Spectral Regularisation: Induced Gravity and the Onset of Inflation

Using spectral regularisation, we compute the Weyl anomaly and express the anomaly generating functional of the quantum effective action through a collective scalar degree of freedom of all quantum vacuum fluctuations. Such a formulation allows us to describe induced gravity on an equal footing with the anomaly-induced effective action, in a self-consistent way. We then show that requiring stability of the cosmological constant under loop quantum corrections, Sakharov's induced gravity and Starobinsky's anomaly-induced inflation are either both present or both absent, depending on the particle content of the theory.


Introduction
It is widely believed that the early stages of our universe were characterised by a phase of an accelerated expansion, dubbed inflation [1][2][3], during which the length scales increased by approximately e 75 times within a relatively short time of less than about 10 6 t Pl (with t Pl denoting the Planck time). To realise such an inflationary era, one usually requires particular initial conditions at the end of the quantum gravity era [4][5][6] and the presence of a (scalar) field, called the inflaton, whose potential satisfies certain properties. It would certainly be more natural to have one of the scalar fields of the theory playing the rôle of the inflaton and the scenario of Higgs-driven inflation [7] even though appealing, it faces several difficulties [8][9][10][11]. Going beyond the minimal Standard Model, there are also difficulties in finding a natural candidate for the inflaton field, as it has been recently shown for the case of hybrid inflation 1 within supersymmetric SO(10) [18]. Hence usually one assumes the presence of an additional -probably within the hidden sector -scalar field with certain appropriate propertiesflatness of its potential and fine-tuned couplings to accommodate the CMB data -so that it can lead to a successful inflationary era. This approach, even though is by construction successful, it is nevertheless not fully satisfactory. Hence, alternative approaches have been proposed and studied in the literature.
One can, for instance, add an R 2 -term in the gravitational action, without changing the particle content of the theory [19,20]. Such a term could appear in the context of quantum gravity corrections as one approaches the Planck scale. It is clearly appealing to study the realisation of an inflationary era through the effects of Quantum Field Theory (QFT) in curved space-time, while keeping the required phenomenological input to its minimum; this is the philosophy we will adopt in this study.
During the very early stages of our universe, matter can be described by a set of massless fields with negligible interactions. Such fields, studied in the context of QFT in curved spacetime, may lead to an inflationary era. More precisely, trace (conformal) anomaly, resulting from the renormalisation of the conformal part of the vacuum action, becomes the dominant quantum effect and can drive an inflationary era in the absence of an inflaton field. Such a proposal was first introduced by Starobinsky [1], then studied by Vilenkin [21] and more recently it has been further investigated by various authors (see for instance Ref. [22,23] and references therein, and Refs. [24,25]).
Even though the Einstein-Hilbert action can be also seen as an induced quantum gravity effect [26,27], one cannot -to our knowledge -find in the literature a consistent mathematical scheme allowing to describe simultaneously induced quantum gravity and anomalyinduced effective action. Standard computation of trace anomaly in curved space-time usually relies on ζ-functional regularisation [28], that does not exploit the ultraviolet cutoff scale, thereby missing the effect of Sakharov's induced gravity. In contrast to that, the frequently used Schwinger's proper time regularisation [29], that gives immediately the Einstein-Hilbert action as a quantum effect, is not suitable to describe the Weyl anomaly, since it leads to a local Weyl noninvariant expression, while anomaly generating functional is however known to be nonlocal [30].
Our aim here is to investigate, in a self-consistent way, whether a model whose only ingredient is QFT can generate a successful inflationary era and moreover imply an induced Einstein-Hilbert action. We will thus revise the onset of inflation driven by the trace anomaly of the quantum effective action, in the absence of both, a "bare" action for gravity and an inflaton field, in the sense of a scalar field incorporated in the model. In other words, we will investigate whether a cosmological arrow of time can result from a purely quantum effect. We will apply spectral regularisation in a classical conformal invariant theory. We will neglect any effects of masses at high energies, supposing that they are all much smaller than the corresponding value of the Hubble parameter. We will show that spectral regularisation allows us to derive the Weyl anomaly generating functional describing both the induced Einstein-Hilbert action and the standard Weyl anomaly. However, spectral regularisation is highly nontrivial. The price we have to pay for capturing the two effects simultaneously, is the introduction of an auxilary field, that can be considered as a collective degree of freedom of vacuum fluctuations of all fields, dual to conformal anomaly, that we call "collective dilaton".
Spectral regularisation was first introduced in a context of chiral, and later also, scale anomalies in Quantum Chromo Dynamics (QCD) in flat space-time [31,32]. It was later applied in the case of fermions moving in a fixed bosonic background in the context of induced gravity and the collective dilaton Lagrangian was computed up to linear order in curvature [33]. More recently, it was extended up to quadratic order in curvature in the context of the bosonic spectral action [34,35]. In the present study, we generalise the spectral regularisation for all quantised fields, in order to study in a systematic way the influence of quantum vacuum fluctuations on the gravitational dynamics. This paper is organised as follows: In Section 2 we derive a mathematical description of anomaly using spectral regularisation. In Section 3 we show how the induced gravitational Einstein-Hilbert action appears in our formalism. Section 4 is devoted to trace anomaly induced inflation. We will show that, requiring stability of the cosmological constant under loop corrections, the condition of Sakharov's induced gravity becomes equivalent to the condition for the existence of a stable inflationary solution. We round up with our conclusions in Section 5. Finally, some technical aspects are presented in the Appendix.

