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The transverse momentum dependent distribution functions in the bag model
H. Avakian,1 A. V. Efremov,2 P. Schweitzer,' and F. Yuan4,5
'Thomas Jefferson National Accelerator Facility, Newport News, VA 23606, U.S.A.
2Joint Institute for Nuclear Research, Dubna, 141980 Russia
3Department of Physics, University of Connecticut, Storrs, CT 06269, U.S.A.
4RIKEN BNL Research Center, Building 510A, BNL, Upton, NY 11973, U.S.A.
5Nuclear Science Division, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, U.S.A.
(Dated: January 2010)
Leading and subleading twist transverse momentum dependent parton distribution functions
(TMDs) are studied in a quark model framework provided by the bag model. A complete set of
relations among different TMDs is derived, and the question is discussed how model-(in)dependent
such relations are. A connection of the pretzelosity distribution and quark orbital angular momen-
tum is derived. Numerical results are presented, and applications for phenomenology discussed. In
particular, it is shown that in the valence-x region the bag model supports a Gaussian Ansatz for
the transverse momentum dependence of TMDs.
PACS numbers: 13.88.+e, 13.85.Ni, 13.60.-r, 13.85.Qk
Keywords: Semi-inclusive deep inelastic scattering, transverse momentum dependent distribution functions
TMDs are a generalization [1 4] of parton distribution functions (PDFs) promising to extend our knowledge of the
nucleon structure far beyond what we have learned from PDFs about the longitudinal momentum distributions of
partons in the nucleon. In addition to the latter, TMDs carry also information on transverse parton momenta and
spin-orbit correlations [5 35]. Here longitudinal and transverse refers to the hard momentum flow in the process, for
example, in deeply inelastic lepton nucleon scattering (DIS) the momentum of the virtual photon.
TMDs (and/or transverse momentum dependent fragmentation functions) enter the description of leading-twist
observables in deeply inelastic reactions [5 7] on which data are available like: semi-inclusive DIS (SIDIS) [36 50],
Drell-Yan process [51 53], or hadron production in e+e- annihilations [54 57].
The interpretation of these data is not straight-forward though. In SIDIS one deals with convolutions of a priori
unknown transverse momentum distributions in nucleon and fragmentation process, and in practice is forced to assume
models for transverse parton momenta such as the Gaussian Ansatz [58 67]. In the case of subleading twist observables,
one moreover faces the problem that several twist-3 TMDs and fragmentation functions enter the description of one
observable [68 77] (we recall that presently factorization is not proven for subleading-twist observables ).
In this situation information from models [74 94] is valuable for several reasons. Models can be used for direct
estimates of observables, though it is difficult to reliably apply the results, typically obtained at low hadronic scales,
to experimentally relevant energies . Another aspect concerns relations among TMDs observed in models [79 83].
Such relations, especially when supported by several models, could be helpful at least for qualitative interpretations
of first data. Furthermore, model results allow to test assumptions made in literature, such as the Gaussian Ansatz
for transverse momentum distributions or certain approximations [95 101].
In addition to such practical applications model studies are of interest also because they provide important insights
into non-perturbative properties of TMDs. In this context the probably most interesting recent observation in models
concerns the pretzelosity distribution function, which in some quark models is related to the difference of the helicity
and transversity distributions  and, so far, in one model to quark orbital momentum  which is, to best of our
knowledge, the first 'rigorous' connection of a TMD and quark orbital angular momentum in a model.
The purpose of this work is to study TMDs in the framework of the MIT bag model. We compute in this model
all leading- and subleading-twist, time-reversal (T-) even TMDs in Sec. II, and address then in Sec. III questions like:
how do relations among TMDs arise in a quark model? How many such relations are there in a model? To which
extent may one expect such relations to be realized in nature? In Sec. IV we establish a connection of pretzelosity
and quark orbital angular momentum in the bag model. In Sec. V we present and discuss the numerical results, using
them, among others, for 'testing' the Gaussian Ansatz or Wandzura-Wilczek-type approximations [98 101]. Finally,
in Sec. VI we present our conclusions. Some of the results presented here were shown in the proceeding .
For convenience and in order to make this work self-contained, in the remainder of this Introduction we include
general definitions of TMDs, and introduce relevant notation.
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Avakian, Harut; Efremov, Anatoly; Schweitzer, Peter & Yuan, Feng. The transverse momentum dependent distribution functions in the bag model, article, January 29, 2010; Berkeley, California. (https://digital.library.unt.edu/ark:/67531/metadc1015520/m1/1/: accessed April 23, 2019), University of North Texas Libraries, Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.