ACAT 2005

Using FORM language for symbolic calculation in CompHEP is described.
We present current status of this project and discuss plans for the future
development.  Also we show some examples for the real processes.

A geometrical way to calculate dimensionally-regulated Feynman 
diagrams is reviewed. In the one-loop N-point case, the 
results can be related to certain volume integrals in 
non-Eulidean geometry. Analytical continuation of the results 
to other regions of the kinematical variables is discussed. As 
an example, the dimensionally-regulated three-point function 
is considered, including all orders of its epsilon-expansion.

The precision control is indispensable for the
large scale computations like done using GRACE, the system for the
automatic Feynman diagram calculations.
Recently, Hitachi Co. Ltd. developed a new library of quadruple and
octuple precision for FORTRAN codes. It has also the function to
report information on lost-bits during the operation. Using this new
library, 1-loop corrections to e+e- -> e+e-tau+tau- were analyzed,
where it was known that the quadruple precision was requested in some
phase space points. The new library not only can locate where the
lost-bits occur but also tells us the precision of results
themselves.

The two dimensional harmonic polylogarithms (GPL-functions) appear
in calculations  of multi-loop integrals. We discuss these functions
in the context  of analytical solutions for master integrals in
the case of massive Bhabha scattering in QED. We derive a set of irreducible
GPLs up to  weight 4, with analytical representations up to weight 3.
Further, we discuss conformal transformations of GPLs, and also
transformations between master integrals in the s- and t- channel.

We discuss the three loop renormalization of QCD in various
nonlinear gauges such as Curci-Ferrari and the maximal abelian
gauge. The anomalous dimensions are determined by paying
respect to the Slavnov-Taylor identities and hence the
renormalization group function of the BRST dimension two
operator is deduced. Given the large number of Feynman
diagrams to be evaluated using the Mincer algorithm written in
Form, the consistency checks on the results are discussed.
The programming issues concerning working with a colour group
split into diagonal and off-diagonal sectors are also considered.

Cuba is a library for multidimensional numerical integration.  
It features four independent algorithms, Vegas, Suave, 
Divonne, and Cuhre. 
All four are general-purpose methods (i.e. do not require a 
particular form of the integrand) and can integrate vector 
integrands.  The Cuba routines can be called from Fortran, 
C/C++, and Mathematica.  Their invocation is very similar and 
it is thus easy to exchange routines for comparison purposes.

An algorithm is proposed to accelerate the calculation of tree processes
by classifying Feynman graphs and by factorizing Feynman amplitudes.
Feynman graphs in a process are classified without duplication based on
graph theoretical method.  As Fenman graphs in each class have a common
vertex, the Feynman amplitudes are factorized in terms of this vertex.
The classification and factorization method are applied recursively
combined with the generation method of Feynman graphs.  Calculations in
Electro-weak theory became 10 -- 100 times faster than the traditional
method for processes with 6 final particles.

We present different low-discrepancy series as
source of quasi-random numbers.
These are useful for numerical integration
in 2 - 15 dimensions since they converge
faster than both equally spaced points (used in one
dimension) and pseudo-random points (used
for many dimensions).
Examples and applications from experimental
and theoretical high-energy physics are given.

In this talk, recent developments in the field of event generators
for high-energy physics will be presented.

CompHEP is a powerful system for calculation in HEP. One of the
advantage of the CompHEP is a possibility to use of user defined QFT models.
However, the model has some restrictions. In particular it can not contain
vertex which depends on scalar function on incoming momentum â~@~S form
factor. One of the non-trivial examples of such model is a SUSY. Authors
describe general approach to implementation of  such objects in
grid-oriented version of CompHEP.

I present the status of the ThePEG project for creating a common
platform for implementing C++ event generators. I also describe briefly how
the new version of Pythia, Herwig and Ariadne are implemented in this
framework.

The package aITALC has been developed for fully automated calculations of
two fermion production at e^+ e^- collider and other similar reactions. We
emphasize the connection and interoperability between the different modules
required for the calculation and the external tools DIANA, FORM and
LOOPTOOLS. Results for e^+ e^- -> e^+ e^-, f \bar{f}, b\bar{s} are
presented.

Higher orders in perturbation theory require the
calculation of Feynman integrals at multiple loops.
We report on an approach to systematically solve
Feynman integrals by means of symbolic summation.
As an example, we discuss recent calculations
of structure functions in deep-inelastic scattering
to three loops.

Discussion of a few selected examples of Feynman diagram generation with
special constraints (most of them have already been used in a practical
calculation, but have not been presented before, to the best of my knowledge).
It will be shown that with a little help QGRAF can be used to solve some types
of problems that may seem to be out of its current reach. Those examples also
serve to expose some weaknesses of the package. The connection between those
weaknesses and the recent (and ongoing) evolution of QGRAF will be roughly
outlined.

