In the Standard Model (SM), all flavour and CP violating phenomena in weak decays are described in terms of quark masses and the four independent parameters in the Cabibbo-Kobayashi-Maskawa (CKM) matrix 12. In particular, there is only one source of CP violation, which is connected to the area of the Unitarity Triangle (UT). A peculiar prediction of the SM, due to the hierarchy among CKM matrix elements, is that CP violation in B_s mixing should be tiny. This property is also valid in models of Minimal Flavour Violation (MFV) 345678, where flavour and CP violation are still governed by the CKM matrix. Therefore, the experimental observation of sizable CP violation in B_s mixing is a clear (and clean) signal of New Physics (NP) and a violation of the MFV paradigm. In the past decade, B factories have collected an impressive amount of data on B_d flavour- and CP-violating processes. The CKM paradigm has passed unscathed all the tests performed at the B factories down to an accuracy just below 10% 91011. This has been often considered as an indication pointing to the MFV hypothesis, which has received considerable attention in recent years. The only possible hint of non-MFV NP is found in the penguin-dominated b → s non-leptonic decays. Indeed, in the SM, the coefficient of the time-dependent CP asymmetry in these channels is equal to the measured with b → decays, up to hadronic uncertainties related to subleading terms in the decay amplitudes. Present data show a systematic, although not statistically significant, downward shift of with respect to 12131415161718192021, while hadronic models predict a shift in the opposite direction in many cases 2223242526272829.

From the theoretical point of view, the hierarchical structure of quark masses and mixing angles of the SM calls for an explanation in terms of flavour symmetries or of other dynamical mechanisms, such as, for example, fermion localization in models with extra dimensions. All such explanations depart from the MFV paradigm, and generically cause deviations from the SM in flavour violating processes. Models with localized fermions 303132, and more generally models of Next-to-Minimal Flavour Violation 33, tend to produce too large effects in ε_K 3435. On the contrary, flavour models based on nonabelian flavour symmetries, such as U(2) or SU(3), typically suppress NP contributions to s ↔ d and possibly also to b ↔ d transitions, but easily produce large NP contributions to b ↔ s processes. This is due to the large flavour symmetry breaking caused by the top quark Yukawa coupling. Thus, if (nonabelian) flavour symmetry models are relevant for the solution of the SM flavour problem, one expects on general grounds NP contributions to b ↔ s transitions. On the other hand, in the context of Grand Unified Theories (GUTs), there is a connection between leptonic and hadronic flavour violation. In particular, in a broad class of GUTs, the large mixing angle observed in neutrino oscillations corresponds to large NP contributions to b ↔ s transitions 36373839.

In this Letter, we show that present data give evidence of a B_s mixing phase much larger than expected in the SM, with a significance of more than 3σ. This result is obtained by combining all available experimental information with the method used by our collaboration for UT analyses and described in Ref. 40.

We perform a model-independent analysis of NP contributions to B_s mixing using the following parametrization 414243444546:

where is the effective Hamiltonian generated by both SM and NP, while only contains SM contributions. The angle β_s is defined as and it equals 0.018 ± 0.001 in the SM (we are using the usual CKM phase convention in which is real to a very good approximation).

We use the following experimental input: the CDF measurement of Δm_s 47, the semileptonic asymmetry in B_s decays 48, the dimuon charge asymmetry from DØ 49 and CDF 50, the measurement of the B_s lifetime from flavour-specific final states 515253545556575859, the two-dimensional likelihood ratio for ΔΓ_s and ϕ_s = 2(β_s - ) from the time-dependent tagged angular analysis of B_s → J/ψϕ decays by CDF 60 and the correlated constraints on Γ_s, ΔΓ_s and ϕ_s from the same analysis performed by DØ 61. For the latter, since the complete likelihood is not available yet, we start from the results of the 7-variable fit in the free-ϕ_s case from Table one of ref. 61. We implement the 7 × 7 correlation matrix and integrate over the strong phases and decay amplitudes to obtain the reduced 3 × 3 correlation matrix used in our analysis. In the DØ analysis, the twofold ambiguity inherent in the measurement (ϕ_s → π - ϕ_s, ΔΓ_s → - ΔΓ_s, cos δ_1,2 → - cos δ_1,2) for arbitrary strong phases was removed using a value for cos δ_1,2 derived from the BaBar analysis of B_d → J/ΨK* using SU(3). However, the strong phases in B_d → J/ΨK* and B_s → J/Ψϕ cannot be exactly related in the SU(3) limit due to the singlet component of ϕ. Although the sign of cos δ_1,2 obtained using SU(3) is consistent with the factorization estimate, to be conservative we reintroduce the ambiguity in the DØ measurement. To this end, we take the errors quoted by DØ as Gaussian and duplicate the likelihood at the point obtained by applying the discrete ambiguity. Indeed, looking at Fig. 2 of ref. 61, this seems a reasonable procedure. Hopefully DØ will present results without assumptions on the strong phases in the future, allowing for a more straightforward combination. Finally, for the CKM parameters we perform the UT analysis in the presence of arbitrary NP as described in ref. 34, obtaining = 0.140 ± 0.046, = 0.384 ± 0.035 and sin 2β_s = 0.0409 ± 0.0038. The new input parameters used in our analysis are summarized in Table 1, all the others are given in Ref. 34. The relevant NLO formulae for ΔΓ_s and for the semileptonic asymmetries in the presence of NP have been already discussed in refs. 346263.

