Vacuum stability in stau-neutralino coannihilation in MSSM

The stau-neutralino coannihilation provides a feasible way to accommodate the observed cosmological dark matter (DM) relic density in the minimal supersymmetric standard model (MSSM). In such a coannihilation mechanism the stau mass usually has an upper bound since its annihilation rate becomes small with the increase of DM mass. Inspired by this observation, we examine the upper limit of stau mass in the parameter space with a large mixing of staus. We find that the stau pair may dominantly annihilate into dibosons and hence the upper bound on the stau mass ($\sim400$ GeV) obtained from the $f\bar{f}$ final states can be relaxed. Imposing the DM relic density constraint and requiring a long lifetime of the present vacuum, we find that the lighter stau mass can be as heavy as about 1.4 TeV for the stau maximum mixing. However, if requiring the present vacuum to survive during the thermal history of the universe, this mass limit will reduce to about 0.9 TeV. We also discuss the complementarity of vacuum stability and direct detections in probing this stau coannihilation scenario.


I. INTRODUCTION
It is well known that about 27% of the global energy budget is dark matter (DM). So far a lot of experimental efforts have been devoted to DM, such as direct detections, indirect detections and collider searches, but its nature still remains elusive. The weakly interacting massive particle (WIMP) is one of the most competitive DM candidates. The lightest neutralino χ 0 1 in the MSSM can serve as WIMP dark matter naturally if the R-parity is conserved.
In general, there exist two generic mechanisms to obtain the correct DM relic density [1][2][3]. One is that the lightest supersymmetric particle (LSP) can pairly annihilate into the SM particles. If the LSPs are wino or higgsino-like, their mass should be heavier than 1 TeV because their annihilation rate is large. If the LSPs are bino-like, they need to annihilate through Z or Higgs funnels or mix with higgsinos/winos to avoid overclosing the universe [4][5][6]. The other is that the LSP is bino-like and coannihilates with some other species (e.g. a stop [7], a stau [8], a wino [9], or gluino [10]) when their masses are nearly degenerate.
In such coannihilation scenarios, the light sparticles are still allowed by the LHC direct searches because they usually produce soft objects in the final states and are difficult to be observed at colliders. The delicated searches have been proposed to probe these compressed scenarios [11][12][13][14][15][16]. On the other hand, with the increase of the LSP mass, the co-annihilation rate becomes small so that the observed relic density will produce an upper limit on the mass of LSP and also its co-annihilating partner.
In this work, we focus on the stau-neutralino coannihilation in a simplified MSSM and attempt to investigate the upper limit of stau mass under the available constraints. Due to the large tau Yukawa contributions to the renormalization group evolution, the staus in some high-scale SUSY models tend to be lighter than other sleptons and may coannihilate with the neutralino DM. This was first noticed in [8] and then was calculated in details in [17,18].
Note that the stau co-annihilation is usually dominated by the processττ * → ff [8,19,20], which requires the mass of stau 400 GeV to satisfy the observed DM relic density 1 . While in the parameter space with a large mixing, the staus will mainly annihilate into dibosons, ττ * → hh/ZZ/W + W − , due to the enhanced couplings between stau and h/Z/W ± . This 1 A recent work on multi-slepton coannhilation without mixing can be found in [21]. may lift the previous upper bound on the stau mass in the coannihilation. However, such a large mixing of staus may induce a new charge-breaking vacuum and affect the vacuum stability because of the tunneling effect [22][23][24][25][26][27][28][29][30][31]. Therefore, it is meaningful to explore the vacuum stability constraint on the stau sector in the stau co-annihilation scenario.
The structure of this paper is organized as follows. In Section II, we recapitulate the calculations of DM relic density of coannihilation and the vacuum stability in the MSSM.
In Section III, we present the numerical results and discussions. Finally, we draw our conclusions in Section IV.

