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FIG. 9: ?H-*He binding energy plot for various lattice spac- ings. The black dot is the physical point and the blue band is the Tjon band according to [23]. The “He errors include statistical and LEC errors while the ?H errors only include LEC errors.

Figure 9 ?H-*He binding energy plot for various lattice spac- ings. The black dot is the physical point and the blue band is the Tjon band according to [23]. The “He errors include statistical and LEC errors while the ?H errors only include LEC errors.

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For details on this, see Ref. [30]. In the LO potential, we include smeared contact interaction operators which are projected on the S-waves and the one-pion-exchange (OPE) potential. The short-range contact interaction reads The coefficients w; represent the improved action includ- ing a stretching factor which connects O(a*) action with O(a?) action to correct the dispersion relation of the nucleon-nucleon system. Explicitly we use
For details on this, see Ref. [30]. In the LO potential, we include smeared contact interaction operators which are projected on the S-waves and the one-pion-exchange (OPE) potential. The short-range contact interaction reads The coefficients w; represent the improved action includ- ing a stretching factor which connects O(a*) action with O(a?) action to correct the dispersion relation of the nucleon-nucleon system. Explicitly we use
Then we have the seven standard NLO contact interac- The NLO contact interaction contribution consists of ten terms totally. First, we have to include two oper- ators which are not NLO operators by power counting but which are necessary for LO corrections. More pre- cisely, for a coarse lattice spacing, these represent the full non-local structure of the TPEP [32] while for finer lat- tices they could be dropped. However, we keep them for consistency. These terms read: in momentum space. For the pion-nucleon coupling con- stant and the pion decay constant, we use the values ga = 1.29 (to account for the Goldberger-Treiman dis- crepancy) and F, = 92.2 MeV. While there are various ways to regularize the long-range part with a Gaussian smearing function in momentum space or an analogue function directly in position space [27], a similar analysis was done in [31] for the NLEFT approach where it was argued that the hard cut-off regularization due to lattice spacing is sufficient up to those we are interested in.
Then we have the seven standard NLO contact interac- The NLO contact interaction contribution consists of ten terms totally. First, we have to include two oper- ators which are not NLO operators by power counting but which are necessary for LO corrections. More pre- cisely, for a coarse lattice spacing, these represent the full non-local structure of the TPEP [32] while for finer lat- tices they could be dropped. However, we keep them for consistency. These terms read: in momentum space. For the pion-nucleon coupling con- stant and the pion decay constant, we use the values ga = 1.29 (to account for the Goldberger-Treiman dis- crepancy) and F, = 92.2 MeV. While there are various ways to regularize the long-range part with a Gaussian smearing function in momentum space or an analogue function directly in position space [27], a similar analysis was done in [31] for the NLEFT approach where it was argued that the hard cut-off regularization due to lattice spacing is sufficient up to those we are interested in.
with €4,5,- the totally antisymmetric Levi-Civita tensor in three dimensions. Finally, we also include the following SO(3) breaking term which allows us to remove lattice artefacts (unphysical partial wave mixing) due to rotational symmetry break- ing. More specifically, this term is tuned to remove the mixing between the °,5)-?.D, and the ?D3-°G3 channels. Unphysical partial wave mixing in higher waves is so small that it can be ignored. More details of the notation can be found App. A.
with €4,5,- the totally antisymmetric Levi-Civita tensor in three dimensions. Finally, we also include the following SO(3) breaking term which allows us to remove lattice artefacts (unphysical partial wave mixing) due to rotational symmetry break- ing. More specifically, this term is tuned to remove the mixing between the °,5)-?.D, and the ?D3-°G3 channels. Unphysical partial wave mixing in higher waves is so small that it can be ignored. More details of the notation can be found App. A.
TABLE I: Summary of fit results with perturbatively im- proved OPE (in units of a) for the perturbative NLO+NNLO analysis at a = 1.97 fm. All LECs are given in lattice units.
TABLE I: Summary of fit results with perturbatively im- proved OPE (in units of a) for the perturbative NLO+NNLO analysis at a = 1.97 fm. All LECs are given in lattice units.
