Multiple $SU(3)$ algebras in shell model and \\ interacting boson model

TL;DR

This paper explores multiple $SU(3)$ algebras in nuclear shell and interacting boson models, revealing different shapes and quadrupole properties, supported by analytical and numerical results across models.

Contribution

It demonstrates the existence of multiple $SU(3)$ algebras in shell and boson models and analyzes their structural and shape implications using various theoretical approaches.

Findings

Different $SU(3)$ algebras generate prolate and oblate shapes.

One $SU(3)$ algebra produces small quadrupole moments.

Results are supported by shell model and interacting boson model analyses.

Abstract

Rotational $SU(3)$ algebraic symmetry continues to generate new results in the shell model (SM). Interestingly, it is possible to have multiple $SU(3)$ algebras for nucleons occupying an oscillator shell $\eta$. Several different aspects of the multiple $SU(3)$ algebras are investigated using shell model and also deformed shell model based on Hartree-Fock single particle states with nucleons in $sdg$ orbits giving four $SU(3)$ algebras. Results show that one of the $SU(3)$ algebra generates prolate shapes, one oblate shape and the other two also generate prolate shape but one of them gives quiet small quadrupole moments for low-lying levels. These are inferred by using the standard form for the electric quadrupole transition operator and using quadrupole moments and $B(E2)$ values in the ground $K=0^+$ band in three different examples. Multiple $SU(3)$ algebras extend to interacting boson model and using $sdg$IBM, the structure of the four $SU(3)$ algebras in this model are studied by coherent state analysis and asymptotic formulas for $E2$ matrix elements. The results from $sdg$IBM further support the conclusions from the $sdg$ shell model examples.

Tables

Table 1. Table 1: Correlation coefficient ζ 𝜁 \zeta between Q ⋅ Q ⋅ 𝑄 𝑄 Q\cdot Q operators with different values for the phases ( α s ​ d , α d ​ g ) subscript 𝛼 𝑠 𝑑 subscript 𝛼 𝑑 𝑔 (\alpha_{sd},\alpha_{dg}) in s ​ d ​ g 𝑠 𝑑 𝑔 sdg shell model m 𝑚 m -particle spaces ( m 𝑚 m is number of nucleons). Note that the total number of single particle states (with spin and isospin) is 60. The ζ 𝜁 \zeta values in column 3 are for m = 4 𝑚 4 m=4 , 8, 12, 20, 30, 40, 50 and 56. See text for other details.

$({\alpha }_{sd},{\alpha }_{dg})$	$({\alpha }_{sd}^{\prime },{\alpha }_{dg}^{\prime })$	$\zeta$
$(-,-)$	$(+,-)$	$0.39,0.36,0.35,0.35,0.35,0.35,0.36,0.39$
	$(-,+)$	$0.14,0.1,0.08,0.08,0.08,0.08,0.09,0.14$
	$(+,+)$	$0.07,0.02,0.01,0.0,0.0,0.0,0.01,0.07$
$(+,-)$	$(-,+)$	$0.07,0.02,0.01,0.0,0.0,0.0,0.01,0.07$
	$(+,+)$	$0.14,0.1,0.08,0.08,0.08,0.08,0.09,0.14$
$(-,+)$	$(+,+)$	$0.39,0.36,0.35,0.35,0.35,0.35,0.36,0.39$

Table 2. Table 2: Ground state or leading S ​ U ​ ( 3 ) 𝑆 𝑈 3 SU(3) irrep ( λ H , μ H ) subscript 𝜆 𝐻 subscript 𝜇 𝐻 (\lambda_{H},\mu_{H}) for a given number m 𝑚 m of identical nucleons and also for a given number m 𝑚 m of nucleons with isospin T = | T Z | 𝑇 subscript 𝑇 𝑍 T=|T_{Z}| . Results are given for the oscillator shell η = 4 𝜂 4 \eta=4 . The ( λ H , μ H ) subscript 𝜆 𝐻 subscript 𝜇 𝐻 (\lambda_{H},\mu_{H}) are given in the table as ( λ H , μ H ) m superscript subscript 𝜆 𝐻 subscript 𝜇 𝐻 𝑚 (\lambda_{H},\mu_{H})^{m} for identical nucleons with m ≥ 2 𝑚 2 m\geq 2 and ( λ H , μ H ) m , T superscript subscript 𝜆 𝐻 subscript 𝜇 𝐻 𝑚 𝑇 (\lambda_{H},\mu_{H})^{m,T} for nucleons with T = | T z | 𝑇 subscript 𝑇 𝑧 T=|T_{z}| and 3 ≤ m ≤ 15 3 𝑚 15 3\leq m\leq 15 ; for odd m 𝑚 m values, 2 ​ T 2 𝑇 2T value given instead of T 𝑇 T value. More complete results are available in Kota-hw .

$\eta =4$: identical nucleons

${(8,0)}^2$,${(10,1)}^3$,${(12,2)}^4$,${(15,1)}^5$,${(18,0)}^6$,${(18,2)}^7$, ${(18,4)}^8$,${(19,4)}^9$,${(20,4)}^{10}$,${(22,2)}^{11}$,${(24,0)}^{12}$,

${(22,3)}^{13}$, ${(20,6)}^{14}$,${(19,7)}^{15}$,${(18,8)}^{16}$,${(18,7)}^{17}$,${(18,6)}^{18}$, ${(19,3)}^{19}$,${(20,0)}^{20}$,${(16,4)}^{21}$,${(12,8)}^{22}$,

${(9,10)}^{23}$,${(6,12)}^{24}$, ${(4,12)}^{25}$,${(2,12)}^{26}$,${(1,10)}^{27}$,${(0,8)}^{28}$,${(0,4)}^{29}$, ${(0,0)}^{30}$

$\eta =4$: even number of nucleons

${(16,0)}^{4,0}$,${(14,1)}^{4,1}$,${(12,2)}^{4,2}$,${(20,2)}^{6,0}$, ${(20,2)}^{6,1}$,${(19,1)}^{6,2}$, ${(18,0)}^{6,3}$,${(24,4)}^{8,0}$,${(25,2)}^{8,1}$,

${(26,0)}^{8,2}$,${(22,2)}^{8,3}$,${(18,4)}^{8,4}$, ${(30,2)}^{10,0}$,${(30,2)}^{10,1}$,${(28,3)}^{10,2}$,${(26,4)}^{10,3}$, ${(23,4)}^{10,4}$,${(20,4)}^{10,5}$,

${(36,0)}^{12,0}$,${(33,3)}^{12,1}$,${(30,6)}^{12,2}$,${(29,5)}^{12,3}$, ${(28,4)}^{12,4}$,${(26,2)}^{12,5}$, ${(24,0)}^{12,6}$,${(36,4)}^{14,0}$,${(36,4)}^{14,1}$,

${(34,5)}^{14,2}$,${(32,6)}^{14,3}$,${(32,3)}^{14,4}$, ${(32,0)}^{14,5}$,${(26,3)}^{14,6}$,${(20,6)}^{14,7}$

$\eta =4$: odd number of nucleons

${(12,0)}^{3,1}$, ${(10,1)}^{3,3}$, ${(18,1)}^{5,1}$, ${(16,2)}^{5,3}$, ${(15,1)}^{5,5}$, ${(22,3)}^{7,1}$, ${(23,1)}^{7,3}$, ${(22,0)}^{7,5}$,

${(18,2)}^{7,7}$, ${(27,3)}^{9,1}$, ${(28,1)}^{9,3}$, ${(26,2)}^{9,5}$, ${(22,4)}^{9,7}$, ${(19,4)}^{9,9}$, ${(33,1)}^{11,1}$, ${(30,4)}^{11,3}$,

${(28,5)}^{11,5}$, ${(27,4)}^{11,7}$, ${(24,4)}^{11,9}$, ${(22,2)}^{11,11}$, ${(36,2)}^{13,1}$, ${(33,5)}^{13,3}$, ${(31,6)}^{13,5}$, ${(30,5)}^{13,7}$,

${(30,2)}^{13,9}$, ${(28,0)}^{13,11}$, ${(22,3)}^{13,13}$, ${(36,6)}^{15,1}$, ${(37,4)}^{15,3}$, ${(35,5)}^{15,5}$, ${(34,4)}^{15,7}$, ${(34,1)}^{15,9}$,

${(30,3)}^{15,11}$, ${(24,6)}^{15,13}$, ${(19,7)}^{15,15}$

Table 3. Table 3: Shell model results for quadrupole moments Q ​ ( J ) 𝑄 𝐽 Q(J) and B ​ ( E ​ 2 ; J → J − 2 ) 𝐵 → 𝐸 2 𝐽 𝐽 2 B(E2;J\rightarrow J-2) values for the ground K = 0 + 𝐾 superscript 0 K=0^{+} band members for a system of 6 protons in η = 4 𝜂 4 \eta=4 shell. Results are given for the four Hamiltonians in Eq. ( 16 ). In the table ( − , − ) (-,-) means we are using the wavefunctions obtained using H Q ( − , − ) superscript subscript 𝐻 𝑄 H_{Q}^{(-,-)} and similarly others. For other details see text.

$J$	$Q\left(J\right)ef{m}^2$
	$(-,-)$	$(+,-)$	$(-,+)$	$(+,+)$
$2_1^{+}$	$-49.18$	$-33.90$	$-1.85$	$13.44$
$4_1^{+}$	$-62.59$	$-40.16$	$-4.71$	$17.72$
$6_1^{+}$	$-68.85$	$-39.78$	$-9.12$	$19.97$
$8_1^{+}$	$-72.48$	$-37.06$	$-14.96$	$20.46$
${10}_1^{+}$	$-74.84$	$-34.57$	$-21.99$	$18.28$
$J$	$B\left(E2;J\to J-2\right)e^{2}f{m}^4$
	$(-,-)$	$(+,-)$	$(-,+)$	$(+,+)$
$2_1^{+}$	$585.97$	$291.34$	$0.42$	$42.20$
$4_1^{+}$	$815.31$	$388.79$	$1.38$	$58.58$
$6_1^{+}$	$853.68$	$377.33$	$3.90$	$61.14$
$8_1^{+}$	$827.24$	$325.68$	$9.24$	$58.93$
${10}_1^{+}$	$760.52$	$254.56$	$18.38$	$53.81$

Table 4. Table 4: Shell model results for quadrupole moments Q ​ ( J ) 𝑄 𝐽 Q(J) and B ​ ( E ​ 2 ; J → J − 2 ) 𝐵 → 𝐸 2 𝐽 𝐽 2 B(E2;J\rightarrow J-2) values for the ground K = 0 + 𝐾 superscript 0 K=0^{+} band members for a system of 6 protons and 2 neutrons in η = 4 𝜂 4 \eta=4 shell. Results are given for the four Hamiltonians in Eq. ( 16 ). In the table ( − , − ) (-,-) means we are using the wavefunctions obtained using H Q ( − , − ) superscript subscript 𝐻 𝑄 H_{Q}^{(-,-)} and similarly others. For other details see text.

