Heavy baryon/meson ratios in relativistic heavy ion collisions

0.25 MB·

by  Yongseok Oh

Heavy baryon/meson ratios in relativistic heavy ion collisions
Yongseok Oh,1,∗ Che Ming Ko,1,† Su Houng Lee,2,‡ and Shigehiro Yasui3,§
1Cyclotron Institute and Physics Department, Texas A&M University, College Station, Texas 77843, U.S.A.
2Institute of Physics and Applied Physics, Yonsei University, Seoul 120-749, Korea
3 High Energy Accelerator Research Organization (KEK), Tsukuba, Ibaraki, 302-0801, Japan
(Dated: January 12, 2009)
Heavy baryon/meson ratios Λc/D0 and Λb/B¯0 in relativistic heavy ion collisions are studied in
the quark coalescence model. For heavy baryons, we include production from coalescence of heavy
quarkswithfreelightquarksaswellaswithboundedlightdiquarksthatmightexistinthestrongly
coupled quark-gluon plasma produced in these collisions. Including the contribution from decays
of heavy hadron resonances and also that due to fragmentation of heavy quarks that are left in
thesystem after coalescence, theresultingΛc/D0 andΛb/B¯0 ratios in midrapidity(y 0.5) from
9 centralAu+Aucollisionsat√s =200GeVareaboutafactoroffiveandten,resp|ec|t≤ively,larger
NN
0 than those given by the thermal model, and about a factor of ten and twelve, respectively, larger
0 than corresponding ratios in the PYTHIA model for pp collisions. These ratios are reduced by a
2
factor of about 1.6 if there are no diquarks in the quark-gluon plasma. The transverse momentum

n dependence of the heavy baryon/meson ratios is found to be sensitive to the heavy quark mass,
a with the Λb/B¯0 ratio being much flatter than the Λc/D0 ratio. The latter peaks at the transverse
J momentum p 0.8 GeV butthe peak shifts to p 2 GeV in the absence of diquarks.
T ≃ T ≃
2
1 PACSnumbers: 25.75.-q,25.75.Cj,25.75.Dw,

]
h I. INTRODUCTION ductionintheintermediatetransversemomentumregion
-t as well. Since the Λc/D0 ratiohas not been measuredin
l heavyioncollisions,itwasassumedinRef.[11]tobethe
c Recent measurements on nonphotonic electrons from
u decays of midrapidity heavy-flavored mesons in central same as the measuredΛ/KS0 ratio or in Ref. [12] to have
n a Gaussian form with a maximum value peaked at the
heavyioncollisionsattheRelativisticHeavyIonCollider
[ transverse momentum p =5 GeV. It was then claimed
(RHIC) have shown that the nuclear modification factor T
1 Re relative to pp collisions is significantly smaller than thatavalueofΛc/D0 1,whichisaboutafactoroften
AA ∼
v theoretical predictions based on radiative energy loss of larger than that in pp collisions at same energy, could
2 resultinanadditional20–25%suppressionofRe apart
heavyquarksinproducedquark-gluonplasma[1,2,3,4]. AA
8 To explain the observed small value of Re or large fromthe suppressiondue to the collisionalenergyloss of
3 AA heavy quarks in quark-gluon plasma. However, it is not
suppression of heavy meson production in relativistic
1
yet known what kind of underlying physics can cause
. heavy ion collisions, other mechanisms for energy loss
1 such a large enhancement of the Λ /D0 ratio.
of heavy quarks in quark-gluon plasma have been pro- c
0
posed[5,6,7,8,9,10]. Also,itwassuggestedthattheen-
9 Recently, enhancement of the Λ /D0 ratio due to the
hancement of charmed baryon production over charmed c
0 existenceof[ud]diquarksinstronglycoupledquark-gluon
v: tmheesoRneproodfutchtieone,leic.et.r,otnhsefΛrocm/Dh0eraavtyiom, wesoounlds s[1u1p,pr1e2s]s. plasma(QGP)wassuggestedbyLeeet al.[20]. Thiswas
i AA based on the idea of possible existence of quasi bound
X This is due to the fact that enhanced charmed baryon
states of quarks and gluons including diquarks in QGP
r production reduces the production of charmed mesons whenthetemperatureisbetweenT and4T [21],where
a and thus their contribution to decay electrons. C C
T isthecriticaltemperatureforthequark-gluonplasma
In fact, experiments at RHIC have shown that in cen- C
to hadronic matter transition. Based on the quark coa-
tral heavy ion collisions there is an enhanced produc-
lescencemodel, itwasshowninRef. [20]thatthe Λ /D0
c
tion of midrapidity baryons in the intermediate trans-
ratiocouldbe enhancedbyafactorof4–8relativetothe
verse momentum region not only in light hadrons but
casewithoutdiquarksinQGP,dependingonthebinding
also in strange hadrons. This enhancement could be
energy of the light diquark at T =175 MeV. The ques-
described by multi-quark dynamics through quark co- C
tion whether diquarks can exist as quasibound states in
alescence or recombination [13, 14, 15], baryon junc-
cold quark matter was raised earlier in Ref. [22], where
tion loops [16, 17, 18], or long-range coherent fields [19].
thequark-diquarkmatterwasclaimedtobeenergetically
Therefore,onemightexpectenhancedheavybaryonpro-
more favorable than the free quark matter. The idea of
enhancement of Λ baryon production in heavy-ion col-
c
lisions was then suggested in Ref. [23]. Based on the
assumptionthatthenumberofdiquarkspresentinacol-
∗Electronicaddress: [email protected]
lision is mainly determined by that in the initial nuclei,
†Electronicaddress: [email protected]
‡Electronicaddress: [email protected] theauthorofRef.[23]consideredtheΛc/Σc ratioandes-
§Electronicaddress: [email protected] timated that it couldbe enhancedby a factor of as large
2
as80. SincetheΣ cannotbedirectlymeasuredinnear- together with both light and heavy quark distribution
c
future heavy-ion experiments, the more phenomenologi- functions. Also described are the calculational meth-
callyaccessibleΛ /D0ratioisthusaddressedinRef.[20]. ods and the contributions from fragmentation of heavy
c
The idea of diquarks [24, 25, 26] has been widely quarks that are not used in coalescence. We then show
usedinhadronphysicsfordescribingthemassspectrum, inthesamesectiontheresultsfortherelativeproduction
electromagnetic properties, and many other properties fractionsofheavyhadronsandtheirpT spectraaswellas
of hadrons (for a review, see, for example, Ref. [27]), the transversemomentum dependence of the ΛQ/H0 ra-
although the properties and even the existence of di- tios. Section V contains the conclusions and discussions.
quarksarestillunderdebate[28]. Amongpossiblesigna-
tures of diquarks in hadrons, the observed large ratio of
Λ /Σ 7ine+e− collisionswaspointedoutinRef.[29], II. HEAVY HADRON PRODUCTION IN
c c
which i∼mplies a significantly larger production of scalar PROTON-PROTON COLLISIONS
diquarks than vector diquarks [30]. (See also Ref. [31].)
