J Aichelin

K Werner Subatech

Core -Corona Model describes the Centrality Dependence of v 2

Keywords:

Event by event EPOS calculations in which the expansion of the system is described by ideal hydrodynamics reproduce well the measured centrality dependence of v 2 / part , although it has been claimed that only viscous hydrodynamics can reproduce these data. This is due to the core -corona effect which manifests itself in the initial condition of the hydrodynamical expansion. The centrality dependence of v 2 / part can be understood in the recently advanced core-corona model, a simple parameter free EPOS inspired model to describe the centrality dependence of different observables from SPS to RHIC energies. This model has already been successfully applied to understand the centrality dependence of multiplicities and of the average transverse momentum of identified particles.

Ever since Drescher et al. [1] have claimed that the elliptic flow, v 2 , observed in the RHIC experiments, is even for the most central collisions at least 25% below the ideal hydrodynamical limit, the centrality dependence of the elliptic flow is considered as one of the key variables for the understanding of ultrarelativistic heavy ion collisions. Before this finding one had concluded from other observables that the Quark Gluon Plasma, created in the energetic reaction, is a perfect liquid [2].

The analysis in [1] is based on the assumption that the centrality dependence of the elliptic flow can be described by a simple formula [3]:

v 2 = v hydro 2 1 1 + K K 0 . ( 1
)
K 0 is a constant, v 2 is the measured elliptic flow, v 2 =< cos2(φ -φ R ) >, where φ (φ R ) is the azimuthal angle of the emitted particle (reaction plane). v hydro 2 is the value of the elliptic flow expected in the hydrodynamical limit. The average is performed over many events which belong to the same centrality class of the reaction.

is the spatial participant eccentricity of the particles in an event at the beginning of the hydrodynamical expansion, defined as = ( σ 2 y -σ 2

x)

2 +4σ 2 xy σ 2 y +σ 2 x with σ 2 x =< x 2 > -< x > 2 , σ 2 y =< y 2 > -< y > 2 , σ xy =< xy > -< x >< y >. It has to be calculated in a theoretical model. K is the Knudsen number defined as 1 K = σ S dN dy c s . ( 2
)
with σ being the partonic cross section, c s being the sound velocity and S being the transverse area = 4π σ 2 x σ 2 y -σ 2 xy . Using a Glauber model to determine S and , the measured [4] (phobos) or calculated values [5] (star) values of dn/dη as well as the experimentally observed v 2 values of the Phobos [6], the Star [7,8] and the Phenix collaboration [START_REF] Shimomura | AIP Conf.Proc[END_REF][START_REF] Shimomura | [END_REF] one can calculate v 2 as a function of dn/dη. The ratio between dn/dη and dn/dy depends on the particle species but is not far from 1. We assume here dn/dη = dn/dy because our conclusions do not depend on the precise value of this ratio. The result of this analysis is shown in fig. 1.

On the first view it seems that the data of different colliding systems at different energies fall in this representation on top of each other and therefore the Knudsen number depends indeed only on 1 S dN dy . However, if one regards the left figure in more detail one realizes that at a given centrality v 2 is almost independent of the beam energy [6]. dn/dη for 200 AGeV is about 1.4 times larger than that observed for 62 AGeV, almost independent of the centrality [4]. Thus the Knudsen number extracted from the 200 AGeV data is 40% smaller than that from the 62 AGeV data. This 40% change of the Knudsen number is hardly visible in the plot which means that this presentation is not very sensitive to the Knudsen number. That for the same value of dn/dη the value of v 2 of the 62 AGeV data is systematically higher than that of the 200 AGeV data is unexpected in this model because it means that at the lower energy the system is closer to the hydrodynamical limit than at the higher energy. However, the error bars are too large to allow for firm conclusions. The scaling of v 2 with dn/dη [7,8] and for 200 AGeV/62 AGeV in comparison with the Phobos data [6].

is questioned by the more recent Star data, displayed on the right hand side, where only statistical errors are shown. The Star collaboration has analyzed also peripheral AuAu events, in contradistinction to the Phobos collaboration. Therefore these data allow a more detailed comparison of both systems. We see for both systems an almost linear dependence of v 2 with dn/dη. The slope for the CuCu system is, however, clearly different from that for AuAu, what one could have guessed already from the Phobos data [1].

By fitting the experimental results with the theoretical curve, eq. 2, one can determine K/(σc s ) and finally the Knudsen number K of each centrality bin, assuming (centrality independent) values for σ and c s . Even for the most central bins v 2 depends on dn/dη. Thus the hydrodynamical limit K=0 is not reached. From the extrapolation towards dn/dη → ∞, corresponding to K → 0, the authors concluded that the observed flow is at least 25 % below the limit for ideal hydrodynamics.

These findings triggered the development of codes using viscous hydrodynamics. Luzum et al. [9] showed in viscous hydrodynamical calculations that different viscosities yield a different centrality dependence of v 2 / part and of < p T >. This dependence they used to identify that viscosity value which describes the 200 AGeV AuAu data. More recently, Song et al. [10] have extended the calculation to the CuCu system.

What has not been realized in this line of arguments is the importance of the initial configuration at the moment when the hydrodynamical expansion starts. The equations of ideal hydrodynamics predict how -for a given equation of state -this initial configuration develops in time under the assumption that the system is locally in equilibrium. Therefore it is expected that different initial conditions give different final distributions. The initial configuration at the beginning of the hydrodynamical expansion is unknown. It is difficult to asses because the time scale for equilibration in the initial state is much faster than predicted by microscopical calculations.

Recently one realized that event by event fluctuations of the initial conditions are not only much stronger than initially expected but also that they manifest themselves in the observables [11,12]. Only the particles deep inside the interaction region, the core particles, collide sufficiently frequent to form a locally equilibrated source whereas those close to the surface, called corona particles, do not come to a local equilibrium and do therefore not take part in the hydrodynamical evolution of the system.

EPOS is a consistent quantum mechanical multiple scattering approach based on partons and strings [13], where cross sections and the particle production are calculated consistently, taking into account energy conservation in both cases (unlike other models where energy conservation is not considered for cross section calculations [14]). A special feature is the explicit treatment of projectile and target remnants, leading to a very good description of baryon and antibaryon production as measured in proton-proton collisions at 158 GeV at CERN [15]. Nuclear effects related to CRONIN transverse momentum broadening, parton saturation, and screening have been introduced into EPOS [16].

In heavy ion collisions (and more recently also in proton-proton collisions) collective behavior is taken into account [17], in the following fashion: the initial scatterings, as described above, lead to the formation of strings, which break into segments, usually identified with hadrons. When it comes to heavy ion collisions, the procedure is modified: one considers the situation at an early proper time τ 0 , long before the hadrons are formed: one distinguishes between string segments in dense areas (more than some critical density ρ 0 segments per unit volume), from those in low density areas. The high density areas are referred to as core, the low density areas as corona [17]. It is important to note that initial conditions from EPOS are based on strings, providing a "flux-tube" like structure in case of individual events (a single flux tube in case of many overlayed events). Based on the four-momenta of the string segments which constitute the core, we compute the energy density ε(τ 0 , x) and the flow velocity v(τ 0 , x).

Having fixed the initial conditions, the system evolves according to the equations of ideal hydrodynamics. Hadronization occurs finally based on the Cooper-Frye prescription. The centrality dependence of v 2 of the EPOS calculation for AuAu at 200 AGeV is compared with the experimental results in fig. 2. We see an almost perfect agreement despite of the fact that EPOS uses ideal hydrodynamics. The centrality dependence is due to the core-corona effect in the initial configuration [17], which manifests itself also in other observables.

Recently we have developed a simple model [18,19] to study the consequences of this core-corona effect and to interpret the very involved EPOS simulations. In this model we define corona particles as those nucleons which have only one initial collision whereas the others are considered as core particles. f core is the fraction of core nucleons which depends on the centrality, the system size and (weakly) on the beam energy. This fraction is determined in the same Glauber model which is used to calculate the eccentricity and the transverse area S. Core particles form a locally equilibrated source whereas corona nucleons are treated like elementary proton-proton collisions with no interactions with their environment. Of course this is a very crude model but it has the advantage that no free parameter is needed in this superposition of pp collisions and a thermalized source. The present experimental error bars do leave little room for an improvement as has been shown in [18,19].

In this simple model we could show that, independent of the system size, the centrality dependence of the multiplicity M i and of the < p i T > of all identified hadrons from SPS to RHIC energies can quantitatively be described by

M i (N part ) (3) = N part [f core • M i core + (1 -f core ) • M i corona ] < p i T > (N part ) (4) = N part [f core • < p i T core > +(1 -f core )• < p i T corona > ].
For our calculation, we fix M core (< p i T core >) by applying eq. 3 (4)to the most central AuAu or PbPb data point. M corona and < p i T corona > are given as half of the multiplicity and the < p i T > measured in pp collisions. Once these parameters are fixed, the centrality dependence of M i and < p i T > is determined by eq. 3 and 4. Especially the centrality dependence of the observables for the lighter CuCu system follows then without any further input. Both, the core -corona model as well as the approach of Drescher et al. [1], assume that the system is not in the hydrodynamical limit, however the reason is different. In the approach of [1] it is assumed that all particles come to the same degree of thermalization independent of whether they are at the surface or in the interior of the interaction zone and that this thermalization is not complete.

