Abstract
We compare three methods to calculate the nucleon–nucleon t-matrix based on the three-dimensional formulation of Golak et al. (Phys Rev C 81:034006, 2010). In the first place we solve a system of complex linear inhomogeneous equations directly for the t-matrix. Our second method is based on iterations and a variant of the Lanczos algorithm. In the third case we obtain the t-matrix in two steps, solving a system of real linear equations for the k-matrix expansion coefficients and then solving an on-shell equation, which connects the scalar coefficients of the k- and t-matrices. A very good agreement among the three methods is demonstrated for selected nucleon–nucleon scattering observables using a chiral next-to-next-to-leading-order neutron–proton potential. We also apply our three-dimensional framework to the demanding problem of proton–proton scattering, using a corresponding version of the nucleon–nucleon potential and supplementing it with the (screened) Coulomb force, taken also in the three-dimensional form. We show converged results for two different screening functions and find a very good agreement with other methods dealing with proton–proton scattering.
Similar content being viewed by others
References
Golak J., Glöckle W., Skibiński R., Witała H., Rozpędzik D., Topolnicki K., Fachruddin I., Elster Ch., Nogga A.: Two-nucleon systems in three dimensions. Phys. Rev. C 81, 034006 (2010)
Glöckle W., Witała H., Hüber D., Kamada H., Golak J.: The three-nucleon continuum: achievements, challenges and applications. Phys. Rep. 274, 107 (1996)
Nogga A., Kamada H., Glöckle W.: Modern nuclear force predictions for the alpha particle. Phys. Rev. Lett. 85, 944 (2000)
Elster Ch., Thomas J.H., Glöckle W.: Two-body T-matrices without angular-momentum decomposition: energy and momentum dependences. Few-Body Syst. 24, 55 (1998)
Fachruddin I., Elster Ch., Glöckle W.: New forms of deuteron equations and wave function representations. Phys. Rev. C 63, 054003 (2001)
Ramalho G., Arriaga A., Peña M.T.: Solution of the spectator equation for relativistic NN scattering without partial wave expansion. Few-Body Syst. 39, 123 (2006)
Caia G., Pascalutsa V., Wright L.E.: Solving potential scattering equations without partial wave decomposition. Phys. Rev. C 69, 034003 (2004)
Rodriguez-Gallardo M., Deltuva A., Cravo E., Crespo R., Fonseca A.C.: Two-body scattering without angular-momentum decomposition. Phys. Rev. C 78, 034602 (2008)
Elster Ch., Schadow W., Nogga A., Glöckle W.: Three body bound state calculations without angular momentum decomposition. Few-Body Syst. 27, 83 (1999)
Liu H., Elster Ch., Glöckle W.: Three-body scattering at intermediate energies. Phys. Rev. C 72, 054003 (2005)
Bayegan S., Hadizadeh M.R., Harzchi M.: Three-nucleon bound state in a spin-isospin dependent three dimensional approach. Phys. Rev. C 77, 064005 (2008)
Bayegan S., Shalchi M.A., Hadizadeh M.R.: Three dimensional calculations of NN bound and scattering states with a chiral potential up to N3LO. Phys. Rev. C 79, 057001 (2009)
Hadizadeh M.R., Tomio L., Bayegan S.: Solutions of the bound-state Faddeev-Yakubovsky equations in three dimensions by using NN and 3N potential models. Phys. Rev. C 83, 054004 (2011)
Glöckle W., Elster Ch., Golak J., Skibiński R., Witała H., Kamada H.: A new treatment of 2N and 3N bound states in three dimensions. Few-Body Syst. 47, 25 (2010)
Golak, J., Topolnicki, K., Skibiński, R., Glöckle, W., Kamada, H., Nogga, A.: A Three-dimensional treatment of the three-nucleon bound state. Few-Body Syst. doi:10.1007/s00601-012-0472-5
Glöckle W., Fachruddin I., Elster Ch., Golak J., Skibiński R., Witała H.: 3N scattering in a three-dimensional operator formulation. Eur. Phys. J. A43, 339 (2010)
Glöckle W.: The Quantum Mechanical Few-Body Problem. Springer, Berlin (1983)
Epelbaum E., Glöckle W., Meißner Ulf-G.: The two-nucleon system at next-to-next-to-next-to-leading order. Nucl. Phys. A 747, 362 (2005)
Epelbaum E.: Few-nucleon forces and systems in chiral effective field theory. Prog. Part. Nucl. Phys. 57, 654 (2006)
Epelbaum E., Hammer H.W., Meißner U.-G.: Modern theory of nuclear forces. Rev. Mod. Phys. 81, 1773 (2009)
Wolfenstein L.: Possible triple scattering experiments. Phys. Rev. 96, 1654 (1954)
Press W., Flannery B., Teukolsky S., Vetterling W.: Numerical Recipes. Cambridge University Press, Cambridge (1989)
Machleidt R.: The meson theory of nuclear forces and nuclear structure. Adv. Nucl. Phys. 19, 189 (1989)
Stadler A., Glöckle W., Sauer P.U.: Faddeev equations with three-nucleon force in momentum space. Phys. Rev. C 44, 2319 (1991)
Epelbaum, E.: Private communication
ScaLAPACK Home Page: http://www.netlib.org/scalapack/scalapack_home.html
Skibiński R., Golak J., Witała H., Glöckle W.: Proton-proton scattering without Coulomb force renormalization. Eur. Phys. J. A40, 215 (2009)
Skibiński R., Golak J., Witała H.: Numerical investigations of the three-dimensional proton-proton screened Coulomb t-matrix. Acta Phys. Polon. B41, 875 (2010)
Vincent C.M., Phatak S.C.: Accurate momentum-space method for scattering by nuclear and Coulomb potentials. Phys. Rev. C 10, 391 (1974)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Golak, J., Skibiński, R., Witała, H. et al. Different Methods for the Two-Nucleon T-Matrix in the Operator Form. Few-Body Syst 53, 237–252 (2012). https://doi.org/10.1007/s00601-012-0480-5
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00601-012-0480-5