Microscopic bell-shape nonlocality: the case of proton scattering off 40 Ca at 200 MeV

. This work is part of an ongoing e ff ort to build microscopically-driven nonlocal optical potentials easily tracktable in scattering codes. Based on the separable ‘ J v H ’ structure proposed recently [1], where the potential can be cast as the product of a radial and nonlocality form factors, we investigate its angular dependence in momentum space. We find that scattering observables have a weak angular dependence between the momentum transfer q = k − k ′ , and K = ( k + k ′ ) / 2. The study is focussed on proton elastic scattering o ff 40 Ca at 200 MeV, where the J v H structure is found to be inadequate. We conclude that any improvement of the J v H structure of the potential can be made to the lowest order in multipole expansions in Kq − representation of the potential.


Introduction
In the early sixties, Perey and Buck (PB) proposed a phenomenological nonlocal optical potential for neutron scattering off spherical nuclei [2].This potential factorizes into a Gaussian nonlocal form factor and a Woods-Saxon radial shape.The PB potential model has been further used as a starting point for numerous phenomenological studies.In addition to its extension to proton projectiles [3], its imaginary component has been made energy dependent [4,5] and also dispersive [6,7,8].
In a previous work [1], we have studied microscopic optical model potentials based on a g-matrix approach [9], where the potential can be reduced to lowest order to the so-called JvH form which relates directly to PB form factor.The work is done in momentum space, where a simplification is obtained with the use of mean momentum K and momentum transfer q instead of the usual k and k ′ relative momenta.This JvH potential is shown to reproduce elastic observables for incident energies up to 40 MeV.
In the present work we explore this microscopicallydriven potential at 200 MeV incident energy.We start from microscopic potential based on a density-dependent infinite nuclear-matter model to represent the in-medium effective N N interaction [10,11,12], folded with proton and neutron densities from Gogny D1S Hartree-Fock cala e-mail: guillaume.blanchon@cea.frculations [13].The Brueckner-Hartree-Fock g-matrix is obtained using AV18 bare interaction [14].We restrict this work to proton elastic scattering off 40 Ca at 200 MeV.In Sec. 2, we define two alternative representations: kk ′ and Kq, named after the coordinate system used.We truncate the angular dependence in the Kq-representation.The truncated potential is then expanded in kk ′ which is suitable to solve the scattering equations.The convergence of this multipolar expansion is assessed in Sec. 3. The underlying idea is to investigate whether lower orders in Kq-expansion retains the leading physics at 200 MeV incident energy.In Sec. 4, we assess the JvH form for the optical potential.Limitations of the JvH form relative to its original potential is illustrated in the context of elastic scattering.
Results from this work show that the Kq-representation exhibits a weak angular dependence between q and K allowing to focus on the term participating to J, v and H terms.This microscopically-driven potential should be tracktable for further phenomenology.Conclusions and perspectives are drawn in Sec. 5.

Expansions
We consider the general structure of the optical potential in momentum space.In all these approaches the optical potential for NA elastic scattering, U(k ′ , k; E), can be cast in the form [1] arXiv:2402.15009v1 [nucl-th] 22 Feb 2024 with n the unit vector perpendicular to the scattering plane given by and σ the spin of the projectile.Here U c and U so represent the central and spin-orbit components of the potential.
From now on, dependences on the energy E are implicit.One can expand the potential in Eq. ( 1) in terms of relative momenta, k and k ′ , and the angle between k and k ′ expressed by u = k • k′ , as In the following, this expansion is referred to as kk ′ −expansion, where l represents the orbital angular momentum [15].This is the form we use to obtain the scattering observables using SWANLOP code [16] that allows treatment of optical potential in momentum space as input.
Let us now define the following alternative variables, with K the mean momentum and q the momentum transfer.We observe that k ′ × k = K × q.As done in Ref. [1], we investigate the structure of the potential in terms of K and q.Accordingly, we introduce the notation, ) Now we consider the following expansions, for the central part and the spin-orbit potentials, respectively.In principle N → ∞, but in practice we choose an upper limit.P m n (w) is the associated Legendre polynomials.These expansions are referred to as Kq−expansion.Restriction of summation to even n values in the central term comes from symmetry considerations.Note that the spin orbit term is summed over odd values.Details of spinorbit derivations in Eq. (8b) can be found in Appendix A.

