D Blaschke
email: blaschke@ift.uni.wroc.pl

T Klähn

R Lastowiecki

F Sandin

How strange are compact star interiors?

Keywords: numbers: 12.38.Lg, 26.60.Kp, 97.60.Jd

We discuss a Nambu-Jona-Lasinio (NJL) type quantum field theoretical approach to the quark matter equation of state with color superconductivity and construct hybrid star models on this basis. It has recently been demonstrated that with increasing baryon density, the different quark flavors may occur sequentially, starting with down-quarks only, before the second light quark flavor and at highest densities also the strange quark flavor appears. We find that color superconducting phases are favorable over non-superconducting ones which entails consequences for thermodynamic and transport properties of hybrid star matter. In particular, for NJL-type models no strange quark matter phases can occur in compact star interiors due to mechanical instability against gravitational collapse, unless a sufficiently strong flavor mixing as provided by the Kobayashi-Maskawa-'t Hooft determinant interaction is present in the model. We discuss observational data on mass-radius relationships of compact stars which can put constraints on the properties of dense matter equation of state.

Introduction

The question of strangeness in compact star interiors is a multifaceted one [1] and has been discussed controversially ever since the first discussion of hyperons [2] and kaon condensates [3,4,5] for the degenerate matter phases of neutron star cores. Present-day state-of-the-art calculations of hyperonic matter within the Brueckner-Bethe-Goldstone scheme [6] reveal the problem that due to the softening of the high-density equation of state (EoS) the maximum mass of hyperonic stars is below the measured mass of binary radiopulsars M BRP = 1.35 ± 0.04 M ⊙ [7]. As a possible solution to this problem a sufficiently early onset of quark deconfinement in neutron star matter at baryon densities n = 0.3 -0.5 fm -3 has been suggested [8]. In the present contribution we will not touch the question of absolutely stable strange quark matter [9] in compact stars but rather focus the discussion on models for hybrid stars with deconfined quark matter cores and recent observational constraints for their masses (M) and radii (R), see [10,11].

The status of the theoretical approach to the neutron star matter equation of state is very different from that for the high-temperature case at low or vanishing net baryon densities, where ab initio lattice QCD simulations provide EoS with almost physical quark masses systematically approaching the continuum limit [12]. This guidance is absent at zero temperature and high baryon number densities, where a variety of models exists on different levels of the microphysical description which make different predictions for the state of matter. A common feature is that the transition from hadronic to quark matter cannot yet be described on a unique footing where hadrons would appear as bound states (clusters) of quarks and their possible dissociation at high densities as a kind of Mott transition [13] like in nonideal plasmas [14] or in nuclear matter [15,16]. Early nonrelativistic potential model approaches [17,18] are insufficient since they cannot accomodate the chiral symmetry restoration transition in a proper way. Therefore, at present the discussion is restricted to so-called two-phase aproaches where the hadronic and the quark matter EoS are modeled separately followed by a subsequent phase transition construction to obtain a hybrid EoS.

Widely employed for a description of quark matter in compact stars are thermodynamical bag models of three-flavor quark matter with eventually even densitydependent bag pressure B(n), as in Ref. [8]. A qualitatively new feature of the phase structure appears in chiral quark models of the Nambu-Jona-Lasinio type which describe the dynamical chiral symmetry breaking of the QCD vacuum and its partial restoration in hot and dense matter, see Ref. [19] for a review. In contrast to bag models, in these approaches at low temperatures the light and strange quark degrees of freedom may appear sequentially with increasing density [20,21,22], so that strangeness may even not appear in the quark matter cores of hybrid stars, before their maximum mass is reached. Once chiral symmetry is restored, a rich spectrum of color superconducting quark matter phases may be realized in dense quark matter, depending on it's flavor composition and isospin asymmetry [23] with far-reaching consequences for hybrid star phenomenology, e.g., M -R relationships and cooling behavior.

We will consider here a color superconducting three-flavor NJL model with selfconsistently determined density dependences of quark masses and scalar diquark gaps, developed in Refs. [24] and [25] which differ in the fact that the former includes the flavor-mixing Kobayashi-Maskawa-'t Hooft (KMT) determinant interaction [26,27] while the latter does not. Only recently it became clear [28] that this flavor mixing is crucial for the possible stability of strange quark matter phases in hybrid stars.

Role of the Kobayashi-Maskawa-'t Hooft (KMT) interaction

Our description of quark matter is based on the grand canonical thermodynamic potential [19,24,25,29,30]

Ω(T, {µ}) = φ 2 u + φ 2 d + φ 2 s 8G S + Kφ u φ d φ s 16G 3 S - ω 2 u + ω 2 d + ω 2 s 8G V + ∆ 2 ud + ∆ 2 us + ∆ 2 ds 4G D - d 3 p (2π) 3 18 n=1 E n + 2T ln 1 + e -En/T + Ω lep -Ω 0 , (1)
where

E n = E n (p, µ; µ Q , µ 3 , µ 8 , φ u , φ d , φ s , ω u , ω d , ω s , ∆ ud , ∆ us , ∆ ds
) are the quasiparticle dispersion relations, obtained by numerical diagonalization of the quark propagator matrix in color-, flavor-, Dirac-and Nambu-Gorkov spaces. The values of the meanfields (order parameters) are obtained from a minimization of Ω(T, {µ}), which is equivalent to the selfconsistent solution of the set of corresponding gap equations. Ω lep is the contribution from leptons (e.g., electrons, muons and the corresponding neutrino flavors) and the subtraction of Ω 0 garantees that in the vacuum Ω(0, 0) = 0. Eq. ( 1) is obtained in the meanfield approximation to the path integral representation of the partition function for the three-flavor NJL model Lagrangian with the interaction channels

L qq = G S 8 a=0 (qτ a q) 2 + (qiγ 5 τ a q) 2 + G V (qiγ 0 q) 2 -K [det f (q(1 + γ 5 )q) + det f (q(1 -γ 5 )q)] , (2)
L qq = G D a,b=2,5,7 (qiγ 5 τ a λ b C qT )(q T Ciγ 5 τ a λ b q), (3)
where τ a and λ b are the antisymmetric Gell-Mann matrices acting in flavor and color space, respectively. The scalar (G S ), diquark (G D ), vector (G V ) and KMT (K) couplings are to be determined by hadron phenomenology, see [31].

