Proton-proton scattering without Coulomb force renormalization R. Skibiński, J. Golak, H. Witała, W. Glöckle

Proton-proton scattering without Coulomb force renormalization

• R.Skibiński, J.Golak, H.Witała, W.Glöckle

• Renormalization and proton-proton scattering observables.

• The screened Coulomb t-matrix properties.

The brief introduction

• The screened Coulomb potential

• → short range methods

• (eg. exponential screening )

• t matrix from the Lippmann-Schwinger equation

• The direct limit R → ∞ of t does not exist

• Renormalization (t → te-2i, =(R,n,E….))

• The renormalization factor is just a phase factor →

• it is not needed for observables ~|t|2

The pp scattering observables at Eplab=13 MeV for the exponential screening with n=4 and different values of R.

The pp scattering observables at Eplab=13 MeV for the exponential screening with R=120 fm and different values of n.

The 3-dimensional screened Coulomb t-matrix at Eplab=13 MeV, n=4, R=120 fm

• ° 5°

The screening limit of the off-the-energy-shell screened Coulomb t-matrix elements

• The pure Coulomb t-matrix off-shell elements (L.P.Kok and H. van Haeringen; PRC21, 512 (1980)):

An example: the off-shell screened Coulomb t-matrix t(q=0.36 fm-1,q’,=45°) at Eplab=13 MeV, n=4

The screening limit of the half-the-energy-shell screened Coulomb t-matrix elements

• The pure Coulomb t-matrix half-shell elements (L.P.Kok and H. van Haeringen; PRL46, 1257 (1981)):

An example: the half-shell screened Coulomb t-matrix t(q0=0.4 fm-1,q’,=45°) at Eplab=13 MeV, n=4

The screening limit of the on-the-energy-shell screened Coulomb t-matrix element

• The pure Coulomb t-matrix on-shell element is achieved after renormalization of the screened Coulomb t-matrix

An example: the on-shell screened Coulomb t-matrix t(q0=0.4 fm-1,q’0=0.4 fm-1,) at Eplab=13 MeV, n=4

Summary

• The screened Coulomb t-matrix can be obtained numerically with high precision

• ― 3-dimensional L-S equation is essential

• ― application in three-body calculations

• The presented numerical results confirm previous analytical studies

• The renormalization is useful to study the screening limit of half- and on-shell t-matrix elements

• The pp scattering observables can be obtained using screened Coulomb potential without renormalization

• Different sets of screening parameters lead to similar results

• More details:

• R.Skibiński, J.Golak, H.Witała, W.Glöckle, Eur. Phys. J. A40, 215 (2009)