Properties of strange vector mesons in dense and hot matter
Andrej Ilner, Daniel Cabrera, Pornrad Srisawad, Elena Bratkovskaya
aa r X i v : . [ h e p - ph ] M a r Properties of strange vector mesons in dense and hot matter
Andrej Ilner , Daniel Cabrera , , Pornrad Srisawad , Elena Bratkovskaya , Institut f¨ur Theoretische Physik, Johann Wolfgang Goethe-Universit¨at Frankfurt am Main, 60438 Frankfurtam Main, Germany Frankfurt Institute for Advanced Studies (FIAS), 60438 Frankfurt am Main, Germany Naresuan University, Faculty of Science, Phitsanulok 65000, Thailand
Abstract
We investigate the in-medium properties of strange vector mesons ( K ∗ and ¯ K ∗ ) in dense and hotnuclear matter based on chirally motivated models of the meson selfenergies. We parameterisemedium effects as density or temperature dependent effective masses and widths, obtain thevector meson spectral functions within a Breit-Wigner prescription (as often used in transportsimulations) and study whether such an approach can retain the essential features of full micro-scopic calculations. For µ B = 0 the medium corrections arise from ¯ K ∗ ( K ∗ ) N scattering and the¯ K ∗ ( K ∗ ) → ¯ K ( K ) π decay mode (accounting for in-medium ¯ K ( K ) dynamics). We calculate thescattering contribution to the K ∗ selfenergy based on the hidden local symmetry formalism forvector meson nucleon interactions, whereas for the ¯ K ∗ selfenergy we implement recent resultsfrom a selfconsistent coupled-channel determination within the same approach. For µ B ≃ K ∗ ( K ∗ ) → ¯ K ( K ) π decaywidth. The emergence of a mass shift at finite temperature is studied with a dispersion relationover the imaginary part of the vector meson selfenergy. Keywords:
Strange vector mesons; Hot and dense nuclear matter; In-medium spectralfunction; Chiral effective models
1. Introduction
The features of strongly interacting matter in a broad range of temperatures and densitieshas been a subject of great interest in the last decades, in connection with fundamental aspectsof the strong interaction such as the nature of deconfinement or the physical mechanism of chiralsymmetry restoration [1]. Hadrons with strangeness, in particular, have been matter of intenseinvestigation regarding the study of exotic atoms [2], the analysis of strangeness production inheavy-ion collisions (HICs) [3, 4, 5] and the microscopic dynamics ruling the composition ofneutron stars [6].Understanding the dynamics of light strange mesons in a nuclear environment has posed achallenge to theoretical models, amongst different reasons, due to the fact that the low-densityapproximation 2 ωV opt ≃ tρ N is unable to to describe the interaction of the ¯ K meson withthe medium, as it is concluded from the phenomenology of kaonic atoms [2]: whereas the ¯ KN scattering amplitude ( T -matrix) is repulsive at low energies, the data require an attractive opticalpotential. The early departure of the ¯ K nuclear potential from the low-density result is tied tothe presence, below the ¯ KN threshold, of the Λ(1405) resonance. The onset of an attractive¯ KN interaction at low densities is a consequence of an upper shift of the excitation energy ofthis resonance induced by Pauli blocking on the intermediate nucleon states [7, 8, 9, 10]. Thelatter changes the real part of the ¯ KN scattering amplitude from repulsive in free space toattractive in a nuclear medium already at very low densities, whereas additional medium effects Preprint submitted to Elsevier. October 8, 2018 uch as the selfenergy of mesons in related coupled channels and the binding of baryons in thenuclear environment bring the Λ(1405) back to (almost) its vacuum position. This peculiarbehaviour of the ¯ K meson interaction with the nuclear medium is successfully described inchirally motivated coupled-channel approaches implementing the exact unitarity of the scatteringmatrix [8, 9, 10, 11] and a selfconsistent evaluation of the kaon selfenergy [10, 12, 13, 14]. Anattractive potential of about 40-60 MeV at normal nuclear matter density is obtained whenselfconsistency is implemented in chiral approaches (approximately half the size when only Pauliblocking is implemented), rather shallow as compared to relativistic mean-field calculations[15] or required by phenomenological analysis of kaonic atom data [2] with density dependentpotentials including non-linearities. Yet, this potential is able to reproduce the data from kaonicatoms [16, 17] and leads to deep K − states in medium and heavy nuclei bound by 30-40 MeVwith sizable widths ∼
100 MeV. In contrast with the former results, it should be noted the recentanalysis in Refs. [18, 19]. Starting from a ¯ KN pseudopotential inspired in the lowest order s -wave amplitude from the meson baryon chiral Lagrangian, constrained by data within the energyregion ≃ V K − ≃ −
80 MeV at threshold). A subsequent fitto kaonic atom data including additional quadratic terms in the potential (mimicking absorptivechannels not included in the single-nucleon interaction) increases the discrepancy by a factor ofthree. There are, however, important differences regarding the zero-energy limit of the proposedscattering potential, the double-pole structure of the emerging Λ(1405) state, the treatment ofthe (off-shell) energy dependence of meson selfenergies and the self-consistent implementationof the kaon potential, as compared to previous determinations (see, e.g., Refs. [12, 14]), whichmay explain the differences found at the level of the one-body interaction and the strongercontribution of the two-body mechanisms at central nuclear densities.The properties of both K and ¯ K close to threshold have also been thoroughly investigatedin HICs at SIS energies [3, 4, 5, 20]. The analysis of experimental data (such as productioncross sections, energy and polar angle distributions) in conjunction with microscopic trans-port approaches has allowed to draw solid conclusions regarding the production mechanisms ofstrangeness, the different freeze-out conditions exhibited by K + and K − mesons and the use of K + as a probe of the nuclear matter equation of state at high densities. A good agreement withdata has been achieved for many different observables when the in-medium properties of kaonsare implemented. Still, a consensus on the influence of the kaon-nucleus potential on the wholeset of experimental data involving antikaon production is still missing, leaving room for a moreelaborate description of the most relevant reactions (e.g., πY → ¯ KN ) within hadronic models.An early motivation for the present study of the properties of strange vector mesons originatesin the substantial experimental activity within the RHIC low energy scan programme and theHADES experiment at GSI, which are currently performing measurements in order to extract thein-medium properties of hadronic resonances and, in particular, of strange vector mesons fromHICs [21, 22]. Recent developments within the last 10-15 years suggest the use of hadronic many-body models together with microscopic transport approaches (particularly those incorporatingoff-shell effects) as a powerful tool to investigate the complicated reactions that take placein HICs. Within this scenario strange vector meson resonances have triggered attention onlyrecently, mostly due to the fact that, unlike their unflavoured partners, they do not decay inthe dilepton channel, making their experimental detection less clean. On the theory side, theproperties of ¯ K ∗ have been studied in dense nuclear matter [23] starting from a model of the¯ K ∗ N interaction within the hidden gauge formalism. In a similar way as for the ¯ K meson, theexcitation of several Y N − components leads to a large broadening of the ¯ K ∗ spectral functionwith a moderate attraction on the quasiparticle energy. The K ∗ , however, has not been studiedwithin this framework. Presumably, the lack of baryonic resonances with positive strangenesscalls for much more moderate medium effects in the S = +1 sector. Still, such information is2mportant for the present experimental program at GSI and for the future FAIR facility. Thepresent study contains novel aspects which will cover part of the missing information at thetheoretical level.In the context of hot hadronic matter with low baryonic content some estimations havebeen done for the φ meson decay at temperatures close to deconfinement based on the chiraldynamics of kaons and anti-kaons in a hot gas of pions [24]. The same scenario with the strangevector mesons K ∗ and ¯ K ∗ is, however, unexplored. In heavy-ion collisions at RHIC and LHCenergies the matter at midrapidity is practically baryon free and dominated in the later stagesby mesons. Presently, there is a big interest in the experimental study of K ∗ , ¯ K ∗ productionat ultrarelativistic energies since the vector strange resonances are considered as a promisingprobe of the hot medium and the quark-gluon plasma (QGP) formation - cf. e.g. the review [15].Therefore having a theoretical asset on the properties of strange vector mesons in hot matter isalso very important for the interpretation of the experimental observations. We conclude fromour study that the K ∗ and ¯ K ∗ change their properties only moderately in hot matter, leavingthe hope that the strange vector mesons might be used as a promising probe of QGP propertiessince they are not strongly distorted by final-state hadronic interactions.With the focus on the implementation of effective hadronic models based on chiral dynamicsin microscopic transport simulations [20, 25], we present in this paper a study of the spectralfunction of strange vector mesons within a Breit-Wigner quasiparticle picture, where the ef-fects of the nuclear environment are encoded in density or temperature dependent masses anddecay widths. An important caveat worth mentioning is the role of pionic medium effects.Whereas these have been extensively studied in dense baryonic matter within hadronic manybody theory, the complexity of microscopic transport simulations makes the implementation ofpion in-medium properties an enormous task, since pions participate in many different reactions,either as product or in the initial state. In addition, the current precision of experimental dataon observables from HICs makes it difficult to resolve whether the role of pion in-medium prop-erties is a determinant factor. In this respect, we constrain ourselves in the present work to themedium effects on strangeness, and we shall assume that pions are on-shell quasiparticles withvacuum-like properties in all explicit calculations involving ¯ K ∗ /K ∗ decays.This work is organised as follows. In Section 2 we introduce the Breit-Wigner approach forthe spectral function and discuss our results for dense nuclear matter in terms of collisional anddecay width effects, emphasising the differences between the K ∗ and the ¯ K ∗ . We present acalculation of the K ∗ selfenergy within the hidden local symmetry formalism whereas for the¯ K ∗ we build upon previous results in the same approach. The high-temperature scenario isexplored in Sec. 3 by implementing kaon dynamics in a pion gas from a chirally constrainedmodel in the decays of K ∗ / ¯ K ∗ s. We also study whether an in-medium mass shift of the K ∗ atfinite temperature appears from the dispersion relation of the in-medium energy dependent K ∗ decay width. Finally, we draw our summary and conclusions in Sec. 4.
