Diffeomorphism Invariance Demands Conformal Anomalies

We study a series of the Wess-Zumino actions obtained by repeatedly integrating conformal anomalies with respect to the conformal-factor field that appear at higher loops. We show that they arise as physical quantities required to make nonlocal loop correction terms diffeomorphism invariant. Specifically, in a conformally flat spacetime $ds^2=e^{2\phi}(-d\eta^2 + d{\bf x}^2)$, we find that effective actions are described in terms of momentum squared expressed as a physical $Q^2 = q^2/e^{2\phi}$ for $q^2$ measured by the flat metric, which recalls the relationship between physical momentum and comoving momentum in cosmology. It is confirmed by calculating the effective action of QED in such a curved spacetime at the 3-loop level using dimensional regularization. The same applies to the case of QCD, in which we show that the effective action can be summarized in the form of the reciprocal of a running coupling constant squared described by the physical momentum. We also see that the same holds for renormalizable quantum conformal gravity and that conformal anomalies are indispensable for formulating the theory.


Introduction
In general, when a symmetry that holds classically is broken by quantum effects, it is called an anomaly. Conformal anomalies imply that even if a classical action has conformal symmetry, it breaks down at the quantum level [1,2,3,4,5,6,7,8,9,10]. Hence, letting Γ be an effective action of the theory, a conformal variation of Γ is a conformal anomaly. That is, the trace of the energy-momentum tensor which vanishes classically becomes nonzero by quantum effects. For this reason, conformal anomalies are also called trace anomalies. This paper will focus on the role of conformal anomalies. As the name implies, conformal anomalies break conformal invariance, but it does not mean that they are anomalous. Rather, we will see that they are physical quantities required to preserve diffeomorphism invariance, and are associated with nonlocality caused by quantization.
The relation between conformal anomaly and diffeomorphism invariance can be seen as follows. Let f be a field that has conformal invariance classically and I be its action in curved spacetime. The line element is defined by ds 2 = g µν dx µ dx ν and the metric field is decomposed into a conformal factor and other modes as then the relation I(f, g) = I(f,ḡ) holds. 1 On the other hand, a path integral measure [df ] g that is diffeomorphism invariant generally depends on the conformal-factor field φ. Therefore, by extracting the φ-dependence of the measure as [df ] g = e iS(φ,ḡ) [df ]ḡ, we rewrite the effective action in the form e iΓ(g) = [df ] g e iI(f,g) = e iS(φ,ḡ) [df ]ḡ e iI(f,ḡ) = e iS(φ,ḡ) e iΓ(ḡ) , that is, Γ(g) = S(φ,ḡ)+Γ(ḡ). 2 Hence, a variation of S by φ gives a conformal anomaly. Now, let us consider a simultaneous transformation φ → φ − ω,ḡ µν → e 2ωḡ µν 1 Depending on the type of the field, it is necessary to rescale the field appropriately to exclude the φ-dependence. 2 When calculating the effective action, normally a background field of f is introduced.
that does not change the full metric g µν , that is, preserves diffeomorphism invariance. Applying it to the above expression of the effective action, the left-hand side is trivially invariant, while the right-hand side is e iS(φ−ω,e 2ωḡ ) e iΓ(e 2ωḡ ) = e iS(φ−ω,e 2ωḡ ) e iS(ω,ḡ) e iΓ(ḡ) .
In the following sections, we will investigate the Wess-Zumino actions that arise at higher loops, mainly focusing on conformal anomalies related to gauge fields in curved spacetime, and see that they are quantities necessary to construct a diffeomorphism invariant effective action. In particular, we will concretely calculate the effective action of QED at the 3-loop level employing dimensional regularization, and show that the φ-dependence is actually involved in a physical momentum measured by the full metric g µν . The same is true for QCD, or Yang-Mills theory. That also applies to quantum conformal gravity, and other series of the Wess-Zumino actions that consists only of the gravitational field will be discussed.
