Two-Loop Scattering Amplitudes from the Riemann Sphere

Two-Loop Scattering Amplitudes from the Riemann Sphere

Yvonne Geyer, Lionel Mason, Ricardo Monteiro, Piotr Tourkine
Mathematical Institute, University of Oxford, Woodstock Road, Oxford OX2 6GG, UK
Theoretical Physics Department, CERN, Geneva, Switzerland
DAMTP, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, UK
Abstract

The scattering equations give striking formulae for massless scattering amplitudes at tree level and, as shown recently, at one loop. The progress at loop level was based on ambitwistor string theory, which naturally yields the scattering equations. We proposed that, for ambitwistor strings, the standard loop expansion in terms of the genus of the worldsheet is equivalent to an expansion in terms of nodes of a Riemann sphere, with the nodes carrying the loop momenta. In this paper, we show how to obtain two-loop scattering equations with the correct factorization properties. We adapt genus-two integrands from the ambitwistor string to the nodal Riemann sphere and show that these yield correct answers, by matching standard results for the four-point two-loop amplitudes of maximal supergravity and super-Yang-Mills theory. In the Yang-Mills case, this requires the loop analogue of the Parke-Taylor factor carrying the colour dependence, which includes non-planar contributions.

preprint: CERN-TH-2016-172, DAMTP-2016-54

I Introduction

The Cachazo-He-Yuan (CHY) formulae provide remarkable tree-level expressions for scattering amplitudes in theories of massless particles, written as an integral over marked points on the Riemann sphere. The integral localises as a sum over the solutions to the scattering equations Cachazo et al. (2014a). This formalism generalizes earlier work of Roiban, Spradlin and Volovich Roiban et al. (2004) based on Witten’s twistor string theory Witten (2004). The CHY formulae themselves originate in ambitwistor string theory Mason and Skinner (2014): this provided a loop-level formulation Adamo et al. (2014); Ohmori (2015) giving new formulae at genus one (torus) Adamo et al. (2014); Casali and Tourkine (2015) and two Adamo and Casali (2015) for type II supergravities in 10 dimensions. In Geyer et al. (2015a, b), we showed how the torus formulae reduce to formulae on a nodal Riemann sphere, by means of integration by parts in the moduli space of the torus. The node carries the loop momentum. We proposed that an analogous reduction was possible at any genus, leading to a new formalism that could become a practical tool in the computation of scattering amplitudes. In the one-loop case, our explicit analysis provided a proof that the formulae from ambitwistor strings reproduce the correct answer. Furthermore, on the nodal Riemann sphere, the formalism is more flexible than on the torus, and the formulae could be extended to a variety of theories with or without supersymmetry. An alternative approach to the one-loop scattering equations was pursued in Cardona and Gomez (2016a, b).

However, one loop is not such a stringent test of the framework, as many difficulties arise only at higher loops. The Feynman tree theorem, for example, shows how to construct one-loop integrands from tree formulae, if massive legs are allowed, and massive legs had already been considered in this context Naculich (2014); an example of our formulae has been reproduced following such an approach Cachazo et al. (2015). However, the situation is more difficult at higher loops despite recent progress inspired by the tree theorem Baadsgaard et al. (2015).

In Geyer et al. (2015a), we gave a brief sketch as to how the loop-level scattering equations are obtained by reduction to the nodal Riemann sphere. In this Letter, we give a precise formulation at two loops. To fix the details of the reduction to the sphere, we use a factorization argument that leads to new off-shell scattering equations. An alternative approach Feng (2016) applies higher-dimensional tree-level rules for the integration of the scattering equations to give diagrams for a scalar theory; however, our aim here is to give a framework that yields loop integrands on a nodal Riemann sphere for complete amplitudes. With this, we adapt genus-two supergravity integrands (type II, ) to a doubly nodal sphere, leading to the correct integrand for the four-point amplitude in maximal supergravity. We then conjecture an adjustment that gives instead a super-Yang-Mills integrand. These are checked both by factorization and numerically. Non-supersymmetric integrands require certain degenerate solutions to the scattering equations (on which the supersymmetric integrands vanish). We characterize these degenerate solutions here, but leave the subtler non-supersymmetric integrands for the future.

