Ultraviolet suppression and nonlocality in optical model potentials for nucleon-nucleus scattering

Abstract

We investigate the role of high momentum components of optical model potentials for nucleon-nucleus scattering and its incidence on their nonlocal structure in coordinate space. The study covers closed-shell nuclei with mass number in the range $4 \le A \le 208$, for nucleon energies from tens of MeV up to 1 GeV. To this purpose microscopic optical potentials were calculated using density-dependent off-shell g matrices in Brueckner-Hartree-Fock approximation and based on Argonne $v_{18}$ as well as chiral 2N force up to next-to-next-to-next-to-leading order. We confirm that the gradual suppression of high-momentum contributions of the optical potential results in quite different coordinate-space counterparts, all of them accounting for the same scattering observables. We infer a minimum cutoff momentum Q, a function of the target mass number and energy of the process, that filters out irrelevant ultraviolet components of the potential. We find that when ultraviolet suppression is applied to Perey-Buck nonlocal potential or local Woods-Saxon potentials, they result with similar nonlocal structure to those obtained from microscopic models in momentum space. We examine the transversal nonlocality, quantity that makes comparable the intrinsic nonlocality of any potential regardless of its representation. We conclude that meaningful comparisons of nonlocal features of alternative potential models require the suppression of their ultraviolet components.

Multipoles of Gaussian form factor

We evaluate

$$\begin{aligned} w_l(b)={b}\int _{-1}^{1} P_l(u)e^{bu}\,du, \end{aligned}$$

(A.1)

with l positive integer. For low $l\le 3$ the evaluation of this integral is direct. For higher values they become tedious but straightforward. In such cases we use symbolic manipulation software to evaluate explicitly the cases $l\le 5$, obtaining

$$\begin{aligned} w_0(b)/2&=\sinh b \end{aligned}$$

(A.2a)

$$\begin{aligned} -{b}\, w_1(b)/2&=\sinh b + b\cosh \,b \end{aligned}$$

(A.2b)

$$\begin{aligned} {b^2} w_2(b)/2&= (3+b^2)\sinh \,b - 3b\cosh \,b \end{aligned}$$

(A.2c)

$$\begin{aligned} -b^3 w_3(b)/2&= (15+b^2)\sinh b \nonumber \\&-(15\,b+2\,b^3)\cosh b \end{aligned}$$

(A.2d)

$$\begin{aligned} b^4 w_4(b)/2&= (105+45\, b^2+b^4)\sinh b \nonumber \\&- (105\,b+10\,b^3)\cosh b \end{aligned}$$

(A.2e)

$$\begin{aligned} -b^5 w_5(b)/2&= (945+420\,b^2+b^4)\sinh b \nonumber \\&-(945\,b+105\,b^3+b^5)\cosh b. \end{aligned}$$

(A.2f)

Factorization by exponentials yields

$$\begin{aligned} w_0(b)&=e^{b} - e^{-b} \end{aligned}$$

(A.3a)

$$\begin{aligned} w_1(b)&=e^{b} \left( 1 - \frac{1}{b} \right) - e^{-b} \left( 1 + \frac{1}{b} \right) \end{aligned}$$

(A.3b)

$$\begin{aligned} w_2(b)&=e^{b} \left( 1 - \frac{3}{b} + \frac{3}{b^2} \right) - e^{-b} \left( 1 + \frac{3}{b} + \frac{3}{b^2} \right) \end{aligned}$$

(A.3c)

$$\begin{aligned} w_3(b)&=e^{b} \left( 1 - \frac{6}{b} + \frac{15}{b^2} - \frac{15}{b^3}\right) \nonumber \\&- e^{-b}\left( 1 + \frac{6}{b} + \frac{15}{b^2} + \frac{15}{b^3}\right) \end{aligned}$$

(A.3d)

$$\begin{aligned} w_4(b)&=e^{b} \left( 1 - \frac{10}{b} + \frac{45}{b^2} - \frac{105}{b^3} + \frac{105}{b^4}\right) \nonumber \\&- e^{-b}\left( 1 + \frac{10}{b} + \frac{45}{b^2} + \frac{105}{b^3} + \frac{105}{b^4}\right) \end{aligned}$$

(A.3e)

$$\begin{aligned} w_5(b)&=e^{b} \left( 1 - \frac{15}{b} + \frac{105}{b^2} - \frac{420}{b^3} + \frac{945}{b^4}- \frac{945}{b^5}\right) \nonumber \\&- e^{-b} \left( 1 + \frac{15}{b} + \frac{105}{b^2} + \frac{420}{b^3} + \frac{945}{b^4}+ \frac{945}{b^5}\right) . \end{aligned}$$

(A.3f)

Here we recognize Bessel polynomials ${y}_n(x)$ given by

$$\begin{aligned} {y}_0(x)&= 1 \end{aligned}$$

(A.4a)

$$\begin{aligned} {y}_1(x)&= x+1 \end{aligned}$$

(A.4b)

$$\begin{aligned} {y}_2(x)&= 3x^2+3x+1 \end{aligned}$$

(A.4c)

$$\begin{aligned} {y}_3(x)&= 15x^3+15x^2+6x+1 \end{aligned}$$

(A.4d)

$$\begin{aligned} {y}_4(x)&= 105x^4+105x^3+45x^2+10x+1 \end{aligned}$$

(A.4e)

$$\begin{aligned} {y}_5(x)&= 945x^5+945x^4+420x^3+105x^2+15x+1 \end{aligned}$$

(A.4f)

Thus,

$$\begin{aligned} w_l(b)= e^{b}{y}_l \left( \textstyle {\frac{-1\,}{b}} \right) -e^{-b}{y}_l \left( \textstyle {\frac{\,1\,}{\,b\,}} \right) . \end{aligned}$$

(A.5)

We note that Bessel polynomials are related to modified Bessel functions of the second kind through

$$\begin{aligned} y_n(x) = \sqrt{\frac{2}{\pi x}} e^{1/x} K_{n+\nicefrac {1}{2}}(1/x). \end{aligned}$$

(A.6)

Furthermore, they satisfy the recursion relation

$$\begin{aligned} y_{n+1}(x) = (2n+3)x\,y_n(x) + y_{n-1}(x). \end{aligned}$$

(A.7)