Some results on Lepton Flavour Universality Violation

Motivated by recent experimental measurements on flavour physics, in tension with Standard Model predictions, we perform an updated analysis of New Physics violating Lepton Flavour Universality, by using the effective Lagrangian approach and in the Z' and S_3 leptoquark models. We explicitly analyze the impact of considering complex Wilson coefficients in the analysis of B-anomalies, by performing a global fit of R_K and R_K*0 observables, together with \Delta Ms and A_CP^mix. The inclusion of complex couplings provides a slightly improved global fit, and a marginally improved \Delta Ms prediction.


Introduction
The observation of flavour violating processes at the LHC would be a definite sign for physics beyond the Standard Model (SM). At present, many interesting measurements on flavour physics are performed at the LHC [1][2][3][4][5][6][7][8][9][10][11][12][13][14]. Some relevant flavour transition processes in order to constrain new physics at the LHC are the leptonic, semi-leptonic, baryon and radiative exclusive decays. Some of these decays allow us to build optimized observables, as ratios of these decays, that are theoretically clean observables and whose measurements are in tension with SM predictions. One example is the case of observables in b → sll transitions. Recently, the LHCb collaboration observed a deviation from the SM predictions in the neutral-current b → s transition [1,2,[5][6][7]11,13], hinting at lepton flavour universality violation effects. The results for ratios of branching ratios involving different lepton flavours are given by [2,11,13], where the first uncertainty is statistical and the second one comes from systematic effects. In the SM R K = R K * 0 = 1 with theoretical uncertainties of the order of 1% [15,16], as a consequence of Lepton Flavour Universality. The compatibility of the above results with respect to the SM predictions is of 2.6 σ deviation in the first case and for R K * 0 , in the low q 2 di-lepton invariant mass region is of about 2.3 standard deviations; being in the central−q 2 of 2.4 σ. A discrepancy of about 3 σ is found when the measurements of R K and R K * 0 are combined [17]. Anomalous deviations were also observed in the angular distributions of the decay rate of B → K * µ + µ − , being the most significant discrepancy for the P 5 observable [1,6]. The Belle Collaboration has also reported a discrepancy in angular observables consistent with LHCb results [18]. In addition, ATLAS and CMS collaborations have presented their updated results for the angular parameters of the B meson decay, B 0 → K 0 µ + µ − [19][20][21]. A great theoretical effort has been devoted to the understanding of these deviations, see for example [15,17,[22][23][24][25][26][27][28][29][30][31][32][33][34][35][36][37][38][39][40] and references therein. From the theoretical side, the ratios R K and R K * 0 are very clean observables; essentially free of hadronic uncertainties that cancel in the ratios [15]. The experimental data has been used to constrain New Physics (NP) models. However, it is still an open question whether the recent experimental data on B-decays at the LHC can be accommodated in the context of some of the New Physics approaches. One useful way to analyze the effects of NP in these observables and to quantify the possible deviations from the SM predictions is through the effective Hamiltonian approach, allowing us a model-independent analysis of new physics effects. In addition, one can compute this effective Hamiltonian in the context of specific NP models. It has been shown that Z and leptoquark models could explain the R K , R K * 0 deviations.
On the other hand, NP models are also severely constrained by other flavour observables, for example in B s −mixing. Recently an updated computation for the B s mesons mass difference in the SM has been presented [41][42][43][44][45], which shows a deviation with the experimental result [45,46]: such that ∆M SM s > ∆M exp s at about 2σ. This fact imposes additional constraints over the NP parameter space. Therefore, a global fit is mandatory when considering all updated flavour observables. A negative contribution to ∆M s is needed to reconcile it with the experimental result, in the context of some NP models (like Z or leptoquarks) it implies complex Wilson coefficients in the effective Hamiltonian of R K , R K * 0 [45] (see also below). To the best of our knowledge, up to now only real Wilson coefficients have been used in global fits of R K and R K * 0 observables together with ∆M s . An effect of introducing complex couplings is the generation of CP asymmetries. The mixing-induced CP asymmetry in the B-sector can be measured through A mix CP ≡ A mix CP (B s → J/ψφ) ≡ sin(φ ccs s ), experimentally it is measured to be [46]: In the SM is given by A mix CP SM = sin(−2β s ) [45,47,48], with β s = 0.01852 ± 0.00032 [49] we obtain A mix CP SM = −0.03703 ± 0.00064, which is consistent with the experimental result (3) at the ∼ 0.5σ level.
The aim of the present work is to investigate the effects of complex Wilson coefficients in the global analyses of NP in B-meson anomalies. We assume a model independent effective Hamiltonian approach and we study the region of NP parameter space compatible with the experimental data, by considering the dependence of the results on the assumptions of imaginary and/or complex Wilson coefficients. We compare our results with the case of considering only real Wilson coefficients. A brief summary of the NP contributions to the effective Lagrangian relevant for b → s transitions and B s -mixing is presented in Section 2, where we also recall the need to consider complex Wilson coefficients in the analysis. In Section 3 we discuss the effects of having imaginary or complex Wilson coefficients on R K observables. The impact of these complex Wilson coefficients in the analysis of B-meson anomalies in two specific models, Z and leptoquarks, is included in Section 4. We consider a global fit of R K and R K * 0 observables, together with ∆M s and CP -violation observable A mix CP in this analysis. Finally, conclusions are given in Section 5.