2. Medium effects in dense nuclear matter
In the present study we adopt the relativistic Breit-Wigner prescription for the strange mesonin-medium spectral function [26], A i ( M, ρ N ) = C π M Γ ∗ i ( M, ρ N ) (cid:0) M − M ∗ i ( ρ N ) (cid:1) + ( M Γ ∗ i ( M, ρ N )) , (1)where C stands for a normalisation constant, which is determined as the spectral function mustfulfil the sum rule R ∞ A i ( M, ρ N ) dM = 1, and i = K/ ¯ K, K ∗ / ¯ K ∗ throughout this work. InEq. (1) we have explicitly written the dependence on the nuclear matter density ρ N to indicatemedium effects. In Sec. 3 this will be replaced by a dependence on the temperature of thethermal bath. We also note that in general the strange meson width Γ ∗ i depends on the (off-shell)3nergy M , thus Eq. (1) departs from the standard Lorentzian parameterisation used in quantummechanics but keeps closer to the actual phenomenology of strange vector meson resonances. Inthe quantum-field theory sense, the latter is mandatory since the decay width of a bosonic state istied to the imaginary part of the associated selfenergy, namely Γ ∗ i ( M, ρ N ) = − Im Π i ( M, ρ N ) /M .The real part of the selfenergy is often reabsorbed in the definition of the in-medium mesonmass, denoted by M ∗ i above, which is a good approximation as long as δM i = M ∗ i − M i ≪ M i (with M i being the nominal mass in vacuum) and Re Π i is not strongly energy dependent. Tobe precise, the in-medium meson mass is defined as the solution of the meson dispersion relationat vanishing momentum, ( M ∗ i ) − M i − Re Π i ( M ∗ i , ρ N ) = 0 . (2)The approximations discussed above typically introduce small violations of the spectral functionsum rule which is why we explicitly keep a normalisation constant.It is also worth mentioning that, in general, the meson selfenergy operator depends separately on the energy and momentum of the particle, as a consequence of the breaking of Lorentz invari-ance induced by the existence of a preferred reference frame, that of the nuclear environment.The parameterisation in Eq. (1) does not take this fact into account. As we shall discuss be-low, we explore different implementations of the in-medium properties based on previous modelcalculations of the selfenergy of strange mesons. As for the momentum dependence, this limita-tion in our parameterisation will be overcome by considering static properties, i.e. by assumingthe strange meson at rest with respect to the nuclear medium. Similarly, the vector nature of¯ K ∗ /K ∗ mesons induces a splitting of their spectral function in longitudinal ( L ) and transverse( T ) modes. We shall restrict ourselves to the case of zero momentum where both the L and T spectral functions become degenerate.We distinguish two sources of medium effects that contribute to the selfenergy of a strangemeson in a nuclear medium: (i) the modification of its dominant decay mode, e.g. ¯ K ∗ → ¯ Kπ , induced by medium effects on the light pseudoscalars (also typically referred to as two-meson cloud effects); and (ii) the quasi-elastic interaction of the strange meson with nucleons,e.g. ¯ K ( ∗ ) N → ¯ K ( ∗ ) N and related absorptive channels. Obviously the first mechanism is onlypresent for strange vector mesons, which have predominant decay modes into ¯ K ( K ) π . In thesame fashion we write the in-medium width Γ ∗ entering Eq. (1) as Γ ∗ = Γ dec + Γ coll , the sumof the width associated to the main decay mode (Γ dec ) plus a collisional width (Γ coll ) emergingfrom the scattering with nucleons. The latter will be either parameterised from existing modelsor else calculated explicitly, whereas the former one can be evaluated straightforwardly as [27]Γ V, dec ( µ, ρ N ) = Γ V (cid:18) µ µ (cid:19) R µ − m π q ( µ, M ) A j ( M, ρ N ) dM R µ − m π M min q ( µ , M ) A j ( M, dM , (3)where j = K, ¯ K , q ( µ, M ) = p λ ( µ, M, M π ) / µ , λ ( x, y, z ) = (cid:2) x − ( y + z ) (cid:3) (cid:2) x − ( y − z ) (cid:3) andwe have explicitly included the subindex V as this expression stands for the ( p -wave) decay ofthe strange vectors ¯ K ∗ and K ∗ . Γ V stands for the vector meson (vacuum) partial decay widthin the considered channel and µ is the nominal resonance mass, particularly Γ K ∗ , ¯ K ∗ = 42 MeVand µ = 892 MeV [28]. We note that the same expression can be derived from the imaginarypart of the lowest order selfenergy diagram in Fig. 1 [23, 29]. Eq. (3) accounts for the in-mediummodification of the resonance width by its decay products. In particular, we consider the factthat kaons and anti-kaons may acquire a broad spectral function in the medium, A j ( M, ρ N ).Pions will be assumed to stay as narrow quasiparticles with vacuum properties in the evaluationof Γ V, dec throughout this work. This choice is, most likely, not realistic at finite nuclear density(we refer, for instance, to the classic references [30, 31]), whereas in hot matter with low baryoniccontent the pion is expected to experience small changes up to temperatures T ≃
100 MeV[32, 33]. We want to focus on the role of medium effects on the strangeness degrees of freedom,4hile sticking to the typical set-up of transport calculations. Thus our results for Γ V, dec should beconsidered as a lower boundary and reference to other calculations accounting for pion propertieswill be made where possible. Finally we note that, for the case of interest, A j ( M,
0) in Eq. (3)is a delta function since the kaon is stable in vacuum with respect to the strong interaction. Ingeneral this may not be the case, e.g., a → ρπ , where the ρ meson decays into two pions mostof the time ( M min then stands for the threshold energy). K ∗ meson The absence of baryonic resonances with S = +1 close to threshold induces milder nuclearmedium effects in the properties of the K meson [12, 34] as compared to the ¯ K meson, whosebehaviour is largely dominated by the Λ(1405) resonance appearing in s -wave ¯ KN scattering.A similar situation is expected to take place for the vector partner of the K , the K ∗ meson.Note that the K ∗ decays predominantly into Kπ , and therefore not only collisional effects butthe in-medium decay width has to be taken into account. However, since the K meson itself isbarely influenced by nuclear matter one can anticipate small changes in Γ K ∗ , dec as compared tothe vacuum case.The collisional selfenergy of the K ∗ has not been evaluated before in the context of effectivetheories of QCD. We take recourse of the hidden local symmetry approach in [35], which hasbeen recently used to study the selfenergy of the ¯ K ∗ in cold nuclear matter [23] (see [36] fora recent review). In this approach, the interaction of light vector mesons with the J P = 1 / + baryon octet is built by assuming that the low-energy amplitudes are dominated by vectormeson exchanges between the baryons and the vector themselves, which couple to each other.At small momentum transfer, the transition potential (tree-level scattering amplitude, c.f. Fig. 1)is completely determined by chiral symmetry breaking. Neglecting corrections of order p/M ,with p ( M ) the baryon three-momentum (mass), the s -wave projection reads V ij = − C ij f (2 √ s − M B i − M B j ) (cid:18) M B i + E i M B i (cid:19) / (cid:18) M B j + E j M B j (cid:19) / ~ε · ~ε ′ ≃ − C ij f ( q + q ′ ) ~ε · ~ε ′ , (4)where q ( q ′ ) stands for the energy of the incoming (outgoing) vector meson with polarisation ~ε ( ~ε ′ ), C ij stand for channel-dependent symmetry coefficients [35], and the latin indices labela specific vector-meson baryon ( V B ) channel, e.g., K ∗ + p . For practical purposes the secondequation is satisfied to a good approximation. Within this approach the K ∗ selfenergy can beevaluated straightforwardly in a tρ approximation (justified for the present study) ,Π coll K ∗ = 12 (cid:0) V K ∗ + p + V K ∗ + n (cid:1) ρ (cid:18) ρ N ρ (cid:19) = 14 ( V + 3 V ) ρ (cid:18) ρ N ρ (cid:19) , (5)where in the second equation V I stands for the potential in isospin basis and we omit thepolarisation vectors for simplicity of notation. By taking q = q ′ ≃ M K ∗ , and using C K ∗ + p = − C K ∗ + n = − f = 93 MeV we obtainΠ coll K ∗ ≃ α M K M K ∗ M K ∗ (cid:18) ρ N ρ (cid:19) , (6)with α = 0 .
