Photoproduction of doubly heavy baryon at the LHeC
Abstract
The photoproduction of doubly heavy baryon, , , and , is predicted within the nonrelativistic QCD at the Large Hadron Electron Collider (LHeC). The production via the photongluon fusing channel and the extrinsic heavy quark channel have been considered, where or stand for heavy or quark and stands for a diquark with given spin and color configurations . The diquark shall fragmentate into baryon with high probability. For and production, equals (in configurations spinsinglet wave and colorsextuplet 6) or (in configurations spintriplet wave and colorantitriplet ) and for production, equals , , , or . A detailed comparison of those channels and configurations on total and differential cross sections, together with their uncertainties, is presented. We find the dominant contributions for production are from extrinsic heavy quark channel and for configuration, while other diquark states can also provide sizable contributions to the production. As a combination of all the mentioned channels and configurations and by taking GeV and GeV, we observe that , , and events can be generated at the LHeC in one operation year with colliding energy TeV and luminosity . Thus, the LHeC shall help us to get more information about the doubly heavy baryon, especially for and .
I Introduction
The heavyquark mass provides a natural hard scale, so the processes involving heavy quarks are perturbative QCD (pQCD) calculable by applying proper factorization theories. Among them, the nonrelativistic quantum chromodynamics (NRQCD) NRQCD () provides a powerful approach to study the properties of doubly heavy hadrons, since their heavy constituent quarks move nonrelativistically in the bound system QWGC1 (); QWGC2 (). Recently, we have studied the photoproduction of meson at the Large Hadron Electron Collider (LHeC) LHeC () within the framework of NRQCD, and it was found that sizable amounts of meson events can be generated lhecbc (). As a step forward, we shall investigate whether sizable amounts of doubly heavy baryon events can also be produced at the LHeC. If so, to compare with future possible data, it could be inversely treated as a platform for testing the effectiveness of NRQCD.
For doubly heavy baryons , , and , only was observed in the fixed target experiment by the SELEX collaboration selex1 (); selex2 (); selex3 (), which however has not been confirmed by other experiments so far. Here and henceforth, for simplicity, we use to denote the doubly heavy baryon , where stands for a light quark , , or , respectively. At present, most of the predictions on the production rate and decay width are well underestimated compared to the SELEX measurements prodxiqq1 (); xicc1 (); xicc2 (); xicc3 (); xicc4 (); xicc5 (); xicc6 (); xicc7 (); fpro1 (); fpro3 (). The unexpected high production rate of baryons may be due to the kinematics features of the SELEX experiment, and could not be described by the production mechanism only Koshkarev:2016rci (). Thus, to understand the properties of doubly heavy baryon better, experimentally, we need to accumulate more events to reduce the statistical error; and theoretically, it is important to make a systematic research on various production mechanism of doubly heavy baryon at different experimental platforms.
A dedicated generator GENXICC GENXICC1 (); GENXICC11 (); GENXICC2 () was developed and has been applied to study the production of doubly heavy baryon at the hadronic colliders prodxiqq2 (); prodxiqq3 (). In 2013, the LHCb collaboration at the LHC performed their first search of , but no significant signal has been found Aaij:2013voa (). Future searches at the LHC with improved trigger conditions and larger data samples should improve the sensitivity significantly. In addition, series predictions on the production of doubly heavy baryon at the high luminosity electronelectron colliders, such as the factory zfactory () and the International Linear Collider (ILC) Djouadi:2007ik (), have also been performed. Sizable amounts of doubly heavy baryon events are expected to be generated in those platforms eexiqq1 (); eexiqq2 (); eexiqq3 (); ggxiqq ().
In addition, the hadronlepton collider may also be a possible machine to probe the properties of the doubly heavy baryons. As clarified in Refs.LHeC (); lhecbc (), the photoproduction mechanism dominates the production of quark at the LHeC, and the doubly heavy baryon can thus be mainly generated via the photoproduction channels and .
Schematic diagrams for the photoproduction of at the LHeC are shown in Fig. 1. The photoproduction of can be divided into three steps. Taking channel as an example, the first step is the production of and pairs, where the heavy quarks and are required to be in the color and spinconfiguration ; The second step is the pair fuses into a binding diquark with certain probability; The third step is the evolution of the diquark into a doubly heavy baryon by grabbing a light quark from the “vacuo” or emitting/grabbing a suitable number of gluons. The first step is perturbatively calculable since the gluon should be hard enough to generate the heavy quarkantiquark pair. For the second step, the transition probability can be described by a nonperturbative NRQCD matrix element. We use and to stand for the matrix elements of the production of a colorsextuplet (6) and a colorantitriplet () diquark, respectively ^{1}^{1}1Here, we don’t distinguish the matrix elements of and states, since the spinsplitting effect is small prodxiqq2 (); eexiqq1 ().. For the third step, one usually assumes the efficiency of evolution from a diquark to a doubly heavy baryon is , referring as the “direct evolution,” Reference ggxiqq () has studied the evolution through direct evolution as well as “evolution via fragmentation” in which the fragmentation function has been taken into account. The authors there have found that the direct evolution is of high precision and is sufficient enough for studying the production of doubly heavy baryon, and thus we adopt the direct evolution in our calculation.
Since the predicted production rate of are much smaller than the SELEX measurements, the authors of Refs.xicc6 (); xicc7 (); GENXICC1 () suggested to take into account the extrinsic and intrinsic charm production mechanism such that to shrink the gaps between theoretical and experimental predictions. It is noted that the intrinsic charm’s contribution to the cross section of channel is less than even if the density of intrinsic component in proton is up to intr1 (); intr2 (). According to our experience on the meson photoproduction and following the suggestion given in Refs.xicc6 (); GENXICC1 (), we shall concentrate our attention on the channels and , in which the intermediate diquark is , , , and , respectively. Other diquark configurations and are forbidden due to the FermiDirac statistics for identical particles.
Ii Calculation technology
ii.1 Basic formulas
For the photoproduction mechanism, the initial photon is emitted from the electron, which can be described by the WeizsckerWilliams approximation (WWA) wwa1 (); wwa2 (); wwa3 (). When considering the extrinsic heavyquark mechanism, we should pay special attention to avoid the “double counting” problem between the and the extrinsic channels. A proper approach to deal with the extrinsic heavy quark is to employ the generalmass variableflavornumber scheme (GMVFNs) GMVFN1 (); GMVFN22 (); GMVFN2 (); GMVFN3 (); GMVFN4 (). According to the pQCD factorization theorem, the cross section for the photoproduction of within the GMVFNs scheme can be written as
(1)  
where is the WWA photon density function, is the parton distribution function (PDF) of parton inside a proton , is the factorization scale, and is the hard cross section for the partonic process . is the nonperturbative matrix element which represents the transition probability from the quark pair to the desired baryon . for (). Since we adopt the direct evolution scheme, we have = or , respectively.
