# Sensitivity of High-Scale SUSY in Low Energy Hadronic FCNC

^{1}

^{2}

^{*}

## Abstract

**:**We discuss the sensitivity of the high-scale supersymmetry (SUSY) at 10–1000 TeV in B

^{0}, B

_{s}, K

^{0}and D meson systems together with the neutron electric dipole moment (EDM) and the mercury EDM. In order to estimate the contribution of the squark flavor mixing to these flavor changing neutral currents (FCNCs), we calculate the squark mass spectrum, which is consistent with the recent Higgs discovery. The SUSY contribution in ϵ

_{K}could be large, around 40% in the region of the SUSY scale 10–100 TeV. The neutron EDM and the mercury EDM are also sensitive to the SUSY contribution induced by the gluinosquark interaction. The predicted EDMs are roughly proportional to $\left|{\u03f5}_{K}^{\text{SUSY}}\right|$. If the SUSY contribution is the level of $\mathcal{O}$ (10%) for ϵ

_{K}, the neutron EDM is expected to be discovered in the region of 10

^{−28}–10

^{−26}ecm. The mercury EDM also gives a strong constraint for the gluinosquark interaction. The SUSY contribution of ΔM

_{D}is also discussed.

## 1. Introduction

Supersymmetry (SUSY) is one of the most attractive theories beyond the standard model (SM). Therefore, SUSY has been expected to be observed at the LHC experiments. However, no signals of SUSY have been discovered yet. The present searches for SUSY particles give us important constraints for SUSY. Since the lower bounds of the superparticle masses increase gradually, the squark and the gluino masses are supposed to be at a higher scale than 1 TeV [1–3]. On the other hand, the SUSY model has been seriously constrained by the Higgs discovery, in which the Higgs mass is 125 GeV [4–6]. Based on this theoretical and experimental situation, we consider the high-scale SUSY models, which have been widely discussed with a great deal of attention [7–22].

If the squark and slepton masses are at the high-scale $\mathcal{O}$ (10–1000) TeV, the lightest Higgs mass can be pushed up to 125 GeV, whereas SUSY particles are out of the reach of the LHC experiment. Therefore, the indirect search of the SUSY particles becomes important in the low-energy flavor physics [23–25].

The flavor physics is also on the a stage in light of LHCb data. The LHCb collaboration has reported new data of the CP violation of the B_{s} meson and the branching ratios of rare B_{s} decays [26–38]. For many years, the CP violation in the K and B^{0} mesons has been successfully understood within the framework of the standard model (SM), the so-called Kobayashi–Maskawa (KM) model [39], where the source of the CP violation is the KM phase in the quark sector with three families. However, a new physics has been expected to be indirectly discovered in the precise data of B^{0} and B_{s} meson decays at the LHCb experiment and the further coming experiment, Belle-II.

There are new sources of the CP violation if the SM is extended to the SUSY models. The soft squark mass matrices contain the CP violating phases, which contribute to the flavor changing neutral current (FCNC) with the CP violation [40]. Therefore, we can expect the SUSY effect in the CP violating phenomena. However, the clear deviation from the SM prediction has not been observed yet in the LHCb experiment [26–38]. Actually, we have found that the CP violation of B^{0} and B_{s} meson systems are suppressed if the SUSY scale is above 10 TeV [41]. On the other hand, the CKM fitter group presented the current limits on new physics contributions of
$\mathcal{O}$ (10%) in B^{0}, B_{s} and K^{0} systems [42]. They have also estimated the sensitivity to new physics in B^{0} and B_{s} mixing achievable with 50 ab^{−1} of Belle-II and 50 fb^{−1} of LHCb data. Therefore, we should carefully study the sensitivity of the high-scale SUSY to the hadronic FCNC.

In this work, we discuss the high-scale SUSY contribution to the B^{0}, B_{s} and K^{0} meson systems. Furthermore, we also discuss the sensitivity to the D meson and the electric dipole moment (EDM) of the neutron and mercury. For these modes, the most important process of the SUSY contribution is the gluino-squark-mediated flavor changing process [43–58]. The CP violation of the K meson, ϵ_{K}, provides a severe constraint to the gluino-squark-mediated FCNC [59,60]. In addition, recent work has found that the chromo-electric dipole moment (cEDM) is sensitive to the high-scale SUSY [61]. It is noted that the upper-bound of the neutron EDM (nEDM) [62] gives a severe constraint for the gluino-squark interaction through the cEDM [63–68]. It is also remarked that the upper bound of the mercury EDM (HgEDM) [69] can give an important constraint [70].

In order to estimate the gluino-squark-mediated FCNC of the K, B^{0}, B_{s} and D mesons, we work in the basis of the squark mass eigenstate with the non-minimal squark (slepton) flavor mixing. There are three reasons why the SUSY contribution to the FCNC considerably depends on the squark mass spectrum. The first one is that the GIM mechanism works in the squark flavor mixing, and the second one is that the loop functions depend on the mass ratio of the squark and gluino. The last one is that we need the mixing angle between the left-handed sbottom and right-handed sbottom, which dominates the ΔB = 1 decay processes. Therefore, we discuss the squark mass spectrum, which is consistent with the recent Higgs discovery. Taking the universal soft parameters at the SUSY breaking scale, we obtain the squark mass spectrum at the matching scale where the SM emerges, by using the renormalization group equations (REGs) of the soft masses. On the other hand, the 6 × 6 mixing matrix between squarks and quarks is taken to be free at the low energy.