22, leading to a positive mass shift (equivalent to a repulsive optical potential) ofabout δM K ∗ ≃
50 MeV at normal matter density ( ρ = 0 .
17 fm − ). The normalisation of the scattering amplitude T (or V ) is such that the differential cross section reads dσd Ω = π MM ′ s q ′ q | T | . S = − KN scattering, thus extending the applicability ofthe low-energy theory. A similar situation was encountered in ¯ K ∗ N scattering [23], as we shalldiscuss in the next section. In the S = +1 sector, the use of unitarised amplitudes also results ina more reliable description of KN scattering data and optical potential [34, 41, 42]. We followthe same reasoning here and together with the Born approximation, Eq. (5), we present resultsfor the K ∗ selfenergy by iterating the leading order potential in a Bethe-Salpeter equation toproduce the full scattering amplitude, schematically T = V + V GT . We note that the solution ofthe Bethe-Salpeter equation is particularly simplified within the chiral effective theory since boththe potential V and the resummed amplitude T can be factorised on-shell, and thus the solutionproceeds by algebraic inversion. It has been shown that the off-shell parts within the integralterm of the equation lead to singularities that can be renormalised by higher-order countertermsand are effectively accounted for by using physical masses and coupling constants. We referto [37, 38] for details about different unitarisation schemes in meson-meson and meson-baryonscattering within Chiral Perturbation Theory.The K ∗ + p state is pure isospin I = 1 and gets single channel unitarisation, whereas the K ∗ + n amplitude is coupled with the K ∗ p state and thus a 2 × T K ∗ + p = V − V G , T K ∗ + n = T K ∗ p = T K ∗ p,K ∗ + n = 12 V − V G , (7)with V = ( q + q ′ ) / f = V K ∗ + p . G stands for the meson-baryon loop function, which can befound, for instance, in [41]. We adopt here the same regularisation scale for G as it was fixedin ¯ KN scattering studies: in a cut-off scheme this corresponds to q max ≃
630 MeV /c [41] or,equivalently, to a subtraction constant of natural size a K ∗ N ≃ − δM K ∗ ( ρ ) ≃
30 MeV[i.e. α ≃ .
13 in Eq. (6)].Several comments regarding the chosen values for f and a K ∗ N are in order. First, previousexperience from ¯ K ( K ) N studies indicates that the agreement with meson-baryon scatteringdata improves when leaving f as a free parameter, and typically an intermediate value between f π and f K as obtained in meson-meson calculations comes out from these analysis [41], f K ≃ . f π = 107 MeV. Note, however, that this responds to a strictly phenomenological motivationin the pseudoscalar sector, where the amount of data available permits to constrain furtherthe model . In the sector of strange vector meson interactions with baryons, the amount ofexperimental data is comparatively much smaller, and insufficient to simultaneously constrainboth f and the subtraction constant a K ∗ N (which in fact are substantially correlated). In absenceof further knowledge one sticks to f = 93 MeV, which in the Hidden Local Symmetry formalismleads to vector self-interactions respecting the KSFR rule and vector meson dominance [35].Still, one may expect a similar renormalisation of f here and we shall estimate this uncertaintyby calculating the K ∗ selfenergy with both f = 93 ,
107 MeV. Second, our choice of a K ∗ N ≃ − KN system which also provide a fair description ofscattering observables in the S = +1 sector such as I = 0 , . Lacking further constraints on the subtraction constant in the vector meson sector,it is also pertinent to study the uncertainty in the K ∗ selfenergy due to this parameter. In We note, for instance, the study of [38] where a rather good agreement with ¯ KN cross sections and thresholdratios were obtained by using f = f = 86 . SU (3) ChPT)and a slightly different subtraction constant. See also the recent analysis in [39] where other possible values of a MB in coupled channels accommodate the ∗ / ¯ K ∗ πK/ ¯ K NK ∗ / ¯ K ∗ Figure 1: Left: The K ∗ ( ¯ K ∗ ) → K ( ¯ K ) π loop selfenergy diagram. Right: Tree level diagram for elastic scatteringbetween a K ∗ ( ¯ K ∗ ) and a nucleon. order to have an educated estimate we take recourse of previous results in ¯ K ( K ) N calculations.The value of the subtraction constant cannot be arbitrarily changed as it is correlated to themomentum scale running in the meson baryon loop (with a characteristic exponential dependence[38]), so that smaller values of, e.g., a KN in absolute magnitude correspond to higher values of µ ≡ q max (in the limiting case a KN = − ≃ − .
39 one has µ → ∞ ). A “natural” valuewindow around − µ in the region of the first (vector) meson resonance, the ρ (770), whereas the “infrared” boundary is better constrained from phenomenology. By usingour unitarised s -wave scattering amplitude for the KN system we have obtained a limiting rangeof variation for a KN of [ − . , − . K + p cross section (below q LAB ≃
800 MeV /c the reaction is elastic and therefore coupled-channel dynamics does not playa role in this case). Comparison to the I = 1 KN phase shifts close to threshold providesmore stringent constraints and best results are obtained within the interval a KN = − . ± . /c . q max . /c ), slightly above − I = 1 scattering length A I =1 KN = − .
