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U(1)_{B_1+B_2-2L_1} mediation for the natural SUSY and the anomalous muon g-2

We propose a U(1)^\prime mediated supersymmetry (SUSY) breaking, in which U(1)^\prime is identified with U(1)_{B_1+B_2-2L_1}. The U(1)_{B_1+B_2-2L_1} gauge symmetry, which is anomaly-free with the field contents of the minimal supersymmetric standard model, assigns \pm 1/3 charges to the first and second generations of the quarks, and \mp 2 to the first generation of the leptons. As a result, the first two generations of squarks acquire masses of about 7 TeV, and the first generation of the sleptons do those of 40 TeV, respectively, in the presence of one or three pairs of extra vector-like matter {{\bf 5},\bar{\bf 5}}. Non-observation on extra colored particles below 1 TeV at the large hadron collider, and also the flavor violations such as \mu^-\rightarrow e^-\gamma are explained. By virtue of such a gauge symmetry, proton stability can be protected. The other squarks and sleptons as well as the gauginos can obtain masses of order 10^{2-3} GeV through the conventional gravity or gauge mediated SUSY breaking mechanism. The relatively light smuon/sneutrino and the neutralino/chargino could be responsible for the (g-2)_\mu deviated from the standard model prediction. The stop mass of \sim 500 GeV relieves the fine-tuning problem in the Higgs sector. Two-loop effects by the relatively heavy sfermions can protect the smallness of the stop mass from the radiative correction by the heavy gluino (\gtrsim 1 TeV). Extra vector-like matter can enhance the radiative corrections to the Higgs mass up to 126 GeV, and induce the desired mixing among the chiral fermions after U(1)_{B_1+B_2-2L_1} breaking.


I. INTRODUCTION
Recently, ATLAS and CMS have announced the discovery of the standard model (SM) Higgs(-like) boson in the 125 − 126 GeV invariant mass range [1,2]. Thus, the SM seems to have been almost completely confirmed as the basic theory describing the nature. So far, any evidence beyond the SM including supersymmetry (SUSY) has not appeared yet at the large hadron collider (LHC). It implies that the theoretical puzzles raised in the SM such as the gauge hierarchy problem still remain unsolved [3].
In fact, 126 GeV is too large for the mass of Higgs boson in the minimal SUSY SM (MSSM), if we conservatively keep the original motivation of introducing SUSY at the EW scale. In the MSSM the observed 126 GeV Higgs mass requires a too heavy stop mass ( m t a few TeV) for enhancing the radiative Higgs mass [4], which compels the soft parameters to be finely tuned to match the Z boson mass at the minimum of the Higgs potential [3]. Thus, an excessively heavy stop mass spoils the status of SUSY as a solution of the fine-tuning problem associated with the SM Higgs boson mass.
For naturalness of the Higgs boson mass, thus, we need a relatively light stop. At the moment, fortunately the stop mass bound is much lower compared to those of the other squarks, m t 500 GeV, which provides just ∆m 2 h | top (71 GeV) 2 through one-loop radiative correction together with the top quark. As a result, we need more ingredients beyond the MSSM for raising the Higgs mass except the stop: for explaining the 126 GeV Higgs mass, ∆m 2 h | new ∼ (89 − 51 GeV) 2 for tanβ = 2 − 50 should be supplemented by extending the MSSM, if the stop mass is taken to be just around 500 GeV in order to minimize the fine-tuning in the Higgs sector, avoiding the experimental bound on it 1 (see, for instance, Refs. [6][7][8]).
The fact that any SUSY particles have not been observed yet at the LHC means that the masses of new colored particles absent in the SM would be quite heavier than 1 TeV. Here we should note that such a bound is still applied only to the first and second generations of squarks and to the gluino: the constraint on the third generation is just around 500 GeV, as mentioned above. Fortunately, such heavy squark masses do not affect much the Higgs mass because of their small Yukawa couplings. Moreover, heavy masses of the first and second generations of squarks are helpful for avoiding the flavor problem expected in the gravity mediation scenario. If we insist on the stop mass being ∼ 500 GeV at the electroweak (EW) scale, however, gluino heavier than 1 TeV at the EW scale would drive the squared mass of the stop negative at a higher energy scale through the renormalization group (RG) effects. It means that the stop becomes much heavier than 500 GeV at low energies, if we take it positive at high energy scales. To avoid color breaking at a higher energy scale, and keep the small stop mass at the EW scale, we need to somehow compensate the heavy gluino effect in the RG equation.