In spite of the lack of concrete consensus on the diquark Before studying the production of heavy hadrons in
nature, it is well-known that the (isoscalar) scalar light heavy ion collisions, we briefly discuss their production
[ud] diquark of coloranti-triplet is the most probabledi- inpp collisionsby using the PYTHIAmodel [38]. For pp
quark state. Evidence for such scalar diquarks has been collisionsat√s=200GeV,thePYTHIAmodelgivesfol-
confirmed by a recent lattice QCD simulation[32]. (See, lowing relative production fractions of charmed hadrons
however, Ref. [33] for other lattice QCD calculation on in midrapidity (y 0.5),1
| |≤
diquarks.)
In diquark models, the Λ is usually considered as a
c
system consisting of a charm quark and a scalar light f(D+) 0.196, f(D0) 0.607,
≃ ≃
[aundd] Σdi∗quiasrmk.odOeln-dtehpeenodtehnetr:htahnedy, mthaeyshtrauvcetuarne aoxfiΣalc- f(Ds+)≃0.121, f(Λc)≃0.076. (1)
c
vector light diquark but they also may contain diquarks This leads to the particle number ratios
[Qq] made of one heavy quark (Q) and one light quark
(q) [34, 35]. In Ref. [36], it is claimed that the ground D0 Λc
3.1, 0.13. (2)
state baryons favor the [Qq] diquark. This is based on (cid:18)D+(cid:19) ≃ (cid:18)D0(cid:19) ≃
pp pp
the observation that the [Qq]q system gives a Λ mass
c
closertothatfromthe three-quarkcalculation,although The above estimated D0/D+ ratio can be understood
ithasalargermassthanthatbasedontheQ[qq]configu- largely by considering resonance decays, namely, decays
ration. Sincethecolor-spininteractionbetweenquarksis of D∗ mesons into D mesons. First, the D∗0 meson de-
inversely proportional to the quark masses, the binding cays into D0 meson by 100% [39] since the decay of D∗0
energiesof[Qq]diquarksaresmallerthanthose ofscalar into D+π− is prohibited by energy conservation.2 Sec-
light [qq] diquarks, and it is not clear whether such [Qq] ond, the D∗+ meson can decay either into D+ or D0.
diquarks can exist, in particular, in QGP. (See Ref. [37] The branching ratios are BR(D∗+ D0π+) 68%and
for a recent discussion on the diquark picture for heavy BR(D∗+ D+X) 32%, where→sum over≈any state
baryons.) X is unde→rstood. If≈the production rates are equal for
In this paper, we calculate the ΛQ/H0 ratios (Q = D and D∗ mesons aside from the factor due to different
b,c and H0 stands for D0 or B¯0) in relativistic heavy spin degeneracies, we then would have
ion collisions based on the quark coalescence model. We
consider the two cases of Λ being a pure three-quark D0 1+3+3 0.68 6.04
Q
× = 3.1, (3)
state and of ΛQ being made of a heavy quark and one (cid:18)D+(cid:19) ≈ 1+3 0.32 1.96 ≈
scalar light diquark. In addition to the direct formation pp ×
ofΛQ andH0,wealsotakeintoaccountthecontribution where the spin degeneracy of D∗ is included. We also
from decays of heavy hadron resonances. Namely, we note that the estimated Λ /D0 ratio is close to that (
c
consider the decay of D∗ mesons for D0 production and 0.16) in the SELEX measurement [40]. ≈
ΣQandΣ∗QbaryondecaysforΛQproduction. Aswehave For bottom hadron production in pp collisions at
discussed before, we assume that ΣQ and Σ∗Q baryons √sNN = 200 GeV, the relative production fractions in
are made of three quarks and that neither vector light
diquarks nor scalar heavy-light diquark exists in QGP.
Thispaperisorganizedasfollows. Inthenextsection,
we briefly discuss heavy hadron production in pp colli- 1 We have confirmed the results reported in Ref. [12]. These
sions. A simple thermalmodel is then usedinSec. III to values are also close to the values quoted by Particle Data
show the role of heavy hadron resonances in the Λ /H0 Group (PDG) [39] for e+e− annihilation at √s = 91 GeV:
ratios. The difference between the ratios for chaQrmed 2 Nf(oDte+t)h≃at0m.2(1D,∗f0()D=0)2≃0070.M54e,Vf,(Dms+(D)∗≃+0).=162,0a1n0dMf(eΛVc,)m≃(D0.00)93=.
hadrons and for bottom hadrons is also discussed. In 1865MeV,m(D+)=1869MeV,whilem(π±)=139.6MeVand
Sec. IV, the coalescencemodel used in this work is given m(π0)=135MeV.
3
midrapidity from the PYTHIA model are collisions is thus about 25% smaller than that in pp col-
lisions.
f(B−)≃0.101, f(B¯0)≃0.101, f(Bs−)≃0.030, For the Λc/D0 ratio, we have, by using mΛc =
f(B∗−)≃0.302, f(B¯∗0)≃0.301, f(Bs∗−)≃0.089, 2285 MeV and mD0 =1865 MeV,
f(Λ ) 0.076, (4)
b ≃ Λc = 2m2ΛcK2(mΛc/TC) 0.24, (9)
leading to3 (cid:18)D0(cid:19) m2 K (m /T ) ≃
0 D0 2 D0 C
Λ /B¯0 0.7, (5)
b where the factor 2 is the spin degeneracy of Λ baryon.
