- Email: [email protected]
Heavy flavor production
Nuclear Physics B (Pro¢. Suppl.) 12 (1990) 219-225 North-Holland
219
HEAVY FLAVOR PRODUCTION John C. COLLINS Physics Department, Illinois Institute of Technology, Chicago, IL 60616, U.S,S.. I summarize the situation for the theory and phenomenology of the production of heavy colored quanta in hadron-hadron comsions.
1 INTRODUCT~N In this talk I consider the production of heavy colored quanta in hadron-hadron collisions. Now, production of such objects involves an internal line that is off-shen by an amount that is at least about the mass of the heavy flavor. Thus there is a hard scattering subprocess in the collision, and one expects to be able to discuss the process within perturbative QCD. The experimentally discovered heavy flavors are so far the charm and bottom quarks. Potential candidates for future discovery include not only the top quark, of course, but also gluinos and squarks, for example. The reasons for discussing these processes include the following: (a) They give a probe of hard scattering at scales O of order me, ms in hadron-hadron collisions, where jets are hard or impossible to measure well. (b) The cross sections are needed when searching for new heavy flavors. (c) Often we have V~ ) MO, where MQ is the mass of the heavy flavor; in that case we probe the small-z region. Perturbative methods are then useful, but we are also probing a region of Regge kinematics. Charm and bottom quarks may provide a better probe of this region than jet production (which is limited to prs above about ~ GeV) and single hadron production (which is dominated by details of jet fragmentation). 2 APPLICABILITY OF PERTURBATIVE QCD The fundamental theorem one uses for heavy flavor production is the factorization theorem. This states that the inclusive cross section is given by a hard scattering cross section convoluted with probability distributions for light flavors in the initial-state hadrons: 1
d'(HAHB " ~ "1"X)-- ~ f
dzA /
dzsJ~qA(ZA,P)fj/s(ZB,o~)d~"1"'higher twist°.
(1),
The large charm cross sections that were measured in early experiments 2 prompted concern that this formula is inapplicable, and alternative mechanisms (intrinsic heavy flavor, flavor excitation etc) were proposed 3. We now know that these processes appear not to exist at the leading twist levelz. The standard proof of fuctorization (as in the Drell-Y~ process4) appears to apply. Complete details are still needed. (This is a comment aboat the state of the art in constructing factorization theorems.) The theorem includes the following properties of the quantities in eq. (1): I. d~ can usefully be expanded in powers of a,(/~ ~- toO). That is, no large logarithms of the kinematic variables appear. 2. A similar property hold for the Lipatov-Altarelli-Parisi kernel that determines the evolution of the parton distributions with the scale #. However: 3. Close to threshold in the hard scattering, there are large corrections, which correspond to Coulomb binding.
0920-5632/90/$03.50(~ Elsevier Science Publishers B.V. (North-Holland)
220
J.C.
Collins/Heavyflavor production
4. For the production of the 3 / 4 , the T, etc, fudge factors are needed. 5. When the transverse momentum of the produced heavy flavor is much larger than its mass, there are contributions from jet fragmentation to its production. These give a factor of Iog(kA/mq) per loop in the hard scattering. The physics here is dear, but we do not yet have a proper formulation of the reorganization needed for the perturbation expansion. 6. If mQ ~ ~/~, as is common, then d~ has a log(~/~/mq) per loop, while the AltarelU-Parisi kernel has a factor log(=e/=). This is the region of small z and of R e ~ e kinematics. Lipatov and coworkerss have shown how to do the leading logarithm resummation here, but a full formalism integrated with the hard scattering formula is not available. The threshold and small-z effects are important in generating a large K-factor, as can be seen from the graphs in8. An especially large K-factor results from applying factorization outside the domain in which it is actually proved. However there are many cases where higher order corrections are large, but one is not particularly close to a kinematic boundary. Even so, these large corrections appear not to be anonymously large, but to be there for specific re~sons. A particular case has recently been investigated by Appel, MacKenzie and StermanT: the z --~ 1 behavior of the Altarem-Parisi kernel is the culprit, even when one is not close to z = 1. 3 HIGHER ORDEIt CALCULATIONS 3 . 1 0 ( c l ~ ) calculations Nason, Dawson and l~.lllss have calculated the O(c~.3) corrections to inclusive heavy quark production in hadron-hadron collisions. Their results for the cross section integrated over the kA of the heavy quark have been available for some time. They have now9 presented results for the cross section differential in k±. These results are important because the give an estimate of the K factor for heavy quark production. Some highlights are: a. Roughly, the K factor for the kj. distribution is constant and equals the K factor for the integrated cross section. This is a useful rule of thumb valid over many kinematic regions (charm to top, fixed target to collider). b. There is a discrepancy, possibly by a factor of as much as 10, between theory and experiment for bottom quark production at UA1 at large k~. See Fig. 1. The errors are large enough that this discrepancy is not very significant at present. But this is an issue to watch. c. The K factor is large, as we have come to expect. For bottom production at current colliders it is about 2. d. There remains substantial sensitivity to the factorization scale Q (alias p). The last two items represent theoretical issues that are important for many other processes as well. Do the large K factors that are often found at reasonably large Q simply signal that perturbation theory is not very good and that perturbative QCD does not have much predictive power for many processes of interest? Or is the coupling actually small enough that perturbation theory is sensible ( a , / f ;~ 0.1), but there are specific physical mechanisms that generate large corrections within the currently used methods of factorization? What is an appropriate way to resolve the ambiguity in the choice of the scale parameter p?
