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Renormalization: general theory

I review the theory of renormalization, as applied to weak-coupling perturbation theory in quantum field theories.


I. INTRODUCTION
Quantum field theories (QFTs) provide a natural framework for quantum theories that obey the principles of special relativity. Among their most striking features are ultra-violet divergences, which at first sight invalidate the existence of the theories. The divergences arise from Fourier modes of very high wave number, and hence from the structure of the theories at very short distances. In the very restricted class of theories called "renormalizable", the divergences may be removed by a singular redefinition of the parameters of the theory. This is the process of renormalization, that defines a QFT as a nontrivial limit of a theory with a UV cut-off.
A very important QFT is the Standard Model, an accurate and successful theory for all the known interactions except gravity. Calculations using renormalization and related methods are vital to the theory's success.
The basic idea of renormalization predates QFT. Suppose we treat an observed electron as a combination of a bare electron of mass m 0 and the associated classical electromagnetic field down to a radius a. The observed mass of the electron is its bare mass plus the energy in the field (divided by c 2 ). The field energy is substantial, e.g., 0.7 MeV when a = 10 −15 m, and it diverges when a → 0. The observed mass, 0.5 MeV, is the sum of the large (or infinite) field contribution compensated by a negative and large (or infinite) bare mass. This calculation needs replacing by a more correct version for short distances, of course, but it remains a good motivation.
In this article, I review the theory of renormalization in its classic form, as applied to weak-coupling perturbation theory, or Feynman graphs. It is this method, rather than the Wilsonian approach reviewed elsewhere in this volume, that is typically used in practice for perturbative calculations in the Standard Model, especially its QCD part.
Much of the emphasis is on weak-coupling perturbation theory, where there are well-known algorithmic rules for performing calculations and renormalization. Applications -see the article on QCD and confinement for some important non-trivial examples -involve further related results, such as the operator product expansion, factorization theorems, and the renormalization group, to go far beyond simple fixed order perturbation theory. * To appear in the Encyclopedia of Mathematical Physics (Elsevier, 2006).
FIG. 1: One-loop approximation to connected and amputated 4-point function, before renormalization.
The construction of fully rigorous mathematical treatments for the exact theory is a topic of future research.

II. FORMULATION OF QFT
A QFT is specified by its Lagrangian density. A simple example is φ 4 theory: where φ(x) = φ(t, x) is a single component hermitian field. The Lagrangian density and the resulting equation of motion, ∂ 2 φ + m 2 φ + 1 6 λφ 3 = 0, are local; they involve only products of fields at the same space-time point. Such locality is characteristic of relativistic theories, where otherwise it is difficult or impossible to preserve causality, but it is also the source of the UV divergences. The question mark over the equality symbol in Eq. (1) is a reminder that renormalization of UV divergences will force us to modify the equation.
The Feynman rules for perturbation theory are given by a free propagator i/(p 2 − m 2 + i0) and a interaction vertex −iλ. Although we will usually work in four spacetime dimensions, it is useful also to consider the theory in a general space-time dimensionality n, where the coupling has energy dimension [λ] = E 4−n . We use "natural units", i.e., withh = c = 1. The "i0" in the propagator i/(p 2 − m 2 + i0) symbolizes the location of the pole relative to the integration contour; it is often written as iǫ.
The primary targets of calculations are the vacuum expectation values of time-ordered products of φ; in QFT these are called the Green functions of the theory. From these can be reconstructed the scattering matrix, scattering cross sections, and other measurable quantities.