≈ c
Since the ratio D∗/D is 1.47, inclusion of D∗ resonance
which is significantly larger than the Λ /D0 ratio given
c decayscauses a strongreduction ofthe Λ /D0 ratio. On
in Eq. (2). c
theotherhand,baryonresonancescanalsocontributeto
theproductionofΛ baryonthroughtheirdecaysandcan
c
enhancetheΛ /D0ratio. Themajorresonancecontribu-
III. THE THERMAL MODEL c
tion to Λ comes from Σ∗(2520) of spin-3/2. In thermal
c c
model, we have
In relativistic heavy-ion collisions, the Λ /D0 and
c
iΛnb/EBq¯s0.r(a2t)ioasnadre(5e)xpfoerctpepdctoollbiseiodniffs.erTenhtisfrcoamntbheosseeegnivuesn- Σ∗c(2520) 2 3 m2Σ∗cK2(mΣ∗c/TC) =1.8, (10)
Λ ≈ × × m2 K (m /T )
ing a simple thermal model which assumes that in rela- c Λc 2 Λc C
tivisticheavyioncollisionscharmedandbottomhadrons
where the factor 2 comes from the difference in the spin
are produced during hadronization of the quark-gluon
degeneraciesof Λ andΣ∗, andthe factor 3 is due to the
plasma and that they are in both thermal and chemi- c c
isospin1 of Σ∗. Since Σ∗(2520)decaysalmost100%into
cal equilibrium at the phase transition temperature T . c c
C
Λ ,inclusionofitscontributiongives180%enhancement
The total charm and bottom numbers are, however, de- c
to the number of Λ . Another important contribution
terminedbyinitialhardscatteringasthermalproduction c
to Λ production is from the decay of Σ ground state,
ofheavyquarksfromcreatedquark-gluonplasmaisneg- c c
Σ (2455) of spin-1/2, via the Σ Λ π decay.4 In ther-
ligible in heavy ion collisions at RHIC [41]. The number c c → c
mal model, its abundance relative to that of Λ is
ofheavyhadronsofmassm insidea fireballofvolumeV c
at T is then given by
C Σc(2455) 3 m2ΣcK2(mΣc/TC) =1.26. (11)
gV Λ ≈ × m2 K (m /T )
N =γQ2π2m2TK2(m/TC), (6) c Λc 2 Λc C
Therefore,the contributionfromΣ (2455)andΣ∗(2520)
whereγ istheheavy-quarkfugacity,g isthedegeneracy c c
Q decays increases the Λ number by a factor of 3.1 and
of the particle, and K is the modified Bessel function. c
2 brings the Λ to D0 ratio close to the naive expectation
For charmed hadrons, we would then have c
given in Eq. (9) based only on directly produced Λ and
(D0/D+) = 1, where the subscript 0 means the c
0 D0, i.e.,
ratio without resonance contributions, if the D∗ D
→
decay is not included. Since
Λ Λ 1+Σ∗(2520)/Λ +Σ (2455)/Λ
c = c{ c c c c}
D∗0 = 3m2D∗K2(mD∗/TC) 1.47 (7) D0 D0(1+1.68D∗/D) (cid:12)(cid:12)(cid:12)ther
D0 m2D0K2(mD0/TC) ≃ Λc 1+1.8+1.26 (cid:12)(cid:12)
0.28. (12)
atT =175MeV,includingdecaysofD∗mesonschanges ≃ (cid:18)D0(cid:19) 1+1.68 1.47 ≈
C 0 ×
the ratio to
The contribution from higher mass charmed reso-
D0 1+(1+0.68) 1.47 nances is negligibly small in the thermal model. For D
× 2.36, (8)
D+ ≈ 1+0.32 1.47 ≈ meson production, the next higher state is D (2420) of
1
× JP = 1+. The branching ratio of the decay of this me-
if we assume that the same branching ratios for D∗ de-
sonisnotwell-knownexceptthatitdecaysintoD∗π but
caytoD asinfreespace. TheD0/D+ ratioinheavy-ion
not into Dπ. If we assume that it decays into D0 via
an intermediate D∗, i.e., D D∗π Dππ, the de-
1
→ →
nominator of Eq. (12) is then modified by the addition
3 We note that the average fractions of bottom mesons and bot-
tom baryons in pp¯annihilation at √s=1.5 TeV has been esti-
matedto bef(bu¯)=f(bd¯)=0.399and f(baryon)=0.092 [39].
If we assume that bq¯ mesons include equal number of B and 4 Inthestrangequarksector,thesmallmassdifferencebetweenΛ
B∗ mesons apart from the spin degeneracy and the number andΣdoesnotallowthestrongdecayofΣintoΛ,andonlyΣ0
→
of bottom baryons is mostly due to Λb, then we would have Λγ is allowed. But in heavy quark sector, the mass difference
Λb/B¯0 1.1inpp¯collisions. between Λc andΣc iswellabovethepionmass.
≈
4
of 1.68 3 0.06 0.3, where the factor 0.06 is the wherethefactors0.35and0.31aretheΣ /Λ andΣ∗/Λ
D (2420×)/D×0 ratio,≃giving thus less than 10% enhance- ratios,respectively. The Λ /B¯0 ratio is sbeenbto be labrgebr
1 b
ment to D0 production. For higher baryon resonances, than the Λ /D0 ratio by a factor of about 5. Compared
c
the next excited state is Λ (2593). Its contribution to to its value in pp collisions, the Λ /B¯0 ratio in thermal
c b
the numerator of Eq. (12) is about 0.2, since there is no model for heavy ion collisions is close to a factor of two
difference in its spin and isospin degeneracy from that larger,similar to the case for the Λ /D0 ratio.
c
of Λ . Thus it again gives an effect less than 5% to Λ Thereareotherhighercharmedandbottomresonances
c c
production. The final value of the Λ /D0 ratio then es- listed by the PDG [39]. These include heavy baryons
c
sentially does not change, i.e., the ratio is modified from that contain strange quarks such as Ξ , Ξ′ , Ω , and
Q Q Q
0.28to0.27,whichisabouttwiceofthatinppcollisions. their resonances. Although most of corresponding bot-
The estimated value of the Λ /D0 ratio from our tombaryonsareyetto be discovered,their existence has
c
simple thermal model is close to the value obtained in beenpredictedbythequarkmodel[46]. Usingthemasses
the more sophisticated thermal or statistical model of listedbythePDGandfromthequarkmodel,wefindthat
Ref. [42], which gives a charmed baryon to meson ratio the yield of these baryons is about 60% of the Λ yield
Q
(cqq)/(cq¯) 0.25. Itisalsoconsistentwiththestatistical in the thermal model and is thus appreciable. Including
modelpred≤ictionsofRef. [43],whichhasΛ /D0 0.2at thesehigherresonancesinthethermalmodelchangesthe
c
∼
the RHIC energy. baryon/mesonratios to
The Λ /B¯0 ratio in the thermal model can be esti-
b
mated in the same way as for the Λc/D0 ratio. The Λc 0.27, Λb 0.86. (17)
main difference between B¯0 and D0 productions is that D0 ≃ B¯0 ≃
thebottomquarkismuchheavierthanthecharmquark.
