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--> --> -->In this work, we will study whether the bottom meson loops mechanism can produce the
The remainder of this paper is organized as follows. In Sec. 2, the theoretical framework is described in detail. In Sec. 3, we provide the theoretical predictions for the decay branching fractions of
2.1.Lagrangians
To calculate the contribution of the mechanism $ {\cal L}_{Z_b\Upsilon\pi} = \sum\limits_{j = 1,2}C_{Z_{bj}\Upsilon(mS)\pi} \Upsilon^i(mS) \langle {Z^i_{bj}}^\dagger u_\mu \rangle v^\mu +\mathrm{h.c.} \,, $ ![]() | (1) |
$ {\cal L}_{Z_b h_b\pi} = \sum\limits_{j = 1,2}g_{Z_{bj}h_b(nP)\pi}\epsilon_{ijk} \langle {Z^i_{bj}}^\dagger u^j \rangle h_b^k +\mathrm{h.c.} \,, $ ![]() | (2) |
$ Z^i_{bj} = \left( {\begin{array}{*{2}c} \dfrac{1}{\sqrt{2}}Z^{0i}_{bj} & Z^{+i}_{bj} \\ Z^{-i}_{bj} & -\dfrac{1}{\sqrt{2}}Z^{0i}_{bj} \end{array}} \right)\,. $ ![]() | (3) |
$\begin{split}{u_\mu } = &{\rm i}\left( {{u^\dagger }{\partial _\mu }u{\mkern 1mu} - {\mkern 1mu} u{\partial _\mu }{u^\dagger }} \right){\mkern 1mu} ,\;\;\;\;u = \exp (\dfrac{{{\rm i}\Phi }}{{\sqrt 2 {F_\pi }}}){\mkern 1mu} ,\\\Phi =& \left( {\begin{array}{*{20}{c}}{\frac{1}{{\sqrt 2 }}{\pi ^0}}&{{\pi ^ + }}\\{{\pi ^ - }}&{ - \frac{1}{{\sqrt 2 }}{\pi ^0}}\end{array}} \right),\end{split}$ ![]() | (4) |
To calculate the box diagrams, we need the Lagrangian for the coupling of the
$ {\cal L}_{\Upsilon HH} = \frac{{\rm i}\, g_{JHH}}{2} \langle J^\dagger H_a {{\sigma}}\cdot \!\overleftrightarrow{\partial}\! \bar{H}_a \rangle + {\rm h.c.}\,, $ ![]() | (5) |
$ {\cal L}_{h_b HH} = \frac{{\rm i}\, g_{1}}{2} \langle h_b^{\dagger i} H_a \sigma^i\bar{H}_a \rangle + {\rm h.c.}\,, $ ![]() | (6) |
$ {\cal L}_{HH\Phi} = \frac{g_\pi}{2} \langle \bar{H}_a^\dagger {{\sigma}} \cdot {{u}}_{ab} \bar{H}_b \rangle -\frac{g_\pi}{2} \langle H_a^\dagger H_b {{\sigma}} \cdot {{u}}_{ba} \rangle, $ ![]() | (7) |
2
2.2.Power counting of loops
Because theThere are five different kinds of loop contributions, namely the box diagrams displayed in Fig. 1(b), (c), triangle diagrams displayed in Fig. 2(a)–(c), and the bubble loop in Fig. 2(d). We analyze them one by one as follows:
![](http://cpc.ihep.ac.cn/fileZGWLC/journal/article/zgwlc/2020/2/PIC/CPC-2019-0317-1.jpg)
![](http://cpc.ihep.ac.cn/fileZGWLC/journal/article/zgwlc/2020/2/PIC/CPC-2019-0317-2.jpg)
