Measuring $CP$ violation and mixing in charm with inclusive self-conjugate multibody decay modes

Time-dependent studies of inclusive charm decays to multibody self-conjugate final states can be used to determine the indirect $CP$-violating observable $A_\Gamma$ and the mixing observable $y_{CP}$, provided that the fractional $CP$-even content of the final state, $F_+$, is known. This approach can yield significantly improved sensitivity compared with the conventional method that relies on decays to $CP$ eigenstates. In particular, $D \to \pi^+\pi^-\pi^0$ appears to be an especially powerful channel, given its relatively large branching fraction and the high value of $F_+$ that has recently been measured at charm threshold.

It is of great interest to search for effects of indirect CP violation in time-dependent studies of neutral charmmeson decays. In the Standard Model indirect CP violation is expected to be well below the current level of experimental precision [1], but many models of New Physics predict enhancements [2]. A very important CPviolating observable is A Γ , which is measured from the difference in lifetimes of the decays of D 0 and D 0 mesons to a CP eigenstate. In this Letter it is shown how inclusive self-conjugate multibody decays that are not CP eigenstates can also be harnessed for the measurement of A Γ , provided that their fractional CP -even content, F + , is known. This new approach has the potential to improve significantly the knowledge of A Γ and has become possible thanks to measurements of F + that have recently begun to emerge from analyses of coherent charm-meson pairs produced at the ψ(3770) resonance [3]. Furthermore, it is explained how exploiting these decays can also provide a corresponding improvement in the precision on y CP , which is an important observable that describes D 0 D 0 oscillations. For the purpose of concreteness the discussion is presented for the example decay D → π + π − π 0 , although the results are valid for all selfconjugate multibody modes. Here and throughout the discussion D indicates a neutral charm meson; this notation is used when it is either unnecessary or not meaningful to specify a flavour eigenstate.

Measurements with CP eigenstates
In the D-meson system the mass eigenstates, D 1,2 , are related to the flavour eigenstates D 0 and D 0 as follows: where the coefficients satisfy |p| 2 + |q| 2 = 1 and The phase convention CP |D 0 = |D 0 is adopted. Indirect CP violation occurs if r CP = 1 and/or φ CP = 0.
Charm mixing is conventionally parameterised by the quantities x and y, defined as where M 1,2 and Γ 1,2 are the mass and width of the two neutral meson mass eigenstates, and Γ the mean decay width of the mass eigenstates. In the chosen convention D 1 is almost CP even. The average of currently available measurements gives x = (0.41 +0.14 −0.15 )% and y = (0.63 +0.07 −0.08 )% [4]. Consider an environment where charm mesons are produced incoherently, such as the LHC, or in the cc continuum or from a b-decay at an e + e − B-factory, and are observed through their decay into a CP eigenstate of eigenvalue η CP . Time-dependent measurements allow the decay widthsΓ andΓ to be determined for mesons produced in the D 0 and D 0 flavour states, respectively. From these quantities the CP -violating observable A Γ and mixing observable y CP may be constructed: Assuming x, y, (r CP − 1/r CP ) and φ CP to be small, and assuming direct CP violation to be negligible, it can be shown [10] that these observables have the following dependence on the underlying physics parameters Expressions that also allow for the contribution of direct CP violation can be found in Ref. [11]. Thus in the limit of CP conservation A Γ vanishes and y CP → y. The average of currently available measurements, dominated by studies based on the CP -even eigenstates K + K − and π + π − , yields A Γ = (−0.058 ± 0.040)% and y CP = (0.866 ± 0.155)% [4]. (Here the A Γ average includes new measurements from the LHCb [5] and CDF [6] collaborations, in addition to the older set of results from LHCb [7], BaBar [8] and Belle [9] that are considered in Ref. [4].) Introducing self-conjugate multibody decays and the CP -even fraction F + The CP content of an inclusive self-conjugate multibody decay, for example D → π + π − π 0 , can be measured with a sample of coherently produced DD pairs at the ψ(3770) resonance, such as that collected by the CLEOc and BESIII experiments. A double-tag technique is employed in which one D meson is reconstructed in the signal decay of interest, and the other in its decay to a CP eigenstate. In such an event, and neglecting any CP violation, the quantum numbers of the ψ(3770) meson means that the CP eigenvalue of the signal decay is fixed. The CP -even fraction of the signal decay is given by ignates the number of decays tagged as CP -even (-odd), after correction for detector inefficiencies and the specific branching fractions of the CP eigenstate tags employed. In this manner F + has been measured for the decay D → π + π − π 0 and found to be 0.968 ± 0.017 ± 0.006, indicating the mode to be almost fully CP even [3].