Effective Hamiltonians and new physics models
The effective Lagrangian for b → s transitions is conventionally given by [50], being O ( ) i ( = e, µ) the operators and C ( ) i the corresponding Wilson coefficients. The relevant semi-leptonic operators for explaining deviation in R K observables, eq. (1), can be defined as, The Wilson coefficients have contributions from the SM and NP, In the present work we analyze the NP contributions C ( ) NP i . The NP contributions to B s -mixing are described by the effective Lagrangian [50]: where C LL bs is a Wilson coefficient. In order to study the allowed NP parameter space we follow the same procedure as given in [45], comparing the experimental measurement of the mass difference with the prediction in the SM and NP. Therefore, the effects can be parametrized as [45], where R loop SM = 1.3397 × 10 −3 [45]. The NP prediction to the CP -asymmetry A mix CP is given by [45,47,48] where β s and R loop SM have been given above. Since ∆M exp s < ∆M SM s (2), eq. (8) tells us that to obtain a prediction of ∆M s closer to ∆M exp s the NP Wilson coefficient C LL bs (7) must be negative (C LL bs < 0). In a generic effective Hamiltonian approach, each Wilson coefficient is independent, and setting C LL bs < 0 has no effect on C NP µ 9 , C NP µ 10 , etc. However, explicit NP models give predictions on the Wilson coefficients which introduce correlations among them. We will concentrate on two specific models that have been proposed to solve the semi-leptonic B s -decay anomalies: Z and leptoquarks.
We start with the Z model that contains a Z with mass M Z and whose extra NP operators can involve different chiralities. The part of the effective Lagrangian relevant for b → sµ + µ − transitions and B s -mixing is given by [45], where d i and α denote down-quark and charged-lepton mass eigenstates, and λ Q and λ L are hermitian matrices in flavour space. When matching the above equation with eqs. (4) and (7) one obtains the expressions for the Wilson coefficients [45], and where η LL (M Z ) > 0 encodes the running down to the bottom mass scale. From (12) it is clear that to obtain a negative C LL bs one needs an imaginary number inside the square (λ Q 23 /(V tb V * ts ) ∈ I), but this is the same factor that appears in C NP µ 9 = −C NP µ 10 in (11). λ L 22 ∈ R, since λ is an hermitic matrix, then it follows that C NP µ 9,10 are imaginary (C NP µ 9,10 ∈ I). Now we focus on leptoquark models. Specifically, we consider the scalar leptoquark S 3 ∼ (3, 3, 1/3). The quantum number in brackets indicate colour, weak and hypercharge representation, respectively. The interaction Lagrangian reads [45] where σ a are the Pauli matrices, ε = iσ 2 , and Q i and L α are the left-handed quark and lepton doublets. In this case, the contribution to the Wilson coefficients C NP µ 9,10 arises at the tree level and is given by [45], For C LL bs the contribution appears at one loop level and can be written as [45,51]: where α = 1, 2, 3 is a lepton family index. Again, in order to obtain C LL bs < 0 in (15), the couplings must comply 3 α=1 y QL 3α y QL * 2α /(V tb V * ts ) ∈ I. If the combinations y QL 3α y QL * 2α /(V tb V * ts ) ∈ I, then the expression in eq. (14) suggests C NP µ 9,10 ∈ I. Of course, the expression (15) is a sum over all generations, so it is possible to set up a model with y QL 32 y QL * 22 /(V tb V * ts ) ∈ R, and to have a cancellation such that the sum in eq. (15) is imaginary, but this would be a highly fine-tuned scenario. If the sum in (15) has an imaginary part, it would be most natural if all its addends have some imaginary part.
Here we have shown two examples of new physics models which justify the choice of imaginary (or complex) values for the Wilson coefficients C NP µ 9,10 . In the next section we take an effective Hamiltonian approach and explore whether an imaginary or complex NP Wilson coefficients can accommodate the experimental R K deviations.

