Right-handed sneutrino dark matter and big-bang nucleosynthesis
Koji Ishiwata, Masahiro Kawasaki, Kazunori Kohri, Takeo Moroi
aa r X i v : . [ h e p - ph ] D ec TU-858ICRR-Report-555-2009-17IPMU 09-0144December, 2009
Right-handed sneutrino dark matterand big-bang nucleosynthesis
Koji Ishiwata ( a ) , Masahiro Kawasaki ( b,c ) , Kazunori Kohri ( a ) ,and Takeo Moroi ( a,c ) a Department of Physics, Tohoku University, Sendai 980-8578, Japan b Institute for Cosmic Ray Research, University of Tokyo, Kashiwa 277-8582, Japan c Institute for the Physics and Mathematics of the Universe, University of Tokyo,Kashiwa 277-8568, Japan
Abstract
We study the light-element abundances in supersymmetric model where the right-handed sneutrino is the lightest superparticle (LSP), assuming that the neutrino massesare purely Dirac-type. In such a scenario, the lightest superparticle in the minimalsupersymmetric standard model sector (which we call MSSM-LSP) becomes long-lived,and thermal relic MSSM-LSP may decay after the big-bang nucleosynthesis starts.We calculate the light-element abundances including non-standard nuclear reactionsinduced by the MSSM-LSP decay, and derive constraints on the scenario of right-handed sneutrino LSP. ith the precise astrophysical observations, it is now widely believed that about 23% ofthe energy density of the present universe is due to dark matter (DM) [1]. The existenceof dark matter, however, raises a serious question to particle physics because there is noviable candidate for dark matter in the particle content of the standard model. To solve thisproblem, many dark-matter models have been proposed so far.In constructing dark-matter model, it is important to understand how dark matter wasproduced in the early universe. In many cases, the thermal freeze-out mechanism is adoptedto produce dark matter particle in the early universe; then, dark matter particle, which isin thermal bath when the cosmic temperature is higher than its mass, freezes out from thethermal bath when the cosmic temperature becomes low.However, the freeze-out scenario is not the only possibility to produce dark matter particlein the early universe. Even if the dark matter particle is very weakly interacting so that itis never thermalized, it can be produced by the decay and scattering of particles in thermalbath. In particular, if the interaction of dark matter is dominated by renormalizable ones,dark-matter production is most effective when the temperature is comparable to the massof parent particle which produces dark matter via the decay or scattering. Thus, if thereheating temperature after inflation is higher than the mass of parent particle, the relicdensity of the dark matter becomes insensitive to the cosmic evolution in the early stage.Such a scenario was originally proposed in [2], where the right-handed sneutrino ˜ ν R insupersymmetric model is shown to be a viable candidate for dark matter. In [2], it was alsoshown that, if ˜ ν R -DM is dominantly produced from the decay and scattering of superparticlesin thermal bath, the primordial abundance of ˜ ν R is determined when the cosmic temperatureis comparable to the masses of superparticles. Then, recently, more general discussion ofsuch a scenario has been given in [3], where a variety of candidates for such very weakly-interacting dark-matter particles have been also considered.If a very weakly interacting particle is dark matter, it is often the case that a long-livedparticle (with lifetime longer than ∼ ν R (and R -even particles) viavery small neutrino Yukawa interaction. Then, decay of the MSSM-LSP after the big-bangnucleosynthesis (BBN) epoch may affect the light-element abundances. Thus, it is importantto check the BBN constraints on the scenario.In this letter, we consider the case where a right-handed sneutrino is the LSP, assumingDirac-type neutrino masses [2]. We study the light-element abundances in such a casein detail, and derive BBN constraints on the mass and lifetime of the MSSM-LSP. Wealso comment on the implication of the sneutrino LSP scenario on the Li overproductionproblem.First, we discuss the model framework that we consider in this letter. The superpotential
For related topics, see also [7].
