The p-h channel

In the p-h energy density, indices of the Pauli matrices are contracted directly with density matrices of particles 1 and 2, and immediately give non-local densities through appropriate traces in Eqs. (20)-(26). However, the relative momentum operators (124) affect both particles at the same time, and hence have to be first recoupled to forms where the two particles are acted upon independently, i.e.,

$${\textstyle{\frac{1}{2}}}\big(\hat{\mbox{{\boldmath {$k$}}}}^{\prime2}+\hat{\mbox{{\boldmath {$k$}}}}^2\big) = {\textstyle{\frac{1}{8}}}\big(\hat{\mbox{{\boldmath {$K$}}}}_1^2+... ..._1 +\mbox{{\boldmath {$\nabla$}}}_2\cdot\mbox{{\boldmath {$\nabla$}}}'_2\big) ,$$ (129)

$${\hat{\mbox{{\boldmath {$k$}}}}}^{\prime *} \cdot\hat{\mbox{{\boldmath {$k$}}}} = {\textstyle{\frac{1}{8}}}\big(\hat{\mbox{{\boldmath {$K$}}}}_1 \c... ..._1 +\mbox{{\boldmath {$\nabla$}}}_2\cdot\mbox{{\boldmath {$\nabla$}}}'_2\big) ,$$ (130)

$${\hat{\mbox{{\boldmath {$k$}}}}}^{\prime *} \times\hat{\mbox{{\boldmath {$k$}}}} = {\textstyle{\frac{1}{4}}}\big(\hat{\mbox{{\boldmath {$K$}}}}_1-\h... ...s \big(\hat{\mbox{{\boldmath {$k$}}}}_2-\hat{\mbox{{\boldmath {$k$}}}}_1\big) ,$$ (131)

$${\hat{\mbox{{\boldmath {$k$}}}}}^{\prime *}\otimes{\hat{\mbox{{\b... ...^{\prime *}+\hat{\mbox{{\boldmath {$k$}}}}\otimes\hat{\mbox{{\boldmath {$k$}}}} = {\textstyle{\frac{1}{4}}}\big(\hat{\mbox{{\boldmath {$K$}}}}_1\ot... ...2 +\hat{\mbox{{\boldmath {$k$}}}}_2\otimes\hat{\mbox{{\boldmath {$k$}}}}_1\big)$$

$$+ {\textstyle{\frac{1}{4}}}\big(\mbox{{\boldmath {$\nabla$}}} _1\ot... ...+\mbox{{\boldmath {$\nabla$}}}'_2\otimes\mbox{{\boldmath {$\nabla$}}} _2\big) ,$$ (132)

$${\hat{\mbox{{\boldmath {$k$}}}}}^{\prime *} \otimes\hat{\mbox{{\b... ...hat{\mbox{{\boldmath {$k$}}}}\otimes{\hat{\mbox{{\boldmath {$k$}}}}}^{\prime *} = {\textstyle{\frac{1}{8}}}\big(\hat{\mbox{{\boldmath {$K$}}}}_1\ot... ...2 +\hat{\mbox{{\boldmath {$k$}}}}_2\otimes\hat{\mbox{{\boldmath {$k$}}}}_1\big)$$

$$+ {\textstyle{\frac{1}{4}}}\big(\mbox{{\boldmath {$\nabla$}}} _1\ot... ...+\mbox{{\boldmath {$\nabla$}}}'_2\otimes\mbox{{\boldmath {$\nabla$}}} _2\big) ,$$ (133)

where

$$\hat{\mbox{{\boldmath {$k$}}}}_1 = {\textstyle{\frac{1}{2i}}}\left(\mbox{{\boldmath {$\nabla$}}}_1-\mbox{{\boldmath {$\nabla$}}}'_1\right) ,$$ (134)

$$\hat{\mbox{{\boldmath {$k$}}}}_2 = {\textstyle{\frac{1}{2i}}}\left(\mbox{{\boldmath {$\nabla$}}}_2-\mbox{{\boldmath {$\nabla$}}}'_2\right) ,$$ (135)

and

$$\hat{\mbox{{\boldmath {$K$}}}}_1 = -i\left(\mbox{{\boldmath {$\nabla$}}}_1+\mbox{{\boldmath {$\nabla$}}}'_1\right) ,$$ (136)

$$\hat{\mbox{{\boldmath {$K$}}}}_2 = -i\left(\mbox{{\boldmath {$\nabla$}}}_2+\mbox{{\boldmath {$\nabla$}}}'_2\right) .$$ (137)

Final results can now be easily obtained by noting that relative momenta (134) lead to the current densities (46) and (50), total momenta (136) lead to derivatives of local densities, and the scalar and tensor products of individual momenta lead to kinetic densities (40), (44), and (48).

