PHYSICAL REVIEW D 99, 085013 (2019)
Hidden and explicit quantum scale invariance
† ‡ Sander Mooij,* Mikhail Shaposhnikov, and Thibault Voumard Institute of Physics, Laboratory for Particle Physics and Cosmology (LPPC), École Polytechnique F´ed´erale de Lausanne (EPFL), CH-1015 Lausanne, Switzerland
(Received 11 February 2019; published 30 April 2019)
There exist renormalization schemes that explicitly preserve the scale invariance of a theory at the quantum level. Imposing a scale-invariant renormalization breaks renormalizability and induces new nontrivial operators in the theory. In this work, we study the effects of such scale-invariant renormalization procedures. On the one hand, an explicitly quantum scale-invariant theory can emerge from the scale- invariant renormalization of a scale-invariant Lagrangian. On the other hand, we show how a quantum scale-invariant theory can equally emerge from a Lagrangian visibly breaking scale invariance renor- malized with scale-dependent renormalization (such as the traditional MS scheme). In this last case, scale invariance is hidden in the theory, in the sense that it only appears explicitly after renormalization.
DOI: 10.1103/PhysRevD.99.085013
I. INTRODUCTION formulation of scale-invariant renormalization uses dimen- sional regularization, but with the renormalization scale μ It is a textbook truth that a classical theory is defined replaced by a field operator of the appropriate mass completely by its Lagrangian whereas for the formulation dimension. No explicit mass scale is introduced in the of a quantum field theory (QFT) one needs a Lagrangian theory, and SI is preserved. and a renormalization procedure for removing the diver- These SI schemes thus have the peculiarity of using gences. This combination can render the symmetries of the dynamical fields to generate renormalization scales. This QFT obscure: if a Lagrangian possesses a symmetry but the leads to a conflict between renormalizability and scale renormalization scheme does not preserve this symmetry, invariance and thus to the introduction of infinite series of the renormalized theory is not explicitly symmetric. This is 1 operators into the Lagrangian which ultimately ensure scale the case with scale invariance (SI), where, for example, the invariance of the quantum theory. The aim of this work is to MS subtraction scheme [24] breaks SI explicitly by show how one can trade these dynamical scales for some introducing an explicit mass scale μ. One may circu- specific modification of the Lagrangian considered, while mvent this problem by using renormalization procedures equipping it with the traditional, constant renormalization designed to preserve the symmetries of a given Lagrangian. scale μ. Renormalization schemes that preserve SI have been This is made possible by the following obvious obser- proposed and give rise to the cancellation of the dilatation vation. If one starts from a given SI Lagrangian equipped anomaly [25,26] [9,27]. In particular, in Ref. [28] an with a dynamical renormalization scale, one can derive a set approach was presented to preserve conformal symmetry of Green’s functions at any chosen loop level. This set of up to first order in perturbation theory. This result has been Green’s functions defines the theory (at the chosen loop extended to any fixed loop order in the elegant work [29]. level). However, the Lagrangian of the QFT that produces It was shown that one can choose conformally invariant this set of Green’s functions is not unique, as a change in counterterms such that the resulting quantum-corrected the renormalization procedure can be compensated by an theory is conformally invariant and finite. The simplest appropriate modification of the Lagrangian. In particular, starting from a SI Lagrangian equipped with SI-preserving *[email protected] † renormalization, one can always find a new Lagrangian [email protected] ‡ equipped with traditional scale-dependent renormalization [email protected] 1 ’ The SI theories may be interesting from the point of view of that will produce the same set of Green s functions at the the hierarchy problem; see e.g., Refs. [1–23]. chosen loop level. Clearly, the Lagrangian in this new theory will need to break SI, in order to compensate for the Published by the American Physical Society under the terms of SI breaking coming from the scale-dependent renormali- the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to zation. The original symmetry of the theory is then hidden the author(s) and the published article’s title, journal citation, in the new Lagrangian, but becomes explicit again when the and DOI. Funded by SCOAP3. theory is renormalized at the chosen loop level with the
