An approach without partial wave expansion to calculate scattering of spin-0 and spin-1/2 particles in high energy regions and
those governed by long range interactions
I. Fachruddin and A. Salam
Zeitschrift für Naturforschung A (2023), DOI: 10.1515/zna-2023-0248 (just accepted)
Abstract
Scattering of spin-0 and spin-1/2 particles is formulated in momentum space based on basis states being not
expanded in partial waves. No sequential calculations with increasing angular momentum are performed to reach physical
convergence, which depends on the scattering energy and the interaction range. Both nonrelativistic and relativistic
cases are described. We put forward for consideration the utilization of this approach. By taking some simple
interaction models we show some advantages in calculations representing those of high energy scattering as well as those
of scattering governed by long range interactions.
Kaon-nucleon scattering in three-dimensional technique
A. Salam and I. Fachruddin
Proceeding of The 4th International Conference on Theoretical and Applied Physics (ICTAP) 2014, October 16-17, 2014, Denpasar, Bali, Indonesia
AIP Conf. Proc. 1719, 030012 (2016), DOI: 10.1063/1.4943707
Abstract
Kaon-nucleon (KN) scattering is formulated in the three-dimensional (3D) momentum space, in which the basis state is not expanded into partial waves.
Based on this basis the Lippmann-Schwinger equation for the T-matrix is evaluated. We obtain as final equation for the T-matrix elements a set of
two coupled integral equations in two variables, which are the momentum’s magnitude and the scattering angle. Calculations for the differential
cross section and some spin observables are shown, for which we employ a hadrons exchange model with the second order contributions only.
A momentum-space formulation without partial wave decomposition for scattering of two spin-half particles
I. Fachruddin and A. Salam
Proceeding of The 4th International Conference on Theoretical and Applied Physics (ICTAP) 2014, October 16-17, 2014, Denpasar, Bali, Indonesia
AIP Conf. Proc. 1719, 030037 (2016), DOI: 10.1063/1.4943732
Abstract
A new momentum-space formulation for scattering of two spin-half particles, both either identical or unidentical,
is formulated. As basis states the free linear-momentum states are not expanded into the angular-momentum states,
the system’s spin states are described by the product of the spin states of the two particles,
and the system’s isospin states by the total isospin states of the two particles. We evaluate the Lippmann-Schwinger
equations for the T-matrix elements in these basis states. The azimuthal behavior of the potential and of the T-matrix
elements leads to a set of coupled integral equations for the T-matrix elements in two variables only,
which are the magnitude of the relative momentum and the scattering angle. Some symmetry relations for the potential
and the T-matrix elements reduce the number of the integral equations to be solved. A set of six spin operators to
express any interaction of two spin-half particles is introduced. We show the spin-averaged differential cross section
as being calculated in terms of the solution of the set of the integral equations.
KN scattering with hadrons exchange potential in 3D technique
A. Salam and I. Fachruddin
Proceedings of The XV International Conference on Hadron Spectroscopy,
November 4-8, 2013, Nara, Japan
Abstract
We have formulated the KN scattering in a three-dimensional (3D) technique. The Lippmann-Schwinger equation is evaluated
in 3D basis states without partial-wave (PW) expansion. The result is a set of two coupled integral equations for T-matrix
elements in two variables, namely the momentum magnitude and the scattering angle. For the KN interaction we employ an
already derived hadrons exchange model with the second order contributions only. The parameters of the model will be later
determined by fitting to the experimental data for the cross section and some spin observables.
NN scattering formulations without partial-wave decomposition
I. Fachruddin and A. Salam
Proceedings of The XV International Conference on Hadron Spectroscopy,
November 4-8, 2013, Nara, Japan
Abstract
Recently a new three-dimensional (3D) formulation for scattering of two spin-half particles, either identical
or unidentical, is presented. In the formulation the free linear-momentum states are not expanded into the
angular-momentum states and the system's spin and isospin states, respectively, are described by the product of the spin
and isospin states of the two particles. We apply this technique to calculate nucleon-nucleon (NN) scattering. But as NN
interaction models are usually being parametrized regarding to the NN total isospin, we take the total isospin state,
instead of the product of the isospin states of the two nucleons, as part of the 3D basis state. We evaluate the
Lippmann-Schwinger equations for NN T-matrix elements in these 3D basis states and end up with a set of coupled integral
equations for the NN T-matrix elements in two variables, namely the magnitude of the relative momentum and the
scattering angle. We show as an example the spin-averaged differential cross section, which can be calculated directly
from the solution of the set of the integral equations. The recently developed 3D formulation also introduces a set of
six spin operators to express any interaction of two spin-half particles. We show the Bonn NN potential in terms of
these six spin operators.
KN scattering in 3D formulation
A. Salam and I. Fachruddin
Few-Body Systems 54, 1625 (2013), DOI: 10.1007/s00601-012-0557-1
Abstract
KN scattering is formulated in three-dimensional (3D) momentum space. A direct product of the
relative-momentum state and the spin state is used as the basis state. The spin quantization axis is chosen along
the z-axis. The interaction for the KN system is assumed to take the Yukawa-type. It consists of two terms,
the central and the spin-orbit one. Calculations for the cross section based on this technique are shown, as well
as comparison with the standard partial-wave calculations.
Scattering of two spin-half particles in a three-dimensional approach
I. Fachruddin
Few-Body Systems 54, 1621 (2013), DOI: 10.1007/s00601-012-0513-0
Abstract
Scattering of two spin-1/2 particles is formulated in a three-dimensional approach
based on a simple three-dimensional momentum-spin basis. Both cases of identical and
nonidentical particles are considered. The azimuthal behaviour of the potential and
of the T-matrix elements leads to a set of integral equations for the T-matrix elements
in two variables, i.e. the momentum magnitude and the scattering angle. Observables
can be directly calculated from these T-matrix elements. Some symmetry relations for
the T-matrix elements reduce the number of equations to be solved.
Scattering of two spinless particles in 3D formulation with coulomb admixtures
F. Maulida and I. Fachruddin
Few-Body Systems 54, 217 (2013), DOI: 10.1007/s00601-012-0354-x
Abstract
Scattering of two spinless charge particles for simple forces including coulomb
admixtures is calculated without partial wave decomposition. The coulomb interaction
being taken is of the type of screened coulomb potential. For the forces range are
not infinite, the standard scattering theory is applied. The differential and total
cross section is calculated and coulomb effects are shown.
Scattering of a spin-1/2 particle off a spin-0 target in a simple three-dimensional basis
I. Fachruddin and A. Salam
Few-Body Systems 54, 221 (2013), DOI: 10.1007/s00601-012-0353-y
Abstract
Scattering of a spin-1/2 particle off a spin-0 target is formulated based on
a simple three-dimensional momentum-spin basis. The azimuthal behaviour of both
the potential and the T-matrix elements leads to a set of integral equations for
the T-matrix elements in two variables only, namely the momentum's magnitude and
the scattering angle. Some symmetry relations for the potential and the T-matrix
elements reduce the number of the integral equations to be solved by a factor of
one half. A complete list of the spin observables in terms of the two-dimensional
T-matrix elements is presented.
Recent developments of a three-dimensional description of the NN system
R. Skibinski, J. Golak, D. Rozpedzik, K. Topolnicki, H. Witala, W. Glöckle, A. Nogga, E. Epelbaum, H. Kamada, Ch. Elster, I. Fachruddin
Few-Body Systems 50, 279 (2011), DOI: 10.1007/s00601-010-0204-7
Abstract
A recently developed three-dimensional formulation of nucleon-nucleon (NN) scattering is briefly
presented. Here the NN t-matrix is represented by six spin-momentum operators accompanied by six scalar
functions of momentum vectors. A numerical example for the NN scattering cross section is given.
Two-nucleon system in three dimensions
J. Golak, W. Glöckle, R. Skibinski, H. Witala, D. Rozpedzik, K. Topolnicki, I. Fachruddin, Ch. Elster, A. Nogga
Phys. Rev. C81, 034006 (2010)
Abstract
A recently developed formulation for treating two- and three-nucleon bound states
in a three-dimensional formulation based on spin-momentum operators is extended to
nucleon-nucleon scattering. Here the nucleon-nucleon t-matrix is represented by
six spin-momentum operators accompanied by six scalar functions of momentum vectors.
We present the formulation and provide numerical examples for the deuteron and
nucleon-nucleon scattering observables. A comparison to results from a standard
partial wave decomposition establishes the reliability of this new formulation.