The first Wess-Zumino action that appears at the 1-loop level is obtained by integrating L (0) (g) with respect to the conformal-factor field from 0 to φ, written as Note here that it is essential that the integrand L (0) (g) is covariant, that is, composed of the full metric g µν . Next, we consider what kind of the Wess-Zumino action appears at higher loops. First, let us simply repeat the definite integral for φ performed in (2.1) n times. Writing the resulting action as S Here, S top is the Wess-Zumino action S (1) that satisfies the consistency condition (1.2). However, others do not satisfy it, that is, they are not the Wess-Zumino actions, because the integrand S (n−1) top to derive S (n) top is no longer a covariant function of the full metric g µν for n > 1.
In order to find the Wess-Zumino action that arises at higher loops, we need to find a diffeomorphism invariant effective action Γ (1) (g) = d 4 x L (1) (g), which can be expressed as where Γ (1) (ḡ) is a loop correction of the theory defined on the metricḡ µν , generally has a nonlocal form. By using the covariant L (1) (g) as the integrand of the φ-integral, we can obtain the second Wess-Zumino action S (2) (φ,ḡ). Furthermore, adding a nonlocal Γ (2) (ḡ) to S (2) (φ,ḡ) appropriately to make a diffeomorphism invariant Γ (2) (g) = d 4 xL (2) (g) and integrating L (2) (g), we obtain S (3) (φ,ḡ). Repeating this procedure, we find top as the term with the highest power of φ.
The diffeomorphism invariant effective action will then be given by using Γ (n) (g) as where the coefficient a n is a function of the coupling constant.
In the following, taking the case of gauge fields as an example, a specific expression of the Wess-Zumino action and the effective action will be given. If the Minkowski metric η µν = (−1, 1, 1, 1) is employed asḡ µν , then Γ(ḡ = η) is the ordinary effective action. For simplicity, let φ be a constant and consider in momentum space. Still, in this example, the covariant structure we want to see is well retained. As for conformal anomalies composed of the gravitational field, we will discuss in the penultimate section.
The tree action of gauge fields is conformally invariant in 4 dimensions and √ −gF a µν F a λσ g µλ g νσ = F a µν F a λσ η µλ η νσ holds, where F a µν is a field strength and a is the index of gauge group. From this, L (0) (q) is given by F a µν F a λσ (q) η µλ η νσ , where q represents momentum measured in Minkowski spacetime. Therefore, we can easily perform the integral of φ, and obtain S (1) (φ, η) as Further performing the φ-integration n times gives A diffeomorphism invariant form of the effective action is obtained by adding a loop correction Γ (1) (η), known to be given by a logarithmic function of momentum, to the first Wess-Zumino action (2.3). It will be given by where q 2 = η µν q µ q ν and it has been known that the logarithmic correction term arises exactly with this coefficient according to that of φ. Q is momentum measured by the full metric g µν , that is, In cosmology, Q is often called physical momentum, while q is called comoving momentum.
In gauge theories, the effective actions up to 2 loops can be written using Γ (1) (g). At 3 loops or more, the second Wess-Zumino action will appear, which is given by Here, we can see that this action satisfies the Wess-Zumino consistency condition (1.2) using the fact that when changing the Minkowski metric as η µν → e 2ω η µν , the momentum squared is changed to q 2 → e −2ω q 2 . Adding a nonlocal loop correction Γ (2) (η) to the above gives where the last logarithm squared term in the first line is Γ (2) (η), and its coefficient is chosen so that the momentum is in the form of Q. Further repeating this procedure gives In the following, we will see that the effective action is yielded actually in this form.

QED Effective Action and Diffeomorphism Invariance
Here, we will calculate the effective action of massless QED in curved spacetime to confirm the form (2.5) at the 3-loop level. As a method to perform the renormalization calculations, dimensional regularization is employed [1,2,3,4,5,6,7,8,9,10,18,19,20,21,22]. This is the only known regularization method that can do high-loop calculations while preserving diffeomorphism invariance as well as gauge invariance.
