M. A. Braun, M. I. Vyazovsky
Ward identities for reggeons are studied in the framework of effective action approach to the QCD in Regge kinematics. It is shown that they require introduction of new contributions not present in the reggeon diagrams initially. Application to vertices RR$\to$RP and RR$\to$RRP are considered and diagrams which have to be added to the QCD ones are found.
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Leading plus next-to leading log results for the Regge limit of massless Yang-Mills theories are reproduced by reggeon diagrams in which the Regge slope $α' \to 0$ and reggeon amplitudes satisfy Ward identity constraints at zero transverse momentum. Using reggeon unitarity together with multiple discontinuity theory a complete set of such diagrams can be constructed. The resulting two-two, one-three and two-four kernels which generalise the Lipatov equation at $O(g^4)$ are determined uniquely.
We show both in 1+1 and 3+1 dimensions that, contrary to the recent suggestions, the contribution of the fermion loop to the polarization tensor is manifestly transverse at finite temperature. Some subtleties associated with the Ward identities at finite temperature are also pointed out.
We use supersymmetric Ward identities to relate multi-gluon helicity amplitudes involving a pair of massive quarks to amplitudes with massive scalars. This allows to use the recent results for scalar amplitudes with an arbitrary number of gluons obtained by on-shell recursion relations to obtain scattering amplitudes involving top quarks.
In the framework of the Electroweak Theory we discuss a class of gauge fixing choices suitable for the calculation of electromagnetic processes. We show in particular that, with our choices, beyond the basic Slavnov-Taylor identities guaranteeing the independence of the physical results of the particular gauge fixing, one also has the standard QED Ward identities which play a well known crucial role in the calculation of electromagnetic processes and, in particular, in the analysis of the electromagnetic radiative corrections.
In a simple abelian spinor field theory, the canonical trace identities for certain axial-vector and axial-scalar operators are reexamined in dimensional regularization, some disagreements with previous results are found and an interesting new phenomenon is observed and briefly discussed.
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.
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…
Anomalies in transverse Ward--Takahashi identities are studied, allowing discussion of the feasibility of anomalies arising in general non-symmetry Ward--Takahashi identities. We adopt the popular Fujikawa's 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 non-symmetry anomalies. Papers that claim the non-existence …
We consider finite superamplitudes of N=1 matter, and use superconformal symmetry to derive powerful first-order differential equations for them. Due to on-shell collinear singularities, the Ward identities have an anomaly, which is obtained from lower-loop information. We show that in the five-particle case, the solution to the equations is uniquely fixed by the expected analytic behavior. We apply the method to a non-planar two-loop five-particle integral.
We study the cancellations among Feynman diagrams that implement the Ward and Slavnov-Taylor identities corresponding to the conserved supersymmetry current in supersymmetric quantum field theories. In particular, we show that the Faddeev-Popov ghosts of gauge- and supersymmetries never decouple from the physical fields, even for abelian gauge groups. The supersymmetric Slavnov-Taylor identities provide efficient consistency checks for automatized calculations and can verify the supersymmetry of Feynman rules and the numerical stability of phenomenological predictions simultaneously.
We discuss the use of BRST symmetry and the resulting Ward identities for orbifold gauge theories as consistency checks in an arbitrary number of dimensions. We verify that both the usual orbifold symmetry breaking and the recently proposed Higgsless symmetry breaking are consistent with the nilpotency of the BRST transformation. Imposing the Ward identities resulting from the BRST symmetry on the 4-point functions of theory, we obtain relations on the coupling constants that are shown to be equivalent to the conditions for tree level unitarity. We present the complete set of these sum rules a…
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 analyze a formulation of QED based on the Wilson renormalization group. Although the ``effective Lagrangian'' used at any given scale does not have simple gauge symmetry, we show that the resulting renormalized Green's functions correctly satisfies Ward identities to all orders in perturbation theory. The loop expansion is obtained by solving iteratively the Polchinski's renormalization group equation. We also give a new simple proof of perturbative renormalizability. The subtractions in the Feynman graphs and the corresponding counterterms are generated in the process of fixing the physica…
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.