The degree of thermalization depends on the centrality. In the core -corona model the particles in the interior of the reaction zone have sufficient collisions to reach complete thermalization whereas those located at the surface do not came to equilibrium at all. They behave like being created in independent NN collisions. In this separation it follows the findings of the EPOS program. The centrality dependence is due to the centrality dependence of f core .

In the core -corona model only the core particles feel the eccentricity of the overlap region. The particles produced by corona nucleons show an isotropic distribution in azimuthal direction. Therefore, in the core -corona model, v 2 / part is expected to be

∝ f core v 2 / part (N part ) = f core (N part ) v hydro 2 . ( 5
)
Because f core (N part ) and part is calculated in the Glauber model there is no free parameter. The core -corona model allows for some immediate predictions. Because f core is similar in central CuCu and AuAu collisions (Fig. 3) we expect that central collisions of AuAu and CuCu show a similar v 2 / part despite of the large difference of N part . Therefore v 2 / part plotted as a function of In Fig. 3 all presently available experimental results are presented in that way. We see that in this presentation the 200 AGeV data of Phenix and Phobos, presented in the bottom part, follow a straight line. The Star data show a small slope. We should stress that for the peripheral CuCu data where the core contains about 15 nucleons we cannot expect that our assumptions are fully justified and one may doubt whether there the core forms really a fully thermalized source. Also the data at 62 AGeV, shown in the top panels, are compatible with our prediction that v hydro 2 / part = v 2 / part (N part )/f core is constant. However, whereas the Phobos data are compatible with the assumptions that v hydro 2 / part is the same for AuAu and CuCu, the Phenix data show a difference of 30% between CuCu and AuAu. This is not understood yet.

Our model can be extended to the lower SPS energies where v 2 / part (N part ) has been measured by the NA49 collaboration [22]. These results, show in fig. 4, are also compatible with a straight line. In conclusions, we have shown that the experimentally observed centrality dependence of v 2 / part (N part ) is well described by the EPOS event generator using ideal hydrodynamics. In contradistinction to other hydrodynamical calculations EPOS creates its initial condition which fluctuates from event by event. Only those regions of the interaction zone which have a density above a threshold density form a locally equilibrated source whereas the others are treated as independent NN collisions. The simple EPOS inspired core -corona model, which has successfully described the centrality dependence of the multiplicity and the average transverse momentum of identified particles, can also quantitatively explain the centrality dependence of v 2 / part (N part ). Only core particles come to a local thermal equilibrium and develop a v 2 whereas the corona particles decay isotropically. The fraction of both classes of particles depends on the centrality and the centrality dependence of the elliptic flow reflects directly this dependence. All parameters of this model are fixed either by other data or by geometry. This results questions the idea to use the centrality dependence to determine the viscosity of a quark gluon plasma.

Figure 1 .

 1 Figure 1. Dependence of v 2 / part on 1 S

Figure 2 .

 2 Figure 2. v 2 / part for AuAu collsions at √ s = 200GeV , as predicted by EPOS, compared with data. The curves are the result of event -by event hydrodynamical calculations with a freeze out temperature of 166 MeV with (solid) and without (dashed) hadronic rescattering.

Figure 3 . v 2 /

 32 Figure 3. v 2 / part (N part )/f core (N part ) as a function of N part for the data at 62 (top) and 200 AGeV (bottom). The data are from refs. [7, 8] (Star) , [6] (Phobos) and [20, 21] (Phenix).

Figure 4 .

 4 Figure 4. v 2 / part (N part )/f core (N part ) as a function of N part for the NA49 data [22] at √ s = 17.2GeV .

  AuAu data. In peripheral CuCu and AuAu collision v 2 / part should be very similar for the same N part . The easiest way to compare our results with experiments is to divide the experimental results by f core (N part ). In this presentation we divided out the centrality dependence and therefore we expect that the data are horizontal straight lines.

	compared to
	1 S	dn dy is in central CuCu data larger as

Acknowledgements: We would like to thank Drs. R. Bellwied, U. Heinz, B. Jacak, J.Y. Ollitrault, S. Shi, R. Snellings, A. Timmings for valuable discussions.


Volker Friese
email: v.friese@gsi.de

Prospects for strangeness and charm measurements with the CBM experiment

The CBM experiment will investigate strongly interacting matter at extreme densities at the future accelerator facility FAIR in Darmstadt, in search for the first-order phase transition from confined to deconfined matter. Key observables of CBM are strange and charmed hadrons. We discuss the performance of the planned experimental setup with respect to the measurement of strange and multi-strange hadrons, open charm and charmonium, based on detailed detector simulations which include a realistic detector response.

The CBM experiment

The interest in the high-density part of the QCD phase diagram and its features, namely the possible first-order phase transition from confined to deconfined matter, the critical endpoint separating the first-order transition region from that of a cross-over at lower densities, and the properties of hadrons in a dense environment, reflect in several experimental activities in accelerator laboratories around the world: the RHIC energy scan programme, the NA61 activities at CERN-SPS, the NICA project at JINR Dubna, and the CBM project at the future facility FAIR in Darmstadt [1]. Amongst them, the fixed-target experiment CBM provides the unique opportunity to study extremely rare probes like charmed hadrons, which are close to their production thresholds at FAIR beam energies (10 -45A GeV). This opportunity is due to the high beam intensities to be delivered by the SIS-300 synchrotron of FAIR (up to 10 9 ions/s), giving rise to interaction rates of up to 10 MHz with a typical 1 % interaction target.

CBM is being designed to measure hadronic, leptonic and photonic observables in a large acceptance, covering the full p t range and rapidities from centre-of-mass rapidity close to beam rapidity. The planned experimental setup is shown in the left panel of Fig. 1. The core tracker (STS), consisting of eight stations of Silicon microstrip detectors, is located between the yokes of a super-conducting dipole magnet. Close to the target, the Micro-Vertex Detector (MVD) provides excellent resolution of displaced vertices from open charm decays. Downstream of the STS, a RICH and several layers of Transition Radiation Detectors (TRD) provide electron identification in a large momentum range. The TRD also serves for global tracking, connecting the tracks reconstructed in the STS with the time-of-flight wall located approximately 10 m downstream of the target. An electromagentic calorimeter after the TOF detector enables the measurement of neutral particles. The setup is completed with a downstream calorimeter (PSD) for event characterisation in terms of centrality and reaction plane angle.

For muon measurements, the RICH will be replaced by an absorber system interlayed with several tracking detector planes, allowing to follow the tracks reconstructed in the main tracker through the setup (right panel of Fig. 1).

In the following, we present results of physics performance studies in this detector system. They are based on the simulation of signals embedded into typical background events generated by UrQMD [2]. If not mentioned otherwise, all results refer to the collision system Au+Au at 25A GeV beam momentum. The results were obtained with full detector response simulation and event reconstruction.

Identification of kaons and particle ratio fluctuations

Charged kaons, the most abundant strange particle species, will be separated from pions and protons by means of a time-of-flight measurement. The TOF wall at about 10 m distance from the target covers approximately 150 m 2 and consists of RPC detectors with pad and strip read-out. The goal of the detector development is to achieve a uniform system resolution of 80 ps, which will allow a clean kaon identification up to 3.5 GeV as demonstrated in Fig. 2, covering the bulk of the produced kaons in the forward rapidity hemisphere and enabling to reliably extract full phase-space kaon yields. Event-by-event fluctuations in the strange to non-strange particle yield ratios are discussed as a possible signature of the QCD critical end-point [4]. We thus investigated the sensitivity of CBM to such kind of fluctuations in terms of the measure σ dyn denoting the relative width of the distribution of the eventwise K/π ratio after geometrical subtraction of the contributions from finite statistics and detector resolution, which are obtained by event mixing. As Fig. 3 shows, the experimental acceptance, track reconstruction and particle identification in CBM do not introduce a significant bias to the measurement; although the mean value and the total width of the input distribution generated by UrQMD change considerably, σ dyn is reproduced quite satisfactorily.

Hyperons and multi-strange di-baryons

Hyperons will be measured by means of the reconstruction of their weak decay topology in the STS. The high accuracy of the track reconstruction in the silicon detectors results in an almost background-free measurement of the decays Λ → pπ -and Ξ -→ Λπ -and a very good signal-to-background ratio for Ω -→ ΛK - [3]. The same method also allows to address the hypothetical multi-strange di-baryons. As an example, we studied the decay (Ξ 0 , Λ) → ΛΛ in the CBM-STS. We assumed a mass at the threshold, a lifetime of 3 cm and a branching ratio of 2 % [5]. For the multiplicity, we employ the prediction of the thermal model (7 • 10 -3 in central Au+Au collisions at 25A GeV beam momentum [6]). After appropriate topological cuts, in particular requiring the Λ daughters not to point back to the interaction point, we arrive at a very clean signal in the invariant-mass spectrum (S/B ≈ 200, see left panel of Fig. 4). The sensitivity limit is only reached at a multiplicity three orders of magnitudes below the thermal model prediction (right panel of Fig. 4). CBM is thus well suited for the search for such exotic objects.

