High-energy evolution to three loops

This is an electronic reprint of the original article. This reprint may differ from the original in pagination and typographic detail. Author(s): Title: Year: Version: Please cite the original version: All material supplied via JYX is protected by copyright and other intellectual property rights, and duplication or sale of all or part of any of the repository collections is not permitted, except that material may be duplicated by you for your research use or educational purposes in electronic or print form. You must obtain permission for any other use. Electronic or print copies may not be offered, whether for sale or otherwise to anyone who is not an authorised user. High-energy evolution to three loops Caron-Huot, Simon; Herranen, Matti Caron-Huot, S., & Herranen, M. (2018). High-energy evolution to three loops. Journal of High Energy Physics, 2018(2), Article 58. https://doi.org/10.1007/JHEP02(2018)058 2018 JHEP02(2018)058 Published for SISSA by Springer Received:August 12, 2017 Revised:December 31, 2017 Accepted:January 26, 2018 Published:February 9, 2018 High-energy evolution to three loops Simon Caron-Huotaand Matti Herranena,b,c aNiels Bohr International Academy and Discovery Center, Blegdamsvej 17, Copenhagen 2100, Denmark bDepartment of Physics, University of Jyv¨askyl¨a, P.O. Box 35 (YFL), FI-40014 University of Jyv¨askyl¨a, Finland cHelsinki Institute of Physics, P.O. Box 64, FI-00014, Helsinki, Finland E-mail: [email protected],[email protected] Abstract: The Balitsky-Kovchegov equation describes the high-energy growth of gauge theory scattering amplitudes as well as nonlinear saturation effects which stop it. We obtain the three-loop corrections to the equation in planar N= 4 super Yang-Mills theory. Our method exploits a recently established equivalence with the physics of soft wide-angle radiation, so-called non-global logarithms, and thus yields at the same time the threeloop evolution equation for non-global logarithms. As a by-product of our analysis, we develop a Lorentz-covariant method to subtract infrared and collinear divergences in crosssection calculations in the planar limit. We compare our result in the linear regime with a recent prediction for the so-called Pomeron trajectory, and compare its collinear limit with predictions from the spectrum of twist-two operators. Keywords: Scattering Amplitudes, Perturbative QCD, Resummation, Supersymmetric Gauge Theory ArXiv ePrint: 1604.07417 Open Access,c The Authors. Article funded by SCOAP3.https://doi.org/10.1007/JHEP02(2018)058 JHEP02(2018)058 Contents 1 Introduction 1 1.1 High-energy scattering, soft gluons, and non-global logarithms 2 2 Notations 5 3 Two-loop evolution: fixing a convenient scheme 6 3.1 One-loop in Lorentz-invariant form 7 3.2 Lorentz invariant slicing of multi-particle phase spaces 8 3.3 Quick rederivation of two-loop evolution 9 3.4 More on virtual corrections 10 4 Three-loop evolution 12 4.1 First ingredient: triple-soft current 12 4.2 Second ingredient: double-soft current and the remainder function 14 4.3 Nested subtractions for virtual contribution 18 4.4 Nested subtractions for triple real contribution 19 4.5 Nested subtractions for renormalization counter-terms 21 4.6 Final result: the three-loop BK equation in planar N= 4 SYM 23 5 Linearized evolution and BFKL Pomeron trajectory 25 5.1 Result for the eigenvalue 28 5.2 Collinear singularities and resummation 29 6 Discussion and conclusion 32 A One-loop correction to the squared double soft current 33 B Doing transverse integrals efficiently 34 B.1 Single-valued functions for the linearized kernel 36 C Eigenvalue in terms of harmonic sums for m= 0 and m= 1 37 1 Introduction In high-energy scattering, the aspect of a particle depends on the energy scale at which it is probed. In hadronic collisions this effect can be seen in the well known energy dependence of parton distribution functions. The energy dependence can be accessed in a more detailed way by looking at less inclusive observables, for example ones probing correlations between very different rapidities, opening a window on the transverse structure of the projectile. – 1 – JHEP02(2018)058 One then encounters another fundamental evolution equation of QCD, the Balitsky-FadinKuraev-Lipatov (BFKL) equation [1,2]. In contrast to other evolution equations, which are typically linear, nonlinear effects can also play a role in rapidity evolution: once scattering at a given impact parameter has reached opacity, it must saturate. A nonlinear evolution equation which incorporates such effects within perturbation theory has been derived by Balitsky and Kovchegov [3,4]. Asymptotically, saturation may occur at distances shorter than the nonperturbative scale Λ−1 QCD, justifying the use of perturbation theory [5,6]. For many observables, such as inclusive jet correlations or deep inelastic scattering, perturbation theory is also justified by the large momentum transfer in the problem (see for example [7,8] and references therein). The need to control higher order corrections, and the need to better understand the theory at finite coupling, motivate a deeper look into the perturbative series. The next-to-leading-order evolution equation has been known for some time [9]. It reproduces, in the appropriate limit, the next-to-leading order BFKL Pomeron trajectory [10,11]. A notable feature is that the degree of nonlinearity and its complexity increases with each new order in perturbation theory. This is a rather unfamiliar situation which makes it unclear how to best formulate the equation at finite coupling. Furthermore, the corrections have turned out to be numerically large. This has been attributed to collinear effects, suggesting a possibility to resum them at higher orders at both the linear (BFKL) and nonlinear level [12–14]. In order to shed light on these issue, and to critically assess the quality of proposed resummations, higher-loop data is clearly highly desirable. The aim of this paper is to initiate a systematic study of the Balitsky-Kovchegov and BFKL equations at three loops and beyond. Specifically, as a first step, we will derive its three-loop (next-to-next-to-leading order) correction in the planar limit of N= 4 super Yang-Mills (SYM). This calculation is made possible by recent conceptual and technological developments in the calculation of scattering amplitudes. Our methods remain however essentially diagrammatic and we expect them to prove applicable to QCD in a next step. The SYM model is an ideal stepping stone for several reasons. First, partial crosschecks are available due to a recent and highly remarkable prediction of the Pomeron trajectory exploiting integrability in this model [15,16]. Such tests are valuable both from the perturbative and integrability perspective. At the nonlinear level, the interactions to be predicted are related to structure constants [17], soon to be within reach of similar methods. Together with the AdS/CFT correspondence at strong coupling [18], these hint at a possible exact description of the Pomeron and its interactions at finite coupling in this model. 1.1 High-energy scattering, soft gluons, and non-global logarithms A modern description of high-energy forward scattering is based on the eikonal approximation: fast projectiles and targets are approximated by null Wilson lines U. More precisely, by a collection of such Wilson lines, reflecting the transverse structure of the colliding objects at the given rapidity scale [3]. It is simple to translate this language to that of classic Regge theory: the reggeized gluon is the state sourced by (the logarithm of) a null Wilson line [19]. – 2 – JHEP02(2018)058 The three-loop calculation in this paper is enabled by a recently established correspondence with the physics of wide-angle soft radiation, sometimes called “non-global logarithms.” Consider the QCD decay of a color singlet state like a virtual photon or Z boson, with energy Q. A representative observable, sensitive to soft wide-angle radiation, is the probability to not find radiation with energy above a cutoff µwithin an exclusion region R(see figure 1). If the cutoff µis low, this probability is small and controlled in the planar approximation (’t Hooft limit Nc→ ∞) by the Banfi-Marchesini-Smye evolution equation [20]: d dlog µU12 =λ 16π2ZdΩ0 4π α12 α10α02 2U12 −2U10U02≡λ 16π2K(1)U12.