Quarkonium Polarization and the Long Distance Matrix Elements Hierarchies using Jet Substructure
QQuarkonium polarization and the long distance matrix elementshierarchies using jet substructure
Lin Dai ∗ and Prashant Shrivastava † Pittsburgh Particle Physics Astrophysics and Cosmology Center (PITT PACC)Department of Physics and Astronomy, Universityof Pittsburgh, Pittsburgh, Pennsylvania 15260, USA Department of Physics, Carnegie Mellon University, Pittsburgh, Pennsylvania, 15213, USA
Abstract
We investigate the quarkonium production mechanisms in jets at the LHC, using the FragmentingJet Functions (FJF) approach. Specifically, we discuss the jet energy dependence of the
J/ψ production cross section at the LHC. By comparing the cross sections for the different NRQCDproduction channels ( S [8]0 , S [8]1 , P [8] J , and S [1]1 ), we find that at fixed values of energy fraction z carried by the J/ψ , if the normalized cross section is a decreasing function of the jet energy,in particular for z > .
5, then the depolarizing S [8]0 must be the dominant channel. This makesthe prediction made in [Baumgart et al., JHEP , 003 (2014)] for the FJF’s also true for thecross section. We also make comparisons between the long distance matrix elements extracted byvarious groups. This analysis could potentially shed light on the polarization properties of the J/ψ production in high p T region. ∗ E-mail:[email protected] † E-mail:[email protected] a r X i v : . [ h e p - ph ] A ug . INTRODUCTION Analyzing quarkonium production in jets provides a new way of probing the physicsinvolved in their production. Recent developments include the LHCb measurements of
J/ψ production in jets [1] and the related analyses [2–4]. A factorization theorem basedon Non-Relativistic QCD (NRQCD) can be used to calculate the cross section for J/ψ production [5, 6]. Due to the large mass of the charm quark ( m c ), the short distanceproduction of the cc pair can be calculated perturbatively while the non-perturbative physicsof the hadronization into a J/ψ is captured by the long distance matrix elements (LDMEs)of the relevant production channels ( S [8]0 , S [8]1 , P [8] J , and S [1]1 ). The predictive power ofthe theory is then predicated on our knowledge of these LDMEs. Different groups haveextracted these matrix elements by using various fits to the data [10–13] but have arrived atvery different values. Currently the NRQCD factorization theorem can consistently fit theunpolarized J/ψ production cross section.The cc pair produced by the fragmentation of a nearly on-shell gluon should inherit thetransverse polarization of the gluon. Due to the spin symmetry of the leading order NRQCDLagrangian, this polarization remains intact during the non-perturbative hadronization pro-cess (up to power corrections) [15, 16]. At leading order in α s , only the S [8]1 channel forthe gluon contributes among the octet channels and since the color octet contribution isexpected to dominate at high p T [17], the J/ψ meson should be produced with significantpolarization at high p T . However this prediction of NRQCD is at odds with the measure-ments of the J/ψ polarization [18–20]. Understanding this polarization puzzle is one of themost important challenges in quarkonium physics [21].A method based on jet substructure techniques to study the different production mecha-nisms of the
J/ψ was proposed in Ref. [22]. By using the properties of the Fragmenting JetFunctions (FJF) [23], it is predicted in Ref. [22] that for a jet of energy E and cone size R ,containing a J/ψ with energy fraction z ( z = E J/ψ /E ), if the FJF is a decreasing functionof the jet energy, then the dominant contribution to the J/ψ production at high p T shouldbe the depolarizing S [8]0 channel and hence, if confirmed by the data, this would resolve thepolarization puzzle.In this work, we investigate how the predictions of the diagnostic tool introduced inRef. [22] are affected by inclusion of the hard scattering effects. To do this, we calculatethe total production cross section for the J/ψ . This should make the comparison of theorywith experiments much simpler since the cross section can be directly measured. In orderto make the distinction between various production channels, we calculate the cross section NRQCD is an effective theory with a double expansion in the relative velocity v of the heavy quark andanti-quark bound state and the strong coupling constant α s [5–9]. For
J/ψ production via gluon fragmentation in NRQCD, the S [1]1 contribution is leading order in the v expansion since the color octet channels are suppressed by v . But the S [1]1 is suppressed relative to the S [8]1 channel by power of α s . The matching onto P [8] J and S [8]0 is down by α s compared to S [8]1 buttheir LDMEs are of the same order as S [8]1 in v . An alternate power counting for charmonium productionis formulated in Ref. [14]. z while in the other case we normalize by using the1-jet inclusive cross section. Additionally we also make comparisons between the LDMEsextracted by various groups.The main result of our paper is that the prediction made in Ref. [22], regarding theshapes of the FJF’s, is also true for the cross section. By using a combination of differentlynormalized cross sections, we can break the degeneracy of the production channels and isolatethe dominant contribution to the J/ψ production at high p T . Our results show that if thenormalized cross section is a decreasing function of the jet energy at large z , in particularfor z > .
