The Higgs boson mass in a natural MSSM with nonuniversal gaugino masses at the GUT scale

We identify a parameter region where the mass of the lightest CP-even Higgs boson resides in $124.4-126.8$ GeV, and at the same time the degree of tuning a Higgsino-mass parameter (so-called $\mu$-parameter) is relaxed above 10% in the minimal supersymmetric standard model (MSSM) with soft supersymmetry breaking terms, by solving the full set of one-loop renormalization group equations numerically. It is found that certain nonuniversal values of gaugino-mass parameters at the so-called grand unification theory (GUT) scale $\sim 10^{16}$ GeV are important ingredients for the MSSM to predict, without a severe fine-tuning, the Higgs boson mass $\sim 125$ GeV indicated by recent observations at the Large Hadron Collider. We also show a typical superparticle spectrum in this parameter region.


Introduction
The low-energy supersymmetry is one of the most promising candidates for a new physics beyond the standard model (SM) of elementary particles due to the absence of quadratic divergences. The supersymmetric partners of SM particles cancel the radiative corrections to the mass of Higgs bosons, then protect the electroweak (EW) scale M EW ∼ 10 2 GeV against the huge corrections. The lightest supersymmetric particle (LSP) can be a candidate for the dark matter required from cosmological observations. Moreover, the minimal supersymmetric standard model (MSSM) predicts that three gauge coupling constants are unified at a high-energy scale around M GUT ≃ 2 × 10 16 GeV, and the electroweak symmetry is broken in a wide range of the parameter space of the MSSM with soft supersymmetry breaking terms, due to logarithmic radiative corrections that cause sizable running of parameters from M GUT to M EW . (See, for a review, Ref. [1].) On a stable vacuum where the EW symmetry is broken successfully, the mass of Z-boson is determined by where µ(M EW ) and m Hu (M EW ) are supersymmetric and soft supersymmetry breaking masses, respectively, for the Higgs field evaluated at M EW . The Higgsino mass is also given by µ and we refer to this parameter as µ-parameter. The above mentioned radiative EW breaking roughly means m 2 Hu (M EW ) < −|µ(M EW )| 2 < 0 even with m 2 Hu (M GUT ) > 0 at M GUT . The observed value m Z = 91.2 GeV indicates |µ(M EW )| ∼ |m Hu (M EW )| ∼ M EW , otherwise a fine-tuning is required between parameters µ and m Hu . One of the guiding principles toward a more fundamental theory beyond the MSSM can be provided by an argument of the naturalness. Because the mass parameters µ and m Hu should have essentially different origins from the viewpoint of supersymmetry, there is no reason that these parameters are closely related.
However, from the recent results in the search for Higgs and supersymmetric particles at the Large Hadron Collider (LHC), the mass of scalar quarks (squarks) in the first and the second generation is indicated above about 1.3 TeV [2], and the allowed region of the mass of the lightest CP-even Higgs boson has been reported in between 124.4 and 126.8 GeV [3]. The latter implies a large radiative correction for the mass of the Higgs boson which is lighter than the Z-boson at the tree-level [4]. In this case the mass of scalar top quarks must be much heavier than M EW which dominantly contribute to such a correction due to a large top Yukawa coupling. These observations indicate that the mass scale of soft supersymmetry breaking parameters tends to be much larger than m Z , which in general cause the fine-tuning problem mentioned above.
It was pointed out in Ref. [5] that certain nonuniversal gaugino masses at M GUT relax the degree of the above mentioned fine-tuning in the MSSM. In this paper, we update the analysis based on the recent experimental data. 1 Because the latest Higgs mass bound shown above indicates a larger value of tan β than the value adopted in Ref. [5], in the following analysis, we solve the full set of one-loop renormalization group equations (RGEs), in contrast to the previous analysis in Ref. [5] where all the Yukawa (and scalar trilinear) couplings are neglected except for those involving only (scalar) top quarks.
The following sections are organized as follows. In Sec. 2, we roughly estimate the effects of certain nonuniversal gaugino masses on the mass of the lightest CP-even Higgs boson and on the degree of tuning the µ-parameter. Based on these implications, in Sec. 3, we perform a full numerical analysis at the one-loop level and identify a parameter region where the Higgs mass resides in 124.4 − 126.8 GeV and the degree of tuning is relaxed above 10 % at the same time. Finally, Sec. 4 is devoted to conclusions and discussions. In Appendix A, we show the boundary conditions for RGEs at the GUT scale adopted in this paper. The relevant RGEs for analyzing the Higgs mass are exhibited in Appendix B.

