Probing Quarkonium Production Mechanisms with Jet Substructure
Matthew Baumgart, Adam K. Leibovich, Thomas Mehen, Ira Z. Rothstein
PProbing Quarkonium Production Mechanisms with Jet Substructure
Matthew Baumgart a , Adam K. Leibovich b , Thomas Mehen c , and Ira Z. Rothstein d1 Department of Physics, Carnegie Mellon University, Pittsburgh, PA 15213 Pittsburgh Particle Physics Astrophysics and Cosmology Center (PITT PACC)Department of Physics and Astronomy, University of Pittsburgh, Pittsburgh, PA 15260 Department of Physics, Duke University, Durham, NC 27708 (Dated: June 27, 2018)We use fragmenting jet functions (FJFs) in the context of quarkonia to study the productionchannels predicted by NRQCD ( S (1)1 , S (8)1 , S (8)0 , P (8) J ). We choose a set of FJFs that give theprobability to find a quarkonium with a given momentum fraction inside a cone-algorithm jet withfixed cone size and energy. This observable gives several lever arms that allow one to distinguishdifferent production channels. In particular, we show that at fixed momentum fraction the indi-vidual production mechanisms have distinct behaviors as a function of the the jet energy. As aconsequence of this fact, we arrive at the robust prediction that if the depolarizing S (8)0 matrixelement dominates, then the gluon FJF will diminish with increasing energy for fixed momentumfraction, z , and z > a Electronic address: [email protected] b Electronic address: [email protected] c Electronic address: [email protected] d Electronic address: [email protected] a r X i v : . [ h e p - ph ] N ov I. INTRODUCTION
Nonrelativistic QCD (NRQCD) is an effective field theory [1] for quarkonium that reproduces full QCD as anexpansion in the relative velocity, v , of the heavy quark and antiquark. This theory has been used to study boththe decay and production of these bound states [2]. Its predictive power is predicated on our knowledge of a set ofnon-perturbative matrix elements that must be extracted from the data. In the case of J/ψ or Υ production there arefour such matrix elements that must be fit at leading order, and thus predictions have mainly been limited to shapesof spectra. NRQCD predictions at NLO in the coupling have been compared to the world data on
J/ψ productionin Refs. [3, 4]. The χ /d.o.f. of 4.42 found in Ref. [3] is higher than one would hope for, but not unexpected givenlarge theoretical uncertainties.Thus, it is perhaps fair to say that we cannot yet claim that NRQCD is correctly describing quarkonium productionwith unqualified success. In particular, one prediction [5] of the theory is that, at asymptotically large transversemomentum, the S state ( J/ψ or Υ) should be purely transverse at leading order. At present the data in both thecharm and bottom sector do not see this trend [6] though the error bars are large, especially in the bottom system.Furthermore the various experiments are not in agreement.It is important to appreciate that concluding that NRQCD is “wrong”, in any sense, is equivalent to saying thatQCD does not properly describe these states. If NRQCD predictions for large p T production are not agreeing withthe data, and we assume that the data is correct, then the only logical alternatives are: (1) the velocity and/or α s expansions are not converging, (2) the fragmentation approximation, along with its expansion in m Q /p T , is wrong,either due to the failure of factorization or the presence of anomalously large power corrections. Let us consider eachof these possibilities in turn. The perturbative corrections in α s (2 m Q ) to the fragmentation function were found to besmall [7]. The possibility that the velocity power counting could not apply to the charmed system [8, 9] is certainly aviable option, though the velocity expansion seems to work relatively well for the decay processes [10]. Moreover, onewould expect for the bottom system that the velocity expansion should converge nicely. It is possible that factorizationis breaking down in the production processes, as all such proofs, at least within the confines of SCET, are lacking atreatment of the factorization breaking “Glauber mode”. Nonetheless, given the success of semi-inclusive predictionsin light hadronic systems, it would be surprising to see a failure in the case of quarkonium.A more conservative guess would be that there is nothing wrong with the theory, but perhaps the values of extractedmatrix elements are sufficiently inaccurate as to change the nature of the polarization prediction. For example, themagnetic spin flip operator could be anomalously large. In any case, to get a better handle on the situation we mustimprove our quantitative understanding of the various production channels associated with the aforementioned matrixelements. The purpose of this paper is to introduce a new tool that will allow for a new extraction of these matrixelements by studying the characteristics of jets within which the quarkonium reside. II. THE FRAGMENTING JET FUNCTION (FJF)