Spectral Regularisation and Collective Dilaton Lagrangian
We will derive a mathematical description of anomaly using spectral regularisation. We will first compute the anomaly and then present the anomaly generating functional. The latter is achieved through the introduction of an auxilary field, that can be considered as a collective degree of freedom of vacuum fluctuations of all fields, dual to conformal anomaly.

Spectral Regularisation: A Brief Overview
Our main aim is to compute the influence of vacuum fluctuations of quantised fields on the dynamics of the metric tensor in the context of QFT with an ultraviolet cutoff.
Since in asymptotically free QFT, the interactions -non-abelian interactions, Yukawa interactions and Higgs self-interactions -can be considered as perturbative, the effect we are interested in is, at leading order, given by one-loop vacuum energy of free fields. However, even this simple approximation may lead, in curved space-time, to non-trivial effects like Sakharov's induced gravity [26] and Starobinsky's anomaly-induced inflation [1].
Let us consider a theory of free quantised fields of various spins moving in a gravitational background. The classical action reads with F µν the field strength, R the scalar curvature, ω mn µ the spin connection and σ mn the generators of the representation of the Lorentz group. Note that N F , N V , N H stand for the number of Dirac four component fermions, gauge vector bosons and real Higgs-like scalars, respectively. The classical action Eq. (2.1) is conformally invariant. This setup may be considered as a good description of the Standard Model (or its generalisations) when all masses are much smaller than the Planck mass and the scalar fields are conformally coupled to gravity.
In order to quantise the theory we follow Faddeev-Popov gauge fixing procedure. In Feynman-t'Hooft gauge (a type of an R ξ gauge, as a generalisation of the Lorentz gauge, with ξ = 1), the action reads The object we are interested in, is a quantum partition function that (up to irrelevant constant) is given by and is formally equal to: (2.8) Note that in a theory with N w F two-component Weyl fermions one should replace N F by N w F /2 in q. (2.8). Each operator O, appearing as det(O) in Eq. (2.8), is of a Laplacian type and unbounded; hence each determinant is infinite, rendering the whole partition function ill-defined. The idea of spectral regularisation [31] is to count eigenvalues of each operator that are smaller than a cutoff scale Λ. If in addition one considers the Euclidean space-time to have finite volume, then the spectrum of each Laplacian becomes discrete and after its truncation one obtains a product of finite number of modes. Hence, the projector on the subspace of eigenfunctions of O with eigenvalues smaller than Λ. The parameter µ is introduced in order to have a dimensionless expression under the sign of determinant and in what follows, we consider Λ = µ; other choices of µ will not affect substantially the regularisation scheme. 2 Although the procedure of spectral regularisation can be easily understood and has nice properties, like preserving gauge invariance and general covariance, technically it is not easy to handle (in contrast to the Schwinger's proper time formalism). Nevertheless, one can address both, induced quantum gravity and anomaly-induced inflation, using spectral regularisation. Indeed, they are both related with Weyl non-invariance of the effective quantum action (or Weyl anomaly), since the classical theory is Weyl invariant. In the following, we compute Weyl anomaly and present the anomaly generating functional.

