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PHYSICAL REVIEW C SEPTEMBER 1996VOLUME 54, NUMBER 3

0556-2

Collision broadening of ther meson in a dropping mass scenario

K. L. Haglin*

National Superconducting Cyclotron Laboratory, Michigan State University, East Lansing, Michigan 48824–1321~Received 22 April 1996!

Vector mesons containing light quarks are thought to have their masses reduced in dense nuclear msacrificing some of their energy to the scalar field which becomes appreciable at finite baryon density. Mcalculations find masses which fall by a couple tens of percents in normal nuclear matter, and by sehundred MeV in dense matter. We estimate the collision rate forr mesons in such a scenario and at finitetemperature. Compared to its free-mass value, the collision rate changes by nearly a factor of 2 both abovbelow, depending on the density. This collision broadening effect could be important for estimates of low-mdilepton production in heavy-ion collisions.@S0556-2813~96!01909-7#

PACS number~s!: 25.75.Dw, 14.40.Cs, 21.65.1f

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Energetic heavy-ion collisions produce matter far frothe ground state and provide an exploratory work placeideas about strong interaction physics modifying particproperties. Restoration of chiral symmetry is an exampwhich could have dramatic effects on the overall dynamof the collision by changing particle masses~Brown-Rhoscaling! @1#, cross sections, propagation, and decay ratSeveral issues remain unresolved. For instance, resonaspectral densities could become degenerate with their chpartners, they could remain separate but mix very strongor they could disappear altogether by effectively melting incontinua. One popular picture is to couple nucleons to scaand vector fields and to couple light vector mesons to tscalar field in a self-consistent manner. This Walecka-typeapproach leads to masses that drop nearly universally wincreasing density@2# and presumably become degenerate,nearly so. However, any deviation from free-space behavis brief and unlikely observable in terms of hadronic signasince it is masked behind layers of complicated dynamand spacetime evolutionary information redistribution. Othe other hand, dilepton signals can, in principle, provideclear snapshot of the isoscalar and neutral isovector-vecmesons since the electromagnetic quanta couple weaklythe hadronic medium. Final spectra exhibit resonance strture which, in the low-mass region, includesr andv me-sons. The natural width of thev being 8.43 MeV dictates alifetime which is too long for decay in the high density paof the evolution and so one turns to ther meson for oppor-tunity.

Recent dielectron measurements have reported anhancement in the low-mass region for 200 GeV/nucleS1Au collisions at the Super Proton Synchrotron as compared with expected yields from hadronic decays@3#. Theenhancement was most clear around 400 MeV mass andpossible explanation among those proposed@4–7# is that ther mass drops in the early, high density stage of the evolutand provides a component to the dielectron signal muchlow its free mass@4#. Later stages of evolution support lowedensities and consequently, higherr meson masses and wilprovide dielectron components all the way up to a free-spa

*Electronic address: haglin@nscl.msu.edu

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r distribution. The question we address here is the relatiimportance of collision broadening in such a picture.

To further motivate and quantify the dropping mass scnario, we imagine a system of baryons, pseudoscalar, vecand axial-vector mesons at temperatureT and baryon densityrB , and we look to models of the in-medium hadronic properties. They include QCD sum-rule approaches@8–11#, ef-fective field theories based largely on particle-hole polariztion effects @12#, quark models@13#, and mean-field orWalecka-type models@14#. A trend they seem to share is thedropping mass at finite density. In the Walecka-type aproach, the nucleons couple to vector and scalar fields whthe light vector mesons couple only to the scalar field sinthey are composed of quark and antiquark giving counterbancing contributions. For given baryon density, the enerdensity« is a function of the scalar fields. By locating theextrema of]«/]s50, one establishes the strength of thscalar field. Then, the constituent quark model@13# togetherwith Weinberg sum rules@15# for relatingr anda1 massesgive, for example,

mN*5mN2gSs, mr*'mr2~2/3!gSs,

ma1* 'ma1

2~2A2/3!gSs, ~1!

where the asterisk quantities are in-medium values andgS isa coupling constant determined by balancing the repulsvector, and attractive scalar influences in order to reprodubulk properties of nuclear matter. Modifications in the surules themselves due to finite temperature and density wnot be included here. While finiteT effects would not changethe results dramatically@16#, finite density effects are lessclear. Ignoring medium modifications to pions and kaonthen leads to a picture of baryon, vector, and axial-vectmeson masses, all dropping with increasing density. At thstage, choosing a specific effectiver mass determines thescalar field which, in turn, determines the density. Forgiven scalar field, all other hadron masses are then demined. It is, therefore, a well-defined question in this modto ask for the collision rate at fixedT as a function of effec-tive r mass frommr 5 770 MeV down to the two-pionthreshold.

