Flavor violating $Z'$ from $SO(10)$ SUSY GUT in High-Scale SUSY

We propose an $SO(10)$ supersymmetric grand unified theory (SUSY GUT), where the $SO(10)$ gauge symmetry breaks down to $SU(3)_c \times SU(2)_L \times U(1)_Y\times U(1)_{X}$ at the GUT scale and $U(1)_X$ is radiatively broken at the SUSY-braking scale. In order to achieve the observed Higgs mass around $126$ GeV and also to satisfy constraints on flavor- and/or CP-violating processes, we assume that the SUSY-breaking scale is $O(100)$ TeV, so that the $U(1)_X$ breaking scale is also $O(100)$ TeV. One big issue in the SO(10) GUTs is how to realize realistic Yukawa couplings. In our model, not only ${\bf 16}$-dimensional but also ${\bf 10}$-dimensional matter fields are introduced to predict the observed fermion masses and mixings. The Standard-Model quarks and leptons are linear combinations of the ${\bf 16}$- and ${\bf 10}$-dimensional fields so that the $U(1)_{X}$ gauge interaction may be flavor-violating. We investigate the current constraints on the flavor-violating $Z'$ interaction from the flavor physics and discuss prospects for future experiments.


I. INTRODUCTION
The Grand Unified Theories (GUTs) are longstanding hypotheses, and continue to fascinate us because of the excellent explanation of mysteries in the Standard Model (SM). The GUTs unify not only the gauge groups but also quarks and leptons, and reveal the origin of the structure of the SM, such as the hypercharge assignment for the SM particles.
The gauge groups in the SM are SU(3) c × SU(2) L × U(1) Y (≡ G SM ). The minimal candidate for the unified gauge group is SU (5), which was originally proposed by Georgi and Glashow [1]. There, quarks and leptons belong to 10-and 5-dimensional representations in SU (5), and the SM Higgs doublet is embedded into 5, introducing additional colored Higgs particle. One big issue is the unification of the SM gauge coupling constants, and it could be realized in the supersymmetric (SUSY) extension. It is well-known that the minimal SU(5) SUSY GUT realizes the gauge coupling unification around 2 × 10 16 GeV, if SUSY particle masses are around 1 TeV [2].
Another candidate for the unified gauge group would be SO (10). It is non-minimal, but it would be an attractive extension because the SO(10) GUT explains the anomalyfree conditions in the SM. Furthermore, all leptons and quarks, including the right-handed neutrinos, in one generation may belong to one 16-dimensional representation in the minimal setup [3].
On the other hand, the GUTs face several problems, especially because of the experimental constraints. One stringent constraint is from nonobservation of proton decay [1,4]. While the GUT scale in the SUSY GUT may be high enough to suppress the proton decay induced by the so-called X-boson exchange, the dimension-five operator generated by the colored Higgs exchange is severely constrained. Another stringent constraint is from the observed fermion masses and mixings. The SU(5) GUT predicts a common mass ratio of down-type quark and charged lepton in each generation. Furthermore, in the SO(10) GUT, the up-type, down-type quarks, and charged lepton in each generation would have common mass ratios if the all matter fields in one generation are embedded in one 16-dimensional representation. The predictions obviously conflict with the observation, and the modifications should be achieved by, for instance, higher-dimensional operators [5], additional Higgs fields [6] and additional matter fields [7].
In this letter, we propose an SO(10) SUSY GUT model, where the realistic fermion masses and mixings may be achieved by introducing extra 10-dimensional matter fields. The SM quarks and leptons come from 10-and 16-dimensional fields, and especially, the right-handed down-type quarks and left-handed leptons in the SM are given by the linear combinations of 10-and 16-dimensional fields. We assume that SO(10) gauge symmetry breaks down to G SM ×U(1) X around 10 16 GeV according to the nonzero vacuum expectation values (VEVs) of SO(10) adjoint fields. Thus, the low-energy effective theory is an U(1) X extension of the SUSY SM with extra matters. The additional gauge symmetry will survive up to the SUSY scale, but we could expect that it is radiatively broken, as the electroweak (EW) symmetry breaking in the minimal supersymmetry Standard Model (MSSM).