First, we analyze the power counting of the box diagrams, namely Fig. 1(b), (c). As shown in Eq. (7), the vertex of
For the triangle diagram Fig. 2(a), the leading
In Fig. 2(b), the leading
The leading
In Fig. 2(d), both the initial and final vertices are proportional to
Therefore, we expect that the ratios of the contributions of the box diagrams, triangle diagram Fig. 2(a)–(c), and the bubble loop Fig. 2(d) are
$ \begin{aligned} \frac{ q_\pi^3}{\nu^3} : \frac{ m_B q_\pi^2}{\nu} : \frac{ q_\pi^3}{\nu} : \frac{ q_\pi^3}{\nu} : m_B q_\pi^2\nu = 1 : \frac{m_B \nu^2}{q_\pi} : \nu^2 : \nu^2 : \frac{m_B \nu^4}{q_\pi}\,, \end{aligned}$ ![]() | (8) |
2
2.3.Tree-level amplitudes and box diagram calculation
The decay amplitude for $ \Upsilon(mS)(p_a) \to h_b(nP)(p_b) \pi(p_c)\pi(p_d) $ ![]() | (9) |
$ s = (p_c+p_d)^2 , \quad t = (p_a-p_c)^2\,, \quad u = (p_a-p_d)^2 \,. $ ![]() | (10) |
$ M_{Zb} = \frac{2 \sqrt{m_{\Upsilon(mS)}m_{h_b(nP)}}}{F_\pi^2}\epsilon_{abj}\epsilon_{\Upsilon(mS)}^a\epsilon_{h_b(nP)}^b\sum\limits_{i = 1,2}m_{Z_{bi}} C_{Z_{bi}\Upsilon(mS)\pi}g_{Z_{bi}h_b(nP)\pi} \bigg\{ p_c^0 p_d^j\frac{1}{t-m_{Z_{bi}}^2}+p_d^0 p_c^j\frac{1}{u-m_{Z_{bi}}^2} \bigg\}\,. $ ![]() | (11) |
Now, we discuss the calculation of the box diagrams. In the box diagrams Fig. 1(b) and (c), we denote the top left intermediate bottom meson as M1, and the other intermediate bottom mesons as M2, M3, and M4, in counterclockwise order. For the pseudoscalar or vector content of
For the tensor reduction of the loop integrals, it is convenient to define
$ \begin{split} M_{\rm{loop}} =& \epsilon_{\Upsilon(mS)}^a\epsilon_{h_b(nP)}^b \Big\{ \epsilon_{abi}q^i A_1+ \epsilon_{abi}q_\perp^i A_2 +\epsilon_{bij}q^i q_\perp^j q_\perp^a A_3 \\&+\epsilon_{bij}q^i q_\perp^j q^a A_4 +\epsilon_{aij}q^i q_\perp^j q_\perp^b A_5 +\epsilon_{aij}q^i q_\perp^j q^b A_6 \Big\}\,. \end{split}$ ![]() | (12) |
The decay width for
$ \Gamma = \int_{s_-}^{s_+} \int_{t_-}^{t_+}\frac{|M_{Zb}+M_{\rm{loop}}|^2 {\rm d}s {\rm d}t}{768 \pi^3 m_{\Upsilon(mS)}^3}\,, $ ![]() | (13) |
$ \begin{split}s_- =& 4m_\pi^2,\\s_+ =& (m_{\Upsilon(mS)}-m_{h_b(nP)})^2,\\ t_{\pm} =& \frac{1}{4s}\Big\{ (m_{\Upsilon(mS)}^2-m_{h_b(nP)}^2)^2\\&-\big[\lambda^{\frac{1}{2}}(s,m_\pi^2,m_\pi^2)\mp \lambda^{\frac{1}{2}}(m_{\Upsilon(mS)}^2,s,m_{h_b(nP)}^2)\big]^2\Big\},\\ \lambda(a,b,c) = & a^2+b^2+c^2-2(ab+ac+bc)\,. \end{split}$ ![]() | (14) |