1s written as W = W MSSM + y ν ˆ L ˆ H u ˆ ν cR , (1)where W MSSM is the superpotential of the MSSM, ˆ L = (ˆ ν L , ˆ e L ) and ˆ H u = ( ˆ H + u , ˆ H u ) areleft-handed lepton doublet and up-type Higgs doublet, respectively. (In this letter, “hat” isused for superfields, while “tilde” is for superpartners.) Generation indices are omitted forsimplicity. In this model, neutrinos acquire their masses only through Yukawa interactionsas m ν = y ν h H u i = y ν v sin β , where v ≃
174 GeV is the vacuum expectation value (VEV) ofthe standard model Higgs field and tan β = h H u i / h H d i . Thus, the neutrino Yukawa couplingis determined by the neutrino mass as y ν sin β = 3 . × − × (cid:18) m ν . × − eV (cid:19) / . (2)Mass squared differences among neutrinos have already been determined accurately by neu-trino oscillation experiments. In particular, the K2K experiment suggests [∆ m ν ] atom ≃ (1 . − . × − eV [8]. In the following discussion, we assume that the spectrum ofneutrino masses is hierarchical, hence the largest neutrino Yukawa coupling is of the orderof 10 − unless otherwise mentioned; we use y ν = 3 . × − for our numerical study. (Weneglect effects of smaller Yukawa coupling constants.) For our study, it is also necessary tointroduce soft supersymmetry (SUSY) breaking terms. Soft SUSY breaking terms relevantto our analysis are L soft = −
12 ( m ˜ B ˜ B ˜ B + m ˜ W ˜ W ˜ W + h . c . ) − M L ˜ L † ˜ L − M ν R ˜ ν ∗ R ˜ ν R + ( A ν ˜ LH u ˜ ν cR + h . c . ) , (3)where ˜ B and ˜ W are Bino and Wino, respectively. We parameterize A ν by using the dimen-sionless constant a ν as A ν = a ν y ν M ˜ L . (4)Notice that a ν is a free parameter and, in gravity-mediated SUSY breaking scenario, forexample, a ν is expected to be O (1). The A ν -term induces the left-right mixing in thesneutrino mass matrix, through which the MSSM-LSP decays in the present case. In thecalculation of mass eigenvalues, however, the mixing is negligible because of the smallnessof neutrino Yukawa coupling constants, and we obtain m ν L ≃ M L + 12 m Z cos 2 β, m ν R ≃ M ν R , (5)where m Z is the Z boson mass. Here and hereafter, we assume that all the right-handedsneutrinos are degenerate in mass for simplicity. In the numerical study, we take the followingmodel parameters: m ˜ ν R = 100 GeV, tan β = 30, and m h = 115 GeV (with m h being thelightest Higgs boson mass). In addition, the Wino mass is related to the Bino mass usingthe GUT relation. 2n the early universe, right-handed sneutrino is never thermalized because of the weaknessof neutrino Yukawa interaction. Although it is decoupled from thermal bath, right-handedsneutrino can be produced in various processes; (i) decay or scattering of MSSM particlesin thermal bath, (ii) decay of MSSM-LSP after freeze-out, and (iii) production in very earlyuniverse via the decay of exotic particles (like gravitino or inflaton). Thereafter, we donatethe contribution of each process as, Ω (Thermal) ˜ ν R , Ω (F.O.) ˜ ν R , and Ω (non-MSSM) ˜ ν R in order. Primarily,right-handed sneutrino is produced through neutrino Yukawa interaction (and the left-rightmixing of sneutrino) dominantly in the following decay processes: ˜ H → ˜ ν R ¯ ν , ˜ H + → ˜ ν R l + ,˜ ν L → ˜ ν R h , ˜ ν L → ˜ ν R Z , ˜ l L → ˜ ν R W − , ˜ B → ˜ ν R ¯ ν , ˜ W → ˜ ν R ¯ ν , and ˜ W + → ˜ ν R l + . In the previouswork, it was shown that right-handed sneutrino can be adequately produced to become darkmatter when the masses of left- and right-handed sneutrino are degenerate at 10 −
20 % with a ν .