$J$	$Q\left(J\right)ef{m}^2$
	$(-,-)$	$(+,-)$	$(-,+)$	$(+,+)$
$2_1^{+}$	$-83.34$	$-50.54$	$-8.84$	$23.96$
$4_1^{+}$	$-106.07$	$-62.96$	$-12.96$	$30.15$
$6_1^{+}$	$-116.68$	$-67.10$	$-17.17$	$32.42$
$8_1^{+}$	$-122.82$	$-67.95$	$-22.17$	$32.70$
${10}_1^{+}$	$-126.82$	$-67.38$	$-28.13$	$32.32$
$J$	$B\left(E2;J\to J-2\right)e^{2}f{m}^4$
	$(-,-)$	$(+,-)$	$(-,+)$	$(+,+)$
$2_1^{+}$	$1687.70$	$629.26$	$17.14$	$140.57$
$4_1^{+}$	$2379.04$	$876.18$	$25.78$	$198.76$
$6_1^{+}$	$2556.85$	$920.91$	$30.98$	$214.70$
$8_1^{+}$	$2580.68$	$899.62$	$36.37$	$218.34$
${10}_1^{+}$	$2521.87$	$841.70$	$42.61$	$215.48$

Table 5. Table 5: Deformed shell model results for quadrupole moments Q ​ ( J ) 𝑄 𝐽 Q(J) and B ​ ( E ​ 2 ; J → J − 2 ) 𝐵 → 𝐸 2 𝐽 𝐽 2 B(E2;J\rightarrow J-2) values for the ground K = 0 + 𝐾 superscript 0 K=0^{+} band members for a system of 6 protons and 6 neutrons (with T = 0 𝑇 0 T=0 ) in η = 4 𝜂 4 \eta=4 shell. Results are given for the four Hamiltonians in Eq. ( 16 ). In the table ( − , − ) (-,-) means we are using the wavefunctions obtained using H Q ( − , − ) superscript subscript 𝐻 𝑄 H_{Q}^{(-,-)} and similarly others. Numbers in the brackets in the second column are exact S ​ U ​ ( 3 ) 𝑆 𝑈 3 SU(3) results for H Q ( − , − ) superscript subscript 𝐻 𝑄 H_{Q}^{(-,-)} . For other details see text.

$J$	$Q\left(J\right)ef{m}^2$
	$(-,-)$	$(+,-)$	$(-,+)$	$(+,+)$
$2_1^{+}$	$-96.67\left(-95.31\right)$	$-65.95$	$-4.63$	$26.1$
$4_1^{+}$	$-123.04\left(-123.12\right)$	$-82.61$	$-7.02$	$33.41$
$6_1^{+}$	$-135.34\left(-135.43\right)$	$-88.65$	$-9.66$	$37.03$
$8_1^{+}$	$-142.45\left(-142.58\right)$	$-90.30$	$-12.91$	$39.24$
${10}_1^{+}$	$-147.09\left(-147.21\right)$	$-89.60$	$-16.86$	$40.63$
$J$	$B\left(E2;J\to J-2\right)e^{2}f{m}^4$
	$(-,-)$	$(+,-)$	$(-,+)$	$(+,+)$
$2_1^{+}$	$2273.96\left(2276.93\right)$	$1069.13$	$4.61$	$164.89$
$4_1^{+}$	$3225.34\left(3229.59\right)$	$1503.04$	$7.43$	$233.99$
$6_1^{+}$	$3506.56\left(3511.13\right)$	$1608$	$9.98$	$254.61$
$8_1^{+}$	$3601.29\left(3605.99\right)$	$1613.12$	$13.44$	$261.78$
${10}_1^{+}$	$3605.81\left(3610.51\right)$	$1565.64$	$18.32$	$262.44$

Table 6. Table 6: Equilibrium shapes for the four S ​ U ​ ( 3 ) 𝑆 𝑈 3 SU(3) algebras in s ​ d ​ g 𝑠 𝑑 𝑔 sdg IBM. For ( α s ​ d , α d ​ g ) = ( − 1 , + 1 ) subscript 𝛼 𝑠 𝑑 subscript 𝛼 𝑑 𝑔 1 1 (\alpha_{sd},\alpha_{dg})=(-1,+1) and ( − 1 , − 1 ) 1 1 (-1,-1) , shown are the β 2 0 superscript subscript 𝛽 2 0 \beta_{2}^{0} and β 4 0 superscript subscript 𝛽 4 0 \beta_{4}^{0} values for both γ 0 = 0 ∘ superscript 𝛾 0 superscript 0 \gamma^{0}=0^{\circ} and 60 ∘ superscript 60 60^{\circ} and they are equivalent.

${\alpha }_{sd}$	${\alpha }_{dg}$	${\beta }_2^0$	${\beta }_4^0$	${\gamma }^0$	$E_{S{U}_{sdg}\left(3\right)}^0$
+1	+1	$\sqrt{20/7}$	$\sqrt{8/7}$	$0^{\circ }$	$-16N^2$
+1	-1	$\sqrt{20/7}$	$-\sqrt{8/7}$	$0^{\circ }$	$-16N^2$
-1	+1	$\sqrt{20/7}$	$-\sqrt{8/7}$	${60}^{\circ }$	$-16N^2$
		$-\sqrt{20/7}$	$-\sqrt{8/7}$	$0^{\circ }$	$-16N^2$
-1	-1	$\sqrt{20/7}$	$\sqrt{8/7}$	${60}^{\circ }$	$-16N^2$
		$-\sqrt{20/7}$	$\sqrt{8/7}$	$0^{\circ }$	$-16N^2$

Table 7. Table 7: Quadrupole moments and B ​ ( E ​ 2 ) 𝐵 𝐸 2 B(E2) values for low-lying states in the ground band for a 10 boson system generated by the four S ​ U ​ ( 3 ) 𝑆 𝑈 3 SU(3) algebras in s ​ d ​ g 𝑠 𝑑 𝑔 sdg IBM. Note that T E ​ 2 superscript 𝑇 𝐸 2 T^{E2} Eq. ( 33 ) is unit-less and therefore Q ​ ( L ) 𝑄 𝐿 Q(L) and B ​ ( E ​ 2 ) 𝐵 𝐸 2 B(E2) ’s in the table are unit-less.

${\alpha }_{sd}$	${\alpha }_{dg}$	$Q\left(2_1^{+}\right)$	$Q\left(4_1^{+}\right)$	$B\left(E2;2_1^{+}\to 0_1^{+}\right)$	$B\left(E2;4_1^{+}\to 2_1^{+}\right)$
+1	+1	$-13.69$	$-17.43$	$45.68$	$64.87$
+1	-1	$-6.15$	$-7.89$	$9.15$	$12.61$
-1	+1	$-2.16$	$-2.98$	$1.05$	$1.59$
-1	-1	$5.38$	$6.55$	$7.33$	$10.51$

Equations

$$L_q^1=2\sum _{\ell }\sqrt{\frac{\ell (\ell +1)(2\ell +1)}{3}}{\left(a_{\ell \frac{1}{2}\frac{1}{2}}^{†}{\tilde{a}}_{\ell \frac{1}{2}\frac{1}{2}}\right)}_q^{1,0,0}.$$

$$Q_q^2=2\sum _{{\ell }_f,{\ell }_i}\frac{⟨\eta ,{\ell }_f\mid \mid Q^2\mid \mid \eta ,{\ell }_i⟩}{\sqrt{5}}{\left(a_{{\ell }_f\frac{1}{2}\frac{1}{2}}^{†}{\tilde{a}}_{{\ell }_i\frac{1}{2}\frac{1}{2}}\right)}^{2,0,0}.$$

$$⟨\eta ,{\ell }_f\mid \mid Q^2\mid \mid \eta ,{\ell }_i⟩=⟨\eta ,{\ell }_f\mid \mid \sqrt{\frac{16\pi }{5}}{Y}^2(\theta ,\varphi )\mid \mid \eta ,{\ell }_i⟩⟨\eta ,{\ell }_f\mid \mid r^2\mid \mid \eta ,{\ell }_i⟩,$$

$$\begin{array}[]{l}\left\langle\eta,\ell\mid\mid\displaystyle\sqrt{\frac{16\pi}{5}}Y^{2}(\theta,\phi)\mid\mid\eta,\ell\right\rangle=-2\displaystyle\sqrt{\displaystyle\frac{\ell(\ell+1)(2\ell+1)}{(2\ell+3)(2\ell-1)}}\;,\\ \left\langle\eta,\ell\mid\mid\displaystyle\sqrt{\frac{16\pi}{5}}Y^{2}(\theta,\phi)\mid\mid\eta,\ell+2\right\rangle=\left\langle\eta,\ell+2\mid\mid\displaystyle\sqrt{\frac{16\pi}{5}}Y^{2}(\theta,\phi)\mid\mid\eta,\ell\right\rangle=\displaystyle\sqrt{\displaystyle\frac{6(\ell+1)(\ell+2)}{(2\ell+3)}}\;.\end{array}$$

$$\begin{array}[]{l}\left\langle\eta,\ell\mid\mid r^{2}\mid\mid\eta,\ell\right\rangle=\displaystyle\frac{2\eta+3}{2}\;,\\ \left\langle\eta,\ell\mid\mid r^{2}\mid\mid\eta,\ell+2\right\rangle=\left\langle\eta,\ell+2\mid\mid r^{2}\mid\mid\eta,\ell\right\rangle=\alpha_{\ell,\ell+2}\displaystyle\sqrt{(\eta-\ell)(\eta+\ell+3)}\;;\\ \alpha_{\ell,\ell+2}=\alpha_{\ell+2,\ell}=\pm 1\;.\end{array}$$

