Table Of ContentReconciling J/ψ production at HERA, RHIC, Tevatron, and LHC with NRQCD
factorization at next-to-leading order
Mathias Butenscho¨n and Bernd A. Kniehl
II. Institut fu¨r Theoretische Physik, Universita¨t Hamburg,
Luruper Chaussee 149, 22761 Hamburg, Germany
(Dated: January 12, 2011)
We calculate the cross section of inclusive direct J/ψ hadroproduction at next-to-leading order
(NLO) within the factorization formalism of nonrelativistic quantum chromodynamics (NRQCD),
including the full relativistic corrections due to the intermediate 1S[8], 3S[8], and 3P[8] color-octet
0 1 J
states. We perform a combined fit of the color-octet (CO) long-distance matrix elements to the
transverse-momentum (p ) distributions measured by CDF at the Fermilab Tevatron and H1
T
1 at DESY HERA and demonstrate that they also successfully describe the pT distributions from
1 PHENIXatBNLRHICandCMSattheCERNLHCaswellasthephoton-protonc.m.energyand
0 (with worse agreement) theinelasticity distributions from H1. This providesa first rigorous test of
2 NRQCDfactorization atNLO.Inallexperiments,theCOprocessesareshowntobeindispensable.
n
PACSnumbers: 12.38.Bx,13.60.Le,13.85.Ni,14.40.Gx
a
J
1 The factorization formalism of nonrelativistic QCD annihilation [8], and direct photoproduction [6]. As for
1
(NRQCD) [1] provides a rigorous theoretical framework hadroproduction at NLO, the CO contributions due to
for the description of heavy-quarkoniumproduction and intermediate 1S[8] and 3S[8] states [4] were calculated as
] 0 1
h decay. This implies a separation of process-dependent well as the complete NLO corrections to χ production,
J
p short-distance coefficients, to be calculated perturba- including the 3S[8] contribution [9].
- tively as expansions in the strong-coupling constant α , 1
p s In order to convincingly establish the CO mechanism
e fromsupposedlyuniversallong-distancematrixelements and the LDME universality, it is an urgent task to com-
h (LDMEs),tobeextractedfromexperiment. Therelative
plete the NLO description of J/ψ hadroproduction by
[ importance of the latter can be estimated by means of
including the full COcontributions atNLO,whichis ac-
velocity scaling rules; i.e., the LDMEs are predicted to
3 tually achieved in this Letter. In fact, because of their
v scale with a definite power of the heavy-quark (Q) ve- highprecisionandtheirwidecoverageandfinebinningin
2 locity v in the limit v ≪ 1. In this way, the theoretical
p , the Tevatron data on inclusive J/ψ production have
T
6 predictionsareorganizedasdoubleexpansionsinα and
s so far been the major source of information on the CO
6 v. Acrucialfeatureofthisformalismisthatittakesinto
LDMEs[10],andtheLHCdatatocomewillbeevenmore
5
account the complete structure of the QQ Fock space,
. constraining. Previous NLO analyses of J/ψ hadropro-
09 nwihteichspiisnsSp,aonrnbeidtablyantghuelasrtamteosmnen=tu2mS+L1,Lt[Jao]tawliathngduelafir- duction [3, 4] were lacking the 3PJ[8] contributions, for
0 which there is no reason to be insignificant. This tech-
momentum J, and color multiplicity a =1,8. In partic-
1 nical bottleneck, which has prevented essential progress
ular, this formalism predicts the existence of color-octet
: in the global test of NRQCD factorization for the past
v (CO) processes in nature. This means that QQ pairs
fifteen years, is overcome here by further improving and
i
X are produced at short distances in CO states and subse- refiningthecalculationaltechniquesdevelopedinRef.[6].
quently evolve into physical, color-singlet (CS) quarko-
r InvokingthefactorizationtheoremsoftheQCDparton
a nia by the nonperturbative emission of soft gluons. In
model and NRQCD [1], the inclusive J/ψ hadroproduc-
the limit v → 0, the traditional CS model (CSM) is re-
tion cross section is evaluated from
covered in the case of S-wave quarkonia. In the case of
J/ψ production,theCSMpredictionisbasedjustonthe
dσ(AB →J/ψ+X) = dxdyf (x)f (y) (1)
3S[1] CS state, while the leading relativistic corrections, Z i/A j/B
1 iX,j,n
of relative order O(v4), are built up by the 1S[8], 3S[8],
0 1 ×hOJ/ψ[n]idσ(ij →cc[n]+X),
and 3P[8] (J =0,1,2) CO states.
