to quote instead values for the $\mu$-independent $\bar l_i$ which are defined as $$\label{eq:deflib}
\bar l_i = \frac{32\pi^2}{\gamma_i}l_i^r(\mu)-\ln\frac{M_\pi^2}{\mu^2}\,.$$ The definition of the renormalized couplings at $O(p^6)$ is $$\hat{C_i} =\frac{(c\mu)^{2(d-4)}}{\hat F^2}
\left(\hat C_i^r(\mu)-\hat \Gamma_i^{(2)}\Lambda^2-\left(\hat \Gamma_i^{(1)}
  +\hat \Gamma_i^{(L)}(\mu)\right)\Lambda\right)\,.$$ The values for all quantities needed for the [$n_f=2$]{} and [$n_f=3$]{} cases can be found in [@Bijnens:1999hw]. We will below quote the $c_i^r$ and $C_i^r$ at a scale $\mu=0.77~$GeV and use the physical pion decay constant $F_\pi=0.0922~$GeV to make the $\hat C_i^r$ dimensionless.

In addition to the operators listed explicitly in the Lagrangians (\[eq:LOlag\],\[eq:NLOlag2\],\[eq:NLOlag3\],\[eq:NNLOlag\]) there are also so-called contact terms. The corresponding coefficients cannot be directly measured in physical quantities involving mesons and are therefore not relevant for phenomenology. Nevertheless, they have in principle well-defined values from Green functions of currents, but depend on the precise definitions of these currents.

Other Lagrangians {#sec:other}
-----------------

In the treatment of radiative corrections for strong and semileptonic processes at low energies, photons and leptons enter as dynamical degrees of freedom. Consequently, additional effective Lagrangians are needed.

Leaving out the kinetic terms for photons and leptons, a single new term arises to lowest order, $O(e^2 p^0)$ [@Ecker:1988te]: $$\label{eq:e2p0}
\mathcal{L}_{e^2p^0} = e^2 {F_0}^4 Z \langle {{\cal Q}}_L^{\rm em} {{\cal 