As a result, the mass difference between B∗ and B is
much smaller than that between D∗ and D because of IV. THE COALESCENCE MODEL
themanifestationofheavyquarkspinsymmetry. Infact,
themassdifferencebetweenD∗andDisabout140MeV, In the thermal model, the relative abundance of par-
while that between B∗ and B is only [39] ticles depends only on their masses. Whether diquarks
exist in QGP thus does not affect the Λ /D0 and Λ /B¯0
m(B∗) m(B) 5325 MeV 5279 MeV=46 MeV. c b
− ≈ − ratios in the thermal model. This is not the case in the
(13)
quark coalescence model as to be shown in the rest of
Therefore, unlike D0 production, B∗ mesons cannot de-
this paper.
cay into B mesons by strong interaction.5 This leads
to a large suppression of B meson production compared
with the case for D meson production. With m(Λ ) =
b A. Formalism
5620 MeV, our simple thermal model then gives
B∗+ Λb AsinthecoalescencemodeldescribedinRefs.[13,47],
0.78, 0.32. (14)
B+ ≈ (cid:18)B¯0(cid:19) ≈ which has been used in Ref. [48] to study the D meson
0
transverse momentum spectrum in relativistic heavy ion
It indeed shows that Λ /B¯0 is somewhat larger than
b 0 collisions, we assume that quarks, antiquarks, and glu-
Λc/D0 0. (cid:0) (cid:1) ons in the produced QGP are uniformly distributed in
(cid:0) Unlik(cid:1)e B production, the production of Λb is affected a fire cylinder of volume V. Their momentum distribu-
by resonance decays. This is because the mass differ- tions are taken to be thermal in the transverse direction
ence between Λb and Σb approaches to a finite value but uniform in midrapidity. For the Wigner functions of
( 195MeV[44])intheinfinite masslimit,althoughthe hadrons, they are expressed in terms of Gaussian func-
m∼ass difference between Σb of spin-1/2 and Σ∗b of spin- tions. Modeling the effect of transverse radial flow of
3/2 becomes smaller as the heavy quark mass increases. the fire cylinder by an effective temperature, the pro-
By using the recent experimental information for the Σb duced heavy meson transverse momentum spectrum is
baryons [45], then given by
M(Σ )=5812 MeV, M(Σ∗)=5833 MeV, (15)
b b dN (2√πσ)3 dN dN
the Λb/B¯0 ratio becomes dpM = gM V Z dp1dp2dp1 dp2
M 1 2
Λb = Λb (1+3 0.35+3 2 0.31) 1.25, ×exp −k2σ2 δ(pM −p1−p2), (18)
B¯0 (cid:18)B¯0(cid:19) × × × ≈ (cid:0) (cid:1)
0
(16) where gM is the statistical factor for colored quark and
antiquarktoformacolorneutralmeson,e.g.,g =1/36
D0
and g = 1/12 for D0 and D∗0, respectively, and sim-
D∗0
ilarly for B¯0 and B¯∗0. The distribution of heavy quarks
5 ThedecayofB∗intoBiscausedbyelectromagneticinteractions, with transversemomentum p1 andlight antiquarkswith
B∗ Bγ,andcanthus bedistinguishedinexperiments. transverse momentum p2 in the fire cylinder frame are
→
5
denotedbydN /dp anddN /dp ,respectively. Thep and the oscillator frequency ω is related to the baryon
1 1 2 2 M
is the transverse momentum of produced heavy meson. charge radius by
The relative transverse momentum k between the heavy
3 1 m +m
quark and light antiquark is defined as r2 = 2 3Q
ch 1
h i 2ωm +m +m (cid:18) m
1 2 3 1
k= m1+1 m2 (m2p′1−m1p′2), (19) + m3m+2m1Q2+ m1m+3m2Q3(cid:19). (24)
where m are quark masses, and p′ and p′ are heavy The root-mean-squared charge radii of Λc and Λb pre-
1,2 1 2
quark and light antiquark transverse momenta, respec- dicted by the quark model have similar values of about
tively, defined in the center-of-mass frame of produced 0.39 fm [50], and they are reproduced by the oscillator
heavy meson. The width parameter σ is related to the frequencies 0.43 GeV and 0.41 GeV, respectively. These
harmonic oscillator frequency ω by σ = 1/√µω, where valuesare25-30%largerthantheonesforheavymesons.
µ = m m /(m +m ) is the reduced mass and can be As we shall show in the next subsection, to convert as
1 2 1 2
related to the size of produced hadrons. Namely, the many heavy quarks of small pT into heavy hadrons via
charge rms radius of produced meson is given by coalescence as possible6 requires, however, smaller oscil-
lator frequencies or larger heavy hadron radii.
r2 = 3 1 Q1m22+Q2m21, (20) For production of ΛQ baryons from the coalescence of
h ich 2µω (m +m )2 heavyquarksand[ud]diquarks,we considerΛQ baryons
1 2
asmadeofaheavyquarkandalightdiquarkonly. Their
with Q being the charge of the i-th (anti)quark. To spectrumcanthusbeobtainedfromEq.(18)byreplacing
i
reproduce the root-mean-squaredchargeradii0.43 fm of g with g′ =1/9. Explicitly, it is written as
M B
D+ and 0.62 fm of B+ as predicted by the light-front
dN (2√πσ )3 dN dN
quark model of Ref. [49], an oscillator frequency ω = B = g′ dq dp dp 1 2
0.33 GeV is needed. dpB B V Z 1 2dp1 dp2
We note that contrary to that for mesons consisting exp k2σ2 δ(p p p ). (25)
× − dq B− 1− 2
of only light quarks, the coalescence formula, Eq. (18), (cid:0) (cid:1)
Thewidth parameterσ orcorrespondingoscillatorfre-
does not require heavy and light quarks to have simi- dq
quency ω is determined by fitting the sizes of heavy
lar momenta to form heavy hadrons. Instead, quarks of dq
baryons used in the three-quark model.
similar velocities have the most probable chance to form
The scalar diquarks can also lead to formation of ex-
hadrons.