J
The greatest success of NRQCD was that it was able where fi/A(x) are the parton distribution functions of
to explain the J/ψ hadroproduction yield at the Fermi- hadron A, hOJ/ψ[n]i are the LDMEs, and dσ(ij →
lab Tevatron [2], while the CSM prediction lies orders cc[n]+X) are the partonic cross sections. Working in
of magnitude below the data, even if the latter is eval- the fixed-flavor-number scheme, i and j run over the
uated at NLO [3, 4]. Also in the case of J/ψ photo- gluon g and the light quarks q = u,d,s and anti-quarks
production at DESY HERA, the CSM cross section at q. ThecounterpartofEq.(1)fordirectphotoproduction
NLOsignificantlyfalls shortofthe data[5,6]. Complete emergesbyreplacingf (x)bythephotonfluxfunction
i/A
NLO calculations in NRQCD were performed for inclu- f (x) and fixing i=γ.
γ/e
sive J/ψ production in two-photon collisions [7], e+e− Wecheckedanalyticallythatallappearingsingularities
2
hOJ/ψ(1S[8])i (4.50±0.72)×10−2 GeV3 from our fit. We checked that our fit results depend
0
hOJ/ψ(3S[8])i (3.12±0.93)×10−3 GeV3 only feebly on the precise location of this cut-off. We
1
hOJ/ψ(3P[8])i (−1.21±0.35)×10−2 GeV5 also verified that exclusion of the H1 data points with
0
p < 2.5 GeV, which might require power corrections
T
TABLE I: NLOfit results for the J/ψ CO LDMEs. neglected here, is inconsequential for our fit. The fit re-
sults for the CO LDMEs corresponding to our default
NLO NRQCD predictions are collected in Table I. In
cancel. As for the ultraviolet singularities, we renormal- Figs. 1(a) and (d), the latter (solid lines) are compared
ize the charm-quark mass and the wave functions of the withtheCDF[17]andH1[14,15]data,respectively. For
external particles according to the on-shell scheme, and comparison, also the default predictions at LO (dashed
the strong-coupling constant according to the modified lines) as well as those of the CSM at NLO (dot-dashed
minimal-subtraction scheme. The infrared (IR) singu- lines)andLO(dottedlines)areshown. Inordertovisual-
larities are canceled similarly as described in Ref. [6]. In izethesizeoftheNLOcorrectionstothehard-scattering
particular,the3P[8] short-distancecrosssectionsproduce crosssections,the LO predictionsareevaluatedwith the
J
two new classes of soft singularities, named soft #2 and same LDMEs. The yellow and blue (shaded) bands in-
soft #3 terms, on top of the soft #1 terms familiar from dicate the theoretical errors on the NLO NRQCD and
CSM predictions, respectively, due to the lack of knowl-
the S-wavechannels. The soft#2 termsdo notfactorize
edge of corrections beyond NLO, which are estimated
toLOcrosssections;theycancelagainsttheIRsingulari-
by varying µ , µ , and µ by a factor 2 up and down
tiesofthevirtualcorrectionsleftoverupontheusualcan- r f Λ
relative to their default values. The µ , µ , and µ de-
cellation against the soft #1 terms. The soft #3 terms r f Λ
pendencies of α , the parton distribution functions, and
cancelagainsttheIRsingularitiesfromtheradiativecor- s
rections to the hOJ/ψ(3S[1])i and hOJ/ψ(3S[8])i LDMEs. the LDMEs, respectively, induced by the renormaliza-
1 1 tion group are canceled only partially, namely through
theorderofthecalculation,bylinearlylogarithmicterms
We now describe our theoretical input and the kine-
appearingintheNLOcorrections. Data-over-theoryrep-
matic conditions for our numerical analysis. We set
resentations of Figs. 1(a) and (d) are given in Figs. 1(b)
m = 1.5 GeV, adopt the values of m , α, and the
c e and (e), respectively. In Figs. 1(c) and (f), the default
branching ratios B(J/ψ → e+e−) and B(J/ψ → µ+µ−)
NLO NRQCD predictions of Figs. 1(a) and (d), respec-
from Ref. [11], and use the one-loop (two-loop) for-
tively, are decomposed into their 3S[1], 1S[8], 3S[8], and
mula for α(nf)(µ), with n = 4 active quark flavors, 1 0 1
s f 3P[8] components. We observe from Fig. 1(c) that the
at LO (NLO). As for the proton parton distribution J