Spectral regularisation: computation of the Weyl anomaly
Let us consider a conformal transformation of the metric tensor: (2.12) Since the classical action Eq. (2.1) is Weyl invariant, the Weyl non-invariant contribution comes out, by definition, from Weyl anomaly. Let us compute the difference between the initial and the Weyl transformed quantum effective action, namely For the fermionic effective action W F , this difference, Eq. (2.13), reads [34,35] where Repeating the same computation for the case of a scalar field, one can easily show that Indeed, under conformal transformation the Laplacian D H transforms as and thus one obtains To complete the computation of the scalar and gauge contributions to the anomaly, Eqs. (2.16) and (2.21) respectively, we will follow the same procedure as in Refs. [32][33][34][35][36].
Let us first perform a decomposition of the projector P Λ : and then do a heat kernel expansion 3 in terms of the heat kernel (Schwinger-De Witt) coefficients [37]: and a n (φ, O) = d 4 x √ g φ a n (O, x) . (2.25) The main advantage of the heat kernel method 4 is that it provides the required information in terms of only a few geometric invariants. Since the ultraviolet divergences in the oneloop effective action are defined by the heat kernel coefficients, the asymptotic expansion, Eq. (2.25), makes sense only when the background field invariants, appearing in the heat kernel coefficients a n (O, x), are smaller than the corresponding powers of the ultraviolet cutoff Λ. This requirement defines the applicability of our approach; we assume this requirement to be satisfied. 5 Since we are working on a manifold without boundary, only even heat kernel coefficients a 2k are non-zero.
Performing the integration over s in Eq.
we obtain (2.27) Table 1. Heat kernel coefficients a 0 and a 2 for free massless fields of various spin; we have calculated them using Ref. [37].
Spin a 0 a 2 0, conformal coupling Table 2. Heat kernel coefficient a 4 for free massless fields of various spin [37].
Spin a b c 0, conformal coupling 3/2 -1/2 -1 1/2, Dirac fermion -9 11/2 6 1, gauge (i.e., with ghosts) 18 -31 18 Using the expansion Eq. (A.12) for the total anomaly we obtain We give in Tables 1 and 2 the values of the heat kernel coefficients a 0 , a 2 and a 4 , respectively, for free massless fields of different spin. Note that C 2 = C µνρσ C µνρσ with C µνρσ the Weyl tensor, and GB stands for the Gauss-Bonnet term, given by GB= Using where the total anomaly Eq. (A.12) reads At this point, one can make two remarks:

Remark 1
We would like to compare results for the trace anomaly obtained via the spectral and the ζ-function regularisations. An infinitesimal anomaly reads 34) and the heat kernel coefficients a 0 , a 2 , a 4 are given in Tables 1 and 2. The A 4 -contribution coincides with the result for anomaly obtained via ζ-function regularisation and the dimensional one [38]. Quadric and quadratic in Λ terms can be interpreted as an ultraviolet divergence and hence subtracted through the addition of the corresponding local counter terms. Indeed, one can define the renormalised effective action with α 1 , α 2 defined in Eq. (2.32). One can easily check (see computations in subsection 3.2) that lim In what follows, we will not subtract the divergent terms and we will keep Λ finite and of order of the Planck scale. We will thus be able to describe simultaneously both, the induced gravitational action and the onset of (trace) anomaly-induced inflation. We will hence conclude that all terms in the Lagrangian, leading to a period of an accelerated expansion of the universe, may be considered as the outcome of a quantum effect. Remark 2 As we will show, fermions play an important rôle in both, induced gravity and anomalyinduced inflation, and hence the case of a purely fermonic anomaly is of special importance. This was studied in Refs. [34,35,39], and the Weyl anomaly generating functional was expressed in terms of a structure very similar to the Chamseddine and Connes bosonic spectral action [40], introduced in a context of noncommutative spectral geometry [41,42] coupled to the collective dilaton. 6 At this point, let us emphasise that the infinitesimal Weyl anomaly, obtained within QFT with spectral regularisation, is the bosonic spectral Lagrangian. Indeed, by definition, where χ(z) is a cutoff function, equal to one at z < 1 and rapidly vanishing at z > 1, and L BS (x) stands for the bosonic spectral Lagrangian, computed via the heat-kernel technique. Performing a similar computation and inserting φ(x) under the sign of the trace, we get 38) and the bosonic spectral Lagrangian reads Expanding Eq. (2.14) up to linear order in φ and taking the functional derivative in the 6 Noncommutative spectral geometry provides a purely geometric explanation for the Standard Model of particle physics [43][44][45][46] and offers a framework to address early universe cosmology [10,[47][48][49][50][51][52].
infinitesimal limit, we obtain: infinitesimal Weyl anomaly ≡ lim where in the last step we used Eq. (2.39).

Anomaly generating functional and collective dilaton
Although -in contrast to proper time regularisation -spectral regularisation does not allow one to compute the partition function explicitly, there is a formalism via the introduction of a collective dilaton that allows one to extract the Weyl non-invariant part of such a regularised determinant, as an integral over an auxilary field φ of some local expression that depends on φ and the background fields. Substituting the conformally transformed metric tensor g µν e 2φ in Eq. (2.8) and integrating over all possible φ(x), one can write the identity (2.41) Since the first term above is the integral over the Weyl group of a Weyl transformed quantity, it is Weyl invariant under the action of the Weyl group, so we denote it by Z inv . Hence, Eq. (2.41) can be rewritten as Clearly, φ stands for a collective degree of freedom of vacuum fluctuations of all fields dual to conformal anomaly, hence the term "collective action". Since all our computations were carried in Euclidean QFT, in order to apply our result in a physical context one should perform a Wick rotation back to Minkowski signature in Eq. (2.31). Hence, for the anomaly generating functional we have 45) and the Minkowskian version of the collective action reads with the coefficients given in Eq. (2.32).
In what follows, we will show that Weyl anomaly in QFT with spectral regularisation reproduces Sakharov's induced gravity, as well as Starobinsky's anomaly-induced inflation. This is the main message of our study.

