Physics Letters B 642 (2006) 62–67

www.elsevier.com/locate/physletb

Colour connection and diquark fragmentationin e+e− → ccqq → h′s process

Wei Han a, Shi-Yuan Li a,b,∗, Zong-Guo Si a, Zhong-Juan Yang a

a Department of Physics, Shandong University, Jinan 250100, PR Chinab Institute of Particle Physics, Huazhong Normal University, Wuhan 430079, PR China

Received 24 January 2006; received in revised form 12 May 2006; accepted 3 August 2006

Available online 15 September 2006

Editor: N. Glover

Abstract

The hadronization effects induced by different colour connections of ccqq system in e+e− annihilation are investigated by a toy model wherediquark fragmentation is employed based on Pythia. It is found that the correlations between the charm baryons and charm antibaryons producedvia diquark pair fragmentation are much stronger, and their momentum spectra are harder than those from the standard colour connection inPythia.© 2006 Elsevier B.V. All rights reserved.

Keywords: Colour connection; Diquark; Hadronization

The hadronization for partons fragmenting into hadrons isvery important, and can only be treated by hadronization mod-els up to now. The colour connection plays a crucial rôle toset the “surface” between the perturbative quantum chromo-dynamics (PQCD) phase and the hadronization one, while thetopic related to the colour connection is beyond the approachof PQCD. It is argued that some clews can be drawn fromanalysing the decomposition of the colour space of the final par-tons produced in PQCD phase [1–3]. For the colour space of theparton system, there are many decomposition ways correspond-ing to various colour connections, respectively (see, e.g., [4,5]).Hence, it is instructive to study the special properties of the finalhadrons, which could give information for the evidence of spe-cific colour connections. Such kinds of investigation have beenmade for various parton systems [6–8]. One interesting and im-portant example is the (q1q2q3q4) system produced via hardcollisions. This is the simplest system where many phenom-ena/mechanisms related to QCD properties could be studied,

* Corresponding author.E-mail addresses: hanwei@mail.sdu.edu.cn (W. Han), lishy@sdu.edu.cn

(S.-Y. Li), zgsi@sdu.edu.cn (Z.-G. Si), yangzhongjuan@mail.sdu.edu.cn(Z.-J. Yang).

0370-2693/$ – see front matter © 2006 Elsevier B.V. All rights reserved.doi:10.1016/j.physletb.2006.08.067

e.g., (re)combination of quarks in producing special hadrons,influence on reconstruction of intermediate particles (like W±)by soft interaction, etc., most of which more or less relate withthe colour connections among the four (anti)quarks. In this Let-ter, the possible colour connections of this system are brieflyreviewed. It is pointed out that the hadronization effects of onekind of the colour connections have never been modelled yet.Its special case (diquark pair fragmentation) in e+e− annihila-tion is investigated in detail.

For q1q2q3q4 system, two of the decomposition ways of itscolour space are given as follows:

(31 ⊗ 3∗

2

) ⊗ (33 ⊗ 3∗

4

) = (112 ⊕ 812) ⊗ (134 ⊕ 834)

(1)= (112 ⊗ 134) ⊕ (812 ⊗ 834) ⊕ · · · ,(31 ⊗ 3∗

4

) ⊗ (33 ⊗ 3∗

2

) = (114 ⊕ 814) ⊗ (132 ⊕ 832)

(2)= (114 ⊗ 132) ⊕ (814 ⊗ 832) ⊕ · · · ,where 3, 3∗, 1, 8 respectively denote the representations triplet,antitriplet, singlet and octet of the group SUc(3), and thesubscripts correspond to the relevant (anti)quarks. In thesetwo colour decompositions, quark and antiquark form a clus-ter/string which fragments into hadrons independently. This

W. Han et al. / Physics Letters B 642 (2006) 62–67 63

Fig. 1. (a) B− → π− + D0 and (b) B− → p + Σ0c processes.

picture is adopted in the popular hadronization models (for de-tail, see [9–12]). The difference between these two decomposi-tions leads to the colour reconnection phenomena as discussedin the processes e+e− → W+W−/Z0Z0 → q1q2q3q4 → 4jets at LEPII [2,8,13–16], etc. One might also notice, in thee+e− → q1q1 + g∗ → q1q1q2q2 process, (q1q1) and (q2q2)cannot be in colour-singlets, while (q1q2) and (q2q1) are usuallytreated as colour-singlets and hadronize independently. Thiscorresponds to the term 114 ⊗ 132 in Eq. (2) [17].

