J Dudek

A G Óź D Ź2

K Mazurek

H Molique
email: herve.molique@ires.in2p3.fr

Mean-field theory of nuclear stability and exotic point-group symmetries

Keywords: numbers: 21.60.-n, 21.10.-k, 21.10.Pc

We formulate the principles of the mean-field theory of nuclear stability employing the point-group, and group-representation theories. The related pointgroup hierarchy of importance in the context of nuclear stability is constructed and discussed. We introduce the notion of the magic-number chains associated with each symmetry -in analogy to the spherical-symmetry nuclear magic-numbers. To prepare the criteria for the experimental search of introduced symmetries we examine the simplified collective rotation-vibration model whose Hamiltonian is invariant under the symmetries in question. We illustrate the construction of the solutions that form at the same time irreducible-representations of the pointgroups in question -in view of formulating the experimental symmetry-criteria through the application of the branching-ratio techniques. Since the criteria may involve very weak transitions whose experimental research may be at the limit of the present-day experiments, the desires may arise, as it was the case in the past, to replace the difficult experiments by an inadequate modelling. In this context we present an alert: The use of oversimplified quantum mechanics exercises in place of experiments and/or microscopic theories is likely to produce meaningless results.

Introduction: Nuclear stability and point-group symmetries

Studying exotic nuclear systems with extreme numbers of neutrons for any fixed proton number represents one of the most important research directions at the large scale nuclear physics facilities world-wide. One of the central issues in this research is the question of stability (thus of life-times) of the investigated nuclear species, the shorter the life-times the bigger the problems with the production and, importantly, detection at a conclusive level of statistics. It therefore becomes clear that realistic modelling of nuclear stability on the theory grounds will play a very important role, in particular, in determining both the strategies as well as the detailed planing of experiments.

The stability of a nuclear system is related to the nucleonic binding, thus to the presence of large gaps in the single-nucleonic spectra: to an approximation, the larger the energy gaps at the Fermi level the stronger the stability. Contemporary effective Hamiltonians such as nuclear mean-field Hamiltonians can generate information of this type in a realistic manner. Since the majority of nuclei are non-spherical, the well established capacity of the mean-field theories to describe the nuclear spontaneous spherical-symmetry breaking, and by the same token, the nuclear deformations, is of considerable advantage. Nuclear shapes are directly related to the spatial properties of the underlying mean-field Hamiltonians, whereas the latter determine the nucleonic energy spectra. This link opens up the possibilities of analyzing the global properties of nucleonic spectra, such as energy-level degeneracies, fluctuations of the level densities etc. through the symmetries of the mean-field Hamiltonians directly. Indeed, through employing powerful group-and group-representation theories that help to predict the most likely presence of gaps in the single particle spectra prior to the detailed calculations we may obtain, as it will be shown, instructive guide-lines for such large scale calculations, cf. section 2, especially table 1 and figure 1.

In this article we formulate the principal lines of the symmetry-based research of stability of exotic nuclear shapes. The term 'exotic' is arbitrarily used to denote any symmetry other than spherical, prolate-, oblate-or triaxial-quadrupole as well as axial-octupole geometries. We present general group-theoretical arguments adapted to the form of the mean-field theory Hamiltonians as well as the collective rotationvibration Hamiltonians. It will be shown that the latter present certain mathematical advantages when formulating the criteria for the experimental identification of the symmetries, but the corresponding signals may be weak and detection difficult.

In this article we discuss a number of Open Problems related to nuclear pointgroup (geometrical) symmetries whose hierarchy is presented in section 2. We have verified on the examples of tetrahedral and octahedral symmetries which, for reasons explained below are on top of the list, that using realistic Hartree-Fock Bogoliubov and phenomenological methods one obtains the high-symmetry solutions for nuclei in numerous areas of the Periodic Table. We believe this is not so much the question of 'Whether?' but 'Where in the Nuclear Chart exotic symmetries can be found?' A number of Open Problems relates to 'What to look for?' In section 3 we present group-theory considerations that may help in the dedicated experimental studies.

Finally, we formulate a few brief comments on the in-adapted approaches to look for the symmetries in nuclear systems which, very unfortunately, are applied by certain authors leading to totally wrong conclusions based on possibly technically correct an experimental input. Interested reader is referred to the article focussing on the complete description of the problem and the underlying techniques in [1].

Symmetry-based mean-field theory of nuclear stability

Let us begin by a few facts from the group-representation theory relating the symmetry of the Hamiltonian, H, and the degeneracy of levels. Let group G = {ĝ 1 , ĝ2 , . . . ĝf } be a symmetry group of H, what implies that a representation D(ĝ) exists such that [D(ĝ), H] = 0, ∀ĝ ∈ G. Suppose that D 1 , D 2 , . . . D r are the irreducible representations of G with dimensions d 1 , d 2 , . . . d r , respectively. Then the spectrum of H splits into multiplets with degeneracies either equal d 1 , or d 2 or ... d r . The spherical symmetry provides a particular example: the irreducible representations are characterised by the single-particle angular momentum quantum numbers j, the implied Hamiltonian block-sub-matrices have the dimensions (2j + 1) × (2j + 1) and the degeneracies are equal to d j = (2j + 1). We will argue that the irreducible representations of the symmetry groups influence spectral properties through two characteristics: both the dimensions {d k : k = 1, . . . r} and the total number of irreducible representations, r. Point groups that express geometrical symmetries in quantum mechanics are classified as 'simple' or 'double', the latter applicable to the Fermion spaces and thus to the mean-field Hamiltonians for nucleons. The names of the double groups are marked with the superscript D. The point-groups of interest in this article are listed in table 1 according to decreasing dimensions of their irreducible representations -as well as the decreasing numbers of irreducible representations. These two characteristics of various groups are compared with the ones of D D 2h , the symmetry of the so-called 'triaxial' nuclei, usually described with the help of quadrupole elongation (β) and triaxiality (γ) deformations. It is suggested in sections 2.1 and 2.2 that the bigger the numbers of irreducible representations and larger their dimensions per given symmetry group, the larger the chances for strong inter-spacing of levels.

Symmetry groups and the effect of a large number of irreducible representations

To our knowledge, there exist no rigorous mathematical argument connecting e.g. the gaps or the maximum-gaps in the energy spectra with the characteristics of the irreducible representations of symmetry groups. We have performed numerous tests using the realistic mean-field models to convince ourselves that the following properties are approximately valid (so far O D h , T D d and C D 4h groups were tested), cf. figure 1. To introduce a reference mechanism let us consider a symmetry group possessing only one two-dimensional or two-dimensional equivalent ‡ irreducible representation leading to Kramers degeneracy. Hamiltonians with the implied symmetry generate two matrix-blocks, [H] and [H * ], and it follows from the hermiticity that the corresponding spectra are identical. With increasing deformation which preserves that symmetry (the nuclear shapes resembling less and less a sphere) the levels have a tendency to fill in the energy window V o tending towards a uniform distribution (figure 1, right).

V o

Energy Eigenvalues

For symmetry groups with r irreducible representations the mechanism of filling in the energy window V o repeats itself within each of the r blocks. Level-density fluctuations leading to relatively irregular spacings are stronger since there are much fewer levels per each block in this case (figure 1 left). The mechanism presented here qualitatively leads to strong non-spherical symmetry "magic" gaps in the single particle spectra with the accompanying strong shell effects that lead to prediction of possible shape-coexisting minima in various areas of the Periodic Table, cf. [6].

Symmetry groups and the effect of dimensions of irreducible representations

Dimensions of the irreducible representations are equal to the degeneracies of related levels, as e.g. (2j + 1)-degeneracy in spherical nuclei. Suppose there are N b bound states in a deformed potential of depth V o . Formally, we can place 2 • N b particles in the potential and the average level spacing d ∼ V o /(N b ). The 2 • N b particles can be put much more 'economically' if several levels carry degeneracies higher than 2 -this is because the same number of particles can be placed on the number of levels N deg which is smaller than N b , so that the average level spacing d deg ∼ V o /(N deg ) will be larger (or considerably larger) than the one in the case of no extra degeneracies.

There are only a few double point groups that have dimensions of their irreducible representations larger than 2. These are first of all the octahedral and tetrahedral groups listed as the first two in table 1. The icosahedral group which belongs to the same category generates regular shapes probably too close to the spherical form to be of interest in the nuclear physics applications; so far, arbitrarily, we have disregarded this group. However, the degeneracy arguments (equivalently: dimensions of irreducible representations) together with the arguments of the large number of irreducible representations place the octahedral and tetrahedral groups on top of the list of interest in the context of strong non-spherical shell-gaps and an implied increase in nuclear stability at the corresponding particle numbers, see below.

An illustration of this qualitative discussion in the context of the realistic nuclear calculations will be given in section 2.4 after having introduced the solution of the non-trivial problem of constructing the mean-field Hamiltonians preserving an a priori given (pre-defined) point-group symmetry.

Construction of the symmetry-preserving mean-field Hamiltonians

Focusing on the mean-field applications, suppose that the considered Hamiltonian depends on an ensemble of parameters {α} specifying nuclear shapes, thus e n = e n (α).

α λµ Y λµ (ϑ, ϕ) , (1)
where function c(α) assures the constant volume condition. Given symmetry group G = {ĝ 1 , ĝ2 , . . . ĝf }; by acting with any of the symmetry operations ĝi on the nuclear surface we must obtain an image coinciding with the original i.e.

λmax λ=2 λ µ=-λ α λµ [ĝ i Y λµ (ϑ, ϕ)] = λmax λ=2 λ µ=-λ α λµ Y λµ (ϑ, ϕ), ∀ ĝi . (2)
Let us introduce the Euler-angle representation of group elements adapted to act on the spherical harmonics: ĝi → Ω i where Ω i ≡ {ω 1 , ω 2 , ω 3 } are the triplets of Euler angles, cf. [START_REF] Cornwell | Group theory in physics[END_REF]; e.g. for a four-fold O z -axis, Ω = {π/2, 0, 0}. One can show, [START_REF] Dudek | [END_REF], that deformation parameters α λµ obeying the symmetry G satisfy in each order λ:

λ µ=-λ (-1) ηi (-1) λ D λ µ µ (Ω i ) -δ µµ α λµ = 0 for i = 1, 2, . . . f, (3)
where η i =0 (or 1) depending on whether the inversion is not (or is) involved. This system of f equations resembles the system of eigenvalue problems with eigenvalues equal to unity. Since equations in (3) are uniform, multiplying their solution ᾱλµ by a constant generates another solution. This property allows to select, e.g. ᾱλµ=0 as an independent parameter and define a 'hyper-curve' ᾱλµ = ᾱλµ (ᾱ λµ=0 ); for higher multipolarites bi-or multi-dimensional hyper-surfaces are obtained provided the solutions of the system in (3) exits at all. In fact, the system in question is over-defined -it contains more equations than unknowns. The fact that solutions (sometimes) exist is a result of the symmetry. Since all the elements of a group can be constructed out of generators, in practice we need to solve only the systems of equations for the generators.