$$\begin{array}[]{l}L^{1}_{q}=2\displaystyle\sum_{\ell}\displaystyle\sqrt{\displaystyle\frac{\ell(\ell+1)(2\ell+1)}{3}}\left(a^{\dagger}_{\ell\frac{1}{2}\frac{1}{2}}\tilde{a}_{\ell\frac{1}{2}\frac{1}{2}}\right)^{1,0,0}_{q}\;,\\ Q^{2}_{q}({\mbox{\boldmath$\alpha$}})=-2(2\eta+3)\,\displaystyle\sum_{\ell}\displaystyle\sqrt{\displaystyle\frac{\ell(\ell+1)(2\ell+1)}{5(2\ell+3)(2\ell-1)}}\left(a^{\dagger}_{\ell\frac{1}{2}\frac{1}{2}}\tilde{a}_{\ell\frac{1}{2}\frac{1}{2}}\right)^{2,0,0}_{q}\\ +\displaystyle\sum_{\ell<\eta}\,2\alpha_{\ell,\ell+2}\;\displaystyle\sqrt{\displaystyle\frac{6(\ell+1)(\ell+2)(\eta-\ell)(\eta+\ell+3)}{5(2\ell+3)}}\left[\left(a^{\dagger}_{\ell\frac{1}{2}\frac{1}{2}}\tilde{a}_{\ell+2,\frac{1}{2}\frac{1}{2}}\right)^{2,0,0}_{q}+\left(a^{\dagger}_{\ell+2,\frac{1}{2}\frac{1}{2}}\tilde{a}_{\ell\frac{1}{2}\frac{1}{2}}\right)^{2,0,0}_{q}\right]\;;\\ {\mbox{\boldmath$\alpha$}}=(\alpha_{0,2},\alpha_{2,4},\ldots,\alpha_{\eta-2,\eta})\;\;\mbox{for}\;\;\eta\;\;\mbox{even}\;,\\ {\mbox{\boldmath$\alpha$}}=(\alpha_{1,3},\alpha_{3,5},\ldots,\alpha_{\eta-2,\eta})\;\;\mbox{for}\;\;\eta\;\;\mbox{odd}\;,\\ {\mbox{\boldmath$\alpha$}}=(\pm 1,\pm 1,\ldots)\;.\end{array}$$

$$\begin{array}[]{l}\left[L^{1}_{q}\,,\,L^{1}_{q^{\prime}}\right]=-\displaystyle\sqrt{2}\left\langle 1q\;1q^{\prime}\mid 1q+q^{\prime}\right\rangle\,L^{1}_{q+q^{\prime}}\;,\\ \left[L^{1}_{q}\,,\,Q^{2}_{q^{\prime}}({\mbox{\boldmath$\alpha$}})\right]=-\displaystyle\sqrt{6}\left\langle 1q\;2q^{\prime}\mid 2q+q^{\prime}\right\rangle\,Q^{2}_{q+q^{\prime}}({\mbox{\boldmath$\alpha$}})\;,\\ \left[Q^{2}_{q}({\mbox{\boldmath$\alpha$}})\,,\,Q^{2}_{q^{\prime}}({\mbox{\boldmath$\alpha$}})\right]=3\displaystyle\sqrt{10}\left\langle 2q\;2q^{\prime}\mid 1q+q^{\prime}\right\rangle\,L^{1}_{q+q^{\prime}}\;.\end{array}$$

$$Q^{2}_{q}({\mbox{\boldmath$\alpha$}})=2\displaystyle\sum_{\ell_{f},\ell_{i}}C^{{\mbox{\boldmath$\alpha$}}}_{\ell_{f},\ell_{i}}\left(a^{\dagger}_{\ell_{f}\frac{1}{2}\frac{1}{2}}\tilde{a}_{\ell_{i}\frac{1}{2}\frac{1}{2}}\right)^{2,0,0}_{q}\;.$$

$$Q\cdot Q=\sum _{i=1}^{m}Q(i)\cdot Q(i)+2\sum _{i<k=1}^{m}Q(i)\cdot Q(k)$$

$$\begin{array}[]{l}\left\langle(j_{a}j_{b})JT\mid Q(1)\cdot Q(2)\mid(j_{c}j_{d})JT\right\rangle_{a.s.m.}=\\ \displaystyle\frac{\left\langle(j_{a}j_{b})JT\mid Q(1)\cdot Q(2)\mid(j_{c}j_{d})JT\right\rangle_{n.a.s.m.}+(-1)^{J+T-j_{c}-j_{d}}\left\langle(j_{a}j_{b})JT\mid Q(1)\cdot Q(2)\mid(j_{d}j_{c})JT\right\rangle_{n.a.s.m.}}{\displaystyle\sqrt{\left(1+\delta_{ab}\right)\left(1+\delta_{cd}\right)}}\;.\end{array}$$

$$\begin{array}[]{l}\left\langle(j_{a}j_{b})JT\mid Q(1)\cdot Q(2)\mid(j_{c}j_{d})JT\right\rangle_{n.a.s.m.}=(-1)^{j_{b}+j_{c}+J}\left\{\begin{array}[]{ccc}j_{a}&j_{b}&J\\ j_{d}&j_{c}&2\end{array}\right\}\\ \times\;\left\langle j_{a}\mid\mid Q\mid\mid j_{c}\right\rangle\,\left\langle j_{b}\mid\mid Q\mid\mid j_{d}\right\rangle\;.\end{array}$$

$$\begin{array}[]{l}\left\langle\eta,\ell_{f},j_{f}\mid\mid Q^{2}({\mbox{\boldmath$\alpha$}})\mid\mid\eta,\ell_{i},j_{i}\right\rangle=(-1)^{\ell_{f}+\frac{1}{2}+j_{i}+2}\\ \times\;\displaystyle\sqrt{5(2j_{i}+1)(2j_{f}+1)}\;\left\{\begin{array}[]{ccc}\ell_{f}&j_{f}&\frac{1}{2}\\ j_{i}&\ell_{i}&2\end{array}\right\}\;C^{{\mbox{\boldmath$\alpha$}}}_{\ell_{f},\ell_{i}}\;.\end{array}$$

$$\epsilon^{{\mbox{\boldmath$\alpha$}}}_{\ell j}=\displaystyle\frac{5}{2\ell+1}\;\displaystyle\sum_{\ell^{\prime}}\left|C^{{\mbox{\boldmath$\alpha$}}}_{\ell\ell^{\prime}}\right|^{2}\;.$$

$$-Q^{2}({\mbox{\boldmath$\alpha$}})\cdot Q^{2}({\mbox{\boldmath$\alpha$}})=-C_{2}(SU^{{\mbox{\boldmath$\alpha$}}}(3))+\displaystyle\frac{3}{4}L\cdot L\;.$$

$$\zeta ({\mathcal{O}}_1,{\mathcal{O}}_2)=\frac{{⟨\widetilde{{\mathcal{O}}_1}\widetilde{{\mathcal{O}}_2}⟩}^m}{\mid \mid {\mathcal{O}}_1\mid {\mid }_m\mid \mid {\mathcal{O}}_2\mid {\mid }_m},$$

$$\begin{array}[]{rcl}H^{(-,-)}_{Q}=-Q^{2}(-,-)\cdot Q^{2}(-,-)\;,\\ H^{(+,-)}_{Q}=-Q^{2}(+,-)\cdot Q^{2}(+,-)\;,\\ H^{(-,+)}_{Q}=-Q^{2}(-,+)\cdot Q^{2}(-,+)\;,\\ H^{(+,+)}_{Q}=-Q^{2}(+,+)\cdot Q^{2}(+,+)\;.\end{array}$$

$$\begin{array}[]{rcl}(\lambda\mu)\longrightarrow L:&&\\ K&=&min(\lambda,\mu),\;min(\lambda,\mu)-2,\;\cdots,0\;\mbox{or}\;1,\\ L&=&K,\;K+1,\;K+2,\;\cdots,\;K+max(\lambda,\mu)\;\;for\;\;K\neq 0\;,\\ L&=&max(\lambda,\mu),\;max(\lambda,\mu)-2,\;\cdots,0\;\mbox{or}\;1\;\;for\;\;K=0\;,\\ (\lambda,\mu)\rightarrow L&\Longleftrightarrow&(\mu,\lambda)\rightarrow L\;.\end{array}$$

$$E(J=L)=-({\lambda }_H^2+{\mu }_H^2+{\lambda }_H{\mu }_H+3({\lambda }_H+{\mu }_H))+\frac{3}{4}L(L+1).$$

$$T^{E2}=Q_q^2(-,-)e_{eff}{b}^2$$

$$\begin{array}[]{l}Q((\lambda,0):J=L)=-\displaystyle\frac{L}{2L+3}(2\lambda+3)\,e_{eff}b^{2}\;,\\ B(E2;(\lambda,0)J=L\rightarrow J-2=L-2)=\displaystyle\frac{5}{16\pi}\left\{\displaystyle\frac{6J(J-1)(\lambda-J+2)(\lambda+J+1)}{(2J-1)(2J+1)}\right\}(e_{eff})^{2}b^{4}\;.\end{array}$$

$$T^{E2}=\left[e_{eff}^p{Q}_q^2(-,-;p)+e_{eff}^n{Q}_q^2(-,-;n)\right]b^2$$

$$\begin{array}[]{l}Q((\lambda,0):J=L)=-\displaystyle\frac{L}{2L+3}2(\lambda+3)\,X_{eff}\,b^{2}\;,\\ B(E2;(\lambda,0)J=L\rightarrow J-2=L-2)=\displaystyle\frac{5}{16\pi}\left\{\displaystyle\frac{6J(J-1)(\lambda-J+2)(\lambda+J+1)}{(2J-1)(2J+1)}\right\}(X_{eff})^{2}\,b^{4}\;;\\ X_{eff}=\displaystyle\frac{e^{p}_{eff}\left(\lambda_{\pi}^{2}+3\lambda_{\pi}+\lambda_{\pi}\lambda_{\nu}\right)+e^{n}_{eff}\left(\lambda_{\nu}^{2}+3\lambda_{\nu}+\lambda_{\pi}\lambda_{\nu}\right)}{\left(\lambda^{2}+3\lambda\right)}\;,\;\;\lambda=\lambda_{\pi}+\lambda_{\nu}\;.\end{array}$$

$$\begin{array}[]{l}L^{1}_{q}=\displaystyle\sum_{\ell}\displaystyle\sqrt{\displaystyle\frac{\ell(\ell+1)(2\ell+1)}{3}}\left(b^{\dagger}_{\ell}\tilde{b}_{\ell}\right)^{1}_{q}\;,\\ Q^{2}_{q}({\mbox{\boldmath$\alpha$}})=-(2\eta+3)\,\displaystyle\sum_{\ell}\displaystyle\sqrt{\displaystyle\frac{\ell(\ell+1)(2\ell+1)}{5(2\ell+3)(2\ell-1)}}\left(b^{\dagger}_{\ell}\tilde{b}_{\ell}\right)^{2}_{q}\\ +\displaystyle\sum_{\ell<\eta}\,\alpha_{\ell,\ell+2}\;\displaystyle\sqrt{\displaystyle\frac{6(\ell+1)(\ell+2)(\eta-\ell)(\eta+\ell+3)}{5(2\ell+3)}}\left[\left(b^{\dagger}_{\ell}\tilde{b}_{\ell+2}\right)^{2}_{q}+\left(b^{\dagger}_{\ell+2}\tilde{b}_{\ell}\right)^{2}_{q}\right]\;;\alpha_{\ell,\ell+2}=\pm 1\;.\end{array}$$

