C.-C Jean-Louis
email: charles.jean-louis@th.u-psud.fr

G Moreau

Dark matter and neutrino masses in the R-parity violating NMSSM

The R-Parity symmetry Violating (RPV) version of the Next-to-Minimal Supersymmetric Standard Model (NMSSM) is attractive simultaneously with regard to the so-called µ-problem and the accommodation of three-flavor neutrino data at tree level. In this context, we show here that if the Lightest Supersymmetric Particle (LSP) is the gravitino, it possesses a lifetime larger than the age of the universe since its RPV induced decay channels are suppressed by the weak gravitational strength. This conclusion holds if one considers gravitino masses ∼ 10 2 GeV like in supergravity scenarios, and is robust if the lightest pseudoscalar Higgs field is as light as ∼ 10 GeV [as may occur in the NMSSM]. For these models predicting in particular an RPV neutrino-photino mixing, the gravitino lifetime exceeds the age of the universe by two orders of magnitude. However, we find that the gravitino cannot constitute a viable dark matter candidate since its too large RPV decay widths would then conflict with the flux data of last indirect detection experiments. The cases of a sneutrino LSP or a neutralino LSP as well as the more promising gauge-mediated supersymmetry breaking scenario are also discussed. Both the one-flavor simplification hypothesis and the realistic scenario of three neutrino flavors are analyzed. We have modified the NMHDECAY program to extend the neutralino mass matrix to the present framework.

I Introduction

In supersymmetry, the superpartner of the graviton, namely the so-called gravitino, plays a central theoretical role as it constitutes the gauge fermion of supergravity theories [1,2]. From the cosmological point of view, the gravitino may be the Lightest Supersymmetric Particle (LSP), depending on the way supersymmetry is broken, and hence constitute a stable dark matter candidate [3]. Interestingly, one could even assume supersymmetric models where the R-parity symmetry [4] is violated. Then the gravitino becomes unstable due to new decay channels into Standard Model (SM) particles induced by the R-Parity Violating (RPV) interactions. However, because of the small gravitational interaction and possibly weak RPV couplings, the gravitino lifetime can still exceed the age of the universe (by several orders of magnitude) in the particular case of RPV bilinear terms [5] remaining thus a realistic dark matter candidate. This scenario can yield a complete cosmological framework also incorporating leptogenesis and nucleosynthesis, as recently shown in Ref. [6]. In such a scenario, the gravitino decays into SM particles lead to specific signatures in high-energy cosmic rays; the produced flux of gamma rays and positrons can even account [7] respectively for the extragalactic component of the excess in the HEAT [8] data. The decaying gravitinos could also constitute an interpretation (see below) of the exotic positron source recently discovered by the PAMELA Collaboration [9].

Another advantage of the existence of RPV interactions is the induced mixing between left-handed neutrinos and neutral gauginos. This mixing mechanism generates neutrino masses economically, without extending the SM field content to additional right-handed neutrino fields. There exists a tension between generating a sufficiently large neutrino Majorana mass scale [requiring strong RPV] and, at the same time, keeping the gravitino lifetime larger than the age of the universe [imposing weak RPV]. Nevertheless, this tension leaves some acceptable windows in the parameter space [5] for example with gravitino masses around 100 GeV in the Minimal Supersymmetric Standard Model (MSSM).

In the present paper, we re-consider the above scenario in the framework of the Next-to-Minimal Supersymmetric Standard Model (NMSSM) [10] (see recent reviews [11] for the phenomenological studies). An increasingly important virtue of the NMSSM is that it improves the 'little hierarchy' problem originating from the requirement of large soft supersymmetry breaking masses compared to the ElectroWeak (EW) scale. It also provides a solution to the so-called µ-problem by arranging the vacuum expectation value (vev) of a new gauge singlet scalar field of order of the supersymmetry breaking scale, so that the µ parameter turns out to be at the EW scale. We have shown in a previous work [12] that in the NMSSM with bilinear RPV terms, two non-degenerate massive neutrino states can emerge at tree level, in contrast with the MSSM case where only one neutrino eigenstate acquires a mass (at tree level). Hence, the global three-flavor neutrino data can be accommodated [13], at tree level, without appealing to loop dynamics which is vulnerable to model-dependent uncertainties.

In such a framework, both the gaugino mass matrix and RPV mixing terms are modified w.r.t. the MSSM case. Furthermore, the NMSSM provides the possibility 1 of reducing significantly w.r.t. MSSM the mass of the lightest pseudoscalar Higgs field [which can constitute partially the gravitino decay final state]. Moreover, the specific gauge singlet scalar field gives rise to new contributions to the gravitino RPV decay. Hence, the double question in the NMSSM on the possibility of generating the correct neutrino mass scale, while still keeping the gravitino as a good dark matter candidate, is relevant and well motivated. We find in this paper that the answer concerning the gravitino stability can be positive for parameters passing the theoretical and phenomenological constraints implemented in the NMHDECAY code [START_REF] Ellwanger | [END_REF] and in particular for gravitino masses as large as m 3/2 = O(10 2 ) GeV [tending to increase the total gravitino width]. This is satisfactory as it corresponds to the typical scale of supersymmetry breaking in supergravity scenarios. In both scenarios where the supersymmetry breaking is mediated purely by gravity [1,2] or partially by gravity and gauge interactions [17] (such classes of hybrid models have been recently motivated in string inspired constructions [18]), the gravitino is the LSP in wide regions of the parameter space and its typical mass is in the range between 100 GeV and 1 TeV. Besides, thermal leptogenesis and universal boundary conditions for gaugino masses at the Grand Unification Theory (GUT) scale restrict m 3/2 < 600 GeV [19] and, on the other hand, the interpretation of the PAMELA positron anomaly imposes m 3/2 200 GeV [20]. Nevertheless, our second result on a gravitino LSP around ∼ 10 2 GeV is that its RPV decay rates lead to fluxes exceeding the total flux measured in last indirect detection experiments. We will finally discuss the pure case, still within the NMSSM, of Gauge-Mediated Supersymmetry Breaking (GMSB) where the LSP gravitino mass verifies typically m 3/2 1 eV [21] and the neutralino mass matrix receives some modifications [22]. Conclusions here are more optimistic.

Let us make some comments on the results obtained in the present paper. First, we have considered all the possible gravitino decays, namely into photon, Z, W bosons and (pseudo)scalar Higgs fields, in contrast with Ref. [5] where only the photon channel was considered. The new kinematically allowed channels open up due to the higher gravitino masses considered here. Secondly, the trilinear RPV couplings are also discussed. Furthermore, we investigate for the first time the realistic case of three flavors of neutrinos. We conclude that the complete RPV mixing obtained in this case does not invalidate the positive results obtained for the simultaneous solution of the neutrino mass and dark matter problems. Finally, the pure neutrino results include the various NMSSM-constraints implemented in NMHDECAY [START_REF] Ellwanger | [END_REF] (that we have modified to include the neutrino components in the neutralino mass matrix) and are derived from a numerical treatment of the full neutral gaugino mass matrix, in contrast with preliminary work in Ref. [12]. The modified version of NMHDECAY thus contains an implementation of the whole neutralino mass matrix which includes, in the present context, some matrix elements induced by the RPV couplings and responsible for the mixing between the higgsinos and neutrinos (see Section II.3).

In addition, we will explore the alternative possibilities of the neutralino and sneutrino as stable LSP dark matter candidates, under the same assumption of the desired neutrino mass spectrum generated through the RPV mixing in the NMSSM. The philosophy is to establish a systematic list of the viable supersymmetric dark matter candidates. Since this topic relies on the particle spectrum, we base our study on a systematic exploration of the parameter space. Our conclusions are also based on new calculations of the neutralino and sneutrino decay channels.

Let us mention previous related works. The neutrino flux from direct gravitino decays has been computed in a simple scenario with bilinear R-parity breaking [23]. The diffuse gamma ray flux was also studied in Ref. [24]. A realization in minimal supersymmetric left-right models within supergravity, with a gravitino LSP dark matter and R-parity breaking, was proposed in Ref. [25] (see also Ref. [26]). Finally, a scenario with R-parity violation in the right-handed neutrino sector was motivated by the PAMELA data [27]. Concerning supersymmetric models for neutrinos -independently of the dark matter problem -RPV versions of the NMSSM have been previously studied in Ref. [28], Ref. [29] and Ref. [30]. Besides, alternative supersymmetric extensions with broken R-symmetries have been proposed in order to address simultaneously the µ-problem and the neutrino mass aspect [31,32,33] (and see Ref. [34] for a gravitino dark matter decay study in this context). RPV supersymmetric scenarios have also been studied within the context of neutrino astrophysics (see e.g. [35,36]). Finally, for studies of gravitino dark matter without RPV couplings, see for instance Ref. [37] for the constrained MSSM or GMSB scenarios.

The paper is organized as follows. In Section II, we elaborate the RPV scenario and describe the corresponding neutralino mass matrix. In the following sections, we discuss subsequently the cases of a sneutrino LSP (Section III), a neutralino LSP (IV) and a gravitino LSP (V). Finally, we conclude in Section VI.

II The RPV version of the NMSSM

II.1 Theoretical framework

We consider the NMSSM which possesses a superpotential containing two dimensionless couplings λ and κ in addition to the usual Yukawa couplings:

W NMSSM = Y u ij Q i H u U c j + Y d ij Q i H d D c j + Y ij L i H d E c j + λSH u H d + 1 3 κS 3 (1)
where Y u,d, ij are the Yukawa coupling constants (i, j, k are family indexes), and

Q i , L i , U c i , D c i , E c i , H u , H d ,
S respectively are the superfields for the quark doublets, lepton doublets, up-type anti-quarks, down-type anti-quarks, anti-leptons, up Higgs, down Higgs, extra singlet under the standard model gauge group. An effective µ term, given by λ s H u H d , is generated via the vev of the scalar component s ( s ) of the singlet superfield S. In case of GMSB [22], there are additional terms that must be added to the NMSSM superpotential (1). Those terms are given by:

W GMSB = ξ F S + µ S 2 (2)
where ξ F (µ ) is a new dimension-two (-one) parameter. These parameters in GMSB models are generated at low energy as S is coupled to the messenger sector.

Recalling that there exists no deep theoretical principle in supersymmetry for the existence of an exact R-parity symmetry [38], we adopt a generic approach by introducing both the bilinear and trilinear RPV terms characteristic of the NMSSM [the usual trilinear RPV couplings of the MSSM are considered in next section]:

W = W NMSSM + µ i L i H u + λ i SL i H u , (3)
where µ i (λ i ) are the dimension-one (dimensionless) RPV parameters.

Within the usual NMSSM, only trilinear couplings with dimensionless parameters (like λ and κ) are kept in the superpotential, while dimensional parameters (like µ) are generated from the vev s . Here the RPV NMSSM superpotential (3), containing a µ i L i H u but no µH u H d term, is assumed to arise in one of the scenarios proposed in [12].

