The form factors of tau ->K pi(eta) nu and the predictions for CP violation beyond the standard model

We study the hadronic form factors of $\tau$ lepton decays $\tau \to K \pi(\eta) \nu$. We compute one loop corrections to the form factors using the chiral Lagrangian including vector mesons. The counterterms which subtract the divergence of the one-loop amplitudes are determined by using background field method. In the vector form factor, $K^\ast$ resonance behavior is reproduced because the diagram with a vector meson propagator is included. We fit the data of the hadronic invariant mass spectrum measured by Belle by determining some of the counterterms of the Lagrangian. Besides the hadronic invariant mass spectrum, the forward-backward asymmetry is predicted. We also study the effect of CP violation of a two Higgs doublet model. In the model, CP violation of the neutral Higgs sector generates the mixing of CP even Higgs and CP odd Higgs. We show how the mixing leads to the direct CP violation of the $\tau$ decays and predict the CP violation of the forward-backward asymmetry.

To predict the direct CP violation of the hadronic τ decays, the strong phase shifts are important quantities and the quantitative prediction on the strong phase shifts is necessary when extracting the weak CP violating phases from the experimentally observed CP asymmetries [6,10,11]. This is a reason why we study the hadronic form factors.
The hadronic form factors for the decay τ → Kπν and SU (2) isospin vector form factors have been studied with various methods. The common future of them is the effects of vector mesons (ρ, K * ) and higher resonances are included. In Ref. [5,20], vector dominance models are studied. One loop corrections to the SU(2) vector form factor are studied with the resonance chiral theory in [21]. In Ref. [22][23][24][25] the τ → Kπν form factors are predicted using the chiral theory combined with the dispersion relations. In our previous study, we use the resonance chiral Lagrangian including the one loop corrections to the self-energy of resonance [10]. We also note in the experimental study [26], the Breit-Wigner form for several resonances is used to fit the data of the hadronic invariant mass spectrum.
In this paper, we use different approach from the previous study [10]. By using the chiral Lagrangian including vector resonance [27], we compute the one loop corrections of pseu- Since Belle and Babar reported the precise measurements of the branching fractions of τ − → K s π − ν [26], τ − → K − π 0 ν [30] and τ − → K − ην [31,32], we can compare our prediction of the hadronic invariant mass distribution with the experimental data. We have determined the finite parts of the coefficients of counterterms so that the hadronic invariant mass spectrum is reproduced.
Once the form factors are fixed, one can use them for predictions of various distributions within the standard model (SM) and beyond. We first compute the angular distribution and the forward-backward asymmetry (FBA) for τ → Kπν and τ → Kην in SM [33]. Furthermore, CP violation of FBA is predicted with a two Higgs doublet model. In type II two Higgs doublet model, within the tree level approximation, the charged Higgs couplings with quarks and leptons are written in terms of Cabibbo-Kobayashi-Maskawa (CKM) matrix. However, if we take into account the one loop corrections to the masses of quarks and leptons due to the neutral Higgs exchanged diagrams, CP violation of the neutral Higgs sector becomes a new source of CP violation of the charged Higgs Yukawa couplings. We show the CP violation of the type II two Higgs doublet model can be probed by the direct CP violation of the τ hadronic decays.
The paper is organized as follows. In section II, we show the hadronic chiral Lagrangian with the vector resonances which are relevant for the form factors. The counterterms are also given. In section III, we compute the form factors. In section IV, we calculate the hadronic invariant mass spectra of the decays τ → Kπ(η)ν. The spectra are compared with the experimental data and the FBAs are predicted. In section V, we explain how CP violation in neutral Higgs sector reveals itself in the charged Higgs Yukawa couplings in a two Higgs doublet model. We also calculate the CP violation of FBA of the hadronic τ decays and the numerical result is presented. Section VI is devoted to discussion and summary. In appendix, we give some details of the derivation of the formulae used in the text.
