Low fine tuning in the MSSM with higgsino dark matter and unification constraints

We examine the issue of fine tuning in the MSSM with GUT-scale boundary conditions. We identify specific unification patterns and mass relations that can lead to a significant lowering of the fine tuning due to gauginos, scalars, and the \mu\ parameter, relative to the simplest unification conditions. We focus on a phenomenologically interesting region that is favored by the Higgs mass and the relic density where the dark matter is a nearly pure higgsino with mass given by \mu~1 TeV while the scalars and gauginos have masses in the multi-TeV regime. There, we find that the fine tuning can be reduced to the level of a few percent. Despite the gluino mass in the ballpark of 2 TeV, resulting mass spectra will be hard to explore at the LHC, but good prospects for detection come from dark matter direct detection experiments. Finally, we demonstrate with a specific example how the conditions and mass relations giving low fine tuning can originate in the context of supergravity and Grand Unified Theories.


Introduction
The recent discovery of the Higgs boson with mass m h 126 GeV [1,2] was universally acclaimed as a great experimental and theoretical success, as it completed the verification of the Standard Model (SM) of particle physics. The mass of the new boson is relatively close to the scale of electroweak (EW) interactions, M Z 91.2 GeV, consistent with the predictions of low-scale supersymmetry (SUSY).
On the other hand, in the Minimal Supersymmetric Standard Model (MSSM) the tree level value of the Higgs mass is bounded from above by M Z , and large radiative corrections are necessary to push the mass up to the measured value. This requires either a large SUSY scale, M SUSY = (mt 1 mt 2 ) 1/2 ∼ several TeV, or nearly maximal stop mixing, |X t |/M SUSY √ 6. While the above conditions can be easily satisfied in the MSSM and other phenomenological models due to the freedom that comes with a large number of parameters, in SUSY scenarios constrained by boundary conditions at the scale of Grand Unification, M GUT , the available parameter space is significantly reduced. Particularly, in the prototypical Constrained MSSM (CMSSM) it was shown that, by employing twoloop calculations of the Higgs mass, the regions consistent with the experimental value are most naturally characterized by M SUSY in the multi-TeV regime [3][4][5]. Only recently it was shown that including three-loop corrections pushes the available parameter space to regions of somewhat lower, albeit still TeV-scale, scalar masses [6,7].
Since the imminent discovery of the Higgs boson was anticipated, it was pointed out in [8][9][10][11] and many other successive papers that the ensuing large values of M SUSY put SUSY at odds with requirements of EW naturalness (for a non-comprehensive list of early articles on this vast topic see ). In short, when SUSY is embedded into a more fundamental theory defined at a very high scale, a large M SUSY implies that the input parameters have to be fine-tuned to uncomfortably high precision to obtain the correct value of the EW symmetry breaking (EWSB) scale, v = 246 GeV, when the parameters are run down to M SUSY through the Renormalization Group Equations (RGEs). However, scenarios of a priori natural SUSY, characterized by the stops with masses 600 GeV and relatively low values of the Higgs/higgsino mass parameter µ [10,[36][37][38][39][40], become increasingly less viable as the limits on third generation squarks from the LHC became more stringent [41][42][43][44][45][46][47]. In addition, the unavoidable requirement of maximal stop mixing, necessary to fit the measured Higgs mass when the stop mass is not too large, typically also leads to high fine tuning [9,[46][47][48].
The latter strategy resonates with the idea that the a priori fine tuning does not necessarily entails an unnatural low-energy theory, but may rather be a result of our ignorance about the physics of the high scale [63], generically called M X hereafter. As a matter of fact, one may turn the argument around and use a "bottom-up" approach to identify mass relations at the high scale that give low fine tuning in regions of low-scale SUSY that are favored by phenomenology and that would otherwise be "generically" characterized by high levels of fine tuning.
It is well known that different high-scale models can lead to different levels of fine tuning. When quantified through the Barbieri-Giudice measure [12,13], ∆ = |∂ ln M 2 Z /∂ ln p 2 |, with p denoting high-scale input parameters, fine tuning indicates excessive sensitivity of the EW scale to variations in the soft SUSY-breaking parameters defined at M X . In general, ∆ increases with ln(M X ) and may become very large for models defined at the unification or Planck scale. However, if certain relations among the high-scale parameters exist due to some new physics at M X , ∆ can be significantly smaller than in the uncorrelated case, as those relations propagate through RGE running and conspire to make the EW scale much less sensitive to M X inputs. This is the known mechanism of "focusing", which has been widely studied in the literature, for scalars (focus-point (FP) SUSY [26-28, 64, 65]) and gauginos alike (non-universal gaugino mass models (NUGM) [29-33, 56, 66-69]). There have also been several attempts to relate the scalar and gaugino sectors, for example in the framework of higher-dimensional brane scenarios [70,71].
A separate issue is the fine tuning due to the parameter µ, which is in most models a SUSY-conserving parameter and appears in the superpotential independently of the soft SUSY-breaking terms. In order to keep the fine tuning due to µ low and at the same time satisfy the condition of EWSB, one generally assumes µ 200 − 300 GeV. A characteristic feature of "low µ" scenarios is, however, an under-abundance of neutralino dark matter with respect to the value measured by PLANCK [72]. While this is not fatal for the model (one can conceive additional particles that can boost the value of the relic density, as explained, e.g., in [73] and references therein), one might feel reluctant to abandon the simplicity of having just one dark matter candidate.
