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P-Wave Charmonium Production

arXiv:hep-ph/9208254v1 26 Aug 1992ANL-HEP-PR-92-63NUHEP-TH-92-16August 1992P-Wave Charmonium Productionin B-Meson DecaysGeoffrey T. BodwinHigh Energy Physics Division, Argonne National Laboratory, Argonne, IL 60439Eric Braaten and Tzu Chiang YuanDepartment of Physics and Astronomy, Northwestern University, Evanston, IL 60208G. Peter LepageNewman Laboratory of Nuclear Studies, Cornell University, Ithaca, NY 14853AbstractWe calculate the decay rates of B mesons into P-wave charmonium states usingnew factorization formulas that are valid to leading order in the relative velocity ofthe charmed quark and antiquark and to all orders in the running coupling constantof QCD.

We express the production rates for all four P states in terms of two non-perturbative parameters, the derivative of the wavefunction at the origin and anotherparameter related to the probability for a charmed-quark-antiquark pair in a color-octetS-wave state to radiate a soft gluon and form a P-wave bound state. Using existingdata on B meson decays into χc1 to estimate the color-octet parameter, we find thatthe color-octet mechanism may account for a significant fraction of the χc1 productionrate and that B mesons should decay into χc2 at a similar rate.

The production rate of quarkonium states in various high energy physics processescan provide valuable insight not only into the interactions between a heavy quark and anti-quark, but also into the elementary processes that produce the Q ¯Q pair. The spin-1 S-waveresonances, like the J/ψ of charmonium, are of special experimental significance because theyhave extremely clean signatures through their leptonic decay modes.

Because a significantfraction of the ψ’s come from the decays of the P-wave χcJ states, an understanding of theproduction of P-wave resonances is necessary in order to understand inclusive ψ production.The P states are also important in their own right because they probe a qualitatively dif-ferent aspect of the Q ¯Q production process. While the S states probe only the productionat short distances of a Q ¯Q pair in a color-singlet state, the P states, as we shall show, alsoprobe the production of a Q ¯Q pair in a color-octet state.One of the simplest production processes for charmonium states is the decay of a Bmeson or baryon.

For B−, ¯B0, ¯Bs, and Λb (but not for ¯Bc), the c and the ¯c that form thecharmonium bound state must both be produced by the decay or annihilation of the b quark.Since this is a short distance process which occurs on a length scale of order 1/Mb, where Mbis the mass of the b quark, it should be possible to apply perturbative QCD to calculate theinclusive decay rate into a particular charmonium state. If we neglect contributions that aresuppressed by powers of Λ/Mb, where Λ is a typical momentum scale for light quarks, thedecay rate of a B hadron is given by the decay rate of the b quark, with the light antiquarkin the B meson and the light quarks in the B baryon treated as noninteracting spectators.In the case of the hadronic and the semileptonic decay rates, the leading corrections aresuppressed by two powers of Λ/Mb (Ref.

[1]). We expect perturbative QCD calculations toyield predictions for the inclusive decay rates of B hadrons into charmonium states that arecomparable in accuracy to the predictions for their semileptonic decay rates.Most previous calculations of the rate for charmonium production in B meson decays[2, 3, 4, 5] have been carried out under the assumption that the production mechanismis the decay at short distances of a b quark into a color-singlet c¯c pair plus other quarksand gluons, with the c and ¯c having almost equal momenta and residing in the appropriateangular-momentum state.

We will refer to this as the “color-singlet mechanism”. It wasassumed that the only nonperturbative input required in the calculation is the nonrelativistic

c¯c wave function (for S states) or its derivative (for P states) at the origin. In this paper wepoint out that this assumption fails for P states, the failure being signaled by the presenceof infrared divergences in the QCD radiative corrections.

The divergences appear becausethere is a second production mechanism for P states that also contributes at leading orderin αs, namely the decay at short distances of the b quark into a color-octet c¯c pair in an Sstate, plus other partons. We will call this the “color-octet mechanism”.

In addition to thederivative of the nonrelativistic wavefunction at the origin, calculations of P-wave productionrequire a second nonperturbative input parameter, the probability for the color-octet c¯c pairto radiate a soft gluon and form a color-singlet P-wave bound state. The two productionmechanisms are summarized below by factorization formulas which are accurate to leadingorder in v2, the square of the relative velocity of the charmed quark and antiquark, and toall orders in the strong coupling constant αs(Mb).

