Typical Computations
I have mainly used the FORTRAN language for numerical computations and
data analysis in high energy pysics, and have worked extensively with many
numerical packages including IMSL, Numerical Recipes, and MINUIT. For
symbolic work, I have used the MAPLE package in most of my computations.
Here I give some examples of typical computations in my research.
Example 1. Multidimensional integration:
As is well known, in perturbative quantum field theory the amplitudes of
the physical processes can be expressed in terms of Feynman diagrams at
different orders of perturbation theory. These diagrams are in turn
expressed in terms of integrals over four-momenta of the intermediate
particles running in the loops of the diagrams. Evaluation of these
integrals is in general a non-trivial task as the integration variables
are four-component vectors. This could lead to complicated computations.
A well known technique to evaluate the loop integrals is know as Feynman
parameterization in which the integrals over four-momenta are rewritten in
terms of ordinary multidimensional integrals which can then be evaluated
using different symbolic and numerical techniques.
I published a paper
[A.H. Fariborz, ``A Form Factor Evaluation in Heavy Meson Decay'',
Maple Technical NewsLetters 2, 52-55 (1995)]
in which Maple was used to evaluate
the form factors for a decay of heavy mesons arising from a loop momentum
integral of quantum field theory.
In this work, different components of a MAPLE code for evaluation of
multidimensional integrals is analysed in detail.
The form factor discussed in this paper, can be expressed in the form of
the follwing double integral
where $D, A, B, L'$ and $L''$ are polynomials in $x_1$ and $x_2$, and $c$
and $c'$ are contants.
I have shown how to apply MAPLE to evaluate this form factor, as well as a
wide range of similar problems. My
article was qualified by the journal as the first paper published in
{\it Maple Technical NewsLetters} oriented towards high energy
physics.
Since publishing my first paper on MAPLE application, I have worked on
different projects involving more complicated multidimensional
integration. For
example, in the evaluation of the induced Yukawa coupling of the light
quarks
[M.R. Ahmady, V. Elias, A.H. Fariborz and R.R. Mendel,
`` Induced Light-Quark Yukawa Couplings as a Probe of Low-Energy
Dynamics in QCD'',
Prog. Theor. Phys. 96, 781 (1996)]
, one has to evaluate family of multidimensional integrals of the form
where $L(s)$ is, for example,
and $c_1$ and $c_2$ are constants.
Pure analytical approach, if not impossible, is not certainly an
efficient way of dealing with such complicated integrals.
I have used different symbolic and numerical techniques to evaluate
similar multidimensional integrals. In particular I have employed
utilities in MAPLE, and have written FORTRAN programs
which incorporate routines from IMSL
(International Mathematical and Statistical Libraries), and Numerical
Recopies, for computing such integrals. Techniques developed in the
paper discussed above are applicable to this more general situation.
Example 2. Simulation of resonance shapes:
In low-energy QCD investigations one has to often incorporate shape of
physical resonances into the computation. This is, however, either not
analytically known, or even the simplest analytical approximations are
not easy to carry through some field theoretical calculations like QCD
sum-rules.
Therefore, at least in certain regimes, it would be interesting to
approximate the
resonance shapes with simple shapes.
For this purpose, I have written MAPLE codes which simulate the famous
Breit-Wigner shape for a resonance in terms of square pulses. This
technique, was used in the QCD sum-rule analysis of the lowest-lying
$\sigma$ meson
[V. Elias, A.H. Fariborz, Fang Shi and T.G. Steele,
`` QCD Sum-Rule Consistency of Lowest-Lying $q {\bar q}$ Scalar
Resonance'', Nucl. Phys. A633, 279 (1998)]
, and was found to provide a reliable approximation to the real
situation. In figure 1, the Breit-Wigner shape is compared with its
pulse approximation generated by a program in MAPLE.
Example 3. Numerical investigation of functional constraints:
In QCD sum-rules one computes mathematical quantities that describe the
hadron under investigation from two different ways -- from the effective
hadron field theory (EFT), and from the underlying quark fundamental field
theory (FFT). These mathematical quantities are in general function of
some parameter ($\tau$). Matching these quantities leads to a
set of non-linear equations which
in general are quite complicated, and certain amount of computational
techniques should be developed to estimate the solutions and study their
stability. These sets of equations can be symbolically expressed as:
where the left hand side is a function of hadron properties (like hadron
mass, hadron decay width, and hadron decay constant). The right hand
side is a function of the parameters in the fundamental Lagrangian (like
QCD mass scale, condensates of elementary fields, instanton size ...) and
in general is a collection of complicated functions including
different special functions and integral functions. This is why it is
not easy to study the above system of constraints for a range of
parameter $\tau$.
