Study of CP violation in D->VV decay at BESIII

In this paper, we intend to study the problem of CP violation in $D$ meson by $D\to VV$ decay mode in which the T violating triple-product correlation is examined. That would undoubtedly be another excellent probe of New Physics beyond Standard Model. For the neutral $D$, we focus on direct CP violation without considering $D^0-\bar D^0$ oscillation. Experimentally, by a full angular analysis one may obtain such CP violating signals, and particularly it is worth mentioning that the upcoming large $D$ data samples at BES-III in Beijing will provide a great opportunity to perform it.

In this paper, we intend to study the problem of CP violation in D meson by D → V V decay mode in which the T violating triple-product correlation is examined. That would undoubtedly be another excellent probe of New Physics beyond Standard Model. For the neutral D, we focus on direct CP violation without considering D 0 −D 0 oscillation. Experimentally, by a full angular analysis one may obtain such CP violating signals, and particularly it is worth mentioning that the upcoming large D data samples at BES-III in Beijing will provide a great opportunity to perform it. The topic of CP violation in the D-meson sector has been the subject of extensive studies involving both charged and neutral D meson decays these years [1][2][3][4][5][6][7][8][9][10][11][12]. In a recent reference [13], they exploited the angular and quantum correlation in the D 0D0 pairs produced through the decay of the ψ(3770) resonance at BES-III to investigate CP violation. They build CP violating observables in e + e − → ψ(3770) → D 0D0 → (V 1 V 2 )(V 3 V 4 ) (V denotes vector meson) to isolate specific New Physics effects in the charm sector [13].
In practice, one can also probe the CP violation in D meson by one D decay without considering the quantum correlation, which is the motivation of this paper. Among the various kinds of D decay modes, D → V 1 V 2 and subsequently decaying to two pseudoscalars for each vector meson is a particularly interesting one in the perspective of the copious kinematics of final state interaction (FSI). As the same case in B meson [14,15], the new type of T violating signal involving so-called triple-product (TP) will emerge by comparing a pair of CP conjugate processes, where TP is composed of the momentum of one vector meson and two polarizations. Assuming CP T invariance, T violating TP asymmetry is equivalent to CP violating. We shall see in this letter such TP asymmetries are related to the helicity components in the angular distribution of the D → V 1 V 2 process. Moreover, performing a full angular analysis is feasible and realistic in experiment.
On the other hand, TP asymmetry (in the latter section we sometimes also refer to TP asymmetry as CP asymmetry assuming CP T invariance) is a sensitive signal of New Physics. As is well known that Standard Model (SM) predictions for CP violation in the charm sector are very small, thus any significant such signals would be exciting. Currently, the BES-III experiment is collecting data at ψ(3770) peak. In the short future, lots of events of D decay will be accumulated, which will provide a great opportunity to perform a full angular analysis to further achieve a valuable information for the question discussed here.
Let us first consider the process D(p) → V 1 (k, ǫ 1 )V 2 (q, ǫ 2 ), where the two vectors V 1 , V 2 are characterized as their four-momenta and polarizations (k, ǫ 1 ) and (q, ǫ 2 ), respectively. We can write the most general invariant amplitude as a sum of three terms that we will call s, d, p [14][15][16][17], where m 1 (m 2 ) is the mass of V 1 (V 2 ), and the scalar coefficients a, b and c are generally complex and can receive contributions from several amplitudes with different phases. Thus, one can parameterize the coefficients as [14] a = j a j e iδsj e iφsj , where a j , b j and c j are the modulus of their corresponding complex quantities, and δ j denotes strong phase (also called unitary phase in Ref. [14]), and φ j is the weak phase which is the necessary condition for occurring of CP violation on the basis of Cabibbo-Kobayashi-Maskawa (CKM) mechanism [18,19] in the SM. Squaring the matrix element of Eq. (1), one obtains Next, using CP T invariance, the matrix element for the antiparticle decayD(p) →V 1 (k, ǫ 1 )V 2 (q, ǫ 2 ) can be written as Note that CP operator leaves strong phases invariant and only changes the sign of weak phase. From Eq. (1) and Eq. (4), we will find that the p wave amplitude in M changes the sign comparing with M, which will induce an interesting property between |M| 2 and |M| 2 .
To be clear, we would square the matrix element for the antiparticle decay in Eq. (4): For D 0 → V 1 V 2 decay, one can define an asymmetry A T with the definite sign for the triple product ( k · ǫ * 1 × ǫ * 2 ) as [14] where the subscript T implies triple products and N denotes the corresponding number of events. Eq. (7) above is actually Similarly, forD 0 →V 1V2 decay,Ā T can also be constructed as the same way. In |M| 2 , a triple-product correlation arises from interference terms involving the p amplitude, and will be present if Im(ac * ) or Im(bc * ) is non-zero. After a simple calculation by inserting Eq. (2) and Eq. (5), we see that Note that a non-zero triple correlation does not necessarily imply CP violation, since final state interactions (FSI) can fake it, namely the strong phase can also produce non-zero A T (orĀ T ) even the weak phases are zero. Yet comparing a triple correlation for CP conjugate transitions allows to distinguish genuine CP violation from FSI effects. Thus we obtain, and So far, a non-zero value A T +Ā T will be undoubtedly a clean signal of CP non-conservation, because there must be at least one non-zero weak phase φ. Here, we also note that if there is only one amplitude contributing to each partial wave, one can simultaneously determine the strong phase difference and weak phase difference from Eq. (10) and Eq. (11).
Experimentally, one can perform a full angular analysis to obtain the above TP asymmetry information because the complex coefficients a, b, c are related to the helicity amplitudes A 0 , A || , A ⊥ as discussed in Refs. [16,20], where where m D is the mass of D meson. Now we turn to the full angular dependence of process D → V 1 V 2 → (P 1 P 2 )(P 3 P 4 ) with P pseudoscalar, after some algebra one can get where we have introduced A ⊥ with definite odd CP eigenvalue and the CP even partners A 0 , A || via with A λ1λ2 denoting the helicity mode of two vector mesons. θ ′ i s (i=1,2) are the angles between the direction of motion of one of the V 1,2 → P P pseudoscalar final states and the inverse direction of motion of the D meson as measured in the V 1,2 rest frame, φ is the angle between the two decay plane of vector mesons in the D rest frame. Figure 1 illustrates the decay kinematics of the process D → V 1 V 2 → (P 1 P 2 )(P 3 P 4 ) in the rest frame of V 1,2 . Eq. (14) is consistent with that from Refs. [21,22].
As discussed previously, the TP asymmetry is connected with the angular dependence, combining Eqs. (10), (11), (12), one can define the following T -odd quantities [14,15], and thus we will derive the CP violating observables, Before this study, there had been attempts to study CP violation of D meson via T violating TP correlation in theoretical viewpoint that differs from our method [12,23,24]. But the only reported experimental search for T -odd asymmetries is from FOCUS in the D 0 → K + K − π + π − and D + S → K S K + π + π − decay modes [25], as listed in Table I. No evidence for a T asymmetry is observed. The large BES-III data sample is expected to provide enhanced sensitivity to possible T violating asymmetries.

