The Master Ward Identity for scalar QED

It is emphasized that for interactions with derivative couplings, the Ward Identity (WI) securing the preservation of a global gauge symmetry should be modified. Scalar QED is taken as an explicit example. More precisely, it is rigorously shown in scalar QED that the naive WI and the improved Ward Identity ("Master Ward Identity", MWI) are related to each other by a finite renormalization of the time-ordered product ("T-product") for the derivative fields; and we point out that the MWI has advantages over the naive WI - in particular with regard to the proof of the MWI. We show that the MWI can be fullfilled in all orders of perturbation theory by an appropriate renormalization of the T-product, without conflict with other standard renormalization conditions. Relations with other recent formulations of the MWI are established.


Introduction
In spinor QED the Master Ward Identity (MWI) expressing global U (1)-symmetry contains all information that is needed for a consistent perturbative BRST-construction of the model, see [8] or [6,Chap. 5]. This "QED-MWI" is a renormalization condition on T -products to be satisfied to all orders of perturbation theory. It reads ∂ µ y T n+1 B 1 (x 1 )⊗· · ·⊗ B n (x n )⊗j µ (y) 0 = − n l=1 δ(y−x l ) T n B 1 (x 1 )⊗· · ·⊗ θB l (x l )⊗· · ·⊗ B n (x n ) 0 , (1.1) where j µ = ψγ µ ψ is the Dirac current, B 1 , . . . , B n are arbitrary submonomials of the interaction L = e j µ A µ , and θ is the charge number operator. The notation (·) means that fermionic field polynomials are converted into bosonic field polynomials by multiplying them with a Grassmann variable. By T (. . . ) 0 we denote on-shell T -products (see below).
There is an essential difference between spinor QED and scalar QED: in the latter, the current to which the electromagnetic potential is coupled, contains first derivatives of the basic fields: 1 j µ := i(φ∂ µ φ * − φ * ∂ µ φ). (

1.2)
It is apriori not evident how to translate the QED-MWI (1.1) to models with derivative couplings, and scalar QED may serve as a prototype of such models.
Our results can be summarized as follows: in Sect. 2 we postulate a naive WI for scalar QED, just by analogy to spinor QED. To fulfil it, an "unnatural" renormalization of the T -product of ∂ µ φ(x) with ∂ ν φ * (y) is required [11]: one has to add ig µν δ(x − y) to ∂ ν ∂ µ ∆ F (x − y). This addition violates the standard renormalization conditions 'Field Equation ' and 'Action Ward Identity'.
In Sect. 3 we work out the MWI for the global U (1)-transformation φ(x) → e iα φ(x) in scalar QED, and find that, compared with the naive WI, it contains an additional term.
In Sect. 5 we prove that the MWI can be fulfilled by an appropriate renormalization of the T -product, which is compatible with the further standard renormalization conditions. In Sect. 6, starting with the time-ordered product "T ", we define in all orders a new timeordered product T induced from the initial finite renormalization ∂ ν ∂ µ ∆ F → ∂ ν ∂ µ ∆ F + ig µν δ, by the inductive Epstein-Glaser method [12]. We prove that the validity of the MWI for T is equivalent to the validity of naive WI for T . In fact, one may continuously interpolate between T and T .
In Sect. 4.2 we prove, in the perturbative approach to scalar QED, that the MWI is equivalent to the so-called "unitary MWI". The latter is an identity, conjectured by Fredenhagen [4], which seems to be well suited for the formulation of symmetries in the Buchholz-Fredenhagen quantum algebra [5].
All proofs are given to all orders of perturbation theory.

Some technical preparations
We use natural units, in particular = 1, and the underlying spacetime is the 4-dimensional Minkowski space M. We work with causal perturbation theory in the formalism where quantum fields are functionals on classical configuration spaces, equipped with a non-commutative product: the star product of the free theory (denoted by "⋆"). Perturbation theory represents interacting fields as formal power series within this algebra, using the time-ordered product of local fields, which is commutative. The prominent mathematical task is the construction of the time-ordered product. For details and conventions, we refer to the book [6], where in particular the conventions for the propagators are fixed in [6,App. B].
For the convenience of the reader, we sketch some basic definitions of the formalism for the model at hand, that is, scalar QED. The configuration space is C = C ∞ (M, C) × C ∞ (M, R 4 ), where the first factor stands for the configurations of the complex scalar field φ and the second for the configurations of the photon field A ≡ (A µ ). The space of fields F is the set of all polynomial functionals on the configuration space satisfying certain properties. More precisely, F ∋ F : C → C is a finite sum of the form F = p,n,l dx 1 · · · dx p dy 1 · · · dy n dz 1 · · · dz l p i=1 where f 0,0,0 ∈ C is constant; and for p+n+l ≥ 1, each expression f µ 1 ... p,n,l is an element of D ′ (M n , C) with compact support, which satisfies a certain wave front set condition, see [6,Def. 1.2.1].
To a far extent, we work on-shell. This means that all functionals F ∈ F are restricted to the space C S 0 of solutions of the free field equations; we indicate this restriction by Algebraically, on-shell fields can be identified with Fock space operators, where the star product of on-shell fields corresponds to the operator product, and the pointwise product of on-shell functionals (i.e., (F 0 · G 0 )[h, a] := F 0 [h, a] · G 0 [h, a] for all F, G ∈ F and (h, a) ∈ C S 0 ) to the normally ordered product, see [6,Thm. 2.6.3]. Throughout this paper, we need only distributions f µ 1 ... p,n,l such that no derivatives of A µ and solely zeroth and first derivatives of φ and φ * appear. So we define P to be the space of polynomials in A µ , φ, φ * , ∂ µ φ and ∂ ν φ * only.
The subspace F loc ⊂ F of local fields is the linear span of the set { B(g) ≡ dx g(x) B(x) B ∈ P, g ∈ D(M) }. For example: Finally, the vacuum expectation value (VEV) of a field F ∈ F is ω 0 (F ) := F [0, 0].

