# Frame (in)equivalence in quantum field theory and cosmology

## Abstract

We revisit the question of frame equivalence in Quantum Field Theory in the presence of gravity, a situation of relevance for theories aiming to describe the early Universe dynamics and Inflation in particular. We show that in those cases, the path integral measure must be carefully defined and that the requirement of diffeomorphism invariance forces it to depend non-trivially on the fields. As a consequence, the measure will transform also non-trivially between different frames and it will induce a new finite contribution to the Quantum Effective Action that we name *frame discriminant*. This new contribution must be taken into account in order to assess the dynamics and physical consequences of a given theory. We apply our result to scalar-tensor theories described in the Einstein and Jordan frame, where we find that the frame discriminant can be thought as inducing a scale-invariant regularization scheme in the Jordan frame.

## 1 Introduction

A fundamental property that all sensible physical theories share is the fact that physical statements cannot depend on the choice of variables we use to describe the physical system, even though there might be a set of variables which have a preference. For example, in Special Relativity we have the notion of different inertial frames associated to observers moving at different relative velocities. Both observers have their own preferred coordinate frames in which to describe events but physical statements are invariant under Lorentz transformations which relate the two frames. Moving to the theory of General Relativity, we demand physical statements to be invariant under quite arbitrary coordinate transformations on space-time. In classical field theory one can also extend the notion of general covariance to field space by demanding that physical statements are independent of the way we parametrise the field variables. Invariably, the equations of motion will appear simpler if we use a certain set of variables, however the physics should be indifferent to this choice.

Quantum Field Theory (QFT) is a different story though, since the formalism is drastically different to classical mechanics. In perturbative QFT we are interested in amplitudes between asymptotic states, which can be obtained by taking variational derivatives of the Quantum Effective Action after performing a path integral over all possible paths with the right boundary conditions. One problem is that the standard definition of the Quantum Effective Action depends on the choice of variables as a consequence of the source term. However since the source is equal to the effective equations of motion, the non-equivalent pieces which arise for this reason do not contribute to on-shell amplitudes used to derive S-matrix elements. More generally, since observables are evaluated for vanishing source this dependence on the choice of variables is innocuous. Indeed one may even overcome this problem off-shell by using the unique effective action [1] which makes use of a covariant source term. However, as we shall see, this is not the end of the story. Even with the vanishing source terms, the path integral measure must also transform in a covariant manner for theories formulated with different field variables to be equivalent. This issue becomes especially subtle in the presence of gravity and whenever extra symmetries are required for the field manifold.

A first indication that the choice of variables can be significant was found in [2] where it was pointed out that in certain scalar-tensor theories it is possible to map anomalous symmetries (scale invariance) to healthy ones (shift symmetry) after a field redefinition. Specifically, theories which are classically scale invariant in the Jordan frame and are related to a theory which enjoys shift symmetry in the Einstein frame. In that case, quantization in the two different sets of variables lead to a different S-matrix due to the appearance of new transition amplitudes only in the Jordan frame, where the scale invariance is anomalous.^{1} In the Einstein frame the shift symmetry remains intact in the quantum theory and consequently no anomaly occurs. Through the example in [2], one can trace the origin of the discrepancy to the existence of the metric as a dynamical degree of freedom, since it is the metric redefinition what eventually leads to the transmutation of symmetries. This prompts us to further investigate the frame dependence of more general scalar-tensor theories and to identify the origin of disparity between different quantum theories.

^{2}

^{3}Related to scalar tensor models are

*f*(

*R*) models where \(\mathcal{L} = f(R)\) which, apart for the case where

*f*(

*R*) is linear, also describe one physical scalar particle coupled to a spin-two graviton.

*f*(

*R*) model as a scalar tensor theory either in the Jordan frame or the Einstein frame.