Axion EFT in the BMHV Scheme: Flavor Currents, Evanescent Operators and Ward Identities
本文系统分析了轴子有效场论在BMHV方案中的重整化问题，重点研究了费米子维五算符的重整化及相关的手性味电流。通过图示计算验证了这些身份在两环阶的有效性，揭示了蒸发算符在保持方案一致性中的关键作用。
Helicity-Resolved Constructibility from Unified Ward Identities
本文证明大质量高自旋粒子的大$z$行为由螺旋度而非总自旋控制，利用统一Ward恒等式消除额外增长，给出任意自旋的螺旋度分辨可构造性判据，并表明非负耦合维度振幅在五点及以上对所有螺旋度组态可构造。
In this talk the gauge symmetry for Wilsonian flows in pure Yang-Mills theories is discussed. The background field formalism is used for the construction of a gauge invariant effective action. The symmetries of the effective action under gauge transformations for both the gauge field and the auxiliary background field are separately evaluated. Modified Ward-Takahashi and background field identities are used in my study. Finally it is shown how the symmetry properties of the full theory are restored in the limit where the cut-off is removed.
We derive spectral representations for the different components of the 4-point function at finite temperature in the real time formalism in terms of five real spectral densities. We explicitly calculate all these functions in QED in the hard thermal loop approximation. The Ward identities obeyed by the 1-loop 3- and 4-point functions in real time and their spectral functions are derived. We compare our results with those derived previously in the imaginary-time formalism for retarded functions in hot QCD, and we discuss the generalization of our results to non-equilibrium situations.
We analyse the recent controversy on a possible Chern-Simons like term generated through radiative corrections in QED with a CPT violating term : we prove that, if the theory is correctly defined through Ward identities and normalisation conditions, no Chern-Simons term appears, without any ambiguity. This is related to the fact that such a term is a kind of minor modification of the gauge fixing term, and then no renormalised. The past year literature on that subject is discussed, and we insist on the fact that any absence of an {\sl a priori} divergence should be explained by some symmetry o…
We present a general study of 3-point functions of conformal field theory (CFT) in momentum space, following a reconstruction method for tensor correlators, based on the solution of the conformal Ward identities (CWIs), introduced in recent works. We investigate and detail the structure of the CWIs and their non-perturbative solutions, and compare them to perturbation theory, taking QED and QCD as examples. Exact solutions of CFT's in the flat background limit in momentum space are matched by the perturbative realizations in free field theories, showing that the origin the conformal anomaly is…
Planar gluon amplitudes in N=4 SYM are remarkably similar to expectation values of Wilson loops made of light-like segments. We argue that the latter can be determined by making use of the conformal symmetry of the gauge theory, broken by cusp anomalies. We derive the corresponding anomalous conformal Ward identities valid to all loops and show that they uniquely fix the form of the finite part of a Wilson loop with n cusps (up to an additive constant) for n=4 and n=5 and reduce the freedom in it to a function of conformal invariants for n>=6. We also present an explicit two-loop calculation f…
We study chiral properties of eigenvalue spectrum for staggered quarks. We present a new method to identify would-be zero modes and nonzero modes using their symmetry and chiral properties. Here, we review the traditional method with HYP improved staggered quarks, and extend it to a completely new method which uses the chiral Ward identities and leakage patterns to achieve the goal.
