Amplitude analysis of the anomalous decay $\eta'\to\pi^+\pi^-\gamma$

In this paper we perform an amplitude analysis of $\eta'\to\pi^+\pi^-\gamma$ and confront it with the latest BESIII data. Based on the final-state interaction theorem, we represent the amplitude in terms of an Omn\'es function multiplied by a form factor that corresponds to the contributions from left-hand cuts and right-hand cuts in the inelastic channels. We also take into account the isospin violation effect induced by $\rho-\omega$ mixing. Our results show that the anomaly contribution is mandatory in order to explain the data. Its contribution to the decay width of $\Gamma(\eta'\to\pi\pi\gamma)$ is larger than that induced by isospin violation. Finally we extract the pole positions of the $\rho$ and $\omega$ as well as their corresponding residues.

The so-obtained knowledge of these form factors is also crucial in the determination of hadronic contribution to the anomalous magnetic moment of the muon [9][10][11], as witnessed by the preparation for the planned experiments at Fermilab [12] and J-PARC [13]. Furthermore, the two processes are helpful for decoding information on the resonances (intermediate states) such as ρ and ω. For instance, the branching ratio of ω → ππ has been extracted in Ref. [5].
a peak shift of about +20 MeV/c 2 in the di-pion mass spectrum, with respect to the expected position of the ρ 0 . This certainly indicates that only a contribution from the ρ is not sufficient. This issue is also discussed in Ref. [22] for B ℓ4 decay. In the analysis of Ref. [23], it is shown that the contribution from the box anomaly could be essential in η ′ → π + π − γ. Later, the significance of the box anomaly was found to be 4σ by the Crystal Barrel (CB) Collaboration [24] with 7400 events, while the L3 Collaboration [25] claimed that the ρ contribution is sufficient to describe the data with less data (2123 ± 53 events). Nonetheless, the CB data is not precise enough to disentangle the effect of ρ − ω interference from others in the line shape, as shown, e.g., by Ref. [26]. Recently, the BESIII Collaboration [27,28] explored the process η ′ → π + π − γ with very high statistics (of about 9.7 × 10 5 events) and the ρ−ω interference is seen for the first time in this decay. Therefore, it is timely to make a refined amplitude analysis of the anomalous decay η ′ → π + π − γ.
In the η ′ → π + π − γ decay, the contribution of the anomaly is significant, and hence can not be simply determined by a tree-level amplitude from the Wess-Zumino-Witten (WZW) [29,30] term. In Ref. [31], this anomalous decay is studied using χPT in combination with a non-perturbative method based on coupled channels. Here, by properly taking into account the effects of finalstate interaction (FSI) and ρ − ω mixing, we aim at obtaining precise information about the anomaly. On the one hand, since the effect of three-body rescattering between the pions and the photon is negligible due to the tiny electromagnetic interaction, a purely strong ππ FSI should be sufficient. The non-perturbative ππ FSI is implemented in a model-independent way, where the contribution corresponding to the unitary cut is represented by an Omnès function. On the other hand, our treatment of ρ− ω mixing is beyond the simplified version employed in Ref. [5]. In our case, the isospin-violating ρ − ω interference is constructed by invoking resonance chiral theory (RχT) [32,[34][35][36][37][38]80] (for earlier attempts using χPT with explicit vector mesons, see e.g., Refs. [39,40]). The approach we use here is a combination of dispersion theory and chiral effective field theory. Similar prescriptions have been widely applied to, e.g., γγ → ππ [41,42], η → 3π [43], φ/ω → πγ * [44], D ℓ3 [45] and B ℓ4 [22]. With our method, it is clearly shown that the anomaly is mandatory in order to describe the experimental data. Besides, it is possible and also interesting to see how much the leading-order (LO) result in the 1/N C expansion of RχT is modified by the ππ rescattering. For a general discussion the role of vector mesons in anomalous processes, see e.g. [46,47]. This paper is organized as follows. In Section II we discuss the formalism for the η ′ → ππγ decay amplitude. The inclusion of the FSI effect is discussed in Section II A, while the isospin-violating form factors are calculated within RχT in Section II B. Section II C provides the final form of the decay amplitude. Section III contains our numerical results. We fit to the experimental data and pin down all the relevant unknown parameters in Section III A. In Section III B, we extract the poles of the ρ and ω resonances as well as their couplings to the η ′ γ states, which are then used to calculate the decay widths. We also discuss the impact of isospin-violating effect on the P -wave phase of π + π − scattering. Finally, we summarize and make conclusions in Section IV. The explicit expressions of the isospin-violating form factors in RχT are relegated to Appendix A.