Although CP violation is neglected in the currently available measurements of F + this assumption introduces negligible bias in the result. Both the Standard Model and theories of New Physics expect direct CP violation in charm decays to be ≤ 10 −3 [12], a prediction which is compatible with existing experimental results [13]. Any effects will therefore be small alongside the measurement precision attainable with the CLEO-c and current BE-SIII data sets. Furthermore, the double-tag analyses performed at these experiments have no sensitivity to indirect CP violation at leading order in (x, y), as the DD system is produced at rest. For the specific case of D → π + π − π 0 , a recent time-integrated high precision analysis by LHCb has revealed no evidence of any direct CP -violating effects [14].
There is a simple relationship between F + and the parameters that describe the intensity and strong-phase variation over the phase space of the decay. The amplitude of a multibody decay such as D → π + π − π 0 is dependent on the final-state kinematics, which can be uniquely defined by the Dalitz plot coordinates s 12 = m 2 (π + π 0 ) and s 13 = m 2 (π − π 0 ). The amplitude of a D 0 decay to a specific final state is given by A D 0 (s 12 , s 13 ) = a 12,13 e iδ12,13 , where the integral of |A D 0 (s 12 , s 13 )| 2 over the full Dalitz plot is normalised to unity. Consider the situation where the Dalitz plot is divided into two bins by the line s 12 = s 13 . The bin for which s 12 > s 13 is labelled −1 and the opposite bin is labelled +1. The parameter K i (K i ) is the flavour-tagged fractional intensity, being the proportion of decays to fall in bin i in the case that the mother particle is known to be a D 0 (D 0 ) meson: The parameter c i is the cosine of the strong-phase difference between D 0 and D 0 decays averaged in bin i and weighted by the absolute decay rate: 13ā12,13 cos(δ 12,13 −δ 12,13 ) A parameter s i is defined in an analogous manner for the sine of the strong-phase difference.
The CP -tagged populations of these bins, N ± i , normalised by the corresponding single CP -tag yields, is given by [15] Here h D is a normalisation factor independent of bin number and CP tag. When there is no direct CP violation in the decay A D 0 (s 12 , s 13 ) =ā 12,13 e iδ12,13 ≡ a 13,12 e iδ13,12 and so Under this assumption, and the identities N ± = i N ± i , and i K i = 1, it follows that in the two-bin case Measurements with inclusive self-conjugate multibody decays Now consider, for an incoherently produced D meson, the time-dependence of a self-conjugate multibody decay. The time evolution of the D 0 to the point (s 12 , s 13 ) is given by A D 0 (t, s 12 , s 13 ) = a 12,13 e iδ12,13 g + (t) + r CP e iφCP a 13,12 e iδ13,12 g − (t), (12) where Ignoring terms of O(x 2 , y 2 , xy) or higher, the rate of decay to that point is proportional to |A D 0 (t, s 12 , s 13 )| 2 = e −Γt a 2 12,13 − a 12,13 a 13,12 r CP Γt × y cos(δ 12,13 − δ 13,12 − φ CP ) + x sin(δ 12,13 − δ 13,12 − φ CP ) .
Integrating this over the two bins of the full Dalitz plot leads to the time-dependent decay probability P(D 0 (t)) = +1 |A D 0 (t, s 12 , s 13 )| 2 ds 12 ds 13 + −1 |A D 0 (t, s 12 , s 13 )| 2 ds 12 ds 13 (14) where use is made of the definitions of c i , s i and the relations given in Eqs. (10) and (11). The time evolution for the D 0 decay to the point (s 12 , s 13 ) is given by A D 0 (t, s 12 , s 13 ) = 1 r CP e −iφCP a 12,13 e iδ12,13 g − (t) + a 13,12 e iδ13,12 g + (t), and hence the time-dependent decay probability for the D 0 decay can be shown to be These expressions contain an additional dilution factor of (2F + −1) in comparison to the CP -eigenstate relations of Eqs. (5) and (6) and are identical in the case when F = 0 or 1. In the limit F + → 0.5 then both observables vanish. It is interesting to note that a similar relationship between the two classes of D decays was found in Ref. [3] when considering the determination of the unitarity triangle angle γ using B ± → DK ± decays.
