Ward Identities Research Articles

Bootstrapping multi-field inflation: non-Gaussianities from light scalars revisited

Primordial non-Gaussianities from multi-field inflation are a leading target for cosmological observations, because of the possible large correlations generated between long and short distances.These signatures are captured by the local shape of the scalar bispectrum. In this paper, we revisit the nonlinearities of the conversion process from additional light scalars into curvature perturbations during inflation.We provide analytic templates for correlation functions valid at any kinematical configuration, using the cosmological bootstrap as a main computational tool. Our results include the possibility of large breaking of boost symmetry, in the form of small speeds of sound for both the inflaton and the mediators.We consider correlators coming from the tree-level exchange of a massless scalar field.By introducing a late-time cutoff, we identify that the symmetry constraints on the correlators are modified. This leads to anomalous conformal Ward identities, and consequently the bootstrap differential equations acquire a source term that depends on this cutoff.The solutions to the differential equations are scalar seed functions that incorporate these late-time growth effects. Applying weight-shifting operators to auxiliary “seed” functions, we obtain a systematic classification of shapes of non-Gaussianity coming from massless exchange.For theories with de Sitter symmetry, we compare the resulting shapes with the ones obtained via the δN formalism, identifying missing contributions away from the squeezed limit. For boost-breaking scenarios, we derive a novel class of shape functions with phenomenologically distinct features in scale-invariant theories.Specifically, the new shape provides a simple extension of equilateral non-Gaussianity: the signal peaks at a geometric configuration controlled by the ratio of the sound speeds of the mediator and the inflaton.

One-loop fermion-photon vertex in arbitrary gauge and dimensions: A novel approach

We compute a one-loop electron-photon vertex with fully off-shell external momenta in an arbitrary covariant gauge and space-time dimension. There exist numerous efforts in literature where a one-loop off-shell vertex is calculated by employing the standard first-order Feynman rules in different covariant gauges and space-time dimensions of interest. The tensor structure which decomposes this three-point vertex into the components transverse and longitudinal to the photon momentum gets intertwined in this first-order formalism. The Ward-Takahashi identity is explicitly invoked to untangle the two pieces and the results are expressed in a preferred basis of 12 spin amplitudes. We propose a novel approach based upon an efficient combination of the first- and second-order formalisms of quantum electrodynamics to compute this one-loop vertex. Among some conspicuous advantages is the fact that this less known second-order formalism separates the spin and scalar degrees of freedom of an electron interacting electromagnetically. More noticeably, the longitudinal and transverse contributions naturally disentangle from the onset in our approach. Moreover, this decomposition leads to identities between one-loop scalar Feynman integrals with higher powers in the propagators and shifted space-time dimensions that can be used to prove the Ward-Takahashi identity at one-loop order without the need to evaluate any Feynman integral. Additionally, this natural decomposition allows us to establish the gauge independence of the Pauli form factor through explicit cancellations of scalar Feynman integrals that depend on the gauge parameter. These cancellations naturally lead to a compact expression for the Pauli form factor in arbitrary dimensions. Wherever necessary and insightful, we make comparisons with earlier works.

Even the photon propagator must break de Sitter symmetry

The propagator for the massless vector field in de Sitter space cannot maintain de Sitter invariance in the general covariant gauge, except in the exactly transverse limit. This is due to a previously overlooked Ward-Takahashi identity that the propagator must satisfy. We construct the propagator that satisfies all the necessary conditions, and that preserves cosmological symmetries and dilation, but breaks spatial special conformal transformations. Our propagator vanishes in the infrared, contrary to previously reported de Sitter invariant results, which is relevant for loop computations.

1/L2 corrected soft photon theorem from a CFT3 Ward identity

Classical soft theorems applied to probe scattering processes on AdS4 spacetimes predict the existence of perturbative 1/L2 corrections to the soft photon and soft graviton factors of asymptotically flat spacetimes. In this paper, we establish that the 1/L2 corrected soft photon theorem can be derived from a large N CFT3 Ward identity. We derive a perturbed soft photon mode operator on a flat spacetime patch in global AdS4 in terms of an integrated expression of the boundary CFT current. Using the same in the CFT3 Ward identity, we recover the 1/L2 corrected soft photon factor derived from classical soft theorems.

Thermal three-point functions from holographic Schwinger-Keldysh contours

We compute fully retarded scalar three-point functions of holographic CFTs at finite temperature using real-time holography. They describe the nonlinear response of a holographic medium under scalar forcing, and display single and higher-order poles associated to resonant QNM excitations. This involves computing the bulk-to-bulk propagator on a piecewise mixed-signature spacetime, the dual of the Schwinger-Keldysh contour. We show this construction is equivalent to imposing ingoing boundary conditions on a single copy of a black hole spacetime, similar to the case of the two-point function. We also compute retarded scalar correlators with stress-tensor insertions in general CFTs by solving Ward identities on the Schwinger-Keldysh contour.

