Strong-force isospin-symmetry breaking in masses of $\bm{N\sim Z}$ nuclei

P. Baczyk, J. Dobaczewski, M. Konieczka, W. Satula, T. Nakatsukasa, and K. Sato

Date: February 3, 2017

Abstract:

Effects of the isospin-symmetry breaking due to the strong interaction are systematically studied for nuclear masses near the $N=Z$ line, using extended Skyrme energy density functionals (EDFs) with proton-neutron-mixed densities and new terms breaking the isospin symmetry. Two parameters associated with the new terms are determined by fitting mirror and triplet displacement energies (MDEs and TDEs) of isospin multiplets. The new EDFs reproduce the MDEs for $T=\frac 12$ doublets and $T=1$ triplets, as well as the staggering of TDE for $T=1$ triplets. The relative strengths of the obtained isospin-symmetry-breaking terms are consistent with the differences in the $NN$ scattering lengths, $a_{nn}$, $a_{pp}$, and $a_{np}$.

Similarity between the neutron-neutron ($nn$), proton-proton ($pp$), and proton-neutron ($pn$) nuclear forces, commonly known as their charge independence, has been well established experimentally already in 1930's, leading to the concept of isospin symmetry introduced by Heisenberg [1] and Wigner [2]. Since then, the isospin symmetry has been tested and widely used in theoretical modelling of atomic nuclei, with explicit violation by the Coulomb interaction. In addition, the nuclear force also weakly violates the isospin symmetry. There exists firm experimental evidence in the nucleon-nucleon ($NN$) scattering data that it also contains small charge-dependent (CD) components. The differences in the $NN$ phase shifts indicate that the nn interaction, $V_{nn}$, is about 1% stronger than the pp interaction, $V_{pp}$, and that the np interaction, $V_{np}$, is about 2.5% stronger than the average of $V_{nn}$ and $V_{pp}$ [3]. These are called charge-symmetry breaking (CSB) and charge-independence breaking (CIB), respectively. In this paper, we show that the manifestation of the CSB and CIB in nuclear masses can systematically be accounted for in extended nuclear density functional theory (DFT).

The charge dependence of the nuclear force fundamentally originates from mass and charge differences between $u$ and $d$ quarks. The strong and electromagnetic interactions among these quarks give rise to the mass splitting among the baryonic and mesonic multiplets. The neutron is slightly heavier than the proton. The pions, which are the Goldstone bosons associated with the chiral symmetry breaking and are the primary carriers of the nuclear force at low energy, also have the mass splitting. The CSB mostly originates from the difference in masses of protons and neutrons, leading to the difference in the kinetic energies and influencing the one- and two-boson exchange. On the other hand, the major cause of the CIB is the pion mass splitting. For more details, see Refs. [3,4].

The isospin formalism offers a convenient classification of different components of the nuclear force by dividing them into four distinct classes. Class I isoscalar forces are invariant under any rotation in the isospin space. Class II isotensor forces break the charge independence but are invariant under a rotation by $\pi$ with respect to the $y-$axis in the isospace preserving therefore the charge symmetry. Class III isovector forces break both the charge independence and the charge symmetry, and are symmetric under interchange of two interacting particles. Finally, forces of class IV break both symmetries and are anti-symmetric under the interchange of two particles. This classification was originally proposed by Henley and Miller [4,5] and subsequently used in the framework of potential models based on boson-exchange formalism, like CD-Bonn [3] or AV18 [6]. The CSB and CIB were also studied in terms of the chiral effective field theory [7,8]. So far, the Henley-Miller classification has been rather rarely utilized within the nuclear DFT [9,10], which is usually based on the charge-independent strong forces.

The most prominent manifestation of the isospin symmetry breaking (ISB) is in the mirror displacement energies (MDEs) defined as the differences between binding energies of mirror nuclei:

$$\mathrm{MDE}=BE\left(T,T_z=-T\right)-BE\left(T,T_z=+T\right).$$ (1)

A systematic study by Nolen and Schiffer [11] showed that the MDEs cannot be reproduced by using models involving Coulomb interaction as the only source of the ISB, see also Refs. [9,12,13]. Another source of information on the ISB is the so-called triplet displacement energy (TDE):

$$\mathrm{TDE}=BE\left(T=1,T_z=-1\right)+BE\left(T=1,T_z=+1\right)$$

$$-2BE\left(T=1,T_z=0\right),$$ (2)

which is a measure of the binding-energy curvature within the isospin triplet. The TDE also cannot be reproduced by means of conventional approaches disregarding nuclear CIB forces, see [14]. In these definitions the binding energies are negative ($BE<0$) and the proton (neutron) has isospin projection of $t_z=-\frac12(+\frac12)$.

