Ward Identities and Chiral Symmetries

The previous pages used the renormalization group to show how a classically scale-free theory can generate a physical mass scale. We now pause before going further into current correlators and many-body examples, because the next layer of structure is controlled by symmetry. Symmetry does more than label states. In a local quantum field theory it becomes a system of identities among correlation functions.

Those identities are called Ward identities. They are the quantum version of current conservation, but with two important upgrades. First, local current conservation inside a time-ordered product includes contact terms when the current collides with another operator. Second, a symmetry of the classical action need not be a symmetry of the quantum measure and regulator. When this happens, the Ward identity has an anomalous term.

Massless fermions make these points vivid. A massless Dirac field has independent phase rotations of its left- and right-moving components. In two dimensions this becomes almost embarrassingly explicit: one chirality depends only on $x^+$ and the other only on $x^-$. The same simplicity also exposes the axial anomaly. The vector current can be conserved as an operator identity in a background gauge field, but the axial current cannot be conserved at the same time.

Required background. Continuous symmetries, generators, and charges supplies the current and the local-parameter argument; localized transformations and Ward–Takahashi identities supplies the distributional contact-term logic used below. Helpful background. Vacuum polarization and gauge-invariant counterterms develops the same transversality condition in perturbative QED.

From Noether currents to Ward identities

Light-cone and current conventions. Most formulas use the global QFT II conventions. In the two-dimensional parts of this page we use

$$x^\pm=x^0\pm x^1, \qquad \partial_\pm={1\over2}(\partial_0\pm\partial_1), \qquad A_\pm=A_0\pm A_1.$$

The metric is mostly minus, $\eta_{\mu\nu}=\operatorname{diag}(+,-)$, and $\epsilon^{01}=+1$. We take

$$\gamma^5=\gamma^0\gamma^1, \qquad P_\pm={1\over2}(1\pm\gamma^5), \qquad \psi_\pm=P_\pm\psi.$$

With these conventions one convenient identity is

$$\gamma^\mu\gamma^5=-\epsilon^{\mu\nu}\gamma_\nu, \qquad j_5^\mu=\bar\psi\gamma^\mu\gamma^5\psi =-\epsilon^{\mu\nu}j_\nu.$$

For Dirac slash notation we use the KaTeX-compatible form

$$p\!\!\!/\equiv\gamma^\mu p_\mu, \qquad q\!\!\!/\equiv\gamma^\mu q_\mu.$$

For a background gauge field we write

$$D_\mu=\partial_\mu-iA_\mu, \qquad \psi\mapsto e^{+i\alpha}\psi, \qquad A_\mu\mapsto A_\mu+\partial_\mu\alpha.$$

The source term generated by $\bar\psi i\gamma^\mu D_\mu\psi$ is therefore $+A_\mu j^\mu$. The displayed anomaly coefficient is for one massless Dirac fermion of positive unit charge.

Start with a field theory whose action is invariant under a continuous global transformation. For a Dirac field the simplest example is

$$\psi(x)\mapsto e^{+i\alpha}\psi(x), \qquad \bar\psi(x)\mapsto \bar\psi(x)e^{-i\alpha}.$$

For the free massive Dirac Lagrangian

$$\mathcal L=\bar\psi(i\gamma^\mu\partial_\mu-m)\psi,$$

Noether’s theorem gives the vector current

$$\boxed{ j^\mu=\bar\psi\gamma^\mu\psi }$$

and the classical equation

$$\partial_\mu j^\mu=0.$$

The quickest way to turn this into a Ward identity is to make the symmetry parameter local, $\alpha\to\alpha(x)$. The action is no longer invariant; instead,

$$\delta S=-\int d^dx\,\partial_\mu\alpha(x)j^\mu(x) =+\int d^dx\,\alpha(x)\partial_\mu j^\mu(x),$$

up to boundary terms. In a path integral, an infinitesimal change of integration variables gives

$$0=\int \mathcal D\Phi\,\delta\left(e^{iS}\mathcal O\right).$$

If the measure is invariant, this becomes a relation between the divergence of the current and the variation of the inserted operator $\mathcal O$. Let $Q_k$ be the charge of $\mathcal O_k$, so $\delta_\alpha\mathcal O_k=+i\alpha Q_k\mathcal O_k$. With the time-ordered Green-function normalization used here, the Minkowski-space identity is

$$\boxed{ \begin{aligned} &\partial_\mu\left\langle Tj^\mu(x)\prod_{\ell=1}^n\mathcal O_\ell(x_\ell) \right\rangle \\ &\qquad=-\sum_{k=1}^nQ_k\delta^{(d)}(x-x_k) \left\langle T\prod_{\ell=1}^n\mathcal O_\ell(x_\ell)\right\rangle. \end{aligned} }$$