A. Final state interaction
Bose and charge conjugation symmetry guarantee that η ′ → π 0 π 0 γ is forbidden and hence we only need to consider the mode η ′ → π + π − γ. As will be explained below, its full amplitude is composed of partial waves with isospin I = 1 and odd angular momentum, i.e. P , F , . . . waves. The isospin violation part induced mainly by the ρ−ω mixing is of isospin I = 0, which will be discussed in the next section. The Lorentz-invariant decay amplitude for η ′ (q) → π + (p + )π − (p − )γ(k) can be written as: where ǫ µ (k, λ) is the polarization of the outgoing photon with helicity λ. The variables of the form factor F λ are chosen to be s ≡ (q − k) 2 and cos θ. Here, θ is the scattering angle in the ππ center of mass frame. The partial-wave decomposition of M λ reads where the normalization factor should be set to N ππ = 2, in accordance with the Bose statistics of identical particles. The isospin decomposition is given by M Jλ (s) = −M 1 Jλ (s)/ √ 2, so that with From the parity conservation one has F 1 Jλ (s) = (−) J F 1 J(−λ) (s). Here we only keep the lowest P -wave and ignore F -and higher partial waves since their contributions are relatively small. Following these constraints we have only one independent partial wave, M 1 1+ (s) or M 1 1− (s). Comparing Eqs. (1-4) one soon finds F + (s, cos θ) = −3/(2 √ 2)F 1 1+ (s). Notice that higher order corrections of QED are negligible compared to hadronic FSI, thus we do not take them into account. Finally we construct our amplitude based on Watson's FSI theorem: with Ω 1 1 (s) the so-called Omnès function and P (s) a polynomial. We will discuss P (s) in Section II C. The Omnès function satisfies the following dispersion relation The function ϕ 1 1 (s) denotes the phase of P -wave elastic ππ → ππ amplitude, which was given in a previous amplitude analysis of ππ scattering [48,49]. In Fig. 1, we show the phase and modulus of the Omnès function. Up to the KK threshold, it has the same phase as that of the decay amplitude F 1 1+ (s).

B. Formalism of RχT
In this section, we will calculate the chiral anomaly contribution as well as the isospin-violating amplitude for η ′ → π + π − γ at LO in the large N C expansion in RχT, with N C the number of colors. The ρ − ω mixing is taken into account following Ref. [38]. The relevant chiral Lagrangian can be written as [34]: The part L GB (4) containing the LO operators of the chiral anomaly is [29,30] L GB (4) where F ≈ 92.2 MeV is the pion decay constant in the chiral limit and Φ is a nonet matrix collecting the pseudoscalar Goldstone bosons: The mixing of the η 8 and η 0 with an angle θ P yields the physical η and η ′ states 2 .
The kinematic part of the vector resonances reads where M V is the mass of the vector resonances in the chiral limit. The matrix field V µν , in antisymmetric tensor representation, incorporates the low-lying vector resonances in a nonet form with λ i (i = 1, · · · , 8) and ½ the standard Gell-Mann matrices and the 3 × 3 unit matrix, respectively. The covariant derivative acting on the vector fields is defined by [32] where v µ and a µ denote external vector and axial-vector fields, respectively. It is worth noting that the photon field A µ can be introduced by setting r µ = l µ = e Q A µ with Q = diag{2/3, −1/3, −1/3}. Furthermore, u = exp{iΦ/( √ 2F )}.
The interaction between the vector resonances and the Goldstone bosons is described by Here, the subscripts denote the chiral orders, the corresponding superscripts imply the numbers of vector resonance. Specifically, the first term reads [32] L 2 We notice that more complicated schemes for η − η ′ mixing involving two mixing angles have been studied in e.g., Refs. [50][51][52][53][54], which is commonly used to describe the two-photon decays well [50,51,54].
Here, F V and G V are unknown coupling constants, that can be fixed from certain resonance decays or from shortdistance QCD constraints, see in Appendix A. The chiral building blocks, f µν + and u µ are given explicitely in Ref. [32]. The last two terms in Eq (14) are of oddintrinsic parity. The pieces relevant to our calculation can be expressed as [34,37] These couplings are defined to be dimensionless. For the explicit expressions of the odd-intrinsic chiral operators we refer the readers to Refs. [34,37]. The values of the parameters, c i , g j and d k , are taken from Ref. [38], and are also given in our Appendix. A.
