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Color-Octet $Jpsi$ Production at Low $p_perp$

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NORDITA-96/18P March 1996hep-ph/9603266Color-Octet J/ψProduction at Low p ⊥Wai-Keung Tang ?Stanford Linear Accelerator Center,Stanford University,Stanford,CA 94309M.V¨a nttinen ?NORDITA,Blegdamsvej 17,DK-2100Copenhagen (February 1,2008)Abstract We study contributions from color-octet quarkonium formation mechanisms to J/ψhadroproduction at low p ⊥.We include transitions of color-octet c ˉc states into “direct”J/ψand into χ1,2which decay radiatively into a J/ψ.Together with earlier work,this calculation constitutes a complete analysis of p ⊥-integrated J/ψproduction at leading twist.We ?nd that the leading-twist

contribution is not su?cient to reproduce the observed production rates and

polarization of the J/ψand χ1,2.Hence there must exist other important

quarkonium production mechanisms at low p ⊥.

Typeset using REVT E X

I.INTRODUCTION

The production and decays of heavy quarkonia have been widely studied using perturba-tive QCD[1].Due to the large masses of the c and b quarks,the production or annihilation of heavy-quark–antiquark pairs takes place at a much shorter timescale than the forma-tion of the bound state.This makes it possible to factorize the transition amplitudes.In hadroproduction reactions,also the initial-state hadronic structure is usually expected to be factorizable from the QˉQ production dynamics in terms of single parton distributions;in other words,quarkonium production is usually taken to be a leading-twist process.

In the simplest approach,one takes the hadroproduction cross section to be a?xed fraction of the integrated QˉQ cross section below the open heavy?avour threshold.This is called the semilocal duality model or color evaporation model.This model reproduces successfully the dependence of the cross sections on the center-of-mass energy of the colli-sion and on the longitudinal momentum of the quarkonium[2].However,many observables such as the relative production rates of di?erent charmonium states,transverse momentum distributions,and quarkonium polarization are not predicted.In a more sophisticated ap-proach,a color-singlet QˉQ pair with the appropriate J P C quantum numbers is produced in the hard process.The quarkonium production amplitude is then written as the convolution of a perturbative amplitude for the production of a QˉQ 2S+1L(1)J pair(the superscript“(1)”

stands for a color-singlet con?guration)and a nonrelativistic wave function.This model,in which only the leading,color-singlet Fock component of the quarkonium wave function is taken into account,is called the color singlet model or charmonium model.

In the color singlet model,the polarization of?nal-state quarkonium is determined by the perturbative dynamics of QˉQ production and by the angular momentum projection in the wave function.Since polarization is relatively insensitive to higher-order corrections,it provides a very good probe of the basic QˉQ production mechanisms.

Experimentally,J/ψpolarization has been measured in?xed-targetπ?N[3–5]andˉp N [5]reactions.The parameterαin the angular distribution1+αcos2θof J/ψdecay dileptons

in the Gottfried-Jackson frame has been measured to beα=0.028±0.004inπ?N reactions andα=?0.115±0.061inˉp N reactions for longitudinal momentum fraction x F>0at125 GeV[5].This corresponds to unpolarized production.The theoretical interpretation of these results is complicated by the fact that the“direct”andχc-decay components of J/ψproduction have not been resolved in the polarization analysis.However,the polarization of theψ′(2S)has been measured inπ?N reactions[6].Theψ′is also produced unpolarized, withα=0.02±0.14for x F>0.25at253GeV.Apart from an overall normalization factor, the direct J/ψcross section is expected to be similar to the totalψ′cross section,because no signi?cant contribution toψ′production from the decays of higher-mass states has been observed.We therefore assume that the direct J/ψcomponent is unpolarized;theχ-decay component should then also be unpolarized.

In Ref.[7],we calculated the leading-twist contribution to the p⊥-integrated cross section σ(π?N→J/ψ(λ)+X),whereλis the helicity of the J/ψ,within the color singlet model. We included contributions from the direct mechanism gg→J/ψ+g and from the radiative decays ofχJ states produced in the reactions gg→χ2,gg→χ1g,qg→χ1q,and qˉq→χ1g. The polarization analysis of the decay contribution is made possible by the electric dipole nature of the decay.Inψ′production,only the direct mechanism needs to be taken into account.

