View source for Algebraic renormalization

<strong><nowiki>Algebraic Renormalization</nowiki></strong> 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 
<strong><nowiki>Power Counting Theorem</nowiki></strong> and
the <strong><nowiki>Quantum Action Principle</nowiki></strong>. The main applications of Algebraic Renormalization
are in the proof of renormalizability of <strong><nowiki>theories with broken rigid symmetries</nowiki></strong> and of
<strong><nowiki>gauge theories</nowiki></strong>. 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|(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|(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 [[#pigsor|(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 
<math>\hbar</math> [[#itzu|(Itzykson C and Zuber J-B,1980)]]. Thus the lowest order 
in <math>\hbar</math>  will correspond to the contribution of the tree
Feynman graphs to the Green functions, whereas the loop graphs
contributing to the higher orders in <math>\hbar</math> 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.
([[#pigsib|Piguet O and Sibold K,1982]];
[[#bonneau|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
[[#thooftveltman|('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 <strong><nowiki>BPHZL</nowiki></strong> and 
<strong><nowiki>Epstein-Glaser</nowiki></strong> (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
[[#pigsib1|(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|Gauge theories]] with [[Axial_anomaly|axial couplings]].
In such a situation, one has to rely on the ''algebraic renormalization'' 
procedure
[[#pigsor|(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|Weinberg S,1960]];
[[#zim-p-counting|Zimmermann W,1968]]),
which states that in <math>D</math>-dimensional QFT, the renormalization 
ambiguities can be  expressed as finite counterterms to the action, of 
dimensions bounded by <math>D</math> 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''
([[#QAP-BPHZL|Lowenstein JH,1971]];
[[#QAP-BPHZL1|Lam YMP,1972]];
[[#QAP-BPHZL11|Lam YMP,1973]];
[[#QAP-BPHZL2|Clark TE and Lowenstein JH,1976]];
[[#QAP-Epstein-Glaser|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
<math>d</math>  is equivalent to the insertion of an (integrated) local
field polynomial of dimension bounded by <math>D+d</math>.

These statements are general results of perturbative renormalization theory, 
which were proved in the 
[[Bogoliubov-Parasiuk-Hepp-Zimmermann_renormalization_scheme|momentum space subtraction 
scheme of Bogoliubov, Parasiuk, Hepp, Zimmermann and Lowenstein (BPHZL)]],
in the minimal dimensional 
regularization scheme
[[#breit-maison|(P. Breitenlohner and D. Maison,1977)]]
and also in the configuration space scheme of Epstein and 
Glaser
[[#epstein-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|(Symanzik K,1970)]] 
and a Sigma Model with fermions
[[#becchi73|(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 <math>\hbar</math>.
A more complex situation was encountered in
[[#piguet-cmp|(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 [[Becchi-Rouet-Stora-Tyutin_symmetry|BRST symmetry]], 
[[#BRS1|(Becchi C, Rouet A and Stora R,1975)]], 
[[#Tyutin|(Tyutin,1975)]],
[[#BRS2|(Becchi C, Rouet A and Stora R,1976)]], 
a general iterative scheme - ''later on called algebraic renormalization'' 
[[#Stora-Houches|(Stora, R,1973)]],
[[#BRS1|(Becchi C, Rouet A and Stora R,1975)]], 
[[#BRS2|(Becchi C, Rouet A and Stora R,1976)]], 
[[#BRS3|(Becchi C, Rouet A and Stora R,1981)]], 
[[#pigsor|(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, 
<math>S</math>-matrix elements or, equivalently, the Green functions from which 
the observables can be reconstructed [[#PCT|(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 <math>\Gamma</math> with respect 
to external fields or parameters.
The [[Bogoliubov-Parasiuk-Hepp-Zimmermann renormalization scheme|''vertex functional'' 
or ''quantum effective action'']] is defined by
:<math>
\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}\,,
</math>
where <math>\left\langle \cdots \right\rangle_{\rm 1PI}</math>
denotes the amputated one-particle irreducible (1PI)
Feynman graph contributions to the Green functions
<math>\left\langle 0\right| T\, \hat\phi_{A_1}(x_1)\cdots \hat\phi_{A_N}(x_N)
\left| 0\right\rangle</math> (Summation over repeated indices is assumed. 
<math>dx</math> denotes the volume integration measure <math>d^Dx</math> of <math>D</math>-dimensional space-time.
Our conventions of units are <math>c=\hbar=1</math> - keeping <math>\hbar</math>
explicit when counting the loop order.)
The arguments of the functional <math>\Gamma</math> are test functions 
(smooth functions) <math>\phi_A</math> (<math>A=1,\cdots,n</math>) in 
one-to-one correspondence with the elementary quantum fields <math>\hat\phi_A</math>
of the theory.
Perturbative expansion according to the number of loops writes as a formal
power series in the order parameter <math>\hbar</math>,
:<math>
\Gamma[\phi]=\sum_{n=0}^\infty \hbar^n \Gamma_n[\phi]\,.
</math>
The zero loop order of the loop expansion is the so-called ''classical action'' 
<math>\Sigma_{\rm class}</math>:
<math>
\Gamma_0[\phi] =: \Sigma_{\rm class}[\phi]\,.
</math>

