Complete Nonrelativistic-QCD Prediction for Prompt Double J/ψ Hadroproduction
aa r X i v : . [ h e p - ph ] S e p DESY 15-104 ISSN 0418-9833
Complete Nonrelativistic-QCD Prediction for Prompt Double
J/ψ
Hadroproduction
Zhi-Guo He and Bernd A. Kniehl
II. Institut f¨ur Theoretische Physik, Universit¨at Hamburg,Luruper Chaussee 149, 22761 Hamburg, Germany (Dated: March 6, 2018)We perform a complete study of prompt double
J/ψ hadroproduction at leading order in thenonrelativistic-QCD factorization framework by including all possible pairings of the c ¯ c Fock states S [8]0 , S [1 , , and P [1 , J with J = 0 , ,
2. We find that the S [8]0 and P [8] J channels of J/ψ and ψ ′ production and the P [1] J and S [8]1 channels of χ cJ production, which have been overlooked so far,greatly dominate at large invariant masses and rapidity separations of the J/ψ pair, and that theirinclusion nearly fills the large gap between previous incomplete predictions within the color-singletmodel and the recent measurement by the CMS Collaboration at the CERN LHC, leaving room fornext-to-leading-order corrections of typical size.
PACS numbers: 12.38.Bx, 12.39.St, 13.85.Ni, 14.40.Pq
The nonrelativistic QCD (NRQCD) [1] factorizationformalism, introduced two decades ago in a seminal workby Bodwin, Braaten, and Lepage [2], nowadays is theonly game in town for the theoretical description ofheavy-quarkonium production and decay, and its exper-imental verification is generally considered to be amongthe most urgent tasks of heavy-quarkonium physics [3].The production cross sections and decay rates are sep-arated into process-dependent short-distance coefficients(SDCs), calculated by expansion in the strong-couplingconstant α s , and universal long-distance matrix elements(LDMEs), which are strongly ordered in size by velocity( v ) scaling rules [4]. The heavy-quark pair may appearin any Fock state n = S +1 L [ a ] J , both as color singlet(CS) a = 1 and color octet (CO) a = 8, thus givingrise to the CO mechanism (COM), while, in the tradi-tional CS model, it is restricted to the CS state shar-ing the spectroscopic quantum numbers S +1 L J with thephysical quarkonium state considered. Despite its aes-thetic simplicity and theoretical rigor, consolidated veryrecently by an all-order proof [5], NRQCD factorizationhas reached the crossroads because the predicted univer-sality of the LDMEs is challenged [6] by recent measure-ments of J/ψ polarization [7] and η c yield [8], which is inthe very focus at the CERN LHC.Our Letter addresses another burning problem ofNRQCD, namely, its seeming failure to describe recentmeasurements of prompt double J/ψ hadroproductionperformed by the LHCb [9] and CMS [10] Collabora-tions at the LHC, and the D0 Collaboration [11] at theFermilab Tevatron. This is a particularly sensitive test-ing ground for NRQCD factorization, which takes effectthere twice, and a topic of old vintage, pioneered byRef. [12] in 1995, which has attracted considerable the-oretical interest since then (see, e.g., Refs. [13–19]), butis much less advanced than single
J/ψ production. Sofar, only the CS contribution due to gg → c ¯ c ( S [1]1 ) andthe CO contribution due to gg → c ¯ c ( S [8]1 ), which re-sembles double fragmentation [see Fig. 1(d)], have beenstudied for direct J/ψ production and also for the feed down from ψ ′ mesons, which requires no extra calculation[12–19]. These calculations of prompt double J/ψ pro-duction, which we henceforth denote as CS ∗ and CO ∗ ,respectively, are incomplete because they lack the S [8]0 and P [8] J contributions to J/ψ and ψ ′ production andthe P [1] J and S [8]1 contributions to χ cJ production, where J = 0 , ,
2. Interestingly,
J/ψ + χ cJ production is for-bidden at O ( α s ) in the CS model by CP conservation,while it is enabled by the COM of NRQCD. Thus, we areled to include a total of (cid:0) (cid:1) − c ¯ c Fock states altogether, as indicated in Table I, out ofwhich only 2 have been considered so far. In our Letter,we demonstrate that NRQCD factorization may be rec-onciled with the experimental data [9–11], leaving roomfor typical next-to-leading-order (NLO) corrections, ifthe previously neglected CO and feed-down channels areproperly included. We thus add another crucial pieceof information to the tantalizing tale of NRQCD factor-ization [2] and point into a new direction, namely therelative O ( α s ) corrections to the next-to-leading-power(NLP) and next-to-next-to-leading-power (NNLP) COprocesses of prompt double J/ψ hadroproduction to beidentified below. If their inclusion turned out to bringthe NRQCD prediction in agreement with the LHC data,which we deem very likely for reasons explained below,this would be an important milestone in the verificationof the COM, which is a key prediction of NRQCD fac-torization. Owing to the predicted LDME universality,double
