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JHEP02(2017)137

Published for SISSA by Springer

Received: January 5, 2017

Accepted: February 6, 2017

Published: February 28, 2017

Heptagons from the Steinmann cluster bootstrap

Lance J. Dixon,

James Drummond,

Thomas Harrington,

Andrew J. McLeod,

Georgios Papathanasiou

a,d

and Marcus Spradlin

SLAC National Accelerator Laboratory, Stanford University,

Stanford, CA 94309, U.S.A.

School of Physics & Astronomy, University of Southampton,

Highﬁeld, Southampton, SO17 1BJ, United Kingdom

Department of Physics, Brown University,

Providence, RI 02912, U.S.A.

DESY Theory Group, DESY Hamburg,

Notkestraße 85, D-22607 Hamburg, Germany

E-mail: lance@slac.stanford.edu, j.m.drummond@soton.ac.uk,

thomas harrington@brown.edu, ajmcleod@stanford.edu,

georgios@slac.stanford.edu, marcus spradlin@brown.edu

Abstract: We reformulate the heptagon cluster bootstrap to take advantage of the Stein-

mann relations, which require certain double discontinuities of any amplitude to vanish.

These constraints vastly reduce the number of functions needed to bootstrap seven-point

amplitudes in planar N = 4 supersymmetric Yang-Mills theory, making higher-loop con-

tributions to these amplitudes more computationally accessible. In particular, dual su-

perconformal symmetry and well-deﬁned collinear limits suﬃce to determine uniquely the

symbols of the three-loop NMHV and four-loop MHV seven-point amplitudes. We also

show that at three loops, relaxing the dual superconformal (

Q) relations and imposing

dihedral symmetry (and for NMHV the absence of spurious poles) leaves only a single am-

biguity in the heptagon amplitudes. These results point to a strong tension between the

collinear properties of the amplitudes and the Steinmann relations.

Keywords: 1/N Expansion, Scattering Amplitudes, Supersymmetric gauge theory

ArXiv ePrint: 1612.08976

Open Access,

 The Authors.

Article funded by SCOAP

doi:10.1007/JHEP02(2017)137

JHEP02(2017)137

as bootstrapping. Aspects of this approach have been applied to theories such as quantum

chromodynamics at one loop [2–4] and more recently at two loops [5–7]. However, the

most powerful applications to date have been to the planar limit of N = 4 super-Yang-

Mills (SYM) theory in four dimensions [8, 9]. Fueled by an increased understanding of the

classes of analytic functions appearing in amplitudes in general quantum ﬁeld theories, as

well as the stringent constraints obeyed by amplitudes in planar N = 4 SYM, it has been

possible to advance as far as ﬁve loops [10–15]. These results in turn provide a rich mine

of theoretical data for understanding how scattering amplitudes behave.

The planar limit of a large number of colors in N = 4 SYM has received a great deal

of attention because of the remarkable properties it exhibits. In addition to superconfor-

mal symmetry it respects a dual conformal symmetry [16–20], and amplitudes are dual to

polygonal light-like Wilson loops [16, 21–26]. Dual (super)conformal symmetry ﬁxes the

four-point and ﬁve-point amplitudes uniquely to match the Bern-Dixon-Smirnov (BDS)

ansatz [27], which captures all the infrared divergences of planar scattering amplitudes.

Starting at six points, the BDS ansatz receives corrections from ﬁnite functions of dual

conformal invariants [25, 26, 28, 29]. The correction to the maximally helicity violating

(MHV) amplitude has traditionally been expressed in terms of a (BDS) remainder func-

tion [10, 12, 25, 26, 30], while the correction to the next-to-maximally helicity violating

(NMHV) amplitude has traditionally been expressed in terms of the infrared-ﬁnite NMHV

ratio function [11, 31–35].

The cluster bootstrap program is built on the idea that certain scattering amplitudes

can be determined order by order in perturbation theory using a set of basic building blocks

known as cluster coordinates [36, 37]. Inspired by the results of refs. [38, 39], the bootstrap

approach developed in refs. [10–15] assumes that the MHV and NMHV amplitudes at each

loop order belong to a particular class of iterated integrals, or generalized polylogarithms.

More speciﬁcally, the L-loop contribution to the remainder and ratio functions is expected

to lie within the space spanned by polylogarithms of weight 2L [40] whose symbols can

be written in terms of cluster A-coordinates. A further constraint on the relevant space

of functions comes from the restriction that only physical branch cuts can appear in the

remainder and ratio functions [41].

To make use of this expectation, in the bootstrap program one ﬁrst constructs a

general linear combination of the above set of functions to serve as an ansatz. Then

one tries to determine all free coeﬃcients in the ansatz by imposing analytic and physical

constraints. This procedure becomes increasingly computationally expensive at higher loop

orders, largely due to the fact that the number of relevant functions increases exponentially

with the weight. It is hoped that one day a constructive procedure for determining these

amplitudes can be developed that does not require constructing the full weight-2L space as

an intermediate step. A promising candidate in this respect is the Wilson loop Operator

Product Expansion (OPE) [41–43] and the Pentagon OPE program [44–50] which provides

ﬁnite-coupling expressions for the amplitudes as an expansion around (multi-)collinear

kinematics. The main challenge in this framework is to resum the inﬁnite series around

these kinematics; there has been progress recently in this direction at weak coupling [51–

53]. Another potential constructive approach could involve the Amplituhedron [54, 55]

– 2 –

JHEP02(2017)137

description of the multi-loop integrand. Perhaps one can extend the methods of ref. [56]

for reading oﬀ the branch-point locations, in order to enable reading oﬀ the entire function.