We assume that SUSY particles in the SUSY SM, except for gauginos, reside around 100 TeV, in order to realize the observed 126 GeV Higgs mass and also to satisfy constraints on flavor-and/or CP-violating processes. This type of setup is called the high-scale SUSY [8]. In the high-scale SUSY, the gauge coupling unification is rather improved when only the gaugino masses are around 1 TeV [9], and the dangerous dimension-five proton decay is suppressed [10]. On the other hand, since tan β (the ratio of the VEVs of the two Higgs doublets in the SUSY SM) is close to one, it is difficult to explain the large hierarchy between top and bottom quarks when all the matter fields are embedded into only 16 representational representations. In our model, the introduction of 10-representational matter fields makes it possible to explain the large hierarchy. In the high-scale SUSY, the UV theory of the SM need not be the MSSM. The U(1) X extension of the SUSY SM with extra matters is an alternative model, motivated by the SO(10) SUSY GUTs.
The mass of the Z ′ boson associated with the gauged U(1) X may be O(100) TeV so that it may be viable in the searches for flavor violations. The right-handed down-type quarks and left-handed leptons in the SM are given by linear combinations of the parts of 10-and 16-dimensional fields. Thus, that generically leads flavor-violating Z ′ interaction and crucial promises against flavor experiments. We will see that tree-level Flavor Changing Neutral Currents (FCNC) induced by the Z ′ boson are generated and they largely contribute to the flavor violation processes: for instance, µ → 3e, µ-e conversion in nuclei, and K 0 − K 0 and d/s mixings. This paper is organized as follows. We introduce our setup of the SO(10) SUSY GUT model in Sec. II. We see not only how to break SO(10), but also how to realize realistic fermion masses and mixings. The conventional seesaw mechanism, in which the Majorana masses for the right-handed neutrinos are much higher than the EW scale, could not work, since the U(1) X gauge symmetry forbids the Majorana masses. We show our solution according to the so-called inverted hierarchy [11] in the Sec. II A. The small parameters could be controlled with the global U(1) P Q symmetry there. In Sec. II B, we discuss the tree-level FCNCs corresponding to the realistic fermion masses and mixings. Sec. III is devoted to the flavor physics induced by the Z ′ interaction. Sec. IV is conclusion and discussion.

II. SETUP OF SO(10) SUSY GUT
The SO(10) gauge group has been considered to unify the three gauge symmetry in the SM. In the simple setup, the SM matter fields are also unified into 16-dimensional representation in the each generation, and the number of Yukawa couplings for the fermions masses is less than in the SM. When the SM Higgs field belongs to 10-dimensional field 10 H , the only Yukawa couplings are where i, j = 1, 2, 3 are defined. This assumption is too strict to explain the observed fermion masses and mixings, even if we include radiative corrections. The observed mass hierarchies are different in the up-type and down-type quarks, and the CKM mixing will be vanishing without other Yukawa couplings. Now, we introduce a 10-dimensional matter field in the each generation in addition to 16-dimensional matter fields. Three SO(10)-singlet matter fields S i are also introduced to achieve the realistic masses of neutrinos. The matter fields 10 i and 16 i are decomposed as the ones in Table I. For convenience, the assignment of SU(5) × U(1) X is also shown in Table I.
Let us show the superpotential relevant to the Yukawa couplings for the matter fields in our model; Here, the 16 and 16-dimensional Higgs fields 16 H and 16 H are introduced to break the U(1) X gauge symmetry in SO (10). We assume that the mass parameters µ BL , µ 10 and µ H are around SUSY scale (m SU SY ) and µ S is much smaller to realize the tiny neutrino masses. It may be important to pursue the origin of the mass scales. In Sec. II A, we show that the global U(1) P Q symmetry may control their mass scales.
