Naturalness in low-scale SUSY models and “non-linear” MSSM

In MSSM models with various boundary conditions for the soft breaking terms (msoft\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m_\mathrm{soft}$$\end{document}) and for a Higgs mass of 126 GeV, there is a (minimal) electroweak fine-tuning Δ≈800\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta \approx 800$$\end{document} to 1000\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1000$$\end{document} for the constrained MSSM and Δ≈500\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta \approx 500$$\end{document} for non-universal gaugino masses. These values, often regarded as unacceptably large, may indicate a problem of supersymmetry (SUSY) breaking, rather than of SUSY itself. A minimal modification of these models is to lower the SUSY breaking scale in the hidden sector (f\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sqrt{f}$$\end{document}) to few TeV, which we show to restore naturalness to more acceptable levels Δ≈80\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta \approx 80$$\end{document} for the most conservative case of low tanβ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tan \beta $$\end{document} and ultraviolet boundary conditions as in the constrained MSSM. This is done without introducing additional fields in the visible sector, unlike other models that attempt to reduce Δ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta $$\end{document}. In the present case Δ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta $$\end{document} is reduced due to additional (effective) quartic Higgs couplings proportional to the ratio msoft/f\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m_\mathrm{soft}/\sqrt{f}$$\end{document} of the visible to the hidden sector SUSY breaking scales. These couplings are generated by the auxiliary component of the goldstino superfield. The model is discussed in the limit its sgoldstino component is integrated out so this superfield is realized non-linearly (hence the name of the model) while the other MSSM superfields are in their linear realization. By increasing the hidden sector scale f\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sqrt{f}$$\end{document} one obtains a continuous transition for fine-tuning values, from this model to the usual (gravity mediated) MSSM-like models.


Introduction
If supersymmetry (SUSY) is realized in Nature, it should be broken at some high scale. A consequence of SUSY breaking is the existence of a Goldstone fermion-the goldstino-and its scalar superpartner, the sgoldstino. The goldstino becomes the longitudinal component of the gravitino which is rendered massive (super-Higgs mechanism), with a mass of order f /M P where √ f is the scale of spontaneous supersymmea e-mail: ghilencea@theory.nipne.ro try breaking in the hidden sector and M P is the Planck scale. Also, the sgoldstino can become massive and decouple at low energies. One interesting possibility is that √ f M P , which represents the case of the so-called low-scale SUSY breaking models that we analyze in this work. Then the longitudinal gravitino component couplings which are those of the goldstino and proportional to 1/ √ f [1][2][3][4][5] are much stronger than the couplings of the transverse gravitino component fields, which are Planck-scale suppressed. The latter vanish in the gravity-decoupled limit and one is left with a goldstino superfield besides the matter and vector superfields of the model. The gravitino is then very light, in the milli-eV range if SUSY breaking is in the multi-TeV region.
In this work we consider a variation of the minimal supersymmetric standard model (MSSM) called "non-linear MSSM" defined in [6] (see also [7][8][9]) in which √ f is a free parameter that can be as low as few times the scale of soft breaking terms in the visible sector, denoted generically m soft . We assume that all fields beyond the MSSM spectrum (if any) are heavier than √ f (including the sgoldstino). Then, at energies of few TeV, E ∼ m soft < √ f we have the MSSM fields and the (non-linear) goldstino superfield (X ) coupled to them. The auxiliary component field F X (with F X ∼ − f ) of X can mediate interactions (∝1/ f ) between the MSSM fields and generate sizeable effective couplings, in particular in the Higgs sector, if √ f is low (few TeV). The study of their implications for the electroweak (EW) fine-tuning is one main purpose of this work. This energy regime can be described by a non-linear goldstino superfield 1 that satisfies X 2 = 0 [8][9][10][11]. This constraint decouples (integrates out) the scalar component of X (sgoldstino), independent of the visible sector details (it depends only on the hidden sector [12][13][14]). The alternative case of a light sgoldstino, one that can mix with the Standard Model (SM) Higgs, was studied in [7,15,16]. At even lower energies, below the sparticle masses one is left with the goldstino fermion coupled to SM fields only, and all supermultiplets are realized non-linearly, i.e. all superpartners are integrated out.