Power counting dictates that at asymptotic values for p ⊥ (cid:29) m Q , quarkonia should be produced by single partonfragmentation. Since the parton initiating the fragmentation is a colored object, the quarkonium will be producedin association with light hadrons. In this paper we will consider a
J/ψ produced within a jet of energy E and conesize R , in which the J/ψ carries a fraction of the jet energy, z . In this situation, a generic cross section is determinedby the convolution of a hard and soft function (and possibly other jet functions, if there are other jets detected in thefinal state) multiplied by a quantity known as the fragmenting jet function (FJF), first introduced in Ref. [14] andfurther studied in Refs. [15–19]. These papers focused on FJFs for light hadrons such as pions. FJFs for particleswith a single heavy quark are studied in Ref. [20]. We show that the FJFs for gluon and charm quark jets containinga J/ψ can be calculated in terms of a set of NRQCD long-distance matrix elements (LDME). In our calculations therelevant LDMEs are: (cid:104)O
J/ψ ( S (1)1 ) (cid:105) , (cid:104)O J/ψ ( S (8)0 ) (cid:105) , (cid:104)O J/ψ ( S (8)1 ) (cid:105) , and (cid:104)O J/ψ ( P (8)0 ) (cid:105) . The spectroscopic notationindicates the quantum numbers of the heavy quarks prior to hadronization. We show that the contribution to theFJF from each of these mechanisms depends differently on z and E and can thus be used to extract the LDME. Ourresults could easily be extended to jets containing other quarkonia states.Since there are many observables associated with jets (angularities [21], broadening [22], jet shape [23], N-subjettiness [24], etc.), one can generate a very large number of new tests of the NRQCD factorization formalism This χ is based on an analysis in which feed down from higher charmonia is ignored. Accounting for these contributions reduces the χ slightly to 3.74. In intermediate ranges of p ⊥ double-parton fragmentation should dominate [11–13]. The phenomenology of double-parton fragmentationhas yet to be performed. The results will apply for the Υ as well. Of course the matrix elements will be different but most of the calculations in this paper arenormalized such that the result is independent of the matrix element. Thus, when we use the term
J/ψ we really mean the generic S state. by applying jet physics techniques to the study of quarkonia produced within jets. Furthermore, studying high p ⊥ quarkonia produced within jets avoids some of the potential theoretical pitfalls that could plague tests of the NRQCDfactorization formalism at small p ⊥ . At the highest p ⊥ available, we expect factorization to hold up to correctionswhich scale as m Q /p ⊥ , and furthermore the α s expansion should be well behaved. A. Operator Definitions
We first briefly review the properties of the FJF [14–19]. We can consider many different production processeswith a quarkonium inside a jet. As an example, consider the two-jet cross section where one of the jets contains anidentified
J/ψ . The factorization theorem [14] for the production cross section for a jet with energy E , cone size R ,and a J/ψ with energy fraction z in a pp collision is schematically of the form d σdE dz = (cid:88) a,b,i,j H ab → ij × f a/p ⊗ f b/p ⊗ J j ⊗ S × G ψi ( E, R, z, µ ) , (1)where H ab → ij is the hard function, f a/p and f b/p are parton distributions functions, J j is the jet function for thejet not containing the J/ψ initiated by a final state parton j , and G ψi is the FJF for the jet containing the J/ψ fragmenting from parton i . S is the soft function. Generically there are two types of jets, unmeasured and measured,in the terminology of Ref. [26]. Unmeasured jet functions describe jets in which only the large light-cone momentum(measured along the jet axis) is known. In measured jets, some aspect of the jet’s substructure has also been measured.For unmeasured jets, soft gluon radiation does not affect the total momentum of the jet (up to power corrections)and therefore these jet functions enter the cross section multiplicatively. For measured jets, the jet substructure maybe sensitive to the soft radiation, therefore it must be convolved with the soft function. For G ψi ( E, R, z, µ ), R , E and z are not affected by soft radiation (up to power corrections) so it also enters the cross section multiplicatively andall of the z dependence is contained in G ψi ( E, R, z, µ ), which enables us to ignore all the other factors in Eq. (1) andfocus on G ψi ( E, R, z, µ ). We can therefore ignore the dependence on the other jet in Eq. 1, or indeed we could look atother processes with a
