Dilepton production in microscopic transport theory with in-medium ρ -meson spectral function
aa r X i v : . [ nu c l - t h ] S e p Dilepton production in microscopic transport theory within-medium ρ -meson spectral function A.B. Larionov, ∗ U. Mosel, and L. von Smekal
1, 21
Institut f¨ur Theoretische Physik, Justus-Liebig-Universit¨at, 35392 Giessen, Germany Helmholtz Research Academy Hesse for FAIR(HFHF), Campus Giessen, 35392 Giessen, Germany
Abstract
We use the microscopic GiBUU transport model to calculate dilepton ( e + e − ) production inheavy-ion collisions at SIS18 energies focusing on the effect of collisional broadening of the ρ -meson. The collisional width of the ρ -meson at finite temperature and baryon density in nuclearmatter is calculated on the basis of the collision integral of the GiBUU model. A systematiccomparison with HADES data on dilepton production in heavy-ion collisions is performed. Thecollisional broadening of the ρ improves the agreement between theory and experiment for thedilepton invariant-mass distributions near the ρ pole mass and for the excess radiation in Au+Auat 1.23 A GeV. We furthermore show that some remaining underprediction of the experimentaldilepton spectra in C+C at 1 A GeV and Au+Au at 1.23 A GeV at intermediate invariant masses ∼ . − . pn bremsstrahlung cross section in a wayto agree with the inclusive dilepton spectrum from dp collisions at 1.25 A GeV. ∗ Corresponding author: [email protected] ONTENTS
1. Introduction 22. The model 53. Spectral function of the ρ -meson in the nuclear medium 134. Results 214.1. Comparison with HADES data 284.1.1. pp and dp collisions 294.1.2. C + C collisions 344.1.3. Ar + KCl collisions 384.1.4. Au + Au collisions 434.1.5. Ag + Ag collisions 454.2. Excess radiation 465. Summary and conclusions 49Acknowledgments 52A. Hadron Numbers 531. Pion numbers 532. Hadron multiplicities 54B. Residual Uncertainties 55References 57
1. INTRODUCTION
The study of in-medium properties of hadrons has been an active field of research over thelast 30 years. The EMC experiment had already shown that the electromagnetic structurefunctions of nucleons change when these are bound in a nucleus. Also the spectral functionsof bound nucleons differ from those of free ones thus reflecting the complicated in-mediuminteractions with other nucleons. On the basis of QCD sum rules Hatsuda and Lee [1]2redicted that the masses of vector mesons should drop significantly inside the nuclearmedium, a prediction that agreed with a similar one by Brown and Rho [2]. In thesepredictions the mass drop was due to the disappearance of the q ¯ q scalar condensate withincreasing nucleon density. These predictions triggered experiments at CERN/SPS to findsignals of the in-medium modifications of ρ meson [3, 4] which, however, did not allowto make a clear distinction between the in-medium mass shift and a possible broadening.Indeed, it had been shown that the QCD sum rules could be fulfilled as well by a significantbroadening of the vector-meson spectral function in the medium [5, 6] without any significantmass shift. Calculations based on a quark model had indeed not yielded any significant masschange [7].The in-medium spectral function of the ρ -meson has been calculated in quite differentapproaches. In Refs. [8–12] purely hadronic resonance models were developed that do notinclude chiral symmetry restoration in the nuclear medium. In Refs. [13, 14] a chirallygauged linear sigma model with quarks was used to calculate ρ and a spectral functionsfrom analytically continued Functional Renormalization Group (aFRG) flow equations atfinite temperature and density. These studies demonstrated the degeneracy of the spectralfunctions of the chiral partners in a way that further supports the ρ broadening essentiallywithout mass shift, as chiral symmetry gets gradually restored at finite temperature and, inparticular, in the vicinity of a chiral critical endpoint at finite chemical potential.Early on dileptons have been used as probes for these in-medium changes of vector mesonsbecause dilepton ( e + e − or µ + µ − ) decays of hadrons are not distorted by hadronic final-stateinteractions and thus open a window to study the decays of short-lived hadronic resonancesinside the nuclear medium. These experiments are summarized and reviewed in [15–17] andmost recently in [18].Measuring e + e − pairs from heavy-ion collisions is in the focus of the experimental pro-gram of the HADES collaboration [19–22]. To disentangle the various contributions to themeasured dilepton spectra remains a major challenge in the extraction and interpretationof dilepton signals from nuclear matter, however. The decays of long-lived particles, mostimportantly the π and η Dalitz decays ( π , η → γe + e − ), which occur long after the breakupof the compressed nuclear configuration, are relatively well known experimentally. More dif-ficult is the evaluation of the bremsstrahlung, pn → pne + e − and π ± N → π ± e + e − , where onehas to rely mostly on theory. The two most prominent dilepton signals from nuclear matter3re the P ∆(1232) Dalitz decay, ∆ → N e + e − , and the ρ -meson direct decay, ρ → e + e − .The latter is of special interest due to the possibility to probe through this decay the ρ -mesonspectral function in the nuclear medium.The HADES results seem to indicate a significant broadening of the ρ meson in Ar+KClat 1.756 A GeV and Au+Au at 1.23 A GeV. On the basis of a so-called coarse-grainedtransport model [23–25], in which local thermal equilibrium is assumed, it was speculatedthat these new experimental results could signal the onset of chiral symmetry restoration[26, 27].In the present paper we instead perform microscopic transport simulations of the dileptonproduction in heavy-ion collisions at SIS18 energies, without invoking thermal equilibriumand thermal radiation which can be questionable at these rather low energies [28]. Thecalculations are based on the Giessen Boltzmann-Uehling-Uhlenbeck (GiBUU) microscopictransport model [29]. The focus of our present study is on the effect of collisional broadeningof the ρ -meson. We calculate the collisional width of the ρ -meson in excited nuclear matterby using the collision term of the transport equation. This width is then added to the free ρ decay width and used to evaluate the ρ spectral function in nuclear matter that is includedin the transport simulations.We analyse the detailed composition of the dilepton spectra, and present the time evolu-tion of the different components. We discuss the interplay between the collisional broadeningof the ρ -meson and its off-shell transport. We provide a systematic comparison of the GiBUUcalculations with available HADES data for the invariant mass, rapidity, and transverse mo-mentum distributions of the dileptons, and we also present an analysis of the dilepton excessradiation.The structure of our paper is as follows: In Sec. 2 we briefly describe the GiBUU trans-port model with particular emphasis on the off-shell propagation of the ρ meson and thedilepton production channels. Sec. 3 contains the formalism used to describe the ρ spectralfunction. We first demonstrate how the spectral function emerges in the splitting of produc-tion and decay processes of the ρ meson. We then discuss the collisional broadening of the ρ meson caused by the resonance production on the nucleons of the Fermi sea, ρN → R . Wepresent our results from the GiBUU transport simulations in Sec. 4, starting with the timeevolution of the density, the temperature, the invariant mass distribution of the ρ -meson,and the different components of the dilepton invariant mass spectrum. We then compare4ur calculations with HADES data for the dilepton observables in p + p collisions at beamenergies of 1.25 GeV, 2.2 GeV and 3.5 GeV, d + p at 1.25 A GeV, C+C at 1 A GeV and2 A GeV, Ar+KCl at 1.76 A GeV, and Au+Au at 1.23 A GeV. For the Au+Au system, wealso compare the calculated particle multiplicities with experimental data. The predictionsfor the dilepton invariant mass spectrum from Ag+Ag at 1.58 A GeV are given. Finally,our summary and conclusions together with a brief outlook are provided in Sec. 5. App. Aaddresses the calculated hadron multiplicities. App. B contains a discussion of the variousuncertainties which may influence our results.
2. THE MODEL
The GiBUU transport model [29] is built on the solution of the coupled set of semi-classical quantum-kinetic equations for the baryons ( N, ∆ , N ∗ , Y, . . . ), respective an-tibaryons ( ¯ N , ¯∆ , ¯ N ∗ , ¯ Y , . . . ) and mesons ( π, η, ρ, ω, K, K ∗ , . . . ). In relativistic kinematics[30] the kinetic equation for the nucleons with fixed isospin projection reads,( p ∗ ) − (cid:20) p ∗ µ ∂ µ + ( p ∗ µ F αµ + m ∗ ∂ α m ∗ ) ∂∂p ∗ α (cid:21) f ∗ ( x, p ∗ ) == Z g s d p ∗ (2 π ) v Z d Ω dσ → d Ω ( f ∗ f ∗ ¯ f ∗ ¯ f ∗ − f ∗ f ∗ ¯ f ∗ ¯ f ∗ ) , (1)where α = 1 , , µ = 0 , , ,
3. The left side of the kinetic equations describes particlepropagation in a self-consistent relativistic mean field (RMF) potential that includes scalar( S ) and vector ( V ) nuclear potentials as well as the Coulomb potential. The collisionintegrals on the right side of the kinetic equations describe two and three-body collisionsas well as resonance decays. Here, we have explicitly included only the expression for theelastic two-body collision integral, for simplicity, where f ∗ ( x, p ∗ ) with x ≡ ( t, r ) is thedistribution function of the nucleons in the kinetic phase space ( r , p ∗ ). It is defined suchthat f ∗ ( x, p ∗ ) g s d rd p ∗ / (2 π ) equals the number of particles in the phase space element d rd p ∗ ; g s = 2 is the nucleon spin degeneracy, p ∗ µ = p µ − V µ its kinetic four-momentumsatisfying the mass-shell condition p ∗ µ p ∗ µ = m ∗ , where m ∗ = m N + S is the Dirac massof the nucleon with mass m N = 0 .
938 GeV in the vacuum. F µν ≡ ∂ µ V ν − ∂ ν V µ is thefield-strength tensor of the vector potential V . In this work the RMF is included only for the baryons while the mesons feel only the Coulomb potential.
5n the collision integral, we have introduced short-hand notations f ∗ n ≡ f ∗ ( x, p ∗ n ) , ¯ f ∗ n ≡ − f ∗ n ( n = 1 , , , p ∗ ≡ p ∗ . The relative velocity of the colliding particles is definedas v = I /p ∗ p ∗ where I = p ( p ∗ p ∗ ) − ( m ∗ m ∗ ) is the M¨oller flux factor. The angulardifferential cross section is defined by dσ → for scattering in the solid angle element d Ω =sin Θ d Θ dφ with polar, Θ, and azimuthal, φ , scattering angles in the center-of-mass (c.m.)frame. The extension of the collision integral to include inelastic channels and broad particlesis rather straightforward (see Sec. 3.3 of Ref. [29] for details), and all these features are alsoincluded in our present GiBUU simulations.The scalar, S , and vector, V µ , mean fields are obtained from the Dirac equation for thenucleon, [ γ µ ( i∂ µ − V µ ) − ( m N + S )] ψ ( x ) = 0 , (2)coupled to the scalar-isoscalar σ -meson, the vector-isoscalar ω -meson, and the electromag-netic field A µ , via S = g σN σ, (3) V µ = g ωN ω µ + e τ ) A µ , µ = 0 , . . . , (4)where τ = +( − )1 for the proton (neutron); e = 1 / √
137 (in natural units with ¯ h = c = 1);and the coupling constants to the meson fields are those of the non-linear Walecka modelin the version NL2 of Ref. [31], i.e. g σN = 8 . g ωN = 7 .
54. The mesonic mean fields arecalculated by solving the Lagrange equations of motion with source terms provided by thebaryon densities and currents (see Sec. 3.1.3 of Ref. [29] for details).The numerical solution of Eq. (1) is based on the test-particle representation of thedistribution function f ∗ ( x, p ∗ ) = (2 π ) g s N N phys N X n =1 δ ( r − r n ( t )) δ ( p ∗ − p ∗ n ( t )) , (5)where N phys is the number of physical particles, while N is the number of test particles perphysical one. Turning off the interaction terms, by setting the right side in Eq. (1) to zero(the Vlasov limit), one obtains the equations of motion for the centroids of the δ -functions r n ( t ) , p ∗ n ( t ), ˙ r n = p ∗ n p ∗ n , (6)˙ p ∗ αn = p ∗ nµ p ∗ n F αµn + m ∗ n p ∗ n ∂m ∗ n ∂r α , (7)6here α = 1 , , µ = 0 , . . .
