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 to the correct behavior of Im σ_N(t) and Im G^V_E,M(t) at the two-pion threshold t₀ = 4m²_π in connection with the nonrelativistic 1/M expansion. We recover the well-known strong enhancement near threshold originating from the nearby anomalous singularity at t_c = 4m²_π-m⁴ _π/M² = 3.98m²_π. In the case of the scalar spectral function Im σ_N(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 Im G^V _E,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.