Current Correlators and Polarization Tensors

The previous page derived Ward identities from local symmetry transformations. We now apply the same logic to one of the most useful objects in quantum field theory: the two-point function of a current. In QED this is the vacuum-polarization tensor. In linear response it is the kernel relating an applied source to an induced current. In two-dimensional field theory it is the quickest way to see why vector gauge invariance and axial current conservation cannot both survive quantization.

The central lesson is that a current-current correlator is not just a separated-point loop diagram. The product $Tj^\mu(x)j^\nu(y)$ is singular at $x=y$, so the full polarization tensor is a distribution. It may contain contact terms supported at coincident points. Those terms are local in position space, polynomial in momentum space, and fixed by the Ward identities we choose to preserve. A calculation that keeps only separated points can therefore get the nonlocal part right while still violating the exact Ward identity.

Thus the useful slogan is

$$\text{conserved vector current} \quad\Longrightarrow\quad \text{transverse polarization tensor},$$

but “polarization tensor” means the regulated loop plus its local counterterms.

Required background. Ward identities and chiral symmetries supplies the source-functional Ward identity and the two-dimensional vector/axial-current relation. Helpful background. Vacuum polarization and gauge-invariant counterterms gives the higher-dimensional one-loop prototype.

Current response from a background source

Source and light-cone conventions. Define the connected source functional by

$$Z[A]=e^{iW[A]} =\left\langle\exp\left(+i\int d^dx\,A_\mu j^\mu\right)\right\rangle_0.$$

This follows from $D_\mu=\partial_\mu-iA_\mu$, with $\psi\mapsto e^{+i\alpha}\psi$ and $A_\mu\mapsto A_\mu+\partial_\mu\alpha$.

The quadratic part of $W[A]$ is

$$W[A]=W[0]+{1\over2}\int_q A_\mu(-q)\Pi^{\mu\nu}(q)A_\nu(q)+O(A^3), \qquad \int_q\equiv\int {d^dq\over(2\pi)^d}.$$

In the two-dimensional sections,

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

For Dirac slash notation we use the KaTeX-compatible form

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

Ratios such as $q_+/q_-$ are insensitive to harmless factors of $2$ in light-cone momentum conventions. We denote the common chiral-bubble normalization by $\kappa$ rather than assigning it an absolute number before specifying how $J_\pm$, $A_\pm$, and the light-cone measure are normalized. For one unit-charge Dirac fermion, the convention-independent covariant normalization is $\Pi^{\mu\nu}=(1/\pi)(\eta^{\mu\nu}-q^\mu q^\nu/q^2)$, up to the overall sign convention for $W$.

Couple a conserved current to a classical background field,

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

Then

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

and

$$\Pi^{\mu\nu}(x-y) ={\delta^2 W[A]\over\delta A_\mu(x)\delta A_\nu(y)}\bigg|_{A=0}.$$

At separated points this is the connected time-ordered correlator,

$$\Pi^{\mu\nu}(x-y)\big|_{x\neq y} =i\langle Tj^\mu(x)j^\nu(y)\rangle_{\rm conn}.$$

As a distribution, however, it can also contain local terms such as derivatives of $\delta^{(d)}(x-y)$. In momentum space these are polynomials in $q$. They are the contact terms.

Expanding $W[A]$ gives

$$\begin{aligned} W[A]=W[0] &+\int_q A_\mu(-q)\langle j^\mu(q)\rangle \\ &+{1\over2}\int_q A_\mu(-q)\Pi^{\mu\nu}(q)A_\nu(q)+O(A^3). \end{aligned}$$

Therefore, if the one-point current vanishes,

$$\langle j^\mu(q)\rangle_A =+\Pi^{\mu\nu}(q)A_\nu(q)+O(A^2).$$

This is why $\Pi^{\mu\nu}$ is a response kernel. Its sign here follows from the action coupling $+A_\mu j^\mu$; alternative source conventions change the displayed response sign but not transversality or the pole structure.

For later use, keep the sign convention separate from the physics. With $H_{\rm ext}=\int U\rho$, a positive potential energy tends to lower the density, so static density response is negative. With a local chemical-potential source $\delta\mu$, the coupling is $-\int\delta\mu\rho$ and the same compressibility appears with the opposite sign.