In Section 2, we discuss the squark and gluino mass spectrum and the squark mixing. In Section 3, we present the formulation of the FCNC with ΔF = 2 in the K, B^{0}, B_{s} and D meson systems together with nEDM and HgEDM. We present numerical results and discussions in Section 4. Section 5 is devoted to the summary. The relevant formulations are presented in Appendices A–D.

## 2. SUSY Spectrum and Squark Mixing

The low-energy FCNCs depend significantly on the spectrum of the SUSY particles, which depend on the model. As is well known, the lightest Higgs mass can be pushed up to 125 GeV if the squark masses are expected to be $\mathcal{O}$ (10) TeV. Therefore, let us consider the heavy SUSY particle mass spectrum in the framework of the minimal supersymmetric standard model (MSSM), which is consistent with the observed Higgs mass. The discussion of how to obtain the SUSY spectrum has been given in [71,72].

We outline how to obtain the SUSY spectrum in our work. The details are presented in Appendix A. At the SUSY breaking scale Λ, we write the quadratic terms in the MSSM potential as:

Then, the Higgs mass parameter m^{2} is expressed in terms of
${m}_{1}^{2}$,
${m}_{2}^{2}$ and tan β as:

After running down to the Q_{0} scale, in which the SM emerges, by the one-loop SUSY renormalization group equations (RGEs) [73], the scalar potential is the SM one as follows:

Here, the Higgs coupling λ is given in terms of the SUSY parameters at the leading order as:

_{t}is the top Yukawa coupling of the SM. The parameters m

_{2}and λ run with the two-loop SM RGEs with the $\overline{\mathrm{MS}}$ scheme [74–76] down to the electroweak scale Q

_{EW}= m

_{H}and then give:

When m_{H} = 125 GeV is placed, λ(Q_{0}) and m^{2} (Q_{0}) are obtained. This input constrains the SUSY mass spectrum of the MSSM. In our work, we take the universal soft breaking parameters at the SUSY breaking scale Λ as follows:

By inputting m_{H} = 125 GeV and taking the heavy scalar mass
${m}_{\mathscr{H}}\simeq {Q}_{0}$ (see Appendix A), we can obtain the SUSY spectrum for the fixed Q_{0} and tan β. The details and numerical results are presented in Appendix A.

Let us consider the squark flavor mixing. As discussed above, there is no flavor mixing at Λ in the MSSM. However, in order to consider the non-minimal flavor mixing framework, we allow the off-diagonal components of the squark mass matrices at the 10% level, which leads to the flavor mixing of order 0.1. We take these flavor mixing angles as free parameters at low energies. Now, we consider the 6 × 6 squark mass matrix ${M}_{\tilde{q}}$ in the super-CKM basis. In order to move the mass eigenstate basis of squark masses, we should diagonalize the mass matrix by rotation matrix ${\mathrm{\Gamma}}_{G}^{(q)}$ as:

_{θ}= cos θ and s

_{θ}= sin θ. Here, θ is the left-right mixing angle between ${\tilde{b}}_{L}$ and ${\tilde{b}}_{R}$, which is discussed in Appendix A. It is remarked that we take ${s}_{12}^{L,R}=0$ due to the degenerate squark masses of the first and second families, as discussed in Appendix A.

The gluino-squark-quark interaction is given as:

^{i}are three left-handed (i = 1, 2, 3) and three right-handed quarks (i = 4, 5, 6). This interaction leads to the gluino-squark-mediated flavor changing processes with ΔF = 2 and ΔF = 1 through the box and penguin diagrams.

The chargino (neutralino)-squark-quark interaction can be also discussed in a similar way.

## 3. FCNC of ΔF = 2

In our previous work [41], we have probed the high-scale SUSY, which is at the 10–50 TeV scale, in the CP violations of K, B^{0} and B_{s} mesons. It is found that ϵ_{K} is most sensitive to SUSY, even if the SUSY scale is at 50 TeV. The SUSY contributions for the time-dependent CP asymmetries of B^{0} and B_{s} with ΔB = 1 are suppressed at the SUSY scale of 10 TeV. Furthermore, the SUSY contribution for the b → sγ process is also suppressed, since the left-right mixing angle, which induces the chiral enhancement, is very small, as discussed in Appendix A. Therefore, we discuss the neutral meson mixing
${P}^{0}-{\overline{P}}^{0}$ (P^{0} = K, B^{0}, B_{s}, D), which are FCNCs with ΔF = 2.

In those FCNCs, the dominant SUSY contribution is given through the gluino-squark interaction. Then, the dispersive part of meson mixing
${M}_{12}^{{P}^{0}}$ (P^{0} = K, B^{0}, B_{s}) is written as:

At first, we discuss the ΔB = 2 process, that is the mass differences
$\mathrm{\Delta}{M}_{{B}^{0}}$ and
$\mathrm{\Delta}{M}_{{B}_{s}}$ and the CP-violating phases ϕ_{d} and ϕ_{s}. In general, the contribution of the new physics (NP) to the dispersive part
${M}_{12}^{q}$ is parameterized as:

^{0}and B

_{s}mixing have given the constraints on (h

_{q}, σ

_{q}) [42], where it is assumed that the NP does not significantly affect the SM tree-level charged-current interaction, that is the absorptive part ${\mathrm{\Gamma}}_{12}^{q}$ is dominated by the decay $b\to c\overline{c}s.$ At present, the NP contributions h

_{q}are 10%–35% and 15%–25%, depending on σ

_{q}for B

^{0}and B

_{s}, respectively. Thus, we can expect the sizable NP contribution of $\mathcal{O}$ (20%). We will discuss whether the high-scale SUSY can fill in the magnitude of the present NP contribution of $\mathcal{O}$ (20%).

