1 Introduction

SFB/CPP-10-122

TTP10-48

MZ-TH/10-44

November 2010

and as a handle on

isospin-violating New Physics

Lars Hofer, Dominik Scherer and Leonardo Vernazza1

Institut für Theoretische Teilchenphysik,

Karlsruhe Institute of Technology, D–76128 Karlsruhe, Germany

[0.4cm] Institut für Theoretische Physik und Astrophysik,

Universität Würzburg, D–97074 Würzburg, Germany

[0.4cm] Institut für Physik (THEP),

Johannes Gutenberg-Universität, D–55099 Mainz, Germany

[0.4cm]

The discrepancy between theory and experiment observed in the difference can be explained by a new electroweak penguin amplitude. Motivated by this result, we analyse the purely isospin-violating decays and , which are dominated by electroweak penguins, and show that in presence of a new electroweak penguin amplitude their branching ratio can be enhanced by up to an order of magnitude, without violating any constraints from other hadronic decays. This makes them very interesting modes for LHCb and future factories. We perform both a model-independent analysis and a study within realistic New Physics models such as a modified--penguin scenario, a model with an additional boson and the MSSM. In the latter cases the new amplitude can be correlated with other flavour phenomena, such as semileptonic decays and - mixing, which impose stringent constraints on the enhancement of the two decays. In particular we find that, contrary to claims in the literature, electroweak penguins in the MSSM can reduce the discrepancy in the modes only marginally. As byproducts we update the SM predictions to and and perform a state-of-the-art analysis of amplitudes in QCD factorisation.

## 1 Introduction

At present flavour physics has entered a new exciting era. The new experiment LHCb and the planned super-B-factories will bring the precision of Standard Model (SM) tests and the scope of searches for New Physics (NP) to unseen heights. Particularly important thereby are flavour-changing neutral current (FCNC) decays, which in the SM are highly-suppressed electroweak loop processes. In this work we present a phenomenological analysis of two hadronic FCNC decays, namely and . We argue that within the next years these decays will become very interesting objects for experimental analyses of the electroweak penguin sector. Up to now, this sector has been tested in hadronic decays only in modes, and the discrepancies found between the SM prediction and experimental measurements is the main motivation for our work.

The four decay channels, first observed by the CLEO experiment in the late 1990s [1, 2], have become by now a classic in flavour physics thanks to the precise measurements by BABAR and BELLE. This is also reflected in the large number of theoretical studies of these decays in the SM and various extensions of it. Charged and neutral mesons can decay to a final state due to a weak process at the partonic level, with . This process is dominated by an FCNC loop governed by the CKM factor and receives, in the case, also a small tree-level contribution involving the smaller CKM factor . The branching fractions are therefore small, of order , and sensitive to new FCNCs arising in extensions of the SM. For this reason they are, together with the corresponding CP asymmetries, important observables for tests of the SM flavour structure and for NP searches.

With the data of the factories having become more and more precise, some discrepancies between measurements and SM predictions have occurred, provoking speculations on a “ puzzle”. To date, the measurements of the branching fractions have fluctuated towards the SM predictions, the latter still suffering from large hadronic uncertainties, and only the CP asymmetries show an unexpected behavior [3, 4, 5] manifesting itself in the quantity

 ΔACP≡ACP(B−→π0K−)−ACP(¯B0→π+K−). (1)

For this observable we find in the framework of QCD factorisation (QCDF)

 ΔACP\lx@stackrelSM=1.9+5.8−4.8% (2)

as the SM prediction, which differs significantly from the experimental value

 ΔACP\lx@stackrelexp.=(14.8±2.8),%. (3)

[6]. Adopting a frequentist approach where we consider a theoretical “error bar” as a range of values definitely containing the true theory result but without assigning any statistical meaning to it [7], this amounts to a discrepancy.

A point which has received much attention in the literature (see e.g. [8] and references therein) is the fact that the formerly observed discrepancies as well as the currently existing anomaly in suggest a violation of the strong isospin symmetry beyond the amount expected in the SM. This has often been interpreted as a hint for enhanced electroweak penguins (EW penguins) [9, 10, 11]. We will give a brief overview and discuss the current status of this topic in section 2.1. Whether the discrepancy in is a hint for NP in EW penguins or a non-perturbative hadronic effect or simply a statistical fluctuation is controversial. The point that we want to make is that, in order to assess this question, it is highly desirable to obtain further information from other hadronic decays which are sensitive to EW penguin contributions. For this reason we study the purely isospin-violating decays and , which are dominated by EW penguins, extending and updating our analysis presented in ref. [12]. If NP in this sector exists at a level where it can explain the puzzle, it could be clearly visible in these purely isospin-violating decays. The upcoming new generation of flavour experiments will have the opportunity to detect these modes for the first time and to measure their branching fractions. The aim of our work is to provide a detailed analysis from the theory side, both in the SM and beyond.

