The exact renormalization group in Astrophysics

The coarse-graining operation in hydrodynamics is equivalent to a change of scale which can be formalized as a renormalization group transformation. In particular, its application to the probability distribution of a self-gravitating fluid yields an "exact renormalization group equation" of Fokker-Planck type. Since the time evolution of that distribution can also be described by a Fokker-Planck equation, we propose a connection between both equations, that is, a connection between scale and time evolution. We finally remark on the essentially non-perturbative nature of astrophysical problems, which suggests that the exact renormalization group is the adequate tool for them.Comment: World Scientific style, 6 pages, presented at the 2nd Conference on the Exact RG, Rome 200

Quark masses in QCD: a progress report

Recent progress on QCD sum rule determinations of the light and heavy quark masses is reported. In the light quark sector a major breakthrough has been made recently in connection with the historical systematic uncertainties due to a lack of experimental information on the pseudoscalar resonance spectral functions. It is now possible to suppress this contribution to the 1% level by using suitable integration kernels in Finite Energy QCD sum rules. This allows to determine the up-, down-, and strange-quark masses with an unprecedented precision of some 8-10%. Further reduction of this uncertainty will be possible with improved accuracy in the strong coupling, now the main source of error. In the heavy quark sector, the availability of experimental data in the vector channel, and the use of suitable multipurpose integration kernels allows to increase the accuracy of the charm- and bottom-quarks masses to the 1% level.Comment: Invited review paper to be published in Modern Physics Letters

Deconfinement and Chiral-Symmetry Restoration in Finite Temperature QCD

QCD sum rules are based on the Operator Product Expansion of current correlators, and on QCD-hadron duality. An extension of this program to finite temperature is discussed. This allows for a study of deconfinement and chiral-symmetry restoration. In addition, it is possible to relate certain hadronic matrix elements to expectation values of quark and gluon field operators by using thermal Finite Energy Sum Rules. In this way one can determine the temperature behaviour of hadron masses and couplings, as well as form factors. An attempt is made to clarify some misconceptions in the existing literature on QCD sum rules at finite temperature.Comment: Invited talk at CAM-94, Cancun, Mexico, September 1994. 21 pages and 8 figures (not included). LATEX file. UCT-TP-218/9

Electromagnetic Form Factors of Hadrons in Dual-Large NcN_c QCD

In this talk, results are presented of determinations of electromagnetic form factors of hadrons (pion, proton, and $\Delta(1236)$) in the framework of Dual-Large $N_c$ QCD (Dual-$QCD_\infty$). This framework improves considerably tree-level VMD results by incorporating an infinite number of zero-width resonances, with masses and couplings fixed by the dual-resonance (Veneziano-type) model.Comment: Invited talk at the XII Mexican Workshop on Particles & Fields, Mazatlan, November 2009. To be published in American Institute of Physics Conference Proceedings Serie

Chiral sum rules and duality in QCD

The ALEPH data on the vector and axial-vector spectral functions, extracted from tau-lepton decays is used in order to test local and global duality, as well as a set of four QCD chiral sum rules. These are the Das-Mathur-Okubo sum rule, the first and second Weinberg sum rules, and a relation for the electromagnetic pion mass difference. We find these sum rules to be poorly saturated, even when the upper limit in the dispersion integrals is as high as $3 GeV^{2}$. Since perturbative QCD, plus condensates, is expected to be valid for $|q^{2}| \geq \cal{O}$$(1 GeV^{2})$ in the whole complex energy plane, except in the vicinity of the right hand cut, we propose a modified set of sum rules with weight factors that vanish at the end of the integration range on the real axis. These sum rules are found to be precociously saturated by the data to a remarkable extent. As a byproduct, we extract for the low energy renormalization constant $\bar{L}_{10}$ the value $- 4 \bar{L}_{10}= 2.43 \times 10^{-2}$, to be compared with the standard value $- 4 \bar{L}_{10} = (2.73 \pm 0.12) \times 10^{-2}$. This in turn implies a pion polarizability $\alpha_{E} = 3.7 \times 10^{-4} fm^{3}$.Comment: October 1998. Submitted to Phys. Lett. B. Latex file plus 7 postscript figure

Up- and down-quark masses from QCD sum rules

The QCD up- and down-quark masses are determined from an optimized QCD Finite Energy Sum Rule (FESR) involving the correlator of axial-vector current divergences. In the QCD sector this correlator is known to five loop order in perturbative QCD (PQCD), together with non-perturbative corrections from the quark and gluon condensates. This FESR is designed to reduce considerably the systematic uncertainties arising from the hadronic spectral function. The determination is done in the framework of both fixed order and contour improved perturbation theory. Results from the latter, involving far less systematic uncertainties, are: \bar{m}_u (2\, \mbox{GeV}) = (2.6 \, \pm \, 0.4) \, {\mbox{MeV}}, \bar{m}_d (2\, \mbox{GeV}) = (5.3 \, \pm \, 0.4) \, {\mbox{MeV}}, and the sum $\bar{m}_{ud} \equiv (\bar{m}_u \, + \, \bar{m}_d)/2$, is \bar{m}_{ud}({ 2 \,\mbox{GeV}}) =( 3.9 \, \pm \, 0.3 \,) {\mbox{MeV}}.Comment: A Mathematica^(C) file pertaining to numerical evaluations is attached as Ancillar

Multipolar expansion of the electrostatic interaction between charged colloids at interfaces

The general form of the electrostatic potential around an arbitrarily charged colloid at an interface between a dielectric and a screening phase (such as air and water, respectively) is analyzed in terms of a multipole expansion. The leading term is isotropic in the interfacial plane and varies with $d^{-3}$ where $d$ is the in--plane distance from the colloid. The electrostatic interaction potential between two arbitrarily charged colloids is likewise isotropic and $\propto d^{-3}$, corresponding to the dipole--dipole interaction first found for point charges at water interfaces. Anisotropic interaction terms arise only for higher powers $d^{-n}$ with $n \ge 4$.Comment: 6 pages, mathematical details adde

Recursive no-envy

In economics the main efficiency criterion is that of Pareto-optimality. For problems of distributing a social endowment a central notion of fairness is no-envy (each agent should receive a bundle at least as good, according to her own preferences, as any of the other agent's bundle). For most economies there are multiple allocations satisfying these two properties. We provide a procedure, based on distributional implications of these two properties, which selects a single allocation which is Pareto-optimal and satisfies no-envy in two-agent exchange economies. There is no straightforward generalization of our procedure to more than two-agents.no-envy, fair allocation, recursive methods, exchange economies