The hard subprocess rates appearing inthese formulas are calculated to leading order in the QCD coupling constant. Estimates ofthe nonperturbative parameter associated with the color-octet mechanism are obtained fromexperimental data on the production rate of χc states in B meson decays.

We find that thecolor-octet mechanism may account for a significant fraction of the observed decay rate intoχc1 and that B mesons should decay into χc2 at a similar rate.The color-singlet and color-octet mechanisms for the production of P-wave quarko-nium states have analogs in the decays of these states. For S-wave resonances, the electro-magnetic and light-hadronic decays proceed through the annihilation at short distances ofthe heavy quark and antiquark in a color-singlet S-wave state.

The decay rates are thenproportional to the square of RS(0), the nonrelativistic wavefunction at the origin, with co-efficients that can be calculated perturbatively as a series in αs(MQ), where MQ is the heavyquark mass. In the case of P-waves, one might expect that the decay should also proceedthrough the annihilation at short distances of the Q ¯Q pair in a color-singlet P-wave state.

Ifthis were the case, the decay rates would be calculable in terms of a single nonperturbativefactor R′P(0), the derivative of the nonrelativistic wavefunction at the origin. While thisexpectation holds true for decay rates into two photons, explicit calculations of the lighthadronic decay rates reveal infrared divergences [6].

It has recently been shown that theseinfrared divergences can be systematically factored into a second nonperturbative parameter,which is proportional to the probability for the quark and antiquark to be in a color-octet

S-wave state at the origin [7]. These results can be summarized by rigorous factorizationformulas for the decay rates of quarkonium states into light hadronic or electromagnetic finalstates [7, 8].

The factorization formulas are valid to leading order in v2, where v is the typicalvelocity of the heavy quark, and to all orders in the QCD coupling constant αs(MQ).The inclusive production rates for charmonium states also obey factorization formulas.To leading order in v2, where v is the typical velocity of the charmed quark relative to thecharmed antiquark, and to all orders in αs(Mb), the inclusive decay rate into ψ satisfies thesimple factorization formula that was assumed in previous work: [2, 3, 4, 5]Γ (b →ψ + X) = G1 bΓ1b →c¯c(3S1) + X,(1)where bΓ1 is the hard subprocess rate for producing a color-singlet c¯c pair with equal momentaand in the appropriate angular-momentum state 2S+1LJ =3S1. It can be calculated per-turbatively as a series in the QCD running coupling constant αs(Mb).

The nonperturbativeparameter G1 in (1) is proportional to the probability for the c¯c pair to form a bound stateand is related to the nonrelativistic wavefunction at the origin RS(0):G1 ≈32π|RS(0)|2M2c. (2)A phenomenological determination that makes use of the electronic decay rate of the ψ givesG1 ≈108 MeV.

It can also be given a rigorous nonperturbative definition, so that it can bemeasured in lattice simulations of QCD [8]. The production rate of the radial excitation ψ′obeys the same factorization formula (1), but with a parameter G1 that is smaller by theratio of the electronic decay rates of ψ and ψ′: G1 ≈43 MeV.In previous work [4], the decays of b quarks into P-wave charmonium states wereassumed to satisfy simple one-term factorization formulas like (1).

Because of the color-octet production mechanism, such formulas are incomplete, and their breakdown is signaledby the appearance of infrared divergences in order αs. The correct factorization formulas forP-wave production rates have two terms:Γ (b →hc + X) = H1 bΓ1b →c¯c(1P1) + X, µ+ 3 H′8(µ) bΓ8b →c¯c(1S0) + X,(3)

Γ (b →χcJ + X) = H1 bΓ1b →c¯c(3PJ) + X, µ+ (2J + 1) H′8(µ) bΓ8b →c¯c(3S1) + X. (4)The factors bΓ1 and bΓ8 are hard subprocess rates for the decay of the b quark into a c¯c pair inthe appropriate angular-momentum state with vanishing relative momentum, the c¯c beingin a color-singlet state for bΓ1 and in a color-octet state for bΓ8.

They can be calculatedperturbatively as series in αs(Mb), with coefficients that are free of infrared divergences.Note that in (4) the only dependence on the total angular momentum quantum numberJ in the second term on the right side lies in the coefficient 2J + 1. The nonperturbativeparameters H1 and H′8 are proportional to the probabilities for a c¯c pair in a color-singletP-wave and a color-octet S-wave state, respectively, to fragment into a color-singlet P-wavebound state.

