Upper Bound On Gluino Mass From Thermal Leptogenesis

Thermal leptogenesis requires the reheating temperature $T_R \gsim 3\times 10^{9}$ GeV, which contradicts a recently obtained constraint on the reheating temperature, $T_R \lsim 10^6$ GeV, for the gravitino mass of 100 GeV-10 TeV. This stringent constraint comes from the fact that the hadronic decays of gravitinos destroy very efficiently light elements produced by the Big-Bang nucleosynthesis. However, it is not applicable if the gravitino is the lightest supersymmetric particle (LSP). We show that this solution to the gravitino problem works for the case where the next LSP is a scalar charged lepton or a scalar neutrino. We point out that there is an upper bound on the gluino mass as $m_{\rm gluino} \lsim 1.8$ TeV so that the energy density of gravitino does not exceed the observed dark matter density $\Omega_{\rm DM}h^2\simeq 0.11$.


Introduction
In the light of experimental data of neutrino oscillations the leptogenesis [1] is the most interesting and fruitful mechanism for explaining the baryon-number asymmetry in the universe.A detailed analysis on the thermal leptogenesis [2] requires the reheating temperature T R > ∼ 3×10 9 GeV, which, however, leads to overproduction of unstable gravitinos [3].Namely, decays of gravitinos produced after inflation destroy the success of Big Bang nucleosynthesis (BBN) [4,5].This problem is not solved even if one raises the gravitino mass m 3/2 up to 30 TeV [6].Thus, the thermal leptogenesis seems to have a problem with the gravity mediation model of supersymmetry (SUSY) breaking.It is, however, pointed out in [7] that this gravitino problem may be solved if the gravitino is the lightest SUSY particle (LSP). 1e show, in this letter, that there is an upper bound on gluino mass, m gluino < ∼ 1.8 TeV, for the above solution to work.This is because the heavier gluino produces more abundantly the gravitino after the inflation and the mass density of the produced gravitino LSP, Ω 3/2 h 2 , exceeds the observed energy density of dark matter, Ω DM h 2 ≃ 0.11 [9].
The above bound on the gluino mass will be tested in the next generation accelerator experiments such as LHC.
We show that nonthermal gravitino production by decay of the next LSP (NLSP) plays a crucial role of determining the upper bound on gluino mass.We find that a consistent NLSP is a scalar charged lepton or a scalar neutrino.Its mass is stringently constrained as m 3/2 < m NLSP < m gluino , where the gravitino mass is in the region of m 3/2 ≃ 10 − 800 GeV.Other candidates for the NLSP, that is, wino, Higgsino, scalar quark (squark) and gluino, are excluded by a new strong constraint from the BBN [10], except for light gluino of mass m gluino < ∼ 70 GeV.However, such a light gluino seems to be excluded by the present accelerator experiments [11,12].
As we have mentioned in the introduction, even if the gravitino is the stable LSP, it does not totally solve the cosmological gravitino problem in the thermal leptogenesis.Namely, we have to constrain its relic abundance consistent with the recent WMAP result [9], −0.0181 .The production rate of the gravitino in the thermal bath depends on the reheating temperature T R and the mass-squared ratio m2 gluino /m 2 3/2 .The resultant relic density of the gravitino is calculated as [13,14] where α 3 (µ) is a gauge coupling constant of SU(3) c at the scale µ ≃ 1 TeV and the gluino mass m gluino is the one given at the weak scale. 2 The above WMAP constraint on Ω 3/2 h 2 ≤ Ω DM h 2 gives an upper bound on the gluino mass at a given reheating temperature T R and a given gravitino mass m 3/2 .
Another important constraint comes from late-time decay of the NLSP into a gravitino and its superpartner.The decay width of the NLSP is approximately given by [13,14] Γ NLSP ≃ 1 48π where m NLSP is the mass of the NLSP and M * = 2.4 × 10 18 GeV the reduced Planck scale.
In terms of the lifetime, it is written as and then the NLSP can decay during or even after the BBN releasing a large amount of energy.Therefore, we have to consider seriously effects of its decay on the BBN to confirm the validity of a given model.
