^{3}

Observables of light hadron decays are analyzed in a chiral Lagrangian model that includes resonance fields of vector mesons. In particular, transition form factors are investigated for Dalitz decays of

Decays of light hadrons play a crucial role in the investigation of the low-energy behavior of quantum chromodynamics (QCD), and are measured extensively in experiments. In particular, Dalitz decays such as

In order to describe the dynamics of light hadrons, we adopt a chiral Lagrangian model that includes vector mesons. In this model, chiral octets and singlets are introduced as a representation of SU(3). There are some models [

This effective dynamics of hadrons is applicable to a variety of phenomena, e.g., hadronic

For vector mesons, we calculate the quantum correction to the self-energies to obtain a one-loop-corrected mass matrix. The mixing matrix, which is an orthogonal matrix to diagonalize the mass matrix, is determined in the procedure of diagonalization. After diagonalizing the mass matrix, the relevant mass eigenstates play the role of resonance fields of

In processes such as

For IP-violating interactions in the model, we introduce operators including SU(3) singlet fields, in addition to the ones suggested in Refs. [

Using the introduced operators, we write formulae for the IP-violating decays of hadrons. In particular, expressions for the (differential) decay widths and electromagnetic TFFs are shown. These formulae are useful for thorough analysis to test the validity of the model.

Since IPV interactions include an antisymmetric tensor, one needs the fourth derivatives on the chiral fields so that the Lagrangian is Lorentz invariant. This is the O(

In this paper, the observables of IP-violating decays are analyzed in our model. For the HLS model, a numerical result for IP-violating decay widths is given in Refs. [

In our analysis,

Using the estimated parameter region, a model prediction for hadron decays is presented in this work. Specifically, we give predictions for (1) the electromagnetic TFFs of

The remaining part of this paper is organized as follows: In

In this section, we introduce a chiral Lagrangian model with vector mesons [

The Lagrangian is divided into three parts in Eq. (

In the following, we present how one-loop counterterms given as

Note that

The counterterms for the self-energy for vector mesons and

The coefficients of the effective counterterms in Eq. (

In Eq. (

In this subsection, we diagonalize the mass matrix for neutral vector mesons and obtain the mass eigenstates that correspond to (

In Eq. (

We calculate the mass of the neutral vector mesons,

The propagator for the neutral vector mesons is denoted as

In the following, we expand the propagator in Eq. (

The denominator of the propagator in Eq. (

We define the wavefunction renormalization of neutral vector meson:

Thus, in the vicinity of the pole mass, the propagator takes the following form:

In this subsection, we derive formulae for the width of the vector meson decay into two pseudoscalars. We compute the one-loop diagrams that are shown in

One-loop-order Feynman diagrams for

The amplitude for

In the isospin limit, one finds that the relations

In the above calculation, the isospin-breaking effect is not taken into account. Using the one-loop-corrected couplings, we obtain the partial decay width for

Using the isospin relation of

In this subsection, we study the decay width of

The effective Lagrangian for the singlet and octet states is given as

Next one can rewrite the Lagrangian in terms of the mass eigenstates using their relations with the octet and singlet states:

Substituting the above equation, one obtains the effective Lagrangian for renormalized mass eigenstates

To evaluate the partial decay width for

In the above Lagrangian, we set the wavefunction renormalization

The

Ignoring the isospin-breaking effect, we note that

In this subsection, mixing between photons and vector mesons is analyzed. The contributing diagrams for

Feynman diagrams for the two-point function of the mixing of photons and vector mesons. Wavy lines show photons while bold wavy lines indicate vector mesons.

The

In the basis of SU(3), the two-point functions on the l.h.s. of Eq. (

One can find that the

One can write the two-point functions in Eq. (

The derivation of Eq. (

In this subsection, the structures of the mixing matrix and decay constants for pseudoscalars are given. We take account of the one-loop correction to both the mixing and the decay constants.

The basis for an SU(3) eigenstate is written in terms of mass eigenstates as

The mixing angles denoted as

For the decay constants of

In this section, we discuss IPV in the model. As well as ChPT, the quantum anomaly of chiral symmetry causes an IP-violating interaction. The expression of the WZW term is given in Eq. (

Since SU(3) singlet fields are contained in the model, IP-violating operators with singlets should be taken into account. We consider such singlet-induced operators within the invariance of SU(3) symmetry. Imposing charge conjugation (C) symmetry, one can obtain the operators in the model:

In Eqs. (

In contrast to our work, the singlet fields are contained as a component of a chiral nonet matrix in Ref. [

The IP-violating interactions in our model are denoted as

In Eq. (

In this subsection, IP-violating decays of

Diagrams contributing to the decay widths for: (a)–(b)

