First order flows for N=2 extremal black holes and duality invariants

We derive explicitly the superpotential W for the non-BPS branch of $N=2$ extremal black holes in terms of duality invariants of special geometry. Although this is done for a one-modulus case (the $t^3$ model), the example gives $Z\ne 0$ black holes and captures the basic distinction from previous attempts on the quadratic series (vanishing C tensor) and from the other $Z=0$ cases. The superpotential W turns out to be a non-polynomial expression (containing radicals) of the basic duality invariant quantities. These are the same which enter in the quartic invariant $I_4$ for $N=2$ theories based on symmetric spaces. Using the flow equations generated by W, we also provide the analytic general solution for the warp factor and for the scalar field supporting the non-BPS black holes.     

Supersymmetric extensions of Schrödinger-invariance

The set of dynamic symmetries of the scalar free Schrödinger equation in d space dimensions gives a realization of the Schrödinger algebra that may be extended into a representation of the conformal algebra in $d+2$ dimensions, which yields the set of dynamic symmetries of the same equation where the mass is not viewed as a constant, but as an additional coordinate. An analogous construction also holds for the spin-$\frac{1}{2}$ Lévy-Leblond equation. An $N=2$ supersymmetric extension of these equations leads, respectively, to a ‘super-Schrödinger’ model and to the $(3|2)$-supersymmetric model. Their dynamic supersymmetries form the Lie superalgebras $\mathfrak{osp}(2|2)⋉\mathfrak{sh}(2|2)$ and $\mathfrak{osp}(2|4)$, respectively. The Schrödinger algebra and its supersymmetric counterparts are found to be the largest finite-dimensional Lie subalgebras of a family of infinite-dimensional Lie superalgebras that are systematically constructed in a Poisson algebra setting, including the Schrödinger–Neveu–Schwarz algebra ${\mathfrak{sns}}^{(N)}$ with N supercharges. Covariant two-point functions of quasiprimary superfields are calculated for several subalgebras of $\mathfrak{osp}(2|4)$. If one includes both $N=2$ supercharges and time-inversions, then the sum of the scaling dimensions is restricted to a finite set of possible values.     

Temperature-dependent cross sections for meson–meson nonresonant reactions in hadronic matter

We present a potential of which the short-distance part is given by one gluon exchange plus perturbative one- and two-loop corrections and of which the large-distance part exhibits a temperature-dependent constant value. The Schrödinger equation with this temperature-dependent potential yields a temperature dependence of the mesonic quark–antiquark relative-motion wave function and of meson masses. The temperature dependence of the potential, the wave function and the meson masses brings about temperature dependence of cross sections for the nonresonant reactions $\pi \pi \to \rho \rho$ for $I=2$, $KK\to K^{*}{K}^{*}$ for $I=1$, $K{K}^{*}\to K^{*}{K}^{*}$ for $I=1$, $\pi K\to \rho K^{*}$ for $I=3/2$, $\pi K^{*}\to \rho K^{*}$ for $I=3/2$, $\rho K\to \rho K^{*}$ for $I=3/2$ and $\pi K^{*}\to \rho K$ for $I=3/2$. As the temperature increases, the rise or fall of peak cross sections is determined by the increased radii of initial mesons, the loosened bound states of final mesons, and the total-mass difference of the initial and final mesons. The temperature-dependent cross sections and meson masses are parametrized.