Chiral Ward-Takahashi Identities Research Articles

Anomalies from an effective field theory perspective

The path-integral measure of a gauge-invariant fermion theory is transformed under the chiral transformation and leads to an elegant derivation of the anomalous chiral Ward-Takahashi identities, as we know from the seminal work of Fujikawa. We present in this work an alternative and illuminating way to calculate the Jacobian in the path-integral measure from the covariant derivative expansion technique used in effective field theory. We present several ways to customize the crucial regularization such that the anomaly is located in the desired current, which is unprecedented within the path integral approach. We are then able to derive, in a transparent and unified way, the covariant, consistent, gravitational, and scale anomalies.

Topological susceptibility in finite temperature ( 2+1 )-flavor QCD using gradient flow

We compute the topological charge and its susceptibility in finite temperature (2+1)-flavor QCD on the lattice applying a gradient flow method. With the Iwasaki gauge action and nonperturbatively $O(a)$-improved Wilson quarks, we perform simulations on a fine lattice with~$a\simeq0.07\,\mathrm{fm}$ at a heavy $u$, $d$ quark mass with $m_\pi/m_\rho\simeq0.63$ but approximately physical $s$ quark mass with $m_{\eta_{ss}}/m_\phi\simeq0.74$. In a temperature range from~$T\simeq174\,\mathrm{MeV}$ ($N_t=16$) to $697\,\mathrm{MeV}$ ($N_t=4$), we study two topics on the topological susceptibility. One is a comparison of gluonic and fermionic definitions of the topological susceptibility. Because the two definitions are related by chiral Ward-Takahashi identities, their equivalence is not trivial for lattice quarks which violate the chiral symmetry explicitly at finite lattice spacings. The gradient flow method enables us to compute them without being bothered by the chiral violation. We find a good agreement between the two definitions with Wilson quarks. The other is a comparison with a prediction of the dilute instanton gas approximation, which is relevant in a study of axions as a candidate of the dark matter in the evolution of the Universe. We find that the topological susceptibility shows a decrease in $T$ which is consistent with the predicted $\chi_\mathrm{t}(T) \propto (T/T_{\rm pc})^{-8}$ for three-flavor QCD even at low temperature $T_{\rm pc} < T\le1.5 T_{\rm pc}$.

Pion masses in two-flavor QCD with η condensation.

We investigate some aspects of two-flavor QCD with mu≠md at low energy, using the leading order chiral perturbation theory including anomaly effects. While nothing special happens at mu=0 for the fixed md≠0, the neutral pion mass becomes zero at two critical values of mu, between which the neutral pion field condenses, leading to a spontaneously CP broken phase, the so-called Dashen phase. We also show that the "topological susceptibility" in the chiral perturbation theory diverges at these two critical points. We briefly discuss a possibility that mu=0 can be defined by the vanishing the "topological susceptibility. We finally analyze the case of mu=md=m with θ=π, which is equivalent to mu=-md=-m with θ=0 by the chiral rotation. In this case, the η condensation occurs at small m, violating the CP symmetry spontaneously. Deep in the η condensation phase, three pions become Nambu-Goldstone bosons, but they show unorthodox behavior at small m that mπ2=O(m2), which, however, is shown to be consistent with the chiral Ward-Takahashi identities.

Vector and axial currents in Wilson chiral perturbation theory

We reconsider the construction and matching of the vector and axial currents in Wilson Chiral Perturbation Theory (WChPT), the low-energy effective theory for lattice QCD with Wilson fermions. In particular, we discuss in detail the impact of the finite renormalization of the currents on their matching from the lattice theory to WChPT.We explicitly show that imposing chiral Ward-Takahashi identities on the currents leads, in general, to additional terms of O(a) in the axial current. We illustrate the impact on physical quantities by computing the pion decay constant to one-loop order in the two flavor theory. Our result differs from previously published ones.

Chiral symmetry in light-cone field theory

An analysis of spontaneously broken chiral symmetry in light-cone field theory is presented. The non-locality inherent to light-cone field theory requires revision of the standard procedure in the derivation of Ward-Takahashi identities. We derive the general structure of chiral Ward-Takahashi identities and construct them explicitly for various model field theories. Gell-Mann-Oakes-Renner relations and relations between fermion propagators and the structure functions of Nambu-Goldstone bosons are discussed and the necessary modifications of the Ward-Takahashi identities due to the axial anomaly are indicated.