Next, we discuss the ΔS = 2 process,
$\mathrm{\Delta}{M}_{{K}^{0}}$ and the CP-violating parameter in the K meson, ϵ_{K}. By the similar parametrization in Equation (11), the allowed region of (h_{K}, σ_{K}) has been estimated in [42]. The NP contribution is at least 50%, although there is the strong σ_{K} dependence. Therefore, it is important to examine carefully the CP violating parameter ϵ_{K}, which is given as follows:

_{K}is the mass difference in the neutral K meson. The effects of ξ ≠ 0 and φ

_{ϵ}< π/4 give a suppression effect in ϵ

_{K}, and it is parameterized as κ

_{ϵ}and estimated by Buras and Guadagnoli [77] as:

In the SM, the dispersive part ${M}_{K}^{12}$ is given as follows,

_{cc,tt,ct}are the QCD corrections [77]. Then, $\left|{\u03f5}_{K}^{\mathrm{SM}}\right|$ is given in terms of the Wolfenstein parameters λ, $\overline{\rho}$ and $\overline{\eta}$ as follows:

Note that $\left|{\u03f5}_{K}^{\mathrm{SM}}\right|$ depends on the non-perturbative parameter ${\widehat{B}}_{K}$ in Equation (15). Recently, the error of this parameter shrank dramatically in the lattice calculations [79]. In our calculation, we use the updated value by the flavor Lattice averaging group [80]:

Let us write down ϵ_{K} as:

In addition to the above FCNC processes, the neutron EDM, d_{n}, arises through the cEDM of the quarks,
${d}_{q}^{c}$, due to the gluino-squark mixing [63–68]. By using the QCD sum rules, d_{n} is given as:

_{q}and ${d}_{q}^{c}$ denote the EDM and cEDM of quarks ${d}_{q}^{c}$ defined in Appendix D. On the other hand, by using the chiral perturbation theory:

Therefore, the experimental upper bound [62]:

The HgEDM can also probe the gluino-squark mixing [70]. The QCD sum rule approach gives [81]:

The experimental upper bound [69]:

At the last step, we discuss the charmsector, which is a promising field to probe for the new physics beyond the SM. The D^{0}−D^{0} mixing is now well established [83] as follows:

_{D}and ΔΓ

_{D}are the differences of the masses and the decay widths between the mass eigenstates of the D meson, respectively, and Γ

_{D}is the averaged decay width of the D meson. Since the SM prediction of ΔM

_{D}at the short distance is much suppressed compared with the experimental value due to the bottom quark loop, the SUSY contribution may be enhanced.

## 4. Results and Discussions

Let us estimate the SUSY contribution of the low-energy FCNC. We calculate the SUSY mass spectrum at Q_{0} = 10, 50, 100, 1000 TeV and interpolate the each mass of the SUSY particle in the region of Q_{0} = 10–1000 TeV. This approximation is satisfied within
$\mathcal{O}$ (10%). Therefore, our numerical results should be taken with the ambiguity of
$\mathcal{O}$ (10%). The mass spectrum at Q_{0} = 10 TeV is presented in Appendix A. See [41,60] for the mass spectrum at Q_{0} = 50 TeV.

Then, we have four mixing angles
${\theta}_{13}^{L(R)}$ and
${\theta}_{23}^{L(R)}$, five phase
${\varphi}_{13}^{L(R)}$,
${\varphi}_{23}^{L(R)}$,
$\varphi $ We reduce the number of parameters by taking
$\mathrm{sin}{\theta}_{ij}^{\mathrm{L}}$ =
$\mathrm{sin}{\theta}_{ij}^{\mathrm{R}}$≡ s_{ij} for simplicity. In the numerical calculations, we scan the phases of Equation (8) in the region of 0 ∼ 2π for fixed s_{ij}, where the Cabibbo angle 0.22 and the large angle 0.5 are taken as the typical mixing. Other relevant input parameters, such as quark masses m_{c}, m_{b}, the CKM parameters V_{us}, V_{cb},
$\overline{\rho}$,
$\overline{\eta}$ and f_{B}, f_{K}, etc., have been presented in our previous paper [57], which are referred from the UTfit Collaboration [84] and PDG [62].

