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

Abstract

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 ${\sigma }_{nd}$.

arXiv comment61 pages, 10 figures, version published in PRC. Some typos and errors in numerical results are fixed compared to the previous version

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