2470-0010=2019=99(8)=085013(9) 085013-1 Published by the American Physical Society MOOIJ, SHAPOSHNIKOV, and VOUMARD PHYS. REV. D 99, 085013 (2019) symmetry-breaking scheme. We will explicitly construct renormalization. This does not limit the type of interactions this new Lagrangian for three examples, at the one-loop that can be renormalized with the dilaton. Any interaction level. that in dimensional regularization gets renormalized with This paper is organized as follows. We begin by the help of μ, now gets renormalized with the help of the reminding the reader about scale-invariant renormalization dilaton’sVEVσ¯. in Sec. II. The first example, given in Sec. III, uses the Although this scale ensures a SI form of the quantum simplest nontrivial scenario one could have, i.e., a theory corrections, it has other nontrivial effects. As in usual with two scalar fields. The second one includes a fermion dimensional regularization, we must redefine the couplings field in order to show how the concept generalizes to of the theory to keep them dimensionless in d ¼ 4 − 2ϵ theories with wave-function renormalization and is dis- dimensions using the renormalization scale. This induces cussed in Sec. IV. We finally discuss the inclusion of new couplings between the dilaton and the other fields. For gravity in a model with nonminimal coupling between example, taking the simple theory gravity and the scalar sector of the standard model (SM) in Z Sec. V. We present our conclusions in Sec. VI. 1 λ 4 1 ¼ 4 ∂ ∂μ − h þ ∂ σ∂μσ ð Þ S d x 2 μh h 4! 2 μ ; 2 II. SCALE-INVARIANT RENORMALIZATION Suppose we take a classically SI theory, that is, an action one makes the replacement in d ¼ 4 − 2ϵ dimensions, invariant under a rescaling of the coordinates xμ → αxμ, using σ ¼ σ¯ þ σˆ −Δ accompanied by a rescaling of the fields ϕ → α ϕ ϕ, where Δ ϕ X∞ ð−1Þn−1 σˆ n ϕ is the canonical mass dimension of the field . It should λ → σ2ϵ=ð1−ϵÞλ ¼ λσ¯ 2ϵ 1 þ 2ϵ be noted that this symmetry is satisfied as long as no σ¯ n¼1 n explicit massive (dimensionful) parameter enters the þ Oðϵ2Þ ð Þ theory, as these do not get rescaled by the dilatation. : 3 However, when renormalizing the theory using, for exam- ple, MS dimensional regularization, one precisely intro- One sees a major consequence of taking a dynamical duces such a mass scale μ to regularize the divergences. renormalization scale: it induces infinite series of new This mass scale explicitly breaks SI. operators in the Lagrangian, suppressed by the VEV of the To remedy this situation [9,25], one replaces μ by an dilaton. σˆ operator with the appropriate mass dimension, as fields will Note that for this series to make sense, the fluctuations σ¯ rescale under a dilatation. No explicit mass parameter then need to be small compared to the background field value . enters the QFT, and thus SI is protected at the quantum This is reflected in the earlier observation that scale- level. In our case, for the sake of simplicity, we will only independent renormalization breaks renormalizability. It use scalar fields. Supposing that we have a theory with a generates operators of mass dimension higher than four, scalar field σ (hereafter, the dilaton), we can choose in d suppressed by appropriate powers of the background field. dimensions The demand that these higher-order operators do not spoil the predictivity of the theory is precisely equivalent to the μ ∝ σ2=ðd−2Þ: ð1Þ demand that the series expansion in Eq. (3) is sensible. Consequently, equipping a Lagrangian with a dynamical In order to use this scale for perturbative computations, σ is renormalization scale can preserve SI but renders the theory required to have a vacuum expectation value (VEV) σ¯.Ifit nonrenormalizable [25]. For a potentially renormalizable does not, Eq. (1) admits no polynomial expansion and the Lagrangian, choosing the scale-invariant prescription then dilaton field cannot be used for perturbative renormaliza- amounts to trading renormalizability for scale invariance at tion. Using a more physical point of view, the VEV is the quantum level, provided the original Lagrangian is SI. required to generate the massive scale necessary for On the other hand, if the Lagrangian is not scale invariant at renormalization and to reproduce the original running of the classical level, then it seems that the scale-invariant the couplings [9,30]. Without the dilaton taking a VEV, prescription would never be of use, as no symmetry is perturbative scale-invariant renormalization is impossible. preserved and renormalizability is lost. However, this It might be that in the absence of a dilaton VEV scale- prescription takes all its meaning when gravity is taken invariant renormalization can be done nonperturbatively, into account. In this case, the theory is always nonrenor- but this is not at all clear yet. See for example Ref. [31] for a malizable, so no desirable property of the theory is lost by lattice approach. On the other hand, it has been shown in choosing a scale-invariant prescription. Refs. [8,19,32] that once gravity is included, there is always The infinite series of operators appearing during the a solution with σ¯ ≠ 0. regularization make it clear that, beginning from a given We note that the dilaton field is an extra field that is Lagrangian, choosing a dynamical renormalization scale added to the model in order to facilitate scale-invariant over a traditional, constant scale amounts to considering a
