Precision theoretical analysis of neutron radiative beta decay to order $O({\alpha }^2/{\pi }^2)$

In the Standard Model (SM) we calculate the decay rate of the neutron radiative ${\beta }^{-}$ decay to order $O({\alpha }^2/{\pi }^2\sim {10}^{-5})$, where $\alpha$ is the fine-structure constant, and radiative corrections to order $O(\alpha /\pi \sim {10}^{-3})$. The obtained results together with the recent analysis of the neutron radiative ${\beta }^{-}$ decay to next-to-leading order in the large proton-mass expansion, performed by Ivanov et al. [Phys. Rev. D 95, 033007 (2017)], describe recent experimental data by the RDK II Collaboration [Bales et al., Phys. Rev. Lett. 116, 242501 (2016)] within 1.5 standard deviations. We argue a substantial influence of strong low-energy interactions of hadrons coupled to photons on the properties of the amplitude of the neutron radiative ${\beta }^{-}$ decay under gauge transformations of real and virtual photons.

Figure 1

The Feynman diagrams, defining the contribution to the amplitude of the neutron radiative ${\beta }^{-}$ decay in the tree approximation to order $e$.

Figure 2

The Feynman diagrams, defining the contribution to the amplitude of the neutron radiative ${\beta }^{-}$ decay of order $e^3$, caused by pure QCD.

Figure 3

The Feynman diagrams, defining the main contribution to the amplitude of the neutron radiative ${\beta }^{-}$ decay of order $e^3$.

Figure 4

The Feynman diagrams, responsible for restoration of the gauge invariance of the Feynman diagrams Figs. 3 and 3.

Figure 5

The Feynman diagrams, defining the main contribution of order $O({\alpha }^2/{\pi }^2)$ to the rate of the neutron radiative ${\beta }^{-}$ decay $n\to p+e^{-}+{\overline{\nu }}_e+\gamma +\gamma$ with one detected and one undetected photon.

Figure 6

The Feynman diagrams, defining the main contribution of the radiative corrections of order $O(\alpha /\pi )$, caused by one-virtual-photon exchanges, to the neutron ${\beta }^{-}$ decay (see Sirlin [5]).