1 Introduction



Flavor physics induced by light from SO(10) GUT

Junji Hisano, Yu Muramatsu, Yuji Omura and Yoshihiro Shigekami

Kobayashi-Maskawa Institute for the Origin of Particles and the Universe,

Nagoya University, Nagoya 464-8602, Japan

[3pt] Department of Physics, Nagoya University, Nagoya 464-8602, Japan

[3pt] Kavli IPMU (WPI), UTIAS, The University of Tokyo, Kashiwa,

Chiba 277-8583, Japan

[3pt] School of Physics, KIAS, Seoul 130-722, Republic of Korea

[3pt] Quantum Universe Center, KIAS, Seoul 130-722, Republic of Korea

In this paper, we investigate predictions of the SO(10) Grand Unified Theory (GUT), where an extra U(1) gauge symmetry remains up to the supersymmetry (SUSY) breaking scale. The minimal setup of SO(10) GUT unifies quarks and leptons into a -representational field in each generations. The setup, however, suffers from the realization of the realistic Yukawa couplings at the electroweak scale. In order to solve this problem, we introduce -representational matter fields, and then the two kinds of matter fields mix with each other at the SUSY breaking scale, where the extra U(1) gauge symmetry breaks down radiatively. One crucial prediction is that the Standard Model quarks and leptons are given by the linear combinations of the fields with two different U(1) charges. The mixing also depends on the flavor. Consequently, the U(1) interaction becomes flavor violating, and the flavor physics is the smoking-gun signal of our GUT model. The flavor violating couplings are related to the fermion masses and the CKM matrix, so that we can derive some explicit predictions in flavor physics. We especially discuss - mixing, - mixing, and the (semi)leptonic decays of and in our model. We also study the flavor violating and decays and discuss the correlations among the physical observables in this SO(10) GUT framework.

1 Introduction

The supersymmetric SO(10) Grand Unified Theory (GUT) is one of the promising candidates for the underlying theory of the Standard Model (SM). The GUT elegantly explains the origin of the SM gauge groups and shows that the SM matter fields can be unified into three-family -representational fields in the minimal SO(10) GUT [1]. In fact, several problems have been pointed out in the framework of the minimal setup, but the supersymmetric GUT deserves to be believed because of the beauty and the elegant explanations of the origins of not only the SM gauge groups but also the electroweak (EW) scale, so that a lot of solutions for the problems have been also proposed so far.

For instance, the unification of the SM matters, i.e. the unification of the Yukawa couplings, is a very attractive hypothesis, but unfortunately the precise experimental measurements of the masses and the CKM matrix require some deviation from the unified Yukawa couplings. One simple solution is to add higher-dimensional operators involving Higgs fields to break the SO(10) and SU(5) gauge symmetries [2].***Introducing additional Higgs fields [3] and additional matter fields [4] have been proposed so far. In the minimal SO(10) GUT, there is only up-type Yukawa coupling, , at the renormalizable level, but realistic Yukawa couplings could be effectively obtained by including such a higher-dimensional operator contribution. However, we have to assume that the additional contributions and are compatible and cancel each other, in order to realize the large mass hierarchy between top and bottom quarks, if is small. , which corresponds to the top quark mass is , and then the effective term should be also for the bottom quark mass.

Another issue is how to achieve the Higgs mass observed around 125 GeV. In the supersymmetric GUT, the EW scale is naturally derived and the lightest Higgs mass is predicted. The lower bound on the predicted Higgs mass is roughly the boson mass and shifted by the supersymmetry (SUSY) breaking scale. In order to achieve the 125 GeV mass, it is known that the SUSY breaking scale should be TeV [5], unless the SUSY spectrum is unique (e.g. see Refs. [6, 7, 8, 9]). In this high-scale SUSY scenario, however, the problem about the Yukawa couplings is revived because such TeV SUSY scale requires small for the 125 GeV Higgs mass. Thus, we have to consider some mechanisms to realize the large mass hierarchy especially between top and bottom quarks, in order to avoid the remarkably large coefficients of higher-dimensional operators.

In Ref. [10], the authors propose an extension of the minimal SO(10) GUT to explain the hierarchy in the high-scale SUSY scenario. In addition to the matter fields, three-family fields are introduced and the realistic Yukawa couplings are achieved by the mixing between two kinds of SU(5) -representational fields originated from and fields respectively. An interesting point is that interaction, predicted by SO(10) gauge symmetry, becomes flavor-dependent because the SU(5) -representational fields carry different U(1) charges [10]. Once we assume that U(1) is radiatively broken at the SUSY scale as the EW scale is, the flavor violating processes triggered by are verifiable in the flavor experiments, such as the LHCb, the Belle II, the COMET and the Mu2e experiments.Introduction of additional matter multiplets at low energy enhances proton decay by -boson exchange, since the gauge coupling constants at the GUT scale become larger [11]. If proton decay is discovered, embedding quarks and leptons to GUT multiplets may be resolved.

