1 Introduction
Abstract

We compute the three-loop contribution to the supersymmetric Yang-Mills planar four-gluon amplitude using the recently-proposed Higgs IR regulator of Alday, Henn, Plefka, and Schuster. In particular, we test the proposed exponential ansatz for the four-gluon amplitude that is the analog of the BDS ansatz in dimensional regularization. By evaluating our results at a number of kinematic points, and also in several kinematic limits, we establish the validity of this ansatz at the three-loop level.

We also examine the Regge limit of the planar four-gluon amplitude using several different IR regulators: dimensional regularization, Higgs regularization, and a cutoff regularization. In the latter two schemes, it is shown that the leading logarithmic (LL) behavior of the amplitudes, and therefore the lowest-order approximation to the gluon Regge trajectory, can be correctly obtained from the ladder approximation of the sum of diagrams. In dimensional regularization, on the other hand, there is no single dominant set of diagrams in the LL approximation. We also compute the NLL and NNLL behavior of the -loop ladder diagram using Higgs regularization.

HU-EP-09/64

BOW-PH-146

BRX-TH-615

Brown-HET-1591

Higgs-regularized three-loop four-gluon amplitude in SYM:

exponentiation and Regge limits

Johannes M. Henn, Stephen G. Naculich111Research supported in part by the NSF under grant PHY-0756518, Howard J. Schnitzer222Research supported in part by the DOE under grant DE–FG02–92ER40706 and Marcus Spradlin333Research supported in part by the DOE under grant DE–DE-FG02-91ER40688
henn@physik.hu-berlin.de, naculich@bowdoin.edu, schnitzr@brandeis.edu, spradlin@het.brown.edu

Institut für Physik

Humboldt-Universität zu Berlin, Newtonstraße15, D-12489 Berlin, Germany


Department of Physics

Bowdoin College, Brunswick, ME 04011, USA



Theoretical Physics Group

Martin Fisher School of Physics

Brandeis University, Waltham, MA 02454, USA



Brown University, Providence, Rhode Island 02912, USA

1 Introduction

Recent years have witnessed significant advances in supersymmetric Yang-Mills theory (SYM) in four dimensions. The AdS/CFT conjecture relating SYM theory to maximally supersymmetric string theory on AdS S reveals various integrable structures. One aspect of recent progress is a greater understanding of the structure of on-shell scattering amplitudes, at both tree level and loop level. For example, based on an iterative structure found at two loops [1], Bern, Dixon and Smirnov (BDS) conjectured an all-loop form of the maximally-helicity-violating planar -gluon amplitude [2], which is believed to be correct for and , but requires modification by a function of cross-ratios for six or more gluons [3, 4, 5, 6, 7].

The simplicity of the BDS form for four- and five-gluon loop amplitudes arises from a hidden symmetry of the planar theory, viz., dual conformal invariance [8]. Dual conformal symmetry is also present in the theory at strong coupling, and can be seen in a string theory description of the scattering amplitudes, which are identified with Wilson loop expectation values in a T-dual AdS space [9]. The dual conformal symmetry of scattering amplitudes is understood as the usual conformal symmetry of the dual Wilson loops. In fact, this scattering amplitude/Wilson loop relation extends to weak coupling as well [10, 11, 3] (for reviews see [12, 13]).

The dual conformal symmetry is anomalous at loop level due to ultraviolet divergences associated with the cusps of the Wilson loops (which correspond to infrared (IR) divergences of the scattering amplitudes). Anomalous Ward identities can be derived for the Wilson loop expectation values [14], whose solution is unique up to a function of conformal cross-ratios. The BDS ansatz satisfies the anomalous Ward identity, therefore it is exact for Wilson loops with four and five cusps, since there are no cross-ratios in these cases. Assuming the Wilson loop/scattering amplitude duality, this implies the validity of the BDS ansatz for four- and five-gluon amplitudes.