So the question arises as to whether there exist ways of reconciling neutralino dark matter with naturalness in SUSY models defined at M X ∼ M GUT . As was mentioned above, recent global analyses of the CMSSM [3][4][5] and the Non-Universal Higgs Model [74] have shown that the Higgs mass and flavor constraints are well satisfied in an extended region of the parameter space with large scalar and gaugino masses, over which the relic density constraint can also be satisfied thanks to a nearly pure higgsino lightest SUSY particle (LSP) with mass m χ ≈ µ ≈ 1 TeV. This region was shown to appear in the phenomenological MSSM [75,76] and the NMSSM [68,69] as well. It provides an attractive solution in the direction of reconciling EW naturalness with neutralino dark matter: it is extended, phenomenologically viable (except for δ (g − 2) µ ), it naturally gives m h 126 GeV and Ω χ h 2 0.12 and, when different focusing mechanisms are assumed to reduce the fine tuning of scalars and gauginos, it is very stable from the fine-tuning point of view. In fact, µ 1 TeV means a value of ∆ 250, which is quite large but not catastrophic.
The goal of this paper is to investigate the extent to which relations among the MSSM parameters defined at M GUT can be used to reduce the level of fine tuning in the ∼ 1 TeV higgsino region. We will show that, if one includes the µ parameter on the same footing as the soft-breaking terms, the specific mass relation that ensues induces a total fine tuning ∆ 20, even when the scalar and gaugino masses are in the multi-TeV regime and µ 1 TeV. We will show that this relation emerges naturally in high-scale physics scenarios well known in the literature, like the supergravity mechanism first introduced by Giudice and Masiero [77] combined with the Missing Partner mechanism [78,79] in Grand Unified Theories (GUTs). While such scenarios have their problems, our goal is to provide an example of how the relations we determine in the bottom-up approach might be used by the model building community in their effort to solve the puzzle of MSSM naturalness. This paper is organized as follows. In Sec. 2 we define the measure we use to quantify the fine tuning and describe the numerical approach to calculate its value. In Sec. 3 we derive the GUT-scale relations that can reduce the fine tuning in the gaugino and scalar sectors of some specific, constrained models. In Sec. 4 we show that there exist experimentally favored regions in the parameter space of these models for which relating µ and the scalar sector can lead to further substantial reduction of the fine tuning. We show that this can be achieved through the Giudice-Masiero mechanism. We give examples of possible physical spectra and phenomenology. In Sec. 5 we show that the Missing Partner mechanism can be employed to complete the picture that leads to the desired fine-tuning properties. Finally we present our summary and conclusions in Sec. 6.

Definition of fine tuning
The criterion of fine tuning is intrinsically subjective and difficult to quantify in a unique way. Different definitions and measures exist in the literature [12,13,16,17,34,35,61,62,[80][81][82] and the amount of fine tuning for a particular model can differ when different measures are employed. Throughout this paper we will use the well-known Barbieri-Giudice measure [12,13] and p i are the defining parameters of the model. We do not assign absolute meaning to the numerical value of ∆, but rather take it as an estimate of the relative fine tuning of considered models. ∆ quantifies the stability of the global minimum of the scalar potential, v 2 ≡ M 2 Z /g 2 (where g 2 ≡ (g 2 1 + g 2 2 )/2 is the average of the U (1) Y and SU (2) L squared gauge couplings), with respect to variations of the input parameters p i . Assuming that softly-broken SUSY is the low-scale remnant of a high-scale theory, like some GUT theory, or supergravity, or some string theory, the p i 's are the parameters of an effective theory defined at the scale of gauge coupling unification, M GUT , and they are renormalized through the RGEs to M SUSY . If the input parameters are independent from one another, ∆ p i must be calculated for each of them separately and ∆ often becomes significant.
The scale M Z is related to the other parameters through the well-known EWSB conditions that come from minimization of the scalar potential. In this paper we will limit ourselves to the regions of tan β ≥ 10, where tan β is the ratio of the Higgs doublets' vacuum expectation values (vev), as it will be clear below that regions of large tan β can more easily show lower levels of fine tuning.
In fact, for large tan β the EWSB conditions read  .3) are the one-loop tadpole corrections to the scalar potential whose expression in terms of physical masses and low-scale soft-breaking parameters is given, e.g., in the Appendix of [57]. The r.h.s. of Eq. (2.2) is approximately independent of tan β and, consequently, of the parameter Bµ.
In a theory defined in terms of a certain number of high-scale masses and trilinear couplings, called p i , that are subsequently run down to M SUSY , one can integrate the RGEs to express the low-scale parameters in terms of the high-scale ones. These expressions take the form of polynomial expansions, e.g., m i (M SUSY ) = ij C ij p i p j , where the coefficients C ij depend on the running of the parameters p i between the two scales [83]. As is common practice (see, e.g., [56]), we will explicitly write down the dependence of m 2 Hu at M SUSY in terms of the GUT-scale masses and trilinear couplings: ( The parameter µ, commonly defined at M SUSY , is related to its GUT-scale value, µ 0 , through RGE running, which depends only on the Yukawa couplings of the third generation, y t,b,τ , and the gauge couplings of the SU (2) L and U (1) Y groups, g 2 and g 1 : On the other hand, ∂m 2 This shows the well-known fact that in the MSSM naturalness requires preferably small values of µ = Rµ 0 . The impact of any trilinear term or mass other than µ depends on the expansion of m 2 Hu (M SUSY ), given approximately by Eq. (2.4), or on the tadpole corrections. This can be seen from Eq. (2.5) by setting ∂µ 2 /∂p 2 i = 0. The Σ u u and other tadpole corrections depend indirectly on the high-scale parameters. Applying the expressions of [57] to the case of small stop mixing, |A t |/M SUSY 1, to which we limit ourselves in this study to minimize the fine tuning due to trilinear terms, Σ u u reads where we used m 2 Numerically, once a procedure for calculating the fine tuning due to each individual term on the r.h.s. of Eq. (2.4) is established, it is very easy to calculate the fine tuning reductions due to eventual cancellations. Let us assume, for example, that some soft parameters p i depend on one "fundamental" parameter p 0 because of new physics beyond the GUT scale: p i = a i p 0 . It is straightforward to see that the fine tuning due to p 0 is just the sum with signs of the individual outputs of FT 3 Fine tuning of the MSSM with GUT constraints In this section we analyze the fine tuning of some popular GUT-constrained models, accounting for the effects of cancellations induced by GUT-scale relations. We start in Sec. 3.1 with the CMSSM, which, as the simplest model of the class, we use as a sort of "meter stick" to measure the reduction of fine tuning when certain non-universal GUT-scale assumptions are considered, as discussed in Sec. 3.2. In Sec. 3.3 we will quantify the fine tuning reduction due to these relations. The likelihood L for a point in the parameter space is evaluated using the χ 2 statistics as a sum of individual contributions from the experimental constraints listed in Table 1. Confidence regions are calculated with the profile-likelihood method from tabulated values of δχ 2 ≡ −2 ln(L/L max ). In two dimensions, 68.3% confidence regions are given by δχ 2 = 2.30 and 95.0% confidence regions by δχ 2 = 5.99. Throughout this study, we present points belonging to the 95% (∼ 2σ) confidence regions in two-dimensional projections of the profile-likelihood.