The current-current correlator is the quadratic response to a background source. Gauge invariance requires the full polarization tensor, including contact terms, to be transverse.

If $A_\mu$ is a background gauge field, vector gauge invariance means

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

Infinitesimally,

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

Since $\alpha(x)$ is arbitrary,

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

Differentiating once more with respect to $A_\nu(y)$ and setting $A=0$ gives

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

In momentum space,

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

This is the transverse Ward identity.

Tensor structure and local counterterms

Lorentz invariance constrains the parity-even two-point tensor to be

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

Transversality gives

$$q_\mu\Pi^{\mu\nu}(q)=\left[A(q^2)+q^2B(q^2)\right]q^\nu=0,$$

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

$$\boxed{ \Pi^{\mu\nu}(q) =\left(q^\mu q^\nu-q^2\eta^{\mu\nu}\right)\Pi(q^2) +\Pi_{\rm loc}^{\mu\nu}(q). }$$

In Euclidean signature the same structure is

$$\Pi_{\mu\nu}^{E}(q) =\left(q^2\delta_{\mu\nu}-q_\mu q_\nu\right)\Pi_E(q^2) +\Pi^{E}_{{\rm loc},\mu\nu}(q).$$

The local part must also be transverse if the vector current is gauged. For example,

$$\Delta W[A]=-{c\over4}\int d^dx\,F_{\mu\nu}F^{\mu\nu}$$

shifts the polarization tensor by a local transverse polynomial,

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

up to the overall sign convention in $W[A]$. By contrast,

$${1\over2}m_A^2\int d^dx\,A_\mu A^\mu$$

would contribute $m_A^2\eta^{\mu\nu}$ and is not transverse. A local photon mass term is forbidden by exact gauge invariance.

A useful checklist is: nonlocal terms can carry physical long-distance information, while local transverse terms encode scheme choices and counterterms. Local non-transverse terms are allowed only when the source is not gauged or when the symmetry is explicitly broken.

This distinction matters later in two dimensions. A massless fermion loop can generate

$$A_\mu(-q)\left(\eta^{\mu\nu}-{q^\mu q^\nu\over q^2}\right)A_\nu(q).$$

This behaves like a mass term after gauge fixing, but it is nonlocal and transverse. It is therefore compatible with gauge invariance.

Ward contraction of the one-loop bubble

For a Dirac fermion with vector current

$$j^\mu=\bar\psi\gamma^\mu\psi,$$

the one-loop current-current correlator is schematically

$$\Pi^{\mu\nu}_{\rm loop}(q) =-i\int_p \operatorname{tr}\left[ \gamma^\mu S(p+q)\gamma^\nu S(p) \right],$$

where

$$S(p)={i\over p\!\!\!/-m+i0}.$$

Contract with $q_\mu$. Since

$$q\!\!\!/=(p\!\!\!/+q\!\!\!/-m)-(p\!\!\!/-m) =S_0^{-1}(p+q)-S_0^{-1}(p),$$

the contracted bubble becomes

$$q_\mu\Pi^{\mu\nu}_{\rm loop}(q) \propto \int_p\operatorname{tr}\left[ \left(S_0^{-1}(p+q)-S_0^{-1}(p)\right) S(p+q)\gamma^\nu S(p) \right].$$

Using $S_0^{-1}S=i$ inside the trace, together with cyclicity of the trace, this reduces to a difference of two one-propagator integrals,

$$q_\mu\Pi^{\mu\nu}_{\rm loop}(q) \propto \int_p\operatorname{tr}\left[\gamma^\nu S(p)\right] - \int_p\operatorname{tr}\left[S(p+q)\gamma^\nu\right].$$

If these integrals were absolutely convergent, the second term could be shifted by $p\mapsto p-q$, and the difference would vanish. But the individual integrals are ultraviolet divergent. Shifting a divergent integral is not a harmless algebraic step; it is a statement about the regulator.

For a regulated divergent integral, shifting the integration variable removes one boundary strip and adds another. Their difference is the elementary origin of local surface terms in Ward-identity manipulations.