3. It can be shown that Eqs. (6) and (7) are equivalent tothe Hamiltonian equations ˙ r n = ∂ε ( r n , p n , t ) ∂ p n , (8)˙ p n = − ∂ε ( r n , p n , t ) ∂ r n , (9)with the single-particle energy defined as ε = V + p ( p ∗ ) + ( m ∗ ) . (10)Particles in the medium can be collision-broadened and/or have already a decay width invacuum. For the propagation of such broad particles, one has to use the off-shell transportimplemented in GiBUU. This is based on using a generalized distribution function that alsoincludes the particle energy as an independent variable [32–35], F ( x, p ) = (2 π ) N N phys N X n =1 δ ( r − r n ( t )) δ ( p − p n ( t )) δ ( p − ε n ( t )) . (11)The time evolution of the centroids r n ( t ) , p n ( t ) , ε n ( t ) is given by the so-called off-shell po-tential (OSP) ansatz [29] which is based on the following equations:˙ r n = (cid:18) − ∂H n ∂ε n (cid:19) − ∂H n ∂ p n , (12)˙ p n = − (cid:18) − ∂H n ∂ε n (cid:19) − ∂H n ∂ r n , (13)˙ ε n = (cid:18) − ∂H n ∂ε n (cid:19) − ∂H n ∂t . (14)Here H n ( ε n , p n , t, r n ) is a (generalized) single-particle Hamilton function as defined below.Eqs. (12) and (13) are obtained by expressing the partial derivatives of the single-particleenergy ε n ≡ ε ( r n , p n , t ) in Eqs. (8) and (9) in terms of those of H n using the self-consistencycondition, ε n = H n ( ε n , p n , t, r n ) . (15)Eq. (14) then follows directly from this self-consistency condition.The single-particle Hamilton function H n is defined such that particle n can be arbitrarilyfar off shell, when its in-medium width Γ n >
0, but becomes an on-shell particle for Γ n = 0.The simplest form of H n that satisfies these requirements is: H n = q m + Re Π + ∆ m n + p n , ∆ m n = − χ n Im Π , (16)7here m phys is the vacuum mass of the physical on-shell particle, Π( ε n , p n , t, r n ) is a retardedself-energy, and χ n is a constant fixed from the initial conditions at the production timeof the test particle n . In the most general case the long-range potential and short-rangecollisional interactions of the particle in the nuclear medium modify, respectively, the realand imaginary parts of the particle self-energy. In particular, the imaginary part of self-energy is related to the width by a usual bosonic formula:ImΠ = − p p n Γ n , (17)where p n = ε n − p n . Since the collisional width of the particle is roughly proportionalto the nucleon density, Eqs. (16) and (17) imply that for the particle with small naturaldecay width (e.g. a pion) the deviation of the particle-mass squared from its on-shell value,i.e. ∆ m n scales with the nucleon density. On the other hand, for particles with a largenatural decay width (e.g. the ρ ), the quantity ∆ m n becomes constant when the particle isemitted to the vacuum, since the decay width depends only on the particle invariant mass(cf. Eq. (55) below). In-particular, this means that the OSP ansatz without collisional widthsis equivalent to the treatment of broad particles with off-shell masses chosen according totheir vacuum spectral functions.With the Hamilton function of Eq. (16), the OSP ansatz is equivalent to solving thetest-particle equations of motion for relativistic off-shell bosons as derived from the retardedGreen function formalism [34]. In the present work, the OSP ansatz is applied to describe the dynamics of the ρ mesonin the nuclear medium; we set Re Π = 0 for simplicity. After time stepping according toEqs. (12) and (13), the single-particle energy ε n is obtained at the new time step by solvingEq. (15) for fixed three-momentum p n .The collision term on the right of the transport equation (1) is modeled geometrically:when the two test particles 1 and 2 are approaching their minimum distance b their collisionis simulated by Monte-Carlo provided b < p σ /π where σ is the total interaction crosssection. To approximate Lorentz covariance, the minimum distance b is calculated in thec.m. frame of the colliding particles assuming straight-line trajectories: b = ( x cm ) − ( x cm · β cm ) ( β cm ) , (18) The off-shell dynamics of vector mesons has first been discussed in Ref. [36] where an ad hoc form of theOSP ansatz with a scalar off-shell potential was introduced to bring off-shell particles back on-shell whenthey leave the nucleus. x cm = x cm − x cm − β cm t cm + β cm t cm (19)is the relative position vector at zero time; β cm and β cm are the particle velocities. Allquantities in Eqs. (18), (19) refer to the c.m. frame of particles 1 and 2. The four vectors( t cmi , x cmi ), i = 1 , t, x i ) inthe computational frame to the c.m. frame of the pair. This exactly corresponds to thecovariant prescription given by Eq. (2b) of Ref. [37]. The determination of the collision timeinstant is a more delicate problem. One can not naively use the collision instant (i.e. the timeof closest approach) in the c.m. frame since this would result in different collision instantsfor particles 1 and 2 in the computational frame. It has been shown in Ref. [37] that thecausality in a collision sequence can be preserved by using collision proper times. The propertime difference between the collision instant and the current instant for particle 1 is∆ τ = ˜ x · ˜ β ( ˜ β ) − ˜ t , (20)where the relative position vector at the zero time is now˜ x = ˜ x − ˜ x + ˜ β ˜ t . (21)All quantities with tilde refer to the rest frame of particle 1. The condition that particles1 and 2 pass their closest-approach distance during the time interval [ t − ∆ t/ t + ∆ t/ t is the time step in the computational frame, then can be written as | ∆ τ γ + ∆ τ γ | < ∆ t , (22)where γ , are the Lorentz factors of particles 1, 2 in the computational frame; and ∆ τ isthe proper time difference for particle 2 defined in its rest frame by Eqs. (20), (21) withreplacement 1 ↔ √ s < .
2) GeV are simulated within theresonance model which treats the interactions of nucleons, their resonances and mesonsexplicitly. At higher invariant energies the PYTHIA model [40] is applied. The model inprinciple also accounts for meson-meson collisions; these are, however, not important atSIS18 energies. Pauli blocking is applied for collisions with nucleons in the final state. For heavy-ion collisions, the c.m. frame of colliding nuclei is normally chosen as the computational frame.The choice of the time t (it was set to zero in actual calculations) does not influence the minimum distance,Eq. (18), since it leads only to the change of the component of x cm along the relative velocity β cm − β cm . • Direct decays of vector mesons, V → e + e − , with V = ρ, ω, φ : For vector mesons thispartial width is calculated based on the vector dominance model (VDM) [41, 42],Γ V → e + e − ( m ) = C V m V m (1 + 2 m e /m ) p − m e /m , (23)where m V and m are the on-shell and off-shell mass of the decaying meson, respectively,and C V = 4 πα / f V . The numerical values used in GiBUU are C ρ = 9 . · − , C ω = 7 . · − , and C φ = 1 . · − [43]. These values have been fit to reproducethe empirical partial widths V → e + e − for the on-shell meson masses. For a collision-broadened meson with a very small off-shell mass m , the applicability of this expressionwith constant C V becomes questionable, however. In particular, far below mass shell,Γ V from Eq. (23) grows artificially strongly with decreasing m until it reaches anunphysically sharp peak very close to the e + e − threshold where it vanishes. • Direct η → e + e − decay: For the partial decay width Γ η → e + e − = Γ tot η BR η → e + e − , whereΓ tot η = 1 . η , the phenomenological upper limit of thebranching ratio BR η → e + e − = 7 · − is adopted [44]. • Dalitz decays A → Be + e − : Using a factorization prescription (c.f. [45]) the followingexpression for the partial width can be obtained, d Γ A → Be + e − dm = Γ A → Bγ ∗ α πm (1 + 2 m e /m ) p − m e /m , (24)where m is the invariant mass of the dilepton pair, and Γ A → Bγ ∗ is the decay width tothe virtual photon in the final state.Meson Dalitz decays: In the case of the pseudoscalar meson P = π , η, η ′ decays P → γe + e − and the ω -meson decay ω → π e + e − the decay width to virtual photon isproportional to the decay width to the real photon [46]:Γ A → Bγ ∗ ( m ) = η sym Γ A → Bγ (cid:18) q Bγ ∗ ( m ) q Bγ ∗ (0) (cid:19) | F AB ( m ) | , (25)where η sym = 2 for pseudoscalar meson decays and η sym = 1 for ω -meson decay is thesymmetry factor, q Bγ ∗ ( m ) = m A − m B m A "(cid:18) m m A − m B (cid:19) − m A m ( m A − m B ) / (26)10s the c.m. momentum of B and γ ∗ , and F AB ( m ) is the A → B transition form fac-tor. The term ( q Bγ ∗ ( m ) /q Bγ ∗ (0) ) reflects the p -wave coupling arising in the effectiveLagrangian description of meson decays [47, 48]. For the partial decay widths to thereal photon the following values are used: Γ π → γγ = 0 . tot π with Γ tot π = 7 .
836 eV,Γ η → γγ = 511 eV, Γ η ′ → γγ = 4 . ω → π γ = 703 keV.For the form factor of the π → γγ ∗ vertex it is enough to use the linear approximationin m : F π γ ( m ) = 1 + b π m (27)with b π = 5 . − [46]. In the η → γγ ∗ vertex, the pole approximation is used: F ηγ ( m ) = (1 − m / Λ η ) − (28)with Λ − η = 1 .
95 GeV − as determined in the NA60 measurements of the low-mass µ + µ − pairs in 158 A GeV In+In collisions [49]. The η ′ → γγ ∗ vertex form factor isneglected. The form factor of the ω → π γ ∗ vertex is adopted from [50]: F ωπ ( m ) = Λ ω [(Λ ω − m ) + Λ ω Γ ω ] / , (29)where Λ ω = 0 .
65 GeV and Γ ω = 75 MeV. Eq. (29) is also in good agreement with theNA60 data [49].∆ Dalitz decay: In the case of the Delta resonance Dalitz decay, ∆(1232) → N e + e − ,we apply Eq. (24) with the partial decay width of Ref. [45],Γ ∆ → Nγ ∗ ( m ) = α
16 ( m ∆ + m N ) m m N [( m ∆ + m N ) − m ] / [( m ∆ − m N ) − m ] / | F ∆ N ( m ) | , (30)where the form factor is set to be constant, F ∆ N ( m ) ≡ F ∆ N (0) = 3 . ∆ → Nγ ∗ (0) = 0 .
66 MeV. • Bremsstrahlung: The pn → pne + e − and pp → ppe + e − bremsstrahlung is included inthe form of the boson exchange model of Ref. [51]. This model takes into accountthe e + e − emission from the internal charged pion exchange line in the pn scattering. An s -wave coupling is forbidden due to the presence of the pseudoscalar meson either in the initial or inthe final state of the A → Bγ ∗ decay. We apply the version with the pion electromagnetic form factor taking into account the direct couplingof the photon to the quark content of the pion, apart from the photon coupling to the ρ meson (the FF2parameterization of [51]). N N → N ∆ reactionfollowed by ∆ → N e + e − incoherently. The N ∗ (1520) Dalitz decays are effectivelyincluded in our calculations via the two-step decay N ∗ → ρN , ρ → e + e − .For the charged pion bremsstrahlung, π ± N → π ± N e + e − , the soft-photon approxima-tion (SPA) [52, 53] is applied, E dσ e + e − d pdm = α π σ el ( s ) mE R ( s ) R ( s ) , (31)with σ el ( s ) = Z −| t | max dt − tm π dσ el ( s, t ) dt ≃ q ( s ) m π σ el ( s ) , (32)where dσ el ( s, t ) /dt is the differential elastic scattering cross section, and q cm ( s ) =[( s + m π − m N ) / s − m π ] / is the c.m. momentum of pion and nucleon. ( E, p ) is thefour-momentum of the e + e − pair in the c.m. system of the colliding pion and nucleon.The reduction of the two-body phase space available for the outgoing pion and nucleonis included in Eq. (31) via the ratio of phase space volumes at the invariant energysquared with ( s ) and without ( s ) emission of the e + e − pair, where s = s + m − √ sE , (33)and R ( s ) = 2 q cm ( s ) / √ s . (34)In the last step of Eq. (32) an isotropic cross section is assumed for simplicity. Pion-nucleon scattering in heavy-ion collision processes at 1-2 A GeV is mostly mediatedby the ∆(1232) resonance, i.e. πN → ∆ → πN . In this case the angular distributionis forward-backward peaked due to the dominant p -wave. Thus, the approximation ofEq. (32) is quite rough. However, given a large overall uncertainty of the SPA at e + e − invariant masses above 100 −
200 MeV [52] the approximation of Eq. (32) seems stillreasonable.Since the cross section for e + e − production in pp collisions at the beam energy of1-3 GeV is quite small, on the µb level, the direct calculation of dilepton production inheavy-ion collisions would be extremely time consuming. Therefore, a so-called shiningmethod [54] is applied. In a given time step dt , the probability of the dilepton decay of12 resonance is P = Γ e + e − dt/γ where Γ e + e − is the partial decay width R → Xe + e − inthe resonance rest frame and γ is the Lorentz factor of the resonance. In the shiningmethod, the dilepton decay of every relevant resonance at every time step is simulatedand the produced e + e − pair carries the weight P which is then used to fill variousstatistical distributions. Note that the actual state of the resonance is not changedafter the dilepton emission, i.e. the resonance is further propagated according to thetest particle equations of motion, Eqs. (12), (13), and participates in collision anddecay processes. If the resonance survives until the end of time evolution, the decaydilepton will have the weight P = Γ e + e − / Γ where Γ is the total in-medium width ofthe resonance.The e + e − bremsstrahlung in pn , pp , and π ± N collisions is simulated in the followingway: If the collision takes place, the probability of the e + e − emission P = σ e + e − /σ is calculated where σ is the total interaction cross section of colliding particles and σ e + e − is the partial cross section of dilepton emission. The produced e + e − pair hasthe weight P while the actual two-body collision is simulated neglecting the dileptonemission.In the case of the pseudoscalar meson Dalitz decays, the polar angle distribution of theoutgoing e − in the rest frame of γ ∗ is sampled according to the distribution d P /d cos Θ ∝ Θ where the z -axis is chosen along the three-momentum of γ ∗ [55]. For all otherdilepton decays an isotropic angular distribution is used, just as for bremsstrahlung in thec.m. system of the colliding pair.