3N scattering in a three-dimensional operator formulation
W. Glöckle, I. Fachruddin, Ch. Elster, J. Golak, R. Skibinski, and H. Witala
Eur. Phys. J. A 43, 339–350 (2010)
Abstract
A recently developed formulation for a direct treatment of the equations for two- and three-
nucleon bound states as set of coupled equations of scalar functions depending only on vector momenta
is extended to three-nucleon scattering. Starting from the spin-momentum dependence occurring as scalar
products in two- and three-nucleon forces together with other scalar functions, we present the Faddeev
multiple scattering series in which order by order the spin degrees can be treated analytically leading to
3D integrations over scalar functions depending on momentum vectors only. Such formulation is especially
important in view of awaiting extension of 3N Faddeev calculations to projectile energies above the pion
production threshold and applications of chiral perturbation theory 3N forces, which are to be most
efficiently treated directly in such three-dimensional formulation without having to expand these forces
into a partial-wave basis.
A new way to perform partial-wave decompositions of few-nucleon forces
J. Golak, D. Rozpedzik, R. Skibinski, K. Topolnicki, H. Witala, W. Glöckle, A. Nogga, E. Epelbaum, H. Kamada, Ch. Elster, and I. Fachruddin
Eur. Phys. J. A 43, 241–250 (2010)
Abstract
We formulate a general and exact method of partial-wave decomposition (PWD) of any nucleon-
nucleon (NN) potential and any three-nucleon (3N) force. The approach allows one to efficiently use
symbolic algebra software to generate the interaction-dependent part of the program code calculating the
interaction. We demonstrate the feasibility of this approach for the one-boson exchange BonnB potential,
a recent nucleon-nucleon chiral force and the chiral two-pion-exchange three-nucleon force. In all cases
very good agreement between the new and the traditional PWD is found. The automated PWD offered
by the new approach is of the utmost importance in view of future applications of numerous chiral N3LO
contributions to the 3N force in three-nucleon calculations.
Treatment of two nucleons in three dimensions
I. Fachruddin, Ch. Elster, J. Golak, R. Skibinski, W. Glöckle, H. Witala
Proceeding of The 19th International IUPAP Conference on Few-Body Problems in Physics, August 31 - September 5, 2009, Bonn, Germany
EPJ Web Conf. 3, 05021 (2010), DOI: 10.1051/epjconf/20100305021
Abstract
We extend a new treatment proposed for two-nucleon (2N) and three-nucleon (3N) bound states
to 2N scattering. This technique takes momentum vectors as variables, thus, avoiding
partial wave decomposition, and handles spin operators analytically. We apply the general
operator structure of a nucleon-nucleon (NN) potential to the NN T-matrix, which becomes a sum
of six terms, each term being scalar products of spin operators and momentum vectors multiplied
with scalar functions of vector momenta. Inserting this expansions of the NN force and T-matrix
into the Lippmann-Schwinger equation allows to remove the spin dependence by taking traces and
yields a set of six coupled equations for the scalar functions found in the expansion of the T-matrix.
A formulation without partial wave decomposition for scattering of
spin-1/2 and spin-0 particles
I. Abdulrahman and I. Fachruddin
Mod. Phys. Lett. A24, 843 (2009)
Abstract
A new technique has been developed to calculate scattering of spin-1/2 and
spin-0 particles. The so called momentum-helicity basis states are constructed
from the helicity and the momentum states, which are not expanded in the
angular momentum states. Thus, all angular momentum states are taken into
account. Compared with the partial-wave approach this technique will then give
more benefit especially in calculations for higher energies. Taking as input a
simple spin-orbit potential, the Lippman-Schwinger equations for the T-matrix
elements are solved and some observables are calculated.
Scattering of spin-zero and spin-half particles in momentum-
helicity basis
I. Fachruddin and I. Abdulrahman
Proceeding of The Second Asian Physics Symposium, November 29-30, 2007, Bandung, Indonesia
Abstract
Scattering of 2 particles of spin 0 and 1/2 is evaluated based on
a basis constructed from the momentum and the helicity states
(the momentum-helicity basis). This shortly called three-dimensional
(3D) technique is a good alternative to the
standard partial wave (PW) technique especially for higher
energies, where PW calculations may become not feasible.
Taking as input a simple spin-orbit potential model we
calculate as an example the spin averaged differential cross
section and polarization.
Three-nucleon scattering at intermediate energies
I. Fachruddin, Ch. Elster, W. Glöckle
Proceeding of The Third Asia-Pacific Conference on Few-Body Problem
in Physics, July 26-30, 2005, Nakhon Ratchasima, Thailand
Abstract
By means of a technique, which does not employ partial wave (PW)
decompositions, the nucleon-deuteron break-up process is evaluated in
the Faddeev scheme, where only the leading order term of the
amplitude is considered. This technique is then applied to calculate
the semi-exclusive proton-deuteron break-up reaction d(p,n)pp
for proton laboratory energies Elab of a few hundred MeV.