The leading-twist,color-singlet contributions to J/ψandψ′production turn out to be dominantly transversely polarized,i.e.σ(λ=±1)>σ(λ=0),withα～0.5for the total J/ψandα～0.3for theψ′and direct J/ψproduced by pions.This is in contrast to the experimental observation of unpolarized production.The model also fails to reproduce the observed relative production rates[8]of the J/ψ,χ1andχ2states.In Ref.[7],we inter-preted these discrepancies as evidence for important higher-twist mechanisms of charmonium production.

However,one could argue that the discrepancies are due to neglecting the higher Fock components in the quarkonium wavefunction.These are systematically included in nonrel-ativistic QCD(NRQCD),an e?ective?eld theory which has been formulated during the

recent years[9,10].NRQCD provides an expansion of quarkonium cross sections and decay widths in terms of the relative velocity v of the heavy quark and antiquark in the bound state.The nonperturbative physics is factorized into an in?nite number of matrix elements that scale in a well-de?ned way in m Q and v.

Color-octet mechanisms have recently been suggested[11–13]as an explanation of the large quarkonium production rates observed at the Tevatron pˉp collider[14,15].Some of these rates exceed color-singlet-model predictions by more than an order of magnitude.

There exist color-octet mechanisms of low-p⊥quarkonium production which are of higher order in v2but lower order inαs compared to the leading color-singlet mechanisms.They might help to reproduce the?xed-target data;the purpose of this paper is to?nd out whether they do.If not,then the?xed-target discrepancies must be due to a breakdown of the leading-twist approximation of QˉQ production.

In a recent work[16],we already analyzed the color-octet contribution toψ′hadroproduc-tion at low p⊥.We found that the leading color-octet contribution is dominantly transversely polarized.Hence,even with the leading color-octet components included,the unpolarized ψ′production data[6]cannot be reproduced.This suggests that there are important higher-twist QˉQ production mechanisms.Theψ′analysis also applies to the direct component of J/ψproduction.In the present paper,we complete our analysis by calculating the contribu-tion to J/ψproduction from the radiative decays ofχ1andχ2produced through color-octet intermediate states.It will turn out that this component,too,is dominantly transversely polarized.Furthermore,determinations of color-octet matrix elements from the analysis of other reactions imply that the magnitude of this component is small compared to the data. We shall conclude that in J/ψas well as inψ′production,new mechanisms are likely to be important at low p⊥.

II.SUMMARY OF COLOR-SINGLET J/ψPRODUCTION CROSS SECTIONS

At leading order inαs and at leading twist,the color-singlet QˉQ production subprocesses are

gg→1S0,3P0,2,(2.1)

gg→3S1+g,3P J+g,(2.2)

gq→3P J+q,(2.3)

qˉq→3P J+g,(2.4)

i.e.they are obtained from the leading-order annihilation subprocesses such as J/ψ→ggg by crossing.Note that the process gg?3P1,with the gluons on mass shell,is forbidden by Yang’s theorem[17].Within the color singlet model,the QˉQ production mechanism could be di?erent from(2.1–2.4);it could e.g.be a higher-twist process,where many partons from the same initial-state hadron participate in the hard scattering.

The radiative decaysχ1,2→J/ψ+γare known experimentally to be a major source of J/ψproduction.They contribute30-40%of the total cross section in both pion-and proton-induced reactions[8].The production of theψ′,on the other hand,is expected to be dominantly due to the direct subprocess(2.2).

The polarization of a3S1state like the J/ψis re?ected in the polar-angle distribution of its decay dileptons in their rest frame.The parameterαin the angular distribution 1+αcos2θis related to the polarized J/ψproduction cross section

σ(λ)=σtot

1+aδλ0

dσ(λ=1)+2dσ(λ=0)+dσ(λ=?1)=?

the beam momentum lies along the z axis and the target momentum lies in the xz plane, with p x(target)≤0.At vanishing transverse momentum of the quarkonium,the Gottfried-Jackson frame is obtained from the laboratory frame by a simple boost,and the choice of the x axis is immaterial.