3.C. Coffins/Heavy flavor production
I00
- -
I0 '~ .I0
I
221
p~ collisions, ~ = 0.63 TeV, lyl< 1.5, kT>k rni" DATA POINTS UAI
m
t.,-,,j
:,4
I.
JD
O.I--
!
IQ. Q.
.oIL
i
b
.001 f
.0001
f
0
I0
20
k"" [GeV]
30
40
Fig. 1. Bottom quark production at UA1. Comparison of theory and experiment, from Ref. 9. The solid line uses DFLM parton distributions, with/, - Fo _ r'-=-.~/m~-I- kA2., ms - 4.75 GeV, and A4 = 260 MeV. The dashed lines represent the range of values obtmned by using 4.5 < m s < 5 GeV, 160 < A4 < 360 MeV, and/~o/2 < / t < 2/,0. 3.2 Large K factor" Nason, Dawson and Elliss,9 present some evidence that there are indeed specific physical mechanisms producing the large K factor. (Hence, an obvious conjecture is that there are ways of resumming the large higher order corrections.) Their evidence comes from looking inside the convolution integral in the factorization formula (1) applied to the cross section integrated over kjL. They write the hard scattering cross section as
6 = factor {.f(o) + g2[~f(,) + ](1) ln(p2/M~)]} '
(2)
where g2 is around 2 for bottom quark production. In Fig. 2 are plotted the values of the coemcients in eq. (2), for the gluon-gluon subprocess, as a function of 1/p - 4M~/.~, with V~ being the centerof-mass energy of the hard scattering subprocess. For bottom quark production at UA1, 1/p can range up to several hundreds. When 1/p is neither large nor close to threshold, the higher order corrections are indeed small. But Coulomb singularities make the corrections larger than the lowest term when p is close to unity. Moreover, when p is large, the lowest order term, which only has a quark exchanged in the ~ e~b.h~nnelgoes to z~,ro~whereas the corrections, which include gluon exchange go to a constant. (The large p behavior is entirely expected.) The numerical effect after weighting by parton densities is shown in Fig. 3, for bottom production at the Ferm|lab collider. The logarithmic scale in T - ~/8 is appropriate, since the integration is:
o = / ~ ~- xa.xx#.
(3)
A substantial contribution to the K factor is being made both by the threshold region and by the R e ~ e region for the hard scattering.
3.C. Coil/as/Heavy flavor production
222
0
-
!
I
-0.1
i I0 i
I I0 i
I IO s
/ I 4
I 5
I/p Fig. 2. Coefficients for eq. (2), from Re/'. 8.
20
I'|
i
I
I
I
I I(~ s
I li~ 2
lie
15
-ii
to
0 "11(~5
-=--I li~ 4
~'-~.--
T : XlX 2
Fig. 3. Lowest order term and corrections for eq. (2),.after weighting by patton fluxes, for m = # = 5 GeV, in p~ collisions at V~ = 1.8 TeV. The parton distributions are EHLQ set 1 with A - 0.2 GeV. The solid line shows the order a.2 contribution. The dotted and dashed lines show the order a,s contributions from the gluon-gluon and qq subprocesses, respectively. The graph is from ltef. 8.
