Contents

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FLAVOUR(267104)-ERC-38

BARI-TH/13-671

Nikhef-2013-008

UT-13-09

The Anatomy of Neutral Scalars with FCNCs in the Flavour Precision Era
Andrzej J. Buras, Fulvia De Fazio, Jennifer Girrbach

Robert Knegjens and Minoru Nagai

[0.4 cm] TUM Institute for Advanced Study, Lichtenbergstr. 2a, D-85747 Garching, Germany

Physik Department, TUM, James-Franck-Straße, D-85747 Garching, Germany

Istituto Nazionale di Fisica Nucleare, Sezione di Bari, Via Orabona 4, I-70126 Bari, Italy

Nikhef, Science Park 105, NL-1098 XG Amsterdam, The Netherlands

Department of Physics, University of Tokyo, Tokyo 113-0033, Japan

Abstract

[10pt] In many extensions of the Standard Model (SM) flavour changing neutral current (FCNC) processes can be mediated by tree-level heavy neutral scalars and/or pseudo-scalars . This generally introduces new sources of flavour violation and CP violation as well as left-handed (LH) and right-handed (RH) scalar () currents. These new physics (NP) contributions imply a pattern of deviations from SM expectations for FCNC processes that depends only on the couplings of to fermions and on their masses. In situations in which a single or dominates NP contributions stringent correlations between and observables exist. Anticipating the Flavour Precision Era (FPE) ahead of us we illustrate this by searching for allowed oases in the landscape of a given model assuming significantly smaller uncertainties in CKM and hadronic parameters than presently available. To this end we analyze observables in and systems and rare and decays with charged leptons in the final state including both left-handed and right-handed scalar couplings of and to quarks in various combinations. We identify a number of correlations between various flavour observables that could test and distinguish these different scenarios. The prominent role of the decays in these studies is emphasized. Imposing the existing flavour constraints, a rich pattern of deviations from the SM expectations in rare decays emerges provided . NP effects in rare decays, except for , turn out to be very small. In they can be as large as the SM contributions but due to hadronic uncertainties this is still insufficient to learn much about new scalars from this decay in the context of models considered here. Flavour violating SM Higgs contributions to rare and decays turn out to be negligible once the constraints from processes are taken into account. But can still be enhanced up to . Finally, we point out striking differences between the correlations found here and in scenarios in which tree-level FCNC are mediated by a new neutral gauge boson .

## 1 Introduction

The recent discovery of a scalar particle with a mass of opened the gate to the unexplored world of scalar particles which could be elementary or composite. While we will surely learn a lot about the properties of these new objects through collider experiments like ATLAS and CMS, also low energy processes, in particular flavour violating transitions, will teach us about their nature. In the Standard Model (SM) and in many of its extensions there are no fundamental flavour-violating couplings of scalars111Unless otherwise specified we will use the name scalar for both scalars and pseudo-scalars. to quarks and leptons but such couplings can be generated through loop corrections leading in the case of transitions to Higgs-Penguins (HP) and in transitions to double Higgs-Penguins (DHP). However, when the masses of the scalar particles are significantly lower than the heavy new particles exchanged in the loops, the HP and DHP look at the electroweak scale as flavour violating tree diagrams. Beyond the SM such diagrams can also be present at the fundamental level, an important example being the left-right symmetric models. From the point of view of low energy theory there is no distinction between these possibilities as long as the vertices involving heavy particles in a Higgs-Penguin cannot be resolved and to first approximation what really matters is the mass of the exchanged scalar and its flavour violating couplings, either fundamental or generated at one-loop level. While all this can be formulated with the help of effective field theories and spurion technology, we find it more transparent to study directly tree diagrams with heavy particle exchanges.

In a recent paper [1] an anatomy of neutral gauge boson ( and ) couplings to quark flavour changing neutral currents (FCNC) has been presented. Anticipating the Flavour Precision Era (FPE) ahead of us and consequently assuming significantly smaller uncertainties in CKM and hadronic parameters than presently available, it was possible to find allowed oases in the landscape of new parameters in these models and to uncover stringent correlations between and observables characteristic for such NP scenarios.

