Conformal anomaly from gauge fields without gauge fixing

We show how the Weyl anomaly generated by gauge fields, can be computed from manifestly gauge invariant and diffeomorphism invariant exact renormalization group equations, without having to fix the gauge at any stage. Regularisation is provided by covariant higher derivatives and by embedding the Maxwell field into a spontaneously broken $U(1|1)$ supergauge theory. We first provide a realisation that leaves behind two versions of the original $U(1)$ gauge field, and then construct a manifestly $U(1|1)$ supergauge invariant flow equation which leaves behind only the original Maxwell field in the spontaneously broken regime.


Introduction
Over a period of some years, a framework has been developed for gauge theory which allows continuum computations without fixing the gauge. This is achieved by utilising the freedom to design manifestly gauge invariant versions of the continuum realisation of Wilson's renormalization group (christened exact RG in ref. [1]). Such manifest gauge invariance was first incorporated into the exact RG in ref. [2], however in the limited context of pure U (1) gauge theory. Following ref. [3] it was generalised and extensively studied first for SU (N ) Yang-Mills theory, then QCD [4] and QED [5,6]. For these gauge theories, regularisation is based on gauge-invariant higher derivatives set at some ultraviolet cutoff scale Λ, supplemented by gauge invariant Pauli-Villars fields [7] with particular flavours and interactions so that their regularisation properties are preserved under RG flow. It was later realised that the resulting structure could be simply understood as arising from spontaneously broken SU (N |N ) super-Yang-Mills theory [8,9]. In this scheme, the original gauge field A 1 µ is joined by a copy gauge field A 2 µ with wrong sign action, and a complex fermionic (i.e. wrong-statistics) gauge field B µ : This extra regularisation works because these degrees of freedom cancel each other, as happens with Parisi-Sourlas supersymmetry [10], at least sufficiently that, together with appropriately chosen covariant cutoff functions, the theory is then regularised to all orders in perturbation theory [11][12][13].
The symmetry is then broken spontaneously along the fermionic directions, endowing the B µ with a mass at the cutoff scale Λ. The computational methods were generalised in refs. [14][15][16][17][18] so that universal results could be extracted in a way which was manifestly independent of the detailed form of the regularisation structure, and such that general group invariants could be handled [19]. Using these techniques, the initial computation of the one-loop β function at infinite N [3] was generalised to finite N [15,18,20,21], then to two loops [19,[22][23][24][25], extended to all loops in refs. [26,27] and to computation of gauge invariant operators in refs. [28,29]. For reviews and further advances see refs. [30][31][32].
In ref. [33] the first steps were made in generalising these ideas so as to yield a manifestly diffeomorphism invariant exact RG for use in quantum gravity. 1 On the one hand the renormalization group structure of quantum gravity is surely of importance [35][36][37][38] and on the other hand one can expect conceptual and computational advances from a framework which allows computations to 1 For an alternative attempt, see ref. [34].
be done while keeping exact diffeomorphism invariance at every stage, i.e. without gauge fixing.
Indeed as shown in ref. [33], it turns out that these computations can then be done without first choosing the space-time manifold and in particular without introducing a separate background metric dependence. A solution to the difficult issue of background independence is thus automatic in this formalism. 2 However, only the flow equation for classical gravity was developed in ref. [33]. In this paper we make the first step towards including manifestly diffeomorphism invariant quantum effects involving gravity.
We will be concerned with the conformal, a.k.a. Weyl or trace, anomaly [46][47][48] 3 generated by gauge fields. Although this does not involve dynamical gravity, it is clearly important to understand how the known universal answer can arise in this framework, i.e. such that gauge invariance is maintained at all stages. Indeed, since the conformal anomaly can be read off from the logarithmically divergent curvature-squared terms at one loop, it is proportional to the signed number of fields (i.e. with fermionic fields appearing with opposite sign). Thus in the usual calculation [46][47][48], the ghost degrees of freedom are indispensable. The manifestly gauge invariant formalism reviewed above, proceeds without ghosts. Thus the question arises: working on a curved spacetime, is gauge fixing now necessary to recover the correct Weyl anomaly or not? As we will see in fact the correct Weyl anomaly is reproduced without gauge fixing. This is thus a dramatic confirmation of a formalism that was developed and tested only in flat space calculations. Needless to say, it is also the first time that a manifestly gauge invariant computation has been achieved for the gauge field contribution to the conformal anomaly.
