Remarks on Constraint Analysis and Vacuum Persistence Amplitudes in Gauge Theories

This review gives an overview of many of the recent developments in understanding the structure of relativistic scattering amplitudes in gauge theories ranging from QCD to super-Yang–Mills theory, as well as (super)gravity. I also provide a pedagogical introduction to some of the basic tools used to organize and illuminate the color and kinematic structure of amplitudes. This is an invited review introducing a special issue of Journal of Physics A: Mathematical and Theoretical devoted to ‘Scattering amplitudes in gauge theories’.

Providing a comprehensive, pedagogical introduction to scattering amplitudes in gauge theory and gravity, this book is ideal for graduate students and researchers. It offers a smooth transition from basic quantum field theory to the frontier of modern research. The book starts with an introduction to the spinor helicity formalism in the context of Feynman rules for tree-level amplitudes. The material covered includes on-shell recursion relations, superamplitudes, symmetries of N=4 super Yang–Mills theory, twistors and momentum twistors, Grassmannians, and polytopes. The presentation also covers amplitudes in perturbative supergravity, 3D Chern–Simons matter theories, and color–kinematics duality and its connection to 'gravity=(gauge theory)x(gauge theory)'. Basic knowledge of Feynman rules in scalar field theory and quantum electrodynamics is assumed, but all other tools are introduced as needed. Worked examples and more than 150 exercises are included. This title is also available as open access on Cambridge Core.

It is important to find nontrivial constraint relations for color-ordered amplitudes in gauge theories. In the past several years, a pure group-theoretic iterative method has been proposed to derive linear constraints on color-ordered amplitudes in SU(N) gauge theories. In this paper, we use the same method to derive linear constraints on four-point gluon amplitudes in SO(N) and Sp(2N) gauge theories. These constraints are derived up to four-loop order. It is found that there are $n=1,6,10,13,16$ constraint relations at $L=0,1,2,3,4$ loop orders in both SO(N) and Sp(2N) cases. Correspondingly, the numbers of independent four-point color-ordered amplitudes are $2,3, 5, 8, 11$ at $L=0,1,2,3,4$ loop orders in both theories.

This issue aims to serve as an introduction to our current understanding of the structure of scattering amplitudes in gauge theory, an area which has seen particularly rapid advances in recent years following decades of steady progress. The articles contained herein provide a snapshot of the latest developments which we hope will serve as a valuable resource for graduate students and other scientists wishing to learn about the current state of the field, even if our continually evolving understanding of the subject might soon render this compilation incomplete.Why the fascination with scattering amplitudes, which have attracted the imagination and dedicated effort of so many physicists? Part of it stems from the belief, supported now by numerous examples, that unexpected simplifications of otherwise apparently complicated calculations do not happen by accident. Instead they provide a strong motivation to seek out an underlying explanation. The insight thereby gained can subsequently be used to make the next class of seemingly impossible calculations not only possible, but in some cases even trivial. This two-pronged strategy of exploring and exploiting the structure of gauge theory amplitudes appeals to a wide audience from formal theorists interested in mathematical structure for the sake of its own beauty to more phenomenologically-minded physicists eager to speed up the next generation of analysis software.Understandably it is the maximally supersymmetric \U0001d4a9 = 4 Yang–Mills theory (SYM) which has the simplest structure and has correspondingly received the most attention. Rarely in theoretical physics are we fortunate enough to encounter a toy model which is simple enough to be solved completely yet rich enough to possess interesting non-trivial structure while simultaneously, and most importantly, being applicable (even if only as a good approximation) to a wide range of 'real' systems. The canonical example in quantum mechanics is of course the harmonic oscillator. In the much more complicated realm of four-dimensional quantum field theories, developments over the past several years have led to the extremely exciting, and already partially realized, prospect of completely solving SYM theory (at least in the planar approximation). This alone is a thrilling prospect for theorists, but the great interest in this subject stems in particular from the fact that this is not some obscure field theory but rather a gauge theory, and hence a close cousin of QCD. As reviewed in several of the articles in this issue, many of the insights and methods developed for SYM theory can be applied, with suitable care, to arbitrary gauge theories.It has occasionally been noted that the study of amplitudes is an experimental science in which expressions for, or empirically observed properties of, various scattering amplitudes serve as the 'data' to be collected and analyzed. The rapid pace of progress is made possible in part by the fact that new data is often available at the click of a mouse. The articles in this issue offer testament to the riches which have been discovered hiding in these data, and there is no doubt that more rewards await theorists with the ambition to seek them out.

The generic googly amplitudes in gauge theory are computed by using the Cachazo-Svrcek-Witten approach to perturbative calculation in gauge theory and the results are in agreement with the previously well-known ones. Within this approach we also discuss the parity transformation, charge conjugation and the dual Ward identity. We also extend this calculation to include fermions and the googly amplitudes with a single quark-anti-quark pair are obtained correctly from fermionic MHV vertices. At the end we briefly discuss the possible extension of this approach to gravity.

