Table Of ContentJ/ψ(ψ′) production at the Tevatron and LHC at O(α4v4) in nonrelativistic QCD
s
Yan-Qing Ma (a), Kai Wang (a), and Kuang-Ta Chao (a,b)
(a) Department of Physics and State Key Laboratory of Nuclear Physics and Technology, Peking University, Beijing 100871, China
(b) Center for High Energy Physics, Peking University, Beijing 100871, China
′
We present a complete evaluation for J/ψ(ψ) prompt production at the Tevatron and LHC at
next-to-leading order in nonrelativistic QCD, including color-singlet, color-octet, and higher char-
monia feeddown contributions. The short-distance coefficients of 3P[8] at next-to-leading order are
J
found to be larger than leading order by more than an order of magnitude but with a minus sign
at high transverse momentum pT. Two new linear combinations of color-octet matrix elements are
1 obtained from the CDF data, and used to predict J/ψ production at the LHC, which agrees with
1 theCMSdata. Thepossibilityof1S[8]dominanceandtheJ/ψpolarizationpuzzlearealsodiscussed.
0 0
2
PACSnumbers: 12.38.Bx,13.60.Le,14.40.Pq
n
a
J Nearly 20 years ago, the CDF Collaboration found a may conclude nothing definite until all important chan-
′
7 surprisingly large production rate of ψ at high p [1]. nels in 1/p expansion are presented. It means the CO
T T
2 To solve the large discrepancy between data and the- channels 1S[8] and 3P[8] should be considered at NLO,
0 J
] oretical predictions, the color-octet(CO) mechanism [2] while the CS channel 3S[1] atnext-to-next-to-leadingor-
h was proposed based on nonrelativistic QCD (NRQCD) 1
der (NNLO) in α . Among these corrections, the com-
p factorization[3]. With the CO mechanism, QQ¯ pairs can s
pleteNNLOcalculationforCSisbeyondthestateofthe
p- be produced at short distances in CO (1S0[8], 3S1[8], 3PJ[8]) art, and the NNLO⋆ method is instead proposed[11], in
e states and subsequently evolve into physical quarkonia which only tree-level diagrams at this order are consid-
h
by nonperturbative emission of soft gluons. It can be
eredandaninfraredcutoffisimposedtocontrolsoftand
[
verified that the partonic differential cross sections at collinear divergences, and the NNLO⋆ contributions are
v2 leading-order (LO) in αs behave as 1/p4T for 3S1[8], and showntobelarge. However,theonly1/p4T leadingcontri-
5 1/p6T for 1S0[8] and 3PJ[8], all of which decrease at high pT bution at NNLO in CS is given by gluon fragmentation,
5 much slower than 1/p8 of the color-singlet (CS) state. whichwasfound[12]tobenegligiblecomparedtotheob-
T ′
6 The CO mechanism could give a naturalexplanation for servedJ/ψ(ψ )productiondata. OtherNNLO contribu-
3 the observedp distributions and largeproductionrates tions may givea 1/p6 term. Ina complete NNLO calcu-
T T
9. of ψ′ and J/ψ [4]. lationwithbothrealandvirtualcorrections,infraredand
0 collineardivergencesareremovedandthese NNLO 1/p6
However, the CO mechanism seems to encounter dif- T
0 ′ contributionsshouldbe smallerthantheNLO1/p6 con-
1 ficulties in explaining the observed J/ψ(ψ ) polariza- T
: tions. Dominatedbygluonfragmentationto3S[8],theLO tribution due to αs suppression. Therefore, to achieve
v 1 ′ a good description for J/ψ(ψ′) production a complete
NRQCD predicts transverse polarization for J/ψ(ψ ) at
Xi high p [4] whereas measurements at the Fermilab Teva- NLO calculation including both CS and CO seems to be
r tron giTve almost unpolarized J/ψ(ψ′)[5]. To exploit the necessary.
