Higgs boson decay to the pair of S- and P-wave $B_c$ mesons

Abstract

We investigate rare production of the pair of S- and P-wave $B_c$-meson in the Higgs boson decay in the perturbative Standard Model and relativistic quark model. Relativistic amplitudes and decay widths are constructed with the account of the relative motion of heavy quarks forming $B_c$ mesons. When constructing the Higgs boson decay amplitudes, the method of projection operators on the S- and P-wave states of quarks is used. Relativistic corrections are expressed in terms of special relativistic parameters and are calculated numerically in the quark model. The dependence of the decay widths of the Higgs boson on various sources of relativistic corrections is investigated.

Appendices

Appendix A: Explicit form of the functions $f_i$, $\tilde{f}_i$, $g_i$, $\tilde{g}_i$ entering the Higgs boson decay widths (30)–(34)

The functions presented here are expressed in terms of mass ratios: $\tilde{r}_1=m_1/M$, $\tilde{r}_2=m_2/M$, $r_5=M_2/M$, $r_6=M_1/M$, $r_3=M_H/\sqrt{M_1M_2}$.

$$\begin{aligned}&f_1=- \frac{3}{4}r_6 + \frac{9}{4}r_5 + \frac{3}{4}\tilde{r}_1\tilde{r}_2^{-1}r_6 + \frac{9}{4}\tilde{r}_1\tilde{r}_2^{-1}r_5 + 3\tilde{r}_1 \nonumber \\&\quad - \frac{3}{2}\tilde{r}_1^2\tilde{r}_2^{-1} - \frac{3}{4}\rho _1r_5 \nonumber \\&\quad - \frac{3}{4}\rho _1\tilde{r}_1\tilde{r}_2^{-1}r_5 + \frac{3}{4}\eta _1r_6- \frac{3}{4}\eta _1\tilde{r}_1\tilde{r}_2^{-1}r_6, \end{aligned}$$

(A1)

$$\begin{aligned}&g_1= \tilde{\omega }_{\frac{1}{2}\frac{1}{2}}\left( \frac{3}{2}\tilde{r}_1 \right) +\tilde{\omega }_{01} \biggl (- \frac{3}{4}\tilde{r}_1\tilde{r}_2^{-1}r_6 - \frac{9}{4}\tilde{r}_1\tilde{r}_2^{-1}r_5 \nonumber \\&\quad + \frac{3}{2}\tilde{r}_1^2\tilde{r}_2^{-1} + \frac{3}{4}\tilde{r}_1^2\tilde{r}_2^{-1}r_6 + \frac{3}{4}\tilde{r}_1^2\tilde{r}_2^{-1}r_5 \biggr ) \nonumber \\&\quad +\tilde{\omega }_{10} \biggl ( \frac{3}{4}r_6 - \frac{9}{4}r_5 - \frac{3}{2}\tilde{r}_1 - \frac{3}{4}\tilde{r}_1r_6 + \frac{3}{4}\tilde{r}_1r_5 \biggr ) \nonumber \\&\quad +\omega _{\frac{1}{2}\frac{1}{2}}( 3\tilde{r}_1 )+\omega _{\frac{1}{2}\frac{1}{2}} \tilde{\omega }_{01}(- \frac{3}{2}\tilde{r}_1 )+ \omega _{\frac{1}{2}\frac{1}{2}}\tilde{\omega }_{10} (- \tilde{r}_1 ) \nonumber \\&\quad +\omega _{01}\biggl (- \frac{1}{4}r_6 + \frac{3}{4}r_5 - \tilde{r}_1 + \frac{1}{4}\tilde{r}_1r_6 - \frac{1}{4}\tilde{r}_1r_5 \biggr )\nonumber \\&\quad +\omega _{01}\tilde{\omega }_{01}\biggl (\frac{1}{4}r_6-\frac{3}{4}r_5 + \tilde{r}_1 -\frac{1}{4}\tilde{r}_1r_6 + \frac{1}{4}\tilde{r}_1r_5 \biggr ) \nonumber \\&\quad +\omega _{10}\biggl ( \frac{1}{4}\tilde{r}_1\tilde{r}_2^{-1}r_6 + \frac{3}{4}\tilde{r}_1\tilde{r}_2^{-1}r_5 \nonumber \\&\quad - \frac{3}{2}\tilde{r}_1 - \frac{1}{4}\tilde{r}_1^2\tilde{r}_2^{-1}r_6 - \frac{1}{4}\tilde{r}_1^2\tilde{r}_2^{-1}r_5 \biggr ) \nonumber \\&\quad +\omega _{10}\tilde{\omega }_{01} \biggl ( \frac{1}{2}\tilde{r}_1 \biggr )+\omega _{10}\tilde{\omega }_{10} \biggl (- \frac{1}{4}\tilde{r}_1\tilde{r}_2^{-1}r_6 - \frac{3}{4}\tilde{r}_1\tilde{r}_2^{-1}r_5 \nonumber \\&\quad + \frac{3}{2}\tilde{r}_1+ \frac{1}{4}\tilde{r}_1^2\tilde{r}_2^{-1}r_6 + \frac{1}{4}\tilde{r}_1^2\tilde{r}_2^{-1}r_5 \biggr ), \end{aligned}$$

