Conclusions

In summary, in this work we derived the Galilean-invariant nuclear N$^{3}$LO pseudopotential with derivatives up to sixth order and found the corresponding N$^{3}$LO EDF, which was obtained by calculating the corresponding HF average energy. Owing to the zero range of the pseudopotential, the number of terms thereof is twice smaller then that of the most general EDF. We found explicit linear relations between the parameters of the pseudopotential and coupling constant of the EDF. These linear relations constitute a set of constraints, which allow for expressing one half of the coupling constants through the other half. As an example of such constraints, we have derived linear relations between the isoscalar and isovector coupling constants. The gauge-invariant form of the pseudopotential was also derived, and all derivations were repeated also for this case.

Table 27: Number of terms of different orders in the pseudopotential (2) and in the EDF up to N$^{3}$LO, evaluated for the conserved Galilean and gauge symmetries. The last four columns show the number of terms in the EDF evaluated by taking into account the additional constraints coming from the relation of the EDF to pseudopotential.

	Pseudopotential		EDF
			Not related to pseudopotential				Related to pseudopotential
			General		Spherical		General		Spherical
Order	Galilean	Gauge	Galilean	Gauge	Galilean	Gauge	Galilean	Gauge	Galilean	Gauge
0	2	2	4	4	2	2	2	2	2	2
2	7	7	14	14	8	8	7	7	7	7
4	15	6	30	12	18	6	15	6	14	6
6	26	6	52	12	32	6	26	6	24	6
N$^{3}$LO	50	21	100	42	60	22	50	21	46	20

We have also analyzed properties of the EDF restricted by imposing the spherical, space-inversion, and time-reversal symmetries, which are relevant for describing spherical nuclei. In this case, by relating the EDF to the pseudopotential, at second, fourth, and sixth order one reduces the numbers of coupling constants only from 8, 18, and 32 to 7, 14, and 24, respectively. Such reduction has two origins: (i) at each order 1, 2, or 4 spin-orbit isovector and isoscalar coupling constants become dependent on one another and (ii) at fourth and sixth order one or two non-spin-orbit coupling constants become linearly dependent on the remaining 13 or 22 ones, respectively. Therefore, in spherical magic nuclei one can expect relatively small effects related to imposing on the EDF the pseudopotential origins, whereas this may have much more important consequences in deformed, asymmetric, odd, and/or rotating nuclei.

Table 27 gives an overview of the results by showing the number of terms of pseudopotential and EDF with Galilean or gauge symmetries imposed.

This work was supported in part by the Academy of Finland and the University of Jyväskylä within the FIDIPRO programme, and by the Polish Ministry of Science and Higher Education under Contract No. N N202 328234.