Triton charge radius to next-to-next-to-leading order in pionless effective field theory

The triton point charge radius is calculated to next-to-next-to-leading order (NNLO) in pionless effective field theory $(\mathrm{EFT}\left(\pi /\right))$, yielding a prediction of $1.14\pm 0.19$ fm (leading order), $1.59\pm 0.08$ fm (next-to leading order), and $1.62\pm 0.03$ fm (NNLO) in agreement with the current experimental extraction of $1.5978\pm 0.040$ fm [Angeli and Marinova, At. Data Nucl. Data Tables 99, 69 (2013)]. The error at NNLO is due to cutoff variation ($\sim 1\%$) within a reasonable range of calculated cutoffs and from a $\mathrm{EFT}\left(\pi /\right)$ error estimate ($\sim 1.5\%$). In addition new techniques are introduced to add perturbative corrections to bound- and scattering state calculations for short-range effective field theories, but with a focus on their use in $\mathrm{EFT}\left(\pi /\right)$.

Figure 1

The top equation shows the LO dressed spin-triplet dibaryon propagator, which can be solved analytically via a geometric series. The solid bar is the bare dibaryon propagator $i/{\mathrm{\Delta }}_t$, the single lines with arrows are nucleon propagators, the cross represents a NLO effective range insertion from $c_{0t}^{\left(0\right)}$, and the star a NNLO correction from $c_{0t}^{\left(1\right)}$.

Figure 2

The coupled-channel integral equations for the LO doublet channel $nd$ scattering amplitude. Single lines represent nucleons and double lines (dashed double lines) spin-triplet (spin-singlet) dibaryons.

Figure 3

The coupled-channel integral equations for the NLO correction to the doublet channel $nd$ scattering amplitude. The cross refers to a single effective range insertion from $c_{0t}^{\left(0\right)}$ or $c_{0s}^{\left(0\right)}$ and the number “1” to the NLO correction to the $nd$ scattering amplitude.

Figure 4

The coupled-channel integral equations for the NNLO correction to the doublet channel $nd$ scattering amplitude. The star refers to an insertion of $c_{0t}^{\left(1\right)}$ or $c_{0s}^{\left(1\right)}$ and the number “2” refers to the NNLO correction to the doublet channel $nd$ scattering amplitude.

Figure 5

The coupled-channel integral equations for the LO triton vertex function, where the triple line is the triton, and the filled circle is the LO triton vertex function.

Figure 6

The coupled-channel integral equations for the NLO correction to the triton vertex function.

Figure 7

The coupled-channel integral equations for the NNLO correction to the triton vertex function.

Figure 8

Diagrammatic representation of the function ${\mathrm{\Sigma }}_0^P\left(E\right)$.

Figure 9

LO dressed triton propagator. The triangle is the dressed triton propagator, and the triple line is the bare triton propagator $i/\mathrm{\Omega }$.

Figure 10

Diagrammatic representation of the function ${\mathrm{\Sigma }}_1^P\left(E\right)$.

Figure 11

Diagrammatic representation of the function ${\mathrm{\Sigma }}_2^P\left(E\right)$.

Figure 12

NLO and NNLO corrections to the triton propagator. The diagram with $h_2$ comes from the kinetic term of the triton auxiliary field.

Figure 13

Diagrams for the LO doublet $S$-wave $nd$ scattering amplitude.

Figure 14

Diagrams for the NLO correction to the doublet $S$-wave $nd$ scattering amplitude. The factor of 2 takes into account the diagram related by time reversal symmetry that is not shown.

Figure 15

Diagrams for the NNLO correction to the doublet $S$-wave $nd$ scattering amplitude. The factors of 2 take into account diagrams related by time reversal symmetry that are not shown.

Figure 16

Diagrams for the LO triton charge form factor. The wavy blue lines represent minimally coupled ${\widehat{A}}_0$ photons.

Figure 17

Diagrams for the NLO correction to the triton charge form factor, where diagrams related by time reversal symmetry are not shown. The diagram in the dashed box is subtracted from the other diagrams to avoid double counting. The photon in diagram (d) is minimally coupled to the dibaryon.

Figure 18

Diagrams for the NNLO correction to the triton charge form factor, where diagrams related by time reversal symmetry are not shown. The diagrams in the dashed boxes are subtracted from the other diagrams to avoid double counting.

Figure 19

Cutoff dependence of the LO, NLO, and NNLO predictions for the triton point charge radius. The pink band is a 15% error estimate for the LO triton point charge radius, the green band is a 5% error estimate for the NLO triton point charge radius, and the blue band is a 1.5% error estimate for the NNLO triton point charge radius. The dotted line is the value extracted from experiment, $1.5978\pm 0.040$ fm [35], and the black lines its error.