Two-nucleon ${}^{1}S_0$ amplitude zero in chiral effective field theory

We present a new rearrangement of short-range interactions in the ${}^{1}S_0$ nucleon-nucleon channel within chiral effective field theory. This is intended to address the slow convergence of Weinberg's scheme, which we attribute to its failure to reproduce the amplitude zero (scattering momentum $\simeq 340$ MeV) at leading order. After the power counting scheme is modified to accommodate the zero at leading order, it includes subleading corrections perturbatively in a way that is consistent with renormalization-group invariance. Systematic improvement is shown at next-to-leading order, and we obtain results that fit empirical phase shifts remarkably well all the way up to the pion-production threshold. An approach in which pions have been integrated out is included, which allows us to derive analytic results that also fit phenomenology surprisingly well.

Figure 1

Full two-dibaryon propagator (solid box) resulting from the nonperturbative dressing of bare dibaryon-1 (dashed box) and dibaryon-2 (plain box) propagators with nucleon bubbles (circles).

Figure 2

$np{}^1{S}_0$ phase shift $\delta$ (in degrees) versus laboratory energy $T_{\mathrm{lab}}=2k^2/m_N$ (in MeV) for $\pi /\mathrm{EFT}$ at LO in our new PC. The (black) solid line shows the analytical result (28) with $\mathrm{\Lambda }\to \infty$, while the (green) band around it represents the evolution of the cutoff from 500 MeV to infinity, with ${\theta }_{-1}=\pm 1$. The (black) squares are the Nijm93 results [63, 70].

Figure 3

$np{}^1{S}_0$ phase shift $\delta$ (in degrees) versus laboratory energy $T_{\mathrm{lab}}=2k^2/m_N$ (in MeV) for $\pi /\mathrm{EFT}$ at NLO in our new PC. The (black) line shows the analytical result (40) with $\mathrm{\Lambda }\to \infty$ and the value of the shape parameter from Ref. [74], while the (green) band around it represents a $\pm 30\%$ variation in this value. The (black) squares are the Nijm93 results [63, 70].

Figure 4

$np{}^1{S}_0$ phase shift $\delta$ (in degrees) versus laboratory energy $T_{\mathrm{lab}}=2k^2/m_N$ (in MeV) for $\chi \mathrm{EFT}$ at LO in our new PC. The narrow (green) band represents the evolution of the sharp cutoff from 600 MeV to 2 GeV. The (black) squares are the Nijm93 results [63, 70].

Figure 5

$np{}^1{S}_0$ shape parameter $P_0^{\left[0\right]}\left(\mathrm{\Lambda }\right)$ (in ${\mathrm{fm}}^3$) versus inverse cutoff $1/\mathrm{\Lambda }$ (in ${\mathrm{GeV}}^{-1}$) for $\chi \mathrm{EFT}$ at LO in our new PC. The straight line represents a linear fit to the numerical values.

Figure 6

$np{}^1{S}_0$ phase shift $\delta$ (in degrees) versus laboratory energy $T_{\mathrm{lab}}=2k^2/m_N$ (in MeV) for $\chi \mathrm{EFT}$ at NLO in our new PC. The narrow (green) band represents the evolution of the cutoff from 600 MeV to 2 GeV. The (black) squares are the Nijm93 results [63, 70].

Figure 7

$np{}^1{S}_0$ phase shift $\delta$ (in degrees) versus laboratory energy $T_{\mathrm{lab}}=2k^2/m_N$ (in MeV) for $\chi \mathrm{EFT}$ at LO and NLO in our new PC from an alternative fitting protocol. The (green) light and (red) dark narrow bands represent, respectively, LO and LO+NLO under a cutoff variation from $600\mathrm{MeV}$ to $2\mathrm{GeV}$. The LO and LO+NLO phase shifts from Ref. [26] have also been displayed; the upper (violet) LO band and the lower (cyan) LO+NLO band come from the same cutoff evolution as before. The (black) squares are the Nijm93 results [63, 70].