Since the decays and are not related to other decay modes via flavour symmetries, the non-perturbative part of their decay amplitudes has to be determined from first principles. This can be achieved using the framework of QCDF [13, 14, 15, 16, 17]. This method amounts to a calculation of the hadronic matrix elements up to corrections of order , where is a typical non-perturbative energy scale of strong interactions. We will use this method throughout the paper in all analyses of decays to light mesons.

The plan of the paper is as follows: In Chapter 2, we discuss the issue of isospin-violation in decays and the phenomenology of and . As a byproduct we provide simple formulas which allow for an easy calculation of various observables concerning these decay modes, taking into account NP effects in EW penguins. Chapter 3 contains a detailed quantitative analysis of and in different scenarios of a model-independent parameterisation of NP in EW penguins. This analysis is performed in light of our present knowledge on EW penguins from other decays, in particular . It is complemented in Chapter 4 with studies of particular extensions of the SM which feature enhanced EW penguins. We conclude in Chapter 5. We keep the main body of the paper free of technicalities and refer the reader interested in technical details to the appendices.

## 2 Isospin-violation in hadronic B decays

### 2.1 The B→πK modes

The decays are dominated by the isospin-conserving QCD penguin amplitude. Nevertheless, they receive small contributions from the tree and the EW penguin amplitude, which are isospin-violating. Combining measurements of the four different decay modes , , and , it is possible to construct observables in which the leading contribution from the QCD penguin drops out, so that they are sensitive to isospin violation.

The mesons participating in decays transform under isospin rotations as

 (¯B0,−B−)1/2,(¯K0,−K−)1/2,(π+,−π0,−π−)1. (4)

Furthermore we can assign isospin to the operators appearing in the effective Hamiltonian

 Heff=GF√2∑p=u,cλ(s)p(C1Qp1+C2Qp2+10∑i=3CiQi+C7γQ7γ+C8gQ8g)+h.c., (5)

which mediates the transitions. Here represents a product of elements of the quark mixing (CKM) matrix, are the so-called current-current operators, are QCD penguin operators, and represent the electromagnetic and chromomagnetic operators and

 Q7=(¯sαbα)V−A∑q32eq(¯qβqβ)V+A, Q8 =(¯sαbβ)V−A∑q32eq(¯qβqα)V+A, (6) Q9=(¯sαbα)V−A∑q32eq(¯qβqβ)V−A, Q10 =(¯sαbβ)V−A∑q32eq(¯qβqα)V−A, (7)

are the EW penguin operators ( denote colours). The latter are of great importance for our work. We define the operators as in [15] so that at leading order. Containing - and -bilinears, the operators ,…, can be distributed among

 Heff=HΔI=1eff+HΔI=0eff (8)

according to the decomposition [18]. Since the QCD penguin operators involve the isosinglet combination , they contribute solely to whereas the other operators give contributions to both parts of . The decays thus follow the isospin pattern

 1/2\lx@stackrelΔI=1,0⟶1⊗1/2=3/2⊕1/2, (9)

implying that all four decay amplitudes can be decomposed into three independent isospin amplitudes, , and with the lower index denoting the total isospin of the final state.

One finds that is dominated by the QCD penguin contribution and thus . To a first approximation, all the decay modes can be described by the amplitude only, dictating the relative size of the branching fractions to be (in the same order as in tab. 1).

