Alexander Schmah
email: a.schmah@gsi.de

Strangeness Measurements at the HADES Experiment

We report on HADES measurements of strange hadrons in the collision systems Ar(1.756 AGeV)+KCl and p+p at 3.5 GeV. Comparisons of K 0 s transverse mass and rapidity spectra to IQMD transport model calculations give a strong hint to a repulsive kaon-nucleon potential. The effect of the potential shows up strongest at very low transverse momenta, which were measured by HADES with high statistics. Statistical model fits show a fair agreement to the particle yields measured in the heavyion data, apart from the Ξ -yield which is off by more than an order of magnitude. Furthermore, first results from the Σ(1385) + reconstruction in p+p reactions are presented, showing an excellent agreement with published data.

Introduction

Heavy-ion collisions at SIS energies (E= 1-2AGeV) allow to create rather dense nuclear systems up to few times the saturation density, providing a favorable environment to study the hadron in-medium properties. The investigation of possible in-medium effects of hadrons, involving strange particles produced in heavy-ion collisions at SIS energies has been pursued during the last two decades and particular efforts have been devoted to the analysis of the K + and K -production. The High Acceptance Di-Electron Spectrometer at the heavy-ion synchrotron SIS at GSI has been shown to be suited for precise measurements of strange hadrons [1]. A wide set of particles with single and double strange quark content, produced at or below the free NN-threshold energies, could be reconstructed in a small-sized heavy-ion reaction system and in p+p collisions. For the first time, at SIS energies, the combined and inclusive identification of sub-threshold produced K + , K -and φ mesons in heavy-ion reactions was carried out [2]. As one of the results, the importance of the φ meson as a source of K -was found to be much more relevant than expected. K 0 S mesons have been measured as well and high statistics has been collected also in the low-m t range for a broad rapidity interval. The detailed study of the m t distribution of K 0 S down to very low transverse masses allows to discriminate more quantitatively among the theoretical calculations performed assuming different strength of the K 0 S -nucleus potential. As an example we show in the first section the comparison with the IQMD model. The measured particle species allows to fit the particle yields by a statistical model. A clear deviation from the estimations by the model are found in case of the Ξ -yield. The baryon chemical potential and the chemical freeze-out temperature was extracted and compared to other results in the T -µ B plane. In addition to the heavy-ion measurements, p+p data at 3.5 GeV have been measured and a large statistics has been collected. One of the goals of the elementary reaction program is the reconstruction of the Λ(1405)/Σ(1385) in various decay channels. The understanding of the Λ(1405) is also a starting point for the search of deeply bound kaonic states [3]. As a first step and result of this analysis the reconstruction of the Σ(1385) + will be presented in the last section.

In medium modification of K 0 mesons in Ar+KCl reactions

The properties of neutral kaons (K 0 S ), produced in heavy-ion collisions, can be studied in order to exctract a kaon-nucleon potential. The in-medium potential is expected to be repulsive for both K + and K 0 . As far as the K 0 in-medium properties are concerned, recent results extracted from pion-induced reactions should be mentioned [4]. High statistics and a large phase space coverage of K 0 S , produced in Ar+KCl reactions at 1.756 AGeV, allows a differential comparison of the HADES data to transport model calculations with and without a kaon-nucleon potential. About 66.000 K 0 S mesons were reconstructed in the decay channel K 0 S → π + + π - with a signal to background ratio varying from 0.3 to 2.0 depending on the transverse momentum and rapidity. Figure 1 shows the integral invariant mass distribution of π ±pairs after cuts on the production and decay vertices were applied in order to suppress background events. The remaining combinatorial background was subtracted by using the mixed event techniques. The mixed event background is shown in Fig. 1 as a grey shaded area. For the efficiency and acceptance corrections full scale simulations were performed by using the GEANT3 package [5].

Fig. 2 (left) shows the transverse mass m t -m K 0 s spectra of K 0 s by empty symbols after background subtraction and correction for acceptance and efficiency for different y c.m. bins. A fit according to

1 m 2 t d 2 N dm t dy = C(y) exp - (m t -m 0 )c 2 T (y) (1)
has been applied to the data. Using this Boltzmann parameterization an inverse slope parameter T B (y) can be determined as a function of the rapidity. Fitting the T B (y) distribution with the function T B (y) = T eff cosh(y) the parameter T eff can be determined. This represents the inverse slope at mid-rapidity and corresponds to an effective temperature at the kinetic freeze-out stage as was found to be 92 ± 2 MeV. The rapidity densitiy distribution was calculated by integrating the fitted Boltzmann distributions from 0 to ∞. The transverse mass spectra in Fig. 2 as well as the dN/dy and T B distributions, shown in Fig. 2 (right), are compared to IQMD transport model calculations. Among other models, the IQMD model was used in the past to compare with KaoS and FOPI data [6]. The IQMD simulations have been calculated assuming a selection of the impact parameter b < 6 fm. This value was fixed by a normalization to the pion yield. Both, the dN/dy and the inverse slope distributions, are better reproduced by a IQMD calculation with a K 0 s -nucleus potential of about 40 MeV. Within the errors the K + meson transverse mass spectra agree with the K 0 s distributions [START_REF] Agakishiev | Medium Modification of K 0 in Relativistic Heavy Ion Collisions[END_REF]. A more detailed comparison of K 0 s p t -spectra to the IQMD model calculations to extract the KN potential strength is shown in [START_REF] Agakishiev | Medium Modification of K 0 in Relativistic Heavy Ion Collisions[END_REF].

!" " ! # $ % & ' " % ( ) * & % # $ & + , & + + % , + % + , % + + , + + , + + , + +- , +

Statistical model fit and chemical freeze-out

In the reaction system Ar+KCl at 1.756 AGeV several particles species were measured within a large kinematic range, allowing to extract their multiplicities with small extrapolation uncertainties. In the past it was shown that the statistical model is able to describe relative particle yields from heavy-ion collisions fairly good [10]. Possible fit parameters of the statistical model are the chemical freeze-out temperature T , the baryon-chemical potential µ B , the volume of the fireball R and the strangeness correlation radius R c . In Fig. 3 (left) a fit of the THERMUS [START_REF] Wheaton | [END_REF] model to the HADES Ar+KCl data is shown. The K ± , K 0 S , π -and Λ hadrons were measured in a large acceptance, whereas the φ and Ξ -were measured only in a small kinematical range. Except for the Ξ -data point all other particle yields were fitted within the error bars by the statistical model. The Ξ -analysis and the comparison to the statistical model is described in detail elsewhere [11].

We extracted the following values for the chemical freeze-out conditions: T ≈ 73 MeV, µ B ≈ 770 MeV, R c ≈ 2.4 fm and R ≈ 4.9 fm. The T and µ B value is plotted in Fig. 3 (right) together with other data from SIS, AGS, SPS and RHIC experiments [12]. The HADES data is consistent with results from the FOPI collaboration at a similar beam energy. The line with a constant mean energy per nucleon with a value of about < E > / < N >≈ 1.1 GeV is in agreement with all SIS data points.

Production of Σ(1385) and Λ(1405) in p+p reactions at 3.5 GeV

In April/May 2007 about 1.2 billion p+p reactions were collected with HADES. As a first level trigger, three particles in the META detector were required. The experimental setup was nearly identical to the one of the Ar+KCl beam time. As an additional detector, a time-of-flight wall was installed 7 m downstream of the target, which covers polar angles from 0.3 to 7 degrees. Since a start detector was not installed, the event start time was recalculated, using the momentum information of at least one identified particle. For this physics analysis the pions and protons were identified via the energy loss information of the multi-wire drift chambers, whereas for the kaons additional information as the energy loss in the time-of-flight detectors and the calculated particle mass were used. One of the analysis goals is the reconstruction of the Λ(1405) in all its decay channels:

Λ(1405) → Σ + + π -→ n + π + + π -, (2)
Λ(1405) → Σ -+ π + → n + π -+ π + , (3)
Λ(1405) → Σ 0 + π 0 → Λ + γ + π 0 . (4)
The particle is well known since many decades but its internal structure is still unknown. Calculations and recent experimental findings show different lineshapes of the Λ(1405) depending on the production and decay channels. The Λ(1405) signal is overlapped in all charged Σπ decay channels by the Σ(1385) 0 resonance, which has identical daughter particles. Even in the Σ 0 π 0 decay channel it is not possible with the actual HADES setup to fully distinguish between the two particles as it was done for example in [13]. To get rid of background an exclusive analysis of the production channel p + p → Λ(1405)/Σ(1385) 0 + K + + p is inevitable. This means a measurement of all charged particles and a reconstruction of intermediate and missing neutral particles using the missing mass and invariant mass technique.

Since the production and decay chain of the Σ(1385) + is comparable to the ones of the Λ(1405), its reconstruction can be used as a benchmark for the Λ(1405) analysis. Furthermore the lineshape of the Σ(1385) + and its cross section is important for the later subtraction of the Σ(1385) 0 signal from the Λ(1405) spectrum. The following production and decay channel was investigated:

p + p → Σ(1385) + + K + + n → (Λ + π + ) + K + + n. (5)
In Fig. 4 the reconstructed missing neutron and the Λ hyperon are shown. Figure 5 shows the invariant mass of Λ + π + after the selection of the missing neutron and the Λ and the subtraction of a simulated phase space background. The reconstructed The non-resonant background is already subtracted. The black rectangles shows the HADES data from p+p reactions in comparison to the normalized data from [14] and [15]. The width and pole mass position corresponds to the Breit-Wigner fit shown in the figure .  Σ(1385) + shows a nice agreement to the lineshapes of previously published data [14,15]. The Λ(1405) analysis was performed in a similar way. First results show the feasibility to reconstruct the resonance in all Σπ decay channels.

