange as in fig. 2.1(4). In the appropriate limit
s Â» Mx > t
(2.9)
the differential cross section is given by
d2a
dtdx
pp'P
where
(2.10)
G^p(t) = Ippipit) 7bbp(0) (211)
with 7 denoting a vertex coupling and Â£ the signature factor introduced in
eq. (2.7). In the case of the reaction
7r p Â» 7TÂ° X
(2.12)
in the triple Regge limit the exchanged reggeons are the p and the pomeron, P,
which is the leading reggeon with the quantum numbers of the vacuum. From
the flatness of total elastic cross sections one infers
c*p(0) ~ 1 .
(2.13)
9
In the tripleRegge limit of reaction (2.12), expression (2.11) is valid as only
the spin nonflip amplitude for the proton contributes.
For large â€”t, where perturbative QCD (pQCD) is applicable, the scattering
amplitude can be obtained by a renormalizationgroupimproved sum over the
leading double logarithms [2,3]. In the largeNc limit the diagrams are only of
the ladder type and for massless quarks the associated BetheSalpeter equation
yields the following for the meson trajectories [4]
â€”tÂ»AQCD
Oin(t) >
where
aQCD(~t) â€” (
/^(n + 3/4)
12/H \
ln(â€”i/Ag^p)/
(2.14)
In contrast, for a fixed coupling constant the leading singularity would be a
branch point at a â€” The running of the coupling causes an accumuÂ¬
lation of trajectories above zero to the left of what would have been a branch
point. The set of trajectories in eq. (2.14) will be referred to as a bundle.
The cross sections for the reaction given in expression (2.12) fitted to a pure
Regge form in ref. 5 implied trajectories which flattened off around t â€” â€” 2
GeV2 to an apparent asymptotic value between â€”.5 and â€”1. The question
then arises as to whether these empirically extracted trajectories essentially
represent the leading bundle, or whether they should be viewed as an effective
trajectory strongly influenced by the softer physics of the subleading ones [6].
Assuming the applicability of pQCD calculations for t Â» AqCD, the first
scenario would be problematic. It would imply an eventual rise of the leading
trajectory to above zero at large â€”t (in quantum mechanics for l > â€”.5 the
trajectories are real and monotonically increasing functions of the energy). This
leads eventually to a large increase in the cross section as â€”t increases, for very
10
o
0.1
0.2
0.4
0.5
0.6
.6
tfGeV2)
Figure 2.2 The diamonds are ours and the others, shifted to the right for
clarity, are ref. 7â€™s effective trajectories for the exclusive process.
large values of s. We, therefore, consider the second scenario as physically more
plausible, and first examine the data for any such indications.
The exclusive interactions of eq. (2.8) were studied in ref. 7. The experiment
was conducted at beam energies of 20.8 , 40.8 , 64.4 , 100.7 , 150.2 and 199.3
GeV, and momentatransfers of up to t = 1.3 GeV2. The trajectories go
through zero at around â€”t ~ .70. Figure 2.2 shows a comparison of our fits for
higher values of â€”t to those of the experimentalists. The values of and errors
associated with the two fits are practically the same. These trajectories were
extracted by fitting the cross sections for all six of the data points at a given
tfGeV2)
11
0.3
0.2
0.1
0
^ Â° 1
â– 4â€”'
5
0.2
0.3
0.4
0.5
0.6
.6
t(GeV 2)
Figure 2.3 Our effective trajectories for the exclusive process: the diamonds
are for the 20.8 < En < 199.3 GeV, the squares for 64.4 < En < 199.3 GeV
and the others for 40.8 < En < 199.3 GeV, where the last two sets of data
are shifted to the right.
t to the Regge form
= P(t)(En.)2a^~2 . (2.15)
We next made fits by excluding first one and then two of the lowest beam
energies from the fits. Assuming that the above beam energies, especially the
lower ones, are not yet probing the Regge limit, restricting the range of these fits
to higher energies should bring a(i) in formula (2.15) closer to the true leading
trajectory. The results of these fits are presented in fig. 2.3. There seems to be
a tendency for the trajectory to rise as the range of energies is moved towards
higher values. This is much less clear for the largest two t values which have
t(GeV2)
12
big error bars attached to them. The rise for the rightmost (on the graph)
values of t is not inconsistent with the assumption that in the region where the
leading trajectory is expected to flatten toward zero, the subleading trajectory
develops a stronger coupling to the scattering amplitude and hence lowers the
effective trajectory.
We next examine the inclusive reactions of eq. (2.12) subject to eqs. (2.10)
and (2.13) in the triple Regge limit, i.e.,
a" = G(t)y1~2aÂ®, y = (1 â€” xF) .
These were tested for (i) s = 200 GeV2 (Ejr_ = 100 GeV) in ref. 8 and for (ii)
s = 400 GeV2 {E^_ = 200 GeV) in ref. 5. Figure 2.4 shows the trajectories
extracted in the last two references. Clearly there is a rise in the trajectory as
the fitted range of xF is brought closer to one, i.e., the range of E 0 is brought
closer to E' from the bottom to the top graph. For the sake of clarity, we
have plotted case (i) separately as well in fig. 2.5.
Although the above effect is heartening in our interpretation of the exÂ¬
tracted trajectories not being the leading one, we cast doubt on the quantitaÂ¬
tive extent of this trajectory lifting. The problem is our inability to reproduce
the experimentalistsâ€™ fits for either of the s values within the associated tolerÂ¬
ance, as shown in fig. 2.6 for the abovementioned two ranges in case (i). In
fact our fits for the two respective ranges of xF in fig. 2.7 do not show any
indication of the rise in the trajectory that could have signaled the approach
to the Regge limit.
Case (ii) is similar although less inconclusive. Here the data available to
us were in rather large xF bins so that for a fixed t only five data points were
13
Figure 2.4 Reference 5â€™s fitted trajectories for the two indicated values of
Â£'7r_: y : .05 * .2 for the top graph and y : .05 â€”> .3 for the bottom one.
available. The top half of fig. 2.8 shows our fits to the two and three data points
14
Figure 2.5 Reference 8â€™s fitted trajectories: y : .05 â€”> .3 for the diamonds
and y : .05 â€”> .2 for the others.
that correspond to the ranges of Kennettâ€™s fits at each t. Comparing this figure
to fig. 2.4 there is the same discrepancy of the results of our fits being lower
than theirs. For the smaller â€”t values there is some indication of a rise in the
effective trajectory but not as unequivocal as in their fits.
We discount the possibility of some deficiency in our fitting routine, which
utilizes Marquardtâ€™s method of minimizing \2, since it reproduced the fits of
ref. 7 as indicated above and it is a routine in common use by highenergy
experimentalists. A more likely situation is that our fits are simply too â€œraw.â€
The subleading effects for which the experimentalists make subtractions from
the data are the ppf (/ has the same quantum numbers as the pomeron but a
15
0.65
0.7
0.75
0.85 
0.95
2.2 2.1
KGeV2)
0.3
0.4
0.5
0.6
f 07
0.8
0.9
1
1.1
25 2.4 2.3 2.2 2.1 2 1.g
t(GeV2)
Figure 2.6 Reference 8â€™s and our (diamonds) fitted trajectories: y : .05 > .3
for the top and y : .05 â€”> .2 for the bottom figures.
t 1 1 1 râ€”
t s=200 GeV2
J L
16
Figure 2.7 Our fitted trajectories at s = 200 GeV2: y : .05 â€”Â» .3 for the
diamonds and y : .05 â€”> .2 for the others, shifted to the right for clarity.
Regge intercept of around .5) effects, low mass resonances and a 27rexchange
cut. However, for the t values discussed above these effects were expected or
assumed to be small and therefore ignored by them. There are also experimenÂ¬
tal subtleties discussed for which corrections need to be made. These include
smearing the data with beam momentum distribution and energy resolution,
and integrating the data over each bin. We included such corrections the best
we understood with no substantial effect on the results. In fact in ref. 8 it
is indicated that energy resolution and beam momentum distribution did not
affect the results appreciably.
17
Figure 2.8 Our fitted trajectories at s = 400 GeV2 for the range y : .05 â€”>
V\, where y^ is .25 and .45 for the diamonds in the top and bottom figures,
respectively, and .15 and .35 for the top and bottom fig. nondiamonds, which
are shifted to the right, respectively.
18
loglO(y)
log (y)
10
Figure 2.9 Differential cross sections at s = 200 GeV2: ref. 8â€™s â€œrawâ€
(dashed) and our (solid) predictions are plotted for y : .05 â€”> .3 in the top
figure and for y : .05 Â» .2 in the bottom one.
19
Figure 2.10 Differential cross sections at s = 200 GeV2: ref. 8â€™s â€œrawâ€
(dashed) and our (solid) predictions are plotted for y : .05 .3 in the top
figure and for y : .05 >â€¢ .2 in the bottom one.
20
Figure 2.11 Differential cross sections at s = 200 GeV2: ref. 8â€™s â€œrawâ€
(dashed) and our (solid) predictions are plotted for y : .05 > .3 in the top
figure and for y : .05 4 .2 in the bottom one.
21
As a means of contrast we give several plots of the experimental cross
section and the predictions of their and our fits for various t bins in figs. 2.9 to
2.11. In most of these their â€œrawâ€ prediction is not close. In the bottom half
of fig. 2.8 we present our fits for two different ranges of xF which are not as
restricted as the previous two ranges, for the sake of having more data points
to fit to. There is again a slight trend for the trajectory to rise although it is
marred by big error bars at larger â€”t values.
We proceed to make a simplistic two power fit of the inclusive data. AsÂ¬
suming that the behavior of the fitted trajectory is due to a stronglycoupled
trajectory subleading to the p, we would have
o" = GooWU  Xf)1'200 + 2G0i(f)(l  if)1Â»10Â» + Gâ€ž(l  if)121" ,
where 0 and 1 refer to the p and the proposed subleading trajectory, respecÂ¬
tively, and the Gâ€™s are as in eq. (2.10). In the next chapter it will be shown
that in the context of potential scattering, a squareroot branch cut behaves
like a fixed pole at the center of the cut for smaller values of the cut. Since
the QCD bundle of trajectories simulates a cut, we assume the bundle can be
represented by a pole at a = 0. Thus we set
a0 = 0 .
Indeed, the t dependence of aQ(t) is not readily available as eq. (2.14) is valid
only for much larger values of â€” t where the perturbative approximation is
valid. The G's are taken to be real and, as a further simplification, we assume
the factorization Gqi = n/GqoGii Thus we fit to the form
cr" = (1 â€” xf)(Gq + G\(l â€” Xp) ai)2 . (216)
22
Table 2.1
Results of a TwoPower Fitting
t
<*1
Gi/Go
X2/d.o.f.
s(r â€” 10)
2.3
1.14 Â± .09
21.6 Â±5.7
3.4
5000
2.9
1.64 Â± .12
26.2 Â±4.1
.75
1000
3.3
â€”1.17 Â± .19
24.5 Â±9.9
1.8
5000
3.7
1.33 Â± .16
61.6 Â±35.
5.3
5000
4.1
1.87 Â± .10
33.4 Â±5.0
2.6
500.
4.8
1.71 Â±.20
39.8 Â± 13.
.15
1000
5.2
1.32 Â± .30
29.6 Â± 18.
0.0
2500
5.7
1.45 Â± .29
47.1 Â±33.
2.5
2000
6.3
2.02 Â± .32
41.3 Â±18.
1.8
500.
Note: Above table for s = 400GeV2; for the first four columns y : .05 â€”> .45
and for the last M2 = 20
t
Â«1
Cl/Go
X2/d.o.f.
s(r = 10)
2.3
.992 Â± .16
29.2 Â±20.
3.5
13000
4.1
1.52 Â± .28
25.7 Â±10.
3.6
1000
4.8
1.71 Â± .36
39.8 Â±21.
0.3
1000
5.2
1.31 Â±.33
29.6 Â± 19.
0.0
2500
6.3
1.87 Â± .36
34.7 Â± 18.
3.0
500.
Note: Above table for s = 400GeF2; for the first four columns y : .05 â€”Â»â€¢ .35
and for the last M2 = 20
The results are presented in the first four columns of tables 2.1 and 2.2 .
Within the error bars, the subleading power a1 does not show a rise as the
range of y is brought closer to the Regge limit. One can estimate the s at which
23
Table 2.2
Results of a TwoPower Fitting
t
ai
Gi/Gq
X2/d.o.f.
s(r = 10)
2.1
1.36 Â± .18
25.4 Â±9.0
.70
1000
2.4
1.04 Â± .16
56.8 Â± 74.
1.5
8500
Note: Above table for s = 200GeV2; for the first four columns y : .05 â€”Â» .45
and for the last M2 = 10
t
Â«1
Cl/Go
X2/d.o.f.
s(r = 10)
2.1
1.18 Â± .33
27.5 Â±23.
.94
2000
2.4
1.06 Â± .22
58.8 Â±93.
1.5
8000
Note: Above table for s = 200GeV2; for the first four columns y : .05 â€”> .29
and for the last M2 = 10
the nexttoleading term in the cross section is r times smaller than the leading
one by
2 (2rG\\V<*\
â–
In the last column of the above two tables some representative estimates of s
are tabulated. The chosen corresponds to xF = .05 in cases (i) and (ii).
Three orders of magnitude for s is an estimate compatible with entries that
have the lower \2 per degree of freedom.
We point out that more sophisticated analysis of the available data is being
considered. However, cleaner experiments probing the Regge limit could be
more decisive in settling the discrepancy between data and pQCD.
