19,042 research outputs found
Practical Algebraic Renormalization
A practical approach is presented which allows the use of a non-invariant
regularization scheme for the computation of quantum corrections in
perturbative quantum field theory. The theoretical control of algebraic
renormalization over non-invariant counterterms is translated into a practical
computational method. We provide a detailed introduction into the handling of
the Slavnov-Taylor and Ward-Takahashi identities in the Standard Model both in
the conventional and the background gauge. Explicit examples for their
practical derivation are presented. After a brief introduction into the Quantum
Action Principle the conventional algebraic method which allows for the
restoration of the functional identities is discussed. The main point of our
approach is the optimization of this procedure which results in an enormous
reduction of the calculational effort. The counterterms which have to be
computed are universal in the sense that they are independent of the
regularization scheme. The method is explicitly illustrated for two processes
of phenomenological interest: QCD corrections to the decay of the Higgs boson
into two photons and two-loop electroweak corrections to the process .Comment: version to be published in Annals of Physic
Dilogarithm Identities in Conformal Field Theory and Group Homology
Recently, Rogers' dilogarithm identities have attracted much attention in the
setting of conformal field theory as well as lattice model calculations. One of
the connecting threads is an identity of Richmond-Szekeres that appeared in the
computation of central charges in conformal field theory. We show that the
Richmond-Szekeres identity and its extension by Kirillov-Reshetikhin can be
interpreted as a lift of a generator of the third integral homology of a finite
cyclic subgroup sitting inside the projective special linear group of all real matrices viewed as a {\it discrete} group. This connection
allows us to clarify a few of the assertions and conjectures stated in the work
of Nahm-Recknagel-Terhoven concerning the role of algebraic -theory and
Thurston's program on hyperbolic 3-manifolds. Specifically, it is not related
to hyperbolic 3-manifolds as suggested but is more appropriately related to the
group manifold of the universal covering group of the projective special linear
group of all real matrices viewed as a topological group. This
also resolves the weaker version of the conjecture as formulated by Kirillov.
We end with the summary of a number of open conjectures on the mathematical
side.Comment: 20 pages, 2 figures not include
High-precision calculation of multi-loop Feynman integrals by difference equations
We describe a new method of calculation of generic multi-loop master
integrals based on the numerical solution of systems of difference equations in
one variable. We show algorithms for the construction of the systems using
integration-by-parts identities and methods of solutions by means of expansions
in factorial series and Laplace's transformation. We also describe new
algorithms for the identification of master integrals and the reduction of
generic Feynman integrals to master integrals, and procedures for generating
and solving systems of differential equations in masses and momenta for master
integrals. We apply our method to the calculation of the master integrals of
massive vacuum and self-energy diagrams up to three loops and of massive vertex
and box diagrams up to two loops. Implementation in a computer program of our
approach is described. Important features of the implementation are: the
ability to deal with hundreds of master integrals and the ability to obtain
very high precision results expanded at will in the number of dimensions.Comment: 55 pages, 5 figures, LaTe
Kira - A Feynman Integral Reduction Program
In this article, we present a new implementation of the Laporta algorithm to
reduce scalar multi-loop integrals---appearing in quantum field theoretic
calculations---to a set of master integrals. We extend existing approaches by
using an additional algorithm based on modular arithmetic to remove linearly
dependent equations from the system of equations arising from
integration-by-parts and Lorentz identities. Furthermore, the algebraic
manipulations required in the back substitution are optimized. We describe in
detail the implementation as well as the usage of the program. In addition, we
show benchmarks for concrete examples and compare the performance to Reduze 2
and FIRE 5.
In our benchmarks we find that Kira is highly competitive with these existing
tools.Comment: 37 pages, 3 figure
Is the third coefficient of the Jones knot polynomial a quantum state of gravity?
Some time ago it was conjectured that the coefficients of an expansion of the
Jones polynomial in terms of the cosmological constant could provide an
infinite string of knot invariants that are solutions of the vacuum Hamiltonian
constraint of quantum gravity in the loop representation. Here we discuss the
status of this conjecture at third order in the cosmological constant. The
calculation is performed in the extended loop representation, a generalization
of the loop representation. It is shown that the the Hamiltonian does not
annihilate the third coefficient of the Jones polynomal () for general
extended loops. For ordinary loops the result acquires an interesting
geometrical meaning and new possibilities appear for to represent a
quantum state of gravity.Comment: 22 page
Renormalization aspects of N=1 Super Yang-Mills theory in the Wess-Zumino gauge
The renormalization of N=1 Super Yang-Mills theory is analysed in the
Wess-Zumino gauge, employing the Landau condition. An all orders proof of the
renormalizability of the theory is given by means of the Algebraic
Renormalization procedure. Only three renormalization constants are needed,
which can be identified with the coupling constant, gauge field and gluino
renormalization. The non-renormalization theorem of the gluon-ghost-antighost
vertex in the Landau gauge is shown to remain valid in N=1 Super Yang-Mills.
Moreover, due to the non-linear realization of the supersymmetry in the
Wess-Zumino gauge, the renormalization factor of the gauge field turns out to
be different from that of the gluino. These features are explicitly checked
through a three loop calculation.Comment: 15 pages, minor text improvements, references added. Version accepted
for publication in the EPJ
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