Generating heavy quarkonia in a perturbative QCD cascade

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NORDITA-96/40P hep-ph/9606472June 28,1996Generating heavy quarkonia in a perturbative QCD cascade Per Ernstr¨o m and Leif L¨o nnblad NORDITA Blegdamsvej 17,DK-2100Copenhagen ?per@nordita.dk,leif@nordita.dk Abstract We present an implementation of heavy quarkonium production within a perturbative QCD cascade based on the Color Dipole Cascade model.We consider the processes most relevant in the context of the ψ′sur-plus at the Tevatron;g →ψ′and c →ψ′in the color-singlet model and g →ψ′through the color-octet mechanism.Our implementation is,however,easily extendible to other quarkonia and other production

mechanisms.

Where comparison is possible we ?nd good agreement with ana-lytical calculations.

We present some suggestions for measurements at the Tevatron that would be sensitive to the shape of the fragmentation functions.Our calculations indicate that such measurements could be used to test the color-octet mechanism solution to the ψ′surplus at the Tevatron.

1Introduction

Recent Tevatron data[1,2]on prompt charmonium(ψandψ′)production at large p T has attracted a lot of attention.

At large p T,charmonium production is believed to be dominated by par-ton fragmentation[3],and the Tevatron data[1,2]does indeed show a1/p5T behavior consistent with the predictions of a fragmentation process.

The large c quark mass m c make,however,predictions possible also for the magnitude of the cross section.Production of charmonium states can be factorized into the hard QCD production of a heavy cˉc pair within a region1/m c,followed by the nonperturbative formation of the charmonium state.In[4],a general factorization scheme was developed to all orders in the relative velocity v of the c andˉc quark in the charmonium state.For S-wave states,treated to leading order in v2,this factorization scheme reduces to the color singlet model[5];the cˉc pair must be produced in a color singlet state with the same spin and angular momentum as the charmonium state. The subsequent nonperturbative formation of the charmonium state can be described by a single process independent parameter.Fixing this parameter from measurements on the electro-magnetic decay of theψandψ′,predictions can be made.

A color singlet model calculation to LO inαs of large p Tψ′production fall, however,more than an order of magnitude below the Tevatron data[1,2,6,7]. Although this is the largest deviation from LO color singlet model predictions seen,it should be noted that large deviations have been observed also at low p T[8,9].

A calculation of promptψproduction[6,7,10]give a similar result,but in this case contributions from P-wave charmonium states decaying toψobscure the interpretation of the data.

2

Braaten et al.[11]have proposed that theψ′surplus at the Tevatron could be explained by a v suppressed production mechanism,usually referred to as the color octet mechanism.The cˉc pair is produced in a color octet3S1 state,and forms theψ′through soft gluon emission or absorption.The new nonperturbative parameter giving the“probability”for the color octet cˉc pair to form theψ′is supressed by v4as compared to the color singlet mechanism. The hard gluon fragmentation production of a color octet cˉc pair is,on the other hand,of lower order inαs as compared to the production of a3S1color singlet cˉc pair.

In addition,the color octet fragmentation process is enhanced by the so called trigger bias e?ect.The fragmentation contribution to the charmonium production cross section can be written as a convolution

dσO(p)= i 10dzdσi(p/z,μ)D i→O(z,μ),(1) of parton production cross sections dσi and fragmentation functions D i→O, describing the fragmentation of a parton i into a charmonium state O.Since the parton production cross sections fall steeply with the parton transverse momentum p T/z,fragmentation processes where the charmonium state takes a large fraction z of the parton momentum are favored.In a strict LO calculation the color octet fragmentation function is proportional toδ(1?z). When leading logs are re-summed,fragmentation functions are softened,but the fact remains that in comparison with other production mechanisms the color octet mechanism is characterized by large z fragmentation.

The arguments in favor of the color octet solution are of a qualitative nature,given a new nonperturbative parameter one can obviously?t to data. It is therefore crucial to?nd independent means to check the hypotheses by Braaten et al..

