New differential equations for on-shell loop integrals
James M. Drummond,
PH-TH Division, CERN, Geneva, Switzerland
LAPTH, Université de Savoie, CNRS
B.P. 110, F-74941 Annecy-le-Vieux Cedex, France
Johannes M. Henn,
Institut für Physik, Humboldt-Universität zu Berlin,
Newtonstraße 15, D-12489 Berlin, Germany
School of Natural Sciences, Institute for Advanced Study,
Department of Physics, Princeton University
Princeton, NJ 08540, USA
We present a novel type of differential equations for on-shell loop integrals. The equations are second-order and importantly, they reduce the loop level by one, so that they can be solved iteratively in the loop order. We present several infinite series of integrals satisfying such iterative differential equations. The differential operators we use are best written using momentum twistor space. The use of the latter was advocated in recent papers discussing loop integrals in super Yang-Mills. One of our motivations is to provide a tool for deriving analytical results for scattering amplitudes in this theory. We show that the integrals needed for planar MHV amplitudes up to two loops can be thought of as deriving from a single master topology. The master integral satisfies our differential equations, and so do most of the reduced integrals. A consequence of the differential equations is that the integrals we discuss are not arbitrarily complicated transcendental functions. For two specific two-loop integrals we give the full analytic solution. The simplicity of the integrals appearing in the scattering amplitudes in planar super Yang-Mills is strongly suggestive of a relation to the conjectured underlying integrability of the theory. We expect these differential equations to be relevant for all planar MHV and non-MHV amplitudes. We also discuss possible extensions of our method to more general classes of integrals.
- 1 Introduction
- 2 Motivation: MHV amplitudes
- 3 Differential equations for loop integrals
- 4 Solving the differential equations
- 5 Conclusion and outlook
- A Twistor differential operators
- B Five-point pentagon and penta-box integrals
In this paper we present new differential equations for on-shell loop integrals. Our main motivation is to develop efficient
methods to determine the loop-level S-matrix in super
Yang-Mills (SYM), which is built from precisely the types
of integrals we discuss here.
The planar theory is believed to be governed by some underlying integrable structure and hence it certainly deserves to have a beautiful and simple S-matrix.
Our differential equations can be viewed as a step towards explaining this simplicity, as they
restrict the integrals contributing to the amplitudes. Our results are at the level of specific integrals and so can also be used for subsets of integrals appearing in less supersymmetric theories.
We also expect our method to be applicable to larger classes of integrals.
The simplicity of the loop integrals implied by the differential equations we find is suggestive of a connection to the expected underlying integrability of planar
SYM, and more concretely, of the underlying symmetries of
its tree-level amplitudes.
It was found that the tree-level
S-matrix in this theory has a hidden symmetry, dual superconformal
symmetry [Drummond:2008vq, Brandhuber:2008pf, Drummond:2008cr].
Together with the ordinary superconformal symmetry
of the theory, it leads to a Yangian symmetry [Drummond:2009fd].
At loop level, a priori the symmetries are broken.
The breaking of the bosonic part of dual superconformal symmetry is well understood
and is controlled by anomalous Ward identities [Drummond:2007au, Drummond:2008vq]
that were initially derived for certain Wilson loops, and are conjectured to hold for the
scattering amplitudes. The Ward identities completely fix the functional form
of the four-point and five-point amplitudes, confirming the ABDK/BDS conjecture for these
cases [Anastasiou:2003kj, Bern:2005iz]. Starting from six points, a modification is required, called
the remainder function [Drummond:2008aq, Bern:2008ap].
An alternative way of understanding the dual conformal symmetry at loop level
is possible by using the massive regulator of [Alday:2009zm], which is inspired by the AdS/CFT correspondence [Alday:2007hr].
This setup is expected to make dual conformal symmetry exact, i.e. unbroken, at loop level.
Given this, it is natural to assume that the loop-level integral basis of SYM should
consist of integrals having this exact symmetry (see [Henn:2010bk] and references therein).
Indeed, the absence of triangle sub-graphs at one loop was confirmed in [Boels:2010mj] and
further support for this conjecture comes from [Bern:2010qa].
A similar understanding of the full Yangian symmetry at loop
level is still lacking and subject to ongoing research [Sever:2009aa, Bargheer:2009qu, Korchemsky:2009hm, Beisert:2010gn].
