Flavor violating Z ′ from SO(10) SUSY GUT in High-Scale SUSY
aa r X i v : . [ h e p - ph ] M a r IPMU–15–0030
Flavor violating Z ′ from SO (10) SUSY GUT in High-Scale SUSY
Junji Hisano,
1, 2
Yu Muramatsu, Yuji Omura, and Masato Yamanaka Department of Physics, Nagoya University, Nagoya 464-8602, Japan Kavli IPMU (WPI), University of Tokyo, Kashiwa, Chiba 277-8583, Japan
Abstract
We propose an SO (10) supersymmetric grand unified theory (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 isradiatively broken at the SUSY-braking scale. In order to achieve the observed Higgs mass around126 GeV and also to satisfy constraints on flavor- and/or CP-violating processes, we assume thatthe 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, notonly -dimensional but also -dimensional matter fields are introduced to predict the observedfermion masses and mixings. The Standard-Model quarks and leptons are linear combinations ofthe - and -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 physicsand discuss prospects for future experiments. . INTRODUCTION The Grand Unified Theories (GUTs) are longstanding hypotheses, and continue to fasci-nate us because of the excellent explanation of mysteries in the Standard Model (SM). TheGUTs unify not only the gauge groups but also quarks and leptons, and reveal the origin ofthe 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 minimalcandidate for the unified gauge group is SU (5), which was originally proposed by Georgiand Glashow [1]. There, quarks and leptons belong to - and -dimensional representationsin SU (5), and the SM Higgs doublet is embedded into , introducing additional colored Higgsparticle. One big issue is the unification of the SM gauge coupling constants, and it could berealized in the supersymmetric (SUSY) extension. It is well-known that the minimal SU (5)SUSY GUT realizes the gauge coupling unification around 2 × GeV, if SUSY particlemasses 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 anomaly-free conditions in the SM. Furthermore, all leptons and quarks, including the right-handedneutrinos, in one generation may belong to one -dimensional representation in the minimalsetup [3].On the other hand, the GUTs face several problems, especially because of the experi-mental 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 decayinduced by the so-called X -boson exchange, the dimension-five operator generated by thecolored Higgs exchange is severely constrained. Another stringent constraint is from theobserved fermion masses and mixings. The SU (5) GUT predicts a common mass ratio ofdown-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 commonmass ratios if the all matter fields in one generation are embedded in one -dimensionalrepresentation. The predictions obviously conflict with the observation, and the modifica-tions should be achieved by, for instance, higher-dimensional operators [5], additional Higgsfields [6] and additional matter fields [7].In this letter, we propose an SO (10) SUSY GUT model, where the realistic fermionmasses and mixings may be achieved by introducing extra -dimensional matter fields.The SM quarks and leptons come from - and -dimensional fields, and especially, theright-handed down-type quarks and left-handed leptons in the SM are given by the linearcombinations of - and -dimensional fields. We assume that SO (10) gauge symmetrybreaks down to G SM × U (1) X around 10 GeV according to the nonzero vacuum expectationvalues (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 surviveup 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 around100 TeV, in order to realize the observed 126 GeV Higgs mass and also to satisfy constraintson 