The likelihood relative to the recent LUX results [91] was calculated by closely following the procedure for the XENON100 likelihood developed in [105] and updated in Sec. 3.2 of [76]. We assume that the number of observed events, o, follows a Poisson distribution about the number of signal+background events, s + b, As can be seen in Eq. (3.1), the systematic uncertainties are accounted for by marginalizing the background prediction with a Gaussian distribution. We use the values given in [91]: o = 3.1 mDRU ee (1 mDRU ee is 10 −3 events/KeV ee /kg/day), b = 2.6 mDRU ee , δb = √ 2 · 0.2 2 + 0.4 2 mDRU ee , and the signal is simulated with MicrOMEGAs in the nuclear recoil energy range of 3-25 keV nr .
Note that, in practice, the constraints on the Higgs mass and Ω χ h 2 are by far the strongest and play the most important role in determining the allowed parameter space. In particular, in the ∼ 1 TeV higgsino region the other constraints are easily satisfied as SUSY particles are very heavy and their contributions to the observables are suppressed. In some regions characterized by neutralinos with a mixed bino/higgsino composition the LUX likelihood plays a significant role and we will point out in the text when this is the case.

CMSSM
As was explained in Sec. 1, SUSY models defined in terms of high-scale boundary conditions are in general characterized by large levels of fine tuning because ∆ ∼ ln(M X /M SUSY ). On the other hand, the induced relations among parameters can translate into regions of low fine tuning due to the focusing mechanism, as is the case of the FP region [26-28, 64, 65] of the CMSSM. Here we consider fine tuning in the CMSSM, which we use as a model of reference for the following cases.
In the CMSSM the fundamental GUT-scale parameters are the unified scalar mass, m 0 , the unified gaugino mass, m 1/2 , the unified trilinear parameter, A 0 , the unified bilinear parameter, B 0 , and the high-scale Higgs/higgsino mass parameter, µ 0 .
To obtain an approximate estimate of the impact of the parameters on the parameter space, one can recast Eq. (2.4) as The coefficient multiplying m 2 0 is the smallest, resulting in general in low scalar fine tuning, with a consequently low total fine tuning in the regions where m 0 is of the order of a few TeV but µ, A 0 , and m 1/2 are not too large (the FP region). However, the focusing in the scalar sector loses its efficiency with increasing m 0 . One finds ∆ m 0 20 for m 0 = 1 TeV, but ∆ m 0 500 for m 0 = 5 TeV.
We scanned the CMSSM parameter space in the following broad ranges for m 0 , m 1/2 : In order to minimize the impact of A 0 and tan β on the total fine tuning (Bµ, as usual, is traded for tan β) we scanned those parameters in the following limited ranges: The choice of a limited range for A 0 and tan β does not affect significantly the distribution of the profile likelihood in the (m 0 , m 1/2 ) plane, with the exception of the stau-coannihilation region [106], which is not allowed in the ranges of (3.4) because it requires large mixing between the stops in order to obtain the right value of the Higgs mass [5,92]. It is known, however, that the stau-coannihilation region of the CMSSM presents large values for ∆ A 0 , so that we do not treat it in this paper. We show in Fig. 1 the distribution in the (m 0 , m 1/2 ) plane of the fine tuning contributions due to (a) m 0 , (b) m 1/2 , (c) A 0 , and (d) µ 0 . All the points satisfy the constraints of Table 1 at 2σ. Due to the choice (3.4) of tan β ranges, the values of ∆ Bµ are below 10 over the whole parameter space and we do not show its distribution.
A few features are immediately visible in Fig. 1: in the region of m 0 4 TeV the dominant contribution to the fine tuning is given by µ 0 , ∆ µ ∼ 500 − 1000, with the exception of a few points at m 0 3 − 4 TeV and m 1/2 1 TeV, for which ∆ µ 100 and All the points satisfy the constraints of Table 1  ∆ m 1/2 is dominant. Those are the points adjacent to the FP region, where µ is lower.
The FP region appears to be nearly excluded in the plots because it is disfavored by the LUX likelihood. Note, however, that this tension can be ameliorated if one includes the theoretical uncertainties due to the nuclear physics Σ πN terms in the likelihood function [7].