The right-hand side is the contact-term part of the Ward identity. Away from all other insertions, the current is conserved. At coincident points, the current generates the symmetry transformation of the operator it hits. In canonical language this is the same statement as

$$[Q,\mathcal O(x)]=-Q_{\mathcal O}\mathcal O(x), \qquad \delta_\alpha\mathcal O(x)=-i\alpha[Q,\mathcal O(x)] =+i\alpha Q_{\mathcal O}\mathcal O(x), \qquad Q=\int d^{d-1}x\,j^0,$$

where the transformation is implemented by $U(\alpha)=e^{-i\alpha Q}$ and $Q_{\mathcal O}$ is the charge appearing in the phase $e^{+i\alpha Q_{\mathcal O}}$. The path-integral contact terms are the spacetime version of this equal-time commutator.

This is why Ward identities are identities of distributions, not merely identities of ordinary functions at separated points. Dropping the contact terms gives a statement that is too weak to determine counterterms and too weak to protect gauge invariance.

For the Dirac two-point function, the vector Ward identity reads

$$\partial_\mu^x\left\langle Tj^\mu(x)\psi(y)\bar\psi(z)\right\rangle =-\delta^{(d)}(x-y)\langle T\psi(y)\bar\psi(z)\rangle +\delta^{(d)}(x-z)\langle T\psi(y)\bar\psi(z)\rangle.$$

The signs simply say that $\psi$ and $\bar\psi$ carry opposite $U(1)$ charges. In momentum space, the same statement becomes the Ward–Takahashi identity

$$\boxed{ q_\mu\Gamma^\mu(p+q,p)=S^{-1}(p+q)-S^{-1}(p), }$$

where $S(p)$ is the full fermion propagator and $\Gamma^\mu$ is the full current vertex.

The Ward–Takahashi identity says that contracting a current insertion with its momentum is equivalent to subtracting the inverse propagator on the two sides of the insertion. This is the diagrammatic form of charge conservation inside correlation functions.

This identity is much stronger than the classical equation $\partial_\mu j^\mu=0$. It constrains loop corrections. In QED it implies, after renormalization, the equality of the vertex and fermion wavefunction renormalization constants in a gauge-invariant scheme. More generally, it is the algebraic reason why gauge invariance organizes counterterms.

Background sources and transverse polarization

A clean way to package Ward identities is to couple a source to the current. Write

$$Z[A]=e^{iW[A]} =\int \mathcal D\Phi\, \exp\left(iS_0+i\int d^dx\,A_\mu j^\mu\right).$$

The current expectation value in the background $A$ is

$$\langle j^\mu(x)\rangle_A=+{\delta W[A]\over\delta A_\mu(x)}.$$

If the source $A_\mu$ is interpreted as a background gauge field, then the local transformation

$$\psi(x)\mapsto e^{+i\alpha(x)}\psi(x), \qquad A_\mu(x)\mapsto A_\mu(x)+\partial_\mu\alpha(x)$$

leaves the source-coupled theory invariant. Therefore

$$W[A+\partial\alpha]=W[A].$$

Expanding to first order in $\alpha$ gives

$$0=\int d^dx\,\partial_\mu\alpha(x){\delta W\over\delta A_\mu(x)} =-\int d^dx\,\alpha(x)\partial_\mu{\delta W\over\delta A_\mu(x)}.$$