They can be given rigorous nonperturbative definitions [8] in terms of matrixelements in nonrelativistic QCD. The parameter H1 is directly related to the nonrelativisticwavefunction for the heavy quark and antiquark:H1 ≈92π|R′P(0)|2M4c.

(5)Its value has been determined phenomenologically from the light hadronic decay rates ofthe χc1 and χc2 to be H1 ≈15 MeV [7]. There is no rigorous perturbative expression forH′8 in terms of the wavefunction RP(r), since a c¯c pair in a color-octet S-wave state canmake a transition to a color-singlet P-wave state through the radiation of a soft gluon, whichis a nonperturbative process.

The parameter H′8 is not related in any simple way to theanalogous parameter H8 that appears in P-wave decays [7], so it cannot be determined fromdata on the decays of P states. A phenomenological determination of H′8 can come only froma production process.

Below we will use experimental data on χc production in B mesondecays to obtain a rough determination of H′8.In the factorization formulas (3) and (4), H′8 and bΓ1 depend on an arbitrary factor-ization scale µ in such a way that the complete decay rate is independent of µ. In orderto avoid large logarithms of µ/Mb in bΓ1, the factorization scale µ should be chosen to beon the order of Mb.

The scale dependence of H′8(µ) is governed by a renormalization group

equation, which to leading order in αs(µ) is [8]µ ddµH′8(µ) ≈1627παs(µ) H1 . (6)The matrix element H1 is independent of the factorization scale µ.

The solution to therenormalization group equation is therefore elementary.For example, for µ < Mc, thesolution isH′8(Mb) = H′8(µ) +" 1627β3ln αs(µ)αs(Mc)!+1627β4ln αs(Mc)αs(Mb)!#H1 ,(7)where βn = (33 −2n)/6 is the first coefficient of the QCD beta function for n flavors ofmassless quarks. In the limit in which Mb is very large, the contribution to H′8 from theperturbative evolution dominates, and one can estimate H′8(Mb) by setting αs(µ) ∼1 andneglecting the constant H′8(µ) in (7).

Taking αs(Mb) ≈0.20 and αs(Mc) ≈0.31, we obtainH′8(Mb) ≈3 MeV. This should be regarded as only a rough estimate, since the physical valueof Mb is probably not large enough for the constant H′8(µ) to be negligible.The subprocesses in the factorization formulas (3) and (4) are decays of the b quarkinto c¯c plus other quarks and gluons.The dominant contributions involve the couplingof the b quark to c¯cs via an effective 4-quark weak interaction [9], which can, by Fierzrearrangement, be put into the formLweak = −GF√2VcbV ∗cs 2C+ −C−3¯cγµ(1 −γ5)c ¯sγµ(1 −γ5)b+ (C+ + C−) ¯cγµ(1 −γ5)T ac ¯sγµ(1 −γ5)T ab!,(8)where GF is the Fermi constant and the Vij’s are elements of the Kobayashi-Maskawa mixingmatrix.The weak interaction that gives the Cabibbo-suppressed transition b →c¯cd isobtained by replacing ¯s by ¯d and V ∗cs by V ∗cd in (8).

The coefficients C+ and C−in (8) areWilson coefficients that arise from evolving the effective 4-quark interaction mediated by theW boson from the scale MW down to the scale Mb. To leading order in αs(Mb) and to all

orders in αs(Mb) ln(MW/Mb), they areC+(Mb) ≈ αs(Mb)αs(MW)!−6/23,(9)C−(Mb) ≈ αs(Mb)αs(MW)!12/23. (10)Taking αs(MW) = 0.116 and αs(Mb) = 0.20, we find that C+(Mb) ≈0.87 and C−(Mb) ≈1.34.

When a b quark decays through the interaction (8), the first term produces a c¯c pairin a color-singlet state, while the second produces a c¯c pair in a color-octet state. The color-singlet coefficient 2C+ −C−is decreased dramatically by renormalization-group evolution,from 1 at the scale MW, to 2C+ −C−≈0.40 at the scale Mb.

