Dynamics of bottom quark production in hadron collisions

Nuclear PhysicsB (Proc. Suppl.) 1B (1988) 425-446 North-Holland, Amsterdam

425

D Y N A M I C S OF B O T T O M Q U A R K P R O D U C T I O N IN H A D R O N COLLISIONS Edmond L. BERGER High Energy Physics Division, Argonne National Laboratory, Argonne, IL 60439 Developments in the dynamics of heavy quark production are discussed. My focus is principally on bottom quark production. I present extensive calculations of c r o s s sections and production spectra for both collider and fixed target energies. Available data are in excellent agreement with expectations of lowest order perturbative quantum chromodynamics. Uncertainties in the theoretical estimates are explored. The paper includes calculations and comments on charm and top quark production. 1. INTRODUCTION Reliable specification of the dynamics of heavy flavor production in hadron collisions is an important challenge for several reasons. From a theoretical perspective, heavy flavor production offers an opportunity to develop and test perturbative quantum chromodynamics (QCD) in a reaction which is of intermediate complexity. There are fewer Born diagrams for heavy flavor production than for hadron jet production at large PT, but more than for the relatively straightforward Drell Yan process. Moreover, heavy flavor production provides an additional variable in that the heavy quark mass may be changed. Experiments on the hadroproduction of charm I have become increasingly precise, bottom quark production has been observed, 2-5 and there was one report 6 (since retracted 7) of a signal consistent with the top quark. Precise understanding of production dynamics is essential for planning experiments to search for B, B mixing. From a practical standpoint, in order to extract a convincing signal for top quark production, it is crucial to understand the bottom quark cross section quantitatively at large transverse momentum, meaning the evaluation of higher order terms in the QCD perturbative expansion. Finally, it is important to assess the trustworthiness of the methods used to describe charm, bottom, and top production since the same procedures are used to estimate the production of supersymmetric, technicolor, and other postulated new particles at energies expected from the Large Hadron CoUider and the Superconducting Super Collider. In this paper I discuss theoretical developments in heavy quark production dynamics as well as recent data. I begin in Section 2 with a review of the second order O (~2) Born diagrams for heavy quark production in QCD and discuss the characteristic features expected in the data if these diagrams dominate the dynamics. In Section 3 I survey properties of data on charm quark production and contrast these with phenomenological expectations. The nuclear A dependence, "leading" charm meson effects, and discrepancy with theoretical expectations of the absolute normalization are evidence of a substantial contribution from non-perturbative effects in the hadroproduction of charm. 0920-5632/88/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

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x 0N

:

o

o 6

FIGURE 1 Born diagrams in QCD for the production of a heavy Q and heavy antiquark Q. Section 4 is devoted to bottom quark production. I summarize available data, discuss production dynamics in detail including the role of the third order O (a~) QCD contributions, explore various sources of uncertainty in the theoretical calculations, and present extensive calculations of cross sections and production spectra at both fixed target and collider energies. Available bottom quark production data at both fixed target and collider energies are in excellent agreement with expectations of lowest order perturbative QCD. This agreement, especially in absolute rate, may be contrasted with the historical difficulties in the case of charm production. The increase in mass of the produced quark, from rn~ ~ 1.2 - 1.8 GeV to rn~ ,~ 5.0 GeV, appears to be a prerequisite for quantitative reliability of the perturbative approach. The present level of agreement should encourage additional experiments. Measurements of xF spectra, PT spectra, b/b momentum correlations, energy dependence, absolute yield, and A dependence would test the theory further, including the role of O (a~) contributions, and constrain uncertainties such as the choice of evolution scale and appropriate mass of the bottom quark. In Section 5 I review the status of our understanding of top quark production and contrast aspects of the expected cross section as a function of top mass at CERN and Fermilab collider energies. Conclusions are collected in Section 6. Further discussion of many points treated here may be found in my Moriond paper. 8 2. BORN DIAGRAMS AND THEIR CHARACTERISTIC FEATURES In perturbative quantum chromodynamics, the lowest-order Born diagrams 9 for heavy flavor production are sketched in Figure 1. These are two-parton to two-patton subprocesses (2 to 2) in which either two gluons fuse (gg) or a light quark (q) annihilates with a light antiquark (~/) to produce a heavy quark (Q) and heavy antiquark (Q) in the final state. The initial partons (q, ~/,g) are constituents of the incident interacting hadrons, one incident parton supplied by each of the incident hadrons. The subprocess cross sections, b(gg ~ QQ) and b(q~l ~ QQ,), associated with the diagrams in Figure 1 are proportional to ~ , where ao(Q ~) is the usual strong coupling strength. Later I will discuss the scale Q2 at which a,(Q ~) is evaluated. The inclusive cross section a(hN --+ QX) for the production of a heavy quark Q in the collision of incident hadron h with target N is expressed as a convolution of the subprocess cross sections with probability densities f~(xi, Q2). Each fi (xi, Q2) represents the probability that a specific incident hadron contains at the scale Q2 a parton of type i with light-front

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fractional momentum xi. The convolution is expressed schematically as

d a ( h g ---} QX) ~-, ~_, a,b

f dxx dx2 fo(xl, Q2)fb(x2, Q2)dO(ab ----}QQ).