Expressions (18) and (19) may be modified to allow for the possible contribution of direct CP violation. In this case the relations in Eq. (10) no longer apply. Direct CP violation adds an additional magnitude and weak phase difference when considering the relations between the amplitude of the D 0 and D 0 decay, and this additional magnitude and phase varies as a function of position in phase space.
With the inclusion of direct CP violation the expression for A eff Γ becomes where r CP and φ CP are unchanged in their meaning and relate only to indirect CP violation, (2F ′ Hence the effect of the additional amplitudes due to direct CP violation are contained within the terms F ′ + and ∆. In the limit of no direct CP violation ∆ → 0, and F ′ + → F + . Since ∆ must be small the third term in Eq. 20 is negligible in comparison to the others.
The equivalent expression for y eff CP becomes

Discussion and conclusions
Measurements of A eff Γ and y eff CP performed with any self-conjugate multibody decay can be used to determine A Γ and y CP , respectively, provided that the CP content of the decay is known. The mode D → π + π − π 0 is a very promising candidate for this purpose since the dilution effects arising from the factor (2F + − 1) in Eqs. (18) and (19) are < 10%, and it possesses a branching ratio that is around 3.5 times higher than that of D → K + K − , the most common CP -eigenstate mode used for these measurements. Therefore this channel offers an opportunity to improve the knowledge of A Γ and y CP significantly, particularly at e + e − experiments such as Belle-II, where the π 0 reconstruction efficiency is good. The relatively abundant four-body decay D → π + π − π + π − also has the potential to be a high impact channel, although this cannot be confirmed until its CP content is measured. The same remarks apply to D → K 0 S π + π − π 0 , which has a branching fraction of over 5% and comprises the CP -odd eigenstates K 0 S η and K 0 S ω as sub-modes. This channel also has the feature of being Cabibbo favoured, which means that it is extremely robust against any pollution from direct CP violation. The extensively-studied decay D → K 0 S π + π − is not suitable for an inclusive treatment, since it has a CP content of F + ∼ 0.5, as is evident from examining the relative proportion of CP -even and CP -odd double-tagged events reported in a CLEO analysis performed to measure the c i and s i parameters [16].
The Belle collaboration has reported a model dependent analysis of the mode D → K 0 S K + K − that measures y CP through comparing the CP -odd and CP -even regions of the Dalitz plot [17]. Studies also exist that fit time-dependent amplitude models to the Dalitz plots of the decays D → K 0 S π + π − and D → K 0 S K + K − in order to determine the mixing and CP violation parameters [18][19][20]. Furthermore, proposals have been made of how to perform model-independent analyses of selfconjugate decays binned in phase space [21,22]. The method advocated in this Letter is novel because it is inclusive, model-independent and suitable for those decays which are dominated by a single CP eigenstate, such as D → π + π − π 0 . Inclusive analyses are experimentally more straightforward since there is no need to account for the position in phase space of each decay, provided that the acceptance is relatively uniform.
As explained in Ref. [3], self-conjugate multibody modes can also be used to measure the unitarity triangle angle γ with B ± → DK ± decays as long as F + is known for the mode under consideration. In cases where no measurement of F + exists from the charm threshold it is possible to obtain this information from a comparison of a measurement of y eff CP and the value of y CP obtained from CP eigenstates, or indeed that of y itself, assuming negligible CP violation in the charm system. This strategy of using charm-mixing observables to help provide input for the γ determination is similar to that already proposed for quasi-flavour specific states [23].
In summary, inclusive measurements of the time evolution of mutibody self-conjugate charm decays offer the possibility to obtain significantly improved sensitivity to CP violation and mixing in the D 0 D 0 system. The observables A eff Γ and y eff CP are simply related to those of the CP eigenstate case, A Γ and y CP , by a dilution factor (2F + − 1), where F + is the fractional CP -even content of the decay. This parameter may be measured in coherently produced DD decays at the ψ(3770). One mode for which F + is known, D → π + π − π 0 , has the potential to yield a more precise determination of A Γ and y CP than is possible with CP eigenstate decays. Several other promising channels exist with relatively high branching fractions and should also be exploited, provided that analyses at the ψ(3770) show them to be dominated by a single CP eigenstate. Alternatively, measurements of y eff CP using these latter channels will allow their CP con-tent to be determined, which is valuable input for the programme to measure the unitarity angle γ. First results using this class of decays are eagerly awaited.