III. ONE-LOOP CALCULATIONS
Low-order graphs for the connected and amputated 4point Green function are shown in Fig. 1. Each one-loop graph has the form where p is a combination of external momenta. There is a divergence from where the loop momentum k goes to infinity. We define the degree of divergence, ∆, by counting powers of k at large k, to get ∆ = 0. In an n-dimensional space-time we would have ∆ = n − 4. The integral is divergent whenever ∆ ≥ 0. Comparing the dimensions of the one-loop and tree graphs shows that ∆ equals the negative of the energy-dimension of the coupling λ. Thus the dimensionlessness of λ at the physical space-time dimension is equivalent to the integral being just divergent. The infinity in the integral implies that the theory in its naive formulation is not defined. With the aid of renormalization group methods, it has been shown that the problem is with the complete theory, not just perturbation theory.
The divergence only arises because we use a continuum space-time. So suppose that we formulate the theory initially on a lattice of spacing a (in space or space-time). Our loop graph is now where the free propagator S(k, m; a) approaches the usual value i/(k 2 − m 2 + i0) when k is much smaller than 1/a, and it falls off more rapidly for large k. The basic observation that propels the renormalization program is that the divergence as a → 0 is independent of p. This is most easily seen by differentiating once with respect to p, after which the integral is convergent when a = 0, because the differentiated integral has degree of divergence −1.
Thus we can cancel the divergence in Eq. (2) by replacing the coupling in the first term in Fig. 1, by the so-called bare coupling Here A(a) is chosen so that the renormalized value of our one-loop graph, exists, at a = 0, with A(a) in fact being real-valued. The factor 3 multiplying A(a) in Eq. (4) is because there are three one-loop graphs, with equal divergent parts. The replacement for the coupling is made in the tree graph in Fig. 1 because at the moment we are only doing a calculation accurate to order λ 2 ; the appropriate expansion parameter of the theory is the finite renormalized coupling λ, held fixed as a → 0. We call the extra term in Eq. (5) a counterterm. The diagrams for the correct renormalized calculation are represented in Fig. 2, which has a counterterm graph compared with Fig. 1.
In the physics terminology, used here, the cutting-off of the divergence by using a modified theory is called a regularization. This contrasts with the mathematics literature, where "regularized integral" usually means the same as a physicist's "renormalized integral".
There is always freedom to add a finite term to a counterterm. When we discuss the renormalization group, we will see that this corresponds to a reorganization of the perturbation expansion and provides a powerful tool for improving perturbatively based calculations, especially in QCD. Contrary to the impression given in some parts of the literature, it is not necessary that a renormalized mass equal a corresponding physical particle mass, with similar statements for coupling and field renormalization. While such a prescription is common and natural in a simple theory like QED, it is by no means required and certainly may not always be best. If nothing else, the correspondence between fields and stable particles may be poor or non-existent (as in QCD).
One classic possibility is to subtract the value of the graph at p = 0, a prescription associated with Bogoliubov, Parasiuk and Hepp (BPH), which leads to In obtaining this from (2), we used a standard Feynman parameter formula, to combine the propagator denominators, after which the integral over the momentum variable k is elementary. We then obtain the renormalized one-loop (4-point and amputated) Green function where s, t, and u are the three standard Mandelstam invariants for the Green function. [For a 2 → 2 scattering process, or a corresponding off-shell Green function, in which particles of momenta p 1 and p 2 scatter to particles of momenta p ′ 1 and p ′ 2 , the Mandelstam variables are defined as s = (p 1 +p 2 ) 2 , t = (p 1 −p ′ 1 ) 2 , and u = (p 1 −p ′ 2 ) 2 .] In the general case, with a nonzero degree of divergence, the divergent part of an integral is a polynomial in p and m of degree D, where D is the smallest positive integer less than or equal to ∆. In a higher space-time dimension, this implies that renormalization of the original, momentum-independent, interaction vertex is not sufficient to cancel the divergences. We would need higher derivative terms, and this is evidence that the theory is not renormalizable in higher than 4 space-time dimensions. Even so, the terms needed would be local, because of the polynomiality in p.

