Exploring Charge and Color Breaking vacuum in Non-Holomorphic MSSM

Non-Holomorphic MSSM (NHSSM) shows various promising features that are not easily obtained in MSSM. However, the additional Non-Holomorphic (NH) trilinear interactions that attribute to the interesting phenomenological features, also modify the effective scalar potential of the model significantly. We derive analytic constraints involving trilinear parameters $A_t'$ and $A_b'$ that exclude global charge and color breaking minima (CCB). Since the analytic constraints are obtained considering specific directions in the multi-dimensional field space, we further probe the applicability of these constraints by exhaustive scan over NH parameter space with two different regimes of $\tan\beta$ and delineate the nature of metastability by considering vacuum expectation values for third generation squarks. We adhere to a natural scenario by fixing Higgsino mass parameter ($\mu$) to a low value and estimate the allowed ranges of NH trilinear parameters by considering vacuum stability and observed properties of Higgs as the determining criteria.


Introduction
The electroweak symmetry breaking (EWSB) and strong interactions are quite successfully explained in the Standard Model (SM) [1] of particle physics. The discovery of the SM-like Higgs boson of mass about 125 GeV at the Large Hadron Collider experiments at ATLAS [2] and CMS [3] marks the end of particle searches within the SM. In such a scenario, studies Beyond the Standard Model (BSM) are motivated by the quest for new physics that is essential for addressing the puzzles that are not resolved in the SM. BSM theories are required to provide a potential solution to the so-called gauge hierarchy problem along with explaining the massive neutrinos and providing suitable particle candidate for Dark Matter (DM) [4][5][6]. Supersymmetry (SUSY) [7][8][9][10]  neutralino in a R-parity conserving scenario is a viable particle candidate for a cold dark matter. However, the absence of any hint of SUSY at the LHC has considerably constrained the MSSM. The requirement of large radiative corrections to Higgs boson mass m h 1 demands heavier stops or larger stop mixing trilinear soft SUSY breaking parameter (A t ). However, such a large A t is severely constrained from the Br(B → X s +γ) limits at large tan β [11]. On the other hand, the well-motivated higgsino dark matter that can easily accommodate the relic abundance data from PLANCK [12] and direct detection cross-section measurements from LUX [13] experiment demands the lightest SUSY particle (LSP) to be about 1 TeV.
Moreover, the results from Brookhaven experiment [14] for the anomalous magnetic moment of muon or (g − 2) µ show about 3σ deviation from the SM predictions creating a scope for a new physics explanation. In the MSSM, a very light smuon (μ) is required to accommodate the (g − 2) µ results within 1σ, in a region where the SUSY contribution (a SUSY µ ) [15][16][17][18][19][20][21] is dominated by theμ − χ 0 1 loops. On the contrary,μ with very low mass is not so friendly with the LHC data [22,23].
Apart from the experimental and phenomenological aspects, there are a few theoretical issues that need to be explored. In view of current experimental constraints, ElectroWeak Fine Tuning (EWFT) is a major concern in the phenomenological MSSM (pMSSM). In general, larger value of the bilinear Higgs mixing parameter of the superpotential (µ) significantly enhances the EWFT [24]. In the MSSM, even in scenarios that are devoid of any Higgsino like LSP, lower bound on |µ| is around 100 GeV via LEP limits on lighter chargino.
Moreover, in a Higgsino DM setup, relic density limits from PLANCK constrain m χ 0 1 to be theory, particularly for the possible appearance of charge and color breaking (CCB) minima [29][30][31][32][33][34][35][36][37][38][39][40][41][42]. Some well-known analytic constraints with simplified assumptions have been used in the MSSM in order to avoid the regions of parameter space where CCB minima is deeper than the desired symmetry breaking (DSB) SM like (SML) vacuum. Consideration of scenarios with global CCB minima, where the DSB minima are stable with respect to tunneling to the deeper CCB minima have led to considerable relaxation of the analytic CCB constraints in the MSSM [43,44]. If local DSB minimum has a large lifetime in regard to quantum tunneling to global CCB minima, the corresponding DSB minima is referred to as 'long-lived' SML vacuum in the literature [45][46][47][48][49][50][51]. In this work, we would probe the vacuum structure of the NHSSM and study the stability of DSB minima using our implementation of complete 1-loop corrected scalar potential in Mathematica and the publicly available package Vevacious [61] which in turn uses CosmoTransitions [62] for determining tunneling time to a deeper vacuum. We would determine the analytic constraints to avoid global CCB vacua using tree-level scalar potential and explore the extent of its applicability. We shall also study the role of different NH parameters in radiative correction to m h 1 in order to identify phenomenologically important regions of NH trilinear SUSY breaking parameters associated with third generation squarks sector viz. A t and A b respectively.
