Unpolarized Quark and Gluon TMD PDFs and FFs at N 3 LO
PPrepared for submission to JHEP
ZU-TH-46/20
Unpolarized Quark and Gluon TMD PDFs and FFs atN LO Ming-xing Luo, , Tong-Zhi Yang, , Hua Xing Zhu, and Yu Jiao Zhu Zhejiang Institute of Modern Physics, Department of Physics, Zhejiang University, Hangzhou,310027, China Complex Systems Division, Beijing Computational Science Research Center, Beijing, 100193,China Department of Physics, University of Z¨urich, CH-8057 Z¨urich, Switzerland
Abstract:
In this paper we calculate analytically the perturbative matching coefficientsfor unpolarized quark and gluon Transverse-Momentum-Dependent (TMD) Parton Distri-bution Functions (PDFs) and Fragmentation Functions (FFs) through Next-to-Next-to-Next-to-Leading Order (N LO) in QCD. The N LO TMD PDFs are calculated by solvinga system of differential equation of Feynman and phase space integrals. The TMD FFs areobtained by analytic continuation from space-like quantities to time-like quantities, takinginto account the probability interpretation of TMD PDFs and FFs properly. The coeffi-cient functions for TMD FFs exhibit double logarithmic enhancement at small momentumfraction z . We resum such logarithmic terms to the third order in the expansion of α s .Our results constitute important ingredients for precision determination of TMD PDFsand FFs in current and future experiments. Keywords:
TMD PDFs, TMD FFs, SCET, N3LO, small- x a r X i v : . [ h e p - ph ] D ec ontents LO coefficients for unpolarized quark and gluon TMDs 6 x expansion of unpolarized TMD coefficients and resummation forTMD FFs 27 x expansion of unpolarized TMD PDFs 274.2 Small- z expansion of unpolarized TMD FFs 284.3 Resummation of small- x logarithms for unpolarized TMD FFs 31 – 1 – Introduction
Understanding the parton structures of hadron is one of the outstanding problem in Quan-tum ChromodyDynamics (QCD). TMD PDFs and FFs describe the distribution of partontransverse momentum inside a hadron in parton scattering or decay. Their knowledge is es-sential to our understanding of the confined motion of parton in nucleons [1–3]. Thanks tofactorization and evolution [4–15], TMD PDFs and FFs also enter high precision theoreticalprediction for a large variety of observables at high energy colliders. From pure theoreticalpoint of view, TMD PDFs and FFs are also interesting since they represent light-cone cor-relation of quantum fields with intrinsic space-like and time-like origin, respectively. Theirtransparent definition in terms of light-cone correlator also allow higher-order perturbativecalculation, from which one can uncover interesting analytic structure of the correlators.In the past decade perturbative calculations for TMD distributions have seen rapiddevelopment. Next-to-Next-to-Leading Order (NNLO) corrections to TMD PDFs are firstobtained by extraction from expansion of Drell-Yan and Higgs p T distribution in the rel-evant kinemtical limit [16, 17]. Direct calculation of TMD PDFs and FFs from theirlight-cone operator definition is difficult due to the existence of unregulated rapidity sin-gularities. Much efforts have been devoted to the inclusion of rapidity regulator and itsassociated rapidity factorization [7, 9, 12, 13, 15, 18–22]. Using rapidity regulators, directcomputation of TMD PDFs and FFs at NNLO have since become available [23–28]. Theprogress in NNLO calculation for TMD distributions have allowed many cutting-edge calcu-lations for phenomenology, including fixed-order calculation at NNLO and beyond [29–34]and resummation for large logarithms at small q T at unprecedented N LL accuracy [34–41]. We also note that perturbative calculation for TMD quantities suitable for latticecalculation [42–45] has also made important progress recently [46].Recently a first step towards N LO TMD distributions have been achieved in [47] bycalculating the unpolarized quark TMD PDFs, extending and significantly improving uponthe methods in [26, 27]. Subsequently, both unpolarized quark and gluon TMD PDFs areobtained in [48], based on an independent method from [49]. In this paper we continueour calculation in [47], and present the N LO results for unpolarized quark TMD FFs, andunpolarized gluon TMD PDFs and FFs. We also present the results for unpolarized quarkTMD PDFs from [47] for completeness.Our method for calculating the unpolarized gluon TMD PDFs follows [47]. In orderto obtain the results for TMD FFs, we adopt a strategy of analytic continuation proposedin [50]. The crucial observation is that although TMD PDFs or FFs are not themselvesanalytic function of momentum fraction, but the building blocks, space-like and time-likesplitting amplitudes, are. At N LO, there are four distinct contributions to TMD PDFsor FFs, namely the triple real part, the double real-virtual part, the double virtual-realpart, and the virtual squared-real part. Ref. [50] shows that (a) the analytic continuationfor the triple real part is trivial since it doesn’t involve loop integrals; (b) the analyticcontinuation for the double real-virtual is also trivial at this order since the continuation ofvirtual loop only generate iπ terms which cancel in the sum with complex conjugate. (c)the analytic continuation of double virtual-real part and virtual squared-real part is non-– 2 –rivial. But they are also simple to calculate since they only involve a one-particle phasespace integral. It is suggested in [50] that one can calculate these two parts using thecorresponding space-like or time-like splitting amplitudes, instead of trying to analyticallycontinue them. Using this approach, Ref. [50] determines the complete time-like splittingfunctions at NNLO from the space-like counterpart, including the off-diagonal P T (2) qg , whichcannot be completely fixed with previous methods [51–53]. With the complete NNLOtime-like splitting functions, Ref. [50] also provides striking evidence for the existence ofa generalized Gribov-Lipatov reciprocity relation between space-like and time-like QCDsplitting functions in both non-singlet and singlet sector through NNLO. In this paper weuse the idea of Ref. [50] to determine not just the splitting functions, but the full time-likeTMD FFs through N LO. Our results for TMD PDFs and FFs are written in terms offamiliar harmonic polylogarithms [54] up to transcendental weight 5 and allow convenientanalytical and numerical manipulation. We provide analytic expression for TMD PDFs andFFs in the threshold limit ( x , z →
1) and high energy limit ( x , z → z terms in TMD FFs to the thirdorder in the expansion of strong coupling, following the method of Vogt [56, 57]. We notethat a different method based on celestial BFKL equation has also been developed to resumthe time-like small- z logarithms to NNLL in [55].The structure of this paper is as follows. In Sec. 2 we give the necessary operatordefinition and renormalization for TMD PDFs and FFs. In Sec. 3 we present the N LOresults for unpolarized quark and gluon TMD PDFs and FFs. Since the expressions arelengthy, we present in the text only a numerical fit to these functions, valid to 0.1 percentin the range of 0 < x , z <
1. The full analytic expressions are provided as ancillary files inthe arXiv submission of this paper. We also give the analytic expressions in the thresholdlimit in this section. In Sec. 4 we investigate the high energy limit of TMD PDFs andFFs, x , z →
0. We provide explicit analytic expressions, showing that TMD FFs are moresingular than TMD PDFs from fixed-order point of view, namely double logs in contrastto single logs. We resum the small z logarithms for TMD FFs to the third order in the theexpansion of α s . We conclude in Sec. 5. In this section we give the necessary operator defintion for unpolarized quark and gluonTMD PDFs and FFs. We also give the renormalization counter terms and specify thezero-bin subtraction.
We begin with the bare TMD PDFs for unpolarized quark and gluon, which can be definedin terms of SCET [58–62] collinear fields B bare q/N ( x, b ⊥ ) = (cid:90) db − π e − ixb − P + (cid:104) N ( P ) | ¯ χ n (0 , b − , b ⊥ ) / ¯ n χ n (0) | N ( P ) (cid:105) , – 3 – bare ,µνg/N ( x, b ⊥ ) = − xP + (cid:90) db − π e − ixb − P + (cid:104) N ( P ) |A a,µn ⊥ (0 , b − , b ⊥ ) A a,νn ⊥ (0) | N ( P ) (cid:105) , (2.1)where N ( P ) is a hadron state with momentum P µ = (¯ n · P ) n µ / P + n µ /
2, with n µ =(1 , , ,
1) and ¯ n µ = (1 , , , − χ n = W † n ξ n is the gauge invariant collinear quark field [63]in SCET, constructed from collinear quark field ξ n and path-ordered collinear Wilson line W n ( x ) = P exp (cid:16) ig (cid:82) −∞ ds ¯ n · A n ( x + ¯ ns ) (cid:17) , and A a,µn ⊥ is the gauge invariant collinear gluonfield with color index a and Lorentz index µ .For sufficiently small b ⊥ , the TMD PDFs in Eq. (2.1) admit operator product expansiononto the usual collinear PDFs, B bare q/N ( x, b ⊥ ) = (cid:88) i (cid:90) x dξξ I bare qi ( ξ, b ⊥ ) φ bare i/N ( x/ξ ) + power corrections , B bare ,µνg/N ( x, b ⊥ ) = (cid:88) i (cid:90) x dξξ I bare ,µνgi ( ξ, b ⊥ ) φ bare i/N ( x/ξ ) + power corrections , (2.2)where the summation is over all parton flavors i . The perturbative matching coefficients I bare qi ( ξ, b ⊥ ) and I bare ,µνgi ( ξ, b ⊥ ) in Eq. (2.2) are independent of the actual hadron N . Inpractical calculations, one can replace the hadron N with a partonic state j . Furthermore,the usual bare partonic collinear PDFs are just φ bare i/j ( x ) = δ ij δ (1 − x ), therefore I bare qi ( x, b ⊥ ) = B bare qi ( x, b ⊥ ) , I bare ,µνgi ( x, b ⊥ ) = B bare ,µνgi ( x, b ⊥ ) (2.3)up to power correction terms.For gluon coefficient functions, one can perform a further decomposition into twoindependent Lorentz structures in d = 4 − (cid:15) dimension, I bare ,µνgi ( ξ, b ⊥ ) = g µν ⊥ d − I bare gi ( ξ, b T ) + (cid:18) g µν ⊥ d − b µ ⊥ b ν ⊥ b T (cid:19) I (cid:48) bare gi ( ξ, b T ) , (2.4)where we have defined two scalar form factors, I bare gi , the unpolarized gluon coefficient func-tions, and I (cid:48) bare gi , the linearly-polarized gluon coefficient functions. They can be projectedout using I bare gi ( ξ, b T ) = g µν ⊥ I bare ,µνgi ( ξ, b ⊥ ) , I (cid:48) bare gi ( ξ, b T ) = 1 d − (cid:20) g µν ⊥ + ( d − b µ ⊥ b ν ⊥ b T (cid:21) I bare ,µνgi ( ξ, b ⊥ ) , (2.5)with b T = − b ⊥ > b T = (cid:113) b T . We focus on unpolarized TMD distributions for thecurrent paper, and the results for the linearly-polarized gluon distribution are left for futurework.