Momentum space CFT correlators of non-conserved spinning operators

We analyse the 3-point CFT correlators involving non-conserved spinning operators in momentum space. We derive a general expression for the conformal Ward identities defining the 3-point functions involving two generic spin s non-conserved operators and a spin 1 conserved current. We give explicit expressions for the 3-point function when the two non-conserved operators have spins 1 and 2 and generic conformal dimensions. We also systematically analyse the divergences appearing in these 3-point functions when the conformal dimensions of the two non-conserved operators coincide.

Shift operators from the simplex representation in momentum-space CFT

We derive parametric integral representations for the general n-point function of scalar operators in momentum-space conformal field theory. Recently, this was shown to be expressible as a generalised Feynman integral with the topology of an (n − 1)-simplex, featuring an arbitrary function of momentum-space cross ratios. Here, we show all graph polynomials for this integral can be expressed in terms of the first and second minors of the Laplacian matrix for the simplex. Computing the effective resistance between nodes of the corresponding electrical network, an inverse parametrisation is found in terms of the determinant and first minors of the Cayley-Menger matrix. These parametrisations reveal new families of weight-shifting operators, expressible as determinants, that connect n-point functions in spacetime dimensions differing by two. Moreover, the action of all previously known weight-shifting operators preserving the spacetime dimension is manifest. Finally, the new parametric representations enable the validity of the conformal Ward identities to be established directly, without recourse to recursion in the number of points.

Soft gluon theorems in curved spacetime

In this paper, we derive a soft gluon theorem in the near horizon region of the Schwarzschild black hole from the Ward identity of the near horizon large gauge transformation. The flat spacetime soft gluon theorem can be recovered as a limiting case of the curved spacetime.

Effects of different 3D QED vertex Ansätze on the critical coupling

We study the semimetal-insulator phase transition in graphene using a Schwinger-Dyson approach. We consider various forms of vertex Ans\"atze to truncate the hierarchy of Schwinger-Dyson equations. We define a Ball-Chiu-type vertex that truncates the equations without violating gauge invariance. We show that there is a family of these vertices, parametrized by a continuous parameter that we call $a$, all of which satisfy the Ward identity. We have calculated the critical coupling of the phase transition using different values of $a$. We have also tested a common approximation in which only the first term in the Ball-Chiu Ansatz is included. This vertex is independent of $a$, and, although it is not gauge invariant, it has been used many times in the literature because of the numerical simplifications it provides. We have found that, with a one-loop photon polarization tensor, the results obtained for the critical coupling from the truncated vertex and the full vertex with $a=1$ agree very well, but other values of $a$ give significantly different results. We have also done a fully self-consistent calculation, in which the photons are backcoupled to the fermion degrees of freedom, for one choice $a=1$. Our results show that when photon dynamics are correctly taken into account, it is no longer true that the truncated vertex and the full Ball-Chiu vertex with $a=1$ agree well. The conclusion is that traditional vertex truncations do not really make sense in a system that does not respect Lorentz invariance, like graphene, and the need to include vertex contributions self-consistently is likely inescapable.

QMeS-Derivation: Mathematica package for the symbolic derivation of functional equations

We present the Mathematica package QMeS-Derivation, available on GitHub. It derives symbolic functional equations from a given master equation. The latter include functional renormalisation group equations, Dyson-Schwinger equations, Slavnov-Taylor and Ward identities and their modifications in the presence of momentum cutoffs. The modules allow to derive the functional equations, take functional derivatives, trace over field space, apply a given truncation scheme, and do momentum routings while keeping track of prefactors and signs that arise from fermionic commutation relations. The package furthermore contains an installer as well as Mathematica notebooks with showcase examples. Program summaryProgram Title: QMeS-DerivationCPC Library link to program files:https://doi.org/10.17632/dzb2z4tshd.1Developer's repository link:https://github.com/QMeS-toolbox/QMeS-DerivationLicensing provisions: GPLv3Programming language: MathematicaNature of problem: Deriving symbolic functional equations starting from a quantum master equation.Solution method: Taking functional derivatives of the modified Slavnov-Taylor identities, the Dyson-Schwinger, functional renormalisation group equation or user defined master equations, taking the trace in field space and applying a truncation, doing a momentum routing for one-loop diagrams.Additional comments including restrictions and unusual features: QMeS operates theory independent and is based on a small number of rules, i.e. the (anti-)commutation relation of (fermions)bosons and general functional derivative rules.