3, or in a case of larger a ν without degeneracy [2]. It is also mentioned that enhancementof right-handed sneutrino production is possible with larger neutrino Yukawa coupling if weconsider the case where neutrino masses are degenerate. Giving an eye on the thermal bathagain, the MSSM-LSP decouples from thermal bath and its number freezes out in the samemanner with usual MSSM, while the number of the other MSSM particles is suppressed byBoltzmann factor in this epoch. However, relic MSSM-LSP, which is assumed to be thenext-to-the-lightest superparticle (NLSP) in this letter, decays to right-handed sneutrinothrough neutrino Yukawa coupling in the late time. In this process, the contribution to theabundance is given as Ω (F . O . )˜ ν R = m ˜ ν R m NLSP Ω (F . O . )NLSP , (6)where m NLSP is the mass of the NLSP and Ω (F . O . )NLSP is the would-be density parameter ofthe relic NLSP (for the case where it does not decay into ˜ ν R ). Lastly, we mention thatthere might be a possibility that right-handed sneutrino is produced directly from an exoticparticle in the very early universe. The abundance of the expected right-handed sneutrinois model-dependent, and we do not discuss further detail of specific model. In this letter, weconsider the scenario that right-handed sneutrino produced in these processes becomes darkmatter. We do not specify which is dominant process to produce right-handed sneutrino.If ˜ ν R is the LSP, it is always the case that the MSSM-LSP becomes long-lived. Becausesome amount of relic NLSP always exists in the early universe, they may cause seriousproblem in BBN; if the relic MSSM-LSP decays during or after the BBN epoch, energeticcharged and/or colored particles are emitted; they cause the photo- and hadro-dissociationprocesses of light elements, which may spoil the success of the standard BBN scenario. Inthe following, we consider three typical candidates for the MSSM-LSP; Bino ˜ B , left-handedsneutrino ˜ ν L , and lighter stau ˜ τ , and study how the ˜ ν R -DM scenario is constrained by theBBN.In the Bino-NLSP case, the Bino dominantly decays as ˜ B → ˜ ν R ¯ ν (and its CP-conjugated3rocess) and its decay rate is given by Γ ˜ B → ˜ ν R ¯ ν = β g π (cid:20) A ν vm ν L − m ν R (cid:21) m ˜ B , (7)where g is the U (1) Y gauge coupling constant and, for the process x → ˜ ν R y , β f is given by β = 1 m x [ m x − m ν R + m y ) m x + ( m ν R − m y ) ] , (8)with m x and m y being the masses of the particles x and y , respectively. When ˜ ν L or ˜ τ isthe NLSP, the NLSP decays by emitting weak- or Higgs-boson if kinematically allowed. Thedecay rates for those processes are given byΓ ˜ ν L → ˜ ν R Z = β π (cid:20) m ν L m ν L − m ν R (cid:21) A ν m ˜ ν L , (9)Γ ˜ ν L → ˜ ν R h = β f π A ν m ˜ ν L , (10)Γ ˜ τ → ˜ ν R W − = β sin θ ˜ τ π (cid:20) m τ m ν L − m ν R (cid:21) A ν m ˜ τ , (11)where m ˜ τ is the stau mass and θ ˜ τ is the left-right mixing angle of stau. (The lighter stau isgiven by ˜ τ = ˜ τ R cos θ ˜ τ + ˜ τ L sin θ ˜ τ .) If the two-body processes are kinematically blocked, theslepton-NLSP decays into three-body final state as ˜ ν L → ˜ ν R f ¯ f and ˜ τ → ˜ ν R f ¯ f ′ (with f and f ′ being standard-model fermions).Now, we are at the position to discuss the BBN constraints on ˜ ν R -DM scenario. We startwith the case where Bino is NLSP. As we have mentioned, the Bino-NLSP dominantly decaysas ˜ B → ˜ ν R ¯ ν . Since ˜ ν R and ν are (very) weakly interacting particles, the BBN constraintsare not so severe if this is the only possible decay mode. However, ˜ B may also decay as˜ B → ˜ ν R ¯ νZ ( ∗ ) and ˜ ν R lW ( ∗ ) , where Z ( ∗ ) and W ( ∗ ) are on-shell or off-shell Z and W bosons(where the “star” is for off-shell particle), respectively, while l is charged lepton. Then,through the decay of Z ( ∗ ) and W ( ∗ ) , quarks and charged leptons are produced. Even thoughthe branching ratio for such processes are phase-space suppressed, they produce sizableamount of hadrons which may significantly affect the light-element abundances. Thus, in ouranalysis, effects of those decay modes are taken into account in deriving the BBN constraints.The light-element abundances also depend on the primordial abundance of the NLSP, andwe adopt the abundance of Bino in the focus-point (or co-annihilation) region [9] Y (focus)˜ B = 9 × − × (cid:16) m ˜ B
100 GeV (cid:17) , (12) In this letter, we consider the case where the Gaugino-Higgsino mixing is so small that its effect isnegligible.
