Takahashi Identities Research Articles

Ward–Green–Takahashi identities and the axial-vector vertex

The colour-singlet axial-vector vertex plays a pivotal role in understanding dynamical chiral symmetry breaking and numerous hadronic weak interactions, yet scant model-independent information is available. We therefore use longitudinal and transverse Ward–Green–Takahashi (WGT) identities, together with kinematic constraints, in order to ameliorate this situation and expose novel features of the axial vertex: amongst them, Ward-like identities for elements in the transverse piece of the vertex, which complement and shed new light on identities determined previously for components in its longitudinal part. Such algebraic results are verified via solutions of the Bethe–Salpeter equation for the axial vertex obtained using two materially different kernels for the relevant Dyson–Schwinger equations. The solutions also provide insights that suggest a practical Ansatz for the axial-vector vertex.

Euclidian Crystals in Many-Body Systems: Breakdown of Goldstone’s Theorem

It is proven via modification of the Nambu–Goldstone theorem, based on the Ward–Takahashi identity that Goldstone’s bosons of the quantum order could be massive. The quantum order parameter (QOP), introduced previously as the instantonic crystal in Euclidian space (Mukhin, J. Supercond. Nov. Magn. 24:1165–1171, 2011), breaks Matsubara’s time translational invariance while possessing zero classical expectation value and vanishing scattering cross-section (“hidden order”). The amplitude of the mass-gap can be calculated from the eigenmodes spectrum of the effective Euclidian action for a particular self-consistent QOP solution in the fermionic repulsive Hubbard model. Theoretical results are discussed in relation with possible linking together of the two phenomena that previously were considered separately, i.e., “hidden order” fingerprints in ARPES measurements and the “neutron resonance” feature in the magnetic spectrum in lightly hole-doped copper oxides.

Minimal Ward—Takahashi Vertices and Light Cone Pion Distribution Amplitudes from the GND Quark Model

The gauge-invariant, nonlocal, dynamical quark model is shown to generate minimal vertices that satisfy the Ward—Takahashi identities and the flat-like form of the light-cone pion distribution amplitudes. Non-flat form amplitudes can be produced only if we take a finite momentum cutoff and include nonzero pion mass corrections or go beyond the minimal vertices. A by-product of our investigation shows that the variable u appearing in light-cone pion distribution amplitudes is just the standard Feynman parameter in the Feynman parameter integrals.

Practical corollaries of transverse Ward–Green–Takahashi identities

The gauge principle is fundamental in formulating the Standard Model. Fermion–gauge-boson couplings are the inescapable consequence and the primary determining factor for observable phenomena. Vertices describing such couplings are simple in perturbation theory and yet the existence of strong-interaction bound-states guarantees that many phenomena within the Model are nonperturbative. It is therefore crucial to understand how dynamics dresses the vertices and thereby fundamentally alters the appearance of fermion–gauge-boson interactions. We consider the coupling of a dressed-fermion to an Abelian gauge boson, and describe a unified treatment and solution of the familiar longitudinal Ward–Green–Takahashi identity and its less well known transverse counterparts. Novel consequences for the dressed-fermion–gauge-boson vertex are exposed.

No one-loop gauge anomalies for a Lorentz-violating quantum electrodynamics: evaluation of the three-photon vertex

Identification of the diagrams that can lead to gauge anomalies in the (minimal) Lorentz- and CPT-violating extension of quantum electrodynamics reveal these to be the electron self-energy and vertex correction (related to the Ward–Takahashi identity), the photon self-energy (related to the vacuum polarization tensor transversality) and the three-photon vertex diagrams. All but the latter were explicitly verified to be free of anomalies to first order in loop expansion. Here we provide this remaining evaluation and verify the absence of anomalies in this process.

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.

Hidden symmetries and equilibrium properties of multiplicative white-noise stochastic processes

Multiplicative white-noise stochastic processes continue to attract attention in a wide area of scientific research. The variety of prescriptions available for defining them makes the development of general tools for their characterization difficult. In this work, we study equilibrium properties of Markovian multiplicative white-noise processes. For this, we define the time reversal transformation for such processes, taking into account that the asymptotic stationary probability distribution depends on the prescription. Representing the stochastic process in a functional Grassmann formalism, we avoid the necessity of fixing a particular prescription. In this framework, we analyze equilibrium properties and study hidden symmetries of the process. We show that, using a careful definition of the equilibrium distribution and taking into account the appropriate time reversal transformation, usual equilibrium properties are satisfied for any prescription. Finally, we present a detailed deduction of a covariant supersymmetric formulation of a multiplicative Markovian white-noise process and study some of the constraints that it imposes on correlation functions using Ward–Takahashi identities.

