Academic literature on the topic 'Perturbative Momentum Shell Renormalization Group (RG)'

In this paper, we investigate the Lindblad dynamics arising due to single particle loss and local dephasing in a d-dimensional nonrelativistic quantum system. Using Schwinger–Keldysh functional techniques, we derive the static mean field equation for the symmetry broken phase. Fluctuations over mean field are studied within the Bogoliubov approximation, and the momentum distribution and effective temperature are calculated. An effective temperature arises which depends on the dephasing rate [Formula: see text] and mean field particle density. Furthermore, a perturbative renormalization group (RG) upto one loop order is used to study the underlying dynamics of the model. The conditions for the system to be perturbative are obtained.

Abstract The B-meson light-cone distribution amplitude is an important non- perturbative quantity arising in the factorization of the amplitudes for many exclusive decays of B mesons, such as B− → γℓ−$$ \overline{\nu} $$ ν ¯ . We reconsider the renormalization-group (RG) equation satisfied by this function and present its solution at next-to-leading order (NLO) in RG-improved perturbation theory in Laplace space and, for the first time, in momentum space and the so-called diagonal (or dual) space. Since the information needed to describe the B decay processes at leading order in ΛQCD/mb is most directly contained in the distribution amplitude in Laplace space evaluated near the origin, we propose a convenient parameterization of this object in terms of a small set of uncorrelated hadronic parameters. Using recent results on the three-loop anomalous dimension for heavy-light current operators, we derive an expression for the convolution integral appearing in the B− → γℓ−$$ \overline{\nu} $$ ν ¯ factorization formula that is explicitly scale independent, and we evaluate this formula at (approximate) NNLO.

Abstract Employing the QCD factorization formalism we compute $$ {B}_u^{-}\to {\gamma}^{\ast}\mathrm{\ell}{\overline{v}}_{\mathrm{\ell}} $$ B u − → γ ∗ ℓ v ¯ ℓ form factors with an off-shell photon state possessing the virtuality of order mb ΛQCD and $$ {m}_b^2 $$ m b 2 , respectively, at next-to-leading order in QCD. Perturbative resummation for the enhanced logarithms of mb/ΛQCD in the resulting factorization formulae is subsequently accomplished at next-to-leading logarithmic accuracy with the renormalization-group technique in momentum space. In particular, we derive the soft-collinear factorization formulae for a variety of the subleading power corrections to the exclusive radiative $$ {B}_u^{-}\to {\gamma}^{\ast }{W}^{\ast } $$ B u − → γ ∗ W ∗ form factors with a hard-collinear photon at $$ \mathcal{O}\left({\alpha}_s^0\right) $$ O α s 0 . We further construct a complete set of the angular observables governing the full differential distribution of the four-body leptonic decays $$ {B}_u^{-}\to {\mathrm{\ell}}^{\prime }{\overline{\mathrm{\ell}}}^{\prime}\mathrm{\ell}{\overline{v}}_{\mathrm{\ell}} $$ B u − → ℓ ′ ℓ ¯ ′ ℓ v ¯ ℓ with ℓ, ℓ′ ∈ {e, μ} and then perform an exploratory phenomenological analysis for a number of decay observables accessible at the LHCb and Belle II experiments, with an emphasis on the systematic uncertainties arising from the shape parameters of the leading-twist B-meson light-cone distribution amplitude in heavy quark effective theory.

A straightforward construction of a local, relativistic quantum field theory (QFT) leads to ultraviolet (UV) divergences and a QFT has to be regularized by modifying its short-distance or large energy momentum structure (momentum regularization is often used in this work). Since such a modification is somewhat arbitrary, it is necessary to verify that the resulting large-scale predictions are, at least to a large extent, short-distance insensitive. Such a verification relies on the renormalization theory and the corresponding renormalization group (RG). In this chapter, the essential steps of a proof of the perturbative renormalizability of the scalar φ⁴ QFT in dimension 4 are described. All the basic difficulties of renormalization theory, based on power counting, are already present in this simple example. The elegant presentation of Callan is followed, which makes it possible to prove renormalizability and RG equations (in Callan–Symanzik's (CS) form) simultaneously. The background of the discussion is effective QFT and emergent renormalizable theory. The concept of fine tuning and the issue of triviality are emphasized.

Chapter 6 describes how the perturbative renormalization group (RG) of quantum field theory has made it possible to derive universal properties of continuous macroscopic phase transitions. The RG initially emerged as a consequence of the necessity to cancel infinities that appear in the perturbative expansion (the renormalization procedure) and the possibility of defining the parameters of the renormalized theory at different momentum scales. Although the field theory RG is now understood to be an asymptotic form, it has made it possible to confirm the Wilson–Fisher fixed point and led to an understanding of universality for a large class of critical phenomena. In the framework of dimensional continuation, zeros of RG beta functions, which correspond to Wilson–Fisher’s fixed points, have made it possible to recover Wilson–Fisher’s epsilon expansion, proving scaling relations and calculating critical exponents. Series summation methods have then been used to generate precise values of exponents.

In this chapter, the notions of dimensional continuation and dimensional regularization are introduced, by defining a continuation of Feynman diagrams to analytic functions of the space dimension. Dimensional continuation, which is essential for generating Wilson–Fisher's famous ϵexpansion in the theory of critical phenomena, and dimensional regularization seem to have no meaning outside the perturbative expansion of quantum field theory (QFT). Dimensional regularization is a powerful regularization technique, which is often used, when applicable because it leads to much simpler perturbative calculations. Dimensional regularization performs a partial renormalization, cancelling what would show up as power-law divergences in momentum or lattice regularization. In particular it cancels the commutator of quantum operators in local QFTs. These cancellations may be convenient but may also, occasionally, remove divergences that have an important physical meaning. It is not applicable when some essential property of the field theory is specific to the initial dimension. For example, in even space dimensions, the relation between γ_S (identical to γ₅ in four dimensions) and the other γ matrices involving the completely antisymmetric tensor ϵμ1···μd, may be needed in theories violating parity symmetry. Its use requires some care in massless theories because its rules may lead to unwanted cancellations between ultraviolet and infrared logarithmic divergences. Explicit calculations at two-loop order in a scalar QFT with a general four-field interaction are performed.