NIKHEF-2013-040

Accessing the Transverse Dynamics and the Polarization of the Gluons inside the Proton at the LHC

Wilco J. den Dunnen,1, ∗ Jean-Philippe Lansberg,2, † Cristian Pisano,3, ‡ and Marc Schlegel1, §1Institute for Theoretical Physics, Universitat Tubingen,Auf der Morgenstelle 14, D-72076 Tubingen, Germany

2IPNO, Universite Paris-Sud, CNRS/IN2P3, F-91406, Orsay, France3Nikhef and Department of Physics and Astronomy, VU University Amsterdam,

De Boelelaan 1081, NL-1081 HV Amsterdam, The Netherlands(Dated: January 30, 2014)

We argue that the study of heavy quarkonia, in particular that of Υ, produced back-to-back with an isolatedphoton in pp collisions at the LHC is the best –and currently unique– way to access the distribution of both thetransverse momentum and the polarization of the gluon in an unpolarized proton. These encode fundamentalinformation on the dynamics of QCD. We have derived analytical expressions for various transverse-momentumdistributions which can be measured at the LHC and which allow for a direct extraction of the aforementionedquantities. To assess the feasibility of such measurements, we have evaluated the expected yields and the relevanttransverse-momentum distributions for different models of the gluon dynamics inside a proton.

PACS numbers: 12.38.-t; 13.85.Ni; 13.88.+e

Introduction.— At LHC energies, the vast majority of hardreactions are initiated by the fusion of two gluons from bothcolliding protons. A good knowledge of gluon densities istherefore mandatory to perform reliable cross-section predic-tions, the archetypal example being the H0 boson production.In perturbative QCD (pQCD), the production cross section ofa given particle is conventionally obtained from the convo-lution of a hard parton-scattering amplitude squared and ofthe collinear parton distribution functions (PDFs) inside thecolliding hadrons, G(x, µ) or f g

1 (x, µ) for the gluon [1]. ThePDF provides the distribution of a given parton in the protonas a function of its collinear momentum fraction x, at a cer-tain (factorization) scale µ. Whereas the scale evolution of thePDFs is given by pQCD, experimental data are necessary todetermine their magnitude (see e.g. [2]).

This collinear factorization, inspired by the parton modelof Feynman and Bjorken, can be extended to take into accountthe transverse dynamics of the partons inside the hadrons. Dif-ferent approaches have been proposed (unintegrated PDF, im-pact factors within kT factorization, etc.). Out of these, theTransverse-Momentum (TM) dependent factorization is cer-tainly the most rigorous with proofs of factorization for a cou-ple of processes [3–6]. The further advantage of the TM De-pendent (TMD) formalism lies in its ability to deal with spin-dependent objects, both for the partons and the hadrons.

Much effort has been made in the last years to extract quarkTMD distributions (TMDs in short) inside a proton from lowenergy data from HERMES, COMPASS or JLab experiments(see e.g. Ref. [7] for recent reviews). On the contrary, nothingis known about the gluon TMDs which rigorously parametrizethe transverse motion of gluons inside a proton. For an unpo-larized proton, these are the distribution of unpolarized glu-ons, denoted by f g

1 , and the distribution of linearly-polarizedgluons, h⊥g

1 [8]. These functions contain fundamental infor-mation on the transverse dynamics of the gluon content of theproton [see the interpretation of h⊥g

1 in Fig. 1 (a-b)]. Theirstudy is also motivated by a recently proposed method to de-

(a) (b)

P2

Pγx1P1 + k1T

x2P2 + k2TPQ

P1Φµνg (x1, k1T , ζ1, µ)

µ ν

Φρσg (x2, k2T , ζ2, µ)

ρ σ

(c)

FIG. 1. Visualization of the gluon polarization in the TM plane fora positive (a) and negative (b) Gaussian h⊥g

1 . [The ellipsoid ma-jor/minor axis lengths in the plane are proportional to the probabilityof finding a gluon with a linear polarization in that direction]. (c)Feynman diagram for p(P1) + p(P2) → Q(PQ) + γ(Pγ) +X via gluonfusion at LO in the TMD-factorization formalism.

termine the spin and parity of the H0 boson [9], which is basedon the linear polarization of gluons that appears for non-zeroparton TM.