Open charm

The measurement of the hadronic decays of open charm in the CBM energy regime constitutes a particular challenge because of the very low multiplicity of charmed hadrons. For a sufficient suppression of the combinatorial background, the detection of the displaced decay vertices within a millimeter or less from the interaction point is indispensable. In CBM, a high-precision Micro-Vertex Detector (MVD) will be devoted to this task, using Monolithic Active Pixel Sensors with a single-hit resolution of about 3 µm and low material budget. Two station of these detectors will be operated at 5 cm and 10 cm from the target, respectively. The read-out speed of the devices limits the maximal interaction rate for open charm measurements to 10 6 /s. Simulations show that with this detector, a secondary-vertex resolution of about 50 µm -80 µm, depending on the decay channel under study, can be achieved (see Table 1). This precision allows a clean separation of signal from background as shown in Fig. 5 for the decay channels D 0 → K -π -π + π + and Λ + c → pK -π + . In terms of signal-to-background ratio, the fourparticle decay channel of the D 0 turns out to be favourable over the two-particle decay, since the geometrical and topological constraint of four tracks coming from the same origin is strict enough to overcome the background from four-particle combinatorics. Remarkably, even the measurement of Λ c seems to be in reach, which would give access to the total charm production.

Figure 6 shows the measured yield of D + → K -π + π + as expected in a typical CBM run of 25 days at 1 MHz interaction rate, as function of transverse momentum and rapidity. The coverage in both kinematical variables is sufficient to extract phasespace integrated yields. Table 1 summarises the properties of the so far studied open charm decay channels, assuming multiplicities as predicted by the HSD [7] model. The statistical model predictions differ from those of HSD, most pronounced so for D + s , the measurement of which would thus allow to discriminate different production scenarios.

Charmonium

CBM will measure charmonium in both the electron and the muon decay channel. For the electron case, the challenge is the separation of electrons from hadrons exploiting both the Cherenkov radiation in the RICH detector and the Transition Radiation in the TRD. The TOF and ECAL detectors can help for pion suppression at low and high momenta, respectively. Our simulations show that pion suppression factors of more than 10 3 can be reached by combining the information from these detectors. Secondary electrons are suppressed by a cut on the transverse momentum. In the resulting invariant-mass spectrum (left panel of Fig. 7), the J/ψ peak is well visible above the remaining background. A measurement of the ψ ' might be feasible, but requires stronger cuts on the lepton candidates. A similar performance is obtained in the muon setup (right panel of Fig. 7). Here, the background mainly consists of wrong matches between tracks in the muon system and tracks reconstructed in the STS.

Challenges

The high interaction rates needed for the charm measurements -up to 10 MHz in the case of charmonium -pose extreme challenges not only to the radiation tolerance, rate capability and speed of the detectors, but also to the data handling. At 10 7 interactions per second, a raw data rate of about 1 TB/s has to be shipped from the front-ends and reduced online to a recordable rate of about 1 GB/s, i. e. by a about a factor of [7] and by the Statistical Model, the branching ratios, the detection efficiency including geometrical acceptance, track reconstruction efficiency and analysis cuts, the secondary-vertex resolution, the resolution in invariant mass, the signal-to-background ratio, and the measured yield in 10 11 central events, corresponding to about 25 days of CBM running at 1 MHz interaction rate. 1,000. This calls for a novel DAQ concept with self-triggered, free-streaming front-end electronics instead of a conventional trigger approach. A prototype of such a system is being developed at GSI and was tested under beam conditions with prototypes of the silicon strip detectors. Online event reconstruction will be performed in a dedicated computer farm consisting of a combination of FPGAs and modern many-core computers.

D 0 + D 0 D 0 D + D + S Λ + c decay channel K -π + + c.c. K -π -π + π + K -π + π + K -K + π + pK -π + < n > HSD 1.
The corresponding reconstruction algorithms are also being optimised with respect to speed, making use of the features of modern computer architectures (SIMD, many core, multi-threading). Currently, an event in the STS (about 600 charged tracks in the acceptance) can be reconstructed in about 50 ms.

Figure 1 .

 1 Figure 1. CBM experimental setup for (left) electron and hadron measurements and (right) muon measurements

Figure 2 .

 2 Figure 2. (Left) Reconstructed squared mass of hadrons as measured in the CBM time-of-flight system versus momentum. (Right) Squared-mass spectrum at p = 3 GeV.

Figure 3 .

 3 Figure 3. Simulated event-by event K/π ratio in full phase space (left) and measured in the CBM acceptance (right)

Figure 4 .

 4 Figure 4. Invariant-mass signal of the decay (Ξ 0 , Λ) → ΛΛ, simulated for central Au+Au collisions at 25A GeV with CBM. The left hand panel shows the signal assuming a multiplicity as predicted by the statistical model. In the right-hand plot, the multiplicity is assumed to be three orders of magnitude smaller. The statistics of 10 12 central events corresponds to 25 days of data taking at 10 MHz interaction

Figure 5 .Figure 6 .

 56 Figure 5. Invariant-mass signals of the decays D 0 → K -π -π + π + (left) and Λ + c → pK -π + (right), obtained from simulation of central Au+Au events at 25A GeV reconstructed with the CBM detector

Figure 7 .

 7 Figure 7. J/ψ signal in the invariant-mass spectrum of electron pairs (left) and muon pairs (right), obtained from simulation of central Au+Au events at 25A GeV. The statistics corresponds to 25 days of data taking at 10 MHz interaction rate.

Table 1 .

 1 Properties of the reconstrution of open charm decays in the CBM experiment in central Au+Au collisions at 25A GeV. Shown are the predicted multiplicities by HSD


N Herrmann
email: herrmann@physi.uni-heidelberg.de

Strangeness production at (sub)threshold energies

come

Strangeness production at (sub)threshold energies

Introduction

Modifications of hadron properties in dense baryonic matter are a current subject of intensive research in hadron physics. Various theoretical approaches [1,2,3] agree qualitatively on predicting, for example, modifications of masses and coupling constants for kaons and anti-kaons. Due to the density dependence of the KN(KN) potential, the K -effective mass is expected to drop, whereas the mass of K + mesons is predicted to rise with increasing density of nuclear matter. Already two decades ago, Kaplan and Nelson [4] pointed out that due to additional attractive interactions with the surrounding nucleons a condensation of anti-kaons (K -) may take place in a dense baryonic environment as encountered in the interior of neutron stars.

The strength of the K --nucleon potential might give rise to a new class of bound states [5,6] with quite exotic properties. These bound states are called kaonic clusters and are being searched for in stopped K --beams [7,8] and in pp -reactions [9]. Due to their predicted large central densities and the large compression reached in heavy ion reaction their production might be favored in heavy ion reactions [10] and therefore a complementary program was started to look for bound multibaryonic strange states in heavy ion reactions [START_REF] Herrmann | Proceedings of EXA05 conference[END_REF].

In the following new data obtained with the FOPI detector in 2008 at the SIS18 accelerator of GSI are shown for the reaction Ni+Ni at an incident energy of 1.91 AGeV elucidating some key aspects of strangeness physics in the threshold energy range.

Experiment: FOPI -Phase III

In order to allow the detection of strange particles the FOPI detector was upgraded with the installation of a new Time-of-Flight barrel covering the polar angle range 35 o < Θ < 60 o based on Multigap -Resistive -Plate -Chambers (MRPC) [START_REF] Schüttauf | [END_REF] in 2006. Since then this detector was used in major data taking runs for which first result become available now. The performance in identifying the emitted particles is depicted for the reaction Ni+Ni at an incident energy of 1.91 AGeV in Fig. 1 where the mass values are shown that are calculated from the momentum reconstructed in the Central Drift Chamber (CDC) and the velocity measured in the MRPC -barrel. The time resolution and the granularity is sufficient to separate positively (negatively) charged kaons with a signal-to-background ratio in excess of 10 when the maximum momentum is limited to 1.0 (0.8) GeV/c, respectively.

Differential kaon flow

The superb kaon identification of the MRPC barrel allows to perform differential flow analysis of these rare particles and thus to continue the effort to determine the in-medium potential for kaons and antikaons [13]. The analysis has progressed to an extend that first surprising data on the differential sideflow of kaons can be shown.

