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Ward Identities and in Dimension Regularization

In Package-X, the fifth Dirac gamma matrix γ5 is defined to anticommute with all other gamma matrices. This implementation is "naive dimensional regularization", and can lead to apparent violations of Ward identities. This tutorial explains how to deal with γ5, enforce Ward identities, and check for anomaly cancellation with Package-X, using the on-shell vertex function in the standard model as an example.


For a given fermion flavor in the standard model (quark or lepton), there are two diagrams contributing to the vertex function:


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The weak neutral and electromagnetic vertex factors for a given fermion are:

where , . Applying the Feynman rules starting at the μ-index and working around counterclockwise for both diagrams, the vertex function is:

We will put the photons on shell , but leave the off its mass shell with a mass . In the computations that follow, the prefactor is dropped.

Initialize Package-X:
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Define the on shell kinematic conditions:
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Initiate integration with LoopIntegrate, and apply the on shell conditions:
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Globally set option ExplicitC0None to LoopRefine, to leave all scalar C-functions symbolic. Apply LoopRefine:
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Note that charge conjugation invariance leads to a result that is independent of the neutral current vector coupling .

This result suffers from two ambiguities. We'll start by addressing the more innocuous problem first.

Ambiguity #1: Vanishing totally antisymmetric tensor

The result we computed in computedVertexFunction is unique only up to a totally antisymmetric rank-5 tensor. A family of different expressions can be obtained by cyclically permuting the elements of the trace (Spur). Observe the effect of cyclically permuting the vertex factor in the first diagram, so that it is on the other side:

Copy the code for diagram1, and manually move the first two arguments of Spur (γμ,gV -gA γ5) to the other side:
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Notice that the coefficient of the very first term 10 gA Q2 has changed. Compute the difference for greater clarity:

Form the difference:
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Notice that this difference is a totally antisymmteric rank-5 tensor that has been twice-contracted into vectors and . Remember that tensors are multilinear maps from vectors to numbers, and a totally antisymmetric rank-5 tensor can map to a nonzero number only when all five vectors are linearly independent. In four dimensional spacetime, up to a maximum of four vectors are linearly independent; the fifth vector can be written as a linear combination of the other four. We should then expect this tensor to be vanishing.

We can confirm this by contracting the difference tensor into three additional vectors , , and , with being the linearly dependent vector with scalar coefficients , , , and .

Use LScalarQ to declare coefficients a, b, c and d as Lorentz scalars:
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Multiply the difference tensor by e1μ, e2ν and e3ρ. Replace vector e3 with the linear combination. Contract the indices, and apply the kinematic on shell conditions:
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These extra pieces may make intermediate results more verbose. But ultimately, when the computed vertex function is embedded into a larger physical calculation where all tensors have been contracted, the extra vanshing contributions generated by Spur will drop out.

Ambiguity #2: Artifical violations of the Ward identity

A more serious problem with the calculation is that it does not satisfy the Ward identities implied by electromagnetic gauge invariance. These identities are:

We can check these by contracting and into the vertex function.

Contract and into the vertex function, and apply the on shell conditions:
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The Ward identities require these expressions to vanish, so something has gone awry in the computation. Let's parametrize the vertex function in the following way:

To diagnose the problem, it is useful to note the superficial UV degrees of divergence of the varous form factors:

Superficial degree of divergence
(Full integral)+1 (linearly divergent)
2 (finite)
, 0 (logarithmically divergent)

Superficial UV degrees of divergence.

Form factors are UV convergent, so their forms are unambiguous. But form factors and are superficially logarithmically divergent. So, their precise forms depend on the UV regulator. Within dimensional regularization they will especially depend on how is treated. Since Package-X's γ5 anticommutes with all other gamma matrices, which is techincally inconsistent within dimensional regularization, we can expect that and are incorrect, leading to the violations of Ward identities.

Next, we will apply Adler's method to correct the form factors and . We will determine them in terms of the unambiguous form factors by requiring that the QED Ward identities be satisfied at the ν and ρ vertices.

Let's start by storing the form factors , by extracting the coefficients of the multiplying tensors. We are not storing and since we will correct them.

Use Coefficient to extract the form factors, and store them as F1F6:
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Define a modified vertex function, with but with form factors G1 and G2 left symbolic.
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Let us now check what conditions the electromagnetic Ward identities impose on form factors G1 and G2:

Contract p1ν and p2ρ into the vertex function to form and , and apply the on shell conditions:
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We now demand that the Ward identities be satisfied by requiring these expressions to vanish. This will imply a condition on the undetermined form factors G1 and G2.

Use Solve to obtain G1 and G2:
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Update correctedVertexFunction with the newly determined G1 and G2:
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We now have the corrected vertex function that satisfies the QED Ward identities at indices ν and ρ:

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This accounts for the limitation of Package-X's implementation of γ5 in dimensional regularization. Next we must check the Ward identity at the μ index by confirming the anomaly cancellation associated with weak neutral currents in the standard model.

Ward identity at the μ index, and anomaly cancellation

The Ward identity associated with weak interactions is more complicated due to its associatation with the spontaneous breakdown of the standard model gauge symmetry. The continuity equation satisfied by the weak neutral current is related to the pseuoscalar density

which in momentum space leads to the desired Ward identity for the vertex function,

In the left hand side, is the vertex function computed above, summed over all standard model fermions .

Contract the correctedVertexFunction with (p1+p2)μ, and store the result as LHS1. This gives the contribution from a single fermion flavor, dependent on gA, Q, and m:
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Consider just one generation of standard model fermions (the minimal set required to ensure anomaly cancellation), consisting of three colors of up and down quarks, the neutrino and the electron. Accounting for their differing charge and masses, the sum gives the LHS of the Ward identity:
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In the RHS of the Ward identity is , the vertex function involving the neutral Goldstone boson (also summed over all standard model fermions ), which needs to be calculated. For a given fermion flavor, the two diagrams contributing to the vertex function are:


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The Goldstone boson coupling to a standard model fermion is:

where . Using the Feynman rules, the vertex function is:

As before, the photons are on shell , and the Goldstone boson is off its mass shell with a mass . To be consistent, we continue to omit the prefactor from the calculations.

Initiate loop integration with LoopIntegrate, and apply the on shell conditions:
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Sum the diagrams to obtain the contribution of a single fermion flavor to the RHS of the Ward identity, and store the result as RHS1:
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Add up the contributions from all fermion flavors in one generation, to obtain the RHS of the Ward identity:
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Check that the Ward identity is satisfied:
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Final result for the vertex function

Verification of the Ward identities at all three vertices confirms the correctness of the calculation obtained with Package-X, despite being computed in naive dimensional regularization. The final result of the vertex function can be constructed from correctedVertexFunction upon summing over all fermion flavors.

Sum correctedVertexFunction over the fermion flavors to obtain the final result:
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Therefore, . Observe the absence of integer constants in the sum which are present in for a single fermion flavor. Its absence is due to anomaly cancellation.