Starting from the New Physics Free determination of ρ and η, we explore, in a model-independent approach, the possible contributions of NP effects to B_d-B_d, B_s-B_s and K⁰-K⁰ mixing. Each of these processes can be parameterized in terms of only two new parameters, which we choose to quantify the difference of the amplitude, in absolute value and phase, with respect to the SM one. Thus in the case of B_q-B_q mixings, we define

where H_eff^SM includes only the SM box diagram, while H_eff^full includes also the NP contributions. In the second equation we also introduce φ_q^SM where φ_d^SM = β and φ_s^SM = -β_s These definitions imply that the mass differences and the CP asymmetry are related to the SM counterparts by

Concerning K⁰-K⁰ mixing, we introduce a single parameter which relates the imaginary part of the amplitude to the SM one,

This definition implies a simple relation for |ε_K|:

• &Delta m_s measurement: this provides the first constraint on the φ_Bs vs C_Bs plane;
• improved measurement of A_SL in B_d decays and new measurement of the CP asymmetry A_CH in dimuon events: these give further possibilities of constraining φ_Bq and C_Bq;
• the use of ΔΓ_s constrains both C_Bs and φ_Bs (see CDF and D0 results), and analogously the ΔΓ_d measurement helps in reducing the uncertainty in C_Bd.

These constraints allow for a complete NP analysis with the fully model-independent determination of CKM parameters ρ and η, obtaining simultaneously predictions on all the NP quantities introduced above.
In the general model-independent analysis we also have to consider Δ F = 1 effects: below are the detailed descriptions of the various constraint treatments.

In principle, the extraction of α from B → ππ, ρπ, and ρρ decays is affected by NP effects in |Δ F|=1 transitions. In the presence of NP in the strong b &rarr d penguins, the decay amplitudes for B mesons decaying into ππ, ρπ, and ρρ are a simple generalization of the SM ones (given for example here): Assuming that NP modifies significantly only the "penguin" amplitude P without changing its isospin quantum numbers (i.e. barring large isospin-breaking NP effects), the only necessary modification amounts to distinguish the penguin complex parameters P and P, which come from the sum of SM and NP contributions in B and B decays and which bring in general different weak and strong phases. The procedure to extract α is the same as in the SM case but the expected bound is weaker due to the presence of two (four) extra parameter(s) for ππ and ρρ (ρπ).

(EPS) [JPG]

Since NP effects in |ΔF|=2 are not included, this constraint has to be interpreted as a bound to α-φ_Bd (see above) once it is used in the extension of the UT analysis including NP in |ΔF|=2.

One can also add the constraint coming from the CP asymmetry in semi-leptonic B decays A_SL, defined as

It has been noted in hep-ph/0202010 that A_SL is a crucial ingredient of the UT analysis once the formulae are generalized as described below, since it depends on both C_Bd and φ_Bd. Infact, we can write:

where Γ₁₂ and M₁₂ are the absorptive and dispersive parts of the B⁰_d-B⁰_d mixing amplitude.

At the leading order, A_SL is independent of penguin operators, but, at the NLO, the penguin contribution should be taken into account. In the SM, the effect of penguin operators is GIM suppressed since their CKM factor is aligned with M₁₂: both are proportional to (V*_tbV_td)². This is not true anymore in the presence of NP, so that the effects of penguins are amplified beyond the SM. We therefore start from the full NLO calculation of hep-ph/0308029, allowing for an additional NP contribution to the penguin term in the |Δ F |=1 amplitude. This introduces two additional parameters C_Pen and φ_Pen that, because of the extra α_s factor, enter as a smearing in the expression of A_SL. The generalized expression of A_SL is given in hep-ph/0509219

BaBar has recently released a new improved A_SL measurement that in association with the quantity A_CH described in the following allows for further constraining the φ_Bd and C_Bd; NP parameters. As can be seen below, the actual measurements of these two observables strongly disfavour the solution with ρ and η in the third quadrant, which now has only 0.4% probability.
Here we show the one-dimensional distributions for this quantity both in the Standard Model fit (left plot) and in the New Physics general scenario (right plot). See the table for the numerical results.

