A short review on recent developments in TMD factorization and implementation

TL;DR

This paper reviews recent theoretical and phenomenological advances in TMD factorization and evolution, highlighting new calculations and tools that are crucial for collider physics and future experiments like the EIC.

Contribution

It provides a comprehensive overview of the latest developments in TMD factorization, including recent perturbative QCD calculations and conceptual tools, accessible to both experts and newcomers.

Findings

Recent progress in TMD factorization theory

New perturbative QCD calculations for TMDs

Potential insights from upcoming collider data

Abstract

In the latest years the theoretical and phenomenological advances in the factorization of several collider processes using the transverse momentum dependent distributions (TMD)has greatly increased. I attempt here a short resume of the newest developments discussing also the most recent perturbative QCD calculations. The work is not strictly directed to experts in the field and it wants to offer an overview of the tools and concepts which are behind the TMD factorization and evolution. I consider both theoretical and phenomenological aspects, some of which have still to be fully explored. It is expected that actual colliders and the Electron Ion Collider (EIC) will provide important information in this respect.

Tables

Table 1. Table 1: Summary of available perturbative calculations of quark TMD distributions and their leading matching at small-b.

		Leading	Twist of	Maximum		Mix
Name	Function	matching	leading	known order	Ref.	with
		function	matching	of coef.function		gluon
unpolarized	$f_1(x,𝒃)$	$f_1$	tw-2	NNLO ($a_s^2$)	Gehrmann et al. (2014); Echevarria et al. (2016b)	yes
Sivers	$f_{1T}^{⟂}(x,𝒃)$	$T$	tw-3	NLO ($a_s^1$)	Boer et al. (2003); Ji et al. (2006a, b); Koike et al. (2008); Kang et al. (2011); Sun and Yuan (2013); Dai et al. (2015); Scimemi and Vladimirov (2018c); Scimemi et al. (2019)***	yes
helicity	$g_{1L}(x,𝒃)$	$g_1$	tw-2	NLO ($a_s^1$)	Bacchetta and Prokudin (2013); Gutierrez-Reyes et al. (2017); Buffing et al. (2018a); Scimemi and Vladimirov (2018c)	yes
worm-gear T	$g_{1T}(x,𝒃)$	$g_1$, $T$, $\Delta T$	tw-2/3	LO ($a_s^0$)	Kanazawa et al. (2016)* Scimemi and Vladimirov (2018c)	yes
transversity	$h_1(x,𝒃)$	$h_1$	tw-2	NNLO($a_s^2$)	Gutierrez-Reyes et al. (2018a)	no
Boer-Mulders	$h_1^{⟂}(x,𝒃)$	$\delta T_{ϵ}$	tw-3	LO ($a_s^0$)	Scimemi and Vladimirov (2018c)	no
worm-gear L	$h_{1L}^{⟂}(x,𝒃)$	$h_1$, $\delta T_g$	tw-2/3	LO ($a_s^0$)	Kanazawa et al. (2016)* Scimemi and Vladimirov (2018c)	no
pretzelosity**	$h_{1T}^{⟂}(x,𝒃)$	–	tw-4	–	–	–

Table 2. Table 2: Notation for TMD anomalous dimensions used in the literature.

	rapidity evolution scale	TMD anomalous dimension	cusp anomalous dimension	vector form factor anomalous dimensions	rapidity anomalous dimension
Echevarria et al. (2013b, 2016c); Scimemi and Vladimirov (2018a, b)	$\zeta$	${\gamma }_F$	$\Gamma$	${\gamma }_V$	$𝒟$
Collins (2013); Aybat and Rogers (2011)	$\zeta$	${\gamma }_F\left(={\gamma }_D\right)$	$\frac{1}{2}{\gamma }_K$	$-{\gamma }_F(g\left(\mu \right);1)$	$-\frac{1}{2}\tilde{K}$
Becher and Neubert (2011); Becher et al. (2012); Gehrmann et al. (2014)	–	–	${\Gamma }_{cusp}$	$2{\gamma }^q$	$\frac{1}{2}{F}_{f\overline{f}}$
Chiu et al. (2012a)	${\nu }^2$	${\gamma }_{\mu }^{f_{⟂}}$	${\Gamma }_{cusp}$	–	$-\frac{1}{2}{\gamma }_{\nu }^{f_{⟂}}$

Equations

$$\frac{d\sigma }{d{Q}^2}$$

$$\frac{d\sigma }{d{Q}^{2}d{q}_T^{2}dy}$$

$$x_1=\frac{\sqrt{Q^2+q_T^2}}{\sqrt{s}}{e}^y{x}_2=\frac{\sqrt{Q^2+q_T^2}}{\sqrt{s}}{e}^{-y}.$$

$$q_T\ll Q,$$

$$\frac{d\sigma }{d{Q}^{2}d{q}_T^{2}dy}$$

$$\displaystyle S(\mbox{\boldmath$k$})$$

$$S_n^T=T_{n(\bar{n})}{S}_n,{\tilde{S}}_{\bar{n}}^T={\tilde{T}}_{n(\bar{n})}{\tilde{S}}_{\bar{n}},$$

$$S_n(x)=P\exp \left[ig{\int }_{-\infty }^{0}dsn\cdot A(x+sn)\right],$$

$$T_n(x_T)=P\exp \left[ig{\int }_{-\infty }^{0}d\tau {\mathbf{l}}_{\perp }\cdot {\mathbf{A}}_{\perp }({\infty }^{+},0^{-},{\mathbf{x}}_T+{\mathbf{l}}_{\perp }\tau )\right],$$

$$T_{\bar{n}}(x_T)=P\exp \left[ig{\int }_{-\infty }^{0}d\tau {\mathbf{l}}_{\perp }\cdot {\mathbf{A}}_{\perp }(0^{+},{\infty }^{-},{\mathbf{x}}_T+{\mathbf{l}}_{\perp }\tau )\right],$$

$${\tilde{S}}_{\bar{n}}(x)=P\exp \left[-ig{\int }_0^{\infty }ds\bar{n}\cdot A(x+\bar{n}s)\right],$$

$${\tilde{T}}_n(x_T)=P\exp \left[-ig{\int }_0^{\infty }d\tau {\mathbf{l}}_{\perp }\cdot {\mathbf{A}}_{\perp }({\infty }^{+},0^{-},{\mathbf{x}}_T+{\mathbf{l}}_{\perp }\tau )\right],$$

$${\tilde{T}}_{\bar{n}}(x_T)=P\exp \left[-ig{\int }_0^{\infty }d\tau {\mathbf{l}}_{\perp }\cdot {\mathbf{A}}_{\perp }(0^{+},{\infty }^{-},{\mathbf{x}}_T+{\mathbf{l}}_{\perp }\tau )\right].$$

$$\tilde{S}({\mathbf{L}}_{\mu },{\mathbf{L}}_{\sqrt{{\delta }^{+}{\delta }^{-}}})={\tilde{S}}^{\frac{1}{2}}({\mathbf{L}}_{\mu },{\mathbf{L}}_{{\delta }^{+}/\nu }){\tilde{S}}^{\frac{1}{2}}({\mathbf{L}}_{\mu },{\mathbf{L}}_{\nu {\delta }^{-}}),$$

$${\mathbf{L}}_{X}\equiv\hbox{ln}(X^{2}\mbox{\boldmath${\mbox{\boldmath$b$}_{T}}$}^{2}e^{2\gamma_{E}}/4).$$

$$\mathcal{D}=\frac{1}{2}\frac{dln\tilde{S}}{d{\mathbf{l}}_{\delta }}{|}_{ϵ-\text{finite}}.$$

