Dr Joana Barbosa

Cláudia Bragada

Sofia Costa-De-Oliveira

; Elisabete

Ricardo ; Acacio

Gonçalves Rodrigues

Cidalia Pina-Vaz

Elisabete Ricardo

Acácio Gonçalves Rodrigues

Cidália Pina-Vaz

Joana Margarida

Moutinho Barbosa

Prof Al Hernâni Monteiro

A new method for detection of Pneumocystis jirovecii using flow cytometry

Keywords: Pneumocystis jirovecii 1, Flow cytometry 2, Epifluorescence microscopy 3, Monoclonal antibodies 4, Pneumonia by Pneumocystis 5 Pneumocystis jirovecii, flow cytometry, epifluorescence microscopy, monoclonal antibodies, 16 pneumonia by Pneumocystis 17 18

Pneumocystis jirovecii is an opportunistic pathogen responsible for severe pneumonia in immunocompromised patients. Its diagnosis has been based upon direct microscopy either by classic staining (Gomori Grocott) or by epifluorescence microscopy (IFS), both time-consuming and low on sensitivity. Our aim was to develop a Flow Cytometric (FC) protocol for detection of P. jirovecii on respiratory samples. In our study, 420 respiratory samples were analysed in parallel by IFS and FC, and compared from clinical diagnosis to its resolution upon specific anti-Pneumocystis therapy. The optimum specific antibody concentration for FC analysis was determined as 10µg/ml, without any cross-reactions to bacteria or fungi. All positive cases detected by IFS were positive by FC, however FC considered 8 samples positive which were classified as negative by routine technique. These samples arose from patients with respiratory symptoms who responded favourably to Pneumocystis specific therapy and were subsequently considered true positive. Using clinical diagnosis as a reference method, FC showed 100% of sensitivity and specificity, whereas IFS showed 90.9% of sensivity and 100% of specificity. According to our results, a new diagnostic approach is now available to detect P. jirovecii in respiratory samples.

Dear Doctor Alex van Belkum

Editor-in-Chief of European Journal of Clinical Microbiology & Infectious Diseases Thank you for your comments and for considering our manuscript EJCMID-D-09-00424R1 for publication. We have attended to all comments and made the recommended changes. About the major comment, for the 420 samples, 130 were from HIV positive patients being 35 (27%) positive by both methods. The percentage of positive samples on non HIV population was 18 % (53). So, we included these data on Material and Methods (line 5page 5) and also compared the positive percentage found on HIV and non HIV population in the Results (line 24, 25page 7). We also introduce a new reference (D'Avignon LC, Schofield CM, Hospenthal DR (2008) Pneumocystis Pneumonia. Semin Respir Crit Care Med 29:132-140) to validate the results found from positive samples of HIV patients. The manuscript was corrected by a native English speaker. We hope that the paper could now be considered acceptable for publication.

Regarding specific comments

-Abstract: We have accepted all your suggestions and made all the necessary changes -Introduction: We have replaced and shortened the sentence according your suggestion -Discussion: We have changed "visual" by "microscopic".

Best regards

INTRODUCTION 1

Pneumocystis jirovecii (previously named Pneumocystis carinii) is an atypical fungus. For 80 years it was 2 considered a protozoan, since its diagnostic form is a cyst, but ribosomal RNA and DNA studies demonstrated namely in paediatrics, hematology -oncology, intensive care, transplantation unit and infectious diseases units 20 [START_REF] Nevez | Nosocomial 24 Pneumocystis jiroveci infections[END_REF]. P. jirovecii may be exhaled to the environment from infected patients, as demonstrated from cyst DNA 21 detection in air filters from hospital rooms of patients who developed PCP [START_REF] Nevez | Nosocomial 24 Pneumocystis jiroveci infections[END_REF].

22

The detection of P. jirovecii in an infected patient is based upon the direct visualization of organisms (cysts, 23 sporocysts or trophozoites) in clinical specimens, such as bronchoalveolar lavage (BAL), induced sputum (IS) 24 or lung biopsy, which is the gold standard specimen [6,11]. Although specific fluorescent staining techniques 25 were developed for assessment under fluorescent microscopy, such methods are very time-consuming, too 26 cumbersome and subject to human error, especially when samples yield a low number of P. jirovecii 27 organisms [11]. Molecular studies such as polymerase chain reaction (PCR) presented a higher sensitivity and 28 variable specificity when compared to microscopic detection of P. jirovecii in BAL [12][13]. However, PCR 1 remains unavailable in most clinical microbiology laboratories and is an expensive technique [14-15]. More 2 so, a variable specificity has been described ranging from 62.5 to 100% depending on the type of specimen 3 [14][START_REF] Robberts | Polymerase chain reaction detection of Pneumocystis jiroveci: 7 evaluation of 9 assays[END_REF]. Distinct applications of Flow Cytometry (FC) in Microbiology have been developed by our group in 4 order to increase the diagnostic sensitivity in clinical samples [16][17][18]. In the present study we have developed 5 and optimized a specific FC protocol for the detection of P. jirovecii on respiratory samples.

28

One hundred microliters of centrifuged sample were stained with serial concentrations (0, 5.0, 10.0, 15.0 and 1 20.0 μl) of specific P. jirovecii mouse monoclonal antibody (Axis -Shield Diagnostics Limited, United 2 Kingdom), followed by dark incubation at 37ºC for 15 minutes; Subsequently, 500 µl of sterile H 2 O were 3 added and centrifugation at 3000g for 5 min was performed; the supernantant was discarded and the pellet 4 ressuspended in 100 μl of sterile H 2 O, vortexed for 30s, transferred to a propylene tube and analyzed by FC.

1

Respiratory samples were considered positive for P. jirovecii when two or more green round formations, 2 namely cysts, were observed on epifluorescence microscopy. By IFS, 340 samples were negative and 80 3 samples were positive for P. jirovecii.

4

Since there is not an available commercial type strain of P. jirovecii, we used a mix of four IFS positive and 5 four IFS negative samples (without cysts) as material for optimization of the Flow Cytometric (FC) protocol.

6

As we increased the specific antibody concentrations, an increase of the Mean Intensity of Fluorescence (MIF) 7 was also clear. Twenty µl of specific-antibody resulted in the highest value of MIF at FL1 although a similar 8 staining pattern was obtained with 10.0 or 15.0 µl (Figure 1). As such, all further experiments were carried out 9 with the lowest concentration sufficient to detect the organisms from the autofluorescence background i.e. 10.0 10 µl.

11

To determine the specificity of the staining, microorganisms usually present in respiratory samples were 12 mixed with negative and positive control samples, which were then incubated with the monoclonal antibody.

13

Although an increase in events was detectable in the presence of high concentrations of bacteria, those events 14 were not selected in our settings due to their smaller size; yeasts appeared near the acquisition gate 15 corresponding to P. jirovecii cysts but displayed much lower fluorescence intensity than that of the stained 16 cysts (Figure 2). 17 Upon optimization of the cytometric protocol, clinical samples were analysed. Sample preparation included 18 mucolytic agent, with and without filtration (30 µm pore size). A greater number of non-fluorescent 19 cytometric events occurred on samples without filtration leading to difficulties in its interpretation. A better 20 separation of stained cysts from debris was obtained upon filtration without signal lost (Figure 3). For FC 21 analysis, we considered positive all the respiratory samples whose events appeared on the acquisition gate R 22 corresponding to P. jirovecii cysts stained with specific antibody (typical example shown on Figure 3).

23

Through FC we found 88 positive samples, 35 of which belonged from HIV patients (27%). Eighty positive 24 cases (19%) were detected by both IFS and FC methods. However, 8 clinical samples previously considered 25 negative by IFS, were found to be positive by FC assay (Table 1). All 8 cases were from HIV positive patients 26 with clinical diagnostic criteria of PCP such as fever, cough, shortness of breath, weight loss associated with 27 pulmonary infiltrates on chest x-ray, and an arterial oxygen low level. In addition, the patients had improved 28 upon specific treatment with trimethoprim despite negative IFS. Considering the clinical criteria as the 1 diagnostic standard, such cases should be considered as true positive. According to our study FC analysis 2 displays 100% sensitivity and specificity versus 90.9% sensitivity and 100% specificity of IFS assay.

3

Consequently, IFS presented a predictive positive value of 100% and a predictive negative value of 97.6%.

16

Since the amount of specific monoclonal antibody is lower than that used for IFS, it has the potential to be 17 more economic.

18

Our study aimed to develop and optimize the cytometric analytical protocol regarding clinical samples. An 19 important condition should be fulfilled: the concentration of the probe should be at its minimum to allow a 20 clear separation of the stained population from all other events; while insufficient staining would not allow a 21 clear separation from debris, excessive staining could result in unspecific fluorescence. Our study faced a 22 number of limitations such as the absence of a commercially available type strain as well as the relative low 23 number of positive samples, often containing very few organisms, which made it difficult to assess the 24 sensitivity limit. Cytometric analysis could be difficult to perform in the presence of organic debris and as 25 such, the filtration step prior to specific antibody staining resulted in an improved discrimination of debris 26 from P. jirovecci organisms. In order to confirm the specificity of the monoclonal antibody used in this study, 27 cross-reactions were investigated using both prokaryotic (E. coli, S. aureus) and eukaryotic microorganisms 28 (C. albicans) mixed in a negative sample for P. jirovecii. An increased number of events was clear although 1 without staining with the specific antibody (only fungi showed a very slightly increase in green fluorescence), 2 demonstrating that the test is quite specific.