Dual conformal symmetry extends to a dual superconformal symmetry [15], which is a symmetry of all tree-level amplitudes [15, 16, 17]. (This dual superconformal symmetry can also be understood from the string theory point of view by means of fermionic T-duality [18, 19, 20].) The dual superconformal symmetry combines with the conventional superconformal symmetry of SYM theory to form a Yangian symmetry [21]. The IR divergences of the loop amplitudes, however, a priori destroy both the ordinary and dual superconformal symmetries (and therefore the Yangian symmetry); this breakdown is explicitly seen in dimensional regularization. Unlike the dual conformal symmetry, the breaking of the ordinary conformal symmetry is not (yet) under control (see however refs. [22, 23, 24] for progress in this direction), and so the role of the Yangian symmetry for loop amplitudes is unclear.

In practice, it is desirable to use a regulator that preserves as many symmetries as possible. Recently, Alday, Henn, Plefka, and Schuster (AHPS) presented a regulator which, in contrast to dimensional regularization, leaves the dual conformal symmetry unbroken [25]. In this approach, the SYM theory is considered on the Coulomb branch where scalar vevs break the gauge symmetry, causing some of the gauge bosons to become massive through the Higgs mechanism. Planar gluon scattering amplitudes on this branch can be computed using scalar diagrams in which some of the internal and external states are massive, regulating the IR divergences of the scattering amplitudes (for earlier references see [9, 26, 27, 28]). The diagrams remain dual conformal invariant, however, provided that the dual conformal generators are taken to act on the masses as well as on the kinematical variables. The diagrams that can appear in the scattering amplitudes are highly constrained by the assumption of (extended) dual conformal symmetry. There is one point in moduli space for which all the lines along the periphery of the diagrams have mass , while the external states and the lines in the interior of the diagram are all massless. This is believed to be sufficient to regulate all IR divergences of planar scattering amplitudes. The original SYM theory is then recovered by taking small.

Using this Higgs regulator, AHPS computed the SYM four-gluon amplitude at one and two loops. They showed that the planar two-loop amplitude satisfies an iterative relation analogous to the one that holds in dimensional regularization [1], and suggested that exponentiation might extend to higher loops in an analog of the BDS ansatz for the four-gluon amplitude:

(1.1)

where is the ratio of the all-orders planar amplitude to the tree-level amplitude, is the cusp anomalous dimension (3.3), and and are analogs of functions appearing in the BDS ansatz in dimensional regularization (3.5). Let us emphasize that the nontrivial content of the BDS ansatz is the statement about the finite terms; the IR singular terms of the amplitude are expected to obey eq. (1.1) (or eq. (3.5)) on general field theory grounds.444In ref. [29] the transition from amplitudes in dimensional regularization to amplitudes where (part of) the IR divergences are regulated by (small) masses was investigated. In this way, the and terms in eq. (1.1) can be understood as arising from a different multiplicative renormalization factor relative to dimensional regularization. It is also conceivable that by adapting the formalism of ref. [29] to the present case one could show that the terms finite as follow from the corresponding formula in dimensional regularization. This would imply that the BDS ansatz can be stated in a scheme-independent way.

In this paper, we explore whether eq. (1.1) continues to hold at three loops and beyond. We compute the three-loop four-gluon amplitude using Higgs regularization, assuming that only integrals invariant under (extended) dual conformal symmetry contribute. Mellin-Barnes techniques are used to evaluate the integrals, some parts of which are computed numerically. We numerically evaluate the results at a number of kinematic points, and obtain explicit expressions in several kinematic limits (e.g., and also the Regge limit ). In every case, our results confirm the expected exponential ansatz (1.1) at the three-loop level.

An important difference between Higgs regularization and dimensional regularization is that in the former, IR divergences take the form of logarithms of whereas in the latter, IR divergences appear as poles in , where . A consequence of this is that, provided eq. (1.1) is valid, the -loop amplitude in Higgs regularization may be computed by simply exponentiating without regard for the terms since they continue to vanish as even when multiplied by logarithms of . In contrast, the BDS ansatz in dimensional regularization (3.5) specifies up to terms that vanish as . In exponentiating , these neglected terms can combine with the IR poles to give rather complicated contributions to the IR-finite -loop amplitude.