CHAPTER 3
THE POTENTIAL ANALOGY
The analogy here is between a relativistic physical process and a non
relativistic unphysical one. In a relativistic 2 â€”>â– 2 scattering the physical Regge
limit corresponds to extreme forward scattering in the schannel due to reggeon
exchange in the Â¿channel. As only one channel is available in nonrelativistic
scattering, the Regge limit we examine corresponds to unphysical elastic scatÂ¬
tering parametrized by the limit
A2
â€”2 1 , z = cos 6 = 1 â€” â€”â€” and A2 = (k â€” k')Â¿ , (3.1)
and E < 0 is the domain of interest, i.e., t o E and zt <> z is the corÂ¬
respondence between the exclusiverelativistic and nonrelativistic cases. The
potential is QCD inspired in its shortdistance behavior, where the running
of oÃqCD is simulated by a shortdistance logarithmic violation of canonical
scaling in the Schrodinger equation. We shall see below that this choice for the
hard physics of the model leads to an accumulation of the leading trajectories,
analogous to eq. (2.14), above a â€” â€”.5 at large negative energies. Thus we conÂ¬
sider the following radial Schrodinger equation for our spherically symmetric
potential (h = c = 1 = 2m)
( â€” dr H 2~^~â€”l~^(r))'^(r) = E\!)(r) , with A = Z h 1 /2 ,
cA  1 (32)
U\2
V(r) =
+
r2ln(l + ^) r
24
25
The coefficient of the extra Coulombic term in eq. (3.2) is fixed by requiring
the large distance behavior of V to be that of an attractive Coulomb potential
of unit strength; that exhausts our freedom to scale the Schrodinger equation.
This provides some degree of solubility to the model.
We will also study the scattering amplitude due to the potential Vq in the
above Schrodinger equation, where
A'2 1
V0  r2 r
(3.3)
simulates the short distance behavior of a fixed coupling constant. As we will
discuss later the scattering amplitude of the latter model has a fixed cut in
the angular momentum plane centered at a = â€” .5 and subleading Regge poles
that are Coulombic in character for â€” E Â» 1. For c, A â€¢< 1, Vq may be conÂ¬
sidered a coarse graining for the short distance behavior of V, whereby a fixed
cut is simulated by the accumulation of the Regge poles at large negative enÂ¬
ergies. Hence, we would expect subleading trajectories in the model with the
potential V that approach the negative integers with large enough â€” E. This
is qualitatively in accord with the largeâ€”t behavior of the effective trajectoÂ¬
ries in fig. 2.4. As indicated in the previous chapter, simple twopower fittings
of the data on the inclusive reaction indicate a subleading power between â€”1
and â€”1.5. Therefore, such a choice for the subleading term in our potential,
although not based on any fundamental property of QCD, is indicative of the
data.
A potential that at large distances behaves like a harmonic oscillator would
render a more faithful representation of QCD. It would imply linear trajectoÂ¬
ries for large E analogous to trajectories that interpolate between qq bound
26
states in the confining limit of QCD. Such asymptotics, however, would create
unnecessary complications in defining the scattering amplitude in our model;
the very large distance behavior of the theory is not expected to contribute sigÂ¬
nificantly to the scattering amplitude at energies that probe the short distance
physics, as we shall substantiate in the next section. Therefore, we consider
the potential given in eq. (3.2) sufficient for the purpose of testing the basic
premise of this paper.
3.1 The Trajectories
3.1.1 The Model Incorporating Asymptotic Freedom
The trajectories of our proposed models can be obtained by numerically
solving the Schrodinger equation. Adding a harmonic oscillator to eq. (3.2),
V(r) â€”Â» V(r) + u2r2 , (3.4)
linearizes the largepositiveenergy limit of the trajectories as in fig. 3.1.
The influence of the soft physics of the resulting model in the short distance
limit can be inferred by comparing the position of the trajectories at moderate
â€”E for representative values of cj2, as given in table 3.1 . For the first set
of values, A = 10_2,c = .125, the parameter is varied from where there is
no harmonic oscillator to an extreme value of u2 = 16, and three moderate
negative energies are picked. We see that the locations of the trajectories are
indifferent to ur to three significant figures. The value J1 = 16 is extreme
because the trajectories it induces at moderate positive energies are practically
flat slightly above the l = â€”.5 line. On the other hand, u>2 Â« .2 induces positive
E trajectories that have a slope of around one; a closer analogy with the strong
27
Figure 3.1 Regge trajectories exhibiting asymptotic freedom and confineÂ¬
ment: the first two nodes for A = .5, c = .125 and w2 = .2 are plotted.
interactions. A similar comparison is shown for A = .5 and c = .125, values
to be of interest later, for two values of u, and another comparison for an
arbitrary value of parameters.
One would expect the same lack of sensitivity to the verylongdistance
(confining) physics for the nexttoleading (Coulomblike here) trajectory, exÂ¬
cept perhaps at eversoslightly more negative energies. If at negative energies
the location of the 5matrix singularities are unaffected by the variation of
a parameter, it is reasonable to expect the partialwave amplitude to behave
similarly. Thus, as advertised earlier, we shall ignore simulating confining traÂ¬
jectories for the main purpose of this work.
28
Table 3.1
Effect of a Confining Potential on the Asymptotically Free Component
u,2
Â¡(5)
i(â€”6)
Â¡(7)
A
c
n
0
.4500
.4507
.4511
102
.125
1
1
.4502
.4507
.4511
10"2
.125
1
9
.4507
.4510
.4513
10"2
.125
1
16
.4509
.4512
.4515
hâ€˜
o
1
to
.125
1
1
.3580
.3587
.3592
io10
1
1
9
.3586
.3591
.3595
10"10
1
1
0
.4690
.4693
.4696
.5
.125
2
.2
.4690
.4694
.4696
.5
.125
2
The numerical trajectories for the model of eq. (3.2) are plotted in fig. 3.2.
The potential becomes more negative with increasing A or c, with the effect of
increasing the binding energy and thus raising the trajectories, as in the figure.
As evident though, there is more sensitivity to a change in c than A, since the
latter only incurs a logarithmic change in the potential.
Let us obtain an analytic expression for the trajectories in the region of
interest. The following transformations are helpful. Defining
R = ln(l/Ar) and ip = e~R/2f , (3.5)
eq. (3.2) can be transformed into the following form
i2 , wmw_ \2j ) W1U1
E
(d2R + V(R))f = \2f,
c
V(R) = 
, (cA  1) ,r E _m c (3.6)
ln(l + eR) A A2 R
In the above Schrodingertype equation, where A2 plays the role of bound
state energy, the Coulombic tail of the â€œpotentialâ€ V(R) implies a logarithmic
29
Figure 3.2 Regge trajectories for the potential V(r): the first node for
A = .5, c = .5, the first three nodes for A = .5, c = .125, and the first two
nodes for A = .01, c = .125 are plotted as dotted, solid and dashed curves,
respectively.
modification of the usual asymptotics ij>(r) Â»â€¢ rl+l near the origin in eq. (3.2),
i.e. for A > 0
f(R) ? e~XRRc!2X =â–º rx+ a ln^ j . (3.7)
Hence, there is a zeroA singularity in the asymptotic behavior of the wave
function of an asymptotically free potential, reminiscent of the threshold sinÂ¬
gularity in the wave function asymptotics of a Coulombtailed potential [2].
Given this correspondence, we would expect an accumulation of trajectories
30
above A = 0 for an asymptotically free potential, analogous to such accumulaÂ¬
tion near threshold of a Coulombtailed potential, i.e.,
E
^2>1 as
A2
oo .
(3.8)
There is, of course, the actual Coulombic behavior far from the origin in
eq. (3.2) which induces the accumulation of the trajectories just below E = 0.
Invoking the theorem on the monotonicity of the trajectories above A = 0,
together with the E â€”> 0_ and A â€”> 0+ behaviors delineated above, gives us
the behavior exhibited in fig. 3.2.
WKB approximation is applicable to a Coulombic potential near threshold
and so is justified in the A Â« 0 region of interest. The eigenvalues An of eq. (3.6)
are obtained as solutions to
1 = j*+ dR\Jâ€”V(R) â€” A2 = (n + 77)tt , (3.9)
where RÂ± are zeros of the integrand and 77 will be determined by examining
the boundary condition across the turning points. In the limit of eq. (3.8) the
wave function is concentrated near the origin, R Â» 1, so that henceforth we
may approximate
I Â« J*+ dR\JWn(R) for Wn{R) = ^ +^e~2R  \2n . (3.10)
We determine the constant rj in eq. (3.9) by examining the exact solution
to the SchrÃ³dinger equation. Near R
v(R)*~ik+(l~~x2)e~2x where x=RR ^
and the corresponding SchrÃ³dinger equation, eq. (3.6), becomes
(~dl~i'2e~2x)f(x) = v2f(x) for u = %^L  A2 . (3.11)
31
The inverse of the transformation of the type in eq. (3.5) yields
(~dy + â€”= (A^)2 t(f{y) if iÂ¡) = e~x/2f, y = e~x/A . (3.12)
u
/, then, has the Bessel functions as its solutions, out of which we seek the one
that is exponentially damped to the left of the turning point R (x â€”>â– â€”oo).
The corresponding solution for f(x) is Macdonaldâ€™s function, Ku(ive~x), with
the following asymptotics
I ixed
â€”ive
fix) = Kâ€ž(iVe~x) 4 V 2iy
y sin(ivx)
for x â€”y â€”oo
(3.13)
IV
for i/<1 ,i>oo .
The integrals relevant to the WKB solution are
rx< 0
rx< u
X\ = iv V e~2x> â€” 1 dx' = iv{ â€” y e~2x â€” 1 + tan1 v e~2x â€” 1)
Jo
rx>0
I2 = iv I V 1 â€” e~^x' dx' = iv( â€” yl â€” e~2x + tanh1 yl  e~^x )
Jo
with
xâ€”1â€”00
X\ > â€”ive x ,
xâ€”loo
Z2 â–º ivx
(3.14)
with the solution in the neighborhood of R having the forms
( (iv\/e~^x â€” 1 )~2 Cexp(Zi) if x < 0
%kb(x) = \ , x (315)
k (ivy/' 1  eâ€œ2a;) 2 (Ce*12 + C+e_l12) if x > 0 .
Matching eq. (3.15) to eq. (3.13) subject to eq. (3.14) gives
CÂ± = CeÂ±l 2 /y/2ivn .
(3.16)
Thus, with p as the momentum, the WKB solution to the right of R_ has the
form
C ' rR
f(R> R) = â– ^ cos
\/2ÃVTTp
(JR_pdx~ I)' (317)
32
In the vicinity of R+ we have
and a conventional analysis of the boundary conditions at that turning point,
by linearizing the potential there, yields
C' ' rR*
f(R < R+) = ^cos (.fR+ pix i) â€¢
Comparison of eqs. (3.17) and (3.18) completes eq. (3.9) with
>1 = 3/4 ,
(3.18)
(3.19)
the value indicative of a barrier at R consistent with the faster than expoÂ¬
nential decay in eq. (3.13) on the classically forbidden side. (The corrections
to eq. (3.19) are higher order than the nextorder corrections to Xn below and
hence will be ignored.)
To evaluate the eigenvalues An notice that in the limits given by eq. (3.8),
the classically allowed region is small compared to the distances characterizÂ¬
ing the turning points and the exponential decay in eq. (3.10) implies that
the maximum of Wn occurs very close to R, as in fig. 3.3. Hence, we may
approximate eqs. (3.9) and (3.10) by
iv t \/l  e 2x' dx' + [  p^dR = (n + 3/4)n ,
JO JR_\x V
with
JiL j _
R_ â€™
IR
(3.20)
R
< 1 .
+
To first order in the above small quantities we can write
1/2
dR Â« (n + 3/4)7r ,
/!7 / /rT\i/2
R w
(3.21)
33
Figure 3.3 V(R) is plotted for c = 1 , A = 10 4 , â€” E = 1.
The x dependence drops out so that the trajectories are given by
4y/2c
"3A7
An , i
â€”P= ln2
VC
A2)
(n + 3/4)7t .
(3.22)
This is the same result that would have been obtained had the expression for
W(R) in eq. (3.10) been approximated by a purely Coulombic one and
a barrier introduced at R. Upon inverting eq. (3.22),
3 (<*'
An, ~ yj'lco.
for
QAP 1 
aQM = !/ln
QM
4 V c
ÃE
V A2
(n + 3/4)7tJ
)â€¢
12/3
(3.23)
eq. (3.23) is the quantum mechanical analog of expression (2.14) in QCD. This
accumulation of trajectories to the left of A' = j2caQM takes the place of the
34
Table 3.2
Asymptotically Free Trajectories
E
A
c
n
awkb
A
AA/A
104
IQ60
10
0
1.56(.897) = .2374
.2389
6.3 x 10"3
104
4 x 10_1Â°
10
0
.617(.819) = .5053
.5170
2.3 x 102
104
IQ60
10
1
.265(.819) = .2166
.2214
2.2 x 10â€œ2
104
O
CO
1
o
rH
5
1
.187(.771) = .1443
.1496
3.5 x 102
104
IQ60
10
3
.265(.698) = .1848
.1974
6.4 x 10~2
103
O
CO
1
o
iH
5
3
.263(.524) = .1370
.1668
1.8 x 101
500
10â€œ12
5
2
.403(.484) = .1953
.2476
2.1 x 10_1
102
10â€œ9
1
0
.208(.592) = .1233
.1408
1.2 x 10_1
102
10â€œ6
5
0
.557(.731) = .4071
.4312
5.6 x 102
branch cut singularity in the fixedcoupling case (to be discussed in the next
section) and in that sense simulates the cut.
We can compare the last equation with some numerically obtained results.
First we specify the range of validity of that equation. The WKB approxiÂ¬
mation is in general more valid for higher node solutions. However, eq. (3.23)
is obtained by an expansion in oÃqM so that the second term in the curly
brackets in eq. (3.23) should be small compared to one. Also, in the particÂ¬
ular case of a Coulombtailed potential, c/R, the important region of motion
is where potential ~ energy, or in our notation, R ~ The semiclassical
approximation applies if this region is large compared to the wavelength of the
particle, 1/^/energy = 1/A, i.e., c> A and from eq. (3.7) the wavefunction
is enhanced for large R. Therefore, eq. (3.23) is improved for larger values of
c.
35
Table 3.2 compares the numerical values of A from solving the correspondÂ¬
ing Schrodinger equation to those obtained from eq. (3.23) for different choice
of the parameters in the range discussed above. In all entries the agreement is
better than around 20%. Compaing entries 1 and 2, we see the sensitivity to
the value of olqM. Comparing the second and third entries indicates the deÂ¬
pendence on the number of nodes. Comparing the first, third and fifth entries
shows that the sensitivity is stronger to the perturbative expansion than to
the number of nodes (the closer the number in paranthesis under the WKB
column is to one, the more justified the perturbative approximation is). ComÂ¬
paring the third and fourth entries indicates the cdependence. In general, the
different entries give an indication of where eq. (3.23) is more valid. It should
be noted that the large orders of magnitude chosen for ^ values are necessary
to satisfy the perturbativity requirement of our expression.
