Cosmology of Stop NLPS and gravitino LSP

In this section we discuss shortly the cosmological bounds on the scenario with stop NLSP and the gravitino DM and LSP, in order to single out the cosmologically preferred parameter space in both RPC and RPV models. Such a cosmologically viable parameter space will be then compared to the LHC reach for direct stop NLSP production we are going to discuss later in this chapter.

Let us consider first of all the effect of a stop NLSP during BBN. The stop is a colored and EM-charged particle and therefore it can disrupt BBN not only through the energy release in the decay, but also because of the bound state effects [101]. In the first case the light element abundances are more strongly affected by hadro-dissociation and therefore the limits are more stringent for hadronically decaying particles like the stop [99, 149]. In the latter case the constraints are independent of the decay channel and just depend on the stop lifetime and density at the time of decay [101,150]. For this reason, such constraints can be applied equally to any of the scenarios we have discussed.

The limits on the abundance Y_χ(τ_χ) = n_χ/s from bound state effects for a hypothetical long-lived strongly interacting massive particle χ, have been computed by Motohiko Kusakabe et al. in [150]. In particular, by requiring that the primordial light element abundances remain within the observed ranges, they obtained the following constraints depending on the particle lifetime τ_χ:

• Yχ <10⁻¹⁸−10⁻¹² (for 30 s< τχ<200−300 s),

• Y_χ <10⁻¹⁸−10⁻²¹ (for 200−300 s < τ_χ<2×10³ s)

• Y_χ <10⁻²¹−10^−22.6 (for 2×10³ s < τ_χ4×10¹⁷ s) .

In the window (30s < τχ < 200−300 s) the most stringent constraint comes from the upper limit on the ⁷Li, while for (200−300 s< τχ <2×10³s) the strongest constraint is due to the upper limit on the B abundance. Finally the upper limit on the ⁹Be abundance determines the bound for longer lifetimes. These constraints are very steep and become quickly dominant over the hadro-dissociation bounds [99, 149]. For lifetimes shorter than approximately 30 s the constraints from bound states disappear and those on hadronic decays are very weak [151] and this corresponds in our two scenarios to a stop mass range:

m˜t≥







451 GeV_{1 GeV}^m3/22/5 RPC

1.4×10⁻⁸ GeV^sin₁₀^{θ/−8cosβ−2} RPV . (3.11)

1Indeed also for the B-mesons in the limit of infinitemb the hadronic matrix elements for the inclusive decay rate tend to one [23].

Chapter 3. Long-Lived stop at LHC 70

Figure 3.2: Contours for the mass fraction of ⁴He, Y_X =n_X/n_b = 0.2419 (red line) and the number ratios of³He/H = 3.1×10⁻⁵ (green line), D/H =3.24×10⁻⁵ and D/H =2.45×10⁻⁵ (black lines),⁶Li/H≈10⁻¹⁰(blue solid line) and⁶Li/H =(7.1±0.7)×10⁻¹²(blue dashed lines),

7Li/H =6.15×10⁻¹⁰ (purple line),⁹Be/H =10⁻¹³ (pink line), B/H = 10⁻¹² (orange line) and C/H =10⁻⁸ (gray line) are shown [150].

We see therefore that BBN does not provide practically any bound on the R-parity violating (RPV) scenario, unless the RPV coupling is very small, below 10⁻¹². For this reason we will only consider in detail the constraints for the RPC case in the following.

The above limits on the abundanceY_χcan be found in Figure 3.2 if we convert the mass frac-tion YX(= nX/nb) intoYχ through the formula: Yχ'10⁻¹⁰YX. Note that the latter formula can be obtained by using the well-known relation s = 7.04nγ along with the approximation η=n_b/n_γ'5×10⁻¹⁰ and bothχ and X represent the same hypothetical long-lived particle.

RPC decay of stop NLSP in cosmology

The relic density of a colored relic like a scalar top and the BBN bounds from hadro-dissociation have been studied in a model independent way in the past by C. Berger et al. in [104]. Below we follow this analysis, but we update the constraints to include also the bound state effects discussed above. In [104] the authors first have considered the simplified case of a single an-nihilation channel ˜t˜t^∗ → gg in the stop Boltzmann equation. Such a choice is motivated by the fact that such channel just depends on the stop mass and its QCD representation, with-out dependence on the rest of the supersymmetric spectrum, and, in addition, it is always the dominant channel, contributing at least 50% of the total annihilation cross-section. It therefore gives the most conservative result since it cannot be suppressed by particular choices of the superparticles spectrum and it provides a reliable upper limit on the stop abundance. In fact, other annihilation channels can only increase the cross section and, therefore, reduce the stop density. In this case the stop abundance is proportional to the stop mass and it reads [104]:

Y˜t(m˜t) =Y˜t(1 TeV) m˜t

1 TeV

, (3.12)

Chapter 3. Long-Lived stop at LHC 71

Figure 3.3: The effect of the Sommerfeld enhancement on the yield from ˜t˜t^∗ →gg: the full line shows the tree-level result, the dashed line the result for σ^av, i.e. applying an averaged Sommerfeld factor, and the dashed–dotted line is forσ^sum, i.e. applying a summed factor. For

more details see [104].

up to logarithmic corrections, since, in general, the mass always appears linearly in the Boltz-mann equation for the stop density² [104].

Perturbation theory is commonly used to calculate annihilation and scattering cross sections, with higher-order terms in the perturbative expansion being neglected. Provided that the theory is not strongly coupled, this is generally a good approximation for relativistic particles, however at low velocities and in the presence of a long-range force (classically, when the potential energy due to the long-range force is comparable to the particles kinetic energy), the perturbative approach breaks down. In the non-relativistic limit, the question of how the long-range potential modifies the cross section for short-range interactions can be formulated as a scattering problem in quantum mechanics, with significant modifications to the cross sections occuring when the particle wave functions are no longer well approximated by plane waves (so the Born expansion is not well-behaved). The deformation of the wave functions due to a Coulomb potential was calculated by Sommerfeld in [152], yielding a∼1/v enhancement to the cross section for short-range interactions (where the long-short-range behavior due to the potential can be factorized from the relevant short-range behavior).

The computation of such a Sommerfeld enhancement [153,154] for the studied channel ˜t˜t^∗ → gg was also performed in [104], where the authors employed two different prescriptions for the higher orders. Note that the Sommerfeld enhancement increases the cross-section at low velocity and can be obtained by resumming over the exchange of a ladder of gauge bosons between the initial particles. It was found that the averaged Sommerfeld factor reduces the tree-level yield by roughly a factor of 2, while the summed Sommerfeld one by roughly a factor of 3, as we can see it in Figure 3.3. We will therefore take the stop abundance from the leading order computation in [104] and vary it by a factor 2-3 to see the effect of both the Sommerfeld enhancement and the additional annihilation channels.

In order to set limits on the RPC model, we compute the stop density as a function of the stop mass from Equation (3.12) and we compare it with the limits in [150]. We determine then

2Recently it has been discussed in [135] that the stau NLSP abundance is better fitted by a dependence given bym^0.9_τ˜ but in the range of masses we are considering such a difference in the exponent has negligible effect.

Chapter 3. Long-Lived stop at LHC 72