Examples of solutions for the tetrahedral and octahedral symmetries are given by equations ( 4)-( 9) below. For instance, for the tetrahedral symmetry the solutions exist only for odd λ ≥ 3, with the exception of λ = 5; we have [4]:

λ = 3 : α 3,±2 ≡ t 3 ; λ = 5 : no solution exists; (4)
Let us notice a property of a certain mathematical elegance: for the multipolarity orders λ ≤ λ max = 9 studied we have the symmetries generated by one-parameter families of (hyper-)curves in the deformation space and moreover, the multipolarities λ are never mixed what simplifies the technical applications of the formalism.

To our knowledge tetrahedral and octahedral symmetries are the only ones for which the problem of generating mean-field Hamiltonians invariant with respect to a predefined symmetry has been examined and solved; the solutions for most of the other point groups in table 1 belong to the domain of Open Problems, although the simple formalism summarized by equations ( 1)-( 9) paves the way towards this goal.

Microscopic Mean-Fields -Hartree-Fock and Density-Functional Hamiltonians.

All what has been said about the symmetry-respecting combinations of the ᾱλµ coefficients in each given order λ for the Woods-Saxon and Yukawa-folded potentials can be translated into the Hartree-Fock and constrained Hartree-Fock formalisms after replacing ᾱλµ by multipole-moment constraints Qλµ . For instance, for the octahedral symmetry of the rank λ = 4 we would search for solutions for which Q4,±4 ≡ -5/14 Q4 , all other moments being either equal to zero or obeying the relations of the type in ( 8)-( 9) for higher permitted λ-ranks.

The geometrical formalism introduced in the preceding section is extremely powerful in studying the symmetries of the self-consistent solutions as it can be seen Table 2. The Skyrme SIII Hartree-Fock-Boglyubov calculation results for tetrahedral solutions in light Rare-Earth nuclei using the zero-range densitydependent mixed pairing force in the pairing channel (based on [START_REF] Dudek | [END_REF]). Columns 3, 4 and 5 give the multipole moments indicated. Column 6 shows the symmetry test based on relation (7); observe that the high precision equality between the results in columns 5 and 6 is a measure of an (almost) perfect symmetry. Let us emphasize that the unprepared user of the Hartree-Fock formalism would not recognize the presence of the higher-level symmetries without relations (4)-( 9), as illustrated through comparison of columns 5 and 6 in table 2 for the tetrahedral and octahedral symmetries. It is a great advantage of the simple approach presented in section 2.3.1 that the results obtained with the e.g. Hartree Fock (and for that sake any other density-functional) formalism can be easily recognized and studied by using the multipole moments of the self-consistent solutions. The underlying geometrical formalism is in fact a common factor between the phenomenological and the densityfunctional methods. Preliminary studies in this direction have been advanced for the octahedral and tetrahedral groups and the interested reader is referred to the papers quoted above. However no systematic calculations for these two groups and no calculations for all other groups listed in the table 1 exist today and the implied studies belong to the Open Problem series.

Z N Q 32 Q 40 Q 44 -Q 40 × 5 14 (b 3/2 ) (b 2 ) (b 2 ) (b 2 )

Group vs. subgroup relations and symmetry-preserving energy minimisation.

It will be instructive to present the role of the weaker i.e. less symmetric (more precisely: a subgroup) in the pair T D d ⊂ O D h groups. The corresponding geometrical shapes are those of a triangle-base 'pyramid' and a 'diamond' shapes, respectively. And yet, the tetrahedral group is a sub-group of the octahedral one: in other words, the shapes which are octahedral-symmetric are also tetrahedral symmetric and the combination of the two classes of shapes may (and according to our tests does) enhance strong shell effects of the over-all tetrahedral symmetry, thus increasing the chances of the presence of the weaker among the two symmetries in atomic nuclei. 4) and (7). It is usually recognized as not intuitive that so qualitatively different shapes belong to the common, 'weaker', tetrahedral symmetry and when combined may decrease considerably a energy of the tetrahedral-symmetric nucleus.

We find it instructive to compare the geometrical differences between shapes generated by the two discussed symmetry groups and yet, when combined together, leading to the symmetry of the subgroup, cf. figure 2. This simple observation is of a very high importance when minimising the energies with e.g. the very performant phenomenological models -and yet systematic studies of this type are only at the beginning, thus enlarging the list of the related Open Problems..

Exotic symmetry groups and generalized magic gap chains

Let us emphasize at this point that the term magic numbers usually employed in relation to the large gaps at spherical shapes corresponding to proton and/or neutron numbers 8, 20, 28, 50, 82 and 126 should by no means be reserved to the O(3) group. To the contrary, we suggest that any point-group symmetry may generate its own series of magic numbers (we call them generalized magic gap chains) -with an important difference compared to the spherical symmetry case. Whereas a given nucleus cannot be more spherical than the other, it may be more (or less) deformed with the deformation belonging to e.g. tetrahedral-symmetry shapes. In other words, we may introduce the notion of tetrahedral-symmetry magic gaps by examining the single-particle spectra in function of multidimensional paths parametrised with the help of variables t 3 -t 9 of (4-6) and/or o λ of various orders. The maximal gaps in the single-particle spectra do depend, individually, on deformation parameters. The gaps at the particle numbers N i correspond to the sets of symmetry-preserving deformation parameters P i ≡ { t3 , t7 , t9 , . . .} i -the generalized magic gap chains for the octahedral group are then characterized by the sets of numbers {N i , O i }.

More generally, consider an arbitrary symmetry-group G with the symmetryconserving parameters ᾱG λ . By examining the single particle spectra in the spaces of { ᾱG λ } we obtain the magic gaps corresponding to the occupation numbers N G i at deformations P G i ≡ { ᾱG λ } i , so that finally the generalized magic gap chains are given by the ensembles of numbers {N G i , P G i }. Properties of this type have been established for the tetrahedral symmetry group in [START_REF] Dudek | [END_REF]- [8] and references therein, but the systematic analysis for the other symmetry groups remains an Open Problem. Tetrahedral Deformation Figure 3. An example of realistic calculations with the phenomenological Woods-Saxon potential (here: Zirconium region) illustrating the mechanism of creating strong "tetrahedral" gaps in the single-particle spectra that can be comparable in size with the "spherical" gaps. The name "tetrahedral deformation" applies to t 3 of (4). The full lines represent solutions belonging to the four-dimensional irreducible representations while the dashed lines -the two remaining inequivalent two-dimensional representations. We use the Nilsson labelling of the curves, the numbers in parentheses being percentages of the dominating (Nilsson) label.

Figure 3 illustrates the powerful impact of the symmetry-considerations on generating large gaps in the realistic single-nucleon spectra. As it can be seen from the figure the tetrahedral-symmetry "magic" gap at Z=40 is comparable with the Z = 50 spherical gap. Using this type of calculations the chain of the tetrahedral magic numbers has been found with Z, N=32, 40, 56, 70, 90 and N=136. Similar diagrams in which the single-nucleon energies are drawn in a way allowing to distinguish between irreducible representations for various symmetry groups, cf. table 1, do not exist in the nuclear physics literature and represent a series of Open Problems for the future.

There exists yet another difference between the nuclear stability associated with spherical magic-gaps as compared to the generalized magic gaps. Whereas spherical configurations offer the most optimal surface-to Coulomb-energy ratio the nonspherical magic gaps may correspond to the forms whose surface-to Coulomb ratio of energies becomes extremely unfavourable -in other words: the corresponding total nuclear energies may lie very high, or, for the deformations large enough there may be no stable minima in the total energy landscapes corresponding to the magic gaps representing only the mean-field (and not the total nuclear energy) properties. These and related stability questions belong to the realm of Open Problems today.

Properties of rotational bands and criteria of detection of symmetries

We will limit our considerations to the coexisting nuclear states with tetrahedral and quadrupole shapes as examples of 'exotic'-and 'usual' non-spherical symmetries, for which nuclear orientations in space and thus rotational bands can be defined. It has been shown using the methods of the preceding Section that the tetrahedral symmetry can be parametrised, to the lowest order, by α 32 multipole deformation under the condition that all other (static) deformations are zero. Moreover, it can easily be shown that at the static tetrahedral minima Q 2,µ =0, whereas all other multipole components α 3,µ =2 lead to non-zero quadrupole moments. Within a simplified interpretation this implies that the E2 transitions within the bands of exact tetrahedral symmetry should vanish whereas in the case of a partial symmetry breaking they could be considerably smaller than the ones in the usual quadrupole-deformed nuclei §. This implies that the usually neglected mechanisms such as e.g. the zero-point motion around the vanishing equilibrium deformations must not be neglected anymore, e.g. the zero-point motion in other octupole degrees of freedom, α 3,µ =2 , as well as the quadrupole ones thus leading to, possibly less hindered, E2 transitions in the tetrahedral bands, cf. (28).

Let us introduce an intrinsic reference frame Σ rot and consider the Hamiltonian:

Σ rot : H = H vib;λ=2 + H vib;λ=3 + H rot . (10)
Since the spherical tensors α λµ are in general complex, the usual way to proceed is to express the vibrational Hamiltonians using real parameters a λµ and b λµ defined by α λµ ≡ a λµ + i b λµ . Above we assume the simplest decoupled-vibration picture, where vibration potentials have the form 1 2 C a λµ (a λµ -a eq. λµ ) 2 and/or 1 2 C b λµ (b λµ -b eq. λµ ) 2 and the single-variable vibration Hamiltonians have the structure

H vib = - h2 2B ∂ 2 ∂ρ 2 + 1 2 C(ρ -ρ eq ) 2 with ρ ↔ a λµ or b λµ (11)
where B ↔ B λµ and C ↔ C λµ play the roles of the inertial mass parameter and stiffness-coefficient, respectively, here considered constant. Equilibrium deformations a eq. λµ and/or b eq. λµ are selected using our extensive Woods-Saxon Strutinsky results. Introducing the vibration frequencies hω ≡ h C/B and simplifying the notation we define η ≡ h/[ √ 2Bω] 1/2 and introduce the usual harmonic-vibration solutions

φ n (ρ) = η/(2 n n! √ π) exp -η 2 ρ 2 /2 H n (ρ), ( 12
)
where H n are Hermite polynomials. We wish to avoid mathematical details at this point and limit ourselves to schematizing the main lines of the approach; the interested reader is referred to a more detailed discussion in [9]. We use the standard definition of the intrinsic reference frame (alternatively: Euler angles) based on the choice of the quadrupole deformations as α 2,+1 = 0 = α 2,-1 and α 2,+2 = α 2,-2 . In such a case the quadrupole vibrations are described in terms § The negative parity bands with the above properties have been found in the literature, cf. a short review of the actual research status of the International TetraNuc (Tetrahedral Nuclei) Collaboration in [10]; they can be seen as tetrahedral-symmetry candidate structures.