Brookhaven National Laboratory (BNL) reported a remarkable result on (g − 2) of the muon (µ) [9] which is deviated from the SM prediction [10,11] by 3.3σ − 3.6σ, Even though the statistical significance is not strong enough in addition to the theoretical uncertainty on the hadronic effects, still this discrepancy may hint new physics beyond the SM. If it originates from SUSY, relatively light smuon/sneutrino and neutralino/chargino are needed to give this order of (g − 2) µ [12,13]. In contrast to the squarks, the mass bounds on sleptons at the LHC are less severe [14]. As pointed out in Ref. [7], moreover, the extra vector-like leptons lighter than ∼ 500 GeV is very helpful for explaining 126 GeV Higgs mass with m t ∼ 500 GeV but without assuming a large mixing between the left-and the right-handed stops. Their order-one Yukawa coupling to the MSSM Higgs boson can enhance the radiative Higgs mass. The Landau-pole problem associated with the order-one Yukawa coupling can be avoided by introducing an extra gauge symmetry, under which only the new vector-like leptons are charged.
In view of the recent experimental data at the LHC, the "effective SUSY" (or "more minimal SUSY") [15,16] and "split SUSY" [17] scenarios look promising. According to the effective SUSY, the first two generations of superpartners are required to be about 5 − 20 TeV in order to avoid the SUSY flavor and SUSY CP problems, while the third ones and gauginos can be lighter than 1 TeV. However, the masses heavier than 22 TeV for the first two generations drive the stop mass squared negative at the EW scale through the two-loop RG effects, if the third ones are lighter than 4 TeV, and if such soft parameters are generated around the grand unification (GUT) scale (≈ 2 × 10 16 GeV) [18]. We note here that such a two-loop RG effect by heavy superpartners on lighter superpartners can be utilized to keep the stop mass squared positive at higher energy scales, when the gluino is also heavy enough ( 1 TeV). Namely, it can compensate the heavy gluino effect on the stop mass we mentioned above, keeping the light stop mass at low energy. In this Letter, we will discuss this possibility.
On the other hand, in the split SUSY all the superpartners of the SM chiral fermions are assumed to be heavy while the superpartners of bosonic particles in the SM, i.e. gauginos and Higgsinos remain relatively light. One such idea realizing the split SUSY is to introduce the so-called Z ′ mediated SUSY breaking [19] or U(1) ′ mediation, in which a U(1) ′ gauge sector plays the role of the messenger sector for SUSY breaking in the visible sector. In U(1) ′ mediation, scalar components of chiral superfields charged under U(1) ′ acquire quite heavy masses of order 100 TeV at one-loop level, while the MSSM gauginos get masses of 10 2−3 GeV at two-loop level. Moreover, by employing a family dependent U(1) ′ charge assignment, one can achieve a hierarchical sfermion spectrum. Thus, e.g. the third family of squarks can be made the lightest [20] or heaviest family [21].
With the original forms of the effective SUSY and split SUSY, however, the discrepancy of (g − 2) µ observed at BNL from the MSSM prediction cannot be accommodated. In this Letter, we attempt to obtain a more desirable spectrum on the superparticles by employing a proper U(1) ′ mediation such that the (g − 2) µ observed at BNL is explained in the SUSY framework. To be consistent with the LHC results, the first two generations of squarks should be made quite heavy while the second generation of sleptons and the stop need to remain light to explain (g − 2) µ and to avoid the fine-tuning problem. Indeed, such a hierarchical spectrum of the superparticles is hard to obtain in the conventional SUSY breaking scenarios. In order to obtain such a spectrum, we need to find a desirable U(1) ′ gauge symmetry useful for the U(1) ′ mediation.