Ii From higher genus to the sphere

The higher genus scattering equations were formulated in the ambitwistor-string framework on Riemann surfaces of genus  Adamo et al. (2014); Ohmori (2015), in terms of a meromorphic 1-form , (the momentum of the string) that solves

(1)

where are marked points on . The solution is written as

(2)

where , span a basis of holomorphic 1-forms on dual to a choice of a-cycles , and are meromorphic differentials with simple poles of residues at and and vanishing -cycle integrals. The dependence on the auxiliary point drops by momentum conservation. The parametrize the zero-modes of and will play the role of the loop momenta.

The genus- scattering equations are a minimal set of conditions on the and moduli of required for to vanish globally. They include conditions

(3)

that set the residues of the simple poles of at the to zero. These fix the locations of on the surface. Once these are imposed, the quadratic differential is holomorphic, so it has further degrees of freedom, corresponding to the moduli of the surface (its shape). We therefore impose another scattering equations to reach . This can be done by writing

(4)

where the only depend on the moduli of and on the kinematics, and setting an independent subset of the to zero. In total, the scattering equations localise the full moduli space integral to a discrete set of points. For , there are precisely ’s, thus we simply set them all to zero and the ambitwistor-string loop integrand reads

(5)

where is a correlator depending on the theory and the holomorphic functions are . We stress that this is a formula for the loop integrand, and the -point -loop amplitude is .

To reduce this expression to one on nodal Riemann spheres, the heuristics described in Geyer et al. (2015a), based on the explicit genus-one calculation, was to integrate by parts (or use residue theorems) in the moduli integral times. This relaxes the delta functions to give measure factors , with the integration by parts yielding residues at the boundary of moduli space where all the chosen -cycles contract to give double points, leaving a Riemann sphere with pairs of double points. This leaves moduli that can be identified with the moduli of points on the Riemann sphere, corresponding to the nodes, modulo Möbius transformations. These moduli are fixed by remaining scattering equations.

On the nodal Riemann sphere , the basis of 1-forms forms descending from the ’s dual to the pinched a-cycles, given by the pairs of double points , is

(6)

so now

(7)

From (4) the coefficient of the double poles at identifies as . Thus the measure factor becomes . Furthermore, the quadratic differential

(8)

now only has simple poles at the and . The off-shell scattering equations were then proposed in Geyer et al. (2015a) to be Res. There are three relations between these equations so that only of them need to be imposed to enforce . The three relations follow from the vanishing of the sum of residues of multiplied by three independent tangent vectors to the sphere.

There is an ambiguity at two loops and higher, however. We could equally well have defined as

(9)

where the are linear combinations of the . We will see that is a better choice where at two loops. This does not change the heuristic argument as it corresponds to replacing the original scattering equations for the moduli by nondegenerate linear combinations thereof. The choice (or equivalently -1) at two loops will be forced upon us by requiring correct factorisation channels.

Thus our formula on the nodal Riemann sphere is

(10)

where , with the index spanning the marked points and the double points. The delta functions enforce the off-shell scattering equations 111with the on the product denoting the omission of three of the delta functions in line with the quotient and the three relations between the . In the first instance, will be taken to be the nodal limit of the higher-genus worldsheet correlator from the ambitwistor string type II supergravity in (together with a cross ratio motivated by factorization). This can be extended to theories for which no higher-genus expression is known, as we will demonstrate explicitly for super-Yang-Mills theory.

Iii The 2-loop scattering equations

We now take with and the double points corresponding to and . The two-loop scattering equations are the vanishing of the residues of

(11)

We adopt the shorthand notation . The scattering equations are then given by

(12)

where . In particular, for , . The equations are not independent, since there are three linear relations between them,

(13)

We will see that follows from the correct factorisation.

iii.1 Poles and factorization

Factorization channels of the integrand are related by the scattering equations to the boundary of the moduli space of the Riemann surface, where a subset of the marked points coalesce. Conformally, these configurations are equivalent to keeping the marked points at a finite distance, but pinching them off on another sphere, connected to the original one at the coalescence point .

When degenerates in this way, the scattering equations force a kinematic configuration where an intermediate momentum goes on-shell Dolan and Goddard (2014), corresponding to a potential pole in the integrand. The pole can thus be calculated as for from

(14)

Note however that whether this singularity is realized in a specific theory depends on the integrand .