J.C. Collins/Heavy IIavorproduction
223
The Regge region (large l/p) becomes even more important for larger s and for smaller quark masses. (Charm production at the SSC comes to mind.) The existence of such a region corresponds, of course, to the fact that the process is at small z. The threshold region will be cut out if even a modest cut of the quark kx is made, as is likely to be done for experimental regions at a collider. 3.3 Scale ambiguity While it is appropriate to set the factorization scale p to be of order the heavy quark mass MO, R is not dear whether it should be MO, 2MO, MQ/2, or whatever. The full physical cross section is independent of the choice of p, but perturbative approximations are not. Cf course, it is not correct to say that exactly one particular value is the right one. But surely one can say that some reasonably small range is the most suitable. Standard methods like FAC1° and PMS u only address the mathematical properties of the perturbative coefficients and do not concern themselves with the physical reasons for the sizes of coefficients. Experience with simple diagrams suggests that a,/Ir is an appropriate expansion parameter with coefficients that at low order are moderate. The plot in Fig. 2 surely suggests that the K factor is large, not because the coefficients of the corrections are intrinsically large, but that :he standard factoriz~tion formula is not powerful enough to separate out the various contributions. The proposal of Brodsky, Lepage and MacKenzie12 goes in the direction I would like. Some calculations I have been doing suggest that a suitable choice of/zH---~ is around a typical virtuality (to within a factor 1.5 or so). (Note that I mean ~ here, rather than MS.) In any event, it should be clear that more work is needed in this area. One should certainly figure out a convenient way of resumming the large corrections near threshold and particularly in the region V~ ) MQ. There is already much work in this last area s - - it is certainly an important topic at this meeting, for it is the subject of the perturbative approach to Regge theory. But I do not yet see how to mesh these results with the specific higher order calculations. 4
PHENOMENOLOGY
We have already seen a possible disagreement between data and theory at large kA for bottom production. But the bulk of bottom production does indeed agree with the perturbative QCD predictions, with the K factor predicted by the theory giving a better fit. In the area of charm production there were well-known disagreements~ between theory and experiment, at least when the Born diagrams are used for the hard scattering. Once one knows, from other processes of the likelihood of a substantial K factor, one must first estimate it. It seems that the best current data does agree with the theory is, with a reasonable value of the charm quark mass, about 1.5 GeV. (Note that this is the renorms~ized mass parameter in the Lagrangian, and not necessarily exactly the correct value to use in a constituent potential model of charmonium.) The mass scale is about as low as one would expect perturbative methods to be applicable in the best circumstances. Since without vertex detectors it is hard to do good experiments, it appears that it may be best to disregard the older experiments. The nuclear A dependence should be explored better. Charm production is sensitive to the giuon distribution at low Q, and at fairly small z, a region not well probed by other experiments, like deep inelastic scattering. Since the process is at best at the low end of the range of applicability of perturbation theory, it is likely to be sensitive to effects that cause the breakdown of simple factorization. In particular, one might suppose that even if the standard perturbative calculations apply on a hydrogen target, parton saturation and shadowing effects, such as are discussed in Ryskin's talk, might become important when one goes to bigger nuclei. That is, charm production should be a good way of probing the margin between the perturbative and totally nonperturbative regimes. In the 1970s data from cosmic ray collisions taken in emulsion chambers appeared to show evidence for charm production 14. The energies are such that in a fixed target experiment charmed mesons
224
J.O. Collins/Heavy Bavor production