In this Letter, we argue that the LHC experiments are ide-ally positioned to extract for the first time the gluon TMDsthrough the study of an isolated photon produced back-to-back with a heavy quarkonium. Furthermore, we show thatthe yields are large enough to perform such extractions withexisting data at

√s = 7 and 8 TeV.

Reactions sensitive to gluon TMDs.— Several processeshave been proposed to measure both f g

1 and h⊥g1 . A poten-

tially very clean probe to extract gluon TMDs is the back-to-back production of a heavy-quark pair in electron-protoncollisions, e p → e QQ X in which the gluon TMDs appearlinearly. Theoretical predictions have been provided at lead-ing order (LO) [10] and next-to-leading order (NLO) [5] inpQCD. However, such a process can only be measured at fu-ture facilities (EIC or LHeC), whose realization is at best adecade away.

arX

iv:1

401.

7611

v1 [

hep-

ph]

29

Jan

2014

2

Another process sensitive to gluon TMDs is the back-to-back isolated photon-pair production in proton collisions,p p → γ γ X [11]. In principle, this process is accessible atRHIC and the LHC but it suffers from a contamination fromquark-induced channels, a huge background from π0-decaysand an inherent difficulty to trigger on such events.

As for the gluon PDF, final states such as a heavy-quarkpair or a dijet [10] should also be ideal candidates to probegluon TMDs. However, once there is a color flow into thedetected final state in the partonic-scattering subprocess, onecannot cleanly separate final state interactions of this colorflow from the non-perturbative TMD objects due to the non-Abelian characteristics of QCD [12]. This leads to a break-down of TMD factorization for processes with colored finalstates. We however note that we do not know of any obviousreason why such complications could be avoided by using an-other approach, relying for instance on unintegrated PDFs orkT factorization with impact factors.

On the other hand, this problem can be avoided in the caseof the production of heavy quarkonia, provided that the heavy-quark pair is produced in a colorless state at short distances asin the color-singlet model [13], and that it is not accompaniedby other –necessarily colorful– partons. The production ofC-even quarkonia (χQ, ηQ) at small TM is one of these caseswhere the factorization is expected to hold as illustrated bystudies both at LO [14] and NLO [15]. At low PQT , the pro-duction of ηQ and χQ0,2 proceeds without the emission of afinal state gluon and the color-octet (CO) contributions [16]are not kinematically enhanced. However, such experimentalmeasurements are particularly difficult since they should bedone at low TM, PQT Q ' MQ, as required by TMD fac-torization. The hard scale of the process, Q, can only be themass of the heavy quarkonium, hence Q ' MQ. The obser-vation of low PQT C-even quarkonia is likely impossible withATLAS and CMS. LHCb may look at these down to PQT ' 1GeV, but an unambiguous determination of the gluon TMDs– free of large power corrections in PQT /Q – requires to reachthe sub-GeV region. Besides, this would not allow one to lookat the scale evolution of the TMDs. Only two ranges can beprobed – close to the charmonium and bottomonium masses.

Back-to-back quarkonium+isolated-photon production.—We propose a novel process to overcome these experimentalcomplications and theoretical limitations: the production of aback-to-back pair of a 3S 1 quarkonium Q (Υ or J/ψ) and anisolated photon, p p→ Q+γ+ X. Compared to the aforemen-tioned processes, it has the advantage of being accessible bythe LHC experiments: only the TM imbalance, qT = PQT +PγT ,has to be small, not the individual TM, for TMD factorizationto apply. In addition, the hard scale of the process Q can betuned by selecting different invariant masses of the Q− γ pair.This allows one to look at the scale evolution of the TMDs andto greatly increase the qT -range where the TMD factorizationapplies with tolerable power corrections.

Previous studies [17–19] have shown that the CO contri-butions to inclusive Q + γ production are likely smaller thanin the inclusive case Q + X (see e.g. [20–22]) . [The case of

J/ψ+γ is however intriguing since a state-of-the art NLO eval-uations using recent NRQCD fits predict negative CO cross-sections [23].] The smallness of CO contributions is crucialsince these would violate the TMD factorization.