3

The results from the 420 clinical samples collected from putative PCP patients revealed that, in accordance to

3

  similarities to fungus [1]. In 2001 this microorganism was officially reclassified as a fungus belonging to the 4 class Archiascomycetes [1], and renamed P. jirovecii [2]. In humans, this pathogen is responsible for an 5 opportunistic infection at the lower respiratory tract, but it can lead to severe Pneumocystis pneumonia (PCP) 6 in immunocompromised patients, particularly in those with AIDS [3-4]. Prior to the AIDS epidemic, P.7 jirovecii had been sporadically reported as a cause of death in malnourished infants and responsible for 8 epidemics in institutions like nursing homes and hospitals [5]. Currently, it represents the most frequent AIDS 9 opportunistic pathogen, although PcP incidence has decreased significantly due to the extensive use of highly 10 active antiretroviral therapy (HAART) and prophylactic treatment [4-6]. PCP may also represent a small 11 fraction of respiratory infections in patients with pulmonary diseases such as chronic obstructive pulmonary 12 disease (DPCO) or displaying immunity conditions like malignancy [7-8]. 13 Over the past fifteen years, P. jirovecii carriage in healthy subjects has been widely demonstrated through the 14 presence of anti -P. jirovecii antibodies in their serum, with serologic analysis being used simply for 15 assessment of the infection epidemiology [9]. PCP however is considered to result from de novo infection 16 rather than from reactivation of a latent infection [6]. It is assumed that human transmission of P. jirovecii 17 occurs through airborne route, as suggested by studies performed in rodent models [6]. Nonetheless, 18 interindividual transmission has been suggested due to the occurrence of PCP case clusters in hospitals, 19

  420 samples, including 380 bronchoalveolar lavage (BAL) and bronchial washing (BW) specimens, 3 and 40 bronchial secretion (BS) aspirates were prospectively collected from patients with different 4 immunological conditions. From those 420 patients, 130 were HIV positive. Samples were kindly provided by 5 the Microbiology Laboratories of several Portuguese hospitals namely Hospital São João (HSJ, Porto, 6 Portugal), Instituto Português de Oncologia Francisco Gentil (IPOFG, Porto, Portugal), Hospital Joaquim 7 Urbano (HJU, Porto, Portugal) and Centro Hospitalar de Coimbra (CHC, Coimbra, Portugal). A maximum of 8 30 mL of BAL or BW, and 2 to 4 mL of BS samples were placed in a sterile container, and frozen at -70°C for 9 further processing for immunofluorescence staining (IFS) and flow cytometry (FC) analysis. 10 SAMPLE PREPARATION 11 For IFS, clinical samples were mixed with mucolityc agent Nacetyl -Lcystein (Merck®) and incubated 12 in a water bath at 37ºC for 20 minutes or until complete dissolution, followed by centrifugation at 3000g for 13 15 min. Subsequently, samples were washed twice with sterile water (H 2 O), centrifuged as above and the 14 sediment ressuspended in a small volume of water which was divided into two portions; one portion was 15 placed on a glass slide, air dried and fixed while the other portion was used for FC analysis. 16 IMMUNOFLUORESCENCE STAINING 17 Smears were stained according to the manufacturer's instructions with the Detect IF TM kit Pneumocystis 18 carinii (Axis -Shield Diagnostics Limited, United Kingdom) and analysed under an epifluorescence 19 microscope Leitz Laborlux K (Leica, NY, U.S.A.). Smears were considered positive for P. jirovecii whenever 20 two or more green cysts were visualized, whether isolated or in a group. 21 OPTIMIZATION OF A FLOW CYTOMETRY PROTOCOL 22 For optimization of the FC protocol, a mix of four positive samples and a mix of four negative samples 23 (evaluated by IFS) were prepared. Samples were divided into two equal aliquots and one of them was passed 24 through a filter with 30 μm pore size (Partec CellTrics®). Twenty microliters of specific enzyme from Detect 25 IF TM kit Pneumocystis carinii (Axis -Shield Diagnostics Limited, United Kingdom) were added to 100 µl of 26 the respiratory sample and incubated at 37ºC for 30 minutes; 500 µl of sterile H 2 O were then added and a 27 subsequent centrifugation was performed at 3000 g for 5 minutes.

  of the cyst suspensions were evaluated on a FACSCalibur Cytometer (BD 7 Biosciences, Sydney) standard model equipped with 3 photomultiplayers (PMTs), standard filters (FL1: BP 8 530/30 nm; FL2: BP 585/42 nm; FL3: LP 650 nm), a 15 mW 488 nm Argon Laser and with cell Quest Pro 9 software (version 4.0.2, BD Biosciences, Sydney). Operating conditions included log scales on all detectors 10 (forward scatter (FSC), side scatter (SSC), and fluorescence detectors (FL1)). Acquisition settings were 11 defined using a non-stained sample (autofluorescence) and adjusting the PMTs voltage to the first logarithmic 12 (log) decade. Instrument controls followed the described standard procedures by the manufacturer. 13 ASSESSMENT OF SENSITIVITY AND SPECIFICITY OF FC STAINING PROTOCOL 14 To assess specificity, positive and negative IFS samples for P. jirovecii were mixed with 1x10 6 cells/ml of 15 different microorganisms, namely Escherichia coli type strain ATCC 35218 from the American Type Culture 16 Collection (ATCC), Staphylococcus aureus ATCC 25923 and Candida albicans ATCC 10231. The mix was 17 stained with the specific P. jirovecii antibody under optimized conditions and analysed by FC. 18 EVALUATION OF CLINICAL RESPIRATORY SAMPLES 19 All clinical samples were analysed both by IFS and by FC according to the previously optimized conditions and 20 the results were subsequently compared. 21 DATA COMPARISON 22 Results of IFS were compared to those obtained by FC. Clinical diagnosis was used as the standard. When 23 results were discrepant, clinical symptoms and signs were considered; patients with oxygen level PaO 2 <70 24 mmHg, an X-ray depicting a diffuse bilateral infiltrate and with a favourable outcome upon specific therapy 25 were considered as true positive cases for PCP. 26 27 RESULTS

  diagnosis of Pneumocystis jirovecii infection has significant clinical relevance since this 2 opportunistic organism can cause severe pneumonia in immunocompromised patients [6]. As this organism 3 cannot be cultured in vitro, the standard diagnostic procedure for detection of P. jirovecii relies upon the direct 4 microscopic examination of the clinical specimen. Using commercialized fluorescein or enzyme-labelled 5 monoclonal or polyclonal Pneumocystis antibodies in respiratory samples, a higher detection sensitivity can be 6 achieved [11]. Considering the increasing complexity of diagnostic laboratory techniques, resulting in higher 7 costs and the need of human expertise, the development of more sensitive and automated methods which allow 8 a rapid assessment of clinical specimens is highly recommended. Flow cytometry (FC) is a method based upon 9 the evaluation of cell fluorescence, frequently shown to be specific and yielding great sensitivity. Moreover, it 10 allows the detection of low threshold limits, difficult to obtain by standard methods [16-19]. The first 11 cytometric study regarding the formally P. carinii was developed by Libertin and colleagues [19] in lung 12 homogenates samples from infected mice and humans, however without practical application. Later assays 13 were performed by Lapinsky and colleagues [20] in order to observe the pharmacological action of drugs upon 14 growth and viability of these microorganisms in animal models. The flow cytometer is an expensive 15 equipment present in most immunology laboratories that could probably be easily shared by microbiologists.

4 1 FIGURE 1 -FIGURE 2 -FIGURE 3

 1123 FIGURE 1 -Dot plot (side scatter (SSC) versus fluorescence intensity (FL1)) of a positive sample containing

TABLE 1 -

 1 Results obtained from 420 respiratory samples for P. jirovecii diagnosis by Immunofluorescence 1 (IFS) and Flow Cytometry (FC).

	2
	3
		IFS	FC
	Positive	80	88
	Negative	340	332

here to download high resolution image Figure 2 Click here to download high resolution image Table Click here to download high resolution image Figure 3 Click here to download high resolution image


A Lavagno
email: andrea.lavagno@polito.it

D Pigato

P Quarati

Nonextensive statistical effects in the hadron to quark-gluon phase transition

We investigate the relativistic equation of state of hadronic matter and quark-gluon plasma at finite temperature and baryon density in the framework of the nonextensive statistical mechanics, characterized by power-law quantum distributions. We study the phase transition from hadronic matter to quark-gluon plasma by requiring the Gibbs conditions on the global conservation of baryon number and electric charge fraction. We show that nonextensive statistical effects play a crucial role in the equation of state and in the formation of mixed phase also for small deviations from the standard Boltzmann-Gibbs statistics.

Introduction

The physics of high energy heavy ion collisions is a goldmine of problems in statistical mechanics and thermodynamics due to a large average number of particles involved and possible phase transition phenomena in the hot and dense fireball created during the collisions [START_REF] Hwa | Quark Gluon Plasma[END_REF]. In relativistic heavy ion collisions the baryon density can reach values of a few times the saturation nuclear density and/or high temperatures. Furthermore, the future CBM (Compressed Baryonic Matter) experiment of FAIR (Facility of Antiproton and Ion Research) at GSI Darmstadt, will make possible to create compressed baryonic matter with a high net baryon density [START_REF] Senger | [END_REF]3,4]. In this direction interesting results have been obtained at low SPS energy and are foreseen at a low-energy scan at RHIC [5,6,7,8,9].

Lattice calculations predict a critical phase transition from hadronic matter to quark-gluon plasma (QGP) at temperature T c of about 170 MeV, corresponding to a critical energy density c ≈ 1 GeV/fm 3 [10]. On the other hand, in dense nuclear matter, baryons are forced to stay so close one to another that they would overlap. At large densities, constituent quarks are shared by neighboring baryons and should eventually become mobile over a distance larger than the typical size of one baryon. This means that quarks become deconfined and that at large densities and/or high temperatures they are the real degrees of freedom of strongly interacting matter instead of baryons. The process of deconfinement and the equation of state (EOS) of quark-gluon matter can in principle be described by quantum chromodynamics. However, in energy density range reached in relativistic heavy-ion collisions, non-perturbative effects in the complex theory of QCD are not negligible [10]. The generated QGP in the early stages of the collisions does not at all resemble a quasi-ideal gas of quarks and gluons because strongly dynamical correlations are present, including long-range interactions [START_REF] Hwa | Quark Gluon Plasma[END_REF]11,12,13,14]. In the absence of a converging method to approach QCD at finite density one often turns to (effective) model investigations [15,16,17,18,19]. Various results from QCD inspired models indicate that, increasing the baryon chemical potential in the phase diagram, a region of non-singular but rapid cross-over of thermodynamic observable around a quasi-critical temperature, leads to a critical endpoint (CEP), beyond which the system shows a first order phase transition from confined to deconfined matter. The existence or exclusion of a CEP has not yet been confirmed by QCD lattice simulations. Actually, there are some extrapolation techniques to finite chemical potentials [20], although the precise location of the CEP is still a matter of debate [21]. For example, in Ref. [START_REF] Fodor | AIP Conf. Prof. No[END_REF], the authors estimate the values T CEP = 162 MeV and µ CEP = 360 MeV. Such a CEP can be in principle detected in future high-energy compressed nuclear matter experiments.

Recently, there is an increasing evidence that the nonextensive statistical mechanics, proposed by Tsallis, can be considered as an appropriate basis to deal with physical phenomena where strong dynamical correlations, long-range interactions and microscopic memory effects take place [START_REF] Tsallis | [END_REF][START_REF] Gell-Mann | Nonextensive Entropy: Interdisciplinary Applications[END_REF][START_REF] Tsallis | Introduction to Nonextensive Statistical Mechanics[END_REF][START_REF] Kodama | [END_REF]. A considerable variety of physical applications involve a quantitative agreement between experimental data and theoretical models based on Tsallis thermostatistics. In particular, in the last years there is a growing interest to high energy physics applications of nonextensive statistics [START_REF] Biró | Topical Issue on Statistical Power Law Tails in High-Energy Phenomena[END_REF] and several authors have outlined the possibility that experimental observations in relativistic heavy ion collisions can reflect nonextensive statistical behaviors [START_REF] Bediaga | [END_REF]29,30,31,32,33,34,35,36,37,38,39,40].