#### 4.1. B^{0} and B_{s} Meson Systems

At first, we examine the SUSY contribution in the ΔB = 2 process. We show the SUSY scale
${m}_{\tilde{Q}}\equiv {Q}_{0}$ dependence of the SUSY contributions of
$\Delta {M}_{{B}^{0}}$ and
$\mathrm{\Delta}{M}_{{B}_{s}}$ in Figure 1a,b, where the experimental central value is shown by the red line. The experimental error bars are the 1% and 0.1% levels for
$\mathrm{\Delta}{M}_{{B}^{0}}$ and
$\mathrm{\Delta}{M}_{{B}_{s}}$, respectively. We take s_{13} = s_{23} = 0.22, 0.5. There is no phase dependence in our predictions. It is found that the SUSY contributions in
$\mathrm{\Delta}{M}_{{B}^{0}}$ and
$\mathrm{\Delta}{M}_{{B}_{s}}$ are at most 1.5% and 0.1% at
${m}_{\tilde{Q}}=10\text{TeV}$, respectively. Namely, the high-scale SUSY cannot explain the NP contributions of h_{d} = 0.1–0.35 and h_{s} = 0.15–0.25, which have been discussed in Equation (11). As
${m}_{\tilde{Q}}$ increases, the SUSY contributions of both
$\mathrm{\Delta}{M}_{{B}^{0}}$ and
$\mathrm{\Delta}{M}_{{B}_{s}}$ decrease approximately with the power of
$1/{m}_{\tilde{Q}}^{2}$. Thus, there is no hope to observe the SUSY contribution in the ΔB = 2 process for the high-scale SUSY. It should be noted that the SM predictions are comparable to these experimental data.

The related phenomena are the CP violations of the non-leptonic decays B^{0} → J/ψK_{S} and B_{s} → J/ψφ. The recent experimental data of these phases are [29,36–38]:

_{d}and φ

_{s}are expressed in terms of the parameters of Equation (11) as [57]:

_{d}(β

_{s}) is the one angle of the unitarity triangle giving by the CKM matrix elements of the SM. However, h

_{d}and h

_{s}in the high-scale SUSY are much suppressed compared with h

_{d}= 0.1–0.35 and h

_{s}= 0.15–0.25 of Equation (11), and one cannot find signals of the high-scale SUSY in the CP violating decays B

^{0}→ J/ψK

_{S}and B

_{s}→ J/ψφ.

#### 4.2. Neutral K Meson System

At the second step, we examine the neutral K meson. We show the SUSY contributions of
${M}_{{K}^{0}}$ and ϵ_{K} versus${m}_{\overline{Q}}\equiv {Q}_{0}$ in Figure 2a,b, where the experimental central value is shown by the horizontal red line. The experimental error bars are the 0.2% and 0.5% levels for
${M}_{{K}^{0}}$ and ϵ_{K}, respectively. Since
${\theta}_{12}^{L,R}=0$, the SUSY flavor mixing arises from the second order of s_{13} × s_{23}, where s_{13} = s_{23} = 0.22, 0.5 are placed.

It is found in Figure 2a that the SUSY contribution in
${M}_{{K}^{0}}$ can be comparable to the experimental value in the case of s_{13} = s_{23} = 0.5, whereas it is suppressed in the case of s_{13} = s_{23} = 0.22 at
${m}_{\tilde{Q}}=10\text{TeV}$. Thus, ΔM_{K}0 constrains the squark mixing of s_{13} and s_{23} around
${m}_{\tilde{Q}}=10\text{TeV}$. When the SUSY scale increases to more than 20 TeV, no SUSY contribution is expected.

On the other hand, ϵ_{K} is very sensitive to the SUSY contribution up to 100 TeV, as seen in Figure 2b. The plot is scattered due to the random phases of the squark mixing. The experimental data of ϵ_{K} constrain the squark mixing and phases considerably. Actually, we have already pointed out that the SUSY contribution in _{K} could be 40% and 35% at
${m}_{\tilde{Q}}=10$, 50 TeV, respectively [41]. It is found that this sizable SUSY contribution still exist up to 100 TeV in this work.

In the SM, there is only one CP violating phase. Therefore, the observed value of φ_{d} in Equation (27), should be correlated with ϵ_{K} in the SM. According to the recent experimental results, it is found that the consistency between the SM prediction and the experimental data of sin φ_{d} and ϵ_{K} is marginal. This fact was pointed out by Buras and Guadagnoli [77] and called the tension between ϵ_{K} and sin φ_{d}. Considering the effect of the SUSY contribution
$\mathcal{O}$ (10%) in ϵ_{K}, this tension can be relaxed even if
${m}_{\tilde{Q}}=10\text{TeV}$. The precise determination of the unitarity triangle of B^{0} is required in order to find the SUSY contribution of this level.

It is noted that the SUSY contribution of both ΔM_{K}0 and ϵ_{K} also decrease approximately with the power of
$1/{m}_{\tilde{Q}}^{2}$ as
${m}_{\tilde{Q}}$ increases up to 1000 TeV.

#### 4.3. The nEDM and HgEDM with ϵ_{K}

The nEDM and HgEDM are also sensitive to the SUSY contribution [61,70]. The gluino-squark interaction leads to the cEDM of quarks, which give the nEDM as shown in Equations (19) and (20). We show the predicted nEDM versus${m}_{\tilde{Q}}$ for the case of the QCD sum rules of Equation (19) in Figure 3a, where the upper bound of |d_{n}| is shown by the red line. The plot is scattered due to the random phases of the squark mixing, as well as in the case of ϵ_{K}. We find that the contributions of EDM, d_{d} and d_{u} occupy around 25% of the neutron EDM. The SUSY contribution is close to the experimental upper bound up to 50 TeV. Since the predicted nEDM depends on the phases of the squark mixing matrix significantly, we plot the nEDM versus$|{\u03f5}_{K}^{\text{SUSY}}|$ in Figure 3b. It is found that the predicted nEDM is roughly proportional to
$|{\u03f5}_{K}^{\text{SUSY}}|$. If the SUSY contribution is the level of
$\mathcal{O}$ (10%) for ϵ_{K}, the nEDM is expected to be discovered in the region of 10^{−27}–10^{−26} cm. On the other hand, if the nEDM is not observed above 10^{−28} cm, the SUSY contribution of ϵ_{K} is below a few %. Thus, there is the correlation between d_{n} and
${\u03f5}_{K}^{\text{SUSY}}$.

