We calculate the intrinsic quark spin contribution to the total proton spin using overlap valence quarks on three ensembles of $2+1$-flavor RBC/UKQCD domain-wall configurations with different lattice spacings. The lowest pion mass of the ensembles is around 171 MeV which is close to the physical point. With overlap fermions and topological charge derived from the overlap operator, we verify the anomalous Ward identity between nucleon states with momentum transfer. Both the connected and disconnected insertions of the axial-vector current are calculated. For the disconnected-insertion part, the cluster-decomposition error reduction (CDER) technique is utilized for the lattice with the largest volume and the error can be reduced by $10\%\sim40\%$. Nonperturbative renormalization is carried out and the final results are all reported in the $\overline{\rm MS}$ scheme at 2 GeV. We determine the total quark spin contribution to the nucleon spin to be $ΔΣ=0.401(25)(37)$, which is consistent with the recent global fitting result of experimental data. The isovector axial coupling we obtain in this study is $g_A^3=1.256(16)(30)$, which agrees well with the experimental value of 1.2723(23).
The one-loop corrections to the supersymmetric Ward-Takahashi identity (WTi) are investigated in the off-shell regime in the Wilson formulation of the discretized N=1 Super Yang-Mills (SYM) theory. The study of the continuum limit as well as the renormalization procedure for the supercurrent are presented.
We derive an Abelian-like Ward identity in color superconducting phase and calculate vertex corrections to the color superconducting gap. Making use of the Ward identity, we show that subleading order contributions to the gap from vertices are absent for gapped excitations.
We present a determination of the strange quark mass for two flavours (nf=2) of light dynamical quarks using the axial Ward identity. The calculations are performed on the lattice using O(a) improved Wilson fermions and include a fully non-perturbative determination of the renormalisation constant. In the continuum limit we find in the MSbar scheme at 2GeV, ms = 111(6)(4)(6)MeV using the force scale r0 = 0.467fm, where the first error is statistical, the second and third are systematic due to the fit and scale uncertainties respectively. Results are also presented for the light quark mass and …
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 quasi-particle mass shell to subleading order in covariant gauge.
The dependence of the effective action for gauge theories on the background field obeys an exact identity. We argue that for Abelian theories the Ward identity follows from the more general background field identity. This observation is particularly relevant for the anomalous Ward identity valid for gauge theories with an effective infrared cutoff as used for flow equations.
We study the question of the Ward identity for "large" gauge invariance in 0+1 dimensional theories. We derive the relevant Ward identities for a single flavor fermion and a single flavor complex scalar field interacting with an Abelian gauge field. These identities are nonlinear. The Ward identity for any other complicated theory can be derived from these basic sets of identities. However, the structure of the Ward identity changes since these are nonlinear identities. In particular, we work out the "large" gauge Ward identity for a supersymmetric theory involving a single flavor of fermion a…
We show that supersymmetry Ward identities contain an anomalous term which takes the form of a surface term in Hilbert space. In the one-instanton sector the anomalous term is the integral of a total rho-derivative where rho is the instanton's size. There are cases where the anomalous term is non-zero, and cannot be modified by subtractions. This constitutes a supersymmetry anomaly. The derivation is based on Feynman rules suitable for any non-perturbative sector of a weakly-coupled, renormalizable gauge theory.
Although noncommutative QED presents a nonabelian structure, it does not present structure constants. In view of this we investigate how Ward identity is satisfied in pair annihilation process and $γγ\to γγ$ scattering in noncommutative QED.