where U is the chiral field which is given as U = exp(2iπ/f ) = ξ 2 . π is SU(3) octet pseudoscalar and B is a constant parameter. η 0 is U(1) A pseudoscalar of which mass is denoted by M 0 and g 2p is the coupling for η 0 − η 8 mixing. The covariant derivative for the chiral field U is given by, where the external gauge field of SU(3) L denoted by A L is introduced. V µ is the vector nonets and α µ is defined as, The form of the mass term of vector mesons is identical to the that of the unitary gauge fixed version of hidden local symmetry approach [27][28][29]. The kinetic term of the vector mesons is not included in the leading order. This treatment is important when including loop corrections in a systematic way. Note that we have added the chiral breaking term by M = diag(m u , m d , m s ) for the pseudoscalar mesons. The chiral breaking term χ in the isospin limit can be written in terms of the masses of π and K mesons as, When computing the form factors for τ → Kη (′) ν, they are sensitive to the mixing angle of η and η ′ , We first summarize the mixing of the octet and singlet pseudoscalar meson at one-loop order. The self-energy correction for η 0 and η 8 sector in one loop is computed with the interaction terms shown in Appendix D, where the tree level mass squared matrix elements M 2 88 and M 2 08 are given by respectively. z 88 − 1, δM 2 08 , and δM 2 88 are one loop corrections and they are given by, where c = 1− . µ denotes renormalization scale. We also introduce the notation; µ η 8 = µ η cos 2 θ 08 + µ η ′ sin 2 θ 08 . t r i (i = 3, 5) and L r i (i = 4, 5, 6, 8) are the finite counterterms which are defined in Eq. (21) and Eq. (23).θ 08 denotes the mixing angle at the leading order and is given by, The self energy in Eq.(5) can be diagonalized with the following transformation, cos θ 08 sin θ 08 − sin θ 08 cos θ 08 where, We use the transformation Eq.(8) when we compute the form factors for τ → Kη (′) ν decays.
From Eq.(9), M 0 and M 08 are written by, Eq.(10) can be used to obtain the input values for M 0 and M 08 from the mass spectrum when the finite counterterms are given. The mixing angle θ 08 including the correction is also given by, When we compute the form factors related to η and η ′ in one-loop order, the mixing angle θ 08 implies one-loop corrected one. The treatment is consistent with the rigorous one-loop computation and the difference is at two loop order.

B. vector meson sector
Now we turn to the vector meson sector of the Lagrangian. The quantum corrections to the chiral Lagrangian with vector mesons have been discussed in several works [21,28,29].
This formula tells us the types of the counterterms which should be added when we carry out N loop order computation. In general, the local counterterms and the finite counterterms can be classified with the number of derivative n d and the number of the vector mesons n V in the Lagrangian. The interaction term with n d derivatives and with n V vector meson fields has the form of, where the Lorents indices are contracted appropriately. F (ξ) denotes some function of the chiral field. The derivatives can act on both the chiral field and the vector field V . Since the number of derivatives of the vertex of counterterms is equal to the superficial degree of divergence ω, the divergence of the N loop order Feynman diagram with N V external vector mesons can be subtracted by the counterterm with the following number of the derivatives and the vector meson legs, In table I, we show (n d , n V ) for a given set of N and N V . The lowest order Lagrangian corresponds to N = 0 case in the table I and it includes the interaction terms of the type, where (n d , n V ) = (1, 1) corresponds to the term of Tr[V µ α µ ] in Eq.(1). The lowest order Lagrangian includes mass term of the vector mesons while the kinetic term is not included. This is in contrast to the approach of [28,29] where the vector boson is treated as gauge v [n d , n V ]. Note that n d + n V = 2n + 2. The total number of the n loop order interaction vertices in the 1 PI diagram is given by Although in N loop order 1 PI diagram consists of the various loop order vertices, the number of the vertices must satisfy the following relation The number of pseudoscalar meson internal propagator I B is written as, Then one can compute the superficial divergence ω of the 1 PI diagram, The last term of Eq. (19) is the number of the derivatives of the diagram. Substituting Eq. (18) and Eq.(17) into Eq. (19), one can show Eq.(12) as, In the same way as the chiral perturbation theory, we rely on the momentum expansion.
