Hadronic Production of the Doubly Heavy Baryon \Xi_{bc} at LHC

Hadronic Production of the Doubly Heavy Baryon at LHC

Abstract

We investigate the hadronic production of the doubly heavy baryon at the large hadron collider (LHC), where contributions from the four -diquark states and have been taken into consideration. Numerical results show that under the condition of GeV and , sizable events about and per year can be produced for the center-of-mass energy TeV and TeV respectively. For experimental usage, the total and the interested differential cross-sections are estimated under some typical - and - cuts for the LHC detectors CMS, ATLAS and LHCb. Main uncertainties are discussed and a comparative study on the hadronic production of , and at LHC are also presented.

PACS numbers: 12.38.Bx, 12.39.Jh, 13.60.Rj, 14.20.Lq, 14.20.Pt

Keywords: hadronic production, doubly heavy baryon, LHC.

I Introduction

The doubly heavy baryons, which represent a new type of objects in comparison with the ordinary baryons, were first predicted by Ref.quark (). These baryons shall offer a good platform for testing various theories and models, such as the quark model, the perturbative Quantum Chromodynamics (pQCD), the nonrelativistic QCD (NRQCD), the potential model and so on. Moreover, to know these baryons well can help us to understand the heavy-flavor physics, the weak interaction, the charge-parity violation, and etc.. A number of heavy baryons were discovered by several experiment collaborations, such as CLEO, Belle and BaBar, ARGUS, SELEX and CDF collaborations, a review on this point can be found in Refs.bspect (); lhc1 (); lhc2 (). However, for the family of double-heavy baryons, only has been observed and reported exp1 (); exp2 (); exp3 (); exp4 (). While due to their smaller production rate, few and have been observed. Even for , its measured production rate and decay width are much larger than most of the theoretical predictions xicc1 (); xicc2 (); xicc3 (); xicc4 (); xicc5 (); xicc6 (); xicc7 (); fpro1 (); fpro2 (); fpro3 (). More data are needed to clarify the present situation.

The CERN Large Hadron Collider (LHC), which is designed to run with a high center-of-mass (C.M.) collision energy up to TeV and a high luminosity up to LHC (), shall be of great help for the purpose. At the present, it is setting up for running with TeV and will result in an integrated luminosity of 10 after its first year of running. Taking into account the prospects of observation and measurement of doubly heavy baryons at LHC, it would be interesting to investigate the properties of these states. In the present paper, we shall first concentrate our attention on the hadronic production of , and then make a comparative study with those of and .

The doubly heavy baryon can be regarded as a combination of the heavy diquark and a light quark twobody (). The dominant mechanism for the hadronic production of baryon is the gluon-gluon fusion mechanism via the process . The gluon-gluon fusion mechanism includes 36 Feynman diagrams similar to the case of the baryon production xicc1 (); xicc2 (); xicc3 (); xicc4 (); xicc5 (); xicc6 (); xicc7 (); genxicc1 (); genxicc2 () and the meson production bc1 (); bc2 (); bc3 (), all of which can be schematically represented by Fig.(1), where and are two momenta for the initial gluons, and are momenta for the two outgoing and , is the momentum of . The intermediate -diquark pair can be in one of the four Fock states, i.e. , , and respectively. More definitely, according to Refs.xicc6 (); genxicc1 (); genxicc2 (), the hadronic production of can be divided into three steps: the first step is the production of a -pair and a -pair that can be calculated by pQCD, since the intermediate gluon should be hard enough to form a heavy quark-antiquark pair. The second step is that the two heavy quarks fusion into a binding -diquark, and the third step is the fragmentation of such diquark into the desired baryon by grabbing a light quark and suitable number of gluons when needs. The second and the third steps are non-perturbative, which can be described by a universal matrix element within the NRQCD framework nrqcd ().

Figure 1: Schematic diagram for the hadroproduction of from the gluon-gluon fusion mechanism , where the dashed box stands for the hard interaction kernel.

The paper is organized as follows. In Sec.II, we give the main idea in dealing with the hadroproduction. Numerical results are presented in Sec.III. And in Sec.IV, we make a discussion on the main uncertainties for hadroproduction and a comparison of the hadronic production of , and . The final section is reserved for a summary.