Recently it was argued that the (unregulated)
loop integrand has the full Yangian symmetry, up to total derivatives [ArkaniHamed:2010kv] .
Indeed the integrand can be recursively constructed via a BCFW type recursion relation [ArkaniHamed:2010kv] (see also [Boels:2010nw]).
Ignoring regularisation, this recursive construction can be written as a sequence of Yangian invariant operations on the basic Yangian
invariant functions [ArkaniHamed:2010kv] (see also [ArkaniHamed:2009vw, Mason:2009qx, Drummond:2010uq, Korchemsky:2010ut] for a discussion of Yangian invariants).
Just as the dual conformal symmetry of the integrand [Drummond:2006rz]
was a hint that there is an anomalous Ward identity [Drummond:2007au],
the existence of the Yangian invariant integrand indicates that there
should be a way to directly understand the breaking of the full symmetry.
Although we do not yet make a direct connection to
the Yangian generators, we find it likely that the underlying Yangian
the differential equations we find.
Further very concrete motivation for our study also comes from recent explicit results for the
hexagonal light-like Wilson loop, which is dual to the six-gluon MHV amplitude [Drummond:2008aq, Bern:2008ap]
(for reviews see [Alday:2008yw, Henn:2009bd] and references therein, and [Mason:2010yk, CaronHuot:2010ek]
for recent developments).
In [Goncharov:2010jf], a remarkably simple form of the six-point remainder function [Drummond:2008aq, Bern:2008ap]
was given, based on previous work [DelDuca:2010zg].
Simple results are also conjectured to hold in special kinematics [DelDuca:2010zp, Heslop:2010kq].
Recently, the analytic six-point Wilson loop result of [DelDuca:2010zg]
could be reproduced in a kinematical
limit by an analytic calculation of the corresponding scattering amplitude,
extending previous calculations for mass-regulated amplitudes [Henn:2010bk, Henn:2010ir].
Related recent work on loop amplitudes
in SYM can be found in [Kosower:2010yk, Alday:2010jz].
See also [Alday:2010zy, Eden:2010ce] for other related recent developments.
The use of differential equations to evaluate loop integrals [Kotikov:1990kg, Kotikov:1991pm, Gehrmann:1999as]
is widespread in the literature,
for a review see chapter 7 of [smirnov2006feynman].
In this approach one considers a set of master integrals,
and an associated family of integrals is
be obtained from by
One differentiates the master integrals
with respect to kinematical invariants or masses.
The result is in general a linear combination
of several integrals. Integral
reduction identities are then applied to re-express the latter
in terms of the set of master integrals.
In complicated cases this step can be non-trivial,
as in general it requires the knowledge of all reduction
In general one obtains a set of first-order differential equations
for the master integrals.
A disadvantage is that in
order to solve for a given integral, one has to deal with all
integrals of a given family. It can also happen
that when computing a finite integral,
intermediate steps contain
divergences that only cancel at the end,
and a regulator has to be used.
The differential equations we obtain here are quite
different in nature. There are two important differences
to the method described above:
our differential equations can be applied directly to a given
integral, without having to know all integrals with
fewer propagators, or any integral reduction identities.
Secondly, they are second-order
equations, and, importantly, they reduce the loop level by one. In other
words, the homogeneous term of the equations
corresponds to lower-loop integrals, and the equations
therefore have an iterative structure.
While the differential equations method known in the literature can be applied in principle to all loop integrals, the differential equations we find are specific to a certain class of integrals. The latter constitute a subset of integrals needed for computing scattering amplitudes in a generic theory. There is reason to believe that in the special case of SYM, all planar loop integrals are constrained by the type of differential equations we find here. Very concretely we observe that the one- and two-loop MHV amplitudes in SYM in the form given in [ArkaniHamed:2010kv], can all be expressed in terms of a single ‘master’ integral, namely
at one- and two-loop order, respectively. The reason is that reduced topologies, such as box, double box or penta-box topologies, can be obtained by taking soft limits of the master integrals. We will give the precise definition of the integrals later. In the six-point case, they were used recently to obtain an analytic result for the remainder function at two loops (in a kinematical limit) by two of the present authors in [Drummond:2010mb]. This is the first time that such an analytical result was obtained directly for the scattering amplitude, and the results also hinted at a certain simplicity of the integrals that had to be computed. We will make this more precise in this paper. As we will show, the integrals depicted in (1.1) satisfy second-order differential equations that reduce their loop degree by one, namely
The one-loop integrals appearing in (1.1) and on the r.h.s. of (1.2)
satisfies a similar equation with simple rational functions as inhomogeneous terms.