The analysis is structured as below. In Sec.2 we very briefly introduce the NHSSM and discuss the analytical CCB constraints and dependence of effective potential on tri-linear NH parameters. We also analyze the role of vacuum expectation values (vevs) of stops (t) and sbottoms (b) in determining the fate of the DSB vacuum while satisfying the Higgs mass constraints. In Sec.3 we present the results and analyze them in the light of theoretical predictions. Finally, we conclude in Sec.4.
The "may be soft" [64] D-term contributions like 1 give rise to the NH terms in the Lagrangian viz. φ 2 φ * and ψψ respectively. Here X and Φ are chiral superfields and SUSY breaking in the hidden sector is driven by vev of an auxiliary field F belonging to X. We note that < F > /M should refer to a weak scale mass which we consider here as the W-boson mass M W . However, in models like pMSSM it can be varied independently. Hence, coefficients of φ 2 φ * and ψψ varies as |<F >| 2 Thus, it is evident that they are highly suppressed in super-gravity type of models. However, in weak scale scenario φ 2 φ * and ψψ are quite significant and appear as the NH trilinear terms and bare Higgsino mass term respectively 1 .
This was discussed or at least pointed out in several works [64-66, 68, 71-75]. Impact of NH parameters on m h 1 , electroweak fine tuning, Br(B → X s + γ) and (g − 2) µ have been analyzed in Ref. [26]. Earlier works involving the analysis of the effects of NH terms include Ref. [76][77][78]. Here we would like to explore the scalar potential of NHSSM with emphasis on the vacuum stability issues. Our prime focus will be on the attribution of the NH terms towards determining the fate of the EWSB vacuum.

CCB in NHSSM
We remind that the MSSM is considered to have only holomorphic soft SUSY breaking terms. The trilinear soft terms, in particular, are given by as follows [8].
We have only shown here the dominant terms involving the third generation of sfermions. It has been shown that in the absence of any gauge singlet it is possible to extend the SUSY breaking soft sector by including NH soft SUSY breaking terms, without aggravating any quadratic divergence [63,66,72]. Thus, the NH soft terms of the NHSSM in general that include trilinear coupling terms as well as a NH higgsino mass (µ ) term are given by [25,75] Thus, the scalar potential at the tree level including the Higgs and the stop fields read as follows. where m 2 2 = m 2 Hu + µ 2 and m 2 1 = m 2 H d + µ 2 and m 2 Hu (m 2 H d ) is the soft squared mass term for up (down) type Higgs. Relevant terms may be added to the above equation to construct the complete scalar potential at the tree level taking into account the other sfermions. In the above expression, vanishing value of A t leads to the generic MSSM scenario. In this analysis we will explore the CCB constraints at tree-level for deriving simple analytic result.
The results from our numerical scans are obtained via Vevacious [61] which includes 1loop corrected potential both at zero temperature and finite temperature. Furthermore, in presence of CCB minima, it computes the tunneling rate from DSB vacuum to the former using CosmoTransitions [62]. However, we consider the scalar potential only at the treelevel in order to obtain the analytic CCB constraints analogous to those used in the MSSM.
One loop correction (in the DR-scheme) [79,80]) at zero temperature is given by where the sum runs over all real scalars, vectors and Weyl fermions that are present in the model with and Q i = 2(1) for charged particles (neutral particles), C i is the color degrees of freedom, s i is the spin of the particle, m i is the mass of the same and Q is the renormalization scale used.