a underlyingphysics,severaleffortshavebeenmade,either At present, NRQCD factorization formalism with the
by introducing new channels[6] or by proposing other CO mechanism is used to describe various processes in
mechanisms[7]. It is a significant step to work out the heavy quarkonium production and decay. While J/ψ
next-to-leading order (NLO) QCD correction for the CS production in two-photon collisions at CERN LEP2[13]
channel, which enhances the differential cross section by and photoproduction at DESY HERA[14] are shown to
about 2 orders of magnitude at high p [8], and changes favor the presence of CO contribution, the J/ψ produc-
T
the polarization from being transverse at LO into longi- tionatB factoriesisdescribedwellusingNLOCSmodel
tudinal at NLO[9]. Although the CS NLO cross section and leaves little room for CO contributions[15]. In order
still lies far below the experimentaldata, it implies that, tofurthertesttheCOmechanism,itisnecessarytostudy
comparedto the α suppression,kinematic enhancement hadroproductionandextractCOlongdistancematrixel-
s
at high p is more important in the current issue. This ements (LDMEs) at NLO.
T
observation is also supported by our recent work[10] for In view of the importance, here we presenta complete
′
χ production,wherewefindtheratioofproductionrates NLO contribution to J/ψ(ψ ) production at the Teva-
c
ofσ /σ canbedramaticallyalteredbytheNLOcon- tronandLHC,includingallimportantCSandCOchan-
χc2 χc1
tributiondue tochangeofthe p distributionfrom1/p6 nels. According to the NRQCD factorization formalism,
T T
atLOto1/p4 atNLOintheCSP-wavechannels. Sowe the inclusive cross section for direct J/ψ production in
T
2
20 1.5
ïï(cid:144)ΣΣddNLOLO -11000 3S1@8D 1S0@8D 3P@J8D 3S1@1D ïï@D@D8831D(cid:144)H@D@DLΣ+ΣndSdS10 001...050 3P@J8D 1S0@8D 3S1@8D Sum
ï@Σ -0.5
d
´
-20 rn -1.0
10 20 30 40 50 10 20 30 40 50
p HGeVL p HGeVL
T T
FIG.1: DependenceofK factors(ratiosofNLOtoLOshort- FIG. 2: NLO short-distance coefficients dˆσ[3P[8]], r dˆσ[1S[8]],
J 0 0
distance coefficients dˆσ) on pT in J/ψ(ψ′) direct production r1dˆσ[3S1[8]], and Sum = r0dˆσ[1S0[8]]+r1dˆσ[3S1[8]] as functions
at theTevatron. of pT at the Tevatron, where r0 = 3.9, r1 = -0.56 and each
contribution is divided by dˆσ[1S[8]]+dˆσ[3S[8]].
0 1
hadron-hadroncollisions is expressed as
when p 3 GeV, which can be seen in Fig. 1. All the
J/ψ T
≈
dσ[pp→J/ψ+X]=Pn dˆσ[(cc¯)n]hmOn2cLni (1)lianr1g/epcTorerxepctainosnisonc.an be attributed to the enhancement
= P R dx1dx2Gi/pGj/p×dˆσ[i+j →(cc¯)n+X]hOnJ/ψi, Since we find 3PJ[8] channels can give a 1/p4T term and
i,j,n have a large K factor, the 3S[8] channel is no longer the
1
where p is either a proton or an antiproton, the indices unique source for high pT contributions. In fact, for the
i,j run over all the partonic species, and n denote the short-distancecoefficientsdefinedinEq.(1)thefollowing
color, spin and angular momentum (L ) of the interme- decomposition holds within an error of a few percent
n
diate cc¯states,including 3S[1], 1S[8], 3S[8] and3P[8] inthe
1 0 1 J dˆσ[3P[8]]=r dˆσ[1S[8]]+r dˆσ[3S[8]], (2)
present issue. Compared with the S-wave channel ob- J 0 0 1 1
