Tree-level MHV Research Articles

Yang-Mills form factors on self-dual backgrounds

The construction of perturbative quantities on non-linear backgrounds leads to the possibility of incorporating strong field effects in perturbation theory. We continue a programme to construct QFT observables on self-dual backgrounds in Yang-Mills theory. The approach works with asymptotic data for fields defined at null infinity mathcal{I} , extending earlier work on Yang-Mills amplitudes on self-dual backgrounds to form factors and further incorporating supersymmetry. Since our analysis is based on reconstruction from data at null infinity, it naturally ties into work on celestial and twisted holography. We study form factors both in pure Yang-Mills and their supersymmetric counterparts in mathcal{N} = 4 SYM, giving a full treatment of mathcal{N} = 4 super-Yang-Mills at null infinity and their self-dual nonlinear backgrounds. We obtain tree-level MHV form factors around these backgrounds using new formulae for lifting operators to twistor space leading to simple dressings of the corresponding form factors around the vacuum. We give brief indications on how to go beyond the MHV sector by introducing dressed versions of the MHV diagram propagator. We discuss generating functionals of the MHV all plus 1-loop amplitude in this context together with its various dual conformal representations.

Celestial Liouville theory for Yang-Mills amplitudes

We consider Yang-Mills theory with the coupling constant and theta angle determined by the vacuum expectation values of a dynamical (complex, zero mass) dilaton field. We discuss the tree-level N-gluon MHV scattering amplitudes in the presence of a nontrivial background dilaton field and construct the corresponding celestial amplitudes by taking Mellin transforms with respect to the light cone frame energies. In this way, we obtain two-dimensional CFT correlators of primary fields on the celestial sphere. We show that the celestial Yang-Mills amplitudes evaluated in the presence of a spherical dilaton shockwave are given by the correlation functions of primary field operators factorized into the holomorphic current operators times the “light” Liouville operators. They are evaluated in the semiclassical limit of Liouville theory (the limit of infinite central charge) and are determined by the classical Liouville field describing metrics on the celestial sphere.

(Chiral) Virasoro invariance of the tree-level MHV graviton scattering amplitudes

In this paper we continue our study of the tree level MHV graviton scattering amplitudes from the point of view of celestial holography. In arXiv:2008.04330 we showed that the celestial OPE of two gravitons in the MHV sector can be written as a linear combination of overline{mathrm{SL}left(2,mathbb{C}right)} current algebra and supertranslation descendants. In this note we show that the OPE is in fact manifestly invariant under the infinite dimensional Virasoro algebra as is expected for a 2-D CFT. This is consistent with the conjecture that the holographic dual in 4-D asymptotically flat space time is a 2-D CFT. Since we get only one copy of the Virasoro algebra we can conclude that the holographic dual theory which computes the MHV amplitudes is a chiral CFT with a host of other infinite dimensional global symmetries including overline{mathrm{SL}left(2,mathbb{C}right)} current algebra, supertranslations and subsubleading soft graviton symmetry. We also discuss some puzzles related to the appearance of the Virasoro symmetry.

Celestial dual superconformal symmetry, MHV amplitudes and differential equations

Celestial and momentum space amplitudes for massless particles are related to each other by a change of basis provided by the Mellin transform. Therefore properties of celestial amplitudes have counterparts in momentum space amplitudes and vice versa. In this paper, we study the celestial avatar of dual superconformal symmetry of mathcal{N} = 4 Yang-Mills theory. We also analyze various differential equations known to be satisfied by celestial n-point tree-level MHV amplitudes and identify their momentum space origins.

MHV gluon scattering amplitudes from celestial current algebras

We show that the Mellin transform of an n-point tree level MHV gluon scattering amplitude, also known as the celestial amplitude in pure Yang-Mills theory, satisfies a system of (n−2) linear first order partial differential equations corresponding to (n−2) positive helicity gluons. Although these equations closely resemble Knizhnik-Zamoldochikov equations for SU(N) current algebra there is also an additional “correction” term coming from the subleading soft gluon current algebra. These equations can be used to compute the leading term in the gluon-gluon OPE on the celestial sphere. Similar equations can also be written down for the momentum space tree level MHV scattering amplitudes. We also propose a way to deal with the non closure of subleading current algebra generators under commutation. This is then used to compute some subleading terms in the mixed helicity gluon OPE.