J/ψ production will then also yield independentconstraints on yield and polarization of single
J/ψ pro-duction.Our Letter also suggests a solution to another impor-tant QCD problem of general interest [20], namely thedouble-parton-scattering (DPS) surplus observed by theD0 Collaboration [11]. In fact, their result for σ eff =( σ J/ψ ) /σ DPS is considerably smaller than the findingsby other experiments [11]. The increase of the single-parton-scattering (SPS) portion σ SPS due to our comple-tion of the NRQCD prediction results in a reduction of σ DPS , which in turn increases σ eff and so places it in the [ a ] [ b ] [ c ] [ d ] FIG. 1: Typical Feynman diagrams for gg → c ¯ c ( m ) c ¯ c ( n ): (a)nonfragmentation type I, (b) nonfragmentation type II, (c)single-fragmentation-like, (d) double-fragmentation-like. ball park of other determinations.So far, the experimental data [9–11], which come astotal cross sections σ tot and distributions in the invari-ant mass M , the transverse momentum P T , and the ra-pidity ( y ) separation | ∆ y | of the J/ψ pair, have mostlybeen compared with CS ∗ predictions, which dominate forsmall values of the J/ψ transverse momentum p T [13–15],while the CO ∗ contributions take over in the large- p T re-gion, for p T &
16 GeV at the LHC [16]. In the LHCb [9]case, the CS ∗ prediction for σ tot , which receives a mod-erate enhancement of relative order O ( α s ) of about 10%[19], is compatible with the measurement, but the onefor the distribution dσ/dM significantly overshoots thedata points close to the J/ψ pair production threshold,even after including the negative corrections of relativeorder O ( v ), which are about −
23% [18]. In the CMS[10] case, the CS ∗ prediction for σ tot , which is enhancedby more than 1 order of magnitude by relative O ( α s )corrections [19], can only account for about 2 / dσ/dP T sig-nificantly differs from the measurement as for the lineshape, and the one for dσ/dM dramatically undershootsthe measurement, by 4 orders of magnitude in the large- M region, for M >
35 GeV. This enormous discrepancyseriously jeopardizes the validity of NRQCD factoriza-tion [2], and it is an important task of general interest toperform a systematic study of all the contributing chan-nels, which is the very purpose of our Letter. In theD0 [11] case, there is also a large gap between the CS ∗ prediction and the experimental result for the SPS crosssection [11].Owing to the factorization theorems of the QCDparton model and NRQCD, the prompt double J/ψ hadroproduction cross section may be evaluated as dσ ( AB → J/ψ + X ) = X i,j,m,n,H ,H Z dx dx × f i/A ( x ) f j/B ( x ) d ˆ σ ( ij → c ¯ c ( m ) c ¯ c ( n ) + X ) × hO H ( m ) i Br( H → J/ψ + X ) × hO H ( n ) i Br( H → J/ψ + X ) , (1)where f i/A ( x ) is the parton distribution function (PDF)of parton i in hadron A , d ˆ σ [ ij → c ¯ c ( m ) c ¯ c ( n ) + X ] is theSDC, hO H ( m ) i is the LDME of H = J/ψ, χ cJ , ψ ′ , andBr( H → J/ψ + X ) is the branching fraction with theunderstanding that Br( H → J/ψ + X ) = 1 if H = J/ψ .Since the q ¯ q -initiated subprocesses are greatly suppressed TABLE I: Scaling with p T and v of dσ/dp T for gg → c ¯ c ( m ) c ¯ c ( n ) times the respective LDMEs and branching frac-tions for the relevant pairings ( m, n ) of c ¯ c Fock states. Notethat P [1] J are counted separately for J = 0 , , m, n ) S [1]1 3 S [8]1 1 S [8]0 3 P [8] J P [1] J S [1]1 /p T v /p T v /p T v /p T S [8]1 · · · v /p T v /p T v /p T v /p T S [8]0 · · · · · · v /p T v /p T v /p T P [8] J · · · · · · · · · v /p T v /p T P [1] J · · · · · · · · · · · · v /p T by the light-quark PDFs [15], we concentrate on gg fu-sion. Because of the smallness of Br( χ c → J/ψγ ) =1 .