5, then the S [8]0 channel dominates at high p T and this prediction should be easilyverifiable with the LHC data. A recent work [2] also proposed using observables similar toours to probe the J/ψ production mechanisms. II. THE FRAGMENTING JET FUNCTIONS
We briefly review the factorization theorem for the production of
J/ψ [23–29] beforemoving onto our main results in the next section. We consider the process pp → dijets at √ s = 13 TeV and integrate over one of the jets, assuming that the other jet contains anidentified J/ψ . The dijet cross section [23] with one jet of energy E , cone size R and a J/ψ in the jet carrying an energy fraction z , is schematically of the form dσdEdz = (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 process, f a/p and f b/p are the parton distribution functions(PDF), J j is the jet function for the jet not containing the J/ψ , S is the soft functionand G ψi ( E, R, z, µ ) is the FJF for the jet containing the
J/ψ . The parton i can be a gluon,charm or an anti-charm (contributions of the other partons are suppressed). We are inter-ested in the E and z dependence of the cross section, which comes from the hard function(including PDFs) and the FJF. We integrate over the jet originating from the parton j sothe jet function J j enters the cross section multiplicatively. The soft function S does notaffect G ψi ( E, R, z, µ ), R , E and z (up to power corrections) [22] and so it also enters thecross section multiplicatively. Hence both the jet function J j and the soft function S givean overall normalization to the cross section and are ignored in the rest of our analysis. InRef. [22], the hard function was not included but here we calculate the normalized crosssection, including both the charm quark and gluon contributions, and account for its E dependence. Ref. [2] differentiates between the NRQCD global fits based on inclusive
J/ψ cross section and suggestsusing the polarization measurements of
J/ψ meson produced in the jets as a way of constraining the heavyquarkonium production mechanisms. J ij ( E, R, z, µ )and the fragmentation function D j → ψ : G ψi ( E, R, z, µ ) = (cid:90) z dyy J ij ( E, R, y, µ ) D j → ψ (cid:16) zy , µ (cid:17)(cid:16) O (cid:16) m ψ E tan ( R/ (cid:17)(cid:17) . (2)The collection of NRQCD based fragmentation functions D j → ψ used in this paper can befound in Ref. [22].Large logarithms in J ij ( E, R, z, µ ) are minimized at the scale µ = 2 E tan( R/ − z )and can be easily resummed using the jet anomalous dimension [27]. But we do not considerthis resummation in this work since for us, 1 − z ∼ O (1) [22]. Instead we evaluate the PDFsand J ij ( E, R, z, µ ) at the jet scale µ J = 2 E tan( R/
2) and evolve the fragmentation functionfrom 2 m c to the scale µ J using the Dokshitzer-Gribov-Lipatov-Altarelli-Parisi (DGLAP)equation, 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 ) , (3)where P i → j ( z/y, µ ) are the QCD splitting functions. We consider mixing between the charmquark and gluon splitting functions only for the S [1]1 channel. To leading order in α s , itcan be shown that [22] G ψi ( E, R, z, µ J )2(2 π ) → D i → ψ ( z, µ J ) + O ( α s ( µ J )) . (4)Later in III B, we will also consider the 1-jet inclusive