Cold Neutron-Deuteron Capture and Wigner-SU(4) Symmetry

We calculate the cold neutron-deuteron ($nd$) capture cross section, $\sigma_{nd}$, to next-to-next-to leading order (NNLO) using the model-independent approach of pionless effective field theory (EFT($\pi\!\!\!/$)). At leading order we find $\sigma_{nd} = 0.315 \pm 0.217$ mb, while the experimental result is 0.508(15) mb [Jurney, Bendt and Browne in Phys. Rev. C 25, 2810 (1982)] for a laboratory neutron velocity of 2200 m/s. At next-to-leading-order (NLO), we show that $\sigma_{nd}$ is sensitive to the low energy constant (LEC), $L_1^{(0)}$, of the two-nucleon isovector current appearing at NLO. A fit of $L_1^{(0)}$ at NLO to the triton magnetic moment yields a NLO prediction of $\sigma_{nd}=0.393 \pm 0.164$ mb, where the error comes from propagating the error from the $L_1^{(0)}$ fit. At next-to-next-to-leading-order (NNLO), we find that a new three-nucleon magnetic moment counterterm is required for renormalization group invariance of both $\sigma_{nd}$ and the triton magnetic moment. Fitting the NNLO correction to $L_1^{(0)}$ (denoted $L_1^{(1)}$) to cold neutron-proton capture ($\sigma_{np}$) yields a NNLO prediction of $\sigma_{nd}=0.447 \pm 0.130$ mb, where the error comes from propagating the error from the $L_1^{(1)}$ fit. We also study different fittings of $L_1^{(0)}$ and $L_1^{(1)}$ to $\sigma_{np}$, $\sigma_{nd}$, and/or the triton magnetic moment. For example, fitting $L_1^{(0)}$ simultaneously to $\sigma_{np}$, $\sigma_{nd}$, and the triton magnetic moment at NLO, and fitting $L_1^{(1)}$ simultaneously to $\sigma_{np}$ and $\sigma_{nd}$ at NNLO, yields $\sigma_{nd} = 0.480 \pm 0.114$ mb and $0.511 \pm 0.042$ mb, respectively, where errors are naively estimated from EFT($\pi\!\!\!/$) power counting. In addition, we discuss how Wigner-SU(4) symmetry may alter the naive EFT($\pi\!\!\!/$) expansion of $\sigma_{nd}$.