$$\begin{array}[]{l}L^{1}_{q}=\displaystyle\sqrt{10}\left(d^{\dagger}\tilde{d}\right)^{1}_{q}\;,\\ Q^{2}_{q}(\alpha_{sd})=\displaystyle\sqrt{2}\,\left[-\displaystyle\frac{\displaystyle\sqrt{7}}{2}\left(d^{\dagger}\tilde{d}\right)^{2}_{q}+\alpha_{sd}\left(s^{\dagger}\tilde{d}+d^{\dagger}\tilde{s}\right)^{2}_{q}\right]\;;\;\;\alpha_{sd}=\pm 1\;.\end{array}$$

$$\begin{array}[]{l}L^{1}_{\mu}=\displaystyle\sqrt{10}\left(d^{\dagger}\tilde{d}\right)^{1}_{\mu}+2\displaystyle\sqrt{15}\left(g^{\dagger}\tilde{g}\right)^{1}_{\mu}\;,\\ Q^{2}_{\mu}(\alpha_{sd},\alpha_{dg})=\displaystyle\sqrt{\displaystyle\frac{3}{4}}\left\{-11\displaystyle\sqrt{\displaystyle\frac{2}{21}}\,(d^{\dagger}\tilde{d})^{2}_{\mu}-2\displaystyle\sqrt{\displaystyle\frac{33}{7}}\,(g^{\dagger}\tilde{g})^{2}_{\mu}\right.\\ \left.+\alpha_{sd}\,4\displaystyle\sqrt{\displaystyle\frac{7}{15}}\;\left(s^{\dagger}\tilde{d}+d^{\dagger}\tilde{s}\right)^{2}_{\mu}+\alpha_{dg}\,\displaystyle\frac{36}{\displaystyle\sqrt{105}}\;\left(d^{\dagger}\tilde{g}+g^{\dagger}\tilde{d}\right)^{2}_{\mu}\right\}\;,\end{array}$$

$$|N;{\beta }_2;\gamma ⟩={\left[N!{\left(1+{\beta }_2^2\right)}^N\right]}^{-1/2}{\left\{s_0^{†}+{\beta }_2\left[\cos \gamma d_0^{†}+\sqrt{\frac{1}{2}}\sin \gamma \left(d_2^{†}+d_{-2}^{†}\right)\right]\right\}}^N,$$

$$H_{S{U}_{sd}(3)}^{{\alpha }_{sd}}=-Q^2({\alpha }_{sd})\cdot Q^2({\alpha }_{sd})$$

$$\begin{array}[]{l}E_{SU^{\alpha_{sd}}_{sd}(3)}\left(N;\beta_{2},\gamma\right)=\left\langle N;\beta_{2},\gamma\mid-Q^{2}(\alpha_{sd})\cdot Q^{2}(\alpha_{sd})\mid N;\beta_{2},\gamma\right\rangle\\ =-\displaystyle\frac{2N^{2}}{\left(1+\beta_{2}^{2}\right)^{2}}\;\left[4\beta_{2}^{2}+\displaystyle\frac{\beta^{4}_{2}}{2}+2\displaystyle\sqrt{2}\,\alpha_{sd}\,\beta_{2}^{3}\,\cos 3\gamma\right]\;.\end{array}$$

$$\begin{array}[]{l}\vspace{3mm}\left|N;\beta_{2};\beta_{4},\gamma\right\rangle=\left[N!\left(1+\beta_{2}^{2}+\beta_{4}^{2}\right)^{N}\right]^{-1/2}\left\{s^{\dagger}_{0}+\beta_{2}\left[\cos\gamma\,\,d^{\dagger}_{0}+\right.\right.\\ \left.\left.\sqrt{\frac{1}{2}}\sin\gamma\left(d^{\dagger}_{2}+d^{\dagger}_{-2}\right)\right]+\frac{1}{6}\beta_{4}\left[\left(5\cos^{2}\gamma+1\right)g^{\dagger}_{0}\right.\right.\\ \left.\left.+\sqrt{\frac{15}{2}}\sin 2\gamma\,\left(g^{\dagger}_{2}+g^{\dagger}_{-2}\right)+\sqrt{\frac{35}{2}}\sin^{2}\gamma\,\left(g^{\dagger}_{4}+g^{\dagger}_{-4}\right)\right]\right\}^{N}\,\left|0\right\rangle\;.\end{array}$$

$$\begin{array}[]{l}E_{SU^{{\mbox{\boldmath$\alpha$}}}_{sdg}(3)}\left(N;\beta_{2},\beta_{4},\gamma\right)=\left\langle N;\beta_{2};\beta_{4},\gamma\mid-Q^{2}({\mbox{\boldmath$\alpha$}})\cdot Q^{2}({\mbox{\boldmath$\alpha$}})\mid N;\beta_{2};\beta_{4},\gamma\right\rangle\\ =\displaystyle\frac{-3N^{2}}{4\left(1+\beta_{2}^{2}+\beta_{4}^{2}\right)^{2}}\;\left[\displaystyle\frac{448}{15}\,\alpha_{sd}^{2}\,\beta_{2}^{2}+\displaystyle\frac{384\displaystyle\sqrt{14}}{35}\,\alpha_{sd}\alpha_{dg}\,\beta_{2}^{2}\;\beta_{4}\right.\\ +\displaystyle\frac{352\displaystyle\sqrt{35}}{105}\,\alpha_{sd}\,\beta_{2}^{3}\,\cos{3\gamma}+\displaystyle\frac{64\displaystyle\sqrt{35}}{21}\,\alpha_{sd}\beta_{2}\,\beta_{4}^{2}\,\cos{3\gamma}+\displaystyle\frac{3456}{245}\,\alpha_{dg}^{2}\,\beta_{2}^{2}\,\beta_{4}^{2}\\ +\displaystyle\frac{1056\displaystyle\sqrt{10}}{245}\,\alpha_{dg}\,\beta_{2}^{3}\,\beta_{4}\,\cos{3\gamma}+\displaystyle\frac{484}{147}\,\beta_{2}^{4}+\displaystyle\frac{192\displaystyle\sqrt{10}}{49}\,\alpha_{dg}\,\beta_{2}\,\beta_{4}^{3}\,\cos{3\gamma}\\ +\left.\displaystyle\frac{880}{441}\,\left(4-{\cos}^{2}3\gamma\right)\,\beta_{2}^{2}\beta_{4}^{2}+\displaystyle\frac{400}{1323}\,\left(16-7{\cos}^{2}3\gamma\right)\,\beta_{4}^{4}\right]\;.\end{array}$$

Full text

I Introduction

Elliott has recognized way back in 1958 that shell model (SM) admits $SU(3)\supset SO(3)$ algebra and this will generate rotational spectra in nuclei starting with the interacting particle picture Ell-58a ; Ell-58b . Following this, $SU(3)$ algebra was developed in considerable detail by various groups and this includes methods to obtain $SU(3)$ irreducible representations (irreps) and $SU(3)$ Wigner-Racah algebra with codes for calculating $SU(3)\supset SO(3)$ and $SU(3)\supset SU(2)\times U(1)$ reduced Wigner coefficients, $SU(3)$ Racah coefficients, $SU(3)$ coefficients of fractional parentage and so on KYbook ; Ko-report ; Dr-1 ; Verg ; DrAk-1 ; DrAk-2 ; Dr-2 ; Dr-3 . By mid 60’s it was recognized that the $SU(3)$ symmetry is good for $1p$ and $2s1d$ shell nuclei but due to the strong spin-orbit force it will be a badly broken symmetry for $1p2f$ shell nuclei and beyond. Hecht, Draayer and others later recognized Hecht-1 ; Hecht-2 ; Hecht-3 ; JPD-1 ; JPD-2 that for heavy deformed nuclei, pseudo-$SU(3)$ based on pseudo spin and pseudo Nilsson orbits will be a useful symmetry and it gave rise to many new results. Very recently, a proxy-$SU(3)$ scheme by Bonatsos, Casten and others Bona-1 ; Bona-2 ; Bona-3 has appeared within SM with definite prediction for prolate dominance over oblate shape in heavy deformed nuclei. This $SU(3)$ model is currently being investigated in more detail. In addition, in the multishell situation again $SU(3)$ appears within the $Sp(6,R)$ model of Rowe and Rosensteel RR-1 ; RR-2 ; RR-3 and this has given rise to the no-core-sympletic shell model Krist-1 ; Krist-2 . Going beyond SM, a major basis for the interacting boson model (IBM) of atomic nuclei is that with $s$ and $d$ bosons the spectrum generating algebra (SGA) is $U(6)$ and it has $SU(3)$ as a subalgebra generating rotational spectrum Iac-76 ; Iac-87 . Similarly, $sdg$IBM KY-rev ; K-sdg , $sdpf$IBM Long-1 ; Long-2 and also IBM-3 with isospin ($T$) and IBM-4 with spin-isospin ($ST$) degrees of freedom Iac-87 ; KS all contain $SU(3)$ symmetry generating rotational spectra. In addition, in IBM-3 and IBM-4 models, $SU(3)$ also appears for isospin ($T$) and spin-isospin ($ST$) degrees of freedom respectively. Similarly, for odd-A nuclei we have $SU^{BF}(3)\times SU^{F}(2)$ symmetry in IBFM model with Nilsson correspondence BK . This extends to $SU(3)$ in IBFFM for odd-odd nuclei KU-1 ; KU-2 and $SU(3)$ in IBF2M for two quasi-particle excitations KY-quasi . With $SU(3)$ generating rotational spectra within both SM and IBM, it is natural to look for new perspectives for $SU(3)$ symmetry in nuclei.

One curious aspect of $SU(3)$ in nuclei is that in a given oscillator shell $\eta$, there will be multiple $SU(3)$ algebras. Very early it is recognized that in SM with $s$ and $d$ orbits there will be two $SU(3)$ algebras parikh but its consequences are not explored in any detail. Similarly, in $sd$IBM there are two $SU(3)$ algebras Iac-87 and they are applied in phase transition studies RMP . Finally, it was also recognized that there will be four $SU(3)$ algebras in $sdg$IBM K-sdg . Except for the $sd$IBM, properties of multiple $SU(3)$ algebras are not investigated in any detail in the past. As we will show, for a given oscillator shell with major shell number $\eta$, there will be $2^{\left[\frac{\eta}{2}\right]}$ number of $SU(3)$ algebras where $\left[\frac{\eta}{2}\right]$ is the integer part of $\eta/2$. In the present paper, following the recent investigation of multiple pairing algebras in SM and IBM Ko-BJP , several different aspects of multiple $SU(3)$’s in SM and IBM are investigated. Now, we will give a preview.

