Three-loop corrections to the soft anomalous dimension in multi-leg scattering

We present the three-loop result for the soft anomalous dimension governing long-distance singularities of multi-leg gauge-theory scattering amplitudes of massless partons. We compute all contributing webs involving semi-infinite Wilson lines at three loops and obtain the complete three-loop correction to the dipole formula. We find that non-dipole corrections appear already for three coloured partons, where the correction is a constant without kinematic dependence. Kinematic dependence appears only through conformally-invariant cross ratios for four coloured partons or more, and the result can be expressed in terms of single-valued harmonic polylogarithms of weight five. While the non-dipole three-loop term does not vanish in two-particle collinear limits, its contribution to the splitting amplitude anomalous dimension reduces to a constant, and it only depends on the colour charges of the collinear pair, thereby preserving strict collinear factorization properties. Finally we verify that our result is consistent with expectations from the Regge limit.

The focus of this paper will be the infrared (IR) structure of a scattering amplitude for n massless partons. More precisely, if the external legs have momenta p i , i = 1..n, with p 2 i = 0, long distance singularities (both soft and collinear) can be factorized as follows M n ({p i } , α s ) = Z n ({p i } , µ, α s ) H n ({p i } , µ, α s ) , (1) where µ is a factorization scale, α s ≡ α s (µ 2 ) is the renormalised D-dimensional running coupling, H n is a finite hard scattering function, and Z n is an operator in colour space that collects all IR singularities in the form of poles in the dimensional regularization parameter = (4 − D)/2. The IR singularities contained in Z n have their origin in loop momenta becoming either soft or collinear to any of the scattered partons (see e.g. ref. [45]). Collinear singularities depend on the spin and momentum of that particle, and decouple from the rest of the process. In contrast, soft (non-collinear) singularities * On leave from the "Fonds National de la Recherche Scientifique" (FNRS), Belgium.
are independent of the spin, but they depend on the relative directions of motion and the colour degrees of freedom of the scattered particles. Hence, soft singularities are sensitive to the colour flow in the entire process, and their structure is a priori rather complex. Nevertheless, they are significantly simpler than finite contributions to the amplitude. They can be computed by considering correlators of products of Wilson-line operators emanating from the hard interaction, following the classical trajectory of the scattered particles and carrying the same colour charge.
Specifically, Z n can be obtained as a solution of a renormalization-group equation as where Γ n is the so-called soft anomalous dimension matrix for multi-leg scattering, and P stands for pathordering of the matrices according to the order of scales λ. We stress that Γ n itself is finite, and IR singularities are generated in eq. (2) owing to the fact that Γ n depends on the D-dimensional coupling, which is integrated over the scale down to zero momentum. The functional form of Γ n is highly constrained, and owing to factorization and the rescaling symmetry of the Wilson line velocities [17][18][19][20], through three loops it must take the form with where −s ij = 2 |p i · p j | e −iπλij , with λ ij = 1 if partons i and j both belong to either the initial or the final state and λ ij = 0 otherwise; T i are colour generators in the representation of parton i, acting on the colour indices of the amplitude as described in ref. [11]; γ K (α s ) is the universal cusp anomalous dimension [7,46,47], with the quadratic Casimir of the appropriate representation scaled out (Casimir scaling of the cusp anomalous dimension holds through three loops [46]; it may be broken by quartic Casimirs starting at four loops); γ Ji are the anomalous dimensions of the fields associated with external particles, which govern hard collinear singularities, currently known up to three loops [28,48]. Equation (4) is known as the dipole formula, and captures the entirety of the soft anomalous dimension matrix up to two loops. In particular, tripole corrections correlating three partons, with colour factors of the form if abc T a i T b j T c k , which could appear starting from two loops, are not present in the soft anomalous dimension at any order because the corresponding kinematic dependence on the three momenta is bound to violate the rescaling symmetry constraints [17][18][19][20]. The first admissible corrections in eq. (3) are then quadrupoles, because four momenta can form conformally-invariant cross ratios, which are invariant under rescaling of any of the momenta. Given that diagrams connecting four lines contribute for the first time at three loops, this is the first order at which contributions to ∆ n in eq. (3) may appear, According to the non-Abelian exponentiation theorem [44] the colour factors must correspond to connected graphs -specifically those of Fig. 1 -and therefore the general form of the quadrupole term at three loops is completely