The isospin-invariant amplitudes receive contributions from various SM quark diagrams. It is only at the level of these diagrams that the pattern of CP violation can be correctly implemented, i.e. that the amplitudes , can be related to their CP-conjugated counterparts , . This suggests an alternative parameterisation of the amplitudes in terms of the topologies of the underlying quark-level transitions [19, 20]:

 A(B−→π−¯K0) ≃ P(1−13rCEW+23rAEW), √2A(B−→π0K−) ≃ P(1+rEW+23rCEW+23rAEW−(rT+rC)e−iγ), A(¯B0→π+K−) ≃ P(1+23rCEW−13rAEW−rTe−iγ), √2A(¯B0→π0¯K0) ≃ −P(1−rEW−13rCEW−13rAEW+rCe−iγ). (10)

This topological parameterisation is illustrated by the corresponding Feynman diagrams for in fig. 1. In eq. (2.1) we have factored out the dominant QCD penguin amplitude and neglected penguin amplitudes suppressed by . The dependence on the weak CKM phase has been made explicit, while strong phases are contained in the ratios which fulfill . These quantities denote corrections from different types of Feynman diagrams: and stem from colour-allowed and colour-suppressed tree diagrams, and from colour-allowed and colour-suppressed EW penguins, respectively. Annihilation via QCD penguin diagrams is absorbed into whereas weak annihilation via EW penguin diagrams is parameterised by and colour-suppressed tree annihilation is neglected. With our QCDF setup explained in Appendix A and the expressions for the ratios given in Appendix B we obtain

 rT = 0.17+0.07−0.06+0.03+0.03−0.10i, rC = 0.07+0.04−0.06+(−0.01)+0.03−0.05i, rEW = 0.13+0.05−0.05+0.02+0.02−0.07i, rCEW = 0.04+0.02−0.03+(−0.01)+0.02−0.03i, rAEW = 0.007+0.002−0.010+(−0.004)+0.011−0.003i. (11)

The result displays the typical features of QCDF predictions, namely small strong phases and large uncertainties of colour-suppressed topologies. The smallness of the reflects the domination of the isospin-conserving QCD penguin and justifies the expansion of physical observables in the . Among the isospin-violating contributions the colour-allowed tree gives the largest corrections followed by the EW penguin which dominates over the colour-suppressed tree. The colour-suppressed EW penguin ratio and especially the EW penguin annihilation ratio are quite small and consequently they have been omitted in most analyses of decays. In particular, the possibility of having NP in the EW penguin annihilation amplitude has to our knowledge not been considered so far. However, we want to point out that such an approximation is not valid in the analysis of CP asymmetries: non-vanishing direct CP asymmetries are caused by the interference of parts of the decay amplitude with different weak and strong phases. Consequently direct CP asymmetries in cannot be generated by the QCD penguin amplitude alone and are automatically sensitive to subleading contributions, encoded in the imaginary parts of the coefficients. These, in turn, are generated in QCDF either perturbatively at or non-perturbatively at . At the colour-suppression of is not present anymore and the  - suppressed can compete as well. Therefore we keep and in our calculation and we will see in later chapters that we can indeed have a large NP contribution in these amplitudes.

One can easily see from eqs. (2.1,2.1) that the two amplitudes involved in differ only by the subdominant contributions , and , all of which are isospin-violating. Turning to the CP asymmetries, one finds in the SM

 ACP(B−→π0K−) ≃ −2Im(rT+rC)sinγ, ACP(¯B0→π+K−) ≃ −2Im(rT)sinγ, (12)

with terms quadratic in the being neglected. Thus the only possible explanation for a large in the SM seems to be a large imaginary part of , i.e. a large absolute value and large strong phase of the colour-suppressed tree amplitude, generated by some hadronic effects at the low scale which can hardly be calculated perturbatively. However, QCDF predicts only a small , insufficient to explain the data, even when all the theory uncertainties are included. Therefore one is tempted to conclude that the discrepancy in is not due to our lack of understanding of strong interactions but due to isospin-violating NP.

For this reason, has been studied in various NP models in recent publications [5, 36, 37, 38, 39, 40, 41, 42, 43, 44]. The main ingredient of these analyses is usually an enhancement of the EW penguin topologies by effects of virtual heavy particles. Such contributions can be included into the amplitudes (2.1) by the replacements

 rEW→rEW+~rEWe−iδ,rCEW→rCEW+~rCEWe−iδ,rAEW→rAEW+~rAEWe−iδ, (13)

where is a new weak phase and are complex numbers including a strong phase. The CP asymmetries then become

 ACP(B−→π0K−) ≃ −2Im(rT+rC)sinγ+2Im(~rEW+23~rCEW+23~rAEW)sinδ, ACP(¯B0→π+K−) ≃ −2Im(rT)sinγ+2Im(23~rCEW−13~rAEW)sinδ, (14)

such that

 ΔACP≃−2Im(rC)sinγ+2Im(~rEW+~rAEW)sinδ (15)

can turn out to be much larger than in the SM. The observed discrepancy can be solved by a or a term comparable in size to the corresponding SM term .

