Proposal for exotic-hadron search by fragmentation functions

0.59 MB·English

by  M. Hirai

KEK-TH-1167
Proposal for exotic-hadron search by fragmentation functions
M. Hirai,1 S. Kumano,2,3 M. Oka,1 and K. Sudoh2
1Department of Physics, H-27, Tokyo Institute of Technology, Meguro, Tokyo, 152-8551, Japan
2Institute of Particle and Nuclear Studies, High Energy Accelerator Research Organization (KEK)
1-1, Ooho, Tsukuba, Ibaraki, 305-0801, Japan
3Department of Particle and Nuclear Studies, Graduate University for Advanced Studies
1-1, Ooho, Tsukuba, Ibaraki, 305-0801, Japan
(Dated: November16, 2007)
It is proposed that fragmentation functions should be used to identify exotic hadrons. As an
8
0 example, fragmentation functions of the scalar meson f0(980) are investigated. It is pointed out
0 that the second moments and functional forms of the u- and s-quark fragmentation functions can
2 distinguish the tetraquark structure from qq¯. By the global analysis of f0(980) production data in
electron-positron annihilation, its fragmentation functions and their uncertainties are determined.
n
Itisfoundthatthecurrentavailabledataarenotsufficienttodetermineitsinternalstructure,while
a
precise data in futureshould beable to identify exotic quarkconfigurations.
J

4 PACSnumbers: 12.39.Mk,13.87.Fh,13.66.Bc
2

] In the hadron mass region below 1 GeV, there are f (980) corresponds to the tetraquark state because the
h 0
p scalar mesons, f0(600), f0(980), and a0(980), whose in- scalartetraquarkmassisabout1.1GeV[8]. Inaddition,
- ternal configurations are not obvious [1]. Their fla- f0(980) used to be considered as a glueball candidate;
p vor compositions could be f (600) = (uu¯ + dd¯)/√2, however, recent lattice QCD calculations rule out such
e 0
h f0(980) = ss¯, a0(980) = ud¯, (uu¯ dd¯)/√2, u¯d in a a possibility because the mass of a 0++ glueball is es-
[ simple quark model by considering−the mass relation, timated about 1700 MeV [9]. The situation of scalar
m m < m . However, this assignment implies a mesons with masses in the 1 GeV region is summarized
v2 mauss∼sequdence, ms(f0(600)) m(a0(980))< m(f0(980)), in Ref. [1]. All the possible f0(980) configurations are
6 which contradicts with the∼observed one, m(f0(600)) < listedinTableIalthoughthenonstrange-qq¯andglueball
1 m(a0(980)) m(f0(980)). If f0(980) and a0(980) are statesseemtobeunlikelyaccordingtotherecentstudies.
∼
8 exotic states such as tetraquark ones, the observedspec- In the following, the notation f indicates the f (980)
0 0
1 trum could be naturally understood. Since f (980) and meson and f (600) is not discussed. There were propos-
0 0
8. a0(980) are experimentally established resonances, they als to find the structure by a φ radiative decay into f0
0 provide a good opportunity to study exotic mesons be- [10,11,12]. Sinceitisanelectricdipoledecay,thewidth
7 yond a naive qq¯-type quark model. should reflect information on its size, namely its inter-
0 First, a brief outline of recent studies is given for the nal structure [10]. The experimental measurements of
:
v f (980)structure. Inasimplequarkmodel,alightscalar VEPP-2M [13] and DAΦNE [14] were reported for the
0
Xi meson f0 with JPC =0++ is identified as a 3P0 quarko- decay φ → f0γ. The data may suggest the tetraquark
niumwiththeflavorstructure(uu¯+dd¯)/√2. However,if picture; however, there are still discussions on their in-
ar t5shu0ec0hsat1rn0o0on0rgdMidneeaVcray,yaqcq¯wcciodortdnhifingigustrovaetviroaynriloiasurasgsets,higeΓon(refed0tifco→arlfc0πa(lπ9c)8u0la=)-, titswerotph-rpeehtγaoγttioon→nd[eπ1c+2a]πy.−wAipndrotohtcheoesfrsfp0in(o9st8shi0be)lewf0aesxmpraeecrsiesmnrteelngytioraenlp.porrTotbhedee
tion−s [2]. The smallexperimentalwidth 40 100MeV [3] as0.205+−00..009853(stat)+−00..114177(syst)keVbythe Bellecollabo-
cannotbeconsistentlyexplainedbysimpleq−uarkmodels. ration [15]. Model calculations indicate 1.3 1.8 keV in
−
the nonstrange qq¯picture; however, the measurement is
The strong-decay width suggests that f (980) should
0 consistent with the ss¯and KK¯-molecule configurations.
not be an ordinary nonstrange qq¯-type meson. The
Fermilab-E791 collaboration measured the decay D+ Therearealsoideastouseelliptic flowandnuclearmod-
π−π+π+ [4], which can proceed via intermediate staste→s, ification ratios in heavy-ion reactions for finding exotic
forexample,D+ f (980)π+ withss¯quarksinf (980). hadron structure [16].