The existence of nonextensive statistical effects should strongly affects the finite temperature and density nuclear EOS [41,42,43]. In fact, by varying temperature and density, the EOS reflects in terms of the macroscopic thermodynamical variables the microscopic interactions of the different phases of nuclear matter. The extraction of information about the EOS at different densities and temperatures by means of heavy ion collisions is a very difficult task and can be realized only indirectly by comparing the experimental data with different theoretical models, such as, for example, fluiddynamical models [START_REF] Russkikh | [END_REF]. Related to this aspect, it is relevant to observe that a relativistic kinetic nonextensive theory [45] and a nonextensive version of a hydrodynamic model for multiparticle production processes have been proposed [46]. Very recently, nonextensive statistical effects on the hadronic EOS have been investigated by means of a Walecka type relativistic mean field model [47]. Furthermore, a nonextensive version of Nambu-Jona-Lasinio model [48] and the effects on color superconducting phase for two quark flavors due to a change to Tsallis statistics have been studied [49].

The main goal of this paper is to study how nonextensive statistical effects influence, from a phenomenological point of view, the nuclear EOS and, as a consequence, the relative phase transition at finite temperature and density reachable in high-energy heavy-ion collisions. Focusing our investigation to lower temperatures and higher baryon chemical potentials than the corresponding CEP values, a mixed phase of hadrons, quarks and gluons can be formed following the Gibbs conditions for the phase equilibrium. If, in general, we consider a substance composed of two conserved "charges", like the baryon number and the isospin charge in heavy ion collisions at finite baryon density, the ratio between the two charges is fixed only as long as the system remains in one of the two pure phases. In the mixed phase, the concentration in each of the regions of one phase or the other may be different. Their values are restricted only by the conservation on the total charge numbers. The essential point is that conservation laws in chemical thermodynamics are global, not local. The main result of this formalism is that, different from the so-called Maxwell construction, the pressure in the mixed phase is not constant and therefore the nuclear incompressibility, for example, does not vanish [50].

Furthermore, the scenario we are going to explore corresponds to the situation realized in heavy ion collisions experiments at not too high energy where finite temperature and high compressed baryon density is reached. In this condition, a not large fraction of strangeness can be produced and, therefore, we will limit ourselves to study the deconfinement transition from hadronic matter into up and down quark matter [51,52,53]. We aspect that, in the range of temperature and density considered, the presence of strange particles does not significantly affect the main conclusions regarding the relevance of nonextensive statistical effects on the nuclear EOS.

The paper is organized as follows. In Section 2, we introduce the basic formalism of the nonextensive statistics. In Section 3, we study the nonextensive hadronic EOS for symmetric and asymmetric nuclear matter and we explore the behavior of meson fields in presence of small deviations from the standard statistics. In Section 4, we investigate nonextensive proprieties of the quark-gluon EOS. In Section 5, we study the hadron to quark-gluon phase transition and the consequent formation of a mixed phase, mainly focusing our study in the variation of the first critical transition density for various set of parameters. Finally, we summarize our conclusions in Section 6.

Basic assumptions in nonextensive statistics

Nonextensive statistical mechanics introduced by Tsallis is a generalization of the common Boltzmann-Gibbs statistical mechanics [START_REF] Tsallis | [END_REF][START_REF] Gell-Mann | Nonextensive Entropy: Interdisciplinary Applications[END_REF][START_REF] Tsallis | Introduction to Nonextensive Statistical Mechanics[END_REF]. It is based upon the introduction of the following entropy

S q [f ] = 1 q -1 1 -[f (x)] q dΩ , f (x) dΩ = 1 , (1)
where f (x) stands for a normalized probability distribution, x and dΩ denoting, respectively, a generic point and the volume element in the corresponding phase space.

Here and in the following we set the Boltzmann and the Planck constant equal to unity.

The real parameter q determines the degree of non-additivity exhibited by the entropy form [START_REF] Hwa | Quark Gluon Plasma[END_REF].

The generalized entropy has the usual properties of positivity, equiprobability, concavity and irreversibility, preserves the whole mathematical structure of thermodynamics (Legendre transformations). In the limit q → 1, the entropic form (1) becomes additive and reduces to the standard Boltzmann-Gibbs entropy

S 1 = -f (x) ln f (x) dΩ . (2)
Peculiarity of the Tsallis generalized thermostatistics is that if we have two statistically independent subsystems A e B, described, respectively, by the individual probability density f (A) and f (B) and we call

f (A+B) (x A , x B ) = f (A) (x A ) f (B) (x B
) the joint probability density of a composite system A + B, the nonadditive (nonextensive) character of S q is summarized in the relation [START_REF] Tsallis | Introduction to Nonextensive Statistical Mechanics[END_REF] S

q [f (A+B) ] = S q [f (A) ] + S q [f (B) ]) + (1 -q)S q [f (A) ] S q [f (B) ] . (3)
In the limit q → 1, the third term in right hand side of Eq.( 3) vanishes and the above equation reduces to the well-known additivity (extensivity) relation of the Boltzmann-Gibbs logarithmic entropy. Here, the word nonextensive should be associated with the fact that the total energy of long-range-interacting mechanical systems is nonextensive, in contrast with the case of short-range-interacting systems, whose total energy is extensive in the thermodynamical sense [START_REF] Tsallis | Introduction to Nonextensive Statistical Mechanics[END_REF].

Second crucial assumption on nonextensive statistics is the introduction of the qmean value (or escort mean value) of a physical observable A(x)

A q = A(x) [f (x)] q dΩ [f (x)] q dΩ . (4)
The probability distribution can be obtained maximizing the measure S q under appropriate constraints related to the previous definition of the q-mean value. In this context, it is important to observe that the Tsallis classical distribution can be seen as a superposition of Boltzmann distributions with different temperatures which have a mean value corresponding to the temperature appearing in the Tsallis distribution. The nonextensive q parameter is related to the fluctuation in the temperature and describes the spread around the average value of the Boltzmann temperature [32].

Following the above prescriptions, it is possible to obtain the associate quantum mean occupation number of particles species i in a grand canonical ensemble. For a dilute gas of particles and for small deviations from the standard statistics (q ≈ 1,) it can be written as [54,55]

n i = 1 [1 + (q -1) β(E i -µ i )] 1/(q-1) ± 1 , (5)
where β = 1/T and the sign (+1) is for fermions and (-1) for bosons. Naturally, for q → 1 the above quantum distribution reduces to the standard Fermi-Dirac and Bose-Einstein distribution. Let us observe that nonextensive statistical effects vanishes approaching to zero temperature. This is the reason for which nonextensive effects could be significantly relevant in high energy heavy ion collision and probably, in the protoneutron star. In addition, in high density quark-gluon matter the color magnetic field remains unscreened (in leading order) and long-range color magnetic interaction should be present at any finite temperature, thus QGP appears to be an ideal candidate for finding some nonextensive behavior. Finally, let us observe that when the entropic q parameter is smaller than one, the above distribution have a natural high energy cut-off which implies that the energy tail is depleted; when q is greater than one, the cut-off is absent and the energy tail of the particle distribution (for fermions and bosons) is enhanced. Hence the nonextensive statistics entails a sensible difference of the power-law particle distribution shape in the high energy region with respect to the standard statistics. In this context, it is relevant to observe that in Ref. [48], the authors postulate a modified quantum distribution function for fermions and bosons at the scope of satisfy the particle-hole symmetry, both for q > 1 and q < 1. In the present work we will focus our study for small deviations from the standard statistics and for values of q > 1, because these values were obtained in several phenomenological studies in high energy heavy ion collisions (see, for example, Ref.s [29,37,38,39,40]). We have explicitly verified that, for the values q > 1 considered in this investigation, the prescription introduced in Ref. [48] does not affect the results that we are going to obtain and, therefore, we adopt the original formulation of nonextensive statistics. Furthermore, it is proper to remember that in a relativistic mean field theory, considered in this investigation, baryons are assumed as Dirac quasiparticles moving in classical meson fields, the field operator are replaced by their expectation values and the contributions coming from the Dirac sea are neglected.

Nonextensive hadronic equation of state

In this Section we study the nonextensive hadronic EOS in the framework of a relativistic mean field theory in which nucleons interact through the nuclear force mediated by the exchange of virtual isoscalar-scalar (σ), isoscalar-vector (ω) and isovector-vector (ρ) meson fields [56,57,58]. As quoted in the Introduction, a similar approach has been studied in Ref. [47] for pure neutron matter and for symmetric nuclear matter (thus, without considering the effects of the ρ meson field) focusing principally the attention to a different range of density and temperature considered in this paper. Here, we are going to study the hadronic EOS at the scope of emphasize several features previously not investigated that result to be crucial for our following purposes.

The Lagrangian density describing hadronic matter can be written as

L = L QHD + L qfm , (6)
where [58]

L QHD = ψ[iγ µ ∂ µ -(M -g σ σ) -g ω γ µ ω µ -g ρ γ µ τ • ρ µ ]ψ + 1 2 (∂ µ σ∂ µ σ -m 2 σ σ 2 ) -U (σ) + 1 2 m 2 ω ω µ ω µ + 1 2 m 2 ρ ρ µ • ρ µ - 1 4 F µν F µν - 1 4 G µν G µν , (7)
and M = 939 MeV is the vacuum baryon mass. The field strength tensors for the vector mesons are given by the usual expressions

F µν ≡ ∂ µ ω ν -∂ ν ω µ , G µν ≡ ∂ µ ρ ν -∂ ν ρ µ , and U (σ) is a nonlinear potential of σ meson U (σ) = 1 3 aσ 3 + 1 4 bσ 4 , (8)
usually introduced to achieve a reasonable compression modulus for equilibrium nuclear matter. Following Ref.s [59,60], L qfm in Eq.( 6) is related to a (quasi) free gas of pions with an effective chemical potential (see below for details).