The one-dimensional analog is

$$I_\Lambda(a)=\int_{-\Lambda}^{\Lambda}dp\,\left[f(p+a)-f(p)\right].$$

For small $a$,

$$I_\Lambda(a) =a\int_{-\Lambda}^{\Lambda}dp\,f'(p)+O(a^2) =a\left[f(\Lambda)-f(-\Lambda)\right]+O(a^2).$$

If $f$ vanishes at both ends, the boundary term disappears. If $f$ approaches different limits, the boundary term survives. A divergent loop integral can behave the same way.

The lesson is not that gauge Ward identities are unreliable. It is that Ward identities are statements about regulated composite operators. A gauge-invariant regulator and a gauge-invariant counterterm prescription define the local terms so that

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

If another classical symmetry is incompatible with this choice, that other symmetry becomes anomalous. This is why the anomaly is often visible as a finite “surface term” in a formally linearly divergent integral.

Vector and axial currents in two dimensions

In two dimensions the axial current is dual to the vector current. With the convention used on the previous page,

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

Suppose vector gauge invariance fixes the current-current correlator to be

$$\Pi^{\mu\nu}_{VV}(q) =C_V\left(\eta^{\mu\nu}-{q^\mu q^\nu\over q^2}\right).$$

For one massless Dirac fermion with the covariant unit-current normalization,

$$C_V={1\over\pi}.$$

The vector Ward identity is automatic:

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

Now dualize one index to form the axial-vector correlator,

$$\Pi^{\mu\nu}_{5V}(q) =-\epsilon^{\mu\rho}\Pi_{\rho}{}^\nu(q).$$

Its divergence is

$$\begin{aligned} q_\mu\Pi^{\mu\nu}_{5V}(q) &=-C_Vq_\mu\epsilon^{\mu\rho} \left(\delta_\rho{}^\nu-{q_\rho q^\nu\over q^2}\right) \\ &=-C_Vq_\mu\epsilon^{\mu\nu}, \end{aligned}$$

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

$$\boxed{ q_\mu\Pi^{\mu\nu}_{5V}(q)=C_V\epsilon^{\nu\rho}q_\rho. }$$

Thus a nonzero transverse vector response implies a nonzero axial divergence. With the source term $+A_\nu j^\nu$, linear response gives $\langle j_5^\mu\rangle_A=+\Pi_{5V}^{\mu\nu}A_\nu+O(A^2)$. In position space, for one positively charged unit Dirac fermion, this gives

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

The regulator has preserved vector gauge invariance. The axial current is anomalous.

Chiral bubbles in light-cone variables

The same conclusion can be seen without gamma-matrix traces. A massless Dirac fermion in two dimensions splits into right- and left-moving components,

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

The chiral currents are

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

The right-moving propagator has the distributional form

$$G_+(p)={i\over p_-+i0\operatorname{sgn}(p_+)},$$

and the left-moving one is

$$G_-(p)={i\over p_++i0\operatorname{sgn}(p_-)}.$$

The sign of the infinitesimal imaginary part is crucial. It is the remnant of the relativistic Feynman prescription $1/(p^2+i0)$ after the massless propagator has been factorized into chiral pieces.

The $J_+J_+$ bubble is proportional to

$$\Pi_{++}(q) \propto \int {dp_+dp_-\over(2\pi)^2} {1\over p_-+i0\operatorname{sgn}(p_+)} {1\over p_-+q_-+i0\operatorname{sgn}(p_++q_+)}.$$

For fixed $p_+$, the two poles in the complex $p_-$ plane lie on the same side unless $p_+$ and $p_++q_+$ have opposite signs. For $q_+>0$, this happens only in the strip

$$-q_+<p_+<0.$$

The width of the strip is $q_+$. The $p_-$ residue gives a factor proportional to $1/q_-$. Therefore the nonlocal dependence is

$$\boxed{ \Pi_{++}(q)=\kappa{q_+\over q_-}, \qquad \Pi_{--}(q)=\kappa{q_-\over q_+}. }$$

The same $\kappa$ appears for both chiralities when their currents are normalized symmetrically. Its numerical value depends on the factors of two absorbed into light-cone coordinates, currents, sources, and integration measure. After translating back to the covariant unit-current convention, the full Dirac response has $C_V=1/\pi$ as stated above.