27% [21], we neglect the contributions from H = χ c .Our analytic results for the CS ∗ and CO ∗ channels agreewith the literature [13, 16].There is a total of 72 Feynman diagrams contribut-ing to the generic partonic subprocess gg → c ¯ c ( m ) c ¯ c ( n ),and representative ones are depicted in Fig. 1. For given m and n , not all of them contribute due to J P C con-servation. According to the scaling dσ/dp T ∝ /p NT andthe topologies of the contributing Feynman diagrams [seeFigs. 1(a)–(d)], we divide the partonic subprocesses into 4categories: (i) NNLP-I, with N = 8, including m = S [1]1 and n = S [1 , , S [8]0 , P [8] J ; (ii) NNLP-II, with N = 8,too, including m, n = S [8]0 , P [8] J , P [1] J ; (iii) NLP, with N = 6, including m = S [8]1 and n = S [8]0 , P [8] J , P [1] J ;and (iv) leading power (LP), with N = 4, including m = n = S [8]1 . While the NNLP-I and NNPL-II sub-processes exhibit the same p T scaling, they differ bythe topologies of the respective Feynman diagrams. Inthe latter case, these are the diffractionlike ones as inFig. 1(b), which allow for large values of | ∆ y | and thusfor an enhancement of the cross section at large valuesof M . Also taking into account the scaling with v of theLDMEs and noticing that Br( χ c , → J/ψγ ) = O ( v )numerically, we roughly estimate the relative importanceof each channel at large values of p T as summarized inTable I.We work at leading order (LO) in the fixed-flavor-number scheme with 3 massless quark flavors and acharm-quark mass of m c = 1 . α (4) s ( µ r ) with asymptotic scale parameterΛ (4) = 192 MeV [22] and the CTEQ5L set of LO pro-ton PDFs [22]. We choose the renormalization and fac-torization scales as µ r = µ f = ξ p (4 m c ) + p T and vary ξ between 1 / J/ψ , χ cJ , and ψ ′ mesons, we adopt the CS val-ues from Ref. [23], evaluated using the Buchm¨uller-Tyepotential, and the CO values from Ref. [24], fitted tosingle J/ψ hadroproduction data at LO in NRQCD. Be-cause of the strong correlations between hO H ( S [8]0 ) i and hO H ( P [8]0 ) i for H = J/ψ, ψ ′ , only the linear combina- LHCbNRQCD LO6 8 10 12 140.010.050.100.501.005.0010.00 M H GeV L d Σ d M H nb (cid:144) G e V L FIG. 2: The M distribution of prompt double J/ψ hadropro-duction measured by LHCb [9] is compared to the full LONRQCD prediction (solid lines). The theoretical uncertaintyis indicated by the shaded (yellow) bands. tions M Hr = hO H ( S [8]0 ) i + r hO H ( P [8]0 ) i /m c could bedetermined in Ref. [24]. Fortunately, these correlationsare very similar in prompt double J/ψ hadroproduc-tion via the NNLP-II and NLP subprocesses. We useBr( χ c → J/ψγ ) = 33 . χ c → J/ψγ ) = 19 . ψ ′ → J/ψ + X ) = 60 .
