On relevance of triple gluon fusion in $J/\psi$ hadroproduction

0.22 MB·English

by  Leszek Motyka

y
On relevance of triple gluon fusion in J/ hadroproduction
Leszek Motyka and Mariusz Sadzikowski†
∗
Institute of Physics, Jagiellonian University, Łojasiewicza 11, 30-348, Kraków, Poland
(Dated: January 21,2015)
A contribution to J/y hadroproduction is analyzed in which the meson production is
mediated by three-gluon partonic state, with two gluons coming from the target and one
gluonfromtheprojectile. Thismechanism involves doublegluondensityinoneofthepro-
tons, hence this contribution enters at a non-leading twist. It is, however, relevant due to
an enhancement factor coming from large double gluon density at small x. We calculate
5
1 the three-gluon contribution to J/y hadroproduction within perturbative QCD in the k -
0 T
2
factorization framework. Results are obtained for differential p -dependent cross-sections
T
n
a for all J/y polarizations and for the sum over the polarization components. The rescatter-
J

0 ing contribution is found to provide a significant correction to the standard leading twist
2
] cross-section at the energies of the Tevatron or the LHC at moderate pT. We suggest J/y
h
p production inproton-nucleus collision asapossible probeofthetriplegluonmechanism.
-
p
e
h
[

1 I. INTRODUCTION
v
5
1
Productionofheavyvectorquarkoniainprotoncollisionshas beenasubjectofintenseexperimental
9
4
and theoretical investigations since CDF found huge excess of the measured cross-sections of prompt
0
1. J/y [1] with respect to the standard theoretical predictions in QCD. Initially the theory predictions
0
5 were based on a leading order collinear QCD approximation, in which the hard matrix element of a
1
: partonic color singlet subprocess gg J/y g gives the dominant contribution. This approximation,
v
→
i
X however,underestimatesbadlythemeasuredcross-sectionandyieldsanincorrect p distributionofthe
T
r
a producedmesons,soitwasnecessarytoconsideralternativemechanismsofquarkoniahadroproduction.
Inparticular,twotheoreticalconceptsreceivedalotofattention: thesocalledcoloroctetmodel(COM)
[2–9]andtheotherapproachbasedonthek -factorization(KTF)scheme[10–15]. Foracomprehensive
T
theoryreviewofvariousapproaches toquarkoniumhadroproductionsee[16].
The COM is based on the fact that the hadron wave functions contain higher Fock components. For
Electronicaddress:[email protected]
∗
†Electronicaddress:[email protected]
2
heavy quarkonia it implies that the heavy quark-antiquark (QQ¯) pair in a heavy meson may be accom-
panied e.g. by one or more gluons. So the QQ¯-pair may appear in the meson in a color octet state [2].
Amplitudes of such octet components are subleading (w.r.t. the standard, color singlet component) in
heavy quark velocity expansion within the Heavy Quark Effective Theory [2], but in the vector meson
hadroproduction the color octet states contributions are enhanced by the corresponding hard partonic
cross-sectionforproductionofsuchstates[3, 4]. Withonlyafew freeparameters thecoloroctetmodel
is quitesuccessful in describing themagnitudeand p -dependence ofthe promptJ/y hadroproduction
T
[6–9],however,itfacessomeproblemswithanaccuratedesriptionoftheproducedmesonspolarization.
Inparticular,predictionsforthemesonpolarizationchangealotwhengoingfromLOtoNLOaccuracy
[7, 9]. Besides, neither LO nor NLO COM estimates of prompt J/y polarization are fully consistent
withthe data. It isfair to add that in theCOM theamplitudesof coloroctet QQ¯ transitionsinto mesons
arenotknownfromthefirstprinciples,andthetheoreticalestimatesofthecross-sectionmagnitudesrely
upon fitted parameters. On the other hand, kinematical dependence of the cross-sections, e.g. the p -
T
dependence,isalreadyagenuineQCDprediction(foreachpartonicsubprocessseparately)andthegood
description of the meson p -dependence within the COM supports strongly validity of this approach.
T
The COM may be also understood within hard factorization theorem in QCD, where universal (inde-
pendent on the environment)fragmentation functions exist of QQ¯ partonic states with various quantum
numbers(includingcoloroctet)intofinal statequarkonia[17].