$$\displaystyle O^{bare}_{q}(x,\mbox{\boldmath$b$}_{T})$$

$$\displaystyle\Phi_{f\leftarrow N}(x,\mbox{\boldmath$b$}_{T})$$

$$\displaystyle O_{q}(x,\mbox{\boldmath$b$}_{T},\mu,\zeta)$$

$$\displaystyle F_{f\leftarrow N}(x,\mbox{\boldmath$b$}_{T};\mu,\zeta)$$

$$\displaystyle R_{f}(\zeta,\mu)=\frac{\sqrt{S(\mbox{\boldmath$b$}_{T})}}{\textbf{Zb}},\quad f=q,g,$$

$$\displaystyle R_{f}(\zeta,\mu)\bigg{|}_{\delta\text{-reg.}}=\frac{1}{\sqrt{S(\mbox{\boldmath$b$}_{T};\zeta)}},$$

$$\displaystyle R^{f}(\zeta,\mu)\bigg{|}_{JCC}=\sqrt{\frac{\tilde{S}(y_{n},y_{c})}{\tilde{S}(y_{c},y_{\bar{n}})\tilde{S}(y_{n},y_{\bar{n}})}}.$$

$$\displaystyle O_{f}(x,{\mbox{\boldmath$b$}_{T}};\mu,\zeta)=\sum_{f^{\prime}}C_{f\leftarrow f^{\prime}}(x,{\mbox{\boldmath$b$}_{T}};\mu,\zeta,\mu_{b})\otimes O_{f^{\prime}}(x,\mu_{b})+\mathcal{O}\left(\frac{{\mbox{\boldmath$b$}_{T}}}{B_{T}}\right),$$

$$\displaystyle F_{f\leftarrow N}(x,{\mbox{\boldmath$b$}_{T}};\mu,\zeta)=\sum_{f^{\prime}}C_{f\leftarrow f^{\prime}}(x,{\mbox{\boldmath$b$}_{T}};\mu,\zeta,\mu_{b})\otimes f_{f^{\prime}\leftarrow N}(x,\mu_{b})+\mathcal{O}\left(\frac{{\mbox{\boldmath$b$}_{T}}}{B_{T}}\right),$$

$$f_{q\leftarrow N}(x)=\frac{1}{2}\sum _X\int \frac{d{\xi }^{-}}{2\pi }{e}^{-ix{p}^{+}{\xi }^{-}}⟨N|\left\{T{\left[{\bar{q}}_i{\tilde{W}}_n^T\right]}_a\left(\frac{{\xi }^{-}}{2}\right)|X⟩{\gamma }_{ij}^{+}⟨X|\bar{T}{\left[{\tilde{W}}_n^{T†}{q}_j\right]}_a\left(-\frac{{\xi }^{-}}{2}\right)\right\}|N⟩.$$

$$\displaystyle\mu^{2}\frac{d}{d\mu^{2}}O_{f}(x,\mbox{\boldmath$b$}_{T})=\frac{1}{2}\gamma^{f}(\mu,\zeta)O_{f}(x,\mbox{\boldmath$b$}_{T})$$

$$\displaystyle\zeta\frac{d}{d\zeta}O_{f}(x,{\mbox{\boldmath$b$}_{T}})=-\mathcal{D}^{f}(\mu,\mbox{\boldmath$b$}_{T})O_{f}(x,{\mbox{\boldmath$b$}_{T}}).$$

$$\frac{{\mathcal{D}}^q}{{\mathcal{D}}^g}=\frac{C_F}{C_A}=\frac{N_c^2-1}{2N_c^2},$$

$$\displaystyle\mu^{2}\frac{d}{d\mu^{2}}\left(-\mathcal{D}^{f}(\mu^{2},\mbox{\boldmath$b$}_{T})\right)=\zeta\frac{d}{d\zeta}\left(\frac{\gamma^{f}(\mu,\zeta)}{2}\right)=-\frac{\Gamma_{cusp}^{f}}{2}.$$

$${\gamma }^f={\Gamma }_{cusp}^f{\mathbf{l}}_{\zeta }-{\gamma }_V^f,$$

$${\mathbf{l}}_X\equiv ln\left(\frac{{\mu }^2}{X}\right).$$

$${\mu }_b^2\frac{d}{d{\mu }_b^2}{O}_f(x,{\mu }_b)=\sum _{f^{\prime }}{P}_{f\leftarrow f^{\prime }}(x)O_{f^{\prime }}(x,{\mu }_b),$$

$$\displaystyle\zeta\frac{d}{d\zeta}C_{f\leftarrow f^{\prime}}(x,\mbox{\boldmath$b$}_{T};\mu,\zeta)=-\mathcal{D}^{f}(\mu,\mbox{\boldmath$b$}_{T})C_{f\leftarrow f^{\prime}}(x,\mbox{\boldmath$b$}_{T};\mu,\zeta),$$

Full text

I Introduction

The knowledge of the structure of hadrons is a leitmotiv for the study of quantum chromodynamics (QCD) for decades. Apart from the notions of quarks and gluons (we call them generically ”partons” in the following), the natural question is how the momenta of these particles are distributed inside the hadrons and how the spin of hadrons is generated. Phenomenologically it is possible to access at this problem only in some particular kinematical conditions, as provided for instance in experiments like (semi-inclusive) deep inelastic scattering, vector and scalar boson production, $\ell^{+}\ell^{-}\rightarrow$ hadrons or jets. I review the basic principle which support this investigation. Let us consider, to start with, the cross section for di-lepton production in a typical Drell-Yan process $pp\rightarrow\ell^{+}\ell^{-}+X$ where $X$ includes all particles which are not directly measured. The cross section for this process can be written formally as

[TABLE]

where $Q^{2}$ is the virtual di-lepton invariant mass, $x_{i}$ are the parton momenta fraction along a light-cone direction or Bjorken variables and $f$ are the parton distribution functions (PDF). The r.h.s. of eq. (1) assumes several notions which, nowadays, can be found in textbooks. In fact a central hypothesis is a clear energy separation between the di-lepton invariant mass and the scale at which QCD cannot be treated perturbatively any more (we call it the hadronization scale $\Lambda\sim{\cal O}(1)$ GeV), that is $Q^{2}\gg\Lambda^{2}$. Given this, one can factorize the cross section in a perturbatively calculable part ${\cal H}$ and the rest. Formula (1) represents just a first term of an ”operator product expansion” of the cross section. The price to pay for this separation is the introduction of a factorization scale $\mu$ which can be used to resum logarithms in combination with renormalization group equations Gribov and Lipatov (1972); Dokshitzer (1977); Altarelli and Parisi (1977). Another aspect, which is remarkable, is that the non-perturbative part of the cross section can be also expressed as the product of two parton distribution functions. This fact has two main consequences: on the one hand, all the non-perturbative information of the process is included in the PDFs; on the other hand, the partons belonging to different hadrons are completely disentangled. In these conditions so the longitudinal momenta of quarks and gluons can be reconstructed non-perturbatively and this fact has given rise to a large investigation whose review goes beyond the purpose of this writing.

The ideal description of the process in eq. (1) however becomes more involved in the case of more differential cross sections  Parisi and Petronzio (1979); Collins and Soper (1982); Collins et al. (1985). So, for instance, one can wonder whether a formula like

[TABLE]

has any physical consistency111 I use the notation {\mbox{\boldmathb}_{T}} for 2-dimensional impact parameter, -b_{T}^{2}={{\mbox{\boldmathb}_{T}}^{2}}\geq 0, $s$ is the center of mass energy of the process,

$x_{1}=\frac{\sqrt{Q^{2}+q_{T}^{2}}}{\sqrt{s}}e^{y}\quad x_{2}=\frac{\sqrt{Q^{2}+q_{T}^{2}}}{\sqrt{s}}e^{-y}.$

. The answer to this question is necessarily more complex then in the case of eq. (1) for the simple fact that a new kinematic scale, $q_{T}$, the transverse momentum of the di-lepton pair, has now appeared. In this article I will concentrate on the description of the case

[TABLE]

which is interesting for a number of observables. The restriction to this kinematical regime represents also a limitation of the present approach which should be overcome with further studies.