It is well-known that with the MSSM chiral matter contents, U(1) B and U(1) L are anomalous for each generation. However, their proper combinations could cancel all the gauge anomalies. For instance, U(1) B−L is anomaly-free for each generation. U(1) B i −B j and U(1) L i −L j (i = j) are also anomaly-free, where i and j denote generation numbers. Of course, all the linear combinations of U(1) B−L , U(1) B i −B j and U(1) L i −L j are also anomaly-free. Hence, U(1) B i −L j (i = j) can be an anomaly-free gauge symmetry. Indeed, U(1) ±(B 1 +B 2 −L 1 −L 3 )+B 3 −L 3 was adopted for U(1) ′ mediation [21] where the smuon/sneutrino can be light enough for an explanation of (g − 2) µ .
In this Letter, we propose a U(1) ′ mediation with which is a linear combination of U(1) B 2 −B 1 and U(1) B 1 −L 1 . By virtue of this gauge symmetry, proton stability can be protected: U(1) B 1 +B 2 −2L 1 disallows all the dimension 4 and 5 baryon number violating operators. 2 Hence, the first two generations of squarks and the selectron are expected to be quite heavy through the U(1) ′ mediation mechanism since they carry nonzero charges of U(1) B 1 +B 2 −2L 1 . On the other hand, the stops and smuons/sneutrino remain relatively light. Their masses could be obtained through the ordinary gravity or gauge mediation mechanism. The paper is organized as follows. In Section II, we introduce the U(1) B 1 +B 2 −2L 1 mediated SUSY breaking, and attempt to obtain the low energy spectrum of superparticles. We also discuss (g − 2) µ . In Section III, we propose a concrete U(1) ′ mediation model. Section IV is a conclusion.
is also a quite interesting gauge symmetry, in which the stop and the smuon/sneutrino can still be light. However, we note that the extra U(1) gauge symmetry allows a heavy right-handed neutrino mass term in the superpotential only for one right-handed neutrino, unless they are broken at a high energy scale. For a successful seesaw mechanism, we intend to cheaply obtain heavy mass terms for at least two right-handed neutrinos, which should be neutral under the extra U(1). It is another motivation to study U(1) B1+B2−2L1 .

II. SUPERPARTICLE SPECTRUM AND (g − 2) µ
We suppose that the visible and the hidden sectors can communicate through a U(1) gauge interaction. The SUSY breaking in the hidden sector, which can be parametrized with a spurion field X = M + θ 2 F , is assumed to generate a mass of the gaugino of U(1) ′ (≡ MZ′) at the scale Λ S [19]: which plays the role of the order parameter of the SUSY breaking effects in the visible sector. The SUSY breaking in the U(1) ′ gauge sector can be transferred to the visible sector [19], inducing the soft masses for the scalar components of the superfields carrying U(1) ′ charges: where Q f stands for the U(1) ′ gauge charge. We note here that it can be designed to solve the flavor changing neutral coupling (FCNC) problem in the gravity mediation scenario. By assigning the same U(1) ′ charges to the families needed to avoid the FCNC problem, we can get the degenerate and diagonal form of mass matrices for those families of superpartners. If m 2 f dominates the soft mass squared over the one induced by gravity mediation, it does not have to be much heavier than 1 TeV to avoid the FCNC. Unlike the original U(1) ′ mediation of Ref. [19], thus, we assume that the soft scalar masses generated through this mechanism are heavy enough just to avoid the experimental bounds on superparticles at the LHC. For Eq. (4) we suppose that MZ′ is of order 10 4 GeV and Λ S ∼ 10 16 GeV. Hence, for instance, M would be of order 10 14 GeV if F ∼ (10 10 GeV) 2 , and M ∼ 10 10 GeV if F ∼ (10 8 GeV) 2 .
As seen in Eq. (4), the RG running of m 2 f between the two energy scales Λ S and MZ′ is dominated by the U(1) ′ gaugino. Below the MZ′ scale, however, the U(1) ′ gaugino is decoupled, and so the RG running of m 2 f is governed only by the RG equations of the relevant MSSM superfields and the U(1) ′ gauge field. Accordingly, below the MZ′ scale the change of m 2 f would become much smaller. The mass splitting between the bosonic and fermionic modes in the chiral matter sector as well as in the U(1) ′ gauge sector generate also the MSSM gaugino masses at two-loop level [19]. Hence, the MSSM gaugino masses are much suppressed compared to the soft masses of the scalars with U(1) ′ charges: where g k (k = 3, 2, 1) denotes the MSSM gauge couplings. These gaugino masses are too light, and so they are dominated by other mediation mechanisms of SUSY breaking so that they are of order 10 2−3 GeV. We will neglect the contributions by Eq. (5) to the MSSM gaugino masses. Superfields (H) } are MSSM singlets. The scalar components of them acquire heavy masses of 7 − 40 TeV depending on their charges (and the full matter contents) through the U(1) B 1 +B 2 −2L 1 mediated SUSY breaking mechanism.
