]>PLB33042S0370-2693(17)30593-210.1016/j.physletb.2017.07.037The Author(s)PhenomenologyFig. 1The Feynman diagrams for the pion weak form factors. The dashed line depicts the pion, the dashed double line and the wavy line describe the vector (axial-vector) field and the photon field, respectively. Diagram (a) contains both the local and nonlocal contributions, whereas diagrams (b)–(e) arise from the nonlocal interaction due to the momentum-dependent dynamical quark mass.Fig. 1Fig. 2The form factors FV(Q2), FA(Q2), and RA(Q2) for the radiative pion decay as functions of Q2. In the left panel the vector form factor is depicted. The middle and the right panels correspond to the axial-vector form factor and the second axial-vector form factor, respectively. The momentum-dependent quark mass defined in Eq. (10) with Eq. (11) was used. The dashed curve presents the local contribution whereas the dot-dashed one draws the nonlocal contribution. The solid curve depicts the total result.Fig. 2Fig. 3Comparisons of the form factors, FV(Q2), FA(Q2) and RA(Q2) with two different types of the momentum-dependent quark mass used. The solid curves draw the form factors derived by using the quark mass from the instanton vacuum given in Eq. (11) (D.P.), whereas the dashed ones depict that by the dipole-type mass given in Eq. (12) (Dipole).Fig. 3Table 1Comparison of the present results with those from other works. “D.P.” stands for the results derive from the instanton vacuum, whereas “Dipole” denotes those obtained from the dipole-type parametrization of the dynamical quark mass.Table 1NJL [47]NL NJL(A) [23]χPT [20]Experimental dataPresent work

D.P.Dipole

FV(0)0.02420.02700.0262(5)0.0258(17) [15]0.02710.0269

aV0.01910.0332(42)0.10(6) [15]0.02870.0280

FA(0)0.02390.01320.0106(36)0.0117(17) [15]0.01320.0116

aA0.0120.0191(61)–0.01920.0193

RA(0)0.059−0.008+0.009 [12]0.04620.0459

Table 2The results of the low-energy constants. “D.P.” stands for the results derive from the instanton vacuum, whereas “Dipole” denotes those obtained from the dipole-type parametrization of the dynamical quark mass.Table 2L9rL10rL9r+L10r

D.P.5.43−3.881.55

Dipole5.41−4.051.36

Table 3The results of the p-pole parameters. “D.P.” stands for the results derive from the instanton vacuum, whereas “Dipole” denotes those obtained from the dipole-type parametrization of the dynamical quark mass.Table 3PVMPVPAMPAPRMPR