If the Bino is the NLSP, its primordial abundance strongly depends on the MSSM parameters. In the Y x ≡ n x /s with n x being the number density of particle x and s the entropy density of the universe.Following the procedure given in [10], we calculate the light-element abundances takingaccount of the hadro-dissociation, photo-dissociation, and p ↔ n conversion processes. Theenergy distribution of the final-state particles are calculated by using the HELAS package[11], and the hadronization processes of colored particles are studied by using the PYTHIApackage [12]. In the Bino-NLSP case, high energy neutrino emitted by the Bino decay mayscatter off background neutrino and generate energetic e ± , which becomes the source ofenergetic photon [13]. In our analysis, we have taken into account the effects of the photo-dissociation process induced by photon from the neutrino injection. (However, we foundthat the neutrino-induced processes are less important compared to other processes.) Oncetheoretical values of the primordial light-element abundances are obtained as functions ofthe mass and the lifetime of the NLSP, we compare them with the observed values of theprimordial abundances. In deriving the constraints on the model, we adopt the followingobservational constraints: • D to H ratio [14, 15]: (n D / n H ) p = (2 . ± . × − . (13) • He mass fraction [16, 17]: Y p = 0 . ± . . (14) • He to D ratio [18, 10]: ( n He /n D ) p < .
83 + 0 . . (15) • Li to Li ratio [19, 20]: ( n Li /n Li ) p < .
046 + 0 .
022 + 0 . . (16) • Li to H ratio [22, 20]:log ( n Li /n H ) p = − . ± .
09 + 0 . . (17) so-called bulk region, the abundance is larger, and is approximately given by Y (bulk)˜ B = 4 × − × (cid:16) m ˜ B
100 GeV (cid:17) . We have checked that the BBN constraints in such a case are almost the same as the focus-point case. Ifwe adopt the abundance in the bulk region, however, Ω (F . O . )˜ ν R becomes larger than the present dark matterdensity if m ˜ ν R = 100 GeV. Thus we will not consider such a case in the following discussion. Asplund et al. reported n Li /n Li = 0 . ± . .
106 and +0 .
35 to theobservational face-values of ( n Li /n Li ) p and log ( n Li /n H ) p , respectively. We expect thatthese systematic errors result from possible depletion in stars through rotational mixing [23]or diffusion [24]. Since both Li and Li are destroyed by depletion process, their systematicerrors are correlated (for more details, see [20]). We note here that the standard BBN isexcluded at more than 4- σ level if we do not adopt the systematic error on Li abundance [25](so-called Li problem). Thus, to derive a conservative constraint, we add these systematicerrors. At the end of this letter, we will comment on implications of the ˜ ν R -DM scenario onthe Li problem.
In Fig. 1, we show the constraint from BBN for the Bino-NLSP case on m ˜ B vs. τ ˜ B plane(with τ ˜ B being the lifetime of Bino). The lifetime is related to the fundamental parametersvia Eq. (7); in particular, τ ˜ B is proportional to a − ν . Taking m ˜ ν L = 1 . m ˜ B , we calculate a ν -parameter. In the figure, un-shaded, lightly shaded, and darkly shaded regions indicate theregion with a ν <
1, 1 < a ν <
10, and a ν >
10, respectively. One can see that the region with m ˜ B .
