Axial charge of the triton from lattice QCD

The axial charge of the triton is investigated using lattice quantum chromodynamics (QCD). Extending previous work at heavier quark masses, calculations are performed using three ensembles of gauge field configurations generated with quark masses corresponding to a pion mass of 450 MeV and a single value of the lattice spacing. Finite-volume energy levels for the triton, as well as for the deuteron and diproton systems, are extracted from analysis of correlation functions computed on these ensembles, and the corresponding energies are extrapolated to infinite volume using finite-volume pionless effective field theory (FVEFT). It is found with high likelihood that there is a compact bound state with the quantum numbers of the triton at these quark masses. The axial current matrix elements are computed using background field techniques on one of the ensembles and FVEFT is again used to determine the axial charge of the proton and triton. A simple quark mass extrapolation of these results and earlier calculations at heavier quark masses leads to a value of the ratio of the triton to proton axial charges at the physical quark masses of $g_A^{{}^3\mathrm{H}}/g_A^p={0.91}_{-0.09}^{+0.07}$. This result is consistent with the ratio determined from experiment and prefers values less than unity (in which case the triton axial charge would be unmodified from that of the proton), thereby demonstrating that QCD can explain the modification of the axial charge of the triton.

Figure 1

Left column: effective mass functions [Eq. (8)] for the proton correlation function on the E24 (top), E32 (center), and E48 (bottom) ensembles. The orange circles and blue squares show effective masses constructed from SP and SS correlation functions, respectively, and the corresponding orange and blue curves display the highest-weight fit contributing to the weighted average of Eq. (6); the light (dark) gray bands show the mass extracted from the weighted average (single highest-weight fit). Right column: masses extracted from each successful fit to the correlators. The opacity of each point is set by the contribution of the fit to the weighted average. The horizontal line and gray band in each figure correspond to the final result for the mass and its total uncertainty.

Figure 2

Effective mass functions and fit summaries for the fits of the triton correlation function on the E24 (top), E32 (center), and E48 (bottom) ensembles. All features are as in Fig. 1.

Figure 3

(a) Mass of the proton as a function of the spatial extent of the lattice geometry, along with the fit to this dependence via Eq. (9). (b) The binding energy of the triton as a function of the spatial extent of the lattice geometry. Gray squares correspond to the binding energies determined in the LQCD calculations, while the blue curve shows the finite-volume dependence in pionless EFT fitted to the LQCD data. The blue point at the right indicates the infinite-volume extrapolation of the binding energy. For comparison, the binding energies of the $pp$ (purple) and deuteron (green) systems obtained via an analogous FVEFT study are also shown.

Figure 4

Binding energy of the triton as a function of the pion mass. The current result is shown in blue, along with the result obtained in a previous calculation [24] using the same action at $m_{\pi }=806\mathrm{MeV}$. The results of Refs. [41, 42] are shown in orange. The physical value of the binding is shown as a red star at the physical pion mass, as indicated by the vertical line. Linear and quadratic fits in $m_{\pi }$ to the blue data are shown as the green and blue bands, respectively.

Figure 5

The left panels show the proton and triton bare effective matrix element functions $g_A^p(t)$ and $g_A^{{}^3\mathrm{H}}(t)$, respectively, calculated on the E32 ensemble. In each figure, the orange (blue) data correspond to the SP (SS) calculations and the corresponding shaded bands illustrate the highest-weight fit. The light (dark) shaded gray bands denote the extracted values of the matrix elements arising from the combined analysis (single highest-weight fit). The right panels show the values obtained for all the successful fits to different time ranges, with the opacity determined by the contribution of the fit in the final weighted average. The horizontal line and gray band in the right panels show the final central value and uncertainty.

Figure 6

The effective ratio function $g_A^{{}^3\mathrm{H}}(t)/g_A^p(t)$, constructed through Eq. (12), which asymptotes to $g_A^{{}^3\mathrm{H}}/g_A^p$ and is independent of axial current renormalization. The orange (blue) data show the SP (SS) correlation functions on the E32 ensemble and the shaded band indicates the extracted value of the matrix element (not a fit to the displayed data).

Figure 7

(a) Determination of the LEC ratio ${\tilde{L}}_{1,A}$ by matching the matrix element defined in Eq. (19), computed in the optimized FVEFT wave function for the E32 volume, to the LQCD result in the E32 volume, which is depicted as the horizontal band. (b) The FVEFT extrapolation of the ratio of triton to proton axial charges to infinite volume.

Figure 8

Ratio of the axial charge of tritium to that of the single nucleon as a function of the pion mass. The result from this work and that of Ref. [18] are shown as the blue points while the physical value [6] is shown in red at the physical pion mass (indicated by the vertical line).

Figure 9

Determination of the EFT two-body couplings, $C_{S,T}$ from the deuteron (left) and $pp$ (right) energies. The horizontal bands correspond to the energy shifts from the LQCD calculations on the E24 (blue), E32 (orange), and E48 (green) ensembles. The curves in each figure show the FVEFT two-body energy shifts for $L\in \{24,32,48\}$ (using the same color scheme) for the EFT cutoff parameter $r_0=0.2\mathrm{fm}$ [49]. The LECs are determined by a simultaneous optimization matching the LQCD constraints from all three volumes to the corresponding curves; the results are shown as the red bands. To guide the eye, the intercept ranges of each band with the corresponding curve are shown as the colored bars.

Figure 10

Extrapolation of the deuteron (green) and $pp$ (purple) energies to the infinite-volume limit. Data points denote lattice results (from Ref. [19]), displaced slightly horizontally for clarity, while the shaded bands and infinite-volume points show the result of the FVEFT extrapolation.

Figure 11

Determination of the EFT three-body couplings $D_0$. As in Fig. 9, the horizontal bands correspond to the energy shifts from the LQCD calculations on the E24 (blue), E32 (orange), and E48 (green) ensembles. The curves in each figure show the corresponding FVEFT energy shifts for the EFT cutoff parameter $r_0=0.2\mathrm{fm}$. The uncertainties on the curves are propagated from the uncertainties on the two-body LECs.