D.P.1.160.843 GeV1.481.05 GeV0.7571.10 GeV

Dipole1.340.870 GeV1.591.05 GeV0.7341.12 GeV

Pion radiative weak decay from the instanton vacuumSang-InShimabssimr426@gmail.comHyun-ChulKimac⁎hchkim@inha.ac.kraDepartment of Physics, Inha University, Incheon 22212, Republic of KoreaDepartment of PhysicsInha UniversityIncheon22212Republic of KoreabResearch Center for Nuclear Physics (RCNP), Osaka University, Ibaraki, Osaka 567-0047, JapanResearch Center for Nuclear Physics (RCNP)Osaka UniversityIbarakiOsaka567-0047JapancSchool of Physics, Korea Institute for Advanced Study (KIAS), Seoul 02455, Republic of KoreaSchool of PhysicsKorea Institute for Advanced Study (KIAS)Seoul02455Republic of Korea⁎Corresponding author.Editor: J.-P. BlaizotAbstractWe investigate the vector and axial-vector form factors for the pion radiative weak decays π+→e+νeγ and π+→e+νee+e−, based on the gauged effective chiral action from the instanton vacuum in the large Nc limit. The nonlocal contributions, which arise from the gauging of the action, enhance the vector form factor by about 20%, whereas the axial-vector form factor is reduced by almost 30%. Both the results for the vector and axial-vector form factors at the zero momentum transfer are in good agreement with the experimental data. The dependence of the form factors on the momentum transfer is also studied. The slope parameters are computed and compared with other works.1IntroductionPion radiative decay π+→e+νeγ provides rich information on the structure of the pion. The decay amplitude for the pion radiative decay consists of two part, i.e., the structure-dependent (SD) part containing the vector and axial-vector form factors of the pion and the inner Bremsstrahlung (IB) part [1–4]. The advantage of studying π+→e+νeγ decay over π+→μ+νμγ is that the IB part is suppressed in the π+→e+νeγ decay [3,4], whereas the corresponding SD part is enhanced due to the helicity. Thus, the π+→e+νeγ decay allows one to get access to the structure of the pion experimentally. The vector form factor FV is related to the decay rate of the π0→γγ decay [1,5] by the vector current conservation, so it was easier to find it using the lifetime of π0. On the other hand, it took many years to measure unambiguously the axial-vector form factors [6–14]. Some years ago, PIBETA Collaboration [15] conducted a precise measurement of the pion weak form factors, reporting the values of the vector and axial-vector form factors respectively as FV(0)=0.0258(17) and FA(0)=0.0117(17). The slope of the vector form factor was also measured: aV=0.10(6), which is defined in the parametrization of the vector form factor FV(q2)=FV(0)/(1−aVq2/mπ2) near q2≈0. There is yet the second axial-vector form factor which comes into play when the photon is virtual. The SINDRUM Collaboration [12] reported the first measurement of the decay π+→e+νee+e− in which the off-mass-shell photon decays into e+e−, and yielded the second axial-vector form factor to be RA(0)=0.059−0.008+0.009.The vector and axial-vector form factors for the pion radiative decay were studied in chiral perturbation theory [16–20], since the experimental data on the axial-vector form factor can be used to determine a part of the low-energy constants that encode information on nonperturbative quark–gluon dynamics. These form factors have been also investigated within various theoretical frameworks: For example, quantum chromodynamics (QCD) sum rules [21], the nonlocal Nambu–Jona–Lasinio (NJL) model [22,23], and in the light-front quark model [24]. Since the photon can be virtual, it is of interest to examine the dependence of the form factors on the momentum transfer. Chiral perturbation theory predicts very mild dependence on the momentum transfer in the range of 0≤q2≤0.018GeV2 [17]. On the other hand, the results for the vector and axial-vector form factors from the light-front quark model start to rise near q2=0 and then fall off drastically as q2 increases [24]. On the contrary, the nonlocal NJL model [23] predicted only the q2 dependence of