II.2 Discussion on the trilinear RPV terms

The most general RPV NMSSM superpotential also includes the renormalizable trilinear RPV interactions:

W RPV = W + λ ijk L i L j E c k + λ ijk L i Q j D c k + λ ijk U c i D c j D c k . (4)
It is remarkable that if the order of magnitude for λ ijk and λ ijk is comparable to the values of the Yukawa coupling constants for the electron and the down quark [Y e , Y d ], the exchange of leptons/sleptons and respectively quarks/squarks in one-loop processes can generate Majorana neutrino masses [39] in agreement with (or few orders below) the oscillation experiment results: 10 -3 eV m ν i 1eV. This occurs for sfermion masses in the vicinity of 10 2 GeV, the order of supersymmetry breaking scale amount required from the gauge hierarchy solution. Now, given the analog structure of the Yukawa interactions (1) and the trilinear RPV terms (4), one can assume that the same (flavor-)structure responsible of the Y e,d suppression (w.r.t. Y t for the top quark) would also characterize the λ ( ) ijk couplings, inducing in turn the wanted tiny neutrino mass scale through one-loop processes. Assuming such low values for the λ ( ) ijk coupling constants (which in general easily pass the various phenomenological constraints [40]), one finds from formulas obtained in [41] that a gravitino ( G) LSP would decay into the three SM fermion final state f i f j f k via the λ ( ) ijk interactions with a sufficiently small width. Quantitatively, for a gravitino mass around m 3/2 ∼ 10 2 GeV, its lifetime would lie in the range 10 22 -10 24 sec which is well above the age of the universe: t 0 3.2 10 17 sec.

As a first conclusion, it is interesting to note that a scenario with trilinear RPV interactions of type λ ( ) ijk addressing simultaneously the neutrino mass and dark matter problems is conceivable. In the following, we will address this double problem within a different supersymmetric scenario where the neutrino mass is generated via bilinear and trilinear RPV couplings of type µ i and λ i , respectively, which do not induce too large decay widths of the dark matter candidate. In contrast with the λ ( ) ijk case, the present framework deserves a more general treatment in the sense that the neutrinos are mixed with neutral gauginos/higgsinos so that the neutrino constraints and the neutralino ones are correlated. In fact, the mixing is generally so small that it does not affect significantly the gaugino/higgsino mass matrix, but the precise values of the several parameters entering this matrix crucially determine the small induced neutrino masses.

Nevertheless, let us finish this part by commenting on the fact that even if the λ ( ) ijk couplings of the order of weak Yukawa couplings cannot induce dangerous G decay channels, they could contribute partially to the neutrino masses through one-loop diagrams. Here, we concentrate on the dominant tree-level contributions to the neutrino masses and hence leave for the future a precise and complete fit of neutrino data at loop-level.

II.3 Neutralino mass matrix

The neutralino mass terms read,

L m χ0 = - 1 2 Ψ 0 T M χ0 Ψ 0 + h.c. ( 5
)
in the basis Ψ 0 T ≡ ( B0 , W 0 3 , h0 d , h0 u , s, ν i ), where h0 u,d (s) are the fermionic components of the superfields H 0 u,d (S) and ν i [i = 1, 2, 3] denote the neutrinos. In Eq. ( 5), the neutralino mass matrix is given, in a generic basis (where νi ≡ v i = 0, µ i = 0 and λ i = 0, as will be discussed later), by

M χ0 = M NMSSM ξ T R p ξ R p 0 3×3 , (6)
where M NMSSM is the neutralino mass matrix corresponding to the NMSSM. For the latter mass matrix, we assume

v i v u,d , so that v 2 = v 2 u + v 2 d + 3 i=1 v 2 i = 2c 2 θ W m 2 Z /g 2 (175GeV) 2 .
In the mass matrix below, s and c stand for sine and cosine, respectively.

M NMSSM =       M 1 0 -m Z s θ W c β m Z s θ W s β 0 0 M 2 m Z c θ W c β -m Z c θ W s β 0 -m Z s θ W c β m Z c θ W c β 0 -µ -λv u m Z s θ W s β -m Z c θ W s β -µ 0 -λv d + 3 i=1 λ i v i 0 0 -λv u -λv d + 3 i=1 λ i v i 2(κ s + µ )       . (7)
Above, M 1 (M 2 ) is the soft supersymmetry breaking mass of the bino (wino), tan We assume for simplicity that λ, κ and the soft supersymmetry breaking parameters are all real.

β = v u /v d = h 0 u / h 0 d (with c β =
In Eq. ( 6), ξ R p is the RPV part of the matrix mixing neutrinos and neutralinos:

ξ R p =     -g v 1 √ 2 gv 1 √ 2 0 µ 1 + λ 1 s λ 1 v u -g v 2 √ 2 gv 2 √ 2 0 µ 2 + λ 2 s λ 2 v u -g v 3 √ 2 gv 3 √ 2 0 µ 3 + λ 3 s λ 3 v u     . (8)
g and g being the SU(2) and U(1) gauge couplings. We restrict ourselves to the situation where v i /v u,d 1 (as before), |µ i /µ| 1 and |λ i /λ| 1 so that (i) no considerable modifications of the NMSSM scalar potential are induced by the additional bilinear and trilinear terms in superpotential (3), (ii) the neutralino-neutrino mixing is suppressed, leading to sufficiently small neutrino masses; it is remarkable, as mentioned above, that the necessary order of magnitude, v i /v d ∼ µ i /µ ∼ λ i /λ ∼ 10 -5 -10 -7 , corresponds typically to the hierarchy between the electron mass and the top quark mass (or equivalently the EW symmetry breaking scale).

II.4 The various (s)lepton mixings

Chargino/neutralino-lepton mixing: The condition for generating two non-vanishing and nondegenerate neutrino mass eigenvalues at tree level is to ensure simultaneously

µ i = 0 and Λ i = 0, (9)
where

Λ i = s (λ i + λ v i v d ).
The respective roles of these effective parameters is reflected e.g. in the sub-matrix (8) mixing the neutrinos with neutral NMSSM states.

Since the H d and L i superfields possess the same gauge quantum numbers, one can freely rotate L α = (H d , L i ) [α = 0, . . . , 3] through SU (4) matrices by a redefinition of fields (for examples of unitary matrix associated to SU(4) transformations, see Ref. [START_REF] Hall | [END_REF] or Ref. [38] in Section 2.1.4). Motivated by the condition (9), we work in a general basis where the µ i L i H u terms are non-vanishing. One could choose a specific basis of L α superfields where the λ i couplings vanish, but then in general (i.e. assuming no particular correlation between the λ i and B i , m2

d,i values) some v i are generated due to the destabilization of the scalar potential by terms linear in νi originating from the RPV soft scalar bilinear terms

B i h u Li , m2 d,i h † d Li 2
after translation of the Higgs fields:

h 0 u → h 0 u + v u / √ 2 (h 0 d → h 0 d + v d / √ 2)
. Reciprocally, a field basis where the v i vanish generally leads to non-zero λ i couplings. In conclusion, the condition ( 9) is generally fulfilled.

Higgs-slepton mixing:

The physical amount of Higgs-slepton mixing is parametrized by the basisindependent angle between the 4-vectors

v α = (v d , v i ) and B α = (B d , B i )
, where B d is the soft parameter entering the biscalar term B d h u h d [38]. We assume this angle sufficiently small so that the additional potentially dangerous decay channels G, χ0

1 → H ± ∓ / h 0 ν and ν → f f, V V [f ≡ fermion, V ≡ Vector boson] play no role in the present context of a long-lived LSP dark matter candidate of type G, χ0

1 or ν.

Charged lepton mixing:

The charged leptons also mix with the Wino (via the v i 's) and the charged higgsino (controlled similarly by the µ i and λ i parameters). However the detail of this mixing also relies on the determination of the precise value of each Yukawa coupling constant Y ij for the charged leptons [see Eq.( 1)] which requires a complete scenario of flavor taking into account all the experimental constraints [from neutrino oscillations and lepton flavor violating reactions] on leptonic mixing angles (U P MNS matrix [43]), a task beyond the scope of the present work. Therefore, we will not explicitly work out the chargino-charged lepton mixing which is expected to be of comparable amount as the neutralino-neutrino mixing involving similar matrix structures and parameter orders [24]. Concerning the present cosmological context, this implies for instance that the width for the decay channel G → W ± ∓ is close to the one for G → Z 0 ν, so that our principal conclusions on the relative stability of the gravitino LSP based on the latter channel will not be modified by the former one 3 .

III Sneutrino LSP

III.1 RPV sneutrino decays

In case of a sneutrino LSP decaying through the RPV couplings of type Λ i and µ i , the sneutrino can exclusively decay into neutrinos via the two types of Feynman diagrams drawn in Fig. (1) or into two charged leptons. The associated decay widths are given below.

For that purpose, we need some definitions. First, the sneutrino LSP, noted ν1 , corresponds to the lightest of the three sneutrino mass eigenstates, keeping in mind that arbitrary non-universal conditions on scalar soft masses and the dependence of sneutrino mass running e.g. on flavor-dependent λ i parameters lead generally to a non-degeneracy of m νi (i = 1, 2, 3) eigenvalues.

We also need to introduce the matrix N αβ which is defined as follows. For convenience, we change the order (w.r.t. Section II.3) of the fields in the weak basis [for the rest of the paper]: Ψ 0 T = ( B0 , W 0 3 , h0 u , h0 d , s, ν i ) where as before the index i = 1, 2, 3, used for compact notations, corresponds to the three flavor states: ν e , ν µ and ν τ respectively. After diagonalization, the eigenstates are ordered in the mass basis according to Υ 0 T = (ν m j , χ0 1 , χ0 2 , χ0 3 , χ0 4 , χ0 5 ) where ν m j denotes the three neutrino mass eigenstates (j = 1, 2, 3) and the five χ0 's are the NMSSM neutralinos. Now, the unitary transformation matrix N αβ acts as Υ 0

α = N αβ Ψ 0 β with α, β = {1, . . . , 8}. B ,W ~u 0 v u s ν 0 ĩ m ν m ν m j i ν ν m ν m v i i j 0 ~3 h Figure 1:
Feynman diagrams for the sneutrino decay channel in two neutrinos, ν → ν m i νm j . We use the effective quantity m i = µ i + λ i s to parametrize a direct µ i mixing term effect combined with a s vev insertion. v u denotes the up Higgs vev insertion and v i symbolizes the sneutrino vev insertion (i = 1, 2, 3).

• Sneutrino decay into two neutrinos ν m i νm j (phase space factors involving neutrino over sneutrino masses are neglected):

Γ(ν 1 → ν m i νm j ) = m ν1 16π 3 k=1 N 1k 1 + δ ij λ k N i3 N j5 + g √ 2cosθ W N j{k+5} U ν m i Z + [i ↔ j] 2 (10)
where i, j = {1, 2, 3} run over the neutrino eigenstate indexes, N 1k are elements of the modeldependent sneutrino basis transformation matrix (1 is for the lightest sneutrino eigenstate and k = {1, 2, 3} is for the sum over sneutrino flavor states), U ν m i Z = -N i1 sinθ W + N i2 cosθ W and '+[i ↔ j]' indicates that the same expression in brackets must be added by switching i with j.

• Sneutrino decay into two charged leptons + i j (phase space factors involving lepton over sneutrino masses are also negligible here):

Γ(ν 1 → + i - j ) = m ν1 16π g 2 3 k=1 N 1k 1 + δ ij U j{k+2} U i W + [i ↔ j] 2 (11)
where U j{k+2} is the equivalent of neutralino rotation matrix N but for charginos (here with k = {1, 2, 3} for the sum over charged lepton flavor states and j = {1, 2, 3} associated to the final charged lepton eigenstate) and similarly

U i W is the equivalent for charginos of U ν m i Z .

