Recent theorems

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Dimensional regularization

Using the Batalin-Vilkovisky formalism, we study the Ward identities and the equations of gauge dependence in potentially anomalous general gauge theories, renormalizable or not. A crucial new term, absent in manifestly nonanomalous theories, is responsible for interesting effects. We prove that gauge invariance always implies gauge independence, which in turn ensures perturbative unitarity. Precisely, we consider potentially anomalous theories that are actually free of gauge anomalies thanks to the Adler-Bardeen theorem. We show that when we make a canonical transformation on the tree-level action, it is always possible to re-renormalize the divergences and re-fine-tune the finite local counterterms, so that the renormalized $\Gamma $ functional of the transformed theory is also free of gauge anomalies, and is related to the renormalized $\Gamma $ functional of the starting theory by a canonical transformation. An unexpected consequence of our results is that the beta functions of the couplings may depend on the gauge-fixing parameters, although the physical quantities remain gauge independent. We discuss nontrivial checks of high-order calculations based on gauge independence and determine how powerful they are.


Phys. Rev. D 92 (2015) 025027 | DOI: 10.1103/PhysRevD.92.025027

arXiv: 1501.06692 [hep-th]

Course on renormalization, taught in Pisa in 2015. (More chapters will be added later.)

Last update: May 9th 2015, 230 pages



1. Functional integral

  • 1.1 Path integral
    • Schroedinger equation
    • Free particle
  • 1.2 Free field theory
  • 1.3 Perturbative expansion
    • Feynman rules
  • 1.4 Generating functionals, Schwinger-Dyson equations
  • 1.5 Advanced generating functionals
  • 1.6 Massive vector fields
  • 1.7 Fermions

2. Renormalization

  • 2.1 Dimensional regularization
    • 2.1.1 Limits and other operations in $D$ dimensions
    • 2.1.2 Functional integration measure
    • 2.1.3 Dimensional regularization for vectors and fermions
  • 2.2 Divergences and counterterms
  • 2.3 Renormalization to all orders
  • 2.4 Locality of counterterms
  • 2.5 Power counting
  • 2.6 Renormalizable theories
  • 2.7 Composite fields
  • 2.8 Maximum poles of diagrams
  • 2.9 Subtraction prescription
  • 2.10 Regularization prescription
  • 2.11 Comments about the dimensional regularization
  • 2.12 About the series resummation

3. Renormalization group

  • 3.1 The Callan-Symanzik equation
  • 3.2 Finiteness of the beta function and the anomalous dimensions
  • 3.3 Fixed points of the RG flow
  • 3.4 Scheme (in)dependence
  • 3.5 A deeper look into the renormalization group

4. Gauge symmetry

  • 4.1 Abelian gauge symmetry
  • 4.2 Gauge fixing
  • 4.3 Non-Abelian global symmetry
  • 4.4 Non-Abelian gauge symmetry

5. Canonical gauge formalism

  • 5.1 General idea behind the canonical gauge formalism
  • 5.2 Systematics of the canonical gauge formalism
  • 5.3 Canonical transformations
  • 5.4 Gauge fixing
  • 5.5 Generating functionals
  • 5.6 Ward identities

6. Quantum electrodynamics

  • 6.1 Ward identities
  • 6.2 Renormalizability of QED to all orders

7 Non-Abelian gauge field theories

  • 7.1 Renormalizability of non-Abelian gauge theories to all orders
    • Raw subtraction

A. Notation and useful formulas

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We define a modified dimensional-regularization technique that overcomes several difficulties of the ordinary technique, and is specially designed to work efficiently in chiral and parity violating quantum field theories, in arbitrary dimensions greater than 2. When the dimension of spacetime is continued to complex values, spinors, vectors and tensors keep the components they have in the physical dimension, therefore the $\gamma $ matrices are the standard ones. Propagators are regularized with the help of evanescent higher-derivative kinetic terms, which are of the Majorana type in the case of chiral fermions. If the new terms are organized in a clever way, weighted power counting provides an efficient control on the renormalization of the theory, and allows us to show that the resulting chiral dimensional regularization is consistent to all orders. The new technique considerably simplifies the proofs of properties that hold to all orders, and makes them suitable to be generalized to wider classes of models. Typical examples are the renormalizability of chiral gauge theories and the Adler-Bardeen theorem. The difficulty of explicit computations, on the other hand, may increase.


Phys. Rev. D 89 (2014) 125024 | DOI: 10.1103/PhysRevD.89.125024

arXiv: 1405.3110 [hep-th]

We reconsider the Adler-Bardeen theorem for the cancellation of gauge anomalies to all orders, when they vanish at one loop. Using the Batalin-Vilkovisky formalism and combining the dimensional-regularization technique with the higher-derivative gauge invariant regularization, we prove the theorem in the most general perturbatively unitary renormalizable gauge theories coupled to matter in four dimensions, and identify the subtraction scheme where anomaly cancellation to all orders is manifest, namely no subtractions of finite local counterterms are required from two loops onwards. Our approach is based on an order-by-order analysis of renormalization, and, differently from most derivations existing in the literature, does not make use of arguments based on the properties of the renormalization group. As a consequence, the proof we give also applies to conformal field theories and finite theories.


Eur. Phys. J. C 74 (2014) 3083 | DOI: 10.1140/epjc/s10052-014-3083-0

arXiv: 1402.6453 [hep-th]

The maximum pole of a diagram with $V$ vertices and $L$ loops is at most $1/\varepsilon^{m(V,L)}$, where $m(V,L)=\min (V-1,L).$ The result holds in dimensional regularization, where $\varepsilon = d-D$, $d$ is the physical dimension and $D$ the continued one. Moreover, vertices are counted treating mass terms and the other non-dominant quadratic terms as “two-leg vertices”.

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14B1 D. Anselmi

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Last update: May 9th 2015, 230 pages

Contents: Preface | 1. Functional integral | 2. Renormalization | 3. Renormalization group | 4. Gauge symmetry | 5. Canonical formalism | 6. Quantum electrodynamics | 7. Non-Abelian gauge field theories | Notation and useful formulas | References

Course on renormalization, taught in Pisa in 2015. (More chapters will be added later.)