An ${\alpha }^2(Z\alpha {)}^{5}m$ Contribution to the Hydrogen Lamb Shift from Virtual Light by Light Scattering

The radiative correction to the Lamb shift of order ${\alpha }^2(Z\alpha {)}^{5}m$ induced by the light by light scattering insertion in external photons is obtained. The new contribution turns out to be equal to $-0.122(2){\alpha }^2(Z\alpha {)}^5\mathrm{/}(\pi n^3)(m_r\mathrm{/}m{)}^{3}m$. Combining this contribution with our previous results we obtain the complete correction of order ${\alpha }^2(Z\alpha {)}^{5}m$ induced by all diagrams with closed electron loops. This correction is $37.3(1)$ kHz and $4.67(1)$ kHz for the $1S$- and $2S$-states in hydrogen, respectively.