We analyzed the polarized cross sections of J/ψproduction through the direct process andχJ decays in Ref.[7],where we found that the color singlet model is insu?cient to explain the existing?xed-target data.In the color singlet model,theχ1and direct J/ψcross sections are predicted to be signi?cantly lower than measured relative to theχ2cross section,which is within a factor K=2–3of the experimental value.In contrast with the observed unpolarized production,the predicted total J/ψcross section is dominantly transversely polarized even if the various contributions are renormalized according to the data.The quantitative results of the calculation of Ref.[7]are listed in Table I together with experimental data and color-octet predictions.

III.COLOR-OCTET J/ψPRODUCTION CROSS SECTIONS

As one calculates quarkonium production cross sections within the NRQCD factorization scheme,one is using two expansions:the perturbative expansion of the short-distance QˉQ production amplitude and the velocity expansion of the long-distance quarkonium formation amplitude.General rules for?nding out the power dependence of the NRQCD matrix elements on m Q and v can be found in Refs.[9,10].At leading order in perturbation theory, i.e.O(α2s),and up to next-to-leading order in the velocity expansion,the subprocesses for leading-twist J/ψproduction through color-octet intermediate states are

qˉq→cˉc 3S(8)1 →J/ψ+gg,(3.1)

gg→cˉc 1S(8)0 →J/ψ+g,(3.2)

gg→cˉc 3P(8)J →J/ψ+g,(3.3)

qˉq→cˉc 3S(8)1 →χJ+g→J/ψ+γ+g.(3.4)

These are illustrated by the Feynman diagrams of Fig.1,where the blob represents a nonper-turbative transition.Their cross sections are proportional to the NRQCD matrix elements

0|O J/ψ8(3S1)|0 ～m3c v7,(3.5)

0|O J/ψ8(1S0)|0 ～m3c v7,(3.6)

0|O J/ψ8(3P J)|0 ～m5c v7,(3.7)

0|OχJ8(3S1)|0 ～m3c v5,(3.8) respectively.Hence the direct J/ψproduction cross sections are proportional toα2s v7,and theχJ cross section is proportional toα2s v5.They are to be compared with the leading color-singlet cross sections

σ(gg→χ2)～α2s v5,

σ(gg→J/ψ+g)～α3s v3,

σ(ij→χ1+k)～α3s v5.(3.9)

The amplitude

A qˉq→cˉc 3P(8)J →J/ψ+g (3.10) is of higher order in v2than the amplitudes of the processes(3.1–3.4),because the lowest-order non-perturbative transition(single chromoelectric dipole)is forbidden by charge con-jugation.Furthermore,the amplitude

A gg→cˉc 3S(8)1 →χJ+g (3.11) is of higher order inαs because the amplitude A gg→cˉc 3S(8)1 vanishes in the leading order.Charge conjugation or Yang’s theorem would not require this amplitude to vanish because the cˉc pair is not in a color-singlet state.

The contribution toσ(hN→J/ψ(λ)+X)from the direct production subprocesses(3.1–3.3)follows immediately from theψ′production analysis of Ref.[16]:

σoctet(hN→direct J/ψ(λ)+X)=O1 0|O J/ψ8(3P1)|0 (3?2δλ0)

+O2 0|O J/ψ8(1S0)|0 (1?δλ0)

+O3 0|O J/ψ8(3S1)|0 (1?δλ0).(3.12) The coe?cients are

5π3α2s

O1=

Φgg/hN(M2/s,μF),(3.14) 24M5

8π3α2s

O3=

x1x2 (3.16) is a parton?ux factor evaluated at the leading-twist factorization scaleμF(since the?nal-state gluons are taken to be soft,the kinematics is essentially that of a2→1subprocess, with?s=M2=4m2c).For ease of reference,we plot in Fig.2the gg and qˉq?ux factors inπ?-proton,proton-proton and antiproton-proton collisions using the GRV-LO parton distributions[18].In accordance with the existing experiments,the integral is over the region x F=x1?x2>0.

