NSUSY fits

We perform a global fit to Higgs signal-strength data in the context of light stops in Natural SUSY. In this case, the Wilson coefficients of the higher dimensional operators mediating g g ->h and h ->\gamma \gamma, given by c_g, c_\gamma, are related by c_g = 3 (1 + 3 \alpha_s/(2 \pi)) c_\gamma/8. We examine this predictive scenario in detail, combining Higgs signal-strength constraints with recent precision measurements of m_W, b->s \gamma constraints and direct collider bounds on weak scale SUSY, finding regions of parameter space that are consistent with all of these constraints. However it is challenging for the allowed parameter space to reproduce the observed Higgs mass value with sub-TeV stops. We discuss some of the direct stop discovery prospects and show how global Higgs fits can be used to exclude light stop parameter space difficult to probe by direct collider searches. We determine the current status of such indirect exclusions and estimate their reach by the end of the 8 TeV LHC run.

the properties of the Higgs if they are to stabilize this scale. The minimalistic scenario for NSUSY focuses on the vestige of the SUSY spectrum which is required to be light in order to keep the fine-tuning of the theory reasonably small. In this case, the most significant impact of new states on Higgs phenomenology is through the presence of light stop states. This scenario is not just motivated by simplicity, but also by the lack of evidence for SUSY to date, indicating that a weak scale SUSY spectrum needs to be non-generic to satisfy collider constraints. The states directly related to naturalness (primarily the stop and Higgsinos) are especially challenging, and model independent collider bounds are weak or non-existent.
Conversely, the study of the impact of an NSUSY scenario on the properties of the Higgs benefits from the enormous effort expended by the experimental collaborations in refining the accuracy and precision of the reported Higgs signal strength measurements. When considering the experimentally resolvable impact of NSUSY, indirect probes, e.g. through a fit to Higgs properties, electroweak precision data and flavor physics may well be more powerful in constraining many minimal scenarios than direct searches for some time. This is the line of reasoning we develop in this paper, where we examine the current constraints on minimal NSUSY from these indirect probes.
The outline of this paper is as follows. We briefly review and introduce NSUSY in Section II.
In Section III we then review the impact of the NSUSY spectra on the properties of the SM Higgs through modifications in the loop-induced h → g g and h → γ γ couplings. Further, in Section IV we work through the constraints set on NSUSY from global fits to Higgs properties in the presence of light stops. We advance such studies by using a more complete global fit (now including 48 signal strength channels, including ICHEP and post ICHEP data updates) 1 . We make use of the fixed relationship between the Wilson coefficients (including the QCD matching corrections) for hγγ and hgg in the case of NSUSY to perform a one-parameter fit and then directly map the allowed parameter space in global Higgs fits into the allowed stop space. Further, we determine 95% confidence level (C.L.) exclusion limits on the stop parameter space derived from Higgs search data. We then consider constraints from BR(B → X s γ) and recent precision measurements of m W at the Tevatron, as these results, which are under excellent theoretical control, are sensitive to light stops. The (statistically insignificant, but interesting) deviations from the SM pre-  dictions in these observables in a weak scale NSUSY scenario could offer some further resolution on the allowed stop parameter space, if NSUSY exists. Ascribing these deviations to the effect of stops in NSUSY, such stop states are consistent with the results of the global fits to Higgs signal strengths, as we will show. Finally, we also take into account direct collider bounds. In Section V we discuss the interplay of these constraints and determine the allowed parameter space that remains. We include the limits coming from the Higgs mass measurement and estimate the degree of fine-tuning incurred, In Section VI we discuss the current exclusion bounds that can be derived using these indirect probes of stop parameter space. To study the future prospects of such limits by the end of 2012, assuming that the experimental error in Higgs signal-strength measurements scales down as ∼ 1/ √ L int , we consider two hypothetical cases: 1) the current pattern of best-fit signal-strength values does not change, and 2) the dataset evolves to converge on the SM expected signal strengths. Finally, in Section VII we conclude.

