New Physics signature in $D^0 (\bar{D}^0)\to f$ effective width asymmetries

In this paper, the approach given in Ref.\cite{Gersabeck:2011xj} is generalized and a new contribution to the effective time-integrated CP asymmetry $A_{\Gamma}^{CP}$ is obtained. We show that the new contribution is directly proportional to the weak phase responsible for the direct CP violation. As an illustration we apply this approach to $D^0 \rightarrow K^- \pi^+ $ decay channel. We show that the large direct CP asymmetry that can be produced in some extensions of the Standard Model such as Left-Rigth models induces a $A_{\Gamma}^{CP}$ around $10^{-4}$ very close to the experimental limit. We also show that the weak phases that appears in the direct CP violation can induce a $A_{\Gamma}^{CP} \neq 0 $ even if the strong phases are not large enough to generate an observable direct CP violation.

CP Violation and time dependence in decay amplitude due to mixing between neutral mesons have been clearly observed in K and B mesons [2].In D mesons, the mixing between D 0 and D0 has been well established [4,[14][15][16][17][18][19].But the situation is more difficult as the slow mixing rate between D 0 and D0 doesn't permit to measure directly the time-dependence of the D 0 , D0 decays.It is only possible to measure the direct CPV and an effective timeintegrated width which usually is assumed to depend on interference between indirect CPV and neutral mesons mixing [20][21][22].In Ref. [1], it has been showed that the situation is much more intricate and that these effective time-integrated widths are also depending on the direct CPV.
In the literature, the study of the interference between direct CPV and mixing has been performed through the introduction of a non-universal weak phase defined as δ f ≡ − arg( Āf /A f ) where A f is the A(D 0 → f ) amplitude [23][24][25].It is important to notice that this weak phase is irrelevant for the direct CPV as direct CPV is proportional to To get an observable direct CPV it is necesary that the amplitude, A f , can be written as the sum of at least two amplitudes with different relative weak and strong phases.So, we generalize the approach adopted in Ref. [1] to take into account the structure of the direct CPV and to show that these relative weak and strong phases are contributing to the CPV in the effective decay widths.Moreover we show that even if the strong phases are too small to generate an observable direct CPV, the relative weak phase will contribute to the CPV in the effective decay widths and can compete with Standard Model contributions in some The paper is organized as follows: in section II, we review the general formalism to treat the interference between D 0 − D0 mixing and different CP violation sources keeping open the possibilities to have direct or indirect CP violation and generalize the approach used in Ref. [1] taking into account the structure of direct CPV.We show that even without CPV in mixing or without observable direct CPV, the New Physics weak phases can contribute to the effective time-integrated asymmetries.These contributions are new and should be included in the experimental data analysis as it could mimic the effects of indirect CPV.In section III, we apply our generalized approach to the Cabibbo favored process Finally we conclude in section IV.
with the normalization condition imposes |p| 2 + |q| 2 = 1.We consider the decay modes where for the last case, it is used the fact that the process D 0 → K + π − is double Cabibbo suppressed compared to D 0 → K − π + which is Cabibbo favored.Using the general formalism for D 0 − D0 mixing, it is possible to compute the respective widths as a function of time where Γ is the mean D 0 width and where one defines In the case of D mesons, these equations can be easily expanded and, to a good approximation, one can keep the terms up to the linear terms in time: Now, one can define the following CP observables: with CP asymmetry can be generated in the presence of the required strong and weak phases.

Let us define
where δ 0,1 are the weak phases and α 0,1 are the strong CP conserving phases.Here we assume that the amplitude A 0 comes from the Standard Model and that A 1 comes from radiative corrections to Standard Model or from New Physics contributions.Thus for |A 1 /A 0 | = ǫ ≪ 1, the corresponding CP asymmetry can be expressed, at first order in ǫ, as: where The second term is a contribution which is directly proportional to direct CP asymmetry.
This contribution has already been noticed in Ref. [1].The first term is new and usually it is not taken into account.It is important to emphasize that the second term will vanish upon setting α ≈ 0. However, for α ≈ 0, the first term can generate a non zero contribution to A f Γ thanks to the weak phase δ.This contribution is given as In such a case, using the experimental limit for the mixing parameters and assuming ǫ to be of order 0.01 and a maximal δ CP violating phase, one obtains which is very close to its experimental limit.These values can be easily obtained in nonmanifest left right models as it has been shown in Ref. [10] If it is possible experimentally to measure the effective time-integrated CP asymmetry and the Direct CP asymmetry, it will be possible to get access to the strong phase via the relation IV. CONCLUSION In this work we have generalized the approach given in Ref. [1] where the interplay between direct and indirect CPV has been studied.Usually, the non-universality in the weak phase φ f is taken into account but the relative weak phase between A f and Āf is irrelevant for direct CPV.We have studied the CP asymmetry A CP Γ in the time-integrated effective widths and the effects of the weak and strong phases needed to generate direct CPV .The interference between this weak phase and the mixing induces a contribution to A CP Γ .We have shown that the A CP Γ gets three and not only two different contributions: • arg(p/q) = 0 or π which is the well known contribution coming from CPV in mixing and should be the same for all D 0 → f decay channels .
• A D CP = 0, where A D CP is any direct CPV as it has been shown in Ref. [1].
• δ = 0 or π where δ is the relative weak phase between two amplitudes contributing to A f or Āf .One important point of our results is that this contribution still exists even if the strong phases responsible for the direct CPV are set to zero which implies This last contribution is usually not taken into account even if in some extension of the Standard Model, this contribution could be as large as the one coming from mixing.For instance, we have shown that for the D 0 → K − π + in a non manifest left right model, a A CP Γ of order 10 −4 can be generated.
In conclusion, the CPV in time-integrated effective widths is very sensitive to both the scal and weak phases of New Physics.Particularly, if the scale of New Physics implies that ǫ is around 0.01 which means typically T eV scale, this new contribution to the CPV in the effective widths could give contributions of the same order or larger than the one expected from CPV in mixing or from Standard Model radiative corrections.
II. GENERAL FORMALISM OF CP VIOLATION IN INTERFERENCE BE-TWEEN D 0 − D0 MIXING AND DIFFERENT CP VIOLATION SOURCES As for the K and B neutral mesons, the mass eigenstates of the neutral D mesones, called |D 1,2 >, with respective masses m 1,2 and total widths Γ 1,2 , are linear combinations of the flavor eigenstates |D 0 > and | D0 > defined as follows: