Analytical coalescence formula for particle production in relativistic heavy-ion collisions

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by  Kai-Jia Sun

Analytical coalescence formula for particle production in relativistic heavy-ion
collisions
Kai-Jia Sun1 and Lie-Wen Chen 1
∗
1Department of Physics and Astronomy and Shanghai Key Laboratory for Particle Physics and Cosmology,
Shanghai Jiao Tong University, Shanghai 200240, China
(Dated: March 29, 2017)
Basedonacovariantcoalescencemodelwithablast-wave-likeparametrizationforthephase-space
configurationofconstituentparticlesatfreeze-out,wederiveanapproximateanalyticalformulafor
the yields of clusters produced in relativistic heavy-ion collisions. Compared to previous existing
formulae, the present work additionally considers the contributions from the longitudinal dimen-
7 sion in momentum space, the relativistic corrections and the finite size effects of the produced
1 clusters relative to the spatial distribution of constituent particles at freeze-out. The new analyt-
0 ical coalescence formula provides a useful tool to evaluate the yield of produced clusters, such as
2 light nuclei from nucleon coalescence and hadrons from quark coalescence, in heavy-ion collisions.
As a first application of the new analytical formula, we explore the strangeness population factor
ar S3 =3ΛH/(3He×Λ/p) based on nucleon/Λ coalescence as well as the production of exotic hadrons
M based on quark coalescence, in central Pb+Pb collisions at √sNN = 2.76 TeV. The results are
compared with thepredictions from othermodels.

7
2 PACSnumbers: 25.75.-q,25.75.Dw

]
h I. INTRODUCTION parametrization [13–17, 21, 22, 31, 32], and the final re-
t sultsarethenobtainedbymulti-dimensionnumericalin-
-
l tegration. Although the numerical calculations can give
c Particle production in relativistic heavy-ion collisions
u is of fundamental importance for many issues in nu- exact results in the coalescence model, the approximate
n analytical formula is extremely useful and has its own
clear physics, particle physics, astrophysics and cosmol-
[ merits, e.g., it can significantly simplify the computa-
ogy. Coalescence model [1–4] provides an important ap-
tionalworkandmostimportantlyitcangivemoretrans-
2 proach to describe the particle production in heavy-ion
v collisionsatbothintermediateandhighenergies. Forin- parent and deep insights into the physics of the coales-
5 cence scenario.
stance, the coalescence model has been successfully and
3 widely applied to describe both light nuclei production In the literature, there indeed exist some analyti-
9
from nucleon coalescence [5–17] and hadron production cal coalescence formulae (see, e.g., [32–34]) for cluster
1
0 from quark coalescence in heavy-ion collisions [18–28]. yields in relativistic heavy-ion collisions, obtained under
. Thequarkcoalescenceprovidesanimportantmechanism some approximations. In particular, the analytical for-
1
for the hadronization of partons produced in relativistic mula reported recently by the ExHIC collaboration in
0
heavy-ion collisions and thus is very useful for under- Refs.[33,34](denotedasCOAL-Exinthe presentwork)
7
1 standing the partonic dynamics as well as the formation isquitegeneralforN-bodyclusterproductionwiththeN
: signals and properties of quark-gluon plasma possibly constituentparticlesatvariousorbitangularmomentum
v
formed in these collisions [28–30]. Since the coalescence states. In the derivation of the COAL-Ex formula, for
i
X probability is based on the overlap of density matrix of simplicity, the integrationof Wigner function in longitu-
r constituent particles in an emission source with Wigner dinal (z-) direction is neglected, namely, the integration
a function of the produced cluster, the predicted cluster in momentum space is only treated in two-dimensional
yield generally depends on the internal structure wave transverse plane although the integration in coordinate
function ofthe cluster. Therefore,the coalescencemodel space is treated three-dimensionally, which leads to a
provides a unique tool to identify the constituent quark problematic feature that the yield of the produced clus-
structure of some exotic hadrons via quark coalescence ter in the COAL-Ex formula increases as a power func-
from studying their yields in relativistic heavy-ion colli- tionofitsroot-mean-square(RMS)radiusrrmswhenrrms
sions [31–34]. becomes large. Even though the single-particle momen-
tumdistributionforaparticleintheemissionsourcemay
To describe particle production in heavy-ion colli-
only depend on its transverse momentum due to boost
sions,thecoalescencemodelisusuallyimplementedwith
invariance [35] in the longitudinal direction, the many-
the phase-space configuration of constituent particles at
body coalescence process cannot be treated merely in
freeze-out in an emission source obtained from either
two-dimensional transverse plane. It is thus of great in-
transport model simulations [5–12, 26, 27] or empirical
teresttoincludethecontributionoflongitudinalmomen-
tum dimension and check its importance.
In the present work, based on a blast-wave-like
∗Correspondingauthor(email: [email protected]) parametrization, which is inherently longitudinal boost
2
invariantbydesign,fortheemissionsourceofconstituent Assuming the temperature of fireball is much smaller
particlesatfreeze-out,wederiveanew analyticalcoales- than the mass of constituent particles, we can then use
cenceformula(denotedasCOAL-SHinthefollowing)by the Boltzmann approximation for f(x,p), and Eq. (1)
treatingthe integrationofWigner functioninfullphase- can then be analytically obtained as [36]
space for the many-body coalescence process. We find
d3N gξτm
that the COAL-SH formula possesses a nice saturation = T 2K (β)2πI (α)rdr, (3)
propertythat whenrrms of the produced cluster is large, pTdpTdydφp (2π)3 Z 1 0
its yield converges to a constant. In addition, we con- where we have β = m cosh[ρ(r)]/T and α =
T
sider the relativistic corrections to leading order in the p sinh[ρ(r)]/T; I (x) and K (x) are the first and sec-
T 0 1
COAL-SH formula. Furthermore, an empirical expres- ond kind of modified Bessel functions, respectively. The
sionisproposedforthecorrectionsoffinitesizeeffectsof radialexpansionofthe fireballwouldleadto a blue-shift
the produced clusters relative to the size of the emission on the transverse spectrum of the emitted particles, and
source for constituent particles. As a first application thus effectively increase the temperature. As a result,
of the new analytical coalescence formula, we explore wecanapproximatelytakeT asaneffectivetemperature
the strangeness population factor S3 =3ΛH/(3He×Λ/p) and set radial flow rapidity ρ to be zero, i.e., α = 0 and
as well as the production of exotic hadrons in central I (0) = 1, and in this case the above formula can be
0
Pb+Pb collisions at √sNN = 2.76 TeV, and compare further simplified to be
the results with the predictions from other models.