To better appreciate this argument consider a nucleus with the quadrupole-deformed minimum of the order of α 20 ≈ 0.3 and the induced collective rotation. The corresponding reduced transition probabilities, B(E2), could be as large as 300 W.u. depending of the nuclear mass while the correction of the zero-point vibrations of the order of a dozen W.u. or so. The same 'correction' applied in the case of the static, nearly-vanishing deformation [and therefore considerably smaller, nearly-vanishing B(E2)] may take the role of the leading term rather then that of a negligible correction.

of two independent real parameters α 20 = a 20 and α 22 = a 22 and it turns out that the octupole vibrations represent seven independent modes, here parametrised with the help of a 30 as well as a 31 , a 32 , a 33 and b 31 , b 32 , b 33 . In the following we would like to construct the negative-parity one-phonon wave-functions to be able to model the negative-parity bands, candidates for the tetrahedral symmetry, in terms of the irreducible representations of the underlying tetrahedral group. With n in [START_REF] Davydov | Quantum Mechanics[END_REF] representing the phonon number, we will introduce the symbol {n} ≡ {n 1 , n 2 ; n 3 , . . . n 9 } in which the first two places are reserved for the quadrupole degrees of freedom and all the others of the octupole ones. We will introduce a basis of product wave-functions

Φ {n} = φ n1 φ n2 × φ n3 . . . φ n9 , (13)
but since the conditions for the negative parity solutions imply the one-phonon structure it follows that n 1 = 0 and n 2 = 0 (zero-phonon quadrupole ground-state) whereas i=9 i=3 n i = 1, with n i = 0 or 1. It will be possible (and convenient) to shorten the notation by replacing {n 1 , n 2 ; n 3 , . . . n 9 } by {00; m}, where m gives the only nonzero index n i that is allowed. For instance, symbol Φ 00;n5 represents the product

(φ n1=0 • φ n2=0 ) × (φ n3=0 • . . . • φ n5=1 • . . . • φ n9=0 ).
Consider an arbitrary solution Ψ to the vibration-part of Hamiltonian in (10); it can be expanded in terms of our basis wave-functions as

Φ = 9 j=3 c {n}j Φ {n}j . (14)
Let P Γ denote the operator projecting from the space of Ψ onto a sub-space of the irreducible representations Γ of G; we will apply it for G = T d , but the considerations remain analogous for any symmetry point-group. According to the well known result

P Γ = dim[Γ] card(G) ĝ∈G χ Γ (ĝ) ĝ, (15)
where card(G) is the number of elements in the group G and χ Γ (ĝ) are the related characters of the group elements. Using the above operator we find the structure of the wave-functions spanning the irreducible representations of the group T d in the 7-dimensional space of one-phonon solutions; up to a normalisation they are given by:

|A1 ∼ Ψ 00;n6 (16)
for the scalar representation Γ = A1,

|T 1; 1 ∼ Ψ 00;n3 , (17) |T 1; 2 ∼ 1 4 (- √ 5Ψ 00;n2 - √ 3Ψ 00;n4 -i √ 5Ψ 00;n5 + i √ 3Ψ 00;n7 ), (18)
|T 1; 3 ∼ 1 4 (+ √ 5Ψ 00;n2 + √ 3Ψ 00;n4 -i √ 5Ψ 00;n5 + i √ 3Ψ 00;n7 ), (19)
for the first 3-dimensional representation Γ = T 1 and

|T 2; 1 ∼ Ψ 00;n1 , (20)
|T 2; 2 ∼ 1 non-zero equilibrium deformation we may introduce the deformation-shift operators, say S, that must preserve the group structure of G -and thus must commute with all the symmetry operations in G. It is possible to show that the only invariant of the G = T d group composed of the α λ=3,µ tensor can be expressed (depending on conventions that are of no importance here) as either b 32 or a 32 . Using the first convention we may demonstrate that the 'tetrahedral-deformation shift' operator with the sought mathematical properties has the form

S(b eq. 32 ) = exp -b eq. 32 ∂ ∂b 32 . (23)
Introducing the above shift we formally break the symmetry under the inversion ¶. In such a case the projection onto the good parity is an option of choice. It is easy to see that the operator 1 2 (1-C i ), where C i denotes the inversion, is the sought projection on the negative-parity irreducible-representations of the considered T d -group and thus by applying the combined projection-and-shift operation

P(b eq. 32 ) ≡ 1 2 (1 -C i ) S(b eq. 32 ) (24)
on the wave functions in [START_REF] Coussement | [END_REF](17)(18)(19)(20)(21)(22) we obtain the sought solutions that belong to the irreducible representations of the group considered. They have the appropriately adjusted equilibrium deformations and are also parity-projected.

We have discussed so far, for the sake of compactness of this presentation, only the main lines of the proposed approach, but the interested reader can convince her/himself that due to the simplicity of the vibrational wave-function basis the calculations of interest can be performed analytically + ; this is a helpful factor for quick estimates but by no means the most important one. As a first example of an application (and using the fact that the exotic deformations considered here are relatively small) we may introduce the general expansion expression for the multipole moment operators

Q rot λν = 3ZR λ 0 4π α λν + λ + 2 2 √ 4π λ 1 λ 2 (2λ1 + 1)(2λ2 + 1) 2λ + 1 × (λ10λ20|λ0) (α λ 1 ⊗ α λ 2 ) λν , (25)
valid for the simplified uniform charge distributions in the nuclear intrinsic ('rotating') reference frame. The laboratory-frame image of this operator is

Q lab λµ = ν D λ µν (Ω) Q rot λν , (26)
where D λ µν (Ω) are the complex-conjugate Wigner functions. Using theorems of the group-representation theory and the fact that the assumed Hamiltonian does not couple the rotations and the vibrations we may construct the final solutions in the group-theory adapted form

Ψ (λ=2,3) Γjσ;JMν (α, Ω) ∼ Φ (λ=2,3) Γjσ (α) R JMν (Ω), (27)
¶ We encounter here a certain mathematical artifice related to the fact that we have chosen to illustrate the parity-projection technique. The use of the latter in this particular case is not really necessary, since the potential energies are in fact always even functions of α λ=3,µ variables. However, the projection technique presented here has certain merits for its applicability in more general situations and we discuss it for that reason. + An introduction of the realistic potentials generated using the mean-field theory leads to the condition V (-α λ=3,µ ) = V (+α λ=3,µ ), (even functions) and the corresponding solutions are not proportional just to a single Hermite-polynomial but rather to a series of such functions and must be obtained numerically.

where α ≡ {α 2 ⊗ α 3 }. Above, R JMν (Ω) represents the rotational part of the wavefunction while quantum numbers JM represent the total angular momentum of the system together with its projection and ν distinguishes among the (2J+1)-solutions of the rotor Hamiltonian that correspond to a given (JM ). The phonons can also be characterised by the associated quantum numbers jm, analogous with respect to JM . In the solutions that were presented in [START_REF] Coussement | [END_REF](17)(18)(19)(20)(21)(22), (and functions Φ (λ=2,3) Γjσ (α) are constructed out of them) the involved three-dimensional irreducible representations contain in general the combinations of the m-components. It then follows that the resulting wave functions need to be labelled by a separate index, σ, enumerating states within the three-dimensional irreducible representations. Finally, the symbol Γ refers to irreducible representations of the group T d , above denoted A1, T 1 and T 2, that possibly belong to different irreducible representations of the SO(3)-group, the latter enumerated with j.

We are going to limit ourselves here to presenting some instructive estimates of the expectation values of selected charge moments using the irreducible-representation form of the vibrational solutions. Calculations using the elements of the approach presented so far and assuming for simplicity that all the η 3µ = η 3 give

A1|Q 20 |A1 = T 1; 1|Q 20 |T 1; 1 = -T 2; 1|Q 20 |T 2; 1 = - ZR 2 0 2π √ 5π • 1 η 2 3 . (28)
This is instructive (although at this stage quite schematic) a result stressing once again the importance of the zero-point motion mechanism in the symmetry considerations.

In words: the above results suggest that the tetrahedral-symmetry solutions contribute to the quadrupole transitions through a dynamical (oscillation) mechanism involving all the octupole components. This is in contrast to the earlier suggestions based on the pure static considerations and implying that the quadrupole transitions in the exact symmetry limit should vanish exactly. Let us observe that the above conclusion is certainly only one on a long list. The issues that remain Open Problems to us are: a. Complete (and not only schematic) calculations of the reduced transition probabilities B(E2; Γ → Γ), B(E2; Γ → Γ ) and: b. The ratios B(E2; Γ → Γ)/B(E1; Γ → gsb) and B(E2; Γ → Γ)/B(E2; Γ → Γ ) together with the related combinations of stretched and non-stretched E2 and E1 transitions. Symmetries act very restrictively on the rules for inter-band transitions depending on the type of an irreducible representation and the transition kind, giving characteristic fully forbidden or retarded transitions, alternatively characteristic branching ratios -that can be used to distinguish one symmetry from another.

Level-mixing models as a totally unsuited tool to search for symmetries

There exists an exercise in quantum mechanics dubbed two-level mixing model, cf. [11] for probably the most rigorous presentation, and also [START_REF] Davydov | Quantum Mechanics[END_REF]. The model has been introduced to nuclear structure physics in the context of rotational-band mixing and the Coriolis mixing, cf. [START_REF] Rowe | Nuclear Collective Motion[END_REF][START_REF] Bohr | [END_REF][START_REF] Casten | Nuclear Structure from a Simple Perspective[END_REF], and references therein. In nuclear structure it has been successful in parametrising certain features in the presence of strong quadrupole deformations and dominating E2 transitions, but in the mean time used gradually less and less properly, its limiting assumptions ignored or presumed valid without verification, no attention payed to parametric instabilities etc.

In testing the symmetries, the signals are expected to be weak and so are the related electromagnetic transition matrix elements expected to be of the order of those which are usually neglected in 'level-mixing' articles known to us. Moreover, all exotic symmetries treated in this text involve non-axial shapes and thus using the assumption of axiality (good K quantum number, the 'usual' assumption) is totally incorrect. Anticipating that in the present context the level-mixing model may again be a tempting option to replace partly or entirely the thorough measurements we briefly add a few more among the most relevant inadequacies of this 'method' in studying symmetries.

Let us emphasize that no calculation in general and the level-mixing calculations in particular should ever be used in place of a measurement. However, in our opinion, much too often the information deduced from the level-mixing calculations forces its way down to the conclusions as if it were the experimental result.

In which respect the quantum-mechanics exercise of two-level mixing poses an Open Problem in nuclear physics of the XXI st Century? We will reformulate this question: "How can we lower the damage to the experimental research caused by the model's in-appropriate use?" In commenting these issues we let ourselves being guided by the observations made during recent discussions, conference presentations and even through recent publications!

In order to establish the presence of a symmetry it is important to verify the symmetry criteria e.g. the ones based on the electromagnetic transition probabilities or, if that is not fully possible, certain specific branching ratios as discussed in section 3. This task involves inevitably weak transitions that may be difficult to measure. The band mixing models have been used in the past to replace the weak experimental signal from, say band B, by the results obtained through a mixing model calculations provided some information about the electromagnetic transitions has been experimentally available for another band, say A, the so-called reference band.