IV. COMPLETE FORMULATION OF RENORMALIZATION PROGRAM
The full renormalization program motivated by example calculations is: • The theory is regulated to cut off the divergences.
• The numerical value of each coefficient in L is allowed to depend on the regulator parameter (e.g., a).
• These dependences are adjusted so that finite results for Green functions are obtained after removal of the regulator.
In φ 4 theory, we therefore replace L by with the bare parameters, Z, m 0 and λ 0 , having a regulator dependence such that Green functions of φ are finite at a = 0. The slightly odd labeling of the coefficients in Eq. (9) arises because observables like cross sections are invariant under a redefinition of the field by a factor. In terms of the bare field φ 0 The unit coefficient of 1 2 (∂φ 0 ) 2 implies that φ 0 has canonical commutation relations (in the regulated theory). This provides a natural standard for the normalization of the bare mass m 0 and the bare coupling λ 0 .
All terms in L have coefficients with dimension zero or larger. This is commonly characterized by saying that the terms L "have dimension 4 or less", which refers to the products of field operators and derivatives in each term. A generalization of the power-counting analysis shows that if we start with a theory whose L only has terms of dimension 4 or less, then no terms of higher dimension are needed as counterterms, at least not in perturbation theory. This is a very powerful restriction on self-contained QFTs, and was critical in the discovery of the Standard Model.
Sometimes it is found that the description of some piece of physics appears to need higher dimension operators, as was the case originally with weak interaction physics. The lack of renormalizability of such theories indicates that they cannot be complete, and an upper bound on the scale of their applicability can be computed, e.g., a few hundred GeV for the four-fermion theory of weak interactions. Eventually this theory was superseded by the renormalizable Weinberg-Salam theory of weak interactions, now a part of the Standard Model, to which the four-fermion theory provides a low-energy approximation for charged current weak interactions.
Certain operators of allowed dimensions are missing in Eq. (9): the unit operator, and φ and φ 3 . Symmetry under the transformation φ → −φ implies that Green functions with an odd number of fields vanish, so that no φ and φ 3 counterterms are needed. Divergences with the unit operator do appear, but not for ordinary Green functions. In gravitational physics, the coefficient of the unit operator gives renormalization of the cosmological constant.
To implement renormalized perturbation theory, we partition L (non-uniquely) as where the free, the basic interaction and the counterterm Lagrangians are The renormalized coupling and mass, λ and m, are to be fixed and finite when the UV regulator is removed. Both the basic interaction and the counterterms are treated as interactions. First we compute "basic graphs" for Green functions using only the basic interaction. The counterterms are expanded in powers of λ, and then all graphs involving counterterm vertices at the chosen order in λ are added to the calculation. The counterterms are arranged to cancel all the divergences, so that the UV regulator can be removed, with m and λ held fixed. The counterterms cancel the parts of the basic Feynman graphs associated with large loop momenta. An algorithmic specification of the otherwise arbitrary finite parts of the counterterms is called a renormalization prescription or a renormalization scheme. Thus it gives a definite relation between the renormalized and bare parameters, and hence a definite specification of the partitioning of L into its three parts. It has been proved that this procedure works to all orders in λ, with corresponding results for other theories. Even in the absence of fully rigorous non-perturbative proofs, it appears clear that the results extend beyond perturbation theory, at least in asymptotically free theories like QCD: see the discussion of the Wilsonian RG elsewhere in this volume.

V. DIMENSIONAL REGULARIZATION AND MINIMAL SUBTRACTION
The final result for renormalized graphs does not depend on the particular regularization procedure. A particularly convenient procedure, especially in QCD, is dimensional regularization, where divergences are removed by going to a low space-time dimension n. To make a useful regularization method, n is treated as a continuous variable, n = 4 − 2ǫ.
Great advantages of the method are that it preserves Poincaré invariance and many other symmetries (including the gauge symmetry of QCD), and that Feynman graph calculations are minimally more complicated than for finite graphs at n = 4, particularly when all the lines are massless, as in many QCD calculations.
Although there is no such object as a genuine vector space of finite non-integer dimension, it is possible to construct an operation that behaves as if it were an integration over such a space. The operation was proved unique by Wilson, and explicit constructions have been made, so that consistency is assured at the level of all Feynman graphs. Whether a satisfactory definition beyond perturbation theory exists remains to be determined.
It is convenient to arrange that the renormalized coupling is dimensionless in the regulated theory. This is done by changing the normalization of λ with the aid of an extra parameter, the unit of mass µ: A natural renormalization procedure is to subtract the pole at ǫ = 0, but it is convenient to accompany this with other factors to remove some universally occurring finite terms. So MS renormalization ("modified minimal subtraction") is defined by using the counterterm where S ǫ def = (4πe −γE ) ǫ , with γ E = 0.5772 . . . being the Euler constant. This gives a renormalized integral (at which can be evaluated easily. A particularly simple result is obtained at m = 0: This formula symptomizes important and very useful algorithmic simplifications in the higher-order massless calculations common in QCD. The MS scheme amounts to a de facto standard for QCD. At higher orders a factor of S ǫ L is used in the counterterms, with L being the number of loops.

