# Position-space cuts for Wilson line correlators

###### Abstract

We further develop the formalism for taking position-space cuts of eikonal diagrams introduced in ref. Laenen:2014jga . These cuts are applied directly to the position-space representation of any such diagram and compute its discontinuity to the leading order in the dimensional regulator. We provide algorithms for computing the position-space cuts and apply them to several two- and three-loop eikonal diagrams, finding agreement with results previously obtained in the literature. We discuss a non-trivial interplay between the cutting prescription and non-Abelian exponentiation. We furthermore discuss the relation of the imaginary part of the cusp anomalous dimension to the static interquark potential.

###### Keywords:

Scattering Amplitudes, Eikonal Approximation, QCDNIKHEF/2015-014 |

ITP-UU-15/05 |

## 1 Introduction

The infrared singularities of gauge theory scattering amplitudes play a fundamental role in particle physics for phenomenological as well as more theoretical studies. Determining the long-distance singularities is necessary for combining the real and virtual contributions to the cross section, as the divergences of the separate contributions only cancel once they are added. Infrared singularities moreover dictate the structure of large logarithmic contributions to the cross section, allowing such terms to be resummed—which is in many cases required in order to obtain reliable perturbative predictions. Beyond their significance to collider phenomenology, long-distance singularities are highly interesting from a theoretical point of view. Among several properties, they have a universal structure among different gauge theories; moreover, their exponentiation properties Yennie:1961ad ; Sterman:1981jc ; Gatheral:1983cz ; Frenkel:1984pz ; Magnea:1990zb ; Magnea:2000ss ; Gardi:2010rn ; Mitov:2010rp ; Gardi:2011wa ; Gardi:2011yz ; Dukes:2013wa ; Dukes:2013gea ; Gardi:2013ita and their relation to the renormalization of Wilson line correlators Polyakov:1980ca ; Arefeva:1980zd ; Dotsenko:1979wb ; Brandt:1981kf ; Korchemsky:1985xj ; Korchemsky:1985xu ; Korchemsky:1987wg allow their perturbative expansion to be explored to all orders, a feat currently unattainable for complete scattering amplitudes.

The basic tool for computing the infrared singularities of any scattering amplitude is provided by the eikonal approximation. In this limit the momenta of the soft gauge bosons emitted between the partons emerging from the hard interaction are neglected with respect to the hard momenta . As a result, each hard parton simply acts as a source of soft gluon radiation and is accordingly replaced by a semi-infinite Wilson line

(1.1) |

which extends from time , when the hard scattering takes place, to infinity along the classical trajectory of the hard parton, traced out by its four-velocity . The long-distance singularities of the scattering amplitude of the hard partons are then encoded in the eikonal amplitude

(1.2) |

which has the same soft singularities as the original amplitude, but is much simpler to compute. An important feature of the eikonal amplitude (1.2) is the fact that it depends on the kinematics only through the angles between the four-velocities (defined through ). Before renormalization, the integrals involved in the loop-level contributions to are thus scale invariant and vanish identically. This in turn allows the infrared singularities at any loop order to be computed by studying the ultraviolet renormalization factor of the Wilson line correlator (1.2) Korchemsky:1985xj ; Korchemsky:1987wg ; Kidonakis:1998nf ; Kidonakis:1997gm ; Becher:2009cu ; Gardi:2009qi . This renormalization factor forms a matrix in the space of color configurations available for the scattering process at hand, referred to as the soft anomalous dimension matrix. In processes involving only two Wilson lines, this matrix reduces to the cusp anomalous dimension, a quantity which has been computed in QCD up to three loops Korchemsky:1987wg ; Kidonakis:2009ev ; Grozin:2014hna . In super Yang-Mills theory, the cusp anomalous dimension is known to three loops Correa:2012nk , and partial results have been obtained at four loops Henn:2012qz ; Henn:2013wfa . For multi-parton amplitudes, the soft anomalous dimension matrix has been computed through two loops for massless Aybat:2006wq ; Aybat:2006mz as well as massive Becher:2009kw ; Ferroglia:2009ep ; Ferroglia:2009ii ; Mitov:2010xw ; Kidonakis:2010dk Wilson lines. Recently, much progress has been made toward the calculation of the soft anomalous dimension matrix at three loops Gardi:2013saa ; Falcioni:2014pka .

