# TMD Evolution: Matching SIDIS to Drell-Yan and W/Z Boson Production

###### Abstract

We examine the QCD evolution for the transverse momentum dependent observables in hard processes of semi-inclusive hadron production in deep inelastic scattering and Drell-Yan lepton pair production in collisions, including the spin-average cross sections and Sivers single transverse spin asymmetries. We show that the evolution equations derived by a direct integral of the Collins-Soper-Sterman evolution kernel from low to high can describe well the transverse momentum distribution of the unpolarized cross sections in the range from 2 to 100 GeV. In addition, the matching is established between our evolution and the Collins-Soper-Sterman resummation with -prescription and Konychev-Nodalsky parameterization of the non-perturbative form factors, which are formulated to describe the Drell-Yan lepton pair and W/Z boson production in hadronic collisions. With these results, we present the predictions for the Sivers single transverse spin asymmetries in Drell-Yan lepton pair production and boson production in polarized and collisions for several proposed experiments. We emphasize that these experiments will not only provide crucial test of the sign change of the Sivers asymmetry, but also provide important opportunities to study the QCD evolution effects.

###### pacs:

## I introduction

Transverse momentum dependent (TMD) parton distributions and fragmentation functions are formally introduced as an extension to the parton model description of nucleon structure and an important tool to calculate hadronic processes. In the last few years, these distribution functions have attracted great attentions in hadron physics community. In particular, the novel single transverse spin asymmetries (SSAs) in semi-inclusive hadron production in deep inelastic scattering processes (SIDIS) observed by the HERMES, COMPASS, and JLab Hall A collaborations, have stimulated much theoretical developments. The TMD factorization provides a solid theoretical framework to understand these spin asymmetries. Moreover, together with the generalized parton distributions (GPDs), the TMDs unveil the internal structure of nucleon in a three dimension fashion, the so-called nucleon tomography. These topics are major emphases in the planed electron-ion collider Boer:2011fh .

An important theoretical aspect of the TMD parton distribution and fragmentation functions is the energy evolution, which was thoroughly studied in the early paper by Collins and Soper Collins:1981uk . This evolution is referred as the Collins-Soper (CS) evolution equation. It has been applied to formulate the perturbative resummation of large double logarithms in hard scattering processes where transverse momentum distribution are measured. The associated resummation is called Collins-Soper-Sterman (CSS) Collins:1984kg resummation, or transverse momentum resummation. In these hard processes, because of two separate scales, there exist large double logarithms in each order of perturbative calculations (originally from a QED calculation by Sudakov Sudakov:1954sw ), and the relevant resummation has to be taken in the calculation Dokshitzer:1978dr ; Parisi:1979se ; Collins:1984kg . For example, in Drell-Yan lepton pair production in collisions, the invariant mass is much larger than the total transverse momentum of the lepton pair , , where perturbative corrections will induce large logarithms . The resummation of these large logarithms are performed by applying the TMD factorization and the CS evolution. Successful applications have been made to study the low transverse momentum distribution of Drell-Yan type of processes in hadronic collisions from fixed target experiments to highest collider energy experiments, such as the Tevatron at Fermilab and the large hadron collider (LHC) at CERN, see, for example, the relevant publications in Refs. Landry:2002ix ; Konychev:2005iy ; Qiu:2000ga ; Kulesza:2002rh ; Catani:2000vq ; Catani:2003zt ; Bozzi:2003jy .

