Renormalization of the one-pion-exchange interaction

A renormalization scheme for the nucleon-nucleon (NN) interaction based on a subtracted T-matrix equation is proposed and applied to the one-pion-exchange potential supplemented by contact interactions. The singlet and triplet scattering lengths are given to fix the renormalized strengths of the contact interactions. With only one scaling parameter (μ), the results show an overall very good agreement with neutron-proton data, particularly for the observables related to the triplet channel. The agreement is qualitative in the 1 S0 channel. Between the low-energy NN observables we have examined, the mixing parameter of the 3S1-3D1 states is the most sensitive to the scale. The scheme is renormalization group invariant for μ → ∞. © 1999 Elsevier Science B.V. All rights reserved.

Reply to 'Lifshitz-point critical behaviour to O(εL 2)'

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

A Monte Carlo study of the anisotropic N=3 Ashkin-Teller model

The phase diagram of an asymmetric N = 3 Ashkin-Teller model is obtained by a numerical analysis which combines Monte Carlo renormalization group and reweighting techniques. Present results reveal several differences with those obtained by mean-field calculations and a Hamiltonian approach. In particular, we found Ising critical exponents along a line where Goldschmidt has located the Kosterlitz-Thouless multicritical point. On the other hand, we did find nonuniversal exponents along another transition line. Symmetry breaking in this case is very similar to the N = 2 case, since the symmetries associated to only two of the Ising variables are broken. However, for large values of the coupling constant ratio XW = W/K, when the only broken symmetry is of a hidden variable, we detected first-order phase transitions giving evidence supporting the existence of a multicritical point, as suggested by Goldschmidt, but in a different region of the phase diagram. © 2002 Elsevier Science B.V. All rights reserved.

Charge dynamics in half-filled Hubbard chains with finite on-site interaction

We study the charge dynamic structure factor of the one-dimensional Hubbard model with finite on-site repulsion U at half-filling. Numerical results from the time-dependent density matrix renormalization group are analyzed by comparison with the exact spectrum of the model. The evolution of the line shape as a function of U is explained in terms of a relative transfer of spectral weight between the two-holon continuum that dominates in the limit U -> infinity and a subset of the two-holon-two-spinon continuum that reconstructs the electron-hole continuum in the limit U -> 0. Power-law singularities along boundary lines of the spectrum are described by effective impurity models that are explicitly invariant under spin and eta-spin SU(2) rotations. The Mott-Hubbard metal-insulator transition is reflected in a discontinuous change of the exponents of edge singularities at U = 0. The sharp feature observed in the spectrum for momenta near the zone boundary is attributed to a van Hove singularity that persists as a consequence of integrability.

Finite-size corrections of the entanglement entropy of critical quantum chains

Using the density matrix renormalization group, we calculated the finite-size corrections of the entanglement alpha-Renyi entropy of a single interval for several critical quantum chains. We considered models with U(1) symmetry such as the spin-1/2 XXZ and spin-1 Fateev-Zamolodchikov models, as well as models with discrete symmetries such as the Ising, the Blume-Capel, and the three-state Potts models. These corrections contain physically relevant information. Their amplitudes, which depend on the value of a, are related to the dimensions of operators in the conformal field theory governing the long-distance correlations of the critical quantum chains. The obtained results together with earlier exact and numerical ones allow us to formulate some general conjectures about the operator responsible for the leading finite-size correction of the alpha-Renyi entropies. We conjecture that the exponent of the leading finite-size correction of the alpha-Renyi entropies is p(alpha) = 2X(epsilon)/alpha for alpha > 1 and p(1) = nu, where X-epsilon denotes the dimensions of the energy operator of the model and nu = 2 for all the models.