We assume that two SO(10) adjoint Higgs fields, 45 H and 45 ′ H , develop nonzero VEVs so that the SO(10) gauge symmetry breaks down to G SM × U(1) X at the GUT scale [12]. The low-energy effective theory is the U(1) X extension of the SUSY SM with 10-and 16dimensional matter fields. The G SM -singlet fields charged under U(1) X , Φ and Φ, which are originated from 16 H and 16 H , should be included there. Φ and Φ would develop the nonzero VEVs as Φ = v Φ and Φ = v Φ around m SU SY , and the U(1) X symmetry is spontaneously broken. For simplicity, we assume that the other fields in 16 H and 16 H have masses at the GUT scale. If they stay at the low energy spectrum, the gauge coupling constants at the GUT scale is not perturbative.
The superpotential in the U(1) X extension of the SUSY SM is given as follows, The effective Yukawa couplings will be deviated from the ones in Eq. (2), because of the higher-order terms involving 45 H and 45 ′ H . * h u is Yukawa coupling for up-type quark including effect of higher-dimensional operators. ǫ d and ǫ e describe the size of higher-dimensional operators for the down-type quarks and charged leptons, which suppressed by 45 H /Λ and 45 ′ H /Λ. After the U(1) X symmetry breaking, the chiral superfieldsD c R i and D ′ c R i (L L i and L ′ L i ) mix each other, and we find the massless modes which correspond to the SM right-handed down-type quarks and left-handed leptons. g ij v Φ and µ 10 ij give the mass mixing between D c R i and D ′ c R i (L L i and L ′ L i ). Eventually, the relevant Yukawa couplings for quarks and * In general, the other parameters such as µ S and µ 10 would be effectively modified by the higherdimensional operators as well. We disregard these extra corrections to the parameters because they are not essential in this discussion.
charged leptons are described as where ψ denotes D c R and L L . ψ and ψ h are massless modes which correspond to the SM matters and the superheavy modes with masses O(m SU SY ), respectively. U ψ is the 6 × 6 unitary matrix, andÛ ψ ,Û ψ h ,Û ′ ψ andÛ ′ ψ h satisfy not only the unitarity condition for U ψ but also the following relation, Using the couplings in Eq. (3), the Yukawa coupling constants for the SM down-type quarks and charged leptons in Eq. (4) are described as In general, the up-type quark Yukawa coupling constants h u ij is given by v sin β (v cos β) is the VEV of the neutral component of H u (H d ) and m u i are the up-type quark masses. We define the diagonalizing matrices V CKM and V e R for (Y d ) ij and (Y e ) ij as below: where m d i and m e i are the down-type quark and the charged lepton masses. Note that we take the flavor basis that the right-handed down-type quarks and left-handed charged leptons are in the mass eigenstates. Then V CKM is the CKM matrix and V e R satisfies V e R = V CKM in the SU(5) limit. The size of higher-dimensional terms is depicted by ǫ d and ǫ e and expected to be small, compared to the third generation, h u 33 = m t /(v sin β). According to Eq. (8), (Û ψ ) ij could be described by the observables as, If ǫ d 11 v sin β is sufficiently smaller than m u , the (1, j) elements ofÛ D c R are too large to satisfy the unitary condition for U ψ . In order to achieve the consistency, the extra term ǫ d 11 v sin β should be larger than O(tan β(V CKM ) 13 m b ).