Electromagnetic form factors and charge radii of pseudoscalar and scalar mesons: A comprehensive contact interaction analysis

We carry out a comprehensive survey of electromagnetic form factors of all light, heavy and heavy-light ground-state pseudoscalar and scalar mesons. Our analysis is based upon a Schwinger-Dyson equations treatment of a vector $\times$ vector contact interaction. It incorporates confinement and ensures axial vector and vector Ward-Takahashi identities are satisfied along with the corresponding corollaries such as the Goldberger-Treiman relations. The algebraic simplicity of the model allows us to compute the form factors at arbitrarily large virtualities of the probing photon momentum squared with relative ease. Wherever possible and insightful, we compare our results for the electromagnetic form factors and the charge radii with those obtained earlier through Schwinger-Dyson equations, lattice and with experimental observations available. We also comment on the scope and shortcomings of the model.

Gauge invariance and finite counterterms in chiral gauge theories

We derive the finite one-loop counterterm required to restore the Ward Identities broken by the regularization scheme in chiral gauge theories. Our result is an analytic expression applicable to a wide class of regularizations satisfying a few general properties. We adopt the background field method, which ensures background gauge invariance in the quantized theory, and focus on renormalizable chiral theories with arbitrary gauge group and fermions in general representations. Our approach can be extended to theories involving scalars, such as the Standard Model, or to non-renormalizable theories, such as the SMEFT. As a concrete application, we work out the finite counterterm at one loop in the Standard Model, within dimensional regularization and the Breitenlohner-Maison-’t Hooft-Veltman prescription for γ5.

Three-point energy correlators and the celestial block expansion

We study the three-point energy correlator (EEEC), defined as a matrix element of a product of three energy detectors at different locations on the celestial sphere. Lorentz symmetry implies that the EEEC can be decomposed into special functions called celestial blocks. We compute three-point celestial blocks in an expansion around the collinear limit, where the three detectors approach each other on the celestial sphere. The leading term is a traditional d – 2-dimensional four-point conformal block, and thus the collinear EEEC behaves like a conformally-invariant four-point function in d – 2 dimensions. We obtain the coefficients of the conformal block decomposition for the collinear EEEC at leading nontrivial order in weakly-coupled \U0001d4a9 = 4 SYM and QCD. These data allow us to make certain all-orders predictions for the collinear EEEC in various kinematic limits, including the OPE limit and the double lightcone limit. We also study Ward identities satisfied by the EEEC and compute contact terms in the EEEC in weakly-coupled \U0001d4a9 = 4 SYM. Finally, we study the celestial block expansion of the EEEC in planar \U0001d4a9 = 4 SYM at strong coupling, determining celestial block coefficients to leading and first subleading order at large λ.

Linear response functions respecting Ward-Takahashi identity and fluctuation-dissipation theorem within the GW approximation

Fundamental equalities, such as the Ward-Takahashi identity (WTI) and the fluctuation-dissipation theorem (FDT), are important in the calculation of the response functions, which are defined as the variations of physical quantities with respect to the external sources. In this paper, the formalism of calculating the response functions according to their original definitions is presented, based on the generalized $GW$ (GGW) method which was developed for the electronic systems including spin-dependent interaction. This formalism automatically ensures the FDT, and is theorectically proved to respect the WTI. By contrast, the commonly used random phase approximation (RPA) within the GGW method violates both the WTI and the FDT, and the Bethe-Salpeter equation (BSE) satisfies the WTI but does not fulfill the FDT. The validity of this methodology is demonstrated on the two-dimensional one-band Hubbard model, and the results show that our formalism makes significant improvements over the RPA formula. Due to the similar computational cost to the BSE, our formalism is expected to be applied to realistic materials.

Semileptonic weak Hamiltonian to in momentum-space subtraction schemes

The CKM unitarity precision test of the Standard Model requires a systematic treatment of electromagnetic corrections for semi-leptonic decays. Electromagnetic corrections renormalize a semi-leptonic four-fermion operator and we calculate the \U0001d4aa(ααs ) perturbative scheme conversion between the scheme and several momentum-space subtraction schemes, which can also be implemented on the lattice. We consider schemes defined by MOM and SMOM kinematics and emphasize the importance of the choice of projector for each scheme. The conventional projector, that has been used in the literature for MOM kinematics, generates an artificial QCD contribution to the conversion factor, which reflects in an unphysical dependence on the lattice matching scale. This can be traced to the violation of a Ward identity that holds in the case QED is neglected. We show how to remedy this by judicious choices of projector. The resulting Wilson coefficients (and operator matrix elements) have greatly reduced scale dependence.

