15 February 2001

Physics Letters B 500 (2001) 99–104www.elsevier.nl/locate/npe

χcJ polarizations at the Fermilab Tevatron

Feng Yuana,b,∗, Kuang-Ta Chaoc,d

a Department of Physics, Peking University, Beijing 100871, People’s Republic of Chinab Institut für Theoretische Physik der Universität, Philosophenweg 19, D-69120 Heidelberg, Germany

c China Center of Advanced Science and Technology (World Laboratory), Beijing 100080, People’s Republic of Chinad Department of Physics, Peking University, Beijing 100871, People’s Republic of China

Received 26 September 2000; received in revised form 17 December 2000; accepted 23 December 2000Editor: T. Yanagida

Abstract

We propose the measurement ofχcJ polarizations at high energy hadron colliders to study heavy quarkonium productionmechanism. We find that the color-singlet model in thekt factorization approach predicts very different behavior forχcJpolarizations at the Fermilab Tevatron compared with the NRQCD predictions in the collinear parton model. In the color-singletkt factorization approach, for bothχc1 andχc2 productions, the helicityh= 0 states dominate over other helicity statesat largepT . These properties are very useful in distinguishing between the two production mechanisms which are related to theinteresting issue ofJψ andψ ′ polarizations, and may provide a crucial test for thekt factorization approach. 2001 ElsevierScience B.V. All rights reserved.

PACS: 12.40.Nn; 13.85.Ni; 14.40.Gx

Studies of heavy quarkonium production in highenergy collisions provide important information onboth perturbative and nonperturbative QCD. In re-cent years, heavy quarkonium production has attractedmuch attention from both theory and experiment. Toexplain theJ/ψ and ψ ′ surplus problem of largetransverse momentum production at Tevatron [1], thecolor-octet production mechanism was introduced forthe description of heavy quarkonium production [2]based on the NRQCD factorization framework [3].During the last few years, extensive studies have beenperformed for the test of this color-octet productionmechanism. However, most recently the CDF col-

* Corresponding author.E-mail address: yf@tphys.uni-heidelberg.de (F. Yuan).

laboration have reported their preliminary measure-ments on the polarizations of the promptly producedcharmonium states [4], which appear not to sup-port the color-octet predictions that the directly pro-duced S-wave quarkonia have transverse polarizationsat largepT [5,6]. In [7,8], the authors considered thefeeddown contributions fromχc decays, and foundthe promptJ/ψ polarization disagree with the CDFdata at largepT by 3 standard deviations. This conflictshows that the heavy quarkonium production mech-anism may be more complicated than we knew be-fore, and further studies on heavy quarkonium pro-duction mechanisms other than the color-octet mech-anism in the collinear parton model are still needed atpresent.

In [9], the authors studiedχcJ hadroproductionat Tevatron in thekt factorization approach [10,11].

0370-2693/01/$ – see front matter 2001 Elsevier Science B.V. All rights reserved.PII: S0370-2693(01)00039-9

100 F. Yuan, K.-T. Chao / Physics Letters B 500 (2001) 99–104

Their results show that only color-singlet contribu-tions can reproduce the Tevatron data onχcJ produc-tion in the kt factorization approach, and the color-octet contributions disagree with the data. However,we note that previous studies in the collinear par-ton model have also given a good description for thedata ofχcJ production at Tevatron in the NRQCDapproach including both color-singlet and color-octetcontributions [13]. In this context, we have two dif-ferent mechanisms, i.e., the color-singletkt factor-ization approach and the NRQCD approach in thecollinear parton model, of which both can success-fully describe the Tevatron data on the largepT χcJproduction rates. So, it is quite urgent now to dis-tinguish between these two mechanisms for under-standing heavy quarkonium production at high en-ergy hadron colliders. For this purpose, we proposehere the measurements ofχcJ polarizations at the Fer-milab Tevatron. From our calculations we find thatthe color-singlet model in thekt factorization ap-proach predicts very different behavior forχcJ polar-izations compared with the NRQCD predictions in thecollinear parton model. In the color-singletkt factor-ization approach, for bothχc1 and χc2 productions,the helicity h = 0 states dominate over other helic-ity states at large transverse momentum. This novelproperty can provide a crucial test for this produc-tion mechanism. Furthermore, theχcJ polarizationmeasurements can help to clarify the present conflictbetween the color-octet predictions and the experi-mental data on promptJ/ψ andψ ′ polarizations atTevatron.