(1.1) This resums large logarithms log Q µ. Here αij ≡1−cos θij 2, and the subscripts denote the angles of outgoing partons; the dipole Uij =1 NcTr[U(θi)U†(θj)] is a function of two angles which can be interpreted (see below) as the trace of a color dipole at angles θiand θj. The basic physics of this equation is that the color flow, and therefore the energy flow, is affected by radiation of an extra gluon at angle θ0. The observable, through the exclusion region R, is encoded by the infrared boundary condition that Uij = 0 when either ior jare in R. Qualitatively, the evolution leads to an increased effective size of the exclusion region, as radiation near the allowed boundaries become more and more in danger of leaking out.1 This equation is mathematically equivalent to the Balitsky-Kovchegov equation, which governs the rapidity dependence of perturbative high-energy scattering near the forward direction. In this context, the trivial fixed point Uij=1 represents a transparent target, which is unstable: by linearizing in the departure (1 −U), which gives the BFKL equation, one finds a growing solution known as the BFKL Pomeron. The nonlinear term then accounts for a class of saturation effects which stop the growth (locally in the transverse plane) toward the attractive, opaque, fixed-point Uij=0. The nonlinear term in both equations share a similar physical origin: in both cases one is interested in the probability that something does not happen, while many possibly complicated things may happen [21]. Indeed, to describe the probability to not radiate in a certain region, one must keep track of all allowed radiation, which is what the nonlinear term of eq. (1.1) produces. Similarly, in near-forward scattering, one measures the probability for a projectile to not be destroyed at a given impact parameter. The two evolutions share other physical similarities: both are dominated by soft gluons, and both feature “opaque” and “transparent” regimes. Given these similarities, it seems natural to expect a relationship between these two problems. The geometry is however different. To establish a rigorous map turns out to require a conformal transformation [22,23], which equates detector measurements at infinity with the physics of a fast particle crossing a Lorentz-contracted target (also known as a shockwave). This had been used notably by Hofman and Maldacena and others to describe detector measurements in conformal field theories [22,24,25] and at the same 1The form (1.1) is valid provided that Ris smooth enough that no jets are forced to be narrow. This is assumed here in order to avoid further subtractions of collinear singularities as in the original setup [20], thereby focusing on soft wide-angle radiation and preserving the most symmetrical form of the equation. – 3 – JHEP02(2018)058 time gain new insight into high-energy scattering. This conformal transformation is just the stereographic projection of a two-sphere onto the transverse impact parameter plane: ZdΩ 4π⇔Zd2z π,1−cos θij 2⇔m2|zi−zj|2,d dlog µ⇔ − d dη.(1.2) Here mis an arbitrary mass scale and ηis rapidity. Under this dictionary, the BanfiMarchesini-Smye equation (1.1) becomes precisely the Balitsky-Kovchegov equation, as was noted early on [26]. In this paper we will exploit this correspondence and work exclusively on the nonglobal logarithm side, which is technically advantageous due to a body of knowledge on the infrared and collinear factorization of amplitudes and cross-sections. This correspondence was emphasized and tested explicitly at two-loops in [23], where the full two-loop BFKL/BK equation (including running coupling effects and non-planar corrections) was re-derived starting from non-global logarithm problem. The evolution equation (1.1) at finite coupling is best viewed as a renormalization group (RG) equation: d dlog µ+β(g2)d dg2−Kσ[U;µ] = 0,(1.3) where the color density matrix, or weighted cross-section σ[U], is defined operationally by weighing each final state parton by a color rotation U(θi) [23] (see also [27]). These color rotations can be understood as Wilson lines U(θi) accounting for the effect of more infrared radiation (these Wilson lines connect the decaying state in the matrix element and its conjugate) [28,29]. In the planar limit, the color factors reduce to products of color dipoles and the color density matrix simplifies to a single function Uij of only two angles, as shown in figure 1, which illustrates the “UU” term in eq. (1.1). In both eqs. (1.1) and (1.3), µis an infrared cutoff below which all radiation is inclusive. In our practical calculation we will work within dimensional regularization to D= 4 −2 ( < 0). Then the cutoff µappears in a renormalization procedure. Following standard procedure, this is equivalent to integrating the RG equation from the deep infrared: σbare[U] = Pexp −Zµ 0 dλ λK(g2(λ))σren[U;µ],(1.4) where, writing g2(λ) = g2(µ)(λ/µ)−2+O(g4) for the running coupling in Ddimensions, one can see that the integral produces 1/ poles. The subtraction then cancels the poles in the bare amplitude so as to make σren[U;µ] finite as →0. That the divergences exponentiate in precisely this way was proved to all orders in ref. [23], exploiting known results on the factorization of soft partons [30,31]. The upshot of eq. (1.4) is that we can use the 1/ poles in the dimensionally regulated weighted cross-section to read off the nonglobal-logarithm/Balitsky-Kovchegov kernel K. Note that this is identical to the standard procedure to extract (ultraviolet) anomalous dimensions of local operators, by using their 1/ poles. The fact that divergences (either infrared or ultraviolet) are controlled by renormalization group equations is of course due to the Wilsonian decoupling between physics at different scales. – 4 – JHEP02(2018)058 U(θ1) U(θ0) U(θ2) * Figure 1. Soft wide-angle radiation: radiation is allowed in some region but excluded in another. To keep track of the allowed radiation we use a color density matrix, defined by applying an angledependent color rotation U(θi) between the matrix element and its conjugate for each final state particle. In the planar limit this configuration reduces to a product of two color dipoles. This paper is organized as follows. In section 2we introduce useful notations for soft currents and phase space integrals. In section 3we revisit the two-loop calculation, improving on previous treatments by introducing a scheme where Lorentz symmetry is manifest at each step; under the correspondence with the Regge limit, this is equivalent to maintaining the conformal symmetry of the BK equation. In section 4we perform the threeloop calculation, paying special attention to the combinatorics of subdivergences and their cancellations, culminating in the final result for the nonlinear equation in subsection 4.6. In section 5we analyze its linearized limit, compute its eigenvalues, and compare it with integrability predictions. Finally section 6contains our concluding remarks. In three appendices, we record the one-loop double soft current squared (appendix A), we detail our algorithm to compute finite angular or transverse integrals (appendix B), and record the three-loop eigenvalue (appendix C). 2 Notations The calculation of Krequires squared matrix elements for emitting soft partons off two color-correlated parents (“dipole”), the so-called soft currents. The evolution equation, just like the soft currents, is universal and does not depend on details of the underlying short-distance process, only on the color charges and angles of the outgoing partons. Final states are then weighted, in the planar limit, by a product of color dipoles (see figure 1). For each such product, it is useful to pull out a universal factor which accounts for its dimensionality and most singular limits. We thus write the contribution from the soft current with nsoft partons to the n-loop cross-section, starting from a parent dipole U12 along directions p1and p2, as: σ(1) 1=Zp0 s12 s10s02 2U10U02 F[1 0 2] ,(2.1a) σ(2) 2=Zp0,p00 s12 s10s000s002 4U10U000U002F[1 0002] ,(2.1b) σ(3) 3=Zp0,p00,p000 s12 s10s000s00000 s0002 8U10U000U00000 U0002F[1 000000 2] ,etc. (2.1c) – 5 – JHEP02(2018)058 Here loops are counted in powers of g2≡g2 YMNc 16π2=αsNc 4π,Uij =1 NcTr[U(θi)U†(θj)], and phase-space integrals are normalized as Rp0≡16π2Rµ2d3−2p0 (2π)3−22p0 0. For the Mandelstam invariants and their multi-index generalizations, we let: sij = 2pi·pj, si(jk)= 2pi·(pj+pk), sijk = 2(pi·pj+pi·pk+pj·pk).(2.2) All invariants will always be positive (timelike), since we assume a color singlet initial state. Naturally for our setup, on-shell momenta will be split into an energy a0and angular parts β0:pµ 0=a0βµ 0where βµ 0= (1,~n0) is null. The Lorentz-invariant phase space measure correspondingly splits into an energy and angular parts: Zp0 =µ2Z∞ 0 2da0(2a0)1−2×Zβ0 ,Zβ0≡Zd2−2Ω0 (4π)1−2.