The simpler setting of the Einstein frame allows for an also straightforward interpretation of the dynamics of the system as a scalar field rolling down the new potential, from which we can derive all relevant inflationary parameters. However, as we have previously pointed out, this is a dangerous step if we want to include quantum effects, since the quantum formulation of both theories might be different in certain cases and we might be missing important physical effects. Indeed, the question of equivalence of scalar-tensor theories in Cosmology has been thoroughly studied in the recent years from many different points of view ([20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37] and references therein). However, most works are focused on the classical and observational aspects of frame equivalence and the few that study the issue at a quantum level find contradicting results. Several works have concentrated on the divergent part of the one-loop the effective action and the corresponding beta functions (for related non-perturbative studies using the functional renormalisation see [38, 39]). These studies show that the divergent part of the effective action generically differs in the two frames by terms proportional to the equations of motion. This was shown in two and four dimensional dilatonic gravity in [21, 40] while in four dimensions this has also been proven in [22] for a wide range of models. In [30] calculations were carried out using the field space-covariant Vilkovisky-DeWitt effective action which guarantees that results are formally independent of parameterisation of the quantum fields (see also [41, 42] for another frame-covariant approach). However one should bear in mind that even if one uses a covariant approach results can still depend on the definition of the geometric objects such as metrics and connections defined on field space.

Motivated by these concerns, which might have consequences for many important inflationary and gravitational models, we wish to revisit the problem of equivalence of Quantum Field Theories. We will do this by giving a proper definition of all the elements involved in the path integral quantization of a given Quantum Field Theory and studying their behaviour under a change of frames. We will find that, as hinted by the previous discussion about anomalies, the source of the apparent inequivalence between the frames is the definition of the path integral measure, which includes the determinant of a metric defined on the field manifold. While this metric is generically field independent for scalars, fermions and vector fields, and thus it can be ignored for perturbative computations, this is no longer the case when gravity enters into the game. The requirement of diffeomorphism invariance of the Quantum Effective Action (even when the metric is just a semi-classical degree of freedom or a external source) forces the integration measure to depend non-trivially on the field variables.

If we want preserve frame equivalence at the quantum level, the measure must also transform non-trivially after a change of frames. However if we first change frames at the classical level and then quantize the resulting theory, the measure will not coincide with the transformed one. The operations of changing frames and quantizing do not commute. Consequently, the corresponding Quantum Effective Actions will differ by a non-vanishing finite piece which is not proportional to the equations of motion. The derivation of this frame discriminant term constitutes the main novel result of this work. It will contribute to 1PI correlation functions and thus it cannot be ignored. Disregarding it represents a different choice of integration measure, and thus a *different Quantum Field Theory*.

This paper is organized as follows. In Sect. 2 we will introduce the concept of frame equivalence both at the classical and quantum level, discussing the state-of-the-art of the discussion and raising some concerns for scalar-tensor theories. In Sect. 3 we will define the path integral and the integration measure for a general theory, keeping in mind the scalar-tensor theories of interest and discussing the transformation of the path integral measure.

We will then present the derivation of the frame discriminant using the background field method in Sect. 4 and we will apply our formalism to scalar-tensor theories in Sect. 5, describing also its relation with the so called *scale-invariant regularization*. Finally, we will summarize and discuss our results and conclusions in Sect. 6. Appendix A will be devoted to prove some statements about our derivation in the presence of gauge invariance.

## 2 Frame equivalence

Frame equivalence is an important assumption for physics to be reliable. It means that the choice of variables used to describe a system should not matter when deriving physical statements, although of course computations might be simpler for some of these choices than for others. The trivial example of this situation is the case of a particle forced to move in a circumference in classical mechanics. The system can be described either by using Cartesian or polar coordinates. The equations are simpler in the latter but physical statements are equivalent and in one-to-one correspondence, provided that we properly transform quantities between different coordinates systems.

*equivalent*if any physical quantity \(A(\Phi )\) satisfies

^{4}One may summerize the situation by noting that it is the non-covariance of (9) which is responsible for (10) but that, since observables are calculated for \(\mathcal{J} = 0\), this can only lead to disparities in intermediate steps in the calculation of correlation functions but not in observables (e.g. the S-matrix).