The colour space of q1q2q3q4 system can also be decom-posed as follows:

(31 ⊗ 33) ⊗ (3∗

2 ⊗ 3∗4

) = (3∗

13 ⊕ 613) ⊗ (

324 ⊕ 6∗24

)

(3)= (3∗

13 ⊗ 324) ⊕ (

613 ⊗ 6∗24

) ⊕ · · · ,where 613 (6∗

24) denotes the representation sextet (antisextet)of the group SUc(3). A special case is that two quarks incolour state 3* attract each other and form a “diquark” whentheir invariant mass is small enough, and similarly two anti-quarks form an antidiquark. For this decomposition, only thewhole four (anti)quarks system can form a colour singlet, hencethey must hadronize altogether. This hadronization picture isquite different from that adopted in the popular hadronizationmodels. Unfortunately, this picture has not been taken into ac-count though it is crucial and exists indeed in many processes.One example is the exclusive B− decay as shown in Fig. 1where b → c + W−∗ → c + d + u1. In this process, the quark–antiquark pairs du1 and cu are in colour-singlet, and naturallyform two mesons π− and D0 (Fig. 1(a)). Here the combina-tion for du and cu1 is also possible. These two combinationscorrespond to the colour connections shown in Eqs. (1) and(2), respectively. The branching ratio of B− → π−D0 has beenmeasured to be (4.98 ± 0.29) × 10−3 [18]. In the meantime,it is very interesting to notice that another kind of colour con-nection corresponding to Eq. (3) exists, i.e., the cd and uu1are in colour state 3∗ and 3, and combine with a d or d fromvacuum to form two baryons Σ0

c and p, respectively. This isa ‘signal channel’ for diquark–antidiquark pair fragmentation,and its branching ratio is (0.45+0.26

−0.19 ± 0.007 ± 0.12) × 10−4

[19]. Additionally, the three body baryonic decay processessuch as B− → Λcpπ− (Br = (2.1 ± 0.7) × 10−4 [18]) canalso be regarded to be induced by diquark fragmentation. An-other example is the doubly-heavy baryon production in highenergy collisions [20,21]. Recently, SELEX Collaboration ob-served the hadronic production of Ξcc at Fermilab [22]. This isdominated by the production of cc pair in PQCD process, and

Fig. 2. One of the four Feynman diagrams for e+e− → ccqq process.

can also give an unambiguous confirmation of the existence ofthe diquark fragmentation.

The aim of this Letter is to investigate the diquark frag-mentation in e+e− → q1q1q2q2 → h′s process. The accumu-lated/accumulating data in various e+e− colliders can providesufficient statistics to study the hadronization of the four quarkstates. In order to specify different colour connections of thefour quark system via measuring the corresponding hadron(s),we focus on the e+e− → ccqq → h′s process, where q rep-resents u, d , or s quark. The ensemble properties related to aclass of hadrons (e.g., charm baryons) are investigated, so thatwe can search for the quantities which could be measured inexperiments. The statistics can also be enhanced by studying agroup of hadrons.

As mentioned above, the hadronization of the colour con-nection displayed in Eq. (3) has not been considered in popularhadronization models. However, one can employ the event gen-erator Pythia [10] based on Lund model [9], to describe thehadronization of the special case that both cq (in 3∗) and cq

(in 3) have small invariant masses so that they can be regardedas diquark and antidiquark, respectively. For this specific case,there are strong constrains on both colour state and phase space.Therefore it is important to investigate the fraction of this spe-cific case, and relevant properties, at parton level.