Because the loop momentum of the pseudoscalar mesons is soft, the expansion is valid. In the Lagrangian, there is no kinetic term for the vector meson at the leading order. The kinetic term is generated as the loop correction of the pseudoscalar mesons. According to Eq.(12), one can extend the chiral counting to the case with vector mesons. In generalized chiral counting, the vector meson field V µ is counted as O(p) and the chiral breaking χ is counted as O(p 2 ). The couterterms for O(p 4 ) are obtained by computing divergent part of the one loop corrections due to pseudoscalar mesons. We use the background field method and the corrections can be computed and the counterterms can be determined so that they are consisitent with chiral symmetry [34]. The outline of the derivation is shown in Appendix A and they are given by, where The coeffcients of the counterterms are splitted into the finite parts and divergent parts as, where, with C U V = 1 ǫ − γ + log 4π. The coefficients k i , t i , Γ i and ∆ i are given in the Table II. From Eq.(21), we extract the effective counterterms which are relevant for the calculation of the form factor of τ → Kπν decay. The effective counterterms which subtract the divergence of the amplitudes which contains a vector meson in Fig.1, can be deduced from the counterterms shown in Eq. (21). They are the counterterms for the self energy of vector mesons, V → P P vertex, and the production amplitude of the vector meson; A L → V and are defined as, where . . , 4) are renormalization constants. The coefficients C i can be written in terms of the coefficients of the counterterms of Eq. (21), The finite parts of the counterterms also satisfy the relations similar to Eq.(26), One can extract the counterterms for the 1P I vertex of the type A L → P P . They are given as, where C 5 and its finite part C r 5 are given by, We briefly comment on an intrinsic parity violating interaction and its contribution to the vector meson self-energy. After quarks are integrated out, the intrinsic parity violating interaction term of two vector mesons and a pseudoscalar meson can be generated. One framework, the resonant contribution is included in the second diagram of Fig.1. We take into account the resonance contribution by using the vector meson propagator with one-loop corrections to self energy. Since the propagator have a pole in complex plane, the effect of the width of resonance is also included. Thus we can reproduce the resonance behavior.
There are three parts of the diagrams of Fig.1. The first one is 1 particle irreducible (1 PI) diagrams and the diagrams with one loop corrections are shown in Fig.2. They correspond to the one loop corrections to W + → K + π 0 vertex. They include all the diagrams which are also present in chiral perturbation within one loop. Their contributions to the matrix element K + π 0 |u L γ µ s L |0 become, where Q = p K + p π and q = p K − p π . ∆ P Q denotes the mass squared difference ∆ P Q = m 2 P − m 2 Q . Σ P Q denotes the sum of the mass squared Σ P Q = m 2 P + m 2 Q . The loop functions I P , χ QP µ , J QP µ are given in Eq.(B1) and Eq.(B3). z K and z π are finite wave function renormalization and they are given as, Including the finite part of the counter terms, the result of the 1 PI part is L r i and C r i are finite parts of the counterterms L i and C i respectively. The function H P Q is written in terms of the functions defined in Eq.(C9) as, J P Q can be found in Eq.(C10) and Eq.(C12). We also introduce the following notations in this paper.
where Y = H, M r and L which also appear in the following equations. The diagram with a vector meson propagator is shown in Fig.1. It includes the diagram with a K * propagator, K * → K + π 0 vertex and W + → K * production amplitude. The self-energy of the propagator, the vertex and the production amplitudes include one-loop corrections. We first compute the one loop corrections to K * → Kπ vertex which are shown in Fig.3.
where ∆ Kη 8J Kη 8 ≡ ∆ KηJKη cos 2 θ 08 + ∆ Kη ′J Kη ′ sin 2 θ 08 . Next, the propagator of the K * meson is obtained by including one loop self energy corrections. Using Eq.(C2), the K * meson propagator is given by iD µρ where D µρ is given by, where the self energy corrections δA and δB in this section are identical to the K * mesons ones given in Eq.(C8), The K * production amplitude with the one loop corrections are shown in Fig.4 and is given by, where, J QP µν is defined as, J η 8 K µν is defined as; J η 8 K µν = J ηK µν cos 2 θ 08 +J η ′ K µν sin 2 θ 08 . Now one can assemble the contribution from the diagram with a K * propagator to the form factor. One can write K * production amplitude of the weak vertex and K * → Kπ decay amplitudes as, where G, H, , E and F are given as, Using the form factors, we obtain, The vector form factor and the scalar form factors are defined as, Then the contribution to the form factors is given as, where, For numerical calculation, we use Eq. (51) The self energy correction of vector meson, δA in the chiral limit can be obtained with where H is given by taking the chiral limit of Q 2 M r f 2 in Eq.(C5) as, ].