Subsubleading soft graviton symmetry and MHV graviton scattering amplitudes

In [8] it was shown that supertranslation and overline{mathrm{SL}left(2,mathbb{C}right)} current algebra symmetries, corresponding to leading and subleading soft graviton theorems, are enough to determine the tree level MHV graviton scattering amplitudes. In this note we clarify the role of subsubleading soft graviton theorem in this context.

Composite operators and form factors in SYM

We construct the most general composite operators of SYM in Lorentz harmonic chiral (≈twistor) superspace. The operators are built from the SYM supercurvature which is nonpolynomial in the chiral gauge prepotentials. We reconstruct the full nonchiral dependence of the supercurvature. We compute all tree-level MHV form factors via the LSZ reduction procedure with on-shell states made of the same supercurvature.

Direct evaluation of n-point single-trace MHV amplitudes in 4d Einstein-Yang-Mills theory using the CHY formalism

In this paper we extend our techniques, developed in a previous paper (Du, etc, JHEP 05(2016)086) for direct evaluation of arbitrary $n$-point tree-level MHV amplitudes in 4d Yang-Mills and gravity theory using the Cachazo-He-Yuan (CHY) formalism, to the 4d Einstein-Yang-Mills (EYM) theory. Any single-trace color-ordered $n$-point tree-level MHV amplitude in EYM theory, obtained by a direct evaluation of the CHY formula, is of an elegant factorized form of a Parke-Taylor factor and a Hodges determinant, much simpler and more compact than the existing formulas in the literature. We prove that our new expression is equivalent to the conjectured Selivanov-Bern-De Freitas-Wong (SBDW) formula, with the help of a new theorem showing that the SBDW generating function has a graph theory interpretation. Together with Ref. (Du, etc, JHEP 05(2016)086), we provide strong analytic evidence for hidden simplicity in quantum field theory.

All tree-level MHV form factors in N $$ \mathcal{N} $$ = 4 SYM from twistor space

We incorporate all gauge-invariant local composite operators into the twistor-space formulation of N=4 SYM theory, detailing and expanding on ideas we presented recently in arXiv:1603.04471. The vertices for these operators contain infinitely many terms and we show how they can be constructed by taking suitable derivatives of a light-like Wilson loop in twistor space and shrinking it down to a point. In particular, these vertices directly yield the tree-level MHV super form factors of all composite operators in N=4 SYM theory.

Recursion in multiplet bases for tree-level MHV gluon amplitudes

We investigate the construction of tree-level MHV gluon amplitudes in multiplet bases using BCFW recursion. The multiplet basis decomposition can either be obtained by decomposing results derived in (for example) the DDM basis or by formulating the recursion directly in the multiplet basis. We focus on the latter approach and show how to efficiently deal with the color structure appearing in the recursion. For illustration, we also explicitly calculate the four-, five- and six-gluon amplitudes.

A simpler prescription for MHV graviton tree amplitudes in superstring theory

We extend the Berkovits-Maldacena prescription for MHV amplitudes of the open superstring to the closed superstring, showing that in the \alpha'=0 limit it reduces to the result of supergravity found recently by Hodges. We also verify that this prescription calculates the correct superstring tree level MHV amplitude for 4 gravitons including \alpha' corrections.

On-shell superamplitudes in $ \mathcal{N} < 4 $ SYM

We present an on-shell formalism for superamplitudes of pure N<4 super Yang-Mills theory. Two superfields, Phi and Phi^+, are required to describe the two CPT conjugate supermultiplets. Simple truncation prescriptions allow us to derive explicit tree-level MHV and NMHV superamplitudes with N-fold SUSY. Any N=0,1,2 tree superamplitudes have large-z falloffs under super-BCFW shifts, except under [Phi,Phi^+>-shifts. We show that this `bad' shift is responsible for the bubble contributions to 1-loop amplitudes in N=0,1,2 SYM. We evaluate the MHV bubble coefficients in a manifestly supersymmetric form and demonstrate for the case of four external particles that the sum of bubble coefficients is equal to minus the tree superamplitude times the 1-loop beta-function coefficient. The connection to the beta-function is expected since only bubble integrals capture UV divergences; we discuss briefly how the minus sign arises from UV and IR divergences in dimensional regularization. Other applications of the on-shell formalism include a solution to the N^{K}MHV N=1 SUSY Ward identities and a clear description of the connection between 6d superamplitudes and the 4d ones for both N=4 and N=2 SYM. We outline extensions to N<8 supergravity.