the vector form factor. The results monotonically decrease as q2 increases. Thus, it is of great importance to investigate the weak form factors for the pion radiative decay and compare them with those from other works.In the present work, we study the three weak form factors of the pion, i.e., the vector form factor, the axial-vector form factor, and the second axial-vector form factor, based on the gauged effective chiral action (EχA) from the instanton vacuum [25–31]. Since the spontaneous breakdown of chiral symmetry is naturally realized from the instanton vacuum, it provides a good framework to investigate properties of the pion, i.e. of the pseudo-Nambu–Goldstone bosons. The quark acquires the dynamical quark mass that is momentum-dependent through the quark zero modes in the instanton background. Moreover, there are only two parameters in this approach, namely, the average instanton size ρ¯≈1/3 fm and average interinstanton distance R¯≈1 fm. Since the average size of instantons is considered as a normalization point equal to ρ¯−1≈0.6 GeV, we can use the model for computing any observables of hadrons and compare the results with those from other theoretical framework such as χPT and lattice QCD, in particular, when a specific scale is involved. These values of the ρ¯ and R¯ were determined many years ago theoretically [25,26] as well as phenomenologically [32,33]. They were also confirmed by various lattice works [34–36]. In Ref. [37], the QCD vacuum was simulated in the interacting instanton liquid model and ρ¯≈0.32fm and R¯≈0.76fm were obtained with the finite current quark mass taken into account.Since we consider the pion mass, we need to introduce the current quark mass. Musakhanov [38,40] improved the EχA derived by Diakonov and Petrov [25], including the current quark mass. In fact, this improvement plays an essential role in understanding the QCD vacuum in the presence of the finite mass of the current quark. In Ref. [41], it was shown that the improved EχA properly described the dependence of the quark and gluon condensates on the current quark mass. Furthermore, the nonlocality arising from the momentum-dependent dynamical quark mass is known to bring out the breakdown of the Ward–Takahashi (WT) identities, that is, the current nonconservation [42–45]. In Refs. [27,28], the gauged EχA was derived from the instanton vacuum, which satisfies the WT identities. We will employ this action in the present work to investigate the weak form factors for pion radiative decay.The structure of the present work is sketched as follows: In Section 2, we will define the three weak form factors of the pion, which will be related to the transition matrix elements of the vector and axial-vector currents. In Section 3, we briefly explain the gauged EχA. In Section 4, we derive the vector and axial-vector form factors, using the gauged EχA. In Section 5, we present the numerical results of the three form factors and discuss them. The final Section is devoted to summary and conclusion.2Weak form factors of the π+→e+νγ decayThe SD part of the pion radiative decay amplitude consists of the weak transition form factors of the pion, i.e., the vector form factor FV(q2), the axial-vector form factor FA(q2), and the second axial-vector form factors RA(q2). They are defined in terms of the transition matrix elements of the vector and axial-vector currents as follows(1)〈γ(k)|V12μ(0)|π+(p)〉=−emπϵα⁎FV(q2)ϵμαρσpρkσ,(2)〈γ(k)|A12μ(0)|π+(p)〉=ieϵα⁎2fπ[−gμα+qμ(qα+pα)Fπ(k2)q2−mπ2]+iϵα⁎emπ[FA(q2)(kμqα−gμαq⋅k)+RA(q2)(kμkα−gμαk2)], where |π+(p)〉 and |γ(k)〉 stand for the initial pion and the final photon states, the transition vector and axial-vector currents are defined respectively as(3)V12μ=ψ¯γμτ1−iτ22ψ,A12μ=ψ¯γμγ5τ1−iτ22ψ, consisting of the quark fields ψ=(u,d), the Dirac matrices γμ and γ5, and the Pauli matrices τi in isospin space. p and k denote respectively the momenta of the pion and the photon, whereas q is the momentum of the lepton pair. The mass of the pion can be obtained from p2=mπ2 with the mass