Transverse Ward-Takahashi Identities Research Articles

The full vertex functions constrained by the longitudinal and transverse Ward–Takahashi identities in massless QED3 theory

In this paper, we find apart from the Ward–Takahashi (WT) identity, the identity between gamma matrices can also constrain the vertex functions in low-dimensional gauge theories. In (1 + 1) dimensions, the identity between gamma matrices gives the identity between vector and axial-vector vertex functions while in (2 + 1) dimensions it leads to the identity between vector and tensor vertex functions. Then, we derive the expressions of the full scalar, vector and tensor vertex functions in (2 + 1) dimensions Quantum Electrodynamics (QED3) by using the longitudinal and transverse WT identities for vector and tensor currents. Furthermore, we find that in the chiral limit with zero fermion masses, the contribution of Wilson line in full vector vertex function is eliminated and the full vector vertex function is strictly expressed in terms of the fermion propagators when using the identity between vector and tensor vertex functions to further constraint the vertex functions.

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.

Transverse Ward-Takahashi identities and full vertex functions in different representations of QED3 * * Supported in part by the National Natural Science Foundation of China (11475085, 11535005, 11690030), the National Major state Basic Research and Development (2016YFE0129300) and the Anhui Provincial Natural Science Foundation (1908085MA15)

We derive the transverse Ward-Takahashi identities (WTI) of N-dimensional quantum electrodynamics by means of the canonical quantization method and the path integration method, and subsequently attempt to prove that QED3 is solvable based on the transverse and longitudinal WTI, indicating that the full vector and tensor vertices functions can be expressed in terms of the fermion propagators in QED3. Further, we discuss the effect of different γ matrix representations on the full vertex function.

The constraint equation of the vertex function of quantum anomaly and Dyson–Schwinger equations in massless QED2 theory

In this paper, we calculate the quantum anomaly for the longitudinal and the transverse Ward–Takahashi (WT) identities for vector and axial-vector currents in QED2 theory by means of the point-splitting method. It is found that the longitudinal WT identity for vector current and transverse WT identity for axial-vector current have no anomaly while the longitudinal WT identity for axial-vector current and the transverse WT identity for vector current have anomaly in QED2 theory. Moreover, we study the four WT identities in massless QED2 theory and get the result that the four WT identities together give the constraint equation of the vertex function of quantum anomaly. At last, we discuss the Dyson–Schwinger equations in massless QED2 theory. It is found that the vertex function of the quantum anomaly has corrections for the fermion propagator and Schwinger model.

Topological index associated with transverse axial vector and vector anomalies in QED**Supported by National Natural Science Foundation of China (10775059)

It is shown that a novel anomaly associated with transverse Ward-Takahashi identity exists for a pseudo-tensor current in QED, and the anomaly gives rise to a topological index of a Dirac operator in terms of an Atiyah-Singer index theorem.

Path-Integral Derivation of the Transverse Axial Vector and Vector Anomalies in QED

It is shown that a novel anomaly associated with transverse Ward-Takahashi identity of axial vector current in QED is derived by using Fujikawa’s method in the path-integral formulation of quantum field theory. Also it is verified that there is no transverse anomaly for the vector current.

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.

Transverse symmetry transformations and the quark-gluon vertex function in QCD

The transverse symmetry transformations associated with the normal symmetry transformations in gauge theories are introduced, which at first are used to reproduce the transverse Ward-Takahashi identities in the Abelian theory QED. Then the transverse symmetry transformations associated with the BRST symmetry and chiral transformations in the non-Abelian theory QCD are used to derive the transverse Slavnov-Taylor identities for the vector and axial-vector quark-gluon vertices, respectively. Based on the set of normal and transverse Slavnov-Taylor identities, an expression of the quark-gluon vertex function is derived, which describes the constraints on the structure of the quark-gluon vertex imposed from the underlying gauge symmetry of QCD alone. Its role in the study of the Dyson-Schwinger equation for the quark propagator in QCD is discussed.

Transverse Ward–Takahashi identity for the fermion-boson vertex in gauge theories

The transverse Ward–Takahashi (W-T) identity for the fermion-gauge-boson vertex in gauge theories is derived by calculating the curl of the time-ordered product of the three-point function including the vector current operator. This provides the constraint on the transverse part of the vertex. By combining the transverse and normal (longitudinal) W-T identity, we obtain the full vector-vertex in terms of the axial-vector and tensor vertices.

Longitudinal and Transverse Ward–Takahashi Identities, Anomaly and Schwinger–Dyson Equation

Based on the path integral formalism, we rederive and extend the transverse Ward–Takahashi identities (which were first derived by Yasushi Takahashi) for the vector and the axial vector currents and simultaneously discuss the possible quantum anomaly for them. Subsequently, we propose a new scheme for writing down and solving the Schwinger–Dyson equation in which the transverse Ward–Takahashi identity together with the usual (longitudinal) Ward–Takahashi identity are applied to specify the fermion–boson vertex function. Within this framework, we give an example of exactly soluble truncated Schwinger–Dyson equation for the fermion propagator in an Abelian gauge theory in arbitrary dimension when the bare fermion mass is zero. It is especially shown that in two dimensions, it becomes the exact and closed Schwinger–Dyson equation which can be exactly solved.