VI. COORDINATE-SPACE
Quantum fields are written as if they are functions of x, but they are in fact distributions or generalized functions, with quantum-mechanical-operator values. This indicates that using products of fields is dangerous and in need of careful definition. The relation with ordinary distribution theory is simplest in the coordinate-space version of Feynman graphs. Indeed in the 1950s, Bogoliubov and Shirkov formulated renormalization as a problem of defining products of the singular numeric-valued distributions in coordinate-space Feynman graphs; theirs was perhaps the best treatment of renormalization in that era.
For example, the coordinate-space version of Eq. (5) is where x and y are the coordinates for the interaction vertices, f (x, y) is the product of external-line free propagators, andS(x − y; m, a) is the coordinate-space free propagator, which at a = 0 has a singularity as (x − y) 2 → 0. We see in Eq. (20) a version of the Hadamard finite-part of a divergent integral, and renormalization theory generalizes this to particular kinds of arbitrarily high-dimension integrals. The physical realization and justification of the use of the finite-part procedure is in terms of renormalization of parameters in the Lagrangian; this also gives the procedure a significance that goes beyond the integrals themselves and involves the full non-perturbative formulation of QFT.

VII. GENERAL COUNTERTERM FORMULATION
We have written L as a basic Lagrangian density plus counterterms, and have seen in an example how to cancel divergences at one-loop order. In this section, we will see how the procedure works to all orders. The central mathematical tool is Bogoliubov's R-operation. Here the counterterms are expanded as a sum of terms, one for each basic 1PI graph with a non-negative degree of divergence. To each basic graph for a Green function is added a set of counterterm graphs associated with divergences for subgraphs. The central theorem of renormalization is that this procedure does in fact remove all the UV divergences, with the form of the counterterms being determined by the simple computation of the degree of divergence for 1PI graphs.
To see the essential difficulty to be solved, consider a two-loop graph like the first one in Fig. 3. Its divergence is not a polynomial in external momenta, and is therefore not canceled by an allowed counterterm. This is shown by differentiation with respect to external momenta, which does not produce a finite result because of the divergent one-loop subgraph. But for consistency of the theory, the one-loop counterterms already computed must be themselves put into loop graphs. Among others, this gives the second graph of Fig. 3, where the cross denotes that a counterterm contribution is used. The contribution used here is actually 2/3 of the total one-loop counterterm, for reasons of symmetry factors that are not fully evident at first sight. The remainder of the one-loop coupling renormalization cancels a subdivergence in another 2-loop graph. It is readily shown that the divergence of the sum of the first two graphs in Fig.  3 is momentum-independent, and thus can be canceled by a vertex counterterm.
This method is fully general, and is formalized in the Bogoliubov R-operation, which gives a recursive specification of the renormalized value R(G) of a graph G: The sum is over all sets of non-intersecting one-particleirreducible (1PI) subgraphs of G, and the notation G| γi→C(γi) denotes G with all the subgraphs γ i replaced by associated counterterms C(γ i ). The counterterm C(γ) Here T is an operation that extracts the divergent part of its argument and whose precise definition gives the renormalization scheme. For example, in minimal subtraction we define We formalize the term inside parentheses in Eq. (23) as: where the prime on the ′ denotes that we sum over all sets of non-intersecting 1PI subgraphs except for the case that there is a single γ i equal to the whole graph (i.e., the term with n = 1 and γ 1 = γ is omitted). Note that, for the MS scheme, we define the T operation to be applied to a factor of constant dimension obtained by taking the appropriate power of µ ǫ outside of the pole part operation. Moreover it is not a strict pole part operation; instead each pole is to be multiplied by S ǫ L , where L is the number of loops, and S ǫ is defined after Eq. (17).
Each counterterm C(γ) is implemented as a contribution to the counterterm Lagrangian. The Feynman rules ensure that once C(γ) has been computed, it appears as a vertex in bigger graphs in such a way as to give exactly the counterterms for subdivergences used in the R-operation. It has been proved that the R-operation does in fact give finite results for Feynman graphs, and that basic power-counting in exactly the same fashion as at one-loop determines the relevant operators.
In early treatments of renormalization, a problem was caused by graphs like Fig. 4. This graph has three divergent subgraphs which overlap, rather than being nested. Within the R-operation approach, such cases are no harder to deal with than merely nested divergences.
The recursive specification of R-operation can be converted to a non-recursive formulation by the forest formula of Zavyalov and Stepanov, later rediscovered by Zimmermann. It is normally the recursive formulation that is suited to all-orders proofs.
Whether these results, proved to all orders of perturbation theory, genuinely extend to the complete theory is not so easy to answer, certainly in a realistic 4-dimensional QFT. One illuminating case is of a non-relativistic quantum mechanics model with a deltafunction potential in a two-dimensional space. Renormalization can be applied just as in field theory, but the model can also be treated exactly and it has been shown that the results agree with perturbation theory.
Perturbation series in relativistic QFTs can at best be expected to be asymptotic, not convergent. So instead of a radius of convergence, we should talk about a region of applicability of a weak coupling expansion. In a direct calculation of counterterms, etc, the radius of applicability shrinks to zero as the regulator is removed. However we can deduce the expansion for a renormalized quantity, whose expansion is expected to have a nonzero range of applicability. We can therefore appeal to the uniqueness of power series expansions to allow the calculation, at intermediate stages, to use bare quantities that are divergent as the regulator is removed.