Imaginary Wilson coefficients and R K observables
Several groups have analyzed the predictions for the ratios (1) based on different global fits [17,28,29,[33][34][35][36][37][38], extracting possible NP contributions or constraining it. As it is well known, an excellent fit to the experimental data is obtained when C NP 9 = −C NP 10 ; corresponding to lefthanded lepton currents. By considering this relation, we investigate the effects of having imaginary Wilson coefficients on R K observables. For the numerical evaluation we use inputs values as given in [52]. The SM input parameters most relevant for our computation are: note that the product V tb V * ts , which appears in the computation of Wilson coefficients in NP models (11), (12), (14), (15) is approximately a negative real number (V tb V * ts −0.04). Figure 1 shows the values of the ratios R K and R K * 0 , in their respective q 2 ranges, when both Wilson coefficients C NP µ 9 and C NP µ 10 are imaginary (Figure 1a) and when they are real (Figure 1b), by assuming that C NP µ 9 = −C NP µ 10 . If these two coefficients are imaginary, in all cases the minimum value for the ratio is obtained at the corresponding SM point C NP µ 9 = −C NP µ 10 = 0. The addition of non-zero imaginary Wilson coefficients results in larger values of R K and R K * 0 , at odds with the experimental values R exp K ( * 0) < R SM K ( * 0) . This behaviour was already pointed out in Ref. [25], where it is shown that the interference of purely imaginary Wilson with the SM vanishes, and therefore they can not provide negative contributions to R K , R K * 0 (see also below). In contrast, as shown  We have done a global fit by including the ratios R K and R K * 0 , and the angular observables P 4 and P 5 [6]. Results are shown in Figure 2. The allowed regions for imaginary values of C NP µ 9 and C NP µ 10 when fitting to measurements of a series of b → sµ + µ − observables are presented in Figure 2a, by assuming all other Wilson coefficients to be SM-like. The numerical analysis has been done by using the open source code flavio 0.28 [53], which computes the χ 2 function with each (C NP µ 9 , C NP µ 10 ) pair. The χ 2 difference is evaluated with respect to the SM point, ∆χ 2 = χ 2 SM − χ 2 min . Then, the pull in σ is defined as ∆χ 2 , in the case of only one Wilson coefficient, and for the two-dimensional case it can be evaluated by using the inverse cumulative distribution function of a χ 2 distribution having two degrees of freedom; for instance, ∆χ Ref. [25] showed that imaginary Wilson coefficients do not interfere with the SM amplitude, an therefore imaginary C NP µ 9,10 can not decrease the prediction for R K , R K * 0 . This is numerically shown in the above analysis, where imaginary Wilson coefficients C NP µ 9,10 are not able to reduce significantly the prediction for R K , R K * 0 . To further investigate this question we have analytically computed a numerical approximation to R K * 0 as a function of C NP µ 9 , C NP µ 10 in the region 1.1 ≤ q 2 ≤ 6.0 GeV 2 . After integration and some approximations regarding the scalar products of final state momenta, we obtain: − 0.3013 Re C NP e 10 + 0.0212|C NP e 9 | 2 + 0.0357|C NP e 10 | 2 (1.1 ≤ q 2 ≤ 6.0 GeV 2 ). (17) We have checked that this approximation reproduces the flavio-computed value of R K * 0 to better than 4% in a large region of the parameter space. Now, if we assume that NP does not affect the electron channel (C NP e 9 = C NP e 10 = 0), it is clear that to obtain R K * 0 < R SM K * 0 one needs to introduce C NP µ 9 and C NP µ 10 with a non-zero real part: the only possible negative contributions come from the Re C NP µ 9 , Re C NP µ 10 terms, whereas the |C NP µ 9 | 2 , |C NP µ 10 | 2 terms have a positive-defined sign, and can not reduce the value of R K * 0 . Thus, purely imaginary values of C NP µ 9,10 contribute only to the modulus (positive-definite) and not to the real part, and can not bring the prediction of R K * 0 closer to the experimental value. In addition, this expression tells us that the better option to reduce the prediction of R K * 0 is using a real negative C NP µ 9 , and a real positive C NP µ 10 . This is actually the result that we wave obtained in our numerical analysis. Fig. 1b shows that, for real Wilson coefficients, the lowest prediction for R K * 0 is obtained for C NP µ 9 = −C NP µ 10 < 0, and Fig.2b shows that the best fit is obtained for negative C NP µ 9 and positive C NP µ 10 . Fig. 1a shows that, in general, imaginary Wilson coefficients give positive contributions to R K , R K * 0 , in accordance with eq. (17). Of course, the full expression is richer than eq. (17), and we expect some deviations, Fig. 2a shows that the best fit point is not the SM (C NP µ 9 = C NP µ 10 = 0), but the best fit regions are centered around it, and the χ 2 min is the same as the SM as explained above. We conclude that, actually, a NP explanation for R K , R K * 0 requires that C NP µ 9 , C NP µ 10 have a non-zero real part, whereas we saw above that NP explanation for ∆M s requires that C NP µ 9 , C NP µ 10 have a non-zero imaginary part. Then, to have a NP explanation for both observables C NP µ