200 GeV is always allowed. This is because, in such a region, the dominant hadronicdecay processes are four-body ones ( ˜ B → ˜ ν R ¯ νq ¯ q and ˜ ν R lq ¯ q ′ ), for which the branching ratiois significantly suppressed by the phase-space factor. On the other hand, when the decayprocesses ˜ B → ˜ ν R ¯ νZ and ˜ ν R lW are kinematically allowed, those three-body decay processeshave sizable branching ratio, resulting in an enhanced production of hadrons. We can seethat the lifetime of Bino is constrained to be smaller than τ ˜ B . sec in such a parameterregion in order not to overproduce deuterium via the hadro-dissociation of He.In the figure, we also plot contours of constant density parameters. Once the primordialabundance of the NLSP is fixed, Ω (F.O.) ˜ ν R is calculated by using Eq. (6). We show the contourof Ω (F . O . )˜ ν R = Ω c = 0 .
228 [1]; the right-hand side of the line is excluded by the overclosureconstraint if we adopt the abundance given in Eq. (12). In studying the ˜ ν R -DM scenario,we should also consider ˜ ν R from the MSSM particles in the thermal bath. Following [2], wecalculate the sneutrino abundance by solving the Boltzmann equation taking account of allthe relevant sneutrino production processes. The contour of Ω (Thermal)˜ ν R = Ω c is shown in Fig.1; the a ν -parameter is determined by using Eq. (7), while the MSSM parameters are takento be m ˜ ν L = 1 . m ˜ B , and µ H = 2 m ˜ B (with µ H being the SUSY invariant Higgs mass). ˜ ν R isoverproduced below the line of Ω (Thermal)˜ ν R = Ω c with the present choice of parameters. Onecan see that the line is well below the constrained region by BBN. In the present choiceof parameters, a relatively large value of a ν is needed unless the masses of ˜ B and ˜ ν R aredegenerate in order to realize Ω (Thermal)˜ ν R = Ω c . However, notice that the relic abundance of˜ ν R depends on various parameters. In particular, Ω (Thermal)˜ ν R becomes larger when the massdifference between ˜ ν R and ˜ ν L becomes smaller because the left-right mixing is enhanced. In As we will discuss later in considering the Li problem, one may adopt a slightly higher value of D toH ratio, (n D / n H ) p = (3 . +0 . − . ) × − [14], and/or that of Li to H ratio, log ( n Li /n H ) p = − . ± . − Li abundance is taken into account).
BBN constraints on the Bino-NLSP case are shown on m ˜ B vs. τ ˜ B plane. The un-shaded,lightly shaded, and darkly shaded regions are for a ν <
1, 1 < a ν <
10, and a ν >
10, respectively.In addition, the contours of Ω (F . O . )˜ ν R = Ω c and Ω (Thermal)˜ ν R = Ω c are also shown (dotted lines). Forthe calculation of Ω (Thermal)˜ ν R , we take m ˜ ν R = 100 GeV, m ˜ ν L = 1 . m ˜ B , µ H = 2 m ˜ B , m h = 115 GeV,and tan β = 30. addition, Ω (Thermal)˜ ν R is also enhanced if we use a larger value of the neutrino Yukawa couplingconstant; it may happen when we adopt the degenerate neutrino masses. Thus, with otherchoices of parameters, the required value of a ν to realize Ω (Thermal)˜ ν R = Ω c changes.Another candidate for the NLSP is the left-handed sneutrino. Such a scenario is attractivein the ˜ ν R -DM scenario because the ˜ ν R abundance is enhanced if the masses of ˜ ν R and˜ ν L becomes closer. When ˜ ν L is the NLSP, its dominant decay process is ˜ ν L → ˜ ν R Z ( ∗ ) and ˜ ν L → ˜ ν R h ( ∗ ) . Thus, colored and/or charged particles are effectively produced via thedominant decay modes. Again, we calculate the light-element abundances taking accountof the hadro-dissociation, photo-dissociation, and p ↔ n conversion processes, and comparethe resultant light-element abundances with observational constraints given in (13) − (17).The relic abundance of left-handed sneutrino is approximated as [27]: Y ˜ ν L ≃ × − × (cid:16) m ˜ ν