We also show the predicted HgEDM versus${m}_{\tilde{Q}}$ for the case of the QCD sum rules of Equation (22) in Figure 4a, where the upper bound of
$|{d}_{{H}_{g}}|$ is shown by the red line. The SUSY contribution is close to the experimental upper bound up to 200 TeV, which is much higher than the one of the nEDM. In Figure 4b, we plot the HgEDM versus$|{\u03f5}_{K}^{\text{SUSY}}|$. It is found that the experimental upper bound of the HgEDM excludes completely
$|{\u03f5}_{K}^{\text{SUSY}}|$, which is inconsistent with the experimental data. If the SUSY contribution is the level of
$\mathcal{O}$ (10%) for ϵ_{K}, the nEDM is expected to be discovered in the region of 10^{−27}–10^{−26} cm. If the HgEDM is not observed above 10^{−29} cm, the SUSY contribution of ϵ_{K} is below a few %. Thus, the mercury EDM gives more significant information for the gluino-squark interaction compared with the neutron EDM.

However, these correlations strongly depend on the assumptions of ${\theta}_{23}^{L}={\theta}_{13}^{L}$ and ${\theta}_{ij}^{L}={\theta}_{ij}^{R}$. The deviation from these relations destroys these correlations. For instance, for the case of ${\theta}_{23}^{L}\gg {\theta}_{13}^{L}$ with ${\theta}_{ij}^{L}={\theta}_{ij}^{R}$, ${\u03f5}_{K}^{\text{SUSY}}$ is much suppressed, whereas the nEDM and HgEDM are still sizable. On the other hand, if ${\theta}_{ij}^{L}\gg {\theta}_{ij}^{R}$ or ${\theta}_{ij}^{L}\gg {\theta}_{ij}^{R}$ is realized, the cEDMs are suppressed, because they require the chirality flipping. In conclusion, careful studies of the mixing angle relations are required to test the correlations between EDMs and ${\u03f5}_{K}^{\text{SUSY}}$.

We should comment on the hadronic model dependence of our numerical result. For both nEDM and HgEDM, we show the numerical result by using the hadronic model of the QCD sum rules in Equations (19) and (22). We have also calculated the EDMs by using the hadronic model of the chiral perturbation theory in Equations (20) and (23). For the neutron EDM, the prediction of the chiral perturbation theory is larger than the one of the QCD sum rule at most of a factor of two. However, for the mercury EDM, the prediction of the QCD sum rule is more than three-times larger compared with the one of the chiral perturbation theory. Thus, predicted EDMs have ambiguity with a factor of 2–3 from the hadronic model.

#### 4.4. $D-\overline{D}$ Mixing

Since the SM prediction of ΔM_{D} at the short distance is
$\mathcal{O}$ (10^{−18}) GeV, which is very small compared with the experimental value due to the bottom quark loop, it is important to estimate the SUSY contribution of ΔM_{D}. The mixing angle
${\theta}_{ij}^{L(R)}$${\theta}_{ij}^{L(R)}$ also appears in the up-type squark mixing matrix, whereas the down-type squark mixing matrix contributes to the K^{0}, B^{0} and B_{s} meson systems induced by the gluino-squark-quark interaction.

We show the SUSY component of ΔM_{D} and x_{D} versus${m}_{\tilde{Q}}$ for s_{13} = s_{23} = 0.22, 0.5 in Figure 5. At the SUSY scale of 10 TeV, the SUSY component may be comparable to the observed value. Although the accurate estimate of the long-distance effect is difficult, Cheng and Chiang estimated x_{D} of order 10^{−3} from the two body hadronic modes [85]. This obtained value is consistent with the experimental one. Therefore, we should take into account the long-distance effect properly in order to constrain the SUSY contribution from ΔM_{D}.

Before closing the presentation of the numerical results, we add a comment on the other gaugino contribution. There are additional contributions to the FCNC induced by chargino exchanging diagrams. The chargino contribution to the gluino one is approximately 10% in the above numerical study of ΔF = 2. Thus, the chargino contributions are the sub-leading ones.

## 5. Summary

We discussed the sensitivity of the high-scale SUSY at 10–1000 TeV in the B^{0}, B_{s} and K^{0} meson systems. Furthermore, we have also discussed the sensitivity to the
$D-\overline{D}$ mixing, the neutron EDM and the mercury EDM. In order to estimate the contribution of the squark flavor mixing to these FCNC, we calculate the squark mass spectrum, which is consistent with the recent Higgs discovery.