(1)

The internal gluons or heavy quark lines in the diagrams of Figure 1 are off-mass-shell by m~, where mQ denotes the mass of the heavy quark. For large enough values of toO, this means that the subprocesses qi/--, QQ and gg ~ Q ~ are dominated by short distance dynamics, and, correspondingly, that an approach based on QCD perturbation theory should provide a valid description of data. 1° Embodied in Eq. (1) is the important assumption of parton model factorization. The integrand is a product of three separate factors: a subprocess cross section, 5(ab --, QQ), and two probability densities which are assumed to be universal, process-independent properties of each of the incident hadrons. Because gluons can be exchanged between the spectator partons and the active partons in either the initial or the final state, the statement of factorization is by no means a trivial assumption. A demonstration of its validityn in QCD exists for massive lepton pair production, tile Drell-Yan process. For h N ---, Q(~X suggestive arguments have been given 1° but not a complete proof. Although the Drell-Yan process and h N ---, Q Q X are similar in some respects, a complicating factor in h N ~ Q Q X is that the heavy quarks carry color charge into the final state, whereas the massive virtual photon in h N --, "7*X is color neutral. The validity of factorization in h N --, QQ,X requires that rnQ be large. Important dynamical features of the 2 to 2 diagrams are these: i. The subprocess cross sections, ~, associated with the diagrams in Figure 1 are proportional to a2o/3, where ~ is the square of the parton-parton center of mass energy. The parton probability densities fi(xi) are rapidly decreasing functions of the fractional momenta xl; ~ = xlx2s; and s is the square of the total hadron hadron collision energy. As a result, the inclusive heavy quark cross section a(hN --* Q(~X) is expected to behave as

a(hN ----}QQX) ~x m-~QFhlv(mQ/V/~).

(2)

In addition to the m~ 2 behavior in Eq. (2), note the scaling function FhN(mQ/v/-d) whose behavior is determined by the parton densities in Eq. (1). It is therefore a rapidly decreasing function of mQ/v/S. (In expressing F(mQ/v/S-) as a scaling function, I have ignored Q2 dependences of a~ and f~(x,Q 2) in Eq. 1.) ii. The Q and Q are produced back-to-back in the parton-parton center of mass system. Heavy flavors are often identified by semi-leptonic decays: Q ~ Q~£tJ. Transverse momentum of the £ resulting from the decay process introduces some smearing, but back-to-back production of the Q and Q means that the decay leptons, g and ~, will also be found approximately back-to-back.

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iii. Because the parton densities fall off rapidly, Q and Q production is predicted to be central in the scaled longitudinal momentum, XF = 2pL/V/~, or the rapidity, y = ln(E + pL)/(E - PL). Here E, pL are the hadron-hadron center of mass energy and longitudinal momentum of the heavy quark. iv. For either a Q or a Q, the transverse momentum distribution, da(hN ~ Q X ) / d p ~ , obtained from the diagrams of Figure 1 has roughly the form

d a ( h g ~ QX)

where m T =

~+m

+

1

(3)

and, as noted, Grin is approximately a scaling function of

mT/V~. Correspondingly, the mean transverse momentum of a heavy quark increases with mQ: (PT,Q) ~ mQ.

(4)

On the other hand, the 2 to 2 diagrams provide a small value for the transverse m o m e n t u m of the QQ pair, (p~i~). In the absence of soft gluon radiation and of intrinsic transverse momenta of the incident partons, one would e x p e c t / p ~ ' ~ / -~ 0. A more realistic estimate of (ppair)is the measured value at rnQo : m ~ of ( p ~ ) i n massive lepton pair production (Drell-Yan) process hN --~ ~f~X.

the

v. For large enough values of 8, the Q and Q will be strongly correlated 12 in rapidity, the Q emerging within two units of y of the Q. At finite values of s, this positive correlation is offset to some extent 12 by kinematic effects. 3. CHARM Data on charm production have been reviewed extensively. 1 I call attention here to a few challenging aspects of the data. Except for an unusually large cross section measured at the CERN-ISR, the energy dependence of the charm cross section agrees with the expectations of simple perturbative QCD. The LEBC-MPS collaboration 13 measured a(pp --* D I D X) at V~ = 38.8 GeV and compared the value to that obtained earlier 14 at v/s : 27.4 GeV. The experimental ratio

a(pp --* DID X, ~

o(pp

D / D x,

= 38.8 GeV) : 24.4 GeV)

=

7 +0'6 .... 1

(5)

is in good agreement with the theoretical expectation of 1.7 to 2.4 for the same ratio. The range of values quoted here for the theoretical ratio reflects uncertainties associated with the choice of the charm quark mass. The overall normalization of the charm quark cross section is a serious problem. At V~ = 38.8 GeV, the LEBC-MPS collaboration measured a(pp ~ DID X) = 34.4 ± 4.2/~b. In perturbative QCD, the calculated cross section a(pp ~ c~ X) is highly sensitive t o the choice of the charm quark mass rnc. As mc is reduced, the estimated value of a(ce) grows

