# Leptonic CP and Flavor Violations in SUSY GUT with Right-handed Neutrinos

FFebruary, 2021

Leptonic CP and Flavor Violationsin SUSY GUT with Right-handed Neutrinos

Kaigo Hirao and Takeo Moroi

Department of Physics, University of Tokyo, Tokyo 113-0033, Japan

Abstract

We study leptonic CP and ﬂavor violations in supersymmetric (SUSY) grand uniﬁedtheory (GUT) with right handed neutrinos, paying attention to the renormalizationgroup eﬀects on the slepton mass matrices due to the neutrino and GUT Yukawainteractions. In particular, we study in detail the impacts of the so-called Casas-Ibarra parameters on CP and ﬂavor violating observables. The renormalization groupeﬀects induce CP and ﬂavor violating elements of the SUSY breaking scalar masssquared matrices, which may result in sizable leptonic CP and ﬂavor violating signals.Assuming seesaw formula for the active neutrino masses, the renormalization groupeﬀects have been often thought to be negligible as the right-handed neutrino massesbecome small. With the most general form of the neutrino Yukawa matrix, i.e., takinginto account the Casas-Ibarra parameters, however, this is not the case. We foundthat the maximal possible sizes of signals of leptonic CP and ﬂavor violating processesare found to be insensitive to the mass scale of the right-handed neutrinos and thatthey are as large as (or larger than) the present experimental bounds irrespective ofthe right-handed neutrino masses. a r X i v : . [ h e p - ph ] F e b Introduction

Even though the standard model (SM) of particle physics successfully explains many ofresults of high energy experiments, the existence of a physics beyond the SM (BSM) hasbeen highly anticipated. Particularly, from particle cosmology point of view, there are manymiseries which cannot be explained in the framework of the SM, like the existence of darkmatter, the origin of the baryon asymmetry of the universe, the dynamics of inﬂation, andso on. Many experimental eﬀorts have been performed to ﬁnd signals of the BSM physics.In the search of the BSM signals, energy and precision frontier experiments are bothimportant. The energy frontier experiments, represented by collider experiments like theLHC at present, may directly ﬁnd and study particles in BSM models, but their discoveryreach is limited by the beam energy. On the contrary, the precision frontier ones may reachthe BSM whose energy scale is much higher than the energy scale of the LHC experiment,although information about the BSM from those experiments may be indirect. Currently,the LHC has not found any convincing evidence of the BSM physics. In such a circumstance,it is important to reconsider the role of precision frontier experiments and study what kindof signal may be obtained from them.In this paper, we study CP and ﬂavor violations in models with supersymmetry (SUSY),and their impacts on on-going and future experiments. Even though the LHC has notfound any signal of SUSY particles with their mass scale of ∼ TeV, the SUSY is still awell-motivated candidate of BSM physics. Taking into account the observed Higgs mass of125 .

10 GeV [1], heavy SUSY particles (more speciﬁcally, heavy stops) are preferred to pushup the Higgs mass via radiative corrections [2–5]. We note that, in a large class of models,SUSY particles can acquire masses of ∼ O (10 − M GUT ∼ O (10 ) GeVis suggested, and also because (ii) right-handed neutrinos are well-motivated to explain theorigin of the active neutrino masses via the seesaw mechanism [11–13]. In SUSY GUT withright handed neutrinos, some of the cosmological mysteries mentioned above may be alsosolved; the baryon asymmetry of the universe may be explained by the leptogenesis scenario[14], while the lightest superparticle (LSP) may play the role of dark matter. Compared tothe SM, the SUSY models contain various new sources of CP and ﬂavor violations. It maycause signiﬁcant CP and ﬂavor violating processes which cannot be explained in the SM. Ifsuch processes are experimentally observed, they can be smoking gun evidences of the BSMphysics, based on which we may study the BSM model behind the CP and ﬂavor violations.It has been well known that the renormalization group eﬀect may induce CP and ﬂavorviolating oﬀ diagonal elements of the slepton mass matrices [15–18]; in the framework of ourinterest, the left and right handed slepton mass matrices are aﬀected by the renormalizationgroup eﬀects from the neutrino Yukawa coupling and the running above the GUT scale M GUT , respectively, even though the lepton ﬂavor is conserved in the Yukawa interaction ofthe minimal SUSY standard model (MSSM). Thus, even though the slepton mass matrices1re universal at some high scale (for example, Planck scale), such universalities are violatedby the renormalization group eﬀects. The eﬀects of the oﬀ diagonal elements of the sleptonmass matrices on CP and/or ﬂavor violating observables have been studied (see, for example,[19–44]). In particular, it has been pointed out that, even if the MSSM particles are out ofthe reach of the LHC experiment, the signal of the MSSM may be observed by on-going orfuture CP or ﬂavor violation experiments. In previous studies, the neutrino Yukawa matrixwas reconstructed by combining the seesaw formula with the active neutrino mass squareddiﬀerences suggested by the neutrino oscillation experiments. Then, the neutrino Yukawacoupling constants are inversely proportional to the square root of the mass scale of theright handed neutrinos. With adopting simple assumption about the neutrino sector, i.e.,the universal masses for the right handed neutrinos as well as a simple mixing structure,the renormalization group eﬀects due to the neutrino Yukawa couplings become irrelevantas the mass scale of the right handed neutrinos becomes smaller. However, as pointed outby Casas and Ibarra (CI) [25], there exist several parameters (which we call CI parameters)which complicate the mixing structure of the neutrino Yukawa matrix.In this paper, we study CP and ﬂavor violating processes paying particular attentionto the eﬀects of the CI parameters, as well as the eﬀects of the non-universality of theright handed neutrino masses, whose eﬀects have not been fully investigated so far. (Forsome discussion about the eﬀect of the CI parameters, see [21, 27, 28, 30, 31, 35, 43]). Theorganization of this paper is as follows. In Section 2, we introduce the model based on whichwe perform our analysis. In Section 3, we show the results of our numerical analysis. Section4 is devoted to conclusions and discussion.