085013-2 HIDDEN AND EXPLICIT QUANTUM SCALE INVARIANCE PHYS. REV. D 99, 085013 (2019) different QFT. As explained before, we will show that one However, let us stress again that, notwithstanding the can always translate this dynamical scale into a modification increasing complexity, scale-invariant renormalization is of the Lagrangian. The new Lagrangian is constructed in well defined up to any fixed loop order, as shown in such a way that, when renormalized at a constant scale, it Ref. [29]. Explicit results up to third order in the loop gives rise to the same Green’s functions produced by the expansion can be found in Ref. [35]. original QFT at a chosen loop level. In the next sections, we We regularize the divergence by adding a counterterm explicitly show how this can be done at the one-loop level for 2 a scalar theory, a Yukawa theory and a theory where gravity is 2ϵ 4 λ 1 λ μ δλh ; δλ ¼ − ln : ð8Þ included, and discuss the generalization to any loop level. ð16πÞ2 ϵ 8πe3=2−γ
III. SCALAR FIELD THEORY This particular choice of counterterm recasts the potential into a simple form when ϵ → 0 Let us consider the simple model with two scalar fields 4 2 4 2 described by the scale-invariant Lagrangian λh λ h h V1 ¼ þ : ð Þ 4! ð16πÞ2 ln σ2 9 1 1 λ L ¼ ∂ ∂μ þ ∂ σ∂μσ − 4 ð Þ 2 μh h 2 μ 4! h : 4 As one can see, the one-loop effective potential2 is explicitly SI: it is free of explicit mass scales. This would To get a SI quantum theory, we choose a renormalization not be so had we used traditional dimensional regulariza- 2 ð −2Þ scale μ ¼ σ = d , where d is the spacetime dimension. tion. A result for more general potentials following the As stated before, in order to make a perturbative expansion, same approach has been given in Ref. [36]. we must impose that SI is broken spontaneously so that the In order to clarify the effects of the dynamical nature of field σ has a nonzero VEV σ¯, i.e., the potential must have a the renormalization scale, we now show that we can start flat direction. from a different potential equipped with another renorm- As we expect no wave-function renormalization at one alization scale, i.e., another QFT, and end up with the same loop for scalar fields, it is sufficient here to consider effective potential. Here we choose the simplest renorm- the effective potential [33,34]. To continue the theory in alization scale one could imagine: a constant one. In other d ¼ 4 − 2ϵ dimensions, we replace words, we return to traditional dimensional regularization. We must then construct a new Lagrangian which will give 2ϵ λ → μ λ; ð5Þ rise to Eq. (9) when renormalized at this constant scale at the one-loop level. which ensures that the coupling λ remains dimensionless. We required at the beginning that the SI must be 2ϵ 2ϵ ð −2Þ When expanding μ ¼ σ = d , we now see that this spontaneously broken, so let us expand Eq. (9) around redefinition introduces an infinite number of new operators the VEV of σ ¼ σ¯ þ σˆ in the theory. Still, as can be seen from Eq. (3), only the 4 2 4 2 2 4 lowest-order operator in σˆ=σ¯ λh λ h h 2λ h σˆ V1 ¼ þ − 1 þ : ð Þ 4! ð16πÞ2 ln σ¯ 2 ð16πÞ2 ln σ¯ 10 4 λσ¯ 2ϵ h ð Þ 4! 6 One can identify two distinct contributions in Eq. (10): the first two terms correspond to a λh4 theory renormalized at a comes at order Oðϵ0Þ. All the operators coming at OðϵÞ constant scale μ ¼ σ¯, whereas the last term corresponds to order are called evanescent, as they disappear when setting an infinite series of operators which ensures SI of the total d ¼ 4. The one-loop effective potential is then potential. From this observation, one can guess that the potential (where the second term carries a factor ℏ that we 4 2 4 2 2ϵ λh λ h 1 λh are suppressing in this work) V1 ¼ μ þ − þ ln 4! 2 ϵ 3=2−γ 2 ð16πÞ 8πe μ 4 2 4 ˜ λh 2λ h σˆ V0 ¼ − 1 þ ð Þ þ OðϵÞ; ð7Þ 4! ð16πÞ2 ln σ¯ 11 where the OðϵÞ term accounts for the contribution of the will give rise to Eq. (9) when renormalized at the constant whole evanescent infinite series. When we send ϵ → 0, the scale σ¯. Indeed, using this constant scale and this new contribution of the infinite series thus vanishes. It is potential, one gets for the one-loop effective potential important to note that this is only true at the one-loop Oðϵ−2Þ level. At higher loop levels, divergences of order 2We will sloppily refer to the potential in Eq. (9) as “effective can appear, and the contribution of OðϵÞ terms in the potential,” even if it contains not only background fields, but also effective potential no longer vanishes when ϵ → 0 [30,35]. quantum fluctuations.