$ |g_{Z_{b}h_b\pi}| = \Bigg\{\frac{6\pi F_\pi^2 m_{Z_b} \Gamma_{Z_b \to h_b\pi}}{|{{p}}_f|^3 m_{h_b}} \Bigg\}^{\frac{1}{2}} \,, $ ![]() | (15) |
$ \begin{split} |g_{Z_{b1}h_b(1P)\pi}^{\rm{naive}}| =& 0.019 \pm 0.003\,, \\ |g_{Z_{b2}h_b(1P)\pi}^{\rm{naive}}| =& 0.021 \pm 0.003\,, \\ |g_{Z_{b1}h_b(2P)\pi}^{\rm{naive}}| =& 0.068 \pm 0.011\,, \\ |g_{Z_{b2}h_b(2P)\pi}^{\rm{naive}}| = &0.077 \pm 0.010\,. \end{split}$ ![]() | (16) |
$ \begin{split} |g_{Z_{b}h_b(1P)\pi}|\simeq 0.057 \,,\quad |g_{Z_{b}h_b(2P)\pi}|\simeq 0.204 \,. \end{split}$ ![]() | (17) |
In the calculation of the box diagrams, the coupling strength
By using the coupling strengths above, we can predict the decay branching fractions of
$ \begin{split} & {{BR}}_{\Upsilon(4S) \to h_b(1P)\pi^+\pi^-}\simeq \big(1.2^{+0.8}_{-0.4}\times10^{-6}\big) \quad {\rm{or}} \quad \big( 0.5^{+0.5}_{-0.2}\times10^{-6}\big) \,, \\ &{{BR}}_{\Upsilon(4S) \to h_b(2P)\pi^+\pi^-}\simeq \big(7.1^{+1.7}_{-1.1}\times10^{-10}\big) \quad {\rm{or}} \quad\big( 2.4^{+0.2}_{-0.1}\times10^{-10}\big)\,. \end{split}$ ![]() | (18) |
$ \begin{split}& {{BR}}_{\Upsilon(4S) \to h_b(1P)\pi^+\pi^-}^{Z_b} = 0.6^{+0.1}_{-0.1}\times10^{-7}\,, \\ &{{BR}}_{\Upsilon(4S) \to h_b(1P)\pi^+\pi^-}^{\rm{Box}} = 0.8^{+0.7}_{-0.3}\times10^{-6}\,. \end{split}$ ![]() | (19) |
In Fig. 3, we plot the distributions of the
![](http://cpc.ihep.ac.cn/fileZGWLC/journal/article/zgwlc/2020/2/PIC/CPC-2019-0317-3.jpg)
The
Owing to the similarity between the bottomonium and charmonium families, we can extend the box diagrams calculation to give a rough estimation of the branch fractions of the
$ \begin{split}& {{BR}}_{\psi(3S) \to h_c(1P)\pi^+\pi^-}^{\rm{Box}} = 2.9\times 10^{-5}\,, \\ &{{BR}}_{\psi(4S) \to h_c(1P)\pi^+\pi^-}^{\rm{Box}} = 4.5\times10^{-3}\,. \end{split} $ ![]() | (20) |
2
A.1.Scalar four-point integrals
For the first topology, as shown in Fig. A1, the scalar integral evaluated for the initial bottomonium at rest (![](http://cpc.ihep.ac.cn/fileZGWLC/journal/article/zgwlc/2020/2/PIC/CPC-2019-0317-4.jpg)
We are grateful to the referees' useful suggestions and constructive remarks, which helped formulate the present version of this manuscript. We are grateful to Martin Cleven for the collaboration in the early stages of this study. We acknowledge Guo-Ying Chen, Meng-Lin Du, and Qian Wang for their helpful discussions, and Feng-Kun Guo for a careful reading of the manuscript and valuable comments.