VIII. RENORMALIZABILITY, NON-RENORMALIZABILITY AND SUPER-RENORMALIZABILITY
The basic power counting method shows that if a theory with conventional fields (at n = 4) has only operators of dimension 4 or less in its L, then the necessary counterterm operators are also of dimension 4 or less. So if we start with a Lagrangian with all possible such operators, given the field content, then the theory is renormalizable. This is not the whole story, as we will see in the discussion of gauge theories.
If we start with a Lagrangian containing operators of dimension higher than 4, then renormalization requires operators of ever higher dimension as counterterms when one goes to higher orders in perturbation theory. Therefore, such a theory is said to be perturbatively nonrenormalizable. Some very powerful methods of cancellation or some non-perturbative effects are needed to evade this result.
In the case of dimension-4 interactions, there is only a finite set of operators given the set of basic fields, but divergences occur at arbitrarily high orders in perturbation theory. If, instead, all the operators have at most dimension 3, then only a finite number of graphs need counterterms. Such theories are called superrenormalizable. The divergent graphs also occur as subgraphs inside bigger graphs, of course. There is only one such theory in a 4-dimensional space-time: φ 3 theory, which suffers from an energy density that is unbounded from below, so it is not physical. In lower space-time dimension, where the requirements on operator dimension are different, there are many more known super-renormalizable theories, some with a very rigorous proof of existence.
All the above characterizations rely primarily on per-turbative analysis, so they are subject to being not quite accurate in an exact theory, but they form a guide to the relevant issues.