In order not to spoil the success of the BBN, we have to satisfy two constraints on the abundance of the NLSP before its decay.One of them comes from the hadronic energy release associated with the NLSP decay.Recently, a detailed analysis on the hadronic effects has been carried out for a wide range of the NLSP lifetime [10].According to this research, even if we take a very conservative bound, the NLSP abundance is constrained as Here, B h is the branching ratio of the NLSP decay into the hadroninc components.Y NLSP is defined as the yield of the NLSP before its decay, and is given by where n γ and n NLSP are the densities of the photon and the NLSP, respectively.Furthermore, if we take the constraint from 6 Li/ 7 Li into account, this constraint becomes much stronger as 15-16) GeV for 10 Another constraint comes from the photo-dissociation of light elements by the NLSP decay.A detailed investigation on this effect was done in Ref. [5], and the result is available in Fig. 2 of Ref. [5] : where B em is the branching ratio into electromagnetic components 3 .
The SUSY standard model (SSM) has various candidates for the NLSP, that is, wino, Higgsino, squark, gluino, bino and scalar lepton (slepton).All candidates besides the bino and the scalar lepton have dominant hadronic decay modes and hence they are subject to the new constraint Eq. ( 4) from the BBN.Our numerical calculation shows that the yields of those particles are 3 For 10 4 sec < ∼ τ NLSP < ∼ 10 6 sec, this constraint is rather weak; 10) GeV [5].
For our purpose, we have calculated the relic densities of the candidates for the NLSP by using micrOMEGAs computer code [15], which includes all possible co-annihilation effects. 4We see that non of them satisfies the cosmological constraint Eq. ( 4) except for the light gluino of mass < ∼ 70 GeV.However, as pointed out in the introduction, this interesting possibility was already excluded by the present accelerator experiments [16]. 5  We should note here that if lifetimes of the NLSP's are shorter than 10 2 sec the constraint from the hadronic effects on the BBN becomes weaker [10] as We find that this weaker constraint can be satisfied when the NLSP mass is smaller than a few TeV (see Eq. ( 7)-( 10)).However, we see from Eq. (1) 6 and (3) that Ω th 3/2 h 2 is always larger than Ω DM h 2 for τ NLSP < ∼ 10 2 sec and T R > ∼ 3 × 10 9 GeV.As a result, no NLSP candidate which have dominant hadronic decay modes satisfies all the constraints, Eq. (11), GeV, simultaneously.On the other hand, the bino and the scalar lepton are still possible candidates for the NLSP, since hadronic energy releases from their decays are very small [17,18] and they are not subject to the strong constraint Eq. ( 4).However, the bino NLSP is not interesting, since its electromagnetic decay violates the constraint Eq. ( 6) as pointed out in [2].Therefore, we concentrate ourselves to the case of the scalar lepton NLSP in the subsequent sections.

Upper bound on the gluino mass with the scalar charged lepton NLSP
There are two candidates for the scalar lepton NLSP.One is the scalar charged lepton and the other the scalar neutrino.The scalar neutrino decays into a gravitino and a neutrino. 4The above numerical expressions for the NLSP abundance depend on parameters of models and may be enhanced by about a factor of 3, which does not, however, affect the following discussion.Here, we have assumed that the annihilation processes of the NLSP's do not take place near poles of some particles.We have neglected also the nonperturbative QCD effects for the gluino annihilation process. 5OPAL and CDF data exclude the existence of (quasi)stable gluino in the mass range 3 GeV < ∼ m gluino < ∼ 23 GeV and 35 GeV < ∼ m gluino < ∼ 130 GeV [16].Thus, the mass of the stable gluino is still allowed between 23 GeV and 35 GeV. 6Here, we set m gluino ≃ m NLSP in Eq. ( 1) for a conservative estimation.
The effects on BBN from the produced high-energy neutrino will be discussed in the next section.