Implications of nonuniversal gaugino masses
The superpotential and soft supersymmetry breaking terms in the MSSM are shown in Eqs. (7) and (8) in Appendix A, respectively. We use the notations and the conventions adopted in Appendix A throughout this paper.
The radiative corrections to the Higgs mass are dominated by loops of scalar top quarks. With an approximation that the mass eigenstates of top squarks are nearly degenerate, the mass of the lightest CP-even Higgs boson is evaluated as [7] with where m t = 165 GeV is the top quark mass, y t ≡ y u 33 is the top Yukawa coupling, a u 33 ≡ y t A u 33 ≡ y t A t is the scalar trilinear coupling involving only top squarks, m 2 is the left-(right-)handed top squark mass square, and t = ln(M 2 st /m t 2 ). Here v = 174 GeV is related to the VEVs of up-and down-type Higgs fields h u and h d as v u = v sin β and v d = v cos β, respectively. As is well known, the value of X t is maximized with |Ã t | ∼ √ 6M st , and then the loop corrections are enhanced in Eq. (2).
If all the Yukawa (and scalar trilinear) couplings are neglected except for those involving only (scalar) top quarks, the one-loop RGEs show that the soft supersymmetry breaking parameters evaluated at the Z-boson mass scale m Z are related to those at M GUT as where the soft parameters without any arguments in the right-handed sides represent those evaluated at M GUT . Here the numerical values of the MSSM gauge couplings and the top Yukawa coupling are chosen in such a way that these values become the observed ones at low energies, and we take tan β = 15 as declared at the end of Appendix A. The numerical values of the coefficients in Eq. (3) suggest |Ã t |/M st < 1 for soft parameters of the same orders of magnitude at M GUT , without any cancellation between terms in the right-handed sides of Eq. (3). This insists that a certain cancellation is required in Eq. (3) in order to enhance radiative corrections in Eq. (2) for obtaining a heavier Higgs mass. We find that the gluino mass-square term with the largest coefficient can be cancelled by the wino mass-square term in m 2 U (m Z ) with the ratio M 2 /M 3 ∼ 4.6/0.21 ∼ 4.8, and then 0 < m U (m Z ) < |A t (m Z )| is realized. This phenomenon can be understood as follows. The gaugino masses and soft scalar masses act as positive and negative driving forces, respectively, in the renormalization group evolution of soft scalar mass square from the GUT to the EW scale. The relevant RGEs are shown in Appendix B. Therefore, the mass square of the left-handed scalar quark m 2 Q tends to increase for a larger wino mass M 2 /M 3 > 1, and then the mass square of the right-handed squark m 2 U tends to decrease with the increasing m 2 Q . The averaged top squark mass M st decreases because the increasing contribution from m 2 Q is dominated by the decreasing one from m 2 U . As we can estimate in Eq. (3), the contribution from the wino mass satisfying 2 M 2 /M 3 5 to reduce m 2 U dominates the one to increase m 2 Q , and then |Ã t |/M st > 1 is achieved. The above rough estimation can be verified numerically. Fig. 1 shows contours of M st /m Q 3 (the left panel) and those of r a ≡ |Ã t |/M st (the right panel) in the (r 1 , r 2 ) plane, evaluated by a numerical solution for the full set of one-loop RGEs with the boundary conditions at the GUT scale shown in Appendix A, where the bino-gluino and the wino-gluino mass ratios at the GUT scale are denoted respectively by The ratios r 1 and r 2 are varied in the following analysis. From Fig. 1 we find M st decreases and then |Ã t |/M st increases as the wino mass parameter M 2 increases, assuring the above rough estimation.
On the other hand, concerning about the fine-tuning, a smaller absolute value of the soft scalar mass square for the up-type Higgs m 2 Hu is favored. A similar relation to the ones in The numerical values of the coefficients suggest that m 2 Hu (m Z ) is of the order of soft parameters for those of the same orders of magnitude at M GUT , without any cancellation between terms in the right-handed side of Eq. (4).