In this section, we introduce the model we consider. We also summarize our convention ofthe model parameters, including CI parameters and GUT phases. To this end, we deﬁnethe couplings and specify the ﬂavor basis we use for each eﬀective theories and explain howthey are related at the matching scales.Eﬀective theories in our model appropriate for the each energy scales lower than M Pl areshown in Fig. 1, where • QEDQCD: QED and QCD • MSSMNR: MSSM with three generations of right-handed neutrinos • SU(5)NR: minimal SU(5) GUT with three generations of right-handed neutrinosAt each renormalization scale Q , we use the relevant eﬀective theory as we explain below.We assume that the eﬀect of the SUSY breaking is mediated to the visible sector(containing the MSSM particles and right-handed neutrinos) at the reduced Planck scale M Pl (cid:39) . × GeV. Then, at the scales between M Pl and M GUT , the model is describedby SU(5)NR. In order to introduce three generations of quarks and leptons, three copies of2EDQCD M t M S SM MSSM M N , , M GUT

MSSMNR M Pl SU(5)NR Q Figure 1: The eﬀective theories used for each regions of the renormalization scale Q .chiral supermultiplets Φ i and Ψ i , which are in the ¯5 and representations of SU(5), re-spectively, are introduced. (Here, the i = 1 − i is composed of the right-handed down-type quark multiplets ¯ D i and thelepton doublets L i , while Ψ i is composed of the quark doublets Q i , the right handed up-typequark multiplets U i and the right-handed charged lepton ¯ E i . The right-handed neutrinos¯ N i of MSSMNR are added as SU(5) singlets Υ i . The MSSM Higgs doublets H u and H d arecontained embedded into H and ¯ H , which are SU(5) and ¯5 representations, respectively.There is also a multiplet which breaks SU(5) symmetry to the SM gauge group. We assumethat a chiral multiplet in the adjoint representation of SU(5), which we call Σ, is responsi-ble for the breaking of the SU(5) symmetry. The vacuum expectation value (VEV) of Σ isdenoted as (cid:104) Σ (cid:105) = diag(2 v GUT , v GUT , v GUT , − v GUT , − v GUT ).We consider the superpotential of SU(5)NR in the following form: W SU(5)NR = W renSU(5)NR + W nonrenSU(5)NR , (2.1) W renSU(5)NR = W matterSU(5)NR + W HiggsSU(5)NR , (2.2)and W matterSU(5)NR = 14 ( f u ) ij Ψ i Ψ j H + √ f d ) ij Ψ i Φ j ¯ H + ( f ν ) ij Υ i Φ j H + 12 ( M Υ ) ij Υ i Υ j , (2.3)where f u , f d , and f ν are 3 × M Υ is 3 × f u and M Υ are symmetric. In Eq. (2.3), the summations over SU(5) indicesare implicit. (We follow [38] for the group theoretical notations.) W SU(5)NR consists of therenormalizable part W renSU(5)NR and the non-renormalizable part W nonrenSU(5)NR . W renSU(5)NR is furthersplit into W matterSU(5)NR (i.e., the superpotential containing the matter sector) and W HiggsSU(5)NR (i.e.,the superpotential for the Higgs sector); W HiggsSU(5)NR is the superpotential containing only theHiggs ﬁeld and Σ. In Eq. (2.3), W matterSU(5)NR contains superpotential responsible for the up-type, down-type and neutrino-type Yukawa terms in the MSSMNR. In addition, in order toexplain the uniﬁcation of the down-type and electron-type Yukawa matrices, we assume that W nonrenSU(5)NR contains a term in the following form: W nonrenSU(5)NR (cid:51) √ M Pl c ij Ψ i ΣΦ j ¯ H. (2.4)Unitary rotations on the family indices can make the coupling matrices to the following3orms: f u = V T ˆ f u ˆΘ q V, (2.5) f d = ˆ f d , (2.6) f ν = ˜ W † ˆ f ν U † ˆΘ l , (2.7) M Υ = ˆ M Υ , (2.8)where ˆ f u , ˆ f d , ˆ f ν and ˆ M Υ are real diagonal matrices. In addition, V and U are unitarymatrices with only a single CP phase and three mixing angles while ˜ W is a general unitarymatrix with additional 5 phases. Furthermore, ˆΘ q and ˆΘ l are diagonal phase matrices andrepresent CP phases intrinsic in SU(5) GUT. Notice that the overall phases of ˆΘ q and ˆΘ l are unphysical because they can be absorbed to ˜ W . Thus, each of ˆΘ q and ˆΘ l contains twoparameters; we parameterize these matrices asˆΘ f = diag(1 , e iϕ f , e iϕ f ) , (2.9)with f = q, l .In the following argument, we take the ﬂavor basis in which the coupling matrices ofSU(5)NR take the forms of Eqs. (2.6) – (2.8) at Q = M GUT . In our discussion, the higherdimensional operator proportional to c is introduced just to guarantee the uniﬁcation of ¯ U i and L i into ¯5 multiplets of SU(5). For simplicity, we assume that c is real and diagonal at Q = M GUT in this basis: c = ˆ c .At the GUT scale M GUT , SU(5)NR couplings are matched to MSSMNR couplings. Thematter sector superpotential of MSSMNR is given in the following form: W matterMSSMNR = ( y u ) ij H u ¯ U i Q j + ( y d ) ij H d ¯ D i Q j + ( y e ) ij H d ¯ E i L j + ( y ν ) ij H u ¯ N i L j + 12 ( M N ) ij ¯ N i ¯ N j , (2.10)where y u , y d , y e and y ν are the MSSMNR Yukawa matrices while M N is the Majorana massmatrix of ¯ N . The MSSMNR chiral multiplets are embedded into the SU(5)NR ones as The hat on matrix symbols indicates that they are diagonal.