J/ψ inside a jet, such as the single-jet inclusive cross section. In this case, there are no otherjet functions and the soft function is only an overall normalization and is therefore irrelevant for our purposes.A generic fragmenting jet function may be defined as a product of operators of the form G ψi = (cid:104) | O int O meas (cid:88) X | X + H (cid:105)(cid:104) O + H | O int | (cid:105) , (2)where O int is some interpolating field for the parton of interest, i . O meas is a measurement operator (a set ofdelta functions) that fixes the measured jet characteristics, such as E, R and z . The operators are manifestly gaugeinvariant. In SCET these operators would involve only fields with the same large momentum (and possibly soft fields)and compose a piece of the factorization theorem (not shown) that is generated at the highest scale Q , which isusually taken to be on the order of the jet energies E . G ψi contains two relevant scales: the invariant mass or energyof the jet and the hadron mass. Thus one can perform a further factorization to separate out these two scales wherethe long distance physics is captured by a fragmentation function, and the short distance physics can be calculatedperturbatively. The resulting form of this second step of factorization can be written as [14] G ψi ( E, R, z, µ ) = (cid:88) j (cid:90) z dyy J ij ( E, R, y, µ ) D j → ψ (cid:18) zy , µ (cid:19) × (cid:34) O (cid:32) m ψ E tan ( R/ (cid:33)(cid:35) , (3)where we have now specialized to the case of interest where the jet energy E is measured for cone size R . Looselyspeaking this function gives the probability of finding a quarkonium whose large momentum fraction, relative to thejet within which it is found, is z . It is possible and indeed likely that there are small invariant mass jets in thedata. However, note that the process is inclusive in the sense that one integrates over all invariant masses up to2 E tan[( R/ x = 1, which is in the If there were a hierarchy then one would have to run these operators from the scale Q to the scale E . In Refs. [14–19] the error scales as Λ instead of m ψ . For our processes, the low energy scale is m ψ , and thus the error scalesdifferently. resonance region. But if we take moments (integrating over x ) that region gets washed out. The differential crosssection near x = 1 is sensitive to the IR but the integrated cross section is not.The operator definition of the quark fragmentation function is [27] given by D j → ψ ( z ) = z π (cid:90) dx + e ix + p − ψ /z N c T r (cid:104) | ¯ n/ q ( x + , , (cid:88) X | X + ψ (cid:105)(cid:104) X + ψ | ¯ q (0) | (cid:105) , (4)where the operator q includes an anti-path ordered Wilson line that renders the matrix element gauge invariant. Asimilar matrix element can be written down for the gluon fragmentation function. What distinguishes the quarkoniumfragmentation function from other cases is that it contains a further subset of scales: the quark mass, the Bohr radius,and the binding energy that scale as 1 , v , and v respectively in units of the quark mass. Furthermore, taking thequark mass scale to be perturbative implies that the constituents are produced at a point, and that the momentumfraction carried by the quarkonium is set perturbatively. This is so even if the pair is produced in an octet state,since the shedding of color occurs via soft multipole emission whose effect on the kinematics is suppressed by anamount of order v , except near the end point z = 1 where these non-perturbative corrections are enhanced and canbe accounted for by the inclusion of a non-perturbative shape function [28]. In general we will present our resultsaway from the end point to avoid the need for such a function. Thus, the fragmentation functions for quarkonium arecalculable up to a set of LDMEs.The matching coefficients J ij ( E, R, z, µ ) can be calculated in perturbation theory. Large logarithms in the J ij ( E, R, z, µ ) are minimized at the scale 2 E tan( R/ − z ). Note that the matching coefficients J ij ( E, R, z, µ ) areindependent of the choice of hadronic final states, and thus we may utilize the results in Ref. [18] for the FJF for lighthadrons for the case at hand.
B. Expressions for the
J/ψ
FJF
We will focus gluon and charm quark fragmentation to
J/ψ . For gluon fragmentation to
J/ψ through c ¯ c pairs, weconsider the S (1)1 , S (8)1 , S (8)0 , and P (8) J quark states. The S (1)1 gluon fragmentation function is leading order inthe v expansion, as the color-octet contributions are suppressed by v . However the gluon color-singlet contributionis suppressed relative to S (8)1 by a power of α s . For charm quark fragmentation to J/ψ , we consider only the S (1)1 contributions because both color-singlet and color-octet mechanisms start at the same order in α s . The ratio ofgluon to charm production cross sections at the LHC is approximately 50, but the ratio of charm quark to gluonfragmentation functions, partially compensates for this suppression. Fragmentation from light quarks is suppressedby one power of α s relative to the S (8)1 gluon fragmentation contribution and shares the octet velocity suppression.The J ij ( E, R, z, µ ) and the relevant fragmentation functions are collected in the Appendix.The convolution in Eq. (3) can be explicitly evaluated using the formula for J gg ( E, R, z, µ ) and J gq ( E, R, z, µ ) inthe Appendix to obtain G ψg ( E, R, z, µ ) = (cid:90) z dyy J gg ( y ) D g → ψ (cid:18) zy , µ (cid:19) + (cid:90) z dyy J gq ( y ) D q → ψ (cid:18) zy , µ (cid:19) (5)= G ψg ( g ) ( E, R, z, µ ) + G ψg ( q ) ( E, R, z, µ ) , where G ψg ( g ) ( E, R, z, µ )2(2 π ) = D g → ψ ( z, µ ) (cid:18) C A α s π (cid:18) L − z − π (cid:19)(cid:19) (6)+ C A α s π (cid:20)(cid:90) z dyy ˜ P gg ( y ) L − y D g → ψ (cid:18) zy , µ (cid:19) +2 (cid:90) z dy D g → ψ ( z/y, µ ) − D g → ψ ( z, µ )1 − y L − y + θ (cid:18) − z (cid:19) (cid:90) / z dyy ˆ P gg ( y ) ln (cid:18) y − y (cid:19) D g → ψ (cid:18) zy , µ (cid:19)(cid:35) , and G ψg ( q ) ( E, R, z, µ )2(2 π ) = T F α s π (cid:20)(cid:90) z dyy [ P qg ( y ) L − y + y (1 − y )] D q → ψ (cid:18) zy , µ (cid:19) (7)+ θ (cid:18) − z (cid:19) (cid:90) / z dyy P qg ( y ) ln (cid:18) y − y (cid:19) D q → ψ (cid:18) zy , µ (cid:19)(cid:35) . In this expression, we have defined L − z = ln (cid:18) E tan( R/ − z ) µ (cid:19) , ˆ P gg ( z ) = 2 (cid:20) z (1 − z ) + + 1 − zz + z (1 − z ) (cid:21) , ˜ P gg ( z ) = 2 (cid:20) − zz + z (1 − z ) (cid:21) ,P qg ( z ) = z + (1 − z ) . This expression shows that the logarithms in G ψg ( E, R, z, µ ) are minimized at the scale µ = 2 E tan( R/ − z ), asfirst pointed out in Ref. [18]. The logarithms of 1 − z are easily resummed using the jet anomalous dimension [18],however, we will not do this resummation in this paper as we consider 1 − z ∼ O (1). We instead set the scale in J gg ( E, R, z, µ ) to be µ J = 2 E tan( R/ m c to the scale µ J .This is done by taking moments of the fragmentation functions, evolving each moment according to its anomalousdimension as obtained from the Altarelli-Parisi equations, and then performing an inverse-Mellin transform.The G ψg ( q ) ( E, R, z, µ ) is present because of mixing with the quark fragmentation function. In principle there shouldbe a sum over all quark flavors. However, the light quark fragmentation function contributes only via fragmentationthrough S (8)1 c ¯ c pairs at O ( α s ) and is subleading to the S (8)1 gluon fragmentation so it will be neglected. Charmquarks and antiquarks can fragment via S (1)1 c ¯ c pairs at O ( α s ), which is lower order than the corresponding gluonfragmentation function. Therefore this mixing must be included.The quark FJF is given by: G ψq ( E, R, z, µ )2(2 π ) = D q → ψ ( z, µ ) (cid:18) C F α s π (cid:18) L − z − π (cid:19)(cid:19) (8)+ C F α s π (cid:20)(cid:90) z dyy (1 − y ) (cid:18) L − y + 12 (cid:19) D q → ψ (cid:18) zy , µ (cid:19) +2 (cid:90) z dy D q → ψ ( z/y, µ ) − D q → ψ ( z, µ )1 − y L − y + θ (cid:18) − z (cid:19) (cid:90) / z dyy ˆ P qq ( y ) ln (cid:18) y − y (cid:19) D q → ψ (cid:18) zy , µ (cid:19)(cid:35) + C F α s π (cid:20)(cid:90) z dyy (cid:16) P gq ( z ) L − y + y (cid:17) D g → ψ (cid:18) zy , µ (cid:19) + θ (cid:18) − z (cid:19) (cid:90) / z dyy log (cid:18) y − y (cid:19) P gq ( y ) D g → ψ (cid:18) zy , µ (cid:19)(cid:35) , where ˆ P qq ( z ) = 1 + z (1 − z ) + ,P gq ( z ) = 1 + (1 − z ) z . For this contribution, as previously mentioned we will only consider the q = c contribution fragmenting via S (1)1 c ¯ c pairs. The mixing contribution of gluon fragmentation into this FJF must also be included. To evaluate G ψi ( E, R, z, µ J )we will use Eqs. (5-8) with our numerically evaluated D i → ψ ( z, µ J ). We see that up to O ( α s ) corrections G ψi ( E, R, z, µ J )2(2 π ) → D i → ψ ( z, µ J ) + O ( α s ( µ J )) , (9) Out[892]= z FIG. 1. The gluon fragmentation functions at µ = 2 m c for S (1)1 (black), S (8)1 (red), S (8)0 (green), P (8) J (blue). Relativenormalization is arbitrary and relevant formulas are found in the Appendix. Out[843]= E (cid:61)
50 GeV E (cid:61)
200 GeV G i z z FIG. 2. The gluon FJF (color coding the same as in Fig. 1) and the charm quark FJF for S (1)1 (purple). which shows that the z distribution of a J/ψ within a jet with energy E and cone size R is approximately equal tothe fragmentation function evaluated at the jet scale µ J = 2 E tan( R/ S (1)1 , S (8)1 , S (8)0 , and P (8) J are very different, this observable has thepower to discriminate between all four gluon-production mechanisms. This can seen from a cursory inspection ofthe expressions for the fragmentation functions given in the Appendix and shown in Fig. 1. Though the dramaticdifferences in these functions are considerably softened by Altarelli-Parisi evolution, we will see that each contributionto G ψg ( E, R, z, µ ) has a different E dependence that varies for fixed z ( cf. Fig. 3). This makes it clear that measurementof G ψg ( E, R, z, µ ) for different momentum fractions has potential to allow independent extraction of all four LDME.In our calculations E and R will always enter in the combination E tan( R/
2) and we will choose R = 0 . S (1)1 (black), S (8)1 (red), S (8)0 (green), and P (8) J (blue) gluon FJFs as well as the S (1)1 charm(purple) FJF for E = 50 GeV and E = 200 GeV. This plot illustrates the discriminating power of the jet observables.For Fig. 2 we have chosen the LDME to be the central values extracted in the fits of Refs. [3, 4]: (cid:104)O J/ψ ( S (1)1 ) (cid:105) =1 .
