Isospin breaking in the phases of the K _e4 form factors

Abstract

Isospin breaking in the K _ℓ4 form factors induced by the difference between charged and neutral pion masses is studied. Starting from suitably subtracted dispersion representations, the form factors are constructed in an iterative way up to two loops in the low-energy expansion by implementing analyticity, crossing, and unitarity due to two-meson intermediate states. Analytical expressions for the phases of the two-loop form factors of the K ^±→π ⁺ π ⁻ e ^± ν _e channel are given, allowing one to connect the difference of form-factor phase shifts measured experimentally (out of the isospin limit) and the difference of S- and P-wave ππ phase shifts studied theoretically (in the isospin limit). The isospin-breaking correction consists of the sum of a universal part, involving only ππ rescattering, and a process-dependent contribution, involving the form factors in the coupled channels. The dependence on the two S-wave scattering lengths $a_{0}^{0}$ and $a_{0}^{2}$ in the isospin limit is worked out in a general way, in contrast to previous analyses based on one-loop chiral perturbation theory. The latter is used only to assess the subtraction constants involved in the dispersive approach. The two-loop universal and process-dependent contributions are estimated and cancel partially to yield an isospin-breaking correction close to the one-loop case. The recent results on the phases of K ^±→π ⁺ π ⁻ e ^± ν _e form factors obtained by the NA48/2 collaboration at the CERN SPS are reanalysed including this isospin-breaking correction to extract values for the scattering lengths $a_{0}^{0}$ and $a_{0}^{2}$, as well as for low-energy constants and order parameters of two-flavour χPT.

Appendices

Appendix A: Mesonic scattering amplitudes

In Sect. 4, we have given expressions of one-loop form factors describing the amplitudes for the decay channels K ⁺→π ⁺ π ⁻ ℓ ⁺ ν _ℓ and K ⁺→π ⁰ π ⁰ ℓ ⁺ ν _ℓ . These expressions involve a certain number of mesonic scattering amplitudes, corresponding to the various intermediate states that can contribute to the unitarity conditions. Apart from the ππ amplitudes, which contribute up to next-to-leading order in the isospin-breaking corrections to the phase shifts in Sect. 5, the amplitudes involving also the other pseudoscalar mesons (kaons and eta) occur only at lowest order and we begin the discussion with the latter. For the ππ amplitudes, we adopt a parametrisation in terms of threshold parameters (scattering lengths, effective ranges), but describe the remaining amplitudes with a parametrisation in terms of subthreshold parameters. They could easily be converted into expressions involving the scattering lengths if necessary. We rely on the definitions given in Sect. 4.1. Since no confusion can arise, in this appendix we denote the relevant kinematical variables by s,t,u and θ, instead of the notation $s,{\hat{t}}, {\hat{u}}$, and ${\hat{\theta}}$ used there.

1.1 A.1 πK and other scattering amplitudes at lowest order

Typically, the amplitudes of interest are of the general form

$$\begin{aligned} A^{a b ; a^\prime b^\prime} (s,t,u)|_{LO} =& \frac{\beta^{a b ; a^\prime b^\prime}}{F_\pi^2} \biggl( s - \frac {1}{3} \varSigma \biggr) + \frac{\alpha^{a b ; a^\prime b^\prime}}{3} \frac{ \varSigma }{F_\pi^2} \\ &{}+ 3 \gamma^{a b ; a^\prime b^\prime} \frac{t - u}{F_\pi^2} , \end{aligned}$$

(A.1)

where $\varSigma\equiv M_{a}^{2} + M_{b}^{2} + M_{a^{\prime}}^{2} + M_{b^{\prime}}^{2}$. This form is preserved under crossing. For instance

$$\begin{aligned} \begin{aligned} &\alpha^{{\bar{a}}^\prime b ; {\bar{a}} b^\prime} = \lambda_a \lambda _{a^\prime} \alpha^{a b ; a^\prime b^\prime} , \\ & \beta^{{\bar{a}}^\prime b ; {\bar{a}} b^\prime} = - \frac{1}{2} \lambda_a \lambda_{a^\prime} \bigl( \beta^{a b ; a^\prime b^\prime} + 9 \gamma^{a b ; a^\prime b^\prime} \bigr) , \\ & \gamma^{{\bar{a}}^\prime b ; {\bar{a}} b^\prime} = \frac{1}{6} \lambda_a \lambda_{a^\prime} \bigl( 3 \gamma^{a b ; a^\prime b^\prime} - \beta^{a b ; a^\prime b^\prime} \bigr) , \end{aligned} \end{aligned}$$

(A.2)

or

$$\begin{aligned} \begin{aligned} &\alpha^{{\bar{b}}^\prime b ; a^\prime{\bar{a}}} = \lambda_a \lambda _{b^\prime} \alpha^{a b ; a^\prime b^\prime} , \\ & \beta^{{\bar{b}}^\prime b ; a^\prime{\bar{a}}} = - \frac{1}{2} \lambda_a \lambda_{b^\prime} \bigl( \beta^{a b ; a^\prime b^\prime} - 9 \gamma^{a b ; a^\prime b^\prime} \bigr) , \\ & \gamma^{{\bar{b}}^\prime b ; a^\prime{\bar{a}}} = \frac{1}{6} \lambda_a \lambda_{b^\prime} \bigl( 3 \gamma^{a b ; a^\prime b^\prime} + \beta^{a b ; a^\prime b^\prime} \bigr) . \end{aligned} \end{aligned}$$

(A.3)

Equivalently, one may write $A^{a b ; a^{\prime}b^{\prime}} (s,t,u)\vert _{LO} = 16 \pi \times [ \varphi_{0}^{a b ; a^{\prime}b^{\prime}} (s) + 3 \varphi _{1}^{a b ; a^{\prime}b^{\prime}} (s) \cos\theta_{ab}]$, with

$$\begin{aligned} \begin{aligned} &\begin{aligned} 16 \pi\varphi_0^{a b ; a^\prime b^\prime} (s) &= \frac{\beta^{a b ; a^\prime b^\prime}}{F_\pi^2} s + \frac{1}{3} \bigl( \alpha^{a b ; a^\prime b^\prime} - \beta^{a b ; a^\prime b^\prime} \bigr) \frac{\varSigma}{F_\pi^2} \\ &\quad - 3 \frac{\gamma^{a b ; a^\prime b^\prime}}{F_\pi^2} \frac {\varDelta_{ab} \varDelta_{a^\prime b^\prime}}{s } , \end{aligned} \\ &16 \pi\varphi_1^{a b ; a^\prime b^\prime} (s) = \frac{\gamma^{a b ; a^\prime b^\prime}}{F_\pi^2} \frac{\lambda ^{1/2}_{ab} (s) \lambda^{1/2}_{a^\prime b^\prime} (s) }{s } , \end{aligned} \end{aligned}$$

(A.4)

and $\varDelta_{ab} \equiv M_{a}^{2} - M_{b}^{2}$, $\varDelta_{a^{\prime}b^{\prime}} \equiv M_{a^{\prime}}^{2} - M_{b^{\prime}}^{2}$.

The subthreshold parameters β ^ab;a′b′ and γ ^ab;a′b′ in these expressions are the same as those defined in Eq. (60), as can be seen upon forming the combinations $\varphi_{P}^{a b ; a^{\prime}b^{\prime}}$ and $\varphi_{S}^{a b ; a^{\prime}b^{\prime}} (s)$, as defined in Eqs. (54) and (55), respectively. Some of these coefficients are given, at lowest order, in Table 11. Those not shown in this table can be obtained through the crossing relations given above. These expressions involve the two mixing angles ϵ ₁ and ϵ ₂, defined in terms of the isospin-breaking matrix elements of the flavour-diagonal octet axial currents [5]. At linear order in isospin breaking, we have

$$\begin{aligned} \begin{aligned} &\langle\varOmega\vert A_\mu^8 (0) \bigl\vert \pi^0(p) \bigr\rangle = i p_\mu\epsilon_1 F_\pi , \\ &\langle\varOmega\vert A_\mu^3 (0) \bigl\vert \eta(p) \bigr\rangle = -i p_\mu\epsilon_2 F_\eta , \end{aligned} \end{aligned}$$

(A.5)

which at lowest order read

$$\begin{aligned} \epsilon_1 = \epsilon_2 = \frac{\sqrt{3}}{4R}, \quad R=\frac{m_d - m_u}{m_s - {\widehat{m}}}. \end{aligned}$$

(A.6)

Finally, one has

$$ 16 \pi\varphi_S^{a b ; a^\prime b^\prime} (0) = \frac{1}{3} \bigl( \alpha^{a b ; a^\prime b^\prime} - \beta^{a b ; a^\prime b^\prime} \bigr) \frac{\varSigma}{F_\pi^2} $$

(A.7)

with the crossing relations

$$\begin{aligned} \begin{aligned} &\begin{aligned}16 \pi\varphi_S^{{\bar{a}}^\prime b ; {\bar{a}} b^\prime} (0) &= \lambda_a \lambda_{a^\prime} \biggl[ 16 \pi\varphi_S^{a b ; a^\prime b^\prime} (0) \\ &\quad + \frac{1}{2} \bigl( \beta^{a b ; a^\prime b^\prime} + 3 \gamma^{a b ; a^\prime b^\prime} \bigr) \frac{\varSigma}{F_\pi^2} \biggr], \end{aligned} \\ &\begin{aligned} 16 \pi\varphi_S^{{\bar{b}}^\prime b ; a^\prime{\bar{a}}} (0) &= \lambda_a \lambda_{b^\prime} \biggl[ 16 \pi\varphi_S^{a b ; a^\prime b^\prime} (0) \\ &\quad + \frac{1}{2} \bigl( \beta^{a b ; a^\prime b^\prime} - 3 \gamma^{a b ; a^\prime b^\prime} \bigr) \frac{\varSigma}{F_\pi^2} \biggr] . \end{aligned} \end{aligned} \end{aligned}$$

(A.8)

Table 11 The parameters β ^ab;a′b′ and γ ^ab;a′b′ corresponding to various lowest-order amplitudes needed for the one-loop expressions of the form factors F ^ab(s,t,u) and G ^ab(s,t,u) discussed in Sect. 4

1.2 A.2 ππ scattering amplitudes

The ππ scattering amplitudes have already been discussed in quite some detail in [37], see in particular Appendix F therein. We need the following lowest-order partial-wave projections:

$$\begin{aligned} \begin{aligned} &\varphi^{+- ; +-}_0(s) = a_{+-} + b_{+-}\frac{s - 4 M_{\pi^\pm }^2}{F_\pi^2} , \\ &\varphi^{+- ; +-}_1(s) = \frac{c_{+-}}{3} \frac{s - 4 M_{\pi^\pm}^2}{F_\pi^2} , \\ & \varphi^{+- ; 00}_0(s) = a_{x} + b_{x}\frac{s - 4 M_{\pi^\pm}^2}{F_\pi^2} ,\\ &\varphi^{+- ; 00}_1(s) = 0 ,\qquad \varphi^{00 ; 00}_0(s) = a_{00} , \\ &\varphi^{00 ; 00}_1(s) = 0 . \end{aligned} \end{aligned}$$

(A.9)

The parameters entering these amplitudes in terms of $a^{0}_{0}$ and $a^{0}_{2}$ (the S-wave scattering lengths in the isospin limit, in the isospin channels I=0 and I=2, respectively) can be obtained at the lowest order upon combining the formulae (3.3), (3.4), and (3.5) of Ref. [37], which gives [$\varDelta_{\pi}=M_{\pi}^{2} - M_{\pi^{0}}^{2}$]

$$\begin{aligned} \begin{aligned} & a_{+-}=\frac{2}{3}a^0_0+ \frac{1}{3} a^2_0-2 a^2_0 \frac{\varDelta_\pi}{M_{\pi}^2} , \\ &b_{+-}=c_{+-} = \frac {1}{24} \frac{F_\pi^2}{M_\pi^2} \bigl( 2 a^2_0 - 5 a^2_0 \bigr), \\ & a_x = - \frac{2}{3}a^0_0 +\frac{2}{3}a^2_0+ a^2_0 \frac{\varDelta_\pi}{M_{\pi}^2}, \\ &b_x=- \frac{1}{12} \frac{F_\pi^2}{M_\pi^2} \bigl( 2 a^2_0 - 5 a^2_0 \bigr), \\ & a_{00}=\frac{2}{3}a^0_0+ \frac{4}{3}a^2_0-\frac {2}{3} \bigl(a^0_0 + 2 a^2_0 \bigr) \frac{\varDelta_\pi}{M_{\pi}^2} . \end{aligned} \end{aligned}$$

(A.10)

