Quantum implications of a scale invariant regularisation

We study scale invariance at the quantum level (three loops) in a perturbative approach. For a scale-invariant classical theory the scalar potential is computed at three-loop level while keeping manifest this symmetry. Spontaneous scale symmetry breaking is transmitted at quantum level to the visible sector (of $\phi$) by the associated Goldstone mode (dilaton $\sigma$) which enables a scale-invariant regularisation and whose vev $\langle\sigma\rangle$ generates the subtraction scale ($\mu$). While the hidden ($\sigma$) and visible sector ($\phi$) are classically decoupled in $d=4$ due to an enhanced Poincar\'e symmetry, they interact through (a series of) evanescent couplings $\propto\epsilon^k$, ($k\geq 1$), dictated by the scale invariance of the action in $d=4-2\epsilon$. At the quantum level these couplings generate new corrections to the potential, such as scale-invariant non-polynomial effective operators $\phi^{2n+4}/\sigma^{2n}$ and also log-like terms ($\propto \ln^k \sigma$) restoring the scale-invariance of known quantum corrections. The former are comparable in size to"standard"loop corrections and important for values of $\phi$ close to $\langle\sigma\rangle$. For $n=1,2$ the beta functions of their coefficient are computed at three-loops. In the infrared (IR) limit the dilaton fluctuations decouple, the effective operators are suppressed by large $\langle\sigma\rangle$ and the effective potential becomes that of a renormalizable theory with explicit scale symmetry breaking by the"usual"DR scheme (of $\mu=$constant).

At the quantum level, the manifest scale symmetry of the action in d = 4 − 2ǫ introduces evanescent couplings ∝ ǫσ/ σ of the hidden to the visible sector 5 (σ: dilaton fluctuations). The SR scheme is thus reformulated as an "ordinary" DR scheme of µ=constant (∝ σ ) plus an additional field (σ) with an infinite series of evanescent couplings to the visible sector.
At the quantum level, such evanescent couplings have physical effects. When these couplings multiply poles of momentum integrals, they generate new (finite or infinite) counterterms, all scale invariant. For example one finds non-polynomial operators generated radiatively, such as φ 2n+4 /σ 2n , n ≥ 1 (but also higher derivative operators suppressed by σ). They can transmit scale symmetry breaking to the visible sector. Such operators can be understood via their Taylor expansion about σ = σ +σ, when they become polynomial. Scale symmetry acts at the quantum level as an organising principle that re-sums the polynomial ones. We shall study closer these operators, since they are important at large φ. Because of their presence, the quantum scale invariant theory is non-renormalizable.
We compute in a manifest scale invariant way the quantum corrections to the scalar potential in two-loop order (diagrammatically) and three-loop (via Callan-Symanzik equation), for a scale-invariant classical theory. The two-loop (three-loop) potential contains effective operators as finite (infinite) counterterms, respectively. In the infrared (IR) decoupling limit of the dilaton (large σ ) effective operators vanish; one then recovers the effective potential and trace anomaly of a renormalizable theory (if classical theory was so) with only classical scale invariance and explicit scale symmetry breaking (SSB) by the "usual" DR scheme of µ =constant (no dilaton). The combined role of quantum scale invariance and enhanced Poincaré symmetry in protecting the scalar mass at large σ is also reviewed.
Since M Planck breaks scale symmetry, this analysis is valid for field values well below this scale. One should extend this study to a Brans-Dicke-Jordan theory of gravity with non-minimal coupling where the dilaton vev σ fixes spontaneously M Planck . We restrict the analysis to a perturbative (quantum) scale symmetry. At very high momentum scales some couplings (e.g. hypercharge) may become non-perturbative, but such scale is above M Planck , where flat space-time description used here fails anyway.
2 From classical to quantum scale invariance 2

.1 Implementing quantum scale invariance
Consider a classical scale invariant action, e.g. a toy model or the SM with vanishing higgs mass parameter, extended by the dilaton σ. We assume that there is no classical interaction between the visible sector (of fields φ j ) and the hidden sector (of dilaton σ). Then The action in d = 4 has an enhanced Poincaré symmetry (P v × P h ) associated with both sectors, which forbids a classical coupling λ m φ 2 j σ 2 . Such coupling can be naturally set to λ m = 0 and remains so at the quantum level 6 "protected" by this symmetry [39].
