Analysis of Dalitz decays with intrinsic parity violating interactions in resonance chiral perturbation theory

Observables of light hadron decays are analyzed in a model of chiral Lagrangian which includes resonance fields of vector mesons. In particular, transition form factors are investigated for Dalitz decays of $V\to Pl^+l^-$ and $P\to \gamma l^+l^-$ $(V=1^-, P=0^-)$. Moreover, the differential decay width of $P\to \pi^+\pi^-\gamma$ and the partial widths of $P\to2\gamma, V\to P\gamma, \eta^\prime\to V\gamma, \phi(1020)\to\omega(782)\pi^0$ and $V\to 3P$ are also calculated. In this study, we consider a model which contains octet and singlet fields as representation of $SU(3)$. As an extension of chiral perturbation theory, we include 1-loop ordered interaction terms. For both pseudoscalar and vector meson, we evaluate mixing matrices in which isospin/$SU(3)$ breaking is taken into account. Furthermore, intrinsic parity violating interactions are considered with singlet fields. For parameter estimation, we carry out $\chi^2$ fittings in which a spectral function of $\tau$ decays, vector meson masses, decay widths of $V\to P\gamma$ and transition form factor of $V\to Pl^+l^-$ are utilized as input data. Using the estimated parameter region in the model, we give predictions for decay widths and transition form factors of intrinsic parity violating decays. As further model predictions, we calculate the transition form factors of $\phi(1020)\to \pi^0l^+l^-$ and $\eta^\prime\to\gamma l^+l^-$ in the vicinity of resonance regions, taking account of the contribution for intermediate $\rho(770)$ and $\omega(782)$.


I. INTRODUCTION
Decays of light hadrons play a crucial role to investigate low-energy behavior of quantum chromodynamics (QCD), and are measured extensively in experiments. In particular, Dalitz decays such as P → γl + l − and V → P l + l − provide rich resources as hadronic observables.
Using these experimental data, we can test the validity of QCD effective theories which include resonances of vector meson. As a recent result, high-precision data of transition form factors (TFFs) for ω → π 0 µ + µ − and η → γµ + µ − are measured by the NA60 collaboration [1] in proton-nucleus (p-A) collisions. Moreover, the measurement of the branching ratio and the TFF of η → γe + e − has been carried out by the BES III collaboration [2].
In order to describe dynamics of light hadrons, we adopt a model of chiral Lagrangian which includes vector mesons. In this model, chiral octets and singlets are introduced as representation of SU (3). There are some models [3][4][5], which incorporate vector mesons and the other resonances. In this study, we develop the framework so that one can include chiral correction to processes in which vector mesons and/or pseudoscalars are involved. On the basis of power counting of superficial degree of divergence, 1-loop order counter terms, which correspond to O(p 4 ), are introduced [6]. Finite parts of the coefficients of the counter terms are estimated in the fitting procedure with the experimental observables in the same way as the chiral perturbation theory (ChPT). Once those parameters are determined, one can predict other observables such as TFFs and decay widths.
This effective dynamics of hadrons is applicable to a variety of phenomena, e.g., hadronic τ decays. As experimental results, a spectral function of τ − → π 0 π − ν decay is measured in the experiments [7][8][9]. As for decays including kaons, a mass spectrum of τ − → K − π 0 ν is observed in the BaBar experiment [10] while one of τ − → K S π − ν is measured in the Belle experiment [11]. In Ref. [12,13], the branching ratios of τ decays including η are reported.
As theoretical study, the spectral function of τ − → K S π − ν decay is fitted with a resonance field of K * (892) − [6]. The review for τ decays is given in Ref. [14].
For vector mesons, we calculate quantum correction to self-energies to obtain a 1-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 as resonance fields of ρ(770), ω(782) and φ(1020). In our formulation, SU(3)/isospin breaking contribution in the self-energies is taken into account in the mixing matrix for vector mesons. We also consider kinetic mixing of neutral vector mesons, which arises from 1-loop correction to the self-energies. Including such mixing contribution, we obtain analytic formulae of the widths for V → P P decays.
Furthermore, we consider mixing between vector meson and photon, which also comes from 1-loop correction to self-energies. As analyzed explicitly in this paper, we find that the V −γ mixing plays a crucial role in processes such as radiative decays. For pseudoscalars, we also take account of quantum correction to self-energies. We use parametrization in which the 1loop correction to mass matrix elements is accounted. The mixing matrix for pseudoscalars is determined so as to diagonalize the 1-loop corrected mass matrix. Using this formulation, we consider SU(3)/isospin breaking in the 3 × 3 mixing matrix for π 0 , η and η .
In processes such as P → 2γ and radiative decays of V → P γ, intrinsic parity (IP) [15] is violated. It is well-known that intrinsic parity violation (IPV) in models with vector mesons is categorized as two types: The first one is the Wess-Zumino-Witten (WZW) term, which results from quantum anomaly of SU(3) symmetry [16,17]. The second one comes from the presence of resonance fields for vector mesons, as originally suggested in the framework of hidden local symmetry (HLS) [18,19].
For IP violating interactions in the model, we introduce operators including SU(3) singlet fields, in addition to ones suggested in Refs. [18][19][20]. As shown in our numerical result, inclusion of the singlet-induced IP violating operators plays an important role in the framework of the octet+singlet scheme, typically for η → 2γ.
Using the introduced operators, we write formulae of the IP violating decays of hadrons.
In particular, the expressions of (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 anti-symmetric tensor, one needs the fourth derivatives on the chiral fields so that the Lagrangian is Lorentz invariant. It is O(p 4 ) contribution.
In contrast to IP conserving part, SU(3) breaking effect for IPV interactions is O(p 4 m 2 π ) and it is one loop effect. To this accuracy, we need to include SU(3) breaking both in IP conserving part and in IPV part. In the first part of our paper, we include SU(3) breaking effect for IP conserving part up to one loop order without introducing SU(3) breaking for IPV interaction. In the last part of the paper, we incorporate SU(3) breaking interactions for IPV part in the tree level.
