<p>We investigate the Ward-Takahashi identity at one-loop in the light-front (LF) formalism for a bound state of fermions. We consider a spinless bound state made up of two fermions in which the Ward-Takahashi identity is satisfied in the covariant formulation. Considering the same system in the light-front formalism, we investigate the proof of Ward-Takahashi identity by integrating the light-front energy component through the identification of the relevant ranges of the longitudinal LF momentum. We elucidate that the pair production diagram plays a crucial role in establishing the Ward-Takahashi identity. We also point out the necessity of taking into account the corresponding zero modes for truly establishing the Ward-Takahashi identity in the LF formalism.</p>

Brazil

Deepesh Bhamre

J.P.B.C. de Melo

Ward-Takahashi identity in the light-front formalism for a bound state of fermions

We derive the new identity in homotopy algebras which directly corresponds to the Schwinger-Dyson equations in quantum field theory. As an application, we derive the Ward-Takahashi identities. We demonstrate that the Ward-Takahashi identities are reproduced in several examples. In general, our formula contains divergence. We mediate this problem by introducing stubs known in the context of string field theory. With the regularization, we can calculate the anomaly such as axial U(1) anomaly in vector-like U(1) gauge theory.

Japan

Keisuke Konosu

Jojiro Totsuka-Yoshinaka

Noether’s theorem and Ward-Takahashi identities from homotopy algebras

This paper aims at investigating the nonperturbative functional renormalization group for tensorial group field theories with nontrivial kinetic action and closure constraint. We consider the quartic melonic just-renormalizable <math><mo>[</mo><mi>U</mi><mo>(</mo><mn>1</mn><mo>)</mo><msup><mo>]</mo><mn>6</mn></msup></math> model and show that due to this closure constraint the first order Ward-Takahashi identity takes the trivial form as for the models with propagators proportional to identity. We then construct the new version of the effective vertex expansion applicable to this class of models, which in particular allows us to close the hierarchical structure of the flow equations in the melonic sector. As a consequence, there are no additional constraints on the flow equations, and then we can focus on the existence of the physical Wilson-Fisher fixed-points in the symmetric phase.

France

Vincent Lahoche

Benin

Bêm-Biéri Barthélemy Natta

Dine Ousmane Samary

No Ward-Takahashi identity violation for Abelian tensorial group field theories with a closure constraint

Anomalies in transverse Ward-Takahashi identities are studied, allowing discussion of the feasibility of anomalies arising in general nonsymmetry Ward-Takahashi identities. We adopt the popular Fujikawa 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 nonsymmetry anomalies. Papers that claim the nonexistence of transverse anomalies are reviewed to find anomalies hidden in their approaches. The role played by transverse anomalies is discussed.

China

Yi-Da Li

Qing Wang

Beyond symmetries: Anomalies in transverse Ward-Takahashi identities

We discuss the relation between the CP violation of the quark mixing and that of the lepton mixing by investigating a CP violating observable, the Jarlskog invariant, as well as the CP violating Dirac phase. The down-type quark mass matrix with three zeros is given in terms of the minimal number of parameters, while the up-type quark mass matrix is diagonal. These quark mass matrices leading to the successful CKM mixing angles and CP violation are embedded in both the Pati-Salam and SU(5) models. The leptonic Jarlskog invariant J  CP  l   (as well as CP violating Dirac phase) is examined for two cases: the neutrino mass matrix is diagonal or non-diagonal, where no additional CP violating phase is introduced apart from the Majorana phases. In the case of the diagonal neutrino mass matrix, the favorable sign of the leptonic CP violation is obtained, however, the magnitude of J  CP  l   is at most   <math> <mi>O</mi> </math>  $$ \mathcal{O} $$ (10−4), which is too small compared with the expected value from the observation −0.02. In the case of the non-diagonal neutrino mass matrix where the tri-bimaximal mixing pattern is taken, we obtain the successful J  CP  l   up to its sign.

Yusuke Shimizu

Kenta Takagi

Shunya Takahashi

Morimitsu Tanimoto

Sign of CP violating phase in quarks and leptons

Abstract We revisit the  model for leptons in light of the new result of NuFIT 3.2. We introduce a new flavon  transforming as an  singlet  or , which couples to both charged leptons and neutrinos in next-to-leading-order operators. The model consists of five parameters: the lightest neutrino mass , the vacuum expectation value of , and three CP-violating phases after inputting the experimental values of  and . The model with the  singlet flavon gives the prediction of  around the best fit of NuFIT 3.2 while staying near the maximal mixing of . Inputting the experimental mixing angles with the  error-bar, the Dirac CP-violating phase is clearly predicted to be –, which will be tested by the precise observed value in the future. In order to get the best-fit value , the sum of three neutrino masses is predicted to be larger than meV. The cosmological observation for the sum of the neutrino masses will also provide a crucial test of our predictions. It is remarked that the model is consistent with the experimental data only for the normal hierarchy of neutrino masses.

South Korea

Sin Kyu Kang

Revisiting the model for leptons in light of NuFIT 3.2 Latest oscillation results from NOvA. An Introduction to Non-Abelian Discrete Symmetries for Particle Physicists

We show that the soft photon, gluon, and graviton theorems can be understood as the Ward-Takahashi identities of large gauge transformation, i.e., diffeomorphism that does not fall off at spatial infinity. We found infinitely many new identities which constrain the higher order soft behavior of the gauge bosons and gravitons in scattering amplitudes of gauge and gravity theories. Diagrammatic representations of these soft theorems are presented.

United States

Yuta Hamada

Gary Shiu

Infinite Set of Soft Theorems in Gauge-Gravity Theories as Ward-Takahashi Identities

We study a model with a down-type SU(2) singlet vector-like quark (VLQ) as a minimal extension of the standard model (SM). In this model, flavor-changing neutral currents (FCNCs) arise at tree level and the unitarity of the Cabibbo–Kobayashi–Maskawa (CKM) matrix does not hold. In this paper, we constrain the FCNC coupling from transitions, especially and processes. In order to analyze these processes we derive an effective Lagrangian that is valid below the electroweak symmetry breaking scale. For this purpose, we first integrate out the VLQ field and derive an effective theory by matching Wilson coefficients up to one-loop level. Using the effective theory, we construct the effective Lagrangian for . It includes the effects of the SM quarks and the violation of CKM unitarity. We show the constraints on the magnitude of the FCNC coupling and its phase by taking account of the current experimental data on , , , and CKM matrix elements, as well as theoretical uncertainties. We find that the constraint from is more stringent than that from ]. We also obtain a bound for the mass of the VLQ and the strength of the Yukawa couplings related to the FCNC coupling of the transition. Using the CKM elements that satisfy the above constraints, we show how the unitarity is violated on the complex plane.

Takuya Morozumi

Hiroyuki Umeeda

Effective theory analysis for vector-like quark model

We study the identity-based solution of Witten’s cubic bosonic open string field theory constructed by Takahashi and Tanimoto, which is claimed to describe the tachyon vacuum. We argue that the observables of the solution coincide with those of the tachyon vacuum using the method proposed by Kishimoto and Takahashi. We also discuss how to treat the kinetic term of the string field theory expanded around it.

Nobuyuki Ishibashi

Comments on Takahashi-Tanimoto’s scalar solution

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.