In Fig. 1 we show MDEs and TDEs calculated fully self-consistently using three different standard Skyrme EDFs; SV$_{\rm T}$ [15,16], SkM$^*$ [17], and SLy4 [18]. Details of the calculations, performed using code HFODD [19,20], are presented in the Supplemental Material [21]. In Fig. 1(a), we clearly see that the values of obtained MDEs are systematically lower by about 10% than the experimental ones. Even more spectacular discrepancy appears in Fig. 1(b) for TDEs - their values are underestimated by about a factor of three and the characteristic staggering pattern seen in experiment is entirely absent. It is also very clear that the calculated MDEs and TDEs, which are specific differences of binding energies, are independent of the choice of Skyrme EDF parametrization, that is, of the isospin-invariant part of the EDF.

Figure 1: (Color online) Calculated (no ISB terms) and experimental values of MDEs (a) and TDEs (b). The values of MDEs for triplets are divided by two to fit in the plot. Thin dashed line shows the average linear trend of experimental MDEs in doublets, defined as $\overline {MDE} = 0.137A + 1.63$ (in MeV). Measured values of binding energies were taken from Ref. [24] and the excitation energies of the $T=1$, $T_z=0$ states from Ref. [25]. Open squares denote data that depend on masses derived from systematics [24].

We aim at comprehensive study of MDEs and TDEs based on extended Skyrme $pn$-mixed DFT [16,19,20] that includes zero-range class II and III forces. We consider the following zero-range interactions of class II and III with two new low-energy coupling constants $t^\mathrm {II}_0$ and $t^\mathrm {III}_0$ [26]:

$$\hat{V}^{\rm {II}}(i,j) = t_0^{\rm {II}}\, \delta\left(\boldsymbol{r}_i - \boldsymbol{r}_j\... ...t{\tau}_3(i)\hat{\tau}_3(j)-\hat{\vec{\tau}}(i)\circ\hat{\vec{\tau}}(j)\right],$$ (3)

$$\hat{V}^{\rm {III}}(i,j) = t_0^{\rm {III}}\, \delta\left(\boldsymbol{r}_i - \boldsymbol{r}_j\right) \left[\hat{\tau}_3(i)+\hat{\tau}_3(j)\right] .$$ (4)

The corresponding contributions to EDF read:

$$\mathcal{H}_{\rm {II}} = \frac{1}{2}t_0^{\rm {II}} (\rho_n^2+\rho_p^2-2\rho_n\rho_p-2\rho_{np}\rho_{pn}\notag$$ (5)

$$-\boldsymbol{s}_{n}^2-\boldsymbol{s}_{p}^2+2\boldsymbol{s}_{n}\cdot\boldsymbol{s}_{p}+2\boldsymbol{s}_{np}\cdot\boldsymbol{s}_{pn}),$$ (6)

$$\mathcal{H}_{\rm {III}} = \frac{1}{2}t_0^{\rm {III}} \left(\rho_n^2-\rho_p^2 - \boldsymbol{s}_{n}^2+\boldsymbol{s}_{p}^2\right),$$ (7)

where $\rho$ and $\boldsymbol{s}$ are scalar and spin (vector) densities, respectively. Inclusion of the spin exchange terms in Eqs. (3) and (4) leads to trivial rescaling of the coupling constants $t^\mathrm {II}_0$ and $t^\mathrm {III}_0$, see [26]. Hence, it can be disregarded.

Contributions of class III force to EDF (6) depend on the standard nn and pp densities and, therefore, can be taken into account within the conventional $pn$-separable DFT approach [9]. In contrast, contributions of class II force (5) depend explicitly on the mixed densities, $\rho_{np}$ and $\boldsymbol{s}_{np}$, and require the use of $pn$-mixed DFT [27,28], augmented by the isospin cranking to control the magnitude and direction of the isospin $(T,T_z)$.

We implemented the new terms of the EDF in the code HFODD [19,20], where the isospin degree of freedom is controlled within the isocranking method [29,30,27] - an analogue of the cranking technique that is widely used in high-spin physics. The isocranking method allows us to calculate the entire isospin multiplet, $T$, by starting from an isospin-aligned state $\vert T,T_z=T\rangle$ and isocranking it around the $y$-axis in the isospace. The method can be regarded as an approximate isospin projection. A rigorous treatment of the isospin symmetry within the $pn$-mixed DFT formalism requires full, three-dimensional isospin projection, which is currently under development.

