Gluon Amplitudes Research Articles

Conformal supergravity tree amplitudes from open twistor string theory

We display the vertex operators for all states in the conformal supergravity sector of the twistor string, as outlined by Berkovits and Witten. These include ‘dipole’ states, which are pairs of supergravitons that do not diagonalize the translation generators. We use canonical quantization of the open string version of Berkovits, and compute N-point tree level scattering amplitudes for gravitons, gluons and scalars. We reproduce the Berkovits–Witten formula for maximal helicity violating (MHV) amplitudes (which they derived using path integrals), and extend their results to the dipole pairs. We compare these trees with those of Einstein gravity field theory.

All tree-level amplitudes in 𝒩 = 4 SYM

We give an explicit formula for all tree amplitudes in = 4 SYM, derived by solving the recently presented supersymmetric tree-level recursion relations. The result is given in a compact, manifestly supersymmetric form and we show how to extract from it all possible component amplitudes for an arbitrary number of external particles and any arrangement of external particles and helicities. We focus particularly on extracting gluon amplitudes which are valid for any gauge theory. The formula for all tree-level amplitudes is given in terms of nested sums of dual superconformal invariants and it therefore manifestly respects both conventional and dual superconformal symmetry.

One-loop gluon amplitude for heavy-quark production at next-to-next-to-leading order

We compute the one-loop QCD amplitude for the process $gg\ensuremath{\rightarrow}Q\overline{Q}$ in dimensional regularization through order ${ϵ}^{2}$ in the dimensional regulator and for arbitrary quark mass values. This result is an ingredient of the next-to-next-to-leading order cross section for heavy-quark production at hadron colliders. The calculation is performed in conventional dimensional regularization using well-known reduction techniques as well as a method based on recent ideas for the functional form of one-loop integrands in four dimensions.

MHV, CSW and BCFW: field theory structures in string theory amplitudes

Motivated by recent progress in calculating field theory amplitudes, we study applications of the basic ideas in these developments to the calculation of amplitudes in string theory. We consider in particular both non-Abelian and Abelian open superstring disk amplitudes in a flat space background, focusing mainly on the four-dimensional case. The basic field theory ideas under consideration split into three separate categories. In the first, we argue that the calculation of alpha'-corrections to MHV open string disk amplitudes reduces to the determination of certain classes of polynomials. This line of reasoning is then used to determine the alpha'^3-correction to the MHV amplitude for all multiplicities. A second line of attack concerns the existence of an analog of CSW rules derived from the Abelian Dirac-Born-Infeld action in four dimensions. We show explicitly that the CSW-like perturbation series of this action is surprisingly trivial: only helicity conserving amplitudes are non-zero. Last but not least, we initiate the study of BCFW on-shell recursion relations in string theory. These should appear very naturally as the UV properties of the string theory are excellent. We show that all open four-point string amplitudes in a flat background at the disk level obey BCFW recursion relations. Based on the naturalness of the proof and some explicit results for the five-point gluon amplitude, it is expected that this pattern persists for all higher point amplitudes and for the closed string.

Gluon Scattering in = 4 Super Yang-Mills at Finite Temperature

We extend the AdS/CFT prescription of Alday and Maldacena to finite temperature T , defining an amplitude for gluon scattering in N = 4 Super Yang-Mills at strong coupling from string theory. It is defined by a lightlike “Wilson loop” living at the horizon of the T-dual to the black hole in AdS space. Unlike the zero temperature case, this is different from the Wilson loop contour defined at the boundary of the AdS black hole metric. Thus at nonzero T there is no relation between gluon scattering amplitudes and the Wilson loop. We calculate a gauge theory observable that can be interpreted as the amplitude at strong coupling for forward scattering of a low energy gluon (E < T ) off a high energy gluon (E T ) in both cutoff and generalized dimensional regularization. The generalized dimensional regularization is defined in string theory as an IR modified dimensional reduction. For this calculation, the corresponding usual Wilson loop of the same boundary shape was argued to be related to the jet quenching parameter of the finite temperature N = 4 SYM plasma, while the gluon scattering amplitude is related to the viscosity coefficient.

