Accessing unpolarized and linearly polarized gluon TMDs through quarkonium production

1.5 MB·

by  Asmita Mukherjee

Accessing unpolarized and linearly polarized gluon TMDs through quarkonium
production
Asmita Mukherjee and Sangem Rajesh∗
Department of Physics, Indian Institute of Technology Bombay, Mumbai-400076, India.
(Dated: January 3, 2017)
We present the study of accessing unpolarized and linearly polarized gluon TMDs in J/ψ and
Υ(1S)productioninunpolarizedproton-protoncollisionatLHC,RHICandAFTERenergies.Non-
relativisticQCDbasedcoloroctetmodel(COM)isusedforestimatingquarkoniumproductionrates
within transverse momentum dependent factorization formalism. A comparison is drawn between
theexperimentaldataandthetransversemomentumdistributionofquarkoniumobtainedinCOM
and color evaporation model.
I. INTRODUCTION h⊥g functionsareofprimeimportance.Inordertoprobe
7 1
1 h⊥g, several processes have been proposed theoretically.
1
0 Amongst the eight leading twist-2 transverse momen- Linear gluon polarization can be determined by measur-
2 tum dependent parton distribution functions (TMDs) ing azimuthal asymmetry in heavy quark pair and dijet
n [1,2,3],fg(x,k )andh⊥g(x,k )aretheonlytwoTMDs production in SIDIS [6], Υ + jet [7] and γγ [8] in pp
1 ⊥ 1 ⊥
a which describe the dynamics of gluons inside an unpo- collision at LHC. h⊥g can also be accessed through the
1
2 J ulanrpizoeldarhizaeddroanndwhliinleeafr1glyanpdolha⊥1rigzerdepgrleusoennst tinhseiddeenasnityuno-f c[1r3o]ssansedctCio-neveonf H(cihgagrsg-beocsoonnju[9g,at1io0n, 1e1v,en1)2]q,uHarigkgosn+iujemt

polarizedhadronrespectively.TMDshavebeenreceiving production [14].
]
h paramount interest in both theoretically and experimen- Inthisproceedingcontribution,wediscusstheJ/ψand
p tallyastheyprovidethe3-dimensionalstructureandspin Υ(1S) production in unpolarized proton-proton collision
- informationofthenucleon.TMDsdependonbothlongi- toshowthatthequarkoniumproductionisalsoapromis-
p
e tudinal momentum fraction (x) and intrinsic transverse ing channel to extract both f1g and h⊥1g. Details of our
h momentum (k ) of the parton whereas usual collinear work can be found in [15, 16]. We estimate the quarko-
⊥
[ parton distribution functions (PDFs) depend only on x. nium production rates using color octet model (COM)

Gauge links are required to define the gauge invariant [15]withintransversemomentumdependent(TMD)[17]
1
v operator definition of TMDs and are process dependent. framework and draw a comparison between the results
9 In general, linearly polarized gluons can be present with color evaporation model (CEM) [16] and experi-
2 even at tree level inside an unpolarized hadron [1] pro- mentaldata.COM,colorsingletmodel(CSM)andCEM
3 vided that the gluons carry transverse momentum w.r.t are the three important models for quarkonium produc-
0
parent hadron. The associated density function of lin- tion, which are successful at different energies. Gener-
0
early polarized gluons, h⊥g is a T-even (time-reversal ally, two scales are involved in quarkonium production
. 1
1 even) distribution and is also even in the transverse mo- [18, 19, 20]. The first one is related to the production
0 mentum. TMDs are nonperturbative objects and have of heavy quark pair with momentum of order M (heavy
7
to be extracted from experiments. Drell-Yan (DY) and quark mass) which is called short distance factor. This
1
: semi-inclusive deep inelastic scattering (SIDIS) are the short distance factor can be calculated in order αs(M)
v two processes which provide the experimental data re- usingperturbationtheory.Thesecondoneisthebinding
i
X lated to the TMDs [4]. In these processes, the intrinsic ofquarkoniumboundstatewhichistakingplaceatscale
r transversemomentum(k⊥)hasanimprintontheexper- of order ΛQCD. This is a nonperturbative process and
a imentally measurable quantities, for instance, azimuthal is denoted with long distance matrix elements (LDME)
asymmetriesandtransversemomentum(p )distribution in factorization expression. The hadronization informa-
T
of the final hadron. Hence, these quantities are very sen- tionisencodedintheLDMEwhichareusuallyextracted
sitive to the TMDs. However, h⊥g and even fg have not byfittingdata.Thenon-relativisticQuantumchromody-
1 1
been extracted yet. Gluon Sivers function (f⊥g) [5] gen- namics(NRQCD)effectivefieldtheory[20]separatesthe
1 short distance and long distance factors systematically.