A comparison with PW calculations is performed at 197 MeV projectile energy.
At the same energy rescattering processes, which are not included in
the 3D calculations yet, are shown to be still important in the full Faddeev
PW calculations, especially for the cross section and the analyzing power
Ay. Next, kinematical relativistic effects are investigated
for projectile energies up to about 500 MeV. At the higher energies,
those relativistic effects start not to be negligible, especially in
the peak of the cross section.
Operator form of 3H (3He) and its spin structure
I. Fachruddin, W. Glöckle, Ch. Elster, A. Nogga
Phys. Rev. C69, 064002 (2004)
Abstract
An operator form of the 3N bound state is proposed. It consists of eight operators
formed out of scalar products in relative momentum and spin vectors, which are
applied on a pure 3N spin 1/2 state. Each of the operators is associated with
a scalar function depending only on the magnitudes of the two relative momenta
and the angle between them. The connection between the standard partial wave decomposition
of the 3N bound state and the operator form is established, and the decomposition of
these scalar function in terms of partial wave components and analytically known
auxiliary functions is given. That newly established operator form of the 3N bound state
exhibits the dominant angular and spin dependence analytically. The scalar functions are
tabulated and can be downloaded. As an application the spin dependent nucleon momentum
distribution in a polarized 3N bound state is calculated to illustrate the use of
the new form of the 3N bound state.
nucl-th/0402015 | Back to Articles
The Nd break-up process in leading order in a three-dimensional approach
I. Fachruddin, Ch. Elster and W. Glöckle
Phys. Rev. C68, 054003 (2003)
Abstract
A three-dimensional approach based on momentum vectors as variables for solving
the three nucleon Faddeev equation in first order is presented. The nucleon-deuteron
break-up amplitude is evaluated in leading order in the NN T-matrix, which is also
generated directly in three dimensions avoiding a summation of partial wave contributions.
A comparison of semi-exclusive observables in the d(p,n)pp reaction calculated in
this scheme with those generated by a traditional partial wave expansion shows
perfect agreement at lower energies. At about 200 MeV nucleon laboratory energies deviations
in the peak of the cross section appear, which may indicate that special care is
required in a partial wave approach for energies at and higher than 200 MeV. The role of
higher order rescattering processes beyond the leading order in the NN T-matrix is
investigated with the result, that at 200 MeV rescattering still provides important
contributions to the cross section and certain spin observables. The influence of
a relativistic treatment of the kinematics is investigated. It is found that
relativistic effects become important at projectile energies higher than 200 MeV.
nucl-th/0307011 | Back to Articles
The proton-deuteron break-up process in a three-dimensional approach
I. Fachruddin, Ch. Elster and W. Glöckle
Mod. Phys. Lett. A18, 452 (2003)
Abstract
The pd break-up amplitude in the Faddeev scheme is calculated by employing a three-dimensional
method without partial wave decomposition (PWD). In a first step and in view of higher energies
only the leading term is evaluated and this for the process d(p,n)pp. A comparison with
the results based on PWD reveals discrepancies in the cross section around 200 MeV. This indicates
the onset of a limitation of the partial wave scheme. Also, around 200 MeV relativistic effects
are clearly visible and the use of relativistic kinematics shifts the cross section peak to
where the experimental peak is located. The theoretical peak height, however, is wrong and
calls first of all for the inclusion of rescattering terms, which are shown to be important in
a nonrelativistic full Faddeev calculation in PWD.
nucl-th/0211069 | Back to Articles
New forms of deuteron equations and wave function representations
I. Fachruddin, Ch. Elster and W. Glöckle
Phys. Rev. C63, 054003 (2001)
Abstract
A recently developed helicity basis for nucleon-nucleon (NN) scattering is applied to
the deuteron bound state. Here the total spin of the deuteron is treated in such a helicity
representation. For the bound state, two sets of two coupled eigenvalue equations are developed,
where the amplitudes depend on two and one variable, respectively. Numerical illustrations
based on the realistic Bonn-B NN potential are given. In addition, an `operator form' of
the deuteron wave function is presented, and several momentum dependent spin densities are
derived and shown, in which the angular dependence is given analytically.