To put the matter the other way around, in order to test eq. (1.1) one need not compute any terms of the Higgs-regulated -loop amplitudes because they cannot make any contribution to the IR-finite part of , whereas to test the BDS ansatz in dimensional regularization, one must compute and higher terms in the lower-loop amplitudes to obtain all the IR-finite contributions to . This is one of several significant advantages that Higgs-regulated amplitudes have over their dimensionally-regulated counterparts.

A second major focus of this paper is the Regge behavior of planar SYM amplitudes using Higgs regularization. One motivation for this is that it presents a different way of examining the iterative structure of the theory, in which some results can be obtained to all orders. Equation (1.1) may be rewritten as

(1.2)

in which the terms have cancelled555A similar cancellation occurs in the Regge limit using dimensional regularization [10, 30, 31, 32]. Subleading-color corrections to the Regge trajectory in dimensional regularization were also considered in ref. [32]. to leave a single . Consequently, exhibits exact Regge behavior

(1.3)

where the all-loop-orders Regge trajectory is

(1.4)

The lowest order term of the trajectory, , gives rise to the leading log (LL) behavior of the -loop amplitude

(1.5)

in the Regge limit, whereas higher-order terms in the cusp anomalous dimension contribute to the NLL (next-to-leading-log), NNLL, etc. pieces of the amplitude (see eq. (4.10)).

There is a subtle question of order of limits that appears when considering the Regge limit. In fact there are two ways this limit can be taken. The first possibility (a) consists in taking the limit first, and then taking the limit . This order is implicit in eq. (1.2). The second possibility (b) consists in taking the limit first, and then taking . A priori, it is not clear that the result will not depend on the way the limit is taken. Explicit calculation shows that no such ambiguity arises for the integrals appearing at one and two loops. At three loops, as we will show, the individual integrals give different contributions in the two limits; the three-loop amplitude , however, is unchanged.

It seems that both ways of taking the limit can be justified, with slightly different interpretations.666There is an analogous question in dimensional regularization of whether to take or first [33], but see the Erratum in v5 [34]. On the one hand, the Regge (a) limit seems more appropriate in order to make contact with results in the massless theory. On the other hand, the Regge (b) limit is natural for scattering amplitudes with masses (see ref. [35] and sec. 4.4 below).

We show in this paper that the leading log behavior (1.5) stems entirely from a single scalar diagram, the vertical ladder, in the Regge (b) limit of the Higgs-regularized loop expansion. (This contrasts with dimensional regularization, in which ladder diagrams do not dominate in the Regge limit; the leading log behavior of the -loop amplitude receives contributions from most of the contributing diagrams. This might have significant implications for recent discussions of multi-Regge behavior [36, 37, 38, 39, 33, 40].) We also compute the NLL and NNLL terms of the -loop vertical ladder diagram. Other Higgs-regularized diagrams also contribute to the NLL, NNLL, etc. terms of the amplitude, although we do not yet have an all-orders characterization of which diagrams contribute to each order in the leading log expansion.

In ref. [41], the LL approximation to the gluon Regge trajectory was computed using a ladder approximation and an alternative IR regulator, in which the external lines are massless while all internal lines of the diagrams are given a common mass (which cuts off the IR divergences)777This procedure is closely related to an off-shell regulator. If all physical states have a common mass arising from a conventional Higgs mechanism, then external lines with are off-shell from this point of view.. In the LL limit, this “cutoff regulator” yields results identical to those found in this paper.888The Higgs regulator considered in this paper does not seem directly applicable to computing the subleading-color contributions of SYM amplitudes, which are not dual-conformal-invariant, and which involve non-planar diagrams. The cutoff regulator might be more suitable in this respect. The reason for this, as we will see, is that the LL approximation to the vertical ladder is insensitive to the masses of the propagators constituting the rungs of the ladder, and this is the only difference between the cutoff and Higgs regulators. We also compute the NLL contribution to the -loop vertical ladder diagram using the cutoff regulator, which differs from the Higgs regulator by scheme-dependent constants.