functions, we use set CTEQ6L1 (CTEQ6M) [12] at LO 3P[8] short-distancecrosssectionofhadroproduction(ex-
J
(NLO), which comes with an asymptotic scale parame- cludingthenegativeLDME)receivessizableNLOcorrec-
ter of Λ(4) = 215 MeV (326 MeV). We evaluate the tions that even turn it negative at pT & 7 GeV. This is,
QCD
however, not problematic because a particular CO con-
photon flux function using Eq. (5) of Ref. [13] with
tributionrepresentsanunphysicalquantitydependingon
the cut-off Q2 = 2.5 GeV2 [15] on the photon vir-
max the choices of renormalization scheme and scale µ and
tuality. As for the CS LDME, we adopt the value Λ
is entitled to become negative as long as the full cross
hOJ/ψ(3S[1])i = 1.32 GeV3 from Ref. [16]. Our de-
1 sectionremainspositive. Such features arefamiliar, e.g.,
fault choices for the renormalization, factorization, and
from inclusive heavy-hadronproduction at NLO [20]. In
NRQCD scales are µ =µ =m and µ =m , respec-
r f T Λ c contrast to the situation at LO, the line shapes of the
tively, where mT = p2T +4m2c is the J/ψ transverse 1S[8] and 3P[8] contributions significantly differ at NLO,
mass. p 0 J
sothathOJ/ψ(1S[8])iandhOJ/ψ(3P[8])imaynowbefitted
Our strategy for testing NRQCD factorization in J/ψ 0 0
independently (see Table I). Besides that, the injection
production at NLO is as follows. We first perform a
ofHERAdata into the fit alsosupports the independent
common fit of the CO LDMEs to the p distributions
measured by CDF in hadroproduction atTTevatron Run determinationofhOJ/ψ(1S0[8])iandhOJ/ψ(3P0[8])i. Notice
II[17]andbyH1inphotoproductionatHERA1[14]and thathOJ/ψ(3P[8])icomesoutnegative,whichisnotprob-
0
HERA2 [15] (see Table I and Fig. 1). We then compare lematic by the same token as above. In compliance with
the p distributions measuredbyPHENIXatRHIC [18] the velocityscalingrulesofNRQCD[1], the COLDMEs
T
and CMS at the LHC [19] as well as the W and z dis- in Table I are approximately of order O(v4) relative to
tributions measured by H1 at HERA1 [14] and HERA2 hOJ/ψ(3S[1])i. We read off from Fig. 1(a) and (d) that
1
[15] with our respective NLO predictions based on these theNLOcorrection(K)factorsforhadro-andphotopro-
CO LDMEs (see Fig. 2). duction range from 1.30 to 2.28 and from 0.54 to 1.80,
ThepT distributionofJ/ψ hadroproductionmeasured respectively, in the pT intervals considered.
experimentally flattens at p < 3 GeV due to nonper-
T
turbative effects, a feature that cannot be faithfully de- WeobservefromFig.2thatourNLONRQCDpredic-
scribed by fixed-order perturbation theory. We, there- tions nicely describe the p distributions from PHENIX
T
fore, exclude the CDF data points with p < 3 GeV [18] (a) and CMS [19] (b) as well as the W distribu-
T
3
×y→mm[]X) B(J/) nb/GeV1110100-21 CCCCCDSSSS,,++F LNCC dOLOOaOt,,a LNOLO CDF Data / NRQCD0126...01230555 4 6 8 1p0T [G1e2V] 14 16 18 20 ×y→mm[]X) B(J/) nb/GeV1110100-21 3133-TC3SSSPoPD[[[[Jt1011888[J Fa 8 ]]]] ,,,, ]l , ,d NNNN NNaLLLLtLLOOOOaOO
+ -2 M + -2
–→ys(ppJ/d/dpT111000--43 √|ys–| <= 01..696 TeV CDF Data / CS 1234500000 –s→yd/dp(ppJ/T111000--43 √|ys–| <= 01..696 TeV
0
4 6 8 10 12 14 16 18 20 4 6 8 10 12 14 16 18 20 4 6 8 10 12 14 16 18 20
(a) p [GeV] (b) p [GeV] (c) p [GeV]
T T T
D 3
60 GeV < W < 240 GeV C 60 GeV < W < 240 GeV
2]GeV101-1 0.3 < z < 0.9 ata / NRQ12..1255 2]GeV101-1 0.3 < z < 0.9
nb/ -2 1 D0.5 nb/ -2
2[dp T10 H 0 1 10 1022[dp T10
+X)/10-3 CCSS,, LNOLO pT2 [GeV2] +X)/10-3
→yJ/10-4 CS+CO, LO SM 56 →yJ/10-4
sd(ep10-5 CHS1 +dCatOa,: NHLEORA1 Data / C 234 sd(ep10-5 H1 data: HERA1
H1 data: HERA2 H1 1 H1 data: HERA2
-6 -6
10 0 10
2 2 2
1 10 10 1 10 10 1 10 10
(d) pT2 [GeV2] (e) pT2 [GeV2] (f) pT2 [GeV2]
FIG. 1: NLO NRQCD predictions of J/ψ hadro- and photoproduction resulting from the fit compared to the CDF [17] and
H1 [14, 15] input data. The coding of the lines in part (f) of the figure is the same as in part (c). The seeming singularity of
the3P[8] contribution in part (c) is an artifact of thelogarithmic scale on thevertical axis..