We consider, in this section, the consequence of the constraints we have discussed in the previous section, taking the lightest scalar charged lepton (probably stau) to be the NLSP. 7Constraints for the stau NLSP come from the photo-dissociation of light elements by the stau decay, and then we determine the upper bound on the gluino mass m gluino from the relic gravitino abundance in the following procedure.(In the slepton NLSP case, the hadronic contribution dominantly comes from three-and four-body decay channels, such as lZ G and lq q G [17].The branching ratios for these modes are highly suppressed as B h = 10 −3 -10 −5 .However, even in this case, if we take the constraint from 6 Li/ 7 Li very seriously, the allowed regions that we will present in the following are likely to be reduced.In the rest of the paper, we take a conservative point of view assuming this constraint to be avoided.) Since the relic abundance of the gravitino Ω th 3/2 h 2 must be smaller than Ω DM h 2 ≃ 0.11, we obtain the upper bound on the gluino mass at a given gravitino mass from Eq. ( 1).
We show the resultant upper bound on m gluino in Fig. 1-(a).We see that this constraint does not provide a correct upper bound on m gluino , since in addition to the thermal relic abundance of the gravitino Ω th 3/2 h 2 , there is a nonthermal contribution from the late-time decay of the stau NLSP, Ω nonT 3/2 h 2 .To determine Ω nonT 3/2 h 2 at a given gravitino mass, we calculate the abundance of the stau NLSP, m stau Y stau , before its decay at a given stau mass m stau .Our numerical result 8is m stau Y stau ∼ 10 −10.3 (m stau /100GeV) 2 for tanβ = 30, where tanβ is a ratio of vacuum expectation values of the two neutral Higgs bosons (H 0 u , H 0 d ), tanβ ≡ H 0 u / H 0 d . 9Then, we can determine the upper bound on the lifetime of the stau NLSP from the Fig. 2 of Ref. [5] for a given m stau , which determines the upper bound on the gravitino mass m 3/2 (see Eq. ( 3)).By reversing this argument, we can obtain the lower bound on the stau mass and hence its abundance, Ω stau h 2 , at a given gravitino mass.By converting this lower bound on Ω stau h 2 to the Ω nonT 3/2 h 2 by Ω nonT 3/2 h 2 = (m 3/2 /m stau )Ω stau h 2 , we obtain the lower bound on the Ω nonT 3/2 h 2 at a given gravitino mass.Finally, we obtain the upper bound on the gluino mass for each set of (m 3/2 , T R ) in order to make the total gravitino relic density Ω 3/2 h 2 = Ω th 3/2 h 2 + Ω nonT 3/2 h 2 not to exceed the WMAP result, Ω DM h 2 ≃ 0.11.The result of the above procedure is given in Fig. 1-(b), which shows the upper bound on the gluino mass for a given gravitino mass.We see that the upper bound reaches 10  GeV from the bottom up, respectively.The dashed line denotes the lower limit on the stau NLSP mass for a given gravitino mass.We include nonthermal relic abundance of the gravitino in the panel (b) (tanβ = 30).
Here, we comment on the falling-off behavior of the upper bound in the region of GeV.This behavior comes from the fact that the nonthermal production of the gravitino from the stau decay becomes dominant.Namely, in the region of m 3/2 > ∼ 200 GeV 10 If one adopts the reheating temperature T R > ∼ 10 10 GeV for the leptogenesis [2], we find the upper bound on the gluino mass to be 600 GeV. 11We also find that the upper bound on m gluino reaches 1.1 TeV at m 3/2 ≃ 160 GeV for T R > ∼ 3×10 9 GeV and tanβ=10.
the gravitino produced in the stau decay dominates over the relic gravitino produced just after the inflation.In this sense the mass density of the gravitino is given by the lowenergy parameters and it is almost independent of the reheating temperature T R .Thus, we find a gravitino DM scenario becomes more attractive in this falling-off region.