The field equations in a mean field approximation are

(iγ µ ∂ µ -(M -g σ σ) -g ω γ 0 ω -g ρ γ 0 τ 3 ρ)ψ = 0 , (9)
m 2 σ σ + aσ 2 + bσ 3 = g σ < ψψ >= g σ ρ S , (10)
m 2 ω ω = g ω < ψγ 0 ψ >= g ω ρ B , (11)
m 2 ρ ρ = g ρ < ψγ 0 τ 3 ψ >= g ρ ρ I , (12)
where σ = σ , ω = ω 0 and ρ = ρ 0 3 are the nonvanishing expectation values of meson fields, ρ I is the total isospin density, ρ B and ρ S are the baryon density and the baryon scalar density, respectively. They are given by

ρ B = 2 i=n,p d 3 k (2π) 3 [n i (k) -n i (k)] , (13)
ρ S = 2 i=n,p d 3 k (2π) 3 M * i E * i [n q i (k) + n q i (k)] , (14)
where n i (k) and n i (k) are the q-deformed fermion particle and antiparticle distributions given in Eq.( 5); more explicitly, in this context, we have

n i (k) = 1 [1 + (q -1) β(E * i (k) -µ * i )] 1/(q-1) + 1 , (15)
n i (k) = 1 [1 + (q -1) β(E * i (k) + µ * i )] 1/(q-1) + 1 . ( 16
)
The nucleon effective energy is defined as

E i * (k) = k 2 + M i * 2 , where M i * = M i -g σ σ.
The effective chemical potentials µ * i are given in terms of the meson fields as follows

µ * i = µ i -g ω ω -τ 3i g ρ ρ , (17)
where µ i are the thermodynamical chemical potentials µ i = ∂ /∂ρ i . At zero temperature they reduce to the Fermi energies

E F i ≡ k 2 F i + M * i 2
and the nonextensive statistical effects disappear. The meson fields are obtained as a solution of the field equations in mean field approximation and the related meson-nucleon couplings (g σ , g ω and g ρ ) are the free parameters of the model. In the following, they will be fixed to the parameters set marked as GM2 of Ref. [58].

The thermodynamical quantities can be obtained from the thermodynamic potential in the standard way. More explicitly, the baryon pressure P B and the energy density B can be written as

P B = 2 3 i=n,p d 3 k (2π) 3 k 2 E * i (k) [n q i (k) + n q i (k)] - 1 2 m 2 σ σ 2 -U (σ) + 1 2 m 2 ω ω 2 + 1 2 m 2 ρ ρ 2 ,( 18
) B = 2 i=n,p d 3 k (2π) 3 E * i (k)[n q i (k) + n q i (k)] + 1 2 m 2 σ σ 2 + U (σ) + 1 2 m 2 ω ω 2 + 1 2 m 2 ρ ρ 2 . ( 19
)
It is important to observe that Eq.s( 14), ( 18) and ( 19) apply to n q i ≡ (n i ) q rather than n i itself, this is a direct consequence of the basic prescription related to the qmean expectation value in nonextensive statistics [START_REF] Tsallis | Introduction to Nonextensive Statistical Mechanics[END_REF]45] (this recipe was not adopted in Ref. [47]). In addition, since all equations must be solved in a self-consistent way, the presence of nonextensive statistical effects influences the many-body interaction mediated by the meson fields.

Especially in regime of low density and high temperature the contribution of the lightest mesons to the thermodynamical potential (and, consequently, to the other thermodynamical quantities) becomes relevant. As quoted before, following Ref. [59], we have included the contribution of pions considering them as a (quasi) ideal gas of nonextensive bosons with effective chemical potentials expressed in terms of the corresponding effective baryon chemical potentials. More explicitly, for π + mesons we have µ π + = µ C ≡ µ p -µ n , where µ C is the electric charge chemical potential. Thus, the corresponding effective pion chemical potential can be written as

µ * π + ≡ µ * p -µ * n = µ p -µ n -2 g ρ ρ , (20)
where the last equivalence follows from Eq. (17). Therefore, the ρ meson field couples to the total isospin density, which receives a contribution from nucleons and pions. Let us start our numerical investigation by considering the behavior of σ, ω and ρ meson fields at a fixed value Z/A = 0.4, for different values of temperature and nonextensive parameter q. Because meson fields have their source in the baryon and scalar density, which are very sensible to the behavior of the mean occupation number, all meson fields appear to be significantly changed in presence of nonextensive effects.

In Fig. 1, we show the σ meson field as a function of the baryon chemical potential µ B . It is interesting to observe that at lower µ B , in presence of nonextensive effects, the value of the meson field is significantly increased for all values of temperature respect to the standard case, the other way round happens at higher µ B . This important feature is due to the fact that, as already remarked in Section 2, for q > 1 and fixed baryon density (or µ B ), the (normalized) mean occupation function is enhanced at high values of its argument and depressed at low values. Being the argument of the mean occupation function x i = β(E * i -µ * i ), in the integration over momentum (energy), at lower µ B (corresponding to lower values of the effective particle chemical potential µ * i ) the enhanced Tsallis high energy tail weighs much more that at higher µ B where depressed low energy effects prevail and the mean occupation number results to be bigger for the standard Fermi-Dirac statistics. Concerning the antiparticle contribution, the argument of n i is x i = β(E * i + µ * i ) and the Tsallis enhancement at high energy tail is favored also at higher µ B . At the same time, higher temperatures (where antiparticle contribution are more relevant) reduce the value of the argument of n i and n i , favoring the extensive distribution. These effects are much more evident for the scalar density ρ S (self-consistently related to the σ meson field) where appears (n i ) q and particle and antiparticle contributions are summed. The same effect involves also the nucleon effective mass M * = M -g σ σ, which becomes, respect to the standard case, smaller for lower values of µ B and bigger for higher values, with very relevant consequences for the hadronic EOS ‡.

In Fig. 2, we report the ω meson field as a function of the baryon chemical potential for different values of temperature and q. In this case the situation is different from the σ meson, because the ω field have its source in the baryon density ρ B where appears n i and particle and antiparticle contributions are subtracted. At lower temperatures (T = 60 MeV), antiparticle contributions are negligible and we have a behavior similar (although less evident) to the σ field. At higher temperatures (T = 120 MeV), the contributions of antiparticle increase and nonextensive effects vanish at higher µ B .

Finally, in Fig. 3, we report the behavior of the ρ meson field which depends from the isospin density (let us remember that we have fixed Z/A = 0.4). Similar arguments as done for the ω meson applies also in this case. The valuable increasing of its absolute value, also for weakly asymmetric nuclear matter, makes ρ meson very relevant in the hadronic EOS, especially at not too large µ B .

In Fig. 4, the total pressure P and energy density are plotted as a function of µ B for different values of temperature and q. The different behavior from P and reflects essentially the nonlinear combinations of the meson fields and the different functions under integration in Eq.s (18) and (19). Concerning the pressure, we have that becomes stiffer by increasing the q parameter. On the other hand, the behavior of the energy density presents features very similar to the σ field one. At low µ B , nonextensive effects make the energy density greater with respect to the standard case. At medium-high µ B , the standard (q = 1) component of the energy density becomes dominant, this effect is essentially due to the reduction of the σ field for q > 1. The intersection point depends, naturally, on the physical parameters of the system.

Nonextensive QGP equation of state

Concerning the nonextensive quark-gluon EOS, due to its simplicity, we adopt the MIT bag model [61]. In this model, quark matter is described as a gas of free quarks and all non-perturbative effects are simulated by the bag constant B which represents the pressure of the vacuum. Following this line, the pressure, energy density and baryon density for a relativistic Fermi gas of quarks in the framework of nonextensive statistics can be written, respectively, as

P q = γ f 3 f =u,d ∞ 0 d 3 k (2π) 3 k 2 e f [n q f (k) + n q f (k)] -B , (21)
q = γ f f =u,d ∞ 0 d 3 k (2π) 3 e f [n q f (k) + n q f (k)] + B , (22)
ρ q = γ f 3 f =u,d ∞ 0 d 3 k (2π) 3 [n f (k) -n f (k)] , (23)
where the quark degeneracy for each flavor is γ f = 6, e f = (k 2 + m 2 f ) 1/2 , n f (k) and n f (k) are the q-deformed particle and antiparticle quark distributions

n f (k) = 1 [1 + (q -1)(e f (k) -µ f )/T ] 1/(q-1) + 1 , ( 24
)
n f (k) = 1 [1 + (q -1)(e f (k) + µ f )/T ] 1/(q-1) + 1 . ( 25
)
Similar expressions for the pressure and the energy density can be written for gluons treating them as a massless q-deformed Bose gas with zero chemical potential. Explicitly, we can calculate the nonextensive pressure P g and energy density g for gluons as

P g = γ g 3 ∞ 0 d 3 k (2π) 3 k [1 + (q -1) k/T ] q/(q-1) -1 , ( 26
) g = 3 P g , (27)
with the gluon degeneracy factor γ g = 16. In the limit q → 1, one recovers the usual analytical expression:

P g = 8π 2 /45 T 4 .
Let us note that, since one has to employ the fermion (boson) nonextensive distributions, the results are not analytical, even in the massless quark approximation. Hence a numerical evaluations of the integrals in Eq.s ( 21)-( 23) and ( 26) must be performed.

In Fig. 5, we report the total pressure as a function of the baryon chemical potential for massless quarks and gluons, for different values of q and at fixed value of Z/A = 0.4. The bag constant is set equal to B 1/4 =190 MeV. In presence of nonextensive effects, as in the case of hadronic phase, the pressure is significantly increased even for small deviations from standard statistics.

The hadron to quark-gluon phase transition

In this Section we are going to investigate the phase transition from hadronic matter to QGP at finite temperature and baryon chemical potential in the framework of nonextensive statistics.

At this scope, we use the Gibbs formalism applied to systems where more than one conserved charge is present [62]. In fact, because we are going to describe the nuclear EOS, we have to require the global conservation of two "charges": baryon number and electric charge. Each conserved charge has a conjugated chemical potential and the systems is described by two independent chemical potentials: µ B and µ C . The structure of the mixed phase is obtained by imposing the following Gibbs conditions for chemical potentials and pressure

µ (H) B = µ (Q) B , µ (H) C = µ (Q) C , (28)
P H (T, µ B , µ C ) = P Q (T, µ B , µ C ) . (29)
Therefore, at a given baryon density ρ B and at a given net electric charge density ρ C = Z/A ρ B , the chemical potentials µ B are µ C are univocally determined by the following equations

ρ B = (1 -χ) ρ H B (T, µ B , µ C ) + χ ρ Q B (T, µ B , µ C ) , (30)
ρ C = (1 -χ) ρ H C (T, µ B , µ C ) + χ ρ Q C (T, µ B , µ C ) , (31)
where ρ

H(Q) B
and ρ

H(Q) C
are, respectively, the net baryon and electric charge densities in the hadronic (H) and in the quark (Q) phase and χ is the fraction volume of quark-gluon matter in the mixed phase. In this way we can find out the phase coexistence region, q 1.00 q 1.05 q 1.10 for example, in the (T, µ B ) plane. We are particularly interested in the lower baryon density (baryon chemical potential) border, i.e. the first critical transition density ρ I cr (µ I cr ), in order to check the possibility of reaching such conditions in a transient state during a heavy-ion collision at relativistic energies.