In Section 2, multiple $SU(3)$ algebras in SM generated by angular momentum operator $L^{1}_{q}$ and quadrupole moment operator $Q^{2}_{q}$ with different signs for the $\ell\rightarrow\ell\pm 2$ matrix elements are identified and the matrix elements for the corresponding $Q\cdot Q$ operators are given. Using these, correlations between different $Q\cdot Q$ operators are studied. In Section 3, Spectra and electric quadrupole ($E2$) properties of these algebras are studied using shell model codes and also deformed shell model based on Hartree-Fock single particle states (called DSM KS ). Used here are examples with 6 protons, 6 protons plus 2 neutrons and 6 protons plus 6 neutrons systems. In Section 4, results for multiple $SU(3)$ algebras in IBM’s (with no internal degrees of freedom for the the bosons) are presented. Finally, Section 5 gives conclusions.

II Phase choice and Multiple $SU(3)$ algebras in shell model

Let us consider the situation where valence nucleons in a nucleus occupying an oscillator shell with major shell number $\eta$. With the spin-isospin degrees of freedom for the nucleons, the spectrum generating algebra (SGA) is $U(4{\cal N})$ and decomposing the space into orbital and spin-isospin ($ST$) parts, we have $U(4{\cal N})\supset U({\cal N})\times SU(4)$. Here, ${\cal N}=(\eta+1)(\eta+2)/2$ and $SU(4)$ is the Wigner’s spin-isospin $SU(4)$ algebra; see for example Manan ; Piet-su41 ; Piet-su42 ; octup ; KS . Also, for a given $\eta$, the the single particle (sp) orbital angular momentum $\ell$ takes values $\ell=\eta$, $\eta-2$, $\ldots$, [math] or $1$. Note that, for nuclei with only valence protons or neutrons $SU(4)$ changes to $SU(2)$ generating spin $S$. As Elliott has established, the orbital $U({\cal N})$ algebra admits $SU(3)$ subalgebra with $U({\cal N})\supset SU(3)\supset SO(3)$ where $SO(3)$ generates orbital angular momentum. The eight generators of $SU(3)$ are the orbital angular momentum operators $L^{1}_{q}$ and quadrupole moment operators $Q^{2}_{q}$. In $LST$ coupling and using fermion creation ($a^{\dagger}$) and annihilation ($a$) operators,

[TABLE]

Note that $\tilde{a}_{\ell-m,\frac{1}{2}-m_{s},\frac{1}{2}-m_{t}}=(-1)^{\ell-m+\frac{1}{2}-m_{s}+\frac{1}{2}-m_{t}}a_{\ell m,\frac{1}{2}m_{s},\frac{1}{2}m_{t}}$ where $m_{s}$ and $m_{t}$ are the $S_{z}$ and $T_{z}$ quantum numbers for a single nucleon. Similarly, the quadrupole operator is

[TABLE]

Closure examination of the reduced matrix element $\left\langle\eta,\ell_{f}\mid\mid Q^{2}\mid\mid\eta,\ell_{i}\right\rangle$ of the quadrupole operator in the orbital space allows us to recognize that there will be multiple $SU(3)$ subalgebras in $U({\cal N})$. We will turn to this now.

As Elliott considered Ell-58a , the quadrupole operator is $Q^{2}_{q}=\sqrt{\frac{4\pi}{5}}\left[r^{2}Y^{2}_{q}(\theta,\phi)+p^{2}Y^{2}_{q}(\theta_{p},\phi_{p})\right]$ with oscillator length parameter $b=1$. For a single shell. this is equivalent to using $Q^{2}_{q}=\sqrt{\frac{16\pi}{5}}\,r^{2}Y^{2}_{q}(\theta,\phi)$. Therefore, the reduced matrix elements of $Q^{2}$ decompose into the radial part and angular part,

[TABLE]

with the angular part given by Brussard ,

[TABLE]

Similarly, the radial matrix elements are

[TABLE]

The phase factor $\alpha_{\ell,\ell+2}$ arises as there is freedom in choosing the phases of the radial wavefunctions of a 3D oscillator. In SM studies, the standard convention is to use $\alpha_{\ell,\ell+2}=-1$ for all $\ell$ octup ; Brussard ; Bertsch . However, Elliott in his $SU(3)$ introductory paper Ell-58a and in $sd$ as well as $sdg$ IBM and IBFM the choice made is $\alpha_{\ell,\ell+2}=+1$ for all $\ell$ Iac-87 ; KY-rev ; Piet ; BK . Thus, in general we have,

[TABLE]

Now, the most important result that can be proved by using the tedious but straight forward angular momentum algebra is that the eight operators (L^{1}_{q},Q^{2}_{q^{\prime}}({\mbox{\boldmath\alpha}})) generate $SU(3)$ algebra independent of the choice of the $\alpha$’s and they satisfy the commutation relations Ell-58a ; octup ,

[TABLE]

Thus, we have multiple $SU(3)$ algebras SU^{{\mbox{\boldmath\alpha}}}(3) in SM spaces generated by the operators in Eq. (6). Clearly for a given $\eta$, there will be $2^{\left[\frac{\eta}{2}\right]}$ number of $SU(3)$ algebras; $\left[\frac{\eta}{2}\right]$ is the integer part of $\eta/2$. Then, we have two $SU(3)$ algebras in $sd$ ($\eta=2$) and $pf$ ($\eta=3$) shells, four $SU(3)$ algebras in $sdg$ ($\eta=4$) and $pfh$ ($\eta=5$) shells, eight $SU(3)$ algebras in $(sdgi)$ ($\eta=6$) and $(pfhj)$ ($\eta=7$) shells and so on. Thus, the first non-trivial situation that is not discussed in literature before is $sdg$ or $\eta=4$ shell with four $SU(3)$ algebras $SU^{(-,-)}(3)$, $SU^{(+,-)}(3)$, $SU^{(-,+)}(3)$ and $SU^{(+,+)}(3)$. Here, {\mbox{\boldmath\alpha}}=(\alpha_{sd},\alpha_{dg}) and $(-,-)$ means $(\alpha_{sd},\alpha_{dg})=(-1,-1)$ and similarly for other choices of $(\alpha_{sd},\alpha_{dg})$. In the reminder of this paper, we will use the example of $\eta=4$ shell to present some results from multiple $SU(3)$ algebras. Before this, we will first consider the quadrupole-quadrupole interaction generated by Q^{2}_{q}({\mbox{\boldmath\alpha}}).

II.1 Matrix elements of Quadrupole-quadrupole interaction from multiple

$SU(3)$ algebras

Investigation of multiple $SU(3)$ algebras in shell model spaces needs firstly the single particle energies (spe) and two-body matrix elements (TBME) of the quadrupole-quadrupole interaction operator Q^{2}({\mbox{\boldmath\alpha}})\cdot Q^{2}({\mbox{\boldmath\alpha}}) for all phase choices $\alpha$ (also the spe and TBME for the simpler $L\cdot L$ operator). The methods for obtaining these are well known Brussard and we will give only the final formulas. In order to derive formulas for the spe and TBME generated by Q^{2}({\mbox{\boldmath\alpha}})\cdot Q^{2}({\mbox{\boldmath\alpha}}) operators, firstly notice that the $Q^{2}_{q}$ operator can be written as,

[TABLE]

The C^{{\mbox{\boldmath\alpha}}}_{\ell_{f},\ell_{i}} follow easily from Eq. (6). From now on we will drop ’2’ and $\alpha$ in Q^{2}_{q}({\mbox{\boldmath\alpha}}) when there is no confusion. For a many particle system,

[TABLE]

where $i$ and $k$ are particle indices and $m$ is number of particles. The first sum generates spe and the second term TBME. Given the shell model single particle $(n\ell j)$-orbits (note that the oscillator shell number $\eta=2n+\ell$), matrix elements of $Q(1)\cdot Q(2)$ in the two-particle antisymmetric states (called a.s.m.) can be written in terms of the matrix elements in the two-particle non-antisymmetric states (called n.a.s.m.) as,

[TABLE]

Using angular momentum algebra it is easy to recognize that,

[TABLE]

The reduced matrix elements $\left\langle\mid\mid Q\mid\mid\right\rangle$ are given by,

[TABLE]

Combining Eqs. (11) and (12) with Eq. (10) and Eq. (9) will give the TBME of the Q^{2}({\mbox{\boldmath\alpha}})\cdot Q^{2}({\mbox{\boldmath\alpha}}) operator. The spe \epsilon^{{\mbox{\boldmath\alpha}}}_{\ell j} of the Q^{2}({\mbox{\boldmath\alpha}})\cdot Q^{2}({\mbox{\boldmath\alpha}}) are simply given by

[TABLE]

An important property of the Q^{2}({\mbox{\boldmath\alpha}})\cdot Q^{2}({\mbox{\boldmath\alpha}}) operator is that it is related to the quadratic Casimir invariant ($C_{2}$) of SU^{{\mbox{\boldmath\alpha}}}(3) in a simple manner,

[TABLE]

The procedure described above will also give the spe and TBME of $L\cdot L$ operator. Let us mention that the eigenvalue of C_{2}(SU^{{\mbox{\boldmath\alpha}}}(3)) over a SU^{{\mbox{\boldmath\alpha}}}(3) irrep $(\lambda\mu)$ is $\lambda^{2}+\mu^{2}+\lambda\mu+3(\lambda+\mu)$. Also, note that the dot product in Eqs. (14) and (9) is with respect to the orbital space.

II.2 Correlation between different $Q\cdot Q$ operators

In order to gain some insight into the differences between different SU^{{\mbox{\boldmath\alpha}}}(3) algebras, we will consider the correlation in $m$ nucleon spaces between different Q({\mbox{\boldmath\alpha}})\cdot Q({\mbox{\boldmath\alpha}}) operators. For this, we will use the example of $\eta=4$ shell giving $(n\ell j)$ to be $(2,0,1/2)$, $(1,2,3/2)$, $(1,2,5/2)$, $(0,4,7/2)$ and $(0,4,9/2)$. In this space, spe and TBME are obtained for Q^{2}({\mbox{\boldmath\alpha}})\cdot Q^{2}({\mbox{\boldmath\alpha}}) operators with {\mbox{\boldmath\alpha}}=(\alpha_{sd},\alpha_{dg})=(+,+),(+,-),(-,+) and $(-,-)$ using the results in Section IIA.

