Ward identities of SUSY and R-symmetry relate n-point amplitudes in supersymmetric theories. We review recent work in which these Ward identities are solved in N=4 SYM and N=8 supergravity. The solution, valid at both tree and loop level, expresses any (Next-to)^K MHV superamplitude in terms of a basis of ordinary amplitudes. Basis amplitudes are classified by semi-standard tableaux of rectangular N-by-K Young diagrams. The SUSY Ward identities also impose constraints on the matrix elements of candidate ultraviolet counterterms in N=8 supergravity, and they can be studied using superamplitude basis expansions. This leads to a novel and quite comprehensive matrix element approach to counterterms, which we also review. This article is an invited review for a special issue of Journal of Physics A devoted to "Scattering Amplitudes in Gauge Theories".
Supersymmetry and R-symmetry Ward identities relate on-shell amplitudes in a supersymmetric field theory. We solve these Ward identities for (Next-to)^K MHV amplitudes of the maximally supersymmetric N=4 and N=8 theories. The resulting superamplitude is written in a new, manifestly supersymmetric and R-invariant form: it is expressed as a sum of very simple SUSY and SU(N)_R-invariant Grassmann polynomials, each multiplied by a "basis amplitude". For (Next-to)^K MHV n-point superamplitudes the number of basis amplitudes is equal to the dimension of the irreducible representation of SU(n-4) corr…
The UV finiteness found in calculations of the 4-point amplitude in $\mathcal N=5$ supergravity at loop order $L=3, 4$ has not been explained, which motivates our study of the relevant superspace invariants and on-shell superamplitudes for both $\mathcal N=5$ and $\mathcal N=6$. The local 4-point superinvariants for $L = 3,4$ are expected to have nonlinear completions whose 6-point amplitudes have non-vanishing SSL's (soft scalar limits), violating the behavior required of Goldstone bosons. For $\mathcal N=5$, we find at $L=3$ that local 6-point superinvariant and superamplitudes, which might …
We show that 10D spinor helicity formalism can be understood as spinor moving frame approach to supersymmetric particles extended to the description of amplitudes. This allows us to develop the spinor helicity formalism for 11D supergravity and a new constrained superfield formalism for 10D SYM and 11D SUGRA amplitudes. We show how the constrained on-shell superfields, one-particle counterparts of the superamplitudes, can be obtained by quantization of massless superparticle mechanics. We make some stages towards the calculation of amplitudes of 10D SYM and 11D SUGRA in this framework. In part…
Celestial amplitude is a new reformulation of momentum space scattering amplitude and offers a promising way for flat holography. In this paper, we study the celestial amplitude in $\mathcal N=4$ Super-Yang-Mills (SYM) theory aiming at understanding the role of superconformal symmetry in celestial holography. We first construct the superconformal generators acting on the celestial superfield which assembles all the on-shell fields in the multiplet together in terms of celestial variables and Grassmann parameters. These generators satisfy the superconformal algebra of $\mathcal N=4$ SYM theory.…
We compute the anomalous two-loop four-point amplitudes in N=4 pure supergravity, using unitarity and the double-copy construction. We also present all terms determined by four-dimensional cuts in two all-multiplicity two-loop anomalous superamplitudes. This result provides the first two-loop n-point gravity amplitude, up to a class of undetermined rational terms, which are absent at four points. We show that a recently proposed finite counterterm cancels these amplitudes to this order. We argue that the counterterm does not spoil the three-loop finiteness of anomalous amplitudes in the N=4 th…
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.
In this paper we show that the one-loop graviton self-energy contribution is ultraviolet finite, without introducing counterterms, and cutoff-free in the framework of causal perturbation theory. In addition, it satisfies the gravitational Slavnov-Ward identities for the two-point connected Green function. The condition of perturbative gauge invariance to second order for loop graphs is proved. Corrections to the Newtonian potential are also derived.
In the 't Hooft-Veltman dimensional regularization scheme it is necessary to introduce finite counterterms to satisfy chiral Ward identities. It is a non-trivial task to evaluate these counterterms even at two loops. We suggest the use of Wilsonian exact renormalization group techniques to reduce the computation of these counterterms to simple master integrals. We illustrate this method by a detailed study of a generic Yukawa model with massless fermions at two loops.