To specify the definition for gluon TMD FFs, it is necessary to specify a reference framefirst. This is in contrast to PDFs, where the incoming hadron provides a canonical reference– 4 –rame for transverse momentum. In the case of TMD FFs, two different frames can bedefined, the hadron frame and the parton frame [12, 26, 27]. In the hadron frame, wherethe detected hadron has zero transverse momentum, an operator definition for gluon TMDFFs can be written down D bare N/q ( z, b ⊥ ) = 1 z (cid:88) X (cid:90) db − π e iP + b − /z (cid:104) | ¯ χ n (0 , b − , b ⊥ ) | N ( P ) , X (cid:105) / ¯ n (cid:104) N ( P ) , X | χ n (0) | (cid:105) , D bare ,µνN/g ( z, b ⊥ ) = − P + z (cid:88) X (cid:90) db − π e iP + b − /z (cid:104) |A a,µn ⊥ (0 , b − , b ⊥ ) | N ( P ) , X (cid:105)(cid:104) N ( P ) , X |A a,νn ⊥ (0) | (cid:105) , (2.6)where P µ = (¯ n · P ) n µ / P + n µ / F bare j/i ( z, b ⊥ /z ) = z − (cid:15) D bare j/i ( z, b ⊥ ) , (2.7)where we denote the bare partonic TMD FFs in the parton frame by F bare j/i . Our N LOresults for TMD FFs will be given in the hadron frame, by choosing the argument of theparton frame coefficient to be b ⊥ /z . The TMD PDFs or TMD FFs, as well as their matching coefficients, contain both UV andrapidity divergences. We adopt dimensional regularization for the UV, and exponentionalregularization [21] for the rapidity divergences. After coupling constant renormalizationin MS scheme, the α s -renormalized matching coefficients still contains overlapping contri-butions between collinear and soft modes which is removed by a zero-bin subtraction [64].After this, the remaining UV divergences are removed by multiplicative renormalizationcounter terms. After UV subtraction and zero-bin subtraction described above, the TMDPDFs or FFs still contain collinear divergence due to the tagged hadron in initial state orfinal state, and the remaining infrared poles are absorbed into the partonic dimensionalregularized collinear PDFs φ ij ( x, α s ) = δ ij δ (1 − x ) − α s π P (0) ij ( x ) (cid:15) + (cid:16) α s π (cid:17) (cid:20) (cid:15) (cid:18)(cid:88) k P (0) ik ⊗ P (0) kj ( x ) + β P (0) ij ( x ) (cid:19) − (cid:15) P (1) ij ( x ) (cid:21) + (cid:16) α s π (cid:17) (cid:20) − (cid:15) (cid:18)(cid:88) m ,k P (0) im ⊗ P (0) mk ⊗ P (0) kj ( x ) + 3 β (cid:88) k P (0) ik ⊗ P (0) kj ( x ) + 2 β P (0) ij ( x ) (cid:19) – 5 – 16 (cid:15) (cid:18)(cid:88) k P (0) ik ⊗ P (1) kj ( x ) + 2 (cid:88) k P (1) ik ⊗ P (0) kj ( x ) + 2 β P (1) ij ( x ) + 2 β P (0) ij ( x ) (cid:19) − (cid:15) P (2) ij ( x ) (cid:21) + O ( α s ) , (2.8)or the FFs d ij ( z, α s ) = δ ij δ (1 − z ) − α s π P T (0) ij ( z ) (cid:15) + (cid:16) α s π (cid:17) (cid:20) (cid:15) (cid:18)(cid:88) k P T (0) ik ⊗ P T (0) kj ( z ) + β P T (0) ij ( z ) (cid:19) − (cid:15) P T (1) ij ( z ) (cid:21) + (cid:16) α s π (cid:17) (cid:20) − (cid:15) (cid:18)(cid:88) m ,k P T (0) im ⊗ P T (0) mk ⊗ P T (0) kj ( z ) + 3 β (cid:88) k P T (0) ik ⊗ P T (0) kj ( z ) + 2 β P T (0) ij ( z ) (cid:19) + 16 (cid:15) (cid:18) (cid:88) k P T (0) ik ⊗ P T (1) kj ( z ) + (cid:88) k P T (1) ik ⊗ P T (0) kj ( z ) + 2 β P T (1) ij ( z ) + 2 β P T (0) ij ( z ) (cid:19) − (cid:15) P T (2) ij ( z ) (cid:21) + O ( α s ) , (2.9)where P ( n ) ij is the ( n + 1)-loop space-like splitting function [65, 66], which are also knownto the same accuracy in a massive environment [67, 68]. P T ( n ) ij is the ( n + 1)-loop time-likesplitting function [50–53], and the symbol ⊗ is denoted as the convolution of two functions f ( z, · · · ) ⊗ g ( z, · · · ) ≡ (cid:90) z dξξ f ( ξ, · · · ) g ( z/ξ, · · · ) . (2.10)The steps above can be summarized as the following collinear factorization formulas1 Z Bi B bare i/j ( x, b ⊥ , µ, ν ) S = (cid:88) k I ik ( x, b ⊥ , µ, ν ) ⊗ φ kj ( x, µ ) , Z Bi F bare j/i ( z, b ⊥ /z, µ, ν ) S = (cid:88) k d jk ( z, µ ) ⊗ C ki ( z, b ⊥ /z, µ, ν ) . (2.11)where B bare i/j ( F bare j/i ) and S ( α s ) are the bare TMD PDFs (FFs) and bare zero-bin softfunction, and Z Bi (see in Sec. C) are the multiplicative operator renormalization constantsfor i = q, g . All the quantities are expressed in terms of the renormalized strong coupling α s . The zero-bin soft function is the same as TMD soft function, which is known to N LOin Ref. [69]. Note also that the time-like TMD soft function is identical to the space-likeone [70, 71] up to this order, so we have a universal soft function up to O ( α s ). LO coefficients for unpolarized quark and gluon TMDs
In this section we give our results for coefficient functions I ij and C ij . We give only anumeric fit to these functions in the paper, but the full analytic expressions can be foundin the ancillary files. For TMD FFs, we give the results for C ij with an argument b ⊥ /z ,which after divided by z are exactly the results in the hadron frame, see Eq. (2.7).– 6 – .1 Renormalization group equations The renormalized coefficient functions obey the following RG equations dd ln µ I ji ( x, b ⊥ , µ, ν ) = 2 (cid:20) Γcusp j ( α s ( µ )) ln νxP + + γ Bj ( α s ( µ )) (cid:21) I ji ( x, b ⊥ , µ, ν ) − (cid:88) k I jk ( x, b ⊥ , µ, ν ) ⊗ P ki ( x, α s ( µ )) , (3.1) dd ln µ C ij ( z, b ⊥ /z, µ, ν ) = 2 (cid:20) Γcusp j ( α s ( µ )) ln zνP + + γ Bj ( α s ( µ )) (cid:21) C ij ( z, b ⊥ /z, µ, ν ) − (cid:88) k P Tik ( z, α s ( µ )) ⊗ C kj ( z, b ⊥ /z, µ, ν ) . (3.2)The rapidity evolution equations are [15, 72] dd ln ν I ji ( x, b ⊥ , µ, ν ) = − (cid:34)(cid:90) b /b T µ d ¯ µ ¯ µ Γcusp j ( α s (¯ µ )) + γ Rj ( α s ( b /b T )) (cid:35) I ji ( x, b ⊥ , µ, ν ) ,dd ln ν C ij ( z, b ⊥ /z, µ, ν ) = − (cid:34)(cid:90) b /b T µ d ¯ µ ¯ µ Γcusp j ( α s (¯ µ )) + γ Rj ( α s ( b /b T )) (cid:35) C ij ( z, b ⊥ /z, µ, ν ) . (3.3)Expanding the perturbative coefficient functions in terms of α s / (4 π ), the solution to theseevolution equations up to O ( α s ) reads, I (0) ji ( x, b ⊥ , µ, ν ) = δ ji δ (1 − x ) , I (1) ji ( x, b ⊥ , µ, ν ) = (cid:32) − Γcusp L ⊥ L Q + γ B L ⊥ + γ R L Q (cid:33) δ ji δ (1 − x ) − P (0) ji ( x ) L ⊥ + I (1) ji ( x ) , I (2) ji ( x, b ⊥ , µ, ν ) = (cid:20) (cid:16) − Γcusp L Q + 2 γ B (cid:17) (cid:16) − Γcusp L Q + 2 γ B + 2 β (cid:17) L ⊥ + (cid:32) ( − Γcusp L Q + 2 γ B + 2 β ) γ R L Q − Γcusp L Q + γ B (cid:33) L ⊥ + ( γ R ) L Q + γ R L Q (cid:21) δ ji δ (1 − x ) + (cid:18) (cid:88) l P (0) jl ⊗ P (0) li ( x )+ P (0) ji ( x )2 (Γcusp L Q − γ B − β ) (cid:19) L ⊥ + (cid:20) − P (1) ji ( x ) − P (0) ji ( x ) γ R L Q − (cid:88) l I (1) jl ⊗ P (0) li ( x ) + (cid:32) − Γcusp L Q + γ B + β (cid:33) I (1) ji ( x ) (cid:21) L ⊥ + γ R L Q I (1) ji ( x ) + I (2) ji ( x ) , I (3) ji ( x, b ⊥ , µ, ν ) = L ⊥ (cid:20) (cid:18) β + 14 (2 γ B − Γcusp L Q ) (cid:19) (cid:88) l P (0) jl ⊗ P (0) li ( x )– 7 – (cid:88) l k P (0) jl ⊗ P (0) lk ⊗ P (0) ki ( x ) + δ ji δ (1 − x ) (cid:18) β (2 γ B − Γcusp L Q )+ 18 β (2 γ B − Γcusp L Q ) + 148 (2 γ B − Γcusp L Q ) (cid:19) + P (0) ji (cid:18) − β (2 γ B − Γcusp L Q ) − β −
18 (2 γ B − Γcusp L Q ) (cid:19) (cid:21) + L ⊥ (cid:20) (cid:18) − β −
12 (2 γ B − Γcusp L Q ) (cid:19) (cid:88) l I (1) jl ⊗ P (0) li ( x )+ 12 (cid:88) l k I (1) jl ⊗ P (0) lk ⊗ P (0) ki ( x ) + 12 (cid:88) l P (0) jl ⊗ P (1) li ( x ) + 12 (cid:88) l P (1) jl ⊗ P (0) li ( x )+ P (0) ji ( x ) (cid:18) − β −
12 (2 γ B − Γcusp L Q ) (cid:19) + δ ji δ (1 − x ) (cid:18) β (2 γ B − Γcusp L Q )+ 12 β (2 γ B − Γcusp L Q ) + 14 (2 γ B − Γcusp L Q )(2 γ B − Γcusp L Q ) (cid:19) + I (1) ji ( x ) (cid:18) β (2 γ B − Γcusp L Q ) + β + 18 (2 γ B − Γcusp L Q ) (cid:19) + P (1) ji ( x ) (cid:18) − β −
12 (2 γ B − Γcusp L Q ) (cid:19) (cid:21) + L ⊥ (cid:20) − (cid:88) l I (1) jl ⊗ P (1) li ( x ) − (cid:88) l I (2) jl ⊗ P (0) li ( x ) − P (0) ji ( x ) γ R L Q − P (2) ji ( x )+ δ ji δ (1 − x ) (cid:18) β γ R L Q + 12 γ R (2 γ B − Γcusp L Q ) L Q + 12 (2 γ B − Γcusp L Q ) (cid:19) + I (1) ji ( x ) (cid:18) β + 12 (2 γ B − Γcusp L Q ) (cid:19) + I (2) ji ( x ) (cid:18) β + 12 (2 γ B − Γcusp L Q ) (cid:19) (cid:21) + δ ji δ (1 − x ) γ R L Q + I (1) ji ( x ) γ R L Q + I (3) ji ( x ) , (3.4)where in I (3) ji we have used γ R = 0 to simplify the expression and I ( n ) ji ( z ) are the scale-independent coefficient functions. We have defined L ⊥ = ln b T µ b , L Q = 2 ln x P + ν , L ν = ln ν µ , b = 2 e − γ E . (3.5)Similarly, the solution to the fragmentation coefficient functions are C (0) ji ( z, b ⊥ /z, µ, ν ) = δ ji δ (1 − z ) , C (1) ji ( z, b ⊥ /z, µ, ν ) = (cid:32) − Γcusp L ⊥ L Q + γ B L ⊥ + γ R L Q (cid:33) δ ji δ (1 − z ) − P T (0) ji ( z ) L ⊥ + C (1) ji ( z ) , C (2) ji ( z, b ⊥ /z, µ, ν ) = (cid:20) (cid:16) − Γcusp L Q + 2 γ B (cid:17) (cid:16) − Γcusp L Q + 2 γ B + 2 β (cid:17) L ⊥ + (cid:32) ( − Γcusp L Q + 2 γ B + 2 β ) γ R L Q − Γcusp L Q + γ B (cid:33) L ⊥ + ( γ R ) L Q + γ R L Q (cid:21) δ ji δ (1 − z ) + (cid:18) (cid:88) l P T (0) jl ⊗ P T (0) li ( z )– 8 – P T (0) ji ( z )2 (Γcusp L Q − γ B − β ) (cid:19) L ⊥ + (cid:20) − P T (1) ji ( z ) − P T (0) ji ( z ) γ R L Q − (cid:88) l P T (0) jl ⊗ C (1) li ( z ) + (cid:32) − Γcusp L Q + γ B + β (cid:33) C (1) ji ( z ) (cid:21) L ⊥ + γ R L Q C (1) ji ( z ) + C (2) ji ( z ) , C (3) ji ( z, b ⊥ /z, µ, ν ) = L ⊥ (cid:20) (cid:18) β + 14 (2 γ B − Γcusp L Q ) (cid:19) (cid:88) l P T (0) jl ⊗ P T (0) li ( z ) − (cid:88) l k P T (0) jl ⊗ P T (0) lk ⊗ P T (0) ki ( z ) + δ ji δ (1 − z ) (cid:18) β (2 γ B − Γcusp L Q )+ 18 β (2 γ B − Γcusp L Q ) + 148 (2 γ B − Γcusp L Q ) (cid:19) + P T (0) ji (cid:18) − β (2 γ B − Γcusp L Q ) − β −