Ii Calculation technology

Within the NRQCD framework, the total hadronic cross-section for the gluon-gluon fusion mechanism can be schematically written as the following factorization form

where (with or ; or ) is the distribution function of parton in hadron . is the factorization scale and is the renormalization scale, and for convenience, we take them to be the transverse mass of , i.e. . stands for the cross-section for the gluon-gluon fusion subprocess, which can be expressed as xicc7 (); genxicc1 (); fpro3 (),

(1)

where the ellipsis stands for the terms in higher orders of , is the relative velocity between the constitute and quarks. or is the perturbative coefficient for producing -diquark in different spin and color configurations respectively. Four matrix elements: , , and characterize the transitions of the -diquark in , , , spin and color configurations into baryon respectively. can be related to the wavefunction of the color anti-triplet diquark as . According to the discussions shown by Ref.fpro3 (), other matrix elements , and are of the same order in as . Since all these matrix elements emerge as overall parameters, we can easily improve our numerical results when we know these matrix elements well. Naively, we take all of them to be to do our estimation genxicc1 (); genxicc2 (); fpro3 ().

To derive analytical squared amplitude of the 36 Feynman diagrams for the hard subprocess is a tedious task, since it contains non-Abelian gluons and massive fermions. In Refs.bc2 (); xicc6 (), the so-called improved helicity amplitude approach has been adopted to derive analytic expressions for the process at the amplitude level. And basing on the obtained sententious and analytical expressions, an effective generator GENXICC genxicc1 (); genxicc2 () for simulating , and events has been accomplished. Here we shall use GENXICC to make a detailed study on the hadronic production of .

Iii Numerical results

To be consistent with the leading-order (LO) hard scattering amplitude, the LO parton distribution function (PDF) of CTEQ group, i.e. CTEQ6L 6lcteq (), and the LO running are adopted in doing the numerical calculation. And for other parameters we adopt the following values xicc2 ():

(2)
- - -   LHC (CMS, ATLAS) LHCb
- \backslashbox or NO cut
GeV 20.90 10.82 16.21 11.12
- GeV 16.01 8.363 12.49 7.941
- GeV 10.72 5.674 8.446 4.886
GeV 5.120 2.618 3.938 2.689
- GeV 4.062 2.094 3.142 2.006
- GeV 2.853 1.489 2.227 1.307
GeV 31.70 16.03 24.20 17.09
- GeV 24.09 12.30 18.52 12.17
- GeV 15.97 8.276 12.42 7.441
GeV 5.502 2.886 4.315 2.896
- GeV 4.280 2.259 3.372 2.096
- GeV 2.941 1.569 2.337 1.325
Table 1: Hadronic cross section (in unit ) for at LHC with TeV. Three typical cuts are adopted. As for the rapidity and pseudo-rapidity cut, we take and for CMS and ATLAS, and for LHCb.
- - -   LHC(CMS, ATLAS) LHCb
- \backslashbox or NO cut
GeV 47.24 21.70 33.43 25.85
- GeV 36.55 16.92 26.04 19.17
- GeV 24.92 11.70 17.95 12.34
GeV 11.55 5.259 8.112 6.250
- GeV 9.255 4.243 6.537 4.822
- GeV 6.607 3.067 4.713 3.269
GeV 70.67 31.80 49.19 38.89
- GeV 54.29 24.65 38.07 28.74
- GeV 36.59 16.85 25.97 18.36
GeV 12.46 5.794 8.909 6.788
- GeV 9.802 4.591 7.049 5.111
- GeV 6.855 3.248 4.975 3.377
Table 2: Hadronic cross section (in unit ) for at LHC with TeV. Three typical cuts are adopted. As for the rapidity and pseudo-rapidity cut, we take and for CMS and ATLAS, and for LHCb.

In TAB.1 and TAB.2, we show the total cross sections for with its -diquark in and states respectively. In these two tables, the results for the C.M. energies TeV and TeV are presented. Total cross sections with typical cuts for ATLAS, CMS and LHCb are adopted ATLAS (); CMS (); LHCb (), e.g. the transverse momentum cut , GeV and GeV, and the rapidity cut and for ATLAS and CMS, and for LHCb are used for the estimation. For CMS, it usually adopts the pseudo-rapidity cut condition around , since the - and - differential distributions for the four diquark states under the two cases of and are close in shape, and their corresponding total cross sections with pseudo-rapidity cut are also close to those with rapidity cut , i.e. , so to short the paper we take the same cut conditions for both ATLAS and CMS. Here the short notation with (n=1,2,3,4) stands for , , and respectively.

Figure 2: -distributions for hadroproduction under ATLAS and CMS rapidity cut , where the left and the right diagrams are for TeV and TeV respectively. The solid, the dashed, the dash-dot-dot, the dash-dot and the short-dash lines stand for the total, that of , , and respectively.
Figure 3: -distributions for hadroproduction under ATLAS and CMS rapidity cut , where the left and the right diagrams are for TeV and TeV respectively. The solid, the dashed, the dash-dot-dot, the dash-dot and the short-dash lines stand for the total, that of , , and respectively.
Figure 4: -distributions for hadroproduction under LHCb pseudo-rapidity cut , where the left and the right diagrams are for TeV and TeV respectively. The solid, the dashed, the dash-dot-dot, the dash-dot and the short-dash lines stand for the total, that of , , and respectively.

