Imprint of SUSY in radiative B-meson decays

We study supersymmetric (SUSY) effects on $C_7(\mu_b)$ and $C'_7(\mu_b)$ which are the Wilson coefficients (WCs) for $b \to s \gamma$ at b-quark mass scale $\mu_b$ and are closely related to radiative $B$-meson decays. The SUSY-loop contributions to $C_7(\mu_b)$ and $C'_7(\mu_b)$ are calculated at leading order (LO) in the Minimal Supersymmetric Standard Model (MSSM) with general quark-flavour violation (QFV). For the first time we perform a systematic MSSM parameter scan for the WCs $C_7(\mu_b)$ and $C'_7(\mu_b)$ respecting all the relevant constraints, i.e. the theoretical constraints from vacuum stability conditions and the experimental constraints, such as those from $K$- and $B$-meson data and electroweak precision data, as well as recent limits on SUSY particle masses and the 125 GeV Higgs boson data from LHC experiments. From the parameter scan we find the following: (1) The MSSM contribution to Re($C_7(\mu_b)$) can be as large as $\sim \pm 0.05$, which could correspond to about 3$\sigma$ significance of New Physics (NP) signal in the future LHCb and Belle II experiments. (2) The MSSM contribution to Re($C'_7(\mu_b)$) can be as large as $\sim -0.08$, which could correspond to about 4$\sigma$ significance of NP signal in the future LHCb and Belle II experiments. (3) These large MSSM contributions to the WCs are mainly due to (i) large scharm-stop mixing and large scharm/stop involved trilinear couplings, (ii) large sstrange-sbottom mixing and large sstrange-sbottom involved trilinear couplings and (iii) large bottom Yukawa coupling $Y_b$ for large $\tan\beta$ and large top Yukawa coupling $Y_t$. In case such large NP contributions to the WCs are really observed in the future experiments at Belle II and LHCb Upgrade, this could be the imprint of QFV SUSY (the MSSM with general QFV).


Introduction
In our analysis we assume that there is no SUSY lepton-flavour violation. We also assume that R-parity is conserved and that the lightest neutralinoχ 0 1 is the lightest SUSY particle (LSP). We work in the MSSM with real parameters, except for the CKM matrix.
In the following section we introduce the SUSY QFV parameters originating from the squark mass matrices. Details about our parameters scan are given in Section 3. In Section 4 we define the relevant WCs and analyze their behaviour in the MSSM with QFV. The conclusions are in Section 5. All relevant constraints are listed in Appendix A.
In this paper we focus on thec L −t L ,c R −t R ,c R −t L ,c L −t R ,s L −b L ,s R −b R ,s R −b L , ands L −b R mixing which is described by the QFV parameters M 2 Qu23 M 2 Q23 , M 2 U 23 , T U 23 , T U 32 , M 2 Q23 , M 2 D23 , T D23 and T D32 , respectively. We will also often refer to the QFC parameter T U 33 and T D33 which induces thet L −t R andb L −b R mixing, respectively, and plays an important role in this study. The slepton parameters are defined analogously to the squark ones. All the parameters in this study are assumed to be real, except the CKM matrix V CKM .

Parameter scan
In our MSSM-parameter scan we take into account theoretical constraints from vacuum stability conditions and experimental constraints from K-and B-meson data, the H 0 mass and coupling data and electroweak precision data, as well as limits on SUSY particle masses from recent LHC experiments (see Appendix A). Here H 0 is the discovered SMlike Higgs boson which we identify as the lightest CP even neutral Higgs boson h 0 in the MSSM. Concerning squark generation mixings, we only consider the mixing between the second and third generation of squarks. The mixing between the first and the second generation squarks is strongly constrained by the K-and D-meson data [23,24]. The experimental constraints on the mixing of the first and third generation squarks are not so strong [25], but we don't consider this mixing since its effect is essentially similar to that of the mixing of the second and third generation squarks. We generate the input parameter points by using random numbers in the ranges shown in Table 1, where some parameters are fixed as given in the last box. All input parameters are DR parameters defined at scale Q = 1 TeV, except m A (pole) which is the pole mass of the CP odd Higgs boson A 0 . The parameters that are not shown explicitly are taken to be zero. The entire scan lies in the decoupling Higgs limit, i.e. in the scenarios with large tan β ≥ 10 and large m A ≥ 1350 GeV (see Table 1), respecting the fact that the discovered Higgs boson is SM-like. It is well known that the lightest MSSM Higgs boson h 0 is SM-like (including its couplings) in this limit. We don't assume a GUT relation for the gaugino masses M 1 , All MSSM input parameters are taken as DR parameters at the scale Q = 1 TeV, except m A (pole), and then are transformed by RGEs to those at the weak scale of Q = µ W for the computation of the WCs C 7,8 (µ W ) and C 7,8 (µ W ) in the MSSM. The masses and rotation matrices of the sfermions are renormalized at one-loop level by using the public code SPheno-v3.3.8 [26,27] based on the technique given in [28].
