One-loop matching for gluon lattice TMDs

Transverse-momentum-dependent parton distributions (TMDs) can be calculated from first principles by computing a related set of Euclidean lattice observables and connecting them via a factorization formula. This work focuses on the leading-power factorization formula connecting the lattice quasi-TMD and continuum Collins TMD for gluons. We calculate the one-loop gluon matching coefficient, which is known to be independent of spin and exhibits no mixing with quarks. We demonstrate that this coefficient satisfies Casimir scaling with respect to the quark matching coefficient at one-loop order. Our result facilitates reliable lattice QCD calculations of gluon TMDs.

At leading power, the factorization formula connecting the continuum limit of quasi-TMDs (lattice) and the physical Collins TMD reads [39]: where f is a continuum TMD,f is the continuum limit of a lattice TMD, C κ i is a perturbative matching coefficient, and the exponential term evolves the so-called Collins-Soper (CS) scale ζ →ζ using the CS kernel γ κ i ζ . The other parameters include x, which is the fraction of the hadron h's longitudinal momentum P that the parton of type i carries; the Fourier-conjugate to the parton's transverse momentum b T ; and a renormalization scale µ.
Here i refers to either a gluon (i = g) or specific quark flavor (i = u, d, s, . . .), where κ i = q is universal for all quarks, but differs from κ g = g for gluons. No quark-gluon or flavor mixing occurs in the factorization relation, which simplifies calculation of gluon TMDs from the lattice [39]. The definition of each of the TMDs in eq. (1.1) will be discussed in detail in section 1.1.
The quasi-to-Collins matching coefficient C q is known for quarks at one loop [26,28,29], for logarithmic terms at two loops [32], and for all next-to-leading-logarithmic (NLL) terms [39]. A key missing ingredient is the matching coefficient for gluons. Gluons are known to be responsible for a large part of internal proton dynamics and would be useful to understand from first principles. In this work, we calculate the one-loop gluon matching coefficient C g .

Definitions of lattice and physical continuum TMDs
A TMD f i/h for gluon or quark flavor i generally is a product of three components: a beam function (hadronic matrix element), a soft function (vacuum matrix element), and their renormalization. The beam and soft matrix elements are defined in terms of Wilson lines: where γ is the Wilson line's path and R is its color representation. Here we focus on gluons, taking R = g for the adjoint representation. Beam functions involve staple-shaped  Figure (a) is adapted from ref. [39] and (b) is adapted from ref. [48].
line paths as shown in figure 1, (1.3) A generic notation for gluon beam functions is given by Here, G µν (b) is the gluon field strength tensor. Gluon fields are spatially separated by b, which is Fourier-conjugate to the struck parton's momentum, h denotes a parent hadron with momentum P , is the UV regulator, and ηv and δ characterize the Wilson line path. See refs. [46,47] for the decomposition of gluon TMDs into independent spin structures. Note that the correlator Ω characterizes the TMDs in b-space. When we define a full TMD in terms of beam and soft functions, we generally make a Fourier transform P · b → x and write where B indicates the correlator in these new coordinates. The gluon soft function is defined as where the trace is over color, and the soft Wilson line is We can now define all off-lightcone gluon TMD schemes in terms of the exact same B and S: (1.8) where the ellipses and Z g UV account for a scheme-dependent renormalization. 1 Each scheme is characterized by a specific choice of the parameters b µ , ηv µ , P µ , and δ µ appearing in the beam and soft functions, as well as the precise nature and order of the lightcone and UV limits in eq. (1.8). The longitudinal staple length η distinguishes physical TMDs (η = ±∞) from lattice TMDs (finite η).
The remainder of section 1.1 specifies the order of limits and beam/soft function arguments that define three major TMD schemes: the continuum Collins TMD, the lattice quasi-TMD, and the continuum Large Rapidity (LR) TMD. We summarize this information in table 1, which also includes values taken on by the four-vectors b µ , v µ , δ µ , P µ .
Collins scheme [6]. The continuum Collins scheme uses spacelike Wilson lines with directions parametrized by the rapidities y A and y B . (See appendix A for our lightcone coordinate conventions.) The Collins gluon TMD for a hadron h moving along n a with rapidity y P is , (1.10) where |b ⊥ | = b T and the beam function is defined as with Ω +ρ+σ Quasi-TMDs [26][27][28][29][30][31][32][33][34][35]39]. Collins TMDs cannot be directly discretized and computed on the lattice, as their Wilson lines' explicit dependence on time induces a sign problem. Instead, one defines numerically-tractable quasi-TMD beam functions using finite-length Wilson lines lying on equal-time paths. The lattice gluon quasi-TMD takes the form Note that for the MHENS scheme, renormalization should be performed before the Fourier transform as Z g UV depends on the longitudinal component of b µ .