We review the structure of gauge theory scattering amplitudes at tree level and describe how a compact expression can be found which encodes all the tree-level amplitudes in the maximally supersymmetric theory. The expressions for the amplitudes reveal a dual superconformal symmetry. We describe how these ideas can be extended to leading singularities and the loop integrand in the planar theory and discuss the appearance of dual conformal symmetry in higher-dimensional gauge theories. This paper is an invited review for a special issue of Journal of Physics A: Mathematical and Theoretical devoted to ‘Scattering amplitudes in gauge theories’.

The googly amplitudes in gauge theory are computed by using the off shell MHV vertices with the newly proposed rules of Cachazo, Svrcek and Witten. The result is in agreement with the previously well-known results. In particular we also obtain a simple result for the all negative but one positive helicity amplitude when one of the external line is off shell.

This paper reviews the recent progress in twistor approaches to Wilson loops, amplitudes and their duality for super-Yang–Mills. Wilson loops and amplitudes are derived from first principles using the twistor action for maximally supersymmetric Yang–Mills theory. We start by deriving the MHV rules for gauge theory amplitudes from the twistor action in an axial gauge in twistor space, and show that this gives rise to the original momentum space version given by Cachazo, Svrček and Witten. We then go on to obtain from these the construction of the momentum twistor space loop integrand using (planar) MHV rules and show how it arises as the expectation value of a holomorphic Wilson loop in twistor space. We explain the connection between the holomorphic Wilson loop and certain light-cone limits of correlation functions. We give a brief review of other ideas in connection with amplitudes in twistor space: twistor-strings, recursion in twistor space, the Grassmannian residue formula for leading singularities and amplitudes as polytopes. This paper is an invited review for a special issue of Journal of Physics A: Mathematical and Theoretical devoted to ‘Scattering amplitudes in gauge theories’.

Dual conformal symmetry has had a huge impact on our understanding of planar scattering amplitudes in super Yang–Mills. At tree level, it combines with the original conformal symmetry generators to a Yangian algebra, a hallmark of integrability, and helps in determining the tree-level amplitudes. The latter are now known in closed form. At loop level, it determines the functional form of the four- and five-point scattering amplitudes to all orders in the coupling constant and gives restrictions at six points and beyond. The symmetry is best understood at loop level in terms of a novel AdS-inspired infrared regularization which makes the symmetry exact, despite the infrared divergences. This has important consequences for the basis of loop integrals in this theory. Recently, a number of selective reviews have appeared which discuss dual conformal symmetry, mostly at tree level. Here, we give an up-to-date account of dual conformal symmetry, focussing on its status at loop level.

Infrared finiteness and analyticity properties of the loop-loop scattering amplitudes in gauge theories

One-loop n-point gauge theory amplitudes, unitarity and collinear limits

We review on-shell and unitarity methods and discuss their application to precision predictions for Large Hadron Collider (LHC) physics. Being universal and numerically robust, these methods are straightforward to automate for next-to-leading-order computations within standard model and beyond. Several state-of-the-art results including studies of (W/Z+3)-jet and (W+4)-jet production have explicitly demonstrated the effectiveness of the unitarity method for describing multi-parton scattering. Here we review central ideas needed to obtain efficient numerical implementations. This includes on-shell loop-level recursions, the unitarity method, color management and further refined tricks.

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 SYM and supergravity. The solution, valid at both tree and loop level, expresses any NKMHV superamplitude in terms of a basis of ordinary amplitudes. Basis amplitudes are classified by semi-standard tableaux of rectangular ×K Young diagrams. The SUSY Ward identities also impose constraints on the matrix elements of candidate ultraviolet counterterms in 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.

We review two novel techniques used to calculate tree-level scattering amplitudes efficiently: MHV diagrams, and on-shell recursion relations. For the MHV diagrams, we consider applications to tree-level amplitudes and focus in particular on the supersymmetric formulation. We also briefly describe the derivation of loop amplitudes using MHV diagrams. For the recursion relations, after presenting their general proof, we discuss several applications to massless theories with and without supersymmetry, to theories with massive particles, and to graviton amplitudes in general relativity. This article is an invited review for a special issue of Journal of Physics A: Mathematical and Theoretical devoted to ‘Scattering amplitudes in gauge theories’.

We review recent progress in the understanding of symmetries for scattering amplitudes in superconformal Yang–Mills theory. It is summarized how the superficial breaking of superconformal symmetry by collinear anomalies and the renormalization process can be cured at tree and loop level. This is achieved by correcting the representation of the superconformal group on amplitudes. Moreover, we comment on the Yangian symmetry of scattering amplitudes and how it inherits these correction terms from the ordinary Lie algebra symmetry. Invariants under this algebra and their relation to the Graßmannian generating function for scattering amplitudes are discussed. Finally, parallel developments in superconformal Chern–Simons theory are summarized. This paper is an invited review for a special issue of Journal of Physics A: Mathematical and Theoretical devoted to ‘Scattering amplitudes in gauge theories’.