The polarization of the directly produced J/ψhas not been measured separately from the polarization of those fromχdecays.However,the fact that theψ′are produced unpolarized [6]lets us expect that the direct J/ψcomponent is also unpolarized.Then,as in theψ′case, only about half of the observed direct production can be due to the strongly transversely polarized contribution(3.12).We discussed the derivation of a quantitative bound on a linear combination of NRQCD matrix elements in Ref.[16].

To evaluate the contribution from the process(3.4),we make use of the fact that both the cˉc 3S(8)1 →χJ+g transition and the radiative decay of theχJ are electric dipole transitions.The necessary formulas are given in the appendix.From the qˉq→cˉc scattering amplitude,

O AB q ˉq →c ˉc =4παs

a A q ˉq →c ˉc 3S (8)1;S z ,a

2=32π2α2s

27M 5δ 1?

M 28

(3.19)

σ q ˉq →c ˉ

c 3S (8)1 →χ2+g →J/ψ(λ)+γ+g =16π3α2s ?s Br (χ2→J/ψ+γ) 0 O χ2

8(3S 1) 0 47?21δλ0

s ,μF

+Φˉq q/hN M 2

27M 5 0|O χ1

8(3S 1)|0

× Br (χ1→J/ψ+γ)3?δ

λ0

3Br (χ2→J/ψ+γ)47?21δ

λ0

values actually imply that the contribution(3.21)is rather small compared to the observed J/ψcross https://www.sodocs.net/doc/3114615620.html,ing 0|Oχ18(3S1)|0 =(9.8±1.3)·10?3(GeV)3[13],αs=0.26and M=3.5GeV,we obtain

λσoctet(hN→χ1,2+g→J/ψ(λ)+γ+g)

=4.5nb q Φqˉq/hN M2s,μF =

1.3nb(π?beam)

0.6nb(p beam)

(3.22)

at E lab(π)=300GeV.These numbers are more than an order of magnitude smaller than the experimental cross sections of72nb,67nb and45nb(with errors of about25%)forπ+,π?and p beams,respectively[8].Hence the color-octet mechanisms cannot reproduce the ?xed-target data.On the other hand,our analysis does not set any constraints that would contradict the color-octet description of other reactions.

IV.SUMMARY

In this paper,we have considered the production of J/ψcharmonium in?xed-target reactions within the factorization scheme of nonrelativistic QCD(NRQCD)[10].We have calculated the contribution from the radiative decays,χ1,2→J/ψ+γ,ofχJ charmonia produced through intermediate color-octet cˉc states at leading twist.The“direct”color-octet component of J/ψproduction is given by our analysis of color-octetψ′production [16].Together with our earlier evaluation of the color-singlet component[7],the present calculation constitutes a complete analysis of leading-twist J/ψproduction at low p⊥.We have included contributions from nonperturbative transitions between intermediate cˉc states and charmonium up to relative order v4,where v is the relative velocity of the charm quark and antiquark in the bound state.The cˉc production amplitudes have in each case been evaluated at the lowest order inαs allowed by the quantum numbers of the intermediate cˉc state.

The results on J/ψproduction obtained in this paper and Refs.[7,16]have been collected

in Table I.In the evaluation of color-octet contributions,we have used the NRQCD matrix elements determined in Refs.[13,21].

Our motivation has been to test both the NRQCD picture of the long-distance quarko-nium formation process and the leading-twist approximation of short-distance QˉQ produc-tion.Predictions of the?nal-state charmonium polarization provide a very good test of the models involved.They are relatively insensitive to higher-order corrections in perturbation theory.Polarization analysis of the long-distance process is made possible by the fact that the emission or absorption of soft gluons does not change the heavy quark spins.

We have shown that the leading-twist contribution to J/ψproduction is dominantly transversely polarized,i.e.α>0in the angular distribution,1+αcos2θ,of the decay J/ψ→?+??in the Gottfried-Jackson frame.Actually,all but one of the individual com-ponents of the theoretical cross section are dominantly transversely polarized.The only exception is the small contribution from the radiative decay of theχ1produced by color-singlet mechanisms,which givesα≈?0.15.Experimentally,on the other hand,it has been observed that S-wave charmonia are produced unpolarized.

Furthermore,the values of NRQCD matrix elements determined from other reactions imply that the normalization of color-octet contributions is small compared to the observed charmonium cross sections.