18 (2 γ B − Γcusp L Q ) (cid:19) (cid:21) + L ⊥ (cid:20) (cid:18) − β −
12 (2 γ B − Γcusp L Q ) (cid:19) (cid:88) l P T (0) jl ⊗ C (1) li ( z )+ 12 (cid:88) l k P T (0) jl ⊗ P T (0) lk ⊗ C (1) ki ( z ) + 12 (cid:88) l P T (0) jl ⊗ P T (1) li ( z ) + 12 (cid:88) l P T (1) jl ⊗ P T (0) li ( z )+ P T (0) ji ( z ) (cid:18) − β −
12 (2 γ B − Γcusp L Q ) (cid:19) + δ ji δ (1 − z ) (cid:18) β (2 γ B − Γcusp L Q )+ 12 β (2 γ B − Γcusp L Q ) + 14 (2 γ B − Γcusp L Q )(2 γ B − Γcusp L Q ) (cid:19) + C (1) ji ( z ) (cid:18) β (2 γ B − Γcusp L Q ) + β + 18 (2 γ B − Γcusp L Q ) (cid:19) + P T (1) ji ( z ) (cid:18) − β −
12 (2 γ B − Γcusp L Q ) (cid:19) (cid:21) + L ⊥ (cid:20) − (cid:88) l P T (1) jl ⊗ C (1) li ( z ) − (cid:88) l P T (0) jl ⊗ C (2) li ( z ) − P T (0) ji ( z ) γ R L Q − P T (2) ji ( z )+ δ ji δ (1 − z ) (cid:18) β γ R L Q + 12 γ R (2 γ B − Γcusp L Q ) L Q + 12 (2 γ B − Γcusp L Q ) (cid:19) + C (1) ji ( z ) (cid:18) β + 12 (2 γ B − Γcusp L Q ) (cid:19) + C (2) ji ( z ) (cid:18) β + 12 (2 γ B − Γcusp L Q ) (cid:19) (cid:21) + δ ji δ (1 − z ) γ R L Q + C (1) ji ( z ) γ R L Q + C (3) ji ( z ) . (3.6)We stress again that due to the chosen argument, the expressions given above are forTMD FFs in the hadron frame. The anomalous dimensions appeared above are identicalto those in space-like case and we have suppressed their dependence on the exact flavor.The logarithms appeared in the fragmentation coefficient functions are defined as L ⊥ = ln b T µ b , L Q = 2 ln P + z ν , L ν = ln ν µ , b = 2 e − γ E , (3.7)which differ from thsoe in Eq. (3.5) only in L Q . Both space-like and time-like coefficientfunctions depend on the rapidity regulator being used. Rapidity-regulator-independent– 9 –MD PDFs and TMD FFs can be obtained by multiplying the coefficient functions withthe squared root of the TMD soft functions S ( b ⊥ , µ, ν ) [26, 27] f ⊥ ,ij ( x, b ⊥ , µ ) = I ij ( x, b ⊥ , µ, ν ) (cid:112) S ( b ⊥ , µ, ν ) ,g ⊥ ,ij ( z, b ⊥ /z, µ ) = C ij ( z, b ⊥ /z, µ, ν ) (cid:112) S ( b ⊥ , µ, ν ) . (3.8) The analytic expressions for the coefficient functions will be provided in the ancillary filesalong with the arXiv submission. In this section we will present their numerical fits.The coefficient functions develop end-point divergences both in the threshold and highenergy limit. We first present here the results for leading threshold limit. The results forhigh energy limit will be discussed in next section. In the z → z → I (2) ij ( z ) = lim z → C (2) ji ( z ) = 2 γ R ,i (1 − z ) + δ ij , lim z → I (3) ij ( z ) = lim z → C (3) ji ( z ) = 2 γ R ,i (1 − z ) + δ ij , (3.9)where γ R are the two(three)-loop rapidity anomalous dimensions [69, 73]. The relationbetween threshold limit and rapidity anomalous dimension has been anticipated in [25, 74,75]. The explicit expressions up to three-loop read [47] I (1) qq ( z ) = C (1) qq ( z ) = 0 , I (2) qq ( z ) = C (2) qq ( z ) = 1(1 − z ) + (cid:20) (cid:18) ζ − (cid:19) C A C F + 22427 C F N f T F (cid:21) , I (3) qq ( z ) = C (3) qq ( z ) = 1(1 − z ) + (cid:20)(cid:18) − ζ − ζ
27 + 40 ζ (cid:19) C A C F N f T F + (cid:18) − ζ ζ + 6392 ζ
81 + 12328 ζ
27 + 154 ζ − ζ − (cid:19) C A C F + (cid:18) − ζ − ζ + 342227 (cid:19) C F N f T F + (cid:18) − ζ − (cid:19) C F N f T F (cid:21) . (3.10)We also found that threshold limit exhibits Casimir scaling up to three loops ( n = 1 , , z → I ( n ) gg ( z ) I ( n ) qq ( z ) = lim z → C ( n ) gg ( z ) C ( n ) qq ( z ) = C A C F . (3.11) The analytic expressions of three-loop coefficient functions contain harmonic polyloga-rithms up to transcendental weight 5. To facilitate straightforward numerical implementa-tion, we provide a numerical fitting to all the coefficient functions. Following Ref. [76], weuse the following elementary functions to fit the results, L x ≡ ln x , L ¯ x ≡ ln(1 − x ) , ¯ x ≡ − x . (3.12)For two loop and three loop coefficient functions, we fit the exact results in the region10 − < x < − − (Numerical evaluation of HPLs are made with the HPL package [77]),and we have set the color factor to numerical values in QCD, i.e. C F = 43 , C A = 3 , T F = 12 . (3.13)– 10 –n more detail, we subtract the x → x → x and (1 − x ) ). Then we fit the remaining terms in the region 10 − < x < − − .Combining the two parts, the fitted results can achieve an accuracy better than 10 − for0 < x <
1. We show below the numerical fitting with six significant digits. The fullnumerical fitting is attached as ancillary files with the arXiv submission. The one loopscale independent coefficient functions are given by I (1) qq ( x ) =2 . x , (3.14a) I (1) qg ( x ) =2 x ¯ x , (3.14b) I (1) gq ( x ) =2 . x , (3.14c) I (1) gg ( x ) =0 , (3.14d)The two-loop scale independent coefficient function are given by (3.15a) I (2) qq (cid:48) ( x ) = 2 . x − . x (cid:0) − . L x − . L x + 1 . L x − . (cid:1) + x (0 . L x − . L x + 5 . L x + 3 . x (cid:0) . L x − . L x − . L x + 1 . (cid:1) + 0 . L x − . L x + 2 . L x − . x + 0 . x + 1 . x , (3.15b) I (2) q ¯ q ( x ) = I (2) qq (cid:48) ( x ) + x (cid:0) . L x − . L x + 391 . L x − . (cid:1) + x (1 . L x + 29 . L x + 207 . L x + 533 . x (cid:0) − . L x − . L x − . L x + 6 . (cid:1) + 0 . L x − . L x + 1 . x − . x − . x − . ,I (2) qq ( x ) = I (2) qq (cid:48) ( x ) + (5 . N f + 14 . x ) + + N f (cid:26) x (cid:0) − . L x + 1 . L x − . (cid:1) + x (cid:0) . L x + 3 . L x + 1 . (cid:1) + x (cid:0) . L x + 1 . L x + 2 . (cid:1) + 0 . L x + 1 . L x + 0 . x − . x + 4 . x − . (cid:27) + x (3 . L x − . L x + 60 . L x − . x (cid:0) − . L x − . L x − . L x + 59 . (cid:1) + (¯ x ) (cid:0) . L ¯ x − . L x (cid:1) − . L x + (¯ x ) (cid:0) . L ¯ x − . L x (cid:1) + 22 . L ¯ x + ¯ x (cid:0) . L x − . L ¯ x − . (cid:1) + x (cid:0) − . L x − .L x − . L x + 1 . (cid:1) − . L x − .L x − .L x + 0 . x − . x − . , (3.15c)– 11 –3.15d) I (2) qg ( x ) = − . . L x + (¯ x ) (cid:0) − . L x + 2 . L x − . L ¯ x (cid:1) + (¯ x ) (cid:0) . L x + 2 . L x − . L ¯ x (cid:1) − . L ¯ x + ¯ x ( − . L x − . L x + 5 .L ¯ x + 58 . x (cid:0) . L x + 11 . L x + 6 . L x − . (cid:1) + 0 . L x − . L x + 11 . L x + 4 . x − . x + 5 . x + x (cid:0) − . L x + 15 . L x + 21 . L x − . (cid:1) + x ( − . L x − . L x − . L x + 200 . , (3.15e) I (2) gq ( x ) = N f (cid:26) − . x (cid:0) . L x + 3 . L ¯ x (cid:1) + ¯ x (cid:0) . L x + 6 . L ¯ x (cid:1) + 0 . L x + 1 . L ¯ x + ¯ x (cid:0) . L x + 4 . L ¯ x + 8 . (cid:1) − . x − . x − . x + 10 . x − . x + 32 . x + 11 . x (cid:27) + 25 . x (cid:0) . L x + 11 . L x + 67 . L x + 124 . (cid:1) + x (cid:0) . L x + 0 . L x + 2 . L x − . (cid:1) + x (cid:0) − . L x + 9 . L x − . L x + 48 . (cid:1) − . L x + 12 . L x − . L x + ¯ x (cid:0) . L x − . L x + 101 . L ¯ x (cid:1) + ¯ x (cid:0) − . L x − . L x − . L ¯ x (cid:1) − . L x − . L x − . L ¯ x + ¯ x (cid:0) − . L x − . L x − . L ¯ x − . (cid:1) − . x + 7 . x − . x − . x , (3.15f) I (2) gg ( x ) = N f (cid:26) . x ) + + 0 . L x + 6 .L x + 24 . L x + x (cid:0) . L x + 3 . L x + 22 . L x + 36 . (cid:1) − .L ¯ x + ¯ x (2 .L ¯ x + 54 . − . x + 25 . x − . (cid:27) + 33 .
585 1(¯ x ) + − . x (cid:0) − . L x + 433 . L x − . L x + 2429 . (cid:1) + x (cid:0) − . L x − . L x − . L x − . (cid:1) + x (cid:0) − .L x − .L x − .L x + 221 . (cid:1) − .L x + 3 .L x − .L x + ¯ x (cid:0) . L ¯ x − . L x (cid:1) + ¯ x (cid:0) − . L x − . L ¯ x (cid:1) − .L x + 6 .L ¯ x + ¯ x (cid:0) .L x − .L ¯ x − . (cid:1) − . x + 160 . x − . x − . x . To present the three-loop scale independent coefficient functions, we first perform thefollowing decompositions, I (3) qq (cid:48) ( x ) = I ∗ qq (cid:48) ( x ) + d ABC d ABC N c I d ( x ) ,I (3) q ¯ q (cid:48) ( x ) = I ∗ qq (cid:48) ( x ) − d ABC d ABC N c I d ( x ) ,I (3) qq ( x ) = I ∗ qq ( x ) + I (3) qq (cid:48) ( x ) ,I (3) q ¯ q ( x ) = C F ( C A − C F ) I ∗ q ¯ q ( x ) + I (3) q ¯ q (cid:48) ( x ) , (3.16)– 12 –here d ABC d ABC = 4Tr[ T A { T B , T C } ]Tr[ T A { T B , T C } ] = ( N c − N c − N c = 403 . (3.17)The numerical fitting of different color structures are given by I ∗ qq (cid:48) ( x ) = N f (cid:26) − . x (cid:0) . L x − . L x + 3 . L x − . L x + 18 . (cid:1) + x ( − . L x + 0 . L x + 0 . L x + 5 . L x − . x ) (cid:0) − . L x − . L x − . L ¯ x (cid:1) + (¯ x ) (cid:0) − . L x + 0 . L x + 1 . L ¯ x (cid:1) + ¯ x ( − . L x − . L x − . L ¯ x − . x ( − . L x − . L x − . L x − . L x + 51 . − . L x − . L x − . L x − . L x + 0 . x − . x + 5 . x (cid:27) + 307 .