Summary and outlook

We presented high resolution K 0 s data from Ar+KCl reactions at 1.756 AGeV in comparison to IQMD transport model calculation. The simulation with a kaon-nucleon potential of about 40 MeV show a good agreement with the data. Statistical model calculations were performed and fitted to the yields of the measured hadrons in the Ar+KCl beam time. The extracted T and µ B values are in good agreement with the world data. In p+p reactions at 3.5 GeV, the reconstruction of the Σ(1385) + was presented and compared to reference data. The result shows the feasibility for an Λ(1405) analysis with HADES. The upgrade of the HADES experiment with the high resolution RPC detectors, an up to ten times faster DAQ system, and improved multi-wire-drift chambers will be completed within this year. An already installed forward wall hodoscope will deliver important information for the event plane reconstruction and the reaction centrality. With this setup it is planned to take data using medium and heavy sized reaction systems from 2010 on. Furthermore, experiments with pion beams are scheduled to study strangeness production in cold matter.

The collaboration gratefully acknowledges the support by BMBF grants 06DR135, 06FY171, 06MT238 T5, and 06MT9156 TP5 (Germany), by the DFG EClust 153, by GSI TMKRUE, by grants MSMT LC07050 and GA ASCR IAA100480803 (Czech Rep.), by grants RII3-CT-2005-515876, FPA2006-09154, and CPAN:CSD2007-00042 (Spain).

Figure 1 .

 1 Figure 1. Invariant-mass distribution of π + -π -pairs (top). The combinatorial background (shaded area) is obtained by the mixed-event technique. The backgroundsubtracted distribution (bottom) shows clearly he K 0 S meson signal (grey area with a Gaussian fit).

Figure 2 .

 2 Figure 2. Left: Transverse mass spectra of K 0 s mesons from Ar+KCl reactions at 1.756 AGeV for different rapidity bins. The overlayed blue histogram shows a IQMD transport model calculation with a kaon-nucleon potential strength of about 40 MeV. The dashed curves are fits by Eq. 1. Right: Rapidity distribution (top) and inverse slope parameters (bottom) as a function of the center-of-momentum rapidity. IQMD transport model calculations without kaon-nucleon potential (green stars) and with potential (open blue circles) are overlayed.

Figure 3 .

 3 Figure 3. Left: Particle multiplicities in Ar+KCl reactions at 1.756 AGeV together with fit by the THERMUS statistical model to the data. Right: Chemical freeze-out temperature as a function of the baryon chemical potential µ B . The shown data ranges from RHIC down to SIS energies. The included HADES data point corresponds to the result of the statistical model fit shown left picture.

Figure 4 .

 4 Figure 4. Left: The missing neutron in the decay is calculated by the missing mass of all charged particles. Right: The invariant mass of proton and π -shows the Λ signal. The grey shaded area shows the cut regions on the two particles for the Σ(1385) + reconstruction.

Figure 5 .

 5 Figure 5. Invariant mass of Λ and π + after a selection of a missing neutron (see text).The non-resonant background is already subtracted. The black rectangles shows the HADES data from p+p reactions in comparison to the normalized data from[14] and[15]. The width and pole mass position corresponds to the Breit-Wigner fit shown in the figure.


Chihiro Sasaki
email: csasaki@ph.tum.de

Implications of Quarkyonic Phase with an alternative pattern of chiral symmetry breaking

We discuss, along with the anomaly matching, a possible phase at finite density where chiral symmetry is spontaneously broken while its center remains unbroken. The quark number susceptibility exhibits a strong enhancement at the restoration point of the center symmetry rather than that of the chiral symmetry. This can be interpreted as the quarkyonic transition which was recently proposed in large N c QCD.

Introduction

Model studies of dense baryonic and quark matter have suggested a rich phase structure of QCD at temperatures and quark chemical potentials being of order Λ QCD . Our knowledge on the phase structure is however still limited and the description of the matter around the phase transitions does not reach a consensus because of the nonperturbative nature of QCD [1]. Possible phases and spectra of excitations are guided by symmetries and their breaking pattern in a medium. Dynamical chiral symmetry breaking and confinement are characterized by strict order parameters associated with global symmetries of the QCD Lagrangian in two limiting situations: the quark bilinear qq in the limit of massless quarks, and the Polyakov loop Φ in the limit of infinitely heavy quarks.

A novel phase of dense quarks, Quarkyonic Phase, was recently proposed based on the argument using large N c counting where N c denotes number of colors [2]: in the large N c limit there are three phases which are rigorously distinguished using Φ and the baryon number density N B . The quarkyonic phase is characterized by Φ = 0 indicating the system confined and non-vanishing N B above µ B = M B with a baryon mass M B . The phase structure in large N c is shown in Figure 1.

A possible deformation of the phase boundaries in Figure 1 together with the chiral phase transition can be described using a chiral model coupled to the Polyakov loop [3]. The Nambu-Jona-Lasinio model with Polyakov loops (PNJL model) has been developed to deal with chiral dynamics and "confinement" simultaneously [4]. The model describes that only three-quark states are thermally relevant below the chiral critical temperature, which is reminiscent of confinement. Figure 2 shows the two transition lines for N c = ∞ and for N c = 3 in the two-flavored PNJL model. In the large N c limit assuming that the system is confined, the gap equations for the order parameters qq and Φ become two uncorrelated equations. Consequently, the quark dynamics carries only a µ dependence and the Polyakov loop sector does only a T dependence. Finite N c corrections make the transition lines bending down. The crossover for deconfinement shows a weak dependence on µ which is a remnant of the phase structure in large N c . One finds that for N c = 3 deconfinement and chiral crossover lines are on top of each other in a wide range of µ. A critical point associated with chiral symmetry appears around the junction of those crossovers. The clear separation of the quarkyonic from hadronic phase is lost in a system with finite N c . Nevertheless, an abrupt change in the baryon number density would be interpreted as the quarkyonic transition which separates meson dominant from baryon dominant regions [5]. In fact, a steep increase in the baryon number density and the corresponding maximum in its susceptibility χ B are driven by a phase transition from chirally broken to restored phase in most model-approaches using constituent quarks. One might then consider the chirally symmetric confined phase as the quarkyonic phase.

The constituent quarks are however unphysical in confined phase. It is not obvious to have a realistic description of hadrons from chiral quarks. In particular, chiral symmetry restoration for baryons must be worked out. Two alternatives for chirality assignment are known [6] and it remains an open question which scenario is preferred by nature: (i) in the naive assignment, dynamical chiral symmetry breaking generates a baryon mass which thus vanishes at the restoration. (ii) in the mirror assignment, dynamical chiral symmetry breaking generates a mass difference between parity partners and the chiral symmetry restoration does not necessarily dictate the chiral partners being massless. If the chiral invariant mass is not very small, the baryon number density is supposed to be insensitive to the quarkyonic transition.

Besides, it seems unlikely that the chirally-restored confined phase is realized in QCD on the basis of the anomaly matching: external gauge fields, e.g. photons, interacting with quarks lead to anomalies in the axial current. Since there are no Nambu-Goldstone bosons in chiral restored phase, the anomalous contribution must be generated from the triangle diagram in which the baryons are circulating. In three flavors, however, the baryons forming an octet do not contribute to the pole in the axial current because of the cancellations [7]. The mirror scenario has nothing to do with this problem because the sign of the axial couplings to the positive and negative parity states are relatively opposite. It is indispensable to any rigorous argument for this taking account of the physics around the Fermi surface, which could lead to a possibility of the chrially restored phase with confinement. The anomaly matching conditions at finite temperature and density are in fact altered, see e.g. [8].

Role of the tetra-quark at finite density

There is a possibility of two different phases with broken chiral symmetry distinguished by the baryon number density. An alternative pattern of spontaneous chiral symmetry breaking was suggested in the context of QCD at zero temperature and density [9,10,11]. This pattern keeps the center of chiral group unbroken, i.e.

SU(N

f ) L × SU(N f ) R → SU(N f ) V × (Z N f ) A , ( 1
)
where a discrete symmetry (Z N f ) A is the maximal axial subgroup of SU(N f ) L × SU(N f ) R . The center Z N f symmetry protects a theory from condensate of quark bilinears qq . Spontaneous symmetry breaking is driven by quartic condensates which are invariant under both SU(N f ) V and Z N f transformation. Although meson phenomenology with this breaking pattern seems to explain the reality reasonably [9], this possibility is strictly ruled out in QCD both at zero and finite temperatures but at zero density since a different way of coupling of Nambu-Goldstone bosons to pseudoscalar density violates QCD inequalities for density-density correlators [12]. However, this does not exclude the unorthodox pattern in the presence of dense matter. In a system with the breaking pattern (1) the quartic condensate is the strict order parameter which separates different chirally-broken phases.

It has been shown that the phase where the symmetry is spontaneously broken due to the higher-dimensional operator is realized as a meta-stable state in an O(2) scalar model [13]. Another interesting observation came out from the Skyrme model on crystal: a new intermediate phase where a skyrmion turns into two half skyrmions was numerically found [14]. This phase is characterized by a vanishing quark condensate qq and a non-vanishing pion decay constant. Although the above non-standard pattern of symmetry breaking (1) was not imposed in the Skyrme Lagrangian, the result could suggest a dynamical emergence of new symmetries in dense environment.