We identify U(1) ′ with U(1) B 1 +B 2 −2L 1 , which is anomaly-free. The charge assignments of U(1) B 1 +B 2 −2L 1 are presented in Table I The masses of 5−10 TeV for the first two squark families and 500 GeV for the third squark family satisfy the LHC mass bounds on the extra colored particles. Such a hierarchical mass spectrum is non-trivial to realize in the constrained MSSM (CMSSM). In our U(1) ′ mediation, a serious fine-tuning problem in the Higgs sector can be avoided due to the relatively light stop. We will briefly discuss how to explain 126 GeV Higgs mass later. The relatively light smuon/sneutrino and neutralino/chargino admit the possibility that (g − 2) µ deviated from the SM prediction is supported from SUSY, as will be discussed later.
The SUSY FCNC problems are associated with the ratios of the off-diagonal to the diagonal components in the soft mass matrices. In the U(1) ′ mediation, where the U(1) ′ charges are given by B 1 + B 2 − 2L 1 , SUSY breaking effects generates degenerate heavy soft masses (5 − 10 TeV) in the diagonal (11) and (22) components in the squark mass matrix. Since the masses induced by the U(1) ′ mediation dominate gravity mediation for the soft masses, the FCNC in the quark sector can be sufficiently suppressed. Also, the U(1) ′ mediation induces very heavy masses (∼ 40 TeV) to the first family of sleptons. Note that the U(1) B 1 +B 2 −2L 1 symmetry forbids the off-diagonal elements M 2 lepton (1i) (i = 2, 3) in the slepton mass matrix. If U(1) B 1 +B 2 −2L 1 survives down to low energies, thus, the FCNC problem in the lepton sector, e.g. such as µ − → e − γ and τ − → e − γ, could not be severe. 3 In supergravity models, the predicted mass relation for the MSSM gauginos is where Q denotes the renormalization energy scale, m 1/2 and g U mean the gaugino mass and gauge coupling at the GUT scale. In Eq. (6), the MSSM gauge couplings are given by where t − t 0 ≡ log(Q/M GUT ), and b k (k = 3, 2, 1) denotes the beta function coefficients of the gauge couplings for SU (3)  In the next section, we will propose a concrete model with v = 3.
We require (500 GeV) 2 m 2 t ≪ (1 TeV) 2 , which can be obtained from m 2 0 (100 GeV) 2 at the GUT scale in the CMSSM (v = 0), if m 1/2 ≈ 300 GeV. In this case, the slepton masses of the second generation become much lighter than 500 GeV. In the case m 1/2 1 TeV, however, the stop mass squared rapidly decreases with energy, and becomes negative at some high energy scale. It means that the stop mass should be much heavier than 500 GeV, if it is taken to be positive at the GUT scale. We will see that such a heavy gluino effect on the RG equation could be compensated with the two-loop effects by very heavy superpartners, maintaining the positive squared mass of the stop all the way up to the GUT scale.