In their excellent presentation entitled "The theory of nuclear level mixing resonant spectroscopy", [START_REF] Coussement | [END_REF], the authors begin by justly emphasizing the fact that the level-mixing models are based by construction on the symmetry breaking mechanism. Indeed, according to Landau-Zener rule, two bands cross because their states differ in symmetry -they repel each other because there is an interaction that causes the state mixing. More precisely: given states φ a and φ b belonging either to two different irreducible representations of the symmetry group of one generating Hamiltonian h g or obtained from two distinct generating Hamiltonians h g and h g . In order that the states mix it is necessary that the mixing interaction v m does not commute, neither with h g nor with at least one of the other two Hamiltonians. In other words, the ' full' Hamiltonian h = h g + v m or h = h g + h g + v m generates solutions of lower symmetry since [h, v m ] = 0, and none of the interacting states has its initial symmetry.

Since the level-mixing Hamiltonian breaks the original symmetries of each of the mixed states, it becomes clear that the band mixing model is from the very beginning not suited to look for symmetries. It then follows that e.g. complicating the same model by adding more and more mixing bands would be multiplying inadequacies and possibly even increasing the existing confusion.

Point-groups symmetries and nuclear stability: Open Problems

In this article we have formulated in fact two strategies in searching for the so-called geometrical (synonymous with the point-group) symmetries in Nuclear Structure. They are based on the group-theory guided discussion of the problems: What are the nuclear geometries that privilege increased stability of the nuclear N-body systems? and: Where in the Periodic Table such symmetries can possibly be manifested? The second strategy addresses the issue of the experimental criteria: What should be measured in order to establish the presence (or absence) of the point-group symmetries?

In discussing the first series of the implied Open Problems we have introduced the notion of the generalized (as compared to the standard, spherical one) shellstructures that can be associated with any symmetry group of the Hamiltoniannot necessarily the 'usual' spherical symmetry. The first observables of interest in this case are the 'magic gap chains' -the series of large energy gaps in the singlenucleon spectra that express the system-stabilizing response of the nuclear mean field to the privileged geometrical shapes: those manifesting the point-group symmetries. On the basis of the earlier publications we have formulated a qualitative criterion suggesting that the symmetries expressed by the point groups which have numerous irreducible representations and/or whose irreducible representations have possibly largest dimensions are likely to produce the strongest non-spherical shell effect. The systematic although in a way still preliminary studies were focussed so far on the tetrahedral and octahedral symmetries, with theoretical effort accompanied by the series of dedicated experiments performed within the TetraNuc collaboration. But examining the whole series of nearly a dozen of point groups of special interest listed in this article is an Open Problem for this domain of research.

The second series of Open Problems is related to the research of observables and the construction of the adequate experimental methods. We have proposed an adaptation of the well known, and again a group-theory based approach, that consists in calculating the electromagnetic transition probabilities in function of the pointgroup symmetry. According to this suggestion the solutions of the collective rotationvibration Hamiltonian whose eigenfunctions are expressed in terms of the irreducible representations of the involved symmetry groups are used to predict the properties of the electromagnetic transitions and related branching ratios. Such transitions vary, often dramatically, depending on the symmetry group involved and can be employed as valuable criteria -however, they were not used so far in the search for symmetries.

We have also re-visited the level-mixing models that can occasionally be used to fit and/or estimate the experimental data otherwise difficult to really measure. Since these models have also been used in the past as substitutes for the experimental research we have re-examined the model's functioning and formulated rather obvious, strong warning: the model's unknowns and the applicability conditions may be more difficult to establish experimentally than the attempted measurement itself.

Figure 1 .

 1 Figure 1. Schematic: 20 levels in a potential well; for comments cf. the text.

2. 3 . 1 .

 31 Phenomenological Mean Field Hamiltonians: Woods-Saxon or Yukawa folded. Consider first a deformed Woods-Saxon or Yukawa-folded mean-field potential, with the nuclear surface Σ represented with the help of spherical harmonics Y λµ as Σ : R(ϑ, ϕ) = c({α}) 1 + λ µ=-λ

Figure 2 .

 2 Figure 2. A nuclear 'pyramid' (left) and a nuclear 'diamond' (right) shapes corresponding to the tetrahedral and octahedral deformations with t 3 = 0.2 and o 4 = 0.2, respectively, cf. (4) and(7). It is usually recognized as not intuitive that so qualitatively different shapes belong to the common, 'weaker', tetrahedral symmetry and when combined may decrease considerably a energy of the tetrahedral-symmetric nucleus.

Table 1 .

 1 Double point groups together with the numbers of their irreducible representations and dimensions. The information in the last column should be read as follows. For instance, in the case of the double octahedral point group O D

h we find 4 two-dimensional and 2 four-dimensional irreducible representations.

  λ = 7 : α 7,±2 ≡ t 7 and α 7,±6 ≡ -11 13 t 7 ;(5) λ = 9 : α 9,±2 ≡ t 9 and α 9,±6 ≡ + 28 198 t 9 , etc.

(6)

Observe the presence of the characteristic geometrical factors which are necessary to guarantee that the symmetry of the nuclear surfaces is obeyed at each value of t 7 , t 9 , etc. The solutions for the octahedral symmetry exist for even λ ≥ 4 and we find λ = 4 : α 4,0 ≡ o 4 and α 4,±4 ≡ -5 14 o 4 ; (7) λ = 6 : α 6,0 ≡ o 6 and α 6,±4 ≡ + 7 2 o 6 , (8) together with λ = 8 : α 8,0 ≡ o 8 and α 8,±4 ≡ + 28 198 o 8 , and α 8,±8 ≡ + 65 198 o 8 , etc.

‡ The mathematical details about the Wigner theorem classifying the irreducible representations as potentially real, pseudo-real and essentially complex, the mathematical terminology that we will not need here, can nevertheless be found in[START_REF] Cornwell | Group theory in physics[END_REF].

(-√3Ψ 00;n2 + √

5Ψ 00;n4 + i √ 3Ψ 00;n5 + i √ 5Ψ 00;n7 ), (21)|T 2; 3 ∼ 1 4 (+ √ 3Ψ 00;n2 -√ 5Ψ 00;n4 + i √ 3Ψ 00;n5 + i √ 5Ψ 00;n7 ), (22)for the remaining one, Γ = T 2, all that for the vibrations around the spherical equilibrium in the 7-dimensional octupole space. To obtain the solutions for the

Acknowledgements

The research programme and associated Open Problems reported here are related to the activities of the TetraNuc (Tetrahedral Nuclei) Collaboration involving 18 institutions worldwide. This informal collaboration supported in part by IN2P3, France, aims at searching for the fingerprints of the tetrahedral symmetry throughout the Nuclear Chart. The authors want to warmly thank all who have taken part in these activities and more specifically the Direction of the IPN-Orsay and D Verney, D Curien, the co-spokesman of the collaboration as well as O Stézowski and Q T Doan at IPN-Lyon, J Gerl at GSI-Darmstadt, R Julin and P Jones at JYFL-Jyvaskyla, F Haas, P Medina and M Richer at IPHC-Strasbourg, M Jentschel, B Lauss and W Urban at ILL-Grenoble, D Tonev at INRNE-Sofia, G de Angelis at LNL-Legnaro, R J Singh at IUAC-New Dehli, L Riedinger at University of Tennessee at Knoxville, D Hartley at U.S. Naval Academy-Annapolis, and last but not least the Direction of the ANL-Argonne, USA and the Direction of the IN2P3, France.


Iouri Belikov
email: iouri.belikov@ires.in2p3.fr

Strangeness and heavy flavours within the ALICE experiment

The ALICE experiment at CERN is now ready to take data in protonproton and ultra-relativistic heavy-ion collisions at LHC. This review starts with a description of the ALICE experimental setup and its excellent tracking and particle identification capabilities. Then, we will discuss the ways to probe the hot and dense medium created at multi-TeV energies by measuring the yields and kinematic characteristics of strange and heavy-flavour particles produced in these collisions.

Introduction

A Large Ion Collider Experiment (ALICE) [1] at CERN is a general-purpose heavy-ion experiment designed to study the physics of strongly interacting matter in nucleusnucleus collisions at the LHC. The centre-of-mass energy per nucleon pair will be √ s NN = 5.5 TeV for the Pb-Pb system, and the energy density reached will be of the order of a few tens GeV/fm 3 . Under these conditions, a deconfined state of quarks and gluons, the quark-gluon plasma, is expected to be formed. In addition to heavy systems, the ALICE Collaboration will study collisions of lower-mass ions and protons (both pp and pA), which will provide the reference data for the nucleus-nucleus collisions, and will also allow for a number of genuine pp and pA physics studies.

The detector consists of a central part (see Fig. 1), which measures hadrons, electrons and photons, and of a forward spectrometer to measure muons. The central part, which covers polar angles from 45 • to 135 • is embedded in the large L3 solenoidal magnet. It consists of: an Inner Tracking System (ITS) of high-resolution silicon detectors; a cylindrical Time-Projection Chamber (TPC); three particle identification arrays of: Time-Of-Flight (TOF) detector, Transition-Radiation Detector (TRD) and a single-arm ring imaging Cherenkov (HMPID); and two electromagnetic calorimeters (PHOS and EMCAL). The forward muon spectrometer (covering polar angles θ = 2 • -9 • ) consists of a complex arrangement of absorbers, a large dipole magnet, and fourteen planes of tracking and triggering chambers. Several smaller detectors for global event characterization and triggering are located at forward angles.

With such a design, ALICE is capable of correlating measurements done at central and forward rapidities.

ALICE detector performance

The apparatus was optimized for a charged-particle density dN ch /dy = 4000 and its performance has been checked in detailed simulations up to dN ch /dy = 8000. The track reconstruction efficiency in the acceptance of TPC is about 80% down to p T = 0.2 GeV/c, and it is limited by particle decays and absorption in the material. For the tracks with p T above 1 GeV/c, the reconstruction efficiency reaches 90 %, which is constrained by the dead zones between the TPC sectors.

The typical momentum resolution obtained with the magnetic field of 0.5 T is 1 % at p T = 1 GeV/c. At lower momenta, it is dominated by the energy-loss fluctuations and multiple scattering in the material. The material budget of ITS and TPC together is about 15 % of X 0 . At high momenta, the momentum resolution is determined by the space-point precision and the quality of the calibration and alignment of the tracking detectors (ITS, TPC and TRD). At p T = 100 GeV/c, the target momentum resolution is 3 -5% for the tracks reconstructed in all the tracking detectors.

An essential part of the ALICE physics program involves the measurements of charm and beauty hadrons. The detection of these short-living particles (as an example, cτ for the D 0 meson is 123 µm) relies on the resolution of the reconstructed track impact parameters with respect to the primary vertex, which has to be of the order of the particle's cτ , or better. In ALICE, the impact parameter resolution is mainly defined by the two innermost pixel layers of ITS. For the events with a number of tracks bigger than 100, the resolution on the transverse impact parameter is below 60 µm for the tracks with a momentum above 1 GeV/c. For the events with lower track multiplicities, the contribution from the precision of the primary vertex reconstruction becomes nonnegligible. But, even in this case, the transverse impact parameter resolution is always better than 100 µm at p T > 1 GeV/c.