These equations imply powerful constraints on the
functional dependence of those integrals.
In the following we will give several examples where
we solve such equations and obtain analytic
answers for two-loop integrals with multiple external legs.
Iterative differential equations of the type (1.2) were previously found
in [Drummond:2006rz] by two of the present authors and Smirnov and Sokatchev
for certain off-shell ladder (i.e. multiple box) integrals, and are closely
related to the equations we find here. We will review the differential equations
of [Drummond:2006rz] in section 3.
The differential equations presented here can be thought of
as a generalisation of the ones used in [Drummond:2006rz] to on-shell integrals.
We find it very convenient to work with momentum twistor variables [Hodges:2009hk]. These are well-suited to
describe planar loop integrals. They solve the momentum conservation and on-shell
constraints and therefore are unconstrained variables. This makes them the
natural variables to use if one is interested in differential equations.
One-loop box integrals were discussed in momentum
twistor space in [Hodges:2010kq, Mason:2010pg].
In the cases where infrared divergences are present we employ the
AdS-inspired mass regulator of [Alday:2009zm]. The latter allows us to stay in
four dimensions and continue to use momentum twistors in those cases as well.
For recent references using momentum twistors and this regulator see [Hodges:2010kq, Mason:2010pg, Drummond:2010mb].
The outline of the paper is as follows. In section 2, we recall the definition of momentum twistor variables and the expression for the one- and two-loop MHV amplitudes in SYM. It is shown that at each loop level there is only one ‘master’ topology. In section 3 we show that certain classes of integrals satisfy second-order differential equations that reduce the loop order by one. We first review the differential equations for off-shell integrals found in [Drummond:2006rz], and then generalise them to the on-shell case. We present two mechanisms for finding such differential equations. We present several infinite classes of integrals satisfying iterative differential equations. In section 4 we give an example for how to solve the differential equations, using certain assumptions about boundary conditions. We give explicit analytical results for several multi-leg integrals at two loops. In section 5 we conclude and comment on several possibilities of extending our method. There are two Appendices. In Appendix A we discuss twistor differential operators that annihilate the integrand of the integrals appearing in the MHV amplitudes, up to anomalies. Appendix B contains the analytic formulas for the one- and two-loop penta-box integrals with “magic” numerator.
2 Motivation: MHV amplitudes
In this section we recall how momentum twistor variables
can be used to describe loop integrals. We also recall
the recently-proposed integral representation for planar
two-loop MHV amplitudes in super Yang-Mills [ArkaniHamed:2010kv].
We show that at each loop order, all integrals contained in
the MHV amplitudes can be thought of as deriving from
a single master topology. Reduced topologies are obtained
by taking soft limits. This is related to the consistency of the
loop integrand with soft limits [Drummond:2010mb].
Let us briefly recall how the momentum twistor variables are related to the standard momentum space variables. Given the incoming light-like momenta of a planar colour-orderd ordered amplitude,
we define the dual coordinates in the usual manner [Broadhurst:1993ib, Drummond:2006rz],
The dual define a light-like polygon in the dual space. On the other hand, a point in dual coordinate space corresponds to a (complex, projective) line in momentum twistor space. Two dual points are light-like separated if the corresponding lines in momentum twistor space intersect at some point in momentum twistor space. Thus the light-like polygon in dual space corresponds to a polygon in momentum twistor space with each line intersecting its two neighbouring lines as each dual point is light-like separated from its two neighbours. The momentum twistors associated to this configuration of light-like lines are defined via the incidence relations,
The momentum twistor transforms linearly under the action of dual conformal symmetry, as indicated by the fundamental index . Moreover the momentum twistors describing the polygon are free variables, in contrast to the dual points which obey the constraints of light-like separation from their neighbours. The dual point is associated to the line described by the pair or for short. The incidence relations (2.5) allow one to express functions of the in terms of momentum twistors. For example we have
where the four-brackets and two-brackets are defined as follows,
The four-brackets are obviously dual conformal invariants while the two-brackets are invariant under just the Lorentz and (dual) translation transformations.