This experimenst→sugg0ested a sizable strange-quar0k com- There arecompelling theoreticalandexperimentalev-
ponent in f0(980). The simplest configuration is a pure idences that the scalar meson f0(980) is not an ordinary
strangequarkoniumss¯for f (980). In addition, since its nonstrange qq¯ meson. However, a precise configuration
0
mass is just below the KK¯ threshold, it could be con- is not determined yet,and a clearexperimentalevidence
sidered as a KK¯ molecule [5]. If two color-singlet states is awaited. It is the purpose of this paper to show that
of K and K¯ are not well separated, it corresponds to the internal structure of exotic hadrons should be de-
a tetraquark state, (uu¯ss¯+dd¯ss¯)/√2, which was origi- termined from their fragmentation functions by noting
nally suggestedin the MIT bag model [6]. Recent QCD- differences in favored and disfavored functions. We in-
sum-rule studies support this idea of a tetraquark state vestigate f0(980) as an example in this work.
[7]. Furthermore, there are lattice-QCD studies that A fragmentation function is defined by a hadron-
TABLE I:Possible f0(980) configurations and their features in fragmentation functions at small Q2.
Type Configuration Second moments Peak positions
Nonstrange qq¯ (uu¯+dd¯)/√2 Ms <Mu <Mg zumax >zsmax
Strangeqq¯ ss¯ Mu <Ms <Mg zumax <zsmax
TetraqGulaurekb(aollr KK¯) (uu¯ss¯+gdgd¯ss¯)/√2 MMuu ∼∼MMss <<∼∼MMgg zzuummaaxx ∼∼zzssmmaaxx
productioncrosssectionandthetotalhadroniccrosssec- different functional forms and their peak positions are
tion: Fh(z,Q2) = 1 dσ(e+e−→hX). Here, the variable different at small Q2 ( 1 GeV2). We express this situ-
z is defined by theσhtoatdron edznergy Eh and the center- ation as zumax < zsmax i∼n Table I. The form of the gluon
of-mass energy √s (= Q2) by z E /(√s/2). The fragmentationfunctionmaynotbesimplycomparedwith
h
≡ the quark processes.
fragmentationoccurs from primary partons, so that it is
expressedbythesumofptheircontributions: Fh(z,Q2)= In the same way, the second moments and functional
C (z,α ) Dh(z,Q2),where indicatestheconvolu- forms are roughly estimated for the tetraquark picture.
tioininitegrasl,⊗f(zi) g(z)= 1dyf⊗(y)g(z/y)/y,Dh(z,Q2) Sincethefragmentationsfromuandsquarksareequally
P ⊗ z i favored processes in this case, their moments and func-
isthefragmentationfunctionofthehadronhfromapar-
R tions forms should be almost the same. The fragmen-
atonndiα(=ius,tdh,esr,u·n··n,ingg),cCoiu(zp,liαnsg)cisoansctoaenffit.cieTnhtefufnacvtoiroend, tations into f0 proceed by creating uu¯ (or dd¯) and ss¯
s pairs as shown in Fig. 2. There are more fragmenta-
fragmentation means a fragmentation from a quark or
tion processes from a gluon, so that the gluon moment
anantiquarkwhichexistsinahadronasaconstituentin
is expected to be larger than the others. In this way, we
aquarkmodel,andthedisfavoredmeansafragmentation obtain the relations, M M < M and zmax zmax,
from a sea quark. The favored and disfavored functions in Table I. Since the fluav∼or cosm∼posigtion of uf is∼simsply
are assigned in the following discussions by considering 0
considered in the above discussions, this relation could
the naive quark configurations in Table I. be also applied to the KK¯ case. However, the KK¯ is
We first consider a possible ss¯ configuration for f0. a loose and extended bound state so that its production
Then, the u- and d-quark fragmentation functions are probability in the fragmentation is expected to be much
disfavored ones and the s-quark function is a favored smaller than that of the tetraquark state.
one. For example, the favored fragmentation from s is Although the nonstrange-qq¯ and glueball configura-
possible if a gluon is radiated from s, and then it splits tions seem to be unlikely according to recent theoreti-
into a ss¯pair to form the f meson as shown in Fig. 1.