cross section. This is calculatedby replacing the FJF in Eq. (1) with the jet function for a cone-type algorithm [30]. TheFJFs are defined in Ref. [23] so that the sum over all possible fragmentations of a partoninto hadrons equals the inclusive jet function. J i ( E, R, µ ) = 12 (cid:88) h (cid:90) dz (2 π ) z G hi ( E, R, z, µ ) . (5)For further details about these calculations we refer the reader to Ref. [22]. Throughout thispaper we choose m c = 1 . R = 0 . III. DISCUSSION OF THE
J/ψ
PRODUCTION MECHANISMS
In this section, we discuss the predictions for
J/ψ production in jets using the LDMEsextracted by various groups and reveal some generic features that are independent of theseextractions. The LDMEs we use in this paper are summarized in Table I. Refs. [11, 12] usea global fit to 194 data points from 26 data sets and predict significant polarization of the The charm quark fragmentation into a
J/ψ is dominated by the S [1]1 channel because the color singletand octet contributions start at same order in α s but the color octet channels are suppressed in the v expansion. O J/ψ ( S [8]1 ) (cid:105) (cid:104)O J/ψ ( S [8]0 ) (cid:105) (cid:104)O J/ψ ( P [8]0 ) (cid:105) /m c (cid:104)O J/ψ ( S [1]1 ) (cid:105)× − GeV × − GeV × − GeV × GeV Bodwin et al. Ref. [10] 1 . ± . . ± . . ± .
51 1 . . ± .
059 4 . ± . − . ± .
10 1 . . ± .
12 8 . ± .
98 0 . ± .
21 1 .
60 80 100 120 140010203040 60 80 100 120 1400.00.20.40.60.81.01.2
FIG. 1. Cross sections for inclusive gluon and charm jets at the LHC. The center of mass energyis √ s = 13 TeV. J/ψ in the high p T region, which contradicts the measurements at the Tevatron [18] and theLHC [19, 20]. The extractions in Refs. [10, 13] focus on the high p T region and attempt tosolve the polarization puzzle. A. Normalized
J/ψ production cross section
To discuss the dependence of
J/ψ production on the associated jet energy, we use anormalized differential cross section defined as d ˜ σ i dEdz ≡ dσ i dEdz (cid:44)(cid:88) i (cid:90) z max z min d z dσ i dEdz , (6)and d ˜ σdEdz ≡ (cid:88) i d ˜ σ i dEdz , (7)where i denotes different J/ψ production channels (i.e., for the gluon initiated jets i ∈{ S [8]0 , S [8]1 , P [8] J , S [1]1 } and for the charm initiated jets i = S [1]1 ), and dσ i /dEdz is definedin Eq. (1).In Eq. (6), z min ( z max ) should not be too close to 0 (1) where the factorization breaksdown. The motivation for studying this normalized cross section is that we want to isolatethe properties of quarkonium fragmentation in jets from the hard process that generatesthe jet initiating parton’s. Fig. (1) shows the energy distributions of the hard process for5 ���������� �� ��� � ��� � � � �� �� � �� �� ��� ��� �������� �� �� ��� ��� �������� �� �� ��� ��� �������� � � �� � � �� �� �� ��� ��� ����������������������������������� �� �� ��� ��� ����������������������������������� �� �� ��� ��� ������������������� � � � � � �� �� � �� �� ��� ��� ������������������ �� �� ��� ��� ������������������ �� �� ��� ��� ������������������ � � �� � � �� �� �� ��� ��� ��������������������������� �� �� ��� ��� ��������������������������������� �� �� ��� ��� ����������������������� FIG. 2. Cross sections for the different production channels at z = 0 .
4, 0 .