For this exercise it is sufficient to consider Maxwell theory, i.e. free U (1) gauge fields. As already mentioned, manifest gauge invariance can be straightforwardly incorporated in flow equations for pure U (1) gauge theory in flat space [2], where in fact only the gauge field A 1 µ appears. Even for manifestly gauge invariant QED, the gauge field degrees of freedom are not altered or supplemented: only the Dirac fields need regularisation with opposite statistics Pauli-Villars partners [5,6]. This is because it is straightforward to regularise a U (1) gauge field gauge invariantly using only a cutoff profile c which is a function of partial derivatives rather than covariant derivatives: (Throughout we will be working with Euclidean signature.) However the arguments above already show that such a framework could not possibly give the correct Weyl anomaly. In fact once we use a non-flat metric (and thus replace the partial derivatives in c −1 with covariant derivatives) we introduce interactions with the metric which destroy the regularisation, since this is then again effectively covariant higher derivative regularisation, which is known to fail at one loop [33,52,53].
Therefore even for pure Maxwell theory, we need a wrong-statistics counterpart to play the rôle of the Pauli-Villars field. Following the same chain of reasoning as reviewed above, in order for this to be embedded in an exact RG framework, we are led to developing versions of spontaneously broken U (1|1) theory for this purpose.
As we will see the wrong-statistics fields that are introduced then ensure the correct Weyl anomaly. In fact the result can be directly compared to a more conventional calculation, although only after rearranging contributions from the wrong-statistics vector and Goldstone fields, reflecting the fact that the supergauge invariance, while spontaneously broken, is nevertheless manifest throughout.
Actually, a wholesale adaptation of the previously developed manifestly gauge invariant methods is not quite what we want, because the A 2 µ sector is left unbroken. For the purposes of flat space computations in non-Abelian Yang-Mills theory, this is not a problem [11-13, 54, 55] because all interactions with this sector are irrelevant, starting with (with indices contracted in some way), and thus the A 2 sector decouples in the continuum limit, providing we work in D ≤ 4 dimensions [13]. For a computation of the conformal anomaly however, and more generally the purely gravitational action at one loop, the A 2 sector will also contribute and thus we expect to find twice the right answer whatever the space-time dimension. 4 We will confirm that this is indeed the case.
While the above framework is enough to work out the Yang-Mills contribution to the pure gravitational action at one loop, its use would clearly be limited beyond this while the unphysical A 2 sector remains as part of the continuum theory. We therefore also build an alternative formulation with spontaneous breaking of both the B µ and the A 2 µ field so that all these fields gain masses at the regularisation scale Λ. This thus leaves only the original unbroken Maxwell field A 1 µ at low energies, which, as we will see, then gives exactly the correct value for the conformal anomaly.

Differential operators
Before proceeding, it is useful to collect together properties of the curved space differential operators that will naturally appear when working with gauge fields. For scalar fields ω, the operator that naturally appears, e.g. as the kernel in the kinetic term, is just the Laplace-Beltrami operator (2.1) (With the sign, it is positive semi-definite.) However for a, e.g. U (1), gauge field 5 the kernel from the simplest action , is the differential operator ∆ T 1 = δd, where d is the exterior derivative, and δ the co-differential. In components: Here we have also introduced the (positive semi-definite) Laplace-de Rham operator which on a one-form is explicitly (coinciding with the Lichnerowicz Laplacian). Abelian gauge invariance (i.e. d 2 = 0) ensures that ∆ T 1 annihilates longitudinal one-forms, as is easily explicitly verified: On the other hand since d and δ commute with (d+δ) 2 , while on a scalar field de Rham = Beltrami: we must have: as is also readily verified using the component formulae. Thus using (2.3), we see that ∆ 1 and ∆ T 1 commute. Ignoring normalisable zero-modes (or working on a manifold which has none), Π L = d 1 (d+δ) 2 δ is a longitudinal projector for one-forms, equivalently Therefore the transverse projector is By d, δ algebra, or using (2.8), we have i.e. ∆ T 1 is just the transverse projection of ∆ 1 . Splitting the transverse eigenmodes of ∆ 1 are the non-zero eigenmodes of ∆ T 1 , while longitudinal eigenmodes of ∆ 1 are eigenmodes of ∆ 0 A L = λA L , since then As a result a trace involving ∆ 1 projected into the transverse modes can be expressed as while the trace over the longitudinal sector is

Manifestly gauge invariant flow equation on a curved spacetime
We give a brief review of manifestly gauge invariant flow equations for Yang-Mills theory, making some minimal adaptations so that they apply to Maxwell theory propagating in a curved spacetime.