From 8660000 input points generated in the scan 72904 points survived all constraints. These are 0.84%. We show these survival points in all scatter plots in this article.
4500 2 4500 2 4500 2 1500 2 1500 2 1500 2 1500 2 1500 2 1500 2 4 WCs C 7 (µ b ) and C 7 (µ b ) in the MSSM with QFV The effective Hamiltonian for the radiative transition b → sγ is given by where G F is the Fermi constant and V tb V * ts is a CKM factor. The operators relevant to b → sγ are and their chirality counterparts where m b is the bottom quark mass, e and g s are the electromagnetic and strong coupling, F µν and G a µν the U (1) em and SU (3) c field-strength tensors, T a are colour generators, and the indices L,R denote the chirality of the quark fields. Here note that the SM contributions to C 2,7,8 (µ W ) are (almost) zero at LO. The WCs C 7 (µ b ) and C 7 (µ b ) at the bottom quark mass scale µ b can be measured precisely in the experiments at Belle II and LHCb Upgrade [1][2][3][4]. We compute C 7 (µ b ) and C 7 (µ b ) at LO in the MSSM with QFV and study the deviation of the MSSM predictions from their SM ones 2 . Following the standard procedure, first we compute C 7,8 (µ W ) and C 7,8 (µ W ) at the weak scale µ W at LO in the MSSM and then we compute C 7 (µ b ) and C 7 (µ b ) by using the QCD RGEs for the 2 Here it is worth to mention that these WCs are related to the photon polarization in radiative B-meson decays. The helicity polarization of the external photon in b → sγ is defined as At LO it is given as [18] P In the SM C 7 (µ b ) is strongly suppressed by a factor m s /m b and hence the photon in b → sγ decay is predominantly left-handed. In principle, the photon polarization can be extracted from the measurement of radiative B-meson decays in the experiments such as Belle II and LHCb Upgrade [1,2,[29][30][31][32][33][34][35][36][37][38].
scale evolution at leading log (LL) level [8] 3 : where We take the NLO formula with 5 flavours for the strong coupling constant α s (µ) for where 3 Here we comment on the RG running of the WCs at LL level. In footnote 5 of Ref. [18], it is argued as follows: In Ref. [39] it has been pointed out that the gluino contribution to the WCs C ( ) 7,8 (µ) is the sum of two different pieces, one proportional to the gluino mass and one proportional to the bottom mass, which have a different RG evolution (i.e. Eqs. (40) and (41) of [39], respectively). However, it has been found that at LO this is equivalent to the usual SM RG-evolution (i.e. Eqs. (13,14) of [18] which correspond to Eq.(10) of the present paper) once the running bottom mass m b (µ 0 ) is used instead of the pole mass m b (pole) in the WCs C ( ) i (µ 0 ), where µ 0 is the high-energy matching scale (e.g. the electroweak scale µ W ).
We have also confirmed this point (fact) independently of Ref. [18]. Here, note that we have used the public code SPheno-v3.3.8 [26,27] in the computation of the WCs C Moreover, just after Eq.(41) in Ref. [39] it is clearly stated that the terms R 7b,g (µ b ) and R 8b,g (µ b ) turn out to be numerically very small with respect to the other terms on the right-hand sides of Eq.(41) for the RG running of the WCs. Here R 7b,g (µ b ) and R 8b,g (µ b ) are linear combinations of the WCs (such as 16,19,20)) of the additional four-quark operators in Eq.(15) of [39], all of which are operators at NLO of QCD. Hence, the effects of the additional four-quark operators onto the RG running of C  at the weak scale µ W for the transitions b R → s L γ L , g L and b L → s R γ R , g R , respectively (see Eqs. (5,6,7)). Here γ L , g L and γ R , g R are left-handed photon, gluon and righthanded photon, gluon, respectively. The photon is emitted from any electrically charged line and the gluon from any colour charged line. For the SM one-loop contributions (X, Y) = (t/c/u, W + ). For the MSSM one-loop contributions (X, Y) = (stop/scharm/sup, chargino), (sbottom/sstrange/sdown, gluino), (sbottom/sstrange/sdown, neutralino) and (t/c/u, H + ), where stop/scharm/sup denotes top-, charm-, up-squark mixtures and so on.