As pointed out in [19], due to the mixing of the light and strange flavor sectors by the KMT interaction, the difference in the critical chemical potentials for the chiral phase transitions in these sectors (which coincide with the onset of 2SC and CFL phases, respectively) gets diminished. This entails that the phase transition between hadronic matter (described by a realistic nuclear EoS, e.g., the DBHF one, see [30]) and superconducting quark matter may eventually proceed directly into the CFL phase, provided the diquark coupling is sufficiently strong, see the left panel of Fig. 1. For consistency with nuclear matter phenomenology, one may check that the corresponding isospin symmetric EoS would not predict a too low transition density and is not in Phase transitions from two-flavor (2SC) to three-flavor (CFL) quark matter are marked by full dots whereas phase transitions from hadronic to quark matter are indicated by open dots. Right: Hybrid equations of state for symmetric matter compared to the flow constraint [32] from heavy-ion collisions. Except η D = 1.35, for which the onset of the 2SC phase has a too low critical density, the parametriztions fulfil the flow constraint, see also [10,11,30]. The 2SC-CFL transition lies outside the tested density region. contradiction with the flow constraint from heavy-ion collisions [32], see the right panel of Fig. 1.

In Fig. 2 we show the M -R and M -n c relationships for nonrotating compact star sequences obtained as solutions of the Tolman-Oppenheimer-Volkoff equations for the hybrid EoS shown in the left panel of Fig. 1 where the diquark coupling η D = G D /G S is a free parameter and the vector coupling is set to η V = G V /G S = 0.1. For sufficiently large η D > 1.2 there is a stable branch of hybrid stars with color superconducting quark matter core. These solutions are more compact and have a higher central density n c than the purely hadronic stars with the same mass. In the parameter range 1.25 < η D < 1.3, mass twin solutions are possible, for which the more compact configuration has a CFL quark matter core. Such solutions have not been possible without the KMT interaction term, when the occurrence of CFL quark matter marks the onset of gravitational instability and thus sets the limit for the maximum mass of the hybrid star [30,[START_REF] Blaschke | Exotic States of Nuclear Matter[END_REF]. The question arises whether other flavor-mixing interactions have to be invoked. It seems necessary in a next step to include channels which appear after Fierz transformation of the KMT interaction and couple chiral condensates with diquark condensates [START_REF] Steiner | [END_REF]. This may lead to a new critical point in the QCD phase diagram [35] and, depending on the sign of the coupling, to a further reduction of the strange chiral condensate which enforces the flavor mixing effect studied here.

The observational constraints for masses and radii are not yet settled. There is a 2 no stable configurations with quark matter exist while for the strong coupling η D = 1.35 there is an early onset of 2SC quark core configurations at M = 0.6 M ⊙ followed by a stable branch of hybrid stars CFL quark matter cores in the mass range 0.9 < M/M ⊙ < 1.46. For comparison recent mass radius constraints are shown coming from RX J1856 [36,11], M13 [37] and ω Cen [38], see also [10].

lower limit for the M -R relation from RXJ 1856.5-3754 [36] which requires either a large radius R > 14 km for a star with M = M BRP or a mass larger than ∼ 2 M ⊙ for a star with 12 km radius. Both is difficult to accomodate with the present EoS. M -R relations for the quiescent binary neutron stars in globular clusters M13 [37] and ω Cen [38] point to rather compact stars as described by those sequences obtained here. The presented microscopic EoS features an intermediate softening of the EoS due to diquark condensation in the density range ∼ 0.4 -0.8 fm -3 and predicts two stable branches in an overlapping range of masses (twins) as a robust feature of compact star M -R relationships. The hypothesis of the existence of a "third family" of supercompact stars is testable in future observational campaigns devoted to determine the masses [39] and M -R relationship for compact stars [40] with high precision and thus to constrain the dense matter EoS.

Conclusions

The effect of the flavor-mixing KMT determinant interaction on the sequential occurrence of superconducting two-and three-flavor quark matter phase transitions in the EoS for cold dense matter has been studied in an NJL-type model. It is found that at sufficiently strong diquark coupling the deconfinement phase transition in neutron star matter leads directly from the hadronic phase to the CFL phase and occurs at low enough density to entail the formation of stable branch of CFL core hybrid stars in the M -R diagram of compact stars. This branch may either continuously join that of normal neutron stars or eventually form a "third family" branch, separated from that of the neutron stars by a sequence of unstable configurations. The possibility of such a characteristic feature of mass twins in the M -R diagram suggested by advanced microscopic QCD motivated hybrid EoS with superconducting dense quark matter phases should be kept in the focus of observational programs to deduce the cold dense EoS exploiting measured M -R relationships for compact stars.

Figure 1 .

 1 Figure1. Left: Equations of state for neutron star matter in beta-equilibrium. The hadronic phase is described by the DBHF EoS (solid line) and the quark matter EoS correspond to the NJL model(1) for different choices of the diquark coupling strength η D = G D /G S for given vector coupling η V = G V /G S = 0.1 and KMT coupling K. Phase transitions from two-flavor (2SC) to three-flavor (CFL) quark matter are marked by full dots whereas phase transitions from hadronic to quark matter are indicated by open dots. Right: Hybrid equations of state for symmetric matter compared to the flow constraint[32] from heavy-ion collisions. Except η D = 1.35, for which the onset of the 2SC phase has a too low critical density, the parametriztions fulfil the flow constraint, see also[10,11,30]. The 2SC-CFL transition lies outside the tested density region.

Figure 2 .

 2 Figure 2. Compact star sequences for different values of the diquark coupling strength η D . For η D = 1.2 no stable configurations with quark matter exist while for the strong coupling η D = 1.35 there is an early onset of 2SC quark core configurations at M = 0.6 M ⊙ followed by a stable branch of hybrid stars CFL quark matter cores in the mass range 0.9 < M/M ⊙ < 1.46. For comparison recent mass radius constraints are shown coming from RX J1856[36,11], M13[37] and ω Cen[38], see also[10].