The color-octet coefficient(C+ + C−)/2 is increased slightly, from 1 at the scale MW, to (C+ + C−)/2 ≈1.10 at thescale Mb. Since the dramatic suppression of 2C+ −C−is due to a cancellation between 2C+and C−, it is sensitive to both the choice Mb for the scale and to higher-order perturbativecorrections to the Wilson coefficients.

This sensitivity can be removed only by calculationsbeyond leading order. The Wilson coefficients C+ and C−have been calculated at next-to-leading order [10], but the calculations are meaningful only when combined with decay ratesthat are also calculated beyond leading order.It is convenient to express all the subprocess rates appearing in the factorizationformulas (1), (3), and (4) in terms ofbΓ0 = |Vcb|2 G2F144πM3b Mc 1 −4M2cM2b!2.

(11)To leading order in αs(Mb), the color-singlet subprocess rates bΓ1 can be extracted fromprevious calculations [2, 3, 4]. The sum of the two subprocess rates that contribute to thedecay into the ψ at leading order isbΓ1b →c¯c(3S1) + s, d= (2C+ −C−)2 1 + 8M2cM2b!bΓ0 ,(12)where we have used |Vcs|2 + |Vcd|2 ≈1.

The subprocess rate that contributes to the decay

into χc1 is [4]bΓ1b →c¯c(3P1) + s, d= 2 (2C+ −C−)2 1 + 8M2cM2b!bΓ0 . (13)The color-singlet subprocess rates that contribute to the production of hc, χc0, and χc2 (the1P1, 3P0, and 3P2 states, respectively) vanish to leading order in αs because of the JP Cquantum numbers of these charmonium states.The rate for direct production of ψ in B meson decay has been calculated to next-to-leading order in αs (Ref.

[5]). Unfortunately, renormalization group effects were treatedincorrectly in that calculation.

The treatment of Ref. [5] was equivalent to using 2C+ −C−= (αs(Mb)/αs(MW))−24/23 and (C++C−)/2 = (αs(Mb)/αs(MW))3/23 for the color-singletand color-octet coefficients at the scale Mb.

This treatment correctly reproduces the termproportional to αs ln(MW/Mb) in the order αs correction, but it fails to reproduce the leadinglogarithms at order α2s and higher. The results of Ref.

[5] were presented only in graphicalform, which prevents us from extracting the correct order-αs contribution to the subprocessrate for ψ production. In using the leading order result (12), one should keep in mind thatthe next-to-leading correction proportional to αs(C++C−)2 may be as important numericallyas the leading term, which is proportional to (2C+ −C−)2.The color-octet subprocess rates bΓ8 appearing in (3) and (4) require a new calculation.The most straightforward method (although not the simplest) is to calculate the infrareddivergent part of the rate for the decay b →c¯csg, with the c¯c in a color-singlet P-wavestate after having emitted the soft gluon.

This rate can be calculated in terms of R′P(0) bymaking use of a covariant formalism [12]. The infrared divergences come from the regionof phase space in which the momentum of the radiated gluon is soft.

In this region, thematrix element can be factored into the amplitude for b →c¯cs, with the c¯c projected ontoan S state, and a term that depends on the gluon momentum. The divergence comes fromintegrating over the phase space of the gluon.

Imposing an infrared cutoffµ on the energyof the soft gluon, one finds that the divergence is proportional to ln(Mb/µ)H1. The final

results for the infrared divergent terms in the decay rates areΓ(b →hc + sg) ∼12π2|Vcs|2(C+ + C−)2 αs ln Mbµ|R′P(0)|2M4cbΓ0 ,(14)Γ(b →χcJ + sg) ∼(2J + 1) 43π2|Vcs|2(C+ + C−)2 αs ln Mbµ|R′P(0)|2M4c 1 + 8M2cM2b!bΓ0 . (15)One can identify the corresponding infrared divergence in the perturbative expression for H′8by neglecting the running of the coupling constant in (6).

The resulting expression for H′8 isH′8(Mb) ∼1627παs ln Mbµ!H1 . (16)Using this identification together with the expression (5) for H1, we find that the color-octetsubprocess rates defined in (3) and (4) arebΓ8b →c¯c(1S0) + s, d= 32 (C+ + C−)2 bΓ0 ,(17)bΓ8b →c¯c(3S1) + s, d= 12 (C+ + C−)2 1 + 8M2cM2b!bΓ0 .