*m*. The corresponding action is

*C*is a constant.

Now we can ask to what extent the two actions (12) and (15) are classically equivalent. If we consider the equations of motion for (12) it is clear that \(\phi =0\) is a solution for all metrics \(g_{\mu \nu }\). However for \(\phi = 0\) the coordinate transformation between the two frames is singular, since it maps to \({\tilde{g}}_{\mu \nu } = 0\) and \({\tilde{\phi }} = \infty \). Thus, equivalence demands the theory to be in the broken phase. As long as we give a vacuum expectation value to the field \(\phi \), both frames are classically equivalent.

^{5}we will generate contributions to the effective action of the generic form

*d*-dimensional, with \(d=4+\epsilon \). This means that, after a scale transformation, the integrand will transform as

^{6}they will generate a new scattering amplitude from the anomalous contribution to the current conservation

*not generated*. This effect distinguishes the frames.

There is an obvious clash here with the conclusion of [1], which claims that the S-matrix must be equivalent in both frames. However, by examining this particular illuminating example of a scale-invariant theory that we have chosen, it is not difficult to see what is the origin of the issue. Going carefully over the derivation described in the previous paragraphs, we see that there are two crucial steps involved in the computation – the introduction of a regularization and the computation of the anomaly. Taking into account that, provided that the transformation between frames is not singular, the action transforms in a proper manner, this clearly isolates the origin of the problem in the measure of the path integral. Indeed, if one goes over the derivation of the Unique Effective Action in [1], it can be seen that although the path integral measure is carefully defined in the paper, it is considered to be the same in any frame and thus to give the same contribution regardless of the choice of variables. We will see in the next section that this is actually not true.

## 3 The functional integral

*before regularisation*, whose definition might vary depending on the parametrization of the degrees of freedom used to construct the perturbative expansion of the path integral. However, we must require this measure to be reparametrisation invariant. This can be achieved by regarding the fields \(\Phi \) as coordinates on the configuration space \({\mathcal {M}}_\Phi \). General covariance in this space thus defines

*a*as a DeWitt index. Consequently the product over

*a*also implies a product over points in spacetime. The factor of \(\sqrt{2\pi }\) appears for normalization purposes. The measure is then parametrized by the metric \(C_{ab}\) in \({\mathcal {M}}_\Phi \), which will be generally curved and is to be understood as a two point function of the ultra-local form

Thus the path integral will not depend only on the action \(S[\Phi ]\) but also on the choice for the metrics \(C_{ab}[\Phi ]\) and \(\eta _{\alpha \beta }[\Phi ]\). For theories such as Yang-Mills, the metrics can be chosen to be independent of the fields without breaking gauge invariance and thus they are not relevant for the computation of correlators in perturbation theory. However, in the presence of gravity, the metrics have to depend on the dynamical fields \(\Phi \) themselves to preserve diffeomorphism invariance [46]. This implies that in a general case we cannot neglect the contribution of the functional measure into the result of the path integral.

*S*satisfies (4) in a trivial manner

*can not affect the physics*; it is just a choice of coordinates on \({\mathcal {M}}_\Phi \). However, what can affect physics is the choice of the metric \(C_{ab}\) and the choice of the metric \(\eta _{\alpha \beta }\). If one were to choose different metrics \(C_{ab}\) and \(\eta _{\alpha \beta }\) then evidently the path integral would be different. Thus, while classically we require that equivalent theories have actions related by (27), quantum mechanically we have the addition requirement that the measures of the theories are equivalent which is satisfied by (28).

The explicit construction of the metric is a subtle issue and different approaches can be found in the literature [46, 47, 48, 49, 50]. A fundamental restriction on the choice of \(C_{ab}\) and \(\eta _{\alpha \beta }\) which we can impose is that they must lead to a BRST invariant measure for gauge theories [46]. However this only dictates that they transform in a covariant manner under a gauge transformation and does not fix their form. Thus to completely fix the measure we must give a prescription which may itself depend on a preferred choice for the field variables (also phrased as a choice of *frame*). Since different prescriptions lead to different path integrals involving the same action, they correspond to *different quantisations* of the same classical theory. In other words we may encounter a situation where (27) holds but (28) is violated.