3. SPECTRAL FUNCTION OF THE ρ -MESON IN THE NUCLEAR MEDIUM The retarded ρ -meson propagator has the following spectral representation [14], G Rµν ( q ) = Z ∞ ds A ( s ) s q g µν − q µ q ν ( q + i − ~q − s , (35)where q is the four-momentum of the ρ -meson, so that the spin-averaged spectral functionof the ρ -meson is given by sgn( q ) A ( q ) = − π Im G R ( q ) , (36)13here G R ( q ) = 13 (cid:16) g µν − q µ q ν q (cid:17) G Rµν ( q ) (37)is the corresponding spin-averaged propagator. The tensor structure in Eq. (35) representsan off-shell extension of the sum over the polarization states of an on-shell ρ -meson withpole mass m ρ , X λ =0 , ± ε ( λ ) µ ε ∗ ( λ ) ν = − g µν + q µ q ν m ρ . (38)Neglecting a possible q -dependence in the real part of the ρ -meson self-energy Π( q ) (themore general case, including polarization dependence of the self-energy is discussed, e.g. inRefs. [8–10]), and writing Im Π( q ) = − p q Γ( p q ), on the other hand, we can make thefollowing Ansatz for the spin-averaged Feynman propagator of the ρ -meson in the nuclearmedium, G F ( q ) = 1 q − m ρ + i p q Γ( p q ) . (39)Here, Γ( p q ) defines the off-shell decay width of the ρ -meson (see below) which vanishesbelow the e + e − threshold, for q < m e . Including the sign function in the correspondingretarded self-energy, Im Π R ( q ) = − sgn( q ) p q Γ( p q ) in (36), this amounts to using thefollowing form for the spectral function, A ( q ) = p q Γ( p q ) /π ( q − m ρ ) + q Γ ( p q ) , (40)with the normalization condition, ∞ Z m e ds A ( s ) = 1 . (41)For later convenience, we also note that for q > m e we may write, | G F ( q ) | = π A ( q ) p q Γ( p q ) . (42)In order to illustrate the use of the spectral function for our purposes consider, for exam-ple, the process a + b → ρ → X . The invariant matrix element of this process is M X ; ab = − X λ =0 , ± M X ; ρ ( λ ) M ρ ( λ ) ; ab q − m ρ + i p q Γ , (43) This definition is equivalent to setting A ( q ) = (2 A T ( q )+ A L ( q )), where A T ( q ) and A L ( q ) are, respectively,the transverse and longitudinal spectral functions defined in Ref. [9]. ρ polarizations in the direct and conjugated amplitudes and performingthe independent summations over λ in the ρ -production and decay amplitudes squared, weobtain | M X ; ab | = | M X ; ρ ( λ ) | | M ρ ( λ ) ; ab | | q − m ρ + i p q Γ | , (44)where an overline means the sum over polarizations of final states and the averaging overpolarizations of initial states. The differential cross section of the process a + b → ρ → X isgiven by the following expression dσ a + b → X = (2 π ) | M X ; ab | d Φ X I ab , (45)where I ab = q ( p a p b ) − p a p b (46)is the M¨oller flux factor and d Φ X = δ (4) ( p a + p b − n X X i =1 p i ) n X Y i =1 d p i (2 π ) E i (47)is the invariant phase space volume element of the final state. Using the formula for thepartial decay width ρ → X in the rest frame of the ρ -meson, d Γ ρ → X = (2 π ) | M X ; ρ ( λ ) | d Φ X p q , (48)we can now rewrite Eq.(45) in a factorized form: dσ a + b → X = d Γ ρ → X Γ σ ab → ρ , (49)so that with Eq. (42) we obtain from Eq. (43), σ ab → ρ = 2 π A (cid:0) ( p a + p b ) (cid:1) | M ρ ( λ ) ; ab | I ab . (50)Here we have thus defined a quantity which has the meaning of a cross section for theproduction of an off-shell ρ -meson in the collision of particles a and b . Indeed, Eq. (50)can be obtained from the standard formula for an ab → ρ process where ρ is treated as afictitious “on-shell” particle with a mass of √ s , σ on − shellab → ρ = 2 πδ ( s − ( p a + p b ) ) | M ρ ( λ ) ; ab | I ab , (51)15hich is then multiplied with a weight given by A ( s ), and integrated over s . Thus, the squareof the invariant mass of the intermediate ρ -meson is distributed according to the spectralfunction A ( s ). Similar relations can be readily derived for any other process mediated byan off-shell ρ -meson. Moreover, using detailed balance, | M ab ; ρ ( λ ) | = (2 S a + 1)(2 S b + 1)2 S ρ + 1 | M ρ ( λ ) ; ab | , (52)Eq. (50) can be transformed into a relativistic Breit-Wigner formula, σ ab → ρ = 2 S ρ + 1(2 S a + 1)(2 S b + 1) 4 πq ab q ΓΓ ρ → ab ( q − m ρ ) + q Γ , (53)where S a,b,ρ are the spins of the involved particles, q ab = I ab / p q is the center-of-mass (c.m.)momentum of the colliding particles a and b , and Γ ρ → ab = | M ab ; ρ ( λ ) | q ab / πq denotes the ρ → ab decay width.The total width of ρ -meson in its rest frame is the sum of the vacuum decay width andthe collisional width in the nuclear medium:Γ = Γ dec + Γ coll . (54)The off-shell decay width Γ dec ( m ) = Γ ρ → ππ ( m )+Γ ρ → e + e − ( m ), where m ≡ p q is the off-shell ρ -meson mass parameter, is dominated by the ρ → ππ channel, for whichΓ ρ → ππ ( m ) = Γ ρ → ππ (cid:18) q cm ( m ) q cm ( m ρ ) (cid:19) m ρ m q cm ( m ρ ) R ) q cm ( m ) R ) , (55)where q cm ( m ) = p m / − m π is the c.m. momentum of the decay pions. In the presentcalculations the following values of the constants are used: m ρ = 775 . ρ → ππ =149 . R = 1 fm (see [56]).The collisional width of the ρ -meson in the nuclear medium is calculated in the semi-classical approximation, i.e. by using the loss term of the collision integral. For simplicity,isospin-symmetric nuclear matter is assumed. This leads to the following expression for thewidth in the ρ meson’s rest frame:Γ coll = γ Lor h v ρN σ ρN i ρ N , (56)where ρ N = ρ n + ρ p is the total nucleon density, σ ρN = ( σ ρn + σ ρp ) / ρ -meson nucleon cross section, v ρN = I ρN /q p N is the relative velocity of the ρ -meson16nd the nucleon, and γ Lor = q /m is the Lorentz factor of the ρ -meson. h . . . i denotes theaveraging over nucleon Fermi motion. The collisional width just defined is given in theso-called low-density approximation which is reflected in the linear dependence on density.At √ s < ∼ ρN cross section is saturated by the resonance production chan-nels. The corresponding partial resonance cross sections ρN → R are given by Eq. (53)with trivial replacements. At √ s ∼ ρN → πN background cross section which absorbs the missing part of the totalphenomenological ρN cross section.In order to explore the dependence of the collisional width on the density and the exci-tation energy of the average h . . . i in Eq. (56), the nucleon momentum p has been sampledby Monte Carlo according to a probability distribution dP ∝ n p d p where n p is the Fermidistribution at some finite temperature: n p = 1exp(( E p − µ ) /T ) + 1 , (57)with E p = p p + m N . The chemical potential µ for the given values of the nucleon density ρ N and temperature T has been determined from ρ N = Z d p (2 π ) n p . (58)We note that the ‘equivalent temperature’ here is introduced only as a parameter to char-acterize the excitation energy.Figs. 1 (a) and (b) show the collisional width of the ρ -meson calculated in ground-state nu-clear matter within the baryon resonance model using the resonance parameters of Ref. [56],i.e. those used in default GiBUU calculations. The set of resonance parameters [56] wasobtained within the multichannel unitarity analysis of πN scattering data. At q = 0, thedominant contributions are given by the resonances with large s -wave couplings to the ρN state: D N ∗ (1520) (Γ L =0 ρN / Γ tot = 21%) and S ∆ ∗ (1620) (Γ L =0 ρN / Γ tot = 25%). As expected,at finite momentum of the ρ -meson, the resonances with non-zero angular momentum cou-pling to the ρN state grow in importance. At q = 0 . N ∗ (1520) See Table A.3 in Ref. [29] for the full list of non strange resonances included in GiBUU. The N ∗ ( I = 1 / ρN → Y K included according to Ref. [57]. They are,however, of minor importance for the present study. -5 -4 -3 -2 -1
0 0.5 1 1.5(a) Γ c o ll / γ Lo r ( G e V ) m (GeV) q=0, ρ N =0.16 fm -3 , T=0 fullS (1535)S (1650)P (1900)D (1520)F (1680)G (2190)S (1620)S (1900)D (1700)D (1940)F (1750)F (1905) -5 -4 -3 -2 -1
0 0.5 1 1.5(b) Γ c o ll / γ Lo r ( G e V ) m (GeV) q=0.6 GeV/c, ρ N =0.16 fm -3 , T=0 -5 -4 -3 -2 -1
0 0.5 1 1.5(c) new res. Γ c o ll / γ Lo r ( G e V ) m (GeV) q=0, ρ N =0.16 fm -3 , T=0 -5 -4 -3 -2 -1
0 0.5 1 1.5(d) new res. Γ c o ll / γ Lo r ( G e V ) m (GeV) q=0.6 GeV/c, ρ N =0.16 fm -3 , T=0 FIG. 1. Collisional width of the ρ -meson in nuclear matter at saturation density and zero temper-ature as a function of the meson mass at momentum q = 0 (panels (a), (c)) and q = 0 . ρ -meson is divided out, i.e. the plotted width isgiven in the rest frame of nuclear matter. remains dominant, however, also the resonances with p -wave coupling to the ρN state, i.e. F ∆ ∗ (1750) (Γ L =1 ρN / Γ tot = 22%) and F ∆ ∗ (1905) (Γ L =1 ρN / Γ tot = 87%) contribute signifi-cantly. We also observe strongly increased contributions of S N ∗ (1650) (Γ L =2 ρN / Γ tot = 3%)and G N ∗ (2190) (Γ L =2 ρN / Γ tot = 29%) at finite momentum of ρ .18 Γ c o ll / γ Lo r ( G e V ) m (GeV) q=0 ρ N =0.32 fm -3 , T=70 MeV ρ N =0.32 fm -3 , T=0 ρ N =0.16 fm -3 , T=0 Γ c o ll / γ Lo r ( G e V ) m (GeV) q=0.6 GeV/c ρ N =0.32 fm -3 , T=70 MeV ρ N =0.32 fm -3 , T=0 ρ N =0.16 fm -3 , T=0 FIG. 2. Collisional width of the ρ -meson as a function of invariant mass in nuclear matter atdifferent densities and temperatures, solid (black): ρ N = 0 .
16 fm − , T = 0; long-dashed (blue): ρ N = 0 .
32 fm − , T = 0; and short-dashed (red): ρ N = 0 .
32 fm − , T = 70 MeV, as a function of themeson mass at momentum q = 0 (a) and q = 0 . ρ -meson is divided out. In the recent Ref. [61], the nucleon resonance parameters have been updated includingboth πN and γN scattering data. In order to assess the influence of these updates, we havealso used the new parameters of the resonances coupled to the ρN channel according toRef. [61]. The collisional width of the ρ -meson with the updated resonance parameters isshown in Figs. 1 (c) and (d). As compared to the default parameters, the most pronouncedchanges occur with the new ones for D (1520), S (1535), and S (1620). The branchingratio of the ( ρN ) S channel decreases from 21% to 14% for D (1520), while it increases from2% to 14% for S (1535). The mass of S (1620) decreases from 1672 MeV to 1589 MeV,while the total width decreases from 154 MeV to 107 MeV. As a result, the collisional widthof ρ meson is slightly larger at small masses with the new resonance parameters.Fig. 2 shows the invariant-mass dependence of the collisional width of the ρ -meson. At T = 0 we observe an approximate scaling Γ coll ∝ ρ N . Increasing the temperature leads to asmearing of the mass dependence. This is because at finite T the range of √ s of the colliding19 -5 -4 -3 -2 -1
0 0.5 1 1.5(a) A ( G e V - ) m (GeV) q=0 ρ N =0.32 fm -3 , T=70 MeV ρ N =0.32 fm -3 , T=0 ρ N =0.16 fm -3 , T=0vacuum -5 -4 -3 -2 -1
0 0.5 1 1.5(b) A ( G e V - ) m (GeV) q=0.6 GeV/c ρ N =0.32 fm -3 , T=70 MeV ρ N =0.32 fm -3 , T=0 ρ N =0.16 fm -3 , T=0vacuum FIG. 3. Spectral function of the ρ -meson in nuclear matter, Eq. (40), at different densities andtemperatures as a function of the meson mass at momentum q = 0 (a) and q = 0 . ρ N = 0 .