II. NATURAL SUSY
Naively, in generic SUSY scenarios motivated as a solution to the hierarchy problem, one expects all superpartners near the electroweak scale, with the soft breaking mass scale M SUSY not higher than O(1) TeV. The experimental picture emerging from the LHC is in growing tension with this expectation. After searching in many typical discovery channels, and reaching a peak sensitivity of roughly O(1.5TeV)/ O(100)'s GeV, for coloured/electroweak SUSY states [19], no statistically significant experimental excess has been reported to date. On the other hand, to avoid destabilizing the electroweak scale when M SUSY v (without fine-tuning), only a minimal set of SUSY particles have to be light ( < ∼ 1 − 2 TeV) [1][2][3][4][5][6][7]. The stop soft masses are directly connected (at one-loop) to the Higgs mass scale (or Z mass) through the sizeable top coupling, so fine-tuning considerations require them to be light. Although sbottoms (b) do not directly affect the fine-tuning of the Z mass,b L is required to be light as it is linked to thet L mass scale by SU(2) L symmetry.
The Higgsino mass coming from the µ term 2 is tied to tree-level contributions to the Higgs mass.
With gaugino masses assumed heavy, of order M SU SY , light charginos and neutralinos are almost pure Higgsinos, with mass given by µ up to corrections that we neglect. To a lesser degree, the gluinosg are also expected to be light due to their contribution to two-loop corrections to the Higgs mass parameter, which leads to the rough estimate mg < ∼ 2mt (4mt) for a Majorana (Dirac) gluino [6]. In practice, we can also decouple these somewhat heavier gluinos in our analysis.
The states just discussed are listed in Table I. These are the states whose impact on Higgs signal strengths and low-energy precision measurements we will focus on in this paper. We will not consider light staus (τ ± ), which can also modify the Higgs decays to photons, see Refs. [8,9] for recent studies. We neglect these states as we assume a moderate value of tan β < ∼ O(10), for whichτ ± effects are negligible compared to the stop contribution. We also neglect the effects of theb on the Higgs mass and in the loop corrections to the Higgs signal strength parameters for the same reason.
Regarding the SUSY Higgs sector, we will consider the decoupling regime in which only one Higgs doublet remains light 3 , while the second doublet, with mass controlled by the pseudoscalar mass m A , has a mass ∼ TeV. In this limit, the couplings of the lightest Higgs h to fermions and gauge bosons approach their SM values, and we will only consider deviations in the loop-induced couplings of the light h to photons and gluons (see next section).
In order to fix our notation, we write now the parts of the low-energy Lagrangian most relevant for our analysis. This Lagrangian is not supersymmetric as it applies below the scale of the heavy SUSY particles (with masses ∼ M SU SY > ∼ 1 TeV). Supersymmetric relations between some couplings are broken and one should introduce different couplings, to be matched to the supersym- 2 In this paper we distinguish signal-strengths with a subscript, µ i for a final state i, from the µ parameter in NSUSY, which carries no subscript. 3 This choice is supported by the results of Ref. [17], where the 2HDM is studied in the light of the Post-Moriond Higgs data and no compelling region in the parameter space of the MSSM consistent with the data was identified, except the decoupling limit. metric theory at the scale M SU SY . In practice, the hierarchy between M SU SY and the electroweak scale is mild and one can neglect most of these breaking effects.
From the superpotential W = µH d · H u + h t Q L · H 2 t c R , the Lagrangian gets the terms where · stands for the SU(2) L product: (1) contains a Dirac mass term µ for Higgsinos, the top Yukawa coupling between top quarks and the Higgs, and the related Higgsino-squark-quark couplings, relevant for the contribution of Higgsino-stops to Br(B → X s γ).
The scalar potential for Higgses and squarks is well known and includes supersymmetric F and D terms and soft-SUSY-breaking terms. We can always perform a rotation of the full Higgs in such a way that H l is the doublet involved in electroweak symmetry breaking while H h does not take a vacuum expectation value. In the Higgs decoupling limit, with large pseudoscalar mass, H h is in fact composed of the heavy fields H 0 , A 0 , H ± , while H l is SM-like and contains the light Higgs h and the Goldstones. The quartic H l coupling determines the light Higgs mass as usual and is the prime example of a coupling that receives sizeable SUSY-breaking corrections (that help in increasing the Higgs mass above its tree level minimal SUSY value below m Z ). Such corrections will be discussed in Subsection V.A.
Finally, the stop masses are given by the mass matrix where we used c 2β = cos 2β, m t is the top mass and MQ L , Mt R , A t are soft SUSY-breaking masses. The stop mixing angle θ t relates the interaction eigenstatest L,R to the mass eigenstates t 1,2 by the rotation  The mixing angle θt is taken in the interval (−π/2, π/2) and defined by with the signs of M 2 RR − M 2 LL and M 2 LR determining the quadrant of 2θt. With this definition of θt, one automatically guarantees mt 1 ≤ mt 2 , with Finally, neglecting sbottom mixing (proportional to m b ), the light sbottom has mass m 2

III. LOOP-LEVEL CORRECTIONS TO HIGGS PROPERTIES IN NSUSY
In this section, we review the loop-level NSUSY corrections to the couplings of h to photons and gluons. 4 The leading correction from stops to the gluon-fusion process is given by [20,21] where Here C g (α s ) is a factor that takes into account higher order QCD corrections-see the discussion below-and the F g functions are defined as follows where See the Appendix for the SM inputs used in determining the numerical value of F SM g above. The QCD correction applied above for the b quark contribution is kept to the value of the correction quoted above, which is determined in the large quark mass limit m q m h . The correction to this QCD correction due to the smaller b quark mass is (very) subdominant in the numerical results.
The couplings g ht iti are given, in the decoupling limit by where sin θt cos θt has been defined in Eq. (3); the t i |t L,R can be directly read from Eq. (4); and we have included the D-term contributions proportional to cos 2β, although their effect is negligible. 5 The sign of i=1,2 g ht iti τ i is positive if the stop sector is dominated by a light eigenstate of pure chirality, and negative if the term in . In the no-mixing case, we expect an enhancement of σ(gg → h). In the maximal-mixing case, the suppression depends on the separation between the two eigenstates.
The decay width Γ(h → γ γ) is also modified by stop loops through the same function in Eq. (10), as the non-Abelian nature of QCD is irrelevant for the leading-order loop function. One finds [20] the correction The SM contribution is given by where F 1 (τ ) = [2 τ 2 + 3τ + 3(2τ − 1)f (τ )] /τ 2 , N c is the number of colours, and Q i is the electric charge with e factored out. The matching correction in this case is given by C γ (α s ), to be further discussed in the next section. 5 Note that the stop contributions have an erroneous overall sign in Ref. [21] which propagated in the original version of this paper.