The Regge (b) limit of the scattering amplitudes can be understood by performing the Higgs regularization at a different point on the Coulomb branch of the theory. In this case, the scattered particles are massive, whereas some of the internal lines remain massless, eliminating the collinear but not the soft IR divergences. We suggest how the Regge behavior of the four-point amplitude can be understood from the cusp anomalous dimension of a Wilson line with a non-light-like cusp.

The paper is organized as follows: In sec. 2, we give a short review of dual conformal symmetry. Exponentiation of the planar four-gluon amplitude in dimensional regularization and in Higgs regularization are discussed in section 3, and the three-loop amplitude in Higgs regularization is computed. The Regge limit of the planar four-gluon amplitude in several regularization schemes is examined in sec. 4, and the first three terms in the leading log expansion of the -loop vertical ladder diagram are computed. Most of the technical details are relegated to three appendices.

2 Short review of dual conformal symmetry

In this section we give a short review of dual conformal symmetry, with a particular focus on four-gluon amplitudes. Hints for dual conformal symmetry first appeared as an observation that the loop integrals contributing to planar four-gluon scattering amplitudes in SYM theory have special properties when written in a dual coordinate space [8].

Let us recall that the full four-gluon amplitude can be decomposed in a trace basis, the coefficients of which are referred to as color-ordered amplitudes [42, 43, 44]. The color-ordered planar (i.e., large ) amplitudes may be written in a loop expansion

(2.1)

in the ’t Hooft parameter [2]

(2.2)

where is Euler’s constant, and with to regulate IR divergences. In SYM theory, loop corrections to the four-gluon amplitude have the same helicity dependence as the tree amplitude, so we factor out the tree amplitude to express the amplitude as a function of the kinematic variables and only999In this paper, we follow the metric conventions of ref. [25], so that is negative for positive CM energy. The amplitude will be real for and both positive.

(2.3)

At one loop, we have

(2.4)

with the one-loop integral given by

(2.5)

where the external states are on-shell: . After a change of variables to a dual coordinate space [8, 45],

(2.6)

one obtains

(2.7)

where now the on-shell conditions read . Note that this change of variables can be most easily done in a graphical way (see fig. 1).

\Gin@PS@raw/PSfrag wherepop(x1)[[0(cc)1 0]](x2)[[1(cc)1 0]](x3)[[2(cc)1 0]](x4)[[3(cc)1 0]](xa)[[4(cc)1 0]](p1)[[5(cc)1 0]](p2)[[6(cc)1 0]](p3)[[7(cc)1 0]](p4)[[8(cc)1 0]](k)[[9(cc)1 0]]10 0 1/Begin PSfraguserdict /PSfragpopputifelse\Gin@PS@raw/End PSfrag \Gin@PS@raw /Hide PSfragPSfrag replacements\Gin@PS@raw/Unhide PSfrag \Gin@PS@raw \Gin@PS@raw ˝ 0/Place PSfrag \Gin@PS@raw \Gin@PS@raw ˝ 1/Place PSfrag \Gin@PS@raw \Gin@PS@raw ˝ 2/Place PSfrag \Gin@PS@raw \Gin@PS@raw ˝ 3/Place PSfrag \Gin@PS@raw \Gin@PS@raw ˝ 4/Place PSfrag \Gin@PS@raw \Gin@PS@raw ˝ 5/Place PSfrag \Gin@PS@raw \Gin@PS@raw ˝ 6/Place PSfrag \Gin@PS@raw \Gin@PS@raw ˝ 7/Place PSfrag \Gin@PS@raw \Gin@PS@raw ˝ 8/Place PSfrag \Gin@PS@raw \Gin@PS@raw ˝ 9/Place PSfrag

Figure 1: One-loop scalar box diagram representing the on-shell integral (2.5) together with its dual diagram (2.7). The numerator factor is not displayed.