In the upper right part of the (m 0 , m 1/2 ) plane, a very large region characterized by a nearly pure higgsino LSP with m χ 1 TeV is present. As discussed in Sec. 1, in this region the relic abundance assumes the correct value and it is also most naturally compatible with m h 126 GeV due to large M SUSY . The fine tuning due to µ 0 , ∆ µ 250, is large but insensitive to varying the CMSSM parameters. The total fine tuning ∆ is dominated by the contributions of multi-TeV scalar and gaugino masses. Note, finally, that Fig. 1(d) shows that our choice of the A 0 range helps maintaining ∆ A 0 well below 100 over large For completeness, we show in Fig. 2 the distribution of the top-Yukawa fine tuning, ∆ yt , in the allowed regions of the CMSSM. As was to be expected, since ∆ ∼ y 2 t (M SUSY /M Z ) 2 ln(M X /M SUSY ), when the scalar and gaugino masses become large ∆ yt can assume values of the order of 10 4 , thus becoming the dominant source of fine tuning.
The main goal of this study is to suitably choose the GUT-scale mass relations of the scalars and gauginos to reduce the overall fine tuning of the SUSY sector with respect to the CMSSM. This is done by extending the mechanism of parameter focusing. Expansions of the form of (2.4) are obtained by integrating the RGEs for the SUSY-breaking parameters and fixing the initial conditions for the gauge and Yukawa couplings at the experimentally determined values. One can thus express the dependence of the EW scale on the values of the SUSY-breaking parameters at the high scale. As will be clear in Sec. 3.2 and the following sections, the relations leading to the lowest levels of fine tuning are those involving parameters characterized by the largest coefficients in (2.4). Searching for such mass relations assumes specific initial values of the gauge and Yukawa couplings, and including the latter in the analysis would make finding efficient focusing mechanisms much more complicated, especially if one wanted to take into account relations suggested by frameworks of family symmetries. Thus, in this paper we have decided to concentrate specifically on the fine tuning of the SUSY sector of the theory.

Non-universal gaugino and scalar masses
As was shown in Sec. 3.1, in the broad ∼ 1 TeV higgsino region the fine tuning is dominated by the contributions of large scalar and gaugino masses, while the contribution due to µ 0 is smaller and stable over the whole region.
The fine tuning in the gaugino sector can be significantly ameliorated in models in which GUT-scale universality is abandoned, i.e. M 1 = M 2 = M 3 at the GUT scale.
Particularly it was shown [29-33, 56, 66, 67] that the ratio M 2 /M 3 2.5 − 3 leads to a significant reduction of the gaugino contribution to the total fine tuning with respect to the case with universality. This can be easily understood by comparing the coefficients of M 2 2 and M 2 3 in Eq. (2.4). NUGM patterns can be generated naturally in the framework of supergravity embedded in some GUT gauge group [107][108][109][110][111]. The source of gaugino masses are nonrenormalizable dimension-5 operators generated by a chiral superfield whose F -term acquires a vev. The field can belong to any irreducible representation in the symmetric product of two adjoints of the gauge group, leading to various fixed mass relations among the gaugino masses. In SU (5) [112] there are four different patterns allowed [107,108]: The corresponding ratios in SO (10) have been obtained and summarized in [113]. Clearly, not all known gaugino mass patterns lead to a reduction of the gaugino fine tuning with respect to the universal case. This can be seen in Fig. 3(a), where we plot the gaugino fine tuning calculated for the non-universal patterns (3.5) of SU (5) and some of the patterns that can be generated in SO (10)  0 and, consequently, ∆ M 3 0. However, Fig. 3(a) also shows that, while the (10 : 2 : 1) and (−5 : 3 : 1) patterns present the same feature, the distribution of fine tuning is somewhat smoother, which can be explained mathematically with the fact that overall |∂ 2 M 2 Z /∂ 2 M 2 3 | is smaller than in the (19/10 : 5/2 : 1) case. We think that a smaller second derivative is an indication of better and less accidental stability of the EW scale, so that we will select the pattern (−5 : 3 : 1) as our case of choice.
In the scalar sector, the amount of fine tuning strongly depends on the assumed boundary conditions at the GUT scale, particularly on the high-scale relation among m 2 Hu , m 2 , and m 2 u 3 [56]. In SU (5) (or in SO (10)) the fermions and the Higgs bosons belong to different representations, so that the corresponding soft-breaking masses are in general unrelated and the fine tuning can become very large. If supergravity-inspired universality conditions are imposed at the high scale, the fine tuning can be reduced to the CMSSM levels shown in Fig. 1(a).
By It is straightforward to see that one can obtain less fine tuning from the scalars than in the CMSSM when m 2 Hu (M GUT ) and m 2 0 are related as For simplicity we will consider b F to be positive. Equation (3.6) is approximate (although it holds rather well over most of the parameter space), but it gives a good estimate of the values of b F that are necessary to reduce the fine tuning with respect to the CMSSM, even for masses in the multi-TeV regime. Note that, remarkably, b F does not deviate substantially from 1, the value corresponding to universal scalar masses. In Fig. 3  Z /∂ 2 m 2 0 | is generally smaller than for the other choices. In Sec. 4.1 we will comment on the possibility of generating non-universality in the scalar sector with supergravity. Alternatively it is possible to generate b F < 1 in the context of the MSSM embedded in a GUT symmetry and in Sec. 5 we give an example of this for SU (5).