Acknowledgments

DB and RL acknowledge the hospitality of the Yukawa Institute for Theoretical Physics Kyoto, partial support by the Yukawa International Program for Quark-Hadron Sciences and discussions during the Workshop program "New Frontiers in QCD" where this work was completed. This work has been supported in part by the Polish Ministry of Science and Higher Education (MNiSW) under grant No. N N 202 2318 37 (DB, TK, RL), by the Russian Fund for Basic research (RFBR) under grant No. 08-02-01003-a (DB), by FNRS, the Belgian fund for scientific research (FS) and by CompStar, a research networking programme of the European Science Foundation.


Irina Sagert
email: sagert@th.physik.uni-frankfurt.de

T Fischer

M Hempel

G Pagliara

J Schaffner-Bielich

F.-K Thielemann

M Liebendörfer

Strange quark matter in explosive astrophysical systems

Explosive astrophysical systems, such as supernovae or compact star binary mergers, provide conditions where strange quark matter can appear. The high degree of isospin asymmetry and temperatures of several MeV in such systems may cause a transition to the quark phase already around saturation density. Observable signals from the appearance of quark matter can be predicted and studied in astrophysical simulations. As input in such simulations, an equation of state with an integrated quark matter phase transition for a large temperature, density and proton fraction range is required. Additionally, restrictions from heavy ion data and pulsar observation must be considered. In this work we present such an approach. We implement a quark matter phase transition in a hadronic equation of state widely used for astrophysical simulations and discuss its compatibility with heavy ion collisions and pulsar data. Furthermore, we review the recently studied implications of the QCD phase transition during the early post-bounce evolution of core-collapse supernovae and introduce the effects from strong interactions to increase the maximum mass of hybrid stars. In the MIT bag model, together with the strange quark mass and the bag constant, the strong coupling constant α s provides a parameter to set the beginning and extension of the quark phase and with this the mass and radius of hybrid stars.

Figure 1. The phase diagram for the quark matter phase transition for SN matter (solid lines) and HI collisions (dashed lines).

Thin lines denote the beginning of the mixed phase, thick lines mark the onset of the pure quark phase.

Introduction:

The future FAIR facility at GSI, Darmstadt, will explore the equation of state (EoS) of strongly interacting matter for intermediate temperatures T and high baryon densities n b around isospin symmetry, that is for proton fractions Y p ∼ 0.5. Supernovae (SNe) and binary mergers hold environments with similar conditions for T and n b but with Y p ≤ 0.3. As will be discussed in the scope of this article, core-collapse SNe with matter at a low value of Y p and dynamical timescales in the range of ms, provide conditions suitable for a phase transition to strange quark matter. Such a scenario was recently studied in [1] applying the MIT bag approach for the EoS of quark matter and using low critical densities for its onset. Simulations with different progenitor models and two different bag constants led to SN explosions accompanied by a significant neutrino burst which can be observed by present and future neutrino detectors [START_REF] Dasgupta | Detecting the QCD phase transition in the next Galactic supernova neutrino burst[END_REF]. In the following, we will introduce in more detail the hybrid EoS used in the above work and analyze its influence on the dynamics of the PNS evolution. We will discuss the compatibility with heavy ion (HI) data and pulsar observations. Furthermore we will include first order corrections from the strong interaction constant α s and study its influence on the maximum mass of the cold hybrid star configurations.

Initial setup for the equation of state

For the hadronic part of the quark-hadron EoS we use the relativistic mean field approach by [START_REF] Shen | [END_REF], while quark matter is described by the MIT bag model. Due to their small current mass, the up and down quarks are treated as massless, while for the strange quark we chose m s = 100MeV which is well within the limits set by the Particle Data Group [4]. If quark masses are fixed and no corrections from the strong coupling constant α s are included, the critical densities for the phase transition are directly given by the bag parameter B for which we applied two values, B 1/4 = 165MeV and B 1/4 = 162MeV. For the construction of the phase transition we choose the Gibbs approach, where a mixed phase of quarks and hadrons is present [5,6]. Figure 1 shows two phase diagrams with the pure hadronic and quark phases and the mixed phase for B 1/4 = 165MeV. As can be seen, the onset of strange quark matter (denoted as uds) for a proton fraction of Y p = 0.3 happens already around saturation density n 0 . For B 1/4 = 162MeV the critical densities are even smaller. However, these values do not contradict with results from HI collisions due to two main reasons. First, supernova dynamics happen on timescales of ms, whereas weak processes operate within 10 -6 -10 -8 s and have therefore enough time to produce strangeness. Consequently, phase transitions can be considered from hadronic to three flavor quark matter in weak equilibrium [START_REF] Mintz | Nucleation of quark matter in protoneutron star matter[END_REF]. For HI collisions, dynamical timescales are much shorter, of the order of 10 -23 s and therefore not long enough for strangeness to be produced and equilibrated by weak interactions. Consequently, for such systems, it seems to be more appropriate to consider a phase transition from hadronic to quark matter composed only of up and down quarks. The higher the number of quark flavours and therefore the number of degrees of freedom, the softer is the EoS in the mixed phase and the lower is the critical density for its onset. The second main difference for SN and HI environments is the proton fraction in the two systems, being Y p ≤ 0.3 for the first and Y p ∼ 0.5 in the second case. Due to the symmetry energy of hadronic matter, its isospin symmetric state is energetically favored. For a proton fraction Y p < 0.5, the energy of hadronic matter is higher and the additional asymmetry pressure stiffens the EoS. This stiffness results in an earlier onset of the mixed phase with its softer EoS. Consequently, a low value of Y p leads to smaller critical densities than for isospin symmetric matter. Figure 1 shows two phase diagrams, for SN environments and HI collisions, for B 1/4 = 165MeV, illustrating that a low onset of quark matter in SN environments is compatible with a high critical density in HI collisions, which, for the chosen B and small T , is up to 5n 0 . However, the exact location of the critical density for different T varies in dependence of the models for the quark and hadron EoSs, or the inclusion of finite size effects. The softening in the mixed phase, as seen in figure 2(a), is caused by the growing quark fraction χ. Figure 2(b) shows the fractions of positive charge Y C in quark and hadronic matter in the pure and mixed phases. In the quark phase Y C is given by (2/3 n u -1/3 n d -1/3 n s ) /n b , whereas n u , n d and n s are the up, down and strange quark number densities. For the hadronic phase, the charge fraction corresponds to the proton fraction Y p . As shown in figure 2(b),Y C in the quark phase can be very low and even negative. Therefore, with increasing χ, the charge fraction in the quark phase can compensate Y C = Y p of hadronic matter, and the latter can consequently approach isospin symmetry towards the end of the mixed phase. At this point, due to the soft EoS in the isospin symmetric hadronic phase and the large number of degrees of freedom, the mixed phase EoS is very soft. However, the vanishing of hadronic degrees of freedom causes a significant stiffening when the pure quark phase sets in. Nevertheless, in the simple MIT bag model, the EoS of the pure quark phase is still much softer than the one for hadronic matter giving low maximum masses for hybrid stars. Up to now, the highest precisely measured mass is the one for the Hulse-Taylor pulsar with 1.4414 ± 0.0002 solar masses M . A new candidate might be the recently studied J1903+0327, a millisecond pulsar with a main sequence star companion. Due to the large eccentricity of the binary system, the advance of periastron can be measured giving a value of 1.67±0.01M for the mass of J1903+0327 [START_REF] Freire | Eccentric Binary Millisecond Pulsars[END_REF]. The mass-radius relations in figure 3(b) show that while hybrid stars for B 1/4 = 165MeV and B 1/4 = 162MeV are above the Hulse-Taylor constraint of 1.44M , their maximum masses are smaller than 1.67±0.01M . A possibility to increase the maximum masses of hybrid stars within the MIT bag model is the inclusion of first order corrections from the strong interaction constant α s [10]. Figure 3(a) shows on the example of B 1/4 = 165MeV how the inclusion of α s corrections shifts the critical density to higher values, increasing the pressure in the mixed and the pure quark phases. A low density for the onset of quark matter can be obtained again by decreasing the value for B. However, the new parameter set for B and α s leads to a stiffer EoS in the mixed and quark phases which results in higher maximum masses of hybrid stars. This is shown in figure 3(b) where the parameter set of B 1/4 = 155MeV with α s = 0.3 leads to a hybrid star maximum mass of ∼ 1.67M and, at the same time, has a similar critical density for the onset of quark matter as B 1/4 = 165MeV with α s = 0.