Physically relevant values of $t^\mathrm {II}_0$ and $t^\mathrm {III}_0$ turn out to be fairly small [26], and thus the new terms do not impair the overall agreement of self-consistent results with the standard experimental data. Moreover, calculated MDEs and TDEs depend on $t^\mathrm {II}_0$ and $t^\mathrm {III}_0$ almost linearly, and, in addition, MDEs (TDEs) depend very weakly on $t^\mathrm {II}_0$ ( $t^\mathrm {III}_0$) [26]. This allows us to use the standard linear regression method, see, e.g. Refs. [31,32], to independently adjust $t^\mathrm {II}_0$ and $t^\mathrm {III}_0$ to experimental values of TDEs and MDEs, respectively. See Supplemental Material [21] for detailed description of the procedure. Coupling constants $t^\mathrm {II}_0$ and $t^\mathrm {III}_0$ resulting from such an adjustment are collected in Table 1.

Table 1: Coupling constants $t^\mathrm {II}_0$ and $t^\mathrm {III}_0$ and their uncertainties obtained in this work for the Skyrme EDFs SV$_{\rm T}$, SkM*, and SLy4.

		SV$_{\rm T}$		SkM*		SLy4
$t^\mathrm {II}_0$ (MeV fm$^3$)		$17\pm5$		$25 \pm 8$		$23 \pm 7$
$t^\mathrm {III}_0$ (MeV fm$^3$)		$-7.4\pm1.9$		$-5.6 \pm 1.4$		$-5.6 \pm 1.1$

In Fig. 2, we show values of MDEs calculated within our extended DFT formalism for the Skyrme SV$_{\rm T}$ EDF. By subtracting an overall linear trend (as defined in Fig. 1) we are able to show results in extended scale, where a detailed comparison with experimental data is possible. In Fig. 3, we show results obtained for TDEs, whereas complementary results obtained for the Skyrme SkM* and SLy4 EDFs are collected in the Supplemental Material [21].

It is gratifying to see that the calculated values of MDEs closely follow the experimental $A$-dependence, see Fig. 2. It is worth noting that a single coupling constant $t^\mathrm {III}_0$ reproduces both $T=\frac 12$ and $T=1$ MDEs, which confirms conclusions of Ref. [9]. In addition, for the $T=\frac 12$ MDEs, the SV$_{\rm T}$ results nicely reproduce (i) changes in experimental trend that occur at $A=15$ and 39, (ii) staggering pattern between $A=15$ and 39, and (iii) disappearance of staggering between $A=41$ and 49 (the f$_{7/2}$ nuclei). We note that these features are already present in the SV$_{\rm T}$ results without the ISB terms, and that adding this terms increases amplitude of the staggering. However, for the SkM* and SLy4 functionals, the staggering of the $T=\frac 12$ MDEs is less pronounced [21]. We also note that all three functionals correctly describe the $A$-dependence and lack of staggering of the $T=1$ MDEs.

It is even more gratifying to see in Fig. 3 that our $pn$-mixed calculations, with the class-II coupling constant, $t^\mathrm {II}_0$, describe absolute values as well as staggering of TDEs very well, whereas results obtained without ISB terms give too small values and show no staggering. Good agreement obtained for the MDEs and TDEs shows that the role and magnitude of the ISB terms are now firmly established.

Figure 2: (Color online) Calculated and experimental values of MDEs for the $T=\frac 12$ (a) and $T=1$ (b) mirror nuclei, shown with respect to the average linear trend defined in Fig. 1. Calculations were performed for functional SV$_{\rm T}$ with the ISB terms added. Shaded bands show theoretical uncertainties. Experimental error bars are shown only when they are larger than the corresponding symbols. Full (open) symbols denote data points included in (excluded from) the fitting procedure.

Figure 3: (Color online) Same as in Fig. 2 but for the $T=1$ TDEs with no linear trend subtracted.

It is very instructive to look at ten outliers which were excluded from the fitting procedure. They are shown by open symbols in Figs. 2 and 3. (i) There are five outliers that depend on masses of $^{52}$Co, $^{56}$Cu, and $^{73}$Rb, which clearly deviate from the calculated trends for MDEs and TDEs. These masses were not directly measured but derived from systematics [24]. (ii) There are two outliers that depend on the mass of $^{44}$V, whose ground-state measurement may be contaminated by an unresolved isomer [33,34,35]. (iii) Large differences between experimental and calculated values are found in MDE for $A=16$, 67 and 69. Inclusion of these data in the fitting procedure would significantly increase the uncertainty of adjusted coupling constants. The former two, (i) and (ii), call for improving experimental values, whereas the last one (iii) may be a result of structural effects not included in our model.

Having at hand a model with ISB strong interactions with fitted parameters we can calculate MDEs for more massive multiplets and make predictions on binding energies of neutron-deficient ($T_z = -T$) nuclei. In particular, in Table 2 we present predictions of mass excesses of $^{52}$Co, $^{56}$Cu, and $^{73}$Rb, whose masses were in AME12 [24] derived from systematics, and $^{44}$V, whose ground-state mass measurement is uncertain. Recently, the mass excess of $^{52}$Co was measured as $-$34361(8) [36] or $-$34331.6(66) keV [37]. These values are in fair agreement with our prediction (1.8 or 2.4$\sigma$ difference with respect to our estimated theoretical uncertainty), even though the difference between them is still far beyond the estimated (much smaller) experimental uncertainties.