In Fig. 2 the first fourier coefficients v 1 [14] are plotted as function of the normalized rapidity for kaon and protons for two different centralities. The direction of the reaction plane was determined by the transverse momentum method [15] making use of particles detected in the forward wall of FOPI only in order to avoid autocorrelations. The data are corrected for the finite reaction plane resolution making use of the Ollitrault formalism [16]. For better comparability with different reaction plane reconstruction methods the v 1 -data were shifted such that the mean v 1 -value at midrapidity is vanishing as required by symmetry. The centrality selection was chosen such that for both centrality bins the proton flow is approximately the same. Thus the event samples differ by the amount of stopping and the maximum maximum density reached within the collision leading to the same sidewards deflection. While the kaon data in the central event sample confirm quantitatively our earlier findings [17,18], the large sideflow in the most peripheral data sample is rather unexpected: at a normalized rapidity of y (0) ≈ 0.5 the antiflow of the kaons is as large in magnitude as the flow of the protons. Tentatively this effect can be attributed to the strong momentum dependence of the kaon potential. This hypothesis awaits its confirmation/rejection by transport model calculations.

In order to compare to the data shown in Fig. 2 model calculations have to take into account the acceptance of the detector, that is limited for this detailed analysis to the acceptance of the CDC covering a polar anlge range of 26. 0 < Θ lab < 120 0 . For kaons additional constraints have to be put on the momenta necessary to obtain sufficient particle identification in the data. In the polar angular range from 30.5 0 < Θ lab < 52 0 (MRPC acceptance) kaons are included up to a momentum of 1.0 GeV, in the range 53 0 < Θ lab < 120 0 (plastic barrel acceptance) the laboratory momentum is limited to 0.5 GeV. The angular limits correspond to the so called 'shifted target' geometry, where the target was placed 40 cm upstream of its nominal position in the FOPI apparatus.

The available data sample also allow the reconstruction of the K --flow. However, this analysis is much more difficult due to the very low yield of K -that is at the 3% level of K + -production. The analysis is in progress.

Λ -baryon reconstruction

Strangeness physics in the threshold energy range is dominated by associate production underlining the importance to investigate hyperons in the same event samples as kaons. Phase space distributions of Λ -baryons are already published from earlier experiments [19]. With the current setup high statistics data samples can be obtained with very high purity as shown in Fig. 3 where a Λ -baryon signal is shown with a signal-to-background ratio of 13. The mean value of the reconstructed invariant mass peak with the current calibration amounts to M = 1.1156 ± 0.0002 GeV in very good agreement with the PDG values. Due to the consistency checks allowed by the presence of the new MRPC barrel the calibration of the CDC could be improved so that a Gaussian width of σ M = 3.7 MeV is now achieved for the Ni+Ni data of the 2008 run. The purities shown in Fig. 3 are necessary for a clean study of Λ -baryon flow and for the correlation analysis presented in the next section.

Proton -Λ -correlations

In our previous high statistics data samples, besides Λ-deuteron [START_REF] Herrmann | Proceedings of EXA05 conference[END_REF] we had analyzed Λ-proton correlations [21,22] showing a clear signal at an invariant mass of M = 2.13 ± 0.02 GeV for the reactions studied so far: Ni+Ni at 1.93 AGeV and Al+Al at 1.91 AGeV. Note that in these analyses the combinatorial background is reconstructed by using the same techniques that were used for the Σ * measurement [20] that could be clearly identified in Λ -charged pion correlations. The existence of a peak structure in the Λ-proton correlation is confirmed by the latest data taken with the FOPI apparatus and shown in Fig. 4. The parameters of the peak structure (mean, width and yield) are consistent within statistical errors with the previous findings [21,22]. The reconstructed yield amounts within the acceptance to 3% of the Λ -yield and is not distributed according to thermal phase space. In Fig. 4 only proton -Λ -pairs with a combined rapidity of y Λp < 0.65 are shown. No significant structure was seen at midrapidity. The location of the peak at M(pΛ) = 2.134 GeV agrees within the resolution with the proton -Σ 0 -threshold at 2.1309 GeV and thus drives the hypothesis that the structure is caused by a deuteron-like configuration of a proton and a hyperon. Similar structures have been reported in elementary hadron reactions [23,24,25]. However, the interpretation of the peak structure is still controversial since strong dependencies on the kinematics were observed. With our accumulated high statistics FOPI data we now confirm the existence of the structure in heavy-ion collisions where essentially no kinematic constraints exist due to the strong thermalization of the reaction systems.

As indicated by the arrow markers in Fig. 4 the structure observed in the heavy ion data is clearly different from the one seen in K --absorption in the FINUDA experiment that is considered a candidate for a deeply bound kaonic state. However, the datasets are not necessarily contradicting each other since a) FINUDA has almost no acceptance for an invariant mass of 2.13 GeV and b) the production yield of deeply bound states could be much lower than the one for the observed deuteron-like strange dibaryon.

Conclusion

New data have been presented making use of the FOPI phase III setup at GSI elucidating the properties of strange particle in dense baryonic matter. For the first time a strong antiflow component for K + is seen for peripheral Ni+Ni reactions at 1.91 AGeV. pointing to a strong momentum dependence of the Kaon-Nucleus potential. The special role of the strange quark is visible in the formation of strange dibaryonic systems. Evidence is presented for the existence of a quasi bound state at an invariant Mass of 2.134 ± 0.004 GeV. Further investigations are necessary to clarify the nature of this state and/or the existence of even heavier multibaryonic strange states. More data are available from the 2009 measurement campaign of FOPI where more than 10 8 events were recorded for the reactions Ni+Pb at an incident energies of 1.91 AGeV and Ru+Ru at 1.65 AGeV opening new possibilities to study hadronic states in the hadronic environment of a heavy-ion collision.

Figure 1 .

 1 Figure 1. Reconstructed masses for a momentum range up to 1.0 GeV/c making use of CDC -RPC matches. Positively charged particles are plotted in red, negative ones are shown in blue.Mass peaks for pions, kaons, proton, deuterons and triton are clearly visible. Positively charged kaons exhibit a signal-to-background ratio of better than 10.

Figure 2 .

 2 Figure 2. Sideflow of kaons and protons in the reaction Ni+Ni at an incident energy of 1.91 AGeV as function of normalized rapidity y (0) = y/y CM -1. The first fourier coefficients v 1 are plotted for two different centrality bins for kaons (left panel) and protons (right panel). Black data points are obtained for peripheral collisions corresponding to a mean impact parameter of 6±1 fm, the red ones for central collisions with a geometrical impact parameter b < 3 fm. The v 1 -values are obtained within the acceptance of the CDC, for details see text.

Figure 3 .

 3 Figure 3.Reconstructed invariant mass of off-vertex pion -proton pairs. A minimal distances of 1.9 (0.6) cm was required for pions (protons) in combination with a pointing angle cut of |∆φ| < 2 o and a transverse flight path of at least 4 cm. In the upper panel 29.000 candidates are reconstructed in 5.6•10 7 events in the reaction Ni+Ni at 1.91 AGeV subtracting the mixed event background (blue histogram) from the correlated pairs (red histogram). The signal distribution (lower panel) exhibits a mass resolution of σ M = 3.7MeV . Within a 2.5 × σ M -interval around the nominal mass (hatched area in upper panel) a signal-to-background ratio of 13 is obtained.

Figure 4 .

 4 Figure 4. Reconstructed invariant mass of Λ -proton pairs. The solid histogram and crosses denote the data (red) and the scaled mixed-events background (blue), respectively (upper panel). Both distributions were normalized over the range indicated by the horizontal arrow. The lower panel shows the difference and an excess at a mean value M (pΛ) = 2.134 ± 0.004 GeV with a width of Γ = 26 ± 14 MeV (statistical errors only). The statistical information (signal strength, significance and signal-over-background ratio) is extracted in the mass range limited by the vertical dashed lines shown in the upper panel.

Confidential: not for distribution. Submitted to IOP Publishing for peer review 7 February 2010

Acknowledgments

This work was supported by the German BMBF under Contract No. 06HD190I, by the Korea Science and Engineering Foundation (KOSEF) under Grant No. F01-2006-000-10035-0, by the mutual agreement between GSI and IN2P3/CEA, by the Hungarian OTKA under Grant No. 47168, within the Framework of the WTZ program (Project RUS 02/021), by DAAD (PPP D/03/44611), by DFG (Projekt 446-KOR-113/76/04) and by the EU, 6th Framework Program, Integrated Infrastructure: Strongly Interacting Matter (Hadron Physics), Contract No. RII3-CT-2004-506078.


'Strangeness in

Keywords: numbers: 12.38.Mh, 14.20.Jn, 14.40.Df, 25.75.-q, 25.75.Nq high multiplicity, deconfinement, strangeness, phi resonance, proton collisions, heavy ion, quark-gluon plasma, ALICE 1.1. High Multiplicity pp Collisions at LHC energies

The ALICE experiment at CERN is primarily focused on quark-gluon plasma production in heavy ion (Pb-Pb) collisions. In addition, high multiplicity events from the LHC pp programme will provide an interesting field of study. As with A-A collisions, high multiplicity could suggest high underlying energy density of the system, possibly exceeding that required for deconfinement. Bjorken first suggested that deconfinement might occur in pp collisions, and evidence for this was sought by the E-735 experiment at Fermilab but no clear signature was seen. The LHC will offer new possibilities for examining this idea owing to the much greater multiplicity reach available at higher energy. The ALICE experiment has powerful triggering tools and particle identification (PID), which place it in a strong position to analyse these events and to measure the observable signatures of deconfinement that may arise. Signatures particularly related to strangeness production will be outlined and the capabilities for such studies at ALICE will be discussed.