085013-3 MOOIJ, SHAPOSHNIKOV, and VOUMARD PHYS. REV. D 99, 085013 (2019) λ 4 2λ2 4 σˆ ˜ 2ϵ h h This is the most general SI Lagrangian containing a scalar V1 ¼ μ − 1 þ 4! ð16πÞ2 ln σ¯ field and a fermion with field operators of mass dimension λ2 4 −1 λ 2 less than or equal to four. þ h þ h As before, we require the SI to be spontaneously broken, 2 ln 3 2−γ 2 ð16π Þ ϵ 8πe = σ¯ although we do not specify how this effect occurs in this þ Oðλ3ÞþOðϵÞ: ð12Þ particular model. We simply assume that the scalar field σ acquires a VEV σ¯, and we define a dynamical renormal- By sending ϵ → 0 and acknowledging that our expansion ization scale as in the previous model, parameter is λ, i.e., that one-loop corrections are relevant up 2 2 to Oðλ Þ, one gets the same effective potential as in Eq. (9) μ ¼ σd−2: ð14Þ by choosing the same δλ counterterm as in Eq. (8). Using this construction, we see that taking a dynamical This mass scale is used to define dimensionless couplings renormalization scale amounts to choosing a Lagrangian in dimension d ¼ 4 − 2ϵ: containing an infinite series of operators suppressed 2ϵ 2ϵ by the VEV of the dynamical renormalization scale. λ → μ λ ¼ σ1−ϵλ; ð15Þ Consequently, when the VEV of the dynamical renorm- alization scale is large, taking the dynamical or the ϵ ϵ → μ ¼ σ1−ϵ ð Þ constant scale yields the same physical predictions. On y y y: 16 the other hand, the physics differ when considering large ¼ 4 − 2ϵ field fluctuations. The d Lagrangian then reads It is interesting to note that V0 is not SI, but when 0 1 λσ4þ2ϵ renormalized at the scale σ¯, one recovers a SI quantum- μ ¯ ¯ 1þϵ L ¼ ∂μσ∂ σ þ iΨ∂Ψ − − yΨσ Ψ; ð17Þ corrected theory. The SI of the theory is thus hidden at the 2 4! tree level, and made explicit after (one-loop) renormaliza- 2 tion: the scale dependence of the quantum corrections where we have dropped Oðϵ Þ terms as they will not compensates the scale dependence of the tree-level contribute to the final results at one loop. Lagrangian. Note however that to find the hidden scale To obtain suitable Feynman rules, one has to expand σ invariance in Eq. (11), one needs to take every term into around its VEV: σ ¼ σ¯ þ σˆ. One can check that this account; an EFT approach based on a truncation at a certain generates a finite number of simple interactions of order 0 order in 1=σ¯ does not suffice. Only the total expression in Oðϵ Þ, and an infinite number of interactions of order OðϵÞ Eq. (11), as opposed to every individual order in 1=σ¯,is and higher. By recollecting every superficially divergent scale invariant. diagram, one can see that, at the one-loop level, up to the The generalization of the construction to other potentials is freedom in the definition of the counterterms, i.e., up to straightforward, as the procedure is similar. It is clear that this finite contributions, the theory that we need to renormalize construction can be applied at any loop level, although it is equivalent to becomes more complicated as early as at the two-loop level. Oðϵ−2Þ 1 λσ¯ 2ϵσ4 As mentioned before, divergences of order lead to μ ¯ ϵ ¯ L ¼ ∂μσ∂ σ þ iΨ=∂Ψ − − yσ¯ ΨσΨ; ð18Þ nonvanishing contributions coming from the evanescent 2 4! operators of Eq. (3) [30,35]. Then, the same contributions should be added to the new Lagrangian to reproduce the where the ϵ powers of the dynamical field are no longer quantum result, in a way similar to what we did at the one- present. Consequently, at the one-loop level, considering a loop level. One also needs to compensate the corrections constant renormalization scale during regularization only generated by the terms needed to reproduce the lower-loop- induces finite errors that can be absorbed in the counter- level results, which can always be done by adding appro- terms. One should note that this simplification cannot be priate operators of higher order in the couplings [29]. done at higher loop levels, as divergences of order Oðϵ−2Þ or lower are generated, which multiply the interactions of IV. YUKAWA THEORY order OðϵÞ. This theory is well known, and renormalization is now To generalize the example we proposed in the last straightforward. In the same way that an effective potential section, we now consider a model which requires a non- can be defined, we define an effective Lagrangian from the trivial wave-function renormalization at the one-loop level. one given in Eq. (18). This effective Lagrangian has the The Lagrangian we write contains a fermion and a dilaton: properties that its diagrams at tree level give the same results as those of the starting Lagrangian at one loop. 1 λσ4 μ ¯ ¯ The effective Lagrangian which satisfies this requirement is L ¼ ∂μσ∂ σ þ iΨ=∂Ψ − − yΨσΨ: ð13Þ 2 4! (as ϵ → 0) in momentum space