The resummation for the hard processes are based on the TMD factorization for these processes Collins:1981uk ; Collins:1984kg ; Ji:2004wu ; Collins:2004nx ; Collins ; Aybat:2011zv ; Aybat:2011ge . Since the definition of the TMDs contains the light-cone singularity Collins:1981uk , the detailed calculations depend on the scheme to regulate this singularity. In the original paper of Collins-Soper Collins:1981uk , an axial gauge has been used. This was followed by a gauge invariant approach in Ji-Ma-Yuan with a slight off-light-cone gauge link in covariant gauge Ji:2004wu (referred as Ji-Ma-Yuan scheme in the following). A new definition for the TMD and the associated soft factor has been proposed in Ref. Collins where a subtraction method was used to regulate the light-cone singularity (referred as Collins-11 in the following), and the phenomenological applications were presented in Refs. Aybat:2011zv ; Aybat:2011ge . Although the TMDs depend on the regularization scheme, the resummation for the physical observables, such as the differential cross sections and the spin asymmetries, is independent on the scheme. We will present detailed discussions in the below between the two formalisms of Ref. Ji:2004wu and Ref. Collins .

To understand the energy evolution of the spin-dependent hard processes, such as the SSA in SIDIS and Drell-Yan lepton pair production, we need to extend the CS and CSS derivation to the interested observables Boer:2001he . The CS evolution equations for the TMDs was extensively discussed in Ref. Idilbi:2004vb , where the evolution kernel was derived for all the leading order TMD quark distributions. In particular, for the so-called -even TMDs, the evolution is exactly the same as the original CS evolution. For the -odd ones, a slightly different form has to be used, but with the same kernel. These evolution equations can be cross checked with the finite order perturbative calculations, which has been shown to yield consistent results Kang:2011mr .

Besides the above developments in the investigation of the TMDs in full QCD, recently an effective theory approach based on the soft-collinear-effective-theory has been applied to the evolution of the TMDs. Several different schemes are proposed in the literature Mantry:2009qz ; Becher:2010tm ; GarciaEchevarria:2011rb ; Chiu:2012ir . It has been shown in Ref. Collins:2012uy that one of the effective theory approach GarciaEchevarria:2011rb (referred as EIS in the following) is equivalent to the Collins-11 formalism Collins .

Although there are different ways to formulate the TMD distribution and fragmentation functions, the energy evolution and resummation for the physical observables (including the differential cross sections and spin asymmetries) will always take the same form as they should be. Therefore, in this paper, we will focus on the energy evolution for the differential cross section and spin asymmetries. Of course, to have a solid prediction for the physical observables, we need to have the TMD factorization proven for the relevant processes. The SIDIS and Drell-Yan lepton pair production in collisions are two examples that a rigorous TMD factorization has been proven.

The main goal of this paper is to make predictions for the Sivers single spin asymmetries in Drell-Yan lepton pair production in collisions from the constraints from the Sivers asymmetries observed in SIDIS from HERMES/COMPASS experiments. The TMD factorization and universality has predicted that the Sivers asymmetries in these two processes differ by a sign, because of difference between the initial/final state interaction Brodsky:2002cx ; Collins:2002kn . The Sivers single spin asymmetries in SIDIS have been observed by HERMES/COMPASS/JLab Hall A collaborations with at the region from 2 to 4 GeV Airapetian:2009ae ; Airapetian:2010ds ; Alekseev:2008aa ; Adolph:2012sp ; Qian:2011py . However, the typical Drell-Yan measurements will be around the region from to GeV compassdy ; fermilabdy ; futurerhic . Therefore, the energy evolution of the associated TMDs is important to carry out a rigorous test of the sign change prediction. Early calculations are based on the TMD factorization, however, without the energy evolution effects in the derivation Vogelsang:2005cs ; Collins:2005rq ; Anselmino:2005ea ; Anselmino:2005an ; Anselmino:2009st ; Kang:2009sm ; Bacchetta:2011gx ; Gamberg:2013kla . Recently, several studies have started to take into account the evolution effects Boer:2001he ; Anselmino:2012aa ; Sun:2013dya ; Aybat:2011ta . In particular, in Ref. Aybat:2011ta , a strong decreasing was found in comparing the SSA in typical Drell-Yan processes to those observed by HERMES/COMPASS. In this paper, we will carefully examine these predictions, and present a consistent calculation for the energy evolution in both spin-average and single-spin dependent cross sections. A brief summary has been published earlier Sun:2013dya .