Introduction

The Large Hadron Collider (LHC) at CERN will collide protons at up to 14TeV cms energy, and Pb ions with a nominal cms energy of 5.5TeV/u. ALICE, A Large Ion Collider Experiment, is designed to study Pb-Pb collisions, but its triggering and PID capabilities provide a unique opportunity to investigate the properties of high multiplicity pp events produced at the LHC. high multiplicity event can indicate that the system had high energy density;

ε B = 1 tA dE T dy = 1 tA dN dy m T ( 1
)
where t is taken to be τ <1fm/c i.e. the formation time, E T and m T are the transverse energy and mass respectively, dN (t)/dy is the multiplicity per unit of rapidity and A is the cross sectional area. LHC pp collisions could reach 5-10GeV/fm 3 -comparable to A-A collisions at RHIC [2]. However, this comparison should be made with caution, as the pp systems have much smaller volume. High multiplicity can also be caused by minijet formation, which affects 50% of events at the Tevatron and is expected to increase with energy [3]. The E735 Collaboration have compared K/π and p/π -ratios from pp events at √ s=1900GeV as a function of multiplicity, reaching only dN/dη≈20 [4]. These measurements were inconclusive due to insufficient statistics, but at LHC energies a large number of pp events with much higher multiplicity can be produced.

Strangeness Production in pp vs A-A systems

Ratios of strange to non-strange particles in A-A collisions (e.g. at SPS [5], RHIC [6]) are enhanced compared with pp collisions. The yields in Pb-Pb collisions at the SPS and Au-Au collisions at RHIC can be described well by a grand-canonical system in equilibrium [7], implying saturation of strangeness upon hadronization of the system [8]. It has been proposed that in pp collisions, strangeness production is suppressed by the limited volume of the colliding system, which requires localised strangeness conservation [9], [START_REF] Kraus | These Proceedings[END_REF]. This canonical suppression, however, does not explain the suppression of the φ meson in pp with respect to A-A collisions [START_REF] Becattini | [END_REF], [12], because the φ has a net strangeness of zero. The γ s factor [START_REF] Rafelski | Hadrons and Quark-Gluon Plasma[END_REF] -which describes the fraction of s and s quarks produced with respect to saturation -is unity in the case where the system reaches equilibrium. The γ s factor affects strange and antistrange quarks equally, and γ s < 1 would be needed to help describe suppression of the φ in pp with respect to A-A collisions in this model.

It is likely that minimum bias pp collisions do not produce a system in full equilibrium. In order for equilibrium to be possible at all, it is necessary that the system has (i) adequate energy density and (ii) a sufficient number of particles to make a statistical ensemble. The increasing density of gluons with decreasing x at HERA suggest that in pp collisions at the LHC systems could have ∼ 30 partons [START_REF] Schukraft | Talk for the ALICE Collaboration[END_REF], which may be sufficient, in combination with the high multiplicity (large energy density) described in Section 1.1, to reach these minimum requirements. In this case, it is possible that a statistical system of a partonic nature could be formed in these collisions. Bjorken first suggested the idea of deconfinement in pp collisions in 1982 [START_REF] Bjorken | [END_REF]. More recently EPOS, a hadronic interaction model which has a core-corona approach to A-A collisions [START_REF] Werner | [END_REF], also points to the production of 'mini-plasma' in high energy density pp collisions.

ALICE at the LHC

ALICE is designed principally for heavy ion studies and is capable of dealing with very high multiplicity events [17]. The central barrel, contained within a large solenoid magnet with 0.5 T field, has an Inner Tracking System (ITS) and Time Projection Chamber (TPC) for tracking, and other detectors for PID and calorimetry. Detectors in the forward region provide triggering, timing and multiplicity information, muon trigger selection and muon tracking. The triggering capabilities and excellent PID of the ALICE detector makes it particularly useful for high multiplicity pp analysis. The Silicon Pixel Detector (SPD) of ALICE makes up the inner two layers of the ITS, providing very precise tracking in the vertex region and fast triggering capabilities. It covers the central region | η |< 1.95 and with ∼10 7 pixels it is designed to handle the extreme multiplicity encountered in A-A collisions, up to 8000 charged particles. In pp collisions, it is used along with forward scintillator detectors to form a first level (L0) minimum bias trigger, but there are many algorithm possibilities for the trigger; for example, one can set up to four tuneable levels of pixel-chip threshold for multiplicity. This is a feature unique to ALICE and provides the opportunity to acquire a large high multiplicity sample during pp collisions. ALICE uses several techniques for charged particle ID [18]. The TPC and Time of flight (TOF) detectors are most important for this, with large acceptance and central coverage. The TPC is a gas-filled chamber used for precise tracking measurements and it provides dE/dx information used for PID of protons, pions and kaons up to ∼1GeV/c. With this basic information, cuts can be made to reduce background in, for example, invariant mass distributions. This study has been limited to a pessimistic scenario, using only TPC PID information with a relatively loose cut of 3σ for kaon selection. The TPC is a relatively slow detector, with a maximum readout rate of around 1kHz, which has an effect on the maximum rate of a high-multiplicity trigger.

Strangeness Analyses with ALICE

Early analyses of strange particles include the relative yields of π/K/p, yields of Λ and Λ, and eventually Ξ and Ξ, as a function of multiplicity. For reasons outlined in the introduction it is clear that the φ yield is also of great interest. The strategy based on analysing a sample collected during the first three months of data taking has been developed. This will provide a first look at high multiplicity and allow for an evaluation of the appropriate next steps over the coming year. To establish the required statistics for each analysis, studies were done assuming multiplicities of at least 5×< dN/dη > would be needed for the high multiplicity sample. Estimates are based on requiring a statistical error of less than 10% for bins below a p T of 3GeV. As high multiplicity samples are difficult to simulate, estimates were made from minimum bias samples. For each case, scaling of significance with dN/dη and number of events was considered in order to obtain the required sample, as shown in Table 1.

Factors in Acquiring a High Multiplicity pp Sample

The reach of multiplicity is not yet known in pp collisions at LHC energies. Different Monte Carlo (MC) models (e.g. PYTHIA [19] and PHOJET [20]) show quite different expected reach for samples of the same size. A reasonable description of the multiplicity distribution can be made using a negative binomial distribution [21], in which case the high multiplicity tail becomes exponential. An early look at LHC data, using four suitable pixel chip multiplicity thresholds, will provide enough statistics in the high tail to allow for an exponential fit. This can be extrapolated to larger statistics and gives the first guess to the potential reach over the period of a few months of data-taking. For interaction rates above about 0.1 interactions/bc, there is a non-negligible contribution to minimum bias data from more than one event in the same bunch crossing. The probability for these 'pile-up' events follows a Poisson distribution, increasing with the interaction rate. With increasing multiplicity, it becomes more likely that an event contains pile-up, as the probability of a single interaction drops faster than pile-up probability. The effect of pile-up events on the multiplicity distribution can be reproduced if the interaction rate is known. As seen in Figure 1(i), using Poisson probabilities and a MC single interaction multiplicity spectrum (b), a distribution for double (c) and triple (d) interactions can be produced, where each pile-up contribution is made up of the sum of multiple single-interaction multiplicities. The effective multiplicity distribution as seen before pile-up removal will be the sum of these contributions (a). Multiple interactions in the same bunch crossing can be removed using offline multiple-vertex finding software [22]. During reconstruction, a primary vertex is found using information from the tracking detectors (ITS, TPC). Once a primary vertex has been found, a further iteration can be used to search for a second vertex. With the right conditions two primary vertices can be distinguished with as little as 750µm separation. The event can then be classed as pile-up and removed. The overall efficiency for this study is assumed 95%. In order to optimise event selection using the high multiplicity trigger, the purity with respect to single interactions needs to be estimated. Using an iterative method based on the interaction rate, the single interaction multiplicity shape can be extracted from the full distribution. With a few million minimum bias events, aided by an exponential fit to the high tail, an estimate of the purity as a function of multiplicity threshold can be produced quickly, indicating the usefulness of a given threshold in selecting 'good' events (see Figure 1(ii).) With this information a suitable threshold can be chosen, balancing loss of statistics and (i).Full (a) and pile-up (c), (d) pixel-chip multiplicity distributions produced from single-interaction only sample (b), using PYTHIA MC with an extended exponential tail, assuming interaction rate of 0.06 ints/bc (colour online). (ii).Purity (fraction of single interactions remaining to total events remaining) as a function of the pixel-chip trigger threshold used. This drop-off describes the loss in purity of the sample and not the reduction in its rate, which drops off exponentially. purity, increased mean dN/dη (7.5 for the minimum bias sample) and suitable rate of the trigger.

