Improved description of di-lepton production in $\tau^{-}\to\nu_{\tau}P^{-}$ decays

In this section we will present, for the sake of simplicity, only the relevant operators in the R$\chi$T Lagrangian needed to compute the form factors, which are given in the next subsection. We will be concise here, for a more extended discussion see eg. ref. [17]. R$\chi$T extends the domain of applicability of Chiral Perturbation Theory [16, 14, 15] ($\chi$PT) by adding the light-flavored resonances as active degrees of freedom.

We start with operators involving no resonances, these being ¹¹1Although these terms also appear in the $\chi$PT Lagrangian, their couplings get shifted in the presence of resonance contributions (see for instance [22, 23, 24]).

$$\mathcal{L}_{0\,Res}=\frac{f^{2}}{4}\langle u^{\mu}u_{\mu}+\chi_{+}\rangle+\mathcal{L}_{WZW}+C_{7}^{W}\mathcal{O}_{7}^{W}+C_{11}^{W}\mathcal{O}_{11}^{W}+C_{22}^{W}\mathcal{O}_{22}^{W}\,,$$

(2)

where the first term is given by the leading $\chi$PT Lagrangian operators of chiral order $p^{2}$ [16, 14, 15], the second one is the anomalous Wess-Zumino-Witten Lagrangian of $\mathcal{O}(p^{4})$ [25, 26] and the last three operators belong to the subleading odd-intrinsic parity sector $\mathcal{O}(p^{6})$ Lagrangian [27]. We neglect operators not included in this Lagrangian. Congruently with refs. [19], [21] and [28], we will not consider any $\mathcal{O}(p^{8})$ contribution whatsoever. In the first term, $f$ is the decay constant in the chiral limit, which we will set to $f=f_{\pi}\sim 92$ MeV, $u^{\mu}$ and $\chi_{+}$ are chiral tensors [29], the former containing derivatives of the $\pi/K$ fields and external spin-one currents and the latter scalar currents involving the previous fields masses squared, $m_{\pi/K}^{2}$, times even powers of such fields.

The equations of motion of the resonances give their classical fields in terms of series of chiral tensors of different order. The resonances are said to be integrated out (tree-level integration) when the classic fields are substituted in favor of chiral tensors in the resonant Lagrangian. Integrating the resonances out using the leading-order terms of the equations of motion very approximately saturates the $\mathcal{O}(p^{4})$ (and leading $\mathcal{O}(p^{6})$) contributions in the even-intrinsic parity sector [12, 13, 19]; therefore, we will not use the non-resonant $\mathcal{O}(p^{4})$ set of operators for the sake of simplicity, since they are considered to yield negligible contributions. Since we will only consider leading-order terms in the resonances equations of motion, the $\mathcal{O}(p^{6})$ chiral low-energy constants in the odd-intrinsic parity sector cannot be saturated upon resonance exchange [28], therefore we have to include the three contributing $C_{i}^{W}\mathcal{O}_{i}^{W}$ terms [27]:

	$\displaystyle\mathcal{O}_{7}^{W}$	$\displaystyle=$	$\displaystyle i\epsilon_{\mu\nu\alpha\beta}\langle\,\chi_{-}f_{+}^{\mu\nu}f_{+}^{\alpha\beta}\,\rangle,$
	$\displaystyle\mathcal{O}_{11}^{W}$	$\displaystyle=$	$\displaystyle i\epsilon_{\mu\nu\alpha\beta}\langle\,\chi_{+}[f_{+}^{\mu\nu},f_{-}^{\alpha\beta}]\,\rangle,$		(3)
	$\displaystyle\mathcal{O}_{22}^{W}$	$\displaystyle=$	$\displaystyle i\epsilon_{\mu\nu\alpha\beta}\langle\,u^{\mu}\{\nabla_{\rho}f_{+}^{\rho\nu},f_{+}^{\alpha\beta}\}\,\rangle\,,$

where the following chiral tensors [29] enter: $\chi_{-}$ gives odd powers of the $\pi/K$ fields with factors involving $m_{\pi}^{2}$ or $m_{K}^{2}$, $\nabla_{\mu}$ is the covariant derivative and includes spin-one left and vector external currents through the connection and $f_{\pm}^{\mu\nu}$ yields the field-strength tensors of the charged-weak or electromagnetic fields.