From eq. (2.7) one can see that for the integral would be invariant under conformal transformations in the dual coordinate space [8, 45]; hence the name “dual conformal symmetry.” Due to infrared divergences one cannot set , and hence the aforementioned symmetry is broken, which is why such integrals were later called “pseudoconformal.” It was found that at least up to four loops all integrals appearing in the four-particle scattering amplitude have this property [46, 10]. All this hinted at some deeper underlying structure.

From the practical point of view, assuming that only pseudoconformal integrals contribute to an amplitude proved to be a useful guiding principle101010Note that there can be subtleties about which integrals should be called pseudoconformal (see e.g. ref. [5]), having to do with the peculiarities of dimensional regularization/reduction. (see e.g. refs. [47, 5]). On the other hand, the inevitable breaking of the symmetry for was not under control. This changed when it was realized that the (finite part of the) logarithm of the amplitude satisfies certain anomalous Ward identities, which were initially derived for Wilson loops and are conjectured to hold for scattering amplitudes as well. Assuming the Ward identities hold for the scattering amplitudes, they explain the correctness of the BDS ansatz for four and five scattered particles [3, 14].

Initially dual conformal symmetry could be applied to the -gluon amplitude for maximally-helicity-violating (MHV) amplitudes only, as can be seen from the fact that the variables in eq. (2.6) do not carry helicity. In ref. [15], it was shown how to incorporate the helicity information and to define dual (super)conformal symmetry for arbitrary amplitudes, both MHV and non-MHV. The predictions of ref. [15] about how dual conformal symmetry is realized at tree and loop level (through an anomalous Ward identity) have by now been checked to one-loop order [48, 49, 50, 51]. The status of dual superconformal symmetry at loop level, which is related to the conventional superconformal symmetry, is under investigation [22, 23, 24].

The symmetries mentioned above can also be seen at strong coupling using the AdS/CFT correspondence. In a groundbreaking paper [9], a prescription for computing scattering amplitudes at strong coupling was given. There, a bosonic T-duality was used that maps the original AdS space to a dual AdS space. Dual conformal symmetry can then be identified with the isometries of the dual AdS space (up to the issue of regularization). It was later shown that the bosonic T-duality can be supplemented with a fermionic T-duality [18, 19, 20], which leads to the counterpart of the dual superconformal symmetry found in the field theory. The analysis of ref. [18] is valid to all orders in the gauge coupling constant; however, just as in the field theory, introducing a regulator may break the symmetry. It would be interesting to address this question in the string theory approach. A somewhat related question is how the helicity dependence of the scattering amplitudes is encoded in the string theory setup. (At strong coupling, it is argued to be an overall factor, which can be ignored, but certainly this is not the case at lower orders in the coupling constant.)

In a recent paper [25], a new regularization of planar amplitudes inspired by the string theory setup of refs. [9, 18] was advocated (see also refs. [26, 27, 28]). In the string theory, besides the usual stack of branes at ( being the radial AdS coordinate, with the AdS radius normalized to unity), further branes are placed at distances . The scattering takes place on the stack of branes. In the field theory, this corresponds to going to the Coulomb branch of SYM. Specifically, one starts with a gauge group and breaks it to . This leads to masses for fields with labels in the , masses for fields with mixed gauge labels, while the fields remain massless. We then consider the scattering of fields with indices in the part of the gauge group and take . (In other words, we drop all diagrams containing loops of particles with gauge indices .) Therefore, the internal labels will all be in the part of the gauge group, while the particles running along the perimeter of all Feynman diagrams will have mixed labels, and hence will have massive propagators. The latter make the integrals IR finite, and there is no need to use dimensional regularization. Of course, strictly speaking one is considering a different theory, but the original theory is approached in the small mass limit.111111Unfortunately, since the requirement of finiteness imposes keeping , one cannot reproduce the non-planar scattering amplitudes of the original theory.

