Supersymmetry with scalar sequestering

Supersymmetric models with a strongly interacting superconformal hidden sector (HS) may drive soft SUSY breaking scalar masses, bilinear soft term B\mu and Higgs combinations m_{H_{u,d}}^2+\mu^2 to small values at some intermediate scale, leading to unique sparticle mass spectra along with possibly diminished finetuning in spite of a large superpotential $\mu$ parameter. We set up a computer code to calculate such spectra, which are then susceptible to a variety of constraints: 1. possible charge-or-color breaking (CCB) minima in the scalar potential, 2. unbounded from below (UFB) scalar potential, 3. improper electroweak symmetry breaking, 4. a charged or sneutrino lightest SUSY particle (LSP), 5. generating m_h~ 125 GeV, 6. consistency with LHC sparticle mass limits, and 7. naturalness. We find this bevy of constraints leaves little or no viable parameter space for the case where hidden sector dynamics dominates MSSM running, even for the case of non-universal gaugino masses. For the case with moderate HS running with comparable MSSM running, and with universal gaugino masses, then the finetuning is ameliorated, but nonetheless remains high. Viable spectra with moderate HS running and with low finetuning and large mu can be found for non-universal gaugino masses.


Introduction
Particle physics models featuring weak scale supersymmetry [1,2] (SUSY) are notable in that they contain a solution to the gauge hierarchy problem (GHP) and are supported by several virtual effects, including the celebrated unification of gauge couplings, a prediction for the Higgs mass within expectations from theory and experiment, and a prediction of a heavy top quark needed for radiative electroweak symmetry breaking (EWSB) well before the top quark was discovered.Even so, the non-appearance of supersymmetric matter at the CERN Large Hadron Collider [3] (LHC) has potentially opened up a different naturalness problem [4]: the little hierarchy problem (LHP) concerning the burgeoning mass gap between the weak scale m weak ∼ m W,Z,h ∼ 100 GeV and the so-called soft SUSY breaking scale m sof t ∼ m sparticles , i.e. why is m weak ≪ m sof t ≳ 1 − 10 TeV?
A potential solution to the LHP comes from examining the explicit connection between m weak and m sof t which arises from minimization of the Higgs (scalar) potential of the Minimal Supersymmetric Standard Model (MSSM): where m H u,d are the Higgs soft SUSY breaking masses, µ is the (SUSY conserving) Higgs mixing term, tan β = v u /v d is the ratio of Higgs field vevs and the Σ u,d u,d terms contain an assortment of loop corrections (expressions may be found in Ref's [5] and [6]).Under practical naturalness [7] -wherein all independent contributions to an observable ought to be comparable to or less than the observable -then only |m Hu | and µ ought to be ∼ m weak whilst other sparticle contributions to the weak scale may be much heavier since their contributions are loop-suppressed.A naturalness measure ∆ EW has been proposed [8] which compares the largest contribution to the right-hand-side of Eq. 1 to m 2 Z /2.Models with ∆ EW ≲ 30 are then presumed practically natural and for such models there is no LHP.
One consequence of natural SUSY is that of all the sparticles, one expects only the higgsinos (with mass ∼ |µ|) to be of order m weak .Higgsino pair production is difficult to observe at LHC due to the small inter-higgsino mass gap [9], and limits on µ vary between ≳ 100 − 200 GeV depending on the mass gap ∆m 21 = m χ0 2 − m 0 χ1 [10].A challenge for LHC Run 3 and high-lumi LHC (HL-LHC) is to either discover light higgsinos or else rule out natural SUSY by excluding the higgsino discovery plane [11].At present, both ATLAS [12] and CMS [13] seem to have ∼ 2σ excesses in the opposite-sign-dilepton-plus-jet-plus-MET (OSDLJMET) signature [14,15]; upcoming new data should either confirm or exclude such signal channels.
Unnatural SUSY with large |µ| ≫ m weak , while possible, seems at first glance highly implausible.However, model builders have proposed a way to remain natural even with |µ| ≫ m weak by discovering models where the combinations m 2 H u,d + µ 2 are driven to be tiny, while |m 2 H u,d | and µ individually can each be large at the weak scale.This method is called scalar sequestering (SS) [16][17][18][19].