The study of factorization Ji et al. (2005); Becher and Neubert (2011); Collins (2013); Echevarria et al. (2012); Chiu et al. (2012a); Echevarria et al. (2013a) has lead finally to the conclusion that actually eq. (2) in not completely correct because the cross section for these kind of processes should instead be of the form

[TABLE]

with $\zeta_{1}\zeta_{2}=Q^{4}$ and $\zeta_{i}$ being the rapidity scales. Formula (4) shows explicitly that the TMD functions $F$ contain non-perturbative QCD information different from the usual PDF, while they still allow to complete disentangle QCD effects coming from different hadrons. These new nonperturbative QCD inputs can be written in terms of well defined matrix elements of field operators which can be extracted from experiments or evaluated with appropriate theoretical tools. These objectives require some discussion, which I partially provide in this text.

The scale $\zeta$ is the authentic key stone of the TMD factorization. Its origin is different from the usual factorization scale $\mu$ and because of this it is allowed to perform a special resummation for this scale. This leads to the fact that a consistent and efficient implementation of the $(\mu,\zeta)$ evolution is crucial for the prediction and extraction of TMDs from data. A possible implementation of the TMD evolution is historically provided by Collins-Soper-Sterman (CSS) Parisi and Petronzio (1979); Collins and Soper (1982); Collins et al. (1985). However a complete discussion of more efficient alternatives has started more recently Aybat and Rogers (2011); Echevarria et al. (2013b); D’Alesio et al. (2014); Scimemi and Vladimirov (2018a, b). The point is that the rapidity scale evolution has both a perturbative and nonperturbative input, as it is actually provided by (derivatives of) an operator matrix element (the so called soft-function). An efficient implementation and scale choice so should separate as much as possible the nonperturbative inputs with different origin inside the cross-sections. This target is not completely realized with the CSS implementation, while it can be achieved with the $\zeta$-prescription discussed in the text. This discussion is also relevant for multiple reasons. In fact various orders in perturbation theory are available already for unpolarized and polarized distribution and, in the future, one expects more results in this respect for many polarized distributions. When dealing with several perturbative orders, the convergence of the perturbative series can be seriously undermined by an inappropriate choice of scales, and this is a well known problem that can affect the theoretical error of any result. A more subtle issue comes from the fact that the evolution corrections can also be of nonperturbative nature. It would be certainly clarifying a scheme in which the nonperturbative effects of the evolution are clearly separated from the instrinsic nonperturbative TMD effects. Such a request results to be important when several extraction of TMD from data are compared and also when a complete nonperturbative evaluation of TMD can be provided.

In the rest of this review I will try to give an idea on how all these problems can be consistently treated, which can be useful also to explore new and more efficient solutions.

II Factorization

The factorization of the cross sections into TMD matrix elements has been provided by several authors and it has been object of many discussions Parisi and Petronzio (1979); Collins and Soper (1982); Collins et al. (1985); Ji et al. (2005); Becher and Neubert (2011); Collins (2013); Echevarria et al. (2012); Chiu et al. (2012a); Echevarria et al. (2013a). We briefly review the main ideas here for the case of Drell-Yan. The process is characterized by two initial hadrons which come from opposite collinear directions and produce two leptons in the final state plus unmeasured radiation. We identify collinear (anti-collinear) light-cone directions $n$ ($\bar{n}$) and $n^{2}=\bar{n}^{2}=0$, $n\cdot\bar{n}=1$ for the momentum of colliding particles. The momentum of collinear particles is $p=(p^{+},p^{-},p_{\perp})$ with $n\cdot p=p^{-},$ $\bar{n}\cdot p=p^{+}$ and $p_{\perp}=p-(n\cdot p)\bar{n}-(\bar{n}\cdot p)n$ and $p^{+}\gg p_{\perp}\gg p_{-}$. The momenta of collinear particles are characterized by the scaling $p\simeq Q(1,\lambda^{2},\lambda)$ where $Q$ is the di-lepton invariant mass and $\lambda$ is a small parameter $\lambda\sim\Lambda_{QCD}/Q$ being $\Lambda_{QCD}$ the hadronization scale. A reversed scaling of momentum is valid for anti-collinear particles, say $p\simeq Q(\lambda^{2},1,\lambda)$. The soft radiation which entangles collinear and anti-collinear particles is homogeneous in momentum distribution (its momentum scales as $p\sim Q(\lambda,\lambda,\lambda)$) and can be distinguished from the collinear radiation only for a different scaling of the components of the momenta. Given this, it is natural to divide the hadronic phase space in regions as in fig. 1. In this picture, the collinear and soft regions are necessarily separated by rapidity and they all share the same energy $p^{2}\sim\Lambda^{2}$.

II.1 Soft interactions and soft factor

Because the soft radiation is not finally measured, its interactions should be included (and resummed) in the collinear parts, which become sensitive to a rapidity scale which acts in a way similar to the usual factorization scale. It is possible to define the soft radiation through a ”soft factor”, that is, by an operator matrix element,

[TABLE]

where we have used the Wilson line definitions Idilbi and Scimemi (2011a, b); Garcia-Echevarria et al. (2011) appropriate for a Drell-Yan process,

[TABLE]

The direct calculation of the soft factor is all but trivial and the way the calculation is performed can influence directly the final formal definition of the transverse momentum dependent distribution used by different authors. In fact a simple perturbative calculation shows that in the soft factor there are divergences which cannot be regularized dimensionally (say, they are not explicitly ultraviolet (UV) or infrared (IR)) which occur when the integration momenta are big and aligned on the light cone directions. The divergences that arise in this configuration of momenta are generically called rapidity divergences and regulated by a rapidity regulator. One can understand the necessity of a specific regulator observing that the light-like Wilson lines are invariant under the coordinate rescaling in their own light-like directions. This invariance leads to an ambiguity in the definition of rapidity divergences. Indeed, the boost of the collinear components of momenta $k^{+}\to ak^{+}$, $k^{-}\to k^{-}/a$ (with $a$ an arbitrary number) leaves the soft function invariant, while in the limit $a\to\infty$ one obtains the rapidity divergent configuration. Therefore the soft function cannot be explicitly calculated without a regularization which breaks its boost invariance. The coordinate space description of rapidity divergences, as well as, the counting rules for them have been derived in Vladimirov (2016, 2018). The nature of the divergences in the soft factor has been studied explicitly in Echevarria et al. (2014a) at one loop and in Echevarria et al. (2016a) at NNLO, which conclude that, once all contributions are included, the soft factor depends only on ultraviolet and rapidity divergences (and IR divergences are present only in the intermediate steps of the calculations, but not in the final result). Different regulators have also shown to be more or less efficient within different approaches to the calculations of transverse momentum dependent distributions. For instance NNLO perturbative calculations for unpolarized distributions, transversity and pretzelosity have been performed using de $\delta$-regulator of Echevarria et al. (2016b); Gutierrez-Reyes et al. (2017, 2018a) while for the recent attempts of lattice calculations off-the-light-cone Wilson lines are preferred Hagler et al. (2009); Musch et al. (2011, 2012); Ji (2013, 2014); Engelhardt et al. (2016); Yoon et al. (2015, 2017); Radyushkin (2017); Orginos et al. (2017); Ji et al. (2018). The discussion of the type of regulator involves usually another issue, which is also important for the complete definition of TMDs. While collinear and soft sectors can be distinguished by rapidity, the choice of a rapidity regulator forces a certain overlap of the two regions which should be removed, in order to arrive to a consistent formulation of the factorized cross section. This is called ”zero-bin” problem in Soft Collinear Effective Theory (SCET) Manohar and Stewart (2007)) and its solution is usually provided in any formulation of the factorization theorem. The amount of the zero-bin overlap is usually fixed by the same soft function in some particular limit although it is generally impossible to define this subtraction in a unique (in the sense of regulator independent) form. Because of this overlap one can find in the literature that the soft function is used in a different way in different formulations of the factorization theorem. The evolution properties of TMDs however are independent of these subtleties and they are the same in all formulations. A possible rapidity renormalization scheme-dependance is traditionally fixed by requiring $R^{-1}SR^{-1}=1$ (for this notation see discussion on sec. II.2).