Another theoretical approach to quarkonia hadroproduction relies on the k -factorization scheme.
T
In this framework, one allows for non-zero transverse momentum of colliding partons. Because of
multiple emissions in the course of QCD evolution the parton momentum may become sizeable if the
evolutionlengthissufficientlylarge. Thiseffectisespeciallypronouncedifgluonsareprobedatsmallx,
which is the case for prompt quarkonia production at the Tevatron or at the LHC. Then the kinemati-
cal distributions of produced quarkonia may be strongly affected by k of incoming gluons. Indeed,
T
inclusion of gluon k in the analysis brings theoretical estimates much closer to the experimental data,
T
both for the magnitude and for the shape of the distributions, already at the lowest order in QCD. This
works rather well for the C-even charmonia, like c , but for the accurate description of Tevatron data
c
on prompt J/y and y production a color octet component was necessary also in the KTF approach
′
[12], although, smaller than in the collinear approximation. For the recent LHC data on prompt char-
monia production, however, a good theoretical description of magnitude, kinematics and polarization
wasfoundwithinKTFwithoutincludingthecoloroctetmechanism[14]. ThisgooddescriptioninKTF
CSMwasachievedbyusingamoreadvancedunintegratedpartondensity,CCFMA0[18]. Still,itisfair
tosaythataglobal,consistentdescriptionofquarkoniaproductionin KTFhas notbeen formulatedyet.
3
It followsfrom this discussionthat theoretical understandingof heavy quarkonia hadroproductionis
not yet fully satisfactory. Polarization description and the need of fitting key parameters are somewhat
weak points of the color octet model, and a global description of J/y , y and c hardroproduction was
′ c
notformulatedyetin theKTFapproach. Therefore itispossible,that bothproposedapproaches arenot
completeand need tobesupplementedby anothermechanism.
The COM and CSM approaches both in the collinear and KTF framework share an important com-
monfeature—itisassumedthatatthepartonlevelthequarkoniaproductionisinitiatedbytwogluons.
At very large energies, however, when the gluon density becomes large, it is expected that sizable cor-
rectionstothispicturemaycomefromsubprocesswithmorethantwoinitialstatepartons,whichwould
beclassified asrescatteringorshadowing/anti-shadowingcontributions. Such effectsareexpectedtobe
particularlyimportantin collisionswithheaviernuclei. Firstestimatesofsuchmultiple-gluoncontribu-
tionsto heavy quarkoniaproductionin pp and pp¯ collisionswere performed by Khoze, Martin, Ryskin
andStirling[19],whoconsideredthree-gluoninitialpartonicstate. Recently,MaandVenugopolan[20]
proposedresummationofrescatteringcontributionswithinthecolorglasscondensate(CGC)formalism
[21–23]. A good description of unpolarized data for J/y and y was achieved down to very low p by
′ T
matching the resummed CGC cross-section with predictions of the collinear NRQCD COM approach
[20,24].
In this paper, followingthe idea of Khoze, Martin, Ryskin and Stirling [19], we shall investigatethe
three gluon mechanism of quarkonia hadroproduction in which hard rescattering plays the key rôle in
color neutralization of the QQ¯ pair. Thus, we shall consider the 3-gluon fusion process at the parton
level: 3g J/y in the color singlet channel instead of the standard color singlet process gg J/y g.
→ →
This three gluon fusion partonic process is, of course, a higher twist correction to the cross-section, as
it couples to a double gluon density in one of the protons. Therefore, one expects it to be suppressed
w.r.t. the standard, leading twist contribution by powers of the hard scale. In high energy collisions,
however, this rescattering term receives significant enhancement due to large gluon density, reflected
by large ratio of double and single gluon density at small-x. Moreover, the matrix elements of the
rescattering term 3g J/y , and the standard one gg J/y g, come at the same order of perturbation
→ →
theory, so no additional a suppression of the rescattering piece occurs in the matrix element. Hence,
s
despite of the higher twist nature, it may be still important. As already stated, such processes were
already proposed in Ref. [19] where estimates were given showing that rescattering might explain the
discrepancybetweenthedataandthepredictionsbasedonthecolorsingletmechanism. Weaddressand
develop this idea in more detail, by performing a complete calculation of the rescattering contribution
in the KTF framework, including an explicit calculation of p distribution of produced quarkonia and
T
4
Φa1a2
A
A A′ A ΦaA1a2 dPSA′ ΦAa3a4∗ A
A
′
α1k,k1 α2k,k2
J/ψ Φ Φ
J/ψ ∗J/ψ
βl,l
B
′
B
B ′ B B
a) Φb1 b) ΦbB1 dPSB′ ΦbB2∗
B
FIG. 1: a) The amplitude M of the J/y hadroproduction in the triple gluon fusion mechanism; b) the dominant
diagraminthesquareoftheamplitude, M 2.