$\tag{A1} \begin{split} J_1^{(0)} \equiv & {\rm i}\int \!\frac{{\rm d}^4l}{(2\pi)^4}\frac{1}{[l^2-m_1^2+{\rm i}\epsilon][(p-l)^2-m_2^2+{\rm i}\epsilon][(l-q_1-q_2)^2-m_3^2+i\epsilon][(l-q_1)^2-m_4^2+{\rm i}\epsilon]}\simeq\frac{-{\rm i}}{16m_1m_2m_3m_4}\\ &\times \int \! \frac{{\rm d}^4l}{(2\pi)^4} \frac{1}{\left[l^0-\frac{{{l}}^2}{2m_1}-m_1+{\rm i}\epsilon\right]\left[l^0-M+\frac{{{l}}^2}{2m_2}+m_2-{\rm i}\epsilon\right]} \frac{1}{\left[l^0-q_1^0-q_2^0-\frac{({\bf{l}}+{{q}})^2}{2m_3}-m_3+{\rm i}\epsilon\right]\left[l^0-q_1^0-\frac{({\bf{l}}-{{{q}}_1})^2}{2m_4}-m_4+{\rm i}\epsilon\right]} . \end{split} $ ![]() |
$\tag{A2} -\frac{\mu_{12}\mu_{23}\mu_{24}}{2m_1m_2m_3m_4}\int \! \frac{{\rm d}^3l}{(2\pi)^3}\frac{1}{ [{{l}}^2+c_{12}-{\rm i}\epsilon][{{l}}^2+2\frac{\mu_{23}}{m_3}{{l}}\cdot {{q}}+c_{23}-{\rm i}\epsilon][{{l}}^2-2\frac{\mu_{24}}{m_4}{{l}}\cdot {{q}}_1+c_{24}-{\rm i}\epsilon]} , $ ![]() |
$ \tag{A3}\begin{split} c_{12} \equiv & 2\mu_{12}\left(m_1+m_2-M\right), \quad c_{23}\equiv 2\mu_{23}\left(m_2+m_3-M+q_1^0+q_2^0+\frac{ {{q}}^2}{2m_3}\right), \\ c_{24} \equiv & 2\mu_{24}\left(m_2+m_4-M+q^0_1+\frac{ {{q}}_1^2}{2m_4}\right), \quad \mu_{ij} = \frac{m_im_j}{m_i+m_j}. \end{split} $ ![]() |
$\tag{A4}\begin{split} J_2^{(0)} =& -\frac{\mu_{12}\mu_{23}\mu_{24}}{2m_1m_2m_3m_4} \int \frac{{\rm d}^3l}{(2\pi)^3}\\&\times\frac{1}{ [{{l}}^2+c_{12}-{\rm i}\epsilon][{{l}}^2+2\frac{\mu_{23}}{m_3}{{l}}\cdot {{q}}+c_{23}-{\rm i}\epsilon][{{l}}^2-2\frac{\mu_{24}}{m_4}{{l}}\cdot {{q}}_2+c_{24}^\prime-{\rm i}\epsilon]} , \end{split}$ ![]() |
$ \tag{A5} c_{24}^\prime \equiv 2\mu_{24}\left(m_2+m_4-M+q^0_2+\frac{ {{q}}_2^2}{2m_4}\right)\,. $ ![]() |
$\tag{A6}\begin{split} J_3^{(0)} \equiv& {\rm i}\int \!\frac{{\rm d}^4l}{(2\pi)^4}\frac{1}{[l^2-m_1^2+{\rm i}\epsilon][(p-l)^2-m_2^2+{\rm i}\epsilon][(p-q_2-l)^2-m_3^2+{\rm i}\epsilon][(l-q_1)^2-m_4^2+{\rm i}\epsilon]}\\ \simeq &\frac{-{\rm i}}{16m_1m_2m_3m_4}\int \! \frac{{\rm d}^4l}{(2\pi)^4} \frac{1}{\left[l^0-\frac{{{l}}^2}{2m_1}-m_1+{\rm i}\epsilon\right]\left[l^0-M+\frac{{{l}}^2}{2m_2}+m_2-{\rm i}\epsilon\right]}\times \frac{1}{\left[l^0+q_2^0-M+\frac{({\bf{l}}+{{{q}}_2})^2}{2m_3}+m_3-{\rm i}\epsilon\right]\left[l^0-q_1^0-\frac{({\bf{l}}-{{{q}}_1})^2}{2m_4}-m_4+{\rm i}\epsilon\right]}. \end{split} $ ![]() |