The method of hidden sector sequestering (HSS) of visible sector operators arises from postulating the existence of a strongly interacting nearly superconformal hidden sector (HS) which is operative between the messenger scale M * (taken to be of order the reduced Planck mass ∼ m P in the case of gravity mediation) and a much lower intermediate scale M int where the superconformal symmetry is broken and SUSY is also broken.This method of sequestering was originally proposed [20] as a means to obtain anomaly-mediated SUSY breaking (AMSB) models [21,22] when geometric sequestering was shown to be difficult to realize [23].
Under HSS, the various soft SUSY breaking terms get squeezed to tiny values via RG running between m * and M int by a power-law behavior: where the exponent Γ includes combinations of classical and anomalous dimensions of HS fields S. Γ is not directly calculable due to the strong dynamics in the HS but is instead assumed to be ∼ 1.For M int ∼ 10 11 GeV and Γ ∼ 1, then the suppression of gravity-mediated soft terms can be ∼ 10 −7 in which case the AMSB soft terms would be dominant.Additional symmetries seemed to be required in order for HSS to be viable; nonetheless, the lesson was that (model dependent) hidden sector effects could potentially modify the assumed running of SUSY model parameters as expected under the MSSM only [24,25].HSS was then found to offer a solution to the needed suppression of various problematic operators.For instance, in gauge mediation [26] the Bµ soft term is expected with Bµ ≫ µ 2 , leading to the famous Bµ/µ problem.HSS could be used to suppress Bµ(M int ) ∼ 0 thus solving the problem [16,27].Also, in gravity mediation, scalar masses arise via hidden sector-visible sector couplings such as where the Q i are visible sector fields and the S are hidden sector fields which acquire an auxiliary field SUSY breaking vev F S ∼ (10 11 GeV) 2 .In gravity-mediation, such operators are unsuppressed by any known symmetry (leading to the SUSY flavor problem), but could be squeezed to tiny values via scalar sequestering.A third application of (scalar) sequestering is to ameliorate the LHP while maintaining large µ values: |µ| ≫ m weak .This case, which is the subject of the present paper, makes use of Eq. 3 to suppress via hidden sector running all scalar masses to ∼ 0. However, in the case where the Giudice-Masiero (GM) mechanism [28] is assumed1 to generate a weak scale value of µ, then the scalar sequestering actually applies to m 2 Q for matter scalars, but to the combinations m 2 H u,d + µ 2 for Higgs scalars.In this case, at the intermediate scale M int , then one expects m 2 Q ∼ 0 but with m 2 H u,d ∼ −µ 2 so that µ can be large whilst the combination m 2 H u,d + µ 2 is small: this has the potential to fulfill the naturalness requirement in Eq. 1 while maintaining large |µ| ≫ m weak since µ 2 and m 2 H u,d are no longer independent.
In this paper, we examine the phenomenology of SUSY models with scalar sequestering.In Sec. 2, we present a brief review of the theory underlying scalar sequestering.Two different theory approaches have emerged: strong scalar sequestering where hidden sector running overwhelms MSSM running [16,17], and moderate scalar sequestering [19], wherein hidden sector running and MSSM running are comparable, leading to quasi-fixed point behavior for the intermediate scale soft term boundary conditions.In Sec. 3, we examine strong SS, dubbed here as the PRS (Perez, Roy and Schmaltz) scheme [17].Here, the intermediate scale boundary conditions are so determinative that only one (or a few) parameters completely determine the SUSY phenomenology.In this case, problems emerge for appropriate electroweak symmetry breaking, vacuum stability, and dark matter physics (with typically a charged LSP and sometimes a left-sneutrino LSP).The latter case with a charged LSP can be dispensed with via either an assumed R-parity violation [30,31] or assumed LSP decays to non-MSSM DM particles such as an axino ã [32].In Sec. 4, we verify these results with parameter space scans in the PRS scheme with and without unified gaugino masses.In Sec. 5, we instead adopt the scheme in [19] -we refer to this scheme as SPM (Stephen P. Martin)-with more limited HS running which is comparable to MSSM running.In this scheme, for the case of unified gaugino masses (UGM), we find that although SS reduces the amount of EW finetuning, significant weak scale finetuning arising from large top-squark masses remains, so that the finetuning problem cannot be said to be eliminated for large µ.However, in the case of non-universal gaugino masses (NUGM) which lead to large stop mixing and m h ≃ 125 GeV, then evidently low finetuning along with appropriate EWSB can be achieved for more moderate values of µ ∼ 1 TeV.Our findings are summarized in Sec. 6.