0
The notations O(g2) and O(g3) indicate the second and
thirdordersofthecouplingconstantg. Inthedisfavored
process from u, there are processes in the order of O(g3)
withoutanO(g2)term,sothatitsprobabilityisexpected
tobesmallerthanthefavoredonefroms. Itleadstothe
relation for the second moments of fragmentation func-
tions: M < M , where M dzzDf0(z). The second
u s i ≡ i
moment M is the energy fraction for f which is cre-
i 0
R
atedfromthe partoni. Inthe same way,fragmentations
occur from a gluon as shown in the figure. Since there
are two processes in O(g2) with a soft gluon radiation, FIG.1: (Color online) Schematicdiagrams for f0 production
in the ss¯configuration.
thesecondmomentforthegluonisexpectedtobelarger
than the others. These considerations lead to the rela-
tionM <M <M in Table I. Sucha naive estimation
u s g
shouldbe acru∼deone,butithasbeenshowntoworkfor
the moments of the pion, kaon, and proton [17], so that
it is also expected to be a reasonable guideline in other
hadrons.
Next, functional forms are discussed in the ss¯picture.
More energy is transferredto f from the initial s in the
0
O(g2) process than the one from the initial u due to an
extra gluon emission in Fig. 1. It means that the frag-
mentation function Df0(z) is distributed in the larger z
s
region in comparison with Duf0(z) because the f0 energy FIG.2: (Color online) Schematicdiagrams for f0 production
is directly proportional to z. Namely, they should have in the tetraquark configuration.
2
calinvestigations,we alsoestimatedpossible relationsin 0.03
Table I. Since the estimation method is essentially the c Q2=1 GeV2,m2,m2
same, derivations are not explained here. If f were a c b
0
nonstrange-qq¯meson, the relations Ms <Mu <Mg and 0.02
zmax >zmax areexpected,whereasthey areM M <
MugThanedfrzsaumgamxe∼ntzasmtiaoxniffiutnwcteiroenas galureebdalelt.erminue∼d bys a zD(z) g b s
global analysis of hadron-production data in e+e− an- 0.01
u
nihilation [18]. There is recent progress on their anal-
ysis. Uncertainties of the fragmentation functions are
determinedinRef. [17],anditwasshownthatthe gluon 0
0 0.2 0.4 0.6 0.8 1
andlight-quarkfunctionshavelargeuncertaintiesforthe z
pion,kaon,andproton. Then,aglobalanalysiswithdata
FIG. 3: (Color online) Determined fragmentation functions
inleptonscatteringandproton-protoncollisionswasalso
reported [19]. This kind of global analysis is suitable for oafndf0z(D98f00)abryetshheowglnobaatlQan2=al1ysGise.VT2h,eanfudntchtieonfusnzcDtiouf0n,szzDDsff00,
finding exotic hadrons by noting the typical features in g c
and zDf0 are at Q2=m2 and m2, respectively.
the favored and disfavored functions. b c b
All the possible configurationsfor f indicate that up-
0
and down-quark compositions are the same; however,
of data. Then, the total number of parameters becomes
they are generally different from the strange-quark and
twelve. The minimum χ2 is obtained χ2/d.o.f.=0.907 in
other ones. Therefore, a natural and model-independent
our analysis.
parametrization is
The determined functions are shown in Fig. 3. It is
Df0(z,Q2)=Df0(z,Q2)=Df0(z,Q2)=Df0(z,Q2) interesting to find that the up- and strange-quark func-
u 0 u¯ 0 d 0 d¯ 0 tionsaredistributed relativelyatlargez,andbothfunc-
=Nf0zαfu0(1 z)βuf0, tions have similar shapes, whereas the gluon, charm-,
u −
Dsf0(z,Q20)=Dsf¯0(z,Q20)=Nsf0zαfs0(1−z)βsf0, azn.dItbmoattyoimnd-qicuaatrekthfuantcbtoiotnhsfuanrectidoinstsr,iDbuft0edanadtDsfm0,alalreer
u s
Df0(z,Q2)=Nf0zαfg0(1 z)βgf0, (1) equally favored ones, which implies that the up-quark
g 0 g −
(and down-quark) is one of main components of f as
Dcf0(z,m2c)=Dcf¯0(z,m2c)=Ncf0zαfc0(1−z)βcf0, wellas the strange-quark. Furthermore,they are pea0ked
Dbf0(z,m2b)=D¯bf0(z,m2b)=Nbf0zαfb0(1−z)βbf0, malmayobsteaatlsothceonssaimdeerepdoainstasnoefvzide(znumceaxfo∼rthzesmtaext)r,awquhaicrhk
whereN ,α ,andβ aretheparameterstobedetermined structure according to Table I. However, if it is judged
i i i
by a χ2 analysis of the data for e++e− f +X [20]. fromtheirsecondmomentratio(M /M =0.43),itlooks
0 u s
→
TheinitialscaleistakenQ2=1GeV2,andthemassesare like the ss¯configuration.