At separated points, the mixed right-left correlator vanishes,

$$\Pi_{+-}^{\rm sep}(q)=0.$$

In the $J_+J_+$ bubble, the $p_-$ contour gives a nonzero residue only when the two Feynman prescriptions put the poles on opposite sides. For $q_+>0$, this occurs in the strip $-q_+<p_+<0$, giving $\Pi_{++}\propto q_+/q_-$.

The nonlocal ratios $q_+/q_-$ and $q_-/q_+$ are robust. A local counterterm cannot change them. What a local counterterm can change is the mixed component $\Pi_{+-}$.

Gauge-invariant completion in two dimensions

With conventional factors of two absorbed into the definitions of $A_\pm$ and $J_\pm$, the global source convention gives

$$S_{\rm int}=+\int d^2x\,\left(A_-J_+ + A_+J_-\right).$$

The separated chiral bubbles give

$$\begin{aligned} W_{\rm sep}[A] ={\kappa\over2}\int_q\bigg[& A_-(-q){q_+\over q_-}A_-(q) \\ &+A_+(-q){q_-\over q_+}A_+(q) \bigg]. \end{aligned}$$

Under the vector gauge transformation

$$\delta A_+(q)=q_+\epsilon(q), \qquad \delta A_-(q)=q_-\epsilon(q),$$

the variation of $W_{\rm sep}$ is local:

$$\delta W_{\rm sep} =\kappa\int_q \epsilon(-q)\left[q_+A_-(q)+q_-A_+(q)\right].$$

This can be canceled by the local contact term

$$W_{\rm ct}[A] =-\kappa\int_q A_+(-q)A_-(q).$$

Thus the gauge-invariant quadratic effective action is

$$\boxed{ \begin{aligned} W_{\rm gi}[A] ={\kappa\over2}\int_q\bigg[& A_-(-q){q_+\over q_-}A_-(q) \\ &+A_+(-q){q_-\over q_+}A_+(q) -2A_+(-q)A_-(q) \bigg]. \end{aligned} }$$

Equivalently,

$$\boxed{ W_{\rm gi}[A] ={\kappa\over2}\int_q F_{+-}(-q){1\over q_+q_-}F_{+-}(q), }$$

where, suppressing the Fourier factor of $i$,

$$F_{+-}(q)=q_+A_-(q)-q_-A_+(q).$$

The separated chiral bubbles give the two nonlocal same-chirality pieces. Vector gauge invariance fixes the local mixed contact term, completing the result into the manifestly gauge-invariant nonlocal action $F_{+-}(1/\Box)F_{+-}$.

This is the cleanest way to see why the mixed polarization component is subtle. The separated diagram gives $\Pi_{+-}^{\rm sep}=0$. The full gauge-invariant distribution contains a local contact term. Without it, the vector Ward identity fails. This is the concrete two-dimensional example of the general warning at the start of the page: separated-point correlators and full response kernels are not the same object.

The expression $F_{+-}(1/q_+q_-)F_{+-}$ should be read as a nonlocal quadratic functional. In ordinary covariant notation it is the two-dimensional version of $F(1/\Box)F$. It is gauge invariant because it is written in terms of $F$, but it is not the same as a local Proca mass $A_\mu A^\mu$.

If $A_\mu$ is made dynamical, the induced action contains the transverse projector

$$A_\mu(-q) \left(\eta^{\mu\nu}-{q^\mu q^\nu\over q^2}\right) A_\nu(q).$$

This is the Schwinger-model mechanism in response-function language: the gauge field acquires a mass scale, but gauge invariance is preserved because the induced term is transverse and nonlocal.

Regulator choice and the anomaly

A gauge-invariant Pauli–Villars definition makes the symmetry choice explicit. Schematically, one replaces

$$\det(iD\!\!\!/)$$

by a regulated ratio such as

$${\det(iD\!\!\!/)\over\det(iD\!\!\!/+M)}.$$

The heavy regulator has the same vector gauge coupling as the light fermion, so vector gauge invariance is preserved. But the mass term $M\bar\chi\chi$ couples left and right movers, so it breaks the axial rotation that assigns opposite phases to $\chi_+$ and $\chi_-$.

The exact vector Ward identity is therefore

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

while the axial Ward identity becomes

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

The anomaly is the finite trace left by the ultraviolet regulator in the contact term of a current-current correlator.

