

 26 Sep 1996

ON THE HADRONIC PRODUCTIONS OF Bc AND B

\Lambda c MESONS

Chao-Hsi CHANG CCAST (World Laboratory), P.O.Box 8730, Beijing 100080, China Institute of Theoretical Physics, Academia Sinica, P.O. Box 2735, Beijing 100080, China

International Center for Theoretical Physics, P.O. Box 586, I-34100 Trieste, Italy

E-mail: ZHANGZX@itp.ac.cn

ABSTRACT With respect to the two theoretical approaches for estimating the hadronic productions of the mesons Bc and B

\Lambda

c : the full ff

4 s calculation and thefragmentation approximation, with a thorough comparative study a quite

remarkable conclusion may be drawn. In hadronic collisions the condition for the applicability of the fragmentation approximation is the transverse momentum PT of the produced Bc and B

\Lambda

c being much greater than the meson masses, andthe higher the energy of the relevant subprocess the higher the

PT is requested.

The mechanisms for the productions are also clarified. In order to show the resultant differences of the two approaches, newly introduced distribution of the fraction of the Bc (or B

\Lambda

c ) meson energy (a measurable observable) in the centerof mass of the subprocess is emphasized.

The reason why we would like to talk about the subject here is three-fold: first of all, the problem is interesting itself; secondly, there were some obscurities due to a few incorrect numerical results and misleadings, and now the situation has been clarified; the third, some fresh results have been achieved.

The heavy flavored _bc meson states have attracted considerable interest due to their interesting properties ( the spectroscopy of _bc meson states1; the weak decays2 and the productions3

\Gamma 18 etc ). The experimental search for these mesons is now under

way at high energy colliders such as the LEP e+e

\Gamma collider (at Z0 resonance) and the

Tevatron _pp collider (at the full energy 1.8 TeV). Various experimental results are expected to come out soon thus to estimate the production cross sections as precisely as possible has become a desirous theoretical task. Fortunately the mesons are weakly bound states and relatively simple, their hadronization is calculable in the framework of perturbative QCD (PQCD) in terms of the wave function obtained by potential model. At LEP being relatively simple, the productions of the pseudoscalar ground state Bc and the vector meson state B

\Lambda

c are dominated by Z

0 decay into a b_b pair,

followed by the `fragmentation' of a _b quark into the Bc or B

\Lambda

c meson

3;4;5, whereas

as first pointed out in Ref.[6], the energetic hadronic productions are much more complicated even only to the lowerst order of PQCD in ff4s and dominated by the subprocess of `gluon-gluon fusion' (gg ! Bc(B

\Lambda c )b_c). In practice, a possible and

alternative way to calculate the hadronic productions is to apply the fragmentation approximation. With the approximation the calculation can then be considerably simplified, as indicated in Ref.[7]. Since then one interesting question is addressed,

1

how well and/or in which kinematic region for the hadronic productions of the Bc and B

\Lambda

c mesons the fragmentation approximation works. Therefore several groups, basedon the same consideration, have recalculated the productions. Namely since Ref.[6]

presented the numerical results for the hadronic productions first, recalculations have been completed and distributed8

\Gamma 15. The calculations for the hadronic productions

of the mesons Bc and B

\Lambda

c to order ff

4 s in PQCD involve very complicated numericalcalculations. At the first stage, not all the calculations were in agreement: in Ref.[8]

an order of magnitude larger result than Ref.[6] is claimed; in Ref.[9] again a result larger than Ref.[6] but smaller than Ref.[8] is obtained; a result similar to Ref.[9] is found by Masetti and Sartogo10. But more recently, it is found by the same authors

of Ref.[9] that a color factor 13 had been overlooked in their previous work and after correcting this factor, a result in agreement with Ref.[6] has been achieved11. In Ref.[12] similar numerical results to Ref.[6] have also independently been obtained. In Ref.[13] different numerical results have been presented, whereas it is difficult to compare directly with others due to different parton distribution functions and different energy scale being adopted in the calculation. Thus in Refs.[14,15], the authors of Ref.[6] as well as their collaborators have re-studied the productions extensively with the updated parton distribution functions19 and made thorough comparisons with others' 11

\Gamma 13. Now at least of the four groups6;14;15;11;12;13, the

numerical results for the cross sections of the subprocess are consistent each others within the uncertainties of the numerical calculations. One may now be confident that the numerical results of the full order ff4s PQCD calculations are in agreement. Thus I will discuss the problem based on the confident results.

From experience, it is naively believed that, when the transverse momentum PT of the meson Bc or B

\Lambda

c is large, the hadronic productions of the mesons aredominated by jet fragmentation, so the fragmentation approximation may become

very valid. However one should carefully examine the approximation not only because it will potentially be used as a further test of the factorization theorem of QCD but also because of special interest in estimating the productions as reliable as possible i.e. as mentioned above to predict the productions precisely so as to guide the discovery of the meson Bc in experiments. To pursue the goal, subsequent comparisons between the PQCD full ff4s calculation and the fragmentation approximation were made by several groups6;14;15;11;13. Though all the results were comparable, different conclusions still were drawn due to different observables in different kenetics regions being emphasized. I will talk about the aspects in detail below.

Based on PQCD to calculate the hadronic productions, the full ff4s order calculation is illustrated in Fig.1a with Fig.1b and the fragmentation approach in Fig.1a and Fig.1c.

