Total polarization effect

**Table 1:** Neutron s.p. energies in $^{ 16}$ O, $^{40}$ Ca, and $^{48}$ Ca (in MeV). The columns show: (a) bare unpolarized s.p. energies in doubly-magic cores, (b) self-consistent s.p. energies obtained from binding energies in one-particle/hole nuclei, Eqs. (21) and (22), with time-even mass and shape polarizations included, (c) as in (b), but with time-odd spin polarizations included in addition, and (d) experimental data taken from Ref. [8]. Time-even, (b)(a), time-odd, (c)(b), and total, (c)(a) polarizations are also shown, along with the differences between the self-consistent and experimental spectra, (c)(d). Positive s.p. energies are shown only to indicate that particular orbitals are unbound in calculations; their values are only very approximately related to positions of resonances. All results have been calculated using the Sly4 $_{L}$ functional.
		bare	T-even	T-even	T-even	T-odd	total	exp.	theory
				pol.	& T-odd	pol.	pol.	[8]	exp
		(a)	(b)	(b)(a)	(c)	(c)(b)	(c)(a)	(d)	(c)(d)
$^{ 16}$ O	$1\nu p_{ 3/2}$	20.57	19.61	0.96	20.29	0.68	0.28	21.84	1.55
	$1\nu p_{ 1/2}$	14.54	13.55	0.99	13.86	0.31	0.68	15.66	1.80
	$1\nu d_{ 5/2}$	6.75	5.83	0.92	5.43	0.40	1.32	4.22	1.21
	$2\nu s_{ 1/2}$	3.78	2.79	0.99	2.30	0.49	1.48	3.35	1.05
	$1\nu d_{ 3/2}$	0.39	1.19	0.80	1.37	0.18	0.98	1.50	0.13
$^{40}$ Ca	$1\nu d_{ 5/2}$	22.01	21.55	0.46	21.87	0.32	0.14	22.39	0.52
	$2\nu s_{ 1/2}$	17.25	16.93	0.32	17.51	0.58	0.26	18.19	0.68
	$1\nu d_{ 3/2}$	15.31	14.84	0.47	14.98	0.14	0.33	15.64	0.66
	$1\nu f_{ 7/2}$	9.58	9.22	0.36	9.00	0.22	0.58	8.62	0.38
	$2\nu p_{ 3/2}$	5.24	4.85	0.39	4.67	0.18	0.57	6.76	2.09
	$2\nu p_{ 1/2}$	3.06	2.66	0.40	2.55	0.11	0.51	4.76	2.21
	$1\nu f_{ 5/2}$	1.38	1.10	0.28	0.99	0.11	0.39	3.38	2.39
$^{48}$ Ca	$1\nu d_{ 5/2}$	22.60	22.02	0.58	22.21	0.19	0.39	17.31	4.90
	$2\nu s_{ 1/2}$	17.60	17.26	0.34	17.78	0.52	0.18	13.16	4.62
	$1\nu d_{ 3/2}$	16.55	15.97	0.58	16.02	0.05	0.53	12.01	4.01
	$1\nu f_{ 7/2}$	9.79	9.23	0.56	9.34	0.11	0.45	9.68	0.34
	$2\nu p_{ 3/2}$	5.54	5.25	0.29	5.12	0.13	0.42	5.25	0.13
	$2\nu p_{ 1/2}$	3.54	3.21	0.33	3.11	0.10	0.43	3.58	0.47
	$1\nu f_{ 5/2}$	1.33	1.25	0.08	1.21	0.04	0.12	1.67	0.46