In this paper, we investigate our predictions of the flavor violating couplings quantitatively and discuss the flavor violating processes relevant to our SO(10) GUT. Especially, all elements of our Flavor Changing Neutral Currents (FCNCs) involving Z become large so that we should carefully check the consistencies with the observables related to the first and second generations: - mixing and lepton flavor violating decays. Besides, we find that the element of the couplings tends to be larger than the others because of the fermion masses, as we will see in Sec. 2.2. Then, we study physics as well: - mixing, and so on. We also show our prediction on motivated by the KOTO experiment. Then, we discuss lepton flavor violations (LFV) in our model. Interestingly, we could find some correlations between the observables of mesons and leptons. Then, we show our predictions for and the -e conversion in nuclei.

Our paper is organized as follows. In Sec. 2, we give a short review on our setup, based on Ref. [10]. Then, we show how well the realistic Yukawa couplings can be achieved and discuss our prediction of the Z FCNCs in Sec. 2.1. In Sec. 3, we study flavor physics in our SO(10) GUT, concentrating on the relevant processes: - mixing, - mixing, , and so on. We give some analyses on processes as well, but we will conclude that gives the strongest bound on our model. We also show the correlation between and LFV decays: and - conversion in nuclei in Sec. 3.3. Then, we see that our model could be tested at the COMET and the Mu2e experiments near future. Finally, we present some results for LFV decays in Sec. 3.4. Sec. 4 is devoted to summary.

2 Overview of the setup

In the minimal setup of the SO(10) GUT, the matter superfields belong to the representation and the Yukawa couplings are described by one matrix, :


where denote the generations and is the chiral superfield for the Higgs. includes all quarks and leptons in each generations, so that it is hard for this minimal setup to describe the mass hierarchies in the each sectors and the CKM matrix.

In Ref. [10], the authors propose a simple setup of the SO(10) GUT to realize the realistic Yukawa couplings at the EW scale. We introduce three -representational chiral superfields () in addition to . Then we write down the additional Yukawa couplings and mass terms for :


is an extra Higgs field to break the remaining U(1) symmetry. In order to sketch our idea, let us focus on the down-type quark sector, assuming that SO(10)-adjoint chiral superfields, and , break SO(10) to G U(1) at the GUT scale. There are two kinds of right-handed down-type quarks which carry different U(1) charges, after the symmetry breaking: , and , which are originated from the and . Involving the scalar component () of the SM singlet in , we find the mass matrixes for the down-type quarks induced by :


where denotes the nonzero VEV of the down-type Higgs doublet belonging to . As we see in Eq. (3), if develops nonzero VEV, and mix with each other and the lightest three down-type quarks can be interpreted as the SM down-type quarks. Note that is charged under U(1), so that non-vanishing VEV of spontaneously breaks U(1).

Let us define the mixing as follows:


where is the right-handed SM quark and is the extra heavy quark. In Eq. (4), the flavor index, , is omitted. is a unitary matrix, and and are matrices that satisfy, for instance,


The mixing unitary matrix, , is fixed by the parameters in the , following Eqs. (3) and (4). Now, let us simply consider the mixing in the limit that are much smaller than and . Then, the left-handed SM quarks are given by . The mixing for the right-handed quarks is given by the equation,


Using the parameters, the Yukawa couplings () to generate the SM down-type quark mass matrix is given by


is expected to explain the up-type SM quark mass matrix, so that matrix should be fitted to realized the mass hierarchy between the up-type and down-type quarks. However, it is difficult for to be realistic because of the relation in Eq. (5). The elements of could be , but cannot be too large because of the unitary condition. As discussed in Ref. [10], the mass hierarchy between top and bottom quarks can be achieved, but the other mass relations and the CKM matrix especially involving the first and second generations require too large , because of the very light up quark mass. In order to complement the suppression factors, one can introduce higher-dimension operators involving and fields and modify the relation in Eq. (7) as


denotes the suppression factor from the ratio between the VEVs of and and the unknown cut-off scale where the higher-dimensional operators are induced. are the Yukawa couplings for the up-type SM quarks and slightly deviated from , because of the higher-dimensional operators. are the free parameters in our model, and assumed to be .