The advantage of this method is that the result does not depend on how to choose the path integral measure. This property comes from the fact that in a 4-dimensional method such as the DeWitt-Schwinger method, conformal anomalies as contributions from the measure are derived by regularizing a divergent quantity δ (4) (0) = x|x ′ | x ′ →x , whereas in dimensional regularization such a quantity identically vanishes due to δ D (0) = d D q = 0, where D is spacetime dimension. In dimensional regularization, conformal anomalies are hidden between D and 4 dimensions, which arise as finite quantities yielded by canceling poles of ultraviolet (UV) divergences with zeros representing deviations of D-dimensional actions from 4 dimensions as First, we summarize some facts concerning with renormalization group (RG) equations. Let e 0 be a bare coupling constant of QED, e be its dimensionless renormalized coupling constant, and Z e be a renormalization factor connecting them. Let Z 1/2 3 be a renormalization factor of the gauge field, then the Ward-Takahashi identity Z e Z 1/2 3 = 1 holds, so that where µ is an arbitrary mass scale introduced to make up for the missing dimension. Since bare quantities do not depend on the scale µ, we obtain Thus, writing asβ e (e) = (e/2)µd(log Z 3 )/dµ, the beta function can be expressed as If the renormalization factor is Laurent-expanded like log Z 3 = ∞ n=1 f n (e)/(D− 4) n , then the beta function is expanded as where µd/dµ = (µde/dµ) ∂/∂e = β e (e, D) ∂/∂e is used. Since this quantity is finite and the pole terms must disappear, we find that the beta function can be expressed using the simple-pole residue as β e (e) = e 2 4 ∂f 1 (e) ∂e (3.1) and the residues satisfy a RG equation for n ≥ 1.
Here, the expansion of the beta function is expressed as Specific values of the coefficients are given by b 1 = 4/3(4π) 2 , b 2 = 4/(4π) 4 , and b 3 = −62/9(4π) 6 [13], but the following discussion proceeds without using these values. For this expression, solving (3.1) and (3.2), we find that the single-and double-pole residues can be expressed as The metric field is set to g µν = e 2φ η µν , as in the latter half of the previous section. Expanding the renormalization factor as Z 3 −1 = ∞ n=1 x n (e)/(D− 4) n as usual, the QED bare action with field-strength F 0µν = Z 1/2 3 F µν is expanded as follows: The residue x n is given by x 1 = f 1 and x 2 = f 2 + f 2 1 /2 and so on using the previously defined f n , thus Note here that the o(e 4 ) term of the double-pole residue x 2 disappears. This represents that the 2-loop self-energy diagram of the gauge field produces a simple pole only, which is a consequence of the gauge invariance, that is, Z e Z 1/2 3 = 1. The first line on the right-hand side of (3.3) gives a normal kinetic term and counterterms of the gauge field. The second and following lines give new terms which do not appear in normal quantum field theory in the flat spacetime. Terms with negative power of D − 4 are set as counterterms for eliminating UV divergences, and terms with zero or positive power are treated as new vertex functions.
A massless fermion field is conformally invariant in any dimension, and we can rescale the field appropriately to eliminate the φ-dependence. Since the result is independent of how to choose the measure as mentioned before, when calculating the effective action, use of the fermion action with the φ-dependence removed simplifies the calculations.
Below, we calculate the effective action of QED in the case of φ = 0 up to 3 loops, and confirm that momentum actually appears in the form of Q (2.4). First, we write down the normal effective action in Minkowski spacetime, which has been calculated as [14] Here, only nonlocal terms are considered as quantum corrections.
There are various contributions to the part depending on φ, namely, the Wess-Zumino action. One is the finite terms derived from the Laurent expansion of the bare action (3.3), given by Others are contributions from finite loop corrections in Figs. 1 and 2. Fig.  1 has a single φ as an external field, and Fig. 2 has φ 2 , in which the solid, wavy, and dotted lines represent the φ-field, gauge field, and fermion field, respectively, and diagrams that include counterterms as subdiagrams are not depicted here. These figures are obtained by inserting the vertex functions containing φ into the 2-and 3-loop self-energy diagrams of the gauge field. Calculations are performed with momentum of φ as zero. They are easily achieved, and adding all these contributions results in Details of each term will be described below.
