Basis truncation and statistical errors in the relativistic approaches to nuclear response

TL;DR

The paper tackles uncertainties in nuclear response theory arising from finite harmonic-oscillator bases and from covariant energy-density functional fits. It systematically extends the fermionic basis to $N_F=50$ and applies RRPA and RTBA to several doubly magic nuclei across multiple multipolarities, highlighting strong continuum and basis-completeness effects on low-spin resonances and soft modes, especially in light, neutron-rich systems. Statistical analysis reveals large uncertainties in monopole strength distributions due to the functional fit, with smaller but non-negligible errors in dipole, quadrupole, and octupole channels. Overall, the study emphasizes the necessity of a complete basis and accurate continuum treatment to reliably extract nuclear matter properties such as compressibility and symmetry energy, and to quantify theoretical uncertainties in nuclear response predictions.

Abstract

Although there exists a clear and, in principle, exact theoretical formulation for the equation of motion for the response of a correlated fermionic system, its numerical implementations for atomic nuclei require feasible approximations. One of the widely accepted approximations is a truncated harmonic oscillator (HO) basis, whose wave functions are used to expand the solutions obtained with realistic interactions. In this work, we extend previously employed HO basis truncated at $N_F$ = 20 fermionic shells to $N_F$ = 50 and perform a systematic study of the effects of such basis increase on nuclear resonances. The relativistic random phase approximation (RRPA) and its extension by the particle-vibration coupling dubbed as relativistic time-blocking approximation (RTBA) are applied to the description of the monopole, dipole, quadrupole, and octupole resonances in $^{48}$Ca, $^{78}$Ni, and $^{132}$Sn, and the RRPA studies are extended to $^{70}$Ca and $^{208}$Pb. A considerable sensitivity of the strength distributions to the HO basis size is found, especially for low-spin resonances in the light neutron-rich nuclei. The effects of the HO basis extension to $N_F$ = 50 are analyzed and linked to the involvement of proton and neutron continuum states and proton quasi-bound states in the strength formation. The obtained results point to the importance of the HO basis completeness and continuum effects in the nuclear response calculations and evaluation of the associated parameters of the nuclear equation of state. In addition, statistical errors in the RRPA strength functions, which emerge from the details of the fitting protocol of the covariant energy density functional employed, are analyzed. It turns out that they are quite substantial for the monopole response, but significantly smaller for the dipole, quadrupole, and octupole ones.

Figures (9)

• Figure 1: The dependence of the single-particle energies of the proton and neutron subsystems of $^{208}$Pb and $^{48}$Ca on the number of fermionic shells $N_F$ employed in the calculations. The calculations have been carried out in steps of $\Delta N_F=10$. Note that for illustration purposes only approximately 25 positive energy single-particle states, which are the lowest in energy at $N_F=20$, are shown. Color boxes in the left panels show the magnitude of the Coulomb barrier.
• Figure 2: ISE0 strength functions $S(E)$ obtained in RRPA for the indicated nuclei. The results obtained with $N_F=20$ and $N_F=50$ are shown by blue and red curves, respectively. Left and right panels show the results with $\Delta = 0.05$ and $\Delta =0.5$ MeV, respectively.
• Figure 3: The same as in Fig. \ref{['Strength-ISGMR']} but for IVE1.
• Figure 4: The same as in Fig. \ref{['Strength-ISGMR']} but for ISE2.
• Figure 5: The same as in Fig. \ref{['Strength-ISGMR']} but for ISE3.