IX. RENORMALIZATION AND SYMMETRIES; GAUGE THEORIES
In most physical applications, we are interested in QFTs whose Lagrangian is restricted to obey certain symmetry requirements. Are these symmetries preserved by renormalization? That is, is the Lagrangian with all necessary counterterms still invariant under the symmetry?
We first discuss non-chiral symmetries; these are symmetries in which the left-handed and right-handed parts of Dirac fields transform identically.
For Poincaré invariance and simple global internal symmetries, it is simplest to use a regulator, like dimensional regularization, which respects the symmetries. Then it is easily shown that the symmetries are preserved under renormalization. This holds even if the internal symmetries are spontaneously broken [as happens with a "wrong-sign mass term", e.g., negative m 2 in Eq. (1)].
The case of local gauge symmetries is harder. But their preservation is more important, because gauge theories contain vector fields which, without a gauge symmetry, generally give unphysical features to the theory. For perturbation theory, BRST quantization is usually used, in which, instead of gauge symmetry, there is a BRST supersymmetry. This is manifested at the Green function level by Slavnov-Taylor identities that are more complicated, in general, than the Ward identities for simple global symmetries and for abelian local symmetries.
Dimensional regularization preserves these symmetries and the Slavnov-Taylor identities. Moreover the Roperation still produces finite results with local counterterms, but cancellations and relations occur between divergences for different graphs in order to preserve the symmetry. A simple example is QED, which has an abelian U(1) gauge symmetry, and whose gauge-invariant Lagrangian is At the level of individual divergent 1PI graphs, we get counterterms proportional to A µ 2 and to (A µ 2 ) 2 , operators not present in the gauge-invariant Lagrangian. The Ward identities and Slavnov-Taylor identities show that these counterterms cancel when they are summed over all graphs at a given order of renormalized perturbation theory. Moreover the renormalization of coupling and the gauge field are inverse, so that e 0 A (0) µ equals the corresponding object with renormalized quantities, µ ǫ eA µ . Naturally, sums of contributions to a counterterm in L can only be quantified with use of a regulator.
In non-abelian theories the gauge-invariance properties are not just the absence of certain terms in L but quantitative relations between the coefficients of terms with different numbers of fields. Even so, the argument with Slavnov-Taylor identities generalizes appropriately and proves renormalizability of QCD, for example. But note that the relation concerning the product of the coupling and the gauge field does not generally hold; the form of the gauge transformation is itself renormalized, in a certain sense.

X. ANOMALIES
Chiral symmetries, as in the weak-interaction part of the gauge symmetry of the Standard Model, are much harder to deal with. Chiral symmetries are ones for which the left-handed and right-handed components of Dirac field transform independently under different components of the symmetry group, local or global as the case may be. Sometimes, some or other of the left-handed or right-handed components may not even be present.
In general, chiral symmetries are not preserved by regularization, at least not without some other pathology. At best one can adjust the finite parts of counterterms such that in the limit of the removal of the regulator, the Ward or Slavnov-Taylor identities hold. But in general, this cannot be done consistently, and the theory is said to suffer from an anomaly. In the case of chiral gauge theories, the presence of an anomaly prevents the (candidate) theory from being valid. A dramatic and non-trivial result (Adler-Bardeen theorem and some nontrivial generalizations) is that if chiral anomalies cancel at the one-loop level, then they cancel at all orders.
Similar results, but more difficult ones, hold for supersymmetries.
The anomaly cancellation conditions in the Standard Model lead to constraints that relate the lepton content to the quark content in each generation. For example, given the existence of the b quark, and the τ and ν τ leptons (of masses around 4.5 GeV, 1.8 GeV, and zero respectively), it was strongly predicted on the grounds of anomaly cancellation that there must be a t quark partner of the b to complete the third generation of quark doublets. This prediction was much later vindicated by the discovery of the much heavier top quark with m t ≃ 175 GeV.

XI. RENORMALIZATION SCHEMES
A precise definition of the counterterms entails a specification of the renormalization prescription (or scheme), so that the finite parts of the counterterms are determined. This apparently induces extra arbitrariness in the results. However, in the φ 4 Lagrangian (for example), there are really only two independent parameters. (A scaling of the field does not affect any observables, so here we do not count Z as a parameter here.) Thus at fixed regulator parameter a or ǫ, renormalization actually just gives a reparameterization of a two-parameter collection of theories. A renormalization prescription gives the change of variables between bare and renormalized parameters, a rather singular transformation when the regulator is removed. If we have two different prescriptions, we can deduce a transformation between the renormalized parameters in the two schemes. The renormalized mass and coupling m 1 and λ 1 in one scheme can be obtained as functions of their values m 2 and λ 2 in the other scheme, with the bare parameters, and hence the physics, being the same in both schemes. Since these are renormalized parameters, the removal of the regulator leaves the transformation well behaved.
Generalization to all renormalizable theories is immediate.

