Charm mixing as input for model-independent determinations of the CKM phase gamma

The coherence factor and average strong phase difference of D0 and D0bar decay amplitudes to the same final state play an important role in the precision determination of the CKM parameter gamma using B- ->DK- and related decay modes. So far, this important input from the charm sector could only be obtained from measurements based on quantum-correlated DDbar pairs produced at the charm threshold. We propose to constrain these parameters using charm mixing, using the large charm samples available at the B factories and LHCb. We demonstrate for the example of D0->Kpipipi that a substantial improvement in the precision of the coherence factor and average strong phase difference can be obtained with this method, using existing data.


Introduction
In this paper we present a new method of constraining the coherence factor and average strong phase difference between D 0 and D 0 decay amplitudes to the same multibody final state [1], using input from charm mixing. Charm threshold data [2][3][4][5][6] provide important input to the measurement of the charge-parity (CP) violating phase γ in B ± → DK ± , B 0 → DK * and similar decay modes 1 [1,[7][8][9][10][11][12], where the details of the analysis depend considerably on the final state of the subsequent D decay [13][14][15][16][17][18]. The importance of charm threshold data in this context results from the well-defined superposition states of D 0 and D 0 accessible with quantum-correlated DD pairs. 1 CP -conjugate decays are implied throughout, unless stated otherwise. D stands for any superposition of D 0 and D 0 .
Charm mixing [19][20][21][22][23][24][25] also provides well-defined D 0 -D 0 superposition states, which can be used in a similar manner. Previous studies indicate that for D decays to self-conjugate final states, like D → K S π + π − and D → K S K + K − , datasets much larger than those currently available are required to significantly improve on the existing constraints from the charm threshold [26]. The interference effects due to D mixing in suppressed decay modes such as D 0 → K + π − π + π − and D 0 → K + π − π 0 are enhanced compared to self-conjugate decays. We propose to exploit this feature, and demonstrate that a substantial reduction of the uncertainty on the coherence factor and average strong phase difference in D → K + π − π + π − is possible with existing data. This paper is structured as follows: In Sections 2 we review the mixing formalism for multibody D decays, building on and extending the treatment presented in [27,28]. We present a unified description of the mixing-induced interference effects in decays to self conjugate and non-self conjugate final states. In Sec. 3 we show how D-mixing can be used to constrain the coherence factor and strong phase difference. Using simulated D → K + π − π + π − decays we demonstrate that a substantial improvement in the precision of the coherence factor and average strong phase difference is possible using existing data; here we are guided by the expected signal yields from LHCb's 2011 and 2012 data taking period. In Sec. 4, we conclude.

Mixing Formalism
In this section we review the charm mixing formalism for multibody decays and its relationship to the interference parameters relevant for the measurement of γ in B ± → DK ± and similar decay modes [26][27][28]. We introduce the complex interference parameter Z f that unifies the formalism for decays to self-conjugate states [2,3,8] and non self-conjugates states [1,4,5]. Z f is also particularly convenient for parameterising the constraints on charm interference effects derived from mixing using suppressed D decay modes, discussed in Sec. 3. Finally, in this section, we identify the important differences in the formalisms conventionally used for charm mixing measurements on one hand, and B ± → DK ± and related measurements on the other.

Charm mixing
The mass eigenstates |D 0 1 and |D 0 2 , with masses M 1 , M 2 and widths Γ 1 , Γ 2 , are related to the flavour eigenstates |D 0 and |D 0 through where p and q are complex numbers that satisfy |q| 2 + |p| 2 = 1. We also define and the usual dimensionless mixing parameters 3) The deviation of | q /p| from 1 is a measure of CP violation (CPV) in Dmixing. The phase φ mix ≡ arg q p is a convention-dependent quantity that is sensitive to CPV in the interference between mixing and decay -usually, a phase convention is chosen where φ mix = 0 in the absence of CPV. In practice we will deal with D mesons that have definite flavour at creation. These evolve over time t to the following superpositions of D 0 and D 0 : where |D 0 (t) refers to a state that was pure D 0 at time t = 0, while |D 0 (t) refers to a state that was purely D 0 at t = 0. The time-dependent functions g − (t) and g + (t) are given by

