The $SU(N)$ Yang-Mills matrix model admits self-dual and anti-self-dual instantons. When coupled to $N_f$ flavors of massless quarks, the Euclidean Dirac equation in an instanton background has $n_+$ positive and $n_-$ negative chirality zero modes. We show that the index $(n_+ - n_-)$ is equal to a suitably defined instanton charge. Further, we show that the path integral measure is not invariant under a chiral rotation, and relate the non-invariance of the measure to the index of the Dirac operator. Axial symmetry is broken anomalously, with the residual symmetry being a finite group. For $N_f$ fundamental fermions, this residual symmetry is $\mathbb{Z}_{2N_f}$, whereas for adjoint quarks it is $\mathbb{Z}_{4N_f}$.
We study the possible quantum anomaly for the transverse Ward-Takahashi relations in four dimensional gauge theories based on the method of computing the axial-vector and the vector current operator equations. In addition to the well-known anomalous axial-vector divergence equation (the Adler-Bell-Jackiw anomaly), we find the anomalous axial-vector curl equation, which leads to the quantum anomaly of the transverse Ward-Takahashi relation for the axial-vector vertex. The computation shows that there is no anomaly for the transverse Ward-Takahashi relation for the vector vertex.
The Adler-Bell-Jackiw (ABJ) anomaly of a 3+1 dimensional QED is calculated in the presence of a strong magnetic field. It is shown that in the regime with the lowest Landau level (LLL) dominance a dimensional reduction from D=4 to D=2 dimensions occurs in the longitudinal sector of the low energy effective field theory. In the chiral limit, the resulting anomaly is therefore comparable with the axial anomaly of a two dimensional massless Schwinger model. It is further shown that the U(1) axial anomaly of QED in a strong magnetic field is closely related to the ``nonplanar'' axial anomaly of a …
The gradient flow exact renormalization group (GFERG) is a formulation of the exact renormalization group that keeps exact gauge invariance. GFERG can keep also modified chiral invariance. We will show that this formulation reproduces the correct axial anomaly in four dimensions.
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.
Imposing the conservation equation of the vector current for a fermion of spin $\frac{1}{2}$ at the quantum level, a gauge anomaly for the fermion coupling with non-Abelian vector and axial--vector fields in six--dimensional curved space is expressed in tensorial form. The anomaly consists of terms that resemble the chiral U(1) anomaly and the commutator terms that disappear if the axial--vector field is Abelian.
In the framework of a gauge invariant continuous and non-perturbative regularization scheme based on the smearing of point like interactions by means of cutoff functions, we show that the axial anomaly, though cutoff independent, depends on the shape of the cutoff functions. The standard value for the strength of the axial anomaly is recovered if we assume that the regularized gauge invariant axial current is in addition local.
We studied the Adler-Bardeen-Bell-Jackiw anomaly in the context of a finite chiral quark model known as the Spectral Quark Model. Within this model, we obtain the general non-local form of the axial vertex compatible with a non vanishing axial coupling, in the chiral limit. The triangle anomaly is computed and we show that the obtained dependence of the axial vertex with the spectral mass is necessary to ensure both finiteness and the correct violation of the chiral Ward-Takahashi identity.
In this work we consider the 2-point Green's functions in (1+1) dimensional quantum electrodynamics and show that the correct implementation of analytic regularization gives a gauge invariant result for the vaccum polarization amplitude and the correct coefficient for the axial anomaly.
We calculate the $U(1)$ axial-anomaly in two and four dimensions using a modified path integral method coupled to a Pauli-Villars regulator field in the noncommutative QED. Pauli-Villars regularization method provides us with unambiguous way to connect the modified path integral formalism with perturbative axial Ward identities at each step of calculations.
The axial anomaly of the noncommutative U(1) gauge theory is calculated by a number of methods and compared with the commutative one. It is found to be given by the corresponding Chern class.
We investigate on the plane the axial anomaly for euclidean Dirac fermions in the presence of a background Aharonov--Bohm gauge potential. The non perturbative analysis depends on the self--adjoint extensions of the Dirac operator and the result is shown to be influenced by the actual way of understanding the local axial current. The role of the quantum mechanical parameters involved in the expression for the axial anomaly is discussed. A derivation of the effective action by means of the stereographic projection is also considered.