Chiral anomaly on a fuzzy 2-sphere

We investigate chiral anomaly for fermions in fundamental representation on noncommutative (fuzzy) 2-sphere. In spite that this system is realized by finite dimensional matrices and no regularization is necessary for either UV or IR, we can reproduce the correct chiral anomaly which is consistent with the calculations done in flat noncommutative space. Like the flat case, there are ambiguities to define chiral currents. We define chiral currents in a gauge-invariant way and a gauge-covariant way, and show that the corresponding anomalous chiral Ward-Takahashi identities take different forms. The Ward-Takahashi identity for the gauge-invariant current contains explicit nonlocality while that for the covariant one is given by a local expression.

Implications of a new effective chiral meson Lagrangian

Based on a recently derived effective chiral meson Lagrangian from the extended SU(2)⊗ SU(2) Nambu–Jona-Lasinio (ENJL) model, in the linear realization of chiral symmetry, we extract to leading order in the 1/ N c expansion all associated relevant three-point functions ρ→ ππ, σ→ ππ, a 1→ ρπ, a 1→ σπ, as well as the amplitude for ππ scattering. We discuss the formal differences of these amplitudes as compared with those derived in the literature and calculate the associated decay widths and scattering parameters. The differences have two origins: 1. new terms, which are proportional to the current quark mass and arise from taking the correct NJL vacuum from the first steps in a proper-time expansion, are present in the Lagrangian; 2. an implemented chiral covariant treatment of the diagonalization in the pseudoscalar–axial-vector sector induces new couplings between three or more mesonic fields. Both effects have been derived from the chiral Ward Takahashi identities, which are fully taken into account at each order of the proper-time expansion.

Linear Model in the Gaussian Functional Approximation

We apply a self-consistent relativistic mean-field variational “Gaussian functional” (or Hartree) approximation to the linear σ model with spontaneously and explicitly broken chiral O(4) symmetry. We set up the self-consistency, or “gap” and the Bethe-Salpeter equations. We check and confirm the chiral Ward-Takahashi identities, among them the Nambu-Goldstone theorem and the (partial) axial current conservation [CAC], both in and away from the chiral limit. With explicit chiral symmetry breaking, we confirm the Dashen relation for the pion mass and partial CAC. We solve numerically the gap and Bethe-Salpeter equations, discuss the solutions’ properties and the particle content of the theory.

Ward-Takahashi identities at finite temperature and phase structure in (2+1)-dimensional chiral Gross-Neveu model.

Chiral Ward-Takahashi identities with composite fields are generalized to finite temperature and applied to investigate the chiral phase transition and phase structure in the (2+1)-dimensional chiral Gross-Neveu model. In terms of these identities, the mass spectra of fermions and bound states and the Goldberger-Treiman relation at finite temperature are obtained. The vertex correction between the fermion and bound states [sigma] is evaluated beyond the leading order in the 1/[ital N] expansion at zero and finite temperatures. With the aid of the gap equation derived from Ward-Takahashi identities, the phase structure is discussed at zero and finite temperatures. It turns out that (i) at zero temperature, the vertex correction is very small and its influence on the phase structure can be neglected and (ii) at nonzero temperature, the infrared divergence in the vertex correction will make the results of the chiral phase transition obtained at the leading order invalid in next to the leading order and the phase structure is in agreement with Coleman's theorem.

Chiral Ward-Takahashi identities and mass spectra in (2+1)-dimensional chiral Gross-Neveu model.

Based on chiral Ward-Takahashi identities with composite fields, we develop a method to investigate fermion mass generation and bound-state spectra in the (2+1)-dimensional chiral Gross-Neveu model.

Path integral for gauge theories with fermions

The Atiyah-Singer index theorem indicates that a naive unitary transformation of basis vectors for fermions interacting with gauge fields is not allowed in general. On the basis of this observation, it was previously shown that the path-integral measure of a gauge-invariant fermion theory is transformed nontrivially under the chiral transformation, and thus leads to a simple derivation of "anomalous" chiral Ward-Takahashi identities. We here clarify some of the technical aspects associated with the discussion. It is shown that the Jacobian factor in the path-integral measure, which corresponds to the Adler-Bell-Jackiw anomaly, is independent of any smooth regularization procedure of large eigenvalues of $D$ in Euclidean theory; this property holds in any even-dimensional space-time and also for the gravitational anomaly. The appearance of the anomaly and its connection with the index theorem are thus related to the fact that the primary importance is attached to the Lorentz-covariant "energy" operator $D$ and that $D$ and ${\ensuremath{\gamma}}_{5}$ do not commute. The abnormal behavior of the path-integral measure at the zero-frequency sector in the presence of instantons and its connection with spontaneous symmetry breaking is also clarified. We comment on several other problems associated with the anomaly and on the Pauli-Villars regularization method.