085013-4 HIDDEN AND EXPLICIT QUANTUM SCALE INVARIANCE PHYS. REV. D 99, 085013 (2019) 2 − 2 2 − 2 y 2 2 p y ¯ p to be renormalized already contains the dilaton field L1 ¼ L þ p σ − ΨpΨ 3ð4πÞ2 ln σ2 2ð4πÞ2 ln σ2 in d ¼ 4. σ4 − 2 This construction can be done at any loop level, and for − ðλ2 − 16 4Þ p any theory admitting a perturbative expansion, although it ð16πÞ2 y ln σ2 becomes complicated at the two-loop level already, for the y3 −p2 same reasons that were discussed in Sec. III. − Ψ¯ σΨ : ð Þ ð16πÞ2 ln σ2 19 We note that if we consider a similar Yukawa interaction, but now including γ5, we run into the same problem of how γ ¼ 4 − 2ϵ This result is SI, which means that choosing a dynamical to define 5 in d as in dimensional regulariza- regularization scale to preserve SI not only works for tion [24]. effective potentials, but also works for full Lagrangians, as Let us finish this section with a quick look at scale- should be expected from the absence of an explicit invariant renormalization when gauge fields are present: mass scale. 1 We now repeat the construction done in the previous L ⊃− μν þ Ψ¯ γμð∂ − ÞϕΨ ð Þ 4 FμνF μ igAμ : 21 section, that is we construct a new QFT which leads to the same effective Lagrangian, but with a constant renormal- Clearly, multiplying g by an appropriate power of the ization scale. This allows us to translate the dynamical dilaton breaks gauge invariance. The solution is to rescale nature of the renormalization scale into a choice of tree- the gauge field such that we get level Lagrangian, while leaving the physical predictions 1 untouched. μν ¯ μ σ L ⊃− FμνF þ Ψγ ð∂μ − iAμÞϕΨ: ð22Þ To do so, one expands Eq. (19) around the VEV of . 4g2 One piece will correspond to the effective Lagrangian obtained by renormalizing Eq. (18) at a constant scale Now the gauge field has mass dimension one, in any μ ¼ σ¯, while the other piece comes from the expansion number of dimensions. In this last theory, we can regularize 2 2ϵ 2 around the VEV. Consequently, similarly to the previous in d ¼ 4 − 2ϵ by setting g → σ1−ϵg . With this explicitly section, we propose for the new Lagrangian gauge-invariant and scale-invariant regularization, one is guaranteed that, just like after dimensional regularization, 2 2 σˆ ˜ μ y the resulting effective potential is gauge invariant on shell, L ¼ L − ∂μσˆ∂ σˆ ln 1 þ 3ð4πÞ2 σ¯ up to any order in the loop expansion. 2 σˆ þ Ψ¯ ∂Ψ iy 1 þ ð4πÞ2 ln σ¯ V. INCLUSION OF GRAVITY 2ðσˆ þ σ¯Þ4 σˆ In this section, we present an explicit example of our þ ðλ2 − 16 4Þ 1 þ ð16πÞ2 y ln σ¯ construction in a model with gravity. As already mentioned in the Introduction, theories including gravity are non- 3 2y σˆ renormalizable from the beginning. Therefore, we lose þ ΨΨ¯ ðσˆ þ σ¯Þ ln 1 þ ; ð20Þ ð16πÞ2 σ¯ nothing in applying scale-invariant renormalization. Let us consider a scale-invariant modification of the SM coupled which is the original Lagrangian to which the contribution in a nonminimal way to gravity [9,25]: coming from the dynamical scale has been explicitly added. 1 This Lagrangian, by the same kind of arguments that were 2 2 R 2 2 L ¼ −ðξσσ þ ξh Þ þ ½ð∂μσÞ þð∂μhÞ given in the example with the scalar fields, will give rise to 2 2 σ¯ − λð 2 − ζ2σ2Þ2 þ L ð Þ Eq. (19) when renormalized at one loop, at the scale . h SM: 23 Corrections coming from the series term are of too high an order in the couplings to be relevant at the one-loop level, Here, σ denotes the dilaton field, h is the Higgs field, ξσ, ξ, λ ζ L and can thus be dropped. Once again, we see that choosing and are dimensionless coupling constants, and SM a field-dependent renormalization scale is equivalent to contains all standard model terms apart from the pure scalar adding infinite series of operators suppressed by the VEV sector. For phenomenology, the Higgs-dilaton potential can of the dilaton. Moreover, we also see that SI is hidden in the be chosen such that after spontaneous symmetry breaking it new Lagrangian, but is recovered at the one-loop level. reduces to the usual SM potential, while the first term in We note that if we add an extra scalar field h to this this Lagrangian effectively plays the role of the Planck Ψ¯ Ψ 2 2 model, with an additional Yukawa interaction yh h ,SI constant: ξσσ¯ → Mp. For simplicity, we neglect further renormalization works out in exactly the same way. Again effects of the dilaton. We are thus left after spontaneous ϵ 1−ϵ we set yh → σ yh to define a dimensionless coupling in symmetry breaking with a theory whose gravity and scalar d ¼ 4 − 2ϵ. It is by no means necessary that the interaction sectors read