We define in the same way the ''insertion'' 
<math>\Delta\cdot\Gamma[\phi]</math> as the 
generating functional of the amputated 1PI graph contributions to the 
Green functions  
<math>\left\langle 0\right| T\, \hat\Delta\, \hat\phi_{A_1}(x_1)
\cdots \hat\phi_{A_N}(x_N)\left| 0\right\rangle</math>,
where <math>\hat\Delta</math> is a composite field operator, a quantum extension
of a (possibly integrated) local functional <math>\Delta</math> of the classical fields 
<math>\phi_A</math>, such that, at zeroth order:
<math>
(\Delta\cdot\Gamma)_0 = \Delta\,.
</math>
More generally one has, at any order, 
:<math>\label{deltagamma}
\Delta\cdot\Gamma=\Delta + {\mathcal O}(\hbar^n\,\Delta)\,\,.
</math>

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

The ''Quantum Action Principle'' (QAP) generalizes this obvious statement to the
full quantum theory: 
:<math>\label{QAP}
\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]\,,
</math>
where <math>\Gamma[\phi,\rho] = \Sigma+{\mathcal O}(\hbar)</math>, and
<math>\Delta</math> is a local insertion of dimension <math>d+D</math> reducing to 
the right-hand side of \eqref{QAP-classical} at the zero-loop (<math>\hbar=0</math>)
order. This result includes the cases where terms non-linear in the source
fields <math>\rho^A</math> 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
[[#bat-vilk|(Batalin IA and Vilkovisky GA,1981)]].

In the case of linear field variations 
<math>\delta\phi_A(x)=(\alpha_A{}^B\phi_B(x)+\beta_A)</math>,
there is no need of the external fields <math>\rho^ A</math>, and \eqref{QAP} reads
:<math>\label{QAP-lin}
\int dx \left(\alpha_A{}^B\phi_B(x)+\beta_A\right)
\frac{\delta\Gamma[\phi]}{\delta\phi_A(x)} 
= \Delta\cdot\Gamma[\phi]\,,
</math>
where again the dimension of <math>\Delta</math> 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 <math>p</math> of the theory, we have
:<math>\label{QAP-param}
\frac{\partial\Gamma[\phi]}{\partial p} = \Delta\cdot\Gamma[\phi]\,,
</math>
where the dimension of <math>\Delta</math> is equal to <math>D</math>, its zeroth order
being equal to <math>\partial\Sigma_{\rm class}/\partial p</math>. 
We note that, although <math>\partial\Sigma_{\rm class}/\partial p</math> may have a dimension <math><D</math>
(e.g., for <math>p</math> being a mass parameter) the quantum insertion in the r.h.s. of \eqref{QAP-param} has dimension
<math>D</math>. This is in fact the source of the scale anomaly, expressed through the renormalization group and Callan-Symanzik 
equations ([[#QAP-BPHZL|Lowenstein JH,1971]], [[#pigsor|(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 <math>s</math> acting on the fields as
:<math>\label{brst-transf}
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\,,
</math>
where the <math>\phi^\alpha</math> are the gauge and matter fields of the
theory, the <math>P_I{}^\alpha</math> partial derivative operators whose coefficients
are local polynomials of the fields <math>\phi</math> and their derivatives,
and the <math>c^A</math> the Faddeev-Popov ghosts.  <math>c^A</math> corresponds to 
the generator <math>T_A</math> of the (super)symmetry group and has its 
statistics opposed to  one of that generator, and the <math>f_{IJ}{}^K</math> are
the structure constants.  Note that one can generalize to an 
infinitesimal generator algebra which closes with field dependent
<math>f_{IJ}{}^K</math>, and  also to the case where this algebra only closes modulo
field equations ("closing on shell").
of the (super)group.  <math>c^A</math> 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 <math>\phi_A</math>, we can rewrite 
\eqref{brst-transf} as
:<math>\label{brst-transf'}
s\phi_A =  P_A[\phi]\,,\quad \qquad \mbox{with}\ s^2=0\,,
</math>
with <math>P_A</math> being local field polynomials.
Denoting by <math>\Sigma_{\rm class}</math> 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
\eqref{brst-transf'} with external fields <math>\rho^A</math>, 
we get the total action
:<math>\label{total-action}
\Sigma=\Sigma_{\rm class} + \int dx\, \rho^A P_A\,.
</math>

