A Novel Approach to Fine-Tuned Supersymmetric Standard Models -- Case of Non-Universal Higgs Masses model

Discarding the prejudice about fine tuning, we propose a novel and efficient approach to identify relevant regions of fundamental parameter space in supersymmetric models with some amount of fine tuning. The essential idea is the mapping of experimental constraints at a low energy scale, rather than the parameter sets, to those of the fundamental parameter space. Applying this method to the non-universal Higgs masses model, we identify a new interesting superparticle mass pattern where some of the first two generation squarks are light whilst the stops are kept heavy as 6TeV. Furthermore, as another application of this method, we show that the discrepancy of the muon anomalous magnetic dipole moment can be filled by a supersymmetric contribution within the 1 {\sigma} level of the experimental and theoretical errors, which was overlooked by the previous studies due to the required terrible fine tuning.


Introduction
Although the discovery of the Higgs boson in July 2012 verifies our thought that the physics up to the electroweak scale should be well described by the standard model (SM) of particle physics [1], the SM itself suffers from the uncomfortably large disparity between the electroweak scale and the fundamental physics scale which is supposedly close to the Planck scale. Supersymmetry (SUSY) has been recognized as a promising candidate to solve this unease. The fact that superparticles have not yet been discovered, however, constrains their mass spectra, if exists: e.g. colored superparticles should weigh at least around 1 TeV [2]. In the minimal supersymmetric standard model (MSSM), the measured Higgs boson mass of about 125 GeV requires large radiative corrections due to supersymmetry breaking (SUSY-breaking) to raise its tree-level mass below the Z boson mass [3]. As the Higgs boson strongly couples to the top-stop sector, this typically requires that the stop mass has to be around 6 TeV or so, unless a SUSY-breaking trilinear coupling is parametrically large [4]. In this case, there will be a little hierarchy between the electroweak scale and SUSY-breaking mass parameters, and thus some amount of fine tuning among these parameters may be requisite in order that the electroweak symmetry breakdown takes place at the correct energy scale.
This situation does not mean that nature rejects SUSY, but implies that we should not have prejudice against the amount of fine tuning. Since SUSY is still a promising candidate for physics beyond the SM, we should study the supersymmetric SM with some amount of fine tuning (FT-SUSY: fine-tuned supersymmetry).
To identify an experimentally viable region or an interesting region of a model, the scatter plot method has been widely used. This method represents a relevant region by a collection of discretized points in the fundamental parameter space, just like a "pointillism". The collection of points is selected from a large number of initially chosen points in the fundamental parameter space to satisfy the experimental (and other) constraints at the experimental scale. However, in a FT-SUSY the relevant region might be too tiny to be represented in this way.
In this paper, we propose a novel approach to a FT-SUSY regardless of the amount of fine tuning.
In Sec.2, we propose a method to identify the relevant region of a FT-SUSY. In contrast to the ordinary top-down renormalization group (RG) picture, in which a point chosen in the fundamental parameter space at the fundamental scale flows to that at the experimental scale, we map a constraint for the parameter space at the experimental scale to that at the fundamental scale. Then, we can directly identify the restricted space by the mapped constraints as the relevant region written in the fundamental parameters. This procedure is like a "coloring". This procedure allows us to identify the whole relevant region as well as its outlines in the fundamental parameter space. Furthermore, the area near an outline can be easily identified as a phenomenologically interesting region, if this outline corresponds to the boundary of a constraint given by an on-going experiment. Since the constraints we map can also include the requirement of a characteristic property, if we choose a suitable requirement, a fine-tuned region is identified.
In Sec.3, to illustrate our idea and to show its efficiency, we apply this procedure to the non-universal Higgs masses model (NUHM) [5] which has the MSSM particle contents with universal SUSY breaking masses except for the Higgs masses at the GUT scale. We identify the experimental viable region of the NUHM and argue its features. We find an interesting region with a new superparticle mass pattern, where some of the first two generation squarks are light (Sec.3.1). This mass pattern is a consequence of the RG running where a negative Higgs mass squared dominantly raises the third generation squark masses due to their rather large Yukawa couplings.
Since this effect never happens in the CMSSM, this region should be one of the characters of the NUHM (Sec.3.2).
In Sec.4, using another application of our method, we find there is a terribly fine-tuned region that explains the anomaly of the muon anomalous magnetic dipole moment (muon g − 2) [6], [7], [8] within the 1 σ experimental and theoretical errors in the NUHM. Furthermore, with sufficiently large tan β, we also show that there is a parameter region explaining the muon g − 2 anomaly with most of the 1st and 2nd generation sfermions light. These regions were overlooked by the previous studies using the scatter plot method which is not practical to find such a tiny and terribly fine-tuned region. This fact shows the power of our approach to a FT-SUSY.