The goal of the present paper is to perform a similar analysis for scalar neutral particles and to investigate whether the patterns of flavour violation in these two different NP scenarios (gauge bosons and scalars) can be distinguished through correlations between quark flavour observables. Already at this stage it is useful to note the following differences in NP contributions to quark flavour observables in these two scenarios:

• While the lower bounds on masses of gauge bosons from collider experiments are at least , new neutral scalars with masses as low as a few hundred are not excluded.

• While in the scenarios in addition to new operators also SM operators with modified Wilson coefficients can be present, in the case of tree-level scalar exchanges all effective low energy operators are new.

• While there is some overlap between operators contributing to processes in and scalar cases after the inclusion of QCD corrections, their Wilson coefficients are very different. Moreover, in transitions there is no overlap with the operators present in models.

• Concerning flavour violating couplings of and the SM Higgs , in the case of the boson large NP effects, in particular in rare decays, are still allowed but then its effects in processes turn out to be very small [1]. In the Higgs case, the smallness of the Higgs coupling to muons and electrons precludes any visible effects from tree-level Higgs exchanges in rare and decays with muon or electron pair in the final state once constraints from processes are taken into account. The corresponding effects in are small but can still be at the level of . Simultaneously tree-level Higgs contributions to transitions can still provide in principle solutions to possible tensions within the SM.

• At first sight the couplings of scalars to neutrinos look totally negligible but if the masses of neutrinos are generated by a different mechanism than coupling to scalars, like in the case of the see-saw mechanism, it is not a priori obvious that such couplings in some NP scenarios could be measurable. Our working assumption in the present paper will be that this is not the case. Consequently NP effects of scalars in , and transitions will be assumed to be negligible in contrast to models, where NP effects in these decays could be very important [1]. As we will see, scalar contributions to although in principle larger than for , and transitions, are found to be small. In they can be as large as the SM contribution but due to hadronic uncertainties this is still insufficient to learn much about scalars from this decay, at least in the context of models considered by us.

In order to have an easy comparison with the anatomy of FCNCs mediated by a neutral gauge bosons presented in [1] the structure of the present work will be similar to the structure of the latter paper but not identical, as rare decays play in this paper a subleading role so that emphasis will be put on and systems. In Section 2 we describe our strategy by defining the relevant couplings and listing processes to be considered. Our analysis will only involve processes which are theoretically clean and have simple structure. Here we will also introduce a number of different scenarios for the scalar couplings to quarks thereby reducing the number of free parameters. In Section 3 we will first present a compendium of formulae relevant for the study of processes mediated by tree-level neutral scalar exchanges including for the first time NLO QCD corrections to these NP contributions. In Section 4 we discuss rare decays, in particular . In Section 5 rare decays are considered. In Section 6 we present a general qualitative view on NP contributions to flavour observables stressing analytic correlations between and observables. In Section 7 we present our strategy for the numerical analysis and in Section 8 we execute our strategy for the determination of scalar couplings in the and systems. We discuss several scenarios for them and identify stringent correlations between various observables. We also investigate what the imposition of the flavour symmetry on scalar couplings would imply. In Section 9 we present the results for rare decays, where NP effects are found to be small. In Section 10 we demonstrate that the contributions of the SM Higgs with induced flavour violating couplings, even if in principle relevant for transitions, are irrelevant for rare and decays with small but still visible effects in . A summary of our main results and a brief outlook for the future are given in Section 11.

## 2 Strategy

### 2.1 Basic Model Assumptions

Our paper is dominated by tree-level contributions to FCNC processes mediated by a heavy neutral scalar or pseudoscalar. We use a common name, , for them unless otherwise specified. When a distinction will have to be made, we will either use and for scalar and pseudoscalar, respectively or in order to distinguish SM Higgs from additional spin 0 particles we will use the familiar 2HDM and MSSM notation: .

Our main goal is to consider the simplest extension of the SM in which the only new particle in the low energy effective theory is a single neutral particle with spin and the question arises whether this is possible from the point of view of an underlying original theory. If the scalar in question is not a singlet, then it must be placed in a complete multiplet, e.g. a second doublet as is the case of 2HDM or the MSSM. However, this implies the existence of its partners in a given multiplet with masses close to the masses of our scalar. In fact in the decoupling regime in 2HDM and MSSM the masses of are approximately degenerate. While breaking effects in the Higgs potential allow for mass splittings, they must be of at most and consequently the case of the dominance of a single scalar is rather unlikely.

It follows then that our scalar should be a singlet. In this case, the scalar-quark couplings of come from the following low energy effective operator

 L=λijLH0Λ¯qiRqjLhSM+h.c. (1)

with denoting the cut-off scale of the low energy theory. After the spontaneous breakdown of the scalar left-handed coupling is given by

 ΔijL(H0)=1√2vΛλijL, (2)

with analogous expression for the right-handed coupling.

Now, in the case of transitions the scalar contributions are governed only by the couplings to quarks and the corresponding Feynman rule has been shown in Fig 1. Here denote quark flavours. Note the following important property

 ΔijL(H0)=[ΔjiR(H0)]∗ (3)

that distinguishes it from the corresponding gauge couplings in which there is no chirality flip.

The couplings are dimensionless quantities but as these are scalar and not gauge couplings they can involve ratios of quark masses and the electroweak vacuum expectation value or other mass scales. While from the SM, 2HDM and MSSM we are used to having scalar couplings proportional to the masses of the participating quarks, it should be emphasized that this is not a general property. It applies only if the scalar and the SM Higgs, responsible for breakdown, are in the same multiplet or a multiplet of a larger gauge group . Then after the breakdown of to , the scalar appears as a singlet of symmetry, with couplings to quarks involving their masses after breakdown. While this is the case in several models, in our simple extension of the SM, it is more natural to think that the involved scalar couplings are unrelated to the generation of quark masses.

In spite of the last statement is useful to recall how the quark masses could enter the scalar couplings. Which quark masses are involved depends on the model. Considering for definiteness the system let us just list a few cases encountered in the literature:

• In models with MFV in which the scalar couplings are just Yukawa couplings one has

 ΔbsL(H0)∝mbv,ΔbsR(H0)∝msv(MFV) (4)

implying that dominate in these scenarios. Note however that using (3) these relations also give

 ΔsbL(H0)∝msv,ΔsbR(H0)∝mbv(MFV) (5)

which implies some care when stating whether LH or RH scalar couplings are dominant. Below we will use the ordering for operators in systems while in the case of rare decays. In the system will be used for both and couplings.

• In non-MFV scenarios the mass dependence in scalar couplings can be reversed

 ΔbsL(H0)∝msv,ΔbsR(H0)∝mbv(non−MFV) (6)

implying that dominate in these scenarios. Correspondingly (5) is changed to

 ΔsbL(H0)∝mbv,ΔsbR(H0)∝msv(non−MFV) (7)

promoting the so-called primed operators in decays.

• There exist also models in which flavour violating neutral scalar couplings do not involve the masses of external quarks. This is the case for the neutral heavy Higgs in the left-right symmetric models analysed in [2] where the scalar down quark couplings are proportional to up-quark masses, in particular . In the case of a manifest left-right symmetry with the right-handed mixing matrix being equal to the CKM matrix one finds

 ΔbsL(H0)=ΔbsR(H0). (8)

Even if in the concrete model analysed in [2] the right-handed mixing matrix equal to the CKM matrix is ruled out by the data, there could be other model constructions in which (8) could be satisfied. Also the LH and RH couplings differing by sign could in principle be possible.