In this paper we continue exploring a notion of cuts of eikonal diagrams (i.e., the diagrams contributing to the eikonal amplitude) introduced in ref. Laenen:2014jga . Applied to any eikonal diagram, the cuts compute the discontinuities of the diagram, in analogy with the Cutkosky rules for standard Feynman diagrams. The discontinuities are in turn readily combined to produce the imaginary part of the diagram, a direct computational method of which is desirable in several contexts. Indeed, collinear factorization theorems for non-inclusive observables were pointed out in refs. Catani:2011st ; Forshaw:2012bi to be violated due to exchanges of Glauber-region (i.e., maximally transverse) gluons. The resulting factorization-breaking terms are purely imaginary and take the form of the non-Abelian analog of the QED Coulomb phase. Therefore, by utilizing the all-order exponentiation property of the eikonal amplitude, the latter could be obtained directly by computing the imaginary part of the exponent. The resulting non-Abelian Coulomb phase Catani:1984dp ; Catani:1985xt may also aid studies of interference effects. The importance of understanding the imaginary part of eikonal diagrams has also recently been highlighted in studies regarding rapidity gaps Forshaw:2006fk ; Forshaw:2008cq . Moreover, cuts of Wilson line correlators are naturally relevant for cross section calculations Korchemsky:1992xv ; Gardi:2005yi .

A cutting prescription for eikonal diagrams may also provide the first step toward extending the modern unitarity method Bern:1994zx ; Bern:1994cg ; Bern:1996je ; Britto:2004nc ; Britto:2005ha ; Anastasiou:2006jv ; Forde:2007mi ; Mastrolia:2009dr ; Kosower:2011ty ; CaronHuot:2012ab to eikonal amplitudes. The development of the unitarity method has led to a dramatic improvement in the ability to compute loop-level (non-eikonal) scattering amplitudes at high multiplicity. In this approach, the loop amplitude is decomposed into a linear basis of loop integrals which are computed independently (for example, by means of Feynman parametrization, or differential equations Gehrmann:1999as ; Henn:2013pwa ). The calculation of the loop amplitude is then reduced to the problem of determining the integral coefficients. This step is performed by applying to both sides of the basis decomposition of the loop amplitude a number of cuts which have the effect of putting the internal lines on shell. In basic unitarity (as opposed to generalized unitarity), the cuts employed measure the discontinuity of the amplitude in its various kinematical channels. Unitarity has proven highly successful, notably in computing one-loop amplitudes with many partons in the final state. It is therefore natural to look for extensions of this method to other physical quantities with a perturbative expansion.

It should be emphasized that Cutkosky rules for eikonal diagrams have been introduced previously in the literature, as a cut prescription applied directly to the momentum-space representation of the diagrams Korchemsky:1987wg . In contrast, the cuts introduced in ref. Laenen:2014jga and further studied here are applied to the position-space representation of the eikonal diagrams. A notion of position-space cuts of non-eikonal diagrams exists in the literature in the form of a cutting equation that follows from Veltman’s largest-time equation 'tHooft:186259 . However, that notion is conceptually different from the position-space cuts in this paper, since the former has the effect of cutting a given diagram in two disconnected subdiagrams while the latter does not. Moreover, in practice, the largest-time equation is typically not applied directly, but rather serves to derive the momentum-space Cutkosky rules, which in turn are used to obtain the imaginary part of a diagram. As already observed in ref. Laenen:2014jga , position-space cuts provide a substantial simplification over momentum-space cuts in the computation of imaginary parts of eikonal diagrams. There has been recent interest in the literature in studying Wilson line correlators in position space, in particular refs. Erdogan:2011yc ; Erdogan:2013bga ; Erdogan:2014gha which investigate the structure of infrared singularities and factorization in position space. Moreover, position-space analogs of generalized unitarity cuts of Wilson line correlators were recently introduced in ref. Engelund:2015cfa .

The structure of this paper is as follows. In section 2 we discuss the origin of the imaginary part of Wilson line correlators from the point of view of causality as well as unitarity. We then show how the imaginary part can be computed from the position- and the momentum-space representations at one loop. In section 3 we review the formula in ref. Laenen:2014jga for the imaginary part of -loop eikonal diagrams containing no internal (i.e., three- or four-gluon) vertices to the leading order in the dimensional regulator . We furthermore discuss the relation of the discontinuities of the diagrams to their imaginary part. In section 4 we apply the formalism to compute the imaginary part of a number of two- and three-loop diagrams and discuss a non-trivial interplay between the cutting prescription and non-Abelian exponentiation. In section 5 we turn to formulas for the imaginary part of eikonal diagrams with internal vertices and provide details on its computation. We give our conclusions in section 6. Appendix A explains our method for computing the principal-value integrals involved in the cutting prescription. In appendix B we present our algorithm for re-expressing multiple polylogarithms in terms of ones with constant indices.

