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Background field method and the cohomology of renormalization

Using the background field method and the Batalin-Vilkovisky formalism, we prove a key theorem on the cohomology of perturbatively local functionals of arbitrary ghost numbers, in renormalizable and nonrenormalizable quantum field theories whose gauge symmetries are general covariance, local Lorentz symmetry, non-Abelian Yang-Mills symmetries and Abelian gauge symmetries. Interpolating between the background field approach and the usual, nonbackground approach by means of a canonical transformation, we take advantage of the properties of both approaches and prove that a closed functional is the sum of an exact functional plus a functional that depends only on the physical fields and possibly the ghosts. The assumptions of the theorem are the mathematical versions of general properties that characterize the counterterms and the local contributions to the potential anomalies. This makes the outcome a theorem on the cohomology of renormalization, rather than the whole local cohomology. The result supersedes numerous involved arguments that are available in the literature.


Introduction
Locality and gauge invariance allow us to prove that the divergences of perturbative quantum field theory can be subtracted in a renormalization-group invariant way, preserving the cancellation of gauge anomalies to all orders when they vanish at one loop.
Common tricks to handle certain difficulties, e.g. to fix the gauge in a local way, consist of extending the set of the physical fields φ to a larger set Φ α , which includes the Faddeev-Popov ghosts C [1], the antighostsC and suitable Lagrange multipliers B for the gauge fixing. Moreover, to keep track of the effects of renormalization on the gauge symmetries, external sources K α are coupled to the transformations of the fields. The extra fields and the sources simplify several arguments and derivations, but enlarge the set of counterterms and potential anomalies. It is then necessary to show that the enlargement has no impact on the physical quantities.
To ease this task, a canonical formalism is introduced, known as Batalin-Vilkovisky formalism [2], which collects the Ward-Takahashi-Slavnov-Taylor (WTST) identities [3] in a compact form and generalizes the BRST symmetry [4]. The basic properties of the gauge symmetries are incorporated into an extended action S(Φ, K) = S c (φ) − R α (Φ)K α , where S c (φ) is the classical action and the functions R α (Φ) are the infinitesimal transformations of the fields Φ α . The great advantage of the BV formalism is that it relates in a simple way the identities satisfied by the action S(Φ, K) to the WTST identities satisfied by the generating functional Γ of the one-particle irreducible correlation functions.
A notion of antiparentheses (X, Y ) for functionals X, Y is defined, where the fields Φ and the sources K are viewed as conjugate variables. Gauge invariance is lifted to a certain identity obeyed by S, the master equation (S, S) = 0, and a certain cohomology. The counterterms and the local contributions to the potential anomalies are characterized by being cohomologically closed, i.e. they are local functionals X that satisfy (S, X) = 0. A local functional is said to be cohomologically exact, or trivial, if it has the form (S, Y ), Y being another local functional. Two local functionals X and Y are said to be cohomologically equivalent if X − Y = (S, Z), Z being another local functional. Throughout this paper, when we speak of local functionals we include the perturbatively local ones. A perturbatively local functional is a functional that can be written as a perturbative expansion where each term of the sum is equal to the spacetime integral of a polynomial function of the fields, the sources and their derivatives, evaluated in the same point.
To show that the enlargement mentioned above does not extend the set of cohomological classes, we must prove that the new local functionals X new that can be built with the extra fields and the sources are all trivial. This problem has been widely studied in the literature.
Kluberg-Stern and Zuber conjectured in ref. [5] that the solution of (S, X) = 0 for local functionals X of vanishing ghost number has the expected form in non-Abelian Yang-Mills theory, i.e. it is the sum of a local functional G(φ) of the physical fields φ plus a trivial term (S, Y ).
Motivated by the Kluberg-Stern-Zuber (KSZ) conjecture, several people, starting from Dixon and Taylor [6] and Joglekar and Lee [7], embarked in the brave task of classifying all the local functionals and operators that are cohomologically closed. A strong motivation was to work out the most general solutions of the Wess-Zumino consistency conditions [8] for the classification of anomalies [9,10,11]. In refs. [12] several results were generalized and formulated in the context of the BV formalism. In ref. [13] the classification was worked out in detail in the physically interesting case of Einstein-Yang-Mills theories. For a review of this approach, see ref. [14].
