we as well provide details of our global fit to the available quasi-elastic deuteron data. In Section IV we present our results, discuss the uncertainty limits, and address possible help from upcoming experiments. Section V is dedicated to the application of the dispersive, data-based, model-independent polarizability calculation described here to the $e$-$H$ and $e$-$D$ systems and its relevance to the isotope shift measurements. Section VI closes the article with a short summary of our findings.

The Basic Formalism
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The diagram that contains the nuclear and hadronic structure-dependent ${\cal O}(\alpha^5)$ correction to the Lamb shift is shown in Fig. \[tpediag\].

![(Color online) Two-photon exchange diagram for the ${\cal O}(\alpha^5)$ correction to the Lamb shift.[]{data-label="tpediag"}](TPEdiag.pdf){height="2cm"}

The lower part of the diagram, the blob containing the nuclear and hadronic structure dependence is encoded in the forward virtual Compton tensor, &&T\^=d\^4xe\^[iqx]{}p|Tj\^(x)j\^(0)|p\
&&=(-g\^+)T\_1(,Q\^2) + T\_2(,Q\^2),where $\hat p^\mu=p^{\mu}-\frac{p\cdot q}{q^2}q^\mu$, $Q^2=-q^2$, $\nu=(p\cdot q)/M_d$ and $M_d$ is the deuteron mass. Following [@Carlson:2011zd], we can write the contribution of the two-photon exchange diagram to the $n\ell$ energy level as &&E\_[n]{}=\^2\_[n]{}(0)d\^4Q\
&&,where a Wick rotation $q_0=iQ_0$ was made, and $\phi^2_{n\ell}(0)=\mu_r^3\alpha^3/(\pi n^3)\delta_{\ell0}$, $\mu_r=mM/(M+m)$ being the reduced mass. $T_{1,2}(\nu,Q^2)$ are even functions of $\nu$ and their imaginary parts are related to 