Quadratic regularization scheme

In the QR scheme, matrix element $V^{{\rm 2B}}_{IMK}$ (10) is replaced by the auxiliary quantity $V^{{\rm 2B,2}}_{IMK}$ defined in Eq. (13) for $n=2$. Again we assume that matrix element $\langle \Psi \vert\hat V_{\rm {2B}} \vert \tilde{\Psi}\rangle$ is regularizable. Inserting expansions (16) and (20) into (13) gives

$$V^{{\rm 2B,2}}_{IMK} = \sum_{I_1 M_1 K_1} \tilde{V}_{I_1 M_1 K_1}... ...M_1 K_1} (\Omega )\, D^{I_2}_{M_2 K_2} (\Omega )\, D^{I_3}_{M_3 K_3} (\Omega ).$$

The integral can be calculated with the aid of the following Clebsch-Gordan series [7]:

$$D^{J_1}_{M_1 K_1} (\Omega ) D^{J_2}_{M_2 K_2} (\Omega ) = \sum_{J... ...J_2 M_2} D^{J_3}_{M_3 K_3} (\Omega ) {\rm {\bf C}}^{J_3 K_3}_{J_1 K_1 J_2 K_2}.$$ (29)

This gives

$$V^{{\rm 2B,2}}_{IMK} = \sum_{I_1 M_1 K_1} \tilde{V}_{I_1 M_1 K_1}^{\rm {2B}} \sum_{I_2 M... ... {\bf C}}^{I'' M''}_{I_2 M_2 I_3 M_3} {\rm {\bf C}}^{I'' K''}_{I_2 K_2 I_3 K_3}$$

$$\times \frac{2I+1}{8\pi^2} \int d\Omega\, D^{I\, ^\star}_{MK} (\Omega )\, D^{I_1}_{M_1 K_1} (\Omega )\, D^{I''}_{M'' K''} (\Omega ).$$ (30)

Eventually, after using expression (23), we obtain

$$V^{{\rm 2B,2}}_{IMK} = \sum_{I_1 M_1 K_1} \left\{ \sum_{I_2 M_2 K_2} c_{I_2 M_2 K_2}^{\r... ...rm {\bf C}}^{I K}_{I_1 K_1 I'' K''} \right\} \tilde{V}_{I_1 M_1 K_1}^{\rm {2B}}$$

$$\equiv \sum_{I_1 M_1 K_1} A^{IMK}_{I_1 M_1 K_1} \tilde{V}_{I_1 M_1 K_1}^{\rm {2B}} ,$$ (31)

with the following selection rules on intermediate summations:

$$\vert I_2 -I_3\vert \leq I'' \leq I_2+I_3 ,$$

$$M_2+M_3 = M'' ,\quad M_2+M_3 = M'',$$

$$M_1+M'' = M ,\quad K_1+K'' = K .$$ (32)

From the practical point of view, it is important to notice that the intermediate summations over $I'',M'',K''$ can be extended as,

$$\sum_{I'' = \vert I_2 -I_3\vert}^{I_2+I_3} \longrightarrow \sum_{I'' = 0}^{2J},$$ (33)

with the added terms being zero owing to properties of the Clebsch-Gordan coefficients. This allows us to change in Eq. (32) the order of summations, and to split the multi-dimensional summations into two independent sets of sums, that is,

$$A^{IMK}_{I_1 M_1 K_1} = \sum_{I'' M'' K''} X^{I''}_{M'' K''} {\rm {\bf C}}^{I M}_{I_1 M_1 I'' M''} {\rm {\bf C}}^{I K}_{I_1 K_1 I'' K''},$$ (34)

where

$$X^{I''}_{M'' K''} = \sum_{I_2 M_2 K_2} c_{I_2 M_2 K_2}^{\rm {{\cal N}}} \sum_{I_3 M_3 K_3} c_{I_3 M_3 K_3}^{\rm {{\cal N}}}$$

$$\times {\rm {\bf C}}^{I'' M''}_{I_2 M_2 I_3 M_3} {\rm {\bf C}}^{I'' K''}_{I_2 K_2 I_3 K_3}.$$ (35)

The trick used above facilitates numerical calculations. It should also be noted that the Clebsch-Gordan coefficients impose selection rules that introduce further simplifications. For example, in Eq. (35), one has $M''=M-M_1$ and $K''=K-K_1$, meaning that corresponding summations are effectively one-dimensional.

The QR scheme proposed above is not the simplest one. An alternative formulation of the scheme, which, in fact, can be generalized to higher orders, can be obtained as follows. By performing a direct decomposition of the $n$th power of the overlap, which is always regular and, therefore, expandable in terms of the Wigner $D$-functions, we obtain:

$${\rm {{\cal N}}}^n (\Omega ) \equiv \langle \Psi \vert \tilde{\Ps... ... \sum_{I''M''K''} c_{I''M''K''}^{\rm {{\cal N},n}} D^{I''}_{M'' K''} (\Omega ).$$ (36)

It is straightforward to show that with the aid of Eq. (37), the QR scheme reduces formally to the LR scheme, that is, the algebraic equations (24)-(25) are valid in the QR theory provided that the coefficients $c_{IMK}^{\rm {{\cal N}}}$ are replaced with $c_{IMK}^{\rm {{\cal N},2}}$. Furthermore, the direct decomposition of ${\rm {{\cal N}}}^n (\Omega )$ seems to be the most natural and numerically most efficient generalization of the proposed regularization scheme to higher values of $n$.

In this work we used the QR scheme as defined through the equations (35)-(36). In the applications we used a strategy similar to the one used for the LR scheme, that is, energies $E_{I=0}$ and sum-rule residuals were investigated as a functions of ${I_{\text{max}}}$, which is the maximum value of angular momentum admitted in the summations in Eqs. (35) and (36).

Figure 5: (Color online) Same as in Fig. 1 but for the SLy4 Skyrme functional. Full squares represent results obtained using our QR method.

First, we applied the QR method to the SIII case, already analyzed in Sec. 2.2 in the context of the LR method, see Fig. 3. In this case, both regularization schemes are fully equivalent and give identical results. This result was, in fact, expected. Indeed, let us observe that regularizability of the theory at certain order $n$ guarantees, see Eq. (17), that the expression:

$$(\langle \Psi \vert\hat V_{\rm {2B}} \vert \tilde{\Psi}\rangle - ... ...rm {2B}} \vert \tilde{\Psi}\rangle}) \langle \Psi \vert \tilde{\Psi}\rangle^m ,$$ (37)

differs from zero at most over a set of measure zero for any $m\geq n$. This, in turn, implies an equivalence of all regularization schemes for $m\geq n$. This fact speaks in favor of our regularization scheme and the result quoted above serves as an independent test of the numerical implementation of the method.

Next, we applied both regularization schemes to the case of the SLy4 functional [28], which features a fractional-power density dependence with $\eta =1/6$. The results are presented in Fig. 5. In this case, the regularization is insufficient to stabilize the energy of the lowest $I$=0 state. By removing contributions coming from uncompensated poles in energy kernels, the QR scheme lowers the energy of the $I=0$ state as compared to the LR scheme, but is unable to fully stabilize the solution. The reason is that, apart from singularities, the fractional power of the density introduces non-analyticities related to branch cuts, which are not removable by our regularization scheme, irrespective of its order $n$.