## 2 Imaginary parts of eikonal diagrams and their physical origin

In this section we will discuss the origin of the imaginary part of Wilson line correlators from the point of view of causality as well as unitarity. These viewpoints are naturally provided by the representation of the correlator in position and momentum space, respectively. We will show how the imaginary part can be computed directly from each of the respective integral representations at one loop.

We adopt the convention that all velocities are outgoing, such that the velocities associated with outgoing and incoming states respectively have positive and negative time components. We will take the gauge group to be and work in Feynman–’t Hooft gauge with spacetime signature. Ultraviolet divergences will be regulated by computing all diagrams in dimensions with . To avoid complications arising from regulating collinear singularities, we take all velocities to be time-like, .

We start our investigations by examining the simplest eikonal diagram,
the one-loop exchange illustrated in figure 1.
In both kinematic regions (a) and (b), the position-space
representation of the diagram is straightforwardly obtained
by direct perturbative expansion^{1}^{1}1See chapter 8
of ref. sterman1993introduction for the position-space
Feynman rules. in of the correlator
(1.2) and takes the form

(2.1) |

where the prefactor is defined as , with denoting the quadratic Casimir of the fundamental representation. Furthermore, have the dimension of time and denote the positions of the attachment points of the soft-gluon propagator on the Wilson lines spanned by the four-velocities and .

The integrations in eq. (2.1) produce an infrared divergence which can be extracted via the change of variables with , where has the dimension of length,

(2.2) |

Indeed, the -integral is has an infrared divergence, owing to the exchange of gluons of increasingly longer wavelength as . This divergence can be regularized in a gauge invariant fashion by introducing an exponential damping factor with , whereby it becomes

(2.3) |

The two diagrams in figure 1 have the same integrand; however, as the external kinematics is taken from the distinct regions and , the integrations will produce distinct results. It is most convenient to compute the diagram in figure 1(b) first and obtain the result for figure 1(a) by analytic continuation as follows. For the diagram in figure 1(b), we may define the deflection angle such that , in terms of which the diagram in figure 1(b) becomes, to the leading order in ,

(2.4) |

Likewise, for the diagram in figure 1(a), we may define the cusp angle such that . The integrated expression for this diagram can thus be obtained from eq. (2.4) by replacing ,

(2.5) |

We observe that the imaginary parts of the eikonal diagrams in figs. 1(a) and 1(b) are respectively non-vanishing and vanishing. Before turning to the question of how the imaginary parts of the diagrams in eqs. (2.4)–(2.5) may be extracted from their integral representation in eq. (2.2), let us consider their physical origin and interpretation.

From the position-space representation (2.1) of the eikonal diagram, the origin of the imaginary part may be understood from a simple causality consideration as follows. As our focus is on computing the imaginary part to the leading order in , the in the propagator exponent may be dropped once the infrared divergence has been extracted. After moreover stripping off real prefactors from eq. (2.1), the integral takes the form,

(2.6) |

Now, for the kinematics corresponding to the diagram in figure 1(a), there are regions within the integration domain where . Here the term becomes relevant and generates an imaginary part. What is happening physically at such times is that the two partons traveling along and become lightlike separated. This is illustrated in figure 2(a). As a result, the phases of their states will change through exchanges of lightlike gluons (or photons)—leading to observable consequences that will be discussed shortly. In contrast, for external kinematics corresponding to the diagram in figure 1(b), the integral in eq. (2.6) has a vanishing imaginary part: the denominator is strictly positive within the region of integration, and the can therefore be dropped. In this situation, the partons are never lightlike separated, as illustrated in figure 2(b), and the phases of their states cannot change through exchanges of lightlike massless gauge bosons.

These observations on the evolution of the phases of the hard-parton states suggest that the imaginary part of the correlator of two Wilson lines defines an interparton potential. Indeed, in the non-relativistic limit, the final and initial two-particle states are related in the interaction picture through time evolution by where denotes the interaction potential. The relation of the correlator to a non-relativistic potential can be made precise in the situation where the pair of energetic particles carry no color charges, as for example in the case of an pair. In Abelian gauge theories, the correlator of two Wilson lines can be written as the exponential of the sum of connected diagrams Yennie:1961ad ,

(2.7) |

where is the one-loop diagram in figure 1, and the additional diagrams contain a single lepton loop connected to the Wilson lines by an arbitrary (even) number of soft-photon exchanges. Using the result for the diagram computed for time-like kinematics in eq. (2.5) (with to recover the Abelian case), the anomalous dimension of the Wilson-line correlator—i.e., the cusp anomalous dimension—evaluates to

(2.8) |

The non-relativistic limit corresponds to the small-angle regime where the two velocities and are nearly collinear, and the relative velocity of the hard leptons thus small. Accordingly, expanding eq. (2.8) around and taking the imaginary part, we find

(2.9) |

We observe that the imaginary part of the cusp anomalous dimension evaluated in time-like kinematics takes the form of the non-relativistic Coulomb potential (the appropriate dimension of energy is acquired after replacing the angle by the distance between the two fermions).

