Spectral functions of isoscalar scalar and isovector electromagnetic form factors of the nucleon at two-loop order

We calculate the imaginary parts of the isoscalar scalar and isovector electromagnetic form factors of the nucleon up to two-loop order in chiral perturbation theory. Particular attention is paid on the correct behavior of Im σN (t) and Im GE,M (t) at the two-pion threshold t0 = 4mπ in connection with the non-relativistic 1/M -expansion. We recover the well-known strong enhancement near threshold originating from the nearby anomalous singularity at tc = 4mπ − mπ/M = 3.98mπ. In the case of the scalar spectral function ImσN (t) one finds a significant improvement in comparison to the lowest order one-loop result. Higher order ππ-rescattering effects are however still necessary to close a remaining 20%-gap to the empirical scalar spectral function. The isovector electric and magnetic spectral functions ImGE,M (t) get additionally enhanced near threshold by the two-pion-loop contributions. After supplementing their two-loop results by a phenomenological ρ-meson exchange term one can reproduce the empirical isovector electric and magnetic spectral functions fairly well. PACS: 12.38.Bx, 12.39.Fe, 13.40.Gp. The structure of the nucleon as probed in elastic electron-nucleon scattering is encoded in four form factors G E,M(t) (with t the squared momentum transfer). The understanding of these form factors is of importance for any theory or model of the strong interactions. Abundant data on the electromagnetic form factors of the nucleon exist over a large range of momentum transfers, 0 < √−t < 6 GeV. Dispersion theory is a tool to interpret these data in a largely model-independent fashion. Each electromagnetic form factor G E,M(t) can be written in terms of an unsubtracted dispersion relation and their imaginary parts are often modeled by a few vector meson poles. This procedure is based on the successful vector meson dominance hypothesis which states that the (virtual) photon couples to hadrons only via intermediate vector mesons. However, as pointed out already in 1959 by Fulco and Frazer [1] such an approach is not in conformity with general constraints from unitarity and analyticity. In particular, the singularity structure of the triangle diagram (left graph in Fig. 1) is not respected at all when using only vector meson poles. As a consequence of a nearby logarithmic singularity at tc = 4m 2 π −mπ/M (on the second Riemann-sheet) the low-mass two-pion continuum has a very pronounced effect on the isovector electric and magnetic spectral functions ImGE,M(t) on the left wing of the ρ(770)-resonance. This becomes particularly visible in the isovector mean square radii of the nucleon. The effect was quantified by Höhler and Pietarinen [2] in their work on the nucleon electromagnetic form factors based on ππ → N̄N partial wave amplitudes. The dispersion relation analysis of the nucleon electromagnetic form factors was recently refined by the Bonn-Mainz group [3, 4] accounting for new electron scattering data and the high-energy constraints from perturbative QCD. The first one-loop calculation of the isovector electromagnetic form factors has been performed in ref.[5] using the fully relativistic version of baryon chiral perturbation theory. The