III.2 Sneutrino stability

It turns out that the strength of effective RPV couplings Λ i / s and µ i necessary to generate sufficiently large neutrino masses, i.e. typically m ν 1 10 -1 -10 -3 eV (in the approximate case of a unique neutrino flavor), induces RPV-like mixings between neutrinos and neutralinos which are too large from the cosmological point of view. Indeed, those mixings translate into matrix elements N i3 ∼ 10 -7 , N j5 ∼ 10 -7 , U ν m i Z ∼ 10 -7 and N j{k+5} ∼ 1 with i, j, k = 1, 2, 3 [for standard neutralino masses m χ0 1 = O(10 2 ) GeV] so that the total sneutrino RPV decay width Γ(ν 1 → ν m 1 νm 1 ) ∼ 10 -15 GeV is well above the critical value of 1. 52 10 -42 GeV [for a sneutrino mass: m ν1 10 GeV and assuming for now N 1k ∼ 1]. It means that the sneutrino lifetime is well below the age of the universe t 0 and thus a sneutrino LSP does not constitute a viable dark matter candidate. A source of width suppression may arise from the generic amount of the matrix element N 11 but its effect is at much of a few orders of magnitude (more suppression would correspond either to a fine-tuning of parameters or to the weak breaking of a certain symmetry which should be described) so that this cannot invalidate the above conclusion. Furthermore, the effect of the additional decay channels into two charged leptons and into (pseudo)scalar Higgs bosons,

h 0 i h 0 j , h 0 i a 0 k or a 0 k a 0 l [i, j = 1, 2, 3, k, l = 2, 3
] (via the soft term sh u Li which is characteristic of the NMSSM with RPV interactions), can only increase the total sneutrino width 4 . 1 → h 0 k ν m j , and pseudoscalar Higgs fields, χ0

IV Neutralino LSP

IV.1 RPV neutralino decays

1 → a 0 k ν m j . The cross indicates which neutralino components are coupled.

• Neutralino decay into a Z boson and a neutrino ν m j (phase space factors involving neutrino masses are neglected):

Γ( χ0 1 → Z 0 ν m j ) = 1 96π 3 i=1 N 4i N ji 2 g 2 cos 2 θ W m χ0 1 1 - m 2 Z m 2 χ0 1 2 1 + m 2 χ0 1 2m 2 Z ( 12
)
where j = 1, 2, 3 labels the neutrino mass eigenstate and i = 6, 7, 8 corresponds to the sum over neutrino flavor states.

• Neutralino decay into the W boson and charged lepton ± j (the dependency on lepton masses, which represents subleading effects, is omitted):

Γ( χ0 1 → W ± ∓ j ) = 1 48π 8 i=6 N 4i U j{i-3} 2 g 2 m χ0 1 1 - m 2 W m 2 χ0 1 2 1 + m 2 χ0 1 2m 2 W ( 13
)
where j = 1, 2, 3 labels the charged lepton mass eigenstate and i = 6, 7, 8 is corresponding to the charged lepton/neutrino flavor states.

• Neutralino decay into scalar Higgs bosons h 0 k and neutrino ν m j :

Γ( χ0 1 → h 0 k ν m j ) = 1 32π 8 i=6 N ji λ {i-5} 2 |N 43 S k3 + N 45 S k1 | 2 m χ0 1 1 - m 2 h 0 k m 2 χ0 1 2 (14)
with j = 1, 2, 3 labeling the neutrino mass eigenstate, i = 6, 7, 8 corresponding to the neutrino flavor states and h 0 k ≡ h 0 1 , h 0 2 , h 0 3 (scalar Higgs mass eigenstates). The rotation matrix S relates the real parts of the neutral Higgs bosons and singlet scalar field to the scalar Higgs mass eigenstates h 0 k (see the precise definition in [START_REF] Ellwanger | [END_REF]). We recall that the parameters λ {i-5} which appear here are the NMSSM specific trilinear parameters of Eq.( 3). We also mention the decays into charged Higgs fields which are expected to be of comparable widths.

• Neutralino decay into pseudoscalar Higgs boson a 0 k and neutrino ν m j :

Γ( χ0 1 → a 0 k ν m j ) = 1 32π 8 i=6 N ji λ {i-5} 2 |N 43 P k3 + N 45 P k1 | 2 m χ0 1 1 - m 2 a 0 k m 2 χ0 1 2 (15)
with j = 1, 2, 3 labeling the neutrino mass eigenstate, i = 6, 7, 8 corresponding to the neutrino flavor states and a 0 k ≡ a 0 1 , a 0 2 (pseudoscalar Higgs mass eigenstates). The rotation matrix P translates the imaginary parts of the neutral Higgs bosons and singlet scalar field into the pseudoscalar Higgs mass eigenstates a 0 k [START_REF] Ellwanger | [END_REF].

IV.2 Neutralino stability

Based on the typical values of RPV parameters λ i , v i and µ i generating a neutrino mass scale m ν 1 10 -1 -10 -3 eV, we find for the parameters involved in the partial widths ( 12), ( 13), ( 14) and ( 15):

N ji ∼ 1, N 4i ∼ 10 -7 , U j{i-3} ∼ 1, λ {i-5} ∼ 10 -6
, N 43 S k3 + N 45 S k1 ∼ 10 -1 -10 -2 and N 43 P k3 + N 45 P k1 ∼ 10 -1 -10 -2 with i = 6, 7, 8 and j = 1, 2, 3 [if neutralino masses are of the order: m χ0 1 = O(10 2 ) GeV, and taking m χ0 1 > m Z ]. We can see that the contributions to the total neutralino decay width of Eqs.( 14) and ( 15) are smaller than Eqs.( 12) and ( 13). This leads [whatever is the (pseudo)scalar Higgs spectrum] to a total neutralino RPV decay width Γ total ( χ0 1 ) ∼ 10 -15 GeV that is several orders of magnitude above the critical value of 1. 52 10 -42 GeV. There are even additional contributions e.g. to the lightest neutralino decay into Z 0 ν m j , of the same order (originating from the Z 0 χ0 i χ0 j coupling), that should slightly increase the total neutralino decay width. Adding the decay channel into H ± , closed in most of the parts of the parameter space, can only increase again the total neutralino width. The conclusion is thus as for the sneutrino case: the neutralino lifetime is much smaller than t 0 and hence it does not represent a possible dark matter LSP candidate.

Nevertheless, a possibility to insure the stability of the lightest neutralino is to restrict to domains of parameter space where the 4 possible decay channels of Eq.( 12), Eq.( 13), Eq.( 14) and Eq.( 15), as well as the channels into charged Higgs bosons, are all kinematically closed [even opening only one channel is sufficient to render the χ0

1 unstable]. Although in the NMSSM the lightest pseudoscar Higgs boson noted a 1 can be much lighter than in the MSSM, we can find realistic regions of the NMSSM parameter space where the lightest neutralino can be simultaneously lighter than the W, Z bosons, the lightest scalar h 0 1 and pseudoscalar a 0 1 , as well as the charged Higgs bosons H ± , thus closing the corresponding RPV channels. By the term 'realistic', we mean that the NMSSM parameters pass the theoretical and phenomenological constraints implemented in the NMHDECAY program [START_REF] Ellwanger | [END_REF] like: (i) the physical minimum of the scalar potential is deeper than the local unphysical minima with h 0 u,d = 0 and/or s = 0 (ii) the running couplings λ, κ, Y b , Y t do not encounter a Landau singularity (iii) the experimental constraints from LEP in the neutralino, chargino and Higgs sectors are effectively satisfied. For instance, using NMHDECAY, we have plotted the masses m χ0 1 , m h 0 1 and m a 0 1 as a function of the κ coupling constant in Fig. (4), for fixed values of the other NMSSM parameters. We see on this figure that for κ 0.05, the lightest neutralino LSP is lighter than the W, Z bosons, the lightest scalar h 0 1 and pseudoscalar a 0 1 . So the scenario of a completely stable χ0 1 LSP can be a priori realized. Note that in such a scenario the RPV couplings can now be chosen freely to satisfy the neutrino constraints, since those couplings have a negligible impact on the neutralino mass spectrum.

We remark that there exist various triangular one-loop processes, exchanging charged leptons/sleptons, quarks/squarks or charged leptons/W bosons (through bilinear or trilinear RPV couplings5 ), which contribute all to the decay channel χ0

1 → γν m i . Such a channel is always kinematically open in the NMSSM, even in the case of the specific spectrum discussed just above, and could thus render

= λ s =530 GeV, M 1 =66 GeV, M 2 =133 GeV, M 3 =500 GeV, A λ =1280 GeV, A κ =0, A t = A b = A τ =-2.5
TeV, m ˜ ± =200 GeV and m q=1 TeV (universally). The A parameters are the trilinear scalar soft supersymmetry breaking parameters which do not affect the neutralino mass matrix. The low values of M i 's allow to get a neutralino LSP. Note also the particularly low values of m a 0 1 , characteristic of the NMSSM. For completeness, we give the employed RPV parameters: Λ 1 / s = 9 10 -7 and µ 1 = 6 10 -6 GeV.

the neutralino unstable the cosmological scale point of view. Assuming that no important destructive interferences occur among the several triangular contributions mentioned above, rough estimates indicate that the suppression provided by RPV terms [reproducing the correct neutrino mass scale] and by loop factors are not sufficient to make the LSP neutralino stable with respect to t 0 (whatever is the neutralino composition). However, precise loop calculations of all contributing reactions, beyond the scope of our study, are necessary to conclude definitively on this aspect, taking into account notably effects of heavy sfermion loop factors. The other important final comment is that three-body decay channels like χ0

1 → + -ν m (via an off-shell Z/W boson [45,46] or an off-shell Higgs boson [45]) are systematically open. From order of magnitude estimates, it tuns out that such channels are expected to render the lightest neutralino clearly unstable.

V Gravitino LSP

V.1 RPV gravitino decays

In the case of a gravitino LSP and RPV couplings of type Λ i and µ i , the gravitino can decay into the EW gauge bosons and the (pseudo)scalar Higgs fields [as shown in Fig.( 5)-( 6)]. We give below the obtained associated partial decay widths. All formulas given for the gravitino width, and above for sneutrino and neutralino widths, result from original calculations. Nevertheless, some of the gravitino decay amplitudes, into γ, Z 0 , W + and h 0 k , were computed with other conventions by the authors of Ref. [20,23,47] in the case of the MSSM (see also Ref. [48] for a more generic calculation approach). Note in particular that within the present framework of the NMSSM, the numerical results for the gravitino decay into the pseudoscalar a 0 1 -which can be quite light there -might differ significantly from the MSSM and thus decrease dangerously the gravitino lifetime. Moreover, within the NMSSM, new contributions arise for the decays G → h 0 k ν m i and G → a 0 k ν m i due to the presence of the s superfield [see the right Feynman diagram of Fig. (6)]. Finally, in the NMSSM, the neutralino mass matrix (and in turn the matrix elements N ij involved in gravitino amplitudes) takes a specific form, especially in the context of the GMSB (see Section II.3). • Gravitino decay into the photon γ and neutrino ν m i -the dependency on the neutrino mass is omitted (see first calculation in Ref. [5]):

0 ~~0 0 3 G ~G γ ν v i i m ν v i i m Z B ,W ~~0 0 3 B ,W
Γ( G → γν m i ) = 1 64π |U ν m i γ | 2 m 3 3/2 M 2 P l ( 16
)
where

U ν m i γ = N i1 cosθ W + N i2 sinθ W , i = 1, 2,
3 labels the neutrino mass eigenstate and 1 (2) corresponds to the B0 ( W 0 3 ) component. M P l is the reduced Planck mass: M P l 2.4 10 18 GeV. • Gravitino decay into the Z boson and neutrino ν m i (see original calculation in Ref. [24]): where i = 1, 2, 3 labels the neutrino mass eigenstate and 1 (2) corresponds to the B0 ( W 0 3 ) component.