To compare our result with those of the other methods, we examine the case that the vector meson dominace (VMD) relation M 2 V = 2g 2 f 2 holds. Then the vector form factor is written as, The result can be compared with the same limit of the form factor in [22], .
The difference of the overall factor − 1 √ 2 is just due to the the definition of the form factors. We observe that in the form factor of [22] , the chiral loop correction denoted by H is exponentiated and appears in the numerator of vector meson propagator while in our approach with the vector dominance assumption, the chiral correction appears in the selfenergy function in the denominator of the vector meson propagator. We also note that the finite counterterms generate linear Q 2 dependence in the form factor. They include the wave function renormalization constant of the vector meson Z (r) V , and the other coefficients of the finite counter terms; C r 3 , C r 4 and C r 5 . One can also compare our result with that of the resonance chiral theory [21]. A difference of the form factor in [21] from Eq.(58) of our result is that the one loop corrections to their form factor depends quadratically on momentum squared Q 4 . This is due to the second derivatives coupling of the vector meson to two pseudoscalars in their anti-symmetric tensor formulation of vector mesons. In contrast to their approach, the vector meson coupling into two pseudoscalar meson coupling includes the first derivative. Therefore, the form factor of the present approach depends on Q 2 linearly. They also consider the loop contribution of all the resonances while in our approach, the vector mesons do not contribute in the loop.

IV. NUMERICAL ANALYSIS IN THE SM
To evaluate the vector and scalar form factors, we fix g, M V and the coefficients of the counterterms by using the decay constants, masses and widthes of the mesons. We also use the hadronic mass spectrum. There are ten parameters, {g, M V , Z r V , C r i , L r 4 , L r 5 }, (i = 1, · · · , 5) to be fixed.
From the matrix elements of the axial currents, we obtain the pion and kaon decay constants [35], Using the ratio of f K /f π , we can write L r 5 as follows, If we assume f = f π , from Eq. (60) L r 4 is expressed as, One can take any renormalization scale µ at around K * meson mass. We specifically choose the value of the particle data group (PDG) [37], namely µ = 895.47MeV. If ) is obtained, L r 4 and L r 5 can be fixed. From the imaginary part of the self energy for K * meson in Eq.(C8), the decay width of K * is given by, where ν Kπ is defined in Eq.(C11). Once M V is determined, g can be fixed with the decay width K * (Γ K * ) and M K * . The relations among Z r V , C r 1 , C r 2 are derived by the conditions for the pole masses of K * and ρ mesons. We define K * and ρ meson masses as the momentum squared (Q 2 ) for which the real parts of the inverse propagators vanish, Solving the above equations, one obtains C r 1 and C r 2 , where, From the condition for the residue of the vector meson propagator (35), Z r V is written as follows, We use ρ meson mass of the PDG value [37]. For K * meson mass and the decay width, we fix them with the hadronic mass spectrum of τ → Kπν. Instead of using g and Z V as the fitting parameters, one can use the decay width Γ K * and the mass M K * .
One can write Z r V , C r 3 , C r 4 and C r 5 in terms of K r 1 , K r 2 , K r 3 and L r 9 with Eqs. (27) and (29). Since K r 3 is related to Z r V with Eq. (27) and Z r V is fixed with Eq. (71), K r 3 is already determined by M K * . We note that the form factor of Eq. (51) depends on the combination C r 3 − 4C r 4 , which is written as, (72) One also notes that C r 5 is written in terms of K r 1 + K r 2 , K r 3 and L r 9 . Therefore we choose {Γ K * , M V , M K * , K r 1 + K r 2 , L r 9 } as fitting parameters in the following analysis. We fit them by using the differential branching fraction of the experimental data [26].
The differential branching fraction for KP ν(P = π, η) is given by, where p K is the momentum of K in the hadronic center of mass (CM) frame. The differential decay distribution for τ − → K s π − ν is shown in Fig. 5. One can see the peak of K * resonance around at Q 2 ≃ 900 MeV.