31 fm, in good agreement withexperiment and previous determinations. In Fig. 2 we summarise our estimation of uncertaintyin the K ∗ selfenergy by plotting the K ∗ mass shift at normal nuclear matter density as a functionof the subtraction constant for two values of f . Within errors we estimate a small, positive massshift for the K ∗ around 5% at ρ N = ρ .Finally, the decay width of the K ∗ is evaluated according to Eq. (3), where we adopt anarrow quasiparticle approximation for the K spectral function, namely A ( M, ρ N ) = 2 M δ [ M − M ∗ K ( ρ N )] (8)with M ∗ K ( ρ N ) = M K + Π K ( ρ N ) and the K selfenergy as determined in [41, 40] within a tρ ap-proximation, Π K ( ρ N ) ≃ . M K ( ρ N /ρ ), and consistent with the findings of [34]. As expected,the K ∗ in-medium width is slightly reduced at finite density as the phase space for the KN system is quenched due to the larger K mass.The spectral function of the K ∗ is depicted in Fig. 3, according to Eq. (1) with the mediumeffects discussed above. At normal matter density the K ∗ quasiparticle peak experiences arepulsive shift corresponding to the mass modification in Eqs. (5,6,7). In spite of a nominalreduction in Γ K ∗ , dec due to the increase of M K at finite density, the shift in the K ∗ mass increasesthe available energy for the Kπ decay and the K ∗ spectral function exhibits practically negligiblechanges in shape. We recall that we have explicitly neglected the effect of pion modificationsin the K ∗ width, which may lead to changes up to a factor of ≃ K ∗ behaves as a quasiparticle with a single-peak distribution function and amodified effective mass as a result of the repulsive K ∗ N interaction. It is worth mentioningthat the HADES Collaboration have recently reported a reduction of the K ∗ yield in Ar+KCLcollisions at 1.76 A GeV as compared to estimations within the UrQMD transport approach [22]. data in the S = − a ¯ KN is not far from ≃ −
2, and no statement is donefor the S = +1 sector. .5 1.6 1.7 1.8 1.9 2.0 2.1 2.2 2.30.0180.0200.0220.0240.0260.0280.0300.0320.034 Dependence of M on a M [ G e V ] |a| f = 0.093 GeV f = 0.107 GeV Figure 2: K ∗ mass shift from Eqs. (2, 6, 7) at normal matter density as a function of the subtraction constant inthe meson-baryon loop function, for two values of the pseudoscalar decay constant f . K* spectral function A ( , ) [ G e V - ] [GeV] free (vacuum) / = 0.5 / = 1.0 / = 1.5 Figure 3: The Breit-Wigner spectral function of the K ∗ as a function of energy for different baryonic densities,including the collisional mass shift from K ∗ N scattering ( f = 107 MeV, a K ∗ N = − .
8) and Kπ width within-medium kaons. .2. ¯ K ∗ meson The properties of ¯ K ∗ have been studied in cold nuclear matter [23] starting from a modelof the ¯ K ∗ N interaction within the hidden gauge formalism, including coupled channels andimposing unitarisation of the scattering amplitudes, as introduced in the previous section. Theset of coupled equations determining the ¯ K ∗ N scattering amplitude and the ¯ K ∗ selfenergy(schematically Π ¯ K ∗ = P ~p n ( ~p ) T ¯ K ∗ N ) were solved selfconsistently accounting for Pauli-blockingeffects, baryonic mean-field potentials and the selfenergy of intermediate meson states. Suchscheme is usually known as G -matrix approach in reference to the in-medium effective T -matrixobtained in Dirac-Brueckner theory.The collisional part of the selfenergy, related to the quasi-elastic reaction ¯ K ∗ N → ¯ K ∗ N andaccounting for absorption channels, induces a strong broadening of the ¯ K ∗ spectral function as aresult of the mixing with two J P = 1 / − states, the Λ(1783) and Σ(1830), which are dynamicallygenerated in a parallel way to the ¯ KN interaction. At ρ N = ρ , it was found that the ¯ K ∗ widthis enlarged beyond 200 MeV, the in-medium ¯ Kπ decay mode contributing about 100 MeV. The¯ K ∗ nuclear optical potential indicates a moderate attraction of about 50 MeV. The quasiparticlepeak, however, exhibits a larger attraction as a consequence of the interference with the collectivemodes above threshold, which are responsible for a considerable energy dependence of the ¯ K ∗ selfenergy. This reflects the limitations of the quasiparticle representation of the spectral functionof the system, which we illustrate further in the following.To construct the spectral function of the ¯ K ∗ according to Eq. (1) we have obtained theeffective in-medium mass and width by solving the ¯ K ∗ dispersion relation utilising the re-sults for the ¯ K ∗ selfenergy obtained in [23], i.e., ω K ∗ − M K ∗ − Re Π ¯ K ∗ ( ω ¯ K ∗ , ρ N ) = 0 andΓ ∗ ¯ K ∗ ( ρ N ) = − Im Π ¯ K ∗ ( ω ¯ K ∗ , ρ N ) /M ¯ K ∗ (no energy dependence is retained in Γ ∗ ¯ K ∗ ). We shalldenote this method as “G-matrix 1”. We note that the results of [23] also account for thein-medium decay width for ¯ K ∗ → ¯ Kπ , thus our parameterisation according to the dispersionrelation with the total selfenergy captures both the collisional and the decay width of the ¯ K ∗ at the quasiparticle energy. We recall that the ¯ K ∗ meson spectral function in a nuclear mediumdepends explicitly on energy and momentum [23], S ¯ K ∗ ( ω, ~q ; ρ N ) = − π Im Π ¯ K ∗ ( ω, ~q ; ρ N ) h ω − ~q − M K ∗ − Re Π ¯ K ∗ ( ω, ~q ; ρ N ) i + [Im Π ¯ K ∗ ( ω, ~q ; ρ N )] . (9)The proposed parameterisation, relying on Eq. (1), is only suitable for ~q = 0, namely A ¯ K ∗ ( M, ρ N ) = 2 C M S ¯ K ∗ ( M,~ ρ N ) . (10)Our results can be seen in Fig. 4, where we compare our parameterisation with the original¯ K ∗ spectral function from [23]. It is evident that a Breit-Wigner form with in-medium proper-ties based on the quasiparticle picture cannot retain the full structure of the original spectralfunction, particularly the modes appearing at the right hand side of the ¯ K ∗ peak, which arerelated to the Λ h and Σ h excitations of the system. The position of the peak in our resultlies close to the ¯ K ∗ quasiparticle peak in the full