Unfortunately, the assumptions under which the Kluberg-Stern-Zuber conjecture holds are too restrictive. In important cases, including the standard model, coupled to quantum gravity or not, there exist nontrivial cohomological classes X new (Φ, K) that depend on the sources K. The reasons are the presence of global symmetries and the U (1) factor in the gauge group. It is well known that the hypercharges of the matter fields are not uniquely fixed (up to the overall normalization) by the tree-level standard-model Lagrangian. The extra terms X new are precisely those associated with the free hypercharges. If some counterterms proportional to X new were generated by renormalization, they would jeopardize the cancellation of gauge anomalies. Indeed, the cancellation of gauge anomalies at one loop imposes further constraints on the hypercharges and often fixes them uniquely [15]. Thus, it is crucial to show that renormalization cannot generate the terms X new , even if such terms are cohomologically allowed. In several situations, this result can be achieved with supplementary ad hoc arguments [16,17]. However, a deeper understanding is most welcome.
It is clear that the cohomology we must consider is not the whole cohomology of the local functionals of the fields and the sources, but the cohomology of renormalization, that is to say the cohomology of the local functionals that can be generated by renormalization. This is a sort of cohomology with constraints. The crucial point is that it is not enough to characterize the counterterms and the local contributions to the potential anomalies as being cohomologically closed. Indeed, they satisfy more restrictive conditions. In this paper, we convert those conditions into mathematical assumptions and prove a general theorem that bypasses most of the involved arguments offered so far in the literature and provides a better understanding of the matter.
In power-counting renormalizable theories, it is relatively easy to list all the local terms and solve the cohomological problem with a few algebraic manipulations. On the other hand, in nonrenormalizable theories, such as the standard model coupled to quantum gravity, the potential counterterms and the local contributions to anomalies can have arbitrarily large dimensions, which makes their cohomological classification rather involved. The problem is equally hard in renormalizable theories, when we include composite fields of higher dimensions. Most of the arguments that can be found in the literature are extremely involved and unfit to become part of a quantum field theory textbook.
The theorem we prove here is simpler and more to the point. We show that the nontrivial sector of the cohomology of renormalization just depends on the physical fields φ and (in the case of functionals of nonvanishing ghost numbers) the ghosts C. Precisely, in general gauge theories whose gauge symmetries possibly include general covariance, local Lorentz symmetry, Abelian gauge symmetries and non-Abelian Yang-Mills symmetries, the solution of the problem (S, X) = 0, where X(Φ, K) is a local functional generated by renormalization, has the form where G and Y (Φ, K) are other local functionals.
The key ingredient of the proof is the use of the background field method [18] and the comparison with the usual, nonbackground approach. The background field method was formulated in the context of the BV formalism by Binosi and Quadri in refs. [19] 1 and by the present author in ref. [22]. The two approaches differ in some respects and highlight different properties. Here we take advantage of the approach of [22], which offers, in particular, an exhaustive characterization of the counterterms and the local contributions to the potential anomalies. Specifically, it allows us to translate the assumption that X is "generated by renormalization" into simple mathematical requirements. In the end, we manage to prove the theorem with a relatively small effort.
The result (1.1) implies that the antighostsC, the Lagrange multipliers B and the sources K cannot alter the cohomology of renormalization and offers a better understanding of why renormalization cannot generate the extra terms X new (Φ, K).
We stress the main differences between the results of this paper and those of the previous literature, in particular refs. [12,13,14]. Those references contain theorems about the algebraic cohomology of local functionals. Precisely, they classify the local solutions X of the cohomological problem (S, X) = 0 from the purely algebraic point of view. In various cases, K-dependent solutions X new (Φ, K) are present, and it is necessary to find ad hoc arguments to prove that they are actually not generated by renormalization. Instead, the theorem proved here goes straight to the point and deals directly with the local functionals that can be generated by renormalization, which satisfy (S, X) = 0 plus a few other assumptions. The K-dependent functionals are automatically excluded from the cohomology, and the results are much easier to prove.
We mention an important application of our theorem, to be presented in detail in a separate publication. The basic tool to prove the cancellation of gauge anomalies to all orders, once they vanish at one loop, is the Adler-Bardeen theorem [23,24], which was recently generalized to nonrenormalizable general gauge theories in ref. [17]. A step of the proof given in [17] requires the knowledge of the cohomological properties satisfied by the counterterms. There, a variant of the KSZ conjecture for functionals of vanishing ghost number was proved using the results of refs. [12,13,14] and some ad hoc arguments. Now, it is possible to upgrade the proof of [17] by using the background field method and the theorem proved here.