The five parameters are determined by fitting the hadronic mass spectrum in the region m K + m π ≤ Q 2 ≤ 1665MeV with 90 bins data. We also use the PDG values [37], m π ± , f π ± , m K 0 , m η , m η ′ , m τ as inputs. The set of parameters leading to the smallest χ 2 /n.d.f value are fixed by where the obtained χ 2 /n.d.f. is 152.3/85. The other parameters are shown in Table III. We also note 1 − M 2 V /(2g 2 f 2 π ) = 0.2688 for this case. It implies that the relation of the vector meson dominance, M 2 V = 2g 2 f 2 π , is slightly violated.    (47), where,  We study τ − → K − ην decay using the parameters fixed with τ − → K s π − ν decay. The form factors for Kη are given in Appendix E. Figure 10 shows the prediction of the decay distribution for τ − → K − ην. It is found that the contribution of vector form factor is dominant. The predicted branching fraction for τ − → K − ην decay is 2.114 × 10 −4 . Since the experimental results are Br(τ − → K − ην) = (1.52 ± 0.08) × 10 −4 [37], our prediction is larger than the experimental data. We note that the predicted branching fractions for τ − → K S π − ν and τ − → K − ην decays are 4.030 × 10 −3 and 1.157 × 10 −4 respectively with the other parameter set of parameters which is obtained by 67 bins data fitting (m K + m π ≤ Q 2 ≤ 1400.5MeV).
We also consider the forward-backward asymmetry [33] for τ → KP ν decay. The double 800 1000 1200 1400 1600 1800 differential rate of the unpolarized τ decay [6] is given by where θ is the scattering angle of kaon with respect to the incoming τ in the hadronic CM frame. The forward-backward asymmetry extracts the interference term of the vector form factor and the scalar form factor. cases are shown in Fig. 11. As can be seen in Fig. 11, the forward-backward asymmetry for Kπ case is large below K * resonance and reaches to 70%. Here the decay distribution for τ − → K s π − ν is identical to that of τ − → K − π 0 ν by taking the limit for ǫ K of K 0 K 0 mixing zero. In Fig. 11, we have evaluated the forward-backward asymmetry for τ − → K − π 0 ν as that for Kπ case.  The well known effect of CP violation of the two Higgs doublet model is CP even and CP odd Higgs mixing [41,42]. In the large limit of the ratio of Higgs VEVs, among three neutral Higgs, the SM like CP even Higgs is decoupled from the other two Higgs bosons.
Therefore, in good approximation, CP even and CP odd Higgs mixing occurs among two Higgs bosons in the sector of the Higgs with the small VEV. We investigate how the CP violating mixing of the neutral Higgs sector leads to some observable effect on charged Higgs Yukawa coupling. We also explicitly show how it generates the direct CP violation of τ decays. For this purpose, we compute one loop corrections to masses of the charged leptons and down type quarks. One finds the one loop corrected mass is flavor diagonal and a small CP violating chiral phase due to the CP even and CP odd Higgs mixing is generated.
To remove the phase of one loop corrected mass, one needs to carry out the chiral rotation.
After the chiral rotation, CP violating phase in charged Higgs sector arises. The phase is due to the CP violation of Higgs sector which is the different origin from Kobayashi Maskawa phase [43].
After all, the relative CP violating phase difference between the charged current interaction of W boson and charged Higgs interaction arises as, The phase φ τ vanishes if CP even and CP odd Higgs mixing angle θ AH vanishes. The phase φ τ can be measured by direct CP violation of τ ± decays. The decays go through the intermediate states W − and H − which are converted to a common hadronic final state (K, π). Schematically, the process goes as, To measure the phase φ τ , the angular analysis of the decay distributions of τ → Kπν is useful. The direct CP violation arises in the interference of two amplitudes with both weak phase difference and strong phase difference. In the τ → Kπν decays, the interference of two amplitudes with different angular momentum of K − π 0 , i.e., l = 1 and l = 0 can take place. The difference of the angular distribution of τ − → K − π 0 ν and its CP conjugate τ + → K + π 0ν is sensitive to the CP violating phase described above. As we have shown in [10], the forward-backward CP asymmetry is a good observable for the CP violation.