Collins TMD (continuum)
Quasi-TMD (lattice) where the bare quasi-beam function reads as (1.14) Here, α, β, ρ and σ are generic Lorentz indices, and N αρβσ is a trivial normalization factor. The factor Z g uv (a,μ, y n − y B ) renormalizes UV divergences from lattice regularization, and Z g uv (µ,μ) matches the result to the MS scheme. The second equality in eq. (1.13) holds for largeP z , and the variableζ is (1.15) Large Rapidity (LR) scheme To facilitate deriving the factorization relation between the quasi-and Collins TMDs, Ref. [39] introduced the Large Rapidity (LR) scheme, which differs from the Collins scheme only by its order of → 0 and y B → −∞ limits. Using Lorentz invariance and the largeP z expansion (−y B 1), one can demonstrate that whenζ = ζ or yP = y P − y B , the quasi-and LR TMDs are equivalent at leading order under a large rapidity expansion [39], i.e., where O(y k B e y B ) are exponentially suppressed contributions.

Calculation of one-loop gluon matching coefficient
We now compute the one-loop quasi-to-Collins matching coefficient C g for gluons. To do so, we actually carry out a simpler equivalent matching calculation: between the LR and Collins schemes. In the course of proving the quasi-to-Collins factorization formula in eq. (1.1), ref. [39] showed that the quasi-to-Collins and LR-to-Collins matching coefficients are identical. Note that this matching calculation focuses on beam functions: the quasi-soft function has been chosen to reproduce the Collins soft function as |η| → ∞. Let us compute the matrix element of the Collins beam function for a free external gluon state with momentum p µ = (p + , 0, 0 ⊥ ), transverse polarization vectors α ⊥ and β ⊥ , as well as color indices a and b. We work in Feynman gauge and dimensional regularization with space-time dimension d = 4 − 2 . As we are interested in the leading power gluon TMDs, the Lorentz indices ρ, σ are both transverse, and for both the LR and Collins schemes µ = ν = +. In our calculation we leave the choice of ρ and σ unspecified, to make clear that our result applies to all spin-dependent gluon TMDs.
At tree-level, we must compute the diagrams in figure 2a, which give where the second term in the square bracket comes from the exchange symmetry of bosonic particles. Taking a Fourier transform b z → xp z , the tree-level gluon beam function becomes xB where we moved the normalization factor 1/x to the left hand side for clarity. We emphasize that to find the matching coefficient, we need only examine terms proportional to δ(1 − x) since the factorization relation does not involve a convolution in x.
The one-loop Feynman diagrams that contribute to the beam function are shown in figure 2. According to Ref. [39], only diagrams containing a rapidity divergence contribute to the matching, significantly simplifying our task. Only figures 2c and 2e could exhibit rapidity divergences, as the other diagrams have Lorentz-covariant loop integrals, which remain unaffected by the large rapidity limit y B → −∞ (except for the transformation of tensor structures). Using the Feynman rules in appendix B, we express figures 2c and 2e as and Ω +ρ+σ(1) g/g,e where the Wilson line takes a staple-shaped path: and we have suppressed the contributions from diagrams with external gluon lines exchanged (α ↔ β, p → −p) for conciseness. We also use a coupling in the MS scheme, which introduced the factors ι = (e γ E /(4π)) .