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 onlythe gaugino masses are around 1 TeV [9], and the dangerous dimension-five proton decayis suppressed [10]. On the other hand, since tan β (the ratio of the VEVs of the two Higgs2oublets in the SUSY SM) is close to one, it is difficult to explain the large hierarchy betweentop and bottom quarks when all the matter fields are embedded into only representationalrepresentations. In our model, the introduction of -representational matter fields makesit possible to explain the large hierarchy. In the high-scale SUSY, the UV theory of the SMneed not be the MSSM. The U (1) X extension of the SUSY SM with extra matters is analternative 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 thatit may be viable in the searches for flavor violations. The right-handed down-type quarksand left-handed leptons in the SM are given by linear combinations of the parts of - and -dimensional fields. Thus, that generically leads flavor-violating Z ′ interaction and crucialpromises against flavor experiments. We will see that tree-level Flavor Changing NeutralCurrents (FCNC) induced by the Z ′ boson are generated and they largely contribute to theflavor violation processes: for instance, µ → e , µ - e conversion in nuclei, and K − K and B d/s − B d/s mixings.This paper is organized as follows. We introduce our setup of the SO (10) SUSY GUTmodel in Sec. II. We see not only how to break SO (10), but also how to realize realisticfermion masses and mixings. The conventional seesaw mechanism, in which the Majoranamasses 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 ac-cording to the so-called inverted hierarchy [11] in the Sec. II A. The small parameters couldbe controlled with the global U (1) P Q symmetry there. In Sec. II B, we discuss the tree-levelFCNCs corresponding to the realistic fermion masses and mixings. Sec. III is devoted to theflavor 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 inthe SM. In the simple setup, the SM matter fields are also unified into -dimensionalrepresentation in the each generation, and the number of Yukawa couplings for the fermionsmasses is less than in the SM. When the SM Higgs field belongs to -dimensional field H , the only Yukawa couplings are W min = h ij i j H (1)where i, j = 1 , , -dimensional matter field in the each generation in addition to -dimensional matter fields. Three SO (10)-singlet matter fields S i are also introduced toachieve the realistic masses of neutrinos. The matter fields i and i are decomposedas the ones in Table I. For convenience, the assignment of SU (5) × U (1) X is also shown inTable I.Let us show the superpotential relevant to the Yukawa couplings for the matter fields inour model; W Y = h ij i j H + f ij i H S j + g ij i j H + µ BL H H + µ H H H + µ ij i j + µ S ij S i S j . (2)3 L U cR E cR ˆ L L ˆ D cR N cR SO (10) SU (5) × U (1) X ( , −
1) ( ¯5 ,
3) ( , − G SM ( , , ) ( ¯3 , , − ) ( , ,
1) ( , , − ) ( ¯3 , , ) ( , , L ′ L D ′ cR L ′ L D ′ cR SO (10) SU (5) × U (1) X ( ¯5 , −
2) ( , G SM ( , , − ) ( ¯3 , , ) ( , , ) ( , , − )TABLE I. Charge assignment for matter fields. Charge assignment for G SM is denoted as ( SU (3) c , SU (2) L , U (1) Y ). U (1) X gauge coupling constant is normalized as g X = g/ √