The factorization theorem to all orders in perturbation theory relies on the peculiar property of Soft function of being at most linear in the logarithms generated by the rapidity divergences. Then it comes natural to factorize it in two pieces Echevarria et al. (2013a), and in turn this feature allows to define the individual TMDs. Using the $\delta$-regulator one can write to all orders in perturbation theory, as well as to all orders in the $\epsilon$-expansion (the UV divergences are regulated in dimensional regularization $d=4-2\epsilon$)Echevarria et al. (2016a).

[TABLE]

where tildes mark quantities calculated in coordinate space, $\nu$ is an arbitrary and positive real number that transforms as $p^{+}$ under boosts and we introduce the convenient notation

[TABLE]

Despite the fact that the soft function is not measurable per se, its derivative provides the so called rapidity anomalous dimension,

[TABLE]

with $\mathbf{l}_{\delta}=\hbox{ln}\left(\mu^{2}/|\delta_{+}\delta_{-}|\right)$. Because of its definition the rapidity anomalous dimension ${\cal D}$ has both a perturbative (finite, calculable) part and a nonperturbative part. This fact should be always taken into account despite the fact that many experimental data are actually marginally sensitive to the nonperturbative nature of the rapidity anomalous dimension. A non-perturbative estimation of the evolution kernel with lattice has been recently proposed in Ebert et al. (2018a) and I expect a deep discussion on this issue in the future. A renormalon based calculation has also provided some approximate value for this nonperturbative contribution Scimemi and Vladimirov (2017).

II.2 TMD operators

Another fundamental ingredient in the formulation of the factorization theorem is represented by the definition of the TMD operators that are involved. We use here the notation of Echevarria et al. (2016b). The TMDs which appear in a Drell-Yan process can be re-written starting from the bare operators (here I consider only the quark case, for simplicity)

[TABLE]

where \xi=\{0^{+},\xi^{-},\mbox{\boldmathb}_{T}\}, $n$ and ${\bar{n}}$ are light-cone vectors ($n^{2}={\bar{n}}^{2}=0,\;n\cdot{\bar{n}}=1$), and $\Gamma$ is some Dirac matrix, the repeated color indices $a$ ($a=1,\dots,N_{c}$ ) are summed up. The representations of the color SU(3) generators inside the Wilson lines are the same as the representation of the corresponding partons. The Wilson lines $\tilde{W}_{n}^{T}(x)$ are rooted at the coordinate $x$ and continue to the light-cone infinity along the vector $n$, where they are connected by a transverse link to the transverse infinity (that is indicated by the superscript $T$). The bare or unsubtracted TMDs are given then by the hadronic matrix elements of the corresponding bare TMD operator:

[TABLE]

These bare operators do not include for the moment any soft radiation and they are just collinear object (one can refer to them as ”beam functions”). Because of boost invariance they can be calculated in principle in any frame. However because of Wilson lines appearing in their definition we have to deal with rapidity divergences and their regularization. The soft interactions can be incorporated in the definition of the TMD through an appropriate ”rapidity renormalization factor” (which takes into account also a solution for the zero bin problem). The final form of the rapidity renormalization factor ($R$ in the following) is dictated by the factorization theorem. The renormalized operators and the TMD are defined respectively as

[TABLE]

and $Z_{q}$ is the UV renormalization constant for TMD operators, and $R_{q}$ the rapidity renormalization factor. Both these factors are the same for particle and anti-particle however they are different for quarks and gluons. These factors also occur in the same way in parton distribution functions and fragmentation functions. The scales $\mu$ and $\zeta$ are the scales of UV and rapidity subtractions respectively. The factor $R_{q}$ is built out of the soft factor and includes also the zero-bin corrections. There is a physical logic in this, because the factor $R$ actually fixes how much soft radiation should be included inside a properly defined TMD. In this respect it is useful to specify in actual calculations how the factor $R$ is derived. For instance in Echevarria et al. (2016b) the authors first remove all rapidity divergences and perform the zero-bin subtraction, and afterwards multiply by $Z$’s, and as a result the $R$ factors depend both on rapidity and renormalization scales.

Different logic has been used by other authors. For instance, in Luebbert et al. (2016), the authors follow the “Rapidity Renormalization Group” introduced in Chiu et al. (2012b, a), which is built in order to cancel the rapidity divergences through renormalization factors from the beam functions and soft factors independently although finally one achieves an equivalent resummation of rapidity logarithms. In Ref. Becher and Neubert (2011); Gehrmann et al. (2012, 2014) for TMDPDFs the soft function is hidden in the product of two TMDs.

I conclude this section providing the actual definition of the rapidity renormalization factor $R$,

[TABLE]

where S(\mbox{\boldmathb}_{T}) is the soft function and Zb denotes the zero-bin contribution, or in other words the soft overlap of the collinear and soft sectors which appear in the factorization theorem Manohar and Stewart (2007); Collins (2013); Echevarria et al. (2012); Echevarria et al. (2013a, 2014b). Depending on the rapidity regularization, the zero-bin subtractions are related to a particular combination of the soft factors. For instance the modified $\delta$-regularization Echevarria et al. (2016a) has been constructed such that the zero-bin subtraction is literally equal to the soft function: \textbf{Zb}=S(\mbox{\boldmathb}_{T}). The definition is non-trivial because it implies a different regularized form for collinear Wilson lines $W_{n({\bar{n}})}(x)$ and for soft Wilson lines $S_{n({\bar{n}})}(x)$. In the modified $\delta$-regularization, the expression for the rapidity renormalization factor is

[TABLE]

and this relation has been tested at NNLO in Echevarria et al. (2016c, a, b). We notice that due to the process independence of the soft function Collins (2013); Echevarria et al. (2012); Echevarria et al. (2013a, 2014b); Collins and Metz (2004), the factor $R_{f}$ is also process independent. In the formulation of TMDs by Collins in Collins (2013) the rapidity divergences are handled by tilting the Wilson lines off-the-light-cone. Then the contribution of the overlapping regions and soft factors can be recombined into individual TMDs by the proper combination of different soft functions with a partially removed regulator. This combination gives the factor $R^{f}$,

[TABLE]

The rest of logical steps remain the same as with the $\delta$-regulator.

An important aspect of factorization is finally represented by the cancellation of unphysical modes, the Glauber gluons. A check of this cancellation has been provided in Collins (2013); Gaunt (2014); Diehl et al. (2016); Boer et al. (2017) and I do not review it here.