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, C NP µ 10 should be general complex numbers. Following this reasoning we have performed a global fit to the semi-leptonic decay observables R K , R K * 0 , P 4 and P 5 using generic complex Wilson coefficients allowing only one free Wilson coefficient. Table 1 shows the best fit values and pulls, defined as ∆χ 2 , for scenarios with NP in one individual complex Wilson coefficient. The primed Wilson coefficients are also included. We found that the best fit of R K and R K * 0 and the angular distributions is obtained for C NP µ  [34]. Choosing complex Wilson coefficients also implies additional constraints from CP -violating observables. This fact has not been considered in the previous analysis. In the next section we  Table 1: Best fit Wilson coefficients complex values to semi-leptonic decay observables R K , R K * 0 , P 4 and P 5 , allowing only one free coefficient at a time. Shown are also the corresponding pulls and R K , R K * 0 predictions. study the consequences of having these coefficients in the analysis of B-meson anomalies on some NP models and we consider a global fit of both the ratios R K and R K * 0 and the angular observables P 4 and P 5 , and also the CP -mixing asymmetry.

B s -mixing and NP models
Several NP models that are able to explain the lepton flavour universality violation effects are constrained by other flavour observables like B s -mixing. In particular the parameter space of Z and leptoquark models are severely constrained by the present experimental results of ∆M s [45]. Besides, as already mentioned, additional constraints emerge from CP -violating observables when considering complex couplings. Ref. [45] argues that nearly imaginary Wilson coefficients could explain the discrepancies with the ∆M s experimental measurement, but a global fit of R K and R K * 0 observables, together with ∆M s and CP -violation observable A mix CP in B s → J/ψφ decays should be performed. In the next subsections we investigate these issues for the case of Z and leptoquark models.