result, as it is determined by the solutionof the dispersion relation, which we account for explicitly. The obtained width, however, islarger than reflected by the full calculation. This is related to the fact that the ¯ K ∗ selfenergy ishighly energy dependent in the vicinity of the quasiparticle energy, due to the presence of theΛ(1783) N − and the ¯ Kπ thresholds, which induce a quick rise of the ¯ K ∗ width. Such energydependence is not kept in our parameterisation and the resulting distribution exhibits a largerwidth which, incidentally, partly makes up for the missing strength in the higher energy region.In the low energy part, however, the broadening of our distribution overestimates the strengthof the spectral function from the full calculation.Alternatively, in order to estimate the effect of the energy dependence in the resonancewidth, we correct the quasiparticle energies from the dispersion relation so as to match the9 = 0.5 = A ( , ) [ G e V - ] = 1.5 Original G-matrix G-matrix 1 G-matrix 2 anti-K* spectral function [GeV]
Figure 4: The spectral function of the ¯ K ∗ for different baryonic densities. The blue solid line corresponds to theoriginal calculation in [23] by Tolos et al. The Breit-Wigner parameterisations with in-medium mass and decaywidth are shown in orange dashed (“G-matrix 1”) and green dash-dotted (“G-matrix-2”) lines, c.f. the text fordetails. The pink solid line in the middle panel corresponds to the explicit evaluation of the ¯ Kπ width within-medium kaons. The vacuum spectral function is also displayed in the upper panel for reference (black thinsolid line). The normalisation of the Breit-Wigner functions is set as to match the results of [23]. K ∗ peak, i.e. ω ¯ K ∗ → ˜ ω ¯ K ∗ = ω ¯ K ∗ + ∆ ω | Im Π ′ ¯ K ∗ (˜ ω ¯ K ∗ , ρ N ) = 0 andΓ ∗ ¯ K ∗ ( ρ N ) = − Im Π ¯ K ∗ (˜ ω ¯ K ∗ , ρ N ) /M ¯ K ∗ . We denote this parameterisation method as “G-matrix 2”.The resulting spectral function, c.f. Fig. 4, now peaks at the same position as the full calculationand exhibits a smaller width in closer agreement with the ¯ K ∗ structure, the reduction in thewidth tied to lower quasiparticle energies leading to a more quenched in-medium phase space.Consequently, the upper region containing the strange collective modes is rather underestimated,as expected since we cannot reproduce multi-mode structures with a simple Breit-Wigner pa-rameterisation. Finally, as an additional test we show in the middle panel of Fig. 4 (solid purpleline) our spectral function with a similar parameterisation of the collisional part of the width,Γ ¯ K ∗ , coll , and explicitly calculating the ¯ Kπ decay width Γ ¯ K ∗ , dec according to Eq. (3). For this wehave taken recourse of the results in [12] for the ¯ K spectral function at normal nuclear density.The resulting distribution is narrower than any of the former proposed methods, which preciselyreflects the contribution of in-medium pions in the ¯ Kπ cloud from [23].We conclude that the complex many-body dynamics present in the ¯ K ∗ spectral function indense nuclear matter, as a result of the interference between several collective excitations ofthe system, cannot be successfully reproduced in a Breit-Wigner scheme. The main featuresof the full dynamics, such as considerable broadening and attractive shift of the quasiparticlemode, of special importance in transport calculations, can be parameterised in terms of a densitydependent effective mass and width according to the dispersion relation in the full microscopicmodel of [23]. Still, other details such as the modification of thresholds in the medium, whichstrongly influences the low-energy behaviour of the spectral function, are also important fromthe point of view of transport simulations since they modify the onset of strangeness productionreactions (e.g. N N → ¯ KX ). Thus for a precise description of these features one is bound torecourse to the results from realistic many-body evaluations.
3. Medium effects in hot hadronic matter
In this section we focus on the scenario of a hot hadronic system with low net content ofbaryons, i. e., µ B ≃
0. Then the medium is mostly constituted by thermal excitations of thelightest hadrons, namely pions, and at next to leading order heavier mesons ( K , ¯ K , η ,...). Sucha set-up corresponds to the late evolution of a high-energy heavy-ion collision, as produced inRHIC and LHC. The µ B = 0 scenario, where the interaction with nucleons sets in at a finitetemperature, is expected to be encountered at low and intermediate energy collisions, to beproduced in the near future at the FAIR and NICA facilities.In a hot, isotopically symmetric pionic medium, both the ¯ K ∗ and K ∗ vector mesons behaveidentically. The medium effects in this case are tied to the modifications of the predominantdecay mode, ¯ K ( K ) π . The in-medium properties of kaons in hot matter have been studied byFuchs et al. [24] in the framework of Chiral Perturbation Theory, together with a phenomeno-logical extension to reach high temperatures (up to T . M π ) based on experimental Kπ phaseshifts. The kaon selfenergy is obtained as the sum of the forward Kπ scattering amplitudes overthe pion momentum distribution function, namelyΠ K ( M K , T ) = − Z T + Kπ ( s, , u )( dn sπ + + dn sπ + dn sπ − ) + Z T − Kπ ( s, , u )( dn sπ + − dn sπ − ) , (11)where dn sπ = d p (2 π ) ω π ( p ) 1exp (cid:16) ω π ( p ) − µ π T (cid:17) − T ± Kπ ( s, t, u ) are related to the amplitudes in isospin basis T IKπ as T / Kπ = T + Kπ + 2 T − Kπ , T / Kπ = T + Kπ − T − Kπ . It was found in [24] that the kaon experiences amoderate broadening and a comparatively small, attractive mass shift up to T = 100-120 MeV,11 .2 0.4 0.6 0.805101520253035 K spectral function A ( M , ) [ G e V - ] M [GeV] T = 0.12 GeV T = 0.15 GeV
Figure 5: The spectral function of the K is shown as a function of the invariant mass M for different temperatures.The blue solid line corresponds to T = 0 .