The paper is organized as follows. In section 2 we build the background field action and work out its relation with the ordinary, nonbackground action. In section 3 we state and prove the theorem, after listing the mathematical assumptions and motivating them physically. The proof is split into four main steps to make the reader better appreciate the tricks we use and the guiding philosophy. Section 4 contains the conclusions. The appendix collects some useful formulas from previous references.

Background field method in the Batalin-Vilkovisky formalism
We assume that the gauge symmetries of the theory are general covariance, local Lorentz symmetry, Abelian gauge symmetries and non-Abelian Yang-Mills symmetries, or a subset of these. What such symmetries have in common is that (i) the infinitesimal gauge transformations of the physical fields φ are linear functions of φ, and (ii) the closure relations are φ independent. These features are crucial, because they endow the background field method with the renormalization properties we need. An example of symmetry that does not obey these assumptions is local supersymmetry.
The Batalin-Vilkovisky formalism is convenient for studying general gauge theories. It is a type of canonical formalism, where the conjugate variables are the fields Φ α and certain external sources K α coupled to the Φ transformations. The fields Φ α and the sources K α have statistics ε α and ε α + 1, respectively, where ε α is to 0 mod 2 for bosons and 1 mod 2 for fermions. A notion of antiparentheses is introduced, where X and Y are functionals of Φ and K, the subscripts l and r in δ l and δ r denote the left and right functional derivatives, respectively, and the integral is over spacetime points associated with repeated indices. The set of fields Φ α = {φ i , C I ,C I , B I } contains the classical fields φ i , the Faddeev-Popov ghosts C I , the antighostsC I and the Lagrange multipliers B I for the gauge fixing. The action S(Φ, K) is a local functional that solves the master equation (S, S) = 0 and coincides with the classical action S c (φ) atC = B = K = 0. The terms that are linear in K α = {K i φ , K I C , K Ī C , K I B } collect the infinitesimal transformations R α (Φ) of the fields. With the gauge symmetries we are considering here, the master equation admits the simple solution which is linear in K. Explicit expressions of the functions R α (Φ) can be found in the appendix. Note that each R α (Φ) is at most quadratic in Φ.
Several operations, such as the gauge fixing, can be performed by means of canonical transformations, which are the transformations Φ, K → Φ ′ , K ′ that preserve the antiparentheses (2.1). They can be derived from a generating functional F (Φ, K ′ ) of fermionic statistics, by means of the formulas Given a functional χ(Φ, K) that behaves as a scalar, we write its transformation law

Background field action
In the framework of the BV formalism, the background field method can be implemented as follows. Let Φ and K denote the background fields and the background sources. We associate background fields with just the physical fields φ and the ghosts C, but not the antighostsC and the Lagrange multipliers B. Thus, which is obtained from (2.2) by adding a background copy with vanishing classical action. Obviously, the action (2.3) satisfies the master equation where the squared antiparentheses are defined as and act on functionals X and Y of Φ, K, Φ and K. The background shift is the canonical transformation generated by 2 which gives the action F b S. The new fields Φ α are called quantum fields and the sources K α are called quantum sources.
Recalling that the functions R i (φ, C) are proportional to C, the transformations (ii) split into the sum of (a) the quantum transformations R i (φ + φ, C) of φ i , which are given by the C-independent contributions, plus Something similar happens to the symmetry transformations R I (C) of the ghosts C. Using the fact that the functions R I (C) depend quadratically on C, the quantum transformations of C I are just R I (C), and the background transformations of C I are C J (δ l R I (C)/δC J ), while the background transformations of C I are R I (C). The background fields have trivial quantum transformations, because they are external fields from the quantum field theoretical point of view.
The background transformations ofC and B remain trivial after the shift due to F b , which is not what we want. We can adjust them by making the further transformation generated by where R IC (C, C) is the antighost background transformation. More explicitly, the last term of (2.6) (together with the minus sign in front of it) is for Yang-Mills gauge symmetries (including the Abelian ones), local Lorentz symmetry and diffeomorphisms, where the indicesâ,b, . . . are local Lorentz indices.
At this point, all the quantum fields transform as matter fields under the background transformations. For example, a Yang-Mills gauge potential A a µ behaves as a vector in the adjoint representation, instead of a connection. Complete formulas of the background transformations are given below.