16 fm − , T = 0; long-dashed (blue) line – ρ N = 0 .
32 fm − , T = 0;short-dashed (red) line – ρ N = 0 .
32 fm − , T = 70 MeV; dot-dashed (magenta) line – ρ N = 0. Thecollisional width is calculated in the resonance + high energy model. ρN pair becomes broader for a fixed four momentum of the ρ . This leads to less pronouncedbaryon resonance structures. Overall, the collisional width is comparable to or even largerthen the vacuum ρ width ( ∼
149 MeV). Thus, significant modifications of the ρ spectralfunction due to collisional broadening can be expected.The resulting in-medium modifications to the spectral function of the ρ -meson are shownin Fig. 3. For a ρ meson at rest, the effects of the nuclear medium are only marginal. Thespike seen near the e + e − threshold is a consequence of the oversimplified description of the ρ width below the 2 π threshold, as discussed earlier in connection with Eq. (23). Moreover,the behaviour of the free ρ width in this low mass region is missing additional contributions,not included here, from ρ → µ + µ − and, in particular, the ρ → π γ decay channel that hasan order of magnitude larger branching ratio as compared to ρ → e + e − [44].In contrast, at finite momentum of the ρ , there is a dramatic enhancement of the spectralstrength at small invariant masses due to the additional collisional width. The effect oftemperature is quite small and more visible for the meson at rest, in form of a moderate20 A ( G e V - ) m (GeV) ρ N =0.16 fm -3 , T=0 vacuumq=0q=0.4 GeV/cq=0.8 GeV/c FIG. 4. Spectral function of the ρ -meson in ground-state nuclear matter for different values ofthe three momentum | q | : 0 – solid (black) line, 0.4 GeV/c – long-dashed (blue) line, 0.8 GeV/c –short-dashed (red) line. The vacuum spectral function is shown by the dot-dashed (magenta) line. broadening of the spectral strength.For comparison with Fig. 8 of Ref. [9], we present in Fig. 4 the spectral function calculatedat three different values of the ρ -meson three momentum. We observe a tendency of abroadening towards lower invariant masses with increasing q . However, our results showa somewhat smaller collisional broadening as compared to the self-energy calculations ofRefs. [9, 11]. In particular, the double-humped structure in the transverse spectral functionobtained there due to the coupling to the N ∗ (1520) resonance is missing here. Given thesimplicity of our resonance model for the in-medium ρ spectral function, on the other hand,the overall agreement with the more sophisticated resonance models of Refs. [9, 11] is quitecompelling.
4. RESULTS
Before comparing our calculations with experimental data in Sec. 4.1 below, we firstconsider the time evolution of some selected observables. Fig. 5 shows baryon density andtemperature in the position of the center-of-mass for central collisions of C+C at 1 A GeV21 ρ B ( f m - ) t (fm/c) Au+Au, 1.23 A GeVC+C, 1 A GeVC+C, 2 A GeVAr+KCl, 1.756 A GeVAg+Ag, 1.58 A GeV T ( M e V ) t (fm/c) Au+Au, 1.23 A GeVC+C, 1 A GeVC+C, 2 A GeVAr+KCl, 1.756 A GeVAg+Ag, 1.58 A GeV
FIG. 5. (Color online) The central baryon density (a) and temperature (b) vs time for Au+Auat 1.23 A GeV (black solid line), C+C at 1 A GeV (blue dashed line), C+C at 2 A GeV (browndotted line), Ar+KCl at 1.756 A GeV (red dash-dotted line), and Ag+Ag at 1.58 A GeV (magentadash-double-dotted line) The impact parameter is set to zero for all systems. and 2 A GeV, Ar+KCl at 1.756 A GeV, Ag+Ag at 1.58 A GeV, and Au+Au at 1.23 A GeV.The temperature has been extracted locally in position space by fitting h p i of the baryonsin the local rest frame of nuclear matter using the Fermi distribution, Eq. (57). Note thatthis is only an effective equivalent temperature, the colliding systems are not necessarilyfully equilibrated at any time. It is also obvious that there is no thermal equilibrium atthe initial inter-penetration stage when the two counter-streaming flows of nucleons onlystart to decelerate each other by elastic and inelastic N N collisions. Thus, the extremelyhigh temperatures at the beginning of the collision must not be considered as real physicalones, but demonstrate that T is just a parameter to fit the non-equilibrium momentumdistribution of the baryons by a Fermi distribution having the same h p i . Earlier studieshave in fact indicated that, at the relatively low bombarding energies considered here, fullthermal equilibrium is not achieved during the high-density phase of the collision. [28, 31].The density evolution looks quite simple and intuitive: it consists of a compression stagefollowed by a plateau behaviour, and finally the expansion of the system. Central baryondensities of up to 2 − ρ are reached; where ρ = 0 .
16 fm − is the nuclear saturation density.22or the Au+Au system, the temperature evolution pattern shows up a bump at t ≃
18 fm/ccorrelated with the end of the density plateau. At these times, the calculated temperatureis about T ∼
80 MeV. We have checked that the sum of the pion and ∆ multiplicities, as anestimate of the total inelastic production, saturates at approximately the same time. Thisimplies that during the density plateau stage the temperature drops mainly due to inelasticproduction. In contrast, at the expansion stage the temperature drops mostly because fastnucleons leave the central zone faster, a feature of kinetic free streaming. A similar behavioris observed for other colliding systems.Fig. 6 displays the time evolution of the ρ -meson invariant mass distribution. At the initialstage of a collision, hard first-chance N N collisions and multistep processes allow to producebaryonic resonances in a broad mass range (in particular, the N ∗ (1520) that dominates the ρ production). Thus, there is not much phase-space limitation for the ρ production here,and a large part of the ρ spectral strength, including the on-shell peak region, is populatedin N ∗ → ρN decays. With increasing time, baryon resonance production becomes governedby soft πN collisions which ultimately leads to smaller invariant masses of the produced ρ ’s. Another effect, which shifts the ρ strength to smaller invariant masses with increasingtime is the ρ → ππ decay, since the Γ ρ → ππ width grows with the invariant mass of the ρ , see Eq. (55). Altogether, this leads to a softening of the ρ -invariant-mass spectrum at t > ∼
20 fm/c, even in calculations with vacuum ρ spectral function.Including the collisional width in the ρ spectral function leads to a softer ρ invariant-massspectrum at the early stage, t < ∼
15 fm/c, due to the spreading of the ρ spectral strengthtowards lower invariant masses, see Fig. 3 (b) above. The collisional width of the ρ -meson hasbeen determined by using the local values of baryon density and temperature calculated onthe spatial grid with step size ∼ . − . ρ ’s as free particles with fixed masses (blue dashed lines in Fig. 6) theexcess of the low-mass ρ ’s ( m < ∼ . ρ ’s graduallymigrate closer to the on-shell peak. Therefore, at late times the off-shell transport, throughthe OSP ansatz of Eqs. (12) – (17), produces ρ mass distributions close to those with vacuum ρ width. These observations are in-line with HSD model calculations [62].Fig. 7 shows the time evolution of the dilepton spectrum in central Au+Au collisions23 Au +
Au, E=1.23 A GeV, b=0 d N ρ / d M ( G e V - ) M (GeV) 0 0.1 0.2 0.3 0.4 0.5 0 0.5 1t=10 fm/c d N ρ / d M ( G e V - ) M (GeV) vac. ρ coll. broad. ρ o.s. pot. d N ρ / d M ( G e V - ) M (GeV) 0 0.5 1 1.5 0 0.5 1t=20 fm/c d N ρ / d M ( G e V - ) M (GeV) 0 0.5 1 0 0.5 1t=25 fm/c d N ρ / d M ( G e V - ) M (GeV) 0 0.5 1 0 0.5 1t=30 fm/c d N ρ / d M ( G e V - ) M (GeV)
FIG. 6. (Color online) Invariant mass per-event spectra of ρ -mesons produced at different momentsin time for central Au+Au ( b = 0 fm) at 1.23 A GeV: Black solid line – vacuum width, blue dashedline – vacuum and collisional width, brown dotted line – vacuum and collisional width with OSPansatz. -8 -7 -6 -5 -4 -3 -2 -1
0 0.2 0.4 0.6 0.8 1 1.2 t=5 fm/c
Au +
Au, E=1.23 A GeV, b=0 d N / d M e + e - ( G e V - ) M e + e - (GeV) total ρ → e + e - ω → e + e - φ → e + e - ω → π e + e - π → e + e - γη → e + e - γ∆ → Ne + e - η → e + e - η ’ → e + e - γ pn Bremspp Brems π N Brems -8 -7 -6 -5 -4 -3 -2 -1
0 0.2 0.4 0.6 0.8 1 1.2 t=15 fm/c d N / d M e + e - ( G e V - ) M e + e - (GeV)10 -8 -7 -6 -5 -4 -3 -2 -1
0 0.2 0.4 0.6 0.8 1 1.2 t=20 fm/c d N / d M e + e - ( G e V - ) M e + e - (GeV) 10 -8 -7 -6 -5 -4 -3 -2 -1
0 0.2 0.4 0.6 0.8 1 1.2 t=25 fm/c d N / d M e + e - ( G e V - ) M e + e - (GeV)10 -8 -7 -6 -5 -4 -3 -2 -1
0 0.2 0.4 0.6 0.8 1 1.2 t=30 fm/c d N / d M e + e - ( G e V - ) M e + e - (GeV)10 -8 -7 -6 -5 -4 -3 -2 -1
0 0.2 0.4 0.6 0.8 1 1.2 after res. dec. d N / d M e + e - ( G e V - ) M e + e - (GeV) FIG. 7. (Color online) Invariant mass per-event spectrum of e + e − pairs produced in Au+Au at1.23 A GeV, b = 0 fm at different time moments. Solid black lines show the total spectrum. Otherlines show the partial contributions of the different production channels as indicated. Calculationsinclude the collisional width of the ρ meson within the OSP ansatz. Full acceptance is assumed.