12 GeV and the orange dashed line corresponds to T = 0 .
15 GeV. Thevertical dotted line indicates the position of the vacuum kaon mass. reaching values of about Γ ∗ K ≃
50 MeV and δM K ≃ −
35 MeV at T = 150 MeV, which is thehighest temperature that we considered here. By parameterising the results from [24] and usingEq. (1) we obtain a kaon spectral function with a single quasiparticle structure that is mildlyshifted to lower energies while becoming substantially broadened only at very high temperatures,c.f. Fig. 5.Next we make use of Eq. (3) to calculate the K ∗ (or ¯ K ∗ ) decay width, related to theimaginary part of the selfenergy. We note that in this case the collisional broadening is alreadyautomatically accounted for since all the relevant dynamics is tied to the coupling to ¯ K ( K ) π .Since there is no further information available on a possible mass shift of the K ∗ in the hotmedium, we evaluate it explicitly from the real part of the selfenergy, which we calculate usinga dispersion relation over the imaginary part, namely:Re Π K ∗ ( µ, T ) = 2 π Z ∞ M π dµ ′ µ ′ µ ′ − µ Im Π K ∗ ( µ ′ , T ) , (12)where we have renamed the vector meson energy as µ to avoid confusion with the kaon energy, M . The integral in Eq. (12) is quadratically divergent and needs to be regularised. This hasbeen done, for instance, in [44] for neutral vector mesons by using a twice subtracted dispersionrelation. Since in the strange sector we lack enough constraints on the subtraction constants wechoose instead to remove the vacuum decay width of the K ∗ ,Re Π K ∗ ( µ, T ) − Re Π K ∗ ( µ,
0) = − π Z ∞ M π dµ ′ µ ′ µ ′ − µ (cid:2) Γ ∗ K ∗ , dec ( µ ′ , T ) − Γ vac K ∗ ( µ ′ ) (cid:3) , (13)which then renders the net medium effect on the physical mass of the particle (the real part ofthe vacuum selfenergy renormalises the bare mass and is not related to medium effects). This is,however, still not enough to guarantee convergence and thus we introduce a phenomenologicalform factor of dipole form, associated to the K ∗ Kπ vertex, F (Λ , µ, M ) = (cid:18) Λ + q ( M K ∗ , M K )Λ + q ( µ, M ) (cid:19) (14)12 .5 1.0 1.5 2.0 2.5-0.0010.0000.0010.0020.0030.0040.005 R e () / ( ) [ G e V ] [GeV] T = 0.06 GeV T = 0.09 GeV T = 0.12 GeV T = 0.15 GeV M [ G e V ] T [GeV] = 1.5 GeV = 1.8 GeV = 2.1 GeV dependence of M
Figure 6: Left: Real part of the K ∗ selfenergy at different temperatures up to T = 150 MeV. Right: K ∗ massshift as a function of the temperature for Λ = 1 . , . , . which enters the calculation of the K ∗ decay width in vacuum and in the medium,Γ ∗ K ∗ ≡ Γ K ∗ , dec ( µ, T ) = Γ (cid:18) µ µ (cid:19) q ( µ , M K ) Z µ − m π dM A ( M, T ) q ( µ, M ) F (Λ , µ, M ) , Γ vac K ∗ ( µ ) = Γ K ∗ (cid:18) µ µ (cid:19) q ( µ, M K ) q ( µ , M K ) F (Λ , µ, M K ) , (15)and cuts off high-energy dynamics which is not expected to be described by Eq. (3). The formfactor employed here was used in [26, 45] in the context of ρ → ππ dynamics in order to improvethe comparison with data in vector-isovector ππ scattering (pion electromagnetic form factor).The ρ decay into ππ probes similar kinematics as the K ∗ which is why we choose the sameform factor here in order to regularise the K ∗ dispersion relation. For our calculation of the K ∗ properties we bracket the uncertainty due to the unconstrained Λ parameter in the rangeΛ = 1 . . . µ ≃ K ∗ width [integrand of Eq. (15)] one can infer the mean value of the momentum q relevant for thedecay process and we find that the form factor is cutting off momenta beyond a typical hadronicscale of 1.5-2 GeV /c .The real part of the K ∗ selfenergy, as obtained from Eq. (13), is shown in Fig. 6 (left panel).It exhibits a sizable energy dependence associated to the quick rise in Γ ∗ K ∗ beyond the threshold,as expected from a p -wave decaying resonance. We note that the absolute size of Re Π ∗ K is below10 MeV and of repulsive nature, practically negligible up to T = 100 MeV and the larger valuesbeing reached at the highest temperature of T = 150 MeV. The right panel of Fig. 6 shows thecorresponding K ∗ mass shift as a function of the temperature for three different values of Λ.The spread between the different values of Λ becomes larger with increasing temperatures. Wenote, however, that the mass shift is in itself already much smaller than M ∗ K , thus renderingthe uncertainty due to the form factor unimportant. Nevertheless we find that, although wehave changed the form factor parameter within a large range of values as compared to typicalhadronic scales, the real part of the selfenergy at the K ∗ mass stays repulsive (does not flipsign), an indication of stability of the dispersion relation.In Fig. 7 (left) we depict the K ∗ in-medium width as a function of the energy at severaltemperatures. The width is slightly enhanced in the lower mass region with increasing tempera-tures due to the attraction experienced by the kaons and their broader spectral function, whichaugments the density of available ¯ K ( K ) π states to decay in. We find that the sensitivity of Γ ∗ K ∗ to the in-medium properties of the kaons is small, in agreement with the findings of [23] in coldnuclear matter, and only at the highest temperatures some broadening is expectable below the K ∗ mass. In Ref. [24] the results for the kaon selfenergy have been used to estimate a factor 313 .6 0.8 1.0 1.20.000.020.040.060.080.100.12 vacuum T = 0.09 GeV T = 0.15 GeV K* in-medium width [ G e V ] [GeV] -7 -6 -5 -4 -3 -2 -1 K* spectral function A ( , ) [ G e V - ] [GeV] vacuum T = 0.09 GeV T = 0.15 GeV A ( , ) [ G e V - ] [GeV] Figure 7: Left: Off-shell in-medium width of the K ∗ in hot matter as a function of energy