Γ( G → Z 0 ν m i ) = 1 64π |U ν m i Z | 2 m 3 3/2 M 2 P l 1 - m 2 Z m 2 3/2 2 1 + 2 3 m 2 Z m 2 3/2 + 1 3 m 4 Z m 4 3/2 (17) s ~h ~u 0 k 0 , a h k 0 ν i i m m G ~k 0 , a h k 0 v u ν i m G
• Gravitino decay into the W boson and charged lepton ∓ i (see original calculation in Ref. [24]):

Γ( G → W + - i ) = 1 32π |U i W | 2 m 3 3/2 M 2 P l 1 - m 2 W m 2 3/2 2 1 + 2 3 m 2 W m 2 3/2 + 1 3 m 4 W m 4 3/2 (18)
where numerically U i W is expected to be of a comparable order to U ν m i Z as already discussed. Similarly, here, i = 1, 2, 3 labels the charged lepton mass eigenstate.

• Gravitino decay into the scalar Higgs boson h 0 k and neutrino

ν m i Γ( G → h 0 k ν m i ) = 1 384π |N i3 S k1 + N i4 S k2 + N i5 S k3 | 2 m 3 3/2 M 2 P l 1 - m 2 h 0 k m 2 3/2 4 (19)
with i = 1, 2, 3 labeling the neutrino mass eigenstate, the numbers 3, 4, 5 corresponding respectively to the h0 u , h0 d , s components and 1, 2, 3 respectively to the real h 0 u , h 0 d , s components.

• Gravitino decay into the pseudoscalar Higgs boson a 0 k and neutrino

ν m i Γ( G → a 0 k ν m i ) = 1 384π |N i3 P k1 + N i4 P k2 + N i5 P k3 | 2 m 3 3/2 M 2 P l 1 - m 2 a 0 k m 2 3/2 4 ( 20
)
still with k ≡ 1, 2. Similarly, i = 1, 2, 3 labels the neutrino mass eigenstate, the numbers 3, 4, 5 correspond respectively to the h0 u , h0 d , s components and 1, 2, 3 respectively to the imaginary h 0 u , h 0 d , s components.

• Gravitino decay into the charged Higgs boson H ± and charged lepton

∓ i Γ( G → H + - i ) = 1 384π |U i2 | 2 m 3 3/2 M 2 P l 1 - m 2 H + m 2 3/2 4 (21)
i = 1, 2, 3 labels the charged lepton mass eigenstate. U i2 represents the components of the three charged lepton eigenstates ± i into the charged higgsino h± .

We note that the partial widths for the charge conjugated final states are equal, which means for instance that Γ( G → γ ν) = Γ( G → γν). Later, we will thus refer to the total gravitino decay width as:

Γ total ( G) = 2[Γ( G → γν m ) + Γ( G → Z 0 ν m ) + Γ( G → W + -) + Γ( G → h 0 ν m ) + Γ( G → a 0 ν m ) + Γ( G → H + -)].
V.2 Gravitino stability

V.2.1 Scenario I: almost all decay channels open

Let us first consider a NMSSM scenario which is a priori dangerous from the cosmological point of view: we choose an heavy gravitino, so that the 5 first of the 6 types of decay channels described above are kinematically open (parameters allowing only the decays into the first h 0 1 and a 0 1 ), and a quite light pseudoscalar a 0 1 (as is possible in the NMSSM), tending to increase the phase space for the partial width Γ( G → a 0 1 ν m 1 ).

First, the considered region of the parameter space must be such that the gravitino is the LSP. We thus take rather large values for the gaugino masses M 1 , M 2 and M 3 :

M 1 =300 GeV, M 2 =600 GeV, M 3 =2
TeV to push the neutralino mass to higher values. For the same reason, we take m ˜ ± =300 GeV and m q=1 TeV. We fix the other NMSSM parameters at: λ=0.3, µ=237 GeV, κ=0.35, A κ =-30 GeV, A t = A b = A τ =-2500 GeV (trilinear soft parameters). We allow tanβ and M A to vary accordingly to a scan performed for tanβ = 4 → 14, M A = 250 → 300 GeV. M A represents the pseudoscalar mass in the MSSM but it is an effective parameter in the NMSSM. It is somewhat equivalent to the second (pseudo)scalar masses in the NMSSM and to the charged Higgs mass which are almost degenerated in this model:

M A m h 0 2 , m a 0 2 , m H ± .
It is related to the other ones via the minimization equations. In the Fortran code NMHDECAY, we can choose between M A or A λ as input parameters. This scan is performed through the NMHDECAY code so that the generic NMSSM constraints mentioned above are satisfied for the selected parameters. As a first step, we present in Fig. (7) the output masses m χ0 1 , m ν1 , m h 0 1 and m a 0 1 , obtained via this scan, as functions of tanβ. Among the possible points of Fig. (7), we choose the one corresponding to tanβ=8 and M A =275 GeV leading to m χ0 1 = 219.5 GeV, m ν =293.5 GeV, m h 0 1 = 116.7 GeV, m h 0 2 = 259.9 GeV, m h 0 3 = 548.8 GeV, m a 0 1 = 25.3 GeV, m a 0 2 = 303.5 GeV and m H ± = 265.5 GeV. The gravitino mass is fixed at m 3/2 =200 GeV so that G is well the LSP. Then the channels G

→ h 0 k ν m , G → a 0 k ν m [with k ≥ 2]
and Γ( G → H ± ∓ ) are kinematically closed but all the other ones are open. Now, in the one lepton flavor approximation we choose the RPV parameter values Λ 1 / s = 2 10 -6 and µ 1 = 10 -5 GeV leading to m 2 ν 1 = 1.04 10 -22 GeV 2 which is reasonable from the point of view of experimental neutrino data.

Finally, for the chosen NMSSM parameters and RPV couplings, the induced RPV neutrino-neutralino mixings give rise to the partial widths Γ( G → γν m ) = 1.95 10 47 10 -45 GeV and Γ( G → a 0 1 ν m ) = 2.08 10 -46 GeV. It is important to note that the relative smallness of Γ( G → γν m ) reflects in particular

-47 GeV, Γ( G → Z 0 ν m ) = 8.80 10 -47 GeV, Γ( G → W + -) = 1.94 10 -46 GeV, Γ( G → h 0 1 ν m ) = 9.
M 1 =300 GeV, M 2 =600 GeV, M 3 =2 TeV, A κ =-30 GeV, A t = A b = A τ =-2500 GeV, m ˜ ± =300 GeV (universal), m q=1
TeV. The points are obtained from a scan performed on the two parameters tanβ = 4 → 14 and M A = 250 → 300 GeV, using the NMHDECAY code [START_REF] Ellwanger | [END_REF]. The chosen RPV parameters are: Λ 1 / s = 2 10 -6 and µ 1 = 10 -5 GeV.

the smallness of the photino component for the neutrino mass eigenstate. The corresponding total gravitino width is Γ total ( G) = 2.00 10 -44 GeV giving rise to a gravitino lifetime τ G 76.3 t 0 . This result illustrates the feature that within the NMSSM, and for RPV couplings that generate the correct neutrino mass scale, a gravitino LSP is stable (with respect to t 0 ) even for a gravitino mass up to 200 GeV typically in agreement with supergravity (or mixed with GMSB) scenarios. Indeed, more generally speaking Γ total ( G) remains of the same order of magnitude as for the above NMSSM parameters, if m ν ∼ 10 -11 GeV (as imposed by data) and m χ0 1 ∼ m 3/2 ∼ 10 2 GeV (as natural in supergravity-like models). The reason, taking e.g. Γ( G → γν m ), is that the orders of m ν and m χ0 1 fix the order of U ν m γ (see Eq.( 6) in Ref. [5]) and in turn the partial gravitino decay width (c.f. Eq.( 16)) once m 3/2 is chosen. Furthermore one expects U ν m Z U W U ν m γ in Eq.( 17)-( 18) [24]. Similarly, the partial decay widths into the (lightest) neutral Higgs bosons are fixed by m h 0 1 , m a 0 1 , m 3/2 as well as the amount of neutrino components in the higgsino and singlino (c.f. Fig. . If kinematically open, the decays into the h 0 2 /a 0 2 Higgs bosons have rates of similar order (for identical parameters). For a a 0 1 boson heavier than in the above set of parameters, Γ( G → a 0 1 ν m ) decreases but the total width remains around ∼ 10 -44 GeV. Finally, if allowed, the channel G → H + -is expected to reach a width of the same order as G → h 0 1 ν m and hence would not significantly modify Γ total ( G). It has been recently [49] pointed out that the widths for G → ± W ∓ → ± f f and G → ν m Z → ν m f f together can reach 10 2 × Γ( G → γν m ) restricting oneself around parameter domains where m χ0

1 ∼ m 3/2 ∼ 10 2 GeV6 . It means that the new three-body decay widths can reach ∼ 10 -45 GeV, from our present results, and this does not modify Γ total ( G) significantly.

V.2.2 Scenario II: a few decay channels open

To be more general than in the approach of Section V.2.1, we consider here the case where the gravitino is the LSP and is lighter than the W boson as well as the (pseudo)scalar Higgs fields [rather than considering some given points of the parameter space]. Then the RPV channel through the process G → γν m i remains open. Imposing the width Γ( G → γν m i ) to be smaller than the critical value of 1. 52 10 -42 GeV leads to the upper bound on the gravitino mass m 3/2 3 TeV for |U γν | ∼ 10 -7 (see Eq.( 16)) as imposed by the neutrino mass scale if m χ0 1 10 2 GeV. For example, let us consider a given point of parameter space for which such a neutralino mass is realized. We take randomly the point of Fig. ). The exact bound resulting from the gravitino lifetime in this precise case is m 3/2 < 2.7 TeV. Taking into account the other allowed channels, G → ± f f and G → ν m f f [f ≡ fermions] (see Ref. [49]), this bound changes to m 3/2 650 GeV -3 TeV as long as m χ0 1 ∼ 10 2 GeV. Since we are in a situation where m 3/2 < m W , this bound is respected. From the general point of view, we conclude that the NMSSM with phenomenologically interesting RPV couplings implies a gravitino LSP which is systematically stable (with respect to t 0 ) if one takes its mass to be smaller than m W , m h 0 1 and m a 0 1 . This quite general conclusion applies to supergravity (or mixed with GMSB) scenarios where m 3/2 ∼ 50 GeV is realistic, as well as to pure GMSB scenarios in which possibly m 3/2 ∼ 1 eV. Indeed, within the GMSB framework, the new form of the neutralino mass matrix (see Section II.3) leads to values of U ν m i γ extremely close to the above case [and hence to same bounds on m 3/2 ] for the realistic neutrino mass scale, if one chooses e.g. the typical characteristic value µ = 100 GeV. Note that in GMSB, the gravitino is clearly automatically lighter than W , h 0 1 and a 0 1 . Note also that the channel G → γν m 1 is systematically open (in the realistic 3 flavor case) since within our neutrino scenario m ν 1 = 0 at the tree level.