However, with so far negative searches for supersymmetry at the TeV scale, the original motivation for SUSY, of solving the hierarchy problem, is sometimes questioned, since the stability at the quantum level of the hierarchy EW scale M P becomes more difficult to respect. Indeed, the EW scale v 2 = −m 2 /λ, where m is a combination of soft masses (m soft ), therefore m ∼ TeV and λ ∼ O (1), an effective quartic Higgs coupling; with an increasing m ∼ m soft , it is more difficult to obtain v = 246 GeV. This tension is quantified by EW scale fine-tuning measures, hereafter denoted generically , with two examples being m , q [17][18][19][20] (early studies in [21][22][23][24][25]) defined as q and m quantify the variation of v under small relative variations of the ultraviolet (UV) parameters γ that denote the SUSY breaking parameters and the (bare) higgsino mass (μ 0 ). m,q are regarded as intuitive measures of the success of SUSY as a solution to the hierarchy problem. For the constrained MSSM, γ denotes the set: m 0 , m 12 , μ 0 , A t , B 0 . For the recently measured Standard Model-like Higgs mass m h ≈ 126 GeV [26][27][28][29], minimal values of m,q in the constrained MSSM are ≈ 800-1000 [30], reduced to ≈500 for non-universal boundary conditions for gauginos. These values are rather far from those regarded by theorists as more "acceptable" (but still subjective) of 10 to 100.
One can ask, however, what relevance such values of the EW fine-tuning have for the realistic character of a model and whether less subjective, model-independent bounds actually exist. Recent results [31][32][33] (based on previous [30,[34][35][36][37]) suggest that there is an interesting link between the EW finetuning and the minimal value of chi-square (χ 2 min ) to fit the EW observables. Under the condition that motivated SUSY of fixing the EW scale v = v(γ ) to its value (246 GeV) and with some simplifying assumptions it was found that there exists a model-independent upper bound q exp(n d f ) [31][32][33]; here n d f is the number of degrees of freedom of the model, n d f = n O − n p with n O the number of observables and n p the number of parameters. Generically, n d f ∼ 10 or so; see for example Table 1 in [32], depending on the boundary conditions of the MSSM-like model. This gives q exp 5 ≈ 150 or so. This is an estimate of the magnitude one should seek for and supports the common view mentioned above that a tuning q ≈ 100 is "acceptable". It should be noted, however, that the nearly exponential depen-dence of minimal m,q ≈ exp(m h /GeV) noticed in [38][39][40][41] and the theoretical error of 2-3 GeV of the Higgs mass [42][43][44] bring an error factor to the "acceptable" value of as large as exp(2) ≈ 7.4 (or exp(3) ≈ 20). Therefore any value of should be regarded with due care. Nevertheless, the above results tell us that a small is preferable.
This view is further confirmed by a less conservative approach, which shows that there is also a link between the EW fine-tuning and the covariance matrix of a model [45,46] in the basis of UV parameters (γ ). This matrix was shown [46] to automatically contain contributions due to the EW fine-tuning w.r.t. parameters γ and, in particular, the trace of its inverse contains a contribution proportional to q . As a result, imposing a fixed, s-standard deviation of the value of chi-square χ 2 from its minimal value χ 2 min , i.e. δχ 2 ≤ s 2 (χ 2 = χ 2 min + δχ 2 ), then requires in the loop order considered that q have an upper bound [46]. This is a modelindependent result and supports our motivation here of seeking models with low .