This relation does not extend to generic non-Abelian gauge theories,
as we will discuss shortly. It does, however, extend to the case of
conformal field theories, such as
super Yang-Mills theory, where the state-operator correspondence relates
Wilson-line operators in Minkowski space to states in
. In radial quantization,
a pair of Wilson lines intersecting at a cusp angle
with the resulting anomalous dimension
is mapped to a pair of static charges in separated by
a distance of with an electrostatic energy^{2}^{2}2The real
part of the cusp anomalous dimension gives rise to an imaginary part of
the electrostatic energy. As argued in ref. Chien:2011wz ,
the resulting non-unitary time evolution is accounted for by the real
radiation of soft and collinear gluons along the Wilson lines. of
Chien:2011wz .
For small values of the cusp angle, the charges on become
closer than the curvature scale, and the electrostatic energy
takes the form of the non-relativistic interquark potential in
flat space Correa:2012nk ; Correa:2012hh . (The non-relativistic
approximation becomes relevant here, as in the small-angle regime
, the relative velocity of the hard partons is small,
as discussed above.)

However, for non-Abelian and non-conformal gauge theories such as QCD,
diagrams containing loop corrections to the soft propagators will
have a dependence on the beta function, thereby explicitly breaking
the scale invariance of the diagram. As a result, in QCD, the imaginary part
of the three-loop cusp anomalous dimension
differs from the static
interquark potential by terms proportional to the beta function Grozin:2014hna .
(This can be seen by comparing the contribution
to , given in eq. (A.2) of
ref. Beneke:1995pq , against the term of the three-loop
static QCD potential^{3}^{3}3Note that in the literature on
the interquark potential, the loop order is often defined
as one less than the standard notion., given in eq. (10) of
ref. Schroder:1998vy .)

Let us now turn to the question of how the imaginary part
of the eikonal diagrams in figure 1
may be obtained from their integral representation in
eq. (2.2) where
the infrared divergence has been extracted. We will restrict
attention to the leading order in the dimensional regulator
, and accordingly drop the in the propagator
exponent. We can then utilize the formula

(2.10) |

where indicates that the Cauchy principal value prescription is to be applied, and the integration bounds and are real numbers. The denominator is a real-valued polynomial in , and the numerator is an arbitrary real-valued function with no poles or branch points inside the integration path. As both integrals on the right-hand side of eq. (2.10) are real, this formula achieves a decomposition into a purely real and purely imaginary part.

Accordingly, at one loop, we define the position-space cut prescription

(2.11) |

in terms of which it is straightforward to obtain the imaginary part of the diagrams in figure 1 to the leading order in . For example, considering the time-like kinematics situation in figure 1(a) and applying the prescription (2.11) to eq. (2.2) with the in the propagator exponent set to zero, we find

(2.12) |

We can integrate out the delta function by use of the identity

(2.13) |

where and respectively denote the discriminant and roots of the polynomial. The roots of the delta function argument in eq. (2.12) are manifestly located inside the domain of the -integration. The result of integrating out the delta function in eq. (2.12) is therefore

(2.14) |

in agreement with eq. (2.5). The calculation for the diagram in figure 1(b) is completely analogous, except that in this case , as both roots are located outside the domain of integration. We therefore find a vanishing imaginary part, in agreement with eq. (2.4). We conclude that in both cases (a) and (b), the cutting prescription (2.11) produces the correct imaginary part. We introduce a graphical notation for the cutting prescription (2.11) in figure 3.

It is natural to ask whether the imaginary part of eikonal diagrams
can also be obtained from their momentum-space representation^{4}^{4}4The momentum-space representation in eq. (2.15)
is straightforwardly obtained from the eikonal Feynman rules.
By Schwinger parametrizing the eikonal propagators in
eq. (2.15) and performing
the resulting Fourier transform, one recovers the
position-space representation in eq. (2.1).

