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

1997 ◽  
Vol 12 (31) ◽  
pp. 5651-5686 ◽  
Author(s):  
Kei-Ichi Kondo
Keyword(s):  
Arbitrary Dimension ◽  
Axial Vector ◽  
Dyson Equation ◽  
Two Dimensions ◽  
Fermion Mass ◽  
Fermion Propagator ◽  
Integral Formalism ◽  
Abelian Gauge ◽  
Abelian Gauge Theory ◽  
Quantum Anomaly

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.

scholarly journals Electroweak SU(2)L × U(1)Y model with strong spontaneously fermion-mass-generating gauge dynamics

2021 ◽  
Vol 2021 (3) ◽  
Author(s):  
Petr Beneš ◽  
Jiří Hošek ◽  
Adam Smetana
Keyword(s):  
Chiral Symmetry Breaking ◽  
Axial Vector ◽  
Higgs Sector ◽  
Dyson Equation ◽  
Mass Splitting ◽  
Fermion Mass ◽  
Goldstone Bosons ◽  
The Standard Model ◽  
Exploratory Stage ◽  
The Masses

Abstract Higgs sector of the Standard model (SM) is replaced by quantum flavor dynamics (QFD), the gauged flavor SU(3)f symmetry with scale Λ. Anomaly freedom requires addition of three νR. The approximate QFD Schwinger-Dyson equation for the Euclidean infrared fermion self-energies Σf(p2) has the spontaneous-chiral-symmetry-breaking solutions ideal for seesaw: (1) Σf(p2) = $$ {M}_{fR}^2/p $$ M fR 2 / p where three Majorana masses MfR of νfR are of order Λ. (2) Σf(p2) = $$ {m}_f^2/p $$ m f 2 / p where three Dirac masses mf = m(0)1 + m(3)λ3 + m(8)λ8 of SM fermions are exponentially suppressed w.r.t. Λ, and degenerate for all SM fermions in f. (1) MfR break SU(3)f symmetry completely; m(3), m(8) superimpose the tiny breaking to U(1) × U(1). All flavor gluons thus acquire self-consistently the masses ∼ Λ. (2) All mf break the electroweak SU(2)L × U(1)Y to U(1)em. Symmetry partners of the composite Nambu-Goldstone bosons are the genuine Higgs particles: (1) three νR-composed Higgses χi with masses ∼ Λ. (2) Two new SM-fermion-composed Higgses h3, h8 with masses ∼ m(3), m(8), respectively. (3) The SM-like SM-fermion-composed Higgs h with mass ∼ m(0), the effective Fermi scale. Σf(p2)-dependent vertices in the electroweak Ward-Takahashi identities imply: the axial-vector ones give rise to the W and Z masses at Fermi scale. The polar-vector ones give rise to the fermion mass splitting in f. At the present exploratory stage the splitting comes out unrealistic.

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

2020 ◽  
Vol 35 (18) ◽  
pp. 2050146
Author(s):  
Yang Yu ◽  
Jian-Feng Li
Keyword(s):  
Vertex Function ◽  
Axial Vector ◽  
Splitting Method ◽  
Vector Current ◽  
Constraint Equation ◽  
Axial Vector Current ◽  
Fermion Propagator ◽  
Schwinger Model ◽  
Quantum Anomaly ◽  
Point Splitting

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.

TRANSVERSE WARD–TAKAHASHI RELATION FOR THE FERMION–BOSON VERTEX FUNCTION IN FOUR-DIMENSIONAL ABELIAN GAUGE THEORY

2007 ◽  
Vol 22 (11) ◽  
pp. 2119-2132 ◽  
Author(s):  
HAN-XIN HE
Keyword(s):  
Gauge Theory ◽  
Momentum Space ◽  
Vertex Function ◽  
Axial Vector ◽  
Space Representation ◽  
Nonlocal Term ◽  
Loop Order ◽  
Abelian Gauge ◽  
Abelian Gauge Theory ◽  
Four Dimensions

In this paper, we give a general formula of the transverse Ward–Takahashi relation for the fermion–boson vertex function in momentum space in four-dimensional Abelian gauge theory, with the complete expression of the integral-term involving the nonlocal axial-vector vertex function, from which the corresponding one-loop expression is derived straightforwardly. Then we deduce carefully this transverse Ward–Takahashi relation to one-loop order in d dimensions, with d = 4 + ∊. The result shows that this relation holds in d dimensions if the combination ελμνργργ5 in d dimensions is replaced with -½{γλ, σμν} by the identity of γ-matrices, and there is no additional piece proportional to (d-4) to include for this relation to hold in four dimensions. This result is further confirmed by an explicit computation of terms in this transverse WT relation to one-loop order. We also discuss the momentum space representation of the nonlocal term in the transverse WT relation.