A. Effective Theory Approach
It is useful to consider the approximation that all of the light NSUSY states are still heavy enough to be integrated out giving local operators. We can then fit to the data directly using the effective Lagrangian 6 where g 1 , g 2 , g 3 are the weak hypercharge, SU(2) gauge and SU(3) gauge couplings and the scale Λ corresponds to the mass of the NSUSY states integrated out. The effects in NSUSY appear at the loop level, so we find it convenient to rescale the Wilson coefficients as c j =c j /(16π 2 ). In this case, using the results of Ref. [23], the effect of the operators in (15) is Herec γ =c W +c B −c W B . We can translate the effect of stops in the language of local operators by inspecting our expressions in Sec. III and those in Eq. (16), These relationships are only approximate in the sense that the limit mt 1,2 m h should be taken in the loop functions r g , r γ to match onto the local operators. If the two stop mass eigenstates can be integrated out simultaneously, the matching can be directly performed by expanding the loop functions in this limit, and one obtains 7 6 We do not include CP violating operators, assuming that all non-SM CP violating phases of the states integrated out are negligible. It has been argued [6] that this can be naturally accomplished in NSUSY when additional assumptions are employed concerning R-parity for example. Our assumption is conservative as the existence of large CP violating phases would only increase the constraints on a NSUSY spectrum. 7 Here we have neglected D term contributions, although we retain the effect of D terms in some of the numerical results presented. These corrections are negligible except for mt 1 < 150 GeV masses. They introduce a (minor) tan β dependence into the definition of F g in the local operator approximation when retained.
While, in this limit, the QCD matching corrections take the simple form These perturbative corrections are the matching corrections due to top squarks in the loops that do not cancel when a ratio is taken with the SM contribution to these loops. This correction factor is obtained in Ref. [24] in the limit where gluino effects and the effect of squark mixing was neglected. 8 This approach also neglects running that would sum large logs if a two stage matching was employed, integrating out each stop eigenstate in sequence. 9 Whether one integrates out the stops and matches onto the local operator approximation or not, there is a relationship between the NP effects on σ gg→h and σ h→γγ that is independent of the stop mass parameters in the minimal NSUSY limit. In the local operator approximation, the relationship is simplyc where we see how the ratio ∼ 3/8 is determined by the stop quantum numbers. This is a consequence of assuming that the only BSM contribution to both the γγ and gg loops comes from stops. This strong relationship will be relaxed in less minimal scenarios. For example, light χ ± 's with mass m 2 χ 1 ∼ µ 2 (in the decoupling limit) would in principle also contribute to the γ γ loops. However, the Higgs couples to the higgsino as hW ±H ∓ and a large mixing between wino and higgsino eigenstates would be required. As we are considering the large gaugino mass limit in NSUSY, M 2 µ, v, this mixing scales as ∼ m 2 W sin 2 β/(M 2 2 ) and is suppressed, so that we can neglect the chargino contribution to hγγ.
We also utilize this effective Lagrangian to examine the issue of efficiency corrections to the µ i when high-dimension operators are present. We find that such efficiency corrections to event rates are very small and neglect them. See the Appendix for details. 8 It has been pointed out that mg → ∞ leads to mixed stop-gluino UV divergences [25] requiring extra counterterms, but this technical requirement is not a barrier to the numerical investigations we perform. The full matching correction is given in Ref. [26]: the gluino contributions and stop mixing effects are a small correction to the ∼ 5% matching correction we consider. 9 There are also perturbative corrections to the matrix element of the local effective operator h G A µ ν G A µ ν . These are common multiplicative factors, as are soft gluon re-summation effects, and cancel in the ratios taken.