100 GeV (cid:17) . (18)The BBN constraints are shown in Fig. 2. We can see that the parameter space is con-strained as τ ˜ ν L . sec (with τ ˜ ν L being the lifetime of ˜ ν L ) by the deuterium overproductionirrespective of m ˜ ν L . This is due to the fact that, if ˜ ν L is the NLSP, production of hadronsoccurs in the dominant decay processes. This is a large contrast to the Bino-NLSP case.If ˜ ν L is the NLSP, its primordial abundance is so small that Ω (F.O.) ˜ ν R < Ω c as far as7igure 2: BBN constraints on the ˜ ν L -NLSP case are shown on m ˜ ν L vs. τ ˜ ν L plane. The un-shaded,lightly shaded, and darkly shaded regions are for a ν <
1, 1 < a ν <
10, and a ν >
10, respectively.In addition, the contour of Ω (Thermal)˜ ν R = Ω c is shown in dotted line. Here, we take m ˜ ν R = 100 GeV, m ˜ B = 1 . m ˜ ν L , µ H = 2 m ˜ B , m h = 115 GeV, and tan β = 30. m ˜ ν L .
10 TeV (for m ˜ ν R = 100 GeV). On the contrary, Ω (Thermal)˜ ν R can be as large as Ω c ; inthe figure, we plot the contour of Ω (Thermal)˜ ν R = Ω c . Here, the a ν -parameter is determined forgiven values of m ˜ ν L and τ ˜ ν L , while the MSSM parameters are taken to be m ˜ B = 1 . m ˜ ν L , and µ H = 2 m ˜ B . One can see that, when m ˜ ν L &
160 GeV, Ω (Thermal)˜ ν R = Ω c can be realized with a ν .
10 (which is marginally consistent with the naive order-of-estimate of the a ν -parameterin gravity-mediated SUSY breaking scenario). Notice that, even with a ν ∼ (Thermal)˜ ν R can be large enough if a larger value of y ν is adopted or if ˜ ν R is produced by thedecay of some exotic particles.Next, we consider the ˜ τ -NLSP case. In this case, because the NLSP is charged, it mayform a bound state with He during the BBN epoch and change the reaction rate [28]. (Suchan effect is called ˜ τ -catalyzed effect.) Consequently, Li abundance may be significantlyenhanced if the lifetime of ˜ τ is longer than ∼ sec. Here, the light-element abundancesare calculated by including the ˜ τ -catalyzed effect. Assuming that ˜ τ is almost right-handed,we approximate the primordial abundance as [27]: Y ˜ τ ≃ × − × (cid:16) m ˜ τ
100 GeV (cid:17) , (19)and calculate the light-element abundance. The numerical result is shown in Fig. 3. Asin the ˜ ν L -NLSP case, the parameter space τ ˜ τ & sec (with τ ˜ τ being the lifetime of ˜ τ )is excluded. In addition, the He is overproduced due to p ↔ n conversion process when8igure 3: BBN constraints on the ˜ τ -NLSP case are shown on m ˜ τ vs. τ ˜ τ plane. The un-shaded,lightly shaded, and heavily shaded regions are for a ν <
1, 1 < a ν <
10, and a ν >
10, respectively.In addition, the contour of Ω (Thermal)˜ ν R = Ω c is shown in dotted line. Here, we take m ˜ ν R = 100 GeV, m ˜ B = 1 . m ˜ τ , m ˜ ν L = 1 . m ˜ τ , µ H = 2 m ˜ B , m h = 115 GeV, tan β = 30, and sin θ ˜ τ = 0 . m ˜ τ &
500 GeV and τ ˜ τ ∼
10 sec. The He constraint becomes more stringent than the˜ ν L -NLSP case because the yield variable used in the ˜ τ -NLSP case is larger. We also showthe line which satisfies Ω (Thermal)˜ ν R = Ω c , taking m ˜ B = 1 . m ˜ τ , m ˜ ν L = 1 . m ˜ τ , µ H = 2 m ˜ B , andsin θ ˜ τ = 0 .