We use the numerical results for C ( ) 7,8 (µ W ) at LO in the MSSM obtained from the public code SPheno-v3.3.8 [26,27], which takes into account the following one-loop contributions to C Here the charginoχ ± 1,2 is a mixture of charged winoW ± and charged higgsinoH ± , the neutralinoχ 0 1,2,3,4 is a mixture of photinoγ, zinoZ and two neutral higgsinosH 0 1,2 , and H + is the charged Higgs boson.    (10)). This is mainly due to the following reasons: 32,33 . Here note that |T U 23,32,33 | can be large due to large Y t (see Eqs. (14,16)) and that |T D23,32,33 | can be large due to large Y b for large tan β (see Eqs. (15,17)). In the following we assume these setups.
• As for the up-type squark -chargino loop contributions to C 7 (µ W ) and C 8 (µ W ) which is the effective coupling for the transition b R → s L γ and b R → s L g, respectively; The b R -ũ 1,2,3 -χ ± 1,2 vertex which contains the b R -t L -H ± coupling can be enhanced by the large bottom Yukawa coupling Y b for large tan β. The s L -ũ 1,2,3 -χ ± 1,2 vertex contains the s L -c L -W ± coupling which is not CKM-suppressed 4 . This vertex contains also the s L -t R -H ± coupling which is enhanced by the large top Yukawa coupling Y t despite the suppression due to the CKM factor V * ts . Hence, the up-type squark -chargino loop contributions to C 7,8 (µ W ) can be enhanced by the large Y b for large tan β and the large Y t , and further by the largec L -t L mixing term M 2 Q23 and the larget L -t R mixing term T U 33 for whichũ 1,2,3 contain a strong mixture ofc L , t L andt R . Important parts of this squark -chargino loop contributions to C 7,8 (µ W ) are schematically illustrated in terms of the mass-insertion approximation in Fig. 2.
• As for the down-type squark -gluino loop contributions to C 7,8 (µ W ); The b R -d 1,2,3 -g vertex which contains the b R -b R -g coupling can be enhanced by the sizable QCD coupling. The s L -d 1,2,3 -g vertex which contains the s L -s L -g coupling can also be enhanced by the QCD coupling. Furthermore, absence of the CKM-suppression factor in this loop diagram results in additional strong enhancement. Therefore, the down-type squark -gluino loop contributions to C 7,8 (µ W ) can be enhanced by the sizable QCD coupling, and further by the largeb R -s L mixing term T D32 for whichd 1,2,3 contain a strong mixture ofb R ands L . Moreover, |T D32 | can be large due to large Y b for large tan β (see Eq. (17)). An important part of this squark -gluino loop contribution to C 7,8 (µ W ) is schematically illustrated in terms of the mass-insertion approximation in Fig. 3(a).
• As for the down-type squark -neutralino loop contributions to couplings with the latter coupling being proportional to Y b can be enhanced by large Y b for large tan β. The s L -d 1,2,3 -χ 0 1,2,3,4 vertex contains the s L -s L -γ/Z couplings. The absence of the CKM-suppression factor in this loop diagram results in additional strong enhancement. Hence, the down-type squark -neutralino loop contributions to C 7,8 (µ W ) can be enhanced by large Y b for large tan β, and further by the largeb R -s L andb L -s L mixing terms (T D32 and M 2 Q23 ), for whichd 1,2,3 contain a strong mixture ofb R -s L andb L -s L . Moreover, |T D32 | controlled by Y b can be large for large tanβ (see Eq. (17)).
• As for the up-type quark - H + coupling can be enhanced by the large top-quark Yukawa coupling Y t despite the suppression due to the CKM factor V * ts . Hence t -H + loop contributions to • As for the up-type squark -chargino loop contributions to C 7 (µ W ) and C 8 (µ W ) which are the effective couplings for the transition b L → s R γ and b L → s R g, respectively; From a similar argument one finds that these loop contributions to C 7,8 (µ W ) should be small due to the very small s-quark Yukawa coupling Y s .
