Maximal Helicity Violating Amplitudes Research Articles

All-order celestial OPE from on-shell recursion

We determine tree level, all-order celestial operator product expansions (OPEs) of gluons and gravitons in the maximally helicity violating (MHV) sector. We start by obtaining the all-order collinear expansions of MHV amplitudes using the inverse soft recursion relations that they satisfy. These collinear expansions are recast as celestial OPE expansions in bases of momentum as well as boost eigenstates. This shows that inverse soft recursion for MHV amplitudes is dual to OPE recursion in celestial conformal field theory.

Constructing EYM amplitudes by inverse soft limit

It is well known that gravity amplitudes in four dimensions can be reconstructed by the inverse soft limit (ISL) method. According to ISL, a tree level n-graviton maximally-helicity-violating (MHV) amplitude is expressed in terms of deformed (n − 1)-graviton amplitudes accompanied by soft graviton factors. On another hand, single- and double-trace tree-level Einstein-Yang-Mills (EYM) MHV amplitudes have been proven to satisfy spanning forest formulas, where each edge in a forest has the same form with a term in soft graviton factor. It is not transparent that the formulas satisfied by EYM amplitudes can be constructed with ISL. In this paper, we construct the single- and double-trace MHV amplitudes in EYM, by the ISL and show that the known formulas can be precisely reproduced. Interesting identities which are based on Schouten identity and characterized by graphs are also introduced.

A note on multi-trace EYM amplitudes in four dimensions

In four dimensions, a tree-level double-trace Einstein-Yang-Mills (EYM) amplitude with two negative-helicity gluons (the (g−, g−)-configuration) satisfies a symmetric spanning forest formula, which was derived from the graphic expansion rule. On another hand, in the framework of Cachazo-He-Yuan (CHY) formula, the maximally-helicity-violating (MHV) amplitudes are supported by the MHV solution of scattering equations. The relationship between the symmetric formula for double-trace amplitudes, and the MHV sector of Cachazo-He-Yuan (CHY) formula in four dimensions is still not clear. In this note, we promote a series of transformations of the spanning forests in four dimensions and then show a systematic way for decomposing the MHV sector of the CHY formula of double-trace EYM amplitudes. Along this line, the symmetric formula of double-trace MHV amplitudes is directly obtained by the MHV sector of CHY formula. We then prove that EYM amplitude with an arbitrary total number of negative-helicity particles (gravitons and gluons) has to vanish when the number of negative- (or positive-) helicity gluons is less than the number of traces.

Feynman Integrals and Scattering Amplitudes from Wilson Loops.

We study Feynman integrals and scattering amplitudes in N=4 super-Yang-Mills theory by exploiting the duality with null polygonal Wilson loops. As the main application, we compute for the first time the symbols of the general double pentagon integrals, which give the finite part of two-loop maximally helicity violating (MHV) amplitudes and finite components of next-to-MHV (NMHV) amplitudes to all multiplicities. The rational parts of the symbol consist of 164 letters, while the algebraic part contains 96 algebraic letters and cancel in MHV amplitudes and NMHV components which are free of square roots.

Six-Gluon amplitudes in planar mathcal{N} = 4 super-Yang-Mills theory at six and seven loops

We compute the six-particle maximally-helicity-violating (MHV) and next-to-MHV (NMHV) amplitudes in planar maximally supersymmetric Yang-Mills theory through seven loops and six loops, respectively, as an application of the extended Steinmann relations and using the cosmic Galois coaction principle. Starting from a minimal space of functions constructed using these principles, we identify the amplitude by matching its symmetries and predicted behavior in various kinematic limits. Through five loops, the MHV and NMHV amplitudes are uniquely determined using only the multi-Regge and leading collinear limits. Beyond five loops, the MHV amplitude requires additional data from the kinematic expansion around the collinear limit, which we obtain from the Pentagon Operator Product Expansion, and in particular from its single-gluon bound state contribution. We study the MHV amplitude in the self-crossing limit, where its singular terms agree with previous predictions. Analyzing and plotting the amplitudes along various kinematical lines, we continue to find remarkable stability between loop orders.

Minitwistors and 3d Yang-Mills-Higgs theory

We construct a minitwistor action for Yang–Mills–Higgs (YMH) theory in three dimensions. The Feynman diagrams of this action will construct perturbation theory around solutions of the Bogomolny equations in much the same way that MHV (maximally helicity violating) diagrams describe perturbation theory around the self-dual Yang Mills equations in four dimensions. We also provide a new formula for all tree amplitudes in YMH theory (and its maximally supersymmetric extension) in terms of degree d maps to minitwistor space. We demonstrate its relationship to the Roiban-Spradlin-Volovich-Witten (RSVW) formula in four dimensions and show that it generates the correct MHV amplitudes at d = 1 and factorizes correctly in all channels for all degrees.