(2.3) For angles we write αij =1−cos θij 2, which runs between 0 and 1. Throughout, we will use the subscripts 0, 00, 000 to index radiated gluons. The various factors of 2 in our definitions have been chosen to simplify limits and preclude unnecessary (log 2)’s in integrated expressions. For example, for one soft gluon, F[1 0 2] is the square of the well-known eikonal soft current. Including the factor TaTa/Nc= 1/2 from the color sum, this evaluates to 2s12 s10s02 F[1 0 2] ≡1 2 pµ 1 p1·p0−pµ 2 p2·p0 2 −→ F[1 0 2] = 1 .(2.4) For two soft partons one needs the square of the double soft current, described for example in [30]. The result after squaring it and including the fermions and scalars of N= 4 SYM can be borrowed from formulas of ref. [23] (section 3), also rederived below in subsection 4.1: F[1 0002] = 1 + s12s000+s10s002−s100s02 2s1(000)s(000)2 .(2.5) One can easily verify that this factorizes in soft limits: F[1 0002] |p0||p00| −−−−−−→ F[1 0 00]F[1 002] = 1, F[1 0002] |p00||p0| −−−−−−→ F[0 002]F[1 0 2] = 1 .(2.6) Our three-loop computation builds on the one-loop corrections to F[1 0002] and the tree-level three-parton amplitude F[1 000000 2], which will be efficiently obtained as described below. 3 Two-loop evolution: fixing a convenient scheme The next-to-leading order correction to the Balitsky-Kovchegov equation was obtained in QCD and N= 4 SYM in [9,32]. It was postulated [33] and verified explicitly that the same kernel governs non-global logarithms [23]. In the latter reference, soft partons were organized in terms of their energies. Because “energy” is not Lorentz invariant, this scheme did not manifest Lorentz invariance, which had to be restored manually through a finite renormalization (guaranteed to exist given the – 6 – JHEP02(2018)058 Lorentz invariance of the underlying theory). This is formally similar to the transformation used to reach to the so-called conformal scheme in the BK literature [32]. Indeed the mapping (1.2) interchanges the Lorentz and conformal symmetry of the two problems. Here we improve on this by using explicitly Lorentz-invariant cutoffs. To fully define the scheme in which our three-loop result will apply, we thus quickly revisit the two-loop calculation. 3.1 One-loop in Lorentz-invariant form The idea is to define the evolution so that its exponentiation generates the emission probability of a soft gluon, in the soft approximation, integrated over a complete phase space region bounded by a Lorentz-invariant cutoff. For example, at one-loop, we define the anomalous dimension K(1) so that its integral (following the first term in the expansion of eq. (1.4)) matches the emission amplitude given in eqs. (2.1a), (2.4): −Zµ 0 dλ λλ−2K(1)U12 ≡Zp0 θ(Q[1 0 2] < µ)s12 s10s02 2U10U02 −2U12,(3.1) where θ(x<y) is a step function forcing xto be smaller than y, and Q2 [1 0 2] ≡s10s02 s12 defines our cutoff. From this definition one can see that Q[1 0 2] is proportional to the energy of the radiated gluon. Physically, Q[1 0 2] is the absolute value of its transverse momentum in a frame where the parents p1and p2are back to back. (This ordering variable has been used in many other contexts, see for example [34].) It is the only Lorentz invariant scale that depends on the direction but not the energies of the parent partons. To find K(1) from the definition (3.1), we simply identify the integration over the energy component of p0(called a0in eq. (2.3)) with that over the ordering scale λ. More precisely, λis proportional, but not equal, to the energy a0, because of the angle dependence of Q[1 0 2]: λ=Q[1 0 2] = 2a0rα10α02 α12 .(3.2) Inserting this change of variable into the right-hand-side of eq. (3.1) using the measure (2.3), and stripping off Rdλ λλ−2on both sides, we thus get: K(1)U12 =Zβ0α12 α10α02 1− 2U12 −2U10U02.(3.3) This of course reproduces the one-loop Banfi-Marchesini-Smye equation recorded in (1.1), except for the in the exponent, which arose because of the angular dependence of the ordering variable Q[1 0 2]. This exponent ensures exact Lorentz invariance in any dimension, not only in the →0 limit2, which is critical to ensure Lorentz invariance of the higher-loop corrections to K[23]. We briefly comment on the inclusion of virtual corrections, which simply add the (−2U12) term to eq. (3.1). This form is determined by the Kinoshita-Lee-Nauenberg (KLN) 2An angular integral Rd2−2Ω0I(β0) is Lorentz invariant if Iis homogenous of degree −(2 −2) in β0. This condition ensures that the rescaling of β0under a boost cancels against the Jacobian of the transformation. – 7 – JHEP02(2018)058 each limit can be accounted for by a single term, hence leaving a finite remainder: F[1 000000 2] = 1 + (1 + P)s12s000+s10s002−s100s02 2s1(000)s(000000)2 +s1000 s000+s10s00000 −s100s0000 2s1(000)s000000  +s12s0000 +s10s0002−s1000 s02 2s1(000000)s(000000)2 +Fsafe [1 000000 2] .(4.4) Here Pthe parity operation {1,0}↔{2,000}. The result we obtain from the four-point integrand matches precisely this form, with the remainder vanishing in all soft limits. For future convenience we write it here as a sum of individually regular pieces: Fsafe [1 000000 2] = (1 + P)(e1+e2) + e3+e4,(4.5) e1=1 4s1(000000)s(00000)2s000000 × s0000 (2s1000 s002+s100(s002−s0002)) −s000(2s100s0002+s1000 (s0002−s002)) +s00000 (2s10s0002−s1(00000)s02 −s12s0(00000))!, e2=s10(s12s00000 +s1000 s002−s100s0002) 4s1(000)s1(000000)s(00000)2 , e3=s100s002 2s1(000)s(00000)2 −s100s002+s100s02 +s1000 s002 2s1(000000)s(000000)2 , e4=s12(s12s000s00000 +s10s0002s000000 ) 4s1(000)s1(000000)s(000000)2s(00000)2 +s12(s000+s00000 −s0000 ) 4s1(000000)s(000000)2 −s12s000 4s1(000)s(000000)2 −s12s00000 4s1(000000)s(000)2 . As a cross-check, we have reproduced numerically the squared soft current (4.4)–(4.5) by a direct Feynman diagram calculation, summing up the gluon, fermion and scalar contributions, and also using the computer package [42]. For convenience, this formula, and others in this paper, is included in computer-readable format in the ancillary text file formulas.txt, attached to the arXiv submission of this paper. 4.2 Second ingredient: double-soft current and the remainder function To obtain the one-loop correction to the double soft current in the simplest way, we take the limit of two soft partons in the known one-loop six-point amplitude. These soft partons can be of any species (gluons, fermions and scalars). Consider for example the case when the two soft gluons have the same helicity. In this case we use the one-loop correction to the MHV amplitude (four positive and two negative helicity gluons), divided by the tree amplitude [43]: 1 cΓ M(1)MHV 6 M(0)MHV 6 = (1 + C2+C4)−2 2+2 log (−s23)(−s56) µ4+ Li21−s123s345 s12s45 +π2 3 −log (−s23) µ2log (−s12)(−s34) µ2(−s56),(4.6) – 14 – JHEP02(2018)058 Figure 3. One-loop virtual correction to double soft current contributing to the cross-section at three loops. where the operation Cis a cyclic rotation by one. The one-loop soft current is obtained by taking the limit where partons 2 and 3 become the soft partons 0 and 00, and subtracting the one-loop correction to the parent four-point amplitude. In this limit, the two coloradjacent partons 1 and 4 define the parent dipole, and the other two decouple, thus giving us the soft current 1 cΓ S(1) [1 0+00+2] S(0) [1 0+00+2] =−2 2+2 log −Q2 [1 0002] µ2−log (−s10)(−s002) µ2(−s12)log (−s000) µ2 +Li2−s100 s10 + Li2−s02 s002+ Li21−s1(000)s(000)2 s12s000+O().(4.7) It is important to note that since all invariants are positive (timelike), the Feynman prescription adds an imaginary part to all logarithms: log(−sij) = log |sij|−iπ. For soft gluons of opposite helicity, as well as for soft fermions and scalars, one needs the NMHV (super)amplitude [44,45]. It may be amusing to note that the two fermions soft current is the same in QCD and N= 4 SYM, since the contributing diagrams are the same. Thus some effective supersymmetry can also be used at one loop in QCD as well. The component formulas are somewhat involved, and in the N= 4 theory further simplifications occur when summing over particle species in the interference with the tree amplitude. For this reason, here we record only the final result of the helicity sum, e.g. the one-loop correction to the squared soft current, in appendix in eq. (A.1): F(1) [1 0002] ≡4s12 s10s000s002−1X h1,h2hS(1)[1 0h100h22]i∗hS(0)[1 0h100h22]i+ c.c..