Early and Long-Term Running Scenarios: Results and Conclusions

A realistic early (first three months) running scenario for the LHC at optimal energy for ALICE pp collisions is assumed for this study. The interaction rate is assumed to be 0.06 interactions/bc, with 24 bunches. Each bunch will orbit at a rate of just over 11kHz. It is likely that 70% of the triggering time will be dedicated to minimum-bias event selection. Within the remaining 30%, other rare triggers may be used, and the total rate must not exceed 100Hz (in order to keep deadtime to a minimum). For this study, it is assumed that the high-multiplicity trigger is allowed a rate of up to 10Hz. It is assumed that a purity of at least 5% will keep contamination at a safe level. PYTHIA MC simulation is used to generate the minimum bias distribution. Table 2 shows the maximum pixel chip multiplicity threshold and corresponding factor increase in < dN/dη > (assumed to be 7.5 for a minimum bias sample) for a trigger threshold in order to obtain the yields given in Table 1 required for a first analysis at high multiplicity. Even in the most statistically demanding analysis, it appears that in a few months a suitable sample of events of almost 7×< dN/dη > can be obtained. For a full year of running a sample

Figure 1 .

 1 Figure 1. (i).Full (a) and pile-up (c), (d) pixel-chip multiplicity distributions produced from single-interaction only sample (b), using PYTHIA MC with an extended exponential tail, assuming interaction rate of 0.06 ints/bc (colour online). (ii).Purity (fraction of single interactions remaining to total events remaining) as a function of the pixel-chip trigger threshold used. This drop-off describes the loss in purity of the sample and not the reduction in its rate, which drops off exponentially.

Table 1 .

 1 The estimated numbers of events needed for each analysis.

	π/K/p Λ/Λ φ	Ξ/Ξ

Table 2 .

 2 For each analysis, estimated statistics required for the high multiplicity samples after 3 months of data-taking; corresponding maximum pixel chip multiplicity threshold; natural rate using each threshold; minimum dN /dη of the resulting sample; factor increase in <dN /dη> compared to minimum bias data.with appropriately high < dN/dη > can be acquired based on what is found. All PID detectors should be available for use and the first look will indicate whether it is more appropriate to collect a sample with much greater statistics to improve errors in high p T bins or to obtain a sample with a much higher multiplicity threshold. Data-taking from pp collisions at the LHC at 900GeV began in November 2009[23] and the outlook for 2010 looks promising.

		π/K/p	Λ/Λ φ	Ξ/Ξ
	N (10 3 Events)	5	50	300 500
	Max. Threshold	255 (5% pure) 236 208 199
	Rate (Hz)	1	2.4	7.2 10.6 (Max.)
	Min. dN /dη	64	59	52 50
	Factor ×<dN /dη> 8.5	7.8	6.9 6.6


Giuseppe Pagliara
email: pagliara@thphys.uni-heidelberg.de

Matthias Hempel

Jürgen Schaffner-Bielich

Quarkhadron mixed phases in protoneutron stars

published or not. The documents may come   L'archive ouverte pluridisciplinaire

Introduction

The possibility of a first order phase transition from nuclear matter to quark matter at the high densities and moderate temperatures reached in Supernova events or, afterwards, in the still hot and lepton rich protoneutron stars, has attracted much attention in the last years [1,2,3,4,5,6,7,8,[START_REF] Drago | AIP Conf. Proc[END_REF][START_REF] Yasutake | [END_REF]12,13]. In particular, possible interesting fingerprints of this phase transition within the neutrino signal emitted from the newly born neutron star, have been proposed in [6,[START_REF] Yasutake | [END_REF]8,9] thus providing a new tool to investigate the properties of the QCD phase diagram at high densities.

The usual way to model the phase transition, after the seminal paper of Glendenning [14], is to impose the Gibbs conditions for a first order phase transition in muticomponent systems. For matter in neutron stars indeed the baryonic charge and the electric charge are conserved globally resulting in an extended mixed phase in which the pressure increases as a function of the baryon density. In turn, this implies the possible existence of a layer of mixed phase embedded between the pure nuclear and pure quark phases in hybrid stars. Within the mixed phase, due to the finite value of the surface tension between hadronic matter and quark matter, different structures appear depending on the density: first the quark pasta phases, immersed in nuclear matter, and then the nuclear pasta phases immersed in quark matter [15,16]. However, as shown in Refs. [17,18], sophisticated calculations which include finite size and charge screening effects together with more recent estimates for the value of the surface tension, have shown that the mixed phase obtained with the Gibbs construction is actually quite similar to the simpler Maxwell construction. Due to the large value of the surface tension, the charged structures have sizes larger than the electron Debye screening length and therefore their net charge is significantly reduced. One is back to the (almost) constant pressure Maxwell construction which implies that the mixed phase cannot be present in neutron stars where the pressure must be a monotonic function of the radius.

In protoneutron stars, however, an additional charge is conserved, the lepton number, during the stage of neutrino trapping. As we will show in the following, due to this additional conservation law, it is possible to form a new type of mixed phase in protoneutron stars. Since this mixed phase is related to the conservation of lepton number, this phase will gradually "disappear" as the neutrinos become untrapped.

The equation of state of mixed phases

We want to present here two extreme cases for the mixed phase in protoneutron star matter: the first, widely considered in the literature, corresponding to a vanishing surface tension (we call it Mixed Phase 1, MP1) and the second, corresponding to a very large value of the surface tension (we call it Mixed Phase 2, MP2). Let us discuss first the structure of the interface between hadronic matter and quark matter: a charge separated interface is formed with a size of the order of the Debye screening length, ∼ 10 fm, with a layer of positively charged, electron depleted, hadronic matter on one side and a layer of quark matter with an excess of the electrons on the other side (as discussed in [19] for the CFL phase). The interface is stabilized by the resulting electric field. Notice that neutrinos, being not affected by the electric field, can freely stream across the interface. Consequently, the lepton number is conserved globally even in the case of a large surface tension.

We consider the "standard" conditions of ProtoNeutron Star (PNS) matter [3]: fixed lepton fraction Y L = (n e + n ν )/n B = 0.4 and fixed entropy per baryon S/N B = 1 where n e , n ν and n B are the electron, neutrino and baryon number densities and S is the entropy. The chemical equilibrium between the different species of particles within the two phases allows to write the following general relations:

µ n = µ H B , µ p = µ H B + µ H C , µ H e = µ H L -µ H C , µ H ν = µ H L ( 1
)
µ u = µ Q B + 2µ Q C 3 , µ d = µ Q B -µ Q C 3 , µ s = µ Q B -µ Q C 3 , µ Q e = µ Q L -µ Q C , µ Q ν = µ Q L ( 2
)
where µ A i (i = n, p, e, ν, u, d, s and A = H, Q) are the chemical potentials of neutrons, protons, electrons, neutrinos and up, down, strange quarks within the hadronic phase (A=H) and the quark phase (A=Q). The chemical potentials associated with globally conserved quantities are equal in the two phases [20]. In presence of a vanishingly small value of the surface tension, the electric charge is conserved globally as the baryonic number and the lepton number, therefore its chemical potential is continuous at the onsets of the phase transition and we can write the Gibbs conditions as follows:

P H (µ H B , µ H C , µ H L , T H ) = P Q (µ Q B , µ Q C , µ Q L , T Q ) (3) (1 -χ)n H C + χn Q C -n e = 0 (4
)

n e + n ν = Y L n B ( 5
) (1 -χ)s H + χs Q = S/Nn B ( 6
)
T H = T Q (7) µ H B = µ Q B , µ H C = µ Q C , µ H L = µ Q L ( 8
)
where

P H,Q , T H,Q , s H,Q and n H,Q C
are the pressure, the temperature, the entropy density and the electric charge density of the hadronic and quark phases respectively. χ is the volume fraction of the quark phase.

In presence of a large value of the surface tension, the electric charge is conserved only locally, therefore its chemical potential is different in the two phases, but the lepton number, as noticed before, is still conserved globally as the baryon number and its chemical potential must be continuous across the phase transition. In this case, we can write the Gibbs conditions as follows:

0 0.
P H (µ H B , µ H C , µ H L , T H ) = P Q (µ Q B , µ Q C , µ Q L , T Q ) (9) n H C -n H e = 0 ( 10
)
n Q C -n Q e = 0 ( 11
) (1 -χ)(n H e + n ν ) + χ(n Q e + n ν ) = Y L n B ( 12
) (1 -χ)s H + χs Q = S/Nn B ( 13
)
T H = T Q (14) µ H B = µ Q B , µ H L = µ Q L ( 15
)
The systems of equations above presented have a general validity. To give some numerical examples, we consider, as customary, two models for strongly interacting matter: one at low density with nucleon degrees of freedom and one at large densities with quark degrees of freedom. For the former, we adopt the relativistic mean field model with the parameterization TM1 [21] and for the latter the MIT bag model. We set the masses of up and down quarks to zero and the mass of the strange quark to 100 MeV. The bag constant B is set to B 1/4 = 165 MeV. The phase transition is then computed by using the systems for the two mixed phases MP1 and MP2. In Fig. 1, we show the equations of state for protoneutron star matter and cold and beta stable matter for the two cases MP1 and MP2. Notice that for PNS matter, the equations of state are quite similar: an extended mixed phase with varying pressure is obtained also in the case of local charge neutrality due to the existence of an additional globally conserved quantities which is the lepton number. In cold and beta stable matter on the other hand, there are only two conserved charged, the baryonic and the electric charge, and

0 2 4 6 8 1 0 R[km] 0 0.2 0.4 0.6 0.8 1 1.2 n B [fm -3 ] MP2-cold MP2-PNS M B =1.7 M sun Figure 2.
Density profiles for a M B = 1.7M sun star in its protoneutron star stage and the cold and beta stable asymptotic state, the case of the mixed phase MP2 is considered. In the protoneutron star a 2km layer of mixed phase is present while in the cold configuration the mixed phase disappears and a sharp interface separating the pure quark and nuclear phase is obtained.

therefore in the case of MP1 one obtains an extended mixed phase but in MP2 the result is the simple constant pressure Maxwell construction. In general, the mixed phase MP2 extends over a smaller range of density and has a lower pressure gradient with respect to MP1, because the requirement of local charge neutrality is more restrictive.