The fields used in the above-mentioned chiral effective Lagrangians are convenient for analyzing transformation properties under the chiral group. Nonetheless, not all of them directly correspond to physical states. In practice, the physical ω(782) and φ(1020) states are related to the octet and singlet components by a mixing angle In the same manner, the ρ − ω mixing due to isospin symmetry violation can be parameterized as with δ the mixing angle. The values of these mixing angles we used are listed in Appendix A 3 .
Eventually, we are in the position to calculate the amplitudes for η ′ → π + π − γ including explicitly the WZW term as well as the isospin violation. Throughout, we assume that the ρ-ω mixing is the dominant isospin breraking effect. The relevant Feynman diagrams are displayed in Fig. 2 and the resulting amplitudes are given in Appendix. A.

C. Isospin-violating form factor
To include the dominant isospin-violating effect, we utilize the following form for the polynomial in Eq. (5), The contributions from left hand cuts (l.h.c) and inelastic right hand cuts (r.h.c) are ascribed to the polynomial in front, and P i.v. (s) stands for the contribution from isospin violation. Following the experimental paper [27], we set with F i.v. tree (s) the isospin-violating form factor given in Eq. (A.3). The phase of ππ rescattering is included by the Omnès function, and the l.h.c and inelastic r.h.c contributions of isospin violation are absorbed in the parameter β 0 . In principle there should be more terms rather than a single β 0 , but in practice we find that one parameter is good enough to describe the data well.
We notice that F i.v. tree (s) has a sizeable imaginary part around √ s = M ω . It modifies the π + π − phase in the vicinity of √ s = M ω . Nevertheless, this is helpful for us to obtain ρ − ω mixing exactly in the π + π − FSI. Previous dispersive analyses [56,57] and also experiments [58][59][60] take only the contribution of ρ in their partial wave analysis, thus the information of isospin-violating part, i.e. the contribution from the ω, is lacking in this region. This will be discussed with more details in the next section.
For the isospin-violating form factor F i.v. tree (s), we only need to calculate the diagrams, (b), (c) and (d), in Fig. 2. The results are given in Appendix A. Finally, the total (d) The Feynmann diagrams of the Wess-Zumino-Witten (WZW) term (Fig.a), and the isospin violation amplitudes (Figs.b-d). The symbol 'V' represents the vector resonances ρ, ω, and φ. The odd-intrinsic parity and isospinviolating vertices are denoted by the circled crosses and black squares, respectively. Note that in Fig (b) the black square also represents an odd-intrinsic parity vertex. amplitude is given by: contains an isospin-violating part (I = 0), though its superscript is labeled by '1', corresponding to I = 1. With this amplitude one can get the formula for the di-pion mass spectrum: with ρ(s) = 1 − 4M 2 π /s. When fitting to the invariant mass spectrum of BESIII [27], we need to multiply it by a normalization factor N . In Ref. [3], the anomaly is obtained at the point where s = t = u = 0, in the chiral limit. Similarly, we define the anomaly as −eF + (0, cos θ). Notice that here θ is a dummy variable in F + (0, cos θ). We have

A. Fit to experimental data
In this section, we fit to the invariant di-pion mass spectrum by BESIII [27]. The decay width of η ′ → π + π − γ provided by the Particle Data Group (PDG) [61] is implemented in our fit to constrain the unknown normalization factor N . The following fits are performed: 1) Fit 1: We ignore isospin violation (β 0 = 0). The best-fit parameters are collected in the second column of Table I. 2) Fit 2: We include the contribution of isospin violation and α 0 is fixed by Eq. (24). The results are shown in the third column in Table I. 3) Fit 3: As in Fit 2 we include the isospin violation, but set α 0 to be a free parameter. The results are shown in the fourth column in Table I. In fact, a chiral matching can be imposed to fix α 0 [62]. At low energies, our amplitude is required to coincide with the one calculated from the LO WZW term [29,30]. The corresponding Feynman diagram is shown in Fig. 2 (a) and its form factor is given in Eq. (A.2). We choose s = 4M 2 π as the matching point to avoid any complication caused by the α 1 term in Eq. (19). After the chiral matching described above, one obtains The value of α 0 is fixed to be 14.37 GeV −3 provided θ P = −21.37 • . Note that Ω 1 1 (4M 2 π ) = 1.159 is a real number. This corresponds to Fit 2. In contrast, if we use double-angles-mixing scheme (DAMS), we obtain α 0 = 15.17 GeV −3 , where the angles and decay constants are taken from 'NNLO Fit-A' of Ref. [53]. In this case α 0 is fairly increased and still faraway from that of Fit 3. In the absence of the high statics data of η → π + π − γ, one can not reach a definite conclusion on how DAMS will improve the calculation.