We turn next to those operators with one resonance field, in either intrinsic parity sector,

$$\mathcal{L}_{1\,Res}=\mathcal{L}_{1\,Res}^{even}+\mathcal{L}_{1\,Res}^{odd}\,.$$

(4)

In turn, the first piece can be further divided according to the quantum numbers of this resonance

$$\mathcal{L}_{1\,Res}^{even}=\sum_{R_{i}=V,A,P}\mathcal{L}_{1\,R_{i}}^{even}\,.$$

(5)

The contributions with one vector resonance read ²²2$V_{\mu\nu}$ (analogously $A_{\mu\nu}$ for axial resonances below) is a matrix in flavor ($u,d,s$) space and we use the antisymmetric tensor formalism for spin-one fields for convenience [12, 13]. [12, 19]

$$\mathcal{L}_{1\,V}^{even}=\frac{F_{V}}{2\sqrt{2}}\langle\,V_{\mu\nu}f_{+}^{\mu\nu}\,\rangle+\frac{i}{2\sqrt{2}}G_{V}\langle\,V_{\mu\nu}[u^{\mu},u^{\nu}]\,\rangle+\frac{\lambda_{V}}{\sqrt{2}}\langle\,V_{\mu\nu}\{f_{+}^{\mu\nu},\chi_{+}\}\,\rangle\,,$$

(6)

where the $V$ field (we assume ideal mixing of neutral mesons) has an analogous flavor structure as the pseudo Goldstone field $\phi$, namely

$$V_{\mu\nu}=\left(\begin{array}[]{ccc}\frac{1}{\sqrt{2}}\left(\rho^{0}_{\mu\nu}+\omega_{\mu\nu}\right)&\rho^{+}_{\mu\nu}&{K^{\star}}^{+}_{\mu\nu}\\ \rho^{-}_{\mu\nu}&\frac{1}{\sqrt{2}}\left(-\rho^{0}_{\mu\nu}+\omega_{\mu\nu}\right)&{K^{\star}}^{0}_{\mu\nu}\\ {K^{\star}}^{-}_{\mu\nu}&{\overline{K}^{\hskip 1.50694pt\star}}^{0}_{\mu\nu}&\phi_{\mu\nu}\end{array}\right).$$

(7)

In eq. (6), the first two operators give the contribution from the coupling of vector resonances to external fields in the chiral limit and the last term gives the flavor-breaking corrections to such couplings. Our $\lambda_{V}=\sqrt{2}\lambda_{6}^{V}$, using the notation in ref. [19]. This last operator is the only one included from the full basis of $\mathcal{O}(p^{4})$ even-intrinsic parity operators in ref. [19] since it is the single one that can contribute to the $U(3)_{V}$ breaking in the $V-\gamma$ coupling. There are, however, two reason to disregard basis of operators in the even-intrinsic parity sector: The operators that are relevant to the process can be dismissed on the basis of resonance field redefinitions³³3Through the redefinition of the vector resonance field $V\to V+g\{V,\chi_{+}\}$ it is possible to cancel the $\lambda_{V}$ operator [19], however, we keep it in order to show the full basis of possible $U(3)_{V}$ breaking operators since we do not consider the full even-intrinsic parity basis of ref. [19]. We will show later that this is consistent, since the short distance constraints give $\lambda_{V}=0$.; if we, however, keep such operators, they will only give subleading contributions to those from the first two operators in eq. (6) with no contribution to $U(3)_{V}$-breaking vertices.

The axial resonance operators present a similar feature and an analogous flavor-space structure to that of the vector mesons. This is, the $\mathcal{O}(p^{4})$ one-resonance even-intrinsic parity operators for axial resonances in ref. [19] can be absorbed through field redefinitions. We will therefore disregard any contribution from this part of the Lagrangian, including the $U(3)_{V}$ breaking terms to the axial-vector resonance coupling to external currents, namely, the $JA$ vertex. The remaining contributions with one resonance field are [12]

$$\mathcal{L}_{1\,A/P}^{even}=\frac{F_{A}}{2\sqrt{2}}\langle A_{\mu\nu}f_{-}^{\mu\nu}\rangle+id_{m}\langle P\chi_{-}\rangle,$$

