Gauge fixing in the path integral (a) Consider the action for pure electrodynamics SED = Z d4x −1 4FµνFµν

QFT II Problem Set 7.

FS 2019 Prof. M. Grazzini https://www.physik.uzh.ch/en/teaching/PHY552/FS2019.html Due by: 8/4/2019

Exercise 1. Remaining Feynman rules for QCD

Starting from the full QCD-Lagrangian with gauge-fixing and Fadeev-Popov term, L_QCD=−1

4F_µν^a F^a,µν + ¯ψ i /D−m ψ− 1

2ξ ∂^µA^a_µ2

−¯c^a

∂^µD^ab_µ

c^b, (1) find the terms of L_int contributing to the gluon-quark and the gluon-ghost vertices and derive the corresponding Feynman rules using path-integral methods.

Exercise 2. Gauge fixing in the path integral

(a) Consider the action for pure electrodynamics S_ED =

Z d⁴x

−1

4F_µνF^µν

.

Perform the gauge fixing via the Faddeev–Popov method, using the non-linear gauge con- dition

G[A,Ω] =∂µA^µ+1

2ζAµA^µ−Ω.

Invert the kinetic operators that appear in the action to find the propagators for the photon field and for the ghost field.

Is the ghost field decoupled in this gauge? Show that in the limit ζ → 0 the modified Lorenz gauge with gauge-fixing parameterξ is restored.

(b) Consider the action for pure Yang–Mills theory SYM =

Z d⁴x

−1

4F_µν^a F^{a µν}

.

Perform the gauge fixing via the Faddeev–Popov method using the axial gauge condition along a fixed four-vectorn^µ

G[A,Ω]^a=n^µA^a_µ−Ω^a.

Invert the kinetic operators that appear in the action to find the propagators for the ghost field and for the gluon field.

Under which condition do ghosts decouple from gluon fields in this gauge? How can this be exploited in practice?

1