. The mass matrix for the neutral particles in the basis of (ν When we admit the large hierarchy between µ S and the other elements, the neutrino mass matrix (m ν ) is given by following Ref. [11].   One may wonder why µ S is so tiny and µ 10,BL,H are O(m SU SY ). We show one mechanism to explain the large mass hierarchy. In order to induce the dimensional parameters in Eq. (2) effectively, let us assign the global U(1) P Q symmetry to the matter and Higgs fields as in Table II. The global U(1) P Q symmetry, under which the SM fields are charged anomalously, has been proposed motivated by the strong CP problem [13]. We introduce SO(10)-singlet fields, P and T , whose U(1) P Q charges are fixed to allow the c P Q P 3 T term in the superpotential. Assuming canonical Käller potential and their soft SUSY breaking terms, the scale potential for P and T is derived from the superpotential as m 2 P and m 2 T are the soft SUSY breaking masses, and they could be estimated as m 2 SU SY . The mass squared would be driven to the negative value due to the radiative corrections, so that the negative mass squared leads the nonzero VEVs of P and T , and breaks U(1) P Q spontaneously. This leads a light scalar, so-called axion, corresponding to the Nambu-Goldstone boson. As discussed in Ref. [14], it is favorable that the U(1) P Q symmetry breaking scale is around 10 12 GeV, to explain the correct relic density of dark matter. That is, Λ should be almost the Planck scale (O(10 19 ) GeV), when m SU SY is O(100) TeV, for instance. On the other hand, the U(1) P Q charge assignment for the other chiral superfields forbids dimensional parameters like µ S and µ 10,BL,H . Using higher dimensional parameters, µ S and µ 10,BL,H are effectively generated after U(1) P Q breaking: ignoring the dimensionless couplings in front of the higher-order couplings. The above estimation tells that µ 10,BL,H = O(m SU SY ) and µ S = m SU SY × O(m SU SY /Λ). If m SU SY ≪ Λ is satisfied, very small µ S , compared to m SU SY , is predicted, and could realize the observed light neutrino masses, as we discussed above.

B. Flavor Violating Gauge Interaction
As we see above, the SM right-handed down-type quarks and left-handed leptons are given by the linear combinations of quarks and leptons in 10-and 16-dimensional matter fields, respectively. Since the fields in 10 and 16 representations carry different U(1) X charges, the SM fields may have flavor-dependent U(1) X interaction.
Let us see it more explicitly. The U(1) X gauge interactions of right-handed down-type quarks and left-handed leptons are described in the interaction basis as where the factors 3 and −2 are U(1) X charges for the fermionic componentsφ i and ϕ ′ i of the chiral superfieldsψ i and ψ ′ i . Z ′ is the U(1) X gauge boson and g X is defined as g X = g/ √ 40 at GUT scale, where g is the SO(10) gauge coupling constant. We have obtained the mass eigenstates for the fermions in Eqs. (5) and (8). Using the unitary matrix U ψ , we define the flavor-violating couplings A ϕ ij for the SM fermions as where ϕ is the fermion component of the chiral superfield ψ in the mass base and denotes right-handed down-type quark (d c R ) and left-handed lepton (l L ). Here we discuss the size of flavor violating couplings A ϕ ij . According to Eq. (11), (Û D c R ) ij and (Û L L ) ij are depicted by the observables in the SM. The flavor violating couplings A ϕ ij depend on the parameters, ǫ d and ǫ e . They are required to satisfy the unitary condition for U ψ , as discussed in Eqs. (11). In other words, they should be sizable in some elements, compared to h u ij = m u i /v cos δ ij , in order to break the GUT relation and to realize realistic mass matrices. Assuming ǫ d ij = ǫ i δ ij , at least ǫ 1 O(10 −5 ) is required to compensate for the small up quark mass.
Let us show one example to demonstrate the size of the flavor violating coupling A Setting the extra parameter to We find that all elements of the flavor violating couplings are O(1), so that we need careful analyses of their contributions to flavor physics, even if the Z ′ boson is quite heavy. Note that the alignment of A l L ij differs from the one of A d c R ij , because of the different mass spectrum between charged leptons and down-type quarks. In any case, however, the size of A l L ij would be also O(1), because of the small electron mass. The detail analysis on the relation between the FCNCs and the realistic mass spectrum will be given in Ref. [15]. In Sec. III, we introduce the flavor constraints relevant to our model and scan the current experimental bounds and future prospects in flavor physics.