Since $\alpha(x)$ is arbitrary,

$$\boxed{ \partial_\mu{\delta W[A]\over\delta A_\mu(x)}=0. }$$

Differentiating this identity once more with respect to $A_\nu$ and then setting $A=0$ gives transversality of the current-current correlator,

$$\partial_\mu^x\Pi^{\mu\nu}(x-y)=0,$$

where

$$\Pi^{\mu\nu}(x-y) =i\langle Tj^\mu(x)j^\nu(y)\rangle_{\rm conn} +\text{possible local contact terms}.$$

In momentum space,

$$\boxed{ q_\mu\Pi^{\mu\nu}(q)=0. }$$

Lorentz invariance then fixes the nonlocal parity-even part of the two-point function to have the form

$$\boxed{ \Pi^{\mu\nu}(q) =\left(q^\mu q^\nu-q^2\eta^{\mu\nu}\right)\Pi(q^2) +\text{local transverse terms}. }$$

This formula is familiar from vacuum polarization, but here the lesson is more general: a conserved current can only generate a transverse response. In gauge theory, this is why a photon mass term $A_\mu A^\mu$ is forbidden by gauge invariance, while a correction to $F_{\mu\nu}F^{\mu\nu}$ is allowed.

There is one subtlety that will matter below. The correlator $\langle Tj^\mu j^\nu\rangle$ by itself is not always the whole $\Pi^{\mu\nu}$. Local contact terms can be added by local counterterms in $W[A]$. They do not affect separated-point current conservation, but they do affect the precise Ward identity at coincident points. Gauge invariance often fixes those local terms.

Chiral symmetry of a massless Dirac field

In even spacetime dimensions, the massless Dirac Lagrangian has an additional classical symmetry. Define $\gamma^5$ so that it anticommutes with every $\gamma^\mu$,

$$\{\gamma^5,\gamma^\mu\}=0.$$

For $m=0$,

$$\mathcal L_0=\bar\psi i\gamma^\mu\partial_\mu\psi$$

is invariant under the axial rotation

$$\psi\mapsto e^{i\beta\gamma^5}\psi, \qquad \bar\psi\mapsto \bar\psi e^{i\beta\gamma^5}.$$

The corresponding axial current is

$$\boxed{ j_5^\mu=\bar\psi\gamma^\mu\gamma^5\psi. }$$

Classically,

$$\partial_\mu j_5^\mu=0 \qquad (m=0).$$

If the mass is nonzero, the axial symmetry is explicitly broken. A direct variation gives

$$\boxed{ \partial_\mu j_5^\mu=2im\bar\psi\gamma^5\psi }$$

in Minkowski signature. The vector current remains conserved because the mass term preserves ordinary phase rotations.

The chiral projectors

$$P_\pm={1\over2}(1\pm\gamma^5)$$

split the fermion into two components,

$$\psi=\psi_++\psi_-.$$

The vector and axial rotations are equivalently independent phase rotations of these two components. With the vector convention above and the axial rotation $\psi\mapsto e^{i\beta\gamma^5}\psi$,

$$\psi_+\mapsto e^{+i\alpha+i\beta}\psi_+, \qquad \psi_-\mapsto e^{+i\alpha-i\beta}\psi_-.$$

The mass term mixes the two chiralities. Schematically,

$$m\bar\psi\psi =m\left(\bar\psi_+\psi_-+\bar\psi_-\psi_+\right),$$

so it is invariant under the vector phase but not under the axial phase. This is the simplest way to remember why a massless fermion has more symmetry than a massive fermion.

For $N$ massless Dirac fermions, the symmetry is enlarged. In four dimensions, ignoring anomalies and gauge interactions for the moment, one has independent flavor rotations of left- and right-handed fermions,

$$U(N)_L\times U(N)_R.$$

Mass terms reduce this to the diagonal vector subgroup unless the mass matrix has degeneracies. This chiral flavor symmetry is one of the organizing principles of low-energy QCD, where quark masses are small compared with the strong scale.