In order to proceed with the analysis of isospin-breaking in the $K^{+}_{e4}$ data, we need to go beyond lowest order, and work out the contributions $\psi^{+- ; +-}_{0,1}(s)$, $\psi^{+-; 00}_{0}(s)$ to the partial-wave projections at next-to-leading order. These can be obtained from Eqs. (4.6), (4.10), (4.15) of Ref. [37], combined with the formulae (F.8)–(F.12) of that same reference. Likewise, the formulae for a ₊₋, a _x , b ₊₋, and b _x need also to be determined at next-to-leading order. In the case of the scattering lengths, this can be achieved upon using the expressions (F.13) and the formulae given in Appendix E of [37].⁵ The result reads

$$\begin{aligned} &{16 \pi \biggl[ a_{+-} - \frac{1}{3} \bigl( 2a_0^0 + a_0^2 \bigr) \biggr]} \\ &{\quad = -32 \pi a_0^2 \frac{\varDelta_\pi}{M_\pi^2}} \\ &{\qquad{} - 16 \pi\frac{1}{6} \bigl( 2 a_0^0 + 7 a_0^2 \bigr) \frac{\varDelta _\pi}{M_\pi^2}} \\ &{\qquad{} + 16 ( \lambda_1 + 2 \lambda_2 ) \frac{\varDelta_\pi }{F_\pi^2} \frac{M_\pi^2}{F_\pi^2}} \\ &{\qquad{} - \frac{8}{27} \frac{\varDelta_\pi}{M_\pi^2} \biggl[ 37 \bigl( a_0^0 \bigr)^2 - \frac{1499}{4} \bigl( a_0^2 \bigr)^2 + 241 a_0^0 a_0^2 \biggr]} \\ &{\qquad{} + \frac{8}{27} L_\pi \biggl[ 53 \bigl( a_0^0 \bigr)^2 - \frac{1153}{4} \bigl( a_0^2 \bigr)^2 + 26 a_0^0 a_0^2 \biggr]} \\ &{\qquad{} + 16 \pi\frac{1}{18} \bigl( 2 a_0^0 - 5 a_0^2 \bigr) \frac{e^2}{32 \pi^2}} \\ &{ \qquad{}\times \bigl[ 3 {\widehat{\mathcal{K}}}^{+-}_1 + {\widehat{\mathcal{K}}}^{+-}_2 - 3 \bigl( {\widehat{ \mathcal{K}}}^{00}_1 + {\widehat{\mathcal {K}}}^{00}_2 \bigr) \bigr]} \\ &{\qquad{} - 32 \pi a_0^2 \frac{e^2}{32 \pi^2} {\widehat{ \mathcal{K}}}^{+-}_3 + \mathcal{O} \biggl( \frac{\varDelta_\pi^2}{M_\pi^4} \biggr) ,} \end{aligned}$$

(A.11)

$$\begin{aligned} &{16 \pi \biggl[ a_x + \frac{2}{3} \bigl( a_0^0 - a_0^2 \bigr) \biggr]} \\ &{\quad = 16 \pi a_0^2 \frac{\varDelta_\pi}{M_\pi^2}} \\ &{\qquad{} + 16 \pi\frac{1}{12} \bigl( 2 a_0^0 + 7 a_0^2 \bigr) \frac{\varDelta _\pi}{M_\pi^2} - 4 \frac{\varDelta_\pi}{F_\pi^2} \frac{M_{\pi}^2}{F_\pi^2} ( 3 \lambda_1 + 4 \lambda_2 )} \\ &{ \qquad{} + \frac{1}{81} \frac{\varDelta_\pi}{M_\pi^2} \bigl[ 644 \bigl( a_0^0 \bigr)^2 - 3895 \bigl( a_0^2 \bigr)^2 + 6212 a_0^0 a_0^2 \bigr]} \\ &{\qquad{} + \frac{16}{27} \biggl[ \frac{M_{\pi^0}^2}{\varDelta_\pi} L_\pi- 1 \biggr] \bigl[ 4 \bigl( a_0^0 \bigr)^2 - 11 \bigl( a_0^2 \bigr)^2 + 16 a_0^0 a_0^2 \bigr]} \\ &{ \qquad{} - \frac{1}{9} L_\pi \bigl[ 12 \bigl( a_0^0 \bigr)^2 + 133 \bigl( a_0^2 \bigr)^2 + 32 a_0^0 a_0^2 \bigr]} \\ &{ \qquad{} - 16 \pi\frac{1}{36} \bigl( 2 a_0^0 - 5 a_0^2 \bigr)} \\ &{\qquad{}\times \frac{e^2}{32 \pi^2} \bigl[ 10 {\widehat{ \mathcal{K}}}^x_1 + {\widehat{\mathcal{K}}}^x_2 - 3 \bigl( {\widehat{\mathcal{K}}}^{00}_1 + {\widehat{ \mathcal {K}}}^{00}_2 \bigr) \bigr]} \\ &{\qquad{} + 16 \pi a_0^2 \frac{e^2}{32 \pi^2} {\widehat{ \mathcal{K}}}^x_3 + \mathcal{O} \biggl( \frac{\varDelta_\pi^2}{M_\pi^4} \biggr),} \end{aligned}$$

(A.12)

When we consider isospin-breaking corrections to the phase shifts at next-to-leading order, we also need a ₊₀, a ₊₊ and a ₀₀ from the universal contributions due to ππ (re)scattering. These scattering parameters enter only next-to-leading-order corrections, and are thus needed at leading order. However, we will also estimate the impact of higher-order effects in the isospin-breaking corrections to the phase shifts by comparing the numerical results obtained using either leading- or next-to-leading-order expressions for these scattering parameters. The latter read

$$\begin{aligned} &{16 \pi \bigl[a_{+0} - a_0^2 \bigr]} \\ &{\quad = -16 \pi a_0^2 \frac{\varDelta_\pi}{M_\pi^2} -\frac{4 \pi}{3} \bigl(2 a_0^0 +7 a_0^2\bigr) \frac{\varDelta_\pi}{M_\pi^2}} \\ &{\qquad{}+4 \frac{\varDelta_\pi}{F_\pi^2}\frac{M_\pi^2}{F_\pi^2}(\lambda _1 + 2 \lambda_2)} \\ &{\qquad{} +\frac{1}{9} \frac{\varDelta_\pi}{M_\pi^2} \bigl[4 \bigl(a_0^0 \bigr)^2 +345 \bigl(a_0^2\bigr)^2 -76 a_0^0 a_0^2 \bigr]} \\ &{\qquad{}+ \frac{1}{27} \biggl[\frac{M_{\pi^0}^2}{\varDelta_\pi} L_\pi -1 \biggr] \bigl[-64 \bigl(a_0^0\bigr)^2 - 544 \bigl(a_0^2\bigr)^2} \\ &{\qquad{}+ 32 a_0^0 a_0^2 \bigr]+ \frac{1}{27} L_\pi \bigl[ 180 \bigl(a_0^0 \bigr)^2 } \\ &{\qquad{} - 1221 \bigl(a_0^2 \bigr)^2 + 384 a_0^0 a_0^2 \bigr]} \\ &{\qquad{} + 16 \pi\frac{1}{36} \frac{e^2}{32 \pi^2} \bigl(2 a_0^0-5 a_0^2\bigr)} \\ &{ \qquad{}\times \bigl[-2 {\widehat{\mathcal{K}}}^x_1 + {\widehat{\mathcal{K}}}_2^x -3 \bigl({\widehat{ \mathcal{K}}}_1^{00} + {\widehat{\mathcal{K}}}_2^{00} \bigr) \bigr]} \\ &{\qquad{}-16 \pi a_0^2 \frac{e^2}{32 \pi^2} {\widehat{ \mathcal{K}}}_3^x + \mathcal{O} \biggl( \frac{\varDelta_\pi ^2}{M_\pi^4} \biggr),} \end{aligned}$$

(A.13)

$$\begin{aligned} &{16 \pi \bigl[a_{++} -2 a_0^2 \bigr]} \\ &{\quad = -32 \pi a_0^2 \frac{\varDelta_\pi}{M_\pi^2} -\frac{8 \pi}{3} \bigl(2 a_0^0 +7 a_0^2\bigr) \frac{\varDelta_\pi}{M_\pi^2}} \\ &{\qquad{}+16 \frac{\varDelta_\pi}{F_\pi^2}\frac{M_\pi^2}{F_\pi^2}(\lambda _1 + 2 \lambda_2)} \\ &{\qquad{} -\frac{2}{27} \frac{\varDelta_\pi}{M_\pi^2} \bigl[52 \bigl(a_0^0 \bigr)^2 -1955 \bigl(a_0^2\bigr)^2 + 436 a_0^0 a_0^2 \bigr]} \\ &{\qquad{}+ \frac{2}{27} L_\pi \bigl[ 308 \bigl(a_0^0 \bigr)^2 - 1993 \bigl(a_0^2 \bigr)^2 + 200 a_0^0 a_0^2 \bigr]} \\ &{\qquad{} + 16 \pi\frac{1}{18} \frac{e^2}{32 \pi^2} \bigl(2 a_0^0-5 a_0^2\bigr)} \\ &{\qquad{}\times \bigl[ {\widehat{\mathcal{K}}}_2^{+-} -3 \bigl( {\widehat{\mathcal{K}}}_1^{+-} +{\widehat{ \mathcal{K}}}_1^{00} + {\widehat{ \mathcal{K}}}_2^{00} \bigr) \bigr]} \\ &{\qquad{}-32 \pi a_0^2 \frac{e^2}{32 \pi^2} {\widehat{ \mathcal{K}}}_3^{+-} + \mathcal{O} \biggl( \frac{\varDelta _\pi^2}{M_\pi^4} \biggr),} \end{aligned}$$

(A.14)

$$\begin{aligned} &{16 \pi \biggl[a_{00} -\frac{2}{3} \bigl(a_0^0 +2 a_0^2\bigr) \biggr]} \\ &{\quad = -16 \pi\frac{2}{3} \bigl(a_0^0 +2 a_0^2\bigr) \frac{\varDelta_\pi}{M_\pi^2}} \\ &{\qquad{}-16 \pi\frac{1}{4} \bigl(2 a_0^0+7 a_0^2\bigr)\frac{\varDelta_\pi}{M_\pi^2}} \\ &{\qquad{} -\frac{1}{9} \frac{\varDelta_\pi}{M_\pi^2} \bigl[5 \bigl(2a_0^0+a_0^2 \bigr)^2 -128\bigl( a_0^0 -a_0^2 \bigr)^2 \bigr]} \\ &{\qquad{}+ \frac{1}{9} L_\pi \bigl[ 100 \bigl(a_0^0 \bigr)^2 +475 \bigl(a_0^2\bigr)^2 + 424 a_0^0 a_0^2 \bigr]} \\ &{\qquad{} + 16 \pi\frac{1}{12} \frac{e^2}{32 \pi^2} \bigl(2 a_0^0-5 a_0^2\bigr) \bigl[{\widehat{\mathcal{K}}}^{00}_1+{ \widehat{\mathcal{K}}}_2^{00} \bigr]} \\ &{\qquad{} -16 \pi\frac{2}{3}\bigl(a_0^0+2 a_0^2\bigr) \frac{e^2}{32 \pi^2} {\widehat{ \mathcal{K}}}_2^{00}} \\ &{\qquad{}- 16\pi\frac{4}{9} \frac{\sqrt{\varDelta_\pi}}{ M_\pi}\bigl(a_0^0-a_0^2 \bigr)^2 + \mathcal{O} \biggl( \frac{\varDelta_\pi^{3/2}}{M_\pi^3} \biggr),} \end{aligned}$$

(A.15)

where the last term in Eq. (A.15) comes from the expansion of $\bar{J}(4 M_{\pi^{0}}^{2})$ in powers of Δ _π (see the expression of a ₀₀ in Eq. (F.13) in Ref. [37]). We have also

$$\begin{aligned} &{\begin{aligned}[b] 16 \pi b_{+-} &= \frac{2 \pi}{3} \frac{F_\pi ^2}{M_\pi^2} \bigl( 2 a_0^0 - 5 a_0^2 \bigr) \\ &\quad- \frac{4}{3} \frac{F_\pi^2}{M_\pi^2} \bigl[ 2 \bigl( a_0^0 \bigr)^2 - 5 \bigl( a_0^2 \bigr)^2 \bigr] \\ &\quad + \frac{10}{3} \frac{\varDelta_\pi}{M_\pi^2} \frac{F_\pi^2}{M_\pi ^2} a_0^2 \bigl( 2 a_0^0 - 5 a_0^2 \bigr) \\ &\quad - \frac{2}{9} \frac{F_\pi^2}{M_\pi^2} \bigl( 2 a_0^0 - 5 a_0^2 \bigr) \bigl( 2 a_0^0 + 7 a_0^2 \bigr) L_\pi \\ &\quad + 16 \pi\frac{F_\pi^2}{M_{\pi}^2} \frac{1}{24} \bigl( 2 a_0^0 - 5 a_0^2 \bigr) \frac{e^2}{32 \pi^2} {\widehat{ \mathcal{K}}}^{+-}_1 \\ &\quad+ \mathcal{O} \biggl( \frac{\varDelta_\pi^2}{M_\pi^4} \biggr) , \end{aligned}} \end{aligned}$$

(A.16)

$$\begin{aligned} &{\begin{aligned}[b] 16 \pi b_x &= - \frac{4 \pi}{3} \frac{F_\pi^2}{M_\pi^2} \bigl( 2 a_0^0 - 5 a_0^2 \bigr) \\ &\quad + \frac{8}{3} \frac{F_\pi^2}{M_\pi^2} \bigl[ 2 \bigl( a_0^0 \bigr)^2 - 5 \bigl( a_0^2 \bigr)^2 \bigr] - 2 \lambda_1 \frac{\varDelta_\pi}{F_\pi^2} \\ &\quad - \frac{1}{18} \frac{\varDelta_\pi}{M_\pi^2} \frac{F_\pi^2}{M_\pi^2} \bigl[ 28 \bigl( a_0^0 \bigr)^2 - 425 \bigl( a_0^2 \bigr)^2 + 100 a_0^0 a_0^2 \bigr] \\ &\quad+ \frac{1}{3} \frac{F_\pi^2}{M_\pi^2} \bigl( 2 a_0^0 - 5 a_0^2 \bigr) \bigl( 2 a_0^0 + 9 a_0^2 \bigr) L_\pi \\ &\quad + \frac{4}{3} \frac{F_\pi^2}{M_\pi^2} \biggl[ \frac{M_{\pi^0}^2}{\varDelta_\pi} L_\pi- 1 \biggr] a_0^2 \bigl( 2 a_0^0 - 5 a_0^2 \bigr) \\ &\quad- 16 \pi\frac{F_\pi^2}{M_{\pi}^2}\frac{1}{12} \bigl( 2 a_0^0 - 5 a_0^2 \bigr) \frac{e^2}{32 \pi^2} {\widehat{ \mathcal{K}}}^x_1 \\ &\quad+ \mathcal{O} \biggl( \frac{\varDelta_\pi^2}{M_\pi^4} \biggr) , \end{aligned}} \end{aligned}$$

(A.17)

whereas b ₊₀=b _x /2.

