Generalized Ward Identities Research Articles

Different versions of soft-photon theorems exemplified at leading and next-to-leading terms for pion-pion and pion-proton scattering

We investigate the photon emission in pion-pion and pion-proton scattering in the soft-photon limit where the photon energy ω→0. The expansions of the π−π0→π−π0γ and the π±p→π±pγ amplitudes, satisfying the energy-momentum relations, to the orders ω−1 and ω0 are derived. We show that these terms can be expressed completely in terms of the on-shell amplitudes for π−π0→π−π0 and π±p→π±p, respectively, and their partial derivatives with respect to s and t. The structure term which is nonsingular for ω→0 is determined to the order ω0 from the gauge-invariance constraint using the generalized Ward identities for pions and the proton. For the reaction π−π0→π−π0γ we discuss in detail the soft-photon theorems in the versions of both Low and Weinberg. We show that these two versions are different and must not be confounded. Weinberg’s version gives the pole term of a Laurent expansion in ω of the amplitude for π−π0→π−π0γ around the phase-space point of zero radiation. Low’s version gives an approximate expression for the above amplitude at a fixed phase-space point, corresponding to nonzero radiation. Clearly, the leading and next-to-leading terms in theses two approaches must be, and are indeed, different. We show their relation. We also discuss the expansions of differential cross sections for π−π0→π−π0γ with respect to ω for ω→0. Published by the American Physical Society 2024

Form factor of the longitudinal component of weak vector current in pion beta decay

The form factor f- of the longitudinal component of the weak vector current in pion β-decay is derived using the generalized Ward identity. The form factor is expressed through the pion mass difference and mean-square radii in the channels with angular momenta J = 1, 0 and isospins T = 1, 2.

Ward identities for scale and special conformal transformations in inflation

We derive the general Ward identities for scale and special conformal transformations in theories of single field inflation. Our analysis is model independent and based on symmetry considerations alone. The identities we obtain are valid to all orders in the slow roll expansion. For special conformal transformations, the Ward identities include a term which is non-linear in the fields that arises due to a compensating spatial reparametrization. Some observational consequences are also discussed.

Longitudinal component of the weak vector current of spin-1/2 nuclear isodoublets

The generalized Ward identity is derived for the vertex function of the weak vector current (VC) of nucleons within an effective theory with account of the isotopic symmetry breaking. The generalized Ward identity establishes a relationship between the form factor of the longitudinal weak VC component and the mass difference and the charge radii of an isotopic doublet on the mass shell. This relationship can be used to evaluate the form factors of both nucleons and nuclei with spin J = 1/2 and isospin I = 1/2. The reactions with large transferred momenta involving isotopic doublets, in which the mass splitting is maximized, offer the greatest promise for measuring the form factor. Numerical estimates of the form factor of the longitudinal weak VC component of a nucleon pair and eight nuclear isodoublets are presented.

Fully developed isotropic turbulence: Symmetries and exact identities.

We consider the regime of fully developed isotropic and homogeneous turbulence of the Navier-Stokes equation with a stochastic forcing. We present two gauge symmetries of the corresponding Navier-Stokes field theory and derive the associated general Ward identities. Furthermore, by introducing a local source bilinear in the velocity field, we show that these symmetries entail an infinite set of exact and local relations between correlation functions. They include in particular the Kármán-Howarth relation and another exact relation for a pressure-velocity correlation function recently derived in G. Falkovich, I. Fouxon, and Y. Oz [J. Fluid Mech. 644, 465 (2010)] that we further generalize.

Finite-frequency current fluctuations and self-consistent perturbation theory for electron transport through a quantum dot

We formulate the problem of electron transport through an interacting quantum dot system in the framework of the self-consistent perturbation theory of the nonequilibrium Green's function and show that the current conservation through the central region is automatically guaranteed owing to the gauge-invariant properties of the resulting Green's functions and the generalized Ward identity. By using a generating functional for the statistics of the nonequilibrium system, we obtain general formulas for calculating the current and the current fluctuations in the presence of arbitrary time-dependent potentials. As an illustration of application, we study the interaction effects on the finite-frequency noise for electron resonant tunneling through an Anderson impurity and obtain an analytical expression for the finite-frequency current noise within the Hartree approximation, which is an extension of the results obtained by Hershfield on zero-frequency shot noise.