Using the point-splitting regularisation, we calculate the axial anomaly in an arbitrary even dimensional Non-Commutative (NC) field theory. Our result is (star) gauge invariant in its {\it unintegrated} form, to the leading order in the NC parameter. Exploiting the Seiberg Witten map, this result gets transformed to the familiar Adler-Bell-Jackiw anomaly in ordinary space-time. Furthermore, using this map, we derive an expression for the unintegrated axial anomaly for constant fields in NC space-time, that is valid to all finite orders of the NC parameter.
We construct the three point function involving an axial vector current and two energy-momentum tensors for four dimensional conformal field theories. Conformal symmetry determines the form of this three point function uniquely up to a constant factor if the necessary conservation conditions are imposed. The gravitational axial anomaly present on a curved space background leads to a non-zero contribution for the divergence of the axial current in this three point function even on flat space. Using techniques related to differential regularisation which guarantee that the energy-momentum tensor…
In this note, we revise the problem of the axial anomaly in the framework of very special relativity following Fujikawa's path integral approach. We show nonperturbatively that no VSR contribution is present in the path integral measure in $(3+1)$-dimensional spacetime. Furthermore, we extend our results to $(1+1)$ dimensions, as well as to the two-dimesional curved spacetime.
We re-examine the historically important decay of the neutral pion into two photons. Schwinger's Equivalence Theorem is confirmed. We then consider radiative corrections to the famous Adler-Bell-Jackiw (ABJ) anomaly. The result depends crucially on a physically motivated regularization scheme. Our approach is largely based on Schwinger's source (dispersion) method.
A BPHZ renormalized form for the master equation of the field antifiled (or BV) quantization has recently been proposed by De Jonghe, Paris and Troost. This framework was shown to be very powerful in calculating gauge anomalies. We show here that this equation can also be applied in order to calculate a global anomaly (anomalous divergence of a classically conserved Noether current), considering the case of QED. This way, the fundamental result about the anomalous contribution to the Axial Ward identity in standard QED (where there is no gauge anomaly) is reproduced in this BPHZ regularized BV…
The problem of the higher-loop contributions to the axial anomaly is reexamined by a new method. We demonstrate that these contributions depend on the order of the calculations. If the divergence of the axial current by nonperturbative Fujikawa method is calculated first and then average it over the photon field in the presence of an external photon source, a nonzero contribution is obtained. However perturbative Feynman diagram method has an uncertainty. Depending on the order of the calculations above mentioned or zero results are obtained.
We complete our study of non-Abelian gauge theories in the framework of Epstein-Glaser approach to renormalization theory including in the model an arbitrary number of Dirac Fermions. We consider the consistency of the model up to the third order of the perturbation theory. In the second order we obtain pure group theoretical relations expressing a representation property of the numerical coefficients appearing in the left and right handed components of the interaction Lagrangian. In the third order of the perturbation theory we obtain the the condition of cancellation of the axial anomaly.
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, $θ^{μν}$, 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.
The generalization of the axial anomaly is considered. It is shown that bilocal axial quark operators on the light cone possess, beside the point-like anomaly, also a light-like anomaly. The consequences for the definition of anomaly-free quark distribution functions and the effect of both gluon coefficient function and splitting kernels in polarized deep inelastic scattering are discussed.
We generalize the dispersive approach to axial anomaly by A.D. Dolgov and V.I. Zakharov to a non-Abelian case with arbitrary photon virtualites. We derive the anomaly sum rule for the singlet current and obtain the $π^0,η,η'\rightarrowγγ^*$ transition form factors. Using them, we established the behavior of a nonperturbative gluon matrix element $\langle 0 |G\tilde{G} |γγ^*(q^2) \rangle$ in both spacelike and timelike regions. We found a significant contribution of the non-Abelian axial anomaly to the processes with one virtual photon, comparable to that of the electromagnetic anomaly. The dua…
Massless QED(1+1) - the Schwinger model - is studied in a covariant gauge. The main new ingredient is an operator solution of the Dirac equation expressed directly in terms of the fields present in the Lagrangian. This allows us to study in detail the residual symmetry of the covariant gauge. For comparison, we analyze first an analogous solution in the Thirring-Wess model and its implication for the axial anomaly arising from the necessity to correctly define products of fermion operators via point-splitting. In the Schwinger model, one has to define the currents in a gauge invariant (GI) way…
Axial anomaly of lattice abelian gauge theory in hyper-cubic regular lattice in arbitrary even dimensions is investigated by applying the method of exterior differential calculus. The topological invariance, gauge invariance and locality of the axial anomaly determine the explicit form of the topological part. The anomaly is obtained up to a multiplicative constant for finite lattice spacing and can be interpreted as the Chern character of the abelian lattice gauge theory.