V.2.3 Scenario II extended to 3 neutrino flavors

There exist sets of NMSSM parameters [passing the NMHDECAY constraints] and RPV couplings which reproduce the squared neutrino mass eigenvalue differences measured in oscillation experiments. More precisely, the updated three-flavor analyzes based on a global fit including results from solar, atmospheric, reactor and accelerator oscillation experiments lead to (at the 3σ level):

Indeed, this complete set of parameters yields the following three neutrino mass eigenvalues at tree level: m ν 1 = 0 eV, m ν 2 = 0.00867 eV, m ν 3 = 0.04670 eV (23) in agreement with the above experimental intervals for ∆m

2 β = 3 i=1 |U ei | 2 m 2 ν i
where U ei is the leptonic mixing matrix [51].

The point of parameter space considered in this section is the same as the one considered in the previous section (i.e. Section V.2.2), namely: the point of Fig.( 4) with κ=0.1, and we still assume a situation where the gravitino LSP has a mass smaller than m W , m h 0 1 and m a 0 1 (scenario II). For this point, and with the RPV couplings of Eq.( 22), we obtain for the gravitino decay channel into a photon: The resulting bound m 3/2 < 2.5 TeV obtained here is close to the bound m 3/2 < 2.7 TeV of previous section, for the channel G → γν m , which means that moving to the three flavor case should not affect significantly the conclusions. Therefore we conclude that, in the case of three neutrino flavors as well, a gravitino LSP is always sufficiently stable (with respect to t 0 ) if its mass is weaker than m W , m h 0 1 and m a 0 1 . This conclusion on the 3 neutrino flavor case is not trivial in the sense that, starting from the 1 flavor situation, the variation of the U ν m 1 γ value when extending to 3 neutrino flavors cannot be easily predicted due to the rich structure of the whole RPV neutralino mass matrix. Moreover, there is no simple argument to deduce the values of U ν m 2 γ and U ν m 3 γ from the U ν m 1 γ element (encoding the neutrinogaugino mixing) because of the multiple mixing types between the different flavors of neutrinos and the various neutralino states. Nevertheless, we have checked that, as expected, one obtains numerically the following hierarchy among the following matrix elements:

U ν m 1 γ = -4.
|N 1j | < |N 2j | < |N 3j | [j = 1, 2]
. This is interpreted by the fact that here the neutrino eigenstate ν m 3 is heavier than ν m 2 (and in turn m ν 2 > m ν 1 ) since the massive neutral gaugino [ B0 , W 0 3 ] components of ν m 3 are larger than in the ν m 2 state (in turn, than in ν m 1 ). 7 Let us note that we have chosen the specific set (22) of RPV parameters to illustrate on an explicit example that the three flavor neutrino data can be reproduced. Nevertheless, the orders of magnitude in Eq.( 22) of the six effective RPV parameters are general in the sense that those are imposed systematically by the experimental ranges for ∆m 2 21 and ∆m 2 31 as soon as m χ0 1 ∼ 10 2 GeV (as occurs in supergravity and GMSB models). Hence the conclusions at the end of this Section V.2.3, which are based on these orders of magnitude, can also be generalized to any region of the parameter space where m χ0 1 ∼ 10 2 GeV.

VI Discussion and conclusions

In the context of the NMSSM with the presence of RPV couplings large enough to generate realistic neutrino masses, a gravitino LSP of mass O(10 2 ) GeV -as appears in supergravity models -is sufficiently stable from the point of view of the age of the universe. Nevertheless, the gravitino lifetime that we obtain is of order ∼ 10 19 sec. Now even in a supergravity situation with m 3/2 80 GeV, where the opened decay channels G → γν m , G → ± f f and G → ν m f f are thus reduced, the total width looses typically four orders of magnitude only. Hence, the HEAT excess in the positron fraction [7] and the exotic positron source apparently detected by PAMELA [20] (which both require, in standard solutions, m 3/2 ∼ 100 -200 GeV and τ G ∼ 10 26 sec) do not seem to be simultaneously explainable by the present RPV scenario reproducing the neutrino masses. The reason is that the neutrino-neutralino mixing necessary to create large enough neutrino masses seems to induce too large gravitino RPV decay widths. Moreover, this result that the gravitino lifetime is systematically smaller than the lifetime needed to explain the PAMELA excess with a gravitino dark matter also means that, in our dark matter scenario, the gravitino decays produce always too large fluxes which are excluded by PAMELA in particular. Hence our conclusion is that a gravitino LSP cannot be a good dark matter candidate if there exist significant RPV mixing terms (reproducing the neutrino masses).

In the case of the existence of a special supergravity scenario -like e.g. some hybrid supergravity-GMSB models -where one could have m 3/2 below ∼ 10 GeV (in order to sufficiently reduce Γ( G → γν m )) and M 1 ∼ M 2 ∼ 10 2 GeV (suppressing the Γ( G → ± f f , ν m f f) contribution [49]), the gravitino lifetime would then be above the PAMELA limit of ∼ 10 26 sec, making G a stable and viable dark matter candidate [with RPV couplings producing the neutrino masses]. Now if the supersymmetry breaking relies instead on a pure GMSB mechanism, the neutralino mass matrix includes new elements and the gravitino mass can decrease typically to eV scale. Then, the gravitino is clearly a stable LSP since we obtain a bound m 3/2 650 GeV -3 TeV from the requirement that its lifetime exceeds the age of the universe, if the photino component of the neutrino [of value comparable to the above supergravity case] is sufficiently large to induce correct neutrino masses. In this GMSB case, the gravitino lifetime is ∼ 10 56 sec. Hence the PAMELA data cannot be explained with a gravitino dark matter. However, the PAMELA flux excess can have e.g. alternative natural and astrophysical explanations, like in electron-positron pairs produced by nearby pulsars [52] 8 . By consequence, the GMSB framework could allow for the existence of a good gravitino dark matter candidate within the RPV-NMSSM.

Finally, we have checked numerically the robustness of the above conclusions on the gravitino stability when extending to the realistic case of three neutrino flavors. To illustrate this, we have selected some RPV-NMSSM parameter sets in agreement with the various constraints implemented in the NMHDECAY code and reproducing the squared neutrino mass eigenvalue differences measured in oscillation experiments.

Concerning the stability of a sneutrino LSP with respect to t 0 , within RPV versions of the NMSSM reproducing the wanted neutrino mass scale, the result is negative. The lightest neutralino LSP is also not expected to be stable with respect to t 0 .

Finally, we comment that all these difficulties one faces for having a viable dark matter candidate in RPV models are to be bridged with the astrophysical problems one encounters in baryogenesis when RPV interactions are indeed turned on to generate the neutrino masses [54].

  cos β and s β = sin β), and µ = λ s . The µ term appears only within the GMSB framework [see Eq.(2)].

  If the LSP is the lightest neutralino and the RPV couplings of type Λ i and µ i are present, then the lightest neutralino can only decay into the W,Z bosons or into the (pseudo)scalar Higgs fields [as illustrated by the Feynman diagrams of Fig.(2)-(3)] if one restricts oneself to the dominant two-body decay channels. The obtained partial decay widths are given as follows.

Figure 2 :

 2 Figure

Figure 3 :

 3 Figure 3: Feynman diagram for the neutralino decay modes into scalar Higgs fields, χ01 → h 0 k ν m j , and pseudoscalar Higgs fields, χ01 → a 0 k ν m j . The cross indicates which neutralino components are coupled.

Figure 4 :

 4 Figure 4: Masses m χ0 1 [purple curve], m h 0 1 [green curve] and m a 0 1 [red curve] as functions of κ. We fix the other NMSSM parameters: tanβ=2, λ=0.7, µ= λ s =530 GeV, M 1 =66 GeV, M 2 =133 GeV, M 3 =500 GeV, A λ =1280 GeV, A κ =0, A t = A b = A τ =-2.5 TeV, m ˜ ± =200GeV and m q=1 TeV (universally). The A parameters are the trilinear scalar soft supersymmetry breaking parameters which do not affect the neutralino mass matrix. The low values of M i 's allow to get a neutralino LSP. Note also the particularly low values of m a 0 1 , characteristic of the NMSSM. For completeness, we give the employed RPV parameters: Λ 1 / s = 9 10 -7 and µ 1 = 6 10 -6 GeV.

Figure 5 :

 5 Figure 5: Feynman diagrams for the gravitino decay processes into a photon, G → γν m i , and a Z boson, G → Z 0 ν m i . v i denotes a sneutrino vev insertion.

Figure 6 :

 6 Figure 6: Feynman diagrams for the gravitino decay processes into scalar Higgs fields, G → h 0 k ν m i , and pseudoscalar Higgs fields, G → a 0 k ν m i . We use the symbolic notation m i = µ i + λ i s combining a direct µ i mixing and a s insertion. v u indicates an up Higgs vev insertion.

Figure 7 :

 7 Figure 7: Masses m χ0 1 [purple points], m ν (universal) [blue points], m h 0 1 [red points] and m a 0 1 [green points] as functions of tanβ. We use: λ=0.3, κ=0.35, µ=237 GeV, M 1 =300 GeV, M 2 =600 GeV, M 3 =2 TeV, A κ =-30 GeV, A t = A b = A τ =-2500 GeV, m ˜ ± =300 GeV (universal), m q=1TeV. The points are obtained from a scan performed on the two parameters tanβ = 4 → 14 and M A = 250 → 300 GeV, using the NMHDECAY code[START_REF] Ellwanger | [END_REF]. The chosen RPV parameters are: Λ 1 / s = 2 10 -6 and µ 1 = 10 -5 GeV.

  (6)) [and thus by m ν and m χ0 1 ]

  (4) with κ=0.1. Then, m χ0 1 = 67.8 GeV, m ν =193.9 GeV (universal), m h 0 1 = 128.7 GeV and m a 0 1 = 83.2 GeV [insuring also that the gravitino decays into the (pseudo)scalar Higgs fields are forbidden]. For this point, one gets U ν m i γ = 1.94 10 -7 with Λ 1 / s = 9 10 -7 and µ 1 = 6 10 -6 GeV which reproduce the correct neutrino mass scale (m 2 ν 1 = 5.73 10 -22 GeV 2

7 . 1 7 µ 1 = 3 10 - 5

 7171105 ≤ ∆m 2 21 ≤ 8.3 [10 -5 eV 2 ] and 2.0 ≤ ∆m 2 31 ≤ 2.8 [10 -3 eV 2 ] [13]. In our notations e.g. ∆m 2 31 = |m 2 ν 3 -m 2 ν 1 |. Values lying in those two intervals arise for instance for the point of Fig.(4) if κ=0.1 and GeV, µ 2 = 1.5 10 -4 GeV, µ 3 = 2.5 10 -4 GeV Λ 1 / s = 8.5 10 -7 , Λ 2 / s = 1 10 -7 , Λ 3 / s = 1.5 10 -7 .

3 γ

 3 74 10 -11 , U ν m 2 γ = -6.61 10 -8 and U ν m = 2.30 10 -7 , recalling the definition U ν m i γ = N i1 cosθ W + N i2 sinθ W where i = 1, 2, 3 labels the neutrino mass eigenstate and 1 (2) corresponds to its B0 ( W 0 3 ) component. Those numbers imply the limit m 3/2 < 2.5 TeV resulting from the condition2 i Γ( G → γ νm i ) < 1.52 10 -42 GeV.

