# From Perturbative to Constructive Renormalization

Vincent Rivasseau
Pages: 346
https://www.jstor.org/stable/j.ctt7zv6ds

1. Front Matter
(pp. i-vi)
(pp. vii-viii)
3. Acknowledgements
(pp. ix-x)
4. ### Part I Introduction to Euclidean Field Theory

• Chapter I.1 The Ultraviolet Problem
(pp. 3-14)

Quantum field theory is an attempt to describe the properties of elementary “point-like” particles in terms of relativistic quantum fields. It is now widely believed to offer a coherent mathematical framework for relativistic models (like the “standardU(1) xSU(2) xSU(3) model”). These models include all the particles and interactions observed up to now except gravity. Therefore, together with general relativity, field theory is the backbone of our current understanding of the physical world. In the future a new, more unifying framework may be adopted, like the currently promising superstring theory, which is a relativistic and quantum description of...

• Chapter I.2 Euclidean Field Theory. The O.S. Axioms
(pp. 15-22)

From the classical work of the founding fathers of axiomatic field theory, we learn that the minimal mathematical requirements or axioms for a field theory are conveniently expressed in terms of the vacuum expectation values of products of the field operators, the “Wightman functions” [SW]. From these quantities one could also, at least in principle, compute more physical quantities like theSmatrix. We could start directly from the Wightman axioms or their Euclidean counterpart, the Osterwalder-Schrader axioms, but we prefer to motivate them first with a brief sketch of their relation to theSmatrix formalism, without any attempt...

• Chapter I.3 The ${\Phi ^4}$ Model
(pp. 23-36)

The simplest interacting field theory is the theory of a one-component scalar bosonic field φ with quartic interaction$g{\varphi ^4}$(${\varphi ^3}$which is simpler looks unstable). In${\mathbb{R}^d}$it is called the$\varphi _d^4$model. Ford= 1, 2, 3 the model is superrenormalizable and has been built by constructive field theory. Ford= 4 it is renormalizable in perturbation theory. Although a constructive version may not exist [Aiz][Frö], it remains a valuable tool at least for a pedagogical introduction to renormalization theory.

Formally the Schwinger functions of the$\varphi _d^4$are the moments of the measure:

$dv = \frac{1} {Z}{e^{( - g/4!)\int {{\varphi ^4} - ({m^2}/2)\int {{\varphi ^2} - (a/2)} \int {({\partial _\mu }\varphi {\text{ }}{\partial ^u}\varphi )} } }}D\varphi$. (1.3.1)

gis...

• Chapter I.4 Feynman Graphs and Amplitudes
(pp. 37-53)

The perturbation expansion is an expansion in powers of the coupling constant. In the case of the Schwinger functions (1.3.4) this means that one writes:

${S_N}({z_1}{z_n}) = \frac{1} {Z}\sum\limits_{n = 0}^\infty {\frac{{{{( - g)}^n}}} {{n!}}} {\left[ {\int {\frac{{{\varphi ^4}(x)}} {{4!}}} } \right]^n}\varphi ({z_1}){\text{. }}{\text{. }}{\text{.}}\varphi ({z_N})d\mu (\varphi )$

It is possible to perform explicitly the functional integral of a polynomial in the fields with respect to a Gaussian measure. The result, called in physics “Wick’s theorem,” gives at any order a sum over “Wick contractions,” i.e., ways of pairing together the fields in (1.4.1). More precisely, we can label the integration variables in (1.4.1) as${x_1}{\text{, }}{\text{. }}{\text{. }}{\text{. , }}{x_n}$, and the 4 fields in a monomial${\varphi ^4}$as${\varphi _1}{\varphi _2}{\varphi _3}{\varphi _4}$. Then we commute the spatial integral...

• Chapter I.5 Borel Summability
(pp. 54-56)

Borel summability is one of many possible substitutes for ordinary summability. It has proved very useful for the analysis of many divergent series met in physics, and provides in particular a natural framework for the study of the perturbative series met in this book. Therefore we include this brief section to recall what it means.

An analytic function inside its domain of analyticity is the sum of its Taylor series, so that all information about this function is embedded in the list of Taylor coefficients at an interior point. Ordinary summation then provides a one to one correspondence (at least...

5. ### Part II Perturbative Renormalization

• Chapter II.1 The Multiscale Representation and a Bound on Convergent Graphs
(pp. 59-73)

In this section we prove in detail a bound which we think is the best pedagogical introduction to multiscale expansions. It is a kind of “uniform” Weinberg theorem. The original Weinberg theorem [We] proves that any graphGfor which each connected subgraphShas a positive superficial degree of convergence$\omega (S)$has indeed a finite amplitude (hence does not require renormalization). From (1.4.7) this means in the case of$\varphi _4^4$that any graphGsuch that each connected subgraphShas at least 6 external lines (5 being forbidden) is finite. Such graphs will be called completely convergent; some...

• Chapter II.2 Renormalization Theory for $\Phi _4^4$
(pp. 74-89)

We start with an overview of the situation and some examples before to introduce the full formalism of renormalization, which unfortunately even in its most recent and transparent versions still involves some heavy notations.

Our first remark is that the proof we gave of the “uniform Weinberg” theorem also gives uniform bounds for many momentum assignments corresponding to graphs which are no longer completely convergent. Indeed the only convergence degrees which appear in a given μ are the$\omega (G_k^i)$’s. This leads us to call a momentum assignment μ for a graphGconvergent if$\omega (g) > 0$for anygalmost local...

