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We propose a novel method to measure flavor oscillations and charge-parity (

The noncoincidence of mass and flavor eigenstates of neutral flavored mesons results in flavor oscillations, which are meson–antimeson transitions that follow an oscillating pattern as a function of time. Flavor oscillations are sensitive probes for non-standard-model physics because virtual massive particles can contribute to the amplitude, possibly enhancing the average oscillation rate or the difference between rates of mesons and those of their respective antimesons. Indeed, the study of flavor oscillations has long been established as a powerful instrument to uncover, or constrain, possible dynamics not described by the standard model.

Oscillations are typically characterized by the dimensionless mixing parameters

The most direct experimental access to the charm-mixing parameters is offered by the analysis of self-conjugate multibody decays, such as

With the large samples of

We propose a novel approach for measuring parameters of oscillation and

In Sec.

Mass eigenstates of neutral-charm mesons are expressed as

We parametrize the

We indicate with

We divide the Dalitz plane into two sets of

As oscillations develop as a function of time, the relative variations of intensities between pairs of bins change depending on the mixing parameters and relevant charm-decay hadronic parameters. The expressions for the event yields integrated over each Dalitz-plot bin

If the probability

For small mixing parameters (

If the probability

In the limit of

The bin-flip approach consists of performing a joint fit of the

In practice, to avoid instabilities of the fit due to

Various Dalitz-plot binning schemes were developed by the CLEO collaboration for measuring the coefficients

The

Iso-

Values of

Correlation coefficients (in %) between the

The bin-flip method is validated using simulated experiments. In Sec.

The sensitivity of the bin-flip method to oscillation and

No mixing (NM), corresponding to

World-average mixing with

Dalitz-plot distribution for

For each scenario we generate an ensemble of

Figure

Distributions of (top) fit residuals and (bottom) pulls on (left)

Values of

Equation

Bin-flip ratio in each Dalitz-plot bin as a function of decay time, in the limit of

In addition, we use the simulated samples to compare the performance of the bin-flip method to those of existing approaches. The customary model-dependent analysis implies a joint maximum likelihood fit to the unbinned decay-time and Dalitz-plot distributions, based on the same amplitude model used in generation. The established model-independent analysis implies a joint maximum likelihood fit to the unbinned decay-time distributions of decays in the 16 Dalitz-plot bins. While evaluating the performance of both standard methods, we keep all parameters fixed except

Expected statistical uncertainties from

Figure

Summary of results from simulated experiments of

We study the dependence of our findings on bin multiplicity by repeating the study with various choices for the number of decay-time bins and of pairs of Dalitz-plot bins. In all tests we consider equipopulated decay-time bins and

Uncertainties on the mixing and

The sensitivity studies of Sec.

Table

Expected statistical sensitivities from

To assess the impact of the limited precision of external constraints on future larger samples of

Expected statistical uncertainties as functions of signal yields from fits to simulated

The above analysis is carried out in the limit of

For the bin-flip method to be applicable to experimental data, effects such as backgrounds, flavor tagging, finite resolutions, and nonuniform efficiency variations across decay time and Dalitz plane need, in principle, to be accounted for. Backgrounds and flavor tagging are not a significant limitation. Using the

The bin-flip method is constructed so as to be insensitive to such effects. To validate this notion, we incorporate in the simulated samples realistic resolution and efficiency effects based on publicly available information from the LHCb and Belle II experiments, which are the environments where this method is most likely to be considered. In both cases we consider experimental effects typical of

For LHCb, we assume a decay-time resolution corresponding to 10% of the

Efficiency (normalized to unity at its maximum) as a function of decay time assumed for

Efficiency (normalized to unity at its maximum) as a function of the Dalitz plot location assumed for

Data are generated using the same amplitude model as for the previous studies. The decay-time resolution is included by smearing the generated decay time with a Gaussian distribution with a width of

The fits are performed with unconstrained

Biases (

These findings show that no accurate knowledge of the decay-time resolution or efficiency variation as a function of decay time and Dalitz-plane position is needed to apply the method. This supports the approach as an expedient and powerful alternative to standard approaches for charm-mixing measurements using

To assess the impact of a bin-flip analysis on the current global knowledge of oscillation and

Confidence regions at the (inner, darker hatching) 68.3% and (outer, lighter hatching) 95.5% confidence level in the two-dimensional space of (left) oscillation parameters (

Although the effect on

We finally emphasize that the alternative additive parametrization proposed for the effects of charm mixing offers superior statistical properties to standard parametrizations and is particularly preferable for combinations, in which central values cannot be assumed to be known.

In summary, we propose the bin-flip method, a model-independent approach to measure parameters of mixing and

The bin-flip method offers 35% better statistical sensitivity, compared to existing model-independent methods, to

The bin-flip method is expected to offer good sensitivity in high-yield multibody decays that receive large contributions from doubly Cabibbo-suppressed amplitudes. In addition to

A bin-flip analysis of the samples of

We are grateful to Andrea Contu and Michal Kreps for earlier involvement in this work, and Jolanta Brodzicka for fruitful discussions and valuable comments. T. G. and N. J. acknowledge support from the Science and Technology Facilities Council (United Kingdom). T. G. and T. P. acknowledge support from the European Research Council under FP7.

In Sec.

Fits suffer from non-Gaussian estimator distributions when the dimensionality of the likelihood or least-squares function depends on the estimated value of one or more parameters. This may happen if all terms sensitive to a parameter of interest involve products with another parameter, or a function of it, that can vanish. The likelihood then becomes scarcely sensitive to the parameter of interest for vanishing values of the multiplication factor, incurring in non-Gaussian estimator distributions. A multiplicative parametrization as

Distributions of (left) fit pull for the