(2.15) |

Such a cutting prescription was provided in ref. Korchemsky:1987wg . Here it was shown that the imaginary part of the one-loop diagram in eq. (2.15) may be obtained by replacing the two eikonal propagators by delta functions,

(2.16) |

This prescription can be thought of as the eikonal limit of the standard Cutkosky rules. It is illustrated in figure 4 below. More explicitly, applying the prescription (2.16) to eq. (2.15), the imaginary part is determined as follows,

(2.17) |

This representation of the imaginary part of the one-loop diagram motivates two remarks.

The first remark concerns the region of momentum space which gives rise to the imaginary part. Defining the light-cone variables and choosing the Lorentz frame in which the transverse components of the velocities vanish, , the support of the delta functions in eq. (2.17) is the region where the momentum of the exchanged gluon is maximally transverse,

(2.18) |

which was identified in ref. Korchemsky:1987wg as the Glauber region Collins:1983ju . This agrees with the discussion in section 1: the imaginary part of eikonal diagrams arises from the exchanges of Glauber-region gluons.

The second remark concerns the physical interpretation of applying the momentum-space cuts (2.16). By writing the delta functions in eq. (2.17) in terms of the plane-wave representation and performing the Fourier transform we find

(2.19) |

We observe that the resulting integration bounds compared to those of the uncut diagram in eq. (2.1) are extended according to . This state of affairs can be simply understood on physical grounds: as the hard partons have been put on shell through the cutting rule (2.16), they are now asymptotic states propagating from to the interaction point.

We see that the position- and momentum-space representations of eikonal diagrams offer complementary points of view on the origin of their imaginary part. To summarize, in the position-space representation, the imaginary part is seen to arise from the exchanges of lightlike soft gauge bosons whose emission and absorption change the phases of the hard-parton states. In contrast, in momentum space, the imaginary part (related to the branch cut discontinuity through eq. (3.14)) arises from the two hard partons going on shell and exchanging Glauber gluons. Thus, the position- and momentum-space representations explain the origin of the imaginary part from the points of view of causality and unitarity, respectively.

The momentum-space cutting prescription in eq. (2.16) has the conceptual advantage of factoring eikonal diagrams into on-shell lower-loop and tree diagrams which in turn can be computed as independent objects. However, the resulting cut diagrams involve integrations over two-, three-, four-, particle phase space, as illustrated in figure 5. In practice, the evaluation of these phase-space integrals poses a substantial computational challenge which limits the applicability of the cut prescription (2.16) for obtaining imaginary parts.

## 3 Position-space cuts of eikonal diagrams without internal vertices

For completeness, in this section we review the derivation presented in ref. Laenen:2014jga of the imaginary part of -loop eikonal diagrams without internal (i.e., three- or four-gluon) vertices to the leading order in the dimensional regulator . We will interchangeably refer to these diagrams as ladder-type diagrams. The basic observation is that in position space these diagrams are iterated integrals, and as a result their imaginary part can be obtained by decomposing the real-line integrations into principal-value and delta function contributions.

In position space, an arbitrary -loop eikonal diagram without internal vertices is composed of soft-gluon propagators, interchangeably referred to here as rungs. Each rung extends between the Wilson lines spanned by any two (possibly identical) external four-velocities where . For the th rung we will denote these four-velocities by and . We let denote the position of the th attachment on the Wilson line spanned by , counting from the hard interaction vertex and outwards, so that , where denotes the total number of soft-gluon attachments on the Wilson line. In addition, for the th rung, we let the variables and record the soft-gluon attachment numbers on the Wilson lines spanned by and , respectively. The -loop eikonal diagram is then defined as the -fold iterated integral

(3.1) |

where the kinematics-independent prefactor is determined by the color structure of the diagram and where it is implied that and . Without loss of generality, we will assume that any rungs with both endpoints attached to the same Wilson line have been integrated out, and we suppress the resulting pole factors in . (The additional factors produced by the integrations, involving epsilonic powers of the remaining variables, will not be of importance here, as our aim is to extract the imaginary part of to the leading order in .)

To extract the imaginary part of from the integral
representation in eq. (3.1)
it turns out to be useful to perform a change of variables which leaves
each soft propagator dependent on a single variable.
To this end, we adopt a change of variables introduced
in ref. Gardi:2011yz . The idea is to first
express the attachment points of the th rung in terms
of “polar” coordinates measuring the distance to the cusp
(in units of the infrared cutoff ) and
essentially measuring the emission
angle of the soft gluon to the Wilson line,^{5}^{5}5For a given rung,
the two endpoints may of course be referred to interchangeably
as left or right. However, for practical calculations,
one particular choice may prove slightly more convenient.
We refer to section 4 for examples.