This viewpoint is deliberately operational. To compute a current correlator, first regulate it; then add the local counterterms required by the symmetry you insist on preserving; only then ask whether another classical Ward identity still holds.

Bridge to many-body response

The contour argument in the chiral bubble will reappear in the next page. A nonrelativistic Fermi gas has particle and hole propagators whose poles can lie on opposite sides of the energy contour. The corresponding loop is nonzero only in restricted kinematic regions near the Fermi surface. The same analytic idea underlies density response, current response, and Fermi-surface instabilities.

So this page is a bridge. In relativistic field theory, current correlators organize Ward identities and anomalies. In many-body theory, they become response functions of a filled sea. In both cases, the physics is controlled by poles, contact terms, and symmetry.

Summary

The current-current correlator is the quadratic response of the connected effective action $W[A]$ to a background source. For a non-anomalous vector current,

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

Lorentz invariance then gives the transverse structure

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

At one loop, the Ward contraction reduces to a difference of shifted integrals. If the integrals are divergent, the shift can leave a finite boundary term. A regulator and local counterterms are therefore part of the definition of the Ward identity.

In two dimensions, chiral current bubbles give

$$\Pi_{++}=\kappa{q_+\over q_-}, \qquad \Pi_{--}=\kappa{q_-\over q_+}.$$

The coefficient $\kappa$ keeps track of light-cone normalization; the covariant unit-current result is $C_V=1/\pi$. The separated mixed correlator vanishes, but gauge invariance fixes a local mixed contact term. The completed answer is

$$W_{\rm gi}[A] ={\kappa\over2}\int_q F_{+-}(-q){1\over q_+q_-}F_{+-}(q).$$

Preserving vector gauge invariance forces the axial Ward identity to contain the anomaly.

Common pitfalls

Ignoring contact terms. The separated-point diagram is not the full polarization tensor. The missing term can be local and still required by a Ward identity.

Shifting divergent integrals without saying how they are regulated. A formal loop-momentum shift is safe only for convergent integrals or for regulators that make the shift legitimate.

Mistaking a transverse nonlocal term for a forbidden local mass. The two-dimensional induced action behaves like a mass after gauge fixing, but it is written with a transverse projector and is gauge invariant.

Trying to conserve vector and axial currents simultaneously in the regulated theory. For the massless two-dimensional Dirac fermion, preserving vector gauge invariance forces the axial current to be anomalous.

Comparing absolute light-cone coefficients before translating conventions. The ratios $q_+/q_-$ and $q_-/q_+$ are invariant under harmless rescalings, but their prefactor is not. Compare the covariant tensor $C_V(\eta^{\mu\nu}-q^\mu q^\nu/q^2)$ or explicitly translate coordinates, currents, sources, and the measure.

Exercises

Exercise 1: Derive transversality from source gauge invariance

Starting from

$$W[A]=W[0]+{1\over2}\int_q A_\mu(-q)\Pi^{\mu\nu}(q)A_\nu(q)+O(A^3),$$

show that invariance under $A_\mu(q)\mapsto A_\mu(q)+q_\mu\alpha(q)$ implies $q_\mu\Pi^{\mu\nu}(q)=0$.

Solution

The variation of the quadratic term is

$$\delta W^{(2)} ={1\over2}\int_q\left[ q_\mu\alpha(-q)\Pi^{\mu\nu}(q)A_\nu(q) + A_\mu(-q)\Pi^{\mu\nu}(q)q_\nu\alpha(q) \right].$$

Using the symmetry of the quadratic kernel under exchange of the two source legs, this is

$$\delta W^{(2)} =\int_q \alpha(-q)q_\mu\Pi^{\mu\nu}(q)A_\nu(q).$$

Since $\alpha$ and $A_\nu$ are arbitrary,

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

Exercise 2: Construct the transverse tensor structure

Assume Lorentz invariance and write

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

Use $q_\mu\Pi^{\mu\nu}=0$ to determine the transverse form.

Solution

Contracting with $q_\mu$ gives

$$q_\mu\Pi^{\mu\nu} =\left[A(q^2)+q^2B(q^2)\right]q^\nu.$$

For arbitrary $q^\nu$, transversality 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 standard tensor structure.