tained in [8, 9, 16], the NLO treatment of 3P[8] is much
J where we find r0 =3.9 and r1 = 0.56 for the Tevatron,
more complicated. Fortunately, using the same method −
and r = 4.1 and r = 0.56 for the LHC. This decom-
0 1
asin[10],weareabletogetacompactexpressionforthe −′
position in direct J/ψ(ψ ) production at the Tevatron is
virtualcorrection,whichisbothtime-savingandnumeri-
shown in Fig.2, where each contribution is divided by
callystableinthefinalstatephasespaceintegration. For dˆσ[1S[8]]+dˆσ[3S[8]] to make it easy to read. As a result,
technical details, we refer readers to Ref.[10]. 0 1
it is convenient to use two linearly combined LDMEs
For numerical results, we choose the same parameters
as in [10] except that here we are restricted to √S = MJ/ψ = J/ψ(1S[8]) + r0 J/ψ(3P[8]) ,
1.96 TeV and yJ/ψ(ψ′) < 0.6 with the Tevatron, while 0,r0 hO 0 i m2chO 0 i
√S =7 TeV an|d yJ/ψ(|ψ′) <2.4 with the LHC. MJ/ψ = J/ψ(3S[8]) + r1 J/ψ(3P[8]) , (3)
Let us first have| a glanc|e at the overall correction be- 1,r1 hO 1 i m2chO 0 i
haviors as presented in Fig. 1. We find the K factor
when comparing theoreticalpredictions with experimen-
of short-distance coefficients dˆσ for 3P[8] channels (the
J tal data for production rates at the Tevatron and LHC.
sumoverJ=0,1,2weightedwithafactorof2J+1byspin Wenotethat,althoughboth J/ψ(3S[8]) anddˆσ[3P[8]]
symmetryinnonrelativisticlimit)islargebutnegativeat hO 1 i J
dependontherenormalizationschemeandthefactoriza-
high p . As explained in [10], the negative value mainly
T tionscaleµ , MJ/ψ doesnot. Thereasonis thatthede-
originatedfrom using the MS scheme when choosing the Λ 1,r1
renormalization scheme for S-wave spin-triplet NRQCD pendenceofhOJ/ψ(3S1[8])iiscanceledbythatofr1,which
LDMEs,anddoesnotaffectthephysicalresult. Another is originatedfrom decomposing dˆσ[3P[8]] at high p with
J T
nontrivialphenomenonisthat,differingfromotherchan- all information for the dependence (here we ignore the
nels, the K factor of 3S[8] channel is almost independent contribution of 3S[1], which decreases quickly at high p
1 1 T
of pT and not larger than 1.3. This can be understood in LO). So r1 should be viewed as r1(MS,µΛ) but for
sincetheαs correctiondoesnotbringanynewkinemati- simplicity we suppress these variables in the expression.
cally enhanced contributions for the 3S[8] channel, and it By fitting the p distributions of prompt ψ′ and J/ψ
1 T
impliestheexpansioninα isundercontroloncethelead- productionmeasuredatTevatron[17]inFig.3andFig.4,
s
ing p (scaling as 1/p4) channel is opened up. We also the CO LDMEs are determined as showing in Table I,
T T
note thatK factorsofallotherchannelsarejust about1 while the CS LDMEs are estimated using a potential
3
pcut hOHi MH MH
GTeV H GeV3 10−21G,r1eV3 10−20G,r0eV3 χ2/d.o.f.
3S@1D Tevatron
1 J/ψ 1.16 0.05±0.02 7.4±1.9 0.33
10 M1 Tevatron 7 ψ′ 0.76 0.12±0.03 2.0±0.6 0.56
LeV 1 TMo0taTlTeveavtartornon 5 Jψ/′ψ 10..1766 00..1167±±00..0054 51..21±±10..33 32..52
G
(cid:144)b TotalLHC
n
-LH 10-1 CDFData TABLE I: Fitted color-octet LDMEs in J/ψ(ψ′) production
+ΜΜ with chosen pcTut. Here r0 = 3.9, r1 = −0.56 are determined
® fromshort-distancecoefficientdecompositionattheTevatron.
