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PHYSICAL REVIEW C, VOLUME 61, 024904Hadronic scattering of charmed mesons
Ziwei Lin, C. M. Ko, and Bin ZhangCyclotron Institute and Physics Department, Texas A&M University, College Station, Texas 77843
~Received 4 May 1999; published 4 January 2000!
The scattering cross sections of charm mesons with hadrons such as the pion, rho meson, and nucleon arestudied in an effective Lagrangian. In heavy ion collisions, rescattering of produced charm mesons by hadronsaffects the invariant mass spectra of both charm meson pairs and dileptons resulting from their decays. Theseeffects are estimated for heavy ion collisions at Super Proton Synchrotron energies and are found to besignificant.
PACS number~s!: 25.75.2q, 13.75.Lb, 14.40.LbN
uc
transabel,haatn
neb
ioundgas
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c
onsntsematcts
morum. IIIavylts
entheI. INTRODUCTION
Recently, experiments on heavy ion collisions at CERSuper Proton Synchrotron~SPS! by the HELIOS3@1# andNA50 @2# Collaborations have shown an enhanced prodtion of dileptons of intermediate masses (1.5,M,2.5GeV!. In one explanation, this enhancement is attributeddilepton production from secondary mesonmeson intetions @3#, while in another it was proposed that dileptofrom charm meson decays could also contribute appreciin this invariant mass region@4#. In the latter case, one of thpresent authors has shown, based on a schematic modeif one assumes that the transverse mass spectra of cmesons become hardened as a result of final state rescing with hadrons, the invariant mass spectrum of dimuofrom decays of charm mesons would also become hardeand more dimuons would then have an invariant masstween 1.5 and 2.5 GeV. Although charm quark productfrom hadronic interactions has been extensively studieding perturbative QCD@5,6#, not much has been done istudying charm meson interactions with hadrons. Knowleof charm meson interactions with hadrons is important,whether charm mesons develop a transverse flow dependhow strongly they interact with other hadrons as they progate through the matter.
In this paper, we shall first introduce in Sec. II an effe05562813/2000/61~2!/024904~7!/$15.00 61 0249
oc
ly
thatrmtersd,
ens
eson

tive Lagrangian to describe the interactions of charm meswith pion, rho, and nucleon. Using the coupling constaand cutoff parameters at the vertices determined eitherpirically or from symmetry arguments, we evaluate the sctering cross section of charm mesons with hadrons. Effeof hadronic scattering on the charm meson transversementum spectrum and the dimuon invariant mass spectfrom charm meson decays are then estimated in Secbased on a schematic model for the time evolution of heion collision dynamics. In Sec. IV, we summarize our resuand discuss the uncertainties involved in the studies.
II. CHARM MESON INTERACTIONS WITH HADRONS
A. Lagrangian
We consider the scattering of charm mesons (D1, D2,D0, D0, D* 1, D* 2, D* 0, and D* 0) with pion, rho, andnucleon. If SU~4! symmetry were exact, interactions betwepseudoscalar and vector mesons could be described byLagrangian
LPPV5 igTr~PVm]mP!1H.c., ~1!
whereP andV represent, respectively, the 434 pseudoscalarand vector meson matricesP51
A2 1p0
A21
h
A61
hcA12
p1 K1 D0
p2 2p0
A21
h
A61
hcA12
K0 D2
K2 K0 2A23
h1hcA12
Ds2
D0 D1 Ds1
23hcA12
2 ,
2000 The American Physical Society041
ZIWEI LIN, C. M. KO, AND BIN ZHANG PHYSICAL REVIEW C 61 024904V51
A2 1r0
A21
v
A61
J/c
A12r1 K* 1 D* 0
r2 2r0
A21
v
A61
J/c
A12K* 0 D* 2
K* 2 K* 0 2A23
v1J/c
A12Ds*
2
D* 0 D* 1 Ds*1
23J/c
A12
2 .
