430 research outputs found
An Explicit Construction of Casimir Operators and Eigenvalues : I
We give a general method to construct a complete set of linearly independent
Casimir operators of a Lie algebra with rank N. For a Casimir operator of
degree p, this will be provided by an explicit calculation of its symmetric
coefficients . It is seen that these coefficients can be
descibed by some rational polinomials of rank N. These polinomials are also
multilinear in Cartan sub-algebra indices taking values from the set . The crucial point here is that for each degree one needs, in
general, more than one polinomials. This in fact is related with an observation
that the whole set of symmetric coefficients is
decomposed into sum subsets which are in one to one correspondence with these
polinomials. We call these subsets clusters and introduce some indicators with
which we specify different clusters. These indicators determine all the
clusters whatever the numerical values of coefficients
are. For any degree p, the number of clusters is independent of rank N. This
hence allows us to generalize our results to any value of rank N.
To specify the general framework explicit constructions of 4th and 5th order
Casimir operators of Lie algebras are studied and all the polinomials
which specify the numerical value of their coefficients are given explicitly.Comment: 14 pages, no figures, revised version, to appear in Jour.Math.Phy
Casimir operators of the exceptional group G2
We calculate the degree 2 and 6 Casimirs operators in explicit form, with the
generators of G2 written in terms of the subalgebra A2Comment: 10 p., MAD/TH/93-05, (LaTex
Evidence for higher order QED in e+ e- pair production at RHIC
A new lowest order QED calculation for RHIC e+ e- pair production has been
carried out with a phenomenological treatment of the Coulomb dissociation of
the heavy ion nuclei observed in the STAR ZDC triggers. The lowest order QED
result for the experimental acceptance is nearly two standard deviations larger
than the STAR data. A corresponding higher order QED calculation is consistent
with the data.Comment: 4 pages, 4 figures, latex, revte
Symmetry and Supersymmetry in Nuclear Pairing: Exact Solutions
Pairing plays a crucial role in nuclear spectra and attempts to describe it
has a long history in nuclear physics. The limiting case in which all single
particle states are degenerate, but with different s-wave pairing strengths was
only recently solved. In this strong coupling limit the nuclear pairing
Hamiltonian also exhibits a supersymmetry. Another solution away from those
limits, namely two non-degenerate single particle states with different pairing
strengths, was also given. In this contribution these developments are
summarized and difficulties with possible generalizations to more single
particle states and d-wave pairing are discussed.Comment: 6 pages of LATEX, to be published in the Proceedings of the "10th
Int. Spring Seminar on Nuclear Physics: New Quests in Nuclear Structure",
Vietri Sul Mare, May 21-25, 201
On characteristic equations, trace identities and Casimir operators of simple Lie algebras
Two approaches are developed to exploit, for simple complex or compact real
Lie algebras g, the information that stems from the characteristic equations of
representation matrices and Casimir operators. These approaches are selected so
as to be viable not only for `small' Lie algebras and suitable for treatment by
computer algebra. A very large body of new results emerges in the forms, a) of
identities of a tensorial nature, involving structure constants etc. of g, b)
of trace identities for powers of matrices of the adjoint and defining
representations of g, c) of expressions of non-primitive Casimir operators of g
in terms of primitive ones. The methods are sufficiently tractable to allow not
only explicit proof by hand of the non-primitive nature of the quartic Casimir
of g2, f4, e6, but also e.g. of that of the tenth order Casimir of f4.Comment: 39 pages, 8 tables, late
Exclusive and inclusive muon pair production in collisions of relativistic nuclei
The exclusive production of one pair in collisions of two
ultra-relativistic nuclei is considered. We present a simple method for the
calculation of the Born cross section for this process based on an improved
equivalent photon approximation. We find that the Coulomb corrections to this
cross section (corresponding to multi-photon exchange of the produced
with the nuclei) are small while the unitarity corrections
(corresponding to the exchange of light-by-light blocks between nuclei) are
large. This is in sharp contrast to the exclusive pair production
where the Coulomb corrections to the Born cross section are large while the
unitarity corrections are small. We calculate also the cross section for the
production of one pair and several pairs in the leading
logarithmic approximation. Using this cross section we find that the inclusive
production of pair coincides in this approximation with its Born
value.Comment: 16 pages, 4 figure
Invariant tensors and Casimir operators for simple compact Lie groups
The Casimir operators of a Lie algebra are in one-to-one correspondence with
the symmetric invariant tensors of the algebra. There is an infinite family of
Casimir operators whose members are expressible in terms of a number of
primitive Casimirs equal to the rank of the underlying group. A systematic
derivation is presented of a complete set of identities expressing
non-primitive symmetric tensors in terms of primitive tensors. Several examples
are given including an application to an exceptional Lie algebra.Comment: 11 pages, LaTeX, minor changes, version in J. Math. Phy
Production of QED pairs at small impact parameter in relativistic heavy ion collisions
The STAR collaboration at RHIC is measuring the production of
electron-positron pairs at small impact parameters, larger than but already
close to the range, where the ions interact strongly with each other. We
calculate the total cross section, as well as, differential distributions of
the pair production process with the electromagnetic excitation of both ions in
a semiclassical approach and within a lowest order QED calculation. We compare
the distribution of electron and positron with the one coming from the cross
section calculation without restriction on impact parameter. Finally we give an
outlook of possible results at the LHC.Comment: 15 pages, 8 figure
Multipair approach to pairing in nuclei
The ground state of a general pairing Hamiltonian for a finite nuclear system
is constructed as a product of collective, real, distinct pairs. These are
determined sequentially via an iterative variational procedure that resorts to
diagonalizations of the Hamiltonian in restricted model spaces. Different
applications of the method are provided that include comparisons with exact and
projected BCS results. The quantities that are examined are correlation
energies, occupation numbers and pair transfer matrix elements. In a first
application within the picket-fence model, the method is seen to generate the
exact ground state for pairing strengths confined in a given range. Further
applications of the method concern pairing in spherically symmetric mean fields
and include simple exactly solvable models as well as some realistic
calculations for middle-shell Sn isotopes. In the latter applications, two
different ways of defining the pairs are examined: either with J=0 or with no
well-defined angular momentum. The second choice reveals to be more effective
leading, under some circumstances, to solutions that are basically exact.Comment: To appear in Physical Review
A search for double beta decays of tin isotopes with enhanced sensitivity
A search for the various double beta decay modes of 124Sn and 112Sn has been
performed on 75 kg.days of data. New half-life limits for excited states in
124Sn have been obtained including a lower limit for the decay into the first
excited 2+ state of 124Te of T_half > 0.87e20 yrs (90% CL) and into the first
excited 0+ state of T_half > 1.08e20 yrs (90% CL). Ground state and excited
state transitions of 112Sn have also been experimentally explored. A limit for
the 2 neutrino double electron capture of T_half > 1.8e19 yrs (90% CL) is
obtained. The non-observation of de-excitation gammas from the 0+ at 1888.5keV
results in a lower half-life limit on the 0 neutrino double electron capture
decay of 112Sn of T_half > 0.8e19 yrs (90% CL), despite a possible resonant
enhancement of the decay rate due to degenerated states.Comment: 6 pages, 7 figures, updated analysis and tex
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