Isospin splitting in the pion-nucleon couplings from QCD sum rules

We use QCD sum rules for the three point function of a pseudoscalar and two nucleonic currents in order to estimate the charge dependence of the pion nucleon coupling constant $g_{\mathrm{NN\pi }}$ coming from isospin violation in the strong interaction. The effect can be attributed primarily to the difference of the quark condensates $\mathrm{〈ūu〉}$ and $\mathrm{〈d̄d〉.}$ Assuming that the ${\pi }^0$ is a pure isostate we obtain for the splitting ${\mathrm{(|g}}_{{\mathrm{pp\pi }}_0}{\mathrm{|-|g}}_{{\mathrm{nn\pi }}_0}{\mathrm{|)/g}}_{{\mathrm{NN}}_{\pi }}$ an interval of ${\text{0.8}}^{*}{10}^{\text{−2}}$ to ${\text{2.3}}^{*}{10}^{\text{−2}},$ the uncertainties coming mainly from the input parameters. In order to obtain the coupling to a physical ${\pi }^0$ we have to take $\mathrm{\pi -\eta }$ mixing into account leading to an interval of ${\text{1.2}}^{*}{10}^{\text{−2}}$ to ${\text{3.7}}^{*}{10}^{\text{−2}}.$ The charged pion nucleon coupling is found to be the average of ${\mathrm{|g}}_{{\mathrm{pp\pi }}_0}|$ and ${\mathrm{|g}}_{{\mathrm{nn\pi }}_0}\mathrm{|.}$ Electromagnetic effects are not included.