# Algebraic renormalization

Post-publication activity

Curator: Olivier Piguet

Algebraic Renormalization deals with the perturbative renormalization of quantum Lagrangian field models with local or rigid symmetries. In short, the aim is to prove that the full perturbative quantum theory has the same invariance properties as the classical theory. Becchi, Rouet and Stora found a general iterative scheme (later on called algebraic renormalization) which reduces the proof of the preservation of symmetries at the level of the radiative corrections (or the determination of all possible breakings, the so-called 'anomalies') to the study of the cohomology of the underlying group. The algebraic renormalization scheme is based on general theorems of perturbative quantum field theory, such as the Power Counting Theorem and the Quantum Action Principle. The main applications of Algebraic Renormalization are in the proof of renormalizability of theories with broken rigid symmetries and of gauge theories. Its use is mostly indicated in cases where no known regularization preserving all the symmetries of the model under study is available.

## Introduction

Since the times of Einstein, Heisenberg, Weyl and Wigner, a physical system is basically defined by giving the set of variables which describe it and the symmetries its dynamics must obey (Gross DJ,1996). In Quantum Field Theory, this means giving first a set of local fields and their transformation laws under the action of the group of the symmetries of the system. One has then to establish the evolution equations - the field equations - of the theory, in such a manner that they are covariant under the symmetry transformations. In classical field theory, this is usually done by finding an action - a local functional of the classical fields of the theory - invariant under the symmetry group.

The scope of this note is Perturbative Quantum Field Theory (PQFT), a perturbative realization of General Quantum Field Theory (Haag R,1996). We consider here theories which, at the classical level, are expressed in the Lagrangian formalism. The main applications are "second order" theories such as the Yang-Mills theories, but "first order theories" such as topological theories may be treated as well (Piguet O and Sorella SP,1995).

Perturbation is understood as a formal expansion of the Green functions - vacuum expectation values of time ordered products of field operators - in the powers of the couplings or, more suitably for our general purpose, in the number of Feynman diagram loops. This number is equal to the order of the expansion in the powers of the Planck constant $$\hbar$$ (Itzykson C and Zuber J-B,1980). Thus the lowest order in $$\hbar$$ will correspond to the contribution of the tree Feynman graphs to the Green functions, whereas the loop graphs contributing to the higher orders in $$\hbar$$ give rise to the so-called radiative corrections.

In quantum field theory, the symmetries are expressed by a set of relations between the Green functions, called Ward identities (WI). In the tree approximation, these yield relations between the coefficients of the various field monomials which compose the most general action. At the level of the radiative corrections, the WI'es are equivalent to a set of relations between the coefficients which parametrize the renormalization ambiguities - the latter appearing as arbitrary finite counterterms in the action. In a renormalizable theory, there is a one-to-one correspondence between the parameters of the classical theory and the renormalization ambiguity parameters, and also a one-to-one correspondence between them and the parameters which are left free after the symmetry requirements are fulfilled. Moreover, in a physically meaningful theory, the number of the physical free parameters must be finite, of course, since they must be measurable by a finite number of experiments. Let us remark that gauge theories generally possess, beyond the physical parameters, non-physical gauge parameters among their free parameters. Since these parameters do not affect the observable output of the theory, their number may be infinite, see e.g. (Piguet O and Sibold K,1982; Bonneau G,1983).

Given a classical theory, with its group of symmetries depending on a finite number of parameters, the construction of a perturbative quantum theory obeying the corresponding WI'es is not always possible: in this case one speaks then of an anomalous theory.

In the cases where perturbation theory can be constructed using a regularization which preserves the symmetries - such as the renowned dimensional regularization used in non-abelian Yang-Mills gauge theories without axial couplings ('t Hooft G and Veltman M,1972), with minimal subtractions - the counterterms generated by the radiative corrections are automatically symmetric, their coefficients corresponding to a redefinition of the physical parameters, a process called multiplicative renormalization. One should also include in this category renormalization schemes such as BPHZL and Epstein-Glaser (see below), which do not use any regularization. Lorentz invariance, for instance, is explicitly preserved by these latter schemes.