XII. RENORMALIZATION GROUP AND APPLICATIONS AND GENERALIZATIONS
One part of the choice of renormalization scheme is that of a scale parameter such as the unit of mass µ of the MS scheme. The physical predictions of the theory are invariant if a change of µ is accompanied by a suitable change of the renormalized parameters, now considered as µ-dependent parameters λ(µ) and m(µ). These are called the effective, or running, coupling and mass. The transformation of the parameterization of the theory is called a renormalization-group (RG) transformation.
The bare coupling and mass λ 0 and m 0 are RGinvariant, and this can be used to obtain equations for the RG-evolution of the effective parameters from the perturbatively computed counterterms. For example, in φ 4 theory, we have (in the renormalized theory after removal of the regulator) dλ d ln µ 2 = β(λ), with β(λ) = 3λ 2 /(16π 2 ) + O(λ 3 ). As exemplified in Eqs.
(18) and (19), Feynman diagrams depend logarithmically on µ. By choosing µ to be comparable to the physical external momentum scale, we remove possible large logarithms in this and higher orders. Thus, provided that the effective coupling at this scale is weak, we get an effective perturbation expansion. This is a basic technique for exploiting perturbation theory in QCD, for the strong interactions, where the interactions are not automatically weak. In this theory the RG β function is negative so that the coupling decreases to zero as µ → ∞; this is the asymptotic freedom of QCD.
A closely related method is that associated with the Callan-Symanzik equation, which is a formulation of a Ward identity for anomalously broken scale invariance. However, RG methods are the actually-used ones, normally, even if sometimes a RG equation is incorrectly labeled as a Callan-Symanzik equation.
The elementary use of the RG is not sufficient for most interesting processes, which involve a set of widely different scales. Then more powerful theorems come into play. Typical are the factorization theorems of QCD, reviewed elsewhere. These express differential cross sections for certain important reactions as a product of quantities that involve a single scale: dσ = C(Q, µ, λ(µ)) ⊗ f (m, µ, λ(µ)) + small correction.
(28) The product is typically a matrix or a convolution product. The factors obey non-trivial RG equations, and these enable different values of µ to be used in the different factors. Predictions arise because some factors and the kernels of the RG equation are perturbatively calculable, with a weak effective coupling. Other factors, such as f in Eq. (28), are not perturbative. These are quantities with names like "parton distribution functions", and they are universal between many different processes. Thus the non-perturbative functions can be measured in a limited set of reactions and used to predict cross sections for many other reactions with the aid of calculations of the perturbative factors.
Ultimately this whole area depends on physical phenomena associated with renormalization.

XIII. CONCLUDING REMARKS
The actual ability to remove the divergences in certain QFTs to produce consistent, finite and non-trivial theories is a quite dramatic result. Moreover, associated with the integrals that give the divergences is behavior of the kind that is analyzed with renormalization-group methods and generalizations. So the properties of QFTs associated with renormalization get tightly coupled to many interesting consequences of the theories, most notably in QCD.
Quantum field theories are actually very abstruse and difficult theories; only certain aspects currently lend themselves to practical calculations. So the reader should not assume that all aspects of their rigorous mathematical treatment are perfect. Experience, both within the theories and in their comparison with experiment, indicates, nevertheless, that we have a good approximation to the truth.
When one examines the mathematics associated with the R-operation and its generalizations with factoriza-tion theorems, there are clearly present some interesting mathematical structures that are not yet formulated in their most general terms. Some indications of this can be seen in the work by Connes and Kreimer, reviewed in another article, where it is seen that renormalization is associated with a Hopf algebra structure for Feynman graphs.
With such a deep subject, it is not surprising that it lends itself to other approaches, notably the Connes-Kreimer one and the Wilsonian one, also reviewed in another article. Readers new to the subject should not be surprised if it is difficult to get a fully unified view of these different approaches.