have an easily measurable macroscopic track length. The lifetimes deduced from cosmic ray data agree with much more recent data from acceleratom. In particular the lifetime of the charged D mesons is lonkm.rby a factor of two to three than the lifetime of the neutral D's. However, the particle physics literature tends to ignore this data. It would be interesting to reexamine it. The rate at first sight appears to be excessively high E 1 charmed particle per 20 to 40 collisions at a beam energy of 20 TeV. The kinematic r e , on is of course of very small x, and the beam may well be mostly composed of heavy nuclei, like iron. (The target is typically carbon, or nitrogen in these co~_m;cray experiments.) 5 DIFFRACTIVE PRODUCTION, ETC There appenm to be no reliable evidence for very large forward or diffractive charm production, and also to be no particularly, good theoretical motivation any more, within QCD. Diffractive production within QCD, if perturbation theory is appropriate for charm production, is just a special case of difl'ractive hard scattering. The data that do exist from UA1 appear to be in the ran~ of reasonable theoretical expectations. ACKNOWLEDGEMENTS This work was supported in part by the U.S. Department of Energy, Division of High Energy Physics, contract DE-FG02-85ER-40235, and by the National Science Foundation, grant Phy-85-07627. I wish to th~nlr the Institutes for Theoretical Physics at Stony Brook for its hospitality during p ~ t of the preparation of this article. REFERENCES 1) J.C. Collins, D.E. Soper and G. Steman, Nucl. Phys. B263, 37 (i~$6) and B308, 833 (1988). 2) A. Kernan and G. van Dalen, Phys. Reports 106, 297 (1984), and references therein. 3) S.J. Brodsky, C. Psterson and N. Sakai, Phys. Rev. D23, 9.?45 (1981); B.L. Combridge, Nucl. Phys. B151, 429 (1979); It. Oderico, Nud. Phys. B209, 77 (1982); V. Barger, F. Halzen and W.Y. Keung, Phys. Rev. D25, 112 (1979); P.D.B. Collins and T.P. Spiller, J. Phys. G10, 1667
(1984). 4) G.T. Bodwin, Phys. Rev. D31, 2616 (1985) and D34, 3932 (1986); J.C. Collins, D.E. Soper and
G. Sterman, Nud. Phys. B261, 104 (1985). 5) L.N. Lipatov, Soy. $. Nucl. Phys. 63, 904 (1986); L.N. Lipatov, in "Perturbative QCD" (A.H. Mueller, ed.) (World Scientific, Singapore, to appear); L.V. Gribov, E.M. Levin and M.G. Ryskin, Phys. Reports 100, 1 (1983). 6) Ref. DEL not yet defined. 7) D. Appeil, G. Sterman and P. Mackenzie, Nucl. Phys. B309, 259 (1988). 8) P. Naso~, S. Dawson and R.K. Ellis, Nud. Phys. B303, 607 (1988). 9) P. Na~n, S. Dawson and R.K. Ellis, preprint ETH-PT/89-2, (Apr. 1989). 10) G. Grunberg, Phys. Lett. 95B, 70 (1980). 11) P.M. Stevenson and H.D. Politzer, Nud. Phys. B277, 758 (1986); P.M. Stevenson, Phys. Rev. D23, 2916 (1981).
3.C. Col/ins/Heavy flavor production
225
12) S.3. Brodsky, G.P. Lepage and P.B. Mackenzie, Phys. Rev. D28, 228 (1983). 13) LEBC-EHS Collaboration (M. Aguilar-Benitez, et al.), Z. Phys. CA0, 321 (1988); ACCMOR Collaboration (S. Barlag, et al.), Z. Phys. C39, 451 (1988). 14) C.M.G. Lattes, Y. Fujimoto and S. Hasegawa, Phys. Reports 65, 151 (1980), T.K. GMsser and G.B. Yodh, Ann. Rev. Nucl. Part. Sci. 30, 475 (1980), and references therein.
219
HEAVY FLAVOR PRODUCTION John C. COLLINS Physics Department, Illinois Institute of Technology, Chicago, IL 60616, U.S,S.. I summarize the situation for the theory and phenomenology of the production of heavy colored quanta in hadron-hadron comsions.
1 INTRODUCT~N In this talk I consider the production of heavy colored quanta in hadron-hadron collisions. Now, production of such objects involves an internal line that is off-shen by an amount that is at least about the mass of the heavy flavor. Thus there is a hard scattering subprocess in the collision, and one expects to be able to discuss the process within perturbative QCD. The experimentally discovered heavy flavors are so far the charm and bottom quarks. Potential candidates for future discovery include not only the top quark, of course, but also gluinos and squarks, for example. The reasons for discussing these processes include the following: (a) They give a probe of hard scattering at scales O of order me, ms in hadron-hadron collisions, where jets are hard or impossible to measure well. (b) The cross sections are needed when searching for new heavy flavors. (c) Often we have V~ ) MO, where MQ is the mass of the heavy flavor; in that case we probe the small-z region. Perturbative methods are then useful, but we are also probing a region of Regge kinematics. Charm and bottom quarks may provide a better probe of this region than jet production (which is limited to prs above about ~ GeV) and single hadron production (which is dominated by details of jet fragmentation). 2 APPLICABILITY OF PERTURBATIVE QCD The fundamental theorem one uses for heavy flavor production is the factorization theorem. This states that the inclusive cross section is given by a hard scattering cross section convoluted with probability distributions for light flavors in the initial-state hadrons: 1
d'(HAHB " ~ "1"X)-- ~ f
dzA /
dzsJ~qA(ZA,P)fj/s(ZB,o~)d~"1"'higher twist°.