J
tions from H1 [14, 15] (c), with most of the data points factorization formalism 15 years ago [1].
fallinginsidethe yellow(shaded)errorband. Inallthese
Weshouldremarkthatourtheoreticalpredictionsrefer
cases,inclusion of the NLO corrections tends to improve
to direct J/ψ production, while the CDF and CMS data
the agreement. The NLO NRQCD prediction of the z
include prompt events and the H1 and PHENIX data
distribution (d) agrees with the H1 data in the inter-
even non-prompt ones, but the resulting error is small
mediate z range, but its slope appears to be somewhat
against our theoretical uncertainties and has no effect
too steep at first sight. However, the contribution due
on our conclusions. In fact, the fraction of J/ψ events
to resolved photoproduction, which is not yet included
originatingfromthefeed-downofheaviercharmoniaonly
here, is expected to fill the gap in the low-z range, pre-
amountstoabout30%[22]forhadroproductionand15%
cisely where it is peaked; the overshoot of the NRQCD
[15]forphotopoprduction,andthefractionofJ/ψevents
prediction in the upper endpoint region, which actually
from B decays is negligible at HERA [15] and RHIC en-
turns into a breakdown at z = 1, is an artifact of the
ergies.
fixed-order treatment and may be eliminated by invok-
ing soft collinear effective theory [21]. We conclude from We thank G. Kramer for useful discussions and B. Ja-
Figs. 1 and 2 that all experimental data sets considered cak and C. Luiz da Silva for help with the comparison
here significantly overshoot the NLO CSM predictions, to the PHENIX data [18]. This work was supported in
by many experimental standard deviations. Specifically, part by BMBF Grant No. 05H09GUE, DFG Grant No.
the excess amounts to 1–2 orders of magnitude in the KN 365/6–1,and HGF Grant No. HA 101.
case of hadroproduction [see Fig. 1(b)] and typically a
Note added. At the final stage of preparing this
factor of 3 in the case of photoproduction[see Fig. 1(e)].
manuscript, after our results were presented at an in-
Ontheotherhand,thesedatanicelyagreewiththeNLO
ternationalconference[23],apreprint[24]appearedthat
NRQCD predictions, apart from well-understood devia-
also reports on a NLO calculation of J/ψ hadroproduc-
tionsinthecaseofthez distributionofphotoproduction
tioninfullNRQCD.Adoptingtheirinputs,wefindagree-
[see Fig. 2(d)]. This constitutes the most rigorous evi- mentwiththeirresults forthe 3S[1], 1S[8], 3S[8], and3P[8]
dencefortheexistenceofCOprocessesinnatureandthe 1 0 1 J
contributions.
LDMEuniversalitysincetheintroductionoftheNRQCD
4
]V 102 PHENIX data ]V 103 CMS data
e e
G G
nb/ 10 CS, LO nb/ 102 CS, LO
[ee) 1 CCSS,+ NCLOO, LO [) 10 CCSS,+ NCLOO, LO
→ CS+CO, NLO →mm CS+CO, NLO
yJ/ -1 yJ/
B(10 B( 1
× ×
X) -2 X) -1
+10 +10
yJ/ yJ/
→pp10-3 →pp10-2
/dp(T10-4 √|ys–| <= 02.0305 GeV /dp(T10-3 √1s.–6 = < 7 | yT| e<V 2.4
sd sd
2 3 4 5 6 7 8 9 10 4 6 8 10 12 14 16 18 20
(a) p [GeV] (b) p [GeV]
T T
0.3 < z < 0.9 H1 data: HERA1
pT2 > 1 GeV2 102 H1 data: HERA2
[]dW nb/GeV10-2 []X)/dz nb 10 CCCCSSSS,,++ LNCCOLOOO,, LNOLO
X)/ CS, LO +
→yJ/+10-3 CCSS,+ NCLOO, LO →yepJ/
(ep CS+CO, NLO sd( 1
sd
H1 data: HERA1 pT2 > 1 GeV2
H1 data: HERA2 60 GeV < W < 240 GeV
-4
10
60 80 100 120 140 160 180 200 220 240 0.3 0.4 0.5 0.6 0.7 0.8 0.9
(c) W [GeV] (d) z
FIG. 2: NLONRQCD predictions of J/ψ hadro- and photoproduction resulting from thefit compared to PHENIX[18], CMS
[19], and H1[14, 15] data not included in the fit.