Quark matter in supernovae

We performed core-collapse simulations of low and intermediate mass Fe-core progenitors in spherical symmetry with two different bag constants B 1/4 = 165MeV and B 1/4 = 162MeV for the quark matter EoS. Our numerical model is based on general relativistic radiation hydrodynamics and three flavor Boltzmann neutrino transport (for details see [11] and references therein). The conditions for the appearance of quark matter, i.e. the beginning of the mixed phase, are already obtained at the Fe-core bounce at central densities close to and slightly above n 0 . However, the produced small quark fraction initially does not influence the SN dynamics and the evolution proceeds like in a normal core collapse supernova. A hydrodynamic shock wave forms, travels outwards and looses energy due to the disintegration of infalling heavy nuclei and production of neutrinos. The latter become observable in the neutrino spectra as a neutrino burst dominated by electron neutrinos as the shock wave propagates across the neutrinospheres (i.e. the neutrino energy and flavor dependent spheres of last scattering). These energy losses turn the expanding shock quickly into a standing accretion shock already ∼ 5ms after bounce. As discussed previously, the reason for the unchanged dynamics are the relative stiffness and similarity of the mixed phase EoS to the hadronic one at small quark fractions and Y p close to isospin symmetry. However, as matter continues to be accreted on the surface of the PNS, the density and temperature in its interior increases and a growing volume enters the mixed phase. The quark matter fraction in the mixed phase rises and the PNS interior moves up to softer regions of the EoS, where it becomes gravitational unstable. A contraction proceeds into a collapse till pure quark matter is reached and the EoS stiffens again due to the disappearance of the additional hadronic degrees of freedom. The collapse halts and a second shock front forms at the phase boundary between the mixed and hadronic phases. This second shock front moves outwards and turns into a shock wave when it reaches the PNS surface where the density drops over several orders of magnitude. Hereby, the decrease in density accelerates the shock wave to velocities of the order of the vacuum speed of light. Shock heating of infalling hadronic matter leads to a lift of degeneracy and an increase in its proton fraction accompanied by the production of anti-neutrinos. As soon as the shock wave propagates over the neutrinospheres, a second neutrino burst, dominated by antineutrinos is released. The delay of this second burst after the first deleptonization burst contains correlated information about the progenitor model, the hadronic and quark EoSs, and the quark-hadron phase transition. For more details, see [1].

Conclusions

Because of the different proton fractions of matter in terrestrial and astrophysical laboratories, such as the future FAIR facility at GSI and supernovae or compact star mergers, on the one hand and their similarities in T and n b on the other, the study of heavy ion collisions and explosive astrophysical scenarios can complement each other in probing the phase diagram of strongly interacting matter, also in regard to the phase transition from hadronic to quark matter. However, the study of possible observable signals and impacts of quark matter in astrophysical systems requires hydrodynamical simulations with an input of an appropriate quark-hadron equation of state. In this article we present such an approach where a quark matter phase transition has been implemented in a hadronic equation of state for a large range of temperatures, proton fractions and densities. Applying the latter to simulations of core-collapse supernovae, we find that a quark matter phase transition can cause the formation of a second shock wave which leads to the explosion of the star, accompanied by a second neutrino burst. The latter is dominated by anti-neutrinos, which can be observed by future and present neutrino detectors. If found, the second neutrino peak can give correlated information about the progenitor mass and the critical density for the onset of quark matter.

Figure 2 .

 2 Figure 2. (a) Hybrid EoS for B 1/4 = 165MeV and α s = 0, together with the EoSs for the pure hadronic and quark phases at Y p = 0.3 and T = 15MeV; (b) Charge fractions in quark and hadronic matter in the mixed phase for a total Y C = 0.3 and temperature T = 15MeV using B 1/4 = 165MeV and α s = 0.

Figure 3 .