Brief review of scalar sequestering
Let us assume a gravity-mediated generation of soft SUSY breaking terms since gauge-mediation gives rise to trilinear soft terms A ∼ 0 and hence requires large, unnatural values of top squarks [33] to generate m h ∼ 125 GeV [34].At some scale m * < m P , the (superconformal) hidden sector becomes strongly interacting.Its coupling to visible sector fields leads to suppression of scalar soft breaking masses and also the bilinear soft term b ≡ Bµ.At some intermediate scale M int , the conformal symmetry is broken and the hidden sector is integrated out of the low energy EFT.Also around this scale, SUSY is broken at a scale Q 2 SU SY ∼ F S .Under gravity-mediation, the following operators give rise to the usual soft terms: and where S is a HS chiral superfield and R is a real product of hidden sector fields with R ∼ S † S + • • • .In addition, for the scalar sequestering model, one assumes the µ term is initially suppressed (by some symmetry?)but then arises via the Giudice-Masiero [28] mechanism at the scale m sof t via The holomorphic terms ( d 2 θ) are protected against renormalization effects by non-renormalization theorems but the non-holomorphic terms are not.The latter terms give rise to scalar masses m 2 ϕ ij and the bilinear soft term Bµ, and will scale between m * and M int as (M int /m * ) Γ where the exponent Γ is related to the anomalous dimension of the S field.
While Γ is not directly calculable since the HS is strongly interacting, under the assumption that Γ is large and positive, e.g.∼ 1, then the factor (M int /m * ) Γ can lead to large suppression of scalar masses and Bµ as compared to gaugino masses, A-terms and µ.However, while µ can remain large under scalar sequestering, the combination m2 gets driven to tiny values by the (M int /m * ) Γ factor.

Scalar sequestered SUSY: PRS boundary conditions
In the PRS scheme [16,17], the SS is assumed to dominate any MSSM running of soft terms.In this case, one expects the usual MSSM running for gaugino masses, A-terms and µ between the high scale m * and the intermediate scale M int , whilst HS effects suppress matter scalar masses m 2 ϕ ij , Bµ and Higgs combinations m 2 H u,d + µ 2 .Thus, (under the assumption of unified gaugino masses) the parameter space of the model is given by where the first three of these are given at the high scale m * .Motivated by gauge coupling unification, we take m * = m GU T , the scale where g 1 and g 2 unify under MSSM running, and where m GU T ≃ 2 × 10 16 GeV.Meanwhile, the matter scalar masses, Bµ and m 2 H u,d + µ 2 are taken to be ∼ 0 at the scale Q ∼ M int .
3.1 Results for M int = 10 11 GeV and A 0 < 0 As an illustration, we show in Fig. 1 the running of soft terms and µ for the case where m 1/2 = −A 0 = 1.5 TeV with µ = 500 GeV (the reason for µ ∼ 500 GeV is to be explained shortly).The pink shaded region shows the superconformal regime, whilst the soft terms run according to MSSM-only RGEs in the left-side unshaded region.We see from frame a) that the matter scalars start running at Q = 10 11 GeV where the squark masses are pulled to large values ≳ 2 TeV due to the influence of the SU (3) gaugino mass M 3 .Left-slepton masses are pulled up by a large SU (2) L gaugino mass M 2 to the vicinity of ∼ 650 GeV at m weak whilst the right slepton masses are pulled up by the U (1) Y gaugino mass M 1 to ∼ 300 GeV.The running of the bilinear b-term is given at the one-loop level by where the one-loop beta function is given by where the f i are Yukawa couplings, the g i are gauge couplings, the a i = A i f i are the reduced trilinear couplings, and the M i are gaugino masses (further RGEs are given in, e.g., Ref. [1], with their two-loop counterparts in Ref. [35]).In our numerical results presented in this paper, we use the full two-loop running of soft terms and gauge and Yukawa couplings.Hu is driven to large negative values at m weak due to the large top-quark Yukawa coupling f t .Also, m H d is driven dominantly by the gaugino mass M 2 to small negative values ∼ −100 GeV at m weak .Frame b) shows the running of trilinear soft terms starting from Q = m GU T .These terms are pushed to large negative values by the respective gauge interactions.In the case of A t , this may help drive stop masses towards tachyonic values and consequently to charge and/or color breaking (CCB) minima in the scalar potential.
A major check on this very constrained PRS scheme is if the EW symmetry is properly broken.Let us recall the (tree-level) conditions for proper EWSB.First, one must check whether the scalar potential indeed does not develop a minimum at h 0 u = h 0 d = 0, the origin of neutral scalar field space.The stability of the critical point satisfying are determined by the nature of the eigenvalues of the matrix of second derivatives of the scalar potential, V , evaluated at the origin of field space.We refer to this matrix of second derivatives as the Hessian.Here, the neutral scalar fields are denoted h 0 u,d .The goal is to have a vacuum whose origin of field space is destabilized, else EWSB fails to occur properly.There are two cases in which this can happen: 1. the origin is a maximum in field space, or perhaps 2. the origin is a saddle point.
To determine the stability of the critical points we find, the type of critical point can be identified using the multivariate second partial derivative test.Case 1 occurs when the determinant of this Hessian is positive, but m 2 Hu + µ 2 < 0 at the SUSY scale; then, the origin of field space will be a maximum.This secondary condition is crucial, meaning the positive determinant alone is insufficient here to determine the nature of the critical point at the origin.When the determinant is positive, but m 2 Hu + µ 2 > 0, then the origin of field space will be a minimum, hence the scalar fields fail to acquire nonzero VEVs and EWSB fails to occur.
Case 2 occurs when the determinant of the Hessian of the scalar potential at the origin with respect to the neutral Higgs scalars is negative, as this implies its eigenvalues have opposite signs, leading to This is often referred to in the literature as the condition for having a maximum at the origin of field space, but is more accurately described as a saddle point.In either case of a maximum or a saddle point, the origin is destabilized, so proper EWSB may yet be achievable, barring failure in the conditions below.In particular, given that m 2 Hu is driven large negative and Bµ is driven small positive, this saddle point condition may not always occur, but maxima sometimes occur instead as in case 1 and must be checked carefully!Secondly, one must check that the scalar potential is bounded from below (vacuum stability) in the D-flat direction h 0 u = h 0 d leading to the requirement that Given that m 2 Hu is large negative, this condition also may be subject to failure, in which case the scalar potential is unbounded from below (UFB).
If an appropriate EWSB occurs, then minimization of the Higgs potential allows one to determine the Higgs vevs v u and v d , with tan β = v u /v d as usual.The minimization conditions can be recast at tree-level as and Usually, in models like mSUGRA, the first of these is used to trade Bµ for tan β and the second is used to determine the magnitude of µ.In the present case, since the boundary condition for Bµ is ∼ 0 at Q = M int , it is not available to determine a unique value of tan β, since the running of the soft parameters depends on the Yukawa couplings which in turn depend on v u and v d , whose values then define tan β.Furthermore, from Eq. 15 we see that µ is not freely available to be determined by the measured value of m Z = 91.2GeV.Thus, the equations 14 and 15 must be used to map out the derived values of m Z in the µ vs. tan β plane.This is shown in Fig. 2 for the case at hand.Here, we see that for large µ values, then m 2