The Ward identity in gauge theory constrains the behavior of the amplitudes. We discuss the Ward identity for amplitudes with a pair of shifted lines with complex momenta. This will induce a recursion relation identical to BCFW recursion relations at the finite poles of the complexified amplitudes. Furthermore, according to the Ward identity, it is also possible to transform the boundary term into a simple form, which can be obtained by a new recursion relation. For the amplitude with one off-shell line in pure Yang-Mills theory, we find this technique is effective for obtaining the amplitude …
Using a recently proposed method arXiv:1910.11597 (Yu Meng et al.), we study the two-photon decay rate of $η_c$ using two $N_f=2$ twisted mass gauge ensembles with lattice spacings $0.067$fm and $0.085$fm. The results obtained from these two ensembles can be extrapolated in a naive fashion to the continuum limit, yielding a result that is consistent with the experimental one within two standard deviations. To be specific, we obtain the results for two-photon decay of $η_c$ as $\mathcal{B}(η_c\rightarrow 2γ)= 1.29(3)(18)\times 10^{-4}$ where the first error is statistical and the second is our …
We present a calculation of the strange and charm quark contributions to the nucleon spin from the anomalous Ward identity (AWI). It is performed with overlap valence quarks on 2+1-flavor domain-wall fermion gauge configurations on a $24^3 \times 64$ lattice with the light sea mass at $m_π = 330$ MeV. To satisfy the AWI, the overlap fermion for the pseudoscalar density and the overlap Dirac operator for the topological density, which do not have multiplicative renormalization, are used to normalize the form factor of the local axial-vector current at finite $q^2$. For the charm quark, we find …
We present a Ward identity for nonlinear sigma models using generalized nonlinear shift symmetries, without introducing current algebra or coset space. The Ward identity constrains correlation functions of the sigma model such that the Adler's zero is guaranteed for $S$-matrix elements, and gives rise to a subleading single soft theorem that is valid at the quantum level and to all orders in the Goldstone decay constant. For tree amplitudes, the Ward identity leads to a novel Berends-Giele recursion relation as well as an explicit form of the subleading single soft factor. Furthermore, interac…
We study the Ward identity for the effective photon-electron vertex summing the ladder diagrams contributing to the electrical conductivity in hot QED at leading logarithmic order. It is shown that the Ward identity requires the inclusion of a new diagram in the integral equation for the vertex that has not been considered before. The real part of this diagram is subleading and therefore the final expressions for the electrical conductivity at leading logarithmic order are not affected.
We give a detailed analysis of the anti-self-adjoint operator contribution to the fluctuation terms in the trace dynamics Ward identity. This clarifies the origin of the apparent inconsistency between two forms of this identity discussed in Chapter 6 of our recent book on emergent quantum theory.
We derive the Ward-Takahashi identity obeyed by the fermion-antifermion-gauge boson vertex in ladder QED in the presence of a constant magnetic field. The general structure in momentum space of the fermion mass operator with external electromagnetic field is discussed. Using it we find the solutions of the ladder WT identity with magnetic field. The consistency of our results with the solutions of the corresponding Schwinger-Dyson equation ensures the gauge invariance of the magnetic field induced chiral symmetry breaking recently found in ladder QED.
The one-loop corrections to the lattice supersymmetric Ward-Takahashi identity (WTi) are investigated in the off-shell regime. In the Wilson formulation of the N=1 supersymmetric Yang-Mills (SYM) theory, supersymmetry (SUSY) is broken by the lattice, by the Wilson term and is softly broken by the presence of the gluino mass. However, the renormalization of the supercurrent can be realized in a scheme that restores the continuum supersymmetric WTi (once the on-shell condition is imposed). The general procedure used to calculate the renormalization constants and mixing coefficients for the local…
The chiral susceptibility is given by the scalar vacuum polarisation at zero total momentum. This follows directly from the expression for the vacuum quark condensate so long as a nonperturbative symmetry preserving truncation scheme is employed. For QCD in-vacuum the susceptibility can rigorously be defined via a Pauli-Villars regularisation procedure. Owing to the scalar Ward identity, irrespective of the form or Ansatz for the kernel of the gap equation, the consistent scalar vertex at zero total momentum can automatically be obtained and hence the consistent susceptibility. This enables ca…
We propose a three-dimensional electromagnetic current operator within light-front dynamics that satisfies a light-front Ward-Takahashi identity for two-fermion systems. The light-front current operator is obtained by a quasi-potential reduction of the four-dimensional current operator and acts on the light-front valence component of bound or scattering states. A relation between the light-front valence wave function and the four-dimensional Bethe-Salpeter amplitude both for bound or scattering states is also derived, such that the matrix elements of the four-dimensional current operator can b…
We verify the QED Ward identity for the two- and three -point functions at non-equilibrium in the HTL limit. We use the Keldysh formalism of real time finite temperature field theory. We obtain an identity of the same form as the Ward identity for a set of one loop self-energy and one loop three-point vertex diagrams which are constructed from HTL effective propagators and vertices.
