A simple Proof of Stolarsky's Invariance Principle

Stolarsky [Proc. Amer. Math. Soc. 41 (1973), 575--582] showed a beautiful relation that balances the sums of distances of points on the unit sphere and their spherical cap $\mathbb{L}_2$-discrepancy to give the distance integral of the uniform measure on the sphere a potential-theoretical quantity (Bj{\"o}rck [Ark. Mat. 3 (1956), 255--269]). Read differently it expresses the worst-case numerical integration error for functions from the unit ball in a certain Hilbert space setting in terms of the $\mathbb{L}_2$-discrepancy and vice versa (first author and Womersley [Preprint]). In this note we give a simple proof of the invariance principle using reproducing kernel Hilbert spaces

Metastable dark matter mechanisms for INTEGRAL 511 keV γ\gamma rays and DAMA/CoGeNT events

We explore dark matter mechanisms that can simultaneously explain the galactic 511 keV gamma rays observed by INTEGRAL/SPI, the DAMA/LIBRA annual modulation, and the excess of low-recoil dark matter candidates observed by CoGeNT. It requires three nearly degenerate states of dark matter in the 4-7 GeV mass range, with splittings respectively of order an MeV and a few keV. The top two states have the small mass gap and transitions between them, either exothermic or endothermic, can account for direct detections. Decays from one of the top states to the ground state produce low-energy positrons in the galaxy whose associated 511 keV gamma rays are seen by INTEGRAL. This decay can happen spontaneously, if the excited state is metastable (longer-lived than the age of the universe), or it can be triggered by inelastic scattering of the metastable states into the shorter-lived ones. We focus on a simple model where the DM is a triplet of an SU(2) hidden sector gauge symmetry, broken at the scale of a few GeV, giving masses of order \lsim 1 GeV to the dark gauge bosons, which mix kinetically with the standard model hypercharge. The purely decaying scenario can give the observed angular dependence of the 511 keV signal with no positron diffusion, while the inelastic scattering mechanism requires transport of the positrons over distances \sim 1 kpc before annihilating. We note that an x-ray line of several keV in energy, due to single-photon decays involving the top DM states, could provide an additional component to the diffuse x-ray background. The model is testable by proposed low-energy fixed target experiments.Comment: 27 pp, 19 figures; v2. minor clarification, added refs; v3. corrected observed rate of positron production, added new section responding to criticisms of arXiv:0904.1025; v4. corrected typos in eqs. (6) and (40

Pair Correlation Function of Wilson Loops

We give a path integral prescription for the pair correlation function of Wilson loops lying in the worldvolume of Dbranes in the bosonic open and closed string theory. The results can be applied both in ordinary flat spacetime in the critical dimension d or in the presence of a generic background for the Liouville field. We compute the potential between heavy nonrelativistic sources in an abelian gauge theory in relative collinear motion with velocity v = tanh(u), probing length scales down to r_min^2 = 2 \pi \alpha' u. We predict a universal -(d-2)/r static interaction at short distances. We show that the velocity dependent corrections to the short distance potential in the bosonic string take the form of an infinite power series in the dimensionless variables z = r_min^2/r^2, uz/\pi, and u^2.Comment: 16 pages, 1 figure, Revtex. Corrected factor of two in potential. Some changes in discussio

Chiral dynamics and peripheral transverse densities

In the partonic (or light-front) description of relativistic systems the electromagnetic form factors are expressed in terms of frame-independent charge and magnetization densities in transverse space. This formulation allows one to identify the chiral components of nucleon structure as the peripheral densities at transverse distances b = O(M_pi^{-1}) and compute them in a parametrically controlled manner. A dispersion relation connects the large-distance behavior of the transverse charge and magnetization densities to the spectral functions of the Dirac and Pauli form factors near the two-pion threshold at timelike t = 4 M_pi^2. Using relativistic chiral effective field theory in the leading-order approximation, we (a) derive the asymptotic behavior (Yukawa tail) of the isovector transverse densities in the "chiral" region b = O(M_pi^{-1}) and the "molecular" region b = O(M_N^2/M_pi^3); (b) perform the heavy-baryon expansion; (c) explain the relative magnitude of the peripheral charge and magnetization densities in a simple mechanical picture; (d) include Delta intermediate states and study the densities in the large-N_c limit of QCD; (e) quantify the spatial region where the chiral components are numerically dominant; (f) calculate the chiral divergences of the b^2-weighted moments of the transverse densities (charge and magnetic radii) and determine their spatial support. Our approach provides a concise formulation of the spatial structure of the nucleon's chiral component and offers new insights into basic properties of the chiral expansion. It relates the information extracted from low-t elastic form factors to the generalized parton distributions probed in peripheral high-energy scattering processes.Comment: 52 pages, 13 figure

Non-linear screening of external charge by doped graphene

We solve a nonlinear integral equation for the electrostatic potential in doped graphene due to an external charge, arising from a Thomas-Fermi (TF) model for screening by graphene's $\pi$ electron bands. In particular, we study the effects of a finite equilibrium charge carrier density in graphene, non-zero temperature, non-zero gap between graphene and a dielectric substrate, as well as the nonlinearity in the band density of states. Effects of the exchange and correlation interactions are also briefly discussed for undoped graphene at zero temperature. Nonlinear results are compared with both the linearized TF model and the dielectric screening model within random phase approximation (RPA). In addition, image potential of the external charge is evaluated from the solution of the nonlinear integral equation and compared to the results of linear models. We have found generally good agreement between the results of the nonlinear TF model and the RPA model in doped graphene, apart from Friedel oscillations in the latter model. However, relatively strong nonlinear effects are found in the TF model to persist even at high doping densities and large distances of the external charge.Comment: 12 pages including 6 figure