• As for the down-type squark -gluino loop contributions to C 7,8 (µ W ); The b L -d 1,2,3 -g vertex which contains the b L -b L -g coupling can be enhanced by the sizable QCD coupling. The s R -d 1,2,3 -g vertex which contains the s Rs R -g coupling can also be enhanced by the QCD coupling. Absence of the CKMsuppression factor in this loop diagram results in additional strong enhancement. Therefore, the down-type squark -gluino loop contributions to C 7,8 (µ W ) can be enhanced by the sizable QCD couplings, and further by largeb L -s R mixing term T D23 for whichd 1,2,3 contain a strong mixture ofb L ands R . Moreover, |T D23 | can be large due to large Y b for large tan β (see Eq. (17)). An important part of this squark -gluino loop contribution to C 7,8 (µ W ) is schematically illustrated in terms of the mass-insertion approximation in Fig. 3(b).
• As for the down-type squark -neutralino loop contributions to couplings with the latter coupling being proportional to Y b can be enhanced by large Y b for large tan β. The s R -d 1,2,3 -χ 0 1,2,3,4 vertex contains the s R -s R -γ/Z coupling. Absence of the CKM-suppression factor in this loop diagram results in additional strong enhancement. Hence, the down-type squark -neutralino loop contributions to C 7,8 (µ W ) can be enhanced by large Y b for large tan β, and further by largeb L -s R andb R -s R mixing terms T D23 and M 2 D23 , for whichd 1,2,3 contain strong mixtures of b L -s R andb R -s R . Moreover, |T D23 | controlled by Y b can be large for large tanβ (see Eq.(17)).
• As for the up-type quark -H + loop contributions to C 7,8 (µ W ); These contributions turn out to be very small due to the very small Y s .
In the following we will show scatter plots in various planes obtained from the MSSM parameter scan described above (see Table 1), respecting all the relevant constraints (see Appendix A).
In Fig. 5 we show scatter plots for C MSSM 7 (µ b ) and C 7 (µ b ). In Fig. 5(a) we show a scatter plot in the Re(C 7 (µ b ))-Im(C 7 (µ b )) plane. We see that the MSSM contribution to Re(C 7 (µ b )) can be as large as ∼ −0.07, which could correspond to an about 4σ New Physics (NP) signal significance in the combination of the future LHCb Upgrade (Phase III) and Belle II (Phase II) experiments (see Figure A.13 of [3]). Note that |Im(C 7 (µ b ))| is very small ( < ∼ 0.004) and that C 7 (µ b ) 0 in the SM. In Fig. 5(b) we show the scatter plot in the Re(C MSSM 7 (µ b ))-Im(C MSSM 7 (µ b )) plane. We see that the MSSM contribution to Re(C 7 (µ b )) can be as large as ∼ −0.05, which could correspond to an about 3σ NP signal significance in the combination of the future LHCb Upgrade (50 f b −1 ) and Belle II (50 ab −1 ) experiments (see Figure 8 of [3]). Note that |Im(C MSSM 7 (µ b ))| is very small ( < ∼ 0.003) and that the MSSM contribution C MSSM 7 (µ b ) can be quite sizable compared to C SM 7 (µ b ) −0.325. In Fig. 5(c) we show a scatter plot in the Re(C MSSM 7 (µ b ))-Re(C 7 (µ b )) plane. We see that the Re(C 7 (µ b )) and Re(C MSSM 7 (µ b )) can be quite sizable simultaneously.
Here we comment on the errors of the data on C 7 (µ b ) and C N P 7 (µ b ). The errors of the data on C 7 (µ b ) and C N P Here we remark the following points: (i) As for the determination of C 7 (µ b ) one can get much more precise information from the fully-inclusive B(B → X s γ) measurement than from the measurement of the exclusive observables such as B(B → K * γ) 5 since the theoretical predictions for the exclusive observables involve hadronic form factors which have large theoretical uncertainties. (ii) The fully-inclusive observable B(B → X s γ) can be measured reliably and precisely at Belle II [1] whereas its measurement is very difficult at LHCb [2]. (iii) As a result, Belle II plays a specially important role in the precise determination (extraction) of C 7 (µ b ) in the near future. As for the experimental errors of the WCs C 7 (µ b ) and C N P 7 (µ b ) obtained (extracted) from the future B-meson experiments, Belle II is now planning to upgrade to accumulate about 5 times larger data (up to ∼ 250 ab −1 ) [40]. If this is realized, the (statistical) uncertainty of the observable data from Belle II could be reduced by a factor of about √ 5.