There are however various cases where no invariant regularization is known, and others where it is indeed known that no such regularization exists. A well known example of the former is provided by the supersymmetric field theories (Piguet O and Sibold K,1986), whereas obvious examples of the latter are given by the theories which have been shown to be anomalous, such as the Gauge theories with axial couplings. In such a situation, one has to rely on the algebraic renormalization procedure (Piguet O and Sorella SP,1995), which essentially aims to show that the lack of symmetry of a regularization and/or renormalization scheme can be exactly compensated by the addition of non-symmetric counterterms to the action. The main tools of algebraic renormalization are:

- A corollary of the Weinberg Power-Counting Theorem (Weinberg S,1960; Zimmermann W,1968), which states that in $$D$$-dimensional QFT, the renormalization ambiguities can be expressed as finite counterterms to the action, of dimensions bounded by $$D$$ if the action defining the Feynman rules has the same dimensional bound. By the dimension of an action - or of any other integral of a local field polynomial - we mean the maximal dimension of the various terms of its integrand in mass units, within a system of units where Planck constant and light velocity are equal to 1. If the functional is a sum of terms, its dimension is defined as that of the highest dimension term.

- The Quantum Action Principle (Lowenstein JH,1971; Lam YMP,1972; Lam YMP,1973; Clark TE and Lowenstein JH,1976; F. Brennecke and M. Dütsch,2009), which states that any infinitesimal variation of the quantum effective action due to a variation of the fields which changes their dimension by the amount $$d$$ is equivalent to the insertion of an (integrated) local field polynomial of dimension bounded by $$D+d$$.

These statements are general results of perturbative renormalization theory, which were proved in the momentum space subtraction scheme of Bogoliubov, Parasiuk, Hepp, Zimmermann and Lowenstein (BPHZL), in the minimal dimensional regularization scheme (P. Breitenlohner and D. Maison,1977) and also in the configuration space scheme of Epstein and Glaser (Epstein H and Glaser V,1973).

These elements where already put in practice by Symanzik and Becchi in some particular examples, such as the O(N) scalar field model with linearly broken symmetry (Symanzik K,1970) and a Sigma Model with fermions (Becchi C,1973), where it was shown that it is possible to exactly compensate all the radiatively induced breakings of the symmetry transformation of the quantum effective action by adding corresponding non-invariant counterterms in the action, at each order in $$\hbar$$. A more complex situation was encountered in (Piguet O,1974), where some of the possible breaking terms could not be compensated by counterterms, and they had to be proved to vanish using a recursive consistency procedure.

Becchi, Rouet and Stora (BRS) then found, together with the discovery of the BRST symmetry, (Becchi C, Rouet A and Stora R,1975), (Tyutin,1975), (Becchi C, Rouet A and Stora R,1976), a general iterative scheme - later on called algebraic renormalization (Stora, R,1973), (Becchi C, Rouet A and Stora R,1975), (Becchi C, Rouet A and Stora R,1976), (Becchi C, Rouet A and Stora R,1981), (Piguet O and Sorella SP,1995). It is based on the two principles stated above. For a power-counting renormalizable model, at the level of the radiative corrections, it reduces the proof of the preservation of the symmetries or the determination of all possible anomalies to the solution of the cohomology of its symmetry group: trivial elements (co-boundaries) correspond to breakings which can be compensated by counterterms, whereas the non-trivial elements are the possible anomalies.

In short, the aim of the algebraic renormalization procedure is to prove that the full perturbative quantum theory has the same symmetry properties as the classical theory, i.e., to prove that the WI'es associated to the symmetries of the theory are satisfied to all orders of perturbation theory and, if this is not possible, to determine the anomalies. This will be explained with some more details in the rest of this note.

## The quantum action principle

In physics, one always formulates basic equations for the fundamental quantities. For a Quantum Field Theory, the fundamental quantities are the observables, field operators, $$S$$-matrix elements or, equivalently, the Green functions from which the observables can be reconstructed (Streater and Wightman,1964). The basic equation is the quantum action principle, a generalization of the classical action principle taking into account the quantum fluctuations, which tells us the response of the theory to (infinitesimal) variations of the external conditions: variations of parameters or external fields.