The complex interference parameter Z f
For the decay amplitudes of a D flavour eigenstate to a particular final state f , or its CP conjugatef , we use the following notation: Here p identifies a point in phase space for the multibody final state f , andp identifies the corresponding point for the CP-conjugate final state, where all final state momenta and charges are reversed. In practice we will integrate over finite phase-space volumes. Following [1] we therefore define 2 We use the symbols dΦ dp and dΦ dp for the density of states at p andp respectively. The integrals containing A(p) and B(p) run over the phase space volume Ω, and the ones containingĀ(p) andB(p) run over the CP conjugate volumē Ω. These volumes can encompass all of phase space, or any part thereof. The interference effects are described by the integrals over the cross terms: Ω A(p)B * (p) dΦ dp dp This defines the complex interference parameter Z f Ω for the final state f over the phase space region Ω, and Zf Ω , its CP-conjugate. For integrals over all phase space we use Z f , omitting the subscript. The magnitude of Z f Ω is between 0 and 1. The phase of Z f Ω represents a weighted average of the phase difference between the two amplitudes over Ω. The parameter Z f is directly related to the coherence factor R f D and average phase difference For binned analyses in decays to self-conjugate final states such as K S π + π − , these interference effects are usually parametrised instead by c i and s i . These are weighted averages of the cosine and the sine of the phase difference between D 0 and D 0 decay amplitudes, taken over a phase-space bin i covering the volume Ω i . This formalism was originally introduced in [8]; we follow the definition of c i and s i used in most subsequent articles [2,3,9,28,29]. The c i and s i parameters are related to the complex interference parameter through We will continue to use Z f Ω as it unifies the formalism for decays to selfconjugate and non-self conjugate states. In terms of the parameters defined above, the time-dependent decay rates are given by (2.14) Assuming that terms of order 3 and higher in the mixing parameters x and y are negligible leads to the following expressions: For the remainder of this article we assume, for simplicity, that CPV in charm is negligible, leading to: |Z f Ω | = |Zf Ω |, A =Ā and B =B (no direct CPV); | q /p| = 1.0 (no CPV in mixing); and arg(Z f Ω q p ) = arg(Zf Ω p q ) (no CPV in the interference between mixing and decay). Following the usual phase convention, we set φ mix to zero in the absence of CPV in the interference between mixing and decay, leading to q /p = 1 and Z f Ω = Zf Ω . With this, the expressions in Eqs. 2.15 -2.18 simplify to with identical expressions for the CP-conjugate processes. Since we have removed all weak phases, arg(Z f ) = −δ f D now represents the average strong phase difference.

Conventions
There are two different definitions of the CP operator in use. The Heavy Flavour Averaging Group (HFAG) [25] uses which is the convention usually adopted for charm analyses. In the context of extracting γ from B → DK decays, it is usual practice to follow the "ADS" convention [15], The choice of convention affects several relevant parameters, which needs to be taken into account when providing charm input to the measurement of γ. The choice of convention decides how the mass eigenstates |D 1 and |D 2 defined in Eq. 2.1 relate to the CP even and odd eigenstates, |D + and |D − . In the HFAG convention |D 1 ≈ |D − and |D 2 ≈ |D + (these relations become exact in the absence of CPV). In the ADS convention it is the other way around. The mixing variables x and y are defined in terms of (approximate) CP eigenstates, where the subscripts + and − label the masses and widths of the predominantly CPeven and CP-odd mass eigenstates, respectively. The formalism detailed above, with the mixing parameters defined in Eq. 2.3, follows the HFAG convention. Changing this to the ADS convention implies a simultaneous change x → −x and y → −y.
The choice of convention also affects the complex interference parameter Z f Ω . To ensure that the same physical CP even or CP odd state corresponds to the same wave function (up to a phase), the |D 0 and |D 0 wavefunctions between the two conventions must be related by where ξ is an arbitrary phase. As Z f Ω ∝ Ω f p |Ĥ|D 0 f p |Ĥ|D 0 * dΦ dp dp, this implies which is equivalent to (2.25) 3. Constraining the Coherence Factor and strong phase difference with D Mixing
In Eq. 2.19, the term linear in t (the "interference term") is sensitive to Ω and ImZ f Ω can be extracted. However, previous studies [26] indicate that datasets much larger than those currently available are required to provide useful constraints on Z f Ω i (or c i and s i ) from mixing using selfconjugate decays such as D → K S π + π − and D → K S K + K − .
We will demonstrate here that significant improvements on Z f can be achieved with existing data for the case where D 0 (t) → f is a "wrong-sign" (WS) decay. This is a decay where A is a doubly Cabibbo-suppressed (DCS) amplitude, such as In practice we will use the RS rate to normalise the WS rate, as this cancels many experimental uncertainties.