A Novel Approach to FT-SUSY
We propose a novel approach to tackle a FT-SUSY. 1 In this approach, we can directly identify the relevant region in the FT-SUSY without being bothered with some amount of fine tuning.
For the sake of simplicity, suppose that a fundamental supersymmetric model, such as a grand unified theory (GUT), can be described as a generic MSSM. The generic MSSM is defined as an effective theory with most general SUSY-breaking soft mass parameters of the particle contents, below the fundamental scale t f of the fundamental model.
This defines a map, On the other hand, the solution of the RG equation [9] which gives correspondence among the parameters of the same theory at different scales can also be considered as a map f RG . 2 Since the fundamental model can be described by the generic MSSM, we consider f RG in the context of the generic MSSM: M gen e is the parameter space at the experimental scale t e = log ( O(100)GeV mz ), and a 1 In fact, the method developed here can apply to many models even without SUSY. However, for ease of explanation, we only apply our method to a FT-SUSY in this paper. 2 For simplicity, we suppose that f RG is a bijection, so that the image satisfies the equality, imf RG ≡ f RG (M gen f ) = M gen e , and the inverse map, f −1 RG , can be defined.
set of g sol i (t e ; t f , g f j ) is the solution of the RG equation [9] in the generic MSSM at t e with an initial condition of a set of parameters, g f j at t f . Suppose thatM gen e denotes the region of interest of the generic MSSM at t e , which may be either a viable region, i.e. the part of the parameter space that Here φ l (g e i ) is a condition function for the generic MSSM parameters at t e , which could either correspond to a fitted function of an experimental constraint or a re- Namely, given a set of the fundamental parameters, G a ∈ M fund f , we apply the RG procedure to obtain the corresponding parameters at t e , and check whether they satisfy the conditions characterizing the region of interest of the generic MSSMM gen e . The ordinary scatter plot method follows this procedure recursively by using sample points, S = G where ψ l (G a ) is a condition function for the fundamental parameter space, expressed as and hence should be equivalent to Eq. (5). However, what we would like to obtain is not the correspondence among points in M fund f andM gen e , but the correspondence between the two condition functions, φ l (g f i ) and ψ l (G a ). Namely, we map the given set of conditions, φ l (g e i ) > 0 that characterizesM gen e , to the corresponding one, we identify asM fund f is the interior of the outlines, our procedure is like a "coloring".
Since the map, f f , is given, what we would like to know is the RG map of the condition function, φ l (g e i ), within the generic MSSM, and we will show how to derive the explicit form of this. One way is to solve the RG equation of the generic MSSM so that we can express g sol i (t e ; t f , g f j ) in terms of the set of the parameters g f j at t f . Alternatively, we can solve the differential equation which follows the RG map of the condition function, Φ l (g j , t) ≡ φ l g sol i (t e ; t, g j ) , by varying t: where β i is the RG beta function for g i [9].
If the perturbative expansion is allowed, a set of linear differential equations for the coefficients, φ i 1 ,i 2 ...in (t), are derived by requiring the vanishing of each Taylor coefficient in Eq.(9). The upper limit of the summation in Eq.(11) comes from the fact that a perturbative RG beta function always contains parameters of total exponents 1. Eq. (11) is the running equation for coefficients of constraint (RECC) . If all the parameters are dimensionful as in our case, Hence the coefficients can be evaluated by numerically solving the derived linear differential equation. 3 We show the explicit derivation of RECC in the generic MSSM in Appendix A.
We also note that in the derivation of RECC, ..jm can even depend on the scale t. This is a convenient fact because we may take a shortcut to derive RECC with some parameters approximately treated as constants. Namely, if possible, we can numerically solve the RG equations for these parameters in advance, and substitute the numerical solutions as constants in the remaining RG equations. Then we can derive RECC from these remaining RG equations which explicitly depend on t. This is what we do in Appendix A.
Solving the corresponding RECCs, we can obtain the set of conditions in terms of > 0, and applying the given map, f f , a set of ψ l (G a ) > 0 is derived from Eq. (7). Therefore the whole region of interest,M fund f , is identified from Eq.(6).
There are two additional advantages in our approach.
Since a boundary of a constraint could correspond to an outline ofM fund f , the viable region near such an outline may be testable if this constraint is given by an ongoing experiment. This implies that a viable region near an outline can be a phenomenologically interesting region. Therefore, checking the boundary profiles, namely the constraints the boundaries correspond to, we can guess some of the phenomenological interesting regions even without any additional requirement of characteristic properties. On the other hand, the boundary profiles of a phenomenologically interesting region in turn suggest the predictions that can be accompanied with the characteristic property.
The second advantage is due to the fact that in our procedure the RG map of a condition function is followed within the MSSM. In fact, we can define the region of interest of the generic MSSM at t f , Since the viable region of the generic MSSM at t f directly responds to the funda- As we have noted, tan β, as well as the other dimensionless couplings, is taken to be a given constant that is not included in the parameter set, Eq. (13). Since the NUHM can be described by the generic MSSM below t f , a set of fundamental parameters in M fund f is related to the parameters in M gen f : Eqs. (14), (15) and (16)