Standard Approach: Proper Time Regularisation
The standard approach to the Sakharov's induced gravity is based on Schwinger's proper time regularisation [27]. In this formalism, one first selects a convenient reference metricg µν and then computes the difference in the one-loop contribution to the effective action which results from comparing two different metrics defined on the same topological manifold. Hence, we consider the difference W [g µν ] − W [g µν ], with W defined as W ≡ − log Z.
Let us write the formal equality and then perform a heat kernel expansion for the Tr (e −sD ) and Tr (e sD ) terms to get where the coefficients a k are the Seeley-De Witt coefficients, universal functions of the spacetime geometry. In order to perform the integration over s in Eq. (3.4) for k = 0, 1 one needs an ultraviolet regulator µ uv ; integration over s for all other values of k, namely for all k > 1, is ultraviolet finite but it requires the infrared regulator µ ir µ uv . It is worth noting that the heat kernel expansion has allowed us to identify the potential divergences. We obtain Let us emphasise that the regulators µ uv and µ ir are not ultaviolet and infrared, respectively, cutoff scales for the spectrum of D; they are attributes to make the regularisation scheme finite. 7 Using Eq. (2.8) with log det = Tr log and Eq. (3.5) we get and The main idea of Sakharov's induced gravity lies in attributing a physical meaning to the ultraviolet cutoff scale, so that it denotes the upper scale for which the considered QFT is a valid effective theory. In this way, it is not necessary to subtract divergences, and setting Λ ∼ M Pl ∼ 10 19 GeV, the term µ 2 uv R can be considered as an induced gravitational action. Hence, starting from a classically Weyl invariant theory, quantisation implied a Weyl non-invariant Einstein-Hilbert action. One may thus conclude that, under proper time regularisation, the Weyl anomaly contains operators of dimension two, in contrast to the (standard) dimensional regularisation or the ζ-function regularisation, where anomaly contains just operators of dimension four.
Nevertheless, the considered proper time regularisation procedure does not reproduce correctly the a 4 -contribution to the anomaly (c.f. Eq. (2.36)), and therefore it cannot be used to investigate the trace anomaly induced inflation. Indeed, substituting in W pt , defined in (3.6) above, the conformally transformed metric tensor e 2φ g µν and then taking the derivative over φ(x), one immediately finds and thus do not contribute in Eq. (3.9). Let us remind to the reader that as we have previously shown (see Remark 1, Eq. (2.33)), the spectral regularisation reproduces correctly the a 4 -contribution to the anomaly. We will next show that it also reproduces correctly the induced Einstein-Hilbert action; it can be thus used to describe both.

The Spectral Regularisation Approach
The effective action is known to be a non-local functional of the metric tensor g µν and in particular, of the Lagrangian density L eff [g µν ], so that does not exist and correspondingly the local collective action S coll , once integrated over φ, captures all non-locality of the Weyl non-invariant part of the effective action. Nevertheless, the terms with coefficients α 1 , α 2 and α 5 in the anomaly, Eq. (2.31), can be generated by local terms in the effective action W eff . Indeed, let us consider (3.14) Comparing Eqs. (2.31) and (3.14), we conclude that the total effective action can be rewritten as where W inv is some Weyl invariant functional of g µν and W nonloc is a non-local functional, generating α 3 and α 4 terms in the collective action Eq. (2.31) (we refer the reader to Ref. [30]). Equivalently, one can say that QFT with spectral regularisation leads to Sakharov's induced gravity: (3.17) and with O R 2 denoting all local and non-local terms responsible for Λ 0 -contributions in the anomaly-induced effective action. The latter is much smaller in the low energy regime (R << Λ) and hence it can be neglected at energies much smaller than the cutoff scale. It however plays a significant rôle during the inflationary era; it will be studied in the next section within the isotropic approximation.
In order to identify the induced Planck mass with the real one at ∼ 10 19 GeV one should impose the cutoff scale Λ at the Planck energy scale. This however automatically leads to a huge value of the induced cosmological constant, namely λ ind ∼ M 4 Pl . One may consider the presence of bare cosmological constant with the opposite sign, namely λ bare ∼ − M 4 Pl and impose the fine-turning: To avoid such a fine-tuning, we will adopt an alternative approach and hence, we will impose the Pauli compensation principle, i.e we require, that the Λ 4 fermonic and bosonic contributions to the vacuum energy cancel each other. The latter implies that the numbers of physical fermonic and bosonic degrees of freedom are equal, namely on the number of scalars, spinors and vectors, so that all quartic divergences cancel. Thus, under spectral regularisation we obtain: (3.20) The above equation, Eq. (3.20), agrees with the one obtained following the Schwinger proper time formalism [27]. It is worth noting that the Pauli compensation condition N H = 2 (N w F − N V ) is not just a property of the spectral regularisation; it holds in all regularisation procedures with an ultraviolet cutoff scale and in that sense it is universal.
In the next section, we will consider a high energy region, but R < Λ 2 , i.e. where the spectral regularisation is still applicable. We will show that, imposing the Pauli compensation condition, the Λ 0 -contribution to the anomaly (which we have neglected here), together with the Λ 2 -contribution, leads automatically to Starobinsky's anomaly-induced inflation.