Lattice studies of QCD at temperatures above the chiral restoration and QCD with truncated low modes of the Dirac operator indicate approximate and explicit $SU(4)$ degeneracies in hadron spectra, respectively. At the same time, the QCD classical action and the path integral measure are not invariant under $SU(4)$. Here we investigate $SU(4)$ transformations in the continuum limit by deriving corresponding Ward identities. We show that, if there is a gap in low-lying modes of the Dirac operator the obtained $SU(4)$ Ward identities are simplified and look like they would be if $SU(4)$ symmetry …
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 first derive the transverse Ward-Takahashi identities (WTI) of 3-dimensional quantum electrodynamics (QED$_3$) by means of the canonical quantization method and the path integration method, and then prove for the first time that QED$_3$ is strictly solvable based on the transverse WTI and the longitudinal WTI, that is, the full vector and tensor vertices functions can be expressed in term of the fermion propagators in QED$_3$. Further, we discuss the effect of different $γ$ matrix representations on the full fermion-boson vertex function. It is found that the full vector vertex function doe…
We outline the pinch technique for constructing gauge invariant Green's functions in gauge theories and derive the Ward identities that must be satisfied by the gauge invariant three boson vertices of the standard model. They are generalizations of their tree level Ward identities and are shown to be crucial for the delicate gauge cancellations of the S-matrix.
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.
Anomalous Ward identities for on-shell amplitudes at the conformal fixed point
在共形固定点的维度正则化 $ϕ^3$ 模型中，证明壳上重整化振幅满足反常共形 Ward 恒等式，反常项由基本场反常量纲和通用共线函数与低阶振幅的卷积构成，比同阶振幅计算更简单且具强预测力。
The exclusive FCNC beauty semileptonic decay B-> ρis studied using Ward identities in a general vector meson dominance framework, predicting vector meson couplings involved. The long distance contributions are discussed which results to obtain form factors and |V_ub|. A detailed comparison is given with other approaches.
We verify Standard Model Effective Field Theory Ward identities to one loop order when background field gauge is used to quantize the theory. The results we present lay the foundation of next to leading order automatic generation of results in the SMEFT, in both the perturbative and non-perturbative expansion using the geoSMEFT formalism, and background field gauge.
It is found that in the SM the Ward-Takahashi(WT) identities of the axial-vector currents and the charged vector currents of fermions are invalid after spontaneous symmetry breaking. The spin-0 components of Z and W fields are revealed from the invalidity of these WT identities. The masses of these spin-0 components are at $10^{14}$GeV. They are ghosts. Therefore, unitarity of the SM after spontaneous symmetry breaking is broken at $10^{14}$GeV.
We derive Ward identities for the Standard Model Effective Field Theory using the background field method. The resulting symmetry constraints on the Standard Model Effective Field Theory are basis independent, and constrain the perturbative and power-counting expansions. A geometric description of the field connections, and real representations for the $\rm SU(2)_L \times U(1)_Y$ generators, underlies the derivation.
Supersymmetric Slavnov-Taylor and Ward identities are investigated in presence of soft and spontaneous symmetry breaking. We consider an abelian model where soft supersymmetry breaking yields a mass splitting between electron and selectron and triggers spontaneous symmetry breaking, and we derive corresponding identities that relate the electron and selectron masses with the Yukawa coupling. We demonstrate that the identities are valid in dimensional reduction and invalid in dimensional regularization and compute the necessary symmetry-restoring counterterms.
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…
Thermal hadron resonances and Ward identities: results for the QCD phase diagram
本文回顾了基于幺正手征微扰理论和Ward恒等式研究有限温度下轻标量共振对手征对称性及$U(1)_A$恢复的作用，并关联QCD相图中的未解问题。
Using reggeon diagrams as a partial implementation of $t$-channel unitarity, $O(g^4)$ corrections to the BFKL evolution equation have been obtained. We describe the spectrum and holomorphic factorization properties of the resulting scale-invariant kernel. For a gauge theory, $t$-channel unitarity can be studied directly in the complex $j$-plane by implementing Ward identity constraints together with the group structure of reggeon interactions. We discuss how both the $O(g^2)$ BFKL kernel and the $O(g^4)$ corrections can then be derived.
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.