In each case, the first term on the right-hand side of the equation corresponds to the leading-order contribution while further contributions are chirally suppressed. The combinations ${\widehat{\mathcal{K}}}^{ab}_{i}$ of electromagnetic low-energy constants appearing in these expressions are given in Eq. (E.4) of [37]. In Eq. (E.3) of the same reference, one also finds the connection between the subthreshold parameters $\lambda^{(1,2)}_{ab}$ in presence of isospin and those in the isospin limit, λ _1,2. In both cases, the ππ subthreshold parameter β defined in the isospin limit is involved. For our purposes, it is only needed at lowest order, where it is related to the two S-wave scattering lengths by

$$ \beta= \frac{4 \pi}{3} \frac{F_\pi^2}{M_\pi^2} \bigl( 2 a^2_0- 5 a^2_0 \bigr) . $$

(A.18)

Appendix B: Subtraction polynomials

The expressions of the one-loop K _ℓ4 form factors obtained in Sect. 4 involve subtraction polynomials $P^{ab}_{F}(s,t,u)$ and $P^{ab}_{G}(s,t,u)$, which themselves depend on a certain number of coefficients $\pi^{ab}_{i, F/G}$, which will be discussed in this appendix.

In the effective theory framework, the subtraction coefficients are given in terms of the low-energy constants and chiral logarithms. It is therefore possible to extract the expressions of the polynomials $P^{ab}_{F}(s,t,u)$ and $P^{ab}_{G}(s,t,u)$ from a comparison of the expressions obtained in Sects. 4.2 and 4.3 with a one-loop computation of the relevant form factors. Although such computations, including isospin breaking, are available, it is only necessary to compute the contributions from the tree diagrams and from the tadpoles, since we have been careful in expressing the unitarity parts in terms of the appropriate loop functions. All we need to take into account from the unitarity parts are the terms generated by the differences between ${\bar{J}}_{ab}$ and $J^{r}_{ab}$, or between M _ab and $M^{r}_{ab}$. As far as the tree-level contributions are concerned, they include those arising from the strong N _f =3 low-energy constants L _i [5], and those coming from the counterterms ${\widehat{K}}_{i}$ describing isospin-breaking corrections [74]. The tadpole contributions are most easily computed upon using the corresponding expression of the one-loop generating functional [5]. We have actually redone this calculation, taking into account the contributions from the mixing angles ϵ ₁ and ϵ ₂.

We want to stress that only tadpoles generated by the kinematic term with two covariant derivatives at lowest order contribute to the K _e4 form factors at one loop. Therefore, their dependence on the meson masses comes from the invariant products of the meson momenta involved, and are not reconstructed from the B ₀ term in the leading-order N _f =3 chiral Lagrangian [5]. According to the chiral counting, these tadpoles contribute only to $\pi_{0,F}^{+-}$, $\pi_{0,G}^{+-}$, and $\pi_{0,F}^{00}$ at the one-loop order. These three quantities will thus require a separate discussion once we have dealt with the remaining subtraction constants.

In order to illustrate the procedure just outlined, let us consider in some detail the case of $\pi_{3,F}^{+-}$, cf. Eq. (70). Here and in what follows, we drop contributions that are of second order in isospin breaking whenever convenient (e.g. quadratic terms in the mixing angles ϵ _1,2). Our expressions and statements will always be exact up to isospin-violating corrections of higher order. Concerning $\pi _{3,F}^{+-}$, we thus find

$$\begin{aligned} \pi_{3,F}^{+-} =& - 2 L_3 \\ &{} + \frac{1}{2} \frac{F_0^2}{F_\pi^2} \bigl( \beta^{+ K^-;+ K^-} + 2 \beta ^{K^- -;K^- -} \\ &{}- 3 \gamma^{+ K^-;+ K^-} \bigr) k_{K \pi} (\mu) \\ &{} - \frac{3}{2} \frac{1}{\sqrt{2}} \frac{F_0^2}{F_\pi^2} \biggl[ \biggl( 1 + \frac{\epsilon_2}{\sqrt{3}} \biggr) \beta ^{+ K^-;0 {\bar{K}}^0} \\ &{} - ( 1 - \sqrt{3} \epsilon_2 ) \gamma^{+ K^-;0 {\bar{K}}^0} \biggr] k_{K^0 \pi^0} (\mu) \\ &{} - \frac{1}{2} \sqrt{\frac{3}{2}} \frac{F_0^2}{F_\pi^2} \biggl[ ( 1 - \sqrt{3} \epsilon_1 ) \beta^{+ K^-;\eta{\bar{K}}^0} \\ &{} + 3 \biggl( 1 + \frac{\epsilon_1}{\sqrt{3}} \biggr) \gamma^{+ K^-;\eta{\bar{K}}^0} \biggr] k_{\eta K^0} (\mu) , \end{aligned}$$

(B.1)

which clearly exhibits the two types of contributions, L ₃ at the tree level (tadpoles are absent in this case) and the terms proportional to the k _ab (μ) coming from the unitarity loops. $\pi_{3,F}^{+-}$ should not depend on the subtraction scale μ scale, as can be checked using the information concerning the values of the coefficients β ^ab;a′b′ and γ ^ab;a′b′ given in Appendix A since L ₃ is not renormalised and

$$ \mu\frac{d}{d \mu} k_{ab} (\mu) = - \frac{1}{16 \pi^2} . $$

(B.2)

Equivalently, we can rewrite the expression for $\pi_{3,F}^{+-}$ in a manner where scale independence is manifest, upon eliminating the last term such as to make appear the differences $k_{K \pi} (\mu) - k_{\eta K^{0}} (\mu)$ and $k_{K^{0} \pi^{0}} (\mu) - k_{\eta K^{0}} (\mu)$ only (the result of this operation is shown below). Concerning the remaining coefficients in the channel with two charged pions, we proceed along similar lines. Besides L ₃, their expressions also involve the renormalised low-energy constants $L_{1}^{r} (\mu)$, $L_{2}^{r} (\mu)$, and $L_{9}^{r} (\mu)$. The scale dependence of the latter, which has been worked out in Ref. [5], can be found in Eqs. (B.13) and (B.14) below. Together with the information on the coefficients β ^ab;a′b′ and γ ^ab;a′b′ in Appendix A, we have checked that these expressions indeed do not depend on the renormalisation scale μ. Making use of this property and the crossing relations between the coefficients β ^ab;a′b′ and γ ^ab;a′b′ to simplify these expressions further, we obtain the following manifestly scale-independent formulae:

$$\begin{aligned} \pi_{1,F}^{+-} =& 10 L_3 + 16 \bigl[ 2 L_1^r (\mu ) - L_2^r (\mu) \bigr] \\ &{}- 6 \frac{F_0^2}{F_\pi^2} \gamma^{+-;K^+ K^-} \biggl[ k_{K \pi} (\mu) - \frac{16}{3} L_2^r (\mu) \biggr] \\ & {}+ \frac{1}{\sqrt{2}} \frac{F_0^2}{F_\pi^2} \biggl[ \beta^{0 -;K^0 K^-} \,{-}\, 6 \biggl( 1 + \frac{\sqrt{3}}{2} \epsilon_2 \biggr) \gamma^{0 -;K^0 K^-} \biggr] \\ &{}\times \biggl[ k_{K^0 \pi^0} (\mu) - \frac{16}{3} L_2^r (\mu) \biggr] \\ &{} + \frac{1}{2} \sqrt{\frac{3}{2}} \frac{F_0^2}{F_\pi^2} \biggl[ ( 1 - \sqrt{3} \epsilon_1 ) \beta^{+ K^-;\eta{\bar{K}}^0} \\ & {} + 3 \biggl( 1 + \frac{\epsilon_1}{\sqrt{3}} \biggr) \gamma^{+ K^-;\eta{\bar{K}}^0} \biggr] \biggl[ k_{\eta K^0} (\mu) - \frac{16}{3} L_2^r (\mu) \biggr] \\ & {}- \frac{F_0^2}{F_\pi^2} \biggl\{ 2 \beta^{+-;+-} \biggl[ k_{\pi \pi} (\mu) - \frac{16}{3} L_2^r (\mu) \biggr] \\ &{} - (1 + 2 \sqrt{3} \epsilon_2 ) \beta^{+-;00} \biggl[ k_{\pi^0 \pi^0} (\mu) - \frac{16}{3} L_2^r (\mu) \biggr] \\ &{} + 4 \beta^{+-;K^+ K^-} \biggl[ k_{KK} (\mu) - \frac{16}{3} L_2^r (\mu) \biggr] \\ &{} - 2 \beta^{+-;K^0 {\bar{K}}^0} \biggl[ k_{K^0 {\bar{K}}^0} (\mu) - \frac{16}{3} L_2^r (\mu) \biggr] \\ &{} - 3 \biggl( 1 - \frac{2}{\sqrt{3}} \epsilon_1 \biggr) \beta ^{+-;\eta\eta} \biggl[ k_{\eta\eta} (\mu) - \frac{16}{3} L_2^r (\mu) \biggr] \\ &{} - 2 \sqrt{3} \biggl( 1 - \frac{\epsilon_1}{\sqrt{3}} + \sqrt{3} \epsilon_2 \biggr) \beta^{+-;\pi^0 \eta} \\ &{} \times \biggl[ k_{\eta\pi^0} (\mu) - \frac{16}{3} L_2^r (\mu) \biggr] \biggr\} , \\ \pi_{2,F}^{+-} =& - 2 L_3 + 2 \biggl[ L_9^r (\mu) - \frac{4}{3} L_2^r (\mu) \biggr] \\ & {}+ 6 \frac{F_0^2}{F_\pi^2} \gamma^{+-;K^+ K^-} \biggl[ k_{K \pi} ( \mu) - \frac{16}{3} L_2^r (\mu) \biggr] \\ &{} - \frac{1}{\sqrt{2}} \frac{F_0^2}{F_\pi^2} \\ &{}\times \biggl[ \beta^{0 -;K^0 K^-} - 6 \biggl( 1 + \frac{\sqrt{3}}{2} \epsilon_2 \biggr) \gamma^{0 -;K^0 K^-} \biggr] \\ & {}\times \biggl[ k_{K^0 \pi^0} (\mu) - \frac{16}{3} L_2^r (\mu) \biggr] \\ &{} - \frac{1}{2} \sqrt{\frac{3}{2}} \frac{F_0^2}{F_\pi^2} \\ &{}\times \biggl[ ( 1 - \sqrt{3} \epsilon_1 ) \beta^{+ K^-;\eta{\bar{K}}^0} \\ &{}+ 3 \biggl( 1 + \frac{\epsilon_1}{\sqrt{3}} \biggr) \gamma^{+ K^-;\eta{\bar{K}}^0} \biggr] \\ & {}\times \biggl[ k_{\eta\pi^0} (\mu) - \frac{16}{3} L_2^r (\mu) \biggr] , \\ \pi_{3,F}^{+-} =& - 2 L_3 \\ &{} - \frac{F_0^2}{F_\pi^2} \bigl( \beta^{+-;K^+ K^-} + 3 \gamma ^{+-;K^+ K^-} \bigr) \\ & {}\times \bigl[ k_{K \pi} (\mu) - k_{\eta K^0} (\mu) \bigr] \\ &{} - \frac{1}{\sqrt{2}} \frac{F_0^2}{F_\pi^2} \biggl[ \beta^{0 -;K^0 K^-} \\ &{} - 6 \biggl( 1 + \frac{\sqrt{3}}{2} \epsilon_2 \biggr) \gamma^{0 -;K^0 K^-} \biggr] \\ & {}\times \bigl[ k_{K^0 \pi^0} (\mu) - k_{\eta K^0} (\mu) \bigr] , \end{aligned}$$