No-pole condition in Landau gauge: Properties of the Gribov ghost form factor and a constraint on the 2d gluon propagator

We study general properties of the Landau-gauge Gribov ghost form factor sigma(p(2)) for SU(N-c) Yang-Mills theories in the d-dimensional case. We find a qualitatively different behavior for d = 3, 4 with respect to the d = 2 case. In particular, considering any (sufficiently regular) gluon propagator D(p(2)) and the one-loop-corrected ghost propagator, we prove in the 2d case that the function sigma(p(2)) blows up in the infrared limit p -> 0 as -D(0) ln(p(2)). Thus, for d = 2, the no-pole condition sigma(p(2)) < 1 (for p(2) > 0) can be satisfied only if the gluon propagator vanishes at zero momentum, that is, D(0) = 0. On the contrary, in d = 3 and 4, sigma(p(2)) is finite also if D(0) > 0. The same results are obtained by evaluating the ghost propagator G(p(2)) explicitly at one loop, using fitting forms for D(p(2)) that describe well the numerical data of the gluon propagator in two, three and four space-time dimensions in the SU(2) case. These evaluations also show that, if one considers the coupling constant g(2) as a free parameter, the ghost propagator admits a one-parameter family of behaviors (labeled by g(2)), in agreement with previous works by Boucaud et al. In this case the condition sigma(0) <= 1 implies g(2) <= g(c)(2), where g(c)(2) is a "critical" value. Moreover, a freelike ghost propagator in the infrared limit is obtained for any value of g(2) smaller than g(c)(2), while for g(2) = g(c)(2) one finds an infrared-enhanced ghost propagator. Finally, we analyze the Dyson-Schwinger equation for sigma(p(2)) and show that, for infrared-finite ghost-gluon vertices, one can bound the ghost form factor sigma(p(2)). Using these bounds we find again that only in the d = 2 case does one need to impose D(0) = 0 in order to satisfy the no-pole condition. The d = 2 result is also supported by an analysis of the Dyson-Schwinger equation using a spectral representation for the ghost propagator. Thus, if the no-pole condition is imposed, solving the d = 2 Dyson-Schwinger equations cannot lead to a massive behavior for the gluon propagator. These results apply to any Gribov copy inside the so-called first Gribov horizon; i.e., the 2d result D(0) = 0 is not affected by Gribov noise. These findings are also in agreement with lattice data.

Untersuchungen zum nichtstörungstheoretischen Renormierungsverhalten der Quanten-Einstein-Gravitation

Deutsche Version: Zunächst wird eine verallgemeinerte Renormierungsgruppengleichung für die effektiveMittelwertwirkung der EuklidischenQuanten-Einstein-Gravitation konstruiert und dann auf zwei unterschiedliche Trunkierungen, dieEinstein-Hilbert-Trunkierung und die$R^2$-Trunkierung, angewendet. Aus den resultierendenDifferentialgleichungen wird jeweils die Fixpunktstrukturbestimmt. Die Einstein-Hilbert-Trunkierung liefert nebeneinem Gaußschen auch einen nicht-Gaußschen Fixpunkt. Diesernicht-Gaußsche Fixpunkt und auch der Fluß in seinemEinzugsbereich werden mit hoher Genauigkeit durch die$R^2$-Trunkierung reproduziert. Weiterhin erweist sichdie Cutoffschema-Abhängigkeit der analysierten universellenGrößen als äußerst schwach. Diese Ergebnisse deuten daraufhin, daß dieser Fixpunkt wahrscheinlich auch in der exaktenTheorie existiert und die vierdimensionaleQuanten-Einstein-Gravitation somit nichtperturbativ renormierbar sein könnte. Anschließend wird gezeigt, daß der ultraviolette Bereich der$R^2$-Trunkierung und somit auch die Analyse des zugehörigenFixpunkts nicht von den Stabilitätsproblemen betroffen sind,die normalerweise durch den konformen Faktor der Metrikverursacht werden. Dadurch motiviert, wird daraufhin einskalares Spielzeugmodell, das den konformen Sektor einer``$-R+R^2$''-Theorie simuliert, hinsichtlich seinerStabilitätseigenschaften im infraroten (IR) Bereichstudiert. Dabei stellt sich heraus, daß sich die Theorieunter Ausbildung einer nichttrivialen Vakuumstruktur auf dynamische Weise stabilisiert. In der Gravitation könnteneventuell nichtlokale Invarianten des Typs $intd^dx,sqrt{g}R (D^2)^{-1} R$ dafür sorgen, daß der konformeSektor auf ähnliche Weise IR-stabil wird.