5, and 0 . dσ i /dEdz ), with thesecond row showing plots normalized to unit area for a better visualization of the shapes, i.e., wemultiply each curve of the first row by an appropriate constant to get the corresponding curve inthe second row. Similar plots for the normalized cross section ( d ˜ σ i /dEdz ) are shown in the thirdand fourth row. The LDMEs are from Butenschoen et al.’s extractions [11]. gluon and charm jets at the LHC . For all the figures in this paper, we fix the center of massenergy to be √ s = 13 TeV.Fig. (2) shows the comparison of the normalized (Eq. (6)) and unnormalized cross sections(Eq. (1)), where the LDMEs from Ref. [11, 12] are used with z min = 0 . z max = 0 . We consider leading order partonic cross sections convoluted with PDF [31, 32], which includes thefollowing processes: gg → gg , gq ( q ) → gq ( q ), qq → gg , gg → cc , gc ( c ) → gc ( c ), cc → cc , c c → c c , cq ( q ) → cq ( q ), cq ( q ) → cq ( q ), qq → cc , cc → cc . ����� �� ��� ���������� �� ��� ���� �� ��� �� �� ��� ��� ������������������������ �� �� ��� ��� ������������������ �� �� ��� ��� ������������������ FIG. 3. Total normalized cross section (i.e. d ˜ σ/dEdz defined in Eq. (7)) with error bands. Red,black, and blue curves correspond to Bodwin et al. [10], Butenschoen et al. [11, 12], and Chao etal.’s [13] extractions, respectively. Corresponding plots for the LDMEs of Ref. [10] and Ref. [13] are shown in appendix Aand B respectively. We would like to emphasize the fact that both the unnormalized andnormalized cross sections are directly measurable in experiments, although the normalizedcross section has a better resolving power than the unnormalized cross section. In particular,the unnormalized cross section is a decreasing function of E for all the production channelsdue to the decreasing nature of the hard process, while the normalized cross section can bean increasing function for certain production channels due to the properties of their FJF’s.A measurement of the normalized cross section (Eq. (6)) for z > .
5, can help identifyboth the dominant channel and the favored set of LDMEs. From Fig. (2), we can see thatif d ˜ σ i /dEdz turns out be a decreasing function of the jet energy for z > .
5, then thedepolarizing S [8]0 should be the dominant channel. We find this result to be true for LDMEextractions of Ref. [10] as well (see appendix A).In Fig. (3), we show the jet energy dependence of the total normalized cross sections(Eq. (7)) based on different LDME extractions. The error bands are purely due to theLDME uncertainties, that is, we consider the uncertainty due to each LDME and sum byquadrature to obtain the total uncertainties . It can be seen in Fig. (3) that as z goes from0 . .
6, the shapes change from an increasing function to a decreasing function. Howeversince different extractions have distinct slopes, this observable has the potential power totest these extractions at the LHC. A different choice of ( z min , z max ) does not change ourarguments as we demonstrate in appendix C.We also consider the possibility that for z > .
3, the contribution of the S [1]1 channelto the J/ψ production is negligible for the p T range considered here [10, 17, 21]. We testthis by ignoring the S [1]1 channel contribution to the normalization and arrive at the same To obtain the error bands corresponding to the extraction from Bodwin et al., we have used the errorcorrelation matrix not shown in the original paper [33]. ����� �� ��� ���������� �� ��� ���� �� ��� �� �� ��� ��� ����������������������� �� �� ��� ��� ����������������������������������� �� �� ��� ��� ����������������������������������� FIG. 4. Total normalized cross section (i.e. d ˆ σ/dEdz defined in Eq. (9)) with error bands. Red,black, and blue curves correspond to Bodwin et al. [10], Butenschoen et al. [11, 12], and Chao etal.’s [13] extractions, respectively. conclusion of S [8]0 being the dominant contribution if the normalized cross section decreaseswith jet energy for z > . B. Normalization using 1-jet inclusive cross section
We now normalize the cross section in such a way that the denominator is independent ofthe LDMEs. This allows us to make a direct comparison of our results to those of Ref. [22].The normalization is defined as d ˆ σ i dEdz ≡ dσ i dEdz (cid:44) dσ J dE , (8)and d ˆ σdEdz ≡ (cid:88) i d ˆ σ i dEdz , (9)where dσ i /dEdz is the same as that in Eq. (1) and dσ J /dE is the 1-jet inclusive crosssection . Note that the z -dependence of Eq. (8) comes only from the G J/ψi ( E, R, z, µ ) inEq. (1).Fig. (4) shows the total
J/ψ production cross section based on Eq. (9). The key featureof this plot is that the arguments given Ref. [22] based on the FJFs are also true for thecross section (see Fig. 6 in Ref. [22]) . Specifically, when z > .