*C*(

*x*,

*y*) is ultra-local, we can determine it up to the choice of a scalar

*s*(

*x*) where

*s*(

*x*) is independent of \(\phi \) we can thus formally perform the functional integral to obtain

*standard*procedure as it is the one which is adopted in practice. A derivation of this prescription starting from the phase space path integral which defines the canonical theory is given in [48].

Thus, the prescription is not unique. Different choices of the preferred space-time metric \({\mathfrak {g}}_{\mu \nu }\) will lead to different path integrals. This choice can then be interpreted as a preferred frame choice since, if we consider two parametrisations of the fields \(\Phi ^a\) and \({\tilde{\Phi }}^a\) which include metrics \(g_{\mu \nu }\) and \({\tilde{g}}_{\mu \nu }\) respectively, then the choices \( {\mathfrak {g}}_{\mu \nu } = g_{\mu \nu }\) and \({\mathfrak {g}}_{\mu \nu } = {\tilde{g}}_{\mu \nu }\) will in general lead to different path integrals. Nonetheless the choice of which field variables we use to carry out the calculation is independent of how we identify \({\mathfrak {g}}_{\mu \nu }\) in order to determine the form of the measure. While the former choice does not affect the physics, the latter choice can be understood as a different quantization which can lead to different physical predictions and thus to different quantum field theories. One can therefore trace the consequences of defining theories with different preferred metrics \({\mathfrak {g}}_{\mu \nu }\) to the additional factor of \(e^{2 \sigma }\). By carefully keeping track of this difference one can then identify a concrete physical difference between the two inequivalent quantum theories.

## 4 The frame discriminant: a background field approach

*S*therefore transforms as a scalar in the sense that

*S*and \({\tilde{S}}\) are the actions before gauge fixing. In particular, we will consider that the spacetime metrics will differ by a non-trivial conformal factor

*G*are inequivalent metrics. Specifically, they differ by the factor \(e^{2\sigma }\) in addition to the expected tensor transformation between frames given in (28).

As we discussed in the previous section, there now comes a choice of which spacetime metric \(g_{\mu \nu }\) or \({\tilde{g}}_{\mu \nu }\) we select as the physical one \({\mathfrak {g}}_{\mu \nu }\), since different choices will lead to inequivalent path integral measures. If we choose \({\mathfrak {g}}_{\mu \nu } = g_{\mu \nu }\) the metric on field space is given by \(C_{ab} = \Lambda ^2 G_{ab}\) which is the natural measure in the \(\Phi \) frame. Alternatively, if we declare the physical spacetime metric to be \({\mathfrak {g}}_{\mu \nu } = {\tilde{g}}_{\mu \nu }\), which is the natural choice in the \({\tilde{\Phi }}\) frame, the field space metric is given by \({\tilde{C}}_{ab} = \Lambda ^2 {\tilde{G}}_{ab}\). However from (47) we see that \(C_{ab}\) and \({\tilde{C}}_{ab}\) are not related by simply a change of coordinates on field space.

*Y*and \({\tilde{Y}}\) are related by

### 4.1 Regularisation and renormalisation

^{7}

*D*. One can show [53, 55] that although the operators differ in the two frames, the coefficients agree such that \( \sqrt{g}B_{4}( e^{-2 \sigma } \Delta ) = \sqrt{g} B_{4}( \Delta )\) and \( \sqrt{g}B_{4}( e^{-2 \sigma } Q) = \sqrt{g} B_{4}( Q)\).