The naive Ward Identity
The most natural candidate for the Ward Identity (WI) expressing global U (1)-symmetry for scalar QED just copies the QED-MWI (1.1) with the charge number operator and with the time-ordered product T that is required to satisfy the basic axioms (i)-(iv) and the renormalization conditions (v)-(viii) listed in Appendix A.1. This yields where B 1 , . . . , B n are arbitrary submonomials of the interaction This Ward identity is a generalization of the one postulated and proved in [11] -the WI there is motivated by gauge invariance of the on-shell S-matrix, that is, invariance under the trans- The task would be to establish the existence of T satisfying (i)-(viii) and (2.2). Below in Sect. 3 we show that (2.2) is only a simplified version of the Master Ward Identity (MWI) expressing U (1)-symmetry; the latter is better suited for models with derivative couplings, and is easier to establish. Particular cases of the WI (2.2) are These identities have an important property: Requiring that T satisfies the axiom "Field Independence", that is, the validity of the causal Wick expansion, the tree diagram part of the first identity, and the other two identities are fulfilled iff the numerical distributiont(∂ ν φ, ∂ µ φ * ) is specified aŝ as one sees by explicit computation. The finite renormalization of the Feynman propagator with two derivatives is admissible in the framework of causal perturbation theory, since the singular order is ω(∂ ν ∂ µ ∆ F ) = 0. The additional term ig µν δ(x − y) has the advantage, that it generates as a necessary finite "counter term" the quartic interaction part, i.e., −e 2 A µ A µ φφ * (as it was first realized in [11]), propagating correctly to higher orders in the inductive Epstein-Glaser construction of T ≡ ( T n ). Indeed, for the S-matrix belonging to T we obtain where the dots contain further terms of order O((eg) 2 ) and all terms of higher orders in (eg). But the addition ig µν δ(x − y) has the disadvantages that it violates the axiom "Field Equation" (FE) and the "Action Ward Identity" (AWI) (generally formulated in Appendix A.1): A proof of the WI (2.2) along the lines of the proof of the QED-MWI in [6, Chap. 5.2.2] would require additional work, because that proof uses essentially that the time-ordered product fulfills the "Field Equation". Instead, an indirect proof via the MWI will be given in Sect. 6.

The Master Ward Identity
Due to the mentioned bad properties of the time-ordered product T and the resulting problems in trying to adapt the proof of the QED-MWI to the WI (2.2), we prefer to work with the complete relevant MWI for scalar QED. The original references for the MWI are [7,9] and [2]. It is a universal formulation of symmetries; it can be understood as the straightforward generalization to QFT of the most general classical identity for local fields that can be obtained from the field equation and the fact that classical fields may be multiplied pointwise. In contrast, the quantum version of the MWI is a renormalization condition with regard to the axioms for the T -product (cf. Appendix A.1). It cannot always be fulfilled due to the well-known anomalies.

Working out the relevant MWI for scalar QED
Generally, the on-shell MWI (see [2,9] and [6,Chap. 4.2]) is derived from the symmetry at hand. It reads where δ Q(y) is a functional differential operator specified by the symmetry, and the time-ordered product T is required to fulfil the axioms (i)-(viii) given in Appendix A.1 and the additional renormalization conditions AWI and FE.
In the case at hand, we study the global U (1)-transformation φ(y) → e iα φ(y) (α ∈ R). Let and the pertinent functional differential operator Introduce a modification θ µ of the charge number operator, and recall that . Then, one verifies straightforwardly that For scalar QED and the symmetry given by the above defined Q, the MWI takes the particular form (cf. [6, Exer. 4.2.6]) for B 1 , . . . , B n ∈ P, by using the AWI. Compared with (2.2), the additional terms (i.e., the terms in the last line) arise from the last term in the formula (3.5) for δ Q(y) B(x). Instead of the identities (2.4) we now obtain by using θ µ j ν = −2ig µν φφ * .
When working with the time-ordered product T satisfying the MWI (3.6) one has to add the quartic interaction part "by hand", that is, one starts the inductive Epstein-Glaser construction of the S-matrix with the following interaction S: (3.8) The addition of the quartic interaction term can be motivated by classical gauge invariance.
In this procedure, the order of the time-ordered product does not agree with the order in the coupling constant (eg); gauge invariance of the S-matrix must hold in each order in (eg) individually.

