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B-Quark Production at Hadron Colliders


SMU HEP 93-08 ANL-HEP-CP-93-66

arXiv:hep-ph/9308316v1 21 Aug 1993

B-Quark Production at Hadron Colliders
S. Riemersma Department of Physics, Southern Methodist University, Fondren Science Building, Dallas, TX 75275-0175, USA and Ruibin Meng High Energy Physics Division, Argonne National Laboratory, Argonne, IL 60439 Abstract Results for b-quark production at hadron colliders, both current and proposed, are presented. Distributions in pt are presented for the TeVatron and 3 SSC. Con?rmation of agreement between the O(αS ) calculations and UA1 3 data is presented, and the discrepancy between the O(αS ) calculations and the CDF results is updated with the most recent data.

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Introduction

Studying B-physics at hadron accelerators requires a good understanding of the total and di?erential cross sections for b-quark production. This knowl? edge gives those involved in BB mixing, rare B decays, and those trying to determine the CKM angles α , β, and γ an idea of how many events they can expect, given the luminosity and the branching ratios. It is particularly important for those studying rare B decays as they set limits on where we can hope to see new physics. For these reasons and others, the complete 3 O(αS ) corrections to heavy-quark production at hadron accelerators were calculated in [1] and [2]. Also three groups [3], [4], [5] have attempted to calculate heavy-quark production using resummation techniques in the small-x kinematic region. These techniques are necessary since the b-quark mass mb √ is small relative to the center-of-mass energies S of the TeVatron and the SSC. While these techniques o?er some hope of obtaining reasonable predictions for b-production at these machines, the current results can best be considered as preliminary. Thus we must turn to perturbative QCD for guidance, as we have no other real choice at this point. However, let us submit a caveat here: ?xed-order perturbative QCD works best when all the scales are roughly comparable, √ √ i.e. s ≈ mb ≈ pt , s being the partonic center-of-mass energy. When we are not in this regime, for example at the TeVatron and the SSC, our predictions will then be less reliable. Bearing this in mind, let us continue to the results section.

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Results

A number of ?xed-target pp experiments have been proposed for HERA, LHC, and SSC. The cross sections given in Table 1. are total cross sections without any cuts applied. The purpose is to give an idea of the overall rate of b-production at these proposed experiments. Note that these cross sections are for inclusive b-production, so if one wants to calculate rates for b- or b-production, one needs to multiply these results by a factor of two.

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Table 1. Cross Sections for Proposed Fixed-Target Experiments. √ 3 S (GeV) Born O(αS ) 43 8.3 nb 17 nb 124 0.32 ?b 0.58 ?b 200 0.89 ?b 1.6 ?b These cross sections were generated using programs created by [2] with the following inputs: mb was chosen to be 4.75 GeV/ c2 , the mass factorization scale M 2 was chosen to be m2 , and the parton distribution set used was b CTEQ1M [6]. We would also like to mention here that similar results have been obtained earlier in in [7] using a similar parton distribution set and our numbers in Table 1. as well as in Table 2. below agree with theirs. From Table 1., we see that the corrections even at these low energies are sizeable. √ For S = 43 GeV, one should probably take into account resummation e?ects at large-x (see [8]). However, at these energies, we expect that the results are fairly accurate. The situation for b-production at the TeVatron, LHC, and SSC is more problematic. We are no longer in a region where we expect ?xed-order perturbative QCD to give experimentally valid results. Nevertheless, the predictions made are worth noting, to get a quantitative idea of which regions in phase space our predictions are lacking and how much of an improvement needs to be made. Having given su?cient warning, we present Table 2., cross sections for the TeVatron, LHC, and SSC. Table 2. Cross Sections for the √ S (TeV) Born 1.8 17 ?b 15.4 92 ?b 40 170 ?b Various Colliders.
3 O(αS ) 37 ?b 270 ?b 550 ?b