Gauge-invariant linear response theory of relativistic Bardeen-Cooper-Schrieffer superfluids

We develop a gauge-invariant linear response theory for relativistic Bardeen-Cooper-Schrieffer (BCS) superfluids based on a consistent-fluctuation-of-the order-parameter (CFOP) approach. The response functions from the CFOP approach satisfy important generalized Ward identities. The gauge invariance of the CFOP theory is a consequence of treating the gauge transformation and the fluctuations of the order parameter on equal footing so collective-mode effects are properly included. We demonstrate that the pole of the response functions is associated with the massless Goldstone boson. Important physical quantities such as the compressibility and superfluid density of relativistic BCS superfluids can also be inferred from our approach. We argue that the contribution from the massless Goldstone boson is crucial in obtaining a consistent expression for the compressibility.

Scalar wave propagation in random amplifying media: Influence of localization effects on length and time scales and threshold behavior

We present a detailed discussion of scalar wave propagation and light intensity transport in three-dimensional random dielectric media with optical gain. The intrinsic length and time scales of such amplifying systems are studied and comprehensively discussed as well as the threshold characteristics of single- and two-particle propagators. Our semianalytical theory is based on a self-consistent Cooperon resummation, representing the repeated self-interference, and incorporates as well optical gain and absorption, modeled in a semianalytical way by a finite imaginary part of the dielectric function. Energy conservation in terms of a generalized Ward identity is taken into account.

Elastic response and Ward identities in stressed nematic elastomers

Nematic elastomers exhibit a rich elastic response to external stresses. Of particular interest is the semisoft response of elastomers with an anisotropy direction (z) frozen in by a double cross-linking process. This response is characterized by a stress-strain curve for stresses along x perpendicular to z that rises initially, exhibits a nearly flat plateau between two critical values of strain, and then rises again. This paper explores elastic response in semisoft elastomers as a function of externally applied strain. It derives general Ward identities for elastic moduli and shows that the elastic modulus measuring response to xz shears vanishes at the boundaries of the semisoft plateau whereas moduli measuring response to shears perpendicular to the xz plane do not. It then calculates all relevant moduli in a simple model of elastomers and verifies the general Ward-identity predictions.

Gauge fixing, BRS invariance and Ward identities for randomly stirred flows

The Galilean invariance of the Navier–Stokes equation is shown to be akin to a global gauge symmetry familiar from quantum field theory. This symmetry leads to a multiple counting of infinitely many inertial reference frames in the path integral approach to randomly stirred fluids. This problem is solved by fixing the gauge, i.e., singling out one reference frame. The gauge fixed theory has an underlying Becchi–Rouet–Stora (BRS) symmetry which leads to the Ward identity relating the exact inverse response and vertex functions. This identification of Galilean invariance as a gauge symmetry is explored in detail, for different gauge choices and by performing a rigorous examination of a discretized version of the theory. The Navier–Stokes equation is also invariant under arbitrary rectilinear frame accelerations, known as extended Galilean invariance (EGI). We gauge fix this extended symmetry and derive the generalized Ward identity that follows from the BRS invariance of the gauge-fixed theory. This new Ward identity reduces to the standard one in the limit of zero acceleration. This gauge-fixing approach unambiguously shows that Galilean invariance and EGI constrain only the zero mode of the vertex but none of the higher wavenumber modes.

Trilinear anomalous gauge interactions from intersecting branes and the neutral currents sector

We present a study of the trilinear gauge interactions in extensions of the Standard Model (SM) with several anomalous extra U(1)'s, identified in various constructions, from special vacua of string theory to large extra dimensions. In these models an axion and generalized Chern-Simons interactions for anomalies cancellation are present. We derive generalized Ward identities for these vertices and discuss their structure in the Stuckelberg and Higgs-Stuckelberg phases. We give their explicit expressions in all the relevant cases, which can be used for explicit phenomenological studies of these models at the LHC.

Thermodynamics and superfluid density in BCS-BEC crossover with and without population imbalance