er
eatosoxirmaeliin
nstr
ted
in
goreond
no
son
nd
ted,The above interaction Lagrangian may be consideredbeing motivated by the hidden gauge theory, in which thare no fourpoint vertices that involve two pseudoscalar msons and two vector mesons. This is in contrast to theproach of using the minimal substitution to introduce vecmesons as gauge particles, where such fourpoint verticeappear. It is, however, known that the two methods are csistent if one also includes in the latter approach the avector mesons, which are unfortunately not known for chahadrons. Furthermore, gauge invariance in the latterproach cannot be consistently maintained if one uses theperimental vector meson masses, empirical meson coupconstants and form factors at interacting vertices. Expandthe Lagrangian in Eq.~1! in terms of the meson fields explicitly, we obtain the following Lagrangians for mesonmeson interactions:
LpDD* 5 igpDD* D* mtW@D~]mpW !2~]mD !pW #1H.c.,
LrDD5 igrDD@DtW~]mD !2~]mD !tWD#rW m,
Lrpp5grpprW m~pW 3]mpW !, ~2!
where the coupling constantsgpDD* , grDD , and grpp arerelated to the coupling constantg via the SU~4! symmetry asshown below in Eq.~4!.
In SU~3!, the Lagrangian for mesonbaryon interactiocan be similarly written using the meson and baryon ma
FIG. 1. Feynman diagrams for~a! Dp, ~b! Dr, and ~c! DNscatterings.02490ase
prdonal
px
ngg
i
ces. The formulation becomes, however, more complicain SU~4! where a more general tensor method is required@7#.The interaction Lagrangians needed for our study thenclude the following:
LpNN52 igpNNNg5tWNpW ,
LDNLc5 igDNLc~Ng5LcD1Lcg5ND!.
In the above we have used the following conventions:
p65p17 ip2
A2,N5~p,n! andD5~D0,D1!.
Again, SU~4! symmetry would relate the above couplinconstants to each other with the introduction of one mparameter, as shown below in Eq.~8!, because there are twSU~4!invariant Lagrangians for pseudoscalar meson abaryon interactions. We also need the following phenomelogical Lagrangian:
LrNN5grNNNS gmtWrW m1 kr2mN smntW]mrW nDN,where values of the coupling constantsgrNN andkr are wellknown as discussed below.
B. Cross sections
In Fig. 1, Feynman diagrams are shown for charm meinteractions with the pion~diagrams 1 to 8!, the rho meson~diagrams 9 and 10!, and the nucleon~diagrams 11 to 13!.Explicit isospin states are not indicated. The spin aisospinaveraged differential cross sections for thet channelandu channel processes can be straightforwardly evaluaand they are given by42
HADRONIC SCATTERING OF CHARMED MESONS PHYSICAL REVIEW C61 024904ds1dt
5grpp
2 gpDD*2
32pspi2
@mr222mp
2 22t1~ t2mp2 !2/mr
2#@mD*2
22mD2 22t1~ t2mD
2 !2/mD*2
#
~ t2mp2 !2
,
ds2dt
51
3
ds1dt
,
ds3dt
5grpp
2 grDD2
32pspi2
~2s1t22mp2 22mD
2 !2
~ t2mr2!2
,
ds9dt
51
3
ds1dt
,
ds10dt
51
9
ds1dt
,
ds11dt
53gpDD*
2 gpNN2
64pspi2
~2t !@mD*2
22mD2 22t1~ t2mD
2 !2/mD*2
#
~ t2mp2 !2
,
ds12dt
51
3
ds11dt
,
ds13dt
53grDD
2 grNN2
32pspi2
2~11kr!2@2su1mN
2 ~s1u!1mD4 2mN
4 #2~s2u!2kr~11kr/21tkr/8mN2 !
~ t2mr2!2
, ~3!ng
sak
te
n
hin
e
es
gree
posinwherepi denotes the initial momentum of the two scatteriparticles in their centerofmass frame.
For schannel processes through charm meson renances, shown by diagrams 4 to 8, the cross section is tto have a BreitWigner form
s5~2J11!
~2s111!~2s211!
