8 No-Core Gamow Shell Model
367
N 3 LO interaction. The fifth and sixth orders have been considered in the N 4 LO and
N 5 LO interactions [18,19]. It has been noticed that the inclusion of the fifth order is
not negligible, as it compensates for the energy overbinding provided by the fourth
order [18]. However, the sixth order of chiral interactions is rather small, which
suggests that convergence could be attained at this level [19]. Further calculations
must be done in order to confirm this assertion.
The chiral effective field theory is in general not renormalizable, even though
some effective field theories can be, as is the case for the pionless effective field
theory [20]. In general, the chiral effective field theory exhibits not only the
regularization scale dependence but also unwanted dependence on the particular
form of regulators [21]. Added to that, it is not clear which three-body and fourbody terms are really important in the chiral expansion of the nuclear interaction
[22].
A pionless chiral effective field theory, satisfying the criteria of a physically
correct theory, in particular the independence on the regularization scale, has
been developed in Ref. [20]. In this approach, the three-body interaction appears
at leading order, but four-body terms might be necessary at higher order [23].
Moreover, the dependence on the number of particles remains to be investigated
[23]. This dependence of chiral effective field theory expansions is not understood
in all presently discussed formulations (see also Ref. [22]).
As the calculation of realistic interactions is based on the fit of a restricted set
of parameters, a statistical analysis of errors made in that process is important to
check the physical content of parameters. The use of Bayesian inference is well
suited for that purpose [24]. Indeed, this approach can identify interesting parameter
domains and better fitting functionals. For example, redundancy has been found in
determining the s-wave phase shifts through the use of Bayesian inference [25].
Using this approach, the range of data, such as energies, can be improved [25, 26],
and the truncation and parameter errors can be combined to make better predictions
[26].
The naturalness priors of the Bayesian formalism also allow to examine the
influence of higher order terms in the expansions provided by the effective field
theory [26]. Those terms can be included in the fit, using an already known
functional or an effective field theory expansion for observables. The estimation of
parameters can account for sources of uncertainty in the extraction of low-energy
constants. They include data uncertainties issued from experiment or numerical
calculations, systematic errors arising from the truncation of effective field theory,
and errors induced by theoretical frameworks considered to calculate observables.
Overfitting and underfitting are also mitigated by Bayesian inference.
A prior based on the naturalness of parameters suppresses spurious linear
combinations of parameters. It also makes sure that low-energy constants do not
over-reproduce experimental data in a necessarily limited theoretical framework.
This occurs through the calculations of posterior probability distributions of lowenergy constants, naturalness being explicitly taken into account. The amount of
information found in the fitted low-energy constants is thus heavily increased.
367
N 3 LO interaction. The fifth and sixth orders have been considered in the N 4 LO and
N 5 LO interactions [18,19]. It has been noticed that the inclusion of the fifth order is
not negligible, as it compensates for the energy overbinding provided by the fourth
order [18]. However, the sixth order of chiral interactions is rather small, which
suggests that convergence could be attained at this level [19]. Further calculations
must be done in order to confirm this assertion.
The chiral effective field theory is in general not renormalizable, even though
some effective field theories can be, as is the case for the pionless effective field
theory [20]. In general, the chiral effective field theory exhibits not only the
regularization scale dependence but also unwanted dependence on the particular
form of regulators [21]. Added to that, it is not clear which three-body and fourbody terms are really important in the chiral expansion of the nuclear interaction
[22].
A pionless chiral effective field theory, satisfying the criteria of a physically
correct theory, in particular the independence on the regularization scale, has
been developed in Ref. [20]. In this approach, the three-body interaction appears
at leading order, but four-body terms might be necessary at higher order [23].
Moreover, the dependence on the number of particles remains to be investigated
[23]. This dependence of chiral effective field theory expansions is not understood
in all presently discussed formulations (see also Ref. [22]).
As the calculation of realistic interactions is based on the fit of a restricted set
of parameters, a statistical analysis of errors made in that process is important to
check the physical content of parameters. The use of Bayesian inference is well
suited for that purpose [24]. Indeed, this approach can identify interesting parameter
domains and better fitting functionals. For example, redundancy has been found in
determining the s-wave phase shifts through the use of Bayesian inference [25].
Using this approach, the range of data, such as energies, can be improved [25, 26],
and the truncation and parameter errors can be combined to make better predictions
[26].
The naturalness priors of the Bayesian formalism also allow to examine the
influence of higher order terms in the expansions provided by the effective field
theory [26]. Those terms can be included in the fit, using an already known
functional or an effective field theory expansion for observables. The estimation of
parameters can account for sources of uncertainty in the extraction of low-energy
constants. They include data uncertainties issued from experiment or numerical
calculations, systematic errors arising from the truncation of effective field theory,
and errors induced by theoretical frameworks considered to calculate observables.
Overfitting and underfitting are also mitigated by Bayesian inference.
A prior based on the naturalness of parameters suppresses spurious linear
combinations of parameters. It also makes sure that low-energy constants do not
over-reproduce experimental data in a necessarily limited theoretical framework.
This occurs through the calculations of posterior probability distributions of lowenergy constants, naturalness being explicitly taken into account. The amount of
information found in the fitted low-energy constants is thus heavily increased.