Using virtually all known experimental techniques, the ALICE detector has excellent particle identification capabilities. Combining the dE/dx from ITS, TPC and TRD, the time-of-flight from TOF, and the Cherenkov angle from HMPID, ALICE can identify charged pions, kaons and protons in the momentum range from 0.2 GeV/c to about 5 GeV/c on a track-by-track basis. Statistically, the charged hadrons can be identified up to a few tens GeV/c using the relativistic rise of dE/dx in TPC. Electrons above 1 GeV/c are identified by transition radiation registered in TRD.

In 2008 and 2009, ALICE has collected a few million cosmic tracks suitable for the calibration and alignment of the detectors. The preliminary results are quite encouraging. The alignment procedures in ITS result in an average impact parameter resolution for cosmic tracks of the order of 50 µm, to be compared with 40 µm observed in Monte Carlo events simulated with perfectly aligned detectors [START_REF] Bombonati | ALICE Internal Note[END_REF]. The dE/dx resolution in TPC is 5.7 %, with the theoretically expected limit of ∼5.5 %. The achieved time resolution in TOF is about 110 ps, which is within the range of 60 -120 ps resolution used in the Monte Carlo performance studies.

Strangeness program

Topological reconstruction of the strange hadrons

Strange particles in ALICE are reconstructed via their decay topologies: charged kaons as kinks, Λ and K 0 s as V0s, and Ξ and Ω as cascade decays. Depending on the actual slope of the p T spectra and taking into account the reconstruction rates estimated with Monte Carlo, ALICE will have enough statistics of kaons and Λ's up to p T = 10 -15 GeV/c, and Ξ and Ω up to p T = 7 -10 GeV/c for both 10 7 central Pb-Pb and 10 9 minimum bias pp collisions (one year of data taking under nominal conditions) [START_REF] Alessandro | [END_REF].

In-medium enhancement and the canonical suppression

If a deconfined state of quarks and gluons is formed in heavy-ion collisions, strangeness production was predicted to be enhanced [4], and this was observed at SPS [5]. This can happen due to several reasons, like for a example lower kinematic threshold for the in-medium production of s-quarks and presence of additional production channels gg → ss, which are absent in the hadronic state. This excess of produced strangeness is preserved at the subsequent hadronization and results in a higher number of strange particles in the final state, than expected in a purely hadronic scenario.

The production of strangeness in heavy-ion collisions has to be compared with that in pp. However, the small systems require the strangeness to be conserved in a smaller correlation volume. In the language of statistical mechanics, the grand canonical ensemble limit (possibly reached in high-multiplicity heavy-ion reactions) is not applicable to the pp collisions. In this case, the canonical ensemble description becomes more appropriate. The lack of the available phase space leads to the canonical suppression of the strangeness production in small colliding systems [6].

The fact that this constraint is removed in heavy-ion collisions is not trivial. If these collisions were nothing else but a simple superposition of independent nucleon-nucleon interactions, the canonical suppression would still be in place, because the correlation volume would still be small. For this not to happen, there must exist a mechanism for long-range communication within the system, as it is expected if the partons become free in a quark-gluon plasma.

The statistical models based on canonical ensemble predict a fast saturation of the enhancement with the event centrality, which is not what was observed at RHIC. Also the magnitude of the enhancement decrease with increasing energy is weaker than it was predicted [7].

Measuring the strangeness enhancement as a function of centrality in nucleusnucleus collisions, and as the function of multiplicity in pp collisions at the LHC energies with ALICE will provide an important piece of information for constraining the models. The studies of the φ meson enhancement and its excitation function are of particular interest, because the production of this particle is not canonically suppressed (due to the ss structure of this meson).

Collective effects at the quark level and mechanisms of strangeness production

Topological reconstruction of the strange particles (K 0 S and Λ) gives us the unique possibility to distinguish between baryons and mesons at the momenta above the reach of the conventional particle identification methods. The Λ/K 0 S ratio measured in Au-Au collisions by STAR shows an excess of the baryon production with respect to the meson production within the range 1.5 < p T < 4.5 GeV/c [8]. This excess increases with the collision centrality. A possible explanation of the phenomenon involves the collective effects at the quark level: the transverse radial flow and recombination.

The recombination models predict that the Ω production in heavy-ion collisions should be dominated by recombination of three thermal quarks. This should result in a thermal shape of the momentum spectrum and absence of correlations with jets. The first prediction was confirmed by the measurements, whereas the second was not. An interesting solution to this Ω puzzle was proposed in [9], suggesting that Ω's are produced not in the jets themselves but rather in the ridge associated with them.

Due to its excellent capabilities for both strange particle and full jet reconstruction, ALICE will be able to check these ideas directly. High statistics of reconstructed strange particles will allow for studying the Λ/K 0 S ratio in pp collisions as a function of multiplicity, which, at the LHC energies, may also start revealing non-trivial collective effects at the partonic level.

In-medium modifications of resonance properties

Resonances with lifetimes comparable to that of the deconfined phase (such as K * (890) 0 , φ(1020)) may change their properties (mass, width) when they are produced in the dense medium. This may happen due to the final state interaction (which can be sorted out by comparing the results obtained in the hadronic and leptonic decay channels) or due to the partial chiral symmetry restoration.

The ALICE detector will be able to register these resonances with sufficient statistics and a mass resolution of the order of a few MeV/c 2 (see Fig. 2).

Heavy-flavour program

Reconstruction of the open-heavy-flavour particles and quarkonia

The D mesons in ALICE will be topologically reconstructed in the central detectors. The two most promising channels are D 0 → K -π + and D + → K -π + π + . The detection strategy to cope with the large combinatorial background is based on the selection of displaced-vertex topologies and good colinearity between the reconstructed D meson momentum and the flight line [START_REF] Alessandro | [END_REF]. An invariant-mass analysis is then used to extract the signal yield. For one year of data taking at nominal luminosity, the accessible p T range for the D 0 is 1-20 GeV/c in Pb-Pb and 0.5-20 GeV/c in pp, with statistical errors better than 15-20 % at high p T . Similar capability is expected for the D + , though at present the statistical errors are estimated only in the range 1 < p T < 8 GeV/c.

The production of open beauty can be studied in the central part of ALICE by detecting the semi-electronic decays of beauty hadrons, mostly B mesons. Given that electrons from beauty have an average impact parameter d 0 ∼ 500 µm and a hard p T spectrum, it is possible to obtain a high-purity sample with a strategy that relies on the electron identification by combining dE/dx (TPC) and transition radiation (TRD) information, and the impact parameter cut reducing the charm-decay component and rejecting misidentified π ± and e ± from Dalitz decays and γ conversions. Beauty production can be measured also in the ALICE muon spectrometer, by analyzing the single-muon p T distribution and the opposite-sign di-muons invariant mass distribution.

ALICE will be able to measure the charmonia and bottomonia production both at mid-rapidity (in di-electron channels) and at forward rapidities (in di-muon channels). In both channels the acceptance extends down to zero transverse momentum, and the mass resolution of ∼90 GeV is sufficient to separate the Υ states (see Fig. 3).

Mass and colour dependence of the in-medium energy losses

When traversing the dense matter created in nucleus-nucleus collisions, the initiallyproduced hard partons loose energy. A substantial part of this energy loss is due to the medium-induced gluon radiation. Experiments at RHIC have shown that the nuclear modification factor of particle distributions, R AA = 1 <N coll > dN AA /dp T dNpp/dp T , is a sensitive observable for the study of the interaction of the hard partons with the medium produced in nucleus-nucleus collisions. Quarks are predicted to loose less energy than gluons (that have a higher colour charge) and, in addition, the "dead-cone effect" is expected to reduce the energy loss of massive quarks with respect to light partons [10,11]. Therefore, one should observe a pattern of progressively decreasing R AA suppression when going from the mostly gluon-originated light-flavour hadrons (h ± or π 0 ) to D and to B mesons: [START_REF] Alessandro | [END_REF] in the di-electron (left panel) and in the di-muon (right panel) channel, in one month of data-taking.

R h AA ≤ R D AA ≤ R B AA .
The possible enhancement, with respect to unity, of ratios between these R AA 's has been suggested [12] as a sensitive observable to test the colour-charge (R D/h = R D AA /R h AA ) and mass (R B/D = R B AA /R D AA ) dependence of parton energy loss. At LHC, due to high production rates expected for heavy flavour particles and also jets, the measurements of the in-medium energy losses can be performed in a new and promising way. With its high potential for detecting the charm and beauty combined with the capabilities for full jet reconstruction, ALICE will address these effects also by studying directly the properties of heavy-flavour-tagged jets.

In addition to the partonic energy loss studies, the precise detection of the opencharm (open-beauty) hadrons is of an essential interest as a natural reference for the quarkonia production measurement.

In medium suppression and regeneration of the quakonium states

In nucleus-nucleus collisions, quarkonium suppression is expected to occur due to the Debye screening of colour charge in the deconfined matter. Such a suppression was already observed at SPS [13]. However, at higher energies (RHIC, LHC) the situation becomes more complicated, because the charmonia can be regenerated in the hot medium by recombination [14,15]. On the other hand, at LHC energies, the statistics of registered Υ's is for the first time expected to be sufficient for detailed suppression studies. In this case, due to the much higher mass, the influence of the regeneration should be negligible.

Thus, the most interesting aspect of the quarkonia studies with ALICE will be understanding the interplay between colour-screening-induced suppression and regeneration for J/ψ production in a medium containing on the order of 100 cc pairs, and measuring the medium effects on the bottomonia resonances.

Conclusions

We have discussed how the measurements of yields and kinematic properties of strange and heavy-flavour particles, abundantly produced at LHC energies, will allow to address several issues of the physics of deconfined matter. They provide tools for testing the thermodynamical equilibrium (by measuring the particle ratios) and collective effects at the parton level (by studying the mechanisms of particle production), tools for observing the possible chiral symmetry restoration (via detection of the in-medium changes of the resonance properties). By measuring the parton energy loss and its predicted mass dependence we will probe the properties of the QCD medium produced in the collisions, and the observations of the sequential dissociation patterns of quarkonia states will provide information about the temperature of the system.

Owing to its excellent tracking, vertexing and particle identification performance, ALICE is well equipped for strangeness and heavy-flavour physics. The low overall material budget of the central detectors makes possible charm and beauty detection down to p T ∼ 0. The experiment has the capabilities to study correlations between strange, charm and beauty particles and jets. The measurements will be done at central and forward rapidities simultaneously.

All these advantages, in combination with high statistics of strange and heavyflavour particles and well developed jets expected at LHC, will allow us to explore in detail this rich field of high-energy physics.

Figure 1 .

 1 Figure 1. Schematic layout of the ALICE detector.

Figure 2 .

 2 Figure 2. Background subtracted invariant-mass spectra of K * (890) 0 (left panel) and φ(1020) (right panel) reconstructed by ALICE in simulated central Pb-Pb events [3].

