Effective field theory for nuclear vibrations with quantified uncertainties

We develop an effective field theory (EFT) for nuclear vibrations. The key ingredients—quadrupole degrees of freedom, rotational invariance, and a breakdown scale around the three-phonon level—are taken from data. The EFT is developed for spectra and electromagnetic moments and transitions. We employ tools from Bayesian statistics for the quantification of theoretical uncertainties. The EFT consistently describes spectra and electromagnetic transitions for ${}^{62}\mathrm{Ni},{}^{98,100}\mathrm{Ru},{}^{106,108}\mathrm{Pd},{}^{110,112,114}\mathrm{Cd}$, and ${}^{118,120,122}\mathrm{Te}$ within the theoretical uncertainties. This suggests that these nuclei can be viewed as anharmonic vibrators.

Figure 1

Cumulative distribution for the $c_2$ coefficients for states up to the two-phonon level in the ensemble of all nuclei studied in this work. The cumulative distributions of the hard-wall and Gaussian priors are also shown for comparison.

Figure 2

Comparison between the normalized energies $E/\omega$ of the one- and two-phonon states as a function of spin $I$ in the ensemble of the nuclei studied in this work. Experimental energies are shown as thick black lines. LO and NLO energies are shown as red crosses and blue diamonds, respectively. Theoretical uncertainties quantified from 68% DOB intervals are shown as shaded and hatched areas at LO and NLO, respectively.

Figure 3

Partial energy spectrum of ${}^{62}\mathrm{Ni}$ up to the three-phonon level. Experimental data [57], shown as black lines, are compared to LO and NLO calculations, shown as red crosses and blue diamonds, respectively. States up to the two-phonon level are shown as thick black lines. Theoretical uncertainties quantified from 68% DOB intervals are shown as shaded and hatched areas at LO and NLO, respectively.

Figure 4

Partial energy spectrum of ${}^{98}\mathrm{Ru}$ (a) and ${}^{100}\mathrm{Ru}$ (b) up to the three-phonon level. Experimental data [58, 59], shown as black lines, are compared to LO and NLO calculations, shown as red crosses and blue diamonds, respectively. States up to the two-phonon level are shown as thick black lines. Theoretical uncertainties quantified from 68% DOB intervals are shown as shaded and hatched areas at LO and NLO, respectively.

Figure 5

Partial energy spectrum of ${}^{106}\mathrm{Pd}$ (a) and ${}^{108}\mathrm{Pd}$ (b) up to the three-phonon level. Experimental data [63, 64], shown as black lines, are compared to LO and NLO calculations, shown as red crosses and blue diamonds, respectively. States up to the two-phonon level are shown as thick black lines. Theoretical uncertainties quantified from 68% DOB intervals are shown as shaded and hatched areas at LO and NLO, respectively.

Figure 6

Partial energy spectrum of ${}^{110}\mathrm{Cd}$ (a), ${}^{112}\mathrm{Cd}$ (b), and ${}^{114}\mathrm{Cd}$ (c) up to the three-phonon level. Experimental data [67, 68, 69], shown as black lines, are compared to LO and NLO calculations, shown as red crosses and blue diamonds, respectively. States up to the two-phonon level are shown as thick black lines. Theoretical uncertainties quantified from 68% DOB intervals are shown as shaded and hatched areas at LO and NLO, respectively.

Figure 7

Partial energy spectrum of ${}^{118}\mathrm{Te}$ (a), ${}^{120}\mathrm{Te}$ (b), and ${}^{122}\mathrm{Te}$ (c) up to the three-phonon level. Experimental data [74, 75, 76], shown as black lines, are compared to LO and NLO calculations, shown as red crosses and blue diamonds, respectively. States up to the two-phonon level are shown as thick black lines. Theoretical uncertainties quantified from 68% DOB intervals are shown as shaded and hatched areas at LO and NLO, respectively.

Figure 8

Comparison between the normalized $B\left(E2\right)$ values for decays from the one- and two-phonon states in the ensemble of the nuclei studied in this work. Experimental $B\left(E2\right)$ values are shown as black lines with error bars. Theoretical uncertainties quantified from 68% DOB intervals are shown as shaded areas.

Figure 9

Reduced electric quadrupole matrix elements in ${}^{106}\mathrm{Pd}$ (a) and ${}^{108}\mathrm{Pd}$ (b). Experimental data, shown as black lines with error bars, are compared LO calculations, shown as red crosses. Theoretical uncertainties from 68% DOB intervals are shown as shaded areas. The left side shows diagonal matrix elements employed in the fit of the LEC constant $Q_1$. The right side shows predictions for the absolute values of the reduced matrix elements governing $E2$ transitions between two-phonon states.

Figure 10

Comparison between data and EFT results for some reduced quadrupole matrix elements in ${}^{114}\mathrm{Cd}$. Experimental data, shown as black lines with error bars, are compared LO calculations, shown as red crosses. Theoretical uncertainties from 68% DOB intervals are shown as shaded areas. The left side shows diagonal matrix elements employed in the fit of the LEC constant $Q_1$. The right side shows predictions for the absolute values of the reduced matrix elements governing $E2$ transitions between two-phonon states.