As for the theoretical errors of the WCs C 7 (µ b ) and C N P 7 (µ b ) obtained (extracted) from the B-meson experiments, there is a sign of promising possibility of significant reduction of the theoretical errors in the future: Very recently M. Misiak et al. performed a new computation of B(B → X s γ) in the SM at the NNLO in QCD [12]. Taking into account the recently improved estimates of non-perturbative contributions, they have obtained B(B → X s γ) = (3.40 ± 0.17) · 10 −4 for E γ > 1.6GeV . Compared with their previous SM prediction B(B → X s γ) = (3.36 ± 0.23) · 10 −4 [11], the theoretical uncertainty is now reduced from 6.8% to 5.0%. Note here that the Figure A.13 and Figure 8 of [3] showing expected errors of C 7 (µ b ) and C N P 7 (µ b ) obtained (extracted) from the future B-meson experiments were made in the year 2017. Hence, in case the significant reduction of the experimental and theoretical errors is achieved in the future, the NP signal significances for Re(C 7 (µ b )) and Re(C N P 7 (µ b )) in the MSSM could be significantly higher than those mentioned above which are about 4 σ NP significances for Re(C 7 (µ b )) and about 3 σ significance for Re(C MSSM 7 (µ b )). Thus, it is very important to improve the precision of both theory and experiment on B-meson physics by a factor about 1.5 or so in view of NP search (such as SUSY search). Therefore, we strongly encourage theorists and experimentalists to challenge this task.
In Fig. 7 we show scatter plots in the T D23 -Re(C 7 (µ b )), T D32 -Re(C 7 (µ b )) and T D33 -Re(C 7 (µ b )) planes. From Fig. 7(a) and Fig. 7 . An appreciable correlation between T D23 and Re(C 7 (µ b )) can be seen in Fig. 7(a). From Fig. 7(c) we see that it can be large for large |T D33 | > ∼ 2 TeV. These behaviors are also consistent with our expectation.
In Fig. 8  (µ b )) < ∼ 0.035 for T U 33 > ∼ 2 TeV. There is a significant correlation between Re(C MSSM 7 (µ b )) and T U 33 , which can be explained partly by the important contribution of Fig. 2(b) (see Eq. (10)). The fewer scatter points around T U 33 = 3.5 TeV is Figure 5: The scatter plot of the scanned parameter points within the ranges given in Table 1 in the planes of (a) Re( again due to the fact that the m h 0 bound tends to be violated around this point. From Fig. 8(d) we see that it can be large (up to ∼ ±0.05) for large tan β ( > ∼ 40). These behaviors are also consistent with our expectation.
In Fig. 9 we show scatter plots in the T D23 -Re(C MSSM 7 (µ b )) plane. We see Re(C MSSM Figure 6: The scatter plots of the scanned parameter points within the ranges given in Table 1 in the planes of (a) T U 23 -Re( can be sizable (up to ∼ ±0.05) for any values of T D23 . We have found that scatter plots in the T D32 -Re(C  Figure 7: The scatter plots of the scanned parameter points within the ranges given in Table 1 in the planes of (a) T D23 -Re(C 7 (µ b )), (b) T D32 -Re(C 7 (µ b )) and (c) T D33 -Re(C 7 (µ b )).