This insists again that a certain cancellation is required in Eq. (4) in order to realize −m 2 Hu (m Z ) ∼ m 2 Z to reduce the fine-tuning required by Eq. (1). We find that the gluino mass-square term with the largest coefficient can be cancelled by the wino mass-square term in

The Higgs mass and the naturalness
Motivated by the above rough estimations based on Eqs. (2), (3) and (4), we search a parameter space of the MSSM with soft terms, especially a region of gaugino mass ratios, where the mass of the lightest CP-even Higgs boson m h resides in 124.4 − 126.8 GeV and at the same time the degree of tuning the µ-parameter is relaxed above 10 %. Because we take tan β = 15 as declared at the end of Appendix A, the effect of the other Yukawa couplings than the top one, especially the bottom Yukawa coupling, can induce sizable corrections, we numerically solve the full set 2 of one-loop RGEs including all the Yukawa couplings. (The complete set of the MSSM RGEs are found, e.g., in Ref. [9] at the two-loop level.) By utilizing the numerical solutions of RGEs, we evaluate the Higgs mass m h based on the mass matrix with one-and two-loop contributions from top and bottom squarks, respectively, derived in Ref. [7]. Here we do not adopt the approximated expression (2) because left-and right-handed top squarks are not degenerate in the favored region 2 M 2 /M 3 5 with the maximal-mixing of top squarks. Although the above rough estimation is based on Eq. (2), it turns out to be valid and consequently a heavier Higgs mass can be obtained in the full numerical analysis as we will see later.
We assume a certain mechanism of supersymmetry breaking which determines soft supersymmetry breaking parameters at the GUT scale, and then the ratios among these parameters are fixed with an accuracy. The µ-parameter in the MSSM superpotential (6) is in general independent to the mechanism of supersymmetry breaking. As a measure of the degree of tuning the µ-parameter at the GUT scale, we adopt a parameter ∆ µ defined by which represents a sensitivity [10] of the Z-boson mass m Z at the EW scale on the µ-parameter at the GUT scale. The degree of tuning to obtain m Z = 91.2 GeV is then estimated as 100 × |∆ −1 µ | %. It seems that, with the current experimental status, |∆ µ | > 100 is inevitable with universal values of soft parameters at the GUT scale, which requires more severe finetuning than the degree of 1 %.
The results from the direct search at the LHC set the lower bounds of the mass of gluino 860 GeV and of the masses of squarks in the first and the second generations 1320 GeV [2]. We adopt the severest bound although it may be lowered in models with some nonuniversal gaugino mass ratios [11]. Taking these stringent bounds and the other experimental bounds for top squarks, neutralinos and charginos [12] into account, we fix numerical values of the other soft parameters than wino and bino masses at the GUT scale as shown in Appendix A, and vary gaugino mass ratios r 1 , r 2 at the GUT scale in the evaluation of the Higgs mass m h and the parameter |∆ µ |. Fig. 2 shows contours of m h [GeV] and 100 × |∆ −1 µ | (%) in the parameter space (r 1 , r 2 ).
From Fig. 2, we find that the mass of the Higgs boson resides in 124.4 − 126.8 GeV in the region 3.0 r 2 5.5 for the wide range of r 1 −3. Moreover we emphasize that the region is overlapped with 5.2 r 2 5.5 where the degree of tuning the µ-parameter is relaxed above 10 %. Therefore we find a parameter region where the Higgs boson mass ∼ 125 GeV indicated by recent LHC observations is predicted in a natural MSSM with certain nonuniversal gaugino masses at the GUT scale. Compared with Ref. [5], here the Higgs boson mass above 115 GeV 2 Although the effects of light generations are negligible, for concreteness, we input numerical values of all the Yukawa couplings determined by Froggatt-Nielsen mechanism [8] at the GUT scale, shown in Appendix A. Also, our calculation covers the situation that the scalar charm quark masses are about 10 2 times higher than scalar top quark masses and the correction to m 2 Hu running is sizable, as can be seen in Appendix B. is achieved by a larger value of tan β = 15, that is one of the reason we employed the full set of MSSM RGEs including all the Yukawa couplings. Although the overlapped region is not so wide, the required accuracy for the ratios of the gaugino masses are not so stringent. There would exist some supersymmetry breaking and the mediation mechanisms that fix the gaugino mass ratios to the above numerical values favored by the naturalness. For example, concrete fixed values of the ratios are shown in Ref. [13] for various breaking and mediation mechanisms. A typical superparticle spectrum at the EW scale and the masses of neutral and charged Higgs bosons are shown in Tables 1 and 2, by setting the gaugino mass ratios (r 1 , r 2 ) = (11, 5.4) inside the favored region found above. From these tables, we find all the experimental lower bounds on the masses of gluinos, neutralinos, charginos, squarks and sleptons as well as neutral and charged Higgs bosons can be satisfied with 12.84 % tuning of the µ-parameter. With this parameter choice, Fig. 3 shows the running of gaugino and soft scalar masses from the GUT to the EW scale. We find that gaugino masses tend to degenerate at low energies, that is a typical signal of this parameter region as discussed in Ref. [5].