Quantum Integrability in Non-Linear Sigma Models related to Gauge/String Correspondences

The Thermodynamic Bethe Ansatz analysis is carried out for the extended-CP^N class of integrable 2-dimensional Non-Linear Sigma Models related to the low energy limit of the AdS_4xCP^3 type IIA superstring theory. The principal aim of this program is to obtain further non-perturbative consistency check to the S-matrix proposed to describe the scattering processes between the fundamental excitations of the theory by analyzing the structure of the Renormalization Group flow. As a noteworthy byproduct we eventually obtain a novel class of TBA models which fits in the known classification but with several important differences. The TBA framework allows the evaluation of some exact quantities related to the conformal UV limit of the model: effective central charge, conformal dimension of the perturbing operator and field content of the underlying CFT. The knowledge of this physical quantities has led to the possibility of conjecturing a perturbed CFT realization of the integrable models in terms of coset Kac-Moody CFT. The set of numerical tools and programs developed ad hoc to solve the problem at hand is also discussed in some detail with references to the code.

Top-quark pair production at hadron colliders

In this thesis we investigate several phenomenologically important properties of top-quark pair production at hadron colliders. We calculate double differential cross sections in two different kinematical setups, pair invariant-mass (PIM) and single-particle inclusive (1PI) kinematics. In pair invariant-mass kinematics we are able to present results for the double differential cross section with respect to the invariant mass of the top-quark pair and the top-quark scattering angle. Working in the threshold region, where the pair invariant mass M is close to the partonic center-of-mass energy sqrt{hat{s}}, we are able to factorize the partonic cross section into different energy regions. We use renormalization-group (RG) methods to resum large threshold logarithms to next-to-next-to-leading-logarithmic (NNLL) accuracy. On a technical level this is done using effective field theories, such as heavy-quark effective theory (HQET) and soft-collinear effective theory (SCET). The same techniques are applied when working in 1PI kinematics, leading to a calculation of the double differential cross section with respect to transverse-momentum pT and the rapidity of the top quark. We restrict the phase-space such that only soft emission of gluons is possible, and perform a NNLL resummation of threshold logarithms. The obtained analytical expressions enable us to precisely predict several observables, and a substantial part of this thesis is devoted to their detailed phenomenological analysis. Matching our results in the threshold regions to the exact ones at next-to-leading order (NLO) in fixed-order perturbation theory, allows us to make predictions at NLO+NNLL order in RG-improved, and at approximate next-to-next-to-leading order (NNLO) in fixed order perturbation theory. We give numerical results for the invariant mass distribution of the top-quark pair, and for the top-quark transverse-momentum and rapidity spectrum. We predict the total cross section, separately for both kinematics. Using these results, we analyze subleading contributions to the total cross section in 1PI and PIM originating from power corrections to the leading terms in the threshold expansions, and compare them to previous approaches. We later combine our PIM and 1PI results for the total cross section, this way eliminating uncertainties due to these corrections. The combined predictions for the total cross section are presented as a function of the top-quark mass in the pole, the minimal-subtraction (MS), and the 1S mass scheme. In addition, we calculate the forward-backward (FB) asymmetry at the Tevatron in the laboratory, and in the ttbar rest frames as a function of the rapidity and the invariant mass of the top-quark pair at NLO+NNLL. We also give binned results for the asymmetry as a function of the invariant mass and the rapidity difference of the ttbar pair, and compare those to recent measurements. As a last application we calculate the charge asymmetry at the LHC as a function of a lower rapidity cut-off for the top and anti-top quarks.