$\tag{A7} \begin{split} -\frac{\mu_{12}\mu_{34}}{2m_1m_2m_3m_4}\int \! \frac{{\rm d}^3l}{(2\pi)^3}\frac{1}{ [{{l}}^2+d_{12}-{\rm i}\epsilon][{{l}}^2-2\frac{\mu_{34}}{m_4}{{l}}\cdot {{{q}}_1}-2\frac{\mu_{34}}{m_3}{{l}}\cdot {{{q}}_2}+d_{34}-{\rm {\rm i}}\epsilon]}\times\left[\frac{\mu_{24}}{[{\bf{l}}^2-2\frac{\mu_{24}}{m_4}{{l}}\cdot {{q}}_1+d_{24}-{\rm i}\epsilon]}+\frac{\mu_{13}}{[{{l}}^2+2\frac{\mu_{13}}{m_3}{{l}}\cdot {{q}}_2+d_{13}-{\rm i}\epsilon]} \right], \end{split}$ ![]() |
$\tag{A8}\begin{split}d_{12}\equiv {2\mu_{12}\left(m_1+m_2-M\right),}\;\;\;\;\;{d_{34}\equiv 2\mu_{34}\left(m_3+m_4-q^0+\frac{ {{q}}_1^2}{2m_4}+\frac{ {{q}}_2^2}{2m_3}\right),}\;\;\;d_{24}\equiv { 2\mu_{24}\left(m_2+m_4-M+q_1^0+\frac{ {{q}}_1^2}{2m_4}\right),}\;\;\;{d_{13}\equiv 2\mu_{13}\left(m_1+m_3-M+q_2^0+\frac{ {{q}}_2^2}{2m_3}\right). }\end{split}$ ![]() |
2
A.2.Tensor reduction
Because the $\tag{A9} \frac{-\mu_{12}\mu_{23}\mu_{24}}{2m_1m_2m_3m_4}\int \! \frac{{\rm d}^3l}{(2\pi)^3}\frac{f(l)}{ [{{l}}^2+c_{12}-{\rm i}\epsilon][{{l}}^2+2\frac{\mu_{23}}{m_3}{{l}}\cdot {{q}}+c_{23}-{\rm i}\epsilon][{{l}}^2-2\frac{\mu_{24}}{m_4}{{l}}\cdot {{{q}}_1}+c_{24}-{\rm i}\epsilon]}, $ ![]() |
$ \tag{A10} J_1^{(1)i} = \frac{-\mu_{12}\mu_{23}\mu_{24}}{2m_1m_2m_3m_4}\int \! \frac{{\rm d}^3l}{(2\pi)^3}\frac{l^i}{[{{l}}^2+c_1-{\rm i}\epsilon][{{l}}^2-2\frac{\mu_{23}}{m_3}{{l}}\cdot {{q}}+c_2-{\rm i}\epsilon][{{l}}^2-2\frac{\mu_{24}}{m_4}{{l}}\cdot {{q}}_1+c_3-{\rm i}\epsilon]} \equiv q^iJ_1^{(1)}+q_{1\perp}^iJ_1^{(2)}\,, $ ![]() |
2
A.3.Amplitudes
We define the scalar integralsWe give the amplitude of the box diagrams for the
$\tag{A11}\begin{split} A_1 = &\frac{8 g_1 g_{JHH}g_\pi^2}{F_\pi^2 {{q}}^2} \bigg\{ {{q}}^2 \Big\{{{{p}}_c}\cdot {{{p}}_d} \big[J1(1,1,3)+J1(2,1,3)+J1(3,1,8)\big]+{{{p}}_c}\cdot {{q}} \big[J1(1,1,9)+J1(1,1,11) -J1(2,1,12)+J1(3,1,9)+J1(3,1,11)\big]+ {{q}}_\perp^2 \big[J1(1,2,9)\\&+J1(1,2,11)-J1(2,2,12)+J1(3,2,9) +J1(3,2,11)\big]\Big\}+{{{p}}_c}\cdot {{q}} \Big\{{{{p}}_c}\cdot {{q}} \big[J1(1,1,9)+J1(1,1,11)-J1(2,1,12)+J1(3,1,9)+J1(3,1,11)\big] \\&+{{{p}}_d}\cdot {{q}} \big[J1(1,1,12)-J1(2,1,9)-J1(2,1,11)+J1(3,1,10)\big]+ {{q}}_\perp^2 \big[J1(1,2,9)+J1(1,2,11) -J1(1,2,12)+J1(2,2,9)+J1(2,2,11)-J1(2,2,12)\\&+J1(3,2,9)-J1(3,2,10)+J1(3,2,11)\big]\Big\}\bigg\}\,, \end{split}$ ![]() |
