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Lepton pairs from thermal mesons

arXiv:hep-ph/9307363v1 29 Jul 1993McGill/93–8TPI–MINN–93/19–TLepton pairs from thermal mesonsCharles Gale∗Physics Department, McGill University3600 University St., Montr´eal, QC, H3A 2T8, CanadaPeter Lichard†Theoretical Physics Institute, University of MinnesotaMinneapolis, MN 55455, USAandDepartment of Theoretical Physics, Faculty of Mathematics and PhysicsComenius University, CS-842 15 Bratislava, Slovakia‡AbstractWe study the net dielectron production rates from an ensemble of thermalmesons, using an effective Lagrangian to model their interaction. The cou-pling between the electromagnetic and the hadronic sectors is done throughthe vector meson dominance approach.For the first time, a complete setof light mesons is considered.

We include contributions from decays of thetype V (PS) →PS (V) + e+ e−, where V is a vector meson and PS is apseudoscalar, as well as those from binary reactions PS + PS, V + V, andV + PS →e+e−. Direct decays of the type V →e+e−are included andshown to be important.

We find that the dielectron invariant mass spectrumnaturally divides in distinct regions: in the low mass domain the decays fromvector and pseudoscalar mesons form the dominant contribution. The pion–pion annihilation and direct decays then pick up and form the leading signalin an invariant mass region that includes the ρ −ω complex and extends upto the φ.

Above invariant mass M≈1 GeV other two-body reactions takeover as the prominent mechanisms for lepton pair generation. These facts willhave quantitative bearing on the eventual identification of the quark–gluonplasma.PACS numbers: 25.75.+r, 12.38.Mh, 13.75.LbTypeset using REVTEX1

I. INTRODUCTIONOne of the ultimate goals of high energy heavy ion physics is the formation and ob-servation of a quark–gluon plasma (QGP) as predicted by QCD. A vigorous experimentalprogram is under way and it is fair to say that this area is one of the most active fields ofcontemporary subatomic physics.

The creation of such a novel state of matter represents aconsiderable challenge, both in its experimental realization and also in the theoretical inter-pretation of the experimental results. The lifetimes involved are of the order of ≈10 fm/cand the detailed dynamics of the collision process may furthermore play an important role,complicating the extraction of a clear signal.

Nevertheless, much progress has been madeboth in theory and in experiment, and we may say that even in the absence of a genuineQGP the study of hot and dense hadronic systems is still a fascinating subject from whicha great deal can be learnt.For a while now, electromagnetic signals have been known as ideal probes of stronglyinteracting matter at high temperatures and densities [1]. This owes to the fact that oncethey are produced, they will travel relatively unscathed from their point of origin to thedetector.

Since production rates are rapidly increasing functions of temperature and density,these electromagnetic signals provide valuable information on the hot and dense phases ofthe reaction. It is hoped that, because of these facts, those signals should constitute preciousaids in the process of analyzing the behaviour of hot quark–gluon matter [2].

As with anypossible experimental signature of the QGP, a great deal of care must go into the calculationof a corresponding “purely hadronic” signal, that is a contribution to the same experimentalobservables from sources other than the deconfined, chiral–symmetric phase. As far as thequark–gluon plasma is concerned, one may refer to these sources as the “background”.In this paper, we are concerned with the thermal rate of dielectron emission only butour treatment is completely general.The source is a hot environment of several mesonspecies: for the first time, we use a rather complete set of mesons, rather than restrictingourselves to the usual pion gas approximation.

The equilibrium assumptions inherent to theapproaches similar to the one been used here have to be carried to their logical conclusion: insuch scenarios, once the temperature has been set one can clearly calculate the populationof species present. These mesons can then interact among themselves, or even decay, toproduce lepton pairs in the final state.

It is important to realize that we deliberately makeno attempt here to connect with experiment because our calculation is rather meant toanswer a well defined theoretical question: what is the electromagnetic emissivity (in thedilepton channel) of a hot hadron gas? To answer this question, we shall proceed along thelines of a similar calculation for photon rates [3].We estimate the rates of producing lepton pairs using relativistic kinetic theory.