In general, a unitary matrix ˜ X can be decomposed as˜ X = e iϕ ˜ X ˆΘ ( L )˜ X X ˆΘ ( R )˜ X , where ϕ ˜ X is the overall phase of the matrix ˜ X , ˆΘ ( L )˜ X and ˆΘ ( R )˜ X are diagonal phase matrices parameterizedby two physical phases, and X is a unitary matrix parameterized by three mixing angles, ϑ , ϑ , and ϑ ,and a single phase δ as X = c c s c s e − iδ − s c − c s s e iδ c c − s s s e iδ s c s s − c c s e iδ − c c − s c s e iδ c c , with c ij = cos ϑ ij and s ij = sin ϑ ij . i = ( U † Q Q, V † ˆΘ ∗ q U † ¯ U ¯ U , ˆΘ l U † ¯ E ¯ E ) i , (2.11)Φ i = ( U † ¯ D ¯ D, ˆΘ ∗ l U † L L ) i , (2.12)Υ i = ( U † ¯ N ¯ N ) i , (2.13)where U Q , U ¯ U , U ¯ D , U L , U ¯ E , and U ¯ N are 3 × Q = M GUT are obtained as U T ¯ U [ y u ] Q = M GUT U Q = (cid:104) ˆ f u V (cid:105) Q = M GUT , (2.14) U T ¯ D [ y d ] Q = M GUT U Q = (cid:20) ˆ f d + 2 v GUT M Pl ˆ c (cid:21) Q = M GUT , (2.15) U T ¯ E [ y e ] Q = M GUT U L = (cid:20) ˆ f d − v GUT M Pl ˆ c (cid:21) Q = M GUT , (2.16) U T ¯ N [ y ν ] Q = M GUT U L = (cid:104) ˜ W † ˆ f ν U † (cid:105) Q = M GUT , (2.17) U T ¯ N [ M N ] Q = M GUT U ¯ N = (cid:104) ˆ M Υ (cid:105) Q = M GUT . (2.18)Based on the above relations, the coupling matrices of the SU(5)NR are determined fromthose of the MSSMNR in our numerical analysis with properly choosing the unitary matrices U Q , U ¯ U , U ¯ D , U L , U ¯ E , and U ¯ N . Notice that the GUT phases ˆΘ q and ˆΘ l can be absorbedinto the deﬁnitions of the MSSMNR superﬁelds and can be removed from the MSSMNRsuperpotential; however, they are physical in SU(5)NR.Now, let us consider the neutrino masses. For this purpose, it is more convenient to usethe ﬂavor basis in which ¯ N i ( i = 1 −

3) become the mass eigenstates (see below). In general,the masses of right-handed neutrinos are diﬀerent. We denote the mass of i -th right-handedneutrino as M N i , with M N ≤ M N ≤ M N . The dimension-ﬁve operator responsible for theMajorana mass terms of the left-handed neutrinos is generated with see-saw mechanism byintegrating out right-handed neutrinos [11–13]. Let us deﬁne the following diagonal matrix:ˆ M N ≡ diag( M N , M N , M N ). Using ˆ M N , the active neutrino mass matrix is given by m ν = v u y Tν ˆ M − N y ν (2.19)where v u is the VEV of H u . In addition, y ν,ij ≡ y ν,ij ( Q = M N i ), with j = 1 −

3, is theneutrino Yukawa coupling constant at Q = M N i , where, again, we are adopting the ﬂavorbasis in which N i is the mass eigenstate. In Eq. (2.19), the running of the Wilson coeﬃcients of the dimension-ﬁve operator is neglected in ouranalysis. θ sin θ sin θ δ CP ∆ m [eV ] ∆ m [eV ] m ν [eV ]0 .

307 0 .