The stability of the DSB vacuum can be significantly affected by thermal corrections to the scalar potential. The 1-loop thermal correction to the potential at temperature 'T' is given by [81] ∆V thermal = 1 The sum in equation 2.6 runs over all degrees of freedom that couple to the scalar fields including the scalar fields themselves. J + (J − ) corresponds to the corrections arising from bosons (Weyl fermions) and m is the mass of the corresponding particle at zero temperature.
J ± ( m 2 T 2 ) asymptotically approaches zero (−∞) as m 2 T 2 approaches ∞ (zero). It is to noted that thermal corrections would always lower the potential [54,61] depending on the magnitude of m 2 T 2 . The complete scalar potential at one loop that includes both zero temperature and thermal corrections, is given by V = V| tree + ∆V rad.corr. + ∆V thermal . (2.8) Analytic CCB constraints in MSSM were derived along particular direction of field space, namely "D-flat" direction [29]. Keeping this in mind, we consider non vanishing vevs for the two higgs scalar and the stops fields. The latter are responsible for the generation of CCB minima. In the direction where |H d | = |H u | = |t R | = |t L | = ζ, the scalar potential (Eq.2.3) reduces to where a = 3y 2 t − g 2 1 +g 2 Minimizing the above potential with respect to ζ and considering non-vanishing ζ, we obtain Hence, the value of the field at the minima is From the reality condition of the roots, we get 9b 2 > 32ac. V min > 0 at the minima implies, aζ 2 +bζ +c > 0. Here we consider a scenario with 4 vevs otherwise, the contribution of A t and A t cannot be taken into account simultaneously. One may also consider the scalar potential excluding the holomorphic trilinear terms in a three vevs scenario with where m 2 2 = m 2 Hu + µ 2 . This is referred as the condition for the existence of a deeper CCB in MSSM which is often approximated as that is the traditionally used CCB constraint associated with stop (t) scalars receiving vevs [29, 30, 32-35, 39, 41]. Similar constraints are used for different sfermion fields. In NHSSM, we consider the following inequality resulting from reality condition 9b 2 > 32ac that implies In contrast to MSSM, here we do analyze the scalar potential considering non-vanishing vevs for all the four scalar fields. The Eq.2.13 simplifies to Eq.2.14 identifies regions of parameter space associated with global CCB minima. Neglecting the contributions from g 2 1 and g 2 2 with respect to y 2 t , the CCB constraint in NHSSM fort fields becomes One can always retrieve the traditional constraint in MSSM (Eq.2.12) from Eq.2.15 assuming A t = 0 and H d = 0. As discussed in Ref. [29], the most stringent CCB constraint is obtained when the relative sign between the trilinear terms in the potential are always considered positive. Thus, in NHSSM the analytic condition predicting an absolutely stable DSB vacuum Along with the absolute stable DSB minima, we also consider long-lived DSB minima, in presence of global CCB vacuum that in turn will increase the valid parameter space. Here one should note that rate of tunneling from DSB false vacuum to such CCB true vacuum is roughly proportional to e −a/y 2 , where a is a constant of suitable dimension that can be determined via field theoretic calculations and y is the Yukawa coupling. The tunneling rate is enhanced for large Yukawa couplings [30,[47][48][49][50][51]. As a result, the third generation of sfermions will be the most important candidate in connection with the presence of potentially dangerous global minima. In models like the MSSM and the NHSSM, the Yukawa couplings vary with tan β which is given by tan β = vu v d . However, unless tan β is very large the relation y t > y b > y τ is by and large satisfied, consistent with the mass hierarchy of the corresponding fermions. Hence, global CCB minima associated witht fields will be most dangerous due to comparatively larger y t . Theb fields may also play an important role in determining the fate of the DSB minima. A large value of the NH parameter A b associated withb significantly modifies m h 1 andb phenomenology particularly for large tan β. In the phenomenologically interesting regions of parameter space, the structure of the scalar potential is modified by the presence of large A b and in such a setup, vacuum stability needs to be re-explored. In this context, one should study the analytic CCB constraint associated with theb fields.