Simplifying the above, where we obtain the Wilson line propagators [n B · (p − k) ± iδ] −1 by integrating over the path parameter s. Note that in the LR and Collins scheme that the staple Wilson line has infinite extent. We introduce the infinitesimal imaginary part ±iδ to ensure that the path integral contribution from s = −∞ properly vanishes in Feynman gauge.
In the y B → −∞ limit, the rapidity divergence is in the first line of eq. (2.6). The second line is suppressed by a factor of e 2y B k 2 T /(p + ) 2 compared to the first line, as k T = |k ⊥ | k + , p + in the power counting, and α, β, ρ, σ =⊥. The last line in eq. (2.6) is Lorentz covariant and so is not affected by y B → −∞; it thus does not contribute to the matching.
After taking a Fourier transform b − → xp + , the first line of eq. (2.6) becomes xB The first term of eq. (2.8) is a plus function in x; it is unaffected by the y B → −∞ limit as long as x = 1. The second term contains the contribution to the matching coefficient.
Likewise, after a Fourier transform b − → xp + , the second line of eq. (2.6) becomes xB Cρσ (1) g/g,e (x, b T , , y P − y B ) (2.9) The first term is again a plus function. The second term vanishes under parity transforms k ± → −k ± . Therefore, the diagrams in figure 2e do not contribute to the matching. All that remains to compute is the term proportional to δ(1 − x) in eq. (2.8). After a change of variables k → p − k, the next step is to evaluate the integral (2.10) To simplify the integral we can make a change of variable k − → e −y B k − and k + → e y B k + which corresponds to a boost to equal-time with k z = (k + − k − )/ √ 2, and yields Taking → 0 before y B → −∞. Here, we can directly integrate: Again the 1/ pole is a UV divergence, and is subtracted for the MS renormalized result. Finally we note that the one-loop Collins soft function is the same for quarks and gluons, so can be obtained from the quark result [29] with the replacement of C F → C A , Here we use the shorthand S g . This result is not affected by the order of → 0 and y B → −∞ limits, i.e., the part of the Wilson line diagrams associated with the rapidity divergence. The wellknown (generalized) Casimir scaling of the cusp anomalous dimension and CS kernel have been validated to four-loop order [49][50][51][52][53][54]. The fact that the full one-loop matching and cusp-anomalous dimension obey Casimir scaling, implies that the full tower of next-toleading-logarithms for C g will obey Casimir scaling as well [39]. This hints at a relationship for quasi-TMD matching coefficients at higher orders.

Conclusion
In this paper, we calculated the one-loop gluon matching coefficient C g in the factorization formula relating quasi and Collins TMDs, marking an important step towards calculating gluon TMDs from lattice QCD. As the matching coefficient is independent of spin structure and there is no quark-gluon mixing, our result enables lattice calculation of all eight leadingtwist gluon TMDs. Given the encouraging first results for quark TMDs, it is reasonable to expect that lattice QCD will provide highly useful nonperturbative input for gluon TMDs and their Collins-Soper evolution, neither of which has yet been extracted from experiments.

Acknowledgments
This work is dedicated to Markus Ebert, who has become more industrious.

A Lightcone coordinate conventions
We work in a frame where the hadron momentum P is close to the lightlike unit vectors which obey n 2 a = n 2 b = 0 and n a · n b = 1. We can decompose a four-vector p µ is where p ± = (p 0 ±p z )/ √ 2 and p µ ⊥ = (0, p x , p y , 0) = (0, p T , 0). Here p µ ⊥ is a Minkowski vector, and p T is the corresponding Euclidean vector with magnitude p T ≡ ( p 2 T ) 1/2 = (−p 2 ⊥ ) 1/2 .