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Relativistic correction to color Octet production at
hadron colliders

###### Abstract

The relativistic corrections to the color-octet hadroproduction at the Tevatron and LHC are calculated up to in nonrelativistic QCD factorization frame. The short distance coefficients are obtained by matching full QCD with NRQCD results for the partonic subprocess , and . The short distance coefficient ratios of relativistic correction to leading order for color-octet states , , and at large are approximately -5/6, -11/6, and -31/30, respectively, for each subprocess, and it is 1/6 for color-singlet state . If the higher order long distance matrix elements are estimated through velocity scaling rule with adopting and the lower order long distance matrix elements are fixed, the leading order cross sections of color-octet states are reduced by about a factor of at large at both the Tevatron and the LHC. Comparing with QCD radiative corrections to color-octet states, relativistic correction is ignored along with increasing. Using long distance matrix elements extracted from the fit to production at the Tevatron, we can find the unpolarization cross sections of production at the LHC taking into account both QCD and relativistic corrections are changed by about of that considering only QCD corrections. These results indicate that relativistic corrections may play an important role in production at the Tevatron and the LHC.

###### pacs:

12.38.Bx,12.39.St,13.85.Ni## I Introduction

Heavy quarkonium is an excellent candidate to probe quantum chromodynamics (QCD) from the high energy to the low energy regimes. Nonrelativistic QCD (NRQCD) factorization formalism was establishedBodwin:1994jh () to describe the production and decay of heavy quarkonium. In the NRQCD approach, the production and decay of heavy quarkonium is factored into short distance coefficients and long distance matrix elements(LDMEs). The short distance coefficients indicate the creation or annihilation of a heavy quark pair can be calculated perturbatively with the expansions by the strong coupling constant . However, the LDMEs, which represent the evolution of a free heavy quark pair into a bound state, can be scaled by the relative velocity between the quark and antiquark and obtained by lattice QCD or extracted from the experiment. is about for charmonium and about for bottomonium. The color-octet mechanism (COM) was introduced here. The heavy quark pair should be a color-singlet (CS) bound state at long distances, but it may be in a color-octet (CO) state at short distances. NRQCD had achieved great success since it was proposed. The COM was applied to cancel the infrared divergences in the decay widths of -wave Huang (); Petrelli:1997 () and -waveHe:2008xb (); Fan:2009cj () heavy quarkonium. However, difficulties were still encountered. The large discrepancy between the experimental data and the theoretical calculation of and unpolarization and polarization production at Tevatron is an interesting phenomenon that can verify NRQCD when solvedCDF:1992 (); arXiv:0704.0638 (). Theoretical prediction with COM contributions was introduced and was found to fit with the experimental data on production at TevatronBraaten:1994 (). However, the CO contributions from gluon fragmentation indicated that the was transversely polarized at large , which is inconsistent with the experimental dataCDF:1992 ().

The next-to-leading order (NLO) QCD corrections and other possible solutions for hadroproduction were calculated to resolve the hadronic production and polarization puzzleCampbell:2007 (); Gong:2008 (). The calculation enhanced the CS cross sections at large by approximately an order of magnitude. However, the large discrepancy between the CS predictions and experimental data remains unsolved. The relativistic correction to CS hadroproduction was insignificantFan:2009zq (). The NLO QCD corrections of COM hadroproduction were also calculated to formulate a possible solution to the long-standing polarization puzzle Chao:2012iv (); Ma:2010jj (); Butenschoen:2011yh (). The spin-flip interactions in the spin density matrix of the hardronization of a color-octet charm quark pair had been examined in Ref.Liu:2006hc (). A similar large discrepancy was found in double-charmonium production at factoriesAbe:2002rb (); BaBar:2005 (); Braaten:2002fi (). A great deal of work had been performed on this area, and these discrepancies can apparently be resolved by including NLO QCD correctionsZhang:2005cha (); Zhang:2006ay (); Gong:2007db (); zhangma08 () and relativistic correctionsBodwin:2006dm (); He:2007te (); Jia:2009np (); He:2009uf (). The data from factories highlight that the COM LDMEs of production may be smaller than previously expectedMa:2008gq (); Gong:2009kp (); Jia:2009np (); He:2009uf (); Zhang:2009ym (). Relativistic corrections have also been studied in Ref.Huang:1996bk () for heavy quarkonium decay, in Ref.Paranavitane:2000if () for photoproduction, in Ref.Ma:2000qn () for production in decay, and in Ref.Bodwin:2003wh () for gluon fragmentation into spin triplet wave quarkonium. More information about heavy quarkonium physics can be found in Refs.Kramer:2001 (); Lansberg:2006dh (); Brambilla:2010cs ().