  2 21 and ∆m2 31 since in the present RPV model one has ∆m2 21 = m 2 ν 2 = 7.51 10 -5 eV 2 and ∆m 2 31 = m 2 ν 3 = 2.18 10 -3 eV 2 . From the cosmological point of view, the neutrino mass eigenvalues in(23) satisfy the bound extracted from WMAP and 2dFGRS galaxy survey (depending on cosmological priors): Finally, the above neutrino mass eigenvalues are perfectly compatible with the limits extracted from the tritium beta decay experiments (95% C.L.): m β ≤ 2.2 eV [Mainz] and m β ≤ 2.5 eV [Troitsk], this effective mass being defined as m

	3 i=1 m ν i	0.7 eV
	[50].

Such a possibility was in particular motivated by interpreting the well-known LEP excess at

2.3σ in the Z + 2b channel -via the production of a light Higgs boson (m h 99 GeV)[14] -but this NMSSM interpretation seems to be excluded by a recent ALEPH analysis[START_REF] Cranmer | Searching Higgs decaying to 4 taus[END_REF].

Similarly, there exists a trilinear RPV soft term, arising in the NMSSM, which involves the singlet scalar field; this one is of the form shu Li in the Lagrangian[29]. Its direct effect on the present study will be discussed in Section III.2.

Besides, note that the contributions to the charged lepton masses should be of the order of magnitude of the neutrino mass scale and in turn negligible compared to the direct Yukawa contributions.

In this part, we study the sneutrino RPV decays for completeness, even if a left-handed sneutrino LSP as a candidate for dark matter has been already excluded by direct dark matter searches for most of the realistic sneutrino mass ranges.

The contributions from trilinear RPV interactions were estimated in Ref.[44].

This ratio result is not expected to be significantly changed when extending the MSSM to the NMSSM.

We also note that, anyway, the mentioned interpretation[7] of the HEAT anomaly predicts an antiproton flux which tends to be too large, although the prediction suffers from significant uncertainties. Besides, electron and positron fluxes from gravitino decays (whatever are the gravitino characteristics) cannot explain[20] both the PAMELA positron fraction and the electron plus positron flux measured by Fermi LAT[53].

Acknowledgments

The authors thank P. E. Larre and Y. Mambrini for useful discussions. This work is supported by the HEPTOOLS network, the A.N.R. TAPAMS and the A.N.R. LFV-CPV-LHC under project NT09 508531.


M.D.; Tomi Juhani Mölkänen
email: tomi.j.molkanen@hus.fi

M.Sc. ; Annina Rostila

M.D Eeva Ruotsalainen

Ph.D ; Mervi Alanne

Ph.D.; Markus Perola

Ph.D Asko Järvinen

Mervi Alanne

Genetic polymorphism of C-reactive protein (CRP) gene and a deep infection focus determine maximal serum CRP level in Staphylococcus aureus bacteremia

Keywords: Staphylococcus aureus bacteremia, C-reactive protein, single nucleotide polymorphisms Staphylococcus aureus bacteremia, C-reactive protein, single nucleotide polymorphisms

CRP is widely used in early detection of sepsis or organ dysfunction. Several SNPs in the CRP gene are shown to be associated with variability of basal CRP. To clarify the effect of SNPs to CRP in systemic infections, we compared genetic and clinical data on patients with Staphylococcus aureus bacteremia(SAB). Six SNPs in the CRP gene (rs2794521, rs30912449, rs1800947, rs1130864, rs1205 and rs3093075) were genotyped in 145 patients and analyzed for associations with CRP and clinical outcomes. We found minor A-allele of triallelic SNP rs30912449 (C>T>A) and presence of a deep infection focus independently associate to the higher maximal CRP during the first week of SAB. Median of the maximal CRP in patients who had the A-minor allele was 282 mg/L [IQR, 169 mg/L) but only 179 mg/L (IQR, 148 mg/L) in patients without this allele (P=0.004) and CRP in patients who had deep infection focus was higher 208 mg/L (IQR, 147mg/L) than in other patients 114 mg/L (IQR, 121 mg/l) (P<0.0001). Mortality, degree of leucocytosis, time to defervescence or number of deep infections were not affected by SNPs. The maximal CRP in SAB was determined by variation in the CRP gene and by presence of deep infection focus.

Answer to question 1.)Each patient was included into the study only once. This has been added in the methods in the paragraph "patient population" on Page 4 lines 20-21.

2. page 7, lines 19-26; Table 2; and online resource supplementary table. I am concerned by the number of statistical tests that were performed -I count 33 statistical tests in the supplementary table andtable 2. By chance alone, one would expect that 1/20 might be found to be significant.

Is it possible to adjust the finding for the number of tests performed? If not, or if this adjustment produces a p-value of >0.05 for the observed significant associations, this should be noted as a limitation of the study.

Answer to question 2.) Multiple comparisons has now been analysed by False-discovery rate (FDR) estimation. It has been included into the statistics section of methods page 6 lines 19-20. Results from FDR analysis have been shown in tables 2 and 4. Answer to question 3.) Table 2 includes only 116 patients because genotyping in laboratory of SNP rs3091244 was successful only in 116 patients. This specific triallelic SNP turned out to be technically much more challenging to sequence compared to all other biallelic SNPs studied. [START_REF] Lobo | C-reactive protein levels correlate with mortality and organ failure in critically ill patients[END_REF]. Is there any information as to the antimicrobial or surgical treatment provided patients (surgical intervention or even the placement of a central venous catheter could certainly increase the CRP)?

Table 2. Why do the data for this

The duration of symptoms prior to the onset of therapy? These are important covariates to consider in assessing CRP levels. If information is not available, these may be important limitations of the study.

Answer to question 4.) Surgical procedures during the first week have been analysed as suggested by the referee and were found not to affect the maximal CRP level. However, we analysed also the effect of presence of a deep infection focus on the maximal CRP level and found this association highly significant. Patients with deep infection foci have been added into table 3 and this information into results on Page 7 lines 13-18 and into Table 4. This finding has been further discussed in a new written part of discussion on Page 4, in paragraph two. We considered this finding of so high importance that it has resulted into modification of our title as well. As a limitation of our study duration of symptoms prior the onset of therapy could have had certain effect on maximal CRP, but this data was not available for analysis. Added in Discussion part paragraph three.

5.

Are there any data available on the genetic characteristics of the S. aureus strains -at a minimum their antibiograms -as characteristics of the bacteria may have been a confounder in the maximum CRP level during the first week of therapy.

Answer to question 5.) The procedures named above in answer 4. and the possible effect of S. aureus strains on maximal CRP levels have been discussed in a new paragraph three in the Discussion part on page 8.

6. Lines 42-45. These sentences are not clear. Was BMI independently associated with maximal CRP in the same multivariate model as the rs3091244 A-minor allele carriership? Table 4 shows that they were apparently in the same model. If so, was their interaction a critical factor in the maximal CRP level -either alone or in the multivariate model? Answer to question 6) When results were re-evaluated Association of BMI or age to maximal CRP appeared not to be significant in the univariate or multivariate analysis, when the limit of statistical significance used was reset to p<0.05.

Due to invaluable comments critical interactions between one explanatory variable: history of intravenous drug abuse (IDU) with both BMI and age were detected and no independent effect of IDU and age to maximal CRP level really existed. In univariate analysis detected association of IDU to maximal CRP level was also due to critical interaction with existing deep infection focus and if IDU was added to multivariate analysis with significant explanatory variables it resulted in instability of the statistical model and no clear benefit was achieved. Due to substantial collinearity with IDU (IDUs were younger and also their BMI was much lower compared to other patients) age and BMI were omitted from the multivariate analysis. And finally determinants for the maximal CRP appeared to be presence of deep infection focus and the A-minor allele of rs30912449.

The unclear sentences have been omitted in the revised version of the manuscript.

7. Table 1. While information about the 145 patients is interesting, it would be more relevant to compare the indicated variables for these 145 patients with the larger group of 430 patients from the original study. This would address the question: How representative of patients generally presenting with SAB were the 145 patients included in this manuscript?

Answer to question 7) The demographic and underlying factors of our subgroup of patients have been compared to the original study population of 430 patients in the Discussion part paragraph three of the manuscript (Page 8). No significant differences between our patient population and the original patient material were found. Unfortunately, due to ethical regulations related to gene sampling all patients in the original data set could not be included into the present analysis. We found the comparison of the two patient populations confounding for the reader in a tabular form. After all, only data from the subgroup of patients is presented in the present manuscript.

Introduction

C-reactive protein (CRP) is rapidly synthesized in liver as a part of acute phase response following infection, injury or trauma [START_REF] Black | C-reactive protein[END_REF]. In clinical practice, rising CRP has been found useful in detection of sepsis or organ dysfunction [START_REF] Povoa | C-reactive protein: A valuable marker of sepsis[END_REF]. Changes in CRP levels have been used to monitor the treatment response, whereas the decline of CRP level is one of the earliest markers of improved condition [START_REF] Povoa | Early identification of intensive care unit-acquired infections with daily monitoring of C-reactive protein: A prospective observational study[END_REF]. Although CRP seems to be a good marker of systemic inflammation and a valuable clinical tool in severe infections [START_REF] Lobo | C-reactive protein levels correlate with mortality and organ failure in critically ill patients[END_REF], its exact role in defence against microbes remains incompletely understood [START_REF] Agrawal | The protective function of human C-reactive protein in mouse models of streptococcus pneumoniae infection[END_REF].

Based on family and twin studies, it is estimated that genetic factors account for 35% to 40% of the variance of the changes in basal CRP levels [START_REF] Pankow | Familial and genetic determinants of systemic markers of inflammation: The NHLBI family heart study[END_REF][START_REF] Retterstol | A twin study of C-reactive protein compared to other risk factors for coronary heart disease[END_REF][START_REF] Macgregor | Genetic effects on baseline values of C-reactive protein and serum amyloid a protein: A comparison of monozygotic and dizygotic twins[END_REF]. The CRP gene is located on chromosome 1q32 and it consists of 2 exons and 1 intron [START_REF] Woo | Characterization of genomic and complementary DNA sequence of human C-reactive protein, and comparison with the complementary DNA sequence of serum amyloid P component[END_REF][START_REF] Walsh | Fine mapping of the human pentraxin gene region on chromosome 1q23[END_REF]. Several SNPs in the CRP gene are shown to be associated with basal CRP level variability [START_REF] Hage | C-reactive protein gene polymorphisms, C-reactive protein blood levels, and cardiovascular disease risk[END_REF]. Moreover, in the Women's Genome Health Study seven genetic loci were shown to be associated with circulating CRP levels in a genome-wide setting, including the CRP locus and six other loci with a trans-effect [START_REF] Ridker | Loci related to metabolic-syndrome pathways including LEPR,HNF1A, IL6R, and GCKR associate with plasma C-reactive protein: The women's genome health study[END_REF]. Some SNPs in the CRP gene might also be associated with an increased risk for ischemic stroke [START_REF] Morita | Association study between C-reactive protein genes and ischemic stroke in japanese subjects[END_REF], for reduced arterial compliance [START_REF] Eklund | C-reactive protein genetics is associated with carotid artery compliance in men in the cardiovascular risk in young finns study[END_REF], or even for risk of getting type 2 diabetes mellitus [START_REF] Zee | C-reactive protein gene variation and type 2 diabetes mellitus: A casecontrol study[END_REF]. However, the causative effect of the cis-acting SNPs on ischemic heart disease remains under some controversy [START_REF] Zacho | Genetically elevated Creactive protein and ischemic vascular disease[END_REF]. CRP gene SNPs have been related to susceptibility or worse outcome in some infections as well as other diseases. A CRP gene poly (GT) allele was associated with susceptibility to invasive pneumococcal disease [START_REF] Roy | Research pointers: Association of common genetic variant with susceptibility to invasive pneumococcal disease[END_REF] and CRP gene promoter region polymorphism rs2794521 with an increased mortality to invasive pneumococcal disease [START_REF] Eklund | Polymorphism of the C-reactive protein gene is associated with mortality in bacteraemia[END_REF]. However, little is known about how the polymorphism affects CRP levels stimulated by infection or trauma. Recently, lower CRP levels after coronary by-pass operation have been shown to be associated to minor allele of the synonymous coding SNP rs1800947 within exon 2 in the CRP gene [START_REF] Perry | C-reactive protein gene variants are associated with postoperative C-reactive protein levels after coronary artery bypass surgery[END_REF]. This data indicates that polymorphisms in the CRP gene might be an important factor in CRP response in severe systemic infections.