40 at GUT scale, where g is SO (10) gauge coupling constant. Here, the and -dimensional Higgs fields H and H are introduced to break the U (1) X gauge symmetry in SO (10). We assume that the mass parameters µ BL , µ and µ H are around SUSY scale ( m SUSY ) 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 theglobal U (1) P Q symmetry may control their mass scales.We assume that two SO (10) adjoint Higgs fields, H and ′ H , develop nonzero VEVsso 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 - and -dimensional matter fields. The G SM -singlet fields charged under U (1) X , Φ and Φ, which areoriginated from H and H , should be included there. Φ and Φ would develop the nonzeroVEVs as h Φ i = v Φ and h Φ i = v Φ around m SUSY , and the U (1) X symmetry is spontaneouslybroken. For simplicity, we assume that the other fields in H and H have masses at theGUT scale. If they stay at the low energy spectrum, the gauge coupling constants at theGUT scale is not perturbative.The superpotential in the U (1) X extension of the SUSY SM is given as follows, W effY = h u ij Q L i U cR j H u + ( h u ij + ǫ d ij ) Q L i ˆ D cR j H d + ( h u ij + ǫ e ij ) ˆ L L i E cR j H d + g ij Φ( D ′ cRi ˆ D cR j + L ′ Li ˆ L L j ) + µ ij ( D ′ cRi D ′ cR j + L ′ Li L ′ L j )+ h ij ˆ L L i N cR j H u + f ij Φ N cR i S j + µ S ij S i S j + µ BL ΦΦ + µ H H u H d . (3)The effective Yukawa couplings will be deviated from the ones in Eq. (2), because of thehigher-order terms involving H and ′ H . ∗ h u is Yukawa coupling for up-type quark includ-ing effect of higher-dimensional operators. ǫ d and ǫ e describe the size of higher-dimensionaloperators for the down-type quarks and charged leptons, which suppressed by h H i / Λ and h ′ H i / Λ.After the U (1) X symmetry breaking, the chiral superfields ˆ D cR i and D ′ cR i ( ˆ L L i and L ′ L i )mix each other, and we find the massless modes which correspond to the SM right-handeddown-type quarks and left-handed leptons. g ij v Φ and µ ij give the mass mixing betweenˆ D cR i and D ′ cR i ( ˆ 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 would be effectively modified by the higher-dimensional operators as well. We disregard these extra corrections to the parameters because theyare not essential in this discussion. W SSMY = h u ij Q L i U cR j H u + Y d ij Q L i D cR j H d + Y e ij L L i E cR j H d + e µ ij ( D cR hi D cR h j + L L hi L L h j ) . (4) D cR i , D cR h i , L L i and L L h i are the chiral superfields of right-handed down-type quarks andleft-handed leptons in the mass bases defined by (cid:18) ˆ ψψ ′ (cid:19) = U ψ (cid:18) ψψ h (cid:19) = (cid:18) ˆ U ψ ˆ U ψ h ˆ U ′ ψ ˆ U ′ ψ h (cid:19) (cid:18) ψψ h (cid:19) , (5)where ψ denotes D cR and L L . ψ and ψ h are massless modes which correspond to the SMmatters and the superheavy modes with masses O ( m SUSY ), respectively. U ψ is the 6 × U ψ , ˆ U ψ h , ˆ U ′ ψ and ˆ U ′ ψ h satisfy not only the unitarity condition for U ψ but also the following relation,0 = g ik v Φ ( ˆ U ψ ) kj + µ ik ( ˆ U ′ ψ ) kj , (6) e µ ij = g ik v Φ ( ˆ U ψ h ) kj + µ ik ( ˆ U ′ ψ h ) kj . (7)Using the couplings in Eq. (3), the Yukawa coupling constants for the SM down-type quarksand charged leptons in Eq. (4) are described as( Y d ) ij = ( h u ik + ǫ d ik )( ˆ U D cR ) kj , ( Y e ) ij = ( ˆ U TL L ) ik ( h u kj + ǫ e kj ) . (8)In general, the up-type quark Yukawa coupling constants h u ij is given by h u ij = m u i v sin β δ ij . (9) 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-typequark masses. We define the diagonalizing matrices V CKM and V e R for ( Y d ) ij and ( Y e ) ij asbelow: ( Y d ) ij = 1 v cos β ( V ∗ CKM ) ij m d j , ( Y Te ) ij = 1 v cos β ( V ∗ e R ) ij m e j , (10)where m d i and m e i are the down-type quark and the charged lepton masses. Note that wetake the flavor basis that the right-handed down-type quarks and left-handed charged leptonsare 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 = m t / ( v sin β ). According to Eq. (8), ( ˆ U ψ ) ij couldbe described by the observables as,( m u i δ ik + ǫ d ik v sin β ) ( ˆ U D cR ) kj = tan β ( V ∗ CKM ) ij m d j , (cid:0) m u i δ ik + ǫ Te ki v