Mass, or binding energy, is one of the most fundamental properties of atomic nuclei. Measuring and modelling nuclear masses has been for many years, and still is, at the center stage of nuclear physics. See Ref. [1] for a recent review. Determination of mass from first principles, viz. quantum chromodynamics, is extremely difficult and only possible in lattice QCD for composite particles like mesons or nucleons [2], and is beyond anything possible or sensible for nuclei. For light nuclei, one can quite accurately calculate nuclear masses by using many-body technics that employ parametrized models of nucleon-nucleon (NN) and NNN interactions, see e.g. Ref. [3]. In these so-called ab initio models, parameters are partly fitted to observables other than mass (like NN phase shifts) and partly to masses (NNN interactions). There are many other, less sophisticated methods to calculate nuclear masses, and all of them include fitting to mass data to a larger or smaller extent. Therefore, there is an extensive history of mass fits in nuclear physics.
Nevertheless, and strangely enough, the history of error analyses of these mass fits is virtually nonexistent (see notable examples in Refs. [4,5]). As a consequence, many mass tables and mass predictions exist in the literature, but there are no estimates of the reliability of these results which would be based on thorough methods of analyzing their uncertainties.
In the present study, we aim at (i) recalling the well-known methods that must be used to analyze errors along with any fits of parameters, and (ii) pointing several particular features of such analyses that are characteristic in applications to mass fits. At present, one cannot overestimate the importance of quantitatively analyzing the predictivity of mass calculations when applied to exotic nuclei far from stability. However, such mass calculations must be accompanied by predictions of their theoretical error bars. Professional error analyses will put predictions on firm grounds--often showing explicitly that such predictions are simply impossible, when they are based on a given model fitted to a given set of masses. On the other hand, they will give quantitative information on how much measuring the mass of the last available isotope (often very difficult) will improve predictivity of models.
As a benchmark number that characterizes mass fits, one has the mass root-mean-squared (rms) deviation, which nowadays does not go below about 0.6MeV [6,1,7]. Down to this level, nuclear models were successfully used to describe nuclear masses, and moreover, they often correctly describe other observables such as charge radii and excitations. In the present study we do not enter into the discussion of which observables, apart from mass, should be used to fit given models to data. Of course, error analyses should be performed when fitting any kinds of observables, although our particular example below concerns only a mass model.
The best Skyrme and Gogny energy-density-functional (EDF) methods [8], fitted to large numbers of nuclei, have resulted in rms deviations of 0.7-1.0MeV from experimental masses. The deviations from experiment are not random, but show systematic patterns [9]. These patterns are a clear sign that the functionals are too simplified, see also Ref. [10]. Systematic methods are needed to improve EDF models by introducing new terms (for example, by using density-dependent coupling constants, see e.g. Refs. [11,12], or higher-order derivative terms [13]) and testing the importance and physical feasibility of the new terms.
Current EDF models typically use 10-14 parameters or coupling constants. Skyrme functionals, for example have clear physical interpretation for all parameters of the functional. However, if the number of model parameters is drastically increased, the meaning and importance of parameters might not always be apparent. To be able to understand the significance of each parameter, clear and efficient methods must be used, as is discussed in this study.