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The survey of ISB corrections in $ 10\leq A \leq 74$ nuclei


Table 2: Results of calculations for the superallowed transitions measured experimentally. Shown are: the empirical $ ft$-values [5]; SV values of $ \delta_{\rm C}$ calculated by projecting from the $ \vert\varphi_{\rm
X}\rangle,\vert\varphi_{\rm Y}\rangle$, and $ \vert\varphi_{\rm Z}\rangle$ Slater determinants, see Sec. 3.1; recommended mean $ \delta_{\rm C}^{{\rm (SV)}}$ corrections (see Sec. 3.3) and the corresponding $ {\cal F}t$-values; empirical $ \delta_{\rm C}^{{\rm (exp)}}$ corrections calculated by using Eq. (23); contributions coming from the individual transitions to the $ \chi^2$ budget in the confidence-level test; mean $ \delta_{\rm C}^{{\rm (SHZ2)}}$ corrections and the corresponding $ {\cal F}t$-values.
Parent $ ft$ $ \quad$ $ \delta_{\rm C}^{\rm (X)}$ $ \delta_{\rm C}^{\rm (Y)}$ $ \delta_{\rm C}^{\rm (Z)}$ $ \quad$ $ \delta_{\rm C}^{{\rm (SV)}}~\strut$ $ {\cal F}t~~~~~\strut$ $ \quad$ $ \delta_{\rm C}^{{\rm (exp)}}~\strut$ $ \chi^2_i$ $ \quad$ $ \delta_{\rm C}^{{\rm (SHZ2)}}~~\strut$ $ {\cal F}t~~~~~\strut$
nucleus (s)   (%) (%) (%)   (%)   (s)        (%)       (%)     (s)     
$ T_z=-1:$                            
$ ^{10}$C 3041.7(43)   0.559 0.559 0.823   0.65(14) 3062.1(62)   0.37(15) 3.7   0.462(65) 3067.8(49)
$ ^{14}$O 3042.3(11)   0.303 0.303 0.303   0.303(30) 3072.3(21)   0.36(06) 0.8   0.480(48) 3066.9(24)
$ ^{22}$Mg 3052.0(70)   0.243 0.243 0.417   0.301(87) 3080.5(75)   0.62(23) 1.9   0.342(49) 3079.2(72)
$ ^{34}$Ar 3052.7(82)   0.865 0.997 1.475   1.11(29) 3056(12)   0.63(27) 3.1   1.08(42) 3057(15)
$ T_z=0: $                            
$ ^{26}$Al 3036.9(09)   0.308 0.308 0.494   0.370(95) 3070.5(31)   0.37(04) 0.0   0.307(62) 3072.5(23)
$ ^{34}$Cl 3049.4(11)   0.809 0.679 1.504   1.00(38) 3060(12)   0.65(05) 48.4   0.83(50) 3065(15)
$ ^{42}$Sc 3047.6(12)   -- -- --   0.77(27) 3069.2(85)   0.72(06) 0.5   0.70(32) 3071(10)
$ ^{46}$V 3049.5(08)   0.486 0.486 0.759   0.58(14) 3074.6(47)   0.71(06) 4.5   0.375(96) 3080.9(35)
$ ^{50}$Mn 3048.4(07)   0.460 0.460 0.740   0.55(14) 3074.1(47)   0.67(07) 3.1   0.39(13) 3079.2(45)
$ ^{54}$Co 3050.8(10)   0.622 0.622 0.671   0.638(68) 3074.0(32)   0.75(08) 2.0   0.51(20) 3078.0(66)
$ ^{62}$Ga 3074.1(11)   0.925 0.840 0.881   0.882(95) 3090.0(42)   1.51(09) 44.0   0.49(11) 3102.3(45)
$ ^{74}$Rb 3084.9(77)   2.054 1.995 1.273   1.77(40) 3073(15)   1.86(27) 0.1   0.90(22) 3101(11)
              $ \overline{{\cal F}t}=$ 3073.6(12)   $ \chi^2 =$ 112.2   $ \overline{{\cal F}t}=$ 3075.0(12)
              $ \vert V_{\rm ud}\vert=$ 0.97397(27)   $ \chi_d^2 =$ 10.2   $ \vert V_{\rm ud}\vert=$ 0.97374(27)
                0.99935(67)           0.99890(67)