Themesonic interactions are modelled with an effective Lagrangian and the coupling of radiationto hadronic matter is done in the vector meson dominance (VMD) approach. The values ofthe coupling constants involved are adjusted so that the experimentally measured radiativedecay widths are reproduced.

We describe the details of our model in section II. We giveresults in section III and finally we end with a discussion in section IV.2

II. THE MODELOur starting point is an ensemble of mesons in thermal equilibrium.

We consider thelightest and thus most abundant strange and non–strange mesons together with their maininteraction channels. This means we shall include: π, η, ρ, ω, η′, φ, K and K∗.

The chargestates are not labelled but all of them are present. This collection can be further divided intwo categories: pseudoscalar (PS) and vector (V) particles.

From this hot meson gas, howdo we calculate what is the lepton pair radiation? It has been shown [4] that the thermalproduction rate for electron–positron pairs is related to the imaginary part of the retardedphoton self energy byE+E−dRd3p+d3p−=2e2(2π)61M4 (pµ+pν−+ pν+pµ−−p+ · p−gµν) ImΠR(γ)µν(k)(2.1)×1eE/T −1.Here p+ and p−are the positron and electron momenta, kµ = (E,⃗k) is the virtual photonmomentum, T is the temperature, and we have set the electron mass to zero (nonzero leptonmass is easy to include).

R is the number of times per unit four-volume an e+e−pair ofinvariant mass M is produced with the specified momentum configuration. Note that theabove equation is perturbative in the electromagnetic interaction only; it is a completelynon–perturbative expression in the strong interaction.Furthermore, we shall make use of the VMD model, which states that the hadronicelectromagnetic current operator is given by the current–field identityJµ = −egρm2ρρµ −egφm2φφµ −egωm2ωωµ.

(2.2)The above expression tells us how the electromagnetic radiation couples to hadronic (inour case mesonic) matter: by first coupling to one of the vector mesons with some couplingconstant. In the above, we have kept the ρ, ω and φ fields, but in some cases we shall tacitlyinclude also higher vector mesons by using phenomenological form factors inspired by data.We further need a model for how the mesons interact among themselves.

For this, we shalluse a simple phenomenological approach, inspired by the chiral properties of low energyQCD. Such classes of phenomenological Lagrangians have been quite successful in the pastin the description of low energy hadronic physics [5].

We are explicitly interested in theinteraction between the different possible combinations of vector (V ) and pseudoscalar (ϕ)fields. For reasons that will become clear shortly we restrict our discussion to the followinginteraction Lagrangians [5]:LintV V ϕ = gV V ϕ ǫµναβ ∂µV ν∂αV βϕ ,(2.3)andLintV ϕϕ = gV ϕϕ Vµϕ↔∂ϕ .

(2.4)In the above, the coupling constants are fitted for each field combination, in a procedure wenow describe. We have a model for how mesons interact among themselves and how they3

interact with the electromagnetic field. With this approach, let us study a simple radiativeprocess like the decay of a vector meson into a pseudoscalar meson and a photon like e.g.ω →π0γ.

In this model, the process goes via the ωρπ vertex, owing to G parity conservationat the strong vertex, and the ρ0 couples to the photon in virtue of the current–field identity.This corresponds to the Feynman diagram of Fig. 1.

The ratio of coupling constants fromEqs. (2.3) and (2.2), in this case gωρπ/gρ, is adjusted so that the correct experimentalradiative decay width [6] Γ(ω →π0γ) is obtained.

Our Lagrangians are then “calibrated”through all the following processes: ρ →πγ, K∗±→K±γ, K∗0( ¯K∗0) →K0( ¯K0)γ,ω→π0γ, ρ0→ηγ, η′→ρ0γ, η′→ωγ, φ→ηγ, φ→η′γ, φ→π0γ. Onerealizes [c.f.

Eqs. (2.2, 2.3, 2.4)] that via this procedure, we can only fix the ratio of strongto “electromagnetic” (vector meson–photon) couplings.

