Generic Loop Effects of New Scalars and Fermions in $b\to s\ell^+\ell^-$, $(g-2)_\mu$ and a Vector-like $4^{\rm th}$ Generation

In this article we investigate in detail the possibility of accounting for the $b\to s\ell^+\ell^-$ and $(g-2)_\mu$ anomalies via loop contributions involving with new scalars and fermions. For this purpose, we first write down the most general Lagrangian which can generate the desired effects and then calculate the generic expressions for all relevant $b\to s$ Wilson coefficients. Here we extend previous analysis by allowing that the new particles can also couple to right-handed Standard Model (SM) fermions as preferred by recent $b\to s\ell^+\ell^-$ data and the anomalous magnetic moment of the muon. In the second part of this article we illustrate this generic approach for a UV complete model in which we supplement the Standard Model by a $4^{\rm th}$ generation of vector-like fermions and a real scalar field. This model allows one to coherently address the observed anomalies in $b\to s\ell^+\ell^-$ transitions and in $a_\mu$ without violating the bounds from other observables (in particular $B_s -\bar B_s$ mixing) or LHC searches. In fact, we find that our global fit to this model, after the recent experimental updates, is very good and prefers couplings to right-handed SM fermions, showing the importance of our generic setup and calculation performed in the first part of the article.


Introduction
While no particles beyond the ones of the Standard Model (SM) have been observed at the LHC (so far), b → s + − data show a coherent pattern of deviations from the SM predictions with a significance of more than 4-5 σ [1][2][3][4][5][6][7][8]. Recently, results of Belle and LHCb presented at Moriond EW 2019 [9,10] confirmed these tensions, even though the significance for the new physics (NP) hypothesis, compared to the SM, did not change notably 1 . In fact, including these new measurements, global fits of the Wilson coefficients governing b → s + − transitions [12][13][14][15][16][17] still find that NP scenarios can describe data much better than the SM, even though the preferences between the different scenarios changed with respect to the previous experimental situation.
Concerning concrete NP models giving the desired pattern in the effective theory with a good fit to data, most analyses focused on scenarios in which the required NP effects are generated at tree-level, either by the exchange of Z vector bosons  or via leptoquarks . Nonetheless, since the size of the NP contribution required to account for current data is of the order of 20% compared to the (loop and CKM suppressed) amplitude of the SM, also new loop effects can in principle suffice for an explanation.
However, the effect here is in most models too small since a quite large NP contribution is needed to account from the tantalizing tension between the measurement [93] and the SM prediction of around 3-4 σ. In fact, Ref. [79] found that it is challenging to account for ∆a µ with TeV scale masses and not too large couplings to muons with a minimal particle content. In general, it has been argued [94] that one needs new sources of electroweak symmetry breaking (EWSB) if one aims at a high scale explanation of the anomalous magnetic moment of the muon. In the context of adding new scalars and fermions to the SM this can be achieved for example by a fourth generation of vector-like leptons coupling to the SM Higgs [94][95][96][97][98][99][100][101][102]. 1 Note that deviations from the SM predictions have been observed in b → cτ ν transitions as well [11].
However, since these tensions cannot be explained by loop effects, we do not discuss them in this article. 2 Box contributions of new vectors and fermions were studied in the context of Z models with vector-like quarks in Ref. [74]. 3 Alternatively, models with large couplings to right-handed top quarks can give the desired effect via a W -loop [82][83][84], as first shown in the EFT context in Ref. [85].
Therefore, we extend in this article the analysis of Ref. [79] to include the possibility of new sources of EW symmetry breaking within the NP sector. For this purpose, an extension of the field content with respect to the minimal one of Ref. [79] is necessary, i.e. more than three new fields need to be added to the SM particle content. In doing so, new couplings to right-handed quarks and leptons are introduced which do not only affect a µ but also lead to different effects in b → sµ + µ − (i.e. lead to solutions other than the purely left-handed C 9 = −C 10 one obtained in Ref. [79]). In fact, while before Moriond 2019 scenarios with left-handed current were in general preferred, now including right-handed contributions (both in quark and leptonic sectors) can even give a better fit to data [12-14, 16, 17].
A UV complete example of such a setup with new scalars and fermions couplings to left-and right-handed SM fermions is a model with a vector-like 4 th generation. With respect to Ref. [103,104], also aiming at an explanation of the b → s + − anomalies, we add not only a 4 th generation of leptons but also of quarks [105][106][107][108]  This article is organized as follows: In Sec. 2 we define our generic setup, in which new scalars and fermions couple to SM quarks and leptons via Yukawa-like interactions.
There, we also provide completely general expressions for the formulae of the relevant Wilson coefficients. We review the corresponding observables together with the current experimental situation in Sec. 3. Our generic approach of Sec. 2 is then applied to a specific UV complete model in Sec. 4, which contains a vector-like fourth generation of fermions and a neutral scalar. We study the phenomenology of this model in detail before we conclude in Sec. 5.