B. Global Fit To Higgs Signal Strengths
In this section we describe our method and results for globally fitting to Higgs signal strength data in the scenario discussed above. Here we only briefly review the fit procedure, the details of our fit method are given in Refs. [27][28][29]. 10 Our fit incorporates the recently released 7 and 8 TeV LHC data [39][40][41][42], and the recently reported Tevatron Higgs results [43]. The data we use is listed in the Appendix. We fit to the available Higgs signal-strength data, for the production of a Higgs that decays into the observed channels i = 1 · · · N ch . Here N ch denotes the number of channels, the label j in the cross section, σ j→h , is due to the fact that some final states are summed over different Higgs production processes, labelled with j. The ij are the efficiency factors for the various production processes producing a final state j to pass experimental cuts. The reported best-fit value of a signal strength we denote byμ i , and the χ 2 we construct is defined as The covariance matrix has been taken to be diagonal with the square of the 1 σ theory and experimental errors added in quadrature giving σ i . We necessarily neglect correlation coefficients as these are not supplied. For the experimental errors we use ± symmetric 1σ errors on the reportedμ i , while for theory predictions and related errors we use the results of the LHC Higgs Cross Section Working Group [44]. The minimum (χ 2 min ) is determined, and the 68.2% (1 σ), 95% (2 σ), 99% (3 σ) best fit regions are plotted as χ 2 = χ 2 min + ∆χ 2 , with the appropriate cumulative distribution function (CDF) defining the corresponding ∆χ 2 .
We first perform a one-parameter fit in terms of the free parameter F g , that depends on the stop sector of NSUSY, and plot χ 2 − χ 2 min in Fig. 1 (left). This χ 2 distribution is directly related to the fit in the broader space of the generic Wilson coefficients of the local operators contributing to h → γγ and gg → h, as shown in Fig. 1 (right), although the match of the C.L. regions is only approximate due to the difference in number of degrees of freedom in the fit. It is not surprising that with the addition of this free parameter, the χ 2 measure is improved as compared to the SM. fit space (defined with the CDF appropriate to each case. This difference accounts for the mismatch in the ∆χ 2 's that define the best-fit regions). Also shown as solid (brown) contours is the enhancement of the µ γ γ signal strength and how such a condition projects into the best fit space.
However, it was by no means guaranteed that the stop line determined by the NSUSY relationship between the Wilson coefficients would pierce the best fit region away from the SM one. This accidental fact allows a ∼ 2σ improvement of the fit. Although this is intriguing, we caution the reader that the interval of Wilson coefficients that intersect the 1 σ best fit region is mapped into a very narrow range of stop mass parameters, corresponding to a tuned area of parameter space. In addition, best-fit regions in (c γ ,c g ) space more distant from the SM point at (0, 0) will generically correspond to lighter states, as their impact scales as 1/Λ 2 . This will represent a further problem for this region.
We can characterize the allowed relationship between the Wilson coefficients that intersect the (1σ) best fit region in a model-independent way, finding that current data is consistent with the following four ranges of the Wilson coefficient ratios, corresponding to the four different best-fit regions in that 2D space [29]. 11 Forc γ > 0: and, forc γ < 0: In the limit of a single field contributing to the Wilson coefficients, thec g /c γ ratio is dictated by the quantum numbers of the field integrated out. 12 Clearly, the study of the possible intersections of such lines with the best-fit regions in the space (c g ,c γ ) for any model (including NSUSY) will become much more important with further refinements in the measurement of Higgs properties.
We discuss some prospects for the improvement of these fits in Section VI. 11 Note that these bounds are approximate in the following sense: for a 1D fixed relationship between the Wilson coefficients, the allowed C.L. regions are slightly different if obtained with the 1D CDF or for the 2D Wilson coefficient case. Again, this effect can be seen in the NSUSY case in Fig. 1 12 See Ref. [45] for a recent study that also emphasizes this point.
For the NSUSY case, the light stop best-fit region occurs for F g ∼ 2; one can see how this space relates to the (c g ,c γ ) plane in Fig. 1, and it corresponds to having the lightest stop mass eigenstate significantly lighter than the second stop eigenstate. Most of this space is already strongly constrained by monophoton searches, as we discuss further below. NSUSY hopes in light of current global Higgs data (when our assumptions are adopted) are based on the consistency of NSUSY in the (c g ,c γ ) parameter space near the SM point (c g ,c γ ) = (0, 0), for larger mt 1 and small F g . Translating the allowed fit space to the space of the stop parameters is very convenient to discuss the interplay with further constraints and direct stop discovery prospects. When we translate the results of the global fit to Higgs signal-strengths to the stop space, we find the best-fit regions shown in maximal mixing (θt = π/4). We will show these canonical parameter choices throughout this paper when examining the global constraint picture. The point in the 1D fit that has a local minimum χ 2 with small F g is mapped into the red dashed line in the two rightmost plots in Fig. 2. It can be checked that the isocontours of fixed value of F g all converge to mt 1 = 0, δm 2 = 4m 2 t / sin 2 (2θt), which explains that, for small values of the mixing angle, the red dashed line corresponds to a large spliting of the stops and for zero mixing the red dashed line does not exist since F g is always negative. As anticipated, the best-fit region at low stop-mass is extremely narrow, and even when a disconnected region in the stop space exists, the region is surely fine-tuned.
Another important challenge for the parameter space of NSUSY scenarios comes from non-SM contributions to magnetic moment operators. Although the contributions to these operators vanish [46] in the pure SUSY limit, NSUSY scenarios are far from this limit by construction. As a result, the reduction in the allowed parameter space due to constraints from Br(B → X s γ) can be significant. Recall that the effective Lagrangian (neglecting light quark masses) is given by [47] The SUSY contributions to the magnetic moment operators are well known [48] and can be applied to the particular NSUSY scenario. The dominant contributions to the Wilson coefficients come from stop-chargino loops: We have only retained the light χ ± 1 with mass m χ ± 1 µ, which is consistent with NSUSY assumptions (the gaugino mass is M 2 > µ > ∼ m W ) . The loop functions F j 7,8 are given in the Appendix. We vary the µ parameter in the range ∼ 100−200 GeV. The lower limit of this range is set by LEP bounds onχ ± [49]; the upper limit by naturalness considerations [6]. In principle, there are also loop contributions from the light χ 0 . However, these contributions can be strongly suppressed in the minimal NSUSY limit we consider. For example, there is no source of breaking of the residual U(1) R symmetry, unless Majorana masses for the gluino are introduced. This symmetry plays a role in suppressing effects of large tan β or proton decay or flavour violating observables which require a flip in chirality, see Refs. [50,51]. We will consider a minimal flavour violating scenario [52][53][54][55][56] when examining the NSUSY spectrum, this can follow from the argument above directly.
In the case of Majorana gluino masses we assume MFV.
We use the results of Ref. [57] for the constraints on the BSM Wilson coefficients C i = C SM i + ∆C i set by the observable BR(B → X s γ) Eγ >1.6 GeV . The contribution of the BSM Wilson coefficients to this observable is given by Here we neglect (∆C i ) 2 terms and have used (implicitly) the input values listed in Ref. [57], where, in particular, the scale µ 0 = 160 GeV was chosen for the SM results. Another assumption used in Ref. [57] is that all other induced BSM operators have Wilson coefficients that satisfy and the NSUSY scenarios we are interested in will have to satisfy this condition when this constraint is strictly applied. Comparing to the current world experimental (HFAG) average given in Ref. [58] BR(B → X s γ) Eγ >1.6 GeV = [3.43 ± 0.21 ± 0.07] × 10 −4 , parameter space. 13 In Fig. 3 we show the interplay of the constraints from the global fit to Higgs data and constraints due to Br(B → X s γ) in minimal NSUSY. We see that there exists consistent parameter space that can pass both experimental tests, primarily at the level of ∼ 2σ in each case.
Large mass splittings scenarios of the stop states when large mixing is present are significantly disfavoured.