(8)

with $P$ a matrix in three-flavors space containing the lightest pseudoscalar resonances. The inclusion of the pseudoscalar resonance is necessary in order to obtain consistent short-distance constraints in $\langle VAP\rangle$ and $\langle VJP\rangle$ Green’s functions [19, 28, 30, 31]. All Feynman diagrams involving these resonances will give $U(3)$ breaking contributions to the amplitude due to the last term in eq. (8). We have neglected other spin-zero resonance contributions (scalar and heavier pseudoscalar resonances [5]), which are not needed for theoretical consistency and are irrelevant phenomenologically. The odd-intrinsic parity contributions to $\mathcal{L}_{1\,Res}$ are [32] ⁴⁴4Since we are only considering operators with one $\pi/K$ field, these constitute a basis. In the general case, the basis is given in ref. [28]. The translation between them can be read from ref. [30].

$$\mathcal{L}_{1\,Res}^{odd}=\sum_{j=1}^{7}\frac{c_{j}}{M_{V}}\mathcal{O}^{j}_{V}+\varepsilon_{\mu\nu\alpha\beta}\langle\kappa_{5}^{P}\{f_{+}^{\mu\nu},f_{+}^{\alpha\beta}\}P\rangle\,,$$

(9)

with the operators

	$\displaystyle\mathcal{O}^{1}_{V}$	$\displaystyle=$	$\displaystyle\varepsilon_{\mu\nu\rho\sigma}\langle\left\{V^{\mu\nu},f^{\rho\alpha}_{+}\right\}\nabla_{\alpha}u^{\sigma}\rangle\,,$
	$\displaystyle\mathcal{O}^{2}_{V}$	$\displaystyle=$	$\displaystyle\varepsilon_{\mu\nu\rho\sigma}\langle\left\{V^{\mu\alpha},f^{\rho\sigma}_{+}\right\}\nabla_{\alpha}u^{\nu}\rangle\,,$
	$\displaystyle\mathcal{O}^{3}_{V}$	$\displaystyle=$	$\displaystyle i\varepsilon_{\mu\nu\rho\sigma}\langle\left\{V^{\mu\nu},f^{\rho\sigma}_{+}\right\}\chi_{-}\rangle\,,$		(10)
	$\displaystyle\mathcal{O}^{4}_{V}$	$\displaystyle=$	$\displaystyle i\varepsilon_{\mu\nu\rho\sigma}\langle V^{\mu\nu}\left[f^{\rho\sigma}_{-},\chi_{+}\right]\rangle\,,$
	$\displaystyle\mathcal{O}^{5}_{V}$	$\displaystyle=$	$\displaystyle\varepsilon_{\mu\nu\rho\sigma}\langle\left\{\nabla_{\alpha}V^{\mu\nu},f^{\rho\alpha}_{+}\right\}u^{\sigma}\rangle\,,$
	$\displaystyle\mathcal{O}^{6}_{V}$	$\displaystyle=$	$\displaystyle\varepsilon_{\mu\nu\rho\sigma}\langle\left\{\nabla_{\alpha}V^{\mu\alpha},f^{\rho\sigma}_{+}\right\}u^{\nu}\rangle\,,$
	$\displaystyle\mathcal{O}^{7}_{V}$	$\displaystyle=$	$\displaystyle\varepsilon_{\mu\nu\rho\sigma}\langle\left\{\nabla^{\sigma}V^{\mu\nu},f^{\rho\alpha}_{+}\right\}u_{\alpha}\rangle.$

In the following, we quote those terms bilinear in resonance fields (we do not display the kinetic terms for the resonances, which can be found in ref. [12], as they do not contribute to the effective vertices).

$$\mathcal{L}_{2\,Res}=\mathcal{L}_{2\,Res}^{even}+\mathcal{L}_{2\,Res}^{odd}\,,$$

(11)

with [19, 33, 34, 35] ⁵⁵5The operator with coefficient $e^{V}_{M}$ allows to account for $U(3)$ breaking effects in the vector resonance masses, in agreement with phenomenology.

$$\mathcal{L}_{2\,Res}^{even}=-e^{V}_{M}\langle V_{\mu\nu}V^{\mu\nu}\chi_{+}\rangle+\lambda_{1}^{PV}\mathcal{O}_{1}^{PV}+\lambda_{2}^{PV}\mathcal{O}_{2}^{PV}+\lambda_{1}^{PA}\mathcal{O}_{1}^{PA}+\sum_{i=1}^{5}\lambda^{VA}_{i}\mathcal{O}^{VA}_{i}\,,$$

(12)

and [28, 32]

$$\mathcal{L}_{2\,Res}^{odd}=\sum_{i=1}^{3}d_{i}\mathcal{O}_{i}^{VV}+\kappa_{3}^{PV}\mathcal{O}_{3}^{PV}\,.$$