Vector mesons can decay into

Although the vector meson propagator is clearly shown in

Decay amplitudes are obtained from the operators in Eqs. (

The pseudoscalar decay width in Eq. (

In the above relation, the ratio of the effective coupling for

In the above relation, the effective coupling is written in terms of model parameters. We use the relations in Eqs. (

In this subsection, an IP-violating process of

Diagram contributing to the IP-violating decay of

The contribution coming from

In this subsection, we evaluate the partial decay widths of the IP-violating process given as

Diagrams contributing to the decay width of

In this subsection, a form factor for the IP-violating modes

Diagrams contributing to the decay width of

In this subsection, a form factor for IP-violating electromagnetic decays for neutral vector mesons is analyzed. Contributing diagrams are exhibited in

Diagrams contributing to the decay width of

The TFF in the above equation are normalized as unity in the limit where the virtual photon goes on-shell.

In this subsection, partial decay widths for

Diagrams contributing to the decay width of

The transition amplitude is given as

In this subsection, differential decay widths for

Diagrams contributing to the decay width for

Using the above equations, one can obtain the differential decay width:

In this section, a phenomenological analysis is carried out in the model. In the following subsection, we perform

In order to carry out the analysis, the following points are addressed:

For the parameter

For

In the expression of

In this subsection, we estimate the parameters in the model with the decay distribution for

The differential branching fraction for

In this paper, we take the tree-level pion decay constant,

Since the tree-level

The result of the decay distribution is shown in

The fitting result of the decay distribution for

In

Numerical values of the parameters in the model.

Hereafter, the values in Eq. (

For the ratio of decay constants of pseudoscalars, we verify whether the model prediction of

In the above result, one can find that the model prediction deviates slightly from the case of the tree-level

In this subsection, we explain how the parameters

At first, we consider the off-diagonal elements of

Imposing the condition for the residue of the vector meson propagator,

Since

The decay widths are given by the imaginary part of the inverse propagators,

In the following, we determine the parameters

The results of the neutral vector meson mass.

Mass | Theory [MeV] | PDG [MeV] |
---|---|---|

Using the above parameters, we have the orthogonal matrix

In this subsection, we estimate the model parameters by using the IP-violating observables for light hadrons. As input data for the

The widths of radiative decays,

In this fitting, the sign of

Partial widths of radiative decays. For

Decay mode | Model [MeV] | PDG [MeV] |
---|---|---|

For parameter estimation, we use observables for pseudoscalars. In particular, the PDG data [

Confidence intervals of the model parameters estimated from the data [

Parameter | |||||
---|---|---|---|---|---|

In Eqs. (

Using the above values, the mixing matrix and the wavefunction renormalizations of pseudoscalars are calculated as

In the following analysis, the parameter values in Eq. (

Here, we discuss a case in which the singlet-induced contribution is absent. If one takes the limit

For parameter estimation of the IP-violating parameters, the ratio of the effective coupling for

The experimental data used in the above

Fitting result and model prediction of the ratio of effective coupling for

Ratio | Model | PDG | Model in the isospin limit |
---|---|---|---|

1 | |||

Partial widths of the radiative decays for vector mesons. For comparison, the PDG data [

Model [MeV] | PDG [MeV] | |
---|---|---|

In the following, TFFs for Dalitz decay of vector mesons are analyzed. In particular, we fit

Fitting results of the TFFs for vector meson decays. For the four cases of fitting, the goodness-of fit is shown. Estimated 1

p-value | ||||||
---|---|---|---|---|---|---|

170.1/144 | ||||||

211.8/151 | ||||||

173.7/144 | 0.046 | |||||

215.4/151 |

In

In the following analysis in this paper, we adopt parameter sets that are estimated from the case without the Lepton-G data. The TFFs obtained in the model, which result from the case without the Lepton-G data, are shown in

Transition form factors versus di-lepton invariant mass: (a)

We determine the IP-violating parameters,

Intrinsic parity-violating parameters estimated in the fittings. For

68.3% C.L. | ||||

99.7% C.L. | ||||

68.3% C.L. | ||||

99.7% C.L. |

In this subsection, predictions of the model are given for the TFFs of Dalitz decays, partial widths, and differential decay widths of IP-violating modes. We utilize the parameter set obtained in the previous subsection.