The starting point of our calculations is to build the correct evolution framework which can describe the known experimental data of the unpolarized cross sections in the associated processes. One has to test the TMD evolution with the unpolarized cross sections before they can be applied to spin-dependent cross sections and the spin asymmetries. This is a very important point, which, unfortunately, is often forgotten in the phenomenological studies.

We will make use of the successful approach in the CSS resummation. In these formulations, a non-perturbative form factor has to be included. We follow the BLNY and KN calculations Landry:2002ix ; Konychev:2005iy , where -prescription of CSS resummation is applied: with the impact parameter. This prescription guarantees that . The non-perturbative form factor takes a form as in the impact parameter space with and the longitudinal momentum fractions of the incoming nucleons carried by the initial state quark and antiquark. The parametrization was fitted to the typical Drell-Yan lepton pair production with and production (). By applying the universality argument, these parameterizations should be able to apply in the SIDIS processes for the associated quark distribution part. However, if we extrapolate the above parameterization down to the typical HERMES/COMPASS kinematics where is around , we can not describe the transverse momentum distribution of hadron production in these experiments (see the discussion in Sec. III D). The main reason is that the logarithmic dependence leads to a strong change around low , which, however, contradicts with the smooth dependence from the experimental observation. It will be interesting to check other forms of non-perturative form factors to see if they can be extrapolated to HERMES/COMPASS energy region Meng:1995yn ; Nadolsky:1999kb . We will come back to this issue in a future publication.

Meanwhile, for moderate variation, there is an alternative approach to apply, from which we can directly solve the evolution by an integral of the kernel from low to high Ji:2004wu . This is, in particular, useful at relative low region, and can be applied to describe the transverse momentum distributions in SIDIS from HERMES/COMPASS experiments and fixed target Drell-Yan lepton pair production experiments Sun:2013dya . This will also help to build a connection to the ultimate CSS resummation in Drell-Yan and production. As illustrated in Fig. 1, in the moderate region (including HERMES/COMPASS kinematics of SIDIS and Drell-Yan process in fixed target experiments), we apply the evolution by a direct integral of the kernel from relative low to relative high . In the high region which covers Drell-Yan lepton pair production and production, we apply the complete CSS resummation with -prescription (following BLNY/KN parameterization of the non-perturbative form factors). In the overlap region, we shall obtain a consistent picture for the transverse momentum distribution of the cross section and the spin asymmetries.

Following this procedure, we will determine the quark Sivers functions
from the HERMES/COMPASS experiments in SIDIS with
-evolution taken into account using direct integral of the kernel.
In particular, we constrain the transverse momentum moments of the
quark Sivers functions, which correspond to the twist-three quark-gluon-quark
correlation functions (so-called Qiu-Sterman matrix elements Efremov:1981sh ; Qiu:pp ).
These are the bases to evaluate the Sivers single spin asymmetries in the
CSS resummation formalism Kang:2011mr . We then
calculate the Sivers asymmetries in Drell-Yan processes with
the constrained Sivers functions.
The consistent check is carried out by comparing the predictions
between the evolutions done with direct integral of the kernel from
to and that with CSS resummation with integral of the kernel
from to . We notice that in the original BLNY parameterization,
there is a strong -dependence (which is correlated to the -dependence)
in the non-perturbative form factor Landry:2002ix . To avoid this strong dependence, we
follow an updated fit by Konychev and Nadolsky Konychev:2005iy
which describes equally well the Drell-Yan and W/Z boson data with a
mild dependence on . We would like to emphasize that the Sivers
asymmetries observed in HERMES/COMPASS experiments mainly focus
on the moderate -region around 0.1, which is also the typical -range
for the Drell-Yan fixed target experiments ^{1}^{1}1Drell-Yan process
at RHIC will be able to probe, for the first time, the wide range of .
This will be important to check the -dependence of the non-perturbative form
factor. Hope this experiment can be carried out soon..