E.L. Berger / Dynamics of bottom quark production

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dramatically, approaching to within a factor of two of experiment for m~ = 1.2 GeV. Unfortunately, theoretical confidence in the answer drops as m~ drops. High twist terms have been neglected in Eq. (1) because they are "suppressed" by a factor of 1/m~ relative to those retained. For m~ ~ 1 GeV, the high twist terms are likely to be comparable in magnitude to those computed, t5 In addition, when mo ~ 1 GeV, neglected higher order terms in the perturbation series, such as terms proportional to a~, are also likely to be as large as the a l terms. The message I draw from the A dependence, 16 leading charm effects, 17 and normalization discrepancy is that non-perturbative and higher,order effects play a big role in the hadroproduction of charm. Theory and data are not closely related. It is releyant to recall that lowest order perturbative QCD does not describe hadronic jet production at Pr ~ 1 to 2 GeV. Disappointing as it may seem, it is not surprising that the simplest 2 --* 2 QCD perturbative approach to charm production does not work reliably either. Purely perturbative estimates of the charm cross section should nevertheless be valid for charm production at large Pr, where, however, the O (a~) contributions are expected to be large (see Section 4.3). 4. B O T T O M QUARK P R O D U C T I O N The mass of the b o t t o m quark is approximately 5 GeV, comparable to the values Pr ~> 5 GeV at which hadron jet physics is described reasonably successfully. We may pose two questions: Should we expect simple QCD perturbation theory to be reliable for bottom quark production, in spite of its shortcomings for charm? Does it describe data successfully? The answer is yes to the first of these, and yes, within a factor of two, to the second. 4.1. Data The data in question have been reported recently by two CERN collaborations. The UA1 collaboration s observes pairs of opposite sign muons at large values of muon tlansverse momentum. After a careful analysis which includes a considerable dose of~Monte Carlo simulation, the group identifies the process ~p --* bb X; b --~ # - X , b --~ iz+X, and reports a cross section at x/s = 630 GeV,

,bX;

_> 5 GeV, I,I < 2.0) = (1.1-4-0.1 + 0.4)/zb.

(6)

Note the restriction on the transverse momentum of the b quark, p~, > 5 GeV, and on the pseudorapidity of the b quark, [~/1 < 2.0. The other measurement was reported by the CERN-WA78 collaboration. 4,5 Directing a 320 GeV 7r- beam on a uranium target, they observe events having three identified muona plus large missing transverse m o m e n t u m (E~ i'' > 50 GeV). These events are associated with the reaction

~r-U ---+bb X; b ~ ~- + e(--, ~,+X) + X;

(7)

~ #+ + ~ ( ~ ~ - X ) + X. If nuclear dependence is assumed to be of the form a oc A ~ with a = 1, the cross section quoted per nucleon is a(~r-N --~ bbX) = 4.5 4- 1.4 4- 1.4 nb. (8)

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Evaluation of the experimental acceptance in the extraction of this cross section is based on assumptions about the XF and PT dependences of b quark production. The assumed value pb) 1 GeV in Ref. 4, while consistent with charm data, is inconsistent with theoretical estimates for bottom quark production. Using instead the value (P~'/ = 2.1 to 2.3 GeV suggested the0retically, 18 as well as the form of the XF distribution computed theoretically, 18 the WA 78 group computes a larger acceptance for b quark production and, correspondingly, a revised, reduced cross section s a(~r-N ~ bbX) = 2.4 ± 0.7 + O.8nb.

(9)

4.2. Production Dynamics Novel mechanisms were introduced during the past decade in attempts to describe data on charm production. While some or all of these may be relevant for charm, with mass mc ~ 1 GeV, they are not judged relevant for the production of bottom quarks with mass mb --~ 5 GeV. For example, the probability that there is an intrinsic heavy quark component 19 in the initial hadron wave function falls as m~ ~. This means that the intrinsic heavy quark provides a higher-twist contribution to a ( h N -~ bX), suppressed by mb ~ relative to the QCD perturbative contribution in Eq. (1). It is relevant for bottom quark production only in special, restricted parts of phase space. Likewise, diffractive 2°, recombination, and final state prehinding contributions are either included in the perturbative expansion 1° or belong to the higher-twist class. Last, but not least, the "flavor excitation" mechanism, 21 gO - ' gO and qQ - , qO, deserves somewhat more detailed discussion. The arbitrarily large cross section associated with "flavor excitation" diagrams continues to be a source of confusion. In computations made with flavor excitation diagrams, the initial heavy quark Q is assumed to b e on its mass-shell. Consequently the scattering amplitudes are divergent, and finite integrated cross sections a ( h N --, bX) are obtained only after arbitrary cutoffs are introduced. Numerical results are obviously sensitive to the choice of this cutoff. The resolution of this confusion lies in the more careful examination of the hard-scattering expansion, m The "flavor excitation" diagrams are included within the class of perturbative 2 --~ 3 subprocesses su~:h as qg -~ qQQ. A summary of this subsection is that perturbative QCD should provide a reliable description of the production of sufficiently heavy quarks. In the next subsection, I address the status of our understanding and quantitative evaluation of the perturbation expansion for a ( h N ---, bX). This involves a discussion of the role of the O(a~) perturbative QCD 2 --, 3 diagrams and the status of their evaluation. Subsequently, I will review other more prosaic sources of uncertainty, show comparisons with data, and provide various predictions. 4.3. Two-to-three subprocesses Examples are shown in Figure 2 of QCD subprocesses in which two partons interact to produce a QQ pair plus a third (light) parton in the final state. Cross sections associated with these diagrams are proportional to a~(Q2). Note that in addition to qq --~ QQ g and gg ---, QQ g, there are subprocesses initiated by gluon-quark interactions, qg ~ O(~ q.