• Chapter II.3 Proof of the Uniform BPH Theorem
(pp. 90-110)

For the proof to work in the multiscale representation, the key point is to extract additional “index space” decay when some almost local subgraphs are divergent; in the biped free case they can only be quadrupeds. This decay should be extracted as in the example of the bubble subgraph treated in the preceding section. We need to give first the precise definition of the subtraction operators which are the equivalents inx-space of Zimmermann’s operators in momentum space; in other words we must give the rule for “adjoints”$\tau _g^*$which generalize (II.2.3–4). Let us fix some forest of quadrupeds...

• Chapter II.4 The Effective Expansion
(pp. 111-122)

Let us summarize the conclusions of the preceding section. The bare expansion does not have a limit when the ultraviolet cutoff is removed. The renormalized expansion has cured this defect, but the price to pay for that is definitely too heavy. The “useless counterterms” make the study of renormalization quite painful (because one has to use forestry and to work always in reduced subgraphsg/Af(g)); they also generate renormalon behavior which puts in danger the constructive program. Intuitively a factor${K^n}$in large order estimates may be compensated by requiring the coupling constant to be very small, but this is...

• Chapter II.5 Construction of “Wrong Sign” Planar $\Phi _4^4$
(pp. 123-143)

In this section we show how to apply the formalism of the effective expansion beyond the sterile level of formal power series, on a simple model which is not a full-fledged field theory but has nevertheless some physical interest.

Even in the effective version without renormalon effects, it is not easy to sum up perturbation theory because of the large number of graphs involved at large order. This divergence of perturbation theory occurs even in 0 dimension, for a single integral$\int {{e^{ - {x^2} - g{x^4}}}} dx$. In the next section we will analyze it in some detail and conclude that we must trade absolute...

• Chapter II.6 The Large Order Behavior of Perturbation Theory
(pp. 144-168)

In this last section on perturbation theory we no longer discuss large order bounds for individual Feynman amplitudes, but consider the more difficult problem of the exact large orderbehaviorof the renormalized perturbation series. We will meet again the problem of the large number of graphs in the ordinary${\varphi ^4}$theory, and the renormalon problem for$\varphi _4^4$, and discuss how they shape the large order behavior of the theory, using the convenient mathematical formalism of the Borel transform introduced in section 1.5. The rigorous results obtained so far are still fragmentary and in our opinion a lot of interesting...

6. ### Part III Constructive Renormalization

• Chapter III.1 Single Scale Cluster and Mayer Expansions
(pp. 171-209)

Cluster and Mayer expansions are key tools in many areas of mathematical physics. Introduced in constructive field theory by Glimm, Jaffe and Spencer to complete the construction of$\varphi _2^4$[GJS], they have been improved or generalized over the years, in particular by Brydges, Battle and Federbush [BrFe][BaFl–3][Bat]. Unfortunately these expansions had for a while a reputation of being heavy to handle. In this section we try to dispel this impression by underlying the main facts behind their convergence. We do not try to present the best techniques or the optimal bounds. Our goal is simply to introduce beginners to...

• Chapter III.2 The Phase Space Expansion: The Convergent Case
(pp. 210-240)

We turn to a description of the natural generalization of the cluster expansion of last section in the case of a model in which the principle of phase space chopping becomes necessary.

What are the models of this type? Consider first the massive${\varphi ^4}$theory in dimensiond. Ford= 2 the renormalization problem reduces to Wick ordering. We have to do some momentum analysis but we can avoid spatial localization, hence a true phase space expansion, as shown in the preceding section. But ford= 3 phase space chopping starts being truly useful [GJ1]. Three dimensional theories...

• Chapter III.3 The Effective Expansion and Infrared $\Phi _4^4$
(pp. 241-271)

In this section we add renormalization to the phase space expansion (more preciselyusefulrenormalization), to obtain finally a tool sufficiently powerful for a non-perturbative investigation of some properties of$\Phi _4^4$. In particular we give the results on the construction of the critical$\Phi _4^4$with fixed ultra-violet cutoff, or infrared$\Phi _4^4$, based on infrared asymptotic freedom [FMRS5], and we include also some corresponding triviality results on ultraviolet$\Phi _4^4$.

In the case of infrared$\Phi _4^4$we start with a bare theory of type (1.3.1) in a finite box$\Lambda$with a fixed ultraviolet cutoff and a mass counterterm:

$dv(\varphi ) = {Z^{ - 1}} \cdot {e^{ - \lambda \int_\Lambda {{\varphi ^4} - (1/2)\int_\Lambda {{m^2}{\varphi ^2}} } }}d{\mu _C}(\varphi )$, (111.3.1)

where...

• Chapter III.4 The Gross–Neveu Model
(pp. 272-288)

The Gross–Neveu model in two dimensions [MW][GrNe] is a model of fermions with a color indexN> 1 and a quartic interaction. The model was introduced in [MW] and shown to be asymptotically free in the ultraviolet direction. In [GN] it was further studied and advocated as a toy model for non-Abelian gauge theories. In particular it was argued in [GN] that physical phenomena like dynamical symmetry breaking and the non-perturbative generation of mass (which is an analogue of the non-perturbative confinement of quarks) could be studied in this model in a much simpler way than in non-Abelian gauge...

• Chapter III.5 The Ultraviolet Problem in Non-Abelian Gauge Theories
(pp. 289-320)

The efforts to understand renormalization theory better should culminate in a rigorous solution of the ultraviolet problem for non-Abelian gauge theories. Most physicists are convinced that the problem is well understood and void of any surprises, because of its asymptotically free character. However there is only one rigorous program of study of this problem completed so far, the one of Balaban [Ba2–9]. This program defines a sequence of block-spin transformations for the pure Yang–Mills theory in a finite volume on the lattice and shows that as the lattice spacing tends to 0 and these transformations are iterated many...

7. References and Bibliography
(pp. 321-332)
8. Index
(pp. 333-336)