Similarly,thecoalescenceformulaforformingΛQ,ΣQ, cited states of ΛQ baryons with jπ = 21− and 23−, which
ΞQ, Ξ′Q, and ΩQ as well as their resonances from three can decay into ΛQ by emitting pions. We note that the
quarks are given by mass differences between jπ = 1− and 3− states are
2 2
small because of heavy quark spin symmetry, e.g., the
ddNpBB = gB(2√π)V6(2σ1σ2)3 Z dp1dp2dp3ddNp11ddNp22ddNp33 lΛow(e2s6t25ex)coiftejdπs=tat3e−s,oafnΛdcthaures hΛa(v2e59a3)moafssjπdiff=er21e−ncaenodf
exp k2σ2 k2σ2 δ(p p p p ), oncly 32 MeV. Alth2ough the number of these resonances
× − 1 1 − 2 2 B − 1− 2− 3
(cid:0) (cid:1) (21) is rather small in the thermal model due to their higher
masses compared to those of ground state Λ baryons,
Q
where the index 3 refers to the heavy quark and indices it is not the case in the coalescence model if diquarks
1 and 2 to light quarks, and the statistical factor gB has exist in QGP. Since excited ΛQ baryons have negative
values1/108,1/36,and1/18forΛ ,Σ ,andΣ∗,respec- parities, the scalar diquark inside them is in a relative
Q Q Q
tively,and1/54,1/18,1/108,and1/36forΞ (Ξ′ ),Ξ∗, P-wave state with respect to the heavy quark. The coa-
Q Q Q
Ω , and Ω∗, respectively. The relative transverse mo- lescence formula for the production of such hadrons is
Q Q
menta defined in the center of mass frame of produced dN 2(2√πσ )3 dN dN
baryon are B = g′ dq dp dp 1 2σ2 k2
dp B 3V Z 1 2dp dp dq
B 1 2
k1 = m1+1 m2 (m2p′1−m1p′2), using the Wig×neerxfpu(cid:0)n−ctkio2nσd2gqi(cid:1)vδe(npiBn−Repf1s.−[5p12,)5,2]. In(2th6e)
1
k = [m (p′ +p′) (m +m )p′]. above,k issimilarlydefinedasinEq.(19)andthe width
2 m +m +m 3 1 2 − 1 2 3
1 2 3 parameterσ istakentohavesamevalueasthatforthe
dq
(22) ground state Λ .
Q
The width parameters σi are σi =1/√µiω, where
µ = m1m2 , µ = (m1+m2)m3 , (23) 6 This is similar to the assumption introduced in the thermal
1 m +m 2 m +m +m modelthatallcharmedquarksareconvertedintoheavyhadrons.
1 2 1 2 3
6
B. Quark distribution functions in QGP is, on the other hand, determined from fitting simulta-
neously measured p spectrum of charmed mesons from
T
For the quark distribution functions in quark-gluon d+Au collisions [57] and of electrons from heavy meson
plasma,we use the thermaldistribution for light quarks, decays in pp collisions.
namely, Including heavy quark energy loss as estimated in
Ref.[8],theresultingheavyquarktransversemomentum
dN g V distribution in midrapidity can be parameterized as
q q
=λ m exp( m /T ), (27)
dp q(2π)3 T − T eff
dNEL dN
Q Q
= L (p), (29)
dp dp Q
where g = 6, m = p2+m2, and p is the transverse
q T q
momentum. Instead oqf the local temperature and trans- with
verse flow velocity, we use an effective temperature T
eff
L = 0.8exp( p/1.2)+0.6exp( p/15),
(=200MeV) andintroduce a normalizationfactor λq to c − −
fix the total number of quarks as in Ref. [53]. Following Lb = 0.36+0.84exp( p/10), (30)
−
Ref. [54], we assume that the phase transition tempera-
for charm and bottom quarks, respectively, and p again
tureisT =175MeVandthevolumeofthequark-gluon
C inGeV.The heavyquarknumbersarethenestimatedto
plasma formedin rapidity y 0.5 is V =1,000fm3 for
centralAu+Au collisionsa|t√|≤s =200GeV atRHIC. be Nc ≃9.23 and Nb ≃0.035. The charmquarknumber
NN usedhereisaboutafactorthreelargerthanthatusedin
This gives the light quark numbers N = N 245 and
u d ≈ Ref. [20] based on perturbative QCD calculations. This,
the strange quark number N 149 if the constituent
lightquarkandstrangequarksm≈assesaretakentobe300 however,doesnotaffecttheΛQ/H0ratios. Fortheheavy
quark masses, we use m =1.5 GeV and m =5.0 GeV.
MeV and 475 MeV, respectively. The resulting normal- c b
ization factors λ for light and strange quarks are then
q
0.95 and 0.81, respectively.
C. Results
Diquarks are also assumed to be in thermal equilib-
rium and their distribution dN /dp is then the same
[ud] TocalculatethespectraofH0 andΛ producedinrel-
as given in Eq. (27) with g = 3. The number of di- Q
[ud] ativistic heavy ion collisions, we consider two scenarios
quarks depends on the diquark mass and is in the range
for Λ production. In the first model, we assume that
Q
of 77 44 with m = 455 600 MeV [20]. If di-
∼ [ud] ∼ the ΛQ baryons have no diquark structure and there are
quarks exist in QGP, the number of free light quarks nodiquarksinQGPeither. Therefore,Λ , Σ ,Σ∗,Ξ ,
Q Q Q Q
is reduced by twice the number of diquarks so that the
Ξ′ , Ξ∗, Ω , and Ω∗ baryons are all formed by three-
total number of light quarks is preserved. This would Q Q Q Q
quarkcoalescence. Wefurtherincludeheavymesonsthat
cause a reduction of meson yields and an enhancement
consist of strange quarks such as D and B and their
of baryon yields and thus induces larger Λ /H0 ratios. s s
Q resonances. Using the same oscillatorfrequency for both
Results to be presented below are obtained by using the
heavy mesons and baryons as shown approximately in
diquark mass m =455 MeV, which is in the range of
[ud] the quarkmodel,we adjustits value toconvertallheavy
diquark mass estimated in Ref. [55]. The sensitivity of quarks of small p into heavy hadrons via coalescence
T
theseresultstothediquarkmasswill,however,bebriefly as in the thermal model. This leads to ω = 0.106 GeV
discussed. for charmed hadrons and ω = 0.059 GeV for bottom
Forheavyquarktransversemomentumdistributionsin hadrons. The resulting D+ and B+ charge radii are
midrapidity,weadoptthoseparameterizedinRef.[8]for 0.74 fm and 1.44 fm, respectively, which are factors of
centralAu+Au collisionsat√sNN =200GeV atRHIC, 1.7and2.3largerthanthose predictedby the light-front
i.e., quark model of Ref. [49]. The requiredsmaller oscillator
frequencies may be partly due to the change in hadron
dN 19.2 1+(p/6)2
c radii in medium [58] and partly due to the incomplete
= ,
dp (1+p/3.7)12(cid:2)[1+exp(0(cid:3).9 2p)] treatmentof production anddecay of hadronresonances
−
dN p 5 in the present study.
b
= 0.0025 1+ exp( p/1.495), (28)
dp (cid:20) (cid:16)16(cid:17) (cid:21) − The second model assumes the diquark structure in
Λ andthe existence ofdiquarks inQGP asin Ref. [20].
