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Matevosyan, Hrayr H.; Bentz, Wolfgang; Cloët, Ian Christopher; Thomas, Anthony William Transverse-momentum-dependent fragmentation and quark distribution functions from the Nambu-Jona-Lasinio-jet model Physical Review D, 2012; 85(1):014021

© 2012 American Physical Society

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11th April 2013

Transverse-momentum-dependent fragmentation and quark distributionfunctions from the Nambu–Jona-Lasinio–jet model

Hrayr H. Matevosyan,1 Wolfgang Bentz,2 Ian C. Cloet,3,1 and Anthony W. Thomas1

1CSSM and ARC Centre of Excellence for Particle Physics at the Tera-scale, School of Chemistry and Physics,University of Adelaide, Adelaide SA 5005, Australia*

2Department of Physics, School of Science, Tokai University, Hiratsuka-shi, Kanagawa 259-1292, Japan†3Department of Physics, University of Washington, Seattle Washington 98195, USA‡

(Received 12 November 2011; published 18 January 2012)

Using the model of Nambu and Jona-Lasinio to provide a microscopic description of both the structure

of the nucleon and of the quark to hadron elementary fragmentation functions, we investigate the

transverse-momentum dependence of the unpolarized quark distributions in the nucleon and of the quark

to pion and kaon fragmentation functions. The transverse-momentum dependence of the fragmentation

functions is determined within a Monte Carlo framework, with the notable result that the average P2? of

the produced kaons is significantly larger than that of the pions. We also find that hP2?i has a sizable z

dependence, in contrast with the naive Gaussian ansatz for the fragmentation functions. Diquark

correlations in the nucleon give rise to a nontrivial flavor dependence in the unpolarized transverse-

momentum-dependent quark distribution functions. The hk2Ti of the quarks in the nucleon are also found tohave a sizable x dependence. Finally, these results are used as input to a Monte Carlo event generator for

semi-inclusive deep inelastic scattering (SIDIS), which is used to determine the average transverse

momentum squared of the produced hadrons measured in SIDIS, namely, hP2Ti. Again, we find that the

average P2T of the produced kaons in SIDIS is significantly larger than that of the pions and in each case

hP2Ti has a sizable z dependence.

DOI: 10.1103/PhysRevD.85.014021 PACS numbers: 13.60.Hb, 12.39.Ki, 13.60.Le, 13.87.Fh

I. INTRODUCTION

Semi-inclusive deep inelastic scattering (SIDIS) has avery rich structure which provides a wealth of observablesfar in excess of the familiar inclusive deep inelastic scat-tering. The two-dimensional picture of a target provided bySIDIS promises many new insights into nucleon and nu-clear structure [1–4]. For example, it has been realized thatSIDIS may shed light on the angular momentum structureof the proton in terms of the spin and orbital angularmomentum of its quarks and gluons [5–7]. It will alsoprovide new information on the in-medium modificationof bound nucleons and deepen our understanding of QCDitself [1–4]. The study of the transverse-momentum distri-bution of hadrons produced in SIDIS [1–4,8–11] is char-acterized by determining the transverse-momentum-dependent (TMD) parton distribution functions (PDFs)and the TMD fragmentation functions.

Early theoretical models of the fragmentation functionshave been constructed in Refs. [12–16] and more recentlythe development of the Nambu–Jona-Lasinio–jet model[17] has provided a framework which automaticallysatisfies the relevant sum rules. Lattice QCD studies ofTMD PDFs are presented in Ref. [18] and the QCD evo-lution of TMD PDFs is discussed in Ref. [19]. Extensive

phenomenological data analysis of transverse momentumin distribution and fragmentation processes was presentedin Ref. [20]. Considerable experimental work has alreadybeen carried out at JLab [21–25], HERMES [26–28], andCOMPASS [29–31], while for an overview of the futureperspectives for this field we refer to the recent review byAnselmino et al. [32].In this work, we present the first microscopic calculation

of the spin-independent TMD quark distribution functionsin the nucleon and the TMD quark to pion and kaonfragmentation functions, where none of the parametersare adjusted to TMD data. The underlying theoreticalframework is the Nambu–Jona-Lasinio (NJL) model[33,34]. While this certainly represents a simplificationof QCD, it has many desirable properties. For example, itis covariant and respects the chiral symmetry of QCD,including its dynamical breaking. Moreover, it describesthe spin and flavor dependence of the nucleon PDFs, aswell as their modification in-medium [35–37]. It alsoproduces transversity quark distributions [38] which arein good agreement with the empirical distributions ex-tracted by Anselmino et al. [39].For the present purpose, the recent developments in the

NJL-jet model [17,40,41], which provides a quark-jet de-scription of the fragmentation process using elementaryfragmentation functions calculated within the standardNJL model, are also critical. This framework provides agood description of the parametrizations of experimentaldata and has been extended to include vector meson

*http://www.physics.adelaide.edu.au/cssm†http://www.sp.u-tokai.ac.jp/‡http://www.phys.washington.edu/

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resonances and nucleons as fragmentation channels. Theuse of Monte Carlo methods to calculate these fragmenta-tion functions has also been implemented and that develop-ment allows us to address a wider array of processes withinthe model, including physical cross-section calculations.

In Sec. II, we present the general formalism for describ-ing transverse-momentum distributions in SIDIS, includ-ing the quark-jet model originally proposed by Field andFeynman. The calculation of the elementary, unintegratedfragmentation functions in the NJL model, which are theinput to the jet model which describes the TMD fragmen-tation functions in quark hadronization, is explained inSec. III. Our model for the TMD quark distribution func-tions in the nucleon is outlined in Sec. IV, where we alsopresent results for the TMD PDFs. Results for the TMDfragmentation functions are discussed in Sec. V and theaverage transverse momentum in the SIDIS process, de-termined using our Monte Carlo event generator, is dis-cussed in Sec. VI. Finally, Sec. VII contains a summaryand outlook.