A very large EW fine-tuning, which increases further with negative searches for SUSY may suggest that we do not understand well the mechanism of SUSY breaking (assuming that SUSY exists not far above the TeV scale). This motivated us to consider the models with low SUSY breaking scale mentioned above and to evaluate their EW finetuning for the recently measured Higgs mass. (An early, pre-LHC study of other models with low SUSY scale is found in [47][48][49].) We examine the values of both m and q in the "non-linear MSSM" [6] which has a low scale of SUSY breaking, √ f ∼ few TeV. The only difference of this model from the usual MSSM is present in the gravitino/goldstino and dark matter sectors. We show that this model can have a reduced fine-tuning compared to that in the MSSM-like models. The reduction is done without additional parameters or extra fields in the "visible" sector, which is unlike other models that reduce EW fine-tuning by enlarging the spectrum. Our results depend only on the ratio m 2 soft / f of the SUSY breaking scale in the visible sector to that in the hidden sector. When √ f is low (few TeV) we are in the region of low-scale-SUSY breaking models (with light gravitino) while at large √ f ∼ 10 10 GeV we recover the MSSM-like models. We thus have an interpolating parameter between these classes of models. The reason why EW fine-tuning is reduced is the additional quartic Higgs interactions mediated by the auxiliary component of the goldstino superfield, as mentioned earlier; these enhance the effective Higgs coupling λ and even increase the Higgs mass already at tree level. We stress that this behavior is generic to low-scale SUSY models.
In the next section we review the model. In Sect. 3 we compute analytically the one-loop corrected Higgs mass including O(1/ f 2 ) corrections from effective operators generated by SUSY breaking. In Sect. 4 we compute at one loop m,q as functions of the SUSY breaking parameters and √ f and then present their numerical values in terms of the one-loop SM-like Higgs mass. For a most conservative case of low tan β and constrained MSSM boundary conditions for the soft terms, we find in "non-linear" MSSM an "acceptable" m ≈ 80 ( q ≈ 120) for √ f = 2.8 TeV and m h ≈ 126 GeV. This value of can be reduced further for non-universal gaugino masses and is well below that in the constrained MSSM (for any tan β) where m,q ∼ 800-1000 [30]. This reduction is done without enlarging the MSSM spectrum (for an example with additional massive singlets see [50,51]).

The Lagrangian in "non-linear" MSSM
The Lagrangian of the "non-linear MSSM" model can be written as [6][7][8][9] L = L 0 + L X + L 1 + L 2 ; (2) L 0 is the usual MSSM SUSY Lagrangian which we write below to establish the notation: κ is a constant canceling the trace factor, and the gauge coupling is g i , i = 1, 2, 3 for U (1) Y , SU (2) L , and SU (3), respectively. Further, L X is the Lagrangian of the goldstino superfield X = (φ X , ψ X , F X ) that breaks SUSY spontaneously and whose Weyl component is "eaten" by the gravitino (super-Higgs effect [52,53]). L X can be written as [8,9] The otherwise interaction-free L X when endowed with a constraint X 2 = 0 [8][9][10][11] describes (on-shell) the Akulov-Volkov Lagrangian of the goldstino [54]; see also [55][56][57][58][59][60][61], with non-linear SUSY. The constraint has a solution φ X = ψ X ψ X /(2F X ) that projects (integrates) out the sgoldstino field which becomes massive and is appropriate for a low energy description of SUSY breaking. Further, F X ∼ − f fixes the SUSY breaking scale ( √ f ) and the breaking is transmitted to the visible sector by the couplings of X to the MSSM superfields, to generate the usual SUSY breaking (effective) terms in L 1 + L 2 (see below). These couplings are commonly parametrized (on-shell) in terms of the spurion field S = m soft θθ where m soft is a generic notation for the soft masses (later denoted m 1,2,3 , m λ i ); however, this parametrization obscures the dynamics of X (offshell effects) relevant below that generates additional Feynman diagrams mediated by F X (Fig. 1). Such effects are not seen in the leading order (in 1/ f ) in the spurion formalism. The off-shell couplings are easily recovered by the formal replacement [8,9] S → m soft f X.
In this way one obtains the SUSY breaking couplings that are indeed identical to those obtained by the equivalence theorem [1][2][3][4][5] from a theory with the corresponding explicit soft breaking terms and in which the goldstino fermion couples to the derivative of the supercurrent of the initial theory. These couplings are generated by the D-terms below: and by the F-terms: with In the UV one can eventually take m = m 0 = m 1 = m 2 , m λ i = m 12 (i = 1, 2, 3) for all gaugino masses, m 2 3 = B 0 m 0 μ 0 (μ ≡ μ 0 in the UV) and these define the "constrained" version of the "non-linear" MSSM, discussed later. For simplicity, Yukawa matrices are not displayed; to recover them just replace above any pair of fields ; similar for the fermions and auxiliary fields, with γ u,d,e 3 × 3 matrices.
