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For an Abelian gauge theory with $A_{\mu }\mapsto A_{\mu }+{\partial }_{\mu }\alpha$, a nonlinear gauge functional $F[A]$ has Faddeev-Popov determinant $\det (\delta F[A+\partial \alpha ]/\delta \alpha )$. This operator can depend on $A$ even though the Adjoint representation of $U(1)$ is trivial. Consequently Faddeev-Popov ghost fields need not decouple in a nonlinear Abelian gauge, unlike a field-independent linear gauge condition.

Use the Minkowski metric, path-integral weight $e^{iS}$ and the Abelian gauge theory transformation $A_{\mu }\mapsto A_{\mu }+{\partial }_{\mu }\omega$. The gauge functional $F[A]=\partial \cdot A+A^2$ varies asThus the Faddeev-Popov operator is ${\mathcal{M}}_A=\square +2A\cdot \partial$. It depends on the gauge field despite the gauge group being Abelian: the ghosts interact because this gauge condition is nonlinear.

Choose the gauge-fixing fermion $\Psi =\int d^{4}x\bar{c}(F[A]+\xi h/2)$. The gauge-fixed action $S=S_{\mathrm{Maxwell}}+s\Psi$ isHere the tensor $F_{\mu \nu }={\partial }_{\mu }{A}_{\nu }-{\partial }_{\nu }{A}_{\mu }$ is distinct from the scalar gauge functional $F[A]$. With $\xi =0$, integrating over $h$ imposes the exact printed constraint. For nonzero $\xi$, eliminating $h=-F[A]/\xi$ instead givesThis version displays the additional gauge-dependent cubic and quartic gauge-field vertices as well as the ghost interaction; the strict condition is its $\xi \to 0$ limit.

For the Fourier transform convention $c(x)=\int d^{4}p(2\pi {)}^{-4}{e}^{-ipx}c(p)$, the quadratic ghost kernel is $p^2$. With the ordering $⟨c(p)\bar{c}(q)⟩$,The term $-2\bar{c}{A}^{\mu }{\partial }_{\mu }c$ has Fourier coefficient $2i{p}_{\mu }$, where $p$ is the incoming ghost momentum. Multiplication by $i$ in the Feynman rule givesThese signs refer to the displayed action, Fourier convention and ghost ordering. Reversing the ghost/antighost convention changes corresponding signs consistently. A closed ghost loop has the additional minus sign from Grassmann variables.

Let the charge parameter depend on position temporarily. Using the gauge-covariant derivative of a charged scalar field, the variation of the matter action is $\delta S=\int d^{4}x{j}^{\mu }{\partial }_{\mu }\alpha$. The resulting Noether current isHere ${\nabla }_{\mu }\bar{\varphi }={\partial }_{\mu }\bar{\varphi }+i{A}_{\mu }\bar{\varphi }$. The potential does not contribute, and $j^{\mu }=-\delta S_{\mathrm{matter}}/\delta A_{\mu }$. Because $A$ has not been rescaled to a canonical kinetic term, there is no factor $e$ in this Noether current or in the matter interaction vertices.

A local change of variables in the path integral gives $\delta ⟨\varphi (y)\bar{\varphi }(z)⟩=⟨\delta (\varphi (y)\bar{\varphi }(z))-(\delta S)\varphi (y)\bar{\varphi }(z)⟩=0$. Integration by parts in the term containing $\partial \alpha$ produces the two opposite contact terms of the Ward identity.

For clarity about phases, transform the current correlator with $e^{i(p-p^{\prime })\cdot x-ip\cdot y+i{p}^{\prime }\cdot z}$ and suppress its overall momentum conservation Dirac delta function. Denote the result by $C^{\mu }(p,p^{\prime })$. The coordinate Ward identity becomesAmputation of the two external quantum field theory propagators then fixes the longitudinal part of the current interaction vertex. Since the action couples $A_{\mu }$ to $-j^{\mu }$, the gauge interaction vertex has the opposite sign. The phase convention for the paper's gauge interaction vertices is ${\Gamma }^{\mu }=i{G}^{\mu }$, where $G^{\mu }$ is the corresponding derivative of the Euclidean quantum effective action.

One can derive the exact one-particle-irreducible correlation function directly, without assuming the entire connected current correlator is one-particle irreducible. Local gauge invariance of the quantum effective action impliesA linear covariant gauge fixing adds a breaking term independent of the scalars, which disappears after the scalar differentiations below. The identity requires a regularization in quantum field theory and counterterms preserving the Abelian gauge theory Ward identity; scalar quantum electrodynamics has no gauge anomaly.