(13)

The operators appearing in the two previous equations are ($h^{\mu\nu}=\nabla^{\mu}u^{\nu}+\nabla^{\nu}u^{\mu}$)

	$\displaystyle\mathcal{O}_{1}^{PV}$	$\displaystyle=$	$\displaystyle i\langle[\nabla^{\mu}P,V_{\mu\nu}]u^{\nu}\rangle,$
	$\displaystyle\mathcal{O}_{2}^{PV}$	$\displaystyle=$	$\displaystyle i\langle[P,V_{\mu\nu}]f_{-}^{\mu\nu}\rangle;$
	$\displaystyle\mathcal{O}_{1}^{PA}$	$\displaystyle=$	$\displaystyle i\langle[P,A_{\mu\nu}]f_{+}^{\mu\nu}\rangle;$
	$\displaystyle\mathcal{O}_{1}^{VA}$	$\displaystyle=$	$\displaystyle\langle[V^{\mu\nu},A_{\mu\nu}]\chi_{-}\rangle,$
	$\displaystyle\mathcal{O}_{2}^{VA}$	$\displaystyle=$	$\displaystyle i\langle[V^{\mu\nu},A_{\nu\alpha}]h_{\mu}^{\alpha}\rangle,$
	$\displaystyle\mathcal{O}_{3}^{VA}$	$\displaystyle=$	$\displaystyle i\langle[\nabla^{\mu}V_{\mu\nu},A^{\nu\alpha}]u_{\alpha}\rangle,$
	$\displaystyle\mathcal{O}_{4}^{VA}$	$\displaystyle=$	$\displaystyle i\langle\nabla^{\alpha}V_{\mu\nu},A_{\alpha}^{\;\nu}]u^{\mu}\rangle,$		(14)
	$\displaystyle\mathcal{O}_{5}^{VA}$	$\displaystyle=$	$\displaystyle i\langle[\nabla^{\alpha}V_{\mu\nu},A^{\mu\nu}]u_{\alpha}\rangle;$
	$\displaystyle\mathcal{O}_{1}^{VV}$	$\displaystyle=$	$\displaystyle\varepsilon_{\mu\nu\rho\sigma}\langle\left\{V^{\mu\nu},V^{\rho\alpha}\right\}\nabla_{\alpha}u^{\sigma}\rangle,$
	$\displaystyle\mathcal{O}_{2}^{VV}$	$\displaystyle=$	$\displaystyle i\varepsilon_{\mu\nu\rho\sigma}\langle\left\{V^{\mu\nu},V^{\rho\sigma}\right\}\chi_{-}\rangle,$
	$\displaystyle\mathcal{O}_{3}^{VV}$	$\displaystyle=$	$\displaystyle\varepsilon_{\mu\nu\rho\sigma}\langle\left\{\nabla_{\alpha}V^{\mu\nu},V^{\rho\alpha}\right\}u^{\sigma}\rangle,$
	$\displaystyle\mathcal{O}_{3}^{PV}$	$\displaystyle=$	$\displaystyle\varepsilon_{\mu\nu\alpha\beta}\langle\{V^{\mu\nu},f_{+}^{\alpha\beta}\}P\rangle.$

There is only one relevant operator with three resonance fields in either parity sector

$$\mathcal{L}_{3\,Res}=i\lambda^{VAP}\langle[V_{\mu\nu},A^{\mu\nu}]P\rangle+\kappa^{PVV}\varepsilon_{\mu\nu\rho\sigma}\langle V^{\mu\nu}V^{\alpha\beta}P\rangle.$$

(15)

Operators with a higer number of resonant fields will not be included, since otherwise one has to include subleading diagrams with loops where some of the internal lines are given by resonances. The present analysis is restricted to tree-level diagrams, which should already capture the leading effects associated with resonance exchange. One-loop diagrams with resonances are expected to be a numerically small correction since these would be subleading in the $1/N_{C}$ expansion [4]. Such corrections will be neglected due to the already sizeable number of parameters involved in the tree-level analysis and the current precision of the experimental data.

