How linear reinforcement affects Donsker’s theorem for empirical processes

A reinforcement algorithm introduced by Simon (Biometrika 42(3/4):425–440, 1955) produces a sequence of uniform random variables with long range memory as follows. At each step, with a fixed probability p∈(0,1), U^n+1 is sampled uniformly from U^1,…,U^n, and with complementary probability 1−p, U^n+1 is a new independent uniform variable. The Glivenko–Cantelli theorem remains valid for the reinforced empirical measure, but not the Donsker theorem. Specifically, we show that the sequence of empirical processes converges in law to a Brownian bridge only up to a constant factor when p1/2 and the limit is then a bridge with exchangeable increments and discontinuous paths. This is related to earlier limit theorems for correlated Bernoulli processes, the so-called elephant random walk, and more generally step reinforced random walks

Fractional excitations in the Luttinger liquid

We reconsider the spectrum of the Luttinger liquid (LL) usually understood in terms of phonons (density fluctuations), and within the context of bosonization we give an alternative representation in terms of fractional states. This allows to make contact with Bethe Ansatz which predicts similar fractional states. As an example we study the spinon operator in the absence of spin rotational invariance and derive it from first principles: we find that it is not a semion in general; a trial Jastrow wavefunction is also given for that spinon state. Our construction of the new spectroscopy based on fractional states leads to several new physical insights: in the low-energy limit, we find that the $S_{z}=0$ continuum of gapless spin chains is due to pairs of fractional quasiparticle-quasihole states which are the 1D counterpart of the Laughlin FQHE quasiparticles. The holon operator for the Luttinger liquid with spin is also derived. In the presence of a magnetic field, spin-charge separation is not realized any longer in a LL: the holon and the spinon are then replaced by new fractional states which we are able to describe.Comment: Revised version to appear in Physical Review B. 27 pages, 5 figures. Expands cond-mat/9905020 (Eur.Phys.Journ.B 9, 573 (1999)

Experimental Tests of Asymptotic Freedom

Quantum Chromodynamics (QCD), the gauge field theory of the Strong Interaction, has specific features, asymptotic freedom and confinement, which determine the behaviour of quarks and gluons in particle reactions at high and at low energy scales. QCD predicts that the strong coupling strength \as decreases with increasing energy or momentum transfer, and vanishes at asymptotically high energies. In this review, the history and the status of experimental tests of asymptotic freedom are summarised. The world summary of measurements of \as is updated, leading to an unambiguous verification of the running of \as and of asymptotic freedom, in excellent agreement with the predictions of QCD. Averaging a set of measurements balanced between different particle processes and the available energy range, results in a new and improved world average of \amz = 0.1189 \pm 0.0010 .Comment: 38 pages, 18 figure

Generalized parton distributions of the pion: modeling, evolution and observable implications