The phase diagram and observables

Assuming the symmetry breaking pattern (1) at finite density, it has been shown that an intermediate phase between chiral symmetry broken and its restored phases can be realized using a general Ginzburg-Landau free energy [15,16]. A 2-quark state M in the fundamental and a 4-quark state Σ in the adjoint representation are introduced as §

M ij = 1 √ 2 σδ ij + iφ a τ a ij , Σ ab = 1 √ 3 χδ ab + 1 √ 2 abc ψ c , (2)
where the flavor indices run (i, j) = 1, 2 and (a, b, c) = 1, 2, 3 and Pauli matrices τ a = 2T a with tr[T a T b ] = δ ab /2. σ and χ represent scalar fields and φ and ψ pseudoscalar fields, and ijk is the total anti-symmetric tensor with 123 = 1. The pion decay constant is read from the Noether current as

F π = σ 2 0 + 8 3 χ 2 0 , (3)
with χ 0 and σ 0 being the expectation values of χ and σ, determined from the gap equations. One can deduce a potential in the mean field approximation, which with an explicit breaking term is obtained as

V (σ, χ) = Aσ 2 + Bχ 2 + σ 4 + χ 4 -hσ + Cσ 2 χ + Dχ 3 + F σ 2 χ 2 . ( 4
)
A phase diagram of this model is shown in Fig. 3. There are three distinct phases § We restrict ourselves to a two-flavor case. A similar potential was considered for a system with 2-and 4-quark states under the symmetry breaking pattern without unbroken center symmetry in [17] where their 4-quark states are chiral singlet and the potential does not include quartic terms fields. characterized by two order parameters: Phase I represents the system where both chiral symmetry and its center are spontaneously broken due to non-vanishing expectation values χ 0 and σ 0 . The center symmetry is restored when σ 0 becomes zero. However, chiral symmetry remains broken as long as χ 0 is non-vanishing, indicated by phase II where the pure 4-quark state is the massless Nambu-Goldstone boson. The chiral symmetry restoration takes place under χ 0 → 0 which corresponds to phase III. The phases II and III are separated by a second-order line, while the broken phase I from II or from III is by both first-and second-order lines. Accordingly, there exist two tricritical points (TCPs) and one triple point. One of these TCP, TCP 2 in Fig. 3, is associated with the center Z 2 symmetry restoration rather than the chiral transition. The other coefficients D and F change the topology of the phase diagram and a TCP 1 turns to be a critical point depending on its sign even for h = 0. With an explicit breaking of chiral symmetry one would draw a phase diagram mapped onto (T, µ) plane as in Fig. 4. The intermediate phase remains characterized by a small condensation |σ 0 | |χ 0 |. One would expect a new critical point associated with the restoration of the center symmetry, CP 2 , rather than that of the chiral symmetry if dynamics prefers a negative coefficient of the cubit term in χ. Multiple critical points in principle can be observed as singularities of the quark number susceptibility.

It has been suggested that a similar critical point in lower temperature could appear in the QCD phase diagram based on the two-flavored Nambu-Jona-Lasinio model with vector interaction [18] and a Ginzburg-Landau potential with the effect of axial anomaly in three flavors [19]. There the interplay between the chiral (2-quark) condensate and BCS pairings plays an important role. In our framework without diquarks, the critical point is driven by the interplay between the 2-quark and 4-quark condensates, where anomalies have nothing to do with its appearance. Besides, the universality class which the critical point in our model belongs to is expected to be different from the anomalyinduced one since spontaneous breaking of U (1) B is not imposed in (1). Appearance of the above intermediate phase seems to have a similarity to the notion of Quarkyonic Phase. The transition from hadronic to quarkyonic world can be characterized by a rapid change in the net baryon number density. In our model this feature is driven by the restoration of center symmetry and is due to the fact that the Yukawa coupling of χ to baryons is not allowed by the Z 2 invariance. Fig. 5 shows an expected behavior of the quark (baryon) number susceptibility which exhibits a maximum when across the Z 2 cross over. This can be interpreted as the realization of the quarkyonic transition in N c = 3 world. How far µ z2 from µ chiral is depends crucially on its dynamical-model description. Thus, the present analysis does not exclude the possibility that both transitions take place simultaneously and in such case enhancement of χ B is driven by chiral phase transition. The phase with χ 0 = 0 and σ 0 = 0 does not seem to appear in the large N c limit [11,12,13]. It would be expected that including 1/N c corrections induce a phase with unbroken center symmetry.

Conclusions

We have discussed the phases in dense QCD from chiral approaches along with the anomaly matching which is a field-theoretical requirement. A possibility of a nonstandard breaking pattern leads to a new phase where chiral symmetry is spontaneously broken while its center symmetry is restored. This might appear as an intermediate phase between chirally broken and restored phases in (T, µ) plane. The appearance of this phase also suggests a new critical point in low temperatures. A tendency of the center symmetry restoration is carried by the net baryon number density which shows a rapid increase indicating baryons more activated, and this is reminiscent of the quarkyonic transition. Dynamical breaking of chiral symmetry SU(N f ) L × SU(N f ) R down to SU(N f ) V × (Z N f ) A should be addressed in microscopic calculations using the Swinger-Dyson equations or Nambu-Jona-Lasinio type models with careful treatment of the quartic operators. The properties of baryons near the chiral phase transition are also an issue to be clarified. Depending on the chirality assignment to baryons, equations of state may be altered. The chirality assignment becomes more involved when one introduces axial-vector mesons [20]. In this case the axial-vector meson gives a non-trivial contribution to the axial couplings and eventually it is not clear that the sign of the axial coupling to the negative parity state does distinguish two scenarios.

In this respect, it attracts an interest that the same sign of the axial couplings to the parity partners is predicted in a top-down holographic QCD model [21] and the lattice QCD with dynamical quarks [22].

Figure 1 .Figure 2 .

 12 Figure 1. The phase diagram in large N c proposed in [2].

Figure 3 .

 3 Figure 3. Phase diagram with C = -1, D = F = 0 and h = 0 [15]. The solid and dashed lines indicate first and second order phase boundaries, respectively. One tricritical point, TCP 1 , is located at (A, B) = (0, 1/4) and another, TCP 2 , at (A, B) = (1/4, -1/8). The triple point represented by T is at (A, B) = (1/8, 0).

Figure 4 .

 4 Figure 4. Schematic phase diagram mapped onto (T, µ) plane. The lines do not distinguish the order of phase transitions. The critical point can appear on the boundary that separates the phase I from III at low µ and/or the phase I from II at intermediate µ [15].

Figure 5 .

 5 Figure 5. The behavior of the baryon number susceptibility as a function of chemical potential.

Acknowledgments

The work of C.S. has been supported in part by the DFG cluster of excellence "Origin and Structure of the Universe".


Giorgio Torrieri

Rene Bellwied

Christina Markert

Gary Westfall

Statistical hadronization phenomenology in K/ fluctuations at ultra-relativistic energies

niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

The idea of modelling the abundance of hadrons using statistical mechanics techniques has a long and distinguished history [1,2,3,4]. In a sense, any discussion of the thermodynamic properties of hadronic matter (e.g. the existence of a phase transition) requires that statistical mechanics be applicable to this system ( through not necessarily at the freeze-out stage).

That such a model can describe quantitatively the yield of most particles, including multi-strange ones, has in fact been indicated by fits to average particle abundances at AGS,SPS and RHIC energies [5,6,[START_REF] Letessier | Hadrons quark -gluon plasma[END_REF]8,9,10].

There are, however, several points of contention to this understanding: Some practitioners,starting from [1] interpret the statistical model results in terms of nothing more than phase space dominance: For a process strongly enough interacting with enough particles in the final state, dynamics "factors out" into a normalization constant, and the final state probabilities are dominated by phase space. If this is the case, the applicability of the statistical model has nothing to do with a genuine equilibration of the system.

Others [5,11] believe that the applicability of the statistical model is a sign of a phase transition, as the chemical equilibration of hadrons signals a regime in which multi-particle processes and high-lying resonances dominate.

Still others think that in soft QCD processes particles are "born in equilibrium" [12], and the applicability of the statistical model to even smaller systems (itself contested [13]) is a fundamental characteristic of QCD.

How can these scenarios be differentiated? One observable "at the heart" of statistical mechanics is event-by-event fluctuations [14,15]. It is a fundamental principle of statistics that variances around averages scale a certain way w.r.t. averages. In our context "Averages" are particle multiplicities per event and fluctuations are event-by-event fluctuations. For macroscopic systems, this principle ensures that fluctuations become negligible and the expectation that the state of the system is the maximum entropy one is nearly certain to be realized.

O (100 -1000) particles is not enough for this to be the case, but, if statistical mechanics applies, one should still see that yields, fluctuations and higher cumulants scale in a way calculable from the partition function. There are,however, some experimental issues specific to heavy ion collisions that need to be explored before this can be transformed into a quantitative test.

"standard" fluctuation calculations assume that the volume ‡ of the system is fixed. Obviously, in heavy ion collisions, the system is not in a box, so this requirement ‡ The applicability and definition of the term "volume" is itself problematic in a dynamical quantum system. Statistical particle production assumes a volume, but leaves unexplained how this volume is defined, eg if acceptance cuts can be considered as "cuts in volume". If particles are created with a boost-invariant flow,vz = z/t, and the thermal parameters are approximately independent of rapidity, than cuts in rapidity are equivalent to cuts in volume. No other kinematic cuts, eg cuts in p T or φ, can be reliably treated in this way, although the effect of these cuts on fluctuations can be partially cancelled out by mixed event subtraction, as discussed later in this work has no reason to hold. The most straight-forward way around this is to construct an ensemble where volume can fluctuate while its canonical conjugate, the pressure, stays fixed [16]. This, however, can also present phenomenological difficulties, since the origin of volume fluctuations in heavy ion collisions is not necessarily statistical, but also geometric (initial state fluctuations) and dynamical (non-zero Knudsen number [17]). Statistical mechanics, therefore, must include these fluctuations in the constraints, since it has no ability to describe them.

Due to our incomplete understanding of how the initial state and dynamics contribute to fluctuations at freeze-out, the best approach is to choose an observable which is insensitive to any volume fluctuations. In the thermodynamic limit, where volume becomes a proportionality constant at the level of the partition function, a tempting observable is the scaled variance of the ratio of two particle multiplicities measured event by event. That this is in fact a good guess can be seen at the level of particle distributions. Assuming we are in the thermodynamic limit, the distribution should separate into a component sensitive to volume V and the other on density,

N 1 /V F 1,2 (N 1,2 ) = g(V )f 1,2 N 1,2 V dV (1)
here f 1,2 depend on thermodynamics and g(V ) on initial geometry and dynamics (note that we left it as a general function, and in general we do not really know what this function is). Now, the probability distribution function of a ratio is

F R = N 1 N 2 = F 1 (N 1 )F 2 (N 2 )δ N 1 N 2 -R (2)
Expanding and substituting α = V N 2 we get

F (R) = g(V ) 2 V dV Independent of R f 1 (αR) f 2 (α) dα (3)
hence, N = g(V ) 2 V dV appears equally in all cumulants. Thus, quantities such as σ N1/N2 are strictly independent of any volume fluctuations §. Note that this is not true, for example, for the Kurtosis, recently proposed as signatures for the critical point [19], since the Kurtosis is not the ratio of two cumulants. The procedure above, however, makes it easy to derive the appropriate volume-independent observable encoding higher cumulants of ratios (µ 4 /σ of the distribution of a ratio, for example, would be volume independent. Here µ 4 is the fourth cumulant and σ the fluctuation).