scholarly journals HOW TO SOLVE QUANTUM NONLINEAR ABELIAN GAUGE THEORY IN TWO DIMENSIONS IN THE HEISENBERG PICTURE

1998 ◽  
Vol 13 (19) ◽  
pp. 3245-3254 ◽  
Author(s):  
NORIAKI IKEDA
Keyword(s):  
Gauge Theory ◽  
Canonical Quantization ◽  
Gravitational Theory ◽  
Two Dimensions ◽  
Field Theories ◽  
Quantum Field Theories ◽  
Operator Formalism ◽  
Abelian Gauge ◽  
Abelian Gauge Theory ◽  
Heisenberg Picture

The new method based on the operator formalism proposed by Abe and Nakanishi is applied to the quantum nonlinear Abelian gauge theory in two dimensions. The soluble models in this method are extended to a wider class of quantum field theories. We obtain the exact solution in the canonical-quantization operator formalism in the Heisenberg picture, so this analysis might shed some light on the analysis of gravitational theory and nonpolynomial field theories.

scholarly journals INFRARED DIVERGENCE IN QED3 AT FINITE TEMPERATURE

1999 ◽  
Vol 14 (18) ◽  
pp. 2921-2947 ◽  
Author(s):  
DOMINIC LEE ◽  
GEORGIOS METIKAS
Keyword(s):  
Finite Temperature ◽  
Statistical Physics ◽  
Order Term ◽  
Dyson Equation ◽  
Infrared Divergence ◽  
Fermion Mass ◽  
Photon Propagator ◽  
Transverse Part ◽  
Transverse Photon ◽  
Range Of Values

We consider various ways of treating the infrared divergence which appears in the dynamically generated fermion mass, when the transverse part of the photon propagator in N flavour QED 3 at finite temperature is included in the Matsubara formalism. This divergence is likely to be an artifact of taking into account only the leading order term in the [Formula: see text] expansion when we calculate the photon propagator and is handled here phenomenologically by means of an infrared cutoff. Inserting both the longitudinal and the transverse part of the photon propagator in the Schwinger–Dyson equation, we find the dependence of the dynamically generated fermion mass on the temperature and the cutoff parameters. It turns out that consistency with certain statistical physics arguments imposes conditions on the cutoff parameters. For parameters in the allowed range of values we find that the ratio r=2* Mass (T=0)/critical temperature is approximately 6, consistent with previous calculations which neglected the transverse photon contribution.

Regge behavior of spontaneously broken non-Abelian gauge theory

1978 ◽  
Vol 17 (2) ◽  
pp. 585-597 ◽  
Author(s):  
J. B. Bronzan ◽  
R. L. Sugar
Keyword(s):  
Gauge Theory ◽  
Abelian Gauge ◽  
Abelian Gauge Theory ◽  
Regge Behavior

scholarly journals Static force potential of a non-Abelian gauge theory in a finite box in Coulomb gauge

2021 ◽  
Vol 103 (5) ◽  
Author(s):  
Tomohiro Furukawa ◽  
Keiichi Ishibashi ◽  
H. Itoyama ◽  
Satoshi Kambayashi
Keyword(s):  
Gauge Theory ◽  
Coulomb Gauge ◽  
Static Force ◽  
Abelian Gauge ◽  
Abelian Gauge Theory ◽  
Force Potential

Non-Abelian gauge theory from the Poisson bracket

2018 ◽  
Vol 33 (30) ◽  
pp. 1850182
Author(s):  
Mu Yi Chen ◽  
Su-Long Nyeo
Keyword(s):  
Gauge Theory ◽  
Poisson Bracket ◽  
Commutation Relation ◽  
Maxwell Equations ◽  
Dynamical Variable ◽  
Gauge Fields ◽  
Abelian Gauge ◽  
Abelian Gauge Theory ◽  
Generalized Poisson ◽  
Lie Algebraic Structure

The Hamiltonian of a nonrelativistic particle coupled to non-Abelian gauge fields is defined to construct a non-Abelian gauge theory. The Hamiltonian which includes isospin as a dynamical variable dictates the dynamics of the particle and isospin according to the Poisson bracket that incorporates the Lie algebraic structure of isospin. The generalized Poisson bracket allows us to derive Wong’s equations, which describe the dynamics of isospin, and the homogeneous (sourceless) equations for non-Abelian gauge fields by following Feynman’s proof of the homogeneous Maxwell equations.It is shown that the derivation of the homogeneous equations for non-Abelian gauge fields using the generalized Poisson bracket does not require that Wong’s equations be defined in the time-axial gauge, which was used with the commutation relation. The homogeneous equations derived by using the commutation relation are not Galilean and Lorentz invariant. However, by using the generalized Poisson bracket, it can be shown that the homogeneous equations are not only Galilean and Lorentz invariant but also gauge independent. In addition, the quantum ordering ambiguity that arises from using the commutation relation can be avoided when using the Poisson bracket.From the homogeneous equations, which define the “electric field” and “magnetic field” in terms of non-Abelian gauge fields, we construct the gauge and Lorentz invariant Lagrangian density and derive the inhomogeneous equations that describe the interaction of non-Abelian gauge fields with a particle.

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