Z f
Ω from the mixing-induced interference of DCS and CF amplitudes In this scenario it is useful to define the ratio of the DCS amplitude (A DCS ) to the CF amplitude (B CF ): Neglecting terms of order 4 or higher in the small quantities x, y and r Df results in the following expression for the ratio of WS to RS decays as a function of the D decay time t: r Ω (t) = r 2 Df + r Df yReZ f Ω + xImZ f Ω Γt + An analysis of the time-dependent decay rate ratio will, through the linear term of Eq. 3.2, provide a measurement of The factor r Df , which also features in the linear term, can be obtained in the same analysis from the 0 th order term of Eq. 3.2, and Γ has been measured very precisely [30]. Taking the D mixing parameters x and y as input, we can translate a measurement of b into constraints in the ReZ f Ω −ImZ f Ω plane. A given value of b corresponds to a line of slope y /x in the ReZ f Ω − ImZ f Ω plane defined by: (3.4)

Sensitivity with existing LHCb datasets
To estimate the precision on Z K3π achievable with current data, we perform a simulation study based on plausible D → K + π − π + π − event yields in LHCb's 3 fb −1 data sample taken in 2011 and 2012. We use the values for the mixing parameters given in Eq. 3.5, and r D = 0.058 based on the WS to RS branching ratio reported in [30]. We generate simulated events according to the full expressions for the decay rates given in Eqs. 2.11 -2.14. To take into account the effect of LHCb's trigger and event selection process, which preferentially selects decays with long D decay times, we apply a decay-time dependent efficiency function ǫ(t) based on that seen in [31]. For this feasibility study, we ignore other detector effects and background contamination. LHCb results for D → K − π + indicate that backgrounds can be controlled sufficiently well even for WS decays [24]. Based on the RS yields reported in [32], and taking into account that for the WS mode tighter selection criteria might be necessary to control backgrounds, we estimate about 8 million RS+WS events in LHCb's 2011-2012 dataset. The exact fraction of WS events depends on the input parameters, in particular on R K3π D ; typically, 8 million RS+WS events correspond to about 30, 000 WS events.
To constrain Z K3π we perform a χ 2 fit to the WS/RS ratio in 10 bins of proper decay time. The bins have variable widths, chosen such that each bin contains a sufficient number of events. Using the same approximations that led to Eq. 3.2, we obtain for the expected WS to RS ratio R (3.6) The fit parameters are r Df , b = yRe(Z f Ω ) + xIm(Z f Ω ), x, and y, where x and y are constrained according to Eq. 3.5.
In a real experiment, the time-dependent efficiency ǫ(t) would not necessarily be known a priory, but it is reasonable to assume that ǫ(t) would be the same for WS and RS decays. We therefore extract its shape from the (simulated) data by dividing the RS decay time distribution (histogrammed in 100 bins) by e −Γt , as shown in Fig. 2(a); the overall normalisation cancels when using ǫ(t) in Eq. 3.6.
A pull study based on generating and fitting 200 simulated data samples, each containing 8 million RS+WS events, shows no evidence of fit biases, and confirms the correct coverage of the confidence intervals obtained from the fit χ 2 .
An example of such a fit is shown in Fig. 2(b). The 8M events have been generated using CLEO-c's central value Z K3π = −0.133−0.301i [5] and include 30.5k WS events. Figure 3 shows 1, 2 and 3σ confidence regions based on 8 million simulated events that have been generated with the illustrative   values Z K3π = −0.3 and Z K3π = 0.9i used also to obtain Fig. 1. Figure 4 shows the constraints for events generated using the CLEO-c central value for Z K3π , in both polar coordinates (i.e. the coherence factor R K3π D = Z K3π and strong phase difference δ K3π D = − arg Z K3π ) and cartesian coordinates (ReZ K3π and ImZ K3π ).