The Higgs potential of two Higgs doublet model with softly broken Z 2 symmetry is given as, where under Z 2 transformation, the Higgs fields transform as, θ 5 is a CP violation parameter of Higgs sector. One may write the vacuum expectation values with three order parameters [44], The three order parameters are determined by the stationary conditions. For large tan β, the solution can be written approximately as, where only the leading terms with respect to the expansion of the soft breaking parameter where a 12 and a 13 are subleading of the expansion of cos β and can be neglected in large tan β limit. Therefore in the limit, one can simply diagonalize 2×2 matrix. For the purpose, one introduces the mixing angle θ AH , where H 2 and H 3 are mass eigen states. The other matrix elements in small cos β limit are, Then one finds the mixing angle is given by, In Fig.12, we have shown the mixing angle θ AH as a function of CP violating parameter θ 5 of the Higgs potential as given by Eq.(92). One can see when the mass splitting of H 2 and H 3 are large, θ AH tends to deviate from the line of θ AH = θ 5 2 , which leads to θ ′ is non-vanishing. Next we compute the one loop corrected mass due to H 2 and H 3 . Yukawa couplings of them to down type quarks and charged leptons can be written as, Note that the Yukawa couplings of H 2 and H 3 have an enhancement factor tan β. The CP violation of the Yukawa couplings are written in terms of the chiral phase, e −iγ 5 θ AH . One defines the one loop corrected masses for down type quarks and charged leptons as, The corrections are evaluated by computing Feynman diagrams Fig.13 and the result is, Now we study the effects of the CP violation of the Higgs mixing on τ lepton decays.
The effective four Fermi interactions from the SM contribution and from the charged Higgs exchange are given by, where the relative phase φ τ − φ d i of charged current interaction due to W − exchanged and charged Higgs interaction H − arises.
The forward-backward CP asymmetry in the two Higgs doublet model can be obtained by replacing the SM scalar form factor with the one including the charged Higgs contribution in Eq.(78), By comparing the forward-backward asymmetry of τ − and τ + , one obtains the direct CP violation [10], where P = π 0 , η. for tan β ≃ 40 [48]. The ratio of the neutral Higgs masses can be constrained from T parameter. T parameter of the present model is computed as [51], where F (m a , m b ) is given by, In Eq.(104), we take the limit; β → π 2 . From Eq.(10.61) of Ref. [52], T New = 0.03 ± 0.11 for the SM Higgs boson mass M h = 117GeV case. We shift the SM reference point for the Higgs mass to M h = 126GeV [53], which amounts to the shift of T New is 3 8πc 2 W log 126 117 ≃ 0.01. Therefore we adopt the following value for T New , With M H + = 600GeV, the constraints on (M H 2 , M H 3 ) plane are shown in Fig. 15.
In Fig. 16 • By using the propagator with the one loop corrected self-energy of the vector mesons, one can reproduce the vector meson intermediate states.
• We fit our theoretical curve of the hadronic invariant mass distribution with that obtained by Belle. By tuning the parameters, we demonstrate that one can fit the hadronic mass distribution up to ∼ 1300 MeV well. Between 1300 MeV and 1500 MeV , our prediction is slightly lower than the experimental data. We also compute the branching fraction τ → Kην, which is consistent with the experimental one.
About the CP violation of the two Higgs doublet model, we study the CP violation of the Higgs sector. The model was invented to explain the large isospin breaking of bottom and top due to large tan β ≃ 40 [40]. CP violation of Higgs potential leads to the mixings of CP even and CP odd Higgs. The Yukawa couplings of down type quarks and charged leptons with the neutral Higgs of the second Higgs doublet are large and are CP violating. We found that; • The CP violation in the neutral Higgs sector leads to the CP violating effect on the quarks and leptons mass matrices through one loop corrections.
• After removing the CP violating phases in the mass matrices, one obtains CP violating phases of the charged Higgs couplings to the down type quarks and charged leptons.