at several temperatures.Middle (Right): Spectral function (in log-scale) of the K ∗ including both the mass shift and in-medium width inhot matter for different temperatures. increase in the φ meson decay width close to the critical temperature. Such a large sensitivityof the φ meson is due to (i) the main decay mode being ¯ KK and (ii) the fact that this decayoccurs slightly above threshold in vacuum. We also note that we have neglected the effect ofBose enhancement at finite temperature. As it has been discussed in [29, 34], for the nominalmasses of the daughter particles Bose enhancement is not relevant at the expected temperaturesin the hadronic phase of a heavy-ion collision. However, it may become important if K and π develop low-energy excitations, which is likely to be the case in hot matter at finite baryonicchemical potential (FAIR conditions) [29].In the middle and right panels of Fig. 7 one can see the resulting spectral function for the K ∗ . As expected from the discussion above the K ∗ peak experiences a very small shift to higherenergies from the interaction with the medium. In spite of the changes in the width, the effectis barely noticeable in the spectral function, since those are moderate and take place in the low-energy region. A logarithmic plot of the spectral function (c.f. Fig. 7 right) shows the additionalstrength below the nominal Kπ threshold. The present result justifies the description of the K ∗ in a hot environment as a narrow quasiparticle (with properties close to vacuum ones) intransport approaches.
4. Summary
We have presented a study of medium effects on the light strange vector mesons in differentscenarios of hot and dense strongly interacting matter, namely ( µ B = 0 , T ≃
0) and ( µ B ≃ , T = 0).In the first case, the K ∗ and ¯ K ∗ mesons behave rather differently, due to the possibility ofexciting near-threshold S = − K ∗ , a mechanism that is absent for the K ∗ due tostrangeness conservation. We have evaluated the collisional part of the K ∗ selfenergy in a chiralunitary approach based on the hidden local gauge symmetry, which was recently applied to de-scribe the interaction between vector mesons and baryons. Similarly to what has been found forthe K meson in previous studies, the K ∗ experiences a mild repulsive mass shift from the inter-action with nucleons, of about 5% at normal nuclear matter density. The decay width of the K ∗ is somewhat modified by the properties of the K in the medium (pionic effects at finite density,however, may provide a more sizable enhancement of the Kπ decay mode [23, 29]). Overall, thespectral function of the K ∗ maintains a single-peak structure, which facilitates its treatment inmicroscopic transport approaches [5], particularly those incorporating off-shell effects [20]. Forthe ¯ K ∗ however, the quasiparticle picture within a Breit-Wigner parameterisation of the spectralfunction is difficult to be reconciled with existing hadronic many-body calculations, which pointat a broad, multi-component structure as a consequence of the mixing with several collectiveexcitations involving strange baryon resonances. In this case the Breit-Wigner parameterisation,with in-medium parameters extracted from the ¯ K ∗ selfenergy, can only describe qualitatively14he behaviour of the ¯ K ∗ spectral function at finite nuclear density. Whereas the characteristicfeatures of the quasiparticle mode are retained, the details of the collective modes excited in thesystem as well as the energy dependence inherent to the opening of in-medium channels, areclearly missed.In the second case (hot hadronic matter with low net baryon content), the major modifi-cations in the ¯ K ∗ ( K ∗ ) properties are tied to their predominant decay mode, ¯ K ( K ) π . Basedon a phenomenological estimation of the kaon selfenergy, consistent with Chiral PerturbationTheory at low energies, we have calculated the in-medium vector meson width at finite tem-perature. The latter increases moderately with temperature at energies below the K ∗ mass upto the highest considered temperature of 150 MeV (the effect being practically negligible upto T .
100 MeV), due to the kaon experiencing a mildly attractive potential and a moderatebroadening from interactions with the (predominantly) pionic gas. The changes in the K ∗ widthare accompanied by a small repulsive mass shift, which we have calculated by means of a dis-persion relation. We conclude that the K ∗ resonances are only mildly sensitive to the hadronicmedium in absence of baryons for the typical temperature range expected in the hadronic phaseof a heavy-ion collision.The present study aims to facilitate the analysis of the in-medium properties of strangeresonances in heavy-ion collisions within microscopic transport approaches, a topic of on-goingexperimental activity within the RHIC low-energy scan programme and the HADES Collabora-tion at GSI. Implementation of in-medium spectral functions in the Breit-Wigner prescriptionprovides a fair description of the underlying many-body dynamics, with the exception of the S = − µ B = 0 it could be mimicked by anincrease of the in-medium decay rate in the case of the K ∗ . Acknowledgements
We acknowledge fruitful discussions with J¨org Aichelin, Wolfgang Cassing, Laura Fabbietti,Christina Markert, Eulogio Oset, `Angels Ramos and Laura Tol´os. This work has been supportedby the Helmholtz International Center for FAIR within the framework of the LOEWE program.D.C. acknowledges support from the BMBF (Germany) under project no. 05P12RFFCQ andA.I. acknowledges financial support from the HGS-HIRe for FAIR and H-QM.
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