Study of Color Octet Matrix Elements Through $J/\psi$ Production in $e^{+}e^{-}$ Annihilation

In this paper, the color-octet long distance matrix elements are studied though the inclusive $J/\psi$ production in $e^{+}e^{-}$ annihilation within the non-relativistic QCD frame. The calculations are up-to next-to-leading order with the radiative corrections and relativistic corrections at the B-factory energy region and the near-threshold region of $4.6\sim5.6~{\rm GeV}$. A constraint of the long distance matrix elements ($\langle^1S_0^{8}\rangle$, $\langle^3P_0^{8}\rangle$) is obtained. Through our estimation, the P-wave color-octet matrix element ( $\langle0|^3P^8_0|0\rangle$ ) should be the order of $0.005m_c^2~{\rm GeV}^3$ or less. The constraint region is not compatible with the values of the long distance matrix elements fitted at hadron colliders.


Contents 1 Introduction
The CO (color octet) mechanism was introduced to describe the production and decay of heavy quarkonium in nonrelativistic quantum chromodynamics(NRQCD) [? ]. Differing from the case in the CS(color singlet) model, in a CO production process the intermediate pair of quark and antiquark can be created at short distances with CO and then formed the non color quarkonium at long distances by emitting or absorbing soft gluons. The processes at short distances are called as short-distance coefficients, which are scaled by the strong coupling constant α s and calculated perturbatively. The processes at long distances are called as LDMEs (long distance matrix elements), which are scaled by the relative velocity v and can not be calculated perturbatively. The LDMEs are universal, process-independence, and must be determined by experimental extraction, potential model, or lattice QCD calculations.
The NRQCD CO mechanism acquire some significant successes since it was proposed. The surplus production for transverse momentum of J/ψ and ψ′ at the Tevatron seems to be a powerful evidence in favor of CO mechanism [? ]. The contribution given by the LO (leading order) calculations in CS model counts for less than 5 percent of the experimental data on the unpolarized cross section. For the CO case, it introduced three extra phenomenological parameters, the LDMEs of CO states 3 S [8] 1 , 1 S [8] 0 , 3 P [8] 0 , which are fitted to the experimental data. Then the result of CO make the gap between CS LO prediction and the experiment data appeared. The calculated of inclusive direct J/ψ photon-production at NLO (next-toleading order) CO cross section at HERA [? ], and in the γ + γ → J/ψ + X production at DELPH [? ]also suggested CO mechanism is indeed realized the production mechanism.
However, the predictions within the CO model to the polarization of J/ψ hadron production are transverse polarization that much more contradicts with unpolarized result measured by the Tevatron[? ? ? ]. Many theoretical efforts were made to solve this puzzle, including the NLO radiative corrections for production [? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ] and polarization [? ? ? ? ? ? ]. And the NLO relativistic corrections to J/ψ hadronic production are considered too [? ? ? ]. But the three CO LDMEs fitting by the three groups were incompatible with each other's [? ? ? ]. To determine these values, several efforts were made including a global fit with the world's unpolarized data [? ], which are inclusive J/ψ production data from various hadron-production, photon-production, two-photon scattering, and e + e − annihilation [? ]. With the fit of the polarization data [? ], and the feed-down contributions [? ] are also considered. Unfortunately, their results were violently incompatible with others. More information about NRQCD and heavy quarkonium physics can be found in Ref. [? ] and the related papers.
The CO mechanism in heavy quarkonium production at e + e − process annihilation are not the same as the hadron-production case. In the e + e − annihilation at B-  [? ] . It seems that the CS may give the main contribution, and leave little room for CO state in the e + e − collider experiments. The CO LDMEs of e + e − collider process may be smaller than that expected at the hadron colliders. So the test of the university to the high-order LDMEs in NRQCD is still interesting and helpful to comprehend the production mechanism of heavy quarkonium, especially, the CO mechanism. At the same time, the exclusive cross sections e + e − → J/ψ + π + π − are measured by Belle [? ] and BESIII [? ].
In this paper, we concentrate on the CO pieces to the inclusive production of J/ψ and Υ production associated with LH (light hadrons) in e + e − annihilation. For the processe + e − → J/ψ + LH, there are two subprocess, e + e − →qq + J/ψ and e + e − →J/ψ + g, at parton level. The former process can be ignored at the energy of the B factories[? ] and we will discuss the latter process only. The octet signature of the latter process emerges near the endpoint region of the energy of J/ψ in the leading order calculation [? ]. Two CO intermediate states 1 S 8 0 and 3 P

