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

Abstract

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 ${\sigma }_N(t)$ and Im $G_{E,M}^V(t)$ at the two-pion threshold $t_0=4m_{\pi }^2$ 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 $t_c=4m_{\pi }^2-m_{\pi }^4/M^2=3.98m_{\pi }^2$. In the case of the scalar spectral function Im ${\sigma }_N(t)$ one finds a significant improvement in comparison to the lowest order one-loop result. Higher order $\pi \pi$-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 $G_{E,M}^V(t)$ get additionally enhanced near threshold by the two-pion-loop contributions. After supplementing their two-loop results by a phenomenological $\rho$-meson exchange term one can reproduce the empirical isovector electric and magnetic spectral functions fairly well.

Comments: 10 pages, 6 figures, submitted to Physical Review C