d3N gξV m
T
= 2m K , (4)
p dp dydφ (2π)3 T 1 T
T T p
II. THEORETICAL FORMALISM (cid:16) (cid:17)
where we denote V =πR2τ as an effective volume. Fur-
thermore,themultiplicityoftheconstituentparticlescan
A. Blast-wave-like parametrization for emission
be integrated out to be
source of constituent particles at freeze-out
dN gξV m
= 4πTm2K . (5)
We assume that particles are emitted from a hyper- dy (2π)3 2 T
(cid:16) (cid:17)
surface Σµ, and then the Lorentz invariant one-particle In the case of m T, according to the asymptotic
momentum distribution is given by behavior of Bessel fu≫nction K (x) in large x limit, i.e.,
ν
dN π 4ν2 1 1
Ed3p = dσµpµf(x,p)= d4xS(x,p), (1) Kν(x)→ 2xe−x(1+ 8x− +O(x2)), (6)
ΣZµ Z r
wecankeeponlythefirsttermtomakeanon-relativistic
where σµ denotes the normal vector of the hypersurface approximation, and then Eq. (5) can be written as
Σµ and pµ is the four-momentum of the emitted con-
dN g
sptiidtiuteynitnpraelratticivleis.tiFcohreatvhye-icolnusctoelrlispiroondsutchtaiotnwaetarmeicdorna-- dy = (2π)3ξe−mT VVp, (7)
sidering in this work, we adopt the longitudinal boost 3
invariance assumption [35] and assume the constituent where Vp = (2πTm)2 reflects the effective volume of
the emission source in momentum space. If we denote
epmaritsisciloens faurnecetmionittceadnattheanfibxeedexpprroepseserdtiamse[3τ60], and the yL = ymax −ymin, V′ = yLV and ξ′ = ξe−mT , then the
number of the emitted constituent particle in rapidity
S(x,p)d4x = m cosh(η y)f(x,p) region [ymin, ymax] is given by
T
−
g
×δ(τ −τ0)τdτdηrdrdφs, (2) N = (2π)3ξ′V′Vp. (8)
wherewe use the longitudinalpropertime τ =√t2 z2,
space-time rapidity η = 1lnt+z, polar coordinate−s (r,
2 t z
φ ), rapidity y = 1ln(E+pz−), transverse momentum B. Analytical coalescence formula
s 2 E−pz
(p ,φ ) and transverse mass m = m2+p2. The
T p T T We consider the case that N constituent particles are
statistical distribution function f(x,p) is given by [37]
p coalesced into a cluster, and the total multiplicity of the
f(x,p)=g(2π) 3[exp(pµu /kT)/ξ 1] 1whereg isspin
− µ ± − cluster can be expressed as
degeneracyfactor,ξisthefugacity,u isthefour-velocity
µ
ofafluidelementinthefireballoftheemissionsourceand N
N = g dN ρW(x ,...,x ;p ,...,p )
T isthecorrespondinglocaltemperature. Inaddition,we c c i c 1 N 1 N
notepµuµ =mT coshρcosh(η y) pT sinhρcos(φp φs) Z (cid:16)iY=1 (cid:17)
gives the energy in local res−t fra−me of the fluid,−and N d3p
preµsde3nσtsµ =theτmtrTancosvsher(sηe−flyow)dηrradpriddiφtys.dTishterisbyumtibonoloρfrtehpe- = gcZ (cid:18)i=1pµidσiµ Eiif(xi,pi)(cid:19)×
Y
fluid element in the fireball with a transverse radius R. ρW(x ,...,x ;p ,...,p ), (9)
c 1 N 1 N
3
where ρW(x ,...,x ;p ,...,p ) is the Wigner density Jˆwe use here is different from that in Ref. [34], but the
c 1 N 1 N
function which gives the coalescence probability, and g physical results should be independent of the choice of
c
is the coalescence factor [2]. Eq. (8) can be used to nor- coordinate transformation. The modulus of determinant
malize the coalescence probability and so Eq. (9) can be of the Jacobimatrix is Jˆ =N−21, and one then has the
| |
rewritten as following identity
Nc = gc(cid:20)i=N1Ni(cid:21)R ρWc QNi=Ni=11pµi(2dgπi3)3σξiµi′VdE3′pViipf,i(xi,pi), N d3xid3pi = d3Rd3PN−1d3qid3ki. (15)
Y i=1 i=1
Y Y
Q (10)
From the above identity, Eq. (11) can be simplified as
where N = gi ξ V V is the multiplicity of the i-th
i (2π)3 i′ ′ p,i
cfoornmstuitlauecnatnpnaorttibclee.anTahlyetiWcailglyneirntfuegnrcattioend ionutthaenadbothvee N g N N V′(2πµ0T)23 ρWc Ni=−11e−2kµT2i,Ti d3qidki3.
resultisusuallyobtainedbyamulti-dimensionnumerical c ≈ c(cid:20)i=1 i(cid:21) R Ni=1QV′Vp,i
Y
integration method (see, e.g., Ref. [15]).
(16)
Q
FollowingRefs.[33,34],weconsidertheproducedclus-
ters are in midrapidity region where one has τ t, and where
the volume element on the hypersurface pµdσ ≃d3p /E
can then be approximated to be d3xid3pi. iFuritµhermiorei, µ = i+1mi+1 ik=1mk, (1 i N 1) (17)
thenon-relativisticBoltzmannapproximationforthedis- i i i+1 m ≤ ≤ −
kP=1 k
tribution function f(x,p) is adopted in the following
derivation, and thus one has Vp,i = (2πTmi)23. We will insattehse(sereedAupcpedendmPixasDsforrelmatoerdedteotatilhse);µrela=tiveNcomordiis-
discussthecorrectionsofrelativisticeffectslater. There- 0 i=1 i
the total mass of constituent particles inside the cluster,
fore, Eq. (10) can be approximated as
P
whichis equalto the restmass of the cluster if the bind-
N g N N ρWc Ni=1e−2pm2Ti,iT cosh(ηi−yi)d3xid3pi. ainrgetehneersgpyatoifalthanedclmusotmerenistunmegeleffcetcetdi;veVv′oalunmde(s2,πrµes0pTe)c23-
c ≈ c(cid:20)i=1 i(cid:21)R Q Ni=1V′Vp,i tively, for the center-of-mass motion of the cluster. To
Y
obtainEq.(16),followingRefs.[32–34],wehaveassumed
(11)
Q
η y , and this can be justified from the fact that the
i i
≈
Since the Wigner function does not depend on the coalescence probability is highly suppressed when the
center-of-mass coordinate of the cluster, a Jacobi trans- relative position or the relative momentum of the con-
formation [6, 7, 26, 31, 32] can be performed to sepa- stituentparticlesinsidetheclusterislarge. Oneseesthat
rate the center-of-masscoordinate and the relative coor- the integrationwith respect to momentum in Eq.(16) is
dinates of constituent particles of the cluster, i.e., three-dimensional, which is different from Refs. [32–34]
where the momentum integration is assumed to be two-
R x P p
1 1 dimensional in transverse plane.
