Stability of the Standard Model vacuum with respect to vacuum tunneling to the Komatsu vacuum in the cMSSM
Hyukjung Kim Ewan D. Stewart Heeseung Zoe







curi951007@gmail.com
Department of Physics, KAIST, Daejeon 305-701, South Korea 


Department of Physics, Izmir Institute of Technology, Gulbahce, Urla 35430, Izmir, Turkiye 



heeseungzoe@iyte.edu.tr
Department of Physics, Izmir Institute of Technology, Gulbahce, Urla 35430, Izmir, Turkiye 





We investigate the stability of the Standard Model vacuum with respect to vacuum tunneling to the Komatsu vacuum, which exists when m_L^2 + m_H_u^2<0, in the cMSSM. Employing the numerical tools ,  and , we scan and constrain the parameter space of the cMSSM up to 10 TeV. Regions excluded due to having a vacuum tunneling half-life less than the age of the observable universe are concentrated near the regions where the Standard Model vacuum is tachyonic and are more stringent at smaller m_0, larger and negative A_0, and larger tanβ. New excluded regions, which satisfy m_h ≃ 125 GeV, are found.


Stability of the Standard Model vacuum with respect to vacuum tunneling to the Komatsu vacuum in the cMSSM
    Heeseung Zoe
    
==========================================================================================================








§ INTRODUCTION


The Minimal Supersymmetric Standard Model (MSSM) <cit.> is one of the most compelling models for physics beyond the Standard Model, as its minimal combination of the Standard Model and supersymmetry not only provides a solution to the hierarchy problem but also predictions that have been confirmed by experiments. First, assuming gauge coupling unification and a supersymmetry mass scale in the range of 10^2 to 10^4 GeV, the MSSM correctly <cit.> predicted <cit.> a relation between the Standard Model gauge couplings. Second, the MSSM has consistently predicted the Higgs mass m_h ≲ 130 GeV since well before the LHC <cit.>. Combined with the experimental bound of m_h ≳ 114 GeV from the Large Electron–Positron Collider (LEP) <cit.>, the MSSM thus predicted the Higgs boson mass within a 10% range of the value observed at the LHC <cit.>.


In the MSSM, the superpartners of the Standard Model particles were expected to be found in the range of 10^2 to 10^4 GeV and have been searched for in experiments such as the Large Hadron Collider (LHC). The observation of the Higgs boson mass at 125 GeV and recent searches have constrained the stop and gluino masses to be above 1 TeV <cit.>. More generally, the LHC has increased the lower bound of the supersymmetric particle masses <cit.>, and the difference between the electroweak scale and the supersymmetric particle mass scale has become a fine-tuning problem known as the little hierarchy problem.

The Muon g-2 experiment conducted by Brookhaven National Laboratory <cit.> and Fermilab <cit.> has measured the anomalous magnetic moment of the muon to 20 significant figures. Theoretical estimates of muon g-2 in the Standard Model derived using electron collision data <cit.> show a 5 σ significance difference with the experiment, but lattice QCD simulation of the Standard Model <cit.> agrees with the experimental results. Which calculation is correct will determine whether the g-2 experiment is evidence of physics beyond the Standard Model.

Despite a large region of parameter space being ruled out by experiments, the MSSM is still a compelling candidate for physics beyond the Standard Model, especially compared to its alternatives, such as extra dimensions <cit.>, composite Higgs <cit.> and cosmological relaxation <cit.>, which lack successful predictions. It is expected to see clear evidence for the MSSM in a future collider that can detect sparticles with masses up to 10 TeV <cit.>. 


The MSSM has a high dimensional scalar field space, whose potential may have multiple vacua <cit.>, which should satisfy the cosmological requirement that the tunneling half-life from the Standard Model vacuum to the other vacua should not be much less than the age of the observable universe.

Calculating tunneling in one-dimensional field space is straightforward, but finding the instanton path in multi-dimensional field space is difficult. However, various numerical packages have been developed over the last decade <cit.> and used to constrain MSSM parameters <cit.>. In <ref>, the authors scan and constrain the cMSSM parameters up to 4 for vacuum tunneling to vacua formed by h_u, h_d, u_3, u̅_3, e_3, e̅_3. In <ref>, the parameter space of the Natural MSSM is constrained using vacuum and thermal tunneling with non-zero h_u, h_d, u_3, u̅_3. No points satisfying the Higgs mass constraint m_h ≃ 125 were excluded by vacuum tunneling, but some were by thermal tunneling. In <ref>, the authors assume neutralino dark matter with stau-coannihilation and the theoretical estimate of muon g-2 and find points which best fit with dark matter abundance, muon g-2, and the Higgs mass and decay rate. These points are tested by vacuum tunneling to the Komatsu vacuum and they are found to be safe. Several other papers <cit.> use vacuum tunneling to further reduce the parameter space that survives after making various assumptions and applying constraints. In this paper, we scan and constrain the full cMSSM parameter space up to 10 by vacuum tunneling to the Komatsu vacuum.