However it is this very combinationwe shall need for our specific application.We now integrate our model for interacting mesons with a dilepton radiation calculation.If we keep a calculation of the photon self–energy at the one–loop level an evaluation of itsimaginary part, as instructed in Eq. (2.1), will yield processes of the type V (PS) →PS (V)γ∗, PS + PS →γ∗, V + PS →γ∗and V + V →γ∗.

Since such tree–level amplitudes canbe readily computed and that our general field–theoretic treatment for dilepton emissionhas been shown to agree with relativistic kinetic calculations (up to temperature–dependenteffects in the form factors, which have been shown to be small [4]) we use the latter ap-proach. Finally note that the two–body channels listed above will kinematically dominatethe contributions of the type V + PS →PS + γ∗, which we shall neglect.

The inclusion ofsuch processes would correspond to evaluation of the photon self–energy beyond the one–loop level. The first attempt to investigate the role of processes with more than two mesonsinvolved has recently been made in [7].

We will return to this point later.The basic relativistic kinetic expression for the dilepton production rate from a processa + b →e+e−is well known and can be written down asRab→e+e−= NZd3pa2Ea(2π)3d3pb2Eb(2π)3d3p+2E+(2π)3d3p−2E−(2π)3 fa fb(2.5)×|M|2 (2π)4 δ4(pa + pb −p+ −p−) .Similarly, we may write a rate equation for the decay process a →b + e+e−:dRa →b + e+e−dM2= NZd3pa2Ea(2π)3d3pb2Eb(2π)3d3p+2E+(2π)3d3p−2E−(2π)3 fa (1 + fb)(2.6)× |M|2 (2π)4 δ4(pa −pb −p+ −p−) δ(M2 −(p+ + p−)2).In the above equations, N is an overall degeneracy factor dependent upon the specificchannel and the f’s are Bose–Einstein mean occupation numbers.These equations are not suitable for numerical evaluation because of the delta functions.However, they can be cast, using standard methods of simplifying phase integrals and thespherical symmetry in momentum space, into an appropriate form. The dilepton productionrate for the process a + b →e+e−becomesRab→e+e−=N(2π)4Z ∞ma dEa pafa(Ea)Z ∞mbdEb pbfb(Eb)(2.7)4

×Z 1−1 dxq(s −s+)(s −s−) σab→e+e−(s),where s ≡M2 = m2a + m2b + 2(EaEb −papbx), s+ = (ma + mb)2 and s−= (ma −mb)2. Thecross section σab→e+e−(s), is obtained by an evaluation of an appropriate Feynman diagram,Fig.

2. The multiple integral in Eq.

(2.7) is evaluated by Monte Carlo methods. While it iscertainly possible to do some of the integrations analytically, we chose the avenue of keepinga relatively transparent integrand and we let the Monte Carlo approach handle the numericalcomplexity.

Moreover, this approach allows us to evaluate any desirable differential dileptonproduction rate easily (see, e.g., [8] or [9]).Similarly, the rate equation for the decay process a →b + e+e−is now:dRa →b + e+e−dM2= N ma(2π)2dΓa →b + e+e−dM2Z ∞madEa pafa(Ea)(2.8)×Z 1−1 dx [1 + fb(Eb)] ,where Eb = (EaE∗b + pap∗bx)/ma, E∗b = (m2a + m2b −M2)/(2ma) and dΓa →b + e+e−/dM2 isthe differential decay width into the appropriate channel. In Eqs.

(2.6) and (2.8) one noticesthe Bose–Einstein final state enhancement, an in–medium effect.We also include the direct decay channels of the form V→e+ e−. As we will show,their contributions are non–negligible.

This is especially true in the case of ρ →e+e−. Onecan show that for such decaysdR V →e+e−dM2=32π2ΓV →e+e−˜Nm3VM2 B(M2)Z ∞M dE f(E)√E2 −M2 ,(2.9)whereB(M2) = βΓtot(M2 −m2V )2 + (mV Γtot)2 .

(2.10)The constant β fixes the normalization of the Breit-Wigner probability density function. Itsvalue is not important here as it enters also the factor˜N =ZdM2mVM3B(M2) ,(2.11)which ensures the correct overall normalization based on the experimental value of the partialdecay width into the dielectron channel, ΓV →e+e−.