0
m =1.43GeVandm =4.3GeV.Thedetailsoftheanal- This conflictis mainly causedby the inaccuratedeter-
c b
ysismethodinthenext-to-leadingorderareexplainedin minationofthe fragmentationfunctions althoughitmay
Ref. [17]. Uncertainties of the determined functions are be understood by admixture of the ss¯ and tetraquark
estimated by the Hessian method [17], which has been configurations. In Fig. 4, the uncertainties of zDf0,
u
used also in the studies of various parton distribution zDf0, and zDf0 are shown at Q2=1 GeV2 together
s g
functions [21, 22]: with the functions themselves. We notice huge uncer-
tainties which are an order of magnitude larger than
[δDf0(z)]2 =∆χ2 ∂Dif0(z,ξ) H−1 ∂Dif0(z,ξ) . the determined functions. If their moments are calcu-
i Xj,k ∂ξj !ξˆ jk ∂ξk (2!)ξˆ lMatsed=,0t.h0e0y27h±av0e.0l1a8r3g,eaenrdroMrs:g =M0u.0=0900±.0001.020±460..F01ro0m7,
these results, the error of the moment ratio is estimated
Here, δDf0(z) is the uncertainty of the fragmentation
i as Mu/Ms = 0.43 6.73, which makes it impossible to
function Dif0(z), ∆χ2 value is taken so that the con- discusstheeffectof±theorderof50%. Inthisway,wefind
fidence level P becomes the one-σ-error range (P = the structure of f cannot be determined by the current
0
0.6826)byassumingthenormaldistributioninthemulti- e+e− data.
parameter space, Hij is the Hessian matrix, ξi is a pa- It is the purpose of this work to point out that struc-
rameter, and ξˆindicates the optimum parameter set. ture ofexotic hadronsshouldbe determinedby the frag-
Thenumberoff dataisverylimitedatthisstage. In mentationfunctions. Accurate measurementsof hadron-
0
fact, available data are merely twenty three. This situ- production cross sections can be used for determining
ation makes the analysis difficult in obtaining the mini- their internal quark and gluon configurations as ex-
mumχ2 point. Thereareirrelevantparameterswhichdo plained in this paper by taking f (980) as an example.
0
not affect the total χ2. We decided to fix three parame- We have shown that ss¯ and tetraquark configurations,
ters at β =1, α =10, and α =10 because of the lack andalsononstrange-qq¯andglueballstates,shouldbedis-
g u s
3
0.3 mined accurately. Second, semi-inclusive f -production
0
Q2= 1 GeV2 data in lepton-proton scattering can be used for distin-
s guishing between up- and strange-quark fragmentations
0.2 becausetheinitialquarkdistributionsaredifferentinthe
) proton. These flavor separations will become possible
z
(
D by future experimental analyses. Our work is a starting
z
0.1 pointforexotichadronsearchbysuggestingtherelations
g u in the second moments and the functional forms and by
indicating the current experimental situation as the un-
certainty bands.
0
0 0.2 0.4 0.6 0.8 1
z Thefragmentationfunctionsoff andtheiruncertain-
0
FIG.4: (Coloronline) Fragmentationfunctions, zDf0,zDf0, ties have been determined by the global analysis of f0
and zDf0, and their uncertainties are shown at Q2=u1 GeVs2. production data. At this stage, the e+e− data are not
g
The uncertainties are shown by theshaded bands. preciseenough;however,accurateexperimentalmeasure-
mentscouldcreateafieldofexotichadronswhicharebe-
yondthenaiveqq¯andqqq typeones. Currently,analyses
tinguished by the second moments and functional forms are in progress by the Belle collaboration[23] to provide
of the favored and disfavored functions. Especially, the accurate fragmentation functions. They are especially
ratio of the u-quark moment to the s-quark one should important because the functions are measured at small
be useful to judge the configuration. Q2 ( M2),sothatscalingviolationcanbeinvestigated
≪ Z
Inordertodetermine the internalstructure,theflavor to find the gluon functions [17]. It is also important to
separationis important especially because the difference have accurate measurements for ordinal mesons such as
between the up- and strange-quark functions is the key φ(1020) and f (1270)in order to establish the f config-
2 0
to find the structure of f . First, charm- and bottom- uration by comparing their favored and disfavored func-
0
quarktaggeddatashouldbeprovidedforf astheyhave tions with the ones of f . We could investigate other
0 0
been obtained for the pion, kaon, and proton. Then, exotic hadrons in the same way by their fragmentation
the charm-andbottom-quarkfunctions shouldbe deter- functions.
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4