III Matching at large $q_{T}$ (or small-$b$)

Once factorization is settled, the phenomenological analysis of data using TMDs need more information to be practicable. While a complete nonperturbative calculation of TMD is not available at the moment one can resort to asymptotic limits of TMDs in order to achieve an approximate intuition of TMDs. It turns out that a valuable information can be achieved in the limit of TMDs at large transverse momentum. In this limit it is possible to ”re-factorize” the TMDs in terms of Wilson coefficient and collinear parton distribution functions (PDF), following the usual rules for operator product expansion (OPE). At operator level we have

[TABLE]

where the symbol $\otimes$ is the Mellin convolution in variable $x$ or $z$ , and $f,\;f^{\prime}$ enumerate the flavors of partons. The running on the scales $\mu$, $\mu_{b}$ and $\zeta$ is independent of the regularization scheme and it is dictated by the renormalization group equations that I will discuss in the next section. Taking the hadron matrix elements of the operators we obtain the small-$b_{T}$ matching between the TMDs and their corresponding integrated functions,

[TABLE]

The integrated functions (that is, the PDFs) depend only on the Bjorken variables ($x$ for PDFs) and the renormalization scale $\mu$, while all the dependence on the transverse coordinate \mbox{\boldmathb}_{T} and rapidity scale is contained in the matching coefficient and can be calculated perturbatively. The definition of the integrated PDFs is

[TABLE]

In order to accomplish the calculation of the matching coefficients one uses eq. (14) on some particular states and solve the system for matching coefficients. For instance for twist-2 TMDs, since we are interested only in the leading term of the OPE, i.e. the term without transverse derivatives, it is enough to consider single parton matrix elements, with $p^{2}=0$. The current status of these calculations for quark distributions is resumed in tab. 1. Less information is generally available in the case of gluon TMDs. Basically the matching coefficients for unpolarized gluons are known at NNLO Scimemi and Vladimirov (2018c) and linearly polarized gluons at NLO Gutierrez-Reyes et al. (2017). In general the TMDs which match onto collinear twist-3 functions are much less known, which reflects the difficulty of the computations. It would be very useful to have a better knowledge of all these less known functions at higher perturbative order before the advent of Electron Ion Collider (EIC). In the rest of this section I focus on unpolarized quark distributions which offer also an important understanding on the power of the TMD factorization. The necessity of a complete NLO estimation of all TMDs is both theoretical and phenomenological. Actually a difficulty of the TMD extraction from data is due to the fact that it is a nontrivial function of two variables (Bjorken $x$ and transverse momentum) so that a complete mapping on a plane is necessary. This target is achievable thanks to the factorization of the cross section and the consequent extraction of the TMD evolution part, which is process independent. A second important information comes from the asymptotic limit of the TMD, which is perturbatively calculable. The simple LO expressions for the TMD in general do not provide much information (they are just constants), so that in order to achieve a wise modeling a NLO calculation is always necessary. The higher order calculations allow also to test the stability with respect to the scales that match the TMD perturbative and nonperturbative parts. For the unpolarized case a study in this sense can be found in Scimemi and Vladimirov (2018a) both for high energy and low energy data. Using a LO calculations one cannot even quantify this error. Finally, another lesson that comes from the analysis of the unpolarized case is that a good portion of the TMD is tractable starting from their asymptotic expansion for large transverse momenta. In any case even a 10$\%$ average precision of the SIDIS cross section at EIC will need a NLO theoretical input.

IV Evolution

The factorization scale dependence of the TMDs can be established starting from their defining operators and from eq. (10),

[TABLE]

in an usual way. The equation (16) is a standard renormalization group equation (which comes from the renormalization of the ultraviolet divergences), the function $\gamma_{F}(\mu,\zeta)$ is called the TMD anomalous dimension and it contains both single and double logarithms. The same eq. (10) can be used to write the running with respect to the rapidity scale, $\zeta$, which is fixed from the knowledge of soft interactions (see discussion in Echevarria et al. (2016a), also in Chiu et al. (2012b)) and comes from the factorization of rapidity divergences (see e.g.  Echevarria et al. (2016c); Vladimirov (2016, 2018)). Given that the soft factor is the same for initial and final states, the rapidity scale evolution is universally valid for TMD parton distribution functions and TMD fragmentation functions, and it is also spin-independent (so it is the same also for TMDs at higher twist),

[TABLE]

The function \mathcal{D}(\mu,{\mbox{\boldmathb}_{T}}) is called the rapidity anomalous dimension and actually one has \mathcal{D}(\mu,{\mbox{\boldmathb}_{T}})\equiv\mathcal{D}(\mu,|{\mbox{\boldmathb}_{T}}|). Several notations for rapidity anomalous dimensions have been used in the literature. The notations $\gamma_{F}$ and $\mathcal{D}$, used in this article, were suggested in Echevarria et al. (2013b). For convenience we list some popular notations and their relation to our notation in the table 2.

One has a different anomalous dimension for quarks and gluons, and the QCD properties of exponentiation implies the so-called Casimir scaling of anomalous dimension $\mathcal{D}$, see Echevarria et al. (2016a),

[TABLE]

which has been checked up to three loops Li and Zhu (2017); Vladimirov (2017).

The consistency of the differential equations (16-17) implies that the cross-derivatives of the anomalous dimension are equal to each other (Echevarria et al. (2016a); Chiu et al. (2012b)),

[TABLE]

From Eq. (19) one finds that the anomalous dimension $\gamma$ is

[TABLE]

where we introduce the notation

[TABLE]

The large-$q_{T}$ expansion of the TMD introduces also another evolution scale, which is needed for the matching Wilson coefficients, that can be obtained by deriving both sides of eq. (14). In the case of the unpolarized TMDs this is provided by the DGLAP222DGLAP is an acronym for Dokshitzer, Gribov, Lipatov, Altarelli, Parisi Gribov and Lipatov (1972); Dokshitzer (1977); Altarelli and Parisi (1977). equations

[TABLE]

where $P$ are the DGLAP kernels for the PDF. Similar equations hold for unpolarized TMD fragmentation functions (at NLO one can check Moch and Vermaseren (2000); Mitov and Moch (2006)). It is useful to recall also the running of the matching coefficient with respect to the rapidity scale (we set $\mu_{b}=\mu$)

[TABLE]

The solutions of these differential equations are

[TABLE]

This defines the reduced matching coefficients $\hat{C}$ whose renormalization group evolution equations are

[TABLE]

with the kernel $K$

[TABLE]

Using these equations one can find the expression for the logarithmical part of the matching coefficients at any given order, in terms of the anomalous dimensions and the finite part of the coefficient at one order lower. It is convenient to introduce the notation for the $n$-th perturbative order:

[TABLE]

Given the knowledge of the coefficient at order $n-1$ one can reconstruct all the terms with $k\neq 0$ at order $n$ in this series. So finally any higher order calculation provides new informations on terms $C_{f\leftarrow f^{\prime}}^{(n;0)}$. A resume of the present status of available calculations is provided in tab. 1 .

V Implementation of TMD formalism and TMD extraction from data

The implementation of TMD formalism and its phenomenological application is not trivial and eq. (4) should be written more carefully in order to describe correctly each single experiment. As an example let me review the case of the study of unpolarized TMD parton distribution functions in Drell-Yan and Z-boson production following Scimemi and Vladimirov (2018b).

Namely I consider the process $h_{1}+h_{2}\to G(\to ll^{\prime})+X$, where $G$ is the electroweak neutral gauge boson, $\gamma^{*}$ or $Z$. The incoming hadrons $h_{i}$ have momenta $p_{1}$ and $p_{2}$ with $(p_{1}+p_{2})^{2}=s$. The gauge boson decays to the lepton pair with momenta $k_{1}$ and $k_{2}$. The momentum of the gauge boson or equivalently the invariant mass of lepton pair is $Q^{2}=q^{2}=(k_{1}+k_{2})^{2}$. The differential cross-section for the Drell-Yan process can be written in the form Drell and Yan (1970); Altarelli et al. (1978)

[TABLE]

where $1/2s$ is the flux factor, $\Delta_{G}$ is the (Feynman) propagator for the gauge boson $G$. The hadron and lepton tensors are respectively

[TABLE]

where $J_{\mu}^{G}$ is the electroweak current. Within the TMD factorization, one obtains the following expression for the unpolarized hadron tensor (see e.g. Tangerman and Mulders (1995))

[TABLE]

where $g_{T}$ is the transverse part of the metric tensor and the summation runs over the active quark flavors. The variable $\mu$ is the hard factorization scale. The variables $\zeta_{1,2}$ are the scales of soft-gluons factorization, and they fulfill the relation $\zeta_{1}\zeta_{2}\simeq Q^{4}$. In the following, we consider the symmetric point $\zeta_{1}=\zeta_{2}=\zeta=Q^{2}$. The factors $z_{ff^{\prime}}^{GG^{\prime}}$ are the electro-weak charges and they are given explicitly in Scimemi and Vladimirov (2018b). The factor $C_{V}$ is the matching coefficient of the QCD neutral current to the same current expressed in terms of collinear quark fields. The explicit expressions for $C_{V}$ can be found in Kramer and Lampe (1987); Matsuura et al. (1989); Idilbi et al. (2006).

