QFT II Solution 7

QFT II FS 2019 Solution 7. Prof. M. Grazzini

Exercise 1. Remaining Feynman rules for QCD

Starting from the full QCD-Lagrangian with gauge-ﬁxing and Fadeev-Popov term,

1 1 2 L = − F a F a,µν + ψ¯iD/ − mψ − ∂µAa − c¯a∂µDabcb , (1) QCD 4 µν 2ξ µ µ

ﬁnd the terms of Lint contributing to the gluon-quark and the gluon-ghost vertices and derive the corresponding Feynman rules using path-integral methods.

Solution. The first term, the pure Yang-Mills Lagrangian, only contributes to the gluon self- interaction we calculated on sheet 6. The gauge fixing term is quadratic in the fields and will therefore not contribute to any interaction. We are left with the second and fourth term, and the corresponding contributions to Lint will become obvious once we write out the covariant derivatives. For the quarks (fundamental representation) we have

a a Dµψi = (∂µδij − igtijAµ)ψj , (S.1) and for the ghosts (adjoint representation)

ab b ab abc c b Dµ c = (∂µδ − gf Aµ)c . (S.2)

With this we ﬁnd

gqq¯ ¯ a a gcc¯ abc µ a c b Lint = gψitijA/ ψj , Lint = −gf (∂ c¯ )Aµc , (S.3) where we shifted the derivative to act on the anti-ghost by means of partial integration. To derive the vertices we expand the exponential in

a a a Z[Jµ, a¯i, ai, η¯ , η ] Z 4 1 δ 1 δ 1 δ 1 δ 1 δ a a a = N exp i d zLint a , , − , a , − a Z0[Jµ, a¯i, ai, η¯ , η ] (S.4) i δJµ i δa¯i i δai i δη¯ i δη

a a a to ﬁrst order, as usual.a ¯i, ai are the sources for the quarks,η ¯ , η for the ghosts and Jµ for the gluon-ﬁeld.

Exercise 2. Gauge ﬁxing in the path integral

(a) Consider the action for pure electrodynamics

Z 1 S = d4x − F F µν . ED 4 µν

Perform the gauge fixing via the Faddeev–Popov method, using the non-linear gauge condition 1 G[A, Ω] = ∂ Aµ + ζA Aµ − Ω. µ 2 µ 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 parameter ξ is restored. (b) Consider the action for pure Yang–Mills theory

Z 1 S = d4x − F a F a µν . YM 4 µν

Perform the gauge fixing via the Faddeev–Popov method using the axial gauge condition along a fixed four-vector nµ a µ a a G[A, Ω] = n Aµ − Ω . 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?

Solution. Recycled from FS17, S6E1. Original author: Simone Lionetti

(a) The gauge-ﬁxing term is 1 G[A] = ∂ Aµ + ζA Aµ − Ω, µ 2 µ

whose gauge variation is, with δAµ = ∂µα,

µ µ µ δG[A] = ∂µ∂ α + ζAµ∂ α = (∂µ + ζAµ)∂ α.

We can express the functional determinant we get as

δG[A] Z Z det = DcDc¯exp −i dx c¯∂2 + ζA ∂µc δα µ

in terms of the ghost/anti-ghost ﬁelds c/c¯. The kinetic term of the action is left untouched, therefore the photon propagator is again 1 GA (p) = g − (1 − ξ)p p /p2, µν p2 − i µν µ ν whereas the ghost propagator is trivially 1 Ggh(p) = . p2 − i

5 The Lagrangian that is obtained from the given gauge condition reads

0 LED = LED + Lgf + Lgh, 1 L0 = − F F µν, ED 4 µν 1 1 L = − [∂ Aµ + ζA Aµ]2, gf 2ξ µ 2 µ 2 µ Lgh =c ¯(∂ + ζAµ∂ )c. (S.5)

Note that this choice introduces gauge-dependent cubic and quartic photon couplings, which are an artefact of gauge fixing and will be exactly cancelled by ghost contributions. For every ζ 6= 0 ghosts do not decouple from the gauge field, and are in fact needed for the described cancellation to take place; in the limit ζ → 0 their interaction with Aµ is negligible, only the ghost kinetic term survives and the c,c ¯ fields may be safely integrated out. We then recover the modified Lorenz gauge with gauge-fixing parameter ξ (also known as Rξ-gauge). (b) The gauge-fixing functional and Lagrangian are 1 G[A]a = nµAa , L = − (nµAa )2. µ gf 2ξ µ a The gauge transformation of the gluon field Aµ is