[1] G.T.Bodwin,E.Braaten,andG.P.Lepage,Phys.Rev. D 62, 094005 (2000); B. A. Kniehl and C. P. Palisoc,
D 51, 1125 (1995); 55, 5853(E) (1997). Eur. Phys. J. C 48, 451 (2006).
[2] P. L. Cho and A. K. Leibovich, Phys. Rev. D 53, 150 [11] ParticleDataGroup,K.Nakamuraet al.,J.Phys.G37,
(1996); 53, 6203 (1996). 075021 (2010).
[3] J.Campbell,F.Maltoni,andF.Tramontano,Phys.Rev. [12] CTEQCollaboration,J.Pumplinetal.,JHEP0207,012
Lett.98,252002 (2007);B.GongandJ.-X.Wang,Phys. (2002).
Rev. Lett. 100, 232001 (2008). J. P. Lansberg, Phys. [13] B. A. Kniehl, G. Kramer, and M. Spira, Z. Phys. C 76,
Lett.B 695, 149 (2011). 689 (1997).
[4] B. Gong, X. Q. Li, and J.-X. Wang, Phys. Lett. B 673, [14] H1 Collaboration, C. Adloff et al., Eur. Phys. J. C 25,
197 (2009). 25 (2002).
[5] M. Kr¨amer, J. Zunft, J. Steegborn, and P. M. Zerwas, [15] H1Collaboration,F.D.Aaronetal.,Eur.Phys.J.C68,
Phys. Lett. B 348, 657 (1995); M. Kr¨amer, Nucl. Phys. 401 (2010).
B459,3(1996); P.Artoisenet,J.Campbell,F.Maltoni, [16] G.T.Bodwin,H.S.Chung,D.Kang,J.Lee,andC.Yu,
andF.Tramontano,Phys.Rev.Lett.102,142001(2009); Phys. Rev.D 77, 094017 (2008).
C. H. Chang, R. Li, and J. X. Wang, Phys. Rev. D 80, [17] CDF Collaboration, D. Acosta et al., Phys. Rev. D 71,
034020 (2009). 032001 (2005).
[6] M. Butensch¨on and B. A.Kniehl, Phys. Rev.Lett. 104, [18] PHENIX Collaboration, A. Adare et al., Phys. Rev. D
072001 (2010). 82, 012001 (2010).
[7] M. Klasen, B. A. Kniehl, L. N. Mihaila, and M. Stein- [19] D. Abbaneo et al. (CMS Collaboration),
hauser, Phys. Rev. Lett. 89, 032001 (2002); Nucl. Phys. arXiv:1011.4193.
B 713, 487 (2005); Phys. Rev.D 71, 014016 (2005). [20] B. A. Kniehl, G. Kramer, I. Schienbein, and H. Spies-
[8] Y.J.Zhang,Y.Q.Ma,K.Wang,andK.T.Chao,Phys. berger, Phys. Rev. D 71, 014018 (2005); Eur. Phys. J.
Rev.D 81, 034015 (2010). C 41, 199 (2005); Phys. Rev. Lett. 96, 012001 (2006);
[9] Y.Q. Ma, K.Wang, and K. T. Chao, arXiv:1002.3987. Phys. Rev.D 77, 014011 (2008).
[10] B. A. Kniehl and G. Kramer, Eur. Phys. J. C 6, 493 [21] S. Fleming, A. K. Leibovich, and T. Mehen, Phys. Rev.
(1999);E.Braaten,B.A.Kniehl,andJ.Lee,Phys.Rev. D 74, 114004 (2006).
5
[22] CDF Collaboration, F. Abe et al., Phys. Rev. Lett. 79, [24] Y. Q. Ma, K. Wang, and K. T. Chao, arXiv:1009.3655.
578 (1997).
[23] M. Butensch¨on and B. A.Kniehl, arXiv:1011.5619.