Equivalent reformulations of the MWI
Some remarks on the notations: in this section we solely work with the time-ordered product T ≡ (T n ), which satisfies the AWI. Thus we may interpret T n as a map T n : F loc → F, and for the S-matrix (A.2) we may write in the sense of formal power series in F . In addition, let B ∈ P and g, α ∈ D(M, R) the function switching the coupling constant and an infinitesimal local U (1)-transformation, respectively. In this section all tensor products are symmetrized tensor products: ⊗ ≡ ⊗ sym .

The MWI as an identity for formal power series
Motivated by the expressions which appear in the MWI (3.6) when integrated out with α(y) j g(x j ) ∈ D(M n+1 , R), we introduce two derivations (i.e., linear maps satisfying the Leibniz rule) on T(F loc ) (by which we mean the completion of the tensor algebra over F loc , see [6, App. A]): θ (α) : T(F loc ) −→ T(F loc ) uniquely specified by δ  An immediate consequence is the relation θ (α). In addition, looking at (3.4)-(3.5), we see that With these tools we can give a more concise equivalent reformulation of the on-shell MWI (3.6): which we understand as an identity for formal power series in F ∈ F loc .
Conservation of the interacting current. As an application of the version (4.4) of the MWI, we study current conservation. For S, G ∈ F loc let be the interacting field to the interaction S and corresponding to G, as defined by Bogoliubov [1]. More precisely, G S,0 is a formal power series in S and to zeroth order in S it agrees with G 0 . For S being the interaction of scalar QED (3.8), we obtain Omitting the arbitrary testfunction α, this can be written as conservation of the interacting electromagnetic current: . Note that J µ is the Noether current belonging to the invariance of the total action under the same global U (1)-transformation φ(x) → e iα φ(x) as in the preceding sections. We recognize a further significant difference to spinor QED: the Noether currents j and J belonging to the free and interacting theory, respectively, are different.

The unitary MWI
The Buchholz-Fredenhagen quantum algebra ("BF-algebra") [5] is an abstract C*-algebra (more precisely: a local net of C*-algebras) which, given the field content and a classical relativistic Lagrangian, encodes the pertinent interactions in QFT. In this generality, the most adequate formulation of symmetries is an open problem. A concrete algebra A fulfilling the defining relations of the BF-algebra belonging to the field content of scalar QED and the Lagrangian For this algebra, the above mentioned problem amounts to the task of finding an equivalent reformulation of the MWI in terms of the maps F loc ∋ F → S(F ) 0 ; in contrast to (4.4), expressions of the type T G ⊗ e iF ⊗ 0 must not appear.
2 By " ⋆ " we mean the algebra, under the star product, generated by members of the indicated set. Also the analogous algebra generated by the off-shell S-matrices (i.e., without restriction to CS 0 ) fits into the definition of the BF-algebra for the same field content and the same Lagrangian L0; however, in view of the MWI, we prefer in the following to work on-shell.
For scalar QED, Fredenhagen has noted that the following conjectured identity [4] (see also [15]) would serve the purpose, which has some analogy to the Schwinger-Dyson equation: where α ∈ D(M, R) (see Remark 4.2 below) and define Note that on the r.h.s. the range of integration is only supp α, that is, a bounded region. The conjecture asserts that the time-ordered product can be renormalized such that We understand (4.9) as identity for formal power series in , F and α, 3 and we will call it the "unitary MWI", because it expresses the MWI in an equivalent way (as we show below) in terms of the S-matrix. Setting F := 0 the unitary MWI reduces to For illustration we explicitly compute δL 0 (α). Taking into account that and the analogous relation for ∂ x φ * α (x), we obtain The following Theorem supports the conjecture: Remark 4.2. Before giving the proof, we point out that α in (4.7) having compact support does not mean that the transformation underlying the unitary MWI is a local gauge transformation. Specifically, A µ is not transformed. The test function α is used to control the dependence of functionals on the scalar field only, and its localization means that the transformation acts non-trivially only in a bounded region. Indeed, the Theorem does not hold true for local gauge transformations, in particular the relation (4.13) becomes obviously wrong. A second reason becomes apparent by looking at the model containing only the electromagnetic field and assuming that α is a local gauge transformation. Then, it holds that δL 0 (α) = 0. Hence, the conjectured formula (4.9) would be trivial for all observables F (i.e., F α = F ), hence worthless.
Proof. Let 0 = β ∈ D(M, R) be arbitrary and let α(x) := a β(x) with a ∈ R. To prove that the MWI (4.4) implies the unitary MWI (4.9), let β be fixed and interpret the l.h.s. of (4.9) as a function of a ∈ R. Since this function is differentiable (as we see from the explicit formulas) and since the unitary MWI holds trivially true for a = 0, it suffices to show that the derivative of this function w.r.t. a vanishes for all a ∈ R -as a consequence of the MWI (4.4).
To prove this we use the following crucial relation for F a := F α : [[Proof of (4.13): we compute the l.h.s. by using d .
Taking into account that δφα(z) δφ(y) = δ(z − y) e iα(z) , which implies , (4.14) and inserting Q(y) = −iφ(y) we obtain the assertion (4.13): where L 0 (f ) a is defined similarly to F a := F α . Note in particular, that the two expressions (4.15) do not depend on the choice of f . By applying the relation (4.13) to L 0 (f ) a , we obtain Equipped with these tools we are able to verify that The derivative can easily be computed: That the unitary MWI implies the MWI is obvious from our procedure: the former yields d da a=0 T e iG(a) ⊗ 0 = 0; after insertion of (4.18) this is the MWI (4.4) for G(0) = F and β, which is the MWI in its full generality, because F ∈ F loc and β ∈ D(M, R) are arbitrary.
The fact that in this proof model-specific information is used only in the verification of the relation (4.13), indicates that the conjecture is valid also for other models with other symmetry transformations of the basic fields -see [4].