In this section we quote our results for the different contributions to the $F_{V}$, $F_{A}$, $A_{2}$ ,$A_{4}$ form factors, for $P=\pi,K$. All resonance propagators are to be understood as provided with an energy-dependent width ($M_{R}^{2}-x\to M_{R}^{2}-x-iM_{R}\Gamma_{R}(x)$, $x=W^{2},k^{2}$) computed within $R\chi T$, using those in refs. [36] ($\rho(770)$), [37] ($K^{*}(892)$) and [38, 39] ($a_{1}(1260)$, including the $KK\pi$ cuts [40]). A constant width will suffice for the very narrow $\omega(782)$ and $\phi(1020)$ mesons (their PDG [41] values will be taken). For the $K_{1}(1270/1400)$ states we will follow [42].

The vector form factors are ($N_{C}=3$ in QCD)

$$F_{V}^{(\pi)}(W^{2},k^{2})=\frac{1}{3f}\left\{-\frac{N_{C}\frac{}{}}{8\pi^{2}}+64m_{\pi}^{2}C_{7}^{W\star}-8C_{22}^{W}(W^{2}+k^{2})\hskip 43.05542pt\right.$$

$$\hskip 43.05542pt+\frac{4{F_{V}^{ud}}^{2}}{M_{\rho}^{2}-W^{2}}\frac{d_{3}(W^{2}+k^{2})+d_{123}^{\star}m_{\pi}^{2}}{M_{\omega}^{2}-k^{2}}$$

$$\hskip 60.27759pt+\frac{2\sqrt{2}F_{V}^{ud}}{M_{V}}\frac{c_{1256}W^{2}-c_{1235}^{\star}m_{\pi}^{2}-c_{125}k^{2}}{M_{\rho}^{2}-W^{2}}$$

$$\left.\hskip 60.27759pt+\frac{2\sqrt{2}F_{V}^{ud}}{M_{V}}\frac{c_{1256}k^{2}-c_{1235}^{\star}m_{\pi}^{2}-c_{125}W^{2}}{M_{\omega}^{2}-k^{2}}\right\},$$

(16)

$$F_{V}^{(K)}(W^{2},k^{2})=\frac{1}{f}\left\{-\frac{N_{C}}{24\pi^{2}}+\frac{64}{3}m_{K}^{2}C_{7}^{W\star}+32C_{11}^{W}\Delta_{K\pi}^{2}-\frac{8}{3}C_{22}^{W}(W^{2}+k^{2})\right.$$

$$+\frac{2F_{V}^{us}\left[d_{3}(W^{2}+k^{2})+d_{123}^{\star}m_{K}^{2}\right]}{M_{K^{\star}}^{2}-W^{2}}\left(\frac{F_{V}^{ud}}{M_{\rho}^{2}-k^{2}}+\frac{1}{3}\frac{F_{V}^{ud}}{M_{\omega}^{2}-k^{2}}-\frac{2}{3}\frac{F_{V}^{ss}}{M_{\phi}^{2}-k^{2}}\right)$$

$$+\frac{2\sqrt{2}F_{V}^{us}}{3M_{V}}\frac{c_{1256}W^{2}-c_{1235}^{\star}m_{K}^{2}-c_{125}k^{2}+24c_{4}\Delta_{K\pi}^{2}}{M_{K^{\star}}^{2}-W^{2}}$$

(17)

$$\left.+\frac{\sqrt{2}\left(c_{1256}k^{2}-c_{1235}^{\star}m_{K}^{2}-c_{125}W^{2}\right)}{M_{V}}\left(\frac{F_{V}^{ud}}{M_{\rho}^{2}-k^{2}}+\frac{1}{3}\frac{F_{V}^{ud}}{M_{\omega}^{2}-k^{2}}-\frac{2}{3}\frac{F_{V}^{ss}}{M_{\phi}^{2}-k^{2}}\right)\right\},$$

where $\Delta_{K\pi}^{2}=m_{K}^{2}-m_{\pi}^{2}$ and we have used the combinations of coupling constants [43]

	$\displaystyle c_{125}=c_{1}-c_{2}+c_{5},$
	$\displaystyle c_{1256}=c_{1}-c_{2}-c_{5}+2c_{6},$
	$\displaystyle c_{1235}=c_{1}+c_{2}+8c_{3}-c_{5},$		(18)
	$\displaystyle d_{123}=d_{1}+8d_{2}-d_{3}.$

$F_{V}^{uD,ss}$ and starred coefficients absorb $U(3)$ breaking contributions induced by $\lambda_{V}$ in eq. (6) and pseudoscalar resonances, respectively. Their expressions are given at the end of this subsection, after eq. (24).