1492 © 1996 The American Physical Society

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54 1493BRIEF REPORTS

Collision rates are most conveniently estimated in a fireball model and from kinetic theory rather than some sort odynamic model. This approach has been carried out torather extensive level forr, v, and f mesons interactingwith pseudoscalar and light vector mesons@17#. Interactionswere parametrized according to the simplest Lagrangians rspecting proper symmetries and then calibrated to give thfree-space decays. The tree-level Feynman graphs wereenumerated and computed while checking unitarity boundand including form factors to account for the composite nature of the particles. Collision rates forr andv were foundto be numerically similar and ranged from 40 to 120 MeV athe temperature varied from 150 to 200 MeV. The completeness of the calculation was crucial to getting thef result of10–25 MeV since there are no strong resonances. But fr and v, even with many nonresonant contributions, theresonances dominated the vector-pseudoscalar cross sectand, consequently, the contributions to the rate. This simpfies things enormously since we can be satisfied with BreiWigner parametrizations and eliminate the need for all thacomplexity. Then, a nonresonantr-r cross section will becomputed as it becomes important for high densities.

Consider the system of a hadronic mixture of pions, kaons,r mesons, and nucleons. Introduction ofh, v, or highermass mesons does not change the results significantly. Gthem equilibrium distributionsf (p,m), having the flexibilityof a chemical potentialm. It is probably true that an equili-brated system is not fully reached for all of these constituenin heavy-ion collisions, but for first estimates it is the mosreasonable thing to assume.r mesons scatter vigorously withpions through thea1(1260) resonance and with kaonsthrough theK1(1270), both of which are axial vectors andcarry isospin 1 and 1/2, respectively. TheK1 mass is as-sumed to scale just like thea1’s. r-r scattering becomesimportant asmr* drops, owing to the increased range of ther-exchange interaction. Finally,r-nucleon scatterings give acontribution which is very much smaller than these, so wwill restrict attention to pions and kaons and otherr mesons.

Ignoring final state suppression or enhancements and aproximating with classical distributions, the average scatteing rate is

Grcoll5(

iexp~m i /T!

grginr*

T2

~2p!4

3Ezmin

`

dzl~z2T2,mr*2 ,mi

2!sr i~s!, ~2!

where the sum runs overi5$p,K,r%, theg’s are degenera-cies,nr* is the particle density of a thermal population ofrmesons having massmr* and zero chemical potential,zmin5(mr*1mi)/T, and l is the usual kinematic function.Vector-pseudoscalar cross sections are taken to be BreWigner functions with energy-dependent widths. As themasses drop, phase space changes which naturally has mfying effects. The simplest Lagrangian respecting both gauginvariance and minimal pion derivatives is chosen for thaxial-vector–vector–pseudoscalar~AVP! interactions. It hasbeen used for photon@18,19# and dilepton@6# production

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estimates and is known to be reliable. The form of the inteaction and the resulting strong decay rates are

LAVP5gAVP]mpW •~VW mn3AW n!,

GA→VP5gAVP2

24pmA2 upW u@2~pV•pP!21mV

2~mP21pW 2!#, ~3!

whereVmn5]mVn2]nVm and pW is the center-of-mass mo-mentum of the decay products. Numerical values for the coplings arega1pr (gK1Kr) 5 16.1 ~11.6! GeV21 with masses1230 ~1273! MeV, in order to give widths of 400~37.8!MeV, respectively. Cross sections are taken to be

sVi54p

pW 2 F mR2GR→Vi

2

~s2mR2 !21~mRGR

tot!2G ~4!

since the spin factor ratio is 1 for bothpr andKr scatteringthrougha1 andK1 resonances, respectively.