erates single spin asymmetry in scattering processes like
In COM [21], the initially produced heavy quark pair
ep↑ and pp↑. In order to understand asymmetries fully,
can be either in color singlet or octet state.
one should have complete knowledge about unpolarized
TMDs since fg sits in the denominator of the asymme-
1
try expression [4]. Therefore, the extraction of fg and
1
II. J/ψ AND Υ(1S) PRODUCTION IN COM
We consider unpolarized proton-proton collision pro-
∗ Thisproceedingisbasedonatalkdeliveredat22ndInternational cess for quarkonium production i.e., p + p →
SpinSymposium,2016,UIUC.;[email protected] J/ψorΥ(1S)+X.Protonisrichofgluonsathighenergy,
2
(cid:104) (cid:105)
henceweconsidertheleadingorder(LO)gluon-gluonfu- w = 1 (k .k )2− 1k2 k2 and k = p −
2M4 ⊥a ⊥b 2 ⊥a ⊥b ⊥b T
sion channel for quarkonium production. Assuming that h
k . The definition of C is given in Eq.(6) of Ref. [15].
the TMD factorization holds good, the differential cross ⊥a n
Herep andyarethetransversemomentumandrapidity
section is given by [15] T
of the quarkonium.
(cid:90)
dσ = dx dx d2k d2k Φµν(x ,k )
a b ⊥a ⊥b g a ⊥a
(1)
III. EVOLUTION OF TMDS
×Φ (x ,k )dσJ/ψ(Υ),
gµν b ⊥b
where, Φµν is the gluon-gluon correlator of unpolarized As per Ref. [14], we assume that the unpolarized and
g
spin-1 hadron, which can be further parametrized in linearlypolarizedgluonTMDsfollowtheGaussianform.
2 In Gaussian parametrization, TMDs are factorized into
terms of leading twist-2 TMDs as the following [1]
productofcollinearPDFstimesexponentialfactorwhich
1 (cid:110) (cid:16)kµkν is a function of only k and Gaussian width.
Φµν(x,k )= − gµνfg(x,k2)− ⊥ ⊥ ⊥
g ⊥ 2x T 1 ⊥ M2
+gµν k2⊥ (cid:17)h⊥g(x,k2)(cid:111). h (2) f1g(x,k2⊥)=f1g(x,Q2)π(cid:104)k1⊥2(cid:105)e−k2⊥/(cid:104)k⊥2(cid:105), (7)
T 2M2 1 ⊥
h
Here f1g and h⊥1g are the unpolarized and linearly polar- h⊥1g(x,k2⊥)= Mh2πf(cid:104)1gk(2x(cid:105),2Q2)2(1r−r)e1−k2⊥r(cid:104)k1⊥2(cid:105), (8)
ized gluon TMDs respectively. M is the proton mass. ⊥
h
The dσJ/ψ(Υ) in Eq.(1) is the partonic differential cross where, fg(x,Q2) is the collinear PDF which follows the
1
section of gg → QQ¯[2S+1L(a)] channel. Using NRQCD, DGLAP evolution equation and r = 2/3 and 1/3 [14]
J
the partonic differential cross section can be factorized values are taken for numerical estimation. The Gaussian
as follows [20, 22] widthsare(cid:104)k2(cid:105)=0.25GeV2 and1GeV2 [14].Inmodel-
⊥
I, we do not take any upper limit for k integration.