q x k p
 1   2   1   2  The denominator in Eq. (16) can be re-expressed as
· =Jˆ · , · =(Jˆ−1)T · ,
 ·   ·   ·   ·  N N 3
 ·   ·   ·   ·  V′Vp,i = V′(2πTmi)2
 q   x   k   p 
 N−1   N   N−1   N  iY=1 iY=1
(12) N 1
3 3 − 3
where x (p ) is the spatial (momentum) coordinate of = N−2V′(2πµ0T)2 V′(2πµiT)2,(18)
j j
i=1
thej-thconstituentparticleandJˆisJacobimatrixwhose Y
property can be found in Appendix D. In particular,the where the factor N−23 comes from the coordinate trans-
center-of-mass position vector of the cluster R and the formation and will be canceled out at last. It should be
relativespatialcoordinatevectorsqi canbeexpressedas noted that to obtain Eq. (18), Eq. (D5) has been used.
We note thatthe contributionsofcenter-of-masscoor-
N m x
R = j=1 j j, (13) dinate inthe numeratoranddenominator(i.e., Eq.(18))
P Nj=1mj of Eq. (16) cancel out, and Eq. (16) can be written as
qi = rPi+i 1 Pij=ij=11mmjxjj −xi+1!. (14) Nc ≈ gc(cid:20)i=N1Ni(cid:21)RNρWc−32QNi=Ni−1=−11e1−V2k′µT2(i2,TiπdT3µqiid)332ki. (19)
Correspondingly,in the moPmentum space, P is the total Y
momentumoftheclusterandk aretherelativemomen- Furthermore, for simplicity, weQassume that the Wigner
i
tum vectors. It should be noted that the Jacobi matrix function of the cluster can be factorized into the
4
Wigner functions of each relative coordinate, i.e., ρW = C. Relativistic corrections
c
Ni=−11ρWc,i, then Eq. (19) can be recast into
In the derivation of Eqs. (24) and (25), a non-
Q 3 N N−1 ρWc,ie−2kµT2i,Ti d3qid3ki relativisticapproximationofthemomentumdistribution
Nc ≈ gcN2(cid:20)i=1Ni(cid:21) i=1 R V′(2πTµi)23 .(20) hrealsatbiveeisnticadeoffpetcetdc.ouHlodwdeevcerre,aistesthhoeuclodableescneontceedptrhoabtabthile-
Y Y
If we adopt the harmonic oscillator assumption for the ityasshowninthestudyfortheexoticΘ+ productionin
potential of constituent particles (see Appendix A), and relativisticheavy-ioncollisions[31]. Itisthus interesting
consideramixedensemblewithadefiniteorbitalangular and important to include this effect in the analytical co-
momentumstate(l)inthelowestenergystatewithn=l, alescence formula. We use grel to denote the relativistic
then the spatial integration of Wigner function can be correction factor.
obtained as Foracluster,itisreasonabletoassumethatthecenter-
of-massmotionshouldobeyrelativisticallycovariantone-
(2π)3 l 2
ρWd3q = ψ˜(k ,θ,φ) particle momentum distribution, while the relative mo-
c,i i 2l+1 i nlm tions of constituent particlesinside the cluster should be
Z m= l(cid:12) (cid:12)
X− (cid:12) (cid:12) approximately non-relativistic because the relative mo-
= (2π)3P(k ), (cid:12) (cid:12) (21)
i (cid:12) (cid:12) mentum and spatial coordinates have to be small, oth-
where (2π)3 comes from the normalization of Wigner erwise the coalescence probability of forming a cluster
function ρWd3q d3k =(2π~)3 with the reduced Plank is highly suppressed. This means that the relativistic
c,i i i
constant ~ taken to be unity, and the ensemble aver- correction of the Wigner function integration should be
R
aged probability distribution is represented by P(k ) = small, and thus the leading order relativistic correction
i
(4(π2σπi2)3)32 ((22σl+i2k1i2))!!le−σi2ki2 with σi2 = µi1w where w is the fre- sdhisoturilbdumtioanin.lyFrocmomEeqsfr.o(m5)athned(o6n)ea-pnadrktiecelpeinmgotmheenletaudm-
quency of the harmonic oscillator (see Appendix A) and
ingordercorrectioninthelatter,oneobtainstheeffective
ictlucsatnerb(esedeeEteqr.m(iDne1d3)byintAheppReMndSixraDd)iu.sA(frtermrst)hoefmthoe- volume in momentum space as Vp =(2πTm)23(1+ 185mT).
Therefore, the relativistic correction factor g can be
rel
mentum integration, the multiplicity of the cluster can
expressed as
be obtained as
1+ 15 T
Nc gcN23 N Ni N−1F(σi,µi,li,T), (22) grel ≈ N (1+8 µ150 T ). (26)
≈ i=1 8 mi
(cid:20)i=1 (cid:21) i=1
Y Y On one hand, it is eaQsy to prove that grel is always less
where F(σ,µ,l,T) is expressed as (see Appendix B)
than one and this leads to a general conclusion that the
F(σ,µ,l,T) relativistic effect tends to decrease the cluster yield. On
= V′(2w(T4)π12σ(21)+32 2wT)(cid:16)2wT2wT+1(cid:17)lG(cid:16)l,(2wT)12(cid:17), (23) ttchhoearrnoetcohtnieoern,hcgaarnendl,tahopenpnerobcaaecnhnseeesgelettochtaeutdn.withyenanmdTitihsemrueclahtilvairsgteicr
with the function G(l,x) defined asa ratioof twohyper-
geometric function (see Appendix C).
D. Cluster finite size effects
Finally. we obtainthe following analyticalcoalescence
formula, i.e.,
In evaluating the spatial part of the integration in
N
N g µ23 Ni Eq. (20), we have assumed that the radius R of the fire-
c ≈ c 0 3 × ball is much largerthan the size of the produced cluster.