3. As one can see, the lifetime becomes longer for a given value of a ν compared tothe case of ˜ ν L -NLSP; this is because we have taken a small value of θ ˜ τ . Even in this case, wecan see that Ω (Thermal)˜ ν R = Ω c can be realized with a ν .
10 in the parameter region consistentwith all the BBN constraints.Finally we comment on the implication of the ˜ ν R -LSP scenario on the so-called Liproblem. As we have mentioned, the standard BBN is excluded at more than 4- σ level ifwe take the face value of the observational constraints on the Li abundance; the theoreticalprediction of the Li abundance becomes significantly larger than the observed value. Eventhough the Li problem does not exist if a significant depletion of Li occurs in stars, thedegree of the depletion has not yet been accurately understood. If one adopts models withsmall depletion, the astrophysical or particle-physics solution to the Li problem is required.It is notable that the Li abundance can be reduced if a long-lived particle decays intohadrons during the BBN epoch [29, 30]. Thus, in the present case, the decay of the NLSPduring the BBN may be a solution to the Li problem. In the following, we will see thatthe Li problem may be solved if ˜ B is the NLSP. (For the cases of ˜ ν L - and ˜ τ -NLSP, the Li problem is hardly solved because the parameter region with the lifetime longer than ∼ sec is (almost) excluded, as shown in Figs. 2 and 3.)9o study the Li problem in the present framework, we neglect the systematic error(i.e., +0 .
35 dex) in the observational constraint on Li abundance. In addition, because theallowed parameter region is sensitive to the observational constraint on Li, we consider twodifferent observational constraints on Li abundance:Low Li : log ( n Li /n H ) p = − . ± .
09 [22] , (20)High Li : log ( n Li /n H ) p = − . ± .
06 [26] . (21)Notice that the low value corresponds to the one given in (17), while the high value is frommeasurement using different method to estimate temperature of the atmosphere in dwarfhalo stars. In addition, because the systematic error in the Li to Li ratio is correlated tothat of Li, we also remove the systematic error from (16):( n Li /n Li ) p < .
046 + 0 . . (22)BBN constraints on the Bino-NLSP case are shown in Fig. 4, using the constraints (20)and (22) (upper panel) or (21) and (22) (lower panel). Here, constraints on (n D / n H ) p , Y p ,and ( n He /n D ) p are unchanged from the previous cases; the D to H ratio given in (13) iscalled “Low D” because of the reason below. As one can see, if we adopt the high value ofthe Li to H ratio, all the light-element abundances can be consistent with the observationalconstraints if 10 sec . τ ˜ B . sec. On the contrary, with the low value of Li abundance,the constraint on the D to H ratio (13) makes it difficult to solve the Li problem. However,this conclusion changes if we adopt a slight systematic error in the Li abundance, or if adifferent observational constraint on the D to H ratio is adopted. Indeed, in some literature,a higher value of the D to H ratio (which is the highest value among the data points for sixmost precise observations [14]) is adopted because D is the most fragile light element and theobserved values might reflect the abundance after suffering from some destruction processes:High D : (n D / n H ) p = (3 . +0 . − . ) × − . (23)(We call this as “High D.”) In Fig. 4, we also present the parameter region consistent withthe constraint (23) using the dotted line. As one can see, with (23), the Li problem can besolved even with the low value of the Li to H ratio.Notice that, in the parameter region where all the light-element abundances becomeconsistent, Ω (Thermal) ˜ ν R becomes much smaller than Ω c if the constraint (20) or (21) is adopted.However, this fact does not imply that the ˜ ν R -LSP scenario cannot solve the Li problem.One possibility is to consider the effects of the decay products of MSSM-LSP after freeze-out;indeed, as shown in the figure, Ω (F.O.) ˜ ν R ≃ Ω c is realized when m ˜ B ∼
400 GeV and τ ˜ B ∼ secwith solving the Li problem.
Acknowledgments:
This work was supported in part by Research Fellowships of the JapanSociety for the Promotion of Science for Young Scientists (K.I.), and by the Grant-in-Aid forScientific Research from the Ministry of Education, Science, Sports, and Culture of Japan,No. 14102004 (M.K.), No. 18071001 (K.K.) and No. 19540255 (T.M.), and also by WorldPremier International Research Center Initiative, MEXT, Japan (M.K. and T.M.).10igure 4: Same as Fig. 1, but with different set of observational constraints. Regions where Li abundance becomes consistent with the observation are shaded.11 eferences [1] G. Hinshaw et al. [WMAP Collaboration], Astrophys. J. Suppl. , 225 (2009).[2] T. Asaka, K. Ishiwata and T. Moroi, Phys. Rev. D , 051301 (2006); Phys. Rev. D , 065001 (2007).[3] L. J. Hall, K. Jedamzik, J. March-Russell and S. M. West, arXiv:0911.1120 [hep-ph].[4] T. Moroi, H. Murayama and M. Yamaguchi, Phys. Lett. B , 289 (1993).[5] J. L. Feng, A. Rajaraman and F. Takayama, Phys. Rev. Lett. , 011302 (2003); Phys.Rev. D , 063504 (2003).[6] K. Ishiwata, S. Matsumoto and T. Moroi, Phys. Rev. D , 035004 (2008).[7] S. Gopalakrishna, A. de Gouvea and W. Porod, JCAP , 005 (2006); J. March-Russell, C. McCabe, M. McCullough, arXiv:0911.4489 [hep-ph].[8] M. H. Ahn et al. [K2K Collaboration], Phys. Rev. D , 072003 (2006).[9] J. L. Feng, S. Su and F. Takayama, Phys. Rev. D , 075019 (2004).[10] M. Kawasaki, K. Kohri and T. Moroi, Phys. Lett. B , 7 (2005); Phys. Rev. D ,083502 (2005).[11] H. Murayama, I. Watanabe and K. Hagiwara, “HELAS: HELicity amplitude subroutinesfor Feynman diagram evaluations,” KEK-91-11.[12] T. Sjostrand et al. , Comput. Phys. Commun. , 238 (2001).[13] M. Kawasaki and T. Moroi, Phys. Lett. B , 27 (1995); T. Kanzaki, M. Kawasaki,K. Kohri and T. Moroi, Phys. Rev. D , 105017 (2007).[14] J. M. O’Meara et al. , Astrophys. J. , L61 (2006).[15] C. Amsler et al. [Particle Data Group], Phys. Lett. B , 1 (2008).[16] Y. I. Izotov, T. X. Thuan and G. Stasinska, arXiv:astro-ph/0702072.[17] M. Fukugita and M. Kawasaki, Astrophys. J. , 691 (2006).[18] J. Geiss and G. Gloeckler, Space Sience Reviews , 3 (2003).[19] M. Asplund et al. , Astrophys. J. , 229 (2006).[20] J. Hisano et al. , Phys. Rev. D , 083522 (2009).[21] A. E. G. Perez et al. , arXiv:0909.5163 [astro-ph.SR].1222] P. Bonifacio et al. , arXiv:astro-ph/0610245.[23] M. H. Pinsonneault, T. P. Walker, G. Steigman and V. K. Narayanan, Astrophys. J. , 180 (2002); M. H. Pinsonneault, G. Steigman, T. P. Walker and V. K. Narayanans,Astrophys. J. , 398 (2002).[24] A. J. Korn et al. , Nature , 657 (2006).[25] R. H. Cyburt, B. D. Fields and K. A. Olive, JCAP , 012 (2008); R. H. Cyburt andB. Davids, Phys. Rev. C , 064614 (2008).[26] J. Melendez and I. Ramirez, Astrophys. J. , L33 (2004).[27] M. Fujii, M. Ibe and T. Yanagida, Phys. Lett. B , 6 (2004).[28] M. Pospelov, Phys. Rev. Lett. , 231301 (2007).[29] K. Jedamzik, Phys. Rev. D , 063524 (2004).[30] D. Cumberbatch et al. , Phys. Rev. D76