D. Electroweak Precision Data
Measurements of m W and other EW precision observables also restrict the allowed parameter space of the NSUSY scenario. Recent measurements of m W at the Tevatron [60,61] are of particular interest. The world average [62] has been refined to (m W ) exp = 80.385 ± 0.015 GeV, with a significant reduction of the quoted error. As recently re-emphasized in Refs. [63,64], precise measurements of the value of m W constrain the allowed parameter space of a weak scale NSUSY spectra when m h is known. This occurs as the allowed custodial symmetry violation that could be present in the sfermion sector is bounded. Note that a global fit to EWPD produces a ∆T constraint that has about twice the error of the constraint used here, which is directly determined from the shift in the W mass.
In this Section, we add this further constraint in the study of the impact of NSUSY spectra on Higgs properties. We use the numerical approximation of the two-loop SM prediction of m W given by Ref. [65] and the method of Ref. [64] . The relevant SUSY correction to the SM prediction 14 of m W is given in Refs. [66][67][68][69] as Neglecting terms proportional to smallb mixing angles, the SUSY contribution is given by and the function F 0 is defined as 13 We have updated our numerical results in V3 of this paper from the result in Ref. [58]   With the choice of input parameters in Table II, we find (m W ) SM = 80.368 ± 0.006 GeV, which constrains NSUSY through the resulting bound (∆m W ) SU SY < ∼ 0.017 ± 0.016 GeV. This translates into a constraint ∆ρ SU SY < ∼ (3.0 ± 2.8) × 10 −4 . This is in good agreement with the result of Ref. [64], which uses the same method, up to small differences in the input parameters.
Note that for θt > π/4 the lightest stop is dominantlyt R compared to the interval we discuss, 0 < θt < π/4, in which it is dominantlyt L . However, in the former case one still obtains a similar constraint space for the ∆m W constraint, with the shift in the SM prediction and the best fit value selecting for degenerate stop states.

The Funnel Region
We show the overlap of the global Higgs fit constraints and the constraints due to m W in Fig. 4.
Note the good degree of consistency between both constraints. This consistency can be traced to the following. The low mass "funnel region" in stop parameter space arises from minimizing the contribution to F g . The condition F g → 0 translates into the relationship .
When the stop and sbottom masses are not widely split, it is an excellent approximation [70] to showing that ∆ρ SU SY 0 has a minimum near At that point the approximation for ∆ρ SU SY 0 written above gives exactly zero. Near the minimum, the result for ∆ρ SU SY 0 is non-zero but suppressed and of order (for mt 2 mt 1 , m t ) which is of the right order of magnitude to match the condition ∆ρ SU SY ∼ 10 −4 . Note that away from the minimum ∆ρ SU SY 0 could be larger by a factor m 2 t 2 /m 2 t 1 (always in the limit mt 2 mt 1 ) which would destroy the compatibility with the measurement of m W .
This slight mismatch between the conditions (34) and (36) gives rise to a non-zero F g : which can nonetheless remain small enough to fit the LHC Higgs data. In this sense, this funnel region is clearly associated with some cancelations due to parameter tuning.