32 GeV , (cid:104)O J/ψ ( S (8)0 ) (cid:105) = 4 . × − GeV , (cid:104)O J/ψ ( S (8)1 ) (cid:105) = 2 . × − GeV , and (cid:104)O J/ψ ( P (8)0 ) (cid:105) = − . × − GeV . Throughout this work we take m c = 1.4 GeV.It is also interesting to study the energy dependence of the fragmentation functions. In Fig. 3 we plot the fourgluon FJFs as a function of energy E for three different values of z using the same color-coding as above. The LDMEof Refs. [3, 4] have again been used to set the normalization of the curves. In order to the make shapes of the curvesmore easily viewable, we have divided the P (8) J by a factor of 5 and the color-singlet contribution has been divided bya factor of 2. The shapes of the energy dependence at different values of z are quite distinct for all four fragmentationfunctions. For example, the P (8) J FJF is an increasing function of energy for all three z values, while the S and the Out[842]= z (cid:61) (cid:61) (cid:61) G g
60 80 100 120 140 160 180 2000.0200.0250.0300.035 60 80 100 120 140 160 180 2000.0120.0130.0140.0150.0160.0170.0180.019 60 80 100 120 140 160 180 2000.0000.0050.0100.015 E (cid:72) GeV (cid:76) E (cid:72) GeV (cid:76) E (cid:72) GeV (cid:76)
FIG. 3. The energy dependence of the four different contributions to the gluon FJF for fixed z = 0 .
3, 0 .
5, and 0 .
8. Colorcoding is the same as in Figs. 1, 4. For readability, we have scaled the P (8) J function down by a factor of 5 and S (1)1 down by2. These plots have been normalized with respect to the total rate and thus do not reflect its underlying energy dependence. color-singlet are decreasing functions of E for z = 0.5 and 0.8, and the S (8)1 is decreasing only for 0.8. Extractionsof the E dependence of the FJF for different values of z should allow one to disentangle the various contributions toquarkonium production. In particular, note that if the lack of polarization is due to an anomalously large S (8)0 , thenwe should see a decrease in the gluon FJF as a function of the jet energy for fixed z , with z > (cid:104) z N (cid:105) ≡ (cid:82) dz z N − G ψg ( E, R, z, µ ), can be calculated analytically using the formulae in theAppendix. Note that this integral diverges if N = 1 because the N = 1 moments of both the Altarelli-Parisi splittingfunction and the matching coefficients J gg ( E, R, z, µ ) have poles at N = 1. This could be cured by resummation oflog z , as implemented for the D g → ψ ( z, µ ) fragmentation function in Ref. [29], but this is beyond the scope of thispaper. The LDME cancel in the ratios of moments, and we plot ratios of successive moments, (cid:104) z N +1 (cid:105) / (cid:104) z N (cid:105) , for N = 2 , , and 4 in Fig. 4. In all columns we have plotted the moment ratios of the S (1)1 FJF (black). We also plotmoment ratios for the S (8)1 FJF (red), P (8) J (blue), S (8)0 FJF (green), and the charm quark FJF (purple), in eachcolumn respectively. Scale uncertainties are included by varying E tan( R/ < µ < E tan( R/ (cid:104) z n +1 (cid:105)(cid:104) z n (cid:105) (cid:12)(cid:12)(cid:12)(cid:12) P (8) J ≈ (cid:104) z n +1 (cid:105)(cid:104) z n (cid:105) (cid:12)(cid:12)(cid:12)(cid:12) S (8)1 > (cid:104) z n +1 (cid:105)(cid:104) z n (cid:105) (cid:12)(cid:12)(cid:12)(cid:12) S (8)0 ≈ (cid:104) z n +1 (cid:105)(cid:104) z n (cid:105) (cid:12)(cid:12)(cid:12)(cid:12) c − quark > (cid:104) z n +1 (cid:105)(cid:104) z n (cid:105) (cid:12)(cid:12)(cid:12)(cid:12) S (1)1 . (10)Note that for the same choice of µ , (cid:104) z n +1 (cid:105)(cid:104) z n (cid:105) (cid:12)(cid:12)(cid:12)(cid:12) P (8) J > (cid:104) z n +1 (cid:105)(cid:104) z n (cid:105) (cid:12)(cid:12)(cid:12)(cid:12) S (8)1 , (11)but once scale uncertainties are included it is hard to distinguish these two moment ratios. The energy dependenceof the moments of the color-octet FJFs is given by (cid:104) z N (cid:105) = ˜ J gg ( E, R, N, µ ) (cid:18) α s ( µ ) α s (2 m c ) (cid:19) γ Ngg /b ˜ D g → ψ ( N, m c ) . (12)When we set µ ≈ E tan( R/ m c to µ J .Making log-log plots of (cid:104) z N (cid:105) we find that that (cid:104) z N (cid:105) ∝ (log E ) F ( N ) where F ( N ) can be extracted from Eq. (12). III. COMPARISON OF VARIOUS LDME EXTRACTIONS
In the final part of this paper, we will discuss what recent extractions of the LDME predict for the gluon FJF. Inaddition to the extractions in Refs. [3, 4], we will consider values of the LDME extracted in two recent papers [30, 31]that attempt to solve the polarization puzzle by focusing exclusively on high p ⊥ production of charmonia at colliderexperiments. The study in Ref. [30] uses a NLO NRQCD calculation to fit the color-octet LDME to inclusive J/ψ production at high p ⊥ and finds values of the LDME that can produce negligible polarization in agreement with thedata. However, these values of LDME are inconsistent with the results of fitting the world data in Refs. [3, 4]. Inparticular, (cid:104)O J/ψ ( S (8)0 ) (cid:105) is larger by a factor of two and (cid:104)O J/ψ ( P (8)0 ) (cid:105) has the opposite sign as the fit in Refs. [3, 4]. S (cid:72) (cid:76) P J (cid:72) (cid:76) S (cid:72) (cid:76) Charm QuarkFragmentation (cid:60) z (cid:62) (cid:144) (cid:60) z (cid:62)
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60 80 100 120 140 160 180 2000.00.20.40.60.81.0 60 80 100 120 140 160 180 2000.00.20.40.60.81.0 60 80 100 120 140 160 180 2000.00.20.40.60.81.0 60 80 100 120 140 160 180 2000.00.20.40.60.81.0
Jet Energy (cid:72)
GeV (cid:76)
Jet Energy (cid:72)
GeV (cid:76)
Jet Energy (cid:72)
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FIG. 4. Ratios of successive moments as a function of the jet energy. See text for explanation.