The theory must be gauge fixed in a background invariant way. This can be achieved by means of a canonical transformation generated by where G Ii (φ, ∂)φ i are the gauge-fixing functions. The operator matrix ζ IJ (φ, ∂) is nonsingular at φ = 0 and symmetric. Invariance under background transformations can be easily ensured, sincē C I and B I transform as matter fields, while φ and the plain derivative ∂ can be combined into the background covariant derivative. For example, we can take Here A a µ , e a µ and g µν denote the background gauge fields, vielbein and metric, respectively, while A a µ , f a µ and h µν are the respective quantum fluctuations. The tensor ζ ab is constant and proportional to the identity in every simple subgroup of the Yang-Mills gauge group, while ζ i are other constants. Finally, D denotes the covariant derivative on the background fields.
The three canonical transformations listed so far can be easily composed using the theorems of ref. [25], recalled in the appendix. In our case, it suffices to use formula (A.3), because the nontrivial sectors of the canonical transformations do not contain any pairs of conjugate variables besides B and K ′ B , on which they depend linearly. The corrections to (A.3) contain second or higher derivatives, as shown in (A.2), so they vanish. The composition gives the generating functional Applying this canonical transformation to the action (2.3), we obtain the background field gauge-fixed action which can be decomposed as the sum of a quantum actionŜ b plus a background actionS b . Precisely, we have Here the tilde sourcesK α coincide with K α apart fromK i φ andK Ī C , which are given bỹ The source-dependent sectors of the actionsŜ b andS b encode the quantum transformations and the background transformations, respectively. The curly R denotes the background transformations of the quantum fields. Those of the antighosts are R IC , and those of φ and C are given by the formula 13) while those of the Lagrange multipliers B are given by (2.14) We recall that the transformations R α and R α obey the identity [22] R which can be easily checked in the three cases (2.7) and is useful to work out the formulas (2.11) and (2.12). For example, in the case of Yang-Mills symmetry the gauge fields (2.4) and (2.9) imply that the background field action satisfies the master equation To prove the splitting, rescale C and K C by τ and 1/τ , respectively. Then,Ŝ b →Ŝ b and S b → τS b , so S b , S b is a quadratic polynomial in τ . Setting the three coefficients of the polynomial to zero gives (2.17). A way to characterize a functional X that is invariant under the background transformations is to say that it satisfies S b , X = 0. In particular, the gauge fermion (2.8) must satisfy S b , Ψ = 0.

Nonbackground action
The nonbackground gauge-fixed action is is the canonical transformation that performs the gauge fixing. The background fields and sources are inert here. We have included them just for comparison with the background field action. To simplify the renormalization, we take a quadratic gauge fermion Ψ ′ , such as For convenience, we define the nonbackground action S nb by making a further background shift through the transformation F b . So doing, we have We easily find the decomposition whereŜ 22) and the sourcesK α with bars coincide with K α apart fromK i φ andK Ī C , which arē The nonbackground action satisfies the master equation which splits into Ŝ nb ,Ŝ nb = Ŝ nb ,S nb = S nb ,S nb = 0. (2.25) The splitting is a direct consequence of (2.21) and (2.22).

Interpolation between the background and nonbackground actions
To switch back and forth between the background and nonbackground approaches we must make the canonical transformation (2.9) and (2.19) give Using again formula (A.3), we easily find the generating function It is convenient to express the canonical transformation (2.26) by means of the componential map C of ref. [25], also recalled in the appendix. We can write Again, this functional does not contain any pairs of conjugate variables besides B and K ′ B , and depends linearly on them. Thus, the expansion (A.4) effectively reduces to (A.6), which gives (2.26).
We can continuously interpolate between the two approaches by introducing a parameter ξ that varies from 0 to 1 and make the canonical transformation generated by C(ξQ), whose inverse is C(−ξQ). We find The form of this transformation is important to simplify some arguments that follow. We define the interpolating action as while the other ξ-dependent tilde sources have expressions that we do not need to report here. It suffices to say that they are linear functions of the quantum fields Φ, apart fromK i φ (ξ) andK I C (ξ), which are quadratic. Moreover, the differencesK α (ξ) − K α andK α (ξ) − K α are independent of the sources K and K other than K I B . In particular, δ r S ξ /δK α = −R α (Φ). Note that the Bdependent terms of the interpolating action (2.30) are quadratic functions of the quantum fields. Moreover, the K Ī C -dependent terms are just 2.32) and the K B -dependent terms are linear in the quantum fields. Obviously, S ξ satisfies the master equation which is a direct consequence of (2.4) and (2.29).