(1),
The large charm cross sections that were measured in early experiments 2 prompted concern that this formula is inapplicable, and alternative mechanisms (intrinsic heavy flavor, flavor excitation etc) were proposed 3. We now know that these processes appear not to exist at the leading twist levelz. The standard proof of fuctorization (as in the Drell-Y~ process4) appears to apply. Complete details are still needed. (This is a comment aboat the state of the art in constructing factorization theorems.) The theorem includes the following properties of the quantities in eq. (1): I. d~ can usefully be expanded in powers of a,(/~ ~- toO). That is, no large logarithms of the kinematic variables appear. 2. A similar property hold for the Lipatov-Altarelli-Parisi kernel that determines the evolution of the parton distributions with the scale #. However: 3. Close to threshold in the hard scattering, there are large corrections, which correspond to Coulomb binding.
0920-5632/90/$03.50(~ Elsevier Science Publishers B.V. (North-Holland)
220
J.C.
Collins/Heavyflavor production
4. For the production of the 3 / 4 , the T, etc, fudge factors are needed. 5. When the transverse momentum of the produced heavy flavor is much larger than its mass, there are contributions from jet fragmentation to its production. These give a factor of Iog(kA/mq) per loop in the hard scattering. The physics here is dear, but we do not yet have a proper formulation of the reorganization needed for the perturbation expansion. 6. If mQ ~ ~/~, as is common, then d~ has a log(~/~/mq) per loop, while the AltarelU-Parisi kernel has a factor log(=e/=). This is the region of small z and of R e ~ e kinematics. Lipatov and coworkerss have shown how to do the leading logarithm resummation here, but a full formalism integrated with the hard scattering formula is not available. The threshold and small-z effects are important in generating a large K-factor, as can be seen from the graphs in8. An especially large K-factor results from applying factorization outside the domain in which it is actually proved. However there are many cases where higher order corrections are large, but one is not particularly close to a kinematic boundary. Even so, these large corrections appear not to be anonymously large, but to be there for specific re~sons. A particular case has recently been investigated by Appel, MacKenzie and StermanT: the z --~ 1 behavior of the Altarem-Parisi kernel is the culprit, even when one is not close to z = 1. 3 HIGHER ORDEIt CALCULATIONS 3 . 1 0 ( c l ~ ) calculations Nason, Dawson and l~.lllss have calculated the O(c~.3) corrections to inclusive heavy quark production in hadron-hadron collisions. Their results for the cross section integrated over the kA of the heavy quark have been available for some time. They have now9 presented results for the cross section differential in k±. These results are important because the give an estimate of the K factor for heavy quark production. Some highlights are: a. Roughly, the K factor for the kj. distribution is constant and equals the K factor for the integrated cross section. This is a useful rule of thumb valid over many kinematic regions (charm to top, fixed target to collider). b. There is a discrepancy, possibly by a factor of as much as 10, between theory and experiment for bottom quark production at UA1 at large k~. See Fig. 1. The errors are large enough that this discrepancy is not very significant at present. But this is an issue to watch. c. The K factor is large, as we have come to expect. For bottom production at current colliders it is about 2. d. There remains substantial sensitivity to the factorization scale Q (alias p). The last two items represent theoretical issues that are important for many other processes as well. Do the large K factors that are often found at reasonably large Q simply signal that perturbation theory is not very good and that perturbative QCD does not have much predictive power for many processes of interest? Or is the coupling actually small enough that perturbation theory is sensible ( a , / f ;~ 0.1), but there are specific physical mechanisms that generate large corrections within the currently used methods of factorization? What is an appropriate way to resolve the ambiguity in the choice of the scale parameter p?
3.C. Coffins/Heavy flavor production
I00
- -
I0 '~ .I0
I
221
p~ collisions, ~ = 0.63 TeV, lyl< 1.5, kT>k rni" DATA POINTS UAI
m
t.,-,,j
:,4
I.
JD
O.I--
!
IQ. Q.