One way would be to look for other processes where the same color octet

3

production mechanism could be important,see eg.[12].Alternatively one might look for characteristics of the color octet production mechanism at the Tevatron.It has eg.been noted thatψ′produced through the color octet mechanism are dominantly transversely polarized[13].

Here we will make use of the fact that aψ′produced in a fragmentation process will be part of a jet.We investigate the possibilities to extract more information on the fragmentation process by detecting the associated jet. At a super?cial level one can identify the jet momentum with the parton momentum and thus get a measure of the fraction z of the parton momentum taken by theψ′.As noted above,large z fragmentation is a characteristic of the color octet production mechanism.

In[14]it was proposed that trigger bias enhanced higher order corrections might explain theψ′surplus within the color singlet model.It was shown explicitly that trigger bias enhanced higher order corrections to light quark fragmentation intoηc dominate over the LO result.Forψandψ′higher order calculations are however lacking.If this proposal is correct one would expect the fragmentation to be intermediate in hardness between the soft LO color singlet mechanisms and the very hard color octet mechanism.

In order to make a quantitative analysis it is necessary to implement the fragmentation processes in a perturbative QCD cascade,allowing the use of measurable variables such as z jet=p Tψ′/p T jet.

In this paper we present an implementation of the LO color octet and singlet production mechanisms relevant forψ′production,but the extension to other mechanisms and other quarkonia is trivial.

In section2we describe the implementation of fragmentation production of quarkonium states in a leading–log QCD shower generator based on the Dipole Cascade Model(DCM),?rst in general and then more speci?cly for the

4

LO color octet and singlet production mechanisms relevant forψ′production.

In section3we give some results of our simulations and discuss the possi-bilities to distinguish between di?erent fragmentation mechanisms based on the shape of fragmentation functions.

Our conclusions are presented in section4,which is followed by an ap-pendix describing the analytical approximations used to verify our imple-mentation.

2The Implementation

The fragmentation production of quarkonia may be viewed as a two step process:1.the hard fragmentation into a color singlet or octet cˉc-pair,and 2.the nonperturbative formation of the quarkonium state.

Here we implement both these steps in conjunction in a hard QCD shower generator,by including splitting functions not into cˉc-pairs but directly into quarkonium states.Thus,the nonperturbative formation of the quarkonium state is implemented at the perturbative level,before normal hadronization sets in.This is motivated by the fact that all hard interactions are already included in the splitting functions.The cˉc-pairs are thus e?ectively non-interacting until the hadronization onset scale.Finally,according to BBL factorization[4],the“probability”that a color octet or singlet cˉc-pair form a quarkonium state is given by process independent parameters(matrix el-ements).BBL factorization is however fully inclusive and does not describe how the cˉc-pair e?ects the hadronization of the remaining partons.In the color singlet case,the cˉc-pair is color disconnected from the other partons in the event and it is certainly reasonable to let the other partons hadronize independently of the cˉc-pair.The color octet case is more troublesome.In our implementation we enforce the radiation of a soft gluon that inherits

5

the color connection of the cˉc-pair to the other partons.That is,we sim-ply implement splitting into the quarkonium state plus a soft gluon.The hadronization is done using the stringfragmentation model in Jetset which is infrared safe w.r.t.soft gluons,suggesting insensitivity to the exact mod-elling of these soft interactions.We will,however,return to this nontrivial question in a forthcoming publication.

In this paper we present an implementation of heavy quarkonium pro-duction in the Ariadne program[15]which implements the Dipole Cascade Model(DCM)[16,17]for QCD cascades.

The DCM di?ers from conventional parton shower models in that gluon emission is treated as dipole radiation from color dipoles between partons.In eg.e+e?annihilation,the?rst gluon,g1,is emitted from the dipole between the initial qˉq pair.An emission of a second softer gluon can then be described in terms of radiation from two independent dipoles,one between the q and g1and one between g1andˉq.Further gluon emissions are given by three independent dipoles etc.