of the pion mπ=139.57 MeV. FV(q2) and FA(q2) are the vector and axial-vector form factors of the pion respectively. The second axial-vector form factor, RA(q2) contributes only when the outgoing photon is virtual (k2≠0). Fπ(k2) is the electromagnetic form factor which gives Fπ(0)=1. Electromagnetic charge radius 〈rπ2〉 and Fπ(k2), were already calculated by one of the authors and his collaborator in this model [46].3Gauged effective chiral action in the presence of external fieldsSince we want to compute the weak form factors of pion radiative decay in this work, we introduce all the relevant external fields in the gauge-invariant manner, i.e., the electromagnetic field vem, the vector fields v, and the axial-vector fields a(4)Seff[vem,v,a,π]=−Spln[iD̸+imˆ+iM(iDL)Uγ5M(iDR)], where the functional trace Sp runs over the space–time, color, flavor, and spin spaces. The current quark mass matrix mˆ is written as diag(mu,md)=m¯1+m3τ3 with m¯=(mu+md)/2 and m3=(mu−md)/2. τ3 is the third component of the Pauli matrix. Note that isospin symmetry is assumed, so m3=0. The covariant derivative Dμ is defined as(5)iDμ=i∂μ+eQˆvemμ+τa2vμa+γ5τa2aμa with the charge operator for the quark fields(6)Qˆ=(2300−13)=16+12τ3. The left-handed and right-handed covariant derivatives in the momentum-dependent dynamical quark mass M(iDL,R) are defined respectively as(7)iDμL=i∂μ+eQˆvμem+τa2vμa−γ5τa2aμa,iDμR=i∂μ+eQˆvμem+τa2vμa+γ5τa2aμa. The momentum-dependent quark mass with the covariant derivatives ensures the gauge invariance of Eq. (4) in the presence of the external fields. In fact, it was shown that the nonlinear pseudo-Nambu–Goldstone boson field is expressed as(8)Uγ5=U(x)1+γ52+U†(x)1−γ52=exp(iγ5fπτ⋅π), where Fπ is the pion decay constant. The pion fields are given as(9)τ⋅π=12(12π0π+π−−12π0). The momentum-dependent dynamical quark mass, which arises from the quark-zero mode of the Dirac equation with the instanton fields, is given by(10)Mf(k)=M0F2(k)f(mf), where M0 is the constituent quark mass at zero quark virtuality, and is determined by the saddle-point equation, resulting in about 350 MeV [25,26]. The form factor F(k) arises from the Fourier transform of the quark zero-mode solution for the Dirac equation with the instanton and has the following form:(11)F(k)=2τ[I0(τ)K1(τ)−I1(τ)K0(τ)−1τI1(τ)K1(τ)], where τ≡|k|ρ¯2. I0,1 and K0,1 denote the modified Bessel functions. In addition to this original form, we also use the dipole-type parametrization of F(k) defined by(12)F(k)=2Λ22Λ2+k2 with Λ=1/ρ¯. As mentioned in Introduction already, the average size of the instanton ρ¯ was determined either phenomenonlogically [32,33] or theoretically [25,26]. In the large Nc limit, the value of ρ¯ was determined to be ρ¯≃0.33 fm [25,26]. When one considers the 1/Nc meson-loop corrections, ρ¯ is modified to be ρ¯≃0.35 fm [28–31]. Lattice QCD yields similar results ρ¯=(0.32–0.36) fm [34–37]. Since we compute in this work the weak form factors of pion radiative decay in the large Nc limit, we will take ρ¯=0.33 fm or Λ=600 MeV. We will compare the results obtained by using the both form factors. The presence of the current quark mass also affects the dynamical one, which was studied in Refs. [38,40] in detail. The additional factor f(mf) describes the mf dependence of the dynamical quark mass, which is defined as [43,39](13)f(mf)=1+mf2d2−mfd. This mf-dependent dynamical quark mass yields the gluon condensate that does not depend on mf. Pobylitsa considered the sum of all planar diagrams, expanding the quark propagator in the instanton background in the large Nc limit [43]. Taking the limit of N/(VNc)→0 leads to f(mf). The parameter d is given as 198 MeV. The mf-dependent dynamical quark mass also explains a correct hierarchy of the chiral condensates: 〈u¯u〉≈〈d¯d〉>〈s¯s〉 [41].4Pion weak form factorsThe matrix elements of the vector and axial vector currents in Eq. (2) are related to the three-point correlation function(14)〈γ(k)|Wμa(0)|πb(p)〉=ϵα⁎∫d4xe−ik⋅x×∫d4yeip⋅yGαρ−1(k)Gπ−1(p)〈0|{Vρem(x)Wμa(0)Pb(y)}|0〉, where Wμa expresses