The string theory setup suggests that the planar amplitudes defined in this way should have an exact, i.e. unbroken, dual conformal symmetry. This is possible because there are now additional terms in the dual conformal generators that act on the Higgs masses. These additional terms come from the isometries of dual AdS space. At one loop, the integral (2.7) is replaced by

(2.8)

now subject to the on-shell conditions , with the identification . As anticipated, is annihilated by the extended form of dual conformal transformations,121212This can be seen most easily by thinking of the masses as a fifth coordinate of the dual coordinates, defining , and considering conformal inversions in this five-dimensional space. For further details, see ref. [25].

(2.9)

where

(2.10)

Note that the integral is finite (for ), and the symmetry is exact (i.e. unbroken), hence there is no anomaly term on the r.h.s. of eq. (2.9). From eq. (2.9) one can deduce that the functional dependence of is [25]

(2.11)

where

(2.12)

Similar restrictions hold for a higher number of external legs. This form of dual conformal invariance in the field theory was checked for a particular four-scalar amplitude at one loop [25]. There it was also shown that assuming this symmetry at two loops leads to an iterative relation similar to that which holds in dimensional regularization.

\Gin@PS@raw/PSfrag wherepop(m1)[[0(cc)1 0]](m2)[[1(cc)1 0]](m3)[[2(cc)1 0]](m4)[[3(cc)1 0]](mn)[[4(cc)1 0]](p1sq)[[5(cc)1 0]](p2sq)[[6(cc)1 0]](p3sq)[[7(cc)1 0]](pnsq)[[8(cc)1 0]](dots)[[9(cc)1 0]]10 0 1/Begin PSfraguserdict /PSfragpopputifelse\Gin@PS@raw/End PSfrag \Gin@PS@raw /Hide PSfragPSfrag replacements\Gin@PS@raw/Unhide PSfrag \Gin@PS@raw \Gin@PS@raw ˝ 0/Place PSfrag \Gin@PS@raw \Gin@PS@raw ˝ 1/Place PSfrag \Gin@PS@raw \Gin@PS@raw ˝ 2/Place PSfrag \Gin@PS@raw \Gin@PS@raw ˝ 3/Place PSfrag \Gin@PS@raw \Gin@PS@raw ˝ 4/Place PSfrag \Gin@PS@raw \Gin@PS@raw ˝ 5/Place PSfrag \Gin@PS@raw \Gin@PS@raw ˝ 6/Place PSfrag \Gin@PS@raw \Gin@PS@raw ˝ 7/Place PSfrag \Gin@PS@raw \Gin@PS@raw ˝ 8/Place PSfrag \Gin@PS@raw \Gin@PS@raw ˝ 9/Place PSfrag

Figure 2: Mass assignment of a generic planar diagram in the Higgs setup. The external lines correspond to on-shell particles with . The internal dashed lines correspond to massless particles. We call particles of mass ‘light’ and particles of mass ‘heavy’ since the former become massless when we consider the equal mass case .

In fig. 2, we illustrate where the Higgs masses appear in a generic -point integral in this setup. Thanks to dual conformal symmetry, we can set all masses equal without loss of generality, in which case all external particles are massless, the particles in the outer loop of a diagram are massive, and all particles on the inside are massless.

One can in principle write down all dual conformal integrals at a given loop order and for a given number of external legs. The amplitude would then be given by a linear combination of these integrals [10]

(2.13)

where the sum runs over all dual conformal integrals131313Here we mean dual conformal integrals in the sense of the Higgs regulator of ref. [25], not the off-shell regulator of ref. [10]. , with the loop order of the integral, and a (rational) coefficient (a number in the case of MHV amplitudes). Further restrictions result from the requirement that the diagrams must arise from a scattering process; for example, at two loops, the one-loop integral (2.8) cannot appear squared. Moreover, there are restrictions on the number of propagators and numerator factors. Finally, integrals that would be formally dual conformal invariant but are divergent despite the introduction of the Higgs regulator should not appear [10].