(4.8) We used the package in [46] to cross-check our expressions. Importantly, as was the case at two loops (and for the MHV example above), this one-loop correction is infrared divergent, while we expect the physics to depend only on renormalized, finite quantities. The standard, MS way to renormalize is to remove the – 15 – JHEP02(2018)058 integral of the infrared anomalous dimension: F(1)ren,MS [1 0002] ≡¯ Pexp Zµ 0 dλ λγIR(λ)F(1)bare [1 0002] (4.9) where at one-loop γIR =g2 YMNc 8π2log |s10s000s002| |s12|µ4for the soft current squared. This is the conventional definition of so-called hard matrix elements in the SCET literature. Although a good starting point, this is however not very convenient for us, because we would like to subtract something which has a simple representation as a phase space integral. The governing physical principle is that the subtraction should match all singularities of the triple-real emission at the integrand level, in all (single-) soft and collinear limits. This ensures that when we add it back later all divergences will cancel cleanly pre-integration. Furthermore we would like a simple analytic form for the integrated subtraction. This can be achieved by defining Lorentz-invariant functions of three angles, like we did in section 3.4, since these automatically integrate to constants. Let us thus consider the general problem of renormalizing an amplitude F[1 23... n]with (n−2) soft partons. We want to renormalize it by adding, say at one-loop, a phase space integral with one additional real parton v: Zp2,...,pn−1F(1)ren [1 2... n]−F(1)bare [1 2... n]≡Zp2,...,pn−1,pv Γv [1 2... n]F(0) [1 2... n].(4.10) There are two constraints. In the limit where vis soft, the integrand should reduce to minus the square of the soft current, G{i v j}≡ − sij sivsvj , emitted from all possible regions, minus that from the parent dipole. In collinear limits there is a similar factorization, but with the important distinction that the parent amplitude must be evaluated with the total momentum (here j= 2, . . . , n −1 and i=j−1, k=j+ 1): Γv [1 2... n]F(0) [1 2... n] pvsoft −−−−→ −G{1v n}+ n−1 X i=1 G{i v i+1}!F(0) [1 2... n], Γv [1 2... n]F(0) [1 2... n] pvkpj −−−→ 1 svj F[i jv k]×F(0) [1 ...(pj+pv)... n]. (4.11) Note that the labels iand kdecouple in the collinear limit. The fact that the argument of the amplitude is shifted to (pj+pv) is the main complication since it precludes a simple multiplicative solution. To solve it, recycling the ingredients in the two-loop subtraction (3.18), we define a three-index operator Γ[i j k]which rescales pjin whatever it multiplies: Γv [i j k]F[1 ...j... n]≡G{i vj k}+G{i jv k}F[1 ...˜ j... n],˜pµ j≡pµ j1 + Q[i v k] Q[i j k].(4.12) Then, it is easy to see that all constraints are simultaneously solved by: Γv [1 2... n]≡1 2 n−1 X i=2 Γv [1 i i+1] + Γv [i−1i n].(4.13) Indeed, the three-index Γv [i j k]only has collinear singularities in one region pvkpj, where the spectator labels iand kdecouple, so the collinear limits work out. In the soft limit it – 16 – JHEP02(2018)058 approaches G{i v j}+G{j v k}−G{i v k}, and using telescopic cancellations one can also see that the first of (4.11) is fulfilled. Using the change of variable (3.17) and the integral (3.20), the renormalization defined at one-loop by eq. (4.10) can be rewritten in a more suggestive form: Fren [1 2... n]=Fbare [1 2... n]×hV[1 2 n]V[2 3 n]···V[n−2n−1n]i×hV[1 2 3]V[1 3 4]···V[1 n−1n]i,(4.14) where V[i j k]=e1 2δ(1)g2(Q[i j k])(4.15) represents the real-emission correction to the soft current squared between legs iand k. We recall that δ(1) ≈2 2(3.20) starts with a double pole, and g2(λ)≈(λ/µ)−2g2(µ) is the D-dimensional running coupling. Physically, the renormalized amplitude is thus obtained by including amplitudes for a sequence of splittings, each with the coupling evaluated at its natural scale. (Either of the sequences in the square brackets would work, but we chose to include both and multiply the exponent by 1 2for symmetry reasons.) The renormalized amplitude Fren is finite for any number of points. At one-loop F(1)ren is obtained from the bare result (A.1) by the simple substitution given in eq. (A.2). It turns out that Fren is closely related to another canonical finite function in N= 4 SYM: the Bern-Dixon-Smirnov remainder function [47]. This is defined by dividing the amplitude by an ansatz ABDS n+2 (essentially an exponential of the one-loop MHV amplitude), which makes it finite and dual-conformal invariant and trivializes its collinear limits. The ansatz has four parameters: three are essentially the constant, order and 2terms in the function called f() in [47], which multiplies the one-loop amplitude in the exponent, while the fourth adds a common multiplicative factor to all n-point amplitudes and cancels out for the soft current. Thus three parameters affect the soft current; comparing with eq. (4.15) it is easy to see that these three parameters are in one-to-one correspondence with the double-pole, single-pole and constant term in δ(1), and that the infrared divergent parts match. The 0 term in our δis slightly different because for five-partons our scheme automatically yields the cusp anomalous dimension F[1 0 2] =γK(see section 5for the higher-loop explanation) whereas by definition the BDS remainder is unity for five-points. Thus, with a somewhat schematic notation, we can express our renormalized soft current directly in terms of the BDS remainder to all loop orders: Fren [1 2 ... n]= (γK)n−2×|Rn+2|2×eγKfn,(4.16) where Rn+2 =An+2/ABDS n+2 is the BDS remainder (e.g. the amplitude An+2 divided by the BDS ansatz ABDS n+2 ) and fnan explicitly known (and finite) function equal to the real part of the difference between the exponent in eq. (4.14) and the squared one-loop MHV soft current (given for n= 4 in (4.7)), which was exponentiated by BDS. As we will prove shortly, only the →0 limit of the finite Fren, and thus also BDS remainder, is needed to get the evolution kernel K. Finally, it is interesting to look at eq. (4.16) in the other direction, going from the soft current to the remainder. We can count the number of variables on which the soft current Fren depends for msoft partons. Each on-shell parton gives 3mdegrees of freedom while – 17 – JHEP02(2018)058 invariance under two Lorentz generators and one independent rescaling of the βiremove 3, giving 3(m−1) invariants. Due to dual conformal symmetry, the k-point BDS remainder Ak/ABDS kin eq. (4.16) depends on 3(k−5) invariants, which is equal since k=m+4. These two numbers agree! That is, dual conformal symmetry implies that the soft limit is not lossy, and we conclude that, through eq. (4.16), the →0 limit of BDS remainder in planar N= 4 SYM uniquely determines the renormalized soft current, and vice-versa. 4.3 Nested subtractions for virtual contribution Given the renormalized amplitude, it is natural to integrate it over relative energies to obtain a contribution to K, with suitable subtractions as was done in section 3.3: Kren [1 0002] ≡g4(Q[1 0002])Z∞ 0 dτ τ2Fren,sub [1 (τβ0)β002] .(4.17) For the subtracted integrand Fren,sub it would be tempting to use again eq. (3.10), but one needs to be more careful and pay due attention to the renormalization scales of the various objects. Indeed, as is clear from the renormalization group equation (1.4), all couplings in the subtractions get evaluated at their private scales Q[i j k], which are distinct from the common overall scale Q[1 0002] that we assign to Kren [1 0002]. In addition, the finite parts of the renormalization (4.14) do not match. The correct loop-level definition, which accounts for all these effects, is rather Fren,sub [1 0002] ≡Fren [1 0002] −θ(Q[1 0 00]<Q[1 002])κ[1 0 00];[1 002]Fren [1 0 00](g2(Q[1 0 00]))Fren [1 002](g2(Q[1 002])) −θ(Q[0 002]<Q[1 0 2])κ[0 002];[1 0 2] Fren [0 002](g2(Q[0 002])) Fren [1 0 2](g2(Q[1 0 2])),(4.18) where all the couplings are to be evaluated in terms of the common one of the overall process: g2(λ)7→ g2(Q[1 0002])λ Q[1 0002] −2. The prefactors, which account for the coupling constants stripped from the two-parton amplitude and for mismatching subtractions of infrared divergences, are κ[1 0 00];[1 002] ≡g2(Q[1 0 00])g2(Q[1 002]) g4(Q[1 0002])e1 2δ(1)g2(Q[1 0 00])+g2(Q[1 002])−g2(Q[0 002])−g2(Q[1 0 2]) =eg2κ(1) +O(), κ(1) = log s10s02 s100s002 log s12s000 s100s02 .(4.19) For the other subprocess we get the same but with −κ(1). Specializing to what we need at three loops, extracting the coefficient of g2(Q[1 0002]) in Fren,sub and using that F(1)ren [1 0 2] = −π2/3 from subsection 3.4, this becomes F(1)ren,sub [1 0002] =F(1)ren [1 0002] −θQ[0 002]<Q[1 0 2]−2π2 3+κ(1) −θQ[1 0 00]<Q[1 002]−2π2 3−κ(1).