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ANL-HEP-CP--87-121 DE88 009981
BENCHMARK CROSS SECTIONS FOR BOTTOM QUARK PRODUCTION*
Edmond L. Berger High Energy Physics Division Argonne National Laboratory Argonne, IL 60439
Abstract
A summary is presented of theoretical expectations for the total cross sections for bottom quark production, for longitudinal and transverse momentum distributions, and for h,T> momentum correlations at Fermilab fixed target and collider energies.
DISCLAIMER
This report was prepared as an account or work sponsored by an agency of the United States Government. Neither the United States Government nor any agency thereof, nor any of their employees, makes any warranty, express or implied, or assumes any legal liability or responsi- bility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Refer- ence herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, or otherwise does not necessarily constitute or imply its endorsement, recom- mendation, or favoring by the United States Government or any agency thereof. The views and opinions of authors expressed herein do not necessarily state or reflect those of the United States Government or any agency thereof.
'Work supported by the U.S. Department of Energy, Division of High Energy Physics, Contract W-31-109- ENG-38. To be published in the Proceedings of the Fermilab Workshop on High Sensitivity Beauty Physics, Nov. 11-14, 1987. Total Cross Sections
For purposes of planning experiments and comparing various proposals, it is useful to begin with a common set of expected cross sections upon which rate estimates are based. In two recent papers,1-2 predictions were presented of bottom quark cross sections at the energies of the Fermilab pp collider and in the momentum region accessible to the fixed target program at Fermilab. In this note, I begin with a brief discussion and comparison of those predictions. I then extract a set of conservative "benchmark" cross sections which I propose be used as a reference standard.
Good theoretical arguments1"4 justify the belief that quantum chromodynamics (QCD) perturbation theory provides reliable predictions for bottom quark cross sections. Principally because the bottom quark mass m» is relatively large, m» =; 5 GeV, non- perturbative, higher-twist, and higher-order perturbative contributions should be substan- tially smaller for bottom quark production than for charm.4
The predictions of Refs. 1 and 2 are based on order a] QCD 2 —*• 2 subprocesses. They should be applicable for total cross sections and for inclusive cross sections differential in rapidity or Feynman Xjr, except for overall multiplicative *K factors" whose size5 for bottom quark production should be modest, in the range K = 2 to 3. Contributions from 3 O (a t) QCD 2 -» 3 subprocesses grow in importance relative to the O (a]) terms when the transverse momentum p? of the bottom quark is large, i.e. pr > m». Thus, for cross sections differential in pr, uK(pr) functions" may become very large for pr "> m». The O (a*) diagrams also provide different event topologies.1
There are prosaic sources of uncertainty in addition to the important role of O (aj) subprocesses. They include choice of the value of m* used in the calculations; of the evolution scale Q* used in evaluating structure functions and a,(Q3); and of structure function parametrizations. These uncertainties have a limited impact on predictions at y/s = 1.8 TeV. However, at fixed target energies, where typical values of the parton fractional momenta are relatively large (x,- ~ 2m»/-v/s), uncertainties associated with m* and Q2 have substantial impact on both the energy dependence of cross sections (e.g. threshold behavior) and the predicted magnitudes. In Figs. 1 and 2 I reproduce two of my figures1 upon which I have superimposed curves from the Ellis-Quigg paper.2 In obtaining my curves, I fixed mj = 5 GeV and varied the evolution scale, whereas Ellis and Quigg fixed the evolution scale and varied mj. Ellis and Quigg used different pion structure functions and a slightly different definition of 2 6 at(Q ). We both used the Duke-Owens set 1 structure functions for the nucleons. In Ref. 1 I discuss the slight differences which arise if different nucleon structure functions are used. Also shown in Fig. 2 is the measurement of the CERN WA78 collaboration,7 in respectable agreement with theoretical expectations.