¢H´ΨBr 10-2 pEernrodresnacreeodnulye.toCroelonro-rsminagllieztat(i3oSn1[1a]n)dLfDaMctoErsizhaOtioHnisacraeleesdtei--
pT 10-3 mated using a potential model result[19].
d
(cid:144)Σ
d
10-4
points on the origination of the small R.
(1) We find the fitted results are not good for data
10-5
5 10 15 20 25 30 with pT < 7 GeV, while the data for pT 7 GeV can
≥
p HGeVL befittedverywellusingthedeterminedLDMEsforboth
T J/ψ and ψ′. We perform a χ2 analysis for comparing
′ theoreticalfitwithexperimentaldatawithdifferentpcut.
FIG. 3: Transverse momentum distributions of prompt ψ T
Values of χ2/d.o.f. decrease rapidly as the cut increas-
production at the Tevatron and LHC. CDF data are taken
from Ref.[17]. Theyellowbandsindicatetheuncertaintydue ingfrom3 GeVto7 GeV,andχ2/d.o.f.becomesalmost
to CO LDMEs. unchanged when pcut is larger. This may be understood
T
as factorization and perturbation expansion may not be
reliable at low p . In Fig. 4 the curvature of observed
T
3S@1D crosssectionis positiveatlargepT but negativeatsmall
1
M p , with a turning point at p 6 GeV. But the the-
1 T T
102 M0 oretical curvature is positive. T≈his implies data below
LV feed-down 7 GeV may not be well explained in this work (even in
e TotalTevatron
(cid:144)bG 10 TotalLHC perturbative QCD) and needs further studying. Never-
Hn CDFData theless, as an alternative choice, we also give the fitted
+-LΜ 1 CMSData result for pcut =5 GeV, for which MJ/ψ is increased by
Μ T 1,r1
Ψ® a factor of 3, while the price paid is χ2/d.o.f. increases
H(cid:144)BrJ 10-1 from 0.33 to 3.5. The results for both pcTut =7 GeV and
´ pcut =5 GeV are shown in Table I.
(cid:144)ΣdpT 10-2 T(2) Feed-down contributions from ψ′ and χcJ to J/ψ
d prompt production are properly considered. Because
10-3
mψ′ and mχcJ are larger than mJ/ψ by only a few hun-
dred MeV, J/ψ is almost motionless in the higher char-
10-4
5 10 15 20 25 30 monium rest frame. So pT of J/ψ can be expressed
′ ′
as p p (m /m ), where p and m are the
pTHGeVL T ≈ T × J/ψ H T H
transversemomentumandmassofthedirectlyproduced
′
highercharmoniumH. LDMEs ofψ aretakenfromTa-
FIG. 4: The same as Fig. 3 but for J/ψ production. The
bleI,whilethatofχ arechosenwithrelativelysmaller
preliminary CMS data, taken from Ref.[18], are compared cJ
values from Ref.[10]. From experimental data in Figs. 3
with thetheoretical prediction.
and 4; and Ref.[10], we see that the prompt production
′
p distributionofJ/ψissteeperthanthatofψ andχ .
T cJ
model result of the wave functions at the origin[19]. In This implies that the subtraction of more feeddown con-
the fit we introduce a pcut and only use experimental tributions will lead to a steeper J/ψ direct production
T
data for the region p pcut. In Figs. 3and 4 and the distribution and hence a smaller R.