Let (with external particles only). The location of the singularities in terms of the external and loop momenta can then be characterized as follows:

  • When , (14) simply gives , the standard factorization channel as for tree amplitudes, where a pole can appear in some intermediate propagator in massless scattering.

  • When , but , we find the (potential) pole as at 1-loop Geyer et al. (2015a), where such poles arise from certain partial fraction relations and shifts in the loop momenta, and coincide with the ‘Q-cut’ poles Baadsgaard et al. (2015).

  • The crucial new configuration at two loops is given by , corresponding to the condition

    (15)

    with as above. As detailed in Baadsgaard et al. (2015), the partial fraction identities and shifts always give a quadratic propagator of the form at two loops; see also Appendix A. Therefore, requiring the correct behaviour under factorisation determines and , or and . While both options are fully equivalent up to reparametrisation of the loop momenta, we will choose the former for the rest of the paper.

    For , this choice leads to a potential pole at . However, for , we are left with an unphysical potential pole at . The requirement that this pole is absent from the final answer will give important restrictions on the integrand .

  • Let us briefly comment on the only other new scenario at 2-loops – to have both . Since their contributions cancel in (14), this just leads to , although now associated to two 1-loop diagrams joined by an on-shell propagator.

The criterion for an integrand to give a simple kinematic pole in the final formula at one of these potential singularities is that should have a pole of order as the marked points coalesce, as described in detail in §4.1 of Geyer et al. (2015b). If the pole has lower degree, the final formula will not have a factorisation pole in this channel. This gives an important criterion for determining the precise forms of possible integrands .

iii.2 Degenerate and regular solutions

The off-shell scattering equations associated to the node have the same functional form, when seen as functions of and respectively. We distinguish between ‘regular solutions’, when are different roots of , and ‘degenerate’ solutions with the same root, for generic momenta. These degenerate solutions can be summarized by the factorization diagrams given in figure 1, and can be understood as forward limits of the ( dimensional) tree-level scattering equations; see Appendix B for details and He and Yuan (2015); Cachazo et al. (2015) for a discussion at one loop.

Case A:

Case B:

Case C:

Figure 1: Different possible worldsheet degenerations.

However, not all solutions of the dimensional scattering equations survive in the forward limit. Consider a degeneration parameter which vanishes in the forward limit. While there are degenerate solutions with , the zero-locus of  (III) excludes them, and thus the two-loop integrands localise on the degenerate solutions with and on

(16)

regular solutions (with ). Moreover, we shall see that the supersymmetric integrands only receive contributions from the regular solutions. As an important consequence, unphysical poles in the form of Gram determinants arising from double roots in are absent for supersymmetric theories: as discussed in Cachazo et al. (2015) and appendix B, these poles can be localised on the degenerate solutions. For non-supersymmetric theories, however, degenerate solutions may contribute, and one must check that contributions with unphysical poles vanish upon loop integration, as detailed in Cachazo et al. (2015) at one loop.

Iv Supersymmetric two-loop amplitudes

We now consider explicit expressions at four points for maximal supergravity and super-Yang-Mills. These expressions are examples of (10) for , . The representation of loop integrands that arises can be connected to a standard Feynman-like representation, after use of partial fractions and shifts in the loop momenta as in Geyer et al. (2015a) at one-loop; see Appendix A.

iv.1 Four-point supergravity integrand

We use the integrand that arises directly from the degeneration of the genus-two (ambitwistor) string Adamo and Casali (2015); D’Hoker and Phong (2002); Berkovits (2006); Berkovits and Mafra (2006). Define

(17)

and

(18)

where cyc(234) is a sum over cyclic permutations. Our prescription for the four-point supergravity integrand is

(19)

where is the standard kinematical supersymmetry prefactor and , so that the supergravity states in the scattering are the direct product of super-Yang-Mills untilded (left) and tilded (right) states.

The cross-ratio is inserted by hand to remove poles in the unphysical factorization channels discussed in the previous section. This is an important aspect of our prescription. We set the relative sign between and to be , consistently with the factors in the scattering equations. This choice implies that the degenerations of the worldsheet at or occur when , where is a partial sum of the external momenta. These physical poles can be realized in the formula. However, the numerator in the cross ratio suppresses unphysical poles of the type , which might have arisen when or .