To evaluate the potential impact of input from charm mixing on the precision of Z K3π , we combine the χ 2 function used to obtain Fig. 4 with CLEO-c's measurement of Z K3π [5]. The CLEO-c results, and the combination with our simulated data, are shown in Fig. 5. The input from charm mixing improves the constraints considerably. The effect is particularly striking at the ≥ 2σ level where there were previously no constraints on δ f D . To quantify these improvements, one-dimensional 68% and 95% confidence intervals for R K3π D and δ K3π D are calculated, following the same procedures as used by CLEO-c [5] to ensure comparable results. The 68% confidence limits are based on a standard χ 2 difference calculation. The same process would lead to 95% confidence limits reaching the edge of the R f D -δ f D parameter space in the CLEO-c measurement. These are therefore obtained using a Bayesian approach with a uniform prior in the physically allowed region of the parameter of interest. The results are summarised in Tab. 1. The constraints from our 8M simulated charm events (with 30k WS events) shrink the existing uncertainties on ReZ K3π and ImZ K3π by a factor of ∼ 1.5, and the 95% CL on ImZ K3π by a factor of two. In terms of polar coordinates, the simulated in-    as well as ReZ K3π and ImZ K3π from simulation, CLEO-c [5], and their combination, at 68% and 95% CL, obtained with two different techniques following [5], as described in the text. The ∆χ 2 method is not suitable for obtaining separate constraints on ReZ K3π and ImZ K3π from the simulated mixing data alone.

Conclusion
Charm mixing is sensitive to the same same charm interference parameters that are relevant to the measurement of γ in B ± → DK ± and related decay modes [1,[7][8][9][26][27][28]. So far, these have only been accessible at the charm threshold [2][3][4][5][6]. The increased precision with which the charm mixing parameters x and y have been measured [6,[19][20][21][22][23][24][25] opens up the possibility of constraining charm interference parameters using charm mixing. However, previous studies indicate that for decays to self-conjugate final states, such as D → K S π + π − and D → K S K + K − , datasets much larger than those currently available are required to significantly improve constraints on the binned complex interference parameters Z f Ω i = c i + is i from charm mixing [26]. On the other hand, in wrong-sign decay modes such as D 0 → K + π − π + π − and D 0 → K + π − π 0 , the mixing-induced interference effects are significantly enhanced compared to self-conjugate decays. This provides greater sensitivity to the complex interference parameter Z f , or, equivalently, the coherence factor R f D = |Z f | and average strong phase difference δ f D = − arg(Z f ) introduced in [1]. While it is interesting to note that useful information can be obtained in this way without additional input, the true power of the method lies in the combination with threshold data. We evaluate the potential of this approach with a simulation study based on estimated D → K + π − π + π − signal yields expected in LHCb's 2011 and 2012 dataset. We do not assume any improvements on external inputs. Our results indicate that charm mixing input from existing LHCb data, when combined with CLEO-c's measurement [5], could substantially reduce the current uncertainty on the coherence factor and average strong phase difference in D → K + π − π + π − . Such a measurement can be expected to have a significant impact on the precision with which the CKM parameter γ can be measured at LHCb, BELLE II, and the LHCb upgrade.