The SUSY contributions in
$\mathrm{\Delta}{M}_{{B}_{s}}$ and ΔM_{Bs} are at most 1.5% and 0.1% at
${m}_{\tilde{Q}}=10\text{TeV}$, respectively. As
${m}_{\tilde{Q}}$ increases, the SUSY contributions of both
$\mathrm{\Delta}{M}_{{B}^{0}}$ and
$\mathrm{\Delta}{M}_{{B}_{s}}$ decrease approximately with the power of
$1/{m}_{\tilde{Q}}^{2}$. Therefore, the SUSY scale increases to more than 10 TeV, and no signal of SUSY is expected. On the other hand, the SUSY contribution in
${M}_{{K}^{0}}$ can be comparable to the experimental value in the case of s_{13} = s_{23} = 0.5, whereas it is suppressed in the case of s_{13} = s_{23} = 0.22 at
${m}_{\tilde{Q}}=10\text{TeV}$. Furthermore, the SUSY contribution in ϵ_{K} could be large, around 40% in the region of the SUSY scale 10–100 TeV. By considering the effect of the SUSY contribution O(10%) in ϵ_{K}, the tension between ϵ_{K} and sin φ_{d} can be relaxed even if the SUSY scale is 100 TeV.

The neutron EDM and the mercury EDM are also sensitive to the SUSY contribution induced by the gluino-squark interaction. The |d_{n}| is expected to be close to the experimental upper bound, even if the SUSY scale is 50 TeV. The predicted nEDM is roughly proportional to
$|{\u03f5}_{K}^{\text{SUSY}}|$. If the SUSY contribution is the level of
$\mathcal{O}$ (10%) for ϵ_{K}, the |d_{n}| is expected to be discovered in the region of 10^{−27}−10^{−26} cm. For the |d_{Hg}|, the SUSY contribution is close to the experimental upper bound up to 200TeV, which is much higher than the one of the nEDM. If the HgEDM is not observed above 10^{−29} cm, the SUSY contribution of ϵ_{K} is below a few %. Thus, the mercury EDM gives more significant information for the gluino-squark interaction compared with the neutron EDM. It may be important to give a comment that these predictions depend strongly on the assumptions of
${\theta}_{23}^{L}={\theta}_{13}^{L}$ and
${\theta}_{ij}^{L}={\theta}_{ij}^{R}$. The deviation from these relations destroys these correlations. In conclusion, careful studies of the mixing angle relations are required to test the correlations between EDMs and
${\u03f5}_{K}^{\text{SUSY}}$. The predicted EDMs have also ambiguity with the factor of 2–3 from the hadronic model.

Since the SM prediction of ΔM_{D} at the short distance is
$\mathcal{O}$ (10^{−18}) GeV, which is very small compared with the experimental value, it is important to estimate the SUSY contribution of ΔM_{D}.

In conclusion, more detailed studies of the K^{0} meson system, the EDMs of the neutron and mercury are required in order to probe the high-scale SUSY at 10–1000 TeV.

## Acknowledgments

This work is supported by JSPS Grant-in-Aid for Scientific Research, 24654062 and 25-5222.

## Author Contributions

Morimitsu Tanimoto and Kei Yamamoto conceived of and discussed ideas of this work, did the calculations and wrote the paper.

## Conflicts of Interest

The authors declare no conflict of interest.

## Appendix

## A. Running of SUSY Particle Masses

In the framework of the MSSM, one obtains the SUSY particle spectrum, which is consistent with the observed Higgs mass. The numerical analyses have been given in [71,72]. At the SUSY breaking scale Λ, the quadratic terms in the MSSM potential are given as:

The mass eigenvalues at the H_{1} and
${\tilde{H}}_{2}\equiv \u03f5{H}_{2}^{*}$ system are given:

Suppose that the MSSM matches the SM at the SUSY mass scale Q_{0} ≡ m_{0}. Then, the smaller one
${m}_{-}^{2}$ is identified to be the mass squared of the SM Higgs H with the tachyonic mass. The larger one
${m}_{+}^{2}$ is the mass squared of the orthogonal combination
$\mathscr{H}$, which is decoupled from the SM at Q_{0}, that is
${m}_{\mathscr{H}}\simeq {Q}_{0}$. Therefore, we have:

_{1}and ${\tilde{H}}_{2}$, β as follows:

Thus, the Higgs mass parameter m^{2} is expressed in terms of
${m}_{1}^{2}$,
${m}_{2}^{2}$ and tan β:

Below the Q_{0} scale, in which the SM emerges, the scalar potential is the SM one as follows:

Here, the Higgs coupling λ is given in terms of the SUSY parameters at the leading order as:

_{t}is the top Yukawa coupling of the SM. The parameters m

_{2}and λ run with the SM renormalization group equation down to the electroweak scale Q

_{EW}= m

_{H}and then give:

It is easily seen that the VEV of Higgs, $\langle H\rangle $, is v, and $\langle \mathscr{H}\rangle =0$, taking account of $\langle {H}_{1}\rangle =v\mathrm{cos}\beta $ and $\langle {H}_{2}\rangle =v\mathrm{sin}\beta $, where v = 246 GeV.

Let us fix m_{H} = 125 GeV, which gives λ(Q_{0}) and m^{2}(Q_{0}). This experimental input constrains the SUSY mass spectrum of the MSSM. We consider the some universal soft breaking parameters at the SUSY breaking scale Λ as follows:

Now, we have the SUSY five parameters, Λ, tan β, m_{0}, m_{1}_{/}_{2}, A_{0}, where Q_{0} = m_{0}. In addition to these parameters, we take µ = Q_{0}. Inputting m_{H} = 125 GeV and taking
${m}_{\mathscr{H}}\simeq {Q}_{0}$, we can obtain the SUSY spectrum for the fixed Q_{0} and tan β.