The leading, planar diagrams of the $1/N_c$ expansion and the usual string description suggest that quarks propagate on the boundary of a two-dimensional world surface. We restrict the quarks to the boundary of the world surface by giving them infinitely large mass on the interior of the surface and zero mass on its boundary and show that in this picture the QCD $θ$--vacua can be represented by the self-intersection number (or equivalently by the first Chern number of the normal bundle) of the surface.
We study the axial anomaly defined on a finite-size lattice by using a Dirac operator which obeys the Ginsparg-Wilson relation. When the gauge group is U(1), we show that the basic structure of axial anomaly on the infinite lattice, which can be deduced by a cohomological analysis, persists even on (sufficiently large) finite-size lattices. For non-abelian gauge groups, we propose a conjecture on a possible form of axial anomaly on the infinite lattice, which holds to all orders in perturbation theory. With this conjecture, we show that a structure of the axial anomaly on finite-size lattices …
We consider a Dirac fermion in a metric-axial-tensor (MAT) background. By regulating it with Pauli-Villars fields we analyze and compute its full anomaly structure. Appropriate limits of the MAT background allows to recover the anomalies of Dirac and Weyl fermions in the usual curved spacetime, obtaining in particular the trace anomaly of a chiral fermion, which has been the object of recent analyses.
We evaluate for arbitrary even dimensions the classical continuum limit of the lattice axial anomaly defined by the overlap-Dirac operator. Our calculational scheme is simple and systematic. In particular, a powerful topological argument is utilized to determine the value of a lattice integral involved in the calculation. When the Dirac operator is free of species doubling, the classical continuum limit of the axial anomaly in various dimensions is combined into a form of the Chern character, as expected.
We show that Weyl semimetals exhibit a mixed axial-torsional anomaly in the presence of axial torsion, a concept exclusive of these materials with no known natural fundamental interpretation in terms of the geometry of spacetime. This anomaly implies a nonconservation of the axial current---the difference in current of left- and right-handed chiral fermions---when the torsion of the spacetime in which the Weyl fermions move couples with opposite sign to different chiralities. The anomaly is activated by driving transverse sound waves through a Weyl semimetal with a spatially varying tilted dis…
Axial anomaly and nesting is elucidated in the context of the inhomogeneous chiral phase. Using the Gross-Neveu models in 1+1 dimensions, we shall discuss axial anomaly and nesting from two different points of view: one is homogeneous chiral transition and the other is the Ferrel-Fulde-Larkin-Ovchinnikov (FFLO) state in superconductivity, which are closely related to each other by way of duality. It is shown that axial anomaly leads to a particular kind of the FFLO state within the two dimensional Nambu-Jona Lasinio model, where axial anomaly is manifested in a different mode. Nesting is a dri…
A careful and complete discussion is given of the renormalization of the singlet axial anomaly equation in a vector-like nonabelian gauge theory such as QCD regularized by ordinary dimensional regularization. Pseudotensorial structures are treated with the 't Hooft-Veltman prescription. A general framework for calculations is developed, and subsequently verified by explicit computations through two loops. This is followed by a discussion of the matrix elements obtained.
Motivated by an apparent puzzle of the light-front vacua incompatible with the axial anomaly, we have considered the two-dimensional massless Schwinger model for an arbitrary interpolating angle of the quantization surface. By examining spectral deformation of the Dirac sea under an external electric field semiclassically, we have found that the axial anomaly is quantization angle independent. This indicates an intricate nontrivial vacuum structure present even in the light-front limit.
We study the chiral effective theory in the presence of QCD vortices. Gauge invariance requires novel terms from vortex singularities in the gauged Wess-Zumino-Witten action, which incorporate anomaly induced currents along the vortices. We examine these terms for systems with QCD axial domain-walls bounded by vortices (vortons) under magnetic fields. We discuss how the baryon and the electric charge conservations are satisfied in these systems through interplay between domain-walls and vortices, which manifests Callan-Harvey's mechanism of the anomaly inflow.