The residual dependence of σ N1/N2 on the average volume V can be in turn eliminated, in the grand canonical ensemble, by focusing on Ψ = N 1 σ N1/N2 dyn , where N 1 and σ dyn are to be measured within the same acceptance. Note that this independence is specific to the grand-canonical ensemble, so should not apply to scenarios where the "enhancement of strangeness" in A-A collisions is due to the transition between the canonical limit in elementary processes and the grand canonical limit in A-A [20]. In this scenario, the "strangeness correlation volume" (V corr not in general equal to the system volume) should regulate Ψ as in Fig. 1 [21] § [18] section IV-C makes the opposite claim based on an expansion around ∆V / V . The flaw in that procedure is that it can not be determined weather terms such as ∆V / V 2 are to be included in the fluctuation or in higher cumulants. The transport analysis in [18] validates our derivation, since strictly no dependence of ratio fluctuations on multiplicity fluctuations is found Thermodynamic limit small V corr A different problem is the effect of a detectors limited acceptance ( Particle (mis)identification, Limited rapidity and momentum resolution, momentum cuts necessary to eliminate jets etc) on fluctuation observables. These are much more difficult to model than averages, and once again an observable needs to be constructed insensitive to them. Hence, the necessity of mixed event subtraction [22]. Mixed events here, are defined as events where no physical correlation from the original event are left in. This means that any correlation seen is due to the imperfection of the detector (the fact that detector acceptance excludes particles of p T > 1 GeV for both event A and B creates a correlation between A and B). To a good approximation, examined in the next paragraph, the measured fluctuation is σ 2 = σ 2 physics + σ 2 acceptance and the mixed event one is σ 2 mix = σ 2 trivial + σ 2 acceptance . Therefore, concentrating on σ 2 dyn = σ 2 -σ 2 mix should eliminate acceptance effects. Two considerations are needed at this point: The first is that this method is not perfect, since limited acceptance spoils not just fluctuations but correlations, and this will not be corrected with mixed event subtraction. For example, in a narrow acceptance detector, resonance kinematics modifies σ 2 physics (weakens the correlation due to the resonance) rather than introducing an additional σ 2 acceptance (HBT would be another source of such physical correlations, although the smallness of the relevant relative momentum makes this largely irrelevant here). There is no perfect solution: One empirical fix is to vary the acceptance, until its large enough that Ψ reaches an asymptotic limit Fig. 2. This limit is the physical value of Ψ. Such a method is not perfect, but it relies on the fact that, for a good event mixing procedure (defined above) limited acceptance should only destroy correlations, not create them. The highest value of the correlation seen by varying the detector acceptance is therefore the physical one.

The second consideration is that, by the definition above (no physical correlations in mixed events), the current algorithms used by experimental collaborations [23,24] for mixed event definition are not perfect. For example, the total multiplicity of mixed events is determined according to the experimentally measured multiplicity distribution, containing all correlations of real events (eg, the π + π -correlation induced by the ρ). Since the abundance of all species are greater in a higher multiplicity event, the autocorrelations contained in the experimentally measured multiplicity distribution will translate into a correlation in the mixed event ratio of any two particles (see Fig 3 for an illustration). This correlation is physical rather than acceptance-induced, and hence should not be there in a "good" mixed event.

A more sound procedure,therefore, would be to normalize π, K and protons separately and construct the mixed event bottom-up. In other words, for each mixed event π, K and p multiplicities should be determined according to the respective experimentally measured multiplicity distributions (uncorrected for acceptance). Then the event should be constructed (the appropriate number of π, K and proton tracks obtained from different events). As physical autocorrelations within the the same particle species of relative momenta larger than O (100)MeV are negligible, this procedure should yield a much better (according to the above definition) mixed event sample. We do not think the multiplicity auto-correlation in mixed events is large, but it is there, and it could be responsible for the deviation between statistical model and the data shown in in the next paragraphs. It will be interesting to see how implementing the mixed event definition presented here will change the results. We now proceed to discuss specific results and conclusions that can be obtained from a study of fluctuation data. We refer to [5,6,[START_REF] Letessier | Hadrons quark -gluon plasma[END_REF]8,9,10,15] for a review of the statistical model. For this work, it is sufficient to repeat that the particle abundances and fluctuations can be calculated from the first and second derivative's a particle's partition function. In the Grand-Canonical limit, this partition function is strictly proportional to volume. Because of this, the fluctuation (∆N ) 2 and the yield N should both scale linearly with volume, and hence the fluctuation of a ratio

σ N1/N2 = (∆N 1 ) 2 N 1 2 + (∆N 2 ) 2 N 2 2 - ∆N 1 ∆N 2 N 1 N 2 + O ∆N N 2 (4)
should scale inversely to the volume V (and be strictly independent of higher cumulants of V , by the earlier derivation). Hence, provided the chemical parameters do not change across systems, Ψ N1 N1/N2 = N 1 σ N1/N2 dyn should be strictly independent of centrality and system size. This requirement is not satisfied by the SPS scan [24] discussed in the last part of the paper, since there µ/T does vary considerably . It is however satisfied by the RHIC upper energies. Thus, the Grand-Canonical statistical model predicts a flat dependence with system size at these energies. The canonical model, on the other hand, would predict a "kink" within the same centrality as when the strangeness correlation volume approaches the thermodynamic limit (Fig. 1).

The results can be seen on Fig. 4 left panel. None of the models come out perfectly, but the Grand-Canonical model is qualitatively much more similar to the data than the canonical one, as no kink is visible there. The discrepancy in Ψ between 200 GeV and 62 GeV, the slight upward trend in centrality, and the lack of scaling between A-A and Cu-Cu should however be closely watched, as the slight increase of fluctuations with multiplicity can not be accounted for by any statistical model (Since we are dealing with ratio fluctuations, a volume fluctuation tuned to KNO scaling, as in [25], will not help here).

Before passing judgement on this topic, however, it will be interesting to see how much of this trend is due to the previously described multiplicity correlation retained in mixed events. Correcting for this effect using an improved mixing procedure would go in the right direction (since physical correlations due to volume would not be cancelled out). An analysis is ongoing to determine their extent. The scaling of fluctuations A scan in centrality and system size within the same energy should however produce the same approximately horizontal bands in Ψ as are seen at RHIC: Each value of √ s, and hence µ/T , should produce a band when scanned in centrality and system size between RHIC and the LHC, where µ/T is virtually the same, should provide further light on this.

Quantitatively, as reported earlier, Ψ is modelled much better with the inclusion of the light-quark non-equilibrium parameter γ q , due to Bose-Einstein enhancement of fluctuations [14]. It remains to be seen weather the measurement of fluctuations of more particles (Fig. 5 left panel) will corroborate this conclusion.

We now turn to a potentially important model-independent use of fluctuations, the constraint on resonance reinteraction between chemical freeze-out and thermal freeze-out [26]. The former can be estimated, to a good approximation, by comparing the fluctuations of same-charged particles (uncorrelated) and opposite charged particles (correlated by K * )

K * 0 K - 3 4 Ψ π - K -/π --Ψ π - K + /π - (5)
Note that both Ψ π - K -/π -and Ψ π - K + /π -are equally affected by the volume correlations present in mixed events, so these should cancel out when the difference is taken. Comparing the fluctuation estimate of K * 0 K -to a direct measurement should yield the amount of K 0 * destroyed by rescattering or regenerated through pseudo-elastic interactions.

The results are shown in Fig. 4 (right panel, resonance values are taken from [27]). Rescattering and regeneration have,within error bar little or no effect on the final abundance of K * s. Either chemical and thermal freeze-out proceed very close to each other, so the amount of reinteraction is negligible (as was concluded in [28] based on [29,30]), or rescattering and regeneration of detectable resonances cancel each other out to the degree of approximation allowed by the error bars (10 -15%). This margin appears somewhat below the estimates from transport models, which are ∼ 40% [31,32]. This discrepancy is not too surprising, since, within the framework of a collectively expanding system, a balance between rescattering and regeneration is unnatural: If the number of reinteractions is small, rescattering generally dominates as was assumed in [29,30]. If the number of rescattering is large, detailed balance would mean K * /K would reequilibrate from the higher chemical freeze-out to the lower thermal freeze-out temperature.

It would therefore be very interesting to investigate weather a transport model such as uRQMD [START_REF] Haussler | Fluctuations in ultra-relativistic heavy ion collisions from Microscopic descriptions[END_REF] or HSD [18] could be tuned to simultaneously reproduce the fluctuation inference and direct measurement of K * 0 K -. Unfortunately, the baroqueness of the resonance decay tree severely limits the feasibilness of such graphic methods, as the right panel of Fig 5 shows. K, π is a special pair of particles in that there is only one type of resonances that decays into both, the K * , and its lightest state is considerably lighter than the heavier ones. No similar definition is possible for Σ * /Λ, ρ/π and φ/K, since the resonance decays equally into all pairs of decay products. For other resonances, cross-contamination destroys any value of fluctuations as a graphic tool of reinteraction time.