We derive and analyze the conformal Ward identities (CWI's) of a tensor 4-point function of a generic CFT in momentum space. The correlator involves the stress-energy tensor $T$ and three scalar operators $O$ ($TOOO$). We extend the reconstruction method for tensor correlators from 3- to 4-point functions, starting from the transverse traceless sector of the $TOOO$. We derive the structure of the corresponding CWI's in two different sets of variables, relevant for the analysis of the 1-to-3 (1 graviton $\to$ 3 scalars) and 2-to-2 (graviton + scalar $\to$ two scalars) scattering processes. The …
The content of two additional Ward identities exhibited by the $U(1)$ Higgs model is exploited. These novel Ward identities can be derived only when a pair of local composite operators providing a gauge invariant setup for the Higgs particle and the massive vector boson is introduced in the theory from the beginning. Among the results obtained from the above mentioned Ward identities, we underline a new exact relationship between the stationary condition for the vacuum energy, the vanishing of the tadpoles and the vacuum expectation value of the gauge invariant scalar operator. We also present…
In the context of the Standard Model we extend the S--matrix pinch technique for non--conserved currents to the case of three boson vertices. We outline in detail how effective gauge invariant three boson vertices can be constructed, with all three incoming momenta {\it off--shell}. Explicit closed expressions for the vertices $γW^- W^+$, $Z W^- W^+$, and $χW^- W^+$ are reported. The three boson vertices so constructed satisfy naive QED--like Ward identities which relate them to the gauge invariant gauge boson self--energies previously constructed by the same method. The derivation of the afor…
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.
We present a field-theoretical method to obtain consistently the equations of motion for small amplitude condensate perturbations in a homogeneous Bose-condensed gas directly in real time. It is based on linear response, and combines the Schwinger-Keldysh formulation of nonequilibrium quantum field theory with the Nambu-Gorkov formalism of quasiparticle excitations in the condensed phase and the tadpole method in quantum field theory. This method leads to causal equations of motion that allow to study the nonequilibrium evolution as an initial value problem. It also allows to extract directly …
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 longit…
We exploit the Ward-Slavnov-Taylor identity relating the 3-gluons to the ghost-gluon vertices to conclude either that the ghost dressing function is finite and non vanishing at zero momentum while the gluon propagator diverges (although it may do so weakly enough not to be in contradiction with current lattice data) or that the 3-gluons vertex is non-regular when one momentum goes to zero. We stress that those results should be kept in mind when one studies the Infrared properties of the ghost and gluon propagators, for example by means of Dyson-Schwinger equations.
We introduce a new local symmetry into the fermion sector of a gauge invariant Lagrangian which may or may not contain a scalar or spontaneous symmetry breaking. The Standard Model in the unitary gauge and QCD are particular cases where this symmetry may apply. We determine the associated vector and axial vector currents and their conservation laws. We show that a single current conservation law may lead to multiple Ward-Takahashi identities. Our results can potentially have important consequences for effective models of low-energy QCD and hadron structure. As an specific example, we discuss t…
We investigate the emergence of a conformal anomaly pole in conformal field theories in the case of the $TJJ$ correlator. We show how it comes to be generated in dimensional renormalization, using a basis of 13 form factors (the $F$-basis), where only one of them requires renormalization $(F_{13})$, extending previous studies. We then combine recent results on the structure of the non-perturbative solutions of the conformal Ward identities (CWI's) for the $TJJ$ in momentum space, expressed in terms of a minimal set of 4 form factors ($A-$ basis), with the properties of the $F$-basis, and show …
Environmental Effects in Post-Minkowskian Dynamics: Effective Field Theory, Feynman Rules, and Ward Identities for Compact Objects in Relativistic Fluids
本文为致密物体在无粘流体中的引力动力学建立流体有效场论，推导费曼规则和广义Ward恒等式，并用于恢复领头阶相对论动力学摩擦力，为后闵可夫斯基计算纳入环境效应奠定基础。