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. [20,21], with SU(3) breaking effect in IP violating interactions. Radiative decays are analyzed with V P γ vertices in Ref. [22]. Moreover, numerical analyses of the TFFs are given in Refs. [23][24][25][26][27][28].
In our analysis, χ 2 fittings are carried out to estimate parameters in the model. As input data in the fittings, the spectral function in τ − → K S π − ν decay measured by the Belle collaboration [11] are used. Furthermore, we also utilize the data of masses for the vector mesons, which are precisely determined in experiments. For parameter estimation of coefficients of IP violating operators, the data of partial widths for radiative decays and TFFs of V → P l + l − decays are used. As shown in the numerical result, one can find a parameter region which is consistent with experimental data for the TFFs.
As discussed in the latter part of this paper, the TFFs for φ → π 0 l + l − and η → γl + l − have a peak region around which di-lepton invariant mass is close to the pole of ω.
Remaining part of this paper is organized as follows: In Sec. II, the model is introduced and 1-loop ordered interactions are given with SU(3) octets and singlets. The quantum correction to self-energies of vector mesons are also shown. Using the 1-loop corrected propagators, we write the width of V → P P decay, including the contribution of kinetic mixing. The V − γ vertex, which arises from 1-loop order interactions, is also shown. The mixing matrix for π 0 , η and η , in which 1-loop correction is accounted, is introduced. In Sec.
III, IP violating interaction terms are given. The formulae of decay widths for IP violating modes are explicitly shown. In Sec. IV, the results of numerical analysis are presented. We show the fitting result of the invariant mass distribution of τ decay. Physical masses of vector mesons are also fitted in this section. Moreover, we estimate coefficients of the IP violating operators, via experimental data of hadron decays. We give the model prediction for decay widths, TFFs and differential decay widths for IP violating decays. In Sec. V, SU(3) breaking effect of IPV interaction is studied. Finally, Sec. VI is devoted to summary and discussion.

II. THE MODEL WITH SU(3) OCTETS AND SINGLETS
In this section, we introduce a model of chiral Lagrangian with vector mesons [6]. In this paper, we extend the previous one so that it includes φ meson and electromagnetic mass of pseudoscalar mesons as follows, where The Lagrangian is divided into three parts in Eq. (1), which consist of the parts of pseudoscalars, vector mesons, and 1-loop order counter terms. As fields of pseudoscalar, the octet matrix and the singlet field are contained in Eq. (2). η 0 is U(1) A pseudoscalar and its mass is given as M 00 . The term denoted as CTrQU QU † in Eq. (2) is the electromagnetic correction to ChPT. This term describes the effect of virtual photon [29], and affects the mass of the charged pseudoscalar. Vector mesons are introduced as SU(3) octet and singlet in Eq. (3). Vector meson matrix for octet is denoted by V µ , and its mass is given as M V , while the field φ 0 µ denotes SU(3) singlet vector meson. In the following, we present how 1-loop counter terms given as L c are introduced for chiral Lagrangian with vector mesons and pseudoscalar singlet. The form of 1-loop counter terms depends on the tree-level Lagrangian and is obtained with power counting of the superficial degree of divergence in the loop calculation. The tree-level Lagrangian is constructed based on the expansion with respect to derivatives and chiral SU(3) breaking. The Lagrangian includes either the second derivatives or an insertion of chiral SU(3) breaking. The interaction Lagrangian which satisfies such criteria is extracted from Eqs. (2,3), Note that L 0 does not include the parts which are written only with vector mesons and singlet pseudoscalars. With L 0 , the divergent parts of the 1-loop correction is extracted and the counter terms are given in Eq. (A1). As proven in App. B, the counter terms satisfy the power counting rule, which enables us to specify the structure of them. Based on the discussion, in Eq. (1), we have included a singlet-octets vector mesons mixing term as a finite counter term.
The counter terms for the self-energy for vector mesons and V − γ mixing can be summarized as the effective counter terms [6], where all the field strength are Abelian part defined by, V and C i (i = 1, 2, 4) are renormalization constants and they are written in terms of the coefficients in Eq. (A1), The coefficients of the effective counter terms in Eq. (15) include the divergent part and the finite part. The finite parts are denoted with suffix (r). Both divergent and finite parts of In Eq. (16), (g ρππ ) tree denotes a tree-level vertex for ρππ coupling. We also define the 1-loop ordered ρππ coupling, A. Neutral vector meson In this subsection, we diagonalize the mass matrix for neutral vector mesons and obtain the mass eigenstates which correspond to (ρ, ω, φ). The mixing matrix between (ρ 0 , ω 8 , φ 0 ) and mass eigenstates determines the interaction among the physical states. The inverse propagator for the vector mesons is, where V I denotes the eigenstate for the mass matrix, The mixing matrix O V relates the mass eigenstates to SU(3) basis in the following, In Eq. (18), where ∆M r K + K 0 = M r K + − M r K 0 . In Eqs. (22,24), Z r V denotes the coefficient of kinetic term of octet vector meson defined as 1 + Z r(1) V . M r P are the loop functions of vector mesons, where µ is a renormalization scale. In the numerical analysis, we fix it as µ = m K * + . The elements in the mass matrix (20) are given by, with We calculate the mass of the neutral vector mesons, ρ, ω and φ. The first term in Eq. (20) is diagonalized as, where δB V (Q 2 ) is a 3 × 3 matrix, The propagator for the neutral vector mesons is denoted as, In the following, we expand the propagator in Eq. (38) with respect to the off-diagonal parts The amplitude for V → P π is written as the sum of the tree-level amplitude and 1-loop correction, whereĝ V P π denotes the tree-level coupling and ∆g V P π is 1-loop correction. We denote q = p P − p π . Firstly, we consider the case that V and P consist of the same quark flavor contents. For V = K * +(0) and P = K +(0) , they are given as, In the isospin limit, one can find the relations,ĝ K * + K 0 π + = √ 2ĝ K * + K + π 0 and ∆g K * + K 0 π + = √ 2∆g K * + K + π 0 , are satisfied. Therefore, the decay width of K * + → K 0 π + is two times larger than that of K * + → K + π 0 . For V = ρ + and P = π + , the couplings are given as, In the above calculation, the isospin breaking effect is not taken into account. Using the 1-loop corrected couplings, we obtain the partial decay width for V → P π, Using the isospin relation of K * decays, one can find, In this subsection, we study the decay width of V → P P , including the effect of kinetic mixing. First, V → π + π − (V = ω, φ) is investigated. Since the two pions in the final state are p wave and form an isotriplet, the decays of ω and φ occur due to isospin breaking.