^{8}After subtracting the counter term we will then have a finite contribution to the effective action given by

*frame discriminant*hereinafter. An equivalent way to quantify the difference between quantising the theory in either frames follows from promoting \(\mu \) to a field dependent scale via

Thus we can conclude that the two theories are inequivalent at the one-loop level and will therefore give different physical predictions, derived from the frame discriminant \(\mathcal{A}\). This means that there is an ambiguity in the quantization of the theory related to the choice of the functional measure. This choice of functional measure can in turn be traced to a choice of which spacetime metric is declared to be the physical one. The frame discriminant \({\mathcal {A}}\) is finite and a function of the fields in the theory, so it will potentially generate new amplitudes that can contribute to S-matrix elements.

As a final note, let us note that our derivation here can be applied to any QFT regardless of its renormalizablity. Even in the case of an EFT, this procedure is well-defined as long as we compare expressions at the same order in the perturbative expansion in terms of the coupling constants. Indeed, in the next section we will show how this piece solves the anomaly problem in the scale invariant scalar-tensor theory that we used as an example in Sect. 2, which is non-renormalizable and thus can be only considered as most as an EFT.

## 5 The frame discriminant in scale invariant scalar-tensor theories

*a different quantum field theory.*

*preferred frame*in which to determine the measure will mean we remain anomaly free even if we ultimately use Jordan frame variables.

^{9}In a more general case we would find terms involving derivatives of \({\tilde{\phi }}\) and terms which vanish on the equations of motion.

^{10}

What is happening here is that the S-matrix, and thus all physical properties, are defined by *the frame in which we define the functional measure*, where we implicitly choose a preferred metric \({\mathfrak {g}}_{\mu \nu }\). In any other frame, the effective action must transform appropriately in order to preserve all physical statements and in particular all S-matrix amplitudes. Since there are no anomalously generated elements in the Einstein frame, our quantization process must preserve this condition in any other frame.

The role of the frame discriminant in this example is precisely to compensate the differences in the finite pieces of the quantum effective action between the two different frames, being those the origin of the scale anomaly. But this also means that the frame in which we choose to start is very important. If instead we were starting from the Jordan frame, where the anomaly is a physical effect, the frame discriminant would give us exactly the opposite effect to what we have shown here – to generate the consequences of the scale anomaly in any other frame, in order to preserve all S-matrix elements. Of course, this effect is not restricted to theories with anomalous currents, but it appears whenever we do a non-linear redefinition of variables which affects the integration measure. In summary, a quantum field theory is not defined solely by the action, but also by the choice of integration measure or equivalently by the choice of preferred frame which selects the form of the measure.

### 5.1 A comment on scale-invariant regularization

*field dependent*, with a parameter

*z*encoding the scheme independence

^{11}inherited from \(\mu \). Moreover, once in the broken phase, which is the only phase in which both frames are even classically equivalent, we have \(\phi =\langle \phi \rangle +\delta \phi \) and therefore

This construction can be found in the literature under the name of *scale-invariant regularization*, motivated by the search of a common solution to the hierarchy and cosmological constant problems altogether [16, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73], as well as to the question of whether scale invariance can be preserved at the quantum level as a fundamental symmetry of Nature. Indeed, if one uses this regularization by substituting \(\mu \) by \({\bar{\mu }}\) everywhere, scale invariance is preserved in the quantum effective action at all orders in the perturbative expansion. Then, both the hierarchy and cosmological constant problems seem to be solved at once thanks to the cancellation of radiative corrections to dimensionful quantities [74, 75]. Afterwards, the spontaneous breaking of the symmetry by \(\langle \phi \rangle \) gives rise to the standard terms plus new interactions. Ways to trigger this spontaneous symmetry breaking from the point of view of cosmology have been also recently explored [76, 77, 78].

