$nd$ scattering and the $A_y$ puzzle to next-to-next-to-next-to-leading order

Polarization observables in neutron-deuteron scattering are calculated to next-to-next-to-next-to-leading order $({\mathrm{N}}^3\mathrm{LO})$ in pionless effective field theory $\left({\mathrm{EFT}}_{\neg \pi }\right)$. At ${\mathrm{N}}^3\mathrm{LO}$ the two-body $P$-wave contact interactions are found to be important contributions to the neutron vector analyzing power, $A_y\left(\theta \right)$, and the deuteron vector analyzing power, $i{T}_{11}\left(\theta \right)$. Extracting the two-body $P$-wave ${\mathrm{EFT}}_{\neg \pi }$ coefficients from two-body scattering data and varying them within the expected ${\mathrm{EFT}}_{\neg \pi }$ theoretical errors provides results that are consistent (at the ${\mathrm{N}}^3\mathrm{LO}$ level) with $A_y$ experimental data at low energies. Cutoff dependence of the ${\mathrm{N}}^3\mathrm{LO}$ correction of the doublet $S$-wave $nd$ scattering amplitude suggests the need for a new three-body force at ${\mathrm{N}}^3\mathrm{LO}$, which is likely one that mixes Wigner-symmetric and Wigner-antisymmetric three-body channels.

Figure 1

Single lines represent nucleons, double lines spin-triplet dibaryons, and double dashed lines spin-singlet dibaryons. Thick solid lines denote a sum over both spin-triplet and spin-singlet dibaryons. The LO $nd$ scattering amplitude is the oval with a “0” inside, and the oval with the $n$ inside is the ${\mathrm{N}}^n\mathrm{LO}$ correction to the $nd$ scattering amplitude. The circle with the $n$ inside is the ${\mathrm{N}}^n\mathrm{LO}$ correction to the dibaryon propagators (see Fig. 2), and the rectangle with the $n$ inside is the $n\mathrm{th}$ order “three-body” correction (see Fig. 3).

Figure 2

Higher order corrections to dibaryon propagators (used in the diagrams of Fig. 1). The NLO ($n=1$) corrections are range corrections from $c_{0t}^{\left(0\right)}$ and $c_{0s}^{\left(0\right)}$. At ${\mathrm{N}}^2\mathrm{LO}$ ($n=2$) the dibaryons receive further range corrections $c_{0t}^{\left(1\right)}$ and $c_{0s}^{\left(1\right)}$ in the $Z$-parametrization, as well as the ${\mathrm{\Delta }}^{\left({\mathrm{N}}^2\mathrm{LO}\right)}$ correction from splitting between the $nn$ and $np$ spin-singlet scattering lengths. The ${\mathrm{N}}^3\mathrm{LO}$ ($n=3$) corrections are from higher order range corrections $c_{0t}^{\left(2\right)}$ and $c_{0s}^{\left(2\right)}$ in the $Z$-parametrization, and shape parameter corrections $c_{1t}$ and $c_{1s}$.

Figure 3

“three-body” contributions to integral equations (used in the diagrams of Fig. 1). The LO terms are nucleon exchange plus in the doublet $S$-wave channel the LO three-body force (dark square). The NLO term is a NLO correction to the LO three-body force. At ${\mathrm{N}}^2\mathrm{LO}$ there are contributions from the two-body $SD$-mixing term (coupling indicated by pale square), the ${\mathrm{N}}^2\mathrm{LO}$ correction to the LO three-body force, $H_{{\mathrm{N}}^2\mathrm{LO}}$, and a new energy-dependent three-body force, $H_2^{{\mathrm{N}}^2\mathrm{LO}}$. The ${\mathrm{N}}^3\mathrm{LO}$ contributions are from the two-body $P$-wave contact interactions (green circle), the ${\mathrm{N}}^3\mathrm{LO}$ correction to the LO three-body force, and the ${\mathrm{N}}^3\mathrm{LO}$ correction to the ${\mathrm{N}}^2\mathrm{LO}$ energy dependent three-body force.

Figure 4

Unboxed diagrams are the integral equations for the ${\mathrm{N}}^3\mathrm{LO}$ contribution to the $nd$ scattering amplitude from the two-body $P$-wave contact interactions. The double lines with a zigzag in the middle are the $P$-wave dibaryon propagator, given by $i/{\mathrm{\Delta }}^{{}^{2R+1}P_J}$. The boxed diagrams represent the equation for the “P” amplitude used in the unboxed integral equations above. The notation “$3_{\mathrm{P}}$” in the oval indicates that this is a ${\mathrm{N}}^3\mathrm{LO}$ correction but one that only involves the two-body $P$-wave contributions.

Figure 5

$nd$ scattering cross section for $E_n=3.0$ MeV with experimental data from Schwarz et al. [37]. The LO prediction (without theoretical errors) is the solid green line, the dashed blue line the NLO prediction (without theoretical errors), and the solid red band the ${\mathrm{N}}^2\mathrm{LO}$ prediction with a 6% error estimate.

Figure 6

The dashed lines are ${\mathrm{EFT}}_{\neg \pi }$ results for $A_y$ for several sets of $C^{{}^{3}P_J}$ coefficients varied by 15% around their central values. Top left: $E_n=1.2$ MeV, experimental data from Neidel et al. [39]. Top right: $E_n=1.9$ MeV, experimental data from Neidel et al. [39]. Bottom: $E_n=3.0$ MeV, the solid line is a PMC calculation from Kievsky et al. using AV-18+UR [38], with experimental data from McAninch et al. [40]. In the following, “$+$” stands for 15 percent above central values given in Eq. (9); “0” is at central value; and “$-$” is 15 percent below central value. The coefficient values ($C^{{}^{3}P_0},C^{{}^{3}P_1},C^{{}^{3}P_2}$) used to produce the curves shown are (from lowest ${\mathrm{EFT}}_{\neg \pi }$ curve to highest ${\mathrm{EFT}}_{\neg \pi }$ curve on the plots): big dots $\left(\text{green}\right)=(+,-,+)$; small dots $\left(\text{blue}\right)=(+,0,+)$; long dash $\left(\text{red}\right)=(0,0,0)$; long dash-dot $\left(\text{purple}\right)=(0,0,+)$; short dash-dot $\left(\text{orange}\right)=(-,0,0)$; double dot $\left(\text{black}\right)=(-,+,+)$.

Figure 7

The solid red line is the ${\mathrm{EFT}}_{\neg \pi }$ prediction (without theoretical error bars) for deuteron polarization observables in $nd$ scattering, the dashed green line PMC calculations using AV18+UR for $nd$ scattering [38], and the dotted blue line PMC calculations using AV18+UR for $pd$ scattering [38]. All experimental data is for $pd$ scattering from Shimizu et al. [41] at a laboratory deuteron energy of $E_d=6.0$ MeV.