On the axial anomaly in Very Special Relativity

In this paper we study the axial anomaly in Very Special Relativity Electrodynamics using Pauli-Villars and dimensional regularization of ultraviolet divergences and Mandelstam-Leibbrandt regularization of infrared divergences. We compute the anomaly in 2 and 4 dimensional space-time. We find that this procedure preserves the vector Ward identity(charge conservation) and reproduce the standard axial anomaly in 2 and 4 dimensions without corrections from VSR. Finally, we show how to obtain the anomaly in the path integral approach.


II. THE MODEL
The Electrodynamics sector of the VSRSM in the Feynman gauge.
The vector current(electric charge conservation) is: The axial vector current is: Both currents are conserved at the classical level [12]. We are interested in computing expectation values of these currents.
To get the Feynman rules we use the expansion of (n.D) −1 both in the currents and the lagrangian. The Feynman rules are listed in Appendix A.

III. TWO DIMENSIONAL AXIAL ANOMALY
In this case we have to compute the expectation value of the axial vector current in a background field A ν . We use the convention of [16], 01 = +1.
Notice that equation (3) is logarithmically divergent and equation (4) is finite. It is easy to check that formally: if shift of the integration variable p → p + k is allowed. Here k is a constant vector. This would be true if the integral (3) would be finite. Introduce a Pauli-Villars particle of massM and define the regularized amplitude: Since Π 5Rµν (M,M , q) is finite, it satisfies the naive Ward identity(electric charge conservation): On the other hand, the axial Ward identity is, formally: if shift of the integration variable p → p + k is allowed.
Therefore the regularized amplitude satisfies: Since the original amplitude is obtained formally as limM →∞ , the axial anomaly is given by: Now, we compute (5). First notice that after computing the trace, the integral is finite. A tipical term containing the vector n µ is of the form: Now we recall an important property of ML prescription. It preserves naive power counting. According to this, Following the same argument, we can easily check that all terms containing n µ vanish when M → ∞. It remains the Lorentz invariant term: Equation (7) is the standard Lorentz invariant result [16]. We want to comment on a previous computation of the anomaly in [12]. There and here, the vector current is conserved, but a different axial anomaly is obtained. This difference may be a result of different normalization conditions [17] or the extra freedom we have when Lorentz symmetry is broken [18] .
It is clear though that the procedure used in [12] does not respect naive power counting of the loop integrals.

IV. FOUR DIMENSIONAL AXIAL ANOMALY
We compute: There are four graphs that contribute to the axial anomaly in four dimensions (Figure 3 -6). Notice that in [14] Figure 5,6 are missing. They are fundamental to satisfy the Ward identity for the vector current(charge conservation) as well as the right computation of the axial anomaly.
To compute the axial anomaly we will use Pauli-Villars regularization and Mandelstam-Leibbrandt prescription to treat infrared divergences. We will follow reference [17].
It is easy to check that formally: Introduce a Pauli-Villars particle of massM and define the regularized amplitude: Since Π 5Rµνδ (M,M , p, q) is finite, it satisfies the naive Ward identity(electric charge conservation): Π 5Rµνδ (M,M , p, q)p δ = 0 Besides, the axial Ward identity formally is,if shift of the integration variable k → k + Q is allowed: The term (13) is convergent and has zero trace in four dimensions. So it vanishes. Therefore the regularized amplitude satisfies: Since the original amplitude is obtained formally as limM →∞ , the axial anomaly is given by: After computing the trace, we use ML prescription to regulate the infrared divergences. A(M, p, q) νδ is ultraviolet finite A nice property of ML prescription is that preserve naive power counting. Using this property, we can easily show that all terms containing n µ in A(M, p, q) νδ are smaller than M −2 for large M , so they do not contribute to the axial anomaly.
A νδ = limM →∞ That is: This is the standard result [17] [16]. We see that Pauli-Villars regularization of ultraviolet divergences and Mandelstam-Leibbrandt regularization of infrared divergences preserve the Ward identity for the vector current(electric charge conservation) as well as the standard anomaly for the axial current, without modification from VSR terms.
The anomaly is: That is: The VSR part of the integral is convergent, using ML prescription, so it is zero, when we take d = 2.
which is the standard result [16]. Following the same reasoning as in Appendix B, we can study the vector Ward identity in four dimensions. If we use dimensional regularization there, then shifting the integration variable k− > k + Q is allowed. So the naive Ward identity for the vector current is satisfied without anomaly.

VII. PATH INTEGRAL DERIVATION OF THE AXIAL ANOMALY
We use the approach of [19]. The generating functional in the presence of an external field A µ is; where the gauge invariant and Sim(2) invariant Dirac operator is For large q and fixed A µ We can expand The integration measure is defined by: Under the change of variables: we get: where the jacobian J is given by: To evaluate it we introduce a gauge invariant and Sim(2) invariant regularization: We can write: Since we take M → ∞, we look at the asymptotic part of the spectrum. It is simpler to evaluate the commutator in the light cone gauge n.A = 0 Then: That is: Then the Adler-Bell-Jackiw anomaly follows.
Notice that we could get this result assuming that the infrared regulator of 1 n.q preserves scaling(naive power counting). To garanty this property we work with the ML prescription, as in the perturbative approach.

VIII. CONCLUSIONS
We have examined the appearance of axial anomalies in VSR electrodynamics, using Pauli-Villars and dimensional regularization of ultraviolet divergences and Mandelstam-Leibbrandt regularization of infrared divergences.
Given that ML preserves naive power counting in loop integrals, we have shown that the usual form for the anomaly of the axial current appears, without corrections from VSR terms. No anomaly is present in the vector current conservation. This computation is at variance from a previous result for the axial anomaly in two dimensions [12],where corrections from VSR terms were found. This difference could be due to different normalization conditions for the anomaly term [17] or some extra freedom that occurs when Lorentz invariance is violated [18]. In any case, our result implies that the procedure of [12] destroys the naive power counting of loop integrals.
In four dimension we find a completely different result compared to [14]. There they claim that the conservation for the vector current has an anomaly and VSR corrections should appear in the anomaly of the axial current. We notice also that Figure 5,6 are lacking in the computation of both anomalies in [14]. Figure 5,6 are crucial to satisfy the Ward identity for the vector current as a procedure in 4d similar to the one explained in Appendix B shows.
We study also the axial anomaly from the point of view of the path integral method. Again ML property of preserving scaling(naive power counting) permits to show that the axial anomaly is the Lorentz invariant one, without corrections from VSR.
Finally we want to recall that M is not the mass of the particle. So if the fermion acquires a VSR mass m even if M = 0, the divergence of the axial current will contain the anomalous term only. )γ 5