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

Due to the nilpotency of <math>s</math> and the hypothesis that the external fields 
<math>\rho^A</math> are <math>s</math>-invariant, the action \eqref{total-action}
is BRST invariant. The classical Ward identity which expresses this invariance takes
the form of a Slavnov-Taylor identity:
:<math>\label{class-slavnov}
{\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)}\,.
</math>

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 <math>\Sigma</math> by an arbitrary integrated local
<math>\tilde{\Sigma}</math>: 
:<math>\label{stabe}
\widehat{\Sigma}=\Sigma + \varepsilon\,\tilde{\Sigma}\,\,,
</math>
where <math>\varepsilon </math> is an infinitesimal parameter and
the integrated local functional <math>\tilde{\Sigma}</math> 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
<math>\varepsilon</math>:
:<math>\label{QAP-classical1}
\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)\,,
</math>
implies, in virtue of \eqref{class-slavnov}, the 
''linearized Ward identity''
:<math>\label{stab-eq}
{\mathcal S}_\Sigma\,\tilde{\Sigma}[\phi]=0\,,
</math>
where we have defined the linearized Slavnov-Taylor operator
:<math>\label{lin-slavnov}
{\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)\,, 
</math>
which obeys the two identities
:<math>\label{s-identities}
{\mathcal S}_F\,{\mathcal S}(F) = 0\quad \mbox{and} \quad
{\mathcal S}_F^2=0\quad\mbox{if}\quad {\mathcal S}(F)=0\,.
</math>


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 
<math>\Gamma(\phi,\rho)=\Sigma(\phi,\rho)+{\mathcal O}(\hbar)</math> 
obeys the same Slavnov-Taylor identity,
:<math>\label{Q-slavnov}
{\mathcal S} (\Gamma) 
=0\,,
</math>
supposed to hold for <math>\Sigma</math>, in 
the tree approximation, and given by \eqref{class-slavnov}.

===The inductive proof===

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

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

Now, let us suppose for a while that <math>{\mathcal A}=0</math>. Then the new action
:<math>\label{non-inv-ct}
\Sigma^{(n)} = \Sigma^{(n-1)} - \hbar^n \hat\Delta
</math>
leads to the new effective action
:<math>
\Gamma^{(n)} = \Gamma^{(n-1)} - \hbar^n \hat\Delta + {\mathcal O}(\hbar^{n+1})\,,
</math>
satisfying the Slavnov-Taylor identity to order <math>n</math>:
:<math>
{\mathcal S} (\Gamma^{(n)}) = {\mathcal O}(\hbar^{n+1})\,.
</math>
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 <math>{\mathcal A}</math> at any step of the
induction. 

It is important to note that the  non-invariant counterterm <math>\hat\Delta</math>
added to the action at each order as shown in Eq. \eqref{non-inv-ct} in order to
reestablish the symmetry is defined up to an invariant:
:<math>
\hat\Delta \to \hat\Delta +\Delta_{\rm inv}\,,
</math>
where <math>\Delta_{\rm inv}</math> is the general solution of 
:<math>\label{inv-Delta}
b\Delta=0
</math>
of dimension 4 and 
ghost number 0. Denoting by <math>\{\Delta_i\}</math> a basis of such invariants, one has
:<math>
\Delta_{\rm inv} = \sum_i c_i \Delta_i\,,
</math>
with arbitrary coefficients <math>c_i</math>. 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. \eqref{inv-Delta} is the same as 
Eq. \eqref{stab-eq}, the basis <math>\{\Delta_i\}</math> is the same as the one 
which spans the possible 
fluctuations of the classical action, and thus the coefficients <math>c_i</math> 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 <math>\gamma^5</math> matrix. 