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As reviewed in Section 2, the O(a~) 2 -+ 2 subprocesses yield QQ pairs in which the Q is back-to-back with the (~ in the parton-parton center of mass system. The 2 --, 3 subprocesses provide new configurations in phase space. Among the possibilities are: Configurations of three hadronic jets in which two of the jets include leptons, from Q --* ~ X and (~ ~ fi X. The Q and (~ and, therefore, the final leptons would not be back-to-back. ii. Situations in which two heavy quarks emerge close together in phase space, balanced

in transverse momentum by a quark or gluon jet. In this case (p~atr/ will be fairly large. iii. Topologies in which one heavy quark emerges at large angle with respect to the beam direction, balanced in transverse momentum by a light parton, with the other heavy quark emitted close to the direction of one of the incident hadrons. This topology would tend to place only one of the final leptons (Q --* #X) in the central region of a typical collider detector. Although the 2 --* 3 subprocesses yield topological configurations which differ from those of the 2 ~ 2 set, one cannot add cross sections, 8(2 --, 3) to 8(2 ~ 2). There are divergences and cancellations to consider. The 2 --* 3 amplitudes are divergent when considered by themselves. These unphysical divergences arise, for example, when the four momentum transfer t vanishes, t --* 0, in Figures 2(c) or 2(e). However, these leading divergent parts of the 2 --* 3 amplitudes are not new contributions. Only the finite non-divergent parts are new. The divergent parts are already included in what is meant by the full (renormalized) 2 --* 2 subprocesses. Indeed, when t is small in Figure 2(e), the exchanged antiquark is close to its mass shell and, therefore, really an antiquark which is already included in the scale dependent antiquark density of one of the incident hadrons. The true QCD subprocess is ~/q --, QQ. Likewise, when t is small in Figure 2(c), the exchanged gluon is close to its mass shell and, therefore, in reality a gluon which is already included in the Q2 dependent gluon density of one of the incident hadrons. The QCD subprocess in this case is the 2 --* 2 reaction gg ~ Q(~. Although the technology is far from trivial for extracting finite contributions from the 2 -* 3 amplitudes, the concept itself is straightforward. The same situation is familiar in deep inelastic lepton scattering which is used to define what is meant by a Q~ dependent quark structure function, qi(x, Q2). In addition to mastery of divergences, specification of the full finite O(a3,) contribution to a(hN ~ bX) requires calculation of interference terms between O(g 4) amplitudes and O(g 2) amplitudes (a, -- g2/41r). This interference contribution may be negative, resulting in cancellations. It has become common to represent the effects of higher-order terms in the perturbative

E.L. Berger / Dynamics of bottom quark production

432

q

(a)

Q

g

g

~ ? "5

o ,o

(b)

(c)

g

(d)

q

5 (e)

FIGURE 2 Examples of QCD subprocesses in order a,.3 QCD expansion in terms of a "K-factor', defined as follows:

d ~ ( h g --, bX) do (2)(hN -~ bX) do (a) ( h N --, bX) = + dp~dxF dp~dxF dp~dxF , do(2)(hN --+ bX)

+o(o:) (10)

On the right hand side of Eq.(10), a {2} represents the O ( a ,2) contribution calculated from the 2 -~ 2 diagrams shown in Figure 1, and a (3} represents the full finite O(aa,) contribution. Tile complete O(aa,) calculation with massive quarks has not been finished. 22 As a result we do not know the size of the residual non-divergent O(aa,} contribution, nor its dependence on XF and PT- Rough approximations exist. For example, in the I S A J E T 23 and E U R O J E T 24 simulations, 2 --+ 3 graphs are included with divergences "removed" by the insertion of a transverse momentum cutoff, p¢,t T --~ 5 or 6 GeV. It is not evident how safe this procedure is. Numerical insensitivity of the answer to the choice of this cutoff is not the issue. Effects of the quark gluon subset of 2 --- 3 graphs, qg --+ QO, q, were studied by R. K. Ellis. 25 For [XF[ not close to 0, a slight reduction of the cross section is predicted,

Kb(s, I FI > o) <

1.

The gluon-gluon subset of 2 --, 3 diagrams, gg --+ gQQ, is believed to be very important for at least two reasons. First, gauge cancellations suppress the magnitude of the 2 -+ 2 process gg --, Q(~. Second, large color factors enhance the cross section for gg --, gg,

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FIGURE 3 Leading pole approximation for the 2 ~ 3 cross section 8(gg ~ gQQ). suggesting that subprocesses such as sketched in Figure 2(c) will provide impressively large K factors. 2e Short of a full O (a,3) calculation, the leading pole approximation sketched in Figure 3 offers the best estimate available at large PT.Q of the effects of the gg --* gQQ diagrams. In this approximation, ~(gg ---. gQQ) is represented by the product of three factors: i) the cross section ~(gg --* gg), ii) the gluon propagator s~ t, where sg is the square of the invariant mass of the intermediate gluon, and iii) tlle splitting function ~_.Q~} for g --~ QQ. One obtains

where p (~,m~) is the multiplicity of QQ pairs in a gluon jet27:

. . . .

a (8,)

1+

-

n , (,~,s,).

(12)

In Eq. (12), ng is a gluon multiplicity factor; 27 p becomes significant for Pr,q ;~ 2mq. Equations (11) and (12) do not provide the expression of most direct interest, which is the full O (a,~) + O (as) cross section as a function of the transverse momentum Pr,o of a single heavy quark. However, they can be used to argue that O (a,s) contributions are potentially very large for PT.Q ~> rnQ. Indeed, the first factor on the right hand side of Eq. (11) is known to be large. At 90 ° in the gg center of mass frame,

a(gg (g9

~ gg) ~- 200.