On tree amplitudes in gauge theory and gravity

The BCFW recursion relations provide a powerful way to compute tree amplitudes in gauge theories and gravity, but only hold if some amplitudes vanish when two of the momenta are taken to infinity in a particular complex direction. This is a very surprising property, since individual Feynman diagrams all diverge at infinite momentum. In this paper we give a simple physical understanding of amplitudes in this limit, which corresponds to a hard particle with (complex) light-like momentum moving in a soft background, and can be conveniently studied using the background field method exploiting background light-cone gauge. An important role is played by enhanced spin symmetries at infinite momentum--a single copy of a "Lorentz" group for gauge theory and two copies for gravity--which together with Ward identities give a systematic expansion for amplitudes at large momentum. We use this to study tree amplitudes in a wide variety of theories, and in particular demonstrate that certain pure gauge and gravity amplitudes do vanish at infinity. Thus the BCFW recursion relations can be used to compute completely general gluon and graviton tree amplitudes in any number of dimensions. We briefly comment on the implications of these results for computing massive 4D amplitudes by KK reduction, as well understanding the unexpected cancelations that have recently been found in loop-level gravity amplitudes.

Quark scattering amplitudes at strong coupling

Following Alday and Maldacena [1], we describe a string theory method to compute the strong coupling behavior of the scattering amplitudes of quarks and gluons in planar = 2 super Yang-Mills theory in the probe approximation. Explicit predictions for these quantities can be constructed using the all-orders planar gluon scattering amplitudes of = 4 super Yang-Mills due to Bern, Dixon and Smirnov [2].

(Non)-supersymmetric marginal deformations from twistor string theory

The tree-level amplitudes in the β-deformed supersymmetric gauge theory are derived from twistor string theory. We first show that a simple generalization of [M. Kulaxizi, K. Zoubos, Marginal deformations of N = 4 SYM from open/closed twistor strings, Nucl. Phys. B 738 (2006) 317, hep-th/0410122] gives us the desired results for all the tree-level amplitudes up to first order in the deformation parameter β. Then we provide a new proposal which matches field theory to all orders of β. With our deformed twistor string theory, it can be shown that integration over connected instantons and disconnected instantons remain equivalent. The tree-level amplitudes in the non-supersymmetric γ-deformed theory can be obtained using a similar deformation of the twistor string. Finally, we find improved twistor string results for the tree-level purely gluonic amplitudes in theories with more general marginal deformations.

Some properties of the Alday–Maldacena minimum

The Alday–Maldacena solution, relevant to the n=4 gluon amplitude in N=4 SYM at strong coupling, was recently identified as a minimum of the regularized action in the moduli space of solutions of the AdS5σ-model equations of motion. Analogous solutions of the Nambu–Goto equations for the n=4 case are presented and shown to form (modulo the reparametrization group) an equally large but different moduli space, with the Alday–Maldacena solution at the intersection of the σ-model and Nambu–Goto moduli spaces. We comment upon the possible form of the regularized action for n=5. A function of moduli parameters za is written, whose minimum reproduces the BDDK one-loop five-gluon amplitude. This function may thus be considered as some kind of Legendre transform of the BDDK formula and has its own value independently of the Alday–Maldacena approach.

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.

Infrared limit of gluon amplitudes at strong coupling

In this Letter, we propose that the infrared structure of gluon amplitudes at strong coupling can be fully extracted from a local consideration near cusps. This is consistent with field theory and correctly reproduces the infrared divergences of the four-gluon amplitude at strong coupling calculated recently by Alday and Maldacena.

Tree and loop amplitudes in open twistor string theory

We compute the one-loop gluon amplitude for the open twistor string model of Berkovits, using a symmetric form of the vertex operators. We discuss the classical solutions in various topologies and instanton sectors and the canonical quantization of the world sheet Lagrangian. We derive the N-point functions for the gluon tree and one-loop amplitudes, and calculate a general one-loop expression for the current algebra.

All one-loop maximally helicity violating gluonic amplitudes in QCD

We use on-shell recursion relations to compute analytically the one-loop corrections to maximally-helicity-violating n-gluon amplitudes in QCD. The cut-containing parts have been computed previously; our work supplies the remaining rational parts for these amplitudes, which contain two gluons of negative helicity and the rest positive, in an arbitrary color ordering. We also present formulae specific to the six-gluon cases, with helicities (-+-+++) and (-++-++), as well as numerical results for six, seven, and eight gluons. Our construction of the n-gluon amplitudes illustrates the relatively modest growth in complexity of the on-shell-recursive calculation as the number of external legs increases. These amplitudes add to the growing body of one-loop amplitudes known for all n, which are useful for studies of general properties of amplitudes, including their twistor-space structure.