However, the effect of these variations on CRP levels in severe infections has not been evaluated. S. aureus is the second most common bloodstream isolate in all age groups and it still carries high overall mortality up to 39% [START_REF] Conterno | Risk factors for mortality in staphylococcus aureus bacteremia[END_REF][START_REF] Soriano | Pathogenic significance of methicillin resistance for patients with staphylococcus aureus bacteremia[END_REF][START_REF] Benfield | Increasing incidence but decreasing in-hospital mortality of adult staphylococcus aureus bacteraemia between 1981 and 2000[END_REF]. In practice, the clinical course of S. aureus bacteremia (SAB) is determined by its complications, particularly by the development of deep infections and thromboembolic events [START_REF] Mylotte | Staphylococcus aureus bacteremia: Predictors of 30-day mortality in a large cohort[END_REF][START_REF] Fowler | Clinical identifiers of complicated staphylococcus aureus bacteremia[END_REF][START_REF] Fatkenheuer | Long-term outcome and quality of care of patients with staphylococcus aureus bacteremia[END_REF]. The reported frequency of deep infection foci in SAB varies greatly, but in recent trial it was over 80% [START_REF] Ruotsalainen | Levofloxacin does not decrease mortality in staphylococcus aureus bacteraemia when added to the standard treatment: A prospective and randomized clinical trial of 381 patients[END_REF].

Therefore, SAB could be a suitable infection to study the possible effect of genetic polymorphisms to frequency of complications and CRP response. To assess the possible contribution of variations in the CRP gene to complications in SAB we genotyped six SNPs in the CRP gene and analyzed their possible correlation to CRP levels and to various clinical outcome parameters.

Methods

Patient population

A total of 430 patients with blood culture positive for S. aureus were enrolled consecutively in 5 university hospitals and in 7 tertiary care hospitals in Finland between January and May 1999, and between January 2000 and August 2002. The trial aimed primarily to examine the potential of two fluoroquinolones (trovafloxacin and levofloxacin) to reduce the high mortality and complication rates in SAB, when added to the standard treatment [START_REF] Ruotsalainen | Levofloxacin does not decrease mortality in staphylococcus aureus bacteraemia when added to the standard treatment: A prospective and randomized clinical trial of 381 patients[END_REF]. The exclusion criteria were: age younger than 18 years, imprisonment, proven or suspected pregnancy, breastfeeding, epilepsy, another bacteremia during the previous 28 days, polymicrobial bacteremia (≥3 microbes), history of allergy to any quinolone antibiotic, previous tendinitis during a fluoroquinolone therapy, prior fluoroquinolone use for more than 5 days before randomization, positive culture for S. aureus only from a central intravenous catheter, meningitis, neutropenia (< 0.5 × 10 9 cells/L) or failure to supply an informed consent. Patients with bacteremia due to methicillin-resistant S. aureus (MRSA) and a S. aureus strain resistant to any fluoroquinolone were also excluded [START_REF] Ruotsalainen | Levofloxacin does not decrease mortality in staphylococcus aureus bacteraemia when added to the standard treatment: A prospective and randomized clinical trial of 381 patients[END_REF]. Each patient was recruited into the study only once to avoid double inclusion in repeated bacteremia. The protocol was approved by the ethics committees of all study sites. Written informed consent was obtained from all patients or their representatives. Permission for genetic blood samples was asked for patients only from Helsinki University Central Hospital. Thus, the genetic blood samples were obtained from 151 of 158 patients from this study site. Blood samples from 3 patients could not be obtained for further analysis and 3 subjects were discarded due to technical problems with insufficient DNA yield. In total, 145 patients were included for this study from Helsinki University Central Hospital.

Definitions

Intravenous drug users (IDUs) were defined as patients who had injected drugs within the past 6 months before randomization based on a history taken on admission. SAB was hospital-acquired if the first positive blood culture was obtained ≥48 hours after admission or the patient was a resident in a long-term care facility or had attended haemodialysis within the preceding 2 months. Prognosis or severity of underlying diseases were characterized as healthy, nonfatal, ultimately or rapidly fatal disease according to the criteria of McCabe and Jackson [START_REF] Jackson | Profiles of pyelonephritis[END_REF]. Infection focus was documented by clinical, bacteriological, radiological, or pathological investigations. Intravenous catheter-associated bacteremia was defined using the guidelines of the Infectious Disease Society of America [START_REF] Mermel | Guidelines for the management of intravascular catheter-related infections[END_REF]. Deep infection focus was classified as endocarditis, pneumonia, deep-seated abscess, osteomyelitis, septic arthritis, meningitis, septic thrombophlebitis, mediastinitis, urinary tract infection, infection of a prosthetic device, or recurrent SAB as described in our previously published article [START_REF] Ruotsalainen | Clinical manifestations and outcome in staphylococcus aureus endocarditis among injection drug users and nonaddicts: A prospective study of 74 patients[END_REF]. Time to defervescence was recorded in days until axillary temperature was < 37.5 ˚C.

Serum CRP and leukocyte measurements

Blood samples for CRP measurement and leukocyte count were conducted on the day of positive blood culture for S. aureus, at randomization and every other day during the first week. Serum concentrations of CRP and leukocyte count were measured in the study site laboratory by standard laboratory methods. The serum or plasma (use of plasma instead of serum begun 18.3.2002) was subjected to automatic immunoturbidimetric analysis using analyzers 917 or Modular PP-analyzer (Hitachi Ltd, Tokyo, Japan) and Tina-quant CRP reagents (Roche Diagnostics, Tina-quant CRP). The normal value of CRP concentration was <10 mg/L for both methods. Leukocytosis was determined if white blood cell count was over 12 × 10 9 cells/L. DNA sequencing and genotyping DNA was extracted using phenol chloroform extraction method. DNA quantity was measured by absorbance at 260 nm and by Pico Green fluorescence assay (PicoGreen dsDNA Quantitation Kit, Molecular Probes, Carlsbad, USA). The detection of SNPs was performed by the analysis of primer extension products from amplified genomic DNA using Sequenom MassArray system with the homogeneous Mass Extension (hME) reaction (Sequenom, San Diego, USA). PCR and extension reactions were designed using MassARRAY design software and were carried out using 10 ng of template DNA. Six common SNPs of the CRP gene (rs27954521, rs3091244, rs1800947, rs1130864, rs1205 and rs3093075) were genotyped in all samples. SNPs were selected as haplotype bin tagging SNPs covering the entire gene as well as large parts of the untranslated regions of the 5´ and 3´ ends of the gene.

Statistics

Statistical analyses were performed using SPSS® version 14.0. (SPSS Inc., Chicago, IL, USA). Within each SNP locus genotypic frequencies were tested against values expected from Hardy-Weinberg proportion. The associations between categorical variables were analyzed by X 2 -test or Fisher´s exact test when appropriate. Preliminary analysis of normality was performed for continuous variables using the Kolmogorov-Smirnov and Shapiro-Wilk tests. For skewed variables heterogeneity of genotype groups was tested with Kruskal-Wallis H test. Linear regression univariate analyses were first performed to analyze association between each explanatory variable and outcome variable. Clinically meaningful explanatory variables showing association in univariate analysis (P<0.05) with the outcome were then included in the multivariate regression in order of strength of their correlation with the outcome variable. When a new explanatory variable was shown to contribute to the variance, influence on the variables already in the model and possible collinearity with them was assessed. All tests were 2-tailed and (P<0.05) were considered significant. For the multiple comparisons, the FDR estimation method was applied to association analyses.

Results

In total, 145 patients with SAB were included into this study. Majority of the patients were male (66%) and their mean age was 52 years (Table 1). A deep infection focus was found in 124 (86%) patients within the 3 months study period.

The genotype distributions of CRP gene SNPs did not deviate from the Hardy-Weinberg equilibrium (Online Resource). Among the six selected haplotype bin tagging SNPs, two are located in the promoter region rs2794521 and rs3091244, one is located in exon 2 rs1800947, one in 3´UTR region rs1130864, and two in 3´ flanking region rs1205 and rs3093075. All these SNPs were found to be in strong linkage disequilibrium (LD) by Haploview 3.2 program [START_REF] Benjamini | Controlling the false discovery rate: A practical and powerful approach to multiple testing[END_REF]. In particular, rs3091244 A-minor allele is in full LD with rs3093075 and rs3091244 T-minor allele with rs1130864. All CRP gene SNPs except rs1205 are within one haplotype block.

When SNPs of the CRP gene were analyzed in additive and recessive genetic models they were found not to affect the CRP levels at the time of S. aureus positive blood culture or 7 days after it. However, the maximal CRP levels during the first week of illness were significantly elevated in patients who had either A-minor allele of rs3091244 or T-minor allele of rs3093075 when compared to patients without minor alleles of these SNPs (Table 2). Due to strong LD between these two SNPs, the results reflect the same signal.

No differences were observed in occurrence of predisposing factors, underlying diseases and clinical manifestations or number of patients with a deep infection focus in patients with one copy of minor allele A of rs3091244 when compared to patients without this minor allele (Table 3). Presence of a deep infection focus and rs3091244 A-minor allele carriage were both found to be significantly associated to maximal CRP level during the first week and this effect persisted in multivariate analysis (Table 4). Associations were retained even after the multiple comparisons with FDR estimation using an FDR threshold <0.10 (Table 4). The median of the maximal CRP level in patients who had a deep infection focus was 208 mg/L (IQR, 147mg/L) but only 114 mg/L (IQR, 121 mg/l) in patients without it (P<0.0001). Other possible explaining factors like age, sex, BMI, surgical intervention or intravenous drug abuse were not found to be significant determinants of the maximal CRP level (Table 4).

Associations between any other parameter and SNPs were not detected when data was analyzed in additive and recessive models. The degree of leukocytosis, time to defervescence, number of any deep infection foci during the first week or three months, the number of patients who had intravenous-catheter-associated bacteremia, or who died within 28 days or three months were not influenced by polymorphisms.