0.1

1

10

100

20 25 30 35

dσ

/dQ

/dY

/d c

os

θC

S x

Br(

Oniu

m →

µ µ

) (f

b/G

eV

)

Qϒ + γ (GeV)

Direct back-to-back Onium + γ at sqrt(s)=14 TeV

µR=µF=moniumT | mQ=monium/2

|Y | < 0.5; |cosθCS| <0.45

q-q (x 100)

g-g

a)

20 25 30 35

QJ/ψ + γ (GeV)

q-q (x 50)

g-g

b)

<O1S

[8]0 (ϒ)>=0.1 GeV

3

<O3S

[8]1 (ϒ)>=0.01 GeV

3

<O1S

[8]0 (J/ψ)>=0.02 GeV

3

<O3S

[8]1 (J/ψ)>=0.002 GeV

3

Color SingletColor Octet

FIG. 2. Different contributions to the production of an isolated pho-ton back-to-back with a) an Υ(1S ) (resp. b) a J/ψ) from g − g andq − q fusion from the CS and CO channels as function the invariantmass of the pair, Q. The curves for the q − q fusion are rescaled bya factor 100 (resp. 50). The CO matrix elements we used are veryclose to those obtained in a recent LO fit of LHC data [24].

As studied in [25], the CO contributions are also suppressedw.r.t. the CS ones when one imposes that the Q−γ pair is pro-duced back-to-back, i.e. dominantly from 2→ 2 processes, al-though the gg fusion CS contribution (Fig. 1c) scales like P−8

QT .Indeed, the P−4

QT (fragmentation) CO contribution only appearsfor qq annihilation –extremely suppressed at LHC energies–and, incidentally, on the order of the pure QED CSM con-tribution (as for J/ψ + W [26]). As regards gg fusion COchannels, they are subleading in PQT , since they come fromquark box and s-channel gluon diagrams, only via C = +1CO states, such as 1S [8]

0 or 3P[8]J . [For the J/ψ, these CO states

are known to be severely constrained if one wants to complywith e+e− inclusive data [27].] To substantiate this, we havecomputed the different CS and CO contributions, see Fig. 2.The CS yield is clearly dominant for the Υ and likely abovethe CO one for the J/ψ at the lowest Q accessible at the LHC(PQT & 10 GeV). It is also clear that this process is purelyfrom gg fusion.

Since QCD corrections to the inclusive production of aquarkonium-photon pair are known to be large for increasingPQT [17, 19], we find it useful to emphasize that the leading-PQT NLO topologies, such as t-channel gluon exchanges, arein fact absorbed in the evolution of the TMD distribution.Moreover, it is clear that topologies with more than 2 particlesin the final state are anyhow suppressed by the back-to-backrequirement since they produce Q−γ pair “near” to each otherrather than “away”. Finally, the isolation of the photon alsofavors the LO contribution, where QCD radiations are strictlyabsent.

3

A further suppression of CO contributions can be achievedby isolating the quarkonium (see [28]) as done for the pho-ton. The isolation should be efficient at large enough PQT

where the soft partons emitted during the hadronization of theCO heavy-quark pair are boosted and energetic enough to bedetected. Experimentally, this would provide an interestingcheck of the CS dominance by measuring the (conventional)qT -integrated cross section which should coincide with theparameter-free CSM prediction. This would also confirm thatdouble-parton scattering contributions are suppressed by theisolation criteria. We emphasize that, according to our evalu-ations, such an isolation is not at all necessary for the Υ case.

Along the same lines, the study of Υ or J/ψ with a Z bo-son is worth some investigation. According to [29, 30], oneexpects their back-to-back production at low PQT to be ba-sically free of CO contributions as well as exclusively fromgluon fusion, thus satisfying the requirements for gluon-TMDextraction. The only drawback is that we might have to waitfor the LHC luminosity increase to get enough events [30].

Analytical expression for the qT -dependent cross section.—We now proceed to the qT differential-cross-section calcu-lation for the reaction of Fig. 1c. At LO in αs, the reac-tion proceeds via the gluon-fusion process, g(k1) + g(k2) →Q(PQ) + γ(Pγ), which involves 6 graphs. The correspondingamplitudeM–obtained as in the collinear case–is squared andconvoluted with the TMD correlator as described in Fig. 1c,i.e.

dσ =(2π)4

8s2

∫d2 k1T d2 k2Tδ

2(k1T + k2T − qT )Mµρ (Mνσ)∗

Φµνg (x1, k1T , ζ1, µ) Φ

ρσg (x2, k2T , ζ2, µ)dR, (1)

where s = (P1 + P2)2 is the hadronic center-of-mass system(c.m.s.) energy squared and the phase space element of theoutgoing particles is denoted by dR. The gluon momentumis decomposed as k1 = x1P1 + k1T − k2