5, the shapes of the curvesare very different for the extraction based on a global fit (black curves) and the other twobased on fit to high p T region (red and blue curves). Since the extractions from the global This includes the contributions of gluon, light quarks, charm and bottom jets. The definition of Eq. (8) is essentially the same as the jet fragmentation function introduced in Ref. [2],except that we have integrated the jet pseudorapidity over the region | η J | < . To facilitate direct comparison of our Fig. (4) to Fig. (6) in Ref. [22], we make plots for z = 0 . , . . �� � � � � � � � � �� �� ��� ��� ��� - ������������ �� �� ��� ��� ��� - ���� - ������������������������ �� �� ��� ��� ��� - ���� - ������������������������ � � � � � � � �� � � � � � � � �� �� ��� ��� ������������������� �� �� ��� ��� ������������������� �� �� ��� ��� ����������������������������������� � � � � � � � � � �� �� ��� ��� ��� - ���� - ���� - ������������������������ �� �� ��� ��� ��� - ���� - �������������������� �� �� ��� ��� ��� - ���� - �������������������� � � � � � �� �� � � �� �� �� ��� ��� ��������������������������������������� �� �� ��� ��� ��������������������������������������� �� �� ��� ��� ��������������������������������������������������� FIG. 5. Comparisons of the production channels for various LDMEs using Eq. (8). Last row showsthe plots normalized to unit area. This is indicated by 1 / ˆ σ i for the cross section label in the fourthrow, which also cancels the LDME dependence of the numerator. fit and high p T fit give rise to different slopes for the J/ψ production cross section, one cantest which set of the LDME extractions are preferred by measuring these slopes. Note thatbecause our results are for the cross section, all the curves have positive values, in contrastto the gluon FJF for the LDMEs of Ref. [13] (shown in Fig. (6) of Ref. [22]) which becamenegative at large energies.In Fig. (5), we plot the E dependence of the individual J/ψ production channels for thedifferent LDMEs using Eq. (8). We find that if the measurements of the observable definedin Eq. (8) results in a cross section which is a decreasing function of the jet energy for z > .
5, then the S [8]0 channel should have an anomalously large contribution to the J/ψ production. The fourth row in Fig. (5), with the curves normalized to unit area, clearly9hows that only S [8]0 channel is a decreasing function of jet energy for z > .