QQ)

(13)

Using a numerical evaluation 27'~s of p (3, m~), we may estimate a K factor of as much as 10 for b quark production at PT.b --~ 50 GeV. Although the K factor estimated above is very large at large Pr, the O (as2) cross section is itself quite small for Pr >> rob. Therefore, for quantities such as do/dxF or a(b~) integrated over all PT, the net K factor should be modest, perhaps no larger than 2. 4.4. Other Sources of Uncertainty As summarized towards the end of the previous subsection, the full K-factor associated with O (a,3) subprocesses has not yet been computed. Computations of the cross section for the Drell-Yan pl'ocess suggest that the K-factor should be about 2 for the overall cross

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E.L. Berger / Dynamics of bottom quark production

section a (hN ~ bbX) and for the inclusive rapidity or Feynman xe distributions, integrated over p~: da(hN ~ bZ)/dy or d a ( h g --, bX)/dxf. For p~ >> rob, however, the K factor could have a very substantial effect on da(hN ~ bX)/dp~. In this subsection I list and discuss more prosaic sources of uncertainty:

Poor knowledge of the gluon density distribution G (x, Q2) in the relevant intervals of x and Q2. One way to evaluate the impact of this uncertainty is to repeat calculations with various parametrizations on the market. 29,~° ii. Choice of the evolution scale Q~. When the full O (a~) calculation is complete, we will have better guidance. For the present, choices include Q~ = m~, 4m~, and 3 (3 is the square of the p a r t o n - p a r t o n center of mass energy). The same choice should be m a d e in the evaluation of a,(Q 2) as in the computation of the quark, antiquark, and gluon structure functions. iii. Choice of the value of rob. I will use a nominal value m b = 5.0 GeV but consider alternatives within the range mb= 5.0 + 0.4 GeV. Changes in the value of the evolution scale Q2 produce several effects. An increase in QZ leads to a decrease in a , , resulting in a reduction of a(hN --~ bbX) oc a,.2. Ots2 (4m~) Ot2

[b]'m2~ --0.68.

(14)

However, this is not the whole story. A larger value of Q2 means that more gluonic evolution takes place, increasing the value of the gluon m o m e n t u m xG(x) density at small x but decreasing it at large x. For x < 0.07, xG(x) rises as Q2 is increased, whereas xG(x) falls with Q~ for x > 0.07. The net effect on the production cross section of a change in Q2 from m~ to 4m~ therefore depends on whether the cross section is sensitive to values of x greater t h a n or less than ~ 0.07. The reaction kinematics determine the values of x~ at which the parton densities are sampled. Very roughly, x~ " 2m~/v'~. As x~ is increased, one of the xi's increases towards 1, and the other decreases. The upshot is that experiments at different energies sample parton densities in different intervals of xi: At typical C E R N and Fermilab fixed target momenta, values of x~ ~> 0.1 are sampled. Therefore, larger values of the evolution scale result in a decrease of the predicted b o t t o m quark cross section. ii. At the energies of the C E R N and FNAL pp colliders, the typical values of xi are near enough to 0.07 so that predicted b o t t o m quark cross sections are fairly insensitive to both the choice of the evolution scale Q2 and the assumed form of the x dependence of the gluon density.

E.L Berger / Dynamics of bottom quark production

435

iii. At the SSC, ~ = 40 TeV, values of xi down to 10 -6 and 10 -~ would be sampled in b o t t o m quark production. Thus, the choice of a larger value for the evolution scale Q2 can lead to very substantial increases in expected yields, sl 1

b

I

i 10.0

10

I

5p -,-- b~X

I

I

I

I

I

I

2 TeV

lyl < 2

630

GeV

1.0

.¢1 1.0

vri=

2000

GeV .Q >,

AI_ Q.

0.1 r

b "o

m b =

b

5.4 GeV

'

Q2=~ 0.1

Q2 - j ~ ffi 6 3 0 GeV

0.01

I

0.01 0

4

8 12 p~in ( G e V )

2 = m b

16

20

-

I -2

I

I

I

I

I

-1

0

1

2

3

y

FIGURE 4

FIGURE 5

Cross sections from O (al) perturbative QCD for b-quark production in protonantiproton collisions for PT > Pr.,. and - 2 < y < 2 as a function of PT.,.. The cross sections are given for the C E R N Collider, V~ - 630 GeV, and for the Fermilab Tevatron, vzJ = 2000 GeV. The result Eq. (6) reported by the UA1 group [Ref. 3] at PT,,,, ----5 GeV is also plotted.

Distributions in rapidity da/dy for pp ~ bX at V~ = 630 GeV and 2 TeV. These results should be multiplied by 2 if one wishes the cross section for either b or ~. For the solid curves, I use rnb = 5.4 GeV, the Duke Owens Set 1 of structure functions, and an evolution scale Q2 = ~. If Q2 is changed to Q2 _ m b 2 the dashed curves are obtained.

4.5. Collider Energies In Figure 4 I present the calculated integral PT spectrum,

v>p~h' dp T

dpy ,

(15)

for pp -* b X at ~ -- 630 GeV and V~ = 2 TeV. The results in Figure 4 include a further restriction on the rapidity y of a b quark: lYl < 2. The theoretical curves 32 are based on the O ( a ,2) subprocesses alone. The b o t t o m quark mass is m b = 5.4 GeV; the Duke-Owens Set 1 structure functions 29 were used; and the Q2 evolution scale was set at Q2 = ~.