4p
pi2
G tot2 BinBout
~s2MR2 !2/s1G tot
2,
whereG tot is the total width of the resonance,Bin and Boutare their decay branching ratios to the initial and final starespectively. We note that diagrams 4 to 7 correspondprocesses through theD2* and D1 resonancesDpDp,DpD* p, D* pDp and D* pD* p, respectively,while diagram 8 represents the processDpDp throughthe D* resonance. Total widths forD2* and D1 resonancesare known, and they areGD
2*0523 MeV, GD
2*1525 MeV,
GD10518.9 MeV, andGD
11528 MeV @8#. For the width of
D* , only an upper limit is known, i.e.,GD* 0,2.1 MeV andGD* 1,0.131 MeV. Studies based on the relativistic potetial model @9# suggest thatGD* 0.42 KeV andGD* 1.46KeV, and we use these values in this paper. The brancratios~BR! are known forD* 1 andD* 0 @8#, but not forD2*andD1. Experimental data show that for bothD2*
0 andD2*1
decays one hasG(Dpch)/G(D* pch);2. SinceD1 decays toD* p instead toDp due to parity conservation, we assum02490oen
s,to

g
B(D1D* p)51, B(D2* D* p)51/3, and B(D2* Dp)52/3, neglecting possible decays ofD1 andD2* to Dr andD* r, respectively@10#.
C. Coupling constants
For coupling constants, we use the empirical valugrpp56.1 @11#, gpDD* 54.4, grDD52.8 @12#, gpNN513.5@13#, grNN53.25, andkr56.1 @14#. From SU~4! symmetry,as assumed in the Lagrangian in Eq.~1!, one would expectthe following relations among these couplings constants:
gpKK* ~3.3!5gpDD* ~4.4!5grKK~3.0!5grDD~2.8!
5grpp
2~3.0!. ~4!
One sees that the empirical values given in parentheses areasonably well with the prediction from SU~4! symmetry.Signs of the coupling constants are not specified as thesible interferences among diagrams 3, 4, and 8 are notcluded. We note that the coupling constantgpDD* is consistent with that determined from theD* width
GD* pD5gpDD*
2 pf3
2pmD*2 ,
wherepf is the momentum of final particles in theD* restframe.43
ro
p
alfo
inla
onstworomaclf
vesec
500his
ions
de
ZIWEI LIN, C. M. KO, AND BIN ZHANG PHYSICAL REVIEW C 61 024904D. Form factors
To take into account the structure of hadrons, we intduce form factors at the vertices. Fortchannel vertices,monopole form factors are used, i.e.,
f ~ t !5L22ma
2
L22t,
whereL is a cutoff parameter, andma is the mass of theexchanged meson. For cutoff parameters, we use the emcal values Lrpp51.6 GeV @11#, LpNN51.3 GeV, andLrNN51.4 GeV @13#. However, there are no experimentinformation onLpDD* and LrDD , and their values are assumed to be similar to those determined empiricallystrange mesons, i.e.,LpDD* 5LpKK* 51.8 GeV, LrDD5LrKK51.8 GeV @11#. For schannel processes, showndiagrams 4 to 8, that are described by BreitWigner formuno form factors are introduced.,
eth
es7
ta
t
uleth
ei
02490
iri
r
,
E. Onshell divergence
The cross sections in Eq.~3! for diagrams 2 and 9(DrD* p) are singular because the exchanged mescan be on shell. Since the onshell process describes astep process, their contribution needs to be subtracted fthe cross section. This can be achieved by taking intocount the medium effects which add an imaginary seenergy to the mass of the exchanged pion as in Ref.@15#. Wetake the imaginary pion selfenergy to be 50 MeV and hachecked that the calculated thermal average of the crosstions do not change much with values between 5 andMeV. We note that there are other ways to regulate tsingularity @16#.
F. Thermal average
We are interested in the thermal averaged cross sectfor the processes shown in Fig. 1. For a process 112314, where the initialstate particles 1 and 2 are bothscribed by thermal distributions at temperatureT, the thermalaveraged cross section is given by^sv&5
Ez0
`
dz@z22~a11a2!2#@z22~a12a2!
2#K1~z!s~s5z2T2!
4~11d12!a1K2~a1!a2K2~a2!.
ons,a
f
umFig.In the above,a i5mi /T ( i 51 to 4!, z05max(a11a2 ,a31a4), d12 is 1 for identical initialstate particles and 0 otherwise, andv is their relative velocity in the collinear framei.e.,
v5A~k1k2!22m12m22
E1E2.
In Fig. 2~a!, we show the results for the thermal averagcross sections as functions of temperature. It is seendominant contributions are fromD and D* scatterings bynucleon,D scattering by pion via rho exchange, andD scattering by rho meson via pion exchange. In obtaining thresult, the rho meson mass is taken at its peak value ofMeV.