Inflation Induced from Trace Anomaly: The Isotropic Approximation
We will explore the dynamics of a metric tensor in the isotropic approximation. The spacetime is considered to be spatially flat, namely g µν = e β(τ ) η µν ; the cases of closed and open universes can be studied along similar lines. Although one should first derive an equation of motion for the metric tensor g µν and only afterwards substitute the conformally flat anzatz, it is possible to avoid the first step following the procedure described in Ref. [53].
Hence, to get equations of motion in the isotropic case for an arbitrary 8 general covariant action W [g µν ] one should [53]: • Firstly, substitute the conformally flat anzatz ds 2 = dt 2 −a(t) 2 d x 2 in the action W [g µν ].
• Secondly, rewrite the result of the substitution in the form where vol is a three-dimensional volume and H stands for the Hubble parameter, H ≡ a/a. In principle, the function I can also depend on higher derivatives of the Hubble parameter, but we will restrict ourselves to the minimal needed case.
• Thirdly, obtain the following equation for the scale factor: which is just the generalisation of the Friedmann equation.

Remark 3
Equation (4.2) is third order in a, while the equation of motion δW/δa = 0 is of fourth order. One can easily check that the above prescription is just a formulation of energy conservation. Indeed, since W [a(t)] in Eq. (4.1) does not depend explicitly on time, Nother's theorem implies conservation of the quantity: If in addition, one imposes that the overall (gravity+fields) energy E vanishes, then the resulting equation will be exactly Eq. (4.2). In our case the action is given by 9 where 10 W is a quantum effective action 1 i log Z with Z defined by Eq. (2.8) and spectral regularisation.
Following the prescription described above, we must substitute the conformally flat anzatz for the metric tensor in comoving coordinates in Eq. (2.31). This is done in two steps: firstly, we substitute the conformally flat anzatz in the conformal coordinates g µν = e 2β(τ ) η µν and secondly, we perform the corresponding change of variables to the comoving frame. Hence, substituting g µν = η µν and φ = β(τ ) in Eq. (2.31), we get 11 Performing the change of variables β(τ ) → a(t) with we arrive to the following expression for the effective action: where Equation (4.9) for the Hubble parameter H(t) is of second order, so we write it as a system of two first order equations, in order to use the phase portrait technique. Hence, We are looking for special points of the vector field on the [H, v]-plane defined by the righthand side of Eq. (4.11). Solving this algebraic equation we find two special points 12 , H 1 and H 2 : describing a slowly expanding universe, and describing a rapidly expanding universe and hence offering a good candidate for an inflationary model. Linearising the system Eq. (4.11) in the vicinity of each special point, we draw the following conclusions: • The rapidly expanding solution H 2 is stable (stable focus on [H, V ] plane).
• The slowly expanding solution H 1 is unstable (unstable focus on [H, V ] plane).
In conclusion, if Pauli compensation condition is satisfied, namely if all quartic divergences cancel each other, then Sakharov's induced gravity leads to Starobinsky's anomalyinduced inflation, and vice versa.