Other theories may admit cohomologically non-trivial breaking terms <math>\mathcal A</math>
in the general solution \eqref{gen-sol-anom} of the consistency condition 
\eqref{cons-cond}, 
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 \eqref{cons-cond} takes the form of 
the Wess-Zumino consistency condition [[#wess-zumino|(Wess and Zumino,1971)]],
the most general gauge anomaly is the [[Axial_anomaly|Adler-Bardeen-Bell-Jackiw
(ABBJ) anomaly]] which, for a simple Lie 
group <math>G</math>, reads
:<math>\label{g-abbj-anom}
{\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)\ ,
</math>
where 
:<math>
D_{abcd} = d_{{\ }ab}^{n}f_{ncd} + d_{{\ }ac}^{n}f_{ndb} +
d_{{\ }ad}^{n}f_{nbc} \ ,
</math>
and <math>d_{abc}</math> is the totally symmetric invariant tensor of rank 3
defined by
<math>
d_{(abc)}=  \frac{1}{2} {\rm Tr} ( \tau_a\{\tau_b,\tau_c\} ) 
</math>.
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 <math>r</math> in \eqref{g-abbj-anom} 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
[[#adl-bard|(Adler SL and Bardeen WA,1969)]]
[[#pigsor|(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
[[#BIM|(Bouchiat, Iliopoulos and Meyer,1972)]],
[[#standard-model|(Kraus E,1998)]], where
the gauge group is U(1)<math>\times</math>SU(2)<math>\times</math>SU(3): there are
three terms like \eqref{g-abbj-anom}, hence three coefficients <math>r</math>,
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
<math>\pi^o \to \gamma\gamma</math> - 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-zumino|(Wess and Zumino,1971)]] - expressed in the BRST context by the condition \eqref{cons-cond}.

==References==


*{{Bibitem article 2|Absence of Higher-Order Corrections in the Anomalous Axial-Vector Divergence Equation.|Physical Review|182(5)|1969|1517-1536|Adler|S|Bardeen|W|preprint=[http://dx.doi.org/10.1103/PhysRev.182.1517 doi:10.1103/PhysRev. 182.1517]|label=adl-bard}}    
*{{Bibitem article 2|Gauge algebra and quantization.|Physics Letters |B102|1981|27-31|Batalin|I A|Vilkovisky|G A|preprint=[http://dx.doi.org/10.1016/0370-2693(81)90205-7 doi:10.1016/0370-2693(81)90205-7]|label=bat-vilk}}
*{{Bibitem article 1|Absence of strong interaction corrections to the axial anomaly in the σ model.|Communications in Mathematical Physics.|33|1973|97-117|Becchi|C|preprint=[http://dx.doi.org/10.1007/BF01645622 doi:10.1007/BF01645622]|label=becchi73|doi=10.1007/bf01645622}}
*{{Bibitem article 3|Renormalization of the Abelian Higgs-Kibble Model.|Communications in Mathematical Physics.|42|1975|127-162|Becchi|C|Rouet|A|Stora|R|preprint=[http://dx.doi.org/10.1007/BF01614158 doi:10.1007/BF01614158]|label=BRS1}}
*{{Bibitem article 3|Renormalization of Gauge Theories.|Annals Phys.|98|1976|287-321|Becchi|C|Rouet|A|Stora|R|preprint=[http://dx.doi.org/10.1016/0003-4916(76)90156-1 10.1016/0003-4916(76)90156-1]|label=BRS2}}
*{{Bibitem article 3|Renormalizable Theories With Symmetry Breaking, in "Field Theory, Quantization and Statistical Physics", E. Tirapegui, Ed.|||1981|3-32|Becchi|C|Rouet|A|Stora|R|label=BRS3}}
*{{Bibitem article 1|Field redefinition and on-shell renormalization.|Nuclear Physics.|B221|1983|178-194|Bonneau|G|preprint=[http://dx.doi.org/10.1016/0550-3213(83)90623-5 doi:10.1016/0550-3213(83)90623-5]|label=bonneau|doi=10.1016/0550-3213(83)90623-5}}
*{{Bibitem article 3|An Anomaly Free Version of Weinberg's Model.|Physics Letters|B38|1972|519-523|Bouchiat|C|Iliopoulos|J|Meyer|P|
preprint=[http://dx.doi.org/10.1016/0370-2693(72)90532-1 doi:10.1016/0370-2693(72)90532-1]|label=BIM|doi=10.1016/0370-2693(72)90532-1}}
*{{Bibitem article 2|Dimensional renormalization and the action principle.|Communications in Mathematical Physics.|52|1977|11-38|Breitenlohner|P|Maison|D|preprint=[http://dx.doi.org/10.1007/BF01609069 doi:10.1007/BF01609069]|label=breit-maison|doi=10.1007/bf01609069}}
*{{Bibitem book 2|The Quantum Action Principle in the Framework of Causal Perturbation Theory,|in "Quantum Field Theory: Competitive Models", B. Fauser, J. Tolksdorf and E. Zeidler, eds.|Birkhäuser Verlag, [arXiv:0801.1408]|2009|Brennecke|F|Duetsch|M|label=QAP-Epstein-Glaser}}
*{{Bibitem article 2|Generalization of Zimmermann's normal-product identity.|Nuclear Physics |B113|1976|109-134|Clark|T E|Lowenstein|J H|preprint=[http://dx.doi.org/10.1016/0550-3213(76)90457-0 doi:10.1016/0550-3213(76)90457-0]|label=QAP-BPHZL2|doi=10.1016/0550-3213(76)90457-0}}
*{{Bibitem article 2|The Role of Locality in Perturbation Theory.|Ann.Inst.Henri Poincaré, Phys.Theor.|A19|1973|211|Epstein|H|Glaser|V|label=epstein-glaser}}
*{{Bibitem article 1|The role of symmetry in fundamental physics.|Proceedings of the National Academy of Sciences.|93|1996|14256-14259|Gross|D J|preprint=[http://dx.doi.org/10.1073/pnas.93.25.14256 doi:10.1073/pnas.93.25.14256]|label=gross|doi=10.1073/pnas.93.25.14256}}
*{{Bibitem book 1|Local Quantum Physics.|Springer-Verlag||1992|Haag|R|preprint=[http://dx.doi.org/10.1007/978-3-642-97306-2 doi:10.1007/978-3-642-97306-2]|label=haag}}
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*{{Bibitem book 2|Quantum Field Theory.|McGraw Hill|New York|1980|Itzykson|C|Zuber|J-B|label=itzu}}
*{{Bibitem article 1|The Renormalization of the Electroweak Standard Model to All Orders.|Annals Phys.|262|1998|155|Kraus|E|
preprint=[http://dx.doi.org/10.1006/aphy.1997.5746 doi: 10.1006/aphy.1997.5746]|label=standard-model}}
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*{{Bibitem article 1|Equivalence Theorem on Bogolyubov-Parasiuk-Hepp-Zimmermann Renormalized Lagrangian Field Theories.|Physical Review|D7|1973|2943|Lam|Y M P|label=QAP-BPHZL11|doi=10.1103/physrevd.7.2943}}
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*{{Bibitem book 2|Renormalized Supersymmetry.|Progress in Physics Vol.12|Birkhäuser Boston|1986|Piguet|O|Sibold|K|label=pigsib1}}
*{{Bibitem book 2|Algebraic Renormalization.|Lecture Notes in Physics Vol. m28|Springer|1995|Piguet|O|Sorella|S P|label=pigsor}}
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*{{Bibitem article 1|High-Energy Behavior in Quantum Field Theory.|Physical Review|118(3)|1960|838-849|Weinberg|S|preprint=[http://dx.doi.org/10.11 03/PhysRev.118.838 doi:10.1103/PhysRev.118.838]|label=weinberg}}
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==Further reading==