MHV amplitudes at strong coupling and linearized TBA equations

The maximally helicity violating (MHV) amplitudes of mathcal{N}=4 super Yang-Mills theory at strong coupling are obtained by solving auxiliary thermodynamic Bethe ansatz (TBA) integral equations. We consider a limit where the TBA equations are linearized for large chemical potentials and masses therein. By solving the linearized equations, we derive analytic expansions of the 6-point MHV amplitudes in terms of the ratio of the chemical potential A and the mass M . The expansions are valid up to corrections exponentially small in A or inversely proportional to powers of A. The analytic expansions describe the amplitudes for small conformal cross-ratios of the particle momenta in a standard basis, and interpolate the amplitudes with equal cross-ratios and those in soft/collinear limits. The leading power corrections are also obtained analytically. We compare the 6-point rescaled remainder functions at strong coupling and at 2 loops for the above kinematics. They are rather different, in contrast to other kinematic regions discussed in the literature where they are found to be similar to each other.

CHY formula and MHV amplitudes

In this paper, we study the relation between the Cachazo-He-Yuan (CHY) formula and the maximal-helicity-violating (MHV) amplitudes of Yang-Mills and gravity in four dimensions. We prove that only one special rational solution of the scattering equations found by Weinzierl support the MHV amplitudes. Namely, localized at this solution, the integrated CHY formula reproduces the Parke-Taylor formula for Yang-Mills amplitudes as well as the Hodges formula for gravitational amplitudes. This is achieved by developing techniques, in a manifestly M\"obius covariant formalism, to explicitly compute relevant reduced Pfaffians/determinants. We observe and prove two interesting properties (or identities), which facilitate the computations. We also check that all the other $(n-3)!-1$ solutions to the scattering equations do not support the MHV amplitudes, and prove analytically that this is indeed true for the other special rational solution proposed by Weinzierl, that actually supports the anti-MHV amplitudes.

Permutation relations of generalized Yangian Invariants, unitarity cuts, and scattering amplitudes

We find a permutation relation among Yangian Invariants -- two Yangian Invariants with adjacent external lines exchanged are related by a simple kinematic factor. This relation is shown to be equivalent to U(1) decoupling and Bern-Carrasco-Johansson (BCJ) relation at the level of maximal helicity violating (MHV) amplitudes. Together with the newly found permutation relation and using unitarity cuts to remove ambiguity in the definition of loop momenta of cut amplitudes due to the nonplanar legs, we propose a systematic way of reconstructing the the integrands of nonplanar MHV amplitudes up to a rational function which vanishes under all possible unitarity cuts. This method when applied to planar diagrams reproduces results from the single cut method. As explicit examples the construction of one-loop double-trace MHV amplitudes of 4- and 5-point interactions are presented using on-shell diagrams. The kinematic factors and the resultant planar diagrams are carefully dealt with using the unitarity cut condition. The first next-to-MHV amplitudes are addressed using generalized unitarity cuts. Their leading singularities can be identified as residues of the Grassmanian integral. This example also serve to demonstrate the power of the newly found relation of Yangian Invariants.

Two-loop six-gluon maximally helicity violating amplitude in maximally supersymmetric Yang-Mills theory

We give a representation of the parity-even part of the planar two-loop six-gluon maximally helicity violating (MHV) amplitude of $\mathcal{N}=4$ super-Yang-Mills theory, in terms of loop-momentum integrals with simple dual conformal properties. We evaluate the integrals numerically in order to test directly the Anastasiou-Bern-Dixon-Kosower/Bern-Dixon-Smirnov all-loop ansatz for planar MHV amplitudes. We find that the ansatz requires an additive remainder function, in accord with previous indications from strong-coupling and Regge limits. The planar six-gluon amplitude can also be compared with the hexagonal Wilson loop computed by Drummond, Henn, Korchemsky, and Sokatchev in arXiv:0803.1466. After accounting for differing singularities and other constants independent of the kinematics, we find that the Wilson loop and MHV-amplitude remainders are identical, to within our numerical precision. This result provides nontrivial confirmation of a proposed $n$-point equivalence between Wilson loops and planar MHV amplitudes, and suggests that an additional mechanism besides dual conformal symmetry fixes their form at six points and beyond.