As pointed out above, it is very interesting to investigate the appoaches. Since the PT distribution decreases very rapidly as PT increases, one would lose a lot of statistics if one had only considered large PT events. Thus the comparitively low

2

PT events should also be considered carefully, as long as they may escape from the experimentally proper cuts. In Ref.[11] by examining the production ratio for B

\Lambda

c toB

c, it is claimed that the fragmentation approximation breaks down even for very large PT , whereas in Ref.[13] by investigating the PT distribution of the Bc meson,

it is claimed that the fragmentation approximation works well if PT exceeds about 5 \Gamma 10 GeV. Whereas we investigated this problem quite early14 and re-examined it more carefully thus a more completed feature has been achieved15.

G

Bc

_c b

Fig 1a

G

Bc

_c b

= + + \Delta \Delta \Delta \Delta \Delta \Delta

Fig 1b

G

Bc

_c b

= + + \Delta \Delta \Delta \Delta \Delta \Delta

Fig 1c

Figure 1: The factorizations for the productions To clarify the problem, let us focus the discussions on the subprocess. To the lowest order (ff4s), there are 36 Feynman diagrams responsible for the dominant gluon fusion subprocess g(k1) + g(k2) ! Bc(p) + b(q2) + _c(q1), where k1, k2, p, q1, and q2 are the respective momenta. As pointed out in Refs.[6,14], of the 36 Feynman diagrams one may split them into five independent "groups" according to the color structure and each of them is gauge invariance for QCD. Furthermore, the contributions from the kinematic region where certain factors of the amplitude for the process are nearly singular; i.e., some of the internal quark lines and gluon lines in the Feynman diagrams

3

are close to mass-shell are substantial even dominant, especially when the c.m.s. energy of the subprocess, p^s, is much larger than the heavy quark mass. Specifically, here for the concerned subprocess and in a special chosen gauge where large number cancellation due to gauge invariance does not happen at all, the possible singularities may arise from the inverse power(s) of the following factors, or their products:

qi \Delta kj; p \Delta kj; (ffip + qi)2; and (kj \Gamma ff1p \Gamma q1)2; (1) which corresponds to the denominators of the quark and gluon propagators appearing in the amplitude and here i; j = 1; 2 and ff1;2 = mc;b(m

c + mb) is the ratio of quark masses.The singularities result in the cross section, for the subprocess and upto the lowest

twist contributions, proportional to 1^s f

2B

c

M 2Bc (where fB

c is the Bc decay constant) and

with some logarithmic correction terms such as ln(^s=M 2Bc) being involved. When the PT of the Bc meson is large, only (ffip + qi)2 can still be small (, m2i ), therefore the fragmentation functions can then be extracted from the most singular part containing the inverse powers of this factor in the square of the amplitude. It then follows that in the large PT region the subprocess is dominated by the fragmentation. However, when the PT of the Bc meson is small, the produced Bc, as well as the b and the _c quarks, can be soft or collinear with the beam. In this region the amplitude is highly singular because two or more of the internal quarks or gluons in certain Feynman diagrams can simultaneously be nearly on-mass-shell. Although this region is a smaller part of the phase space, these nearly singular Feynman diagrams make a substantial contribution and dominate to the cross section in the region. In fact, in the square of the amplitude we can isolate all the terms which contribute to the lowest twist cross sections using singularity power counting rules20. When p^s AE MBc the lowest twist contributions dominate, while the higher twist contributions are suppressed by a factor m2=^s at least and small. We can decompose the terms which contribute to the lowest twist cross sections into three components: One is due to the fragmentation contribution, which dominates the large PT region, and the second is due to the non-fragmentation, which comes from the other singular parts as discussed above (those in which two or more quarks or gluons are nearly on-mass-shell). The non-fragmentation dominates in the smaller PT region. As for the third, the regular parts, they contribute the smooth "background" only and small, thus they are not so interesting. The contributions from the first two components are quite clearly distinguishable in the PT distribution of the subprocess, particularly at large p^s.

In order to show the factor quantatively, the cross sections for the dominant subprocess gg ! Bc(B

\Lambda

c ) + \Delta \Delta \Delta versus PT (without any PT cut) are calculated bythe two approaches and at four energies for the colliding gluons: 30GeV, 60GeV,

100GeV and 200GeV. The results are plotted in Fig. 2 (Fig. 2a-2d). Here the solid lines indicate those for the full QCD ff4s calculations; the dotted lines indicate those

4

for the fragmentation approximation. Note that in the calculations here the values of the parameters ffs = 0:2, mb = 4:9 GeV, mc = 1:5 GeV, MBc = 6:4 GeV and fBc = 480 MeV are taken, moreover in the fragmentation calculations, in order to reduce the error caused by the phase space integrations, we directly used the squared matrix elements from which the fragmentation functions are derived, rather than the fragmentation functions themselves.Error: /rangecheck in --repeat--
Operand stack:
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%interp_exit .runexec2 --nostringval-- --nostringval-- --nostringval-- 2 %stopped_push --nostringval-- --nostringval-- --nostringval-- false 1 %stopped_push 2 3 %oparray_pop 2 3 %oparray_pop 2 3 %oparray_pop 2 3 %oparray_pop .runexec2 --nostringval-- --nostringval-- --nostringval-- 2 %stopped_push --nostringval-- --nostringval-- --nostringval-- --nostringval--
Dictionary stack:
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