Dimensional continuation, regularization, minimal subtraction (MS). Renormalization group (RG) functions

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 γS (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.

Eigenvalue spectrum and scaling dimension of lattice $$ \mathcal{N} $$ = 4 supersymmetric Yang-Mills

We investigate the lattice regularization of $$ \mathcal{N} $$ N = 4 supersymmetric Yang-Mills theory, by stochastically computing the eigenvalue mode number of the fermion operator. This provides important insight into the non-perturbative renormalization group flow of the lattice theory, through the definition of a scale-dependent effective mass anomalous dimension. While this anomalous dimension is expected to vanish in the conformal continuum theory, the finite lattice volume and lattice spacing generically lead to non-zero values, which we use to study the approach to the continuum limit. Our numerical results, comparing multiple lattice volumes, ’t Hooft couplings, and numbers of colors, confirm convergence towards the expected continuum result, while quantifying the increasing significance of lattice artifacts at larger couplings.

Quantum anomalies: A few physics applications

Chapter 17 exhibits various examples where classical symmetries cannot be transferred to quantum theories. The obstructions are characterized by anomalies. The examples involve chiral symmetries combined with currents or gauge symmetries, leading to chiral anomalies. In particular, anomalies lead to obstruction in the construction of theories. In particular, the structure of the Standard Model of particle physics is constrained by the requirement of anomaly cancellation. Other applications, like the relation between electromagnetic pi0 decay and the axial anomaly, are described. Anomalies are related to the Dirac operator index, leading to relations between anomaly and topology. To prove anomaly cancellation beyond perturbation theory, one can use lattice regularization. However, the definition of lattice chiral transformations is non–trivial. It is based on solutions of the Ginsparg–Wilson relation.

Symmetries: From classical to quantum field theories

Chapter 16 deals with the important problem of quantization with symmetries, that is, how to implement symmetries of the classical action in the corresponding quantum theory. The proposed solutions are based on methods like regularization by addition of higher order derivatives or regulator fields, or lattice regularization. Difficulties encountered in the case of chiral theories are emphasized. This may lead to obstacles for symmetric quantization called anomalies. Examples can be found in the case of chiral gauge theories. Their origin can be traced to the problem of quantum operator ordering in products. A non–perturbative regularization, also useful for numerical simulations, is based on introducing a space lattice. Difficulties appear for lattice Dirac fermions, leading the fermion doubling problem. Wilson’s fermions provide a non–chiral invariant solution. Chiral invariant solutions have been found, called overlap fermions or domain wall fermions.

4D N = 1 SYM supercurrent on the lattice in terms of the gradient flow

The gradient flow [1–5] gives rise to a versatile method to construct renor-malized composite operators in a regularization-independent manner. By adopting this method, the authors of Refs. [6–9] obtained the expression of Noether currents on the lattice in the cases where the associated symmetries are broken by lattice regularization. We apply the same method to the Noether current associated with supersymmetry, i.e., the supercurrent. We consider the 4D N = 1 super Yang–Mills theory and calculate the renormalized supercurrent in the one-loop level in the Wess–Zumino gauge. We then re-express this supercurrent in terms of the flowed gauge and flowed gaugino fields [10].