The relatively light smuon/sneutrino (the second generation of the sleptons) and neutralino/chargino could be responsible for the deviation of the muon g − 2 from the SM prediction [12,[21][22][23]. In the MSSM, the muon g − 2 is contributed by the sneutrino/chargino loops and the smuon/neutralino loops. For a large µ, i.e. µ 2 ≫ m 2 µL,R , the contribution by the smuon/neutralino loops dominates that by the the sneutrino/chargino loops [12,21]. If M 2 , µ ∼ 500 GeV, and the sleptons' masses are quite smaller than this value, i.e. ≪ 500 GeV, however, the sneutrino/chargino loop becomes dominant [12,23]: where m µ is the muon mass. The loop function I + (x, y) is defined as where x ≡ M 2 2 / m 2 µL and y ≡ µ 2 / m 2 µL . Here, we assume m 2 µL , m 2 νL ≫ M 2 W and set m 2 µL ≈ m 2 νL . For 2 < x, y < 5, I + (x, y) is in the range of 1.0 − 1.1. Thus, ∆(g − 2) µ ≈ 2.6 × 10 −9 can be achieved for a large tanβ [ 20] and a small M 2 µ [ (400 GeV) 2 ]. Since an excessively large µ could make the fine-tuning in the Higgs sector more serious, we take the case that the sneutrino/chargino loop is dominant. Now let us explore the conditions for m 2 t ≈ (500 GeV) 2 and m 2 µ ≈ (400 GeV) 2 at the EW scale. First, we consider the RG behavior of the heavier soft mass squareds, namely, the mass squareds of the first two generations of squarks and the first generation of sleptons, which are charged under U(1) B 1 +B 2 −2L 1 . The RG evolution of such heavy soft mass squareds are mainly governed by the U(1) B 1 +B 2 −2L 1 gaugino, because it is much heavier (∼ 10 4 GeV) than other superpartners: Here, g 2 Z ′ (t) and MZ′(t) are given by g 2 Z ′ 0 and MZ′ 0 indicate the boundary values of g 2 Z ′ (t) and MZ′(t) at the GUT scale, and b Z ′ is the beta function coefficient of the U(1) B 1 +B 2 −2L 1 gauge coupling. Eq. (10) is integrable. The analytic solution of Eq. (10) is For a large b Z ′ , g 2 Z ′ (t) rapidly drops down, and so m 2 f (t) quickly approaches a constant (= Q 2 f m 2 0 ) as in the case considered in Ref. [18]. In our case, b Z ′ = 314 9 by the charge assignment in Table I. Thus, we will ignore the second term of Eq. (11).
Next, we discuss the RG behavior of the lighter soft mass squareds, namely, the third generation of squarks, and the second (and third) generation(s) of sleptons. We require M 2 ≈ 400 GeV at the EW scale for explaining (g − 2) µ . Accordingly, the gluino mass should be 1.2 TeV at low energy, as discussed above. Since the mass of gluino [M 3 (t) = m 1/2 g 2 3 (t)/g 2 U ] as well as the masses of the first two generations of squarks [ m 2 where the terms with Q 2 q,l correspond to the two-loop effects by the heavy superpartners [18]. squarks [sleptons]. Thus, the two generations of (s)quarks and one generation of (s)leptons make contributions to the two-loop effects in Eq. (12). Here, we inserted the expression m 2 f (µ) ≈ Q 2 f m 2 0 in Eq. (11). The above approximations are possible because m 2 0 , M 2 3 ≫ m 2 t , and g 2 3 is of order unity at low energies. Using Eqs. (6) and (7), the above equations are integrable. Requiring m 2 t ≈ (500 GeV) 2 m 2 µ ≈ (400 GeV) 2 , and M 3 ≈ 1.2 TeV at TeV scale, thus, we have Here m 2 0 indicates a common boundary value of m 2 t (t) and m 2 µ (t) at the GUT scale, which could be generated e.g. through the gravity mediated SUSY breaking mechanism. As discussed above, the boundary value of the gluino mass, m 1/2 is given by 631 GeV for v = 1 and 1. So far we have ignored the Yukawa couplings. They might be important around the GUT scale, since the m 2 0 terms in Eq. (12) or m 2 f (t) in Eq. (11) vanish at the boundary. However, even the top quark Yukawa coupling decreases with energy, eventually down to about 0.5 (0.2) for tanβ = 20 and v = 1 (v = 3) at the GUT scale. Thus, the m 2 0 (m 2 0 ) turns out to be just a bit larger (smaller) than the above estimation, when the Yukawa couplings are also considered in the RG equations.