3.1.2 The Model with Fixed Coupling
For the potential Vq of eq. (3.3) the partialwave scattering amplitude,
and hence its singularities, can be written in a closed form. The â€” ^ part
of eq. (3.3) can be combined with the centrifugal term to yield the effective
potential, A2/r2, for
Ac = VA2  A/2 , lc = Xc~\. (3.24)
The solution to the corresponding Schrodinger equation is then that for a
Coulomb potential with the A â€”>â€¢ Ac replacement. The asymptotic form of the
wavefunction is then
ip(r) > 2 sin (fcr   In2kr  lc + Â¿Coul(/c, k)J , k = \ÃE ,
36
Figure 3.4 Regge trajectories for the potential VQ(r): the first two nodes
and the branch cut (series of flat lines) for A' = .1 are shown.
from which the following partialwave 5matrix can be extracted
S,(k) = e^CouL^c*7^*') = â€”+ ? ~ ^ eÂ¿7r(A"Ac) . (3 25)
r(Aef^ + ^) 1
The above 5matrix satisfies the generalized unitarity condition
5/*(fc*) SÂ¡(k) = 1 ,
and has a fixed squareroot branch cut whose discontinuity does not vanish for
large magnitudes of the energy; the asymptotic strength of the cut singularity is
given by the exponential factor. That factor is the 5matrix for a ^ potential;
the energyindependence a reflection of the potentialâ€™s scale invariance.
37
Figure 3.5 The Mandelstam contour and the Smatrix singularities it enÂ¬
closes.
The expression in eq. (3.25) also has simple Regge poles at
Aâ€ž = Â±J(ni + Â¿)2 + A'2,
as in fig. 3.4, with the corresponding residues
\n ( 1 i i
(â€”l)n(â€”A + ^n)
Pn(k) = . . exp
(3.26)
in(XnTyJx2n A'2)] , (3.27)
n!AnT(n)
where the +(â€”) branch corresponds to â€”n â€” j ^ being positive (negative).
3.2 The Scattering Amplitude: The FixedCoupling Case
3.2.1 MandelstamSommerfeldWatson Transform and the Regge Limit
We first give a brief heuristic derivation of the ReggeWatson continuation,
in the 2plane, of the partialwave expansion using the Mandelstam represenÂ¬
tation [9], mainly for application to potential Vq given in eq. (3.3). For this
38
purpose, the unique analytic continuation of the 5; for the potentials under
consideration is the one obtained by simply taking / in the radial Schrodinger
equation to be a complex number. Consider, then, the following integral in the
Aplane
c = fdX ^^(sx5Wi)q_x_5(z)= [L+lÂ°Â° + [
2nkJc sin 7rA V A 5 ) x 5 JlÃoo Js<
semi circ
(3.28)
where the Q's are the Legendre functions of the second kind, 5/ are the 5
matrix elements as before and the righthand side of eq. (3.28) defines the
contour to be as in fig. 3.5.
The singularities of the above integrand are at the poles of QÂ¡ for negative
integer l, possible poles and branch cuts of the 5matrix, and the zeros of
sin(7r/). Therefore, if (3 denotes the residue of a pole of 5 and N the nearest
integer to the right of L, rearranging terms results in
' 00 A
j E ,c*)
1=0 poles(n)
v LUlÃœ I Q
? JS, , rL+ioo c
1 (Sl.5(k) ~ 1)^l.5(~2:) + / + /
nK â€¢'Lâ€”ioo Js<
semi circ
(3.29)
The left hand side of the last equation is the partialwave expansion of the scatÂ¬
tering amplitude with a finite domain of absolute convergence in the 2plane,
e.g., within the Lehmann ellipse for a superposition of Yukawa potentials.
The domain of convergence can be extended to the whole complex 2plane by
discarding the part of the contour along the semicircle at infinity
39
The expression most suited for our purpose, given the asymptotic behavior
of the Q's below, is obtained by pushing the background line to infinity, L â€”Â» oo,
which makes its line integral vanish in the Regge limit, with all singularities
in the Aplane now contributing. Thus we may ignore the background terms to
write for the scattering amplitude
A(kâ€™Z) = I Â£
all poles
/
lrrkJc
dX
A
2tt k J Cuts sin n A
(SA.5Â«  l)
(3.30)
00
~k Ei1)'1 R.sW  Sl.5Â«]
1=0
Because of the branch cut in expression (3.25) the SÂ¡ do not possess the ManÂ¬
delstam symmetry necessary for the last term in eq. (3.30) to vanish. That
term cancels the poles at integer values of A of the first term on the righthand
side of eq. (3.30). Thus the poles of A coincide with those of Q_x _ 5 which
are at the physical (nonnegative integer) values of l.
We would like to specialize expression (3.30) to Vq of eq. (3.3). Substitution
of eqs. (3.25) (3.27) in (3.30) and repeated use of wellknown Gammafunction
identities yields
g7f/2A; /Â¿7T\ > T) 77 pinXn
Mkâ€™z) = 81,1 (t) B0" â€” r("  2p)QKsi*')
71=0 V
r A' \eiirA
+ 2 A L Â¡toÃÃ ^WA*  A2 ) irCiv/A^  A2  p)2 Q.x.Jz) d\
7T J 2k
 E1 sin(27T\//2A'2)r(V//2A'2  p) T(x//2A'2 p) Qt_5(z)
K 1=0
â€” + Ac ( Am ,
fÂ°r P~Ãk~\' A" = Â±\/(P'Â»)2 + A'2 â€¢
(3.31)
40
The last line of eq. (3.31) labels the contributions from Regge poles, the fixed
cut and the Mandelstam terms, respectively.
We need impose the further restriction
0 < A' < .5 (3.32)
on the above formula. That there is trouble above that value can be seen from
the partialwave expansion of the scattering amplitude, given on the lefthand
side of eq. (3.29) together with eqs. (3.24) and (3.25). Terms for which A/2 > A2
will be illdefined as they correspond to points that are on the branch cut.
This is a reflection of a wellknown quantum mechanical result: if a potential
behaves as =^ near the origin, for A/2 > A2 the ground state of the particle
is at E = â€” oo, i.e., it tends to fall towards the origin.
The approach to the Regge limit, \z\ Â» oo, is determined by the asympÂ¬
totics of the Q's and the relative contribution from each of the terms in
eq. (3.31). As the series expansion for the Legendre functions of the second
kind is given by
= T(l/ + l)y/Ã± ^ (% + l)fc(% + g)fc â€”2k
() r(,+ 3)(2^+lÂ¿ *!(â€ž+ Â§)fc
where (a)k = at(a + 1)... (a + k  1) , (a)0 = 1 ,
(3.33)
in the extreme Regge limit the righthand most singularity in the Aplane would
dominate. In general, the presence of a leading branch point (or a pole if S is
a positive integer) at A7, i.e.,
(A(E)  A')1"^) ,
results in the Regge behavior of the scattering amplitude
Mkx>
A(k,z) * 0(k)(z)x3 .
(3.34)
41
Figure 3.6 The solid line is Q0(z) and the dashed its asymptotic expression.
In the model at hand, the particular hierarchy of singularities that would be
most relevant to us occurs for â€” E > 1. There the leading singularity of the 5
matrix is the fixed cut with a tip below A = ^ and the subleading trajectories
flatten off not far below the cut in a Coulomblike manner. Then we could
examine the interplay between the hard physics of the cut to the softer one of
the poles for signs of the effect we conjectured in the QCD experiments.
The asymptotics of the Q's can give us an indication of the scale relative to
which the onset of the Regge limit is to be measured. Given the series expansion
in eq. (3.33), the main contributions to the scattering amplitude at moderate
values of 2 will come from Qu's of smaller orders, approximately â€” 1 < v < 2.5
for the case at hand, for which at 2 Â« 5 the asymptotic value is not off
42
Figure 3.7 The solid line is Q_ Q(z) and the dashed its asymptotic expresÂ¬
sion.
by more than 5% from the exact one. The approach to asymptopia for the
Legendre functions is plotted in figs. 3.6 and 3.7. Thus, an order of magnitude
would be a conservative scale for z against which to measure the onset of the
Regge regime.
3.2.2 Analytical Estimates
Before giving a numerical evaluation of the scattering amplitude, let us give
an approximation to the various terms in eq. (3.31) for a range of parameters
that will be of interest later on. This is also useful as such approximation
may be evaluated for parameter values that would boggle the computer. As
X' increases towards j, the contribution of the cut becomes more and more
43
pronounced relative to that of the poles in approaching the Regge limit due to
two effects. On the one hand, the separation between the tip of the cut and the
location of the poles in the Aplane increases (i.e., higher power of 2 from the
cut than the pole), and on the other, the strength (width) of the cut increases
relative to the residue of the pole. The dependence on is rather dramatic as
we shall see below.
Here we restrict ourselves to smaller values of A', say A' < .15, a range
that will be argued to be of most relevance to us, later. To expand eq. (3.26)
in powers of A/ we require values of the energy not too close to â€”1 (where the
leading trajectory merges into the cut) say â€” E > 10. To first approximation
then the trajectories are those of a pure Coulomb potential, and the sum over
poles in eq. (3.31) is the scattering amplitude thereof. But that amplitude is
known in closed form, namely the Rutherford amplitude. Hence,
A _ 1 . e(A'2)
'"tfni + iA 2 i +00 ),
(3.35)
r(! + Ã¡)
where the nextorder correction is not needed below. It is a rather fortunate
fact about the Coulomb potential that the exact amplitude has a Regge form
in terms of the momentum transfer A2, i.e., the power of (1 â€” 2) in the
last equation is just the leading trajectory, so that the Regge limit is attained
within an order of magnitude for 2. This is certainly commensurate with our
expectation based on the asymptotics of the Legendre functions. We have yet
to examine the other terms.
We now restrict ourselves to the asymptotic limit of the Qâ€™s to see how far
away from this the Regge asymptopia is situated. So, for now, we work in the
range
X' < .15 and â€” 2 > 10 .
(3.36)
44
In that case, the contribution of the cut integral, Ac, in eq. (3.31) can be
approximated as
 / A, ^x>2 ~ x2 exp (iA[?r ~i in(_8z)])dx
A,2e^ /I i \ 2/ xi ^i(A/k*ln(8*)])
V^A; 1 V2 2fc/U A'[7t â€” Â¿ln(â€”8z)] '
(3.37)
Taking the large\z\ limit of the last expression,
a:Â»l l~2~
J\(x) > \ â€” sin(rr
V 7TX
(3.38)
results in
Ac(k,z)
V(ln8+t7r) ,rfl
1 V2
Â¿)2(z)A'â€œbni(2) . (3.39)
The logarithmic correction to the tipofthecut power behavior of the scatÂ¬
tering amplitude is characteristic of branch point singularities, as indicated in
eq. (3.34). The relation that characterizes the Regge limit by examining the
nexttoleading correction to eq. (3.38) is
A'ln(z) > .375 .
(3.40)
This is an energyindependent criterion and demonstrates a sharp dependence
of the Regge asymptopia on the location of the cut. For a value of interest,
A' = .1 =Â» (2) ~ 1015 to 1030 , (3.41)
if we want the subleading term in the cross section between 5 to 10 times, reÂ¬
spectively, smaller than the Regge term. This is rather astronomical compared
to our oneorderofmagnitude scale for the Regge limit of the poles. Although
the perturbative approximation would not be valid there, as a rough estimate
45
for A' Â« .5 eq. (3.40) yields an order of magnitude for the Regge limit. This is
not unexpected as when approaching our upper limit on A' we are approaching
a pole in the Legendre function in the integrand in eq. (3.31), and we already
expect a more rapid approach to Reggeland for pole singularities.
The contribution of the third term on the right hand side of eq. (3.31),
Am, is negligible in the region given by eq. (3.36) provided the energy is not
too close to â€”1, say conservatively â€” E > 10,
Am ~ 0 ,
which has been verified numerically. As stated previously the contribution of
that term is only significant near values of energy corresponding to integerÂ¬
valued Regge poles An. It drops sharply away from such points as it contains
Legendre functions of comparatively large orders. So the approach of the scatÂ¬
tering amplitude to the Regge regime is mainly dominated by the two conÂ¬
tributions discussed above. For values of â€”2 that are not extremely large the
zerothorder A'expansion of the integral in eq. (3.31) would suffice:
\i2pir/2k .i 7 v
<342>
the region of validity for which can be deduced from the argument of the Bessel
function in eq. (3.37)
A'h^â€”8z) < 1 .
The order of the Legendre function in eq. (3.42) implies that for smaller
values of A', the â€œReggeâ€ behavior of the branch cut is approximated by a
simple pole located at the center of the cut, A = 0, up to sizeable values of 2,
e.g., up to 2 ~ 103 for A' = .1. This aspect of the weakcoupling limit of the
46
potential may also be understood by expanding the Smatrix in eq. (3.25) in
powers of X' to obtain, for p defined in eq. (3.31),
This expression has a fixed simple pole at A = 0 and no branch cuts. Therefore,
in eq. (3.30) we do away with the integral but now add the fixed pole to the
first term on the righthand side of it. The residue can be read off from the
last equation and the resulting expression is eq. (3.42).
This, of course, is suggestive of the Born approximation for a V(r) =
â€”A2/r2 potential, with eq. (3.1),
(3.44)
The large1A: and \z\ limit of eqs. (3.44) and (3.42) coincide. By â€œlargefcâ€
we mean â€”E Â« 100 and â€œlargezâ€ means an order of magnitude as usual.
Although, in the extreme Regge limit the Born approximation and eq. (3.42)
break down, this analysis indicates that they are valid for at least a couple
of orders of magnitude in 2 . This lends credence to the calculations of ref.
10 where an improved Born calculation was used to estimate the hard physics
at large Xp, as long as such calculation is valid in the applied range of â€” t.
Again, the above result has a sharp dependence on A'; for X' close to .5 it
is the rightmost branch point that rapidly dominates the integral. Also note
that from eq. (3.42) (and of course eq. (3.44)) the weakcoupling amplitude is
proportional to the coupling A/2, whereas in the Regge limit, eq. (3.39), that
factor is much enhanced.