In this paper, the effects of relativistic corrections to the COM hadroproduction at Tevatron and LHC were estimated based on NRQCD. The short distance coefficients were calculated up to . Many free LDMEs were realized at , which were estimated according to the velocity scaling rules of NRQCD with velocity ().

The paper is organized as follows. In Sec. II, the frame of calculation
is introduced for the relativistic correction of both the - and -wave
states in NRQCD frame.
Section III provides the numerical result. Finally, a brief
summary of this work is presented.

## Ii Relativistic Corrections of Cross Section in NRQCD

We only consider direct production at high energy hadron colliders, which contributes to the prompt cross section. The differential cross section of direct production can be obtained by integrating the cross sections of parton level as the following expression:

(1) |

where is the parton distribution function(PDF), and is the parton momentum fraction denoted the fraction parton carried from proton or antiproton. The sum is over all the partonic subprocesses including

As shown at the beginning of this paper, under the NRQCD frame, the computation to cross section of each subprocess can be divided into two parts: short distance coefficients and LDMEs:

(2) |

On the right-hand side of the equation, the cross section is expanded to sensible Fock states noted by the subscript . , i.e., short distance coefficients, which describe the process that produces intermediate in a short range before heavy quark and antiquark hadronization to the physical meson state. Here we use initial partons to mark the short distance coefficients for different subprocesses. are the long distance matrix elements that represent the hadronization evolutes to the CS final state by emitting soft gluons. are local four fermion operators. The factor of is introduced to make dimensionless.

In this section, our calculation on the differential cross section for this process in the NRQCD factorization formula is divided into three parts, namely, kinematics, long distance matrix elements, and short distance coefficients.

### ii.1 Kinematics

We denote the three relative momenta between heavy quark and antiquark as , with , in rest frame, where is the mass of charm quark and is the three relative velocity of quark or antiquark in this frame. Thus, the momenta for the quark and antiquark are expressed asHe:2007te (); Ma:2000 (); fourmomenta ()

(3) |

where is the rest energy of both the quark and antiquark, and is the invariable mass of . When boosting to an arbitrary frame,

(4) |

where is the four momenta of , and receives the boost from .

The Lorentz invariant Mandelstam variables are defined as

with the relationship . Here, s is independence. To expand , in terms of , we can first write down , in the center of initial partons mass frame:

(5) |

where are Lorentz invariants of independence and satisfies . These relations between and are also satisfied when boosting to arbitrary frame. In our subsequent calculation and result, we adopt () to represent () directly for simplification.

The FeynArts feynarts () package was used to generate Feynman diagrams and amplitudes, and the FeynCalc Mertig:an () package was used to handle amplitudes. The numerical phase space was integrated with Fortran.

### ii.2 Long Distance Matrix Elements

According to NRQCD factorization, the differential cross section of each partonic subprocess up to next order in to CS state and CO states , , , can be expressed as

(6) | |||||

In this expression, relativistic correction parts, denoted as ””, can easily be distinguished from LO, denoted as ””. Ref. corresponds to CS, and Ref. corresponds to CO. In addition, each differential cross section to different Fock states should be divided in short distance coefficient part and LDMEs. We can introduce to express the short distance coefficient of the LO cross section, corresponding to for relativistic correction. Many LDMEs are presented , all of which are denoted by and for the LO and relativistic correction term respectively. The explicit expressions of the ten four-fermion operators areBodwin:1994jh ()

(7) |

where and are the Pauli spinors describing anticharm quark creation and charm quark annihilation, respectively. is the color matrix. is the Pauli matrices and is the gauge-covariant derivative with . is used as the notation for the symmetric traceless component of a tensor: . Here we have