A brief comment is in order concerning expected "nuclear effects" of which there are several.1 The different x dependences of the up quark and down quark densities mean that the cross section of a neutron is different from that of a proton. There are also modifications of structure functions associated with the nuclear bound state,8 the "EMC effect"; shadowing corrections, if x <> 0.1; and, finally, broadening of px spectra.
The most easily calculated nuclear effect, and the only one included in Figs. 1 and 2, is that associated with the neutron/proton ratio. For pA scattering, the cross section per nucleon is identical to that for pp —*• QQ X. In other words, for pA scattering there is no nuclear target effect in heavy flavor production associated with the different x dependences of the up and down quark densities in neutrons and protons.1 This conclusion rests on the assumption of SU(2) symmetry of the antiquark densities fCjv(i) = dpi(x)), the usual assumption of isospin symmetry (e.g. up(x) = dn(x)), and the assumption that the gluon densities in protons and neutrons are identical.
For pN and x~N collisions there is a difference between the cross sections. The cross sections per nucleon for QQ production from a nucleus are expected to be smaller than the cross sections for production from proton targets.1 For a target N with an equal number of protons and neutrons, the magnitude of the effect in the case of r~N —• QQ X amounts to factors of 0.68,0.80, and 0.87 at laboratory momenta of 200,400, and 600 GeV. For the Ellis-Quigg results in Fig. 2, the target N contains an equal number of neutrons and protons, whereas I computed cross sections per nucleon for ir~U —• bbX. The cross sections per nucleon for ir~U are very slightly smaller than those for w~D, Correspondingly the calculations presented here for ir~U may be used for comparisons with data for all A in the range D < A < U. The sensitivity of predictions to the choices of m» and Q2 was examined in some detail in Refs. 1 and 2. Over the fixed target momentum range shown in Fig. 1, the choice of Q2 = m2 as the evolution scale results in cross sections o(pN —» bbX) larger than for Ql = 4m2 or Q2 = s. The difference between cross sections for Q2 — s and 2 2 Q = rn is a factor of 3.3 at piab = 400 GeV and 3.1 at pi,b = 1000 GeV. Over the range 400 < pub < 1000 GeV, the expected cross section a{pN —» bbX) is decreased by about a factor of two when the b quark mass is increased from m» = 5.0 GeV to m$ = 5.4 GeV, and increased by about a factor of two if the 6 quark mass is decreased from 5.0 GeV to 4.6 GeV. For ir~U —* bbX, shown in Fig. 2, predicted cross sections differ by factors of
2.4, 2.3, and 2.3 at pi,b = 200, 400, and 600 GeV for three choices of the evolution scale Q*. Note the tradeoff between m« and Q2. For example, agreement with the data7 at 2 320 GeV/c in Fig. 2 can be achieved either with (mb =; 5.0 GeV and Q = ml) or with 2 (mt a 4.6 GeV and Q = s).
The spread of predicted values shown in Figs. 1 and 2 is perhaps deceptively large. I note in particular that the current quark mass appropriate for perturbative calculations is most likely smaller than one-half the mass of the T. Therefore, m& < 5 GeV is a good bet. I propose that a good set of benchmark cross sections is obtained by selecting values 2 2 intermediate between the curves marked m» = 4.7 and mt = 5.0 GeV for Q = 4m . These are still conservative in the sense that Q2 may be less than 4m\ but is not likely to be greater. Following this procedure, I obtain the values listed in Tables 1 and 2.
Table 1: Benchmark Cross Sections for pN —* bbX
W.bfGeV/c) *(nb) 400 0.3 if 500 0.8 if 600 1.6 if 700 2.7 K 800 4.0 K Table 2: Benchmark Cross Sections for n N —• bbX
Pub(GeV/c) a(nb) 300 1.6 K 400 3.3 K 500 5.6 if 600 8.2 if
In these tables I have indicated that all values should be multiplied by a if factor associated with O (aj) contributions. For rate estimates, the conservative choice would be if = 1. However, it is not unrealistic to suppose if = 2 to 3 for pN collisions with a , somewhat smaller value for irN collisions (since the O (a*) gg —* QQsubprocess accounts for less than one-half of the n~N cross section over the momentum range indicated in the table). For those planning experiments, note that tr(pN -* bhX) at 800 GeV is about equal to a(ir~N —* bbX) at 450 GeV. Another significant advantage of pion beams is apparent in the xj? spectra, da/dxp, shown below.