One problem with this model is that only gluon radiation is modelled and the process of splitting a gluon into a qˉq pair is not included in a natural way. In[18]it was shown that this process can be included without leaving the dipole picture by taking the ordinary splitting kernel for g→qˉq and dividing it in two equal parts describing the contribution from each connecting dipole to the splitting separately.This procedure works very well and describes cor-rectly the amount of secondary heavy quarks produced at LEP[19]although some theoretical questions have been raised[20].

The process of g→O+g will here be treated in the same way as the g→qˉq one.The splitting kernel d g→O(z,S),which is related to the fragmentation

6

function through

D g→O(z,μ)= μ20dS

,the transverse mass

S

and rapidity of the quarkonium in the dipoles center of mass,W is the total mass of the emitting dipole.

For eg.a q–g dipole there are now several competing processes that can occur;a gluon can be emitted according to the standard DCM,the initial gluon can be split into a qˉq pair of di?erent?avors and it can be split into a quarkonium plus gluons according to several di?erent mechanisms.This competition is as usual governed by the Sudakov formfactor mechanism using the transverse momentum squared as ordering variable.Hence the probabil-ity P i(k2⊥)of the process i to occur at a scale k2⊥is given by

P i(k2⊥)=d i(k2⊥)e? k2⊥,max k2⊥ j d j(p2⊥)dp2⊥(3) where d i(k2⊥)is the splitting kernel with the dependence on rapidity inte-grated out.

7

It is now straight forward to implement basically any mechanism for frag-mentation production of any heavy quarkonium.In this paper we discuss three mechanisms forψ′production:

?Color-Octet mechanism g→cˉc(3S(8)1)→ψ′±soft gluons.A perturba-tive transition of a gluon into a collinear cˉc pair in a color octet3S1 state,followed by a nonperturbative transition intoψ′involving emis-sion and/or absorption of soft gluons.As discussed above we implement this process as g→ψ′+g,with one soft?nal state gluon.Since the gluons are soft,the splitting kernel is essentially a delta-function,

παs 0|Oψ′8(3S1)|0

d g→ψ′(z,S)=

with one within the framework of Lund string fragmentation.The color-singlet gluon fragmentation function is given as a double integral in[22].Performing one of the integrals we?nd an analytical expression for the?ctitious g→ψ′+g splitting kernel.The color singlet parameter |R(0)|2relevant also for the last production mechanisms is taken to be

0.231GeV3.

?Color-Singlet mechanism for c→ψ′+c.Here the whole splitting kernel, given in[23],is assigned to the only dipole connected to the c quark.

The scale is the runningαS is,as in the other cases,taken to be m2⊥. 3What comes out

The Ariadne program is interfaced to Pythia and Jetset[24,25]for handling of the hard sub-process and hadronization respectively.We use this to generateψ′production events at the Tevatron.In?g.1we see the resulting p Tψ′spectrum for the di?erent production https://www.wendangku.net/doc/8a13577601.html,ing the same parameters as in[11],we reproduce approximately their result where the Octet mechanism totally dominates the cross section for high p Tψ′,and we also reproduce the measured p Tψ′→μ+μ?spectrum[1].

The di?erence between the Octet and Singlet mechanisms is larger in our case than in[6,11]even after taking into account the small variations in the parameters used.The generated spectra from Ariadne does,however,agree with the analytical approximation in the appendix at large p T.At small p T we do not expect these to agree because there the simple assumption in the analytical calculation that the gluon spectrum behaving like1/p5T breaks down.Also e?ects of initial-state QCD radiation,implemented through the so-called Soft Radiation Model[26]in Ariadne,are absent in the analytical