sin β (cid:1) ( ˆ U L L ) kj = tan β ( V ∗ e R ) ij m e j . (11)If ǫ d v sin β is sufficiently smaller than m u , the (1 , j ) elements of ˆ U D cR are too large tosatisfy the unitary condition for U ψ . In order to achieve the consistency, the extra term ǫ d v sin β should be larger than O (tan β ( V CKM ) m b ).5 . Neutrino Mass Let us briefly mention the neutrino sector in our model. W effY in Eq. (3) includesneutral particles after the EW symmetry breaking. They reside in the neutral compo-nents of SU (2) L doublets { ˆ L L i , L ′ L i , L ′ Li } and the singlets { N R i , S i } . Let us decom-pose ˆ L L i , L ′ L i and L ′ Li as the charged and neutral ones: ˆ L TL i = (ˆ ν L i , ˆ E L i ), L ′ TL i =( ν ′ L i , E ′ L i ) and L ′ LTi = ( ν ′ Li , E ′ Li ). The mass matrix for the neutral particles in thebasis of (ˆ ν L i , N cR i , ν ′ L i , ν ′ Li , S i ) is M ν = h ij v sin β g ij v Φ h ij v sin β f ij v Φ µ ij g ij v Φ µ ij f ij v Φ µ S ij . (12)When we admit the large hierarchy between µ S and the other elements, the neutrino massmatrix ( m ν ) is given by ( m ν ) ij = ( hf − µ S f − h ) ij (cid:18) v sin βv Φ (cid:19) , (13)following Ref. [11]. For instance, v Φ = O (100) TeV and v sin β = O (100) GeV lead O (1)-eVneutrino masses, if µ S is O (1) MeV and h and f are O (1). The masses of the other neutralelements are O ( m SUSY ), and the phenomenology has been well investigated in Ref. [11]. i H H H i S i P TSO (10)
16 10 16 16 10 1 1 1 U (1) P Q U (1) P Q symmetry.
One may wonder why µ S is so tiny and µ ,BL,H are O ( m SUSY ). We show one mechanismto explain the large mass hierarchy. In order to induce the dimensional parameters inEq. (2) effectively, let us assign the global U (1) P Q symmetry to the matter and Higgsfields as in Table II. The global U (1) P Q symmetry, under which the SM fields are chargedanomalously, 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 T termin the superpotential. Assuming canonical K¨aller potential and their soft SUSY breakingterms, the scale potential for P and T is derived from the superpotential as V P Q = (cid:12)(cid:12)(cid:12) c P Q Λ P (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) c P Q Λ P T (cid:12)(cid:12)(cid:12) + m P | P | + m T | T | . (14) m P and m T are the soft SUSY breaking masses, and they could be estimated as m SUSY .The mass squared would be driven to the negative value due to the radiative corrections, sothat the negative mass squared leads the nonzero VEVs of P and T , h T i ∼ h P i ∼ p Λ | m SUSY | , (15)6nd breaks U (1) P Q spontaneously. This leads a light scalar, so-called axion, correspondingto the Nambu-Goldstone boson. As discussed in Ref. [14], it is favorable that the U (1) P Q symmetry breaking scale is around 10 GeV, to explain the correct relic density of darkmatter. That is, Λ should be almost the Planck scale ( O (10 ) GeV), when m SUSY is O (100) TeV, for instance.On the other hand, the U (1) P Q charge assignment for the other chiral superfields forbidsdimensional parameters like µ S and µ ,BL,H . Using higher dimensional parameters, µ S and µ ,BL,H are effectively generated after U (1) P Q breaking: µ = h P ih T i Λ , µ BL = h P i Λ , µ H = h T i Λ , µ S = h P ih T i Λ , (16)ignoring the dimensionless couplings in front of the higher-order couplings. The aboveestimation tells that µ ,BL,H = O ( m SUSY ) and µ S = m SUSY × O ( m SUSY / Λ). If m SUSY ≪ Λis satisfied, very small µ S , compared to m SUSY , is predicted, and could realize the observedlight 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 givenby the linear combinations of quarks and leptons in - and -dimensional matter fields,respectively. Since the fields in and representations