Two-dimensional chiral components

Two dimensions are special because a massless Dirac equation separates into two one-dimensional propagation equations. With the light-cone conventions in the note above, the free massless action can be written as

The labels $+$ and $-$ are convention labels, not universal names for “right” and “left.” What matters physically is the equation of motion: one component propagates along constant $x^-$ and the other along constant $x^+$. When comparing references, translate the light-cone convention before comparing signs in the axial current.

$$S_0=\int d^2x\, \left(i\psi_+^\dagger\partial_-\psi_+ +i\psi_-^\dagger\partial_+\psi_-\right).$$

The equations of motion are

$$\boxed{ \partial_-\psi_+=0, \qquad \partial_+\psi_-=0. }$$

Therefore

$$\psi_+=\psi_+(x^+), \qquad \psi_-=\psi_-(x^-)$$

on shell. The two chiral densities are

$$J_+=\psi_+^\dagger\psi_+, \qquad J_-=\psi_-^\dagger\psi_-.$$

They obey

$$\partial_-J_+=0, \qquad \partial_+J_-=0$$

classically. The vector and axial currents are the sum and difference of the chiral currents. In the convention $j_5^\mu=-\epsilon^{\mu\nu}j_\nu$, the axial current is not an independent object in two dimensions; it is the Hodge dual of the vector current.

The two chiral components are independent when $m=0$. The vector rotation acts with the same phase on both components, while the axial rotation acts with opposite phases. A Dirac mass couples the two chiralities and breaks the axial symmetry.

The free propagator of a chiral component has a characteristic distributional form. Up to a conventional normalization,

$$\langle T\psi_+(x)\psi_+^\dagger(0)\rangle \propto {1\over x^+-i0\operatorname{sgn}(x^-)}.$$

The pole prescription remembers time ordering. It also hides a contact term. The useful identity is

$${1\over x^+-i0\operatorname{sgn}(x^-)} =\mathcal P{1\over x^+}+i\pi\operatorname{sgn}(x^-)\delta(x^+),$$

so

$$\partial_-{1\over x^+-i0\operatorname{sgn}(x^-)} =2\pi i\,\delta(x^+)\delta(x^-).$$

This is the Lorentzian cousin of

$$\partial_{\bar z}{1\over z}=\pi\delta^{(2)}(z).$$

The lesson is small but important: a chiral propagator looks holomorphic away from coincident points, but its derivative is not zero as a distribution. Contact terms are precisely what make Ward identities true.

The Thirring interaction as an auxiliary gauge field

The massless Thirring model is the two-dimensional current-current theory

$$\mathcal L_{\rm Th} =\bar\psi i\gamma^\mu\partial_\mu\psi -{g\over2}j_\mu j^\mu, \qquad j^\mu=\bar\psi\gamma^\mu\psi.$$

In light-cone variables this is, up to harmless factors of two,

$$\mathcal L_{\rm Th} =i\psi_+^\dagger\partial_-\psi_+ +i\psi_-^\dagger\partial_+\psi_- -gJ_+J_-.$$

The interaction couples the two chiral densities, but it does not mix the fields $\psi_+$ and $\psi_-$ the way a mass term would. It preserves the vector and axial $U(1)$ symmetries. Power counting makes $g$ dimensionless, so this is a marginal current-current deformation rather than a relevant mass perturbation. Symmetry and power counting alone do not prove gaplessness; the exact solution, equivalently bosonization to a free compact scalar with a coupling-dependent radius, shows that the massless one-flavor Thirring model lies on a gapless line of fixed points. Bosonization and sine-Gordon/Thirring duality develops that equivalence later in the course.

This conclusion is specific to the Abelian one-flavor interaction written above. With several flavors there are inequivalent ways to contract flavor and chiral indices. Non-Abelian current-current or Gross–Neveu interactions can have nonzero beta functions and dynamically generate a scale. The number of flavors by itself is therefore not enough information: the operator and its symmetry representation must also be specified.

A useful rewriting introduces an auxiliary vector field $A_\mu$ with no kinetic term:

$$\mathcal L =\bar\psi i\gamma^\mu\partial_\mu\psi +A_\mu j^\mu+{1\over2g}A_\mu A^\mu.$$

The equation of motion for $A_\mu$ is algebraic,

$$A^\mu=-g j^\mu.$$

Substituting back gives

$$+A_\mu j^\mu+{1\over2g}A_\mu A^\mu =-{g\over2}j_\mu j^\mu.$$

Thus the current-current interaction is equivalent to the exchange of an auxiliary vector field.

