Narrow Results

A systematic study of recursive renormalization of Feynman amplitudes is carried out both in Euclidean and in Minkowski configuration spaces. For a massless quantum field theory (QFT), we use the technique of extending associate homogeneous distributions to complete the renormalization recursion. A homogeneous (Poincaré covariant) amplitude is said to be convergent if it admits a (unique covariant) extension as a homogeneous distribution. For any amplitude without subdivergences — i.e. for a Feynman distribution that is homogeneous off the full (small) diagonal — we define a renormalization invariant residue. Its vanishing is a necessary and sufficient condition for the convergence of such an amplitude. It extends to arbitrary — not necessarily primitively divergent — Feynman amplitudes. This notion of convergence is finer than the usual power counting criterion and includes cancellation of divergences.

The exponential decay of lattice Green functions is one of the main technical ingredients of the Bałaban’s approach to renormalization. We give here a self-contained proof, whose various ingredients were scattered in the literature. The main sources of exponential decay are the Combes–Thomas method and the analyticity of the Fourier transforms. They are combined using a renormalization group equation and the method of images.

In the context of 5D N = 1 supersymmetric models compactified on S₁/Z₂ or S₁/(Z₂×Z₂′) orbifolds and with brane-localized superpotential, higher derivative operators are generated radiatively as one-loop counterterms to the mass of the (brane or zero mode of the bulk) scalar field. It is shown that the presence of such operators which are brane-localised is not related to the mechanism of supersymmetry breaking considered (F-term, discrete or continuous Scherk–Schwarz breaking) and initial supersymmetry does not protect against the dynamical generation of such operators. Since in many realistic models the scalar field is commonly regarded as the Higgs field, and the higher derivative operators seem a generic presence in orbifold compactifications, we stress the importance of these operators for solving the hierarchy problem.

Relativistic corrections are investigated to the Gamow–Teller (GT) sum rule with respect to the difference between the β_- and β₊ transition strengths in nuclei. Since the sum rule requires the complete set of the nuclear states, the relativistic corrections come from the anti-nucleon degrees of freedom. In the relativistic mean field approximation, the total GT strengths carried by the nucleon sector is quenched by about 12% in nuclear matter, while by about 8% in finite nuclei, compared to the sum rule value. The coupling between the particle-hole states with the nucleon–antinucleon states is also discussed with the relativistic random phase approximation, where the divergence of the response function is renormalized with use of the counterterms in the Lagrangian. It is shown that the approximation to neglect the divergence, like the no-sea approximation extensively used so far, is unphysical, from the sum-rule point of view.

Inspired by the ideas from topological field theory it is possible to rewrite the supersymmetric charges of certain classes of extended supersymmetric Yang–Mills (SYM) theories in such a way that they are compatible with the discretization on a Euclidean space–time lattice. Such theories are known as maximally twisted SYM theories. In this review we discuss the construction and some applications of such classes of theories. The one-loop perturbative renormalization of the four-dimensional lattice SYM is discussed in particular. The lattice theories constructed using twisted approach play an important role in investigating the thermal phases of strongly coupled SYM theories and also the thermodynamic properties of their dual gravitational theories.

The history of renormalization is reviewed with a critical eye, starting with Lorentz's theory of radiation damping, through perturbative QED with Dyson, Gell-Mann and Low, and others, to Wilson's formulation and Polchinski's functional equation, and applications to "triviality," and dark energy in cosmology.

Finite Unified Theories (FUTs) are N = 1 supersymmetric Grand Unified Theories (GUTs) which can be made finite to all-loop orders, based on the principle of reduction of couplings, and therefore are provided with a large predictive power. We confront the predictions of an SU(5) FUT with the top and bottom quark masses and other low-energy experimental constraints, resulting in a relatively heavy SUSY spectrum, naturally consistent with the nonobservation of those particles at the LHC. The light Higgs boson mass is automatically predicted in the range compatible with the Higgs discovery at the LHC. Requiring a light Higgs boson mass in the precise range of M_h = 125.6 ±2.1 GeV favors the lower part of the allowed spectrum, resulting in clear predictions for the discovery potential at current and future pp, as well as future e⁺e^- colliders.

In this review, we present a derivation of the on-shell renormalization conditions for scalar and fermionic fields in theories with and without parity conservation. We also discuss the specifics of Majorana fermions. Our approach only assumes a canonical form for the renormalized propagators and exploits the fact that the inverse propagators are nonsingular in $𝜀=p^2-m_n^2$, where $p$ is the external four-momentum and $m_n$ is a pole mass. In this way, we obtain full agreement with commonly used on-shell conditions. We also discuss how they are implemented in renormalization.

We review the recent theoretical progress in the construction and solution of the evolution equations which govern the scale dependence for the twist-3 structure and fragmentation functions of the nucleon.

We review the development of the large $N$ method, where $N$ indicates the number of flavours, used to study perturbative and nonperturbative properties of quantum field theories. The relevant historical background is summarized as a prelude to the introduction of the large $N$ critical point formalism. This is used to compute large $N$ corrections to $d$-dimensional critical exponents of the universal quantum field theory present at the Wilson–Fisher fixed point. While pedagogical in part the application to gauge theories is also covered and the use of the large $N$ method to complement explicit high order perturbative computations in gauge theories is also highlighted. The usefulness of the technique in relation to other methods currently used to study quantum field theories in $d$-dimensions is also summarized.