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The charge asymmetry A_CH in dimuon events is a rather unique constraint since it depends on ρ and η and on all ΔB=2 NP parameters (C_Bd, φ_Bd, C_Bs, and φ_Bs). D0 Collaboration has recently announced a new result for this observable. The dimuon charge asymmetry A_CH can be defined as:

using the notation as in the D0 result where the definition and the measured values for the P parameters can be found. We have:

where we have assumed equal semi-leptonic widths for B_d andn B_s mesons, f_d = 0.397 ± 0.010 and f_s = 0.107 ± 0.011 are the production fractions of B_d and B_s mesons respectively and the χ_q and χ_q are given in equation 6 in . They contain the dependence (through equations 7 in the same ref.) on the ΔB=2 NP parameters (C_Bd, φ_Bd, C_Bs, and φ_Bs) as well as the possible NP contributions to ΔB=1 penguins (C_q^Pen, φ_q^Pen).
Here we show the one-dimensional distributions for this quantity both in the Standard Model fit (left plot) and in the New Physics general scenario (right plot) See the table for the numerical results.

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In presence of New Physics, the measurement of ΔΓ_q is related to ρ and η, and to the NP parameters C_Bq and φ_Bq through the value of Δm_q: you can find the relation here (eq. 7 in hep-ph/0605213) A simultaneous use of Δm_q and of the bound from ΔΓ_q allows to constrain the phase of the mixing even without a direct measurement of the mixing phase. This is particularly important in the case of the B_s sector, waiting for the measurement on the time-dependent CP asymmetry in B_s → J/ψ φ Since the available experimental measurements are not directly sensitive to the phase of the mixing amplitude, they are actually a measurement of &Delta&Gamma_q cos2(φ_Bq- β_q) in the presence of NP.

We add here the prediction on the quantities previously described: A_SL, A_CH, ΔΓ_q/Γ_q. These are obtained from the fully model-independent fit without using the given quantities. The values represent the allowed ranges in this generalized framework. For comparison also the SM values are given together with the experimental measurements.

We list below the result of the fully model-independent fit in terms of UT quantities including possible NP contributions as described above. These values represent the allowed ranges in this generalized framework. We found only one favoured solution, corresponding to the result of the Standard Model fit, while the second region, corresponding to the UT with its vertex in the third quadrant and implying sizable New Physics effects in the B_d sector, present in the previous version of this analysis is now excluded at 95% probability.

(EPS) [JPG]	(EPS) [JPG]	(EPS) [JPG]
(EPS) [JPG]	(EPS) [JPG]	(EPS) [JPG]
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Because of the abundance of experimental information, one can determine stringent bounds to NP parameters, simultaneously to the determination of the UT parameters given above. The following plots show what we obtain for C_Bd and φ_Bd.

(EPS) [JPG]	(EPS) [JPG]
RESULT: CBd = 1.04 ± 0.34	RESULT: φBd = (-4.4 ± 2.1)o
(EPS) [JPG]	(EPS) [JPG]

We can also write

and given the p.d.f. for C_Bd and φ_Bd, we can derive the p.d.f. in the (A_NP/A_SM) vs φ_NP plane. The result is shown in the figure above on the right.

We see that the NP contribution can be substantial if its phase is close to the SM phase, while for arbitrary phases its magnitude has to be much smaller than the SM one. Notice that, with the latest data, the SM (φ_Bd=0) is disfavoured at 68% probability due to a slight disagreement between sin 2β and |V_ub/V_cb|. This requires A_NP ≠ 0 and φ_NP ≠ 0. For the same reason, φ_NP > 90^o at 68% probability and the plot is not symmetric around φ_NP = 90^o.

Because of the abundance of experimental information, one can determine stringent bounds on NP parameters, simultaneously with the determination of the UT parameters and the B_d-related NP parameters given above. The following plot shows what we obtain for C_Bs and φ_Bs. Thanks to the good precision of the new new Δm_s measurement and the good precision on ξ from lattice QCD, the bound on C_Bs is already more precise than the bound on C_Bd. Also the A_CH and ΔΓ_s measurements can now provide the first constraints on φ_Bs.