Here we comment on charge and color breaking minima [14]. If we take A t larger than M st , there is a possibility that such minima appear along the direction satisfying |q 3 | = |ũ 3 | = |h u | in the field space. The minima exist unless the following condition, is satisfied. It has been noted that the parameters yielding r a ≃ √ 6 are dangerous with the parameters satisfying m Q 3 = m U 3 due to the inequality (6) [15]. On the other hand, in our analysis, the important region √ 5 r a √ 7 shown in Fig. 1 appears with the parameters satisfying |A t (m Z )| ∼ m Q 3 (m Z ) and 0 < m U 3 (m Z ) < m Q 3 (m Z ), and then the inequality (6) Table 2: The masses of neutral and charged Higgs bosons as well as the gaugino masses at the EW scale for (r 1 , r 2 ) = (11, 5.4). The degree of tuning the µ-parameter, 100 × |∆ −1 µ | (%), is also shown.
Finally, we mention about the next to minimal supersymmetric standard model (NMSSM). When a singlet chiral multiplet is added, the MSSM superpotential can be modified to have a positive correction to the Higgs mass square at the tree level. There is, however, another negative contribution to the Higgs mass square induced by a mixing with the singlet field. The latter negative contribution also becomes larger in general when the parameters are chosen to make the former positive contribution larger. This fact makes the analysis of the NMSSM complicated (see for a review, Ref. [16]). We would not be able to make a definitive statement, at the current stage, that the NMSSM is better than the MSSM from the viewpoint of both the Higgs mass and the naturalness, even though there have been interesting studies concerning those issues in the NMSSM [17].

Conclusions and discussions
By solving the full set of one-loop RGEs numerically for all the parameters in the MSSM with soft supersymmetry breaking terms, we identified a parameter region where the mass of the lightest CP-even Higgs boson resides in 124.4 − 126.8 GeV indicated by the recent LHC results, and at the same time the degree of tuning the µ-parameter is relaxed above 10 %. The region is characterized by certain nonuniversal values of gaugino mass parameters at the GUT scale as indicated in Ref. [5] before the LHC observations. We have confirmed that, even after the LHC results, the main suggestions in Ref. [5] are still valid with a larger value of tan β based on more accurate analyses than those in Ref. [5]. We also derived a superparticle spectrum for a typical parameter choice inside the identified region. We find all the experimental lower bounds on the masses of gluinos, neutralinos, charginos, squarks and sleptons as well as neutral and charged Higgs bosons can be satisfied. One of the outstanding features of the spectrum is the degenerate gaugino masses at low energies as a consequence of a particular choice of gaugino mass ratios at the GUT scale favored by the naturalness, as mentioned in Ref. [5]. It would be interesting to study cosmological features in detail in this parameter region of the MSSM 3,4 .
One of the important guiding principles for the physics beyond the MSSM is provided by the argument of its naturalness. The EW scale is unstable under a tiny numerical deviation of the µ-parameter when a severe tuning is required to realize the observed mass of W and Z bosons. Therefore the underlying theory free from such a fine-tuning is desirable, which becomes a strong guiding principle when we study particle physics models at a more fundamental level. We adopted Eq. (5) as the measure of the degree of tuning. The measure can be extended to include all the other soft supersymmetry breaking parameters like, e.g., in Ref. [22]. However, we have assumed that the ratio between the breaking parameters are fixed with an enough accuracy by a concrete supersymmetry breaking and the mediation mechanism. Even in this situation, the supersymmetric parameter µ would be independent to the other parameters, and the issues of the fine-tuning should be concerned about the µ-parameter.