• The effect is studied in the forward-backward CP asymmetry of τ → Kπν decay. The order of the asymmetry is 10 −6 ∼ 10 −7 . The smallness of the asymmetry comes from the fact that the CP violation is loop induced effect. We give the outline of the derivation of the counterterms. To derive the counterterms, we use the background field method so that the calculation of the counterterms is consistent with chiral symmetry [34], [35]. For the purpose, we first write the chiral Lagrangian in terms of the fields which are decomposed into the background fields and the quantum fields based on Eq.(1). We decompose the fields into the background field and quantum fluctuation as follows; where ξ = exp(i π f ) and π denotes the background pseudoscalar octet fields. ∆ denotes the quantum fluctuation. η 0 is the background field for singlet pseudoscalar and ∆ 0 is its quantum part. We also introduceᾱ ⊥ andᾱ defined as Using the notations given above, we write the Lagrangian including the background field and the quantum parts, If we suppress the quantum fluctuation as ∆ → 0 and ∆ 0 → 0, then Eq.(A3) equals to Eq.(1). We note that under the chiral transformation, ξ transforms non-linearly as, while ∆ transforms linearly as, We also noteᾱ andᾱ ⊥ transform as, We treat the vector meson V as the background field and it transforms as where χ ± is defined as, Since the background fields (ξ, η 0 ) satisfy the equations of motion the first variation with respect to ∆ and ∆ 0 vanishes. Introducing the octet component field ∆ a as ∆ = 8 a=1 ∆ a T a , one can write the quadratic part of the action in terms of the quantum parts ∆ A = (∆ 0 , ∆ a ) as, D AB is a diffrential operator and a 9 × 9 matrix for the nonet space. with, where, and, We also define, and The effective action including one loop corrections is given by By introducing, 9 × 9 matrix, one can write D AB as; The divergent part of one loop correction can be easily computed with the heat kernel method [34] [36]. The counterterms can be also obtained with, where a 2 (x) is given by, The trace for 9 × 9 matrix can be converted to trace for 3 × 3 matrix which leads to the conterterms of Eq.(21).

Appendix B: 1 loop functions
Here we summarize the one loop functions which appear in Eq. (30).

Appendix C: Two point function of vector mesons
Let us determine the coefficient of the counterterms C 1 , C 2 and Z V from the renormalization for self-energy of vector mesons. The vector mesons couplings with pseudo scalar mesons in L are, The quantum field for pseudoscalar octet ∆ is denoted byπ. One can parameterize the inverse propagators of vector fields as, We study the two point functions for ρ + and K * + mesons.
where δA V and δB V denote the one loop corrections including the contribution from the counterterms. For the ρ meson, they are given as, where µ P = m 2 P 32π 2 f 2 ln m P 2 µ 2 . M r P (P = π, K) are the loop functions for π mesons and K mesons defined as, (C5) Z r V (µ), C r 1 (µ) and C r 2 (µ) are finite parts of the renormalization constants defined by, with C U V is the divergent part of the dimensional regularization, where ǫ = 2 − d 2 and γ is Euler constant. The self energy corrections to K * meson are given as, where M r P Q and L P Q are the same functions as the ones defined in Ref. [34], where k P Q = (µ P −µ Q )f 2 ∆ P Q .J P Q is a one loop scalar function of pseudo scalar mesons with masses m P and m Q , Above the threshold; Q 2 ≥ (m P + m Q ) 2 , it is given by, where, while below the threshold (m P − m Q ) 2 ≤ Q 2 ≤ (m P + m Q ) 2 , with Σ = m 2 P + m 2 Q and ∆ P Q = m 2 P − m 2 Q .
In the following equations, δA and δB imply δA K * and δB K * , respectively. In this appendix, we give the equations of the form factors for τ → Kην and τ → Kη ′ ν. The vector and scalar form factors are given as the sums of the contribution of 1 PI diagram and K * resonance contribution.
The contribution of the 1 PI diagrams for τ → Kην form factors is computed as, Using the K * → Kη decay amplitude, the contribution to the form factor is given by, The form factor for τ → Kη ′ ν is also given as, Kη ∆ Kη ′ cos 2 θ 08 +J Kη ′ sin 2 θ 08 ) . (E10) Using the K * → Kη ′ decay amplitude, the contribution to the form factor is given by,