Protonproton scattering without Coulomb force renormalization

1
Proton-proton scattering without Coulomb force
renormalization

• R.Skibinski, J.Golak, H.Witala, W.Glöckle
• Renormalization and proton-proton scattering
  observables.
• The screened Coulomb t-matrix properties.

2
The brief introduction

• The screened Coulomb potential
• ? short range methods
• (eg. exponential screening
  )
• t matrix from the Lippmann-Schwinger equation
• The direct limit R ? 8 of t does not exist
• Renormalization (t ? te-2if, ff(R,n,E.))
• The renormalization factor is just a phase factor
  ?
• it is not needed for observables t2

3
The t-matrix for proton-proton scattering -
inclusion of the 3-dimensional screened Coulomb
t-matrix
Partial wave decomposition of t
The tjNC and tjC can be obtained with partial
wave decomposition (up to jjmax) from
4

• The tC can be obtained without partial wave
  decomposition from the 3-dim form of the
  Lippmann-Schwinger equation, where screened
  potential
• enters via
• For the exponential screening
• Typically we use a grid of 95 q,q and 130 x,x
  points
• For ngt1 only numerical solution
• Filons integral formula
• additional interpolations over Q
• The alternative, rigorous, way to obtain
  pp-observables is using the Vincent-Phatak
  method.

5
The pp scattering observablesat Eplab13 MeV for
the exponential screening with n4 and different
values of R.
6
The pp scattering observablesat Eplab13 MeV for
the exponential screening with R120 fm and
different values of n.
7
The 3-dimensional screened Coulomb t-matrix at
Eplab13 MeV, n4, R120 fm

• Q45 Q5

8
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))

with
and
9
An example the off-shell screened Coulomb
t-matrix t(q0.36 fm-1,q,Q45)
at Eplab13 MeV, n4
10
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))

Renormalization
with
(g0.5772)
11
An example the half-shell screened Coulomb
t-matrix t(q00.4 fm-1,q,Q45) at Eplab13 MeV,
n4
12
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

13
An example the on-shell screened Coulomb
t-matrix t(q00.4 fm-1,q00.4 fm-1,Q)
at Eplab13 MeV, n4
14
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.Skibinski, J.Golak, H.Witala, W.Glöckle, Eur.
  Phys. J. A40, 215 (2009)