The axial anomaly in lattice gauge theories has a topological nature when the Dirac operator satisfies the Ginsparg-Wilson relation. We study the axial anomaly in Abelian gauge theories on an infinite hypercubic lattice by utilizing cohomological arguments. The crucial tool in our approach is the non-commutative differential calculus~(NCDC) which makes the Leibniz rule of exterior derivatives valid on the lattice. The topological nature of the ``Chern character'' on the lattice becomes manifest in the context of NCDC. Our result provides an algebraic proof of Lüscher's theorem for a four-dimen…
The $SU(2)_A \times U(2)_V$-symmetric chiral linear sigma model in the presence of the axial anomaly is studied in the local-potential approximation of the Functional Renormalization Group (FRG). The renormalization group (RG) flow is investigated in a truncation which reproduces recent results for the $U(2)_A \times U(2)_V$-symmetric model in the limit of vanishing axial anomaly strength. We search for the conjectured O(4) fixed point in the presence of the $U(1)_A$ anomaly and analyze its stability properties.
We give a very simple derivation of the Atiyah-Patodi-Singer (APS) index theorem and its small generalization by using the path integral of massless Dirac fermions. It is based on the Fujikawa's argument for the relation between the axial anomaly and the Atiyah-Singer index theorem, and only a minor modification of that argument is sufficient to show the APS index theorem. The key ingredient is the identification of the APS boundary condition and its generalization as physical state vectors in the Hilbert space of the massless fermion theory. The APS $η$-invariant appears as the axial charge o…
In 3+1 dimensional spacetime, two vector gauge anomalies are known: The chiral anomaly and the mixed axial-gravitational anomaly. While the former is well documented and tied to the presence of a magnetic field, the latter instead requires a nonzero spacetime curvature, which has made it rather difficult to study. In this work, we show that a quantum anomaly arises in the second-order electrical response for zero magnetic field, which creates a dc-current that can be either longitudinal or transverse to electric field. Consequently, the continuity equation for the chiral current is not conserv…
The axial anomaly arising from the fermion sector of $\U(N)$ or $\SU(N)$ reduced model is studied under a certain restriction of gauge field configurations (the ``$\U(1)$ embedding'' with $N=L^d$). We use the overlap-Dirac operator and consider how the anomaly changes as a function of a gauge-group representation of the fermion. A simple argument shows that the anomaly vanishes for an irreducible representation expressed by a Young tableau whose number of boxes is a multiple of $L^2$ (such as the adjoint representation) and for a tensor-product of them. We also evaluate the anomaly for general…
The axial anomaly in a quasi-one-dimensional (quasi-1D) chiral p-wave superfluid model, which has a epsilon_{x} p_{x}+i epsilon_{y} p_{y}-wave gap in 2D is studied. The anomaly causes an accumulation of the quasiparticle and a quantized chiral current density in an inhomogeneous magnetic field. These effects are related to the winding number of the gap. By varying the parameters epsilon_{x} and epsilon_{y}, the model could be applicable to Sr2RuO4 near the second superconducting transition point, some quasi-1D organic superconductors and the fractional quantum Hall state at nu = 5 / 2 Landau l…
This work presents a first study of a radiative calculation for the gravitational axial anomaly in the massless Abelian Higgs model. The two loop contribution to the anomalous correlation function of one axial current and two energy-momentum tensors, <A_alpha(z) T_μν(y) T_ρσ(x)>, is computed at an order that involves only internal matter fields. Conformal properties of massless field theories are used in order to perform the Feynman diagram calculations in the coordinate space representation. The two loop contribution is found not to vanish, due to the presence of two independent tensor struct…
We demonstrate that the amplitude $<ργ|\partial_ν(\bar qγ_νγ_5 q)|0>$ does not vanish in the limit of zero quark masses. This represents a new kind of violation of the classical equation of motion for the axial current and should be interpreted as the axial anomaly for bound states. The anomaly emerges in spite of the fact that the one loop integrals are ultraviolet-finite as guaranteed by the presence of the bound-state wave function. As a result, the amplitude behaves like $\sim 1/p^2$ in the limit of a large momentum $p$ of the current. This is to be compared with the amplitude $<γγ|\partia…
We analyze the chiral p-wave superconductor in the low temperature region. The superconductor has a epsilon_{x} p_{x} + i epsilon_{y} p_{y}-wave gap in two dimensional space (2D). Near the second superconducting transition point, the system could be described by a quasi-1D chiral p-wave model in 2D. The axial anomaly occurs in such a model and causes an accumulation of the quasiparticle in an inhomogeneous magnetic field. The effect is related to the winding number of the gap.