1T/(x1s)P2, with k1T

a 4-vector perpendicular to both P1 and P2. The transversecomponents of k1T in the c.m.s. are denoted by the 2-vectork1T . The longitudinal momentum fractions are given by x1 =

q·P2/P1·P2 and x2 = q·P1/P1·P2, where q = PQ + Pγ. Thegluon-TMD correlator for an unpolarized proton is defined as

Φµνg (x, kT , ζ, µ) ≡

∫d(ξ·P) d2ξT

(xP·n)2(2π)3 ei(xP+kT )·ξ×

〈P|Fnνa (0)

(Un[–]

[0,ξ]

)ab

Fnµb (ξ)|P〉

∣∣∣∣ξ·P′=0

= − 12x

gµνT f g

1 −(kµT kνT

M2p

+ gµνT

k2T

2M2p

)h⊥ g

1

+ suppr., (2)

where gµνT = gµν − (Pµ1Pν

2 + Pµ2Pν

1)/P1·P2, Mp is the protonmass and the gauge link Un[–]

[0,ξ] is needed to render the matrixelement gauge invariant. It runs from 0 to ξ via −∞ alongthe n direction. [n is a timelike dimensionless 4-vector withno transverse components such that ζ2 = (2n·P)2/n2.] Thecorrelator is parametrized by the two gluon TMDs discussedabove, f g

1 (x, kT , ζ, µ) and h⊥ g1 (x, kT , ζ, µ) [8] and by terms that

are suppressed in the high-energy limit. In principle, Eqs. (1)and (2) also contain soft factors [6, 31, 32], but with the ap-propriate choice of ζ1ζ2 ∼ 2.3s, one can minimize their con-tribution, at least up to NLO in αs.

The structure of the TMD cross section is then found to be

dσdQdYd2qT dΩ

=C0(Q2 − M2

Q)

s Q3D

F1 C

[f g1 f g

1

]+ F3 cos(2φCS )

C[w3 f g

1 h⊥g1 + x1↔ x2

]+F4cos(4φCS)C

[w4h⊥g

1 h⊥g1

]+ O

(q2

T

Q2

),

(3)

where dΩ = dcos θCS dφCS is expressed in terms of Collins-Soper angles [33] and where Q, Y and qT are the invariantmass, the rapidity and the TM of the pair –the latter two to bemeasured in the hadron c.m.s. frame. The overall normaliza-tion is given by C0 = 4α2

sαeme2Q|R0(0)|2/(3M3

Q), where R0(0)is the quarkonium radial wave function at the origin and eQ

the heavy quark charge. The F factors and the denominator Dare given by

F1 = 1 + 2α2 + 9α4 + (6α4 − 2) cos2 θCS + (α2 − 1)2 cos4 θCS ,

F3 = 4α2 sin2 θCS , F4 = (α2 − 1)2 sin4 θCS ,

D =((α2 + 1)2 − (α2 − 1)2 cos2 θCS

)2, (4)

where α ≡ Q/MQ. The convolution is defined as

C[w f g] ≡∫

d2 k1T

∫d2 k2T δ

2(k1T + k2T − qT )×w(k1T , k2T ) f (x1, k2

1T ) g(x2, k22T ), (5)

in which the longitudinal momentum fractions are evaluatedat x1,2 = exp[±Y] Q/

√s. The weights appearing in the convo-

lutions are given by

w3 ≡q2

T k22T − 2(qT ·k2T )2

2M2pq2

T

,

w4 ≡ 2

k1T ·k2T

2M2p− (k1T ·qT )(k2T ·qT )

M2pq2

T

2

− k21T k2

2T

4M4p. (6)

We propose the measurement of 3 TM spectra, normalizedand weighted by cos nφ for n = 0, 2, 4:

S(n)qT≡

∫dφCS cos(n φCS) dσ

dQdYd2 qT dΩ∫dq2

T

∫dφCS

dσdQdYd2 qT dΩ

, (7)

where we will take the q2T integration in the denominator up to

(Q/2)2. These spectra separate out the the 3 terms in Eq. 3:

S(0)qT

=C[ f g

1 f g1 ]∫

dq2T C[ f g

1 f g1 ],S(2)

qT=

F3 C[w3 f g1 h⊥g

1 + x1 ↔ x2]

