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    "Hard-scattering" approach to very hindered magnetic-dipole transitions in quarkonium

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    For a class of hindered magnetic dipole (M1M1) transition processes, such as Υ(3S)ηb+γ\Upsilon(3S)\to \eta_b+\gamma (the discovery channel of the ηb\eta_b meson), the emitted photon is rather energetic so that the traditional approaches based on multipole expansion may be invalidated. We propose that a "hard-scattering" picture, somewhat analogous to the pion electromagnetic form factor at large momentum transfer, may be more plausible to describe such types of transition processes. We work out a simple factorization formula at lowest order in the strong coupling constant, which involves convolution of the Schr\"odinger wave functions of quarkonia with a perturbatively calculable part induced by exchange of one semihard gluon between quark and antiquark. This formula, without any freely adjustable parameters, is found to agree with the measured rate of Υ(3S)ηb+γ\Upsilon(3S)\to \eta_b+\gamma rather well, and can also reasonably account for other recently measured hindered M1M1 transition rates. The branching fractions of Υ(4S)ηb()+γ\Upsilon(4S)\to \eta_b^{(\prime)}+\gamma are also predicted.Comment: v3; 5 pages, 1 figure and 1 table; title changed, presentation improve

    An algorithm for determining copositive matrices

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    In this paper, we present an algorithm of simple exponential growth called COPOMATRIX for determining the copositivity of a real symmetric matrix. The core of this algorithm is a decomposition theorem, which is used to deal with simplicial subdivision of T^={yΔmβTy0}\hat{T}^{-}=\{y\in \Delta_{m}| \beta^Ty\leq 0\} on the standard simplex Δm\Delta_m, where each component of the vector β\beta is -1, 0 or 1.Comment: 15 page
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