(4.20) – 18 – JHEP02(2018)058 The critical conceptual point here is that we won’t need the O() terms in this expression. This is because the combination in eq. (4.18), in which all objects are defined to all orders in , is precisely the one which vanishes to all order in near the endpoints τ→0 and τ→ ∞ (this follows from the factorization properties of the bare amplitudes Fbare). This precludes any / effect. The extension to higher loops is clear: one just includes more terms in the expansion of δ. Also we expect only minor changes in the presence of a nontrivial β-function as in full QCD, where g2(λ) will now be a series in g2(Q[1 0002]). 4.4 Nested subtractions for triple real contribution We now turn to the fully real contribution to K(3), which is given by the IR divergent part of triple-real emission, minus the subdivergences associated with iterations of K(1) and K(2). The basic idea is to write the subtractions as phase space integrals with step functions, exploiting (3.9) and its higher-multiplicity generalizations. In this way all energy sub-divergences (with fixed angles, as appropriate since the angles are fixed by the color rotations U) will cancel under the integration sign. To write the result concisely, we recursively define subtracted integrands Fsub, generalizing eq. (3.10). Introducing the abbreviations [X][Y]≡Fsub [X]Fsub [Y]θ(Q2 [X]<Q2 [Y]),[X][Y][Z]≡Fsub [X]Fsub [Y]Fsub [Z]θ(Q2 [X]<Q2 [Y]<Q2 [Z]), these are defined as: Fsub [1 0 2] ≡F[1 0 2] = 1,(4.21a) Fsub [1 0002] ≡F[1 0002] −[1 0 00][1 002] −[0 002][1 0 2],(4.21b) Fsub [1 000000 2] ≡F[1 000000 2] −[1 0 00][1 00000 2] −[0 00000][1 0000 2] −[00000 2][1 0002] −[1 000000][1 000 2] −[0 00000 2][1 0 2] −[1 0 00][1 00000][1 000 2] −[00000 2][0 002][1 0 2] −[0 00000][1 0 000][1 000 2] −[0 00000][0 000 2][1 0 2] −[1 0 00][00000 2][1 002] −[00000 2][1 0 00][1 002].(4.21c) The structure is straightforward: there is one subtraction for each possible subprocess (consistent with the planar structure), and the unsubtracted F’s are given in eq. (2.5) and (4.4). Intuitively, the Fsub’s are a device to compute the logarithm of F: the preceding equations can be generated (and generalized to all orders) by formally solving the equation PeRFsub =RF, order by order in the number of emitted partons. As shown in section 3, what is relevant for the evolution is the integral over relative energies: K(3) [1 000000 2] ≡Z∞ 0 dτ τ dτ0 τ04Fsub [1 (τβ0)(τ0β00)β000 2].(4.22) Thanks to the pattern of subtractions, and to the factorization of soft currents (see eqs. (2.6) and (4.3)), Fsub [1 000000 2] vanishes in all soft limits and its energy integral at fixed angles is absolutely convergent at all orders in . One might worry that the step functions make it tricky to integrate in practice, but in fact they always multiply trivial measures like dτ/τ. Furthermore, the explicit expression (4.4) naturally splits into several individually – 19 – JHEP02(2018)058 convergent pieces. For example, the piece Fsafe doesn’t contain any step function and converges by itself. The pieces from the “1” in F[1 0 2],F[1 0002] and F[1 000000 2] contain multiple step functions, but all share the trivial measure dτ/τ dτ0/τ0and so immediately integrate to logarithms. Finally, the five nontrivial subtractions in (4.4) naturally combine with the remaining terms in (4.21c), to produce five individually convergent integrals. So our problem is reduced to computing finite energy integrals; these produce functions of transcendental weight 2. A good, systematic way to compute such integrals is the differential equation method described B.3The most difficult integrals are contained within Fsafe. One of them, in particular, coming from the first line below eq. (4.5), cannot be written simply in terms of the angular distances αij, but requires associated spinors (βα˙ β i≡λα i˜ λ˙ β): f1≡Z∞ 0 dτ τ dτ0 τ04e1(τβ0, τ0β00, β000 ) (4.23) = 2Re 1 + α00000 h0 2i[2 1] α0002h0 00i[001]−α002h0 000i[0001]Li21−α100α0002 α1000 α002−Li21−α0000 α002 α000α0002 +Li2−[1 0][00000] [1 000][0 00]−Li2−h1 0ih00000i h1 000ih0 00i+ log α10α00000 α1000 α000log α0002h0 00i[001] α002h0 000i|[0001]. Here we have used a commonly used notation for the Lorentz-invariant spinor products: hi ji=αβλα iλβ jand [i j] = ˙α˙ β˜ λ˙α i˜ λ˙ β jwith antisymmetric. (Under the stereographic projection (1.2), these map respectively to: hi ji= (zi−zj) and [i j] = (¯zi−¯zj).) The other integrals are more elementary and produce at most dilogarithms of cross-ratios of α’s. To give the final result we define the five cross-ratios: u1≡α12α000 α100α02 , u2≡α12α00000 α1000 α002 , u3≡α12α0000 α1000 α02 , v1≡α10α002 α100α02 , v2≡α100α0002 α1000 α002 . Then the triple-real integral gives K(3) [1 000000 2] =1−u3 1−v1v2   2Li21−1 v1v2−2Li21−1 v1−2Li21−1 v2 + log v1log v2+ log(v1v2)log(u1u2)−3 2log u3   + (u1u2−u1v2−u2v1+v1+v2−u1−u2+u3)Li21−1 v1v2−ζ2 +3 log u1log u2−3 2log2u3+ (1 + P)(f+f1),(4.24) where f1is the special function in eq. (4.23), Pexchanges labels (1,0) and (2,000) and acts 3For energy integrations the method is considerably simpler than for the transverse integrals illustrated in appendix, because partial fractions and integration-by-parts in one variable are more elementary and the final contributions are given from boundary terms instead of contact terms. – 20 – JHEP02(2018)058 on cross-ratios as (u1, v1)↔(u2, v2), and: f=1−u1 1−v12Li21−1 v1+ log v1log u2 v2−1 2log u1 +1 + v2−u2Li21−1 v2−Li21−1 v1v2 +1−u1 u3−v1u2log v1u2 u3log u2 v2−3 2log u1 u3−2Li21−v1u2 u3.(4.25) 4.5 Nested subtractions for renormalization counter-terms The final ingredient is the “add” part from the “add and subtract” game that led to the renormalized amplitude (4.18). These also have three angular integrations but one fewer color dipole U. In addition there are similar pieces inherited from lower loops, for example from the term with just U100U002in the two-loop evolution. It is useful to devise a notation for such terms, like G{1 0002}, wherein the underlined index represents the angle from which a Wilson line is omitted and the curly bracket highlights the presence of a virtual parton. This is why we’ve split the virtual correction into two terms (G{1v0 2}+G{1 0v2}) in eq. (3.18), because these two end up with different color structures and so get exponentiated at different scales: Q[1 v0 2] 6=Q[1 0v2]. Similarly, to exponentiate the three-loop kernel (as would be needed for a putative four-loop calculation), we would need to specify where vfits within the color structures of Γv [1 0002], which determines the relevant scale Q. Thus although not strictly necessary here, it is useful to account for that information because it helps show the internal logic. Thus we organize the “add” terms into three color structures: K(3)U12 3 angles =Zβ0,β00,β000          K(3) [1 000000 2] α12 α10α000α00000 α0002(−2U10U000U00000 U0002) +K(3)add {1 000000 2} α12 α100α00000 α0002(−2U100U00000 U0002) +K(3)add {1 000000 2} α12 α10α0000 α0002(−2U10U0000 U0002) +K(3)add {1 000000 2} α12 α10α000α002(−2U10U000U002)          .(4.26) Here we only show the terms in K(3) with three angular integrations, two has been dealt with in subsection 4.3 and one will be dealt with shortly. The underlined index shows the variable whose Wilson line and radiator factor are omitted. The angular functions are the integrals over relative energies of corresponding Gsub’s, K(3)add {1 000000 2}=Z∞ 0 dτ τ dτ0 τ0Gsub {1 (τβ0)(τ0β00)β000 2},etc. (4.27) The Gsub’s contain two ingredients. First, there is the difference between the renormalized Fren,sub in eq. (4.18) and the corresponding bare expression: Gv [1 0002] ≡Γv [1 0002]Fsub [1 0002] + (Γv [1 0 00]+ Γv [1 002])θ(Q[1 0 00]<Q[1 002]) + (Γv [0 002] + Γv [1 0 2])θ(Q[0 002]<Q[1 0 2]).(4.28) – 21 – JHEP02(2018)058 Second, there is the subtraction of everything inherited from virtual corrections at lower loops: from the U12 term at one-loop (3.3), and from the (U10U02 +U100U002) part of twoloop (3.13). To allow their subsequent exponentiation, the result is to decomposed into 3 color structures: Gsub {1v0002}+Gsub {1 0v002}+Gsub {1 000v2}≡Gv [1 0002] −lower subtractions .(4.29) A simple systematic color decomposition for the subtractions can be done as follows. Whenever, in a subprocess, both indices adjacent to vare the same as in the considered Gsub, we weight this contribution by 1; when only one index is shared, we weight by 1 2, and when none is shared, we weight by 0. For example, consider the following term coming from the real part of K(1) times the virtual part of K(2): Fsub [1 0 00]Gsub {1v002}θ(Q[1 0 00]<Q[1 v002]).