Following a similar method as adopted fort, one gets the CCB constraint forb to be Unlike the case oft, here we cannot neglect g 1 and g 2 with respect to y b , since y b is not very large. In the limit of MSSM, in three vevs scenario with non-vanishing vevs for H d ,b L and b R , with H u = 0, we get the well-known CCB constraint forb, namely Thirdly, in a phenomenological analysis of the pMSSM, tan β varies over a wide range. Thus, y t and y b vary and g 1 and g 2 may not be always negligible with respect to Yukawa couplings.
Lastly, this work goes beyond an absolute vacuum stability by accommodating long-lived scenarios, demanding a detail numerical analysis. In our work using Vevacious [61] we probe the vacuum structure to find the DSB minima. We explore a four vev scenario considering non-vanishing vevs fort L ,t R along with H d and H u . In the other part of our analysis we consider a different four vev combination involvingb L ,b R , H d and H u , in order to gauge the role ofb fields in determining the fate of the DSB minima. In both the above cases, we try to estimate the applicability of the analytic constraints. Finally, we analyze a more involved scenario with six vevs where non-zero vevs were attributed tob L andb R along witht L , t R , H d and H u . Additionally, a varying tan β, changes the values of v u and v d that in turn modify the yukawa couplings. The vacuum structure thus depends on tan β. We consider two different zones of tan β to explore the dependency.

NH parameters and their impact on m h 1
Besides affecting vacuum stability, the NH trilinear parameters play a significant role in phenomenology where left-right mixing of sfermions are important. Additionally, the NH higgsino mass parameter µ is important in processes dominated by higgsino and it controls the overall higgsino phenomenology. The NH parameters significantly modify the Br(B → X s + γ) and radiative corrections to m h 1 . The role of A t together with µ in the radiative corrections of m h 1 has been studied in Ref. [26] in the low scale NHSSM (pNHSSM). Here we would like to probe the contribution of the NH parameters towards the radiative corrections to m h 1 in order to identify phenomenologically interesting regions of NH parameters for exploring vacuum stability. The radiative correction to m h 1 due to stop-loop is given by Here X t = A t − (µ + A t ) cot β andm t stands for the running top-quark mass that includes electroweak, QCD and SUSY QCD corrections [82]. Apart from this, particularly for large tan β, A b the NH trilinear soft parameter associated with the bottom squarks, has significant bearing on m h 1 . The contribution from sbottom-loop is In the MSSM, the contribution fromb loops is important only for large tan β and µ, sincem b is significantly smaller thanm t . Hence, in a MSSM scenario motivated by low EWFT, where µ is predominantly very small, the contribution to m h 1 due to theb is negligible, although the effect is tan β enhanced. The situation is quite different in the NHSSM. Here for a large value of A b , ∆m 2 h,bottom is albeit small but not negligible compared to ∆m 2 h,top because of the associated tan β enhancement. Thus even in a scenario motivated by low EWFT, the radiative correction to m h 1 fromb loops can be nonnegligible in the NHSSM. However, the contribution may be negative or positive depending on the value of X b .

Results
We explore the nature of the scalar potential in a semi-analytic approach that demonstrates the dependency on NH parameters. We study the degree of applicability of the analytic CCB constraints viz. Eqs. 2.16 and 2.17 in the NHSSM. Considering the contributions of A t and A b to m h 1 , we analyze the stability of the DSB minima using Vevacious [61]. SARAH [83][84][85][86] and SPheno [87][88][89][90][91] were used for model building and spectrum generation respectively.
The spectra generated by SPheno include complete two-loop corrections to Higgs mass even from NH parameters [92,93]. The numerical study is performed in three different set-ups of non-vanishing vevs. First, we exclusively probe the stability of DSB minima in presence of global CCB minima associated with non-zero vevs oft L andt R . Here, we consider a four-vev scenario with non-zero vevs for H d , H u ,t L andt R . Then, we explore the role of Figure 1. ∆V represents the depth of the potential at the deeper CCB vacuum with respect the to the field origin. Fig.1(a) shows the variation of ∆V with y t A t fort vevs scenario. Fig.1 b L andb R in rendering the DSB minima unstable. Finally, we analyze a combined multiplevev scenario with non-vanishing vevs for six scalar fields viz. H d , H u ,t L ,t R ,b L andb R . y t and y b will significantly affect the vacuum stability against CCB minima. Hence, we divide our analysis into two tan β regimes, viz. 5 ≤ tan β ≤ 10 and 40 ≤ tan β ≤ 50.