The total Lagrangian L defines the model discussed in detail in [6]. The only difference from the ordinary MSSM is in the supersymmetry breaking sector. In the calculation of the on-shell Lagrangian we restrict the calculations up to and including 1/ f 2 terms. This requires solving for F φ of matter fields up to and including 1/ f 2 terms and for F X up to and including 1/ f 3 terms (due to its leading contribution which is − f ). In this situation, in the final Lagrangian no kinetic mixing is present at the order used. 2

The Higgs masses at one loop in "non-linear" MSSM
From the Lagrangian L one obtains the Higgs scalar potential of the model 3 : What is interesting in the above Higgs potential is the presence of the first term in the second line of V , absent in MSSM, which is generated by the diagrams in Fig. 1. Therefore, quartic Higgs terms are generated by the dynamics of the goldstino superfield and are not captured by the usual spurion formalism in the MSSM. The impact of these terms for phenomenology is important and analyzed below, for when √ f ∼ few TeV. When √ f is very large which is the case of MSSM-like models, these terms are negligible and thus not included by the spurion formalism. The ignored higher order terms O(1/ f 3 ) involve non-renormalizable h 6 1,2 interactions in V and are not considered here. 4 Finally, the radiatively corrected m 1,2,3 and μ in V depend on the scale (hereafter denoted t) while the term δ|h 2 | 4 is generated at one loop by top-stop Yukawa couplings. We thus neglect other Yukawa couplings and our one-loop analysis is valid for low tan β; including two-loop leading log effects δ is 2 We stress that at energy scales below m soft , similar constraints to that used for X (X 2 = 0) can be applied to the MSSM superfields themselves and correspond to integrating out the massive superpartners [8,9]. 3 In the standard notation for a two- 4 Effective operators in the Higgs sector in the SUSY context were discussed in the past [49,[62][63][64][65][66][67][68][69]. where M 2 t = mt 1 mt 2 , and g 3 is the QCD coupling and A t is the dimensionless trilinear top coupling. 5 The minimum conditions of the potential can be written with the notation 6 : The correction to the effective quartic Higgs coupling λ, due to the soft terms (m 1,2,3 ) has implications for the Higgs mass and EW fine-tuning. This positive correction could alleviate the relation between v 2 and m 2 : indeed, with m ∼ O(1 TeV) and λ ∼ O (1), v can only be of order O(1 TeV) as well. This brings about a tension between the EW scale and soft terms (∼m) which cannot easily be separated from each other; this tension is encoded by the EW fine-tuning measures, discussed in Sect. 4. Increasing λ can alleviate this tension, with impact on the EW fine-tuning. Such a correction to λ also arises in models with high scale breaking in the hidden sector, so it is present even in usual MSSM but is extremely small in that case since then √ f ∼ 10 10 GeV. Here we consider √ f ∼ few TeV, which is safely above the current lower bound of ≈700 GeV [6,49,58,70].
The two minimum conditions of the scalar potential lead to where 5 More exactly A t = A u /m 0 with A u as in Eq. (7). 6 Also β where we used the notation of footnote 3 and s β = sin β,  Fig. 1 The diagrams that generate the new quartic effective Higgs couplings in V, Eq. (9). The coefficients c 1,2,3 are generated by L 1 , L 2 . F X is the auxiliary component of X that breaks SUSY. The left (right) diagrams are generated by D (F) terms in the action, while the middle one is a mixture of both. These interactions are important in low-scale SUSY breaking models while in the MSSM they are strongly suppressed since F X is large) There is a second solution for m 2 1,2 at the minimum (with minus in front of √ w 0 ) which, however, is not a perturbation of the MSSM solution and is not considered below (since it brings a shift proportional to f of the soft masses, which invalidates the expansion in m 2 1,2 / f ). The mass of the pseudoscalar Higgs is, including a oneloop correction (due to δ): with upper (lower) sign corresponding to m h (m H ) and the correction m 2 h, It is illustrative to take the limit of large tan β on m 2 h,H with m A fixed. One finds where we ignored the tan β dependence of δ. Due to the O(cot 2 β) suppression, Eq. (20) is valid even at smaller tan β ∼ 10. In this limit a significant increase of m h to 120 or even 126 GeV is easily achieved, driven by classical effects alone with μ near TeV (and eventually small quantum corrections, δ ∼ 0.5). Such an increase due to μ is thus of SUSY origin, even though the quartic Higgs couplings (O(1/ f 2 )) giving this effect involved the soft masses m 1,2,3 . These combined to give, at the EW minimum, the μ-dependent increase in Eq. (20). For large f one recovers the MSSM value of m h,H , at one loop. Equations (17) and (18) are used in Sect. 4 to analyze the EW fine-tuning as a function of m h .

