Three-loop soft anomalous dimensions in QCD

I present results for soft anomalous dimensions through three loops for many QCD processes. In particular, I give detailed expressions for soft anomalous dimensions in various processes with electroweak and Higgs bosons as well as single top quarks and top-antitop pairs.


Introduction
The calculation of higher-order soft-gluon corrections in perturbative QCD requires calculations of soft anomalous dimensions, Γ S , for the corresponding processes [1]. The current state-of-the-art for Γ S for many processes is three loops. In this paper, I present results for Γ S for various processes at hadron colliders. These include processes with W , Z, γ, and H bosons, as well as single-top and top-pair production, and 2 → 3 processes involving top quarks produced in association with electroweak or Higgs bosons.
Soft-gluon corrections are very important because they are typically large and they dominate the perturbative corrections for a multitude of processes, especially those involving top quarks. We consider partonic processes p a + p b → p 1 + p 2 + · · · and define s = (p a + p b ) 2 , t = (p a − p 1 ) 2 , u = (p b − p 1 ) 2 and s 4 = s + t + u − m 2 i . At partonic threshold s 4 → 0, and the soft corrections at order α n s involve logarithmic terms of the form ln k (s 4 /M 2 )/s 4 , with M a hard scale and k ≤ 2n−1. In order to resum these soft corrections in the (differential) cross section at NLL, NNLL, and N 3 LL accuracy, we need to calculate soft anomalous dimensions at, correspondingly, one loop, two loops, and three loops.
If we take transforms of the cross section, with transform variable N , then we can write a factorized expression as where the ψ and J functions describe collinear emission from incoming and outgoing partons, H a b→12··· is a short-distance hard function, and S ab→12··· is a soft function which describes softgluon emission [1] and which satisfies the renormalization group equation The soft anomalous dimension Γ a b→12··· S controls the evolution of the soft function which gives the exponentiation of logarithms of N in the resummed cross section. For a recent review of soft anomalous dimensions for many QCD processes, see Ref. [2].
In the case of the production of heavy-quark pairs, with mass m, we can also write the above expressions in terms of β = tanh(θ /2) = 1 − (4m 2 /s), and denote them by Γ (n) β cusp . If eikonal line i represents a massive quark and eikonal line j a massless quark, then we have simpler expressions. At one loop Γ , and at three loops If both eikonal lines are massless, then Γ massless

Γ S for some simple processes
For processes with trivial color structure, the soft anomalous dimension is very simple. In fact Γ S vanishes for the following: Drell-Yan processes qq → γ * , qq → Z; W -boson production via qq → W ± ; Higgs production via bb → H and g g → H; electroweak-boson pair production qq → γγ, qq → Z Z, qq → W + W − ; production of two different electroweak bosons qq → γZ, Also, for Deep Inelastic Scattering (DIS), lq → lq with subprocess qγ * → q, we have at one loop: Γ s); and at three loops: More generally, when all external lines in a process are massless, then Γ (2) S is proportional to Γ (1) S [10], but this is not true for processes with massive lines. Furthermore, at three loops for multi-leg scattering there are contributions from four-parton correlations [11].

Γ S for large-p T W , Z, γ, H production
Let V denote a W or Z boson or a photon or a Higgs boson. The soft anomalous dimension for these processes is a simple function (not a matrix) [12][13][14] (see also [2]).
For the processes q g → W ± q , q g → Zq, q g → γq, and b g → H b, we have at one loop: Γ ; and at three loops: Γ . The same Γ S also describes the reverse processes such as γq → q g.
For the processes qq → W ± g, qq → Z g, qq → γg, and bb → H g, we have at one loop: ; and at three loops: . The same Γ S also describes the reverse processes such as γg → qq.
For single-top t-channel production, Γ bq→tq S is a 2 × 2 matrix [15,18,19]. Using a t-channel singlet-octet color basis, the matrix elements are at one loop , at two loops and at three loops where the X (3)bq→tq i j denote unknown terms from four-parton correlations in the last three matrix elements at three loops. It is important to note that due to the color structure of this process, only the first three-loop matrix element, Γ For single-top s-channel production, Γ qq →tb S is also a 2 × 2 matrix [15,16,19]. Using an schannel singlet-octet color basis, we have at one loop , at two loops and at three loops where the X , contributes to the N 3 LO soft-gluon corrections. For associated tW production the soft anomalous dimension is a simple function [15,17,19]. At one loop at two loops and at three loops The same soft anomalous dimension applies for the process b g → t H − , and for the FCNC processes, via anomalous top-quark couplings, q g → t Z, q g → t Z , and q g → tγ.
For top-antitop pair production via the qq → tt channel, Γ qq→tt S is a 2 × 2 matrix and we use an s-channel singlet-octet color basis. At one loop for qq → tt , and at two loops , Γ (2)qq→tt 21 At three loops for qq → tt we can write the last matrix element as denotes unknown three-loop contributions from four-parton correlations. The other matrix elements are also not fully known at three loops, but they have an analogous structure to that at two loops (essentially, replace (2)'s by (3)'s in the superscripts as well as replace K 2 's by K 3 's, and add X terms for unknown contributions).
For top-antitop pair production via the g g → tt channel, Γ g g→tt S is a 3 × 3 matrix, and we use a color basis c 1 = δ a b δ 12 , c 2 = d a bc T c 12 , c 3 = i f abc T c 12 . We have At one loop for g g → tt and at two loops , Γ (2)g g→tt S 31 , Γ (2)g g→tt S 32 . At three loops for g g → tt, we can write the 22 matrix element as where X (3)g g→tt 22 denotes unknown three-loop contributions from four-parton correlations. The other matrix elements are, again, also not fully known at three loops, but they have an analogous structure to that at two loops.

Γ S for tqH, tqZ, tqγ, tqW production
We consider processes bq → tq H as well as bq → tq Z, bq → tq γ, bq → tqW − , qq → tq W + . We use a t-channel singlet-octet color basis, and we further define s = (p 1 + p 2 ) 2 , t = (p b − p 2 ) 2 , u = (p a − p 2 ) 2 . All these processes have the same soft anomalous dimension matrix [23]. We have at one loop , at two loops and at three loops where the X (3)bq→tq H i j denote unknown terms in the last three matrix elements which, however, do not contribute to the soft-gluon corrections at N 3 LO. We next consider the processes qq → tbH as well as qq → tbZ, qq → tbγ, qq → tbW − , qq → tq W + , which all have the same soft anomalous dimension matrix [23], and we use an s-channel singlet-octet color basis. We have at one loop , at two loops Γ (2) qq →tbH S 11 = K 2 Γ (1) qq →tbH S 11 where the X (3)qq →tbH i j denote unknown terms in the last three matrix elements which, however, do not contribute to the soft-gluon corrections at N 3 LO.

Conclusion
Soft anomalous dimensions are fundamental in describing soft-gluon emission in QCD processes. In this contribution, I presented results for soft anomalous dimensions for many processes through three loops. These results are needed in calculations of high-order corrections.
Funding information This material is based upon work supported by the National Science Foundation under Grant No. PHY 2112025.