Proof of the Master Ward Identity
In this section we prove the first main result of this paper, to wit, that the MWI (3.6) can be satisfied by a finite renormalization of the T -product for all B 1 , . . . , B n ∈ P 0 , the set of L := e jA + e 2 A 2 φ * φ , j µ and all submonomials of these two field polynomials, except e jA and e 2 A 2 φ * φ and the individual parts of j µ (i.e., φ∂ µ φ * and φ * ∂ µ φ) separately. These exceptions are justified by the fact that physically relevant are only L and j µ , that is, only the sums of the individual parts. A priori we are interested in T -products of arbitrary many factors L and j only. That we prove the MWI also for all their submonomials, is a byproduct of our method of proof, since the latter uses the causal Wick expansion. Note that all B j ∈ P 0 are eigenvectors of θ; we will use the notation b j B j := θB j .
We proceed in analogy with the proof of the QED-MWI in [6, Chap. 5.2.2], which relies on [8,App. B], and in addition we use specific arguments for scalar QED given in [13]. Starting with a T -product fulfilling all other renormalization conditions (including the AWI and the Field Equation) and proceeding by induction on n, the anomalous term (i.e., the possible violation of the MWI) is given by The task is to remove ∆ n B 1 (x 1 ), . . . ; y 0 by a finite, admissible renormalization of T n+1 B 1 (x 1 )⊗ · · ·⊗B n (x n )⊗j µ (y) 0 . By "admissible" we mean that the basic axioms and the above mentioned renormalization conditions are maintained.
Step 1: Similarly to [6, Exer. 5.1.7] one shows that where the time t ∈ R is arbitrary and [ · , · ] ⋆ denotes the commutator w.r.t. the star product.
As it become clear below in (5.6), a necessary condition for the asserted MWI (3.6) is charge number conservation, which is a generalization of the relation (5.2) to time-ordered products of order n ≥ 2, explicitly: It is quite easy to show that (5.3) is a renormalization condition on T n (which is weaker than the MWI), and that it can be fulfilled and is compatible with all other renormalization conditions. So we assume in the following steps, that the T -products satisfy also (5.3). Note that we work here with the charge number operator θ only; θ µ does not play any role here.
Step 2: In this step one proves For a given configuration (x 1 , ..., x n ) ∈ M n let O ⊂ M be an open double cone with x 1 , . . . , x n ∈ O; in addition let g be an arbitrary test function satifying g| O = 1. Since the support of ∆ n B 1 (x 1 ), . . . , B n (x n ); y is contained in the thin diagonal x 1 = . . . = x n = y, we may write where we have integrated out the δ-distributions. Compared with [6, eqn. (5.2.20)], there is an additional term appearing in the last line. However, due to g| O = 1, this term vanishes. So we may continue as in that reference: using causal factorization of the T -product, ∂ µ j µ 0 = 0 and spacelike commutativity one derives that Looking at (5.5), we see that charge number conservation (5.3) implies the assertion (5.4).
Step 3: Following the proof of the QED-MWI we list some structural properties of the anomalous term ∆ n (· · · ) defined in (5.1), for the validity of these properties see the above mentioned references.
First, ∆ n (· · · ) satisfies the causal Wick expansion: where the sum runs over all submonomials B l of B l (where 1 ≤ l ≤ n), and If one of the B j 's is linear in the basic fields, e.g., B 1 = ∂ a φ with a ∈ N 4 , the axiom FE determines uniquely T n+1 ∂ a φ(x 1 ) ⊗ · · · ⊗ j µ (y) , that is, there is no freedom to remove ∆ n ∂ a φ(x 1 ) ⊗ · · · by a finite renormalization of this T -product. However, as verified in [6,Exer. 4.3.3], the validity of the axiom FE implies that ∆ n (· · · ) vanishes in this case. Therefore, on the r.h.s. of (5.7) the sum is restricted to submonomials B l of B l which are at least quadratic in the basic fields for all l.