The theorem
In this section we state and prove the main theorem of this paper. Since the parameter ξ is introduced by means of the canonical transformation (2.28), the action S ξ satisfies the differential equation whereQ(Φ, Φ, K) coincides with Q(Φ, Φ, K). Note that the sources have no primes in the last expression. We use different symbols for the two functionalsQ and Q, since the natural variables ofQ are the fields and the sources without primes, while the natural variables of Q are the fields without primes and the sources with primes. The result (3.1) can be proved as follows. The nonbackground action S nb = C(ξQ)S ξ obviously satisfies ∂S nb /∂ξ = 0. By formula (A.1), recalled in the appendix, the transformed action S ξ satisfies (3.1) as long asQ coincides with the derivative ∂C(ξQ)/∂ξ, after the sources with primes are expressed in terms of the fields and the sources without primes. This fact, which can be checked directly in our case, is a general property of the componential map [25]. Indeed,Q plays the role of the Hamiltonian associated with a fictitious time evolution parametrized by ξ and C(ξQ) solves the Hamilton-Jacobi equation.

Assumptions
We consider local functionals X ξ and Y ξ of Φ, K and Φ that satisfy the equations We use the notation X 0 , X 1 to denote the functional X ξ at ξ = 0 and ξ = 1, respectively. We assume that (i) X 0 just depends on Φ + Φ and K.
(iii) X ξ and Y ξ are independent of B, KC and K B .
Let g denote the ghost number of X ξ . Then, Y ξ has ghost number g − 1.
For future use, we show that assumption (ii) can be replaced by (ii ′ ) X 1 is a sum of terms that contain g background ghosts C at φ = C = 0. When we set φ = C = 0, no objects of positive ghost number remain, besides C. If (ii ′ ) holds, we can drop all objects that have negative ghost numbers inside X 1 , which meansC and all the sources K except KC. Since (iii) tells us that X 1 does not depend on B and KC, (ii) follows. On the other hand, if (ii) holds, X 1 just depends on φ and C at φ = C = 0. Then it must be a sum of terms that contain g background ghosts C, because it has ghost number g. This implies (ii ′ ). Now we explain the meaning of the assumptions listed above. In most applications, the functionals X ξ and Y ξ are originated by the counterterms and the local contributions to the potential anomalies. In particular, in the case g = 0 the functional X ξ typically comes from the renormalization of the action, while Y ξ comes from the renormalization of the average Q .
Without entering into details, we recall that the formulas (3.2) and (3.3) are typical consequences of the regularized and partially renormalized versions of the equations of gauge invariance and gauge independence [16], which are where Γ R is the renormalized Γ functional andQ R is the renormalizedQ. When Γ R andQ R are just partially renormalized, say up to and including n loops, the equations of gauge invariance and gauge independence give conditions on the (n + 1)-loop counterterms, which have the forms (3.2) and (3.3). Condition (i) is dictated by the properties of the nonbackground action (2.20), encoded in the formulas (2.22) and (2.21). If the regularized action S reg nb has a similar structure, that is to say whereS nb is unmodified, then the renormalized action also has this structure. Note thatS nb is just made of external fields, so it cannot contribute to the nontrivial Feynman diagrams. Moreover, the counterterms do not depend on Φ and Φ separately, but only on their sum, as specified in (i). Finally,S nb is not renormalized, for this very reason. Similarly, conditions (ii) and (ii ′ ) are dictated by the properties of the background field side. Indeed, we know that the action S b of formula (2.10) splits into (2.11) plus (2.12). If the regularized action S reg b has a similar structure, that is to say the counterterms just depend on Φ, φ and K, so they satisfy property (ii ′ ) for g = 0. Note thatS b , which is linear in the quantum fields, does not contribute to any nontrivial one-particle irreducible diagrams. It is also not renormalized, because it vanishes at C = 0, while the counterterms are independent of C. The case g = 1 is also important, because it concerns the potential gauge anomalies. When the dimensional regularization -or any regularization that embeds the dimensional one -is used, the gauge anomalies are encoded in the functional S reg b , S reg b = Ŝ reg b ,Ŝ reg b + 2 Ŝ reg b ,S b (for the proof see, for example, the appendix of ref. [24]), so they are linear functions of C. Indeed, must be proportional to C at φ = C = 0, because it has ghost number 1 and no other fields or sources of positive ghost numbers are present in that case. This ensures that the local contributions to the potential anomalies satisfy property (ii ′ ) for g = 1.
Condition (iii) is also suggested by the properties of renormalization. If the regularization preserves the basic properties of the structure of S ξ , then it is easy to show that the counterterms have the same properties. In particular, we can arrange the regularization so that no nontrivial one-particle irreducible diagrams can be constructed with external legs of types B, KC and K B . It is even easier to ensure that X ξ and Y ξ are independent of K, since the K-dependent terms of the action are just made of external fields and do not even need to be regularized.