The quantum action principle can be represented by equations for the functional derivatives of the vertex functional $$\Gamma$$ with respect to external fields or parameters. The vertex functional or quantum effective action is defined by $\Gamma[\phi] = \sum_{N=1}^\infty\,\frac{1}{N!}\int dx_1\cdots dx_N\, \phi_{A_1}(x_1)\cdots \phi_{A_N}(x_N) \left\langle \hat\phi_{A_1}(x_1)\cdots \hat\phi_{A_N}(x_N)\right\rangle_{\rm 1PI}\,,$ where $$\left\langle \cdots \right\rangle_{\rm 1PI}$$ denotes the amputated one-particle irreducible (1PI) Feynman graph contributions to the Green functions $$\left\langle 0\right| T\, \hat\phi_{A_1}(x_1)\cdots \hat\phi_{A_N}(x_N) \left| 0\right\rangle$$ (Summation over repeated indices is assumed. $$dx$$ denotes the volume integration measure $$d^Dx$$ of $$D$$-dimensional space-time. Our conventions of units are $$c=\hbar=1$$ - keeping $$\hbar$$ explicit when counting the loop order.) The arguments of the functional $$\Gamma$$ are test functions (smooth functions) $$\phi_A$$ ($$A=1,\cdots,n$$) in one-to-one correspondence with the elementary quantum fields $$\hat\phi_A$$ of the theory. Perturbative expansion according to the number of loops writes as a formal power series in the order parameter $$\hbar$$, $\Gamma[\phi]=\sum_{n=0}^\infty \hbar^n \Gamma_n[\phi]\,.$ The zero loop order of the loop expansion is the so-called classical action $$\Sigma_{\rm class}$$$\Gamma_0[\phi] =: \Sigma_{\rm class}[\phi]\,.$

We define in the same way the insertion $$\Delta\cdot\Gamma[\phi]$$ as the generating functional of the amputated 1PI graph contributions to the Green functions $$\left\langle 0\right| T\, \hat\Delta\, \hat\phi_{A_1}(x_1) \cdots \hat\phi_{A_N}(x_N)\left| 0\right\rangle$$, where $$\hat\Delta$$ is a composite field operator, a quantum extension of a (possibly integrated) local functional $$\Delta$$ of the classical fields $$\phi_A$$, such that, at zeroth order$(\Delta\cdot\Gamma)_0 = \Delta\,.$ More generally one has, at any order, $\tag{1} \Delta\cdot\Gamma=\Delta + {\mathcal O}(\hbar^n\,\Delta)\,\,.$

Let us consider an infinitesimal field variation $$\delta\phi_A(x)=P_A[\phi](x)$$, where $$P_A[\phi](x)$$ are local functionals of dimensions $$d+d_{\phi_A}$$, and let us couple it to a classical, i.e., non-dynamical, source field $$\rho^A(x)$$, adding to the classical action a term $$\Sigma_{\rm source} = \int dx\, \rho^A P_A\phi$$. The total action $$\Sigma=\Sigma_{\rm class}+\Sigma_{\rm source}$$ then gives rise to a $$\rho$$-dependent effective action $$\Gamma[\phi,\rho]$$. It is clear that the variation of the classical action $$\Sigma_{\rm class}$$ under this field variation is given by $\tag{2} \delta\Sigma_{\rm class}[\phi] = \left. \int dx \frac{\delta\Sigma}{\delta\rho^A(x)} \frac{\delta\Sigma}{\delta\phi_A(x)}\right\vert_{\rho=0}\,,$ the right-hand side being a local functional of dimension $$d+D$$.

The Quantum Action Principle (QAP) generalizes this obvious statement to the full quantum theory: $\tag{3} \int dx \frac{\delta\Gamma[\phi,\rho]}{\delta\rho^A(x)} \frac{\delta\Gamma[\phi,\rho]}{\delta\phi_A(x)} = \Delta\cdot\Gamma[\phi,\rho]\,,$ where $$\Gamma[\phi,\rho] = \Sigma+{\mathcal O}(\hbar)$$, and $$\Delta$$ is a local insertion of dimension $$d+D$$ reducing to the right-hand side of (2) at the zero-loop ($$\hbar=0$$) order. This result includes the cases where terms non-linear in the source fields $$\rho^A$$ are present in the action. This happens when the algebra of the infinitesimal symmetry generators closes only on shell, i.e., in the classical theory, modulo field equations (Batalin IA and Vilkovisky GA,1981).