[ m 2 0 , m 2 0 ] chosen in the above might determine also the soft mass squareds of the Higgs, m 2 hu and m 2 h d , and so affect the Z boson mass at the minimum of the Higgs scalar potential. However, a mechanism introduced in the next section for explaining 126 GeV Higgs mass (with vector-like leptons {L,L c ;N,N c } [7]) are also involved there. Another free parameter, µ should also be considered. Moreover, one can take other boundary boundary values different from m 2 0 for m 2 hu (t) and m 2 h d (t). In this Letter, we don't discuss their RG behaviors, and the EW symmetry breaking in details.

III. THE MODEL
Unlike the first and second generations of (s)quarks and the first generation of (s)leptons, the other MSSM superfields do not carry U(1) B 1 +B 2 −2L 1 gauge charges. Hence the mixing between the first (last) two and the third (first) generations in the quark (lepton) sector is impossible, if U(1) B 1 +B 2 −2L 1 remains unbroken. In this section, we will describe how the desired mixing in the quark and lepton sectors can be induced.
We introduce a global U(1) PQ symmetry. The global charges for the superfields neutral under U(1) B 1 +B 2 −2L 1 are listed in Table II. The renormalizable superpotential consistent with the gauged U(1) B 1 +B 2 −2L 1 and the global U(1) PQ symmetries is where the "y"s denote the dimensionless Yukawa coupling constants, and "M ij " (= M ji ) means the Majorana masses for the two right-handed neutrinos neutral under U(1) B 1 +B 2 −2L 1 . The Majorana mass terms of the right-handed neutrinos and the MSSM µ term break U(1) PQ explicitly. We will explain later how they can be generated with the desired sizes. Note that there are no Dirac and Majorana mass terms for ν c 1 . Only with the two heavy right-handed neutrinos, however, the seesaw mechanism and leptogenesis (through the CP violating phase of heavy neutrinos) are still possible [24].
In order to break U(1)  Tables I and II. Then, the following superpotential is also allowed:  14) can be replaced by the nonrenormalizable terms with P 2 /M P or Q 2 /M P [25], which respect the U(1) PQ symmetry (for a recent discussion, see Ref. [26]). Here M P denotes the reduced Planck mass (≈ 2.4 × 10 18 GeV). The superfields P and Q get VEVs of order 10 10 GeV, breaking the U(1) PQ into the discrete Z 2 symmetry, e.g. through the superpotential W ⊃ κΣ(P Q − M 2 I ) [27], where Σ is a superfield, κ a dimensionless coupling, and M I indicates a mass parameter of order 10 10 GeV. 4 Thus, P 2 /M P and Q 2 /M P can be of order 10 2−3 GeV. The remaining Z 2 symmetry is identified with the matter parity in the MSSM, which forbids the R-parity violating terms. The Majorana masses M ij in Eq. (14) also can be replaced by P .
The first and the other generations of sleptons are not mixed even in the mass eigen basis, because their masses are quite hierarchical [(40 TeV) 2 -(O(0.1) TeV) 2 ] and the U(1) B 1 +B 2 −2L 1 breaking scale is low enough in this model. Moreover, the mass matrix of the charged leptons also remains block-diagonal, as mentioned above. Accordingly, µ − → e − γ cannot arise through mediation by superparticles. The relatively light (ν µL ,μ L ) andν µR can be responsible for the deviation of (g − 2) µ , as discussed above. Large mixing among the left-handed neutrinos doesn't affect it.
Since the charge assignment of U(1) B 1 +B 2 −2L 1 is not universal, the U(1) B 1 +B 2 −2L 1 gauge boson could give rise to lepton flavor violations. As seen in Eq. (11), however, the gauge coupling g Z ′ is quite small at low energy, because U(1) B 1 +B 2 −2L 1 survives down to low energy. Other FCNC effects by such a gauge boson can also be adequately suppressed, if the U(1) B 1 +B 2 −2L 1 breaking scale is above a few TeV [8].