The Thirring interaction may be decoupled by a Hubbard–Stratonovich field $A_\mu$. Because $A_\mu$ has no bare kinetic term, integrating it out is exact and returns the four-fermion current-current interaction; the overall sign is a convention for $g$ and for the auxiliary-field quadratic term.

This rewriting is not just cosmetic. Once fermions are integrated out, the effective action for $A_\mu$ is a determinant,

$$e^{iW[A]}=\int \mathcal D\psi\mathcal D\bar\psi\, \exp\left(i\int d^2x\,\bar\psi i\gamma^\mu(\partial_\mu-iA_\mu)\psi\right),$$

or formally

$$W[A]=-i\operatorname{Tr}\log\left(i\gamma^\mu(\partial_\mu-iA_\mu)\right).$$

Expanding $W[A]$ in powers of $A$ generates current correlators. Gauge invariance of $W[A]$ is exactly the vector Ward identity. The next page computes this quadratic term carefully; here we only need the conceptual point: local counterterms in $W[A]$ decide which Ward identities are preserved.

This last sentence is the practical bridge to anomalies. The separated-point determinant may look symmetric under several classical transformations, but the local terms required to define it can privilege one symmetry over another. For a dynamical gauge field, preserving the gauge symmetry is not optional.

Vector Ward identity versus axial Ward identity

For a massless two-dimensional Dirac fermion, the vector and axial currents are related by

$$j_5^\mu=-\epsilon^{\mu\rho}j_\rho.$$

Define the vector two-point function

$$\Pi^{\mu\nu}(q)=i\int d^2x\,e^{iq\cdot x} \langle Tj^\mu(x)j^\nu(0)\rangle_{\rm conn} +\text{contact terms}.$$

If vector gauge invariance is imposed, then

$$q_\mu\Pi^{\mu\nu}(q)=0.$$

In two dimensions, the one-loop result has the nonlocal structure

$$\Pi^{\mu\nu}(q) ={1\over\pi}\left(\eta^{\mu\nu}-{q^\mu q^\nu\over q^2}\right)$$

for one unit-charge Dirac fermion, up to convention-dependent local signs from the definition of $W[A]$. This tensor is transverse:

$$q_\mu\Pi^{\mu\nu}(q)=0.$$

Now form the mixed axial-vector correlator

$$\Pi_5^{\mu\nu}(q) =i\int d^2x\,e^{iq\cdot x} \langle Tj_5^\mu(x)j^\nu(0)\rangle_{\rm conn}.$$

Using $j_5^\mu=-\epsilon^{\mu\rho}j_\rho$ gives

$$\Pi_5^{\mu\nu}(q)=-\epsilon^{\mu\rho}\Pi_\rho{}^\nu(q).$$

Therefore

$$q_\mu\Pi_5^{\mu\nu}(q) =-q_\mu\epsilon^{\mu\rho}{1\over\pi} \left(\delta_\rho{}^\nu-{q_\rho q^\nu\over q^2}\right).$$

The second term vanishes because $q_\mu\epsilon^{\mu\rho}q_\rho=0$. Hence

$$\boxed{ q_\mu\Pi_5^{\mu\nu}(q) ={1\over\pi}\epsilon^{\nu\rho}q_\rho. }$$

Because the vector source enters as $+A_\nu j^\nu$, linear response gives $\langle j_5^\mu\rangle_A=+\Pi_5^{\mu\nu}A_\nu+O(A^2)$. Thus, suppressing the Fourier factor of $i$,

$$q_\mu\langle j_5^\mu\rangle_A =+{1\over\pi}\epsilon^{\nu\rho}q_\rho A_\nu =-{1\over\pi}\epsilon^{\rho\nu}q_\rho A_\nu.$$

In position space this is the axial anomaly in a background gauge field:

$$\boxed{ \partial_\mu\langle j_5^\mu\rangle_A =-{1\over\pi}\epsilon^{\mu\nu}\partial_\mu A_\nu =-{1\over2\pi}\epsilon^{\mu\nu}F_{\mu\nu}. }$$

Thus the vector current can be conserved, or the axial current can be conserved, but not both in a gauge-invariant quantum theory with a background vector field. For a dynamical gauge field, preserving vector gauge invariance is mandatory; the axial symmetry becomes anomalous.