Finally, we should note that the shaded regions in the Fig. 1 are conservative ones, thus, they are not always allowed for a given value of (m stau , m gluino ), since we use the lower bound on m stau at a given gravitino mass to estimate Ω nonT 3/2 h 2 .Alternatively, we can obtain the upper bound on m gluino for a given m stau in the following procedure.As discussed above, we can obtain the Ω stau h 2 and the upper bound on m 3/2 at a given m stau .Then, we search the upper bound on m gluino which satisfies Ω th 3/2 h 2 + Ω nonT 3/2 h 2 < ∼ Ω DM h 2 ≃ 0.11 for each given m stau within the above gravitino mass bound.We show in Fig. 2 the allowed parameter region in the (m stau , m gluino ) plane. 4 Upper bound on the gluino mass with the scalar neutrino NLSP In this section, we consider the constraints on another candidate for the NLSP, a scalar neutrino.A constraint on the scalar neutrino NLSP comes from the destruction of the light elements produced at the BBN epoch, which is caused by the high energy neutrino injection from the scalar neutrino decay.The high energy neutrino scatters off the background neutrino and produce a lepton-antilepton pair, then it produces many soft photons through electromagnetic cascade processes and destructs the light elements.A detailed analysis on the effects of the high energy neutrino injection was made in Ref. [18] for the case of the scalar neutrino LSP.We convert the constraint on the reheating temperature in the Fig. 2 in [18] to the NLSP abundance as Our numerical result for the abundance of the scalar neutrino is where subscript ν denotes the scalar neutrino.We see that the scalar neutrino of mass < ∼ 3 TeV satisfies the cosmological constraint Eq. ( 12).As we will see in the following discussion, this constraint is not significant to determine the upper bound on the gluino mass which is given by the requirement for the total gravitino relic density not to exceed the WMAP result, Ω DM h 2 ≃ 0.11.
To obtain the upper bound on the gluino mass, let us remember that the lower bound on the scalar neutrino mass is fixed (by the definition of the NLSP) as m ν > m 3/2 at a given gravitino mass. 12Then, it determines the lower bound on the relic abundance of the scalar neutrino, Ω ν h 2 , at a give gravitino mass.As in the previous section, this lower bound on Ω ν h 2 give rise to lower bound of the Ω nonT h 2 at a given gravitino mass.Finally, we obtain the upper bound on the gluino mass for each set of (m 3/2 , T R ) in order to make the total gravitino relic density Ω 3/2 h 2 = Ω th 3/2 h 2 + Ω nonT 3/2 h 2 not to exceed the WMAP result, Ω DM h 2 ≃ 0.11.
The result of the above procedure is given in Fig. 3

Conclusions
A recent detailed analysis [10] of the hadronic effects of the gravitino decay on the BBN leads to a stringent constraint on the reheating temperature of the inflation as T R < ∼ 10 6 GeV.This constraint contradicts the condition for the thermal leptogenesis, that is T R > ∼ 3 × 10 9 GeV.A solution to this serious problem is provided [7] if the gravitino is stable LSP.However, it depends on nature of the NLSP if this solution works or not.We have shown in this letter that the consistent candidate for the NLSP is a scalar charged lepton or a scalar neutrino.We have found that there are upper bounds on the gluino mass < ∼ 1.3 TeV and 1.8 TeV, for the former and the latter case, 13 respectively, so that the density of the gravitino do not exceed the observed dark matter density Ω DM h 2 ≃ 0.11 for T R > ∼ 3 × 10 9 GeV.In the present analysis we have used the perturbative calculation in evaluating the yield of relic gluino.However, it should be kept in mind that the nonperturbative QCD dynamics may increase the annihilation cross section of gluino which decreases the yield of gluino NLSP [11].For instance, if it increases the annihilation cross section by a factor of 100, the gluino of mass < ∼ 300 GeV satisfies the BBN constraint Eq. ( 4).In this case the reheating temperature T R can be easily taken above T R ≃ 3 × 10 9 GeV.

Figure 1 :
Figure1: The upper bound on the gluino mass at a given gravitino mass.The solid lines show the upper bounds on the gluino mass for the reheating temperature T R = 10 10 GeV, 3 × 10 9 GeV and 10 9 GeV from the bottom up, respectively.The dashed line denotes the lower limit on the stau NLSP mass for a given gravitino mass.We include nonthermal relic abundance of the gravitino in the panel (b) (tanβ = 30).

Figure 2 :
Figure2: The upper bound on the gluino mass at a given stau mass (tanβ = 30).The solid lines show the upper bounds on the gluino mass for the reheating temperature T R = 10 10 GeV, 3 × 10 9 GeV and 10 9 GeV from the bottom up, respectively.The dashed line denotes the lower limit on the gluino mass at a given stau mass, m gluino = m stau .