(A2)

$$\begin{aligned}&f_2=\frac{3}{4}r_6 - \frac{1}{4}r_5 - \frac{3}{4}\tilde{r}_1\tilde{r}_2^{-1}r_6 - \frac{1}{4}\tilde{r}_1\tilde{r}_2^{-1}r_5 - \tilde{r}_1\nonumber \\&\quad + \frac{1}{2}\tilde{r}_1^2\tilde{r}_2^{-1} + \frac{1}{4}\rho _1r_5 + \frac{1}{4}\rho _1\tilde{r}_1\tilde{r}_2^{-1}r_5 - \frac{1}{4}\eta _1r_6 \nonumber \\&\quad + \frac{1}{4}\eta _1\tilde{r}_1\tilde{r}_2^{-1}r_6, \end{aligned}$$

(A3)

$$\begin{aligned}&g_2=\tilde{\omega }_{\frac{1}{2}\frac{1}{2}}\left( - \frac{1}{2}\tilde{r}_1 \right) + \tilde{\omega }_{01}\biggl ( \frac{3}{4}\tilde{r}_1\tilde{r}_2^{-1}r_6 + \frac{1}{4}\tilde{r}_1\tilde{r}_2^{-1}r_5\nonumber \\&\quad - \frac{1}{2}\tilde{r}_1^2\tilde{r}_2^{-1}- \frac{1}{4}\tilde{r}_1^2\tilde{r}_2^{-1}r_6 - \frac{1}{4} \tilde{r}_1^2\tilde{r}_2^{-1}r_5\biggr ) \nonumber \\&\quad + \tilde{\omega }_{10}\biggl (- \frac{3}{4}r_6 + \frac{1}{4}r_5 + \frac{1}{2}r_1 + \frac{1}{4}\tilde{r}_1r_6 - \frac{1}{4}\tilde{r}_1r_5\biggr ) + \omega _{\frac{1}{2}\frac{1}{2}} ( \tilde{r}_1 ) \nonumber \\&+ \omega _{\frac{1}{2}\frac{1}{2}}\tilde{\omega }_{01} \biggl ( \frac{1}{2}\tilde{r}_1\biggr ) + \omega _{\frac{1}{2}\frac{1}{2}}\tilde{\omega }_{10} (- \tilde{r}_1 )\nonumber \\&\quad + \omega _{01}\left( - \frac{3}{4}r_6 + \frac{1}{4} \right. \left. r_5 - \tilde{r}_1 + \frac{1}{4}\tilde{r}_1r_6 -\frac{1}{4}\tilde{r}_1r_5 \right) \nonumber \\&\quad + \omega _{01}\tilde{\omega }_{01}\left( \frac{3}{4}r_6 -\frac{1}{4}r_5 + \tilde{r}_1- \frac{1}{4}\tilde{r}_1r_6 + \frac{1}{4}\tilde{r}_1r_5 \right) \nonumber \\&\quad + \omega _{10}\biggl (\frac{3}{4}\tilde{r}_1\tilde{r}_2^{-1}r_6 + \frac{1}{4}\tilde{r}_1\tilde{r}_2^{-1}r_5 + \frac{1}{2}\tilde{r}_1 - \frac{1}{4} \tilde{r}_1^2\tilde{r}_2^{-1}r_6 \nonumber \\&\quad - \frac{1}{4}\tilde{r}_1^2\tilde{r}_2^{-1}r_5 \biggr ) + \omega _{10}\tilde{\omega }_{01}\left( \frac{1}{2}\tilde{r}_1 \right) + \omega _{10}\tilde{\omega }_{10}\biggl (-\frac{3}{4}\tilde{r}_1\tilde{r}_2^{-1}r_6 \nonumber \\&\quad - \frac{1}{4}\tilde{r}_1\tilde{r}_2^{-1}r_5 - \frac{1}{2}\tilde{r}_1+ \frac{1}{4}\tilde{r}_1^2\tilde{r}_2^{-1}r_6 + \frac{1}{4}\tilde{r}_1^2\tilde{r}_2^{-1}r_5 \biggr ), \end{aligned}$$