Differentiate with respect to $\varphi (y)$ and $\bar{\varphi }(z)$, then set all background fields to zero. The two-point derivative is the inverse exact quantum field theory propagator. Use the Fourier transform convention $\varphi (x)={\int }_p{e}^{ip\cdot x}\varphi (p)$, $\bar{\varphi }(x)={\int }_{p^{\prime }}{e}^{-i{p}^{\prime }\cdot x}\bar{\varphi }(p^{\prime })$, with the photon momentum $p^{\prime }-p$ incoming. This givesAt tree level $G^{\mu }=-(p+p^{\prime }{)}^{\mu }$, so ${\Gamma }^{\mu }=-i(p+p^{\prime }{)}^{\mu }$, and the identity reduces to $-i(p^2-p^{\prime 2})=i(p^{\prime 2}-p^2)$.

To obtain the two-photon scalar Ward identity, differentiate the same functional Ward identity also with respect to $A_{\nu }(w)$. In terms of Euclidean action derivatives it readsThis construction automatically includes the seagull vertex: the Noether current depends on $A$, with $\delta j^{\mu }(x)/\delta A_{\nu }(w)=-2{\delta }^{\mu \nu }{\delta }^4(x-w)|\varphi (x){|}^2$. It cannot be discarded when differentiating a current insertion.

Take the first photon momentum to be incoming $k$, and the second incoming $p^{\prime }-p-k$. On the first scalar leg the contact term shifts $p$ to $p+k$; on the other it shifts $p^{\prime }$ to $p^{\prime }-k$. ConsequentlyWith ${\Gamma }^{\mu \nu }=i^2{G}^{\mu \nu }=-G^{\mu \nu }$, the result isThere is a momentum-routing sign error in the printed second identity. With the scalar momentum convention of the first identity, its first shifted argument must be $p^{\prime }-k$, not $k-p^{\prime }$. This is already forced at tree level: the seagull vertex has $G^{\mu \nu }=2{\delta }^{\mu \nu }$ and hence ${\Gamma }^{\mu \nu }=-2{\delta }^{\mu \nu }$. The corrected right-hand side is $-2k^{\nu }$, matching the left-hand side, whereas the printed right-hand side is $-2p^{\prime \nu }$. Changing to an all-incoming scalar convention would also change the first identity, so it does not fix both printed formulas simultaneously.

Let $G$ be a simple Lie group, let $T^a$ be Hermitian matrices for a finite-dimensional unitary representation $R$, and normalizeA matter field $\psi$ transforms locally as ${\psi }^{\prime }(x)=U(x)\psi (x)$, where $U(x)=e^{ig{ϵ}^a(x)T^a}$. An ordinary derivative of $\psi$ does not transform covariantly because it differentiates $U$. Introduce a gauge field $A_{\mu }=A_{\mu }^a{T}^a$ and the gauge covariant derivativeDemanding $D_{\mu }^{\prime }{\psi }^{\prime }=U{D}_{\mu }\psi$ determines the Yang-Mills gauge transformationTo first order in $ϵ$,

The gauge field strength is defined by $[D_{\mu },D_{\nu }]=-ig{F}_{\mu \nu }$:or, in components,Covariance of the commutator givesThe commutator term distinguishes Yang-Mills theory from an Abelian gauge theory and produces cubic and quartic gauge-boson interactions.

For a Dirac field of mass $m$ in $R$, the Lagrangian isEquivalently, the gauge term is proportional to $-\mathrm{tr}(F_{\mu \nu }{F}^{\mu \nu })$. The cyclic property of the trace and $F_{\mu \nu }^{\prime }=U{F}_{\mu \nu }{U}^{-1}$ make it invariant. Unitarity gives ${\bar{\psi }}^{\prime }=\bar{\psi }{U}^{-1}$, while $D_{\mu }^{\prime }{\psi }^{\prime }=U{D}_{\mu }\psi$, so both the matter kinetic term and mass term are invariant. A complex scalar $\varphi$ in a unitary representation may instead be coupled throughprovided the scalar potential $V$ is $G$-invariant.

The simplicity assumption means that the Lie algebra $\mathfrak{g}$ is nonabelian and has no proper nonzero ideal of a Lie algebra. Its Adjoint representation is therefore irreducible, and every invariant symmetric bilinear form is proportional to the Killing form. Consequently the pure gauge kinetic term has one overall gauge coupling for a simple factor. The theory has no independent Abelian gauge direction; if the gauge algebra were a direct sum of simple and Abelian ideals, each factor could instead carry its own coupling. A simple group may still have a discrete center, but this does not add a gauge boson because gauge bosons are indexed by the Lie algebra.

For $\delta A_{\mu }={\partial }_{\mu }\omega$, the real quadratic gauge functional varies by $(\square +2A\cdot \partial )\omega$. Its Faddeev-Popov operator depends on the gauge field, producing a ghost-gauge vertex even in an Abelian gauge theory. With $\Psi =\int \bar{c}(F[A]+\xi h/2)$, a consistent left BRST symmetry convention gives $s\Psi =hF+\xi h^2/2-\bar{c}(\square +2A\cdot \partial )c$. The auxiliary-field integral at $\xi =0$ imposes the exact condition; finite $\xi$ gives a weighted gauge condition. This real functional is distinct from a complex quadratic Abelian gauge condition.