912 + x (1 . L x + 1 . L x + 92 . L x + 24 . L x + 724 . L x + 168 . x ( − . L x − . L x − . L x + 188 . L x − . L x + 147 . x ) (cid:0) − . L x + 13 . L x − . L x + 333 . L ¯ x (cid:1) + (¯ x ) (cid:0) . L x + 1 . L x + 24 . L x + 192 . L ¯ x (cid:1) + ¯ x (0 . L x + 0 . L x + 6 . L x + 38 . L ¯ x + 123 . x (cid:0) . L x + 2 . L x + 4 . L x − . L x − . L x + 167 . (cid:1) − . L x + 6 . L x − . L x + 86 . L x − . L x + − . L x − . x + 10 . x − . x , (3.18a)(3.18b) I d ( x ) = 32 . (cid:26) − x (cid:0) − . L x + 199731 .L x − . × L x + 1 . × L x − . × L x + 7 . × (cid:1) − (¯ x ) (cid:0) . L x + 121 . L ¯ x (cid:1) − (¯ x ) (cid:0) − . L x − . L ¯ x (cid:1) − ¯ x (cid:0) . L x − . L ¯ x − . (cid:1) − x (cid:0) − . L x + 3 . L x + 0 . L x − . (cid:1) − . L x + 0 . L x − . L x − . L x − . L x + 8847 . x − .x − . − x (cid:0) − . L x − . L x − .L x − . × L x − . × L x − . × (cid:1) (cid:27) , – 13 – ∗ qq ( x ) = (cid:0) − . N f + 154 . N f + 140 . (cid:1) x ) + + N f (cid:26) x (cid:0) . L x − . L x + 13 . L x − . (cid:1) + x (cid:0) − . L x − . L x + 0 . L x + 20 . (cid:1) + x ( − . L x − . L x − . L x − . − . L x − . L x − . L x − . x − . x − . x + 15 . (cid:27) + N f (cid:26) x (cid:0) − . L x + 57 . L x − . L x + 1183 . L x − . (cid:1) + x (3 . L x + 37 . L x + 212 . L x + 982 . L x + 2012 . . L x + (¯ x ) (cid:0) . L x + 1 . L x − . L ¯ x (cid:1) + 10 . L x + (¯ x ) (cid:0) . L x + 5 . L x − . L ¯ x (cid:1) − . L ¯ x + ¯ x ( − . L x − . L x + 48 . L ¯ x + 439 . x (cid:0) . L x + 12 . L x + 16 . L x + 43 . L x + 164 . (cid:1) + 0 . L x + 11 . L x + 59 . L x + 187 . L x − . x + 82 . x − . (cid:27) + x (15 . L x − . L x + 195 . L x + 1220 . L x − . L x + 14892 . x ( − . L x − . L x − . L x − . L x − . L x − . − . L x + (¯ x ) (cid:0) − . L x − . L x + 111 . L ¯ x (cid:1) − . L x + (¯ x ) (cid:0) − . L x − . L x + 413 . L ¯ x (cid:1) + 637 . L ¯ x + ¯ x ( − . L x + 9 . L x − . L ¯ x − . x (cid:0) − . L x − . L x − . L x + 90 . L x + 400 . L x + 265 . (cid:1) − . L x − . L x − . L x − . L x − . L x + 9 . x − . x + 1033 . , (3.18c)(3.18d) I ∗ q ¯ q ( x ) = 9 N f (cid:26) x (cid:0) − . L x + 69 . L x − . L x + 2193 . L x − . (cid:1) + x ( − . L x + 8 . L x + 179 . L x + 1428 . L x + 4299 . . x ) L ¯ x + 0 . x ) L ¯ x + ¯ x (0 . L ¯ x + 0 . x (cid:0) . L x + 0 . L x + 4 . L x + 11 . L x − . (cid:1) − . L x − . L x − . L x − . L x − . x + 688 . x + 1 . (cid:27) + 94 (cid:26) x ( − . L x + 15741 . L x − .L x + 944859 .L x − . × L x + 5 . × ) + x ( − . L x − . L x − . L x − .L x − . × L x − . × ) + 3 . x ) L ¯ x − . x ) L ¯ x + ¯ x ( − . L ¯ x − . x (cid:0) . L x − . L x − . L x − . L x − . L x + 105 . (cid:1) − . L x + 1 . L x + 10 . L x + 17 . L x + 6 . L x − . x + 37870 . x + 15 . (cid:27) , – 14 –3.18e) I (3) qg ( x ) = N f (cid:26) . x (cid:0) . L x − . L x + 120 . L x − . L x + 453 . L x + 442 . (cid:1) + x (cid:0) − . L x − . L x − . L x − . L x − . L x +424 . (cid:1) − . L x − . L x + (¯ x ) (cid:0) . L x − . L x + 116 . L x − . L ¯ x (cid:1) + 1 . L x + (¯ x ) (cid:0) − . L x − . L x − . L x − . L ¯ x (cid:1) + 22 . L ¯ x + ¯ x (0 . L x + 2 . L x + 3 . L x − . L ¯ x − . x ( − . L x − . L x − . L x − . L x − . L x − .
7) + 0 . L x + 1 . L x + 14 . L x + 67 . L x + 216 . L x − . x − . x + 11 . x (cid:27) − . x (cid:0) − . L x + 3952 . L x − .L x + 221224 .L x − .L x + 594756 . (cid:1) + x (cid:0) − . L x − . L x − . L x − . L x − .L x + 96171 . (cid:1) − . L x + 2 . L x + 14 . L x + (¯ x ) (cid:0) − . L x + 1791 . L x − . L x + 107809 .L x − .L ¯ x (cid:1) − . L x + (¯ x ) (cid:0) − . L x − . L x − . L x − . L x − .L ¯ x (cid:1) − . L ¯ x + ¯ x (cid:0) . L x + 1 . L x − . L x − . L x + 85 . L ¯ x + 2491 . (cid:1) + x (10 . L x + 65 . L x + 175 . L x − . L x − . L x − . ) − . L x + 5 . L x − . L x − . L x − . L x + − . L x − . x + 1274 . x + 4328 . x , – 15 – (3) qg ( x ) = N f (cid:26) . x (cid:0) . L x − . L x + 27 . L ¯ x (cid:1) + ¯ x (cid:0) − . L x − . L x − . L ¯ x (cid:1) − . L x − . L x − . L ¯ x + ¯ x (cid:0) − . L x − . L x − . L ¯ x − . (cid:1) +0 . x +0 . x + 0 . x + 21 . x − . x − . x − . x (cid:27) + N f (cid:26) − . x (cid:0) . L x + 10 . L x + 8704 . L x + 1416 . L x + 43827 . L x + 34568 . (cid:1) + x (cid:0) − . L x − . L x − . L x − . L x − . L x + 31014 . (cid:1) + x (cid:0) . L x + 1 . L x + 25 . L x − . L x + 333 . L x − . (cid:1) − . L x − . L x − . L x − . L x − . L x + 66 . L x + 299 . x + ¯ x (cid:0) . L x − . L x + 8354 . L x − . L ¯ x (cid:1) + ¯ x (cid:0) . L x − . L x − . L x − .L ¯ x (cid:1) + 2 . L x + 28 . L x + 127 . L x + 392 . L ¯ x + ¯ x (cid:0) . L x + 48 . L x + 198 . L x + 471 . L ¯ x + 467 . (cid:1) − . x + 1147 . x − . x (cid:27) − . x (cid:0) . L x + 710 . L x + 16404 . L x + 6246 . L x + 151636 .L x + 35826 . (cid:1) + x (cid:0) − . L x − . L x − . L x + 596 . L x + 20659 . L x + 30141 . (cid:1) + x (cid:0) − . L x + 3 . L x − . L x + 1422 . L x + 285 . L x + 20018 . (cid:1) + 10 . L x − . L x + 890 . L x − . L x + 10247 . L x + 923 . L x + 2860 . L x + 9121 . x + ¯ x (cid:0) . L x + 121 . L x + 3663 . L x − . L x + 32059 .L ¯ x (cid:1) + ¯ x (cid:0) − . L x − . L x − . L x − . L x − . L ¯ x (cid:1) − . L x − . L x − . L x − . L x − . L ¯ x + ¯ x (cid:0) − . L x − . L x − . L x − . L x − . L ¯ x − . (cid:1) − . x + 3445 . x − . x , (3.18f)– 16 –3.18g) I (3) gg ( x ) = − . x + 66758 . x − . × x + (cid:0) . L x + 24477 . L x + 63961 . L x + 546586 .L x + 910511 .L x + 495749 . (cid:1) x + (cid:0) . L x + 1945 . L x + 27126 . L x + 189429 .L x + 615133 .L x + 439670 . (cid:1) x + (cid:0) − . L x − .L x − . L x + 7123 . L x + 13857 . L x + 19813 . (cid:1) x + 28 . L x − .L x + 3055 . L x − .L x + 2488 . L x − . L x + (cid:26) − . x + 0 . x − . x + (cid:0) . L x − . L x + 4 . L x − . L x + 21 . (cid:1) x + (cid:0) − . L x − . L x − . L x − . L x + 148 . (cid:1) x + (cid:0) . L x − . L x − . L x − . L x − . (cid:1) x + 0 . L x − . L x − . L x + 0 . L x − . x ) + − . L x + 6 . L ¯ x + ¯ x (cid:0) . L x + 1 . L x − . L ¯ x (cid:1) + ¯ x (cid:0) . L x − . L x − . L ¯ x − . (cid:1) + ¯ x (cid:0) . L x + 0 . L x + 0 . L ¯ x (cid:1) + 41 . − . x (cid:27) N f + 315 . (cid:20) x (cid:21) + + 44694 .L x + 1142 . L ¯ x + ¯ x (cid:0) − . L x − . L x + 11535 . L ¯ x (cid:1) + ¯ x (cid:0) − .L x − . L x − . L ¯ x − . (cid:1) + ¯ x (cid:0) . L x − . L x + 51848 . L ¯ x (cid:1) + (cid:26) − . x + 1878 . x − . x + (cid:0) . L x + 47 . L x + 19161 . L x − . L x + 136127 .L x + 25910 . (cid:1) x + (cid:0) − . L x − . L x − . L x − .L x − . L x + 22497 . (cid:1) x + (cid:0) . L x + 50 . L x + 377 . L x + 915 . L x + 274 . L x + 22082 . (cid:1) x − . L x − . L x − . L x + 10 . L x − . L x + 64 . L x + 347 .
079 1(¯ x ) + − . L x − . L ¯ x + ¯ x (cid:0) − . L x + 181 . L x − . L x + 6458 . L ¯ x (cid:1) + ¯ x (cid:0) − . L x − . L x + 14 . L x + 470 . L ¯ x + 4782 . (cid:1) + ¯ x (cid:0) . L x + 69 . L x + 968 . L x + 5625 . L ¯ x (cid:1) − .
15+ 176 . L x + 788 . x (cid:27) N f + 31575 . . L x + 7128 . L x + 23355 . x . Following the same approach, we give in this subsection the results for TMD FFs. Theone-loop scale-independent coefficient functions are given by C (1) qq ( z ) = z (6 . L z − . z (10 . L z − . z (5 . L z + 5 . . L z + 8 . ¯ z − . z + 0 . z − . z − . , (3.19a)– 17 – (1) gq ( z ) = 10 . L z z − . L z + z (5 . L z − . − . ¯ z + 8 . , (3.19b) C (1) qg ( z ) = z (2 . − .L z ) + z (4 .L z − . ) + 2 .L z , (3.19c) C (1) gg ( z ) = − . + z (15 . L z − . z ( − . L z − . − .L z + 24 .L z z − . z + 1 . z − . z + 12 . ¯ z + z (48 .L z + 11 . . (3.19d)The two-loop scale-independent coefficient functions are given by (3.20a) C (2) qq ( z ) = C (2) q (cid:48) q ( z ) + N f (cid:26) . x ) + + z (cid:0) . L z − . L z + 10 . (cid:1) + z (cid:0) . L z − . L z − . (cid:1) + z (cid:0) . L z − . L z − . (cid:1) + 0 . L z − .L z − . z − . z + 0 . z − . z − . (cid:27) + 14 . x ) + + ¯ z (cid:0) . L ¯ z − . L z (cid:1) + ¯ z (cid:0) . L z − . L ¯ z (cid:1) + 7 . L z − . L ¯ z + ¯ z (cid:0) − . L z + 47 . L ¯ z + 139 . (cid:1) + z (cid:0) − . L z − . L z + 31 . L z − . (cid:1) + z (cid:0) − . L z − . L z − . L z + 42 . (cid:1) + z (cid:0) − . L z + 6 .L z − . L z + 146 . (cid:1) − . L z − . L z + 75 . L z + 0 . z − . z + 85 . z − . , (3.20b) C (2)¯ q (cid:48) q ( z ) = C (2) q (cid:48) q ( z ) , (3.20c) C (2)¯ qq ( z ) = C (2) q (cid:48) q ( z ) + z (cid:0) − . L z + 26 . L z + 241 . L z + 645 . (cid:1) + z (cid:0) . L z + 0 . L z − . L z + 18 . (cid:1) − . L z − . L z − . L z − . z − . z + 20 . z − . z − . z (cid:0) . L z − . L z + 664 . L z − . (cid:1) , (3.20d) C (2) q (cid:48) q ( z ) = − .
773 + z (cid:0) . L z + 0 . L z − . L z + 3 . (cid:1) + z (cid:0) . L z − . L z − . L z + 11 . (cid:1) + z (cid:0) . L z − . L z − . L z + 35 . (cid:1) + 4 . L z − . L z − .L z + 7 . L z + 3 . L z − . z + 1 . z − . z + 0 . z − . z , – 18 –3.20e) C (2) gq ( z ) = 1290 .