There are two major contributions to the isospin breaking amplitude. The first one is due to a partial component of isotriplet state (ρ 0 ) in the mass eigenstate of ω(φ). This effect is incorporated as the mixing matrix of the neutral vector mesons. Another contribution comes from the non-vanishing decay amplitude for isosinglet due to isospin breaking. In our model, incomplete cancellation between 1-loop diagram of charged kaon and one of neutral kaon leads to such contribution. The decay amplitudes for octet states ρ 0 , ω 8 and a singlet state φ 0 are given as, The effective Lagrangian for the singlet and octet states is given as, where coupling constants are defined 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 To evaluate the partial decay width for V I → P P , kinetic mixing in the decay process, i.e., V I → V J → P P , should be taken into account. Using the renormalized fields, we can express the kinetic mixing terms, In the above Lagrangian, we set the wave function renormalization Z I = 1, since δB V IJ (I = J) is 1-loop order contribution.
The T -matrix elements for V → P P decays are, . Including the contribution of kinetic mixing, the effective couplings in Eq. (61) are given as, Ignoring the isospin breaking effect, we note that T [ρ 0 → π + π − ] is the same as the amplitude of ρ + → π + π 0 which was studied in the previous subsection. In Eqs. (62-64), the second terms denote the kinetic mixing effects for V I → V J → P P decay process and M J (J = 1, 2, 3) is the eigenvalue for the vector meson mass matrix, which differs from the physical masses, m ρ , m ω or m φ . However, within the accuracy, one can set M J = m J since their difference arises from only the wavefunction renormalization. One can obtain the partial widths for V → P P decay, where g eff V P P is the coupling associated with Eqs. (62-64).

C. Mixing between photon and vector meson
In this subsection, the mixing between photon and vector mesons is analyzed. The conversion vertex is denoted as, In the basis of SU (3), the two-point functions in the l.h.s. of Eq. (68) are given as, One can find that g µν part in Eq. (70) is related to the matrix elements of the 1-loop corrected neutral vector meson masses in Eqs. (26,27,30), One can write the two-point functions in Eq. (69), The derivation of Eq. (78) is shown in App. D. Thus, the mixing vertices for V − γ in Eq.

D. Pseudoscalar
In this subsection, the structures of a mixing matrix and decay constants for pseudoscalars are given. We take account of 1-loop correction to both mixing and the decay constants.
The basis for an SU(3) eigenstate is written in terms of mass eigenstates as, where O denotes an orthogonal matrix which diagonalizes a mass matrix of pseudoscalars. √ Z in Eq. (81) is a matrix which canonically rescales 1-loop corrected kinetic terms for pseudoscalars. The result of 1-loop correction to the mass terms for charged particles is summarized in App. E, while one to the mass matrix for neutral particles is shown in App.
For decay constants of π + and K + , we also consider the 1-loop quantum correction. As stated in App. G, the ratio of a pion decay constant to one for kaon is determined with wave function renormalization of pseudoscalars [6], where c is defined as,

III. INTRINSIC PARITY VIOLATION
In this section, we discuss IPV in the model. As well as ChPT, quantum anomaly of chiral symmetry causes an IP violating interaction. The expression of the WZW term is given in Eq. (H1). In addition to this operator, IP violating interaction terms, which come from the resonance field of vector mesons, are introduced. Subsequently, we write the formula of widths, TFFs, differential widths for IP violating decays.

A. Intrinsic parity violating operators with vector mesons
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 invariance of SU(3) symmetry. Imposing the charge conjugation (C) symmetry, one can obtain the operators in the model, where 0123 = − 0123 = +1 and, In Eqs. (86-88), L 1−3 are introduced in Refs. [18,19] while L 4 is considered in Ref. [20]. We introduced L 5−10 , which are written with singlets of η 0 or φ 0 . In Eqs. (86-95), we required that the operators should be Hermite.
In contrast to our work, the singlet fields are contained as a component of chiral nonet matrix in Ref. [20] and L i (i = 5 − 10) is not included in that work. Chiral SU (3)  The IP violating interactions in our model are denoted as, In Eq. (102), the coefficients of the operators, c IP i (i = 1−10), are free parameters. As carried out in Sec. IV, these parameters are estimated via experimental data which are sensitive to IPV. In the following subsections, with the interaction in Eq. (102), the formulae of IP violating decay modes are explicitly written.  vector meson are shown as, where i, I and J run from 1 to 3 and c − 34 = c IP 3 − c IP 4 . In Eqs. (103-104, 106), fields of mass eigenstate are denoted for vector mesons as (V 1 , V 2 , V 3 ) = (ρ, ω, φ) and for pseudoscalars as (P 1 , P 2 , P 3 ) = (π 0 , η, η ), respectively. The coefficients, χ iI in Eq. (103), describes the vertex of each component, e.g., χ 11 for ρπ 0 γ vertex and χ 12 for ωπ 0 γ vertex. Note that the vertex coefficient of ρωπ 0 is proportional to θ 121 + θ 112 in Eq. (104) since an operator with I = 1, J = 2 and another operator with I = 2, J = 1 give the same amplitude. The coefficients of the vertices in Eqs. (103-106) are given as, Vector mesons can decay into P γ directly with the operator in Eq. (103). V P γ vertex is absent in Ref. [20] since the relation c IP 3 = c IP 4 is adopted. Meanwhile, IP violating V V P operator in Eq. (104) also causes V → P γ with the V − γ conversion vertex in Eq. (79).