generically either the vector current or the axial-vector current defined in Eq. (3). The operators in the correlation function represent the electromagnetic current, vector (axial vector) current, and pion-field operators, respectively. Gαρ(k), Gπ(p) stand for the propagators of the photon and the pion, respectively. Then, the matrix element (14) can be directly derived from the gauged effective chiral action given in Eq. (4)(15)〈γ(k)|Wμa(0)|πb(p)〉=ϵα⁎∫d4xe−ik⋅x×∫d4yeip⋅yδ3Seff[vem,w,π]δvαem(x)δwμa(0)δπb(y)|vem,w,π=0. The three-point correlation function in Eq. (15) consists of five Feynman diagrams drawn in Fig. 1. In the case of the vector form factor, only diagram (a) contributes to it, whereas all other diagrams vanish because of the trace over spin space. On the other hand, all the diagrams contribute to the axial-vector form factors. Note that diagram (a) contains the contributions from both the local and nonlocal terms, while all other diagrams arise only from the nonlocal terms on account of the momentum-dependent dynamical quark mass.4.1Vector form factorWe first deal with the vector form factor of the pion. Having computed Eq. (15) explicitly, we obtain the matrix element of the vector current (W=V)(16)〈γ(k)|Vμ12|π+(p)〉=−i42eNc3fπϵα⁎∫d4l(2π)4M(ka)M(kb)DaDbDc×[εμαρσ(M¯akbρkcσ+M¯bkcρkaσ+M¯ckaρkbσ)−εμβρσkaβkbρkcσ(M(kb)Mα(kb)+M(kc)Mα(kc))+εαβρσkaβkbρkcσ(M(ka)Mμ(ka)+M(kc)Mμ(kc))], where Nc denotes the number of colors. M¯i is the sum of the dynamical and current quark masses M¯i=m+M(ki). The momenta ki are defined as ka=l+q2+k2, kb=l−q2−k2, kc=l−q2+k2, and q=p−k. Di are given as Di=(ki2+M¯i2). Mμ(ki) represents Mμ(ki)=∂M(ki)/∂kiμ. Equation (16) corresponds to diagram (a) in Fig. 1 and there is no contribution from diagrams (b)–(e) in the case of the vector form factor, as mentioned previously. Considering the transverse relation ϵ⁎⋅p=ϵ⋅p=0, we can extract the vector form factors, comparing Eq. (1) with Eq. (16). Thus, the pion vector form factor is obtained finally as(17)FV(Q2)=FVlocal(Q2)+FVNL(Q2), where FVlocal(Q2) and FVNL(Q2) stand for the local and nonlocal contributions(18)FVlocal(Q2)=42NcMπ3fπ(p⋅k)2∫d4l(2π)4M(ka)M(kb)DaDbDcpμkν×[M¯a(kbμkcν−kcμkbν)+M¯b(kcμkaν−kaμkcν)+M¯c(kaμkbν−kbμkaν)],(19)FVNL(Q2)=42NcMπ3fπ(p⋅k)2∫d4l(2π)4M(ka)M(kb)DaDbDc×[−(M′(kb)+M′(kc))(p⋅k)2(ϵ⁎⋅l)(ϵ⋅l)+(M′(ka)+M′(kc))(εμγδλlμϵγpδkλ)×(εαβρσϵα⁎kaβkbρkcσ)], where M′(ki) is the derivative of the dynamical quark mass with respective to the squared momentum M′(ki)=∂M′(ki)/∂ki2. The momentum transfer Q2 is defined to be positive definite, i.e., Q2=−q2.In fact, one can easily see from Eq. (19) that the terms with M′(ki) are derived from the expansion of the dynamical quark mass with respect to the covariant derivative given in Eq. (7). Thus, those terms with M′(ki) are the essential part in obtaining the vector and axial-vector form factors with the corresponding gauge invariance preserved. If the dynamical quark mass is taken to be independent of the quark momentum, then M′(ki) is equal to zero. It indicates that the nonlocal contributions to the vector form factor vanish such that the results are the same as those derived from the local chiral quark model (χQM). However, one has to introduce the regularization to tame the divergence arising from the quark loop in the local χQM. In this sense, the momentum-dependent dynamical quark mass plays also a role of a certain regularization.4.2Axial-vector form factorsThe transition matrix element of the axial-vector current (W=A) given in Eq. (2) is obtained as(20)〈γ(k)|Aμ12|π+(p)〉=−i42eNcfπϵα⁎∫d4l(2π)4∑i=aeFμα(i), where Fμα(i) corresponds to diagram (i), which can be explicitly expressed as(21)Fμα(a)=M(ka)M(kb)DaDbDc[δμα{M¯akb⋅kc−M¯bkc⋅ka+M¯cka⋅kb+M¯abc}+{−M¯a(kbμkcα+kcμkbα)+M¯b(kaμkcα+kcμkaα)+M¯c(kaμkbα−kbμkaα)}−(M′(ka)kaμ−M′(kc)kcμ){(kb⋅kc+M¯bc)kaα−(kc⋅ka+M¯ca)kbα−(ka⋅kb+M¯ab)kcα}+(M′(kb)kbα+M′(kc)kcα){−(kb⋅kc−M¯bc)kaμ+(kc⋅ka−M¯ca)kbμ−(ka⋅kb+M¯ab)kcμ}−(M′(ka)kaμ−M′(kc)kcμ)(M′(kb)kbα+M′(kc)kcα)×{M¯akb⋅kc−M¯bkc⋅ka−M¯cka⋅kb−M¯abc}],Fμα(b)=M(ka)M(kb)DaDcM′(kb)kbαM(kb)×[−{M¯c+M′(ka)(ka⋅kc+M¯ac)}kaμ+{M¯a+M′(kc)(ka⋅kc+M¯ac)}kcμ],Fμα(c)=M(ka)M(kb)DbDcM′(ka)kaμM(ka)×[−{M¯c−M′(kb)(kb⋅kc−M¯bc)}kbα−{M¯b−M′(kc)(kb⋅kc−M¯bc)}kcα],Fμα(d)=M(ka)M(kb)DaDbM(kc)(M′(kc)M(kc))2(ka⋅kb+M¯ab)kcμkcα,Fμα(e)=M(ka)M(kb)1DcM¯c(M′(ka)M′(kb)M(ka)M(kb))kaμkbα. Here, M¯ij=M¯aM¯b