Nature of the NHSSM scalar potential
Now we focus on the NH parameter space to identify the nature of CCB vacuum in detail, by assigning vevs tot andb fields separately. This is essential in order to identify the dangerous directions in the field space while studying the role A t and A b in modifying the vacuum structure of the model. Since pNHSSM is principally motivated by naturalness, we choose µ = 200 GeV for the entire analysis. Fig.1 demonstrates the variation of ∆V with the NH trilinear parameters for the vacuum deeper than the DSB one, where ∆V represents the depth of the scalar potential with respect to the origin. Fig.1(a) (1(b)) corresponds to thet (b) vevs scenario. The red (blue) color refers to tan β = 50 (tan β = 5). We keep mt L , mt R and mb R fixed at 2 TeV. The MSSM trilinear term y t A t (y b A b ) is fixed at 2 TeV (0). The soft parameter B µ in Eq.2.3 is kept fixed at 2 × 10 5 GeV 2 . The relevant NH parameters 3 for the scenarios witht andb vevs are given by the following ranges. (3.1) A t and A b clearly play a significant role in lowering the scalar potential. It is to be noted further, that the range of y b A b is kept small. This is because large A b leads to very large SUSY threshold corrections to bottom yukawa coupling and thus, the spectrum generator SPheno aptly reports error. Recently, the effect of such corrections to y b has been studied in the context of vacuum stability in the MSSM [55,56]. For the given choice of parameters, the depth of the DSB vacuum is many orders shallower than the depth at the deeper CCB vacuum. Furthermore, we observe from Fig.1(a) and 1(b) that depth of the potential at the CCB vacuum is deeper for tan β = 5 case. Stop sector being the major contributor to the effective potential (y t A t = 2 TeV gives large contribution even for vanishingly small NH parameters), the effect gets enhanced for tan β = 5 compared to tan β = 50 due to large y t for the former case. We also observe from Fig.1(a) and 1(b) that the depth of the potential at tan β = 5 for the case withb vev is a bit deeper compared to thet vev one. This is due to the additional contribution arising from sbottom sector for large A b in theb case.
It is to be noted in Fig.1(b) that there are two distinct blue lines for A b < 0. This is purely a numerical artifact due to the fact that the minimization routine Minuit used in Vevacious sometimes misses out some of the minima during random scan. Thus, in some cases, it reports a different CCB vacuum for similar parameter sets. In the framework of Vevacious, the vevs are bounded by a hypercube of length 20Q. We keep the renormalization scale Q at a value √ mt L mt R all through this work.
In the Eqns. 2.16 and 2.17 for the constraints on non-holomorphic parameters, all the vevs are considered to be equal. However, this is not a realistic scenario. In Fig.2(a) and vevs for stops (sbottoms) represent CCB minima. For the stop vevs ( Fig.2(a)), we choose H 0 u = H 0 d direction and for the sbottom vev ( Fig.2(b)), we choose H 0 d = 0 direction in order to capture the global CCB minima and the DSB vacuum. In the subsequent sub-sections, we analyze the CCB vacua due to stop fields, sbottom fields and finally, combination of both stop and sbottom fields on a broader region of parameter space compared to the one presented above.

CCB minima associated with stop fields
Keeping an eye on Eq.2.16 we define M 2 t = m 2 1 + m 2 2 + m 2 t L + m 2 t R − 2B µ and A t 2 = (|A t | + |µ| + |A t |) 2 . These variables will play a crucial role in determining the applicability of the analytic constraint of Eq.2.16. We also define similar variables 2 to be used while studying the CCB constraints arising out of non-vanishing vevs ofb fields.
We probe the vacuum stability of the model for both smaller (5 ≤ tan β ≤ 10) and larger values (40 ≤ tan β ≤ 50) of tan β. We analyze in a minimal four-vev set-up considering non-zero vevs for the Higgs and the stops. We vary the relevant parameters in the following  The red (black) points refer to thermally excluded (quantum mechanically unstable) vacuum. ranges.