The framework of calculation
This section introduces the framework of calculation for the relativistic corrections and the radiative corrections within the NRQCD factorization frame.
The amplitude for e + e − →J/ψ( 1 S 8 0 / 3 P 8 J ) + g is given by the following expression in NRQCD[? ? ] where the factor √ 2M H originates from the relativistic normalization factor [? ]. The d m are the short-distance coefficients and O m are the NRQCD operators which are defined as The short-distance coefficients can be obtained by matching the cc production process in NRQCD with that in full QCD.
The result in the second line of above expression was obtained applying the relativistic normalization of CO cc where E q = m 2 c + q 2 is the energy of c(c) in the J/ψ rest frame. Then the short-distance coefficients d m can be calculated by (2.5) Using Eq.(2.5) and Eq.(2.1), one will get the final expressions of the NRQCD amplitude where the LDMEs q 2m are defined as J/ψ|O m |0 2m / J/ψ|O 0 |0 2m . The perturbative QCD process M(cc( 1 S 8 0 / 3 P 8 J )+g) can be computed using the projection operator method where the normalization factor N L satisfy Where ǫ is the orbit polarization vector for P -wave state. The π 8 is the octet color projector and defined by π 8 = 3i;3j|8a = √ 2T a . The spin projectors P ssz for spin-singlet and spintriplet are given as the bellow expressions in terms of the momenta of the c andc in covariant form We can set p c = P/2 + q and pc = P/2 − q in the calculations, where P is the barycenter momentum of the cc pair, etc. the J/ψ's momentum.

The
Now, we first consider the amplitude expanded to next leading order in v 2 . We expand the J/ψ mass as[ and substitute this expression into Eq.(2.6) to expand to next leading order with omitting the higher-order terms To evaluate these expressions, we expand M cc + g / E q to the order of |q| 2 for 1 S 8 0 state (and |q| 3 for 3 P 8 J state). Notice that, in the amplitude expressions, |q| emerges in the forms of E q and the four momentum q. We now expand them to the order of |q| 3 . It's easy to abstain To expand q, we first write down the expression of J/ψ momentum P and q in the rest frame of J/ψ, where n is the unit direction vector. When bosting to an arbitrary frame, q is dependant of E q , etc. |q|, generally, so we can expand q as (2.14) Form Eq.(2.7), in the amplitude calculation, it needs to integrate over the azimuthal angle of q and it is convenient to compute the tensor integral as bellow Actually, since the terms of derivative of ε Lz and the tensor in the amplitude, it is not easy to compute the next leading order amplitude generally. In the cross section expansion, it is convenient to compute the derivative of squared amplitudes instead of that make the derivative then square the amplitudes. We now evaluate the differential cross sections taking advantage of the expansion of the amplitudes and Eq.(2.11) where the subscripts S, P note it's for 1 S 8 0 or 3 P 8 J states, and the means obtaining the sum of NRQCD amplitudes square M 2 over the final-state color and polarization and the average over the ones of the initial states. dφ 2 is the two-body phase space and using Eq.(2.10), we can also expand it to the next leading order as where λ(x, y, z) = x 2 + y 2 + z 2 − 2(xy + yz + xz). The scalar parameter r is defined as The final expressions of the NLO differential cross sections for the 1 S 8 0 and 3 P 8 J state are given as: ( 2.19) For the e + + e − → J/ψ + LH process, the total differential cross section can be computed by summing the dσ S and dσ P . Our LO results are consistent with Ref. [? ]. The k factors of NLO differential cross sections to the LO ones are shown as (2.21) And x represents cos 2 θ. With the approximation r → 0, the radios, the short-distance coefficients of relativistic corrections to leading order coefficients, are −5/6, −13/10, −11/10, and −7/10 for 1 S 8 0 , 3 P 8 0 , 3 P 8 1 , and 3 P 8 2 states, respectively. These results are consistent with Ref. [? ]. The contributions from phase space are determined by −r/(1 − r) v 2 , which are at the order of O(rv 2 ). We also calculated the NLO QCD corrections [? ]. The process e + e − → Υ + LH can be treated as the process e + e − → J/ψ + LH directly.
With the selection of the matrix elements v 2 ranging from 0.1 to 0.3, We could estimate the contributions from the relativistic corrections reduce the cross sections with α s corrections by 5% to 14%. The experimental data for the total cross section of the non-cc inclusive J/ψ production measured by Belle is [? ] σ[e + e − → J/ψ + X non−cc ] = 0.43 ± 0.13 pb (3.2) .
and got an upper limit as shown in Eq.(3.4). With the negative corrections from relativistic effects, the limit value can enhance by a factor by 8% to 20% and reaches to 2.4 × 10 −2 GeV 3 , i.e. Then we compare the region the LDMEs in Eq.(3.4) with the elements fitted with the hadron production processes in the paper [? ? ? ]. Butenschon and Kniehl fitted the inclusive J/ψ production from KEKB, LEPII, RHIC, HERA, Tevatron, and LHC with data selected as p T > 1 GeV for photon production and two-photon scattering and p T > 3 GeV for hadron production [? ]. Chao, Ma, Shao, Wang, and Zhang fitted the production and polarization of J/ψ from the Tevatron with p T > 7 GeV [? ]. Gong, Wan, Wang, and Zhang fitted the production and polarization of prompt J/ψ, ψ(2S), and χ cJ from the Tevatron and LHC [? ]. We show the value of LDMEs of J/ψ and ψ ′ in Table 1. And the LDMEs of 3 S 1 1 are set as : We can find that most of the LDMEs fitted with the hadron production processes does not in the region of Eq.(3.4). The σ[J/ψ + LH] measured by Belle [? ] give a very strong restriction to the LDMEs. We give the cross section as a function of energy in the following graphs. For all the figures below the solid lines present leading order cross section, the dash-dot lines are the O(α s , v 2 ) correction cross section, and the dash lines present the O(α s ) correction cross section. The blue area show the uncertainty of α s and v 2 where α s = 0.245 ± 0.03 and v 2 = 0.2 ± 0.1. Firstly we give the short distance of the cross section as a function of energy without any LDMEs in Figure 1. From this figure we know the short distance of the cross section 3 P [8] J is 10 times or more great than 1 S [8] 0 . So the fitted LDMEs of 3 P