The notation of propagators for neutral vector meson is given as, where J = 1, 2, 3 is assigned with the propagator of ρ, ω and φ, respectively. In the calculation of the V − γ conversion decay V → P V * → P γ, the term proportional to D J L (0) vanishes since the momentum product Q µ Q ν is eliminated when multiplied with antisymmetric tensor. Consequently, the conversion process V → P V * → P γ is proportional to the contribution from the metric tensor part of intermediate vector mesons. With Eq. (78), one can find that the following relation is satisfied, Decay amplitudes are obtained from the operators in Eqs. (103-106) as, where c + 34 = c IP 3 + c IP 4 and c 69 = 2c IP 6 + gc IP 9 . The coefficient of neutral meson decay amplitude denoted as X iI in Eq. (119) includes the factor ( Eqs. (122 ,123), we assume that the wave function renormalization of charged vector meson is equal to one for neutral vector meson, i.e., Z ρ + = Z ρ and √ Z K * = √ Z K * + , which is valid in the isospin limit. One can write the partial decay width of V → P γ and P → V γ with X's in Eqs. (119, 122-124) as, The pseudoscalar decay width in Eq. (129) is analogous to one for V → P γ given in Eqs.
(125-128) and its coefficient is different by a factor 1/3 which comes from spin average of vector meson. One can find that Γ[V I → P i γ] and Γ[ρ + → π + γ] in Eqs. (125-126) provide the relation, In the above relation, the ratio of the effective coupling for V I → P i γ to one for ρ + → π + γ is written in terms experimental data on r.h.s. Using Eqs. (125-126), we can rewrite l.h.s.
in Eq. (130), In the above relation, the effective coupling is written in terms of model parameters. We use the relations in Eqs. (130, 131) for the numerical analysis of χ 2 fitting.
In this subsection, an IP violating process of φ → ωπ 0 is analyzed. The contributing operator to φ → ωπ 0 is given in Eq. (104), and the diagram is shown in Fig. III transition amplitude of φ → ωπ 0 is written as, The contribution coming from V − γ conversion is negligible since it gives rise to O(α) correction. In Eq. (133), the factor of wave function renormalization of external vectors is included. Thus, the partial decay width of φ → ωπ 0 is, In this subsection, we evaluate partial decay widths of the IP violating process given as P i → 2γ. The IP violating interaction terms yield contribution to P γγ vertex as, where the first term proportional to 1/8π 2 in Eq. (136) implies the contribution from the WZW term. Diagrams of the decay of P i → 2γ are given in Fig. III.3. With the operators in Eqs. (79, 103, 135), the transition amplitude of P i → 2γ is written as, where c 6−9−10 = c IP 6 /g 2 + c IP 9 + 4c IP 10 . The partial decay width of P i → 2γ is given as, In this subsection, a form factor for IP violating modes P i → γl + l − is obtained. the differential decay width is written in terms of the TFF as, where s denotes the squared invariant mass in di-lepton system while θ indicates an angle between P i and l + in the di-lepton rest frame. The model prediction for the TFF is, (143) In this subsection, a form factor for IP violating electromagnetic decays for neutral vector mesons is analyzed. Contributing diagrams are exhibited in Fig. III.5. The differential decay width for V → P l + l − is written in terms of the TFF in the following form, where θ is an angle between V I and l + in the di-lepton rest frame. As the model prediction, the TFF is obtained as, The TFF in the above equation are normalized as unity in the limit where virtual photon goes on-shell. 6. V → P π + π − In this subsection, partial decay widths for V → P π + π − are analyzed. Interaction terms for the process are, where c 123 = c − 12 + 2c IP 3 . The diagrams for the decay of V I → P i π + π − are given in Fig. III.6. Propagators for ρ ± are formulated in the following form as, The transition amplitude is given as, where s +0 , s −0 and s +− are squared invariant masses for the π + P i , π − P i and π + π − system, respectively. s +− is kinematically related with the other variables as s +− = m 2 The formula of the partial decay width is obtained as, where θ(H) denotes step function, and the integral regions are common for s +0 and s −0 .
In this subsection, differential decay widths for P → π + π − γ are calculated. The diagrams contributing to this process are given in Fig. III.7. The transition amplitude for the process Diagrams contributing to the decay width for P i → π + π − γ.
Using the above equations, one can obtain the differential decay width, where s denotes the squared invariant mass in π + π − system and θ implies the angle between π + and γ in the rest frame of π + π − .

IV. NUMERICAL ANALYSIS
In this section, phenomenological analysis is carried out in the model. In the following subsection, we perform χ 2 fittings in order to estimate the parameters in the model. As input data in the fittings, the following data are utilized: (1) the spectral function of τ decay, (2) the masses of vector mesons and (3) the IP violating decay widths, the masses of pseudoscalars and the TFFs of V → P l + l − . Subsequently, using the parameters estimated from the aforementioned observables, we give the prediction of the model. Specifically, the results are presented for Dalitz distributions and partial decay widths of IP violating modes.
In order to carry out the analysis, the following points are addressed: • For the parameter c defined in Eq. (85), we take f = f π .
A. Parameter fit In this subsection, we estimate parameters in the model with the decay distribution for To evaluate the decay distribution, we use the procedure similar to the method in Ref. [6]. Throughout the analysis, we take isospin limit in the decay distribution.
The differential branching fraction for KP ν (P = π, η) is given as, where p K is the momentum of K in the hadronic center of mass (CM) frame. The vector and scalar form factors are written in App. I. In order to compare the model prediction with the Belle data, we use the method in Ref. [6]. Including the overall normalization, the differential width in Eq. (160) is rewritten as, where N tot denotes the observed number of events for τ decay while 11.5 MeV indicates the width of bins in the Belle experiment. We carry out the χ 2 fitting based on Eq. (161), which represents the expected number of events in the model.
In this paper, we take the tree-level pion decay constant, f , as a parameter. Since the effect of the K * resonance is important in the decay mode, we choose the mass and the decay width of K * meson as fitting parameters. Additionally, the octet vector meson mass and the finite parts of 1-loop ordered coefficients, K r 1 + K r 2 and L r 9 , are also free parameters. To summarize, (M V , K r 1 + K r 2 , L r 9 , M K * , Γ K * , f ) are the relevant fitting parameters in this mode. These six parameters are estimated from 90 bins of the data in the region M K + M π ≤ Q 2 ≤ 1665 MeV. As a result of fitting, the parameters are determined as, where the obtained χ 2 min /n.d.f. is 147.1/84. The correlation matrix of (M V , K r 1 +K r 2 , L r Since the tree-level K * Kπ coupling is proportional to   In Table I, we show other parameters which are also determined through Eqs. (162, 163).