The photon density function depicted by the WWA is expressed as wwa1 (); wwa2 (); wwa3 ()
(2) 
where , and are photon and electron energies. is the fine structure constant and is the electron mass. and are given by
(3) 
where the electron scattering angle cut is determined by experiment wwa4 (); wwa5 ().
The subtraction term in Eq.(1) is defined as
(4) 
where is the quark distribution function within an onshell gluon and it can be expanded order by order in . At the order, is given by
(5) 
where is the splitting function.
The partonic hard cross section can be written as
(6) 
where denotes the average of the spin and color states of initial particles and the sum of the color and spin states of all final particles, and represents the final body phase space element,
(7) 
and is the total hard scattering amplitude
(8) 
where runs over the related Feynman diagrams.
ii.2 Feynman diagram and amplitude
There are totally Feynman diagrams for the subprocess () and Feynman diagrams for the subprocess (). As for the subprocesses and , there are another 24 and 4 diagrams coming from the exchanging of two identical quark lines inside the quark pair. Practically, those diagrams are the same as the diagrams without exchanging, since we have set the relative velocity between the two quarks to be zero, i.e., we have set for the production of by applying the nonrelativistic approximation. There is a factor of for the square of the amplitude due to the two identical quarks inside the diquark. Thus, we only need to calculate the and diagrams for and subprocesses, and multiply a factor of at the cross section level. Besides, there is another factor of for the subprocess coming from the two identical open antiquarks in the final 3body phase space. The amplitudes for and can be obtained directly from the Feynman diagrams. In order to describe the bound system of the doubly heavy baryon, we should apply the spin and colorprojection operators on the amplitude of quark pair. For a detailed description on how to apply the projection operators on the amplitude of quark pair and the calculation of the color factor of the production of heavy baryon, one can refer to Refs.lhecbc (); xicc6 ().
To implement the calculation, the FeynArts package FA () is used to generate the Feynman diagrams and amplitudes; The FeynCalc FC (); FC1 () and FeynCalcFormLink packages FL () are used to handle the Dirac trace and algebraic calculations. Numerical integrations over 2 and 3body phase spaces are performed by using the VEGAS VEGAS () and FormCalc formcalc () packages.
Iii Numerical results and discussions
iii.1 Input parameters
The matrix element is related to the Schrdinger wave function at the origin, potential (). According to the velocity scaling rule of NRQCD fpro3 (), the colorsextuplet matrix element is at the same order of , and we take the usual choice of to do our calculation. Since is an overall factor, one can improve our results once more accurate is known. The wave functions at the origin together with the heavy quark masses are taken as follows xicc2 (); potential ():
The electron mass GeV is adopted and the finestructure constant is fixed as . The electron scattering angle cut is chosen as 32 mrad which is consistent with the choices of Refs.theta1 (); theta2 (). The renormalization and factorization scales are set to be the transverse mass of , , where with being the mass of . Here , which ensures the gauge invariance of the hard scattering amplitude. The PDF of the incident quark in hadron is taken as CT10NLO CT10NLO (), and correspondingly, the nexttoleading order running with () is adopted.
iii.2 Basic results
To shorten the notation, we denote the cross sections for the production of and via the channels as and , where respectively. We use and to represent the production of and . We use to represent the production of or , since and are forbidden.
  TeV  TeV  TeV  TeV  