(cid:20)Yi=1mi2(cid:21) However,forsome loosely boundclusters suchas 3H, its
Λ
NiY=−11V′(2wT()4w21π()123+ 2wT)(cid:16)2wT2wT+1(cid:17)liG(cid:16)li,(2wT)12(cid:17). irvnramlRuseecfoa.fn[3in9bt]eetgharsaattlaitorhnge.elIaanrsgt4eh.i9vsafwlmuoer[ko3,f8w]r.rRemIsdtewhniaollstedbetechereneacssoherortwehcne-
(24)
tion factor ofthis cluster finite size effect by g . When
size
SinceV′ =VyL,wecanexpressthemultiplicity perunit rrRms approaches to 0, gsize should approach to 1.
rapidity as The cluster size correction factor gsize cannot be
analytically obtained, but it can be calculated by a
dNc g µ23 N ddNyii multi-dimensional numerical integration [15]. Based on
dy ≈ c 0 3 × the multi-dimensional numerical integration, we pro-
(cid:20)iY=1 mi2 (cid:21) pose the following empirical expression for g for N-
size
NYi=−11V(2wT()412wπ(1)32+ 2wT)(cid:16)2wT2wT+1(cid:17)liG(cid:16)li,(2wT)12(cid:17). partgicle((x2,N≤)N ≤1−6()2c+oa0le.6sc(Nenc−e,1i).0e..5,)x1.3e−2x2, (27)
(25) size ≈ 1+(N 1)x2
−
5
where x = rrms is assumed to be in range [0, 0.5]. For For x 1, one can show that (l,x) is very close to 1
R ≫ S
the same x, we note g is larger for 3-particle coales- for all l. Taking l =1 as an illustration, one has
size
cence than for 2-particle coalescence, but the difference
is relatively small compared with their own values. This (1,x)= x2 G(1,x)= x2+ 31 1, (32)
empirical expression can be applied to central heavy-ion S 1+x2 x2+1 ∼
collisions at both RHIC and LHC energies. For coales-
for x 1. This feature is consistent with the results
cence of constituent particles with different masses, we ≫
in Ref. [40] where the orbital angular momentum sup-
noticethatthemassdifferencetendstodecreasethevalue
pression factor is shown to approach to unity in high
of g , but the correction is not significant.
size temperature limit (large T). This is a little bit different
fromthe resultofthe COAL-Exformula (Eq.(29)) from
which one can see the value of the suppression factor is
III. DISCUSSIONS
2 for l=1 at the limit of x 1.
3 ≫
Combining the relativistic correction factor g and
rel
the cluster size correction factor gsize with Eq. (25), we B. Saturation with the cluster size
obtain the following full analytical coalescence formula
for the cluster multiplicity per unity rapidity, i.e.,
According to Eq. (D13) in Appendix D, the value of
ddNyc ≈ grelgsizegcµ023(cid:20)iY=N1 mddNyi32ii(cid:21)× tohofart2wmhTeo≫rnr2imc1sofvoscarilllulaaertgooefrctflhrueesqtpuereronsdciuzyecwe(dbiucstluintshvteeerrrs.remUlysnopdfretorhpteohcretluiloismnteaitrl
isstillassumedtobesmallerthanthefireballradiusR),
NYi=−11V(2wT()412wπ(1)32+ 2wT)(cid:16)2wT2wT+1(cid:17)liG(cid:16)li,(2wT)12(cid:17). tthoebCeOAL-SHformulaEq.(28) canbe further simplified
(28) dNc g g g µ32 N ddNyii N−1 (4π)32 . (33)
Wisoend,etnhoeteCtOhAeLab-Eoxvefoforrmmuulala(ai.se.C,OEAq.L(-S18H).oFforRceof.m[p3a4r])- dy ≈ rel size c 0(cid:20)iY=1 mi32 (cid:21) iY=1 V(2T)23
reads Theaboveexpressionindicatesthatthe dNc willbesatu-
dy
ratedto a constantatthe limitoflarge(small)r (w).
ddNyc ≈ gcµ023(cid:20)iY=N1 mddNyi23ii(cid:21)× fIanctpoarrtgiscizuelaarp,pfrooraRche≫s torrumns,itythaencdluEsqte.r(3s3iz)ebceocrromrmescetsion
NiY=−11V((14wπ+)232wT)(cid:16)2wT2wT+1(cid:17)li(2(li2+li)1!!)!!. (29) ddNyc ≈ grelgcµ032(cid:20)iY=N1 mddNyi23ii(cid:21)NiY=−11V((42πT))3223 (34)
One can see that compared with the formula COAL-
whichmeansthatthe sizeofthe producedclusterhasno
Ex, besides the additional factors g and g , the new
rel size influence on its yields in the limits 2T w and R
formulaCOAL-SHdisplaysverysimilarstructurebutdif- ≫ ≫
r .
rms
ferent dependence on the w parameter and the orbital For the COAL-Ex formula, in the limit of 2T 1
angularmomentumquantumnumbers,whichwillbedis- w ≫
for large cluster size, the dNc does not saturate with the
cussed in the following. dy
increment (decrease) of r (w). For example, in the
rms
case of two-particle coalescence, the dNc is proportional
dy
A. Suppression from orbital angular momentum to r [33, 34]. As we will see in the following, this
rms
difference can lead to quite different predictions for the
yield ratio 3H/3He between the COAL-Ex and COAL-
In the COAL-SH formula (Eq. (28)), the suppression Λ
SH formulae since the hypertriton 3H is a very loosely
factorfromnon-zeroorbitalangularmomentumquantum Λ
number l is bound state with a huge radius of about 4.9 fm [38].
x2 l
(l,x)= G(l,x), (30)
S 1+x2 IV. APPLICATIONS
(cid:16) (cid:17)
with x = (2wT)12. From the property of G(l,x) (see Ap- In this section, we first apply COAL-SH and COAL-
pendix C), one can easily show
Ex to investigate the strangeness population factor
S =3H/(3He Λ/p), and then to study the production
(l,x)<1, (31) 3 Λ ×
S of some exotic hadrons, in central Pb+Pb collisions at
indicatingthesuppressionfeatureduetothefiniteorbital √s = 2.76 TeV. The results from thermal (statisti-
NN
angular momentum quantum number l. cal) model are also included for comparison.
6
A. Strangeness population factor S3 ggssiizzee((33ΛHHe)) = 0.60, which leads to S3 = 0.54 ∼ 0.57,
consistent with the measured value 0.60 0.13(stat.)