The axial form factors are ⁶⁶6We note two mistakes in writing $F_{A}^{(\pi)}$ in ref. [5], see Appendix A. The result written here agrees with the one in ref. [44] for $k^{2}\to 0$.

$$F_{A}^{(\pi)}(W^{2},k^{2})=\frac{F_{V}^{ud}}{2f}\frac{F_{V}^{ud}-2G_{V}-m_{\pi}^{2}\frac{4\sqrt{2}d_{m}}{M_{\pi^{\prime}}^{2}}(\lambda_{1}^{PV}+2\lambda_{2}^{PV})}{M_{\rho}^{2}-k^{2}}$$

$$-\frac{F_{A}}{2f}\frac{F_{A}-2m_{\pi}^{2}\frac{4\sqrt{2}d_{m}}{M_{\pi^{\prime}}^{2}}\lambda_{1}^{PA}}{M_{a_{1}}^{2}-W^{2}}+\frac{\sqrt{2}}{f}\frac{F_{A}F_{V}^{ud}}{M_{a_{1}}^{2}-W^{2}}\frac{\lambda_{0}^{\star}m_{\pi}^{2}-\lambda^{\prime}k^{2}-\lambda^{\prime\prime}W^{2}}{M_{\rho}^{2}-k^{2}},$$

(19)

$$F_{A}^{(K)}(W^{2},k^{2})=-\frac{F_{A}}{2f}\frac{F_{A}-2m_{K}^{2}\frac{4\sqrt{2}d_{m}}{M_{K^{\prime}}^{2}}\lambda_{1}^{PA}}{M_{K_{1}}^{2}-W^{2}}$$

$$+\left[\frac{\sqrt{2}F_{A}}{2f}\frac{\lambda^{\star}_{0}m_{K}^{2}-\lambda^{\prime}k^{2}-\lambda^{\prime\prime}W^{2}}{M_{K_{1}}^{2}-W^{2}}+\frac{F_{V}^{us}\left(F_{V}^{us}-2G_{V}+m_{K}^{2}\frac{4\sqrt{2}d_{m}}{M_{K^{\prime}}^{2}}(\lambda_{1}^{PV}+2\lambda_{2}^{PV})\right)}{4f}\right]$$

$$\times\left(\frac{F_{V}^{ud}}{M_{\rho}^{2}-k^{2}}+\frac{1}{3}\frac{F_{V}^{ud}}{M_{\omega}^{2}-k^{2}}+\frac{2}{3}\frac{F_{V}^{ss}}{M_{\phi}^{2}-k^{2}}\right),$$

(20)

$$A_{2}^{(\pi)}(W^{2},k^{2})=\hskip 284.16577pt$$

$$\frac{2}{f}\left(G_{V}+\frac{2\sqrt{2}m_{\pi}^{2}d_{m}}{M_{\pi^{\prime}}^{2}}\lambda_{1}^{PV}+\frac{\sqrt{2}F_{A}}{M_{a_{1}}^{2}-W^{2}}W^{2}(\lambda^{\prime}+\lambda^{\prime\prime})\right)\frac{F_{V}^{ud}}{M_{\rho}^{2}-k^{2}},$$

(21)

$$A_{2}^{(K)}(W^{2},k^{2})=\left(\frac{G_{V}}{f}+\frac{2\sqrt{2}m_{K}^{2}d_{m}}{M_{K^{\prime}}^{2}}\frac{\lambda_{1}^{PV}}{f}+\frac{\sqrt{2}F_{A}}{M_{K_{1}}^{2}-W^{2}}\frac{W^{2}(\lambda^{\prime}+\lambda^{\prime\prime})}{f}\right)$$

$$\hskip 120.55518pt\times\left(\frac{F_{V}^{ud}}{M_{\rho}^{2}-k^{2}}+\frac{1}{3}\frac{F_{V}^{ud}}{M_{\omega}^{2}-k^{2}}+\frac{2}{3}\frac{F_{V}^{ss}}{M_{\phi}^{2}-k^{2}}\right),$$

(22)

$$A_{4}^{(\pi)}(W^{2},k^{2})=\frac{2}{f}\frac{F_{V}^{ud}}{M_{\rho}^{2}-k^{2}}\left[\frac{G_{V}}{W^{2}-m_{\pi}^{2}}+\frac{2\sqrt{2}d_{m}m_{\pi}^{2}\lambda_{1}^{PV}}{M_{\pi^{\prime}}^{2}\left(W^{2}-m_{\pi}^{2}\right)}+\frac{\sqrt{2}F_{A}(\lambda^{\prime}+\lambda^{\prime\prime})}{M_{a_{1}}^{2}-W^{2}}\right],$$

