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We evaluate by means of lattice QCD calculations the low-energy constant$\ell_{7}$ which parametrizes strong isospin effects at NLO in $\rm{SU}(2)$chiral perturbation theory. Among all low-energy constants at NLO, $\ell_{7}$is the one known less precisely, and its uncertainty is currently larger than$50\%$. Our strategy is based on the RM123 approach in which the latticepath-integral is expanded in powers of the isospin breaking parameter $\Deltam= (m_{d}-m_{u})/2$. In order to evaluate the relevant lattice correlators wemake use of the recently proposed rotated twisted-mass (RTM) scheme. Within theRM123 approach, it is possible to cleanly extract the value of $\ell_{7}$ fromeither the pion mass splitting $M_{\pi^{+}}-M_{\pi^{0}}$ induced by strongisospin breaking at order $\mathcal{O}\left((\Delta m)^{2}\right)$ (massmethod), or from the coupling of the neutral pion $\pi^{0}$ to the isoscalaroperator $\left(\bar{u}\gamma_{5}u + \bar{d}\gamma_{5} d\right)/\sqrt{2}$ atorder $\mathcal{O}(\Delta m)$ (matrix element method). In this pilot study welimit the analysis to a single ensemble generated by the Extended Twisted MassCollaboration (ETMC) with $N_{f}=2+1+1$ dynamical quark flavours, whichcorresponds to a lattice spacing $a\simeq 0.095~{\rm fm}$ and to a pion mass$M_{\pi}\simeq 260~{\rm MeV}$. We find that the matrix element methodoutperforms the mass method in terms of resulting statistical accuracy. Ourdetermination, $\ell_{7} = 2.5(1.4)\times 10^{-3}$, is in agreement andimproves previous calculations.

First direct lattice calculation of the chiral perturbation theory low-energy constant $\ell_7$