2F1∫dq2

T C[ f g1 f g

1 ],

S(4)qT

=F4 C[w4h⊥g

1 h⊥g1 ]

2F1∫dq2

T C[ f g1 f g

1 ]. (8)

4

2 4 6 8 10qT HGeVL

0.005

0.010

0.020

0.050

0.100

SqT

H0LHGeV

-2L

Set B

KMR

CGC

Gaussian

(a)

2 4 6 8 10qT HGeVL

-0.0010

-0.0008

-0.0006

-0.0004

-0.0002

SqT

H2LHGeV

-2L

Set B + max

KMR + max

CGC

Gaussian + max

(b)2 4 6 8 10

qT HGeVL

0.0001

0.0002

0.0003

0.0004

SqT

H4LHGeV

-2L

Set B + max

KMR + max

CGC

Gaussian + max

(c)

FIG. 3. Model predictions for Υ + γ production at Q = 20 GeV, Y = 0 and θCS = π/2 at√

s = 14 TeV for (a) S(0)qT , (b) S(2)

qT and (c) S(4)qT .

It is remarkable to note that the sole measurement of S(0)qT ,

i.e. of the cross section integrated over φCS , allows for a cleandetermination of the unpolarized gluon TMD, f g

1 , since h⊥g1

does not enter S(0)qT . If the distributions S(2)

qT or S(4)qT can also be

measured, then the linearly-polarized gluon distribution, h⊥g1 ,

is also accessible.

Numerical results and discussions.— In the literature theterm unintegrated gluon distribution (UGD) is widely used,particularly in the field of small-x physics, where it has beenstudied e.g. in the framework of the Color Glass Condensate(CGC) model [34–37]. UGDs also appear in kT -factorizationapproaches and as the solution of the CCFM equation [38].These UGDs are not identical to the gluon TMDs appearing inthe TMD factorization formula. For instance, the Weizsacker-Williams distribution that appears in the CGC model doeshave the same operator structure as Eq. 2, but with a lightlikegauge link. The regularization of the rapidity divergence isthus different. The CCFM equation does not rely on a gauge-invariant-operator definition; its connection with the TMDformalism is certainly less trivial than sometimes asserted, asin e.g. [39, 40]

We nonetheless think that these UGDs form justifiableAnsatze for gluon TMDs. These are in any case sufficientto evaluate S(n)

qT and address the experimental requirements tolearn genuinely new information on the gluon transverse dy-namics, i.e. on the underlying physics involved in it.

For f g1 , we have taken the Set B0 solution to the CCFM

equation with an initial distribution based on the HERA datafrom [41, 42], the KMR parametrization from [43] and theCGC model prediction from [34–37]. The first two depend ona factorization scale, taken to be the invariant mass of the pair,Q, whereas the last one depends on a saturation scale taken asQs = (x0/x)λQ0, with λ = 0.29, x0 = 4 · 10−4 and Q0 = 1GeV [44]. We have also used a simple Gaussian parametriza-tion, as done in [45] to describe the intrinsic gluon TM, butwith a scaled-up width of 〈p2

T 〉 = (2.5 GeV)2. The resultingTM distribution are shown in Fig. 3a.

For h⊥g1 , we use the CGC model prediction of [36, 37]

and the maximal value from the positivity constraint |h⊥g1 | ≤

2M2p/k2

1T f g1 [8]. The latter Ansatz is accordance with kT -

factorization in which a full gluon polarization is implicit. Theresulting S(2,4)

qT are plotted in Fig. 3b and Fig. 3c.

From Fig. 3a, we first conclude that measuring S(0)qT in bins

of 1 GeV should suffice to get a first determination of theshape of the unpolarized gluon distribution. As regards S(2)

qT

and S(4)qT , whose magnitude is obviously smaller, one can inte-

grate them over q2T (up to (Q/2)2) to get the first experimental

verification of a nonzero linearly-polarized gluon distribution.S(2)

qT is here the most promising as we obtain for the integrateddistribution −2.9%, −2.6%, −2.5% and −2.0% for the Gauss,CGC, SetB and KMR Ansatz respectively, whereas for then = 4 distribution we obtain 1.2%, 0.7%, 0.6%, and 0.3%for the Gauss, SetB, KMR and CGC model respectively. Wenote that the qT -integrated cross section for Υ + γ produc-tion in Fig. 2 is about 100 (50) fb/GeV at Q = 20 GeV for√

