Skyrme interaction energy functional

The Skyrme interaction [168,169] is a zero-range local force that depends on relative momenta up to the second-order. The complete list of terms giving its matrix element in the position-spin-isospin representation, including the tensor components [171,176], reads

$$\hat{V}(\mbox{{\boldmath {$r$}}}'_1 s'_1t'_1 \mbox{{\boldmath {$r... ..._2s'_2t'_2,\mbox{{\boldmath {$r$}}}_1s_1t_1\: \mbox{{\boldmath {$r$}}}_2s_2t_2) = \Big\{t_0 ( \hat{\delta}^{\sigma} + x_0 \hat{P}^\sigma ) + {\text... ...yle{\frac{1}{2}}}(\mbox{{\boldmath {$r$}}}_1+\mbox{{\boldmath {$r$}}}_2)\right)$$

$$+ {\textstyle{\frac{1}{2}}} t_1 ( \hat{\delta}^{\sigma} + x_1 \hat{... ...ldmath {$k$}}}} \cdot\hat{{\mathsf S}}\cdot\hat{\mbox{{\boldmath {$k$}}}} \Big]$$ (117)

$$+ t_2 ( \hat{\delta}^{\sigma} + x_2 \hat{P}^{\sigma}) {\hat{\mbox{{... ...{\delta}^{\tau} - \hat{P} ^{\sigma}\hat{P} ^{\tau}P^M\right) \hat{\delta}_{12},$$

where

$$\hat{\delta}^{\sigma}_{s'_1s'_2s_1s_2} = \delta_{s'_1s_1}\delta_{s'_2s_2},$$ (118)

$$\hat{\delta}^{\tau} _{t'_1t'_2t_1t_2} = \delta_{t'_1t_1}\delta_{t'_2t_2},$$ (119)

and

$$\hspace*{-2.5em} \hat{P}^\sigma_{s_1's_2's_1s_2}\! = {\textstyle{\frac{1}{2}}}(\hat{\delta}^{\sigma}_{s'_1s'_2s_1s_2} ... ...ox{{\boldmath {$\sigma$}}}}_{s_2's_2}) \!= \delta_{s'_1s_2}\delta_{s'_2s_1} ,\!$$ (120)

$$\hspace*{-2.5em} \hat{P}^\tau _{t_1't_2't_1t_2}\! = {\textstyle{\frac{1}{2}}}(\hat{\delta}^{\tau}_{t'_1t'_2t_1t_2} +\... ..._1't_1}\circ\hat{\vec {\tau}}_{t_2't_2}) \!= \delta_{t'_1t_2}\delta_{t'_2t_1} ,$$ (121)

are the spin and isospin unity and exchange operators, respectively, and

$$\hat{\mbox{{\boldmath {$S$}}}}_{s_1's_2's_1s_2} = \hat{\mbox{{\boldmath {$\sigma$}}}}_{s_1's_1}\delta_{s_2's_2} + \hat{\mbox{{\boldmath {$\sigma$}}}}_{s_2's_2}\delta_{s_1's_1} ,$$ (122)

$$\hat{{\mathsf S}}_{s_1's_2's_1s_2}^{ab} = {\textstyle{\frac{3}{2}}}\big(\hat{\mbox{{\boldmath {$\sigma$}}}}... ... {$\sigma$}}}}_{s_1's_1}^b \hat{\mbox{{\boldmath {$\sigma$}}}}_{s_2's_2}^a\big)$$

$$- \delta_{ab}\hat{\mbox{{\boldmath {$\sigma$}}}}_{s_1's_1}\cdot \hat{\mbox{{\boldmath {$\sigma$}}}}_{s_2's_2}$$ (123)

are two-body vector and tensor spin operators, respectively. The relative momentum operators,

$$\hat{\mbox{{\boldmath {$k$}}}} = {\textstyle{\frac{1}{2i}}}\left(\mbox{{\boldmath {$\nabla$}}} _1-\mbox{{\boldmath {$\nabla$}}} _2\right) ,$$ (124)

$$\hat{\mbox{{\boldmath {$k$}}}}' = {\textstyle{\frac{1}{2i}}}\left(\mbox{{\boldmath {$\nabla$}}}'_1-\mbox{{\boldmath {$\nabla$}}}'_2\right) ,$$ (125)

act on the delta functions in $\hat{\delta}_{12}$,

$$\hspace*{-1.5em} \hat{\delta}_{12}(\mbox{{\boldmath {$r$}}}'_1\mbox{{\boldmath {$r$}}}'_2,\mbox{{\boldmath {$r$}}}_1\mbox{{\boldmath {$r$}}}_2) = \delta(\mbox{{\boldmath {$r$}}}'_1\!-\!\mbox{{\boldmath {$r$}}}_1... ...h {$r$}}}_2) \delta(\mbox{{\boldmath {$r$}}} _1\!-\!\mbox{{\boldmath {$r$}}}_2)$$

$$= \delta(\mbox{{\boldmath {$r$}}}'_1\!-\!\mbox{{\boldmath {$r$}}}_2... ...{$r$}}}_1) \delta(\mbox{{\boldmath {$r$}}} _2\!-\!\mbox{{\boldmath {$r$}}}_1) .$$ (126)

This action has to be understood in the standard sense of derivatives of distributions.

Whenever the Skyrme interaction (117) is inserted into integrals, like in Eqs. (73)-(75), the integration by parts transfers the derivatives onto appropriate variables in the remaining parts of integrands.

Numbers $P^M$ are equal to $+1$ or $-1$ depending on whether in a given term the power of momentum $\hat{\mbox{{\boldmath {$k$}}}}$ is even or odd, respectively. Skyrme interaction written in the form of the integral kernel (117) is explicitly antisymmetric with respect to exchanging left or right pairs of variables pertaining to particles 1 and 2.

The Skyrme HFB energy density can be calculated by inserting the Skyrme interaction (117) directly into expressions (74), (75), and (72). Results for the p-h channel were published by many authors, see, e.g., Refs. [169,177,172,175], although often some terms of interaction (117) were neglected and/or restricted symmetries were used. Results for the p-p channel were previously published with tensor terms and the proton-neutron mixing neglected [5]. Here we aim at presenting the complete set of results.

Calculations leading to expressions for the Skyrme energy density are tedious, but can be efficiently performed by noting two simplifying facts. First, the two-body spin operators obey conditions,

$$\hat{\mbox{{\boldmath {$S$}}}} \hat{P}^\sigma = \hat{\mbox{{\boldmath {$S$}}}} ,$$ (127)

$$\hat{{\mathsf S}}\hat{P}^\sigma = \hat{{\mathsf S}} ,$$ (128)

and hence only terms up to linear in spin and isospin Pauli matrices appear in the antisymmetrized interaction. Second, the Pauli matrices in (117) pertain to the p-h coupling channel, while the momenta - to the p-p coupling channel. Hence, calculations may become very easy once a common, p-h or p-p, coupling channel is used for all the elements of interaction. This requires either recoupling momenta to the p-h channel, or recoupling the Pauli matrices to the p-p channel. To this end, we separately consider the p-h end p-p energy densities.