Below we work with the canonical dilaton σ related to the actual Goldstone by σ = σ e τ , so that it transforms in a "standard" way under scaling while τ transforms with a shift The most general potential for σ allowed by scale invariance in d = 4 is then κ 0 e 4τ ∼ λ σ σ 4 . But Poincaré symmetry in the dilaton sector demands a flat potential, so λ σ = 0 [40]. Demanding spontaneous scale symmetry breaking σ = 0 means "we live" along a flat direction. This is in the end a tuning of the cosmological constant and is present anyway in e.g. TeV supersymmetry. The details of how σ acquires a vev are not relevant below. At the quantum level it is natural to use the dilaton to generate dynamically the subtraction scale ∝ σ in order to preserve scale symmetry during quantum calculations [29]. We use DR in d = 4 − 2ǫ, then the only possibility dictated by dimensional arguments 7 is with z is an arbitrary dimensionless parameter (scaling factor); it keeps track of the vev of σ after SSB. The d = 4 potential V (φ j ) of the visible sector is then analytically continued and becomes a function of σ! This ensures the d = 4 couplings remain dimensionless in d = 4 − 2ǫ and can be used for perturbative calculations. Therefore, the visible (φ j ) and hidden (σ) sectors have evanescent couplings dictated by the scale symmetry alone of the (regularized) action in d = 4 − 2ǫ. To see these couplings expand (4) in powers of ǫ (loops) and then in terms of fluctuationsσ about the vev σ of σ: where 6 Technically β λm ∝ λm at two-loop [36]. 7 µ has mass dimension one, while σ and σ have dimension (d − 2)/2.
A scale invariant regularization is then re-expressed as an ordinary DR scheme with µ = µ 0 plus an extra field (σ) with (infinitely many) evanescent couplings eq. (5). Since the lhs is scale invariant, so is the rhs if one does not truncate the expansion in field fluctuations. In practice one can still use a truncated expansion (see below). From eq.(5) one can read the new vertices of evanescent interactions ∝ ǫ n (n ≥ 1), betweenσ and φ j and the Feynman rules of the scale invariant quantum theory 8 . While these interactions vanish in d = 4 or in the dilaton decoupling limit (η → 0), at the loop level have physical effects. At quantum level, a coupling proportional to ǫ n , (n ≥ 1) in an amplitude can bring new corrections to it when multiplying the poles 1/ǫ k of the integrals over loop momenta. One generates finite quantum corrections (if n = k) or new poles/counterterms (n < k) beyond those of the theory with µ =constant. If n = k, a scattering amplitude that involves the dilaton depends only on the couplings of initial d = 4 theory, without new parameters needed (counterterm couplings). This can be used to set strong lower bounds on the scale σ .
Since the new couplings are suppressed, η ∼ 1/ σ , the counterterms are higher dimensional. They must however respect the scale symmetry of the lhs of eq.(5); one can then "restore" this symmetry "broken" by working with the truncation of the rhs expression, by simply replacing 1/ σ → 1/σ in their expression. Therefore, the new counterterms of the theory are suppressed by powers of σ and are non-polynomial in fields; log-terms in σ are also possible, however (see later).
For example, for V (φ) = λφ 4 /4!, a first counterterm is found by inserting a single internal line ofσ in an amplitude, which brings a factor (ǫ/ σ ) 2 ; if this multiplies a 1/ǫ 3 pole from a three-loop momentum integral it generates a 1/ǫ pole and a corresponding counterterm φ 6 /σ 2 for the 6-point amplitude (φ 6 ) [37]. By the same argument, finite quantum corrections appear at two-loops (if due to dynamics of σ) or even one-loop (due to scale symmetry alone).
Since the theory is scale invariant and so it has no dimensionful couplings, diagrams that would otherwise be proportional to masses automatically vanish. Then the only possibility to construct scale invariant d = 4 counterterms that are suppressed by powers of σ is to involve appropriate powers φ n , n > 4 and higher derivatives of φ andσ. Therefore the new counterterms are found on dimensional grounds as n,m≥0 where the derivatives act in all possible ways in the numerator. This includes the dilatondilaton scattering (∂ µ σ) 4 /σ 4 (see a−theorem [41]) which emerges at three-loops. We see that quantum scale-invariant theories are non-renormalizable [37], unlike their counterpart with µ =constant which is not quantum scale invariant but is renormalizable (if initial d = 4 action was so). The latter case is recovered in the limit of large σ , when fluctuationsσ decouple, see eq.(5). This picture also applies to gauge theories.