49 + ¯ z (cid:0) − . L z + 37 . L z − . L ¯ z (cid:1) + ¯ z (cid:0) . L z + 23 . L z − . L ¯ z (cid:1) + 1 . L z − . L z − . L ¯ z + ¯ z (cid:0) . L z + 21 . L z − . L ¯ z − . (cid:1) + z (cid:0) − . L z − . L z − . L z + 15 . (cid:1) + z (cid:0) − . L z − . L z − . L z + 304 . (cid:1) + z (cid:0) − . L z + 84 . L z + 151 . L z − . (cid:1) − . L z + 21 . L z + 357 . L z + − . L z − . L z + 156 . L z − . z − . z − . z + 53 . z ,C (2) qg ( z ) = − . N f (cid:26) ¯ z (cid:0) . L ¯ z − . L z (cid:1) + ¯ z (cid:0) . L z + 0 . L ¯ z (cid:1) + 0 . L z + 1 . L ¯ z + ¯ z (cid:0) − . L z − . L ¯ z + 4 . (cid:1) + z (cid:0) − . L z + 2 . L z + 0 . (cid:1) + z (cid:0) . L z − . L z − . (cid:1) + z (cid:0) − . L z +7 . L z +7 . (cid:1) + 0 . L z − . L z − . z + 0 . z − . z − . (cid:27) + ¯ z (cid:0) . L z − . L z + 13 . L ¯ z (cid:1) + ¯ z (cid:0) − . L z − . L z + 41 . L ¯ z (cid:1) − . L z − . L z + 1 . L ¯ z + ¯ z (cid:0) . L z + 8 . L z − . L ¯ z − . (cid:1) + z (cid:0) − . L z − . L z − . L z + 226 . (cid:1) + z (cid:0) − . L z − . L z − . L z − . (cid:1) + z (cid:0) . L z + 2 . L z − . L z − . (cid:1) + 8 . L z − . L z − . L z + 16 .L z + 8 .L z − . z − . z − . z + 33 . z , (3.20f)– 19 – (2) gg ( z ) = N f (cid:26) . x ) + − . − . z L ¯ z − . z L ¯ z + 2 .L ¯ z + z (cid:0) . L z − . L z + 15 . L z + 0 . (cid:1) + z (cid:0) . L z + 7 . L z − . L z + 49 . (cid:1) + z (cid:0) . L z − . L z − .L z + 83 . (cid:1) + 9 . L z + 7 . L z − .L z + − . L z − . L z + 3 . z + 37 . z − . z + 0 . z − . z (cid:27) + 33 .
585 1(¯ x ) + + 1795 .
49 + ¯ z (cid:0) . L z + 287 . L ¯ z (cid:1) + ¯ z (cid:0) . L z + 5 . L ¯ z (cid:1) + 36 .L z − .L ¯ z + ¯ z (cid:0) − .L z + 216 .L ¯ z + 444 . (cid:1) + z (cid:0) − . L z − . L z − . L z + 4605 . (cid:1) + z (cid:0) − . L z − . L z − . L z − . (cid:1) + z (cid:0) − .L z − .L z + 357 . L z − . (cid:1) − .L z + 33 .L z + 1292 . L z + − .L z − .L z + 62 . L z − . z + 24 . z − . z + 5728 . z . (3.20g)– 20 –he three-loop scale-independent coefficient functions are given by C (3) qq ( z ) = C (3) q (cid:48) q ( z ) + N f (cid:26) .
257 1(¯ x ) + + ¯ z (cid:0) . L z + 0 . L z − . L ¯ z (cid:1) + ¯ z (cid:0) − . L z − . L z − . L ¯ z (cid:1) − . L z − . L z + 13 . L ¯ z + ¯ z (cid:0) . L z − . L z − . L ¯ z + 57 . (cid:1) + z (cid:0) − . L z + 33 . L z − . L z + 661 . L z − . (cid:1) + z (cid:0) − . L z + 30 . L z + 191 . L z + 337 . L z + 2071 . (cid:1) + z (cid:0) − . L z − . L z + 50 . L z + 131 . L z − . (cid:1) − . L z + 4 . L z + 115 . L z − . L z + 0 . z − . z + 300 . z + 138 . (cid:27) + N f (cid:26) − . x ) + + z (cid:0) . L z − . L z + 8 . L z + 0 . (cid:1) + z (cid:0) . L z − . L z + 5 . L z − . (cid:1) + z (cid:0) . L z − . L z − . L z − . (cid:1) + 0 . L z − . L z + 8 . L z − . z − . z + 0 . z − . z + 9 . (cid:27) + 140 .
136 1(¯ x ) + + ¯ z (cid:0) . L z − . L z + 6271 . L z − . L ¯ z (cid:1) + ¯ z (cid:0) − . L z − . L z − . L z − . L ¯ z (cid:1) + 34 . L z + 5 . L z − . L ¯ z + ¯ z (cid:0) − . L z + 32 . L z + 1865 . L ¯ z − . (cid:1) + z (cid:0) . L z + 409 . L z + 1340 . L z + 9286 . L z − . L z + 19119 . (cid:1) + z (cid:0) . L z − . L z − . L z − . L z − . L z + 17171 . (cid:1) + z (cid:0) . L z − . L z + 235 . L z − . L z + 998 . L z − . (cid:1) + 0 . L z − . L z − . L z − . L z + 3285 . L z − . z + 420 . z − . z + 133 . , (3.21a)– 21 – (3)¯ qq ( z ) = C (3)¯ q (cid:48) q ( z ) + N f (cid:26) ¯ z (cid:0) − . L z − . L z − . L ¯ z (cid:1) + ¯ z (cid:0) − . L z − . L z − . L ¯ z (cid:1) + ¯ z (0 . L ¯ z + 0 . z (cid:0) − . L z − . L z − . L z − . L z − . (cid:1) + z (cid:0) − . L z − . L z − . L z − . L z − . (cid:1) + z (cid:0) . L z − . L z − . L z + 14 . L z − . (cid:1) − . L z + 1 . L z + 3 . L z + 13 . L z + 15 . z − . z + 1650 . z + 32 . (cid:27) + ¯ z (cid:0) . L z − . L z + 54960 .L z − .L ¯ z (cid:1) + ¯ z (cid:0) − . L z − . L z − .L z − . L ¯ z (cid:1) + ¯ z ( − . L ¯ z − . z (cid:0) − . L z + 2695 . L z − . L z + 99236 .L z − .L z + 137119 . (cid:1) + z (cid:0) . L z + 265 . L z + 2452 . L z + 12096 . L z − . L z + 124275 . (cid:1) + z (cid:0) − . L z − . L z + 109 . L z + 142 . L z − . L z − . (cid:1) + 2 . L z + 24 . L z − . L z + 103 . L z + 333 . L z − . z + 6288 . z − .z + 422 . , (3.21b)(3.21c) C (3) q (cid:48) q ( z ) = C ∗ q (cid:48) q ( z ) + d ABC d ABC N c C d ( z ) , (3.21d) C (3)¯ q (cid:48) q ( z ) = C ∗ q (cid:48) q ( z ) − d ABC d ABC N c C d ( z ) , – 22 – ∗ q (cid:48) q ( z ) = N f (cid:26) . z (cid:0) . L z + 0 . L z − . L ¯ z (cid:1) + ¯ z (cid:0) . L z − . L z − . L ¯ z (cid:1) + ¯ z (cid:0) − . L z − . L z − . L ¯ z − . (cid:1) + z (cid:0) . L z + 0 . L z − . L z + 6 . L z + 1 . (cid:1) + z (cid:0) − . L z − . L z + 16 . L z − . L z + 12 . (cid:1) + z (cid:0) . L z − . L z + 23 . L z + 10 . L z − . (cid:1) + 0 . L z − . L z − . L z + 11 . L z + 6 . L z + 4 . L z + 5 . L z − . z − . z + 0 . z − . z (cid:27) + 2356 .
76 + ¯ z (cid:0) . L z + 0 . L z + 60 . L z + 14 . L ¯ z (cid:1) + ¯ z (cid:0) − . L z + 1 . L z − . L z − . L ¯ z (cid:1) + ¯ z (cid:0) . L z − . L z − . L z + 61 . L ¯ z + 90 . (cid:1) + z (cid:0) . L z + 5 . L z + 123 . L z + 176 . L z + 923 . L z + 28 . (cid:1) + z (cid:0) − . L z + 22 . L z + 1 . L z − . L z + 790 . L z + 5 . (cid:1) + z (cid:0) − . L z + 85 . L z − . L z − . L z + 3982 . L z − . (cid:1) + 5 . L z − . L z + 79 . L z − . L z + 273 . L z + − . L z − . L z − . L z − . L z + 712 . z − . z + 33 . z − . z , (3.21e) C d ( z ) = ¯ z (cid:0) − . L z − . L ¯ z (cid:1) + ¯ z (cid:0) − . L z + 16 . L ¯ z + 62 . (cid:1) + z (cid:0) . L z + 34 . L z + 132 . L z + 595 . L z + 587 . L z + 520 . (cid:1) + z (cid:0) . L z − . L z + 134 . L z + 383 . L z + 246 . L z + 472 . (cid:1) + z (cid:0) . L z − . L z + 967 . L z − . L z + 1419 . (cid:1) + 7 . L z − . L z − . L z + 706 . L z + 521 . L z + 4 . z − . z − . z − .
97 + ¯ z (cid:0) − . L z − . L ¯ z (cid:1) , (3.21f)– 23 – (3) gq ( z ) = N f (cid:26) − .
48 + ¯ z (cid:0) . L z − . L z + 228 . L z − . L ¯ z (cid:1) + ¯ z (cid:0) − . L z − . L z − . L z − . L ¯ z (cid:1) − . L z − . L z + 31 . L z + 126 . L ¯ z + ¯ z (cid:0) − . L z − . L z − . L z + 162 . L ¯ z + 913 . (cid:1) + z (cid:0) . L z + 11 . L z + 203 . L z + 290 . L z + 1265 . L z + 630 . (cid:1) + z (cid:0) − . L z − . L z − . L z − . L z − . L z + 559 . (cid:1) + z (cid:0) − . L z + 6 . L z + 24 . L z − . L z + 915 . L z + 599 . (cid:1) + 8 . L z − . L z − . L z − . L z + 1025 . L z + − . L z − . L z − . L z − . L z + 531 . z − . z + 46 . z − . z (cid:27) − .
8+ ¯ z (cid:0) . L z − . L z + 1485 . L z − . L z + 10384 . L ¯ z (cid:1) + ¯ z (cid:0) − . L z − . L z + 5 . L z − . L z − . L ¯ z (cid:1) − . L z + 7 . L z + 115 . L z − . L z − . L ¯ z + ¯ z (cid:0) − . L z − . L z + 119 . L z + 1075 . L z − . L ¯ z − . (cid:1) + z (cid:0) − . L z − . L z − . L z − . L z − . L z − . (cid:1) + z (cid:0) . L z + 45 . L z + 819 . L z + 722 . L z + 15767 . L z − . (cid:1) + z (cid:0) . L z − . L z + 829 . L z + 21325 . L z − . L z + 29876 . (cid:1) − . L z + 149 . L z + 1805 . L z + 20822 .L z − . L z + 384 .L z + 3473 . L z + 5160 . L z + 6813 . L z + 37045 . L z + 7355 . z + 49 . z − . z + 14561 . z , (3.21g)– 24 – (3) qg ( z ) = − . z + 216 . z − . z + (cid:0) − . L z + 166 . L z − . L z + 2353 . L z − .L z + 452 . (cid:1) z + (cid:0) . L z + 308 . L z + 318 . L z + 6648 . L z + 38837 . L z + 74 . (cid:1) z + (cid:0) − . L z − . L z − . L z − . L z + 8857 . L z − . (cid:1) z − . L z + 16 . L z − . L z − . L z − . L z − . L z − . L z − . L z + (cid:26) − . z + 0 . z − . z + (cid:0) − . L z + 1 . L z − . L z + 9 . (cid:1) z + (cid:0) . L z − . L z + 7 . L z − . (cid:1) z + (cid:0) − . L z + 5 . L z − . L z − . (cid:1) z − . L z + 0 . L z − . L z − . L z +1 . L ¯ z + ¯ z (cid:0) − . L z − . L z +8 . L ¯ z (cid:1) + ¯ z (cid:0) . L z − . L z + 1 . L ¯ z (cid:1) + ¯ z (cid:0) . L z +2 . L z − . L ¯ z − . (cid:1) − . L z +9 . (cid:27) N f +275 . L ¯ z + ¯ z (cid:0) − . L z − . L z − . L z − . L z − . L ¯ z (cid:1) + ¯ z (cid:0) . L z + 28 . L z + 139 . L z − . L z − . L ¯ z − . (cid:1) + ¯ z (cid:0) . L z + 138 . L z + 3798 . L z + 4087 . L z + 17552 . L ¯ z (cid:1) + 1017 . L z + (cid:26) − . z + 150 . z − . z + (cid:0) . L z + 4 . L z + 1004 . L z + 571 . L z + 4318 . L z + 5231 . (cid:1) z + (cid:0) − . L z − . L z − .L z − . L z − . L z + 4714 . (cid:1) z + (cid:0) . L z − . L z − . L z − . L z + 89 . L z − . (cid:1) z − . L z + 1 . L z + 0 . L z + 11 . L z + 0 . L z + 26 . L z + 273 . L z − . L ¯ z + ¯ z (cid:0) − . L z − . L z − . L z + 179 . L ¯ z + 330 . (cid:1) + ¯ z (cid:0) . L z − . L z − . L z − . L ¯ z (cid:1) + ¯ z (cid:0) . L z − . L z + 1456 . L z − . L ¯ z (cid:1) + 99 . L z − .