3 Exponentiation of the four-gluon amplitude

In this section, we recall the exponentiation of the four-point amplitude of SYM theory, first using dimensional regularization to regulate the IR divergences, and then using the Higgs regularization described in sec. 2.

3.1 Exponentiation in dimensional regularization

On the basis of ref. [1], Bern, Dixon, and Smirnov conjectured [2] that the dimensionally-regularized all-loop orders amplitude (2.3) satisfies

(3.1)

where is given by eq. (2.4), , and

(3.2)

with the cusp and collinear anomalous dimensions [52] given by141414We use the notation of ref. [2]. Note that , where is also widely used in the literature.

(3.3)
(3.4)

The constants and and the terms in eq. (3.1) are not known a priori. The BDS ansatz (3.1) may be re-expressed as

(3.5)

where [2]

(3.6)

Overlapping soft and collinear IR divergences are responsible for the pole in eq. (3.5).

While the BDS ansatz (3.5) implies that the IR-finite part of the logarithm of the amplitude is simply expressible in terms of and and a set of constants , , and , the same is not true of the -loop amplitudes themselves. For example, eq. (3.5) implies [1]

Because of interference between the positive and negative powers of in , the and terms [53] in depend on more complicated functions (polylogarithms) of and , which are present in the and terms [1] of . In general will receive contributions from the coefficients of positive powers of in all lower-loop amplitudes.

3.2 Exponentiation in Higgs regularization

The Higgs mechanism reviewed in section 2 can be used as a gauge-invariant regulator of the IR divergences in the planar massless theory, with the IR divergences appearing as terms in the amplitude, and any terms that vanish as are dropped. One could ask whether, when regulated in this way, the four-point loop amplitude satisfies iterative relations similar to the BDS ansatz for the dimensionally-regulated amplitude. In ref. [25], it was suggested that the analog of eq. (3.5) is

(3.8)

where is the cusp anomalous dimension (3.3), and and are the analogs of and , but need not be the same functions since they are scheme-dependent [25]. Overlapping soft and collinear IR divergences151515In sec. 4.4, we will see that only soft divergences contribute in the Regge limit, giving a single logarithmic divergence. are responsible for the double logarithms in eq. (3.8).

The -loop amplitudes are obtained by exponentiating eq. (3.8). In contrast to dimensional regularization, there is no interference between the IR-divergent terms and the terms in eq. (3.8) since such terms vanish for order by order in the coupling constant. Hence the amplitudes are simply expressed in terms of products of and and a set of constants , , and . For comparison with later calculations, we explicitly write the predictions for the first few loop amplitudes, with and ,

(3.9)
(3.10)
(3.11)

where we have explicitly set and .

The dual-conformal integrals that contribute through two loops are (see figs. 1 and 3)

(3.12)

These integrals were computed in ref. [25], and the exponential ansatz (3.8) was verified to two-loop order. To determine the values of the constants in eqs. (3.9) and (3.10), it is sufficient to evaluate eq. (3.12) at . Defining , one finds [25]

(3.13)
(3.14)

As discussed in ref. [25], these are consistent with eqs. (3.9) and (3.10) provided

(3.15)

The expression for is consistent with eq. (3.3).

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Figure 3: Two- and three-loop four-point dual conformal diagrams. and are ladder diagrams; is the tennis-court diagram. Numerator factors (including a loop-momentum-dependent factor for the tennis court) are omitted. See eqs. (A.20) and (A.23) for explicit formulae for the integrals.

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Figure 4: Three-loop four-point dual conformal diagrams, with numerator factors omitted. Both integrals require a factor of in the numerator in order to be dual-conformally-invariant.