Our arguments here seem to suggest that this regularization can be also understood as a consequence of choosing the Einstein frame as our preferred frame, thus forcing the scale anomaly to be absent to satisfy equivalence, thanks to the contributions of the frame discriminant. In the literature about frame equivalence and scale invariant regularization (see e.g [79, 80, 81] and references therein) this is normally described in terms of two different regularization prescriptions – *prescription I* refers to taking the renormalization scale \(\mu \) to be constant in the Einstein frame and field-dependent in the Jordan frame, while *prescription II* represents the opposite situation. This would correspond, in our language, to choose the preferred metric in the Einstein frame (*prescription I*) or in the Jordan frame (*prescription II*) in total agreement with previous results.

The fact that a scale invariant renormalization procedure corresponds to a non-standard quantisation with a scale invariant measure has been observed in [66], where the idea was to have a renomalisation scheme that preserves exact local scale invariance (i.e. Weyl invariance) by the introduction of a dilaton i.e. the field \(\phi \). In this case one can view the dilaton as an auxiliary field, with local scale invariance being ‘fake’, since one can always gauge fix the dilaton to be a constant. From the view point of frames, gauge fixing the dilaton is tantamount to going to the Einstein frame, where the shift symmetry is now local and enforces the action to be independent of \({\tilde{\phi }}\), since the shift transformation is now \({\tilde{\phi }}(x) \rightarrow {\tilde{\phi }}(x) + C(x)\).

## 6 Discussion and conclusions

In this paper we have studied the problem of frame equivalence of a given Quantum Field Theory. While in classical physics it can be easily proven that stationary trajectories map to stationary trajectories under a non-singular change of variables (of frame), Quantum Field Theory requires the extra ingredient of defining the path integral measure. In the case of scalars, fermions and Yang-Mills fields, the integration measure is typically field independent,^{12} but it is not the case anymore if we want to preserve diffeomorphism invariance when the metric is a dynamical degree of freedom. When we quantize a theory in the Einstein frame, where the matter is minimally coupled to gravity and the scalar has a canonical kinetic term, the measure will depend on the metric alone. However the measure obtained by quantizing a theory in the Jordan frame will depend on the scalar field in addition to the metric. What we have established in this paper is that, even after transforming the measure to take account of the Jacobian (a purely mathematical operation), the measures are not equivalent. The frame where we choose to define the path integral matters, and defining the measure in different frames leads to *different Quantum Field Theories*. Of course one could simply insist that the measures in both frames are equivalent, however this is only possible if the quantization in one of the frames would be non-standard.

*any*physical conclusion of the theory. If for some reason we however want to describe it in a different set of variables, perhaps for symmetry or interpretation reasons, then we must carry on the effect of changing frames in the integration measure, together with transforming the action. However the resulting effective action will differ from the one which would result from choosing the second frame as the preferred frame to define the measure. We have shown that this difference can be evaluated in a way which is close to Fujikawa’s method for the trace anomaly [82] and that it reduces, in the case of conformal rescalings of the metric, to the need of adding a

*frame discriminant*contribution to the one-loop Quantum Effective Action in the transformed frame

^{13}

*preferred frame*where we define the integration measure is the Einstein frame, our results can be summarized in the fact that the local part of the one-loop renormalized quantum effective action will read in both frames

That is, when changing frames one should not only transform the divergences of the theory but also promote the renormalization scale \(\mu \) to be *field dependent* precisely by a conformal transformation.^{14} This statement can be actually found in previous literature as a way to preserve the predictions of Higgs [79] and Higgs-Dilaton [16] inflation or under the name of *scale invariant regularization*. Here we give an extra formal justification to this procedure from the request of frame equivalence of the path integral formulation.

We have shown, in particular, that the introduction of the frame discriminant for scalar-tensor theories solves the problem pointed out in [2] with the action (12), whose naive quantization generates a scale anomaly in the Jordan frame which is absent in the Einstein frame. Inclusion of the frame discriminant precisely compensates this effect and enforces the effective action and all S-matrix elements in both frames to agree.