In the same manner, we can discuss the lepton sector. If the SU(5) relation is respected approximately, the Yukawa couplings () for the charged lepton masses are given by . The experimental results, however, require slight SU(5) symmetry breaking effects. Then we introduce


where is the matrix which satisfies the relation in Eq. (5), replacing with . In principle, and ( and ) are different from each other, because the effective couplings generated by the VEVs of and are different. We could expect that the corrections of the higher-dimensional operators are sufficiently small in the effective and couplings, and then it would be reasonable to assume


In this case, the realistic Yukawa couplings are achieved by .

2.1 Requirements for the realistic Yukawa couplings

The up-type quark Yukawa couplings are defined as follows, without loss of generality:


where is the VEV of the up-type Higgs doublet and are the up-type quark masses, respectively. According to Eqs. (8) and (9), we find the equations which should be satisfied by the mixing parameters and coefficients of higher-dimensional operators:


where is the VEV of the down-type Higgs doublet and () are the down-type quark (lepton) masses, respectively. is the unitary matrix and identical to the CKM matrix () in the SU(5) limit. The other constraints on the matrices, and , are from Eq. (5) and the purturbativity.

Note that heavy modes are integrated out around the U(1) breaking scale, and then in Eqs. (8) and (9) are generated. In order to compare our predictions with the observed values of quark and lepton masses and CKM matrix, we need include the RG corrections from the U(1) breaking scale ( TeV) to the low scale, e.g. the EW scale ().

We evaluate the realistic Yukawa couplings at the U(1) breaking scale () from the central values of the experimental measurements summarized in Table 1. There are three scales relevant to our scenario: , gluino mass (around 1 TeV), and . First, we evolve the input parameters in Table 1 into the ones at the scale. We use Mathematica package RunDec [12] to evaluate the running quark masses. We translate lepton pole masses to running masses at the scale, following Ref. [13]. In our analysis, the up-type Yukawa coupling is defined as the diagonal form at , using the up-type quark masses. The down-type Yukawa coupling is given by the CKM matrix and the down-type quark.In fact, we can multiply arbitral unitary matrices to define the Yukawa couplings. When we match our predictions with the realistic Yukawa couplings, we do not take such degrees of freedom into account. Next, we drive the Yukawa matrices from the scale to TeV, using the SM RG running at the two-loop level [13]. We assume that all gaugino mass reside around 1 TeV, so that we convert the scheme into the scheme at 1 TeV according to Ref. [14] and drive the Yukawa matrices from TeV scale to TeV scale, including the gaugino contributions. In our scenario, the other SUSY particles reside around 100 TeV. As a result, we obtain the following values at 100 TeV:




Note that the quark and lepton masses, (), at 100 TeV are obtained, multiplying the running Yukawa couplings by GeV. at 100 TeV are given by Eqs. (11), (12), and (13), taking into account. In the next subsection, and are calculated, using the obtained and the relations in Eqs. (12) and (13).

0.5110 MeV [15] 0.22543 [16]
105.7 MeV [15] 0.8227 [16]
1.777 GeV [15] 0.1504 [16]
(2 GeV) 4.8 MeV [15] 0.3540 [16]
(2 GeV) 95 MeV [15] 91.1876(21) GeV [15]
4.18 GeV [15] GeV [15]
(2 GeV) 27.5 [15] 0.23126(5) [15]
1.275 GeV [15] 1.1663787(6) GeV [15]
173.21 GeV [15] 1/137.036 [15]
Table 1: The input parameters in our analysis. The CKM matrix, , is written in terms of , , and [15].

2.2 Flavor violating couplings

As we see in Eq. (4), the SM right-handed down-type quarks and left-handed leptons are given by the linear combinations of the parts of and in the SO(10) GUT. We consider the scenario that an extra U(1) symmetry remains up to the SUSY breaking scale. Then, we find that the particles from and carry different U(1) charges corresponding to the representations of SO(10). In fact, the U(1) charges of and are and , respectively, and the ones of and are and [10]. The U(1) symmetry breaking is triggered by the nonzero VEV of , and causes the mixing between the different-U(1)-charged fields. Consequently, the interaction becomes flavor violating as follows:


where , and are the mass eigenstates of the left-handed quarks, right-handed up-type quarks and right-handed charged leptons. Note that is not the mass eigenstate. This mixes with the Z boson, as mentioned below. are given by


Assuming the SU(5) relation in Eq. (10), and satisfy

Figure 1: Our predictions for (left) and (right). The coefficients of higher-dimensional operators satisfy (red) and (blue).
Figure 2: Our predictions for (left) and (right). The coefficients of higher-dimensional operators satisfy (red) and (blue).