5. Hence averification of our results in this and the previous section will give strong evidence in favorof the depolarizing S [8]0 channel being dominant at high p T and provide a clear explanationfor the lack of polarization in the J/ψ production at high p T . Note that in the fourth row ofFig. (5), the LDME dependence gets canceled due to normalization to unit area and so theprediction for S [8]0 channel being dominant at high p T is independent of any specific LDMEextractions.To conclude this section, we mention a few things about the normalization conventionsin Eq. (6) and Eq. (8). First of all, both the normalizations can be directly tested inexperiments. Also since both the numerator and denominator of Eq. (6) depend on theLDMEs, they are statistically correlated and hence the width of error bands in Fig. (3) isreduced. However, Eq. (8) does not have such a correlation since the jet cross section usedfor the normalization is independent of the LDMEs. Indeed, if we look at Bodwin et al.’sextraction near z = 0 . E = 100 GeV, the ratio of the width of error band to the centervalue is ∼
4% in Fig. (3) and ∼
30% in Fig. (4). On the other hand, in both Fig. (3) andFig. (4), the shapes of blue and red curves (high p T fit) are in contrast to the black curve(global fit). IV. CONCLUSION
In this paper, we have looked at the total cross section for
J/ψ production at the LHC byusing the FJF approach. We make comparisons between the different NRQCD productionchannels for the
J/ψ . We show that if for z > . S [8]0 should be the dominant production channelat high p T . We find this to be true for two sets of normalized cross sections. Our resultsconfirm that the prediction made in Ref. [22] regarding the decreasing nature (with E ) of theFJF for S [8]0 channel, does not change by inclusion of the hard scattering effects. Using ournormalized cross sections, one can also test which set of the LDME extractions are favored. ACKNOWLEDGMENTS
The authors would like to thank Adam Leibovich and Ira Rothstein for their guidance andcomments on the manuscript. We would also like to thank James Russ for useful discussions.LD was supported in part by NSF grant PHY-1519175.10 ppendix A: Unnormalized and Normalized cross sections for Bodwin et.al
Fig. (6) shows the unnormalized (Eq. (1)) and normalized cross section (Eq. (6)) forBodwin et al.’s LDME extractions [10]. The P [8] J channel contribution is negative, which isa feature of these LDMEs as it leads to a cancellation between the S [8]1 and P [8] J channels,making the depolarizing S [8]0 the dominant production channel of J/ψ for z > . ������� �� ��� � ��� � � � �� �� � �� �� ��� ��� ��� - � - ������ �� �� ��� ��� ��� - � - ������ �� �� ��� ��� ��� - ������ � � �� � � �� �� �� ��� ��� ����������������������������������� �� �� ��� ��� ����������������������������������� �� �� ��� ��� ������������������� � � � � � �� �� � �� �� ��� ��� ��� - ��� - ��������������� �� �� ��� ��� ��� - ��������������� �� �� ��� ��� ��� - ������������ � � �� � � �� �� �� ��� ��� ��������������������������������������� �� �� ��� ��� ��������������������������������������� �� �� ��� ��� ����������������������� FIG. 6. Unnormalized and normalized cross sections for Bodwin et al. extractions [10]. Theconventions followed are same as in Fig. (2). ppendix B: Unnormalized and Normalized cross sections for Chao et.al Fig. (7) shows the unnormalized (Eq. (1)) and normalized cross section (Eq. (6)) forChao et al.’s LDME extractions [13]. Similar to Bodwin et al., these LDMEs result in acancellation between the S [8]1 and P [8] J channels. ���� �� ��� � ��� � � � �� �� � �� �� ��� ��� ��� - � - ������ �� �� ��� ��� ��� - � - ����� �� �� ��� ��� ��� - ����� � � �� � � �� �� �� ��� ��� ����������������������������������� �� �� ��� ��� ����������������������������������� �� �� ��� ��� ������������������� � � � � � �� �� � �� �� ��� ��� ��� - � - ����� �� �� ��� ��� ��� - ���� �� �� ��� ��� ��� - ���� � � �� � � �� �� �� ��� ��� ���������������������������� �� �� ��� ��� ���������������������������� �� �� ��� ��� ��������������������������������� FIG. 7. Unnormalized and normalized cross sections for Chao et al. extractions [13]. The conven-tions followed are same as in Fig. (2). ppendix C: Insensitivity to z min and z max Comparison of the normalized cross sections (Eq. (6)) for different values of z min and z max is shown. This confirms that the discussion in section III A is not sensitive to ( z min , z max )since the shapes of different LDMEs do not change. For validity of the factorization formulaEq. (1), we don’t pick z min too close to 0 and z max too close to 1. ������ �� ��� ���������� �� ��� ���� �� ��� ( ������� ) ( ������� ) ( ������� )( ������� ) ( ������� ) ( ������� ) �� �� ��� ��� ��������������������� �� �� ��� ��� ��������������� �� �� ��� ��� ��������������������� �� �� ��� ��� ����������������������������� �� ��� ��� ������������������ �� �� ��� ��� ������������ �� �� ��� ��� ������������������ �� �� ��� ��� �������������������������� �� ��� ��� ��������������� �� �� ��� ��� ������������ �� �� ��� ��� ��������������� �� �� ��� ��� ������������������������ z min , z max ) = (0 . , .
8) and the dashed curves ( z min , z max ) =(0 . , . ppendix D: Normalization using only color octet channels Fig. (9) shows the cross section for the different
J/ψ production channels based on theLDMEs in Ref. [10] and Ref. [11, 12] with the contribution of S [1]1 channel ignored in Eq. (6),i.e., setting (cid:104)O J/ψ ( S [1]1 ) (cid:105) to 0. Since S [8]0 channel (green curves) has very different slopes forthe two LDMEs, if the S [8]0 channel dominates at high p T , then one can distinguish betweenthese two extractions. We don’t include Chao et al.’s extractions [13] because it gives riseto a negative total cross section and so one can not ignore the color singlet contribution. �� �� ��� ��� ��� - ���� �� �� ��� ��� ��� - � - ����� �� �� ��� ��� ��� - � - ����� ������ �� ��� �� �� ��� ��� ���������������������������� �� �� ��� ��� ���������������������������� �� �� ��� ��� ���������������������������� ������ �� ��� �� �� ��� ��� ������������ �� �� ��� ��� ������������������������ �� �� ��� ��� ��������������������� ����������� �� ��� �� �� ��� ��� ��������������������������������������� �� �� ��� ��� ����������������������� �� �� ��� ��� ���������������������������� ����������� �� ��� FIG. 9. Cross section normalized by ignoring the S [1]1 channel contribution in Eq. 6. The secondand fourth row are obtained by normalizing the curves in the first and third row to unit arearespectively. ppendix E: Lower z plots Fig. (10) shows the
J/ψ production cross section (Eq. (8)) at lower z values for all thethree LDME extractions [10–13] used in this paper. � �� � � � � � � � � �� �� ��� ��� ��� - ���� - �������������������� �� �� ��� ��� ��� - ���������������� �� �� ��� ��� ��� - ������������ � � � � � � � �� � � � � � � � �� �� ��� ��� ��������������� �� �� ��� ��� ��������������������������� �� �� ��� ��� ������������������� � � � � � � � � � �� �� ��� ��� ��� - ���� - ���������������� �� �� ��� ��� ��� - ���������������� �� �� ��� ��� ��� - ���� - ���� - ������������������������ � � � � � �� �� � � �� �� �� ��� ��� ����������������������� �� �� ��� ��� ��������������������������������������� �� �� ��� ��� ��������������������������������������� FIG. 10. Lower z plots for the cross section (Eq. (8)). The conventions followed are same as thosein Fig. (5).
1] R. Aaij et al. (LHCb), Phys. Rev. Lett. , 192001 (2017), arXiv:1701.05116 [hep-ex].[2] Z. B. Kang, J. W. Qiu, F. Ringer, H. Xing, and H. Zhang, Phys. Rev. Lett. , 032001(2017), arXiv:1702.03287 [hep-ph].[3] R. Bain, L. Dai, A. K. Leibovich, Y. Makris, and T. Mehen, Phys. Rev. Lett. , 032002(2017), arXiv:1702.05525 [hep-ph].[4] I. Belyaev, A. V. Berezhnoy, A. K. Likhoded, and A. V. Luchinsky, (2017), arXiv:1703.09081[hep-ph].[5] G. T. Bodwin, E. Braaten, and G. P. Lepage, Phys. Rev.
D51 , 1125 (1995), [Erratum-ibid.D 55, 5853 (1997)], arXiv:hep-ph/9407339 [hep-ph].[6] W. E. Caswell and G. P. Lepage, Phys.Lett.
B167 , 437 (1986).[7] N. Brambilla, A. Pineda, J. Soto, and A. Vairo, Nucl.Phys.
B566 , 275 (2000), arXiv:hep-ph/9907240 [hep-ph].[8] M. E. Luke, A. V. Manohar, and I. Z. Rothstein, Phys.Rev.