There would exist some supersymmetry breaking and the mediation mechanisms that fix the gaugino mass ratios [13] to the favored numerical values by the naturalness. One of such candidates is a mirage mediation model [23], where the gaugino mass ratio at the GUT scale is determined by the ratio of contributions to gaugino masses from the modulus and the anomaly mediated supersymmetry breaking [24]. Issues of fine-tuning in this model are studied in Ref. [25]. Another and more general origin of nonuniversal gaugino masses is moduli-mixing gauge kinetic functions, which appear even at the tree level in a certain effective supergravity action. For example, in some superstring models, such moduli-mixings appear from nontrivial D-brane configurations where the gaugino mass ratios are determined by, e.g., numbers of windings, intersections and magnetic fluxes of D-branes [26] (see, for a reivew, Ref. [27]). Even without mentioning superstrings, in a minimal extension of the MSSM with a single extra dimension, namely, in five-dimensional supergravity models, the situation is similar and the gaugino mass ratios can be determined by data of the very special manifold governing the structure of N = 2 vector multiplets [28]. In all the cases with moduli-mixing gauge kinetic functions, the mechanism of moduli stabilization is important to determine the gaugino mass ratios [29].
The LHC is now exploring the parameter space of the MSSM and the other supersymmetric models. It would be possible that the recent and near-future observations guide us in a direction toward a more fundamental theory of the nature through the results obtained here.

A Boundary conditions at the GUT scale
In this appendix we show numerical values of parameters in the MSSM with soft supersymmetry breaking terms selected as a boundary condition at the GUT scale, M GUT = 2 × 10 16 GeV, for our numerical analysis of the full set of MSSM one-loop RGEs.
For the chiral multiplets Q i , U i , D i , L i and E i carrying the ith generation of the left-handed quark doublet, the right-handed up quark, the right-handed down quark, the left-handed lepton doublet and the right-handed electron, and those H u and H d containing the up-and the downtype Higgs doublet, respectively, the MSSM superpotential is given by where y u,d,e ij are Yukawa coupling constants, and µ is the Higgsino mass parameter referred to as µ-parameter. The summations over the generation indices i, j = 1, 2, 3 are implicit. The soft supersymmetry breaking terms are defined by as well as the parameter Bµ in Eq. (8) are called soft supersymmetry breaking parameters. Note that the MSSM singlet chiral multiplet carrying a right-handed neutrino could be introduced without affecting the basic results of this paper, which is omitted just for simplicity.
As for supersymmetric parameters, three gauge coupling constants g 3 , g 2 and g 1 for SU(3) C , SU(2) L and U(1) Y gauge groups in the MSSM, respectively, are given at the GUT scale as which can be realized by the Froggatt-Nielsen mechanism [8] or quasi-localized matter fields in five-dimensional spacetime [30,31] yielding observed masses and mixings of quarks and charged leptons at the EW scale. Here ǫ = 0.225 represents the magnitude of mixing by Cabibbo angle. Note that these numerical values of the elements involving light generations are just for concreteness of the numerical evaluation, and these concrete values for the light generations are not essential for our results. The basic results are valid for the other ansatz of Yukawa matrices that generate observed quark and lepton masses and mixings as long as it coincides with a value of tan β 10 as will be selected in Eq. (9).
The parameters in soft supersymmetry breaking terms are selected as follows. The gluino mass is chosen as which is almost the lowest value satisfying the experimental lower bound at the EW scale. Note that the gaugino mass ratios r 1 and r 2 , namely the bino mass M 1 and the wino masses M 2 , are varied at the GUT scale with the above fixed gluino mass M 3 in our numerical analysis. The soft scalar masses and the scalar trilinear couplings at the GUT scale are fixed as Although we adopt vanishing off-diagonal elements of the soft scalar mass matrices for simplicity, these essentially do not affect the main results of this paper. We choose the following ratio: where v u = v sin β and v d = v cos β are the VEVs of up-and down-type Higgs fields, h u and h d , respectively, and v = 174 GeV. The smaller value of tan β makes the Higgs mass tend to be below 120 GeV in the MSSM. The larger value makes the hierarchy between v u and v d severer, that is unfavorable form the viewpoint of the naturalness. The µ-parameter in the MSSM superpotential (7) as well as the Bµ-parameter in the soft supersymmetry breaking terms (8) is determined at the GUT scale in such a way that the correct Z boson mass is realized at the EW scale with the above set of the other parameters. Note that the numerical values of these parameters changes as r 1 and r 2 vary.

B RGEs relevant to the Higgs mass
In this appendix, we show the one-loop RGEs relevant to our discussion for the mass of the lightest CP-even Higgs boson: where t = ln (E/M GUT ) is a logarithmic energy scale measured by M GUT = 2 × 10 16 GeV, and 1 is a 3 × 3 unit matrix. The complete set of the MSSM RGEs are found, e.g., in Ref. [9] at the two-loop level.