**Table 2:** Same as in Table 1 but for $^{90}$ Zr, $^{132}$ Sn, and $^{208}$ Pb.
		bare	T-even	T-even	T-even	T-odd	total	exp.	theory
				pol.	& T-odd	pol.	pol.	[8]	exp
		(a)	(b)	(b)(a)	(c)	(c)(b)	(c)(a)	(d)	(c)(d)
$^{90}$ Zr	$1\nu f_{ 7/2}$	23.16	22.83	0.33	22.94	0.11	0.22	14.76	8.18
	$1\nu f_{ 5/2}$	17.07	16.72	0.35	16.74	0.02	0.33	13.05	3.69
	$2\nu p_{ 3/2}$	17.52	17.30	0.22	17.44	0.14	0.08	12.74	4.70
	$2\nu p_{ 1/2}$	15.46	15.26	0.20	15.35	0.09	0.11	12.37	2.98
	$1\nu g_{ 9/2}$	12.08	11.75	0.33	11.81	0.06	0.27	11.69	0.12
	$2\nu d_{ 5/2}$	6.73	6.59	0.14	6.52	0.07	0.21	7.20	0.68
	$3\nu s_{ 1/2}$	4.93	4.70	0.23	4.44	0.26	0.49	5.78	1.34
	$2\nu d_{ 3/2}$	3.99	3.62	0.37	3.59	0.03	0.40	4.77	1.18
	$1\nu g_{ 7/2}$	3.75	3.75	0.00	3.74	0.01	0.01	4.62	0.88
$^{132}$ Sn	$2\nu d_{ 5/2}$	11.72	11.48	0.24	11.56	0.08	0.16	9.10	2.46
	$3\nu s_{ 1/2}$	9.46	9.28	0.18	9.59	0.31	0.13	7.55	2.04
	$1\nu h_{11/2}$	7.66	7.30	0.36	7.33	0.03	0.33	7.42	0.09
	$2\nu d_{ 3/2}$	9.11	8.91	0.20	8.95	0.04	0.16	7.17	1.78
	$2\nu f_{ 7/2}$	2.01	2.00	0.01	1.95	0.05	0.06	2.29	0.34
	$3\nu p_{ 3/2}$	0.17	0.26	0.09	0.31	0.05	0.14	1.31	1.62
	$1\nu h_{ 9/2}$	0.95	0.79	0.16	0.77	0.02	0.18	0.91	1.68
	$3\nu p_{ 1/2}$	0.97	1.08	0.11	1.12	0.04	0.15	0.72	1.84
	$2\nu f_{ 5/2}$	0.82	0.84	0.02	0.88	0.04	0.06	0.35	1.23
$^{208}$ Pb	$2\nu f_{ 7/2}$	12.02	11.85	0.17	11.90	0.05	0.12	9.96	1.94
	$1\nu i_{13/2}$	9.52	9.29	0.23	9.30	0.01	0.22	8.92	0.38
	$3\nu p_{ 3/2}$	9.23	9.09	0.14	9.17	0.08	0.06	8.12	1.05
	$2\nu f_{ 5/2}$	9.03	8.89	0.14	8.91	0.02	0.12	7.78	1.13
	$3\nu p_{ 1/2}$	8.11	8.01	0.10	8.06	0.05	0.05	7.72	0.34
	$2\nu g_{ 9/2}$	3.19	3.19	0.00	3.16	0.03	0.03	3.73	0.57
	$1\nu i_{11/2}$	1.53	1.65	0.12	1.67	0.02	0.14	3.11	1.44
	$3\nu d_{ 5/2}$	0.50	0.46	0.04	0.43	0.03	0.07	2.22	1.79
	$4\nu s_{ 1/2}$	0.56	0.65	0.09	0.80	0.15	0.24	1.81	2.61
	$2\nu g_{ 7/2}$	0.08	0.10	0.02	0.11	0.01	0.03	1.35	1.46
	$3\nu d_{ 3/2}$	0.69	0.76	0.07	0.78	0.02	0.09	1.33	2.11

In Tables 1 and 2 we list the neutron s.p. energies calculated in six doubly-magic nuclei for the SLy4 $_{L}$ interaction. The bare s.p. energies (a) are compared to those calculated from total energies, Eqs. (21) and (22), with the mass and shape (b) or mass, shape, and spin (c) polarizations included. In order to remove ambiguities associated with occupancy of the valence particle (hole), binding energies of odd-

nuclei were calculated by blocking the lowest (highest)

orbitals at oblate (prolate) shape for particle (hole) orbitals. The blocked orbitals were selected by performing cranking calculation with angular-frequency vector parallel to the symmetry axis. Such a cranking does not affect total energy or wave function, but splits spherical multiplets into orbitals having good projections of the angular momentum on the symmetry axis. Calculations were performed by using the code HFODD (v2.30a) [49,50,51,52] for the spherical basis of $N_{\text{sh}}=14$ harmonic-oscillator shells.

As seen by comparing columns (b) and (a) of Tables 1 and 2, the energy shifts caused by the time-even polarization effects with respect to bare s.p. spectra are almost always positive, both for particle and hole states. A few exceptions occur only for large-

unfavored ( $j=\ell-1/2$ ) SO partners in heavy nuclei. These shifts clearly decrease in magnitude with increasing mass, from about 1MeV in $^{ 16}$ O to below 0.25MeV in $^{208}$ Pb. As discussed in Sec. 3.1, they are mainly caused by the mass-polarization effects related to the center-of-mass correction. Indeed, shifts of s.p. energies (25), calculated for the six doubly-magic nuclei of Tables 1 and 2, are 0.87, 0.40, 0.36, 0.20, 0.14, and 0.09MeV, respectively.

It is also clearly visible that shifts of particle states are systematically smaller than those of hole states, i.e., the time-even polarizations tend to slightly decrease shell gaps.

The time-odd polarization effects systematically shift the hole states down and particle states up in energy, i.e., they result in an increase of shell gaps, cf. also Fig. 1. This result is at variance with that obtained within the RMF approach [4], where the time-odd fields corresponded to magnetic properties driven by the Lorentz invariance, while here they are determined by experimental values of the Landau parameters [21,22]. We note here that in recent derivations of the time-odd coupling constants within the relativistic point-coupling model [53], one obtains values of the Landau parameters compatible with experimental values. Shifts of s.p. energies due to the time-odd polarization effects also decrease with mass, from about

0.7(+0.5)MeV in $^{ 16}$ O to below

0.1(0.15)MeV in $^{208}$ Pb for hole (particle) states.

The total effect of combined time-even and time-odd polarizations results in adding up the shifts for particle states and subtracting those for hole states. In this way, the total shifts of particle and hole states become mostly positive and (apart from light nuclei) comparable in magnitude, with quite small net effects on shell gaps. They also decrease with increasing mass, from up to 1.5MeV in $^{ 16}$ O to below 0.25MeV in $^{208}$ Pb. Altogether, polarization effects turn out to be significantly smaller than those obtained in previous estimates. Although in quantitative analysis they cannot at all be neglected, discrepancies with experimental data (last columns in Tables 1 and 2) are still markedly larger in magnitude. Therefore, bare s.p. energies can be safely used, at least in all studies that do not achieve any better overall agreement with data.