Non-universality and fine tuning in the allowed parameter space
Let us now turn to phenomenologically viable models and show how the conditions derived in Sec. 3.2 affect the fine tuning in the parameter space allowed by the constraints of  Hu (M SUSY ) −1 TeV 2 , which is typical of the ∼ 1 TeV higgsino region, can only be obtained for much larger m 0 when b F < 1, as this effectively leads to a strong reduction of the coefficient multiplying m 2 0 with respect to (3.2). Conversely, if all GUT-scale scalar masses are kept equal, not all gaugino mass pattern can produce physical or phenomenologically acceptable regions. Some lead to no EWSB over large parts of the parameter space, as is the case of the (−5 : 3 : 1) pattern; others are physically viable but cannot produce regions that satisfy the constraints of Table 1, as . Note that it is in general lower than in the CMSSM, see Fig. 1(b). All points satisfy the constraints of Table 1 at 2σ. is the case of the (10 : 2 : 1) pattern in the ranges of (3.

3) and (3.4).
Thus, non-universality conditions both in the scalar and gaugino sectors can not only be seen as instrumental to obtaining lower levels of fine tuning, but also as favoring phenomenologically acceptable regions extended over a large parameter space.
In Fig. 4 we show the distribution of ∆ m 0 in the (m 0 , M 3 ) plane of a specific NUGM, (−5 : 3 : 1), for three different choices of b F < 1. The parameter space was scanned in the ranges of (3.3) and (3.4), where m 1/2 was replaced by M 3 , and the additional assumption of m Hu (M GUT ) = b F m 0 was adopted. In all the cases the points satisfying the constraints at 2σ belong to the ∼ 1 TeV higgsino region. One can see that the allowed region of parameter space moves towards larger m 0 with decreasing b F , as explained above, while remaining very weakly dependent on M 3 . The selected b F values consistently allow one to identify regions of the parameter space with ∆ m 0 10. In Fig. 5 we show the distribution of ∆ M 3 in the (m 0 , M 3 ) plane for the three gaugino mass patterns that give the lowest fine tuning in Fig. 3(a): (a) (19/10 : 5/2 : 1), (b) (10 : 2 : 1), and (c) (−5 : 3 : 1). Different mass relations between gauginos at the GUT scale can have different phenomenological implications, as the latter depend on the lowscale ratios between the bino, wino and higgsino masses (see, e.g., [114] for an early study). However, for any chosen gaugino pattern, the value of b F barely affects the distribution of gaugino fine tuning. We adopt, in Fig. 5, b F = 0.94. We scanned the parameter space over the ranges given in (3.3) and (3.4), where m 1/2 was replaced by M 3 . Again, in the three selected cases the points satisfying the constraints at 2σ belong to the ∼ 1 TeV higgsino region.
Interestingly, in all three cases ∆ M 3 is consistently below 100 and can even reach values 10 over broad regions of the allowed parameter space, in striking contrast with the case of gaugino universality (CMSSM, Fig. 1(b)), for which ∆ m 1/2 ∼ > 1000 in the ∼ 1 TeV higgsino region. In Fig. 5(a) the region where the fine tuning is the lowest is restricted to a horizontal strip about M 3 2 TeV. This behavior was explained in Sec. 3.2 and it has to do with the fact that ∂M 2 Z /∂M 2 3 0 .

µ term
We have shown in the previous section that, if one assumes certain relations among the input parameters resulting from high-scale physics, the mechanism of focusing induces significantly lower gaugino and scalar fine tuning than in the CMSSM.
In the region of moderate-to-large tan β and relatively small GUT-scale trilinear terms, the only remaining issue is the impact of the µ parameter. Without any further assumption, from the point of view of reducing fine tuning, µ should probably be as small as possible, see Eq. (2.7). The phenomenology of this case has been extensively treated in the literature [58][59][60][115][116][117]. As was mentioned in Sec. 1, "low µ" scenarios generally imply an underabundance of dark matter, as the LSP is a quite light higgsino-like neutralino. A higgsino LSP at ∼ 1 TeV, on the other hand, does satisfy the constraints from PLANCK, but produces a somewhat larger level of fine tuning.
Several mechanisms that can relate the µ parameter to the other soft terms have been known in the literature, the most popular probably being the one invented by Giudice and Masiero [77] in supergravity, which relates µ 0 to the scalar soft masses. We recall the main points of the mechanism in Sec. 4.1 and we show that the ensuing relation between µ 0 and m 0 can be used to significantly reduce the overall level of fine tuning of the ∼ 1 TeV higgsino region. We present some phenomenological properties of these low fine-tuning scenarios in Sec. 4.2.
Alternative mechanisms that can similarly reduce the fine tuning by relating µ to other parameters through the EWSB conditions have been developed, e.g., in the context of the NMSSM [52,53]. The Giudice-Masiero mechanism [77] gives an elegant framework to solve the µ problem of the MSSM [118], in the context of local supersymmetry-breaking, or N = 1 supergravity [119,120].
The minimal implementation entails a set of visible-sector superfields,Ĉ i ,Ĉ † i , and at least one set of hidden-sector superfields,ĥ,ĥ † . If a SUSY-conserving bilinear term is forbidden in the superpotential of the visible sector, a naturally small µ proportional to the gravitino mass can be generated through interactions with hidden-sector fields.
In brief, theĈ i ,Ĉ † i ,ĥ, andĥ † are related by the minimal Kähler metric, where M Pl 2×10 18 GeV is the reduced Plank mass and λ is an arbitrary (for our purpose) coupling constant. is the gravitino mass and a = M X /M Pl . If the µ term is forbidden in the visible-sector superpotential by some symmetry, as is the case, e.g., in the Missing Partner mechanism [78,79] that we come back to in Sec. 5, the coupling with the hidden-sector field in Eq. (4.1) generates an "effective" µ, µ eff , in the superpotential, related to the gravitino mass [77], Since the same mechanism also generates masses for the scalar fields, m 0 = m 3/2 , µ 0 (≡ µ eff ) becomes dynamically related to m 0 , where C h is a constant determined by the physics of the hidden sector. Equation (4.4) can be used together with Eq. (2.6) to relate the value of the parameter µ at M SUSY with the common scalar mass, where now the effects of the hidden-sector physics are bundled with the running in the rescaled constant c H .