9% [21].Prior to performing detailed comparisons with mea-surements, we expose some general features of our re-sults. (a) Among the NNLP-I subprocesses, no kinematicenhancements are found relative to the CS ∗ channel, sothat all the other channels are suppressed as O ( v ) bythe LDMEs. (b) Although the p T scaling of the NNLP-II subprocesses is as unfavorable as that of the NNLP-Iones, their SDCs may be about 50–200 times larger thanthat of the CS ∗ channel. (c) The contribution of the NLPsubprocesses may also exceed that of the CS ∗ channel,e.g., for p T >
20 GeV under CMS kinematic conditions.(d) At large values of M , the M scalings and the corre-sponding p T scalings of the 4 types of subprocesses arethe same, but the differential cross sections dσ/dM ofthe NNLP-II, NLP, and LP subprocesses may be morethan 1 order of magnitude larger than that of the CS ∗ channel. Observations (b)–(d) indicate that the combi-nation of the CS ∗ and CO ∗ contributions, dσ ∗ , may notbe a good approximation to the full NRQCD result, dσ ,especially at large values of M . (e) As expected fromidentical-boson symmetry and the J/ψ + χ cJ suppres-sion mentioned above, the relative importance of the χ cJ ( ψ ′ ) feed-down contribution is reduced (increased) withrespect to prompt single J/ψ hadroproduction.The LHCb Collaboration [9] measured σ tot at center-of-mass (CM) energy √ s = 7 TeV requiring p T <
10 GeVand 2 . < y < . J/ψ mesons to find σ LHCbtot = (5 . ± . ± .
1) nb. Our corresponding LONRQCD predictions are σ ∗ tot = 12 . +4 . − . nb, which issomewhat larger than in Refs. [18, 19] because of dif-ferent choices of m c , LDMEs, PDFs, and scales, and σ tot = 13 . +5 . − . nb, which is about 2.6 times largerthan the LHCb result. To better understand the ori- CMSNRQCD LOCS * NNLP - IINLPLP10 20 30 40 50 60 70 8010 - - - H GeV L d Σ d M H nb (cid:144) G e V L FIG. 3: The M distribution of prompt double J/ψ hadropro-duction measured by CMS [10] is compared to the full LONRQCD prediction (solid lines), its NNLP-II (dotted lines),NLP (dot-dashed lines), and LP (long-dashed lines) compo-nents, and the LO CS ∗ contribution (dashed lines). The theo-retical uncertainty in the LO NRQCD prediction is indicatedby the shaded (yellow) bands. gin of this excess, we consider in Fig. 2 the LHCb andfull LO NRQCD results differential in M . We observethat the theoretical prediction systematically overshootsthe experimental data in the threshold region, where M . M . Near the J/ψ pair production threshold,multiple soft-gluon emissions spoil the perturbative treat-ment, relativistic corrections are nonnegligible [25], and σ tot ∝ m − c [18], which amplifies the theoretical uncer-tainty. All these effects are likely to render a LO NRQCDanalysis inappropriate there.The CMS data [10] were taken at the same CM en-ergy, but are subject to a y -dependent low- p T cut andcover a more central y range than the LHCb data, asspecified in Eq. (3.3) of Ref. [10]. They yield σ CMStot =(1 . ± . ± .
13) nb. Our LO NRQCD predictionsare σ ∗ tot = 0 . +0 . − . nb and σ tot = 0 . +0 . − . nb, whichis still 1 order of magnitude smaller than the CMS mea-surement. The NNLP-I, NNLP-II, NLP, and LP contri-butions to the central value of σ tot are 97, 13, 27, and14 fb, respectively. I.e., over 36% of σ tot is made up bythe NNLP-II, NLP, and LP processes; about one half ofthis contribution comes as feed down from χ cJ mesons,via J/ψ + χ cJ and χ cJ + χ cJ . Therefore, the CS ∗ approx-imation is bound to be insufficient, even after includingthe O ( α s ) corrections [19]. To substantiate this state-ment, we also consider the scaling dσ/dp T ∝ /p NT . Inthe CS ∗ channel, we have N = 8 at LO and N = 6 atNLO [19, 26]. Similarly, the NNLP-II and NLP processesat NLO are expected to have N = 6 and N = 4, respec-tively, and are thus likely to produce sizable enhance-ments as well. Correction factors of 5–10, which appearplausible, would eliminate the discrepancy between theCMS measurement of σ tot and the NRQCD prediction.The CMS Collaboration also measured the differen-tial cross section in bins of M and | ∆ y | . As mentionedabove, the O ( α s )-corrected CS ∗ prediction for dσ/dM [19] dramatically undershoots the CMS data at largevalues of M , by about 2 and 4 orders of magnitude inthe two outmost bins 22 GeV < M <