We present the SUSY mass spectrum at Q_{0} = 10 TeV. The input parameter set and the obtained SUSY mass spectra at Q_{0} are summarized in Table A1, where we use
${\overline{m}}_{t}\left({m}_{t}\right)$ = 163.5 ± 2 GeV [62,84]. These parameter sets are easily found from the work in [71].

Input at Λ and Q_{0} | Output at Q_{0} |
---|---|

at Λ = 10^{17} GeV | ${m}_{\tilde{g}}=12.8\text{TeV},\phantom{\rule{0.2em}{0ex}}{m}_{\tilde{W}}=5.2\text{TeV},\phantom{\rule{0.2em}{0ex}}{m}_{\tilde{B}}=2.9\text{TeV}$ |

m_{0} = 10 TeV | ${m}_{{\tilde{b}}_{L}}={m}_{{\tilde{t}}_{L}}=12.2\text{TeV}$ |

m_{1}_{/}_{2} = 6.2 TeV | ${m}_{{\tilde{b}}_{R}}=14.1\text{TeV},\phantom{\rule{0.2em}{0ex}}{m}_{{\tilde{t}}_{R}}=8.4\text{TeV}$ |

A_{0} = 25.803 TeV | ${m}_{{\tilde{s}}_{L},{\tilde{d}}_{L}}={m}_{{\tilde{c}}_{L},{\tilde{u}}_{L}}=15.1\text{TeV}$ |

at Q_{0} = 10 TeV | ${m}_{{\tilde{s}}_{R},{\tilde{d}}_{R}}\simeq {m}_{{\tilde{c}}_{R},{\tilde{u}}_{R}}=14.6\text{TeV,}{m}_{\mathscr{H}}=13.7\text{TeV}$ |

µ = 10 TeV | ${m}_{{\tilde{\tau}}_{L}}={m}_{{\tilde{v}}_{\tau L}}=10.4\text{TeV,}\phantom{\rule{0.2em}{0ex}}{m}_{{\tilde{\tau}}_{R}}=9.3\text{TeV}$ |

tan β = 10 | $\begin{array}{c}{m}_{{\tilde{\mu}}_{L},{\tilde{e}}_{L}}={m}_{{\tilde{v}}_{{\mu}_{L}},{\tilde{v}}_{{e}_{L}}}=10.8\text{TeV,}\phantom{\rule{0.2em}{0ex}}{m}_{{\tilde{\mu}}_{R},{\tilde{e}}_{R}}=10.3\text{TeV}\\ {X}_{t}=-0.22,\phantom{\rule{1em}{0ex}}{\lambda}_{H}=0.126\end{array}$ |

As seen in Table A1, the first and second family squarks are degenerate in their masses; on the other hand, the third ones split due to the large RGE effect. Therefore, the mixing angle between the first and second family squarks vanishes, but the mixing angles between the first-third and the second-third family squarks are produced at the Q_{0} scale. The left-right mixing angle between
${\tilde{b}}_{L}$ and
${\tilde{b}}_{R}$ is given as:

## B. Squark Contribution in the ΔF = 2 Process

The ΔF = 2 effective Lagrangian from the gluino-sbottom-quark interaction is given as [86]:

^{0}, b, d), (B

_{s}, b, s), (K

^{0}, s, d). The L, R denote (1 ± γ

_{5})/2, and a, b are color indices. Then, the ${P}^{0}\text{-}{\overline{P}}^{0}$ mixing, M

_{12}, is written as:

The hadronic matrix elements are given in terms of the non-perturbative parameters B_{i} as:

The Wilson coefficients for the gluino contribution in Equation (B1) are written as [86]:

Here, we take (i, j) = (1, 3), (2, 3), (1, 2) which correspond to B^{0}, B_{s} and K^{0} mesons, respectively. The loop functions are given as follows:

If ${x}_{I}^{\tilde{g}}\ne {x}_{J}^{\tilde{g}}({x}_{I,J}^{\tilde{g}}={m}_{{\tilde{d}}_{I,J}}^{2}/{m}_{\tilde{g}}^{2})$,

$$\begin{array}{l}{g}_{1[1]}({x}_{I}^{\tilde{g}},{x}_{J}^{\tilde{g}})=\frac{1}{{x}_{I}^{\tilde{g}}-{x}_{J}^{\tilde{g}}}\left(\frac{{x}_{I}^{\tilde{g}}\mathrm{log}{x}_{I}^{\tilde{g}}}{{({x}_{I}^{\tilde{g}}-1)}^{2}}-\frac{1}{{x}_{I}^{\tilde{g}}-1}-\frac{{x}_{J}^{\tilde{g}}\mathrm{log}{x}_{J}^{\tilde{g}}}{{({x}_{J}^{\tilde{g}}-1)}^{2}}+\frac{1}{{x}_{J}^{\tilde{g}}-1}\right)\\ {g}_{2[1]}({x}_{I}^{\tilde{g}},{x}_{J}^{\tilde{g}})=\frac{1}{{x}_{I}^{\tilde{g}}-{x}_{J}^{\tilde{g}}}\left(\frac{{x}_{I}^{\tilde{g}}\mathrm{log}{x}_{I}^{\tilde{g}}}{{({x}_{I}^{\tilde{g}}-1)}^{2}}-\frac{1}{{x}_{I}^{\tilde{g}}-1}-\frac{{x}_{J}^{\tilde{g}}\mathrm{log}{x}_{J}^{\tilde{g}}}{{({x}_{J}^{\tilde{g}}-1)}^{2}}+\frac{1}{{x}_{J}^{\tilde{g}}-1}\right)\end{array}$$If ${x}_{I}^{\tilde{g}}={x}_{J}^{\tilde{g}}$