  The signal of Υ states, reconstructed by ALICE

	) 2 Entries/(Events*GeV/c	1 1.2 1.4 0.8	-6 10 ×								160 180 200 220 240 140 120				/ ndf LWidth 2 χ GSigma M1 Area1 M2 Area2 / ndf 2 χ LWidth GSigma M1 Area1 M2 Area2 M3 Area3 M3 Area3 Cbkg Cbkg	9.571 / 43 0.00953 ± 0.01149 ± 0.006 ± 9.529 0.03015 0.09726 3.98 ± 67.32 0.02 ± 10.09 2.18 ± 18.64 9.571 / 43 0.00953 ± 0.03015 0.01149 ± 0.09726 0.006 ± 9.529 3.98 ± 67.32 0.02 ± 10.09 2.18 ± 18.64 0.03 ± 10.43 1.9 ± 10.2 0.03 ± 10.43 1.9 ± 10.2 274.2 ± 1872 274.2 ± 1872
		0.6									100
											80
		0.4									60
											40
		0.2									20
		0	8.6	8.8	9	9.2	9.4	9.6	9.8	Mass e+e-(GeV) 10 10.2 10.4	8 0	8.5	9	9.5	) 10 10.5 11 11.5 (GeV/c µ µ M 2
					Figure 3.


W Florkowski

R Ryblewski

Strangeness in Quark Matter 2009, Buzios

Keywords: numbers: 25.75.-q, 25.75.Dw, 25.75.Ld

We consider a physical scenario for ultra-relativistic heavy-ion collisions where, at the early stage, only transverse degrees of freedom of partons are thermalized, while the longitudinal motion is described by free streaming. When the energy density of the partonic system drops to a certain critical value, the partons hadronize and the newly formed hadronic system freezes out. This sudden change is described with the help of the Landau matching conditions followed by the simulations done with THERMINATOR. The proposed scenario reproduces well the transverse-momentum spectra, the elliptic flow coefficient v 2 , and the HBT radii of pions and kaons studied at RHIC (Au+Au collisions at √ s NN = 200 GeV). It also reproduces quite well the transverse-momentum spectra of hyperons.

The spacetime evolution of matter produced in the ultra-relativistic heavy-ion collisions is described most successfully by the perfect-fluid hydrodynamics [START_REF] Kolb | Quark-Gluon Plasma 3[END_REF][START_REF] Huovinen | Quark-Gluon Plasma 3[END_REF][START_REF] Shuryak | [END_REF]. With the initial conditions inferred from the Glauber model, the hydrodynamic approach describes very well the hadron transverse-momentum spectra and the elliptic flow coefficient v 2 . With the Gaussian initial conditions, the hydrodynamic approach describes also the HBT radii [4].

Unfortunately, the use of the hydrodynamics is still challenged by the problem of early thermalization -the correct description of the data requires an early initialization time for hydrodynamics [5], which implicitly assumes a very early equilibration time. A possible solution to this problem was suggested in Ref. [6], where the idea was proposed that only transverse degrees of freedom are thermalized, while the longitudinal motion is described by free streaming (a physical picture explained naturally in the string models [7,8]). In this framework, called below the transverse hydrodynamics, one can obtain the parton transverse-momentum spectra and v 2 which are consistent with the data [6,9] §. In this paper we generalize this approach by inclusion of the sudden-hadronization transition modeled with the help of the Monte Carlo code THERMINATOR [12].

The equations of the transverse hydrodynamics follow from the energy-momentum conservation law, ∂ µ T µν 2 = 0, with the energy-momentum tensor [13]

T µν 2 = n 0 τ [(ε 2 + P 2 ) U µ U ν -P 2 (g µν + V µ V ν ) ] , (1)
where τ = √ t 2 -z 2 is the longitudinal proper time and n 0 describes the density of transverse clusters in rapidity. The clusters are identified with the groups of partons having the same rapidity. They are two-dimensional (2D) objects described by the 2D thermodynamic variables: ε 2 , P 2 , s 2 and T 2 (2D energy density, pressure, entropy density, and temperature, respectively). The 2D thermodynamic variables satisfy the standard thermodynamic identities: ε 2 + P 2 = T 2 s 2 , dε 2 = T 2 ds 2 , and dP 2 = s 2 dT 2 (the baryon chemical potential is set equal to zero). The energy-momentum tensor (1) includes also the two four-vectors,

U µ = (u 0 cosh η, u x , u y , u 0 sinh η), V µ = (sinh η, 0, 0, cosh η), (2)
where u µ = u 0 , u ⊥ , 0 is the hydrodynamic flow in the plane z = 0, while η = 1/2 ln((t + z)/(t -z)) is the spacetime rapidity. The four-vectors U µ and V µ satisfy the following normalization conditions:

U µ U µ = 1, V µ V µ = -1, U µ V µ = 0.
The four-vector U µ corresponds to the flow four-velocity in the standard hydrodynamics. On the other hand, the term V µ V ν in (1) is responsible for vanishing of the longitudinal pressure, i.e., in the local rest-frame of the fluid element, where U µ = (1, 0, 0, 0) and V µ = (0, 0, 0, 1), we have

T µν = n 0 τ     ε 2 0 0 0 0 P 2 0 0 0 0 P 2 0 0 0 0 0     . (3)
It should be emphasized that such a structure of the energy-momentum tensor appears in the theory of the color glass condensate and glasma for τ 1/Q s , where Q s ∼ 1 GeV is the saturation scale expected at RHIC [14,15]. This gives support for consideration of the transverse hydrodynamics as the appropriate description of the early evolution of matter.

The equations of the transverse hydrodynamics are solved numerically with the equation of state

P 2 = ε 2 /2 = ν g T 3 2 ζ(3)/(2π)
, where ν g = 16 [13]. Our analysis is restricted to the central rapidity region (z ≈ η ≈ 0), where the created system may be treated as boost-invariant. The initial conditions assume that the 2D initial energy density at τ = τ i = 1 fm is proportional to the mixture of the wounded-nucleon density, ρ W ( x ⊥ ), and the binary-collision density, ρ B ( x ⊥ ),

ε 2 (τ i , x ⊥ ) = ν g T 3 2 (τ i , x ⊥ ) ζ(3) π ∝ 1 -κ 2 ρ W ( x ⊥ ) + κρ B ( x ⊥ ) , (4)
where, following the PHOBOS studies of the centrality dependence of the hadron production [16], we take κ = 0.14. The normalization constant required in (4) determines the 2D initial central temperature of the system, T 2 i = T 2 (τ i , 0). The Landau matching conditions have the form

T µν 2 U ν = T µν 3 U ν , (5)
where T µν 3 is the standard energy-momentum tensor of the perfect-fluid hydrodynamics

T µν 3 = (ε 3 + P 3 )U µ U ν -P 3 g µν . (6)
Here ε 3 and P 3 are the three-dimensional (3D) energy density and pressure. Equations ( 5) and (6) give

n 0 τ ε 2 = ε 3 , (7)
which should be supplemented by the requirement of the entropy growth,

n 0 τ s 2 ≤ s 3 , (8)
where s 3 is the 3D entropy density. Dividing both sides of Eqs. ( 7) and ( 8) one obtains

T 2 ≥ 3 ε 3 2 s 3 . (9)
In our calculation, ε 3 , P 3 , and s 3 are interpreted as the quantities characterizing the hadron resonance gas in equilibrium at the 3D temperature T 3 f . This means that the use of the Landau matching conditions is a formal scheme to describe the fast hadronization process; if the energy density of the system drops to ε 3 (T 3 f ), the transverse gluonic clusters change into a locally isotropic resonance gas.

The considered model has three free parameters: T 2 i , T 3 f , and n 0 (the initial time τ i is always set equal to 1 fm). In the fitting procedure we first assume n 0 = 1 and try to adjust T 2 i and T 3 f in such a way that a good description of the data is achieved. In this step, the energy-conservation condition ( 7) is used as the only constraint. In the next step we rescale the 2D temperature in such a way that the equality T 2 = 3 ε 3 /(2 s 3 ) holds at the initial time τ = τ i . In this way we make certain that the condition ( 9) is fulfilled on the whole transition hypersurface. The rescaling of the temperature changes the energy density on the 2D side, however, this may be compensated by the appropriate change of n 0 (the flow profile is not affected by the rescaling of the temperature). In this way we are able to satisfy finally the conditions (7) and (8).

Our description of the hadronization process is quite radical. Yet, we add to it another radical assumption: the evolution of the isotropic (locally equilibrated) hadronic system is so short that the hadronization process may be followed immediately by freeze-out. Technically, this assumption is realized by the Monte-Carlo simulations done with THERMINATOR, where the 3D hadronization temperature T 3 f is identified with the freeze-out temperature and the hadronization hypersurface becomes equivalent to the freeze-out hypersurface.

Certainly, our method used to switch from the purely transverse hydrodynamic expansion to the locally isotropic system of hadrons and further to freeze-out is very much simplified. In particular, one expects that the isotropization process has a gradual character. More elaborate approaches would describe this kind of transformation using kinetic theory or dissipative hydrodynamics, for example, see [17,18,19]. Alternatively, one may think of several direct developments of our scheme, for example, the transition criterion may be changed and/or the time evolution of the isotropic phase may be extended. Such developments are the subject of our current studies. Anyway, in our opinion such more realistic smooth changes may be effectively replaced by the sudden but delayed processes.

T 2 i 590 MeV T 3 f 165 MeV Τ i 1. fm n 0 0.43 Κ 0.14 Π K Figure 2.
Elliptic flow coefficient of pions and kaons (solid line) compared to the experimental PHENIX data (dashes) [22].

The results of our numerical calculations with the initial profiles corresponding to the non-central (c = 20-30%) Au+Au collisions at the beam energy √ s NN = 200 GeV are shown in Figs. 1234. The values of the parameters used in the calculations are:

T 2 i = 590 MeV, T 3 f = 165
MeV, and n 0 = 0.43 . We observe that our model reproduces well the transverse-momentum spectra of pions, kaons, and protons, Fig. 1, and also the pion+kaon elliptic flow, Fig. 2, both measured in Au+Au collisions by PHENIX [21,22]. The values of ∆ indicate the average relative difference between the experimental data and the model results.

In Fig. 3 we show the model pionic HBT radii (solid lines) compared with the experimental STAR data (dots) [23]. The radii were calculated with the help of These parameters are slightly different from those used in [20], where we did not study the production of strangeness and the equation of state was of the form P 2 = ε 2 /2 = νgT 3 2 /(2π). THERMINATOR which uses a two particle method with Coulomb corrections [25].

The overall agreement with the data is quite good. The theoretical radii are by about 20% smaller than the experimental values, but their k T dependence is very well reproduced. Moreover, the model ratio R out /R side matches very well the experimental result. Slightly smaller theoretical values indicate that the real isotropization and freeze-out processes take longer time. This is, of course, an expected feature.

The high freeze-out temperature obtained in our model suggests that one may be able to reproduce correctly the spectra of hyperons, as in the case of the singlefreeze-out model [26]. The hyperons, due to their small interaction cross sections, are believed to decouple rather early at the temperature close to the typical chemical freeze-out temperature. Indeed, our calculations presented in Fig. 4 show that the hyperon transverse-momentum spectra are reproduced reasonably well.