Generic Setup and Wilson Coefficients
In this section we define our generic setup and calculate completely general 1-loop expressions for contributions to b → s processes and the anomalous magnetic moment of the muon.
As outlined in the introduction, in the spirit of Refs. [78,79] we add to the SM particle content a NP sector with vector-like fermions Ψ A and new scalars Φ M such that b → sµ + µ − transitions can be generated via box diagrams, as depicted in Fig. 1. In this respect, we generalize the previous analysis of Ref. [79] by including in addition couplings of new particles to SU (2) singlet SM fermions. Moreover, we do not impose limitations on The diagram on the left is generated in models in which the fermions couples only to SM quarks or only to SM leptons, which corresponds to type a). The diagram on the right refers to models with scalars connecting b to s and µ to µ, i.e. type b).
the number of fields added to the SM and allow for couplings of the new sector to the SM Higgs.
In order to generate box diagrams as the ones shown in Fig. 1 it is necessary that either the scalars Φ M,N or the fermions Ψ A,B couple both to quarks and leptons, corresponding to case a) and b), respectively. This means that in diagrams of type a) the amplitudes (before using any Fierz identities) have the structure (sΓb)(μΓµ), while in type b) amplitudes of the form (μΓb)(sΓµ) are generated. Here, Γ denotes an arbitrary Dirac structure. Since semi-leptonic operators are commonly given in the form (sΓb)(μΓµ), Fierz identities must be used in case b) in order to transform the expressions to this standard basis. We give the relevant Fierz identities in Appendix A.
The Yukawa-like couplings of new scalars Φ M and fermions Ψ A to bottom/strange quarks and muons can be parameterized completely generically (below the EWSB scale) by the Lagrangian Here Ψ A and Φ M have to be understood as generic lists containing in principle an arbitrary number of fields, meaning that A and M also include implicitly SU (2) and color indices.
Therefore, the couplings L s,b AM and R s,b AM are generic matrices in (A-M ) space with the restriction that U (1) EM and SU (3) are respected 4 . This Lagrangian will not only affect b → sµ + µ − transitions but also unavoidably generate effects in B s −B s mixing, b → sγ decays, the anomalous magnetic moment of the muon a µ as well as Z couplings and decays to SM fermions. Furthermore, b → sνν processes and D 0 −D 0 mixing can give relevant constraints once SU (2) invariance at the NP scale is imposed. Therefore, all these processes have to be taken into account in a complete phenomenological analysis. In order to perform such an analysis, the Wilson coefficients of the relevant effective Hamiltonian must be known. We will calculate them in the following subsections.
where we have defined and  (3) representations that can give non-zero effects via photonand gluon-penguin diagrams to b → sµ + µ − transitions. χ γ denotes the resulting group factor for the former contribution, while χ g andχ g represent the resulting group factors for the latter.
In the equations above, the labels A, B, M and N denote the particle (in case of several representations) and also include SU (2) components, while the sum over SU where m b is the b quark mass. Q Φ M and Q Ψ A are the electric charges of the NP fields Φ M and Ψ A , respectively. The conservation of electric charge imposes that Note that the terms proportional to F 7 , G 7 and F 9 in Eqs. (2.12)-(2.14) stem from the diagram where the photon couples to the scalar Φ M , while the terms proportional to F 7 , G 7 and G 9 stem from the diagram where the photon couples to the fermion Figure 2. Photon-penguin diagrams contributing to b → sγ transitions and a µ .
Similarly, the gluon-penguin generates where the color factors χ g andχ g for the different possible SU (3) representations are given in Tab. 2.
The contribution of Z-penguins to C ( ) 9,10 is given in Sec. 2.6 together with a discussion of Z decays.

b → sνν
As stated at the beginning of this section, b → sν µνµ processes have to be taken into account once SU (2) invariance at the NP scale is imposed. This implies that, in the generic description in Eq. (2.1), one has to replace the left-handed muon fields with neutrinos. The box diagrams generating b → sν µνµ are therefore obtained from The effective Hamiltonian describing this process reads (following the conventions of Ref. [110]) where The resulting WCs are: where the normalization factor N has been introduced in Eq. (2.11), and the loop functions F (x, y, z) and G(x, y, z) are defined in Appendix B. The colour factor χ is the same as for b → sµ + µ − transitions and is given in Tab. 1 for the different representations.