9

CDF data total ψ′Singlet c →ψ′Singlet g →ψ′Singlet g →ψ′Octet p ⊥ψ′(GeV)

d σ/d p ⊥ψ′→μ+μ?(p b /G

e V )252015105010?6

10?410?21

Figure 1:The transverse momentum distribution of ψ′→μ+μ?production at the Tevatron for di?erent production mechanisms compared to CDF data [1].Full line is the color octet mechanism,dotted line is the sum of the color singlet mechanisms,and the long-dashed and short-dashed lines are the gluon and c-quark initiated singlet mechanisms respectively.The histograms were generated with Ariadne and the smooth lines are from the analytical approximation in the Appendix.

calculation.In addition,Ariadne implements a more complete treatment of the parton evolution,in contrast to the analytical calculations,which,like earlier numerical calculations,uses the approximative though asymptotically correct AP evolution.The full evolution equation and a discussion of the approximation made in the use of the AP evolution equation can be found in [6].

In ?g.2we show the generated z -spectrum for 20

where z is now p T ψ′/?p

T and ?p T is the transverse momentum in the hard subprocess.It is clear that the z –spectrum is much harder for the octet mechanism as expected.This is also seen in ?g.3,where the average z is shown as a function of p T ψ′.Here we also ?nd a good agreement with the

10

c →ψ′Singlet g →ψ′Singlet g →ψ′Octet z

1/N d N /d z 10.80.60.40.204

3

2

1

0Figure 2:The z -distribution for 20

c →ψ′Singlet g →ψ′Singlet g →ψ′Octet p ⊥ψ′ z

50

4030201001

0.8

0.6

0.4

0.2

0Figure 3:The average z as a function of p T ψ′for di?erent production mech-anisms.The smooth lines are from the analytical approximation in the ap-pendix and the (due to low statistics)broken lines were generated with Ari-adne .

11

with fragmentation and MI with fragmentation generated z z

1/N d N /d z 10.80.60.4

0.208

6

4

2

0Figure 4:The z distribution for 20

With the availability of ψ′production in an event generator we can now also study hadronization e?ects.To study eg.the z -spectrum experimentally one could sum up the transverse energy E T jet in a cone in the pseudorapidity–azimuth plane around the detected ψ′,de?ning z jet =p T ψ′/E T jet .In ?g.4we compare the z -spectrum for such a procedure with the generated spectrum in ?g.2for the octet mechanism.We note that just adding hadronization,a smearing of the jet is introduced and some of the jet energy falls outside the cone,which makes the spectrum considerably harder.Adding also multiple interactions to model the underlying event [27]in Pythia makes it softer again.In ?g.5we see that the di?erences in hardness between the octet and singlet mechanisms remains after all hadronization e?ects although the 12

c →ψ′Singlet g →ψ′Singlet g →ψ′Octet z

1/N d N /d z 10.80.60.4

0.204

32

1

0Figure 5:The z distribution for 20

shapes are somewhat distorted.4Conclusions

We have implemented di?erent quarkonium production mechanisms in the Ariadne event generator.Besides giving a more accurate treatment of the initial–and ?nal–state parton evolution than recent analytical approxima-tions,such a generator is an essential tool for understanding experimental corrections and hadronization e?ects in case one wants to study non-inclusive aspects of high p T onium production at e.g.the Tevatron.

In interpreting the results it is important to keep in mind the status of the implementations of the di?erent production mechanisms:

?The implementation of the color singlet c →ψ′+c fragmentation is unproblematic and trustworthy.

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?The color singlet g→ψ′+gg fragmentation has been implemented as g→ψ′+g.The error introduced by this simpli?cation is believed to be small,and in the future the full g→ψ′+gg splitting will be implemented.

?The color octet g→cˉc(3S(8)1)→ψ′±soft gluons,has been implemented as g→ψ′+one soft gluon,and the e?ects of the physical O(v2)width of the fragmentation function has not been investigated.In the future the sensitivity of the results to the modelling of the soft interactions will be studied.