''For other applications of the algebraic renormalization method:''
*{{Bibitem book 5|Applications of noncovariant gauges in the algebraic renormalization procedure.|World Scientific|Singapore, New Jersey, Hong Kong|1998|Boresch, A; Emery|S|Moritsch|O|Schweda|M|Sommer|T|Zerrouki|H|}}

''For the applications to the Standard Model:'' 
*{{Bibitem article 1|The renormalization of the electroweak standard model to all orders.|Annals Phys.|262|1998|155|Kraus|E|preprint=[http://dx.doi.org/10.1006/aphy.1997.5746 doi: 10.1006/aphy.1997.5746]}}
*{{Bibitem article 3|Practical algebraic renormalization.|Annals of Physics|288|2001|197-248|Grassi|P A|Hurt|T|Steinhauser|M|preprint=[http://dx.doi.org/10.1006/aphy.2001.6117 doi: 10.1006/aphy.2001.6117]}}
*{{Bibitem article 3|The Algebraic method.|Nuclear Physics|B610|2001|215-250|Grassi|P A|Hurt|T|Steinhauser|M|preprint=[http://dx.doi.org/10.1016/S0550-3213(01)00303-0 doi: 10.1016/S0550-3213(01)00303-0]|label=Grassi2}}

''For anomalies:''
*{{Bibitem book 1|Anomalies in quantum field theory.|International Series of Monographs on Physics|Oxford University Press|2001|Bertlmann|R A|}}

''For the Batalin-Vilkovisky renormalization scheme:''
*{{Bibitem book 1|Batalin-Vilkovisky Lagrangian quantization,|in "Strings and symmetries", 1991: proceedings.  N. Berkovitz, H. Itoyama, K. Schoutens, A. Sevrin, W. Siegel, P. van Nieuwenhuizen, J. Yamron. River Edge, N.J., eds.|World Scientific|1992|Van Proyen|A}}

 


[[Category:Quantum and Statistical Field Theory]]