As it is argued in [1] Groebner bases form the most universal algorithmic
tool for reduction of loop integrals determined by the recurrence relations
derived from the integration by parts method. These recurrence relations can
be considered as generators of an ideal in the ring of finite-difference
polynomials.

In this talk we present a Maple package for computing a Groebner basis of the
ideal. The built-in algorithm is based on the use of Janet-like monomial
division and will be presented in a separate talk [2].
The package is a modified version of our earlier package oriented to
commmutative and linear differential algebra and based on Janet involutive
division algorithm [3].

The modified version is specialized to linear difference ideals and uses
Janet-like division [4] which is more efficient than Janet division.
We illustrate the package by some one-loop examples

[1] V.P.Gerdt. Groebner Bases in Perturbative Calculations,
Nuclear Physics B (Proc. Suppl.) 135, 2004, 232-237. URL:
http://arXiv.org/hep-ph/0501053.

[2] V.P.Gerdt. An Algorithm for Reduction of Loop Integrals. A talk at ACAT-05.

[3] Yu.A.Blinkov, V.P.Gerdt, C.F.Cid, W.Plesken and D.Robertz.
The Maple Package "Janet": I. Polynomial Systems, Computer Algebra in
Scientific Computing / CASC 2003, V.G.Ganzha, E.W.Mayr, and E.V.Vorozhtsov,
eds., Institute of Informatics, Technical University of Munich, Garching, 2003,
pp.31--54.; II. Linear Partial Differential Equations, ibid., pp.41--54.

[4] V.P.Gerdt. Janet-like Groebner Bases. Submitted to
MEGA-05 (Porto Conte, Alghero, Sardinia, May 27th - June 1st, 2005).

As it was recently shown in [1], determining the output
of a quantum computation is equivalent to counting the number of solutions
of a certain set of polynomials defined over the finite field Z_2.
In this talk we present a C# package that allows a user for an input quantum
circuit to generate a set of multivariate polynomials over Z_2 whose total
number of solutions in Z_2 determines the output of the quantum computation
defined by the circuit. Our program has a user-friendly graphical interface
and a built-in base of the basic elements, i.e., quantum gates and wires.
The user can easily assemble an input circuit from those elements.

The generated polynomial system can further be converted to the canonical
involutive form by applying efficient algorithms described in [2]. The
involutive form is generally more appropriate for counting the number of
common roots of the polynomials.

[1] Christopher M. Dawson et al. Quantum computing and polynomial
equations over the finite field Z_2. arXiv:quant-ph/0408129, 2004.
[2] Gerdt V.P. Involutive Algorithms for Computing Groebner Bases.
Proceedings of the NATO Advanced Research Workshop "Computational
commutative and non-commutative algebraic geometry" (Chishinau,
June 6-11, 2004), IOS Press, to appear.

We report on the status of our project of parallelization of the
symbolic manipulation program FORM.
We have now a parallel version of FORM running on Cluster- and
SMP-architectures. This version can be used to run arbitrary FORM
programs in parallel.

An introduction to the C++ library GiNaC will be given. GiNaC
extents the C++ language by new objects and methods for the representation
and manipulation of arbitrary symbolic expressions. Features and
applications of GiNaC will also be highlighted.

We consider the recent progress in simulating light chiral fermions in QCD
on the lattice. We discuss various approaches to implementing an exact chiral
symmetry on the lattice using different 4- or 5-dimensional formulations
of QCD. This provides a theoretical framework within which we can compare the
algorithmic alternatives for their implementation.

To estimate the multi-jet production cross sections
as well as their characteristic distributions is a difficult
task. Perturbation theory based on Feynman graphs runs into computational
problems, since the
number of graphs contributing to the amplitude grows like n!.
Color and helicity summation is additional
source of computational inefficiencies. In order
to overcome the computational obstacles recursive methods
can be used. A computational algorithm based on such
recursive equations will be  presented. The amplitude
computed by using
recursive equations result in a computational cost
growing asymptotically as 3^n. Additionally color and helicity structure
are appropriately transformed so the Monte Carlo summation is used in
those cases as well.

New developments concerning the extension of the parallelized DICE 
are presented in this paper. DICE is a general purpose multidimensional 
numerical integration package. In general, there are two approaches for
the parallelization, "Data Parallelism" way and "Function Parallelism" way.
We had already developed the parallelization code using "Data Parallelism"
way and reported it in ACAT2002 in Moscow.  
Here, we will present the preliminary result of the implementation of 
parallelized DICE in an another approach, "Function Parallelism" way.