The polarized cross section formulas forχcJ hadro-production in the NRQCD approach have been cal-culated in [8,12,13] (including both color-singlet andcolor-octet processes). In this Letter, we will calculatethe color-singletχcJ polarized cross sections in thektfactorization approach. We will not include the color-octet processes in this approach, because their contri-butions toχcJ production disagree with the Tevatrondata in shape [9].

The kt -factorization approach differs greatly fromthe conventional collinear approximation because ittakes the nonvanishing transverse momenta of thescattering partons into account. The conventionalgluon densities are replaced by the unintegrated gluondistributions which depend on the transverse momen-tumkt . In the calculations, for every 4-momentaki we

make a Sudakov decomposition as

(1)ki = αip1 + βip2 + �kiT ,wherep1 andp2 are the momenta of the incominghadrons. In the high energy limit, we havep2

1 = 0,p2

2 = 0, and 2p1 · p2 = s, wheres is the c.m. energysquared.αi andβi are the momentum fractions ofp1andp2, respectively.kiT is the transverse momentum,which satisfies

(2)kiT · p1 = 0, kiT · p2 = 0.

For the momenta of the incident gluonsq1 andq2, wehave the following decompositions [11]

(3)q1 = x1p1 + q1T , q2 = x2p2 + q2T .

That is to say, the longitudinal component ofq1 (q2) isonly in the direction of light-like vectorp1 (p2).

Using the above defined Sudakov variables, wecan express the polarized cross sections forχcJhadroproduction in the following form

dσ(pp̄→ χ

(λ)cJ X

)

= 1

64× 16π

dαχ

αχd2q1T d

2q2Tf (x1;q2

1T )

q21T

(4)× f (x2;q22T )

q22T

|A(λ)0 (q1T , q2T )|2q2

1T q22T

,

where λ denotes the helicity ofχcJ , and αχ isthe momentum fraction ofp1 carried byχcJ . TheχcJ transverse momentumpT comes from the sumof the transverse momenta ofq1 and q2 as, �pT =�q1T + �q2T . The amplitude squared|A(λ)0 |2 describesχcJ (with helicity λ) production in the gluon–gluonfusion processesg + g → χ

(λ)cJ . To calculate these

helicity amplitudes, we need the polarization sumsfor individual helicity levels ofχcJ . For χc1, thelongitudinal and transverse polarization sums can bewritten in the following covariant forms

(5)∑λ=0

ε(λ)α ε(λ)∗β = PLαβ,

(6)∑|λ|=1

ε(λ)α ε(λ)∗β = PTαβ = Pαβ − PLαβ,

where

(7)Pαβ = −gαβ + pαpβp2 .

F. Yuan, K.-T. Chao / Physics Letters B 500 (2001) 99–104 101

And in the laboratory frame (the helicity frame),PLαβis expressed as [6]

PLαβ = (p ·Q)2(p ·Q)2 −M2s

(pα

M− M

p ·QQα)

(8)×(pβ

M− M

p ·QQβ),

whereM = 2mc is the mass ofχcJ , andQ= p1 + p2is the sum of the initial hadron 4-momenta. Forχc2,the polarization sums for individual helicity levels(λ = 0,1,2) can also be expressed in terms ofPαβ ,PTαβ , andPLαβ [12]. With these polarization sums, wecan calculate the production cross sections for indi-vidual helicity states ofχcJ , which are more involvedand will be presented elsewhere. We have checkedthat these cross section formulas can numerically re-produce the results of [9] for the inclusive productionrates ofχcJ at Tevatron after summing up all helicitystates contributions.