Figs. 1, 2 and 3 show our predictions. In the all figures of this paper, is fixed at and the results in Eqs. (14) and (15) are used. In this calculation we assume that is the CKM matrix. The red (blue) points correspond to arbitral complex satisfying ().

Fig. 1 shows our prediction for and , which face the stringent bounds from - mixing. If we assume the GUT relation in Eq. (18), those elements are constrained by and - conversion in nuclei as well. As we see in Fig. 1, large is predicted, so we carefully study the physics and physics in Sec. 3.

Let us comment on the mixing to realize the realistic Yukawa coupling. In the left panel of Fig. 1, is approximately estimated as , i.e. . This means that the SM down quark mainly comes from the 10-representational fields of SO(10). The reason is as follows. We have introduced the higher dimensional operators, suppressed by , in order to compensate the small up quark mass. In fact, the contribution to the element of the up-type quark mass matrix, denoted by , is larger than the up quark mass. Then, the down quark mass is roughy given by the suppressed according to Eq. (12).

On the other hand, it seems that - and -representational fields mix with each other in the second and third generations, as in Figs. 1 and 2. is relatively smaller than the other off-diagonal elements, but could be according to the sizable . We find that tend to be larger than the other FCNC couplings, in Figs. 1 and 2. This is because is proportional to the down-type quark masses, and , so roughly speaking, the ratios of and are and , respectively, although the dependences of the quark masses and the CKM elements on are not so simple. When is small, the approximate expressions for the flavor violating couplings are

Figure 3: Our predictions for SU(5) relation. The coefficients of higher-dimensional operators satisfy (red) and (blue). Green diamond shows the value of each mass ratio of .

These properties are the same for and then we expect that the ratio between and is predictive even if Eq. (10) is failed. When is small, the ratio is expected to be . Our prediction of the ratio is shown in Fig. 3. These figures show that these ratios tend to be close to the green diamond, which satisfies , in the case with small . Especially, the convergence is remarkable in the elements, .

In addition, is the U(1) gauge boson, but not the mass eigenstate because of mass mixing between and boson denoted by . The mass mixing is generated by the U(1)-charged Higgs doublets [10]:


where is approximately estimated as


We have to include this effect, when we discuss the phenomenology in our model.

Note that a scalar from also has flavor changing Yukawa couplings with the SM fermions and the heavy extra fermions, but the left-handed down-type quarks (right-handed lepton) can be indentified as the heavy fermions, because of the EW symmetry. Then, the flavor violating processes involving the scalar are loop-suppressed and negligibly small in our scenario.

3 Flavor Physics

In this section, we investigate the flavor violating signals predicted by our SO(10) GUT, based on the setup discussed above. One of the important predictions is that there are tree-level FCNCs involving and . Moreover, all elements of the FCNCs could be , corresponding to the higher-dimensional operators. This means that we have to seriously check the consistency with the flavor violating processes concerned with the first and second generations, such as - mixing and , because the processes are the most sensitive to the new physics contributions. Besides, we find that element of the coupling becomes larger than the other, so that we investigate the impact of our model on physics, as well.

The SUSY particle contributions to FCNCs are suppressed by loop factors due to the parity. However, when the flavor violation in squark mass terms is maximal, the SUSY contributions to the system may not be negligible even if squark masses are TeV. Then, we ignore them for simplicity.

First, we study the constraints from the processes in and systems in the next subsection, and then let us discuss the consistency of our model with the observations of the LFV decays in Sec. 3.3. We also study the processes, although the constraints are mild.

497.611(13) MeV [15] 5.3663(6) GeV [15]
156.1(11) MeV [18] 5.2795(3) GeV [15]
0.764(10) [18] 227.7 6.2 MeV [18]
3.484(6) MeV [15] 190.6 4.6 MeV [18]
[15] 1.33(6) [18]
BR % [15] 1.26(11) [18]
(1.238(2)) s [15] 0.55 [20]
(5.116(21)) s [15] 1.012 [39]
1.87(76) [19] s
0.5765(65) [20]
0.496(47) [21]
Table 2: The input parameters relevant to our analyses in flavor physics.

3.1 processes

In the SM, CP violation is caused by the CP phase in the CKM matrix. CP violating processes as well as flavor violating processes are strongly suppressed by the GIM mechanism, and the SM predictions are usually very tiny. The flavor processes of meson are no exception. In fact, the SM prediction of - mixing is quite small, but it is consistent with the experimental observations, although there are still sizable theoretical uncertainties in the SM predictions. In other words, large new physics contributions to the physics conflict with the experimental results, and then the strong constraints should be taken into account. Similarly, we can derive the new physics constraints from - and - mixing.

