^{*}

^{†}

^{‡}

^{3}.

We analyze

The search for physics beyond the Standard Model (SM) has brought about a search for

In the recent past, the controversial measurements

Identifying the dominant source of

In contrast to the above-mentioned approaches, the assumption that FSI is the dominant source of

The paper is organized as follows. In the next section we write the amplitudes for all the decays considered. In Sec.

It is customary to describe weak decay amplitudes in terms of topologies of Wick contractions (or renormalization-group-invariant combinations thereof). Notice that any Wick contraction, as defined in Refs.

In a previous study

Other attempts have been made previously to study

The

To expand the FSI description we need a phase,

The extension to the CA and DCS final states requires the introduction of additional sources of

Next, we observe that the ratio of the branching fractions of the DCS to the CA decays of

With this parametrization we arrive at the following amplitudes for the SCS, CA and DCS amplitudes. Although we present the amplitudes with

In summary, the

For the phases generated by FSI in the different isospin amplitudes:

A mixing angle

The

SCS modes [to be multiplied by

CA modes [to be multiplied by

DCS modes [to be multiplied by

The CKM factors are to be kept explicit and hence the amplitudes given in Eqs.

The

The asymmetries consist of three contributions. The first contribution comes from the terms proportional to

The total amplitude for the SCS decays where we consider

The

Much progress has been made in the measurement of

Recently, LHCb has released the analysis of

We do not use the measurement of the individual asymmetries

Measurements of

In the recent past some theoretical effort has been put on estimating

The only SCS channel for which the

The BESIII Collaboration has performed the first measurements of

We use

The fit results are presented in Table

The branching fractions that were used in the fit

The fit value of the parameters and their correlations. (top) Fit values of the parameters.

We find two equivalent solutions for the parameters in the

Since the phases coming from final state interactions should be interpreted as being generated by rescattering due to the presence of resonances, the phases should follow a distinct pattern determined by the masses of the resonances corresponding to particular isospin quantum numbers. The spectrum of the masses of these scalar resonances can be inferred upon by using the Gell-Mann-Okubo mass formula

The strongest constraint, by far, on

The correlations between

Predictions of

It is important to note here that the

Finally, we make some predictions from our fit. The branching fraction of the decay mode

We also predict the relative strong phase between the amplitudes of the modes

One can also define rate asymmetries involving interference of CA and DCS decays of the neutral

Fit results for

Estimates of

We present a prediction of

While our parametrization is well motivated by

It can be shown that

Fit results for

As a second test of our parametrization we propose the correlation between the

Correlations between asymmetries (in %) as given in Eq.

With Belle II starting up and LHCb having built a very strong charm program over the past few years, it is instructive to see what these measurements will mean in terms of constraining the penguin amplitudes. While the measurements of the branching fractions are expected to improve significantly too, this will not additionally constrain the penguin amplitudes directly. However, the ratio

To keep the analysis simple and on the more conservative side we do not take into account any improvement in the measurement of the branching fractions. We project the central values of the

Numbers used to generate the constraints on

In Fig.

Fit results for

The main purpose of this work is to take advantage of the high precision reached by the measurements of the branching ratios in two particle final states consisting of kaons and (or) pions of the pseudoscalar charmed particles to deduce the predictions of the Standard Model for the

So in this work we extend the formalism presented in

We perform a fit of the parameters to the branching fractions and

Within the ambit of our work we find reasonable success in trying to parametrize

We succeed in describing the measured branching fractions by invoking

The values of the FSI phases, when considering the negative solutions, fall nicely along the pattern of the expected mass ordering of the resonance from the presence of which these FSI phases are generated. This also fixes the imaginary parts, which are relevant for the

Once we relate the 15 in the

We show that amongst the current measurement of

With the correlations between the asymmetries and the direction pointed at by the data we can propose methods for validating our

When we choose the negative solution for the phases, we also predict

In this framework of

A. P. would like to acknowledge partial support from ERC Ideas Starting Grant No. 279972 “NPFlavour” while most of this work was being done. We would like to thank Alessandra Pugliese and Maurizio Lusignoli for fruitful discussions during the initial stages of the work. We would like to thank Luca Silvestrini and Enrico Franco for their help and support for several aspects of this work.

The fit of the parameters to the branching fractions and

The marginalized posterior distributions of the parameters from the fit as given in Table

For recent alternate approaches, see Refs.

The approximate selection rule that leads to the suppression of the simultaneous creation of

The average corresponds to the one presented by HFLAV post-Moriond 2019.

The predicted value has the opposite sign for the positive solutions of the strong phases.

The posterior distributions of all three observables are non-Gaussian and hence, the error bars have been interpreted as the rms of the distributions.

The predicted value is the same for both the solutions of the strong phases, negative and positive.