Aspects of Nonperturbative Heavy Quark Physics

Abstract

The properties of charm and bottom quarks are an interesting corner of Quantum Chromo-Dynamics (QCD) due to the fact that their masses are much heavier than the typical QCD interaction energy ΛQCD. Due to this scale separation, it is possible to describe these heavy quarks by Effective Field Theories (EFTs) that simplify their equations of motion, make explicit additional symmetries that only appear for heavier quark masses, and simplify the theoretical calculations required for predictions. By discretising these EFTs in a lattice regularisation, nonperturbative calculations of observables of interest become possible. This thesis presents progress towards systematically controlled calculations of two such observables: the Spectator Effect contributions to the inclusive decay rates of b-hadrons, and the real-time dynamics of fermions propagating in a thermal medium. Standard EFT calculations in Lattice-QCD proceed by expressing observables as sums over perturbatively computed Wilson coefficients and nonperturbative matrix elements that can be calculated by path integral monte-carlo methods. Though it is possible to carry out this procedure within a regulator-independent renormalization scheme, in practice almost all such decompositions are computed in the modified minimal subtraction scheme MS which is only defined for the dimensional regulator (DR), due to its simplicity. Computing such observables therefore requires a matching between lattice regularised operators and operators renormalized in MS. In Chapter 2, both the dimensional regulator (DR) and the lattice regulator are reviewed, with a particular emphasis on techniques needed for calculations carried out in later sections. An interesting subtelty in DR is the need to introduce d-dimensional counterparts to the Dirac γ-matrices, which a-priori are only well defined in integer number of dimensions. This analytic continuation is of practical importance as it introduces additional Evanescent Operators (Sec. 2.1.4) that have physical consequences. In Sec. 2.1.5, traces of d-dimensional γ-matrices were related to Tutte polynomial evaluations [4], presenting a new graph-theoretic interpretation of the dimensionally regulated γ-matrices. One strategy of renormalizing lattice-regulated operators into MS involves first renormalizing into a regulator independent scheme, before perturbatively matching between the regulator independent scheme and MS. In Chapter 3, regulator independent position-space (X-space) schemes for renormalizing operators defined in the leading order Heavy Quark Effective Theory (HQET) are proposed [3]. Compared to other regulator independent renormalization schemes such as RI-xMOM, X-space schemes have the benefit that they are gauge invariant. The next to leading order matching calculations between X-space and MS are presented for heavy-light and heavy-light-light multiplicatively renormalizable operators, as well as ∆Q = 0 and ∆Q = 2 four quark operators relevant for heavy hadron decays and mixing, where Q refers to the static quark number. Due to their heavy masses, hadrons containing heavy quarks decay via the weak nuclear force. Experimental measurements of these lifetimes provide precision determinations of the fundamental parameters of the Standard Model. The Heavy Quark Expansion expresses the inclusive lifetimes of heavy hadrons in terms of matrix elements of HQET operators of increasing dimension. The Spectator Effects are contributions due to the ∆Q = 0 four-quark operators, where the light quark degrees of freedom within a heavy hadron participate in the decay. In Chapter 4, a Lattice-QCD determination of the static decay constant f HQET B and the isospin-nonsinglet portion of the Spectator Effect matrix elements for heavy-light mesons is presented. Fits of bare matrix elements were performed for three different lattice spacings, and renormalized with the schemes proposed in Chapter 3 before a continuum limit is taken. Due to the heavy masses mQ of the heavy quarks, it is possible to find temperatures T approximately satisfying a hierarchy ΛQCD ≪ T ≪ mQ. At these temperatures, QCD undergoes a deconfinement transition into the Quark-Gluon-Plasma (QGP) phase where the light degrees of freedom are no longer confined, and instead screen the long-range colour forces. The heavy quarks however are not thermalised, and act as probes of the QGP. Further understanding of the QGP requires first principles simulations of the heavy quark dynamics at finite temperature, however such calculations are difficult due to the enormous size of the Hilbert space. Variational approximations of the Hilbert space encode wavefunctions within a few parameters, and provide a practical method to simulate many particle systems. As a testcase, the variational approach was applied for the first time to simulate fermions at finite temperature in a simple QFT: the 1+1d U(1) gauge theory known as the massive Schwinger model. Both the real-time dynamics of string like states, and the properties of the thermal state were studied, and such variational methods are shown to be promising approaches to the more difficult case of a heavy quark effective theory in QCD.