### 2.2 Scenarios for Scalar Couplings

In order to take these different possibilities into account and having also in mind that scalar couplings could be independent of quark masses, we consider the following four scenarios for their couplings to quarks keeping the pair fixed:

1. Left-handed Scenario (LHS) with complex and ,

2. Right-handed Scenario (RHS) with complex and ,

3. Left-Right symmetric Scenario (LRS) with complex ,

4. Left-Right asymmetric Scenario (ALRS) with complex ,

with analogous scenarios for the pair . For rare decays in which the ordering is used, the rule (3) has to be applied to each scenario. For physics this is not required. In the course of our paper we will list specific examples of models that share the properties of these different scenarios. We will see that these simple cases will give us a profound insight into the flavour structure of models in which NP is dominated by left-handed scalar currents or right-handed scalar currents or left-handed and right-handed scalar currents of the same size. We will also consider a model in which both a scalar and a pseudoscalar with approximately the same mass couple equally to quarks and leptons. Moreover we will study a scenario with underlying flavour symmetry which will imply relations between and couplings and interesting phenomenological consequences.

The idea of looking at NP scenarios with the dominance of certain quark couplings to neutral gauge bosons or neutral scalars is not new and has been motivated by detailed studies in concrete models like supersymmetric flavour models [3], LHT model with T-parity [4, 5] or Randall-Sundrum scenario with custodial protection (RSc) [6]. See also [7, 8]. Also our recent analysis of tree-level FCNCs mediated by and in [1] demonstrates this type of NP in a transparent manner.

### 2.3 Scalar vs Pseudoscalar

It will turn out to be useful to exhibit the differences between the scalar and pseudoscalar spin 0 particles, although one should emphasize that in the presence of CP violation, the mass eigenstate propagating in a tree-diagram is not necessarily a CP eigenstate. Therefore, generally the coupling to appearing at many places in our paper can have the general structure

 L=12¯μ(Δμ¯μS(H0)+γ5Δμ¯μP(H0))Hμ (9)

where generalizing the Feynman rule in Fig. 1 to charged lepton couplings we have introduced:

 Δμ¯μS(H)=Δμ¯μR(H)+Δμ¯μL(H),Δμ¯μP(H)=Δμ¯μR(H)−Δμ¯μL(H). (10)

is real and purely imaginary as required by the hermiticity of the Hamiltonian which can be verified by means of (3).

The expressions for various observables will be first given in terms of the couplings and and can be directly used in the case of the scalar particle being CP-even eigenstate, like in 2HDM or MSSM setting . However, when the mass eigenstate is a pseudoscalar , implying , it will be useful to exhibit the i which we illustrate here for the system:

 ΔbsL(A)=−i~ΔbsL(A),ΔbsR(A)=+i~ΔbsR(A),Δμ¯μP(A)=i~Δμ¯μP(A). (11)

Here the flavour violating couplings are still complex, while is real.

The following useful relations follow from (3) and (11):

 ΔsbR(A)=i[~ΔbsL(A)]∗,ΔsbL(A)=−i[~ΔbsR(A)]∗. (12)

As far as transitions are concerned this distinction between scalar and pseudoscalar mass eigenstate is only relevant in a concrete model in which the relevant couplings are given in terms of fundamental parameters. However, as in our numerical analysis we will treat the flavour violating quark-scalar couplings as arbitrary complex numbers to be bounded by observables it will not be possible to distinguish a scalar and pseudoscalar boson on the basis of transitions alone. On the other hand, when rare decays, in particular , are considered there is a difference between these two cases as the pseudoscalar contributions interfere with SM contribution, while the scalar ones do not. Consequently the allowed values for and will differ from each other and we will find other differences. Finally, if both scalar and pseudoscalar contribute to tree-level decays and have approximately the same mass as well as couplings related by symmetries, also their contributions to processes differ. We will consider a simple example in the course of our presentation.