(23)

$$A_{4}^{(K)}(W^{2},k^{2})=\frac{1}{f}\left(\frac{G_{V}}{W^{2}-m_{K}^{2}}+\frac{2\sqrt{2}d_{m}m_{K}^{2}\lambda_{1}^{PV}}{M_{K^{\prime}}^{2}\left(W^{2}-m_{\pi}^{2}\right)}+\frac{\sqrt{2}F_{A}(\lambda^{\prime}+\lambda^{\prime\prime})}{M_{K_{1}}^{2}-W^{2}}\right)$$

$$\hskip 120.55518pt\times\left(\frac{F_{V}^{ud}}{M_{\rho}^{2}-k^{2}}+\frac{1}{3}\frac{F_{V}^{ud}}{M_{\omega}^{2}-k^{2}}+\frac{2}{3}\frac{F_{V}^{ss}}{M_{\phi}^{2}-k^{2}}\right),$$

(24)

It is worth to notice that by replacing the $P$ propagator in $A_{2}^{(P)}$ and $A_{4}^{(P)}$ with the massless pole propagator, one recovers the linear dependence between both form factors, thus getting a congruent expression with those in reference [19]. Therefore, the short-distance constraints obtained in this reference can be used as shown there if the Weinberg’s sum rules are imposed. We, however, do not make use of these sum rules, as $F_{V/A}$ are fitted to data (see discussion in sections 3.3 and 4.2).

We introduced the short-hand notation

	$\displaystyle F_{V}^{ud}\equiv F_{V}+8m_{\pi}^{2}\lambda_{V},$
	$\displaystyle F_{V}^{us}\equiv F_{V}+8m_{K}^{2}\lambda_{V},$		(25)
	$\displaystyle F_{V}^{ss}\equiv F_{V}+8(2m_{K}^{2}-m_{\pi}^{2})\lambda_{V},$

for the shifts appearing also in [17]. ⁷⁷7As mentioned in section 3.1, a similar shift can be introduced in $F_{A}$, however, the operator responsible for such shift can be absorbed through axial resonance field redefinitions [19].

We also used [33]

	$\displaystyle-\sqrt{2}\lambda_{0}^{\star}=4\lambda_{1}^{\star}+\lambda_{2}+\frac{\lambda_{4}}{2}+\lambda_{5},$
	$\displaystyle\sqrt{2}\lambda^{\prime}=\lambda_{2}-\lambda_{3}+\frac{\lambda_{4}}{2}+\lambda_{5}$		(26)

and

$$\sqrt{2}\lambda^{\prime\prime}=\lambda_{2}-\frac{\lambda_{4}}{2}-\lambda_{5}.$$

We employed several starred coefficients including $U(3)$ breaking contributions, as given below:

	$\displaystyle\lambda_{1}^{\star}=\lambda_{1}-\frac{\lambda^{VAP}d_{m}}{M_{P}^{2}},$
	$\displaystyle C_{7}^{W\star}=C_{7}^{W}+\frac{\kappa_{5}^{P}d_{m}}{M_{P}^{2}},$
	$\displaystyle c_{3}^{\star}=c_{3}+\frac{\kappa_{3}^{PV}d_{m}M_{V}}{M_{P}^{2}},$		(27)

implying

	$\displaystyle c_{1235}^{\star}=c_{1}+c_{2}+8c_{3}^{\star}-c_{5},\;\mathrm{and}$		(28)
	$\displaystyle d_{2}^{\star}=d_{2}+\frac{\kappa^{VVP}d_{m}}{2M_{P}^{2}},$		(29)

yielding

$$d_{123}^{\star}=d_{1}+8d_{2}^{\star}-d_{3}.$$

(30)

We have first shown here the correction to $\lambda_{1}$ appearing in $\lambda_{1}^{\star}$, while the remaining starred couplings were already introduced in ref. [17].

We will follow the scheme explained in ref. [42] to account for the mixing between the $K_{1}(1270)=K_{1L}$ and the $K_{1}(1400)=K_{1H}$ states. This amounts to replacing, in eq. (3.2), $(M_{K_{1}}^{2}-W^{2})^{-1}\to\mathrm{cos}^{2}\theta_{A}(M_{K_{1H}}^{2}-W^{2})^{-1}+\mathrm{sin}^{2}\theta_{A}(M_{K_{1L}}^{2}-W^{2})^{-1}$, with mixing angle $\theta_{A}\in[37,58]^{\circ}$.