∆B = ∆S = 2 Processes
The presence of L b,s AM and R b,s AM implies NP contributions to the B s −B s mixing which, using the conventions of Refs. [111,112], is governed by The box diagrams contributing to these above operators are shown in Fig. 3. Using the Lagrangian from Eq. (2.1), one obtains the following results for the coefficients: I  3  3  1  1  1  0   II  1  133  0  1   III  3  3  8    In the context a UV complete model, SU (2) invariance imposes at the high scale that couplings to left-handed up-type quarks are related to the couplings to left-handed downtype quarks via CKM rotations. Therefore, working in the down-basis, the "minimal" effect generated in D 0 −D 0 is induced by the couplings

Anomalous Magnetic Moment of the Muon
The anomalous magnetic moment of the muon (a µ ≡ (g − 2) µ /2) and its electric dipole where m µ is the muon mass, χ aµ is the colour factor given in Table 4, and Q Φ M and Q Ψ A are the electric charges of the NP fields Φ M and Ψ A , respectively. Analogously to photonpenguin contributions to b → s transitions, the conservation of electric charge imposes that

Modified Z Couplings
Here, we study the effects of our new particles on modified Z couplings, i.e. on Zμµ, Zbb, Zss and Zsb couplings, both for off-and on-shell Z bosons 5 . We define the form-factors where f = {b, s, µ}, g 2 is the SU (2) gauge coupling, θ W the Weinberg angle and q is the Z momentum. Moreover, with being the Z couplings to SM fermions at tree-level. The relevant Feynman diagrams are shown in Fig. 4. We write the coupling of the Z boson to the new scalars and fermions as where we have introduced the notation a where the loop functions are defined in Appendix B, and the colour factor Table 2) and χ Z = χ aµ (see Table 4) for f = µ. Here we have set the masses and momenta of the external fermions to 0 and expanded up to first order in q 2 over the NP scale. If one is considering data from Z decays, Eq. (2.35) has to be evaluated to q 2 = m 2 Z while for processes with an off-shell Z (like b → s + − ) one has to set q 2 = 0.
Note that in the absence of EW symmetry breaking in the NP sector, the contribution of the self-energies cancel the one of the genuine vertex correction and Eq. (2.35) vanishes for q 2 = 0. Therefore, as noted above, g Ψ,L AB , g Ψ,R AB and g Φ M N are only meaningful after EWSB and it is not possible to relate them purely to SU (2)×U (1) quantum numbers. In a specific UV model with a known pattern of EWSB, rotation matrices can be used to relate the couplings before and after the breaking. Consequently, the cancellation of UV divergences (present in some of the loop functions in Eq. (2.35)) is only manifest after summation over SU (2) indices, due to a GIM-like cancellation originating from the unitarity of the rotation matrices. We will give a concrete example of this in Sec. 4.
The form-factors in Eq. (2.32) includes Zsb couplings generating contributions to C ( ) 9,10 Note that these contributions are lepton flavour universal and therefore cannot account for R K and R K * . However, a mixture of lepton flavour universal and violating contributions is phenomenologically interesting [115], especially in the light of the recant Belle and LHCb measurements [12,15]. In a similar fashion, Zsb couplings will also generate the following Finally, if SU (2) invariance at the NP scale is imposed, the new scalars and fermions couple also the neutrinos. Hence, contributions to Z → νν and W → µν will arise as well.
Concerning Z → νν, g ν L (q 2 = m 2 Z ) can be straightforwardly extracted from Eq. (2.35) by appropriate replacements and the same is true concerning W µν couplings.