Actually, one may treat θ P as a free parameter, while α 0 is always fixed by using Eq. (24) during the fit procedure. A good fit can be obtained with θ P ≈ −10.9 ± 0.5 • , which is compatible with the determinations in Refs. [3,[63][64][65]. However, the resulting decay widths involving η ′ , Γ η ′ →ργ , Γ η ′ →ωγ and Γ φ→η ′ γ , are now deviated about 30-60 percents from the previous determinations in Ref. [38]. Thus, we fix θ P at −21.37 • [38] and set α 0 free. This is Fit 3. Note that here Eq. (24) is not implemented as a constraint. In Fit 3 the BESIII data can be well described and, furthermore, the previous results in Ref. [38] for the above-mentioned decay widths are untouched. For comparison, the invariant mass spectrum, based on the fitted values of the parameters from Fit 1, Fit 2 and Fit 3, are shown simultaneously in Fig. 3. Note that the Crystal Ball data points are superimposed on the plot.   For Fit 1, the invariant mass spectrum is represented by the blue dotted line in Fig. 3. Obviously, there is no ρ-ω mixing structure appearing in the energy region √ s ∈ [0.76, 0.8] GeV. This is not surprising as we do not take the isospin violation into account. The resulting decay width of η ′ → π + π − γ is 57.3 ± 8.6 keV is absolutely in agreement with the value given by PDG. For Fit 2, the fit to the BESIII data is much better. The invariant mass spectrum of Fit 2 is almost indistinguishable from that of Fit 3, thus we do not show it in Fig. 3. However, the decay width of η ′ → π + π − γ, which is 38.9 ± 8.2 keV now, deviates from the value of PDG [61] by 32%. The anomaly is estimated to be 4.36 GeV −3 . For Fit 3, the fit is rather good. The fit quality is improved both in the low energy region and the ρ − ω mixing region, compared to Fit 2. This is reasonable as, by tuning the free parameter α 0 , the amplitude at low energies can be adjusted, and the same holds for the isospin violation part in the high-energy region. We notice that α 0 is shifted by 28% compared to that of Fit 2, which also quantifies how much the anomaly can affect the fit. Consequently, the anomaly is 5.61 ± 0.29 GeV −3 , shifted by 29% from the one given by matching to tree-level amplitude of the WZW term. Since the tree-level amplitude is calculated in the large N C expansion, a typical 1/N C correction is reasonable. Considering also the improved fit quality of Fit 3 with respect to Fit 2, the correction of order 1/N C to the tree-level amplitude of WZW term in anomalous decay process, is not only reasonable but also necessary. The normalization factor in Fit 3 is similar to the one in Fit 1 but decreases a lot compared to that of Fit 2. The reason is that, when α 0 and α 1 increase, the normalization factor has to decrease so as to compensate for the amplitude F 1 1+ (s). Comparing the quality of these fits, The BESIII data is from [27] and Crystal Ball data from [24]. The bottom figure corresponds to the fit to the experiment data of ππ → ππ P -wave. The Cern-Munich data is from [58], and the olive and light grey bands in the low energy region are from Refs. [66][67][68][69].
we consider Fit 3 to be the reference result.