T ≥ T
following analysis, we prefer to use pcut =7 GeV. (3) Errors come from other sources. Varying renor-
T
J/ψ J/ψ malization and factorization scales from m /2 to 2m ,
We find the ratio R = M /M is determined to T T
be as small as 0.007. Based o1,nr1this0fi,rt0, we may conclude where mT = p4m2c +p2T typically changes both M1J,/rψ1
that the direct J/ψ production could be dominated by and MJ/ψ by 30% (Table I). However, the ratio R is
the 1S[8] channel in the chosen experimental p region. almost0i,nr0dependent of changing scales, because the de-
0 T
To achieve this conclusion, we emphasize the following pendencebetweentwoLDMEscancelseachother. Vary-
4
ing the charm quark mass mc can change the values of [2] E. Braaten and S. Fleming, Phys. Rev. Lett. 74, 3327
both LDMEs and R, and the dependence of R on m is (1995), [arXiv:hep-ph/9411365].
c
approximately R ∝ m2. Thus choosing m = 1.5 0.1 [3] G. T. Bodwin, E. Braaten and G. P. Lepage, Phys.
c c ± Rev. D 51, 1125 (1995), D 55, 5853 (E) (1997),
may cause an error of 20% for R.
[arXiv:hep-ph/9407339].
So,usingtheTevatrondataofJ/ψpromptproduction
[4] M. Kr¨amer, Prog. Part. Nucl. Phys. 47, 141 (2001),
for pT 7 GeV or even pT 5 GeV, we find very small [arXiv:hep-ph/0106120]; See also N. Brambilla et al.,
≥ ≥
values for R, or equivalently, MJ/ψ MJ/ψ (see Table arXiv:hep-ph/0412158.
I). If we make a simple assump1t,ri1on≪that0,trh0e smallness [5] A. A. Affolder et al. [CDF Collaboration], Phys. Rev.
of MJ/ψ is not due to accidental cancellation between Lett. 85, 2886 (2000), [arXiv:hep-ex/0004027]; A. Abu-
1,r1 lencia et al. [CDF Collaboration], Phys. Rev. Lett. 99,
J/ψ(3S[8]) and J/ψ(3P[8]) , we would have an order 132001 (2007), [arXiv:0704.0638].
hO 1 i hO 0 i
of magnitude estimate for the three LDMEs [6] P.Artoisenet,J.P.LansbergandF.Maltoni,Phys.Lett.
B 653, 60 (2007), [arXiv:hep-ph/0703129]; Z. G. He,
hOJ/ψ(3S1[8])i≈hOJ/ψ(3P0[8])i/m2c ≪hOJ/ψ(1S0[8])i. R. Li and J. X. Wang, Phys. Rev. D 79, 094003 (2009),
[arXiv:0904.2069].
This would lead to a nontrivial result that J/ψ direct
[7] G. C. Nayak, J. W. Qiu and G. Sterman, Phys.
production is dominated by the 1S0[8] channel, hence J/ψ Lett. B 613, 45 (2005), [arXiv:hep-ph/0501235]; Phys.
ismainlyunpolarized,whichagreeswiththepolarization Rev. D 72, 114012 (2005), [arXiv:hep-ph/0509021];
measurement[5]. H.HaberzettlandJ. P.Lansberg, Phys.Rev.Lett.100,
WehavealsocomparedourpredictionforpromptJ/ψ 032006 (2008), [arXiv:0709.3471].
[8] J. M. Campbell, F. Maltoni and F. Tramontano, Phys.
productionwiththeCMSdatainFig.4andagoodagree-
Rev. Lett.98, 252002 (2007), [arXiv:hep-ph/0703113].
ment is achieved.
′ [9] B. Gong and J. X. Wang, Phys. Rev. Lett. 100, 232001
As for ψ , since the difference between two LDMEs is (2008), [arXiv:0802.3727]; Phys. Rev. D 77, 054028
not as large as that of J/ψ, Mψ′ may be dominant at (2008), [arXiv:0805.2469].
nottoohighp ;hence, ψ′ may1b,re1transverselypolarized [10] Y. Q. Ma, K. Wang and K. T. Chao, arXiv:1002.3987.