s = 14(7) TeV. The 40 fb−1 of integrated luminosity alreadycollected at 7 + 8 TeV should be sufficient to measure the qT

shape of S (0)qT , while S (2)

qT could be measured in a single qT -bin.Conclusion.— The production of an isolated photon back-

to-back with a –possibly isolated– quarkonium in pp colli-sions is the ideal observable to study the transverse dynamicsand the polarization of the gluons in the proton along the linesof TMD factorization. The requirement for a heavy quarko-nium in the final state suppresses quark-initiated reactionsmaking it a very clean probe of the gluon content of the pro-ton, whereas the large scale set by the invariant mass of thepair allows a TMD-factorized description over an extensiverange of qT and hence an extraction of the gluon TMDs in thisrange. The expected yields at the LHC experiments are largeenough to get the first experimental verification of a nonzerogluon polarization in unpolarized protons and to finally deliverthe first extraction of the gluon TMDs in the proton.

We thank D. d’Enterria, V. Kartvelishvili, C. Lorce, A. Signori, L.Szymanowski, S. Wallon and J.X. Wang for useful discussions. Thiswork was supported in part by the French CNRS, grants PICS-06149Torino-IPNO & PEPS4AFTER2, by the German Bundesministeriumfur Bildung und Forschung (BMBF), grant no. 05P12VTCTG, andby the European Community under the Ideas program QWORK(contract 320389).

∗ wilco.den-dunnen@uni-tuebingen.de† lansberg@ipno.in2p3.fr‡ c.pisano@nikhef.nl§ marc.schlegel@uni-tuebingen.de

5

[1] R. Brock et al., Rev. Mod. Phys. 67 157, (1995).[2] P.M. Nadolsky et al., Phys. Rev. D 78, 013004 (2008);

A.D. Martin et al., Eur. Phys. J. C 63, 189 (2009); R.D. Ballet al., Nucl. Phys. B838, 136 (2010).

[3] J.C. Collins, Foundations of Perturbative QCD, CambridgeUniversity Press, Cambridge, 2011; S.M. Aybat, T.C. Rogers,Phys. Rev. D 83, 114042 (2011); J.C. Collins, T.C. Rogers,Phys. Rev. D 87, 3, 034018 (2013); M.G. Echevarria, A. Idilbiand I. Scimemi, JHEP 1207, 002 (2012).

[4] X. D. Ji, J. -p. Ma and F. Yuan, Phys. Rev. D 71, 034005 (2005).[5] R. Zhu, P. Sun and F. Yuan, arXiv:1309.0780 [hep-ph].[6] J. P. Ma, J. X. Wang and S. Zhao, Phys. Rev. D 88, 014027

(2013).[7] V. Barone, F. Bradamante and A. Martin, Prog. Part. Nucl. Phys.

65, 267 (2010); U. D’Alesio and F. Murgia, Prog. Part. Nucl.Phys. 61, 394 (2008).

[8] P. J. Mulders and J. Rodrigues, Phys. Rev. D 63, 094021 (2001).[9] D. Boer, W. J. den Dunnen, C. Pisano, M. Schlegel and

W. Vogelsang, Phys. Rev. Lett. 108, 032002 (2012); D. Boer,W. J. den Dunnen, C. Pisano and M. Schlegel, Phys. Rev. Lett.111, 032002 (2013); W. J. den Dunnen, D. Boer, C. Pisano,M. Schlegel and W. Vogelsang, arXiv:1205.6931 [hep-ph];W. J. den Dunnen and M. Schlegel, arXiv:1310.4965 [hep-ph];W. J. den Dunnen, arXiv:1311.1048 [hep-ph].

[10] D. Boer, P.J. Mulders and C. Pisano, Phys. Rev. D. 80, 094017(2009); C. Pisano, D. Boer, S. J. Brodsky, M. G. A. Buffing andP. J. Mulders, JHEP 1310, 024 (2013).

[11] J. -W. Qiu, M. Schlegel and W. Vogelsang, Phys. Rev. Lett. 107,062001 (2011).

[12] J.C. Collins and J.W. Qiu, Phys. Rev. D 75, 114014 (2007);T.C. Rogers and P.J. Mulders, Phys. Rev. D 81, 094006 (2010).