The rest of the paper is organized as follows. In Sec.II, we present a brief review on the theory of low transverse momentum hard processes as a self-contained introduction. In particular, we will present the detailed derivations of our previous publication of Ref. Kang:2011mr . We will also discuss various TMD factorizations. In Sec.III, we discuss the TMD evolution and resummation in the context of the transverse momentum dependent differential cross sections and the Sivers single spin asymmetries. We will illustrate the incompatibility between the BLNY parameterization of the CSS resummation and the HERMES/COMPASS measurements of the distribution in SIDIS process. We will also discuss in detail our approach to calculate the transverse momentum distribution in this kinematic region, and compare to the experimental data on multiplicity distribution in SIDIS from HERMES/COMPASS experiments and Drell-Yan fixed target experiments, and demonstrate that our approach consistently describe these data with the TMD evolution taken into account. In Sec.IV, we extend the evolution effects to the Sivers single spin asymmetries measured by the HERMES/COMPASS collaborations, and perform a combined analysis. In Sec. V, we present the predictions for the Sivers asymmetries in Drell-Yan lepton pair productions in the planed experiments, and production at RHIC. We demonstrate the matching between two different calculations. With this, we show the results for the proposed Drell-Yan experiments. We will emphasize the test of the sign change between SIDIS and Drell-Yan, and highlight the ability to separate the flavor dependence by combining Drell-Yan/W measurements with SIDIS results as well. In Sec. VI, we summarize our paper, and discuss further developments.

## Ii Theory Review of Low Transverse Momentum Hard Processes

In this section, we present a brief review of the theory background for low transverse momentum hard processes. Under the context of this paper, the hard processes are hadronic processes with two separate scales: the invariant mass of virtual photon and the transverse momentum of observed particles for lepton pair in Drell-Yan process or for final state hadron in SIDIS.

TMD factorization applies in the kinematic region of low transverse momentum: . As mentioned in the Introduction, large double logarithms will arise from perturbative gluon radiation. These large logs have been demonstrated in the single transverse spin dependent differential cross sections as well Ji:2006ub ; Kang:2011mr . In the following, we will summarize these calculations, and, in particular, present detailed derivations of our previous publication Kang:2011mr . We will start with the low transverse momentum Drell-Yan lepton pair productions for both spin averaged and spin-dependent cross sections. We then examine the TMD factorization. Finally, we extend the discussions to the SIDIS process.

### ii.1 Low Transverse Momentum Drell-Yan

In the Drell-Yan lepton pair production in collisions, we have

(1) |

where and represent the momenta of hadrons and , and for the transverse polarization vector of , respectively. We further assume hadron moving in the direction. Light-cone momentum is defined as . Therefore, is dominated by its plus component, whereas by its minus component. The single transverse spin dependent differential cross section can be expressed as

(2) |

where and are transverse momentum and rapidity of the lepton pair, with , and is defined as . When , the structure function can be formulated in terms of the TMD factorization where the quark Sivers function is involved Brodsky:2002cx ; Collins:2002kn , whereas when it can be calculated in the collinear factorization approach in terms of the twist-three quark-gluon-quark correlation functions Ji:2006ub ; Efremov:1981sh ; Qiu:pp . It has been shown that the TMD and collinear twist-three approaches give the consistent results in the intermediate transverse momentum region: Ji:2006ub ; bbdm . This consistency allows us to separate into two terms Collins:1984kg ,

where the first term dominates in the region, while the second term dominates in the region of and . The latter is obtained by subtracting the the leading term of from the full perturbative calculation. In this paper, we focus on the low transverse momentum region, where a TMD factorization is appropriate. We will review how perturbative corrections modify the differential cross sections, in particular, from the large logarithms in fixed order calculations. The results for up to one-loop corrections will be shown.