From eqs. (3.35) and (3.42) we can get a rough estimate of the 2 at which
the cut dominates over the pole. This estimate is strongly energy dependent,
47
since the Regge pole term on the right side of eq. (3.31) has an extra factor of
1/k suppression relative to the cut integral, in the large1 k  limit. To have the
hard physics say around 5 to 10 times larger than the softer contribution, an
orderofmagnitude estimate would be
A' = .1 , E = 10 =â–º 2 ~ 105 . (3.45)
For more negative energies,
(z)1_i Â« , for  E > 100 , (3.46)
is a similar estimate for A' < .1.
In short, in the presence and for a judicious choice of a branch point singuÂ¬
larity, there is a pronounced hierarchy of scales. The scale at which the leading
effect (the center of the cut) is dominant at moderate binding energies is orders
of magnitude separated from the analogous scale for the nexttotheleading efÂ¬
fects (the poles); and the scale at which the leading branch point dominates is
even farther removed.
3.2.3 Numerics
We can now numerically study certain behavior of the scattering amplitude
without the approximations of the previous section. The ratio of the cut amÂ¬
plitude, Ac to that of the pole as a function of z is plotted in fig. 3.8. The solid
curves validate the estimations of eqs. (3.45) and (3.46). Even for the relatively
large value of = .25 it takes â€”2 ~ 100 for the hard physics to contribute five
times more to the cross section (ten times more to the scattering amplitude)
than the soft.
48
A more accurate indication of the relative contribution of the hard physics
is given by fig. 3.9, where the ratio of the amplitude of the cut to the total
amplitude is plotted as a function of energy for two different values of 2. The
rise with energy of this ratio due to the ^ suppression of the pole contribution
relative to that of the cut is more dramatic for the larger values of the coupling.
In ref. 10 an improved Born approximation of the hard contribution to the
process of eq. (2.12) at â€” t = 4 GeV2 yielded the estimate of
^ Â« 2.5 x 10V*(2/*) (3.47)
7TU
for y = (1  xF). For the values of y~l = 20 and 7 with Â« .2 at the relevant
scale, the estimate is for the hard part to be 2% and .5% of the experimental
49
Figure 3.9 ^f2 for â€” 2 = 10 (solid) and â€”2 = 5 (dashed).
cross section, respectively. Although there is no isomorphism of parameters
between the triple Regge limit of the inclusive interactions and the Regge limit
of our quantum mechanical model, a rough correspondence between y~l and
â€”2, seems to indicate, from fig. 3.9, that the toy model with analogous behavior
corresponds to a value of A' < .1, actually closer to A' Â« .05. The sharp rise
with energy of this ratio in this toy model is probably not a characteristic
of QCD. This difference may be improved when we consider the model with
asymptotic freedom, as we shall discuss below.
The E and 2range of validity of the Born approximation to the cut was
discussed in the previous section. Figure 3.10 is a plot of the ratio of the exact
amplitude to the Born one for the full potential VÃ³(r) for different values of 2
50
Figure 3.10 The exact/Born ratioofamplitudes for the model of Vo(r)
for A' = .1 (solid) and = .05 (dashed) at the indicated 2 values.
and A'. It exhibits a stronger sensitivity to variations in energy than z. Also
the dependence on 2 is mainly due to the rapid approach of the pole terms in
the scattering amplitude to their Regge asymptotics.
For a comparison of the model with the empirical data we have fitted the
differential cross section to an effective power, i.e.,
^ = G(E)y~2aeS with y~x = (3.48)
in fig. 3.11 for two different ranges of y. For A' = .15 the effective trajectory is
practically no lower than the bottom of the cut at l = .65. The corresponding
behavior for A' = .1 is distinctly different. Although it also rises with energy,
51
Figure 3.11 Effective and Coulombic Regge trajectories: the solid curves
are for the range : 8 â€”â–º 15, the dashed one at the top is for
y1 : 4 â€”>â€¢ 10.5 and the dashed one at the bottom is the n = 0 Coulomb
trajectory for A' = .1.
at moderate energy values it is closer to the subleading trajectory, the plotted
Coulomb trajectory, than even the bottom of its associated cut at l = â€”.6. It
is quite suggestive of the empirical effective trajectories in fig. l.A. We note,
however, that although there is a rise in the fitted trajectory as the range of y
is shifted up, it does not appear as dramatic as in fig. l.A and might well have
been hidden had we associated errors with our fits. This is more in tune with
our fittings of the cross sections discussed in the last chapter. We see again
that values of A' not higher than . 1 are more commensurate with the numbers
extracted from the strong interactions (empirically or theoretically).
52
3.3 The Scattering Amplitude: The Asymptotically Free Case
We need to select values for the parameters of the potential V(r). We
discussed earlier the indifference of the left end of the Regge trajectories to the
parameters that characterize the right end. Had we linearized our trajectories
at large positive energies, a natural choice would then have been for A Â« .3
That is we take the dimensionless quantity (trajectory slope/A) to correspond
our model to QCD. We are thus contradicting our earlier arguments for the
irrelevance of the confining aspect to the physics of asymptotic freedom, only
to the extent that an external mechanism is invoked to relate two disparate
scales of our model. The mechanism, of course, has to do with chiral symmetry
breaking in QCD. We choose
A = .5
as a close enough value since we already noted the insensitivity of the trajecÂ¬
tories in fig. 3.2 to an order of magnitude variation in A.
Had we been motivated by economizing the number of parameters in our
model, we might have picked c = This would have summarized the whole
potential in one term. However, such a choice does not lead to the behavior
observed in QCD, since for A of order one, the scattering amplitude at modÂ¬
erate E is domianted by the leading bundle (as can be seen from a born
approximation). Hence we proceed differently. To fix a value for c we compare
eq. (3.23) to (2.14). Taking 12/11 and numbers of similar order as 1 in the
latter equation, for c = ^ the two expressions would coincide. This is because
at that value the shortdistance limit of the potential is
V(r) 4 â€”1/r2 ln(A2r2) ,
(3.49)
53
similar to the asymptotically free limit of QCD VqCD(r) â€”Â» 1/r \n(AQCDr2).
However, we require a further correspondence.
The analog of relation (3.32) in the relativistic case would be a < 1 (for
a â€”a/r potential), beyond which the system is unstable with respect to pair
production. In QCD at the point where perturbation theory is taken to break
down, i.e., oÃqCD = 1, the potential mimics a fixedcoupling Coulombic one
at the threshold of stability. Taking this as a proper guide, the coefficient of V
in eq. (3.49) should be adjusted from 1 to In that case, in eq. (3.2) we set
c= 1/8 .
We point out that since the short distance behavior of V(r) is unaffected if c
and the power of Ar are changed in a correlated manner, we also examined the
trajectories generated by the potential
Vi(r) = 
.25
r2ln(l + 4r) r
This uplifted the trajectories in the â€œflatâ€ domain only slightly so that the
general results presented below are unaffected.
To apply eq. (3.30) to V(r) requires the trajectories and residues for A' < 0.
These correspond to irregular solutions of the Schrodinger equation which can
only be obtained by an analytic continuation of the regular ones. As these soÂ¬
lutions are not available to us in closed form we do not pursue this approach.
Rather we use our original paradigm of the accumulation of trajectories simuÂ¬
lating a squareroot branch cut to approximate the model for V by the model
for VQ. This we argue is a valid approximation in the region of interest because
the nexttoleading term in the potential near the origin is and, since
54
Table 3.3
â€œCut Profileâ€ for A = .5 , câ€” .125
E
Aeff
4.1
.1051
5.1
.1012
6.1
.09855
7.1
.09687
8.1
.09552
9.1
.09450
10.1
.09348
20.0
.08910
cA
bic ones generated by Vq for large negative energies. Around â€”E â€” 1 this
approximation is not valid so we restrict ourselves to â€” E > 5.
An estimate for the effective cut associated with the bundle of trajectories
can be obtained by the ratio of level separations predicted by eq. (3.23). We
call this a â€œcut profileâ€ for the bundle of trajectories and present it in table
3.3 , from which the reason for our earlier focus on A' Â« .1 should be clear. As
the effective cut decreases with energy we expect a more subdued dependence
on energy than exhibited in fig. 3.9.
Figure 3.12 shows the fraction of the total amplitude that is contributed
by the bundle of trajectories around A = 0 given the above cut profile, at
â€”z = 10. (The curves above and below it are for the same quantity had the
cut remained fixed at its corresponding values at the two ends of the energy
range.) The ratio is essentially flat slightly below .1 in the plotted range. The
55
Figure 3.12 \^Â¡2â€™ the solid curve is for V(r) with A = .5 and c = .125, the
dotted and dashed curves are for Vo(r) with â€” .1015 and .0891, respectively.
corresponding effective trajectory would, therefore, be even flatter (and lower)
than that in fig. 3.11 for A' = .1, requiring higher values of energy (for a given
range of z) to display an upward turnabout. This, we suspect, is a closer picture
of what takes place in QCD.
The behavior of the scattering amplitude in the neighborhood of â€”E = 1
where the relevant trajectory of fig. 3.2 is in transition from a Coulombic mode
to the asymptotically free limit, is an indicator of how the hard physics departs
from its softer progenitor. Taking the large2 limit of the Q's in eq. (3.30) for
the parialwave amplitude, An,
AnÂ« >>y(TAn+,;5hl(^",
fcsui7rAn T(An + 1) v â€™
(3.50)
56
Figure 3.13 The Coulomb/exact2ratio for the â€œresidue functionâ€ in the
partialwave scattering amplitude: the plot is for the first node and â€œexactâ€
refers to the model of V(r) with A = .5 , c = .125.
we call the coefficient of (â€”z)1" in the above expression the â€œresidue functionâ€
and examine the departure of that from a Coulombic behavior around and
beyond the transition region. The /Ts above can be obtained from the formula
[U]
gi7rAnjj.
KloÂ°drfn(r)/r2 â€™
where /n(r) are the Jost solutions of the Schrodinger equation defined by the
boundary condition
/(r) r^0Â°> ei(kr+Â±\n2kr) _
57
The ratiosquared of the residue function of a Coulomb potential to that
in eq. (3.50) is plotted in fig. 3.13 for the first node. The transition region is
around â€”E & . 7 where the plotted ratio is below one and shows a rapid rise
with energy.
3.4 Summation
To the extent that our choice for the parameters of the model incorporating
asymptotic freedom was justified and that its bundle of trajectories simulates
the squareroot branch singularity of the fixedcoupling model, the analytic and
numerical results are highly supportive of our conjecture regarding the behavior
of the QCD trajectories. Although this substantiation can not be made exact,
as inclusive strong interactions are described by a richer set of parameters
than our model, the various plots support the correspondence of QCD with
the quantum mechanical model of effective coupling less than .1. These results
also support calculations of ref. 10 both in their range of applicability and the
values obtained.
CHAPTER 4
RENORMALIZATION GROUP STUDY OF THE
STANDARD MODEL AND AN APPLICATION
The renormalization group (RG) equations describe the evolution with
scale of renormalized parameters of a theory. The standard model parameÂ¬
ters can thus be extrapolated from scales at which their values are known to
scales where they are subjected to constraints of a hypothetical extension of
the standard model [12]. These techniques have helped rule out some simpler
extensions of the standard model. An example is the ruling out of simple grand
unified extensions of the standard model based on the absence of the gauge
coupling unification at grandunified scales [13]. No doubt such conclusions
based on evolution over widely separated scales can be sensitive to the initial
values of certain parameters. Below we list our choice for the initial values [14].
We employ the modifiedminimalsubtraction (MS) scheme where the running
couplings are unphysical, and we collect the 0 functions of the standard model
in Appendix A.
From the decoupling theorem we expect the physics at energies below a
given mass scale to be independent of the particles with masses higher than this
threshold. Therefore for a correct interpretation of these running couplings we
must take into account the thresholds [15,16,17]. For the electroweak threshold
we use one loop matching functions[17] with the two loop beta functions valid
in the standard model regime. These matching functions are obtained in MS
renormalization by integrating out the heavy gauge fields in such a way that
58
59
the remaining effective action is invariant under the residual gauge group [16].
At the electroweak threshold, near M\y, we integrate out the heavy gauge fields
and the top quark. Below this threshold there is an effective SU(3)C x U(1)EM
theory. Thresholds in this region are obtained by integrating out each quark
to one loop at a scale equal to its physical mass. At these scales the one loop
matching functions in the gauge couplings vanish and the threshold dependence
appears through steps in the number of quark flavors[18] as the renormalization
group scale passes each physical quark mass.
4.1 The Initial Values
The values of a^{Mz) and q2(Mz), the U(l) and SU(2) running coupling
constants at Mz, are obtained from measurements of the QED coupling conÂ¬
stant and the Weinberg angle from a fit to all neutralcurrent data. They are
ax(Mz) = .01698 Â± 0.00009
(4.1)
a2(Mz) = .03364 Â± 0.0002 .
The value of as has been more difficult to determine. Due to the larger valÂ¬
ues of the coupling at lower energies, renormalization scheme dependences of
the truncated perturbation series can be nonnegligible. We use the Gaussian
weighted average of the <*s determined from e+ e~ annihilation to hadrons, T
decay, deepinelastic scattering and jet production in e+ e~ collision, to obtain
as(Mz) = 113 Â± .004 .
The Yukawa matrices were diagonalized at each step of the numerical proÂ¬
cedure in order to relate the matrix elements to the known parameters such as
60
the quark masses and the CKM mixing angles both as input and output. We
choose the following parametrization of the unitary CKM matrix
/ ci sic3 sis3 \
V = I siC2 C1C2C3  S2S3et6 C1C2S3 + S2C3eIÂ¿ I , (4.2)
\ 81*2 C1S2C3 + C2S3el6 Cis2s3  c2c3elÃ¡ /
with Sj â€” sin 9i and ct â€” cos 0Â¿, i â€” 1,2,3. The experimental ranges of these
parameters are
/ 0.97470.9759 0.2180.224 0.0010.007 \
V = 0.2180.224 0.97340.9752 0.0300.058 . (4.3)
V 0.0030.019 0.0290.058 0.99830.9996/
From the above two equations we arrive at the following bounds on the
mixing angles
0.2188 < sin 0i < 0.2235 ,
0.0216 < sin 02 < 0.0543 , (4.4)
0.0045 < sin 03 < 0.0290 .