We can consider integrals over the space of lines in momentum twistor space . As we have discussed, this is equivalent to an integration over points in dual space. For example, the two-mass hard integral of Fig. 1 is given by
where the first expression is written in momentum space and the second one in dual coordinates. In twistor notation, this becomes
The integral (2.8) as written is infrared divergent, so its proper definition should involve a regulator. We will use the the AdS regularisation introduced in [Alday:2009zm]. The latter allows us to stay in four dimensions and regulates the infrared divergences by masses that in turn are generated by a Higgs mechanism. For actual calculations we will use the regularisation where all masses are equal. In particular this means the outermost propagators in the planar loop integrals that we are studying are modified as follows,
In twistor space this has the effect that each of the propagator factors becomes
When doing this, there is a choice of adding terms to the
integrand. This is certainly relevant if one wishes to obtain the
exact -dependence of amplitudes on the Coulomb branch of
SYM. In the cases considered here we are mostly interested in the
case where is small. Unless an integral diverges linearly as one can
then drop such numerator terms. This is the case for all integrals considered here.
Let us now discuss how such integrals appear in scattering amplitudes in SYM. At one loop, the MHV amplitudes are usually represented (or more precisely, their parity-even part) as a sum over so-called two-mass easy box integrals [Bern:1994zx]. In [ArkaniHamed:2010kv], an alternative form was given that uses pentagon integrals with certain twistor numerators. The formula given in [ArkaniHamed:2010kv] is a sum over the integrals shown in Fig. 2. Their integrand is given by
We use the notation for the integrand and for the function obtained after integration111As was mentioned before, some of the integrals are in fact infrared divergent and require a regulator. In this section we mostly discuss properties of the (unregulated) loop integrand. It was argued in [ArkaniHamed:2010kv] that a correct expression for the loop integrand on the Coulomb branch of SYM [Alday:2009zm], up to terms, can be obtained by adding masses to propagators on the perimeter of the diagams., i.e.
where the number of points is implicit and appears only in the condition .
A comment is in order here regarding the numerator factors: In [ArkaniHamed:2010kv] both one-loop and two-loop amplitudes are written using numerators with wiggly lines representing
The amplitude is
obtained by changing wiggly lines to dashed lines that stand for
, etc. In our discussion we do not distinguish
between an integral and the parity conjugate integral because their
difference is a parity odd integral which
integrates to .
Therefore, in any diagram we can change all (but not just some) wiggly lines to dashed lines
and vice-versa. In the following we will not distinguish between integrals with all
dashed lines and those with all wiggly lines.
We now argue that all integrals appearing in the one-loop MHV amplitudes can be thought of as deriving from the pentagon integral. It is clear that pentagon integrals with lower number of legs can be obtained from a generic pentagon integral by taking soft limits. Therefore we only have to show how to obtain the box integrals of equation (2.9). As we will see presently, they are also obtained by taking soft limits. Indeed, consider the pentagon integral of Fig. 2 with , so that only one external leg enters its rightmost corner. The soft limit of corresponds to letting
Applying this limit to the integrand of the pentagon integral given in (2.12), we find
one reproduces exactly the integrand of the “two-mass hard” box.
In this sense we can say that the one-loop
MHV amplitude is built from a single master integral.
We will show in section 3 that the latter satisfies
a differential equation.
The situation is almost identical at the two-loop level. The expression given in [ArkaniHamed:2010kv] involves only the integrals shown in Fig 3. Recall that we denote the integrand of loop integrals by and the functions obtained after integration by . For example, the first integral shown in Fig. 3 has the following definition,
with the normalisation and
where the number of points is implicit and appears only in the condition .
The double pentagon integrals defined in (2.18) are in fact infrared finite [ArkaniHamed:2010kv].
Some of the pentagon and all of the box integrals we consider are infrared divergent and
their definition is understood with the mass regulator in place, which leads to the modifications discussed above.
From the one-loop case
it is clear that the penta-box and double box integrals shown
in Fig. 3 can be obtained from the double pentagon integral
by taking subsequent soft limits.