D61 , 074025 (2000), arXiv:hep-ph/9910209 [hep-ph].[9] I. Z. Rothstein, P. Shrivastava, and I. W. Stewart, (Unpublished).[10] G. T. Bodwin, H. S. Chung, U.-R. Kim, and J. Lee, Phys. Rev. Lett. , 022001 (2014),arXiv:1403.3612 [hep-ph].[11] M. Butenschoen and B. A. Kniehl, Phys. Rev.
D84 , 051501 (2011), arXiv:1105.0820 [hep-ph].[12] M. Butenschoen and B. A. Kniehl, Mod. Phys. Lett.
A28 , 1350027 (2013), arXiv:1212.2037.[13] K. T. Chao, Y. Q. Ma, H. S. Shao, K. Wang, and Y. J. Zhang, Phys. Rev. Lett. , 242004(2012), arXiv:1201.2675 [hep-ph].[14] S. Fleming, A. K. Leibovich, and I. Z. Rothstein, Phys.Rev.
D64 , 036002 (2001), arXiv:hep-ph/0012062 [hep-ph].[15] P. Cho and M. B. Wise, Phys.Lett.
B346 , 129 (1995), arXiv:hep-ph/9411303 [hep-ph].[16] M. Beneke and I. Z. Rothstein, Phys.Lett.
B372 , 157 (1996), [Erratum-ibid. B389, 769 (1996)],arXiv:hep-ph/9509375 [hep-ph].[17] M. Beneke and I. Z. Rothstein, Phys.Rev.
D54 , 2005 (1996), [Erratum-ibid. D54, 7082 (1996)],arXiv:hep-ph/9603400 [hep-ph].[18] A. Abulencia et al. (CDF), Phys. Rev. Lett. , 132001 (2007), arXiv:0704.0638 [hep-ex].[19] S. Chatrchyan et al. (CMS), Phys. Lett. B727 , 381 (2013), arXiv:1307.6070 [hep-ex].[20] R. Aaij et al. (LHCb), Eur. Phys. J.
C73 , 2631 (2013), arXiv:1307.6379 [hep-ex].[21] N. Brambilla et al., Eur.Phys.J.
C71 , 1534 (2011), arXiv:1010.5827 [hep-ph].[22] M. Baumgart, A. K. Leibovich, T. Mehen, and I. Z. Rothstein, JHEP , 003 (2014),arXiv:1406.2295 [hep-ph].[23] M. Procura and I. W. Stewart, Phys. Rev.
D81 , 074009 (2010), [Erratum-ibid. D83, 039902(2011)], arXiv:0911.4980 [hep-ph].[24] X. Liu, Phys.Lett.
B699 , 87 (2011), arXiv:1011.3872 [hep-ph].[25] A. Jain, M. Procura, and W. J. Waalewijn, JHEP , 035 (2011), arXiv:1101.4953 [hep-ph].[26] A. Jain, M. Procura, and W. J. Waalewijn, JHEP , 132 (2012), arXiv:1110.0839 [hep-ph].[27] M. Procura and W. J. Waalewijn, Phys. Rev.
D85 , 114041 (2012), arXiv:1111.6605 [hep-ph].[28] A. Jain, M. Procura, B. Shotwell, and W. J. Waalewijn, Phys.Rev.
D87 , 074013 (2013), no.7,arXiv:1207.4788 [hep-ph].[29] C. W. Bauer and E. Mereghetti, arXiv:1312.5605 [hep-ph].
30] S. D. Ellis, A. Hornig, C. Lee, C. K. Vermilion, and J. R. Walsh, JHEP , 101 (2010),arXiv:1001.0014 [hep-ph].[31] R. K. Ellis, W. J. Stirling, and B. R. Webber,
QCD and Collider Physics (Cambridge Uni-versity Press, 2003).[32] A. D. Martin, W. J. Stirling, R. S. Throne, and G. Watt, Eur.Phys.J.
C63 , 189 (2009),arXiv:0901.0002 [hep-ph].[33] G. T. Bodwin, private communication., 189 (2009),arXiv:0901.0002 [hep-ph].[33] G. T. Bodwin, private communication.