(8) |

It should be noted that

To NLO in , we can ignore terms and set

So there are four CO LDMEs for -wave, four CO LDMEs for -wave and two CS LDMEs at NLO in . The LDMEs of heavy quarkonium decay may be determined by potential modelBodwin:2007fz (); Bodwin:2006dm (), lattice calculationsBodwin:1996tg (), or phenomenological extraction from experimental dataFan:2009zq (); Guo:2011tz (). But it is very difficult to determine the production of CO LDMEs. Recently, two groups fitted CO LDMEs to NLO in . It is

(10) |

with data of production and polarization at at Tevatron in Ref.Ma:2010jj () and

(11) |

with data of production at at Tevatron and at HERA in Ref.Butenschoen:2011yh (). The two series CO LDMEs are not consistent with each other. For the three CO wave LDMEs , it is hard to determine. To simplify the discussion of the numerical result, it is assumed that

(12) |

At the same time, we can estimate the relation between their order from the Gremm-Kapustin relation Gremm:1997dq () in the weak-coupling regime

(13) |

where is the mass of charm quark that appears in the NRQCD actions and is the pole mass of charm quark. This equation was given only for CS in Ref.Gremm:1997dq (). This is the same with Ref.Bodwin:2003wh (), and we can get . If we select and , we can get .

After those presses, there are three CO LDMEs in the numerical calculation.

### ii.3 Short distance coefficients calculation

The short distance coefficients can be evaluated by matching the computations of perturbative QCD and NRQCD:

(14) |

The covariant projection operator method should be adopted to compute the expression on the left-hand side of the equation. Using this method, spin-singlet and spin-triplet combinations of spinor bilinears in the amplitudes can be written in covariant form. For the spin-singlet case,

(15) |

For spin-triplet case, the expression is defined as

(16) |

where denotes the polarization vector of the spin-triplet state. In our calculation, Dirac spinors are normalized as .

The differential cross section of each state then satisfies:

(17) |

where means sum over the final state color and polarization and average over initial states. According to this expression and Eq.(8), expanding the cross section to next leading order of is to expand the amplitude squared on the right side of the above expression to .

Next, we prepare to expand the short distance coefficients to the next order in . First, we expand each Fock state amplitude, including the -wave and -wave states, in terms of the relative momentum :

(18) |

(19) |

(20) |

The factor comes from the relativistic normalization of state. Odd power terms of four-momentum vanish in either the -wave or the -wave amplitudes, where and are inclusive production amplitudes to triplet and singlet , respectively.

In evaluating the amplitudes in power series in , it needs to be integrated over the space angle to . We can obtain the following replacements to extract the contribution of fixed power of :

For -wave case:

(21) |

For -wave case:

(22) |

where and is the orbital polarization vector of -wave states. Subsequently, by multiplying the complex conjugate of the amplitude, the amplitude squared up to the next order can be obtained:

(23) | |||||

(24) |

(25) | |||||

Any term, which is in the order of , must not be missed to obtain the correction up to the order of . In the three expressions above, the first term on the right side of each equation can be expressed in terms of kinematics variables . Here is dependence and should be expanded by Eq.(II.1). The sum of terms in the order of in the first term as well as all the second term is the contribution of the next leading order. Orbit polarization sum and spin-triplet polarization sum are equal to . According to the expression of mentioned above, the dependence of only appears in the denominator which equals to and only contains even powers of four momentum . So in the computation of unpolarized cross section to next order of as in Eqs.(23,25), expanding the polarization vector in order of is to handle the sum expression .

Therefore, the differential cross section in Eq.(6) takes the following form:

(26) | |||||

The explicit expressions of the short distance coefficients to the relativistic correction of CO states and , for partonic processes , and are relegated to the Appendix. The result of our relativistic correction of is consistent with that of Ref.Fan:2009zq () and was not given in this paper.

## Iii numerical result and discussion

We adopt the gluon distribution function CTEQ6 PDFsPumplin:2002vw (). And the charm quark is set as . The ratios of the short distance coefficient between LO and its relativistic correction at the Tevatron with and at the LHC with or are presented in Fig.1. The ratios of at the Tevatron and at the LHC are very close at large . In the large limit,

(27) |

where is the mass. Then we can expand the short distance coefficients with . The ratios of first order in the expansion are