We now test exponentiation (3.8) at the three-loop level, i.e., the prediction (3.11). Since an explicit calculation of higher-loop amplitudes using the Higgs regulator is not (yet) available, we will start from the assumption (2.13) that only dual-conformal integrals contribute. Equation (2.13) requires two ingredients: the set of dual-conformal integrals and their coefficients. In order to identify the allowed set of dual conformal integrals it is helpful to use the dual notation and graphs introduced in section 2. (For more details and examples, see refs. [8, 46, 10, 47, 54].)

There are four dual conformal integrals that can in principle appear [54] in a four-point three-loop amplitude: , and . The first two, depicted in fig. 3, are natural dual conformal analogs of the integrals appearing in the three-loop amplitude computed in dimensional regularization [2]. The last two, depicted in fig. 4, are absent in dimensional regularization. We will assume that these are the only integrals required in the Higgs regularization, with the same coefficients as in the dimensional regularization result. In general it is not valid to take a result computed in one regularization and transpose it to a different regularization; here this procedure can be justified a posteriori (as we discuss below) by imposing that the IR singular terms of the amplitude obey the relation eq. (3.8) as required on general field theory grounds.161616It would be desirable to determine the coefficients of these integrals using a unitarity-based method. Indeed, once a basis of integrals has been established, in our case using dual conformal symmetry, (generalized) unitarity cuts are a powerful tool to compute the coefficients of the integrals (see e.g. refs. [55, 56, 57, 47, 58]).

Given these assumptions we write

(3.16)

In appendix A, Mellin-Barnes (MB) representations for the integrals appearing in eq. (3.16) are derived. The small limit of these MB integrals is extracted using the same method as in ref. [25]. At the kinematic point , we find

where the decimal coefficients are approximations obtained by numerical integration of MB integrals. We find that the three-loop amplitude (3.16) at is consistent with the exponential ansatz (3.11) provided that

(3.19)

together with eq. (3.15). We note that is indeed the correct three-loop cusp anomalous dimension (3.3).

Having obtained the coefficients from the case, we are now in a position to test the full consistency of eq. (3.16) with eq. (3.11). In order to do this, we have evaluated the coefficients of the small expansion of the integrals and numerically for various values of the kinematical variables . We have found agreement with eq. (3.11) within the numerical accuracy of the calculation.

We assumed above that the coefficients of the integrals and vanish in Higgs regularization, as they do in dimensional regularization. We have nevertheless evaluated these integrals in the small limit using MB methods, and found that

(3.20)
(3.21)

so that, if present, they could be simply accommodated in the BDS ansatz at three loops by redefining171717Such a change could be detected when checking eq. (3.8) at higher orders in perturbation theory. and in eq. (3.11). We will therefore not discuss them further in this paper.

Equation (3.11) predicts the three-loop amplitude for arbitrary values of and . We would like to present a further test of eq. (3.11) in the limit , i.e. . This is Regge limit (a) discussed in the introduction. In order to find formulas for our integrals in Regge limit (a), we perform the small and subsequently the limit in the MB integrals. In this way, we obtain an expression in terms of powers of and , whose kinematic-independent coefficients are either numbers or (relatively simple) MB integrals. Where necessary we evaluate the latter numerically.

Let us first collect the results for the one- and two-loop diagrams [25]. The one-loop box diagram gives

(3.22)

For the two-loop horizontal ladder diagram (cf. fig. 3), one has

while for the two-loop vertical ladder

Substituting these results into eq. (3.12), we find agreement with eqs. (3.9) and (3.10) with the coefficients given in eq. (3.15). (The limit does not affect the form of eqs. (3.9) and (3.10).)

We now turn to the three-loop diagrams. For the three-loop horizontal ladder diagram (cf. fig. 3), we obtain181818The coefficients of the terms were obtained numerically, and replaced by their probable rational equivalents.

(3.25)

while for the vertical three-loop ladder, we obtain

For the tennis court diagram in the orientation shown in fig. 3 we find,

while for the tennis court in the transposed orientation, we obtain