In the next section, in order to extract the couplings of the resonances, the P -wave ππ scattering amplitude is needed in addition to the decay amplitude of η ′ → π + π − γ described above. To get the P -wave ππ scattering amplitude with isospin I = 1, we adopt the following representation with the c i unknown constants. The two constants, c 0 and c 1 , are fixed by the relevant threshold parameters: scattering length ∼ 0.0387 ± 0.0012 M −3 π and slope parameter ∼ 0.0051 ± 0.0026 M −5 π [57]. The Cern-Munich data [58], CFDIV amplitude [57], and the amplitude from the Roy equation analysis [  shown in the bottom panel of Fig. 3. Its analytic continuation to the complex s-plane, confronted with that of the Roy equation analysis [56], is shown in Fig. 4. We do not plot the amplitude on the upper half of s-plane, as it is readily obtainable from the one on the lower half of s-plane according to the Schwarz reflection principle. The contribution of l.h.c to the shade region of s-plane, as shown in Fig. 4, is properly implemented by fitting to the data as well as the amplitude of Roy equation in the presence of crossing symmetry. We see that our amplitudes On the left side are real and imaginary parts of our amplitudes, and on the right side are those from Roy equation [56].
are quite similar to the ones from the Roy equation analysis on the complex s-plane. The distribution of contours is in good agreement with each other. Moreover, their gradient variations, the shading of the color from blue to red, are compatible. Nevertheless, amplitudes on the edge of the domain, shown in Fig. 4, are less consistent with a difference ≤ 0.1. This means our T 1 P (s) amplitude is constrained rather well on the complex plane, allowing for a reliable extraction of the poles and residues.

B. Couplings of g η ′ V γ
With a specific amplitude, the couplings of a resonance are defined by the residues of the pole on the complex splane. Based on the results of Fit 3, the absolute values of the couplings g η ′ V γ , V ∈ {ρ, ω}, extracted on the appropriate Riemann sheets are compiled in Table III.
The definition of the couplings on the appropriate Riemann sheet is given as where 'II' denotes the second Riemann sheet. M 1 1+ is the η ′ → ππγ decay amplitude given in Eq. (2) and T 1 P (s) is the P -wave ππ → ππ scattering amplitude from Eq. (25). What those couplings mean in terms of decay width is provided as with α the usual QED fine structure constant.
To obtain the couplings of ω, we adopt the standard Blatt-Weisskopf barrier factor representation [42] with Q(s) = 1 + q 2 /(s − 4M 2 π ) and q is chosen to be 1 GeV. The mass, width and ππ-mode branching ratio of the ω are taken from PDG [61]: M ω = 782.65±0.12 MeV, Γ ω = 8.49 ± 0.08 MeV and BR ω→ππ = 0.0153 ± 0.0012. Notice that the width is rather small and thus we can ignore its energy dependence 5 . To proceed, we define the coupling g η ′ ωγ through 5 In a more dedicated way, one can use the standard Breit-Wigner formalism to represent the ππ → ω → ππ amplitude [42]: .
Here 1, 2, 3 represents the πγ, ππ and πππ channels, respectively. Following it one can extract out the pole and residue on the (-,-,-) plane. As we have checked, the results obtained in this way are quite the same as what we obtained in Table III.  where s ω = M 2 ω − iM ω Γ ω and BW[ω, s] is the Breit-Wigner representation given in Eq. (A.7).
To get the anomaly contribution to the width of Γ(η ′ → π + π − γ), for simplicity we use F 1 1+ (0) instead of F 1 1+ (s), when integrating over √ s in Eq. (22). The differential decay width of the mode η ′ → ωγ → π + π − γ is written as The contribution through the intermediate ρ meson is the same as Γ(η ′ → ργ) since ρ decays into ππ to hundred percent, see also Eq. (32). Finally, we obtain the poles and couplings, based on Fit 3, and compute the decay widths with the help of Eqs. (27), (28) and (31). The results are shown in Table III. Our uncertainty is from the fit, combined the error from MINUIT and the systematic one: the correlation between the coefficients, see Eqs. (21,25), and the uncertainty of the phase in the Omnès function. Our estimation shows that the systematic error dominates the uncertainty.
We notice that the pole position of ρ is compatible with the one obtained by Padé approximants in [70]. The ρ pole position is shifted a bit compared to the Breit-Wigner mass and width given by PDG [61], while this is not the case for ω. The reason is that ρ is much wider and the pole is farther away from the real axis, thus 'narrow resonance approximation' is not good enough to describe the amplitude. With the pole locations one can extract out the residues and thus determine the contribution to Γ(η ′ → π + π − γ) from each resonance, separately.