(3.2) |

After this change of variables, the diagram takes the form

(3.3) |

where the soft propagators are defined as

(3.4) |

and where the nesting of the integrations is encoded in , defined through

(3.5) |

We observe that the soft propagators’ dependence on the radial coordinates has scaled out in eq. (3.3), and that each propagator now depends only on a single variable . This turns out to be particularly advantageous for the purpose of extracting the imaginary part of the diagram, as this circumvents the need to divide a higher-dimensional domain of integration into subdomains characterized by supporting a specific number of propagator roots.

Now we extract the overall infrared divergence of the diagram by setting and then applying the following sequence of substitutions

(3.6) |

where the variables have the dimension of length and the are dimensionless. The -loop eikonal diagram then becomes

(3.7) |

where the infrared divergence of the diagram is now absorbed into the kernel

(3.8) |

Here denotes the result of applying the substitutions (3.6) to eq. (3.5). In analogy with section 2, we have here regulated the infrared divergence in a gauge invariant way through the exponential damping factor with . Eq. (3.8) contains in addition any potential ultraviolet subdivergences of the diagram (generated by the nesting function ).

Having brought the -loop eikonal diagram in the form (3.7), we now turn to extracting its imaginary part. Restricting our attention to the leading order in the dimensional regulator , we will drop the dependence of the soft propagators on ,

(3.9) |

where denotes the degree of divergence of the diagram, .

To compute the imaginary part of eq. (3.9), we start by observing that eq. (3.8) is manifestly purely real. As a result, the Feynman ’s are the only source of imaginary parts of Eq. (3.9). Each of the -integration paths can therefore be decomposed into a principal-value part and small semicircles around the propagator poles. Given that the integrand takes purely imaginary values in the regions close to the poles and is real-valued on the remaining domain of integration, the resulting terms (which each involve integrations) will be either purely real or purely imaginary.

To collect the imaginary contributions, we define the cut propagator

(3.10) |

and furthermore -fold cutting operator

(3.11) |

The action of this operator is to replace the propagators that depend on the specified variables by delta functions and to place a principal-value prescription on the integrals over the remaining variables. To simplify notation, we here dropped the indices on the (cut) propagators: and .

The imaginary part of any -loop eikonal diagram without internal vertices can then be written, to the leading order in ,

(3.12) |

This is the central formula of our approach Laenen:2014jga . The formula (3.12) is illustrated schematically for a generic ladder diagram in figure 6 below.

We note that the decomposition of the line integrations in eq. (3.9) into principal-value and delta function contributions immediately shows that the imaginary part of the integrated expression for the eikonal diagram will have transcendentality weight one less than the real part. This follows from the fact that the delta functions will map the rational integrand to a rational expression after being integrated out. Thus, compared to the real-part contribution with principal-value integrals, the weight is dropped by one.

It is natural to ask about the relation of the imaginary part of the eikonal diagram to the discontinuities in its various kinematic channels. This in turn leads us to ask for an appropriate set of variables in terms of which to express integrated results. A good choice of variables turns out to be given by the exponentials of the cusp angles,

(3.13) |

with the cusp angles defined through . Expressed in terms of the -variables, the eikonal diagram has branch cuts located on the real line and satisfies Schwarz reflection, . As a result, the discontinuities of the diagram give rise to the imaginary part through the relation

(3.14) |

Here, the step functions account for the fact that the
imaginary part has vanishing contributions from channels
with space-like kinematics .
(This follows from the fact that propagators stretched
between mutually space-like eikonal lines have vanishing cuts,
as will be explained below eq. (3.16).)
We will see an explicit example of this in section 4.3
where we study a diagram that depends on two distinct
cusp angles in purely time-like as well as mixed
time- and space-like kinematics.

In section 4 we will work out examples of
how eq. (3.12) is used in practice to compute
the imaginary part of ladder-type eikonal diagrams.
To this end it will be useful to record the following
partial-fractioned expressions, setting ,

(3.15) |

where the prefactor is the rational expression , and the denominator roots are given by

(3.16) |

We note that in the upper case of eq. (3.16), the roots satisfy , whereas in the lower case they satisfy . Since the delta functions in eq. (3.11) are integrated over the interval , we may thus infer that the eikonal diagram will only have contributions to its imaginary part from channels with time-like kinematics , as encoded in eq. (3.14). This is in agreement with the causality considerations of section 2.