Experimental Constraints on Wilson Coefficients
In this section we review the experimental situation and the resulting constraints on the Wilson coefficients calculated in the previous section.
First of all, the contributions of scalar operators are helicity-enhanced in the B s → µ + µ − branching ratio with respect to the O 10 contribution of the SM. This results in the bound [129] which excludes sizable contributions to scalar operators (unless there is a purely scalar quark current) and leads to from the updated one-parameter fit of Ref. [130]. Therefore, we neglect the effects of scalar operators in semi-leptonic B since they anyway cannot explain the corresponding anomalies.
Moreover, the inclusive b → sγ decay strongly constrains the magnetic operators.
From [131,132], in the limit of vanishing C 7,8 6 , we have leading to Here, we used C 7,8 at a matching scale of 1 TeV as input. Again, these constraints are so stringent that the effect of C 7,8 on the flavour anomalies can be mostly neglected.
As explained in Sec. 2.2, SU (2) invariance implies the presence of contributions to B → K ( * ) νν decays as well. Since there is no experimental way to distinguish different neutrino flavours in these decays, one measures the total branching ratio which we normalize to its SM prediction [110]: In the case of a K in the final state one has κ ≡ −2, while for the K * one gets κ = 1.34(4) [110]. The current experimental limits at 90% C.L. are [134] R νν K < 3.9 , R νν K * < 2.7 . (3.6)

Neutral Meson Mixing
The where R i (µ b ) is related to the matrix element of the operators Q i in Eq. (2.22) at the scale µ b by the relation The coefficients C i and C i are the ones in Eqs. (2.23)-(2.28), computed at the NP scale µ H . The matrix in operator space η ij (µ b , µ H ) encodes the QCD evolution from the high scale µ H to µ b , which we calculated numerically for a reference scale µ H = 1 TeV [135].
The matrix elements in Eqs. (3.7)-(3.8) have been computed by a N f = 2 + 1 lattice simulation [136], which found values consistent with the N f = 2 calculation [137] and recent sum rules results [138]. It is worth mentioning that FLAG-2019 [139] only provides a lattice average for B s |Q 1 (µ b )|B s , which is however dominated by the N f = 2 + 1 results from Ref. [136]. Therefore, we decided to employ the results from Ref. [136] in Eqs. (3.7)- The experimental constraint therefore reads [140]  Analogously to the B s system, D 0 −D 0 mixing is constrained by the mass difference of neutral D 0 mesons [140]: Unfortunately, a precise SM prediction is still lacking in this sector but one can constrain the NP contribution by assuming that not more than the total mass difference is generated by it.

Anomalous Magnetic Moment of the Muon
From the experimental side, this quantity has been already measured quite precisely [93], but further improvements by experiments at Fermilab [143] and J-PARC [144] (see also [145]) are expected in the future. On the theory side, the SM prediction has been improved continuously . The current tension between the two determinations accounts to (3.11)

Z Decays
The main experimental measurements of Z couplings have been performed at LEP [170] (at the Z pole). We extract from the model independent analysis of Ref. [171] the values for the NP contributions 7 neglecting cancellations and correlations.

th Generation Model
In this section we propose a model with a vector-like 4 th generation of fermions and a new complex scalar. This will also allow us to apply and illustrate the generic findings of the previous section to a UV complete model and study the effects in b → s + − data and a µ . 7 Z couplings in Ref. [171] are defined with opposite sign with respect to our conventions [114].

Lagrangian
The Lagrangian for our 4 th generation model is obtained from the SM one by adding a 4 th vector-like generation [103,104] and a neutral scalar where i is a family index and h the SM Higgs doublet. The charge assignments for the new vector-like fermions Ψ = Ψ L + Ψ R with P L,R Ψ = Ψ L,R and the new scalar Φ are After EWSB, mass matrices for the new fermions are generated where M U,D,E are non-diagonal mass matrices (4.4) 8 We did not assume a Z2 symmetry because this would allow the scalar Φ to be real and lead to crossed boxes in b → s + − , canceling the desired effect there.
Here the subscripts 1 and 2 denote the SU (2) component of the doublet. We diagonalize these mass matrices by performing the field redefinitions which resembles Eq. (2.1) for the special case of our 4th generation model. Thus identify

