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A microscopic calculation of ground state entanglement for the XY and Heisenberg models shows the emergence of universal scaling behavior at quantum phase transitions. Entanglement is thus controlled by conformal symmetry. Away from the critical point, entanglement gets saturated by a mass scale. Results borrowed from conformal field theory imply irreversibility of entanglement loss along renormalization group trajectories. Entanglement does not saturate in higher dimensions which appears to limit the success of the density matrix renormalization group technique. A possible connection between majorization and renormalization group irreversibility emerges from our numerical analysis.

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We construct a parametrization of deep-inelastic structure functions which retains information on experimental errors and correlations, and which does not introduce any theoretical bias while interpolating between existing data points. We generate a Monte Carlo sample of pseudo-data configurations and we train an ensemble of neural networks on them. This effectively provides us with a probability measure in the space of structure functions, within the whole kinematic region where data are available. This measure can then be used to determine the value of the structure function, its error, point-to-point correlations and generally the value and uncertainty of any function of the structure function itself. We apply this technique to the determination of the structure function F 2 of the proton and deuteron, and a precision determination of the isotriplet combination F 2 [p-d]. We discuss in detail these results, check their stability and accuracy, and make them available in various formats for applications.April 2002

A U L (3) ⊗ U R (3) low-energy effective lagrangian for the nonet of pseudogoldstone bosons that appear in the large N c limit of QCD is presented including terms up to four derivatives and explicit symmetry breaking terms up to quadratic in the quark masses. The one-loop renormalization of the couplings is worked out using the heat-kernel technique and dimensional renormalization. The calculation is carried through for U L (n l ) ⊗ U R (n l ), thus allowing for a generic number n l of light quark flavours. The crucial advantages that the expansion in powers of 1/N c bring about are discussed. Special emphasis is put in pointing out what features are at variance with the SU L ⊗SU R results when the singlet η ′ is included in the theory.The pattern of the lowest-lying states in the spectrum of strong interactions uncovers an approximate continuous symmetry of nature, the so-called chiral symmetry, which is spontaneously broken. The octet of pseudoscalar particles -π, K, and η -, with masses much smaller than those of the next excited states -the octet of vector particles ρ, ω and K * , the baryons-, are the accepted candidates for pseudo-goldstone bosons associated to the spontaneous breaking of the symmetry.This approximate symmetry is well incorporated in QCD as three of the quarks happen to be light. In the (chiral) limit of vanishing m u , m d , m s the QCD lagrangian has the symmetry freedom of arbitrarily rotating with unitary matrices the quark field components in the space of flavours (u,d,s), independently in the Left and the Right sectors of chirality eigenstates. The symmetry group is U L (3) ⊗ U R (3) and is explicitly broken by the light quark masses; if it had not, the lightest mesons would indeed have been massless particles. This breaking is small, though, since the light quark masses are much smaller than the typical hadronic scale of a few hundred MeV. (This is certainly so for the u and d quarks ( 2 < m u < 8 and 5 < m d < 15 MeV) and still approximately verified for the heavier s-quark ( 100 < m s < 300 MeV), [1]).Empirically, however, only the SU L (3) ⊗ SU R (3) symmetry subgroup, spontaneously broken to SU L+R (3), is manifest: instead of a nonet of light pseudoscalars only an octet is observed. Of the remaining U L (1) ⊗ U R (1), the vector part provides the conserved baryon number current, whereas the axial U A (1) does not seem reflected at all in the spectrum, either as a conserved quantum number or as a goldstone boson. The first possibility would imply that all massive hadrons would appear in parity doublets and this is not what is observed. On the other hand, by its quantum numbers the η ′ would be the ninth candidate for goldstone boson; but if the U A (1) were realized in the Goldstone mode and explicitly broken only by the same quark mass terms that break the SU L (3) ⊗ SU R (3) one would expect that the ninth pseudo-goldstone boson would have a mass similar to the pion (to the masses in the octet): actually, a singlet pseudoscalar meson ought to exist with a mass smaller than √ 3m...

We compute critical exponents in a Z 2 symmetric scalar field theory in three dimensions, usingWilson's exact renormalization group equations expanded in powers of derivatives. A nontrivial relation between these exponents is confirmed explicitly at the first two orders in the derivative expansion. At leading order all our results are cutoff independent, while at next-to-leading order they are not, and the determination of critical exponents becomes ambiguous. We discuss the possible ways in which this scheme ambiguity might be resolved.

Scheme independence of exact renormalization group equations, including independence of the choice of cutoff function, is shown to follow from general field redefinitions, which remains an inherent redundancy in quantum field theories. Renormalization group equations and their solutions are amenable to a simple formulation which is manifestly covariant under such a symmetry group. Notably, the kernel of the exact equations which controls the integration of modes acts as a field connection along the flow.

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