(B.3)

and

$$\begin{aligned} \pi_{1,G}^{+-} =& - 2 L_3 \\ &{} - \frac{2}{3} \frac{F_0^2}{F_\pi^2} \bigl( \beta^{+-;K^+ K^-} + 3 \gamma ^{+-;K^+ K^-} \bigr) \\ &{}\times\bigl[ k_{K \pi} (\mu) - k_{\eta K^0} (\mu) \bigr] \\ &{} - \frac{1}{3\sqrt{2}} \frac{F_0^2}{F_\pi^2} \\ &{}\times \biggl[ 2 \biggl( 1 + \frac{\sqrt{3}}{2} \epsilon_2 \biggr) \beta^{0 -;K^0 K^-} \\ &{}-3 ( 7 + 2 \sqrt{3} \epsilon_2 ) \gamma^{0 -;K^0 K^-} \biggr] \\ &{}\times \bigl[ k_{K^0 \pi^0} (\mu) - k_{\eta K^0} (\mu) \bigr] \\ & {} - 2 \frac{F_0^2}{F_\pi^2} \bigl\{ \gamma^{+-;+-} \bigl[ k_{\pi\pi} (\mu) - k_{\eta K^0} (\mu) \bigr] \\ &{}+ 2 \gamma^{+-;K^+ K^-} \bigl[ k_{KK} (\mu) - k_{\eta K^0} (\mu) \bigr] \\ & {}- \gamma^{+-;K^0 {\bar{K}}^0} \bigl[ k_{K^0 {\bar{K}}^0} (\mu) - k_{\eta K^0} (\mu) \bigr] \bigr\} , \\ \pi_{2,G}^{+-} =& + 2 L_3 \\ &{} + \frac{2}{3} \frac{F_0^2}{F_\pi^2} \bigl( \beta^{+-;K^+ K^-} + 3 \gamma ^{+-;K^+ K^-} \bigr) \\ &{}\times \bigl[ k_{K \pi} (\mu) - 4 L_9^r (\mu) \bigr] \\ &{} + \frac{1}{3\sqrt{2}} \frac{F_0^2}{F_\pi^2} \\ &{}\times \biggl[ 2 \biggl( 1 + \frac{\sqrt{3}}{2} \epsilon_2 \biggr) \beta^{+-;K^+ K^-} \\ &{}- 3 ( 7 + 2 \sqrt{3} \epsilon_2 ) \gamma^{+-;K^+ K^-} \biggr] \\ &{}\times \bigl[ k_{K^0 \pi^0} (\mu) - 4 L_9^r (\mu) \bigr] \\ &{} + \frac{1}{2} \sqrt{\frac{3}{2}} \biggl[ ( 1 - \sqrt{3} \epsilon_1 ) \beta^{+ K^-;\eta{\bar{K}}^0} \\ &{} - \biggl( 1 + \frac{\epsilon_1}{\sqrt{3}} \biggr) \gamma^{+ K^-;\eta{\bar{K}}^0} \biggr] \\ & {}\times \bigl[ k_{\eta K^0} (\mu) - 4 L_9^r (\mu) \bigr] , \\ \pi_{3,G}^{+-} =& + 2 L_3 \\ &{} - \frac{1}{3} \frac{F_0^2}{F_\pi^2} \bigl( \beta^{+-;K^+ K^-} + 21 \gamma^{+-;K^+ K^-} \bigr) \\ &{}\times \biggl[ k_{K \pi} (\mu) - \frac{16}{3} L_2^r (\mu) \biggr] \\ &{} + \frac{1}{3\sqrt{2}} \frac{F_0^2}{F_\pi^2} \\ &{}\times \biggl[ 2 \biggl( 1 + \frac{\sqrt{3}}{2} \epsilon_2 \biggr) \beta^{+-;K^+ K^-} \\ &{}- 3 ( 7 + 2 \sqrt{3} \epsilon_2 ) \gamma^{+-;K^+ K^-}\biggr] \\ & {}\times \biggl[ k_{K^0 \pi^0} (\mu) - \frac{16}{3} L_2^r (\mu) \biggr] \\ &{} + \frac{1}{2} \sqrt{\frac{3}{2}} \frac{F_0^2}{F_\pi^2} \\ &{}\times \biggl[ ( 1 - \sqrt{3} \epsilon_1 ) \beta^{+ K^-;\eta{\bar{K}}^0} \\ &{} - \biggl( 1 + \frac{\epsilon_1}{\sqrt{3}} \biggr) \gamma^{+ K^-;\eta{\bar{K}}^0} \biggr] \biggl[ k_{\eta K^0} (\mu) - \frac{16}{3} L_2^r (\mu) \biggr] . \end{aligned}$$

(B.4)

In the channel with two neutral pions, some coefficients vanish due to Bose symmetry, see Eq. (73). For the remaining ones, we obtain the manifestly scale-independent expressions

$$\begin{aligned} \pi_{1,F}^{00} =&+ 2 ( 5 + 2 \sqrt{3} \epsilon_2) L_3 + 16 \bigl[ 2 L_1^r (\mu) - L_2^r (\mu) \bigr] \\ &{} - \frac{F_0^2}{F_\pi^2} \biggl\{ 3 (1 + 2 \sqrt{3} \epsilon_2 ) \gamma^{00;K^+K^-} \\ &{}\times \biggl[ k_{K\pi^0} (\mu) - \frac{16}{3} L_2^r (\mu) \biggr] \\ &{} + {\sqrt{2}} \biggl[ \beta^{0 -;K^0 K^-} + 6 \biggl(1 - \frac{\sqrt{3}}{2} \epsilon_2 \biggr) \gamma^{0 -;K^0 K^-} \biggr] \\ &{}\times \biggl[ k_{K^0 \pi} (\mu) - \frac{16}{3} L_2^r (\mu ) \biggr] \\ &{} - \frac{\sqrt{3}}{2} \biggl( 1 - \frac{\epsilon_1}{\sqrt{3}} + \sqrt{3} \epsilon_2 \biggr) \\ &{}\times\bigl( \beta^{0 K^-;\eta K^-} + 3 \gamma^{0 K^-;\eta K^- } \bigr) \\ &{}\times \biggl[ k_{\eta K} (\mu) - \frac{16}{3} L_2^r (\mu) \biggr] \\ &{} - 2 \beta^{00;+-} \biggl[ k_{\pi\pi} (\mu) - \frac{16}{3} L_2^r (\mu) \biggr] \\ &{}- 4 \beta^{00;K^+ K^-} \biggl[ k_{KK} (\mu ) - \frac{16}{3} L_2^r (\mu) \biggr] \\ &{} + 2 \beta^{00;K^0 {\bar{K}}^0} \biggl[ k_{K^0 {\bar{K}}^0} (\mu) - \frac{16}{3} L_2^r (\mu) \biggr] \biggr\} , \\ \pi_{2,F}^{00} =& - 2 \bigl[ 4 L_2^r (\mu) - 3 L_9^r (\mu) \bigr] - 2 \biggl( 1 - \frac{2}{\sqrt{3}} \epsilon_2 \biggr) L_3 \\ &{} + \frac{F_0^2}{F_\pi^2} \biggl\{ 3 (1 + 2 \sqrt{3} \epsilon_2 ) \gamma^{0 0 ;K^+K^-} \\ &{}\times \bigl[ k_{K\pi^0} (\mu) - 4 L_9^r (\mu) \bigr] \\ &{} + {\sqrt{2}} \biggl[ \beta^{0 -;K^0 K^-} + 6 \biggl(1 - \frac{\sqrt{3}}{2} \epsilon_2 \biggr) \gamma^{0 -;K^0 K^-} \biggr] \\ &{}\times \bigl[ k_{K^0 \pi} (\mu) - 4 L_9^r (\mu) \bigr] \\ & {} - \frac{\sqrt{3}}{2} \biggl( 1 \,{-}\, \frac{\epsilon_1}{\sqrt{3}} \,{+}\, \sqrt{3} \epsilon_2 \biggr) \bigl( \beta^{0 K^-;\eta K^{{-}}} \\ &{}+ 3 \gamma^{0 K^-;\eta K^- } \bigr) \\ & {}\times \bigl[ k_{\eta K} (\mu) - 4 L_9^r (\mu) \bigr] \biggr\} , \\ \pi_{3,G}^{00} =& + 2 \biggl( 1 - \frac{2}{\sqrt {3}} \epsilon_2 \biggr) L_3 \\ &{} + \frac{F_0^2}{F_\pi^2} \biggl\{ \frac{1}{3} (1 + 2 \sqrt{3} \epsilon_2 ) \bigl( \beta^{0 0;K^+ K^- } - 6 \gamma^{0 0;K^+ K^-} \bigr) \\ &{}\times \biggl[ k_{K\pi^0} (\mu) - \frac{16}{3} L_2^r (\mu) \biggr] \\ &{} - {\sqrt{2}} \biggl[ \frac{2}{3} \biggl( 1 - \frac{\sqrt{3} \epsilon_2}{2} \biggr) \beta^{0 +;K^+ {\bar{K}}^0} \\ & {}+ (7 - 2 \sqrt{3} \epsilon_2 ) \gamma^{0 +;K^+ {\bar{K}}^0} \biggr] \\ & {} \times \biggl[ k_{K^0 \pi} (\mu) - \frac{16}{3} L_2^r (\mu) \biggr] \\ &{} + \frac{\sqrt{3}}{2} \biggl( 1 - \frac{\epsilon_1}{\sqrt{3}} + \sqrt{3} \epsilon_2 \biggr) \bigl( \beta^{0 K^-;\eta K^-} - \gamma^{0 K^-;\eta K^-} \bigr) \\ &{}\times \biggl[ k_{\eta K} (\mu) - \frac{16}{3} L_2^r (\mu) \biggr] \biggr\} . \end{aligned}$$

(B.5)

Let us now consider the coefficients $\pi^{ab}_{0,F/G}$ left aside so far. Their expressions exhibit the same decomposition

$$ \pi^{ab}_{0,F/G} = \bigl( \pi^{ab}_{0,F/G} \bigr)_T + \bigl( \pi ^{ab}_{0,F/G} \bigr)_L ,\quad ab = +-, 00 , $$

(B.6)

into a contribution from tree and tadpole terms (chiral logarithms from tadpoles are now present), and a contribution from the unitarity loops, as describe before. Starting with the former, we find

$$\begin{aligned} &{ \bigl( \pi_{0,F}^{+-} \bigr)_{T} (\mu)} \\ &{\quad =- 64 M_{\pi}^2 L_1^r (\mu) + 8 M_K^2 L_2^r (\mu)} \\ &{\qquad{}+ 2 \bigl( M_K^2 - 8 M_{\pi}^2 \bigr) L_3 + 8 (6 {\widehat{m}} - m_s ) B_0 L_4^r (\mu)} \\ &{\qquad{} + e^2 F_0^2 \biggl[ - \frac{4}{3} \bigl( {\widehat{K}}_1^r (\mu) - 11 { \widehat{K}}_2^r (\mu) \bigr)} \\ &{\qquad{} + \frac{4}{9} \bigl( 2 {\widehat{K}}_5^r (\mu) + 11 {\widehat{K}}_6^r (\mu) \bigr) + 2 {\widehat{K}}_{12}^r (\mu) \biggr]} \\ &{\qquad{} + \frac{1}{256 \pi^2} \biggl[ 10 M_{\pi}^2 \ln \frac{M_{\pi}^2}{\mu^2}} \\ &{\qquad{}+ ( 11 - 2 \sqrt{3} \epsilon_2 ) M_{\pi^0}^2 \ln\frac{M_{\pi^0}^2}{\mu^2}} \\ &{\qquad{}+ 2 M_{K}^2 \ln\frac{M_{K}^2}{\mu^2} + 4 M_{K^0}^2 \ln\frac {M_{K^0}^2}{\mu^2}} \\ &{\qquad{} - 3 \biggl( 1 - \frac{2}{\sqrt{3}} \epsilon_1 \biggr) M_{\eta}^2 \ln\frac{M_{\eta}^2}{\mu^2} \biggr],} \end{aligned}$$

(B.7)

$$\begin{aligned} &{\bigl( \pi_{0,G}^{+-} \bigr)_{T} (\mu)} \\ &{\quad = - 2 M_K^2 L_3 - 8 (2 {\widehat{m}} + m_s ) B_0 L_4^r (\mu)} \\ &{\qquad{} + e^2 F_0^2 \biggl[ - \frac{4}{3} \bigl( {\widehat{K}}_1^r (\mu) + { \widehat{K}}_2^r (\mu) \bigr)} \\ &{\qquad{}+ 4 \bigl( 2 {\widehat{K}}_3^r (\mu) + {\widehat{K}}_4^r (\mu) \bigr)} \\ &{\qquad{} - \frac{4}{9} \bigl( 4 {\widehat{K}}_5^r (\mu) - 5 {\widehat{K}}_6^r (\mu) \bigr) + 2 {\widehat{K}}_{12}^r (\mu) \biggr]} \\ &{\qquad{} + \frac{1}{256 \pi^2} \biggl[ 6 M_{\pi}^2 \ln \frac{M_{\pi}^2}{\mu^2} + ( 5 + 2 \sqrt{3} \epsilon_2 ) M_{\pi^0}^2 \ln\frac{M_{\pi^0}^2}{\mu^2}} \\ &{\qquad{} + 6 M_{K}^2 \ln\frac{M_{K}^2}{\mu^2} + 4 M_{K^0}^2 \ln\frac {M_{K^0}^2}{\mu^2}} \\ &{\qquad{} + 3 \biggl( 1 - \frac{2}{\sqrt{3}} \epsilon_1 \biggr) M_{\eta}^2 \ln\frac{M_{\eta}^2}{\mu^2} \biggr] ,} \end{aligned}$$

(B.8)

and

$$\begin{aligned} &{\bigl( \pi_{0,F}^{00} \bigr)_T} \\ &{\quad = - 64 M_{\pi^0}^2 L_1^r (\mu) + 8 M_K^2 L_2^r (\mu)} \\ &{\qquad{}+ 2 \biggl[ \biggl( 1 - \frac{2}{\sqrt{3}} \epsilon_2 \biggr) M_K^2 - 8 \biggl( 1 + \frac{2}{\sqrt{3}} \epsilon_2 \biggr) M_{\pi^0}^2 \biggr] L_3} \\ &{\qquad{} + 8 \bigl[ 6 {\widehat{m}} - m_s -2 \sqrt{3} \epsilon_2 ( 2 {\widehat{m}} + m_s) \bigr] B_0 L_4^r (\mu)} \\ &{\qquad{}- 8 \biggl( \frac{1}{R} - 4 \frac{\epsilon_2}{ \sqrt{3}} \biggr) ( m_s - {\widehat{m}}) B_0 L_5^r (\mu)} \\ &{\qquad{} - e^2 F_0^2 \biggl[ \frac{4}{3} \bigl( {\widehat{K}}_1^r (\mu) + { \widehat{K}}_2^r (\mu) \bigr)} \\ &{\qquad{} + \frac{4}{9} \bigl( {\widehat{K}}_5^r (\mu) + {\widehat{K}}_6^r (\mu) \bigr) - 2 {\widehat{K}}_{12}^r (\mu) \biggr]} \\ &{\qquad{} + \frac{1}{256 \pi^2} \biggl[ 2 ( 11 + 26 \sqrt{3} \epsilon_2 ) M_{\pi}^2 \ln\frac{M_{\pi }^2}{\mu^2}} \\ &{\qquad{}- ( 1 + 4 \sqrt{3} \epsilon_2 ) M_{\pi^0}^2 \ln\frac {M_{\pi^0}^2}{\mu^2}} \\ &{\qquad{} - 4 ( 1 - 2 \sqrt{3} \epsilon_2 ) M_{K}^2 \ln\frac{M_{K}^2}{\mu^2}} \\ &{\qquad{}+ 2 ( 5 - 2 \sqrt{3} \epsilon_2 ) M_{K^0}^2 \ln\frac{M_{K^0}^2}{\mu^2}} \\ &{\qquad{} - 3 \biggl( 1 - 2 \frac{ \epsilon_1}{\sqrt{3}} + 2 \sqrt{3} \epsilon_2 \biggr) M_{\eta}^2 \ln \frac{M_{\eta}^2}{\mu^2} \biggr] .} \end{aligned}$$