Exercise 3: Locate the chiral bubble support strip

For $q_+>0$, consider the chiral bubble integral

$$I(q)=\int {dp_+dp_-\over(2\pi)^2} {1\over p_-+i0\operatorname{sgn}(p_+)} {1\over p_-+q_-+i0\operatorname{sgn}(p_++q_+)}.$$

Explain why only the strip $-q_+<p_+<0$ contributes and why $I(q)\propto q_+/q_-$.

Solution

For fixed $p_+$, the poles in the $p_-$ plane are

$$p_-=-i0\operatorname{sgn}(p_+), \qquad p_-=-q_- - i0\operatorname{sgn}(p_++q_+).$$

If $p_+$ and $p_++q_+$ have the same sign, the poles lie on the same side of the real $p_-$ axis, so the contour can be closed in the empty half-plane and gives zero.

For $q_+>0$, the signs are opposite only when

$$-q_+<p_+<0.$$

In that strip the $p_-$ residue is proportional to $1/q_-$. The remaining $p_+$ integral gives the width of the strip,

$$\int_{-q_+}^{0}dp_+=q_+.$$

Hence

$$I(q)\propto {q_+\over q_-}.$$

Exercise 4: Complete the chiral bubbles gauge-invariantly

Show that

$${(q_+A_- - q_-A_+)^2\over q_+q_-} = A_-{q_+\over q_-}A_- + A_+{q_-\over q_+}A_+ -2A_+A_-.$$

Solution

Expand the numerator:

$$(q_+A_- - q_-A_+)^2 =q_+^2A_-^2+q_-^2A_+^2-2q_+q_-A_+A_-.$$

Dividing by $q_+q_-$ gives

$$A_-{q_+\over q_-}A_- + A_+{q_-\over q_+}A_+ -2A_+A_-.$$

This is the gauge-invariant completion of the two separated chiral bubbles.

Exercise 5: Recover the axial divergence from vector response

Let

$$\Pi_{VV}^{\mu\nu}(q) =C_V\left(\eta^{\mu\nu}-{q^\mu q^\nu\over q^2}\right), \qquad \Pi_{5V}^{\mu\nu}(q)=-\epsilon^{\mu\rho}\Pi_{\rho}{}^\nu(q).$$

Show that

$$q_\mu\Pi_{5V}^{\mu\nu}(q)=C_V\epsilon^{\nu\rho}q_\rho.$$

Solution

Compute

$$q_\mu\Pi_{5V}^{\mu\nu} =-C_Vq_\mu\epsilon^{\mu\rho} \left(\delta_\rho{}^\nu-{q_\rho q^\nu\over q^2}\right).$$

The second term vanishes because

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

Thus

$$q_\mu\Pi_{5V}^{\mu\nu} =-C_Vq_\mu\epsilon^{\mu\nu}.$$

Using $q_\mu\epsilon^{\mu\nu}=-\epsilon^{\nu\rho}q_\rho$, we obtain

$$q_\mu\Pi_{5V}^{\mu\nu}=C_V\epsilon^{\nu\rho}q_\rho.$$

References

• W. Pauli and F. Villars, “On the Invariant Regularization in Relativistic Quantum Theory”, Reviews of Modern Physics 21 (1949) 434–444.
• J. Schwinger, “Gauge Invariance and Mass. II”, Physical Review 128 (1962) 2425–2429.
• Y. Takahashi, “On the Generalized Ward Identity”, Il Nuovo Cimento 6 (1957) 371–375.
• 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), lectures on Ward identities, regularization, and two-dimensional fermions.
• A. M. Polyakov, Gauge Fields and Strings, Harwood Academic Publishers (1987), discussions of two-dimensional field theory, gauge invariance, and anomalies.
• M. D. Schwartz, Quantum Field Theory and the Standard Model, Cambridge University Press (2014), chapters on vacuum polarization, Ward–Takahashi identities, and anomalies.
• M. Srednicki, Quantum Field Theory, Cambridge University Press (2007), sections on vacuum polarization, QED Ward identities, and anomalies.
• S. Weinberg, The Quantum Theory of Fields, Vols. I–II, Cambridge University Press (1995–1996), chapters on current correlators, gauge-theory renormalization, and anomalies.
• A. Zee, Quantum Field Theory in a Nutshell, 2nd ed., Princeton University Press (2010), discussions of vacuum polarization, two-dimensional fermions, and anomalies.