We elucidate chirality production under parity breaking constant electromagnetic fields, with which we clarify qualitative differences in and out of equilibrium. For a strong magnetic field the pair production from the Schwinger mechanism increments the chirality. The pair production rate is exponentially suppressed with mass according to the Schwinger formula, while the mass dependence of chirality production in the axial Ward identity appears in the pesudo-scalar term. We demonstrate that in equilibrium field theory calculus the axial anomaly is canceled by the pseudo-scalar condensate for a…
We investigate the axial Ward identity (AWI) for massive fermions in strong magnetic fields. The divergence of the axial-vector current is computed at finite temperature and/or density with the help of a relation between the polarization and anomaly diagrams in the effective (1+1) dimensions realized in the lowest Landau level (LLL). We discuss delicate interplay between the vacuum and medium contributions that determines patterns of the spectral flow in the adiabatic limit and, more generally, the diabatic chirality production rate. We also establish an explicit relation between the AWIs from…
A new formula for the composite Higgs boson mass is given, based on the Ward-Takahashi identity and the Schwinger-Dyson(SD) equation. In this formula the dominant asymptotic solution of the SD equation yields a correct answer, in sharp contrast to the Partially Conserved Dilatation Current(PCDC) approach where the sub- and sub-sub-dominant solutions should be taken into account carefully. In the gauged Nambu-Jona-Lasinio model we find M_H \simeq \sqrt{2}M for the composite Higgs boson mass M_H and the dynamical mass of the fermion M in the case of the constant gauge coupling(with large cut off…
We derive the anomalous conformal Ward identities for ${\cal N}=4$ SYM Wilson loops on polygon-like contours with edges formed by circular arcs. With a suitable choice of parameterisation they are very similarly to those for local correlation functions. Their solutions have a conformally covariant factor depending on the distances of the corners times a conformally invariant remainder factor depending, besides on cross ratios of the corners, on the cusp angles and angles parameterising the torsion of the contours.
Off-shell Ward identities in non-abelian gauge theory continue to be a subject of active research, since they are, in general, inhomogeneous and their form depends on the chosen gauge-fixing procedure. For the three-gluon and four-gluon vertices, it is known that a relatively simple form of the Ward identity can be achieved using the pinch technique or, equivalently, the background-field method with quantum Feynman gauge. The latter is also the gauge-fixing underlying the string-inspired formalism, and here we use this formalism to derive the corresponding form of the Ward identity for the one…
Sources of isocurvature perturbations and large non-Gaussianities include field degrees of freedom whose vacuum expectation values are smaller than the expansion rate of inflation. The inhomogeneities in the energy density of such fields are quadratic in the fields to leading order in the inhomogeneity expansion. Although it is often assumed that such isocurvature perturbations and inflaton-driven curvature perturbations are uncorre- lated, this is not obvious from a direct computational point of view due to the form of the minimal gravitational interactions. We thus compute the irreducible gr…
In order to study partial restoration of the chiral symmetry at finite density, we investigate the density corrections of the chiral condensate up to next-leading order of density expansion using the chiral Ward identity and an in-medium chiral perturbation theory. In our study, we assume that all the in-vacuum quantities for the pion, the nucleon and the pi N interaction are determined and focus on density expansion of the in-medium physical quantities. We perform diagrammatic analysis of the correlation functions which provide the in-medium chiral condensate. This density expansion scheme sh…
We investigate non-Abelian gauge theories within a Wilsonian Renormalisation Group approach. The cut-off term inherent in this approach leads to a modified Ward identity (mWI). It is shown that this mWI is compatible with the flow and that the full effective action satisfies the usual Ward identity (WI). The universal 1-loop beta-function is derived within this approach and the extension to the 2-loop level is briefly outlined.