Whole Viable Region and Phenomenologically Interesting Regions
Solving the RECCs, which is derived from the given 1loop RG equations in [9] as in Appendix A, we evaluate the Taylor coefficients of the conditions, Eqs. (19), (20) and (21)  represents the low energy parameters given in Table.1 There is a new interesting region near the red solid line in Fig.1, the boundary of which is given by the mass bound of the first two generation up-type squarks. We note that the stops are kept heavy to reproduce the measured Higgs boson mass.
We call this region an inverted light squark (ILSQ) region. Here "inverted" stands for the new superparticle mass pattern characterizing this region where the first two generation squarks are light, contrary to the ordinary light stop.

Mechanism for the inverted light squark
In this subsection we will explain how the characteristic mass pattern in the ILSQ region is generated in spite of the universal sfermion mass condition, Eq. (14), and the requirement of heavy stops, Eq.(20). Since these two conditions are imposed at two different scales, the RG running should be essential. The RG equation for a right-handed up-type squark mass is, where i represents 12 or 3, and Y is the hypercharge, −2/3 [9]. y t is the top-Yukawa Hd is large and negative, the same argument applies to down-type squarks and sleptons.
In summary, the inverted hierarchy can be generated by the Yukawa contribution through the RG running due to a negative and large Higgs mass squared parameter.
Furthermore, we would like to explain the sfermion mass splitting within a generation. This originates in the gauge interactions, especially for U(1) Y gauge symmetry. In particular, the value of S in Eq.(23) can largely deviate from zero in the NUHM contrary to the CMSSM case where S is always zero. We find that when our mechanism of the ILSQ works, S is non-zero in order to avoid the unstable electroweak vacuum, unless tan β is substantially large. We show this by reductio ad absurdum. Suppose that S = 0 is satisfied with a large and negative m 2 Hu0 , namely m 2 Hu0 = m 2 Hd0 ≪ 0 at the GUT scale. Since X t dominates over the other terms in Eq.(22), the RG equations of m 2 Hu and m 2 H d are also controlled by the large and negative Higgs masses [9], For not so large tan β, the property of S < 0 in the ILSQ region makesũ 12 the lightest squark as in Table.1, because it has the least hypercharge. The requirement of S < 0 is also the reason why the approximately concentrating point in the righthand side of Fig.1 is far away from the viable region. In fact, the approximately concentrating point lies on the line of S = 0 where the first two generation sfermion masses can be small without splitting. On the other hand, since S ∼ 0 may be allowed when tan β is sufficiently large, in this case there is a possibility to have an ILSQ region including most of the first two sfermion masses just above the experimental bounds. This is the case in Sec.4.2.
As we have discussed, the ILSQ region has a new superparticle mass pattern essentially related to the non-universal Higgs masses. Therefore, this mass pattern should be one of the most characteristic phenomenon of the NUHM, as the CMSSM never realize this.