In addition to the SM corrections, the processes are caused by the tree-level FCNCs of and in our model. The induced operators are


where the each operator is given by


and the Wilson coefficient is estimated as


and can be derived by exchanging in with and respectively. Note that the SM correction appears in the (), which are the coefficients of the operators that consist of only left-handed quarks, instead of the right-handed in : for example, . The CP-phase appears in the -element of the CKM matrix in the SM. In our model, the FCNCs, , are generally complex, so that the CP-violating processes strongly constrain our interaction.

In our analyses on flavor physics, we set TeV (500 TeV), which corresponds to TeV (36 TeV) and [10]. is fixed at to achieve the 125 GeV Higgs mass [5].

3.1.1 process

Based on Ref. [17], we investigate the upper bound on the interaction from the - mixing. The physical observables on the mixing are denoted by and , which are evaluated as


and are given by the observations: and . is generated by the - mixing and decomposed as follows in our model:


is the contribution, and then it is given by


The matrix element, , can be extracted from the SM prediction, because the only difference is the chirality. is the Wilson coefficient derived from Eq. (24) and the RG correction. The running correction is studied in Appendix A.

The SM prediction is described as


and denote and , respectively. correspond to the NLO and NNLO QCD corrections [19, 20, 21]. The values we adopt are summarized in Table 2. The functions, and , are shown in Appendix B.

The physical observables in - mixing are experimentally measured well. On the other hand, the SM predictions still suffer from the large uncertainty from the matrix element and the CKM matrix. Using the central values in Table 1 and Table 2, we draw our predictions for the deviations of and from the SM predictions.

Figure 4: Our predictions for and with TeV (left) and TeV (right). The coefficients of higher-dimensional operators satisfy (red) and (blue). Black dashed, solid and dotted line show the deviation from SM by , and , respectively.

Compared to the SM predictions, and , the deviations are defined as


It is difficult to draw the exclusion limits in terms of and , because of the large uncertainties of the SM predictions. In Ref. [22], the CKM fitter group shows that the experimental upper bounds on and are at most O(30) %. It will be developed up to O(20) % at the Belle II experiment [22].

In Fig. 4, our predictions for the deviations of and are shown in the cases with TeV (left) and TeV (right). The black dashed, solid and dotted line show the deviation from SM by , and , respectively. In our model, largely departs from the SM prediction, even if is TeV. Then, we have to consider the consistency with , whenever we discuss the other observables.

3.1.2 process

We now derive our predictions of - and - mixing, as well as - mixing. The observables relevant to the mixing are mass differences denoted by and . They are influenced by and as follows:


where is given by the top-loop contribution:


The input parameters used in our analyses are shown in Table 2. depicts . The SM predictions still have large uncertainties dominated by the errors of hadronic mixing matrix elements and the CKM matrix elements, so that it would be difficult to draw the new physics constraints as well. Recently, the Fermilab and MILC Collaborations have shown their results on the SM predictions of and [23] and about 10 % errors are still inevitable. The LHCb and Belle II experiments will improve the measurement, as discussed in Ref. [22].

In our model, is large compared to the other elements, so that our model may be tested by , although the deviation is relatively smaller than the - mixing because of the size of the SM prediction. Fig. 5 shows our predictions for the deviations of and in the cases with TeV (left) and TeV (right). These deviation are defined as the same manner in Eq. (29). If is around TeV, could reach 10 %, which maybe cause the tension with the current measurement [22]. In these figures, all points satisfy .

Figure 5: Our predictions for and with TeV (left) and TeV (right). The coefficients of higher-dimensional operators satisfy (red) and (blue). In these figures, we only show the points that is satified.

3.2 processes

The interaction deviates the SM predictions in the rare decays of and mesons. The KOTO, Belle II and LHCb experiments will develop the measurements of the rare decays and give some hints to new physics. In this section, we study the (semi) leptonic decays of meson and the leptonic decays of and . The processes we especially study here are , measured by the KOTO experiment, , and .

3.2.1 processes

The processes, such as the rare meson decays, play a crucial role in testing our model. The effective Hamiltonian which causes the tree-level flavor changing is given by the exchanging and boson exchanging through the - mixing:


where denotes and is the chirality of the fermions () (). at is described as


where is defined as


Note that the second term is approximately evaluated as according to Eq.(21), in the limit . are given by Eq. (16) as