Table 3: Similar as in Table 2, except for the unmeasured transitions.
Parent $ \delta_{\rm C}^{\rm (X)}$ $ \delta_{\rm C}^{\rm (Y)}$ $ \delta_{\rm C}^{\rm (Z)}$ $ \quad$ $ \delta_{\rm C}^{\rm (SV)}$ $ \quad$ $ \delta_{\rm C}^{\rm (SHZ2)}$
nucleus (%) (%) (%)   (%)   (%)
$ T_z=-1:$              
$ ^{18}$Ne 2.031 1.064 1.142   1.41(46)   0.72(30)
$ ^{26}$Si 0.399 0.399 0.597   0.47(10)   0.529(77)
$ ^{30}$S 1.731 1.260 1.272   1.42(26)   0.98(21)
$ T_z=0: $              
$ ^{18}$F 1.819 0.956 0.987   1.25(42)   0.42(24)
$ ^{22}$Na 0.255 0.255 0.535   0.35(14)   0.216(86)
$ ^{30}$P 1.506 0.974 1.009   1.16(27)   0.60(20)
$ ^{66}$As 0.956 0.925 1.694   1.19(38)   0.64(12)
$ ^{70}$Br 1.654 1.479 1.429   1.52(18)   1.10(52)

The results of our calculations are collected in Tables 2 and 3, and in Fig. 6. In addition, Fig. 7 shows the differences, $ \delta_{\rm C}^{({\rm SV})}-
\delta_{\rm C}^{({\rm HT})}$, between our results and those of Ref. [3]. In spite of clear differences between SV and HT, which can be seen for specific transitions including those for $ A=10$, 34, and 62, both calculations reveal a similar increase of $ \delta_{\rm C}$ versus $ A$, at variance with the RPA calculations of Ref. [13], which also yield systematically smaller values.

Figure 6: (Color online) ISB corrections to the superallowed $ 0^+ \rightarrow 0^+$ $ \beta$-decays calculated for (a) $ T_z= -1 \rightarrow T_z =0 $ and (b) $ T_z= 0 \rightarrow
T_z =1 $ transitions. Our adopted values from Table 2 (circles with error bars) are compared with ISB corrections from Refs. [3] (dots, shaded band marks errors) and [13] (triangles).
\includegraphics[angle=0,width=0.8\columnwidth]{deltaC.fig06.eps}

Figure 7: Differences between the ISB corrections to the twelve accurately measured superallowed $ 0^+ \rightarrow 0^+$ $ \beta$-transitions (excluding $ A$=38) calculated in this work with SV and those of HT [3]. Circles and dots mark the $ T_z= -1 \rightarrow T_z =0 $ and $ T_z= 0 \rightarrow
T_z =1 $ decays, respectively. The errors, calculated as $ \sqrt{(\Delta \delta_{\rm C}^{\rm (SV)})^2 + (\Delta \delta_{\rm
C}^{\rm (HT)})^2}$, are shown by shaded bands.
\includegraphics[angle=0,width=0.7\columnwidth,clip]{deltaC.fig07.eps}

The ISB corrections used for further calculations of $ V_{\rm ud}$ are collected in Table 2. Let us recall that our preference is to use the averaged corrections and that the $ ^{38}$K $ \rightarrow$$ ^{38}$Ar transition has been disregarded. All other ingredients needed to compute the $ {\cal F}t$-values, including radiative corrections $ \delta_{\rm R}^\prime$ and $ \delta_{\rm NS}$, are taken from Ref. [3], and the empirical $ ft$-values are taken from Ref. [5]. For the sake of completeness, these empirical $ ft$-values are also listed in Table 2.

Figure 8: The matrix element $ \vert V_{\rm ud}\vert$ deduced from the superallowed $ 0^+ \rightarrow 0^+$ $ \beta$-decay (dots) for different sets of the $ \delta_{\rm C}$ corrections calculated in: (a) Ref. [3]; (b) Ref. [13] with NL3 and DD-ME2 Lagrangians; this work, using the averaged values of $ \delta_{\rm C}$ with (c) SV and (d) SHZ2 functionals. Gray circles show ISB corrections with SV calculated at fixed shape-current orientations $ X$, $ Y$, and $ Z$ (from left to right). Triangles mark values obtained from the pion-decay [39] and neutron-decay [10] studies. The open circle shows $ \vert V_{\rm ud}\vert$ deduced from the $ \beta$-decays in the $ T=1/2$ mirror nuclei [40]
\includegraphics[width=0.9\columnwidth]{deltaC.fig08.eps} .