One can quantify to what extent a relation like (4.5) can reduce the fine tuning in the ∼ 1 TeV higgsino region. By using Eqs. (3.6), (3.7), and (4.5), one can recast Eq. (2.5), In the µ 1 TeV higgsino region m 0 and c H are related by Eq. (4.5). By adjusting b F and c H , one can in principle always find a region of parameter space for which the fine tuning induced by the value of m 3/2 achieves any desired size, even for large m 0 . In Fig. 6 we show the distribution of ∆ m 0 , Eq. (4.6), in the (c H , b F ) plane for the three NUGM choices that lead to the lowest gaugino fine tuning: (a) (19/10 : 5/2 : 1), (b) (10 : 2 : 1), and (c) (−5 : 3 : 1). All of the points belong to the respective ∼ 1 TeV higgsino regions and satisfy the constraints of Table 1 at 2σ. In all the cases, the dark crescent-shaped areas with ∆ m 0 10 show the values that a (c H , b F ) pair can adopt to substantially reduce the fine tuning. The difference between the three plots is almost entirely due to the phenomenology: as Fig. 5 shows, the position of the ∼ 1 TeV higgsino region in the (m 0 , M 3 ) plane strongly depends on the chosen gaugino mass pattern, and this leads to differences in the allowed c H ranges. However, it is clear that c H and b F are related to each other in a way largely independent of the gaugino mass pattern.
It is quite a striking coincidence, on the other hand, that the majority of points presenting low fine tuning in Figs. 6(a) and 6(c), and a significant fraction of the ones with low fine tuning in Fig. 6(b), lie right inside the region allowed by the phenomenological constraints, which features the correct Higgs mass and 1 TeV higgsino dark matter.
Note also that it is necessary to have b F < 1 to obtain low levels of m 0 fine tuning, similarly to the cases where µ 0 is a fundamental parameter, discussed in Sec. 3.3. The desired GUT-scale splitting between m Hu and the remaining scalar masses could be obtained, for instance, if one considers a non-flat Kähler metric in the visible sector, instead of Eq. (4.1). Alternatively, the splitting could be generated through quantum gravitational corrections above the GUT scale [121], although it is far from clear to what extent such a procedure could by carried out in a well-defined and controllable way.
If one neglects the gravitational quantum corrections, one way to control the amount of m Hu /m 0 splitting is to employ RGE running above the GUT scale, if a larger GUT gauge group breaks down to the SM group at M GUT . Additional symmetries can be then assumed to forbid the µ term in the visible sector. A possible solution of this kind in the context of SU (5) is the subject of Sec. 5.
In Fig. 7(b) we compare the fine tuning of these three cases (small violet dots) with that of the NUGM case first shown in Fig. 4(b) (green crosses), which was featuring the same gaugino mass pattern, b F = 0.93 to cancel the scalar-sector fine tuning, and µ 0 as an independent parameter of the theory. We also show the fine tuning of the CMSSM (blue dots) for comparison. The ∆ of the case with b F = 0.93 is dominated by ∆ µ 250, typical of the ∼ 1 TeV higgsino region. One can see that relating µ 0 to m 0 reduces the fine tuning by approximately one order of magnitude if c H is chosen so as to give a ∼ 1 TeV higgsino region at m 0 4 TeV. For different c H values, which would lead to higgsino regions with larger m 0 , the fine tuning can still be 5 times smaller than ∆ 250. In contrast, the fine tuning of the ∼ 1 TeV higgsino region can be reduced, for the largest m 0 values shown here, by 15-20 times relative to the CMSSM.
Obviously, the scenario that shows the better prospects is the one characterized by lighter sparticles, shown in Fig. 8(a). Even in that case, though, the requirement of good relic density narrows down the neutralino mass to m χ 1 TeV, a value that will provide a challenge for observation of other superpartners at the LHC, as it strongly limits the transverse momentum of the charged and colored SUSY particles produced in collisions.
From this perspective, it does not seem surprising that SUSY particles have not been observed so far at the LHC and we fear that, if naturalness happened to be encoded in SUSY the way we analyzed in this paper, there will probably be little chance to see sparticles even in future runs.
Rather than at the LHC, the best prospects for observation of this kind of scenarios come from dark matter direct detection experiments, particularly at 1-tonne detectors like XENON1T [122]. It has been shown, see e.g., [76], that there are good prospects for future detection of an m χ 1 TeV neutralino. We present in Table 2 the values of the spinindependent neutralino-proton cross section for the points of lowest ∆ in the three cases given above.
Unfortunately, since these scenarios have approximately all the same m χ and the same higgsino composition, even upon detection at 1-tonne detectors it will be hard to distinguish one case from another, so that the values of the scalar masses will have to be inferred by other means. However, even if the values shown in Table 2 are subject to some theoretical uncertainties, they seem to indicate that an eventual detection at XENON1T is more likely for the cases of larger c H (smaller m 0 ).

Low fine tuning from high-scale effects
We have shown in Sec. 4 that a simple relation between µ 0 and m 0 can increase the naturalness of the ∼ 1 TeV higgsino region of the MSSM. We assumed that the origin of Eq. (4.4) lay in some high-scale physics or hidden-sector mechanism, and we referred to Giudice-Masiero as the most widely known example. However, a relation like (4.4) will not make the model more natural without additional conditions. As Fig. 6 shows, b F also must assume a value that conspires with the particular choice of c H , and this value certainly cannot be equal 1. Finally, there is the fine tuning of the gaugino sector that can be kept under control if one assumes certain non-universality condition for the gaugino masses. We have shown in Sec. 3.2 that, in the context of GUT unification, some patterns emerging from SU (5) are among the most efficient in this sense.