In the following, we clarify how the parameters in Table I are determined. In order to obtain the ρππ coupling, we note that the decay width of K * is given by the imaginary part of the self-energy [6], where ν P π (Q 2 ) is defined in Eq. (52). Solving Eq. (164) with respect to g ρππ , one can fix the ρππ coupling since (M K * , Γ K * ) are determined from the fitting result in Eq. (162). Moreover, g is also obtained from the definition of the ρππ coupling in Eq. (17). Since the large error of M V propagates to g, the error of g increases. In Table I we also give the value of (g ρππ ) tree in Eq. (16). One can find that (g ρππ ) tree and g ρππ are deviated from each other. This is because the tree-level decay constant denoted as f given in Eq. (162) is deviated from PDG value, f π = 92.2 MeV. In order to calculate L r 4 and L r 5 , we use the following pion and kaon decay constants [30] with obtained f , where in the above expressions, f represents the tree-level parameter given in Eq. (162).
The coefficients of 1-loop ordered interaction, (C r 3 − 4C r 4 , C r 5 ), are also determined from the procedure similar to one of Ref. [6]. Wavefunction renormalizations for π 3 and η 8 are calculated from Eq. (F9) and Eq. (F10), respectively. If one fixes the parameters as the best fit values in Eq. (162), the wavefunction renormalizations are, Hereafter, the values in Eq. (167) are referred to as best fit values of the wavefunction renormalizations for pseudoscalars.
For the ratio of decay constants of pseudoscalars, we verify whether the model prediction the PDG data [32] are, .
In the above result, one can find that the model prediction is slightly deviated from the case of the tree-level ρππ coupling in Eq. (84). This is because the estimated value of f is deviated from the experimental value of f π . However, up to the 99.7% confidence interval of the model prediction, it is shown that the central value of the PDG data [32] is included.

Mass and width of vector mesons
In this subsection, we explain how the parameters, C r 1 , C r 2 , Z r V ,ĝ 1V and M 0V are fixed, and evaluate the vector meson mass, the renormalization constant and the decay width.
At first, we consider the off-diagonal elements of V µ , i.e., ρ + , K * + and K * 0 to obtain the parameters, C r 1 , C r 2 and Z r V . We define the masses of ρ + and K * + mesons as the momentumsquared for which real parts of inverse propagators vanish, where δA ρ + and δA K * + are shown in Eqs. (C15) and (C19), respectively. Solving the above equations, we have C r 1 and C r 2 , Imposing the condition for the residue of the vector meson propagator, ResD(Q 2 = m 2 K * + ) = 1, we have Since ∆A V only depends on g ρππ , C r 1 and Z r V can be fixed by g ρππ . On the other hand, C r 2 is related to two parameters, g ρππ and M V .
The decay widths are given by the imaginary part of the inverse propagators, where I = ρ + , K * + , K * 0 , where ∆A I is defined in Eqs. (C16, C18, C20). The renormalization constants Z i are estimated as follows, In the following, we determine the parametersĝ 1V Using the above parameters, we have the orthogonal matrix O V which diagonalizes vector where ω − φ mixing angle is (40.59 ± 0.04) • . The wave function renormalization of the neutral vector meson and the eigenvalues for the mass matrix are obtained as follows,

Intrinsic parity violating decays
In this subsection, we estimate model parameters by using the IP violating observables for light hadrons. As input data of χ 2 fittings, experimental data of decay widths and Dalitz distributions are used. We also utilize experimental values of masses for pseudoscalars to estimate parameters in the mass matrix.
The widths of radiative decays, proportional to the IP violating parameter (gc + 34 ) 2 . In order to estimate this parameter, we consider the following statistic, where δΓ PDG denotes experimental errors of the widths. As a result of the fitting, we find that the minimum of Eq. (180) is χ 2 min /d.o.f. = 1.08/1, which results in the estimated parameter as, In this fitting, the sign of c + 34 is not fixed since the widths in Eqs. (126-128) depend on square of this parameter. In Table III, the widths calculated in the model are compared with the PDG values [32]. The model prediction for Γ[K * + → K + γ] is also given in Table III. For the PDG value [32] of Γ[K * + → K + γ], we adopt the full width of K * + obtained from tau decays.
One finds 3.4σ discrepancy between the model prediction and the experimental value of the width for K * + → K + γ. Since the K * 0 K 0 γ coupling in Eq. (124) is two times larger than one for K * + K + γ in Eq. (123), the widths are related as Γ[ However, this relation is not valid for the present PDG values [32] so that the deviation arises.
For parameter estimation, we use observables for pseudoscalars. In particular, the PDG data [32] for masses of π 0 , η and decay widths of P → 2γ (P = π 0 , η, η ) are adopted. In order to constrain parameters in the model, we consider the following system of equations, where the left-handed sides in these equations denote the model expressions. Solution to Eqs.
(182-186) leads to estimated values for the parameters given as (c 6−9−10 ,ĝ 2p , ∆ EM , M 2 88 , M 2 80 ). This procedure of solving the equations is carried out in the following way: provided that the PDG data [32] obey Gaussian distributions, the right-handed sides in Eqs. (182)(183)(184)(185)(186) are generated as Gaussian data. For ( Z π 1 , Z π 2 ), we use the parameter list obtained from the fitting of tau decays, which is summarized in Table I . (187) In the following analysis, the parameter values in Eq. (189) are referred to as best fit values for the mixing matrix elements.
Here, we discuss a case in which singlet-induced contribution is absent. If one takes the limit c 6−9−10 → 0, the partial width of η becomes Γ[η → 2γ] = 7 × 10 −5 MeV. This value is much smaller than the experimental data, Γ PDG [η → 2γ] = (4.4 ± 0.3) × 10 −3 MeV. Hence, one notices that the presence of singlet-induced IP violation is necessary in the framework of the singlet+octet scheme.