[13] C-H. Chang, Nucl. Phys. B172, 425 (1980); R. Baier and R.Ruckl, Phys. Lett. B 102, 364 (1981); R. Baier and R. Ruckl,Z. Phys. C 19, 251 (1983).

[14] D. Boer and C. Pisano, Phys. Rev. D 86, 094007 (2012).[15] J.P. Ma, J.X. Wang, S. Zhao, Phys. Rev. D 88, 014027 (2013).[16] G. T. Bodwin, E. Braaten, G. P. Lepage, Phys. Rev. D 51, 1125

(1995), 55, 5853(E) (1997); P. L. Cho, A. K. Leibovich, Phys.Rev. D 53, 150 (1996); P. L. Cho, A. K. Leibovich, Phys. Rev.D 53, 6203 (1996).

[17] R. Li and J. -X. Wang, Phys. Lett. B 672, 51 (2009).[18] C. S. Kim, J. Lee, H. S. Song, Phys. Rev. D 55, 5429 (1997).

[19] J. P. Lansberg, Phys. Lett. B 679, 340 (2009).[20] N. Brambilla, et al. Eur. Phys. J. C 71, 1534 (2011).[21] J. P. Lansberg, Eur. Phys. J. C 61, 693 (2009).[22] J. P. Lansberg, Int. J. Mod. Phys. A 21, 3857 (2006).[23] R. Li and J. -X. Wang, arXiv:1401.6918 [hep-ph].[24] R. Sharma and I. Vitev, Phys. Rev. C 87 4, 044905 (2013) .[25] P. Mathews, K. Sridhar and R. Basu, Phys. Rev. D 60, 014009

(1999).[26] J. P. Lansberg and C. Lorce, Phys. Lett. B 726, 218 (2013).[27] Y. J. Zhang, Y. Q. Ma, K. Wang, K. T. Chao, Phys. Rev. D 81,

034015 (2010); Z. G. He, Y. Fan, K. T. Chao, Phys. Rev. D 81,054036 (2010).

[28] A. C. Kraan, AIP Conf. Proc. 1038, 45 (2008); D. Kikola, Nucl.Phys. Proc. Suppl. 214, 177 (2011).

[29] S. Mao, M. Wen-Gan, L. Gang, Z. Ren-You and G. Lei, JHEP1102, 071 (2011) [Erratum-ibid. 1212, 010 (2012)].

[30] B. Gong, J. -P. Lansberg, C. Lorce and J. Wang, JHEP 1303,115 (2013).

[31] X. -D. Ji, J. -P. Ma and F. Yuan, JHEP 0507, 020 (2005).[32] P. Sun, B.-W. Xiao and F. Yuan, Phys. Rev. D 84, 094005

(2011).[33] J. C. Collins and D. E. Soper, Phys. Rev. D 16, 2219 (1977).[34] F. Dominguez, B. -W. Xiao and F. Yuan, Phys. Rev. Lett. 106,

022301 (2011).[35] F. Dominguez, C. Marquet, B. -W. Xiao and F. Yuan, Phys. Rev.

D 83, 105005 (2011).[36] F. Dominguez, J. -W. Qiu, B. -W. Xiao and F. Yuan, Phys. Rev.

D 85, 045003 (2012).[37] A. Metz and J. Zhou, Phys. Rev. D 84, 051503 (2011).[38] B. Andersson et al., Eur. Phys. J. C 25, 77 (2002). J. R. Ander-

sen et al., Eur. Phys. J. C 35, 67 (2004); J. R. Andersen et al.,Eur. Phys. J. C 48, 53 (2006).

[39] F. Hautmann and H. Jung, arXiv:1312.7875 [hep-ph].[40] A. V. Lipatov, G. I. Lykasov and N. P. Zotov, Phys. Rev. D 89,

014001 (2014).[41] H. Jung, hep-ph/0411287.[42] H. Jung, et al., Eur. Phys. J. C 70, 1237 (2010).[43] M. A. Kimber, A. D. Martin and M. G. Ryskin, Phys. Rev. D

63, 114027 (2001).[44] F. Gelis, E. Iancu, J. Jalilian-Marian and R. Venugopalan, Ann.

Rev. Nucl. Part. Sci. 60, 463 (2010).[45] K. Sridhar, A. D. Martin and W. J. Stirling, Phys. Lett. B 438,

211 (1998).