II. TRANSVERSE MOMENTUM INTHE NJL-JET MODEL

The kinematics of semi-inclusive hadron production,lN ! l0hX, is illustrated schematically in Fig. 1, where alepton with momentum l scatters on a target, by emitting avirtual photon with momentum q that hits a quark withinitial momentum k. As usual, the z axis is chosen tocoincide with the direction of the photon’s momentum,where the target has its momentum in the negative zdirection. The transverse momenta in the process—labeledwith a subscript T—are defined with respect to this z axis,so that the photon and target have no transverse-momentum component (�N collinear kinematics). Theangle between the lepton scattering plane and the quarkscattering plane is denoted as ’. We allow for the struck

quark in the target to carry a transverse momentum kT.Some of this transverse momentum is then transferred tothe hadrons emitted by the quark.The kinematics of the quark fragmentation process is

depicted in Fig. 2. The emitted hadron h carries a trans-verse momentum PT with respect to the z axis which canbe decomposed into two contributions. First, the quarktransfers a fraction of its transverse momentum kT to thehadron and second the hadron also acquires a momentumtransverse to the direction of the quark’s momentum, P?.Up to corrections of order Oðk2T=Q2Þ, the following rela-tion holds [42]:

PT ¼ P? þ zkT: (1)

This relation allows one to probe the quark transversemomentum inside a nucleon by measuring the z depen-dence of the emitted hadron’s transverse momentum hP2

Ti,provided hP2

?i is independent of z. However, in the NJL-jetmodel framework we find that hP2

?i is strongly z dependentand this z dependence is also observed at COMPASS [31].A recent analysis of the HERMES data [27] was performedin Ref. [20], where a Gaussian ansatz for the TMD quarkdistribution and fragmentation functions was assumed andan average was performed over the quark flavor and type ofhadron detected. Using a fit region of 0:2< z < 0:7, theyextracted the following results for the average transversemomentum squared [20]:

hk2Ti ¼ 0:38� 0:06 GeV2; (2)

hP2?i ¼ 0:16� 0:01 GeV2: (3)

The latest iteration of the NJL-jet model [41] employsMonte Carlo simulations to calculate the integrated quarkfragmentation functions. It assumes that the initial highenergy quark emits hadrons in a cascade-like process,schematically depicted in Fig. 3. At every emission vertex,we choose the type of emitted hadron h and its fraction ofthe light-cone momentum z of the fragmenting quark, byrandomly sampling the corresponding elementary quark

FIG. 1 (color online). Illustration of the three-dimensionalkinematics of SIDIS. The photon momentum defines the z axisand the struck quark has initial transverse momentum kT in thenucleon, with respect to the z axis.

z

xy

ph

kT

P

PT

kT

k

q Nucleon

FIG. 2 (color online). Illustration of the kinematics of SIDIS,where the final transverse momentum of the produced hadronwith respect to the z axis is denoted by PT, which is related to theinitial quark transverse momentum in the nucleon kT and thatgenerated in the fragmentation process P? by Eq. (1).

MATEVOSYAN et al. PHYSICAL REVIEW D 85, 014021 (2012)

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fragmentation (splitting) functions, dhqðzÞ, that are calcu-

lated within the NJL model. In each elementary fragmen-tation process, we record the flavor of the initial and finalquarks and the type of the emitted hadron, we also note thelight-cone momentum fraction of the initial quark trans-ferred to the hadron and that left to the final quark. Thefragmentation chain is stopped after the quark has emitteda predefined number of hadrons, NLinks. We repeat thecalculation NSims times, with the same initial quark flavor,q, until we have sufficient statistics for the emitted had-rons. The fragmentation functions are then extracted bycalculating the average number of hadrons of type h, withlight-cone momentum fraction z to zþ�z, which wedenote by hNh

qðz; zþ�zÞi. The fragmentation function in

the domain ½z; zþ�z� is then given by

DhqðzÞ�z ¼ hNh

qðz; zþ�zÞi �P

NSims

Nhqðz; zþ �zÞNSims

: (4)

In this work, we extend the NJL-jet model to include thetransverse-momentum dependence of the emitted hadronsin the fragmentation process. This is achieved by usingTMD elementary quark fragmentation functions at thehadron emission vertices and by keeping track of thetransverse momenta of all the particles in the process.Our goal is to calculate the TMD fragmentation function,Dh

qðz; P2?Þ, using its probabilistic interpretation. That is, the

probability of a quark q to emit a hadron hwith a fraction zof its light-cone momentum and P? is given byDh

qðz; P2?Þdzd2P?.

We calculate elementary (one-step) TMD splitting func-

tions, dhqðz; p2?Þ, using the NJL model, where p? denotes

the transverse component of the hadron’s momentum withrespect to the parent quark, as illustrated in Figs. 3 and 4. Ineach step of the Monte Carlo simulation of the quarkcascade emission, we randomly sample the type, thelight-cone momentum fraction, z, and the transversemomentum, p?, of the emitted hadron using as the proba-bility distribution the elementary TMD splitting functionsof the quark, where the elementary probability is

dðz; p2?Þdzd2p?. Schematically, the quark emission pro-

cess is depicted in Fig. 4, where the z0 axis denotes thedirection of the original parent quark’s 3-momentum. The

vectors k and k0 denote the 3-momentum of an arbitraryquark in the cascade chain before and after hadron emis-sion with transverse components k? and k0?, respectively.The emitted hadron’s momentum is labeled by ph, whereits transverse component with respect to k and the z0 axis isdenoted by p? and P?, respectively. P? is obtained usingthe relation P? ¼ p? þ zk?, analogous to that in Eq. (1).The recoil transverse momentum of the final quark, k0?, iscalculated from momentum conservation in the transverseplane, namely,

k? þ P? þ k0?: (5)

The TMD fragmentation function is then calculated afterthe trivial integration over the polar angle of P? in thetransverse plane, that is

Dhqðz;P2

?Þ�z��P2?¼hNh

qðz;zþ�z;P2?;P

2?þ�P2

?i

�P

NSimsNh

qðz;zþ�z;P2?;P

2?þ�P2

?ÞNSims

:

(6)

The model can easily accommodate the initial transversemomentum of the quark, for example, with respect to thedirection of the virtual photon in SIDIS (see Fig. 1). Ourgoal is to describe the average transverse momentum of thehadrons produced in different reactions. The differentialcross section for SIDIS up to terms of order Oðk2T=Q2Þ canbe written as [42]

d3�lN!l0hX

dxdzdP2T

�Xq

e2qZ

d2kTqðx; k2TÞDhqðz; P2

?Þ

� Xq

e2q ~Dhqðx; z; P2

TÞ; (7)

where P? and PT are related by Eq. (1) and qðx; k2TÞ are theTMD quark distribution functions of the target. Thus, forSIDIS, we can use the TMD quark distribution functions torandomly sample the initial transverse momentum of thequark to calculate the relevant number density of the

qQ

p

FIG. 3 (color online). NJL-jet model including transversemomentum.

FIG. 4 (color online). Quark elementary fragmentation kine-matics, for an arbitrary hadron emission in the cascade chain.The z0 axis is defined by the direction of the 3-momentum of theoriginal parent quark.

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produced hadrons. In this work, we use TMD valencequark distributions in the nucleon—calculated within theNJL model—to determine the average transverse momen-tum of the produced hadrons with respect to the directionof the virtual photon, that is hP2

Ti, by calculating thecorresponding probability densities ~Dh

qðx; z; P2TÞ, using an

expression analogous to Eq. (6). In this way, we obtain aself-consistent description of the entire process in theregime where the virtual photon samples the valence quarkcomponent of the target, that is, when the struck quarkhas x * 0:3. The type of target and the allowed range of xin the Monte Carlo simulation can be matched to thosemeasured in any particular experiment.

In this article, we only consider the production of pseu-doscalar mesons, that is, the pions and kaons, as a first stepin determining the TMD fragmentation functions.Eventually, we will also include the vector mesons andnucleon-antinucleon channels, as done for the integratedfragmentation functions in Ref. [41].

III. ELEMENTARY TMD FRAGMENTATIONFUNCTIONS

In this section, we evaluate the ‘‘elementary’’ fragmen-tation functions of quarks to hadrons as a ‘‘one-step’’process in the NJL model, using light-cone coordinates.1

The NJL model which we use includes only four pointquark interactions in the Lagrangian, with up, down, andstrange quarks (see, for example, Refs. [43–45] for detailedreviews of the NJL model). In the present work, we use thenotation introduced in our previous studies [40,41].

The elementary fragmentation function for quark, q, toemit a meson, m, carrying light-cone momentum fraction,z, and carrying transverse momentum, p?, is depicted inFig. 5. In the frame where the fragmenting quark has zerotransverse momentum, but a nonzero transverse-momentum component �p?=z with respect to the direc-tion of the produced hadron [1,17], the unregularizedelementary TMD fragmentation functions to pseudoscalarmesons are given by

dmq ðz;p2?Þ

¼�Cmq

2g2mqQ

z

2

Z dkþdk�ð2�Þ4 Tr½S1ðkÞ�þS1ðkÞ�5

�ð6k� 6pþM2Þ�5��ðk��p�=zÞ2��ððp�kÞ2�M22Þ

¼ Cmq

16�3g2mqQz

p2?þ½ðz�1ÞM1þM2�2

½p2?þzðz�1ÞM2

1þzM22þð1�zÞm2

m�2:

(8)

The trace is over Dirac indices only and the subscripts onthe quark propagator, S1ðkÞ, and constituent masses, M1

andM2, denote quark flavors. Quark flavor is also indicatedby the subscripts q andQ, where a meson of typem has thequark flavor structure m ¼ q �Q and mm denotes the mesonmass. The corresponding isospin factor and quark-mesoncoupling constant are labeled by Cm

q and gmqQ, respec-

tively, and are determined within the NJL model [40,41].The integrated elementary splitting function is obtainedfrom the elementary TMD splitting function via integrationover p?, that is,

dmq ðzÞ ¼Z

d2p?dmq ðz; p2?Þ

¼ Cmq

2g2mqQz

Z d2p?ð2�Þ3

� p2? þ ½ðz� 1ÞM1 þM2�2

½p2? þ zðz� 1ÞM2

1 þ zM22 þ ð1� zÞm2

m�2:

(9)

The probability densities dmq ðzÞ are then obtained by multi-

plying a normalization factor so thatP

m

R10 dzd

mq ðzÞ ¼ 1.

The isospin and momentum sum rules are then satisfiedautomatically [40,41].Previously, we employed the Lepage-Brodsky (LB)

regularization scheme to calculate loop integrals such asthat in Eq. (9). This method puts a sharp cutoff on theinvariant mass squared, M2

12, of the particles in the finalstate (see Refs. [17,40,41,46] for a detailed description asapplied to the NJL-jet model). The maximum invariantmass of the two particles in the loop,�12, is determined by

M212 � �2

12 �� ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

�23 þ�2

1

qþ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�2

3 þ�22

q �2; (10)

where �1 and �2 denote the masses of the particles in theloop and �3 denotes the 3-momentum cutoff, which isfixed in the usual way by reproducing the experimentalpion decay constant. For a light constituent quark mass ofM ¼ 0:4 GeV, the corresponding 3-momentum cutoff is�3 ¼ 0:59 GeV. The strange constituent quark mass isdetermined by reproducing the experimental kaon mass,giving the value Ms ¼ 0:61 GeV and the correspondingquark-meson coupling constants are g�qQ ¼ 4:23 and

gKqQ ¼ 4:51.

FIG. 5 (color online). Feynman diagram describing the ele-mentary quark to hadron fragmentation functions.

1We use the following LC convention for Lorentz 4-vectorsðaþ; a�;a?Þ, a� ¼ 1ffiffi

2p ða0 � a3Þ, and a? ¼ ða1; a2Þ.