(18)We now turn to the phenomenological applications of our results. Among the correc-tions to the factorization formulas for B hadron decays are the effects of spectator quarksand antiquarks, which are suppressed by powers of Λ/Mb (Ref.

[1]). It is clear from decaysof D mesons that spectator effects are much more important for total decay rates than forsemileptonic decays.

Our predictions for the inclusive decay rates into charmonium statesshould therefore be more reliable if they are normalized to the semileptonic decay rate, in-stead of the total decay rate. These branching ratios should be identical for B−, ¯B0, ¯Bs,and Λb, even if their lifetimes differ substantially.

To leading order in αs, the semileptonicdecay rate is [11]Γ(b →e−¯νe + X) = |Vcb|2 G2F192π3M5b F(Mc/Mb) ,(19)

where F(x) = 1 −8x2 −24x4 ln x + 8x6 −x8. Approximating the ratio Mc/Mb of the heavyquark masses by the ratio MD/MB ≈0.35 of the corresponding meson masses, we find thatthe phase space factor is F(Mc/Mb) ≈0.41 and that the semileptonic decay rate (19) is|Vcb|2(Mb/5.3GeV)5(3.9 · 10−11GeV).

The Kobayashi-Maskawa factor |Vcb|2, as well as theextreme sensitivity to the value of the bottom quark mass Mb, cancels in the ratio betweenthe charmonium and semileptonic decay rates.The leading order QCD prediction for the inclusive decay rate into ψ is obtained byinserting (12) into (1). To take into account phase space restrictions as accurately as possible,we approximate the ratio Mc/Mb of quark masses by Mψ/(2MB) ≈0.29.

Normalizing to thesemileptonic decay rate, we obtain the ratioR(ψ) ≡Γ(b →ψ + X)Γ(b →e−¯νe + X) ≈6.8 (2C+ −C−)2 G1Mb. (20)Multiplying by the observed semileptonic branching fraction for B mesons of 10.7%, we findthat the predicted inclusive branching fraction for ψ is 0.23%.

There are large theoreticaluncertainties in this result because order-αs corrections to the color-singlet Wilson coefficient2C+ −C−may be significant and because the perturbative corrections to the subprocess rate(12) proportional to αs(C+ + C−)2 need not be small compared to the leading-order term,which is proportional to (2C+−C−)2. These uncertainties can be removed only by a completecalculation of the subprocess rate bΓ1(b →ψX) to order αs.

There are additional theoreticalerrors due to uncertainties in the quark masses Mc and Mb, relativistic corrections of order v2,and corrections of order Λ2/M2b . However these uncertainties are probably small comparedto the perturbative errors.For the P states, we define ratios of the inclusive charmonium and semileptonic decayrates, analogous to the ratio R(ψ) defined in (20).

We take into account the phase spacerestrictions in the decay rate into charmonium as accurately as possible by using half thebound state mass for Mc and the B meson mass for Mb, so that Mc/Mb varies from 0.32 to0.34 for the various P states. For the mass of the hc, we have taken the center of gravity of

the χcJ states. The resulting leading order QCD predictions for the ratios are thenR(hc) ≈14.7 (C+ + C−)2H′8(Mb)Mb,(21)R(χc0) ≈3.2 (C+ + C−)2H′8(Mb)Mb,(22)R(χc1) ≈12.4 (2C+ −C−)2 H1Mb+ 9.3 (C+ + C−)2H′8(Mb)Mb,(23)R(χc2) ≈15.3 (C+ + C−)2H′8(Mb)Mb.

(24)These predictions are subject to the same uncertainties as the prediction for ψ given in (20).The ratios predicted above are for the direct production of charmonium states in thedecay of the B hadron. The branching fractions that are measured directly by experimentinclude indirect production from cascade decays of higher charmonium states.

The branchingfractions for direct production can only be obtained by making assumptions about the cas-cade decays. The inclusive branching fractions that have been measured are (1.12 ± 0.16)%for ψ [13], (0.46±0.20)% for ψ′ [13], (0.54±0.21)% for χc1 [14], and an upper bound of 0.4%for χc2 [14].

The branching fractions for the cascade processes ψ′ →ψ + X, ψ′ →χc1 + γ,and χc1 →ψ+γ are approximately 57%, 9%, and 27%, respectively [13]. If one assumes thatthere are no other cascade processes that give significant contributions, then the branchingfractions for direct production by B meson decay are (0.71±0.20)% for ψ, (0.46±0.20)% forψ′, and (0.50±0.21)% for χc1.