085013-5 MOOIJ, SHAPOSHNIKOV, and VOUMARD PHYS. REV. D 99, 085013 (2019) Z 2 þ ξ 2 4 4 pffiffiffiffiffiffi Mp h 1 μ 2 2 2 Mp S ¼ d x −g − R þ ∂μh∂ h Prescription II : μ ¼ M ; μ ¼ : ð31Þ J 2 2 J p E M2 þ ξh2 p λ − ð 2 − 2Þ2 ð Þ 4 h v : 24 It is not clear which choice should be used without knowing Note that this Lagrangian can describe Higgs inflation the physics at the Planck scale [39]. As we see, one always (when ξ ≫ 1) [37,38]. produces a dynamical renormalization scale when renorm- One can get rid of the nonminimal coupling of the Higgs alizing, be it in the Jordan or in the Einstein frame. Still, as field to gravity by making a conformal transformation to long as one sticks to one definite prescription, physics will the Einstein frame always be the same in the Jordan and in the Einstein frame [40–43].3 We can thus choose the frame in which we want 2 gˆμν ¼ Ω gμν; ð25Þ to work. In our case, we choose the Einstein frame, as gravity is in a canonical form and its backreaction can be with neglected in light of the corrections coming from the SM loops [45,46]. We will repeat here the constructions done in 2 2 the previous sections to clarify the differences between the Mp þ ξh Ω2 ¼ : ð26Þ two prescriptions, by translating the field-dependent M2 p renormalization scale into a choice of the potential. This allows one to directly see the differences that arise from This transformation yields a noncanonical kinetic term for choosing one or the other prescription. χ the Higgs field. A scalar field with a canonical kinetic The effective potential for the Einstein frame has been term can be defined as computed at one loop in Ref. [47]: counting the Higgs, two sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi W’s, the Z and the three colors of the top quark, one dχ Ω2 þ 6ξ2h2=M2 obtains, up to finite terms, ¼ p; ð27Þ dh Ω4 4 2 6 4 2 giving an Einstein-frame action ¼ þ mH mH þ mW mW V1 V0 2 ln 2 2 ln 2 Z 64π μ 64π μ pffiffiffiffiffiffi 2 M 1 4 2 4 2 ¼ 4 −ˆ − p ˆ þ ∂ χ∂μχ − ðχÞ ð Þ 3m m 3m m SE d x g R μ V0 28 þ Z Z − t t ; ð Þ 2 2 64π2 ln μ2 16π2 ln μ2 32 ˆ where R is computed using gˆμν and
λ where m stands for the mass of the subscripted particle field 2 2 2 2 4 2 2 V0ðχÞ¼ ðhðχÞ − v Þ : ð29Þ μ ¼ ð þ ξ Þ 4ΩðχÞ4 in the Einstein frame and Mp= Mp h . Notice that the contribution coming from the other quarks is negligible in light of mt. It is worth noting that although Notice that the conformal transformation (25) not only μ2 changes the gravitational part of the Lagrangian, but also is not purely dynamical, the regularization procedure is Ω−1 still well defined. Notice that here we took the liberty of induces changes in the other terms. The net effect is a μ rescaling of every mass scale in the theory. Supposing that considering as field independent when computing the effective potential. This amounts to neglecting finite we renormalize the Jordan-frame Lagrangian at a scale μJ, the equivalent scale in the Einstein frame is then corrections in the final result, coming from the infinite series of operators of order OðϵÞ which appears when μE ¼ μJ=ΩðχÞ, making the renormalization scale field dependent. A natural choice for the renormalization scale expanding the dynamical scale around Mp.Aswehave seen in the previous sections, this can be done only at the appears in our theory: the Planck mass Mp. We thus get two different prescriptions for renormalization, depending on one-loop level. At higher loop levels, divergences of higher which frame is chosen to have the natural scale. The first order lead to infinite contributions coming from the one takes a constant scale in the Einstein frame, evanescent operators.
2 2 2 2 2 3Note however that even if functional prescriptions (as a Prescription I : μ ¼ Mp þ ξh ; μ ¼ Mp; ð30Þ J E function of N) for all inflationary observables are equal in both frames, evaluating them at a given number of e-folds before the whereas the second one uses the constant scale in the end of inflation does not have the same meaning in both frames: Jordan frame NJ ≠ NE [44].