Apart from one can also construct other observables from the data which are sensitive to isospin violation, for example certain ratios of branching fractions. Even though tensions with experimental data in these observables raised the formulation of a ” puzzle” in the first place [8, 9, 11, 45, 46, 47, 48], in the meantime these quantities are in reasonable agreement with the SM predictions. However, they serve as important constraints for NP in EW penguins and we define and discuss them in Appendix B. Note in particular that the quantity defined there, which is the difference of the two remaining CP asymmetries not appearing in , probes the same combination of and as . Unfortunately, data on and especially on are not good enough yet to gain any information from these observables. Experimental results and SM predictions for the observables are given in tab. 1.

The main problem which makes it difficult to single out a possible NP contribution in decays is evident from (2.1): the colour-allowed EW penguin contributions and colour-suppressed tree contributions enter the amplitudes in (2.1) exclusively in the combination

 rEW−rCe−iγ. (16)

This implies that colour-allowed EW penguins and colour-suppressed trees are inextricably linked with each other, reflecting the fact that the topological parameterisation contains some redundancy. Physical effects found in any experiment cannot unambiguously be attributed to one or the other partner of this topology pair. A new EW penguin contribution can be probed only in one of the four physical combinations

 Re(~rEW)cosδ−Re(rC)cosγ+Re(rEW), Im(~rEW)sinδ−Im(rC)sinγ, Im(~rEW)cosδ−Im(rC)cosγ+Im(rEW), Re(~rEW)sinδ−Re(rC)sinγ. (17)

Therefore, probing is challenged by the large hadronic uncertainties in the QCDF prediction for , which can mimic or hide such a NP signal. One possible way to constrain is the approximate flavour symmetry which relates it to a corresponding topology. Using this symmetry it has been found that current data on violation in is also in disagreement with the SM, independently of , and can be explained by adding to the amplitude [42].

The perspective of our work is the following: In order to find out whether the discrepancy really is a manifestation of isospin-violating physics beyond the SM, one should also study other observables on which such a kind of NP could have a large impact and see whether similar effects appear in measurements of these observables. Our proposal in this work is to test the hypothesis of isospin-violating NP by looking at processes which are highly sensitive to it, namely purely isospin-violating decays.

### 2.2 Purely isospin-violating Bs decays

EW penguin contributions to hadronic decays are usually overshadowed by the larger QCD penguins. This problem can be avoided if one succeeds in probing exclusively the part of the effective Hamiltonian which is orthogonal to the QCD penguin operators. To achieve this for , we had to single out the part of the transition in eq. (9) by combining different isospin-related decay modes, for example by considering the observable . Our proposal now is to consider decays to which QCD penguins do not contribute at all, i.e. pure decays, where no such procedure is needed.

There are no two-body decays of the or meson with this property. In these cases the final state would have to be a pure isospin state which cannot be constructed out of two mesons. The meson, on the other hand, is an isosinglet and it can decay as

 0\lx@stackrelΔI=1⟶0⊗1=1. (18)

The final state must consist of an isospin triplet, i.e.  or , and an isosinglet, i.e. a meson with the flavour structure . In order to avoid complications stemming from -mixing, we restrict ourselves to the vector-meson which is to a good approximation a pure state. This leaves us with the two channels

.

So far only an upper limit exists [49] and no detailed theory analysis of has been published. Only the SM branching fractions and CP asymmetries have been calculated in general surveys on decays to light mesons [16, 17]. In addition, has been suggested as a tool to measure via the mixing-induced CP asymmetry [50]. Since in the era of LHCb and super B-factories these two processes will become interesting objects for tests of isospin-violation and potential NP we will in the following study their phenomenology in full detail, in the SM and beyond.

In the SM only three basic topologies are present in these decays, depicted in fig. 2:

• EW penguins

• CKM- and colour-suppressed tree diagrams

• Singlet-annihilation diagrams.