436

E.L. Berger / Dynamics of bottom quark production

At v/s = 630 GeV, I compute a(ly I < 2, PT _> 5 GeV) --= 1.0 ~b in fine agreement with Eq. (6). The calculated cross section is increased by 15% for a b quark mass of 5.0 GeV. Likewise, the EHLQ Set 1 structure functions 3° also yield a 15% increase. Changing the scale Q2 from ~ to m~ yields an increase of 50%. Since all these changes increase the predicted cross section, it is tempting to suggest that the UA1 experimental cross section is somewhat underestimated. For a(ly I < 2, PT _~ 5 GeV) I estimate an overall theoretical uncertainty of a factor 2 in either direction, including the effect of higher order corrections. Integrating over all y and PT, I compute cross sections a(pp ~ bb X) = 3.7 #b at V~ = 630 GeV and 15.4 #b at v ~ = 2 TeV. The agreement of d a t a and theory in Figure 4 is important confirmation of QCD in an area in which there had been doubts. Further experimental results are needed. An essential feature of the QCD prediction is that the heavy quarks are produced with large transverse m o m e n t u m , PT "~ mQ. In order to verify this, one would like to know t h a t a (lYl < 2, PT > PT,,,,,,) does not continue to rise as PT decreases, but levels off when PT < mQ, as shown in Figure 4. Another important feature of the QCD prediction is that heavy quarks are produced predominantly in the central rapidity region, illustrated in Figure 5. Note in Figure 5 t h a t a decrease in the evolution scale Q2 from ,~ to m~ leads to a slight increase ( ~ 20 to 30%) in the cross section at V~ = 630 GeV but a slight decrease at V~- = 2 TeV. For b o t h the Duke-Owens and EHLQ Set 1 structure functions, the gluon density at the reference scale Q02 behaves as xG(s, Q~) ---* constant as x ~ 0. On the other hand, for the EHLQ Set l-A, x G ( x , Q~) ~ x -1/2 as x ~ 0. This very different behavior at small x has a strong influence on predicted rapidity distributions at SSC energies. 31 However, there is little sensitivity at C E R N and FNAL collider energies. For example, at V~ = 2 TeV and evolution scale Q2 = 4m~, I find that the EHLQ Set 1 and EHLQ Set 1-A structure functions yield the same values for d a / d y at y = 0, whereas the EHLQ Set 1-A prediction exceeds t h a t of the EHLQ Set 1 by only a factor of 1.7 at lYl = 3. For comparison with ISR data 3~ at v/~ = 62.4 GeV, I compute a(pp ~ bbX) = 44.6, 78.1, and 140 nb for the choices m b = 5.4, 5.0, and 4.6 GeV. I set Q2 = m~ for these calculations at V~ = 62.4 GeV. 4.6. Fixed Target Energies The study of b o t t o m quark production at fixed target energies is interesting for several reasons. Tests may be made of QCD predictions over a broad range of energies, transverse m o m e n t a , and scaled longitudinal m o m e n t u m xF, as well as with a variety of beams and nuclear targets. A large sample of data would permit studies of B , / 3 mixing, lifetimes, and, possibly, of CP violation in the B meson system. In this section I present expectations TM for the b o t t o m quark cross section based on O (a~) perturbative QCD. I use rnb = 5.0 GeV as the nominal value of the b o t t o m quark mass, but I comment on changes associated with variations over the range 4.6 < mb < 5.4 GeV. I base most calculations on the Duke Owens set 1 structure functions for the

E.L Berger / Dynamics of bottom quark production

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Cross section per nucleon a (~r-U --+ bb X ) as a function of the laboratory m o m e n t u m of the incident ~r-. Results are presented for three choices of the evolution scale, Q2 = 3, 4rnb2, and mb2. The m e a s u r e m e n t by the WA78 collaboration at 320 GeV (Eq. 9) is also shown.

s t r u c t u r e functions. R e s u l t s are s h o w n for three choices of the evolution scale, Q2 = 3,

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proton. T h e Duke Owens set 2 and the EHLQ set are also considered. In Figure 6 I present the cross section o(pp --, bbX) as a function of laboratory mom e n t u m . Over the fixed target m o m e n t u m range shown, the choice of Q2 = m~ as the evolution scale results in cross sections larger than for Q2 = 4m~ or Q2 = }. The difference between cross sections for Q2 = } and Q2 = ra~ is a factor of 3.3 at/~ab = 400 GeV and 3.1 at P]ab = 1000 GeV. The two estimates become equal at ~/~ ~- 1000 GeV, above which the choice Q2 : } yields the greater answer, for reasons discussed in Section 4.4. Because the gluon density is harder in the Duke Owens set 2, use of this set results in larger cross sections. For example, with Q2 = 3, o(pp --, bbX) is increased by factors of 1.6, 1.4, and 1.25 at/hab = 400, 600, and 800 GeV. Over the range 400 < Plab < 1000 GeV, the expected cross section o(pp --~ bbX) is decreased by about a factor of two when the b quark mass is increased from m b = 5.0 GeV to tab = 5.4 GeV, and increased by about a factor of two if the b quark mass is decreased from 5.0 GeV to 4.6 GeV.