The same current-current loop tests both vector and axial Ward identities in two dimensions. A gauge-invariant choice of local counterterms preserves $q_\mu\Pi^{\mu\nu}=0$ and leaves a finite contact term in the epsilon-rotated axial identity.

There is a useful diagrammatic way to see why a contradiction can occur. Contracting the vacuum polarization graph with $q_\mu$ gives

$$q_\mu\Pi^{\mu\nu}(q) \sim \int {d^dp\over(2\pi)^d}\, \operatorname{tr}\left[ \gamma^\nu S(p+q)q\!\!\!/ S(p) \right].$$

Since

$$q\!\!\!/=S^{-1}(p+q)-S^{-1}(p),$$

one is tempted to write

$$q_\mu\Pi^{\mu\nu}(q) \sim \int {d^dp\over(2\pi)^d}\, \operatorname{tr}\,\gamma^\nu\left[S(p)-S(p+q)\right] =0$$

by shifting the integration variable. This argument is legitimate only if the integral is sufficiently convergent or if the regulator respects the shift. In anomalous diagrams, the formal shift changes the boundary of the regulated integration region. A finite surface term can survive.

A one-dimensional caricature is

$$\int_{-\Lambda}^{\Lambda}dp\,[f(p+q)-f(p)] \simeq q\int_{-\Lambda}^{\Lambda}dp\,f'(p) =q[f(\Lambda)-f(-\Lambda)].$$

If the boundary term remains finite as $\Lambda\to\infty$, the naive shift fails. The anomaly is a regulated version of precisely this phenomenon.

Summary

A continuous symmetry gives a current. In a quantum field theory, the corresponding conservation law becomes a Ward identity: the divergence of the current inside a correlation function is zero away from insertions, but has contact terms at coincident points. In momentum space, the vector Ward identity for a fermion current is

$$q_\mu\Gamma^\mu(p+q,p)=S^{-1}(p+q)-S^{-1}(p).$$

Coupling the current to a background field turns the same statement into gauge invariance of $W[A]$. Differentiating gauge invariance gives transversality of current correlators,

$$q_\mu\Pi^{\mu\nu}(q)=0.$$

Massless fermions have chiral symmetries. In two dimensions, the massless Dirac fermion splits into independent chiral components $\psi_+(x^+)$ and $\psi_-(x^-)$. The Thirring interaction couples their densities but does not create a mass term, so it preserves the vector and axial $U(1)$ symmetries. Its exact solution is a gapless line; this conclusion is stronger than the symmetry and power-counting argument alone.

The quantum theory adds the crucial caveat: not every classical symmetry survives regularization. In two dimensions, preserving vector gauge invariance forces the axial current to satisfy

$$\partial_\mu j_5^\mu=-{1\over2\pi}\epsilon^{\mu\nu}F_{\mu\nu}$$

for one unit-charge massless Dirac fermion. This is the first anomaly in the course. Notice the logic: the anomaly is not a failure of algebra at separated points; it is a statement about the regulated contact terms of composite currents. The next page studies those current-current correlators and local counterterms more explicitly.

Common pitfalls

Replacing a Ward identity by $\partial_\mu j^\mu=0$. Inside time-ordered products, contact terms are part of the statement and generate the symmetry action on coincident insertions.

Demanding both vector and axial conservation after regularization. The two currents can coexist classically, but a regulator may not preserve both. In a gauge theory, vector gauge invariance is the identity that must be preserved.

Treating $j_5^\mu=-\epsilon^{\mu\nu}j_\nu$ as equality of Ward identities. The identity means that the same correlator is tested in two different tensor directions. A transverse vector correlator generally has an anomalous axial divergence.

Calling a current-current interaction a mass term. The Thirring coupling is marginal in two dimensions; the mass term $\bar\psi\psi$ is a relevant perturbation that mixes chiralities and breaks axial symmetry. Gaplessness follows from the exact solution, not from this distinction by itself.