(4.30) We place half of this term into Gsub {1v0002}and half into Gsub {1 0v002}, because voccurs between 1 and 00. Using these rules to generate the subtractions recursively, using the same notation as in eq. (4.21c) (writing {a b . . . c} ≡ Gsub {a b... d}and inserting a step function between each bracket, either curly or square), then gives Gsub {1v2}≡G{1v2}=−s12 s1vsv2 (4.31a) Gsub {1v0 2}≡Γ{1v0 2}[1 0 2] −{1v0}− 1 2{1v2}[1 0 2],(4.31b) Gsub {1 0v2}≡Γ{1 0v2}[1 0 2] −{0v2}− 1 2{1v2}[1 0 2],(4.31c) Gsub {1v0002}≡Γ{1v0002}[1 0002] + Γ{1v0 00}+1 2Γ{1v002}[1 0 00][1 002] + Γ{1v0 2}[0 002][1 0 2] −1 2[1 0 00]{1v002}−{1v0 00}[1 002] −[0 002]{1v0 2} −{1v0}− 1 2{1v2}[1 0002] + [1 0 00][1 002] + [0 002][1 0 2](4.31d) −[1 0 00]1 2{1v00}− 1 2{1v2}[1 002] −[0 002]{1v0}− 1 2{1v2}[1 0 2] , Gsub {1 0v002}≡Γ{1 0v002}[1 0002] +Γ{1 0v00}+1 2Γ{1v002}[1 0 00][1 002] + Γ{0v002}+1 2Γ{1 0v2}[0 002][1 0 2] −1 2[1 0 00]{1v002}−{1 0v00}[1 002] −1 2[0 002]{1 0v2}−{0v002}[1 0 2] −{0v00}[1 0002] + [1 0 00][1 002] + [0 002][1 0 2] −[1 0 00]1 2{1v00}[1 002] −[0 002]1 2{0v2}[1 0 2] ,(4.31e) Gsub {1 000v2}≡PGsub {2v000 1},(4.31f) where Pis the parity (10)↔(002). This looks messy, but the upshot is that the internal logic is straightforward and the terms can be automatically generated to any desired order. (Formally, the terms can be generated by series-expanding the schematic formula – 22 – JHEP02(2018)058 PeR(Fsub+Gsub)=R(F+G)eRK|U12 .4) This generalizes the subtractions used at two loops: the energy integral of Gsub {1v0 2}matches the U100U002term in eq. (3.13). Although we haven’t defined the individual Γ{1v0002}(only their sum Γv [1 0002]) we expect that a definition exists which will make the energy integrals converge absolutely for each of the color structure Gsub {1v0002}, as this is certainly the case for the sum which is all we need here at three loops. Furthermore, by construction, the collinear singularities of K(3)add cancel exactly those of K(3), to all orders in , so it is apparent that O() corrections to any kernel are not needed. Thus we only need to compute the finite integral (4.26) with = 0. The integrated result turns out to be somewhat inelegant, so we decided to replace it by a simpler counterterm with the same collinear singularity. From inspection of the triple-real result (4.24), we find divergences as 0k00or 00k000, and also in the double scaling limit 0k00k000, but not when one or two partons become collinear to 1 or 2. A simple counter-term which removes the divergence as 00k000 is: K(3)c.t. [1 000000 2] =1 + α12α000 α10α002−α100α02 log α10α002 α100α02 +3 2log α12α000 α100α02 log α12α02α2 00000 α1000 α0000 α2 002 . (4.32) To construct an integral that is also absolutely convergent in double collinear limits, we can easily play with the color structures, exploiting that Uij →1 when ikj. Arranging for each color factor to separately fulfill the KLN theorem (vanishing when Uij = 1), the full three-loop evolution is then written as (4.34c) below, where the difference compared to subsection 4.3 is simply: K(3) [1 0002] −K(3)ren [1 0002] =Zβv(1 + P)α002 α00vαv2 K(3)c.t. [1 000v2] +K(3)add {1v0002}+K(3)add {1 0v002}+K(3)add {1 000v2} with Pthe symmetry (10)↔(200). This is again an absolutely convergent integral which can be done at = 0, using the methods of appendix B. We find a surprisingly compact result: −1 + α12α000 α10α002−α100α02 log2α12α000 α100α02 + 4ζ2log α10α002 α100α02 −11 6log3α12α000 α100α02 . (4.33) Its simplicity (compared with eqs. (4.31)) suggests that an even simpler organization of the subtractions could exist. Adding this result to the energy integral of the one-loop remainder function (4.20), (A.1), (A.2), computing using the same method explained above, we thus obtain the part of the evolution with two angular integrals. 4.6 Final result: the three-loop BK equation in planar N= 4 SYM In summary, we have computed the three-loop correction to the Balitsky-Kovchegov rapidity evolution equation (or equivalently Banfi-Marchesini-Smye equation for non-global logarithms) in planar N= 4 SYM, in terms of absolutely convergent integrals over squared 4The fully virtual correction eRK|U12 to the parent dipole U12 appears on the right-hand side since the soft currents Fare defined to act on the bare amplitude; this is also the reason why {1v2}appears with opposite sign wherever it does. – 23 – JHEP02(2018)058 LO NLO NNLO LO resum -�-�-� � � � � ��� ��� ��� ��� ��� ��� ��� ��� ν �(�� ν) Figure 5. The BFKL eigenvalue for m= 0 along the real νaxis at various orders for λ=g2 YMNc= 6. Convergence near the maximum is visibly slower than away from it. The “resummation of leading-order” is defined below eq. (5.16). observed already at two loops and explained in terms of nearby singularities in the complex plane at iν =±1 [12]. In short, these singularities are related to the collinear limit of BFKL, where the scaling dimension ∆ = 2+iν = 3 of the exchanged state coincides with that of twist-two operators: ∆ = 2 + j+γ(j) with jclose to 1, e.g. the operators entering the DGLAP equation. As is common for two-level quantum systems, this crossing of two energy levels [18] gets resolved as depicted in figure 6: j≈1 + ∆−3±p(∆ −3)2+ 32g2 2,∆ = 2 + iν. (5.16) At small g2≡g2 YMNc 16π2, one branch choice gives the near-horizontal BFKL trajectory while the other gives the 45◦twist-two (DGLAP) trajectory. (The square root formula follows easily by solving ∆(j)≈j+ 2 + 8g2 j−1for the j, within the overlapping regime of validity of BFKL and DGLAP g2 |j−1|  1 where the anomalous dimension γ(j) can be approximated by its leading pole.) It was shown that, expanding the square root to order g4, reduces by half the magnitude of the two-loop corrections to the intercept j(0,0) (if one also includes the complex conjugate singularity at iν =−1) [12]. The “LO resummation” curve in figure 5, called “scheme 2” in ref. [12], thus shows the LO trajectory plus eq. (5.16) minus its O(g2) expansion. (It would be useful to develop a NLO resummation and we leave it as an open problem for the future.) The formula (5.16), expanded to three loops, turns out to not predict very well the three-loop correction to the intercept j(0,0) ≈1 + 11.09g2−84.08g4−2543.05g6+O(g8). In fact it gets even the sign wrong. By looking at the singular terms in Fclose to the pole we can try to understand why: F0,ν iν→1 −−−→ 8g2 δ−64g4 δ3+g61024 δ5−512ζ2 δ3−576ζ3 δ2−464ζ4 δ+ regular + O(g8),(5.17) – 30 – JHEP02(2018)058 BFKL Eq.(5.16) DGLAP -�-� � � � ��� ��� ��� ��� ��� �ν  Δ -� �(�� ν) Figure 6. Level repulsion between the Pomeron and DGLAP trajectories for m= 0 as a function of scaling dimension, illustrating the ν=±isingularities. (LO expressions plotted with λ=g2 YMNc = 1.) LO NLO NNLO -�-�-� � � � � ���� ���� ���� ���� ���� ���� ���� ���� ν �(�� ν) Figure 7. The BFKL eigenvalue for m= 1 along the real νaxis at various orders for λ=g2 YMNc = 6. where δ= 1 −iν. Comparing with eq. (5.16), we find that the leading pole 1024g6/δ5is exactly as predicted (as it had to). Setting δ= 1, the subleading poles however also give a numerically large contribution to the intercept 2F, so truncating to the leading pole does not give a good approximation to the intercept. However, summing up all the singular terms in eq. (5.17), one finds that about 80% of the three-loop correction to the intercept is reproduced. A heuristic explanation is that the contributions from the next singularities, at iν =±3, are suppressed by their distance. Interestingly, all polar terms at L-loops can be obtained from the L-loop DGLAP equation. (See for example [59,60].) From the higher-loop DGLAP equation one can get – 31 – JHEP02(2018)058 nonsingular terms in the expansion (5.17), see for example eq. (21) of [16]. We have verified that our result (5.15)–(C.3) agrees with all these constraints.7 We conclude that the physical picture of [12], that large corrections to the intercept originate from the iν =±1 collinear singularities, is consistent with the three-loop trajectory we obtained, although the full polar part, predicted by DGLAP (as opposed to just the leading pole), must be retained. In general it would be very interesting to find a way to make full use of the DGLAP information at a given loop order Finally, we comment on the Mellin transform of the level-crossing formula (5.16) back to coordinate space. The transform produces a Bessel function: Z+∞ −∞ dν 2π|z|iν−1p(iν −1)2+ 32g2= 32g2J1(4g√2 log |z|) 4g√2 log |z|.(5.18) The right-hand side has appeared in coordinate space and momentum space resummations [13,14], so it is nice to see how it arises form the familiar two-level crossing formula (5.16). 