How to Classify Three-Body Forces -- and Why

To add 3-body forces when theory and data disagree is untenable when predictions are required. For the ``pion-less'' Effective Field Theory at momenta below the pion-mass, I provide a recipe to systematically estimate the typical size of 3-body forces in all partial waves and orders, including external currents~[1]. It is based on the superficial degree of divergence of the 3-body diagrams which contain only two-body forces and the renormalisation-group argument that low-energy observables must be insensitive to details of short-distance dynamics. Na\"ive dimensional analysis must be amended as the asymptotic solution to the leading-order problem depends for large off-shell momenta crucially on the partial wave and spin-combination considered. The typical strength of most 3-body forces turns out weaker than expected, demoting many to high orders. As application, the thermal cross section of $nd\to t\gamma$ bears no new 3-body force~[2], besides those fixed by the triton binding energy and $nd$ scattering length in the triton channel: $0.485(\mathrm{LO})+0.011(\mathrm{NLO})+0.007(\mathrm{NNLO}) ];\mathrm{mb}=[0.503\pm0.003]\;\mathrm{mb}$, converges and compares well with data, $[0.509\pm0.015]\;\mathrm{mb}$. Potential models list $[0.49\dots0.66]\;\mathrm{mb}$, depending on the 2-nucleon potential and inclusion of the $\Delta(1232)$. [1] H.W.~Grie{\ss}hammer: Nucl.~Phys.~\textbf{A760} (2005) 110 [2] H.~Sadeghi, S.~Bayegan and H.W.~Grie{\ss}hammer: Phys.~Lett.\textbf{B643} (2006), 263.