35 GeV and35 GeV < M <
80 GeV, respectively. In Fig. 3, we con-front these CMS data with our full LO NRQCD resultalso showing the LO CS ∗ , NNLP-II, NLP, and LP con-tributions for reference. We observe that the previouslyneglected NRQCD contributions greatly help to fill thegap between data and theory. After their inclusion, theLO NRQCD predictions are only about 4 and 30 timessmaller than the CMS data in the last two bins, wherethe NNLP-II, NLP, and LP processes are approximatelyequally important.At LO, M , p T , and | ∆ y | are not independent of eachother, but related by M = 2 p m c + p T cosh( | ∆ y | / M distributionmay be understood from the | ∆ y | distribution, which isshown in Fig. 4. We observe from Fig. 4 that the CS ∗ contribution to dσ/d | ∆ y | peaks near | ∆ y | = 0, whichimplies that the bulk of the CS ∗ contribution to dσ/dM at M ≫ m J/ψ arises from the large- p T region, with p T ≈ M/
2, where the cross section is already very small.On the other hand, Fig. 4 tells us that the inclusion of theresidual LO NRQCD contributions renders the | ∆ y | dis-tribution significantly broader, which in turn allows forthe moderate- p T region to feed into the large- M bins soas to increase dσ/dM there by orders of magnitude. De-tailed inspection of the SDCs reveals that the broadeningof the dσ/d | ∆ y | peak about | ∆ y | = 0 is produced by thepseudodiffractive topologies of Feynman diagrams, witha t -channel gluon exchange, like those in Figs. 1(b)–(d).Although the agreement between the CMS measurementof dσ/dM and the NRQCD prediction is dramatically im-proved by the inclusion of the missing LO contributions,there remain appreciable gaps, of roughly 1 order of mag-nitude, in the outmost bins in Fig. 3. Because of theirslower falloff with p T in connection with the minimum- p T cut, the NLO corrections to those new CO and feed-down contributions, which lie beyond the scope of ourpresent analysis, are likely to further improve the situ-ation. That the CMS kinematic conditions give rise tolarge NLO corrections may also be understood from the P T distribution in Fig. 2(c) and Table 4 of Ref. [10] byobserving that only 19% of σ tot arise from the lowest bin P T < P T = 0. A good part ofthis bin and all the other bins require the radiation of anadditional parton, which only comes at NLO. This alsoexplains why the CS ∗ prediction for σ tot receives such asizable O ( α s ) correction [19].The LHCb [9] and CMS [10] measurements involveboth SPS and DPS contributions. The D0 Collabora-tion [11] attempted to separate them in their measure-ment of prompt double J/ψ production in p ¯ p collisions at √ s = 1 .
96 TeV with p T > | η | < .
0, where η is the J/ψ pseudorapidity, to find σ D0SPS = (70 ± ±
22) fband σ D0DPS = (59 ± ±
22) fb. The central SPS result
