Three-loop soft anomalous dimensions for top-quark processes

I present results for soft anomalous dimensions through three loops for several processes involving the production of top quarks. In particular, I discuss single-top and top-pair production. I also present some numerical results for double-differential distributions in $t{\bar t}$ production through approximate N$^3$LO.


Introduction
The inclusion of soft-gluon corrections in theoretical predictions for top-quark processes is required for better accuracy, and it involves calculations of soft anomalous dimensions. The first calculations at one loop were done in the mid 90's [1], but two-loop calculations appeared much later. The current state-of-the-art has been extended to three loops for some processes.
For partonic processes f 1 (p 1 ) + f 2 (p 2 ) → t(p t ) + X , we define a kinematical threshold variable s 4 = s + t + u − i m 2 i where s = (p 1 + p 2 ) 2 , t = (p 1 − p t ) 2 , and u = (p 2 − p t ) 2 . At partonic threshold s 4 → 0, and the soft-gluon corrections involve logarithms of the form [ln k (s 4 /m 2 t )/s 4 ] + with k ≤ 2n − 1 at perturbative order α n s . We define transforms of the partonic cross section asσ(N ) = (ds 4 /s) e −N s 4 /sσ (s 4 ), with transform variable N . The factorized expression for the cross section is [1] where H f 1 f 2 →t X is an N -independent hard function, S f 1 f 2 →t X is a soft function [1], while the ψ i and J i describe collinear emission from initial-and final-state particles [2].
The soft function S f 1 f 2 →t X satisfies the renormalization group equation where the soft anomalous dimension Γ controls the evolution of S f 1 f 2 →t X , which gives the exponentiation of logarithms of N in the resummed cross section. The resummation of these soft 1 arXiv:2105.14375v1 [hep-ph] 29 May 2021 corrections at NNLL accuracy requires knowledge of two-loop soft anomalous dimensions while at N 3 LL accuracy it requires three-loop soft anomalous dimensions. The resummed cross sections may be expanded at finite order and produce, upon inversion to momentum space, physical predictions.

Cusp anomalous dimension
The cusp anomalous dimension, Γ cusp , involves two eikonal lines, and it is a basic ingredient in calculations of soft anomalous dimensions for partonic processes. For two lines with momenta p i and p j , the cusp angle is θ = cosh −1 (p i · p j / p 2 i p 2 j ), and we write the perturbative series In the case of two heavy quarks, this can be written in terms of the speed β = tanh(θ /2) as At two loops, we have [4] where This can be written in terms of β and denoted as Γ cusp . The three-loop result [5,6] can be written as [3,6] where K (3) and C (3) have long expressions. Again, the result can be expressed in terms of β. If eikonal line i represents a massive quark, with mass m i , and eikonal line j a massless quark, then we find simpler expressions. At one loop, Γ If both eikonal lines are massless, then Γ massless

Single-top t-channel production
The soft anomalous dimension for t-channel single-top production, Γ bq→tq S , is a 2 × 2 matrix in color space. We use a t-channel singlet-octet color basis. The one-loop [7,10,11] and twoloop [10,11] results are well known.
At three loops, we have The first element, i.e. the "11" element, of the matrix at three loops was calculated in [11]. Due to the relatively simple color structure of the hard matrix for this process, it is the only threeloop element that contributes to the N 3 LO soft-gluon corrections. Here we have also provided three-loop results for the other three matrix elements up to unknown terms from four-parton correlations, which are denoted as X

Single-top s-channel production
We continue with results for the s-channel, for which Γ qq →tb S is a 2 × 2 matrix, and we use an s-channel singlet-octet color basis. The one-loop [7,8,11] and two-loop [8,11] results are, again, well known.
At three loops, we have Again, the "11" element of the matrix at three loops was calculated in [11] and is the only three-loop element to contribute to the N 3 LO soft-gluon corrections. We have also provided in the above equation three-loop results for the other three matrix elements up to unknown terms from four-parton correlations, which are denoted as X

Associated tW production
The soft anomalous dimension for tW production has only one element (not a matrix). It is known at one loop [7,9], two-loops [9], and three loops [11]. The three-loop result is [11]

Top-antitop pair production
We continue with top-antitop pair production which can proceed via the qq → tt and the g g → tt channels.
In the qq → tt channel, Γ qq→tt S is a 2 × 2 matrix, and we use an s-channel singlet-octet color basis. Here we will concentrate on the "22" matrix element which at one-loop contributes already to the soft-gluon corrections at NLO. At one loop, this element is [1,12] Γ (1)qq→tt 22 while at two loops it is [3,12] Γ (2)qq→tt 22 At three loops, we find the following expression: where X (3)qq→tt S 22 denotes unknown three-loop contributions from four-parton correlations. The other three-loop matrix elements are not fully known either but have analogous structure to that at two loops (see also [3]).
In the g g → tt channel, Γ g g→tt S 22 is a 3 × 3 matrix, and we use the color basis c 1 = δ ab δ 12 , c 2 = d a bc T c 12 , c 3 = i f a bc T c 12 . At one loop for g g → tt, the "22" matrix element is [1,12] Γ (1)g g→tt S 22 while at two loops it is [3,12] Γ (2)g g→tt S 22 At three loops, we find the expression where X (3)g g→tt S 22 denotes unknown three-loop contributions from four-parton correlations. As an application of the soft-gluon formalism at NNLL accuracy, in Figure 1 we show top-quark double-differential distributions in p T and rapidity with soft-gluon corrections through approximate N 3 LO [13]. The theoretical predictions describe very well the CMS data at 13 TeV [14].

Conclusion
I have presented results for soft anomalous dimensions at one, two, and three loops. The cusp anomalous dimension was discussed first followed by results for the soft anomalous dimensions in single-top production and in top-antitop pair production.
Funding information This material is based upon work supported by the National Science Foundation under Grant No. PHY 1820795.