A Short Review on Recent Developments in TMD Factorization and Implementation

Table 1

Summary of available perturbative calculations of quark TMD distributions and their leading matching at small-b.

Name	Function	Leading matching function	Twist of leading matching	Maximum known order of coef. function	Ref.	Mix with gluon
unpolarized			tw-2	NNLO ()	[4, 5]	yes
Sivers			tw-3	NLO ()	[6–14]	yes
helicity			tw-2	NLO ()	[6, 15–17]	yes
worm-gear T		, ,	tw-2/3	LO ()	[18]  [6]	yes
transversity			tw-2	NNLO()	[19]	no
Boer-Mulders			tw-3	LO ()	[6]	no
worm-gear L		,	tw-2/3	LO ()	[18]  [6]	no
pretzelosity		–	tw-4	–	–	–

The calculation is done in the momentum space. The result is given for the moments of distribution.
The pretzelosity can in principle be a twist-2 observable; however its twist-2 matching coefficient has been found to be zero up to NNLO [19]. Therefore one can conjecture that pretzelosity is actually a twist-4 observable. Some arguments in favor of this can also be found in [20].
The quark Sivers function at NLO has a long story [8–13]. A complete calculation is now available in [14].