In Fig. 6, we report the pressure as a function of baryon chemical potential; as before, we have set Z/A = 0.4 and B 1/4 = 190 MeV. We can see that at T = 60 MeV, in the range of the considered µ B , the system is in a pure hadronic phase even for the nonextensive index q = 1.1. At T = 90 and 120 MeV, we have both the first and the second transition critical density for the considered values of q. In presence of nonextensive effects, the values of the critical densities result to be sensibly reduced with respect to the standard case. This matter of fact is more evident in Fig. 7, where we report the pressure at T = 90 MeV as a function of baryon density (in units of nuclear saturation density ρ 0 = 0.153 fm -3 ) (left panel) and energy density (right panel). It is interesting to observe that pressure as a function of baryon density (or energy density) is stiffer in the pure hadronic phase for q > 1 but appears a strong softening in the mixed phase. This feature results in significant changes in the incompressibility and may be particularly important in identifying the presence of nonextensive effects in high energy heavy ion collisions experiments. Related to this aspect, let us observe that possible indirect indications of a significative softening of the EOS at the energies reached at AGS have been discussed several times in the literature [START_REF] Russkikh | [END_REF]63,64,65,50].

T = 60 MeV ρ I cr /ρ 0 ρ II cr /ρ 0 µ I cr [MeV] µ II cr [MeV] q = 1.
In Table 1, we report the critical baryon densities and baryon chemical potentials at the beginning (index I) and at the end of the mixed phase (index II) for different values of temperature and q.

In Fig. 8, it is reported the phase diagram in the plane T -ρ B for different values of q. The curves labelled with the index I and II represent, respectively, the beginning and the end of the mixed phase. For q > 1, both the first and the second critical densities are sensibly reduced, even if the shape of the mixed phase is approximately the same. Related to this aspect, let us mention that the simplest version of the MIT bag model, considered in this investigation, appears to be not fully appropriate to describe a large range of temperature and density. To overcome this shortcoming, a phenomenological approach can therefore be based on a density or temperature dependent bag constant [50,66,67,68]. Moreover, as discussed in the Introduction, in regime of high temperature and small baryon chemical potential the first order phase transition may end in a (second order) critical endpoint with a smooth crossover. These features cannot be incorporated in the considered mean field approach. In our investigation, because we are focusing to nonextensive statistical effects on the nuclear EOS, instead of introducing additional parameterizations, we work with a fixed bag constant and limit our analysis to a restricted range of temperature and density, region of particular interest for high energy compressed nuclear matter experiments. Let us now explore in more details the variation of the first transition baryon density ρ I cr as a function of different physical parameters. In Fig. 9, we report the dependence of ρ I cr as a function of Z/A for different values of q (y axis in logarithmic scale). It is interesting to note a significant reduction of ρ I cr in presence of nonextensive statistics; as in the previous cases, this effect increases with the temperature. The dependence of the first transition baryon density as a function of Z/A is essentially a consequence of the ρ meson field behavior in the hadronic phase because it is directly connected with the isospin density of the system (as appears from Eq.( 12)). In this context, let us observe that, at fixed value of q, ρ I cr is significantly reduced by decreasing Z/A only at lower temperatures (T = 60 MeV) while, as expected, at higher temperatures (T = 120 MeV) the transition baryon density becomes very low and its isospin dependence becomes negligible, also in the framework of nonextensive statistics. This matter of fact is in according to the results of Fig. 3 where at low baryon chemical potentials (or baryon densities) the ρ meson field becomes almost constant and its absolute value significantly decreases.

In Fig. 10 (left panel), we show the first critical baryon density as a function of the bag constant for different values of nonextensive parameter q. Obviously, by increasing the bag constant we have a corresponding increasing of ρ I cr . However, this effect depends on the temperature and nonextensive parameter q. Finally, in the right panel of Fig. 10, we show the variation of ρ I cr as a function of the nonextensive index q for different values of temperature and B 1/4 = 190 MeV. At T = 60 MeV, we can see only a little reduction in the first critical density also for large deviations from the standard statistics; on the other hand, the reduction becomes more pronounced at larger temperatures. Variation of the first transition baryon density as a function of the net electric charge fraction Z/A for different temperatures and values of q (q = 1, solid lines; q = 1.05, short dashed lines; q = 1.10, long dashed lines).

Conclusions

To summarize, we have studied the main features of the nuclear EOS in the hadronic and quark-gluon phase and the possible formation of a consequent mixed phase in presence of nonextensive statistical effects. We have focused our investigation in regime of finite temperature and baryon chemical potential, reachable in high-energy heavy-ion collisions, for which the deconfinement phase transition can be still considered of the first order. From a phenomenological point of view, the nonextensive index q is considered here as a free parameter, even if, actually should not be treated as such because, in principle, it should depend on the physical conditions generated in the reaction, on the fluctuation of the temperature and be related to microscopic quantities (such as, for example, the mean interparticle interaction length, the screening length and the collision frequency into the parton plasma). We have restricted our investigation for small deviations from the standard statistics and for values q > 1 because, as quoted in the Introduction, these values were obtained in several phenomenological studies in high energy heavy ion collisions. In this context, it is relevant to observe that by fitting experimental observable at q > 1, the temperature (or slope) parameter T is usually minor of the one obtained in the standard Boltzmann-Gibbs statistics (q = 1) [29,37]. This feature is also present in the considered nuclear equation of state because, at fixed energy per particle E/N, we obtain for q > 1 lower values of temperature respect to the standard case. Moreover, let us remember that, in the diffusional approximation, a value q > 1 implies the presence of a superdiffusion among the constituent particles (the mean square displacement obeys to a power law behavior x 2 ∝ t α , with α > 1) [69].

In the first part of the work, we have investigated the hadronic equation of state and the role played by the meson fields in the framework of a relativistic mean field model which contains the basic prescriptions of nonextensive statistical mechanics. We have shown that, also in presence of small deviations from standard Boltzmann-Gibbs statistics, the meson fields and, consequently, the EOS appear to be sensibly modified. In the second part, we have analyzed the QGP proprieties using the MIT Bag model and also in this case the EOS becomes stiffer in presence of nonextensive effects. Finally, we have studied the proprieties of the phase transition from hadronic matter to QGP and the formation of a relative mixed phase by requiring the Gibbs conditions on the global conservation of baryon number and electric charge fraction. We have seen that nonextensive effects play a crucial role in the deconfinement phase transition. Moreover, although pressure as a function of baryon density is stiffer in the hadronic phase, we have shown that a strong softening in the mixed phase takes place in presence of nonextensive statistics. Such a behavior implies an abruptly variation in the incompressibility and could be considered as a signal of nonextensive statistical effects in high energy heavy ion collisions.

Figure 1 .

 1 Figure 1. The σ meson field as a function of baryon chemical potential for different values of temperature (in units of MeV) and q.

Figure 2 .Figure 3 .

 23 Figure 2. The ω meson field as a function of the baryon chemical potential for different values of temperature and q.

Figure 4 .

 4 Figure 4. Pressure (left panel) and energy density (right panel) versus baryon chemical potential for different values of temperature and q.

Figure 5 .

 5 Figure 5. Pressure of the quark-gluon phase as a function of baryon chemical potential for different values of temperature and q.

Figure 6 .

 6 Figure 6. Pressure versus baryon chemical potential in the mixed phase for different values of temperature and q.

Figure 7 .

 7 Figure 7. Pressure as a function of baryon density (left panel) and energy density (right panel) in the mixed phase for different values of q. The temperature is fixed at T = 90 MeV.

Figure 8 .

 8 Figure 8. Phase diagram T -ρ B for different values of q. The curves with index I and II indicate, respectively, the beginning and the end of the mixed phase.

Figure 9 .

 9 Figure 9. Variation of the first transition baryon density as a function of the net electric charge fraction Z/A for different temperatures and values of q (q = 1, solid lines; q = 1.05, short dashed lines; q = 1.10, long dashed lines).

Figure 10 .

 10 Figure 10. Variation of the first transition baryon density as a function of the bag constant (left panel) and nonextensive index q (right panel) for different temperatures.

Table 1 .

 1 Critical baryon densities and baryon chemical potentials at the beginning (index I) and at the end (index II) of the mixed phase for different values of temperature and nonextensive parameter q.

	00	5.75	9.10	1503	1569
	q = 1.05	5.56	8.88	1472	1537
	q = 1.10	5.33	8.65	1437	1502
	T = 90 MeV ρ I cr /ρ 0 ρ II cr /ρ 0 µ I cr [MeV] µ II cr [MeV]
	q = 1.00	3.41	5.09	1123	1170
	q = 1.05	2.77	4.46	1034	1068
	q = 1.10	1.91	3.69	916	927
	T = 120 MeV ρ I cr /ρ 0 ρ II cr /ρ 0 µ I cr [MeV] µ II cr [MeV]
	q = 1.00	0.45	1.93	588	616
	q = 1.05	0.20	1.33	383	396
	q = 1.10	0.08	0.71	184	201

‡ In Ref.[47], the nucleon effective mass as a function of temperature always diminishes respect to standard statistics, this behavior is a consequence of the fact that it is plotted only at ρ B = 0.


Z J Ajaltouni

E Di Salvo

Searching for New Physics by means of Relative Phases and Moduli of Decay Amplitudes

Keywords: numbers: Nos.: 11.30.Er, 11.80.Et, 13.25.-k, 13.30.Eg

We illustrate some tests which may shed light on physics beyond the Standard Model. Some of them are especially sensitive to possible signals of new physics.

Introduction

Hunt to physics beyond the Standard Model (SM), commonly named new physics (NP), has become particularly lively and intense since the announcement of the LHC start [1][2][3][4][START_REF] Wilczek | Invited talk at EPS-HEPP meeting[END_REF][START_REF] Nath | [END_REF][START_REF] Buras | Talk at EPS-HEPP meeting[END_REF][START_REF] Kalinowski | [END_REF][9]. Indeed, although the SM is consistent with a wealth of data [START_REF] Bigi | Production, Properties and Interactions of Mesons[END_REF], it is unsatisfactory for several reasons (see [1][2][3][4][START_REF] Wilczek | Invited talk at EPS-HEPP meeting[END_REF][START_REF] Nath | [END_REF] for recent reviews).

One of the sectors where people look constantly for signals of NP is constituted by processes involving CP violations. Indeed, the Cabibbo-Kobayashi-Maskawa (CKM) scheme, embedded in the SM, is in good agreement with a great deal of present data, however we look for a dynamical description of the effect. In searching for NP, especial attention is being paid to those decays which have shown possible discrepancies with the SM predictions [START_REF] Ishikawa | [END_REF][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29][30], stimulating explanations outside the SM and proposals for further investigations in analogous decays. Among them we highlight decays, like B → πK [31][32][33][34][35][36][37][38] and B → ΦK ( * ) [39,40,41], involving penguin b → s transitions [42,43,44,45,46], to which SUSY and double Higgs models contribute significantly [42,43].

The possibility of alternative models is also studied in other processes, namely in rare semi-leptonic decays [47,48,49,50,51], in neutral meson mixings [52,53,54,55,56] and in various hadronic decays [57][58][59][60][61][62][63][64][65][66][67][68][69][70][71][72][73]. These lasts may concern, e. g., Uspin symmetry violations [43,69], or the unexpectedly large transverse polarization of vector mesons V in B → V V decays [45,70,71,72,73].