We address the thermodynamics, density profiles, and superfluid density of trapped fermions undergoing BCS-BEC crossover. We consider the case of zero and finite population imbalance and apply the local density approximation (LDA) to include the trap potential. Our approach represents a fully self-consistent treatment of ``pseudogap effects.'' These effects reflect the distinction between the pair formation temperature ${T}^{*}$ and the pair condensation temperature ${T}_{c}$. As a natural corollary, this temperature difference must be accommodated in the fermionic excitation spectrum ${E}_{\mathbf{k}}$ to reflect the fact that fermions are paired at and above ${T}_{c}$. It is precisely this natural corollary which has been omitted from all other many body approaches in the literature. At a formal level, we show how enforcing this corollary implies that pairing fluctuation or self-energy contributions enter into both the gap and the number equations; this is necessary in order to be consistent with a generalized Ward identity. At a less formal level, we demonstrate that we obtain physical results for the superfluid density ${n}_{s}(T)$ at all temperatures. In contrast, previous work in the literature has led to nonmonotonic, or multivalued or discontinuous behavior for ${n}_{s}(T)$. Because it reflects the essence of the superfluid state, we view the superfluid density as a critical measure of the physicality of a given crossover theory. In a similarly unique fashion, we emphasize that in order to properly address thermodynamic properties of a trapped Fermi gas, a necessary first step is to demonstrate that the particle density profiles are consistent with experiment. Without a careful incorporation of the distinction between the pairing gap and the order parameter, the LDA-derived density profiles (of the unpolarized gas) tend to exhibit sharp features at the condensate edge, which are never seen experimentally in the crossover regime. The lack of demonstrable consistency between theoretical and experimental density profiles, along with problematic behavior found for the superfluid density, casts doubt on previous claims in the literature concerning quantitative agreement between thermodynamical calculations and experiment.

Vertex renormalization of weak interactions and Cooper-pair breaking in cooling compact stars

Below the critical temperature of superfluid phase transition baryonic matter emits neutrinos by breaking and recombination of Cooper pairs formed in the condensate. The weak vector and axial-vector vertices and the neutrino loss rates via pair breaking are modified by strong interactions in nuclear medium. We study these modifications nonperturbatively by summing infinite series of particle-hole loops in $S$-wave superfluid neutron matter. The interactions in the particle-hole channel are described within the Landau Fermi-liquid framework with the Landau parameters derived from the microscopic theory. The $S$-wave superfluid is described within the BCS theory. We derive the renormalized three-point vector and axial-vector vertices and the complete polarization tensor of matter and its low momentum transfer expansion. The leading-order term in this expansion and the associated neutrino losses arise at $O({q}^{2})$, consistent with the $f$-sum rule. The neutrino emission rate due to the pair breaking is parametrically suppressed compared to its one-loop counterpart by the ratio of the neutron recoil energy to the temperature, which is of order $5\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}3}$. The approximations to the normal and anomalous self-energies that guarantee the conformity of the theory with the generalized Ward identities are established.

The condensate 〈(A µ 2 )〉 in commutative and noncommutative theories

It is shown that gauge invariance of the operator \int dx tr(A_{\mu}^{2}-\frac{2}{g \xi} x^{\nu} \theta_{\mu\nu} A^{\mu}) in noncommutative gauge theory does not lead to gauge independence of its vacuum condensate. Generalized Ward identities are obtained for Green's functions involving operator \underset{\Omega \to \infty}{lim}\frac{1}{\Omega} \int\limits_{\Omega} dx tr(A_{\mu}^{2}) in noncommutative and commutative gauge theories.

Theory of strong localization effects of light in disordered loss or gain media

We present a systematical theory for the interplay of strong localization effects and absorption or gain of classical waves in three-dimensional, disordered dielectrics. The theory is based on the self-consistent Cooperon resummation, implementing the effects of energy conservation and its absorptive or emissive corrections by an exact, generalized Ward identity. Substantial renormalizations are found, depending on whether the absorption or gain occurs in the scatterers or in the background medium. We find a finite, gain-induced correlation volume which may be significantly smaller than the scale set by the scattering mean-free path, even if there are no truly localized modes. Possible consequences for coherent feedback in random lasers as well as the possibility of oscillatory in time behavior induced by sufficiently strong gain are discussed.

Symmetries in a Constrained System with a Singular Higher-Order Lagrangian

A simple algorithm to construct the generator of gauge transformation for a constrained canonical system with a singular higher-order Lagrangian in field theories is developed. Based on phase-space generating functional of Green function for such a system, the generalized canonical Ward identities under the non-local transformation have been deduced. For the gauge-invariant system, based on configuration-space generating functional, the generalized Ward identities under the non-local transformation have been also derived.The conservation laws are deduced at the quantum level. The applications of the above results to the gauge invariance massive vector field and non-Abelian Chern–Simons(CS) theories with higher-order derivatives are given, a new form of gauge-ghost proper vertices, and Ward–Takahashi identity under BRS transformation and BRS charge at the quantum level are obtained. In the canonical formulation one does not need to carry out the integration over canonical momenta in phase-space path integral as usually performed.

Generalized Ward identity and gauge invariance of the color-superconducting gap

We derive a generalized Ward identity for color-superconducting quark matter via the functional integral approach. The identity implies the gauge independence of the color-superconducting gap parameter on the quasiparticle mass shell to subleading order in the covariant gauge.