In this section we address the issue of obtaining an appropriate value of b F at the GUT scale in the context of SU (5), maintaining consistency with the Giudice-Masiero mechanism. We consider a minimal supergravity scenario in which supersymmetry breaking is transmitted to the visible sector at the generic scale M X . M X can be identified with the GUT scale, M GUT 2 × 10 16 GeV (as it happens in the CMSSM), but also with any scale above, up to the reduced Plank scale, M Pl 2 × 10 18 GeV. We assume that  Table 2: The values of the spin-independent neutralino-proton cross section, σ SI p , for the points shown in Fig. 8. For mχ 1 TeV, the sensitivity of LUX is estimated to reach ∼ 2 × 10 −9 pb by the end of 2014 [122] and the one of XENON1T should reach ∼ 2 × 10 −10 pb by the end of 2017 [123].
between M GUT and M X the theory has an extended gauge symmetry, which we choose to be SU (5) for consistency with requirements of low gaugino fine tuning in the MSSM. Thus, the high-scale parameters are renormalized when evolved down to the GUT scale, leading to non-universal scalar masses at M GUT , which can in turn lead to the appropriate low fine tuning if b F assumes the correct value.
We additionally assume that the doublet-triplet splitting problem of SU (5) is solved by the Missing Partner mechanism [78,79], which can be used here to forbid the µ term in the superpotential of the visible sector superfields. An anomalous U (1) A symmetry with a nonvanishing trace over the charges of the matter superfields is required to assure that the EW Higgs doublets are massless. Such a symmetry can be either global [124] or local [125]. In the latter case, a non-zero Fayet-Iliopoulos term ξ ∝ Tr Q M 2 X is generated [126][127][128][129][130] and some of the scalar fields can obtain a non-zero vev of order ∼ √ ξ . We refer here to the model introduced in [125], but alternative choices could be employed to make our point, as long as they are characterized by two important ingredients: a) an extended gauge group compatible with one of the gaugino mass patterns that lead to low fine tuning and b) a symmetry to forbid the µ term in the superpotential. One could consider, for example, the use of a discrete R-symmetry instead of an anomalous U (1), but a series of no-go theorems [131] prevent this choice in a simple 4-dimensional setup.
The superfield contents of the theory is the following [78]: MSSM matter superfields are indicated withψ 10 andφ 5 as they belong to the 10 and5 representation, respectively. Gauginos and gauge bosons belong to the 24 adjoint (and we indicate them byλ), Higgs doublets belong to the 5 and5 (ĥ 5 ,ĥ5) and they are accompanied by two color triplets. There are also additional Higgs multiplets in the 50 (θ 50 ) and the 50 (θ 50 ), as well as one in the 75 (Σ) that breaks SU (5) at the GUT scale. Finally, there is a gauge singletŜ that breaks U (1) A at a scale M .
The most general renormalizable superpotential invariant under SU (5) × U (1) A takes the form: with standard meaning of the symbols. The coupling constants are of order 1 and the mass term µ Σ is of the order of M GUT , the scale where SU (5) is spontaneously broken.
The U (1) A charges can be chosen to forbid the bilinear term µ Hĥ5ĥ5 [124,125]: (5.2) In fact, in order to have a relation between µ 0 and m 0 , we assume that the µ term of the superpotential is effectively generated by interaction with the hidden sector via the Giudice-Masiero mechanism. Note that not only the term µ Hĥ5ĥ5 , but alsoŜĥ 5ĥ5 is forbidden in Eq. (5.1) by the U (1) A symmetry. It has been shown [125] that this solution is protected against Planck scale corrections to all orders, since non-renormalizable terms of the form ∼ 1/(M k+n−1 Pl )Ŝ kΣnĥ 5ĥ5 are not allowed. Finally, the soft SUSY-breaking part of the Lagrangian reads: The U (1) A symmetry is spontaneously broken when the singlet S develops a vev and the Higgs multiplets θ 50 and θ 50 receive a mass, M = λ S S . The SU (5) symmetry is broken down to the MSSM SU (3) × SU (2) L × U (1) Y when the field Σ develops a vev, v Σ M GUT . As a consequence, the Higgs fields and gauge bosons associated with the broken generators acquire masses of the order of v Σ . The masses of the color triplets of h 5 and h5 are generated via a "seesaw" kind of mechanism, since they mix with the corresponding triplets of θ 50 . The resulting effective triplet mass is The mass M H 3 must satisfy the limits from proton stability, obtained by suppressing the dimension-five operators generated by the color triplet [132][133][134][135][136]. Given the present experimental 90% C.L. lower bound on the proton partial lifetime in the decay mode p → K +ν , τ (p → K +ν ) > 2.3 × 10 33 yrs [88], it can be derived [136] that Recalling that v Σ M GUT , Eq. (5.5) effectively constrains the mass M to The doublet components of h 5 and h5 do not mix with any other field and they thus remain massless, unless the Giudice-Masiero mechanism is used to generate the mass term for the EW Higgs doublets. The choice of Kähler metric given in Eq. (4.1) leads to the universal boundary conditions in the scalar sector at the scale M X , The RGEs for the couplings and soft-SUSY breaking parameters are given in Eqs. (A.1) of Appendix A. One can use Eqs. (A.1) to derive approximate expressions for the following soft masses at the scale M GUT : where t = ln(M GUT /M X ) and we have assumed that g 1 A is small enough so that its contribution can be neglected in a first approximation. We also neglected the trilinear terms, which are relatively small with respect to m 0 , see relation (3.4).