carry different U (1) X charges, theSM 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-typequarks and left-handed leptons are described in the interaction basis as L g = − ig X (3 ˆ ϕ i /Z ′ ˆ ϕ i − ϕ ′ i /Z ′ ϕ ′ i ) , (17)where the factors 3 and − U (1) X charges for the fermionic components ˆ ϕ i and ϕ ′ i of thechiral superfields ˆ ψ i and ψ ′ i . Z ′ is the U (1) X gauge boson and g X is defined as g X = g/ √ g is the SO (10) gauge coupling constant. We have obtained the masseigenstates for the fermions in Eqs. (5) and (8). Using the unitary matrix U ψ , we define theflavor-violating couplings A ϕij for the SM fermions as L g = − ig X ϕ i (cid:16)
5( ˆ U † ψ ˆ U ψ ) ij − δ ij (cid:17) /Z ′ ϕ j ≡ − ig X A ϕij ϕ i /Z ′ ϕ j , (18)where ϕ is the fermion component of the chiral superfield ψ in the mass base and denotesright-handed down-type quark ( d cR ) and left-handed lepton ( l L ).Here we discuss the size of flavor violating couplings A ϕij . According to Eq. (11), ( ˆ U D cR ) ij and ( ˆ U L L ) ij are depicted by the observables in the SM. The flavor violating couplings A ϕij depend on the parameters, ǫ d and ǫ e . They are required to satisfy the unitary conditionfor U ψ , as discussed in Eqs. (11). In other words, they should be sizable in some elements,compared to h uij = m u i /v cos δ ij , in order to break the GUT relation and to realize realisticmass matrices. Assuming ǫ d ij = ǫ i δ ij , at least ǫ & O (10 − ) is required to compensate forthe small up quark mass.Let us show one example to demonstrate the size of the flavor violating coupling A d cR ij .Assuming ǫ & O (10 − ) and ǫ = ǫ = 0, A d cR ij is approximately estimated as A d cR ij ≈ βm d i m d j | ǫ v sin β | ( V CKM ) i ( V ∗ CKM ) j − δ ij . (19)7etting the extra parameter to ǫ = 5 × − , A d cR ij is estimated as (cid:16) A d cR ij (cid:17) ≈ − . . . . . . . . − . . (20)We find that all elements of the flavor violating couplings are O (1), so that we need carefulanalyses 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 cR ij , because of the different massspectrum between charged leptons and down-type quarks. In any case, however, the sizeof A l L ij would be also O (1), because of the small electron mass. The detail analysis on therelation between the FCNCs and the realistic mass spectrum will be given in Ref. [15]. InSec. III, we introduce the flavor constraints relevant to our model and scan the currentexperimental bounds and future prospects in flavor physics. C. Gauge Coupling Unification
Before phenomenology, let us briefly comment on the gauge coupling unification and thepredicted Z ′ coupling ( g X ). As well-known, the MSSM miraculously achieves the unificationof the three SM gauge couplings, if at least gaugino masses are close to the EW scale. Weassume the SUSY mass spectrum, where gauginos reside around the TeV-scale and the otherSUSY particle masses are around 100 TeV. It is shown in Ref. [9] that the unification ofthe gauge coupling constants is improved compared with the MSSM with the SUSY particlemasses O (1) TeV.Once we determine the SO (10) gauge coupling at the GUT scale according to the gaugecoupling unification, the U (1) X gauge coupling g X ( µ ) is derived with the renormalizationgroup equation at the one-loop level as4 πα − X ( µ ) = 4 πα − G ×