$$\begin{array}{l}{g}_{1[1]}({x}_{I}^{\tilde{g}},{x}_{I}^{\tilde{g}})=\frac{({x}_{I}^{\tilde{g}}+1)\mathrm{log}{x}_{I}^{\tilde{g}}}{{({x}_{I}^{\tilde{g}}-1)}^{3}}+\frac{2}{{({x}_{I}^{\tilde{g}}-1)}^{2}}\\ {g}_{2[1]}({x}_{I}^{\tilde{g}},{x}_{I}^{\tilde{g}})=\frac{2{x}_{I}^{\tilde{g}}\mathrm{log}{x}_{I}^{\tilde{g}}}{{({x}_{I}^{\tilde{g}}-1)}^{3}}+\frac{{x}_{I}^{\tilde{g}}+1}{{({x}_{I}^{\tilde{g}}-1)}^{2}}\end{array}$$

Taking account of the case that the gluino mass is much smaller than the squark mass scale Q_{0}, the effective Wilson coefficients are given by using the RGEs for higher-dimensional operators in Equation (B1) at the leading order of QCD as follows:

For the parameters ${B}_{i}^{(d)}(i=2-5)$ of B mesons, we use values in [87] as follows:

On the other hand, we use the most updated values for ${\widehat{B}}_{1}^{(d)}$ and ${\widehat{B}}_{1}^{(s)}$ as [84]:

For the parameters ${B}_{i}^{K}(i=2-5)$, we use the following values [88],

For the parameters ${B}_{i}^{D}(i=1-5)$ we use the following values [89,90],

## C. The Loop Functions F_{i}

The loop functions ${F}_{i}({x}_{\tilde{g}}^{I})$ are given in terms of ${x}_{\tilde{g}}^{I}={m}_{\tilde{g}}^{2}/{m}_{{\tilde{d}}_{I}}^{2}(I=3,6)$ as follows:

## D. EDM and Chromo-EDM of Quarks

We present the EDM of the strange quark from the gluino contribution as the typical example [86]:

Including the RGE effect of QCD [91], the cEDM of the strange quark is given as:

On the other hand, the EDM operator is mixed with the cEDM operator during RGE evolution. Then, one obtains:

The EDMs and cEDMs of the down- and up-quarks induced by the gluino interaction are also given by the similar formulas.

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**Figure 1.**The SUSY components of (

**a**) $\mathrm{\Delta}{M}_{{B}^{0}}$ and (

**b**) $\mathrm{\Delta}{M}_{{B}_{s}}$ versus${m}_{\tilde{Q}}$ for s

_{13}= s

_{23}= 0.22 (cyan) and 0.5 (blue). The horizontal red line denotes the experimental central value.

**Figure 2.**The SUSY components of (

**a**) $\mathrm{\Delta}{M}_{{K}^{0}}$ and (

**b**) |ϵ

_{K}| versus${m}_{\tilde{Q}}$ for s

_{13}= s

_{23}= 0.22 (cyan) and 0.5 (blue). The horizontal red line denotes the experimental central value.

**Figure 3.**The the neutron EDM versus (

**a**) ${m}_{\tilde{Q}}$ and (

**b**) versus$|{\u03f5}_{K}^{\text{SUSY}}|$ for s

_{13}= s

_{23}= 0.22 (cyan) and 0.5 (blue) for the case of the QCD sum rule. The horizontal red line denotes the experimental upper bound of |d

_{n}|, and the vertical one is the experimental central value of |ϵ

_{K}|.

**Figure 4.**The the mercury EDM versus (

**a**) ${m}_{\tilde{Q}}$ and (

**b**) versus$|{\u03f5}_{K}^{\text{SUSY}}|$ for s

_{13}= s

_{23}= 0.22 (cyan) and 0.5 (blue) for the case of the QCD sum rule. The horizontal red line denotes the experimental upper bound of |d

_{Hg}|, and the vertical one is the experimental central value of |ϵ

_{K}|.

**Figure 5.**The SUSY component of (

**a**) ΔM

_{D}and (

**b**) x

_{D}versus${m}_{\tilde{Q}}$ for s

_{13}= s

_{23}= 0.22 (cyan) and 0.5 (blue). The horizontal red line denotes the experimental central value.

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Tanimoto, M.; Yamamoto, K.
Sensitivity of High-Scale SUSY in Low Energy Hadronic FCNC. *Symmetry* **2015**, *7*, 689-713.
https://doi.org/10.3390/sym7020689

**AMA Style**

Tanimoto M, Yamamoto K.
Sensitivity of High-Scale SUSY in Low Energy Hadronic FCNC. *Symmetry*. 2015; 7(2):689-713.
https://doi.org/10.3390/sym7020689

**Chicago/Turabian Style**

Tanimoto, Morimitsu, and Kei Yamamoto.
2015. "Sensitivity of High-Scale SUSY in Low Energy Hadronic FCNC" *Symmetry* 7, no. 2: 689-713.
https://doi.org/10.3390/sym7020689