(A4)

$$\begin{aligned}&f_3=- \frac{1}{4}r_6 - \frac{1}{4}r_5 - \frac{1}{4}\tilde{r}_1\tilde{r}_2^{-1}r_6 + \frac{1}{4}\tilde{r}_1\tilde{r}_2^{-1}r_5 ), \end{aligned}$$

(A5)

$$\begin{aligned}&g_3=\tilde{\omega }_{01} \left( \frac{1}{4}\tilde{r}_1\tilde{r}_2^{-1}r_6 - \frac{1}{4}\tilde{r}_1\tilde{r}_2^{-1}r_5 \right) \nonumber \\&\quad +\tilde{\omega }_{10} \left( \frac{1}{4}r_6 + \frac{1}{4}r_5 \right) + \omega _{01} \left( - \frac{1}{12}r_6 - \frac{1}{12}r_5 \right) \nonumber \\&\quad + \omega _{01}\tilde{\omega }_{01}\left( \frac{1}{12}r_6 + \frac{1}{12}r_5 \right) \nonumber \\&\quad + \omega _{10} \left( -\frac{1}{12}\tilde{r}_1\tilde{r}_2^{-1}r_6 + \frac{1}{12}\tilde{r}_1\tilde{r}_2^{-1}r_5 \right) \nonumber \\&\quad + \omega _{10}\tilde{\omega }_{10} \left( \frac{1}{12}\tilde{r}_1\tilde{r}_2^{-1}r_6 - \frac{1}{12}\tilde{r}_1\tilde{r}_2^{-1}r_5\right) , \end{aligned}$$

(A6)

$$\begin{aligned}&f_4= \frac{3}{4}r_6 - \frac{1}{4}r_5 - \frac{3}{4}\tilde{r}_1\tilde{r}_2^{-1}r_6 - \frac{1}{4}\tilde{r}_1\tilde{r}_2^{-1}r_5 - \tilde{r}_1\nonumber \\&\quad + \frac{1}{2}\tilde{r}_1^2\tilde{r}_2^{-1} + \frac{1}{4}\rho _1r_5 + \frac{1}{4}\rho _1\tilde{r}_1\tilde{r}_2^{-1}r_5 - \frac{1}{4}\eta _1r_6\nonumber \\&\quad + \frac{1}{4}\eta _1\tilde{r}_1\tilde{r}_2^{-1}r_6, \end{aligned}$$

(A7)

