Electromagnetic characteristics of $A\le 3$ physical and lattice nuclei

We analyze the quark-mass dependence of electromagnetic properties of two- and three-nucleon states. To that end, we apply the pionless effective field theory (EFT) to experimental data and numerical lattice calculations which simulate QCD at pion masses of 450 and 806 MeV. At the physical pion mass, we postdict the magnetic moment of helium-3, ${\mu }_{{}^3\mathrm{He}}=-2.13$ nNM (natural nuclear magneton), and the magnetic polarizability of deuterium, ${\beta }_{\text{D}}=7.{3310}^{-2} {\mathrm{fm}}^3$. Magnetic polarizabilities of helium-3, ${\beta }_{{}^3\mathrm{He}}=9.{710}^{-4} {\mathrm{fm}}^3$, and the triton, ${\beta }_{{}^3\mathrm{H}}=8.{210}^{-4} {\mathrm{fm}}^3$, are predictions. Postdictions of the effective theory for the magnetic moments are found consistent with QCD simulations at 806 MeV pion mass, while our EFT result ${\beta }_{\text{D}}=2.{9210}^{-2} {\mathrm{fm}}^3$ was not extracted from the lattice. The deuteron would thus be relatively pliable compared to a three-nucleon state for which we postdict ${\beta }_{{}^3\mathrm{H}}=3.{910}^{-5} {\mathrm{fm}}^3$. At $m_{\pi }=450\mathrm{MeV}$, the magnetic moment of the triton is predicted, ${\mu }_{{}^3\mathrm{He}}=-2.15\left(5\right)$ nNM, based on a conjecture of its binding energy, $B_{{}^3\mathrm{H}}\cong 30$ MeV. For all three pion masses, we compare the point-charge radii of the two- and three-nucleon bound states. The sensitivity of the electromagnetic properties to the Coulomb interaction between protons is studied in anticipation of lattice calculations with dynamical QED.

Figure 1

Cutoff dependence of the LECs $c_S$ (solid line, left $y$ axis), $c_T$ (dashed line, left $y$ axis), and $c_{pp}$ (short-dashed line, right $y$ axis) for three pion masses, $m_{\pi }=137$ MeV (red), $m_{\pi }=450$ MeV (blue), and $m_{\pi }=806$ MeV (green).

Figure 2

Cutoff dependence of the LECs $l_1$ (solid line), and $l_2$ (dashed line) for three pion masses, $m_{\pi }=137$ MeV (red), $m_{\pi }=450$ MeV (blue), and $m_{\pi }=806$ MeV (green). The values for $m_{\pi }=137$ MeV, and $m_{\pi }=806$ MeV are fitted to experimental and LQCD date respectively. The $m_{\pi }=450$ MeV values are results of an interpolation.

Figure 3

Interdependence of two-body LECs consistent with the magnetic moments of the deuteron (green, horizontal), the triton (blue, negative slope), ${}^3\mathrm{He}$ (red, positive slope), and the magnetic-field contribution to the dinucleon energy splitting $\delta E_{{}^3{S}_1,{}^1{S}_0}$ (purple, vertical) at $m_{\pi }=806\mathrm{MeV}$. The bandwidth resembles the total lattice uncertainty in the respective observable. The black lines marks the LEC values which yield the experimental deuteron, triton, and 3-helium magnetic moments at the physical pion mass.

Figure 4

The magnetic moment of the triton as a function of its binding energy for $m_{\pi }=806$ MeV [gray ($\mathrm{\Lambda }=4{\mathrm{fm}}^{-1}$) $\to$ black ($\mathrm{\Lambda }=15{\mathrm{fm}}^{-1}$)] and $m_{\pi }=450$ MeV (blue, $\mathrm{\Lambda }=8{\mathrm{fm}}^{-1}$). Vertical dashed lines mark the deuteron-neutron thresholds at $B_D=19.5\mathrm{MeV}$ and $B_D=14.4\mathrm{MeV}$, respectively. LO results with one-body-current coupling (dotted lines) are compared with NLO values (solid lines) which consider also the two-body-current coupling $l_1,l_2$. Asymptotic limits are indicated with arrows, for $B_{{}^3\mathrm{H}}\to B_{\text{D}}$: ${\mu }_{{}^3\mathrm{H}}\to 1.196\mathrm{nNM}$ (450 MeV), ${\mu }_{{}^3\mathrm{H}}\to 1.472\mathrm{nNM}$ (806 MeV); and $B_{{}^3\mathrm{H}}\to \infty$: ${\mu }_{{}^3\mathrm{H}}\to 2.70\mathrm{nNM}$ (450 MeV), ${\mu }_{{}^3\mathrm{H}}\to 3.119\mathrm{nNM}$ (806 MeV).

Figure 5

Regulator dependence of the point-charge radii for the two- (left panel) and three-nucleon (right panel) bound states at the physical (red), 450 MeV (blue), and 806 MeV (green) pion masses. Solid lines refer to one-proton systems, i.e., the deuteron, $np$, and triton. Results for two-proton systems are shown with (dash-dotted) and without (dashed) electrostatic repulsion between the protons. For $A=2$, the lines indicated with an arrow correspond to singlet $np$ state, while the lower solid lines mark the triplet deuteron.

Figure 6

Regulator dependence of the magnetic polarizability EFT calculations for ${}^3\mathrm{He}$ and triton $m_{\pi }=137$ MeV.

Figure 7

Regulator dependence of the magnetic polarizability EFT calculations for ${}^3\mathrm{He}$ and triton $m_{\pi }=806$ MeV.