One-loop potential
Let us first review the quantum corrections to the potential in a scale invariant toy model at one-loop, before going to higher loops. Consider L below in d = 4 for a scalar φ In d = 4 − 2ǫ the potential becomesṼ (φ, σ) of eq.(4) with V as above, so φ and σ do interact as dictated by scale symmetry of analytically continued L. The one-loop potential is then where c 0 = 4πe 3/2−γ E .M 2 s are field-dependent (masses) 2 , eigenvalues of the second derivatives matrixṼ αβ , with α, β = φ, σ. One eigenvalueM 4 σ ∝ ǫ 2 , thus it cannot generate counterterms at one-loop. Then It is important to note that the factor µ 2ǫ is a function of σ (see eq. (3)) and maintains scale invariance 9 in d = 4 − 2ǫ. Here we work in the minimal subtraction scheme (MS). Thus the (scale-invariant) counterterm is Then the one-loop potential in d = 4 is where we took the limit ǫ → 0. Note that U has acquired a dependence on σ at quantum level (under the log) and for this reason its expression is now scale invariant (in d = 4). Since dimensionless z keeps track of the presence of σ , the one-loop beta function β which is identical to the result for the case µ =constant 10 . The Callan-Symanzik (CS) 9 This can also be relevant if one wanted to define and use instead a non-minimal subtraction scheme. 10 Unlike in theories with no dilaton (with explicit SSB by quantum corrections), β λ = 0 is not a necessary equation for a scale-invariant theory [33] is easily verified: Consider now the limit when the dilaton decouples. For this Taylor expand the potential for σ = σ +σ whereσ are field fluctuations. The result is Forσ ≪ σ , ∆U = 0 and we recover the Coleman-Weinberg result in a d = 4 renormalizable theory obtained in the usual DR scheme of µ =constant(= z σ ) with explicit SSB (no dilaton). Obviously, the CS equation is still respected. One then proceeds to impose boundary conditions to define the quartic self-coupling at φ = σ : λ σ = ∂ 4 U/∂φ 4 | φ= σ , as usual.
The analysis is very similar if more fields φ j are present, of potential V (φ j ). The result is found from eqs.(15), (16) by replacing V φφ by the eigenvalues of matrix V ij = ∂ 2 V /∂φ i ∂φ j and summing over them. Again the dilaton does not contribute counterterms at one-loop, but enforces the scale invariance of U (via ln σ). The second term in the CS equation in (14) is now a sum over all quartic couplings in V . Including fermions and gauge bosons is immediate by extending the sum over field dependent masses, with appropriate factors.

Two-loop potential
The two-loop correction to the potential of φ can be written as with the diagrams below computed from the background field method 11 The vertices and propagators in these diagrams receive evanescent corrections from the dilaton field, as seen from the background field expansion. We Taylor expand condition for having scale symmetry in our case here [32,33] since the spectrum is extended to include a dilaton (spontaneous SSB); thus a non-zero β λ does not mean the theory cannot be scale invariant. 11 We use the approach of [36] but without a classical coupling λm φ 2 σ 2 .
The propagators, obtained from the inverse of the matrix (p 2 δ αβ −Ṽ αβ ), also acquire ǫdependent shifts. We retain all these corrections up to and including O(ǫ 2 ); these can multiply the poles of the loop integrals (1/ǫ 2 or 1/ǫ) to generate finite quantum corrections 12 , as discussed in Section 2.1. Here we shall identify these corrections. One finds with µ 2ǫ a function of σ which maintains the scale invariance in d = 4 − 2ǫ, see eq.(3). The counterterm is scale invariant and in the MS scheme is given by and From these and with the coefficients Z λ = 1 + δ λ , Z φ = 1 + δ φ and since λ B = µ 2ǫ λ Z λ Z −2 φ , dλ B /d(ln z) = 0, one obtains the two-loop corrected beta function β λ is identical to that of the φ 4 theory with µ =constant (no dilaton) [44][45][46]. No new poles (i.e. counterterms) are generated at two-loop beyond those of the theory with µ=constant.