| |
thepolarization composition. We find that the rescattering correction is a significant contributionto the
prompt J/y cross-section especially at moderate p and should be included in a precise description
T
of the data. It should be stressed that the rescattering correction complements in natural way both the
COM and the standard KTF approach, providing a kinematics- and process-dependent mechanism of
color neutralization. Also, we expect that the rescattering mechanism should be strongly enhanced and
thus–evenmoreimportantfor heavyquarkoniaproductionin collisionswithnuclei.
The paper is organized as follows: in Sec. II we introduce the kinematics of the process and the
notation, in Sec. III we derive analytic formulae for the amplitudes and cross-sections of the direct
quarkonium production by rescattering, in Sec. IV we give numerical estimates for the cross sections
and conclusionsaregiveninSec. V. Technicalpieces ofderivationsaregivenintheAppendix.
II. KINEMATICS
We consider direct inclusive production of heavy quarkonia in high energy proton–(anti-) proton
collisionsand focus on a process mediated by partonicsubprocess with tree incominggluons: 3g V.
→
In general, V can be any heavy vector quarkonium, J/y , y , ¡ and so on, but we choose the best
′
measured J/y production as the reference process. The three-gluon fusion contribution enters as a
higher twist correction to the direct vector quarkonium production. The notation for the four-momenta
5
of incoming protons (p , p ), the outgoing meson, p, outgoing hadronic final states (p for A and p
A B ′A ′ ′B
forB) isexplainedintheleftpanel ofFig. 1.
′
WeanalyzetheprocesswithinperturbativeQCD inthehighenergy approximation,usingtheframe-
workofk -factorization. In thehighenergy limit,onlytheleadingpowerofs=(p +p )2 is retained.
T A B
In this approximationthe proton masses may be neglected, p2 = p2 =0, and s 2p p . The masses
A B ≃ A· B
ofoutgoingstatesare denoted by p2 =M2 , p2 =M2 , p2 =M2 for J/y and hadronicfinal states A
J/y A A B B ′
′ ′ ′ ′
and B emergingfrom thetarget andtheprojectilecorrespondingly.
′
The diagram in Fig. 1a illustrates the partonic topology of the amplitude for direct vector quarko-
nium production with three intermediate gluons. The four-momenta of intermediate gluon couplings
to p A and p B transitions are denoted by k , k and l correspondingly. The four-momentum
A ′ B ′ 1 2
→ →
conservationimposesconstraintson thefour-momenta,
p = p +k +k , p = p +l, p=k +k +l. (1)
A A 1 2 B B 1 2
′ ′
WeemploytheSudakov parameterizationofthefour-momenta,
p =a p +b p +p , p=a p +b p +p , (2)
A(B) A(B) A A(B) B A(B) A B
′ ′ ′ ′ ⊥ ⊥
where the transverse directions are in the plane orthogonal to the direction of the collision axis in the
CMSframe. Then, thekinematicconstraintslead to:
b A = p2a+Ms A2′, a B = p2b+Ms A2′, b = p2+a Ms J2/y . (3)
A B
Theprocess ofJ/y productionisanalyzed inthekinematicaldomainwherea =O(1),b =O(1)and
A B
consequentlyb a O(1/s). Thus,thegluonmomentasatisfy
A B
∼ ∼
k k +k (1 a )p p , l (1 b )p p (4)
1 2 A A A B B B
≡ ≈ − − ′⊥ ≈ − − ′⊥