The Komatsu vacuum <cit.> lies in the direction 

    μ H_u L_k = λ_d^ijd̅_iQ_jL_k + λ_e^ije̅_i L_jL_k

along which the μ term contribution to LH_u's mass squared is cancelled. At large values of LH_u, the D-term constrains |L| ≃ |H_u| and if

    m_L^2+m_H_u^2 < 0

the potential descends into a deep vacuum, where it is stabilized by higher-order terms such as supersymmetric neutrino mass terms.

In sec:model, we present the potential used in this paper and briefly review the calculation of vacuum tunneling. In sec:methods, we illustrate the tunneling calculation process, including a brief explanation of the numerical tools:  <cit.>,  <cit.> and  <cit.>. In sec:results, we plot the region excluded by vacuum tunneling to the Komatsu vacuum in cMSSM parameter space. We summarize the results and suggest future work in sec:summary.




§ MODEL




        
The MSSM superpotential is

    W = λ_u^iju̅_i Q_jH_u - λ_d^ijd̅_i Q_jH_d -  λ_e^ije̅_iL_jH_d + μ H_uH_d

There are two types of Komatsu vacuum: quark Komatsu vacuum μ H_u L_k =  λ_d^ijd̅_i Q_j L_k and lepton Komatsu vacuum μ H_u L_k =  λ_e^ije̅_i L_j L_k. We have sampled the tunneling rate for the quark Komatsu vacuum and found it to be consistently less than that for the lepton Komatsu vacuum. Thus, we restrict to the lepton Komatsu vacuum

    μ H_u L_k = λ_e^ije̅_i L_j L_k


Note that L_jL_k=0 if j = k, so we need to consider at least two lepton generations. We neglect the off-diagonal Yukawa couplings and reduce the field space by the following three criteria. First, we build the potential relevant to the Standard Model and lepton Komatsu vacua by using  H_u, H_d, L and e̅, and setting Q=u̅=d̅=0. 
Second, we choose the combination of lepton generations which gives the largest tunneling rate. When the Yukawa coupling λ_e^ij is large, e̅_iL_j can be small but still cancel the μ contribution to LH_u's mass squared, resulting in a saddle point close to the origin and a large tunneling rate. Hence, we use λ_e^33e̅_3L_3 and set e̅_1=e̅_2=0. On the other hand, a larger Yukawa coupling renormalises the corresponding slepton mass squared to smaller values at low energy, increasing the tunneling rate. Hence we set L_1= 0 and use L_2.
Third, we use SU2 gauge freedom to set h_u^+=0. Then e_2=ν_3=h_d^-=0 follows because the fields have no destabilizing linear terms. Thus, the field configuration we use in this paper is

    H_u = 
    [   0; h_u ]
    , 
    H_d = [ h_d;   0 ]
    , 
    L_2 = [ ν_2;   0 ]
    , 
    L_3 = [   0; e_3 ]
    , e̅_3=e̅_3

and other fields zero.

At tree-level our potential is 

    V_tree   = m_H_u^2|h_u|^2+m_H_d^2|h_d|^2
    +m_L_2^2|ν_2|^2+m_L_3^2|e_3|^2
    +m_e̅_3^2|e̅_3|^2 
       +(a_e^33e̅_3e_3h_d-Bμh_uh_d)+c.c.
    +|λ_e^33e̅_3h_d|^2+|μh_u-λ_e^33e̅_3e_3|^2 
       +18g_1^2(|h_u|^2-|h_d|^2-|ν_2|^2-|e_3|^2+2|e̅_3|^2)^2 + 18g_2^2(|h_u|^2-|h_d|^2-|ν_2|^2+|e_3|^2)^2



The tunneling rate to the Komatsu vacuum is calculated by the well-known instanton method <cit.>, which gives a tunneling rate

    Γ = Ae^-S_E
 
where the prefactor A comes from the measure of the path integral and includes fluctuations around the instanton, and S_E is the Euclidean action given by 

    S_E=2π^2 ∫_0^∞ρ^3 dρ[12h_ab∂ϕ^adρ∂ϕ^bdρ+ Vϕ]

where ρ is the radial distance in Euclidean space. The field space vector

    ϕ = (h_u, h_d, ν_2, e_3, e̅_3)

and h_ab is the metric on field space. The equation of motion for the instanton tunneling from false vacuum ϕ_f to true vacuum ϕ_t is

    d^2ϕ^adρ^2+3ρdϕ^adρ=h^ab∂ V∂ϕ^b

with the boundary conditions d ϕd ρ(0) =0, ϕ(0) = ϕ_t and ϕ(∞)=ϕ_f.   

 








§ METHOD


	


 §.§ Numerical tools




We use  <cit.> to generate the one-loop corrected potential and source file for  <cit.>. We calculate the supersymmetric particle spectrum with , including masses and Yukawa couplings at a two-loop level. We use  <cit.> to calculate the multi-field tunneling. It decomposes the instanton equation into parallel and perpendicular to the path and searches for the instanton path, which is the solution for both parallel and perpendicular equations.