Moreover, also time reversal violation (TRV) is used as a probe for NP [74,75,76]. TRV is commonly regarded as the counterpart of CP violation, in view of the CPT theorem. It is valid under very mild assumptions and not contradicted by any experiments, even supported by stringent tests [77]. Up to now direct TRV has been observed only in the CPLEAR experiment [78]. Alternatively, TRV may be revealed by the presence, in a hadronic two-body weak decay amplitude, of a "weak" phase, besides the one produced by strong final state interactions (FSI) [57,[START_REF] Marshak | Theory of Weak Interactions in Particle Physics[END_REF][START_REF] Bigi | CP Violation, Cambridge Monographs on Particle Physics, Nuclear Physics and cosmology[END_REF][START_REF] Wolfenstein | [END_REF]. However, in this case, experimental uncertainties of the "strong" phases create serious problems in singling out the "weak" one [START_REF] Marshak | Theory of Weak Interactions in Particle Physics[END_REF]82]. Incidentally, this phase occurs in the SM via the CKM scheme. correlations [57,[START_REF] Wolfenstein | [END_REF]83,84,85,86], which may be studied in sequential decays. In fact, comparing a triple product with the CP-conjugated one is sensitive to such violations [86]. It could even be more effective than CP violations in probing NP [86].

Furthermore, if a sequential decay involves more spinning particles, like B → φφ or Λ b → J/ψΛ, CP violations, which are usually investigated by means of differences between partial decay widths, could equally well be studied through angular analyses [57]. These studies could be even more convenient [87]. Indeed, angular analyses allow the determination of moduli and relative phases of helicity amplitudes [57,60,88,89,90,91]. Thus, in principle, even CPT violations could be detected [92].

However, until now such observables have not been exploited as an alternative or complementary tool for probing NP. The aim of the present note is to suggest alternative tests for the presence of NP, based on helicity amplitudes, in decays like those mentioned before.

To this end, we derive theoretical consequences of TRV. In this way we propose tests suitable for two-body sequential decays, which can be explored by experiments like those recently suggested or realized for CP violations [START_REF] Ishikawa | [END_REF][12][13][14][15][16][17][18][19][39][40][41].

Section 2 is devoted to deducing the condition for time reversal invariance (TRI) of decay amplitudes. In section 3 we introduce some asymmetries, which allow to apply tests to data. In section 4 we make a few remarks. Finally, our conclusions are presented in section 5.

A condition for TRV

We focus on hadronic two-body decays of the type

R 0 → R 1 R 2 .
(

) 1
Here R 0 is the original resonance and R 1 and R 2 the decay products, with spin J, s 1 and s 2 respectively.

Our condition for TRV is derived by extending the standard treatment of TRI for two-body decays [START_REF] Marshak | Theory of Weak Interactions in Particle Physics[END_REF][START_REF] Bigi | CP Violation, Cambridge Monographs on Particle Physics, Nuclear Physics and cosmology[END_REF] to the case where more than one non-leptonic decay mode is involved [93]. If (1) is a weak decay, the relative, rotationally invariant amplitude reads, at first order in the weak coupling constant,

A J λ 1 λ 2 = f out |H w |JM . (2)
Here H w is the weak hamiltonian, |f out a shorthand notation for the final two-body angular momentum eigenstate |JMλ 1 λ 2 , M the component of the spin of R 0 along the z-axis of a given frame and λ 1 and λ 2 the helicities of, respectively, R 1 and R 2 in the rest frame of R 0 . Assume H w to be TRI, i. e.,

T H w T † = H w . ( 3
)
Here T is the Time Reversal (TR) operator. Then, taking into account the antilinear character of T and the rotational invariance of the amplitude, we get[94]

A J λ 1 λ 2 = f in |H w |JM * . (4)
Inserting a complete set of "out" states yields

A J λ 1 λ 2 = n f in |n out * n out |H w |JM * . (5)
The only terms which survive in this sum correspond to the decay modes of R 0 ;

furthermore the non-leptonic decay modes give the main contribution, since they involve a much greater coupling constant than the semi-leptonic decay modes. Now we relax the limitation of the state |f in to a two-body one; moreover we express the "out" states in terms of the S-matrix. This is unitary and, owing to TRI ‡ , also symmetric with respect to angular momentum eigenstates [START_REF] Martin | Elementary Particle Theory[END_REF]. Then eq. ( 5) can be rewritten as

A m = n S mn A * n . (6)
Here, omitting spin and helicity indices,

A m = m out |H w |R 0 (7) ‡
We neglect weak contributions to scattering.

and

S mn = m in |S|n in . (8)
It is worth noting that eq. ( 6) coincides with eq. ( 12) of ref. [93]. The S-matrix is block-diagonal [START_REF] Wolfenstein | [END_REF], since not all hadronic states are connected to one another by strong interactions. In particular, such blocks are characterized by flavor [START_REF] Wolfenstein | [END_REF].

A solution to eq. ( 6) can be obtained by diagonalizing the S-matrix. To this end we recall a result by Suzuki [93], that is,

S mn = k O mk e 2iδ k O T kn , (9)
where O is an orthogonal matrix and the δ k are strong phase-shifts. Then the solution to eq. ( 6) is given by

A m = n a n O mn e iδn . ( 10
)
Here a n are real amplitudes representing the effects of weak interactions on the decay process. Obviously complex values of one or more such amplitudes -that is, "weak" phases -would imply TRV. This result generalizes the one where only elastic scattering is allowed between the decay products [START_REF] Marshak | Theory of Weak Interactions in Particle Physics[END_REF].

Before considering applications of our model independent result, two important remarks are in order. Firstly, the quantities O mn and δ n depend on strong interactions and therefore are invariant under parity inversion, P, and under charge conjugation, C. Secondly, if a local field theory is assumed -like the SM -, a nontrivial phase of at least one a n implies also CP violation, owing to CPT symmetry. In particular, this phase is related to the phase of the CKM matrix in the SM.

Proposed tests for NP

The problem we are faced with is to determine quantities which are somewhat sensitive to the "weak" phases, so as to compare them with SM predictions. Unfortunately, according to the result found in the previous section, we cannot determine those phases. First of all, the elements of the matrix O are not known, at best one can elaborate models, like ref. [93]. Secondly, the amplitudes A n may be determined up to a phase per decay mode. In fact, a decay of the type (1), with spinning and unstable decay products, allows to determine, through angular distribution, polarizations and polarization correlations [57,60,88,89,90,91], all products of the type

A λ 1 λ 2 A * λ 1 λ 2
. We stress that such observables may be determined by means of a sequential decay, which is sensitive to non-diagonal elements of the density matrix of the decay products [START_REF] Byers | [END_REF]96,97]. Moreover we recall a previous paper [91], where we showed a method for extracting such products from the observables of the decay

Λ b → ΛV. ( 11
)
These products allow, in turn, to infer all moduli of the amplitudes and their phases relative to a given amplitude, taken as a reference. In this connection, it is worth recalling that recent measurements of sequential decays of beauty resonances to two vector mesons have led to determining moduli and relative phases of decay amplitudes in the transversity representation [16,18,39,40,41], linearly related to the helicity representation. Although such quantities do not allow to determine "weak" phases, they may hide signatures for NP, as we shall see in the following subsections. Indeed, we shall present three different types of tests for singling out possible signatures of physics beyond the SM.

Tests for general two-body decays

Define the following two asymmetries:

A CP = Φ λ 1 λ 2 -Φ -λ 1 -λ 2 Φ λ 1 λ 2 + Φ -λ 1 -λ 2 , (12)
A M = |A J λ 1 λ 2 | 2 -|A J -λ 1 -λ 2 | 2 |A J λ 1 λ 2 | 2 + |A J -λ 1 -λ 2 | 2 . ( 13
)
Here the barred symbols refer to charge-conjugate amplitudes. Moreover

Φ λ 1 λ 2 = tan -1 (A J λ 1 λ 2 A J * λ 01 λ 02 ) (A J λ 1 λ 2 A J * λ 01 λ 02 ) = sin -1 (A J λ 1 λ 2 A J * λ 01 λ 02 ) |A J λ 1 λ 2 ||A J λ 01 λ 02 | = cos -1 (A J λ 1 λ 2 A J * λ 01 λ 02 ) |A J λ 1 λ 2 ||A J λ 01 λ 02 | . ( 14
)
are relative phases; λ 1 and λ 2 are the helicities of the decay products in the overall center-of-mass, as introduced in sect. 2. Lastly A J λ 01 λ 02 denotes the reference amplitude. If at least one of the amplitudes a n is complex, the above asymmetries, (12) and (13), will be generally nonzero. In particular, this is true if CPT symmetry is assumed.

The experimental values of such asymmetries -analogous to observables proposed by other authors [86] -have to be compared to SM predictions. In any case, if A CP or A M is nonzero, we may conclude that CP is violated, but not TR.

Tests for particular decays

A particularly intriguing case is represented by two-body decays that fulfil the condition

A M = 0. ( 15
)
This corresponds to one of the following two situations: i) the sum ( 10) consists of just one term, ii) or the "weak" amplitude a n (or at least its phase § ) can be factored out of that sum.

The latter case is typical of a unique basic quark process contributing to all decay modes of a given block of the S-matrix [START_REF] Wolfenstein | [END_REF]93]. For example, some of the b → s transitions are dominated in the SM by the penguin diagram [42,43,44,45,46], the tree diagram being forbidden. In this case the sum (10) amounts to

A m = a n O mn e iδn , (16)
a being the common "weak" amplitude. The sum in eq. ( 16) concerns only strong interactions and is related to the absorptive part of the amplitude of a weak decay to intermediate states, followed by strong interactions that lead to the final state [START_REF] Wolfenstein | [END_REF].

To be precise, condition (15) does not necessarily imply eq. ( 16), however it is rather unlikely that, with different "weak" amplitudes a n in the sum [START_REF] Bigi | Production, Properties and Interactions of Mesons[END_REF], the amplitude A m and its CP-conjugated have the same modulus square. Of course this is not § In the following we do not consider this case, which appears quite unrelistic.

true if CPT symmetry is assumed. Then, if condition ( 15) is satisfied, we assume that the decay is driven by a single dominant diagram, for example penguin or tree.

Among penguin dominated decays, especially interesting for NP are the following decay modes of B, B s and Λ b , in part already studied, either theoretically [43,44,45,46] or experimentally [39,40,41]:

B → φK * , K * K * ; ( 1 7
)

B s → φφ, J/ψK * 0 , ( 18
) Λ b → Λφ. ( 19
)
These decays are dominated by the penguin diagram, driven by the subprocess

b → s + qq. ( 20
)
Under condition (15), it makes sense to define also the following two asymmetries:

A C = Φ λ 1 λ 2 -Φ λ 1 λ 2 Φ λ 1 λ 2 + Φ λ 1 λ 2 , (21)
A P = Φ λ 1 λ 2 -Φ -λ 1 -λ 2 Φ λ 1 λ 2 + Φ -λ 1 -λ 2 . ( 22
)
If condition ( 15) is fulfilled, and simultaneously at least one of the asymmetries ( 12), ( 21) and ( 22) is different from zero, we may conclude that traces of NP are evident.