545 0 . . π . × − . × − . × − Table 1: Model parameters used in our numerical analysis [1].At the energy scales lower than M N , the model is described by the MSSM and one canalways work in the basis in which y e is diagonal. In such a basis, m ν takes the followingform. m ν = U ∗ PMNS ˆ m ν ˆΘ M U † PMNS , (2.20)where U PMNS is the Pontecorvo-Maki-Nakagawa-Sakata (PMNS) matrix [45, 46] with threemixing angles θ , θ , θ , and a Dirac CP phase δ CP , while diagonal phase matrix ˆΘ M con-tains Majorana phases. The overall phase of ˆΘ M is unphysical, and we adopt the conventionsuch that ˆΘ M = diag(1 , e iϕ M , e iϕ M ). Furthermore, ˆ m ν is a real diagonal matrix contain-ing mass eigenvalues of left-handed neutrinos. In Table 1, we summarize the parameters inˆ m ν and U PMNS used in our numerical analysis. Here, we assume the normal hierarchy forneutrino masses and take m ν = 1 . × − eV .Comparing Eq. (2.19) and Eq. (2.20), y ν can be expressed in the following form, i.e., theso-called Casas-Ibarra parameterization [25]: y ν = 1 v u ˆ M N R ˆ m ν ˆΘ M U † PMNS , (2.21)where R is an arbitrary complex and orthogonal matrix: RR T = . (2.22)The two parameterizations of the neutrino Yukawa matrices, Eq. (2.17) and Eq. (2.21), areequivalent. The number of parameters in both parameterizations are summarized in Tables 2and 3. (Notice that the overall phase of the unitary matrix ˜ W , as well as the phase matricesˆΘ q and ˆΘ l , cannot be determined from the low energy observables. We call them “GUTphases.”)The complex orthogonal matrix R can be decomposed into the product of a real orthog-onal matrix O and a hermitian and orthogonal matrix H as R = OH . Let us deﬁne n ( θ, φ ) ≡ (sin θ cos φ, sin θ sin φ, cos θ ) , (2.23)and ( A ( n )) ij ≡ (cid:15) ijk n k . (2.24)Then, H can be expressed as H ( r, n ) = e irA ( n ) = nn T + cosh r (cid:0) − nn T (cid:1) + i sinh rA ( n ) , (2.25) Square root of a diagonal matrix is understood to be applied to each diagonal elements. l ˜ W U ˆ f ν total2 9 4 3 18Table 2: The number of real degrees of freedom in the neutrino Yukawa matrix in theparameterization of Eq. (2.17). ϕ ˜ W ˆΘ l ˆΘ M U PMNS ˆ m ν O H total1 2 2 4 3 3 3 18Table 3: The number of real degrees of freedom in the neutrino Yukawa matrix in CIparameterization, where ϕ ˜ W is the overall phase of the matrix ˜ W .and hence is parameterized by 3 real parameters ( r, θ, φ ). We can derive another usefulexpression of H ( r, n ). For this purpose, we introduce a complex vector˜ n ≡ √ n (cid:48) − in (cid:48)(cid:48) ) , (2.26)where n (cid:48) and n (cid:48)(cid:48) are an arbitrary set of 2 unit vectors such that (cid:104) n, n (cid:48) , n (cid:48)(cid:48) (cid:105) forms a right-handed orthonormal basis of R . One can easily check that (cid:104) n, ˜ n, ˜ n ∗ (cid:105) forms an orthonormalbasis of C . With these vectors, H ( r, n ) can be expressed as H ( r, n ) = e r ˜ P ( n ) + P ( n ) + e − r ˜ P ∗ ( n ) , (2.27)where P ( n ) ≡ nn T , ˜ P ( n ) ≡ ˜ n ˜ n † , ˜ P ∗ ( n ) ≡ ˜ n ∗ ˜ n T . (2.28)Here P , ˜ P , and ˜ P ∗ are orthogonal projection matrices onto C n , C ˜ n , and C ˜ n ∗ , respectively.Note that ˜ P depends only on n and not on the choice of n (cid:48) and n (cid:48)(cid:48) . When r (cid:29)

1, the ﬁrstterm in the right-hand side of Eq. (2.27) dominates.Many of the previous analysis of the ﬂavor violations have not paid signiﬁcant attentionto the eﬀect of the CI parameters, taking R = (see, however, [21, 27, 28, 30, 31, 35, 43]). Inaddition, it has been often assumed that the right-handed neutrino masses are degenerate,i.e., M N = M N = M N . With these simpliﬁcations, y ν = v u ( ˆ M N ˆ m ν ) U † PMNS . However,as we will see in the following, parameters in R may signiﬁcantly aﬀect the CP and ﬂavorviolating observables. Now, let us numerically evaluate the CP and ﬂavor violating observables. Our primarypurpose is to study the eﬀects of CI parameters on electron electric dipole moment (EDM)and branching ratios of lepton ﬂavor violating (LFV) processes. Thus, for simplicity, weassume that the soft SUSY breaking parameters satisfy the so-called mSUGRA boundary7onditions; the soft scalar mass-squared parameters at the Planck scale Q = M Pl are assumedto be universal (and are equal to m ), and tri-linear scalar couplings (so-called A -terms) areproportional to corresponding Yukawa couplings (with the proportionality constant of a ).In our analysis, we calculate the MSSM parameters at the mass scale of MSSM super-particles (which we call MSSM scale). Here are remarks about our calculation: • The input SM parameters related to low energy observables are g a , ˆ y u , ˆ y d , V CKM , ˆ y e , ˆ m ν , U PMNS , (3.1)where g a (with a = 1 −