E. Collider Bounds
In this Section we describe some of the current collider bounds on the NSUSY spectrum and how they pertain to the stop parameter space of interest found in previous sections. The most studied and stringent bounds on NSUSY come from missing energy signatures. However, including R-parity or not in a weak scale NSUSY spectrum is not dictated directly by naturalness. Due to this, we will mostly restrict our attention to more robust collider constraints. We note however that with the absence (to date) of missing energy signatures, several searches for R-parity violating phenomenology (without significant missing energy) [19] in the multijet and multilepton final states are now ongoing. These studies could further constraint the interesting parameter space that we isolate in this study.
The main features of NSUSY relate to the Higgsino and Stop sectors. As we have discussed before, if Higgsinos get their mass solely from the µ term, we expect a very small mass splitting between the χ ± , χ 0 , of the order of O(v µ/M 1,2 ). Here M 1,2 are the gaugino masses that are taken to be large M 1,2 v in minimal NSUSY. In the deep-Higgsino region, the χ ± → χ 0 + X decay occurs with no hard-p T objects to tag on. Then, only monojet/monophoton searches at LEP were sensitive to such decays, leading to a bound mχ± > 92 GeV [49] . One can obtain more information on the stop sector by looking at searches of a single photon plus missing energy [78,79]. Although those searches were performed in the context of large extra-dimensions and dark matter effective theories, a straightforward re-interpretation in terms of SUSY can be done [80], within the assumption that the stop and the neutralino are separated by ≤ 30 GeV. This is indeed the case in a significant amount of the low-mass region preferred by the Higgs fit, namely for mt ≤ 120 GeV. Since the Higgsino is heavier than 90 GeV, the splitting betweenχ andt is right at the best sensitivity point. The mass bound obtained in Ref. [80] is 150 GeV (95% C.L.). This strongly constrains/excludes some of the low mass stop parameter space most preferred in the global Higgs fits. Monojet searches could also be used in the low mt 1 region of parameter space [81], and one could also use top precision measurements (spin correlations) to potentially rule out this area of parameter space [75].
Finally, we note that gluino-assisted stop production searches rely on a light gluino around a TeV, which is in some tension with the combination of multijets+leptons+missing energy [82,83], which leads to a bound of 1050 GeV. This gluino bound is rather model independent, as intermediateχ ± ,χ 2 to leptons or off-shell stops do not change the bound considerably [84,85]. 16 Nevertheless, a gluino heavy enough to disable this search is allowed in a minimal NSUSY scenario and the stop limits from these gluino-assisted studies do not (as yet) directly constrain the parameter space of interest.

V. COMBINING CONSTRAINTS
It is of interest to address how a NSUSY scenario globally fits the increasingly rich dataset.
In past sections, we have shown the interplay of various precision constraints; in this Section we perform a global χ 2 fit to Higgs signal-strength data, constraints due to precision measurements of m W and Br(B → X s γ). We fix tan β = 10 and show in Fig. 5 such global fit results for the three mixing cases we have considered in this paper, to illustrate the allowed global fit space.
The results in Fig. 5 show a good fit to the data. We are fitting to fifty observations: forty-eight Higgs signal-strength measurements, as well as ∆m W and Br(B → X s γ). For the three mixing cases θt = 0, π/12, π/4 and µ = 100, 200 GeV we find, fitting to the two stop parameters mt 1 , The interplay of the constraints that allows a good fit is of interest.
The best-fit region in the Higgs signal-strength fit at very low masses, mt 1 ∼ 100 GeV, (See Fig.2 (right)) is essentially ruled out due to mono-photon searches and constraints from ∆m W and Br(B → X s γ). When combining constraints, one finds a best fit to the Higgs signal-strength data with a larger mt 1 (with less mass splitting for larger θt) in a manner that also improves agreement with the small deviations from the SM predictions in ∆m W for large regions of parameter space.
However, the significance of this observation is currently marginal: the SM gives a comparable fit to these observables with Note that for the plots where θt = 0, the best fit point is greater than 1 TeV, and this is why the point is absent. However we note again that the χ 2 is a shallow function in the degenerate stop mass best fit region.

