Fujikawa's Method Research Articles

Closer look at the matching condition for radiative QCD θ parameter

In this paper, we scrutinize a radiatively generated QCD θ parameter at the two-loop level based on both full analytical loop functions with the Fock-Schwinger gauge method and the effective field theory approach, using simplified models. We observe that the radiatively generated θ parameters at the low energy scale precisely match between them. It provides validity to perturbative loop calculations of the QCD θ parameter with the Fock-Schwinger gauge method. Furthermore, it is also shown that the ordinary Fujikawa method for the radiative θ parameter by using θ¯\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\overline{\ heta} $$\\end{document} = −arg det Mqloop\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {\\mathcal{M}}_q^{\ extrm{loop}} $$\\end{document} does not cover all contributions in the simplified models. But, we also find that when there is a scale hierarchy in CP-violating sector, evaluation of the Fujikawa method is numerically sufficient. As an application, we calculate the radiative θ parameter at the two-loop level in a slightly extended Nelson-Barr model, where the spontaneous CP violation occurs to solve the strong CP problem. It is found a part of the radiative θ parameters cannot be described by the Fujikawa method.

Berry phase in the phase space worldline representation: The axial anomaly and classical kinetic theory

The Berry phase is analyzed for Weyl and Dirac fermions in a phase space representation of the worldline formalism. Kinetic theories are constructed for both at a classical level. Whereas the Weyl fermion case reduces in dimension, resembling a theory in quantum mechanics, the Dirac fermion case takes on a manifestly Lorentz covariant form. To achieve a classical kinetic theory for the non-Abelian Dirac fermion Berry phase a spinor construction of Barut and Zanghi is utilized. The axial anomaly is also studied at a quantum level. It is found that under an adiabatic approximation, which is necessary for facilitating a classical kinetic theory, the index of the Dirac operator for massless fermions vanishes. Even so, similarities of an axial rotation to an exact non-covariant Berry phase transform are drawn by application of the Fujikawa method to the Barut and Zanghi spinors on the worldline.

Investigation of Pontryagin trace anomaly using Pauli-Villars regularization

In this paper, we investigate the Pontryagin trace anomaly for chiral fermions in a general curved background using Pauli-Villars regularization. We use both Feynman diagram method and Fujikawa's method to calculate the parity-odd contribution (Pontryagin term). Our result indicates that the trace anomaly of energy-momentum tensor for chiral fermions has Pontryagin term P=i1536π2ϵνσκλRρμνσRρμκλ which agrees with the work of Bonora et al. [1].

A note on broken dilatation symmetry in planar noncommutative theory

A riveting study of Ward-Takahashi identities is presented for a broken dilatation or scale symmetry in a generalized quantum Hall system. On using the “Peierls substitution” scheme, it is shown that noncommutativity between spatial coordinates emerges naturally at a large magnetic field limit. Thereafter, we derive a path-integral action for the corresponding noncommutative quantum system and discuss the equivalence between the considered noncommutative system and the generalized Landau problem thus rendering an effective commmutative description. We then derive an expression for the unintegrated scale or dilatation anomaly for the generalized Landau system using Fujikawa's method and is subsequently renormalised. In fact, we identify the Ward-Takahashi identities associated with broken dilatation symmetry to be anomalous which is a purely quantum effect induced from the noncommutative structure between spatial coordinates and therefore non-intuitive in nature.

Beyond symmetries: Anomalies in transverse Ward-Takahashi identities

Anomalies in transverse Ward--Takahashi identities are studied, allowing discussion of the feasibility of anomalies arising in general non-symmetry Ward--Takahashi identities. We adopt the popular Fujikawa's method and rigorous dimensional renormalization to verify the existence of transverse anomalies to one-loop order and any loop order, respectively. The arbitrariness of coefficients of transverse anomalies is revealed, and a way out is also proposed after relating transverse anomalies to Schwinger terms and comparing symmetry and non-symmetry anomalies. Papers that claim the non-existence of transverse anomalies are reviewed to find anomalies hidden in their approaches. The role played by transverse anomalies is discussed.

Anomaly of a gauge theory under rescaling of the fields

We determine the anomaly associated to an arbitrary scaling of the fields in a quantum gauge theory without making use of the Fujikawa method. We show that this anomaly is dependent on the spin term present in the action and at one loop can be directly extracted from the spin contribution to the one loop effective action. Our results can be readily applied to any gauge theory, supersymmetric or not and agree with previous determination for supersymmetric gauge theories based on the Fujikawa method.

On scalaron decay via the trace of energy-momentum tensor

In some inflation scenarios such as R2 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 a 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 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.

Atiyah-Patodi-Singer index theorem for domain-wall fermion Dirac operator

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.

Atiyah-Patodi-Singer index from the domain-wall fermion Dirac operator

The Atiyah-Patodi-Singer(APS) index theorem attracts attention for understanding physics on the surface of materials in topological phases. The mathematical set-up for this theorem is, however, not directly related to the physical fermion system, as it imposes on the fermion fields a non-local boundary condition known as the "APS boundary condition" by hand, which is unlikely to be realized in the materials. In this work, we attempt to reformulate the APS index in a "physicist-friendly" way for a simple set-up with $U(1)$ or $SU(N)$ gauge group on a flat four-dimensional Euclidean space. We find that the same index as APS is obtained from the domain-wall fermion Dirac operator with a local boundary condition, which is naturally given by the kink structure in the mass term. As the boundary condition does not depend on the gauge fields, our new definition of the index is easy to compute with the standard Fujikawa method.

