content_image/406512.jpg content_image/406513.jpg content_image/406514.jpg content_image/406515.jpg content_image/406509.jpg content_image/406510.jpg
for both nucleon and nuclear targets, the nuclear dependence of TMD parton distribution functions will come from the nuclear dependence of the higher-twist parton matrix elements. Under the “maximal two gluon approximation”, the TMD quark distribution in a nucleus can be expressed as a convolution of the TMD quark distribution f N q ( x, ⃗ ℓ$_{⊥}$ ) in nucleon and a Gaussian broadening [24] , 

<img src='content_image/406512.jpg'>

The broadening width Δ$_{2}$$_{F}$ or the total average squared transverse momentum broadening, 

<img src='content_image/406513.jpg'>

is given by the quark transport parameter, 

<img src='content_image/406514.jpg'>

where ρ A $_{N}$( ξ$_{N}$ ) is the spatial nucleon number density inside the nucleus and f N g ( x ) is the gluon distribution function in a nucleon, 

<img src='content_image/406515.jpg'>

A summation over the gluon’s color index is implied in the above definition of the gluon distribution function. 

One can generalize the above approach to nuclear modification of higher-twist TMD parton distributions. In this paper, we will study the case of twist-3 TMD quark distributions which determines the azimuthal asymmetry 〈 cos φ 〉 of the unpolarized semi-inclusive deep-inelastic scattering (SIDIS) cross section defined with respect to the leptonic plane. Such asymmetry at large transverse momentum arises predominately from hard gluon bremsstrahlung in perturbative QCD [28, 29]. Similar processes are also responsible for the azimuthal angle dependence of the Drell-Yan dilepton production cross sections [30] and multiple parton scattering in p + A collisions can lead to a twist-4 nuclear dependence [31, 32]. In the small transverse momentum k$_{⊥}$ ≤ 1 GeV/c region, however, the asymmetry is shown to arise mainly from the intrinsic transverse momentum of the quarks in the target nucleon or nucleus [33, 34] and is related to the higher-twist TMD parton distribution functions [35–38]. We will focus on the twist-3 contribution to the azimuthal dependence, proportional to ( k$_{⊥}$/Q ) cos φ , in this paper. Here Q 2 = − q 2 and q is the four momentum transfer from lepton to target. Since the coefficient of the azimuthal asymmetry is related to twist-3 TMD parton distributions, one can take the same approach as in the case of the twist-2 TMD parton distribution to study the nuclear dependence of the twist-3 contribution to the azimuthal asymmetry of SIDIS cross sections. 

The remainder of this paper is organized as follows. In Sec. II, we will review the calculation of the azimuthal dependence of SIDIS up to twist-3 contributions in terms of the TMD parton distributions. In Sec. III, we extend the study of nuclear dependence of TMD parton distributions to the case of other TMD parton correlation functions, including twist-3. Using these results, we calculate the general nuclear dependence of 〈 cos φ 〉 . We will then illustrate the nuclear dependence with an ansatz of the TMD parton distributions in a Gaussian form and discuss the effect of transverse momentum smearing in the fragmentation. A summary will be given in Sec. IV. 

## II. AZIMUTHAL ASYMMETRY IN SIDIS 

We consider semi-inclusive deep inelastic scattering e − + A → e − + q + X with unpolarized beam and nucleus or nucleon target. The differential cross section, 

<img src='content_image/406509.jpg'>

can be expressed as a product of the leptonic tensor, 

<img src='content_image/406510.jpg'>