In section 4 we will make extensive use of the fact that the result for an eikonal diagram in time-like kinematics can be immediately obtained from the space-like result by analytic continuation of the cusp angle. To see this, let us first recall that the soft propagator takes the same form (3.4) in space- and time-like kinematics when expressed in terms of , owing to our convention that all velocity vectors are outgoing. However, once expressed in terms of the relative angle , it takes the respective forms

(3.17) |

where we dropped the in the space-like case, as the propagator roots are located outside the range of . Comparison of these expressions shows that we can map space-like to time-like kinematics by means of the analytic continuation

(3.18) |

or equivalently, in terms of ,

(3.19) |

## 4 Examples

The aim of this section is to apply the formalism reviewed in section 3 to compute the imaginary part of a number of ladder-type eikonal diagrams. The main point to be addressed here concerns the evaluation of the principal-value integrals involved in the -fold cuts in eq. (3.11).

### 4.1 The non-planar two-loop ladder diagram

As a first example we will consider the non-planar two-loop ladder diagram, illustrated in figure 7 below. This diagram contains no ultraviolet subdivergence and therefore only has a simple pole in the dimensional regulator . In agreement with the observations at the end of section 3, the diagram will only have an imaginary part for time-like kinematics . We therefore restrict our attention to this case. Since the diagram contains only one cusp angle, we will drop the subscripts for convenience and define as well as .

The non-planar two-loop ladder diagram has the position-space representation

(4.1) |

where the prefactor is given by 3.7). This is achieved through the change of variables in eq. (3.2), followed by that in eq. (3.6), . To compute the imaginary part of this diagram, our first task is to write it in the form of eq. (

(4.2) |

After these transformations the diagram takes the desired form,

(4.3) |

where the kernel , upon the additional change of variable , is given by

(4.4) |

By comparing eqs. (4.1) and (4.3), we see that the effect of the first two transformations of eq. (4.2) is to leave each soft propagator dependent on a single variable. The effect of the last change of variable is to extract the overall infrared divergence of the diagram.

To facilitate the computation of the cuts in eq. (3.12) we will first evaluate the integral . The -integral in eq. (4.4) may be performed in terms of the hypergeometric function. The primitive has the -expansion

(4.5) |

and so has the -expansion

(4.6) |

Substituting this result for into eq. (4.3), we can write the non-planar two-loop ladder diagram in the convenient form

(4.7) |

where is finite, given to leading order in as,

(4.8) |

Here we have dropped the dependence of the soft propagators on and furthermore rewritten the integrand of to make its symmetry under manifest. (This was achieved by changing variables on the second term arising from eq. (4.6).)

As the prefactor of in eq. (4.7) is real, it factors out on both sides of eq. (3.12), yielding the formula

(4.9) |

where we recall that is defined in eq. (3.11) and replaces the propagator depending on the specified variable by a delta function and places a principal-value prescription on the integral over the remaining variable. The two cuts on the right-hand side are equal because the integrand of eq. (4.8) is symmetric under the interchange of and . (This also follows from the symmetry of the original diagram.) Thus, it suffices to compute , given by

(4.10) |

Inserting the partial-fractioned expressions for and given in eq. (3) and performing the trivial integral over produces

(4.11) |

where the propagator roots are given in the upper part of eq. (3.16).

We are now confronted with the task of evaluating principal-value integrals. As such integrals do not immediately take the form of iterated integrals, our strategy for evaluation will be to write them as differences of iterated integrals, which in turn are readily expressible in terms of multiple polylogarithms. The basic observation is that the principal-value integral equals the corresponding full integral minus the imaginary part of the latter, cf. eq. (2.10).

As a simple illustration, let us consider the evaluation of the following principal-value integral,

(4.12) |

The full integral evaluates to by definition, cf. eqs. (A.1)–(A.2). Its imaginary part arises from the pole of the integrand and is extracted by localizing the integration variable,

(4.13) |

where we used in the last step that the pole is located inside the range of integration, in agreement with the discussion below eq. (3.16). Thus we arrive at

(4.14) |

In this simple example we could have computed the real part more directly,

(4.15) |

However, an extension of this direct approach to higher-weight cases requires the use of a sequence of functional identities which in practice is case-dependent and thus not applicable in a systematic way. In contrast, the above method relies only on a construction of the imaginary part which can be derived systematically as demonstrated in appendix A.