Convergence
We now explore the ability of different Kq−truncations to reproduce the full potential result in the context of the test case: proton elastic scattering off 40 Ca at 200 MeV.First of all, we verify that the exact results are reproduced when using high values of N .
In Fig. 1, we present the calculated differential cross sections obtained from full potential as in Eq. ( 1).We also show results considering the cases N = 0, and N = 2 from Eqs. (8a) and (8b).We observe that the agreement between the full calculation and that obtained for N = 0, is already remarkable.The result for N = 2 overlaps perfectly to the eye with the full calculation.For the sake of scrutinity, we include an inset where the differential cross section is plotted in linear scale, to strengthen this statement.In Table 1, we summarize the calculated reaction cross sections considering the full representation of the potential and its expressions up to N = 0 and 2. The difference between the full potential and the case of N = 0 is about 0.4 %, reaching agreement up to four significant figures in the case of N = 2.These results validate considering the former case as starting point.

JvH and beyond
We have shown that the optical potential in the Kq−representation exhibits a weak dependence on the angle between K and q.Actually, the truncation up to N = 0 is reasonably good to retain the leading features present in the original potential in kk ′ −representation.This feature justifies to focus on the N = 0 truncated expansion of the potential.In this case Eqs.(8a) and (8b) become U so (K, q, w) ≈ U (so) Note that both terms are independent of w, namely the angle between K and q.Following Ref. [1] we define The term U c (0, 0) is directly related to the volume integral through, J (c) = (2π) 3 U c (0, 0).The same construction is applied to the spin-orbit term.Below E = 40 MeV, V (c) and V (so) are K-independent [1].This is the key point to recover the Perey-Buck separability between a nonlocal form factor and a radial shape.Therefore we introduce the radial form factors, V (so) (K, q) ≈ V (so) (0, q) ≡ v (so) (q) .( In this way one gets the JvH form of the potential, The behavior of the nonlocality of the microscopic optical potential is illustrated in Fig. 2, where we plot the nonlocality form factor H as a function of K. Black and red curves correspond to the central and spin-orbit nonlocality form factors extracted from the microscopic optical potential, respectively.Solid and dashed curves correspond to their real (Re) and imaginary (Im) components, respectively.Here we also include the nonlocality form factor proposed by Perey-Buck [2], namely where the nonlocality parameter is taken as β = 0.85 fm.We observe two features worth mentioning.First, the microscopic nonlocality form factor is complex, with a small imaginary contribution at the origin but increasing in magnitude as K increases.Second, the curvature of H at the origin is a measure of the range of the nonlocality.Thus, the nonlocality of the spin-orbit term is smaller than that in the central part.Interestingly, the curvature of the central potential and PB appear comparable.
In Fig. 3, we show results for the ratio-to-Rutherford differential cross sections σ/σ R , analyzing power A y and spin-rotation Q as functions of the momentum transfer q.The data are taken from Refs.[17] and [18].Red solid curves represent results based on the full potential.Black dashed curves correspond to Kq−representation considering N = 0.These results illustrate the close correspondence between the full potential and its Kq−representation up to N = 0, whereas clear differences are evidenced when compared to the JvH form.Thus, the origin of these differences can be traced back to the use of Eqs.(11) and (12), where the K dependence is neglected.This observation indicates that any improvement of the JvH factorization would require corrections in K at this level.Work along this line is underway.

Conclusions
The main result of this work is that Kq−representation of the optical model potential retains the main physics at its zero-th multipolar order.This feature is unveiled from a microscopic optical model potentials based on realistic bare nucleon-nucleon potentials.Although the JvH form has been validated at low incident energies in Ref. [1], its limitations above 100 MeV can be corrected after a close scrutiny of the K dependance in Eqs.(11) and (12).The long term aim of this effort is to provide guidance to phenomenomogical constructions of optical potentials resulting from detailed microscopic approaches.

A Spin-orbit term
We describe in details how to get the expansion for the spin-orbit contribution in Eq. (8a).We start from the

Fig. 1 :
Fig. 1: Ratio-to-Rutherford differential cross section as a function of the momentum transfer for p + 40 Ca at 200 MeV.Red curves correspond to the full potential, whereas solid and dashed black curves represent results in Kq−repesentation truncated at N = 0 and 2, respectively.The inset shows a close-up in linear scale.

Fig. 3 :
Fig.3: Ratio-to-Rutherford differential cross section, analyzing power and spin rotation as functions of the momentum transfer for p + 40 Ca elastic scattering at 200 MeV.Red solid curves are based on the full potential.Black dashed curves are based on its Kq−representation of order zero.Black solid curves represent results based on its JvH factorization.The data are taken from Refs.[17] and[18].

Table 1 :
Reaction cross sections for p+ 40 Ca at 200 MeV for the full potential scattering calculation, the N = 0 and N = 2 truncations in the Kq−representation.