General results
To compute the EW fine-tuning we use two definitions for it already shown in Introduction: where γ = m 0 , m 12 , A t , B 0 , μ 0 for the constrained "nonlinear" MSSM. In the following we evaluate m , q at the one-loop level in our model. Using Eq. (12), which give m 2 = m 2 (γ , β) and λ = λ(γ , β), one has a general result for γ 2 which takes into account that tan β depends on γ via the second min condition in Eq. (12). The result is [47,48] Using these expressions, one obtains m and q . Let us first consider the limit of large tan β, so the first relation in Eq. (12) becomes where s = 1 if γ = μ 0 ; s = 0 if γ = μ 0 , and μ, m 2 are functions of the scale. 7 If also f is large, one recovers the MSSM corresponding expression (ignoring a tan β dependence of δ): which is interesting on its own. For the EW symmetry breaking to exist one must have m 2 2 + μ 2 < 0 and therefore γ 2 of the "non-linear MSSM" is smaller than in the MSSM with similar UV boundary conditions for parameters γ . Indeed, in this case the ratio r of γ 2 to that in a MSSM-like model denoted 0 is smaller than unity: r ≈ 1/2 if δ ≈ 0.8, |m 2 2 |/ f ≈ 0.35, and r ≈ 1/3 if δ ≈ 0.8, |m 2 2 |/ f ≈ 0.5 with √ f above the TeV scale (recall |m 2 2 |/ f < 1 for convergence and δ ∼ 0.5-1). So for a large tan β the EW fine-tuning associated to each UV parameter is smaller relative to the MSSM and the same can then be said about overall m and q . This reduction is actually more significant, since for the same point in the parameter space the Higgs mass is larger in 7 As we shall detail shortly for the case of the constrained MSSM. the "non-linear" MSSM than in the MSSM alone, already at the tree level. Indeed, we saw in Eq. (20) that even in the absence of loop corrections one can easily achieve m h ≈ 120 GeV, without the additional, significant finetuning "cost", present for m h > 115 GeV in the MSSM. This "cost" is ∼ exp(δm h /GeV) due to loop corrections needed to increase m h by δm g in MSSM models; 8 for the same m h the reduction is then expected to be by a factor ∼ exp(120-115) ∼ 150 relative to the constrained MSSM case. Then our m,q can be smaller by this factor and r is also much smaller than unity when evaluated for the same m h . Finally, fixing m h to its measured value is a very strong constraint on the parameter space, which, once satisfied, allows other EW constraints to be automatically respected [30], so this conclusion is unlikely to be affected by them.
Let us mention that in MSSM-like models the EW finetuning is usually reduced as one increases tan β for a fixed m h (all the other parameters allowed to vary) [38][39][40][41]. This is because at large tan β additional Yukawa couplings effects (down sector) are enhanced and help the radiative EW symmetry breaking (thus reducing ), while at small tan β this effect is suppressed [30]. The situation is similar to the above "non-linear" MSSM model. 9

The constrained "non-linear" MSSM
The reduction of the EW fine-tuning in our model can be illustrated further by comparing it with that in the constrained MSSM (CMSSM) with universal UV scalar mass m 0 and gaugino mass m 12 and including only the top-stop Yukawa coupling correction. In that case one has where we made explicit the dependence of the soft masses m 1,2,3 and μ and of the coefficients σ i on the momentum scale t = ln 2 U V /q 2 induced by radiative corrections; σ i also depend on tan β and so do the soft masses. The high scale boundary conditions are chosen such as σ 1,2,3,4,5,6 (0) = 0,

The large tan β case
This regime was already discussed in the general case in Sect. 4.1. A numerical analysis of this case involves additional Yukawa couplings of the "down" sector not included in our V and is beyond the goal of this paper. However, we can still provide further insight for the constrained "non-linear MSSM". From Eq. (25), one has m 2 2 is given in Eq. (28) and, since m 2 2 < 0, the absolute values of the above 's and then of m,q are smaller than those in the limit f → ∞ when one recovers the constrained MSSM model (at large tan β). So fine-tuning is reduced as already argued in the general discussion.