Finally, the term $Y$ denotes the power corrections to the TMD factorization theorem (to be distinguished from the power corrections to the TMD operator product expansion). The $Y$-term is of order $q_{T}/Q$ and is composed of TMD distributions of higher dynamical twist and in principle it can also include factorization breaking terms. These contributions appear each time the condition in eq. (3) is broken. It is a subtle issue to quantify exactly the magnitude of the ratio $q_{T}/Q$ where the $Y$-terms become important. A phenomenological study in Scimemi and Vladimirov (2018b) and a more formal study in the large-$N_{c}$ limit (that is, the limit of large number of colors) in Balitsky and Tarasov (2018) have found a reasonable upper value $(q_{T}/Q)_{max}\sim 0.2$. A study which takes into account the structure of operators in the type of corrections has been started in Ebert et al. (2018b).

In general the $Y$-terms should be included when the di-lepton invariant mass is of order a few GeV (this is the case for instance of HERMES experiment and, perhaps to a possibly less extent, COMPASS) or when the experimental precision is extreme (as it possibly happens with ATLAS experiment). This is issue is important phenomenologically and involves the study of cross sections with the inclusion of factorization breaking contributions. Some recent suggestion have appeared in Collins et al. (2016); Gamberg et al. (2018) which have still to be tested phenomenologically. One should remark however that the implementation of these factorization breaking correction strongly depends on the fact that the factorized part of the cross section is correctly realized and phenomenologically tested. More studies on this issue are necessary in the future.

Evaluating the lepton tensor, and combining together all factors one obtains the cross-section for the unpolarized Drell-Yan process at leading order of TMD factorization, in the form Collins et al. (1985); Davies et al. (1985); Ellis and Veseli (1998); Becher and Neubert (2011); Echevarria et al. (2012); Collins (2013)

[TABLE]

where $y$ is the rapidity of the produced gauge boson. The factor $\mathcal{P}$ is a part of the lepton tensor and contains information on the fiducial cuts. This factor provides important information on the actual measured leptons and should be always included when the relative experimental information is provided.

The evaluation of this cross section requires a correct implementation also of the evolution and perturbative information of the TMDs. In the rest of this section I dedicate particular emphasis to the evolution parts making the point that passing from the all-order formal knowledge of the factorized cross-section to the finite-order practical usage requires the discussion of some subtle points.

V.1 The treatment of TMD evolution

The TMD evolution is resumed by the following equations

[TABLE]

and on the right hand side of these equation we have omitted the reference to flavor $f$ for simplicity. The TMD anomalous dimension $\gamma_{F}(\mu,\zeta)$ contains both single and double logarithms and the anomalous dimension $\gamma_{V}$ refers to the finite part of the renormalization of the vector form factor, see tab. 2. The function \mathcal{D}(\mu,{\mbox{\boldmathb}_{T}}) is the rapidity anomalous dimension, resulting from the TMD factorization of rapidity divergences and actually depends only on (\mu,|{\mbox{\boldmathb}_{T}}|). It is remarkable that eq. (V.1) cannot fix the logarithmic part of $\mathcal{D}$ entirely, but only order by order in perturbation theory, because the parameter $\mu$ is also responsible for the running of the coupling constant. It has been shown Parisi and Petronzio (1979); Korchemsky and Sterman (1995); Scimemi and Vladimirov (2017) that the perturbative series for $\mathcal{D}$ is asymptotical and it has a renormalon pole, whose contribution is significant at large-$b$. Therefore, the rapidity anomalous dimension $\mathcal{D}$ is generically a non-perturbative function, which admits a perturbative expansion only for small values of the parameter |{\mbox{\boldmathb}_{T}}|. One can compare this with the situation in conformal field theory, where the coupling constant is independent on $\mu$, the rapidity anomalous dimension is linear in logarithms of $\mu_{b}$ and maps to the soft anomalous dimension by conformal transformation Vladimirov (2017, 2018).

The double-evolution equation of the TMDs can be formulated as in Scimemi and Vladimirov (2018b) using a two-dimensional vector field notation. The procedure consists in introducing a convenient two-dimensional variable which treats scales $\mu$ and $\zeta$ equally,

[TABLE]

where the dimension of the scale parameters is explicitly indicated and the bold font means the two-dimensional vectors. Then one defines the standard vector differential operations in the plane $\bm{\nu}$, namely, the gradient and the curl

[TABLE]

The TMD anomalous dimensions can be all included in a vector evolution field \mathbf{E}(\bm{\nu},{\mbox{\boldmathb}_{T}}),

[TABLE]

Here and in the following, we use the vectors $\bm{\nu}$ as the argument of the anomalous dimensions for brevity, keeping in mind that \mathcal{D}(\bm{\nu},{\mbox{\boldmathb}_{T}})=\mathcal{D}(\mu,{\mbox{\boldmathb}_{T}}), $\gamma_{F}(\bm{\nu})=\gamma_{F}(\mu,\zeta)$, etc. In other words, the anomalous dimensions are to be evaluated on the corresponding values of $\mu$ and $\zeta$ defined by value of $\bm{\nu}$ in eq. (33). The TMD evolution equations (V.1) and the evolution factor $R$ in this notation have the form

[TABLE]

Using this formalism, eq. (V.1) are equivalent to the statement that the evolution flow is irrotational,

[TABLE]

The irrotational vector fields are conservative fields, and they can be presented as a gradient of a scalar potential,

[TABLE]

i.e. $U$ is the evolution scalar potential for TMD. According to the gradient theorem any line integral of the field $\mathbf{E}$ is path-independent and equals to the difference of values of potential at end-points. Therefore, the solution for the $R$ factor in eq. (36) is

[TABLE]

and $\nu_{1,2}$ are the first and second components of the vector $\bm{\nu}$ in eq. (33), and the last term is an arbitrary $b$-dependent function.

We recall for completeness the perturbative expansions of all these quantities starting from the running of the coupling constant $a_{s}=g^{2}/(4\pi)^{2}$,

[TABLE]

where $\beta_{0}=\frac{11}{3}C_{A}-\frac{2}{3}N_{f}$. The ultraviolet anomalous dimensions read

[TABLE]

The leading coefficients in these expansions are $\Gamma_{0}=4C_{F}$ and $\gamma_{1}=-6C_{F}$ for the quark. In the gluon case, they are $\Gamma_{0}=4C_{A}$ and $\gamma_{1}=-2\beta_{0}$. For the collection of higher order terms see e.g. appendix D in Echevarria et al. (2016b). The perturbative series for the rapidity anomalous dimension $\mathcal{D}$ is

[TABLE]

where $d^{(n,k)}$ are numbers. Note, that using eq. (V.1) the coefficients $d^{(n,k)}$ with $k>0$ are expressed in the terms of $d^{(i,0)}$, $\Gamma_{i}$ and the coefficients of $\beta$-function. The leading terms of $\mathcal{D}$ are $d^{(1,1)}=\Gamma_{0}/2$ and $d^{(1,0)}=0$. The explicit expressions for $d^{(n,k)}$ up to $n=3$ can be found in Vladimirov (2018).