C. Gauge Coupling Unification
Before phenomenology, let us briefly comment on the gauge coupling unification and the predicted Z ′ coupling (g X ). As well-known, the MSSM miraculously achieves the unification of the three SM gauge couplings, if at least gaugino masses are close to the EW scale. We assume the SUSY mass spectrum, where gauginos reside around the TeV-scale and the other SUSY particle masses are around 100 TeV. It is shown in Ref. [9] that the unification of the gauge coupling constants is improved compared with the MSSM with the SUSY particle masses O(1) TeV.
Once we determine the SO(10) gauge coupling at the GUT scale according to the gauge coupling unification, the U(1) X gauge coupling g X (µ) is derived with the renormalization group equation at the one-loop level as where α X = g 2 X /(4π) and α G = g 2 (Λ G )/(4π) are defined and Λ G is the unification scale. b X is fixed by the number of U(1) X -charged particles from µ to Λ G . In our scenario, right-handed neutrinos, additional three 10s of SO(10), and the U(1) X breaking Higgs fields as well as MSSM particles contribute to b X between m SU SY and Λ G , so that they lead b X = 426. At the scale µ = 100 TeV, g X is estimated as g X (100 TeV) = 0.073, where the GUT scale and the gauge coupling with m SU SY = 100 TeV are given by Note that the introduction of additional matter fields increases the gauge coupling constant at the GUT scale α G . Furthermore, heavier gaugino masses than the EW scale decrease the GUT scale Λ G . This means that the proton decay rate may be enhanced in our model [9,16], and could be tested at the future proton decay searches.

III. FLAVOR PHYSICS
As discussed in the subsection II B, the tree-level FCNCs involving the Z ′ boson may be promised in our model. The flavor changing couplings denoted by A ϕ ij could be O(1) in the all elements, as we see in Eq. (20). Here, we sketch the relevant constraints on the flavor-violating Z ′ interactions and give prospects for future experiments.
In our model, the SUSY SM Higgs doublets are charged under U(1) X , so that their nonzero VEVs contribute to the Z ′ mass (m Z ′ ) as well as the SM gauge bosons. The U(1) X charges of Higgs doublets are ±2 respectively, and then the mass mixing between Z and Z ′ is generated by the VEVs as well. The mixing angle between Z and Z ′ is approximately estimated as where g Z is the gauge coupling of Z boson and m Z is the Z boson mass. sin θ is about 3.4 × 10 −7 when Z ′ mass and coupling are fixed at m Z ′ = 100 TeV and g X = 0.073. Since the mixing is quite small as long as the Z ′ mass is O(100) TeV, we treat with Z and Z ′ as the fields in the mass basis and discuss the mixing effect up to O(θ 2 ). The gauge interactions of Z and Z ′ and SM fermions are given by where J µ SM is the SM weak neutral current, and J µ GUT is defined by The fermions in J µ GUT describe the fermionic components of the MSSM chiral superfields in the mass base denoted by the capital letters. The neutral current J µ GUT may significantly contribute to flavor violating processes: B 0 d/s -B 0 d/s and K 0 -K 0 mixings, flavor-violating decays, and µ-e conversion in nuclei. Below, we summarize the constraints relevant to the Z ′ interaction, and discuss the predictions in flavor physics. Note that we ignore contribution from SUSY flavor violating processes, because the sfermion masses are O(100) TeV.

A. Flavor Violating Decays of Leptons
First, let us discuss the contributions to flavor violating decays of leptons. There are two types of flavor violating decays in the presence of Z ′ FCNCs: one is three-body flavor violating decays l j → l i l k l k and the other is radiative flavor violating decays l j → l i γ. With the Z ′ FCNCs, the three-body flavor violating decays occur at the tree level, while the radiative flavor violating decays occur at the loop level. The radiative flavor violating decays have smaller rates by O(10 −3 ) than the tree-level decays. If flavor violating interactions stem from both left-and right-handed lepton (quark) sector, there might be a strong enhancement in radiative flavor violating decays via a chirality flip on an internal heavy fermion [17]. In our model, however, there exists no such an enhancement because only left-handed lepton (right-handed quark) have the flavor violating interactions. Hence we focus on the threebody flavor violating decays.
