# Semi-Inclusive Deep Inelastic Scattering in Wandzura-Wilczek-type approximation

###### Abstract

We present the complete cross-section for the production of unpolarized hadrons in semi-inclusive deep-inelastic scattering up to power-suppressed terms in the Wandzura–Wilczek-type approximation which consists in systematically assuming that –terms are much smaller than –correlators. We compute all twist-2 and twist-3 structure functions and the corresponding asymmetries, and discuss the applicability of the Wandzura–Wilczek-type approximations on the basis of available data. We make predictions that can be tested by data from Jefferson Lab, COMPASS, HERMES, and the future Electron-Ion Collider. The results of this paper can be readily used for phenomenology and for event generators, and will help to improve our understanding of the TMD theory beyond leading twist.

###### Keywords:

Wandzura–Wilczek approximation, semi-inclusive deep-inelastic scattering, transverse momentum dependent distribution and fragmentation functions, spin and azimuthal asymmetries, leading and subleading twistJLAB-THY-18-2775 Department of Physics, University of Connecticut, Storrs, CT 06269, U.S.A. Thomas Jefferson National Accelerator Facility, Newport News, VA 23606, U.S.A. Joint Institute for Nuclear Research, Dubna, 141980 Russia Institut für Theoretische Physik, Universität Regensburg, 93040 Regensburg, Germany CERN, 1211 Geneva 23, Switzerland Department of Physics, New Mexico State University, Las Cruces, NM 88003-001, USA Department of Theoretical Physics, University of the Basque Country UPV/EHU, 48080 Bilbao, Spain, and IKERBASQUE, Basque Foundation for Science, 48013 Bilbao, Spain Department of Physics, University of Connecticut, Storrs, CT 06269, U.S.A.

## 1 Introduction

A great deal of what is known about the quark-gluon structure of nucleons is due to studies of parton distribution functions (PDFs) in deep-inelastic reactions. Leading-twist PDFs tell us how likely it is to find an unpolarized parton (described by PDF , ) or a longitudinally polarized parton (described by PDF , ) in a fast-moving unpolarized or longitudinally polarized nucleon, which carries the fraction of the nucleon momentum. This information depends on the “resolution (renormalization) scale” associated with the hard scale of the process. Although the PDFs and continue being the subject of intense research (small-, large-, helicity sea and gluon distributions) they can be considered as rather well-known, and the frontier has been extended in the last years to go beyond the one-dimensional picture offered by those PDFs.

One way to do this consists in a systematic inclusion of transverse parton momenta , whose effects manifest themselves in terms of transverse momenta of the reaction products in the final state. If these transverse momenta are much smaller than the hard scale of the process, the formal description is given in terms of transverse momentum dependent distribution functions (TMDs) and fragmentation functions (FFs), which are defined in terms of quark-quark correlators Kotzinian:1994dv (); Mulders:1995dh (); Boer:1997nt (); Goeke:2005hb (); Bacchetta:2006tn (). Both of them depend on two independent variables: in the case of TMDs, on the fraction of nucleon momentum carried by the parton and intrinsic transverse momentum of the parton, while in the case of FFs, on the fraction of the parton momentum transferred to the hadron and the transverse momentum of the hadron acquired during the fragmentation process. Being a vector in the plane transverse with respect to the light-cone direction singled out by the hard momentum flow in the process, allows us to access novel information on the nucleon spin structure through correlations of with the nucleon and/or parton spin. The latter is a well-defined concept for twist-2 TMDs interpreted in the infinite momentum frame or in the lightcone quantization formalism.

One powerful tool to study TMDs are measurements of the semi-inclusive deep-inelastic scattering (SIDIS) process. By exploring various possibilities for the lepton beam and target polarizations unambiguous information can be accessed on the 8 leading-twist TMDs Boer:1997nt () and, if one assumes factorization, on certain linear combinations of the 16 subleading-twist TMDs Goeke:2005hb (); Bacchetta:2006tn (). It is important to stress that this information could not have been obtained without advances in target polarization techniques employed in the HERMES, COMPASS and JLab experiments Stock:1994vv (); Crabb:1997cy (); Goertz:2002vv (). Complementary information can be obtained from the Drell–Yan process Arnold:2008kf (), and annihilation Metz:2016swz ().

In QCD the TMDs are independent functions. Each TMD contains unique information on a different aspect of the nucleon structure. Twist-2 TMDs have partonic interpretations. Twist-3 TMDs give insights on quark-gluon correlations in the nucleon Miller:2007ae (); Burkardt:2007rv (); Burkardt:2009rf (). Besides positivity constraints Bacchetta:1999kz () there is little model-independent information on TMDs. An important question with practical applications is: do useful approximations for TMDs exist? Experience from collinear PDFs encourages to explore this possibility: the twist-3 and can be respectively expressed in terms of contributions from twist-2 and , and additional quark-gluon-quark () correlations or current-quark mass terms Wandzura:1977qf (); Jaffe:1991ra () (the index does not include gluons for , and other chiral-odd TMDs below). We shall refer to the latter generically as –terms, keeping in mind one deals in each case with matrix elements of different operators. The –correlations contain new insights on hadron structure, which are worthwhile exploring for their own sake, see Jaffe:1989xx () on .