Our result here is however not restrained to scale-invariant theories, scalar-tensor theories or even to conformal transformations (although this is the most typical situation in literature) but it applies to any Quantum Field Theory where the metric is a dynamical degree of freedom and a change of frame is performed. This includes, among others, several models of inflation [4, 14, 16, 84, 85, 86], higher derivative [87, 88], Lovelock [89] and *F*(*R*) gravity [90], the relation between the string frame and the Einstein frame [27, 91], and the Weyl invariant formulations of Unimodular Gravity [92, 93]. If we want to extract dependable conclusions from the Quantum Effective Action on any of these theories, we must add the frame discriminant contribution whenever we perform a change of variables. Otherwise we might be missing important physical effects that could strongly modify our conclusions.

There are three main questions open for future research following the work in this paper. First, it would be useful to extend our arguments here beyond the one-loop approximation. In particular, it would be interesting to understand if the relation between frame equivalence and scale invariant renormalization holds at all orders, providing thus a complete justification for the use of the latter. More broadly one should establish a consistent effective field theory incorporating the choice of the measure and all of its consequences.

On the other hand, it is reasonable to ask if there is any physical compelling argument to prefer one frame over another. Taking into account that the operators in the action and those generated by radiative corrections differ in different frames, one could think that the choice must be influenced by the UV completion of the models that we are studying. Indeed, if we had such completion at our disposal, the procedure to obtain a low energy effective field theory would be unique and it would single out a preferred expression for the action and variables to use. It would be thus interesting to understand if we can actually make a reasoning on the opposite direction. If we can use our results here to pinpoint a given action as preferred, this could give us information on the shape of the UV completion of our theory, which in particular might be relevant to understand new features of Quantum Gravity.

Finally, it would be useful to have an explicitly frame invariant effective action, following the spirit of the Unique Effective Action of Vilkovisky, including the frame discriminant as a built-in feature. This can be achieved by properly incorporating a frame invariant integration measure for the path integral into the definition of the effective action as we have outlined here.

## Footnotes

- 1.
The textbook example of the triggering of new S-matrix elements by anomalous currents is the decay of the neutral pion in two photons due to the axial anomaly in chiral perturbation theory.

- 2.
Here and throughout we write the Euclidean Lagrangians. The corresponding Lorentzian Lagrangian comes with a relative minus sign for each term.

- 3.
- 4.
- 5.
Here we use dimensional regularization for simplicity of the discussion and computations, since it is a standard tool in QFT. However, any other regularization will unavoidably lead to the same conclusions.

- 6.
The definition of the S-matrix depends on the uniqueness of asymptotic states, which is only possible if the space-time is globally hyperbolic [45]

- 7.
- 8.
The classical example of this is the Coleman-Weinberg potential [56], where \(M_\mathrm{phys}\) will be a combination of the mass and vacuum expectation value of the scalar field.

- 9.
On the other hand, the divergence proportional to \({\tilde{R}}_{\mu \nu \alpha \beta }{\tilde{R}}^{\mu \nu \alpha \beta }\) is a topological invariant [58].

- 10.
Here the subscript

*J*simply denotes which variables we are using while the tilde indicates that the preferred frame is the Einstein frame. - 11.
- 12.
The exception is when the kinetic term in the action is non-canonical, for example in the case of a non-linear sigma model.

- 13.A particular theory of this kind of important relevance is Higgs Inflation [14] where with \(\lambda \) and
*v*being the self-coupling and vacuum expectation value of the Standard Model Higgs boson. - 14.
One can check that, provided that the metric transforms as \({\tilde{g}}_{\mu \nu }=e^{2\sigma }g_{\mu \nu }\) and after choosing a chart of coordinates, any energy scale of the theory must transform as \({\tilde{E}}=e^{-\sigma }E\) by dimensional analysis.

## Notes

### Acknowledgements

We are grateful to Fedor Bezrukov, Christopher T. Hill, Roberto Percacci, Sergey Sibiryakov and Anna Tokareva for discussions and/or e-mail exchange. We also wish to thank Mikhail Shaposhnikov and Sander Mooij for useful comments on a previous version of this text. Our work has received support from the Tomalla Foundation and the Swiss National Science Foundation.

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