Generic loop effects of new scalars and fermions in b → sℓ+ℓ− and a vector-like 4th generation

In this article we investigate in detail the possibility of accounting for the b → sℓ+ℓ− anomalies via box 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 → 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 → sℓ+ℓ− 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 4th generation of vector-like fermions and a real scalar field. This model allows one to coherently address the observed anomalies in b → sℓ+ℓ− transitions and in aμ without violating the bounds from other observables (in particular Bs − B¯s\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\overline{B}}_s $$\end{document} 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.


JHEP06(2019)118 1 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.
In this context, box contributions of heavy new scalars and fermions 2 (also within multi Higgs doublet models with right-handed neutrinos [75][76][77]) have been shown to be a viable option [78][79][80][81]. 3 Furthermore, an explanation of the anomalies in b → s + − via loop effects allows for interesting connections to Dark Matter [86][87][88][89][90][91][92] and typically leads to correlated imprints on other observables like the anomalous magnetic moment of the muon (a µ ). 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].
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

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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 refs. [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] to the SM. However, instead of adding a Z boson we supplement the model by a neutral scalar to get the desired loop-contributions. Furthermore, one can forbid the dangerous mixing effect between the SM fermions and the new vector-like ones by assigning U(1) changes to the new particles (resembling R-parity in the MSSM).
This article is organized as follows: in section 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 section 3. Our generic approach of section 2 is then applied to a specific UV complete model in section 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 section 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 figure 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 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 figure 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 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).
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.
The dimension-6 operators governing b → sµ + µ − and b → sγ transitions are contained in the effective Hamiltonian:

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with e being the electron charge, α EM the fine structure constant and g s the SU(3) gauge coupling. The primed operators are obtained by interchanging L and R. NP contributions from box diagrams will generate effects in C where we have defined  I  3  1  1  1  1  13  1  1  II  1333  3  3  1  3  1  III  3  8  8  8  8  83  8  4/3  IV  8333  3  3  8  3  4/3  V3  3  3  333 33 2 Table 1. Table of  and 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 (3) 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   (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.
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 Ψ A . Similarly, the gluon-penguin generates where the color factors χ g andχ g for the different possible SU(3) representations are given in table 2.
The contribution of Z-penguins to C ( ) 9,10 is given in section 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 figure 1 by replacing muons with neutrinos.

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The effective Hamiltonian describing this process reads (following the conventions of ref. [110]) 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 table 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 with The box diagrams contributing to these above operators are shown in figure 3. Using the Lagrangian from eq. (2.1), one obtains the following results for the coefficients: The loop functions F (x, y, z) and G(x, y, z) are defined in appendix B and the colour factors χ BB andχ BB are given in table 3  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" I  3  3  1  1  1  0  II  1  133  0  1  III  3  3  8  8 1/36 7/12 IV 8   (3) representations that can give a non-zero effect to a µ . χ aµ denotes the resulting group factor. effect generated in D 0 −D 0 is induced by the couplings (2.29)

Anomalous magnetic moment of the muon
The anomalous magnetic moment of the muon (a µ ≡ (g − 2) µ /2) and its electric dipole moments (d µ ) we find from the diagrams in figure 2 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 governing Zf f interactions as [114] − 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, being the Z couplings to SM fermions at tree-level. The relevant Feynman diagrams are shown in figure 4. We write the coupling of the Z boson to the new scalars and fermions as where we have introduced the notation a , and with generic couplings g Ψ,L,R AB and g Φ M N which can only be determined in a UV complete model in which also the couplings of the new particles to the SM Higgs are known. Using the generic Lagrangian from eq. (2.1), one obtains the following results for the coefficients 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. 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 section 4.
The form-factors in eq. (2.32) includes Zsb couplings generating contributions to C 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.
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.

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As explained in section 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)-(3.8). The SM value The experimental constraint therefore reads [140] computed with the values from ref. [136] for B s |Q 1 (µ b )|B s . This value shows a slight tension with the SM as first outlined in refs. [141,142]. The tension would be reduced if one considered the results for the matrix element from ref. [138]: however, in this case

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one should rely on a separate computation for the decay constant, while in ref. [136] both quantities are computed together.
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 ∆a µ = a exp µ − a SM µ ∼ 270(85) × 10 −11 . (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 .

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

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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 where M U,D,E are non-diagonal mass matrices Here the subscripts 1 and 2 denote the SU(2) component of the doublet. We diagonalize these mass matrices by performing the field redefinitions  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.

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with the same electric charges as up-type and down-type quarks (charged-leptons and neutrinos), respectively.
The rotations introduced at eq. (4.5) lead to the following Lagrangian for the interactions in the broken phase which resembles eq. (2.1) for the special case of our 4th generation model. Thus identify Here we worked in the down-basis for the SM quarks which means that CKM matrices V ij appear in vertices involving up-type quarks. The first two columns of the above Lagrangian involves couplings with down-type quarks and charged leptons and can be directly matched on the Lagrangian in eq. (2.1) for the case of only one scalar, i.e. Φ M ≡ Φ and Ψ A ≡ {Ψ D I , Ψ E I }. The presence of L u i I (L ν i ) resembles the fact, mentioned in section 2, that left-handed couplings to down-quarks (leptons) lead via SU(2) to couplings to left-handed up-quarks (neutrinos). In addition couplings to right-handed up-quarks R u i I appear in our model which are however not relevant for our phenomenology.