185 + 13 . L z − . L z + 1 . L z − . z (cid:27) N f + 9611 . − . L z − . L z − . L z − . L z + 1652 . z , (3.21h)– 25 – (3) gg ( z ) = 13587 . z − .z + 4 . × z + (cid:0) − . L z − . L z − .L z − .L z − . × L z − . × (cid:1) z + (cid:0) . L z + 4555 . L z + 63933 . L z + 367397 .L z + 694258 .L z − . × (cid:1) z + (cid:0) . L z + 2099 .L z + 1739 . L z + 121402 .L z − . L z + 190560 . (cid:1) z − . L z + 380 .L z + 176 .L z + 3912 . L z + 839 . L z + 61497 . L z + (cid:26) − . z − . z + 3 . z + (cid:0) − . L z + 3 . L z − . L z + 51 . L z − . (cid:1) z + (cid:0) . L z − . L z − . L z + 104 . L z + 35 . (cid:1) z + (cid:0) . L z − . L z + 51 . L z + 58 . L z − . (cid:1) z + 1 . L z − . L z − . L z + 5 . L z − . x ) + − . L ¯ z + ¯ z (cid:0) . L z − . L z − . L ¯ z (cid:1) + ¯ z (cid:0) . L z − . L z − . L ¯ z − . (cid:1) + ¯ z (cid:0) . L z + 1 . L z − . L ¯ z (cid:1) + 113 . L z + 240 .
465 + 0 . L z + 1 . L z + 37 . L z − . z (cid:27) N f + 315 .
306 1(¯ x ) + − . L ¯ z + ¯ z (cid:0) − . L z − . L z − . L ¯ z (cid:1) + ¯ z (cid:0) − .L z − . L z + 12927 . L ¯ z + 3854 . (cid:1) + ¯ z (cid:0) . L z − . L z − . L ¯ z (cid:1) − . L z + (cid:26) − . z + 5156 . z − .z + (cid:0) . L z + 751 . L z + 35545 . L z + 5384 . L z + 246456 .L z + 81634 . (cid:1) z + (cid:0) − . L z − . L z − . L z − .L z − . L z + 71845 . (cid:1) z + (cid:0) − . L z + 344 . L z + 647 . L z − . L z + 7537 . L z − . (cid:1) z + 32 . L z − . L z − . L z − . L z − . L z − . L z + 347 .
079 1(¯ x ) + + 61 . L ¯ z + ¯ z (cid:0) − . L z − . L z − . L z − . L ¯ z (cid:1) + ¯ z (cid:0) − . L z + 4 . L z + 66 . L z + 0 . L ¯ z + 2403 . (cid:1) + ¯ z (cid:0) . L z − . L z + 3741 . L z − . L ¯ z (cid:1) − . L z + 100 . − . L z + 136 . L z + 909 . L z − . L z + 905 . z (cid:27) N f − . + 864 .L z + 8008 .L z + 15340 . L z + 23082 . L z + 76348 . L z + 19111 . z . (3.21i)– 26 – Small x expansion of unpolarized TMD coefficients and resummationfor TMD FFs In this section we give the results for TMD PDFs and FFs expanded in high energy limit,namely x, z →
0. A striking difference between space-like TMD PDFs and time-like TMDFFs is that there is only single logarithmic enhancement in each order of perturbative ex-pansion in TMD PDFs, while in TMD FFs it becomes double logarithmic enhancement.We also resum the small- z logarithms in TMD FFs to Next-to-Next-to-Leading Logarith-mic (NNLL) accuracy in this section. x expansion of unpolarized TMD PDFs Using the analytic expression we obtained, it is straightforward to obtain the small- x expansion. At leading power in the expansion, the results read xI (2) qq ( x ) = xI (2) qq (cid:48) ( x ) = xI (2) q ¯ q ( x ) = xI (2) q ¯ q (cid:48) ( x ) = 2 C F T F (cid:18) − ζ (cid:19) ,xI (3) qq ( x ) = xI (3) qq (cid:48) ( x ) = xI (3) q ¯ q ( x ) = xI (3) q ¯ q (cid:48) ( x )=2 T F (cid:20) (cid:18) ζ ζ − (cid:19) C A C F ln x + (cid:18) − ζ + 5129 ζ + 323 ζ − (cid:19) C F + (cid:18) ζ
81 + 120 ζ + 920 ζ − (cid:19) C A C F + (cid:18) − ζ − ζ (cid:19) C F N f T F (cid:21) . (4.1) xI (2) qg ( x ) =2 C A T F (cid:18) − ζ (cid:19) ,xI (3) qg ( x ) =2 T F (cid:20) (cid:18) ζ ζ − (cid:19) C A ln x + (cid:18) ζ − ζ − (cid:19) C A N f T F + (cid:18) − ζ + 512 ζ ζ − (cid:19) C A C F + (cid:18) ζ
81 + 1096 ζ ζ − (cid:19) C A + (cid:18) − ζ − ζ (cid:19) C F N f T F (cid:21) , (4.2) xI (2) gq ( x ) = C A C F (cid:20) ζ + 48 ζ − (cid:21) + 44827 C F N f T F ,xI (3) gq ( x ) = C A C F (cid:20) ζ ln x + (cid:18) − ζ − ζ − ζ + 7558481 (cid:19) ln x − ζ + 3203 ζ ζ − ζ − ζ − ζ + 33361354 (cid:21) + C A C F (cid:20) − ζ ζ + 88 ζ − ζ + 368 ζ + 24323 ζ − (cid:21) + C A C F N f T F (cid:20)(cid:18) − ζ + 64 ζ + 142481 (cid:19) ln x − ζ + 8329 ζ + 16 ζ – 27 – 68548243 (cid:21) + C F (cid:20) ζ ζ − ζ + 592 ζ − ζ − ζ + 4673 (cid:21) + C F N f T F (cid:20)(cid:18) ζ − ζ − (cid:19) ln x + 977681 ζ − ζ − ζ − (cid:21) + C F N f T F (cid:20) − ζ − (cid:21) . (4.3) xI (2) gg ( x ) = C A (cid:20) ζ + 48 ζ − (cid:21) + 48427 C A N f T F − C F N f T F ,xI (3) gg ( x ) = C A (cid:20) ζ ln x + (cid:18) − ζ − ζ − ζ + 7558481 (cid:19) ln x − ζ + 128 ζ ζ − ζ − ζ − ζ + 1572769243 (cid:21) + C A N f T F (cid:20)(cid:18) − ζ + 3203 ζ + 156881 (cid:19) ln x − ζ − ζ + 10409 ζ + 535048729 (cid:21) + C A C F N f T F (cid:20)(cid:18) ζ − ζ − (cid:19) ln x + 1014427 ζ + 256 ζ − ζ − (cid:21) + C A N f T F (cid:20) − ζ − (cid:21) + C F N f T F (cid:20) ζ − ζ + 10249 ζ + 12 (cid:21) + C F N f T F (cid:20) − ζ (cid:21) . (4.4)We note that a LL prediction for the small- x expansion has been given in [78]. After fixinga typo in that paper, we find full agreement with its LL prediction for both quark andgluon TMD PDFs at small x . It would be very interesting to extend the formalism of[78] beyond LL and compare with the data presented here. z expansion of unpolarized TMD FFs To facilitate small- z resummation for TMD FFs, we shall consider the coefficient functionsin flavor singlet sector below, since non-singlet TMD FFs are at most logarithmic divergent,but not power divergent in the z → s )coefficient functions can be written as a matrix, (cid:98) C s ( z ) = (cid:32) (cid:101) C qq ( z ) 2 N f C qg ( z ) C gq ( z ) C gg ( z ) (cid:33) , (4.5)where (cid:101) C qq ( z ) = C qq ( z ) + C ¯ qq ( z ) + ( N f − C q (cid:48) q ( z ) + C ¯ q (cid:48) q ( z )) , (4.6)and C ij ( z ) are scaleless coefficient functions as appeared in the solutions of RG equation(3.6). We thank Simone Marzani for communicating with us the typo in Eq. (40) of [78]. – 28 –n contrast to TMD PDFs, which contribute a single logarithm at each perturbativeorder in the small- x , TMD FFs in the singlet sector develop small- z double logarithms,lim z → z (cid:98) C skj ( z ) = lim z → z ∞ (cid:88) n =1 a ns (cid:98) C s ( n ) kj ( z ) ∼ ∞ (cid:88) n =1 a ns (cid:18) n (cid:88) m =1 ln n − m z (cid:19) , (4.7)where a s = α s / (4 π ) is our perturbative expansion parameter. The small- z data for quarkfragmentation in the singlet sector are (non-singlet sector results are suppressed in the z → z (cid:98) C s (1) qq ( z ) =0 ,z (cid:98) C s (2) qq ( z ) =2 N f C F T F (cid:18)
323 ln z + 163 ln z + 163 ζ − (cid:19) ,z (cid:98) C s (3) qq ( z ) =2 N f C A C F T F (cid:20) − z − z + (cid:18) − ζ − (cid:19) ln z + (cid:18) ζ − ζ + 11312243 (cid:19) ln z − ζ + 112 ζ − ζ + 512156729 (cid:21) +2 N f C F T F (cid:20) z + (cid:18) − ζ + 1289 ζ + 12889 (cid:19) ln z − ζ + 329 ζ + 627227 ζ − (cid:21) + 2 C F N f T F (cid:20) z + 1289 ln z + (cid:18) − ζ + 512081 (cid:19) ln z − ζ − (cid:21) . (4.8) z (cid:98) C s (1) gq ( z ) =8 C F ln z ,z (cid:98) C s (2) gq ( z ) = C A C F (cid:20) −
803 ln z − z + (32 ζ + 12) ln z − ζ − ζ + 312827 (cid:21) + C F (cid:20) (48 − ζ ) ln z (cid:21) ,z (cid:98) C s (3) gq ( z ) = C A C F (cid:20)
32 ln z + 781627 ln z + (cid:18) − ζ (cid:19) ln z + (cid:18) − ζ + 11843 ζ + 86089 (cid:19) ln z + (cid:18) ζ + 31363 ζ + 1408 ζ − (cid:19) ln z + 7456 ζ ζ ζ + 58703 ζ − ζ + 2194427 ζ − (cid:21) + C A C F (cid:20)(cid:18) ζ − (cid:19) ln z + (cid:18) − ζ + 18883 ζ − (cid:19) ln z + (cid:18) ζ − ζ − ζ + 7019 (cid:19) ln z + 68003 ζ + 992 ζ ζ + 8303 ζ − ζ − ζ + 101419 (cid:21) + C A C F N f T F (cid:20) − z + 371281 ln z + (cid:18) ζ + 15281 (cid:19) ln z + (cid:18) − ζ − ζ − (cid:19) ln z + 6889 ζ + 7043 ζ + 1086481 ζ − (cid:21) + C F (cid:20)(cid:18) ζ − ζ + 2083 (cid:19) ln z + (cid:18) ζ − ζ − ζ − (cid:19) ln z – 29 – ζ − ζ ζ + 796 ζ + 28883 ζ + 608 ζ − (cid:21) + C F N f T F (cid:20) − z − z + (cid:18) − ζ − (cid:19) ln z + (cid:18) − ζ − ζ + 206027 (cid:19) ln z − ζ − ζ − ζ + 75770729 (cid:21) . (4.9) z (cid:98) C s (1) qg ( z ) =0 ,z (cid:98) C s (2) qg ( z ) =2 N f C A T F (cid:20)
323 ln z + 163 ln z + 163 ζ − (cid:21) ,z (cid:98) C s (3) qg ( z ) =2 N f C A T F (cid:20)(cid:18) − ζ − (cid:19) ln z + (cid:18) − ζ + 2563 ζ + 14360243 (cid:19) ln z − z − z − ζ + 9769 ζ − ζ + 152392243 (cid:21) +2 N f C A C F T F (cid:20)(cid:18) ζ − (cid:19) ln z + (cid:18) ζ − ζ + 10009 (cid:19) ln z + 304027 ζ + 329 ζ − ζ + 51427 (cid:21) + 2 C A N f T F (cid:20)(cid:18) − ζ (cid:19) ln z + 89681 ln z − z − ζ − ζ + 25627 (cid:21) + 2 C F N f T F (cid:20)(cid:18) − ζ (cid:19) ln z + 128081 ln z + 512081 ln z − ζ + 1289 ζ + 10304729 (cid:21) . (4.10) z (cid:98) C s (1) gg ( z ) =8 C A ln z ,z (cid:98) C s (2) gg ( z ) = C A (cid:20)(cid:18) − ζ (cid:19) ln z −
803 ln z − z − ζ − ζ + 326827 (cid:21) + C A N f T F (cid:20) −
163 ln z − z + 55627 (cid:21) + C F N f T F (cid:20)
323 ln z + 163 ln z − (cid:21) ,z (cid:98) C s (3) gg ( z ) = C A (cid:20)(cid:18) − ζ (cid:19) ln z + (cid:18) − ζ + 416 ζ + 1479227 (cid:19) ln z + (cid:18) ζ + 56329 ζ + 2104 ζ − (cid:19) ln z + 32 ln z + 800827 ln z − ζ + 448 ζ ζ − ζ + 3256 ζ + 4336 ζ − (cid:21) + C A N f T F (cid:20)(cid:18) ζ + 2281681 (cid:19) ln z + (cid:18) − ζ + 2569 ζ − (cid:19) ln z + 35227 ln z + 515227 ln z − ζ + 31529 ζ + 240 ζ − (cid:21) + C A C F N f T F (cid:20)(cid:18) − ζ − (cid:19) ln z + (cid:18) − ζ − ζ + 92560243 (cid:19) ln z − z − z + 385627 ζ − ζ − ζ + 849464729 (cid:21) – 30 – C A N f T F (cid:20) z + 147281 ln z + 3776243 ln z − ζ − (cid:21) + C F N f T F (cid:20)(cid:18) ζ − (cid:19) ln z + 643 ln z + 27529 ζ + 8323 ζ − (cid:21) + C F N f T F (cid:20) − z − z + 18560243 ln z + 2569 ζ + 104512729 (cid:21) . (4.11)We note that while both (cid:98) C sqi and (cid:98) C sgi has double logarithmic expansion in the small- z limit,the power of leading logarithmic terms of (cid:98) C sqi is lower by 1 than the corresponding leadinglogarithmic terms of (cid:98) C sgi . x logarithms for unpolarized TMD FFs In this subsection, we shall derive the all-order resummation at NNLL accuracy (resum-mation of the highest three logarithms) in Eq. (4.7), following an idea proposed in [56].To this end, we begin with the unrenormalized version of collinear factorization formula inEq. (2.11) for singlet TMD FFs (see (4.6) for the definition of singlet combination), F si/j ( z, (cid:15) ) = 1 Z Bj F s, bare i/j ( z, (cid:15) ) S = (cid:88) k d sik ⊗ C skj ( z, (cid:15) ) , (4.12)where the convolution is in z . Note that F si/j ( z, (cid:15) ) is a quantity to which the usual strongcoupling renormalization, zero-bin subtraction and operator renormalization have beenperformed. However renormalization of collinear FFs has not been performed, which iswhy we still keep the (cid:15) dependence in (4.12). From now on we concentrate on the scale-independent part of the coefficient functions by setting all the scale logarithms to zero. Wecan do this because the scale logarithms depends either on anomalous dimension, whichhas no z dependence, or on time-like splitting functions, whose small- z behavior is knownto NNLL accuracy [57]. It proves convenient to work in Mellin- N space also, which isdefined as F ( N , (cid:15) ) = M [ F ( z, (cid:15) )] := (cid:90) dz z N − F ( z, (cid:15) ) , (4.13)where N = N −