In Figs. 10 and 11 we show contours of Re(C 7 (µ b )) around the benchmark point P1 in various parameter planes. Fig. 10(a) shows contours of Re(C 7 (µ b )) in the T U 23 -T U 32 plane. We see that Re(C 7 (µ b )) is sensitive to both T U 23 and T U 32 , especially to T U 23 , increases quickly with the increase of T U 23 and T U 32 (< 0), as is expected, and can be as large as about -0.07 in the allowed region. We also see that it is large (−0.07 < ∼ Re(C 7 (µ b )) < ∼ − 0.04) respecting all the constraints in a significant part of this parameter plane. From Fig. 10(b) we see that Re(C 7 (µ b )) is also fairly sensitive to T U 33 and can be as large as ∼ −0.08. From Fig. 10(c) we find that Re(C 7 (µ b )) is very sensitive to tanβ, especially for 4500 2 4500 2 4500 2 1500 2 1500 2 1500 2 1500 2 1500 2 1500 2 large T U 23 > 0, as expected, and can be as large as ∼ −0.07. As can be seen in Fig. 10(d), Re(C 7 (µ b )) is sensitive to M 2 U 23 , especially for large T U 23 > ∼ 2.5 TeV, as expected, and is large (−0.08 < ∼ Re(C 7 (µ b )) < ∼ − 0.04) respecting all the constraints in a significant part of this parameter plane. Fig. 11(a) shows contours of Re(C 7 (µ b )) in the T D23 -T D32 plane. It is fairly sensitive to T D23 and mildly dependent on T D32 as is expected partly from the contribution of Fig. 3(b) (see Eq.(10)), can be as large as ∼ −0.06 in the allowed region, and is large (−0.058 < ∼ Re(C 7 (µ b )) < ∼ − 0.046) respecting all the constraints in a significant part of this parameter plane. From Fig. 11(b) we see that Re(C 7 (µ b )) is also rather sensitive to T D33 and can be as large as ∼ −0.06 in the allowed region. As can be seen in Fig. 11(c), Re(C 7 (µ b )) is very sensitive to tanβ and also sensitive to T D23 for large tanβ > ∼ 70, as expected, and is sizable (−0.05 < ∼ Re(C 7 (µ b )) < ∼ − 0.04) respecting all the constraints in a significant part of this parameter plane. From Fig. 11(d) we find that Re(C 7 (µ b )) is very sensitive to M 2 D23 , and is sizable (−0.05 < ∼ Re(C 7 (µ b )) < ∼ − 0.04) respecting all the constraints in a significant part of this parameter plane.
In Figs. 12 and 13 we show contour plots of Re(C MSSM 7 (µ b )) (i.e. the MSSM contributions to Re(C 7 (µ b ))) around the benchmark point P1 in various parameter planes. Fig. 12(a) shows contours of Re(C MSSM 7 (µ b )) in the T U 23 -T U 32 plane. We see that Re(C MSSM 7 (µ b )) is sensitive to T U 23 and T U 32 : |Re(C MSSM 7 (µ b ))| quickly increases with the increase of T U 23 and T U 32 as is expected. We find also that Re(C MSSM 7 (µ b )) can be as large as about -0.05 in the allowed region and is sizable (−0.05 < ∼ Re(C MSSM 7 (µ b )) < ∼ − 0.04) respecting all the constraints in a significant part of this parameter plane. From Fig. 12(b) we see that Re(C MSSM Table 3: Physical masses in GeV of the particles for the scenario of Table 2.   (µ b )) < ∼ − 0.04) respecting all the constraints in a significant part of this parameter plane. From Fig. 12(c) we find that Re(C MSSM 7 (µ b )) is very sensitive to tanβ and T U 23 as expected, quickly increases with increase of tanβ and T U 23 (> 0), and can be as large as ∼ −0.05 in the allowed region. As can be seen in Fig. 12(d),   Table 2. The "X" marks P1 in the plots. The red hatched region satisfies all the constraints in Appendix A. The definitions of the bound lines are the same as in Fig. 10. In addition to these the blue solid lines and the green solid lines show the ∆M Bs bound and the vacuum stability bound on T D23 , respectively.  Table 2. The "X" marks P1 in the plots. The red hatched region satisfies all the constraints in Appendix A. The definitions of the bound lines are the same as those in Fig. 10.  Table 2. The "X" marks P1 in the plots. The red hatched region satisfies all the constraints in Appendix A. The definitions of the bound lines are the same as those in Fig. 11.

Conclusions
We have studied SUSY effects on C 7 (µ b ) and C 7 (µ b ) which are the Wilson coefficients for b → sγ at b-quark mass scale µ b and are closely related to radiative B-meson decays. The SUSY-loop contributions to the C 7 (µ b ) and C 7 (µ b ) are calculated at LO in the Minimal Supersymmetric Standard Model with general quark-flavour violation. For the first time we have performed a systematic MSSM parameter scan for the WCs C 7 (µ b ) and C 7 (µ b ) respecting all the relevant constraints, i.e. the theoretical constraints from vacuum stability conditions and the experimental constraints, such as those from K-and B-meson data and electroweak precision data, as well as recent limits on SUSY particle masses and the 125 GeV Higgs boson data from LHC experiments. From the parameter scan, we have found the following: • The MSSM contribution to Re(C 7 (µ b )) can be as large as ∼ ±0.05 which could correspond to about 3σ significance of NP (New Physics) signal in future Belle II and LHCb Upgrade experiments.