We will demand that the different form factors have an asymptotic behaviour in agreement with QCD [45, 46]. Specifically, we will require their vanishing for large $\lambda$ in the $\lim_{\lambda\to\infty}F_{V}(\lambda W^{2},0)$ and $\lim_{\lambda\to\infty}F_{V}(\lambda W^{2},\lambda k^{2})$ cases. We will do this first in the chiral limit and then at $\mathcal{O}(m_{P}^{2})$ ⁸⁸8Since we are considering a complete basis of chiral symmetry breaking operators at order $m_{P}^{2}$, we neglect higher-order chiral corrections., paralleling the discussion in ref. [17] for the neutral pseudoscalar transition form factors. In this way, we find the following relations:

• •

  $F_{V}^{(\pi)}(W^{2},k^{2})$, $\mathcal{O}(m_{P}^{0})$:

  	$\displaystyle C_{22}^{W}$	$\displaystyle=$	$\displaystyle 0\,,$		(31)
  	$\displaystyle c_{125}$	$\displaystyle=$	$\displaystyle 0\,,$		(32)
  	$\displaystyle c_{1256}$	$\displaystyle=$	$\displaystyle-\frac{N_{C}M_{V}}{32\sqrt{2}\pi^{2}F_{V}}\,,$		(33)
  	$\displaystyle d_{3}$	$\displaystyle=$	$\displaystyle-\frac{N_{C}M_{V}^{2}}{64\pi^{2}F_{V}^{2}}\,.$		(34)

• •

  $F_{V}^{(\pi)}(W^{2},k^{2})$, $\mathcal{O}(m_{P}^{2})$:

  	$\displaystyle\lambda_{V}$	$\displaystyle=$	$\displaystyle-\frac{64\pi^{2}F_{V}}{N_{C}}C_{7}^{W\star}\,,$		(35)
  	$\displaystyle c_{1235}^{\star}$	$\displaystyle=$	$\displaystyle\frac{N_{C}M_{V}e^{V}_{m}}{8\sqrt{2}\pi^{2}F_{V}}+\frac{N_{C}M_{V}^{3}\lambda_{V}}{4\sqrt{2}\pi^{2}F_{V}^{2}}.$		(36)

• •

  $F_{V}^{(K)}(W^{2},k^{2})$, $\mathcal{O}(m_{P}^{0})$: Same constraints as for the $\pi$ case, since both form factors ⁹⁹9$F_{A}^{P}$ and $A_{2,4}^{P}$ form factors are also identical in this limit for $P=\pi$ or $K$, obviously. are identical in the $U(3)$ symmetry limit.

• •

  $F_{V}^{(K)}(W^{2},k^{2})$, $\mathcal{O}(m_{P}^{2})$:

  $\displaystyle C_{11}^{W}$

  $\displaystyle=$

  $\displaystyle\frac{N_{C}\lambda_{V}}{64\pi^{2}F_{V}}\,.$

  (37)

For the sake of predictability and in order to further constrain the parameters in the form factor, we use the $VVP$ Green function, $\Pi_{VVP}(r^{2},p^{2},q^{2})$, constraints [28] obtained from the high-energy behaviour when $r^{2}\to\infty,p^{2}\to\infty$ and $q^{2}\to\infty$ and matching to the Operator Product Expansion (OPE) leading terms in the chiral and large-$N_{C}$ limits. These give

	$\displaystyle c_{125}=c_{1235}=0,\hskip 107.63855ptc_{1256}=-\frac{N_{C}M_{V}}{32\sqrt{2}\pi^{2}F_{V}},$
	$\displaystyle\kappa_{5}^{P}=0,\hskip 73.19421ptd_{3}=-\frac{N_{C}M_{V}^{2}}{64\pi^{2}F_{V}^{2}}+\frac{F^{2}}{8F_{V}^{2}}+\frac{4\sqrt{2}d_{m}\kappa_{3}^{PV}}{F_{V}},$
	$\displaystyle C_{7}^{W}=C_{22}^{W}=0,\hskip 142.08289ptd_{123}=\frac{F^{2}}{8F_{V}^{2}}.$		(38)