Given an operator ${\cal O}$ acting in $m$ particle spaces (${\cal O}$ is assumed to be real), its trace over the $m$ particle space is $\left\langle\left\langle{\cal O}\right\rangle\right\rangle^{m}=\sum_{\gamma}\,\left\langle m,\gamma\mid{\cal O}\mid m,\gamma\right\rangle$. Note that $\left.\left|m,\gamma\right.\right\rangle$ are $m$-particle states. Similarly, the $m$-particle average is $\left\langle{\cal O}\right\rangle^{m}=[d(m)]^{-1}\left\langle\left\langle{\cal O}\right\rangle\right\rangle^{m}$ where $d(m)$ is $m$-particle space dimension. Using the spectral distribution method of French CFT ; KH-10 , a geometry can be defined CFT with norm (or size or length) of an operator ${\cal O}$ given by $\mid\mid{\cal O}\mid\mid_{m}=\sqrt{\left\langle\tilde{{\cal O}}\tilde{{\cal O}}\right\rangle^{m}}$; $\tilde{{\cal O}}$ is the traceless part of ${\cal O}$. Following this, given any two operators ${\cal O}_{1}$ and ${\cal O}_{2}$, the correlation coefficient

[TABLE]

gives the cosine of the angle between the two operators. Thus, ${\cal O}_{1}$ and ${\cal O}_{2}$ are same within a normalization constant if $\zeta=1$ and they are orthogonal to each other if $\zeta=0$ KH-10 . Most recent application of norms and correlation coefficients is in understanding the structure of multiple pairing algebras in shell model Ko-BJP .

Applying Eq. (15), we have calculated $\zeta$ between the operators $Q^{2}(\alpha_{sd},\alpha_{dg})\cdot Q^{2}(\alpha_{sd},\alpha_{dg})$ and $Q^{2}(\alpha^{\prime}_{sd},\alpha^{\prime}_{dg})\cdot Q^{2}(\alpha^{\prime}_{sd},\alpha^{\prime}_{dg})$ for all possible combinations of $\alpha$’s and $(\alpha^{\prime})$’s. Some results for $\zeta$ are given in Table I. It is seen from the table that $Q^{2}(-,-)\cdot Q^{2}(-,-)$ is strongly correlated with $Q^{2}(+,-)\cdot Q^{2}(+,-)$. Similarly, the $Q\cdot Q$’s with $(\alpha_{sd},\alpha_{dg})=(+,+)$ and $(-,+)$ are strongly correlated. However, the correlations between other pairs of $Q^{2}\cdot Q^{2}$ are quite small. Thus, $SU^{(-,-)}(3)$ and $SU^{(+,-)}(3)$ are expected to give similar results but quite different from $SU^{(+,+)}(3)$ and $SU^{(-,+)}(3)$. This is seen in the results of detailed calculations presented in the next section. It is important to stress that all the four SU^{{\mbox{\boldmath\alpha}}}(3) algebras generate the same spectrum for H({\mbox{\boldmath\alpha}})=Q^{2}(\alpha_{sd},\alpha_{dg})\cdot Q^{2}(\alpha_{sd},\alpha_{dg}) independent of $(\alpha_{sd},\alpha_{dg})$. We will consider these in more detail in the following.

III Results for Spectra, quadrupole moments and $E2$ transition strengths

from SM and DSM

With the $sdg$ example, we have four $Q\cdot Q$ Hamiltonians,

[TABLE]

In this section we will present the results generated by these four $H$’s for the yrast levels, quadrupole moments $Q_{2}(J)$ of these levels and the $B(E2)$’s along the yrast line for $J$ up to 10. Used for this purpose are the Antoine shell model code Anton and also the deformed shell model (DSM) based on Hartree-Fock states KS . DSM is particularly important for bringing out shape information in a transparent manner and also it is useful for larger particle numbers where SM calculations are impractical. We will test the SM results with analytical results derived using $SU(3)$ algebra and also test DSM using SM results. We will first present some analytical results from $SU(3)$ algebra.

III.1 Analytical results from $SU(3)$ algebra

With $SU(3)$ symmetry of the $H_{Q}$ Hamiltonians, the shell model space for a $m$ nucleon system decomposes into $SU(3)$ irreducible representations (irreps) due to the equivalence between $H_{Q}$ and $C_{2}(SU(3))$ as given by Eq. (14). If we have identical nucleons (protons or neutrons), the ground band belongs to the leading $SU(3)$ irrep $(\lambda_{H},\mu_{H})$ with spin $S=0$ and $J=L$ for even $m$ (similarly with $S=1/2$ for odd $m$). It is easy to write a formula for obtaining $(\lambda_{H},\mu_{H})$ as given in Kota-hw . The irreps for $m$ identical nucleons in $\eta=4$ shell are given in Table II. Similarly, for $m$ nucleons with isospin $T$, we need to consider the lowest spin-isospin $SU(4)$ irrep allowed for this system Manan ; Piet-su42 and this will then give $(\lambda_{H},\mu_{H})$ Kota-hw . The irreps $(\lambda_{H},\mu_{H})$ for $m$ nucleons with $T=|T_{z}|$ are given in Table II. The eigenstates of $H_{Q}$ are $\left|m;(\lambda_{H}\mu_{H})KL;S:JT\right\rangle$ and the $(\lambda_{H}\mu_{H})\rightarrow L$ reduction is well known giving,

[TABLE]

It is easy to see that the energies of the yrast levels in a even $m$ system (assuming spin $S=0$) are given by,

[TABLE]

In the examples presented ahead in the present paper we will only consider even $m$ systems with $(\lambda_{H}\mu_{H})=(\lambda 0)$ and then $\lambda$ is even. A $(\lambda,0)$ irrep with $\lambda$ even, as seen from Eq. (17), generates the ground band with $J=0$, $2$, $4$, $\ldots$, $\lambda$. The ground state energy $E_{gs}=(\lambda^{2}+3\lambda)$ and the energies of the $J$ levels with respect to $E_{gs}$ are just $3J(J+1)/4$. In addition, if we choose the $E2$ transition operator to be the $Q$ of one of the $H_{Q}$, then formulas for $Q_{2}(J)$ and $B(E2)$ will be simple for the $(\lambda,0)$ irrep of the corresponding $SU(3)$ algebra. Just as it is considered in SM and DSM codes, we will take the $E2$ operator $T^{E2}$ for identical nucleon systems to be

[TABLE]

where $b$ is the oscillator length parameter and $e_{eff}$ is effective charge. Then, analytical formulas for the quadrupole moments $(Q(J))$ of the yrast levels and $B(E2)$’s among them follow from the simple $SU(3)$ algebra for the eigenstates obtained for $H_{Q}^{(-,-)}$ as they belong to $SU^{(-,-)}(3)$. Using the results in Ell-58a ; BK , we have for $H_{Q}^{(-,-)}$ in Eq. (16) with $T^{E2}$ in Eq. (19),

[TABLE]

However, for systems with valence protons and neutrons, the $E2$ transition operator is taken to be

[TABLE]

where $e^{p}_{eff}$ and $e^{n}_{eff}$ are proton and neutron effective charges. Again, using eigenstates obtained for $H_{Q}^{(-,-)}$ as they belong to $SU^{(-,-)}(3)$ and the $T^{E2}$ in Eq. (21), a simple formula is obtained for $Q(J)$ and $B(E2)$’s in the situation where the ground band is given by $\left|(\lambda_{\pi},0)(\lambda_{\nu},0)(\lambda_{\pi}+\lambda_{\nu},0)K=0,L,S=0,J=L\right\rangle$ for a system with protons ($\pi$) and neutrons ($\nu$). Now, carrying out the $SU(3)$ algebra using the mathematical formulation and analytical results given in JPD-1 ; Hecht-65 ; Mill ; Verg we have,

[TABLE]

Tests of Eqs. (18), (20) and (22) are carried out using SM and DSM in the next three subsections.

It is important to stress that in the event we use the eigenstates of other H^{{\mbox{\boldmath\alpha}}}_{Q}, the ground band generated by them will belong to the $(\lambda 0)$ irrep of the corresponding SU^{{\mbox{\boldmath\alpha}}}(3). However, then the $Q$’s in $T^{E2}$ in Eqs. (19) and (21) are no longer generators of these SU^{{\mbox{\boldmath\alpha}}}(3)’s and hence the formulas in Eqs. (20) and (22) will not apply. In this situation, we have to use Q^{2}_{q}(-,-)=Q^{2}_{q}({\mbox{\boldmath\alpha}})+\Delta Q and $\Delta Q$ follows easily from Eq. (6). Then, one has to carry out the $SU(3)$ tensorial decomposition of $\Delta Q$ with respect to SU^{{\mbox{\boldmath\alpha}}}(3) and use the $SU(3)$ Wigner-Racah algebra as described for example in JPD-1 ; Hecht-65 ; Mill ; Verg for obtaining the matrix elements of $\Delta Q$ in the $\left|(\lambda 0)K=0,L\right\rangle$ states. This exercise is postponed to a future publication and instead we will present results of full (without any truncation) SM results along with some DSM results in the next two subsections and only DSM results in the third subsection. In addition, to gain more insight into the other SU^{{\mbox{\boldmath\alpha}}}(3) algebras, we will use the asymptotic formulas for quadrupole moments and $B(E2)$’s in $sdg$IBM in Section IV.

III.2 SM and DSM results for multiple $SU(3)$ algebras: $(sdg)^{6_{p}}$

example

In our first example, we have analyzed a system of 6 protons in $\eta=4$ shell, i.e. $(sdg)^{6_{p}}$ system by carrying out SM calculations using the four $H_{Q}$ Hamiltonians in the full SM space (matrix dimension in the $m$-scheme is $\sim 10^{5}$) using the Antoine code. For this system, the leading $SU(3)$ irrep (see Table II) is $(18,0)$ with $S=0$. Then, Eq. (18) gives $E_{gs}=378$ and SM calculations for all four $H_{Q}$’s are in agreement with this $SU(3)$ result. Also, in the SM results the excitation energies of the yrast $J$ states or ground band members ($J=0,,2,4,6,\ldots$) are seen to follow for all the four $H_{Q}$’s the $3J(J+1)/4$ law as given by $SU(3)$. Thus, it is verified by explicit SM calculations that all the four $H_{Q}$’s give $SU(3)$ symmetry. Though the energy spectra are same, the wavefunctions of the yrast $J$ states are different. This is established by calculating $Q(J)$ and $B(E2)$’s for the ground band members using $T^{E2}$ given by Eq. (19). In all the calculations, $e_{eff}=1e$ and $b^{2}=A^{1/3}\,fm^{2}$ with $A=86$ are used. Results from SM for the four $H_{Q}$’s are given in Tables III. It is easy to see that the results for $H_{Q}^{(-,-)}$ are in complete agreement with the results the $SU(3)$ formulas given by Eq. (20). This is expected as $T^{E2}$ in Eq. (19) is a generator of $SU^{(-,-)}(3)$ generated by $H_{Q}^{(-,-)}$. However, the results from the other three $H_{Q}$’s are quite different and do not follow the $SU(3)$ results in Eq. (20) as the $T^{E2}$ chosen is not a generator of the $SU(3)$’s generated by the three $H_{Q}$’s. It is seen from Tables III that the results for $Q(J)$ and $B(E2)$’s from $H_{Q}^{(+,-)}$ are closer to those from $H_{Q}^{(-,-)}$ and this is consistent with the correlation coefficients shown in Table II. The $B(E2)$’s from $H_{Q}^{(-,+)}$ are much smaller in magnitude. Moreover, $H_{Q}^{(-,-)}$ generates prolate shape and $H_{Q}^{(+,+)}$ oblate as seen clearly from Table III. Quadrupole moments show that $H_{Q}(+,-)$ and $H_{Q}(-,+)$ also generate prolate shapes but the deformation from $H_{Q}(-,+)$ is quite small for the low-lying levels. To gain more insight into these results, we have performed DSM calculations using the four $H_{Q}$’s with results as follows.