Indeed, according to the CKM scheme, the "weak" phase is independent of hadron helicities, therefore the asymmetries defined above are predicted to be zero. A nonzero value of one of such asymmetries would be a signature of NP. Helicity dependent "weak" phases could be realized, for example, in the case of flavor changing neutral currents, by interference between Z (or Z ) exchange and Higgs exchange: the former selects only left-handed quarks, while the latter accepts equally left-and right-handed ones.

Our method can help to resolve ambiguities illustrated in ref. [98], about singling out NP contributions in penguin diagrams.

Aside from that, an indication of NP may come from a decay mode which, according to the SM, is governed by a single (penguin or tree) graph and yet it does not satisfy condition (15). A somewhat analogous procedure was suggested by Datta and London [86]. In this connection we recall that interference between a tree diagram and a NP contribution, if present, would be probably detected [99] thanks to the new facilities.

Interference between two "weak" amplitudes

Different, but equally interesting cases are constituted by the decay modes which according to the SM are driven by interference between the tree and the penguin diagram [15,25,43,60,72,73]. Typical decays of this type are given by

B → J/ψK * , ρK * , ωK * ; ( 2 3 )
B s → J/ψφ, J/ψK * 0 ; ( 2 4 )
Λ b → ΛJ/ψ, Λρ, Λω. (25)
In these cases eq. ( 16) has to be replaced by

A m = a t T m + a p P m , (26)
where a t(p) is the "weak" amplitude for the tree (penguin) diagram and

T m (P m ) = n t(p)
O mn t(p) e iδn t(p) (27) describes the effects of the strong interactions in each such diagram. Here n t(p) runs over the intermediate strong states available in a decay described by a tree (penguin)

diagram. The two spectra do not coincide, but may have some overlap. Among the decays above, those which involve the J/ψ resonance in the final state are dominated by the two quark sub-processes (20) and

b → c + cs, (28)
the latter corresponding to the tree diagram. The asymmetry ( 13) is expected to be different from zero, its size giving information on the relative weight of the two amplitudes. If, as in the case of decay (24), the penguin contribution is very weak, one can apply the tests described in the previous subsection: if A M is quite small and at least one of the other asymmetries, concerning relative phases, is significantly different from zero, this is an indication of NP.

A further test, similar to the one suggested in subsection 3.2, can be stated by following a procedure similar to the one suggested in ref. [88]. In that article the authors propose to infer, from angular distribution and polarization of B → V V decays, the relative "weak" phase of the penguin amplitude to the tree one, assuming that such a phase is independent of vector meson helicities. If this assumption is not confirmed by data, we may conclude that NP traces are present.

Remarks

At this point some remarks and comments are in order.

A) We have already observed at the beginning of sect. 3 that recent data yield moduli and relative phases of decay amplitudes. Our tests may be applied, provided data become available for the antiparticle decays as well.

B) Such data have shown finite relative phases [16,18,39,40,41], which implies sizeable "strong " phases produced by FSI, in qualitative agreement with the estimations by Wolfenstein [82]. These phases may be greater than the "weak" ones, which may be an obstacle in the detection of TRV. However, this difficulty is overcome by our method: as recalled at the end of sect. 2, the "strong " phases are eliminated in the numerators of asymmetries (12), owing to C and P invariance of strong interactions. Similar suggestions were given by other authors [37,57,[START_REF] Wolfenstein | [END_REF]83,84,86].

C) The application of our method demands a particular care in decays of neutral, spinless resonances to neutral particles, like some of the B 0 and B s decays mentioned in the previous subsection, especially if the decay products constitute a CP eigenstate [60,87]. In such cases particle-antiparticle mixing complicates the extraction of the parameters needed for our tests. A useful tool for splitting the angular distribution into CP-even and CP-odd contributions is the transversity representation [87].

Moreover a suitable quantity for revealing CP violations, and possibly NP, is defined in ref. [86].

D) The condition of vanishing of the asymmetry A M , which if fulfilled could reveal clear signatures of NP, is opposite to the one demanded for detecting CP violations, which needs interference among amplitudes, and therefore at least two different weak decay amplitudes of the type a n in the sum [START_REF] Bigi | Production, Properties and Interactions of Mesons[END_REF].

Conclusions

We have illustrated a theoretical condition for TRV in non-leptonic two-body decays.

This suggests some kinds of tests for possible probes of NP, to be applied to decays to unstable, spinning decay products. The tests may be realized by means of standard analyses. We propose to apply them particularly to those decay modes for which hints of NP have been already found [START_REF] Ishikawa | [END_REF][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29][30], or/and suggestions for new investigations have been given [20][21][22][23][24][25][26][27][28][29][30][31][32][33][34][35][36][37][38]. Especially effective appear the tests proposed in subsections 3.2 and 3.3, since nonzero values of asymmetries would imply NP.

Obviously, the feasibility of the tests proposed is conditioned by statistics. Such tests appear to be not so unrealistic, given the wealth of b -b pairs to be produced at LHC per year (10 12 ). The asymmetries proposed do not derive significant contribution from the SM; but, if NP is present, and it is realized according to some of the alternative models, it could be detected.

Acknowledgments

The authors are grateful to their friend J. Orloff for stimulating discussions. One of us (EDS) is also indebted to his friend A. Di Giacomo for useful suggestions.


Abderrahim Adahchour

Pierre Descouvemont

Three-body structure of 18 Ne in a microscopic model

Keywords: numbers: 21.60.Gx, 23.50.+z, 27.20.+n

Three-body structure of 18Ne in a microscopic model

Abderrahim Adahchour, Pierre

Introduction

The simultaneous emission of two protons (diproton decay) for nuclei close or beyond the proton drip line is well established since many years [1]. The interest for this decay mode has been recently revived with the availability of intense radioactive beams [2,3]. Diproton radioactivity has been observed in some nuclei, such as 17 Ne [4] or 19 Mg [5]. This decay mode is, in general, in competition with the sequential decay, where the two protons are emitted through an intermediate nucleus with charge Z -1.

Recently, diproton decay of the 1 -excited level of 18 Ne (6.15 MeV) has been reported by the 17 F+p reaction [6] and by Coulomb excitation of a 18 Ne beam on a Pb target [7]. The total width, including one and two-proton decay, was measured to be 50 ± 5 keV [6]. The two-proton partial width is 21 ± 3 eV assuming a diproton emission, and 57 ± 6 eV assuming a three-body decay.

The theoretical analysis of this mechanism requires specific models, able to account for a three-body structure of the nucleus. In addition, the theoretical treatment of threebody resonances is a delicate problem [8,9]. Various calculations of diproton emission have been done in the shell model [10] or in a non-microscopic three-body model [8].

Here we focus on the 1 -state of 18 Ne, and use a microscopic cluster theory, based on an 16 O+p+p description within the hyperspherical formalism [11,12]. This approach has been already applied to bound states of some nuclei, such as 6 He, 9 Be or 12 C [13,14]. In a microscopic model, all nucleons are included in the Hamiltonian, and the Pauli principle is exactly taken into account. The model is based on a nucleon-nucleon interaction, and therefore presents a fairly strong predictive power. Of course, the Schrödinger equation cannot be solved exactly for large nucleon numbers, and the cluster approximation is used [START_REF] Wildermuth | A Unified Theory of the Nucleus[END_REF].

The treatment of resonances requires an additional tool, taken here as the Analytical Continuation in the Coupling Constant (ACCC, see Ref. [START_REF] Kukulin | Theory of Resonances, Principles and Applications[END_REF]). This method provides an estimate of the energy and width of a resonance. Its main advantage is that it is based on bound-state calculations only, which do not need to account for scattering boundary conditions. The ACCC method was successfully applied in the same microscopic threecluster model to describe 5 H and 5 He (T = 3/2) three-body resonances [START_REF] Adahchour | [END_REF]. Of course, the ACCC method only provides the total width of a resonance. A separation between 1p and 2p decays is not feasible in this framework.

Section 2 is devoted to a brief overview of the theory. We describe the microscopic cluster model, and give the main properties of the ACCC method. In Section 3, we present the theoretical 18 O and 18 Ne spectra, and we focus on the proton width of 1 - (6.15 MeV) resonance in 18 Ne. Concluding remarks are discussed in Section 4.

Theoretical background

The microscopic cluster model

We use here a microscopic description [START_REF] Wildermuth | A Unified Theory of the Nucleus[END_REF], where the Hamiltonian of the system is given by

H = A i=1 T i + A j>i=1 V ij , (1)
T i being the kinetic energy of nucleon i, and V ij a nucleon-nucleon effective interaction. For large nucleon numbers (A ≥ 12), and in particular to investigate resonances, the Schrödinger equation associated with Hamiltonian (1) cannot be solved exactly. We use here the cluster approximation [START_REF] Wildermuth | A Unified Theory of the Nucleus[END_REF], where the A nucleons are assumed to be grouped in clusters. In the present case, 18 Ne is described by a three-cluster wave function, with an 16 O+p+p structure. In a schematic notation, the total wave function reads

Ψ = Aφ O φ p φ p G(r 1 , r 2 ), (2)
where φ O and φ p are 16 O and proton wave functions, defined in the harmonic-oscillator model, and G(r 1 , r 2 ) is a relative function depending on the Jacobi coordinates r 1 and r 2 (see Fig. 1). In (2), A is the A-body antisymmetrization operator which enables an exact account of the Pauli principle, not only inside the 16 O core, but also between the core and the external nucleons. The center-of-mass motion is exactly factorized. The wave function (2) is defined in the hyperspherical formalism, well adapted to three-body systems. Jacobi coordinates r 1 and r 2 are therefore transformed into the hyperangle α and hyperradius ρ (see Refs. [11,12] for details). For a given spin J and parity π of 18 Ne, the wave function is expanded as

Ψ JMπ = K x y LS Ψ JMπ K x y LS , ( 3
)
where K is the hypermomentum, x and y are the orbital momenta associated with the Jacobi coordinates, and L and S represent the total angular momentum and intrinsic spin, respectively. The summation over K must be truncated at a maximum value, denoted as K max . Of course, the 18 O wave functions have the same three-cluster structure. Owing to the charge symmetry of the nuclear force, only the Coulomb interaction is changed in (1). The extension of three-body models to a microscopic version has been proposed in Ref. [13]. In the Generator Coordinate Method (GCM), the partial wave functions (3) are expanded over a Gaussian basis, which enables to express them as linear combinations of Slater determinants. Each component of Eq. ( 3) is therefore written as