25t 1.23 A GeV. At early times ( t ∼ pn bremsstrahlung. Then we observe a dramatic increase of the ρ → e + e − component thatquickly becomes dominant at M e + e − = 0 . − M e + e − < ∼ . πN bremsstrahlung contribution is quiteclose to that of the ∆ Dalitz one.Individual views of the time evolution of the most important partial components ofthe dilepton mass spectrum are provided in Fig. 8. We observe that the pn and πN bremsstrahlung components practically saturate at 15 fm/c when the primary stoppingis over and the system reaches the highest compression state (see Fig. 5 above). The∆ → N e + e − component at larger dilepton invariant masses, M e + e − > ∼ . ρ → e + e − decay component is much slower which reflects themultistep processes of the ρ production, mostly mediated by the N ∗ (1520). The spectrumaround the pole mass of the ρ is practically saturated at 30 fm/c since the life-time ofthe ρ meson at the pole mass is only 1.3 fm/c. Thus, large-mass ρ ’s decay very quicklyafter decoupling from the fireball. However, the ρ ’s with masses only slightly above the2 π threshold are long-lived in vacuum. Thus, their dilepton decays continue until quite latetimes, on the order of ∼
60 fm/c. Note that the OSP ansatz leads to almost vanishing ρ massspectrum below 2 m π at late times, except for a peak at extremely small invariant massesdue to the growing partial width Γ ρ → e + e − towards small invariant masses (see Fig. 6). Thisexplains the behaviour of the ρ → e + e − component in the dilepton mass spectrum below2 m π .The Dalitz decays of the π and η mesons have large branching fractions. Thus, theircontributions are almost entirely dominated by time scales much larger than the GiBUUevolution time. While this is equivalent to the summation of the decays of all produced π ’sand η ’s at the end of the GiBUU time evolution, for consistency we also show in Fig. 8 thesmall contributions of π and η decays that occur during the GiBUU time evolution.The final ρ → e + e − component of the dilepton invariant mass spectrum is shown togetherwith its sub-components in Fig. 9. Indeed, the decays of the D N ∗ (1520) resonance are themain source of ρ ’s providing the largest contribution both in the total integrated spectrumand in the intermediate invariant mass region 0 . − . -8 -7 -6 -5 -4 -3 -2 -1
0 0.2 0.4 0.6 0.8 1 1.2 pn Brems
Au +
Au, E=1.23 A GeV, b=0 d N / d M e + e - ( G e V - ) M e + e - (GeV) final15 fm/c10 fm/c5 fm/c -8 -7 -6 -5 -4 -3 -2 -1
0 0.2 0.4 0.6 0.8 1 1.2 π N Brems d N / d M e + e - ( G e V - ) M e + e - (GeV) final15 fm/c10 fm/c5 fm/c -8 -7 -6 -5 -4 -3 -2 -1
0 0.2 0.4 0.6 0.8 1 1.2 ∆ → Ne + e - d N / d M e + e - ( G e V - ) M e + e - (GeV) final20 fm/c15 fm/c10 fm/c5 fm/c -8 -7 -6 -5 -4 -3 -2 -1
0 0.2 0.4 0.6 0.8 1 1.2 ρ → e + e - d N / d M e + e - ( G e V - ) M e + e - (GeV) final30 fm/c15 fm/c10 fm/c5 fm/c -8 -7 -6 -5 -4 -3 -2 -1
0 0.2 0.4 0.6 0.8 1 1.2 η → e + e - γ d N / d M e + e - ( G e V - ) M e + e - (GeV) final30 fm/c25 fm/c20 fm/c15 fm/c10 fm/c -8 -7 -6 -5 -4 -3 -2 -1
0 0.2 0.4 0.6 0.8 1 1.2 π → e + e - γ d N / d M e + e - ( G e V - ) M e + e - (GeV) final30 fm/c25 fm/c20 fm/c15 fm/c10 fm/c FIG. 8. (Color online) Time evolution of the most important partial components of the e + e − invari-ant mass spectrum from Au+Au at 1.23 A GeV, b = 0 fm: pn bremsstrahlung, πN bremsstrahlung,∆(1232) Dalitz decay, ρ → e + e − decay, η Dalitz decay, and π Dalitz decay. Calculations includethe collisional width of the ρ meson within the OSP ansatz. Full acceptance is assumed. -8 -7 -6 -5 -4 -3 -2 -1
0 0.2 0.4 0.6 0.8 1 1.2 1.4 d N / d M e + e - ( G e V - ) M e + e - (GeV) Au +
Au, E=1.23 A GeV, b=0total ρ → e + e - D (1520)higher res. ππ FIG. 9. (Color online) Dilepton invariant mass spectrum produced in ρ → e + e − decays fromAu+Au at 1.23 A GeV b = 0 fm. Solid line – total spectrum. Dashed and dotted lines – partialcontributions from ρ ’s produced in decays of N ∗ (1520) resonance and higher resonances, respec-tively. Dot-dashed line – partial contribution from ππ → ρ process. Calculations are done includingthe collisional width of ρ -meson within the OSP ansatz. Full acceptance is assumed. resonances coupled to the ρN final state (cf. Fig. 1 above) and the ππ collisions provide thedominant contribution at higher invariant masses > ∼ . ππ sub-component does not vanish below 2 m π . This is related to the off-shell dynamics of the ρ meson that changes its invariant mass. As discussed after Eq. (17), the OSP ansatz hasthe effect that a ρ meson moving towards higher density regions tends to shift away fromthe mass shell (and vice versa). The HADES collaboration has measured inclusive dilepton spectra at SIS18 energies forthe following systems: p + p collisions at beam energies of 1.25 GeV [63], 2.2 GeV [64], and28.5 GeV [65], d + p collisions at beam energy of 1.25 A GeV [63], C+C at 1 A GeV [20] and2 A GeV [19], Ar+KCl at 1.76 A GeV [21], and Au+Au at 1.23 A GeV [22]. Recently themeasurements have also been performed for Ag+Ag at 1.58 A GeV although the data havenot being published yet.Below, if not specially mentioned, the calculated spectra are smeared according to theHADES detector resolution and filtered through the HADES acceptance filter. After that,the proper angular and momentum cuts are taken into account. For the Au+Au system at1.23 A GeV the dedicated acceptance filter does not exist yet. Thus, we have applied forthat system the filter for d + p at 1.25 A GeV where the magnetic field setting is similar [66].After filtering the opening angle cut Θ e + e − > ◦ and restrictions on the e − and e + momenta0 . < p e ± < . π multiplicity”(we will refer to this as the pseudo neutral-pion multiplicity in App. A), N π , experimentallydefined as N π ≡ ( N π + + N π − ) / , (59)with the charged pion multiplicities N π ± obtained by extrapolation to the full solid angle.In the following comparisons we do not use this normalization but instead compare with thedilepton data themselves since these are measured only in a limited acceptance window. Amore detailed discussion of pion numbers is given in App. A. pp and dp collisions Fig. 10 shows the invariant mass e + e − spectra from p + p collisions at 1.25 GeV, 2.2 GeV,and 3.5 GeV, as well as d + p collisions at 1.25 A GeV. The experimental spectra from p + p collisions are very well described by our GiBUU transport model simulations.In the case of d + p collisions, following Ref. [63] only n + p collisions were taken into29 -5 -4 -3 -2 -1
0 0.2 0.4 0.6 0.8 1 d σ / d M e + e - ( µ b / G e V ) M e + e - (GeV) p + p, E=1.25 GeVtotal ρ → e + e - π → e + e - γ∆ → Ne + e - pp Brems -5 -4 -3 -2 -1
0 0.2 0.4 0.6 0.8 1 1.2 1.4 d σ / d M e + e - ( µ b / G e V ) M e + e - (GeV) p + p, E=2.2 GeV total ρ → e + e - ω → e + e - ω → π e + e - π → e + e - γη → e + e - γ∆ → Ne + e - η → e + e - pp Brems -5 -4 -3 -2 -1
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 d σ / d M e + e - ( µ b / G e V ) M e + e - (GeV) p + p, E=3.5 GeV total ρ → e + e - ω → e + e - φ → e + e - ω → π e + e - π → e + e - γη → e + e - γ∆ → Ne + e - η → e + e - pp Brems -5 -4 -3 -2 -1
0 0.2 0.4 0.6 0.8 1 d σ / d M e + e - ( µ b / G e V ) M e + e - (GeV) d + p, E=1.25 A GeVtotal, corr.total ρ → e + e - π → e + e - γη → e + e - γ∆ → Ne + e - η → e + e - pn Brems FIG. 10. (Color online) Invariant-mass differential cross section of dilepton production in p + p collisions at the beam energies 1.25, 2.2, and 3.5 GeV, and in d + p collisions at beam energy1.25 A GeV. Thick solid (black) lines show the total calculated cross sections. Other lines show thepartial contributions of the different production channels as indicated. For the d + p reaction, thecross section of the pn bremsstrahlung component (see Eq. (60)) was thereby corrected in orderfor the total (thick black line) cross section to agree with experiment. The total cross withoutthis correction is shown as the thin solid (black) line for comparison. Experimental data are fromRefs. [63–65]. In the d + p reaction, the n + p collisions were exclusively selected by detecting thefast forward spectator proton. η production isbelow threshold in N N collisions at 1.25 GeV, the η Dalitz component for d + p collisionsat 1.25 A GeV is predominantly due to neutron Fermi motion. While the introduction of acoupling of the virtual photon to an exchanged charged pion led to a significant increase ofthe mass spectrum around M e + e − ≈
500 MeV [51] this is still not sufficient to describe theexperimental dilepton yield in this region.The remaining discrepancy might be due to a very simple OBE model used to describe
N N scattering. Also, at the invariant masses near the quasi-free threshold M max e + e − = √ s NN − m N = 0 .
545 GeV the high-momentum part of the deuteron wave function which is subjectto light-cone corrections [68, 69] might become important (and is missing in the elementarycross section used here). In the present work we therefore decided to tune the elementary p + n cross section to the experimental d + p data at 1.25 A GeV by multiplying the pn bremsstrahlung component of the dilepton spectrum by the factor, f ( M ) = C wM /b (exp[( a − M ) /d ] + 1)(exp[( M − b ) /d ] + 1) + 1 , (60)with dilepton invariant mass M in GeV, C = 1 . d = 0 . a = 0 . b = 0 .
55, and w = 3 . pn bremsstrahlung component (shown as dotted magenta line in the figure) by the factor f ( M ) of Eq. (60). For comparison, the corresponding total spectrum without this rescaling isalso shown as thin black line. Once fixed phenomenologically from the elementary reaction,we then use the same factor f ( M ) with the same parameters also for all other cross sections,such as momentum distributions, for heavy-ion collisions at beam energies around 1 A GeV.Fig. 11 displays the transverse momentum differential cross section of e + e − productionin pp collisions at 2.2 GeV in different invariant mass windows where the dominant contri-butions are π → γe + e − (a), η → γe + e − (b), and ρ → e + e − (c). The ρ → e + e − decay and pp bremsstrahlung also provide the two main contributions at large p t ’s in the intermediatemass window (b). There is a good overall agreement of the calculated dilepton p t spectrawith experiment. 31 -3 -2 -1
0 0.2 0.4 0.6 0.8 1 (a) d σ e + e - / dp t e + e - [ µ b / ( G e V / c ) ] p te + e - (GeV) p + p, E=2.2 GeV, M e + e - < 0.15 GeVtotal ρ → e + e - ω → π e + e - π → e + e - γη → e + e - γ∆ → Ne + e - pp Brems -4 -3 -2 -1
0 0.2 0.4 0.6 0.8 1 (b) d σ e + e - / dp t e + e - [ µ b / ( G e V / c ) ] p te + e - (GeV) p + p, E=2.2 GeV, 0.15 GeV < M e + e - < 0.45 GeVtotal ρ → e + e - ω → e + e - ω → π e + e - η → e + e - γ∆ → Ne + e - pp Brems -4 -3 -2 -1
0 0.2 0.4 0.6 0.8 1 (c) d σ e + e - / dp t e + e - [ µ b / ( G e V / c ) ] p te + e - (GeV) p + p, E=2.2 GeV, M e + e - > 0.45 GeVtotal ρ → e + e - ω → e + e - ω → π e + e - η → e + e - γ∆ → Ne + e - η → e + e - pp Brems FIG. 11. (Color online) Transverse momentum differential cross section of dilepton production in p + p collisions at the beam energy 2.2 GeV in the invariant mass intervals M e + e − < .
15 GeV(a), 0 .
15 GeV < M e + e − < .
45 GeV (b), and M e + e − > .
45 GeV (c). The thick solid (black) lineshows the total calculated cross section. The other lines show the partial contributions of differentproduction channels as indicated. Experimental data are from Ref. [64]. -9 -8 -7 -6 -5 -4 -3 -2
0 0.2 0.4 0.6 0.8 1 vac. ρ (a) d N / d M e + e - ( G e V - ) M e + e - (GeV) C + C, E=1 A GeV total, corr.total ρ → e + e - ω → e + e - ω → π e + e - π → e + e - γη → e + e - γ∆ → Ne + e - η → e + e - pn Bremspp Brems π N Brems -9 -8 -7 -6 -5 -4 -3 -2
0 0.2 0.4 0.6 0.8 1 in-med. ρ (b) d N / d M e + e - ( G e V - ) M e + e - (GeV) C + C, E=1 A GeV
FIG. 12. (Color online) Invariant mass spectrum of dileptons produced in C+C collisions at1 A GeV calculated with vacuum (a) and in-medium (b) ρ spectral function. Thin and thicksolid lines show the total calculated spectrum before and after correction of the pn bremsstrahlungcomponent (see Eq.(60)), respectively. Other lines show the partial contributions of the differentproduction channels to the total spectrum as indicated. Experimental data are from Ref. [20]. .1.2. C + C collisions Fig. 12 shows the dilepton invariant mass spectrum from C+C collisions at 1 A GeV.At small invariant masses the spectrum is saturated by the π Dalitz decay component. At M e + e − > ∼ . ρ → e + e − decay. At intermediateinvariant masses in the range M e + e − ≃ . − . → N e + e − , η → γe + e − , pn bremsstrahlung, and ρ → e + e − . Thus, theintermediate region is quite complex and the disagreement with experiment might be causedby any one of these four components, or an accumulated effect from inaccuracies in severalof these four. However, we observe that just using the tuned pn bremsstrahlung componentof Eq.(60) solves the problem of missing yield in the intermediate region of invariant mass(thick solid line).The physical effect of collisional broadening of the ρ meson on the other hand is stillrather weak for this light C+C system. However, including the collisional width of the ρ meson allows to reduce statistical fluctuations in the ρ → e + e − component of the spectra at M e + e − < m π , producing a smoother behaviour of this component. Nevertheless, significantstatistical fluctuations of the ρ → e + e − component still persist in the transverse momentumand rapidity distributions of dileptons at small M e + e − , as seen in Figs. 13 (a), 14 (a), 16 (a),and 17 (a) below.The transverse momentum and rapidity distributions of dileptons produced in C+C col-lisions at 1 A GeV are shown in Fig. 13 and 14, respectively. In the lowest invariant massrange ( M e + e − < .