Soft-exactness of tree-level gluon and graviton MHV amplitudes

We discuss and present derivations of the must succinct forms for tree-level MHV gluon and MHV graviton scattering amplitudes, through appending soft-factors to lower-point MHV scattering amplitudes. For gluons, we see that the canonical spinor-helicity variables are sufficient to absorb all of the gauge redundancy needed to define transverse gluon polarization vectors. Correspondingly, the tree level gluon MHV scattering amplitudes can be cleanly and efficintly written in terms of spinor-helicity variables–and are described by the famous Parke-Taylor amplitude. However, the canonical spinor-helicity variables do not automatically enforce momentum conservation, and thus are not sufficiently redundant to absorb all of the differomorphism invariance that must be built into the graviton soft-factors. Nonetheless, appending graviton soft-factors to lower-point graviton MHV amplitudes yields the simplest known form for MHV graviton amplitudes. We briefly discuss new variables, better suited to describe perturbative scattering amplitudes in quantum gravity.

The kinematic algebra from the self-dual sector

We identify a diffeomorphism Lie algebra in the self-dual sector of Yang-Mills theory, and show that it determines the kinematic numerators of tree-level MHV amplitudes in the full theory. These amplitudes can be computed off-shell from Feynman diagrams with only cubic vertices, which are dressed with the structure constants of both the Yang-Mills colour algebra and the diffeomorphism algebra. Therefore, the latter algebra is the dual of the colour algebra, in the sense suggested by the work of Bern, Carrasco and Johansson. We further study perturbative gravity, both in the self-dual and in the MHV sectors, finding that the kinematic numerators of the theory are the BCJ squares of the Yang-Mills numerators.

Dual superconformal symmetry of scattering amplitudes in [formula omitted] super-Yang–Mills theory

We argue that all scattering amplitudes in the maximally supersymmetric N = 4 super-Yang–Mills theory possess a new, dual superconformal symmetry which extends the previously discovered dual conformal symmetry of MHV amplitudes. To reveal this property we formulate the scattering amplitudes as functions on the appropriate dual superspace. Rewritten in this form, all tree-level MHV and next-to-MHV amplitudes exhibit manifest dual superconformal symmetry. We propose a new, compact and Lorentz covariant formula for the tree-level NMHV amplitudes for arbitrary numbers and types of external particles. The dual superconformal symmetry is broken at loop level by infrared divergences. However, we provide evidence that the dual conformal anomaly of the MHV and NMHV superamplitudes is the same and, therefore, their ratio is dual conformally invariant. We show this explicitly for the six-particle amplitudes at one loop. We conjecture that these properties hold for all, MHV and non-MHV, superamplitudes in N = 4 SYM both at weak and at strong coupling.

One-loop supergravity amplitudes from MHV diagrams

We discuss the calculation of one-loop amplitudes in supergravity using MHV diagrams. In this approach, one-loop amplitudes are expressed as sums of MHV diagrams, which incorporate large numbers of Feynman diagrams. In contrast to MHV amplitudes of gluons in Yang–Mills, tree-level MHV amplitudes of gravitons are not holomorphic in the spinor variables. In order to extend these amplitudes off shell, and use them as vertices to build loops, we introduce certain shifts for the spinor variables associated with the loop momenta. Using this off-shell prescription, we rederive in a compact way the four- and five-point MHV amplitudes of gravitons at one loop, in complete agreement with known results. We also discuss the case of one-loop MHV amplitudes with an arbitrary number of gravitons.

MHV Vertices And Tree Amplitudes In Gauge Theory

As an alternative to the usual Feynman graphs, tree amplitudes in Yang-Mills theory can be constructed from tree graphs in which the vertices are tree level MHV scattering amplitudes, continued off shell in a particular fashion. The formalism leads to new and relatively simple formulas for many amplitudes, and can be heuristically derived from twistor space.