In the case of linear field variations $$\delta\phi_A(x)=(\alpha_A{}^B\phi_B(x)+\beta_A)$$, there is no need of the external fields $$\rho^ A$$, and (3) reads $\tag{4} \int dx \left(\alpha_A{}^B\phi_B(x)+\beta_A\right) \frac{\delta\Gamma[\phi]}{\delta\phi_A(x)} = \Delta\cdot\Gamma[\phi]\,,$ where again the dimension of $$\Delta$$ is controlled by the dimension of the left-hand side.

Finally, if the variation of the effective action is caused by the variation of a parameter $$p$$ of the theory, we have $\tag{5} \frac{\partial\Gamma[\phi]}{\partial p} = \Delta\cdot\Gamma[\phi]\,,$ where the dimension of $$\Delta$$ is equal to $$D$$, its zeroth order being equal to $$\partial\Sigma_{\rm class}/\partial p$$. We note that, although $$\partial\Sigma_{\rm class}/\partial p$$ may have a dimension $$<D$$ (e.g., for $$p$$ being a mass parameter) the quantum insertion in the r.h.s. of (5) has dimension $$D$$. This is in fact the source of the scale anomaly, expressed through the renormalization group and Callan-Symanzik equations (Lowenstein JH,1971, (Piguet O and Sorella SP,1995).

## Classical Ward identities and stability

In theoretical physics the Noether theorem plays a fundamental role in expressing the equivalence between two basic concepts of physics, namely that of symmetry and that of conservation law. In the classical theory, a symmetry - some of these symmetries may be supersymmetries - is expressed by the covariance of the field equations or, equivalently, by the invariance of the action under a group of rigid (global) transformations of the dynamical variables. On the other hand local or gauge symmetries are needed, in the so-called gauge theories, for their physical interpretation in terms of a well defined quantum theory. In both cases, local or rigid, the symmetry is expressed by a set of relations between the Green functions of the theory, known as Ward identities.

All the symmetries - rigid or local - may be expressed by a nilpotent BRST operator $$s$$ acting on the fields as $\tag{6} s\phi^\alpha = P_I{}^\alpha[\phi]c^I\,,\qquad s c^I =f_{JK}{}^I c^Jc^K\,, \qquad \mbox{with}\ s^2=0\,,$ where the $$\phi^\alpha$$ are the gauge and matter fields of the theory, the $$P_I{}^\alpha$$ partial derivative operators whose coefficients are local polynomials of the fields $$\phi$$ and their derivatives, and the $$c^A$$ the Faddeev-Popov ghosts. $$c^A$$ corresponds to the generator $$T_A$$ of the (super)symmetry group and has its statistics opposed to one of that generator, and the $$f_{IJ}{}^K$$ are the structure constants. Note that one can generalize to an infinitesimal generator algebra which closes with field dependent $$f_{IJ}{}^K$$, and also to the case where this algebra only closes modulo field equations ("closing on shell"). of the (super)group. $$c^A$$ is a constant or local field, depending on the rigid or local nature of the corresponding symmetry. The first equality may be interpreted as the infinitesimal symmetry transformations, the ghosts playing the role of the infinitesimal parameters.

Joining together the gauge and matter fields with the ghosts into the set of fields $$\phi_A$$, we can rewrite (6) as $\tag{7} s\phi_A = P_A[\phi]\,,\quad \qquad \mbox{with}\ s^2=0\,,$ with $$P_A$$ being local field polynomials. Denoting by $$\Sigma_{\rm class}$$ the classical action - e.g. a gauge invariant Yang-Mills action with gauge invariant couplings to matter fields plus a gauge-fixing term of the form of a BRST variation - and adding terms coupling the BRST variations (7) with external fields $$\rho^A$$, we get the total action $\tag{8} \Sigma=\Sigma_{\rm class} + \int dx\, \rho^A P_A\,.$

This action is invariant under a discrete U(1) symmetry corresponding to ghost number conservation. Ghost number $$g$$ is defined by setting $$g=0$$ for the gauge and matter fields, $$g=1$$ for the ghost fields $$c^I$$, the total ghost number of the action owing to be equal to 0.