Although we work in the three-body sector, we are still able to do an exact calculation using the Lanczos method. Therefore we extend the former analysis of the 2N sector to the triton, which means that we cannot work in radial coordinates anymore but we calculate the spectrum in three dimensions where we also include three-body po- tentials, which were reviewed in [1]. They consist of a three-body contact interaction, a one- and a two-pion exchange interaction. These various terms read: The latter equation describing the TPEP among three
Although we work in the three-body sector, we are still able to do an exact calculation using the Lanczos method. Therefore we extend the former analysis of the 2N sector to the triton, which means that we cannot work in radial coordinates anymore but we calculate the spectrum in three dimensions where we also include three-body po- tentials, which were reviewed in [1]. They consist of a three-body contact interaction, a one- and a two-pion exchange interaction. These various terms read: The latter equation describing the TPEP among three
FIG. 1: LO (squares) and NNLO (circles) neutron-proton phase shifts and mixing angles for a = 1.97 fm. The NPWA is given by the solid line. particles can be split up into three parts which read
FIG. 1: LO (squares) and NNLO (circles) neutron-proton phase shifts and mixing angles for a = 1.97 fm. The NPWA is given by the solid line. particles can be split up into three parts which read
FIG. 2: LO (squares) and NNLO (circles) neutron-proton phase shifts and mixing angles for a = 1.64 fm. The NPWA is given by the solid line.
FIG. 2: LO (squares) and NNLO (circles) neutron-proton phase shifts and mixing angles for a = 1.64 fm. The NPWA is given by the solid line.
TABLE II: Triton binding energy predictions at LO, N2LO, N2LO+EM, N2LO+EM-+3N and the fit parameter cp. The energy errors in brackets are due to the uncertainties of the LECs. *At this order, the triton binding energy is a fit parameter.
TABLE II: Triton binding energy predictions at LO, N2LO, N2LO+EM, N2LO+EM-+3N and the fit parameter cp. The energy errors in brackets are due to the uncertainties of the LECs. *At this order, the triton binding energy is a fit parameter.
FIG. 3: LO (squares) and NNLO (circles) neutron-proton phase shifts and mixing angles for a = 1.32 fm. The NPWA is given by the solid line.
FIG. 3: LO (squares) and NNLO (circles) neutron-proton phase shifts and mixing angles for a = 1.32 fm. The NPWA is given by the solid line.
FIG. 4: LO (squares) and NNLO (circles) (both including Coulomb) proton-proton ‘So phase shift for a = 1.97 fm, a = 1.64 fm and a = 1.32 fm. The NPWA is given by the solid line.
FIG. 4: LO (squares) and NNLO (circles) (both including Coulomb) proton-proton ‘So phase shift for a = 1.97 fm, a = 1.64 fm and a = 1.32 fm. The NPWA is given by the solid line.
FIG. 5: Finite volume extrapolation for the triton and a lat- tice spacing of a = 1.32 fm. The triangles are included in the fit while the circles are not due to higher order finite volume effects.
FIG. 5: Finite volume extrapolation for the triton and a lat- tice spacing of a = 1.32 fm. The triangles are included in the fit while the circles are not due to higher order finite volume effects.
TABLE IV: *He binding energy prediction at LO, N2LO, N2LO+EM, N2LO+EM-+3N. The first brackets give the sta- tistical error while the latter ones give the errors due to un- cerainties of the LECs. All binding energies are given in MeV.
TABLE IV: *He binding energy prediction at LO, N2LO, N2LO+EM, N2LO+EM-+3N. The first brackets give the sta- tistical error while the latter ones give the errors due to un- cerainties of the LECs. All binding energies are given in MeV.
FIG. 8: *He time extrapolation for lattice spacing of a = 1.32 fm. The triangles are included in the fit while the circles are excluded due to bad statistics. FIG. 7: *He time extrapolation for lattice spacing of a = 1.64 fm. The triangles are included in the fit while the circles are excluded due to bad statistics.
FIG. 8: *He time extrapolation for lattice spacing of a = 1.32 fm. The triangles are included in the fit while the circles are excluded due to bad statistics. FIG. 7: *He time extrapolation for lattice spacing of a = 1.64 fm. The triangles are included in the fit while the circles are excluded due to bad statistics.
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PhysicsNuclear PhysicsLattice Field TheoryEffective Field TheoryNuclear Force
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