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In loop integrals containing two particles, we assign alight-cone momentum fraction x (of the initial particle’slight-cone momentum) to the particle with mass �1 andconsequently a light-cone momentum fraction 1� x forthe particle with mass mu2. Then, in the frame where theinitial particle’s transverse momentum is zero, the invariantmass of the two particles in the loop can be expressed as

M212 ¼

�21 þ p2

?x

þ�22 þ p2

?1� x

: (11)

The relation in Eq. (10), when applied to the integral inEq. (9), yields a sharp cutoff in the integral over thetransverse momentum, namely,

p2? � P 2

? � zð1� zÞ� ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

�23 þ�2

1

qþ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�2

3 þ�22

q �2

� ð1� zÞ�21 � z�2

2: (12)

A consequence of LB regularization is that it restricts thecorresponding regularized functions to a limited range of z,namely, 0< zmin � z � zmax < 1, where zmin and zmax aredetermined by imposing the condition P 2

? � 0 in Eq. (12).These range limitations depend on the masses of hadronsand quarks involved. For example, the z limits are veryclose to the endpoints (z ¼ 0 and z ¼ 1) for quark splittingfunctions to pions, but are further from these endpoints forheavier hadrons like kaons. The plots depicted in Fig. 6show the limited range for the normalized splitting func-tions of a u quark to �þ and Kþ, calculated using LBregularization.

In this work, we employ a slightly modified version ofthe LB regularization, which replaces the sharp cutoff ofthe invariant mass squared in the integrals, namely,�ð�2

12 �M212Þ, by a dipole regulator:

G12ðp2?Þ �

1

½1þ ðM212=�

212Þ2�2

: (13)

A physical motivation for this regularization scheme is thatit gives a pion quark distribution that at large x behaves

approximately as ð1� xÞ2:6 forQ2 ¼ 16 GeV2, which is ingood agreement with the recent reanalysis of Aicher et al.which finds ð1� xÞ2:34 at the same scale [47]. Using thisdipole cutoff version of the LB regularization scheme(LB-DIP), we fix the model parameters by reproducingthe experimentally measured hadronic properties, such asf� and the kaon mass to determine the cutoff as �3 ¼0:773 GeV and the strange constituent quark mass be-comes Ms ¼ 0:59 GeV. The corresponding quark-mesoncoupling constants are g�qQ ¼ 4:24 and gKqQ ¼ 4:52. The

quark distribution functions calculated with LB-DIP regu-larization satisfy both the number and momentum sumrules and allow us to set the model scale at Q2

0 ¼0:2 GeV2 in the usual way by comparing the evolvedpion distribution function with that obtained from experi-ment. This procedure is discussed in detail in Ref. [40].The plots in Fig. 6 clearly show that the z range of the

splitting functions calculated using LB-DIP allows for asmooth continuation of the corresponding splitting func-tions calculated using LB regularization to the endpointsz ¼ 0 and z ¼ 1. The plots in Fig. 7 present results for thefragmentation functions of a u quark to �þ and Kþ using

FIG. 6 (color online). The normalized integrated splittingfunctions for a u quark to �þ and Kþ, calculated using LBand LB-DIP regularizations with the same light constituentquark mass of M ¼ 0:4 GeV.

Q2=4 GeV2

z D

+ u(z

)

0.2

0.4

0.6

0.8

z0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

HKNS

DSS

NJL-jet

Q2=4 GeV2

z D

K+

u(z

)

0

0.05

0.10

0.15

0.20

0.25

0.30

z0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

HKNS

DSS

NJL-jet

FIG. 7 (color online). The integrated fragmentation functionsfor a u quark to �þ (upper) and Kþ (lower), calculated using theLB-DIP regularization with a light constituent quark mass ofM ¼ 0:4 GeV and evolved from the model scale to Q2 ¼4 GeV2. The results are compared to phenomenological parame-trizations of experimental data from Ref. [48] (HKNS) andRef. [49] (DSS). The shaded area represents the uncertaintiesin the HKNS results.

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the LB-DIP regularization scheme. We use the QCD evo-lution code of Ref. [50] at next-to-leading order to evolveour model results from the scale Q2

0 ¼ 0:2 GeV2 to Q2 ¼4 GeV2. We find a slightly better description of the em-pirical parametrizations compared to our earlier work[40,41], especially in the region where z is close to 1.Previously, artifacts of the LB regularization did not allowa good description in this domain.

IV. TMD QUARK DISTRIBUTIONSIN THE NUCLEON

The TMD quark distributions in the nucleon are againdetermined by utilizing the NJL model. The nucleon boundstate is described by a relativistic Faddeev equation thatincludes both scalar and axial-vector diquark correlations,where the static approximation is used to truncate the quarkexchange kernel [35]. The relevant terms of the NJL inter-action Lagrangian are

L I ¼ Gsð �c�5C�2�A �c TÞðc TC�1�5�2�Ac Þ

þGað �c��C�i�2�A �c TÞðc TC�1���2�i�Ac Þ;

(14)

where C ¼ i�2�0 and �A ¼ffiffi32

q�A (A 2 2, 5, 7) are the

color �3 matrices [35]. The strength of the scalar and axial-vector diquark correlations in the nucleon are determinedby the couplings Gs and Ga, respectively. To regularize theNJL model for the calculation of the nucleon, we choosethe proper-time scheme, with an infrared and ultravioletcutoff, labeled by �IR and �UV, respectively. This schemeenables the removal of unphysical thresholds for nucleondecay into quarks, and hence simulates an important aspectof confinement [51–53]. This simulation of quark confine-ment has also been shown to provide a natural saturationmechanism for nuclear matter in the NJL model [53].

The proper-time regularization scheme is not used forthe fragmentation functions because the emitted hadronsare not confined. Therefore, the confining nature of theproper-time regularization is not appropriate in this case.However, for consistency between both regularizationschemes we use the same light constituent quark massand fix the UV cutoff so as to reproduce the pion decayconstant.

