Five-point Amplitudes Research Articles

Amplitudes and correlators to ten loops using simple, graphical bootstraps

We introduce two new graphical-level relations among possible contributions to the four-point correlation function and scattering amplitude in planar, maximally supersymmetric Yang-Mills theory. When combined with the rung rule, these prove powerful enough to fully determine both functions through ten loops. This then also yields the full five-point amplitude to eight loops and the parity-even part to nine loops. We derive these rules, illustrate their applications, compare their relative strengths for fixing coefficients, and survey some of the features of the previously unknown nine and ten loop expressions. Explicit formulae for amplitudes and correlators through ten loops are available at: http://goo.gl/JH0yEc.

Role of Möbius constants and scattering functions in Cachazo-He-Yuan scalar amplitudes

The integrations leading to the Cachazo-He-Yuan (CHY) double-color $n$-point massless scalar amplitude are carried out one integral at a time. M\"obius invariance dictates the final amplitude to be independent of the three M\"obius constants $\sigma_r, \sigma_s, \sigma_t$, but their choice affects integrations and the intermediate results. The effect of the M\"obius constants, the two colors, and the scattering functions on each integration is investigated. A systematic way to carry out the $n-3$ integrations is explained, each exposing one of the $n-3$ propagators of the Feynman diagrams. Two detailed examples are shown to illustrate the procedure, one a five-point amplitude, and the other a nine-point amplitude.

The pentabox Master Integrals with the Simplified Differential Equations approach

We present the calculation of massless two-loop Master Integrals relevant to five-point amplitudes with one off-shell external leg and derive the complete set of planar Master Integrals with five on-mass-shell legs, that contribute to many $2\to 3$ amplitudes of interest at the LHC, as for instance three jet production, $\gamma, V, H +2$ jets etc., based on the Simplified Differential Equations approach.

Two-loop superstring five-point amplitude andS-duality

The low-energy limit of the massless two-loop five-point amplitudes for both type IIA and type IIB superstrings is computed with the pure spinor formalism and its overall coefficient determined from first principles. For the type IIB theory, the five-graviton amplitude is found to be proportional to its tree-level counterpart at the corresponding order in $\alpha'$. Their ratio ties in with expectations based on S-duality since it matches the same modular function $E_{5/2}$ which relates the two-loop and tree-level four-graviton amplitudes. For R-symmetry violating states, the ratio between tree-level and two-loop amplitudes at the same $\alpha'$-order carries an additional factor of $-3/5$. Its S-duality origin can be traced back to a modular form derived from $E_{5/2}$.

Some tree-level string amplitudes in the NSR formalism

We calculate tree level scattering amplitudes for open strings using the NSR formalism. We present a streamlined symmetry-based and pedagogical approach to the computations, which we first develop by checking two-, three-, and four-point functions involving bosons and fermions. We calculate the five-point amplitude for massless gluons and find agreement with an earlier result by Brandt, Machado and Medina. We then compute the five-point amplitudes involving two and four fermions respectively, the general form of which has not been previously obtained in the NSR formalism. The results nicely confirm expectations from the supersymmetric $F^4$ effective action. Finally we use the prescription of Kawai, Lewellen and Tye (KLT) to compute the amplitudes for the closed string sector.

Two-loop five-point amplitudes of super Yang-Mills and supergravity in pure spinor superspace

Supersymmetric integrands for the two-loop five-point amplitudes in ten-dimensional super Yang–Mills and type II supergravity are proposed. The kinematic numerators are manifestly local and satisfy the duality between color and kinematics described by Bern, Carrasco and Johansson. Our results are expected to reproduce the integrated two-loop amplitudes in dimensions D < 7. The UV divergence in the critical dimension D = 7 matches the low-energy limit of the corresponding superstring amplitudes and is written in terms of SYM tree amplitudes.

Symmetries and Feynman rules for the Ramond sector in open superstring field theory

We examine the symmetries of the action suppelemented by the constraint in the WZW-type open superstring field theory. It is found that this pseudo-action has additional symmetries provided we impose the constraint after the transformation. Respecting these additional symmetries, we propose a prescription for the new Feynman rules for the Ramond sector. It is shown that the new rules reproduce the well-known on-shell four- and five-point amplitudes with external fermions.