Supersymmetry, chiral symmetry and the generalized BRS transformation in lattice formulations of 4D [formula omitted] SYM

In the context of the lattice regularization of the four-dimensional N=1 supersymmetric Yang–Mills theory (4D N=1 SYM), we formulate a generalized BRS transformation that treats the gauge, supersymmetry (SUSY), translation and axial U(1) (U(1)A) transformations in a unified way. A resultant Slavnov–Taylor identity or the Zinn-Justin equation gives rise to a strong constraint on the quantum continuum limit of symmetry breaking terms with the lattice regularization. By analyzing the implications of the constraint on operator-mixing coefficients in the SUSY and the U(1)A Ward–Takahashi (WT) identities, we prove to all orders of perturbation theory in the continuum limit that, (i) the chiral symmetric limit implies the supersymmetric limit and, (ii) a three-fermion operator that might potentially give rise to an exotic breaking of the SUSY WT identity does not emerge. In previous literature, only a naive or incomplete treatment on these points can be found. Our results provide a solid theoretical basis for lattice formulations of the 4D N=1 SYM.

Ward–Takahashi identities for the colored Boulatov model

Ward–Takahashi (WT) identities of the colored Boulatov model are derived using a generic unitary field transformation. In a specific instance, this generic transformation turns out to be a symmetry of the interaction so that particular classes of reduced WT identities for that symmetry are consequently identified and interpreted.

STRONG COUPLING LIMITS AND QUANTUM ISOMORPHISMS OF THE GAUGED THIRRING MODEL

We have studied the quantum equivalence in the respective strong coupling limits of the bidimensional gauged Thirring model with both Schwinger and Thirring models. It is achieved following a nonperturbative quantization of the gauged Thirring model into the path-integral approach. First, we have established the constraint structure via the Dirac's formalism for constrained systems and defined the correct vacuum–vacuum transition amplitude by using the Faddeev–Senjanovic method. Next, we have computed exactly the relevant Green's functions and shown the Ward–Takahashi identities. Afterwards, we have established the quantum isomorphisms between gauged Thirring model and both Schwinger and Thirring models by analyzing the respective Green's functions in the strong coupling limits, respectively. A special attention is necessary to establish the quantum isomorphism between the gauged Thirring model and the Thirring model.

A field-theoretic approach to non-equilibrium work identities

We study non-equilibrium work relations for a space-dependent field with stochastic dynamics (model A). Jarzynski's equality is obtained through symmetries of the dynamical action in the path-integral representation. We derive a set of exact identities that generalize the fluctuation–dissipation relations to non-stationary and far-from-equilibrium situations. These identities are prone to experimental verification. Furthermore, we show that a well-studied invariance of the Langevin equation under supersymmetry, which is known to be broken when the external potential is time dependent, can be partially restored by adding to the action a term which is precisely Jarzynski's work. The work identities can then be retrieved as consequences of the associated Ward–Takahashi identities.

Chiral symmetry breaking in monolayer graphene by strong coupling expansion of compact and non-compact U(1) lattice gauge theories

Due to effective enhancement of the Coulomb coupling strength in the vacuum-suspended graphene, the system may turn from a semimetal into an insulator by the formation of a gap in the fermionic spectrum. This phenomenon is analogous to the spontaneous breaking of chiral symmetry in the strong-coupling relativistic field theories. We study this “chiral symmetry breaking” and associated collective excitations on graphene in the strong coupling regime by taking U(1) lattice gauge theory as an effective model for graphene. Both compact and non-compact formulations of the U(1) gauge action show chiral symmetry breaking with equal magnitude of the chiral condensate (exciton condensate) in the strong coupling limit, while they start to deviate from the next-to-leading order in the strong coupling expansion. Phase and amplitude fluctuations of the order parameter are also investigated: in particular, a mass formula for the pseudo-Nambu–Goldstone mode ( π-exciton), which is analogous to Gell–Mann–Oakes–Renner relation for the pion in quantum chromodynamics (QCD), is derived from the axial Ward–Takahashi identity. To check the applicability of the effective field theory description, typical energy scales of fermionic and bosonic excitations are estimated by identifying the lattice spacing of the U(1) gauge theory with that of the original honeycomb lattice of graphene.

Two-body Current Operators and Deuteron Form Factors Within the Light-Front Hamiltonian Dynamics

The deuteron electromagnetic properties are investigated within the Light-Front Hamiltonian dynamics using one-body and two-body currents. The dynamical nature of the latter is generated within a Yukawa model from an analysis of the Light-Front current that acts on the three-dimensional valence component and fulfills the Ward–Takahashi identity. Preliminary results for the magnetic moment and the probability of the nonvalence component are shown.

