Pauli-villars Regularization of Supergravity Coupled to Chiral and Yang-mills Matter

It is shown that the one-loop quadratic divergences of standard supergravity can be regulated by the introduction of heavy PauliVillars fields belonging to chiral and abelian gauge multiplets. The resulting one-loop correction can be interpreted as a renormalization of the Kähler potential. Regularization of the dilaton couplings to the Yang-Mills sector requires special care, and may shed some light on chiral/linear multiplet duality of the dilaton supermultiplet. ∗This work was supported in part by the Director, Office of Energy Research, Office of High Energy and Nuclear Physics, Division of High Energy Physics of the U.S. Department of Energy under Contract DE-AC03-76SF00098 and in part by the National Science Foundation under grant PHY–90–21139. In extracting the phenomenological implications of an underlying supergravity theory, the quadratic divergences arising at one loop have often been considered [1]. However, the coefficients of the quadratically divergent terms are unreliable in the absence of a manifestly supersymmetric regularization procedure [2], [3]. The purpose of this Letter is to describe such a procedure. The one-loop effective action S1 is obtained from the term quadratic in quantum fields when the Lagrangian is expanded about an arbitrary background: Lquad(Φ,Θ, c) = − 1 2 ΦZ ( D̂ Φ +HΦ ) Φ + 1 2 Θ̄Z (i 6 DΘ −MΘ)Θ + 1 2 c̄Z ( D̂ c +Hc ) c+O(ψ), (1) where the column vectors Φ,Θ, c represent quantum bosons, fermions and ghost fields, respectively, and ψ represents background fermions that we shall set to zero throughout this paper. The fermion sector Θ includes a C-odd Majorana auxiliary field α that is introduced to implement the gravitino gauge fixing condition. The full gauge fixing procedure used here is described in detail in [4], [5]. Then the one loop bosonic action is given by S1 = i 2 Tr ln ( D̂ Φ +HΦ ) − i 2 Tr ln (−i 6 DΘ +MΘ) + i 2 STr ln ( D̂ c +Hc ) = i 2 STr ln ( D̂ +H ) + T−, (2) where T− is the helicity-odd fermion contribution which contains no quadratic divergences, and the helicity-even contribution is determined by D̂ Θ +HΘ ≡ (−i 6 DΘ +MΘ) (i 6 DΘ +MΘ) . (3) The field-dependent matricesH(φ) and D̂μ(φ) = ∂μ+Γμ(φ) are given in [4], [5], where the logarithmically divergent contributions have been evaluated. Explicitly evaluating (2) with an ultraviolet cut-off Λ and a massive Pauli-Villars sector with a squared mass matrix of the form M PV = H PV (φ) + ( μ ν ν μ ) ≡ H + μ + ν, |ν| ∼ μ ≫ H ∼ H, 1 gives, with H ′ = H +H : 32πS1 = − ∫ dxdpSTr ln ( p + μ +H ′ + ν ) + 32π (S ′ 1 + T−) = 32π (S ′ 1 + T−)− ∫ dxdpSTr ln ( p + μ )