a a a b c δAµ = ∂µα + gfbc Aµα , therefore we can write the functional determinant as δG[A] Z Z det = DcDc¯exp −i dx c¯anµ(δ ∂ + gf Ac )cb . δα ab µ abc µ The kinetic term for the ghost ﬁeld leads to the propagator δ Ggh(p) = ab . ab n · p Finding the appropriate i prescription for denominators that are linear in p is a nontrivial problem (see e.g. Pokorski p. 132–133) and depends on whether n is space-like, light-like or time-like: the issue is therefore ignored in these solutions. Switching to momentum space, the kinetic term for the gluon ﬁeld is

Z d4p 1 − Aa (−p)KµνAb (p), (2π)4 2 µ ab ν where the kinetic operator reads

µν 2 −1 Kab = δab gµνp − pµpν + ξ nµnν . We now look for a Green function of the kinetic operator, which in momentum space is deﬁned by µν bc c µ Kab Gνρ = δaδρ . ab ab The colour structure is trivial, Gµν = δ Gµν. For the Lorentz part we can write down the most general tensor structure as

Gρν = Gggρν + Gpppρpν + Gnnnρnν + Gnpnρpν + Gpnpρnν,

6 2 2 where Gg, Gpp, Gnn, Gnp and Gpn are functions of the Lorentz scalars p , n and n · p that we want to determine. The left hand side of equation ((b)) then becomes

ρν 2 −1 ρν ρ ν ρ ν ρ ν ρ ν KµρG = (gµρp − pµpρ + ξ nµnρ)(Ggg + Gppp p + Gnnn n + Gnpn p + Gpnp n ) (S.6) 2 ν ν ν ν ν = p Ggδµ + Gpppµp + Gnnnµn + Gnpnµp + Gpnpµn ν 2 ν ν ν 2 ν − Ggpµp + Gppp pµp + Gnn(n · p)pµn + Gnp(n · p)pµp + Gpnp pµn −1 ν ν 2 ν 2 ν ν + ξ Ggnµn + Gpp(n · p)nµp + Gnnn nµn + Gnpn nµp + Gpn(n · p)nµn (S.7) 2 ν ν = {Ggp }δµ − {Gg + Gnp(n · p)}pµp 2 −1 2 ν + {Gnnp + ξ [Gg + Gnnn + Gpn(n · p)]}nµn 2 −1 2 ν ν + {p Gnp + ξ [Gpp(n · p) + Gnpn ]}nµp − {Gnn(n · p)}pµn . (S.8)

ρ Requiring that this expression be equal to δµ for any vectors p and n, i.e. setting the coefficients of the first tensor structure in curly brackets to 1 and all others to zero, yields 5 conditions for 5 unknowns. We thus find 1 ξp2 + n2 1 G = ,G = − ,G = G = − ,G = 0, g p2 pp (n · p)2p2 np pn (n · p)p2 nn which gives the propagator in axial gauge

δab p p p n + p n Gab = g − (ξp2 + n2) µ ν − µ ν ν µ . µν p2 µν (n · p)2 (n · p)

Note that, because the kinetic operator is symmetric under the exchange of its Lorentz indices, its inverse retains this property as conﬁrmed by Gnp = Gpn. We could have restricted the search of its inverse to this type of functions from the start: in this case we would have found 5 equations for 4 unknowns, one of which would have been just a consistency check. We observe that in the limit ξ → 0 the gauge is ﬁxed by the condition

µ a n Aµ = 0,

µν and the propagator inherits this property as nµGab = 0. In this limit, often one chooses a light-like reference vector, n2 = 0, so that the pµpν term in the propagator drops completely and many calculations become easier. Concerning the decoupling of ghosts, the only term µ a b c µ a of interaction for the ghosts is n fabcc¯ Aµc which is zero when n Aµ = 0 i.e. in the limit that was just considered. The presence of (n · p) in denominators, however, makes calculations very tough to manage beyond tree level, to the point that it would seem impossible to exploit the axial gauge. Luckily it is possible to consistently pick a different gauge for internal and external particles. One may thus choose axial gauge for external legs, which does not produce any nasty propagator in amplitude calculations, and use a different gauge (e.g. Feynman) for all internal lines. While such a procedure does not get rid of ghosts in loops, it allows at least to forget about diagrams with initial or final state ghosts.