dσJ/ψ(Υ) =(cid:88)dσˆ[gg →QQ¯(n)](cid:104)0|OnJ/ψ(Υ) |0(cid:105) (3) An upper limit kmax =(cid:112)(cid:104)k⊥2(cid:105) [27] is con⊥siadered for k⊥a
n integrationinmodel-II.Theanalyticalexpressionsofdif-
The first term in the right hand side of Eq.(3) was given ferentialcrosssectionsformodel-Iandmodel-IIaregiven
in [15] that describes the production of heavy quark and inSec-(III)[15].AspointedoutinRef.[4],inordertoex-
anti-quarkpairinadefinitequantumstateanditcanbe plainhighpT spectrumonehastoconsiderthefullTMD
calculated in order α . Spin, orbital angular momentum evolution approach which was derived in impact param-
s
and color quantum numbers are denoted with n. After eter space (b⊥). The Fourier transformations of gluon-
forming the heavy quark pair, its quantum numbers will gluon correlator in b⊥ and k⊥ space are
be readjusted to form a color singlet quarkonium state (cid:90)
by emitting or absorbing soft gluons. This process is ab- Φ(x,b⊥)= d2k⊥e−ik⊥.b⊥Φ(x,k⊥), (9)
sorbed in (cid:104)0 | OJ/ψ(Υ) | 0(cid:105) (LDME) which is nonpertur-
n
bative. All possible configurations of heavy quark pair 1 (cid:90)
in different quantum states are taken into account for Φ(x,k⊥)= (2π)2 d2b⊥eik⊥.b⊥Φ(x,b⊥). (10)
quarkonium production which is represented with sum-
mation over n in Eq.(3). In line with Ref. [22, 23], we The gluon correlator in b⊥ space is given by [12]
consider only the color octet states 1S0, 3P0 and 3P2 1 (cid:110) (cid:16)2bµbν
which have dominant contribution in charmonium and Φg(x,b )= gµνfg(x,b2)− ⊥ ⊥
⊥ 2x T 1 ⊥ b2
bottomonium production. The LDME numerical values ⊥ (11)
(cid:17) (cid:111)
ofthesecoloroctetstatesareextractedinRef.[24,25,26], −gµν h⊥g(x,b2) .
T 1 ⊥
whicharetabulatedin[15].Afterintegratingw.r.tx ,x
a b
and k in Eq.(1) and following the steps in Ref. [15], In TMD evolution approach, TMDs depend on both
⊥b
one can obtain the differential cross section as renormalization scale µ and auxiliary scale ζ which was
dσff+hh dσff dσhh introducedtoregularizetherapiditydivergences.Renor-
= + , (4) malizationgroup(RG)andCollins-Soper(CS)equations
dyd2p dyd2p dyd2p
T T T are obtained by taking scale evolution w.r.t the scales µ
where andζ.AftersolvingtheseequationsoneobtainstheTMD
dσff C (cid:90) evolution expressions of TMDs in b⊥ space [17, 28, 29].
= n d2k fg(x ,k2 )fg(x ,k2 ), (5) The differential cross section expressions of Eq.(4) in
dyd2p s ⊥a 1 a ⊥a 1 b ⊥b
T TMD evolution approach are given by [15]
dyddσ2hphT = Csn (cid:90) d2k⊥awh⊥1g(xa,k2⊥a)h⊥1g(xb,k2⊥b), ddy2σdpf2Tf = C2sn (cid:90)0∞b⊥db⊥J0(pTb⊥)f1g(xa,c/b∗) (12)
(6) ×fg(x ,c/b )R R ,
1 b ∗ pert NP
3
and 1.0
d2σhh C C2 (cid:90) ∞ ff(cid:45)I
dydp2 = 2nsπ2A b⊥db⊥J0(pTb⊥)αs2(c/b∗) 0.8 ff(cid:43)hh(cid:45)I
T 0 ff(cid:45)(cid:72)II(cid:76)
(cid:76)
(cid:90) 1 dx (cid:18)x (cid:19) (cid:45)2 ff(cid:43)hh(cid:45)(cid:72)II(cid:76)
××(cid:90)xx1ba dxxx2121(cid:18)xxx2ba1 −−11(cid:19)ff11gg((xx21,,cc//bb∗∗))RpertRNP 2ΣdpGeVT 00..46(cid:144)(cid:72) k(cid:166)2 (cid:61)1GeV(cid:72)2(cid:76) (cid:72) (cid:76)
(13) d (cid:88) (cid:92)
1
where Rpert and RNP are the perturbative and nonper- (cid:45)Σ
