Conformal invariance of TMD rapidity evolution

The TMDs are defined as matrix elements of quark or gluon operators with attached light-like gauge links (Wilson lines) going to either +∞ or −∞ depending on the process under consideration. It is well known that these TMD operators exhibit rapidity divergencies due to infinite light-like gauge links and the corresponding rapidity/UV divergences should be regularized. There are two schemes on the market: the most popular is based on CSS [2] or SCET [6] formalism and the second one is adopted from the small-x physics [7, 8]. The obtained evolution equations differ even at the leading-order level and need to be reconciled, especially in view of the future EIC accelerator which will probe the TMDs at values of Bjorken x between small-x and x ∼ 1 regions.


Introduction
The TMDs (for a review, see Ref. [1]) are defined as matrix elements of quark or gluon operators with attached light-like gauge links (Wilson lines) going to either +∞ or −∞ depending on the process under consideration. It is well known that these TMD operators exhibit rapidity divergences due to infinite light-like gauge links and the corresponding rapidity/UV divergences should be regularized. Known schemes are based on CSS or SCET formalism. An alternative scheme is inspired by the small-x physics [2,3]. The obtained evolution equations differ even at the leading-order level and need to be reconciled, especially in view of the future EIC accelerator which will probe the TMDs at values of Bjorken x between small-x and x ∼ 1 regions.
A good starting point is to obtain conformal leading-order evolution equations. In our case, since TMD operators are defined with attached light-like Wilson lines, formally they will transform covariantly under the subgroup of full conformal group which preserves this light-like direction. However, as we mentioned, the TMD operators contain rapidity divergencies which need to be regularized. At present, there is no rapidity cutoff which preserves conformal invariance so the best one can do is to find the cutoff which is conformal at the leading order in perturbation theory.
In higher orders, one should not expect conformal invariance since it is broken by running of QCD coupling. However, if one considers corresponding correlation functions in N = 4 SYM, one should expect conformal invariance. After that, the results obtained in N = 4 SYM theory can be used as a starting point of QCD calculation. Thus, the idea is to find TMD operator conformal in N = 4 SYM and use it in QCD.
In the next sections we report results published in Ref. [7].

Conformal invariance of TMD operators
For simplicity, we will consider the gluon operators with light-like Wilson lines stretching to −∞ in "+" direction. The gluon TMD (unintegrated gluon distribution) is defined as [4] where |P⟩ is an unpolarized target with momentum p ≃ p − (typically proton) and n = ( 1 is a light-like vector in "+" direction. Hereafter we use the notation where [x, y] denotes straight-line gauge link connecting points x and y: To simplify one-loop evolution we multiplied F µν by coupling constant. Since the gA µ is renorminvariant we do not need to consider self-energy diagrams (in the background-Feynman gauge). Note that z − = 0 is fixed by the original factorization formula for particle production [1] (see also the discussion in Ref. [5,6]).
The algebra of full conformal group SO(2, 4) consists of four operators P µ , six M µν , four special conformal generators K µ , and dilatation operator D. It is easy to check that in the leading order the following 11 operators act on gluon TMDs covariantly [7] while the action of operators P + , M +i , and K + do not preserve the form of the operator (2.2). The corresponding group consists of transformations which leave the hyperplane z − = 0 and vector n invariant. Those include shifts in transverse and "+ ′′ directions, rotations in the transverse plane, Lorentz rotations/boosts created by M −i , dilatations, and special conformal transformations . As we noted, infinite Wilson lines in the definition (2.2) of TMD operators make them divergent. As we discussed above, it is very advantageous to have a cutoff of these divergencies compatible with approximate conformal invariance of tree-level QCD. The evolution equation with such cutoff should be invariant with respect to transformations described above.

TMD factorization and one-loop evolution in the Sudakov region
It is convenient to consider as a starting point the simple case of TMD evolution in the socalled Sudakov region corresponding to small longitudinal distances. First, let us specify what we call a Sudakov region. A typical factorization formula for the differential cross section of particle production in hadron-hadron collision is [1,8] dσ where η = 1 2 ln q + q − is the rapidity, D f /h (x, z ⊥ , η) is the TMD density of a parton f in hadron h, and σ ( f f → H) is the cross section of production of particle H of invariant mass m 2 H = q 2 ≡ Q 2 in the scattering of two partons. (One can keep in mind Higgs production in the approximation of point-like gluon-gluon-Higgs vertex). The Sudakov region is defined by Q ≫ q ⊥ ≫ 1GeV since at such kinematics there is a double-log evolution for transverse momenta between Q and q ⊥ . In the coordinate space, TMD factorization (3.1) looks like where O i j ,Õ i j , and F i,a are defined in Ref. [7]. Here p A = s 2 n + ⊥ . As we mentioned, TMD operators exhibit rapidity divergencies due to infinite light-like gauge links. The "small-x style" rapidity cutoff for longitudinal divergencies is imposed as the upper limit of k + components of gluons emitted from the Wilson lines. As we will see below, to get the conformal invariance of the leading-order evolution we need to impose the cutoff of k + components of gluons correlated with transverse size of TMD in the following way: Similarly, the operatorÕ is defined with with the rapidity cutoff for β integration imposed as θ σ  where we suppress arguments z 1 ⊥ and z 2 ⊥ since they do not change during the evolution in the Sudakov regime. The first two terms in the kernel K come from the "production" diagram in Fig.  1a while the last two terms from "virtual" diagram in Fig. 1b. The approximations for diagrams in Fig. 1 leading to Eq. (3.5) are valid as long as k + ≫ z + 12 z 2 12 ⊥ which gives the region of applicability of Sudakov-type evolution.
Evolution equation (3.4) can be easily integrated using Fourier transformation (see [7] for details) and one easily obtains where we introduced notationᾱ s ≡ α s N c 4π . It should be mentioned that the factor 4γ E is "schemedependent": if one introduces to α-integrals smooth cutoff e −α/a instead of rigid cutoff θ (a > α), the value 4γ E changes to 2γ E . It is easy to see that the r.h.s. of Eq. (3.6) transforms covariantly under all transformations (2.3) except Lorentz boost generated by M +− (see [7] for details).

Conclusions and Outlook
The first result is that the 11-parameter subgroup of SO(2, 4) formed by generators (2.3) formally leaves TMD operators invariant.
The second result is related to the fact that conformal invariance is violated by the rapidity cutoff (even in N = 4 SYM). We have studied the TMD evolution in the Sudakov region of intermediate x and demonstrated that the rapidity cutoff used in small-x literature preserves all generators of our subgroup except the Lorentz boost which is related to the change of that cutoff.
Our main outlook is to try to connect to small-x region, first in N = 4 SYM and then in QCD. As we mentioned above, although the TMD evolution in a small-x region is conformal with respect to SL(2,C) group, and our evolution (3.6) is also conformal (albeit with respect to different group of which SL(2,C) is a subgroup), the transition between Sudakov region and small-x region is described by rather complicated interpolation formula [9] which is not conformally invariant. Our hope is that in a conformal theory one can simplify that transition using the conformal invariance requirement. The study is in progress.