SYMMETRY PRINCIPLE PRESERVING AND INFINITY FREE REGULARIZATION AND RENORMALIZATION OF QUANTUM FIELD THEORIES AND THE MASS GAP

Through defining irreducible loop integrals (ILI's), a set of consistency conditions for the regularized (quadratically and logarithmically) divergent ILI's are obtained to maintain the generalized Ward identities of gauge invariance in non-Abelian gauge theories. The ILI's of arbitrary loop graphs can be evaluated from the corresponding Feynman loop integrals by adopting an ultraviolet (UV) divergence preserving parameter method. Overlapping UV divergences are explicitly shown to be factorizable in the ILI's and be harmless via suitable subtractions. A new regularization and renormalization method is presented in the initial space–time dimension of the theory. The procedure respects unitarity and causality. Of interest, the method leads to an infinity free renormalization and meanwhile maintains the symmetry principles of the original theory except the intrinsic mass scale caused conformal scaling symmetry breaking and the anomaly induced symmetry breaking. Tadpole graphs of Yang–Mills gauge fields are found to play an essential role for maintaining manifest gauge invariance via cancellations of quadratically divergent ILI's. Quantum field theories (QFT's) regularized through the new method are well defined and governed by a physically meaningful characteristic energy scale (CES) Mc and a physically interesting sliding energy scale (SES) μs which can run from μs ~ Mc to a dynamically generated mass gap μs = μc or to μs = 0 in the absence of mass gap and infrared (IR) problem. For Mc → ∞, the initial UV divergent properties of QFT's are recovered and well-defined. In fact, the CES Mc and SES at μs = μc play the role of UV and IR cutoff energy scales respectively. It is strongly indicated that the conformal scaling symmetry and its breaking mechanism play an important role for understanding the mass gap and quark confinement. The new method is developed to be applicable for both underlying renormalizable QFT's and effective QFT's. It also leads to a set of conjectures on mathematically interesting numbers and functional limits which may provide deep insights in mathematics.

Renormalization Group and Ward Identities in Quantum Liquid Phases and in Unconventional Critical Phenomena

By reviewing the application of the renormalization group to different theoretical problems, we emphasize the role played by the general symmetry properties in identifying the relevant running variables describing the behavior of a given physical system. In particular, we show how the constraints due to the Ward identities, which implement the conservation laws associated with the various symmetries, help to minimize the number of independent running variables. This use of the Ward identities is examined both in the case of a stable phase and of a critical phenomenon. In the first case we consider the problems of interacting fermions and bosons. In one dimension general and specific Ward identities are sufficient to show the non-Fermi-liquid character of the interacting fermion system, and also allow to describe the crossover to a Fermi liquid above one dimension. This crossover is examined both in the absence and presence of singular interaction. On the other hand, in the case of interacting bosons in the superfluid phase, the implementation of the Ward identities provides the asymptotically exact description of the acoustic low-energy excitation spectrum, and clarifies the subtle mechanism of how this is realized below and above three dimensions. As a critical phenomenon, we discuss the disorder-driven metal-insulator transition in a disordered interacting Fermi system. In this case, through the use of Ward identities, one is able to associate all the disorder effects to renormalizations of the Landau parameters. As a consequence, the occurrence of a metal-insulator transition is described as a critical breakdown of a Fermi liquid.

Algebraic Aspects of the Background Field Method

The background field method allows the evaluation of the effective action by exploiting the (background) gauge invariance, which in general yields Ward identities, i.e., linear relations among the vertex functions. In the present approach an extra gauge fixing term is introduced right at the beginning in the action and it is chosen in such a way that BRST invariance is preserved. The background effective action is considered and it is shown to satisfy both the Slavnov–Taylor (ST) identities and the Ward identities. This allows the proof of the background equivalence theorem with the standard techniques. In particular we consider a BRST doublet where the background field enters with a non-zero BRST transformation. The rationale behind the introduction of an extra gauge fixing term is that of removing the singularity of the Legendre transform of the background effective action, thus allowing the construction of the connected amplitudes generating functional Wbg. By using the relevant ST identities we show that the functional Wbg gives the same physical amplitudes as the original one we started with. Moreover we show that Wbg cannot in general be derived from a classical action by the Gell–Mann–Low formula. As a final point of the paper we show that the BRST doublet generated from the background field does not modify the anomaly of the original underlying gauge theory. The proof is algebraic and makes no use of arguments based on power-counting.