and M¯ijk=M¯aM¯bM¯c.In order to pick up the axial-vector form factors from Eq. (2), it is convenient to introduce an arbitrary vector ξμ⊥ that satisfies the following properties, ξ⊥⋅ξ⊥=0, ξ⊥⋅q=0, and ξ⊥⋅k≠0. Then, the axial-vector form factor FA(Q2) and the second axial-vector form factor RA(Q2) can be derived as(22)FA(Q2)=−42Ncmπfπ(q⋅k)∫d4l(2π)4∑i=aeFμα(i)[ϵμϵα⁎−ξμ⊥kαξ⊥⋅k],(23)RA(Q2)=42Ncmπfπ(ξ⊥⋅k)2∫d4l(2π)4∑i=aeFμα(i)ξμ⊥ξα⊥.As in the vector form factor, the local contribution to the axial-vector form factors comes from the first and second terms of Fμα(a) in Eq. (21).5Results and discussionWe are now in a position to discuss the numerical results for the weak form factors of the pion radiative decay. Since the present framework is fully relativistic, the Breit-momentum frame will be used. There are no adjustable parameters in the present work. We will take the original values M0=350 MeV and ρ¯=0.33 fm from Refs. [25,26]. The pion decay constant fπ can be computed within the model and is obtained to be fπ=93 MeV.Fig. 2 draws the results of the pion form factors for pion radiative weak decay. In general, the form factors decrease monotonically, as Q2 decreases. As discussed in the previous section, it is essential to consider the nonlocal contribution to preserve the corresponding gauge invariances, since the electromagnetic and vector currents should be conserved. As shown in the left panel of Fig. 2, the nonlocal part contributes to the pion vector form factor by almost about 20%.The results for the axial-vector form factor is depicted in the middle panel of Fig. 2 as a function of Q2. Note that, however, the nonlocal contribution behaves very differently from the case of the vector form factor. In fact, it turns out negative, so that the final result for the form factor is reduced by about 30%, which implies that it is indeed crucial to consider the nonlocal part in computing the axial-vector form factor. As will be discussed later, it is very important to take into account the nonlocal part to describe the experimental data at Q2=0.The second axial-vector form factor RA(Q2) comes into play, when the momentum of the photon is virtual. That is, one can get access to it by π+→e+νee+e− decay in which the virtual photon is annihilated into e+ and e−. As shown in the right panel of Fig. 2, the Q2 dependence of RA(Q2) is similar to FA(Q2). However, the nonlocal contribution is relatively small and positive. Moreover, it starts to increase as Q2 increases, which makes Q2 dependence slightly milder than those of the vector and axial-vector form factors. Note that the nonlocal contribution becomes saturated as Q2 further increases.In Fig. 3, we compare two different results of the pion weak form factors, employing the two different forms of the dynamical quark mass given in Eq. (11) and Eq. (12), respectively. The dynamical quark mass with the dipole-type parametrization yields almost the same results for the vector and second axial-vector form factors. On the other hand, it gives a smaller result for the axial-vector form factor by around 12% in comparison with that from the instanton vacuum.In Table 1, we list the results for the form factors at Q2=0 and slope parameters that are defined as(24)FV(Q2)=FV(0)1+aVQ2mπ2,FA(Q2)=FA(0)1+aAQ2mπ2, where aV and aA denote the slope parameter for the vector and axial-vector form factors, respectively. The values of the vector and axial-vector form factors at Q2=0 are respectively given as FV(0)=0.0271, FA(0)=0.0132, and RA(0)=0.0462 when the dynamical quark mass from the instanton vacuum is used. The dipole-type parametrization yields FV(0)=0.0269, FA(0)=0.0116, and RA(0)=0.0459. As expected from the results for the form factors shown in Fig. 3, the values of FA(0) from