V.2 Formal treatment of TMD evolution in the truncated perturbation theory

The evolution field presented in the previous section is conservative only when the full perturbative expansion of the evolution equations is known. In practice only a few terms of the evolution are calculated, so that it is important to understand in which sense the evolution field remains conservative. Using the Helmholtz decomposition, the evolution field is split into two parts

[TABLE]

The field $\tilde{\mathbf{E}}$ is irrotational, the field $\mathbf{\Theta}$ is divergence-free and they are orthogonal to each other

[TABLE]

with the notation $\text{curl}(\mathbf{curl})=\nabla^{2}$. Then, one can write the irrotational field $\tilde{\mathbf{E}}$ as the gradient of a scalar potential

[TABLE]

and only this part of the evolution is conservative.

Instead, the divergence-free part in two-dimensions can be written as the vector curl (see eq. (34)) of another scalar potential

[TABLE]

The curl of the evolution field can be calculated using the definitions (V.1),

[TABLE]

The function $\delta\Gamma$ can be calculated order by order in perturbation theory. For instance at order $N$ one finds

[TABLE]

is the $\beta$-function with first $n$ terms removed. For instance, we have

[TABLE]

In these expressions the $\beta$-function is not expanded because in applications it can be of different perturbative order with respect to the rest of anomalous dimensions.

The immediate consequence of the fact that the evolution field $\mathbf{E}$ is no more conservative is that the evolution factor R[{\mbox{\boldmathb}_{T}};\bm{\nu}_{f}\to\bm{\nu}_{i}] is dependent on the path chosen to join the initial and final points $\bm{\nu}_{i},\;\bm{\nu}_{f}$ and this fact introduces a theoretical error which can be dominant in certain implementations of the evolution kernels. The difference between two solutions evaluated on different paths is

[TABLE]

where ${P_{1}\cup P_{2}}$ is the closed path built from paths $P_{1}$ and $P_{2}$ and $\Omega(P_{1}\cup P_{2})$ is the area surrounded by these paths. Using the independence of $\delta\Gamma$ on the variable $\zeta$, eq. (52) becomes

[TABLE]

where $\zeta_{1,2}(\mu)$ is the $\zeta$-component of the path $P_{1,2}$ at the scale $\mu$. This equation shows that the difference between paths becomes bigger with largely separated rapidity scales $\zeta_{i}$.

V.3 Restoring path independence of evolution

The path independence of the evolution is crucial for the implementation of the perturbative formalism, as its absence can derive into uninterpretable extractions of TMDs or big theoretical errors. The path independence can be achieved observing that

[TABLE]

should hold order by order in perturbation theory. Once this is realized it is possible to define null-evolution lines in the $(\mu,\zeta)$ plane, which coincide with equipotential lines, and the evolution of TMD takes place only between two different lines. I resume here two possible solutions to this problem, following Scimemi and Vladimirov (2018b).

V.4 Improved $\mathcal{D}$ scenario

In the literature one can find a typical way to implement the evolution that one can call the improved $\mathcal{D}$ scenario which includes the Collins-Soper-Sterman formalism Collins and Soper (1981); Collins (2013); Aybat and Rogers (2011); Scimemi and Vladimirov (2018a); Echevarria et al. (2013b); Chiu et al. (2012a); Li et al. (2016). In this scenario one chooses a scale $\mu_{0}$ such that

[TABLE]

In this way one obtains

[TABLE]

and the scalar potential $\tilde{U}$ is obtained from eq. (40) replacing $\mathcal{D}$ by eq. (56),

[TABLE]

The TMD evolution factor $R$ depends explicitly on $\mu_{0}$

[TABLE]

The situation in this scenario can be visualized in fig. 2. Choosing a conventional value for $\mu_{0}$ corresponds to choosing a point where evolution flips from path 1 and path 2 in this figure. The differences that can appear in the extraction of TMDs which depend on the choice of $\mu_{0}$ can be numerically large, so that the selection of this scale can cause also some problems when a sufficient precision is required.

V.5 Improved $\gamma$ scenario

The presence of the intermediate scale $\mu_{0}$ is not unavoidable in the implementation of the TMD evolution. In fact the integrability condition eq. (54) can be restored by changing the anomalous dimension $\gamma_{F}$ to a modified value $\gamma_{M}$ such that

[TABLE]

The corresponding scalar potential $\tilde{U}$ is derived replacing $\Gamma\to\Gamma-\delta\Gamma$,

[TABLE]

Using the definition of $\delta\Gamma$ and integrating by parts one obtains

[TABLE]

and the corresponding solution for the evolution factor reads

[TABLE]

These expressions should be completed with the resummation of $\mathcal{D}$ by means of renormalization group eq. (56) as it is not implicitly included in this scenario.

V.6 $\zeta$ prescription and optimal TMDs

From the discussion of the previous section, and using a correct implementation of TMD evolution it is clear now that TMDs defined on the same equi-potential/null-evolution curves (that we call $\bm{\omega}(\bm{\nu}_{B},b)$) are the same, that is

[TABLE]

when the scales $\bm{\nu}_{B}$ and $\bm{\nu}_{B}^{\prime}$ belong to the same null-evolution curve. As a consequence the point $\bm{\nu}_{B}$ in the $(\zeta,\mu)$ plane simply represents a label which defines a null-evolution curve, but it does not enter the function $F(x,b_{T};\bm{\nu}_{B})$ explicitly. The evolution of the TMDs occurs only when two TMDs do not belong to the same null-evolution curve. In this case

[TABLE]

where $\zeta_{\mu}$ is defined such that (\mu_{i},\zeta_{\mu_{i}}(\bm{\nu}_{B},{\mbox{\boldmathb}_{T}}))\in\omega(\bm{\nu}_{B},b_{T}) and (\mu_{f},\zeta_{f})\not\in\omega(\bm{\nu}_{B},{\mbox{\boldmathb}_{T}}) . In order to minimize the evolution effect and so to have a more stable prediction/extraction of TMDs the initial and final scales should be selected with care. The final point of the rapidity evolution, $\zeta_{f}$, is as usual dictated by the hard subprocess. On the contrary, the initial value of the rapidity scale $\zeta_{i}$ should be chosen depending on the input for the non-perturbative behavior of the TMD distribution. In practice it is convenient to match the TMD distribution to the corresponding collinear distribution. This matching guarantees the agreement of the model to its asymptotic behavior in the limit of high transverse momentum, and determines a significant part of the TMD distribution. The expression for small-$b$ matching has the form

[TABLE]

where $f$ is PDF or FF, and $C$ is the Wilson coefficient function. The coefficient function includes the dependence on {\mbox{\boldmathb}_{T}} within the logarithms $\mathbf{L}_{\mu}$ and $\mathbf{L}_{\sqrt{\zeta}}$. In this way, the initial scales $(\mu_{i},\zeta_{i})$ explicitly enter in the TMD modeling. Traditionally, see e.g. Collins (2013); Aybat and Rogers (2011); Bacchetta et al. (2017), many studies use $\zeta_{i}=\mu_{i}^{2}$, however this choice has serious drawbacks, because it leaves uncanceled logarithmic factors in the coefficient function which get larger and larger at small transverse momentum. More prescriptions are then used to solve this problem, like the $b^{*}$ prescription Collins (2013). On top of this an eventual usage of $b^{*}$ prescription in the evolution factor spoils the distinction between the non-perturbative part of the evolution and the intrinsic non-perturbative transverse momentum dependence of the partons inside the hadrons.

The $\zeta$-prescription suggested in Scimemi and Vladimirov (2018a, b) provides an attempt to improve the stability of the perturbative series and to separate the modeling of the TMD distribution from the factorization procedure. In non-$\zeta$-prescription formulation the TMD distribution has a $\mu$-dependence that is typically related to the scale $b$. Thus the evolution, and hence non-perturbative modification of $\mathcal{D}$, is somehow incorporated into the model for the TMD distribution. This fact makes difficult and sometimes impossible the comparison among different TMD non-perturbative estimations such as lattice or low-energy effective theories. Even in the extraction of TMDs from data one would like to have information on the nonperturbative part of the evolution kernel and the intrinsic nonperturbative TMD initial distribution independently.