For parameter estimation of the IP violating parameters, the ratio of the effective coupling for V P γ to one for ρ + π + γ in Eq. (131) are compared with experimental values. Model parameters are estimated from the following statistic, The experimental data used in the above χ 2 are extracted from PDG [32] where the correlation coefficient of these parameters is 0.12. Predictions for effective coupling ratios of Γ[V I → P i γ] to Γ[ρ + → π + γ] for (i, I) = (1, 1), (1, 3), (2, 1), (2, 3) are given in Table   V. Furthermore, the prediction for the decay widths of V I → P i γ are shown in Table VI. is shown while the model predictions are given for (i, I) = (1, 1), (1, 3), (2, 1), (2,3). For comparison, the experimental data extracted from the PDG data [32] are also shown. In the fourth column, the model prediction in the isospin limit is displayed with wave function renormalizations set as unity.
Mixing angles for vector meson are defined as cos θ 08

Ratio
Model PDG Model in the isospin limit    [32] are also shown.
we also fit χ 2 in the case without the Lepton-G data [34]. This fitting analysis leads to , which is a slightly improved result. In this case, we find that the best fit values and the errors of (c IP 3 , c IP 5 , c IP 6 , c IP 7 ) are almost identical to ones in the case with the Lepton-G data. For each fitting, χ 2 min /d.o.f., corresponding p-values and the estimated parameters are summarized in Table VII. As a result In Table VII, one can find that the errors of c IP 6 and c IP 7 are large. This is because the contributions of c IP 6 and c IP 7 are suppressed by either isospin breaking or the η − η mixing angle for |F ωπ 0 | 2 , |F φπ 0 | 2 and |F φη | 2 in Eq. (146). To improve the precisions of c IP 6 and c IP 7 , the experimental errors of the TFFs should be reduced, especially for |F φη | 2 .
In the following analysis in this paper, we adopt parameter sets which 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 Fig. IV.2. One can see that best fit curves for c + 34 < 0 and c + 34 > 0 are slightly deviated from one another in φ → ηl + l − whereas the two predictions mostly overlap with each other for ω → π 0 l + l − and φ → π 0 l + l − .

B. Model prediction
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 Fig. IV.3, the model predictions for P i → γl + l − (i = 1, 2, 3) are given. We show the result for the two cases, c + 34 < 0 and c + 34 > 0, respectively. For c + 34 > 0, one can find a discrepancy between the model prediction and the precise data obtained by the NA60 collaboration [1]. Thus, we do not give a further result of analysis for the case of c + 34 > 0 since this case is disfavored.
In The substantial error of φ → ωπ 0 comes from gθ 123 in Eq. (133), which is proportional to a φωπ 0 coupling. Using the best-fit mixing matrix for mesons, one can obtain the coupling, where each term has a comparable contribution to gθ 123 . In Eq. (194), the errors of g, c IP 6 and c IP 7 in Tables I and VII Table I and c + 34 < 0, one can obtain c + 34 = (−1.5 ± 0.4) × 10 −2 , which leads to uncertainty of Γ[η → π + π − γ].
For comparison, the data measured by the WASA-at-COSY collaboration [39], which are originally given in arbitrary unit, are also shown for η → π + π − γ. For η → π + π − γ in (a) and In (c), one can find a resonance region around E γ ∼ 160MeV. This is because the photon energy in the rest frame of η is related to π + π − invariant mass as E γ = (M 2 η − s +− )/2M η , which indicates that E γ = 164.9MeV(159.1MeV) corresponds to the pole which arises from intermediate ρ (ω).
In Fig. IV.5, we present the numerical result for the Dalitz distributions of V I → P i l + l − for (i, I) = (1, 1), (2, 1), (2,2), (3,3). Since these modes are not measured yet, it is expected that one can test the validity of the model via future experiments.
In Fig. IV.6, predictions for a branching ratio for ρ 0 → π 0 π + π − and decay widths of V I → π 0 π + π − (I = 2, 3) are shown. Varying the value of gc 123 , we estimate error bands of the model prediction. For simplicity, we do not account uncertainty which arises from parameters in the vector meson propagators in Eq. (41). We find that if one fixes gc 123 ∼ 0.35, the predictions for V I → π 0 π + π − (I = 1, 2, 3) are consistent with the PDG data [32].
In the vicinity of the peak region, plots of the TFFs are exhibited for Dalitz decays in Fig. IV.7. The partial contributions from ρ, ω, and interference between ρ and ω are also indicated. In (a) and (c), the predictions in 68.3% C.L. are shown for TFFs of φ → π 0 l + l − and η → γl + l − , respectively. In (b) and (d), the best fit predictions, in which the model parameters are fixed, are given for the two modes. For both φ → π 0 l + l − and η → γl + l − , we find that the contribution from ω pole is dominant around the region of resonance. It is shown that the partial contribution of interference between ρ and ω is not negligible. In particular, for (b), one can see that the contribution of the interference is sizable.
Using Eq. (67), we obtain the decay widths for ω → ππ, φ → K + K − and φ → K 0K 0 which are shown in Table X. We should note that the leading contribution of the decay is a one-loop level and isospin breaking amplitude. About the φ → K + K − and φ → K 0K 0 , they are smaller than the experimental values. However the discrepancy depends on the choice of f π /f K and its deviation from unity leads to two loop order effect. If the ratio is modified properly, one can obtain theoretical predictions which are in good agreement with the experimental results. We also note that the ratio of the decay widths of φ → K + K − and φ → K 0K 0 deviates from unity for both theoretical prediction and experimental result.
This implies the presence of the isospin breaking contribution. We note the ratio of the two decay widths is in good agreement between theory and experiment: With tree-level formulae, one may not explain the width of K * and ρ simultaneously, while the 1-loop formulae in Eqs. (51, 53) can reproduce both of them within errors. In are shown. The 1-loop corrected formulae include the parameters in Table I so that these parameters lead to the sizable errors in the 1-loop prediction for the widths which are given in Table XI.      10 000  and γ in the rest frame of π + π − for decays of η and η , respectively. For comparison, the data measured by the WASA-at-COSY collaboration [39] are shown as red circles in (a) and (b). For both (a) and (b), the vertical axis on the left side denotes the physical differential width while one on the right side shows arbitrary unit. See the text for a detailed explanation of units in which the differential widths of η → π + π − γ are calculated.