Since the flavour-structure of and excludes their production via gluon-exchange, annihilation can only contribute if the meson (the flavour singlet) in the final state is produced from gluons and the second meson comes from weak (as depicted in fig. 2) or electromagnetic interactions. Since the is colour-neutral and it is odd under charge-conjugation, at least three gluons are needed, so that the singlet-annihilation amplitude is formally of higher-order in and does not appear in QCDF at the next-to-leading order [16]. However, annihilation topologies in general do not factorise and cannot be calculated perturbatively, because the exchanged gluons may be soft. This means that we can, from a theoretical point of view, only rely on the suppression of these contributions by , where is a non-perturbative scale, and by . This leads to the expectation that both the tree and the EW penguin amplitudes can receive corrections of from singlet-annihilation. However, we can also argue from a phenomenological point of view that -production from three gluons is suppressed by the OZI rule [51, 52, 53, 54] and should thus be only a small effect, even though this rule is theoretically not well understood. In short, our reasoning leads us to the conclusion that in order to test NP in , we have to look for new effects which are much larger than this intrinsic uncertainty.

In all our calculations of we use the full QCDF decay amplitudes, see refs. [16, 17]. However, since these are quite involved, we now quote simple approximative formulas which can be used as building blocks for an easy calculation of various observables such as branching fractions, CP asymmetries and polarisation fractions. Neglecting singlet-annihilation we can parameterise the amplitudes in analogy to eq. (2.1) as

 √2A(¯Bs→ϕM2)=PM2EW(1−rM2Ce−iγ), (19)

with representing a , a longitudinal or a with negative helicity. The positive helicity amplitude can be neglected in the SM because of its  – suppression. We have factored out the EW penguin amplitude anticipating its dominance over the colour-suppressed tree represented by the tree-to-penguin ratio . A new contribution to the amplitudes of the form (13) would also enter the amplitude (19) modifying it as

 √2A(¯Bs→ϕM2)=PM2EW(1−rM2Ce−iγ+~rM2EWe−iδ) (20)

where contains a strong phase and is the weak phase introduced in (13). If we assume the new contribution to be of the order of the SM EW penguin, as required by a solution of the “-puzzle”, we have and expect a large enhancement of the branching fractions, up to an order of magnitude. In order to obtain the same effect within the SM one would have to assume an even larger enhancement of the soft non-perturbative physics entering the colour-suppressed tree topology in .

Choosing a phase convention such that is real, we find

 (21)

for the isotriplet meson being , longitudinal and with negative helicity, respectively. We further have

 rπC = 0.41+0.37−0.41−0.13+0.30−0.30i, rρ,0C = 0.39+0.35−0.39−0.13+0.28−0.29i, rρ,−C = 0.21+0.49−0.46+0.15+0.45−0.45i. (22)

Inserting these numbers into eq. (19) we obtain a good approximation of the SM amplitudes for the decays. Replacing in eq. (19) yields the corresponding CP-conjugated amplitudes ( decays). Subsequently one can use the formulas in Appendix A.4 to convert the amplitudes into physical observables. In section 3.1 we extend these prescriptions to physics beyond the SM.

One should keep in mind that the numbers above are calculated using state-of-the-art values for the non-perturbative input parameters, summarised in Appendix A.3. They are based on lattice QCD, QCD sum rules and experimental data. Since our knowledge on these parameters is hopefully going to improve in the future it is desirable to have an additional parameterisation of the decay amplitudes where the non-perturbative input can be changed. We find the dominant sources of theory uncertainties to be (ordered by importance)

• the form factors and ,

• the CKM angle ,

• the non-factorisable spectator-scattering amplitudes, parameterised by the complex number and the first inverse moment of the -meson light-cone distribution amplitude.

The remaining uncertainties, stemming from decay constants, Gegenbauer moments, quark masses and CKM parameters, are much less important so we do not need to display them explicitly. Setting the less important theory parameters to their default values we arrive at the following approximate expressions for the quantities in eqs. (21,2.2):

 PπEW = 17.0ABs→ϕ0(0)⋅10−9,rπC = −0.12i−0.02+0.01GeV(1+XH)ABs→ϕ0(0)λBs Pρ,0EW = 26.2ABs→ϕ0(0)⋅10−9,rρ,0C = −0.13i−0.02+0.01GeV(1+XH)ABs→ϕ0(0)λBs Pρ,−EW = 6.6FBs→ϕ−(0)⋅10−9,rρ,−C = 0.14i−0.06−0.02GeV(1−XH)FBs→ϕ−(0)λBs. (23)

The tree topologies suffer from the large spectator-scattering uncertainties due to a strong cancellation between the leading order and QCD vertex corrections. Again one can insert these formulas into eq. (19), this time with arbitrary values and uncertainties for the form factors and spectator-scattering parameters, and use the definitions in Appendix A.4 to calculate physical observables. CP conjugation again amounts to replacement .