Returning to eq. (4.11) and evaluating the principal-value integrals following the steps outlined above, the result for the cut is readily expressed in terms of multiple polylogarithms,

(4.16) |

where we refer to eqs. (A.1)–(A.2) for definitions. In this expression, the multiple polylogarithms depend on the propagator roots through both their indices and their arguments. This expression can in turn be rewritten in terms of polylogarithms with constant indices by exploiting the Hopf algebra structure of multiple polylogarithms, which encodes the plethora of functional identities within this class of functions Goncharov:2005sla ; Goncharov:2010jf ; Brown:2011ik ; Duhr:2011zq ; Duhr:2012fh ; Anastasiou:2013srw . Utilizing this algebraic structure, we have implemented the steps required to achieve the desired functional form as a general algorithm. We refer to appendix B for further details. The algorithm leaves a simplified form of eq. (4.1) expressed in terms of harmonic polylogarithms which, using eq. (A), can be simplified further into classical polylogarithms,

(4.17) |

With this result, we can now immediately obtain the imaginary part of the two-loop ladder from eq. (4.9), recalling that the two cuts are equal. We find

(4.18) |

and we recall that multiplying the infrared pole
cf. eq. (4.7) onto both sides of
eq. (4.18) gives the imaginary part of
the original diagram . This completes
the evaluation of the imaginary part of the
non-planar two-loop ladder diagram to the leading
order in .

As a crosscheck of the result in eq. (4.18),
we can alternatively compute the imaginary part of the non-planar
two-loop ladder by evaluating the diagram for space-like
kinematics , in which case it will
be purely real (cf. the discussion at the end of
section 3), and subsequently
perform the analytic continuation to time-like kinematics.
We refer to the end of section 3
for a more detailed discussion of analytic continuations.

To the leading order in , the two-loop ladder is given by eq. (4.8), although we must bear in mind that for space-like kinematics the propagator roots are given by the lower case of eq. (3.16). Inserting into eq. (4.8) the expressions for given in eq. (3), the diagram readily evaluates into multiple polylogarithms,

(4.19) |

The tilde on the left-hand side serves to remind us that
the expression for the diagram on the right-hand side is valid for
space-like kinematics.
We can use the algorithm in appendix B
to recast this representation in terms of polylogarithms
with constant indices. In fact, the two-loop ladder
diagram can be expressed in terms of classical polylogarithms,^{6}^{6}6Note that in the representation in the second line of
eq. (4.20),
the polylogarithms and logarithms have the respective
arguments and . This form has the advantage
of being particularly compact as well as convenient
for the purpose of performing the analytic continuation
which then maps
the arguments of the (poly)logarithms to the vicinity
of the branch cuts. (In general, computer algebra software,
such as Mathematica augmented with the package
HPL Maitre:2005uu ; Maitre:2007kp ,
does not detect the monodromy of
after tracing out a complete circle around the branch point.)

(4.20) |

We can now find the result for the two-loop ladder diagram in time-like kinematics by performing the analytic continuation on eq. (4.20). Under the analytic continuation, the rational function picks up a minus sign, while the polylogarithms transform according to

(4.21) |

Applying these replacements to eq. (4.20) we find the following result for the non-planar two-loop ladder with time-like kinematics,

(4.22) |

We observe that the imaginary part of eq. (4.22) agrees with the result found in eq. (4.18), as expected. We conclude that the cutting prescription for the two-loop ladder stated in eq. (4.9) produces the correct imaginary part. The cutting prescription (4.9) is illustrated in figure 8.

### 4.2 Three-loop non-planar ladder diagram

To demonstrate that the principal-value integrals involved in the -fold cuts in eq. (3.11) can indeed be evaluated in non-trivial cases, we consider in this section the three-loop ladder diagram illustrated in figure 9. This diagram also represents an example of an eikonal diagram with multiple-cut contributions to its imaginary part (in the case at hand, a triple cut). As in section 4.1, we take , in order to have a non-vanishing imaginary part, and set .

The position-space representation of the diagram in figure 9 takes the form

(4.23) |

where . To compute the imaginary part of this diagram from eq. (3.12), our first task is to bring it into the form of eq. (3.7). This is achieved through the changes of variables in eq. (3.2) (with ), followed by the sequence of substitutions in eq. (3.6), setting and for convenience.

After these transformations, the diagram takes the form

(4.24) |

where the kernel is given by

(4.25) |

The arguments of the step functions simplify after rescaling the integration variables according to and . As we are interested in computing the imaginary part of only to the leading order in , we may set to zero in the -integral. Performing the -integral in terms of hypergeometric functions and subsequently expanding in , we find the expression

(4.26) |

valid to the leading order in . We observe that eq. (4.2) is symmetric under the interchange of and ; hence the integrand of the full diagram in eq. (4.24) is as well. (In other words, interchanging the two parallel gluon lines in figure 9 leaves the diagram invariant.) This observation implies that and thereby reduces the number of independent cuts to be computed.