The five parameters of our NJL model for the nucleonare the light constituent quark mass, M, the regularizationparameters �IR and �UV, and the couplings Gs and Ga.These are determined by fixing M ¼ 0:4 GeV, �IR ¼0:24 GeV, and then reproducing the nucleon mass, piondecay constant, and the nucleon axial coupling via Bjorkensum [37,54]. Strange quarks are not yet included in ourmodel for the nucleon.

The leading-twist spin-independent TMD distribution ofthe quarks of flavor q in the nucleon is defined via thecorrelator [6,55]

Qðx; kTÞ ¼ pþ Z d�d�Tð2�Þ3 eixpþ�e�ikT :�T

� hN; Sj �c qð0Þ�þW ðÞc q

� ð�; TÞjN; Sijþ¼0j; (15)

where W ðÞ is a gauge link connecting the two quarkfields, which are labeled by c q. In QCD, this gauge link is

nontrivial for �T � 0, however at the level of approxima-tion that we are working at, this gauge link equals unity inthe NJL model. Our states are normalized using the non-covariant light-cone normalization, namely,

Xq

hN; Sj �c qð0Þ�þc qð0ÞjN; Si ¼ 3: (16)

Equation (15) can be expressed in terms of two TMD quarkdistribution functions, namely,

Q ðx; kTÞ ¼ qðx; k2TÞ �"�þijkiTS

jT

Mq?1Tðx; k2TÞ; (17)

where the first TMD PDF integrated over kT gives thefamiliar unpolarized quark distribution function and thesecond TMD PDF, known as the Sivers function [56,57], istime-reversal odd and is zero at the level of approximationincluded in this work.To determine the TMD quark distributions in this model,

it is convenient to express them in the form [58,59]

qðx; k2TÞ ¼ �iZ dkþdk�

ð2�Þ4 �

�x� kþ

pþ

�Tr½�þMqðp; kÞ�;

(18)

where Mqðp; kÞ is the quark two-point function in the

bound nucleon. Therefore, within any model that describesthe nucleon as a bound state of quarks, the quark distribu-tion functions can be associated with a straightforwardFeynman diagram calculation.The Feynman diagrams considered here are given in

Fig. 8, where the first diagram represents the so-calledquark diagram and the second the diquark diagram. Thesingle line in each diagram represents a quark propagatorwhich is the solution to the gap equation and the double

FIG. 8 (color online). Feynman diagrams which give the un-polarized TMD quark distribution functions in the nucleon. Thesingle line represents the quark propagator and the double linethe diquark t matrix. The shaded oval denotes the quark-diquarkvertex function, obtained from a relativistic Faddeev equationand the operator insertion has the form �þ�ðx� kþ

pþÞ 12 ð1� �3Þ.

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line is the diquark t matrix obtained from the Bethe-Salpeter equation. The vertex functions represent thesolution to the nucleon Faddeev equation. The resultingdistributions have no support for negative x and thereforethis is essentially a valence quark picture. By separating theisospin factors, the spin-independent u and d TMD quarkdistributions in the proton can be expressed as

uðx; k2TÞ ¼ fsq=Nðx; k2TÞ þ 13f

aq=Nðx; k2TÞ þ 1

2fsqðDÞ=Nðx; k2TÞ

þ 56f

aqðDÞ=Nðx; k2TÞ; (19)

dðx; k2TÞ ¼ 23f

aq=Nðx; k2TÞ þ 1

2fsqðDÞ=Nðx; k2TÞ

þ 16f

aqðDÞ=Nðx; k2TÞ: (20)

The superscripts s and a refer to the scalar and axial-vectorterms, respectively, the subscript q=N implies a quarkdiagram and qðDÞ=N a diquark diagram. Explicit expres-sions for the functions in Eqs. (19) and (20) are given in theAppendix.

Results for the u- and d-quark TMD quark distributionsfunctions in the proton are illustrated in Fig. 9. The Q2

scale to which these results correspond is not determinedby the model. Previously, for the familiar spin-independentPDFs we fitted the valence u-quark distribution in theproton to the empirical result at some large Q2 scale, thisgives a model scale of Q2

0 ¼ 0:16 GeV2 [35,37,38] in the

proper-time regularization scheme. Rigorous comparisonwith the experimental data requires QCD evolution of themodel TMD PDFs, which is left for future work. Here, wejust show the results as they emerge from our model, theexact scale of which is not so important for this purpose.When QCD evolution is included, both the TMD PDF andTMD fragmentation function model scales must be equalwhen determining observables like SIDIS cross sections.The integral of these TMD PDF results over kT gives thefamiliar spin-independent quark distributions functions,which satisfy the baryon number and momentum sumrules. The Bjorken x and k2T dependence in these expres-sions is not separable, and therefore the Gaussian ansatz forthe TMD quark distributions, namely, that they can bewritten in the form

qðx; k2TÞ ¼ qðxÞ e�k2T=hk2T i

�hk2Ti; (21)

is not possible for our TMD PDF results. The Bjorken xdependence of hk2Ti for our proton TMD quark distributionresults is illustrated in Fig. 10, where

hk2TiðxÞ �Rd2kTk

2Tqðx; k2TÞR

d2kTqðx; k2TÞ: (22)

If the x and k2T dependence of our TMD quark distributionswere separable, then the curves in Fig. 10 would be con-stants, however we find that hk2Ti has about a 20% variationover the domain of Bjorken x. We also find that the x

)2 (GeV

2T

k

00.1

0.20.3

0.40.5 x

0 0.10.2

0.30.4 0.5

0.60.7 0.8

0.9 1

x u

0

0.5

1

1.5

2

2.5

3

3.5

)2 (GeV

2T

k

00.1

0.20.3

0.40.5 x

0 0.10.2

0.3 0.40.5

0.6 0.70.8

0.9 1

x d

0

0.2

0.4

0.6

0.8

1

1.2

1.4

FIG. 9 (color online). Results for the u (upper) and d (lower)TMD quark distributions in the proton.

<k2 T

> (

GeV

2 )

0

0.1

0.2

0.3

x0 0.2 0.4 0.6 0.8 1.0

ud

FIG. 10 (color online). The Bjorken x dependence of hk2Ti.Diquark correlations in the nucleon give rise to the quark flavordependence.