determined by the four-parton case, where there are only two independent cross ratios: with where the sum runs over all permutations of the set {2, 3, 4}. Note that the terms in the sum are not all independent, because of the antisymmetry of the structure constants and the Jacobi identity. The function F is independent of the colour degrees of freedom and only depends on two conformal cross ratios. Note that at three loops, ∆ n is independent of the details of the underlying theory and completely determined by soft gluon interactions. In particular, this implies that ∆ is the same in QCD and in N = 4 Super Yang-Mills, and it is therefore expected to be a pure polylogarithmic function of weight five. Moreover, its functional form has been constrained by considering collinear limits and the Regge limit [17][18][19][20][21][22][23][24][25][26], but it has so far remained unclear whether three-loop corrections to the dipole formula are present. The purpose of the present paper is to compute ∆ (3) n . We will present the complete functional form of F, hence determining soft singularities of any massless multi-leg amplitude at three loops. As ∆  the contribution to the soft anomalous dimension is the coefficient of that pole, which is finite for each of the diagrams in Fig. 1 (they have no subdivergences) and can be evaluated in D = 4 dimensions. Next, we observe that the integrals over the positions of the threeand four-gluon vertices give rise to one-and two-loop off-shell four-point functions, for which we derive a multifold Mellin-Barnes (MB) representation. After integration over the four Wilson lines, we obtain a MB representation of each of the connected graphs for the general non-lightlike case, depending on the velocities through the cusp angles γ ij ≡ 2β i · β j / β 2 i β 2 j . In order to proceed, we use standard techniques [50] to perform a simultaneous asymptotic expansion for γ ij → −∞ corresponding to the lightlike limit, where we neglect any term suppressed by powers of 1/γ ij . After this procedure, we obtain a collection of lower-dimensional MB integrals. The remaining MB integrals are then converted into parametric integrals using the techniques of ref. [51], which can be performed using modern integration techniques [52]. The sum over all connected graphs is expressible as a linear combination of products of logarithms of cusp angles γ ij and single-valued harmonic polylogarithms [53,54] with arguments z andz, related to the conformally-invariant cross ratios (5) by A few comments are in order at this point: First, we observe that individual connected graphs are not pure functions, but they involve pure functions of weight five multiplied by rational functions in z andz. These rational functions, however, cancel in the sum over all connected graphs, leaving behind a pure function of weight five, in agreement with the expectation that scattering amplitudes in N = 4 Super Yang-Mills are uniformly transcendental. Second, we see that the sum over all connected graphs cannot be written as a function of conformallyinvariant cross ratios alone, but it still explicitly depends on cusp angles γ ij . We observe, however, that the residual dependence on the cusp angles, which violates rescaling symmetry in the lightlike limit, is only logarithmic: after applying the Jacobi identity, the sum over all connected graphs can be written in the form, ∆ where Q is a polynomial in logarithms of cusp angles, while the polylogarithmic function F con. (z,z) depends exclusively on the invariant cross ratios. Since ∆ 4 only depends on conformally-invariant cross ratios, rescaling invariance must be restored when combining the contribution from Q with the contributions from the non-connected graphs in Fig. 2. Some of these webs have been computed in ref. [38] in the general, nonlightlike case, and the calculation of the remaining ones is nearing completion [55]. It turns out, however, that explicit results for all non-connected graphs are not required in order to determine ∆  (with coefficients that are zeta values), and that no zeta value of weight greater than three can appear. We can then simply write down the most general Bose-symmetric combination of products of logarithms of cusp angles of weight five satisfying these constraints and require that rescaling invariance is restored when combining it with Q in eq. (10). Upon requiring in addition that the sum of connected and non-connected graphs is consistent with the behaviour of a two-to-two scattering amplitude in the Regge limit [23,56], we find that there is a unique function of the type described above that we can add to eq. (10), accounting for the sum over the contributions of all non-connected webs.
Putting everything together, we find the following simple result for the three-loop correction to the soft anomalous dimension where the function F (z) is given by and L w (z) are Brown's single-valued harmonic polylogarithms (SVHPLs) [53] (see also ref. [57]). Note that we kept implicit the dependence of these functions onz.
SVHPLs can be expressed in terms of ordinary harmonic polylogarithms (HPLs) [54] in z andz. The result for ∆ (3) 4 in terms of HPLs is attached in computer-readable format to this paper.