We keep y t A t fixed at 2 TeV, and set all other holomorphic soft trilinear parameters to zero. All scalar mass parameters excluding mt L are fixed at 2 TeV. Since we study the vacuum structure with specific motive of investigating the role of NH parameters, we do not scan over holomorphic trilinear parameters. Other fixed parameters are same as that of Fig.1(a). Fig.3 shows the vacuum stability profile in X t − m h 1 plane. In spite of having no influence on vacuum stability, µ is scanned over a moderate range. This is due to the fact that it contributes to m h 1 via chargino loop [94,95]. The green and blue points are associated with stable and long-lived DSB minima. They are collectively referred to as safe vacua. The black and the red colored regions are excluded since the DSB minima associated with them are rendered unstable with respect to quantum tunneling to deeper CCB minima and thermal effects respectively. Fig.3(a) and 3(b) correspond to low and high tan β regimes.
In Fig.3(a), we see stable points (green) cluster around Xt Ms  The color codes are same as that for Fig.3. in X t − m h 1 plane is qualitatively very much similar to Fig.3(a). Fig.3(b) appear to be a bit different from Fig.3(a). This is because mass of the Higgs boson gets boosted for chosen value of y t A t (2 TeV) and tan β = 50. There are no quantum mechanically short-lived points (black) for Fig.3(b), whereas there are large number of black points in Fig.3(a). Fig.4 shows the dependence of m h 1 on the y t A t for tan β scanned over two different domains. Fig.4(a) shows the stability profile for 5 ≤ tan β ≤ 10. The spread of m h 1 for particular value of y t A t is due to the effect of variation of mt L and tan β on the radiative corrections arising out of stop loops (Eq. 2.19). Fig.4(b) shows the similar plot for 40 ≤ tan β ≤ 50. As discussed in Sec.2, A t contribution in the radiative correction to m h 1 viat loop is tan β suppressed. As a result the effect of y t A t on m h 1 is more prominent in low tan β region. Thus, for Fig.4(b) we see that the variation of m h 1 is significantly small compared to Fig.4(a). In both the plots, the central region characterized by comparatively lower |y t A t | is associated with stable DSB vacua, whereas large |y t A t | region is associated with unsafe DSB minima. In between the two regions, there exists a small zone associated with long-lived states (blue). We find that for low tan β and |y t A t | > ∼ 3 TeV the DSB minima becomes unsafe and they are excluded via quantum tunneling or thermal effects. On the contrary, for 40 ≤ tan β ≤ 50, comparatively large values of |y t A t | (< 4 TeV), is allowed in the region of parameter space associated with safe vacuum. This is due to the fact that the term in the potential associated with A t is suppressed for large tan β. Fig.5 shows the vacuum stability profile in y t A t − mt 1 plane for same set of scans. As before, Fig.5(a) (Fig.5(b)) represents the case with 5 ≤ tan β ≤ 10 (40 ≤ tan β ≤ 50). We observe from both the plots that mt 1 could be rather large in the region of metastability triggered by large |y t A t |. We have kept mt R fixed at 2 TeV. Hence large mt L along with large |y t A t | facilitate the appearance of deeper CCB vacua by inducing large mixing betweent L and t R . Thus, mt 1 becomes large for large |y t A t |. We also notice that due to tan β suppression, the spread of green points in Fig.5(b) is more compared to Fig.5(a). Furthermore, the flat green edge (mt L ≈ mt 1 ≈ 500 GeV) is also broader compared to Fig.5(a) for the same reason.

CCB minima associated with sbottom fields
As discussed in Sec.2 the global CCB minima associated witht fields are more dangerous with respect to the tunneling rate due to large yukawa coupling (y t ). Theb fields may also become important particularly for large tan β. Hence, in this section, we study the role of y b A b associated withb L andb R in determining the fate of DSB minima. We probe the vacuum stability of the model for both smaller (5 ≤ tan β ≤ 10) and larger values (40 ≤ tan β ≤ 50) of tan β. Considering the effects on vacuum stability, we vary the relevant parameters in the following ranges.