So, in summary, at each loop level the integrals appearing
in the MHV amplitudes can be thought of as deriving from
a single master topology. We will show in section 3
that the latter integral satisfies a second-order differential
equation that reduces its loop order by one.
The relationship between the master integrals and the reduced
integrals works at the level of the integrand. When the integration
is taken into account, in some cases infrared divergences can
lead to a non-commutativity of the soft limit and the regulator limit.
We adopt the point of view that in those cases the relation to the
master integral still implies a certain simplicity of the reduced
integral. We show this explicitly for IR-divergent penta-box integrals
in section 3, which satisfy the same type of differential equations.
Let us briefly discuss how many functionally different integrals of this class exist. The most general double pentagon integral of the type shown in Fig. 3 is the one where four of its external legs are doubled (when more legs are added, the function does not change). This is possible for the first time at , e.g. . In general, we can consider all integrals with zero, one, or two momenta flowing into these corners. Because of the symmetries of the pentagon, there are different possibilities. However, because of the finiteness of the double pentagon integrals, all of them can be obtained from the most general case by taking soft limits. Finally, there are penta-box integrals and double box integrals of the type . The latter are all infrared divergent and therefore depend on the regulator .
3 Differential equations for loop integrals
3.1 Finite integrals
The ladders (or scalar boxes) provide a first example of a class of integrals which satisfy the type of differential equations we are interested in. To understand the differential equations for the ladder integrals it is convenient to use the dual coordinate notation. For the integrals appearing in the two-loop amplitude discussed in section 2 we will pass to the momentum twistor notation. We will begin with the one-loop box as an example, using dual variables to express the momenta,
Note that the points are generic points without any light-like separations. The -loop version of this diagram is shown in Fig. 4. Since the integral (3.19) is covariant under conformal transformations of the coordinates, it can be expressed in terms of a function of the two available cross-ratios,
Thus we have
where the function is known [Usyukina:1993ch, Broadhurst:1993ib, Isaev:2003tk]. In fact the function is best expressed in terms of the variables and defined by222Note that and are real and independent in Minkowski signature whereas they are complex conjugate to each other in Euclidean signature.
Explicitly it is given by
The integral satisfies a simple second order differential equation. The reason for this is that acting with the Laplace operator on one of the external points, say, produces a delta function under the integral [Drummond:2006rz],
This has the effect of localising the integral completely, giving a simple second-order equation,
On the other hand [Drummond:2006rz], acting on the form of given in (3.21) one obtains
where is a second-order differential operator,
The main points we wish to stress are that the action of the operator removes the loop integration, leaving a rational function behind. The existence of a simple equation also means that the underlying function is a relatively simple pure transcendental function of degree two.
The one-loop integral we have been discussing is just the first in an infinite sequence of ladder integrals. These integrals (along with a large class of equivalent integrals [Drummond:2006rz]) exhibit an iterative structure,
As before the integrals are invariant under conformal transformations of the and so are given in terms of functions of the the two cross-ratios,
As before the function is best expressed in terms of the variables and ,
The fact that the one-loop box satisfies a differential equation guarantees that all the ladder integrals do. One can see that Laplace operator is effectively acting only on a one-loop box subintegral which we have already seen reduces to a rational function. Thus the operation reduces the loop order of the ladder integral,
The functions appearing in the explicit expressions exhibit corresponding differential equations
The integrals involving twistor numerators can also satisfy differential equations. We will begin with integrals which are similar to the ladders of the previous section. We would like to consider the finite pentaladder integrals, beginning with one of the one-loop pentagon integrals discussed in section 2. We will write in dual coordinate notation to begin with and later move to the momentum twistor notation. The integral we would like to consider is the seven-point one-loop pentagon integral, which is shown for arbitrary number of loops in Fig. 5(a),
Note that we have not normalised the integral so (3.35) is dimensionful. The point is the magic complex point which is null-separated from every point on the null lines and . It is one of the solutions to
The other solution is its parity conjugate.
Just like the ladder integrals, the pentagon we are considering is conformally covariant so we can write it in the following way
where and are the two non-vanishing conformal cross-ratios,
We have chosen to make a factor of explicit on the RHS of (3.37) for later convenience.