The assumption (3.2) is actually necessary for just one value of ξ, since then equation (3.3) implies (3.2) for every ξ. Indeed, taking the derivative of S ξ , X ξ with respect to ξ and using (3.1) and (3.3), we get ∂ ∂ξ S ξ , X ξ − S ξ , X ξ ,Q = 0. (3.5) This equation can be easily integrated [16,22,25]. The result is that the ξ dependence of S ξ , X ξ is just due to the canonical transformation generated by C(−ξQ). Thus, if S ξ , X ξ vanishes for some ξ, it vanishes for all ξ.

Statement
We want to prove that there exist local functionals G(φ, C) and χ ξ (Φ, Φ, K, ξ) such that Moreover, we show that χ ξ is independent of B, KC and K B , while χ ξ ({0, 0,C, B}, Φ, K, 1) = 0 and χ ξ (Φ, Φ, K, 0) just depends on Φ + Φ and K. We also find the explicit expression of χ ξ . Note that G is independent of ξ. Taking the squared antiparentheses · · · of both sides of (3.6) with S ξ and using (2.33) and (3.2), we get S ξ , G = 0. In the case g = 0, the functional G is C independent and gauge invariant.
For convenience, we divide the proof into four steps.
Step 1 of proof: Interpolation between the background and nonbackground sides Consider the functionals where w is a constant parameter such that w 2 = 0. If the ghost number g of X ξ is odd, we can take w = ̟, where ̟ is constant and anticommuting. If g is even, we can take w = ̟̟ ′ , where ̟ ′ is constant, anticommuting and different from ̟. We introduce the parameter w to make the first order of the w Taylor expansion exact, which simplifies several arguments. We have used the subscript R in (3.7), because in several practical applications X ξ and Y ξ are counterterms, while S Rξ andQ Rξ are renormalized functionals. Combining S ξ , S ξ = 0 with (3.2) and (3.1) with (3.3), we get (3.8) which are the analogues of the equations in (3.4). The second formula implies that the canonical transformation generated by (3.10) interpolates between the nonbackground and background values of the action S Rξ , which are S nb + wX 0 ≡ S nbR and S b + wX 1 ≡ S bR , respectively. In compact notation, we have (3.11) To check that (3.9) implies (3.8), it is sufficient to apply formula (A.1) to (3.11) and express ∂F ξ /∂ξ in terms of the fields and the sources without primes. The composition formulas recalled in the appendix allow us to write the relation 12) where F Y ξ denotes the canonical transformation generated by Indeed, formula (A.2) reduces to C = A + B in this case. All the corrections vanish, since the nontrivial part of C(ξQ) does not depend on any sources K ′ , K ′ other than K ′ B , while Y ξ is B independent. Now we compare the background and nonbackground "renormalized" actions S bR and S nbR . Evaluating (3.11) at ξ = 1, we get the equalitŷ (3.14) where the fields and the sources with primes are related to those without primes by the canonical transformation F ξ with ξ = 1, whilê Using (2.28) and (3.9), we find, among other things, Φ α′ = Φ α and Step 2 of proof: The background field trick The basic trick of the background field method is to switch the quantum fields off and resume them later with the help of the background fields. Now we explain how this trick is implemented when the background field method is used together with the BV formalism, and we show how this helps us achieve our goal. First, we need to express both sides of equation (3.14) in terms of the fields Φ, Φ and the sources K ′ , K ′ . Once this is done, we set φ = C = K ′ = 0 and keep φ, C,C, B and K ′ as independent fields and sources. We denote the fields Φ α′ and the sources K α , K α obtained by applying these operations by Φ α′ 0 and K α0 , K α0 , respectively. We find Moreover, (3.15), (2.11) and assumption (ii) givê Consider the canonical transformation {φ, C,C, B},K → Φ ′′ , K ′ defined by the generating functional Using (3.16) and the other transformation rules not reported in that formula, we find which, with the help of (3.18), turn the identity (3.17) intô (3.20) This is the key result we need. Now we elaborate it further and express it in ways that make its contents more transparent.