### 2.4 Steps

Let us then outline our strategy for the determination of flavour violating couplings to quarks and for finding correlations between flavour observables in the context of the simple scenarios listed above. Our strategy will only be fully effective in the second half of this decade, when hadronic uncertainties will be reduced and the data on various observables significantly improved. It involves ten steps including a number of working assumptions:

Step 1:

Determination of CKM parameters by means of tree-level decays and of the necessary non-perturbative parameters by means of lattice calculations. This step will provide the results for all observables considered below within the SM as well as all non-perturbative parameters entering the NP contributions. As is presently poorly known, it will be interesting in the spirit of our recent papers [1, 9, 2] to investigate how the outcome of this step depends on the value of with direct implications for the necessary size of NP contributions which will be different in different observables.

Step 2:

We will assume that the ratios

 Δμ¯μS,P(H)MH (13)

for scalar and pseudoscalar bosons have been determined in pure leptonic processes and that the scalar couplings to neutrinos are negligible. The properties of these couplings have been discussed above. In principle these ratios can be determined up to the sign from quark flavour violating processes and in fact we will be able to bound them from the present data on but their independent knowledge increases predictive power of our analysis. In particular the knowledge of their signs allows us to remove certain discrete ambiguities and is crucial for the distinction between LHS and RHS scenarios in decays. Of course, in concrete models like 2HDM or supersymmetric models these couplings depend on the fundamental parameters of a given model.

Step 3:

Here we will consider the system and the observables

 ΔMs,Sψϕ,B(Bs→μ+μ−),Aμ+μ−ΔΓ,Ssμ+μ−, (14)

where and can be extracted from the time-dependent rate [10, 11]. Explicit expressions for these observables in terms of the relevant couplings can be found in Sections 3 and 4.

Concentrating in this step on the LHS scenario, NP contributions to these three observables are fully described by

 ΔbsL(H)MH=−~s23MHe−iδ23,Δμ¯μS,P(H)MH, (15)

with the second ratio known from Step 2. Here and it is found to be below unity but it does not represent any mixing parameter as in [12]. The minus sign is introduced to cancel the minus sign in in the phenomenological formulae listed in the next section.

Thus we have five observables to our disposal and two parameters in the quark sector to determine. This allows to remove certain discrete ambiguities, determine all parameters uniquely for a given and predict correlations between these five observables that are characteristic for this scenario.

Step 4:

Repeating this exercise in the system we have to our disposal

 ΔMd,SψKS,B(Bd→μ+μ−),Sdμ+μ−. (16)

Explicit expressions for these observables in terms of the relevant couplings can be found in Sections 3 and 4.

Now NP contributions to these three observables are fully described by

 ΔbdL(H)MH=~s13MHe−iδ13,Δμ¯μS,P(H)MH, (17)

with the last one known from Step 2 and bounded in Step 3. Again we can determine all the couplings uniquely for a given . Our notations and sign conventions are as in Step 3 with but no minus sign as has no such sign.

Step 5:

Moving to the system we have to our disposal

 εK,KL→π0ℓ+ℓ−,KL→μ+μ−, (18)

where in view of hadronic uncertainties the last decay on this list will only be used to make sure that the existing rough bound on its short distance branching ratio is satisfied. Unfortunately tree-level neutral Higgs contributions to and are expected to be negligible, but this fact by itself offers an important test and distinction from tree level neutral gauge boson exchanges where these decays could still be significantly affected [1]. Also the decays are subject to considerable hadronic uncertainties and their measurements are not expected in this decade. Yet, as they are known to be sensitive to NP effects it is of interest to consider them as well and compare the scalar case with the case of models [1].

In the present paper we do not study the ratio , which is rather accurately measured but presently subject to much larger hadronic uncertainties than observables listed in (18). Yet, it should be emphasized that is important for the tests of FCNC scenarios as it is very sensitive to any NP contribution [13, 14, 15].

Explicit expressions for the observables in the system in terms of the relevant couplings can be found in Sections 3 and 5.

Now NP contributions to these observables are fully described by

 ΔsdL(H)MH=−~s12MHe−iδ12,Δμ¯μS,P(H)MH (19)

The ratios involving muon couplings are already constrained or determined in previous steps. Consequently, we can bound quark couplings involved by using the data on the observables in (18). Moreover we identify certain correlations characteristic for LHS scenario. and the minus sign is chosen to cancel the one of .