Parton level analysis At leading order QCD, the differ-ential cross section for e+e− → ccqq process (Fig. 2) can bewritten as

(4)dσ = 1

2sdLips4|M|2,

where s is the total energy square, Lips4 represents the 4-particle phase space and |M|2 is the spin averaged-summedinvariant amplitude square. At the energy of B factories(e.g.,

√s = 10.52 GeV), we calculate the joint distribution

64 W. Han et al. / Physics Letters B 642 (2006) 62–67

Fig. 3. The joint distribution d2Ndm1 dm2

for e+e− → ccuu.

d2N/(dm1 dm2) where m1 and m2 respectively represent theinvariant mass of cq and cq . In our calculation, we take q = u

as an example, and set the precise structure constant to be1/137, the strong coupling constant αs = 0.2, the quark massesmc = 1.5 GeV/c2 and mu,d = 0.33 GeV/c2. The correspond-ing results are shown in Fig. 3. It is important to notice that theproduction probability is not quite small when m1 and m2 areclose to the diquark mass.1 Especially, when the luminosity islarge, the absolute event number with diquark pair productioncould be large enough to be measured. Employing the KEKB factory integrated luminosity (525.828 fb−1 until 23 De-cember 2005) we obtain the number of the events with m1,m2 � 2.5 GeV/c2 to be around 3 × 104. The number of eventsfor q = d or q = s is at the same order.

It is well known that in the e+e− → γ ∗ → f f process, thepolar angle distribution of the massless spin 1/2 fermion (f/f )is predicted to be 1 + cos2 θ . For the 2-jet events initialised bye+e− → qq at energies much smaller than MZ0 , this will beapproximately reproduced for the local parton hadron duality.For the two clusters cq and cq fragmentation, the event shapeis very close to 2-jet event. If the polar angle distribution inthis case is much different from the lowest order one, it can beexpected as a possible signal to identify diquark pair fragmen-tation at B factory energy. The corresponding distributions withm1, m2 � 2.5 GeV are shown in Fig. 4. Obviously the distrib-ution is quite different from 1 + cos2 θ , where θ refers to theangle between the cluster cq momentum and the e− momen-tum.

Before investigating the hadronization effects induced bydifferent colour connections, we discuss the theoretical uncer-tainties caused by quark mass, parton shower, etc. For charmquark mass, it is general to take the value between 1.2 GeV/c2

to 1.8 GeV/c2, and for light quark mass, the same value asthat in Pythia is adopted. For e+e− → ccqq (q = u,d) process,we calculate its cross section σ(mq) with respect to light quarkmass mq . At the same time, we investigate the production crosssection σ(mc) of e+e− → cccc. The ratios for σ(mi)/σ (mi0)

1 The diquark mass is around 2 GeV/c2 set by Pythia.

Fig. 4. Normalised polar angle distribution of diquark cq (solid histogram). Thedashed curve is for the normalised 1 + cos2 θ distribution.

(i = q, c) with mq0 = 0.33 GeV/c2 and mc0 = 1.5 GeV/c2

are shown in Fig. 5(a). One can find that the cross sectionfor e+e− → ccqq process is not quite sensitive to the lightquark mass with charm mass fixed. This implies that the smalllight quark mass does not introduce any more dangerous sin-gularities. The differential distributions 1/σ dσ/dMcq/cc aredisplayed in Fig. 5(b). It is easy to find that the shape ofthe distributions for e+e− → ccqq and that for e+e− → cccc

(Fig. 5(b)) are quite similar, and a similar study on doublycharm diquark fragmentation for four charm process might alsobe plausible. We also compare our results for the cross sectionof e+e− → cccc with those given in [23], and find completelyagreement. The cross section for four quark production in e+e−annihilation at high energies (e.g., LEP1 and LEP2) may beaffected by the gluon radiation, which is simulated by partonshower process in Pythia. Fortunately, at B factory energies,√

s ∼ 10 GeV, the largest energy of each quark is about 5 GeV.If it could radiate gluons further, its virtuality will quickly reachthe lower limit that perturbative QCD can be applied safely.As a result, the parton shower process at B factory energieswould not play an important rôle as that at LEP energies. Whenone investigates physics for the e+e− → h′s process at LEP oreven higher energies, the parton shower process must be takeninto account. Unfortunately, till now, it is still unclear how todescribe the gluon radiation from diquark pair in e+e− annihi-lation. As the first step of the investigation, we concentrate onthe physics at the B factory energy, where the influence of gluonradiation seems to be small and could be neglected.