3) are gauge coupling constants, while V CKM is the Cabibbo-Kobayashi-Maskawa (CKM) matrix. For the boundary conditions of SM couplingsat the top mass scale M t , we follow [47]. The parameters in the CKM matrix aretaken from [1] and set at M t ; the lightest neutrino mass, which cannot be determinedfrom current neutrino oscillation experiments, is set to be 1 × − eV. For the left-handed neutrino mass eigenstates and the PMNS matrix, we use the values given inTable 1 at the scale of right-handed neutrino masses, neglecting the renormalizationgroup running of the neutrino mass and mixing parameters below the mass scale ofthe right-handed neutrinos. At M t , 2-loop and 3-loop SM thresholds are included. • In addition, we ﬁx other input parameters: m , a , M / , tan β, sgn( µ ) , ˆ M N , ˆΘ l , ˆΘ M , O, H, (3.2)where M / is SU(5) gaugino mass at the Planck scale, tan β is the ratio of the VEVsof up- and down-type Higgs bosons, and µ is supersymmetric Higgs mass parameter.(Here, we assume that µ is real.) For simplicity, we take M / = m , a = 0 andsgn( µ ) = +. • The SM parameters at the MSSM scale is obtained by using the SM renormalizationgroup equations (RGEs). In our analysis, the MSSM scale is deﬁne as the geometricmean of the stop mass eigenvalues, M S = √ m ˜ t m ˜ t ; in our numerical calculation, M S is determined iteratively (see the following arguments). At M S , SUSY thresholdcorrections to the Higgs quartic coupling constant λ , gauge couplings and the topYukawa coupling are included [48]. • The MSSM parameters at the mass scale of the right-handed neutrino masses areobtained by using SOFTSUSY package [49]. The couplings in the MSSM and those ofthe MSSMNR are matched at the tree level. Notice that each right-handed neutrinodecouples from the eﬀective theory at M N i , and we use the eﬀective theory without N i for the scale Q < M N i ; here, N i is deﬁned in the basis in which it becomes the masseigenstate of M N ( Q = M N i ). • In order to study the running between the mass scale of the right-handed neutrinos andthe GUT scale, we modify SOFTSUSY package with including the coupling constantsrelated to right-handed neutrinos. The RGEs of MSSMNR can be obtained in [32].8

The Yukawa matrices of the MSSMNR and those of the SU(5)NR are matched byusing Eqs. (2.14) – (2.18) at the GUT scale; in our numerical calculation, we take M gut = 2 × GeV. Other parameters are also matched accordingly. At the GUTscale, threshold corrections on down-type and charged-lepton type Yukawa matrices ofMSSMNR are imposed following [44] while tree level matching conditions are adoptedfor other dimensionless couplings and soft masses. • In order to take into account the running above the GUT scale, we also implementthe RGEs in the SU(5)NR. Here, we assume that the particle content of the SU(5)NRis minimal; Φ i , Ψ i , Υ, H , ¯ H , and Σ, as well as the gauge multiplet. The RGEs ofthe SU(5) gauge coupling constant and the gaugino mass are obtained based on thisparticle content. In addition, for simplicity, we assume that the interactions of Σ areso weak that their eﬀects on the running are negligible. • The SUSY breaking parameters at the Planck scale are set by using the parameters m , a , and M / .For RGEs, we use 2-loop (SM and MSSM) and 1-loop (MSSMNR and SU(5)NR). However,for the calculation of the Higgs quartic coupling constant, we follow [48] and solve 3-loopRGEs between M t and M S .With the modiﬁed SOFTSUSY package explained above, we calculate the MSSM param-eters at Q = M S as follows. For the consistency of boundary conditions at M t and at M Pl ,we iterate on M S and M H .1. We ﬁrst ﬁx the set of input parameters given in Eq. (3.2). Then, we adopt the tempo-rary values M S = m and M H = 126 GeV.2. The boundary conditions for SM couplings at Q = M t are determined with followingthe procedure given in [47], using M t , Higgs mass M H , W -boson mass M W , Z -bosonmass M Z , the strong coupling constant α s ( M Z ) and ﬁne structure constant α ( M Z ) asinput parameters.3. We solve the RGE runnings of the Yukawa coupling constants and the gauge cou-pling constants from M t to M Pl and the Planck-scale values of those parameters aredetermined.4. We set the boundary conditions for soft SUSY breaking parameters at M Pl and runthem down to M S .5. At M S , the SM Higgs quartic coupling λ is calculated from tan β , with including theSUSY threshold corrections. The µ -parameter and B -parameter are determined fromthe tree level electroweak symmetry breaking (EWSB) condition, which depends ontan β and Higgs soft masses. The Higgs quartic coupling at Q = M t is also determined.6. Renew M S from stop masses and M H from SM couplings.9. Iterate the steps from 2 to 6 until M S and M H converge.Once the MSSM parameters at Q = M S are ﬁxed, CP and LFV observables are calculatedby using those parameters. After EWSB, relevant operators of our interest are given by L eEDM = − id e ψ e σ µν γ ψ e F µν , (3.3) L µ → eγ = −