± ±
The strangeness population factor S3 was first sug- 0.21(syst.). To fit the measured central value 0.60, one
gested in Ref. [41] and it is expected to be a good repre- can introduce a multi-freeze-out between nucleon and Λ
sentation of the local correlation between baryon num- with an earlier Λ freeze-out [16].
ber and strangeness [13, 42]. Therefore, it may pro- For the formula COAL-Ex, S can be obtained as
3
vide valuable information of deconfinement in relativis-
tic heavy-ion collisions. The S3 was measured to be S = ((mp+mn+mΛ)mp)32 (w3He+2T)2w3He. (37)
t0r.a6l0it±y)0P.1b3+(sPtabt.c)o±llis0io.2n1s(saytst√.)sfor c=en2t.r7a6l T(0e-V10[%43]c.enIt- 3 ((mp+mn+mp)mΛ)32 (w3ΛH+2T)2 w3ΛH
NN
should be noted that while there is negligible feed-down One can see that compared with the expression (35)
fromheavierstatesinto3ΛHand3He,theΛandparesig- based on the COAL-SH formula, the expression (37)
nificantlyinfluencedbyfeed-downfromdecaysofexcited from the COAL-Ex formula does not have correction
baryonic states. When we calculate S3 within the coa- factors grel and gsize but includes an additional factor
lescence model, the contributions to the yields of Λ and w /w = 8.1. As a result, S based on COAL-
3He 3H 3
p from electromagnetic and weak decays are excluded. Ex is laΛrger than about 7 and thus the COAL-Ex for-
Within the formulaCOAL-SH,S3 canbe expressedas mula significantly overestimates the observed S3 fac-
tor value around 0.6 for central Pb+Pb collisions at
S3 = ((((mmpp++mmnn++mmpΛ))mmΛp))2233 ((ww33ΛHHe++22TT))22 √sNN =2.76 TeV.
g (3H) g (3H)
rel Λ size Λ (35) B. Production of exotic hadrons from quark
×g (3He)g (3He)
rel size coalescence
(w +2T)2 g (3H) g (3H)
= 0.845 3He rel Λ size Λ ,
(w3H+2T)2 grel(3He)gsize(3He) In the following, we focus on the production of ex-
Λ
otic hadrons from quark coalescence in central Pb+Pb
where w3He = 0.013 GeV and w3ΛH = 1.6×10−3 GeV collisions at √sNN = 2.76 TeV. Three kinds of exotic
are the corresponding harmonic oscillator frequencies of hadrons with four quark flavors (i.e., u, d, s, and c) are
3He and 3H, respectively, and their values are obtained considered, namely, the exotic mesons that could be in
Λ
from the rrms of 3He (1.76 fm [44]) and 3ΛH (4.9 fm [38]) four-quark states, the exotic baryons that could be in
through Eq. (D13). The relativistic correction of S3 due five-quark states, and the exotic dibaryons that could
to grel (Eq. (26)) can be simply neglected since one has be in six-quark or eight states. The detailed properties,
grel(3He) ≈ grel(3ΛH). Because 3ΛH has a much larger including mass, isospin, spin, parity, orbital angular mo-
rrms than 3He, one has to take account of the cluster mentum quantum number, quark configuration and de-
size correction factor gsize. By neglecting the relativistic cay modes, of these exotic hadrons can be found in Ta-
correction, S3 can be expressed as ble IV of Ref. [34]. For the calculations with the COAL-
SH formula, the cluster size correction factor g is ne-
(0.013+2T)2 g (3H) size
S 0.845 size Λ glectedsince the RMS radiiof the exotic hadrons we are
3 ≈ (0.0016+2T)2gsize(3He) considering here are small (less than about 2 fm) based
0.013 g (3H) on the harmonic oscillatorfrequencies determined in the
0.845(1+ )2 size Λ . (36)
≈ 2T g (3He) following.
size
Forthequarkcoalescencemodelcalculations,following
Now, the difference for the prediction of S between the Refs. [33, 34], the masses of u(d), s and c constituent
3
thermalmodelandthecoalescencemodelbecomesclear: quarks are taken to be m = 0.3 GeV, m = 0.5 GeV
u,d s
thefactor0.845inEq.(36)correspondstotheprediction and m = 1.5 GeV. The temperature is set to be T =
c
ofthermalmodelatLHC energy[45], andthe remaining 0.154 GeV [46, 47], and we note that a small (e.g., 20%)
partsinEq.(36)thuscorrespondtothe correctionsfrom variance of the temperature value does not change our
coalescence model. conclusion. In order to apply the formulae COAL-SH
In general, the freeze-out temperature T lies between and COAL-Ex, one also needs the information on the
0.1 to 0.2 GeV in Pb+Pb collisions that we are con- harmonic oscillatorfrequencies (w for hadrons with only
sidering here, then 0.845(1+ 0.013)2 is in the range of u(d) quarks, w for hadrons with strange hadrons, and
2T s
0.9 0.95, and one can see the temperature depen- w for charmed hadrons), the fireball volume (V), the
c
∼
dence is quite weak for this part. Compared with the number of constituent quarks(N for u(d) quarks, N
u(d) s
thermal model, the largest correction in the coalescence for s quarks and N for c quarks).
c
model comes from the size effect due to different sizes For COAL-SH, the values of frequencies w, w and w
s c
of 3H and 3He. The radius R of the fireball at freeze- canbe deducedfromthe hadronsize throughEq.(D13).
Λ
out in central Pb+Pb collisions at √s =2.76 TeV is In particular, the values of w and w have been deter-
NN s
about 19.7 fm [16], and according to Eq. (27), one has mined to be w = 0.184 GeV and w = 0.078 GeV in
s
7
Ref. [17]. For the value of w , we obtain w = 0.087 uesfortheformulaCOAL-Ex,wethenobtainthefireball
c c
GeV by using the value 0.546 fm for the RMS radius of volume V =6.06 103 fm3, the constituent quark num-
×
Ω [48]. With w = 0.184 GeV, w = 0.078 GeV and bers N =3 267, N =3 156 and N =3 20.
ccc s u(d) s c
× × ×
wc = 0.087 GeV, we obtain reasonable RMS radius val- For the thermal model (see Refs. [33, 34] for details),
ues for various hadrons,namely, 0.84 fm for protons (p), the baryonand strange chemical potentials are set to be
0.87fmforφmesons,1.1fmforΞ−,1.0fmforΩ−,1.2fm zero. And by fitting the measured yields of p, Λ, φ, Ξ−,
forΛ,and0.98fmforD0. Finally,theparametersN , Ω and D0, we obtain the temperature T =0.153 GeV,
u(d) −
Ns and Nc can be obtained from fitting the measured the fireball volume V = 6.45 103 fm3, and the charm
×
yields (dN/dy at midrapidity) of p [49], Λ [50], φ [51], fugacity γ =61.6.
c
Ξ [52], Ω [52], and D0 [53].