Z
is computed at loop level to be negative.For smaller µ, then typically m Z is of order the TeV scale.For a given value of tan β, one can choose µ near the edge of the gray excluded region where m Z ∼ 100 GeV.For Fig. 1, we have chosen tan β = 10 which then fixes µ ∼ 500 GeV.
Unfortunately, for all choices of µ and tan β shown in the plane, we find the scalar potential to be UFB in the D-flat direction.

Results
for A 0 > 0 In Fig. 3, we show a PRS point that does develop appropriate EWSB where M int = 4 × 10 11 GeV and m 1/2 = A 0 = 1 TeV.For tan β = 21.25,we find µ ≃ 1.8 TeV.In this case, with A 0 = 1 TeV, we see from frame b) that the A i parameters are all positive for large Q, with A t and A b becoming small and then negative around Q ≲ 10 10 GeV.This feeds into the b parameter evolution causing b to run at Q < M int to negative values until the large negative A i terms cause it to turn up and become positive around Q ∼ m weak , aiding in appropriate EWSB.While this model does develop a viable EW vacuum, the slepton masses evolve only to m E i ∼ 250 GeV at Q ∼ m weak so that slepton masses are well below both the µ and M 1 terms.Thus, for this point we have a charged slepton as the lightest SUSY particle.The derived sparticle mass spectra for this case are shown in Fig. 4. In the case shown, with the MSSM-only as the low energy EFT, then one would expect a charged stable relic, and dark matter wouldn't be dark.One can circumvent this situation by adding extra particles or interactions to the low energy EFT.An example of the former would be to add a Peccei-Quinn (PQ) sector with an axino ã as the LSP so that ẽR → eã.In this case, one would get a potentially long-lived but unstable slepton and one must avoid collider and other constraints on such objects.The slepton lifetime would depend on the assumed value of the PQ scale f a .An example of added interactions would be to postulate broken R-parity so that the slepton LSP decays to SM particles.Then one must explain why some RPV couplings are substantial whilst others are very small, as required by proton stability bounds [30,31].4 Parameter space scans: PRS scheme