Starting with the same model space, sp energies and two-body interaction, in DSM one solves Hartree-Fock (HF) sp equations self-consistently assuming axial symmetry. The lowest-energy prolate or oblate intrinsic state for the nucleus in question is then obtained. The various excited intrinsic states then are obtained by making particle-hole ($p$-$h$) excitations over the lowest-energy intrinsic state (lowest configuration). Carrying out angular momentum projection from each intrinsic state and performing band mixing, orthonormalized $\left|JK\right\rangle$ states are obtained. See KS for full details and many applications of DSM. Latest application of DSM is to dark matter studies Sahu . In the present DSM calculations, only the lowest intrinsic state is considered. It is found that the four $H_{Q}$’s generate the same HF sp spectrum and it is same as shown in Fig. 1 ahead except for some scale factors. The lowest intrinsic state is obtained by putting two protons each in the $1/2_{1}$, $1/2_{2}$ and $3/2_{1}$ states. The intrinsic quadrupole moments (in units of $b^{2}$) for $H_{Q}^{(-,-)}$, $H_{Q}^{(+,-)}$, $H_{Q}^{(-,+)}$ and $H_{Q}^{(+,+)}$ are $+35.85$, $24.4$, $1.83$ and $-9.63$ respectively. Thus, $H_{Q}^{(-,-)}$ generates prolate shape and $H_{Q}^{(+,+)}$ generates oblate shape in agreement with SM. It is important to emphasize that the intrinsic quadrupole moments are calculated using $T^{E2}=Q^{2}_{q}(-,-)\,b^{2}$ as the quadrupole operator. The ground state energy for the 6 proton system is found to be, for all the four $H_{Q}$’s same as the exact $SU(3)$ values within less than 1% deviation. The energies of the yrast $J$ states from the ground state are also same for four $H_{Q}$’s and they follow the $3J(J+1)/4$ law. Similarly, the results for $Q(J)$’s and $B(E2)$’s are essentially same as the SM values. For example for $H_{Q}^{(-,-)}$, the $Q(J)$ values (in $efm^{2}$ unit) are $-49.04$, $-62.38$, $-68.56$, $-72.07$ and $-74.29$ for $J=2$, $4$, $6$, $8$ and $10$ respectively. The corresponding $B(E2;J\rightarrow J-2)$ values (in $e^{2}fm^{4}$ unit) are $582.78$, $810.44$, $848.78$, $822.26$ and $755.63$ respectively. Thus, for larger particle systems where SM calculations are not possible, one can use with confidence DSM for further insight into the results from the four $H_{Q}$’s, i.e. from multiple $SU(3)$ algebras and this is used in Section III-D.

III.3 SM results for multiple $SU(3)$ algebras:

$(sdg)^{(6_{p},2_{n})T=2}$ example

In our second example, we have considered a system of 6 protons and 2 neutrons in $\eta=4$ shell, i.e. $(sdg)^{6_{p},2_{n}}$ system and carried out SM calculations using the four $H_{Q}$ Hamiltonians in the full SM space (dimension in the $m$-scheme is $\sim 2\times 10^{7}$) using Antoine code. For this system, the leading $SU(3)$ irrep (see Table II) is $(26,0)$ with $S=0$ and $T=2$. Then, Eq. (18) gives $E_{gs}=754$ and SM calculations for all four $H_{Q}$’s is in agreement with this $SU(3)$ result. Also, in the SM results the excitation energies of the yrast $J$ states or ground band members ($J=0,,2,4,6,\ldots$) are seen to follow for all the four $H_{Q}$’s the $3J(J+1)/4$ law as given by $SU(3)$. Thus, it is again verified by explicit SM calculations that all the four $H_{Q}$’s give $SU(3)$ symmetry. The wavefunctions of the yrast $J$ states are investigated by calculating $Q(J)$ and $B(E2)$’s for the ground band members using $T^{E2}$ in Eq. (21). In all the calculations, $e^{p}_{eff}=1.5e$, $e_{eff}^{n}=0.5e$ and $b^{2}=A^{1/3}\,fm^{2}$ with $A=88$ are used. Note that the ground $(26,0)$ irrep arises from the strong coupling of the $(18,0)$ irrep for the $6$ protons (see the previous Section) and the $(8,0)$ irrep for the two neutrons. Therefore, formulas in Eq. (22) will apply for the states from $H_{Q}(-,-)$. Results from SM for the four $H_{Q}$’s are given in Tables IV. It is easy to see that the results for $H_{Q}^{(-,-)}$ are in complete agreement with the formulas in Eq. (22). This is expected as the proton and neutron parts of $T^{E2}$ in Eq. (21) are generators of $SU^{(-,-)}(3)$ for protons and neutrons respectively. However, the results from the other three $H_{Q}$’s are quite different as in the previous $(sdg)^{6_{p}}$ example. Again, it is seen from Tables IV that the results for $Q(J)$ and $B(E2)$’s from $H_{Q}^{(+,-)}$ are closer to those from $H_{Q}^{(-,-)}$. The $B(E2)$’s from $H_{Q}^{(-,+)}$ and $H_{Q}^{(+,+)}$ are much smaller in magnitude. Moreover, $H_{Q}^{(-,-)}$ generates prolate shape and $H_{Q}^{(+,+)}$ oblate as in the previous example. Finally, let us mention that we have also carried out DSM calculations for this example and they are all in agreement with SM results.

III.4 DSM results for multiple $SU(3)$ algebras:

$(sdg)^{(6_{p},6_{n})T=0}$ example

In our final example we have considered a system of 12 nucleons with $T=0$ in $\eta=4$ shell, i.e. $(sdg)^{(6_{p},6_{n})T=0}$ system. Here the dimension in the $m$-scheme in SM is $\sim 10^{10}$ and therefore SM calculations are not possible with our computational facilities. Thus, in this example DSM gives the predictions for four $H_{Q}$’s and only for $H_{Q}^{(-,-)}$ we have exact $SU(3)$ results (they will be same as SM results if performed) from Section III.A. Carrying out DSM calculations for this system, it is found that the four $H_{Q}$’s generate the same HF sp spectrum as shown in Fig. 1. Using the lowest intrinsic shown in Fig. 1, it is seen from the intrinsic quadrupole moments for the four $H$’s that $H_{Q}^{(-,-)}$ generates prolate shape and $H_{Q}^{(+,+)}$ generates oblate shape in agreement with SM. The ground state energy for the system is found to be $-1402.4$ for all four $H_{Q}$’s against the exact $SU(3)$ value $-1404$ giving less than $1\%$ deviation. Note that the $SU(3)$ irrep for the ground band is $(36,0)$ and this generated by the irrep $(18,0)$ for the 6 protons and $(18,0)$ for the 6 neutrons. The energies of the yrast $J$ states are also same for four $H_{Q}$’s and they are also within $1\%$ deviation from the $3J(J+1)/4$ law. Turning to $Q(J)$ and $B(E2)$’s, in the calculations used are $e^{p}_{eff}=1.5e$, $e_{eff}^{n}=0.5e$ and $b^{2}=A^{1/3}\,fm^{2}$ with $A=92$. Note that the ground $(36,0)$ irrep arises from the strong coupling of the $(18,0)$ irreps of the $6$ protons and the 6 neutrons. Therefore, formulas in Eq. (22) will apply for the states from $H_{Q}(-,-)$. DSM results for $H_{Q}^{(-,-)}$, as shown in Table V are in complete agreement with the formulas in Eq. (22) as expected. However, the results from the other three $H_{Q}$’s are quite different as in the previous $(sdg)^{6_{p}}$ and $(sdg)^{6p,2n}$ examples. Again, it is seen from Tables V that the results for $Q(J)$ and $B(E2)$’s from $H_{Q}^{(+,-)}$ are closer to those from $H_{Q}^{(-,-)}$. The $B(E2)$’s from $H_{Q}^{(-,+)}$ and $H_{Q}^{(+,+)}$ are much smaller in magnitude. Moreover, $H_{Q}^{(-,-)}$ generates prolate shape and $H_{Q}^{(+,+)}$ oblate as in the previous examples. Thus, the results in Tables III-V are generic results for the four $H_{Q}$’s.

IV Multiple $SU(3)$ algebras in interacting boson model

In the interacting boson models with $sd$ ($\ell=0,2$) or $sdg$ ($\ell=0,2,4$) bosons (and their appropriate generalizations to $pf$, $sdgi$ etc.), the eight operators (L^{1}_{q},Q^{2}_{q}({\mbox{\boldmath\alpha}})) are

[TABLE]

Note that $b^{\dagger}$ and $b$ are boson creation and annihilation operators and $\tilde{b}_{\ell m}=(-1)^{\ell-m}b_{\ell-m}$. Again, after some tedious angular momentum algebra, it is easy to prove that for all choices of $\alpha_{\ell,\ell+2}=\pm 1$, Eq. (7) is valid and therefore giving a $SU(3)$ algebra for each choice of the $\alpha$’s. With $\alpha_{\ell,\ell+2}$ taking $+1$ or $-1$ value, for a given $\eta$ there will be $2^{[\eta/2]}$ number of $SU(3)$ algebras in IBM’s just as in SM. It is important to stress that $\alpha_{\ell,\ell+1}=+1$ for all $\ell$ values is the standard choice in $sd$IBM and $sdg$IBM. As an example, in $sd$IBM with $\eta=2$, the $(L^{1}_{q},Q^{2}_{q})$ operators generating multiple $SU(3)$ algebra are,

[TABLE]

giving two SU^{{\mbox{\boldmath\alpha}}}(3) algebras. In $sd$IBM they are discussed in the context of quantum phase transitions (QPT) RMP . The $\alpha_{sd}=+1$ and $-1$ generate prolate and oblate shapes respectively as discussed ahead. In $sdg$IBM with $\eta=4$ there will be four SU^{{\mbox{\boldmath\alpha}}}(3) algebras generated by,

[TABLE]

with$\alpha_{sd}=\pm 1$ and $\alpha_{dg}=\pm 1$.