12 ¯ ψ e σ µν ( a L P L + a R P R ) ψ µ F µν , (3.4)where F µν is the ﬁeld strength tensor of the photon, ψ µ and ψ e are ﬁeld operators of muonand electron, respectively, and d e , a L , and a R are coeﬃcients. d e is the electron EDM whilethe decay rate of µ → eγ process is give byΓ( µ → eγ ) = m µ π ( | a L | + | a R | ) . (3.5)For the LFV decay processes of τ , like τ → µγ and τ → eγ , the operator is like that given inEq. (3.4) with ﬁeld operators being properly replaced. For the detail about the calculationof the electron EDM and the LFV decay rates, see, for example, [20, 37].Before showing our numerical results, it is instructive to use the leading-log approxima-tion for the understandings of qualitative behaviors. Above M GUT , f u contributes to theoﬀ-diagonal elements of the right-handed selectron mass matrix m ˜ e and f ν contribute tothat of the left-handed slepton mass matrix m ˜ l . Below M GUT , there is no extra violationproduction in m ˜ e from Yukawa interactions but m ˜ l still acquires oﬀ-diagonal elements fromneutrino-type Yukawa interactions. Assuming a universality of the right-handed neutrinomasses, the leading-log approximation gives( m l ) ij (cid:39) m δ ij − π (3 m + a )( ˆΘ ∗ l y † ν y ν ˆΘ l ) ij log (cid:18) M Pl M N R (cid:19) , (3.6)( m e ) ij (cid:39) m δ ij − π (3 m + a )( V T ˆ f u V ∗ ) ij log (cid:18) M Pl M GUT (cid:19) , (3.7)where M N R is the universal right-handed neutrino mass. The oﬀ-diagonal elements of m l are approximately proportional to the corresponding elements of ˆΘ ∗ l y † ν y ν ˆΘ l . When the r parameter is sizable we can ﬁndˆΘ ∗ l y † ν y ν ˆΘ l (cid:39) e r v u tr (cid:16) ˜ P ( n ) O T ˆ M N O (cid:17) ˜ U ˆ m ν ˜ P ( n ) ˆ m ν ˜ U † + O ( e r ) , (3.8)where ˜ U ≡ ˆΘ ∗ l U PMNS ˆΘ ∗ M . The CI parameters may enhance the oﬀ diagonal elements of m l because the magnitude of y † ν y ν is proportional to e r . From Eqs. (3.6) and (3.8), onecan see that the renormalization group eﬀects on m l is suppressed when the mass scale ofthe right-handed neutrinos becomes smaller. Thus, when the eﬀects of the CI parameters10 √ g (cid:48) ˜ e R − ( m e ) , ˜ τ R µ tan βm τ ˜ τ L − ( m l ) , ˜ e L g (cid:48) √ ˜ B ¯ e e − g ˜ ν eL − ( m l ) , ˜ ν µL √ m µ v cos β ˜ W + ˜ W − − gv sin β/ √ ˜ h + u ˜ h − d e ¯ µ Figure 2: Examples of the mass insertion contributions to d e (left) and to Br( µ → eγ ))(right).are neglected, the CP and ﬂavor violations due to the renormalization group eﬀects arehighly suppressed when the mass scale of the right handed neutrinos is much smaller than ∼ GeV. With the CI parameters, this may not be the case. One can see that, whenthe r parameter is larger than ∼

1, the renormalization group eﬀects can be sizable evenwhen the right-handed neutrinos are relatively light. In the following, we will see that theenhancement due to the CI parameters can indeed enhance the electron EDM and LFVdecay rates.The oﬀ diagonal elements of the slepton mass matrix become the sources of CP and ﬂavorviolations. Although we numerically calculate the electron EDM and LFV decay rates in themass basis, with which the eﬀects of the oﬀ diagonal elements are taken into account at allorders, it is suggestive to consider the mass insertion method to understand the behaviors ofthe results. Fig. 2 shows the examples of the diagrams contributing to the electron EDM and µ → eγ process in the mass insertion approximation. In fact, when tan β (cid:29)

1, the dominantcontributions to µ → eγ originates from a diagram with a mass insertion of ( m ˜ l ) , . For theelectron EDM, the dominant contribution is from a diagram with mass insertions of ( m ˜ e ) , and ( m ˜ l ) , , if there’s no CP phase in µ parameter (see Fig. 2). For the choice of parameterswe adopt in the following analysis, we found that the diagrams shown in Fig. 2 becomedominant for the electron EDM and the decay rate for the process µ → eγ .Now, we show the results of our numerical calculations. Unless otherwise stated, we taketan β = 8 and m = 10 TeV, which give M H (cid:39)

126 GeV. We neglect the eﬀects of Majoranaphases and simply set ˆΘ M = . The three types of structures of ˆ M N and O are adopted: • (U) Universal: O T ˆ M N O = M N R . (3.9) • (H) Hierarchical: O T ˆ M N O = M N R − −

00 0 1 . (3.10)11 �� - �� �� - �� �� - �� �� - �� ��� - �� �� - �� �� - �� �� - �� Figure 3: The dependence of d e on GUT phase ϕ l with r = 0 (left) or r = 2 . β = 8, m = 10 TeV, M N R = 10 GeV, ( θ, φ ) = ( π , M = and ϕ l = 0. • (IH) Inverse hierarchical: O T ˆ M N O = M N R −

00 0 10 − . (3.11)The Yukawa couplings may blow up if the CI parameter r is too large. In such a case, theperturbative calculation becomes unreliable. In order to avoid the blow up of the Yukawacouplings, we impose the following constraints on the neutrino Yukawa couplings at anyrenormalization scale: tr( f † ν f ν )(4 π ) < , tr( y † ν y ν )(4 π ) < . (3.12)In Fig. 3, we show how the electron EDM depends on the GUT phases ϕ l , adopting thestructure (U) of right-handed neutrino masses and M N R = 10 GeV. Here, we take R = (left) and ( r, θ, φ ) = (2 . , π/ ,