As in Table 1., no cuts were applied and the input parameters chosen were 3 the same. We see rather large increases when the O(αS ) corrections are included. The ’K-factors’ are 2.2, 2.9, and 3.2 for the TeVatron, LHC, and SSC, respectively. The size of these ’K-factors’ might give one cause to worry, however they are slightly misleading since the massless t-channel exchanges 3

3 present in the O(αS ) corrections are absent in the Born approximation calculation. A better indication of the convergence should be found in comparing 4 3 the O(αS ) results with the O(αS ) corrections. We were also presented with a list of cuts from various experimental groups, and what was settled upon was the following: for CDF, we were asked for pseudorapidities |η| < 1 and pt > 4 GeV/c in the central region. The D0 cuts were |η| < 3.4 and pt > 5 GeV/c in the central region. In the forward region at the TeVatron, the request was for 2.5 < |η| < 5.5 and pt > 1.5 GeV/c. At the SSC, the central region was determined to be |η| < 2.5 and pt > 10 GeV/c, and the forward region given was 1.5 < |η| < 5.5 and pt > 1.5 GeV/c. The calculations are done with cuts in rapidity not pseudorapidity, but the di?erence should be small. Table 3. shows the results for these cuts.

Table 3. Cross Sections with Cuts Implemented. CDF Central 7.2 ?b D0 Central 13 ?b TeVatron Forward 7.0 ?b SSC Central 62 ?b SSC Forward 300 ?b

The forward region results include the sum of the positive and negative rapidity results. The result for the central SSC region seems low until one considers the large pt -cut made. Also, the large rapidity coverage of D0 helps considerably in enlarging the cross section. For additional enlightenment, we have plotted dσ/dpt versus pt for the central and forward regions for both the TeVatron and the SSC. Before we discuss the dσ/dpt plots we would also refer interested readers to [7] for rapidity distributions giving additional useful information. In Figure 1., we see that the expanded rapidity coverage of D0 makes the cross section larger by a factor of two over CDF rather uniformly over the entire pt -range. Most of the cross section lies in the low-pt range. Therefore if one could lower the pt -cut, the event increase would be sizeable. For these plots, we have chosen M 2 = p2 + m2 . Also, these plots were produced by running the programs for t b the Born approximation pt -distributions and multiplying by the ’K-factors’ previously introduced; 2.2 for the TeVatron plots and 3.2 for the SSC plots. The justi?cation for this was 1) time was of the essence and the higher-order calculations would have taken a day each to compute and 2) in discussions [9], it was revealed that the higher-order calculations generally raise the Born 4

approximation results by a fairly uniform amount across the entire pt -range. Figure 2. shows a dramatic fall-o? in the forward region as pt increases, again with most of the cross section in the low-pt region. In the low-pt range, the cross section is reduced by a factor of three to ?ve compared to the central region. depending on the cut made. Turning to the SSC, Figure 3. shows that by imposing a pt -cut of 10 GeV/c, most of the cross section is lost in the central region. At large pt , we ?nd that the contribution is still appreciable. Finally, in the forward region, Figure 4. reveals the large-pt region is again still signi?cant, but again the majority of the cross section comes from the low-pt region. The loss of cross section as pt increases is not so dramatic as it is in the forward region at the TeVatron.

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Conclusions

What can we conclude from these results? First, the ?xed-target results are probably solid, since we can see from Figure 5. the results from UA1 [10] 3 are in good agreement with the O(αS ) results, and the energies for the ?xedtarget experiments are lower than that of UA1. Looking at Figure 6., we 3 compare the O(αS ) calculations of 1 ,2 with the 1988-89 and 1992-93 results of CDF [11]. Some of these data are still preliminary, of course, but it appears that the data do not ?t the calculation. From the ?gure caption we see that we are o? by a about a factor of 2.6. But we have some consolation because the shape is approximately correct, although a slightly steeper distribution as discussed in [12] would ?t better. This factor of 2.6 will only be magni?ed when we look at the results for the SSC. Clearly, we have a problem. 4 What are the possible solutions? Calculate the O(αS ) corrections and see what di?erence that makes. That is an enormous endeavor and would take years. Try to make further headway on the small-x front. This is possible but large uncertainties remain. As an example, one interesting mechanism to accomodate the CDF data shown in Figure 6. is to alter the form of the gluon distribution in the small-x region [12]. But for a ’ballpark estimate’ that probably is not too bad, why not do the following: try σexp = σ0 e(K?1) , (3.1)