Conclusions
The aim of this study was to relate Sakharov's induced gravity to the anomaly-induced effective action and thus obtain Starobinky's anomaly-induced inflation. Imposing spectral regularisation with the cutoff scale Λ in a classically Weyl invariant theory, we computed the anomaly and expressed the anomalous part of the quantum effective action through the quantised single collective scalar degree of freedom of all quantum vacuum fluctuations, dubbed as the collective dilaton field φ, described by the local action Eq. (2.31).
It is worth noting that the condition of stability of the cosmological constant under Λ 4 -corrections, namely N H = 2 (N w F − N V ), appears naturally within our procedure. Our approach allowed us to treat the Sakharov's induced gravity on equal footing with the Starobinsky's anomaly-induced action, in a self-consistent way. More precisely, we found that Provided the stability condition is satisfied, Sakharov's induced gravity and anomaly induced action leading to Starobinsky's anomaly-induced inflation appear simultaneously if N w The fact that QFT gave rise to the Einstein-Hilbert action and the onset of an inflationary era, in the absence of an inflaton field, may indicate that the cosmological arrow of time results from quantum effects in a classically Weyl-invariant theory. At this point, it is worth emphasising that while our model can account for the onset of inflation and the first inflationary stage, it cannot describe the graceful exit and the subsequent evolution of the universe to its present state. In order to explain the graceful exit one should go beyond our massless particle approximation, as was done for intance in Ref. [54].

A Gauge-Ghost's Contribution: Some Computational Details
Starting from the Maxwell action for gauge fields we perform a Faddeev-Popov quantisation procedure, adding the gauge fixing term and the ghost action S gh The overall gauge fixed Maxwell-ghost action then reads The partition function describing a contribution of the quantised vector fields and ghosts to the vacuum energy is given by the functional integral Since both operators D vec and D gh are unbounded, the last equality is formal and thus we perform a spectral regularisation. Following the same approach as for the fermonic and scalar cases, we introduce the cutoff scale Λ and the two projectors in order to truncate the spectrum of the D vec and D gh operators, respectively. The regularisation is based on replacing the unbounded operators D vec and D gh by the truncated operators D Λ vec and D Λ gh , respectively, denoted by in the determinants appearing in the partition function Eq. (A.7). Hence, the regularised partition function reads We are interested in computing the contribution of the quantised vector fields and ghosts to the anomaly S coll . We will impose the above discussed regularisation and use Eq. (A.10) for the regularised partition function. Note that we denote a quantity Q[g µν ] computed on the transformed metric tensor g µν e 2φ by In contrast to the fermonic and scalar cases, the next step in the computation of the anomaly S coll is not a trivial task, because both operators D dec and D gh do not transform in a covariant way, namely where the "bad terms" are given by "bad terms" ≡ 2 Tr vec P Λ vec (D vec ) −1 φ µ ∇ λ − φ λ ∇ µ − φ λ µ + 2 Tr gh P Λ gh (D gh ) −1 (φ µ ∇ µ ) .
(A. 16) In what follows we will show that the "bad terms" cancel. Let us first introduce a complete set Φ n , n = 1, 2... of orthonormal eigenfunctions of the ghost operator D gh , as D gh Φ n = λ n Φ n with d 4 x √ gΦ n Φ m = δ nm . (A.17) One can easily check that the set of functions ξ µ n ≡ ∇ µ Φn √ λn satisfies (D vec ) µ ν ξ ν n = λ n ξ µ n with d 4 x √ gξ nµ ξ µ m = δ nm , (A. 18) namely it forms an orthonormal basis in a space of longitudinal eigenvectors of the the operator D vec . Let us also introduce the orthonormal set of transversal eigenvectors of (D vec ) µ ν B ν n = β n B µ n with d 4 x √ gB nµ B µ m = δ nm and ∇ µ B µ n = 0 , (A. 19) so the set {ξ µ n , B µ m } with n, m = 1, 2, · · · forms a basis in the space of all gauge potentials. The gauge contribution to the "bad terms" is 20) and the ghost contribution to the "bad terms" reads 2 Tr P Λ gh (D gh ) −1 (φ µ ∇ µ ) = 2 n: λn≤Λ Clearly, the "bad terms" cancel mode by mode. Hence, the final answer for the gauge-ghost contribution to the anomaly is where the cutoff function χ is the Heaviside step-function