CMSNRQCD LOCS * NNLP - IINLPLP0 1 2 3 410 - È D y È d Σ d D y H nb L FIG. 4: As in Fig. 3, but for the | ∆ y | distribution. exceeds the LO CS ∗ prediction σ ∗ tot = 51 . S [8]0 and P [8] J channels and the feeddown from χ cJ mesons considered here, to yield a 28%enhancement, which establishes nice agreement. The sit-uation might change again after including NLO correc-tions. The cutoff-regularized real radiative corrections ofrelative order O ( α s ) to the CS ∗ contribution were con-sidered in Ref. [26].From the comparisons in three different experimentalenvironments, we conclude that, in the small- p T regionand away from the J/ψ pair production threshold, theCS ∗ calculation provides a reasonable approximation tothe full NRQCD result and acceptable descriptions ofthe measurements [9–11]. However, at large values of M and | ∆ y | , the CS ∗ contribution to the full NRQCDprediction is small against those due to the NNLP-II,NLP, and LP processes, which have been neglected sofar. In fact, their inclusion reduces the gap between theCS ∗ result and the CMS data [10] in the outmost M and | ∆ y | bins by several orders of magnitude, but leaveroom for NLO corrections of typical size. Should theNLO NRQCD prediction, which is yet to be calculated,agree with the CMS data, then this would provide strongevidence in favor of the COM.Prompt double J/ψ hadroproduction also serves as auseful laboratory to probe the DPS mechanism [20]. Re-portedly, (46 ± ∗ approximation, then the DPS contri-bution dominates for | ∆ y | > . | ∆ y | distribution [20]. However, includingthe residual NRQCD contributions, due to the NNLP-II,NLP, and LP processes, on top of the CS ∗ contributionrenders the | ∆ y | distribution of SPS much broader, asmay be seen in Fig. 4 for CMS kinematic conditions,leaving less room for DPS in agreement with other mea-surements [11]. In other words, the relative importanceof SPS and DPS extracted from experimental data deli-cately depends on the quality of the NRQCD prediction,and any conclusions concerning the significance of DPSare premature before the NLO corrections to all the rel- evant channels are taken into account. [1] W. E. Caswell and G. P. Lepage, Phys. Lett. B , 437(1986).[2] G. T. Bodwin, E. Braaten, and G. P. Lepage, Phys. Rev.D , 1125 (1995); , 5853(E) (1997).[3] N. Brambilla et al. (Quarkonium Working Group), Eur.Phys. J. C , 1534 (2011); , 2981 (2014).[4] G. P. Lepage, L. Magnea, C. Nakhleh, U. Magnea, andK. Hornbostel, Phys. Rev. D , 4052 (1992).[5] G. C. Nayak, arXiv:1506.02593 [hep-ph].[6] M. Butenschoen and B. A. Kniehl, Mod. Phys. Lett. A , 1350027 (2013).[7] M. Butenschoen and B. A. Kniehl, Phys. Rev. Lett. ,172002 (2012).[8] M. Butenschoen, Z.-G. He, and B. A. Kniehl, Phys. Rev.Lett. , 092004 (2015).[9] R. Aaij et al. (LHCb Collaboration), Phys. Lett. B ,52 (2012).[10] V. Khachatryan et al. (CMS Collaboration), J. High En-ergy Phys. 09 (2014) 094.[11] V. M. Abazov et al. (D0 Collaboration), Phys. Rev. D , 111101(R) (2014).[12] V. Barger, S. Fleming, and R. J. N. Phillips, Phys. Lett.B , 111 (1996).[13] C.-F. Qiao, Phys. Rev. D , 057504 (2002).[14] R. Li, Y.-J. Zhang, and K.-T. Chao, Phys. Rev. D ,014020 (2009).[15] C.-F. Qiao, L.-P. Sun, and P. Sun, J. Phys. G , 075019(2010).[16] P. Ko, J. Lee, and C. Yu, J. High Energy Phys. 01 (2011) 070.[17] A. V. Berezhnoy, A. K. Likhoded, A. V. Luchinsky, andA. A. Novoselov, Phys. Rev. D , 094023 (2011); ,034017 (2012).[18] Y.-J. Li, G.-Z. Xu, K.-Y. Liu, and Y.-J. Zhang, J. HighEnergy Phys. 07 (2013) 051.[19] L.-P. Sun, H. Han, and K.-T. Chao, arXiv:1404.4042[hep-ph].[20] C. H. Kom, A. Kulesza, and W. J. Stirling, Phys. Rev.Lett. , 082002 (2011); S. P. Baranov, A. M. Snigirev,N. P. Zotov, A. Szczurek, and W. Sch¨afer, Phys. Rev. D , 034035 (2013).[21] K. A. Olive et al. (Particle Data Group), Chin. Phys. C , 090001 (2014).[22] H. L. Lai, J. Huston, S. Kuhlmann, J. Morfin, F. Ol-ness, J. F. Owens, J. Pumplin, and W. K. Tung (CTEQCollaboration), Eur. Phys. J. C , 375 (2000).[23] E. J. Eichten and C. Quigg, Phys. Rev. D , 1726(1995).[24] E. Braaten, B. A. Kniehl, and J. Lee, Phys. Rev. D ,094005 (2000).[25] A. P. Martynenko and A. M. Trunin, Phys. Rev. D ,094003 (2012).[26] J.-P. Lansberg and H.-S. Shao, Phys. Rev. Lett. ,122001 (2013).[27] C.-F. Qiao and L.-P. Sun, Chin. Phys. C37