I have developed codes in MAPLE which study the above system of
constraints and investigate their stability.
Example 4. Multiparameter minimization:
Effective field theory approach to low-energy QCD is a powerful method
to gain insights into the hadronic properties and their internal quark
substructure
[D. Black, A.H. Fariborz, F. Sannino and J.Schechter,
``Evidence for a Scalar $\kappa (900)$ Resonance in $\pi K$ Scattering'',
Phys. Rev. D58, 054012 (1998);
D. Black, A.H. Fariborz, F. Sannino and J.Schechter, ``Putative Light
Scalar Nonet'', Phys. Rev. D59, 074026 (1999);
A.H. Fariborz and J. Schechter,
``$\eta'\rightarrow \eta\pi\pi$ Decay as a Probe of a Possible Lowest
Lying Scalar Nonet'', Phys. Rev. D60, 034002 (1999)].
In this approach one computes certain physical amplitudes, like
the scattering amplitude for example, and matches the result to the
experimental data. In this matching, some physical quantities like
the mass and decay width of an unknown resonance, can be taken as free
parameters and be determined in a fit of the theoretical
prediction to the experimental data. The theoretical amplitudes found in
the effective theories are usually a complicated function of the resonance
properties and their fit to the experimental data requires a careful
numerical analysis.
I have extensively worked on FORTRAN codes that perform such numerical
fits using the MINUIT package. Specific challenges in this type of
numerical work includes performing a reasonable error analysis, and
filtering out the unphysical regions in a multidimensional parameter
space. The Following two figures [from paper D. Black, A.H. Fariborz and
J.Schechter, ``Mechanism for a next-to-lowest lying scalar meson nonet'',
Phys. Rev. D61, 074001-1-10 (2000); in which the decay properties of a
scalar meson a0(1450) to different channels is investigated] show examples
of numerical search in a two-dimensional parameter space (A A') for
regions that theory agrees with experiment. Without such numerical
simulation, extracting this information would have not been possible.
Regions in the AA' parameter space consistent
with the currently available experimental estimates on the decay widths of
a0(1450). Points
on the ellipse are consistent with the total decay width of a0(1450).
Red and blue regions respectively represent
points consistent with
the experimental ratio
Gamma [a0(1450) -> K K-bar ] / Gamma
[a0(1450) -> pi eta] = 0.88\pm 0.23, and
Gamma [a0(1450) -> pi eta' ] / Gamma [a0(1450) ->
pi eta] = 0.35 +- 0.16.
Regions in the AA' parameter space consistent with the current
experimental and theoretical estimates on the decay widths of
a0(1450), K0*(1430), a0(980) and kappa(900).
Points on the ellipse are consistent with the total decay width of
a0(1450).
Squares and circles respectively represent points
consistent with
the experimental ratio
Gamma [a0(1450) -> K K-bar ] / Gamma
[a0(1450) -> pi eta ] = 0.88 +- 0.23, and
Gamma [a_0(1450) -> pi eta' ] / Gamma [a0(1450) ->
pi eta] = 0.35 +- 0.16.
Example 5. Neutrino experimental data analysis:
Any theoretical model provides estimates of different quantities
which would be eventually matched to their corresponding experimental
estimates. However, the numerical matching of theoretical predictions to
experimental data is not always straightforward, and sometimes a great
amount of effort is required to perform such data analyses. In the case
of neutrino physics, among other difficulties, there are two challenges in
the data analysis. First, dealing with numbers which
could be different by many orders of magnitude, and as a result,
different
sources of round off errors could make the numerical analysis quite
complex. Second, depending on the theoretical model, there
are usually many unknown parameters and therefore one has to work within a
large parameter space.
In collaboration with Syracuse University group, in an initial phase of
an ongoing collaboration aimed to understand the recent experimental data
on neutrino oscillation, I wrote FORTRAN codes that reproduce, for the
purpose of testing and fine-tuning the programs, the allowed regions in
the parameter space of physical quantities consistent with other
theoretical investigations. In particular, the allowed parameter spaces
determined by LSND, KARMEN and CHOOZ experiments are reproduced by these
programs. The effect of neutrino energy distribution on the probability of
oscillation, which in its simplest form can be expressed by a Gaussian
distribution, is numerically studied in these programs. I have also
developed symbolic codes in MAPLE that study the texture of the neutrino
mass matrix and the resulting probability of oscillation.