Discussion and conclusions

The new mixed phase here proposed, MP2, is present as long as neutrinos are trapped. After the complete deleptonization and cooling of the star it becomes a constant pressure mixed phase which cannot appear in neutron stars. This implies that the deleptonization drives a gradual modification of the structure of the star: for sufficiently large masses of the protoneutron stars, one has a core of pure quark matter, a layer of mixed phase MP2, and then a layer/crust of nuclear matter. The layer of mixed phase disappears during deleptonization and, at the end of the deleptonization stage, a sharp interface will separate the pure quark and the pure nuclear phase, see Fig. 2. Notice that the mixed phase MP2 has quite different properties with respect to MP1: in MP2 the two phases are locally charge neutral, therefore no Coulomb lattice with charged finite structures of the two phases can form. This implies also that only spherical pasta structures can be formed since the 1-D and 2-D structures can take place only in the presence of Coulomb interactions. Moreover the charge neutral structures have macroscopic sizes contrary to the MP1 case where the optimal size of the structures is limited by the Coulomb energy. We expect that within the MP2 mixed phase the process of diffusion of neutrinos can be significantly different with respect to the MP1 case: for instance, due the large sizes of the structures within MP2, no coherent scattering of neutrinos with pasta structures can take place [22], as the neutrino wavelength is of the order of tens of fermi. A detailed simulation of neutrino transport within this new mixed phase would be extremely interesting for the possible implications on the neutrino signal of the changes of the structure of the star during deleptonization. Also the motion and the interactions of the drops/bubbles within the mixed phase, in presence of turbulence, might represent an interesting source of gravitational waves. A more detailed study, which includes also superconducting quark matter [23,24,25], is presently in progress.

  The equations of state for protoneutron star matter and cold and beta stable matter for the two types of mixed phases MP1 and MP2. The dots indicate the onset and the end of the mixed phases. The mixed phases for protoneutron star matter MP1 and MP2 are quite similar. For cold and beta stable matter, MP2 coincides with a constant pressure Maxwell construction.

		MP1-PNS
		MP1-cold
		MP2-PNS
	100	MP2-cold
	P[MeV/fm 3 ]
	1
		1	0.2	0.3	0.4	0.5	0.6
				n B [fm	-3 ]
	Figure 1.

Acknowledgments:

The work of G. P. is supported by the Deutsche Forschungsgemeinschaft (DFG) under Grant No. PA 1780/2-1. M. H. acknowledges support from the Graduate Program for Hadron and Ion Research (GP-HIR). J. S. B. is supported by the DFG through the Heidelberg Graduate School of Fundamental Physics. This work was also supported by CompStar, a Research Networking Programme of the European Science Foundation.


Tim Gershon

∆Γ d : A Forgotten Null Test of the Standard Model

Keywords: numbers: 13.25.Hw, 12.15.Hh, 11.30.Er

The recent measurement of an anomalous like-sign dimuon asymmetry by the D0 collaboration has prompted theoretical speculation on possible sources of physics beyond the Standard Model that may affect lifetimes and lifetime differences in neutral B meson systems. One observable that deserves closer attention is the width difference in the B 0 d system, ∆Γ d . Since the Standard Model prediction for this quantity is well below 1 %, it serves as a "null test" whereby the measurement of a larger value would cleanly reveal the presence of new physics. Methods to measure ∆Γ d at current and future experiments are reviewed and an attractive new approach is proposed.

Although improved experimental measurements are certainly necessary, the D0 result has prompted a great deal of theoretical speculation on sources of physics beyond the Standard Model ("new physics") that could be responsible for the effect. One interesting possibility is that new physics may cause the neutral B meson lifetimes and lifetime differences to differ from their SM values [9][10][11][12]. Such scenarios -in particular the possibility that the width differences ∆Γ q (q = d, s) between the eigenstates of the effective weak Hamiltonian in the B 0 q -B0 q systems may be enhanced above their SM values -have not previously been widely considered. In particular, with little exception [13], the utility of ∆Γ d to test the SM has been largely overlooked.

To test new physics models that affect ∆Γ q it is necessary to have improved measurements of the parameters of the neutral B meson systems, and to have precise SM predictions to compare against. A useful class of tests is those referred to as "null tests", in which the Standard Model prediction is vanishingly small compared to the experimental sensitivity. The search for CP violation in neutral B meson mixing itself falls into this class of tests. Among the CP -conserving mixing parameters, however, there is also one parameter which is very small in the SM, namely ∆Γ d , for which the prediction is [

3] ∆Γ SM d ∆M SM d = 52.6 +11.5 -12.8 × 10 -4 , (1)
∆Γ SM d = 26.7 +5.8 -6.5 × 10 -4 ps -1 , (2)
∆Γ SM d Γ SM d = 40.9 +8.9 -9.9 × 10 -4 , (3)
where the second and third results are obtained using the experimental values of the mass difference and average lifetime in the [14]. This involves an assumption that there are no new physics contributions to ∆M exp

B 0 d -B0 d system, ∆M exp d = (0.507 ± 0.004) ps -1 and 1/Γ exp d = τ (B 0 d ) exp = (1.530 ± 0.009) ps
= Γ SM d .
The former is indeed confirmed to good precision by global fits to the Cabibbo-Kobayashi-Maskawa (CKM) [15,16] Unitarity Triangle (see, for example, Refs. [17] and [18]). In any case, plausible new physics effects in the values of ∆M d and Γ d cannot change the conclusion that

∆Γ SM d /Γ SM d O(10 -2
). It is striking that there are relatively few measurements of ∆Γ d listed by the Particle Data Group [14] and the Heavy Flavour Averaging Group [START_REF] Barberio | Heavy Flavor Averaging Group[END_REF]. The world average, based mainly on a single measurement from BaBar [START_REF] Aubert | [END_REF]21] is sign(Re λ CP )∆Γ d /Γ d = 0.009 ± 0.037. BaBar use a notation in which sign(Re λ CP ) is expected to be +1 in the SM (where it is, to a good approximation, equal to sign(cos 2β), where β is one of the CKM Unitarity Triangle angles). As discussed below, ∆Γ d may be quite challenging to measure, but nevertheless its determination should be possible for the existing B-factory experiments, BaBar [22] and Belle [23], the Tevatron experiments CDF [24] and D0 [25], the CERN LHCb experiment [26] and any future Super Flavour Factory [27][28][29]. Even discounting any potential sensitivity to new physics, ∆Γ d is a fundamental parameter of the neutral B meson system and should be measured as precisely as possible.

As a brief digression, it is interesting to consider the situation in the D 0 -D0 system (see Ref. [30] for a de-Confidential: not for distribution. Submitted to IOP Publishing for peer review 26 August 2010 tailed review). Before the first evidence of charm oscillations was discovered in 2007 [31,32], the SM values of the mixing parameters x D = ∆m D /Γ D and y D = ∆Γ D /2Γ D were generally believed to be O(10 -3 ). However, theoretical re-evaluations have shown that values as large as the experimental measurements, which are O(10 -2 ), cannot be ruled out within the Standard Model. To resolve the situation, more precise measurements and improved theoretical calculations are required. The theoretical sit-uation for the neutral B mesons is, however, somewhat better compared to that for charm mesons, due to the different regime with regard to QCD effects.

