^{1,2}

^{3,4}

^{5,6}

^{3}.

The production of a hard dijet with small transverse momentum imbalance in semi-inclusive DIS probes the conventional and linearly polarized Weizsäcker-Williams (WW) transverse momentum dependent (TMD) gluon distributions. The latter, in particular, gives rise to an azimuthal dependence of the dijet cross section. In this paper we analyze the feasibility of a measurement of these TMDs through dijet production in DIS on a nucleus at an electron-ion collider. We introduce the

Building an electron-ion collider (EIC) is one of the key projects of the nuclear physics community in the U.S. The main purpose of an EIC is to study the gluon fields of QCD and provide insight into the regime of nonlinear color field dynamics

In this paper we focus on the small-

The WW TMD gluon distributions, and in particular the distribution of linearly polarized gluons, appears in a variety of processes. This includes production of a dijet or heavy quark pair in hadronic collisions

At leading order in

given by ^{1}

Here and below the transverse two dimensional component of a three-dimensional vector

The transverse momenta of the produced quark and antiquark are given by

Power corrections to Eqs.

In Eqs.

The average

Since^{2}

A measurement of the

Equations

The linearly polarized and conventional gluon distributions^{3}

We only consider the forward gluon distributions in this paper. In the nonforward case the general decomposition of the WW GTMD involves additional independent functions on the right-hand side of Eq.

The functions

Our event generator described in the following Sec.

In this section we discuss the relation of ^{4}

Recall that this average is performed with normalized weights

The cross section for dijet production in electron-nucleus scattering is given by the product of the virtual photon fluxes of the electron with the

Note that Eqs.

For given

The goal of the event generator

In what follows, we will often refer to the acceptance-rejection method (ACM) of generating random variables from a given probability distribution; although this method is fairly basic, it nevertheless proved sufficient for generating the required number of events on a single processor in a reasonable amount of time.

In order to make the MC generator computationally feasible we have adopted the following simplifying assumptions and approximations:

The dependence of the cross section on the atomic number ^{5}

Throughout the paper we refer to the saturation scale for a dipole in the fundamental representation.

The Wilson lines in the field of the target at

The Wilson lines are used to compute the dependence of

Using ACM based on the cross section summed with respect to polarizations,

The virtual photon may have either longitudinal or transverse polarization; it is assigned by sampling a random number

Using ACM and the differential cross section for the photon polarization defined previously we generate a sample for

Using the obtained

Finally, the longitudinal momenta are given by
^{6}

Our convention here is that the longitudinal momentum of the virtual photon is positive. This is the most common convention in the theoretical literature.

The sampled kinematic variables and the corresponding numerical value for the cross section are then passed to

The momentum assignments

The reference frames: (a) The laboratory frame. In the laboratory frame, the electron and the proton have zero transverse momenta; the energy of the electron (proton) is

In this section we show the distribution of dijet events over various kinematic variables. The target is assumed to be Au with

Distributions of photon virtuality

(a) The contributions of transverse vs. longitudinal photon polarizations as functions of

In this section, based on the theoretical foundation outlined above, we present a detailed study of the feasibility, requirements, and expected precision of measurements of the azimuthal anisotropy of dijets at a future electron-ion collider (EIC). We find that, at an EIC

In order to verify the feasibility we have to show that (i) the anisotropy described by

All studies presented here were conducted with electron beams of 20 GeV and hadron beams with 100 GeV energy resulting in a center-of-mass energy of

Figure

Kinematic range in

The partons from

Figure

Figure

Comparison of

Figure

The reconstructed dijets reflect the original anisotropy at the parton level remarkably well despite the dijet spectra not being fully corrected. The loss in dijet yield, mostly due to loss of low-

In our studies we noted the momentous correlation of the angle

While

The presence of underlying event activity is key to answering the question if one can achieve a clear separation between the products of the hard partonic interaction and the beam remnants. For that reason, one usually labels an event as “2+1 jets” if it has two jets coming from the hard partonic interaction, with the “+1” indicating the beam remnants. The diagram in Fig.

Photon-gluon fusion processes that contributes to the 2+1 jet signal cross section.

While dijet studies have been successfully conducted in

In our

As discussed in Sec.

Azimuthal asymmetry in reconstructed dijet events from

Figure

In order to derive the distribution of linearly polarized gluons via Eqs. ^{7}

The expression for

Our strategy is to perform a combined five-parameter fit of all three components to the full data sample: The signal for longitudinal polarization (

Figure

Result of a fit of combined signal and background to a data sample obtained in

Table

Relative error on the extracted

Our current proof of principle analysis relied on a variety of simplifications and approximations as our main focus was on the reconstruction of relatively low

First, a more realistic modeling of the impact parameter dependence of the thickness of the target nucleus would be useful. This is due to the fact that cuts on the minimal

Another improvement is to include running coupling corrections to the dijet cross section and to small-

The measurement of the distribution of linearly polarized gluons via the

One should also account for the Sudakov suppression, which arises due to the presence of the two scales

Given that the light-cone momentum fraction of the target partons is not very small even at the highest energies envisaged for an EIC it would be important to also account for the

As the electron-ion collider projects progresses detector concepts will become more refined. Once the design of the envisioned multipurpose detector(s) are finalized the feasibility study discussed in this paper should be repeated using detailed detector effects (acceptance, resolution) and include full unfolding procedures that would improve over the simple corrections used in this work. There is an increasing interest in jet studies at an EIC that could potentially lead to improved jet finding procedures tailored to the specific kinematics and energies relevant for this study.

This paper presents a study of the feasibility of measuring the conventional and linearly polarized Weizsäcker-Williams (WW) transverse momentum dependent (TMD) gluon distributions at a future high-energy electron-ion collider via dijet production in deeply inelastic scattering on protons and nuclei at small

The jet transverse momentum

We thank Elke-Caroline Aschenauer, Jin Huang, Larry McLerran, Tuomas Lappi, Elena Petreska, Andrey Tarasov, Prithwish Tribedy, and Pia Zurita for useful discussions. A.D. gratefully acknowledges support by the DOE Office of Nuclear Physics through Grant No. DE-FG02- 09ER41620; and from The City University of New York through the PSC-CUNY Research grant 60262-0048. V.S. thanks the ExtreMe Matter Institute EMMI (GSI Helmholtzzentrum für Schwerionenforschung, Darmstadt, Germany) for partial support and their hospitality. T.U.'s work was supported by the Office of Nuclear Physics within the U.S. DOE Office of Science.

In any frame the ratio of plus momenta of quark and virtual photon is given by

In particular, in the Breit frame (