(B.9)

A few remarks are in order at this stage. First, one notices that in the isospin limit these expressions differ from the corresponding ones in Refs. [41, 42] by the contribution proportional to L ₄ and by the absence of a contribution proportional to L ₅ [a contribution proportional to L ₅ appears in $( \pi_{0,F}^{00} )_{T}$, but it vanishes in the isospin limit, and is identically zero at lowest order, where $\epsilon _{1,2} = \sqrt{3}/(4R)$]. These differences are entirely due to the choice of the overall normalisation in Eqs. (67) and (71), with F ₀ instead of F _π in the denominator. This explains also the differences in the tadpole contribution. Secondly, we also note the appearance of the low-energy constants ${\widehat{K}}_{i}^{r} (\mu)$, already present in equations (A.11), (A.12) and (A.13). Finally, once the normalisation issue is accounted for, the contributions proportional to L ₄ are written in terms of ${\widehat{m}}B_{0}$ and m _s B ₀, which is their actual form when computed from the chiral Lagrangian. Usually, these contributions are directly expressed in terms of the meson masses, but we refrain from following this practice: as we will show shortly, and for reasons similar to those already discussed in the introduction [cf. Eqs. (1)–(3)], the expression of these terms is again dictated by the contributions coming from the unitarity loops, $( \pi_{0,F}^{+-} )_{L}$, $( \pi _{0,G}^{+-} )_{L}$, and $( \pi_{0,F}^{00} )_{L}$. Turning to the latter, we obtain the somewhat lengthy expressions

$$\begin{aligned} &{ \bigl( \pi_{0,F}^{+-} \bigr)_{L} (\mu)} \\ &{\quad = \frac{F_0^2}{F_\pi^2} \biggl[ \frac{1}{2} \bigl( \beta^{+ K^-;+ K^-} - 2 \beta^{K^- -;K^- -} - 3 \gamma^{+ K^-;+ K^-} \bigr)} \\ &{\qquad{}\times\bigl(M_K^2 + 2 M_\pi^2\bigr)} \\ &{ \qquad{} + \bigl( - \beta^{+ K^-;+ K^-} + 6 \gamma^{K^- -;K^- -} + 4 \gamma^{+ K^-;+ K^-} \bigr)} \\ &{\qquad{}\times\bigl(M_K^2 - M_\pi^2 \bigr)} \\ &{\qquad{} + 16 \pi F_\pi^2 \varphi_S^{+ K^-;+ K^-}(0) } \\ &{\qquad{} - 32 \pi F_\pi^2 \varphi_S^{K^- -;K^- -} (0) \biggr] k_{K\pi} (\mu)} \\ &{\qquad{} + \frac{F_0^2}{F_\pi^2} \frac{1}{\sqrt{2}} \biggl\{ \frac{3}{2} \biggl[ - \biggl( 1 + \frac{\epsilon_2}{\sqrt{3}} \biggr) \beta ^{+ K^-;0 {\bar{K}}^0}} \\ &{ \qquad{} + ( 1 - \sqrt{3} \epsilon_2 ) \gamma^{+ K^-;0 {\bar{K}}^0} \biggr] \bigl(M_K^2 + 2 M_\pi^2\bigr)} \\ &{ \qquad{} - \biggl[ 9 \biggl( 1 + \frac{\epsilon_2}{\sqrt{3}} \biggr) \gamma ^{+ K^-;0 {\bar{K}}^0} - ( 1 - \sqrt{3} \epsilon_2 ) \beta^{+ K^-;0 {\bar{K}}^0} \biggr]} \\ &{ \qquad{}\times\bigl(M_{K^0}^2 - M_{\pi^0}^2\bigr)} \\ &{\qquad{} - ( 1 - \sqrt{3} \epsilon_2 ) \gamma^{+ K^-;0 {\bar{K}}^0} \bigl(M_{K}^2 - M_{\pi}^2\bigr)} \\ &{ \qquad{} - 3 \biggl( 1 + \frac{\epsilon_2}{\sqrt{3}} \biggr) 16 \pi F_\pi^2 \varphi_S^{+ K^-;0 {\bar{K}}^0} (0) \biggr\} k_{K^0 \pi^0} (\mu)} \\ &{\qquad{} + \frac{F_0^2}{F_\pi^2} \sqrt{\frac{3}{2}} \biggl\{ - \frac{1}{2} \biggl[ ( 1 - \sqrt{3} \epsilon_1 ) \beta^{+ K^-;\eta{\bar{K}}^0}} \\ &{\qquad{} + 3 \biggl( 1 + \frac{\epsilon_1}{\sqrt{3}} \biggr) \gamma^{+ K^-;\eta{\bar{K}}^0} \biggr] \bigl(M_K^2 + 2 M_\pi^2\bigr)} \\ &{ \qquad{}+ \biggl[ 3 ( 1 - \sqrt{3} \epsilon_1 ) \gamma^{+ K^-;\eta{\bar{K}}^0} + \biggl( 1 + \frac{\epsilon_1}{\sqrt{3}} \biggr) \beta^{+ K^-;\eta{\bar{K}}^0} \biggr]} \\ &{\qquad{}\times\bigl(M_\eta^2 - M_{K^0}^2 \bigr)} \\ &{\qquad{} + \biggl( 1 + \frac{\epsilon_1}{\sqrt{3}} \biggr) \gamma^{+ K^-;\eta{\bar{K}}^0} \bigl( M_K^2 - M_\pi^2 \bigr)} \\ &{\qquad{} - ( 1 - \sqrt{3} \epsilon_1 ) 16 \pi F_\pi^2 \varphi_S^{+ K^-;\eta{\bar{K}}^0} (0) \biggr\} k_{\eta K^0} (\mu)} \\ &{\qquad{} + 16 \pi F_0^2 \biggl[ (1 + 2 \sqrt{3} \epsilon_2 ) \varphi_S^{+-;00} (0) k_{\pi^0 \pi^0} (\mu)} \\ &{ \qquad{} - 2 \varphi_S^{+-;+-} (0) k_{\pi\pi} (\mu) - 4 \varphi_S^{+-;K^+ K^-} (0) k_{KK} (\mu)} \\ &{ \qquad{} + 2 \varphi_S^{+-;K^0 {\bar{K}}^0} (0) k_{K^0 {\bar{K}}^0} (\mu)} \\ &{\qquad{} + 3 \biggl( 1 - \frac{2}{\sqrt{3}} \epsilon_1 \biggr) \varphi _S^{+-;\eta\eta} (0) k_{\eta\eta} (\mu)} \\ &{ \qquad{} + 2 \sqrt{3} \biggl( 1 - \frac{\epsilon_1}{\sqrt{3}} + \sqrt {3} \epsilon_2 \biggr) \varphi_S^{+-;0\eta} (0) k_{\eta\pi^0} (\mu) \biggr] ,} \\ \end{aligned}$$

(B.10)

$$\begin{aligned} &{ \bigl( \pi_{0,G}^{+-} \bigr)_{L} (\mu)} \\ &{\quad =\frac{F_0^2}{F_\pi^2} \biggl[ - \frac{1}{2} \bigl( \beta^{+ K^-;+ K^-} + 2 \beta^{K^- -;K^- -} + \gamma^{+ K^-;+ K^-} \bigr)} \\ &{ \qquad{} \times\bigl(M_K^2 + 2 M_\pi^2\bigr)} \\ &{ \qquad{} + \bigl( \beta^{+ K^-;+ K^-} + 6 \gamma^{K^- -;K^- -} - 4 \gamma^{+ K^-;+ K^-} \bigr)} \\ &{\qquad{}\times\bigl(M_K^2 - M_\pi^2\bigr)} \\ &{\qquad{} - 16 \pi F_\pi^2 \varphi_S^{+ K^-;+ K^-} (0)} \\ &{\qquad{} - 32 \pi F_\pi^2 \varphi_S^{K^- -;K^- -} (0) \biggr] k_{K \pi} (\mu)} \\ &{\qquad{} + \frac{F_0^2}{F_\pi^2} \frac{1}{\sqrt{2}} \biggl\{ \frac{3}{2} \biggl[ \biggl( 1 + \frac{\epsilon_2}{\sqrt{3}} \biggr) \beta ^{+ K^-;0 {\bar{K}}^0}} \\ &{\qquad{} + \frac{1}{3} ( 1 - \sqrt{3} \epsilon_2 ) \gamma^{+ K^-;0 {\bar{K}}^0} \biggr] \bigl(M_K^2 + 2 M_\pi^2\bigr)} \\ &{ \qquad{} + \biggl[ 9 \biggl( 1 + \frac{\epsilon_2}{\sqrt{3}} \biggr) \gamma ^{+ K^-;0 {\bar{K}}^0} - ( 1 - \sqrt{3} \epsilon_2 ) \beta^{+ K^-;0 {\bar{K}}^0} \biggr]} \\ &{ \qquad{} \times\bigl(M_{K^0}^2 - M_{\pi^0}^2\bigr)} \\ &{\qquad{} + ( 1 - \sqrt{3} \epsilon_2 ) \gamma^{+ K^-;0 {\bar{K}}^0} \bigl(M_{K}^2 - M_{\pi}^2\bigr)} \\ &{ \qquad{} + 3 \biggl( 1 + \frac{\epsilon_2}{\sqrt{3}} \biggr) 16 \pi F_\pi^2 \varphi_S^{+ K^-;0 {\bar{K}}^0} (0) \biggr\} k_{K^0 \pi^0} (\mu)} \\ &{\qquad{} + \frac{F_0^2}{F_\pi^2} \sqrt{\frac{3}{2}} \biggl\{ \frac{1}{2} \biggl[ ( 1 - \sqrt{3} \epsilon_1 ) \beta^{+ K^-;\eta{\bar{K}}^0}} \\ &{ \qquad{} - \biggl( 1 + \frac{\epsilon_1}{\sqrt{3}} \biggr) \gamma^{+ K^-;\eta{\bar{K}}^0} \biggr] \bigl(M_K^2 + 2 M_\pi^2\bigr)} \\ &{ \qquad{} - \biggl[ 3 ( 1 - \sqrt{3} \epsilon_1 ) \gamma^{+ K^-;\eta{\bar{K}}^0} + \biggl( 1 + \frac{\epsilon_1}{\sqrt{3}} \biggr) \beta^{+ K^-;\eta{\bar{K}}^0} \biggr]} \\ &{\qquad{} \times\bigl(M_\eta^2 - M_{K^0}^2 \bigr)} \\ &{\qquad{} - \biggl( 1 + \frac{\epsilon_1}{\sqrt{3}} \biggr) \gamma^{+ K^-;\eta{\bar{K}}^0} \bigl(M_K^2 - M_\pi^2 \bigr)} \\ &{ \qquad{} + ( 1 - \sqrt{3} \epsilon_1 ) 16 \pi F_\pi^2 \varphi_S^{+ K^-;\eta{\bar{K}}^0} (0) \biggr\} k_{\eta K^0} (\mu) ,} \end{aligned}$$

(B.11)

and

$$\begin{aligned} &{\bigl( \pi_{0,F}^{00} \bigr)_L (\mu)} \\ &{\quad = - \frac{F_0^2}{F_\pi^2} (1 + 2 \sqrt{3} \epsilon_2 )} \\ &{\qquad{}\times \biggl[ \frac{1}{2} \bigl( \beta^{0 K^-;0 K^-} + 3 \gamma^{0 K^-;0 K^-} \bigr) \bigl(M_K^2 + 2 M_{\pi^0}^2\bigr)} \\ &{ \qquad{} + \bigl( \beta^{0 K^-;0 K^-} + 2 \gamma^{0 K^-;0 K^-} \bigr) \bigl(M_K^2 - M_{\pi^0}^2 \bigr)} \\ &{ \qquad{} + 16 \pi F_\pi^2 \varphi_S^{0 K^-;0 K^-} (0) \biggr] k_{K\pi^0} (\mu)} \\ &{\qquad{} + {\sqrt{2}} \frac{F_0^2}{F_\pi^2} \biggl\{ - \frac{3}{2} \biggl[ \biggl( 1 - \frac{\epsilon_2}{\sqrt{3}} \biggr) \beta^{0 K^-;- {\bar{K}}^0}} \\ &{\qquad{} - (1 + \sqrt{3} \epsilon_2 ) \gamma^{0 K^-;- {\bar{K}}^0} \biggr] \bigl(M_K^2 + 2 M_{\pi^0}^2 \bigr)} \\ &{ \qquad{} + \biggl[ (1 + \sqrt{3} \epsilon_2 ) \beta^{0 K^-;- {\bar{K}}^0} - 9 \biggl( 1 - \frac{\epsilon_2}{\sqrt{3}} \biggr) \gamma^{0 K^-;- {\bar{K}}^0} \biggr]} \\ &{ \qquad{}\times \bigl(M_{K^0}^2 - M_\pi^2\bigr)} \\ &{\qquad{} - (1 + \sqrt{3} \epsilon_2 ) \gamma^{0 K^-;- {\bar{K}}^0} \bigl(M_K^2 - M_{\pi^0}^2 \bigr)} \\ &{\qquad{} - 3 \biggl( 1 - \frac{\epsilon_2}{\sqrt{3}} \biggr) 16 \pi\varphi _S^{0 K^-;- {\bar{K}}^0} (0) \biggr\} k_{K^0 \pi} (\mu)} \\ &{\qquad{} + \sqrt{3} \biggl( 1 - \frac{\epsilon_1}{\sqrt{3}} + \sqrt{3} \epsilon_2 \biggr) \frac{F_0^2}{F_\pi^2}} \\ &{\qquad{} \times \biggl[ \frac{1}{2} \bigl( \beta^{0 K^-;\eta K^-} + 3 \gamma^{0 K^-;\eta K^-} \bigr)} \\ &{\qquad{}\times \bigl(2 M_\eta^2 - 3 M_K^2 - 2 M_{\pi^0}^2 \bigr)} \\ &{ \qquad{} + \gamma^{0 K^-;\eta K^-} \bigl(M_K^2 - M_{\pi^0}^2 \bigr)} \\ &{ \qquad{} - 16 \pi\varphi_S^{0 K^-;\eta K^-} (0) \biggr] k_{\eta K} (\mu)} \\ &{\qquad{} + \frac{F_0^2}{F_\pi^2} \biggl[ - (1 + 2 \sqrt{3} \epsilon_2 ) 16 \pi F_\pi^2 \varphi_S^{00;00} (0) k_{\pi^0 \pi^0} (\mu)} \\ &{ \qquad{} + 32 \pi F_\pi^2 \varphi_S^{00;+-} (0) k_{\pi\pi } (\mu)} \\ &{ \qquad{} + 64 \pi F_\pi^2 \varphi_S^{00;K^+ K^-} (0) k_{KK} (\mu)} \\ &{ \qquad{} - 32 \pi F_\pi^2 \varphi_S^{00;K^0 {\bar{K}}^0} (0) k_{K^0 {\bar{K}}^0} (\mu)} \\ &{\qquad{} - 3 \biggl( 1 - \frac{2}{\sqrt{3}} \epsilon_1 \biggr) 16 \pi F_\pi ^2 \varphi_S^{00;\eta\eta} (0) k_{\eta\eta} (\mu)} \\ &{ \qquad{} - 2\sqrt{3} \biggl( 1 - \frac{\epsilon_1}{\sqrt{3}} + \sqrt{3} \epsilon_2 \biggr) 16 \pi F_\pi^2 \varphi_S^{00;0\eta} (0) k_{\pi ^0 \eta} (\mu) \biggr] .} \\ \end{aligned}$$