40 + b X ln (cid:18) Λ G µ (cid:19) , (21)where α X = g X / (4 π ) and α G = g (Λ G ) / (4 π ) are defined and Λ G is the unification scale. b X isfixed by the number of U (1) X -charged particles from µ to Λ G . In our scenario, right-handedneutrinos, additional three s of SO (10), and the U (1) X breaking Higgs fields as well asMSSM particles contribute to b X between m SUSY and Λ G , so that they lead b X = 426. Atthe scale µ = 100 TeV, g X is estimated as g X (100 TeV) = 0 . , (22)where the GUT scale and the gauge coupling with m SUSY = 100 TeV are given byΛ G = 8 . × GeV , α G = 0 . . (23)Note that the introduction of additional matter fields increases the gauge coupling con-stant at the GUT scale α G . Furthermore, heavier gaugino masses than the EW scale decreasethe 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.8 II. FLAVOR PHYSICS
As discussed in the subsection II B, the tree-level FCNCs involving the Z ′ boson maybe promised in our model. The flavor changing couplings denoted by A ϕij could be O (1)in the all elements, as we see in Eq. (20). Here, we sketch the relevant constraints on theflavor-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 theirnonzero VEVs contribute to the Z ′ mass ( m Z ′ ) as well as the SM gauge bosons. The U (1) X charges of Higgs doublets are ± Z and Z ′ is generated by the VEVs as well. The mixing angle between Z and Z ′ is approximatelyestimated as sin θ ≃ g X g Z m Z m Z ′ , (24)where g Z is the gauge coupling of Z boson and m Z is the Z boson mass. sin θ is about3 . × − when Z ′ mass and coupling are fixed at m Z ′ = 100 TeV and g X = 0 . Z ′ mass is O (100) TeV, we treat with Z and Z ′ asthe fields in the mass basis and discuss the mixing effect up to O ( θ ).The gauge interactions of Z and Z ′ and SM fermions are given by L = − i ( g Z cos θJ µ SM + g X sin θJ µ GUT ) Z µ − i ( g X cos θJ µ GUT − g Z sin θJ µ SM ) Z ′ µ , (25)where J µ SM is the SM weak neutral current, and J µ GUT is defined by J µ GUT = A l L ij l Li γ µ l L j − A d cR ij d Ri γ µ d R j + e Ri γ µ e R i − q Li γ µ q L i + u Ri γ µ u R i . (26)The fermions in J µ GUT describe the fermionic components of the MSSM chiral superfields inthe mass base denoted by the capital letters. The neutral current J µ GUT may significantlycontribute to flavor violating processes: B d/s - B d/s and K - K mixings, flavor-violating de-cays, 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 contributionfrom 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 aretwo types of flavor violating decays in the presence of Z ′ FCNCs: one is three-body flavorviolating 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 theradiative flavor violating decays occur at the loop level. The radiative flavor violating decayshave smaller rates by O (10 − ) than the tree-level decays. If flavor violating interactions stemfrom both left- and right-handed lepton (quark) sector, there might be a strong enhancementin radiative flavor violating decays via a chirality flip on an internal heavy fermion [17]. Inour 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 three-body flavor violating decays. 9et us discuss the µ → e process. The current upper bound on the branching ratio of µ → e is 1 . × − [18] and future experimental limit is expected to be 1 . × − [19].In this model the branching ratio of µ → e is evaluated as follows,BR( µ → e ) = 1 . × − (cid:16) g X . (cid:17) (cid:18)
100 TeV m Z ′ (cid:19) (cid:12)(cid:12)(cid:12) A l L (cid:12)(cid:12)(cid:12) (cid:26) . (cid:12)(cid:12)(cid:12) − . A l L (cid:12)(cid:12)(cid:12) (cid:27) . (27)This is below the current experimental bound as long as m Z ′ is O (100) TeV. It is alsoimportant to emphasize that BR( µ → e ) in our scenario has an additive structure in lastbracket, and our prediction may yield to the stringent bound. If we assume m Z ′ = 100 TeVand A l L = −
2, the Mu3e experiment will cover (cid:12)(cid:12)(cid:12) A l L (cid:12)(cid:12)(cid:12) . . B. µ - e Conversion in Nuclei