First, we write out the results from the 2-loop self-energy (2LSE) and 3loop self-energy (3LSE) diagrams of the gauge field required in the following calculations of V (3.5), which are given by where F 2 is an abbreviation for F µν F λσ (q)η µλ η νσ and the calculation is The counterterms in the first line of (3.3) are designed to eliminate these UV divergences. As mentioned before, 2LSE has no double poles and also 3LSE has no triple poles. For 3LSE, there are diagrams containing one fermion loop and two fermion loops, and the double pole arises only from diagrams with two fermion loops. Moreover, all divergences are local due to renormalizablility, that is, nonlocal ones such as (1/ǫ) × log(q 2 /µ 2 ) do not appear. Fig. 1-(1) shows the insertion of the vertex function −2ǫφ (−F 2 /4) into an internal line of the gauge field in the 2LSE diagram, so that the simple pole of (3.6) and ǫ in the vertex function cancel out to be finite. It gives the first o(e 4 ) term of (3.5). Fig. 1-(2) is a 2LSE diagram with the vertex function x 1 φ (−F 2 /4) inserted. From this diagram, the third nonlocal term with a single φ is obtained as a finite term. At the same time, a simple-pole divergence (−x Fig. 1-(3) shows a 3LSE diagram with the vertex function −2ǫφ (−F 2 /4) inserted. The ǫ at the vertex acts to the double pole of 3LSE, generating a simple-pole divergence −x (6) 2 (1/ǫ) φ (−F 2 /4), and also cancels out the simple pole, resulting in a finite term 2x (6) 1 φ (−F 2 /4), which is the local o(e 6 ) term with a single φ in (3.5). Here, the sum of the divergences from Fig. 1-(2) and   Figure 1: Finite vertex contributions with a single φ and two external background gauge fields.
2 /2, has the opposite sign of it. The last term in (3.5) with φ 2 is derived by summing all finite contributions in Fig. 2. Fig. 2-(1) represents a 2LSE diagram with the vertex function −ǫx 2 φ 2 (−F 2 /4). Adding these yields the last term of (3.5) with coefficient x Adding all the contributions, we find that the effective action can be written in terms of the physical momentum (2.4) as follows: Recalling F µν F λσ η µλ η νσ = √ −gF µν F λσ g µλ g νσ in 4 dimensions, this is a covariant form.

Effective Actions of QCD and Quantum Gravity
Since the basic structure of the Wess-Zumino action for conformal anomaly does not depend on the gauge group, the same holds for QCD, or Yang-Mills theory, as for QED. Using dimensional regularization, the bare action of the Yang-Mills gauge field is also expanded like (3.3), thus the effective action can be calculated in the same way. Since diffeomorphism invariance is guaranteed, it will be expressed in terms of the physical momentum Q (2.4).
Therefore, let g be a coupling constant of QCD and its beta function be then the effective action will be given by 3 The background field method [15] is convenient for calculating a gauge-invariant effective action. When normalizing the gauge field as usual so that the kinetic term does not depend on the coupling constant, a renormalization factor of the coupling constant, Zg, and that of the background gauge field, Z

1/2
A , satisfy the same identity ZgZ 1/2 A = 1 as in QED. This also represents that the background gauge field normalized as in (4.1) does not receive renormalization.
The difference from the previous section is that the beta function is negative and that the gauge field is normalized to factor out 1/g 2 for the following discussion.
In the case of QCD, we can introduce a running coupling constant that becomes small at high energy. It can be expressed as for Q ≫ Λ QCD , where Λ QCD is a physical energy scale of QCD. Rewriting the effective action (4.1) using the running coupling constant results in the simple form This reflects the fact that the effective action is a RG invariant. For details, see Appendix. This expression infers that when the physical momentum Q becomes less than Λ QCD andḡ(Q) diverges, the kinetic term of the gauge field vanishes, that is, the dynamics disappear and confinement will occur. The conformal anomaly composed of the gravitational field is also determined from diffeomorphism invariance, thus the form of relevant quantities is basically unchanged apart from the coefficient whether the gravitational field is quantized or not. The essential difference between them will be described last from a viewpoint of symmetry based on recent research.