The integral runs over the allowed massrange.In the above equations, mV is the vector meson mass and Γtot is its total decay width.For the narrow resonances (ω, φ) the latter is taken constant but the ρ0 width is given itsproper mass dependence.III. RESULTSThe decay channels considered have already been listed: they are the same radiativedecay reactions V (PS) →PS (V) + γ, as used to fix the couplings constants of our La-grangians, with the obvious substitution: γ →γ∗.

The V + PS →e+e−amplitudes can all5

be obtained from the decay reaction amplitudes by crossing symmetry. We list the entrancechannels anyway for completeness.

They are: ω π0, ρ π, φ π0, ω η, φ η, ρ0 η, ω η′, φ η′,ρ η′, ¯K∗K and K∗¯K. For each of the PS + PS and V + V reaction, we follow the followingapproach: their “bare” amplitude is calculated, squared, and finally multiplied by a formfactor obtainable from experimental data on e+e−annihilation.The topic of form factors deserves here a short discussion.

Of course no information ontime–like form factors is available through the analysis of meson radiative decays into realphotons. With respect to this issue, we have followed a simple prescription.

The time–like electromagnetic form factor of charged pion is experimentally very well known [10] andsome experimental information exists also about those of both charged and neutral kaons[11]. In our calculations of π+π−, K+K−, and K0 ¯K0 annihilation rates we have used arecent parametrization [12] of these quantities.

The vector mesons annihilation channelshave been given the same form factors as their corresponding (by strangeness and isospin)pseudoscalar counterparts. In the case of decays and V + PS reactions, whenever the Gparity and isospin conservation laws allowed a coupling only to the ρ0 and its recurrences,the charged pion electromagnetic form factor [12] was used.

In the other cases, we havestuck with a form factor equivalent to a simple pole corresponding to the lightest permittedvector meson. Our way of normalizing coupling constants by means of the radiative decaywidths leads us to a belief that this conservative choice of form factors does not introducetoo much uncertainty.

We made only one exception from the simple rules sketched above.In the case of the reaction ρ + π →e+e−the rules would lead to a simple ω–pole. It wouldbe a rather bad approximation because the threshold of this reaction lies below the positionof the φ–resonance, which thus becomes extremely important.

We take therefore a two–poleformula with the relative weight between the ω and φ contributions same as in the kaonisoscalar form factor FS = (FK+ + FK−)/2 [12].We have performed our thermal hadronic calculations at three temperatures: 100, 150and 200 MeV. We feel that those reflect a range of energies that is somewhat reasonable, bycurrent theoretical standards.The results for V (PS) →PS (V) e+e−at a temperature of 150 MeV are shown onFig.

3. Not all the decays are shown, but only the dominant ones.

Coupling constantsarguments aside, the largest contributions will come from the radiative channels where aheavy meson decays into a light one and a lepton pair. This is precisely what is observed onFig.

3. The largest contribution up to invariant mass ≈0.65 GeV is from ω →π0 e+e−.Over this range, ρ →π e+e−represents the next–to–leading contribution and the otherdecays are at least an order of magnitude lower.

Above 0.65 GeV invariant mass, the onlydecay with phase space left is φ→π0 e+e−.Note that the widths for the radiativedecays of the ω and ρ0 are comparable and are two orders of magnitude larger than thatfor φ →π0γ [6]. Dalitz decay (e.g.

η→γe+e−) is of higher order in α and can thusbe neglected. However, this argument alone is not totally convincing as one could imaginethat the η could be massively produced at such temperatures.

We have therefore performeda calculation of the contribution from eta Dalitz decay to thermal electron pair yield, usingthe VMD prescription for dΓ/dM2 [13] with updated coupling constants. We have found itin fact to be orders of magnitude smaller than the channels discussed above.For the pseudoscalar–pseudoscalar reactions, on Fig.4 we display a plot of all thecontributions, again at T = 150 MeV.