The $\zeta$-prescription consists in a special choice of $\zeta_{i}$ value as a function of $\mu$ and $b$. A TMD distribution in the $\zeta$-prescription reads

[TABLE]

where $\zeta_{\mu}$ is defined such that (\mu_{i},\zeta_{\mu_{i}}(\bm{\nu}_{B},{\mbox{\boldmathb}_{T}}))\in\omega(\bm{\nu}_{B},{\mbox{\boldmathb}_{T}}), that is, the function of \zeta_{\mu}({\mbox{\boldmathb}_{T}}) draws a null-evolution curve in the $(\mu,\zeta)$ plane. The value of $\zeta_{i}$ is selected such that the initial scales TMD distribution $(\mu_{i},\zeta_{\mu_{i}})$ belong to a particular curve. Note that in this way we have a line of equivalent initial conditions, provided by all points $(\mu_{i},\zeta_{\mu_{i}})$ which belong to the same evolution curve.

In order to provide an initial point for the evolution it is convenient to re-write eq. (65) specifying the scales,

[TABLE]

where $\mu_{\text{OPE}}$ is an intrinsic scale for the expansion of the TMD in terms of Wilson coefficients and PDFs and it is a free parameter. The values of $\mu_{\text{OPE}}$ are restricted by the values of $\mu$ spanned by the defining null-evolution curve. In accordance to the general structure of the evolution plane one finds the following restrictions on the parameter $\mu_{\text{OPE}}$

[TABLE]

It is clear that the last case is preferable, since the model of TMD distribution is completely unrestricted. Additionally, only this case has a unique definition. The choice of $\mu_{\text{saddle}}$ as the initial point is so optimal and consistent with the re-expression of TMDs using PDFs. This choice determines the optimal TMD distribution and its related special null-evolution curve. The definition of the initial point is therefore non-perturbative, unique and scale-independent. In such a way one can denote the optimal TMD simply as F(x,{\mbox{\boldmathb}_{T}}).

The implementation of the optimal TMD configuration is compatible with other well-known requests for the evolution and an example of plot for the $R$-factor is given in fig. 3. For instance, at large-$b$ the shape of the rapidity anomalous dimension is non-perturbative and unknown, which is confirmed also a by an explicit renormalon calculation  Scimemi and Vladimirov (2017); Korchemsky and Sterman (1995) (see also Becher and Bell (2014)). So, at large-$b$ the expression for $\mathcal{D}$ should be extracted from data fitting, while at small-$b$ it should match the perturbative expression. In principle there are several possibilities to account for this effect. For instance one can introduce a simple ansatz like the modification

[TABLE]

where b=|{\mbox{\boldmathb}_{T}}| and $b_{\max}$ is a parameter, such that $b_{\text{max}}<\bar{b}$ as suggested a long ago in Collins and Soper (1982),

[TABLE]

as part of the $b^{*}$ prescription Collins (2013). Let us stress that the choice of a $b^{*}$ can be admissible separately for the evolution factor and that eq. (71) does not imply $b^{*}$-prescription for the whole TMD distribution. With the choice $b_{\text{max}}<\bar{b}$ the saddle point is always in the observable region, which allows to determine the optimal TMD. The expression for the cross-section with the optimal TMD definition is particularly compact and reads

[TABLE]

where the evolution exponent can be given by the equivalent expressions

[TABLE]

In eq. (74), the scale $\mu_{\text{saddle}}$ is $b$-dependent, and defined by the equation

[TABLE]

and the universality of the TMD evolution allows this construction for all types of TMD distribution. The derivation of the saddle point using formula (76) is in practice done numerically, so that an efficient method to extract it or to approximate this point should be discussed as in Scimemi and Vladimirov (2018b). A technical discussion of this issue is beyond the point of this paper.

The optimal TMD definition allows in general a more self contained and organized discussion of theoretical errors. The absence of an intermediate scale $\mu_{0}$, remove one artificial source of error while ensuring the path independence of the final result. In this way it is possible to directly compare $\mathcal{D}_{\text{NP}}$ from different extractions and models. The definition also removes the question of the low-energy normalization point $\mu_{i}$. In fact the low-energy normalization is defined ”non-perturbatively” and uniquely by eq. (76). This fact has the important consequence that the perturbative order of the evolution is completely unrelated to the perturbative order of matching of the TMD on the respective collinear functions. Because the evolution factor is known often at higher orders then the Wilson coefficient matching factors, it is possible to fully use all the available perturbative information in whatsoever TMD extraction. Another important consequence is that in order to compute the theoretical uncertainty of TMDs one is left only with the variation of $\mu_{f}$ and $\mu_{\rm OPE}$. The fact that the number of varied scales is different from more standard analysis does not necessarily imply a reduction of theoretical errors. The error in fact reshuffles in $\mu_{f}$ and $\mu_{\rm OPE}$ but the descriptions appear now more coherent. One can appreciate this effect in fig. 4. In this figure one compares for the ATLAS experiment a standard method to test the dependence on the scales, and thus the stability of the perturbation theory prediction, multiplying each scale by a parameter Nadolsky et al. (2001); Bozzi et al. (2011); D’Alesio et al. (2014); Scimemi and Vladimirov (2018a) and varying the parameters nearby their central value. E.g. in the notation of Scimemi and Vladimirov (2018a), one changes scales as

[TABLE]

and checks the variations of $c_{i}\in(1/2,2)$. The variation of all these four parameters is consistent with a non-optimal definition of TMDs, while in the optimal case only the variation of $c_{2}$ and $c_{4}$ is necessary.

VI Conclusions

The formulation of factorization theorems in terms of TMDs is a first fundamental step for the study of the structure of hadrons and the origin of spin. The use of the effective field theory appears essential to correctly order the QCD contributions. Properties of TMDs like evolution and their asymptotic limit at large values of transverse momentum can be systematically calculated starting from the definition of correct operators and the evaluation of the interesting matrix elements. A key point for the renormalization of TMDs is represented by the so called soft matrix element which is common in the definition of all spin dependent leading twist TMD.

Still, all this is just a starting point for the study of TMDs. In fact a correct implementation of evolution requires a control of all renormalization scales that appear in the factorization theorem. I have described here some of these possibility putting the accent on some recent interesting developments which, at least theoretically allow a better control of the resummed QCD series. The understanding of factorization allows also to precisely define the range of ideal experimental conditions where this formalism can be applied. A full analysis of present data using all the theoretical information collected so far is still missing and it will certainly be an object of research in the forthcoming years. The formalism described in this work is the one developed for unpolarized distributions. However the evolution factors are universal, that is, they are the same for polarized and un-polarized leading twist TMDs and they are valid in Drell-Yan, SIDIS experiments and $e^{+}e^{-}$ colliders, where the factorization theorem applies. All this formalism is expected to be tested on data in the near future. Nevertheless a lot of perturbative and non-perturbative information is still missing. Giving a look at tab. 1 one can see that for many TMD one has only a lowest order perturbative calculation which should be improved in order to have a reliable description of data. While, the information on the non-perturbative structure of TMD is still poor and still driven by phenomenological models, it is important to implement the TMD formalism in such a way that perturbative and non-perturbative effects are well separated. And among the non-perturbative effects, one should be able to distinguish the ones of the evolution kernel from the rest. In the text I have discussed a possible solution to this problem. Some prominent research lines which possibly will deserve more attention in the future include the cases where hadrons are measured inside the jets, see for instance Kang et al. (2016, 2017a, 2017b) or outside a jet (say, hadron-jet interactions) Gutierrez-Reyes et al. (2018b); Liu et al. (2018); Buffing et al. (2018b) and lattice.

Acknowledgements.

I would like to thank Alexey Vladimirov and Daniel Gutierrez Reyes for discussing this paper. I.S. is supported by the Spanish MECD grant FPA2016-75654-C2-2-P.