A. Fine-tuning and the Higgs Mass
It is also interesting to examine if the Higgs mass could be consistent with its observed value in the allowed parameter space of the fit, without large fine-tuning. We have included the large radiative corrections to the Higgs mass expected after SUSY breaking [87] by using the public code FeynHiggs [88,89] (for alternatives, see [90]), which includes all one-loop corrections and the dominant two-loop effects [91]. This gives a precise enough determination of m h (with an error of a few GeV), provided the hierarchy in the stop masses is not so large that further re-summation of logarithms is needed [92]. Precision is needed here because the soft masses required to give a large enough Higgs mass are exponentially sensitive to m h .  17 We take the other sfermion states to be ∼ 2 TeV in this analysis, however the Higgs mass band is insensitive to the particular mass values of these states for tan β ∼ 10 and sfermion masses in the TeV range.
masses [88]. We see in Fig. 6, left plot, that the best-fit region with sub-TeV stop masses and zero mixing is not consistent with the Higgs mass condition, as was to be expected. Larger stop masses with large mixing can only produce the observed Higgs mass value when a worse fit is considered, although the degree of fine-tuning in these parameter regions is a concern.
It is well known that physics beyond the minimal supersymmetric version of the SM can easily give contributions to the Higgs quartic coupling which controls the Higgs mass. This can happen through new sectors coupled to the Higgs in such a way that they add at tree-level new supersymmetric F -term or D term (or even susy-breaking) contributions to the Higgs quartic even in the Higgs decoupling limit (see [93][94][95]  It is difficult to define a uniquely compelling fine-tuning measure in an effective theory, or argue what degree of fine-tuning is clearly unacceptable. Fine-tuning considerations are necessarily dependent on the UV physics of the EFT and so one can only make a rough estimate in a lowenergy effective theory. We could follow the definition of fine-tuning measure of Refs. [96,97], which quantifies the tuning by measuring the sensitivity of the electroweak scale (as given by e.g. the Z mass) with respect to changes in the fundamental UV parameters. However, as we want to keep an open mind about what detailed UV physics completes the NSUSY scenario, we will content ourselves with a different estimate of tuning that simply compares the value of the Z mass with known loop contributions to it (in our case those coming from stop corrections), requiring that the latter are not much bigger than m Z . That is, defining the associated fine-tuning will be 1 part in ∆ Z . Restricting ourselves to the moderate value of tan β ∼ 10 and assuming that stops loops are the dominant contribution to this fine-tuning mea-sure, one finds (see e.g. Ref. [98]) The scale Λ is associated with new states required to cut off the logarithmic divergence in the effective theory, and is associated with the messenger scale that transmits SUSY breaking from a hidden sector. The choice of this scale is UV dependent, offering some further caution on the interpretation of the results. For numerical purposes we consider this scale to be Λ ∼ 10, 20 TeV consistent with recent choices in the literature [6]. The fine-tuning contours are overlaid in Fig. 6 in this case (for related work, see Ref. [99]).
Finally, we also note that, if the "funnel region" continues to offer a good fit to the data, such stop masses are very difficult to probe directly in collider searches. A continued deviation consistent with such stop parameters in the global Higgs fits could be the leading experimental indication of the presence of stop states in this exceptional region of parameter space. We discuss the prospects for such future fits in the next Section.

B. Dark Matter Relic density constraints
In this Section, we have not imposed any constraints on the spectrum related to achieving the correct Dark Matter (DM) relic density. We briefly note that typically, Higgsino dark matter is disfavored. The annihilation rate is too large, leading to a small relic density [100]. In the deep Higgsino region, the chargino is also Higgsino-like and is very close in mass to the neutralino.
In this case, coannhilations between charginos and neutralinos become important [101], as well as coannhihilations with the stops, see Refs. [102,103]. Moreover, pure Higgsino DM may be disfavoured by XENON100, see for example a study in the framework of CMSSM [104,105], and also as a possible explanation of FERMI-LAT and PAMELA observations as discussed in Ref. [106]. This situation could be ameliorated if the DM relic density was not entirely due to the neutralino, but there was another relic component, or if the neutralino was not the LSP, but decayed to gravitino, or through RPV interactions. In any case, the focus of this paper is not higgsino DM phenomenology, and we leave a thorough study of the cosmological, indirect and direct detection constraints on pure higgsino dark matter to future work. The same fit to Higgs data does rule out a small portion of the stop parameter space, comparable to the monophoton exclusion. Note however, that those two exclusions, from monophoton searches and from Higgs data, have different degrees of SUSY model dependence. The monophoton exclusion is based on missing energy signatures due to assumed R-parity conservation, whereas the Higgs data exclusion does not assume it, nor any particular stop decay chain.
In the experimental searches, stops are assumed to decay to χ ±,0 . In the case of NSUSY, those electroweak states are degenerate higgsinos, leading in both cases to missing energy in association with a b-jet or a top,t The first decay chain corresponds to the direct sbottom search [63,[107][108][109][110]. and 800 GeV. The vertical lines correspond to interpreting direct sbottom searches as at → bχ ± decay, leading to the same mass limit.
unless it itself comes with some boost. For mt ∼ 300 GeV, this would need some requirement on radiation jets. We illustrate the reduction of missing energy as one increases the value of µ in Fig. 8. Similarly, direct detection of charginos is difficult, due to the degeneracy with the neutralino and small cross sections.
Direct sbottom production searches can be translated into stop production searches when the stop decays to b and χ ± , and this search is more encouraging. The ATLAS collaboration performed a search for sbottom squarks, resulting in a 95% C.L. upper limit mb 1 > 390 GeV for neutralino masses below 60 GeV with L = 2.05fb −1 [111], but would probably be extended to larger values of m χ 0 with the 2012 dataset, being able to cover the case mχ0,± < ∼ 200 GeV. Note that direct sbottom searches assume BR(b → bχ 0 )=1 but can be re-scaled for lower branching ratios. Moreover, pushing the sbottom limits, within NSUSY, is also an indirect probe of the stop sector due to the custodial relations shown in Section IV D. This interplay between direct and indirect limits on stops from direct sbottom production searches is illustrated in Fig. 8, where the maximal mixing case is shown. Interestingly, if the lightest stop is L-dominated, the combination of sbottom and custodial violation limits could be more constraining than the re-interpretation of direct searches in terms of stop decays into charginos. As we do not know the chiral admixture of the stops and sbottoms, using both strategies would give the best sensitivity. Since sbottom direct detection searches do not suffer many of the problems that affect the advance in collider stop searches, this further supports the observation that direct sbottom searches may be the best direct access to stop parameter space in the near term [63]. Higgs search data also provides a powerful insight into stop parameter space. In particular they offer experimental reach into the large µ region that is such a challenge for collider searches. In Fig. 9, we show the region of the higher dimensional operator space excluded at 95% C.L. from current Higgs searches. 18 Translating this exclusion into the stop parameter space does not lead in general to direct lower mass bound that is independent of δm (when θt = 0) due to the existence of the "funnel region" where a cancelation of the stop contributions to F g can occur. However, by jointly imposing the condition that the Br(B → X s γ) constraint is within its 2 σ allowed region, we can define more stringent current exclusion regions for approximately mass degenerate stops.
By studying how the Higgs data will scale with more luminosity, we can also project expected exclusions in the stop space by the end of the 8 TeV LHC run. We find the results shown in Fig 10. Note that these exclusions are in the region of µ ∼ 100 − 200 GeV in NSUSY and do not 18 See Ref. [29] for more details on the exclusion methodology. require a missing energy tag. This makes indirect studies of stop exclusions from Higgs search data broadly applicable to many SUSY models and these bounds have significant reach into the large µ region. Conversely, collider based searches will be severely challenged to reach the large µ space in direct stop searches due to the fact that the signal region is increasingly similar to the tt SM backgrounds with small missing energy.

