Deep Inelastic Scattering in Improved Lattice QCD. I. The first moment of structure functions

We present the complete 1-loop perturbative computation of the renormalization constants and mixing coefficients of the operators that measure the first moment of deep inelastic scattering structure functions, employing the nearest neighbor improved lattice QCD action. The interest of using this action in Monte Carlo simulations lies in the fact that all terms which in the continuum limit are effectively of order $a$ ($a$ being the lattice spacing) have been proven to be absent from on-shell hadronic lattice matrix elements. Because of the complexity of the calculations, we have checked the analytical expression of all Feynman diagrams using Schoonschip. To this end we have developed a suitable code designed to automatically carry out all the necessary lattice algebraic manipulations, starting from the elementary building blocks of each diagram. We have found discrepancies with some of the published numbers, but we are in agreement with the known results on the energy-momentum tensor.Comment: 59 pages, plain LaTeX + Feynman.tex (complete postscript file available upon request to [email protected]), Preprint Roma1 978-93 and ROM2F 93/38 (some numerical mistakes have been corrected in Sects. 2.2 and 8

Fully Unintegrated Parton Correlation Functions and Factorization in Lowest Order Hard Scattering

Motivated by the need to correct the potentially large kinematic errors in approximations used in the standard formulation of perturbative QCD, we reformulate deeply inelastic lepton-proton scattering in terms of gauge invariant, universal parton correlation functions which depend on all components of parton four-momentum. Currently, different hard QCD processes are described by very different perturbative formalisms, each relying on its own set of kinematical approximations. In this paper we show how to set up formalism that avoids approximations on final-state momenta, and thus has a very general domain of applicability. The use of exact kinematics introduces a number of significant conceptual shifts already at leading order, and tightly constrains the formalism. We show how to define parton correlation functions that generalize the concepts of parton density, fragmentation function, and soft factor. After setting up a general subtraction formalism, we obtain a factorization theorem. To avoid complications with Ward identities the full derivation is restricted to abelian gauge theories; even so the resulting structure is highly suggestive of a similar treatment for non-abelian gauge theories.Comment: 44 pages, 69 figures typos fixed, clarifications and second appendix adde

Polarized qqˉ→Z+q \bar{q} \rightarrow Z +Higgs amplitudes at two loops in QCD: the interplay between vector and axial vector form factors and a pitfall in applying a non-anticommuting γ5\gamma_5

We consider QCD corrections to two loops for the polarized amplitudes of $q{\bar q}\to Z +$ Higgs boson. First we show how the polarized amplitudes of $b \bar{b} \rightarrow Z h$ associated with a non-vanishing $b$-quark Yukawa coupling and a scalar or pseudoscalar Higgs boson $h$ can be built up solely from vector form factors (FF) of properly grouped classes of diagrams, bypassing completely the need of explicitly manipulating $\gamma_5$ in dimensional regularization (up to a few "anomalous", i.e., triangle diagrams). We determine the contributions of the triangle diagrams in the heavy top limit. We present the analytic results of the vector FF and the triangle-diagram contributions to the axial vector FF, which are sufficient for deriving the two-loop QCD amplitudes for $b \bar{b} \rightarrow Z h$ with a CP-even and CP-odd Higgs boson $h$. We derive the respective Ward identity for these amplitudes, which are subsequently verified to two-loop order in QCD using these FF. In addition, the FF of a class of corrections to $q \bar{q} \rightarrow ZH$ proportional to the top-Yukawa coupling are obtained analytically to two-loop order in QCD in the heavy-top limit using the Higgs-gluon effective Lagrangian where the top quark is integrated out. We address a pitfall that occurs when applying the non-anticommutating $\gamma_5$ prescription to this class of contributions that has been overlooked so far in the literature. We attribute this issue to the fact that the absence of certain heavy-mass expanded diagrams in the infinite-mass limit of a scattering amplitude with an axial vector current depends on the particular $\gamma_5$ prescription in use.Comment: A few typos fixed along with minor typographic edits, references in the manuscript and the ReadMe.txt updated, matched with the version accepted by JHE

QCD in heavy ion collisions

These lectures provide a modern introduction to selected topics in the physics of ultrarelativistic heavy ion collisions which shed light on the fundamental theory of strong interactions, the Quantum Chromodynamics. The emphasis is on the partonic forms of QCD matter which exist in the early and intermediate stages of a collision -- the colour glass condensate, the glasma, and the quark-gluon plasma -- and on the effective theories that are used for their description. These theories provide qualitative and even quantitative insight into a wealth of remarkable phenomena observed in nucleus-nucleus or deuteron-nucleus collisions at RHIC and/or the LHC, like the suppression of particle production and of azimuthal correlations at forward rapidities, the energy and centrality dependence of the multiplicities, the ridge effect, the limiting fragmentation, the jet quenching, or the dijet asymmetry.Comment: Based on lectures presented at the 2011 European School of High-Energy Physics, 7-20 September 2011, Cheile Gradistei, Romania. 73 pages, many figure

Algorithms and techniques for finding canonical differential equations of Feynman integrals