V. SU(3) BREAKING EFFECT IN IPV INTERACTION
In the previous sections, we include the SU(3) breaking effects from the intrinsic parity conserving part. These effects are order of O(p 4 m 2 π ). The full SU(3) breaking effect for IPV processes up to this order comes from one loop diagrams and also from SU(3) breaking IPV vertex. The latter interactions are studied in [20]. Below we study the SU(3) breaking effect from the IPV vertex. We focus on the processes ρ → πγ and K * → Kγ and show how these terms improve the predictions compared with those without SU(3) breaking terms.
One loop corrections and the renormalization of the divergence is beyond the scope of this paper. We consider the following SU(3) breaking IPV interaction terms whereˆ are spurion field for SU(3) breaking, ∆L 3 and ∆L 4 are newly introduced and the others are studied in [20]. Below we assume that is proportional to current quark mass matrix diag.(m u , m d , m s ). Using the Lagrangian, one can compute the effective interactions for ρ → πγ and K * → Kγ. To obtain the interactions, not only the direct interaction of V → P γ but also the contribution from V → V 0 P → γP is included.
where m s = ms mu and m d = m d mu . gĉ + 34 , a, b and c are defined as, We note that c contributes to all four modes in the same manner and the strength of the contribution is proportional to Tr(QM ) = 2mu−(m d +ms) 3 .
In our numerical calculation, we fit all four modes and determine the parameters.
With the determined parameters, we reproduce the central values of PDG decay widths for K * → Kγ and ρ → πγ shown in Table III [18][19][20]. In order to realize the experimental data in the framework including the singlets and octets, we found that the singlet-induced operators play an important role; if L i (i = 5 − 10) were absent in the model, Γ[η → 2γ] would become much smaller than the observed value in the experiments.
Using the introduced IP violating operators, we obtained the analytic formulae for the IP violating (differential) decay widths. In particular, the widths of P → V γ, V → P γ, φ → ωγ, P → 2γ, V → P π + π − are given. Moreover, the electromagnetic TFFs of P → γl + l − and V → P l + l − are also obtained. Additionally, the formula of the differential width for P → π + π − γ is also shown.
For parameter estimation, we used precise data of spectrum function for τ − → K S π − ν measured by the Belle collaboration [11]. Furthermore, the PDG data [32] of physical masses of charged vector mesons, ρ + and K * + are used for parameter estimation of the coefficient of 1-loop order interaction terms. We also estimated the model parameters which appear in the mass matrix of neutral vector mesons by using the PDG data [32] of m ρ , m ω and m φ . Since the masses of vector mesons are precisely measured in the experiment, the model parameters in the mass matrix are estimated with smaller uncertainty.
The 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 [32] of widths for radiative decays. Specifically, the experimental data of Γ[K * → Kγ] and the effective coupling ratios of V 0 V 0 P 0 to ρ + π + γ are used. We also considered constraints on a mass matrix and a mixing matrix of pseudoscalars. To obtain a parameter region which is consistent with the masses and Γ[P → 2γ], we solved the system of equations to realize the PDG data [32]. Furthermore, χ 2 fitting for the TFFs measured in the experiments [1,[33][34][35][36][37][38] is carried out. We found that the goodness-of-fit is improved if one does not use the input data measured by the Lepton-G experiment. Hence, we adopted the parameter set estimated without their data.
Using the estimated model parameters, we gave the model predictions for IP violating decays. In particular, we found that the electromagnetic TFFs of η → γl + l − , η → γl + l − are consistent with the experimental data for c + 34 < 0. The partial widths of P → γl + l − , P → π + π − γ, φ → ωπ 0 and V → P l + l − are calculated, none of which result in significant deviation from the experimental data up to 99.7% C.L. For the differential widths of η → π + π − γ and η → π + π − γ, the model predictions are given. The differential width of η → π + π − γ is compared with the data measured by the WASA-at-COSY collaboration [39]. Here, no significant deviation is found in this result. The predictions are also obtained for the TFFs of ρ → π 0 l + l − , ρ → ηl + l − , ω → ηl + l − and φ → η l + l − , which are expected be observed in future experiments. The model predictions for Br[ρ 0 → π 0 π + π − ], Γ[ω → π 0 π + π − ] and Γ[φ → π 0 π + π − ] are also presented. We found that these IP violating observables are consistent with the PDG values [32]. In the vicinity of resonance region, the TFFs for φ → π 0 l + l − and one for η → γl + l − are analyzed. It is shown that the ω pole is dominant in the peak region for both TFFs. We also found that the contribution of the interference between ρ and ω is non-negligible in the peak region. It is shown that the theoretical prediction for Γ[φ → depends on two-loop ordered uncertainty. Our framework, which includes O(p 4 ) contribution both in IP conserving part and in IPV part, can not explain simultaneously decay widths of K * + → K + γ and K * 0 → K 0 γ. As the possible solutions of the problem, we study the O(p 4 m 2 π ) contribution. In addition to SU(3) breaking interactions for IPV part of Ref. [20] , we include two new terms. With these terms, we can explain the decay widths for all four modes of ρ → πγ and K * → Kγ. In contrast to the treatment of Ref. [20], we assume that SU(3) breaking is proportional to current quark mass. Ref. [6]. In this work, we only consider the corrections due to SU(3) octet pseudoscalars.