However the accuracy with which V is known does not constrain sinÂ¿. A set
of angles {0i, 02,<5} was chosen that falls within the ranges quoted above.
It is also not clear at what scale should the above initial values be considered
known. However, since for the whole range of initial values the running of the
mixing angles is quite flat, in accordance with their being related to ratios of
quark masses, the choice for that scale is not crucial.
The physical values of the lepton masses are known to a very good precision,
resulting in the running values
me(l GeV) = 0.496 MeV ,
m^( 1 GeV) = 104.57 MeV , (4.5)
mT( 1 GeV) = 1.7835 GeV .
The running top quark mass is chosen arbitrarily, although consistent with
the (thenavailable) bound on the physical mass = ^^^GeV. For the
61
physical Higgs boson mass the bound 91 < Mfj < ITeV is used consistent
with observations and perturbativity. The running and physical masses of the
top and the Higgs are related thru
Mt
_1 ,
mt(Mt) 3 7T
16.11
1 = 1
(4.6)
and
Hi1) â€” +
(4.7)
where Mu i = 1,..., 5 represent the masses of the five lighter quarks and S(fi)
contains the radiative corrections as presented in Appendix B. The masses of
the five lighter quarks are discussed separately.
For the vacuum expectation value of the scalar field the following well
known value suffices:
v = (V2Gn)~2 = 246.22 GeV
The quark masses are a special class of low energy standard model paramÂ¬
eters in that the renormalized quantity which appears in the Lagrangian does
not have a direct physical analog. Since quarks are not observed as physical
states, their masses do not correspond to poles of a physical propagator, interÂ¬
est. In the past decade a variety of techniques have been developed and utilized
to extract quark masses from the observed hadronic spectrum. Below, we shall
briefly recount some such techniques. Furthermore, we shall present new values
for the heavy quark masses based on the application of our numerical technique
to three loops.
62
The light quark masses are the ones least accurately known. They are deÂ¬
termined by a combination of chiral perturbation techniques and QCD spectral
sum rules (QSSR). In the former case the light quark masses are directly exÂ¬
pressible in terms of the parameters of the explicit SU(2) and SU(3) chiral
symmetry breakings. One then considers an expansion of the form [19]
Mbaryon a "b i>mlight T â€¢â€¢â€¢
for the mass of a baryon from the octet and one of the form
2
Mmeson = ^mlight +
(4.8)
(4.9)
for a typical member of the pseudoscalar octet. A parameter measuring the
strength of the breaking of the more exact SU(2) chiral symmetry in compariÂ¬
son with the SU(3) one is the ratio
ms â€” m'
It ,
md  mu
(4.10)
where
m'= j (,mu+md) â–
(4.11)
To lowest order in isospin splittings, this translates in the meson sector into
R K ~ Ml
mkÂ°  Mh '
and in the baryon sector into three different determinations of R,
R ___~ mn) ~ ~ Ma)
MnMp
R  Mn) + f(Ms  Ma)
Mz:  Ms o
R memr
M% â€” Mv+
(4.12)
(4.13)
63
To make R compatible with all the above mass splittings one has to conÂ¬
sider higher order corrections in Eqs. (4.8) and (4.9). Here infrared divergences
emerge as one is expanding about a ground state containing NambuGoldstone
bosons. Once such singularities are removed within the context of an effective
chiral Lagrangian, one finds the following as the optimum value of R
R = 43.5 Â±2.2. (4.14)
Together with the ratic[20]
^7 = 25.7 Â± 2.6 , (4.15)
m
also determined by applying eq. (4.9) to the physical masses of 7r, rj and K,
they imply the following renormalization group invariant mass ratio
md ~ mu
2m'
= 0.28 Â±0.03 .
(4.16)
QCD spectral sum rules are obtained in an attempt to relate the observed
low energy spectrum to the parameters describing the high energy domain
where perturbation theory becomes applicable to the quarkgluon picture [21]
. One starts by considering the twopoint correlation functions for the vector
Vij = and axial vector A^ = ^'7^75^ quark currents
i iÃ©xeâ€œ11
= (Â«V Â¡r?2)ngv(Â«2) + I'V'ngW2),
â€™ (4.17)
i J d^xe1^ *
= (
where i, j = u, d, s are the quark flavors. The current divergences satisfy
dHVij(x) = *(â„¢i ~ mj) ,
dVAij(x) = Kmi + mj) A(xh5^j(x): .
(4.18)
64
The spectral functions Im{II(g2)} obey certain sum rules based on how their
analyticity properties are formulated. Among the QSSRs in vogue are the usual
dispersion relations based on a Hilbert transform
(4.19)
The Laplace transform sum rule is obtained by applying the inverse Laplace
operator L to the last expression,
(4.20)
for r a constant. The longknown finite energy sum rule (FESR) is obtained
by applying the Cauchy theorem to n(z)
(4.21)
where n is any integer, and the moment sum rules are obtained by taking the
nth derivative of eq. (4.19)
(4.22)
The left hand sides of Eqs. (4.19) through (4.22) follow from the high enÂ¬
ergy calculations to which various perturbative and nonperturbative correcÂ¬
tions have been found, while the right hand sides represent the low energy asÂ¬
pect, such as the hadronic vacuum polarization measured in e+e_ â€”> hadrons.
Applied to the light quarks these sum rules imply [20]
rhu + rhd = 240 + 2.5 MeV .
(4.23)
65
Together with eq. (4.16) they reduce to
rhu = 8.7 Â± 0.8 MeV ,
(4.24)
md = 15.4 Â± 0.8 MeV .
The parameter m is a renormalization group invariant which to three loops is
related to the MS running mass parameter m{n) via [22]
"Â»(/*) =*(/%â€”(n))_7l//Jl{ 1 + lf(j 
7T Pi Pi P2 K
+ i[Ã¡(2i _ 11 )2 _ ^(71 _ 72. (4 25.
+ 2 lpj(Pi 02> pfPi h] (4'25)
+ *(21 _ 22w2i(â€ž))2l
0101 03mxW,t
where the and the 7Â¿ are the coefficients of the beta functions for <*s and
m given in appendix A. From eq. (4.24) and eq. (4.25) to two loops, one may
infer the following values
mu{ 1 GeV) = 5.2 Â± 0.5 MeV ,
(4.26)
md{ 1 GeV) = 9.2 Â± 0.5 MeV .
In applying expression (4.24) it should be kept in mind that the continuity
of m(/i) across a quark mass threshold requires rh to depend on the effective
number of flavors at the relevant scale. The strange quark mass is determined,
averaging the value extracted from Eqs. (4.23) and (4.15) with those derived
using eq. (4.24) and the various QSSR values for mu + ms to be[23]
rhs = 266 Â± 29 MeV ,
corresponding to the running value
(4.27)
ms(l GeV) = 194 Â± 4 MeV . (4.28)
For the heavier quarks, charm and bottom, one can make a more precise
prediction. Here the nonrelativistic boundstate approximation may be apÂ¬
plied. The physical mass M(q2 = M2) appearing in the Balmer series may
66
be identified with the gauge and renormalization scheme invariant pole of the
quark propagator
S{q) = z(q)[y â– q  M(q2)]~l .
Corresponding to the above pole mass is its Euclidean version, m(â€”q2), which
although renormalization group invariant, is not gauge invariant and therefore,
not physical. The Euclidean mass parameter is the one often employed in the
J/xj) and T sum rules, as it minimizes the radiative corrections in such sum
rules. In the Landau gauge the two are related to two loops according to [24]
ra(M2) = M(MÂ¿)
13^104
7T
(4.29)
Once the pole mass is determined from the Euclidean one, the running mass
at the pole mass is obtained to three loops via
m(q2 = M2)
M(q2  M2)
1 + fâ€”^ + tf(Ssjrb2
(4.30)
where K = 13.3 for the charm and K â€” 12.4 for the bottom quarks[25].
From the J/xj> and T sum rules the following values have been extractec(20]
mc(q2 = M2) = 1.26 Â± 0.02 GeV ,
(4.31)
mb(q2 = Mjj) = 4.23 Â± 0.05 GeV .
To obtain an accurate value for the corresponding pole masses, we applied our
solution routine to eq. (4.29), with the above values inserted and the three
loop beta function for as given in appendix A to selfconsistently obtain the
following pole masses
Mc(q2 = M2) = 1.46 Â± 0.05 GeV ,
Mb(q2 = Mjj) = 4.58 Â± 0.10 GeV .
(4.32)
67
Recently[26], new values for the charm and bottom pole masses have been
extracted from CUSB and CLEO II by analysis of the heavylight, B and B*,
D and D* meson masses, and the semileptonic B and D decays with the results
Mc{q2 = M2) = 1.60 Â± 0.05 GeV ,
Mb(q2 = MÂ¡) = 4.95 Â± 0.05 GeV .
A weighted average of the values in Eqs. (4.32) and (4.33)yields
Mc(q2 = Ml) = 1.53 Â± 0.04 GeV ,
(4.33)
(4.34)
Mb(q2 = MÂ¡) = 4.89 Â± 0.04 GeV .
The running masses at the corresponding pole masses then follow from eq. (4.30)
mc(Mc) = 1.22 Â± 0.06 GeV ,
(4.35)
mb{Mb) = 4.32 Â± 0.06 GeV .
With these taken as initial data along with the value of the strong coupling at
M% quoted earlier, we run (to three loops) the masses and as to obtain the
following values at the conventionally preferred scale of 1 GeV
mc( 1 GeV) = 1.41 Â± 0.06 GeV ,
(4.36)
mj,(l GeV) = 6.33 Â± 0.06 GeV .
Our numerical approach does not make any more approximations than the
ones assumed in the beta functions and the mass equations used, and is thereÂ¬
fore more commensurate with our program than using the â€œperturbatively
integratedâ€ /3 functions. Thus we shall adopt the above values. It should be
stressed that at the low scales under consideration the three loop o;s corrections
we have included in our mass and strong coupling beta functions are often comÂ¬
parable to the two loop ones and hence affect the accuracy of our final values
noticeably. Nevertheless, it should be noted that the above expressions relating
68
the various mass parameters are not fully loop consistent as to our knowledge
eq. (4.29) has only been computed to two loops.
It should be pointed out that although we opted for the QSSR extracÂ¬
tion of masses, there are rival models, such as the nonperturbative potential
models, which predict appreciably higher values of the heavy quark masses
than the ones quoted here. These models, however, are not as fundamental as
the approach considered here, and their connection to field theory is rather
problematic.
4.2 Results
The results of numerically integrating the (3 functions of the standard model
parameters from 1 GeV to the Planck mass are displayed in the following
nine figures. Unless otherwise stated, = M# = lOOGeV for these plots.
In fig. 4.1 the evolution of the inverse of each of the three gauge couplings
are shown, assuming the standard model as an effective theory in the desert
up to the Planck scale. The â€œGUT triangleâ€ signifying the absence of grand
unification is evident. Here, the differences between one and two loop evolution
appear in the high energy regime, and, for the strong coupling constant, at low
energies, where it is stronger, as well. In fig. 4.2 these same inverse couplings are
displayed, this time including the associated uncertainties in their values. We
note, as is well known, that the uncertainties do not fill in the â€œGUT triangleâ€.
Figures 4.3 thru 4.5 display the one and twoloop evolution of the light
mass fermions (me, mu, and mÂ¿), the intermediate mass fermions (raÂ¿Â¿ and
ms), and the heavy mass fermions (mT, mc, and mÂ¿), respectively. Evidently,
the largest differences between one loop vs. two loop evolution occur in the
bottom, charm, and strange quark masses in these cases. Figure 4.6 is the plot
69
log10(M)
Figure 4.1 Running of the inverse gauge couplings with their central value
used as initial data.
for the self quartic coupling A and the top Yukawa coupling yt for (Mt = 100
GeV, Mh = 100 GeV) and for (Mt = 200 GeV, MH = 195 GeV). We have
studied the effects of changing Mt and Mjj values in our analyses of the running
of the other parameters. For any Mt between 100 GeV and 200 GeV, varying
Mjj, while maintaining perturbativity and vacuum stability, did not affect
appreciably the evolution of any of the other parameters. However, varying Mt
itself showed a significant difference in the running of the heavier quarks.
This is illustrated in figs. 4.7 and 4.8 which are similar to figs. 4.4 and 4.5
except that Mt = 200 GeV and Mjj = 195 GeV. Note that in fig. 4.8 the
intersection point between the bottom quark and the r lepton moves down to
a lower scale for this case of a larger top quark mass. This is expected since
70
logio(M)
Figure 4.2 Running of the inverse gauge couplings at two loops: the pair of
dotted lines denote the experimental errors for the central value (solid line).
from eq. A. 10 one can see that the bottom type Yukawas decrease with an
increasing top Yukawa. In contrast, in the SUSY GUT case the bottom type
Yukawa /3 function is such that this crossing point is shifted toward a higher
scale with increasing top mass. In an SU(5) SUSY GUT model, the equality of
the bottom and r Yukawas at the scale of unification will be used in the next
chapter to get bounds on the top and Higgs masses.
Figure 4.9 displays the running of the CKM angles using the initial data
sin 6\ = 0.2206, sin 62 â€” 0.0298, and sin #3 = 0.0106. We have also taken
S = 90Â° which corresponds to the case of maximal CP violation. As mentioned
previously, the evolution curves for these angles are effectively flat.
71
1Â°Sio(m)
Figure 4.3 Light quark and lepton massses.
In the present case of the standard model, we find that two loop running
of the parameters does at times improve on the one loop running. We have
tabulated the differences of several parameters in their one versus two loop
values at various scales, for the cases (MÂ¿ = 100 GeV, Mjj = 100 GeV) and
(Mt â€” 200 GeV, Mjj = 195 GeV). Table 4.1 illustrates the difference one
loop vs. two loop running make in the ratio mÂ¿/mr, for the three scales 102
GeV, 104 GeV, and 1016 GeV. As expected the difference between one and
two loop results is more pronounced at higher scales. Over all these scales the
difference is never greater than 10%. We note that the ratio becomes equal
to one well below the scale of grand unification as noted in the discussion of
figs. 4.5 and 4.8. Table 4.2 presents a similar comparison for the top Yukawa.