(B.12)

As mentioned above, the sums in (B.6) should no longer depend on the renormalisation scale μ. The scale dependences of $( \pi_{0,F}^{+-})_{T} (\mu)$, $( \pi_{0,G}^{+-} )_{T} (\mu)$, and $( \pi_{0,F}^{00})_{T} (\mu)$ can be worked out through

$$\begin{aligned} \begin{aligned} &\mu\frac{d}{d \mu} L_i^r (\mu) = - \frac{1}{16 \pi^2} \varGamma_i , \\ &\mu\frac{d}{d \mu} e^2 {\widehat{K}}_i^r ( \mu) = - \frac{1}{16 \pi^2} \frac{\varDelta_\pi}{F_0^2} {\widehat{\varSigma}}_i , \end{aligned} \end{aligned}$$

(B.13)

where the coefficients Γ _i were worked out in [5], and the coefficients ${\widehat{\varSigma}}_{i}$ can be obtained from the coefficients Σ _i of Ref. [74], as explained in Ref. [37]:

$$\begin{aligned} \begin{aligned} & \varGamma_1 = \frac{3}{32} , \qquad \varGamma_2 = \frac{3}{16} , \qquad \varGamma_3 = 0 , \\ & \varGamma_4 = \frac{1}{8} , \qquad \varGamma_5 = \frac{3}{8} , \qquad \varGamma_9 = \frac{1}{4} , \\ & {\widehat{\varSigma}}_1 = 0 , \qquad {\widehat{\varSigma}}_2 = \frac{1}{2} , \qquad {\widehat{\varSigma}}_3 = 0 , \\ & {\widehat{\varSigma}}_4 = 1 , \qquad {\widehat{\varSigma}}_5 = 0 , \qquad {\widehat{\varSigma}}_6 = \frac{3}{4} , \\ & {\widehat{\varSigma}}_7 = 0 , \qquad {\widehat{\varSigma}}_8 = \frac{1}{2} , \qquad {\widehat{\varSigma}}_9 = 0 , \\ & {\widehat{\varSigma}}_{10} = \frac{3}{4} , \qquad {\widehat{\varSigma}}_{11} = 0 , \qquad {\widehat{\varSigma}}_{12} = 0 . \end{aligned} \end{aligned}$$

(B.14)

The corresponding expressions for $\mu(d / d \mu) ( \pi _{0,F}^{+-} )_{L} (\mu)$, $( \pi_{0,G}^{+-})_{L} (\mu)$ and $( \pi_{0,F}^{00} )_{L} (\mu)$ can be directly read off from Eqs. (B.10), (B.11), and (B.12), respectively, upon replacing k _ab (μ) by −1/(16π ²). In order to complete this exercise, the lowest-order expressions of the various quantities $\varphi_{S}^{ab ; a^{\prime}b^{\prime}} (0)$ are requested, in addition to the information already provided by Table 11. These expressions can be found in Table 12. The lowest-order expressions of the masses and mixing angles are also needed, see Refs. [5] and [74]. Scale invariance is achieved through identities which, e.g. in the case of $\pi_{0,F}^{+-}$ and at lowest order, relates the combination $8 (2 {\widehat{m}} + m_{s} ) B_{0}$ to a certain combination of masses and parameters describing the lowest-order scattering amplitudes, which multiply the chiral logarithms arising from the tadpole and loop contributions. Instead, one may use this identity in order to rewrite $\pi _{0,F}^{+-}$ in an equivalent form, which is explicitly scale independent but with no explicit reference to B ₀ any longer

$$\begin{aligned} \pi_{0,F}^{+-} =& - 64 M_{\pi}^2 \biggl[ L_1^r (\mu) - \frac{3}{4} L_4^r (\mu) \biggr] \\ &{} + 8 M_K^2 \biggl[ L_2^r ( \mu) - \frac{3}{2} L_4^r (\mu) \biggr] + 2 \bigl( M_K^2 - 8 M_{\pi}^2 \bigr) L_3 \\ &{} - \biggl[ \frac{4}{3}e^2F_0^2 \bigl( {\widehat{K}}_1^r (\mu) - 11 {\widehat{K}}_2^r (\mu) \bigr) \\ & {} - \frac{4}{9}e^2F_0^2 \bigl( 2 {\widehat{K}}_5^r (\mu) + 11 {\widehat{K}}_6^r (\mu) \bigr) - 2 e^2F_0^2 {\widehat{K}}_{12}^r (\mu) \\ &{} + 88\varDelta_\pi L_4^r (\mu) \biggr] \\ &{}+ 10 M_{\pi}^2 \biggl( \frac{1}{256 \pi^2} \ln \frac{M_{\pi}^2}{\mu ^2} - L_4^r (\mu) \biggr) \\ &{} + ( 11 - 2 \sqrt{3} \epsilon_2 ) M_{\pi^0}^2 \biggl( \frac{1}{256 \pi^2} \ln\frac{M_{\pi^0}^2}{\mu^2} - L_4^r (\mu) \biggr) \\ &{} + 2 M_{K}^2 \biggl( \frac{1}{256 \pi^2} \ln \frac{M_{K}^2}{\mu^2} - L_4^r (\mu) \biggr) \\ &{} + 4 M_{K^0}^2 \biggl( \frac{1}{256 \pi^2} \ln \frac{M_{K^0}^2}{\mu ^2} - L_4^r (\mu) \biggr) \\ &{} - 3 \biggl( 1 - \frac{2}{\sqrt{3}} \epsilon_1 \biggr) M_{\eta}^2 \biggl( \frac{1}{256 \pi^2}\ln\frac{M_{\eta}^2}{\mu^2} - L_4^r (\mu) \biggr) \\ &{} + \frac{F_0^2}{F_\pi^2} \biggl[ \frac{1}{6} \beta^{+-;K^+ K^-} \bigl( 7 M_K^2 - 19 M_\pi^2 \bigr) \\ &{}- \frac{17}{2} \gamma^{+-;K^+ K^-} \\ & {} \times\bigl( M_K^2 - M_\pi^2 \bigr) - 16 \pi F_\pi^2 \varphi_S^{+-;K^+ K^-} (0) \biggr] \\ & {}\times \bigl[ k_{K\pi} (\mu) - 8 L_4^r (\mu) \bigr] \\ &{} + \frac{F_0^2}{F_\pi^2} \frac{1}{\sqrt{2}} \biggl\{ \biggl[ - \beta^{0 -;K^0 K^-} \\ &{} + 6 \biggl( 1 + \frac{\sqrt{3}}{2} \epsilon_2 \biggr) \gamma^{0 -;K^0 K^-} \biggr] \bigl(M_K^2 + 2 M_\pi^2\bigr) \\ & {} + \frac{1}{6} ( 1 - \sqrt{3} \epsilon_2 ) \bigl[ \beta^{0 -;K^0 K^-} + 3 \gamma^{0 -;K^0 K^-} \bigr] \\ &{}\times \bigl(M_K^2 - M_\pi^2\bigr) \\ & {} + 2 \bigl[ \beta^{0 -;K^0 K^-} + 3 \sqrt{3} \epsilon_2 \gamma^{0 -;K^0 K^-} \bigr] \bigl(M_{K^0}^2 - M_{\pi^0}^2\bigr) \\ & {} + \frac{3}{2} \biggl( 1 + \frac{\epsilon_2}{\sqrt{3}} \biggr) \bigl[ \beta^{0 -;K^0 K^-} - 3 \gamma^{0 -;K^0 K^-} \bigr] \\ & {} \times\bigl(M_K^2 + M_{K^0}^2 + M_\pi^2 + M_{\pi^0}^2 \bigr) \\ & {} + 3 \biggl( 1 + \frac{\epsilon_2}{\sqrt{3}} \biggr) 16 \pi F_\pi^2 \varphi_S^{0 -;K^0 K^-} (0) \biggr\} \\ &{} \times \bigl[ k_{K^0 \pi^0} (\mu) - 8 L_4^r (\mu) \bigr] \\ & {}+ \frac{F_0^2}{F_\pi^2} \sqrt{\frac{3}{2}} \biggl\{ - \frac{1}{2} \biggl[ ( 1 - \sqrt{3} \epsilon_1 ) \beta^{+ K^-;\eta{\bar{K}}^0} \\ & {} + 3 \biggl( 1 + \frac{\epsilon_1}{\sqrt{3}} \biggr) \gamma^{+ K^-;\eta{\bar{K}}^0} \biggr] \bigl(M_K^2 + 2 M_\pi^2\bigr) \\ &{} + \biggl[ 3 ( 1 - \sqrt{3} \epsilon_1 ) \gamma^{+ K^-;\eta{\bar{K}}^0} \\ & {} + \biggl( 1 + \frac{\epsilon_1}{\sqrt{3}} \biggr) \beta^{+ K^-;\eta{\bar{K}}^0} \bigl(M_\eta^2 - M_{K^0}^2 \bigr) \biggr] \\ & {}+ \biggl( 1 + \frac{\epsilon_1}{\sqrt{3}} \biggr) \gamma^{+ K^-;\eta{\bar{K}}^0} \bigl( M_K^2 - M_\pi^2 \bigr) \\ & {} - ( 1 - \sqrt{3} \epsilon_1 ) 16 \pi F_\pi^2 \varphi_S^{+ K^-;\eta{\bar{K}}^0} (0) \biggr\} \\ &{}\times \bigl[ k_{\eta K^0} (\mu) - 8 L_4^r (\mu) \bigr] \\ &{} + \frac{F_0^2}{F_\pi^2} \biggl\{ (1 + 2 \sqrt{3} \epsilon_2 ) 16 \pi F_\pi^2 \varphi_S^{+-;00} (0) \\ &{}\times \bigl[ k_{\pi^0 \pi^0} (\mu) - 8 L_4^r (\mu) \bigr] \\ & {} - 32 \pi F_\pi^2 \varphi_S^{+-;+-} (0) \bigl[ k_{\pi\pi} (\mu) - 8 L_4^r (\mu) \bigr] \\ & {} - 64 \pi F_\pi^2 \varphi_S^{+-;K^+ K^-} (0) \bigl[ k_{KK} (\mu) - 8 L_4^r (\mu) \bigr] \\ & {} + 32 \pi F_\pi^2 \varphi_S^{+-;K^0 {\bar{K}}^0} (0) \bigl[ k_{K^0 {\bar{K}}^0} (\mu) - 8 L_4^r (\mu) \bigr] \\ & {} + 48 \pi F_\pi^2 \biggl( 1 - \frac{2}{\sqrt{3}} \epsilon_1 \biggr) \varphi_S^{+-;\eta\eta} (0) \\ &{}\times \bigl[ k_{\eta\eta} (\mu) - 8 L_4^r (\mu) \bigr] \\ & {} + 32 \pi\sqrt{3} \biggl( 1 - \frac{\epsilon_1}{\sqrt{3}} + \sqrt {3} \epsilon_2 \biggr) \varphi_S^{+-;0\eta} (0) \\ & {}\times \bigl[ k_{\eta\pi^0} (\mu) - 8 L_4^r (\mu) \bigr] \biggr\} . \end{aligned}$$

(B.15)