Q
with the transverse momentum p in unit of GeV. These In this case, ΛQ baryons are formed by diquark–heavy-
distributionswereobtainedfromheavyquarkp spectra quark coalescence in addition to three-quark coales-
T
from pp collisions at same energy by the number of bi- cence.7 Together with the ΣQ and Σ∗Q resonance con-
nary collisions ( 960)in Au+Au collisions. For bottom tributions as well as those from Ξ , Ξ′ , Ξ∗, Ω , and
∼ Q Q Q Q
quarks, it was taken from the upper limit of the uncer-
tainty band of the pQCD predictions in Ref. [56] for pp
collisions,asnoexperimentalmeasurementsareavailable
at RHIC. The charm quark p spectrum in pp collisions 7 Strictly speaking, the three quarks should coalesce to form the
T
7
Charmed hadrons Bottom hadrons
Model D0 D+ Ds Λc Ξc/Ξ′c/Ωc B¯0 (B−) B¯∗0 (B∗−) Bs0 Bs∗0 Λb Ξb/Ξ′b/Ωc
PYTHIA model 0.607 0.196 0.121 0.076 0.101 0.3015 0.030 0.089 0.076
Thermal model 0.435 0.205 0.179 0.118 0.063 0.111 0.229 0.049 0.113 0.096 0.062
Coalescence model (three-quarkmodel) 0.348 0.113 0.113 0.288 0.138 0.053 0.158 0.027 0.080 0.316 0.155
(ground state contribution) 0.051 0.051 0.0264 0.028 0.066 0.052 0.155 0.026 0.079 0.032 0.073
(fragmentation) 0.043 0.014 0.009 0.005 0.001 0.003 0.0003 0.001 0.001
Coalescence model (diquark model) 0.282 0.091 0.123 0.378 0.126 0.044 0.131 0.031 0.092 0.385 0.142
(ground state contribution) 0.039 0.039 0.028 0.016 0.059 0.040 0.120 0.029 0.088 0.019 0.066
(diquarkcontribution) 0.205 0.192
(fragmentation) 0.048 0.015 0.010 0.005 0.004 0.011 0.002 0.004 0.003
TABLE I: Relative production fractions of midrapidity heavy hadrons produced in central Au+Au collisions at √sNN = 200
GeV.Theheavy-strangebaryon(ΞQ,Ξ′Q, andΩQ) yieldissuppressed inthePYTHIAmodelandthusneglected in thiswork.
Ω∗ resonances, which are formed only from three-quark i.e.,
Q
coalescence as they do not contain the scalar diquark
sintrculucdtuedre,uesxincgitetdheΛPQ-(w12a−v)eadnidquΛaQrk(–32h−e)avbya-rqyuoanrskacroeaallesso- (cid:18)DΛc0(cid:19) = 0.05exp −(pT −4.0)2/8.0 +0.1,
fr (cid:2) (cid:3)
cence.8 As in the three-quark coalescence model, the os- Λ
b
cillator frequencies or width parameters are determined (cid:18)B¯0(cid:19) = 0.75, (31)
by requiring that all low p heavy quarks are converted fr
T
to hadrons. Besides the common oscillator frequency ω
with p in unit of GeV.
formesonsandthree-quarkbaryons,theadditionaloscil- T
lator frequency for the heavy-quark and diquark system
is fixed by reproducing the same Λ radius used in the
Q
1. Relative production fractions of heavy hadrons and
three-quark configuration. Then we have ω = 0.09 GeV
for charmed hadrons and ω = 0.049 GeV for bottom ΛQ/H0 ratios
hadrons, leading to 0.81 fm and 1.58 fm for the D+
and B+ charged radii, respectively. These values lead Our results for the relative production fractions of
to ω = 0.053 GeV and 0.029 GeV, respectively, for heavy hadrons produced in central Au+Au collisions at
dq
charmed and bottom baryons consisting of diquarks. In √s =200GeVarepresentedinTable Itogetherwith
NN
obtaining the contribution of resonances to the ground thosefromthePYTHIAmodelandthethermalmodel. It
state H0 and Λ spectra, we neglect for simplicity the isseenthattherelativeproductionfractionofD0 among
Q
small momentum shift in their decays. allproduced charmedhadronsin the three-quarkcoales-
cencemodelis0.35,whileitis0.44and0.61forthether-
mal model and the PYTHIA model, respectively. This
fraction reduces to 0.28 in the diquark model. The rela-
tive production fraction of B¯0 among all produced bot-
tom hadrons also shows a similar behavior, namely, it is
0.101,0.097,0.052,and0.043inthePYTHIAmodel,the
In both models, remaining heavy quarks that are not
thermal model, the three-quark model, and the diquark
convertedto hadrons via coalescence,whichmostly have
model, respectively.
high p , are converted to heavy hadrons by fragmenta-
T For the Λ /H0 ratios, we find that without resonance
tion. This is achieved by assuming that a heavy quark Q
of transverse momentum p fragments into Λ and H0 contributionΛc/D0 =0.55inthethree-quarkmodeland
T Q Λ /D0 =2.2inthediquarkmodel. Theseresultsaredif-
witharatiosimilartothatgivenbythePYTHIAmodel, c
ferentfromthoseofRef.[20],namely,theΛ /D0islarger
c
andthe enhancement of this ratio due to the presence of
diquarks is about 4 while it was about 8 in Ref. [20].
These differences come from the different values for the
ΛQ indiquarkstructure. Herewesimplyapproximatethis pro-
ductionprocessbythree-quarkcoalescencetoformnon-diquark oscillator frequencies or width parameters as well as the
ΛQ. This thus gives an upper bound for the yield of Λc from fact that in Ref. [20] the number of free quarks in the
threeindependent quarksinQGPaspointedoutinRef.[20]. quark-gluonplasmais notreducedby the presenceofdi-
8 In general, coalescence of three independent quarks can also quarks. For bottom hadrons, we obtain Λ /B¯0 = 0.6 in
form negative parity baryons. Since this process has a smaller b
the three-quarkmodel,whichincreasesbya factorof3.5
contribution than that due to the production of positive parity
baryons[59],itisnotconsideredinpresentwork. to Λb/B¯0 =2.1 in the diquark model.