Let us now briefly discuss the main features of the final result of eqs. (11) and (12). First, we note that while F (z) is defined everywhere in the physical parameter space, it is only single-valued in the part of the Euclidean region (the region where all invariants are negative) where z and z are complex conjugate to each other. Single-valuedness ensures that ∆ 4 has the correct branch cut structure of a physical scattering amplitude [57,58]: it is possible to analytically continue the function to the entire Euclidean region while the function remains real throughout [59]. Next note that if one considers F (z) as a function of two independent variables z andz (not a complex conjugate pair) this function has branch points for z andz at 0, 1 and ∞. Crossing symmetry, i.e., crossing some momenta from the final to the initial state, is realized in a very simple way by taking monodromies around these points.
Next, let us discuss the symmetries of the final answer for the three-loop corrections to the soft anomalous dimension. Bose symmetry is realised on the cross ratios by the action of the group S 3 which keeps the momentum p 1 fixed and permutes the remaining three momenta, cf. eq. (8). This group naturally acts on the space of SVH-PLs by change of arguments generated by the transformations (z,z) → (1 −z, 1 − z) and (z,z) → (1/z, 1/z). We note that geometrically this symmetry simply acts by exchanging the three singularities at z ∈ {0, 1, ∞}. Moreover, it is known that the space of all HPLs, and hence also SVHPLs, is closed under the action of this S 3 , giving rise to functional equations among HPLs, i.e., relations among HPLs with different arguments. As a consequence, it is possible to express all the terms in eq. (11) in terms of SVHPLs with argument z.
Besides the action of this group S 3 , there is a second symmetry group Z 2 acting on the space of SVHPLs. Indeed, the definition of (z,z) in eq. (9) is invariant under the exchange z ↔z, and hence the function F (z) must be invariant under this transformation, i.e., F (z) must be an even function: F (z) = F (z). This symmetry is realised on the space of SVHPLs by the operation of reversal of words, namely, if w is a word made out of 0's and 1's, and w the reversed word, then we have L w (z) = L w (z) + . . ., where the dots indicate terms proportional to multiple zeta values. Even functions then correspond to 'palindromic' words (possibly up to multiple zeta values), and indeed we see that eq. (12) is 'palindromic'.
Let us now comment on the momentum conserving limit of ∆ 4 , which is of particular interest because it corresponds to two-to-two massless scattering. In this limit we havez = z = s 12 /s 13 = −s/(s + t). It follows that for two-to-two massless scattering F (z) can be expressed entirely in terms of HPLs with indices 0 and −1 depending on s/t, in agreement with known results for on-shell three-loop four-point integrals [60][61][62].
Finally, let us comment on the behavior of ∆ (3) 4 in the limit where two partons become collinear. A well known property of an n-parton scattering amplitude is that the limit where any two coloured partons become collinear can be related to an n − 1 parton amplitude: where P = p 1 + p 2 , and p j are the momenta of the (n − 2) non-collinear partons. The splitting amplitude Sp (p 1 , p 2 , {p j }) is an operator in colour space which captures the singular terms for P 2 → 0. All elements in eq. (13) have infrared singularities, and these must clearly be related. In the planar limit Sp only depends on the quantum numbers of the collinear pair [63] to all orders in perturbation theory, a property that also holds beyond the planar limit at two loops. Assuming that this property is general, refs. [19,22] used infrared factorization to derive the relation between the respective anomalous dimensions, and concluded that ∆ must vanish in all collinear limits if the splitting amplitude is to remain independent of the colour and kinematic variables of the non-collinear partons. We find that this expectation is invalidated by our calculation, and we observe that ∆ [[T 1 · T 2 , T 1 · T j ] , T 2 · T k ] 3 2 ζ 4 log P 2 2x(1 − x) p j · p k (P · p j ) (P · p k ) − ζ 5 − 2ζ 2 ζ 3 , which explicitly displays a correlation of the colour and directions of the (n − 2) non-collinear partons. In deriving eq. (14) we used the Jacobi identity to write the result in a manifestly symmetric way, and expressed it as a commutator of generators, where j and k run over all non-collinear partons. The interpretation of eq. (14) is rather striking: starting from three-loops splitting amplitudes probe, via soft-gluon interactions, the quantum numbers of the hard partons in the rest of the process. To conclude, we computed all connected graphs contributing to the soft anomalous dimension in multiparton scattering and determined the first correction going beyond the dipole formula. These corrections correlate directly the momentum and colour degrees of freedom of four partons. The final result is remarkably simple: it is expressed in terms of single-valued harmonic polylogarithms of uniform weight five. In the collinear limits we find an unexpected behaviour: the splitting amplitude anomalous dimension at three loops probes both the colour and kinematic behaviour of the rest of the process. It will be interesting to explore the physical consequences of this effect in the future.