.oIL
i
b
.001 f
.0001
f
0
I0
20
k"" [GeV]
30
40
Fig. 1. Bottom quark production at UA1. Comparison of theory and experiment, from Ref. 9. The solid line uses DFLM parton distributions, with/, - Fo _ r'-=-.~/m~-I- kA2., ms - 4.75 GeV, and A4 = 260 MeV. The dashed lines represent the range of values obtmned by using 4.5 < m s < 5 GeV, 160 < A4 < 360 MeV, and/~o/2 < / t < 2/,0. 3.2 Large K factor" Nason, Dawson and Elliss,9 present some evidence that there are indeed specific physical mechanisms producing the large K factor. (Hence, an obvious conjecture is that there are ways of resumming the large higher order corrections.) Their evidence comes from looking inside the convolution integral in the factorization formula (1) applied to the cross section integrated over kjL. They write the hard scattering cross section as
6 = factor {.f(o) + g2[~f(,) + ](1) ln(p2/M~)]} '
(2)
where g2 is around 2 for bottom quark production. In Fig. 2 are plotted the values of the coemcients in eq. (2), for the gluon-gluon subprocess, as a function of 1/p - 4M~/.~, with V~ being the centerof-mass energy of the hard scattering subprocess. For bottom quark production at UA1, 1/p can range up to several hundreds. When 1/p is neither large nor close to threshold, the higher order corrections are indeed small. But Coulomb singularities make the corrections larger than the lowest term when p is close to unity. Moreover, when p is large, the lowest order term, which only has a quark exchanged in the ~ e~b.h~nnelgoes to z~,ro~whereas the corrections, which include gluon exchange go to a constant. (The large p behavior is entirely expected.) The numerical effect after weighting by parton densities is shown in Fig. 3, for bottom production at the Ferm|lab collider. The logarithmic scale in T - ~/8 is appropriate, since the integration is:
o = / ~ ~- xa.xx#.
(3)
A substantial contribution to the K factor is being made both by the threshold region and by the R e ~ e region for the hard scattering.
3.C. Coil/as/Heavy flavor production
222
0
-
!
I
-0.1
i I0 i
I I0 i
I IO s
/ I 4
I 5
I/p Fig. 2. Coefficients for eq. (2), from Re/'. 8.
20
I'|
i
I
I
I
I I(~ s
I li~ 2
lie
15
-ii
to
0 "11(~5
-=--I li~ 4
~'-~.--
T : XlX 2
Fig. 3. Lowest order term and corrections for eq. (2),.after weighting by patton fluxes, for m = # = 5 GeV, in p~ collisions at V~ = 1.8 TeV. The parton distributions are EHLQ set 1 with A - 0.2 GeV. The solid line shows the order a.2 contribution. The dotted and dashed lines show the order a,s contributions from the gluon-gluon and qq subprocesses, respectively. The graph is from ltef. 8.
J.C. Collins/Heavy IIavorproduction
223
The Regge region (large l/p) becomes even more important for larger s and for smaller quark masses. (Charm production at the SSC comes to mind.) The existence of such a region corresponds, of course, to the fact that the process is at small z. The threshold region will be cut out if even a modest cut of the quark kx is made, as is likely to be done for experimental regions at a collider. 3.3 Scale ambiguity While it is appropriate to set the factorization scale p to be of order the heavy quark mass MO, R is not dear whether it should be MO, 2MO, MQ/2, or whatever. The full physical cross section is independent of the choice of p, but perturbative approximations are not. Cf course, it is not correct to say that exactly one particular value is the right one. But surely one can say that some reasonably small range is the most suitable. Standard methods like FAC1° and PMS u only address the mathematical properties of the perturbative coefficients and do not concern themselves with the physical reasons for the sizes of coefficients. Experience with simple diagrams suggests that a,/Ir is an appropriate expansion parameter with coefficients that at low order are moderate. The plot in Fig. 2 surely suggests that the K factor is large, not because the coefficients of the corrections are intrinsically large, but that :he standard factoriz~tion formula is not powerful enough to separate out the various contributions. The proposal of Brodsky, Lepage and MacKenzie12 goes in the direction I would like. Some calculations I have been doing suggest that a suitable choice of/zH---~ is around a typical virtuality (to within a factor 1.5 or so). (Note that I mean ~ here, rather than MS.) In any event, it should be clear that more work is needed in this area. One should certainly figure out a convenient way of resumming the large corrections near threshold and particularly in the region V~ ) MQ. There is already much work in this last area s - - it is certainly an important topic at this meeting, for it is the subject of the perturbative approach to Regge theory. But I do not yet see how to mesh these results with the specific higher order calculations. 4