The results of our analysis are summarized in Table 2. We see that the phase deviates from zero at 3.7σ. We comment below on the stability of this significance. In Fig. 1 we present the two-dimensional 68% and 95% probability regions for the NP parameters and , the corresponding regions for the parameters and , and the one-dimensional distributions for NP parameters. Notice that the ambiguity of the tagged analysis of B_s → J/Ψϕ is slightly broken by the presence of the CKM-subleading terms in the expression of Γ₁₂/M₁₂ (see for example eq. (5) of ref. 63). The solution around ~ -20° corresponds to ~ -50° and ~ 75%. The second solution is much more distant from the SM and it requires a dominant NP contribution ( ~ 190%). In this case the NP phase is thus very well determined. The strong phase ambiguity affects the sign of cos ϕ_s and thus Re , while Im ~ - 0.74 in any case.

Before concluding, we comment on our treatment of the DØ result for the tagged analysis and on the stability of the NP fit. Clearly, the procedure to reintroduce the strong phase ambiguity in the DØ result and to combine it with CDF is not unique given the available information. In particular, the Gaussian assumption can be questioned, given the likelihood profiles shown in Ref. 61. Thus, we have tested the significance of the NP signal against different modeling of the probability density function (p.d.f.). First, we have used the 90% C.L. range for ϕ_s = [-0.06, 1.20]° given by DØ to estimate the standard deviation, obtaining ϕ_s = (0.57 ± 0.38)° as input for our Gaussian analysis. This is conservative since the likelihood has a visibly larger half-width on the side opposite to the SM expectation (see Fig. 2 of Ref. 61). Second, we have implemented the likelihood profiles for ϕ_s and ΔΓ_s given by DØ, discarding the correlations but restoring the strong phase ambiguity. The likelihood profiles include the second minimum corresponding to ϕ_s → ϕ_s+π, ΔΓ → -ΔΓ, which is disfavoured by the oscillating terms present in the tagged analysis and is discarded in our Gaussian analysis. Also this approach is conservative since each one-dimensional profile likelihood is minimized with respect to the other variables relevant for our analysis. It is remarkable that both methods give a deviation of from zero of 3 σ (the 3 σ ranges for are [-88, -48]° ∪ [-41, 0]° and [-88, 0]° for the two methods respectively). We conclude that the combined analysis gives a stable evidence for NP, although the precise number of standard deviations depends on the procedure followed to combine presently available data.

To illustrate the impact of the experimental constraints, we show in Fig. 2 the p.d.f. for obtained without the tagged analysis of B_s → J/Ψϕ or including only CDF or DØ results. Including only the CDF tagged analysis, we obtain < 0 at 97.7% probability (2.3σ). For DØ, we show results obtained with the Gaussian and likelihood profile treatment of the errors. In the Gaussian case, the DØ tagged analysis gives < 0 at 98.0% probability (2.3σ), while using the likelihood profiles < 0 at 92.8% probability (1.8σ). Finally, it is remarkable that the different constraints in Fig. 2 are all consistent among themselves and with the combined result. We notice, however, that the top-left plot is dominated by the measurement of while favours positive , although with a very low significance. For completeness, in Table 2 we also quote the fit results for , and for ΔΓ_s/Γ_s.

In this Letter we have presented the combination of all available constraints on the B_s mixing amplitude leading to a first evidence of NP contributions to the CP-violating phase. With the procedure we followed to combine the available data, we obtain an evidence for NP at more than 3σ. To put this conclusion on firmer grounds, it would be advisable to combine the likelihoods of the tagged B_s → J/Ψϕ angular analyses obtained without theoretical assumptions. This should be feasible in the near future. We are eager to see updated measurements using larger data sets from both the Tevatron experiments in order to strengthen the present evidence, waiting for the advent of LHCb for a high-precision measurement of the NP phase.

It is remarkable that to explain the result obtained for ϕ_s, new sources of CP violation beyond the CKM phase are required, strongly disfavouring the MFV hypothesis. These new phases will in general produce correlated effects in ΔB = 2 processes and in b → s decays. These correlations cannot be studied in a model-independent way, but it will be interesting to analyse them in specific extensions of the SM. In this respect, improving the results on CP violation in b → s penguins at present and future experimental facilities is of the utmost importance.

During the review procedure of this Letter, results based on new data were presented by the Tevatron experiments, as well as a combination of Tevatron results on the tagged angular analysis of B_s → J/ψϕ. However these updates are all unpublished. Furthermore, the likelihoods required by our analysis are not publicly available except for the new DØ analysis with no assumption on strong phases 64. For the sake of completeness, we quote = (-19 ± 8)° ∪ (-69 ± 7)° ([-36, -5]° ∪ [-83, -54]° at 95% probability), obtained using this new likelihood for the DØ tagged angular analysis of B_s → J/ψϕ. Clearly, we no longer need to manipulate the DØ likelihood to remove the strong phase assumption and to account for the non-Gaussian shape as described above. Remarkably, this updated result is well compatible with the results of this Letter, confirming a deviation from the SM at the level of ~3σ (99.6% probability). More recent experimental results seem to confirm the effect discussed in this Letter. We will include them in future analyses as soon as they become available.

We are much indebted to M. Rescigno for triggering this analysis and for improving it with several valuable suggestions. We also thank G. Giurgiu, G. Punzi and D. Zieminska for their assistance with the Tevatron experimental results. We acknowledge partial support from RTN European contracts MRTN-CT-2006-035482 "FLAVIAnet" and MRTN-CT-2006-035505 "Heptools". M.C. is associated to the Dipartimento di Fisica, Università di Roma Tre. E.F. and L.S. are associated to the Dipartimento di Fisica, Università di Roma "La Sapienza".