A twistor approach to one-loop amplitudes in [formula omitted] supersymmetric Yang–Mills theory

We extend the twistor string theory inspired formalism introduced in hep-th/0407214 for calculating loop amplitudes in N = 4 super-Yang–Mills theory to the case of N = 1 (and N = 2 ) super-Yang–Mills. Our approach yields a novel representation of the gauge theory amplitudes as dispersion integrals, which are surprisingly simple to evaluate. As an application we calculate one-loop maximally helicity violating (MHV) scattering amplitudes with an arbitrary number of external legs. The result we obtain agrees precisely with the expressions for the N = 1 MHV amplitudes derived previously by Bern, Dixon, Dunbar and Kosower using the cut-constructibility approach.

One-loop gauge theory amplitudes in [formula omitted] super Yang–Mills from MHV vertices

We propose a new, twistor string theory inspired formalism to calculate loop amplitudes in N = 4 super-Yang–Mills theory. In this approach, maximal helicity violating (MHV) tree amplitudes of N = 4 super-Yang–Mills are used as vertices, using an off-shell prescription introduced by Cachazo, Svrcek and Witten, and combined into effective diagrams that incorporate large numbers of conventional Feynman diagrams. As an example, we apply this formalism to the particular class of supersymmetric MHV one-loop scattering amplitudes with an arbitrary number of external legs in N = 4 super-Yang–Mills. Remarkably, our approach naturally leads to a representation of the amplitudes as dispersion integrals, which we evaluate exactly. This yields a new, simplified form for the MHV amplitudes, which is equivalent to the expressions obtained previously by Bern, Dixon, Dunbar and Kosower using the cut-constructibility approach.

One-Loop Maximal Helicity Violating Amplitudes in N=4 Super Yang-Mills Theories

One-loop maximal helicity violating (MHV) amplitudes in N=4 super Yang-Mills (SYM) theories are analyzed, using the prescription of Cachazo, Svrcek, and Witten (CSW). The relations between leading N_c amplitudes A_{n;1} and sub-leading amplitudes A_{n;c} obtained by the CSW prescription are found to be identical to those obtained from conventional field theory calculations. Combining with existing results, this establishes the validity of the CSW prescription to one-loop in the calculation of MHV amplitudes in N=4 SYM theories of finite N_c.

An Early Route to MHV Tree Amplitudes

In the perturbative expansion of a gauge theory, large numbers of Feynman amplitudes combine to produce mathematically simple (and elegant) expressions. So many people had long suspected that deeper symmetry structures were involved, but those structures remained tantalizingly beyond reach until recently. The Search and Discovery story in the July 2004 issue of Physics Today (page 19) gives a lucid and detailed explanation of the evolving understanding of symmetry structures that underlie Feynman amplitudes. Identification of the connection between the maximally helicity-violating (MHV) amplitudes for the N = 4 super Yang–Mills theory and supertwistor space was a crucial ingredient in the recent developments.I would like to share the reasoning that led me, some years ago, to make an early proposal, 1 1. V. P. Nair, Phys. Lett. B 214, 215 (1988). https://doi.org/10.1016/0370-2693(88)91471-2 which unfortunately was not mentioned in the Physics Today story. I noticed that the Parke–Taylor formula for the MHV amplitudes involved the inverse of scalar products of spinor momenta that could be related to free fermion propagators (or current correlations of a Wess-Zumino-Witten theory) on the complex projective space CP 1. The space arises naturally (as the fiber) in twistor space. Earlier, Edward Witten had observed that the constraints defining the N = 4 super Yang–Mills theory could be nicely interpreted in supertwistor space. 2 2. E. Witten, Phys. Lett. B 77, 394 (1978). https://doi.org/10.1016/0370-2693(78)90585-3 Putting these observations together, I wrote a formula for MHV amplitudes in N = 4 Yang–Mills theory in twistor language. Conformal symmetry plays an important role. Witten’s recent work is more comprehensive, generalizing this complicated series of relationships to string theory and non-MHV amplitudes, and leading to many beautiful results that, for the special case of MHV amplitudes, are identical to mine. The work I’ve just described may be useful for practical calculations; but beyond that use, I hope there will emerge a new organizational principle for perturbation theory other than expansion in terms of Feynman amplitudes.REFERENCESSection:ChooseTop of pageREFERENCES <<1. V. P. Nair, Phys. Lett. B 214, 215 (1988). https://doi.org/10.1016/0370-2693(88)91471-2 , Google ScholarCrossref, CAS2. E. Witten, Phys. Lett. B 77, 394 (1978). https://doi.org/10.1016/0370-2693(78)90585-3 , Google ScholarCrossref© 2004 American Institute of Physics.