The denominator D used in the above formulas is +y 2 cos 2β + y 1 sin 2β)(y 1 cos 2β In the above expressions we introduced the notations: The expressions for γ 2 simplify considerably if one turns off the quantum corrections to the soft terms (σ 1,2,...,6 = 0, σ 7,8 = 1). We checked that in the limit of large f , γ 2 recover the analytical results for fine-tuning at one loop found in [62] for the constrained MSSM (plus corrections O(1/ f 2 )). One also recovers from the above expressions for γ 2 the results in Eq. (29).

Numerical results
Using the results in Eqs. (31) to (37) we evaluated m and q for fixed values of the SUSY breaking scale in the hidden sector √ f for tan β ≤ 10, subject to the EW constraints (for a discussion of these, see [30]). Note that imposing the Higgs mass range of 126 ± (2 to 3) GeV (to allow for the theoretical error [42][43][44]) automatically respects these constraints [30]. For a rapid convergence of the perturbative expansion in 1/ f of the Lagrangian we demanded that m 2 soft / f < 1/4, For m h = 126 GeV, minimal m ≈ 80 and q ≈ 120, while in the corresponding constrained MSSM minimal values (for tan β < 55), m ∼ q ≈ 800-1000, too large to be shown here; for details see figures 1-8 in [30]. The wide range of values for m h was chosen only to display the tan β dependence and to allow for the 2-3 GeV theoretical error of m h [42][43][44]  where m soft stands for SUSY breaking terms. 11 The results are shown in Figs. 2, 3, and 4.
For m h = 126 GeV we find minimal values of m ≈ 80 and q ≈ 120 for √ f = 2.8 TeV (Fig. 2) and m ≈ 105 and q ≈ 145 for √ f = 3.2 TeV (Fig. 3). These values of √ f are well above the current lower bound of ≈700 GeV [6,49,58,70]. As one increases tan β for a given m h , m or q decreases, as shown by the color encoding corresponding to fixed tan β in Figs. 2 and 3; this is also valid in the MSSM as seen in Figures 3, 4, 5 in the first reference in [38][39][40][41]. These values for fine-tuning are already "acceptable" and significantly below the minimal values in the constrained MSSM where for m h ≈ 126 GeV, m,q ≈ 800-1000, see , obtained after scanning over all 2 ≤ tan β ≤ 55.
The reduced values of m and q are due to the fact that m h is significantly above that of the constrained MSSM already at the classical level, see Eqs. (17) to (20) for δ = 0, where values of 120-126 GeV are easily achieved, so only very small quantum corrections are actually needed (unlike in the MSSM). This is a consequence of the (classically) increased effective quartic Higgs coupling. Also notice that minimal values of m and q have a similar dependence on m h and are only mildly different in size, as also noticed for the MSSM [30].  Fig. 4 The dependence of minimal m (left) and q (right) on m h (GeV) for different √ f , for fixed tan β = 10 with the other parameters allowed to vary. We allowed a ±2 GeV (theoretical) error for m h [42][43][44] Fig. 4). This exponential behavior is characteristic for MSSM-like models due to (large) quantum corrections to the Higgs mass [38][39][40][41]. Relaxing the UV universality boundary condition for the gaugino masses reduces m,q further, similar to the MSSM [23,30,74,75], by a factor of ≈2 from the values given by the curves in Fig. 4. Thus, values of √ f of up to 5-6 TeV can still give an EW fine-tuning of about ∼100, for the low tan β regime considered here.