Due to the nilpotency of $$s$$ and the hypothesis that the external fields $$\rho^A$$ are $$s$$-invariant, the action (8) is BRST invariant. The classical Ward identity which expresses this invariance takes the form of a Slavnov-Taylor identity: $\tag{9} {\mathcal S}(\Sigma)= 0\,,\quad\mbox{where}\quad {\mathcal S}(F) := \int dx\, \frac{\delta F}{\delta\rho^A(x)} \frac{\delta F}{\delta\phi_A(x)}\,.$

At the quantum level, these identities imply relations between the divergences of Green functions and thus between the counterterms which render the theory finite. From these relations one derives the generic form of the counterterms. Such analysis is based on a loop-wise expansion of perturbation theory. However, before entering this discussion, a study of the stability of the classical action should be performed. In this study we are interested in determining the most general classical action compatible with the power-counting and the symmetries of the theory. In order to do so, one perturbs the total classical action $$\Sigma$$ by an arbitrary integrated local $$\tilde{\Sigma}$$: $\tag{10} \widehat{\Sigma}=\Sigma + \varepsilon\,\tilde{\Sigma}\,\,,$ where $$\varepsilon$$ is an infinitesimal parameter and the integrated local functional $$\tilde{\Sigma}$$ has the same quantum numbers (dimension, discrete symmetries) as the classical action. Then, requiring that the perturbed action obeys the Slavnov-Taylor identity at order $$\varepsilon$$: $\tag{11} \int dx\, \frac{\delta(\Sigma+\varepsilon\,\tilde{\Sigma})}{\delta\rho^A(x)} \frac{\delta(\Sigma+\varepsilon\,\tilde{\Sigma})}{\delta\phi_A(x)}= {\mathcal O}(\varepsilon^2)\,,$ implies, in virtue of (9), the linearized Ward identity $\tag{12} {\mathcal S}_\Sigma\,\tilde{\Sigma}[\phi]=0\,,$ where we have defined the linearized Slavnov-Taylor operator $\tag{13} {\mathcal S}_F := \int dx\,\Biggl( \frac{\delta F}{\delta\rho^A(x)} \frac{\delta}{\delta\phi_A(x)} + \frac{\delta F}{\delta\phi_A(x)} \frac{\delta}{\delta\rho^A(x)}\Biggr)\,,$ which obeys the two identities $\tag{14} {\mathcal S}_F\,{\mathcal S}(F) = 0\quad \mbox{and} \quad {\mathcal S}_F^2=0\quad\mbox{if}\quad {\mathcal S}(F)=0\,.$

Two important points should be emphasized. Firstly, for the quantum theory the stability corresponds to the fact that the radiative corrections - the Ward identities being supposed to hold at this stage - can be reabsorbed by a redefinition of the initial parameters of the theory. Second, in the case in which there exists an invariant regularization scheme, the stability requirement is a necessary and sufficient condition to ensure the renormalizability of the theory. Otherwise, i.e., in a more general situation, if no invariant regularization procedure is available, the stability becomes only a necessary condition, and one has to the sure that the symmetries of the theory are anomaly free.

## Quantum Ward identities and anomalies

The purpose is to show - if possible - that the effective action $$\Gamma(\phi,\rho)=\Sigma(\phi,\rho)+{\mathcal O}(\hbar)$$ obeys the same Slavnov-Taylor identity, $\tag{15} {\mathcal S} (\Gamma) =0\,,$ supposed to hold for $$\Sigma$$, in the tree approximation, and given by (9).