Because ∆ n B 1 (x 1 ), . . . , B n (x n ); y is supported on the thin diagonal, the pertinent VEV  Step 4: Obviously, the finite renormalization removes the anomalous term d n (B 1 , . . . , B n ). Looking at the causal Wick expansion of ∆ n (5.7) we conclude: performing the finite renormalization (5.13) for all B 1 , . . . , B n ∈ P 0 being at least quadratic in the basic fields, the MWI (3.6) is proved for all B 1 , . . . , B n ∈ P 0 , provided that all these finite renormalizations are admissible, that is, they maintain the basic axioms and the renormalization conditions (v)-(viii), AWI and FE. This is obvious or easy to check for nearly all of these axioms; for example, that t n+1 (. . . , j µ )+ (−i) n u µ n (. . .) fulfills the axiom Scaling degree follows from (5.12). There is only one exception: if at least one of the B j 's is a current j ν , it is not clear whether the finite renormalization (5.13) preserves the invariance of t n+1 (B 1 , . . . , B l , j ν 1 , . . . , j ν k , j µ )(x 11 − y, . . . , x 1l − y, x 21 − y, . . . , x 2k − y) (5.14) (where l + k = n) under the permutations of the entries pertaining to the currents, that is, under (x 2r , ν r ) ↔ (y, µ) for all 1 ≤ r ≤ k. This permutation invariance is required by the basic axiom (iii) Symmetry. Taking additionally into account that u µ n (B 1 , . . . , B n ) is not uniquely determined by d n (B 1 , . . . , B n ) (one may add to u µ n some u µ n with ∂ y µ u µ n = 0), the remaining task can be formulated as follows: for any B 1 , . . . , B l ∈ P 0 \ { j ν } being at least quadratic in the basic fields, with l ≤ n − 1, and satisfying (where dim j ν = 3 is used) we have to find distributions u µ n (B 1 , . . . , B l , j ν 1 , . . . , j ν k ) having the same permutation symmetries and the same Lorentz covariance properties as t n+1 (B 1 , . . . , B l , j ν 1 , . . . , j ν k , j µ ), and which fulfil the equation (5.9) for d n (B 1 , . . . , B l , j ν 1 , . . . , j ν k ) given by (5.11).
For some n-tuples (B 1 , . . . , B l , j ν 1 , . . . , j ν n−l ) satisfying (5.15), we know that d n (B 1 , . . . , B l , j ν 1 , . . . , j ν n−l ) = 0 due to charge number conservation (CNC) or Furry's theorem (FT). In detail: FT Charge conjugation is a linear operator β C : F → F which is given by the relations where η C ∈ { z ∈ C |z| = 1 } is a fixed number, and by Charge conjugation invariance is the condition on the T -product. One verifies that this is an additional renormalization condition, which can be fulfilled such that all other renormalization conditions are preserved.
Furry's theorem is a consequence of (5.17), obtained by using that ω 0 • β C = ω 0 . It states: There remain the n-tuples listed in the following table, that we shall study case-by-case. In each case, the distribution u µ n (· · · ) for the finite renormalization in (5.13) stands for a Lorentz tensor of rank ≥ 2 according to the entries B 1 , . . . , B n . It is a multiple of the total δ-distribution in all arguments in all cases except Case 1, where the scaling degree admits two derivatives. Therefore, we begin with the simpler cases 3 and 2, before we turn to the more delicate case 1. We use the labelling of the x variables (i.e., of the arguments of B 1 , . . . , B n ) indicated in (5.14). , j ν 1 , j ν 2 , j ν 3 1 2a , A∂φ, A 2 φ * , j ν 1 3 Cases 3: By (5.9)-(5.12), the renormalization is a Lorentz tensor u µν n of rank 2. The only possibility is u µν n (x 11 − y, . . . , x 2 − y) = C g µν δ(x 11 − y, . . . , x 2 − y), for some C ∈ C. Obviously, u µν n (. . . , x 2 − y) is invariant under (ν, x 2 ) ↔ (µ, y), hence, the finite renormalization (5.13) is admissible in this case.