In order to examine how ∆Γ d may be determined experimentally, consider the time-dependent decay rates of a neutral B meson that is initially (at time ∆t = 0) tagged as B0

q or B 0 q to a final state f . Assuming CP T conservation, and neglecting small corrections, these are given by [33]

Γ B0 q →f (∆t) = N f e -|∆t|/τ (B 0 q ) 4τ (B 0 q ) cosh ∆Γ q ∆t 2 + S f sin(∆m q ∆t) -C f cos(∆m q ∆t) + A ∆Γ f sinh ∆Γ q ∆t 2 , (4)
Γ B 0 q →f (∆t) = N f e -|∆t|/τ (B 0 q ) 4τ (B 0 q ) cosh ∆Γ q ∆t 2 -S f sin(∆m q ∆t) + C f cos(∆m q ∆t) + A ∆Γ f sinh ∆Γ q ∆t 2 , (5)
where

S f = 2 Im(λ f ) 1 + |λ f | 2 , C f = 1 -|λ f | 2 1 + |λ f | 2 , (6)
A ∆Γ f = - 2 Re(λ f ) 1 + |λ f | 2 , and λ f = q p Āf A f . (7)
Note that (S f )

2 +(C f ) 2 + A ∆Γ f 2
= 1 by definition. The parameters Āf and A f are the complex amplitudes for the decay of a B0 q and a B 0 q to the final state f , respectively. The parameters q and p describe the eigenstates of the effective weak Hamiltonian of the B 0 q -B0 q system (in the absence of CP violation in mixing, |q/p| = 1). The constant N f is a normalisation factor. The equations above are identical for B meson production in any environment, but in coherent production of B B pairs at e + e -B factory colliders the quantum entanglement allows flavour tagging at any time so that the range of ∆t is -∞ < ∆t < ∞, whereas B mesons produced in hadron collisions (or at higher energy e + e -colliders) are tagged at their production point (0 < ∆t < ∞). Note that in contrast to the CP -violating asymmetry parameters S f and C f , the parameter A ∆Γ f appears with the same sign in both B0

q and B 0 q decay time distributions. Focussing now on the B 0 d system, and assuming that

y d = ∆Γ d /2Γ d 1, the following substitutions can be made cosh ∆Γ d ∆t 2 ≈ 1 , (8)
sinh ∆Γ d ∆t 2 ≈ y d τ (B 0 d ) ∆t , (9)
where terms of O(y 2 d ) have been neglected. The best channels to measure ∆Γ d with are therefore those with large branching fractions and large, well-known values of A ∆Γ f . Experimentally, it is also desirable to use final states that are advantageous from the point of view of minimising systematic errors -specifically, those with relatively clean signals and with accessible control samples with similar topologies. These considerations suggest that decays mediated by the b → ccs quark-level transition and involving J/ψ mesons in the final state will be most suitable.

Since the term involving A ∆Γ f is the same for B0 d and B 0 d decays, y d can be determined in untagged analyses (i.e. without determining the initial flavour of the decaying B meson). With the approximations above, the untagged decay rate is

Γ B0 d →f (∆t) + Γ B 0 d →f (∆t) ∝ e -|∆t|/τ (B 0 d ) 1 + A ∆Γ f y d τ (B 0 d ) ∆t . (10)
Noting that e X(1+ ) ≈ e X (1 + X ), this result shows that (i) the effect of y d = 0 is that the effective lifetimes measured in decays to different final states (with different values of A ∆Γ f ) can differ; (ii) any attempt to measure y d from a single final state will suffer from large systematic uncertainties.

Bearing this in mind, there are four main possible approaches that can be used to determine y d .

1. From the difference in effective lifetime between decays to CP -eigenstates and decays to flavourspecific (or quasi-flavour-specific) final states.

This is the approach used by BaBar [START_REF] Aubert | [END_REF]21], where the CP -eigenstate sample is dominated by J/ψ K 0 S , and the quasi-flavour-specific sample is dominated by D ( * )-h + (h = π, ρ, a 1 ). Semileptonic decays (D ( * )-l + ν) have also been proposed as a high-statistics flavour-specific control sample [13]. Flavour-specific final states have A ∆Γ f = 0 (the equality is only approximate for quasiflavour-specific states), while A ∆Γ J/ψ K 0 S = cos 2β to a good approximation in the SM.

However, if the reconstruction and vertexing requirements differ between the two samples, there is potential for systematic biases. An ideal approach is therefore to compare the lifetime distributions of J/ψ K 0 S and the topologically similar but flavour-specific final state J/ψ K * 0 , with K * 0 → K + π -. Since these final states are advantageous for hadron colliders, extremely high statistics will be available in the near future from LHCb (see, for example, Refs. [34,35]). Note the strong analogy with the measurement of y D from the comparison of effective lifetimes in D 0 → K + K -and D 0 → K -π + [32,36,37].

2. From the difference in effective lifetime between decays to suppressed and favoured final states.

Another method which has been used to powerful effect to determine charm mixing parameters is the comparison of suppressed (D 0 → K + π -) to favoured (D 0 → K -π + ) decays [31,38,39]. A tagged analysis is needed to disentangle the contributions and maximise the statistical sensitivity. The analogy in the B 0 d system is the D ( * )± π ∓ final states. However, while the charm system provides efficient and effective tagging using the decay D * + → D 0 π + , there is no equivalent for the B mesons. Therefore, this approach does not appear statistically competitive. It is, however, worth noting that terms involving y d should be considered in order not to bias measurements of the combination of CKM Unitarity Triangle angles sin(2β + γ) from these decays [40][41][42][43].

From the difference in effective lifetime between

CP -even and CP -odd components of self-conjugate vector-vector final states.

Final states that differ only by having opposite CPeigenvalues have opposite values of A ∆Γ f . Hence one possibility is to compare the lifetime distributions between, say, the J/ψ K 0 S and J/ψ K 0 L final states. An alternative is to use angular analysis to disentangle the components with each CP -eigenvalue in self-conjugate vector-vector final states. This is the approach which is being used to measure ∆Γ s in B 0 s → J/ψ φ decays [44,45]. The equivalent final state for the B 0 d is J/ψ K * 0 , where the subsequent decay K * 0 → K 0 S π 0 must be used. The requirement on the K * 0 decay unfortunately complicates the analysis due to the reduced statistics and the less clean final state (note, however, that this decay chain has been used for measurements of cos(2β) [46,47]). Analyses using B 0 d → D * + D * -or J/ψ ρ 0 are possible but are not particularly attractive from the point of view of statistics.

From the difference in effective lifetime between

CP -even and CP -odd components of self-conjugate multibody final states.

The contributions from amplitudes with each possible CP -eigenvalue for decays to self-conjugate multibody final states can be disentangled using Dalitz plot analysis. However, a tagged analysis is necessary to maximise the statistical sensitivity. In the charm system, such analyses have been carried out using D 0 → K 0 S π + π - [48][49][50]. Appropriate channels that correspond to b → ccs transitions include D + D -K 0 S and D 0 D0 K 0 S , but the high multiplicity of the final states reduces the available statistics [51]. Larger event yields are or will be available in B 0 d decay channels such as K 0 S K + K - [52,53], K 0 S π + π -[54, 55] and D CP π + π -(where D CP denotes that the neutral D meson must be reconstructed in a CP eigenstate). However, the lower branching fractions of these decays compared to the b → ccs transitions, as well as the inevitable systematic uncertainties due to Dalitz plot model dependence, make these analyses less attractive for the measurements of y d . Note, that these final states are, however, interesting for determinations of cos(2β) [56,57].

The most promising approach to determine ∆Γ d therefore appears to be from the difference in effective lifetimes between decays to the CP -eigenstate J/ψ K 0 S and the flavour-specific final state J/ψ K * 0 . If the vertex position is reconstructed identically (specifically, using only the J/ψ decay products, typically µ + µ -) in the two cases then the largest potential source of systematics should cancel almost exactly. Care will be required to ensure that reconstruction and selection requirements on the K 0 S → π + π -and K * 0 → K + π -do not induce a bias on the lifetime distributions, but this does not appear to present a significant obstacle to the analysis. Similarly, potential systematic effects due to the different background composition in the two channels should be manageable. One potentially dangerous systematic effect would be an asymmetry, a prod , between the production rates of B0 d and B 0 d . If non-zero, the cancellation of terms in Eqs. ( 4) and (5) will not be exact, so that the untagged rate of Eq. (10) will include an additional factor in the parentheses of a prod (S f sin(∆m d ∆t) -C f cos(∆m d ∆t)). This can be handled either by fixing a prod based on independent control samples, or by allowing the presence of such a term in the fit to data. Asymmetries in the reconstruction of J/ψ K * 0 events (or direct CP violation in the decay to this final state) would similarly introduce a cos(∆m d ∆t) dependence in the lifetime distribution, which can be handled in the same way.

In summary, the parameter ∆Γ d , which describes the width difference between the eigenstates of the effective weak Hamiltonian in the B 0 d -B0 d system, is of interest to test the Standard Model and potentially to corroborate the recent evidence for anomalous effects in the B system. Few measurements exist, and current experiments have the potential to improve the existing bounds. Further improvement will be possible at the LHCb experiment, where a newly proposed method based on the difference between lifetime distributions for the untagged decays B 0 d → J/ψ K 0 S and B 0 d → J/ψ K * 0 , K * 0 → K + π -appears to be the most promising approach to reach high precision.

I am grateful to Sheldon Stone and Jacques Lefrancois for comments that provoked this work, and to Paul Harrison and Guy Wilkinson for comments on the manuscript.

[arXiv:0809.0872[hep-ph]].

This work is supported by the Science and Technology Facilities Council (United Kingdom) and by the European Research Council under FP7.