As explained in the introduction, this will actually yield a U (1) × U (1) theory, where the second copy has wrong sign action. From this we can nevertheless extract the one-loop pure gravitational contribution. Then in sec. 8 we will give an improved flow equation which leaves behind only a single physical Maxwell gauge field.
Recall that the basic idea is that the flow of the Boltzmann measure exp(−S) should be a total functional derivative, i.e. for some generic fields φ: (corresponding to the statement that each RG step is equivalent to an infinitesimal field redefinition [3,56]. Importantly, this ensures that the partition function Z = Dφ exp(−S), and hence the physics derived from it, is invariant under the RG flow. Working with a fixed background metric g µν and in general D dimensions, we will show how to generalise the previous formulations for Yang-Mills while preserving this crucial feature, and in such a way that the solution remains straightforwardly calculable.
As we sketched in the introduction the previous formulations were developed over a number of years to cope with the most general cases. For our purposes we can closely follow the one set out in ref. [18]. Indeed we will see that the flow equation still takes the generic form S being the so-called seed action, and g being the gauge coupling which, since we work in general D spacetime dimensions, has mass dimension 2 − D/2. The coupling has been factored out so that it plays the rôle of a loop counting parameter: the loop expansion of the effective action being given by where S 0 is the classical effective action, S 1 the one-loop correction, and so on. This ensures that (super-)gauge invariance is manifestly maintained at each order in g. As a consequence the super-covariant derivative is given by The bilinear functional operator that generates the tree-level contributions is manifestly supergauge invariant: as is the linear functional that generates the loop corrections: These expressions are exactly as in ref. [18]. 6 We now explain what the various terms mean. Unlike in ref. [18], we are interested in regularising U (1) gauge theory. We therefore need to use a gauge field A valued as a generator of U (1|1). The gauge field therefore takes the same form as given in eqn. (1.1), except that here each field is thus a single component rather than a matrix and, unlike in the SU (N |N ) case, there is no resulting restriction to strA = 0. The same goes anyway for the superscalar field which will inflict spontaneous symmetry breaking on the supergauge invariance, breaking it down to the diagonal U (1) × U (1) carried by the bosonic gauge fields A i µ , while supplying cutoff size masses to the complex fermionic B µ field which thus turns into a gauge invariant Pauli-Villars regulator field. As explained in ref. [18], an elegant way to impose that the spontaneous symmetry breaking scale tracks the cutoff scale Λ, is to take C to be dimensionless, so we will do the same here. Then we can choose a potential so that the effective vacuum expectation value can be [18]: at any scale Λ. Following previous convention, we write the third Pauli matrix as This matrix appears frequently also as a result of the supergroup symmetry, for example through the supertrace: strX := tr(σX), (3.11) X being a supermatrix and str being the supergroup invariant version the trace. The result of (3.9) is precisely to give a mass ∼ Λ to the off-diagonal entries in A i.e. to the complex B field.
Since for the U (1|1) theory, A is not subject to a constraint, both A and C functional derivatives are freely acting and are thus defined as follows: 12) or in components δ δC the supergauge functional derivatives being defined in the same way. The advantage of this definition is that the U (1|1) invariance remains manifest, for example we have: where X and Y are arbitrary constant supermatrices [8,18].
In order to maintain local supergauge invariance in (3.2) it is then only necessary to ensure that the bi-local kernels˙ AA (x, y) and˙ CC (x, y) are suitably covariantized by including A interactions, after which an invariant is constructed by taking an overall supertrace. This is essentially the meaning of the curly brackets. In fact it proved helpful to extend the definition so that 7 where X(x) and Y (y) are supermatrix fields produced by the functional derivatives in (3.6), is a new kernel that simplifies calculations in the broken phase, and {· · · } A stands for the gauge covariantization just described. In flat space, the most general form of gauge covariantization is described in ref. [18], following [7,8,15], and can be couched in terms of a path integral over Wilson lines. We will not need the details for this paper. However we will need a covariantization to cope with a non-trivial metric. The most general case can again be couched in terms of an integral over Wilson lines for the Levi-Civita connection, as remarked in ref. [33]. In this latter paper we however made the simplest choice, promoting space-time partial derivatives to covariant derivatives in a prescribed way (corresponding implicitly to some specific choice of measure for the Wilson lines). We will do something similar in this paper. Thus for a scalar flat-space kernel: we make the replacement where ∆ 0 is the Laplace-Beltrami operator introduced in (2.1). Following the framework of e.g.