Figure 9: Similar as in Fig. 8 except for the unitarity condition (20).
\includegraphics[angle=0,width=0.9\columnwidth]{deltaC.fig09.eps}

In the error budget of the resulting $ {\cal F}t$-values listed in Table 2, apart from errors in the $ ft$ values and radiative corrections, we also included the uncertainties estimated for the calculated values of $ \delta_{\rm C}$, see Sec. 3.3. To conform with HT, the average value $ \overline{{\cal F}t} = 3073.6(12)$s was calculated by using the Gaussian-distribution-weighted formula. However, unlike HT, we do not apply any further corrections to $ \overline{{\cal F}t}$. This leads to $ \vert V_{\rm ud}\vert = 0.97397(27)$, which agrees very well with both the HT result [3], $ \vert V_{\rm ud}^{{\rm (HT)}}\vert = 0.97418(26)$, and the central value obtained from the neutron decay $ \vert V_{\rm ud}^{(\nu )}\vert = 0.9746(19)$ [10]. A survey of the $ \vert V_{\rm ud}\vert$ values deduced by using different methods is given in Fig. 8. By combining the value of $ \vert V_{\rm ud}\vert$ calculated here with those of $ \vert V_{\rm us}\vert = 0.2252(9)$ and $ \vert V_{\rm ub}\vert = 0.00389(44)$ of the 2010 Particle Data Group [10], one obtains

$\displaystyle \vert V_{\rm ud}\vert^2 + \vert V_{\rm us}\vert^2 + \vert V_{\rm ub}\vert^2 = 0.99935(67),$ (20)

which implies that the unitarity of the first row of the CKM matrix is satisfied with a precision better than 0.1%. A survey of the unitarity condition (20) is shown in Fig. 9.

It is worth noting that by using $ \delta_{\rm C}$ values corresponding to the fixed current-shape orientations ( $ \vert\varphi_{\rm X}\rangle$, $ \vert\varphi_{\rm Y}\rangle$, or $ \vert\varphi_{\rm Z}\rangle$) instead of their average, one still obtains compatible results for $ \vert V_{\rm ud}\vert$ and unitarity condition (20), see Figs. 8 and 9. Moreover, the value of $ \vert V_{\rm ud}\vert$ obtained by using SHZ2 is only $ \approx$0.024% smaller than the SV result, see Table 2. This is an intriguing result, which indicates that an increase of the bulk symmetry energy - that tends to restore the isospin symmetry - is partly compensated by other effects. The most likely origin of this compensation mechanism is due to the time-odd spin-isospin mean fields, which are poorly constrained by the standard fitting protocols of Skyrme EDFs [41,42,43]. For instance, if one compares the Landau-Migdal parameters characterizing the spin-isospin time-odd channels [41,42,43] of SV ( $ g_0 = 0.57$, $ g_0^\prime = 0.31$, $ g_1 = 0.46$, $ g_1^\prime = 0.46$) and SHZ2 ( $ g_0 = 0.27$, $ g_0^\prime = 0.30$, $ g_1 = 0.47$, $ g_1^\prime = 0.47$) one notices that these two functionals differ by a factor of two in the scalar-isoscalar Landau-Migdal parameter $ g_0$.

Figure 10: ISB corrections for $ ^{14}$O $ \rightarrow ^{14}$N and $ ^{14}$N $ \rightarrow ^{14}$C superallowed $ 0^+ \rightarrow 0^+$ $ \beta$-decays calculated for a set of SV-based Skyrme forces with systematically varied $ x_0$ parameter, which affects the bulk asymmetry energy coefficient $ a_{sym}$ and spin-spin fields. The arrows indicate $ x_0$ values corresponding to SV and SHZ2.
\includegraphics[width=0.8\columnwidth]{deltaC.fig10.eps}
To illustrate the compensation mechanism related to the bulk symmetry energy and $ g_0$, in Fig. 10 we plot $ \delta_{\rm C}$ for the $ ^{14}$O $ \rightarrow ^{14}$N $ \rightarrow ^{14}$C super-allowed $ 0^+ \rightarrow 0^+$ transitions as functions of the bulk symmetry parameter $ a_{sym}$ for a set of SV-based Skyrme forces with systematically varied $ x_0$ parameter. At a functional level, $ x_0$ affects only two Skyrme coupling constants (see e.g. Appendix A in Ref. [20]):
$\displaystyle C_1^\rho$ $\displaystyle =$ $\displaystyle -\frac{1}{4} t_0 \left( \frac{1}{2} + x_0 \right) - \frac{1}{24} t_3 \left( \frac{1}{2} + x_3 \right)\rho_0^\alpha ,$ (21)
$\displaystyle C_0^s$ $\displaystyle =$ $\displaystyle -\frac{1}{4} t_0 \left( \frac{1}{2} - x_0 \right) - \frac{1}{24} t_3 \left( \frac{1}{2} - x_3 \right) \rho_0^\alpha.$ (22)

The coupling constant $ C_1^\rho$ influences the isovector part of the bulk symmetry energy [44] while $ C_0^s$ affects $ g_0$. The ISB correction in Fig. 10 exhibits a minimum indicating the presence of the compensation effect. Similar effect was calculated for the $ A=34$ transitions. Hence, it is safe to state that our exploratory calculations are indicative of the interplay between the symmetry energy and time-odd fields.


next up previous
Next: Confidence level test Up: ISB corrections to the Previous: Theoretical uncertainties and error
Jacek Dobaczewski 2012-10-19