In

Transition form factors versus di-lepton invariant mass: (a)

In

Partial decay widths of IPV decay modes. As model predictions, we give the estimated ranges of

Decay mode | Model |
Model |
Exp. [MeV] |
---|---|---|---|

— | |||

— | |||

— | |||

— | |||

— | |||

— | |||

— |

In

Plots of differential decay widths of

In

Prediction of the model for TFFs: (a)

In

Plots of model prediction: (a) branching ratio of

In the vicinity of the peak region, plots of the TFFs are exhibited for Dalitz decays in

Transition form factors versus di-lepton invariant mass in the vicinity of the resonance regions: (a)

Using Eq. (

The results of the decay widths for

Decay mode | Theory [MeV] | PDG [MeV] |
---|---|---|

With tree-level formulae, one may not explain the width of

The results of the decay widths for

Decay mode | Theory [MeV] | PDG [MeV] |
---|---|---|

In the previous sections, we have included the SU(3)-breaking effects from the intrinsic parity-conserving part. These effects are of the order of

We note that

In our numerical calculation, we fit all four modes and determine the parameters:

With the determined parameters, we reproduce the central values of the PDG decay widths for

The IP-violating phenomena of light hadrons are investigated in a chiral Lagrangian model including vector mesons. We introduced suitable tree-level interaction terms, which include singlet fields of vector mesons and pseudoscalars. Power counting of a superficial degree of divergence enables us to specify the one-loop-order interaction Lagrangian in the presence of the tree-level part. With introduced interactions, the one-loop correction to the self-energies of vector mesons is analyzed. Using the one-loop-corrected mass matrix, we obtained expressions for the physical masses and the mixing matrix of

For pseudoscalars, we took account of the one-loop correction to the mass matrix. The physical states of

On the basis of a framework incorporating octet and singlet fields, the IP-violating operators are introduced within SU(3) invariance. We constructed

Using the introduced IP-violating operators, we obtained analytic formulae for the IP-violating (differential) decay widths. In particular, the widths of

For parameter estimation, we used precise data on the spectrum function for

Numerical analyses of IP-violating decay widths, the TFFs for electromagnetic decays, are carried out in the model. In order to estimate the IP-violating parameters, we utilized the PDG data [

Using the estimated model parameters, we gave model predictions for IP-violating decays. In particular, we found that the electromagnetic TFFs of

Our framework, which includes the

We thank H. Tagawa for helpful discussion. This work is supported by JSPS KAKENHI Grant Numbers JP16H03993 and JP17K05418 (T.M.), and JP24540272, JP26247038, JP15H01037, JP16H00871 and JP16H02189 (H.U.).

Open Access funding: SCOAP

The counterterms are computed with the one-loop correction of the SU(3) singlet pseudoscalar in Ref. [

In

The coefficients of the counterterms:

In this appendix, we show the power-counting rule, which is used to classify the interaction Lagrangian in Eq. (

The first two terms of Eq. (

Substituting Eq. (

The ultraviolet divergence can occur when

The counterterms that subtract the divergence also satisfy the above condition on the number of external lines (

Note that

The coefficients of the counterterms | ||||
---|---|---|---|---|

Next we study the power counting of the interaction terms for singlet vector mesons. In contrast to octet vector mesons, the chiral invariant interaction of the singlet vector meson with an octet pseudoscalar with the first derivative vanishes:

Therefore there is no tree-level interaction for the singlet vector meson. The interaction of the singlet vector meson with the chiral-breaking term,

In this appendix, we study self-energy corrections to

Self-energy corrections to

We treat the mixing in Eq. (

In Eqs. (

In Eqs. (

We write the inverse propagators of vector mesons as

In this section, we show that the metric tensor part of the two-point functions for the

Meanwhile, the diagonalization of the mass matrix leads to

In the above equation, the matrix elements for

Plugging Eqs. (

In this appendix, the radiative correction to charged pseudoscalar masses is discussed. A background field method is used to evaluate the chiral loop correction [

In Eq. (

In Eq. (

Using the transformation in Eq. (

In Eqs. (

One can clarify that the one-loop masses are renormalization-scale invariant. Therefore, we find that the following equation is satisfied:

In this appendix, the radiative correction to pseudoscalar masses is evaluated for neutral particles. As in the previous section, the background field method is used to evaluate the quantum correction. We consider the framework in which the chiral octet loop correction is taken into account. The masses and kinetic terms of pseudoscalars in the one-loop-corrected effective Lagrangian are written as

In Eq. (

The matrix in Eq. (

The transformation in Eq. (

In the above mass matrix, the one-loop-corrected masses are denoted by primes. We ignore quadratic terms with respect to the small quantities so that the one-loop-corrected masses in Eq. (

Comparing Eqs. (

Using Eqs. (

Provided that the physical masses

In this appendix, the one-loop-corrected decay constants are analyzed for charged pseudoscalars. The decay constants are defined by parametrizing the matrix elements as

One can find that one-loop-corrected decay constants are related to the wavefunction renormalization in Eq. (

Equation (

In this appendix, we give the expression for the WZW term. As suggested in Ref. [

The expressions given in Eqs. (

The vector form factors for

Since, under CP transformation, the charged currents are related to each other as follows:

In the isospin limit, we also obtain the relations

Using Eqs. (

The contribution to the form factors is divided into two parts. One of them comes from the 1PI diagrams and the other comes from the diagrams that include the propagator of the

Each contribution to the form factors is given below (see also Ref. [

We obtain