Proceeding in the same way with $\pi_{0,G}^{+-}$ and $\pi_{0,F}^{00}$, we obtain

$$\begin{aligned} \pi_{0,G}^{+-} =& - 2 M_K^2 L_3 - \biggl[ \frac{4}{3}e^2 F_0^2 \bigl( {\widehat{K}}_1^r ( \mu) + {\widehat{K}}_2^r (\mu) \bigr) \\ &{} - 4 e^2 F_0^2 \bigl( 2 {\widehat{K}}_3^r (\mu) + {\widehat{K}}_4^r (\mu) \bigr) \\ &{} + \frac{4}{9}e^2 F_0^2 \bigl( 4 {\widehat{K}}_5^r (\mu) - 5 {\widehat{K}}_6^r (\mu) \bigr) \\ &{} - 2e^2 F_0^2 {\widehat{K}}_{12}^r (\mu) + 40\varDelta_\pi L_4^r (\mu) \biggr] \\ &{} + 6 M_{\pi}^2 \biggl( \frac{1}{256 \pi^2} \ln \frac{M_{\pi}^2}{\mu ^2} - L_4^r (\mu) \biggr) \\ &{} + ( 5 + 2 \sqrt{3} \epsilon_2 ) M_{\pi^0}^2 \biggl( \frac{1}{256 \pi^2} \ln\frac{M_{\pi^0}^2}{\mu^2} - L_4^r (\mu) \biggr) \\ & {}+ 6 M_{K}^2 \biggl( \frac{1}{256 \pi^2} \ln \frac{M_{K}^2}{\mu^2} - L_4^r (\mu) \biggr) \\ &{} + 4 M_{K^0}^2 \biggl( \frac{1}{256 \pi^2} \ln \frac{M_{K^0}^2}{\mu ^2} - L_4^r (\mu) \biggr) \\ &{} + 3 \biggl( 1 - \frac{2}{\sqrt{3}} \epsilon_1 \biggr) M_{\eta}^2 \biggl( \frac{1}{256 \pi^2} \ln \frac{M_{\eta}^2}{\mu^2} - L_4^r (\mu) \biggr) \\ & - \frac{F_0^2}{F_\pi^2} \biggl[ \frac{1}{2} \bigl( 5 \beta^{+-;K^+ K^-} + 3 \gamma^{+-;K^+ K^-} \bigr) M_K^2 \\ &{} + \frac{3}{2} \bigl( \beta^{+-;K^+ K^-} - \gamma^{+-;K^+ K^-} \bigr) M_\pi^2 \\ &{} + 3 \cdot16 \pi F_\pi^2 \varphi^{+-;K^+ K^-}_S (0) \biggr] \bigl[ k_{K \pi} (\mu) - 8 L_4^r (\mu) \bigr] \\ &{} + \frac{F_0^2}{F_\pi^2} \frac{1}{\sqrt{2}} \biggl\{ \frac{1}{3} \biggl[ 2 \biggl( 1 + \frac{\sqrt{3}}{2} \epsilon_2 \biggr) \beta ^{0 -;K^0 K^-} \\ &{} - 3 ( 7 + 2 \sqrt{3} \epsilon_2 ) \gamma^{0 -;K^0 K^-} \biggr] \bigl(M_K^2 + 2 M_\pi^2\bigr) \\ &{} - 2 \bigl( \beta^{0 -;K^0 K^-} + 3 \gamma^{0 -;K^0 K^-} \sqrt{3} \epsilon_2 \bigr) \bigl(M_{K^0}^2 - M_{\pi^0}^2\bigr) \\ &{} - \frac{1}{6} ( 1 - \sqrt{3} \epsilon_2 ) \bigl( \beta^{0 -;K^0 K^-} + 3 \gamma ^{0 -;K^0 K^-} \bigr) \\ &{}\times \bigl(M_{K}^2 - M_{\pi}^2\bigr) \\ &{} - \frac{3}{2} \biggl( 1 + \frac{\epsilon_2}{\sqrt{3}} \biggr) \bigl( \beta^{0 -;K^0 K^-} - 3 \gamma ^{0 -;K^0 K^-} \bigr) \\ &{} \times\bigl( M_{K}^2 + M_{K^0}^2 + M_{\pi}^2 + M_{\pi^0}^2 \bigr) \\ &{} - 3 \biggl( 1 + \frac{\epsilon_2}{\sqrt{3}} \biggr) 16 \pi F_\pi^2 \varphi_S^{0 -;K^0 K^-} (0) \biggr\} \\ &{} \times \bigl[ k_{K^0 \pi^0} (\mu) - 8 L_4^r (\mu) \bigr] \\ &{} + \frac{F_0^2}{F_\pi^2} \sqrt{\frac{3}{2}} \biggl\{ \frac{1}{2} \biggl[ ( 1 - \sqrt{3} \epsilon_1 ) \beta^{+ K^-;\eta{\bar{K}}^0} \\ &{} - \biggl( 1 + \frac{\epsilon_1}{\sqrt{3}} \biggr) \gamma^{+ K^-;\eta{\bar{K}}^0} \biggr] \bigl(M_K^2 + 2 M_\pi^2\bigr) \\ &{} - \biggl[ 3 ( 1 - \sqrt{3} \epsilon_1 ) \gamma^{+ K^-;\eta{\bar{K}}^0} \\ &{} + \biggl( 1 + \frac{\epsilon_1}{\sqrt{3}} \biggr) \beta^{+ K^-;\eta{\bar{K}}^0} \biggr] \bigl(M_\eta^2 - M_{K^0}^2 \bigr) \\ &{} - \biggl( 1 + \frac{\epsilon_1}{\sqrt{3}} \biggr) \gamma^{+ K^-;\eta{\bar{K}}^0} \bigl( M_K^2 - M_\pi^2 \bigr) \\ &{} + ( 1 - \sqrt{3} \epsilon_1 ) 16 \pi F_\pi^2 \varphi_S^{+ K^-;\eta{\bar{K}}^0} (0) \biggr\} \\ &{}\times \bigl[ k_{\eta K^0} (\mu) - 8 L_4^r (\mu) \bigr] , \end{aligned}$$

(B.16)

and

$$\begin{aligned} \pi_{0,F}^{00} =& - 64 M_{\pi^0}^2 \biggl[ L_1^r (\mu) - \frac{3}{4} L_4^r (\mu) \biggr] \\ &{} + 8 M_K^2 \biggl[ L_2^r ( \mu) - \frac{3}{2} L_4^r (\mu) \biggr] \\ &{} + 2 \biggl[ \biggl( 1 - \frac{2}{\sqrt{3}} \epsilon_2 \biggr) M_K^2 - 8 \biggl( 1 + \frac{2}{\sqrt{3}} \epsilon_2 \biggr) M_{\pi^0}^2 \biggr] L_3 \\ & {}- \biggl[ \frac{4}{3}e^2 F_0^2 \bigl( {\widehat{K}}_1^r (\mu) + {\widehat{K}}_2^r (\mu) \bigr) \\ & {} + \frac{4}{9} e^2 F_0^2 \bigl( {\widehat{K}}_5^r (\mu) + {\widehat{K}}_6^r (\mu) \bigr) - 2e^2 F_0^2 {\widehat{K}}_{12}^r (\mu) \\ & {} - 8\varDelta_\pi L_4^r (\mu) \biggr] \\ &{} + 2 ( 11 + 26 \sqrt{3} \epsilon_2 ) M_{\pi}^2 \biggl( \frac{1}{256 \pi^2} \ln\frac{M_{\pi}^2}{\mu^2} - L_4^r (\mu) \biggr) \\ & {}- ( 1 + 4 \sqrt{3} \epsilon_2 ) M_{\pi^0}^2 \biggl( \frac {1}{256 \pi^2} \ln\frac{M_{\pi^0}^2}{\mu^2} - L_4^r (\mu) \biggr) \\ & {}- 4 ( 1 - 2 \sqrt{3} \epsilon_2 ) M_{K}^2 \biggl( \frac{1}{256 \pi ^2} \ln\frac{M_{K}^2}{\mu^2} - L_4^r (\mu) \biggr) \\ &{} + 2 ( 5 - 2 \sqrt{3} \epsilon_2 ) M_{K^0}^2 \biggl( \frac{1}{256 \pi ^2} \ln\frac{M_{K^0}^2}{\mu^2} - L_4^r (\mu) \biggr) \\ & {}- 3 \biggl( 1 - 2 \frac{ \epsilon_1}{\sqrt{3}} + 2 \sqrt{3} \epsilon_2 \biggr) M_{\eta}^2 \\ &{}\times \biggl( \frac{1}{256 \pi^2}\ln\frac{M_{\eta }^2}{\mu^2} - L_4^r (\mu) \biggr) \\ & {}- \frac{F_0^2}{F_\pi^2} (1 + 2 \sqrt{3} \epsilon_2 ) \biggl[ \frac{1}{6} \beta^{0 0;K^+ K^-} \bigl(M_K^2 - M_{\pi ^0}^2 \bigr) \\ &{} + 16 \pi F_\pi^2 \varphi_S^{0 K^-;0 K^-} (0) \biggr] \bigl[ k_{K\pi^0} (\mu) - 8 L_4^r ( \mu) \bigr] \\ & {}+ {\sqrt{2}} \frac{F_0^2}{F_\pi^2} \biggl\{ \biggl[ \beta^{0 -;K^0 K^-} \,{+}\, 6 \biggl(1 \,{-}\, \frac{\sqrt {3}}{2} \epsilon_2 \biggr) \gamma^{0 -;K^0 K^-} \biggr] \\ &{}\times \bigl(M_K^2 + 2 M_{\pi^0}^2 \bigr) \\ &{} - 2 \bigl[ \beta^{0 -;K^0 K^-} + 3 \sqrt{3} \epsilon_2 \gamma^{0 -;K^0 K^-} \bigr] \bigl(M_{K^0}^2 - M_\pi^2\bigr) \\ &{} - \frac{1}{6} (1 + \sqrt{3} \epsilon_2 ) \bigl[ \beta^{0 -;K^0 K^-} - 3 \gamma^{0-;K^0 K^-} \bigr] \\ &{}\times \bigl(M_K^2 - M_{\pi^0}^2 \bigr) \\ &{} - \frac{3}{2} \biggl( 1 - \frac{\epsilon_2}{\sqrt{3}} \biggr) \bigl[ \beta^{0 -;K^0 K^-} + 3 \gamma^{0-;K^0 K^-} \bigr] \\ & {}\times \bigl(M_K^2 +M_{K^0}^2 + M_\pi^2 + M_{\pi^0}^2 \bigr) \\ &{} - 3 \biggl( 1 - \frac{\epsilon_2}{\sqrt{3}} \biggr) 16 \pi F_\pi^2 \varphi_S^{0 -;K^0 K^-} (0) \biggr\} \\ &{}\times \bigl[ k_{K^0 \pi} (\mu) - 8 L_4^r (\mu) \bigr] \\ & {}+ \sqrt{3} \biggl( 1 - \frac{\epsilon_1}{\sqrt{3}} + \sqrt{3} \epsilon_2 \biggr) \frac{F_0^2}{F_\pi^2} \\ &{} \times \biggl[ \frac{1}{2} \bigl( \beta^{0 K^-;\eta K^- } + 3 \gamma^{0 K^-;\eta K^-} \bigr) \\ &{} \times\bigl(2 M_\eta^2 - 3 M_K^2 - 2 M_{\pi^0}^2 \bigr) \\ &{} + \gamma^{0 K^-;\eta K^-} \bigl(M_K^2 - M_{\pi^0}^2 \bigr) \\ &{} - 16 \pi\varphi_S^{0 K^-;\eta K^-}(0) \biggr] \bigl[ k_{\eta K} (\mu) - 8 L_4^r (\mu) \bigr] \\ & {}+ \frac{F_0^2}{F_\pi^2} \biggl\{ - (1 + 2 \sqrt{3} \epsilon_2 ) 16 \pi F_\pi^2 \varphi_S^{00;00} (0) \\ &{}\times \bigl[ k_{\pi^0 \pi^0} (\mu) - 8 L_4^r (\mu) \bigr] \\ & {} + 32 \pi F_\pi^2 \varphi_S^{00;+-} (0) \bigl[ k_{\pi\pi} (\mu) - 8 L_4^r (\mu) \bigr] \\ &{} + 64 \pi F_\pi^2 \varphi_S^{00;K^+ K^-} (0) \bigl[ k_{KK} (\mu) - 8 L_4^r (\mu) \bigr] \\ &{} - 32 \pi F_\pi^2 \varphi_S^{00;K^0 {\bar{K}}^0} (0) \bigl[ k_{K^0 {\bar{K}}^0} (\mu) - 8 L_4^r (\mu) \bigr] \\ &{} - 3 \biggl( 1 - \frac{2}{\sqrt{3}} \epsilon_1 \biggr) 16 \pi F_\pi ^2 \varphi_S^{00;\eta\eta} (0) \\ &{}\times \bigl[ k_{\eta\eta} (\mu) - 8 L_4^r (\mu) \bigr] \\ & {}- 2\sqrt{3} \biggl( 1 - \frac{\epsilon_1}{\sqrt{3}} + \sqrt{3} \epsilon_2 \biggr) 16 \pi F_\pi^2 \varphi_S^{00;0\eta} (0) \\ &{}\times \bigl[ k_{\pi^0 \eta} (\mu) - 8 L_4^r (\mu) \bigr] \biggr\} . \end{aligned}$$

(B.17)

Table 12 The quantities $\varphi_{S}^{a b ; a^{\prime}b^{\prime}} (0)$ corresponding to various lowest-order amplitudes appearing in the one-loop expressions of the form factors F ^ab(s,t,u) and G ^ab(s,t,u) discussed in Sect. 4

Appendix C: Indefinite integrals of loop functions

In this appendix, we give integrals of the loop function ${\bar{J}}_{ab}(t)$ that are useful for the computation of the one-loop partial-wave projections in Sect. 5. We need the following (indefinite) integrals:

$$\begin{aligned} &{\int dt \bigl\{ t^n {\bar{J}}_{ab} (t) ; {\bar{J}}_{ab} (t) ; K_{ab} (t) ; M_{ab} (t) ; L_{ab} (t) - t M_{ab} (t) \bigr\}} \\ &{ \quad \equiv \bigl\{ \mathcal{J}_{ab}^{(n)} (t) ; \mathcal{J}_{ab} (t) ; \mathcal{K}_{ab} (t) ; \mathcal{M}_{ab} (t) ; \mathcal{L}_{ab} (t) \bigr\} ,} \end{aligned}$$