The zero-order (density-dependent) p-h coupling constants of the energy density (78) are expressed by the Skyrme force parameters as

$$C_{0}^{\rho} = \phantom{-}{\textstyle{\frac{3}{8}}} t_0 + {\textstyle{\frac{3}{48}}}t_3 \rho_0^\alpha(\mbox{{\boldmath {$r$}}}) ,$$ (138)

$$C_{0}^{s} = \phantom{-}{\textstyle{\frac{1}{8}}} t_0(2x_0-1) + {\textstyle{\frac{1}{48}}}t_3(2x_3-1)\rho_0^\alpha(\mbox{{\boldmath {$r$}}}) ,$$ (139)

$$C_{1}^{\rho} = - {\textstyle{\frac{1}{8}}} t_0(2x_0+1) - {\textstyle{\frac{1}{48}}}t_3(2x_3+1)\rho_0^\alpha(\mbox{{\boldmath {$r$}}}) ,$$ (140)

$$C_{1}^{s} = - {\textstyle{\frac{1}{8}}} t_0 - {\textstyle{\frac{1}{48}}}t_3 \rho_0^\alpha(\mbox{{\boldmath {$r$}}}) ,$$ (141)

and the second-order coupling constants are given in Table 1. One can immediately see that the gauge-invariance conditions (107) are fulfilled. This is so because the momentum-dependent terms of the Skyrme interaction obey the Galilean invariance [172,175]

Since seven Skyrme force parameters define twenty four second-order p-h coupling constants, in the resulting Skyrme energy density there is a high degree of dependency. First, as is well-known [178], a single spin-orbit parameter $W_0$ determines all four spin-orbit coupling constants $C_t^{\nabla{J}}$ and $C_t^{\nabla{j}}$, for $t$=0 and 1. Second, four Skyrme parameters, $t_1$, $x_1$, $t_2$, and $x_2$, uniquely determine four coupling constants $C_t^{\Delta\rho}$ and $C_t^{\tau}$, for $t$=0 and 1. Third, two tensor Skyrme parameters, $t_e$ and $t_o$, uniquely determine either isoscalar or isovector coupling constants, $C_t^{\nabla{s}}$ and $C_t^{F}$. Once such a rôle of the seven Skyrme parameters is fixed, values of the remaining coupling constants are also uniquely fixed.

Table 1: Second-order coupling constants of the p-h energy density (78) as functions of parameters of the Skyrme interaction (117), expressed by the formula: $C=\frac{A}{192}(at_1+bt_1x_1+ ct_2+dt_2x_2+et_{\mbox{\rm\scriptsize{e}}}+ft_{\mbox{\rm\scriptsize{o}}}+gW_0)$.

	$A$		$a$	$b$	$c$	$d$	$e$	$f$	$g$
$C_0^{\Delta\rho}$	3		$-$9	0	5	4	0	0	0
$C_0^{\tau}$	12		3	0	5	4	0	0	0
$C_0^{J0}$	$-$4		$-$1	2	1	2	10	30	0
$C_0^{J1}$	$-$6		$-$1	2	1	2	$-$5	$-$15	0
$C_0^{J2}$	$-$12		$-$1	2	1	2	1	3	0
$C_0^{\nabla{J}}$	48		0	0	0	0	0	0	$-$3
$C_0^{\Delta s}$	3		3	$-$6	1	2	6	$-$6	0
$C_0^{T}$	12		$-$1	2	1	2	$-$2	$-$6	0
$C_0^{j}$	$-$12		3	0	5	4	0	0	0
$C_0^{\nabla{s}}$	18		0	0	0	0	3	$-$3	0
$C_0^{F}$	72		0	0	0	0	1	3	0
$C_0^{\nabla{j}}$	48		0	0	0	0	0	0	$-$3
$C_1^{\Delta\rho}$	3		3	6	1	2	0	0	0
$C_1^{\tau}$	12		$-$1	$-$2	1	2	0	0	0
$C_1^{J0}$	$-$4		$-$1	0	1	0	$-$10	10	0
$C_1^{J1}$	$-$6		$-$1	0	1	0	5	$-$5	0
$C_1^{J2}$	$-$12		$-$1	0	1	0	$-$1	1	0
$C_1^{\nabla{J}}$	48		0	0	0	0	0	0	$-$1
$C_1^{\Delta s}$	3		3	0	1	0	$-$6	$-$2	0
$C_1^{T}$	12		$-$1	0	1	0	2	$-$2	0
$C_1^{j}$	$-$12		$-$1	$-$2	1	2	0	0	0
$C_1^{\nabla{s}}$	18		0	0	0	0	$-$3	$-$1	0
$C_1^{F}$	72		0	0	0	0	$-$1	1	0
$C_1^{\nabla{j}}$	48		0	0	0	0	0	0	$-$1