The production mechanisms we have implemented here are,after a sim-ple change of normalization relevant also forψproduction.In this case it is necessary,however,to take into account also contributions from P-wave states decaying toψ.It is therefore important to implement also P-wave production mechanisms.

There are also other experiments where the fragmentation production of quarkonium states is important.One interesting example is Z0decays at LEP.Here the situation is quite di?erent as compared to large p T production. There are more quarks than gluons available for fragmentation,and there is no trigger bias enhancement.Consequently other production mechanisms might be dominant and measurements would be more sensitive to the low z shape of fragmentation functions.

We have shown that it might be possible to measure the z-distribution of the fragmentation function at the Tevatron and from that learn about the production mechanism.

This result is of course dependent on our very crude approximation of the experimental environment.But with the implementation presented here,the determination of optimal choices of jet de?nitions and experimental cuts is

14

clearly facilitated and it does not seem unlikely that the z spectrum,if mea-sured at the Tevatron,could be used to better understand the quarkonium production mechanism.

Acknowledgement.We are greatful for discussions with Paul Hoyer, Michelangelo Mangano and Mikko V¨a nttinen.

15

Appendix

Here we present a very simple calculation that exhibits the gross features of charmonium production at large p T.

We start out from a rapidity integrated version of eq.(1)

dσψ′(p T)

(p T/z,μ)D g→ψ′(z,μ).(5)

dp i T

We choose the factorization scale asμ=p T/z,and note that large logs log(4m2c/μ2)can be resummed by AP evolution[28]of the fragmentation functions.We calculate the parton production cross sections as convolutions of hard subprocess cross sections(∝p?4T)and parton distributions,and?nd that the parton production cross sections fall roughly as p?5T.Consequently we take dσi/dp i T≈c i/p i T5in eq.(5)and?nd

dσψ′

z5D i→ψ′(z,p i T/z)≈ i c i

p T5

QCD shower generator implementation,we include mixing explicitly and use the LO charm production cross section to avoid double counting.

For moments(n>1)the AP evolution equations can be solved analyti-cally.The solution has the form:

D(n)g→ψ′(μ)= αs(μ)

αs(μ0) d(n)

c ?(n)D(n)g→ψ′(μ0)+δ(n)D(n)c→ψ′(μ0) (7)

D(n)c→ψ′(μ)= αs(μ)

αs(μ0) d(n)

g ?(n)D(n)c→ψ′(μ0)?ρ(n)D(n)g→ψ′(μ0) ,(8)

withρ(n)δ(n)=?(n)(1??(n)).For the moments n relevant here,the mixing parametersρ(n),δ(n)and?(n)are all small,and we may putρ(n)and?(n) to zero.Within the color singlet model theδ(n)D(n)c→ψterms in eq.(7)are important however since as noted above the c quark fragmentation function is much larger than the gluon fragmentation function.For simplicity we choose the initial scales for charm and gluon fragmentation and the mixing onset scale all to2m c and ignore?avor thresholds(re?nements are simple). Combining eq.(6),(7)and(8)we?nd

dσψ′

p5T αs(p T)

p5T αs(p T)

Similarly we can calculate the mean value of z in the convolution in eq.(5).Super?cially identifying z with p Tψ′/p T jet,this gives us a hint of the possibilities to use the form of fragmentation functions to distinguish between di?erent production mechanisms.More importantly,it can be used to check our QCD cascade implementation of quarkonium production.

For convenience we give numerical values for the moment evolution pa-rametersδ(n),d(n)g,d(n)c for4(5)quark?avors,as well as the values we have used for the moments of the color singlet fragmentation functions D(n)c→ψ′and D(n)g→ψ′:

n6

0.030(0.028)0.018(0.017)

2.18(2.46)2.68(

3.00)

0.97(1.05)1.17(1.27)

1.2·10?57.7·10?6

1.8·10?71.1·10?7

For the moments of the color octet gluon fragmentation function we have used the value D(n)g→ψ′(2m c)≈4.2·10?5,independent of the moment n.

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