TheχcJ polarizations can be measured by studyingthe photon’s angular distribution in theχcJ rest framein the decay processesχcJ → J/ψγ . These angulardistributions have the following form

(9)dΓ (χcJ → J/ψγ )

d cosθ∝ 3

2(3+ α)(1+ α cos2 θ

),

whereθ is the angle between the photon’s 3-momen-tum in χcJ rest frame and theχcJ 3-momentum inthe laboratory frame.α is the polarization parameter(angular distribution parameter). Forχc1, α is definedas [12]

(10)α = 2− 3ρ

2+ ρ ,where

(11)ρ = dσ(χ(|λ|=1)c1 )

dσ(χc1).

Forχc2, the polarization parameter is [12]

(12)α = −6− 3η− 12τ

10− η− 4τ,

where

(13)η= dσ(χ(|λ|=1)c2 )

dσ(χc2), τ = dσ(χ

(|λ|=2)c2 )

dσ(χc2).

For numerical calculations, we choose the uninte-grated gluon distribution of [14] which can well fit

the F2(x,Q2) data over a wide range ofx andQ2,

and we set the scalesµ2 for the strong coupling con-stant αs(µ2) in the amplitude squared|A0|2 to beq2

1T for the interaction vertex associated with the in-cident gluonq1, and q2

2T for the vertex associatedwith q2 [9,15].

We first display in Fig. 1 the production ratio ofχc1 to χc2 at the Tevatron as a function ofpT , R =σ(χc1)/σ (χc2). The solid line is for the color-singletprediction in thekt factorization approach, and thedotted-dashed line for the NRQCD prediction in thecollinear parton model. For comparison, in this fig-ure we also plot the results for other two cases inthe collinear parton model: the color-singlet predic-tion as the dotted line and the color-octet predictionas the dashed line. However, we must note that nei-ther the color-singlet contributions nor the color-octetcontributions alone can describe the Tevatron dataon χcJ productions [13], and in this collinear partonmodel only the NRQCD predictions (including boththe color-singlet and the color-octet contributions) canmake sense to describe the Tevatron data onχcJ pro-ductions. From Fig. 1, we can see that theR ratioincreases aspT increases in the color-singlet modelkt factorization approach, and its value approaches to2.0 at large transverse momentum, which means thatat largepT χc1 production dominates overχc2 pro-

Fig. 1. The production ratio ofχc1 to χc2 as a function ofpT .The solid line is for the color-singletkt factorization approachprediction, the dotted-dashed line for the NRQCD prediction in thecollinear parton model, the dotted and dashed lines are, respectively,for the color-singlet and color-octet predictions alone in the collinearparton model.

102 F. Yuan, K.-T. Chao / Physics Letters B 500 (2001) 99–104

duction. In contrast, the NRQCD approach predictsthe R ratio to be much smaller, and its value ap-proaches to 0.6 at largepT . This is because, at largetransverse momentum the color-octet gluon fragmen-tation (the3S

(8)1 channel) dominates theχcJ produc-

tions in NRQCD, which leads toχcJ production ratesasσ(χc0) : σ(χc1) : σ(χc2)= 1 : 3 : 5 (consistent withour numerical calculations at largepT ). This figureshows that the difference onR ratio between the color-singletkt factorization approach and the NRQCD ap-proach in the collinear parton model is distinctive atsufficiently largepT .

We then study theχcJ polarizations at Tevatron.With the polarized cross section formulas, we can cal-culate the production rates for definite helicity statesof χcJ , and get the angular distribution parameterα.From our numerical calculations, we find that thecolor-singletkt factorization approach predicts verydifferent behavior forχcJ polarizations at the Fermi-lab Tevatron compared with the NRQCD predictionsin the collinear parton model. In the color-singletktfactorization approach, for bothχc1 andχc2 produc-tions, the helicity h = 0 states dominate over otherhelicity states at large transverse momentum. Thisproperty has distinguished consequence to the decayangular distribution parameterα for χcJ → J/ψγ

processes. These results are displayed in Figs. 2–4.Forχc1 → J/ψγ , at largepT the color-singletkt fac-torization approach predictsα around 0.8 while theNRQCD approach in the collinear parton model pre-dictsα around 0.2. Forχc2 → J/ψγ , the differencebetween these two mechanisms are more distinctive.The color-singletkt factorization approach predictsα(χc2 → J/ψγ ) to be negative (down to−0.6) atlargepT , while the NRQCD approach in the collinearparton model predictsα to be positive (around 0.3).

