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Renormalization-group flow

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

Last update: May 9th 2015, 230 pages

Contents:

Preface

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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I study some classes of RG flows in three dimensions that are classically conformal and have manifest $g \rightarrow 1/g$ dualities. The RG flow interpolates between known (four-fermion, Wilson-Fischer, $\phi_3^6$) and new interacting fixed points. These models have two remarkable properties: $i$) the RG flow can be integrated for arbitrarily large values of the couplings g at each order of the $1/N$ expansion; $ii$) the duality symmetries are exact at each order of the $1/N$ expansion. I integrate the RG flow explicitly to the order ${\cal O}(1/N)$, write correlators at the leading-log level and study the interpolation between the fixed points. I examine how duality is implemented in the regularized theory and verified in the results of this paper.

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Nucl.Phys. B658 (2003) 440 | DOI: 10.1016/S0550-3213(03)00174-3

arXiv:hep-th/0210123

I review my explanation of the irreversibility of the renormalization-group flow in even dimensions greater than two and address new investigations and tests.

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Acta Phys.Slov. 52 (2002) 573

arXiv:hep-th/0205039

I study various properties of the critical limits of correlators containing insertions of conserved and anomalous currents. In particular, I show that the improvement term of the stress tensor can be fixed unambiguously, studying the RG interpolation between the UV and IR limits. The removal of the improvement ambiguity is encoded in a variational principle, which makes use of sum rules for the trace anomalies $a$ and $a’$. Compatible results follow from the analysis of the RG equations. I perform a number of self-consistency checks and discuss the issues in a large set of theories.

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J.Math.Phys. 43 (2002) 2965-2977 | DOI: 10.1063/1.1475766

arXiv:hep-th/0110292

I derive a procedure to generate sum rules for the trace anomalies $a$ and $a’$. Linear combinations of $\Delta a = a_{UV}-a_{IR}$ and $\Delta a’ = a’_{UV}-a’_{IR}$ are expressed as multiple flow integrals of the two-, three- and four-point functions of the trace of the stress tensor. Eliminating $\Delta a’$, universal flow invariants are obtained, in particular sum rules for $\Delta a$. The formulas hold in the most general renormalizable quantum field theory (unitary or not), interpolating between UV and IR conformal fixed points. I discuss the relevance of these sum rules for the issue of the irreversibility of the RG flow. The procedure can be generalized to derive sum rules for the trace anomaly $c$.

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JHEP 0111:033 (2001) | DOI: 10.1088/1126-6708/2001/11/033

arXiv:hep-th/0107194

A flow invariant is a quantity depending only on the UV and IR conformal fixed points and not on the flow connecting them. Typically, its value is related to the central charges a and c. In classically-conformal field theories, scale invariance is broken by quantum effects and the flow invariant $a_{UV}-a_{IR}$ is measured by the area of the graph of the beta function between the fixed points. There exists a theoretical explanation of this fact. On the other hand, when scale invariance is broken at the classical level, it is empirically known that the flow invariant equals $c_{UV}-c_{IR}$ in massive free-field theories, but a theoretical argument explaining why it is so is still missing. A number of related open questions are answered here. A general formula of the flow invariant is found, which holds also when the stress tensor has improvement terms. The conditions under which the flow invariant equals $c_{UV}-c_{IR}$ are identified. Several non-unitary theories are used as a laboratory, but the conclusions are general and an application to the Standard Model is addressed. The analysis of the results suggests some new minimum principles, which might point towards a better understanding of quantum field theory.

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Class.Quant.Grav. 18 (2001) 4417-4442 | DOI: 10.1088/0264-9381/18/21/304

arXiv:hep-th/0101088

I study a class of interacting conformal field theories and conformal windows in three dimensions, formulated using the Parisi large-$N$ approach and a modified dimensional-regularization technique. Bosons are associated with composite operators and their propagators are dynamically generated by fermion bubbles. Renormalization-group flows between pairs of interacting fixed points satisfy a set of non-perturbative $g \leftrightarrow 1/g$ dualities. There is an exact relation between the beta function and the anomalous dimension of the composite boson. Non-Abelian gauge fields have a non-renormalized and quantized gauge coupling, although no Chern-Simons term is present. A problem of the naive dimensional-regularization technique for these theories is uncovered and removed with a non-local, evanescent, non-renormalized kinetic term. The models are expected to be a fruitful arena for the study of odd-dimensional conformal field theory.

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JHEP 0006 (2000) 042 | DOI: 10.1088/1126-6708/2000/06/042

arXiv:hep-th/0005261

I discuss the properties of the central charges $c$ and $a$ for higher-derivative and higher-spin theories (spin 2 included). Ordinary gravity does not admit a straightforward identification of c and a in the trace anomaly, because it is not conformal. On the other hand, higher-derivative theories can be conformal, but have negative $c$ and $a$. A third possibility is to consider higher-spin conformal field theories. They are not unitary, but have a variety of interesting properties. Bosonic conformal tensors have a positive-definite action, equal to the square of a field strength, and a higher-derivative gauge invariance. There exists a conserved spin-2 current (not the canonical stress tensor) defining positive central charges $c$ and $a$. I calculate the values of $c$ and $a$ and study the operator-product structure. Higher-spin conformal spinors have no gauge invariance, admit a standard definition of $c$ and $a$ and can be coupled to Abelian and non-Abelian gauge fields in a renormalizable way. At the quantum level, they contribute to the one-loop beta function with the same sign as ordinary matter, admit a conformal window and non-trivial interacting fixed points. There are composite operators of high spin and low dimension, which violate the Ferrara-Gatto-Grillo theorem. Finally, other theories, such as conformal antisymmetric tensors, exhibit more severe internal problems. This research is motivated by the idea that fundamental quantum field theories should be renormalization-group (RG) interpolations between ultraviolet and infrared conformal fixed points, and quantum irreversibility should be a general principle of nature.

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Class.Quant.Grav. 17 (2000) 2847-2866 | DOI: 10.1088/0264-9381/17/15/301

arXiv:hep-th/9912122

I review recent results on conformal field theories in four dimensions and quantum field theories interpolating between conformal fixed points, supersymmetric and non-supersymmetric. The talk is structured in three parts: $i$) central charges, $ii$) anomalous dimensions and $iii$) quantum irreversibility.

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PoS (trieste99) 013

arXiv:hep-th/9910255

Some recent ideas are generalized from four dimensions to the general dimension $n$. In quantum field theory, two terms of the trace anomaly in external gravity, the Euler density $G_n$ and $\Box^{n/2-1}R$, are relevant to the problem of quantum irreversibility. By adding the divergence of a gauge-invariant current, $G_n$ can be extended to a new notion of Euler density, linear in the conformal factor. We call it pondered Euler density. This notion relates the trace-anomaly coefficients $a$ and $a’$ of $G_n$ and $\Box^{n/2-1}R$ in a universal way ($a=a’$) and gives a formula expressing the total RG flow of a as the invariant area of the graph of the beta function between the fixed points. I illustrate these facts in detail for $n=6$ and check the prediction to the fourth-loop order in the $\phi^3$-theory. The formula of quantum irreversibility for general n even can be extended to $n$ odd by dimensional continuation. Although the trace anomaly in external gravity is zero in odd dimensions, I show that the odd-dimensional formula has a predictive content.

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Nucl.Phys. B567 (2000) 331-359 | DOI: 10.1016/S0550-3213(99)00479-4

arXiv:hep-th/9905005

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14B1 D. Anselmi
Renormalization

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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.)