− −
TABLEII:dN/dy at midrapidityof p,Λ, φ,Ξ−, Ω− and D0
TABLEI:ModelparametersofCOAL-SH,COAL-Exandthe incentralPb+Pbcollisions at√s=2.76TeVpredictedfrom
thermal model, i.e., freeze-out temperature T (GeV), fire- COAL-SH, COAL-Ex and the thermal model. The experi-
ball volume V (103 fm3), harmonic oscillator frequency w mental data and thecoalescence factor gc are also included.
(GeV), ws (GeV) and wc (GeV), baryon chemical potential
µB, strangeness chemical potential µS, and charm fugacity Hadron p φ Ξ− Ω− Λ D0
γc, for central Pb+Pb collisions at √s=2.76 TeV. gc 33×223 32×322 33×223 33×423 33×223 32×122
Exp. 34 13.8 3.34 0.58 26 8.4
T V Nu/3 Ns/3 Nc/3 w ws wc
3 1.77 0.25 0.1 3 -
COAL-SH 0.154 13.8 462 196 27 0.184 0.078 0.087 ± ± ± ± ±
COAL-SH 34.4 13.8 3.23 0.64 26.3 8.4
COAL-Ex 0.154 6.06 267 156 20 0.55 0.519 0.385
COAL-Ex 36.2 13.8 3.16 0.71 22.5 8.4
T V µB µS γc
Thermal 31.1 16.7 3.48 0.60 19.3 8.4
Thermal 0.153 6.45 0 0 61.6
Table I summarizes the model parameters for COAL-
For the yields of p, Λ, φ, Ξ , Ω and D0 in cen-
− −
SH, COAL-Ex and the thermal model. Table II displays
tral Pb+Pb collisions at √s = 2.76 TeV, it should
NN
the predictions fromCOAL-SH,COAL-Exandthe ther-
be noted that the weak decays have already been cor-
mal model, for the yields of p, Λ, φ, Ξ , Ω and D0
rected in the experimental data, but not for the strong − −
in central Pb+Pb collisions at √s = 2.76 TeV. Also in-
decays and electromagnetic decays. In order to compare
cluded in Table II are the corresponding experimental
with experiment results, we have to include the contri-
dataandthe coalescencefactorsg due to spinandcolor
butionsfromstrongandelectromagneticdecaysinquark c
degrees of freedom. From Table II, one can see COAL-
coalescence model. Following Refs. [17, 33, 34], we as-
sume the relations Nmeasured = N + 1N + SH gives a best fit, COAL-Ex also gives a nice fit, while
Λ(1115) Λ(1115) 3 Σ(1192) the thermal model does not fit the Λ yield well. We
(0.87 + 0.11)N = 7.44N , Nmeasured =
3 Σ(1385) Λ(1115) p note that COAL-SH can give a similar good description
Np + N∆++(1232) + 12N∆+(1232) + 21N∆0(1232) = 5Np, forthe yields ofp,Λ, φ,Ξ− andΩ− asthe morerealistic
NΞm−easured =NΞ−+ 21NΞ(1530) =3NΞ−, and NDm0easured = multi-dimensionalnumericalintegrationmethod,andthe
ND0 + ND∗0 + 0.677ND+∗ = 6.0ND0. For φ and Ω−, latter has been shown to describe very well the spectra
we assume no strong and electromagnetic decay correc- and yields of these hadrons [17]. This implies that the
tions, and thus Nφmeasured = Nφ and NΩm−easured = NΩ−. COAL-SH formula can be served as a very useful tool
Here NΛ(1115), Np, NΞ−, Nφ, NΩ− and ND0 represent to evaluate the yield of clusters produced in relativistic
the corresponding hadron multiplicity obtained directly heavy-ion collisions.
from the quark coalescence model. Shown in Tab. III are the predicted exotic hadron
For COAL-SH, by fitting the measured yields of p, Λ, yieldsfromCOAL-SH,COAL-Exandthethermalmodel.
φ, Ξ , Ω and D0, we obtain the fireball volume V = The properties of these exotic hadrons can be found in
− −
1.38 104 fm3, the constituent quark numbers N = Table IV of Refs. [34]. In Tab. III, most of the predicted
u(d)
×
3 462, N =3 196 and N =3 27. yields from COAL-SH and COAL-Ex are very close and
s c
× × ×
For the formula COAL-Ex, we find it cannot reason- the differences are within about factor two. However,
ably fit the measured yields of p, Λ, φ, Ξ , Ω and there arealsosome casesthatthe difference betweenthe
− −
D0 if the frequency parameters w, w and w are taken COAL-SH and COAL-Ex predictions is larger than fac-
s c
to have the same values as those in the COAL-SH for- tor two. This is mainly due to different suppressions of
mula. In Refs. [33, 34], for central Au+Au collisions non-zero orbital angular momentum states of the exotic
at √s = 200 GeV and central Pb+Pb collisions at hadrons within these two formulae. Compared with the
NN
√sNN = 5 TeV, these frequencies were assumed to be predictions from thermal model, most of the results of
free parameters and determined to be w = 0.55 GeV, COAL-SHaremuchsmallerbyaboutoneorder,andeven
w =0.519GeVandw =0.385GeVbyreproducingthe smaller by about two orders for some cases. However,
s c
yieldsofw(782),ρ(770),Λ(1115)andΛ (2286)predicted when the masses of the exotic hadrons, such as a (980),
c 0
from the thermal model. If we take these frequency val- X(3872), Z+(4430) and Λ(1450), are much larger than
8
the total mass of their constituent quarks, their yields (i.e., qqqsq¯). It is interesting to see that the predicted
from COAL-SH are larger than those from the thermal yields of these hadrons in exotic four(five)-quark states
model. are smaller by about two orders than those in normal
In particular, it is suggested that the mesons f (980), two(three)-quark states, indicating that measuring the
0
a (980), D (2371) and X(3872) could be either in nor- yields of these hadrons in central Pb+Pb collisions at
0 s
maltwo-quarkstatesorinexoticfour-quarkstates. Simi- √s = 2.76 TeV will be potentially useful to identify the
larly,thebaryonΛ(1405)couldbeeitherinnormalthree- internal quark configuration of these hadrons.
quarkstate(i.e.,uds(L=1))orinexoticfive-quarkstate
TABLE III: dN/dy at midrapidity for some exotic hadrons with various quark configurations in central Pb+Pb collisions at
√s=2.76 TeV predicted from COAL-SH,COAL-Exand thethermal model.