VII. CONCLUSIONS.
In    When we use the value α s (m t ) we determine the value from NLO running in QCD finding the central value α s (m t ) = 0.1080.

Efficiency corrections due to Higher Dimensional Operators
We seek to draw as precise a conclusion as possible about the Higgs signal strength fit to higher dimensional operators with Wilson coefficientsc g ,c γ in this paper. The effect of these higher dimensional operators on the signal strength parameters can have two forms. Directly the σ gg→h and Γ h→γ γ rates can be effected, as characterized by our rescaling the data with the effects ofc g ,c γ consistently. This effect is what we fit to using our global fit procedure.
There is also a further effect that higher dimensional operators can have on the event yields that lead to the µ i . The higher dimensional operators can alter the shape of the final kinematic distributions (that sum over more than one production mechanism), leading to a further correction, as a function of the Wilson coefficient, on the signal strength parameter. Thec g ,c γ do not affect the shape of the kinematic distributions due to of an individual production process, such as gg → h → γ γ, however, they do affect the relative proportions of gg versus Higgstrahlung and VBF initiated Higgs production. This later effect is an "efficiency correction" on the change in the number of expected events expected due to a different effective efficiency for passing the cuts of the experimental analyses when higher dimensional operators are present. This effect is not captured in our direct fit toc g ,c γ and should be quantified for precise conclusions.
We have examined the effect of these efficiency corrections through simulating the effect of those operators in the 7 TeV run data in the p p → h → γγ (0j, 1j, 2j). We added the shift due to the operators in the hgg and hγγ effective vertices using Feynrules [118] into an UFO model [119] of MadGraph5 [120]. We generated samples at 7 TeV with a parton level cut on |η γ | < 3.5 and p T,γ > 20 GeV. The parton level events generated by MadGraph5 are passed to Pythia [121] to simulate the effects of parton showering, and then to Delphes [122] for a fast detector simulation.
We use the generic LHC parameters for Delphes, and reconstruct jets with the anti-k T algorithm using 0.5 for jet cone radius. We cannot simulate all the characteristics of photons in the different bins, especially the converted/unconverted nature of the photon. Instead, we check the effect of the new physics on the basic selection cuts described in Table III. Those are on top of the detector effects simulated by Delphes, including isolation and energy-momentum smearing.
ATLAS cuts CMS cuts |η γ 1,2 | < 2.37 |η γ 1,2 | < 2.5 p γ 1 T > 40 GeV, p γ 2 T > 25 (30) GeV p γ 1 T > m γγ /3, p γ 2 T > m γγ /4 m γγ = 125(126) ± 3 GeV m γγ = 125 ± 3 GeV Those cuts are then applied to the signal, containing the SM and the new operators. In Fig. 11 we show the effect of the operators, c γ , c g . The effect on the cross section is sizeable, whereas the effect on the efficiencies due to basic p T , η and ∆m γγ cuts is moderate (less that 1%) except when the cross section drops due to a cancellation between the SM and new physics contributions.
The effect on efficiencies for the relation between c γ and c g given in Eq. 20 is very similar to Fig. 11, namely of the order < ∼ 1 %. Further the corrections stay at this level when both operators are present or when the effects at 8 TeV are simulated as we have explicitly verified. Due to this small correction we neglect effects due to efficiency corrections due to higher dimensional operators in this paper.  Table. III. Both figures correspond to √ s = 7 TeV.

Br(B → X s γ) Loop Functions
The loop functions in Br(B → X s γ) are given by [48,57,123] Note the sign correction in F 1 8 (x) when comparing to the previous version of this paper.   from this choice see Ref. [29,124]. Also shown is the combinedμ and error as a vertical green band and the SM expected signal strength as a black vertical line atμ = 1.