## Path Integral Methods in Quantum Field Theory

- R. J. Rivers
- 1987

1 Excerpt

- Published 2008

We obtain a series expansion for the one loop fermion contribution to the effective potential evaluated for constant gauge potentials in the SU(2) theory with a massive fermionic doublet. The series converges for electric fields bounded in terms of the magnetic fields and the gauge potentials. It turns out that fermion pair creation may be absent (to order h̄) for arbitrary strong pure electric fields, with a proper choice of the gauge potentials. Almost fifty years ago, J. Schwinger obtained [1] in the context of Q.E.D. the one loop fermion contribution to the electromagnetic effective Lagrangian for the special case of constant fields. It seems there are no similar results concerning Yang-Mills fields. Our intention here is to consider the analogous problem for the SU(2) theory. We shall assume constant values throughout Minkowski space for the gauge potentials; non-abelianity obviously assures this does not necessarily implies the vanishing of the field strengths. We shall restrict to the case of a fermionic spin j = 1/2 multiplet. Calculations prove to significantly simplify for this representation due to the properties of the Pauli matrices. See the next footnote. The general case will be analyzed in a subsequent paper. 1 We begin by writing the basic Lagrangian as L = − 4 F a μνF aμν + iψγDμψ −mψ̄ψ, (1) with F a μν = ∂μA a ν − ∂νAμ + εabcAμAν , (2) Dμ = ∂μ − iAμ σa 2 , (3) where a, b, c = 1, 2, 3 denote the group indices (summation over repeated indices is understood). Aμ are the gauge potentials, including the coupling constant of the theory. ψ represents the fermion doublet, with m > 0 the corresponding mass. εabc is the Levi-Civitta antisymmetric tensor and σa are the Pauli matrices. The one loop fermionic contribution to the effective action Seff for A a μ follows most directly in the path integral formalism by integrating out [2] the ψ field Seff = −i lnDet ((γ(Pμ − Aμ σa 2 )−m), (4) where Det is taken in the functional sense and Pμ are the translation operators in the coordinate space. m tacitly incorporates the iǫ prescription. Using the charge conjugation matrix CγμC −1 = −γ μ and [Pμ, A a ν ] = 0, (5) one can eliminate [3] the γ matrices to obtain Seff = − i 2 lnDet ((Pμ −Aμ σa 2 ) −m). (6) We use next the ln Det=Tr Ln formula combined with the Schwinger representation for the logarithm. When tracing over space-time coordinates, translational invariance allows separation of the four-volume factor ∫ dx; the trace in the spinorial space yields simply a 4 factor; tracing over the group indices amounts to evaluate, after expanding the square in eq. (6), the

@inproceedings{Nicolaevici2008OneLF,
title={One loop fermion contribution to the effective potential for constant SU(2) Yang-Mills fields},
author={Nistor Nicolaevici},
year={2008}
}