1. Small- z logarithms becomes poles in ¯ N under Mellin transformation, M (cid:20) z ln k z (cid:21) ≡ (cid:90) dz z N − z ln k z = ( − k k !( N − k +1 = ( − k k ! N k +1 . (4.14)In Mellin space the unrenormalized collinear factorization formula in Eq. (4.12) becomes (cid:32) F sqi ( N , (cid:15) ) F sgi ( N , (cid:15) ) (cid:33) = (cid:98) d s ( N , (cid:15) ) · (cid:32) (cid:98) C sqi ( N , (cid:15) ) (cid:98) C sgi ( N , (cid:15) ) (cid:33) , (4.15)where (cid:98) d s ( N , (cid:15) ) = (cid:32) (cid:98) d sqq ( N , (cid:15) ) (cid:98) d sqg ( N , (cid:15) ) (cid:98) d sgq ( N , (cid:15) ) (cid:98) d sgg ( N , (cid:15) ) (cid:33) . (4.16)– 31 –he collinear FFs in MS scheme evolve with time-like splitting functions. In Mellinmoment space it reads dd ln µ (cid:98) d s ( N , (cid:15) ) = 2 (cid:98) d s ( N , (cid:15) ) · (cid:98) γ T ( N ) , (4.17)where (cid:98) γ T ( N ) is the time-like singlet splitting function in Mellin space. Its complete NNLOresults can be found in [50], see also [51–53].The crucial observation of [56] is that unrenormalized collinear functions have specificsingular behavior in the limit of small- z in dimensional regularization. In the case of TMDFFs, we can write down an general ansatz at small z , F s ( n ) g/i ( z, (cid:15) ) = 1 (cid:15) n − n − (cid:88) l =0 z − − n − l ) (cid:15) ( c (1 ,l,n ) gi (cid:124) (cid:123)(cid:122) (cid:125) LL + (cid:15)c (2 ,l,n ) gi (cid:124) (cid:123)(cid:122) (cid:125) NLL + (cid:15) c (3 ,l,n ) gi (cid:124) (cid:123)(cid:122) (cid:125) NNLL + . . . ) , F s ( n ) q/i ( z, (cid:15) ) = 1 (cid:15) n − n − (cid:88) l =0 z − − n − l ) (cid:15) ( c (1 ,l,n ) qi (cid:124) (cid:123)(cid:122) (cid:125) LL + (cid:15)c (2 ,l,n ) qi (cid:124) (cid:123)(cid:122) (cid:125) NLL + (cid:15) c (3 ,l,n ) qi (cid:124) (cid:123)(cid:122) (cid:125) NNLL + . . . ) , (4.18)where c (1 ,l,n ) gi is the leading term in the (cid:15) expansion and small- z expansion, whose knowl-edge correspond to LL resummation as labeled in (4.18), and similarly for other terms.Precisely, for (cid:98) C sgi ( z ) the LL series correspond to α ns ln n − z terms, while NLL correspondto α ns ln n − z , and NNLL to α ns ln n − z . For (cid:98) C sqi ( z ) the corresponding power of ln z islower by 1. We have verified this general ansatz through explicit N LO calculation fromits operator definition in (2.6) and (2.7). In Mellin space the corresponding ansatz reads F s ( n ) g/i ( N , (cid:15) ) = 1 (cid:15) n − n − (cid:88) l =0 N − n − l ) (cid:15) ( c (1 ,l,n ) gi + (cid:15)c (2 ,l,n ) gi + (cid:15) c (3 ,l,n ) gi + . . . ) , F s ( n ) q/i ( N , (cid:15) ) = 1 (cid:15) n − n − (cid:88) l =0 N − n − l ) (cid:15) ( c (1 ,l,n ) qi + (cid:15)c (2 ,l,n ) qi + (cid:15) c (3 ,l,n ) qi + . . . ) . (4.19)Equations (4.18) or (4.19) provides a way to resum all the large logarithms of z . Specifi-cally, if one knows all the c ,l,ngi for all l and n , then one can do LL resummation for F sg/i , andsimilarly for NLL and NNLL resummation. In this paper instead of working out the con-stants for all n , we provide results for n up to 15, which is sufficient for phenomenologicalpurpose.We note that the neglected terms in (4.19) are higher orders in (cid:15) and in N . To facilitateeasy extraction of the constants, it is convenient to define a “small- z ” weight:[ N ] = 1 , [ (cid:15) ] = 1 , [numbers] = 0 , (4.20)such that c ( m,l,n ) gi corresponds to the weight − (2 n − − m −
1) = m − n − N expansion for F s ( n ) g/i . For NNLL resummation, only m = 1 , , l and n , it turns out that only finite number of input is needed. Let us analyse Eq. (4.19)– 32 –n more detail, taking F s ( n ) g/i ( N , (cid:15) ) as an example. To LL accuracy, one need to determine n unkown coefficients c (1 ,l,n ) gi with l = 0 , · · · n −
1. The lowest order of (cid:15) appearing in theansatz F s ( n ) g/i ( N , (cid:15) ) is (cid:15) − n +1 . Therefore expanding (cid:15) to order( − n + 1) + ( n −
1) = − n (4.21)gives us n conditions and is enough to determine the n unknowns at LL accuracy. Toacheive NNLL accuracy, one only needs two more power of (cid:15) expansion, that is we needto know F s ( n ) g/i ( N , (cid:15) ) to order (cid:15) − n +2 . Similar analysis shows that we also need to know F s ( n ) q/i ( N , (cid:15) ) to order (cid:15) − n +2 to achive NNLL accuracy.Having this important information in hand, let us contrentrate on the right hand sideof Eq. (4.15), which generates the necessary (cid:15) poles. The dimensionally regularized partonicFFs (cid:98) d s ( N , (cid:15) ) in MS scheme is purely divergent in (cid:15) and can be determined easily by solvingEq. (4.17) order by order in α s , d ln (cid:98) d s ( N , (cid:15) ) = da s − (cid:98) γ T ( N ) a s ( (cid:15) + (cid:80) ∞ n =0 a sn +1 β n ) . (4.22)where we have traded d ln µ for da s with the help of the (4 − (cid:15) )-dimension beta function, da s d ln µ = − (cid:15)a s − a s ∞ (cid:88) n =0 a sn +1 β n . (4.23)Assuming (cid:98) d s ( N , (cid:15) ) = 1 + (cid:80) ∞ k =1 a ks (cid:98) d sk and (cid:98) γ T ( N ) = (cid:80) ∞ k =0 a sk +1 (cid:98) γ Tk , the solutions can beworked out order by order. For example, at first four orders we have (cid:98) d s = − (cid:98) γ T (cid:15) , (cid:98) d s = − (cid:98) d s · (cid:98) γ T − (cid:98) γ T (cid:15) + β (cid:98) γ T (cid:15) , (cid:98) d s = β (cid:98) d s · (cid:98) γ T + β (cid:98) γ T + β (cid:98) γ T (cid:15) + − (cid:98) d s · (cid:98) γ T − (cid:98) d s · (cid:98) γ T − (cid:98) γ T (cid:15) − β (cid:98) γ T (cid:15) , (cid:98) d s = − β (cid:16) β (cid:98) d s . (cid:98) γ T + β (cid:98) γ T + 2 β (cid:98) γ T (cid:17) (cid:15) + β (cid:98) d s . (cid:98) γ T + β (cid:98) d s . (cid:98) γ T + β (cid:98) d s . (cid:98) γ T + β (cid:98) γ T + β (cid:98) γ T + β (cid:98) γ T (cid:15) + − (cid:98) d s . (cid:98) γ T − (cid:98) d s . (cid:98) γ T − (cid:98) d s . (cid:98) γ T − (cid:98) γ T (cid:15) + β (cid:98) γ T (cid:15) . (4.24)As explained above, to NNLL accuracy, we need to determine the coefficients of the poleterms in F s ( n ) g ( q ) /i to order (cid:15) − n +2 . Since the matching coefficients (cid:98) C sg ( q ) i ( N , (cid:15) ) in (4.15) mustbe finite in the limit of (cid:15) →
0, we need to determine the coefficients of the pole terms in (cid:98) d sn also to order (cid:15) − n +2 . Also recall that (cid:98) d sn are function of N and we are only interestedin the small N limit. For NNLL resummation we only need to keep the terms in (cid:98) d sn with“small- z ” weight up to 2 − n , see Eq. (4.20). We also note that the lowest weight term in (cid:98) γ Tn is 1 − n , corresponding to the leading 1 /N n − pole. Using these information it canbe shown that the required inputs for (cid:98) d sn to achieve NNLL accuracy are β and (cid:98) γ T , , . One– 33 –an check that this is indeed the case frm Eq. (4.24), where explicit examples for n ≤ d ( N , (cid:15) ) to any order in α s , and thefact that the coefficient function (cid:98) C sg ( q ) i is finite in the (cid:15) → F sg ( q ) /i , order by order in α s . In summary the input data we need to achive NNLL accuracy are β ,γ T , γ T , γ T , F s (0) g ( q ) /i = 1 , F s (1) g ( q ) /i to (cid:15) , F s (2) g ( q ) /i to (cid:15) . (4.25)Note that although we have the explicit results for F s (3) g ( q ) /i from direct calculation, they arenot needed for predicting the small- z logarithms at NNLL accuracy. Rather they can beused as a check for our resummation.Following the approach outlined above, we have determined the coefficients of (cid:15) polesin F s ( n ) g ( q ) /i up to (cid:15) − n +2 for n ≤
15. From these coefficients we solve for the LL to NNLLconstants defined in (4.18). Expanding the results in (cid:15) gives us the resummed NNLL seriestruncated at order α s , which should be sufficient for phenomenology. We have checkedthat a truncation of perturbative series at α s leads to a less that 1% relative uncertainty.The analytic expressions for the truncated resummed perturbative series can be found inthe ancillary files of this paper.At LL, we are able to find the generating function for the series (cid:98) C sgq ( N ) | LL = ∞ (cid:88) n =1 a ns N − n C F C n − A A n , (4.26)with A n = ( − n n Γ (cid:0) n + (cid:1) Γ (cid:0) (cid:1) Γ( n + 1) . The series can be resummed analytically, leading to an closed form expression for LLresults, (cid:98) C sgg ( N ) | LL = C A C F (cid:98) C sgq ( N ) | LL = (cid:18) C A a s N (cid:19) − / − . (4.27)It’s interesting to note that Eq. (4.27) coincides with that of the transverse coefficientfunctions for semi-inclusive e + e − annihilation [56, 79].In Fig. 1 and 2 we plot the fixed-order coefficient functions and with different ordersof small- z resummation. We use N f = 5 throughout the calculations. We note that evenat N LO, the effects of resummation is important for z < − . In summary we presented calculations for the unpolarized quark and gluon TMD PDFsand FFs at N LO in QCD. The unpolarized quark TMD PDFs at N LO have already been– 34 – ���� ����� ����� � - ��������������� α � = ����� �� - � ����� ����� ����� � - ���� α � = ����� Figure 1 . Coefficient functions for quark TMD FFs. Shown in the plots are fixed-order results atNLO, NNLO and N LO, as well as adding the higher-order resummation contributions truncatedto order α s . ����� ����� ����� � - ����� α � = ����� �� - � ����� ����� ����� � - � - �������� α � = ����� Figure 2 . Coefficient functions for gluon TMD FFs. Shown in the plots are fixed-order results atNLO, NNLO and N LO, as well as adding the higher-order resummation contributions truncatedto order α s . reported in Ref. [47]. The rest of the calculations are new. Unpolarized quark and gluonTMD PDFs have also been calculated in [48] recently using an independent method, whoseresults are in full agreement with ours .Our calculations for TMD PDFs are based on the method proposed in [47], which inturns is based on decomposition of light-cone correlators into phase space integration ofcollinear splitting amplitudes of different multiplicities. The advantage of this decomposi-tion is that it allows better understanding of the analytic continuation property of TMDPDFs and FFs [50]. Using the analytic continuation prescription of [50], we successfullyobtained the N LO TMD FFs from corresponding TMD PDFs, without the need to com-pute everything from scratch. Our results open the avenue for precision phenomenology ofTMD physics at N LO in perturbative QCD.We also provide threshold and high energy asymptotics of TMD PDFs and FFs throughN LO by expanding the corresponding analytic expressions. The high energy (small- z )limit of TMD FFs features double logarithmic enhancement, in contrast to the single Except for a minor error in an anomalous color factor in [47], which has been corrected in the arXivversion of that paper. – 35 –ogarithmic enhancement of TMD PDFs. We resum the small- z logarithms through NNLLaccuracy using a method proposed in [56]. The resummation leads to better behavedperturbative convergence for z < − .Our method of calculation is general and is not limited to unpolarized distribution.Works towards to polarized TMD distributions at N LO are in progress.