$\tag{A12}\begin{split} A_2 =& \frac{8 g_1 g_{JHH}g_\pi^2}{F_\pi^2} \Big\{{{{p}}_c}\cdot {{{p}}_d} \big[J1(1,2,3)+J1(2,2,3)+J1(3,2,8)\big]+{{{p}}_c}\cdot {{q}} \big[J1(1,1,9)+J1(1,1,11) -J1(2,1,12)+J1(3,1,9)+J1(3,1,11)\big]\\&+{{{p}}_d}\cdot {{q}} \big[J1(1,1,12)-J1(2,1,9)-J1(2,1,11) +J1(3,1,10)\big]+ {{q}}_\perp^2\big[J1(1,2,9)+J1(1,2,11)-J1(1,2,12)+J1(2,2,9)\\&+J1(2,2,11) -J1(2,2,12)+J1(3,2,9)-J1(3,2,10)+J1(3,2,11)\big]\Big\}\,, \end{split}$ ![]() |
$\tag{A13}\begin{split} A_3 =& -\frac{8 g_1 g_{JHH}g_\pi^2}{F_\pi^2 {{q}}^2} \Big\{ {{q}}^2 \big[-J1(1,1,9)+J1(1,1,11)-J1(1,1,12)-J1(1,2,2)+J1(1,2,9)+J1(1,2,10) +J1(1,2,12)-J1 (2,1,9)+J1(2,1,11)-J1(2,1,12)\\&+J1(2,2,2)-J1(2,2,10)-J1(2,2,11) -J1(3,1,9)-J1(3,1,10)+J1(3,1,11)-J1(3,2,2)+J1(3,2,9)+J1(3,2,10)+J1 (3,2,12)\big] \\&+{{{p}}_c}\cdot {{q}} \big[J1(1,2,9)-J1(1,2,11)+J1(1,2,12)+J1(2,2,9)-J1(2,2,11)+J1(2,2,12) +J1(3,2,9)+J1(3,2,10)-J1(3,2,11)\big]\Big\}\,, \end{split}$ ![]() |
$\tag{A14}\begin{split} A_4 =& \frac{8 g_1 g_{JHH}g_\pi^2}{F_\pi^2 {{q}}^4} \Big\{ {{q}}^4 \big[J1(1,1,2)-J1(1,1,10)-J1(1,1,11)-J1(2,1,2)+J1(2,1,9)+J1(2,1,10) +J1(2,1,12)+J1 (3,1,2)-J1(3,1,11)-J1(3,1,12)\big]\\&+ {{q}}^2 {{{p}}_c}\cdot {{q}} \big[J1(1,1,9)-J1(1,1,11) +J1(1,1,12)-J1(1,2,9)+J1(1,2,11)-J1(1,2,12)+J1 (2,1,9)-J1(2,1,11)+J1(2,1,12) -J1(2,2,9)\\&+J1(2,2,11)-J1(2,2,12)+J1(3,1,9)+J1(3,1,10)-J1(3,1,11)-J1(3,2,9) -J1(3,2,10)+J1(3,2,11)\big)- ({{{p}}_c}\cdot {{q}})^2 \big[J1(1,2,9)-J1(1,2,11)\\&+J1(1,2,12)+J1(2,2,9) -J1(2,2,11)+J1(2,2,12)+J1(3,2,9)+J1(3,2,10)-J1(3,2,11)\big]\Big\}\,, \end{split}$ ![]() |
$\tag{A15}\begin{split} A_5 =& \frac{8 g_1 g_{JHH}g_\pi^2}{F_\pi^2 {{q}}^2} \Big\{ {{q}}^2 \big[-J1(1,1,6)+J1(1,2,8)+J1(1,2,10)-J1(2,1,6)+J1(2,2,6)-J1(2,2,8) -J1(2,2,10)-J1 (3,1,3)\\&+J1(3,1,6)-J1(3,1,8)+J1(3,2,3)+J1(3,2,12)\big]+{{{p}}_c}\cdot {{q}} \big[J1(1,2,6)+J1(2,2,6)+J1(3,2,3)-J1(3,2,6)+J1(3,2,8)\big]\Big\}\,, \end{split}$ ![]() |
$\tag{A16}\begin{split} A_6 =& -\frac{8 g_1 g_{JHH}g_\pi^2}{F_\pi^2 {{q}}^4} \Big\{ {{q}}^4 \big[J1(1,1,6)-J1(1,1,8)-J1(1,1,10)+J1(2,1,8)+J1(2,1,10)-J1(3,1,6) +J1(3,1,8)-J1 (3,1,12)\big]\\&+ {{q}}^2 {{{p}}_c}\cdot {{q}} \big[J1(1,1,6)-J1(1,2,6)+J1(2,1,6)-J1(2,2,6)+J1(3,1,3) -J1(3,1,6)+J1(3,1,8)-J1(3,2,3)+J1 (3,2,6)-J1(3,2,8)\big]\\&-({{{p}}_c}\cdot {{q}})^2 \big[J1(1,2,6)+J1(2,2,6) +J1(3,2,3)-J1(3,2,6)+J1(3,2,8)\big]\Big\}\,. \end{split} $ ![]() |