The anomalous generation of chirality with mass effects via the axial Ward identity and its dependence on the Schwinger mechanism is reviewed, utilizing parity violating homogeneous electromagnetic background fields. The role vacuum asymptotic states play on the interpretation of expectation values is examined. It is discussed that observables calculated with an in-out scattering matrix element predict a scenario under Euclidean equilibrium. A notable ramification of which is a vanishing of the chiral anomaly. In contrast, it is discussed observables calculated under an in-in, or real-time, fo…
For conformal field theories, it is shown how the Ward identity corresponding to dilatation invariance arises in a Wilsonian setting. In so doing, several points which are opaque in textbook treatments are clarified. Exploiting the fact that the Exact Renormalization Group furnishes a representation of the conformal algebra allows dilatation invariance to be stated directly as a property of the action, despite the presence of a regulator. This obviates the need for formal statements that conformal invariance is recovered once the regulator is removed. Furthermore, the proper subset of conforma…
We argue that the topological charge of nonabelian gauge theory is unphysical. To show this statement, we use the Adler-Bardeen theorem and the Becchi-Rouet-Stora-Tyutin symmetry which are warranted by the perturbative finiteness of the chiral anomaly, thus being free of Gribov ambiguity. In addition to the original argument using the unphysical gauge field component collinear to the spatial derivative of the gauge function, we show the unobservability of the topological charge using the Ward-Takahashi identity. We then present the consequences of this finding and show the consistency with man…
We report a quark spin calculation from the anomalous Ward identity with overlap fermions on 2+1 flavor dynamical fermion configurations with light sea quark masses. Such a formulation decomposes the divergence of the flavor-singlet axial-vector current into a quark pseudoscalar term and a triangle anomaly term, flavor by flavor. A large negative contribution from the anomaly term is observed and it is canceled within errors by the contribution from the pseudoscalar term in the disconnected insertion in the heavy quark region. On the other hand, net negative contributions are obtained for the …
We consider the constraints of the Slavnov-Taylor identity of the IR behaviour of gluon and ghost propagators and their compatibility with solutions of the ghost Dyson-Schwinger equation and with the lattice picture.
The contribution to the muon anomalous magnetic moment from the fermion triangle loop diagrams connected to the muon line by a photon and a $Z$ boson is reanalyzed in the unitary gauge. With use of the anomalous axial-vector Ward identity, it is shown that the calculation in the unitary gauge exactly coincides with the one in the 't Hooft-Feynman gauge. The part which arises from the ordinary axial-vector Ward identity corresponds to the contribution of the neutral Goldstone boson. For the top-quark contribution, the one-parameter integral form is obtained up to the order of $m_μ^2/m_Z^2$. The…
These lecture notes give an introduction to the algebraic renormalization of the Standard Model. We start with the construction of the tree approximation and give the classical action and its defining symmetries in functional form. These are the Slavnov-Taylor identity, Ward identities of rigid symmetry and the abelian local Ward identity. The abelian Ward identity ensures coupling of the electromagnetic current in higher orders of perturbation theory, and is the functional form of the Gell-Mann-Nishijima relation. In the second part of the lectures we present in simple examples the basic prop…
We present a computation of the charge and the magnetic moment of the neutrino in the recently developed electro-weak Background Field Method and in the linear $R_ξ^L$ gauge. First, we deduce a formal Ward-Takahashi identity which implies the immediate cancellation of the neutrino electric charge. This Ward-Takahashi identity is as simple as that for QED. The computation of the (proper and improper) one loop vertex diagrams contributing to the neutrino electric charge is also presented in an arbitrary gauge, checking in this way the Ward-Takahashi identity previously obtained. Finally, the cal…
The Ward-Takahashi identity, reflecting local gauge invariance, is perturbatively verified for a boson model in light front field theory. A careful integration over the light front energy, corresponding to exactly taking into account pair terms, which are the contributions of the zero longitudinal momentum mode, is crucial to obtain this result. Furthermore, the one-loop boson form factors are calculated for arbitrary off-shell momenta.