(5.21)
Hence, in cases 2a and 2b we have accomplished that u µν 1 ν 2 ν 3 n has the needed permutation symmetries to be an admissible finite renormalization.
In the case 2c, the symmetry of u µν 1 ν 2 ν 3 n under permutations of the three currents requires C 1 = C 2 = C 3 , which is not already secured by (5.21). To complete the proof for case 2c, we claim that is invariant under x 23 ↔ y. To verify this claim, we insert the definition of d n (5.11). Up to a global prefactor i n we obtain for ∂ x 23 ν 3 d n the following sum of terms: Obviously, the terms (5.23) and (5.26) are individually invariant under x 23 ↔ y. To show this for the sum of the remaining terms, we insert the MWI to order (n − 1), which holds by induction: We see that (5.27) is separately invariant under x 23 ↔ y and that the sum (5.28)+(5.29) also has this symmetry. Hence, the asserted symmetry of (5.22) holds indeed true. So we know that 0 = ∂ x 23 ν 3 ∂ y µ u µ n (L, . . . , A ν 1 φ * φ, j ν 2 , j ν 3 )(x 11 − y, . . . , x 21 − y, . . . , x 23 − y) − (x 23 ↔ y). Inserting the formula (5.19) for u µν 1 ν 2 ν 3 n into this expression and taking into account that C 2 = C 3 (5.21), we obtain that C 1 = C 2 (= C 3 ). Hence, also in the case 2c, u µν 1 ν 2 ν 3 n is an admissible finite renormalization. where u µν n is a Lorentz tensor of rank 2 which is a polynomial in derivatives of the δ-distribution of order ≤ 2. Since terms ∼ ε µναβ ∂ x i α ∂ x j β are ruled out by their antisymmetry, u µν n must be of the form for some a ij , b ij , c 0 ∈ C and with i, j ∈ {11, ..., 1m, We have to show that u µν n is invariant under (ν, x 2 ) ↔ (µ, y). Obviously, for the c 0 -term this holds true; hence we may omit this term in the following.
Since the l.h.s. of (5.30) is invariant under permutations of x 11 , . . . , x 1m , we may assume that u µν n shares this permutation symmetry. We now write down all possible contributions with two derivatives to u µν n satisfying this symmetry: where k, l ∈ {11, ..., 1m}. Obviously, these 9 differential operators -each one applied to the δ-distribution in (5.31) -are linearly independent, hence they form a basis of a vector space. We now give a different set of 9 differential operators, whose elements have a much simpler behaviour under (ν, x 2 ) ↔ (µ, y): where again k ∈ {11, ..., 1m}. By using we can express all elements of the (old) basis (5.32) as linear combination of the new terms (5.33); therefore, the latter are also a basis of the same vector space. Under (ν, x 2 ) ↔ (µ, y) all differential operators in the group (1) (first line of (5.33)) are individually invariant, hence the pertinent contributions to u µν n are admissible finite renormalizations.
To treat the remaining four terms in group (2) (second line of (5.33)), we proceed analogously to (5.22): we claim that To verify this, we insert (5.30) into (5.34): obviously, the ∂ x 2 ∂ y t n+1 (. . . , j, j)-term and the ∂ x 2 ∂ y δ(y − x 2 ) t n (. . . , φ * φ) -term fulfil the claim individually. To show this for the remaining term, we use the MWI to order (n − 1): from which we see that also this term satisfies the claim (5.34). We conclude that the contribution from the four terms in group (2) (second line of (5.33)) satisfies Working out this condition, we obtain the relation Thus eliminating C 1 , we find that the remaining anomaly has the most general form At this point, we exploit the freedom to change u µν without changing ∂ y µ u µν . This allows us to replace C 2 (. . . ) + C 3 (. . . ) + C 4 (. . . ) by which still cancels the anomaly, and enjoys the required symmetry under (y, µ) ↔ (x 2 , ν).