ref. [33], the factor of 1/ √ g is inserted to give the kernel the correct overall density of weight -1 so that combined with the √ g factors from the two functional derivatives in (3.6) and after integrating over x and y (without further factors √ g) the result is clearly generally covariant. (Note that ∇ α commutes with 1/ √ g which can have either x or y dependence, and thus the kernel is symmetric as assumed.) Recognising that the vector kernel˙ AA is associated with one-forms, the elegant choice is to make this a function of the Laplace-de Rham operator ∆ 1 , cf. (2.5). We will see in the remainder of the paper how this choice ensures that computations remain almost as simple as their flat-space counterparts. As in ref. [18], we discard the terms where the left-most C functional derivative in (3.7) hits the C decorations in (3.17). This can be imposed by a limiting procedure [3], see also [7,8].
S is used to determine the form of the classical effective kinetic terms and the kernels˙ . It therefore has to incorporate the covariant higher derivative regularisation and allow the spontaneous symmetry breaking we require. As we will review shortly, the kernels˙ are determined by the requirement that after spontaneous symmetry breaking, the two-point vertices of the classical effective action S 0 , andŜ can be set equal. This is imposed as a useful technical device, since it allows classical vertices to be immediately solved in terms of already known quantities.

Kernels and two-point vertices in a curved background
In this paper we are interested only in the one loop contribution to pure gravity. This arises by first solving for the classical action S 0 . For this we extract the 1/g 2 part of (3.2), using (3.3) and (3.4): Since the above gives structurally the same equations for the flow of the two-point vertices as in ref. [18], we can therefore follow closely in this section the derivations given there. We thus split the supermatrix fields into their block (off-)diagonal components Choosing a single supertrace form forŜ, we need to determine the differential operators that form the two-point verticesŜ AA µν ,Ŝ BB µν ,Ŝ BDσ µ ,Ŝ CC andŜ DD [18]. In each case the superscript gives the order of the fields as they appear in the supertrace (up to cyclicity), for exampleŜ BDσ µ (∇) sits in the seed action as the term where we have used cyclicity of the supertrace to put σ first. The flow equations resulting from (where the last two follow from the third by spontaneously broken supergauge invariance) and where the kernels˙ are also of the same form except that following the standard convention in gravitation, see e.g.

eqns. (2.2) and (2.3)
, the indices on the differential operatorsŜ AA ,Ŝ BB ,˙ AA , and˙ BB have been suppressed where it is unambiguous to do so, and in the last line of (4.5) we recognise that the first vertex that appears on the right hand side needs now to be distinguished from the second (as discussed below).
The only changes to the solutions found in ref. [18] are thus induced by the covariantizations of the seed-action two-point vertices required to cope with the background metric g µν . We are free to choose these. Thus we set andŜ where λ > 0 is a constant dimensionless parameter [18], and c i = c(∆ i /Λ 2 ) andc i =c(∆ i /Λ 2 ) are cutoff functions [18] of the appropriate Laplace-de Rham operator. Similarlŷ where we used (2.8). The second version follows from integration by parts in (4.4), followed by cycling the supertrace and anticommuting σ. In ref. [18] it was not needed, since in flat space the two coincide.
Substituting (4.8), (4.9) and (4.10) into (4.5), yields the kernels, and thus the integrated kernels defined via˙ The integration constant is determined by ensuring that the corresponding vanish for large eigenvalue. We now show that we get for the (integrated) kernels, the obvious covariantization of the results found in [18]. The first two equations in (4.5) are solved by straightforward integration: where the second is the inverse ofŜ AA in the transverse space. Multiplying the third equation in (4.5) by the transverse projector Π T , isolates the inverse of Π TŜ BB in the transverse space. Substituting for the vertices and rearranging the last equation in (4.5) gives where BB 0 ≡ BB (∆ 0 ), i.e. (4.13) with ∆ 1 replaced by ∆ 0 . The above formulae are indeed direct maps of the results in ref. [18].