Hadronization For the e+e− → ccqq process, when m1and m2 near the mass threshold of diquark, we employ Pythiato investigate the differences in final hadron states correspond-ing to two kinds of colour connections. One is that cq and cq

form diquark-antidiquark pair, and hadronize altogether. In thepicture of Lund model, diquark and antidiquark are in 3∗ and3 colour state, respectively. As a result, one colour string willbe stretched between them, and the string will fragment intohadrons at the end. The other case is that cq and cq form twocolour singlet clusters (strings). These two strings will frag-ment into hadrons independently. This hadronization picture iswidely adopted in the popular event generators, and hereafterwe denote it as ‘2-string fragmentation’. The hadronization of

W. Han et al. / Physics Letters B 642 (2006) 62–67 65

Fig. 5. (a) The dependence of cross section on quark mass for e+e− → ccqq process (solid line) and e+e− → cccc process (dashed line). (b) The normaliseddistribution 1/σ dσ/dM for e+e− → ccqq process (solid line) and e+e− → cccc process (dashed line).

Fig. 6. (a) Xp (= ppmax

, with pmax =√

s4 − m2 ) spectra of Λc at

√s = 10.52 GeV, from diquark fragmentation (solid line), 2-string fragmentation (dashed line)

and e+e− → cc → Λc + X (dotted line). (b) Comparing with data [24]. The histograms correspond to three different baryon production scenarios in Pythia: solidline for ‘diquark’, dotted line for ‘simple popcorn’, dashed line for ‘advanced popcorn’.

these two cases can be treated by the corresponding subrou-tines in Pythia. For simplicity, we fix the invariant masses ofthe (anti)quark pair cq (cq) to be the (anti)diquark mass usedin Pythia. The results obtained under this simplification canbe considered as a good approximation of the average overa reasonable (hence small) range of cq/cq cluster mass. It isstraightforward to adopt that the quantities we calculate in thissection is not sensitive to the cluster mass in a small rangearound the diquark mass set by Pythia. By a similar argument,we also assume the inner orbital angular momentum of the cq

(cq) cluster to be 0. However, for treating the second kind ofcolour connection, the inner relative movements between c (c)and q (q) are still not fixed and generated with the relativeweight given by the differential cross section at parton leveldiscussed above.

In the diquark pair fragmentation process, it is straightfor-ward that diquark tends to fragment into baryon, which is ingeneral the leading particle. If the charm baryons from thediquark fragmentation are detected, their momentum spectrashould be harder than those from the 2-string fragmentation,or those from the general e+e− → cc → h′s process. This fea-ture can be displayed by the momentum (fraction) spectra ofΛc at

√s = 10.52 GeV (Fig. 6 (a)). Recently, the momentum

spectra of various charm hadrons have been measured by BelleCollaboration [24]. To make full use of the available data andto identify whether the diquark fragmentation events exist or

not, we must make clear to what extent Pythia can reproducethe data. For charm meson production, we find that the pre-dictions of Pythia agree to the available data quite well (alsosee [25]). For our aim, we concentrate on the investigation ofcharm baryon (e.g., Λc) production. In Pythia, there are threescenarios to describe baryon production, i.e., diquark, simplepopcorn and advanced popcorn. Their predictions can be con-sistent with each other provided that the relevant parametersare tuned (Fig. 6(b), the parameters all take default values ex-cept PARJ(1) = 0.2, PARJ(18) = 0.19 and PARF(192) = 0for advanced popcorn). From Fig. 6(b), one can find that thedistributions of Λc predicted by standard Pythia are slightlysofter than the data, while the momentum spectrum obtainedby diquark fragmentation is slightly harder. It is also interest-ing to notice that for the diquark fragmentation process, thesmall momentum region is slightly enhanced for the mass ef-fect (Fig. 6(a)). This phenomenon is also indicated by the datashown in Fig. 6(b), though the statistical error is too large togive a definite conclusion. Because of the phase space limit, theprobability of diquark pair production may not be large. As aresult, the predictions for Λc production including the diquarkpair production will not conflict with the available data, and aportion of diquark fragmentation events is favoured.