The two-loop potential we find is (24) is an interesting result. First, U is scale invariant. The last two terms in U are new, finite two-loop corrections in the form of non-polynomial operators (φ 6 /σ 2 , φ 8 /σ 4 ,...) and cannot be removed by a different subtraction scheme. These terms are independent of the dimensionless subtraction parameter z and bring corrections beyond those obtained for µ =constant (of explicit SSB). Their presence is easily understood in the light of the 12 New 1/ǫ poles from (ǫ − shifts) × 1/ǫ 2 do not emerge here, unless a classical mixing φ − σ exists. 13 The Clausen function Cl2 is defined as Cl2[x] = − x 0 dθ ln |2 sin θ/2|. discussion in Section 2.1. The field-dependent masses entering the loop calculation, as eigenvalues of the second derivative matrixṼ αβ , contain terms suppressed by µ 2 ∼ σ 2 , since the sole dependence on σ isṼ ∼ σ ǫ . This explains the presence of positive powers of σ only in the denominators of the non-polynomial terms. Even the simplest quantum scale invariant theory is then non-renormalizable (unlike the case with µ =constant which is renormalizable but not quantum scale invariant).
The non-polynomial terms can be larger than the "standard" two-loop correction; they are comparable in size for φ ∼ σ. Higher loops are expected to generate more such operators of larger powers and with new couplings (if they are counterterms 14 ). They are relevant if one is interested in the stability of the potential at large field values φ ∼ σ . The non-polynomial terms vanish in the limit φ ≪ σ.
The result in eq. (24) can be Taylor expanded about the vev of σ using σ = σ +σ. Retaining only the leading term corresponds to decoupling the dilaton. Then Ignoring O(1/ σ ) terms, eq.(26) is the "standard" two-loop result obtained for µ=constant (no dilaton, explicit SSB) in MS scheme [42], more exactly for µ = z σ . The difference between eq.(24) and eq.(26) is made of higher dimensional operators suppressed by large σ ; these suppressed terms are responsible for maintaining manifest scale invariance of (24).
The Callan-Symanzik equation is also respected in the non-scale-invariant case, eq.(26), where µ=constant (µ = z σ , explicit SSB). This is obvious from the above check because z is tracking exactly this scale and the non-polynomial terms in (24) are z-independent 16 .

Three-loop potential
In this section we use the three-loop Callan-Symanzik equation for the scalar potential to identify the three-loop correction to the potential without doing the diagrammatic calculation. As in the two-loop case, this correction is a a sum of two terms V (3) +V (3,n) . V (3) is the "usual" three-loop correction obtained with µ =constant (no dilaton) [42,43], but with the formal replacement µ→ z σ; V (3,n) is a new correction that contains non-polynomial terms. To find these we use the three-loop counterterms for this theory nicely computed in [37] δL 3 is scale-invariant in d = 4 − 2ǫ (as it should) because µ depends on σ, eq.(3). The terms φ 6 /σ 2 and φ 8 /σ 4 are expected since they were present as finite operators at two-loop; also in the MS scheme, giving γ With λ B 6 = µ 2ǫ (σ)λ 6 Z λ 6 Z −3 φ Z σ , etc, and with (d/d ln z) λ B 6 = 0, we find Both beta functions have a two-loop part (hereafter denoted β (2,n) λ 6,8 ∼ 1/κ 2 ) that is absent if λ 6,8 = 0 in the classical Lagrangian, which is our case here 17 ; then the three-loop part (hereafter β (3,n) is induced by λ alone. These beta functions enter in the CS equations in the presence of λ 6 and λ 8 , due to their associated counterterms. In their presence, eq.(29) is unaffected, but eq.(30) is modified such as V is now replaced by and β (2,n) λ 6,8 are also included in the first term under the big bracket of (30). Using these and "new" V above, one immediately sees that (30) is verified for non-zero λ 6,8 .
Further, there is a CS equation at order λ 4 for (V (3) +V (3,n) ), which we divide into two CS equations, eqs. (37) and (40) below. One equation is for the "usual" correction V (3) and is identical to that obtained for µ =constant (= z σ ) We integrate (37) to find V (3) up to an unknown "constant" of integration term ∝ Q Q can be read from the "usual" three-loop computation at µ =constant [42] in MS scheme: where β pressed by σ, e.g. φ 6 /σ 2 , etc. The last term in the lhs with λ j → λ would bring a term ∝ φ 4 which cannot be cancelled, being the only one of this structure. Then the only way to respect the above field-dependent condition is that β (3,n) λ = 0. This is also seen from (40) in the decoupling limit of large σ . Therefore, the three-loop beta function in the quantum scale invariant effective theory is just that of the theory with µ =constant 18,19 . We then integrate eq.(40) using the replacement ln z → (−1/2) ln(V φφ /(z σ) 2 ) which fixes the "constant" of integration in a scale invariant way. We find 20 V (3,n) is correct up to a possible additional presence of a scale invariant z−independent three-loop finite (non-polynomial) term (λ 4 /κ 3 ) φ 10 /σ 6 that cannot be captured by the CS differential equation but only in the diagrammatic approach. In the limit of large field σ and similar to V (2,n) at two-loop, V (3,n) → 0, leaving "usual" V (3) as the sole three-loop correction to the potential, with only a log-dependence on σ.