TRANSVERSE-MOMENTUM-DEPENDENT FRAGMENTATION . . . PHYSICAL REVIEW D 85, 014021 (2012)

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dependence of hk2Ti for the u and d quarks differs some-what, with the d quarks having slightly larger hk2Ti for themajority of Bjorken x.

Figure 11 illustrates our TMD quark distribution resultsat particular x values and compares them to a Gaussianansatz fit for the same x slice. The Gaussian ansatz resultsare obtained by a least squares fit of the TMD factor inEq. (21) to our ratios qðx; k2TÞ=qðxÞ calculated in the NJLmodel, using hk2Ti of Eq. (21) as the only fit parameter foreach value of x. The fitted value of this parameter isapproximately 20% smaller than the value of hk2Ti calcu-lated with our model distribution functions. In the leastsquares fit, we included values of k2T up to 4 GeV2 and thecurves in Fig. 11 indicate that such a fit to a single Gaussianis reasonable only for a limited k2T region, for a single valueof x.

V. TMD FRAGMENTATION FUNCTION RESULTS

In this section, we present NJL-jet model results for theTMD fragmentation functions. The number of emittedhadrons in the decay chain is set to NLinks ¼ 6, which is

sufficient to accurately obtain the pion and kaon fragmen-tation functions in the domain z * 0:02. We solve for thefragmentation of u, d, and s quarks to pions and kaons,utilizing Monte Carlo simulations and the expression inEq. (6), similar to our previous calculations of the inte-grated fragmentation functions detailed in Ref. [41]. Thecomputational challenge for the Monte Carlo simulationsis to obtain sufficient statistics and this becomes signifi-cantly more difficult when we include the transverse-momentum dependence, because now the number of binsbecomes quadratic in the size of the discrete bin size (takento be 1=500 both for z and transverse momentum, in thecorresponding units). Furthermore, the extent of the bins inthe transverse-momentum direction was extended to6 GeV2, in order to avoid any notable numerical artifactsarising from the limited range of transverse momentum. Toovercome the numerical challenge, our software platformwas developed to allow for parallel generation of theMonte Carlo quark decay cascades, with different seeds

x=0.4

u(x,

k2 T)

/ u(x

)

10 6

10 4

10 2

1

k2T (GeV2)

0 1 2 3 4

NJL, <k2T>= 0.17 GeV2

Gauss Fit, <k2T>= 0.13 GeV2

x=0.8

u(x,

k2 T)

/ u(x

)

10 6

10 4

10 2

1

k2T (GeV2)

0 1 2 3 4

NJL, <k2T>= 0.22 GeV2

Gauss Fit, <k2T>= 0.18 GeV2

FIG. 11 (color online). Results for the TMD u-quark distribu-tion in the proton for fixed x slices, where the upper plot hasx ¼ 0:4 and the lower plot is for x ¼ 0:8. Also plotted areindividual fits to the TMD quark distribution using theGaussian ansatz of Eq. (21) for each x, with hk2Ti the single fitparameter in each case.

)2

(GeV

2P00.1

0.20.3

0.40.5

z0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

+ π uz

D

0

0.2

0.4

0.6

0.8

1

1.2

1.4

)2

(GeV

2P00.1

0.20.3

0.40.5

z0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

- π uz

D

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

FIG. 12 (color online). TMD fragmentation functions for a uquark to �þ and ��. The upper figure illustrates the favoredcase, which peaks at relatively large z, while the unfavored case,shown in the lower figure, peaks at much smaller z.

MATEVOSYAN et al. PHYSICAL REVIEW D 85, 014021 (2012)

014021-8

for their random number generators. The results were latercombined to produce the high statistics solutions. Thecomputations were facilitated on the small computer clus-ter at the Special Research Centre for the SubatomicStructure of Matter (CSSM) that consists of 11 machineswith Intel Core i7 920 quad core CPUs running on theLinux Fedora Core 11 operating system and GCC 4.4. Atypical calculation of fragmentation for a given quark typetakes about 12 hours with 44 parallel processors.

Results for the TMD favored and unfavored fragmenta-tion functions for a u quark to � and K mesons areillustrated in Figs. 12 and 13. In each case, the favoredTMD fragmentation functions have more support at large z,while the unfavored results are peaked at smaller z. It is alsoevident that the kaon fragmentation functions fall off moreslowly in P2

? than the corresponding pion fragmentation

functions. The drop in each of the fragmentation functionsfor z & 0:02 is a consequence of choosing NLinks ¼ 6,which means that in the Monte Carlo simulation there

is a vanishingly small probability of emitting hadronswith z < 0:02.The Gaussian ansatz is widely used to describe the

traverse momentum dependence of both quark distributionand fragmentation functions. In particular, the TMD frag-mentation function of a quark q emitting a hadron h isoften modeled by

Dhqðz; P2

?Þ ¼ DhqðzÞ e

�P2?=hP2

?i

�hP2?i

; (23)

where DhqðzÞ is the corresponding integrated fragmentation

function and hP2?i is the average transverse momentum of

the produced hadron h, defined by

hP2?iðzÞ �

Rd2P?P2

?Dhqðz; P2

?ÞRd2P?Dh

qðz; P2?Þ

: (24)

In analyses that assume a Gaussian ansatz for the TMDfragmentation functions, it is usual to assume that hP2

?idoes not depend on z, the type of hadron, h, or the quarkflavor, q. These assumptions will be tested against theNJL-jet TMD fragmentation functions.The results in Fig. 14 depict the average transverse

momenta of � and K mesons produced by a u quark.These plots show that the average transverse momenta ofthe hadrons are relatively flat versus z in the region 0:3<z< 0:6, however they have a significant dependence on thetype of the hadron. We find that the average transversemomentum of the kaons is significantly larger than that ofthe pions.The curves in Fig. 15 depict the TMD fragmentation of a

favored u ! �þ process for z ¼ 0:8 and an unfavored s !Kþ process for z ¼ 0:2. Also presented are least squaresfits to the fragmentation functions for particular z slicesusing the Gaussian ansatz of Eq. (23), with hP2