The different contributions add up to a signal in6

which the only apparent structures are associated with the ρ(770) and the φ, with a slightshoulder at the ρ(2150). The peak in the pion form factor at the ρ(1700) is washed out bythe kaon contributions.We show the V + V contributions on Fig.

5. Above threshold, the sum of these processesoutshine the PS + PS ones by roughly an order of magnitude.

The structure at M = 2.15GeV owes to the corresponding excitation of the ρ.The V + PS reactions are quite numerous, we show the brighter dilepton sources on Fig.6, again for T = 150 MeV. The dominant channels are ω + π0, ρ + π and ρ0 + η.

The kaonchannels are not shown but are roughly the size of the π + ρ contribution. The strongestsignal is from ω + π0, over the entire invariant mass range considered here.

Recall from ourdiscussion of the decays that the radiative decay widths of the ρ and the ω are quite large.Finally, the total rate corresponding to the sum of all processes discussed so far is shownon Fig. 7, along with a curve representing the π+π−contribution only.

We also show thenet direct decay contribution, summing ρ →e+e−, ω→e+e−and φ →e+e−. Theradiation from these channels turns out to be quite important.

The signal from the decayreaction ρ →e+e−closely resembles the pion annihilation spectrum, which in retrospectis quite reasonable.In all cases (decays, PS + PS, V + V, V + PS) our findings at T = 100 and 200 MeVare qualitatively similar, with a global shift in the rate. For these temperatures we thereforepresent only the total rates (see Fig.

8).IV. DISCUSSIONUp to now, thermal calculations of the variety discussed in this paper have rarely gonebeyond a pure pion gas approximation, usually concentrating on the annihilation channel[14].The contribution from thermal meson decays has been considered previously [15].To our knowledge it is the first time that extensive mesonic reactions have been included,together with direct decays.Comparing the individual contributions from different processes (Figs.

3–6) to the totaldilepton rate (Fig. 7) one sees that the dilepton invariant mass spectrum naturally dividesin several parts.

At low masses, the decay channels clearly dominate the entire spectrum.The crossover to the pion–pion annihilation and direct decay signal occurs just above 0.5GeV (at the lower temperature, T = 100 MeV, this crossing point is shifted closer to thetwo–pion threshold). Already at M≈1 GeV, the total rate dominates over the pion gasapproximation result by an approximate factor of 3.

At M = 1.5 GeV, those rates differby a little more than an order of magnitude. The difference increases with larger invariantmasses.

One also sees that the net rate at the vector meson positions is also larger thanin the straight π+ −π−scenario, owing principally to direct decays and also form factoreffects. Probably the most striking conclusion of our work is that the “usual” pion resultsfor lepton pair production calculation holds rather poorly over all regions of invariant massesconsidered in this work.

This statement is true for all temperatures studied here.Thus, the rate for M>∼1 GeV is approximately one order of magnitude larger in ourcalculation than in “conventional” meson background calculations. This enhancement is alsopresent in the momentum structure of the lepton signal: Fig.

9 is a plot of Ed3R/d3p for7

lepton pair invariant masses between 1.1 and 3 GeV. These findings should have importantimplications in connection with the plasma signal identification.

The conventional windowfor thermal lepton pairs of plasma origin is mφ

However, before any more quantitative statements can be made, it is imperative tocomplement our calculations with a dynamical model of some sort, in order to make contactwith genuine observables. Work in this direction is in progress.It is of interest to compare the rates obtained with other similar calculations.

Somerecent interest has been devoted to the emission of lepton pairs from pionic bremsstrahlungprocesses [17]. It was concluded that the radiation from the external pion lines in pion–pioncollisions would be a dominant contribution to the low mass lepton spectrum.

Comparingwith pion bremsstrahlung calculations at T = 150 MeV, we realize that the low mass sig-nal is the same magnitude as the net meson decay contribution. Correcting the pion–pionbremsstrahlung rate for the Landau–Pomeranchuk effect [18] will cut this pion signal by somefactor.

This factor is only ≈2 for low invariant masses and T = 150 MeV [19]. This correc-tion also goes down as invariant mass grows.