Assuming that the NRQCD factorization scheme provides a complete description of quarkonium formation,the reason for the discrepancies must lie in the choice of cˉc production mechanisms.Hence there should exist important higher-twist mechanisms of cˉc production at small p⊥.

Since the mass of the b quark is signi?cantly larger than the c quark mass,all the approx-imations involved in the calculation–perturbation theory,the velocity expansion,and the leading-twist approximation–are expected to work better for bottomonium.Unfortunately, the existing bottomonium production data is insu?cient to test our predictions.

APPENDIX A:DIPOLE TRANSITIONS

1.Electric dipole transitions

The polarized cross section of J/ψproduction via the radiative decay of a χJ charmonium state is

σ(ij →χJ +X →J/ψ(λ)+γ+X )

1 |A (ij →χJ +X →J/ψ(λ)+γ+X )|2=1

2πdLips(χJ ,X )dLips(J/ψ,γ)

M χJ Γtot (χJ )δ(M 2χJ ?p 2χJ ).(A3)

In the electric dipole approximation,the χJ decay amplitude is written as

A (χJ (J z )→J/ψ(λ)+γ(μ))=

L z S z

JJ z |L z S z A (c ˉc (L z ,S z )→J/ψ(λ)+γ(μ))= L z S z

JJ z |L z S z NδS z λ??L z (μ)=N JJ z |J z ?λ,λ ??J z ?λ(μ),(A4)

where the symbol δS z λexpresses the heavy quark spin conservation and the normalization factor is

N = 24πM J/ψM 2χJ

Γ(χJ →J/ψ+γ)

Using the results

dLips(J/ψ,γ)=M2χ

J?M2J/ψ

δij,(A7)

where d?is a solid angle element in the charmonium rest frame,we obtain

σ(ij→χJ+X→J/ψ(λ)+γ+X)

2.Chromoelectric dipole transitions

Analogously with eq.(A8),we write the cross section ofχJ production via the chromo-electric dipole”decay”of a color-octet cˉc[3S1]state as

σ ij→cˉc 3S(8)1 →χJ(J z)+g =1

δ 1?M2 a A ij→cˉc 3S(8)1;S z,a 2,(A9) where 0|OχJ8(3S1)|0 is a matrix element of nonrelativistic QCD and

A ij→cˉc 3S(8)1,S z,a =√2√

σ ij→cˉc 3S(8)1 →χJ+g→J/ψ(λ)+γ+g

=Br(χJ→J/ψ+γ)1

δ 1?M2 a A ij→cˉc 3S(8)1;S z,a 2.(A11)

REFERENCES

?Work supported in part by Department of Energy contract DE–AC03–76SF00515and DE–AC02–76ER03069.

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FIG.1.The Feynman diagrams which describe the leading color-octet mechanisms of J/ψand χJ production.The blob represents a nonperturbative transition.The dashed line indicates the color-octet intermediate state.

FIG.2.Gluon-gluon and quark-antiquark?ux factors,Φgg and q(Φqˉq+Φˉq q),plotted as a function ofτ=?s/s=x1x2using the GRV-LO parton distributions[18].(a)π?-proton reac-tions,(b)proton-proton reactions,(c)antiproton-proton reactions.Solid line:Φgg(μF=M). Dotted line:Φgg(μF=M/2).Dashed line: q[Φqˉq+Φˉq q](μF=M).Dash-dotted line: q[Φqˉq+Φˉq q](μF=M/2).We used M=3.5GeV.

TABLE I.Leading-twist color-singlet and color-octet contributions to J/ψproduction inπ?N collisions at300GeV shown together with experimental data[5,8].The cross sections have been integrated over x F>0and normalized byσexp(all J/ψ)=178±21nb.The dependence of theo-retical contributions on the strong coupling constantαs and on the relative velocity v of the quark and antiquark in the bound state is indicated.We used the GRV-LO parton distributions[18]. Observed process

theoretical subprocesses Scalingσ/σexp(all J/ψ)α

cc [ S1 ]

]

ψ+

Fig. 1

Color-Octet $Jpsi$ Production at Low $p_perp$

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