To conclude, quantum scale invariance demands the presence of non-polynomial operators. This symmetry arranges them in a series expansion in powers of φ/σ that contributes to the scalar potential. Each of these operators is actually an infinite sum of polynomial operators (in fields), after a Taylor expansion about σ = σ +σ. V (2,n) , V (3,n) , ∆V are relevant for the behaviour of the potential at large φ ∼ σ and are suppressed at φ ≪ σ.

More operators
Having seen the scale invariant non-polynomial operators generated at loop level, it is of interest to see their role if they are included in the action already at classical level, as in where we ignore similar higher order terms. The last term breaks the enhanced Poincaré symmetry (P v ×P h ) only mildly, since this symmetry is restored at large σ. In a consistent setup like Brans-Dicke-Jordan theory of gravity, this operator suppressed by σ ∼ M Planck could mediate gravitational interactions of the two sectors. Such operator is also generated when going from Jordan to Einstein frame, after a conformal transformation 21 . The one-loop computation of the potential proceeds as before and has three contributions, all scale invariant. First, there is a one-loop contribution similar to that in eq. (12) with V φφ replaced by the (two) field-dependent (masses) 2 which are eigenvalues of the matrix of second derivatives of V above wrt φ and σ, then sum over these. 18 Therefore we have β  [44][45][46]. 19 This is also consistent with Zσ = 1 at three-loops. A three-loop wavefunction correction to σ generated by a coupling ǫσφ 4 would then be proportional to ∝ ǫ 2 × (1/ǫ 2 ), so no new poles emerge in this order. 20 "Constants" of integration φ 6 /σ 2 , φ 8 /σ 4 , φ 4 are not allowed, being "fixed" in (32), ((38), (39) for φ 4 ). 21 We ignore here the effect of φ on the vev of σ.
A second contribution to the potential exists. The two field-dependent masses derived fromṼ of eq.(4) with V as above have a correction O(ǫ) induced by λ 6 ; when this multiplies 1/ǫ of eq.(9), it generates a finite correction V (1,n) ∝ λ 6 already at one-loop Finally, there are also one-loop counterterms, of the form (Z λp − 1) λ p φ p /(p σ p−4 ), where p = 6, 8, 10, 12 and where Z λp = 1+γ λp /(κǫ) and γ λ 6 = 9λ, γ λ 8 = 56λ 6 /λ 8 , γ λ 10 = 20λ 2 6 /λ 10 , γ λ 12 = 3λ 2 6 /λ 12 . Therefore the potential has a third contribution V (1,n) and ∆V are similar to V (2,n) , V (3,n) , ∆V found in the previous section, except that they are generated at one-loop, due to non-zero λ 6 . The one-loop beta functions of λ p are with p as above and they vanish if λ 6 = 0. We checked that the one-loop CS equation is again verified in the presence of these operators. For large σ , dilaton fluctuations are suppressed and the above corrections to the potential vanish, to leave the "usual" result (first contribution above), obtained in the renormalizable theory with µ constant (= z σ ). The generalisation to more operators in the classical action is immediate.

Symmetries, regularisations and mass hierarchy
From the above examples, we see that a combination of quantum scale invariance and enhanced Poincaré symmetry [39] of the two sectors can ensure a protection of the mass corrections to φ against a quadratic dependence on the scale of symmetry breaking σ (the only UV physical scale here). No term such as λ φ 2 σ 2 = λ σ 2 φ 2 + · · · was generated at the quantum level in the potential, with λ the higgs self-coupling 22 ; if present this would have required the usual SM-like fine-tuning of λ. Further, if one introduces a classical "mixing" coupling λ m , with a tree-level term λ m φ 2 σ 2 which would break the enhanced P v × P h symmetry, this would require a tuning of λ m (rather than λ) upon replacing σ → σ +σ, in order to keep the correction to the mass of φ under control. But such tuning of λ m is natural and needs to be done only once at the classical level, since the beta function β λm ∼ λ m at one-loop [22,30,35,39] and two-loops [36]. Further, for large σ the nonpolynomial operators that broke the P v × P h symmetry vanish and this symmetry and its "protective" role (on λ m ) are restored. Therefore, this protection remains true in the presence of non-polynomial operators e.g. λ 6 = 0. 23 The SR scheme used here is based on the DR scheme which may be considered unsuitable to capture the quadratic UV-scale dependence of the scalar (mass) 2 . It is important to note, however, that in our approach any scale is generated by fields vevs after spontaneous SSB. The field-dependence (e.g. counterterms, etc) of the quantum corrected action is not affected by the regularisation and is actually dictated by the symmetries of the theory (including scale invariance), which our SR scheme respects (unlike DR). Therefore, the dependence of the quantum action on the mass scales (generated by these fields vev's) cannot be affected. The UV behaviour of the mass of φ i.e. its dependence on σ (our physical UV scale), is thus not affected by a regularisation that respected all symmetries of the theory 24 .