### ii.2 Perturbative Contribution in the Small Region

To study the QCD dynamics, in particular, to understand the scale evolution of the TMDs, it is illustrative to have a perturbative calculation for the above quantities at small limit. It is straightforward to write down the leading Born diagram contributions to and ,

(3) |

where , , and
are the integrated quark and antiquark distribution functions. The single
transverse spin asymmetry comes from the quark Sivers function
^{2}^{2}2Transverse-momentum
moment of the Sivers function defined in Bacchetta:2004jz as differs from
by a normalization factor . In this paper, follows the definition of
Ref. Ji:2006ub ..
The Sivers function follows the Trento convention Bacchetta:2004jz . Since it
is process dependent, we adopt that in the Drell-Yan process
to calculate the transverse-momentum moment, which is also defined as twist-three
quark-gluon-quark correlation function,

(4) | |||||

where and , while , is the light-cone gauge link to make the above definition gauge invariant.

At one-loop order, the gluon radiation contribution comes from real and virtual diagrams. The real diagrams have been calculated in the literature Ji:2006ub , and we can write down the results as Ji:2006ub

(5) | |||||

(6) | |||||

where and , and we have kept the (with represents the transverse dimension in the dimension regulation) term in the above calculations. After Fourier transformation into the impact parameter space, this will lead to a finite contribution. In the above results, we only keep the soft and hard gluon pole contributions in the channel for the single spin asymmetry calculations. All other contributions can be formulated similarly.

Applying the Fourier transform formulas we listed in the Appendix, we obtain the following result for the real gluon radiation contribution to at one-loop order,

(7) | |||||

where and a common integral as that in Eq. (6) has been omitted for simplicity. To arrive the above result, we have applied the subtraction scheme with . This is different from the used in the Collins book Collins (see the discussions below).

The above result contains collinear and soft divergences. The soft divergences shall be cancelled by the virtual diagrams, whereas the collinear divergences absorbed into the renormalization of the parton distributions. The virtual diagrams contributes

(8) |

Adding them together, we will have

(9) | |||||

where is the quark splitting kernel.

Similarly, for the single-spin dependent cross section, we have for the real diagram contributions,

(10) | |||||

where we have simplified the expression by integrating out the partial derivative in Eq. (6).
The last second line comes from the second term in the Fourier transform
formula of Eq. (98) in the Appendix ^{3}^{3}3This term accounts
for the partial difference between previous calculations of Refs. Vogelsang:2009pj ; Kang:2008ey ; Zhou:2008mz
and Ref. Braun:2009mi on the splitting kernel for . In particular, after adding
a similar contribution in the calculation of Ref. Vogelsang:2009pj , it can be shown that
the derivation in Ref. Vogelsang:2009pj agrees with that in Ref. Braun:2009mi ..
The virtual contribution has the similar form as Eq. (8). After adding them together,
we find that the total contribution at one-loop order,

(11) | |||||

where is the same as above, and the splitting kernel for the Sivers function can be written as

(12) | |||||

which agrees with recent calculations for the splitting kernel for the part involved in the above calculations Braun:2009mi ; Schafer:2012ra ; Kang:2012em ; Ma:2012xn .

In the above results, the one-loop corrections Eqs. (9,11) clearly demonstrate large logarithms. To resum these large logs, we need to apply the TMD factorization, and solve the relevant evolution equation. Although the different TMD schemes have been used in the literature, the final evolution for the structure functions remain the same. First, we examine the TMD factorization for the perturbative calculations at one-loop order.

### ii.3 Sivers Quark Distributions and TMD factorization

To demonstrate the factorization, we calculate the TMD quark and antiquark distributions, and show that the collinear part of the structure functions calculated in the last subsection can be absorbed into these TMD distributions. It has been known that, however, there is scheme-dependence in the TMD definitions. Therefore, the hard factors will also depend on which scheme you choose to calculate the TMDs. The scheme dependence comes from the fact that the TMD distributions have light-cone singularity, and different ways to regulate this singularity define different schemes of the TMD distributions.