For the Tevatron collider I compute a(pp -+ blX) = 14/tb at yfi - 1.8 TeV. This estimate is based on the same parameters (m» = 5.4 GeV, Q7 = s) which provide good agreement1 with the CERN UAl data.9 Since reasonable changes in nn, Q2, and structure functions all tend to increase the prediction, I quote
o{pp -+ bbX) = 14 to 30 jib at y/a = 1.8 TeV.
For a top quark of mass 100 GeV, I compute a(pp -* iiX) = 0.1 nb at y/s = 1.8 TeV.
Signal to Background
To obtain a crude estimate of the signal to noise ratio, one may divide the bottom quark cross sections listed above by the hadronic total cross sections. Doing so, I obtain 1. Tevatron Collider
a(pp) 80mb
2. pA at 800 GeV/c
^ (Q 8 tQ , 3 x 1( (p) 40 mb A2/3 v
3. ir~A at 600 GeV/c
(6 to 10 nb)A . , „„_,.„„. = ^—TTiTT" ^(2to4x 10 7)A1'3 25mbA*/s v ;
In computing the ratios for fixed target experiments I have used the fact that the total cross section on a nuclear target A scales as A2/3, whereas the bottom quark cross section is predicted to grow linearly with A. For Uranium, A1!* a 6.2, whereas for Silicon
Longitudinal Momentum Distributions and Correlations
Bottom quarks are produced predominantly in the central region of rapidity, as is illustrated in Fig. 3. However, the distribution is broad at ^/s = 2 TeV, with a full width at half maximum of 6 units (i.e. y from -3 to +3). The predicted doubly differential distribution10 cPa/dyidyt is shown in Fig. 4. Here j/i is the rapidity of a bottom quark, say, and yj is that of the associated antibottom. Positive correlations10 are evident, in that (t/j) increases with yi, but the doubly differential distributions are broader than is often assumed. A summary is presented in Table 3. One practical conclusion is that broad coverage in rapidity is necessary in order to observe both the b and associated b at collider energies. Table 3: Average value of 1/2 vs. yit full width at half maximum (FWHM), and average of 2 the absolute value of the difference t/i — j/2 for the distribution d a/dy\dy2 for pp —> bbX a.ty/s = 2 TeV.
In Fig. 5 I present the scaled longitudinal momentum distributions dafdxp for the inclusive processes n~N —* bX and pN -* bX at 600 GeV/c. I have defined xp simply as xp = ipLqfy/sy where, px,g is the longitudinal momentum of Q in the hadron-hadron center of mass system. There is a pronounced asymmetry of the distribution in xp for n~U —• bX; as a crude approximation, it is symmetric about xp = +0.1. Since the acceptance of fixed target experiments is often restricted to Xp > 0, the asymmetry evident in Fig. 5 suggests that ir~ beams are more efficient sources of 6 quarks than proton beams. There is very little change in the shapes of the xp distributions when the laboratory momentum is changed from 300 to 800 GeV/c.