Here, two loops represent a smaller correction with the difference at all scales
72
0 25
1 Loop
i
0.20
2 Loop
Mt=100 GeV

0.00
o
5
10
15
20
logio(M)
Figure 4.4 Intermediate quark and lepton massses.
always being less than 5%. Lastly, Table 4.3 displays the same analysis for
for the case Mt = Mfj = 100 GeV. We observe no appreciable deviation from
the tabulated values for any Mt < 200 GeV (except in the low energy regime
where the difference is at most ~ 4%).
At scales < Mz, the inclusion of two loops is important in the evolution
of the strong coupling (and of the quark masses). Indeed, we find that the
pure QCD three loop contribution is also significant and therefore include it in
the running of the strong coupling and of the quark masses in the low energy
region. As seen in this table, the combined two and three loops in the low
energy regime account for a 17% difference at 1 GeV in as.
73
log io(m)
Figure 4.5 Heavy quark and lepton massses.
Although in the cases considered in these last two tables there does not
appear to be a significant difference in two loop over one loop evolution at
scales above Mg, the first table does show a 10% difference at the scale, 1016
GeV. We expect two loop effects to be more important when the theory is
extended, e.g., to include supersymmetry and/or grand unification.
The effects of using a naive step approximation vs. a proper treatment of
thresholds are numerically unimportant for the cases discussed above. Indeed
they are less important than the two loop effects. We note however, that the
inclusion of nonnaive thresholds effects is significant in the numerical analysis
of extensions of the standard model.
74
lÂ°gio(/Ã)
Figure 4.6 Top Yukawa and scalar quartic couplings at two loops.
4.3 Top Quark and Higgs Boson Masses
The renormalizationgroup technique is now applied to extract bounds on
the mass of the top quark and the Higgs boson in a minimal supersymmetric
extension of the standard model (MSSM) with minimal Higgs structure in
the context of a grand unified theory (GUT) [27]. The MS renormalization
group equations for the standard model and the MSSM [28] are numerically
integrated to evolve the parameters of the model to the Planck scale. As before
full account of the Yukawa sector is taken by diagonalizing the Yukawa matrices
at every step of the numerical routine.
If the standard model is the low energy manifestation of some yet unknown
GUT or of a possible supersymmetric (SUSY) extension thereof, the three
75
E
Ã©
logio (m)
Figure 4.7 Intermediate quark and lepton massses.
logio(M)
Figure 4.8 Heavy quark and lepton massses.
76
co
Cb
Â£
â™¦ pH
w
C\3
Cb
a
â€¢pH
w
a
*w
Figure
Table 4.1
mb/mT
102GeV
104GeV
1016GeV
one loop
1.938
1.499
.8326
two loop
1.937
1.463
.7962
Note: For the above Mt = lOOGeV
102GeV
104GeV
1016GeV
one loop
1.868
1.392
.6647
two loop
1.769
1.285
.6047
Note: For the above Mt = 200GeV
77
Table 4.2
Vt
102GeV
104GeV
1016GeV
one loop
.7879
.6076
.2830
two loop
.7873
.5940
.2701
Note: For the above Mt = lOOGeV
102GeV
104GeV
1016GeV
one loop
1.133
.9780
.7145
two loop
1.143
.9700
.6816
Note: For the above Mt = 200GeV
Table 4.3
lGeV
102GeV
104GeV
1016GeV
one loop
two loop
.3128
.3788
.1118
.1117
.07103
.07039
.02229
.02208
couplings #3, g2, and g\ corresponding to the standard model gauge groups,
SU(3)C x SU(2)W x U(1)Y, should meet at some large grand unification scale.
Using the accepted values and associated errors of these couplings we observe
unification in the SUSYGUT case but not in the pure GUT case, as noted by
several groups[13,29] (see fig. 4.10). However this should not be viewed as proof
of supersymmetry since given the values of c*i, 0:2, Â«3 at some scale, and three
unknowns (the value of a at the unification scale, the unification scale, and an
78
^SUSY â€” ^ ^eV
log10(/Â¿)
Figure 4.10 Running of the inverse couplings: dotted pair of lines are the
experimental errors for the central value (solid line).
extra scale such as the SUSY scale) there is always a solution. The surprising
aspect of the analysis of ref. 13 is the numerical output, namely a low SUSY
scale, MsuSYi and a perturbative solution below the Planck scale which does
not violate proton decay bounds.
Furthermore, in the context of a minimal GUT there are constraints on the
Yukawa couplings at the scale of unification. First we restrict ourselves to an
SU(5) SUSYGUT where and yT, the bottom and r Yukawa couplings, are
equal at unification. The crossing of these renormalization group flow lines is
sensitive to the physical top quark mass, M*. This can be seen in the downtype
79
Yukawa renormalization group equation (above MguSYi fÂ°r example), from
which we extract the evolution of since the top contribution is large and
appears already at the one loop order through the uptype Yukawa dependence:
=77TTY
dt 16tt^ (4.37)
â€œ (jf^l + 3?2 + 1 â–
where Yude are matrices of Yukawa couplings. Demanding that their crossing
point be within the unification region determined by the gauge couplings allows
one to constrain MÂ¿. This yields an upper and lower bound for Mt which is
rather restrictive. Our treatment of the thresholds below the SUSY scale was
discussed towards the beginning of the chapter. For the threshold at MguSYi
the matching condition is the naive one of simple continuity, due to the lack of
knowledge about the superparticle spectrum. We take this scale to be variable
to account for this ignorance.
We consider the simplest implementation of supersymmetry and run the
couplings above Msusy one loop The superpotential for the supersymmetÂ¬
ric theory is:
W = $uQYuuc + $dQYddc + $dÂ¿YeÃ©c + ^dÃ©u , (4.38)
where the hat denotes a chiral supermultiplet. We assume the MSSM above
MsuSYi and a model with a single light Higgs scalar below it. This is done by
integrating out one linear combination of the two doublets at MgusYi thereby
leaving the orthogonal combination in the standard model regime as the â€œHiggs
doubletâ€ :
$(sm) = Â®dcos/? + $usin/? ,
(4.39)
80
where Ã> = Ãt2$*, and where tan/? is also the ratio of the two vacuum exÂ¬
pectation values (vu/vÂ¿) in the limit under consideration. This sets boundary
conditions on the Yukawa couplings at MguSYâ€¢ Furthermore, in this approxÂ¬
imation the quartic self coupling of the surviving Higgs at the SUSY scale is
given by:
Kmsusy) = 4(01 + #2)cos2(2/2) â€¢ (440)
This correlates the mixing angle with the quartic coupling and thereby gives
a value for the physical Higgs mass, M}jiggs. Using the experimental limits on
the Mjjiggs further constrains some of the results. By using the renormalization
group we take into account radiative corrections to the light Higgs mass and
hence relax the tree level upper bound, Mfjiggs ~ Mg.
We determine the bounds on and MÂ¡jiggs by probing their dependence
on /?. In SUSYSU(5), tan/? is constrained to be larger than one in the one light
Higgs limit. It seems natural to us to require that yt > y\, up to the unification
scale, thereby yielding an upper bound on tan /?. The initial values at Mz for
the gauge couplings are taken to be:
ai = 0.016887 Â±0.000040 ,
a2 = 0.03322 Â± 0.00025 ,
<*3
0.109
+0.004
0.005
9
(4.41)
where GUT normalization for a\ is used. We use the set of four quark running
masses defined at 1 GeV by the Particle Data book: mu â€” 5.6 MeV, mÂ¿ = 9.9
MeV, ms = 199 MeV, and mc = 1.35 GeV. For the bottom mass we use
the Gasser and Leutwyler bottom mass value of 5.3 GeV at 1 GeV which
translates into a physical mass of Mf, = 4.6 GeV [19]. To probe the dependence
of our results on we also study the case of M^ = 5 GeV, the typical value
81
obtained from potential model fits for bottom quark bound states [30]. We
also investigate the effect of varying MgjjsYâ€¢ Given the values of the gauge
couplings, we find unification up to a SUSY scale of 8.9 TeV, and as low as
Mw, below which we did not investigate for empirical reasons.
From fig. 4.10 we determine that the lower end scale, MqUT, of the unifiÂ¬
cation region corresponds to an Â«3 value of 0.104 at Mz, while the higher end
scale, MgUT, corresponds to a value of 0.108 at Mz for Â«3. We find that the
unification region is insensitive to the range of top, bottom, and Higgs masses
considered. In our analysis of the bounds for Mt, the values for oq and c*2 are
chosen to be the central values since their associated experimental uncertainÂ¬
ties are less significant than for Â«3. Demanding that y^ and yT cross at MqUT
and taking 0:3 = 0.104 then sets a lower bound on Mt Correspondingly, deÂ¬
manding that yfj and yT cross at MgUT and taking 03 = 0.108 yields an upper
bound on Mt These bounds are found for each possible value of /3.
Figure 4.11 shows the upper and lower bound curves for both Mt and
MHiggs 35 a function of Â¡3 and for M$usy = 1 TeV and M^ = 4.6 GeV.
When applicable we use the current experimental limit of 38 GeV on the light
supersymmetric neutral Higgs mass, to determine the lowest possible Mt value
consistent with the model. We find 139 < Mt < 194 GeV and 44 < Mniggs <
120 GeV. We investigated the sensitivity of these results on MguSY iu the
range, 1.0 Â± 0.5 TeV. It is found that the bounds on Mt are not modified, but
the upper bound on the Higgs is changed to 125 GeV, and the lower bound
drops below the experimental lower bound.
For Mi, = 5.0 GeV, we see an overall decrease in the top and Higgs mass
bounds: 116 < Mt < 181 GeV, MjjtggS <111 GeV. Varying MgusY as above
82
MsusY â€” 1 TeV, Mb â€” 4.6 GeV
>
0)
O
V)
oo
00
SC
s
CX
o
P (deg.)
Figure 4.11 and M}j as a function of the mixing angle /3: the highest
(lowest) curves are for highest (lowest) value of q3 consistent with unification
as per fig. 4.10.
modifies the upper bound on Mjjiggs to 115 GeV. We display the results of our
analysis for the extreme case, MguSY = 8.9 TeV, in fig. 4.12, with = 4.6
GeV. This only significantly changes the upper bound on Mjjiggs to 144 GeV
compared to the MgjjsY = 1 TeV case.
We have also run yt up to the unification region and compared it with
and yT to see what the angle (3 must be for these three couplings to meet [31],
as in an SO(IO) or Eg model with a minimal Higgs structure. It is clear that
this angle is precisely our upper bound on (3 as described earlier. In fig. 4.13 we
display yt/yb at the GUT scale as a function of tan /3 for MguSY â€” 1 TeV and
83
Msusy = 89 TeV* Mb = 4*6 GeV
>
0)
o
V)
W>
w>
X
cx
o
tj
s
/? (deg.)
Figure 4.12 Mt and M# as a function of the mixing angle (3 for the upper
bound on MguSY
for the two bottom masses we have considered. If we demand that the ratio
be one we can determine the mixing angles for the low and high ends of the
unification region. Then going back to fig. 4.11 we find, as expected, a much
tighter bound on the masses of the top and of the Higgs. Indeed, for MÂ¡, = 4.6
GeV, we have 49.40 < tan (3 < 54.98, which yields 162 < Mt < 176 GeV and
106 < Mjjiggg <111 GeV. When MÂ¡Â¡ = 5.0 GeV, we obtain 31.23 < tan/3 <
41.18, which gives 116 < Mt < 147 GeV and 93 < MHiggs < 101 GeV.
We point out several simplifications made. We have not implemented the
supersymmetric two loop beta functions and the corresponding thresholds.
The effects of soft SUSY breaking terms were not investigated. Also, we have
yt(^GUT)/
84
â– ^â– susy â€” 1 TeV
tan/5
Figure 4.13 ^ vs tan /3 for two distinct bottom masses (solid and dashed
curves), and highest and lowest values of (high and low curves) consistent
with unification as per fig. 4.10.
integrated out all the supersymmetric particles at the same scale. Given the
relative crudeness of the approximations in this paper, it is remarkable that
the experimental bounds on the pparameter were satisfied. The top quark has
since been observed at Fermilab with a mass of 174 Â± lOGeV.
CHAPTER 5
CONCLUSIONS
We have studied mainly two aspects of standard model phenomenology.
The discrepancy between a recent calculation of the leading QCD meson traÂ¬
jectory and the relatively old experimental data on such trajectory is conjecÂ¬
tured to be due to strong subleading effects in the scattering amplitude at the
explored energies. The extraction of trajectories is subjected to a new analysis
for both exclusive and inclusive pexchange processes. In both the exclusive
and s = 400 GeV2 inclusive cases restricting the fit range to be closer to the
Regge limit induces a slight rise in the trajectory around, but above, the t
value where it crosses zero. Although the quantitative extent of this rise is noÂ¬
ticeably less than that displayed by the experimentalists, it is consistent with
our conjecture.
To estimate the energies needed to reach the Regge limit, a fairly simple
minded twopower fit of the scattering amplitude is made, with the leading
power fixed at the predicted effective value of zero. The subleading term shows
a strong coupling to the amplitude and the resulting estimate is for a beam
energy of three orders of magnitude to probe the Regge limit.
To have a soluble model to directly test the behavior of the trajectories and
the scattering amplitude a potential model that incorporates asymptotic freeÂ¬
dom is studied. The resulting accumulation of the leading trajectories mimics
that of pQCD. The model is then approximated by one where the accumulation
85
86
of trajectories is replaced by a fixed squareroot branch cut at every energy. For
parameter values that arguably correspond to QCD, the resulting scattering
amplitude is strongly influenced by the subleading effects at moderate energies
and the effective trajectory it implies is well below the leading singularity.
The other aspect of our work is a twoloop order renormalization group
study of the standard model subject to a full accounting of the Yukawa sector
and a more sophisticated treatment of threshold effects. The inclusion of two
loop corrections was shown to be more significant than that of the nonnaive
threshold effects. Assuming a desert between the electroweak scale and typiÂ¬
cal GUT scales, the three gaugecouplings failed to unify. However, extending
the RG equations of the standard model by minimal supersymmetry, induced
coupling unification at moderate values of the SUSY scale. Demanding the
equality of the tau and bottom masses at the resulting unification scale in the
extended model yielded top and Higgs mass predictions, the former of which
has subsequently been verified at Fermilab.