The axial anomaly of Ginsparg-Wilson fermion operator $D$ is discussed in general for the operator $R$ which enters the chiral symmetry breaking part in the Ginsparg-Wilson relation. The axial anomaly and the index of $D$ as well as the exact realization of the Atiyah-Singer index theorem on the lattice are determined solely by the topological characteristics of the chirally symmetric operator $D_c$ in the chiral limit $ R \to 0 $.
In order to treat loops in the Lorentz-violating QED model, we present a derivation of the QED axial anomaly that specifically highlights the infrared origin of the effect. This is done using dimensional regularization while treating $γ_5$ as a spectator. This enables us to revisit aspects in the fermionic sector of Lorentz-violating QED which have analogous structure to the axial anomaly. In particular, it is shown that both the Chern-Simons and photon mass operators are not induced in the one loop effective action. At lowest order in the Lorentz-violating parameter, we can define a non-relat…
We study the 2D Vector Meson model introduced by Thirring and Wess, that is to say the Schwinger model with massive photon and massless fermion. We prove, with a renormalization group approach, that the vector and axial Ward identities are broken by the Adler-Bell-Jackiw anomaly; and we rigorously establish three widely believed consequences: a) the interacting meson-meson correlation equals a free boson propagator, though the mass is additively renormalized by the anomaly; b) the anomaly is quadratic in the charge, in agreement with the Adler-Bardeen formula; c) the fermion-fermion correlatio…
The 1+1 dimensional massive Dirac equation is solved exactly in light-cone coordinates for $x^+ > 0$ and $x^- > -L$, in the presence of an arbitrary $x^+$ dependent electric field. Our solution resolves the ambiguity at $p^+ = 0$. We also obtain the one loop rate for pair production for a positive electric field, compute the expectation values of the vector and axial vector currents, and recover the well known anomaly $e^2 E/π$ in the divergence of the latter. A final intriguing result is that the theory seems to exhibit a phase transition in the limit of infinite $L$.
We study the transition form factors of the light mesons in the kinematics, where one photon is real and other is virtual. Using the dispersive approach to axial anomaly we show that the axial anomaly in this case reveals itself as a collective effect of meson spectrum. This allows us to get the relation between possible corrections to continuum and to lower states within QCD method which does not rely on factorization hypothesis. We show, relying on the recent data of BaBar Collaboration, that the relative correction to continuum is quite small, and small correction to continuum can dramatica…
The induced axial current and the chiral anomaly are studied in the normal phase of magnetized relativistic matter. A special attention is paid to the role of the chiral shift parameter Delta, leading to a relative shift of the longitudinal momenta in the dispersion relations of opposite chirality fermions. In the Nambu-Jona-Lasinio model, it is shown directly from the form of the gap equation that Delta necessarily exists in the normal phase in a magnetic field. By making use of the gauge invariant point-splitting regularization, we then show that the presence of Delta essentially modifies th…
We derive the expression of the abelian axial anomaly in the so-called multi-Weyl and triple-point crossing semimetals. No simplifying restrictions are assumed on the symmetry of the spectrum. Three different computation methods are considered: the perturbative quantum field theory procedure which is based on the evaluation of the one-loop Feynman diagrams, the Nielsen-Ninomiya method, and the Atiyah-Singer index argument. It is shown that the functional form of the axial anomaly does not depend on the Lorentz symmetry, but it is determined by the gauge structure group. We discuss the stabilit…
We evaluate the axial anomaly for the general Ginsparg-Wilson fermion operator $D = D_c (\Id + R D_c)^{-1}$ with $R = r \Id$. For any chirally symmetric $D_c$ which in the free fermion limit, is free of species doubling and behaves like $i γ_μp_μ$ as $p \to 0$, the axial anomaly $\tr[γ_5 (R D) (x,x)]$ for U(1) lattice gauge theory with single fermion flavor is equal to $e^2/(32 π^2) ε_{μνλσ} F_{μν}(x) F_{λσ}(x+\hatμ+\hatν)$ plus terms which are higher orders and/or non-perturbative contribtuons. The $F \tilde{F}$ term is r-invariant and has the correct continuum limit.
The conservation of the vector current and the axial anomaly responsible for the $π_0 \to γγ$ decay amplitude are obtained in leading order within the Taylor-Lagrange formulation of fields considered as operator-valued distributions. As for gauge theories, where this formulation eliminates all divergences and preserves gauge-symmetry, it is shown that the different contributions can be evaluated directly in $4$-dimensional space-time, with no restrictions whatsoever on the four-momentum of the internal loop, and without the need to introduce any additional non physical degrees of freedom like …