II. STAU CO-ANNIHILATION AND VACUUM STABILITY
The time evolution of a stable particle is described by Boltzmann equation where n (n eq ) denotes the number density of dark matter (in thermal equilibrium) and H(T ) is the Hubble expansion rate. σv denotes the thermal averaged annihilation cross section of two DM. In the early universe, n = n eq , DMs were in the equilibrium. With expansion of the universe, the number density n has been decreasing until DM are frozen out of the thermal equilibrium.
More generally, when coannihilations become important, there are several particle species i, with different masses, and internal degree of freedom g i . Each specie has its own number density n i and equilibrium number density n eq i . In this case, the rate Eq.(1) still applies, provided n (n eq ) is interpreted as the total (equilibrium) number density n = N i=1 n i (n eq = N i=1 n eq i ). Then, one can have following evolution equation, Define where and With the above definitions and the condition n i /n ≈ n eq i /n eq , Eq.(2) can be written as where the effective cross section is given by For the stau coannihilation, only the lightest bino-like neutralino and stau are involved in the calculation of Eq. (7). When their mass difference is small enough, the annihilation cross sections of co-annihilating partners may contribute to σ ef f v significantly. In particular, the coupling of stau with the SM Higgs boson can be greatly enhanced by the large mixing parameter, which is given by With this in mind, one can obtain the annihilation cross section of co-annihilating partners, This clearly shows such an annihilation process can become dominant when (A τ −µ tan β) mτ L , mτ R . Similarly, the processesτ Rτ * R → W W/ZZ would also be enhanced due to the longitudinal contribution.
As mentioned above, a large mixing of staus may induce the vacuum instability. The classical stability of the electroweak-breaking vacuum requires that the smaller eigenvalue of the stau mass matrix is positive. However, as long as the left-right mixing is large enough, the scalar potential still has possibility to develop a global minimum different from the electroweak-breaking minimum, even though the classical condition is satisfied. We use the program Vevacious [32] to find the global minima of one-loop effective potential and determine whether the vacuum is stable or not. Firstly, Vevacious input the minimization conditions of the tree-level potential for the program HOM4PS2 [33] to find all tree-level minima. Then these minima are used as starting points for gradient-based minimization (by Minuit [34] through PyMinuit [35]) of full one-loop potential with thermal corrections at a given temperature. If a minimum with lower potential energy than the desired symmetrybreaking (DSB) vacuum, CosmoTransitions [36] is then called to calculate the tunneling time from the false DSB vacuum to the true vacuum. The generic expression for one-loop effective potential energy function used in Vevacious is given by where Here φ i is the real scalar (a complex scalar field can be written as two separate real scalars) in four space-time dimensions. V counter has a polynomial of the same degree in the same fields as V tree except that its coefficients are the renormalization dependent finite parts of the appropriate counterterms [37][38][39][40]. For a given field configuration Φ, the V mass term has the form where m 2 n (Φ) are field-dependent squared masses and s is the spin of involved field. The index n runs over all of the real scalars, fermions, and vector degrees of freedom in the theory. Q is the renormalization scale and k n is constant that depend on the details of the renormalization scheme. In the DR scheme [41,42], k n is 3/2 for all degrees of freedom while in the M S scheme, k n is 3/2 for scalars and Weyl fermions, but 5/6 for vectors.
The expression for the decay rate per unit volume Γ/V for a false vacuum at zero temperature is given as [43,44] where A is a factor which is related to the ratio of eigenfunctions of the determinants of the action's second functional derivative. S 4 is four dimensional bounce action that mainly contributes to the decay rate of the false vacuum. Due to the suppression of e −S 4 , A is much less important than S 4 for the decay rate. If A is assumed to be about (100GeV) 4 , S 4 must be at least 400 in order to make Γ/V be roughly the age of the known Universe to the fourth power. For a given path through the field configuration space from the false vacuum to the true vacuum, one can solve the equations of motion for a bubble of true vacuum that has critical size in an infinite volume of false vacuum [36,43] where T is the universe temperature and S 3 is bounce action integrated over three dimensions. The non-tunneling probability P (T i , T f ) between the temperature T i and T f is given by While the surviving probability is calculated by looking for optical temperature T opt , which appears at the minimum tunneling probability between the vacuum evaporation temperature T 0 and the starting temperature of color/charge vacuum breaking T crit . More details of this method is described in [45]. In our numerical calculations, we implement Vevacious-1.2.03 interfering with CosmoTransitions-2.0 to calculate the tunneling rate for our parameter space.

