59 research outputs found
Determinental formulae for complete symmetric functions
AbstractWe translate Goulden's combinatorial proof of the Jacobi-Trudi identity into the language of lattice paths and then use the Gessel-Viennot technique to prove a general identity between a determinant involving complete symmetric functions and a sum of skew Schur functions
General Solutions for Tunneling of Scalar Fields with Quartic Potentials
For the theory of a single scalar field with a quartic potential
, we find semi-analytic expressions for the Euclidean action in
both four and three dimensions. The action in four dimensions determines the
quantum tunneling rate at zero temperature from a false vacuum state to the
true vacuum state; similarly, the action in three dimensions determines the
thermal tunneling rate for a finite temperature theory. We show that for all
quartic potentials, the action can be obtained from a one parameter family of
instanton solutions corresponding to a one parameter family of differential
equations. We find the solutions numerically and use polynomial fitting
formulae to obtain expressions for the Euclidean action. These results allow
one to calculate tunneling rates for the entire possible range of quartic
potentials, from the thin-wall (nearly degenerate) limit to the opposite limit
of vanishing barrier height. We also present a similar calculation for
potentials containing terms, which arise in the
one-loop approximation to the effective potential in electroweak theory.Comment: 17 pages, 6 figures not included but available upon request, UM AC
93-
Compatibility, multi-brackets and integrability of systems of PDEs
We establish an efficient compatibility criterion for a system of generalized
complete intersection type in terms of certain multi-brackets of differential
operators. These multi-brackets generalize the higher Jacobi-Mayer brackets,
important in the study of evolutionary equations and the integrability problem.
We also calculate Spencer delta-cohomology of generalized complete
intersections and evaluate the formal functional dimension of the solutions
space. The results are applied to establish new integration methods and solve
several differential-geometric problems.Comment: Some modifications in sections 6.1-2; new references're adde
Axions and the Strong CP Problem
Current upper bounds of the neutron electric dipole moment constrain the
physically observable quantum chromodynamic (QCD) vacuum angle . Since QCD explains vast experimental data from the 100 MeV
scale to the TeV scale, it is better to explain this smallness of
in the QCD framework, which is the strong \Ca\Pa problem. Now,
there exist two plausible solutions to this problem, one of which leads to the
existence of the very light axion. The axion decay constant window, $10^9\
{\gev}\lesssim F_a\lesssim 10^{12} \gev{\cal O}(1)\theta_1F_a\gtrsim 10^{12}\theta_1<{\cal O}(1)$,
axions may constitute a significant fraction of dark matter of the universe.
The supersymmetrized axion solution of the strong \Ca\Pa problem introduces its
superpartner the axino which might have affected the universe evolution
significantly. Here, we review the very light axion (theory,
supersymmetrization, and models) with the most recent particle, astrophysical
and cosmological data, and present prospects for its discovery.Comment: 47 pages with 32 figure
Lattic path proofs of extended Bressoud-Wei and Koike skew Schur function identities
Our recent paper provides extensions to two classical determinantal results of Bressoud and Wei, and of Koike. The proofs in that paper were algebraic. The present paper contains combinatorial lattice path proofs
Quarks in the instanton liquid-like picture of the QCD vacuum
We review a broad range of approaches to the problem of light quarks propagating through the instanton liquid-like vacuum of the Quantum Chromodynamics. Numerical and analytical techniques are presented
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