The contributions to the decay width of Γ(η ′ → π + π − γ), induced by intermediate ρ and ω as well as anomaly 'A', are given as follows: 56.6 ± 5.3 keV through ρ, 67.5 ± 16.0 eV through ω, and 3.34 ± 0.35 keV through the anomaly. The contact isospin-violating term, shown in Fig. 2 (b), contributes to the anomaly part too. However, its contribution is tiny and hence neglected. It is found that the ρ dominates the contribution and the anomaly contributes more than the ω. This is not surprising as the ρ dominates the P -wave and the anomaly contributes as a background in the whole energy region, while the ω only acts in a small region and rarely decays into ππ.
For the decay widths of η ′ → V γ, we obtain Γ(η ′ → ργ) = 56.6±5.3 keV, while in LO RχT [38] it is 53.7 keV. Likewise, Γ(η ′ → ωγ) = 4.41 ± 1.04 keV (or 4.10 ± 0.97 through residue in Eqs. (27,30)), while in LO RχT it is 5.12 keV. Since in LO RχT the uncertainty is roughly 1/3 when truncating the large N C expansion, the widths obtained here and in LO RχT are compatible if the errors are taken into account. This shows us how much the ππ FSI affects the strong interaction between the lightest vector and pseudoscalar mesons. Compared to the predictions given by RχT, the FSI effect is sizable but still within the uncertainty.
In addition, we find that in the whole kinematical region the phase of π + π − based on Fit 3 is in good agreement with the one based on Fit 1, i.e., the phase of T 1 P (s), except for the energy region around √ s = M ω , see Fig. 5. Obviously, there is a bump around the ω mass when taking into account the ρ − ω mixing. Note that there is no such bump for π ± π 0 , as the ω is an isospin singlet. It should be noted that for η ′ → π + π − γ the l.h.c (mainly from the a 2 ) and inelastic r.h.c (above KK threshold) are too far away from the energy region we are working. Thus the polynomial P (s), in which the l.h.c and r.h.c are absorbed, should be smooth in the physical energy region. This suggests that, to include isospin violation from the ρ − ω mixing in the line shape of η ′ → π + π − γ, it is more convenient to change the phase of π + π − in the Omnés function than to include a complicated complex form factor in P (s) function. We notice that there is only one data point from [59] located at around √ s = M ω , but it was obtained with low resolution. Most experiments only take into account the isospin-conserved part (the contribution from the ρ) [58][59][60]. To reach a more definite conclusion on the phase caused by isospin violation, more accurate experimental measurement in the √ s = M ω region and more careful theory analysis are required. Here we just point out that, through our amplitude analysis of the η ′ → π + π − γ data provided by Ref. [27], there should be a bump of the π + π − P -wave phase around the energy point √ s = M ω .

Acknowledgment
We are grateful to Zhi-Hui Guo for helpful discussions. We also wish to thank Chu-Wen Xiao for a careful reading of our manuscript and for his suggestions. This work is supported in part by the Deutsche The amplitude of η ′ → π + π − γ is given by For the form factor F (s), the tree-level WZW anomaly term gives The isospin-violating form factor is and K + 6 sin 2 δ(2 cos θ P + √ 2 sin θ P ) + cos 2 δ (4 cos(2θ V + θ P ) G Rη ′ (Q 2 , s) = (g 1 + 2g 2 − g 3 ) (s − 2m 2 π ) +g 2 (−Q 2 + 2m 2 π + m 2 η ) + (2g 4 + g 5 ) m 2 π , Notice that Γ V (0) = 0. Here θ V is the ω−φ mixing angle, δ the ρ − ω mixing angle and θ P the η − η ′ mixing angle. We follow the construction of vector resonance off-shell widths in Ref. [76], where the parametrization of Γ ρ (q 2 ) is employed as with F the pion decay constant. For ω, φ width we use constant decay widths. The resonance parameters are given by PDG [61], and all other parameters can be found in Fit 4 of Ref. [38]. For convenience we compile them in and QCD one can find the constraints on the unknown couplings [34,37]: We notice that these constraints are not the same as a more complicated one [77], where heavier pseudoscalar resonances are included. Since we only focus on the physics below √ s = M η ′ , we do not take the heavier resonances into consideration. It would be interesting to note that our F V is closer to F V = √ 3F [78,79] rather than F V = √ 2F [80]. The former constraint is from the combined analysis of axial-vector current with the contribution from three pseudoscalars (τ → KKπν τ ) and τ → γP ν τ (or V P P ), while our constraints are based on the analysis of vector current with the contribution from three pseudoscalars (e + e − → π + π − π 0 /η). A more careful study is needed for a good understanding of the short-distance QCD constraint.