APPENDIX A
(3 FUNCTIONS FOR THE STANDARD MODEL
In this appendix we compile the renormalization group (3 functions of
the Standard Model. These have appeared in one form or another in various
sources. We have endeavored to confirm their validity through a comparative
analysis of the literature. Our main source is ref. 28. Following their convenÂ¬
tions,
Â£ =Ql^Yu^ur + dR + IL<& Ye^eR + h.c.
1 (A.l)
 jA($*4.)2 ,
where flavor indices have been suppressed, and where QL and iL are the
quark and lepton SU(2) doublets, respectively,
Â«â€˜(so â€™4=fc) â€¢
$ and $ are the Higgs scalar doublet and its SU(2) conjugate:
= (A.3)
uR, dR, and eR are the quark and lepton SU(2) singlets, and Yu Â¿e are the
matrices of the uptype, downtype, and leptontype Yukawa couplings.
The (3 functions for the gauge couplings are
^ _ V' _ 9?
dt 167T2 (167T2)2 (167T2)2 (A.4)
xIV{C,â€žYâ€ž*Yâ€ž + CldYÂ¿Yd + CJeYe*Ye} ,
87
88
where t = \nn and l = 1,2,3, corresponding to the gauge group SU(3)c x
SU(2)l x U(1)Y of the Standard Model. The various coefficients are defined to
be
, _ 4 1
1 3 9 10 â€™
22 4 1
g ng g >
4
63 = 11 â€œ 3 ng ,
(A.5)
(bkl) =
and
(Q/) =

/ 19
T3
3
3
44
V 15
1 11 \
3 30
49 3
T 2
4 T/
with / = w, d, e,
with rig = ^ny/.
In the Yukawa sector the (3 functions are
(A.6)
(A.7)
_ / 1 ^(1) ,
dt A6tr2^u^e
i: )y ,
(l67T2)2Pudâ€™eJ Uâ€™dâ€™e
where the one loop contributions are given by
ÃP = ?(Yâ€ž*Yâ€ž  YÂ¿tYd) + Y2(S)
.17 9 9 2 n 2\
â€œ (^#1 + 4*2 + s93) >
/?/ = 2(YdtY,iYâ€žtYâ€ž) + Y2(S)
 (jffl + JÃ2 +893) 1
41) = 5YetYe + y2(s)j(9? + 9),
(A.8)
(A.9)
with
Y2(S) = Ti{3Yâ€žtYâ€ž + 3YÂ¿Yd + YeÂ»Ye} ,
(A10)
89
and the two loop contributions are given by
ffP = Â¿(Y^Y,,)2  YjYuYdWd  'Yj'YjYJYu + ^(Yj'Y,,)2
+ Y2(S)( jY/Yj  jYâ€ž*Yâ€ž)
 Xi(S) + Â¿A2  2A(3Yâ€ž*Yâ€ž + Yd*Yd) + (^gÂ¡ + ^lgÂ¡ + 169)Yâ€ž*Yâ€ž
 " Tefi + 16Â»3>y
 Â¿SiÃ2 + yfffisl  (:j  nj)S2 + 9sM ~ (5  ^Â«Â«)S3 
0f = j(YdfYÃ¡)2  Yd*YdYâ€žtYâ€ž  ÃYâ€žtYâ€žY,Â¡tYd + j(Yâ€ž*Yâ€ž)2
+ Y2(S)(jYutYu  jY/Yj)
 X*(S) + jA2  2X(3Yd
 <Â£Â«?  Â¿Â»2 + 16^Yâ€œ,Yâ€œ + 5y*  <5Â® + Ã©n*>Â«*
27 o 2 31 2 2
' 20giff2 + 15^3
,35
(T ~ nd)92 + 99293 ~ (
,404 80
â€”â€”n
4 v 3 9
pP = Â¿(Ye*Ye)2  ?Y2(S)YetYe  Xi(S) + 2a2  6AYe*Y(
,387 2 135 2\Â»r +.1
+ (~Â¿7T9i + ~^92)^e^e
g)9Â¡
80
5 . . , 51 11 .4 27 o 2 ,35 . 4
+ 2n(S) + <200 + Jng)gi + 20M2 ~ (T â€œ nÂ°)92
(A.11)
with
and
F4(5) â€”(â€”Pi + pi + 8p3)Tr{Yu^Yu}
+ (\s! + \Ã± + Â«slWY^Yj}
+ f(9l+SÂ§)'&{Y,.tYe} ,
X4(S) =jTV{3(Yâ€žtYâ€ž)2 + 3(YdtYd)2 + (Ye*Ye)2
 jYâ€ž*Yâ€žYdtYd} .
(A12)
(A.13)
In the Higgs sector we present 0 functions for the quartic coupling and the
vacuum expectation value of the scalar field. Here we correct a discrepancy in
90
the one loop contribution to the self quartic coupling of ref. 28
dX
dt
1 +
r/5
(2)
167T2 ^ (167T2)2 ^
where the one loop contribution is given by
(A.14)
â€”12A2  {g\ + 9g%)X + ^(^01 + + 9%)
+ 4Y2(S)\4H(S) ,
with
and the two loop contribution is given by
jj.7? I 3r2^ '2
/42)=78A3 + 18(^ + 3^)A2
(A.15)
H(S) = Tr{3(YutYu)2 + 3(Yd+Yd)2 + (YetYe)2} , (A.16)
, ,313 . 4 117 2 2 9 ,229 O Un
~ [ 10ng)92 + 1^~9i92 + ^(J + 2ng)9\
,497 0 , 6 3.97 8.2 4 9 ,239 40 . 4 2
+
 + f %)9?  64SlV{(Yâ€žtYâ€ž)2 + (Y<,tYd)2}
 ^fTr{2(Yâ€žtYâ€ž)2  (Y^Y,,)2 +3(Ye*Ye)2}  ^Y4(S)
+ 10A[ + ^2/2 + 893)Tr{Yu^Yu} + (^ffi + + %I)â€iâ€™r{Y,/Y.;}
+ j(9l + i22)Tr{YefYe} ]
+ fffll + 219)Tr{Yâ€žtYâ€ž} + (\g\ + ^TUY/Y,,}
+ (J9Ã + HÂ«2WYe'Ye} ]
 24A2Y2(S)  AH (.S') + 6ATr{Yâ€žtYâ€žY/Y,/[
+ 20Tr{3(Yâ€žtYâ€ž)3 + 3(Y/Yd)3 + (YetYe)3}
 12TY{Yâ€žtYâ€ž(Yâ€ž'Yâ€ž + Yd*Yd)YjYd] .
(A.17)
The Â¡3 function for the vacuum expectation value of the scalar field is
dlnu 1 /1 \ 1
7Uj +
dt
167T2
(167T2)2
7
(2)
(A.18)
91
where the one loop contribution is given by
7(1> = +s!)  r2(s),
(A19)
and the two loop contribution is given by
(o\ , 1 .9 23 4 221 4 27
7<2> =X4(S)   A + ^ â–  jÃ³
40
27 2 2
Sl
(A.20)
â€” 5[ (â€”9, + ,,32 + 16
+ (Â«? + fffl + l%3)Tr{Y,,tYrf}
+ 5(9l+S2)Tr{YetYe)l.
These expressions were arrived at using the general formulas provided in ref.
28 for the anomalous dimension of the scalar field, choosing the Landau gauge.
In the low energy regime the effective theory is SU(3)c x U(1)em We
employ the general formula of ref. 32 to arrive at the /3 functions for the
respective gauge couplings:
dS3 ,2, , , n] 93
and
(47t)2
r38. . . go
+ [TK + nd)  102]^
,8 2 ff'e1 5033.
+ l9nâ€œ+9nâ€˜Â¡W + [^â€œ(â€™!u + "
325 , 28571 gÂ¡
~^(nu + nd) ]^,
de .16 4 4 . e3
5=[Tnâ€ž+â€žd+rÂ¡]^
r64 4 . e5
+ [27â€™!â€+27"â€˜i + 4"'W
(A.21)
r64
e^3
râ€” 16 7
+ [y"Â« + jnd](4jr)4 ,
(A.22)
92
where nu, nÂ¿, and nÂ¡ are the number of uptype quarks, downtype quark, and
leptons, respectively. In eq. (A.21) we have also included the three loop pure
QCD contribution to the 0 function of <73 [33].
For the evolution of the fermion masses we used ref. 34 It is known that
there is an error in their printed formula [35]. Using the corrected expression,
we compute the following mass anomalous dimension. The fermion masses in
the low energy theory then evolve as follows:
dm
(A.23)
where the / and q refer to a particular lepton or quark, and where
333 #3
, OOP ^o
(9) (47t)6
The superscripts 1 and 3 superscripts refer to the U(1)em and SU(3)c contriÂ¬
butions, respectively. Explicitly, the above coefficients are given by
7(U) â€œ _6<5(/,9)
7(Z) = 0
T?Â®> = â€œ8
(A.25)
333 2.140. x9 , 2216
7(?) = 3[17("â€œ + "<*> + d60C(3) + â€” )K + nd)  3747] ,
93
where Q(i,q) is the electric charge of a given lepton or quark, and Â£(3) =
1.2020569... is the Riemann zeta function evaluated at three. In the mass
anomalous dimension for the quarks above, we have also included the three
loop pure QCD contribution 7333 [33].
APPENDIX B
THE EXPRESSION FOR 6(h)
In ref. 36 the radiative corrections term 5(h) from eq. (4.7) is derived. In
this appendix, we present its explicit form as it appears in this reference except
for some minor notational changes. In the following, s and c refer to sin 9\y
and cosÃ©fyp, respectively. Also, Â£ is defined to be the ratio
G M2
Â¿(Â¿o = f^Â§{Â£/i(^) + mri+c'fM >
(B.l)
where the various functions are defined as follows:
/l (Â£,/*) =6 In
3 l7,lv 7(c*. , 2 , 9/25 7T
*Â«./â€¢)  ^  2^] + 4 + 2Z(i) + 4e^(d) + ^
Ml
â€˜Ml
+ 12c2 Inc2  22(1 + 2c2)  3^t[2Z{^Â±) + 4In
Â° ml vMe m
Mk
i
5] ,
M) =6In ^2 [1 + 2c4  4^L]  6Z(i)  12c4Z(^)  12c4 Inc2 + 8(1 + 2c4)
. . M+ Mf Mf â€ž
+ 24â€”r In â€”T  2 + Z(â€”^)1 ,
ML M2 ^M2^
M,2
Mj
h
(B.2)
with
2Atan 1(1 /A) for(2 > ,
1,
Z(z)
Aln[(l + A)/(l â€” A)] for(z <
A=\l4z\* .
(B.3)
94
APPENDIX C
SELFDUAL SOLITONS IN A CHERNSIMONS
MODEL WITHOUT A BROKEN VACUUM
We know of two ways of associating particles with fields. One is through secÂ¬
ond quantization and the other through the soliton solutions of field equations.
The latter afford a particle interpretation because they are localized solutions
whose energy density is concentrated in a fairly welldefined region. Outside
this region the fields quickly assume their vacuum values. Here we present a
novel soliton model in 2+1 dimensions that satisfies Bogomolâ€™nyitype equaÂ¬
tions but unlike other known models of this kind does not possess a broken
vacuum.
Abelian ChernSimons (CS) scalar theories have been under recent study
for their novel solitonic solutions and for their use as LandauGinzburg deÂ¬
scriptions of anyon superconductivity. The interest in the former aspect of
these models has been fuelled mainly by the discovery of topological and non
topological selfdual charged solitons. Selfdual solutions, in the Bogomolâ€™nyi
sense, obey first order equations, consistent with the second order equations of
motion, that characterize stationary points of the energy, and typically imply
an N = 2 supersymmetric Lagrangian behind the model. Because their charge,
flux and angular momentum properties give these solitons anyonic features,
they could be pertinent to studies of anyon superconductivity.
In this regard our solitons possess certain interesting features. The soluÂ¬
tions considered here are nontopological solitonvortices whose spectrum has a
95
96
â€œbandgapâ€ structure and from which the electric field is expelled. These feaÂ¬
tures are reminiscent of those associated with an ideal superconducting anyon
gas. Although in the present model the gauge field is perturbatively nondy
namical and there are no explicit selfinteractions for the scalar fields, these
properties are achieved by introducing an effective nonrenormalizable P and
Tviolating coupling of the scalar and gauge fields.
C.l The Model
Nonminimal gauge kinetic terms have been studied in various contexts.
Such couplings have appeared, for instance, in lowenergy effective supersymÂ¬
metric models. In a solitonic context they have been utilized in the nontopo
logical soliton bag models where the YangMills kinetic term is multiplied by a
â€œdielectic functionâ€ to incorporate renormalization effects, and in front of the
Maxwell term in some 2+1 dimensional ChernSimons inspired models with
selfdual solitons. Scalar fields have been coupled to the CS term either for the
sake of studying the solitons of the resulting theory, or as part of a Landau
Ginzburg description of anyon superconductivity.
Here we add the following interaction
7 K
C! = e^tfidnA^Dpt + c.c. , (C.l)
to a minimal scalar ChernSimons Lagrangian,
Â£ = \Dâ€ž4>12 + â€ž)Aâ€ž +C vm
k (C.2)
= lÂ¿VI2 + 5(m  M VMÃ.4,, + 8â€žx)  vm ,
where k is real, Dp = dp + ieAp, (f> = e*e*, and the metric is (1 ). C'
may be regarded as a gauge invariant attempt to make the coefficient of the
CS term, which has the dimension of mass, dynamical.
97
The EulerLagrange equations of motion for the Lagrangian in (C.2) are
 Du4>  c.c.) + ^evp,p(Dp(j))*Dp(j)  k\4>\ievp'pdpAp + mke^^dpAp â€”
 ie(4>*Dâ€)  (j)\2euppdpAp + mkel'pâ€˜pdpAp = 0,
(C.3)
where the last two lines are mere rewritings of each other, and
^epupDpDvDpcj>  DpDp  V\02) = 0 . (C.4)
The Noether current associated with the U(l) global symmetry of the LaÂ¬
grangian, jpQeth = (SC/Sdp(f))5(f) + c.c., is given by,
f = ie(
,k (c5)
= mke^dvAp  ,
using eq. (C.3) in the last step. Thus we have the characteristic chargeflux
relation,
Q = mkA> , (C.6)
where
Q
jo d2x
and $ =
/
Fi2 (TX
(C.7)
are the charge and flux, respectively.