It is common to parameterize the distribution dafdxp with the functional form
da ._ dxp v
The results shown in Fig. 5 are not well represented this way. A gaussian form seems more suitable. For pU -* bX, I use
and find Ap = 0.24. For ?r~J7, I use
da
and obtain Ar a 0.30 for XF > 0. Presented in Figs. 6 and 7 are predictions for pp —>• bX and pp —*• bbX at 800 GeV/c. The predicted doubly differential distribution10 in the Feynman xp variables of the b and 2 b quarks, d a/dxFidxF2, is shown in Fig. 7 as a function of xF2 for three selections on Xpi- The kinematic constraint XFI+XF2 < 1 is apparent in these results, as it should be. To first approximation, the width in XFI of dto/dxpidxpi is nearly the same as that of the single particle distribution dajdxp, shown in Fig. 6. Otherwise, there is a slight anticorrelation effect in the xp variable, summarized in Table 4; (xpi) decreases as xpi is increased.
Table 4: Average XFZ VS. XFI for pp —» bbX at pi»b = 800 GeV/c.
Xpt (xFi) 0.0 0.0 0.2 -0.020 0.4 -0.038 0.6 -0.074
It is important to remark that the longitudinal momentum distributions discussed here are distributions of the quarks, not of mesons or baryons. Final state interactions4 of the 6 or S quarks with spectator partons of the beam or target hadrons may modify these expectations, perhaps resulting in "leading particle" effects for which there is some evidence in the case of charm production.11 Bjorken has used a combination of dynamic and kinematic arguments to estimate these modifications.12
Transverse Momentum Distributions
For either a heavy Q or Q, the transverse momentum distribution do[kN QX)/dp\ is predicted to follow the approximate form
dolhN^QX) 1 a
where my = py + nig and, as noted, GhN is approximately a scaling function of mT/->/s. Correspondingly, the mean transverse momentum of a heavy quark increases with
(PT.Q) ~ mQ.
On the other hand, the QCD 2 —» 2 subprocesses provide a small value for the transverse momentum of the pair, {?$•"")• When the 2 -> 2 diagrams are dominant, one also expects that the Q and Q are produced back-to-back in the parton center-of-mass system. This implies that the distribution dtr/dA.
As remarked earlier, O (aj) subprocesses are expected to gain in importance when PT > rriQ. For pr "> mq, they provide a large correction to the O (a]) cross section and may generate significant values of (p?-*")- Since the O (a*) cross section is small at large PT, the effect of the O («J) terms on the K factor is not large for the overall cross section,5 integrated over pr- However, if attention is focussed on pj ^ mt, as it must be in searches for the top quark, the O (aj) subprocesses become very significant and provide, inter alia, bb events in which A
Restricting myself to the O (a*) contributions, I compute the average values of trans- verse momentum presented in Table 5; (pr) is essentially independent of xjr for |xjr| < 0.6. The values are an underestimate because I have ignored intrinsic transverse momentum of the incident partons and broadening of the pr distribution associated with O (aj) contri- butions. As indicated in the table, there is some growth of (px) with energy.
Table 5: Values of (p\.) in GeV at xF = 0.
| 320 GeV/c 600 GeV/c 800 GeV/c ir~U 2.32 2.61 2.74
PN 1.86 2.24 2.41
For pp -* bX at y/s = 1.8 TeV, I compute (pr) = 5.0 GeV at y = 0; the value drops to 4.2 GeV at \y\ = 3. References
1. E. L. Berger, Argonne report ANL-HEP-PR-87-90, Nucl. Phys. B, Proceedings Supplements: Proceedings of the Topical Seminar on Heavy Flavors, San Miniato, Italy, May, 1987, and Argonne report ANL-HEP-CP-87-53 (June, 1987) Proc. XXII Rencontre de Moriond, 1987.
2. R. K. Ellis and C. Quigg, Fermilab report FN-445 (2013.000).
3. J. C. Collins, D. E. Soper, and G. Sterman, Nucl. Phys. B263, 37 (1986).
4. S. J. Brodsky, J. F. Gunion.and D. E. Soper, Phys. Rev. D36, 2710 (1987).
5. P. Nason, S. Dawson, and R. K. Ellis, Fermilab report FERMILAB-Pub-87/222-T (Dec. 1987).
6. D. Duke and J.Owens, Phys. Rev. D30, 49 (1984).
7. WA78 Collaboration, M. G. Catanesi et al., Phys. Lett. B187,431 (1987); P. Pistilli, Nucl. Phys. B, Proceedings Supplements: Proceedings of the Topical Seminar on Heavy Flavors, San Miniato, Italy, May, 1987.