The case of constrained "non-linear" MSSM at small tan β ≤ 10, for which we found "acceptable" values for m,q , is the most conservative scenario. We saw in Figs. 2 and 3 that for the same m h a larger tan β reduces fine-tuning and this behavior continues to tan β ∼ 40-50. Then additional Yukawa couplings also play a significant role at larger tan β and reduce fine-tuning further by improving the radiative EW symmetry breaking for the same m h (this is because radiative EW symmetry breaking effects are enhanced relative to opposite, QCD ones that increase fine-tuning [38][39][40][41]). We thus expect that for the case of large tan β with additional Yukawa couplings included the values quoted here for m , q be maintained or reduced further. Unlike other attempts to reduce the EW fine-tuning, the present case has the advantage that it does not introduce new states in the visible sector. However, there still is a "cost" at the phenomenological level. In models with a TeV scale for SUSY breaking, the gravitino is very light (milli-eV) and the usual MSSM-like account for dark matter (as due to the LSP) cannot apply. This is a standard problem for models with a low scale of SUSY breaking, and alternative dark matter candidates need to be considered (the axino [76], or the axion [77]; for a review see [78]).

Conclusions
The significant amount of EW fine-tuning present in the MSSM-like models for m h ≈ 126 GeV has prompted an increased interest in finding ways to reduce its value. This is motivated by the fact that is usually regarded as a measure of the success of SUSY in solving the hierarchy problem. Additional reasons to seek a low exist, from the relation of the EW fine-tuning to the variation δχ 2 about the minimal chi-square χ 2 min and the s-standard deviation upper bound on δχ 2 usually sought in the data fits. Reducing can indeed be achieved, but it usually requires the introduction of additional fields in the visible sector, beyond those of the original model. For example, one can consider MSSM-like models with additional, massive gauge singlets present, extra gauge symmetries, etc.
Another point of view is that a large EW fine-tuning may indicate a problem with our understanding of supersymmetry breaking. Motivated by this we considered the case of MSSM-like models with a low scale of supersymmetry breaking in the hidden sector, √ f ∼ few TeV. As a result of this, sizeable quartic effective interactions are present in the Higgs potential, generated by the exchange of the auxiliary field of the goldstino superfield. Such couplings are proportional to the ratio of the soft breaking terms m soft in the visible sector to the SUSY breaking scale √ f of the hidden sector. Thus, such couplings are significant in models with √ f ∼ few TeV and are negligible when √ f is large, which is the usual MSSM scenario. These couplings have significant implications for the Higgs mass and the EW finetuning. This behavior is generic in low-scale SUSY models.
For the most conservative case of a constrained "nonlinear" MSSM model and at low tan β, we computed the level of EW scale fine-tuning measured by two definitions for ( m , q ). We examined m,q as a function of the SM-like Higgs mass, in the one-loop approximation for these quantities. The results show that for m h ≈ 126 GeV, fine-tuning is reduced from minimal values of ≈800-1000 in the constrained MSSM to more acceptable values of ∼80-100 in our model with √ f ∼ 2.8-3.2 TeV. These values for are expected to be further reduced by considering non-universal gaugino masses. We argued that a similar reduction of is expected at large tan β in our model. For larger √ f , usually above 10 TeV, one recovers the case of MSSM-like models. Unlike other similar studies, the reduction of was possible without additional fields in the visible sector and depends only on the ratio(s) m 2 soft / f . One may consider the intriguing possibility of increasing simultaneously one of the soft masses m soft (say m 0 ) and √ f , with their ratio fixed (this could keep unchanged the leading corrections O[(m 2 soft / f ) 2 ] for the Higgs mass and ). This is relevant if no superpartners are found near the TeV scale.
We assumed that in our case the sgoldstino was massive enough and integrated out, by using the superfield constraint that decouples it from the low energy. Corrections to our result can then arise from the scalar potential for the sgoldstino that depends on the structure of its Kähler potential (which gives mass to it) and the superpotential in the hidden sector. Another correction can arise from future experimental constraints that may increase the lower bounds on the value of √ f , currently near ≈700 GeV, if no supersymmetry or other new physics signal is found.