Spontaneous Breakdown of Conformal and Chiral Invariance
Abstract
We discuss the implications of a theory in which scale and chiral invariance are spontaneously broken, and the dilaton appears as a mixture of the two isoscalar members of the scalar nonet. The usual assumptions for the conformal properties of the axialvector current constrain the lowenergy behavior of the spin2 form factor, F_{1}(t), of the pionic matrix element of the stressenergy tensor. In the limit of scale invariance, we find F_{1}'(0)=F_{σπ}'(0)F_{σπ}(0), where F_{σπ}(t) is the axialvector form factor obtained from the coupling of the dilaton to a pion via the axialvector current, and the prime denotes differentiation. This relation connects the assumptions of f dominance of F_{1}(t) and A_{1} dominance of F_{σπ}(t). Using the method of collinear dispersion relations, we estimate the effects of violation of scale invariance. A result previously obtained in the limit of scale invariance becomes F_{σ}F_{σπ}(m_{σ}^{2})f_{π}~=12, where f_{π} is the decay constant of the pion, and F_{σ} couples the dilation to the vacuum via the stressenergy tensor. Similar corrections to the scaleinvariant prediction for F_{1}'(0) are calculated. The magnitudes of the corrections are controlled by the A_{1}σπ coupling constant. According to the usual estimates of this constant, the predicted width of the dilaton is compatible with the AdlerWeisberger sum rule for ππ scattering and phenomenological estimates of the σNN coupling constant. While the relation for F_{1}'(0) obtained in the limit of scale invariance is compatible with the assumption of f dominance, the effects of symmetry breaking are large. In the real world, we find that f dominance is a poor approximation, a conclusion which is supported by recent estimates of the fNN coupling constants. We discuss the relation of our work to the magnitude of parameters measuring symmetry violation in the energy density. Our interpretation of a recent result of Cheng and Dashen is that scale invariance is spontaneously broken, and chiral SU(2)×SU(2) is a much better symmetry than SU(3).
 Publication:

Physical Review D
 Pub Date:
 June 1971
 DOI:
 10.1103/PhysRevD.3.3152
 Bibcode:
 1971PhRvD...3.3152C