Superconformal Ward identities for N=1 supersymmetric quantum field theories in four dimensions are convenienty obtained in the superfield formalism by combining diffeomorphisms and Weyl transformations on curved superspace. Using this approach we study the superconformal transformation properties of Green functions with one or more insertions of the supercurrent to all orders in perturbation theory. For the case of two insertions we pay particular attention to fixing the additional counterterms present, as well as to the purely geometrical anomalies which contribute to the transformation beha…
In this paper the stability and the renormalizability of Yang-Mills theory in the Background Field Gauge are studied. By means of Ward Identities of Background gauge invariance and Slavnov-Taylor Identities the stability of the classical model is proved and, in a regularization independent way, its renormalizability is verified. A prescription on how to build the counterterms is given and the possible anomalies which may appear for Ward Identities and for Slavnov-Taylor Identities are shown.
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 theori…
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 illustrate the importance of mass scales and their relation in the specific case of the linear sigma model within the context of its one loop Ward identities. In the calculation it becomes apparent the delicate and essential connection between divergent and finite parts of amplitudes. The examples show how to use mass scales identities which are absolutely necessary to manipulate graphs involving several masses. Furthermore, in the context of the Implicitly Regularization, finite(physical) and divergent (counterterms) parts of the amplitude can and must be written in terms of a single scale…
~It is shown that the quantum massive non-Abelian field theory established in the former papers is renormalizable. This conclusion is achieved with the aid of the Ward-Takahashi identities satisfied by the generating functionals which were derived in the preceding paper based on the BRST-symmetry of the theory. By the use of the Ward-Takahashi identity, it is proved that the divergences occurring in the perturbative calculations for the massive gauge field theory can be eliminated by introducing a finite number of counterterms in the effective action. As a result of the proof, it is found that…
Application of the background-field method yields a gauge-invariant effective action for the electroweak Standard Model, from which simple QED-like Ward identities are derived. As a consequence of these Ward identities, the background-field Green functions are shown to possess very desirable theoretical properties. The renormalization of the Standard Model in the background-field formalism is studied. A consistent on-shell renormalization procedure retaining the full gauge symmetry is presented. The structure of the counterterms is shown to greatly simplify compared to the conventional formali…
Fixing the Renormalization of Inflationary Loops via Ward Identities
Institute of Theoretical Physics, Chinese Academy of Sciences（中国科学院理论物理研究所）；School of Physical Sciences, University of Chinese Academy of Sciences（中国科学院大学物理科学学院）；School of Fundamental Physics and Mathematical Sciences, Hangzhou Institute for Advanced Study, University of Chinese Academy of Sciences（中国科学院大学杭州高等研究院基础科学与数学学院）
We initiate a systematic study of the consequences of (super)conformal symmetry of massless scattering amplitudes. The classical symmetry is potentially broken at the quantum level by infrared and ultraviolet effects. We study its manifestations on the finite hard part of the scattering process. The conformal Ward identities in momentum space are second-order differential equations, difficult to analyze. We prefer to study superconformal symmetry whose generators are first-order in the momenta. Working in a massless N=1 supersymmetric Wess-Zumino model, we derive on-shell superconformal Ward i…
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…
We discuss the application of two-particle-irreducible (2PI) functional techniques to gauge theories, focusing on the issue of non-perturbative renormalization. In particular, we show how to renormalize the photon and fermion propagators of QED obtained from a systematic loop expansion of the 2PI effective action. At any finite order, this implies introducing new counterterms as compared to the usual ones in perturbation theory. We show that these new counterterms are consistent with the 2PI Ward identities and are systematically of higher order than the approximation order, which guarantees t…
Essential to QCD applications of the operator product expansion, etc., is a knowledge of those operators that mix with gauge-invariant operators. A standard theorem asserts that the renormalization matrix is triangular: Gauge-invariant operators have `alien' gauge-variant operators among their counterterms, but, with a suitably chosen basis, the necessary alien operators have only themselves as counterterms. Moreover, the alien operators are supposed to vanish in physical matrix elements. A recent calculation by Hamberg and van Neerven apparently contradicts these results. By explicit calculat…
We derive the conformal constraints satisfied by classical vertices of a (Einstein) Gauss-Bonnet theory around flat space, in general dimensions and at $d=4$ (4d EGB). In $4d$ EGB they are obtained by a singular limit of the integral of the Euler-Poincarè density. Our analysis exploits the relation between this theory and the conformal anomaly action, which allows to uncover some interesting features of the GB vertex at cubic and quartic level. If we introduce a conformal decomposition of the metric, the resulting theory can be formulated in two different versions, which are regularization dep…