We keep y t A t fixed at 2 TeV, and set all other holomorphic soft trilinear parameters to zero.
All scalar mass parameters excluding mt L and mb R are fixed at 2 TeV. Fig.7(a) shows the stability profile in y b A b − m h 1 plane for low tan β (5 < tan β < 10). As expected, we see that the DSB minima are safe for most of the values of A b . This is consistent with the discussion in Sec.2, as for small tan β the NH terms associated withb are less dominant. It appears   Fig.7(a) shows the stability profile in y b A b − m h 1 plane for 5 < tan β < 10 corresponding to scan of Eq.3.4. Fig.7(b) is a similar plot for 40 < tan β < 50. The color codes for stability profile is same as that for Fig.3.
that most of the metastable points are actually long-lived and thus, these are considered safe. For low tan β, y b is very small and mixing betweenb L andb R would only be enhanced if difference between mt L and mb R is rather small. Hence the long-lived (blue) points have small difference between mt L and mb R . Fig.7(b) shows the stability profile for large tan β. Here we see that large values of y b A b > 0 have deeper CCB vacua compared to the corresponding DSB ones. The region y b A b < 0 hardly appear to possess metastable points. This is because the SUSY threshold corrections to y b for the chosen value of µ = 200 GeV increases y b very significantly for A b < 0 compared to A b > 0. Thus, a particle mass spectrum calculated by SPheno contains a DSB vacuum consistent at two-loop level. The model parameters deduced at this setup may not necessarily yield a similar DSB vacuum when one-loop correction to scalar potential is employed. This makes Vevacious reassign the input vacuum to a rolled down vacuum configuration other than the DSB one determined from the spectrum file. In our analysis, we are not considering such kind of particle spectra where two-loop corrections change the structure of the potential very significantly. This results in hardly any metastable points for A b < 0. We further notice that m h 1 spreads over a larger range for large tan β. This is due to the fact that the contribution of |A b | to m h 1 is enhanced by tan β (Eq.2.20).
In Fig.8, we plot the stability profile Fig.8(a) (Fig.8(b)) corresponds to small (large) tan β. For both tan β regimes, the analytic constraint seems to be unnecessary since almost all of the region of parameter space that was scanned over, corresponds  − m h 1 plane for 5 < tan β < 10. Fig.8(b) is similar plot for 40 < tan β < 50. The color codes for stability profile is same as that for Fig.4. It is evident from the plots that the analytic expression regarding CCB constraint Eq.2.17 is unimportant for small tan β, unlike the case of large tan β (see text for details).  is not always negligible with respect to 1. Thus, the factor 3 does not arise.

CCB minima associated with both stop and sbottom fields
In this section we study a more involved scenario characterized by the non-vanishing vevs for all the third generation squark fields along with Higgs fields. We vary the relevant parameters in the following ranges. associated with them, in the y t A t − y b A b plane, for low and high regime of tan β. Fig.9 corresponds to the scan of Eq.3.5. It is evident that the effect of A t is more prominent for low tan β, whereas A b plays a crucial role for large tan β.
We keep y t A t fixed at 2 TeV and µ at 200 GeV. All sfermion mass parameters except mt L and mb R , are fixed at 2 TeV. Fig.9 shows the stability profile in the y t A t − y b A b plane. The color codes for stability profile are same as that for Fig.3. Fig.9(a) and 9(b) correspond to low and high tan β regimes respectively. It is evident from Fig.9(a) that for low tan β, the effect of A b towards determining the fate of the DSB vacuum is negligible and the vacuum stability is primarily determined by A t . This is consistent with the discussion of the scalar potential of NHSSM (see Sec.2). Both the plots identify a central region near This region is associated with absolute stable DSB vacuum, shown in green in the plots.
