Fujikawa method

Abstract: Using perturbation theory in the Euclidean (imaginary time) formalism as well as the nonperturbative Fujikawa method, we verify that the chiral anomaly equation remains unaffected in the presence of nonzero chemical potential, μ. We extend our considerations to fermions with exact chiral symmetry on the lattice and discuss the consequences for the recent Bloch-Wettig proposal for the Dirac operator at finite chemical potential. We propose a new simpler method of incorporating μ and compare it with the Bloch-Wettig idea.

Abstract: In some inflation scenarios such as $R^{2}$ inflation, a gravitational scalar degrees of freedom called scalaron is identified as inflaton. Scalaron linearly couples to matter via the trace of energy-momentum tensor. We study scenarios with a sequestered matter sector, where the trace of energy-momentum tensor predominantly determines the scalaron coupling to matter. In a sequestered setup, heavy degrees of freedom are expected to decouple from low-energy dynamics. On the other hand, it is non-trivial to see the decoupling since scalaron couples to a mass term of heavy degrees of freedom. Actually, when heavy degrees of freedom carry some gauge charge, the amplitude of scalaron decay to two gauge bosons does not vanish in the heavy mass limit. Here the quantum contribution to the trace of energy-momentum tensor plays an essential role. This quantum contribution is known as trace anomaly or Weyl anomaly. The trace anomaly contribution from heavy degrees of freedom cancels with the contribution from the ${\it classical}$ scalaron coupling to a mass term of heavy degrees of freedom. We see how trace anomaly appears both in the Fujikawa method and in dimensional renormalization. In dimensional renormalization, one can evaluate the scalaron decay amplitude in principle at all orders, while it is unclear how to process it beyond the one-loop level in the Fujikawa method. We consider scalaron decay to two gauge bosons via the trace of energy-momentum tensor in quantum electrodynamics with scalars and fermions. We evaluate the decay amplitude at the leading order to demonstrate the decoupling of heavy degrees of freedom.

Abstract: We propose an extended definition of the regularized Jacobian which allows the calculation of anomalies using parameter-dependent regulators in the Fujikawa approach. This extension incorporates the basic Green's function of the problem in the regularized Jacobian, allowing us to interpret a specific regularization procedure as a way of selecting the finite part of the Green's function, in complete analogy with what is done at the level of the effective action. In this way we are able to consider the effect of counterterms in the regularized Jacobian in order to relate different regularization procedures. We also discuss the ambiguities that arise in our prescription due to some freedom in the place where we can insert the regulator, using charge-conjugation invariance as a guiding principle. The method is applied to the case of vector and axial-vector anomalies in two- and four-dimensional quantum electrodynamics. In the first situation we recover the standard family of anomalies calculated by the point-splitting regularization prescription. We also study in detail an alternative choice in the position of the regulator and we calculate explicitly all the currents that generate the families of anomalies that we are considering. Next we extend the calculation to four dimensions, using the samemore » prescriptions as before, and we compare the results with those obtained from the point-splitting calculation, which we also perform in the case of the vector anomaly. A discussion of the relation among the results obtained by different regularization prescriptions is given in terms of the allowed counterterms in the regularized Jacobian, which are highly constrained by the requirement of charge-conjugation invariance.« less

Abstract: Recently, the Atiyah-Patodi-Singer(APS) index theorem attracts attention for understanding physics on the surface of materials in topological phases. Although it is widely applied to physics, the mathematical set-up in the original APS index theorem is too abstract and general (allowing non-trivial metric and so on) and also the connection between the APS boundary condition and the physical boundary condition on the surface of topological material is unclear. For this reason, in contrast to the Atiyah-Singer index theorem, derivation of the APS index theorem in physics language is still missing. In this talk, we attempt to reformulate the APS index in a "physicist-friendly" way, similar to the Fujikawa method on closed manifolds, for our familiar domain-wall fermion Dirac operator in a flat Euclidean space. We find that the APS index is naturally embedded in the determinant of domain-wall fermions, representing the so-called anomaly descent equations.