To date, six- and seven-point amplitudes have been computed in the cluster bootstrap

program through the study of so-called hexagon and heptagon functions. Both helicity

conﬁgurations of the six-point amplitude have been determined through ﬁve loops [15],

while the MHV seven-point amplitude has been determined at symbol level through three

loops [12]. The seven-point NMHV amplitude has not yet received attention in the boot-

strap program, but it has been calculated through two loops using slightly diﬀerent meth-

ods [57]. Surprisingly, bootstrapping the seven-point remainder function has thus far proven

to be conceptually simpler (i.e. requiring the imposition of fewer constraints) than boot-

strapping its six-point counterpart. The collinear limit of the seven-point remainder func-

tion must be nonsingular and a well-deﬁned hexagon function. This requirement is so

restrictive that it entirely determines the two-loop heptagon remainder function, up to an

overall scale. It similarly determines the three-loop remainder function, once the full impli-

cations of dual superconformal symmetry are taken into account [12]. The corresponding

hexagon remainder function symbols may then be obtained by taking a collinear limit.

In a recent breakthrough [15], the classic work of Steinmann [58, 59] on the compatibil-

ity of branch cuts in diﬀerent channels has been used to supercharge the hexagon function

bootstrap program. The Steinmann relations dramatically reduce the size of the func-

tional haystack one must search through in order to ﬁnd amplitudes, putting higher-loop

amplitudes that were previously inaccessible within reach. In this paper we reformulate

the heptagon bootstrap of ref. [12] to exploit the power of the Steinmann relations. With

their help, we are able to fully determine the symbol of the seven-point three-loop NMHV

and four-loop MHV amplitude in planar N = 4 SYM, using only a few simple physical

and mathematical inputs. In a separate paper [60], we will investigate various kinematical

limits of these amplitudes in more detail, including the multi-Regge limit [29, 61–71], the

OPE limit [41–48], and the self-crossing limit [72, 73]. In this paper, we study one of the

simpler limits, where the NMHV seven-point amplitude factorizes on a multi-particle pole.

This paper is organized as follows. In section 2 we begin by reviewing the general

structure of seven-particle MHV and NMHV (super)amplitudes, and diﬀerent schemes for

subtracting their infrared divergences. Section 3 discusses the essential ingredients of the

amplitude bootstrap for constructing heptagon functions, which are believed to describe the

nontrivial kinematical dependence of these amplitudes. Section 4 focuses on the additional

physical constraints that allow us to single out the MHV or NMHV amplitude from this

space of functions.

Our main results, including the analysis of the general space of heptagon symbols, and

the determination of the three-loop NMHV and four-loop MHV amplitude symbols, are

presented in section 5. Section 6 describes a sample kinematical limit, the behavior of the

NMHV amplitude as a multi-particle Mandelstam invariant vanishes. Finally, section 7

contains our conclusions, and discusses possible avenues for future study.

Many of the analytic results in this paper are too lengthy to present in the manuscript.

Instead, computer-readable ﬁles containing our results can be downloaded from [74].

– 3 –

JHEP02(2017)137

2 Seven-particle scattering amplitudes

2.1 MHV: the remainder function

In planar N = 4 SYM, n-particle amplitudes are completely characterized by the color-

ordered partial amplitudes A

, which are the coeﬃcients of speciﬁc traces Tr(T

· · · T

)

in the color decomposition of the amplitudes. The MHV helicity conﬁguration has precisely

two gluons with negative helicity and (n−2) with positive helicity (in a convention where

all particles are outgoing). The MHV amplitude is encoded in the remainder function R

which is deﬁned by factoring out the BDS ansatz A

BDS

[27] (reviewed in appendix A):

MHV

= A

BDS

exp [R

] . (2.1)

The BDS ansatz captures all the infrared and collinear divergences [75–77] in the pla-

nar amplitude, so the remainder function is infrared ﬁnite. It is also invariant under

dual conformal transformations [16–19, 21]. Moreover, since the BDS ansatz accounts for

collinear factorization to all orders in perturbation theory [27], the n-point remainder func-

tion smoothly tends to the (n−1)-point remainder function in its collinear limits, a fact

that will prove to be an important ingredient in the bootstrap program.

In the deﬁnition (2.1), R

is the ﬁnite-coupling (or all-loop) remainder function. Here

we will be interested in its perturbative expansion. For any function F of the coupling,

we denote the coeﬃcients of its perturbative expansion with a superscript according to

the deﬁnition

F =

∞

L=0

(L)

, (2.2)

where g

= g

Y M

N/(16π

), g

Y M

is the Yang-Mills coupling constant, and N is the num-

ber of colors. Elsewhere in the literature, the coupling constant a = 2g

is often used.

The L-loop contribution to the remainder function, R

(L)

, is expected to be a weight-2L

iterated integral.

The remainder function vanishes for the four- and ﬁve-particle amplitudes, because

dual conformally invariant cross ratios cannot be formed with fewer than six external

lightlike momenta (in other words, the BDS ansatz is correct to all loop orders for n = 4

or 5) [25, 26, 28]. The ﬁrst nontrivial case, the six-point remainder function, has been

successfully computed at two loops [38], three loops [10, 30, 57], four loops [78] and recently

ﬁve loops [15]. At seven points, the remainder function has been computed at two loops [57,

79–81] and its symbol has been computed at three loops [12]. The symbol of the four-loop

seven-point MHV remainder function R

(4)

is one of the main results of this paper.

2.2 NMHV: the ratio function and R-invariants

Beyond the MHV case, scattering amplitudes in SYM theory are most eﬃciently organized

by exploiting the (dual) superconformal symmetry [31] of the theory, as reviewed in ref. [82].

In a nutshell, one starts by packaging the on-shell particle content of the theory into a

single superﬁeld Φ with the help of four Grassmann variables η

, whose index transforms

in the fundamental representation of the SU(4) R-symmetry group. In other words, all

– 4 –

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