?i the singlefitting parameter for each z. The plots in Fig. 15 indicatethat such a fit to a single Gaussian is reasonable only for alimited P2

? region. Also, because hP2?i has a significant z

dependence, the Gaussian ansatz for the entire TMD frag-mentation function offers at best a crude approximation to

)2

(GeV

2P00.1

0.20.3

0.40.5

z0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

+ K u

z D

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

)2

(GeV

2P00.1

0.20.3

0.40.5

z0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

- K u

z D

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

FIG. 13 (color online). TMD fragmentation functions for a uquark to Kþ and K�. The upper figure illustrates the favoredcase, which peaks at relatively large z, while the unfavored case,shown in the lower figure, peaks at much smaller z.

u h<P

2>

(G

eV2 )

0

0.1

0.2

0.3

0.4

z0 0.2 0.4 0.6 0.8 1.0

+

-

K+

K-

FIG. 14 (color online). The averaged transverse momentum of� and K mesons emitted by a u quark.

TRANSVERSE-MOMENTUM-DEPENDENT FRAGMENTATION . . . PHYSICAL REVIEW D 85, 014021 (2012)

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the full results. The corresponding average transverse mo-menta obtained from the Gaussian fits are smaller thanthose obtained directly using the relation in Eq. (24).

VI. AVERAGE TRANSVERSE MOMENTA IN SIDIS

For the SIDIS process, we have created a Monte Carloevent generator that can calculate the physical cross sec-tion. In future work, this will enable us to analyze therelative importance of the different aspects of the processand the implications of the constraints set in individualexperiments. We use it to determine the average transversemomentum of the produced hadrons (at the model scale)observed in a SIDIS experiment, namely, hP2

Ti, which isdefined as

hP2Tiðx; zÞ �

Rd2PTP

2T~Dhqðx; z; P2

TÞRd2P? ~Dh

qðx; z; P2TÞ

: (25)

The function ~Dhqðx; z; P2

TÞ is defined in Eq. (7). The crossesin Fig. 16 represent results for hP2

Ti acquired by�þ mesonsin a SIDIS hadronization process, where the virtual photonstrikes a valence u quark in a proton carrying a light-cone

momentum fraction of x ¼ 0:4. We also plot as the dash-dotted line hP2

?i, which is the average transverse momen-

tum that the�þ mesons acquire in the quark fragmentationprocess. Recall that the transverse momentum P? is de-fined relative to the direction of the original fragmentingquark, while PT is relative to the direction of the photonmomentum, these transverse momenta are related byEq. (1). For the factorization of the SIDIS cross sectiongiven in Eq. (7), it can be shown that hP2

Ti is given by

hP2Tiðx; zÞ ¼ hP2

?iðzÞ þ z2hk2TiðxÞ: (26)

As an additional check on the Monte Carlo calculation, inFig. 16 we plot the result obtained from Eq. (26) as thesolid line and find that it agrees perfectly with that obtainedfrom the Monte Carlo event generator for the SIDIS crosssection. We also find that both hP2

?i and hP2Ti illustrated in

Fig. 16 have a sizable z dependence.Illustrated in Fig. 17 are results for the average trans-

verse momentum acquired by � and K mesons in thehadronization process in SIDIS, where the struck quark is

u +, z=0.8

D(z

,P2)

/ D(z

)

10 6

10 4

10 2

1

P2 (GeV2)

0 1 2 3 4 5 6

NJL-jet, <P2 >= 0.148 GeV2

Gauss Fit, <P2 >= 0.117 GeV2

s K+, z=0.2

D(z

,P2)

/ D(z

)

10 6

10 4

10 2

1

P2 (GeV2)

0 1 2 3 4 5 6

NJL-jet, <P2 >= 0.396 GeV2

Gauss Fit, <P2 >= 0.365 GeV2

FIG. 15 (color online). Normalized TMD fragmentation forthe favored u ! �þ process for z ¼ 0:8 (upper) and the unfa-vored s ! Kþ process for z ¼ 0:2 (lower). Also depicted arefits to the fragmentation functions using the Gaussian ansatz ofEq. (23), with hP2

?i as the single fitting parameter.

u +, x=0.4

<P

2 T>

(G

eV2 )

0

0.1

0.2

0.3

z0 0.2 0.4 0.6 0.8 1.0

<P2 >

<P2 >+z2<k2 >

<P2T>

FIG. 16 (color online). The averaged transverse momentum of�þ mesons in SIDIS produced on a u quark in a proton withlight-cone momentum fraction x ¼ 0:4.

u h, x=0.4<P

2 T>

(G

eV2 )

0

0.1

0.2

0.3

0.4

z0 0.2 0.4 0.6 0.8 1.0

+

-

K+

K-

FIG. 17 (color online). The averaged transverse momentum of� and K mesons in SIDIS on a u quark in the proton with light-cone momentum fraction x ¼ 0:4. The unfavored fragmentationfunctions rapidly approach zero for large z and this causes thedramatic changes in hP2

Ti at large z illustrated in this figure.

MATEVOSYAN et al. PHYSICAL REVIEW D 85, 014021 (2012)

014021-10

a u quark in a proton with light-cone momentum fractionx ¼ 0:4. The rapid approach to zero for the unfavoredfragmentation functions in Fig. 17 is a consequence ofthe large z behavior of the unfavored hP2

?i illustrated in

Fig. 14, which also rapidly approach zero. The HERMESexperimental results for hP2

Ti measured in SIDIS on adeuterium target [27] are of comparable size to our resultsshown in Fig. 17. We do not plot these HERMES resultsbecause the kinematic range is too different for a quanti-tative comparison. The average transverse momentum ofthe kaons is larger than that of the pions at the lowQ2 scaleof the model. Our model includes only the valence quarksin the proton, which should be the dominant component atx ¼ 0:4.