Terribly Fine-Tuned but Important Region
In Sec.3, we have presented the whole viable region of the NUHM by solving the experimental constraints in terms of the fundamental parameters which can be obtained from the method advocated in Sec.2. We have found a new phenomenologically interesting region near the boundary corresponding to a squark mass bound. However, an interesting region with a characteristic property is not necessarily around the outlines, rather inside the viable region. This is particularly the case when the characteristic property requires a fairly large amount of fine tuning among the parameters. Notice that it would be difficult for the scatter plot method to identify such a fine-tuned region.
In this section we will show how to analyze a large and complicated viable region to find out the characteristic properties localized inside of it, and argue this analysis has an advantage when applied to a FT-SUSY.
To illustrate the idea, consider here the muon anomalous magnetic dipole moment (muon g−2), α µ . The muon g−2 anomaly is a hint of new physics, as the discrepancy between the theoretical and experimental values exceeds the 3 σ level of these errors [6], [7], [8] In the generic MSSM at least three light superparticles, weigh around 300GeV, are needed to generate a large enough contribution [10], [11]. Therefore, the region of the generic MSSM,M gen e , that fills the discrepancy of the muon g − 2, should be characterized by three parameters with very tiny values compared to the stop mass parameters around 6TeV. Therefore, if exists, the region of interest of the NUHM now,M fund f , should be fine-tuned from the whole viable region in the previous section.
The particular diagram we consider for the SUSY contribution to the muon g − 2 is made of a loop including a bino and smuons of both chiralities [10], [11]. Following [10], we obtain an approximated formula of the SUSY contribution to the muon g −2, This rough approximation will be corrected by fitting the results evaluated in Feyn-Higgs 2.11.2 [4] by varying the overall coefficient of this function.

How large can the muon g − 2 be in the NUHM?
In order to clarify whether the NUHM can explain the muon g − 2 anomaly, an efficient way is to evaluate the maximal value of the muon g − 2 in the NUHM.
This immediately draws a conclusion of whether the muon g − 2 anomaly can be explained. Therefore we impose a condition, including the maximal NUHM correction, is within the 1 σ (2 σ, 3 σ) level error of the observed value. The black dot, (8700GeV, 2010GeV), represents the low energy parameters given in Table.2.
in addition to Eqs.(19)-(21). In fact, we can vary all the free parameters to maximize the muon g − 2, however, for the illustrative purpose we only vary one parameter. We find the maximized muon g − 2 can exceed 18.1 × 10 −10 with tan β 25, namely the NUHM is able to fill the discrepancy of the muon g − 2 within the 1 σ level error.
In fact, there are two kinds of regions, namely type 1 and type 2 regions, depending on the lightest particle in the loop diagram corresponding to Eq.(29). The type 1 region explains the muon g − 2 anomaly with eitherẽ 12 orL 12 as the lightest particle in this diagram, while the type 2 region has bino as the lightest particle. Fig.2 corresponds to the former one, while the region of type 2 appears for tan β 55 as we will show in Sec.4.2. Table. 2: The low energy parameters corresponding to the black dot, (M 0 , µ 0 , A 0 )=(2010GeV, 8700GeV, 0GeV), in Fig.2 The factor 10 and 7.5 in the right-hand side are fitted experimentally using FeynHiggs 2.11.2 [4].
Also shown are the boundary profiles.
If the muon g − 2 anomaly is explained by the region in Fig.3, from the boundary profiles some of light smuons, selectrons, right-handed stau and A-Higgs might be within the reach of the forth coming experiments. On the other hand, Fig.4 implies that many superparticles might be within the reach of the forth coming experiments, as many boundaries of the experimental constraints are within the dis-tances of O(TeV) from this interesting region. A set of low energy parameters corresponding to the type 2 region is given in Table.3. We have confirmed that the Higgs mass and the muon g − 2 anomaly are explained using FeynHiggs 2.10.2 [4] with an input of these parameters. We have also confirmed that the parame- . Therefore the mechanism of the ILSQ in Sec.3.2 can be applied. As noted, a negative S−term is required to stabilize the electroweak vacuum when tan β is not substantially large. Neglecting RG running due to gauginos,ũ 12 becomes the lightest sfermion in most of the cases as in Table.1. This excludes a smuon around 300GeV, as it should be heavier thanũ 12 which is bounded below by 1TeV. If we would like to decrease the smuon masses in the ILSQ region, what we should do is to increase the gluino mass. The RG effect from a heavy gluino, as the third term in Eq.(22), raises the masses of squarks universally from those of sleptons.
However the universal gaugino mass condition, Eq.(15), implies a heavy bino. Hence, the lightest particle in the loop diagram corresponding to Eq.(29) becomes a smuon which can be seen from the boundary profiles in Fig. 3. This is the case for type 1 region.
The type 2 region where bino is the lightest particle in the loop diagram corresponding to Eq.(29) can be realized with sufficiently large tan β. This is consistent with the argument noted in Sec.3.2: S ∼ 0 can be realized with large tan β. In this case, we do not need a heavy gluino to raise the squark masses and the bino can be as light as the smuons. Furthermore, all the 1st and 2nd generation sfermions will be light because small gaugino masses and S− term imply the degeneration of the sfermions in the first two generations, and the mechanism of ILSQ works for all of them. This is actually the case in Table.3.
As we can see in Fig.2, Fig.3, and Fig.4, the relevant regions have small sizes due to fine tuning. In fact, fine tuning is already alleviated in these figures as we have reduced a free parameter that is fine-tuned contrary to the stop mass scale by solving Eq.(30) or Eq.(31). On the other hand, after the discovery of the Higgs boson the muon g − 2 correction in the NUHM is discussed in several studies [13] by using has low energy parameters given in Table.3 the scatter plot method. In these studies, the tiny regions we found in this paper had been overlooked, as the "pointillism" is not practical to find such a tiny and terribly fine-tuned region. Therefore, we have shown that the method advocated in this paper has a strong advantage in identifying a fine-tuned region and this should be an efficient approach to a FT-SUSY.