Recalling that X 1 is independent of K,C and B at φ = C = 0, while δ r S b /δK i φ and δ r S b /δK I C vanish there, we get 21) which states that X 1 (0, Φ, 0) is invariant under background transformations. The conclusion obviously also applies to S c (φ). The background invariance of the gauge fermion (2.8) implies that the second term on the right-hand side of (3.20) is also invariant under background transformations. Thanks to these facts, we can easily check, from the right-hand side of formula (3.20), thatŜ nbR does satisfy the master equation (Ŝ nbR ,Ŝ nbR ) = 0. It is convenient to relabel the fields {φ, C,C, B} as Φ α , the sourcesK α as K α and the fields Φ α′′ as Φ α′ . Then formulas (3.19) and (3.20) tell us that the canonical transformation 23) andF −1 b is the canonical transformation that undoes the background shift. Precisely,F −1 b is the inverse of the transformation generated bỹ which is obtained from (2.5) by reducing the set of fields and sources and downgrading the background fields Φ to the role of mere spectators.
Making the further canonical transformation F ζ , where In this derivation, it is helpful to recall that the last term of F ζ with φ → φ is invariant under the background transformations. Thus, F ζ simply cancels the second term on the right-hand side of (3.23). Finally, the canonical transformation generated by [check (2.6)] In the last step, we have used (2.2). Collecting the results found so far, we get More explicitly, the canonical transformation Φ, K → Φ ′ , K ′ generated by F F −1 ζF nm is such that Note that formula (3.21) can be recast in the form (S, X 1 (0, {φ, C}, 0)) = 0. (3.25) Together with (S, S) = 0, this formula and (3.24) give back (Ŝ nbR ,Ŝ nbR ) = 0.
Let G i (φ, C) denote a basis of local functionals of ghost number g that satisfy (S, G i ) = 0 and are constructed with the physical fields φ and the ghosts C. For g = 0 they are just the usual gauge invariant local functionals. We have the expansion (3.26) where τ i are constants, and formula (3.24) giveŝ . (3.27) This formula relates the background field functionals with the nonbackground functionals. With the help of a few other manipulations, we can work out the interpolation between the two sides and conclude the proof of the theorem.
Step 4 of proof: Interpolation of the result Using (2.28) and (3.9), it is easy to check the composition rule F =F YF , where F is the transformation encoded in (3.19), whileF andF Y are the transformations generated bȳ respectively. Using this property and collecting the canonical transformations made so far, we can write the identity (3.27) in the compact form Applying the composition rules recalled in the appendix, in particular formula (A.3), it is easy to check thatF is the reduced version of (2.18). Thus, formula (3.30) simplifies tõ The terms independent of w giveF ′−1 gfF −1 bŜ nb = S, (3.32) which can be easily verified. Indeed, by formula (2.21), if we undo the background shift and the gauge fixing on the actionŜ nb , we obtain the starting action S. The terms of (3.31) that are proportional to w can be better read after acting on both sides withF YF ′ gf , which gives Define the functional whereK φ is defined in formula (2.23). Then from (3.33) we can writẽ At this point, we start looking at the functionals as functionals of Φ, Φ, K, K again. We where we have used F −1 bS nb , Υ = 0, which is due to the fact that F −1 bS nb just depends on Φ, K, while Υ just depends on Φ, K. Applying F b to both sides of (3.35), we obtain (3.36) Using this result together with formula (3.11), we find Now we apply F −1 ξ to both sides and use S Rξ = S ξ + wX ξ from the first formula of (3.7) and (3.12). Noting that C(−ξQ) leaves G i (φ + φ, C + C) invariant, because it does not affect the fields besides B I , we find Using (3.13) and (2.29), we observe that Finally, formula (3.37) gives where Formula (3.39) coincides with (3.6) once we set G(φ, C) = i τ i G i (φ, C). Using (2.28), (3.34), (3.38) and (2.31), we find This functional is local and independent of B, KC, K B and K. Moreover, χ ξ (Φ, Φ, K, 0) just depends on Φ + Φ and K. Finally, χ ξ ({0, 0,C, B}, Φ, K, 1) = 0. This concludes the proof of the theorem.
Observe that χ ξ satisfies the equation which can be proved by using (A.1) on (3.40) and noting that C(−ξQ)Q =Q. Formulas (3.39) and (3.26) tell us that, in the end, the nontrivial cohomological content of the functional X ξ is just X 1 (0, Φ, 0), which can be worked out with the background field method by evaluating Feynman diagrams that just have background fields φ and background ghosts C on their external legs.

Conclusions
We have studied the cohomology of the local functionals of arbitrary ghost numbers generated by renormalization in quantum field theories whose gauge symmetries are general covariance, local Lorentz symmetry, non-Abelian Yang-Mills symmetries and Abelian gauge symmetries. The case of ghost number 0 is important to characterize the divergent parts of Feynman diagrams. The case of ghost number 1 is important for the local contributions to the potential anomalies.