085013-6 HIDDEN AND EXPLICIT QUANTUM SCALE INVARIANCE PHYS. REV. D 99, 085013 (2019)
Noticing that preserved scale invariance of the theory at the quantum ’ 2 2 level. By requiring the one-loop-level Green s functions to m ¼ m be equal, we have shown that taking this dynamical ln 4 2 2 ln 2 Mp=ðMp þ ξh Þ Mp renormalization scale is equivalent to considering a new ∞ X ð−1Þn−1 ξh2 n potential equipped with the vacuum expectation value of þ ð33Þ the dynamical scale as a renormalization scale. This new n M2 n¼1 p potential contains an infinite number of new operators and using the results obtained in the preceding sections, one suppressed by powers of the vacuum expectation value of can deduce that a physically equivalent potential is the dynamical scale. We have found that the dilatation symmetry of the new potential is hidden at the tree level, 1 ˜ 4 4 4 4 and becomes explicit again when quantum corrections are V0 ¼ V0 þ ðm þ 6m þ 3m − 12m Þ 64π2 H W Z t taken into account. We have done this construction explic- ∞ X ð−1Þn−1 ξh2 n itly at the one-loop level, and argued that it can be done at × ; ð34Þ any loop level and for any potential. n M2 n¼1 p We then approached a scale-invariant Yukawa model, 2 2 containing a fermion and a scalar field with a nonzero equipped with the constant renormalization scale μ ¼ Mp. This potential, when renormalized at the scale μ2 ¼ M2 in vacuum expectation value. This model has a nontrivial p wave-function renormalization at the one-loop level, and the Einstein frame gives rise to Eq. (32). permits a generalization of the construction. Choosing the From Eq. (34), one sees that choosing one or the other prescription amounts to adding an infinite series of operators scalar field as a dynamical renormalization scale, we pffiffiffi showed that the one-loop quantum-corrected theory suppressed by powers of Mp= ξ to the Lagrangian. If we choose ξ ≫ 1, as in Higgs inflation, we see that, conse- remains scale invariant. We then translated this renormal- quently, both prescriptions correspond to identical ization scale into a modification of the original Lagrangian, Lagrangians in the small-field regime (h
085013-7 MOOIJ, SHAPOSHNIKOV, and VOUMARD PHYS. REV. D 99, 085013 (2019)
ACKNOWLEDGMENTS The authors thank Mario Herrero-Valea, Alexander Monin, Kengo Shimada and an anonymous referee for helpful discussions. This work was supported by the European Research Council (ERC-AdG-2015) Grant No. 694896. The work of M. S. was supported partially by the Swiss National Science Foundation.
[1] C. Wetterich, Fine tuning problem and the renormalization [19] P. G. Ferreira, C. T. Hill, and G. G. Ross, Scale-independent group, Phys. Lett. 140B, 215 (1984). inflation and hierarchy generation, Phys. Lett. B 763, 174 [2] W. A. Bardeen, On naturalness in the standard model, (2016). Fermilab Report No. FERMILAB-CONF-95-391-T. [20] A. Karam and K. Tamvakis, Dark matter from a classically ð3Þ 94 [3] F. Vissani, Do experiments suggest a hierarchy problem, scale-invariant SU X, Phys. Rev. D , 055004 (2016). Phys. Rev. D 57, 7027 (1998). [21] P. G. Ferreira, C. T. Hill, and G. G. Ross, Weyl current, [4] K. A. Meissner and H. Nicolai, Conformal symmetry and scale-invariant inflation and Planck scale generation, Phys. the Standard Model, Phys. Lett. B 648, 312 (2007). Rev. D 95, 043507 (2017). [5] M. Shaposhnikov, Is there a new physics between electro- [22] A. Ghoshal, A. Mazumdar, N. Okada, and D. Villalba, weak and Planck scales, arXiv:0708.3550. Stability of infinite derivative Abelian Higgs models, Phys. [6] R. Foot, A. Kobakhidze, K. L. McDonald, and R. R. Volkas, Rev. D 97, 076011 (2018). A solution to the hierarchy problem from an almost [23] F.Grabowski, J. H. Kwapisz, and K. A. Meissner, Asymptotic decoupled hidden sector within a classically scale invariant safety and Conformal Standard Model, arXiv:1810.08461. theory, Phys. Rev. D 77, 035006 (2008). [24] G. ’t Hooft and M. J. G. Veltman, Regularization and [7] G. F. Giudice, Naturally speaking: The naturalness criterion renormalization of gauge fields, Nucl. Phys. B44, 189 and physics at the LHC, in Perspectives on LHC Physics, (1972). edited by, Gordon Kane and Aaron Pierce (World Scientific, [25] F. Englert, C. Truffin, and R. Gastmans, Conformal invari- Singapore, 2008) pp. 155–178. ance in quantum gravity, Nucl. Phys. B117, 407 (1976). [8] M. Shaposhnikov and D. Zenhausern, Scale invariance, [26] C. Wetterich, Cosmology and the fate of dilatation sym- unimodular gravity and dark energy, Phys. Lett. B 671, 187 