Wilson Coefficients
With these conventions we can now easily derive the Wilson coefficients within our model which can be directly obtained from the results of Sec. 2. In order to simplify the expressions, we will assume M Q = M d ≡ m D and M L = M e ≡ m E and only take into account couplings to b, s and µ in Eq. (4.1): Concerning SU (2) breaking effects the couplings λ D L,R and λ E L,R related to the down and charged leptons sector, respectively, can be relevant. However, concerning λ D L,R recall that from Section 3.1 that experimental data suggests very small values for C S,P and C 7,8 . In our model this can be achieved by assuming λ D L,R = 0 9 . In this limit the mass matrix M D in Eq. (4.4) is diagonal and the corresponding rotation matrices W D R(L) in Eq. (4.6) are equal to the identity, which implies With this setup, we obtain the following non-vanishing couplings in the quark sector of the Lagrangian in Eq. (4.7): (4.12) The expressions of Wilson coefficients for b → s processes simplify to: 14)   and L s 1 → L u 1 within Γ L . In the charged-lepton sector SU (2) breaking effects (encoded in λ E L,R ) can give a sizable chiral enhancement of the NP effect in a µ (see Eq. (2.30)) such that the long-standing anomaly in this channel can be addressed. In general one can parametrize the rotation matrices as , (4.20) leading to In our analysis we will consider a simplified setup with λ E R = −λ E L ≡ λ E that maximizes the effect in a µ (which at leading order in v is proportional to λ E R − λ E L ). In this approximation we have for where we have assumed real values for the couplings, implying a vanishing d µ . Let us stress that the contributions proportional to vλ E , coming from SU (2) breaking terms, is chirally enhanced can give a sizable effect that can explain the a µ anomaly.
where the simplified loop function F Z (x E ) has been defined in Appendix B. The results for Z → bb couplings can be easily obtained by suitable substitutions. Note that in our approximation of λ D L,R = 0 the correction to the Zsb vertex vanishes at q 2 = 0. Note that the UV divergences cancel as required, once for the couplings in Eq. (2.34) the relations

Phenomenology
We are now ready to consider the phenomenology of our 4 th generation model. For this purpose we will perform a combined fit to all the relevant and available experimental data, as briefly reviewed in Sec. 3. We perform this fit using the publicly available HEPfit assuming real values for all couplings. Note that the small value for λ E is obtained from the fit due to its correlation with |Γ R µ |. As can be seen from the combined posterior distribution of these 2 parameters shown in Fig. 10, higher values of λ E would require lower values of |Γ R µ |, which is disfavored by the current fit to b → s + − data. We observe that it is extremely important to allow for a right-handed coupling Γ R µ together a mixing coupling λ E in the muon sector such that a µ can be explained. This 10 Nearly degenerate masses mΦ mE are also welcome in the light of the dark matter relic density since the stable Φ is a suitable DM candidate. In fact, for mΦ = 450 GeV, 450 ≤ mE ≤ 520 GeV the model allows for an efficient annihilation such that one does not over-shoot the matter density of the universe for order one Γ couplings. 11 A detailed study recasting these MSSM analysis for our model has been performed in Refs. [101,102], finding mE mΦ = 450 GeV as an allowed solution. However, the situation changes significantly if one did not allow the presence of a coupling of the vector-like leptons to the SM Higgs. As shown in the right panel of Fig. 5, with λ E = 0, it is not possible to obtain couplings that are perturbative and capable to give a satisfactory explanation of the anomalous magnetic moment of the muon at the same time.
The presence of Γ R µ ameliorates the tension, but it is still not sufficient by itself to address the anomaly.
Also in the quark sector right-handed couplings are needed to address the B anomalies without spoiling at the same time the measurement for ∆M s . This is particularly evident by looking at the left panel of Fig. 6, where the region allowed by both b → sµ + µ − transitions and B s −B s is shown. Indeed, if one performs a separate fit to b → sµ + µ − transitions and ∆M s as shown in the right panel of Fig. 6, it is evident that the two channels are incompatible as long as one assumes a vanishing coupling to right-handed bottom and strange quarks, i.e. Γ R = 0.
The preference for non-zero couplings (i.e. beyond the SM effects) is in general driven by ∆a µ , the angular analyses of B → K * µ + µ − and B s → φµ + µ − , the branching fraction of B s → µ + µ − and the ratios R K and R K * . On the other hand, the experimental constraints coming from b → sγ and B → K ( * ) νν and ∆M Bs set bounds on Γ L,R and |Γ L µ | that are less stringent than the ones obtained by the inclusion of the aforementioned channels involving b → s transitions in our setup with λ D L,R = 0. Analogously, the constraints from Z → µ + µ − are found to give negligible constraints on |Γ L,R µ |. Concerning D 0 −D 0 mixing, we recall that Eq.  All the predictions for these observables are compatible at the 1σ level with their experimental measurements described in Sec. 3, except for R K * which is compatible only at the ∼ 2σ level. However, this is expected both from the global fit and from our specific model: As can be seen from the right panel of Fig. 6, there is no overlap of the 1σ regions from b → sµ + µ − data and ∆M s in our model. Furthermore, since our model only allows for NP in muons (neglecting Z, γ penguin effects), some small tensions are generated since flavour conserving b → sµ + µ − data prefers a smaller value of R K than the one recently measured.