15 GeV), the spectra are saturated by the π Dalitz decay, with the othercontributions suppressed by more than one order of magnitude. in the higher invariant masswindow (0 .
15 GeV < M e + e − < .
50 GeV), the composition of the dilepton spectra is morecomplex. The small- p t part is governed by the ∆ Dalitz decays while at large p t ’s there arecomparable contributions of η Dalitz decay, ρ → e + e − decay, and pn bremsstrahlung. Wesee again that using the tuned pn bremsstrahlung improves the description of the experi-mental data. This illustrates that the pn bremsstrahlung is an essential component in thespectra and has to be quantitatively brought under control.The dilepton invariant mass spectrum in C+C collisions at 2 A GeV is shown in Fig. 15. The π → e + e − γ component extends slightly above m π due to the smearing by the detector resolution. Here we show the cases with the in-medium ρ only. Calculations with vacuum ρ produce practicallyindistinguishable spectra, except for the somewhat stronger statistical fluctuations in the ρ → e + e − components at small invariant masses. -8 -7 -6 -5 -4 -3 -2
0 0.2 0.4 0.6 0.8 1 in-med. ρ (a) d N e + e - / dp t e + e - [ ( G e V / c ) - ] p te + e - (GeV) C + C, E=1 A GeV, M e + e - < 0.15 GeV total corr.total ρ → e + e - π → e + e - γη → e + e - γ∆ → Ne + e - pn Bremspp Brems π N Brems -8 -7 -6 -5 -4
0 0.2 0.4 0.6 0.8 1 in-med. ρ (b) d N e + e - / dp t e + e - [ ( G e V / c ) - ] p te + e - (GeV) C + C, E=1 A GeV, 0.15 GeV < M e + e - < 0.50 GeV FIG. 13. (Color online) Transverse momentum distributions of dileptons produced in C+C colli-sions at 1 A GeV in the invariant mass intervals M e + e − < .
15 GeV (a), 0 .
15 GeV < M e + e − < . ρ spectral functions. Thin and thick solid linesshow the total spectrum before and after correction of the pn bremsstrahlung component (seeEq.(60)), respectively. Other lines show partial contributions of the different production channelsas indicated. Experimental data are from Ref. [70]. -8 -7 -6 -5 -4 -3
0 0.5 1 1.5 2 in-med. ρ (a) d N e + e - / d Y e + e - Y e + e - C + C, E=1 A GeV, M e + e - < 0.15 GeV total corr.total ρ → e + e - π → e + e - γη → e + e - γ∆ → Ne + e - pn Bremspp Brems π N Brems -8 -7 -6 -5
0 0.5 1 1.5 2 in-med. ρ (b) d N e + e - / d Y e + e - Y e + e - C + C, E=1 A GeV, 0.15 GeV < M e + e - < 0.50 GeV FIG. 14. (Color online) Rapidity distributions of dileptons produced in C+C collisions at 1 A GeVin the invariant mass intervals M e + e − < .
15 GeV (a), 0 .
15 GeV < M e + e − < .
50 GeV (b).Calculations were done with in-medium ρ spectral functions. The total spectrum before and aftercorrection of the pn bremsstrahlung component (see Eq.(60)) is shown by thin and thick solidlines, respectively. Other lines show partial contributions of the different production channels asindicated. Experimental data are from Ref. [70]. -9 -8 -7 -6 -5 -4 -3 -2
0 0.2 0.4 0.6 0.8 1 1.2 vac. ρ (a) d N / d M e + e - ( G e V - ) M e + e - (GeV) C + C, E=2 A GeV total ρ → e + e - ω → e + e - φ → e + e - ω → π e + e - π → e + e - γη → e + e - γ∆ → Ne + e - η → e + e - η ’ → e + e - γ pn Bremspp Brems π N Brems -9 -8 -7 -6 -5 -4 -3 -2
0 0.2 0.4 0.6 0.8 1 1.2 in-med. ρ (b) d N / d M e + e - ( G e V - ) M e + e - (GeV) C + C, E=2 A GeV
FIG. 15. (Color online) Invariant mass spectra of dileptons produced in C+C collisions at 2 A GeVcalculated with vacuum (a) and in-medium (b) ρ spectral function. The thick solid (black) lineshows the total calculated spectrum. Other lines show partial contributions of the different pro-duction channels to the total spectrum as indicated. Experimental data are from Ref. [19]. M e + e − the spectrum is dominatedby the π Dalitz and ρ → e + e − decays, respectively. However, in contrast to the same systemat lower energy, now the intermediate mass region is almost saturated by the η Dalitz decays.There are also rather strong contributions of the ω → e + e − and φ → e + e − decays in theinvariant mass regions near their pole masses. We observe an overall quite perfect agreementbetween our GiBUU results and the experimental data.Fig. 16 shows the transverse momentum distributions of dileptons from C+C collisions at2 A GeV in the three invariant mass windows with dominant π Dalitz, η Dalitz, and ρ → e + e − decays, respectively, in the order of increasing M e + e − . It is interesting to compare thesespectra to those from p + p collisions at 2.2 GeV in Fig. 11. The leading components in eachmass window experience sharp cutoffs in the p + p case while they are falling exponentiallywith p t in the C+C case. This difference is largely caused by Fermi motion inside the carbonnuclei. The experimental data in all three invariant mass windows are described very well.The rapidity distributions of the dileptons from C+C collisions at 2 A GeV are shownin Fig. 17. They are not symmetric around mid-rapidity ( Y = 0 .
9) due to the experimentalacceptance. The experimental data are well described, except for the intermediate invariantmass region dominated by the η Dalitz decay where our calculations underestimate thedilepton yield at forward rapidities. A possible reason is the oversimplified description of η production in decays of N ∗ (1535) which are modeled isotropically in the resonance restframe. The dilepton invariant mass spectrum from Ar+KCl collisions at 1.756 A GeV is shownin Fig. 18. For this colliding system, similar to C+C at 2 A GeV, the spectrum at M e + e − ≃ . − . η Dalitz decays. There are also clearly visible peaks dueto the ω → e + e − and φ → e + e − decays. In calculations with vacuum ρ , we furthermore seea pronounced shoulder at the ρ pole mass which is smoothed by the collisional broadening ofthe ρ . However, some overestimation of the data at the ρ pole mass remains. Fig. 19 displaysthe transverse mass distributions of the dileptons from Ar+KCl collisions at 1.756 A GeV.The calculated m t spectra agree with experimental data in all invariant mass windows, The transverse collective flow effect is expected to be small in C+C system. In the calculations we have replaced KCl by Ar. -8 -7 -6 -5 -4 -3 -2
0 0.2 0.4 0.6 0.8 1 in-med. ρ (a) d N e + e - / dp t e + e - [ ( G e V / c ) - ] p te + e - (GeV) C + C, E=2 A GeV, M e + e - < 0.15 GeV -8 -7 -6 -5 -4
0 0.2 0.4 0.6 0.8 1 in-med. ρ (b) d N e + e - / dp t e + e - [ ( G e V / c ) - ] p te + e - (GeV) C + C, E=2 A GeV, 0.15 GeV < M e + e - < 0.55 GeV -9 -8 -7 -6 -5
0 0.2 0.4 0.6 0.8 1 in-med. ρ (c) d N e + e - / dp t e + e - [ ( G e V / c ) - ] p te + e - (GeV) C + C, E=2 A GeV, M e + e - > 0.55 GeV total ρ → e + e - ω → e + e - ω → π e + e - π → e + e - γη → e + e - γ∆ → Ne + e - η → e + e - pn Bremspp Brems π N Brems
FIG. 16. (Color online) Transverse momentum distributions of dileptons produced in C+C colli-sions at 2 A GeV in the invariant-mass intervals M e + e − < .
15 GeV (a), 0 .
15 GeV < M e + e − < .
55 GeV (b), and M e + e − > .
55 GeV (c). Calculations were done with in-medium ρ spectralfunctions. Thick solid (black) lines show the total calculated cross sections, other lines the partialcontributions of different production channels as indicated. The fluctuations of the ρ → e + e − component in the lowest invariant mass interval (a) are purely statistical. Experimental data arefrom Ref. [71]. -8 -7 -6 -5 -4 -3
0 0.5 1 1.5 2 in-med. ρ (a) d N e + e - / d Y e + e - Y e + e - C + C, E=2 A GeV, M e + e - < 0.15 GeV -7 -6 -5
0 0.5 1 1.5 2 in-med. ρ (b) d N e + e - / d Y e + e - Y e + e - C + C, E=2 A GeV, 0.15 GeV < M e + e - < 0.55 GeV -9 -8 -7 -6
0 0.5 1 1.5 2 in-med. ρ (c) d N e + e - / d Y e + e - Y e + e - C + C, E=2 A GeV, M e + e - > 0.55 GeV total ρ → e + e - ω → e + e - ω → π e + e - π → e + e - γη → e + e - γ∆ → Ne + e - η → e + e - pn Bremspp Brems π N Brems
FIG. 17. (Color online) Rapidity distributions of dileptons produced in C+C collisions at 2 A GeVin the invariant-mass intervals M e + e − < .
15 GeV (a), 0 .
15 GeV < M e + e − < .
55 GeV (b), and M e + e − > .
55 GeV (c). Calculations are done with in-medium ρ spectral function. Thick solid(black) lines show the total calculated cross sections, other lines show the partial contributions ofdifferent production channels as indicated. The fluctuations of the ρ → e + e − component in thelowest invariant mass interval (a) are purely statistical. Experimental data are from Ref. [71]. -8 -7 -6 -5 -4 -3 -2
0 0.2 0.4 0.6 0.8 1 1.2 vac. ρ (a) d N / d M e + e - ( G e V - ) M e + e - (GeV) Ar + KCl, E=1.756 A GeV total ρ → e + e - ω → e + e - φ → e + e - ω → π e + e - π → e + e - γη → e + e - γ∆ → Ne + e - η → e + e - η ’ → e + e - γ pn Bremspp Brems π N Brems -8 -7 -6 -5 -4 -3 -2
0 0.2 0.4 0.6 0.8 1 1.2 in-med. ρ (b) d N / d M e + e - ( G e V - ) M e + e - (GeV) Ar + KCl, E=1.756 A GeV total ρ → e + e - ω → e + e - φ → e + e - ω → π e + e - π → e + e - γη → e + e - γ∆ → Ne + e - η → e + e - η ’ → e + e - γ pn Bremspp Brems π N Brems
FIG. 18. (Color online) Invariant mass spectra of dileptons produced in Ar+KCl collisions at1.756 A GeV calculated with vacuum (a) and in-medium (b) ρ spectral function. Thick solid(black) line shows the total calculated spectrum. Other lines show the partial contributions ofthe different production channels to the total spectrum as indicated. Experimental data are fromRef. [21]. -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1
0 0.5 1 1.5 m t - / d N / d m t ( G e V - / ) m t (GeV) Ar + KCl, E=1.756 A GeV M ee < 0.13 GeV0.13 GeV < M ee < 0.30 GeV0.30 GeV < M ee < 0.45 GeV (x0.1)0.45 GeV < M ee < 0.65 GeV (x0.01)M ee > 0.65 GeV (x0.001)vac. ρ in-med. ρ FIG. 19. (Color online) Transverse mass spectra of dileptons produced in Ar+KCl collisions at1.756 A GeV in different invariant mass windows as indicated. Calculations with vacuum and in-medium ρ are shown by solid and dotted lines, respectively. Experimental data are from Ref. [21]. except for the highest one ( M ee > .