Relation between the Master Ward identity and its simplified version
Our aim is to generally relate the time-ordered products T and T and to establish that the validity of the MWI for T (3.6) is equivalent to the validity of the WI for T (2.2) -to all orders and including all loop diagrams. We assume that B 1 , . . . , B n ∈ P are eigenvectors of θ (2.1), 1) and that each of these field polynomials contains at most one derivated basic field, that is, Obviously these two assumptions are true for all elements of the set P 0 , for which we have proved the validity of the MWI in Sect. 5.

Complete definition of the finite renormalization.
In [11] it was investigated how the addition ig µν δ to ∂ µ ∂ ν ∆ F propagates to higher orders in the inductive Epstein-Glaser construction of the sequence ( T n ). This was done there only for tree-like diagrams; more precisely, diagrams consisting of two components which are connected only by one internal φ-line with two derivatives, and the consequences of the addition ig µν δ to this line were studied. By virtue of the Main Theorem of Renormalization ( [6, Eq. (3.6.25)] and Thm. A.3), we are able to give a general definition of the higher orders T ≡ ( T n ) -in particular, the inner φ-line with two derivatives may be part of a loop. In fact, we shall do more, by showing in Thm. 6.3 the equivalence of a one-parameter family of Ward identities for a family of time-ordered products T c , continuously interpolating between T = T c=0 and T = T c=1 . The stronger result for all c ∈ R was suggested by the analogous result [16,Sect. 3.3] found at tree-level (where all renormalizations are fixed by the time-ordered 2-point functions of the derivative fields, and the absence of anomalies can be seen explicitly) in a different but presumably equivalent setup of scalar QED, using "string-localized" potentials.
To interpolate between the time-ordered products T and T , we multiply the addition ig µν δ(x− y) in the finite renormalization (2.6) by a number c ∈ R, and denote the interpolating timeordered product by T c . To formulate completely the so-modified finite renormalization -in particular its consequences for the higher orders in the inductive Epstein-Glaser construction of T c ≡ ( T c,n ) -we work with a finite renormalization map Z c ≡ (Z (k) c ), which is an element of a version of the Stückelberg-Petermann renormalization group (defined in Appendix A.2), and we will use the Main Theorem of Renormalization (given also in that Appendix in Theorem A.3).
To first order an element Z c of the Stückelberg-Petermann group is given by Z To second order, T c,2 differs from T 2 only by the finite renormalization ∂ ν ∂ µ ∆ F → ∂ ν ∂ µ ∆ F + c ig µν δ in the connected tree-diagram part; hence we define .
For later purpose we note that For n ≥ 3 the difference between T c,n and T n is only the one coming from the propagation of Z (2) c to higher orders; hence the higher orders of Z c vanish, that is, We point out: Z (2) c does not fulfil the AWI and the property "Field Equation", because T 2,c violates these relations. Comparing with the definition of the Stückelberg-Petermann group in the mentioned references, the Z c defined above is an element of a modified version of that group; this is explained in detail in parts A.2-A.3 of the Appendix.
The equality (6.13) can be understood in terms of Feynman diagrams: there are diagrams contributing to the second factor on the r.h.s. of (6.12), i.e. T (⊗ k L(y k )) ⊗ j µ (x) 0 , in which the field vertex x is connetced to an interaction vertex y k by an internal φ-line with two derivatives and this line is not part of any loop. The addition ig µν δ to this line in these diagrams generates the additional term 2eg(x)(A µ φ * φ) S (x) 0 of J µ S (x) 0 . From the identity (6.13) and ∂ x µ J µ S (x) 0 = 0 (4.6), we see that j µ (g, L),0 is conserved: Alternatively, this result can directly be obtained, i.e., without using J µ S ; namely, from the WI for T (2.2), by proceeding analogously to the derivation of ∂ x µ J µ S (x) 0 = 0 from the MWI (4.4). Hence, with regard to the interacting electromagnetic current the WI (2.2) and the MWI (3.6) (or (4.4)) contain the same information. This result can strongly be generalized -this is the topic of the next subsection.
6.2 The MWI for T (3.6) and the WI for T (2.2) are equivalent We are now coming to the second main result of this paper. Theorem 6.3. Let a time-ordered product T be given and let a time-ordered product T c be defined in terms of T and Z c by (6.3)-(6.5) and (6.8). Then, for all B j ∈ P satisfying the assumptions (6.1) and (6.2), the validity of the MWI (3.6) for T is equivalent to the validity of the following c-dependent WI for T c -to all orders n ∈ N and for all c ∈ R: For c = 1 the assertion (6.15) agrees with the WI (2.2) and for c = 0 with the MWI (3.6). In particular we obtain with θ µ defined in (3.4). Expressing T c in terms of T on the l.h.s. of (6.15), we get two types of terms: in the first type j µ does not appear in the argument of any Z (2) c , in the second type it does and, hence, we may use (6.17): wherel means that l is omitted in the pertinent set. Now we insert the MWI (3.6) into (6.18): for the θ-terms (displayed in (6.20)) we use (6.1) and (6.4)-(6.6), the latter imply For the θ µ -terms (displayed in (6.21)) we take into account that θ µ ζ(B 1 , B 2 ) = 0, which follows from (6.2), and we reorder the summations. So we obtain: Finally, we reexpress T in terms of T c . Due to (6.2) and (6.4)-(6.5), it holds that Hence, we obtain So we see that the sum of the terms (6.19) and (6.21) is equal to i n+1 ·[(c − 1)-term on the r.h.s. of the assertion (6.15)]. And, the expression (6.20) is equal to δ(y − x l ) T c,n B 1 (x 1 ) ⊗ · · · ⊗ (θB l )(x l ) ⊗ · · · ⊗ B n (x n ) 0 , by using (6.1). WI (6.15) for T c =⇒ MWI (3.6) for T : First we verify that, for B j 's satisfying (6.2), the "inverse" of Z c ∈ R (see Def. A.2 for the definition of R) is Y c ∈ R given by Y Therefore, T n can be expressed in terms of ( T c,k ) 1≤k≤n and Y c by the formula (6.8): T and T c are mutually exchanged and Z c is replaced by Y c . With this, the assertion (i.e., the MWI (3.6) for T ) can be verified by essentially the same computation as in the above proof of the reversed statement: to compute ∂ y µ T n+1 · · · ⊗ j µ (y) we first express T n+1 in terms of T c , then we use the WI (6.15) for T c and finally we reexpress T c in terms of T .