Note added : A few days after this paper appeared, an independent calculation for unpo-larized quark and gluon TMD FFs at N LO was submitted to arXiv [80]. During privatecommunication, the Authors of [80] uncovered a minor error in one of their routine foranalytic continuation. After fixing it, they found full agreement with our results.
Acknowledgments
We thank Duff Neill and Simone Marzani for useful discussion. We also thank MarkusEbert, Bernhard Mistlberger, Gherardo Vita for useful correspondence. This work wassupported in part by the National Science Foundation of China under contract No. 11975200and No. 11935013. T.Z.Y. also want to acknowledge the support from the Swiss NationalScience Foundation (SNF) under contract 200020-175595.
A QCD Beta Function
The QCD beta function is defined as dα s d ln µ = β ( α s ) = − α s ∞ (cid:88) n =0 (cid:16) α s π (cid:17) n +1 β n , (A.1)with [81] β = 113 C A − T F N f ,β = 343 C A − C A T F N f − C F T F N f ,β = (cid:18) C A
27 + 44 C F (cid:19) N f T F + (cid:18) − C A C F − C A
27 + 2 C F (cid:19) N f T F + 2857 C A . (A.2) B Anomalous dimension
For all the anomalous dimensions entering the renormalization group equations of variousTMD functions, we define the perturbative expansion in α s according to γ ( α s ) = ∞ (cid:88) n =0 (cid:16) α s π (cid:17) n +1 γ n , (B.1)where the coefficients for quark are given byΓcusp =4 C F , – 36 –cusp = (cid:18) − ζ (cid:19) C A C F − C F T F N f , Γcusp = (cid:20) (cid:18) ζ − ζ − (cid:19) C A C F + (cid:18) ζ − (cid:19) C F (cid:21) N f T F + (cid:18) − ζ ζ ζ + 4903 (cid:19) C A C F − C F N f T F ,γ S =0 ,γ S = (cid:20)(cid:18) − ζ ζ (cid:19) C A + (cid:18) − ζ (cid:19) T F N f (cid:21) C F ,γ S = (cid:18) − ζ ζ + 6325 ζ
81 + 658 ζ − ζ − ζ − (cid:19) C A C F + (cid:18) ζ − ζ
27+ 4160729 (cid:19) C F N f T F + (cid:18) − ζ − ζ
27 + 48 ζ + 11842729 (cid:19) C A C F N f T F + (cid:18) − ζ − ζ − ζ + 171127 (cid:19) C F N f T F .γ R =0 ,γ R = (cid:20)(cid:18) − ζ (cid:19) C A + 11227 T F N f (cid:21) C F ,γ R = (cid:20) (cid:18) − ζ − ζ
27 + 20 ζ (cid:19) C A N f T F + (cid:18) − ζ ζ + 3196 ζ
81 + 6164 ζ
27+ 77 ζ − ζ − (cid:19) C A + (cid:18) − ζ − ζ + 171127 (cid:19) C F N f T F + (cid:18) − ζ − (cid:19) N f T F (cid:21) C F . (B.2)Since cusp and soft and rapidity anomalous dimensions exhibit Casimir scaling, the corre-sponding anomalous dimensions for gluon could be obtained by multiplying in above with C A /C F .The beam anomalous dimensions do not exhibits Casimir scaling, thus should be listseparately. The beam anomalous dimensions for quark are γ B =3 C F ,γ B = (cid:20)(cid:18) − ζ + 24 ζ (cid:19) C F + (cid:18)
176 + 44 ζ − ζ (cid:19) C A + (cid:18) − − ζ (cid:19) T F N f (cid:21) C F ,γ B = (cid:20) (cid:18) − ζ
27 + 400 ζ ζ + 40 (cid:19) C A C F + (cid:18) ζ − ζ ζ − (cid:19) C F (cid:21) N f T F + (cid:18) ζ ζ − ζ ζ − ζ ζ + 1514 (cid:19) C A C F + (cid:18) ζ − ζ − (cid:19) × C F N f T F + (cid:18) ζ − ζ − ζ + 40 ζ − (cid:19) C A C F + (cid:18) − ζ ζ + 18 ζ +68 ζ + 144 ζ − ζ + 292 (cid:19) C F . (B.3)– 37 –he beam anomalous dimensions for gluon are γ B = 113 C A − T F N f ,γ B = C A (cid:18)
323 + 12 ζ (cid:19) + (cid:18) − C A − C F (cid:19) N f T F ,γ B = C A (cid:18) − ζ − ζ ζ + 553 ζ + 5363 ζ + 83 ζ + 792 (cid:19) + C A N f T F (cid:18) − ζ − ζ − ζ − (cid:19) + 589 C A N f T F − C A C F N f T F +2 C F N f T F + 449 C F N f T F . (B.4)The cusp anomalous dimension Γ cusp can be found in [65]. The beam anomalous dimension γ B is related to the soft anomalous dimension γ S [82] and the hard anomalous dimensions γ H [83–85] by renormalization group invariance condition γ B = γ S − γ H . The rapidityanomalous dimension γ R can be found in [69, 73]. Note that the normalization here differfrom those in [69] by a factor of 1 / C Renormalization Constants
The following constants are needed for the renormalization of zero-bin subtracted [64] TMDPDFs through N LO, see e.g. Ref. [26, 27]. The first three-order corrections to Z B and Z S are Z B = 12 (cid:15) (cid:0) γ B − Γ cusp0 L Q (cid:1) ,Z B = 18 (cid:15) (cid:18) (Γ cusp0 L Q − γ B ) + 2 β (Γ cusp0 L Q − γ B ) (cid:19) + 14 (cid:15) (cid:0) γ B − Γ cusp1 L Q (cid:1) ,Z B = 148 (cid:15) (cid:0) γ B − Γ cusp0 L Q (cid:1) (cid:18) β + 6 β (cid:0) − γ B + Γ cusp0 L Q (cid:1) + (cid:0) − γ B + Γ cusp0 L Q (cid:1) (cid:19) + 124 (cid:15) (cid:18) β (cid:0) − γ B + 4Γ cusp0 L Q (cid:1) + (cid:0) β − γ B + 3Γ cusp0 L Q (cid:1) (cid:0) − γ B + Γ cusp1 L Q (cid:1) (cid:19) + 16 (cid:15) (cid:18) γ B − Γ cusp2 L Q (cid:19) Z S = 1 (cid:15) Γ cusp0 + 1 (cid:15) (cid:0) − γ S − Γ cusp0 L ν (cid:1) ,Z S = 12 (cid:15) (Γ cusp0 ) − (cid:15) (cid:18) Γ cusp0 (3 β + 8 γ S ) + 4(Γ cusp0 ) L ν (cid:19) − (cid:15) (cid:0) γ S + Γ cusp1 L ν (cid:1) + 14 (cid:15) (cid:18) Γ cusp1 + 2(2 γ S + Γ cusp0 L ν )( β + 2 γ S + Γ cusp0 L ν ) (cid:19) ,Z S = 16 (cid:15) (Γ cusp0 ) − (cid:15) (Γ cusp0 ) (cid:0) β + 4 γ S + 2Γ cusp0 L ν (cid:1) + 136 (cid:15) Γ cusp0 (cid:18) β + 45 β (cid:0) γ S + Γ cusp0 L ν (cid:1) +9 (cid:16) Γ cusp1 + 2 (cid:0) γ S + Γ cusp0 L ν (cid:1) (cid:17) (cid:19) + 136 (cid:15) (cid:18) − β Γ cusp0 − β (cid:0) γ S + Γ cusp0 L ν (cid:1) – 38 – β (cid:16) cusp1 + 9 (cid:0) γ S + Γ cusp0 L ν (cid:1) (cid:17) − (cid:20) Γ cusp1 (cid:0) γ S + 9Γ cusp0 L ν (cid:1) +2 (cid:16) (cid:0) γ S (cid:1) + 6Γ cusp0 γ S + 12Γ cusp0 (cid:0) γ S (cid:1) L ν + 6 (Γ cusp0 ) γ S L ν + (Γ cusp0 ) L ν (cid:17) (cid:21)(cid:19) + 118 (cid:15) (cid:18) cusp2 + 3 (cid:0) β (cid:0) γ S + Γ cusp0 L ν (cid:1) + (cid:0) β + 6 γ S + 3Γ cusp0 L ν (cid:1) (cid:0) γ S + Γ cusp1 L ν (cid:1)(cid:1) (cid:19) − γ S + Γ cusp2 L ν (cid:15) . (C.1)Keep in mind that the anomalous dimensions appeared above depends on the flavor, theyshould be replaced by the corresponding values in Sec B. We also remind the reader thatthe renormalization constants are formally identical for TMD PDFs and TMD FFs, thelogarithms appeared above should be replaced by their corresponding values in each case,and we have L ⊥ = ln b T µ b , L ν = ln ν µ , (C.2)with b = 2 e − γ E for both TMD PDFs and TMD FFs.For TMD PDFs, L Q = 2 ln x P + ν , (C.3)while for TMD FFs, L Q = 2 ln P + z ν . (C.4) References [1] D. Boer et al.,
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