New measurements of the beam normal single spin asymmetry in the electron elastic and quasielastic scattering on the proton and deuteron, respectively, at large backward angles and at hQ 2 i ¼ 0.22 ðGeV=cÞ The exchange of two hard virtual photons in the elastic electron-nucleon scattering beyond the one-photon exchange Born approximation has been the subject of recent investigation [1,2]. Two complementary methods, a determination from a measurement of a differential cross section using unpolarized electrons (Rosenbluth separation) and the measurement of the polarization transfer to the proton final state, gave significantly different results forThe two-photon exchange has been addressed as the explanation for such a discrepancy. Other observables in which the two-photon exchange physics plays a role are: neutron form factors, resonance electroproduction, the pion form factor and the elastic electron-nucleus cross section, in particular deuteron and 3 He [1]. The exchange of two virtual photons implies the excitation of nucleon intermediate states offering the possibility of testing models of hadronic structure [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22]. In contrast to previous measurements at forward [23,24] and close to right angles [25], the A4 measurements presented in this work have been performed at large backward angles. The A4 measurements explore a new important parameter region of the virtualities of the exchanged virtual photons, involved in the theoretical calculation of the nucleon intermediate states, see Fig. 2 from Ref. [11]. Moreover, in contrast to the SAMPLE measurement [26], at similar backward scattering angle, the A4 measurements have improved the precision by a factor ∼4. The A4 measurements have been performed at larger beam energies, probing the πN and Δ resonance inelastic intermediate states.The two-photon exchange physics affects the parity violating asymmetry through the normalization to the electromagnetic amplitude [27] and it is of special relevance in precision measurements of the weak charge of the proton at low Q 2 like Q weak [28] and P2 [29]. It is also useful in the theoretical determination of the γZ box diagrams which also takes part in the proton weak charge radiative corrections [30][31][32][33].Two-photon exchange effects can be observed in radiative corrections to the cross section, the ratio in e − p and e þ p elastic scattering cross sections, the depolarization tensor, and the target and beam normal single spin asymmetries [34][35][36][37]. Six generalized form factors depending on two kinematic variables parametrize the elastic electron-proton scatteringG M ðs; Q 2 Þ,G E ðs; Q 2 Þ, [10]. There are also partonic calculations which resort to generalized parton distributions of the nucleon and are able to resolve the discrepancy at large Q 2 [12,16]. The T-odd observables target and beam normal spin asymmetries arise from the interference of the one-photon and two-photon exchange amplitudes, see Fig. 1 in Ref. [39]. The target normal spi...