The striking observation is that the –terms in and are small: theoretical mechanisms predict this Balla:1997hf (); Dressler:1999hc (); Gockeler:2000ja (); Gockeler:2005vw (), and in the case of data confirm or are compatible with these predictions Abe:1998wq (); Anthony:2002hy (); Airapetian:2011wu (). This approximation (“neglect of –terms”) is commonly known as Wandzura–Wilczek (WW) approximation Wandzura:1977qf (). The possibility to apply this type of approximation also to TMDs has been explored in specific cases in Kotzinian:1995cz (); Kotzinian:1997wt (); Kotzinian:2006dw (); Avakian:2007mv (); Metz:2008ib (); Teckentrup:2009tk (); Tangerman:1994bb (). In both cases, PDFs and TMDs, one basically assumes that the contributions from –terms can be neglected with respect to –terms. But the nature of the omitted matrix elements is different, and in the context of TMDs one often prefers to speak about WW-type approximations.

The present work is the first study of all SIDIS structure functions up to twist-3 in a unique approach. Our results are of importance for measurements performed or in preparation at HERMES, COMPASS, Jefferson Lab (JLab) with 12GeV beam-energy upgrade, or proposed in the long-term (Electron Ion Collider), and provide helpful input for the development of Monte Carlo event generators Avakian:2015vha ().

Our predictions, whether confirmed or not supported by current and future experimental data, will in any case provide a useful benchmark, and call for dedicated theoretical studies to explain (i) why the pertinent –terms are small or (ii) why they are sizable. In either case our results will deepen the understanding of –correlations, pave the way towards testing the validity of the TMD factorization approach at subleading twist, and help us to guide further developments.

In this work, after introducing the SIDIS process and defining TMDs and FFs (Sec. 2), we shall introduce the WW(-type) approximations, and review what is presently known about them from experiment and theory (Sec. 3). We will show that under the assumption of the validity of these approximations all leading and subleading SIDIS structure functions are described in terms of a basis of 6 TMDs and 2 FFs (Sec. 4), and review how these basis functions describe available data (Sec. 5). We will systematically apply the WW and/or WW-type approximations to SIDIS structure functions at leading (Sec. 6) and subleading (Sec. 7) twist, and conclude with a critical discussion (Sec. 8). The Appendices A and B contain technical details. In App. C we describe an open-source package implemented in mathematica Mathematica () (already available) and Python (to be released in the near future) that is made publicly available on github.com: https://github.com/prokudin/WW-SIDIS

## 2 The SIDIS process in terms of TMDs and FFs

In this section we review the description of the SIDIS process, define structure functions, PDFs, TMDs, FFs and recall how they describe the SIDIS structure functions.

### 2.1 The SIDIS process

The SIDIS process is sketched in Fig. 1. Here, and are the momenta of the incoming lepton and nucleon, and are the momenta of the outgoing lepton and produced hadron. The virtual-photon momentum defines the z-axis, and points in the direction of the x-axis from which azimuthal angles are counted. The relevant kinematic invariants are

(1) |

Note that we consider the production of unpolarized hadrons in DIS of charged leptons (electrons, positrons, muons) at in the single-photon exchange approximation, where denotes the mass of the electroweak gauge boson. In addition to , , the cross section is also differential in the azimuthal angle of the produced hadron, the square of its momentum component perpendicular with respect to the virtual-photon momentum. The cross section is also differential with respect to the azimuthal angle characterizing the overall orientation of the lepton scattering plane around the incoming lepton direction. The angle is calculated with respect to an arbitrary reference axis, which in case of transversely polarized targets is chosen to be the direction of . In the DIS limit , where the latter is the azimuthal angle of the spin-vector defined as in Fig. 1. It is convenient to define the unpolarized lepton–quark scattering subprocess cross section

(2) |

To leading order in the SIDIS cross-section is given by

(3a) | |||

Here is the structure function due to transverse polarization of the virtual photon (sometimes denoted as ), and we neglect corrections in kinematic factors and a structure function (sometimes denoted as ) arising from longitudinal polarization of the virtual photon (and another structure function , see below). The structure functions (and asymmetries) also depend on via the scale dependence of TMDs and FFs, which we do not show in formulas throughout this work. | |||

At subleading order in the expansion one has | |||

(3b) |

Neglecting corrections, the kinematic prefactors are given by

(4) |

and the asymmetries , are defined in terms of structure functions , as follows

(5) |

Here the first subscript denotes the unpolarized beam (longitudinally polarized beam with helicity ). The second subscript refers to the target, which can be unpolarized (longitudinally or transversely polarized with respect to virtual photon). The superscript “weight” indicates the azimuthal dependence with no index indicating an isotropic angular distribution of the produced hadrons.

In the partonic description the structure functions in (3a) are “twist-2.” Those in (3b) are “twist-3” and contain a factor in their definitions, see below, where is the nucleon mass. In our treatment to accuracy we neglect two structure functions due to longitudinal virtual-photon polarization, which contribute at order in the partonic description of the process, one being and the other contributing to the angular distribution Bacchetta:2006tn ().