6 Discussion and conclusion In this paper we have computed, for the first time, the evolution equation which resums large rapidity logarithms in forward scattering to three loops in a gauge theory. Our main results are the full nonlinear equation (4.34) in planar supersymmetric Yang-Mills theory, its linearization (5.6), characterizing the BFKL Pomeron in impact parameter space, as well as its eigenvalue, the Pomeron Regge trajectory, described in appendix C. This result is a first step toward the analogous QCD result, and by itself can already be used to assess the convergence of perturbation theory and its proposed resummations, and shed light on nonlinear saturation effects at finite coupling. This computation was made possible thanks to a recently established correspondence with the resummation of large so-called non-global logarithms, which occur when soft radiation is excluded from a fixed angular region. This correspondence is helpful because it makes available a body of knowledge on the factorization of infrared and collinear divergences, and at a conceptual level it defines in a clear way the evolution equation to all loop orders. This allowed us to derive a systematic subtraction method for nested subdivergences, embodied in eq. (4.21), such that all energy integrals at fixed angle become convergent. We then dealt with collinear subdivergences and real-virtual cancellations in a second step, by multiplying and dividing by the corrections to the single soft current as in eq. (4.14). Therefore, although we set up our calculation in dimensional regularization and some divergent intermediate objects appeared, we find that in the end the evolution equation depends only on the →0 limit of physical scheme-independent quantities like the the BernDixon-Smirnov remainder (4.16)! This opens a new possibility to relate an object with the topology of the cylinder, the BFKL Pomeron, to the integrable system appearing in planar scattering amplitudes [61]; graphically speaking, this cuts the cylinder into two half-pipes. 7Compared to eq. (21) of [16] (version 1), we have ω7→ −ω, to match with the generally accepted convention ω=j−1 that we are following. – 32 – JHEP02(2018)058 As a highly nontrivial test, of both our calculation and of the integrability approach, we have compared our extracted Pomeron trajectory (5.15) with the recent predictions for m= 0 in [15,16], and found perfect agreement! We have also found perfect agreement, in the collinear limit, with the prediction from anomalous dimensions of twist-two operators. The trajectory for other transverse angular momenta m, and nonlinear interactions, are new predictions which it would be very interesting to check within the integrability approach. It is important to clarify the 1/Nccounting in which our result is valid. The projectile is assumed to be made of a finite ∼N0 cnumber of Wilson lines, but whose expectation values across the target can be finite, 1 NcTr[U1U† 2]∼1. This asymmetric setup, motivated for example in proton-nucleus collisions, is the same as that for which the Balitsky-Kovchegov equation is strictly derived. In the context of AdS/CFT this counting would apply to e.g. a light probe of a black hole. This is also a well-defined setup and in fact it would be interesting to work out the nonlinear terms in the Balitsky-Kovchegov equation at strong coupling λ, including perhaps 1/√λstringy effects. The linear terms, which govern correlators of light operators with large but not-to-large energies (before the onset of saturation, such that 1−U∼sj0−1/N2 c1) have already been identified with graviton exchange [18]. There are several directions in which this work could be extended. One is to go beyond the planar limit at weak coupling, where the two-loop corrections have recently become available [23,62,63]. Interesting new physical effects appear at three loops in the nonplanar sector, for example the 4→2 reggeon transition which “closes the Pomeron loop” and restores the symmetry between the target and projectile would first be seen there (see for example [19]). Through the KLN theorem, the three-loop evolution could also independently predict from real corrections, and thus test, the recent result for three-loop soft anomalous dimension [64]. Another direction is towards QCD: technically, our setup gives direct access to the evolution equation for non-global logarithms, which in QCD will differ from rapidity evolution by terms proportional to the β-function. These could thus be calculated subsequently by calculating matter loops on both sides of the correspondence. Acknowledgments SCH would like to thank Kolya Gromov and Vitaly Velizhanin for discussions regarding the integrability prediction for the trajectory. SCH’s work was supported partly by the Danish FNU Grant No. 126152 and by the Danish National Research Foundation (DNRF91). MH is supported by the Villum Foundation Grant No. YIP/VKR022599. A One-loop correction to the squared double soft current Here we record the interference of the tree and one-loop double soft current, defined in eq. (4.8), obtained from the soft limit of the six point amplitude as explained in the text. 2F(1)bare [1 0002] =s10s02 −s100s002 s1(000)s(000)2 −s12s000log s10s(000)2 s1(000)s002log s12s000 s1(000)s(000)2  +(1 + P)s12s000−s100s02 −s10s002 s100s(000)2 log s10 s1(000)log s12s000 s1(000)s002 – 33 – JHEP02(2018)058 +2F(0) [1 0002] 4 log () s10 s1(000) log s002 s(000)2 −1 2log2 s10s002s12s000 s2 1(000)s2 (000)2 !−2π2 3+X! −2Li21−s10 s1(000)−2Li21−s002 s(000)2 −2Li21−s12s000 s1(000)s(000)2 +2π2 3 + log2s10s(000)2 s1(000)s002+ log s12s000 s10s002log s10s002 s1(000)s(000)2 ,(A.1) with X=−2cΓ 2(Q2 [1 0002]/µ2)−+ 2π2+O(), and the parity operation P:{1,0}↔{2,00}. Here all analytic continuations have been performed, so the logarithms are all real for timelike (positive) invariants, as is the case for our application. The infrared divergences are contained in the factor Xbut in practice all we will need is the fully renormalized form factor, defined in eq. (4.14), which is finite and obtained by a simple substitution: F(1)ren [1 0002] =F(1)bare [1 0002] with X7→ 1 4log2α12α10α000 α2 100α002 +1 4log2α12α000α002 α10α2 02 .(A.2) B Doing transverse integrals efficiently Two-dimensional integrals can be done extremely efficiently with the differential equation method. Here we elaborate on our implementation, emphasizing the simplifications related to the fact that all the integrals are absolutely convergent and done directly in 2 dimensions. We first illustrate the method on the integral g1(y, ¯y) = −Zd2z π|1−y|2 |1−z|2|y−z|2log |y|2 |z−1−y|2|z|2,(B.1) which occurs at two-loops when obtaining the translation-invariant kernel H(y) (5.10). The idea is to differentiate with respect to yand add a total derivative with respect to z to remove derivatives of rational factors. Indeed, using the relevant identity: d dy +d dz 1−z 1−y|1−y|2 |1−z|2|y−z|2= 0,(B.2) one readily gets that d dyg1(y, ¯y) = −Zd2z πd dy +d dz 1−z 1−y|1−y|2 |1−z|2|y−z|2log |y|2 |z−1−y|2|z|2.(B.3) We “win” because the derivatives commutes with the rational factor and hits the logarithm, producing a simpler integral. An important subtlety is that the left-hand-side of eq. (B.2) is singular and so the equation is only strictly valid for generic z. There are additional contact terms given by the “holomorphic anomaly” d dy 1 ¯y−¯z=d d¯y 1 y−z=πδ2(y−z).(B.4) This can be understood from the two-dimensional Poisson equation ∂z∂¯zlog(z¯z) = πδ2(z). These terms would be absent in dimensional regularization but appear because we insist – 34 – JHEP02(2018)058 to work with = 0 (see [54] for four-dimensional examples). In the example (B.3), the contact terms are at z= 1 and z=ybut the logarithm turns out to vanish on both, so these can be dropped. Evaluating the derivative then gives simply d dyg1(y, ¯y) = 1 yZd2z π (1 + y) z(1 + y−z) (1 −¯y) (1 −¯z)(¯y−¯z).(B.5) To finish, one can repeat the same procedure, inserting a variant of eq. (B.2) to differentiate the integral with respect to y(and/or ¯y). Now only the contact term contributes and a general result obtained this way is Iab,cd =Zd2z π (a−b) (z−a)(z−b) (¯c−¯ d) (¯z−¯c)(¯z−¯ d)= log |a−d|2|b−c|2 |a−c|2|b−d|2,(B.6) which gives (using the vanishing at y=−1 to fix the integration constant) d dyg1(y, ¯y) = 1 y×2 log(y¯y)−→ g1(y, ¯y) = log2(y¯y).