Let us discuss the µ → 3e process. The current upper bound on the branching ratio of µ → 3e is 1.0 × 10 −12 [18] and future experimental limit is expected to be 1.0 × 10 −16 [19]. In this model the branching ratio of µ → 3e is evaluated as follows, This is below the current experimental bound as long as m Z ′ is O(100) TeV. It is also important to emphasize that BR(µ → 3e) in our scenario has an additive structure in last bracket, and our prediction may yield to the stringent bound. If we assume m Z ′ = 100 TeV and A l L 11 = −2, the Mu3e experiment will cover A l L 12 0.1.
We also evaluate the branching ratios of other lepton flavor violating decays, and we find that they are also much below the current experimental upper bounds.

B. µ-e Conversion in Nuclei
The flavor violating coupling A l L 12 also gives rise to the µ-e conversion process. The SINDRUM-II experiment, which searched for the µ-e conversion signal with the Au target, gave the upper limit on the branching ratio: BR(µ − Au → e − Au) < 7 × 10 −13 [20]. The DeeMe [21] and the COMET-I [22] will launch soon and they aim to reach to O(10 −15 ) for the branching ratio with different targets. Furthermore, COMET-II and Mu2e [23] are planed to improve the sensitivity up to O(10 −17 ) † .
In our model, the branching ratio for the Au target is predicted as [24] BR(µ − Au → e − Au) = 2.2 × 10 −13 g X 0.073 which is close to the current upper bound at the SINDRUM-II. The branching ratio for the Al target, which is a candidate target of COMET, Mu2e, and PRISM experiments, is evaluated as The branching ratios for the other materials could be estimated as O(10 −13 ) as well, so that we expect that our model could be proved in the future experiments.

C. Neutral Meson Mixing
The Z ′ FCNCs contribute to the mass splitting and CP violation in neutral meson systems. The UTfit collaboration analyzes the experimentally allowed ranges for the effective † It is discussed that the sensitivity might be improved to O (10 −(18-19) ) in the PRISM experiment [22]. couplings of 4-Fermi interactions [25]. We obtain the limits on the Z ′ interaction as follows: The measurement of K 0 -K 0 oscillation is a strong probe on both real and imaginary part of (A d c R 12 ) 2 . Especially, the CP violation gives a sever constraint on the FCNC as we see in Eq. (30), so that the Z ′ mass has to be heavier than a few PeV, if A

IV. CONCLUSION AND DISCUSSION
We have proposed an SO(10) SUSY GUT, where the SO(10) gauge symmetry breaks down to SU(3) c ×SU(2) L ×U(1) Y ×U(1) X at the GUT scale and U(1) X is radiatively broken at the SUSY-breaking scale. In order to achieve the observed Higgs mass around 126 GeV and also to satisfy constraints on flavor-and/or CP-violating processes, we assume that the SUSY-breaking scale is O(100) TeV, so that the U(1) X breaking scale is also O(100) TeV. In order to realize realistic Yukawa couplings, not only 16-dimensional but also 10-dimensional matter fields are introduced. The SM quarks and leptons are linear combinations of the 16and 10-dimensional fields so that the U(1) X gauge interaction may be flavor violating. We investigate the current constraints on the flavor violating Z ′ interaction from the flavor physics and discuss prospects for future experiments. Our model could be tested in the flavor experiments, especially searches for the µ-e conversion processes, even if the Z ′ mass is O(100) TeV.
In this paper, we did not mention the GUT mass hierarchy problem such as the doublettriplet splitting problem. In fact, there is another mass hierarchy between the singlet of 16 H and the other components of 16 H in our model. The Z ′ mass is given by the VEV of the singlet, while other components reside around the GUT scale. We need more careful study on physics at the GUT scale to complete our discussion.