We can already announce at this stage that the results in physics turned out to be much less interesting than in the and systems and we summarize them separately in Section 9.

Step 6:

As all parameters of LHS scenario have been fixed in the first five steps we are in the position to make predictions for the following processes

 B→Xsℓ+ℓ−,B→Kℓ+ℓ−,B→K∗ℓ+ℓ− (20)

and test whether they provide additional constraints on the couplings. Again as in the case of and also the transitions are expected to be SM-like which provides a distinction from the gauge boson mediated tree-level transitions [1].

Step 7:

We repeat Steps 3-6 for the case of RHS. We will see that in view of the change of the sign of NP contributions to and decays the structure of the correlations between various observables will distinguish this scenario from the LHS one. Yet, as we will find out, by going from LHS to RHS scenario we can keep results of Steps 3-5 unchanged by interchanging simultaneously two big oases in the parameter space that we encountered already in our study of the model [12] and models [1]. This LH-RH invariance present in Steps 3-5 can be broken by the transition in (20). They allow us to distinguish the physics of RH scalar currents from LH ones. As only RH couplings are present in the NP contributions in this scenario, we can use the parametrization of these couplings as in (15), (17) and (19) keeping in mind that now RH couplings are involved.

Step 8:

We repeat Steps 3-6 for the case of LRS. In the case of tree-level gauge boson contributions the new features relative to the previous scenarios is enhanced NP contributions due to the presence of LR operators in transitions. Yet, in the scalar case, the matrix elements of SLL and SRR operators present in previous scenarios are also significant larger than the SM ones and the addition of LR operators has a more modest effect than in the gauge boson case. However, one of the important new feature is the vanishing of NP contributions to and decays. As the LH and RH couplings are equal we can again use the parametrization of these couplings as in (15), (17) and (19) but their values will change due to different constraints from transitions. Also in this step transitions can play an important role.

Step 9:

We repeat Steps 3-6 for the case of ALRS. Here the new feature relatively to LRS are non-vanishing NP contributions to , including CP asymmetries. Again the transitions will exhibit their strength in testing the theory in a different environment: NP contributions to observables due to the presence of LR operators. As the LH and RH couplings differ only by a sign we can again use the parametrization of these couplings as in (15), (17) and (19) but their values will change due to different constraints from transitions.

Step 10:

One can consider next the case of simultaneous LH and RH couplings that are unrelated to each other. This step is more challenging as one has more free parameters and in order to reach clear cut conclusions one would need a concrete model for couplings or a very involved numerical analysis [7, 8, 16]. A simple model in which both a scalar and a pseudoscalar with approximately the same mass couple equally to quarks and leptons has been recently presented in [17] showing that the structure of correlations can be quite rich. We refer to this paper for details.

Once this analysis of contributions is completed it will be straightforward to apply it to the case of the SM Higgs boson with flavour violating couplings. Yet, we will see that this case is less interesting than the case of with flavour violating couplings.

## 3 ΔF=2 Processes

### 3.1 Preliminaries

In the SM the dominant top quark contributions to processes are described by flavour universal real valued function given as follows ():

 S0(xt)=4xt−11x2t+x3t4(1−xt)2−3x2tlogxt2(1−xt)3 . (21)

In other CMFV models is replaced by a different function which is still flavour universal and is real valued. This implies very stringent relations between various observables in three meson system in question which have been reviewed in [18].

In the presence of tree-level contributions the flavour universality is generally broken and one needs three different functions

 S(K),S(Bd),S(Bs), (22)

to describe and systems. Moreover, they all become complex quantities. Therefore CMFV relations are generally broken. In introducing these functions we will include in their definitions the contributions of operators with , and Dirac structures.

The derivation of the formulae listed below is so simple that we will not present it here. In any case, the compendium of relevant formulae given below and in next sections is self-contained as far as numerical analysis is concerned.

### 3.2 Master Functions Including H Contributions

Calculating the contributions of to transitions it is straightforward to write down the expressions for the master functions in (22) in terms of the couplings defined in Fig. 1.