IV.1 Geometry of multiple $SU(3)$ algebras in $sd$IBM and $sdg$IBM

In order to have some insight into the multiple $SU(3)$ algebras in IBM, let us examine the geometric shapes generated by them using coherent states. Starting with $sd$IBM, the coherent state is

[TABLE]

where $\beta_{2}\geq 0$ and $0^{\circ}\leq\gamma\leq 60^{\circ}$. Now, let us consider the $SU(3)$ Hamiltonian

[TABLE]

and $-Q^{2}(\alpha_{sd})\cdot Q^{2}(\alpha_{sd})=-C_{2}(SU^{\alpha_{sd}}(3))+\frac{3}{4}L\cdot L$. It is important to note that $H_{SU_{sd}(3)}$ generates the same spectrum for the two choices of $\alpha_{sd}$. In the $N\rightarrow\infty$ limit, the coherent state expectation value of $H_{SU_{sd}(3)}$ is given by

[TABLE]

Minimizing the $SU(3)$ energy functional $E_{SU_{sd}(3)}\left(N;\beta_{2},\gamma\right)$ gives the equilibrium solutions $(\beta_{2}^{0},\gamma^{0})$ to be $\beta_{2}^{0}=\sqrt{2}$ and $\gamma^{0}=0^{\circ}$ for $\alpha_{sd}=+1$ and $\gamma^{0}=60^{\circ}$ for $\alpha_{sd}=-1$. Also, for both situations the equilibrium energy is $-4N^{2}$ and this is same as the large $N$ eigenvalue of $-C_{2}(SU(3))$ in the h.w. $(2N,0)$ irrep [also for the lowest weight $(0,2N)$ irrep]. Note that the eigenvalue of $C_{2}(SU(3))$ in a $SU(3)$ irrep $(\lambda\mu)$ is simply $\lambda^{2}+\mu^{2}+\lambda\mu+3(\lambda+\mu)$. Also, the formula in Eq. (28) is good in the limit $N\rightarrow\infty$ and in this limit $L\cdot L$ will not contribute as only terms of the order of $N^{2}$ will survive. Thus, $\alpha_{sd}=\pm 1$ will give prolate and oblate solutions and these results for $sd$IBM are well known Iac-87 ; RMP .

First non-trivial situation happens with $sdg$IBM and for this we will consider the three parameter coherent state used in KY ; Piet in terms of $(\beta_{2},\beta_{4},\gamma)$ parameters for a $N$ boson system,

[TABLE]

Note that $\beta_{2}\geq 0$, $-\infty\leq\beta_{4}\leq+\infty$ and $0^{\circ}\leq\gamma\leq 60^{\circ}$ respectively. Using the results given KY ; KY-rev , the $SU(3)$ energy functional is given by

[TABLE]

Note that {\mbox{\boldmath\alpha}}=(\alpha_{sd},\alpha_{dg}). Minimizing $E_{SU_{sdg}(3)}\left(N;\beta_{2},\beta_{4},\gamma\right)$ with respect to $\beta_{2}$, $\beta_{4}$ and $\gamma$ will give the equilibrium (ground state) shape parameters ($\beta_{2}^{0}$, $\beta_{4}^{0}$, $\gamma^{0})$ and the corresponding equilibrium energy $E^{0}_{SU_{sdg}(3)}$. Results are given in Table 6.

As seen from the Table 6, the four values of $(\alpha_{sd},\alpha_{dg})$ generate four combinations of $(\beta^{0}_{2},\beta^{0}_{4},\gamma^{0})$. These can be easily understood from the symmetries under $\beta_{2}\rightarrow-\beta_{2}$, $\beta_{4}\rightarrow-\beta_{4}$ and $\gamma=0^{\circ}\rightarrow 60^{\circ}$. We have for example $E(\beta_{2},\beta_{4},\gamma;\alpha_{sd}=1,\alpha_{dg}=1)=E(\beta_{2},-\beta_{4},\gamma;\alpha_{sd}=1,\alpha_{dg}=-1)$, $E(\beta_{2},\beta_{4},\gamma;\alpha_{sd}=1,\alpha_{dg}=1)=E(\beta_{2},\beta_{4},\gamma+60^{\circ};\alpha_{sd}=-1,\alpha_{dg}=-1)=E(-\beta_{2},-\beta_{4},\gamma;\alpha_{sd}=-1,\alpha_{dg}=1)=E(-\beta_{2},\beta_{4},\gamma;\alpha_{sd}=-1,\alpha_{dg}=-1)$. These also show that the solutions with $\gamma=60^{\circ}$ can be changed to $\gamma=0^{\circ}$ with $\beta_{2}\rightarrow-\beta_{2}$ as given in Table 6. More importantly, for all the four solutions, the $E^{0}_{SU_{sdg}(3)}=-16N^{2}$. This energy value is same as the large $N$ eigenvalue of $-C_{2}(SU(3))$ in $(4N,0)$ irrep. This then implies that the internal structure of the $(4N,0)$ irrep is different for the four solutions as discussed ahead. The energy functional is shown in Fig. 2 as a function of $\beta_{2}$ and $\beta_{4}$ for $\gamma=0^{\circ}$ and $60^{\circ}$ for the four choices of $(\alpha_{sd},\alpha_{gd})$.

IV.2 Large $N$ results for quadrupole moments and $B(E2)$’s

For further understanding of the four solutions for $SU_{sdg}(3)$, we have examined quadrupole moments and $B(E2)$ values in the ground $K=0$ band generated by the four solutions in Table 6. Note that the intrinsic state structure for the $K=0$ ground band is

[TABLE]

where $x_{0}=\sqrt{1/5}$, $x_{2}=\beta_{2}^{0}/\sqrt{5}$ and $x_{4}=\beta_{4}^{0}/\sqrt{5}$ with $\gamma^{0}=0^{\circ}$. It is easy to construct the angular momentum projected states $\left|N;K=0,L,M\right\rangle$ and calculate quadrupole moments $Q(L)$ and $B(E2;L\rightarrow L-2)$ for the ground band. The formulation for these is given in detail in Kuyu and valid to order $1/N^{2}$ where $N$ is the boson number. Then we have,

[TABLE]

In Eq. (32), the $t_{\ell^{\prime},\ell}$ are the coefficients in the $E2$ transition operator and they are chosen as,

[TABLE]

See Eq. (25) for $Q^{2}_{q}(\alpha_{sd}=+1,\alpha_{dg}=+1)$. Using the $T^{E2}$, the solutions in Table 6 and Eq. (32), results are obtained for $Q(2^{+}_{1})$, $Q(4^{+}_{1})$, $B(E2;2^{+}_{1}\rightarrow 0^{+}_{1})$ and $B(E2;4^{+}_{1}\rightarrow 2^{+}_{1})$ for a 10 boson system and the results are given in Table 7. It is seen that the $SU^{(+,+)}(3)$ and $SU^{(+,-)}(3)$ are closer generating prolate shape and $SU^{(-,-)}(3)$ generating oblate shape. The $SU^{(-,+)}(3)$ though generates prolate shape, the quadrupole moments are very small. Thus, $sdg$IBM substantiates the general structures observed in $sdg$ shell model examples presented in Section III.

V Conclusions

Multiple $SU(3)$ algebras appear in both shell model and interacting boson model spaces and they open a new paradigm in the applications of $SU(3)$ symmetry in nuclei. In the first detailed attempt made in this paper, using three $(sdg)$ space examples in SM, we showed that the four $SU(3)$ algebras in this space exhibit quite different properties with regard to quadrupole collectivity as brought out by the quadrupole moments $Q(J)$ and $B(E2)$’s in the ground $K=0$ band in even-even systems (see Tables III-V). The SM and DSM calculations are restricted to the examples with the leading $SU(3)$ irrep of the type $(\lambda 0)$. The prolate, oblate and intermediate structures from the four $SU(3)$ algebras found using SM and DSM is further substantiated by coherent state analysis and asymptotic formulas for quadrupole moments and $B(E2)$’s in the ground band in $sdg$IBM. Also, the results from $Q(J)$ and $B(E2)$’s for the four $SU(3)$ algebras are consistent with the correlation coefficients between the four different $Q.Q$ operators in the $sdg$ space of SM. Results in Tables III-V and VII may be useful in finding empirical examples for multiple $SU(3)$ algebras in $sdg$ and larger SM spaces and in $sdg$IBM.

Going beyond the present investigations, in future the structure of the low-lying $\gamma$ (also $\beta$) band generated by the multiple $SU(3)$ algebras will be investigated using SM and DSM. Here, we need to deal with the $SU(3)$ integrity basis operators that are 3 and 4-body, as the leading $SU(3)$ irrep in general will be of the type $(\lambda\mu)$ with $\mu\neq 0$ JPD-1 . For example, as seen from Table II, for $8$ nucleons with $T=0$ the leading $SU(3)$ irrep is $(24,4)$. Let us add that the method for dealing with 3-body operators in DSM was described in KS . In addition, applications of the $H_{Q}$’s in Eq. (16) to quantum phase transitions (QPT) may give new insights. For example, using H=\sum_{{\mbox{\boldmath\alpha}}}c_{\mbox{\boldmath\alpha}}Q^{2}({\mbox{\boldmath\alpha}})\cdot Q^{2}({\mbox{\boldmath\alpha}}) and varying the parameters c_{\mbox{\boldmath\alpha}}, it is possible to study QPT; for a similar study using multiple pairing algebras in SM and IBM see Ko-BJP . Also studies using Q^{2}({\mbox{\boldmath\alpha}},p)\cdot Q^{2}({\mbox{\boldmath\alpha}}^{\prime},n) with {\mbox{\boldmath\alpha}}\neq{\mbox{\boldmath\alpha}}^{\prime} and $p$ ($n$) denoting protons (neutrons) will be of interest; results of such a study in $sd$IBM are known DB-82 . In $sdg$IBM a more general CS in terms of $(\beta_{2},\beta_{4},\gamma,\gamma_{4},\delta_{4})$ given in Ydd ; Piet-cs ; Roz may prove to be important in understanding further the four $SU(3)$ algebras in this model. Also, it is possible to examine the properties of $\beta$ and $\gamma$ bands in this model using the results in Kuyu ; Kuyu-2 . All these will be addressed and the results will be reported in a future publication.

Acknowledgments

RS is thankful to SERB of DST of Government of India for financial support.