Discussion

To our knowledge, this is the first study to investigate the association between CRP gene SNPs and CRP levels in acute phase of SAB. Polymorphisms in the CRP locus are shown to have strong influence on low basal CRP levels [START_REF] Hage | C-reactive protein gene polymorphisms, C-reactive protein blood levels, and cardiovascular disease risk[END_REF] but reports on the effect of these SNPs on high CRP values during systemic infections are few. We assessed the genotypes of six SNPs in the CRP gene in 145 clinically evaluated SAB patients. We demonstrated that SNP rs3091244 in the promoter region of the CRP gene was clearly associated with higher maximal CRP elevation during the first week of illness. This gene region is important in the transcriptional activity of the CRP gene [START_REF] Szalai | Single-nucleotide polymorphisms in the C-reactive protein (CRP) gene promoter that affect transcription factor binding, alter transcriptional activity, and associate with differences in baseline serum CRP level[END_REF]. Patients who were rare minor A-allele carriers of SNP rs3091244 had significantly higher maximal CRP levels as compared to patients who were not. This genetic variant, however, did not have an effect on initial CRP level measured on the day of S. aureus positive blood culture or the subsequent decline of CRP level during the treatment. Neither did it affect the outcome of the disease as measured by mortality, number of deep infections or time to defervescence.

We observed significantly higher maximal CRP levels among the patients who had deep infection focus during the first week of SAB as compared to those who did not. We were not able to find this result in previous literature. However, the importance of deep infection foci on the prognosis has been shown [START_REF] Mylotte | Staphylococcus aureus bacteremia: Predictors of 30-day mortality in a large cohort[END_REF][START_REF] Fowler | Clinical identifiers of complicated staphylococcus aureus bacteremia[END_REF][START_REF] Fatkenheuer | Long-term outcome and quality of care of patients with staphylococcus aureus bacteremia[END_REF]. Surgical procedures or even placement of a central venous catheter (CVI) might increase CRP levels but this was not observed in our study population. In univariate analysis surgical interventions were not associated to elevated CRP levels and only one of the patients had CVIs placed during the first week.

Our study population was a subgroup of patients in a larger study on S. aureus bacteremia which might have an effect on the results. However, the demographic factors and underlying diseases did not differ between our patient group and the original larger patient population [START_REF] Ruotsalainen | Levofloxacin does not decrease mortality in staphylococcus aureus bacteraemia when added to the standard treatment: A prospective and randomized clinical trial of 381 patients[END_REF]. Inadequate antibiotic treatment during the first week has not affected the results as all patients received microbiologically verified effective treatment already from the blood culture day [START_REF] Ruotsalainen | Levofloxacin does not decrease mortality in staphylococcus aureus bacteraemia when added to the standard treatment: A prospective and randomized clinical trial of 381 patients[END_REF]. As a limitation of the study data on duration of symptoms prior the onset of therapy is not available and was not analysed. Infections due to various S. aureus bacterial strains might have different effects on the CRP production although no data on this exists. The effect of variations in S. aureus strains in our results seems, however, not likely as the strains were highly variable [START_REF] Ruotsalainen | Methicillin-sensitive staphylococcus aureus bacteraemia and endocarditis among injection drug users and nonaddicts: Host factors, microbiological and serological characteristics[END_REF].

In contrast to our results, SNP rs2794521 in the promoter region of the CRP gene has been reported to be associated with higher mortality in bacteremia due to Streptococcus pneumoniae [START_REF] Eklund | Polymorphism of the C-reactive protein gene is associated with mortality in bacteraemia[END_REF]. We could not observe that this SNP would have an impact on outcome or CRP levels in SAB. Whether this relation to mortality counts only for S. pneumoniae or is due to other confounding factors remains to be elucidated. It has to be notified that no relationship of polymorphisms in the CRP gene and mortality was observed in the same study in bacteremia due to other microbes, namely α-hemolytic streptococci or Escherichia coli [START_REF] Eklund | Polymorphism of the C-reactive protein gene is associated with mortality in bacteraemia[END_REF]. The minor allele of SNP rs1800947 in the CRP gene was recently shown to be associated to significantly lower CRP levels in patients recovering from coronary by-pass surgery [START_REF] Perry | C-reactive protein gene variants are associated with postoperative C-reactive protein levels after coronary artery bypass surgery[END_REF]. In our study, this same genotype did not seem to have any effect on CRP levels but the relatively small patient population might not make it possible to observe such an effect in our study. Other studies on CRP gene polymorphism in severe acute infections are not available.

Interestingly, this same triallelic SNP rs3091244, which we found to be associated with maximal CRP levels in SAB has been related to high basal CRP levels as well [START_REF] Szalai | Single-nucleotide polymorphisms in the C-reactive protein (CRP) gene promoter that affect transcription factor binding, alter transcriptional activity, and associate with differences in baseline serum CRP level[END_REF][START_REF] Kovacs | A novel common single nucleotide polymorphism in the promoter region of the C-reactive protein gene associated with the plasma concentration of C-reactive protein[END_REF][START_REF] Kathiresan | Contribution of clinical correlates and 13 C-reactive protein gene polymorphisms to interindividual variability in serum C-reactive protein level[END_REF][START_REF] Danik | Influence of genetic variation in the C-reactive protein gene on the inflammatory response during and after acute coronary ischemia[END_REF]. Although reduction of CRP levels has been shown to be clinically beneficial and is associated with positive cardiovascular outcomes [START_REF] Ridker | Clinical application of C-reactive protein for cardiovascular disease detection and prevention[END_REF], associations of polymorphisms in CRP gene with development of chronic diseases like type 2 diabetes [START_REF] Zee | C-reactive protein gene variation and type 2 diabetes mellitus: A casecontrol study[END_REF][START_REF] Wolford | A C-reactive protein promoter polymorphism is associated with type 2 diabetes mellitus in Pima indians[END_REF] or ischemic heart disease remain under some controversory.

In our study, previously reported associations of CRP gene SNPs on the unexpectedly low levels of CRP could not be seen which might be due to rather small sample size in our study. However, this might also be due to healing of infection having a greater impact on the decline of CRP as compared to genetic variability. This further suggests that in studies concerning impact of CRP gene SNPs on basal CRP level, individuals with any infections should be carefully excluded from the patient population.

To answer the question whether our finding is specific for a CRP response in SAB requires further investigation.

However, already at this time one can draw certain conclusions. Our study can help to understand why individuals have so different maximal responses in CRP levels under seemingly similar conditions in bacteremia and why exceptionally high CRP does not necessarily predict the clinical course of SAB. Polymorphisms were not associated with CRP levels in the beginning of SAB. These levels have more variable factors than maximal CRP. CRP measurements started when positive blood culture was achieved and period of time from the onset of symptoms and first blood sampling varied between patients. This may have influenced our results.

Polymorphisms did not seem to have an effect on the decline of CRP levels and this encourages continued use of CRP measurement as a clinical tool in individual surveillance of recovery.

In conclusion, our study reminds to interpret exceptionally high CRP values with caution and to avoid comparing them between patients. The results of this study suggest that disease severity as assessed by occurrence of a deep infection focus and genetic variability in CRP gene might be the factors that determine the maximal CRP level in SAB. The same polymorphism that was associated with high basal CRP levels was found to be related to significantly higher maximal CRP concentrations in SAB.

  Table only include 116 patients while the remainder of the genotypic analyses include 133 -144 patients (see supplementary table)?

Table 1

 1 Characteristics and underlying diseases of 145 patients with Staphylococcus aureus bacteremia NOTE. Data are no. (%) of patients, unless otherwise indicated.

	a During 3 months or
	b during 6 months preceding the positive blood culture.
	c Chronic obstructive pulmonary disease, bronchial asthma and
	other pulmonary disease.
	d Patients were grouped modifying original classification [27]

Table 2

 2 Distribution of SNP rs3091244 genotypes in 116 patients and median CRP levels (mg/L) and interquartile range (IQR) on the first week of Staphylococcus aureus bacteremia CRP values within each additive or recessive model were analysed by Kruskal Wallis H test.

		Genotype		CRP b		CRP ce		CRP d
	SNP a	distribution	n	Day 0	P-value	Day 7	P	Max	P-value
	rs3091244 Additive
		TT	21	158 (206)		20 (74)		182 (167)
		TC	55	171 (166)		44 (84) e		193 (157)
		TA	4	222 (364)	0.550	54 (124)	0.594	299 (202)	0.023
		CC	28	157 (109) e		33 (65)		163 (102)
		CA	8	224 (186)		54 (62)		280 (188)
		A-minor
		allele recessive
		TT+TC+CC	104	166 (160) e		42 (70) e		179 (148)
		TA+CA	12	224 (186)	0.111	47 (64)	0.773	282 (169)	0.004*

a National Center for Biotechnology Information SNP database accession number. b Median CRP (mg/L) and (IQR) when positive blood culture for S.aureus was achieved. c Median CRP (mg/L) and (IQR) at the end of the first week of SAB. d Highest median CRP (mg/L) and (IQR) of the first week of SAB . e One CRP level measurement is missing from value of median. * Multiple P-value testing FDR <0.10

c Chronic obstructive pulmonary disease, bronchial asthma and other pulmonary disease.

d Patients were grouped modifying original classification [START_REF] Jackson | Profiles of pyelonephritis[END_REF]. Please find enclosed our revised manuscript entitled "Genetic polymorphism of C-reactive protein (CRP) gene and a deep infection focus determine maximal serum CRP level in Staphylococcus aureus bacteremia". We are grateful for the valuable comments of the referees which have all lead to changes in manuscript. One of the suggestions of referee 1 led us to reanalyse one part of the statistics which has resulted in a modification of title of our manuscript. We sincerely consider this reanalysis to have improved the manuscript and its importance for the scientific community.

In detail following changes have been made into the manuscript.

Referee #1.

1) Each patient was included into the study only once. This has been explained in the methods in the paragraph "patient population" on Page 4 lines 20-21.

2) Multiple comparisons has been analysed by False-discovery rate (FDR) estimation. It has been included into the statistics section of methods page 6 lines 19-20. Results from FDR analyses have been shown in tables 2 and 4.

3) Table 2 includes only 116 patients because genotyping in laboratory of SNP rs3091244 was successful only in 116 patients. This specific triallelic SNP turned out to be technically much more challenging to sequence compared to all other biallelic SNPs studied.

4) Surgical procedures during the first week have been analysed as suggested by the referee and were found not to affect the maximal CRP level. However, we analysed also the effect of presence of a deep infection focus on the maximal CRP level and found this association highly significant. Patients with deep infection foci have been added into table 3 and this information into results on Page 7 lines 13-18 and into Table 4. This finding has been further discussed in a new written part of discussion on Page 4, in paragraph two. We considered this finding of so high importance that it has resulted into modification of our title as well.

5) The procedures named above in 4) and the possible effect of S. aureus strains on maximal CRP levels have been discussed in a new paragraph three in the Discussion part ort page 8.

6) The unclear sentences have been omitted in the revised version of the manuscript.

7) The demographic and underlying factors of our subgroup of patients have been compared to the original study population of 430 patients in the Discussion part paragraph three of the manuscript (Page 8). No significant differences between our patient population and the original patient material were found. Unfortunately, due to ethical regulations related to gene sampling all patients in the original data set could not be included into the present analysis. We found the comparison of the two patient populations confounding for the reader in a tabular form. After all, only data from the subgroup of patients is presented in the present manuscript.