For further investigation, we calculate thrust distribution1/N dN/dT for the diquark fragmentation, 2-string fragmen-tation and e+e− → cc → h′s process (Fig. 7(a)). One can find

66 W. Han et al. / Physics Letters B 642 (2006) 62–67

Fig. 7. Thrust distributions of three cases at√

s = 10.52 GeV. (a) Diquark fragmentation (solid line), 2-string fragmentation (dashed line) and e+e− → cc → h′s(dotted line). (b) Same as (a) except that we select 2-jet events, with JADE algorithm and ycut = 0.12.

Table 1ΛcΛc flavor correlations√

s e+e− → cc → h′s Diquark fragmentation 2-string

10.52 GeV 0.057 0.76 0.04491.19 GeV 0.081 0.72 0.081

that the thrust distribution of the diquark fragmentation is much‘thinner’ than those of the other cases (Fig. 7(a)). Aware that thediquark fragmentation events are mostly 2-jet like, we comparethe thrust distributions of 2-jet events for the three cases. Weuse the JADE algorithm with ycut = 0.12. In this case, around90% diquark fragmentation events are 2-jet. The properties aresimilar to those without jet constrain (Fig. 7(b)). Accordingto these discussions, further experimental investigation of thecolour connections shown in Eq. (3) can be done by selecting2-jet charm events at the B factory, and choosing those withlarge thrust (say, larger than 0.85).

It is natural that in the fragmentation of diquark-antidiquarkpair, a baryon is produced together with an antibaryon. Thatmeans, the correlation between charm and anticharm baryonsin the diquark pair fragmentation is much stronger than that inthe single charm quark fragmentation (either the 2-string frag-mentation or the general e+e− → cc → h′s). We know that, inthe latter two cases, the production probability for charm mesonis much larger than that for charm baryon. This phenomenoncan be illustrated by the baryon (B) antibaryon (B) correlation,defined as,

(5)A(BB) = 2N(BB)

N(B) + N(B),

where N(BB), NB and NB respectively denote the probabilityfor finding BB pair, B and B . We take the correlation of Λc andΛc as an example in Table 1. One can notice that the correlationin the diquark pair fragmentation process is stronger than theother cases by about 10 times. This correlation is not sensitive tothe gluon radiation of the diquark pair so we also give the resultsat Z0 pole. This quantity could be a good probe for the diquarkpair fragmentation. The results for other charm baryons, suchas Σc, Ξc and the corresponding spin-3/2 particles are similar.

To summarise, we investigate the hadronization effects in-duced by different colour connections of four quark system(e.g., ccqq) in e+e− annihilation. It is interesting to point out

that besides the normal colour structure with two colour sin-glets cq and qc which fragment into hadrons independently inthe popular models, there exists another kind of colour struc-ture, as shown in Eq. (3), which has to be taken into accountin some cases. For that colour connection, needless to say,the hadronization of these four quarks should be treated as awhole system, i.e., they interact among each other during thehadronization process. In this Letter, considering an extremelimit case that cq and cq form diquark–antidiquark pair, wepropose a toy model based on Pythia to describe the hadroniza-tion of them. It is found that the results originated from diquarkfragmentation are significantly different from those of the ordi-nary colour connections. However because of the phase spacelimit, the hadronization effect from diquark pair fragmentationis not very large. To understand the hadronization mechanisminduced by different colour connections of final partons pro-duced perturbatively, further theoretical and experimental in-vestigations are still necessary.

Acknowledgements

This work is supported in part by NSFC, NCET of MoE,PR China, and Huo YingDong Foundation. The authors thankall of the members in Theoretical Particle Physics Group ofShandong University for their helpful discussions. The Author(Si) would like to thank the hospitality of the Phenomenol-ogy research group of Department of Physics, University ofWisconsin–Madison during his visit.

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