We conclude this section quoting our QCDF results for the SM values of the observables. As for the CP-averaged branching fractions we obtain

 (24)

For comparison we also quote the approximate result according to (23):

 (25)

The smallness of the SM branching ratios compared to other hadronic decays is due to the absence of QCD penguins and non-suppressed tree-level contributions. The measurement of these branching fractions is thus challenging and has not been achieved yet. However, we will show in later chapters that NP in EW penguins has the chance to enhance the BRs by up to an order of magnitude, such that this measurement is a very interesting project. We expect that LHCb will be able to measure while the mode is more suitable for a super B-factory where a full reconstruction can cure the notorious difficulties with the identification of neutral pions. In case of a strong enhancement should also be visible in the Tevatron data [55]. The branching ratio is dominated by the longitudinal polarisation state as can be seen in

 Br(¯Bs→ϕLρ0L)=3.7+2.5−0.7⋅10−7 (26)

and the longitudinal polarisation fraction

 fL=0.84+0.08−0.11. (27)

As stated above, one of the main sources of uncertainty in the QCDF predictions is the form factor . It can in principle be eliminated by considering the ratios

 Br(¯Bs→ϕρ0)Br(¯Bs→ϕπ0)=2.83+0.35−0.23,Br(¯Bs→ϕLρ0L)Br(¯Bs→ϕπ0)=2.38+0.10−0.08. (28)

NP could still be visible in these ratios because in many scenarios it enters and in different ways. The cancellation of also occurs in the ratios

 Br(¯Bs→ϕπ0)Br(¯Bs→ϕϕ) = 0.007+0.008−0.004, Br(¯Bs→ϕρ0)Br(¯Bs→ϕϕ) = 0.020+0.023−0.010,Br(¯Bs→ϕLρ0L)Br(¯Bs→ϕϕ)=0.017+0.019−0.009. (29)

There however this gain is compensated by additional uncertainties arising from the QCD-penguin-dominated decay . The experimental benefit in these last ratios is that at LHCb absolute branching ratios cannot be measured because the absolute number of mesons is unknown. Finally, we find the direct CP asymmetries to be very uncertain:

Due to the smallness of the branching ratios, these CP asymmetries are also difficult to access experimentally, therefore we will not consider them any further.

## 3 Model-independent analysis

In the previous chapter we proposed to test the hypothesis of NP in the EW penguin sector, as suggested by the discrepancy in the observable , by a measurement of the decays . In this chapter we support our proposal by a quantitative analysis pursuing the following strategy: We parameterise NP in EW penguins in a model-independent way by adding corresponding terms to the Wilson coefficients . By performing a -fit we determine the NP parameters in such a way that they describe well the data. In particular they should allow for a solution of the discrepancy. Further hadronic decays like are used to impose additional constraints at the level. With respect to the resulting fit we study the decays and quantify a potential enhancement of their branching fractions. Note that such an exhaustive analysis, correlating different hadronic decay modes with sensitivity to isospin violation, is only possible if hadronic matrix elements are calculated from first principles like in the framework of QCDF. A method based on flavour symmetries, as it has been used in most studies of decays so far, could not achieve this. In particular, the decays , which are our main interest, are not related to any other decay via so their branching fractions cannot be predicted in this way.

### 3.1 Modified EW penguin coefficients

In the SM the Wilson coefficients obey the hierarchy at the electroweak scale. This is because receives -enhanced contributions from -penguin and box diagrams in contrast to , while are generated for the first time at two-loop level due to their colour structure. For our model-independent analysis we consider arbitrary NP contributions to the coefficients and as well as to their mirror counterparts and . Normalizing the new coefficients to the SM value defined in eq. (59) in the appendix, we have

 C(′)NP7,9(MW)=CLO9(MW)q(′)7,9,q(′)7,9=|q(′)7,9|eiϕ(′)7,9, (31)

where are new weak phases. The coefficient contains the parts of enhanced by and , as explained in Appendix A.1. There we also describe the scheme which we use for the renormalisation-group evolution. Applying it to the NP coefficients leads to the low-scale values displayed in tab. 2. They can be compared to the dominant SM coefficient .

In our analysis we will study several different scenarios. First, we consider the cases where only one of the coefficients , , , is different from zero. This means we assume the dominance of an individual NP operator as it has also been done for example in ref. [41]. Second, we consider the possibilities of having ,