 3 Figure 3. (a) The inclusion of first order corrections from the strong interaction constant α s for quarks in the MIT bag model leads to an increase in the critical density. This can be compensated by reducing the value of B. The reduction in B and inclusion of α s results in a higher pressure in the mixed and quark phases and therefore a higher hybrid star maximum mass (b).

Acknowledgement The project was funded by the Swiss National Science Foundation grant. no. PP00P2-124879/1 and 200020-122287 and the Helmholtz Research School for Quark Matter Studies, the Italian National Institute for Nuclear Physics, the Graduate Program for Hadron and Ion Research (PG-HIR), the Alliance Program of the Helmholtz Association (HA216/EMMI) and the DFG through the Heidelberg Graduate School of Fundamental Physics. The work of G. P. is supported by the Deutsche Forschungsgemeinschaft (DFG) under Grant No. PA 1780/2-1. The authors are additionally supported by CompStar, a research networking program of the European Science Foundation, and the Scopes project funded by the Swiss National Science Foundation grant. no. IB7320-110996/1.


D Meloni

S Morisi
email: morisi@ific.uv.es

E Peinado
email: epeinado@ific.uv.es

Fritzsch neutrino mass matrix from S 3 symmetry

whether they are published or not. The documents may come

Introduction

Although there is a robust evidence that neutrinos are mixed, many aspects of the neutrino physics are not clearly understood yet. Among them, the comprehension of the values of the masses and mixing and the differences with respect to the quark sector are an open problem whose solution seems to be quite far from being found. Recent data from neutrino oscillations produced the following results: 0.36 ≤ sin 2 θ 23 ≤ 0.67 0.27 ≤ sin 2 θ 12 ≤ 0.38 sin 2 θ 13 < 0.053 ,

and 2.07 × 10 -3 eV 2 ≤ ∆m 2 atm ≤ 2.75 × 10 -3 eV 2 , 7.03 × 10 -5 eV 2 ≤ ∆m 2 sol ≤ 8.27 × 10 -5 eV 2 ,

(2) at 99.73% confidence level [1] (see [2] for other recent interpretations of the neutrino data).

We have only hints coming from cosmological observations that the absolute values of the neutrino masses should be less than 1 eV [3]. In the quark sector the situation is quite different: not only the masses and the hierarchy in the up and down sectors are better known but also the mixing angles are well measured and strongly differ from the neutrino ones. A successful ansatz to reproduce these features in the quark sector is the Fritzsch-like texture [4], where both the up and down quark mass matrices have a simple form

M =    0 A 0 A * C B 0 B * D    . (3)
Such a matrix (already described in, e.g., [5]) gives the well know relation

tan θ 12 = m 1 m 2 , (4)
which predicts the Cabibbo angle whose small value is a consequence of the strong hierarchy in the masses. A texture as in eq.( 3) can also be employed for the Majorana neutrino mass matrix; this is a particular case of the class of two-zero texture [6] which, together with the two relations ∆m atm = m 2 3 -m 2 1 and ∆m sol = m 2 2 -m 2 1 , fix the absolute neutrino mass scale as suggested in [7]. Unlike the quark sector, the solar and atmospheric angles can be large due to the fact that in the neutrino sector the hierarchy is not so strong.

Although a vast class of Fritzsch-like textures (and their phenomenological consequences) has been already studied in the literature, in this paper we propose a leptonic model based on the permutation symmetry S 3 which naturally gives rise to a Fritzsch-type neutrino Majorana mass matrix (and, in addition, to a nearly diagonal charged leptons). At tree level, the tau lepton acquires a mass via the spontaneous electroweak symmetry breaking (ESB) driven by one S 3 doublet and two S 3 singlets, whereas the electron and the muon remain massless. Higher order operators, mediated by just one Standard Model (and S 3 ) scalar singlet (called the flavon) are responsible for m e , m µ = 0. In the neutrino sector, the Majorana mass matrix is generated by dimension five [8] and six operators.

The paper is organized as follows: in the next section we introduce the model; the scalar potential is studied in Sec.3; the lepton and neutrino mass matrices are introduced in Secs.4 and 5, respectively whereas their phenomenological consequences are discussed in Sec.6. Sec.7 is devoted to our conclusions.

The model

We propose a model based on S 3 , the group of permutations of three objects, which is the smallest non-Abelian discrete group. S 3 contains one doublet irreducible representation and two singlets. This feature is useful to separate the third family of fermions from the other two and has been already used for model building [9]. For pioneers papers see [10] (and also references in [11]).

The group S 3 has two generators S and T satisfying the following relations:

S 2 = T 3 = (ST ) 2 = 1 .
(

) 5
One possible realization is the so-called "T-diagonal" basis where

S = 0 1 1 0 T = ω 0 0 ω 2 , (6)
with ω = e i2/3π . The tensor products involving pseudo-singlets are given by 1 × 1 = 1 and 1 × 2 = 2 while the product of two doublets is 2 × 2 = 2 + 1 + 1 which, in terms of the components of the two doublets A = (a 1 , a 2 ) T and B = (b 1 , b 2 ) T in the T-diagonal basis, are as follows:

a 1 b 2 + a 2 b 1 ∈ 1 a 1 b 2 -a 2 b 1 ∈ 1 a 2 b 2 a 1 b 1 ∈ 2 . ( 7
)
The product 2 * × 2 is similar to 2 × 2 with the exchange of a 1 ↔ a 2 .

Construction of the model

The Higgs sector is extended from one SU(2) L -doublet to two SU(2) L -doublets, H D = (H 1 , H 2 ) belonging to a doublet irreducible representation of S 3 and other two SU(2) L doublets, H S and H S , belonging to singlet representations of S 3 . We also introduce an electroweak scalar singlet χ which turns out to be relevant to give a non-vanishing electron and muon masses. In order to have nearly diagonal charged lepton mass matrix we assume two further parity symmetries, so that the global discrete symmetry group of the model is

G = S 3 ⊗ Z 5 ⊗ Z 2 .
The matter assignment under G ⊗ SM is summarized in Tab.1.

fields

L D = L 1,2 L 3 l R D = l R 1,2 l R 3 H D H S H S χ SU L (2) 2 2 1 1 2 2 2 1 Y -1 -1 -2 -2 1 1 1 0 S 3 2 1 2 1 2 1 1 1 Z 2 + - + - + + -+ Z 5 ω 2 ω ω ω 2 ω 3 ω 4 ω 4 ω 2
Table 1: Matter assignment of the model. ω is the Z 5 charge ω = e i2/5π and Y is the SM hypercharge in the convention Y = 2(Q -T 3 ), where T 3 is the third component of the SM SU( 2) doublets.

The scalar potential

The most general Higgs potential invariant under G × SM is as follows:

V = µ 1 H † S H S + µ 2 (H † D H D ) 1 + µ 3 H † S H S + µ 4 |χ| 2 + λ 1 |χ| 4 +(λ 2 H † D H D + λ 3 H † S H S + λ 4 H † S H S )|χ| 2 + λ 5 [(H † D H D )] 2 + λ 6 [(H † D H D ) 1 ] 2 +λ 7 [(H † D H D ) 2 ] 2 + λ 7 (H † D H † D ) 1 (H D H D ) 1 + λ 8 (H † S H S ) 2 +λ 9 (H † D H D ) 1 H † S H S + λ 9 (H † D H S ) 2 (H † S H D ) 2 + λ 9 ((H † D H S ) 2 2 + h.c.)+ +λ 10 (H † D H D ) 1 H † S H S + λ 10 (H † D H S ) 2 (H † S H D ) 2 + λ 10 ((H † D H S ) 2 2 + h.c.)+ +λ 11 (H † D H † D ) 2 (H D H D ) 2 + λ 12 (H † S H S ) 2 + +λ 13 H † S H S H † S H S + λ 13 (H † S H † S H S H S + h.c.) + λ 13 H † S H S H † S H S (8)
where we used the subscripts 1, 1 and 2 to refer to the S 3 contractions when necessary and, for any Higgs fields, H = -iτ T 2 H * . In the case of real vev's, that is

H D = (v 1 , v 2 ), H S = v S , H S = v S , χ = v χ , (9)
the potential can be written as4

V = (λ 11 + λ 5 + λ 6 )(v 4 1 + v 4 2 ) + v 4 S λ 8 + v 4 S λ 12 + v 2 S v 2 S λ 13 + v 2 S µ 1 + v 2 S µ 3 + + (µ 4 + v 2 S λ 3 + v 2 S λ 4 ) χ 2 + λ 1 χ 4 + (v 2 S λ 10 + v 2 S λ 9 + µ 2 + λ 2 χ 2 ) (v 2 2 + v 2 1 )+ + (2v 2 2 v 2 1 (λ 5 -λ 6 + λ 7 )) . (10)
The minima of V are found solving the minimizing equations:

∂V ∂v 1 = 2v 1 λ 2 v 2 χ + λ 10 v 2 S + λ 9 v 2 S + 2v 2 1 (λ 11 + λ 5 + λ 6 ) + 2v 2 2 (λ 5 -λ 6 + λ 7 ) + µ 2 = 0, ∂V ∂v 2 = 2v 2 λ 2 v 2 χ + λ 10 v 2 S + λ 9 v 2 S + 2v 2 2 (λ 11 + λ 5 + λ 6 ) + 2v 2 1 (λ 5 -λ 6 + λ 7 ) + µ 2 = 0, ∂V ∂v S = 2v S λ 3 v 2 χ + λ 10 (v 2 1 + v 2 2 ) + 2v 2 S λ 8 + v 2 S λ 13 + µ 3 = 0, ∂V ∂v s = 2v s λ 4 v 2 χ + λ 9 (v 2 1 + v 2 2 ) + 2v 2 S λ 12 + v 2 S λ 13 + µ 1 = 0, ∂V ∂vχ = 2v χ µ 4 + 2λ 1 v 2 χ + λ 2 (v 2 1 + v 2 2 ) + λ 3 v 2 S + λ 4 v 2 S = 0 . ( 11
)
The second equation is satisfied for v 2 = 0. From the remaining equations we can easily get the vevs of the other scalars in terms of the couplings of the Higgs potential; in particular, a solution with v 1 = 0 can be found and the vev alignment of the S 3 Higgs doublet assumes the structure:

H D = (v, 0) . ( 12
)
For this vev configuration, it is possible to find a huge region of the Higgs parameter space where the eigenvalues of the Hessian of the potential are all positive and therefore where the Higgs potential has a local minimum. Note that a solution of the form H D = (0, v) is physically equivalent to eq.( 12), producing the same phenomenology in the charged lepton and neutrino sectors. In fact it corresponds to the exchange of L 1 with L 2 . We also verified numerically that, in the large parameter space where eq.( 12) is a minimum, other solutions like H D = v (1, 1) do not produce positive definite Hessian. The mass spectra of the Higgs particles will be discussed elsewhere.

Leptons

The most general Lagrangian invariant under G × SM is given by:

L = y 1 Λ L D H D l R D χ * + y 2 Λ L D H S l R D χ + y 3 L 3 H S l R 3 , ( 13
)
where Λ is the cut-off scale. Higher order terms only appear at O(1/Λ 2 ) and will be considered negligible for our discussion. From eq.( 12) the charged lepton mass matrix is:

M l =    y 2 Λ v S v χ 0 0 y 1 Λ vv χ y 2 Λ v S v χ 0 0 0 y 3 v S    . ( 14
)
When v χ is equal to zero only the τ lepton is massive. The electron and muon masses are generated by the vev of the scalar χ and are then suppressed by the large scale Λ. The matrix M l M † l has three distinct eigenvalues that can be identified with the squared charged fermion masses as:

m 2 e = ε 2 2 v 2 y 2 1 + 2v 2 S y 2 2 -vy 1 v 2 y 2 1 + 4v 2 S y 2 2 m 2 µ = ε 2 2 v 2 y 2 1 + 2v 2 S y 2 2 + vy 1 v 2 y 2 1 + 4v 2 S y 2 2 ( 15
)
m 2 τ = v 2 S y 2 3
where we introduced the short-hand notation ε = v χ /Λ. We see that for ε 1, the hierarchy among the τ and the lightest charged leptons is easily reproduced although the latter, in absence of any fine-tuning among the Yukawas and/or the Higgs vevs, are expected to be of the same order of magnitude. We address this question in the next section. The mass matrix for the charged leptons can be written in terms of the physical lepton masses as:

M l =    √ m e m µ 0 0 -m µ (1 -me mµ ) √ m e m µ 0 0 0 m τ    , (16)
and the squared matrix M l M † l is then diagonalized by:

U L =      1 q 1+ me mµ -me mµ 1 q 1+ me mµ 0 me mµ 1 q 1+ me mµ 1 q 1+ me mµ 0 0 0 1      ∼    1 -0.07 0 0.07 1 0 0 0 1    . ( 17
)

Neutrino

The neutrino masses are generated by non-renormalizable operators of dimension 5 and 6 invariant under the group G × SM5 :

Λ • L ν = y ν 1 (L D L D ) 1 ( HD HD ) 1 + y ν 2 (L D L D ) 2 ( HD HD ) 2 + y ν 3 (L D HD ) 1 (L D HD ) 1 + y ν 4 (L D HD ) 1 (L D HD ) 1 + y ν 5 (L D HD ) 2 (L D HD ) 2 + y ν 6 (L D L D ) 1 ( HS HS ) 1 χ * /Λ + y ν 6 (L D L D ) 1 ( H S H S ) 1 χ * /Λ + ( 18
)
y ν 7 L 3 L 3 ( HD HD ) 1 χ/Λ + y ν 8 L 3 L 3 HS HS + y ν 8 L 3 L 3 H S H S + y ν 9 (L D HD ) 1 L 3 H S ,
where we assumed that the large energy scale which suppresses these operators is of the same order of the cutoff scale Λ. The only operators of dimension six are those proportional to y ν 6 = y ν 6 + y ν 6 and y ν 7 . From eq.( 19) the neutrino mass matrix is as follows:

M ν =    0 2y ν 6 (v 2 S + v 2 S )v χ /Λ 0 2 y ν 6 (v 2 S + v 2 S )v χ /Λ (y ν 2 + y ν 3 + y ν 4 )v 2 y ν 9 vv S 0 y ν 9 vv S y ν 8 (v 2 S + v 2 S )    ≡    0 b 0 b a c 0 c d    , (19)
where y ν 8 = y ν 8 + y ν 8 . Before discussing the phenomenological consequences of such a matrix, it is useful to get an estimate of the relevant Yukawa parameters and a relation among the vevs v and v S . Comparing eqs.( 14) and ( 16) and using the parameterization in eq.( 19) we get:

√ m e m µ = y 2 v S ε m µ 1 - m e m µ = y 1 v ε m τ = y 3 v S a = y ν v 2 b = 2 y ν 6 (v 2 S + v 2 S ) ε ,
where we assumed that y ν 2 + y ν 3 + y ν 4 = y ν . We assume

v > ∼ v s ∼ O(100) GeV, ε ∼ O(10 -2 ) then if y 1 ∼ O(10 -1 ), y 2 ∼ O(10 -2 ), y 2 ∼ O(10 -2 ),
we have the correct charged lepton mass hyerarchies. We then consider the b/a ratio

b a = 2 y ν 6 ε(v 2 S + v 2 S ) y ν v 2 v S . ( 20
)
We numerically verified that a b ∼ O( 1) so the Yukawa parameters must satisfy

13 (cos θ ν 23 sin θ ν 12 +cos θ ν 12 sin θ ν 13 sin θ ν 23 e iδν ) . ( 23
)
Our model is compatible with the normal mass ordering only because the ratio |µ 2 | 2 /|µ 3 | 2 is always less than 1; expanding it up to second order in sin θ ν 13 we get:

|µ 2 | 2 |µ 3 | 2 = cot 2 θ ν 12 cot 2 θ ν 23 sin 2 θ ν 13 + O(s ν3 13 ) , (24)
and we checked that higher order corrections do not modify our statement. The mass differences are written as:

∆m 2 sol = m 2 1 cos θ ν 23 (cos θ ν 12 cos θ ν 23 csc 2 θ ν 12 -2 cos δ ν cot θ ν 12 sin θ ν 13 sin θ ν 23 ) D ν , (25)
and

∆m 2 atm = m 2 1 -1 + cos 2 θ ν 12 cos 2 θ ν 13 cot 2 θ ν 13 sin 2 θ ν 23 D ν , (26)
where

D ν = cos
) 28
which will be used below to constrain the physical θ 13 and δ.

Phenomenology

To study the phenomenological implication of our model, it is necessary to relate the parameters in eq.( 22) to the physical ones. This can be achieved introducing the rotations from the charged lepton sector described in Sec.4; the resulting mapping is a set of implicit relations that are quite cumbersome and will not be explicitly presented here. We limit ourselves to describe the procedure which allows us to extract the predictions of our model. The lepton mixing is defined by V = U † L U ν and we can write