Exercises

Exercise 1: Derive the contact-term Ward identity

Let a field $\Phi_k$ have charge $Q_k$ under the global $U(1)$ convention used on this page, so that

$$\delta\Phi_k=-i\alpha Q_k\Phi_k.$$

Use a local parameter $\alpha(x)$ to derive the Ward identity for

$$\left\langle Tj^\mu(x)\Phi_1(x_1)\cdots\Phi_n(x_n)\right\rangle.$$

Solution

Under a local transformation, the action changes by

$$\delta S=\int d^dx\,\partial_\mu\alpha(x)j^\mu(x) =-\int d^dx\,\alpha(x)\partial_\mu j^\mu(x).$$

The path integral is invariant under a change of variables:

$$0=\left\langle i\delta S\,T\Phi_1\cdots\Phi_n\right\rangle +\left\langle T\delta(\Phi_1\cdots\Phi_n)\right\rangle.$$

The operator variation is

$$\delta(\Phi_1\cdots\Phi_n) =-i\sum_{k=1}^n\alpha(x_k)Q_k\Phi_1\cdots\Phi_k\cdots\Phi_n.$$

Writing $\alpha(x_k)=\int d^dx\,\alpha(x)\delta^{(d)}(x-x_k)$ and using the arbitrariness of $\alpha(x)$ gives

$$\boxed{ \begin{aligned} &\partial_\mu^x\left\langle Tj^\mu(x)\prod_{\ell=1}^n\Phi_\ell(x_\ell) \right\rangle \\ &\qquad=-\sum_{k=1}^n Q_k\delta^{(d)}(x-x_k) \left\langle T\prod_{\ell=1}^n\Phi_\ell(x_\ell)\right\rangle \end{aligned} }.$$

The divergence of the current is supported only at operator insertions and generates their charge transformations. Alternative definitions that attach factors of $i$ to Green functions move corresponding factors between the two sides without changing this content.

Exercise 2: Construct the transverse current tensor

Assume Lorentz invariance and current conservation in a parity-even theory. Show that a two-point function of conserved currents in momentum space has the form

$$\Pi^{\mu\nu}(q)=\left(q^\mu q^\nu-q^2\eta^{\mu\nu}\right)\Pi(q^2)$$

up to local transverse terms.

Solution

The most general parity-even rank-two tensor built from $q^\mu$ and $\eta^{\mu\nu}$ is

$$\Pi^{\mu\nu}(q)=A(q^2)\eta^{\mu\nu}+B(q^2)q^\mu q^\nu.$$

Current conservation gives

$$0=q_\mu\Pi^{\mu\nu}(q) =A(q^2)q^\nu+B(q^2)q^2q^\nu.$$

For generic $q$, this requires

$$A(q^2)=-q^2B(q^2).$$

Therefore

$$\Pi^{\mu\nu}(q)=B(q^2)\left(q^\mu q^\nu-q^2\eta^{\mu\nu}\right).$$

Renaming $B(q^2)=\Pi(q^2)$ gives the stated form. Polynomial transverse pieces correspond to local gauge-invariant counterterms.

Exercise 3: Derive the axial anomaly from vector response

In two dimensions, suppose the vector current correlator of one massless Dirac fermion is

$$\Pi^{\mu\nu}(q)={1\over\pi} \left(\eta^{\mu\nu}-{q^\mu q^\nu\over q^2}\right).$$

Using $j_5^\mu=-\epsilon^{\mu\rho}j_\rho$, compute $q_\mu\Pi_5^{\mu\nu}(q)$.

Solution

The mixed axial-vector correlator is

$$\Pi_5^{\mu\nu}(q)=-\epsilon^{\mu\rho}\Pi_\rho{}^\nu(q).$$

Thus

$$q_\mu\Pi_5^{\mu\nu}(q) =-{1\over\pi}q_\mu\epsilon^{\mu\rho} \left(\delta_\rho{}^\nu-{q_\rho q^\nu\over q^2}\right).$$

The second term vanishes because

$$q_\mu\epsilon^{\mu\rho}q_\rho=0.$$

Therefore

$$q_\mu\Pi_5^{\mu\nu}(q) =-{1\over\pi}q_\mu\epsilon^{\mu\nu}.$$

Since $q_\mu\epsilon^{\mu\nu}=-\epsilon^{\nu\rho}q_\rho$, we find

$$\boxed{ q_\mu\Pi_5^{\mu\nu}(q)={1\over\pi}\epsilon^{\nu\rho}q_\rho. }$$

This is the momentum-space axial anomaly in the convention used on this page.