PHENOMENOLOGY
We have already seen a possible disagreement between data and theory at large kA for bottom production. But the bulk of bottom production does indeed agree with the perturbative QCD predictions, with the K factor predicted by the theory giving a better fit. In the area of charm production there were well-known disagreements~ between theory and experiment, at least when the Born diagrams are used for the hard scattering. Once one knows, from other processes of the likelihood of a substantial K factor, one must first estimate it. It seems that the best current data does agree with the theory is, with a reasonable value of the charm quark mass, about 1.5 GeV. (Note that this is the renorms~ized mass parameter in the Lagrangian, and not necessarily exactly the correct value to use in a constituent potential model of charmonium.) The mass scale is about as low as one would expect perturbative methods to be applicable in the best circumstances. Since without vertex detectors it is hard to do good experiments, it appears that it may be best to disregard the older experiments. The nuclear A dependence should be explored better. Charm production is sensitive to the giuon distribution at low Q, and at fairly small z, a region not well probed by other experiments, like deep inelastic scattering. Since the process is at best at the low end of the range of applicability of perturbation theory, it is likely to be sensitive to effects that cause the breakdown of simple factorization. In particular, one might suppose that even if the standard perturbative calculations apply on a hydrogen target, parton saturation and shadowing effects, such as are discussed in Ryskin's talk, might become important when one goes to bigger nuclei. That is, charm production should be a good way of probing the margin between the perturbative and totally nonperturbative regimes. In the 1970s data from cosmic ray collisions taken in emulsion chambers appeared to show evidence for charm production 14. The energies are such that in a fixed target experiment charmed mesons
224
J.O. Collins/Heavy Bavor production
have an easily measurable macroscopic track length. The lifetimes deduced from cosmic ray data agree with much more recent data from acceleratom. In particular the lifetime of the charged D mesons is lonkm.rby a factor of two to three than the lifetime of the neutral D's. However, the particle physics literature tends to ignore this data. It would be interesting to reexamine it. The rate at first sight appears to be excessively high E 1 charmed particle per 20 to 40 collisions at a beam energy of 20 TeV. The kinematic r e , on is of course of very small x, and the beam may well be mostly composed of heavy nuclei, like iron. (The target is typically carbon, or nitrogen in these co~_m;cray experiments.) 5 DIFFRACTIVE PRODUCTION, ETC There appenm to be no reliable evidence for very large forward or diffractive charm production, and also to be no particularly, good theoretical motivation any more, within QCD. Diffractive production within QCD, if perturbation theory is appropriate for charm production, is just a special case of difl'ractive hard scattering. The data that do exist from UA1 appear to be in the ran~ of reasonable theoretical expectations. ACKNOWLEDGEMENTS This work was supported in part by the U.S. Department of Energy, Division of High Energy Physics, contract DE-FG02-85ER-40235, and by the National Science Foundation, grant Phy-85-07627. I wish to th~nlr the Institutes for Theoretical Physics at Stony Brook for its hospitality during p ~ t of the preparation of this article. REFERENCES 1) J.C. Collins, D.E. Soper and G. Steman, Nucl. Phys. B263, 37 (i~$6) and B308, 833 (1988). 2) A. Kernan and G. van Dalen, Phys. Reports 106, 297 (1984), and references therein. 3) S.J. Brodsky, C. Psterson and N. Sakai, Phys. Rev. D23, 9.?45 (1981); B.L. Combridge, Nucl. Phys. B151, 429 (1979); It. Oderico, Nud. Phys. B209, 77 (1982); V. Barger, F. Halzen and W.Y. Keung, Phys. Rev. D25, 112 (1979); P.D.B. Collins and T.P. Spiller, J. Phys. G10, 1667
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12) S.3. Brodsky, G.P. Lepage and P.B. Mackenzie, Phys. Rev. D28, 228 (1983). 13) LEBC-EHS Collaboration (M. Aguilar-Benitez, et al.), Z. Phys. CA0, 321 (1988); ACCMOR Collaboration (S. Barlag, et al.), Z. Phys. C39, 451 (1988). 14) C.M.G. Lattes, Y. Fujimoto and S. Hasegawa, Phys. Reports 65, 151 (1980), T.K. GMsser and G.B. Yodh, Ann. Rev. Nucl. Part. Sci. 30, 475 (1980), and references therein.