T
in this region. However,it should be noted that Mψ′ is [11] P.Artoisenetetal.,Phys.Rev.Lett.101,152001(2008),
1,r1 [arXiv:0806.3282]; J. P. Lansberg, Eur. Phys. J. C 61,
always a combination o′f hOψ′(3S1[8])i and hOψ′(3P0[8])i at 693 (2009), [arXiv:0811.4005].
NLO; thus, whether ψ is transversely polarized at high [12] E. Braaten, M. A. Doncheski, S. Fleming and
p is unclear and needs further studying. M. L. Mangano, Phys.Lett. B 333, 548 (1994).
T
In summary, we calculate J/ψ(ψ′) prompt production [13] M. Klasen, B. A. Kniehl, L. N. Mihaila and
at the Tevatron and LHC at (α4v4), including all CS, M. Steinhauser, Phys. Rev. Lett. 89, 032001 (2002),
O s [arXiv:hep-ph/0112259].
CO, and feeddown contributions. A large K factor of
[14] P. Artoisenet, J. M. Campbell, F. Maltoni and
P-wave CO channels at high pT results in two linearly F. Tramontano, Phys. Rev. Lett. 102, 142001 (2009),
combinedLDMEsMJ/ψ(ψ′) andMJ/ψ(ψ′),whichcanbe [arXiv:0901.4352]; C. H. Chang, R. Li and J. X. Wang,
extracted at NLO fro0,mr0 the Tevatr1o,rn1data. Because of Phys. Rev. D 80, 034020 (2009), [arXiv:0901.4749];
the steep shape of experimental J/ψ prompt production M.ButenschoenandB.A.Kniehl,Phys.Rev.Lett.104,
072001 (2010), [arXiv:0909.2798].
data, we get a very small MJ/ψ, which might indicate
1,r1 [15] Y. Q. Ma, Y. J. Zhang and K. T. Chao, Phys. Rev.
the possibility that CO 1S[8] dominates J/ψ direct pro- Lett. 102, 162002 (2009), [arXiv:0812.5106]; B. Gong
duction. If this is the case0, J/ψ will be mainly unpolar- and J. X. Wang, Phys. Rev. Lett. 102, 162003 (2009),
[arXiv:0901.0117]; Y.J. Zhang, Y.Q. Ma, K. Wang,
ized, which may provide a possible solution to the long-
and K.T. Chao, Phys. Rev. D81, 034015 (2010),
standing J/ψ polarization puzzle.
[arXiv:0911.2166]; Y. J. Zhang and K. T. Chao, Phys.
We thank C. Meng and Y.J. Zhang for helpful dis- Rev. Lett.98, 092003 (2007), [arXiv:hep-ph/0611086].
cussions, and B. Gong and J.X. Wang for useful com- [16] B. Gong, X. Q. Li and J. X. Wang, Phys. Lett. B 673,
munications. This work was supported by the National 197 (2009), [arXiv:0805.4751].
Natural Science Foundation of China (No.10721063,No. [17] D. E. Acosta et al. [CDF Collaboration], Phys. Rev. D
11021092,No.11075002)andthe MinistryofScience and 71, 032001 (2005), [arXiv:hep-ex/0412071]; T. Aaltonen
et al. [CDF Collaboration], Phys. Rev. D 80, 031103
Technology of China (No.2009CB825200).
Note added. Soon after this work was submitted for (2009), [arXiv:0905.1982].
[18] N. Leonardo, PoS ICHEP2010, 207 (2010).
publication, a similar study appeared[20], and for all
[19] SeetheB-T modelin E. J. Eichten andC. Quigg, Phys.
color-singlet and octet channels in J/ψ direct hadropro- Rev. D 52, 1726 (1995), [arXiv:hep-ph/9503356].
duction their short-distance coefficients are consistent [20] M.ButenschoenandB.A.Kniehl,Phys.Rev.Lett.106,
with ours. 022003 (2011), [arXiv:1009.5662].
[1] F. Abe et al. [CDF Collaboration], Phys. Rev. Lett. 69,
3704 (1992).