A.1 Axioms for the time-ordered product
Both time-ordered products T and T , used in the main text, satisfy the following definition: fulfilling certain axioms -the basic axioms and the renormalization conditions. The former read: (i) Linearity: T n is linear (that is, multilinear in (B 1 , . . . , B n )); (ii) Initial condition: T 1 B(x) = B(x) for any B ∈ P ; (iii) Symmetry: For all permutations π of (1, . . . , n) it holds that (iv) Causality. For all B 1 , . . . , B n ∈ P, T n fulfills the causal factorization: We work with the following renormalization conditions: (v) Field independence: and similarly for δ δφ * (z) and δ δA µ (z) . This axiom is equivalent to the requirement that T n satisfies the causal Wick expansion, see [6,Chap. 3.1.4].
(vi) * -structure: To formulate this axiom, we introduce the S-matrix to the interaction it is the generating functional of the time-ordered products: The axiom * -structure reads where G ⋆−1 is the inverse w.r.t. the star product of G ∈ F. For a real interaction (i.e., g j = g j , B * j = B j for all j) this axiom requires unitarity of the S-matrix.
(vii) Poincaré covariance: where (Λ, a) → β Λ,a is the natural representation of P ↑ + on F (see [6,Chap. 3.1.4]). An immediate consequence of translation covariance is that the C-valued distributions The time-ordered product T , underlying Sects. 3, and 5, fulfils additionally the following two renormalization conditions. AWI Action Ward Identity: which implies that T n can be interpreted as a map and analogously for φ replaced by φ * or A µ .

A.2 Stückelberg-Petermann renormalization group R and Main Theorem of Renormalization
In this paper we work with that version of the Stückelberg-Petermann renormalization group (SP-RG) that describes finite renormalizations of time-ordered products satisfying the renormalization conditions (v)-(viii) given in the preceding Sect., however, they may violate the AWI and the FE. Due to the absence of the AWI, the arguments of the elements of the SP-RG cannot be written as local functionals, as it is done in [3,10]  Z ≡ (Z (n) ) ∞ n=1 : being linear (that is, multilinear in (B 1 , . . . , B n )) and symmetric in the sense that for all permutations π of (1, . . . , n). In addition, the maps Z (n) are required to satisfy the following properties for all B, B 1 , . . . , B n ∈ P and for all n ≥ 1: (1) Lowest order: (2) Locality: the support (in the sense of distributions) of every Z (n) B 1 (x 1 ) ⊗ · · · lies on the thin diagonal, that is, (3) Field independence: and similarly for δ δφ * (z) and δ δA µ (z) . This property is equivalent to the validity of the (causal) Wick expansion for Z (n) .
(b) Conversely, given a time-ordered product T fulfilling the axioms and an arbitrary Z ∈ R, the sequence of maps T ≡ ( T n ) ∞ n=1 defined by (A.8) satisfies also the axioms for a timeordered product.
If one selects from the relation (A.8) the terms of order n in the B j 's for a Z ∈ R satisfying Z (k) = 0 ∀k ≥ 3 (as it holds for Z c (6.7)), then one obtains precisely the equation (6.8).
In this paper we only prove part (b) of this Theorem and only for the particular family of elements Z c of the SP-RG, given in (6.3)-(6.7); this is done in section A.4.
A.3 Verification that the concretely given Z c lies in R Z (1) c is uniquely determined by the defining property (1) of the SP-RG R. In this section we verify that Z (2) c , concretely given in (6.3)-(6.5), satisfies the defining properties for Z (n) c given above; this implies then that Z c := (Z  = i n T c,k ⊗ k j=1 B j (x j ) ⋆ T c,n−k ⊗ n r=k+1 B r (x r ) .
Renormalization conditions. The validity of Field independence and Poincaré covariance for T c follows straightforwardly from the corresponding properties of T and Z c . The * -structure axiom for T c can equivalently and much simpler be expressed in terms of the pertinent retarded product R c , to wit R c,n−1,1 ⊗ n−1 j=1 B j (x j ); B(x) * = R c,n−1,1 ⊗ n−1 j=1 B * j (x j ); B * (x) .
This claim follows immediately from the same formula for R (i.e., the retarded product belonging to T ) and the * -structure property of Z c , when working with the Main Theorem formula for ( R c , R) (see [6,Exer. 3.6.11] for the latter formula).
To prove that T c satisfies the axiom Scaling degree, first note that, up to permutations of (B 1 (x 1 ), . . . , B n (x n )) and the prefactor i |P | , every summand of δ(x j − x r+j ) (A.9) for some 0 ≤ r ≤ n/2. Next note that for B 1 , B 2 ∈ P hom it holds that ζ(B 1 , B 2 ) ∈ P hom and that dim ζ(B 1 , B 2 ) ≤ dim B 1 + dim B 2 − 4.
Using additionally the Scaling degree axiom for T and the formulas sd δ(x j − x r+j ) = 4 and sd(f 1 ⊗ f 2 ) = sd(f 1 ) + sd(f 2 ), we see that the scaling degree of the expression on the r.h.s. of (A.