Modern day particle physics relies on perturbative Quantum Field Theory. Beyond the leading order, the computation of the appearing Feynman integrals is one of the most important and often also most complicated steps necessary to extract predictions from the theory. A prominent method for the latter is to derive a set of ordinary differential equations satisfied by the Feynman integrals and then subsequently transforming them into canonical form where the solution in terms of known functions can be obtained in a straightforward manner. In this thesis, we consider the crucial task of finding the appropriate basis change that transforms the set of differential equations into canonical form. To this end, we first discuss some of the basic properties of Feynman integrals and the special functions appearing in the solutions. We then show how the transcendental weight of these functions constitutes an essential guiding principle in the search for members of the canonical basis. In particular, we review the algorithmic determination of so-called dlog integrals and their leading singularities. Further, we provide a summary of heuristic methods that, in many cases, prove to be sufficient for finding a canonical basis with little effort. Through a simple example, we then also review how so-called balance transformations can be used to reach the properties of the canonical form step by step. As the main result of the thesis, we present a new algorithm for attaining the canonical form starting from a single canonical integral. In addition, this algorithm makes it possible to test the transcendental weight properties of individual integrals and can therefore also be seen as complementary to the other methods described in this thesis. Several univariate examples, as well as a multivariate example are used to demonstrate the power and flexibility of the algorithm and our public implementation. Finally, we use the methods discussed in the thesis in three state-of-the-art applications and highlight how our algorithm can find the canonical form in cases where existing methods fail to provide an answer. This includes differential equations with more than 500 basis integrals and a matrix involving elliptic functions.Die moderne Teilchenphysik stützt sich auf die störungstheoretische Quantenfeldtheorie. Jenseits der führenden Ordnung ist die Berechnung der auftretenden Feynman-Integrale einer der wichtigsten und oft auch kompliziertesten Schritte, die notwendig sind, um Vorhersagen aus der Theorie zu gewinnen. Eine häufig verwendete Methode zur Berechnung der Feynman-Integrale besteht darin, einen Satz gewöhnlicher Differentialgleichungen abzuleiten, die von den Feynman-Integralen erfüllt werden, und diese anschließend in eine kanonische Form zu transformieren, bei der die Lösung in Form bekannter Funktionen auf einfache Weise erhalten werden kann. Diese Arbeit befasst sich mit der entscheidenden Aufgabe, die geeignete Basisänderung zu finden, die den Satz der Differentialgleichungen in die kanonische Form bringt. Zu diesem Zweck werden zunächst einige der grundlegenden Eigenschaften von Feyn\-man-Integralen und der speziellen Funktionen, die in den Lösungen vorkommen, erläutert. Anschließend erörtern wir, wie das transzendentale Gewicht dieser Funktionen ein wesentliches Leitprinzip bei der Suche nach Teilen der kanonischen Basis darstellt. Insbesondere gehen wir auf die algorithmische Bestimmung der sogenannten dlog-Integrale und ihrer ``leading singularities'' ein. Außerdem geben wir einen Überblick über heuristische Methoden, die sich in vielen Fällen als ausreichend erweisen, um mit geringem Aufwand eine kanonische Basis zu finden. Anhand eines einfachen Beispiels wird anschließend gezeigt, wie man mit Hilfe von sogenannten ``balance''-Transformationen Schritt für Schritt zu den Eigenschaften der kanonischen Form gelangt. Als Hauptergebnis der Arbeit stellen wir einen neuen Algorithmus vor, mit dem die kanonische Form ausgehend von einem einzigen kanonischen Integral erreicht werden kann. Dieser Algorithmus ermöglicht es ebenfalls, die transzendentalen Gewichtseigenschaften einzelner Integrale zu testen und kann daher auch als Ergänzung zu den anderen in dieser Arbeit beschriebenen Methoden betrachtet werden. Anhand mehrerer univariater Beispiele, sowie eines multivariaten Beispiels, wird die Leistungsfähigkeit und Flexibilität des Algorithmus und der öffentlich zugänglichen Implementierung demonstriert. Schließlich verwenden wir die in dieser Arbeit diskutierten Methoden in drei hochmodernen Anwendungen und zeigen, wie unser Algorithmus die kanonische Form in Fällen finden kann, in denen bestehende Methoden nicht anwendbar sind. Dazu gehören unter anderem Differentialgleichungen mit mehr als 500 Basisintegralen und eine Matrix mit elliptischen Funktionen

The physics of exclusive reactions in QCD: Theory and phenomenology

The modern formulation of exclusive reactions within Quantum Chromodynamics is reviewed, the emphasis being placed on the pivotal ideas and methods pertaining to perturbative and non-perturbative topics. Specific problems, related to scale locality, infrared safety, gluonic radiative corrections (Sudakov effects), and the role of hadronic size effects (intrinsic transverse momentum), are studied. These issues are more precisely analyzed in terms of the essential mechanisms of momentum transfer to a hadron while remaining intact. Different factorization schemes are considered and the conceptual lacunas are pointed out. The quite technical subject of renormalization-group evolution is given a detailed account. By combining analytical and numerical algorithms, the one-gluon exchange nucleon evolution equation is diagonalized and next-to-leading eigenfunctions are calculated in terms of Appell polynomials. The corresponding anomalous dimensions of trilinear quark operators are found to form a degenerate system whose envelope shows logarithmic large-order behavior. Selected applications of this framework are presented, focusing on the helicity-conserving elastic form factors of the pion and the nucleon. The theoretical constraints imposed by QCD sum rules on the moments of nucleon distribution amplitudes are used to determine a whole spectrum of optional solutions. They organize themselves along an ``orbit'' characterized by a striking scaling relation between the form-factor ratio $R=|G_{\rm M}^{\rm n}|/G_{\rm M}^{\rm p}$ and the projection coefficient $B_{4}$ on to the corresponding eigensolution. The main reasons for the failure of the present theoretical predictions to match the experimental data are discussed and workable explanations are sketched.Comment: 112 pages; 12 tables; 35 embedded figures as PS/EPS files; RevTex styles used. Article based on thesis for Dr. nauk (sci.) phys.-math. degree, successfully defended at BLThP, JINR, Dubna (1997); Bochum Reports RUB-TPII-20/96, RUB-TPII-07/99. Published in Eur. Phys. J. direct C7, 1-109 (1999