Universal gaugino masses
In order to search for viable weak scale SUSY spectra in the PRS scheme, we implement a scan over the PRS parameter space: • M int : 10 6 → 10 14 GeV.4: The resulting mass spectrum with a characteristic slepton as the LSP for the PRS scheme with m 1/2 = 1 TeV, A 0 = m 1/2 , and µ = 1.8 TeV.The spectrum was produced using SoftSUSY v4.1.17[36] and slhaplot [37].
Our code then scans over values of (µ, tan β) leading to m Z ∼ 91.2 GeV.We then check for CCB minima, points that are UFB, and appropriate EWSB.For points that pass all criteria with appropriate EWSB, we then check for a neutral or a charged LSP.
Our first results are shown in Fig. 5 where we show scan points a) in the A 0 vs. M int plane and b) in the A 0 vs. m 1/2 plane.From frame a), we see that only the yellow points satisfy all EWSB constraints, although all the surviving points have a slepton as the LSP.In particular, the A 0 < 0 points almost all have either CCB minima (for large negative A 0 ) or else an UFB potential.For A 0 > 0, then the scalar potential is better behaved but frequently does not have appropriate EWSB.The scan points with appropriate EWSB are much more prominent at large M int and large m 1/2 .
In Fig. 6, we show our scan points in the m 1/2 vs. µ plane.Here, we see some structure where µ ∼ 2m 1/2 is favored.These qualitative features were also found by Perez, et al. in Ref. [17] where most of their parameter space was excluded by EWSB constraints except for large M int where they also found µ ∼ 2m 1/2 and for their lone sample point, they also obtained a slepton as the LSP.
Given our overall scan results in the PRS scheme, we find the strong scalar sequestering scenario (with unified gaugino masses) rather difficult (but not impossible) to accept: the bulk of p-space points have problematic EWSB and any surviving points have a charged LSP thus requiring new particles and/or new interactions to evade cosmological constraints on charged relics from the Big Bang.One possibility to try to circumvent the slepton-LSP problem in the PRS scheme is to appeal to NUGMs, by dialing down either M 1 or M 2 from their unifed values until either the bino or the wino becomes the LSP.The computed sparticle mass spectra are shown in Fig. 7 in frame a) for varying M 1 and in frame b) for varying M 2 .From frame a), we see that as M 1 diminishes, the lightest neutralino mass m χ0 1 does indeed decrease (moving from unified gaugino masses on the right to small M 1 on the left as shown by the lavender dashed curve).However, as M 1 decreases, then upward RGE pull on m Ei (right-slepton soft mass of generation i) from the U (1) Y gaugino also diminishes, and ultimately m 2 E 1,2 go tachyonic around M 1 ∼ 0.23m 1/2 .Note in this case that the stau soft mass remains larger due to a large negative X τ term in the m 2 E 3 RGE owing to large negative m 2 H d .This is shown in Fig. 8 which shows the soft mass running for a case with small M 1 compared to m 1/2 .This behavior where the bino fails to become LSP in the PRS scheme with small M 1 appears rather general when we scan over all M 1 values (to be shown shortly).
Likewise, in frame b), we take M 2 to be its unified value on the right-side of the plot, and then dial its value down to try to gain a wino as LSP.Around M 2 ∼ 0.58m 1/2 , the m χ ± 1 and m χ 0 1 mass curves coincide, showing that the lightest neutralino has gone from bino to wino.However, in this case, the right-sleptons remain LSP until M 2 ∼ 0.35m 1/2 whence the left sleptons, and particularly here the left-sneutrino, becomes LSP.Left-sneutrinos have direct detection cross sections for scattering on Xe nuclei of σ(ν eL Xe → νeL Xe) of ∼ 4.5 × 10 −23 cm 2 [38], about 23 orders of magnitude large than current LZ limits [39], and so are excluded as dark matter.
For somewhat lower values of M 2 , then Bµ runs to very small values, leading to a UFB scalar potential.This behavior also seems rather general from our PRS scan with NUGMs.

Scan over PRS scheme with NUGMs
For completeness in our search for viable weak scale SUSY spectra in the PRS scheme, we can adopt the case of non-universal gaugino masses and scan over this expanded parameter space: • M int : 10 6 → 10 14 GeV.
Similar to above, our code then finds pairs of (µ, tan β) leading to m Z ∼ 91.2 GeV.We then check for CCB minima, points that are UFB, and appropriate EWSB.For points that pass all criteria with appropriate EWSB, we then check for a neutral or a charged LSP along with LHC constraints on the gluino mass and lightest stop mass.As discussed above, one may try to dial down the M 1 (GUT) parameter to obtain a neutralino LSP, though this leads to both CCB and EWSB issues in this model.The issue of a charged LSP persists as in the UGM case, though it is possible to accommodate a sneutrino LSP in some cases, when M 2 (GUT) < M 1 (GUT).However, this scenario is severely ruled out due to direct dark matter detection constraints.
Our non-universal gaugino mass scan results are demonstrated in Fig. 9 where we show scan points a) in the A 0 vs. M int plane and b) in the M 1 (GUT)/M 2 (GUT) vs. M 3 (GUT) plane.Even with NUGMs, we do not find any points where EWSB is appropriately broken but without a charged slepton or left-sneutrino LSP.