438

E.L. Berger / Dynamics of bottom quark production

Experiments are often done with nuclear targets in an effort to enhance the overall event rate, expected to grow roughly linearly with baryon number A: a ~ aoA 1. Several types of "nuclear effects" modify this simple expectation. First, there are distortions associated with the different x dependences of the up and down quark densities. Second, there are modifications associated with the nuclear bound state 3s, the "EMC effect". Third, shadowing 36 must be considered if the production kinematics are such that the Bjorken x of the parton from the nuclear target satisfies xN <~ 0.1. Finally, there should be some nuclear broadening of the transverse m o m e n t u m distribution. 37 I will not comment further here on either shadowing or PT broadening. Pinning down an EMC type effect in heavy quark production is difficult. It is "only" a =E15% distortion of the quark m o m e n t u m distribution, an effect easily submerged in other more significant uncertainties, notably the choice of the appropriate evolution scale, Q2. The most easily calculated nuclear effect is that associated with the n e u t r o n / p r o t o n ratio. Considering only effects associated with the n/p ratio, I find that the cross section per nucleon for pA --~ QQ X 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. This conclusion rests on the assumption of SU(2) s y m m e t r y of the antiquark densities (~n(x) -- [iN(X)), the usual assumption of isospin s y m m e t r y (e.g. up(x) = d,,(x)), and the assumption that the gluon densities in protons and neutrons are identical. For p N and l r - N collisions there is a difference between the cross sections. ~ The cross sections per nucleon for Q(~ production from a nucleus are expected to be smaller than the cross sections for production from proton targets. 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 m o m e n t a of 200, 400, and 600 GeV. The cross sections per nucleon for ~r-U are very slightly smaller than those for r - D . Correspondingly the calculation.s presented here for ~r-U may be used for comparisons with d a t a for all A in the range D < A < U. It is perhaps worth emphasizing here that if a(1r-A --* bbX) is measured for several values of A and the results are then extrapolated to A = l, the effect of the n / p ratio will lead to a cross section at A -- 1 smaller than that for a(~r-p -~ bbX). This expectation m a y be contrasted with data for charm production in which the extrapolated cross section is greater than the proton cross section. 1 For the quark and antiquark structure functions of the pion I use Q2 dependent tributions consistent with data 38 on massive lepton pair production. In ~ - N ~-~ bbX, ~q -~ bb contribution accounts for 79%, 68%, and 53% of the cross section at 200, 400, 600 GeV. This implies that uncertainties in knowledge of the gluon density in the ~rnot the dominant uncertainties in the estimates of the cross section.

disthe and are

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o ( ~ - p ---, bbX) at 400 GeV. Another significant advantage of pion beams is apparent in the XF spectra, da/dXF. In Figure 8 I present the scaled longitudinal m o m e n t u m distribution da/dxF. There is a pronounced asymmetry of the distribution in XF for ~ - U ~ bX. At XF = +0.4 dG/dxF is a factor of about two greater than at x F ----- - - 0 . 4 , and at XF = +0.6 it is a factor of about 18 greater than at xF = --0.6. As a crude approximation, the distribution da/dxF for Ir-U --~ b X at 320 GeV is symmetric about 37F = +0.1. The distribution d a / d x F for pU ~ b X is symmetric about XF = O. For all XF < O, da/dxF(pU) is about a factor of 10 smaller than dcr/dxF(Tr-U) at 320 GeV. However, for XF > 0 the ratio a ( l r - U ) / a ( p U ) grows rapidly with xF, from -~ 18 at .TF = 0 . 2 , t o ' ~ 50 at XF = 0.4 and ~- 350 at XF = 0.6. There is very little change in the shapes of the XF distributions when the laboratory m o m e n t u m is

440

E . L Berger / Dynamics of bottom quark production

changed from 320 to 600 GeV/c. It is common to parameterize the distribution da/dXF with the functional form da dXF ¢x

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4.7. S u m m a r y - - B o t t o m Quark Production Because the b o t t o m quark mass is large, m b - 5 GeV, QCD perturbation theory is expected to provide quantitatively reliable predictions for production cross sections. Nuinerical uncertainties are associated with choices of the quark mass, the evolution scale Q~, and the overall K factor connected with O (a,s) and higher contributions in the perturbative expansion. At the CERN and Fermilab eollider energies, the overall numerical uncertainty from all these sources is approximately a factor of two in either direction. The numerical sensitivity to the choices of mb and the evolution scale is greater at fixed target energies where structure functions are sampled at larger values of x. I estimate an uncertainty of a factor of two in either direction associated with variations in mb over the range m b = 5.0 + 0.4 GeV, a further uncertainty of a factor of 3 in pp and of a factor of 2 in ~rN collisions associated with the choice of evolution scale, and perhaps an additional factor of 2 attributable to higher order QCD contributions. Available data at both collider and fixed target energies are in excellent agreement with quantitative expectations. This suggests that further data especially at fixed target energies could tightly constrain the choices of mb and Q2. 5. P R O S P E C T I N G FOR T O P For quarks such as top (t) whose mass is of order row~3 or greater, an additional production mechanism becomes important. Besides the purely hadronic subprocesses qr/--* t~ and gg --* tt, it is necessary to consider production via the intermediate vector bosons: qr/--* W -+ tb and q~/-* Z ° -* tt. Cross sections are shown in Figure 10 as a function of rnt. At V~ -- 630 GeV, QCD subprocesses are dominant for mt < 40 GeV and rnt > 78 GeV, whereas the W --* tb mechanism wins for 40 < m t < 78 GeV. At V~ = 2 TeV, the QCD subprocesses are dominant for all values of mr. The region m, >~ 50 GeV is of particular interest in view of the observations of B, ]~ mixing. The W cross section increases by about a factor of 3 from V~ --- 630 GeV to 2 TeV, whereas the tt cross section grows by a much bigger factor (~ 20 at m t = 60 GeV). Taking both effects into account, we see that the expected increase in yield for top quarks is about a /actor o / 7 for 40 ~ rnt ~< 70 GeV. The FNAL advantage becomes increasingly spectacular as mt is increased above rnt ~> 70 GeV. For rnt = 40 GeV, the expected integral PT spectrum is shown in Figure 11 for top quark production at v ~ = 630 GeV and V~ = 2 TeV. These calculations include only

442

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Solid (dashed) lines show the expected integral PT spectrum for top (bottom) quark production at v ~ = 630 GeV and V~ = 2 TeV as a function of the minimum transverse momentum of the heavy quark. The rapidity of the heavy quark is restricted to lYl < 2. For these calculations I use m t = 40 GeV, mb = 5 GeV, and the evolution scale Q~ = m~.