Decay mode
A(%) At last, we consider the potential sensitivity on the CP violating observables A and A ′ at BES-III. From Eq. (7), for a small asymmetry, there is a general result that it's error is approximately estimated as 1/ √ N total , where N total is the total number of events observed. At BES-III, with an integrated luminosity of 20 fb −1 at ψ(3770) peak, about 72 × 10 6 D 0D0 pairs will be collected with four year's running [26,27]. Table II lists some promising channels to search for T asymmetry for both neutral and charged D decays and the corresponding expected statistical errors are estimated. The projected efficiencies are extracted from Ref. [26] and branching ratios are obtained from Ref. [28].
In table II, the branching fractions with asterisks have not been measured yet, but some estimates combining naive factorization and models for FSI are available from Refs. [29][30][31][32]. Note that in table II, the estimated efficiencies are average value for the various partial waves by assuming that the magnitude of the longitudinal polarization is half of the decay rate. In the future, a careful measurements at BES-III about the efficiency and fraction for each partial wave are suggested. A more realistic analysis requires a likelihood fit to the full angular dependence of the D → V 1 V 2 → (P 1 P 2 )(P 3 P 4 ) mode. Systematics will arise from the mis-reconstruction as V 1 V 2 of the events that actually come from other resonances or nonresonance D → P 1 P 2 P 3 P 4 background contributions. In view of the sizable width of the vector resonances, we expect that these systematics will dominate the final result. Their precise estimate in the BES-III experiment is beyond the scope of this paper. However, as pointed out in Ref. [24] by Bigi, the four body decay of D → P 1 P 2 P 3 P 4 both with and without intermediate states can all be used to probe T asymmetry in the frame work of TP.
In conclusion, we studied the CP violation in D → V V decay mode in which the T violating triple-product correlation is examined. That would undoubtedly be another excellent probe of new physics beyond the SM. The CP violating observables in connection with angular distribution are constructed. For neutral D decays, we neglect the CP violation induced by D 0 −D 0 oscillation. Experimentally, by doing a full angular analysis one may obtain such CP violating signals, and particularly it is worth mentioning that the upcoming large D data sample at BES-III will provide a great opportunity to perform it. The sensitivities for CP violating observables are estimated by assuming 20 −1 fb data-taking at ψ(3770) peak at BES-III.
One of authors, X. W. Kang, wish to thank Rong-