nucl-th/0101009 | Back to Articles
Nucleon-nucleon scattering in a three-dimensional approach
I. Fachruddin, Ch. Elster and W. Glöckle
Nucl. Phys. A689, 507c (2001)
Abstract
The nucleon-nucleon t-matrix is calculated directly as function of two vector momenta for
different realistic NN potentials. The angular and momentum dependence of the full amplitude
is studied and NN observables are calculated.
nucl-th/0104027 | Back to Articles
Nucleon-nucleon scattering in a three dimensional approach
I. Fachruddin, Ch. Elster and W. Glöckle
Phys. Rev. C62, 044002 (2000)
Abstract
The nucleon-nucleon (NN) t-matrix is calculated directly as function of two vector momenta
for different realistic NN potentials. To facilitate this a formalism is developed for solving
the two-nucleon Lippmann-Schwinger equation in momentum space without employing a partial wave
decomposition. The total spin is treated in a helicity representation. Two different
realistic NN interactions, one defined in momentum space and one in coordinate space,
are presented in a form suited for this formulation. The angular and momentum dependence of
the full amplitude is studied and displayed. A partial wave decomposition of the full amplitude
it carried out to compare the presented results with the well known phase shifts provided by
those interactions.
preprint (pdf) | Back to Articles
Helicity formalism for NN scattering without partial wave decomposition
I. Fachruddin and W. Glöckle
Few-Body Systems Suppl. 12, 1-4 (2000)
Abstract
At intermediate energies it appears advantageous to work without partial wave decomposition
and use directly the momentum vectors. We derived a finite set of coupled Lippmann-Schwinger
equations using helicities. They can be used for any type of realistic NN forces, which are
given in operator form. As an example we have chosen the Bonn OBEPR NN potential.
nucl-th/9910045 | Back to Articles
A three-dimensional momentum space formulation for the NN system and the Nd break-up process
I. Fachruddin
PhD thesis (2003), supervisor: Prof. Dr. W. Glöckle,
Ruhr-Universität-Bochum, Bochum, Germany
Abstract
A technique to perform few-nucleon calculations in momentum space without employing partial wave (PW)
decompositions is developed. It is shortly called the 3D technique, and is intended to be a viable
alternative to the successful PW technique, especially for higher energies.
The development begins with the nucleon-nucleon (NN) system, where antisymmetric momentum-helicity
basis states are defined, which are constructed using momentum vector states and helicity states of NN total spin.
Appropiate for the momentum-helicity basis a set of six independent operators is defined to express any NN potential, which is given in operator form. Representative potentials are the modern AV18 and Bonn-B potentials, which are used in this work. The 3D technique is applied to both NN scattering and the deuteron. Comparison with PW calculations is performed to test the formulation and numerical realization of the 3D technique. In addition for the deuteron, an "operator form" of the deuteron wave function is presented, allowing for investigating some momentum dependent spin densities with analytic angular behavior.
The development and application of the 3D technique is extended to the Nd break-up process, in which
the Faddeev's scheme is used. In this work only the leading term of the full Nd break-up amplitude is
considered to describe the process at higher energies, beyond ~ 200 MeV projectile laboratory energy.
For simplicity the deuteron state is kept being expanded in its partial wave components, s and d waves.
The leading term of the amplitude in the 3N basis states is derived in a so called physical representation,
where spins and isospins of the individual nucleons are taken, and the spins are quantized along an arbitrary
but fixed z axis. For the NN system a connection between the physical representation and that in
the momentum-helicity basis has been obtained. This leads then to an expression for the leading term of
the full Nd break-up amplitude in terms of the NN T-matrix element in the momentum-helicity basis,
with which 3N observables are calculated. These are the differential cross section, the neutron polarization,
the proton analyzing power and the polarization transfer coefficients. The formulation is applied to
the (p,n) charge exchange reaction in the semi-exclusive pd break-up process d(p,n)pp for energies
up to ~ 500 MeV. For energies below ~ 200 MeV comparison with PW calculations is performed, as well as with
the ones taking the full Nd break-up amplitude to see rescattering effects. Finally relativistic kinematics is
introduced into the formulation, allowing to observe some relativistic effects. Comparisons are also performed
with experimental data.
Results in this work show that the 3D technique has proven to be a good alternative to the PW decomposition
and appears to be necessary at higher energies, where the PW technique is no longer feasible. In contrast to
the PW decomposition the 3D technique requires much less algebraic work. For lower energies, where
the PW calculations are still reliable, the 3D calculations show perfect agreement with the PW calculations.