Conclusion
Following the original idea of Englert et al and using a perturbative approach, we examined the quantum implications of a regularization scheme that preserves the scale invariance of the classical theory. To this purpose, we demanded that the analytical continuation of the theory to d = 4 − 2ǫ preserves the scale symmetry of the d = 4 action. This is possible under the additional presence of a dilaton field (σ), the Goldstone mode of scale symmetry breaking. This field is classically decoupled from the visible sector, following an enhanced Poincaré symmetry of the two sectors, but there are nevertheless quantum effects.
The scale invariance in d = 4 − 2ǫ and the dilaton it demands have two main effects: a) introduce new "evanescent" interactions (∝ ǫ) which have quantum consequences; b) generate the subtraction scale µ ∼ σ after spontaneous scale symmetry breaking. As a result, a scale invariant regularisation is re-formulated into an ordinary DR scheme of µ =constant (∝ σ ) plus an additional field (dilaton) with an infinite series of evanescent couplings to the visible sector. When evanescent interactions multiply the poles of loop integrals, new quantum corrections (finite or infinite counterterms) are generated, not present in the quantum version of the same theory regularized with µ =constant (i.e. no dilaton, explicit breaking). These corrections, which also include log-like terms in the potential (such as ln σ already at one-loop!), are scale-invariant. They have effects such as transmission of scale symmetry breaking after its spontaneous breaking in the dilaton (hidden) sector, or dilaton-dilaton scattering.
The scalar potential was computed at two-loops by direct calculation and at three loops by integrating its Callan-Symanzik equation. The result is scale invariant. It contains new log-like corrections (in the dilaton σ) similar to those obtained by naively replacing µ → σ in the result obtained in the "usual" DR scheme with µ =constant. In addition, depending on the details of the classical theory, scale invariant non-polynomial effective operators are also generated from one-or two-loops onwards, in a series of the form φ 4 × (φ/σ) 2n . These 24 To appreciate the role of d = 4 enhanced Poincaré symmetry, consider a different scale-invariant regularization that violates the Pv×P h symmetry. For V = λφ 4 /4!, use a momentum "cutoff" regularisation: k 2 ≤ σ 2 ; σ is a hidden sector field with σ the scale of new physics. At one-loop ∆V ∝ σ 2 0 d 4 k ln 1+λφ 2 /(2k 2 ) = λφ 2 σ 2 + ... This term (absent in our case) requires the "usual" order-by-order fine-tuning of self-coupling λ.
operators are important for large field values φ ∼ σ and can be comparable to "standard" log terms of the loop corrections; the beta functions of their couplings were also computed.
These operators are a generic presence and can be understood via their Taylor series expansion about the scale σ = 0 of spontaneous SSB, when they become polynomial. Scale symmetry acts at the quantum level as an organising principle that re-sums the polynomial ones. Therefore, maintaining at the quantum level the scale symmetry of the classical action makes the theory non-renormalizable. In the decoupling limit of the dilaton these operators vanish and one recovers the quantum result of a renormalizable theory with explicit SSB (if classical theory was renormalizable).
The role of the quantum scale symmetry and enhanced Poincaré symmetry in protecting a mass hierarchy m 2 φ ≪ σ 2 was reviewed. This protection cannot be affected by working in a regularisation ultimately based on a DR scheme, because all scales and thus hierarchy thereof are generated by vev's of the fields present in the quantum corrected action (after spontaneous SSB); its counterterms i.e. fields dependence are dictated by the symmetries of the theory (including scale symmetry), that our regularisation respects (unlike DR), hence the aforementioned protection. This remains true in the presence of the non-polynomial terms (i.e. despite non-renormalizability) since at large σ the enhanced Poincaré symmetry is restored. The study can be extended to gauge theories.