Scalar sequestered SUSY: SPM approach
In the SPM approach [19], it is noticed that there exist bounds on the scaling dimension Γ such that Γ is positive but not too large, with Γ ∼ 0.3 maximally [40][41][42].In this case, the superconformal running may be much less, and comparable to the MSSM running.Let us denote the dimension 1 soft breaking terms as m 1 and dimension 2 soft terms as m 2 .Then, after several field rescalings, the dimension-1 terms (the M i , a ijk and µ) run according to while dimension-2 terms (matter scalars m 2 ϕ ij , mH U,d and b) run as   where the β M SSM are the usual MSSM beta functions and t = log(Q/Q 0 ) where Q is the energy scale and Q 0 is a reference scale.For the superconformal regime with M int < Q < m * , then Γ ̸ = 0 but for Q < M int , then the superconformal symmetry is broken and integrated out, and Γ → 0.
An intriguing effect in this case is that the m 2 2 terms can run until Γm /Γ.Approximate expressions for the fixed point values are given by SPM [19], but will not be repeated here.Thus, the m 2 2 terms tend to approach their quasifixed point values as Q → M int instead of zero, as in the PRS scheme.This behavior helps to ameliorate the problems of the PRS scheme with respect to EWSB.
The approach to the quasifixed point values are shown in Figs. 10 and 11 for several m 2 2 cases (some of these results verify similar plots by SPM but are presented again here for the benefit of the reader [19]).In Fig. 10, we show running of a) the bilinear soft term B ≡ b/µ, b) running of m ũR and M 3 and c) running of m ẽR and M 1 .In all frames, we take m 1/2 = 4.5 TeV, M int = 10 11 GeV with all matter and Higgs soft masses (and bilinear b) set to m 0 (m 2 0 ).A different curve is plotted for different values of m 0 ranging from 1 GeV to 10 TeV.The values of A 0 and µ are solved for in each individual curve.Blue curves are for µ > 0 while purple curves are for µ < 0. From frame a), we see the quasi-fixed point for b/µ is largely small but negative.From frame b), we see that the typical squark mass approaches (in this case) a quasi-fixed point value around ∼ 2 TeV rather than zero as in the PRS scheme; this helps avoid CCB minima in the SPM scheme.Squark soft masses are then pulled to large values at m weak due to the large value of M 3 which is chosen.From frame c), we see that typical slepton masses approach ∼ 1 TeV at M int in SPM rather than zero as in PRS.This helps avoid slepton LSP issues in the SPM scheme.
In Fig. 11, we show in frame a) the running of sign( m2 Hu ) | m2 Hu |.This also runs to a quasi-fixed point, in this case around ∼ 2 TeV.In frame b), we show the individual running of m 2 Hu and µ.While µ runs nearly flat as expected, m 2 Hu runs at first to large negative values, and then below M int runs nearly flat.This example may assuage concerns that the running of m 2 Hu below M int may destroy its correlation with µ so that the two terms regain some measure of independence: this doesn't seem to happen.In frame c), the running of m2 H d is shown and again it runs to a quasi-fixed point around 2 TeV.
In the SPM scheme, the m 2 2 values approach (but do not exactly meet) their quasifixed point values at Q = M int , so that the boundary conditions at Q = M int are no longer fixed.Thus, to generate a workable model, we must expand the parameter space from the PRS scheme.For SPM, therefore, one must reintroduce the various m 2 2 boundary conditions at Q = m * , and we will take After checking for appropriate EWSB, and then employing the EWSB minimization conditions, one can again solve for the derived value of m Z .This is shown in Fig. 12 where we show colorcoded regions of m Z in the A 0 vs. µ(GU T ) plane for m 0 = 10 TeV, m 1/2 = 4.5 TeV and tan β = 15.From the plot, one sees that there is no unique solution for m Z ≃ 91.2 GeV but rather two disconnected regions depending on the sign of µ, with a different µ value being obtained for each choice of A 0 .