(C.1)

with n=1,2. From their definitions, it follows that the combination

$$\begin{aligned} &{ 6 \mathcal{L}_{ab} (t) - \varSigma_{ab} \mathcal{J}_{ab} (t) + \varDelta_{ab} \mathcal{K}_{ab} (t) + \frac{1}{2} \mathcal{J}_{ab}^{(1)} (t)} \\ &{ \quad - 2 \varDelta_{ab}^2 {\bar{J}}_{ab}^\prime(0) t + \frac{t^2}{96 \pi^2}} \end{aligned}$$

(C.2)

of these functions is a constant. In the kinematical range of interest for us, t<(M _a +M _b )², so that the loop function ${\bar{J}}_{ab}(t)$ is real and can be expressed as

$$\begin{aligned} {\bar{J}}_{ab} (t) =& \frac{1}{16\pi^2} \biggl\{1 - \frac{\varDelta_{ab}}{t} \ln\frac{M_a}{M_b} + \frac{\varSigma_{ab}}{\varDelta_{ab}}\ln \frac{M_a}{M_b} \\ &{} + \frac{M_a M_b}{\varSigma_{ab} - M_a M_b [ \chi_{ab} (t) + \chi ^{-1}_{ab} (t) ]} \\ &{}\times \biggl[ \chi_{ab} (t) - \frac{1}{\chi_{ab} (t)} \biggr] \ln \chi _{ab} (t) \biggr\} , \end{aligned}$$

(C.3)

with

$$\begin{aligned} \chi_{ab} (t) = \begin{cases} \displaystyle{\frac{\sqrt{(M_a + M_b)^2 - t} - \sqrt{(M_a - M_b)^2 - t}}{\sqrt{(M_a + M_b)^2 - t} + \sqrt{(M_a - M_b)^2 - t}}}, \\ \quad [t < (M_a - M_b)^2], \\ \displaystyle{\frac{\sqrt{(M_a + M_b)^2 - t} - i \sqrt{t - (M_a - M_b)^2 }}{\sqrt{(M_a + M_b)^2 - t} + i \sqrt{t - (M_a - M_b)^2}},} & \\ \quad[(M_a - M_b)^2 < t < (M_a + M_b)^2], \end{cases} \end{aligned}$$

(C.4)

so that 0≤χ _ab (t)≤1 when t<(M _a −M _b )² and |χ _ab (t)|=1 for (M _a −M _b )²<t<(M _a +M _b )². Notice also the identity

$$ t=M_a^2 + M_b^2 - M_a M_b \biggl[ \chi_{ab} (t) + \frac{1}{\chi _{ab} (t)} \biggr] . $$

(C.5)

The possibility to perform the required integrals hinges on finding a function $\mathcal{F}_{ab} (x)$ that satisfies

$$\begin{aligned} \frac{d}{d x} \mathcal{F}_{ab} (x) =& \frac{1}{x} \ln \frac {M_a}{M_b} \cdot \frac{M_a M_b (x - {\frac{1}{x}} )}{\varSigma _{ab} - M_a M_b (x + {\frac{1}{x}} )} \\ &{} - \frac{\ln x}{x} \frac{\varDelta_{ab}}{\varSigma_{ab} - M_a M_b (x + {\frac{1}{x}} )} . \end{aligned}$$

(C.6)

Such a function can indeed be found and reads

$$\begin{aligned} \mathcal{F}_{ab} (x) =& H_{1,0} \biggl( \frac{M_a}{M_b} x \biggr) - H_{1,0} \biggl( \frac{M_b}{M_a} x \biggr) + \ln \frac{M_a}{M_b} \ln x , \end{aligned}$$

(C.7)

where

$$ H_{1,0} (x) = - {\rm Li}_2 (x) - \ln x \ln(1-x) $$

(C.8)

belongs to the family of functions known as harmonic polylogarithms [75]. The function H _1,0(x) is defined in the complex plane, with a cut along the negative real axis. Notice that, despite what the expression (C.8) might suggest, there is no problem along the positive real axis for x>1: the discontinuities of the functions ${\rm Li}_{2} (x)$ and lnxln(1−x) along this line compensate each other exactly. This can also be inferred from the equivalent expression

$$ H_{1,0} (x) = {\rm Li}_2 \biggl( \frac{1}{x} \biggr) - \ln x \ln (x-1) + \frac{1}{2} \ln^2 x - \frac{\pi^2}{3} , $$

(C.9)

which follows from the properties of the dilogarithm function.

With these definitions, we obtain (see also Appendix A of [37])

$$\begin{aligned} &{16 \pi^2 \mathcal{J}_{ab}^{(2)} (t)} \\ &{\quad = \biggl( \frac{4}{3} + \frac{\varSigma_{ab}}{\varDelta_{ab}} \ln\frac {M_a}{M_b} \biggr) \frac{t^3}{3} - \biggl( \varDelta_{ab} \ln\frac{M_a}{M_b} + \frac{\varSigma_{ab}}{6} \biggr) \frac{t^2}{2}} \\ &{\qquad{} - \bigl( M_a^4 + M_b^4 + 10 M_a^2 M_b^2 \bigr) \frac{t}{6}} \\ &{\qquad{} + \frac{M_a M_b}{6} \bigl[ 2 t^2 - \varSigma_{ab} t - \bigl( M_a^4 + M_b^4 + 10 M_a^2 M_b^2 \bigr) \bigr]} \\ &{\qquad{}\times \biggl( \chi_{ab} (t) - \frac{1}{\chi_{ab} (t)} \biggr) \ln \chi_{ab} (t)} \\ &{\qquad{} + M_a^2 M_b^2 \varSigma_{ab} \ln^2 \chi_{ab} (t) ,} \end{aligned}$$

(C.10)

$$\begin{aligned} &{ 16 \pi^2 \mathcal{J}_{ab}^{(1)} (t)} \\ &{\quad = \biggl( \frac{3}{2} + \frac{\varSigma_{ab}}{\varDelta_{ab}} \ln\frac {M_a}{M_b} \biggr) \frac{t^2}{2} - \biggl( \frac{\varSigma_{ab}}{2} + \varDelta_{ab} \ln\frac{M_a}{M_b} \biggr) t} \\ &{\qquad{}+ \frac{M_a M_b}{2} ( t - \varSigma_{ab} ) \biggl( \chi_{ab} (t) - \frac{1}{\chi_{ab} (t)} \biggr) \ln\chi_{ab} (t)} \\ &{\qquad{} + M_a^2 M_b^2 \ln^2 \chi_{ab} (t) ,} \end{aligned}$$

(C.11)

$$\begin{aligned} &{ 16 \pi^2 \mathcal{J}_{ab} (t)} \\ &{\quad = \biggl( 2 + \frac{\varSigma_{ab}}{\varDelta_{ab}} \ln\frac{M_a}{M_b} \biggr) t} \\ &{\qquad{} + M_a M_b \biggl( \chi_{ab} (t) - \frac{1}{\chi_{ab} (t)} \biggr) \ln \chi_{ab} (t)} \\ &{\qquad{} + \frac{\varSigma_{ab}}{2} \ln^2 \chi_{ab} (t)} \\ &{\qquad{} + \varDelta_{ab} \biggl[ H_{1,0} \biggl( \frac{M_a}{M_b} \chi_{ab} (t) \biggr) - H_{1,0} \biggl( \frac{M_b}{M_a} \chi_{ab} (t) \biggr)} \\ &{\qquad{} + \ln\frac{M_a}{M_b} \ln\chi_{ab} (t) \biggr] ,} \end{aligned}$$

(C.12)

$$\begin{aligned} &{ 16 \pi^2 \mathcal{K}_{ab} (t)} \\ &{\quad = \frac{\varDelta_{ab}}{2} \biggl\{ \frac{\varDelta_{ab}}{t} \ln\frac{M_a}{M_b}} \\ &{ \qquad{} - M_a M_b \biggl( \chi_{ab} (t) - \frac{1}{\chi_{ab} (t)} \biggr) \frac{\ln\chi_{ab} (t)}{t}} \\ &{\qquad{} - \frac{1}{2} \ln^2 \chi_{ab} (t) \biggr\}} \\ &{\qquad{} - \frac{\varSigma_{ab}}{2} \biggl[ H_{1,0} \biggl( \frac{M_a}{M_b} \chi_{ab} (t) \biggr) - H_{1,0} \biggl( \frac{M_b}{M_a} \chi_{ab} (t) \biggr)} \\ &{\qquad{} + \ln\frac{M_a}{M_b} \ln\chi_{ab} (t) \biggr] ,} \end{aligned}$$

(C.13)

$$\begin{aligned} &{ 16 \pi^2 \mathcal{M}_{ab} (t)} \\ &{\quad = \biggl( \frac{2}{9} + \frac{1}{12} \frac{\varSigma_{ab}}{\varDelta_{ab}} \ln \frac{M_a}{M_b} \biggr) t} \\ &{\qquad{} + \frac{M_a M_b}{12} \biggl( 1 + 4 \frac{\varSigma_{ab}}{t} - 2 \frac {\varDelta_{ab}^2}{t^2} \biggr)} \\ &{\qquad{} \times \biggl( \chi_{ab} (t) - \frac{1}{\chi_{ab} (t)} \biggr) \ln \chi_{ab} (t)} \\ &{\qquad{} + \frac{\varSigma_{ab}}{8} \ln^2 \chi_{ab} (t) - \frac{\varDelta_{ab}^2}{6 t} \biggl( 1 + 3 \frac{\varSigma_{ab}}{\varDelta _{ab}} \ln\frac{M_a}{M_b} \biggr)} \\ &{\qquad{}+ \frac{\varDelta_{ab}^3}{6t^2} \ln\frac{M_a}{M_b}} \\ &{\qquad{} + \frac{\varDelta_{ab}}{4} \biggl[ H_{1,0} \biggl( \frac{M_a}{M_b} \chi_{ab} (t) \biggr) - H_{1,0} \biggl( \frac{M_b}{M_a} \chi_{ab} (t) \biggr)} \\ &{\qquad{} + \ln\frac{M_a}{M_b} \ln\chi_{ab} (t) \biggr] .} \end{aligned}$$

(C.14)

As far as the ranges of the integrations in the expressions (81) and (83) are concerned, we may note that $0 < (M_{K^{0}} - M_{\eta})^{2} < {t_{-}^{c}} (s,s_{\ell}) \le{t_{+}^{c}} (s,s_{\ell}) \le(M_{K} - M_{\pi})^{2} < (M_{K^{0}} - M_{\pi^{0}})^{2}$, whereas $0 < (M_{K} - M_{\eta})^{2} < {t_{-}^{n}} (s,s_{\ell}) \le{t_{+}^{n}} (s,s_{\ell}) \le(M_{K^{0}} - M_{\pi})^{2} < (M_{K} - M_{\pi^{0}})^{2}$.

Appendix D: Numerical representation

The previous appendices as well as the main part of this article provide all the elements needed to compute the isospin-breaking corrections to the difference $\delta_{0}^{0}-\delta _{1}^{1}$ as measured in K ⁺→π ⁺ π ⁻ ℓ ⁺ ν _ℓ . It is also useful to provide a numerical approximation of the (lengthy) expression for the inputs described in Sect. 6.2. Δ _IB(s,s _ℓ ) is approximated at the level of 0.03 mrad for $a_{0}^{0}$ between 0.18 and 0.30, $a_{0}^{2}$ between −0.06 and −0.03 and s _ℓ between 0 and $(M_{K}-\sqrt {s})^{2}$ by the following expression:

$$\begin{aligned} \begin{aligned}[b] \varDelta_{\mathrm{IB}}(s,s_\ell)&=\sum_{i,j,k,l} c_{ijkl} \biggl(\frac{a_0^0}{0.22} \biggr)^i \biggl( \frac {a_0^2}{-0.045} \biggr)^j \\ &\quad\times \biggl(\sqrt{\frac{s}{4 M_\pi^2}-1} \biggr)^k \biggl(\frac{s_\ell }{4 M_\pi^2} \biggr)^l, \end{aligned} \end{aligned}$$

(D.1)

with the values of the coefficients c _ijkl given in Table 13.

Table 13 Coefficients for the numerical approximation of the central value for the isospin-breaking correction Δ _IB

The uncertainty δΔ _IB(s,s _ℓ ) induced by the variation of the inputs is approximated at the level of 0.01 mrad for the same range of $a_{0}^{0}$, $a_{0}^{2}$ and s _ℓ by the following expression:

$$\begin{aligned} \delta\varDelta_{\mathrm{IB}}(s,s_\ell) =&\sum_{i,j,k,l} d_{ijkl} \biggl(\frac{a_00}{0.22} \biggr)^i \biggl(\frac {a_02}{-0.045} \biggr)^j \\ &{}\times \biggl(\sqrt{\frac{s}{4 M_\pi^2}-1} \biggr)^k \biggl(\frac{s_\ell }{4 M_\pi^2} \biggr)^l, \end{aligned}$$

(D.2)

with the values of the coefficients d _ijkl given in Table 14. We recall that s _ℓ is constrained to lie within 0 and $(M_{K}-\sqrt {s})2\leq(M_{K}-2M_{\pi})2$, so that 0≤s _ℓ /(4M _π 2)≤0.6. The size of the coefficients c _ijkl and d _ijkl indicate that the central value Δ _IB is insensitive to the value of s _ℓ in the range considered here, whereas δΔ _IB exhibits a very mild dependence.

Table 14 Coefficients for the numerical approximation of the uncertainty for the isospin-breaking correction δΔ _IB