and fromthelastterm of(1) onegets
a 1 a , b 1 b , p = p p . (5)
A B A B
≈ − ≈ − ⊥ − ′⊥− ′⊥
In the Sudakov parameterization of the gluon momenta k =a kp +b kp +k , l =a lp +b lp +l
i i A i B i A B
⊥ ⊥
(seeFig. 1a)onehasrelations:
a k+a k =a , b k+b k =O(1/s), a l =O(1/s), b =b . (6)
1 2 1 2 1
6
III. THETRIPLEGLUONCONTRIBUTION:ANALYTICFORMULAE
A. Thetriplegluonamplitude
TheamplitudeinFig. 1a reads
−iM = 21!SBb1µ1(pB,rB,B′,l)dµl12s 1 Z (d24p k)14Ssb11ra11ar 22(p,e ;l,k1,k2,p)dr 1kn 21kd2r 2n 2SAa1na12n 2(pA,rA,A′,k1,k2)
1 2
(7)
b b a a
whereS 1 = p ,l J 1(0) p ,r andS 1 2 (p ,r ,A,k ,k )givetheamplitudesoffindinggluonwith
Bµ1 h B′ | µ1 | b Bi An 1n 2 A A ′ 1 2
color b , four-momentuml and two gluons of colors a ,a and momentak ,k =k k respectively in
1 1 2 1 2 1
−
thedecayproductsofastruckproton. Theincomingprotonpolarizationsaredenotedbyr . Thevertex
A,B
Ssb1ra1ar 2(l,k1,k2;p,e ) describes the amplitude of J/y production with momentum p and polarization
1 1 2
e . In the standard Regge kinematics of high energy scattering at a small momentum transfer one can
approximatepolarization tensors of gluons (in the Feynman gauge) as dµ1s 1 2pµ1ps 1/s and dr in i
≈− A B ≈
r n
2p ip i/s fori=1,2. Then, onecan writetheamplitude(7)inthefollowingform
− A B
2s 1 d2k
iM = F b1(p ,r ,B;a ,b ,l) 1 F a1a2(p ,r ,A;a ,b ;k ,k k )
− (2p )4 B B B ′ l l l2Z k2(k k )2 A A A ′ k k 1 − 1
1 1
−
F b1a1a2(a ,b ;l,k ,k k ;p,e ), (8)
× J/y k l 1 − 1
where the impact factors (at the amplitude level) for a p B +g, p A +2g and 2g+g J/y
B ′ A ′
→ → →
transitionsread correspondingly:
µ
p 1
F b1(p ,r ,B ;a ,b ,l) = Sb1 (p ,r ,B;l) A , (9)
B B B ′ l l Bµ1 B B ′ s
pn 1pn 2
F a1a2(p ,r ,A;a ,b ;k ,k k ) = db kSa1a2 (p ,r ,A,k ,k k ) B B , (10)
A A A ′ k k 1 − 1 Z 1 An 1n 2 A A ′ 1 − 1 s
s r r
p 1p 1p 2
F Jb/1ay 1a2(a k,b l;l,k1,k−k1;pe ) = Z da k1Ssb11ra11ar 22(l,k1,k−k1;p,e ) B sA A . (11)
ThetotalcontributionofthreegluonstoJ/y productionamplitudecomesfromadiagramdepictedin
Fig.1aandan‘upside-down’diagramwiththesametopology,butwithtwogluonscouplingtothelower
vertex p B. The contribution of the other diagram, M may be obtained from M, corresponding
B ′ ′
→
to the diagram in Fig. 1a, by exchange of kinematical variables: a b . It is important to note that the
↔
interference of these two amplitudes, M and M is a subleading effect. This is because terms M M
′ ∗
and M M are driven by the QCD evolution of a four-gluon t-channel state (two ladders in the large
′∗ ′
N limit) emerging from one proton and a two-gluon t-channel state from the other proton (see Fig.
c
2a) whereas the interference terms M M and M M are driven by subleading three-gluon t-channel
∗ ′ ′∗
states from both sides, see Fig. 2b. Hence, at large energies and for large scales, the interference terms,
7
a) b)
FIG.2: Topologiescontributing totherescattering correctionintheJ/y cross-section : a)thetopologywiththree
two-gluonladders (leading), b)thetopology oftwothree-gluon BKPstates(suppressed).
showninFig. 2b,may beneglectedandthecontributiontotheJ/y productioncross-sectionfromthree
intermediategluonsds(cid:181) M 2+ M 2.