U ν = U L V , (29)
where V is parametrized in the standard form as in eq. ( 22) replacing s ν ij and c ν ij with the physical (that is measurable) s ij = sin θ ij and c ij = cos θ ij , respectively. Taking the ratio of U ν 23 and U ν 33 from eq. ( 29) we find an expression for s ν 23 in terms of the physical parameters θ 12 , θ 23 , θ 13 and the Dirac phase δ (and the corrections from the charged leptons). In the same way, always using eq. ( 29), we can express s ν 12 and s ν 13 as a function of θ 12 , θ 23 , θ 13 , δ; finally, δ ν is the argument of the element (13) of the matrix U L V . In this way we have all the parameters θ ν 12 , θ ν 13 , θ ν 23 and δ ν as a function of the neutrino mixing angles θ 13 , θ 12 , θ 23 and the phase δ. These relations can be inserted into eq. ( 28) to get an implicit connection among the mixing parameters and α, which is a characteristic of our model. Also the lightest mass eigenstate can be related to the same parameters and ∆m 2 atm using eq. ( 26). In the left panel of Fig. 1 we show the dependence of sin 2 θ 13 as a function of δ taking θ 12 , θ 23 and α inside their experimental ranges. In particular, the solid line represents the 1σ correlation when also the other parameters are left free to vary in their 1σ allowed ranges quoted in [1], whereas the 2σ correlation is represented by the dot-dashed line. Finally, dashed line is the relation obtained when θ 12 , θ 23 and α are fixed to their best fit values. We also included the upper limit on sin 2 θ 13 at 3σ (upper horizontal dashed line) and the best fit value of ref. [1] (lower horizontal dashed line). We can see that, even considering the 2σ uncertainty, the predicted values for sin 2 θ 13 are different from zero so that, to a very good accuracy, our model is compatible with deviation from θ 13 = 0 for any value of the CP violating phase. The precise value of θ 13 , however, relies on the assumed magnitude for δ; in particular, the CP conserving case δ = 0 is the most promising one to allow large θ 13 (even above the current limits) whereas around δ ∼ ±π we get the smaller θ 13 allowed in our model. It is interesting to observe that, in the case of maximal CP violation7 and for the other oscillation parameters to their best fit values, the predicted sin 2 θ 13 is fully compatible with the best fit value obtained in [1], sin 2 θ 13 ∼ 0.01. Notice that, in the case of diagonal charged lepton mass matrix, the pattern of the θ 13 -δ correlation would have been quite similar, as it can be seen investigating the right panel of Fig. 1. The fact that the corrections coming from U L in eq.( 17) are as large as the values of sin 2 θ 13 is responsible for lowering the allowed θ 13 for δ ∼ ±π. For maximal CP violation at the best fit point, the Jarlskog invariant [14] is as follows: J = c 12 s 23 c 2 13 s 12 s 23 s 13 sin δ = 0.023 .

The next observable we want to discuss is the effective mass m ee entering in the neutrinoless double beta decay. In the basis where the charged leptons are diagonal, m ee is nothing but the (11) element of the neutrino mass matrix. According to eq.( 19), this should vanishes as long as the rotation in the charged leptons is proportional to the identity matrix. Since this is not the case, a non-vanishing m ee is generated by the rotation (17) and it is expected to be small because of the smallness of its off-diagonal entries. This is what we can observe in Fig. (2), where we plot the model predictions for m ee as a function of the lightest neutrino mass m 1 . For m 1 below O(10 -2 ) eV we get |m ee | ∼ 10 -3 eV and then outside the range of future experimental sensitivities. We also see that the allowed range for the lightest neutrino mass is around 10 -3 -10 -2 eV ; this is because, as already mentioned in the introduction, the Fritzsch texture gives a correlation between θ ν 128 and the ratio m 1 /m 2 that, together with the two measured square mass differences, fix the absolute neutrino scale in this range. [1]. Right panel: the same as the left panel but assuming exactly diagonal charged lepton mass matrix.

Conclusion

In this paper we have studied a leptonic model based on the discrete S 3 permutation flavor symmetry. We extended the scalar sector of the Standard Model by introducing three more Higgs doublets and one scalar singlet. We have carefully studied the problem of the minimization of the potential and found that a solution of the form (v, 0) for the Higsses in the We also show the allowed regions for the normal hierarchy (gray band). The two dashed horizontal lines represent the experimental sensitivity of some of the forthcoming experiments while the dashed vertical line is the upper limit for the sum of the absolute neutrino masses from cosmological data. For references to experiments see [15][16][17][18][19].

S 3 doublet representation is a viable minimum of the potential. With such a minimum, we obtain a two-zero Fritzsch-texture for the neutrino mass matrix and a nearly diagonal and hierarchical charged lepton mass matrix. As a consequence of the two zeros of the Fritzsch texture, we get a strong correlation between the reactor angle θ 13 and the Dirac CP phase δ. In particular, for δ ∼ ±π/2 we predict sin 2 θ 13 ≈ 0.01, a value which is very close to the best fit value quoted in [1]. Beside the reactor angle, we also investigated the prediction for the effective mass m ee governing the rate of the 0νββ decay, founding m ee ≈ 10 -3 eV , one order of magnitude less than the sensitivities of the future experiments.

Figure 1 :

 1 Figure1: Left panel: correlation among δ and sin 2 θ 13 as obtained in our model. The 1σ result, obtained varying the other oscillation parameters also in their 1σ allowed ranges, is showed with solid lines, whereas the 2σ result is showed with the dot-dashed line. The dashed line is the relation obtained when θ 12 , θ 23 and α are fixed to their best fit values. Horizontal lines represent the upper limits on sin 2 θ 13 (upper dashed line) and the best fit values (lower dashed line) from[1]. Right panel: the same as the left panel but assuming exactly diagonal charged lepton mass matrix.

Figure 2 :

 2 Figure 2: Model predictions for |m ee | as a function of the lightest neutrino mass m 1 .We also show the allowed regions for the normal hierarchy (gray band). The two dashed horizontal lines represent the experimental sensitivity of some of the forthcoming experiments while the dashed vertical line is the upper limit for the sum of the absolute neutrino masses from cosmological data. For references to experiments see[15][16][17][18][19].

Whereλ i = λ i + λ i + λ i for i =9, 10, 13.

Dimension 7 operators can be built, for instance, adding the singlet χ 2 or doublets H † H to the previous d = 5 operators and will then be neglected.

We have used the short-hand notation s ν ij = sin θ ν ij and c ν ij = cos θ ν ij .

Maximal CP violation can be observed in incoming experiments T2K and NOνA, see for instance[12,13].

Note that in the case of diagonal charged leptons the angle θ 12 corresponds exactly to θ ν 12 and therefore from eqs. (4) and (23), the solar mixing angle does not depend on the absolute scale of neutrino mass m 1 , while in our case this relation acquires a small correction proportional to m e /m µ .

8 Acknowledgments Work of SM and EP supported by the EC contract UNILHC PITN-GA-2009-237920, by the Spanish grants FPA2008-00319 and CDS2009-00064 (MICINN) and PROMETEO/2009/091 (Generalitat Valenciana) and by European Commission Contracts MRTN-CT-2004-503369 and ILIAS/N6 RII3-CT-2004-506222. D.M. was supported by the Deutsche Forschungsgemeinschaft, contract WI 2639/2-1. D.M. also acknowledges the AHEP Group of Valencia for their hospitality during the earlier stages of this work.