This does not mean, however, that such resonances are useless for constraining reinteraction time, since both fluctuations [14] and resonances [29,30] provide a tight constraint on all statistical models. If a statistical model consistently describes fluctuations, but over-predicts resonance abundances, it could be taken as an indication of a long reinteraction times with detailed balance. A casual look at experimental data [27] shows this does not seem to be the case, since most resonances are under-predicted by statistical models. γ q =1.,T=160 MeV

Statistical hadronization phenomenology in

γ q =1.6,T=140 MeV Ψ p,π + Ψ p,Κ - Ψ Λ,π + Ψ Ξ 0 ,π + Ψ K + /K - Ψ π + ,π - Ψ Ν 1, Ν 2 =Ν 2 (σ dyn N 1 /N 2 ) 2 Ψ K -,π -Ψ K + ,π - Ψ Ξ 0 ,π - Ψ p,Κ + Resonances Fluctuations Figure 5.
Equilibrium and non-equilibrium predictions from fluctuations (left panel) and resonances (middle panel) with the statistical parameters taken from [39] (The γq = 1 values are similar to [5]). In the right panel, Full symbols show the N * /N ratio inferred from the correlation in the two models, while empty symbols show, wherever possible, the estimate from σ dyn comparisons

We now turn to the energy scan seen at SPS energies. We first warn the reader not to take any quantitative statements we make here too seriously, since to study the scaling with volume requires the same acceptance region from all particles (see footnote on page 2), as well as for the yield and fluctuation measurements. The analysis in [24] does not meet this requirement, since the acceptance for K and π is considerably different, and the acceptance for yields is different still (note that the earlier analysis in [START_REF] Torrieri | [END_REF] failed to consider this). We await for measurements where this issue is resolved (for experimental reasons they are easier to perform at collider energy scans [35,36,37]) to see how big it is of an issue for the SPS results. [38] has done a very useful job of showing that the scaling of fluctuations with √ s reflects, to a good approximation, that of yields. Of course, in the statistical model an exact scaling of σ K/π dyn with K , π in different energy systems is more complicated than any of the models studied in [38], since it reflects changes in T, µ and possibly γ q,s with √ s. To try to see how the statistical model fits within the analysis of [38] we parametrize the best fit set of parameters in [39,40,41] and [6] (in the second case T, µ B ,in the first T, µ B , γ s,q using the exponent α π

1/2-α K 1/2+α σ K/π dyn = C = π 1/2 K 1/2 σ K/π dyn Ref erence (6)
where C is a constant adjusted from data (Note that all volume dependence cancels out in the definition of α, as expected in the statistical model). α is simply extracted out of the SHARE estimates [15] α

= ln C -ln σ K/π dyn -(ln π + ln K) /2 SHARE ln K -ln π | SHARE (7)
We have performed this procedure with SPS data, taking the parameters from the equilibrium and non-equilibrium values of the analysis in [39], and using the K and π expectations from [38] to fix the normalization. The result is seen in Fig. 6 left panel, with full symbols using the lowest SPS energy as reference and empty symbols using the maximum SPS energy. The conclusions to be drawn from this exercise are two-fold: The first is that none of the scalings examined in [38] is really compatible with the statistical model. The rapid change in µ B , γ q,s really makes the statistical scaling across √ s non-trivial. The second is that the scaling is still close enough to Poissonian that the dependence of the fit parameters on √ s washes away if just K, π are considered. Note that, however, as Fig 6 shows, these scaled values will be somewhat different from the SHARE prediction.

Note, however, that in this work we did not consider abundances of particles other than K,π. Since, as shown in [14], a yield and a fluctuation are enough to remove any ambiguity between T and γ q . Had the acceptance window considered here been the same for K and π measurements, this would yield a tight prediction for the value of protons and hyperons. Taking the normalization constant we fitted with [15] (We want to emphasize this is not a reliable assumption, we did this for illustrative purposes) we computed the particle yields using the equilibrium and non-equilibrium parameters in [39,40,41,6]. The results are shown in the right hand panel of Fig. 6. As can be seen, a quantitative test of both yields and fluctuations differentiates between the equilibrium and non-equilibrium model much more than a test based on yields and fluctuations alone. We look forward, therefore, to perform this analysis on data-sets where the acceptance of the two particles in the ratio is the same, and hence protons and hyperons can also be measured within this acceptance.

In conclusion, we have provided an overview of the current results of fluctuations of particle ratios, and tried to interpret them within the framework of the statistical model. None of the models considered here does a perfect job of describing the data, through models based on Grand-Canonical ensembles better capture the scaling, and the introduction of γ q bring the theory closer to experiment. A different mixed event subtraction is however needed to confirm these results. Comparing the effect of resonances inside particle fluctuations with the directly measured resonances shows that the resonance abundance present at chemical freeze-out seems to remain until the final decoupling, in contrast to current estimates from transport models. While the approximate Poissonian scaling found in the data is expected of the statistical model for all thermal values, the strong variation of µ/T with √ s prevents an exact scaling with energy. A centrality scan at lower √ s, as well as a uniform acceptance for both particles in the measured ratio are necessary before the scaling of fluctuations with energy can be unamibiguously interpreted.

Figure 1 .

 1 Figure 1. (left) Fluctuations dependence in canonical scenario [21] (right) Strangeness fluctuations in the canonical scenario[20]

Figure 2 .

 2 Figure 2. Correlation as a function of acceptance

Figure 3 .Figure 4 .

 34 Figure 3. The experimental multiplicity measure used to normalized mixed events (left panel) and an example of the systematic correlation this procedure generates (right panel) in a ratio between two generic N 1 and N 2 particles

  K/π fluctuations at ultra-relativistic energies8

Figure 6 .

 6 Figure 6. Left and center panel: Scaling of K/π fluctuations with yields. The left panel shown the α exponent, defined as in Eq. 6. Middle panel: The fluctuations observed and calculated with such scaling. Right panel: Yields calculated in the equilibrium and non-equilibrium models assuming the normalization used in the respective calculations corresponds to the physical volume

This work was supported in part by the Offices of NP and HEP within the U.S. DOE Office of Science, the U.S. NSF. G.T. acknowledges the financial support received from the Helmholtz International Center for FAIR within the framework of the LOEWE program (Landesoffensive zur Entwicklung Wissenschaftlich-Ökonomischer Exzellenz) launched by the State of Hesse. G.W. acknowledges the financial support given by the Alexander Von Humboldt foundation research award, and the hospitality provided by the JW Goethe Universitat during the time this work was performed. We thank M.Hauer and Tim Schuster for very helpful discussions, and Tim Schuster and Volker Koch for providing the file used to make Fig. 6.


P Belli

R Bernabei
email: rita.bernabei@roma2.infn.it

F Cappella

R Cerulli

F A Danevich

A 'angelo

A Di Marco

A Incicchitti

F Nozzoli

V I Tretyak

Search for 2β decay of cerium isotopes with CeCl 3 scintillator

Keywords: 29.40.Mc, 23.40.-s CeCl 3 scintillator, double beta decay, low background

Double beta processes in

Ce, 138 Ce and 142 Ce have been searched for by exploiting the active source approach with the help of a 6.9 g CeCl 3 crystal as a scintillator for the first time at the Gran Sasso National Laboratory (LNGS) of the INFN (Italy). The total measurement time is 1638 h; even such a small exposure has allowed the achievement of some improved half-life limits on 2β decay processes in these isotopes at the level of 10 16 -10 18 yr. In particular (90% C.L.):

2 ( 136 Ce)≥ 8.8×10 16 yr, T 2νεβ + 1/2 ( 136 Ce) ≥ 2.4×10 16 yr, T 2ν2K 1/2 ( 136 Ce) ≥ 3.2 × 10 16 yr, T 2ν2K 1/2 ( 138 Ce) ≥ 4.4 × 10 16 yr, and T 2ν2β -1/2 ( 142 Ce) ≥ 1.4 × 10 18 yr. Possible future perspectives are also briefly addressed.

INTRODUCTION

The neutrinoless (0ν) double beta (2β) decay of atomic nuclei (A, Z) → (A, Z ± 2) + 2e ∓ is forbidden in the Standard Model (SM) since it violates the lepton number by two units [1]; however, it is predicted in many SM extensions. The double beta decay experiments are considered the best way to determine an absolute scale of the neutrino mass, to probe the nature of the neutrino, to establish the neutrino mass hierarchy, to search for the existence of right-handed admixtures in the weak interaction, and to test some other effects beyond the SM. During the last two decades the experimental sensitivity to the 0ν2β -decay mode reached levels up to 10 23 -10 25 yr [1].

In the SM the two-neutrino (2ν) double beta decay is an allowed rare process; to-date it is the rarest decay observed in direct laboratory experiments. It was detected only for 10 nuclides, and the corresponding half-lives are in the range of 10 18 -10 24 yr [2,3].

So far the experimental investigations in this field have been mainly focused on the 2β -decays, while the results for double positron decay (2β + ), electron capture with positron emission (εβ + ), and capture of two electrons from the atomic shells (2ε) are much more modest (the most sensitive experiments gave limits on the 2ε, εβ + and 2β + processes at level of 10 17 -10 21 yr [2,4,5]). This is due to: i) the lower energy releases in the 2ε, εβ + and 2β + processes with respect to those in the 2β -decay (which imply lower probabilities for the processes and make difficult the background suppression); ii) the natural abundances of 2β + isotopes (typically lower than 1% with only few exceptions). Nevertheless, efforts are in progress in this direction since the studies of the neutrinoless 2ε and εβ + decays could help to investigate a contribution of right-handed admixtures in weak interaction to the neutrinoless 2β -decay [6].

An interesting case is the Cerium; in fact, it offers three double beta decaying isotopes: 136 Ce, 138 Ce and 142 Ce (see Table 1), and the recent development of new scintillating materials containing Ce also allows the exploitation of the efficient "source=detector" approach. In particular, the 136 Ce is a very interesting isotope since the high energy release allows the 2β + decay mode, which is energetically possible only for 6 candidate-nuclei [2]. Moreover, some resonant neutrinoless 2ε captures in 136 Ce to the excited states of the 136 Ba are also energetically allowed (see e.g. [7]).

Up to now, the possible 2β decays of Ce isotopes were searched for only in 4 experiments: three of them exploited the active source approach with CeF 3 [8,9] and GSO(Ce) crystals [10], and one the passive source approach with HPGe detector using the same CeCl 3 as in the present work [11]. The processes were not detected and only limits on half lives were set; for 136 Ce and 138 Ce they were in the range of 10 15 -10 17 yr; for 142 Ce they were in the range of 10 17 -10 19 yr.

In this paper we report the results obtained in the investigation of the 2β processes in Ce isotopes using for the first time the CeCl 3 crystal as a scintillator to realize the "source=detector" approach.

Table 1 summarizes the main information about the possible 2β decays in the CeCl 3 crystal.