The effect of SU(3) R external gauge boson is included. The counter terms in 1-loop order are, In Eq. (A1), the contribution from singlet pseudoscalar is omitted in the coefficients of Γ 6 , Γ 8 and ∆ 2 . We have also corrected the sign of k 9 and k 10 in Ref. [6].  In this appendix, we show the power counting rule which is used to classify the interaction Lagrangian in Eq. (1) and counter terms in Eq. (A1). Since we treat the electromagnetic correction due to the term proportional to C in Eq. (1) only within tree level, in the following power counting, we do not take this term into account. Since we employ the loop expansion due to pseudoscalar octet, the Lagrangian is organized as follows, where we denote L (n) as n loop contribution. We first evaluate the superficial degree of divergence of the n loop diagram of Nambu-Goldstone bosons using the interaction part of the tree level Lagrangian, The first two terms of Eq.(B2) denote the interaction with the second derivatives among the Nambu-Goldstone bosons. The third term with the first derivative is the interaction between SU (3) where N 2 is the number of the vertex with second derivatives and N I denotes the numbers of the propagators of pseudoscalar octets in the internal line. It is related to the total number of the vertex (N v ) and the number of loop (N L ) as follows, where N v is Substituting Eq.(B4) with Eq.(B5) into Eq. (B3), one obtains the following formula, The ultraviolet divergence can occur when ω ≥ 0 and we obtain the following condition which the divergent diagrams satisfy, The counter terms which subtract the divergence also satisfy the above condition on the number of the external lines (N η 0 , N V 8 ) and the powers of B which correspond to N χ . Let us examine the types of the counter terms which are required within one-loop calculation by setting N L = 1. Then the superficial degree of divergence is Note that the ω is equal to the number of the derivatives ω 0 included in the counter terms.
In Table XIII, we show ω 0 (≥ 0), N χ ;the powers of B, N η 0 and N V 8 in each 1-loop counter term. We classify each counter term in Eq. (A1) according to these numbers and show their coefficients.
Next we study the power counting of the interaction terms for singlet vector meson. In contrast to the octet vector mesons, the chiral invariant interaction of the singlet vector meson to the octet pseudoscalars 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 chiral breaking term is classified as the one loop level interaction since this term also includes a vector meson with the first derivative and the chiral breaking M , which has a structure similar to the one loop effective counter terms in Eq. (B11) for vector meson octets given below, The coefficients of the counter terms 4 0 0 0 L 1 , L 2 , L 3 , L 9 , L 10 , H 1 , K 1 , K 2 , K 3 , K 6 , K 7 , K 8 , K 9 , K 10 3 0 0 1 K 1 , K 2 , K 3 , K 6 , K 7 , K 8 , K 9 , K 10 Appendix C: 1-loop correction to self-energy for K * + , K * 0 and ρ + In this appendix, we study self-energy corrections to K * +0 mesons and charged ρ meson taking SU(3) breaking into account. The interaction Lagrangian for V → P P is given as, where ∆ denotes the quantum fluctuation for the pseudoscalar octet in the background field method [6]. The isospin breaking leads to π 3 − η 8 mixing and the Feynman diagrams of the self-energy for K * +0 are shown in Fig term is given by the following Lagrangian, where Q µν is defined in Eq. (21) and M K denotes M K + or M K 0 and one uses the following 1-loop function, = Q µν 4M r P Q − 2 3 λ + g µν (4L P Q − 2λΣ P Q − 2(µ Q + µ P )), In Eqs. (C5, C6), λ denotes the ultraviolet divergence defined in Eq. (A7). In Eq. (C6), L P Q and M r P Q are functions given below, In Eqs. (C8, C9),J P Q is a 1-loop scalar function of pseudoscalar mesons with masses M P and M Q . Above the threshold Q 2 ≥ (M P + M Q ) 2 , it is given by, while below the threshold (M P − M Q ) 2 ≤ Q 2 ≤ (M P + M Q ) 2 , We write inverse propagators of vector mesons as, where the metric part of the inverse propagator consists of the sum of tree-level mass M V and loop correction δA V . Using loop functions defined, we add the isospin breaking corrections in Fig. C.1 to the calculation given in Ref. [6]. We also take account of the mass differences of K + − K 0 and π + − π 0 , which were not considered in the previous study. The self-energy corrections to K * +0 and ρ + mesons are obtained as, In this section, we show that the metric tensor part of the two-point functions for the V − A mixing satisfies the relation in Eq. (78). Multiplying Eq. (77) by O T V , one can find, Meanwhile, the diagonalization of the mass matrix leads to, In the above equation, the matrix elements for (i, j) = (1, I), (2, I) indicate the following relations, Plugging Eqs. (D3, D4) into Eq. (D1), one can find that the relation in Eq. (78) is satisfied.
Appendix E: 1-loop correction to self-energy for π + , K + and K 0 In this appendix, the radiative correction to charged pseudoscalar masses is discussed.
Background field method is used to evaluate the chiral loop correction [44,45]. Kinetic terms and 1-loop corrected masses in effective Lagrangian are given as, In Eq. (E1), we denote P f as the pseudoscalar in original flavor basis and the coefficient of the kinetic term is, In Eq. (E1), normalization of the kinetic term is slightly deviated from unity due to 1-loop correction. In order to canonically normalize Z P in Eq. (E1), one should implement the transformation in the following form as, evaluate the quantum correction. We consider the framework in which chiral octet loop correction is taken into account. Masses and kinetic terms of pseudoscalars in 1-loop corrected effective Lagrangian are written as, (kinetic terms rescaled) (mass eigenstate) In Eq. (F1), the coefficient of kinetic terms is given as a 3 × 3 matrix, The transformation in Eq. (F8) relates the basis in Eq. (F1) to one given in Eq. (F2). Thus, the kinetic terms are canonically normalized in Eqs. (F2-F3). One diagonalizes the mass where |K 0 = −CP |K 0 . Since ∆S = ∆Q = −1 rule holds, one finds the following relation, One defines the vector form factors forK 0 π − (K − π 0 ) and its CP conjugate states, Since under CP transformation, the charged currents are related to each other as follows, the following relations among the form factors are derived, (I5) In the isospin limit, we also obtain the relations, Using Eq. (I2), Eq, (I3), Eq. (I5) and Eq. (I6), one can relate the form factor of K s π − of Eq. (I2) to that of K + π 0 , K s π − |sγ µ u|0 = K + π 0 |ūγ µ s|0 .
The contribution to the form factors is divided into two parts. One of them comes from 1 PI diagrams and the other comes from the diagrams which include the propagator of K * meson, We obtain δB K * and δA K * in the above equations by taking the isospin limit of Eq. (C16) and Eq. (C15) and they are given respectively as follows, where ∆A K * is given by,