$$\begin{aligned}&g_4=\tilde{\omega }_{\frac{1}{2}\frac{1}{2}} \left( - \frac{1}{2}\tilde{r}_1 \right) + \tilde{\omega }_{01}\biggl ( \frac{3}{4}\tilde{r}_1\tilde{r}_2^{-1}r_6 + \frac{1}{4}\tilde{r}_1\tilde{r}_2^{-1}r_5 \nonumber \\&\quad - \frac{1}{2}\tilde{r}_1^2\tilde{r}_2^{-1}- \frac{1}{4}\tilde{r}_1^2\tilde{r}_2^{-1}r_6 - \frac{1}{4}\tilde{r}_1^2\tilde{r}_2^{-1}r_5 \biggr ) \nonumber \\&\quad +\tilde{\omega }_{10}\left( - \frac{3}{4}r_6 + \frac{1}{4}r_5 + \frac{1}{2}\tilde{r}_1 + \frac{1}{4}\tilde{r}_1r_6 - \frac{1}{4}\tilde{r}_1r_5 \right) \nonumber \\&\quad + \omega _{\frac{1}{2}\frac{1}{2}} ( \tilde{r}_1 )+ \omega _{\frac{1}{2}\frac{1}{2}}\tilde{\omega }_{01}( \frac{1}{2}\tilde{r}_1 ) + \omega _{\frac{1}{2}\frac{1}{2}}\tilde{\omega }_{10}( - \tilde{r}_1 ) \nonumber \\&\quad + \omega _{01}\biggl ( - \frac{3}{4}r_6 + \frac{1}{4}r_5 - \tilde{r}_1 + \frac{1}{4}\tilde{r}_1r_6 - \frac{1}{4}\tilde{r}_1r_5 \biggr )\nonumber \\&\quad + \omega _{01}\tilde{\omega }_{01} \left( \frac{3}{4}r_6 - \frac{1}{4}r_5 + \tilde{r}_1 - \frac{1}{4}\tilde{r}_1r_6 + \frac{1}{4}\tilde{r}_1r_5 \right) \nonumber \\&\quad +\omega _{10}\biggl ( \frac{3}{4}\tilde{r}_1\tilde{r}_2^{-1}r_6 + \frac{1}{4}\tilde{r}_1\tilde{r}_2^{-1}r_5 + \frac{1}{2}r_1 - \frac{1}{4}\tilde{r}_1^2\tilde{r}_2^{-1}r_6 \nonumber \\&\quad - \frac{1}{4}\tilde{r}_1^2\tilde{r}_2^{-1}r_5 \biggr ) + \omega _{10}\tilde{\omega }_{01} \left( \frac{1}{2}\tilde{r}_1 \right) \nonumber \\&\quad + \omega _{10}\tilde{\omega }_{10}\biggl (-\frac{3}{4}\tilde{r}_1\tilde{r}_2^{-1}r_6 - \frac{1}{4}\tilde{r}_1\tilde{r}_2^{-1}r_5 - \frac{1}{2}\tilde{r}_1\nonumber \\&\quad + \frac{1}{4}\tilde{r}_1^2\tilde{r}_2^{-1}r_6 + \frac{1}{4}\tilde{r}_1^2\tilde{r}_2^{-1}r_5 \biggr ), \end{aligned}$$

(A8)

$$\begin{aligned}&f_5=- \frac{3}{2}r_6 + \frac{1}{2}r_5 - \frac{3}{2}\tilde{r}_1\tilde{r}_2^{-1}r_6 - \frac{1}{2}\tilde{r}_1\tilde{r}_2^{-1}r_5 + 2r_1 \nonumber \\&\quad + \tilde{r}_1^2\tilde{r}_2^{-1} - \frac{1}{2}\rho _1r_5 + \frac{1}{2}\rho _1\tilde{r}_1\tilde{r}_2^{-1}r_5 + \frac{1}{2}\eta _1r_6 \nonumber \\&\quad + \frac{1}{2}\eta _1\tilde{r}_1\tilde{r}_2^{-1}r_6, \end{aligned}$$

(A9)