• The MSSM contribution to Re(C 7 (µ b )) can be as large as ∼ −0.08 which could correspond to about 4σ significance of NP signal in future Belle II and LHCb Upgrade experiments.
• These large MSSM contributions to the WCs are mainly due to (i) large scharm-stop mixing and large scharm/stop involved trilinear couplings T U 23 , T U 32 and T U 33 , (ii) large sstrange-sbottom mixing and large sstrange-sbottom involved trilinear couplings T D23 , T D32 and T D33 , and (iii) large bottom Yukawa coupling Y b for large tan β and large top Yukawa coupling Y t .
Moreover, we have pointed out the following: • It is very important to reduce the (theoretical and experimental) errors of the WCs C 7 (µ b ) and C N P 7 (µ b ) obtained (extracted) from the future experiments at Belle II and the LHCb Upgrade. An improvement in precision of both theory and experiment by a factor about 1.5 or so would be very important in view of NP search (such as SUSY search). Therefore, we strongly encourage theorists and experimentalists to challenge this task.
• On the other hand, it is also very important to reduce the theoretical errors of the MSSM contributions to the WCs C 7 (µ b ) and C 7 (µ b ) by performing higher order computations such as those at NLO/NNLO level.
In case such large New Physics contributions to the WCs, i.e. such large deviations of the WCs from their SM values, are really observed in the future experiments at Belle II and the LHCb Upgrade, this could be the imprint of QFV SUSY (the MSSM with general QFV) and would encourage to perform further studies of the WCs C 7 (µ b ) and C MSSM SM prediction 9 . In our scenario with heavy sleptons/sneutrinos with masses of about 1.5 TeV the MSSM loop contributions to a µ are so small that they can not explain the discrepancy between the new data and the SM prediction. Therefore, in the context of our scenario, this discrepancy should be explained by the loop contributions of another new physics coexisting with SUSY.
In addition to these we also require our scenarios to be consistent with the following experimental constraints: Table 5: Constraints on the MSSM parameters from the K-and B-meson data relevant mainly for the mixing between the second and the third generations of squarks and from the data on the h 0 mass and couplings κ b , κ g , κ γ . The fourth column shows constraints at 95% CL obtained by combining the experimental error quadratically with the theoretical uncertainty, except for B(K 0 L → π 0 νν), m h 0 and κ b,g,γ . • The LHC limits on sparticle masses (at 95% CL) [62][63][64][65][66]: We impose conservative limits for safety though actual limits are somewhat weaker than those shown here. In the context of simplified models, gluino masses mg 2.35 TeV are excluded for mχ0 1 < 1.55 TeV. There is no gluino mass limit for mχ0 1 > 1.55 TeV. The 8-fold degenerate first two generation squark masses are excluded below 1.92 TeV for mχ0 1 < 0.9 TeV. There is no limit on the masses for mχ0 1 > 0.9 TeV. We impose this squark mass limit on mũ 3 and md 3 . Bottom-squark masses are excluded below 1.26 TeV for mχ0 1 < 0.73 TeV. There is no bottom-squark mass limit for mχ0 1 > 0.73 TeV. Here the bottom-squark mass means the lighter sbottom mass mb 1 . We impose this limit on md 1 sinced 1 ∼b R (see Table 4). A typical top-squark mass lower limit is ∼ 1.26 TeV for mχ0 1 < 0.62 TeV. There is no top-squark mass limit for mχ0 1 > 0.62 TeV. Here the top-squark mass means the lighter stop mass mt 1 . We impose this limit on mũ 1 sinceũ 1 ∼t R (see Table 4). For sleptons/sneutrinos heavier than the lighter charginoχ ± 1 and the second neutralinõ χ 0 2 , the mass limits are mχ± 1 , mχ0 2 > 0. 74  2 limits for mχ0 1 > 0.72 TeV. For mass degenerate selectronsẽ L,R and smuons µ L,R , masses below 0.7 TeV are excluded for mχ0 1 < 0.41 TeV. For mass degenerate stausτ L andτ R , masses below 0.39 TeV are excluded for mχ0 1 < 0.14 TeV. There is no sneutrinoν mass limit from LHC yet. Sneutrino masses below 94 GeV are excluded by LEP200 experiment [24].