These two effects combine to produce significant depolarization of the
J/ψ . In Ref. [31], the calculations are performedin the leading-power fragmentation approximation and logarithms of p ⊥ /m c are resummed by using Altarelli-Parisiequations for the fragmentation functions. The fitted LDME are similar to those found in Ref. [30] in the sense that (cid:104)O J/ψ ( S (8)0 ) (cid:105) is by far the largest matrix element and (cid:104)O J/ψ ( P (8)0 ) (cid:105) again has opposite sign as that extracted fromfits to the world data. In Ref. [31], the errors on (cid:104)O J/ψ ( S (8)1 ) (cid:105) and (cid:104)O J/ψ ( P (8)0 ) (cid:105) are essentially 100% so the extractedmatrix elements are consistent with zero. This analysis suggests that the production of J/ψ at large p ⊥ is dominatedby c ¯ c pairs in a S (8)0 state rather than S (8)1 . It should be noted that the quoted errors in the extracted LDMEin Refs. [3, 4] are considerably smaller than those in Refs. [30, 31]. However, the presence of nontrivial correlationsbetween the uncertainties in [31] allows us to make a much sharper prediction for the gluon FJF than is naivelysuggested by the large individual error bars [32]. In all of these extractions, there is a hierarchy between matrixelements that are supposed to have the same velocity scaling. However, it is generated by anomalously small matrixelements not anomalously large ones.In Fig. 5, we compare the predictions for the gluon FJF at E = 50 GeV and E = 200 GeV using the resultsfrom the fits to the LDME in Refs. [3, 4, 30, 31]. The gluon FJF is the sum over all contributions, color-singlet aswell as color-octet. The color-singlet matrix element is chosen to be 1.32 GeV in Refs. [3, 4, 31] and 1.16 GeV inRef. [30]. We use the LDME extracted in the original fit and the error bands are the result of adding in quadraturethe uncertainties for the LDME quoted in Refs. [3, 4, 30]. We supplement the uncertainty given in Ref. [31] with the E (cid:61)
50 GeV E (cid:61)
200 GeV G g z z FIG. 5. The gluon FJF at fixed energy for the LDME extracted in Refs. [3, 4] (gray), Ref. [30] (blue), and Ref. [31] (red). z (cid:61) (cid:61) (cid:61) G g
60 80 100 120 140 160 180 2000.000.050.100.150.200.250.30 60 80 100 120 140 160 180 2000.000.050.100.15 60 80 100 120 140 160 180 200 (cid:45) E (cid:72) GeV (cid:76) E (cid:72) GeV (cid:76) E (cid:72) GeV (cid:76)
FIG. 6. The gluon FJF at fixed momentum fraction for the LDME extracted in Refs. [3, 4] (gray), Ref. [30] (blue), andRef. [31] (red). These plots have been normalized with respect to the total rate. full correlation matrix provided by one of the authors [32]. No other theoretical uncertainty is included. The grayband with black borders is the prediction using the LDME extracted in Refs. [3, 4], the red band uses the matrixelements extracted in Ref. [31] and the blue band uses the matrix elements extracted in Ref. [30]. Fig. 6 showsthe energy dependence at fixed momentum fraction for the different determinations. We see that for z >
IV. CONCLUSIONS
We have demonstrated that by studying the characteristics of jets arising from quarkonium production, we candisentangle the various production channels. There are a multitude of ways of analyzing such events. Here we havechosen to measure the energy and cone angle of the jet, but one could consider other observables such as the invariantmass. Within our choice of variables (
E, R ) we found that a particularly discriminating tool is the measurement of theenergy dependence at fixed momentum fraction as shown in Figs. 3 and 6. A robust prediction of our analysis is thatfor z > z should decrease as function of energy if the lack of transverse polarization in thedata is due to the dominance of the S (8)0 LDME over the other color octet matrix elements for high- p ⊥ production.Further information can be gathered by calculating the normalized cross section, in which case one could constrainthe sum of the matrix elements.0 ACKNOWLEDGMENTS
TM and AKL acknowledge support from the ESI workshop, Jets in Quantum Field Theory, where this work wasinitiated. MB acknowledges the Center for Future High Energy Physics at IHEP where a portion of this work wascompleted. We thank Andrew Hornig, Wouter Waalewijn, Massimiliano Procura, Geoffrey Bodwin, and ChristianBauer for useful discussions and James Russ for comments on the manuscript. AKL was supported in part by theNational Science Foundation under Grant No. PHY-1212635. TM was supported in part by the Director, Office ofScience, Office of Nuclear Physics, of the U.S. Department of Energy under grant numbers DE-FG02-05ER41368.IZR and MB are supported by DOE DE-FG02-04ER41338 and FG02-06ER41449.