III. RESULTS AND DISCUSSIONS
In our study, the relevant model parameters come from soft SUSY-breaking parameters of stau sector, tan β, bino mass parameter M 1 and higgsino mass parameter µ. We fix tan β = 50 and take a common mass parameter M SU SY = 5 TeV for other irrelevant sparticle masses and pseudo-scalar mass M A for simplicity. In order to achieve the coannihilation efficiently, we require the mass difference between stau and bino-like LSP lies in the range of 10 GeV. We use spectrum generator SPheno-4.0.3 [46] to produce SLHA files and Micromegas-5.0.4 [47] to calculate DM relic density. There is no tachyon allowed in our spectrum. Also, we require (mτ L = mτ R ) with other two non-maximal mixing (mτ L = 2mτ R , 2mτ L = mτ R ). We vary the mass parameters mτ L and µ within the ranges: 100 GeV < mτ L < 3 TeV and 100 GeV < µ < 10 TeV and tan β is fixed to be 50. It can be seen that the vacuum stability gives a strong bound on the parameter space of stau coannihilation. If requiring the vacuum is long-lived and not thermally excluded, one can obtain the upper bound on the lighter stau mass mτ 1 900, 700, 750 GeV for mτ L = mτ R , mτ L = 2mτ R and 2mτ L = mτ R , respectively. This is because that the interaction between stau and LSP is enhanced most for maximal mixing in three cases so that the corresponding main annihilation channelτ 1τ * 1 → hh has larger cross section than other two cases and allows for heavier stau mass. Besides, it should be mentioned that the thermal correction play an important role in the vacuum stability.
The reason is that for contribution from solitons that are O(3) cylindrical, it mainly from the states which have enough energy to easily pass the barrier, but it is usually suppressed by boltzmann factor e −E/T . Only for a sufficient high temperature, this contribution would become important. We checked the best transition temperature and found it as high as few hunderds GeV comparing the height of barrier ∼TeV 4 . Finally we also check our results at zero temperature with the fitting formula from [48] and found that they are well consistent.
factor of Ωχ0 where Ω P L h 2 is the relic density measured by Planck satellite [53]. From Fig. 2, we can see that almost all samples can escape the existing limits from PandaX-II(2017), XENON1T(2017) and LUX(2017). The projected LUX-ZEPLIN experiment will be able to improve the current sensitivity of LUX by about two orders of magnitude, and thus cover all of stable samples. On the other hand, it is worth to noticing that there are plenty of points below the neutrino floor, which is beyond the sensitivity of direct detections.
Fortunately, all these points can be excluded by the constraint of vacuum stability. We note here that although only the µ tan β appears in the calculation of the dark matter relic and vacuum instability, the dark matter search results mainly relies on the value of Higgsino mass parameter µ and its mixing with bino. For a smaller tan β, the maximum of the dark matter mass is essentially not changed, however, the dark matter searches results can be much relaxed if we have a much larger µ parameter.

IV. CONCLUSION
In this work, we studied the vacuum stability constraint on the stau-neutralino coannihilation in the MSSM. With the increase of the LSP mass, the observed relic density will produce an upper limit on the mass of LSP and also its co-annihilating partner stau. We noticed that the main annihilation channel of stau coannihialtion isτ 1τ * 1 → hh in the parameter space with a large mixing of staus, which can relax the upper bound on the stau mass obtained from the annihilation channelτ 1τ * 1 → ff . Under the constraint of DM relic density, we found that the lighter stau mass should be less than about 900 GeV to guarantee the vacuum stability. Besides, we noted that the vacuum stability can play a complimentary role in probing the stau coannihilation scenario as to direct detections.