From studies of the conformal anomalies applying Hathrell's RG method to QED [7] and QCD [8] in curved spacetime, it has been shown that at least in these theories, gravitational part of the conformal anomalies are classified into only two [10,20]: the Weyl tensor squared, and the Euler density predicted by Riegert [16], where G 4 = R 2 µνλσ − 4R 2 µν + R 2 is the normal Euler density. Since the modification is total-divergence, the volume integral of E 4 is the same as that of G 4 . The corresponding gravitational action is given by Here, t is a coupling constant that controls dynamics of traceless tensor fields in renormalizable quantum conformal gravity [17,18,19,20,21,22]. On the other hand, since the Euler term does not have a kinetic term of the gravitational field, b is not an independent coupling constant, but it gives a pure counterterm for eliminating divergences proportional to G 4 . 4 The Weyl tensor squared (4.3) satisfies √ −g C 2 µνλσ = √ −ḡC 2 µνλσ in 4 dimensions when the metric field is decomposed as in (1.1), where the curvature with the bar is that composed of the metricḡ µν . Furthermore,ḡ µν is expanded by introducing the traceless tensor field h µ ν asḡ µν = (ĝe h ) µν = g µλ (δ λ ν + h λ ν + · · · ), whereĝ µν is a background metric. If the background spacetime is taken to be the flat and the field is normalized as h µ ν → th µ ν , then the Weyl action in I G (4.5) that gives the kinetic term of the traceless tensor field is expanded as , where χ µ = ∂ ν h ν µ and the indices are contracted by η µν . Then, the effective action of the quantum gravity can be calculated as an ordinary quantum field theory defined in the flat spacetime.  [10,19,20]. The coefficient χn can be determined in order by solving the Hathrell's RG equations, and first three have been calculated as χ1 = 1/2, χ2 = 3/4, and χ3 = 1/3. The Riegert action and its series, as discussed later, are generated by Laurent-expanding a bare action bGD. 5 As in footnote 3, if the effective action is calculated by introducing the background traceless tensor fieldĥ µ ν asĝµν = (e tĥ ) µ ν , then diffeomorphism invariance demands that a renormalization factor of the field, Z 1/2 h , and that of the coupling constant, Zt, satisfy the identity ZtZ 1/2 h = 1. Diffeomorphism invariance also demands that a non-renomalization theorem holds for φ because this field is treated exactly without introducing a coupling constant for it [18,19,20,21,22].
The Weyl part has a structure similar to the gauge field. Now, L (0) (g) is √ −g C 2 µνλσ , thus the first Wess-Zumino action is given by Further integrating it, we obtain S (n) µνλσ that satisfies the recursion relation (2.2), and then L (n) (g), which is a covariant form containing it, will be given by log n (Q 2 /µ 2 ) √ −g C 2 µνλσ in momentum space, as in (2.5). Since the beta function of the coupling constant t is known to be negative, we can express the logarithmic part using a running coupling constantt 2 (Q) as in QCD, which is obtained by replacing g with t and, accordingly, renaming Λ QCD to Λ QG , then the effective action will be given by replacing t 2 in (4.5) witht 2 (Q) [18,19,22]. 6 This suggests that whent(Q) becomes larger at the new physical scale Λ QG , the 4th-derivative conformal gravity dynamics disappears.
Next, we consider the Wess-Zumino action for the modified Euler density (4.4). In this case, if the argument is made with φ as a constant as before, the essence will be lacking, so it is treated as a function in coordinate space here. Its first Wess-Zumino action is then given by [16] S (1) The integration of φ is performed using the relation where √ −g∆ 4 is a conformally invariant differential operator defined by for for any scalar A and B. The action (4.6), called 6 Note that in dimensional regularization, a D-dimensional Weyl tensor squared has the structure √ −g C 2 µνλσ = √ −ḡ e (D−4)φC 2 µνλσ similar to the gauge field squared.
the Riegert action, works as a kinetic term of the conformal-factor field [25,26,27,17]. The first Wess-Zumino action can be written in a diffeomorphism invariant form as Γ (1) This action has a nonlocal structure, but does not contain the scale µ. Also, unlike other Wess-Zumino actions, it appears even in the zeroth order of the coupling constant. Further integrating the first Wess-Zumino action yields Using the self-adjointness of∆ 4 and d 4 x √ −ḡ∆ 4 A = 0, we can show that this action satisfies the recursion relation (2.2). Now, we consider a diffeomorphism invariant effective action containing S (n) R top , which will appear in higher-loop corrections. To begin with, Γ R is a quantity that appears even in the zeroth order of the coupling constant, thus it does not involved the scale µ, as shown above. However, if corrections by the coupling constant t are added, the logarithmic correction will be accompanied. From these considerations and paying attention to log ∆ 4 = −4φ + log∆ 4 , we deduce the following diffeomorphism invariant action that contains S (n) R top as a local part: Here the logarithmic part suggests that the effective action will be written in terms of the running coupling constantt(Q).