Appendix A: Formulae For Matching Coefficients, Fragmentation Functions, Moments
In this appendix we collect the basic formulae needed for the calculation. The matching coefficients J ij ( E, R, z, µ )are calculated in Ref. [18]: J gg ( E, R, z, µ )2(2 π ) = δ (1 − z ) + α s ( µ ) C A π (cid:20)(cid:18) L − π (cid:19) δ (1 − z ) + ˆ P gg ( z ) L + ˆ J gg ( z ) (cid:21) , (A1) J qq ( E, R, z, µ )2(2 π ) = δ (1 − z ) + α s ( µ ) C F π (cid:20)(cid:18) L − π (cid:19) δ (1 − z ) + ˆ P qq ( z ) L + ˆ J qq ( z ) (cid:21) , (A2) J gq ( E, R, z, µ )2(2 π ) = α s ( µ ) T F π (cid:104) P qg ( z ) L + ˆ J gq ( z ) (cid:105) , (A3) J qg ( E, R, z, µ )2(2 π ) = α s ( µ ) C F π (cid:104) P gq ( z ) L + ˆ J qg ( z ) (cid:105) , (A4)where L = ln[2 E tan( R/ /µ ], andˆ J gg ( z ) = (cid:40) ˆ P gg ( z ) ln z z ≤ / − z + z ) z (cid:16) ln(1 − z )1 − z (cid:17) + z ≥ / , (A5)ˆ J qq ( z ) = 12 (1 − z ) + (cid:40) ˆ P qq ( z ) ln z z ≤ / z ) (cid:16) ln(1 − z )1 − z (cid:17) + z ≥ / , (A6)ˆ J gq ( z ) = z (1 − z ) + P qg ( z ) (cid:26) ln z z ≤ / − z ) z ≥ / , (A7)ˆ J qg ( z ) = z P gq ( z ) (cid:26) ln z z ≤ / − z ) z ≥ / . (A8)There are five NRQCD fragmentation functions. The S (8)1 gluon fragmentation function is given by [33] D S (8)1 g → ψ ( z, m c ) = πα s (2 m c )24 m c (cid:104) O ψ ( S (8)1 ) (cid:105) δ (1 − z ) , (A9)and the S (1)1 gluon fragmentation function is [34, 35] D S (1)1 g → ψ ( z, m c ) = 5 α s (2 m c )864 π (cid:104) O ψ ( S (1)1 ) (cid:105) m c (cid:90) z dr (cid:90) (1+ r ) / r + z ) / z dy − y ) ( y − r ) ( y − r ) (cid:88) i =0 z i (cid:32) f i ( r, y ) + g i ( r, y ) 1 + r − y y − r ) (cid:112) y − r ln y − r + (cid:112) y − ry − r − (cid:112) y − r (cid:33) , (A10)1where f ( r, y ) = r (1 + r )(3 + 12 r + 13 r ) − r (1 + r )(1 + 3 r ) y − r (3 − r − r + 7 r ) y + 8 r (4 + 3 r + 3 r ) y − r (9 − r − r ) y − r + 3 r ) y + 8(6 + 7 r ) y − y ,f ( r, y ) = − r (1 + 5 r + 19 r + 7 r ) y + 96 r (1 + r ) y + 8(1 − r − r − r ) y +16 r (7 + 3 r ) y − r ) y + 32 y ,f ( r, y ) = r (1 + 5 r + 19 r + 7 r ) − r (1 + r ) y − − r − r − r ) y − r (7 + 3 r ) y + 4(5 + 7 r ) y − y ,g ( r, y ) = r (1 − r )(3 + 24 r + 13 r ) − r (7 − r − r ) y − r (17 + 22 r − r ) y +4 r (13 + 5 r − r ) y − r (1 + 2 r + 5 r + 2 r ) y − r (3 − r − r ) y +8(1 − r − r ) y ,g ( r, y ) = − r (1 + r )(1 − r )(1 + 7 r ) y + 8 r (1 + 3 r )(1 − r ) y +4 r (1 + 10 r + 57 r + 4 r ) y − r (1 + 29 r + 6 r ) y − − r − r ) y ,g ( r, y ) = r (1 + r )(1 − r )(1 + 7 r ) − r (1 + 3 r )(1 − r ) y − r (1 + 10 r + 57 r + 4 r ) y + 4 r (1 + 29 r + 6 r ) y + 4(1 − r − r ) y . The integrals over r and y must be done numerically. The S (8)0 gluon fragmentation function is given by [35–37] D S (8)0 g → ψ ( z, m c ) = 5 α s (2 m c )96 m c (cid:104)O ψ ( S (8)0 ) (cid:105) (cid:0) z − z + 2(1 − z ) log(1 − z ) (cid:1) , (A11)and the P (8) J gluon fragmentation function is given by D P (8) J g → ψ ( z, m c ) = 5 α s (2 m c )12 m c (cid:104)O ψ ( P (8)0 ) (cid:105) (A12) × (cid:18) δ (1 − z ) + 1(1 − z ) + + 13 − z − z ) − (1 − z )(8 − z )8 (cid:19) . Here we have summed over J = 0 , , (cid:104)O ψ ( P (8) J ) (cid:105) = (2 J + 1) (cid:104)O ψ ( P (8)0 ) (cid:105) . The S (1)1 charm quarkfragmentation function is [34], D S (1)1 c → ψ ( z, m c ) = 32 α s (2 m c )81 (cid:104) O ψ ( S (1)1 ) (cid:105) m c ( z − ( z − z (5 z − z + 72 z − z + 16) . (A13)The moments of the color-octet gluon fragmentation functions can be computed analytically. Defining˜ D g → ψ ( N, m c ) = (cid:90) dzz N − D g → ψ ( z, m c ) , (A14)we have ˜ D S (8)1 g → ψ ( N, m c ) = πα s (2 m c )24 m c (cid:104) O ψ ( S (8)1 ) (cid:105) , (A15)˜ D S (8)0 g → ψ ( N, m c ) = 5 α s (2 m c )96 m c (cid:104)O ψ ( S (8)0 ) (cid:105) (cid:20) N + N ( N + 1) ( N + 2) − H N N ( N + 1) (cid:21) , (A16)˜ D P (8) J g → ψ ( N, m c ) = 5 α s (2 m c )12 m c (cid:104)O ψ ( P (8)0 ) (cid:105) (A17) × (cid:20)
188 + 191 N + 49 N + 4 N N + 1) ( N + 2) − N + 10 N + 134 N ( N + 1) H N (cid:21) . The fragmentation function is evolved using the standard DGLAP evolution, µ ∂∂µ D i ( z, µ ) = α s ( µ ) π (cid:88) j (cid:90) z dyy P i → j ( z/y, µ ) D j ( y, µ ) . (A18)2These equations are solved analytically in moment space, and then the fragmentation functions at the scale µ J areobtained by numerically evaluating the inverse Mellin transform. In our calculations, q = c and the mixing betweenthe gluon and c quark fragmentation function is only relevant for the S (1)1 channel.It is useful to have analytic expressions for the moments of the matching coefficients; these are given by:˜ J gg ( E, R, N, µ ) = (cid:90) dz z N − J gg ( E, R, z, µ )2(2 π ) = 1 + α s C A π (cid:18) L + P Ngg L + H N − − π
24 + H N − , + 2 G N − + F N − − F N − + F N − F N +1 (cid:19) , (A19)˜ J qq ( E, R, N, µ ) = (cid:90) dz z N − J qq ( E, R, z, µ )2(2 π ) = 1 + α s C F π (cid:18) L + P Nqq L + H N − + H N +1 − π
24 + H N − , + H N +1 , G N − + G N +1 (cid:19) , (A20)˜ J gq ( E, R, N, µ ) = (cid:90) dz z N − J gq ( E, R, z, µ )2(2 π ) = α s T F π (cid:18) P Nqg L + 1( N + 1)( N + 2) + F N +1 − F N + 12 F N − (cid:19) , (A21)˜ J qg ( E, R, N, µ ) = (cid:90) dz z N − J qg ( E, R, z, µ )2(2 π ) = α s C F π (cid:18) P Ngq L + 12( N + 1) + F N − − F N − + 12 F N (cid:19) , (A22)where H N is the harmonic number, H N, is the generalized harmonic number of order 2, and P Nij , F N , and G N aregiven by F N = 2 N + 1 − H N +1 + N (cid:88) j =1 j j − log 2 , (A23) G N = N (cid:88) j =1 j j − N (cid:88) k =1 k k (cid:88) j =1 j j − log 2 , (A24) P Ngg = 2 (cid:18) − H N + 1 N − − N + 1 N + 1 − N + 2 (cid:19) , (A25) P Nqq = − H N +1 + 1 N + 1 N + 1 , (A26) P Ngq = N + N + 2 N ( N − , (A27) P Nqg = N + N + 2 N ( N + 1)( N + 2) . (A28)Note that F N = O (cid:18) N (cid:19) ,G N = π
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