In the conclusions we state that the transverse-hydrodynamics model supplemented with the sudden isotropization transition and freeze-out is able to describe uniformly the soft pion and kaon data (transverse-momentum spectra, v 2 , and the HBT radii). This brings further evidence that the assumption of the very fast 3D equilibration of matter produced in the relativistic heavy-ion collisions may be relaxed -an earlier study showed that the RHIC data may be explained by the model where the standard hydrodynamic phase is preceded by the parton free streaming [27]. Our analysis of the HBT radii indicates, however, that the ultimate 3D equilibration is necessary to reproduce the experimentally measured value of R long .

Figure 1 .

 1 Figure1. Model transverse-momentum spectra of pions, kaons, and protons (solid lines) compared to the experimental PHENIX data (dots)[21].

Figure 3 .

 3 Figure 3. Model results for the pionic HBT radii (solid lines) compared to the experimental STAR data [23] (dots). The radii are plotted as the functions of the average transverse momentum of the pion pair kT . The model parameters are the same as in Figs. 1 and 2.

Figure 4 .

 4 Figure 4. Model results for hyperons compared to the experimental STAR data [24] (dots). The model parameters are the same as in Figs. 1 and 2.

‡ Supported in part by the Polish Ministry of Science and Higher Education, grant N202 034 32/0918. § We note that our formulation differs from the formalism of the transverse hydrodynamics introduced originally in [10,11]. For a more detailed discussion of this issue see [6].


G S Denicol

T Kodama

T Koide

The effect of shear and bulk viscosities on elliptic flow

In this work, we examine the effect of shear and bulk viscosities on elliptic flow by taking a realistic parameterization of the shear and bulk viscous coefficients, η and ζ, and their respective relaxation times, τ π and τ Π . We argue that the behaviors close to ideal fluid observed at RHIC energies may be related to non-trivial temperature dependence of these transport coefficients.

Introduction

Dissipative hydrodynamics has been applied to relativistic heavy ion collisions to understand the viscous nature of the quark gluon plasma (QGP), in particular, to explain the experimental data of the collective flow parameter v 2 by including the effect of shear viscosity [1].

One of the objectives of these attempts is to determine η/s, where η is the shear viscosity coefficient and s is the entropy density, by comparing hydrodynamical calculations of v 2 with the measured values. However, in addition to the uncertainties on the initial condition for hydrodynamics and the breakdown of the traditional Cooper-Frye freeze-out procedure in dissipative hydrodynamics [2,3], there are several difficulties in these attempts. One should note that these studies are mainly based on three assumptions; 1) the effects of other irreversible currents, such as the bulk viscous pressure and heat conductivity, are smaller than those of the shear stress tensor, 2) η/s does not strongly depend on temperature and 3) the effects of shear viscosity on observables are mostly characterized by one transport coefficient, η/s.

In this work, we would like to discuss the validity of these assumptions. First, η/s is in general a function of temperature. While the exact temperature dependence of this ratio is still unknown, its qualitative behavior is somehow well understood. As was shown in Ref. [4], η/s exhibits a complex temperature dependence showing a minimum near the QCD phase transition. Thus, it is not clear that such a temperature dependence can be approximately replaced by a constant value.

Second, we remark that the effects of the shear stress tensor (or any other irreversible current) cannot be characterized only by η. This might be the case for a fluid described by the relativistic Navier-Stokes (RNS) equation. However, as was shown in Refs. [5,6], hydrodynamical equations which violate causality, such as the RNS equations, are unstable and cannot be used for any application. To solve the problem of acausality and instability, we have to introduce (at least) one more transport coefficient called the relaxation time τ R , which represents the retardation effect in the formation of the irreversible current. Then it is natural to expect that the relaxation time should affect the behavior of v 2 or any other observable.

Because of the relaxation time, a large η does not necessarily induce a large shear stress tensor. To illustrate this, let us consider the retardation effect of an arbitrary irreversible current J which is induced by a thermodynamic force F . For the sake of simplicity, we consider a constant F . Then, the current is given by

J = DF (1 -e -t/τ R ), (1)
where D and τ R are the transport coefficients for this respective irreversible current. One can see that if τ R is small enough compared to the typical scale of the hydrodynamic evolution, J is simply proportional to DF . However, if τ R is not small, we have to consider the competition of D and τ R to quantify the dissipative effects. As a matter of fact, the time scale of the hydrodynamic evolution in relativistic heavy-ion collisions is of the order of 10 fm. On the other hand, as is shown in Ref. [7], the relaxation time of the shear viscosity τ π can also reach values around 10 fm. In this sense, we should not ignore the effect of τ π in the analysis of v 2 .

In this paper, we calculate the effect of the shear and bulk viscosities on elliptic flow by taking a realistic parameterization of the shear and bulk viscous coefficients, η and ζ, and the corresponding relaxation times, τ π and τ Π . We argue that the behaviors close to ideal fluid observed at RHIC energies may be related to non-trivial temperature dependence of these transport coefficients.

Model

In this work, we will use the memory function method [8,9], which is one of the formulations of relativistic dissipative hydrodynamics. For a general metric, the conservation of energy and momentum is expressed as

D µ T µν = 0, (2)
where D µ denotes the covariant derivative. We use the Landau definition of the four-fluid velocity u µ in which case the energy-momentum tensor is decomposed as

T µν = εu µ u ν -(p + Π) ∆ µν + π µν
, where, ε, p, Π and π µν are the energy density, pressure, bulk viscous pressure and shear stress tensor, respectively. The projection operator is defined as usual, ∆ µν = g µν -u µ u ν .

In the memory function method, any irreversible current J is induced by a thermodynamic force F as

τ R u µ D µ J + (1 + τ R D µ u µ )J = F. ( 3
)
This equation is derived to take into account time retardation effects and the extensivity of hydrodynamic variables. One can regard this equation as the generalized version of the Maxwell-Cattaneo equation. It should be emphasized that the existence of the non linear term τ R D µ u µ is essential to obtain a stable theory as is discussed in Ref. [9]. By applying this equation to the cases of the shear stress tensor and the bulk viscous pressure, we obtain

τ π ∆ µνλρ u α D α π λρ + π µν = ησ µν -τ π π µν θ, (4)
τ Π u α D α Π + Π = -ζθ -τ Π Πθ, (5)
where σ µν = D <µ u ν> and θ = D µ u µ are the thermodynamic forces. We use the traditional notation

A µν = ∆ µναβ A αβ where ∆ µναβ = 1 2 ∆ µα ∆ νβ + ∆ µβ ∆ να -2 3
∆ µν ∆ αβ is the symmetric traceless double projection operator.

The above hydrodynamical equation are solved numerically in three spatial dimensions. We use the Smoothed Particle Hydrodynamics (SPH) method. See Refs. [9,10] for details. We use approximately 80000 SPH particles with a smoothing parameter h = 0.6 fm [10].

For simplicity, we use a factorized initial energy density profile into its longitudinal and transverse parts, as proposed by Ref. [11]. The transverse part is parameterized as in Ref. [2]. The longitudinal part is taken as in Ref. [11], with the choice of parameters η Gaus = 0.8 and η flat = 4. The initial value of the irreversible currents will be taken as zero and the impact parameter is always set as b = 7.5 fm. We use the equation of state (EoS) introduced in [2]. As for the freeze out, we used the ordinary Cooper-Frye formula [10]. However, we did not consider non-equilibrium corrections to the one-particle distribution function, because, as was pointed out in Refs. [2,3], this correction gives rise to unphysical behaviors and is not reliable. However, this effect can be important and should be investigated in future works.

As mentioned in the introduction, the temperature dependence of the transport coefficients is very important. The microscopic formula to calculate these quantities is given in Ref. [7]. However, the application of this formula is still limited to leading order perturbative calculations. Fortunately, in this approximation η and ζ become equivalent to the ones calculated in the Green-Kubo-Nakano formalism and we can use these results. In this paper, we use the lattice QCD calculations [12] for the QGP phase and we use the results from Ref. [13] for the hadron phase. These calculations for η/s were fitted and the result is shown in Fig. 1(left panel). We can see that η/s shows a minimum around T ∼ 180 MeV. For the bulk viscosity coefficient ζ/s, we use the parameterization showed in Ref. [2] with a critical temperature of T ∼ 180 MeV.

Similarly, in leading order calculations, the microscopic formula predicts a simple relation for the shear viscosity, τ π = η/P . For the bulk viscosity, there is still no result and we simply assume τ Π = 9ζ/( -3P ). Since τ π (τ Π ) is linear with the transport coefficient η (ζ), the critical behavior of η (ζ) is transfered to that of τ π (τ Π ). Thus, τ π will have a minimum around the QCD phase transition while τ Π will display a maximum. In this sense, the complex temperature dependences of the relaxation times are induced by those of η and ζ.

Results

In Fig. 2(left panel), we show v 2 (p T ) for different choices of η/s and neglecting bulk viscosity. These calculations were made for thermal pions with a freeze-out temperature of T f = 130 MeV. The solid line corresponds to the case where η/s has the temperature dependence shown in Fig. 1(left panel) whereas the dashed, dashed-dotted and dotted lines correspond to constant values of η/s of 0.08, 0.12 and 0.16, respectively. One can see that the elliptic flow calculated with the temperature dependent η/s can be well reproduced by the one calculated with an effective constant shear viscosity coefficient η ef f /s = 0.12.

The quantity η ef f /s is very close to the minimum of η/s. This can be explained by the behavior of the relaxation time, shown in Fig. 1(right panel). As mentioned in the introduction, τ π characterizes the time scale for the velocity gradients to be converted to viscosity. Because the fluid expands with comparable time scale with τ π , the magnitude of the shear stress tensor does not become large despite the large values of η/s achieved throughout the hydrodynamical evolution.

It should be noted that η ef f /s is not bounded by the minimum of η/s. Let us consider another set of parameters, τ π = 3η/P . In this case, the effective viscosity η ef f /s becomes smaller than the minimum value of η/s. This result shows that the value of η ef f /s is not a direct measurement of η/s. It can come from the combined effect of η/s and τ π , and the small values of η ef f /s found at RHIC can just as well be a manifestation of the large relaxation time in the hadronic and QGP phases. Whether η ef f /s can be used to quantify the value of viscosity at the phase transition or not, depends on the magnitude of the relaxation time in these regions.

In Fig. 2(left panel), we show v 2 (p T ) considering the combined effects of bulk and shear viscosities, with realistic transport coefficients. The solid line corresponds to the calculation with only shear viscosity, the dashed line to the calculation with only bulk viscosity and the dashed-dotted line to the calculation with both bulk and shear viscosities. We see that the effect of bulk viscosity is rather small despite the large values of ζ/s near the phase transition. This can be explained as an effect of the relaxation time τ Π , again. Since τ Π also becomes large there, the values of ζ/s do not generate a large bulk viscous pressure. Since ζ decreases very quickly outside the phase transition region, we cannot see considerable effects of the bulk viscosity.

We remark that all the calculations showed in this work were done in three spatial dimensions. However, we were mainly interested in v 2 at zero rapidity, where we confirmed that the Bjorken scaling ansatz is a good approximation. Thus, all the results showed here are approximately equivalent to (2+1)D hydrodynamical calculations with the Bjorken scaling for the longitudinal direction. This happened mainly because of our choice of initial condition, which displays a plauteu in energy density in the longitudinal direction. Studies with different initial conditions, specially in the longitudinal direction, will be done in future.