In the squark mass matrix, the diagonal components, (1, 1) and (2, 2) are degenerate with a squared mass of (7 TeV) 2 , while the off-diagonal components, (j, 3) and (3, j), where j = 1, 2, remain zero due to U(1) B 1 +B 2 −2L 1 . The (1, 2), (2, 1), and (3, 3) can be filled dominantly by the gravity mediation effect, which are quite suppressed compared to the (1, 1) and (2, 2) components. After diagonalization in the fermionic quarks sector, (1, 2), (2, 1), and (j, 3), (3, j) can be induced also by the mixing effect. The (1, 2) and (2, 1) components affect e.g. K-K mixing. The amplitude of K-K mixing by the squark mixing is roughly estimated as [28] where m 2 q ≈ (7 TeV) 2 , and ∆ m 2 q denotes the off-diagonal component of the squark mass matrix. Since the SM still explains well the observed data, Eq. (16) should be smaller than the SM prediction, M SM KK ≈ α 2 2 sin 2 θ c cos 2 θ c (m 2 c /M 4 W ), where θ c stands for the Cabibbo mixing angle. The condition Note that when m q = 500 GeV, this estimation on ∆ m 2 q / m 2 q (≈ δ d 12 ) provides slightly stronger constraint (≪ 4 × 10 −3 ) than those of Ref. [29]. In order to suppress the squark mixing effect, hence, the off-diagonal element (1,2) of the squark mass matrix should be much smaller than 5.6 percent of the diagonal one [i.e. ≪ (1.6 TeV) 2 ] when m q = 7 TeV. If the mixings among the d-type quarks are given fully by the CKM (or a similar order mixing matrix) and the elements induced by the gravity mediation is smaller than (1 TeV) 2 , this constraint can be satisfied.
Finally, let us discuss the Higgs sector. The simplest way to raise the Higgs mass in the framework of the MSSM is to consider the large A-term [3,30], assuming (A t −µcotβ) 2 / m t ≈ 6, where A t indicates the A-term coefficient corresponding to the top quark Yukawa coupling. However, the relation (A t − µcotβ) 2 / m 2 t ≈ 6 would be a fine-tuning relation. Even in this case, we need one pair of {5, 5} (i.e. v = 1) to induce the full mixing among the three generations of the SM chiral fermions as discussed above.
Actually, the Higgs mass could easily be raised at tree level by promoting the MSSM µ term to the renormalizable superpotential λSh u h d ,à la the next to MSSM, introducing a new singlet S together with a new dimensionless coupling λ. In fact, this approach was taken in the original suggestion of the U(1) ′ mediation [19]. For maintaining the perturbativity of the model all the way up to the GUT scale, however, λ and tanβ should be small enough: 0.6 λ 0.7 and 1 tanβ 3 [31]. As seen in the above, however, a large tanβ is needed to explain the central value of the observed (g − 2) µ . 5 In this Letter, we will consider another method to raise the Higgs mass. As pointed out in Ref.  Tables I and II are regarded as being neutral. 6 Due to the new gauge symmetry, we can take a relatively large Yukawa couplings between the new leptons and the Higgs field, avoiding a blowup of the Yukawa coupling below the GUT scale. Of course one can identify the new gauge symmetry with U(1) B 1 +B 2 −2L 1 . In this case, however, the charges of the new vectorlike leptons should be small enough, say ± 1 40 (rather than ∓2) so that their soft masses result in about 500 GeV. Alternatively, one can introduce a new gauge symmetry such as SU (2), which is not related to a mediation mechanism. Then, {L,L c ;N ,N c } are regarded as the SU(2) doublets. Here, we comment on the latter case.
The superpotential of the Higgs sector with such fields is given by where y (′) h andμ L,N,H (μ L μ N ) are dimensionless and dimensionful parameters. {N H ,N c H } are also SU(2) doublets, but neutral under the SM gauge symmetry. They are the spontaneous breaking sector of SU(2): their negative soft mass squareds are assumed, breaking SU(2) above TeV scale. For the gauge coupling unification, two pairs of {D ′ , D c′ } need to be accompanied with their relatively heavy mass terms ( a few TeV), even if we do not 5 The error is still 3.3 σ − 3.6 σ, and it is premature to exclude all the NMSSM models, not accommodating the BNL (g − 2) µ . 6 In principle, {L, L c } can be identified with {L,L c }. small stop and smuon/sneutrino masses against the quantum correction by the heavy gluino (≈ 1.2 TeV). By introducing extra vector-like matter, the radiative corrections to the Higgs mass can be enhanced up to 126 GeV, and the desired mixings among the SM chiral fermions can be generated after U(1) B 1 +B 2 −2L 1 breaking.