We have evaluated our formula numerically and checked that it matches the known result for this amplitude Bern et al. (1998),

(20)

The planar and non-planar double-box integrands are written down in the “shifted” representation in Appendix A. It is also possible to check the factorization of this formula explicitly.

iv.2 Four-point super-Yang-Mills integrand

There is no fully well defined ambitwistor model that would give a first principle derivation of a super-Yang-Mills integrand, however the tree and one-loop results motivated us to postulate the following expression

(21)

where is again the standard kinematical supersymmetry prefactor. The new and crucial ingredient is the extension to two loops of the Park-Taylor factor, . In Geyer et al. (2015a), we presented the analogous object at one loop. We will comment later on the general form, and first focus on the explicit formula for the four-point two-loop case, including the non-planar (NP) contributions,

(22)

where is the rank of the gauge group and denote the colour traces. We have, for the planar part

with

where stands for . For the double-trace contribution, we have

with

where perm(12,34) denotes the eight permutations , and . The remaining contribution is determined by the ones already given, as seen in Naculich (2012),

(23)

We checked numerically that our proposal matches the known result of Bern et al. (1997). For instance, using the colour decomposition of eq. (IV.2), we get for the planar part

(24)

The two-loop Parke-Taylor formula is non-trivial, and may seem hard to extend for higher multiplicity or loop order. We propose, however, that in general it can be computed from the correlator of a current algebra on the Riemann sphere, which was our procedure at four points. This extends the tree-level result of Nair (1988) and, more generally, follows by analogy to the heterotic string Gross et al. (1985), where gauge interactions have a closed string-like nature as in ambitwistor string theory Mason and Skinner (2014). The sum over states at a node of the Riemann sphere translates into a sum over the Lie algebra index of two additional operator insertions per loop momentum. To eliminate the contributions from the unwanted poles, we drop Parke-Taylor terms that have orderings where loop momentum insertions appear with alternate signs as in , keeping only terms with orderings of the type ; here denote any external particles. For instance, we keep contributions such as , but discard terms like . This achieves the same effect as the cross-ratio appearing in the supergravity integrand. Moreover, we only include contributions with a single cyclic structure, e.g. (123456), and discard contributions with subcycles, e.g. (123)(456). These properties can be verified in the expressions above. Our two-loop Parke-Taylor expressions should be applicable to Yang-Mills theories with or without supersymmetry.

V Discussion

We have obtained scattering equation formulae for two-loop integrands on the Riemann sphere, following the heuristic reduction of genus-two ambitwistor string formulae by integration by parts on the moduli space of Riemann surfaces as in Geyer et al. (2015a). Our analysis is not a rigorous derivation from the genus-two ambitwistor string formulae of Adamo and Casali (2015), and in particular does not fix the parameter in the off-shell scattering equations. Nevertheless, we have seen that factorization fixes the ambiguity in , and this choice leads to correct two-loop integrands for maximally supersymmetric theories. A more refined analysis of the ambitwistor string degeneration should uniquely fix the scattering equations and the details of the integrands (such as the cross ratio in the supergravity case). It would also give us the tools to address the higher-loop case, where we expect different boundary contributions (e.g. at three loops “mercedes” vs. “ladder” graphs) in the integration by parts on the moduli space associated to different classes of scattering equations.

There are clearly many other challenges. It should also be possible to obtain integrands for non-supersymmetric theories, as we did in Geyer et al. (2015b) at one loop. These will in principle also have support on the degenerate solutions to the two-loop scattering equations, which we studied here; see Bjerrum-Bohr et al. (2016); Bosma et al. (2016); Cardona et al. (2016) and references therein for recent work on the scattering equations. For both supersymmetric and non-supersymmetric Yang-Mills and gravity at higher points, we need to understand the higher-loop analogues of the CHY Pfaffians on the Riemann sphere with and without supersymmetry. The extent of supersymmetry should be determined by the particular sum over spin structures, as in Geyer et al. (2015b) at one loop. More generally, we would like to extend our results to a new formalism, where our formulae arise directly as correlation functions of vertex operators on the nodal Riemann sphere. A natural question is then what type of quantum field theories admit such a formulation; there are CHY formulae and ambitwistor string models for a variety of theories Cachazo et al. (2014b); Ohmori (2015); Casali et al. (2015). Finally, it would be important to clarify the relation of these ideas to full string theory, which has been the subject of recent works Siegel (2015); Huang et al. (2016); Casali and Tourkine (2016).