Concluding remarks

We have shown that due to the interplay between the effects of viscosity and relaxation time, it is possible to fit the behavior of collective flow in terms of one unique effective shear viscosity η ef f /s. However, its value depends critically on the evolution of the system. Specifically, the value is affected by the time span for which the system remains in different phases. For the LHC energies, for example, we would expect a different value for the effective viscosity since the system will stay longer in the quak gluon plasma phase. However, the details of such changes are hard to calculate due to the uncertainties on η. We also remark that the uncertainties on the relaxation time are even larger since not so many effort have been applied to calculate the temperature dependence of this transport coefficient (unlike the η/s case).

We acknowledge discussions with J. Noronha, P. Huovinen, R. Lacey and D. H. Rischke. This work has been supported by CNPq, FAPERJ, CAPES, PRONEX and the Helmholtz International Center for FAIR within the framework of the LOEWE program (Landesoffensive zur Entwicklung Wissenschaftlich-Okonomischer Exzellenz) launched by the State of Hesse.

Figure 1 .

 1 Figure 1. Our parameterization of the shear viscosity η/s(left panel) and the shear relaxation time τ π (right panel).

Figure 2 .

 2 Figure 2. The elliptic flow v 2 (p T ) calculated for different choices of η/s (left panel) and calculated including the effects of bulk viscosity (right panel).


J D Tapia Takaki
email: daniel.tapia.takaki@cern.ch

The Color Glass Condensate framework in p+p and Pb+p interactions at the ALICE experiment

It is believed that the Color Glass Condensate (CGC) can describe the increase of parton density at very high energies or small values of the Bjorken-x. In this paper, the principal features of the CGC framework are outlined. Special attention is given to the prospects of observing gluon saturation effects at the LHC. In particular, heavy flavour and charmonium production are discussed in terms of physics performance studies of the ALICE Forward Muon Spectrometer.

Introduction

By colliding heavy nuclei at high energy the CERN Large Hadron Collider (LHC) is expected to create a new phase of hadronic matter, known as the Quark Gluon Plasma (QGP). Charmonium suppression was one of the main pieces of evidence for CERN's claim to have produced a deconfined matter state of quarks and gluons at SPS energies [1,2]. By studying QGP signatures such as this, it was soon realised that a good description of less complex systems (p+p and p+A) is essential. Cold nuclear matter effects such as initial state energy loss and shadowing have been studied to quantify whether absorption is also present [3]; it has been suggested that initial state effects have been observed at RHIC energies [4,5]. At the LHC several colliding systems can be studied in a new energy regime. Additionally, the large predicted cc and b b cross sections promise a systematic study of their production mechanism. Moreover, the range of Bjorken-x values to be reached is unique. In particular, ALICE (A Large Ion Collider Experiment) [6] will probe deep into novel x values, as low as about 10 -6 for the J/ψ. As the ALICE experiment can measure heavy flavour decays and quarkonia at both central and forward rapidity regions [6], a comparison between these measurements is foreseen during the forthcoming years. In this paper only results in the forward region are described.

Small-x physics is connected to the subject of gluon saturation. In the absence of gluon saturation, QCD evolution equations (DGLAP [8] and BFKL [9]) are applied to the gluonic parton distribution. The Color Glass Condensate (CGC) effective theory [10] takes saturation effects explicitly into account. It imposes non-linear corrections to these equations leading to the JIMWLK [11,12] or the BK ones [13,14]. Ultimately, a systematic study of small-x physics might provide, for a given colliding system, an experimental estimate of the saturation scale Q 2 s (x) [10] which is believed to be responsible for controlling initial cold matter effects.

In this paper, the prospects of observing gluon saturation effects via the measurements of heavy flavour decays and charmonium will be discussed. In particular, simulations based on the CGC are presented, compared to those given by a version of the PYTHIA [15] event generator tuned for NLO pQCD predictions [16], with the inclusion of shadowing. Unless otherwise stated, such a tuning will be referred to as MNR+EKS98 (for more details see [6,7]).

Prepared simulations and models used

In brief, the CGC description described in [17,18] is used. In such an approach, the heavy quark pair cross-section in a p+Pb collision can be written as:

dN pP b QQ dyd 2 p t = f (p) g (x 1 ) ⊗ ϕ(x 2 ) ⊗ dN sat gg→QQ dyd 2 p t , (1)
where

f (p) g
is the standard gluon distribution in the proton as a function of the momentum carried by the parton (x 1 or x 2 ), for which the CTEQ6L0 parameterisations [19] are used. ϕ(x 2 ) is a function that encodes the necessary information about the gluon distribution in a saturated nucleus. The evolution of this object with x 2 is determined by solving numerically the BK equation, and its initial condition (at the moderate value of x 0 = 10 -2 ) is taken from the McLerran-Venugopalan model [10]. dN sat gluons→QQ /dyd 2 p t is the heavy quark spectrum resulting from gluon fusion processes into heavy quark QQ pairs, evaluated by taking into account the re-scattering effects that are relevant in a saturated regime. It is important to keep in mind that this formula is valid only when x 2

x 1 , since any possible high density effect in the proton has been ignored. Additionally, as described in the references given, the region of large x 2 (> 10 -2 ) should be avoided. The convention used is that the projectile and the target represent hadrons coming from the positive and negative rapidity, respectively. The values for c and b quark masses are set to 1.2 GeV/c 2 and 4.75 GeV/c 2 , respectively.

Furthermore, the collision geometry has been simplified by omitting effects due to the impact parameter; this approximation amounts to considering a nucleus of constant thickness, which may not affect any hard probe production. α s is treated as a constant having the value of 0.15. Q 2 s,p , Q 2 s,A are evaluated at x 0 (= 10 -2 ), taking the following values: 0.33 GeV 2 , 1.93 GeV 2 , respectively. A number of constant pre-factors are also missing in the code used for obtaining the heavy quark yield in the CGC approach. For this reason, our spectra were normalised using the total yield of p+p collisions at √ s= 8.8 TeV as given by the MNR+EKS98. In the future, a proper normalisation will be taken into account. Despite the large spread of the absolute cross-section values for different sets of inputs, the ratios obtained using MNR+EKS98 calculations depend significantly less on the choice of the parameters. Therefore, they provide a baseline reference for making comparisons to CGC results.

Heavy flavour decays

In this paper, the approximation just described is studied in the context of the ALICE experiment. In particular, heavy quark pairs produced in the Forward Muon Spectrometer [20] acceptance originate from the interaction of a low-x parton from the projectile and a higher-x parton from the target. Thus, all quark pairs produced within the detector acceptance are not very sensitive to the large-x extrapolation of the projectile gluon distribution. The rapidity was defined in the laboratory frame. The centre-of-mass-frame shift with respect to the laboratory, of δy = -0.47 for Pb+p collisions, was taken into account. Notice that it constrains the rapidity window where the R P b-p can be measured; the ratio will correspond to muons with y ∈ [-2.97,-4] for p+p collisions and muons with y ∈ [-2.5,-3.53] for Pb+p collisions. Nuclear modification factors (R A-p ) are defined as the ratio between the yield in p+A collisions to the yield in p+p collisions, normalised by the number of binary collisions ‡.

To obtain the results presented in this section, the AliRoot [21] coding ‡ Although not discussed here, the study of multiple scattering effects is also envisaged by measuring the R p+P b ratio. implementation of generated c and b quarks, as given by the CGC, was required. Only heavy quarks have been generated, thus the following results do not take any background into account, e.g. from muons originating from hadronic sources (pions and kaons). We acknowledge that these decays should be taken into account in our analysis to make it somewhat more realistic; this background contributes mostly to the low p t region (p t <1.5 GeV/c) [7]. Both the quark fragmentation processes and the heavy hadron decays were computed through their semi-muonic channels. Notice also that the frontal absorber of the spectrometer does not allow low momentum muons to be detected (∼ p<3-4 GeV/c). Transverse momentum and rapidity distributions of muons from c and b quarks were calculated for both models (MNR+EKS98 and CGC) in p+p and Pb+p collisions at √ s = 8.8 TeV [22]. As is shown in Figure 1, the R P b-p ratio was obtained for muons coming from beauty; from charm, and for their sum. For a p t below 2.5 GeV/c, muons from charm dominate in both models. Figure 1 shows that a depletion in the R P b-p ratio versus p t is more pronounced for the CGC than in MNR+EKS98 simulations. This is due to saturation effects. A full physics performance study is needed as a next step in this analysis.

The J/ψ meson

The QQ production is given by the CGC. The J/ψ production cross-section was calculated following the Color Evaporation Model [23] approach. As in Section 3, only numerical simulations are presented without discussing full simulation results. Q 2 s,p , Q 2 s,A have taken the following values: 0.17 GeV 2 and 2.0 GeV 2 , respectively. Notice that the values for Q 2 s,p and Q 2 s,A are different from those presented in the previous section. New saturation values were estimated from analysing recent RHIC data [24]. It was not possible to use these new values in the earlier sections because these fine-tuned parameters were not available at the time that study was carried out. These model parameters were varied systematically (not shown) observing no major changes. As expected these results are α s dependent, as this variable is proportional to the degree of gluon saturation. Figure 2 shows the R P b+p ratio versus rapidity for different values of transverse momentum: 2.5 GeV/c (dashed line), 5 GeV/c (dotted line) and 10 GeV/c (dash-dotted line). A large suppression is expected for all momenta, and, at low p t , all predictions are lower than those from EPS09 [25] calculations (∼ 0.8) [26]. Notice that an increase of R p+P b at large values of rapidity is expected due to multiple scattering, resulting from the small c quark mass. As mentioned before, these results are not valid for x 2 (P b) >0.01; further theoretical developments might also be considered to verify the validity of QQ production at small-x values (see [27]).

Summary

Both CGC and MNR+EKS98 numerical results on single muon production were obtained in p+p and Pb+p interactions at √ s=8.8 TeV. At low p t , the CGC model shows a stronger suppression for charm quarks than for beauty, owing to the saturation itself. By performing numerical CGC simulations alone it was also discussed that the J/ψ meson offers an additional and very promising way to study small-x physics at the LHC. Further work is being carried out to study full detector performance effects for these analyses.

Figure 1 .

 1 Figure 1. The nuclear modification factor R P b-p is presented for both CGC and MNR+EKS98 calculations on the left and right hand side, respectively. These results are obtained at the generation level only. See text for more details.

Figure 2 .

 2 Figure 2. J/ψ's R P b+p ratio versus rapidity for different values of transverse momentum: 2.5 GeV/c (dashed line), 5 GeV/c (dotted line) and 10 GeV/c (dashdotted line). These results are obtained from numerical CGC simulations only and were calculated at √ s = 8.8 TeV.

Acknowledgments

We are indebted to F. Gelis for many useful discussions on this subject. This work is partly funded by "consortium physique des deux infinis" (P2I). The results given in this paper were obtained thanks to the use of the "Grille de Recherche d'Ile de France" (GRIF-IPNO).