The RGE-driven mass difference between the Higgs soft mass and the other scalars at the GUT scale is given by the ratio ∆ 2 5 (M GUT )/m 2 10 (M GUT ) and is entirely determined by four parameters: the superpotential trilinear Higgs and Yukawa couplings, λ H and Y D , the scale difference, t, and the ratio of the common gaugino to the common scalar mass, M 5 /m 0 . Assuming that Y D λ H one can write: In the three cases shown in Fig. 7(a) the first condition is satisfied (m 0 > M 3 , M 5 ≈ M 3 after SU (5) breaking). The second relation also holds, due to both the requirement of proton stability and the large Higgs representations present in the model. The other mass ratios are essentially stable under the running since the Yukawa couplings Y U and Y D are smaller than λ H and they couple to sfermions through the Higgs fields belonging to much smaller representations.
By using Eqs.  2). Thus, the values of M X we obtained also help respect the A 0 ranges chosen in our scans, see the first of (3.4). 1 The scale of supersymmetry breaking could be, in principle, moved up if one allows for λ H = λH . In particular, if λH = 2.3 (and λ H = 0.53) M X can be taken equal to the perturbativity scale, M pert . Such a choice, however, results in a very strong renormalization of the mass ∆ 2 5 , as it is affected by λH , leading to ∆5 0.1 m 10 at the GUT scale. The latter would imply m H d (M GUT )/m 0 0.1, a condition that often leads to no EWSB at the low scale.
Note, finally, that the mechanism described here can be used beyond the need of relating µ 0 and m 0 . As was shown in Sec. 3, even by accepting the presence of an independent µ parameter, b F 0.93 is required to reduce the fine tuning of the scalar sector with respect to the CMSSM. To generate the right b F in that case, however, it might be enough to use a model characterized by smaller representations of the gauge group, as it happens, e.g., in minimal SU (5) [137]. In that case one could run the parameters further up above M GUT , to energies very close to M Pl . However, large trilinear terms would have then to be forbidden by other means, or their relation to the scalar masses by considered in the calculation of the overall fine tuning due to m 3/2 .

Summary and conclusions
In this paper we investigated the issue of fine tuning in the MSSM with GUT-scale boundary conditions. We analyzed several popular cases: the CMSSM, models with non-universal gaugino masses, and models with non-universal scalar masses. We employed the widely accepted Barbieri-Giudice measure, ∆, to quantify the fine tuning, and we identified the GUT-scale relations among the input parameters that led to small ∆. We focused in particular in the region of the parameter space characterized by the correct Higgs mass, production and decay rates in good agreement with the experimental value, and for which the relic density constraint is satisfied thanks to a nearly pure higgsino LSP at m χ 1 TeV. This region is also characterized by stop masses in the multi-TeV regime, so that its fine tuning can be very large if one assumes the simplest unification conditions like in the CMSSM.
We showed that the mechanism of parameter-focusing along RGE running is, for certain specific GUT-relations, very efficient in lowering the fine tuning of the scalar and gaugino sectors in the ∼ 1 TeV higgsino region with respect to the case with universal masses. In particular, very low fine tuning can be obtained in the gaugino sector if one adopts the SU However, even when the above patterns are enforced, there still remains the issue of a µ parameter too large to give an appreciably small ∆, as µ 1 TeV implies ∆ of the order of a few hundreds. This may be one of the reasons why in the literature the ∼ 1 TeV higgsino region has so far been neglected in studies of low fine tuning. We showed in this paper that the issue can be approached in a way no different from the other soft-breaking parameters if one assumes the existence of a relation among µ and some other soft masses at the GUT scale. In particular we showed that, if µ and the GUT-scale scalar mass m 0 are linearly related, the µ fine tuning becomes greatly reduced with respect to the case where µ and the soft masses are unrelated. We quantified the improvement in fine tuning and we finally provided one example of how a µ 0 ∼ m 0 relation can be generated, at least at the tree level: by combining the known Masiero-Giudice mechanism in supergravity and the Missing Partner mechanism in SU (5) GUT.
We point out here that the example we employ is not free of problems, as is often the case in supergravity where the tree-level relations that lead to universality are spoiled by large and mostly uncontrollable radiative corrections. Moreover, it would be interesting to investigate the consistency of our parameter ranges with the measured value of the cosmological constant in a supergravity-inspired theory like this one, an issue that goes beyond the scope of this paper. It is quite possible that the real high-scale theory will eventually turn out to be different from the simplified case discussed here, it might be perhaps based on some string theory.
More importantly, one can argue that by introducing the parameters c H and b F the problem of fine tuning in the MSSM is not solved, but rather shifted to the framework of a higher scale theory. Particularly, even if the EW scale remains stable under the changes of the GUT-defined MSSM soft terms, which is in itself an achievement worth the effort, the fine tuning will most likely be reintroduced under variations of the continuous fundamental parameters of the underlying theory. The problem can be avoided if a symmetry or some other dynamical mechanism is responsible for the constants that relate the scalar and µ sectors. Thus the effort in this direction should continue.
What we gave here is a set of guidelines to the realizations of high-scale physics that are allowed if the effective low-scale theory is not supposed to be excessively fine-tuned. As was pointed out in Sec. 4.1, it is striking that the values of c H and b F that emerge from requiring a higher level of naturalness are in many occasions also the ones that are favored by independent phenomenological constraints.

A One-loop RGEs for SU(5) × U(1)
We present in this appendix the one-loop RGEs for the superpotential and soft SUSYbreaking parameters of the SU (5) × U (1) gauge group considered in Sec. 5. We limit ourselves to the third generation for the Yukawa and A terms. The results were obtained using the formulas of [83] for the β functions for a generic gauge group and the numerical package Susyno [138]. In all equations we define t = ln(Q/M GUT ).