0) (right). As shown in Fig. 3, the electron EDM is sensitiveto ϕ l . This is because, through the renormalization group eﬀects, ϕ l aﬀects the complexphase of ( m ˜ l ) , which the electron EDM is (approximately) proportional to. We can seethat the position of the peak is shifted with the introduction of the CI parameters becausethey contain CP phases. In the following analysis, in order to (approximately) maximize theelectron EDM, we tune the GUT phase ϕ l so that the contribution of the mass insertiondiagram shown in Fig. 2 (left), which gives the dominant contribution to the electron EDMin most of the parameter region in our study, is maximized. We have also studied how theelectron EDM depends on the phase ϕ l , and conﬁrmed that such a dependence is weak. If r is large enough, other mass-insertion diagrams with multiple insertions of ( m ˜ l ) ij become non-negligible. On the other hand, as y ν is small, the diagrams other than the left one of Fig. 2 become sizable. �� - �� - �� - �� - �� - �� - �� - �� - �� - �� �� � �� � �� � ����� - �� - �� - �� - �� - �� - �� - �� - �� - �� - � �� � �� � �� � ����� ����� ������������ ��� Figure 4: Higgs mass (black), electron EDM (green) and Br( µ → eγ ) (orange) on ( m , tan β )plane with r = 0 or r = 2 .

2. Here, we take M N R = 10 GeV, ( θ, φ ) = ( π , M = and ϕ l = 0. ϕ l is chosen to maximize electron EDM. The black lines are for M H =123 , , , ,

127 GeV from left to right. The numbers in the ﬁgure are log d e (green) and log Br( µ → eγ ) (orange). The green and orange dashed lines are experimentalbounds on the electron EDM and Br( µ → eγ ), respectively.Fig. 4 shows the contours of constant Higgs mass, maximized electron EDM and Br( µ → eγ ) on ( m , tan β ) plane, taking M N R = 10 GeV. In the ﬁgure, we also show the contourson which d e and Br( µ → eγ ) become equal to the current experimental upper bounds; theupper bound on d e is given by ACME as [50] d e < . × − e cm , (3.13)while the upper bound on the branching ratio for µ → eγ process is given by MEG experimentas [51] Br( µ → eγ ) < . × − . (3.14)The left plot is for the case of R = , while the right one is for the case of ( r, θ, φ ) =(2 . , π/ , d e and Br( µ → eγ ). This is because, as we increase the r parameter,the Yukawa couplings can become larger (see Eq. (3.8)), which enhances the renormalizationgroup eﬀects on ( m l ) ij . Because the electron EDM and Br( µ → eγ ) are sensitive to theoﬀ-diagonal elements of slepton mass squared matrices, the proper introduction of the CIparameters has signiﬁcant impact on the CP and ﬂavor violating observables.In Fig. 5, we show contours of constant maximized electron EDM and Br( µ → eγ ) on( M N R , r ) plane, taking ( θ, φ ) = ( π/ , µ → eγ ) are insensitiveto the scale and the structure of the right-handed neutrino mass matrix. This is becausethe enhancement of the neutrino Yukawa couplings due to the factor of e r compensates thesuppression due to the smallness of the right-handed neutrino mass (see Eq. (3.8)). Fig. 6shows the contours of constant Br( τ → eγ ) and Br( τ → µγ ). They are also enhanced by CIparameters but are fairly below the current experimental bounds.These ﬁgures show our main conclusion that the leptonic CP and ﬂavor violating signalsthrough the renormalization group eﬀects can be sizable irrespective of the mass scale ofright-handed neutrinos. This is a contrast to the case without taking into account the eﬀectsof the CI parameters; without the CI parameters, the neutrino Yukawa coupling constantsbecome tiny when the right-handed neutrinos are much lighter than ∼ GeV. In otherwords, we have a chance to observe the leptonic CP and/or ﬂavor violating signals fromthe renormalization group eﬀect even when the right handed neutrino masses are relativelysmall.Fig. 7 shows how large the r parameter can be on ( θ, φ ) plane, taking (U) universalright-handed neutrinos with M N R = 10 GeV. With the choice of parameters adopted inFig. 7, r is required to be smaller than about 2 . − .

3. Using the maximal possible valueof r given in Fig. 7, we calculate the CP and ﬂavor violating observables. In Fig. 8, weshow maximized electron EDM, Br( µ → eγ ), Br( τ → eγ ) and Br( τ → eγ ). We can see thatsome of the observables are suppressed at particular points on the ( θ, φ ) plane and that thepoints of the suppressions are correlated for diﬀerent observables. These are because, at thepoints of the suppressions, two of ( m ˜ l ) ij (with i (cid:54) = j ) are simultaneously suppressed whilethe others are sizable. This can be understood as follows. When r is large, the oﬀ-diagonalelements of m ˜ l can be approximated as( m ˜ l ) ij ∝ u i u ∗ j , (3.15)where (see Eq. (3.8)) u ( θ, φ, ϕ M k , ϕ l k ) ≡ ˆΘ ∗ l U PMNS ˆΘ ∗ M ˆ m ν ˜ n. (3.16)Thus, one of the elements u i becomes accidentally small, ( m ˜ l ) ij ( j = 1 −