VII. CONCLUSIONS AND OUTLOOK

In this work, we extended the NJL-jet model to includethe transverse-momentum dependence in the quark hadro-nization process. This was achieved using TMD elemen-tary fragmentation functions and by keeping track of thequark’s recoil transverse momentum in the hadron emis-sion cascade. We modified the LB regularization scheme toremove artifacts that limit the z range of the splittingfunctions, and this in turn improved our description ofthe integrated fragmentation functions. The TMD frag-mentation functions for u, d, and s quarks to pions andkaons were determined using a Monte Carlo approach. Theaverage P2

? of the produced kaons was found to be sig-

nificantly larger than that of the pions and in both caseshP2

?i had a sizable z dependence. The high statistical

precision needed for these calculations was achievedthrough parallel computing on the small computer farmat CSSM.

The TMD quark distribution functions in the protonwere also determined using the NJL model. In this case,we used the proper-time regularization scheme, becausethis method simulates important aspects of confinement.Our TMD PDF results when integrated over kT give ourearlier results for the familiar spin-independent quark dis-tribution functions [35], whose moments satisfy the baryonnumber and momentum sum rules. We found that theaverage k2T of the quarks in the nucleon have a significantx dependence and therefore the familiar Gaussian ansatzfor the TMD PDFs produces only a crude approximation toour full TMD PDF results.

Finally, using the TMD quark distribution functions forthe nucleon and the results for the TMD fragmentationfunctions, we constructed a Monte Carlo event generatorfor the SIDIS process. Using this Monte Carlo event gen-erator, we determined the average transverse momentum ofthe hadrons, hP2

Ti, produced in SIDIS. These results are of asimilar magnitude to those extracted from experiment,even at our relatively low model scale. As a cross checkfor this SIDIS Monte Carlo event generator, we comparedour results for hP2

Ti with those obtained using Eq. (26),

finding perfect agreement. We find that the hP2Ti of the

produced kaons is significantly larger than that of thepions, which is not apparent in the current experimentalmeasurements.An interesting extension of our model would be to

include the vector meson and nucleon antinucleon emis-sion channels. This extension has already been completedin our previous work on the integrated fragmentationfunctions. It would also be intriguing to consider thespin-dependent effects in the hadronization process, inparticular, to calculate the Collins fragmentation function.Further, using the NJL description of nucleon structure wewill be able to develop a self-consistent description of thespin-dependent effects in SIDIS reactions.

ACKNOWLEDGMENTS

This work was supported by the Australian ResearchCouncil through Grants No. FL0992247 (AWT),No. CE110001004 (CoEPP), and by the University ofAdelaide.

APPENDIX: NUCLEON TMD PDF EXPRESSIONS

The u and d valence TMD quark distribution functionsin the proton are given by

uvðx; k2TÞ ¼ fsq=Nðx; k2TÞ þ 13f

aq=Nðx; k2TÞ þ 1

2fsqðDÞ=Nðx; k2TÞ

þ 56f

aqðDÞ=Nðx; k2TÞ; (A1)

dvðx; k2TÞ ¼ 23f

aq=Nðx; k2TÞ þ 1

2fsqðDÞ=Nðx; k2TÞ

þ 16f

aqðDÞ=Nðx; k2TÞ: (A2)

The individual quark diagrams terms have the form

fsq=Nðx;k2TÞ¼21ZNZs

16�3ð1�xÞ

Zd�½1þ�x½ðMNþMÞ2�m2

s���e��½k2Tþxðx�1ÞM2

Nþxm2sþð1�xÞM2�; (A3)

faq=Nðx;k2TÞ¼�ZaZN

16�3ð1�xÞ

Zd�½ð2

2�223�223Þ

�ð1þ�x½ðMN�MÞ2�m2a�Þ

�1223�xMMN�e��½k2Tþxðx�1ÞM2

Nþxm2aþð1�xÞM2�;

(A4)

and the diquark diagrams terms are given by

fsqðDÞ=Nðx; k2TÞ ¼ZZ 1

0dydz�ðx� yzÞfsq=Dðz; k2TÞ

� fsq=Nð1� yÞ; (A5)

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faqðDÞ=Nðx; k2TÞ ¼ZZ 1

0dydz�ðx� yzÞfaq=Dðz; k2TÞ

� faq=Nð1� yÞ: (A6)

The diquark TMD quark distributions are

fsq=Dðx; k2TÞ ¼3Zs

4�3

Zd�½1þ �xð1� xÞm2

s�� e��½k2Tþxðx�1Þm2

sþM2�; (A7)

faq=Dðx; k2TÞ ¼3Za

�3xð1� xÞ

Zd�½1þ �xð1� xÞm2

a�� e��½k2Tþxðx�1Þm2

aþM2�: (A8)

The constituent quark, scalar diquark, axial-vector diquark,and nucleon masses in these expressions have the valuesM ¼ 0:4 GeV, ms ¼ 0:687 GeV, ma ¼ 1:03 GeV, andMN ¼ 0:94 GeV. The weight factors in the nucleonFaddeev vertex [35,60,61] and its normalization are givenby ð1; 2; 3Þ ¼ ð0:429; 0:0244;�0:445Þ and ZN ¼ 29:9,

respectively. Finally, the pole residues of the scalar andaxial-vector diquark t matrices are Zs ¼ 14:5 and Za ¼3:82, respectively. Integrating over kT in these expressionsgives the familiar spin-independent quark distributionfunctions, the moments of which satisfy the baryon numberand momentum sum rules.These expressions have been derived using the proper-

time regularization scheme, which in practice means tomake the substitution

1

Xn¼ 1

ðn� 1Þ!Z 1

0d��n�1e��X

! 1

ðn� 1Þ!Z 1=�2

IR

1=�2UV

d��n�1e��X; (A9)

where X denotes the denominator function in a loop inte-gral after Feynman parameterization and Wick rotation.The infrared and ultraviolet cutoffs have the values �IR ¼0:240 GeV and �UV ¼ 0:645 GeV, respectively.

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