In terms of the quantities introduced in section 2 we have
The integral satisfies a differential equation very similar to the equation for the ladder integrals,
This equation translates into a second order p.d.e. for the function ,
Just as for the ladders we can define multi-loop pentaladder integrals via an iterative structure. For example we can consider the integrals, see Fig. 5(a),
Equivalently we can write them using the ladder integrals,
As before, we will use conformal symmetry in the variables to write the integral in the form
Now applying the Laplace operator gives the following differential equation for ,
where we define .
We can also consider pentaladder integrals with a massive corner on the pentagon subintegral, see Fig. 5(b). At one loop the integral takes the form
Here the point is light-like separated from . It is convenient to write this integral as
where the three cross-ratios are
We remark that the function discussed before can be obtained from by taking the (smooth) soft limit .
Following the same logic as before we obtain the equation
A similar analysis holds for the -loop case defined by
Writing the integral as a function of the cross-ratios,
we find the differential equation,
3.1.3 Pentaladders reloaded
In the case of the pentaladder integrals we can also arrive at the differential equation by considering the momentum twistor representation of the integrals. A clue to constructing the right operator comes from the fact that the Laplacian naively annihilates the integrand,
Of course we have seen that this does not imply that the integral itself is annihilated by the Laplace operator because there is an anomaly in the form of the delta function as in (3.24).
Let us recall that if one writes the one-loop finite pentagon integral using momentum twistors it takes the form,
It is possible to construct an operator acting on the momentum twistor variables which also annihilates the integrand . Let us first introduce some notation to deal with twistor derivatives.
We define a twistor derivative as
It acts trivially on four-brackets, . The normalized integrals are homogenous in all external twistors, which can be also written as
Let us do a trivial exercise which will be important in the following discussion. Let us start with the rational function of four-bracket, and act on it with the operator . We immediately get
where we used Shouten identity . Now, we see that the dependence of twistor is just through the four-bracket . If we now act with the operator , then we get zero, therefore
Most of the equations we will derive later are based on the same principle.
In the case of the pentagon integral the analysis above leads us to define the following operator
The operator naively annihilates the integrand ,
We have written ‘naive’ here because, just as for the Laplace operator, we must remember that we are going to perform an integration and there may be anomalies like the delta function left behind. In fact we can easily convince ourselves that there are indeed anomalous terms in place of the naive zero on the RHS of (3.61). Let us express the action of the operator on the integral function . We recall that the cross-ratios take the following form in terms of momentum twistors,
Acting with the operator on the integral function we find
We already know from acting with the Laplace operator that the RHS of (3.63) is not zero. Indeed the second-order derivative of is according to (3.41). Writing the equation without the factors of on both sides leads us to the following relation,
This equation can then be used whenever we find the one-loop pentagon as a subintegral. In particular one can derive the differential equations for the multi-loop pentaladder integrals. As an example we discuss here the finite two-loop pentabox integral,
Applying the operator and using the identity (3.64) we find
By using cyclic identities one finds that the four-brackets on the LHS can be written in terms of and and we arrive at the equation
exactly as derived from the Laplace operator in (3.45) in the case . We have rederived the equation for the two-loop finite pentaladder to simplify the equations but the derivation for loops is essentially identical and leads to (3.45) for general .
In summary, we have derived the second-order equations (3.45) by acting with the twistor differential operator on the
pentagon sub-integral, thanks to equation (3.64).
What this implies is that whenever we have an integral with the pentagon integral as a sub-integral, we can use this mechanism and generate
a second-order equation by using (3.64). Importantly, this is also possible in cases where one cannot apply the Laplace operator,
as e.g. for the double pentagon integrals that we will discuss in section 3.1.4.
Before doing this, we are going to discuss the generalisation where the pentagon sub-integral has a massive corner.
In the case of the pentaladders with a massive corner one again finds certain operators which annihilate the integrand. Let us consider the integrand of the one-loop pentagon with three massive legs,
with . The integrand is annihilated by certain twistor operators,
Applying these operators to the explicit function we actually find that they annihilate it, i.e. there is no anomaly associated to this operator on . In fact we will see that is given explicitly by [refNimanew]333Here we are assuming that all dual distances are spacelike. One must carefully analytically continue to the regions where some of them become timelike.
where the cross-ratios take the form
We remark that the function describing the seven-point integral can be obtained from by taking a (smooth) soft limit,