8
Including resonances enhances both ΛQ and H0 pro- Λ / D0
c
duction. We find that in the three-quark model, about 101 2.0
7re0s%onoafncDe0daencadyas.boTuhte9i0m%poorftΛanQcearoefprerosodnuacnecdesthirsoaulgsho Λc tdhiqreuea-rqku marokd melodel (c)
10-1 PYTHIA
seen in the diquark model. In particular, the diquark
2) 1.5
contribution through negative parity ΛQ resonances is V
non-negligible. The number of negative parity ΛQ reso- Ge10-3
nanceofspin-1/2turnsouttobeabouthalfoftheground 1/ (a)
(
srteasotenaΛnQcesbaarreyoanlmnousmtdbeegr.eneSriantceeinΛmQ(a12s−s,)thaendnuΛmQb(e23r−o)f dp TT1100-50 D0 1.0
p
negativeparityΛQ resonancesis abouttwotimes thatof π2
tdheegegnreoruancyd.sTtahties sΛhQoualdftebretcaoknintrgasintetdo wacitchoutnhtetthheersmpianl dN / 10-2 0.5
modelwhichpredictssuppressedyieldsofnegativeparity (b)
Λ resonancesasaresultoftheirlargermasses. Asare-
Q 10-4 0.0
sult,wefindthatabout70%ofD0 andabout75%ofΛ 0 1 2 3 4 5 0 1 2 3 4 5 6
Q
p (GeV) p (GeV)
are produced through resonance decays in the diquark T T
model.
The contribution from heavy quark fragmentation is FIG.1: Spectraof(a)Λc and(b)D0,and(c)theratioΛc/D0
in midrapidity (y 0.5) for central Au+Au collisions at
small,particularlytotheratiosofparticlenumbers. This
| | ≤
is because we have assumed that most hadrons are pro- √sNN = 200 GeV. Solid lines are for the three-quark model
anddashedlinesareforthediquarkmodel. Resultsfromthe
duced by coalescence of low p heavy quarks. The frag-
T PYTHIA model are shown by filled circles.
mentation contribution is thus non-negligible only for
high p hadrons, which constitute less than 8% of pro-
ducedTtotal heavy hadrons. Λ / B0
b
The final Λ /H0 ratios after including fragmentation 12.0
Q
and resonance contributions are 10-3 Λ three-quark model
b diquark model (c)
Λc/D0 =0.83, Λb/B¯0 =6.00, (32) 2) 10-4 PYTHIA 10.0
V10-5
e 8.0
for the three-quark model, and G
1/10-6 (a)
Λc/D0 =1.34, Λb/B¯0 =8.79, (33) (T10-7 6.0
p
d 0
for the diquark model. Therefore, the enhancement fac- pT10-4 B
π 4.0
tor for these ratios due to diquarks in QGP is about 1.6 2
for both charmed and bottom hadrons. N / 10-5
OurresultsthusindicatethattheΛc/D0(Λb/B¯0)ratio d10-6 (b) 2.0
inheavyioncollisionspredictedbythecoalescencemodel
is about a factor of 6.4 (8.6) larger than the predictions 10-7 0.0
0 1 2 3 4 5 0 1 2 3 4 5 6
of the PYTHIA model in Section II and about a factor p (GeV) p (GeV)
T T
of 3.1 (7.0) larger than that from the thermal model in
Section III. The existence of diquarks in QGP further FIG.2: SameasFig.1for(a)Λb spectrum,(b)B¯0 spectrum,
enhances this factor to 10.3 (12.6) and 5.0 (10.2), com- and (c) theΛb/B¯0 ratio.
pared to those from the PYTHIA and thermal models,
respectively.
hadronspectraduetotheflatterbottomquarkspectrum
than the charm quark spectrum.
2. Transverse momentum spectra of heavy hadrons Compared with measured p¯/π− and Λ/K0 ratios [60,
S
61, 62], the obtained Λ /H0 ratios show very different
Q
The transverse momentum spectra of Λ and H0 are behavior. Although the Λ /D0 ratio has a similar shape
Q c
showninFigs.1and2forcharmedandbottomhadrons, as the p¯/π− and Λ/K0 ratios, which could be approx-
S
respectively,togetherwiththetransversemomentumde- imated by a Gaussian shape as assumed in Ref. [12], it
pendence of the Λ /H0 ratio. Solid lines are results hasamuchlargerwidthandisthusmuchflatterthanthe
Q
for the three-quark model and dashed lines are those for lightbaryon/mesonratios. Furthermore,theΛ /B¯0ratio
b
the diquarkmodel. Comparedwith the PYTHIA results is even flatter than the Λ /D0 ratio. This behavior can
c
shownby filled circles, the enhancement ofΛ yield and alsobefoundinthespectraofΛ andH0. Namely,while
Q Q
the suppression of H0 yield are evident. The bottom thespectraofΛ andB¯0haveverysimilarp dependence
b T
hadron spectra are seen to be harder than the charmed (Fig.2), thoseofΛ andD0 showsomewhatdifferentp
c T
9
dependence (Fig. 1). All these behaviors originate from ioncollisions. We have,therefore,includedthe contribu-
thelargemassofheavyquarks. Intheinfinitemasslimit, tion from resonances in estimating these ratios.
the momentum of a heavy hadron is completely carried We have determined the width parameters of hadron
by the heavy quark, and the role of light quarks or di- Wigner functions used in the coalescence model by re-
quarksisjusttogivedifferentproductionprobabilityde- quiringthatthemajorityofheavyhadronsatlowp are
T
pending on the heavy hadron wavefunction. Therefore, formedfromcoalescence,whichissimilartothatassumed
theΛQ/H0ratiointheinfiniteheavymasslimitbecomes in the thermal model. The remaining heavy quarks are
a constantand shows no momentum dependence. In the then converted to heavy hadrons via fragmentation as
case of the Λc/D0 ratio, finite mass effects are not negli- in pp collisions. The resulting Λc/D0 and Λb/B¯0 ratios
gible but its dependence on the transversemomentum is in central Au+Au collisions at √s = 200 GeV are
NN
less strong than in the Λ/K0 ratio. This is clearly seen found to be 1.34 and 8.79, respectively, which are about
S
in pp collisions as shown by the PYTHIA model results 10.3and12.6largerthanthepredictionsofthePYTHIA
in Figs. 1 and 2. Although the coalescence model shows model, and about a factor of 5.0 and 10.2 larger than
a pT dependence of the ΛQ/H0 ratio, this dependence those of the thermal model. The enhancement of the
weakens as the heavy quark mass increases. Λ /H0 ratiodue to diquarksis, however,only about1.6
Q
We also find that inclusion of diquarks in QGP causes for both charmed and bottom hadrons, which is smaller
ashiftofthepeakpositionsintheΛQ/H0 ratiostolower than that predicted in Ref. [20]. The difference mainly
values of pT. In the three-quark model, the Λc/D0 ratio arises from the different constraints used here and in
peaksatp 2GeV,whichshiftstop 0.8GeVinthe Ref. [20] for determining the oscillator frequencies or
T ≃ T ≃
diquarkmodel. Thechangeofthepeakpositioniscaused width parameters in the hadron Wigner functions and
byboththereductionoftheD0 mesonspectrumandthe from maintaining the same total light quark numbers in
enhancement of the Λc spectrum in the diquark model. both the three-quark model and diquark model in the
Wefind,however,thatthelatterhasalargereffectonthe present study.