Acknowledgements

We would like to thank Tim Adamo, Zvi Bern, Eduardo Casali and David Skinner for discussions. We also thank NORDITA, Stockholm, and the Isaac Newton Institute for Mathematical Sciences, Cambridge, for hospitality and financial support from EPSRC grant EP/K032208/1 during the program GTA 2016. YG is supported by the EPSRC Doctoral Prize Scheme EP/M508111/1, LJM by the EPSRC grant EP/M018911/1, and the work of PT is supported by STFC grant ST/L000385/1.

Appendix A Shifted integrands for planar and non-planar double boxes

In this section, we give the representation of double-box integrands with shifted loop momenta that appear in the formalism of the loop-level scattering equations Geyer et al. (2015a). Consider the planar double-box integrand with standard quadratic propagators,

(25)

where we use the notation and . The choice of canonical loop momenta arising from the shifts requires that this integrand is split into two different contributions – one for which is in the middle of the box, (), and one for which or is, (). In the case (), we obtain 9 terms, corresponding to applying the partial fraction identity and shifts to the 3 factors containing only , and to the 3 factors containing only . This gives

(26)

with

In case (), take to be in the middle propagator of the box first, as in

(27)

There are again 9 terms, because there are 3 factors with and 3 factors with . After symmetrising also in the choice of versus , we get

(28)

with given as above and

The total contribution from the planar double-box is obtained after the further symmetrisation of the loop momentum choices:

(29)

where the numerical factor takes into account the symmetrisations over three types of shifts and the four types of loop momentum choices.

The non-planar double-box is analogous. Starting with case (C) where there are three propagators with ,

(30)

we obtain

(31)

with

The other case, (D), is when there three propagators with or , say :

(32)

Then we get

(33)

with

Finally, the total contribution from the non-planar double-box is

(34)

Appendix B Analysis of the degenerate solutions

In this section, we give a more detailed analysis of the two-loop scattering equations and their solutions. As at one loop He and Yuan (2015); Cachazo et al. (2015), the key is to study the dimensional (massless) tree-level scattering equations for additional particles, then reduce to dimensions and take the forward limit:

(35)

In particular, while this procedure reconstructs the two-loop scattering equations on the nodal Riemann sphere  (III), it retains enough information of the massive scattering equations (in dimensions) to analyse the different classes of solutions. The main incentive for this study is an unphysical pole arising from double roots in the loop scattering equations. We will see explicitly how this pole can be reduced to a specific subset of the solutions, which do not contribute for the supersymmetric theories discussed in this Letter.

At two loops, our starting point are thus the -dimensional massless scattering equations for particles with momenta ,

(36)

with , where we have suggestively indexed the particles that will give rise to the loop momentum under the forward limit by and . In particular, we take the external particles to only have components in dimensions, and we denote this dimensional part of by respectively. It is now always possible to choose the remaining components of such that the scattering equations reduce to

(37a)
(37b)
(37c)

Note in particular that in the forward limit (governed by a parameter ), where

(38)

these equations smoothly limit onto the two-loop scattering equations  (III). However, as first pointed out in He and Yuan (2015) at one loop, not all their solutions have a smooth limit as well – the zero-locus of  (III) excludes a subset of the solutions. To see this, first recall that we distinguish two different classes of solutions: since the two-loop scattering equations have the same functional form as functions of respectively, there are both ‘regular solutions’ with localising on different roots of and ‘degenerate solutions’, where . Moreover, perturbing around the soft limit, the degenerate solutions come in three variations, see figure 1:

  • case A:     , but (or , but )

  • case B:     and , but

  • case C:     .

For each degeneration of the nodal Riemann sphere, we distinguish furthermore between two types of solutions, depending on the rate of coalescence of . For the soft limit parameter as above, they behave as

regular:
type I:
type II:

While the type II solutions contribute for the dimensional tree-level scattering equations , the zero-locus of  (III) excludes them, and thus the two-loop integrands localise on the type I solutions and the regular solutions.

Case A. To see this explicitly, let us perturb around the forward limit and focus on the case A. Both these solutions and the type B solutions bear a close resemblance to one-loop He and Yuan (2015), and our discussion will proceed in analogy. We take the forward limit  (38), where is a fixed vector with , and moreover