Semileptonic weak Hamiltonian to mathcal{O} (ααs) in momentum-space subtraction schemes

The CKM unitarity precision test of the Standard Model requires a systematic treatment of electromagnetic and strong corrections for semi-leptonic decays. Electromagnetic corrections require the renormalization of a semileptonic four-fermion operator. In this work we calculate the mathcal{O} (ααs) perturbative scheme conversion between the overline{textrm{MS}} scheme and several momentum-space subtraction schemes, which can also be implemented on the lattice. We consider schemes defined by MOM and SMOM kinematics and emphasize the importance of the choice of projector for each case. The conventional projector, that has been used in the literature for MOM kinematics, generates QCD corrections to the conversion factor that do not vanish for α = 0 and which generate an artificial dependence on the lattice matching scale that would only disappear after summing all orders of perturbation theory. This can be traced to the violation of a Ward identity that holds in the α = 0 limit. We show how to remedy this by judicious choices of projector, and define two new schemes overline{textrm{RI}} -MOM and overline{textrm{RI}} -SMOM. We prove that the Wilson coefficients in the new schemes are free from pure QCD contributions, and find that the Wilson coefficients (and operator matrix elements) have greatly reduced scale dependence. Our choice of the overline{textrm{MS}} scheme over the traditional W-mass scheme is motivated by the fact that, besides being more tractable at higher orders, unlike the latter it allows for a transparent separation of scales. We exploit this to obtain renormalization-group-improved leading-log and next-to-leading-log strong corrections to the electromagnetic contributions and study the (QED-induced) dependence on the lattice matching scale.

Patterns of gauge symmetry in the background field method

The correlation functions of Yang–Mills theories formulated in the background field method satisfy linear Slavnov–Taylor identities, which are naive generalizations of simple tree level relations, with no deformations originating from the ghost-sector of the theory. In recent years, a stronger version of these identities has been found to hold at the level of the background gluon self-energy, whose transversality is enforced separately for each special block of diagrams contributing to the gluon Schwinger–Dyson equation. In the present work we demonstrate by means of explicit calculations that the same distinct realization of the Slavnov–Taylor identity persists in the case of the background three-gluon vertex. The analysis is carried out at the level of the exact Schwinger–Dyson equation for this vertex, with no truncations or simplifying assumptions. The demonstration entails the contraction of individual vertex diagrams by the relevant momentum, which activates Slavnov–Taylor identities of vertices and multi-particle kernels nested inside these graphs; the final result emerges by virtue of a multitude of extensive cancellations, without the need of performing explicit integrations. In addition, we point out that background Ward identities amount to replacing derivatives of propagators by zero-momentum background-gluon insertions, in exact analogy to standard properties of Abelian gauge theories. Finally, certain potential applications of these results are briefly discussed.

Anomalies from an effective field theory perspective

The path-integral measure of a gauge-invariant fermion theory is transformed under the chiral transformation and leads to an elegant derivation of the anomalous chiral Ward-Takahashi identities, as we know from the seminal work of Fujikawa. We present in this work an alternative and illuminating way to calculate the Jacobian in the path-integral measure from the covariant derivative expansion technique used in effective field theory. We present several ways to customize the crucial regularization such that the anomaly is located in the desired current, which is unprecedented within the path integral approach. We are then able to derive, in a transparent and unified way, the covariant, consistent, gravitational, and scale anomalies.

Symmetry preserving contact interaction treatment of the kaon

A symmetry preserving regularization procedure for dealing with the contact interaction model is proposed in this work. This regularization procedure follows a series of consistency conditions that are necessary to maintain gauge symmetry. Under this regularization, proofs for the preservation of the Ward-Takahashi identities are given and the loop integrals in the contact interaction model are systematically computed. As an application example, the kaon electromagnetic form factor and ${K}_{l3}$ transition form factor are computed, and self-consistent results are obtained. Since the proposed regularization properly handles the divergence, one is freed from the inconsistencies caused by the regularization and can concentrate more on the physical discussion.

Non-Abelian Gauge Theories with Composite Fields in the Background Field Method

Non-Abelian gauge theories with composite fields are examined in the background field method. Generating functionals of Green’s functions for a Yang–Mills theory with composite and background fields are introduced, including the generating functional of vertex Green’s functions (effective action). The corresponding Ward identities are obtained, and the issue of gauge dependence is investigated. A gauge variation of the effective action is found in terms of a nilpotent operator depending on the composite and background fields. On-shell independence from the choice of gauge fixing for the effective action is established. In the study of the Ward identities and gauge dependence, finite field-dependent BRST transformations with a background field are introduced and employed on a systematic basis. On the one hand, this involves the consideration of (modified) Ward identities with a field-dependent anticommuting parameter, also depending on a non-trivial background. On the other hand, the issue of gauge dependence is studied with reference to a finite variation of the gauge Fermion. The concept of a joint introduction of composite and background fields to non-Abelian gauge theories is exemplified by the Gribov–Zwanziger theory, including the case of a local BRST-invariant horizon, and also by the Volovich–Katanaev model of two-dimensional gravity with dynamical torsion.