(B.7) This result can be easily confirmed by numerical integration. A critical point to emphasize is that the factor 1/y in eq. (B.5) had to be pulled out in front of the integral before taking the second derivative. If the derivative were allowed to act on that factor, one would gain nothing from it. Only properly normalized integrals simplify upon taking derivatives. A simple criterion to identify properly normalized integrals is that all the Poincar´e residues of their rational factors should be constant (these are often called leading singularities). These are simply the double residue with respect to zfollowed by ¯z, of the rational factors in the integrand, with zand ¯ztreated as independent complex variables. This property is easily verified in eqs. (B.1) and (B.6). Its significance is that it ensures that derivatives of the rational factors have vanishing Poincar´e residues, which is needed for them to be total derivatives which simplify upon integration by parts as in eq. (B.2). See for instance refs. [53,54] for other applications of this criterion. The procedure to decompose an integral into properly normalized ones is essentially partial fractions. When the denominators do not couple zand ¯z, it is in fact literally partial fraction in these two variables, one after the other. But the integrals we need also contain in the denominator an irreducible quadratic form Q(z, ¯z), which is harder to partial-fraction out. We illustrate this with the other integral appearing in H(2)(y), coming from the second term in the kernel (3.12): Zd2z π 1 |1−z|2|y−z|21 + |y|2 Qlog |1−z|2|y−z|2 |1 + y−z|2|z|2, Q =|1−z|2|y−z|2−|1+y−z|2|z|2. Despite appearances, Qis a quadratic form in z, ¯z. The leading singularities of the rational factor can be computed and found to be linear combinations of 1/[(1 −y)(1 + ¯y)] and 1/[(1 + y)(1 −¯y)], so the decomposition into properly normalized integrals will require two terms. To illustrate the result of the partial-fraction method to be detailed shortly, one indeed finds that the integral can be rewritten exactly as: 1 (1 −y)(1 + ¯y)Zd2z π y(1 −y)(1 + ¯y) (1 −z)(y−z)Qlog |1−z|2|y−z|2 |1 + y−z|2|z|2+ (y↔¯y).(B.8) – 35 – JHEP02(2018)058 This rewriting of the integrand is a purely algebraic identity. The upshot is that all residues have been pulled out and the leading singularities of the rational factor inside the integral are only ±1. One then expects, and finds, that the derivative d/dy of the integral is a total derivatives in zand ¯z. One does not need to make any clever guess to find this total derivatives: in practice we simply write down an ansatz with a polynomial numerator in z and ¯zand solve for the coefficients. We obtain for example the identity: d dy +d dz 1−z 1−y+d dz (2z−1−y) + d d¯z(2¯z−1−¯y)¯y(1 + y)(1 −z)(y−z) y(1 −y)(y+ ¯y)(1 + y¯y) ×y(1 −y)(1 + ¯y) (1 −z)(y−z)Q= 0 , again up to contact terms arising from the holomorphic anomaly (at z=yand z= 1). Plugging this into the integral (B.8) thus gives its y-derivative in terms of contact terms and the simpler integral (B.6). The derivative can then be easily integrated, and the integration constant again is fixed by the vanishing at y=−1, yielding the two-loop linearized kernel: H(2)(y) = 4 log(y¯y)2 (1 −y)(1 −¯y)+ 8ReLi2(y)−Li2(¯y) + Li2(−y¯y)−log(y¯y) log 1−¯y 1+y¯y+1 2ζ2 (1 + y)(1 −¯y). (B.9) This agrees precisely with the result in eq. (105) of Balitsky& Chirilli 0710.4330. Chief advantages of this method are its speed and uniform applicability. Indeed, the basic steps (integration by parts and partial fractions) are algebraic and independent of the transcendental weight of the functions being integrated. That is, the same code we used to do the two-loop integral H(2)(y) as just described, automatically also worked for H(3)(y) and would presumably work at higher orders as well (producing the result as an iterated integral). To conclude, we elaborate on partial fractions in the presence of the quadratic form Q in the denominator. The main step is to exploit the geometry to create a complete basis. For the integrals involving Q, there are other singularities on the 8 lines z= 0,1, y, 1+yand ¯z= 0,1,¯y, 1+¯y, each line intersecting the quadric Qat two points. However the geometry is a bit degenerate and there are only 8 intersections: (z, ¯z) = {(1,0),(0,1),(0,¯y),(y, 0),(1+ y, 1),(1,1+¯y),(y, 1+¯y),(1+y, ¯y)}. A complete basis of rational functions is then obtained by writing 8 objects na (z−a)Qwhere a∈ {0,1, y, 1 + y}and the numerators naare linear in ¯z and chosen to leave only one Poincar´e residue nonzero (and equal to 1). A ninth integral √y¯y/Q (accounting for a residue at infinity), together with simpler integrals with only linear denominators, complete the basis. Once a basis is fixed, partial fraction identities like (B.8) follow simply from computing Poincar´e residues, a fast operation. The other integrals needed in this paper, involving the triple-real functions (4.33) and (5.5), were dealt with in a similar way, although their geometry is somewhat simpler (no square roots appeared in these cases). B.1 Single-valued functions for the linearized kernel The linearized kernel (5.5) is a weight 3 function of one complex variable; the preceding method produces it in the form of an iterated integral, whose integration constants could – 36 – JHEP02(2018)058 be easily fixed from the limit x→1. Its symbol turns out to be made of the five letters x, 1−x, ¯x, 1−¯xand 1 −v=x+ ¯x−x¯x. At transcendental weight 3, we found only three nontrivial single-valued functions with such symbol, in terms of which the three-loop linearized kernel K(3)lin(x) in (5.6) is compactly written: O1= 2Li3(x) + Li3(¯x)−2ζ3−log uLi2(x) + Li2(¯x),(B.10a) O2= 2Li3(1 −x) + Li3(1 −¯x)−2ζ3−log vLi2(1 −x) + Li2(1 −¯x),(B.10b) O3=Li3¯x x(¯x−1)+ Li3x(¯x−1) ¯x+1 2Li2¯x x(¯x−1)−Li2x(¯x−1) ¯x ×log(1 −x)(1 −¯x)−4Li3(x)−2Li3(1 −x) + log(x¯x)Li2(x) + 1 6log3(1 −x) −1 2log2(1 −x)log(x)−log(¯x)−1 4log2(1 −x) log(1 −x)(1 −¯x) + ζ2log(1 −x) −(x↔¯x).(B.10c) Although not manifest from these formulas, these functions have no branch cut on the complex plane where ¯x=x∗. This can be confirmed by series-expanding around singular points such as x= ¯x= 0 and x= ¯x= 1, where to all orders one finds only singlevalued logarithms of log(x¯x) or log(1−x)(1−¯x) but never log(x) nor log(1−x) separately. Furthermore, there are no singularities along 1 −v= 0 (which traces a unit circle with center at x= 1). We note that the same five letters are also singularities of the two-loop kernel, so it is natural to conjecture that no other letters appear in K(L)lin(x) to any order in perturbation theory in planar N= 4 SYM. Its translation-invariant projection H(L)(y), defined by the integration (5.10), can then be obtained by applying the algorithm detailed above, which implies that at most the ten letters dlog ny, ¯y, 1±y, 1±¯y, y + ¯y, 1+y¯y,√y+i√¯y √y−i√¯y,1+i√y¯y 1−i√y¯yocan appear in its symbol (all of which do indeed appear at three loops). C Eigenvalue in terms of harmonic sums for m= 0 and m= 1 Here we give explicit expressions for the 3-loop Pomeron trajectory, given in coordinate space in eq. (5.6), in Mellin space using the harmonic sums Sa(N) = N X i=1 (sign a)i i|a|, Sa1,...,an(N) = N X i=1 (sign a)i i|a|Sa2,...,an(i).(C.1) This defines the sums for integer Nand the Mellin transform produces their analytical continuation from even N. Using standard algorithms [55], we have converted the Mellin integral projected onto transverse angular momentum m= 0, eq. (5.15), to harmonic sums – 37 – JHEP02(2018)058 with argument N=−1+iν 2: F(1) 0,ν =−4S1, F(2) 0,ν = 8S3−16S−2,1+ 8ζ23S−1+ 3 log 2 + S1−6ζ3,(C.2) F(3) 0,ν 32 =−S5+ 2S−4,1−S−3,2+ 2S−2,3−S2,−3−2S3,−2+ 4S−3,1,1+ 4S1,−3,1+ 2S1,−2,2 +2S1,2,−2+ 2S2,1,−2−8S1,−2,1,1+ζ2S1S2−3S−3+ 2S−2,1−4S1,−2−49 2ζ4S1 +7ζ32S1,−1+ 2(S1−S−1) log 2 −S−2−log22+ (8ζ−3,1−17ζ4)S−1−S1+log 2 −1 2ζ3S2+ 4ζ5−6ζ2ζ3+ 8ζ−3,1,1.(C.3) Here ζ−3,1≈0.087786 and ζ−3,1,1≈ −0.009602 are multi-zeta values. This result is in precise agreement with [15]. The Pomeron trajectory is the sum of Fm,ν and Fm,−ν, see eq. (5.11). For m6= 0 our result is new. For m= 1, for example, the Mellin transform can be expressed in terms of harmonic sums now with argument N=iν 2, giving the Odderon Regge trajectory: F(1) 1,ν =−4S1,F(2) 1,ν 8=N−1(S−2+ζ2)−N−2S1+S3+ζ2S1+1 2ζ3,(C.4) F(3) 1,ν 16 =N−1(−3S−4+ 2S−3,1+ 2S−2,2+ 2S1,−3+ 4S2,−2−8S−2,1,1+ 4S1,−2,1−8S1,1,−2) +N−22S3−S−3−2S−2,1+4S1,−2+4ζ2S1−5ζ3+N−3(4S1,1−4S−2−S2−3ζ2) +N−1ζ2(−2S2 1−6S−2) + ζ3(7S−1+ 3S1)−9ζ4+ (3N−4−11 2ζ4)S1−2S5 −ζ2ζ3−3ζ5.(C.5) This is regular and in fact vanishes at ν= 0, in accordance with the all-order result (5.8). Other values of mcan be evaluated numerically using the attached Mathematica notebook. Open Access. This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited. References [1] E.A. Kuraev, L.N. Lipatov and V.S. Fadin, The Pomeranchuk Singularity in Nonabelian Gauge Theories,Sov. Phys. 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