8. E. L. Berger and F. Coester, Ann. Rev. Nucl. Part. Sci. 37, 463 (1987).
9. UA1 Collaboration, C. Albajar et al., Phys. Lett. B186, 237 (1987).
10. E. L. Berger, Argonne report ANL-HEP-PR-87-113, Nov. 1987, submitted to Phys. Rev. D. 11. M. Aguilar-Benitez et al., Phys. Lett. 161B, 400 (1985) and Phys. Lett. 123B, 98 and 103 (1983).
12. J. D. Bjorken, "Hadronization of Heavy Quarks" and "QCD: Hard Collisions are Easy and Soft Collisions are Hard", to be published in the Proc. of the Advanced Research Workshop on QCD Hard Hadronic Processes, St. Croix, 1987. Figure Captions
Fig. 1 Cross section per nucleon a (pN —* bbx) from O (a2) perturbative QCD as a func- tion of laboratory momentum of the incident proton for various choices of bottom quark mass and evolution scale Q2. These curves are taken from Ref. 1 except for those at m» = 4.7 and 5.3 GeV which are from Ref. 2.
Fig. 2 Cross section per nucleon a (ir~N —* bb X\ as a function of the laboratory momentum of the incident n~. Results are presented for various choices of the bottom quark mass mt and Q2. These curves are taken from Ref. 1 except for those at mi = 4.7 and 5.3 GeV which are from Ref. 2. The measurement by the WA78 collaboration (Ref. 7) at 320 GeV/c is also shown.
Fig. 3 Distribution in rapidity dajdy for pp —» bX at y/s = 2 TeV. These results should be multiplied by 2 if one wishes the cross section for either borb. I use m» = 5.4 GeV, the Duke Owens Set 1 structure functions, and an evolution scale Q2 = S.
2 Fig. 4 Two particle rapidity distribution d a/dyidyJ as a function of yt for various choices of j/i for pp -* blX at V« = 2 TeV.
Fig. 5 Scaled longitudinal momentum dependence, da/dxp, of the inclusive cross sections per nucleon for ir~U —* bX and pU —> bX &t laboratory momentum 600 GeV/c.
Fig. 6 Scaled longitudinal momentum dependence, da/dxp, of the inclusive cross section for pp -» bX at 800 GeV/c. Here m» = 5.0 GeV, Q2 = mj, and the Duke-Owens Set 1 structure functions are used.
Fig. 7 Doubly differential distribution &2afdxp\dxpi for pp —* bSX at 800 GeV/c.
10 10 _ pN __». bbX
M3 .O 1.0
mb= 4.7 GeV nib = 5.3 GeV
0.1 1 400 600 800 1000 P.ab
Fig. 1 7T"U bbX
10.0
mb * 4.7 GeV
mb = 5.3 GeV
0.01 200 400 600 800 P,ab
Fig. 2 i i i i i r
= 2 TeV
1.0
>, 0.1 •o B T3
0.01 \- E pp -*- bX
mb = 5.4 GeV \
I I j I I I -6 -2 o y PP bbX
f mb = 5.4 GeV; ,/s = 2 TeV 1.0
0.1 I- 6 CM •o
0.01
0.001 -6 -4 -2 6
Fig. 4 100
t/ mb = 5.0 GeV \ \ : 2 m 2 Q = b \ f \ 0.01 i | I I A -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 XF
Fig. 5 100
10
XI c.
X
0.1
I 0.01 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 XF
Fig. 6 100
0.01 -0.6 -0.4 -0.2 0 0.2 0.4 0.6
Xp2
Fig. 7