EXPERIMENTAL SET-UP AND MEASUREMENTS

CeCl 3 crystal scintillator

In recent years scintillating materials containing Cerium have attracted much attention in the developments of new fast scintillators with high density, large light yield, perfect linearity and good energy resolution for applications in fundamental and applied nuclear and particle physics, and in several other fields. In fact -due to the electric dipole allowed 5d-4f transitions in the atomic shell -the Ce 3+ ions yield fast (10-100 ns) scintillation in the 300-500 nm wavelength range [14,15]. Scintillating crystals, where Ce is not a dopant but a main constituent, were also studied such as CeF 3 , CeBr 3 , and CeCl 3 .

In particular, the detector used in the present measurements is a cylindrical 13 × 13 mm CeCl 3 crystal (6.9 g mass), housed -because of its hygroscopicity -in a combined plastic and copper container ( 16 × 20 mm external size). Previous measurements were carried out on the detector radiopurity by using a HPGe detector deep underground at LNGS [11], and main scintillation features were studied in ref. [16].

The present data taking has been performed by equipping the crystal with a 7.6 × 10 cm Tetrasil-B light guide to further reduce the contribution to the background of the used 3 low background photomultiplier (PMT). The detector has been installed deep underground ( 3600 m w.e.) in the low background DAMA/R&D set-up [5,17] at LNGS. It was surrounded by Cu bricks and sealed in a low radioactive air-tight Cu box continuously flushed with high purity nitrogen gas (stored deeply underground for a long time) to avoid presence of residual environmental Radon. The copper box was surrounded by a passive shield made of 10 cm of high purity Cu, 15 cm of low radioactive lead, 1.5 mm of cadmium and 4/10 cm polyethylene/paraffin to reduce the external background. The whole shield has been closed inside a Plexiglas box, also continuously flushed by high purity nitrogen gas.

The signals from the PMT have been recorded -over a time window of 4000 ns and a sampling time of 1 GSample/s -by a 8 bit DC270 Acqiris transient digitizer (T D).

Calibration

The energy scale and the energy resolution of the CeCl 3 scintillator detector have been measured by using the following γ sources: i) 133 Ba (81 keV and 356 keV); ii) 22 Na (511 keV and 1275 keV); iii) 137 Cs (662 keV); iv) 60 Co (1173 keV and 1333 keV); see Fig. 1. The linearity of the energy calibration is shown in Fig. 2.

The dependence of the energy resolution of the CeCl 3 detector on the energy can be fitted by the function: σ γ /E γ = a + b/ E γ , where E γ is the energy of the γ quanta in keV; in particular, for the present assembling: σ γ /E γ = 0.0121 + 0.237/ E γ .

Pulse shape discrimination capability

The pulse shape discrimination (PSD) capability between α and γ(β) particles has been investigated in ref. [16] with the crystal directly coupled to the PMT; the pulses, recorded by a 2.5 GSample/s T D over a time window of 500 ns, were analysed and a good discrimination capability was found for the CeCl 3 crystal.

In the present work the possibility to apply the PSD to the production data collected with the CeCl 3 crystal in the present assembling (light guide and a 1 GSample/s T D) has been investigated. In particular, we use the Optimal Digital Filter Method [START_REF] Gatti | Nuclear Electronics[END_REF]. For each signal f (t), the numerical characteristic of its shape (shape indicator, SI) was defined as SI = f (t k ) × P (t k )/ f (t k ), where the sum is over the time channels k, starting from the origin of signal and averaging up to 70 ns. The f (t k ) is the digitized amplitude (at the time t k ) of a given signal. The weight function P (t) was defined as:

P (t) = {f α (t) -f γ (t)}/{f α (t) + f γ (t)},
where f α (t) and f γ (t) are the reference pulse shapes for α particles and γ quanta, respectively, obtained by summing up shapes of a few thousand γ or α events. To build the γ reference pulse we considered the data collected with the 22 Na and 60 Co sources, while for the α we considered the data collected in the production runs in the 1.5 -2.3 MeV energy range 2 , where the α contribution to the experimental energy distribution is dominant (see later). The distributions of the shape indicator for the two classes of events (energies above 1 MeV in γ scale) are depicted in Fig. 3; they are well described by Gaussian functions. Although there is still a statistical separation between the two populations in the present assembling of the CeCl 3 crystal, it is poorer than in ref. [16]; this can be ascribed to the presence of the light guide, which reduces the far UV component of the scintillation light, and to the smaller sampling frequency. Thus, cautiously in the following analyses we will not profit of it.

Internal contaminations of the CeCl 3

In ref. [16] the residual contaminations in the CeCl 3 crystal were investigated and discussed in details; moreover, the activity of the CeCl 3 detector (including housing) was also previously investigated in ref. [11] on the basis of HPGe γ spectrometry. The present larger exposure and the used approach improve such an investigation in some aspects. The energy distribution, measured deep underground in the present low background set-up during the 1638 h of data taking, is shown in Fig. 4. [16], where also the main components were shown. The background fit is well in agreement with the experimental data with exception just of the lowest energy bins; for this reason in the determination of the results on the double electron capture decay channels of 136 Ce and 138 Ce isotopes this fit has not been applied.

The measured background has a similar behaviour as in [16] and is dominated by the internal residual contaminants; in particular, the events below 1400 keV are mainly due to the radioactive decays of 138 La and to the β decays of radioactive isotopes in the 235 U chain, to which the α particles observed above 1400 keV belong. This has also allowed the determination of the relative light yield for α particles as compared with that for γ quantathe so called α/β ratio3 : α/β = 0.070 + 0.050

× E α -0.0026 × E 2 α
, where E α is the energy of the α particles in MeV (E α between 5.5 and 7.5 MeV).

In particular, the energy distribution of Fig. 4 has been analysed in order to more precisely determine the 235 U and 138 La content in the crystal, following the same procedures as in ref. [16] where the energy spectrum was compared with Monte Carlo expectations; the values obtained here are given in Table 2. The contents of 238 U and of 232 Th have been determined by analysing time correlated events in the T D time window (4000 ns) as described in the following.

During the present live time 14 double pulses events have been identified; two examples of them have already been shown in Fig. 7 of ref. [16]. These double pulses events cannot be ascribed to random coincidence; in fact, considering the mean rate measured above the energy threshold of 50 keV and the effective T D time window ( 3650 ns), the expected number of random coincidences during the present data taking is: 8.3 × 10 -5 .

The measured events can mainly be ascribed to α-α time correlated events produced by α decay of 219 Rn into 215 Po followed by 215 Po α decay (T 1/2 ∼1.781 ms) into 211 Pb (isotopes of the 235 U chain). In particular, the first pulse in each event can be due to the 219 Rn α decay (79.4% α of 6.819 MeV, 12.9% α of 6.553 MeV and 7.5% α of 6.425 MeV), while the second one can be due to the 215 Po α decay ( 100% α of 7.386 MeV). Moreover, the number of the measured time correlated events ( 14) is in agreement with the expected number of α-α events in the effective T D time window: 16 α-α events, which can be derived from the 235 U measured contamination in the detector (see Table 2). Thus, upper limits on Bi-Po events can be set.

A Bi-Po event from 232 Th chain is given by the β decay of 212 Bi into 212 Po followed by the 212 Po α decay (T 1/2 =299(2) ns; 100% α of 8.785 MeV), while a Bi-Po event from 238 U chain is given by the β decay of 214 Bi into 214 Po followed by the 214 Po α decay (T 1/2 =164.3(2.0) µs; 100% α of 7.687 MeV). The detection efficiency in the effective T D time window is 100% for Bi-Po events of the 232 Th chain and 1.5% for those of the 238 U chain.

Considering the expected α-α events from the 235 U residual contamination as background with respect to the Bi-Po events, the Bayesian approach [START_REF] Barnett | [END_REF] allows the estimation of the upper limit on the number of Bi-Po events: 6.61 (90% C.L.) from both 238 U and 232 Th chains. Thus, the activities: a212 Bi-212 P o ≤ 0.16 mBq/kg and a214 Bi-214 P o ≤ 11 mBq/kg (90% C.L.) are derived; then, considering the chains in equilibrium, the upper limits on the 232 Th and on the 238 U contaminations are achieved (see Table 2). The obtained limits and values are compatible and more precise than those of ref. [11,16] with the only exception of the 238 U residual contamination, which was significantly higher in the HPGe measurements [11]; this can be ascribed -as already discussed also in ref. [16] -to a few mBq 238 U residual contamination in the housing materials (copper, optical window, etc.) which are of commercial quality.

The investigation on the 2β decay modes

As discussed in the following, there are no clear peculiarities in the energy distribution measured by the CeCl 3 detector, which could be interpreted as double beta decay of Cerium isotopes. Therefore, only lower half-life limits can be set according to the formula:

lim T 1/2 = N • η • t • ln 2/ lim S,
where N is the number of potentially 2β unstable nuclei, η is the detection efficiency, t is the measuring time, and lim S is the number of events of the effect searched for which can be excluded at a given confidence level. The values of η in the considered energy interval, of S/η (see later) and of lim S for each considered decay mode are summarized in Table 3.

The response functions of the CeCl 3 detector for the 2β processes have been simulated with the help of the EGSnrc code [START_REF] Kawrakow | The EGS Code System[END_REF]; the initial kinematics of the particles emitted in the decays has been generated with the DECAY0 event generator [START_REF] Ponkratenko | [END_REF]. Similar procedure has been used to evaluate the background contribution from the measured residual contaminations. Preliminarily the experimental sensitivity to each considered double beta decay mode is estimated by using the so-called 1σ approach where the excluded number of the events due to the effect searched for is estimated simply as the square root of the number of background counts in the chosen energy window; if the expected signal is a peak the considered energy interval is ± 1 σ around the expected signal peak. Notwithstanding its simplicity, this method gives the right scale of the experimental sensitivity. Then, a suitable refined determination of the experimental limit is performed.