In quantum theory of gravity, the total energy-momentum tensor must vanish as an equation of motion of quantum gravitational fields. It implies that conformal invariance is realized as diffeomorphism invariance at the quantum level. The Riegert action (4.6) then arises rather as necessary to restore the conformal invariance. This conformal symmetry is a gauge symmetry that appears only when gravity is quantized. 7 It means that all theories with different backgrounds connected to each other by conformal transformations are gauge equivalent, that is, background-metric independent. Whether it exists or not is the difference between quantum gravity and quantum field theory in curved spacetime. It has been shown that under this symmetry, all of ghost modes become gauge variant, namely, unphysical, even at the UV limit, thus they are confined in quantum spacetime [28,29,30,22].

Conclusion
In this paper, we examined the Wess-Zumino actions at higher loops that are obtained by integrating the conformal anomalies with respect to the conformal-factor field. The consistency condition (1.2) that the Wess-Zumino actions should satisfy was derived from the fact that the effective action is diffeomorphism invariant, and a series of the Wess-Zumino actions satisfying the condition was constructed by repeating the integration. We have seen that they arise to make the nonlocal loop correction terms diffeomorphism invariant and that the effective action is described in terms of the physical momentum (2.4). Thus, conformal anomalies are indispensable quantities to preserve diffeomorphism invariance, so that we must always incorporate them when considering quantum field theory in curved spacetime and quantum gravity.
As a specific example, the QED effective action in a conformally flat spacetime was calculated at the 3-loop level. It was done by employing dimensional regularization that obviously preserves both gauge and diffeomorphism invariances. In this method, the result does not depend on the choice of the path integral measure. Instead, the information of conformal anomalies is included between D and 4 dimensions, thus it requires careful handling of finite quantities that appear when the poles and zeros of D − 4 cancel out. In this way, we confirmed that a series of the Wess-Zumino actions is actually realized.
The same will hold true for the effective action of QCD from gauge symmetry and diffeomorphism invariance. Here we have shown that it can be summarized in the form of the reciprocal of the running coupling constant squared described in the physical momentum. We also examined the effective action of renormalizable quantum conformal gravity. The Wess-Zumino action obtained by integrating the Weyl tensor squared has the same structure as that of the gauge field, thus the Weyl part of the effective action will be written in terms of the running coupling constant as well. Furthermore, we considered a series of the Wess-Zumino actions concerning the Euler density, and deduced that the corresponding effective action will be given by the Riegert action with logarithmic nonlocality (4.7) that can be regarded as a realization of the running coupling constant.
Finally, in QCD and the quantum gravity, we have seen that the effective action will be summarized in a form in which the kinetic term disappears when the running coupling constant diverges, as in (4.2). Under this consideration, we can construct a simplified model of strong coupling dynamics occurring at low energy by approximating the running coupling constant as a coordinate-dependent average. Applying such a mean field approximation to the quantum gravity, we can describe a spacetime transition that the conformal gravity dynamics disappear at the energy scale Λ QG predicted to be about 10 17 GeV, below the Planck scale, and shift to Einstein gravity [23,24,22]. One of the aim of this paper is to theoretically reinforce those achievements.

A Running Coupling Constant and Effective Action
In this appendix, we show the relationship between the effective action and the running coupling constant. Here, more generally, the beta function is expanded up to and higher-order coefficients are taken to be zero. Integrating (A.1) gives dg ′ β g (g ′ ) .
The lower limit of integration,ḡ(C), is a constant, that is, a RG invariant satisfying µdḡ(C)/dµ = 0. Here, it is decided so that the integration constant C becomes − log Q. If we write it asḡ(Q) and let it be a function that satisfiesḡ(µ) = g, then is obtained. The functionḡ(Q) is called the running coupling constant.
Performing the integration up to the 6th order yields where the right-hand side is defined by and the constant (β 1 /β 0 ) log β 0 has been added to both sides for convention.
Since the left-hand side of (A.2) is a RG invariant, the energy scale Λ QCD defined through this equation is also so, namely, it is a physical scale.
Here, we consider the effective action up to o(g 6 ). Seeing up to o(g 4 ), it agrees with (4.1).