   	


 §.§ Tunneling to Komatsu vacuum


The calculation process is illustrated in tunneling_process. We work in the context of the cMSSM <cit.>, which simplifies MSSM parameters into five parameters: the universal scalar mass m_0, the ratio between the MSSM Higgs vacuum expectation values tanβ, the universal gaugino mass m_1/2, and the universal trilinear coupling A_0, as well as the sign of μ. The scanning range for the cMSSM parameters is shown in scanning_range. 

        


We start by setting the scanning range of the cMSSM parameters.  checks whether the Standard Model vacuum is correctly broken and generates a SUSY spectrum. Next, the existence of the Komatsu vacuum is checked by the sign of m_L^2+m_H_u^2. If the Komatsu vacuum exists, we move to find the first estimate of the escape point.

We look for the escape point of the bounce solution for potentials unbounded from below in the following manner. First, we find the saddle point and set ϕ_* = 5 ×max{ h_u, h_d, ν_2, e_3, e̅_3}_ saddle to choose the initial scanning range for finding the escape point. We find the minimum of the potential within the range ϕ^a < ϕ_* for each a by using the  function in . If the Standard Model vacuum is found as the minimum, we continuously extend the scanning range to ϕ^a < n ϕ_* for some positive integer n until we find a proper minimum, which we take as a first estimate of the escape point. The process of finding the first estimate of the escape point is shown in location_finding_process. Finally, we calculate the bounce action with  and save the results, including the location of the Standard Model vacuum, the escape point and the value of the tunneling action.

        


        	
	


§ RESULTS


The constraints on the cMSSM parameter space derived by considering vacuum tunneling from the Standard Model vacuum to the lepton Komatsu vacuum are plotted in fig:plots. We set the prefactor A in tunneling_rate to ^4, and the action threshold for determining dangerous tunneling to 410, which corresponds to a tunneling half-life of the order of the age of the observable universe. We set the spread of the Higgs mass constraint as 1, which is the theoretical uncertainty of the Higgs mass calculation in  <cit.>.


[htbp]

    < g r a p h i c s >


    < g r a p h i c s >


    < g r a p h i c s >

Constraints from vacuum tunneling to the lepton Komatsu vacuum in cMSSM parameter space. Left to right: tanβ = 10, 20, 30, 40, 50; bottom to top: m_0 = 0.1, 0.2, 0.5, 1, 2, 5, 10 TeV. 0pt3pt3pt0pt  tachyonic Standard Model vacuum; 0pt3pt3pt0pt, 0pt3pt3pt0pt  S<410;   0pt3pt3pt0pt, 0pt3pt3pt0pt S>410;   0pt3pt3pt0pt, 0pt3pt3pt0pt no Komatsu vacuum; 0pt3pt3pt0pt no electroweak symmetry breaking; 0pt3pt3pt0pt, 0pt3pt3pt0pt, 0pt3pt3pt0pt m_h = 125.25 ± 1; 0pt3pt3pt0pt, 0pt3pt3pt0pt, 0pt3pt3pt0pt m_h ≠ 125.25 ± 1.






There are four notable trends in the tunneling constraints. First, regions excluded by tunneling (yellow) are located near the regions where the Standard Model is tachyonic (red). The sparticles are close to tachyonic near the red regions, so the height of the saddle is lower, reducing the action. Second, the constraints are stronger for smaller m0 because the mass squareds are smaller, lowering the height of the saddle. Third, the constraints are stronger for larger tanβ since λ_e is larger allowing e̅_iL_j to be smaller, but still cancel the μ contribution to mass squared of LH_u, resulting in a saddle point located nearer the origin and a smaller action. Fourth, the constraints are stronger for large and negative A_0. For negative A_0, the trilinear couplings are larger in magnitude at low energy, which renormalises the mass squareds to be smaller at low energy, lowering the height of the saddle.

Our main findings are as follows. For tanβ≤ 30, only a few points near the tachyonic region are excluded by tunneling and they don't satisfy the Higgs mass constraint. However, for tanβ = 40, 50, we find a region, enlarged in fig:plots_const_tlp, satisfying the Higgs mass constraint but excluded by tunneling, which has not been reported previously.

We have also scanned using the one-loop corrected potential but the tunneling results are not significantly different compared to those of tree-level potential tunneling. We did not consider the full one-loop corrections to the tunneling <cit.>.




§ SUMMARY


In this paper, we have constrained the cMSSM parameter space up to 10 by requiring the vacuum tunneling half-life to the lepton Komatsu vacuum to be greater than the age of the observable universe. The results in fig:plots show that the tunneling constraints are significant only at tanβ =40,50 and m_0 < 2, and the excluded regions are situated near where the Standard Model is tachyonic. fig:plots_const_tlp enlarges the region satisfying the Higgs mass constraint but excluded by tunneling. The constraints from vacuum tunneling are not strong, but they are model-independent and robust. We expect to see stronger constraints from thermal tunneling, but they will be less robust as they will depend on the cosmological history.




§ ACKNOWLEDGEMENTS

This work is supported by TÜBİTAK-ARDEB-1001 program under project 123F257. 


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