59 research outputs found

    Determinental formulae for complete symmetric functions

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    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

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    For the theory of a single scalar field φ\varphi with a quartic potential V(φ)V(\varphi), 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 φ4lnφ2\varphi^4 \ln \varphi^2 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

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    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

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    Current upper bounds of the neutron electric dipole moment constrain the physically observable quantum chromodynamic (QCD) vacuum angle θˉ1011|\bar\theta| \lesssim 10^{-11}. Since QCD explains vast experimental data from the 100 MeV scale to the TeV scale, it is better to explain this smallness of θˉ|\bar\theta| 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, $10^9\ {\gev}\lesssim F_a\lesssim 10^{12} \gevfora for a {\cal O}(1)initialmisalignmentangle initial misalignment angle \theta_1,hasbeenobtainedbyastrophysicalandcosmologicaldata.For, has been obtained by astrophysical and cosmological data. For F_a\gtrsim 10^{12}GeVwith GeV with \theta_1<{\cal O}(1)$, 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

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    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

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    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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