Z fit
From now on, a global fit of R K and R K * 0 observables, ∆M s and CP -violation observable A mix CP is included in our analysis. Figure 3 shows the fits on the Z mass M Z and the imaginary coupling λ Q 23 (setting λ L 22 = 1) imposed by b → sµ + µ − decays and B s -mixing. The red lines (dotted, dash-dotted) correspond to the fit using only semi-leptonic B-meson decays, i.e. b → sµ + µ − as in Figure 2 plus the branching ratios BR(B s → µ + µ − ) and BR(B 0 → µ + µ − ). The best fit region is the one below the curves; dotted lines: ∆χ 2 = 1, dash-dotted lines: ∆χ 2 = 4. Blue lines (solid, dashed) correspond to the fit to B s -mixing observables ∆M s and A mix CP . The best fit region is the one between the lines; solid  (12), with a ∆M s prediction close to the experimental value (2), while the contributions to |C NP µ 9,10 | decrease as M −1 Z (11). Since imaginary couplings worsen the R K , R K * 0 , the larger M Z provides better predictions for them, bringing them closer to the SM value. The best fit ∆χ 2 grows very slowly with growing allowed M Z . When complex couplings are allowed we found that ∆χ 2 = 5.19; being the pull 4.52σ. Table 2 summarizes the best fit values for λ Q 23 and M Z , and corresponding pulls, to R K and R K * 0 observables, ∆M s and A mix CP ; considering real, imaginary and complex Wilson coefficients. Results for the above observables in each scenario are included in this table. It is clear that R K and R K * 0 observables prefer real Wilson coefficients, as expected. For real couplings the description is better than the SM, with a pull of 4.36σ but it does not improve the prediction for ∆M s . Contrary, to improve the prediction for ∆M s imaginary couplings are required in the Z model, however the pull with respect the SM is residual. When allowing generic complex couplings (third column in Table 2) we find that best fit point is close to the best fit point using only real couplings (first column in Table 2), and the pull with respect the SM improves slightly (4.52σ versus 4.36σ), and the predictions for the observable are also close to the pure real couplings case, showing a slight improvement in the prediction for ∆M s . We conclude that, in the framework of Z models, R K − R K * 0 observables are better described than in the SM, with a pull  Table 2: Best fits, and corresponding pulls, to R K , R K * 0 , ∆M s and A mix CP ; considering real, imaginary and complex Wilson coefficients on the Z model. Shown are also the corresponding predictions for semi-leptonic decay observables R K , R K * 0 ; ∆M s and A mix CP .
of a similar imaginary part for the coupling Im(λ Q 23 ) −0.0019 improves slightly the fit, with a slight improvement for the ∆M s prediction.

Leptoquark fit
The leptoquark model has three independent couplings contributing to ∆M s (15). For the global fits we will assume that the dominant coupling is the muon coupling y QL 32 y QL * 22 , which is the one contributing to R K , R K * 0 (14). The fits on the S 3 leptoquark mass M S 3 and the imaginary coupling y QL 32 y QL *

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imposed by b → sµ + µ − decays and B s -mixing are presented in Figure 4. The observables used in the respective fits are the same as in Figure 3. The red lines (dotted, dashdotted) correspond to the fit using only semi-leptonic B-meson decays, i.e. b → sµ + µ − plus the branching ratios BR(B s → µ + µ − ) and BR(B 0 → µ + µ − ). The best fit region is the one below the curves; dotted lines: ∆χ 2 = 1, dash-dotted lines: ∆χ 2 = 4. Blue lines (solid, dashed) correspond to the fit to B s -mixing observables ∆M s and A mix CP . The best fit region is the one between the lines; solid lines ∆χ 2 = 1, dashed lines ∆χ 2 = 4. The green regions are the combined global fit: dark region ∆χ 2 = 1, medium ∆χ 2 = 4 and light ∆χ 2 = 9. The SM scenario was preferred by the b → sµµ fit (y QL 32 y QL *

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= 0). The leptoquark fit to B s -mixing observables absolute minimum is similar to the one in Z models, it is located at M S 3 = 7.2 TeV, y QL 32 y QL * 22 = 1.12 i, with a SM pull of ∆χ 2 = 1.14. This point is situated in the narrow fringe region, with a value for the Wilson coefficient of C LL bs = −3.39 × 10 −3 . In the bulk region the minimum is at M S 3 = 50 TeV, y QL 32 y QL * 22 = 1.72 i, with the essentially same SM pull of ∆χ 2 = 1.14, and a Wilson coefficient The best fits for the leptoquark model considered in this work, and corresponding pulls, to R K , R K * 0 , ∆M s and A mix CP ; considering real, imaginary and complex Wilson coefficients are summarized in Table 3. In this model the best global fit, in the M S 3 region of our analysis, and when considering only imaginary y QL 32 y QL * fit couplings, and observable predictions, and the pulls improve slowly. The situation is similar than in the Z case: by allowing larger M S 3 masses the best fit coupling reaches an asymptotic straight line, where the contribution to ∆M s is constant (15), whereas the contribution to |C NP µ 9,10 | (14) decreases as M −1 S 3 , however this asymptotic behaviour is not attained until large leptoquark masses M S 3 100 TeV, where it follows the line: y QL 32 y QL * 22 i (4.04 × 10 −2 × M S 3 / TeV − 0.908). In the region M S 3 50 − 100 TeV the fit pull improves marginally. However, considering complex couplings the best global fit emerges at M S 3 = 9.79 TeV and y QL 32 y QL *