The ratio of the direct production rates of ψ′ and ψ is consis-tent, within the large error bars, with the ratio of their electronic widths, which is 0.40 [13].However the result for ψ is several standard deviations larger than the prediction of (20).The discrepancy could be due to order-αs corrections to the Wilson coefficient 2C+ −C−orto corrections to the subprocess rate (12) proportional to αs(C+ + C−)2. Alternatively, itcould be due to contributions from cascade decays into ψ and ψ′ from higher charmoniumstates.

Studies of the momentum distribution [15] and the polarization [2, 4] of the ψ’s couldhelp determine whether cascades from states other than ψ′ and χc1 are important.The measurement of the rate for B meson decay into χc1 (Ref. [14]) makes it possibleto extract a phenomenological value for the unknown matrix element H′8.Dividing the

branching fraction for direct decay of the B meson into χc1 by the semileptonic branchingfraction (10.7 ± 0.5)% [13], we find the branching ratio (23) to be 0.047 ± 0.020. With thechoice Mb = 5.3 GeV, the color-singlet term on the right side of (23) is 0.006.

Attributingthe remainder to the color-octet term, we find that H′8(Mb) ≈(4.8 ± 2.3) MeV. The quotederror is due to the experimental error in the branching ratio only, and does not include thetheoretical error due to the potentially large perturbative corrections to the Wilson coefficient2C+ −C−and to the color-singlet subprocess rate bΓ1.

One can also use (24) together withthe upper bound on the branching fraction into χc2 to obtain the upper bound H′8(Mb) < 2.7MeV. This bound is consistent with the determination from χc1 production, given the largeerror bar.

The large theoretical uncertainty in our determination of H′8 could be reduceddramatically by calculating the order-αs corrections to the P-wave subprocess rates. In theabsence of these calculations, one should regard our determination of H′8 as only a roughestimate.The color-octet mechanism has a dramatic effect on the pattern of the produc-tion rates for P-wave charmonium states.Taking the value H′8(Mb) ≈2.5 MeV, whichis consistent with both the determination from decays into χc1 and with the upper boundfrom decays into χc2, we find that the relative production rates predicted by (21)-(24) arehc : χc0 : χc1 : χc2 = 1.3 : 0.3 : 1 : 1.3.

A previous calculation of the decay rates of B mesonsinto charmonium states [4], which did not take into account the color-octet mechanism, gavethe relative rates hc : χc0 : χc1 : χc2= 0 : 0 : 1 : 0. The observation of χc2 productionat a rate comparable to that for χc1 production would be a dramatic confirmation of thecolor-octet production mechanism and would provide a direct measurement of the matrixelement H′8(Mb).P-wave production provides unique information on the production of heavy-quarkpairs and on their binding into quarkonium.

In this paper we have outlined a consistentfactorization formalism for describing these processes in QCD. In addition to the familiarcolor-singlet production mechanism, the factorization formulas take into account the color-octet mechanism, in which a heavy quark-antiquark pair is created at short distances in acolor-octet S-wave state and subsequently fragments by a nonperturbative process into acolor-singlet P-wave bound state.

The inclusive production rates for all four P states can

be calculated in terms of two nonperturbative inputs: a parameter H1 that is related tothe derivative of the wavefunction at the origin and a second parameter H′8 that gives theprobability for the fragmentation from the color-octet S-wave state. We have applied thisformalism to the production of P-wave charmonium in decays of B hadrons and have usedexperimental data to obtain a crude estimate of the parameter H′8.

An accurate determina-tion of H′8 requires perturbative calculations beyond leading order as well as more accurateexperimental data on χc production in B meson decays.The accurate determination ofthis parameter would be very useful because the production rates for P-wave charmoniumstates in photoproduction, leptoproduction, hadron collisions, and other high energy physicsprocesses satisfy factorization formulas involving the same two nonperturbative parametersH1 and H′8 that appear in B meson decay. Knowledge of these two parameters would, inprinciple, allow one to compute all of the inclusive P-wave charmonium production rates.This work was supported in part by the U.S. Department of Energy, Division of HighEnergy Physics, under Contract W-31-109-ENG-38 and under Grant DE-FG02-91-ER40684,and by the National Science Foundation.

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