Exercise 4: Integrate out the Thirring auxiliary field

Show that integrating out the auxiliary field in

$$\mathcal L_A=\bar\psi i\gamma^\mu\partial_\mu\psi +A_\mu j^\mu+{1\over2g}A_\mu A^\mu$$

reproduces the Thirring interaction $-(g/2)j_\mu j^\mu$.

Solution

The $A_\mu$ equation of motion is

$$+j^\mu+{1\over g}A^\mu=0,$$

so

$$A^\mu=-gj^\mu.$$

Substitute this into the $A$-dependent part of the Lagrangian:

$$+A_\mu j^\mu+{1\over2g}A_\mu A^\mu =-g j_\mu j^\mu+{1\over2g}g^2j_\mu j^\mu.$$

Hence

$$+A_\mu j^\mu+{1\over2g}A_\mu A^\mu =-{g\over2}j_\mu j^\mu.$$

The auxiliary-field path integral gives the same result, up to an irrelevant Gaussian normalization.

Exercise 5: Expose the chiral propagator contact term

For the massless two-dimensional chiral propagator

$$G_+(x)={1\over x^+-i0\operatorname{sgn}(x^-)},$$

use

$${1\over x^+-i0\operatorname{sgn}(x^-)} =\mathcal P{1\over x^+}+i\pi\operatorname{sgn}(x^-)\delta(x^+)$$

to show that $\partial_-G_+(x)$ is a contact term.

Solution

The principal-value term has no $x^-$ dependence, so

$$\partial_-\mathcal P{1\over x^+}=0.$$

The second term gives

$$\partial_-\left(i\pi\operatorname{sgn}(x^-)\delta(x^+)\right) =i\pi\left(2\delta(x^-)\right)\delta(x^+).$$

Therefore

$$\boxed{ \partial_-G_+(x)=2\pi i\,\delta(x^+)\delta(x^-). }$$

Thus the chiral propagator is annihilated by $\partial_-$ away from the origin, but not as a distribution. This is exactly the kind of contact term that appears in Ward identities.

References

• S. Coleman, “Quantum sine-Gordon equation as the massive Thirring model”, Physical Review D 11 (1975) 2088–2097.
• D. J. Gross and A. Neveu, “Dynamical Symmetry Breaking in Asymptotically Free Field Theories”, Physical Review D 10 (1974) 3235–3253.
• Y. Takahashi, “On the Generalized Ward Identity”, Il Nuovo Cimento 6 (1957) 371–375.
• W. E. Thirring, “A Soluble Relativistic Field Theory”, Annals of Physics 3 (1958) 91–112.
• J. C. Ward, “An Identity in Quantum Electrodynamics”, Physical Review 78 (1950) 182.

Further reading

• S. Coleman, Lectures of Sidney Coleman on Quantum Field Theory, B. Gin-ge Chen, D. Derbes, D. Griffiths, B. Hill, R. Sohn, and Y.-S. Ting (eds.), World Scientific (2019), especially the lectures on symmetries, Ward identities, and two-dimensional fermions.
• A. M. Polyakov, Gauge Fields and Strings, Harwood Academic Publishers (1987), especially the discussions of two-dimensional field theory and anomalies.
• M. D. Schwartz, Quantum Field Theory and the Standard Model, Cambridge University Press (2014), chs. 14 and 30.
• M. Srednicki, Quantum Field Theory, Cambridge University Press (2007), sections on continuous symmetries, Ward identities, and anomalies.
• S. Weinberg, The Quantum Theory of Fields, Vols. I–II, Cambridge University Press (1995–1996), chapters on symmetries, current algebra, and anomalies.
• A. Zee, Quantum Field Theory in a Nutshell, 2nd ed., Princeton University Press (2010), discussions of chiral symmetry, bosonization, and anomalies.