Local integrands for the five-point amplitude in planar N=4 SYM up to five loops

Integrands for colour ordered scattering amplitudes in planar N=4 SYM are dual to those of correlation functions of the energy-momentum multiplet of the theory. The construction can relate amplitudes with different numbers of legs. By graph theory methods the integrand of the four-point function of energy-momentum multiplets has been constructed up to six loops in previous work. In this article we extend this analysis to seven loops and use it to construct the full integrand of the five-point amplitude up to five loops, and in the parity even sector to six loops. All results, both parity even and parity odd, are obtained in a concise local form in dual momentum space and can be displayed efficiently through graphs. We have verified agreement with other local formulae both in terms of supertwistors and scalar momentum integrals as well as BCJ forms where those exist in the literature, i.e. up to three loops. Finally we note that the four-point correlation function can be extracted directly from the four-point amplitude and so this uncovers a direct link from four- to five-point amplitudes.

Scattering equations and virtuous kinematic numerators and dual-trace functions

Inspired by recent developments on scattering equations, we present a constructive procedure for computing symmetric, amplitude-encoded, BCJ numerators for n-point gauge-theory amplitudes, thus satisfying the three virtues identified by Broedel and Carrasco. We also develop a constructive procedure for computing symmetric, amplitude-encoded dual-trace functions (tau) for n-point amplitudes. These can be used to obtain symmetric kinematic numerators that automatically satisfy color-kinematic duality. The S_n symmetry of n-point gravity amplitudes formed from these symmetric dual-trace functions is completely manifest. Explicit expressions for four- and five-point amplitudes are presented.

Color-kinematics duality and the Regge limit of inelastic amplitudes

We investigate tree-level five-point amplitudes in scalar-QCD expressed in terms of Sudakov variables and find the equivalent "gravitational" counterparts using the color-kinematics duality proposed by Bern, Carrasco, and Johannson. Taking the multi-Regge limit in the gravitational amplitudes, we show that those pieces in the coupling of two reggeized gravitons to one on-shell graviton directly stemming from the double copy of the vertex for two reggeized gluons to one on-shell gluon are universal and properly reproduced by the duality.

Graviton emission in Einstein-Hilbert gravity

The five-point amplitude for the scattering of two distinct scalars with the emission of one graviton in the final state is calculated in exact kinematics for Einstein-Hilbert gravity. The result, which satisfies the Steinmann relations, is expressed in Sudakov variables, finding that it corresponds to the sum of two gauge invariant contributions written in terms of a new two scalar - two graviton effective vertex. A similar calculation is carried out in Quantum Chromodynamics (QCD) for the scattering of two distinct quarks with one extra gluon in the final state. The effective vertices which appear in both cases are then evaluated in the multi-Regge limit reproducing the well-known result obtained by Lipatov where the Einstein-Hilbert graviton emission vertex can be written as the product of two QCD gluon emission vertices, up to corrections to preserve the Steinmann relations.

SU(N) group-theory constraints on color-ordered five-point amplitudes at all loop orders

Color-ordered amplitudes for the scattering of n particles in the adjoint representation of SU(N) gauge theory satisfy constraints arising solely from group theory. We derive these constraints for n=5 at all loop orders using an iterative approach. These constraints generalize well-known tree-level and one-loop group theory relations.

Five-point amplitudes inN=4super-Yang-Mills theory andN=8supergravity

We present the complete integrands of five-point superamplitudes in $\mathcal{N}=4$ super-Yang-Mills theory and $\mathcal{N}=8$ supergravity, at one and two loops, for four-dimensional external states and $D$-dimensional internal kinematics. For $\mathcal{N}=4$ super-Yang-Mills theory we give the amplitudes for general gauge group---including all nonplanar contributions. The results are constructed using integral diagrams that manifestly satisfy the conjectured duality between color and kinematics, providing additional nontrivial evidence in favor of the duality for multipoint and multiloop amplitudes. We determine the ultraviolet poles by integrating the amplitudes in the dimensions where logarithmic divergences first occur. We introduce new kinematic prefactors which offer a convenient decomposition of the external state structure of the nonplanar five-point amplitudes in the maximally supersymmetric theories to all loop orders.