3) are all suppressed,resulting in the correlation of the suppression points shown in Fig. 8. For example, if u is close to 0, ( m ˜ l ) , and ( m ˜ l ) , , and hence d e , Br( µ → eγ ) and Br( τ → eγ ), becomessimultaneously suppressed; for the present choice of parameters, this happens when ( θ, φ ) (cid:39) (0 . π, . π ) and (0 . π, . π ). We have studied the leptonic CP and ﬂavor violating observables, i.e., the electron EDM d e and the branching rations of lepton ﬂavor violating decays Br( l i → l j γ ), in the minimal14 �� - �� - �� - �� - �� - �� �� �� �� �� �� �� �� �� ���� ( � ) - �� - �� - �� - �� - �� - �� �� �� �� �� �� �� �� �� ����� ( � ) - �� - �� - �� - �� - �� - �� - �� - �� - �� �� �� �� �� �� �� �� �� ������ ( �� ) �������� ��� Figure 5: The electron EDM (green) and Br( µ → eγ ) (orange) on ( M N R , r ) plane for thestructures (U), (H) and (IH) of O T ˆ M N O . We take tan β = 8, m = 10 TeV, ( θ, φ ) = ( π/ , M = and ϕ l = 0. ϕ l is chosen to maximize d e . The numbers in the ﬁgure are log d e (green) and log Br( µ → eγ ) (orange). 15 �� - �� - �� - �� - �� - �� �� �� �� �� �� �� �� �� ���� ( � ) Figure 6: Br( τ → eγ ) (red) and Br( τ → µγ ) (yellow) on ( M N R , r ) plane for the case (U)of universal right-handed neutrinos. Other parameters are same as those in Fig. 5. Thenumbers are log Br( τ → eγ ) (red) and log Br( τ → µγ ) (yellow). ������ ������ ������ ������ ������ ����� Figure 7: The maximized r parameter under the perturbativity constraints (upper-left).Here, right-handed neutrinos are (U) universal with M N R = 10 GeV. tan β = 8, m =10 TeV, ˆΘ M = and ϕ l = 0. ϕ l is chosen to maximize electron EDM.16 ���� - ���� - ���� - ���� - ���� - ���� - ������ �������� ��� - ���� - ���� - ���� - ���� - ���� - ���� - ���� - ������ - ���� - ���� - ���� - ���� - ������ - ���� - ���� - ���� - ���� - ������ Figure 8: Contours of constant electron EDM (upper-left), Br( µ → eγ ) (upper-right), andBr( τ → l j γ ) (lower), calculated with the maximal possible value of r shown in Fig. 7. Themodel parameters are the same as those used in Fig. 7.17upersymmetric SU(5) GUT with three right-handed neutrinos. We paid particular attentionto the eﬀects of the CI parameters R = OH ( r, θ, φ ) in the neutrino Yukawa matrix, whichhas not been studied extensively before. With the assumption of the universality boundaryconditions for soft SUSY breaking masses, we have calculated the electron EDM and Br( l i → l j γ ) with varying CI parameters as well as the MSSM and GUT parameters. Imposing Higgsmass constraints as well as other constraints from low-energy observations, we have studiedhow the CP and ﬂavor violating observables behaves.In SUSY models, the oﬀ-diagonal elements of the slepton mass matrices are induced byrenormalization group eﬀects in particular when there exists right-handed neutrinos withsizable neutrino Yukawa couplings or when quarks and leptons are uniﬁed into same mul-tiplets of GUT. The oﬀ-diagonal elements of the slepton mass matrices become sources ofthe leptonic CP and ﬂavor violating observables, i.e., the electron EDM and Br( l i → l j γ ).Without taking into account the eﬀects of the CI parameters, eﬀects of the right-handedneutrinos on the renormalization group runnings become irrelevant if the mass scale of theright-handed neutrinos is small; this is because, assuming the seesaw formula for the activeneutrino masses, the neutrino Yukawa coupling is suppressed as the right-handed neutrinobecomes lighter. Eﬀects of the CI parameters may compensate such an eﬀect, and we foundthat the maximal possible values of d e and Br( l i → l j γ ) are insensitive to the structure ofright-handed neutrino masses ˆ M N and the orthogonal matrix O . Especially, there are pointswhere 2 of 3 independent oﬀ-diagonal elements are simultaneously suppressed. Therefore,experimental studies of all the CP and ﬂavor violating observables are important to probethe model.One interesting implication of our analysis should be on the leptogenesis scenario [14], inwhich the lepton asymmetry generated by the decay of the right-handed neutrino is convertedto the baryon asymmetry of the universe. In a simple leptogenesis scenario, the mass scale ofthe lightest right-handed neutrino is required to be larger than ∼ − GeV [52, 53], whileit should be smaller than the reheating temperature after inﬂation in order not to dilute thegenerated baryon asymmetry. The total amount of the baryon asymmetry generated by theleptogenesis scenario depends on the detailed structure of the neutrino Yukawa couplingsand neutrino mass matrix. The detailed analysis of the leptonic CP and ﬂavor violatingobservables in connection with the leptogenesis scenario is left for a future work [54].In this paper, we have concentrated on leptonic CP and ﬂavor violations. In SUSY GUTwith right-handed neutrinos, however, it is also notable that sizable oﬀ-diagonal elementsof squark mass matrices may be also generated via the renormalization group eﬀects. Inparticular, above the GUT scale, the neutrino Yukawa interactions aﬀect the renormalizationgroup runnings of the left-handed sdown mass matrix; such an eﬀect should also be sensitiveto the CI parameters. The renormalization group eﬀects on the squark mass matrices, aswell as hadronic CP and ﬂavor violations in connection with such eﬀects, will be studiedelsewhere [54]. 18 cknowledgment

The work of TM is supported by JSPS KAKENHI Grant Nos. 16H06490 and 18K03608.

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