We address in a recent gauge model of unparticles the issues that are important for consistency of a gauge theory, i.e., unitarity and Ward identity of physical amplitudes. We find that non-integrable singularities arise in physical quantities like cross section and decay rate from gauge interactions of unparticles. We also show that Ward identity is violated due to the lack of a dispersion relation for charged unparticles although the Ward-Takahashi identity for general Green functions is incorporated in the model. A previous observation that the unparticle's (with scaling dimension d) contri…
We consider a lattice formulation of the four dimensional N=1 Wess-Zumino model in terms of the Ginsparg-Wilson relation. This formulation has an exact supersymmetry on the lattice. The lattice action is invariant under a deformed supersymmetric transformation which is non-linear in the scalar fields and it is determined by an iterative procedure in the coupling constant to all orders in perturbation theory. We also show that the corresponding Ward-Takahashi identity is satisfied at fixed lattice spacing. The calculation is performed in lattice perturbation theory up to order $g^3$ (two-loop) …
Using diagrammatic methods, we show how the Ward identity can be used to constrain the ladder kernel in transport coefficient calculations. More specifically, we use the Ward identity to determine the necessary diagrams that must be resummed using an integral equation. One of our main results is an equation relating the kernel of the integral equation with functional derivatives of the full self-energy; it is similar to what is obtained with 2PI effective action methods. However, since we use the Ward identity as our starting point, gauge invariance is preserved. Using power counting arguments…
Recently, Bern et al observed that a certain class of next-to-planar Feynman integrals possess a bonus symmetry that is closely related to dual conformal symmetry. It corresponds to a projection of the latter along a certain lightlike direction. Previous studies were performed at the level of the loop integrand, and a Ward identity for the integral was formulated. We investigate the implications of the symmetry at the level of the integrated quantities. In particular, we focus on the phenomenologically important case of five-particle scattering. The symmetry simplifies the four-variable proble…
We solved with a numerical procedure the HTL improved ladder DS equation for the retarded fermion self-energy function $Σ_R$ to study the spontaneous generation of fermion mass in thermal QCD/QED, and studied the gauge-dependence of the solution within a general covariant gauge where the gauge parameter $ξ$ is any constant number. With the numerical solutions thus obtained, we found the followings; i) The fermion wave function renormalization function $A(P)$ always deviates largely from unity even at the momentum where the mass is defined, thus the corresponding solutions explicitly contradict…
By applying the Ward identity found by Weinberg two new relations of the amplitude of $a_{1}\rightarrowρπ$ with other physical quantities have been found.
In the case of massless current quarks we find that the breaking of chiral symmetry usually triggers the generation of an anomalous magnetic moment for the quarks. We show that the kernel of the Ward identity for the vector vertex yields an important contribution. We compute the anomalous magnetic moment in several quark models. The results show that it is hard to escape a measurable anomalous magnetic moment for the quarks in the case of spontaneous chiral symmetry breaking.
Using the Batalin-Vilkovisky formalism, we study the Ward identities and the equations of gauge dependence in potentially anomalous general gauge theories, renormalizable or not. A crucial new term, absent in manifestly nonanomalous theories, is responsible for interesting effects. We prove that gauge invariance always implies gauge independence, which in turn ensures perturbative unitarity. Precisely, we consider potentially anomalous theories that are actually free of gauge anomalies thanks to the Adler-Bardeen theorem. We show that when we make a canonical transformation on the tree-level a…
Consequences of a symmetry, e.g.\ relations amongst Green functions, are renormalization scheme independently expressed in terms of a rigid Ward identity. The corresponding local version yields information on the respective current. In the case of spontaneous breakdown one has to define the theory via the BRS invariance and thus to construct rigid and current Ward identity non-trivially in accordance with it. We performed this construction to all orders of perturbation theory in the abelian Higgs model as a prelude to the standard model. A technical tool of interest in itself is the use of a d…
Building on our recent derivation of the Ward-Schwinger-Dyson equations for the cubic interaction model, we present here the first steps of their resurgent analysis. In our derivation of the WSD equations, we made sure that they had the properties of compatibility with the renormalisation group equations and independence from a regularisation procedure which was known to allow for the comparable studies in the Wess-Zumino model. The interactions between the transseries terms for the anomalous dimensions of the field and the vertex is at the origin of unexpected features, for which the effect o…