Computer simulation studies of vector spin glasses

Die vorliegende Doktorarbeit befasst sich mit klassischen Vektor-Spingläsern eine Art von ungeordneten Magneten - auf verschiedenen Gittertypen. Da siernbedeutsam für eine experimentelle Realisierung sind, ist ein theoretisches Verständnis von Spinglas-Modellen mit wenigen Spinkomponenten und niedriger Gitterdimension von großer Bedeutung. Da sich dies jedoch als sehr schwierigrnerweist, sind neue, aussichtsreiche Ansätze nötig. Diese Arbeit betrachtet daher den Limesrnunendlich vieler Spindimensionen. Darin entstehen mehrere Vereinfachungen im Vergleichrnzu Modellen niedriger Spindimension, so dass für dieses bedeutsame Problem Eigenschaften sowohl bei Temperatur Null als auch bei endlichen Temperaturenrnüberwiegend mit numerischen Methoden ermittelt werden. Sowohl hyperkubische Gitter als auch ein vielseitiges 1d-Modell werden betrachtet. Letzteres erlaubt es, unterschiedliche Universalitätsklassen durch bloßes Abstimmen eines einzigen Parameters zu untersuchen. "Finite-size scaling''-Formen, kritische Exponenten, Quotienten kritischer Exponenten und andere kritische Größen werden nahegelegt und mit numerischen Ergebnissen verglichen. Eine detaillierte Beschreibung der Herleitungen aller numerisch ausgewerteter Gleichungen wird ebenso angegeben. Bei Temperatur Null wird eine gründliche Untersuchung der Grundzustände und Defektenergien gemacht. Eine Reihe interessanter Größen wird analysiert und insbesondere die untere kritische Dimension bestimmt. Bei endlicher Temperatur sind der Ordnungsparameter und die Spinglas-Suszeptibilität über die numerisch berechnete Korrelationsmatrix zugänglich. Das Spinglas-Modell im Limes unendlich vieler Spinkomponenten kann man als Ausgangspunkt zur Untersuchung der natürlicheren Modelle mit niedriger Spindimension betrachten. Wünschenswert wäre natürlich ein Modell, das die Vorteile des ersten mit den Eigenschaften des zweiten verbände. Daher wird in Modell mit Anisotropie vorgeschlagen und getestet, mit welchem versucht wird, dieses Ziel zu erreichen. Es wird auf reizvolle Wege hingewiesen, das Modell zu nutzen und eine tiefergehende Beschäftigung anzuregen. Zuletzt werden sogenannte "real-space" Renormierungsgruppenrechnungen sowohl analytisch als auch numerisch für endlich-dimensionale Vektor-Spingläser mit endlicher Anzahl von Spinkomponenten durchgeführt. Dies wird mit einer zuvor bestimmten neuen Migdal-Kadanoff Rekursionsrelation geschehen. Neben anderen Größen wird die untere kritische Dimension bestimmt.

Tensorial group field theories

In questa tesi il Gruppo di Rinormalizzazione non-perturbativo (FRG) viene applicato ad una particolare classe di modelli rilevanti in Gravit`a quantistica, conosciuti come Tensorial Group Field Theories (TGFT). Le TGFT sono teorie di campo quantistiche definite sulla variet`a di un gruppo G. In ogni dimensione esse possono essere espanse in grafici di Feynman duali a com- plessi simpliciali casuali e sono caratterizzate da interazioni che implementano una non-localit`a combinatoriale. Le TGFT aspirano a generare uno spaziotempo in un contesto background independent e precisamente ad ottenere una descrizione con- tinua della sua geometria attraverso meccanismi fisici come le transizioni di fase. Tra i metodi che meglio affrontano il problema di estrarre le transizioni di fase e un associato limite del continuo, uno dei pi` u efficaci `e il Gruppo di Rinormalizzazione non-perturbativo. In questo elaborato ci concentriamo su TGFT definite sulla variet`a di un gruppo non-compatto (G = R) e studiamo il loro flusso di Rinormalizzazione. Identifichiamo con successo punti fissi del flusso di tipo IR, e una superficie critica che suggerisce la presenza di transizioni di fase in regime Infrarosso. Ci`o spinge ad uno stu- dio per approfondire la comprensione di queste transizioni di fase e della fisica continua che vi `e associata. Affrontiamo inoltre il problema delle divergenze Infrarosse, tramite un processo di regolarizzazione che definisce il limite termodinamico appropriato per le TGFT. Infine, applichiamo i metodi precedentementi sviluppati ad un modello dotato di proiezione sull’insieme dei campi gauge invarianti. L’analisi, simile a quella applicata al modello precedente, conduce nuovamente all’identificazione di punti fissi (sia IR che UV) e di una superficie critica. La presenza di transizioni di fasi `e, dunque, evidente ancora una volta ed `e possibile confrontare il risultato col modello senza proiezione sulla dinamica gauge invariante.