65 GeV) where our calculations produce softer m t spectra and overestimate the yields, in line with the excess observed in Fig. 18.Similar results (with vacuum ρ ) are obtained in the SMASH model [25]. The HSDmodel [72] produces a somewhat better description of the dilepton invariant mass yieldfrom Ar+KCl near the ρ/ω pole masses with a similar qualitative effect of the collisionalbroadening of the ρ meson. In Refs. [23–26, 73], the coarse graining approach has beenapplied for the description of heavy-ion collisions at SIS18 energies. This is a hybrid approachbased on the assumption of local thermal equilibrium with parameters T, µ extracted fromthe microscopic transport calculations. The microscopic transport contribution of the vector-meson decays is then (partly) substituted by the dilepton emission from the thermal systemwith in-medium spectral functions. Coarse-grained transport simulations describe the entiredilepton invariant mass spectrum for Ar+KCl very well. Note however that establishinglocal thermal equilibrium at SIS18 energies can be a delicate issue [28], especially for lightcolliding systems, which needs verification on a case by case basis. While in our present42alculations we also resort to an equivalent local temperature in evaluating the ρ spectralfunction for simplicity, to avoid numerically expensive calculations of ρ collision rates fromthe actual dynamics, in contrast to Refs. [23–26, 73] on the other hand, we entirely relyon our microscopic transport simulations in the computation of dilepton emission from thenon-equilibrium system. The Au+Au system at 1.23 A GeV is currently in the focus of studies by the HADEScollaboration (cf. [22, 74] and refs. therein). This is the heaviest system measured so far.Therefore, one expects the deviations due to the various in-medium effects from the super-position of quasi-free
N N collisions as reference to be stronger than in the other collidingsystems. As we have already seen, inclusive dilepton production is influenced by many reac-tion processes. Since the production of mesons π , η, ρ, ω decaying into dileptons is mediatedby baryon resonances, it is especially important to have baryon resonance production, ab-sorption and decay in the nuclear medium well under control. For this purpose the GiBUUmodel has been extensively applied to various reactions with nuclear targets, such as heavy-ion, (anti-)proton, pion, photon, electron, and neutrino induced (semi-)inclusive productionprocesses [29].For comparison, the dilepton invariant-mass spectrum in Au+Au at 1.23 A GeV withvacuum ρ spectral function is shown in Fig. 20 (a) with the relativistic and (c) with theSkyrme-like mean fields (see App. B for detail). The resonance structure in the ρ → e + e − component is clearly visible in either case. Compared to that, in Fig. 20 (b) and (d), we seethat the collisional broadening of the ρ meson is strong enough in this system, to remove theshoulder in the spectrum near the ρ pole mass and hence, to yield better agreement withdata. As before in this energy range, however, without applying the correction factor ofEq. (60) to the pn bremsstrahlung component we would again underpredict the data in theintermediate invariant-mass region. This lends further support to the hypothesis that thereis indeed some strength missing in the pn bremsstrahlung component at these low energies.As already mentioned in Sec. 3, the set of resonances of Ref. [56] is used in GiBUU bydefault. The nucleon resonance parameters have been recently updated in Ref. [61]. Theinfluence of these new resonance parameters on the collisional width of the ρ -meson has43 -8 -7 -6 -5 -4 -3 -2 -1
0 0.2 0.4 0.6 0.8 1 1.2 vac. ρ (a) d N / d M e + e - ( G e V - ) M e + e - (GeV) Au +
Au, E=1.23 A GeV total, corr.total ρ → e + e - ω → e + e - φ → e + e - ω → π e + e - π → e + e - γη → e + e - γ∆ → Ne + e - η → e + e - η ’ → e + e - γ pn Bremspp Brems π N Brems -8 -7 -6 -5 -4 -3 -2 -1
0 0.2 0.4 0.6 0.8 1 1.2 in-med. ρ (b) d N / d M e + e - ( G e V - ) M e + e - (GeV) Au +
Au, E=1.23 A GeV -8 -7 -6 -5 -4 -3 -2 -1
0 0.2 0.4 0.6 0.8 1 1.2 vac. ρ , Skyrme(c) d N / d M e + e - ( G e V - ) M e + e - (GeV) Au +
Au, E=1.23 A GeV -8 -7 -6 -5 -4 -3 -2 -1
0 0.2 0.4 0.6 0.8 1 1.2 in-med. ρ , new res.(d) d N / d M e + e - ( G e V - ) M e + e - (GeV) Au +
Au, E=1.23 A GeV
FIG. 20. (Color online) Invariant-mass spectra of dileptons produced in Au+Au collisions at1.23 A GeV calculated with vacuum (a), (c) and in-medium (b), (d) ρ spectral function. Thecalculation with the Skyrme-like potential is shown in panel (c). Panel (d) shows the results ob-tained with updated resonance parameters. Thick solid black lines show the total spectra obtainedfrom GiBUU output with multiplying the pn bremsstrahlung component by the factor given inEq. (60). Thin solid black lines show the same spectra without this correction. The other linesshow the partial contributions of different production channels to the total spectra as indicated.Experimental data are from Ref. [22]. ρ meson isslightly larger at small masses with the new resonance parameters. Comparing the resultingdilepton invariant-mass spectrum in Fig. 20 (d) with that obtained from the analogouscalculation with the default resonance parameters in Fig. 20 (b), we observe that the overalleffect of the updated resonance parameters is rather small, with an only moderate increasein the ρ → e + e − component slightly further improving the agreement with experiment atlow invariant masses.Overall, the agreement of the calculations with the experimental data is very good. Inparticular, also the π Dalitz-decay component at small masses is in excellent agreement withthe data. This indicates that the number of produced neutral pions is correctly described.It is, therefore, surprising to see that the calculated total number of pions produced issignificantly larger than the number of experimentally measured pions, as shown in App. A.The latter discrepancy seems to be fairly model-independent since other generators giveroughly the same overestimate (see Fig. 7 in [74]). We will discuss this problem further inApp. A.
In the recent Ref. [75], the SMASH (+ coarse graining) model predictions for the dileptoninvariant mass spectra from Ag+Ag at the beam energy 1.58 A GeV were given, indicatingpractically full coincidence of the Ag+Ag and Au+Au spectra. Fig. 21 displays our predic-tions for the total dilepton invariant mass spectrum and its partial ρ → e + e − componentfor Ag+Ag at 1.58 A GeV. The corresponding spectra for Au+Au at 1.23 A GeV are alsoshown for comparison. In the π Dalitz region, the spectra for Ag+Ag and Au+Au are al-most identical since the (true) π multiplicities per event are quite close: 9.7 for Ag+Ag vs11.7 for Au+Au. However, at larger invariant masses the spectra from Ag+Ag and Au+Audiffer significantly. In the intermediate mass range, M e + e − ≃ . − . ρ -dominated region, M e + e − > ∼ . ∼
30% for Ag+Ag relative to45 -8 -7 -6 -5 -4 -3 -2 -1
0 0.2 0.4 0.6 0.8 1 1.2 in-med. ρ d N / d M e + e - ( G e V - ) M e + e - (GeV) Ag+Ag, 1.58 A GeVAu+Au, 1.23 A GeVAg+Ag, ρ → e + e - Au+Au, ρ → e + e - FIG. 21. (Color online) Dilepton invariant-mass spectra for Ag+Ag at 1.58 A GeV (black lines)and Au+Au at 1.23 A GeV (blue lines). Solid and dashed line show the total spectra and theirpartial ρ → e + e − components, respectively. Calculations were done with the in-medium ρ spectralfunction. The centrality selection is 0 −
40% for both systems, and modeled in the sharp cutoffapproximation for the impact parameter, with b < . b < . Au+Au in the ρ mass region, is more robust. We have checked that a similar enhancementpresents also in calculation with vacuum ρ spectral function. A small, although statisticallysignificant, difference between predictions of GiBUU and SMASH models is most probablydue to different resonance parameters. In-medium effects, such as multiple scattering, secondary particle interactions and, pos-sibly, modifications of the vacuum spectral properties of the ρ are also being discussed interms of the excess radiation, that is the dilepton spectrum without η and ω decay con-46ributions, and with the N N reference spectrum subtracted. The latter describes dileptonradiation in the picture of first-chance
N N collisions and is defined as follows (see [21, 22]): dN ref dM e + e − = (cid:18) c pp σ tot pp dσ ppe + e − dM e + e − + c np σ tot np dσ npe + e − dM e + e − (cid:19) A part . (61)Here, c pp and c np are the fractions of pp + nn and np collisions, respectively, calculable asfollows: c pp = ( P pp + P nn ) σ tot pp ( P pp + P nn ) σ tot pp + P np σ tot np , (62) c np = P np σ tot np ( P pp + P nn ) σ tot pp + P np σ tot np , (63)where P pp = ( Z/A ) , P nn = ( N/A ) , and P np = 2 ZN/A are the probabilities that arandomly chosen N N pair will be, respectively, pp , nn or np one for symmetric ( A, Z )+(
A, Z )nuclear collisions. The total pp and np cross sections calculated using GiBUU at E lab =1 .
25 GeV are σ tot pp = 48 mb and σ tot np = 39 mb, in a good agreement with empirical data [44].The effective participant number can be determined as the ratio of the π multiplicities inthe studied AA and in the N N collision, i.e. A part = N AAπ N NNπ , (64)where N NNπ = c pp σ tot pp σ ppπ + c np σ tot np σ npπ . (65)Note that it is assumed in Eqs. (61) - (65) that the total cross sections, the dilepton andthe π production cross sections in pp and nn collisions are the same. For dileptons thisassumption is obviously quite rough. It is needed here, however, because Eq. (61) is usedin the experimental analysis as well [21, 22]. For internal consistency, we apply Eq. (61)with all quantities calculated from GiBUU. To avoid some possible misunderstanding, inthis subsection the π multiplicity refers properly to the charge neutral pions, and not tothe average charged pion multiplicity of Eq. (59).Fig. 22 shows the excess dilepton radiation spectrum dN excess dM e + e − = dN AA dM e + e − − dN ref dM e + e − (66)calculated in full acceptance. Here dN AA /dM e + e − is the invariant mass spectrum in AA collisions with the η and ω decay components removed. In contrast to the smoothly dropping47 -7 -6 -5 -4 -3 -2 d N e xc e ss / d M e + e - ( G e V - ) M e + e - (GeV) Au +
Au, E=1.23 A GeVvac. ρ in-med. ρ in-med. ρ , pn brems. corr. FIG. 22. (Color online) The invariant mass spectrum of excess dileptons produced in Au+Aucollisions at 1.23 A GeV calculated according to Eq. (66). The dashed (blue) versus the thick andthin solid (black) lines depict calculations with vacuum ρ width versus in-medium ρ width withand without correction of the pn bremsstrahlung according to Eq. (60), respectively. Calculationswere done without filtering, i.e. in full acceptance. Experimental data are from Ref. [22]. experimental spectrum with dilepton invariant mass, the calculation with vacuum ρ showsup a bump at the ρ pole mass and a valley in the intermediate mass region. The collisionalbroadening of the ρ meson improves the agreement with experiment, although the deviationstill remains. The correction of the pn bremsstrahlung by Eq. (60) further improves theagreement in the intermediate mass region.The system mass dependence of the excess radiation can be studied by using the yieldratio [21, 22] R AA = dN AA dM e + e − (cid:18) dN ref dM e + e − (cid:19) − . (67)The η contribution is subtracted in both, the AA and the reference, spectra. At low e + e − invariant masses, where the dilepton spectrum is saturated by the π Dalitz decays, oneobtains R AA = 1 in full acceptance. Taking into account the HADES acceptance in thecalculation of dilepton spectra while keeping the full π multiplicities in the definition of48he participant number, Eq. (64), results in deviations from unity of R AA in the π Dalitzregion.Fig. 23 shows the yield ratio for the different colliding systems. For C+C at 1 and2 A GeV, following Ref. [21], the calculations have been performed within Ar+KCl accep-tance, both for the spectrum from heavy-ion collision and for the reference spectrum. Forthe Au+Au system, the d + p acceptance filter at 1.25 A GeV has been applied, while the pp and np components of the reference spectrum have been calculated with pp and dp ac-ceptance filters at 1.25 A GeV, respectively. Our calculations correctly reproduce the maintrend present in the HADES data, i.e. the enhanced dilepton yields at intermediate invariantmasses with respect to the yield at small invariant masses for the heavy colliding systems,Ar+KCl and Au+Au. Again, the correction of the pn bremsstrahlung improves the agree-ment with the C+C data at 1 A GeV and the Au+Au data at 1.23 A GeV. The figure alsoshows that in the mass-region between about 0.2 and 0.4 GeV not only in-medium effectscontribute, but that there is also a strong sensitivity to the pn -bremsstrahlung.
5. SUMMARY AND CONCLUSIONS
We have performed microscopic transport calculations of dilepton ( e + e − ) production inheavy-ion collisions at E beam = 1 − pA reactions [43]. The main model inputs such as resonance parameters, elementary crosssections, and dilepton production channels are the same as in Ref. [43]. As compared toRef. [43], the present calculations include the pp and np bremsstrahlung described in theframework of the boson-exchange model [51] that provides a somewhat better descriptionof the d + p data (cf. our Fig. 10 and Fig. 3 of Ref. [43] where the np bremsstrahlung iscalculated in the soft-photon approximation).The most important novel feature of the present work is the self-consistent descriptionof the ρ -meson spectral function in the nuclear medium. Self-consistency here implies thatthe collisional width of the ρ -meson that enters the spectral function is calculated from thecollision term of the transport equation, i.e. it includes the contribution of the ρN collisions. In the previous microscopic transport studies of dilepton production [43, 62, 72] the collisional broadeningof the ρ -meson has been included in the linear density approximation. (a) R AA M e + e - (GeV) C+C, 1 A GeVin-med. ρ in-med. ρ , pn Brems. corr. (b) R AA M e + e - (GeV) C+C, 2 A GeVin-med. ρ (c) R AA M e + e - (GeV) Ar + KCl, 1.756 A GeVin-med. ρ (d) R AA M e + e - (GeV) Au + Au, 1.23 A GeVin-med. ρ in-med. ρ , pn Brems. corr. FIG. 23. The yield ratio of Eq. (67) normalized at the experimental value for M e + e − ≃ .