$$\begin{aligned}&g_5= \tilde{\omega }_{\frac{1}{2}\frac{1}{2}} (- \tilde{r}_1 ) + \tilde{\omega }_{01}\biggl ( \frac{3}{2}\tilde{r}_1\tilde{r}_2^{-1}r_6 + \frac{1}{2}\tilde{r}_1 \tilde{r}_2^{-1}r_5 \nonumber \\&\quad - \tilde{r}_1^2\tilde{r}_2^{-1} - \frac{1}{2}\tilde{r}_1^2\tilde{r}_2^{-1}r_6 - \frac{1}{2}\tilde{r}_1^2\tilde{r}_2^{-1}r_5 \biggr ) \nonumber \\&\quad + \tilde{\omega }_{10} \left( \frac{3}{2}r_6 - \frac{1}{2}r_5 - \tilde{r}_1 - \frac{1}{2}\tilde{r}_1r_6 + \frac{1}{2}\tilde{r}_1r_5 \right) + \omega _{\frac{1}{2}\frac{1}{2}} ( - 4\tilde{r}_1 )\nonumber \\&\quad + \omega _{\frac{1}{2}\frac{1}{2}}\tilde{\omega }_{01}( \tilde{r}_1 ) + \omega _{\frac{1}{2}\frac{1}{2}}\tilde{\omega }_{10}( 2\tilde{r}_1 ) \nonumber \\&\quad + \omega _{01} ( - \frac{3}{2}r_6 + \frac{1}{2}r_5 - 2\tilde{r}_1 + \frac{1}{2}\tilde{r}_1r_6 - \frac{1}{2}\tilde{r}_1r_5 ) \nonumber \\&\quad + \omega _{01} \tilde{\omega }_{01} \left( \frac{3}{2}r_6 - \frac{1}{2}r_5 + 2\tilde{r}_1 - \frac{1}{2}\tilde{r}_1r_6 + \frac{1}{2}\tilde{r}_1 r_5 \right) + \nonumber \\&\quad \omega _{10} \biggl ( - \frac{3}{2}\tilde{r}_1\tilde{r}_2^{-1}r_6 - \frac{1}{2}\tilde{r}_1\tilde{r}_2^{-1}r_5 - r_1 \nonumber \\&\quad + \frac{1}{2}\tilde{r}_1^2\tilde{r}_2^{-1}r_6 + \frac{1}{2}\tilde{r}_1^2\tilde{r}_2^{-1}r_5 \biggr ) + \omega _{10}\tilde{\omega }_{01} ( \tilde{r}_1 ) \nonumber \\&\quad + \omega _{10}\tilde{\omega }_{10}\biggl ( \frac{3}{2}\tilde{r}_1\tilde{r}_2^{-1}r_6 + \frac{1}{2}\tilde{r}_1\tilde{r}_2^{-1}r_5 \nonumber \\&\quad + \tilde{r}_1 - \frac{1}{2}\tilde{r}_1^2\tilde{r}_2^{-1}r_6 - \frac{1}{2}\tilde{r}_1^2\tilde{r}_2^{-1}r_5 \biggr ). \end{aligned}$$

(A10)

The functions $\tilde{f}_i$, $\tilde{g}_i$ can be obtained from $f_i$, $g_i$ as a result of the replacement $\tilde{r}_1\leftrightarrow \tilde{r}_2$, $\tilde{\omega }_{ij}\leftrightarrow \tilde{\omega }_{ji}$.

Appendix B: The construction of an effective Hamiltonian

Using potential (33), we construct an effective model of the quark interaction of the Schrödinger-type and calculate the relativistic parameters included in the decay widths. The first step in the construction of the model is related to the rationalization of the kinetic energy operator as follows:

$$\begin{aligned}&T=\sqrt{\mathbf{p}^2+m_1^2}-m_1+\sqrt{\mathbf{p}^2+m_2^2}-m_2\nonumber \\&\quad = \frac{\mathbf{p}^2}{\sqrt{\mathbf{p}^2 +m_1^2}+m_1}+\frac{\mathbf{p}^2}{\sqrt{\mathbf{p}^2+m_2^2}+m_2} \nonumber \\&\quad \approx \frac{\mathbf{p}^2}{E_1+m_1}+\frac{\mathbf{p}^2}{E_2+m_2} \quad = \frac{\mathbf{p}^2}{2\tilde{m}_1}\nonumber \\&\quad \qquad \qquad \qquad \qquad +\frac{\mathbf{p}^2}{2\tilde{m}_2}=\frac{\mathbf{p}^2}{2\tilde{\mu }}, \end{aligned}$$

(B1)

where we change relativistic particle energies $\varepsilon _{1,2}(\mathbf{p})$ by their effective values $E_{1,2}$ so that $M_{B_c}=E_1+E_2$ or what is the same $\mathbf{p}^2\rightarrow \mathbf{p}^2_{eff}$. The effective quark masses $\tilde{m}_{1,2}$ are used further in the program for the numerical solution of the Schrödinger equation. $\mathbf{p}^2_{eff}$ should be considered as a new parameter which effectively accounts for relativistic corrections.

Another part of the relativistic corrections in the Breit Hamiltonian is determined by the term:

$$\begin{aligned} \Delta \tilde{U}=-\frac{2\alpha _s}{3m_1m_2r}\left[ \mathbf{p}^2-\frac{d^2}{dr^2}\right] . \end{aligned}$$

(B2)

In order to replace it by the effective term containing the power-like potentials, we use the approximate bound state wave functions which can be written for S- and P-wave states as follows:

$$\begin{aligned} \Psi _{{\mathcal {S}}}(\mathbf{r})= & {} \frac{\beta ^{3/2}}{\pi ^{3/4}}e^{-\frac{1}{2}\beta ^2r^2},~~ \Psi _{{\mathcal {P}}}(\mathbf{r})\nonumber \\= & {} \sqrt{\frac{8}{3}}\frac{\beta ^{3/2}}{\pi ^{3/4}}\beta re^{-\frac{1}{2}\beta ^2r^2}Y_{1m}(\theta ,\phi ). \end{aligned}$$