Conclusions
In this paper, we proposed a novel and efficient approach to the supersymmetric models with some amount of fine tuning, in which the commonly used approach of scatter plot is inefficient and sometimes even fails to find relevant regions in the parameter space of superparticle masses with the limited number of plotted points. The essential idea of our approach is to directly map the (experimental or other) constraints at low energy to those in the fundamental parameter space. We Another application of our method is to identify, within the NUHM model, the existing but tiny region in the parameter space, where the SUSY contribution explains the discrepancy of the muon g − 2 within the 1 σ level of experimental and theoretical errors. The price to pay is the terrible fine tuning among the parameters, and therefore the previous studies with the conventional scatter plot method failed to find this region, drawing misleading conclusions. This example illustrates the power of our method in particular when the required fine tuning is severe. The relevance of our approach will even increase when the forth coming experiments will give null results in superparticle searches and more fine tuning will be required to correctly produce the electroweak scale.
The method advocated in this paper has a variety of applications, some of which was given in [14] and also will be discussed elsewhere.

A RECCs for the generic MSSM
We would like to show the derivation of RECCs in the generic MSSM with given dimensionless couplings at t e . In fact, the following argument can apply to the derivation of RECCs at any loop order with given RG equation at the same order.
The parameters g f i and constants of the generic MSSM are classified by their dimensions: (32) a k 2,i (y j ), a kl 1,i (y j ), b k i (y j ) and c i (y j ) are given functions that can be derived from loop calculations in the generic MSSM [9]. The summation is understood.
Firstly, we can solve the RG equation for dimensionless constants numerically with given y i at t e . Substituting this numerical solution for y i , the (effective) RG equation for dimensionful parameters become, Secondly, using Eq.(33), we can derive the RECCs for condition functions, corresponding to simplified experimental mass bounds of a scalar and a fermion, respectively. Dimensional analysis allows us to guess the forms of the mapped con-dition functions at an arbitrary scale, t, to be Therefore, employing Eq.(9) the RECCs for Eqs.(35) and (36) are derived as