where σ0 is the Born cross section, K is the appropriate ’K-factor,’ and σexp is the expected cross section. In the case of the TeVatron, σ0 = 17 5

microbarns and K = 2.2. We would get σexp = 56 microbarns. For the SSC, σ0 = 170 microbarns and K = 3.2. Here σexp = 1.5 millibarns. The distributions would also have the factor e(K?1) multiplying the lowest-order distributions. This is of course rather ad hoc, but the results look reasonable. More theoretically valid calculations are still well o? in the distance, and the numbers are needed now. Finally, in the course of many discussions [13], it was decided that approximate cross section ?gures for each of the colliders, current and proposed, should be provided so that an estimate of B-physics event rates could be made. Toward that end, we present Table 4., a compilation of cross section ?gures that should be correct within a factor of two. Table 4. Cross Section Figures for Reference. √ σ S 43 GeV 20 nb 124 GeV 0.5 ?b 200 GeV 2 ?b 1.8 TeV 100 ?b 15.4 TeV 0.5 mb 40 TeV 1 mb

The numbers for the lower energies were arrived at essentially by rounding 3 the results of the O(αS ) calculation. The 1.8 TeV result was derived in the following way: we took the fact that the curve that ?ts the data of CDF is 3 2.6 times the O(αS ) result. Multiplying the 37 microbarns by the factor of 2.6, we get a convenient number of 100 microbarns for the TeVatron with no cuts. The numbers for the LHC and the SSC were based upon various estiamtes obtained using various parton distribution sets. They were also agreed upon in [13] and further detailed discussions about the uncertainties can be found in [7], [13]. Acknowledgements. The authors would like to thank Jack Smith for his careful reading of the manuscript. This work was supported by the Lightner Sams Foundation, Inc. and the U.S. Department of Energy, Division of High Energy Physics, Contract W-31-109-ENG-38.

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References
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(1993); E. Berger, R. Meng, and J.-W. Qiu, Proceedings XXVI Int. Conf. on High Energy Physics Dallas, 1992; E. Berger, R. Meng, and W.-K. Tung, Phys. Rev. D46 (1992) 1859. [13] E.M. Levin, J. Smith, and W.-K. Tung, private communication.

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Fig. 1. dσ/dpt vs. pt for the kinematic cuts imposed for the CDF collaboration (solid line) and the D0 collaboration (dashed line) in the central region. Fig. 2. dσ/dpt vs. pt for the kinematic cuts imposed in the forward region at the TeVatron. Fig. 3. dσ/dpt vs. pt for the kinematic cuts imposed in the central region at the SSC. Fig. 4. dσ/dpt vs. pt for the kinematic cuts imposed in the forward region at the SSC. √ Fig. 5. σ vs. pmin for S = 630 GeV with |y| < 1.5. The data are taken t from Table 2. of [10]. The high curve was run with mb = 4.5 GeV/c2 , and M = mb /2. The middle curve was run with mb = 4.75 GeV/c2 , and M = mb . The low curve was run with mb = 5.0 GeV/c2 , and M = 2mb . CTEQ1M distribution functions were used. √ Fig. 6. σ vs. pmin for S = 1.8 TeV with |y| < 1. The high solid curve was t run with mb = 4.5 GeV/c2 , and M = mb /2. The middle solid curve was run with mb = 4.75 GeV/c2 , and M = mb . The low solid curve was run with mb = 5.0 GeV/c2 , and M = 2mb . CTEQ1M distribution functions were used. The data with the thick error bars are taken from the 88-89 and the thin error bars from the 92-93 runs of CDF [11]. The dashed curve is the middle solid curve multiplied by a factor of 2.6.

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