Double β processes in 136 Ce

As already mentioned above, the 136 Ce can decay through 2β + , εβ + and 2ε. The used CeCl 3 crystal contains N 136 = 3.12 × 10 19 136 Ce nuclei. The physics for the 136 Ce is rather rich. Two positrons could be emitted in its 2β + decay with energy up to 375 keV; thus, only the ground state of 136 Ba can be populated. The annihilation of these positrons will give rise to four γ's of 511 keV. In the εβ + decay, not only the ground state but also the first excited level of 136 Ba at 818.5 keV can be populated. Finally, in the 2ε capture, many excited 0 + and 2 + levels of 136 Ba can be populated with the subsequent emission of different γ quanta in the de-excitation process.

The experimental sensitivity to the 2ν2β + decay mode has preliminarily been determined by the 1σ approach. In this case one gets: S/η ≤ 422 events in the energy interval 596 -840 keV, which is the energy interval of interest for the decay mode (see Fig. 5); this gives a sensitivity:

T 2ν2β + 1/2
≥ 9.6 × 10 15 yr (68% C.L.). Then, a refined analysis is made: the energy distribution is fitted in the energy range 596-840 keV by the background model, which takes into account the measured residual contaminations, and the expected signal behaviour as evaluated by EGSnrc code. A χ 2 /n.d.f. = 60.0/60 = 1.00 is obtained (see Fig. 5). This approach gives a total number of events, which could be ascribed to the 2ν2β + decay: S/η = -(197±393) counts. This corresponds (in accordance with the Feldman-Cousins procedure [22]) to lim S/η = 465(221) counts at 90%(68%) C.L., giving for the considered process the half-life limit: In the case of the 0ν2β + decay mode the maximum sensitivity has been obtained in the energy region around 375 keV. The 1σ approach in the energy interval 366-384 keV gives S/η ≤ 93 events which corresponds to: T 0ν2β + 1/2 ≥ 4.4 × 10 16 yr (68% C.L.). Then, the experimental data in the 320-436 keV energy interval has been fitted following the same procedure described above, obtaining a χ 2 /n.d.f. = 38.1/28 = 1.36 (see Fig. 5). The total number of events which could be ascribed to the 0ν2β + decay is: S/η = -(99 ± 81) counts which corresponds to lim S/η = 55(16) counts at 90%(68%) C.L. Thus, the following half-life limit is obtained:

T 2ν2β +
T 0ν2β + 1/2
( 136 Ce) ≥ 0.74(2.53) × 10 17 yr 90%(68%) C.L.

In the εβ + decay, various particles are emitted: positron, X rays and Auger electrons from de-excitations in atomic shells, and γ quanta and/or conversion electrons from de-excitation of a daughter nucleus and from positron annihilation.

The maximum sensitivity to study the 2νεβ + decay is obtained in the energy interval 200 -1000 keV (see Fig. 6). The 1σ approach in this energy interval gives: S/η ≤ 164 events, and T 2νεβ + 1/2 ≥ 2.5 × 10 16 yr (68% C.L.). Then, the energy spectrum has been fitted in the same energy range according to the aforementioned procedure, obtaining a χ 2 /n.d.f. = 257/199 = 1.29 (see Fig. 6). The total number of events which could be ascribed to the 2νεβ + decay mode is: S/η = -(102 ± 160) counts which corresponds to lim S/η = 171(75) counts at 90%(68%) C.L. and, thus, to: For the 0νεβ + decay, the 1σ approach was applied in the energy interval 1371 -1423 keV (± 1 σ around the expected signal peak at 1397 keV), however in this case the tail of the nearby composite background structure affect this estimate. Anyhow, in this way we obtain S/η ≤ 26 events, which corresponds to T 0νεβ + 1/2 ≥ 1.6 × 10 17 yr (68% C.L.). Then, the energy spectrum was fitted in energy range 1328 -1504 keV following the aforementioned procedure, obtaining a χ 2 /n.d.f. = 17.1/21 = 0.81 (see Fig. 6). The total number of events which could be ascribed to the 0νεβ + decay mode is S/η = -(53 ± 55) counts; this corresponds to lim S/η = 46 (16) counts at 90%(68%) C.L., and to:

T 0νεβ + 1/2 ( 136 Ce) ≥ 0.88(2.53) × 10 17 yr 90%(68%) C.L.
As regards the case of 2ε decay, we have studied just the case of the double electron capture from K shell (binding energy: 37.44 keV). In particular, the 2ν2K gives a structure around 75 keV (sum of the binding energies of the two electrons from K shell). The 1σ approach in the energy range 72 -78 keV (± 1 σ around the expected signal peak) gives S/η ≤ 39 events which corresponds to: T 2ν2K 1/2 ≥ 1.0 × 10 17 yr (68% C.L.). Then, the energy spectrum was fitted in the energy range 56 -92 keV by the background model (here -due to the small energy window and the quite smooth energy spectrum -it is assumed as a straight line) and the signal simulated with EGSnrc code (see Fig. 7); a χ 2 /n.d.f. = 7.4/6 = 1.2 is obtained. The total number of events which could be ascribed to the 2ν2K decay mode is: S/η = (53 ± 44) counts which corresponds to lim S/η = 125(97) counts at 90%(68%) C.L., and to:

T 2ν2K 1/2 ( 136 Ce) ≥ 3.2(4.
2) × 10 16 yr 90%(68%) C.L. In the 0ν2ε process, in addition to the particles described above some other particle(s) should be emitted to take away the energy released instead by the two neutrinos in the 2ν2ε process. Usually a (bremsstrahlung) gamma quantum is assumed, and in the following we also suppose deexcitation by one γ quantum. Its energy will be equal to

E γ = Q -E b1 -E b2
, where E b1 and E b2 are the binding energies of the first and of the second captured electron on the corresponding atomic shells. For the 136 Ce this bremsstrahlung gamma quantum will have energy at least of 2344 keV. In the present experimental set-up the low mass of the used CeCl 3 crystal gives a very low peak efficiency for the detection of this gamma quantum of the order of 0.01%, which makes its detection very difficult. However the 0ν2K can also be identified by the peak at 75 keV, taking into account the different detection efficiency with the respect to the 2ν2K process. In particular, rescaling by efficiencies the result of the previous fit we obtain for the 0ν2K decay mode: S/η = (58 ± 48) counts; this corresponds to lim S/η = 137(106) counts at 90%(68%) C.L. and to:

T 0ν2K
1/2 ( 136 Ce) ≥ 3.0(3.8) × 10 16 yr 90%(68%) C.L.

Double electron capture in 138 Ce

As already mentioned in Introduction, the 138 Ce isotope can decay only through 2ε with Q = 693 (10) keV. Its natural abundance is δ = 0.251%; thus, the used CeCl 3 crystal contains N 138 = 4.23 × 10 19 nuclei.

We have studied the 2ε decay of the 138 Ce analysing the case of the K shell electron capture. In fact, the process of relaxation in the atomic shell of the 138 Ba after the 2ν2K capture in 138 Ce is exactly the same as for the 136 Ce. Thus, we can use our previous calculations for the 136 Ce just correcting for the different number of 138 Ce nuclei. It gives the following limit:

T 2ν2K
1/2 ( 138 Ce) ≥ 4.4(5.7) × 10 16 yr 90%(68%) C.L.

The 0ν2K capture in 138 Ce is also similar to the 0ν2K process in 136 Ce. However, the energy of the emitted additional γ quantum (618 keV) is lower for the 138 Ce than for the 136 Ce (2344 keV). This leads to slightly lower efficiency for the detection of the 75 keV peak. The final values for the T 1/2 limit is the following: In the 2β -decay the particles emitted are electrons and neutrinos (in the mode 2ν); thus, to detect this process with a good efficiency it is needed that the decaying isotope is inside the detector itself as in the present case.

T 0ν2K 1/
The signal expected for the 2ν2β -decay mode is continuous with a maximum around 400 keV (see Fig. 8, left). The maximum sensitivity to study this decay mode is achieved in the energy interval 200 -1000 keV (see Fig. 8, left). The 1σ approach in this energy interval gives: S/η ≤ 165 events, and T 2ν2β - 1/2 ≥ 1.5 × 10 18 yr (68% C.L.). Then, the energy spectrum has been fitted in the same energy range by the background model and by the 2β -signal simulated with EGSnrc code, obtaining a χ 2 /n.d.f. = 257/199 = 1.28 (see Fig. 8, left). The total number of events which could be ascribed to the 2ν2β -decay mode is: S/η = -(92 ± 158) counts which corresponds to lim S/η = 176(80) counts at 90%(68%) C.L. and, thus, to:

T 2ν2β - 1/2 ( 142 Ce) ≥ 1.4(3.0) × 10 18 yr 90%(68%) C.L.
For the 0ν2β -decay, the 1σ approach was applied in the energy interval 1391 -1443 keV (± 1 σ around the expected signal peak at 1417 keV), however in this case the tail of the nearby composite background structure affects this estimate. Anyhow, in this way we obtain S/η ≤ 19 events, which corresponds to T 0ν2β - 1/2 ≥ 1.3×10 19 yr (68% C.L.). Then, the energy spectrum was fitted in the energy range 1320 -1496 keV by the background model and the signal simulated with EGSnrc code, obtaining a χ 2 /n.d.f. = 15.3/21 = 0.73 (see Fig. 8, right). The total number of events which could be ascribed to the 0ν2β -decay mode is S/η = -(26 ± 36) counts; this corresponds to lim S/η = 36(15) counts at 90%(68%) C.L., and to:

T 0ν2β - 1/2
( 142 Ce) ≥ 0.67(1.62) × 10 19 yr 90%(68%) C.L.

Conclusions

A search for double beta processes in Cerium isotopes has been realized by using for the first time a small CeCl 3 crystal scintillator to realize the "source=detector" approach. In spite of the small exposed mass (6.9 g) and of the limited time of measurements (1638 h) some new improved half-life limits have been obtained as summarized in Table 4, where also the results of the previous most sensitive experiments are given for comparison. This supports the interest in realizing larger mass and longer exposure deep underground with new CeCl 3 crystal scintillators exploiting the active source technique. A further relevant topic is the future preliminary selection of all the materials and of the growing/handling procedures as well as the development of chemical/physical purification techniques in order to improve the radiopurity of the crystal and housing materials. In particular, the 138 La contamination could be related mainly with the growing process; in fact, the crystal used here was realized just after the growth of a LaCl 3 scintillator. Moreover, the use of a larger T D sampling frequency and of different optical read-out should be considered in order to have the possibility to effectively apply the PSD for internal α background reduction when of interest. In addition, the 2β + and εβ + processes can give strong signature when coincidences with other close detectors are investigated. It should be also stressed an advantage of much better energy resolution of CeCl 3 crystal scintillator (FWHM ≈ 5% at 662 keV γ line of 137 Cs) in comparison with f.e. CeF 3 crystal scintillators (≈ 18% at the same energy) [8].