Conclusions and Outlook
In this article we have studied in details the possibility that the intriguing anomalies in b → s + − processes are explained via box diagrams involving new scalars and fermions.
Within this setup we have generalized previous analysis [78,79,81]  Our 4 th generation model is also interesting since Φ is a viable (stable) Dark Matter candidate. We briefly showed that if the mass of Φ is close to the one of the vectorlike leptons, the correct relic density can be obtained while respecting the limits from Dark Matter direct detection. However, a more detailed investigation in the future seems worthwhile.
We conclude observing that our formalism can be directly applied to b → d + − transitions. In fact, it leads to correlated effects in Kaon physics [178] if one aims at explaining the slight tensions in B → πµ + µ − [179] simultaneously with the ones in b → s + − data. Furthermore, our setup and results can also be used for addressing the tension between theory and experiment in / [180,181] 12 . Here, as a special case of our generic approach, the MSSM has already been studied with the conclusion that it can provide a valid explanation of the anomaly [184][185][186].
Note added: After publication of this article, the g-2 collaboration at Fermilab released a new measurement of a µ [187], which together with the theory consensus of Ref. [188] leads to a tension of 4.2 σ. Furthermore, the LHCb collaboration updated the measurements of R(K) [189] and released the first measurement of P 5 in B + → K + * µµ [190]. Note that this does not change the conclusions of our article and the impact on the numeric is very small.
However, this gives additional support to our 4 th generation model which can account for (g − 2) µ and B-anomalies.

Acknowledgments
We

A Fierz Identities
Here we list the Fierz identities for spinors used in the computations. With i, j, k and l representing Dirac indices here we find where P L,R = (1 ∓ γ 5 ) /2 and σ µν = i 2 [γ µ , γ ν ]. When dealing with diagrams with crossed fermion lines, one needs Fierz identities involving charge conjugation matrices. Here, exchanging the second and the third Dirac index we find with the charge conjugation matrix defined as C = iγ 0 γ 2 .

B Loop Functions
Here we list the dimensionless loop functions introduced in Sections 2 and 4. The loop functions appearing in box diagrams that involve four different masses are defined as In the presence of only three different masses in the loop, one gets the functions while, in the presence of only two different masses in the loop, one gets The loop functions appearing in photon-and gluon-penguin diagrams are defined as which in the equal mass limit read Finally, the loop functions for the calculation of Z-penguins are defined as where we have defined div ε = ∆ ε − log m 2 µ 2 and It is interesting to notice that the following relations hold between particular limits of the penguin induced functions: Figure 7. Crossed box diagrams contributing to b → sµ + µ − transitions. The diagram on the left appears in models with real scalars, while the one on the right can be constructed in models with Majorana fermions. I  3  1  1  1  I  1  1  3  1  1   III  3  8  8  8  III  8  8  3  8 4/3 Moreover, it is useful to define the limit Finally, the equal mass limits read (B.12)

C Real Scalars and Majorana Fermions
If the NP fields have the appropriate quantum numbers they can be either real scalars or Majorana fermions. If this is the case, crossed diagrams as shown in Fig. 7 can be constructed and contribute to b → sµ + µ − transitions in addition to the ones shown in Fig. 1. Similarly, there are contributions from crossed boxes to B s −B s mixing (in addition to the ones in Fig. 3) arising due to the diagrams in Fig. 8.  Fig. 7) the Wilson coefficients are given by  Figure 8. Box diagrams contributing to B s −B s mixing. The diagram on the left is relative to models with real scalars, while the one on the right refers to models with Majorana fermions.  from Eq. (2.1) with the crossed fermion contributions, one obtains the following results for the coefficients: Figure 9. Crossed box diagrams contributing to b → sµ + µ − transitions. The diagram appears when a complex scalar couples to b, s quarks and its conjugate couples to the muons.

D Crossed Diagrams with Complex Scalars
There is also the possibility that a complex scalar couples to the down-type quarks whereas its hermitian conjugated version couples to muons. This means that the Lagrangian in Eq. (2.1) takes a slightly different form, namely Also this Lagrangian generates a contribution to b → sµ + µ − via the diagram shown in

E Posterior Distributions
Here we show the 1D marginalized posterior distributions of the parameters from the global fit described in Sec. 4.1, together with the 2D combined correlations between these parameters. The results are summarized in Fig. 10. We recall that, for all couplings Γ, we imposed |Γ| ≤ 1.5 such that perturbativity is satisfied. The consequences of such this choice are evident in the posterior distributions of Γ L , |Γ L µ | and |Γ R µ |, which are truncated because of this reason.  Table 7. Table of