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= 0.16 + 0.18i, with ∆χ 2 = 5.17 (4.50σ). It is clear that only imaginary couplings do not improve the results; they cannot explain the R K ( * ) anomaly. However, when complex couplings are considered, we found a better global fit of R K , R K * 0 observables. The best fit point M S 3 is much larger than the real couplings case, and the real part of the coupling is a factor four larger. The imaginary part of the coupling is similar to the real part. The pull with respect the SM is marginally better in the case of complex couplings (4.50σ versus 4.46σ). The predictions for the B-meson physics observables are similar than in the real couplings case.
If one relaxes the condition y QL 33 y QL * 23 y QL 31 y QL * 21 0 then the leptoquark contributions to ∆M s (15) and C NP µ 9,10 (14) are no longer correlated, it would be possible to choose: a purely real coupling to muons, such that it fulfills the first column of Table 3; a vanishing coupling for electrons, such that it does not contribute to R K , R K * 0 ; and a complex coupling for taus, such that y QL 33 y QL * 23 + y QL 32 y QL *

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is purely imaginary, and provides a good prediction for ∆M s like in the second column of Table 3. Of course, this would be a quite strange arrangement for leptoquark couplings! Another option would be to take an specific model construction for the relations among the leptoquark couplings, and make a global fit on these parameters. This analysis is beyond the scope of the present work.

Conclusions
In this work, we have updated the analysis of New Physics violating Lepton Flavour Universality, by using the effective Lagrangian approach and also in the Z and leptoquark models. By considering generic complex Wilson coefficients we found that purely imaginary coefficients do not improve significantly B-meson physics observable predictions, whereas complex coefficients (Table 1) do improve the predictions, but with a similar pull than using only real coefficients [34]. We have  Table 3: Best fits, and corresponding pulls, to R K , R K * 0 , ∆M s and A mix CP ; considering real, imaginary and complex Wilson coefficients on the S 3 leptoquark. Shown are also the corresponding predictions for semi-leptonic decay observables R K , R K * 0 ; ∆M s and A mix CP .
analyzed the impact of considering complex Wilson coefficients in the analysis of B-meson anomalies in two specific models: Z and leptoquarks, and we have presented for the first time a global fit of R K and R K * 0 observables, together with ∆M s and CP -violation observable A mix CP when this complex couplings are included in the analysis. We confirm that real Wilson coefficients cannot explain the B s -mixing anomaly; but also only imaginary Wilson coefficients cannot explain the R K , R K * 0 anomaly. Contrary, complex couplings offer a slightly better global fit. For complex couplings the predictions for R K , R K * 0 and ∆M s are similar than for real couplings (Tables 2, 3). For Z models the best fit in both cases is obtained for M Z 1.1 TeV, a negative real part of the coupling Re(λ Q 23 ) −0.002, with possibly a similar imaginary coupling part Im(λ Q 23 ) −0.0019. For leptoquark models the situation is different, for real couplings the best fit is obtained for small positive couplings ( y QL 32 y QL * 22 0.04) and M S 3 5.6 TeV, for complex couplings the real part of the coupling is a factor 4 larger (Re(y QL 32 y QL * 22 ) 0.16), with a similar imaginary part (Im(y QL 32 y QL * 22 ) 0.18), and a much larger leptoquark mass (M S 3 = 9.79 TeV). One can obtain better fits in the leptoquark models by relaxing the assumption on the leptoquark couplings, or providing specific models for leptoquark couplings, this analysis is beyond the scope of the present work. In summary, new physics Z or leptoquark models with complex couplings provide a slightly improved global fit to B-meson physics observables as compared with models with real couplings.