Models COAL-SH COAL-Ex COAL-SH COAL-Ex Thermal
Quark configuration 2q/3q/6q 2q/3q/6q 4q/5q/8q 4q/5q/8q -
Mesons:
f0(980) 35.5,3.98(ss¯) 7.24,1.14(ss¯) 5.90 10−2 5.54 10−2 6.85
× ×
a0(980) 72.4 19.6 8.54 10−2 13.7 10−2 20.6
× ×
K(1460) - - 4.03 10−1 3.35 10−1 10.0 10−1
× × ×
Ds(2371) 3.38 10−1 1.31 10−1 3.76 10−3 7.52 10−3 2.16 10−1
× × × × ×
Tcc - - 6.28 10−4 9.04 10−4 5.17 10−3
× × ×
X(3872) 5.37 10−2 6.34 10−3 6.28 10−4 9.04 10−4 3.26 10−3
× × × × ×
Z+(4430) - - 5.35 10−4 2.68 10−4 1.01 10−4
× × ×
Baryons:
Λ(1405) 3.06 0.75 4.35 10−2 3.46 10−2 1.36
× ×
Θ+ - - 3.76 10−2 0.86 10−2 6.75 10−1
× × ×
K¯KN - - 2.14 10−2 0.55 10−2 1.44 10−1
× × ×
D¯N - - 2.64 10−3 6.28 10−3 2.56 10−2
× × ×
D¯∗N - - 3.86 10−3 1.32 10−3 2.36 10−2
× × ×
Θcs - - 1.22 10−3 1.12 10−3 1.63 10−2
× × ×
Dibaryons:
H 6.30 10−4 5.32 10−4 - - 5.37 10−3
× × ×
K¯NN 3.91 10−3 8.46 10−3 4.49 10−5 3.01 10−5 5.71 10−3
× × × × ×
ΩΩ 3.49 10−6 4.56 10−6 - - 1.47 10−5
× × ×
Hc++ 1.00 10−4 3.52 10−4 - - 1.10 10−3
× × ×
D¯NN - - 2.12 10−6 1.18 10−6 8.24 10−5
× × ×
Furthermore, the exotic dibaryon K¯NN could be ei- ful to determine the quark configuration of these exotic
ther in six-quark state (i.e., qqqqqs(L = 1)) or in eight- hadrons. It should be noted that these exotic hadrons
quark state (i.e., qqqqqqsq¯) with the latter (eight-quark couldbe inmolecularstatesandthe formulaeCOAL-SH
state) having a yield smaller than that of the former and COAL-Ex can also be used to evaluate the molecu-
(six-quark state) by two orders. In particular, based on lar state yield if the freeze-out information is known for
the COAL-SH calculations, the yield (dN/dy at midra- nucleons, Ξ, Ω, Ξ , K, K¯, D, D¯, D , D¯ and D , and
c ∗ ∗ 1
pidity) of the exotic dibaryon K¯NN is 3.91 10 3 for this is beyond the scope of the presentwork and may be
−
six-quark state and 4.49 10 5 for eight-qu×ark state. pursued in future.
−
×
These yields are measurable in central Pb+Pb colli-
sions at √s = 2.76 TeV at LHC via the strong decay
K¯NN ΛN. V. CONCLUSION
→
The above results based on COAL-SH and COAL-Ex
demonstratethattheyieldsofvariousexotichadronssen- Based on a blast-wave-like parametrization for phase-
sitively depend on the quark configurationand therefore space configuration of constituent particles at freeze-out
theyieldmeasurementofthesehadronsispotentiallyuse- in relativistic heavy-ion collisions, we have derived a
9
new approximate analytical formula COAL-SH for clus- Acknowledgments
ter yield within the covariant coalescence model. The
new analytical coalescence formula COAL-SH improves We are grateful to Che Ming Ko for helpful discus-
some aspects of the existing formulae by treating the in- sions. This work was supported in part by the Major
tegration of Wigner function in the many-body coales- State Basic Research Development Program (973 Pro-
cence process in full phase-space, and thus has a good gram) in China under Contract Nos. 2015CB856904
saturation property when the rrms of the produced clus- and 2013CB834405,the National Natural Science Foun-
ters increases. The corrections of relativistic effects and dation of China under Grant Nos. 11625521, 11275125
the finite size effects of the produced cluster relative to and 11135011,the Program for Professor of Special Ap-
the emission source are also consideredin the COAL-SH pointment (Eastern Scholar) at Shanghai Institutions of
formula. Higher Learning, Key Laboratory for Particle Physics,
We have applied COAL-SH to investigate the Astrophysics and Cosmology, Ministry of Education,
strangeness population factor S3 =3ΛH/(3He×Λ/p) from China, and the Science and Technology Commission of
nucleon/Λcoalescenceandthe exotic hadronproduction Shanghai Municipality (11DZ2260700).
from quark coalescence in central Pb+Pb collisions at
√s =2.76TeV.Theresultshavebeencomparedwith
NN
the prediction from the analytical coalescence formula Appendix A: Quantum 3-dimension isotropic
COAL-Ex derived by the ExHIC collaboration and the harmonic oscillators
thermal model. Our results indicate that the COAL-SH
formula can reasonably describe the measured S3 factor We consider a 3-dimension isotropic quantum har-
while the COAL-Exformulapredicts atoolargevalueof monic oscillator with the potential of V(r) = 1µω2r2,
2
the S3 factor. It is also indicated that the cluster finite where µ (ω) is the mass (frequency) of the oscillator.
size effect is important for the S3 evaluation due to the ThewavefunctioncanbeobtainedbysolvingScho¨dinger
large size of the hypernucleus 3H. equation and the solution reads
Λ
For the exotic hadron production in central Pb+Pb
cmoolldiseilonpsaraatm√esteNrNs b=y2fi.7tt6inTgeVth,ewyeiehladvseodfepte,rΛm,inφe,dΞthe, ψ(r,θ,φ)nlm =Nslrle−2rσ22Lsl+21 σr22 Ylm(θ,φ), (A1)
− (cid:18) (cid:19)
Ω and D0. We have found that COAL-SH can nicely
−
fit the experiment data of these yields but COAL-Ex wheren=2s+listheprincipalquantumnumberrelated
cannot reasonably fit the data if the harmonic oscillator totheenergyE =~ω(n+3), l istheorbitalangularmo-
2
frequencies are determined from the RMS radii of nor- mentum quantum number, m = l,...,l is the magnetic
−
mal hadrons. By adjusting the values of the harmonic quantum number, and s is a non-negative integer rep-
oscillator frequencies, the COAL-Ex can reasonably fit resenting radial excitation. N = 12s+l+2s!σ−2l−3
the measured yields. The thermal model can also fit the sl π (2s+2l+1)!!
r
measured yields except that the Λ yield cannot be well is a normalized constant with σ2 = q1 , and Ll+21(x)
reproduced. µw s
is the generalized Laguerre polynomials with order of s.