E.L. Berger / Dynamics of bottom quark production

443

the O (a,2) QCD contributions. A comparison of Figures 4 and 11 illustrates the fact that (p~) ::~ (p~), in accord with Eq. (4). For the expected large values of ra,, the 2 --* 3 0 ( a , s) QCD contributions are much less important than for bottom quark production. With rnt = 40 GeV, x/s = 630 GeV, and CERN's integrated luminosity of 690 nb -1, approximately 100 events are predicted from each of pp ~ t t X ;

t --* ~vb

and

(19) pp --* W ± X ;

W + --* tb, t --. I~b.

For a sufficiently massive t quark, the decay muon from t --,/~vb will often be well separated in phase space from the hadrons associated with the b quark decay. Therefore, an appealing signal from t --* pt~b is an isolated ~ plus at least two hadronic jets: one jet from the b decay, and other jets from either the [ quark (tt mechanism) or the b quark (W --* tb mechanism). Unfortunately for the identification of the top signal, there are other substantial sources of isolated muons accompanied by jets, including W + jet production, and b~ and ce production. The magnitude of the potential background from bottom quark production is illustrated in Figure 11. For tile parameters chosen, the integral PT spectrum for production of a bottom quark with Pr,b > 40 GeV at ~ = 630 GeV is a factor of 4 greater than that for a top quark with PT,t ~ O. At ~ -- 2 TeV the factor drops to about 2. I choose to compare a(pT,b _> 40 GeV) with a(pT,t _> 0)because a 40 GeV bottom quark should yield a decay muon of about the same momentum as a top quark produced at PT,t = 0. A more selective PT cut improves signal to background but at the expense of event rate: e.g. a(pT,b >_ 60 GeV) is less than a(pT,t >_ 20 GeV). Note that the results in Figure II represent the O ( ~ ) contributions only. As discussed in Section 4.3, the O (a,2) estimate should be reliable for top production for the PT values shown in Figure I i but may underestimate bottom quark production by as much as a factor of 10 for PT,b ~> 50 GeV. Using isolation criteria plus selections against bbg events, the UAI collaboratione'2s identified six events in the 1983 data sample (II0 nb -I) consistent with production and semi-leptonic decay of a top quark whose mass lies in the range 30 < rat < 50 GeV. As reported in detail as this meeting,7 an analysis of the complete data sample fails to confirm a signal in excess of expected backgrounds. Using their evaluation of the predicted QCD cross section for pp --~ t t X , the UAI collaboration quotes a lower limit rat :> 55 GeV at 95% confidence level (CL), whereas for one-half of this estimate, m t > 44 GeV at 95% CL (rat > 55 GeV at 90% CL). Further progress in identifying the top quark in hadron data requires solid understanding of the expected magnitude of the 2 to 3 cross section and of the topological distribution in phase space of the final jets from the 2 --, 3 processes ab --~ bbg, as well as reliable quantitative models of the semileptonic decays b --, # X and c --~ # X for large p~el. 6. CONCLUSIONS I summarized my overall perspective in the Introduction. In the case of charm it is

E.L. Berger / Dynamics of bottom quark production

444

desirable to develop a consistent experimental picture of hadroproduction dynamics: A dependence, including its variation with XF; absolute value of a(ce); XF spectra; and PT dependence for PT > 1 GeV. However, there is already evidence that the theoretical interpretation of these results will not be simple. Because mc and (PT,c) a r e small, non-perturbative and higher order perturbative effects are likely to be large. Purely perturbative calculations should be reliable at large PT.¢ where, however, O (a,s) contributions will be large. A solid understanding of charm production at large PT is essential for precise experimental determination of bottom quark cross sections. Bottom quark production appears to offer a valuable new process in which to test perturbative QCD in detail. Data are becoming available and agree within limited numerical uncertainties with predictions. Because mb is large, theory is expected to be quantitatively reliable for all PT,b- A full O ((~) calculation is promised. 22 A systematic program of fixed target measui,ements appears fully warranted: absolute yields, energy dependence, XF and Pr dependences, bib momentum correlations, as well as nuclear A dependence. The top quark remains elusive. Is it hiding in the 690 nb -1 of collider data at CERN, or must we await the factor of 7 to 10 increase in yield promised at ACOL or at the Fermilab Tevatron? The clear intellectual challenge is to identify a signal in the presence of very substantial bb and bbg backgrounds. ACKNOWLEDGMENT I have benefitted from several discussions with John C. Collins. This work was supported by the U.S. Department of Energy, Division of High Energy Physics, Contract W31-109-ENG-38. REFERENCES 1)

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32)

E. L. Berger, J. C. Collins, and D. Soper, Phys. Rev. D35 (1987) 2272.

33)

M. Basile et al., Lett. Nuovo Cimento 31 (1981) 97; D. Drijard et al., Phys. Lett. 108B (1982) 361.

34)

Complementary calculations are reported by R. K. Ellis and C. Quigg, Fermilab report FN-445 (2013.000).

35)

For a recent review of data and interpretations, see E. L. Berger and F. Coester, Argonne report ANL-HEP-PR-87-13, to be published in Ann. Rev. of Nucl. Part. Sci. 37 (1987) 463.

36)

See, for example, J. Qiu, Columbia University report CU-TP-367.

37)

G. T. Bodwin, S. J. Brodsky, and G. P. Lepage, Phys. Rev. Lett. 47 (1981) 1799; C. Michael and G. Wilk, Z. Phys. C10 (1981) 169.

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J. Badier et al., Z. Phys. C18 (1983) 281; B. Betev et al., Z. Phys. C28 (1985) 15; C. B. Newman et al., Phys. Rev. Lett. 42 (1979) 951; S. Palestini et al., Phys. Rev. Lett. 55 (1985) 2649.