Twice the manifestly gauge invariant conformal anomaly
From (3.2), (3.3) and (3.4), and the equality of classical and seed-action two-point vertices, we have that the purely gravitational part of the one-loop effective action is computed from The functional derivatives in (3.7) are evaluated using (3.16), after expressing the block (off)diagonal fields in terms of the originals via (4.3). The supergroup contribution is then ± 1 2 (strσ) 2 = ±2, i.e. the signed number of each flavour. Combined with the sign in (5.1) and the 1/2 from (3.7), we thus have that the purely gravitational piece satisfies: where Tr stands for a space-time trace, taking into account the relevant Lorentz representation.
Thus for A this is a trace over transverse modes, for B a trace over all vector modes, and for C and D it is a trace over scalar modes. The bosonic contributions are straight-forward to simplify using the equations of the previous section: Noting the first equation in (4.14), we have while using the first equation of (2.3) and cyclicity of the spacetime trace, Combining the last terms in the above two equations givesŜ BB 0˙ BB 0 , which again simplifies. Thus, substituting everything back into (5.2), we can trivially integrate with respect to Λ. Also cancelling Tr ln Λ 2 between D and C sectors, we thus get From (2.15), the third term on the right hand side is just subtracting the longitudinal B contribution. Indeed using (2.14), we can alternatively write: In this form we recognise that the result coincides with twice what would be produced by more conventional calculational methods, reflecting the fact that we have two copies A i of the U (1) gauge field. Thus the first trace is the contribution of two transverse vector fields regularised by covariant higher derivatives and Pauli-Villars. The second trace coincides with twice the Jacobian from the change of variables A µ = A T µ + ∇ µ ω, again regulated by covariant higher derivatives and a Pauli-Villars field.
Using (2.14) we can map (5.8) to a calculation which is even closer to a standard textbook exposition. By replacing the transverse trace by a trace over the full vector representation we get: Now from (2.5) the first term is the one loop contribution for two U (1) gauge fields in Feynman gauge, regulated by covariant higher derivatives and a Pauli-Villars field, whereas the second two terms can be identified with the ghost contributions regulated by covariant higher derivatives and Pauli-Villars fields (with the option to choose different parameters for the regularisation in the second ghost action).
Either way we see that, following now standard treatments, for example computing the Schwinger-Dewitt coefficients in a heat kernel expansion (see e.g. [48]), will yield twice the trace anomaly contribution from massless vector fields: where C is the Weyl tensor, and b = 1/10 and b = −31/180 the standard values.
6 Twice the contribution to the gravitational beta functions The coefficients b and b also appear in the gravitational beta functions induced by the gauge fields.
These are obtained by taking a derivative with respect to Λ of (5.8) obtaining the traces with the functions Evaluating the traces (6.1) using the early time heat kernel expansion up to second order in curvature we have and where B n (∆ i ) are the traced heat kerne coefficients for the operators ∆ 0 and ∆ 1 and Q m [W i ] are functionals of the the functions (6.2). Explicitly the heat kernel coefficients are given by For m > 0 the Q m [W i ] functionals are given by the scheme dependent integrals whereas the Q 0 functionals are given by which are independent of the choice of cutoffs c andc. Consequently we find that the logarithmic terms give the trace anomaly with b = 1/10 and b = −31/180 as in (5.9). The scheme dependent coefficients are given by are non-universal since they depend on the form of the cutoff functions c andc and determine the running of the vacuum energy and the Newton's constant.

Spontaneous symmetry breaking by the vector representation
We have just seen that we get twice the desired gravitational contribution, because the A 2 part of the regularisation structure remains massless. We now repair this problem with the regularisation.