A triple-vector interaction for ther ’s is modeled as@20#

LVVV52g

2VW mn•~VW

m3VW n!, ~5!

whereg is taken to begrpp wheregrpp2 /4p 5 2.9. Both

s- and t-channel contributions are included and monopoform factors

f ~ t !5L22mr

2

L22t~6!

are attached in the usual way tot-channel vertices suppressing high momentum transfers. The cutoff is taken to be madependent

L5L0Smr*

mrD , ~7!

with L0 5 1.8 GeV. The unstable nature of ther calls for afinite width

Gr85Gr8→pp1Gr8→rr ~8!

in the propagator, where

Gr8→pp52

3

grpp2

4p

ukW u3

s~9!

and

Gr8→rr51

24

grpp2

4p

upW umr*

4 @s~mr*21pW 2!24mr*

4#. ~10!

In Eqs. ~9! and ~10!, r8 has invariant massAs and thepp

(rr) center-of-mass momentum iskW (pW ).The scattering rate of Eq.~2! is now a function of tem-

perature, chemical potentials, and hadron masses. Here,assume chemical equilibrium with a single potentialmp , andwith the r chemical potential determined solely from itStrictly speaking, this induces different charged and neutr masses. Here, we ignore those differences and loosely c

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1494 54BRIEF REPORTS

sider an isospin average. Then, as mentioned previouchoosing a value of effectiver mass determines the effectivemasses for the other hadrons as well.

In Fig. 1 we show the rate atT 5 150 MeV as a functionof ther mass choosing two values of pion chemical potent~0 and 130 MeV!. In the free-mass extreme and zero chemcal potential case, we can check against ther collision rateof ;40 MeV computed in Ref.@17# with more sophisticatedmeans. The two are consistent to within 5%, so the presapproach is quite satisfactory. As ther mass drops from itsfree-space value of 770 MeV, the collision rate rises tomaximum of 75 MeV for a mass around 425 MeV. Fromhere it drops back to 40 MeV at 300 MeV mass. Thus, fthe lightestr mesons considered here, the collision broadeing is clearly not important. On the other hand, for intermdiate r masses the collision rates reach 75–200 MeV dpending on the pion chemical potential. Under thecircumstances, collision broadening is most certainly impotant.

We recompute at a higher temperature of 200 MeV ashow the results in Fig. 2. The rate for a free mass and zchemical potential is;140 MeV. This is about 15% higherthan the result of Ref.@17# and is due to the differences in thecross sections used here. The peak in the rate seen attemperature is largely smeared out since the average scaing energies are higher here and are, therefore, able to acthe resonance regions more easily. The key elements fortermining the rate are the positions and widths of the res

FIG. 1. Collision rate forr mesons at fixed temperatureT 5150 MeV as a function of the effectiver mass. Two different val-ues of pion chemical potential are used. The solid and dascurves come frommp 5 0 and 130 MeV, respectively.

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nances compared to the averageAs for individual scatter-ings. In this higher temperature result, the resonancesaccessed equally well down to 500 MeV mass and from thstart to drop off the tail of the resonance distributions.T5 150 MeV, on the other hand, as the mass drops to ab425 MeV, the resonance is accessed more favorably andin fact, narrower, thereby giving a boost to the rate. Finalthe increase in rate as the mass drops below 375 MeV isto the onset of importantr-r scattering.

We have computed the collision rate of ther meson inhot and dense hadronic matter as a function of the effecr mass or alternatively and equivalently, as a functiondensity. At T 5 150 MeV and low densities, the rate iGr

free mass;40 MeV. As the density increases and the madrops, the rate rises to about twiceGr

free mass, peaks, and thendrops to about halfGr

free massnear the two-pion thresholdSince mass resolution in the current dilepton experimentroughly 10%,r meson collision rates of a few tens of MeVare not too important since the two effects~resolution andcollision broadening! add in quadrature. At higher temperatures, we have seen the rates rising to more than 100 toMeV depending whether a chemical potential is introducBoth values are significant and suggest that collision broening could be quite important for model calculationslow-mass dileptons when quantitative interpretation of eperiment is attempted.

I thank C. M. Ko for asking the questions which led tthis study. This work was supported by the National ScienFoundation under Grant No. PHY-9403666.

ed

FIG. 2. Same as previous figure but withT5200 MeV.

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