Yet the Standard Model of particle physics is so far the most successful theory ever conceived in science, with countless of achievements in the description of Nature; still a number of phenomena remains to be properly understood. A paradigmatic example is confinement. Paving the road towards the understanding of such essential feature thus requires a detailed knowledge of hadrons' inside. The main subject covered in this thesis is precisely the study of hadron structure. Nonetheless this is a really broad field. We thus set sights on an exemplary case: The pions. As mesons, they are potentially simpler to describe that other hadrons like the proton. At the same time, as the pseudo Nambu-Goldstone modes associated to the dynamical breakdown of chiral symmetry, its description is of uttermost relevance for the comprehension of the origin of mass in Nature. We thus address the problem of assessing pions' structure, from first principles up to its observable manifestations. This work is thus divided into two parts: First, the issue of a formal description of hadron's structure is addressed. Relying on the role played by Compton scattering as an essential window into hadrons' inside, we review its formal treatment as understood in the generalized Bjorken limit, leading to the introduction of generalized parton distributions (GPDs). Those parametrize the amplitudes for deeply virtual Compton scattering (DVCS) on hadrons to take place, yielding a unique source of information about the way hadrons are built up from elementary constituents. In the second chapter of this dissertation we review the formal definition and primary properties of generalized parton distributions, together with the main strategies allowing for their evaluation. We identify two essential attributes to be fulfilled: Positivity and polynomiality; a task which, regardless, conventional approaches fail to accomplish. In the third chapter of this thesis we face the problem of obtaining models for pion GPDs which fulfill, by construction, these two features. Following an approach to the description of bound-state systems in quantum field theory based on continuum Schwinger methods, we find the hypothesis of decoupling between longitudinal and transverse degrees of freedom at the level of parton dynamics to be associated with the dynamical breakdown of chiral symmetry. On that assumption, a novel family of generalized parton distributions within the DGLAP region is derived and showed to exhibit two striking characteristics: They satisfy the positivity constraints, and are built from the sole knowledge of parton distribution functions. From that point on we exploit the covariant extension strategy, allowing us to find the corresponding ERBL GPDs, such that polynomiality is also fulfilled by construction. Armed with models for pion GPDs that are complete, in the sense that they satisfy every necessary property, the second part of this work thus exploits them in the derivation of their observable manifestations, allowing for a practical assessment of pions' structure as well their benchmarking. To this end, the fourth chapter pursues the necessary evaluation of scale-evolution for GPDs through an effective approach which encompasses some non-perturbative aspects of the procedure. The results hint the crucial role played by gluons in building pions' up, their associated distributions being commensurate with those for quarks. We thus round-off this dissertation by exploiting the evolved models to deliver predictions on event-rates and beam-spin asymmetries as they could be measured at forthcoming electron-ion colliders. The results reveal that indeed DVCS on pions is expected to be measurable at future facilities; and reveals that gluons dominate the response of pions subjected to DVCS, identifying a sign inversion in the beam-spin asymmetries as clear signal for pinning-down the regime for gluon dominance.Pese a la ingente cantidad de resultados derivados del Modelo Estándar de la física de partículas aún existe una gran cantidad de fenómenos que no llegamos a entender completamente. Un ejemplo es el del confinamiento. Un paso esencial para su comprensión es la descripción de la estructura hadrónica. Esta es precisamente la temática de esta tesis, donde se estudia la estructura de piones mediante el formalismo de las distribuciones generalizadas de partones (GPDs, por sus siglas en inglés). Como mesones, los piones deberían ser más sencillos de describir que otros hadrones como el protón. Más aún, debido a su naturaleza dual como estados ligados pero también como bosones Goldstone asociados a la rotura dinámica de simetría quiral, los piones son una pieza fundamental para comprender el origen de la masa en sistemas físicos. Por todo esto esta tesis aborda la descripción de la estructura piónica a primeros principios y evalúa sus manifestaciones en experimentos. El trabajo está dividido en dos partes. Empezando por abordar el problema de la descripción de la estructura hadrónica desde una perspectiva general, el primer capítulo revisa los procesos de scattering Compton en este sentido. Así, la identificación del límite de dispersión profundamente virtual permite la introducción de las GPDs como objectos fundamentales para la descripción de scattering Compton profundamente virtual (DVCS, en inglés). El segundo capítulo se dedica entonces a un análisis detallado de estas, empezando por su definición, propiedades y una discusión detallada sobre toda la información que codifican sobre la estructura hadrónica. En este proceso se encuentra que las llamadas propiedades de "polinomialidad" y "positividad" juegan un papel central en la construcción de modelos de GPDs realistas, y es por tanto satisfacerlas en todo caso. En este sentido es importante destacar que las estrategias convencionales para el cálculo de GPDs no consiguen este objetivo. Con ello, el tercer capítulo de la tesis aborda el problema de construir modelos de GPDs para piones que satisfagan, por construcción, ambas propiedades. Siguiendo una estrategia basada en el método de las ecuaciones de Dyson-Schwinger para la descripción de estados ligados, se encuentra que la hipótesis de desacoplo entre los grados de libertad longitudinales y transversales de la dinámica partónica está íntimamente relacionada con la restauración de la simetría quiral. Explotando dicha hipótesis derivamos una nueva familia de GPDs en la región DGLAP que satisfacen la condición de positividad y para cuya construcción únicamente es necesario el conocimiento de las llamadas funciones de distribución de partones. Partiendo de ese resultado, la estrategia de extensión covariante permite obtener la correspondiente región ERBL, obteniendo por primera vez modelos de GPDs que satisfacen al mismo tiempo y por construcción las condiciones de positividad y polinomialidad. Con el fin de arrojar luz a nuestro entendimiento sobre la estructura de los piones, así como la contrastación de este estudio, la segunda parte del trabajo parte de los mencionados modelos y evalúa sus manifestaciones en los resultados obtenidos en futuros experimentos. Para ello, el capítulo cuatro se encarga de describir la necesaria evolución de GPDs con la escala de renormalización; implementando una estrategia efectiva capaz de abarcar efectos puramente no perturbativos de la interacción fuerte. Los resultados ponen de manifiesto el importante papel del contenido gluónico en la construcción de la estructura de piones. Finalmente, el quinto capítulo describe el cálculo de número de eventos y asimetrías en futuros aceleradores electrón-ion, mostrando que efectivamente estas instalaciones deberían ser capaces de medir DVCS en piones e identificando la inversión en las correspondientes asimetrías como una clara señal experimental capaz de delimitar el régimen en el que los gluones constituyen la parte dominante en la estructura piónica

To d , or not to d : recent developments and comparisons of regularization schemes

We give an introduction to several regularization schemes that deal with ultraviolet and infrared singularities appearing in higher-order computations in quantum field theories. Comparing the computation of simple quantities in the various schemes, we point out similarities and differences between them

Gauge Theories in Particle Physics: A Practical Introduction, Volume 1

Volume 1 of this revised and updated edition provides an accessible and practical introduction to the first gauge theory included in the Standard Model of particle physics: quantum electrodynamics (QED).The book includes self-contained presentations of electromagnetism as a gauge theory as well as relativistic quantum mechanics. It provides a uniq

Effects of dark sector interactions on active-sterile neutrino mixing

Revised version. Minor spelling errors corrected.This thesis presents methods and theory for exploring new physics by using neutrinos as the portal. A thorough introduction on the state of neutrino physics in, and beyond the Standard Model framework, is presented. Anomalies arise from short baseline neutrino oscillation experiments, which may hint towards a fourth neutrino with a mass at the eV-scale. This is explained by the addition of a dark, Abelian symmetry group. The new symmetry is broken at the MeV-scale. To obtain sizeable active-sterile mixing, a hierarchy between Yukawa couplings is required. Majorana masses are included for righthanded neutrinos, which by the see-saw mechanism, provide a spectrum of light neutrino masses. The dark sector neutrinos are allowed to couple to the active neutrinos by Yukawa couplings which form a non-diagonal Yukawa matrix. The interplay between neutrino masses, Yukawa couplings, mixing angles and vacuum expectation values, is studied to provide solutions for the short baseline anomalies while remaining within experimental limits. Simple parameter scans are employed to further investigate the structure of the model. A parameter region compatible with global fits of reactor neutrino data is presented. When applied to bounds arising from resonant leptogenesis, the model confines right-handed neutrino masses in the 1TeV-100TeV range.Masteroppgave i fysikkPHYS399MAMN-PHY