The flavor violating coupling A l L also gives rise to the µ - e conversion process. TheSINDRUM-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) < × − [20]. TheDeeMe [21] and the COMET-I [22] will launch soon and they aim to reach to O (10 − )for the branching ratio with different targets. Furthermore, COMET-II and Mu2e [23] areplaned to improve the sensitivity up to O (10 − ) † .In our model, the branching ratio for the Au target is predicted as [24]BR( µ − Au → e − Au) = 2 . × − (cid:16) g X . (cid:17) (cid:18)
100 TeV m Z ′ (cid:19) (cid:16) A l L (cid:17) (cid:12)(cid:12)(cid:12) . A d cR (cid:12)(cid:12)(cid:12) , (28)which is close to the current upper bound at the SINDRUM-II. The branching ratio forthe Al target, which is a candidate target of COMET, Mu2e, and PRISM experiments, isevaluated asBR( µ − Al → e − Al) = 6 . × − (cid:16) g X . (cid:17) (cid:18)
100 TeV m Z ′ (cid:19) (cid:16) A l L (cid:17) (cid:12)(cid:12)(cid:12) . A d cR (cid:12)(cid:12)(cid:12) . (29)The branching ratios for the other materials could be estimated as O (10 − ) as well, so thatwe 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 sys-tems. 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]. Z ′ interaction as follows: − . × − < (cid:16) g X . (cid:17) (cid:18) m Z ′ (cid:19) Im[( A d cR ) ] < . × − , (30) (cid:16) g X . (cid:17) (cid:18) m Z ′ (cid:19) (cid:12)(cid:12)(cid:12) Re[( A d cR ) ] (cid:12)(cid:12)(cid:12) < . , (31) (cid:16) g X . (cid:17) (cid:18) m Z ′ (cid:19) (cid:12)(cid:12)(cid:12) ( A d cR ) (cid:12)(cid:12)(cid:12) < , (32) (cid:16) g X . (cid:17) (cid:18) m Z ′ (cid:19) (cid:12)(cid:12)(cid:12) ( A d cR ) (cid:12)(cid:12)(cid:12) < . × . (33)The measurement of K - K oscillation is a strong probe on both real and imaginary partof ( A d cR ) . Especially, the CP violation gives a sever constraint on the FCNC as we see inEq. (30), so that the Z ′ mass has to be heavier than a few PeV, if A d cR possesses O (1) CPphase. IV. CONCLUSION AND DISCUSSION
We have proposed an SO (10) SUSY GUT, where the SO (10) gauge symmetry breaksdown to SU (3) c × SU (2) L × U (1) Y × U (1) X at the GUT scale and U (1) X is radiatively brokenat the SUSY-breaking scale. In order to achieve the observed Higgs mass around 126 GeVand also to satisfy constraints on flavor- and/or CP-violating processes, we assume that theSUSY-breaking scale is O (100) TeV, so that the U (1) X breaking scale is also O (100) TeV. Inorder to realize realistic Yukawa couplings, not only -dimensional but also -dimensionalmatter fields are introduced. The SM quarks and leptons are linear combinations of the -and -dimensional fields so that the U (1) X gauge interaction may be flavor violating. Weinvestigate the current constraints on the flavor violating Z ′ interaction from the flavorphysics and discuss prospects for future experiments. Our model could be tested in theflavor experiments, especially searches for the µ - e conversion processes, even if the Z ′ massis O (100) TeV.In this paper, we did not mention the GUT mass hierarchy problem such as the doublet-triplet splitting problem. In fact, there is another mass hierarchy between the singlet of H and the other components of H in our model. The Z ′ mass is given by the VEV of thesinglet, while other components reside around the GUT scale. We need more careful studyon physics at the GUT scale to complete our discussion. ACKNOWLEDGMENTS
This work is supported by Grant-in-Aid for Scientific research from the Ministry of Educa-tion, Science, Sports, and Culture (MEXT), Japan, No. 24340047 (for J.H.), No. 23104011(for J.H. and Y.O.), and No. 25003345 (for M.Y.). The work of J.H. is also supported byWorld Premier International Research Center Initiative (WPI Initiative), MEXT, Japan.11
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