We define first the relevant CKM factors

 λ(K)i=V∗isVid,λ(d)t=V∗tbVtd,λ(s)t=V∗tbVts, (23)

and introduce

 g2SM=4GF√2α2πsin2θW=1.78137×10−7GeV−2. (24)

The master functions for are then given as follows

 S(M)=S0(xt)+ΔS(M)≡|S(M)|eiθMS (25)

with receiving contributions from various operators so that it is useful to write

 ΔS(M)=[ΔS(M)]SLL+[ΔS(M)]SRR+[ΔS(M)]LR. (26)

The contributing new operators are defined for the system as follows [19, 20]

 QLR1 = (¯sγμPLd)(¯sγμPRd), (27a) QLR2 = (¯sPLd)(¯sPRd). (27b)
 QSLL1 = (¯sPLd)(¯sPLd), (28a) QSRR1 = (¯sPRd)(¯sPRd), (28b) QSLL2 = (¯sσμνPLd)(¯sσμνPLd), (28c) QSRR2 = (¯sσμνPRd)(¯sσμνPRd), (28d)

where and we suppressed colour indices as they are summed up in each factor. For instance stands for and similarly for other factors. For mixing our conventions for new operators are:

 QLR1 = (¯bγμPLq)(¯bγμPRq), (29a) QLR2 = (¯bPLq)(¯bPRq), (29b)
 QSLL1 = (¯bPLq)(¯bPLq), (30a) QSRR1 = (¯bPRq)(¯bPRq), (30b) QSLL2 = (¯bσμνPLq)(¯bσμνPLq), (30c) QSRR2 = (¯bσμνPRq)(¯bσμνPRq). (30d)

In order to calculate the SLL, SRR and LR contributions to we introduce quantities familiar from SM expressions for mixing amplitudes

 T(Bq)=G2F12π2F2Bq^BBqmBqM2W(λ(q)t)2ηB, (31)
 T(K)=G2F12π2F2K^BKmKM2W(λ(K)t)2η2, (32)

where are QCD corrections and known SM non-perturbative factors.

Then

 T(K)[ΔS(K)]SLL=−(ΔsdL(H))22M2H[CSLL1(μH)⟨QSLL1(μH,K)⟩+CSLL2(μH)⟨QSLL2(μH,K)⟩] (33)

with the SRR contribution obtained by replacing L by R. Note that this replacement only affects the coupling as the hadronic matrix elements being evaluated in QCD remain unchanged and the Wilson coefficients have been so defined that they also remain unchanged. For LR contributions we find

 T(K)[ΔS(K)]LR=−ΔsdL(H)ΔsdR(H)M2H[CLR1(μH)⟨QLR1(μH,K)⟩+CLR2(μH)⟨QLR2(μH,K)⟩]. (34)

Including NLO QCD corrections [20] the Wilson coefficients of the involved operators are given by

 CSLL1(μ)=CSRR1(μ) =1+αs4π(−3logM2Hμ2+92), (35) CSLL2(μ)=C% SRR2(μ)=αs4π(−112logM2Hμ2+18), (36) CLR1(μ) =−32αs4π, (37) CLR2(μ) =1−αs4π3N=1−αs4π. (38)

Next

 ⟨Qai(μH,K)⟩≡mKF2K3Pai(μH,K) (39)

are the matrix elements of operators evaluated at the matching scale and are the coefficients introduced in [19]. The dependence of cancels the one of and of so that does not depend on . It should be emphasized at this point that in contrast to gauge boson couplings the couplings are scale dependent and consistently with the NLO calculation in [20] they are defined here at . In our numerical calculations we will simply set .