Bethe–Salpeter equations: mesons beyond the rainbow-ladder truncation

We investigate masses of light mesons from a coupled system of Dyson–Schwinger (DSE) and Bethe–Salpeter equations (BSE), taking into account dominant non-Abelian, sub-leading Abelian, and dominant pion cloud contributions to the dressed quark-gluon vertex. The axial-vector Ward–Takahashi identity preserving Bethe–Salpeter kernel is constructed and the spectrum of light mesons calculated. Our model goes significantly beyond the rainbow-ladder. We find that sub-leading Abelian corrections are further dynamically suppressed, and that our results supersede early qualitative predictions from simple truncation schemes.

The quark number susceptibility in the hard-thermal-loop approximation

With the help of the vector Ward–Takahashi identity we derive a general formula for the quark number susceptibility (QNS) which expresses the QNS as an integral expression only involving the full quark propagator at finite temperature and chemical potential. The QNS at finite temperature and zero chemical potential is calculated with the dressed quark propagator in the hard-thermal-loop (HTL) approximation. A comparison of our result with the results of the QNS in the HTL approximation in previous literatures is given.

SUSY WT identity in a lattice formulation of 2D [formula omitted] SYM

We address some issues relating to a supersymmetric (SUSY) Ward–Takahashi (WT) identity in Sugino's lattice formulation of two-dimensional (2D) N=(2,2)SU(k) supersymmetric Yang–Mills theory (SYM). A perturbative argument shows that the SUSY WT identity in the continuum theory is reproduced in the continuum limit without any operator renormalization/mixing and tuning of lattice parameters. As application of the lattice SUSY WT identity, we show that a prescription for the Hamiltonian density in this lattice formulation, proposed by Kanamori, Sugino and Suzuki, is justified also from a perspective of an operator algebra among correctly-normalized supercurrents. We explicitly confirm the SUSY WT identity in the continuum limit to the first nontrivial order in a semi-perturbative expansion.

Inclusion of vertex corrections for superconductivity in gauge-invariant self-consistent approximations

Based on the Ward–Takahashi identities representing both charge and velocity local conservation laws, we provide a scheme for constructing the vertex function Γˆ in the self-consistent iteration loop to determine the self-energy Σˆ for superconductivity. For a singlet superconductor without spin-dependent interaction, the relationship between Γˆ and Σˆ in this iteration loop is mainly controlled by the electronic compressibility of the system, which is directly related to the velocity of the Nambu-Goldstone mode.

Some issues in a gauge model of unparticles

We address in a recent gauge model of unparticles the issues that are important for consistency of a gauge theory, i.e., unitarity and the Ward identity of the physical amplitudes. We find that non-integrable singularities arise in physical quantities like the cross section and the decay rate from the gauge interactions of unparticles. We also show that the Ward identity is violated due to the lack of a dispersion relation for charged unparticles although the Ward–Takahashi identity for general Green functions is incorporated in the model. A previous observation that the contribution of the unparticle (with scaling dimension d) to the gauge boson self-energy is a factor (2−d) of the particle’s self-energy has been extended to the Green function of triple gauge bosons. This (2−d) rule may be generally true for Green functions for any number of points of the gauge bosons. This implies that the model would be trivial even as one that mimics certain dynamical effects on gauge bosons in which unparticles serve as an interpolating field.

Revisiting the vector and axial-vector vacuum susceptibilities

We re-investigate the vector and axial-vector vacuum susceptibilities by taking advantage of the vector and axial-vector Ward–Takahashi identities. We show analytically that, in the chiral limit, the vector vacuum susceptibility is zero and the axial-vector vacuum susceptibility equals three fourths of the square of the pion decay constant. Besides, our analysis reproduces the Weinberg sum rule.

REMARK ON THE CONSISTENCY OF THE LADDER APPROXIMATION AND THE RAINBOW APPROXIMATION OF DYSON–SCHWINGER EQUATIONS OF QCD

We study the consistency of the ladder approximation and the rainbow approximation of the Dyson–Schwinger equations of QCD. By considering the non-Abelian property of QCD, we show that the QED-type Ward–Takahashi identity is not required for the rainbow–ladder approximation of QCD. It indicates that there does not exist any internal inconsistency in the usual rainbow–ladder approximation of QCD. In addition, we propose a modified ladder approximation which guarantees the Slavnov–Taylor identity for the quark–gluon vertex omitting the ghost effect in the approximation.