(B3)

The parameter $\beta$ entering here is chosen in such a way that the calculated value of $R_S(0)$ and $R'_P(0)$ is obtained. The operator $\mathbf{p}^2$ is changed by its nonrelativistic expression: $\mathbf{p}^2=2\mu \left( E_{nr}+\frac{4\alpha _s}{3r}-Ar-B\right)$, where nonrelativistic energy $E_{nr}$ is obtained by the numerical solution of the Schrödinger equation in the nonrelativistic approximation.

In the case of S-wave states it is necessary to change the $\delta$-like term of the potential by known smeared $\delta$-function of the Gaussian form:

$$\begin{aligned} \tilde{\delta }(\mathbf{r})=\frac{b^3}{\pi ^{3/2}}e^{-b^2r^2} \end{aligned}$$

(B4)

with the additional parameter b which defines the hyperfine splitting in the $(b{\bar{c}})$ system.

The second step in constructing an effective model of quark interaction is connected with the angular averaging of the spin-orbit and spin-spin terms. This averaging is performed separately for the considered S- and P-states [49]. In this case, we use a basis transformation of the following form:

$$\begin{aligned} \Psi _{SLFM_F}= & {} \sum _{J}(-1)^{s_1+s_2+L+F}\sqrt{(2S+1)(2J+1)}\nonumber \\&\quad \times \Biggl \{ \begin{array}{ccc} L&{}~s_1&{}~J\\ s_2&{}~F&{}~S \end{array} \Biggr \}\Psi _{Js_2FM_F}, \end{aligned}$$

(B5)

where $\mathbf{J}=\mathbf{L}+\mathbf{s_1}$ is the total nomentum of quark 1, $\mathbf{S}=\mathbf{s}_1+\mathbf{s}_2$, $\mathbf{F}=\mathbf{S}+\mathbf{L}=\mathbf{J}+\mathbf{s}_2$. For diagonal matrix elements we have the expression:

$$\begin{aligned}&<Js_2FM_F|[(\mathbf{L}{} \mathbf{s}_2)+3(\mathbf{s}_1\mathbf{n})(\mathbf{s}_2\mathbf{n})-(\mathbf{s}_1\mathbf{s}_2)]|Js_2FM_F>\nonumber \\&\quad = \frac{L(L+1)}{J(J+1)}[F(F+1)-J(J+1)-s_2(s_2+1)]. \end{aligned}$$

(B6)

The matrix elements of the spin-orbit $T_1=\mathbf{L}{} \mathbf{s}_2$ and spin-spin interactions $T_2=3(\mathbf{s}_1\mathbf{n})(\mathbf{s}_2\mathbf{n})-(\mathbf{s}_1\mathbf{s}_2)$ off-diagonal in J with $F=1$ are determined as follows:

$$\begin{aligned}&<J's_2FM_F|T_1|Js_2FM_F>\nonumber \\&\quad = (-1)^{-J-F-s_2+L+3/2+J'}\sqrt{(2J'+1)(2J+1)} \nonumber \\&\quad \times \sqrt{(2s_2+1)(s_2+1)s_2(2L+1)(L+1)L} \Biggl \{ \begin{array}{ccc} J&{}~s_2&{}~F\\ s_2&{}~J'&{}~1 \end{array} \Biggr \}\nonumber \\&\quad \times \Biggl \{ \begin{array}{ccc} L&{}~J'&{}~1/2\\ J&{}~L&{}~1 \end{array} \Biggr \}. \end{aligned}$$

(B7)

As a result $<J's_2FM_F|T_1|Js_2FM_F>=2<J's_2FM_F|T_2|Js_2FM_F>=-\frac{\sqrt{2}}{3}$. After angular averaging an effective Hamiltonians are obtained describing the states $^3S_1$, $^1S_0$, $^3P_J$, $^1P_1$.

Finally, the third final step is related to the numerical solution of the Schrödinger equation with obtained effective Hamiltonians. We use a calculation program in the system Mathematica [45]. The obtained numerical results are presented in Table 1. They are in good agreement with previous calculations [33, 36, 37].