′
| | | |
B. Theuncorrelated triplegluoncross-section
The simplest model of the two-gluon distribution in the proton assumes lack of correlations of the
gluons in the transverse plane. This means that the double gluon density is proportional to a prod-
uct of independent single gluon distributions. Below we estimate the triple gluon contribution to J/y
hadroproductionin thisscenario.
The contribution to J/y cross-section, corresponding to the diagram shown in Fig. 1b takes the
followingform:
2 d4l 1
ds pp→J/y X = (2p )8Z (2p )4(l2)2F b2,1pb2(a l,b l,l) (12)
d4k d2k d2k
1 3 F a1a2a3a4(a ,b ,k ,k k ;k ,k k )
× Z (2p )4Z k2(k k )2Z k2(k k )2 4,p k k 1 − 1 3 − 3
1 1 3 3
− −
×F Jb/1ay 1a2(a k,b l;l,k1;,k−k1,p;e )F Jb/2ay 3a4∗(a k,b l;l,k3,k−k3,p;e )
d4p
(2p )4d (l+k p) q (p )2pd (p2 M2 ) +(U L),
× − (2p )4 0 − J/y ↔
where the last term describes contribution from the graph where upper and lower vertices from Fig. 1b
areinterchanged. Thepomeron–protonimpactfactor isdefined by theformula
µ µ
F b2,1pb2(a l,b l,l)= 12 (cid:229) (cid:229) Z dF B′(2p )4d (l−pB+pB′)SBb1µ1SBb2µ∗2 pA1spA2 , (13)
rB B′
8
ǫ1,b1,β1,l1 ǫ2,b2,β2,l2
dα
R
B B
ΦbB1 dPS ΦbB2∗
B′
FIG.3: Theunintegrated gluondistribution.
whereas
F a1a2a3a4(a ,b ,k ,k k ;k ,k k )= 1(cid:229) (cid:229) dF (2p )4d (k p +p ) (14)
4,p k k 1 − 1 3 − 3 2 Z A′ − A A′
rA A′
×F aA1a2(pA,rA,A′;a k,b k;k1,k−k1)F Aa3a4∗(pA,rA,A′;a k,b k,k3,k−k3)
givesthetwo-pomeron–protonimpactfactor(2P-p). Inthelastlinetheenergy-momentumconservation
deltafunctionin theJ/y vertexgivesa a ,b b .
k l
≈ ≈
Several remarks arein orderhere:
1. Thepomeron–protonimpact factorF (seeFig. 3)integratedoverlongitudinalvariablesgives
2,p
(2p )3
da db F b1b2(a ,b ,l)= f(b ,l2)d b1b2, (15)
Z l l 2,p l l N2 1
c
−
where b b follows from the energy-momentum conservation in J/y vertex and f(b ,l2) is an
l
≈
unintegratedgluondistribution.
2. The2P–pimpactfactorintegratedoverlongitudinalvariableb canbedecomposeinthefollowing
k
way(seeAppendixforthederivation):
da db F a1a2a3a4(a ,b ,k ,k k ;k ,k k )=
Z k k 4,p k k 1 − 1 3 − 3
R2 (2p )7
sh d a1a3d a2a4f(a ,k2)f(a ,(k k )2)S2(k k )
(N2 1)2s 1 − 1 1− 3
c
−
+d a1a4d a2a(cid:2)3f(a ,k2)f(a ,(k k )2)S2(k +k k)+d a1a2d a3a4... , (16)
1 1 1 3
− −
(cid:3)
where a a follows from the energy-momentum conservation in J/y vertex. The last term in
k
≈
parenthesis does not contribute to the cross-section due to the color factor contraction with the
J/y vertex. S(k) is a symmetricfunctionpeaked around k=0 and R is theShuvaev factor[25]
sh
introduced to account for the fact that longitudinal variables of the gluons are not equal and the
off-diagonalgluondistributionsenterthecross-section.
9
3. TheJ/y triplegluonvertexreads [26]:
da1a2b1
F Jb/1ay 1a2(a ,b ;l,k1;,k−k1,p;e ) = g3 N VJ/y (a ,b ;l,k1,k2;e ), (17)
c
e (xp +l )
VJ/y (a ,b ;l,k1,k2;e ) = 4p mcgJ/y −l2+(∗k· +kA )2+⊥ 4m2
(cid:20) 1 2 c
e p (x 4k1 k2)+e l
+ ∗· A − ys· ∗· ⊥ ,
l2+(k1 k2)2+4m2c #
−
wherek =k k .