Bordering the green central region, there exists a small zone associated with long-lived DSB unstable via quantum tunneling or thermal effects. Despite the similarity in the nature of the two plots, there are some striking differences that arise due to the two different ranges of tan β that are considered in our analysis. Fig.9(a) constrains |y t A t | to be within 2.0 TeV for stable points, whereas there is almost no restriction imposed on y b A b when safe vacua (both stable and long-lived points) are considered. Fig.9(b) reveals that the stable points can appear for wider range of |y t A t | compared to that in the low tan β limit, since its effect is more prominent for low tan β. Thus, the zone |y t A t | < 4 TeV is easily accommodated in the model. Demanding absolute stability one requires |y b A b | < ∼ 1.5 TeV, however, that may be relaxed by a small amount via inclusion of long-lived states. We further notice that the metastability of the DSB vacuum is mostly governed by large y t A t . However, there are metastable points (mostly long-lived) even for smaller y t A t . The origin of such metastable points is attributed to sbottom vevs. Now we impose the experimental limits for m h 1 while considering a 3 GeV window resulting into the following range [96][97][98][99][100][101][102].

GeV
(3.6) The above uncertainty essentially arises from scale dependence, problems in computing higher order loop and renormalization scheme related dependencies. Imposing the above constraint of Eq.3.6 on Fig.9, results into Fig.10. Fig.10(a) shows that a large region of parameter space associated with y t A t > 2 TeV is excluded via the limits on m h 1 . As expected, A t plays a significant role in the radiative corrections to m h 1 for low tan β. For large tan β, as shown in Fig.10(b), A b is also important in regard to vacuum stability but the radiative corrections to m h 1 keeps it well within the range of Eq.3.6. This is consistent with the results obtained from fig.7(b).

Conclusion
Even after the first few years of running of the LHC, SUSY signal is yet to be observed. Along with the theoretical analysis, we follow a semi-analytic approach using complete 1-loop corrected effective potential implemented via Mathematica to investigate the influence of the NH terms on the scalar potential of the model. We find that, the terms associated with A t and A b effectively lower the scalar potential of the model. This lowering essentially leads to the appearance of global CCB minima depending on the values of the NH parameters. We also present an exhaustive scan over relevant region of parameter space by using the dedicated package Vevacious that includes both zero temperature effects and finite temperature corrections to the NHSSM potential.
We divide our Vevacious based analysis into three different parts depending on the fields whose vevs are allowed to be non-vanishing. First, in order to probe the effect of the stop fields, non-vanishing vevs are attributed to stops and higgs fields. The effect of A t is more prominent for low tan β, where the former is confined in a quite small range via the requirement of safe vacua. On the other hand, for large tan β, a comparatively wider range of A t is allowed. This is due to the fact, that the contribution of A t in the scalar potential is tan β suppressed. The region of parameter space with adequate value of m h 1 , is associated with both safe and dangerous vacua. m h 1 becomes important in a CCB related study, simply because a right zone of mass of the higgs boson require large values of trilinear parameters A t and A t that are sensitive to CCB analyses. The results obtained from Vevacious based analysis show the presence of a significant region of long-lived states. Hence the region of parameter space having safe vacua can be extended considerably which is otherwise ruled out by analytical CCB constraint.
Exploring the model for CCB minima associated with the sbottom fields, we find that the CCB constraint is hardly effective in constraining A b for small values of tan β. For large tan β, there are long-lived states even if the analytical CCB constraint is satisfied. We also find that most of the region is safe against deeper CCB vacua arising from sbottom vevs.
In a more realistic scenario, stop and sbottom fields together modify the scalar potential.
In order to probe this effect, all the third generation squarks along with the higgs fields are assumed to have non-vanishing vevs. We try to locate the safe vacuum in the y t A t − y b A b plane. The origin (A t = A b = 0) corresponds to the MSSM scenario where the NH terms are absent. There exists a region of absolute stable vacuum around A t = 0 = A b . This zone is bounded by regions that are characterized by the presence of long-lived DSB minima.
Beyond this, almost the entire y t A t − y b A b plane is excluded due to the instability of DSB minima. A wider range of y t A t is allowed for large tan β, but it is confined in a quite smaller range for low tan β. Moreover, the allowed range of m h 1 has visible impact for the case with low tan β and A t > 0. On the other hand, most of the region with large tan β is consistent with such a range.

Acknowledgement
AD would like to thank Indian Association for the Cultivation of Science for infrastructural support. JB is partially supported by funding available from the Department of Atomic Energy, Government of India for the Regional Centre for Accelerator-based Particle Physics