Case with UGMs
In the SPM paper, Martin has plotted out sample spectra for two cases, one with unified gaugino masses (UGM) and one with non-unified gaugino masses (NUGM).For the case of UGM, he shows sparticle mass spectra vs. m 1/2 for m 0 = m 1/2 and also for m 0 = 2.5m 1/2 , with tan β = 15 and with b = m 2 1/2 , where A 0 and µ are solved for.For the m 0 = m 1/2 case, he always finds a right selectron as the LSP (as do we), so that either additional R-parity violating interactions or lighter DM particles (such as axino) are needed to avoid charged stable relics from the early universe.In the case of m 0 ≳ 2.5m 1/2 , then the bino can become LSP.In Fig. 13, we reproduce these results for the case of m 0 = 2.5m 1/2 .The region between the pink shaded boundaries has 123 GeV < m h < 127 GeV (as computed here using FeynHiggs [43]).Typically, in such cases with heavy sparticles in the multi-TeV range and a bino as the LSP, the thermally-produced neutralino relic density Ω χ h 2 ≫ 0.12.However, from Fig. 13 we do see that since slepton masses are very nearly equal to m(bino), then coannihilation is available to reproduce the measured DM relic density.We also see that the higgsino mass ≃ µ is very large, varying from ∼ 5 − 10 TeV over the range of m 1/2 shown.This would make the model very unnatural under the conservative ∆ EW measure.However, the point here is that a mechanism is now present to drive the combination m 2 Hu + µ 2 to small values, thus potentially ameliorating the LHP.
Since we have now arrived at acceptable spectra for the case of scalar sequestering in the SPM scheme, we next want to check whether it really solves the LHP.In Fig. 14, we compute in frame a) the top five signed contributions to the naturalness measure ∆ EW .The largest contributions come from µ and m Hu (weak), which are seen here as the blue and red curves.These lie in the ∼ 10 4 range in magnitude, making the model highly finetuned under ∆ EW .In frame b), we define a revised finetuning measure ∆ ′ EW , which is the same as ∆ EW except that now m 2 Hu + µ 2 and m 2 H d + µ 2 are combined into single entities since they are now dependent (due to the CFT running above Q = M int ).In frame b), we see the top five contributions to ∆ ′ EW .In this case, the Σ u u ( t1,2 ) terms and m2 Hu terms are largest, typically of order ∼ 10 3 .Thus, we find that although the SPM scheme in the UGM case has reduced finetuning, it is still found to be highly finetuned, mainly due to the large lightly-mixed top-squark masses contributing to the radiative corrections Σ u u ( t1,2 ).

Case with NUGM
Along with the UGM case, SPM also considers the case with NUGMs.This case is motivated by obtaining a large stop mixing element A t which can enhance m h → 125 GeV via maximal stop mixing rather than too large of stop masses.This can be achieved with M 3 ≪ M 2 while adjusting M 1 so that the bino remains as the LSP.In Fig. 15, we show the weak scale sparticle mass spectra in the SPM scheme with NUGMs.We plot vs. m 0 where M 3 = 1.2 TeV, M 2 = 4 TeV, and M 1 = 2 TeV (all M i defined at Q = m GU T ).Our calculations match well with the results of SPM.From the plot, we see that for low m 0 we still get a slepton as the LSP (this time, it is the τ -slepton τ1 ).For higher values of m 0 , then sfermion masses increase as expected and for m 0 ≳ 6 TeV one obtains m l ≳ m(bino) and so we get a bino as the LSP.Also, with M 3 (m GU T ) only 1.2 TeV, then squarks and sleptons are much lighter than in Fig.In Fig. 16 we compute the top five signed contributions to the finetuning measures a) ∆ EW and b) ∆ ′ EW for the same parameters as in Fig. 15.From frame a), we see that the m Hu and µ contributions to ∆ EW are opposite sign but with absolute values ∼ 10 3 so that the spectra are finetuned under ∆ EW .However, the SS of m 2 H u,d + µ 2 means these quantities are no longer independent and instead ∆ ′ EW should be used.From frame b), we see the top five contributions to ∆ ′ EW are typically of order ∼ 10: thus, this case of the SPM scheme with NUGMs seems natural even with higgsino masses of ∼ 2.3 TeV.(The breaks in the curves of frame b) occur due to different contributions to Eq. 1 vying to be within the top five.)

Conclusions
Supersymmetric models with m h ∼ 125 GeV which obey LHC search constraints but have low ∆ EW qualify as permitting a SUSY solution to the GHP while avoiding the LHP: they are electroweak natural and are typified by the presence of light higgsinos with mass m H ≲ 350 GeV.Such light higgsinos are actively being searched for by ATLAS and CMS and indeed both experiments have some small excesses in the OSDLJMET signal channels.However, here we have investigated numerically the proposition whether hidden sector scalar sequestering can yield a solution to the LHP even with large µ ≫ 350 GeV.The key element is to assume a nearly superconformal hidden sector coupled to the visible sector which leads to driving scalar masses, the Bµ term and the Higgs combinations m 2 H u,d + µ 2 → 0 at some intermediate scale where the conformal symmetry is broken and the hidden sector is integrated out.A key ingredient is that the SUSY µ parameter is generatd via the GM mechanism.In such a situation, then HS dynamics pushes m 2 H u,d ∼ −µ 2 so these quantities are no longer independent.Then a revised naturalness measure ∆ ′ EW must be used where the now dependent terms m 2 H u,d and µ 2 are combined.
We investigated two schemes.
• The strong scalar sequestering where scalar masses, Bµ and Higgs combinations are driven to (nearly) zero at the intermediate scale (PRS scheme).This scheme has trouble developing appropriate electroweak symmetry breaking; and when it does, it develops a slepton as the LSP.Via thorough scans of parameter space, with both UGMs and NUGMs, we find no viable spectra with appropriate EWSB but without a slepton as the LSP.Some of the problematic slepton LSPs can be avoided by introducing new sparticle or interactions which allow one to evade cosmological constraints on charged stable relics from the Big Bang.
• The second scheme labeled as SPM introduces moderate scalar sequestering with nonnegligible MSSM running so that scalar masses, Bµ and Higgs combinations run to quasifixed points rather than zero.In the SPM scheme for UGMs, spectra with neutral stable LSPs can be found, and the finetuning can be reduced, but not eliminated.For the case of NUGMs, then m h ∼ 125 GeV can be found with sparticles beyond LHC bounds and with low EW finetuning as found from the ∆ ′ EW measure.The latter spectra can have rather heavy higgsinos since the sequestering leads to non-independent Higgs soft terms and the µ term.Determining the consequences of the viable SPM scheme with NUGMs at colliding beam experiments is a topic for future work.
weak mainly by the second term of Eq. 11.Meanwhile, with µ = 500 GeV, the sign(m 2 H u,d ) * |m H u,d | soft terms begin at −500 GeV and m 2