For the experimental measurement, it may be dif-ficult to distinguish betweenχc1 and χc2 contribu-tions in the observation of the photon’s angular dis-tributions in the decay processesχcJ → J/ψγ . So, itmay be more useful to give the angular distributionsin χcJ → J/ψγ with bothχc1 andχc2 taken into ac-count. We plot this result in Fig. 4. From this figure,we find that the color-singletkt factorization approachpredictsα for χcJ → J/ψγ being about 0.5 whilethe NRQCD approach in the collinear parton modelpredictsα around 0.25 at large transverse momentum.

Fig. 2. The polarization parameterα for χc1 → J/ψγ as a functionof pT . The definitions of the curves are the same as in Fig. 1.

Fig. 3. The same as in Fig. 2 but forχc2.

Fig. 4. The same as in Fig. 2 but forχcJ → J/ψγ with bothχc1andχc2 taken into account.

F. Yuan, K.-T. Chao / Physics Letters B 500 (2001) 99–104 103

The difference between these two mechanisms is stilldistinctive.

Finally, we note that the polarized cross section for-mulas forχcJ production can also be used to predictthe polarization ofJ/ψ which comes fromχcJ feed-down decays.J/ψ polarization can be measured bythe lepton’s angular distribution in theJ/ψ rest framein J/ψ → µ+µ− decay process. These distributionshave a similar form to those forχcJ decays,

(14)dΓ (J/ψ →µ+µ−)

d cosθ∝ 3

2(3+ α)(1+ α cos2 θ

),

where θ is the angle between the 3-momentum ofthe lepton inJ/ψ rest frame and the 3-momentumof J/ψ in the laboratory frame.α is the polarizationparameter, and is equal to

(15)α = 3ξ − 2

2− ξ ,where ξ is the ratio of the transversely polarizedto the totalJ/ψ , which can be calculated by usingthe polarized cross sections forχcJ production [12].In Fig. 5 we give theJ/ψ polarization from thefeeddown contributions ofχcJ decays. Again, we findthat the color-singletkt factorization approach predictsα(J/ψ → µ+µ−) being about 0.5 while the NRQCDapproach in the collinear parton model predictsαaround 0.25 at largepT .

At the Tevatron, the CDF collaboration has mea-sured the inclusive production cross sections ofχcJstates, which contribute about 30% of promptJ/ψ

Fig. 5. The polarization parameterα for J/ψ → µ+µ− for J/ψcoming fromχcJ decays.

production in a wide range ofpT [1]. TheχcJ statescan be identified with a photon plus theJ/ψ whichdecays into a muon pair. The production cross sec-tion of χcJ is found to be comparable to that ofJ/ψ ,and is not small. Unfortunately, the present statisticsof Tevatron run I is not high enough to allow separatepolarization measurements for this part ofJ/ψ (fromχc decays) from the directJ/ψ production andψ ′decay’s contribution [4]. However, with the upgradeTevatron run II, the luminosity will be increased by afactor of more than 8, we will have much more datafor J/ψ andχcJ production, so it will be feasible todistinguish between different contributions to promptJ/ψ polarizations, and then to measure theχcJ polar-izations.

In conclusion, in this Letter we have calculatedthe χcJ polarizations at high energy hadron collid-ers in the color-singletkt factorization approach andthe NRQCD approach in the collinear parton model.We find the difference on the polarization parame-ters for χcJ and their decay productsJ/ψ betweenthese two approaches are distinctive at large trans-verse momentum. Therefore,χcJ polarizations can beused to study these two production mechanisms athadron colliders and may provide important informa-tion on heavy quarkonium polarization mechanisms.Especially, the color-singletkt factorization approachpredicts thatχcJ productions are dominated by the he-licity h= 0 states at large transverse momentum. Thisunique property may provide a crucial test for this pro-duction mechanism.

Acknowledgements

This work was supported in part by the NationalNatural Science Foundation of China, the Ministryof Education of China, and the State Commission ofScience and Technology of China.

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