We discuss our recent results regarding chiral and $U(1)_A$ restoration, both from the formal point of view of QCD Ward Identities (WI) and from an Effective Theory analysis provided by $U(3)$ Chiral Perturbation Theory (ChPT) at finite temperature. Our results lead to relevant conclusions regarding the behavior of chiral partners (in terms of susceptibilities) in the limit of exact restoration and provide useful results for lattice analysis. In addition, it helps to understand the temperature dependence of lattice screening masses in terms of quark condensate combinations. The U(3) ChPT calcu…
In contrast to the conventional treatment of gauge theories, in the background-field method the Ward identities for connected Green functions are not violated by Dyson summation of self-energies in finite orders of perturbation theory. Thus, Dyson summation does not spoil gauge cancelations at high energies which are ruled by the Goldstone-boson equivalence theorem. Moreover, in the background-field method the precise formulation of the equivalence theorem in higher orders (including questions of renormalization) is simplified rendering actual calculations easier. Finally, the equivalence theo…
We re-investigate Adler's PCAC relation in the presence of an external electromagnetic field within the framework of QCD coupled to external fields. We discuss pion electroproduction within a tree-level approximation to chiral perturbation theory and explicitly verify a chiral Ward identity referred to as the Adler-Gilman relation. We critically examine soft-momentum techniques and point out how inadmissable approximations may lead to results incompatible with chiral symmetry. As a result we confirm that threshold pion electroproduction is indeed a tool to obtain information on the axial form …
Working within the linear sigma model at finite temperature, we construct effective one-loop vertices and propagators satisfying the chiral Ward identities, for small pion momentum and mass compared to the sigma mass. We obtain an expansion for the pion dispersion relation up to second order in the parameter m_pi^2/4pi^2f_pi^2. At leading order, this expansion reproduces the dispersion curve obtained in chiral perturbation theory. We also study pion damping and compute the mean free path for pions traveling in the medium before forming a sigma resonance. These results are also in good agreemen…
Long distance effects are studied in the rare exclusive semileptonic $B_{(d,s)}\to V \ell^+ \ell^-$ decays, where $V$ denotes a $K^*$ or $ϕ$ meson. The form factors, which describe the meson transition amplitudes in the effective Hamiltonian approach, are calculated by means of Ward identities, experimental constraints and extrapolated within a general vector meson dominance framework. These form factors are then compared to the ones obtained in Lattice QCD simulations, with Light Cone Sum Rules and a Dyson-Schwinger equation approach. Additionally, the $B_d\to K^* \ell^+ \ell^-$ and $B_s\to ϕ…
We develop one-loop effective vertices and propagators in the linear sigma model at finite temperature satisfying the chiral Ward identities. We use these in turn to compute the pion dispersion relation in a pion medium for small momentum and temperatures on the order of the pion mass at next to leading order in the parameter m_pi^2/4pi^2f_pi^2 and to zeroth order in the parameter (m_pi/m_sigma)^2. We show that this expansion reproduces the result obtained from chiral perturbation theory at leading order. The main effect is a perturbative, temperature-dependent increase of the pion mass. We fi…
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…
In this paper we calculated the n-point hard-thermal-loop (HTL) vertex functions in QCD/QED for n= 2, 3 and 4 in the physical representation in the real time formalism (RTF). The result showed that the n-point HTL vertex functions can be classified into two groups, a) those with odd numbers of external retarded indices, and b) the others with even numbers of external retarded indices. The n-point HTL vertex functions with one retarded index, which obviously belong to the first group a), are nothing but the HTL vertex functions that appear in the imaginary time formalism (ITF), and vise versa. …
The gauge principle is fundamental in formulating the Standard Model. Fermion--gauge-boson couplings are the inescapable consequence and the primary determining factor for observable phenomena. Vertices describing such couplings are simple in perturbation theory and yet the existence of strong-interaction bound-states guarantees that many phenomena within the Model are nonperturbative. It is therefore crucial to understand how dynamics dresses the vertices and thereby fundamentally alters the appearance of fermion--gauge-boson interactions. We consider the coupling of a dressed-fermion to an A…
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