This will then leave the pion bremsstrahlungto compete with the decay channels contribution, up to the two–pion annihilation threshold.In this inquiry, we have pursued the same goals as a similar photon production calculation[3]. Our main aim has been to identify the most important dilepton production processeswhich operate in a hadron gas.We considered only the decays and reactions with theminimal possible number of hadrons: one in the decay final states, none in the final statesof two-initial-hadron reactions.

These processes are believed, on the basis of the phase-space and order-of-interaction arguments, to be dominant here. The reactions of this kind(2 →0 hadrons) do not operate in real photon production due to restrictions from energy-momentum conservation.

However, the dominant reactions for photon production a + b →c + γ can produce virtual photons as well. It is clear that they would populate preferablythe low-mass region.

Even there they would be probably negligible, as it was shown for thecase of π + π →π+ dilepton in [7]. But one cannot exclude surprises.

The latter processamplifies, together with the three pion annihilation channel, the omega peak in dileptonspectrum. This may in turn serve as a signature of a hadron gas creation [7].

It has alsobeen pointed out that the A1 meson could have a significant influence on the real photonyield [20], through the process π ρ→A1→π γ. This conjecture has been carefullyanalyzed in a recent paper [21].

The reflection in the thermal dilepton sector is certainlyworth studying as well. Three body initial state processes a + b + c →e+e−may alsocontribute significantly in the high invariant mass region [7].

We intend to study all thesepoints in detail in future work.ACKNOWLEDGMENTSWe would like to acknowledge the warm hospitality of the Theoretical Physics Instituteof the University of Minnesota, where this work was started. This work was supported inpart by the Natural Sciences and Engineering Research Council of Canada, by the FCAR8

fund of the Qu´ebec Government and by a NATO Collaborative Research grant. The stayof P.L.

at the University of Minnesota was supported by the U.S. Department of Energyunder Contract No. DOE/DE-FG02-87ER-40328; travel expenses were borne by the grantMˇSMˇS SR 01/35.9

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FIGURESFIG. 1.

The Feynman diagram for the radiative decay ω →π0γ in the model described in thetext.FIG. 2.

A “generic” two–body amplitude with lepton pairs in the final state. The different{a, b} combinations we consider are enumerated in the text.

The vector meson V is chosen throughisospin and G parity arguments.FIG. 3.

Differential rate for lepton pair production via vector or pseudoscalar meson decay.The dashed line represents the contribution from ω→π0e+e−, the dashed–dotted line is therate from ρ →πe+e−. The dotted line is the process φ →π0e+e−.

The structure in the latterchannel is due to the ρ(770). The solid line is the sum of all the decay processes, including thosenot listed in this caption but enumerated in the main text.FIG.

4. Rate from PS + PS type reactions.

The dashed line is the rate for the pion annihilationprocess. The dotted curve represents the contribution from K+ + K−.

The dashed–dotted lineis the rate from K0 ¯K0 annihilation. The solid line is the sum of the PS + PS processes.FIG.

5. Rate from V + V type reactions.

The dashed curve is the ρ+ + ρ−contribution. Thedashed–dotted and dotted curves represent charged and neutral K∗annihilation, respectively.

Thesolid curve is the sum of the V + V contributions.FIG. 6.

Rate from V + PS type reactions. The dashed curve is the rate from ω + π0.

Thedashed–dotted curve is the contribution from ρ + π and the dotted curve is the rate from ρ0 + η.Again, the solid line is a sum of all V + PS processes, as enumerated in the text.FIG. 7.

The solid line is the total rate at T = 150 MeV from all processes discussed in thetext. The dashed line is the pion–pion annihilation contribution only.

The short–dashed curverepresents the contribution from direct vector meson decays.FIG. 8.

Same caption as Fig. 7 but for the temperatures T = 100 MeV (lower curves) andT = 200 MeV (higher curves).FIG.

9. The lepton pair momentum spectrum, Ed3R/d3p for lepton pair invariant massesbetween 1.1 and 3 GeV.

This lower bound is chosen so as to exclude the φ peak. The full curverepresents the contribution from all processes described in this work.

The dashed curve is thepion–pion annihilation contribution only.11

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