Integrability and transcendentality

We derive the two-loop Bethe ansatz for the twist operator sector of gauge theory directly from the field theory. We then analyse a recently proposedperturbative asymptotic all-loop Bethe ansatz in the limit of large spacetime spin at largebut finite twist, and find a novel all-loop scaling function. This function obeys theKotikov–Lipatov transcendentality principle and does not depend on the twist. Under theassumption that one may extrapolate back to leading twist, our result yields an all-loopprediction for the large spin anomalous dimensions of twist 2 operators. Thelatter also appears as an undetermined function in a recent conjecture of Bern,Dixon and Smirnov for the all-loop structure of the maximally helicity violatingn-point gluon amplitudes of gauge theory. This potentially establishes a direct link between the worldsheet and the spacetimeS matrix approach. A further assumption for the validity of our prediction is thatperturbative BMN (Berenstein–Maldacena–Nastase) scaling does not break down atfour-loop level or beyond. We also discuss how the result gets modified if BMN scaling doesbreak down. Finally, we show that our result qualitatively agrees at strong coupling with aprediction of string theory.

Bootstrapping one-loop QCD amplitudes with general helicities

The recently developed on-shell bootstrap for computing one-loop amplitudes in nonsupersymmetric theories such as QCD combines the unitarity method with loop-level on-shell recursion. For generic helicity configurations, the recursion relations may involve undetermined contributions from nonstandard complex singularities or from large values of the shift parameter. Here we develop a strategy for sidestepping difficulties through the use of pairs of recursion relations. To illustrate the strategy, we present sets of recursion relations needed for obtaining $n$-gluon amplitudes in QCD. We give a recursive solution for the one-loop $n$-gluon QCD amplitudes with three or four color-adjacent gluons of negative helicity and the remaining ones of positive helicity. We provide an explicit analytic formula for the QCD amplitude ${A}_{6;1}({1}^{\ensuremath{-}},{2}^{\ensuremath{-}},{3}^{\ensuremath{-}},{4}^{+},{5}^{+},{6}^{+})$, as well as numerical results for ${A}_{7;1}({1}^{\ensuremath{-}},{2}^{\ensuremath{-}},{3}^{\ensuremath{-}},{4}^{+},{5}^{+},{6}^{+},{7}^{+})$, ${A}_{8;1}({1}^{\ensuremath{-}},{2}^{\ensuremath{-}},{3}^{\ensuremath{-}},{4}^{+},{5}^{+},{6}^{+},{7}^{+},{8}^{+})$, and ${A}_{8;1}({1}^{\ensuremath{-}},{2}^{\ensuremath{-}},{3}^{\ensuremath{-}},{4}^{\ensuremath{-}},{5}^{+},{6}^{+},{7}^{+},{8}^{+})$. We expect the on-shell bootstrap approach to have widespread applications to phenomenological studies at colliders.

Diagrammatic proof of the BCFW recursion relation for gluon amplitudes in QCD

We present a proof of the Britto-Cachazo-Feng-Witten tree-level recursion relation for gluon amplitudes in QCD, based on a direct equivalence between BCFW decompositions and Feynman diagrams. We demonstrate that this equivalence can be made explicit when working in a convenient gauge. We exhibit that gauge invariance and the particular structure of Yang-Mills vertices guarantees the validity of the BCFW construction.

A comparison of efficient methods for the computation of Born gluon amplitudes

We compare four different methods for the numerical computation of the pure gluonic amplitudes in the Born approximation. We are in particular interested in the efficiency of the various methods as the number n of the external particles increases. In addition we investigate the numerical accuracy in critical phase space regions. The methods considered are based on (i) Berends-Giele recurrence relations, (ii) scalar diagrams, (iii) MHV vertices and (iv) BCF recursion relations.

All-multiplicity one-loop corrections to maximum-helicity-violating amplitudes in QCD

We compute the complete one-loop corrections to the simplest class of QCD gluon amplitudes, those with two color-adjacent opposite-helicity external particles. We present results for an arbitrary number of external legs. This is the first all-$n$ computation of amplitudes that enter into collider observables at next-to-leading order. The computation uses the recently-developed on-shell recursion relations for the rational parts, along with older unitarity-based results for the cut-containing terms.

A deformation of twistor space and a chiral mass term inN= 4 super Yang-Mills Theory

Super twistor space admits a certain (super) complex structure deformation that preserves the Poincare subgroup of the symmetry group PSL(4|4) and depends on 10 parameters. In a previous paper [hep-th/0502076], it was proposed that in twistor string theory this deformation corresponds to augmenting N=4 super Yang-Mills theory by a mass term for the left-chirality spinors. In this paper we analyze this proposal in more detail. We calculate 4-particle scattering amplitudes of fermions, gluons and scalars and show that they are supported on holomorphic curves in the deformed twistor space.

A direct proof of the CSW rules

Using recursion methods similar to those of Britto, Cachazo, Feng and Witten (BCFW) a direct proof of the CSW rules for computing tree-level gluon amplitudes is given.