metry, Nucl. Phys. B302, 668 (1988). (2009). [27] M. E. Shaposhnikov and I. I. Tkachev, Quantum scale [9] M. Shaposhnikov and D. Zenhausern, Quantum scale invariance on the lattice, Phys. Lett. B 675, 403 (2009). invariance, cosmological constant and hierarchy problem, [28] R. Armillis, A. Monin, and M. Shaposhnikov, Spontane- Phys. Lett. B 671, 162 (2009). ously broken conformal symmetry: Dealing with the trace [10] L. Alexander-Nunneley and A. Pilaftsis, The minimal scale anomaly, J. High Energy Phys. 10 (2013) 030. invariant extension of the Standard Model, J. High Energy [29] F. Gretsch and A. Monin, Perturbative conformal symmetry Phys. 09 (2010) 021. and dilaton, Phys. Rev. D 92, 045036 (2015). [11] T. Hur and P. Ko, Scale Invariant Extension of the Standard [30] C. Tamarit, Running couplings with a vanishing scale Model with Strongly Interacting Hidden Sector, Phys. Rev. anomaly, J. High Energy Phys. 12 (2013) 098. Lett. 106, 141802 (2011). [31] M. E. Shaposhnikov and I. I. Tkachev, Quantum scale [12] C. Wetterich, Where to look for solving the gauge hierarchy invariance on the lattice, Phys. Lett. B 675, 403 (2009). problem, Phys. Lett. B 718, 573 (2012). [32] J. Garcia-Bellido, J. Rubio, M. Shaposhnikov, and D. [13] M. Farina, D. Pappadopulo, and A. Strumia, A modified Zenhausern, Higgs-Dilaton cosmology: From the early to naturalness principle and its experimental tests, J. High the late universe, Phys. Rev. D 84, 123504 (2011). Energy Phys. 08 (2013) 022. [33] S. R. Coleman and E. J. Weinberg, Radiative corrections as [14] M. Heikinheimo, A. Racioppi, M. Raidal, C. Spethmann, the origin of spontaneous symmetry breaking, Phys. Rev. D and K. Tuominen, Physical naturalness and dynamical 7, 1888 (1973). breaking of classical scale invariance, Mod. Phys. Lett. A [34] R. Jackiw, Functional evaluation of the effective potential, 29, 1450077 (2014). Phys. Rev. D 9, 1686 (1974). [15] C. D. Carone and R. Ramos, Classical scale-invariance, the [35] D. M. Ghilencea, Quantum implications of a scale invariant electroweak scale and vector dark matter, Phys. Rev. D 88, regularisation, Phys. Rev. D 97, 075015 (2018). 055020 (2013). [36] D. M. Ghilencea, Manifestly scale-invariant regularization [16] I. Quiros, Scale invariance and broken electroweak sym- and quantum effective operators, Phys. Rev. D 93, 105006 metry may coexist together, arXiv:1312.1018. (2016). [17] C. Wetterich, Inflation, quintessence, and the origin of mass, [37] D. S. Salopek, J. R. Bond, and J. M. Bardeen, Designing Nucl. Phys. B897, 111 (2015). density fluctuation spectra in inflation, Phys. Rev. D 40, [18] A. Karam and K. Tamvakis, Dark matter and neutrino 1753 (1989). masses from a scale-invariant multi-Higgs portal, Phys. Rev. [38] F. L. Bezrukov and M. Shaposhnikov, The Standard Model D 92, 075010 (2015). Higgs boson as the inflaton, Phys. Lett. B 659, 703 (2008).
085013-8 HIDDEN AND EXPLICIT QUANTUM SCALE INVARIANCE PHYS. REV. D 99, 085013 (2019)
[39] M. Shaposhnikov and K. Shimada, Asymptotic scale [45] F. Bezrukov and M. Shaposhnikov, Standard Model Higgs invariance and its consequences, arXiv:1812.08706. boson mass from inflation: Two loop analysis, J. High [40] N. Makino and M. Sasaki, The density perturbation in the Energy Phys. 07 (2009) 089. chaotic inflation with nonminimal coupling, Prog. Theor. [46] D. P. George, S. Mooij, and M. Postma, Quantum correc- Phys. 86, 103 (1991). tions in Higgs inflation: The Standard Model case, [41] D. P. George, S. Mooij, and M. Postma, Quantum correc- J. Cosmol. Astropart. Phys. 04 (2016) 006. tions in Higgs inflation: The real scalar case, J. Cosmol. [47] F. L. Bezrukov, A. Magnin, and M. Shaposhnikov, Standard Astropart. Phys. 02 (2014) 024. Model Higgs boson mass from inflation, Phys. Lett. B 675, [42] M. Postma and M. Volponi, Equivalence of the Einstein and 88 (2009). Jordan frames, Phys. Rev. D 90, 103516 (2014). [48] J. Fumagalli and M. Postma, UV (in)sensitivity of Higgs [43] S. Karamitsos and A. Pilaftsis, Frame covariant non- inflation, J. High Energy Phys. 05 (2016) 049. minimal multifield inflation, Nucl. Phys. B927,219 [49] Y.Hamada, H. Kawai, Y.Nakanishi, and K. y. Oda, Meaning (2018). of the field dependence of the renormalization scale in Higgs [44] A. Karam, T. Pappas, and K. Tamvakis, Frame-dependence inflation, Phys. Rev. D 95, 103524 (2017). of higher-order inflationary observables in scalar-tensor [50] K. Falls and M. Herrero-Valea, Frame (In)equivalence in theories, Phys. Rev. D 96, 064036 (2017). quantum field theory and cosmology, arXiv:1812.08187.
085013-9