TABs.(1,2) show that all the four diquark states and can provide sizable contributions to hadroproduction. Moreover, one may observe . And the total cross section for the scalar diquark states is about of that of the vector diquark states . Differential cross sections versus - are drawn in Figs.(2,3,4), where the results for the three typical rapidity or pseudo-rapidity cuts , and are presented and these curves show the relative importance of the four diquark states clearly.

The LHC has been first running at TeV with luminosity from 30th March 2010, and its integrated luminosity is 10 /yr. Based on the total cross sections shown in TAB.1, one can estimate that about events per year can be produced under the condition of GeV and . When the C.M. energy and the luminosity are reached up to 14 TeV and as designed, then the integrated luminosity will be changed to 100 /yr, one can estimate that about events per year can be produced under the condition of GeV and .

Figure 5: The -distributions for production with various in LHC , where the left and the right diagrams are for TeV and TeV. The Solid line corresponds to the full production without , the dashed, the dash-dot, the dotted, the dash-dot-dot and the short dashed lines are for GeV, GeV, GeV, GeV GeV respectively. All the curves are the sum of all the four diquark states.
Figure 6: The -distributions for production under various - and - cut, where the left and the right diagrams are for TeV and TeV respectively. The solid, the dashed, the dash-dot-dot and the short dash lines stand for no-cut, , and respectively. All the curves are the sum of all the four diquark states.

Next, we draw the - and - distributions under some typical - and - cuts in Figs.(5,6), where each curve stands for the sum of all the four diquark states and . The results for TeV and TeV are presented accordingly. Firstly, as shown by Fig.(5), there is an obvious platform within the region of , where dominant contributions to the cross section are there. To show this point clearly, we define a ratio , where stands for the total cross section without -cut and stands for some particular value. Then we obtain and for TeV, and and for TeV. Secondly, as shown by Fig.(6), the -distribution under the case of drops faster than the cases with other -cuts, especially in the large regions. This implies that if the same larger cut (e.g. GeV) is imposed at the colliders 1, ATLAS and CMS are better than LHCb for studying properties, since more events can be produced and measured at ATLAS and CMS.

Iv Discussions

iv.1 Main uncertainties for hadroproduction

To be more useful experimentally, we make a simple discussion on the uncertainties for hadroproduction. For the present LO estimation, the uncertainty sources include the non-perturbative matrix elements, the factorization scale , the constitute quark masses and , PDF and etc..

Numerically, it is found that similar to the hadronic production of , and that have been done in the literature, the LO PDFs like MRST2001L mrst () and CETQ6L 6lcteq () only lead to small difference to the total cross section that is less than . So we shall fix the PDF to be CTEQ6L to do our discussion. Moreover, all the non-perturbative matrix elements emerge as overall parameters, then we can easily improve our numerical results when we know these matrix elements well. In the following, we shall concentrate our attention on the uncertainties caused by the factorization scale , and the constitute quark masses and .

-    A    B    A    B    A    B
- \backslashboxC.M. Energy or
TeV 3.957 5.674 5.889 8.446 3.403 4.886
- TeV 8.477 11.70 12.99 17.95 8.893 12.34
TeV 1.078 1.489 1.612 2.227 0.939 1.307
- TeV 2.298 3.067 3.529 4.713 2.432 3.269
TeV 6.135 8.276 9.200 12.42 5.471 7.441
- TeV 12.90 16.85 19.85 25.97 13.94 18.36
TeV 1.104 1.569 1.646 2.337 0.941 1.325
- TeV 2.360 3.248 3.617 4.975 2.455 3.377
Table 3: Hadronic cross section (in unit ) of at LHC for two typical energy scales A and B. GeV, and for CMS and ATLAS, and for LHCb are adopted for the estimation.

In TAB.3, we present the total cross sections for two typical factorization scales, i.e. type A: with , and type B: . Other parameters are fixed to be their center values. Here, GeV, and the rapidity cut and for ATLAS and CMS, and for LHCb are adopted for the estimation. It is found that the cross section differences caused by these two factorization scales is about for the four diquark states and respectively, which is a comparatively large effect.

Next, we investigate the uncertainties of and in ‘a factorization way’. More explicitly, when focusing on the uncertainty from , we let it be a basic input varying in a possible range GeV with all the other parameters being fixed to their center values, e.g. GeV, and . Similarly, when discussing the uncertainty caused by , we vary within the region of GeV with all the other parameters being fixed to be their center values.