#### ii.3.1 Ji-Ma-Yuan Scheme

In the Ji-Ma-Yuan scheme, the light-cone gauge link in the TMD definition is chosen to be slightly off-light-cone, with . Similarly, for the TMD for the antiquark distribution, was introduced, with . Because of the additional and , there are additional invariants: , , and . The TMD quark distributions of a polarized proton is defined through the following matrix:

(13) | |||||

with the gauge link

(14) |

This gauge link goes to , indicating that we adopt the definition for the TMD quark distributions for the Drell-Yan process. Keeping only the unpolarized quark distribution and the Sivers function, we have the following expansion for the matrix :

(15) |

where is the TMD distribution in an unpolarized proton, is the Sivers function, and is a hadron mass, used to normalize and to the same mass dimension.

First, the soft factor has been calculated,

(16) |

The calculations of the TMDs in the Ji-Ma-Yuan is straightforward, and we find that the quark distribution can be written as,

(17) |

where and in the impact parameter space,

(18) | |||||

The virtual diagram contributes,

(19) |

Adding them together, we have

(20) | |||||

Similar expression can be written for the antiquark distribution. According to the TMD factorization, we can subtract the quark distribution, antiquark distribution and the soft factor, and obtain the hard factor,

(21) |

Applying these results, we have the following result for the hard factor,

(22) |

where has been used to simplify the hard factor.

We can follow the same procedure to calculate that for the Sivers asymmetry. The TMD quark Sivers function can be written as,

(23) | |||||

We note a factor of (-2) in the last term, which is different from that in Ref. Ji:2006ub . This comes from a sub-leading expansion contribution from the soft-pole and hard-pole diagrams, which was omitted in Ref. Ji:2006ub . This term will contribute to the collinear singularity when Fourier transforming into the impact parameter space. Adding the virtual diagram contributions, we will have total result in the impact parameter space,

(24) | |||||

at one-loop order. By subtraction, we obtain the hard factor for the Sivers single spin asymmetry in Drell-Yan process,

(25) |

This is an important result, as it shows that the hard factor is spin-independent.

#### ii.3.2 Collins-11

In 2011, Collins introduces a new definition for the TMDs, where the soft gluon and light-cone singularities are subtracted in the TMDs from the beginning. As a result, there is no soft factor in the factorization formula, which is absorbed into the definition of PDF.

From its definition, we find that the real diagram contribution can be written as Collins

(26) |

where is defined as with the rapidity cutoff to regulate the light-cone singularity. The virtual diagram for only contributes to the counter terms,

(27) |

where, to be consistent, we have followed the prescription of subtraction of Ref. Collins .

Therefore, the total quark distribution can be written as,

(28) | |||||

We notice that an additional term of shall be added to the above equation if we use the subtraction method of the last subsection, see, also the detailed discussions in Ref. Collins:2012uy . To calculate the hard factor in this scheme, we apply the factorization

(29) |

The hard factor can be calculated Collins ,

(30) |

We notice that the different subtraction method will lead to different hard factors in Collins-11 scheme for the TMD definition Collins:2012uy . In particular, the subtraction used in the last sub-section will add additional term of in the above hard factor. This is because the Collins-11 definition of the TMD distribution, there is a double pole in the UV divergence in the dimensional regulation for the virtual diagram Collins:2012uy .

For the Sivers function, the calculations can follow similarly. The real diagram contribution for the quark Sivers function can be calculated in the Collins-11 definition,

(31) | |||||

Virtual diagram is the same as the unpolarized case, and the total quark Sivers function in -space,

(32) | |||||

where, again, we have followed the prescription for subtraction in Collins-11 definition of the TMDs. Again, we find that the hard factor can be calculated