07 GeVas a function of the dilepton invariant mass. Panels (a), (b), (c), and (d) correspond to C+C at1 A GeV, C+C at 2 A GeV, Ar+KCl at 1.756 A GeV, and Au+Au at 1.23 A GeV. The calculationsinclude collisional broadening of the ρ meson. For C+C at 1 A GeV and Au+Au at 1.23 A GeVthe thick and thin solid lines show, respectively, the result with and without the pn bremsstrahlungcorrection factor in Eq. (60). Experimental data are from Refs. [21, 22]. ρ -meson. The applied procedure is basedon using the density and h p i of the baryons in the local rest frame of the baryonic matter.Such a procedure is expected to be accurate enough for the highly-compressed state of thebaryonic matter formed in the central zone of a heavy-ion collision.In order to describe the propagation of the off-shell ρ -meson in the presence of collisionalbroadening, we have applied the relativistic off-shell potential ansatz where the deviation ofthe actual off-shell particle mass squared from the pole mass squared is proportional to thetotal width of the particle, with the constant of proportionality defined at the productiontime of the particle. This allows to recover the vacuum spectral function of the particlewhen it is emitted to the vacuum outside the baryonic matter.We compared the results of GiBUU calculations with HADES data on the dilepton in-variant mass spectra for p + p at E beam = 1 .
25, 2.2, and 3.5 GeV, d + p at 1.25 A GeV,C+C at 1 and 2 A GeV, Ar+KCl at 1.756 A GeV, and Au+Au at 1.23 A GeV, and alsoprovided our predictions for Ag+Ag at 1.58 A GeV. The model calculations agree with allHADES data at E beam ≃ p + p at 1.25 A GeV are alsodescribed quite well. However, for d + p at 1.25 A GeV, C+C at 1 A GeV, and Au+Au at1.23 A GeV there is a systematic underprediction of the dilepton yield in the intermediateinvariant-mass range, M e + e − ≃ . − . ρ -meson smears out the peak in the dileptoninvariant-mass spectrum near the ρ pole mass and increases the dilepton yield in the interme-diate invariant-mass range. The effect is most clearly visible in the heaviest system Au+Au,while it does practically not influence the dilepton spectra for C+C and only weakly changesthat for Ar+KCl. The overall strength of the ρ collisional broadening alone, however, is notsufficient to completely account for the missing strength in the intermediate invariant massregion in Au+Au at 1.23 A GeV.This motivated us to adjust the n + p bremsstrahlung cross section at E beam = 1 .
25 GeVby a dilepton invariant-mass dependent factor so as to describe the inclusive dilepton datafor d + p at 1.25 A GeV. Multiplying the n + p bremsstrahlung component in C+C at1 A GeV and Au+Au at 1.23 A GeV by the same factor, without further adjustments,then leads to good agreement with the data on the inclusive dilepton invariant-mass spectra51n these heavier systems as well. Further circumstantial evidence for the increased n + p bremsstrahlung cross section in vicinity of 1 A GeV beam energy is also provided by animproved agreement of the excess dilepton yield (see Fig. 22) and yield ratios (see Figs. 23(a), (d)) with the experimental data.While the suggested enhancement of the n + p bremsstrahlung is thus effective in im-proving the intermediate mass dilepton yields of all systems in the range of 1 − . η production near threshold which isextremely difficult to constrain from experimental data in p + p and p + n collisions withhigh precision. This thus introduces an additional uncertainty in the input cross sections.Therefore, the missing dilepton yield in d + p collisions at 1.25 A GeV might be also atleast partly attributed to missing strength in the η Dalitz decay component. The detailedtheoretical analysis of the exclusive dp → e + e − npp fast cross sections [76], which is beyondthe scope of our present work, would be needed to further test the suggested enhancementof the n + p bremsstrahlung component. The availability of precisely determined exclusive dp → γe + e − npp fast cross sections would also be useful for better constraining the differentpartial components of the dilepton invariant-mass spectra.Last but not least, the processes involving the deuteron, which would be sub-thresholdfor the corresponding p + p or n + p collisions on a free target proton at the same beamenergy per nucleon, depend on the deuteron wave function at high momenta. The latteris subject to significant relativistic corrections, as follows from the light-cone description ofthe deuteron [68, 69]. This possibility was not considered in the present work and remainsto be studied in future. ACKNOWLEDGMENTS
We thank Tetyana Galatyuk, Volker Metag, Mark Strikman, and Janus Weil for stimulat-ing discussions and interest in this work. The support by the Frankfurt Center for ScientificComputing is gratefully acknowledged. This work was financially supported by the GermanFederal Ministry of Education and Research (BMBF), Grant No. 05P18RGFCA.52 ppendix A: Hadron Numbers1. Pion numbers
At the end of Sect. 4.1.4 we have briefly mentioned a problem connected with the mea-sured vs. calculated pion numbers. We, therefore, now list in Table I the calculated pseudoneutral-pion multiplicities, Eq. (59), together with the corresponding HADES data.
TABLE I. Pseudo neutral π multiplicities N π as defined by Eq. (59) from GiBUU for variouscolliding systems. The results for Au+Au and Ag+Ag are geometrically weighted in the impactparameter ranges that corresponds to 0 −
40% centrality which are b < . b < . N π exp. Ref.Au+Au, 1.23 A GeV 13.1 8 . ± .
52 [22]C+C, 1 A GeV 0.53 0 . ± .
08 [20]C+C, 2 A GeV 1.01 1 . ± .
16 [77]Ar+KCl, 1.76 A GeV 4.2 3 . ± .
25 [21]Ag+Ag, 1.58 A GeV 10.1
While the calculated pseudo neutral-pion multiplicities for C+C at 1 A GeV and 2 A GeVagree with the experimental numbers very well, and those for Ar+KCl still reasonablywell, the theoretical values for the Au+Au system lie about 50% above the experimentallydetermined values. Also the charged pion multiplicities listed in Table II below overestimateHADES measurements by about 50%, i.e. just by the same factor as for the pseudo neutralpions for Au+Au at 0 −
40% centrality. In our calculations, the numbers of true neutral pions, 17.6 at 0 −
10% and 11.7 at 0 − −
10% and 13.1 at 0 −
40% centrality. We can thus exclude this difference as the main The charged pion multiplicities for GiBUU reported in Ref. [74] are slightly different because of the defaultmode of GiBUU using Skyrme-like baryonic mean fields. π Dalitz region and the pseudoneutral-pion multiplicity.The fact that the virtual photons from π Dalitz decays are in agreement with the ex-perimentally measured dileptons in the corresponding invariant-mass range, while the totalpion yields are not, thus remains a puzzle.Any simple mechanism to reduce the pion yields would inevitably also reduce the π Dalitz contribution to the dilepton invariant-mass spectra, which, however, agrees very wellwith the data, see Fig. 20. Our calculations using a medium-dependent suppression ofresonance excitations from [78] have indeed shown that effect. Such a medium-dependentsuppression therefore then requires an additional explanation of missing dilepton yield inthe π Dalitz region. One possibility might be a shortcoming of the acceptance filter usedin our calculations. A designated acceptance filter for Au+Au would certainly help to closein on such a possibility in the future.
2. Hadron multiplicities
As a further benchmark test, Fig. 24 shows particle multiplicities in central collisions ofAu+Au at 1.23 A GeV. The calculated multiplicities are weighted with the impact parameterin the range b = 0 − . .
6) fm for the 0 − N corr .p = N GiBUU p − N exp d − N exp t − N exp He − N est He = 118 , (A1)where N GiBUU p = 168 is the calculated proton multiplicity; N exp d = 28 . ± . N exp t = 8 . ± . N exp He = 4 . ± . He measured exper-imentally [80]. The estimated multiplicity of α particles, N est He ≃ . N exp He, according toEOS data for central Au+Au collisions at 1 A GeV (see Ref. [81] and refs. therein). Table IIsummarizes calculated and measured particle multiplicities. The corrected proton multiplic-ity still overestimates the experimental value by ∼ -4 -3 -2 -1 p d π - π + t He η Λ K + K K - φ y i e l d Au +
Au, E=1.23 A GeVdataGiBUUGiBUU, protons corrected
FIG. 24. (Color online) Particle multiplicities per event in Au+Au collisions at 1.23 A GeV( √ s NN = 2 . missing heavier cluster contributions in Eq. (A1). The detailed analysis of cluster productionis obviously a difficult problem that is not in the focus of this work.The calculated η multiplicity agrees very well with the experimental value. This impliesthat the η → e + e − γ decay component of the dilepton invariant spectra is described correctly. Appendix B: Residual Uncertainties
One important aspect of the GiBUU transport model is the self-consistent baryonic mean-field potential that can be provided either by a Skyrme-like energy density functional or bythe relativistic mean-field (RMF) Lagrangian of the non-linear Walecka model. In orderto estimate the uncertainty caused by the specific choice of mean-field model, we havealso performed calculations using the soft momentum-dependent (SM) Skyrme-like potential[85] with an incompressibility or bulk modulus of nuclear matter at normal density, K =215 MeV. This is quite close to the value K = 210 MeV for the NL2 version [28] of the RMFmodel used in the majority of calculations presented in this work.55 ABLE II. Particle multiplicities per event for Au+Au at E beam = 1 .
23 A GeV. For protons thecorrected multiplicity, Eq. (A1), is given in parentheses.particle N GiBUU N exp centrality ref. p
168 (118) 77 . ± . −
10% [80] π −
25 17 . ± . −
10% [74] π +
15 9 . ± . −
10% [74] η . ± .
056 0 −
20% [82]Λ 0.16 (8 . ± . · − −
10% [83] K + . ± . · − −
10% [84] K S . ± . · − −
10% [83] K − . · − (3 . ± . · − −
20% [84] φ . · − (1 . ± . · − −
20% [84]
While the equation of state is therefore almost the same for SM and RMF NL2, theenergy dependence of the optical potential is quite different (see Figs. 2 and 3 of Ref. [29]),however. This is a well-known effect of the original RMF model that leads to too repulsiveSchroedinger-equivalent potentials at high momenta [30]. Note, however, that around beamenergy of 1 GeV the both potentials are close to the phenomenological one. Thus, we thinkthat the influence of different momentum dependencies in SM and RMF NL2 on heavy-ioncollisions at ∼ √ s free , used in the calculations ofthe baryon-baryon cross sections also differ between the Skyrme-like and RMF modes ofcalculation. In particular, we see from Fig. D69 of Ref. [29] that the N N cross sections aredialed at larger invariant energies in the RMF mode than in the SM mode (we have alsochecked this by direct comparison of the √ s free distributions in the baryon-baryon collisionsfor Au+Au at 1.23 A GeV). One therefore also expects differences in the ∆ resonanceproduction between the two modes of calculation. Dilepton invariant-mass spectra calculatedwith the RMF versus the SM mean fields are compared in Figs. 20 (a) and (c). Indeed, the56 → N e + e − components differ somewhat between the two modes: in the RMF mode(Fig. 20 (a)) there is a second maximum near M e + e − = 0 . M e + e − . These agreeswith our expectation of effectively more energetic N N → N ∆ collisions in the RMF mode.Overall, however, the difference between results obtained with the RMF and the Skyrmepotential is not relevant.Also note that the ∆ potential in the GiBUU default Skyrme-like mode is scaled by afactor of 2/3 relative to the nucleon potential, corresponding to results from the ∆-holemodel. In contrast, the RMF calculation was run with the scalar and vector potentialsacting on the ∆ assumed to be the same as the nucleonic ones. We have checked, however,that changing the scaling factor of the ∆ potential from 2 / → πN ∆ dynamics largely dominates the inelastic processes at SIS18 energies. Here, theuncertainties come from at least two sources: First, the ∆ resonance experiences collisionalbroadening in nuclear medium as well. We have checked by using the potential model of∆ spreading of Ref. [86] that the effect of the in-medium ∆ width is mainly a moderatedecrease of the second peak in the ∆ Dalitz decay component, while the total invariant-mass spectrum of dileptons remains practically unchanged. As already discussed in App. Aany in-medium change of resonance production cross sections would also affect the π Dalitzdecay contribution in the dilepton spectrum.Finally, there also remains an uncertainty related to the centrality selection. For thedilepton spectra calculations in the Au+Au system at 1.23 A GeV we have applied a sharpcutoff approximation, i.e. the spectra are geometrically weighted in the impact parameterrange b < . −
40% centrality [79]. The Glauber Monte-Carlomodel on the other hand produces an impact-parameter distribution smeared by ∼ ± ∓
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