### The inductive proof

The procedure is inductive: supposing the identity to hold at order $$n-1$$ in the loop expansion, one has to show that it holds at order $$n$$. One denotes by $$\Gamma^{(n-1)}$$ the solution of (15) up to the order $$n-1$$, constructed with the Feynman rules originating from a local action $$\Sigma^{(n-1)}$$ and the use of a well defined renormalization procedure, such as BPHZ, Dimensional Regularization with Minimal Subtractions, Epstein-Glaser, etc. The action $$\Sigma^{(n-1)}$$ has the same form as the total classical action $$\Sigma$$, but its coefficients are power series in $$\hbar$$, determined up to the order $$n-1$$. Applying the QAP (3) to the solution $$\Gamma^{(n-1)}$$ yields ${\mathcal S} (\Gamma^{(n-1)}) = \hbar^n\Delta\cdot\Gamma^{(n-1)} = \hbar^n\Delta + {\mathcal O}(\hbar^{n+1})\,,$ where the second equality follows from (1). $$\Delta$$ is a priori the most general in\-teg\-rat\-ed local functional, of dimension $$D$$ and ghost number 1. It may be restricted by other conditions, such as Lorentz symmetry or any other symmetry which is respected by the renormalization procedure. Let us call $${\mathcal F}_g$$ the space of local functionals thus defined, of dimension $$D$$, but with arbitrary ghost number $$g$$. $$\Delta$$ is a generic element of $${\mathcal F}_1$$.

However, applying the first identity in (14) to the functional $$\Gamma^{(n-1)}$$, one finds ${\mathcal S}_{\Gamma^{(n-1)}} \Delta\cdot\Gamma^{(n-1)}=0\,,$ where $${\mathcal S}_{\Gamma^{(n-1)}}$$ is the linearized Slavnov-Taylor operator (13) associated to $$\Gamma^{(n-1)}$$. At the lowest non-vanishing order, equal to $$n$$, the last equation leads to the BRST consistency condition $\tag{16} b\Delta = 0\,,$ where $$b := {\mathcal S}_\Sigma$$ is the zeroth order linearized Slavnov-Taylor operator (13), which is nilpotent, $$b^2=0$$, due to the second identity in (14). To solve (16) is a cohomology problem for the co-boundary operator $$b$$. The general solution of (16) may be written as $\tag{17} \Delta = {\mathcal A} + b\hat\Delta\,.$ $$\hat\Delta$$ is an arbitrary element of $${\mathcal F}_0$$, and $${\mathcal A}$$ an arbitrary element of the cohomology of $$b$$ in the space $${\mathcal F}_1$$, i.e., an element which satisfies the cocycle condition $$bA=0$$, but is not a co-boundary: it cannot be written as $$b\hat\Delta$$ for any $$\hat\Delta\in F_0$$.

Now, let us suppose for a while that $${\mathcal A}=0$$. Then the new action $\tag{18} \Sigma^{(n)} = \Sigma^{(n-1)} - \hbar^n \hat\Delta$ leads to the new effective action $\Gamma^{(n)} = \Gamma^{(n-1)} - \hbar^n \hat\Delta + {\mathcal O}(\hbar^{n+1})\,,$ satisfying the Slavnov-Taylor identity to order $$n$$: ${\mathcal S} (\Gamma^{(n)}) = {\mathcal O}(\hbar^{n+1})\,.$ This concludes the recursive proof of the Slavnov-Taylor identity to all orders, valid under the condition of never meeting a cohomologically non-trivial term $${\mathcal A}$$ at any step of the induction.

It is important to note that the non-invariant counterterm $$\hat\Delta$$ added to the action at each order as shown in Eq. (18) in order to reestablish the symmetry is defined up to an invariant: $\hat\Delta \to \hat\Delta +\Delta_{\rm inv}\,,$ where $$\Delta_{\rm inv}$$ is the general solution of $\tag{19} b\Delta=0$ of dimension 4 and ghost number 0. Denoting by $$\{\Delta_i\}$$ a basis of such invariants, one has $\Delta_{\rm inv} = \sum_i c_i \Delta_i\,,$ with arbitrary coefficients $$c_i$$. These arbitrary coefficients must be fixed in such a way that they hold order by order in perturbation theory by a nonsingular system of normalization conditions. The fixation of these coefficients by a nonsingular set of normalization conditions is analogous to the process of fixing the free constants that appear in the solution of a differential equation.

Since Eq. (19) is the same as Eq. (12), the basis $$\{\Delta_i\}$$ is the same as the one which spans the possible fluctuations of the classical action, and thus the coefficients $$c_i$$ correspond to the renormalization of the parameters of the classical theory.