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 section 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) where the (simplified) loop function are defined in appendix B. In addition there are contributions to the C 1 analogue in D 0 −D 0 mixing obtained by substituting L b 1 → L c 1 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 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

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• a µ (see eq. (2.30)) 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 25) and g Φ = 0 are used. Thus the finiteness of the result can be traced by to the unitarity of the matrices W .

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 section 3. We perform this fit using the publicly available HEPfit package [172], performing a Markov Chain Monte Carlo (MCMC) analysis employing the Bayesian Analysis Toolkit (BAT) [173]. Let us first choose specific values for the masses of the scalar Φ and the fermions Ψ. As observed in ref. [81] a large splitting between the scalar mass and the vector-like lepton mass with respect to the vector-like quark masses is welcome to suppress the relative effect in ∆m Bs . Since the vector-like quarks should not be too light anyway because of direct LHC searches [174,175] we choose m Φ m E 450 GeV 10 and m D = 3.15 TeV, corresponding to x E,L 1 and x D 50. These values are well beyond the reach of direct searches at LHC: concerning m E,L the bounds come from Drell-Yan production of the new 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.

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fermions which are subsequently decaying in the neutral scalar and SM leptons. Therefore, the collider signature is similar to the one of MSSM slepton [176,177]. 11 Turning to the coupling of the new scalars and fermions to quarks and muons, we assume a flatly distributed priors within the range |Γ| ≤ 1.5 such that perturbativity is respected. The marginalized posterior probability distribution for all NP couplings, together with the correlations among them, can be found in appendix E. In the rest of this section we will focus on one particular benchmark point, that we selected because it lies within all the 1σ regions of the combined posterior distributions for the NP couplings (see figure 10). The benchmark values are 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 figure 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 can be seen from the fit in the (|Γ L µ |, |Γ R µ |)-plane from the left panel of figure 5. In the case with λ E = 0.0015, corresponding to the benchmark point reported in eq. (4.26), one can see that it is possible to explain the deviation in a µ by means of couplings of order unity. 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 figure 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 figure 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 figure 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  All the predictions for these observables are compatible at the 1σ level with their experimental measurements described in section 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 figure 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] to include couplings of the new particles to right-handed SM fermion and calculated the completely general expressions for the Wilson coefficients governing b → s processes (b → s + − , b → sνν, b → sγ and B s −B s mixing). In addition, we have computed the effects in a µ and Z → µ + µ − which unavoidably arise in such scenarios. Furthermore, we have proposed a UV complete model containing a 4 th vector-like generation of fermions and a new scalar, which is capable of explaining b → s + − data and a µ . We applied the formula derived in our generic setup (see section 2) to this model, illustrating their usefulness. In the following phenomenological analysis of our 4 th generation model (see section 4.3) we came to the conclusion that the b → s + − anomalies and a µ can be explained simultaneously. As a benchmark point which can achieve this, and is consistent with direct LHC searches, we have used 450 GeV for the vector-like fermions and for the new scalar and m D = 3.15 TeV for the vector like quarks (with order one couplings). We have observed that right-handed couplings in the muon sector allow to address the long-standing anomaly in a µ without having to require too large lepton couplings, if one allows for interactions of the vector-like fermions with the SM Higgs. Interestingly,

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due to the new results from LHCb and BELLE [9,10] the global fit to b → s + − data now prefers non-zero right-handed couplings to quarks and leptons as well, justifying the importance of our generalization of previous analysis performed in this work.
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].

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, ex-

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changing 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 (B.1) which in the equal mass limit read 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

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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: Moreover, it is useful to define the limit Finally, the equal mass limits read (B.12) 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

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 figure 7 can be constructed and contribute to b → sµ + µ − transitions in addition to the ones shown in figure 1. Similarly, there are contributions from crossed boxes to B s −B s mixing (in addition to the ones in figure 3) arising due to the diagrams in figure 8.  figure 7) the Wilson coefficients are given by BBχ   M  BB   I  3  3  1  1  1  0  Real Φ  II  1  1  3  3  0  1  Majorana Ψ  III  3  3  8  8

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 JHEP06(2019)118 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.  eq. (2.1) takes a slightly different form, namely

E Posterior distributions
Here we show the 1D marginalized posterior distributions of the parameters from the global fit described in section 4.2, together with the 2D combined correlations between these parameters. The results are summarized in figure 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.  Figure 10. The 1D marginalized posterior distributions of the parameters from the fit described in section 4.2, together with the 2D correlations between them. The green, red and orange regions correspond to 68%, 95% and 99% probability regions, respectively.
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