[8]
J must be small. The contribution from 3 S [1] 1 are small, so it is ignored here. The cross section of the e + e − → cc( 1 S 8 0 ) + g and e + e − → cc( 3 P 8 J ) + g and the total cross section with the LDMEs fitted by three groups are listed in Table 1. Figure 2 and 3 show the cross section as a function of the energy of center of mass system, and the value of LDMEs are corresponding to the first line and the seconde line in Table 1 which were fitted by Butenschoen and Kniehl [? ]. In Figure 2 the LDMEs are fitted without the feed down contribution. But in Figure 3, the feed down contribution are taken into account. These two group of LDMEs have a negative value of 3 P 8 J . For the short distance of P-wave is very large, the total cross section give negative values. Figure 4 is the cross section correspond the LDMEs fitted by Gong, et al.[? ] in the last line of the Table 1. They consider the contribution from ψ(2s), and the LDMEs of 3 P 8 J is small and negative. The total cross section have a greater uncertainty than other groups. Figure 5 and 6 are the cross section with the LDMEs fitted by Chao, et al.[? ] correspond the third, the fourth, and the fifth lines in Table 1. The LDMEs in the third line of the Table 1 have a positive value of 3 P 8 J corresponding to Figure 5. And the other two line are two special cases, which are show in Figure 6. The total cross section have a larger value.
We give the value of the cross section in Table 2 correspond to the three groups LDMEs. From the table we know the contribution of 3 S 1 1 is very small as in Figure 1. The contribution from CS is small and can be neglected. However the short distance of the the process e + e − → J/ψ( 1 S 8 0 ) + g at BESIII in the Table 2 is about 500. When we product the LDMEs of the cross section of e + e − → J/ψ( 1 S 8 0 ) + g the value is enough to satisfy the experiment for all the three groups, and they do not need to consider the contribution from 3 P 8 J any more. On the other hand, the short distance of 3 P 8 J channel is so large up to 10 4 , so the fitted LDMEs of 3 P 8 J must be very small close to 0. We product the LDMEs of 3 P 8 J which fitted by the   Figure 1. The cross section of inclusive CO J/ψ production. three groups to the short distance part, show in Figure 2, 3, 4, 5, 6 and Table 2. A lot of sets of the cross sections are negative. We can see the LDMEs of 3 P 8 J not only small but also have a great uncertainty. So we can conclude that the LDMEs of P-wave are very small.

Summary
The NLO radiative corrections and relativistic corrections to the inclusive heavy quarkonium production in e + e − → J/ψ + LH and e + e − → Υ + LH are calculated within NRQCD frame. And the cross section with different the long matric elements has been given. We find that the inclusive heavy quarkonium production process in e + e − annihilation can give a strong restrictions on the CO LDMEs. And the LDMEs of P-wave channel for J/ψ(ψ ′ ) would be very small. More measurement of J/ψ(ψ ′ ) and Υ production associated with light hadron at BESIII, Belle, and BelleII can give more information about the CO LDMEs. J piece, and the total cross section, respectively. The LDMEs are chosen as the first row in Table 1 Figure 5. The cross section of inclusive CO J/ψ production without the feed down contribution in LDMEs. The LDMEs are chosen as the "set1" row in Table 1 Figure 6. The cross section of inclusive CO J/ψ production without the feed down contribution in LDMEs. The LDMEs are chosen as the "set2" and "set3" row in Table 1