To summarize, in a renormalizable theory, the power-counting together with its symmetries assure its stability under the radiative corrections.

### Gauge anomalies

The situation described in the preceding Section is necessarily realized in all theories which admit an invariant regularization, such as the gauge theories in four-dimensional space-time without coupling involving the Dirac $$\gamma^5$$ matrix.

Other theories may admit cohomologically non-trivial breaking terms $$\mathcal A$$ in the general solution (17) of the consistency condition (16), called gauge anomalies or consistent anomalies, and a careful examination of their coefficients is necessary in order to know if they are really present. For instance, for the gauge theories in four dimensions, where the consistency condition (16) takes the form of the Wess-Zumino consistency condition (Wess and Zumino,1971), the most general gauge anomaly is the Adler-Bardeen-Bell-Jackiw (ABBJ) anomaly which, for a simple Lie group $$G$$, reads $\tag{20} {\mathcal A}=r\, \epsilon_{\mu\nu\rho\sigma} \int dx\,c_{Ia}\partial^\mu\left( d^{abc} \partial^\nu A_b^\rho A_c^\sigma + \frac{D^{abcd}}{12}A_b^\nu A_c^\rho A_d^\sigma \right)\ ,$ where $D_{abcd} = d_{{\ }ab}^{n}f_{ncd} + d_{{\ }ac}^{n}f_{ndb} + d_{{\ }ad}^{n}f_{nbc} \ ,$ and $$d_{abc}$$ is the totally symmetric invariant tensor of rank 3 defined by $$d_{(abc)}= \frac{1}{2} {\rm Tr} ( \tau_a\{\tau_b,\tau_c\} )$$. For a general Lie group, there is one such term for each simple group factor, but we shall keep the case of a simple group for simplicity. The coefficient $$r$$ in (20) is a calculable quantity which depends on the parameters of the theory. Its value in the one-loop approximation is easy to calculate and depends only on which representations of the gauge group are chosen in order to describe the fermionic content of the theory. The representations may be chosen in such a way that this one-loop coefficient vanishes. The Bardeen non-renormalization theorem (Adler SL and Bardeen WA,1969) (Piguet O and Sorella SP,1995) then ensures that the anomaly coefficient vanishes at all orders of perturbation theory. A relevant example of such an anomaly cancellation is provided by the Standard Model of particle interactions (Bouchiat, Iliopoulos and Meyer,1972), (Kraus E,1998), where the gauge group is U(1)$$\times$$SU(2)$$\times$$SU(3): there are three terms like (20), hence three coefficients $$r$$, which vanish due to an exact cancellation between the lepton and quark contributions.

### Consistent versus covariant anomalies

In a Quantum Field Theory, anomalies appear as the quantum breakdown of invariance properties which are present in the classical theory. The anomalies can be divided into mainly two groups: the covariant anomalies and the consistent anomalies. The latter are the gauge anomalies discussed above. The former type of anomaly consists in the quantum breaking of the conservation laws of some currents. They are called covariant, in the context of gauge theories, since these currents, which correspond to the conservation of physical charges, are gauge invariant. As opposed to the gauge anomalies, which must be absent in order to preserve the physical interpretation of the theory, the presence of covariant anomalies does not hurt the theory. To the contrary, they may lead to physical, observable effects. An important example is provided, in gauge theories of massless fermions, by the Axial Anomaly, which breaks the conservation law of gauge invariant axial currents, which at the classical level are the conserved Noether currents of some axial symmetry. Its physical consequence is a non-vanishing amplitude for the decay $$\pi^o \to \gamma\gamma$$ - the decay of a neutral pion to a pair of photons. More details about covariant anomalies can be found in Axial anomaly.

On the other hand, the presence of a consistent anomaly in a gauge theory reveals a conflict between renormalizability and unitarity, i.e., a theory with consistent anomalies is inconsistent from a perturbative point of view. These anomalies are characterized by the fact that they satisfy a consistency condition - the Wess-Zumino consistency condition (Wess and Zumino,1971) - expressed in the BRST context by the condition (16).