Central pseudopotential depending on the sum of relative momenta

In a recent paper [3], we studied the regularized finite-range interaction as an application of the ET principles to low-energy nuclear phenomena. To give an illustrative version of such effective theory, we considered a restricted form of the finite-range pseudopotential, neglecting the SO and tensor parts, and treating the remaining central part as depending only on the sum of relative momenta. In the spherical-tensor pseudopotential, this amounts to considering only terms of the form

$$\hat{V}_{v_{12} 0}^{N,\bar{t}}= \frac{1}{2}i^{v_{12}} \left[ \left[\left(K'_{11} + K_{11} \right)... ...0 \hat{S}_{v_{12} 0}\right]_{0} \left(\hat{P}^{\tau}\right)^{\bar{t}} \nonumber$$

$$\times \left(1-\hat{P}^{M}\hat{P}^{\sigma}\hat{P}^{\tau}\right) \delta(\bm{r}'_1-\bm{r}_1)\delta(\bm{r}'_2-\bm{r}_2)g_a(r) ,$$ (93)

with $N$=0, 2, 4, and 6.

Dependence on the sum of relative momenta only, that is, on $\left[\left(K'_{11} + K_{11} \right)^N \right]_0$, is a crucial feature to obtain a functional in the form of an expansion in the length scale $a$ of the regularized delta, which we used for the study of the pseudopotentials in the ET framework. Indeed, the series in $a$ is obtained by acting with the relative-momentum operators directly and only on $g_a(r)$, which is possible, because the sums of relative momenta do commute with the locality deltas. This fact can be explicitly demonstrated by calculating the action of $K'_{11}+K_{11}$ on the locality deltas, that is,

$$\Big[K'_{11}+K_{11}\Big] \delta(\bm{r}'_1-\bm{r}_1)\delta(\bm{r}'... ... =\Big[\bm{k}'+\bm{k}\Big] \delta(\bm{r}'_1-\bm{r}_1)\delta(\bm{r}'_2-\bm{r}_2)$$

$$~~ = \frac{1}{2i}\Big[ \bm{\nabla}'_1-\bm{\nabla}'_2+\bm{\nabla}_1-\bm{\nabla}_2\Big] \delta(\bm{r}'_1-\bm{r}_1)\delta(\bm{r}'_2-\bm{r}_2)$$

$$~~ = \frac{1}{2i}\Big[ \delta\,'(\bm{r}'_1-\bm{r}_1)\delta (\bm{r}'_2-\bm{r}_2) -\delta (\bm{r}'_1-\bm{r}_1)\delta\,'(\bm{r}'_2-\bm{r}_2)$$

$$~~~~~~~ -\delta\,'(\bm{r}'_1-\bm{r}_1)\delta (\bm{r}'_2-\bm{r}_2) +\delta (\bm{r}'_1-\bm{r}_1)\delta\,'(\bm{r}'_2-\bm{r}_2)\Big] \equiv 0.$$ (94)

As a consequence, pseudopotentials defined by (94) are strictly equivalent to ordinary local potentials given by a series of powers of Laplacians acting on the regularized delta $g_a(r)$ [3].

This particular restriction of the relative-momentum operators gives rise to a set of constraints on parameters of the general second-, fourth-, and sixth-order pseudopotential (2), expressed in spherical or Cartesian forms, which we list below.

At second order, corresponding to the conditions

$$t^{(2)}_2 = - t^{(2)}_1, \quad x^{(2)}_2 = x^{(2)}_1, \quad y^{(2)}_2 = y^{(2)}_1, \quad x^{(2)}_2 = x^{(2)}_1,$$ (95)

valid for combination of the Cartesian relative momenta [3], we get the constraints,

$$C_{11,00}^{11,\bar{t}} = C_{00,00}^{20,\bar{t}},$$ (96)

$$C_{11,20}^{11,\bar{t}} = C_{00,20}^{20,\bar{t}}.$$ (97)

The analogous relations at fourth and sixth order are found by applying the binomial expansion of the term $\left[\left(K'_{11} + K_{11} \right)^N \right]_0$ for $N$ = 4, 6 respectively. At fourth order for $v_{12}$ = 0, 2 and $t^{(4)}_j=0$ for $j>2$, we find

$$C_{11,v_{12}0}^{31,\bar{t}} = 4 \, C_{00,v_{12}0}^{40,\bar{t}},$$ (98)

$$C_{20,v_{12}0}^{20,\bar{t}} = 3 \, C_{00,v_{12}0}^{40,\bar{t}},$$ (99)

$$C_{22,v_{12}0}^{22,\bar{t}} = 3 \, C_{00,v_{12}0}^{40,\bar{t}},$$ (100)

and at sixth order we have,

$$C_{11,v_{12}0}^{51,\bar{t}} = 6 \, C_{00,v_{12}0}^{60,\bar{t}},$$ (101)

$$C_{20,v_{12}0}^{40,\bar{t}} = 15 \, C_{00,v_{12}0}^{60,\bar{t}},$$ (102)

$$C_{22,v_{12}0}^{42,\bar{t}} = 15 \, C_{00,v_{12}0}^{60,\bar{t}},$$ (103)

$$C_{31,v_{12}0}^{31,\bar{t}} = 10 \, C_{00,v_{12}0}^{60,\bar{t}},$$ (104)

$$C_{33,v_{12}0}^{33,\bar{t}} = 10 \, C_{00,v_{12}0}^{60,\bar{t}},$$ (105)

whereas all the other parameters $C_{\tilde{n} \tilde{L},v_{12} S}^{\tilde{n}' \tilde{L}', \bar{t}}$ with $S\neq$0 are set to zero.