Analytical formula for three points sinusoidal signals amplitude estimation errors

In this note, we show that the amplitude estimation of sinusoidal signals proposed in Wu and Hong [Wu, S.T., and Hong, J.L. (2010), ‘Five-point Amplitude Estimation of Sinusoidal Signals: With Application to LVDT Signal Conditioning’, IEEE Transactions on Instrumentation and Measurement, 59, 623–630] is a particular case of Vizireanu and Halunga [Vizireanu, D.N, and Halunga, S.V. (2011), ‘Single Sine Wave Parameters Estimation Method Based on Four Equally Spaced Samples’, International Journal of Electronics, 98(7), pp. 941–948]. An analytical formula for amplitude estimation errors as effects of sampling period deviation is obtained.

Linear relations between $ \mathcal{N} \geqslant {4} $ supergravity and subleading-color SYM amplitudes

The IR divergences of supergravity amplitudes are less severe than those of planar SYM amplitudes, and are comparable to those subleading-color SYM amplitudes that are most subleading in the 1/N expansion, namely O(1/epsilon^L) for L-loop amplitudes. We derive linear relations between one- and two-loop four-point amplitudes and one-loop five-point amplitudes of N = 4, 5, and 6 supergravity and the most-subleading-color contributions of the analogous amplitudes of N = 0, 1, and 2 SYM theory, extending earlier results for N = 8 supergravity amplitudes. Our work relies on linear relations between N >= 4 supergravity and planar SYM amplitudes that were recently derived using the double-copy property of gravity, and color-kinematic duality of gauge theories.

One-loop SYM-supergravity relation for five-point amplitudes

We derive a linear relation between the one-loop five-point amplitude of $ \mathcal{N} = 8 $ supergravity and the one-loop five-point subleading-color amplitudes of $ \mathcal{N} = 4 $ supersymmetric Yang-Mills theory.

Effective action for the Einstein–Maxwell theory at order RF4

We use a recently derived integral representation of the one-loop effective action in the Einstein–Maxwell theory for an explicit calculation of the part of the effective action containing the information on the low energy limit of the five-point amplitudes involving one graviton, four photons and either a scalar or spinor loop. All available identities are used to get the result in a relatively compact form.

Simplifying the tree-level superstring massless five-point amplitude

We use the pure spinor formalism to obtain the supersymmetric massless five-point amplitude at tree-level in a streamlined fashion. We also prove the equivalence of an OPE identity in string theory with a subset of the Bern-Carrasco-Johansson five-point kinematic relations, and demonstrate how the remaining BCJ identities follow from the different integration regions over the open string world-sheet, therefore providing a first principles derivation of the (supersymmetric) BCJ identities.

The LHC String Hunter's Companion (II): Five-particle amplitudes and universal properties

We extend the study of scattering amplitudes presented in “The LHC String Hunter's Companion” [1] to the case of five-point processes that may reveal the signals of low mass strings at the LHC and are potentially useful for detailed investigations of fundamental Regge excitations. In particular, we compute the full-fledged string disk amplitudes describing all 2 → 3 parton scattering subprocesses leading to the production of three hadronic jets. We cast our results in a form suitable for the implementation of stringy partonic cross sections in the LHC data analysis. We discuss the universal, model-independent features of multi-parton processes and point out the existence of even stronger universality relating N-gluon amplitudes to the amplitudes involving N − 2 gluons and one quark–antiquark pair. We construct a particularly simple basis of two functions describing all universal five-point amplitudes. We also discuss model-dependent amplitudes involving four fermions and one gauge boson that may be relevant for studying jets associated to Drell–Yan pairs and other processes depending on the spectrum of Kaluza–Klein particles, thus on the geometry of compact dimensions.

The one-loop open superstring massless five-point amplitude with the non-minimal pure spinor formalism

We compute the massless five-point amplitude of open superstrings using the non-minimal pure spinor formalism and obtain a simple kinematic factor in pure spinor superspace, which can be viewed as the natural extension of the kinematic factor of the massless four-point amplitude. It encodes bosonic and fermionic external states in supersymmetric form and reduces to existing bosonic amplitudes when expanded in components, therefore proving their equivalence. We also show how to compute the kinematic structures involving fermionic states.