III. ESTIMATES OF RESCATTERING EFFECTS
As shown in the schematic model of Ref.@4#, if one assumes that charm mesons interact strongly in the final shadronic matter, then their transverse mass (m') spectra andpair invariant mass spectra would become harder thaninitial ones as a result of the appreciable transverse flowthe hadronic matter. Dilepton decays of charm mesons wothen lead to an enhanced yield of intermediatemass ditons in heavy ion collisions. In this section, we estimateeffects of hadronic rescattering on charm mesonm' spectraand the invariantmass distribution of dileptons from thdecays in heavy ion collisions at SPS energies.dat
e70
te
heofldpe
r
To characterize the scattering effects on charm meswe first determine the squared momentum transfer tocharm meson when it undergoes a scattering processD1X1D2X2. In the rest frame ofD1, the squared momentum othe final charm mesonD2 is given by
p025
@~mD11mD2!22t#@~mD12mD2!
22t#
~2mD1!2
FIG. 2. Thermal average~a! ^sv& and ~b! ^svp02& of charm
meson scattering cross sections as functions of temperature. Nbers labeling the curves correspond to the diagram numbers in1.44

ytimcs
sso
re
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y.
nti
y
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di
in
dde
ionsthe, as
cat
s
s an
2.1
forpion,mehearm
me
HADRONIC SCATTERING OF CHARMED MESONS PHYSICAL REVIEW C61 024904for tchannel processes with four momentum transfert. Foruchannel processes, one replacesu for t in the above expression.
We determine the total number of collisions suffered bcharm meson from its scattering cross sections and theevolution of the hadron densities. In the charm meson loframe, we assume that the density evolution of hadroninversely proportional to the proper time, i.e.,
r~t!}1
t. ~5!
Neglecting the effect of transverse expansion on the denevolution, the total number of scatterings for a charm meis then
N5Et0
tFsvrdt5svr0t0lnS tFt0D
5svr0t0lnS tFt0coshyD.svr0t0lnS R'm't0p' D ,which leads to the following thermal average of the squatotal momentum transfer due to scatterings:
^pS2&5^Np0
2&5F (i 5p,r,N, . . .
^svp02& ir i0Gt0lnS R'm't0p' D .
~6!
In obtaining the above result, we have assumed the sinitial and final proper times for the time evolution of diffeent particle species that are involved in the scattering. Eqtion ~6! shows that the relevant quantity is^svp0
2& instead ofthe usual sv&. We show in Fig. 2~b! this thermal averagefor all scattering channels considered in the present studis seen that the dominant contributions to^sv& remain important for^svp0
2& and the process involvingD* scatteringby rho meson via pion exchange also becomes significa
Summing up contributions from the scattering channelsFigs. 1~a!, 1~b!, and 1~c! separately, and simply dividing b2 to account for the average overD and D* , we obtain, atT5150 MeV,
^svp02&.1.1, 1.5, and 2.7 mb GeV2
for p, r, andN scatterings with charm mesons, respectiveFor central Pb1Pb collisions at SPS energies, the initi
particle numbers can be obtained from Ref.@17#, i.e., thereare 500p, 220r, 100v, 80 h, 180N, 60 D, and 130 higherbaryon resonances. The initial densities at central rapican then be estimated usingr0t0.(dN/dy)/(pRA
2).N/(4pRA
2). For a conservative estimate on the scattereffect, we only includep, r and nucleon. The initial densities for pion, rho meson and nucleon are thus
r0t0.0.79, 0.35, and 0.28 fm22,
respectively. Equation~6! then gives02490ae
alis
ityn
d
e
a
It
.n
.
ty
g
^pS2&.@^svp0
2&prp01^svp02&rrr0
1^svp02&NrN0#t0lnS R'^m'&t0^p'& D
.~1.130.7911.530.3512.730.28!/103 ln16.7
.0.61 ~GeV2!. ~7!
In the above, we have takent051 fm and R'.RA.1.2A1/3 fm. We have also used the relationsp'&;A^p'2 &.A2mTeff and ^m'&.m1Teff as given by Eq.~A2! in the Appendix. Since the charm mesonTeff increasesas a result of the rescatterings,^pS
2& needs to be determineselfconsistently. However, because of the logarithmicpendence shown in Eq.~7!, ^pS
2& is not very sensitive to thevalue ofTeff , and we have takenTeff5200 MeV in obtainingthe above numerical results. We note that...
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