Experimental collaborations often define asymmetries in terms of counts . This means the kinematic prefactors and are included in the numerators or denominators of the asymmetries which are averaged over within experimental kinematics. We will call the corresponding asymmetries . For instance, in the unpolarized case one has

(6) |

where denotes the total (–averaged) number of counts and the dots indicate further kinematic variables in the kinematic bin of interest (which may also be averaged over). It would be preferable if asymmetries were analyzed with known kinematic prefactors divided out on event-by-event basis. One could then directly compare asymmetries measured in different experiments and kinematics, and focus on effects of evolution or power suppression for twist-3. In practice, often the kinematic factors were included. We will define and comment on the explicit expressions as needed.

For completeness we remark that after integrating the cross section over transverse hadron momenta one obtains

(7a) | |||||

(7b) |

where (and analogous for the other structure functions)

(8) |

and the asymmetries are defined as

(9) |

The connection of “collinear” SIDIS structure functions in (7a, 7b) to those known from inclusive DIS is established by integrating over and summing over hadrons as

(10a) | ||||||

(10b) | ||||||

(10c) | ||||||

(10d) |

where signals the twist-3 character of . Notice that has no DIS counterpart due to time-reversal symmetry of strong interactions, and terms suppressed by are consequently neglected throughout this work including the twist-4 DIS structure function .

### 2.2 TMDs, FFs and structure functions

TMDs are defined in terms of light-front correlators

(11) |

where the Wilson-lines refer to the SIDIS process Collins:2002kn (). For a generic four-vector we define the light-cone coordinates with . The light-cone direction is singled out by the virtual-photon momentum and transverse vectors like are perpendicular to it. In the virtual-photon–nucleon center-of-mass frame, the nucleon and the partons inside it move in the –lightcone direction, while the struck quark and the produced hadron move in the –light-cone direction. In the nucleon rest frame the polarization vector is given by with .

The 8 leading-twist TMDs Boer:1997nt () are projected out from the correlator (11) as follows ( blue: T-even TMDs, red: T-odd TMDs; all TMDs depend on , , renormalization scale and carry a flavor index which we do not indicate for brevity):

(12a) | |||||

(12b) | |||||

(12c) | |||||

and the 16 subleading-twist TMDs Mulders:1995dh (); Bacchetta:2006tn () are given by | |||||

(12d) | |||||

(12e) | |||||

(12f) | |||||

(12g) | |||||

(12h) | |||||

(12i) |

where . The indices refer to the plane transverse with respect to the light-cone, and . Dirac-structures not listed in (12a–12i) are twist-4 Goeke:2005hb (). Integrating out transverse momenta in the correlator (11) leads to the “usual” PDFs known from collinear kinematics Ralston:1979ys (); Jaffe:1991ra (), namely at twist-2 level

(13a) | |||||

(13b) | |||||

(13c) | |||||

and at twist-3 level | |||||

(13d) | |||||

(13e) | |||||

(13f) |

Other structures drop out either due to explicit –dependence, or due to the sum rules Bacchetta:2006tn ()

(14) |

imposed by time reversal constraints.

Fragmentation functions are defined through the following correlator [10] (where denotes the transverse momentum of the produced hadrons acquired during the fragmentation process with respect to the quark)

(15) |

In this work we will consider only unpolarized final-state hadrons. If the produced hadron moves fast in the light-cone direction, the twist-2 FFs are projected out as

(16a) | |||||

(16b) | |||||

and at twist-3 level | |||||

(16c) | |||||

(16d) | |||||

(16e) | |||||

(16f) |

The FFs depend on , , renormalization scale, quark flavor and type of hadron which we do not indicate for brevity. Integration over transverse hadron momenta leaves us with , , while the other structures drop out due to their dependence.

The structure functions in Eqs. (3a, 3b) are described in the Bjorken limit at tree level in terms of convolutions of TMDs and FFs. We define the unit vector and use the following convolution integrals (see Appendix B.1 for details)

(17) |

where is a weight function which in general depends on and . The 8 leading–twist structure functions are

(18a) | |||||

(18b) | |||||

(18c) | |||||

(18d) | |||||

(18e) | |||||

(18f) | |||||

(18g) | |||||

(18h) |

At subleading–twist we have the structure functions

(19a) | |||||

(19b) | |||||

(19c) | |||||

(19d) | |||||

(19e) | |||||

(19f) | |||||

(19g) | |||||

(19h) | |||||

where and . The tilde-functions are defined in terms of -correlators, see Sec. 3.2. The weight functions are defined as

(20) |

and . In the index indicates the (maximal) power with which the corresponding contribution scales, and index (if any) distinguishes different types of contributions at the given order . Notice that twist-3 structure functions in Eqs. (19a–19h) contain an explicit factor . We also recall that we neglect two structure functions (denoted in Bacchetta:2006tn () as and ) due to longitudinal virtual-photon polarization, which are of order in the TMD partonic description.