turbative parts of the evolution kernel. 0.2
(cid:40) (cid:90) Q dµ(cid:18) (cid:18)Q2(cid:19) (cid:19)(cid:41) 0.0
R =exp −2 Alog +B 0.0 0.5 1.0 1.5 2.0 2.5 3.0
pert µ µ2
c/b∗ p GeV
T
(a)
(cid:40) (cid:41)
(cid:104) Q (cid:105) 3.5 (cid:72) (cid:76)
R =exp − 0.184log +0.332 b2
NP 2Q ⊥
0 3.0 ff(cid:45)I
ff(cid:43)hh(cid:45)I
Here A and B are the anomalous dimensions of the
(cid:76) ff(cid:45)(cid:72)II(cid:76)
evolution kernel and TMDs respectively and these have (cid:45)2V 2.5 ff(cid:43)hh(cid:45)(cid:72)II(cid:76)
perturbative expansion [15]. We used the b∗ prescrip- Ge (cid:72) (cid:76)
tion to avoid the Landau poles by freezing the scale as 2.0(cid:72) (cid:72) (cid:76)
b∗(b⊥) = (cid:114)1+(cid:16)b⊥bmb⊥ax(cid:17)2. In the nonperturbative regime 2ΣdpT 1.5(cid:144) (cid:88)k(cid:166)2(cid:92)(cid:61)0.25GeV2
where b is very large, the evolution kernel cannot be d
⊥ (cid:45)1 1.0
calculated using perturbation theory. Hence, the evolu- Σ
tionkernelinthisregimeismodeledasR [28].Wehave
NP 0.5
consideredthesamenonperturbativefactorR forboth
NP
unpolarized and linearly polarized gluon TMDs.
0.0
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4
p GeV
T
IV. NUMERICAL RESULTS
(b)
(cid:72) (cid:76)
We calculated the transverse momentum (p ) distri- FIG. 1. (color online) Differential cross section (normalized)
T
bution of J/ψ and√Υ(1S) in unpolarized√proton-proton of J/ψ a√nd Υ(1S) production√in pp → J/ψ(Υ(1S))+X at
collisionatLHC( s=7TeV),RHIC( s=500GeV) L√HCb ( s = 7 TeV), RHIC ( s = 500 GeV) and AFTER
√
( s = 115 GeV) energies using DGLAP evolution approach
andAFTER( s=115GeV)energies.Quarkoniumpro-
for (a) (cid:104)k2(cid:105) = 1 GeV2 and (b) (cid:104)k2(cid:105) = 0.25 GeV2 at r = 2
duction rates are estimated using NRQCD version COM ⊥ ⊥ 3
. The solid (ff-(I)) and dot dashed (ff-(II)) lines are obtained
within TMD factorization framework. Color octet states
by considering unpolarized gluons in Model-I and Model-II
such as 1S , 3P and 3P of initially produced heavy
0 0 2 respectively. The dashed (ff+hh-(I)) and tiny dashed (ff+hh-
quark pair are taken into account for quarkonium pro- (II)) lines are obtained by taking into account unpolarized
duction. The masses of J/ψ and Υ(1S) are considered gluons plus linearly polarized gluons in Model-I and Model-
3.096 and 9.398 GeV respectively. m = 1.5 GeV and II respectively. See the text for ranges of rapidity integration
c
m = 4.8 GeV are taken for charm and bottom quark [15].
b
masses respectively. MSTW2008 [39] is used for gluon
PDFs.Q=M (quarkoniummass)isconsideredforscale
of the gluon PDFs in DGLAP evolution. Quarkonium differential in p is normalized with total cross section
T
p distribution is obtained by integrating rapidity in the as a result we obtain the p spectrum which is indepen-
T T
range of y ∈ [2.0,4.5], y ∈ [−3.0,3.0] and y ∈ [−0.5,0.5] dentofcenterofmassenergyandquarkoniummass.The
for LHCb, RHIC and AFTER respectively. The conven- obtained p spectrum in DGLAP evolution approach in
T
tion in the figures as follows. “ff”and “ff+hh”represent model-I and model-II are compared in FIG.1 at r =2/3.