For the yields of exotic hadrons, most of the pre-
Y (θ,φ)isthesphericalharmonicfunctionwhichisnor-
lm
dictions from COAL-SH and COAL-Ex (with adjusted malized as Y 2dΩ = 1. The RMS radius r can
lm rms
harmonic oscillator frequencies) are very close to each | |
then be obtained via the following expression
otherwithinaboutfactortwo. COAL-SHandCOAL-Ex R
may give significantly different predictions for the exotic 3+2(l+2s)
r2 = ψ 2r2d3r = σ2. (A2)
hadrons with non-zero angular momentum states as the rms | | 2
two formulae have different suppressions of non-zero an- Z
gular momentum states. Compared with the predictions The wave function in momentum (k) representation
fromthermalmodel,mostoftheresultsofCOAL-SHare reads
aboutoneordersmallerorevenmoreexceptfora (980),
X(3872), Z+(4430) and Λ(1450), for which the h0adron ψ˜(k,θ,φ)nlm = Nslklσ2l+3e−k22σ2Lkl+21(k2σ2)Ylm(θ,φ).
masses are much larger than the total mass of their con-
(A3)
stituent quarks.
The new formula COAL-SH can be served as a useful Foranensemble havingthe lowestenergywith agivenl,
tooltostudyparticleproductioninrelativisticheavy-ion theaveragedprobabilitydistributioninmomentumspace
collisions. It should be pointed out that for the COAL- can be obtained as
SH formula, we have assumed the size of the emission
l
source for constituent particles at freeze-out should be 1
P(k) = ψ˜(k,θ,φ) 2
largerthanthatoftheproducedcluster,andthusitcan- 2l+1 | llm|
not be simply applied to some large-size cluster produc- mX=−l
tteiornestiningsmisaslulecowlillilsiboenesxypstloermedliiknetphpecnoelalirsifountu.rTe.his in- = (4(π2σπ2)3)23 ((22lσ+2k12))!l!e−σ2k2, (A4)
10
and P(k) is normalized as P(k)d3k = 1. To obtain From the series expansion, the following properties of
Eq. (A4), the following Unso¨ld’s theorem has been used G(l,x) can be easily proved, i.e.,
R
l G(l,x) > 1, (C3)
1 1
Y (θ,φ)2 = . (A5)
lm l
2l+1 | | 4π l! 1 1
m= l G(l,x) < =(1+ )l. (C4)
X− k!(l k)!x2k x2
k=0 −
X
Appendix B: Function F(σ,µ,l,T) For x2 1, G(l,x) is thus very close to unity.
≫
The function F(σ,µ,l,T) is defined as
Appendix D: Jacobi transformation and the RMS
(2π)3P(k)e−2kµT2Td3k radius of clusters
F(σ,µ,l,T) = , (B1)
R V′(2πµT)23 The Jacobi matrix Jˆ for the coordinate transforma-
tion,definedinEqs.(12),(13)and(14),hassomespecial
where k = ksinθ is the transverse momentum. With
T properties. The Jacobi matrix Jˆ is introduced to sepa-
Eq. (A4), the k-dependent part can be integratedout as
rate the center-of-mass and the relative coordinates of a
follows
many-body system since the Wigner function does not
(2π)3P(k)e−k22sµinT2θ2πk2sinθdkdθ depend on the center-of-mass coordinate and it can be
F(σ,µ,l,T)=
R V′(2πµT)23 expCroenssseidderinintgerNmsinodfetpheendreelnatticvoencsotiotrudeinntatpeasr.ticles with
= (2l(+4π1σ)!2!)V32′((22σπ2µ)Tl )23 Z k2le−k2(σ2+s2inµ2Tθ)2πk2sinθdkdθ mosacsilslamtoir(siw=ith1,s2a,m3,e..f.r,eNqu)einncy(3wN,-tdhiemteontsailopno)thenatrimaloennic-
(4πσ2)32(2σ2)l2π1Γ(l+1+ 1) π sinθdθ ergy of the system can be expressed as
= 2 2
(2l+1)!!V′(2πµT)3/2 Z0 (σ2+ s2inµ2Tθ)l+23 V = 1 N (m w2r2)= w2rTMˆr
(4πσ2)32σ2l π2 sinθdθ −2 i=1 i i − 2
= X
V′(2µT)3/2 Z0 (σ2+ s2inµ2Tθ)l+23 = w2r′T(J−1)TMˆJ−1r′ = w2r′TMˆ′r′,(D1)
− 2 − 2
(4πσ2)23(2µTσ2)l π2 sinθdθ
= , (B2) where Mˆ is the mass matrix which is diagonal and r is
V′ Z0 (2µTσ2+sin2θ)l+32 the new coordinate vector. The elements of transform′a-
where the Γ-function identity Γ(n + 1) = √π(2n)! has tion matrix Jˆ−1 is
2 4nn!
been used. With the function G(l,x) (see Appendix C), Jˆi−,11 = 1 (i=1,...,N)
F can finally be obtained as
AFp(pσe,µnd,li,xTC):=Or×Vbi(cid:16)′t(a22lµµ2TaTµnσσT(g224σu)+π2l21aσ(1r21(cid:17))m+32lGo2m(cid:16)µlTe,n(σ2t2µu)Tmσf2a)c12t(cid:17)o(.rB3) TheJJJˆˆˆniii−−−,,,eikk111w===mass−mP(1akjt−m=ri1kxPmMˆijjm=r′01iismkaj−lks)or1d(iiak−gi=o1n2a,(l.i(a.o.n=,tdNh2et;,rhk.w.e.>i,rsNeedi))).(uDce2d)
mass can be expressed as
The orbitalangularmomentum factor G(l,x) is anin- µ =Mˆ = (Jˆ 1)TMˆ (Jˆ 1)
tegration which can be done analytically as follows i−1 i′i − ij jk − ki
j k
XX
π
G(l,x) = x(1+x2)l+1 2 sinθdθ = mk(Jˆ−1)ki(Jˆ−1)ki , (D3)
Z0 (sin2θ+x2)l+32 Xk
F [1,l+ 3,3, 1 ] from which one can see that µ is the total mass and µ
= 2 1 2 2 2 1+x2 , (C1) 0 i
F [1,3,3, 1 ] is the reduced mass related to the constituent particles
2 1 2 2 2 1+x2 in the corresponding relative coordinates, i.e.,
where F is the hypergeometric function. G(l,x) can
2 1 µ = m (Jˆ 1) (Jˆ 1) = m ,
also be expanded as 0 k − k1 − k1 k
k k
G(l,x)= l k!(ll! k)!(2k+11)x2k. (C2) µi = i+i 1miX+1i+1ik=m1mk. (1≤i≤NX−1).(D4)
Xk=0 − kP=1 k
 P