Further improvements in sensitivity can be reached by using enriched 136 Ce, increasing the detection efficiency by using larger CeCl 3 detector, and developing CeCl 3 scintillators with lower level of radioactive contamination. An experiment involving ≈100 kg of crystals enriched in 136 Ce to 20% (5 × 10 25 nuclei of 136 Ce) could reach over 5 years of measurements the half-life sensitivity T 1/2 ≈ 10 25 yr (supposing zero background). Such a sensitivity could contribute to our understanding of the neutrino mass mechanism and right-handed currents in neutrinoless processes [6]. Particular interest is offered by the 2ν2ε decay mode of the 136 Ce, whose half-life is expected to be in the range 10 18 -10 22 yr by theoretical estimates [2]; thus, this process should be observed in such an experiment.

Figure 1 :Figure 2 :

 12 Figure 1: Energy distributions measured by the CeCl 3 detector in the assembling with light guide when using the following γ sources: (top left) 133 Ba; (top right) 22 Na; (bottom left) 137 Cs; (bottom right) 60 Co. See text.

Figure 3 :

 3 Figure3: (Color online) -Shape Indicator distributions of the events collected above 1 MeV: α particles (red, lower SI), γ particles (blue, higher SI).

Figure 4 :

 4 Figure 4: Energy distribution measured deep underground during 1638 h live time inside the multicomponent low background passive shield; the superimposed curve (color online) is the overall background fit estimated as in Figs. 9, 11 of ref.[16], where also the main components were shown. The background fit is well in agreement with the experimental data with exception just of the lowest energy bins; for this reason in the determination of the results on the double electron capture decay channels of 136 Ce and 138 Ce isotopes this fit has not been applied.

1 / 2 (Figure 5 :

 125 Figure5: (Color online) (Left) Energy distribution around 698 keV where it is expected the signal from the 2ν2β + decay mode of 136 Ce following the detection of the two positrons and of one of the four γ's emitted in the annihilation process. The dashed line (red) shows the best-fit curve and the histogram presents the experimental data; there is also shown (in dot-dashed blue) the obtained 90% C.L. limit multiplied by a factor twenty. (Right) Energy distribution around 375 keV where it is expected the signal from the 0ν2β + decay mode of 136 Ce when the annihilation γ's escape the detector. The dashed line (red) shows the best-fit curve and the histogram presents the experimental data; there is also shown (in dot-dashed blue) the obtained 90% C.L. limit multiplied by a factor ten.

Figure 6 : 2 (

 62 Figure 6: (Color online) (Left) Energy distribution in the 200 -1000 keV energy interval where it isexpected the signal from the 2νεβ + decay mode of 136 Ce following the detection of one positron and of the X-rays and Auger electrons. The dashed line (red) shows the best-fit curve and the histogram presents the experimental data; there is also shown (in dot-dashed blue) the obtained 90% C.L. limit multiplied by a factor fifty. (Right) Energy distribution around 1397 keV where it is expected the signal from the 0νεβ + decay mode of 136 Ce when the annihilation photons escape the detector. The dashed line (red) shows the best-fit curve and the histogram presents the experimental data; there is also shown (in dot-dashed blue) the obtained 90% C.L. limit multiplied by a factor three. T 2νεβ + 1/2 ( 136 Ce) ≥ 2.4(5.4) × 10 16 yr 90%(68%) C.L.

Figure 7 :

 7 Figure7: (Color online) Energy distribution around 75 keV where it is expected the signal from the 2ν2K decay mode of 136 Ce following the detection of the X-rays and Auger electrons as well as the signal from the 0ν2K decay mode. The horizontal dashed line (red) shows the best-fit curve and the histogram presents the experimental data; there is also shown in dot-dashed (blue) line the obtained 90% C.L. limit multiplied by 2 on the 2ν2K and on 0ν2K decay modes (both for 136 Ce and 138 Ce).

Figure 8 :

 8 Figure 8: (Color online) (Left) Energy distribution in the 200 -1000 keV energy interval where it is mainly expected the signal from the 2ν2β -decay mode of 142 Ce following the detection of the electrons . The dashed line (red) shows the best-fit curve and the histogram presents the experimental data; there is also shown (in dot-dashed blue) the obtained 90% C.L. limit multiplied by a factor fifty.(Right) Energy distribution around 1417 keV where the signal from the 0ν2β -decay mode of 142 Ce is expected. The dashed line (red) shows the best-fit curve and the histogram presents the experimental data; there is also shown (in dot-dashed blue) the obtained 90% C.L. limit multiplied by a factor three.

Table 1 :

 1 Potentially 2β active nuclides present in CeCl 3 crystals.

		Energy	Isotopic	Decay	Number of nuclei
	Transition	release	abundance	channels	in 100 g of CeCl 3
		(keV) [12]	(%) [13]		crystal
	136 Ce→ 136 Ba	2419(13)	0.185(0.002) 2ε, εβ + , 2β +	4.52 × 10 20
	138 Ce→ 138 Ba	693(10)	0.251(0.002)	2ε	6.13 × 10 20
	142 Ce→ 142 Nd 1416.7(2.1) 11.114(0.051)	2β -	2.72 × 10 22

Table 2 :

 2 Radioactive contamination of the CeCl 3 crystal. The upper limits (90% C.L.) arise from the analysis of the Bi-Po events. The chains have been considered in equilibrium.

	Decay Nuclide Activity (mBq/kg) Contamination
	Chain			from the chain
	232 Th	228 Th	< 0.16	< 40 ppt
	235 U	227 Ac	=284(2)	= 3.52(0.03) ppb
	238 U	226 Ra	< 11	< 0.9 ppb
		138 La	=862(31)	=948(35) ppb

Table 3 :

 3 Values of the interesting quantities for the calculation of the T 1/2 limits on the various decay channels (see later). The overall uncertainty on each η value does not exceed 5%.

	Transition	Decay Decay	Energy	Efficiency	S/η	S lim /η
		channel mode range (keV)	η		90%(68%) C.L.
	136 Ce → 136 Ba	2β +	0ν	366-384	48%	-(99 ± 81)	55(16)
			2ν	596-840	17%	-(197 ± 393)	465(221)
		β +	0ν	1328-1504	54%	-(53 ± 55)	46(16)
			2ν	200-1000	86%	-(102 ± 160)	171(75)
		2K	0ν	56-92	79%	(58 ± 48)	137(106)
			2ν	56-92	87%	(53 ± 44)	125(97)
	138 Ce → 138 Ba	2K	0ν	56-92	73%	(64 ± 44)	151(117)
			2ν	56-92	87%	(53 ± 44)	125(97)
	142 Ce → 142 Nd	2β -	0ν	1320-1496	84%	-(26 ± 36)	36(15)
			2ν	200-1000	85%	-(92 ± 158)	176(80)

  As already mentioned in Introduction, the 142 Ce isotope can decay through 2β -with Q = 1416.7 keV. The natural abundance is δ = 11.114%, thus the used CeCl 3 crystal contains N 142 = 1.87 × 10 21 nuclei.

	2 ( 138 Ce) ≥ 3.6(4.7) × 10 16 yr	90%(68%) C.L.
	3.3 2β decays of 142 Ce

Table 4 :

 4 Half-life limits on 2β processes (ground state to ground state transitions) in Cerium isotopes at 90%(68%) C.L. Theoretical values for 0ν mode are given for the effective neutrino mass of 1 eV. More theoretical results can be found in compilations[2]. • 10 17 > (6.9) • 10 17[9] 2.4 • 10 29 [6] 2.7 • 10 29 [23] 2ν > 0.9(1.8) • 10 16 > 1.8 • 10 16 [10] 5.2 • 10 31 [6] εβ + 0ν > 0.9(2.5) • 10 17 > 3.8 • 10 16 [10] 4.7 • 10 26 [6] 4.0 • 10 26 [23] 2ν > 2.4(5.4) • 10 16 > 2.6 • 10 15 [11] 9.2 • 10 23 [6] 6.0 • 10 23 [24] 2K 0ν > 3.0(3.8) • 10 16 > 6.0 • 10 15 [10] -2ν > 3.2(4.2) • 10 16 > 2.7 • 10 16 [8] 3.2 • 10 18 -6.4 • 10 19 [25] (3.2 -5.1) • 10 21 [26] 9.6 • 10 21 [24] 138 Ce → 138 Ba 2K 0ν > 3.6(4.7) • 10 16 > 1.9 • 10 15 [11] -2ν > 4.4(5.7) • 10 16 > 3.7 • 10 16 [8] ≥ 2.1 • 10 26 [27] 142 Ce → 142 Nd 2β - 0ν > 0.7(1.6) • 10 19 > (1.5) • 10 19 [9] 2.8 • 10 24 [28] 2ν > 1.4(3.0) • 10 18 > 1.6 • 10 17 [10] 2.2 • 10 20 -4.2 • 10 21 [28] 2.3 • 10 23 [29]

	Transition	Decay Decay	Exp. T 1/2 limit (yr)	Theoretical
		channel mode	Present work	Previous results	T 1/2 (yr)
	136 Ce → 136 Ba	2β +	0ν	> 0.7(2.5)

The α energies are given in terms of calibrations with γ sources, that is in MeV electron equivalent. See also later in the figures.

The α/β ratio is defined as the ratio of α peak position in the energy scale measured with γ sources to the energy of α particles (E α ). Because γ quanta interact with the detector by β particles, we use the more convenient term "α/β" ratio.

Acknowledgments

The authors would like to express gratitude to the referees for their careful reading of the manuscript and their useful comments.