  
        
        
        
        
Total  0.67  880.61  46.29  1.17  1285.28  76.40  4.10  2955.53  235.15  5.75  3781.33  322.63 
We present the cross sections for the photoproduction of at the electronhadron colliders under various production channels in Table 1. Here, to show how the cross section depends on the electronhadron colliding energy, we present the numerical results at two colliders with four colliding energies, i.e., or for LHeC which corresponds to or and LHeC (), and or for the future circular collider based collider (FCC) which corresponds to or and fcc ().
Table 1 shows

For the same production channel, the diquark state provides the largest contribution for the production cross section of . For the and production, the colorsextuplet diquark state gives about contribution to the total cross section of the same production channel. For the production, contributions from other diquark states are also sizable, especially, the cross section of diquark state is close to that of . Thus a careful discussion of all diquark configurations are helpful for a sound prediction of the doubly heavy baryon production.

In addition to the usually considered channel, the extrinsic heavy quark mechanism via the channel shall also provide sizable contribution to the production cross section. For example, the channel is even dominant over the channel for the production. This dominance, as shall be shown later, is caused by the dominance of the channel in low region.

By summing up all the mentioned diquark configurations and production channels, we obtain , which agrees with the observation of the doubly heavy baryon production at the ILC ggxiqq ().

The total cross section increases with the increment of electronproton colliding energy, differing to the doubly heavy baryon production at the ILC ggxiqq (), where the total cross section decreases with the increment of photonphoton colliding energy. However, it is found that at the partonic level, the production cross sections for the and behave closely, both of which shall decrease with the increment of the subprocess colliding energy. For example, we present the dependence of the cross sections for the subprocess in Fig.2.
We have found that the photoproduction of are similar under various electronproton collision energies. In the following, we take as an explicit example to show the photoproduction of in detail.
We present the transverse momentum () distributions for the photoproduction of in Fig.3. The distribution for each channel has a peak for and then drops down logarithmically. The distributions of the and channels descend more quickly than those of channels in high region. For the production via the same diquark configuration, the channel dominates the channel in small region, which explains sizable total cross section of channel as shown in Table 1. We also observe that the distributions of decrease more slowly than those of and with the increment of , and thus the events may be comparable to and at high region, although the total cross section of is much smaller than those of and .
We present the rapidity () distributions for the photoproduction of in Fig.4. The asymmetry of the rapidity distributions of clearly shows that the dominant contribution appears in the region of . The axis is defined in the direction of the electron beam, thus the fact of implies the parton from the proton is more energetic than the photon, and most of events tend to be produced in the direction of the proton beam.
  GeV  GeV  GeV  

  
      
      
      
      
Total  42.28  17.79  6.83 
  

  
      
      
      
      
Total  20.42  34.95  41.61 
We present the cross sections under various and cuts in Tables 2 and 3. Table 2 shows the cross section of and under various of cuts are more sensitive than that of , which is consistent with Fig. 3. Table 2 shows notable and production rate can be generated even with a large cut.
  

  
  
  
  
  
Total  0.43  547.91  25.23 
Finally, we draw the differential cross sections for various diquark configurations and production channels in Fig.5, where . Here, , , and are four momenta of the photon, proton, and respectively. For elastic/diffractive events, is close to jpsinlo (), besides, at low region, the resolved effect should be taken into consideration H1 (). For the prediction of inelastic direct photoproduction, a proper cut should be taken. We take , which accounts for a clean sample of inelastic direct photoproduction jpsinlo (); H1 (). The cross sections for various diquark configurations and production channels under this cut are listed in Table 4.
iii.3 Theoretical uncertainties
In this subsection, we make a discussion on the uncertainties caused by the quark mass, the quark mass, the renormalization (factorization) scale and the electron scattering angle cut . All the other parameters are fixed as their center values when discussing the uncertainty from in variation of one parameter.
  GeV  GeV  GeV  GeV  

  
        
        
        
        
Total  0.67  1188.83  58.15  0.67  660.78  37.39  0.81  880.61  44.03  0.56  880.61  48.18 
We present the cross sections at the TeV LHeC for GeV and GeV in Table 5. By adding those two errors in quadrature and summing up all the diquark configurations and production channels together, we obtain
  

  
    
    
    
    
Total  0.68  961.24  33.75  0.66  814.47  49.98 
To discuss the scale uncertainty, we set the factorization scale as the renormalization scale, . In addition to the scale choice of , we adopt two other scale choices and for discussing the scale uncertainty. The scale uncertainty for each diquark configuration and production channel is presented in Table 6. The scale uncertainties for the total cross sections are , , and for , , and , respectively, and we can reduce the scale dependence if we know the terms of the pQCD series through a nexttoleading order calculation pmc3 (); pmc5 ().
  mrad  mrad  

  
    
    
    
    
Total  0.61  820.29  42.86  0.73  940.96  49.72 
As a final remark, we make a discussion on the uncertainties from the electron scattering angle cut . For the purpose, we set and mrad. The results are shown in Table 7. Within those choices, Table 7 shows the uncertainties from are for , for , and for . The small uncertainty from shows the WWA is appropriate choice for our calculation.
Iv Summary
In the paper, the photoproduction of doubly heavy baryon at the LHeC via the channels and has been studied within the framework of NRQCD.
We have found the extrinsic heavy quark mechanism via provides a significant production rate comparing to the channel , regardless of the suppressions from the heavy quark PDFs. There are four spinandcolor diquark configurations for the production of doubly heavy baryons, i.e., and . It was found that the diquark state provides dominate contribution to the production. While other diquark states can also provide sizable contributions to the