peakpositionintheΛc/D0ratio. Sincetheenhancement The enhancement due to diquarks would have been
ofΛc duetolightdiquarksismoreappreciableinthelow larger if we do not take into account the resonance con-
pT region, the peak of the Λc/D0 ratio in the diquark tribution. This is because of the enhancement of ΛQ
model thus appears at a lower value of p compared to baryons from Σ and Σ∗ decays, whose production is
that in the three-quark model. The ΛbT/B¯0 ratio also not affected by Qthe preseQnce of diquarks in QGP. This
showsashiftinthepeakposition,i.e.,fromp 3.5GeV is different from the production of strange Λ to which Σ
T ≃
inthethree-quarkmodelto2GeVinthediquarkmodel, can not decay by strong interactions. Furthermore, in-
but it is still much flatter than the Λc/D0 ratio. cluding the resonance contribution increases the Λb/B¯0
For a more realistic momentum distribution of heavy ratio much more than the Λ /D0 ratio because B∗ can-
c
hadrons in relativistic heavy ion collisions after their not decay into B by strong interactions as a result of
production from QGP, their production and annihila- heavy quark spin symmetry. Finally, this model leads to
tion due to collisions of and with surrounding particles the reduction of D0 and B¯0 mesons by factors of about
should be investigated by a transport model like the 2.2 compared to the PYTHIA model.
ampt model [63, 64]. However, cross sections for reac- Contrary to the PYTHIA model, we have found that
tions involving open charmed hadrons are small [65, 66], the yields of heavy baryons with strange quark(s) are
and it has been shown in Ref. [51] that the effects of notsmall. Their yield was estimatedto be about60%of
multiple scattering would be small with such small cross the Λ yield in the thermal model. In the coalescence
Q
sections. Since this workis atthe levelofanexploratory model, this is about 50%and35%of the Λ yield in the
Q
study, such an elaborated investigation is not yet called three-quark model and diquark model, respectively.
for.
We have also compared the transverse momentum de-
pendence of the Λ /H0 ratios with that of measured
Q
Λ/K0 ratio and found that the Λ /H0 ratios have a
S Q
V. CONCLUSIONS AND DISCUSSIONS weakerdependenceonthetransversemomentumbecause
of the massive quarks inside heavy hadrons. In particu-
Inthispaper,wehavestudiedtheΛQ/H0 (Λc/D0 and lar, this leads to a very weak dependence of the Λb/B¯0
Λ /B¯0)ratiosinheavyioncollisions,particularlytheen- ratio on the transverse momentum.
b
hancement of these ratios due to coalescence of heavy We have further found that the ΛQ/D0 ratio peaks
quarks with light quarks as well as diquarks that might at p 2 GeV in the three-quark model and at p
T ≃ T ≃
exist in the produced QGP. We have also considered the 0.8 GeV in the diquark model. Therefore, the enhance-
resonance decay effects which are shown to be impor- ment of heavy baryon production over heavy meson
tant for understanding the particle production ratios in production due to diquarks can mostly be observed at
pp collisions. Our simple estimate based on the thermal low p region. We have also estimated that the ratio
T
model as well as predictions of more sophisticated sta- Λc/Σ0c ≃ 23, which is less than that of Ref. [23] by a
tistical models [42, 43] show that resonance decays also factor of 4.
∼
playsacrucialroleforheavyhadronproductioninheavy Theseresultsarebasedonadiquarkmassof445MeV
10
or a binding energy of 145 MeV. Because of the thermal that the color-spin interaction for color-sextet diquark
factor, the number of diquarks decreases as the diquark of spin-1 is, although not strong, attractive. (See, e.g.,
mass increases. This would reduce both Λ and Λ pro- Ref [33].) This diquark, of course, cannot form a color-
c b
duction and increase that of D0 and B¯0, reducing thus singlet baryon with one heavy quark, and color-sextet
the Λ /H0 ratios. For example, if the diquark mass is diquarks are thus disfavored in phenomenological mod-
Q
550MeV,theenhancementoftheΛ /D0 ratioduetodi- els[31]. However,color-sextetdiquarksinQGP canpro-
c
quarks would reduce from 1.6 to 1.3. The peak position duce color-singletbaryonsby first forming color-octetor
in the ratio changes, however,very little. color-decupletbaryonsandthenneutralizingtheir colors
In studying resonance production in the coalescence by interacting with quarks and gluons in QGP. There-
model, we have assumed that the effect due to energy fore,if weassume the existence of(weakly-bound)color-
mismatchbetweenquarksandproducedhadronissmall. sextet vector diquarks in QGP, it may bring in addi-
Since the coalescence model can be viewed as formation tionalenhancement ofΛ baryonproduction. Since this
Q
of bound states from interacting particles in the system mechanism is very model-dependent, the existence and
with energy mismatchbalanced by other particles in the properties of such diquark states in QGP should be un-
system, this would be reasonableif the energy mismatch derstood before estimating their contributions to heavy
is small. Otherwise, the correction factor can, in princi- hadron production.
ple,bedeterminedbyevaluatingthetransitionprobabil-
ity in the presence of other particles. Also, the coales-
cencemodelcanbefurtherimprovedbyimposingenergy
Acknowledgments
conservation using Breit-Wigner type spectral functions
fortheproducedhadrons[67]. Thiswouldnaturallylead
to suppressed production rate due to the energy mis- We would like to thank Zi-Wei Lin for help with the
match between quarks and produced hadron. Improving HIJING/PYTHIAsimulations. Thisworkwassupported
the coalescence model, therefore, deserves further stud- bytheUSNationalScienceFoundationunderGrantsNo.
ies. PHY-0457265and PHY-0758155,the Welch Foundation
One may also consider contributions to heavy hadron under Grant No. A-1358,and the Korea ResearchFoun-
production from other diquark states. It is well-known dation under Grant No. KRF-2006-C00011.
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