these two different forms of the dynamical quark mass are around 12% different each other. The results are in good agreement with the experimental data FVexp=0.0258(17) and FAexp=0.0117(17) [15]. It is also of great interest to compare the present results with those from other works. Unterdorfer and Pichl [20] analyzed the vector and axial-vector form factors of pion radiative decay, combining the results from χPT with a large Nc expansion and experimental data on other decays. The results are obtained as FV(0)=0.0262(5) and FA(0)=0.0106(36), which are in good agreement with the present results. Courtoy and Noguera [47] employed the NJL model to study the π photo-transition amplitude and derived from it the pion form factors as FV(0)=0.0242 and FA(0)=0.0239. So, the result of the vector form factor is comparable with that of the present work whereas that of the axial-vector form factor is two times larger than this one.The results of Ref. [23] are especially interesting, since the nonlocal NJL model used in Ref. [23] has several aspects in common with the present model. In Ref. [23], three different parameter sets were adopted, among which the results with set A are compared with the present ones. Those from Ref. [23] with set A are listed in Table 1 and are in good agreement with the present results except for the slope parameters. It implies that the vector and axial-vector form factors they obtained fall off more slowly than the present ones. What is interesting is that the value aV=0.032, which is derived from the empirical fit to π0→γγ⁎ experimental data in Ref. [23], is in good agreement with that of the present work.In χPT, the values of FA(0) and RA(0) are given in terms of the low-energy constants (LECs), L9 and L10(25)FA(0)=42mπfπ(L9+L10),RA(0)=42mπfπL9. We obtain the values from the present numerical calculation as Table 2.It is also of interest to extract the parameters for the parametrization of the vector and axial-vector form factors for π radiative weak decays. In lattice QCD, the p-pole parametrization for a form factor is often utilized [48,49], which is different from the typical parametrization given in Eq. (24). Then, the present three transition form factors can be parametrized as(26)FV(Q2)=FV(0)(1+Q2pVmpV2)pV,FA(Q2)=FA(0)(1+Q2pAmpA2)pA,RA(Q2)=RA(0)(1+Q2pRmpR2)pR, where the results of the corresponding parameters are listed in Table 3.6Summary and conclusionIn the present work, we aimed at investigating the form factors for pion radiative weak decays, based on the gauged effective chiral action derived from the instanton vacuum. We computed the vector and axial-vector transition form factors FV(Q2), FA(Q2), and RA(Q2), employing the momentum-dependent dynamical quark mass from the instanton vacuum and that with the dipole-type parametrization. The nonlocal contributions, which arise from the gauging of the effective chiral action, enhance the vector form factor by about 20%, whereas they reduce the axial-vector form factor FA(Q2) by about 30%. The nonlocal terms influence the second axial-vector form factor marginally. The difference between the results from the instanton vacuum and those with the dipole-type dynamical quark mass is almost the same except for the axial-vector form factor for which the result with the dipole-type parametrization is about 12% smaller than that from the instanton vacuum. The present results were compared with the experimental data and were found to be in good agreement with the data except for the slope parameter aV. We also derived the low-energy constants L9r and L10r. Finally, we parametrized the form factors, using the p-pole parametrization, which can be used to compare the present results with those from the lattice data.It is also of interest to consider other types of the form factors for pion radiative decay such as tensor transition form factors within the present framework. These tensor form factors may give a clue about a right direction beyond the Standard model. 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