the quarkonium distribution obtained by taking into ac- Thequarkoniump spectrumhasbeenmodulatedsignif-
T
countonlyunpolarizedgluonsandlinearlypolarizedplus icantly by taking into consideration of linearly polarized
unpolarized gluons respectively. gluons along with the unpolarized gluons in the scatter-
FIG.1 represents the p spectrum of J/ψ and Υ(1S) ingprocess.Theeffectoflinearlypolarizedgluonsismore
T
which is estimated in COM. In FIG.1, the cross section inmodel-IIcomparedtomodel-I.InFIG.2,theestimated
4
ratios of J/ψ → e+e− and Υ(1S) → µ+µ− channels re-
104 spectively. J/ψ and Υ(1S) states can be produced from
LHC √s=7 TeV CEM-ff higher mass excited states. However, we have considered
CEM-ff+hh only the direct production of quarkonium in this arti-
103 COM-ff
COM-ff+hh
2V) CMS 1.6<y<2.4 100
e ALICE 2.5<y<4
G
b/ 102 LHCb 2<y<2.5 LHC √s=7 TeV ff-CEM
n ff+hh-CEM
y) ( 2V) ff-COM
2PdT 101 Ge10−1 ffAT+LhAhS-C 0O<My<1.2
2dσ/(d ) (nb/ CLHMCSb | y2|<<y2<.42.5
100 dy10−2
T
2
P
d
10−1 σ/(
0 1 2 3 4 5 6 7 8 d10−3
μ
(a) PT (GeV) Bμ

10−4
102 0 2 4 6 8 10 12
RHIC √s=200 GeV ffff-+ChEhM-CEM PT (GeV)
2V) 101 ffff-+ChOhM-COM
e FIG. 3. (color online). Differential cross section of Υ(1S) at
G STAR-2016 |y|<1 √
nb/ 100 STAR-2013 |y|<1 LHCb ( s = 7 TeV) as function of pT in pp → Υ(1S)+X
P) (T PHENIX |y|<0.35 usingTMDevolutionapproach.Dataaretakenfrom[36,37,
yd10−1 38]. The rapidity in the range 2.0<y<2.5 is chosen [15].
d
PT
π
2dσ/(210−2 cle. In general, LO calculation is insufficient to explain
Bee10−3 full pT spectrum. It may be possible to explain high pT
spectrum by adding NLO calculation with LO.
10−4
0 1 2 3 4 5 6 7 8
(b) PT (GeV) V. CONCLUSION
FIG. 2. (color online). Differential cross section of J/ψ at
√ √ We studied the transverse momentum (p ) distribu-
(a) LHCb ( s = 7 TeV) and (b) RHIC ( s = 200 GeV) T
tionofJ/ψandΥ(1S)inunpolarizedproton-protoncolli-
as function of p in pp → J/ψ +X using TMD evolution
T
sionwithinTMDfactorizationformalism.NRQCDbased
approach. Data are taken from [30, 31, 32] and [33, 34, 35]
for LHC and RHIC respectively. The rapidity in the range color octet model is employed to estimate the quarko-
2.0<y <2.5 and −0.35<y <0.35 is chosen for LHCb and niumproductionrates.ThequarkoniumpT spectrumhas
RHIC energies respectively [15]. beenmodulatedbythepresenceoflinearlypolarizedglu-
ons inside unpolarized proton and is in good agreement
with LHCb and RHIC data. Hence, quarkonium produc-
p spectrumofJ/ψinTMDevolutionapproachatLHCb tion offers a good possibility to probe both unpolarized
T
andRHICenergiesinCOMandCEMarecomparedwith and linearly polarized gluon TMDs.
data. Experimental data is taken from Ref. [30, 31, 32]
and Ref. [33, 34, 35] for LHCb and RHIC experiments
respectively.InFIG.3,p spectrumofΥ(1S)usingTMD
T
ACKNOWLEDGEMENT
evolution approach in COM and CEM is compared with
data [36, 37, 38]. The production rates are in good ac-
curacy with data up to low p for both J/ψ and Υ(1S), SRacknowledgesIITBombayandspinsymposiumor-
T
however, COM is slightly over estimated. In FIG.2 and ganizersforfinancialsupporttoattendthe22nd Interna-
FIG.3, B (0.0594) and B (0.0248) are the branching tional Spin Symposium, 2016, UIUC.
ee µµ
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