(EPS) [JPG]	(EPS) [JPG]
RESULT: CBs = 1.04 ± 0.29	RESULT: φBs = (-77 ± 16) U (-20 ± 11) U (9 ± 10)o
(EPS) [JPG]	(EPS) [JPG]

As in the case above of the B_d-B_d mixing, we can define in the same way A^s_NP/A^s_SM and φ^s_NP as functions of C_Bs and φ_Bs and given the p.d.f. for C_Bs and φ_Bs, we can derive the p.d.f. in the (A^s_NP/A^s_SM) vs φ_NP plane. The result is shown in the figure above on the right and we have exclusion regions for the first time, thanks to the new measurements described above (A_CH and Δ&Gamma_s).

We'd like to point out an interesting correlation between the values of C_Bd and C_Bs that can be seen in the figure below. This correlation that is present in the general analysis, is due to the fact that lattice QCD determines quite precisely the ratio ξ² of the matrix elements entering B_d and B_s mixing amplitudes.

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Because of the abundance of experimental information, one can determine stringent bounds on NP parameters, simultaneously with the determination of the UT parameters and the B_d-related NP parameters given above. The following plot shows what we obtain for C_εK.

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RESULT: CεK = 0.87 ± 0.14

It is possible to generalize the full UTfit beyond the Standard Model to all those NP models characterized by Minimal Flavour Violation, i.e. having quark miking ruled only by the Standard Model CKM couplings. In fact, in this case no additional weak phases are generated and several observables entering into the Standard Model fit (the tree-level processes and the measurement of angles through the use of time dependent CP asymmetries) are not affected by the presence of New Physics. The only sizable effect we are sensitive to is a shift of the Inami-Lim function of the top contribution in meson mixing. This means that in general ε_K and Δm_d cannot be used in a common SM and MFV framework, but any New Physics contribution disappear in the case of Δm_d/Δm_s. So, simply removing the information related to ε_K and Δm_d from the full UTfit one can obtain a more precise determination of the Universal Unitarity Triangle, which is a common starting point for the Standard Model and any MFV model.

(EPS) [JPG] Result of UUT fit on the (ρ, η) plane

(EPS) [JPG] ρ = 0.153 ± 0.030 from UUT fit	(EPS) [JPG] η = 0.347 ± 0.018 from UUT fit	(EPS) [JPG] PREDICTIONS: α = (91.3 ± 4.8)o from UUT fit
(EPS) [JPG] PREDICTIONS: β = (22.3 ± 0.9)o from UUT fit	(EPS) [JPG] PREDICTIONS: γ = (66.3 ± 4.8)o from UUT fit	(EPS) [JPG] PREDICTIONS: sin2βs = 0.037 ± 0.002 from UUT fit

One has to notice that the precision on ρ and η is in practise the same than in the full UTfit. This means that one can go forward in the study of MFV, using the two neglected informations (ε_K and Δm_d) to bound the scale of New Physics. In fact, the expected contribution is a shift of S₀, the Inami-Lim function associated to top contribution in box diagrams. The shift can than we translated in terms of the tested energy scale for New Particles, using a simple dimensional argument.

where δS₀ is the shift, a is a parameter related to Wilson coefficients of the effective Hamiltonian, Λ is the New Physics scale and Λ₀=Y_tsin²(θ_W)M_W/α ∼ 2.4 TeV is the reference EW scale. One can extract Λ₀ in two different scenarios:

• MFV models with one Higgs doublet or low/moderate tanβ. In this case, the shift δS₀ is a universal factor for all the mixing processes we are considering (K⁰-K⁰, B⁰-B⁰, and B_s-B_s). The information coming from those mixing processes that are ignored in UUT can be used to bound this universal shift, once the CKM parameters are fixed by the UUT.
• MFV models with large values of tanβ. In this case, it is still true that one can use ε_K and Δm_d to bound the New Physics shift. But, in this case, one cannot simply relate the effect on K⁰-K⁰ to the corresponding quantity in B physics (which remains common to B⁰-B⁰ and B_s-B_s).

Here we show the result in terms of the output distribution of δS₀ (δS₀^B and δS₀^K) in the case of models with low/moderate (large) values of tanβ and we give the output value in terms of the tested energy scales, quantified at the 95% Probability.

(EPS) [JPG] δS0 = -0.16 ± 0.32	(EPS) [JPG] δS0B = 0.05 ± 0.67	(EPS) [JPG] δS0K = -0.18 ± 0.37
Λ > 5.5 TeV @95% Prob. for small tanβ	Λ > 5.1 TeV @95% Prob. for large tanβ