2 1
−
Substituting equations (13) through (17) into (12) performing delta function integrations and color
factor contractions one arrives at the final formula for the rescattering correction to polarization depen-
dentdifferentialJ/y hadroproductioncross-section,
d3ds lnppb →dJp/2y X = p2Nc3N(Nc2c−−41)2Rs 2sefhf Z d2kd2k1a 3s f(b ,(p(−pk)k2))2fk(1a2(,kk12)kf(1a)2,(2k−k1)2)
⊥ − −
VJ/y (a ,b ;k1,k k1,p k;e ) 2+(a b ,pA pB)(cid:0), (cid:1) (18)
× − − ↔ ↔
(cid:12) (cid:12)
(cid:12) (cid:12)
where the last term describes contribution from the graph as in Fig. 1b where four gluons are attached
to the p instead of p vertex. In the derivation one needs to evaluatean integral d2k S2(k ) which,
B A
R ⊥ ⊥
assuming the Gaussian form of S˜(b), may be related to the inverse of multiple scattering parameter,
d2k S2(k )=8p 2s 1.
−eff
R ⊥ ⊥
C. Triplegluoncross-section withgluoncorrelations
Formula (18) describes the distributions of transverse positions of two-gluons coming from one of
the protons as being independent, i.e. uncorrelated, in the transverse plane. At large p of the meson
T
one should, however, expect some correlations of these distribution to emerge in the course of QCD
evolution of double gluon density. In this evolution [27], a single parton ladder may split into two
ladders,emerging at thesameimpactparameter, thatintroducescorrelations oftheparton positions.
AdiagramthatdescribesthegluonpartonsplittingintotwogluonsisshowninFig.4. Thiscontribu-
tion to J/y hadroproduction was discussed in [19] in the collinear limit and using only an approximate
estimate of QCD evolution of the double gluon state. This correlated rescattering correction is po-
tentially important as it is expected to lead to a less steep p dependence of the resulting differential
T
cross-sectioncomponentthantheoneobtainedwiththeuncorrelated gluondistributions. Wecalculated
this amplitude with full k dependence of the three gluons that enter the J/y vertex, an assuming that
T
10
FIG.4: Thecorrelated triplegluoncontribution.
the initial gluon in a upper vertex, that acts as a source of the double gluon, is collinear with the parent
proton. Afterstandardstepsweobtainedthefollowingcross-section,
d3s pp→J/y X = a 3sa 2s 9 Nc2−4 g(x ,µ)dx d2kd2k1d2k2f(b ,(p−k)2)
dlnb dp2 p 4 4Nc3(Nc−1)Z Z (p−k)4k12(k−k1)2k22(k−k2)2
⊥ ×VJ/y (a ,b ;k1,k−k1,p−k;e )VJ∗/y (a ,b ;k2,k−k2,p−k;e ). (19)
Ananalogousformulaisvalidalsoforaquarkfromthetargetreplacingthegluonasasourceofthetwo
correlated t-channel gluon after the suitable replacement of the pdfs and the color factor. This formula
providesthelowestorderapproximationtothecorrelated rescatteringamplitude,whichisonlyan input
amplitude for the evolution of the two-gluon state from the parent gluon to the J/y vertex. Since the
typical evolutionlength in rapidity is sizable for thisevolutionone expects a potentiallylarge effects of
theQCDevolutiontooccur. Thetreatmentofthisevolutionintheframeworkofthesmallxresummation
is possiblebut it is a highly nontrivial task and it is beyond the scope of the present analysis. However,
we estimated numerically the lowest order contribution of (19) and found it to be significantly smaller
thantheuncorrelatedrescatteringcontributionof(18). Thusinthepresentanalysisthiscontributionwill
bedisregarded.
IV. NUMERICALRESULTS
A. Triplegluoncorrections in pp¯and ppcollisions.
In the numerical evaluations the unintegrated gluon densities were used derived from the CT10
collinear gluon density [30] using Kimber-Martin-Ryskin (KMR) scheme [31] with the factorization