Alternative derivative expansion in Functional RG and application

We give a brief review of the Functional Renormalization method in quantum field theory, which is intrinsically non perturbative, in terms of both the Polchinski equation for the Wilsonian action and the Wetterich equation for the generator of the proper verteces. For the latter case we show a simple application for a theory with one real scalar field within the LPA and LPA' approximations. For the first case, instead, we give a covariant "Hamiltonian" version of the Polchinski equation which consists in doing a Legendre transform of the flow for the corresponding effective Lagrangian replacing arbitrary high order derivative of fields with momenta fields. This approach is suitable for studying new truncations in the derivative expansion. We apply this formulation for a theory with one real scalar field and, as a novel result, derive the flow equations for a theory with N real scalar fields with the O(N) internal symmetry. Within this new approach we analyze numerically the scaling solutions for N=1 in d=3 (critical Ising model), at the leading order in the derivative expansion with an infinite number of couplings, encoded in two functions V(phi) and Z(phi), obtaining an estimate for the quantum anomalous dimension with a 10% accuracy (confronting with Monte Carlo results).

Analytical properties of horizontal visibility graphs in the Feigenbaum scenario

Time series are proficiently converted into graphs via the horizontal visibility (HV) algorithm, which prompts interest in its capability for capturing the nature of different classes of series in a network context. We have recently shown [B. Luque et al., PLoS ONE 6, 9 (2011)] that dynamical systems can be studied from a novel perspective via the use of this method. Specifically, the period-doubling and band-splitting attractor cascades that characterize unimodal maps transform into families of graphs that turn out to be independent of map nonlinearity or other particulars. Here, we provide an in depth description of the HV treatment of the Feigenbaum scenario, together with analytical derivations that relate to the degree distributions, mean distances, clustering coefficients, etc., associated to the bifurcation cascades and their accumulation points. We describe how the resultant families of graphs can be framed into a renormalization group scheme in which fixed-point graphs reveal their scaling properties. These fixed points are then re-derived from an entropy optimization process defined for the graph sets, confirming a suggested connection between renormalization group and entropy optimization. Finally, we provide analytical and numerical results for the graph entropy and show that it emulates the Lyapunov exponent of the map independently of its sign.

Horizontal visibility graphs generated by type-I intermittency

The type-I intermittency route to (or out of) chaos is investigated within the horizontal visibility (HV) graph theory. For that purpose, we address the trajectories generated by unimodal maps close to an inverse tangent bifurcation and construct their associatedHVgraphs.We showhowthe alternation of laminar episodes and chaotic bursts imprints a fingerprint in the resulting graph structure. Accordingly, we derive a phenomenological theory that predicts quantitative values for several network parameters. In particular, we predict that the characteristic power-law scaling of the mean length of laminar trend sizes is fully inherited by the variance of the graph degree distribution, in good agreement with the numerics. We also report numerical evidence on how the characteristic power-law scaling of the Lyapunov exponent as a function of the distance to the tangent bifurcation is inherited in the graph by an analogous scaling of block entropy functionals defined on the graph. Furthermore, we are able to recast the full set of HV graphs generated by intermittent dynamics into a renormalization-group framework, where the fixed points of its graph-theoretical renormalization-group flow account for the different types of dynamics.We also establish that the nontrivial fixed point of this flow coincides with the tangency condition and that the corresponding invariant graph exhibits extremal entropic properties.

Orbital currents and charge density waves in a generalized Hubbard ladder

We study a generalized Hubbard model on the two-leg ladder at zero temperature, focusing on a parameter region with staggered flux (SF)/d-density wave (DDW) order. To guide our numerical calculations, we first investigate the location of a SF/DDW phase in the phase diagram of the half-filled weakly interacting ladder using a perturbative renormalization group (RG) and bosonization approach. For hole doping 6 away from half-filling, finite-system density-matrix renormalizationgroup (DMRG) calculations are used to study ladders with up to 200 rungs for intermediate-strength interactions. In the doped SF/DDW phase, the staggered rung current and the rung electron density both show periodic spatial oscillations, with characteristic wavelengths 2/delta and 1/delta, respectively, corresponding to ordering wavevectors 2k(F) and 4k(F) for the currents and densities, where 2k(F) = pi(1 - delta). The density minima are located at the anti-phase domain walls of the staggered current. For sufficiently large dopings, SF/DDW order is suppressed. The rung density modulation also exists in neighboring phases where currents decay exponentially. We show that most of the DMRG results can be qualitatively understood from weak-coupling RG/bosonization arguments. However, while these arguments seem to suggest a crossover from non-decaying correlations to power-law decay at a length scale of order 1/delta, the DMRG results are consistent with a true long-range order scenario for the currents and densities. (c) 2005 Elsevier Inc. All rights reserved.