Up until now we have treated the two diagonal entries in (1.1) equally, using A = d + A, where d + is defined in (4.3). Splitting this further down to A = A 1 σ + + A 2 σ − , where σ ± = (1 ± σ)/2, and we want to give a mass to A 2 while leaving A 1 massless. Therefore we must spontaneously break the σ − direction while leaving σ + direction unbroken. That is not possible using only commutators anything. We pause briefly to sketch why the latter property would have led to an issue. In SU (N |N ), the supergauge field can be alternatively expanded as A = A 0 1 + A A T A , where the generators T A are both traceless and supertraceless since strA = 0 has forbidden the appearance of a σ term. When interactions are built on commutators, this furthermore implies that A 0 appears nowhere in the action, resulting in the "no-A 0 " shift symmetry δA 0 µ (x) = λ µ (x) [13,18] which then needs to be imposed as a consistency condition. (The alternative procedure of redefining the Lie bracket in the gauge sector to exclude terms ∝ 1 leads to equivalent consistency conditions [13,18].) The second problem with breaking the SU (N |N ) symmetry using a representation (7.2) is that A 0 will now couple to the action exclusively through such terms. It would therefore work as a Lagrange multiplier field and force an unpromising non-linear constraint. This issue is analogous the problems which arise in regularised SU (N |N ) theory, if one attempts to impose that the matrix scalar field (3.8) is supertraceless [13].
is the real part of C. Since from (3.5), we have that in unitary gauge the kinetic term for C reduces to The choice of sign for the kinetic term, thus provides the right sign mass for both B and A 2 , and shows that C R is a regulator field with the wrong-sign action. (Expanding the supertrace one sees that ∇B∇B is the order that appears in the kinetic term with positive sign. 9 ) As before [18], we can ensure this Higgs field gets a mass term that tracks the cutoff, by making it dimensionless and assuming an appropriate potential. The minimal Lagrangian would be where the functional derivatives with respect to C andC are just the functional derivatives with respect to the components except that the functional derivative in δS/δC := δ r S/δC should be regarded as acting on the action (and thus also Σ g ) from the right, so as not to introduce unnecessary signs into the Grassmann components. 10 The covariantization of the kernels replaces (3.17) with where W m therefore now propagates a vector representation, and where˙ CC m thus propagates the matrix (∼ 'adjoint') representation. This tensor-product type C decoration can be understood as arising from supergauge invariance (compare footnote 6): The choice of sign in (7.4) is already implicit in the previous formulation, as can be seen by taking the right hand column of (3.8) and forming the supertrace. 10 Equivalently one takes a left-derivative using the bottom row of (3.12), and includes an overall minus sign for the overall supertrace, consistent with the identification in footnote 9.
i.e. as before the C decoration in (8.4) can be exchanged for longitudinal terms in (8.3).
The rest of the definition of the flow equation is as in sec. 3. In particular, functional derivatives do not act on the terms that decorate the kernels, only on the relevant action S or Σ g . Clearly the resulting flow equation manifestly preserves local U (1|1) invariance.
9 Kernels and two-point vertices for the Maxwell flow equation  .7), the A 2 and C I kernels coincide with these, and the new unbroken sector has adopted the old A and C expressions: As before, the seed action is our choice (subject to preservation of all the required symmetries, in particular spontaneously broken U (1|1) invariance), and requiring that the two-point vertices of the classical effective action and the seed action can be set equal, then determines the kernels. Thê S kinetic terms for the vector fields follow from covariant higher derivative regularisation of the super-field strength squared [18], while the C-sector is a similarly regularised version of (7.5). By adjusting normalisations, we can arrange for the seed action vertices to closely parallel our previous expressions. In fact we can get exactly (4.9) and (4.10) for B and D, while (4.8) can be adopted by A 1 and C R :Ŝ That leaves only the A 2 and C I kinetic terms and the C I A 2 mixing term. These are constrained by spontaneously broken U (1|1) invariance, and with our choice of normalisations can be taken to coincide with the BD sector: In view of the matches (9.1) we already found for the kernels, we see that the flow equation for these two-point vertices coincide with (4.5) in the sense that the first equation is now forŜ C R C R , the second forŜ A 1 A 1 , and the last three again apply to the BD sector but also get copied over to the A 2 C I sector using the maps (9.1) and (9.3). The solutions for the integrated kernels are thus already given in (4.12) -(4.14), where now we should rename CC as C R C R , AA as A 1 A 1 and recognise that A 2 A 2 = BB and C I C I = DD .

Manifestly gauge invariant conformal anomaly
There is almost nothing left to do. Clearly equation (5.2) is replaced by but using the identifications of the previous section we see that the A 2 and C I parts just cancel half of the B and D parts, and thus this becomes in the old notation: i.e. exactly half the result in (5.2). Therefore we obtain half the expression in (5.8), (5.9) and (6.8), i.e. precisely the standard trace anomaly, however here computed by maintaining manifest gauge invariance at every stage.