Ψ JMπ K x y LS = f Jπ K x y LS (R) Φ JMπ K x y LS (R) dR, (4)
where Φ JMπ (R) are projected Slater determinants, and f Jπ (R) the generator functions. Both depend on the generator coordinate R associated with the hyperradius ρ. In practice this integral is replaced by a sum over a finite set of R values (typically 8 ∼ 12 values). An approximate solution of the Schrödinger equation can be obtained by the variational principle, using matrix elements of the Hamiltonian (1) between GCM basis functions. In Ref. [13] we have shown that matrix elements between projected Slater determinants involve seven-dimensional integrals, which are performed numerically. The main part of the calculation is the evaluation of the potential matrix elements. A twobody matrix element between unprojected basis functions involves quadruple sums over individual orbitals [START_REF] Brink | Proc. Int. School "Enrico Fermi" 36[END_REF]. In the applications considered so far [13,14], this microscopic three-body model was limited to s clusters (nuclei such as 6 He, 9 Be or 12 C were studied). Under these conditions the development of the model to the three-body continuum was proposed recently [START_REF] Damman | [END_REF], and applied to α+n+n unbound states. This extension requires large values of the generator coordinate (typically values up to 25 fm must be considered) and, consequently, a large basis. The calculation was made possible by the use of semianalytical techniques to evaluate the matrix elements. This approach is much faster than the numerical procedure. In contrast, the present calculation involves 16 O as core nucleus, where p orbitals are included. In that case a semi-analytical treatment raises important difficulties, and a numerical approach must be used. However, the presence of p orbitals considerably increases the computer time. In a Cartesian representation, the number of individual orbitals is N o = 6 (1 s and 3 p orbitals for the 16 O core, and 1 for each external nucleon) whereas it is N o = 3 when only s-clusters are involved. The quadruple sums mentioned before contain N 4 o terms, and the computer times therefore increase by a factor 16 when going from an α core to an 16 O core. We have tested Monte-Carlo methods [20] to evaluate the seven-dimensional integrals, but this approach turns out to be not well adapted since many partial waves (K x y LS) are treated simultaneously. It was not possible to achieve a good accuracy for all partial waves simultaneously, and therefore standard quadratures have been adopted.

The ACCC method

In contrast with the approach of Ref. [START_REF] Damman | [END_REF] which provides three-body phase shifts at any energy, the ACCC method [START_REF] Kukulin | Theory of Resonances, Principles and Applications[END_REF] is limited to resonances. However it provides resonance properties from bound-state calculations, which do not require very large values of the generator coordinate (here R max = 12 fm, see Sect. 3.1). Its application to microscopic calculations has been shown in the past to be very efficient [21,[START_REF] Adahchour | [END_REF]. We briefly give here the main principles and refer the reader to Ref. [START_REF] Kukulin | Theory of Resonances, Principles and Applications[END_REF] for more detail.

To apply the ACCC method, the Hamiltonian is written as

H(λ) = H 0 + λ H 1 , ( 5
)
where λ is a linear parameter, and where H 1 is supposed to be attractive. In cluster models, the nucleon-nucleon effective interaction often involves a linear parameter which has a standard value, but can be slightly modified to reproduce important properties of the system, such as a resonance energy. This parameter is the natural choice for the linear parameter λ.

Let us define λ 0 as k(λ 0 ) = 0, where k is the wave number (throughout the paper, energies are determined from the three-body threshold). For negative energies (i.e. λ > λ 0 ), the wave number is imaginary and is parametrized in a Padé approximant of degree (M, N) as

k(x) = i c 0 + c 1 x + • • • + c M x M 1 + d 1 x + • • • + d N x N , (6)
where x = √ λ -λ 0 is real. The calculation is performed for M + N + 1 values of λ, each of them providing a bound-state energy. This procedure allows to determine the Padé coefficients c j and d j .

For a resonance, the physical value λ is lower than λ 0 . Then the resonance energy E R and width Γ are obtained by using (6) for an imaginary value of x. One defines the complex energy E as

E = h2 k 2 2m N = E R -iΓ/2, ( 7
)
m N being the nucleon mass. This method only needs bound-state calculations, but is limited to the total width of a resonance. The associated wave function is not available in this approach. In addition it requires a high accuracy of the variational basis. In particular, the sensitivity with respect to the Padé parameters (M, N) should be checked by considering various sets of λ values.

The 18 O and 18 Ne nuclei

Conditions of the calculation

The 16 O wave function φ O (see Eq. ( 2)) is defined in the shell model, with harmonicoscillator orbitals involving the oscillator parameter b. This parameter determines the properties of the 16 O ground state. It value is chosen as b = 1.6 fm, which corresponds to a compromise between the minimum energy and the experimental radius. The nucleonnucleon interaction is taken as the Volkov force V2 [22], which involves the exchange parameter M (the standard value is M = 0.6). Here it is adjusted on the ground-state energy of 18 O for positive parity (M = 0.620), and on the 1 -(6.15 MeV) level of 18 Ne for negative parity (M = 0.589). Reproducing the energy of a resonance is crucial to derive realistic values for its width. The Coulomb interaction is treated exactly. We also include a zero-range spin-orbit force [23]. The amplitude is adopted as S 0 = 40 MeV.fm 5 , a standard value in this mass region. The generator coordinates R have been selected from 1.5 fm to 12 fm with a step of 1.5 fm. The summation over the hypermoment K [see Eq. ( 3)] has been truncated at K max = 20. As we will show below, this value provides an accurate convergence.

Energy spectra

The 18 O and 18 Ne energy spectra are displayed in Fig. 2, and compared with experiment [24]. In both nuclei, the excitation energy of the 2 + 1 state is underestimated by about 1 MeV. This problem is common to most nuclei in this mass region. The level density beyond 4 MeV excitation energy is quite large, in particular for 18 O. This property represents a challenge for microscopic models, since experimental states may have a different cluster structure. However most of the GCM states can be unambiguously assigned to experimental candidates. In particular we do reproduce the 2 + 2 state. A 0 + level is predicted near E x ≈ 7 MeV. Most likely it can be assigned to the experimental 0 + 3 state since the 0 + 2 state is expected to present an α cluster structure [25]. In negative parity the lowest GCM state is 1 -; other negative-parity states are at higher excitation energies. The 1 - 1 experimental level is known to have an α+ 14

C/α+ 14 O 0+ 0+ 0+ 0+ 2+ 2+ 2+ 2+ 3+ 3+ 3+ 4+ 4+ 4+ 0- 1- 1- 1- 2- 2- 3- 3- 4- 0+ 2+ 2+ 1- 1- 2- -12 -10 -8 -6 -4 0+
0+ 0+ 0+ 2+ 2+ 2+ 2+ 3+ 3+ 4+ 4+ 4+ 1- 1- 1- 3- 3- 0+ 2+ 2+ -4 -2 0 2 GCM exp GCM exp 18 O 18 Ne E (MeV)
Figure 2. 18 O and 18 Ne spectra (experimental data are taken from Ref. [24]). The zero energy is taken as the 16 O+n+n and 16 O+p+p thresholds, respectively.

cluster structure [25] and cannot be described by the present model which includes 16 O as a core. The first theoretical 1 -state therefore corresponds to the 1 - 2 experimental level at 6.15 MeV.

In Fig. 3, we analyze the energy convergence of various states as a function of K max . In positive parity, the convergence is reached near K max ∼ 12. The energy difference between K max = 20 and K max = 18 is lower than 1 keV. As expected for unbound states, the 1 -resonance presents a slower convergence. Going up to large K max values (i.e. larger than 15) is necessary to obtain precise properties.

Decay of the 1 -(6.15 MeV) state in 18 Ne

The ACCC method has been applied to determine the width of the 1 - 2 state in 18 Ne. As mentioned in Sect. 2.2, the ACCC presents some sensitivity with respect to the conditions of the calculation [i.e. the degree of the Padé approximant, and the λ values used to parametrize the wave number (6)]. Consequently convergence tests must be performed, and the resonance properties cannot be determined without some uncertainty. In particular, the width, which is zero at negative energies, is somewhat sensitive to the conditions of the calculation.

In Fig. 4, we show the energy and width for various degrees (M = N ) of the Padé approximant (6). In addition, different sets of λ values have been adopted and defined as where ∆λ is a constant step. Variations of the resonance properties against N and ∆λ provide an estimate of the uncertainty, which is unavoidable with the ACCC method. Small N values do not give accurate Padé approximants. Conversely, when N is large (typically beyond N ≈ 10), round-off errors become more and more important. The energy is very stable and corresponds to the value obtained in the bound-state approximation, i.e. by a simple diagonalization of the Hamiltonian matrix. The width, related to the imaginary part of the wave number is somewhat less stable. Fig. 4 provides E R = 1.67 ± 0.01 MeV and Γ = 40 ± 10 keV. The energy has been adjusted by the nucleon-nucleon interaction, and is of course very close to experiment. Conversely the width is an output of the model, and is in fair agreement with experiment (50 ± 5 keV, see Ref. [6]).

λ n = λ 0 + n∆λ, (8)

Conclusion

This work represents a first attempt to describe the three-body decay of 18 Ne in a microscopic approach. The GCM is known for a long time to be well adapted to the spectroscopy of light nuclei, and to low-energy reactions. Its application to unbound states is however a challenge, as it raises several difficulties. A realistic treatment requires a three-body model. In addition, the unstable nature of the initial state requires to go beyond the bound-state approximation, and to complement the GCM by an additional tool, able at dealing with unbound states. We have used here the ACCC method, which is based on standard variational calculations. Even if the resonance properties cannot be derived without some uncertainty, the method should be considered as a very powerful tool.

The GCM width is found as Γ = 40 ± 10 keV, in good agreement with the experimental value Γ = 50 ± 5 keV. This result demonstrates that the theoretical description of the 1 - 2 unbound state is realistic. Of course the ACCC method only provides the total width. Determining the (small) diproton branching ratio would require to use more sophisticated methods, providing the partial widths as well.

Other proton-rich nuclei with a marked three-body structure, such as 17 Ne or 19 Mg, might be considered in the future. The 16 O+p+p system is however simpler since the 16 O core is a closed-shell nucleus. The hypersperical formalism applied to the GCM requires seven-dimension integrals for the matrix elements. The numerical evaluation of these integrals need very long computer times, and going beyond this limitation represents a challenge for the future.

Figure 1 .

 1 Figure1. Three-cluster structure of18 Ne with the Jacobi coordinates r 1 and r 2 .

Figure 3 .

 3 Figure 3. Convergence of various 18 Ne states as a function of the maximum hypermomentum K max .

Figure 4 .

 4 Figure 4. Energy and width of the 1 -(6.15 MeV) resonance in 18 Ne, as a function of the degree of the Padé approximant (M = N ). The curves correspond to different sets of λ values [see Eq. (8)].

Acknowledgments

This text presents research results of the Belgian program P6/23 on interuniversity attraction poles initiated by the Belgian-state Federal Services for Scientific, Technical and Cultural Affairs.