Path-integral approach to scale anomaly at finite temperature

We derive the relativistic thermodynamic scale equation using imaginary-time path integrals, with complex scalar field theory taken as a concrete example. We use Fujikawa's method to derive the scaling anomaly for this system using a matrix regulator. We make a general scaling argument to show how for anomalous systems, the $\beta$ function of the vacuum theory can be derived from measurement of macroscopic thermodynamic parameters.

Path-integral derivation of the nonrelativistic scale anomaly

In this paper we calculate the scale anomaly for a quantum field theoretic 2D-nonrelativistic Bose gas with contact interactions using Fujikawa's method, both in vacuum and in many-body systems. The use of path integrals for these problems is novel and motivated by a recently developed path-integral framework for addressing questions about scaling in these systems. A natural class of regulators is found that produces the correct value of the anomaly traditionally calculated via other methods, e.g., diagrammatically via the $\ensuremath{\beta}$ function.

Chiral viscoelastic response in Weyl semimetals

We investigate the viscoelastic response of the electronic degrees of freedom in three-dimensional Weyl semimetals and obtain the effective action for this response. This action is derived by adapting Fujikawa's method on chiral anomaly in curved space for the most general Hamiltonian of Weyl semimetals. We clarify that the Weyl semimetal phase will give rise to the nontrivial chiral viscoelastic effect due to the perturbations of the tetrad field in the elastic media, which indicates that the crystal dislocations in Weyl semimetals will generate the energy-momentum current along the direction of dislocations which is analogous to the chiral magnetic effect predicted in electromagnetic response. And this nontrivial viscoelastic response can also be used to detect the properties of Weyl semimetals in the future.

An Anomaly Associated with Ward—Takahashi Identity for Pseudo-Tensor Current in QED

We study the quantum anomaly for the transverse Ward—Takahashi relations in the four-dimensional gauge theory QED by using Fujikawa's method in which the anomaly is fundamentally a variation of the functional integral measure under transformation. A regulator which leads to a finite result for the anomaly is proposed. It is shown that a novel anomaly associated with transverse Ward—Takahashi identity of pseudo-tensor current is derived due to a set of infinitesimal transverse transformation of field variables.

Anomalies at finite density and chiral fermions

Using perturbation theory in the Euclidean (imaginary time) formalism as well as the non-perturbative Fujikawa method, we verify that the chiral anomaly equation remains unaffected in the presence of nonzero chemical potential, $\mu$. 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 $\mu$ and compare it with the Bloch-Wettig idea.

Noncommutative [formula omitted] theories, the axial anomaly, Fujikawa's method and the Atiyah–Singer index

Fujikawa's method is employed to compute at first order in the noncommutative parameter the U(1)A anomaly for noncommutative SU(N). We consider the most general Seiberg–Witten map which commutes with hermiticity and complex conjugation and a noncommutative matrix parameter, θμν, which is of “magnetic” type. Our results for SU(N) can be readily generalized to cover the case of general nonsemisimple gauge groups when the symmetric Seiberg–Witten map is used. Connection with the Atiyah–Singer index theorem is also made.

Chiral anomaly for local boundary conditions

It is known that in the zeta function regularization and in the Fujikawa method chiral anomaly is defined through a coefficient in the heat kernel expansion for the Dirac operator. In this paper we apply the heat kernel methods to calculate boundary contributions to the chiral anomaly for local (bag) boundary conditions. As a by-product some new results on the heat trace asymptotics are also obtained.

Nonplanar anomalies in noncommutative gauge theories with adjoint matter fields

Using path integral method (Fujikawa's method) we calculate anomalies in noncommutative gauge theories with fermions in the bi-fundamental and adjoint representations. We find that axial and chiral gauge anomalies coming from non-planar contributions are derived in the low noncommutative momentum limit $\widetilde{p}^{\mu}(\equiv \theta^{\mu\nu}p_{\nu}) \to 0$. The adjoint chiral fermion carries no anomaly in the non-planar sector in $D=4k (k=1,2,...,)$ dimensions. It is naturally shown from the path integral method that anomalies in non-planar sector originate in UV/IR mixing.

Conformal anomalies in noncommutative gauge theories

We calculate conformal anomalies in noncommutative gauge theories by using the path integral method (Fujikawa's method). Along with the axial anomalies and chiral gauge anomalies, conformal anomalies take the form of the straightforward Moyal deformation in the corresponding conformal anomalies in ordinary gauge theories. However, the Moyal star product leads to the difference in the coefficient of the conformal anomalies between noncommutative gauge theories and ordinary gauge theories. The \ensuremath{\beta} (Callan-Symanzik) functions which are evaluated from the coefficient of the conformal anomalies coincide with the result of perturbative analysis.

CHIRAL GAUGE ANOMALIES ON NONCOMMUTATIVE MINKOWSKI SPACE–TIME

Chiral gauge anomalies on noncommutative Minkowski space–time are computed and have their origin elucidated. The consistent form and the covariant form of the anomaly are obtained. Both Fujikawa's method and Feynman diagram techniques are used to carry out the calculations.

OPERATOR REGULARIZATION, SCALE AND CONFORMAL ANOMALIES FOR THE LANDAU PROBLEM

Adopting the Fujikawa method wherein anomalies are calculated with regulated Jacobians, the Schwinger expansion for [Formula: see text] where [Formula: see text] and [Formula: see text] are operators is used herein to obtain the scale and conformal anomalies for the Landau problem. This approach differs from the conventional method in which a change of basis from eigenstates of the regulator to plane waves is adopted; however the anomalies worked out here match exactly with the results obtained earlier by the author with the usual method.