- \backslashboxC.M. Energy or            
TeV
- TeV
TeV
- TeV
TeV
- TeV
TeV
- TeV
Table 4: Hadronic cross section (in unit ) of at LHC with varying GeV. Other parameters are fixed to be their center values. GeV, and for CMS and ATLAS, and for LHCb are adopted for the estimation.
- \backslashboxC.M. Energy or            
TeV
- TeV
TeV
- TeV
TeV
- TeV
TeV
- TeV
Table 5: Hadronic Cross section (in unit ) of at LHC with varying GeV. Other parameters are fixed to be their center values. GeV, and for CMS and ATLAS, and for LHCb are adopted for the estimation.

We present the total cross sections for with varying or for C.M. energies TeV and TeV in TAB.4 and TAB.5. Here, GeV, the rapidity cut and for ATLAS and CMS, and for LHCb are adopted for the estimation. Quantitatively, it can be found that the total cross sections decreases with the increment of or , which can be roughly explained by the smaller production phase space for larger quark masses. And from TAB.4 and TAB.5, one may observe that the cross sections are more sensitive to the value of than . When increases or decreases by the step of GeV, the cross section of changes around for the four diquark states and . While for the case of , when increases or decreases by step of GeV, the cross section of decreases or increases by for the four diquark states and .

iv.2 A comparison of the hadronic production of , and

Figure 7: The -distributions for , and production with GeV in LHC, where the left and the right diagrams are for TeV and TeV. The Solid, the short dash and the dash-dot lines are for , and respectively. All the curves are the sum of all the s-wave diquark states.
Figure 8: The -distributions for , and production with rapidity cut condition in LHC, where the left and the right diagrams are for TeV and TeV. The Solid, the short dash and the dash-dot lines are for , and respectively. All the curves are the sum of all the s-wave diquark states.
-
- TeV TeV TeV TeV TeV TeV
38.11 69.40 16.7 28.55 0.503 1.137
9.362 17.05 3.72 6.315 0.100 0.226
Total 47.47 86.45 20.42 34.87 0.603 1.363
Table 6: Comparison of the total cross section (in unit ) for the hadronic production of , and at TeV and TeV, where and stand for the combined results for the diquark in spin-triplet and spin-singlet states respectively. In the calculations, we adopt GeV and .

To be useful reference, we make a comparison of the hadronic production of , and at LHC. The total cross sections are presented in TAB.6, where and stand for the results for the diquark in spin-triplet and spin-singlet states respectively. More explicitly, for hadronic production of , one needs to consider the contributions from the two diquark states and . As for hadronic production of , one needs to consider the contributions from the two diquark states and . While for the case of , one needs to consider the contributions from the four diquark states and .

From TAB.6, one can see that the total cross section of is at the same order of that of , i.e. it is about and of that of for TeV and TeV respectively. While, the total cross section of is only and of that of for TeV and TeV respectively. Then, similar to the case of that has been measured by the SELEX experiment at TEVATRON exp1 (); exp2 (), it would be possible for be fully studied at LHC.

We draw the - and - distributions for , and production under the case of GeV and in Figs.(7,8), where each curve includes the sum of all the mentioned S-wave diquark states. Fig.(8) shows that production cross section of is smaller than that of in the lower region, however it will dominant over that of when GeV.

V Summary

We have analyzed the hadronic production of via the dominant gluon-gluon fusion mechanism at LHC with the center-of-mass energy TeV and TeV respectively. For experimental usage, the total and the interested differential cross-sections have been estimated under typical cut conditions for the LHC detectors CMS, ATLAS and LHCb.

Numerical results show that about and events per year can be produced for TeV and TeV under the condition of GeV and . This indicates that can be observed and studied at LHC. Main uncertainties for the estimation have been discussed and a comparative study on the hadronic production of , and at LHC with TeV and TeV have also been presented. As for the total production cross section under the case of GeV, we have , however the differential cross-section of will dominant over that of when GeV.

In the above, we have not distinguished the light components in the baryon. More subtly, as for the production of , after the formation of the heavy -diquark, it will grab a light anti-quark (with gluons when necessary) from the hadron collision environment to form a colorless double heavy baryon with the relative possibility for the light quark as pythia (), i.e. to form the baryons , or . More precisely, when the diquark is produced, it will fragment into with probability, with probability and with probability accordingly. If enough events can be accumulated at LHC, then one may have chances to study the or separately from their decay products.

Acknowledgments: This work was supported in part by the Fundamental Research Funds for the Central Universities under Grant No.CDJXS11102209, by Natural Science Foundation of China under Grant No.10805082 and No.11075225 and by Natural Science Foundation Project of CQ CSTC under Grant No.2008BB0298.

Footnotes

  1. Since the events move very close to the beam direction cannot be detected by the detectors directly, so such kind of events cannot be utilized for experimental studies in common cases.

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