Goldstone Fields in the Superstring Theory

If global supersymmetry is broken by gaugino condensation in the hidden sector of the E₈ × E₈^' heterotic superstring theory after compactification, then the auxiliary field F_B of the modulus B ≡ (B_r, B_i) attains a finite value, while that of the dilaton A ≡ (A_r, A_i) vanishes, F_A = 0, the Goldstone fermion being the modulino ∼ {B}, the spin-½ component of the complex chiral supermultiplet {B} ≡ (B,∼ {B}). The Goldstone boson of the scale symmetry that is broken when the radius of the internal space is fixed at a constant value is B_r, which is determined from the goldstino Lagrangian, compared term by term with the superstring Lagrangian, including higher-derivative gravitational terms {R}² and {R}⁴, after linking the space-time curvature to the energy-momentum tensor of the goldstino via the Einstein equations. This non-linear formulation of supersymmetry, due to Volkov and Akulov, is expressed in terms of the goldstino alone, whose Lagrangian contains a negative cosmological constant, which can be cancelled by the super-Higgs mechanism of Deser and Zumino to make the gravitino massive and break supersymmetry at the level m_s = √ {√ {6}m_3/2/k }≈ 2.5(1.39/A_r)^1/4 (B_r /B_r ^H)^1/2× 10¹⁸ GeV, while m_3/2 ≈ (1.39/A_r) ^1/2(B_r /B_r ^H× 10¹⁸ GeV. Here, the modulus has been scaled to the Hagedorn value for the heterotic superstring theory, B_r ^H = 1/4π ² T² Hα ' = 2.914, and A_r, identified with the inverse square of the tree-level gauge coupling, has been scaled to the calculation in the minimal supersymmetric standard model due to Weinberg, that g^-2 = 1.39 at the unification mass M_X = 2.2 × 10¹⁶ GeV, assuming three generations of elementary particles and two Higgs doublets. In the presence of gravitino condensation in the internal space, however, there is an arbitrary additional contribution to the cosmological constant, facilitating reduction of m_s to 100 TeV, say, and m_3/2 to 1 eV.