Similarly for systems we have

 T(Bq)[ΔS(Bq)]SLL=−(ΔbqL(H))22M2H[CSLL1(μH)⟨QSLL1(μH,Bq)⟩+CSLL2(μH)⟨QSLL2(μH,Bq)⟩] (40)
 T(Bq)[ΔS(Bq)]LR=−ΔbqL(H)ΔbqR(H)M2H[CLR1(μH)⟨QLR1(μH,Bq)⟩+CLR2(μH)⟨QLR2(μH,Bq)⟩], (41)

where the Wilson coefficients are as in the system and the matrix elements are given by

 ⟨Qai(μH,Bq)⟩≡mBqF2Bq3Pai(μH,Bq). (42)

For SRR contributions one proceeds as in the system.

Finally, we collect in Table 1 central values of . They are given in the -NDR scheme and are based on lattice calculations in [21, 22] for system and in [23] for systems. For the system we have just used the average of the results in [21, 22] that are consistent with each other. As the values of the relevant parameters in these papers have been evaluated at and , respectively, we have used the formulae in [19] to obtain the values of the matrix elements in question at . For simplicity we choose this scale to be but any scale of this order would give the same results for the physical quantities up to NNLO QCD corrections that are negligible at these high scales. The renormalization scheme dependence of the matrix elements is canceled by the one of the Wilson coefficients.

In the case of tree-level SM Higgs exchanges we evaluate the matrix elements at as the inclusion of NLO QCD corrections allows us to choose any scale of without changing physical results. Then in the formulae above one should replace by the SM Higgs mass and by . This also means that the flavour violating couplings of SM Higgs are defined here at . The values of hadronic matrix elements at in the -NDR scheme are given in Table 2.

### 3.3 Basic Formulae for ΔF=2 Observables

The mass differences are given as follows:

 ΔMd=G2F6π2M2WmBd|λ(d)t|2F2Bd^BBdηB|S(Bd)|, (43)
 ΔMs=G2F6π2M2WmBs|λ(s)t|2F2Bs^BBsηB|S(Bs)|. (44)

The corresponding mixing induced CP-asymmetries are then given by

 SψKS=sin(2β+2φBd),Sψϕ=sin(2|βs|−2φBs), (45)

where the phases and are defined by

 Vtd=|Vtd|e−iβ,Vts=−|Vts|e−iβs. (46)

. The new phases are directly related to the phases of the functions :

 2φBq=−θBqS. (47)

Our phase conventions are as in [1] and our previous papers quoted in this work. Consequently . On the other hand the experimental results are usually given for the phase

 ϕs=2βs+ϕNP (48)

so that

 Sψϕ=−sin(ϕs),2φBs=ϕNP. (49)

Using this dictionary the most recent result for from the LHCb analysis of CP-violation in decay implies [24]

 2|βs|−2φBs=0.001±0.104, (50)

that is close to its SM value. But the uncertainties are still sufficiently large so that it is of interest to investigate correlations of with other observables in the system.

For the CP-violating parameter and we have respectively

 εK=κεeiφε√2(ΔMK)exp[I(MK12)],ΔMK=2R(MK12), (51)

where

 (MK12)∗=G2F12π2F2K^BKmKM2W[λ2cη1xc+λ2tη2S(K)+2λcλtη3S0(xc,xt)]. (52)

Here, is a real valued one-loop box function for which explicit expression is given e. g. in [25]. The factors are QCD corrections evaluated at the NLO level in [26, 27, 28, 29, 30]. For and also NNLO corrections are known [31, 32]. Next and [33, 34] takes into account that and includes long distance effects in and .

In the rest of the paper, unless otherwise stated, we will assume that all four parameters in the CKM matrix have been determined through tree-level decays without any NP pollution and pollution from QCD-penguin diagrams so that their values can be used universally in all NP models considered by us.

## 4 Rare B Decays

### 4.1 Preliminaries

These decays played already for many years a significant role in constraining NP models. In particular was instrumental in bounding scalar contributions in the framework of supersymmetric models and two Higgs doublet models (2HDM). Recently a very detailed analysis of the decay including the observables involved in the time-dependent rate has been presented [17]. Below, after recalling the relevant effective Hamiltonian that can be used for other transitions, we will summarize the final formulae for the most important observables in