The energymomentum tensor of the model has the same form as that of
one without the LeviCivitized terms,
Tpv â€” (Dp)*Dv + (Du(Â¡))*Dp(f) â€” gpi,(\Da(f)\2 â€” V) . (C.8)
Thus, for D = (y  ieÃ),
the energy is given by,
E = I d1x(\D\2 + e2(A0 + a0x)2<Â¿*2 + (SoM)2 + V) .
(C.9)
98
From eq. (C.3) we obtain the constraint equation
(Â¿0 + dox) = Â¡^Â¡2 V x (M~  vxM2)  + Â¿^Â¡2B â€¢ (C10>
Using eq. (C.10), the identity
D(Â¡>\Â£ = \(Di Â± iD2){Â¿ Â± eB\(f>\2 =F x (4>*D  ex.) ,
(C.ll)
and ignoring the surface term from the last term above, we are left with
E = Jd2x{(do0)2 + pi Â± iD2)\2
+ [m* * ((Ã â€œ â„¢)l02) + W2 " 2m> Â± 26 101
2Uh2
.4e0
4e0
+ V  ^02 j Â± 2me .
For the particular choice
r2i 2.2
(C.12)
2eâ€˜
V â€” Mz\f , where M = â€”â€” ,
(C.13)
and a fixed flux the Bogomolâ€™nyi bound is saturated by demanding
Wl = 0 , (C.14)
(DiÂ±iD2) = 0 , (C.15)
BM2  2m) + V X {{A  VxM2) = =F \\2 â– (C.16)
e
Since to every selfdual (+ sign) solution there corresponds an antiselfdual
(â€” sign) one with the same \\ but \ and A of opposite sign, one need only
catalog solutions for the positive choice. Since there is no symmetry breaking
potential, perturbing around the 0 = 0 vacuum we shall find two propagating
degrees of freedom of mass M, with the gauge field being nondynamical.
99
Furthermore, it is clear that a nonzero m is necessary to obtain nonvanishing
Emim
Emin â€” Â±2me$ = Â±â€”M , (C.17)
e
from eq. (C.12), and for the B field to have zero vacuum expectation value
according to eq. (C.16). Equation (C.17) has the same form as the analogous
CSH one with the correspondence 2m (vev)2. This identification is further
strengthened by the fact that at least for rotationally symmetric solutions,
studied in the next section, m needs to be positive. Then the choice of sign
in eq. (C.12) is correlated with that of the flux, in order to obtain Emin > 0.
These solitons are similar to the ChernSimonsHiggs ones in the following
respects: eq. (C.17) implies that the mass per unit charge of the soliton and
the scalar field coincide, the presence of the term mkB/(2e0) in eq. (C.12)
implies a ringlike structure for the magnetic field of the soliton and provides
stability in the sense that it cannot collapse to the 0 = 0 solution as the trivial
configuration carries no flux.
Equation (C.15) implies
e(Al + dl\) â€” Ee^dj In \\ , (C.18)
which leaves us with
eB = =f V2 In\\ â– (C.19)
Now, from Eqs. (C.10) and (C.16) we get
e(A0 + d0X) = M , (C.20)
which together with (C.18) imply a vanishing electric field in the selfdual limit,
Ã‰ = 0 .
(C.21)
100
Equations (C.16), (C.18) and (C.19) give us
(4>\2  2m) v2 In \\2 + V2
which, upon defining the mass parameter Â¿Â¿ = M2/m, reduces to
V2 In 0  â€”0 V2 0 + M2 = 0. (C.23)
m
It is amusing that simultaneously taking the weak limit of the C! term and the
strong limit of the ChernSimons coupling in eq. (C.2), i.e. k2 ~ m1 â€”>â€¢ 0,
reduces the above equation to the Liouville one for positive and nonsingular
0. We now fix m to be finite and define
101 = y/m/ and s = sgn(m) , (C.24)
so that we are left with
s V2 In /  / v2 / + M2f2 = 0 . (C.25)
C.2 Specific Solutions
Restricting ourselves to the rotationally symmetric static solutions, which
are of the form
0 = yf\~m\f{r)eme ,
 0
eA = (n â€” a(r)) ,
r
eq. (C.10) becomes
eMfAq = r 1(a'f(lsf 2) + af'^j ,
which makes eq. (C.9) turn into
r0Â° ' f/2 , f2a2
E
= 27rlml^ {/'
+
+ M2/2
(C.26)
(C.27)
+
~Â¿[a'f(\sf 2) + a/']2}rdr ,
(C.28)
101
with the selfduality conditions (C.15) and (C.16) given by
f = Â±aA,
r
If2 Â») + if = Â±(M/)2
r r
Substituting eq. (C.29) in eq. (C.30) gives
^(/2s) = Â±/2(m2^)
(C.29)
(C.30)
(C.31)
We shall now confine ourselves to the upper sign in Eqs. (C.29) through (C.31),
since as already mentioned there is a onetoone correspondence between the
solutions corresponding to each choice of sign.
The finiteness of the energy in eq. (C.28) implies
/ â€”Â» 0 and a â€”> â€”a as r â€”> oo
(C.32)
for a a constant, and the regularity of the solutions at the origin requires
nf 0 and a n as r â€”> 0
(C.33)
We now consider the large and short distance behavior of the rotationally
symmetric solutions to eq. (C.25). As we shall elaborate shortly, numerically
there is a lower bound of 1.75 on a. Power series expansion for r â€”> oo, then,
leaves us with
/ = A(Mr)~a
1~S4(al)2(Afr) ^ 1) + sl(Mr) 2â€œ
+
A4
(Mr)4^1) + 0((Mr)x)
16(q â€” l)4
a = rdr In f = a + sn/ ^ ^(Mr)"2^"1)  s^â€”(Mr)~2a
2(orl)
A4
8(q l)3
(Mr)_4(a_1) + 0((Mr)x) ,
(C.34)
102
where x = â€”6a+ 6 for 1.75 < a < 2 while x = â€”4a+ 2 for a > 2. When r â€”Â» 0
and n = 0, we have
f = C0
a =
1 +
2(
4(
(Mr)4
Â°Â°2 (Mr)2 + ^W/lJÃ(Mr)4 + 0((Mr)6)
16(C02  sf
1  Co,Icf2 y:1 (Mr>2+Â°Â«Mr)4)
4(
(C.35)
To the order we are expanding, the case r * 0 and n / 0 leaves us with two
possibilities. For n = lwe have
/ = C^Mr)
1 + S!+(Mr)2 + ^(SC,2  s)(Mr)4 + 0((Mr)6)
4 lo
a = l + ^(Mr)2 + ^(5Ci2/2  s)(Mr)4 + 0((Mr)6) ,
(C.36)
while for n > 1 we end up with
/ = Cn(Mr)n
nCn2
a = n + sâ€”â€”
P 2
Orj
1 + s:ij(Mr)2n  s 2
4 4(n+l)2
r* 2
(Mr)
2n
r1 2
''71
2(n + 1)
(Mr)
2n+2
(Mr)2n+2 + 0((Mr)4n) ,
+ 0((Mr)4n) .
(C.37)
Assume s = â€”1. Then from Eqs. (C.35) â€”(C.37) it is clear that in a suffiÂ¬
ciently small neighborhood of the origin a is positive. In that case as r evolves
a will not change sign. For if a becomes sufficiently small while still positive
the sign of a1 according to eq. (C.31) will be positive so that a cannot get
arbitrarily close to zero. eq. (C.29) would then imply that / is a monotonically
increasing function of r so that the boundary condition in expression (C.32)
may not be fulfilled. Thus we fix
s â€” 1 .
(C.38)
103
We can now employ arguments similar to those of ref. 37, to discuss the
critical values of Cnâ– For n = 0 the critical value is one, for which expression
(C.35) becomes illdefined. For Cq > 1, / > 1 and a > 0 in a sufficiently small
neighborhood of the origin and from eq. (C.29) / will remain greater than
one as r evolves as long as a stays positive. This implies that a will remain
positive, as for sufficiently small but still positive a the sign of a' in eq. (C.31)
will be positive and a can not get arbitrarily close to zero. So / will be a
monotonically increasing function of r, as dictated by eq. (C.29), that cannot
satisfy its boundary condition at oo. For n ^ 0, in a small enough neighborhood
of the origin / and a are positive. If Cn is chosen too large / will be one at
some r â€” rq for which a(ro) is still positive. From eq. (C.29), at r = tq, /
will be an increasing function. Then invoking the argument on the sign of a!
in eq. (C.31) one would infer that a cannot get arbitrarily close to zero and
so shall remain positive, forcing / to be unbounded. Hence, 0 < Cn < C
where the lower bound corresponds to the trivial solution. Examples of n = 0
and n = 2 solutions are shown in fig. C.l. Figure C.2 shows a plot of C^r as a
function of n. It has a shape very similar to that in the CSH case. Numerical
results indicate the following bounds on a,
1.75 < a < 2, n = 0
(C.39)
n + l 1 ,
where the lower bound corresponds to the Cn â€”>â– limit and the upper
bound to the Cn â€”> 0 limit. This is to be contrasted with the CSH case, where
only a lower bound of a > n + 2, corresponding to the Cn â€”Â»â€¢ 0 limit, is placed,
so that solitons of different vorticity can have overlapping energies. Here, for
each vorticity n, only a flux between (47r/e)(n +1/2) and (47r/e)(n +1) can be
accomodated.
104
n=0, C0 = .750
n=2, C2 = .015
Mr
Figure C.l Profile of representative soliton solutions
15
20
cr
105
Figure C.2 log10(C^r) as a function of n.
To obtain an expression for the angular momentum of our soliton, let 7q,
denote Tq1 collectively. The angular momentum, then, is given by
s
J = / drx f x Tq
(C.40)
From eq. (C.8) we get
J = 2/rf2;r e2(A0 + d0X)\(f>\2rx (Avx) +(d0\
After plugging in from eq. (C.18), (C.14) and (C.20) in the above, we end up
with the simple expression
J = M J d2x v â€¢ (r\(t>\2) + 2M
= 2M J d2x 02
(C.42)
106
once the surface term has been disposed of, which in the rotationally symmetric
case amounts to demanding rf â€”Â» 0 in the r â€”Â» oo limit. The last expression
has the same form as that in the nonrelativistic CSH model of ref. 38. For the
rotationally symmetric case, with p = Mr,
J = 4rppdp (C.43)
eÂ¿ Jo
where the last integral varies from zero to n + .8 (obtained numerically), in
each band.
C.3 A Note
Finally, as a sidenote we would like to mention another instance where the
C' term in expression (C.l) can be utilized to obtain an interesting selfdual
solution. To start, consider the following expression containing C!
\D^\2 + m~le^P(t>*D^DvDp(t> , (C.44)
which when augmented by a coupling of the Maxwell term to the scalar field
of the form
1012Ftu,F>"' ,
would yield
A = (Dp Â± m~lepUpDuDp)(t>\2 . (C.45)
A nontrivial vacuum for this Lagrangian, which is also nonrenormalizable,
would be = constant and Ap a pure gauge. The action corresponding to
C\ is minimized by the selfdual equation
{Dp Â± m lepUpDvDp)(f) = 0 ,
(C.46)
107
which has a solution of the form
(Â¡)(x) = (j)(xo) exp ^ â€” ie /, (C.47)
where P denotes a path from xq to x. As written, the phase in expression
(C.47) may in general be path dependent, not corresponding to the particuar
solutions saught here. The ones of interest for us correspond to the integrand
in eq. (C.47) being a â€œpure gaugeâ€, i.e.
(Am + d^x) Â± m~le^pdvAp = 0 , (C.48)
for
ie\ = In . (C.49)
For the â€œgaugeinvariant vector potentialâ€ Ap + dpXi eQ (C.48) is analogous
to that of the selfdual Lagrangian of ref. 39 was shown in ref. 40 be equivalent
to the topologically massive gauge theory of ref. 41. To check this, one need
only contract eq. (C.48) with the LeviCivita symbol and take the divergence
thereof, to obtain
(pvpdvAp = Â±rrCxdvFvp , (C.50)
which is the equation of motion of the model of ref. 41. Thus topologically
massive gauge theory corresponds to the selfdual solution of C\ in eq. (C.45).
C.4 Comments
The nonrelativistic limit of eq. (C.2) subject to the proper constraints is
the socalled JackiwPi model, which is also the nonrelativistic limit of the
ChernSimonsHiggs model, is extendable to possess Galilean supersymmetry
and has its own selfdual solutions. It has been checked that the model of
108
eq. (C.2) is the bosonic part of a N = 1 supersymmetric theory, but evidently
not N = 2. Although selfdual models typically have the latter supersymmetry
behind them, the absence here is probably due to the nonrenormalizability of
the model.
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I certify that I have read this study and that in my opinion it conforms to
acceptable standards of scholarly presentation and is fuily adequate, in scope
and quality, as a dissertation for the degree of Doctor of Philosophy.
L
4
/> o
UP 4
Charles B. Thorn, Chair
Professor of Physics
I certify that I have read this study and that in my opinion it conforms to
acceptable standards of scholarly presentation and is fully adequate, in scope
and quality, as a dissertation for the degree of Doctor of Philosophy.
f f / 
Pierre'Kamond
Professor of Physics
I certify chat I have read this study and that in my opinion it conforms to
acceptable standards of scholarly presentation and is fully adequate, in scope
and quality, as a dissertation for the degree of Doctor of Philosophy.
Richard D. Field
Professor of Physics
I certify that I have read this study and that in my opinion it conforms to
acceptable standards of scholarly presentation and is fully adequate, in scope
and quality, as a dissertation for the degree of Doctor of Philosophy.
c'L
Pierre Sikivie
Professor of Physics
I certify that I have read this study and that in my opinion it conforms to
acceptable standards of scholarly presentation and is fully adequate, in scope
and quality, as a dissertation for the .degree of Doctor of PjailcsGp>hy.
J/* {/(Â¡IjLJUrt
fâ€”vtÂ» â€”
y juico j_/. Aeesimg f
pes it. Kees
ofessor of Mathematics /
This dissertation was submitted to the Graduate Faculty of the De partment
of Physics in the College of Liberal Arts and Sciences and to the Graduate
School and was accepted as partial fulfillment of the requirements for the degree
of Doctor of Philosophy.
May 1996
Dean, Graduate school
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