Using the Batalin-Vilkovisky formalism and the background field method, we have proved that a closed local functional can always be written as a local functional that just depends on the physical fields φ and the Faddeev-Popov ghosts C, plus an exact functional, which is equal to the BV antiparentheses between the action and another local functional. Every term that depends nontrivially on the antighostsC, the Lagrange multipliers B for the gauge fixing and/or the sources K coupled to the symmetry transformations of the fields is cohomologically trivial.
The basic idea of the proof is to interpolate between the background field approach and the usual, nonbackground approach by means of canonical transformations. This allows us to take advantage of the virtues of both approaches, and highlight, among other things, that the counterterms and the local contributions to the potential anomalies are not just cohomologically closed, but satisfy more restrictive conditions. We have managed to translate those conditions into simple mathematical assumptions and obtain a general theorem. The result supersedes numerous involved arguments that exist in the literature and offers a better understanding of the matter. It can be used to upgrade the recent proof [17] of the Adler-Bardeen theorem in nonrenormalizable theories.

Appendix. Useful formulas
In this appendix we collect a few reference formulas that are used in the paper.
The functional − R α (Φ)K α that appears in (2.2) reads Hereâ,b, . . . are local Lorentz indices, while C µ -C µ -B µ , Câb-Câb-Bâb and C a -C a -B a denote the ghosts, the antighosts and the Lagrange multipliers of diffeomorphisms, local Lorentz symmetry and Yang-Mills symmetry, respectively. Moreover, ϕ are scalar fields and ψ L are left-handed fermions, while T a and T a are the anti-Hermitian matrices associated with their representations. A helpful identity tells us that if Y (Φ, K) is a functional that behaves as a scalar [i.e. such that Y ′ (Φ ′ , K ′ ) = Y (Φ, K)] under the canonical transformation Φ, K → Φ ′ , K ′ , generated by the functional F (Φ, K ′ ), then we have [26,22] whereF ζ (Φ, K) = F ζ (Φ, K ′ (Φ, K)) and F ζ (Φ, K ′ ) = ∂F/∂ζ. In this formula, the ζ derivative of each functional is evaluated while the natural arguments of the functional are kept constant.
Precisely, Φ ′ and K ′ are constant in the ζ derivative of Y ′ , while Φ and K are constant in the ζ derivative of Y , and Φ and K ′ are constant in the ζ derivative of F . In ref. [25], various formulas for the manipulation of canonical transformations at the level of their generating functions have been given. In particular, if F A (q, P ) = q i P i + A(q, P ) and F B (q, P ) = q i P i + B(q, P ) are the generating functions of two canonical transformations q, p → Q, P , the generating function of the composed transformation F C = F B F A is written as F C (q, P ) = q i P i + C(q, P ), where the function C(q, P ) is expressed as a sum of monomials built with A, B and their derivatives A i 1 ···in = ∂ n A(q, P ) ∂P i 1 · · · ∂P in , B i 1 ···in = ∂ n B(q, P ) ∂q i 1 · · · ∂q in .
The first contributions to C(q, P ) are These results extend straightforwardly to the BV formalism. When we compose the canonical In the applications of this paper, due to the simple structures of the functionals A and B, formula (A.2) effectively reduces to Several operations on canonical transformations can be handled more practically by means of the componential map C [25], which expresses the generating function F (q, P ) in terms of a another function X(q, P ) as C(X(q, P )), with the expansion C(X) = I + X + 1 2 X i X i + 1 3! X ij X i X j + X j X i j X i + X ij X i X j + · · · , (A.4) where X i 1 ···in j i ···jm ≡ ∂ n+m X(q, P ) ∂q i 1 · · · ∂q in ∂P j 1 · · · ∂P jm .
The advantage of the componential map C(X) is that it satisfies the Baker-Campbell-Hausdorff (BCH) formula, like the exponential map e ad(X) , where ad(X)Y = {X, Y } is the adjoint map and {X, Y } are the Poisson brackets of X and Y . Precisely, if we define X△Y from the BCH formula where the product on the left-hand side is the composition of the canonical transformations. In particular, the inverse of C(X) is just C(−X). Again, the generalization to the BV formalism is straightforward and, due to the simple structures of the functionals X(Φ, K ′ ), in most applications of this paper the expansion (A.4) of the componential map effectively reduces to C(X(Φ, K ′ )) = Φ α K ′ α + X(Φ, K ′ ) +