Table 2: Mass excesses of $^{52}$Co, $^{56}$Cu, $^{73}$Rb, and $^{44}$V obtained in this work and compared with those of AME12 [24]. Our predictions were calculated as weighted averages of values obtained from MDEs and TDEs for all three used Skyrme parametrizations. The AME12 values derived from systematics are labelled with symbol #.

		Mass excess (keV)
Nucleus		This work		AME12 [24]
$^{52}$Co		$-$34450(50)		$-$33990(200)#
$^{56}$Cu		$-$38720(50)		$-$38240(200)#
$^{73}$Rb		$-$46100(80)		$-$46080(100)#
$^{44}$V		$-$23770(50)		$-$24120(180)

Figure 4: (Color online) Coupling constants $t^\mathrm {II}_0$ (a) and $t^\mathrm {III}_0$ (b), together with their average values and uncertainties. Panel (c) shows ratios of coupling constants $t^\mathrm {II}_0$/ $t^\mathrm {III}_0$ compared with the value of Eq. (7).

Assuming that the extracted CSB and CIB effects are, predominantly, due to the ISB in the $^1S_0$ channel we can relate ratio $t^\mathrm {II}_0$/ $t^\mathrm {III}_0$ to the experimental scattering lengths. The reasoning follows the work of Suzuki et al. [10], which assumed a proportionality between the strengths of CSB and CIB forces and the corresponding scattering lengths [38], that is, $V_{CSB} \propto \Delta a_{CSB}=a_{pp}-a_{nn}$ and $V_{CIB} \propto \Delta a_{CIB}=\frac12(a_{pp}+a_{nn})-a_{np}$, which, in our case, is equivalent to $t^\mathrm{III}_0\propto-\frac12 \Delta a_{CSB}$ and $t^\mathrm{II}_0\propto \frac13 \Delta a_{CIB}$. Assuming further that the proportionality constant is the same, and taking for the experimental values $\Delta a_{CSB}=1.5\pm0.3\mathrm{~fm}$ and $\Delta a_{CIB}=5.7\pm0.3\mathrm{~fm}$ [38], one gets:

$$\frac{t_0^{\rm {II}}}{t_0^{\rm {III}}}=-\frac23\frac{\Delta a_{CIB}}{\Delta a_{CSB}} = -2.5 \pm 0.5.$$ (8)

From the values of coupling constants given in Table 1, we obtain their ratios as $t^\mathrm {II}_0$/ $t^\mathrm {III}_0$ = $-2.3\pm0.9$, $-4.5\pm1.8$, and $-4.1\pm1.5$ for the SV$_{\rm T}$, SkM*, and SLy4 EDFs, respectively. Fig. 4 summarizes these values in comparison with the estimate in Eq. (7). As we can see, the values determined by our analysis of masses of $N\simeq Z$ nuclei with $10\leq A\leq 75$ agree very well with estimates based on properties of the $NN$ forces deduced from the $NN$ scattering experiments.

In summary, we showed that the $pn$-mixed DFT with added two new terms related to the ISB interactions of class II and III is able to systematically reproduce observed MDEs and TDEs of $T=\frac 12$ and $T=1$ multiplets. Adjusting only two coupling constants $t^\mathrm {II}_0$ and $t^\mathrm {III}_0$, we reproduced not only the magnitudes of the MDE and TDE but also their characteristic staggering patterns. The obtained values of $t^\mathrm {II}_0$ and $t^\mathrm {III}_0$ turn out to agree with the $NN$ ISB interactions ($NN$ scattering lengths) in the $^1S_0$ channel. We predicted mass excesses of $^{52}$Co, $^{56}$Cu, $^{73}$Rb, and $^{44}$V, and for $^{52}$Co we obtained fair agreement with the recently measured values [36,37]. To better pin down the ISB effects, accurate mass measurements of the other three nuclei are very much called for.

This work was supported in part by the Polish National Science Center under Contract Nos. 2014/15/N/ST2/03454 and 2015/17/N/ST2/04025, by the Academy of Finland and University of Jyväskylä within the FIDIPRO program, by Interdisciplinary Computational Science Program in CCS, University of Tsukuba, and by ImPACT Program of Council for Science, Technology and Innovation (Cabinet Office, Government of Japan). We acknowledge the CIS Swierk Computing Center, Poland, and the CSC-IT Center for Science Ltd., Finland, for the allocation of computational resources.