Figure 1 :
Figure 1: Running of soft terms and −µ in the PRS scalar sequestering scheme for m 1/2 = 1.5 TeV, A 0 = −m 1/2 , and µ = 500 GeV.We also take the intermediate scale M int = 10 11 GeV.In frame a) we show running scalar masses and the µ term, while in frame b) we show the running trilinear soft terms.

Figure 2 :
Figure 2: Computed value of m Z in the µ GU T vs. tan β plane for the PRS BM point with m 1/2 = −A 0 = 1.5 TeV, and M int = 10 11 GeV.

Figure 3 :
Figure 3: Running of soft terms and µ in the PRS scalar sequestering scheme for m 1/2 = 1 TeV, A 0 = m 1/2 , and µ = 1.8 TeV.We take the intermediate scale M int = 4 × 10 11 GeV.In frame a) we show running scalar masses and µ term while in frame b) we show the running trilinear soft terms.

Figure 5 :
Figure 5: Scan over the PRS parameter space with UGMs in the a) A 0 vs. M int plane and b) the A 0 vs. m 1/2 plane.

Figure 6 :
Figure 6: Scan over the PRS parameter space with UGMs in the m 1/2 vs. µ plane.

Figure 7 :
Figure 7: The SUSY mass spectrum vs. GUT-scale gaugino mass parameters M 1 , M 2 in the PRS model varied below m 1/2 .In both frames, the spectrum at the far right is similar to the spectrum seen in Fig. 4. In frame a) the mass spectrum as M 1 (GUT) is varied below m 1/2 to zero is plotted.The neutralino never becomes the LSP, as the selectron and smuon remain lighter until CCB minima are realized.In frame b) we display the mass spectrum as M 2 (GUT) is varied below m 1/2 to zero.Near M 2 ∼ 0.4m 1/2 , the sneutrino briefly becomes the LSP before the Higgs potential becomes unbounded from below due to a lack of running in the b = Bµ parameter.Thus, a neutralino LSP cannot be achieved here.In both frames, we take m 1/2 = M 3 (GUT) = A 0 = 1 TeV, M int = 4 • 10 11 GeV, and tan(β) = 21.25.

Figure 8 :
Figure 8: Example RGE running of the soft masses from Fig. 7a) demonstrating the CCB nature of a point with m 2e R < 0 with M 1 (GUT) ∼ 0.15m 1/2 .Though the left-handed slepton states (red) evolve to moderate values, the right-handed slepton states of the first two generations evolve to be negative at the SUSY scale due to the small value of M 1 .

Figure 9 :
Figure 9: Scan over the PRS parameter space with NUGMs in the a) A 0 vs. M int plane and b) the M 1 (GUT)/M 2 (GUT) vs. M 3 (GUT) plane.

2 2 ≃ β M SSM m 2 which defines a quasifixed point for the m 2 2 running at m 2 2 ≃
−β M SSM m 2

Figure 10 :
Figure 10: Running of a) B ≡ b/µ, b) m U 1 and M 3 and c) m E 1 and M 1 from Q = m GU T to Q = m weak under the SPM scheme with Γ = 0.3.

Figure 11 :
Figure 11: Running of a) m2 Hu , b) m 2 Hu and µ and c) m2 H d from Q = m GU T to Q = m weak under the SPM scheme with Γ = 0.3.

Figure 12 :
Figure 12: Color-coded regions of the derived value of m Z in the µ vs.A 0 plane for tan β = 15 with m 0 = 10 TeV and m 1/2 = 4.5 TeV.