Naturalness, Conformal Symmetry and Duality

We reconsider the naturalness from the viewpoint of effective field theories, motivated by the alternative scenario that the standard model holds until a high-energy scale such as the Planck scale. We propose a calculation scheme of radiative corrections utilizing a hidden duality, in the expectation that the unnaturalness for scalar masses might be an artifact in the effective theory and it could be improved if features of an ultimate theory are taken in.


Introduction
It might be a good time to reconsider various concepts concerning the Higgs boson mass m h and the physics beyond the standard model (SM), on the basis of recent experimental results at LHC.
The discovery of the Higgs boson at LHC with the observed value m h 126GeV [1,2] is quite suggestive. It rekindles the question whether m h is a natural parameter or not [3,4,5]. Furthermore, evidences of new physics such as supersymmtry (SUSY), compositeness and extra dimensions have not yet been discovered, and this fact can be the turning point of the particle physics, because the gauge hierarchy problem [6,7] would be revisited.
Therefore, it is interesting to reexamine the validity of concepts relating m h and the physics beyond the SM, from various aspects. In this paper, we reconsider the naturalness of m h from the viewpoint of effective field theories including the SM. Our study is motivated by the alternative scenario that the SM (modified with massive neutrinos) holds until a high-energy scale such as the Planck scale M Pl [8,9]. We expect that the unnaturalness for scalar masses might be an artifact in the effective theory, and it could be improved if features of an ultimate theory are taken in and ingredients of the effective theory are enriched. We reanalyze radiative corrections on scalar masses using the φ 4 theory, and propose a calculation scheme utilizing a hidden duality.
The outline of this paper is as follows. In the next section, we review the naturalness and its relevant symmetries. We give a suggestion for the subtraction of quadratic divergences by presenting a calculation scheme, in consideration of a duality relating integration variables, in Sect. 3. In the last section, we give conclusions and discussions.

Naturalness and conformal symmetry
Let us first recall the concept of naturalness. According to 't Hooft [4], the naturalness is based on the dogma that "at any energy scale µ, a physical parameter or set of physical parameters a i (µ) is allowed to be very small only if the replacement a i (µ) = 0 would increase the symmetry of the system." We refer to this type of parameter as a natural parameter.
We discuss the naturalness of fermion masses and scalar masses, from the viewpoint of low-energy effective theories such as the quantum electrodynamics (QED), the φ 4 theory and the SM.

Naturalness of fermion masses
The electron mass m e is listed as a natural parameter in the QED. When we set m e = 0, the classical global chiral symmetry appears. Here, the chiral symmetry is the invariance of action integral under the different phase changes for Weyl fermions ψ L and ψ R with the different chiralities, i.e., ψ L → e i θ L ψ L and ψ R → e i θ R ψ R , (θ L = θ R ). Note that the chiral symmetry is broken down, even in the massless case m e = 0, such that in the presence of the axial U (1) anomaly. In the massive case, the U (1) vector current defined as j µ V = ψγ µ ψ = ψ † L σ µ ψ L + ψ † R σ µ ψ R is conserved, and the U (1) axial vector current defined as j where δm e represents the radiative corrections on the tree level mass m e .
The δm e at the one-loop level is given by where α ≡ e 2 /(4π) and Λ is a cutoff scale. In the limit of m e → 0, δm e also vanishes. This feature holds for a higher order corrections, and the chiral symmetry is not broken down perturbatively (although it is broken down anomalously without threatening the consistency of theory). Hence, the chiral symmetry is regarded as a powerful concept to control quantum corrections.
A classical conformal symmetry also appears in the limit of m e → 0, and is broken down in the presence of anomaly. For instance, the scale invariance is broken down as where β α is the β function for α. In the QED, the conformal symmetry seems to play a same role as the chiral symmetry does.
In the SM, the chiral symmetry has a superior quality to the conformal symmetry, because the chiral symmetry such as SU (2) L ×U (1) Y becomes a local one and is not broken down either perturbatively or anomalously, and the conformal symmetry is broken down not only with the negative mass squared of Higgs doublet explicitly, but also in the presence of anomalous terms. The chiral gauge symmetry is broken down spontaneously with the vacuum expectation value of the Higgs boson v = 246GeV, and fermions ψ f acquire masses m f = y f v / 2 via the Higgs mechanism, where y f are Yukawa coupling constants. On the other hand, the global chiral symmetry enhances in the limit of y f → 0. In this way, the smallness of fermion masses, comparing with a high-energy scale M U such as the gravitational scale M ≡ M Pl / 8π = 2.4 × 10 18 GeV, stems from the smallness of v , comparing with M U . Furthermore, the smallness of fermion masses except for the top quark mass, comparing with the weak gauge boson mass M W = g v /2, originates from the smallness of y f , comparing with the SU (2) L gauge coupling constant g . Hence, it is considered that the chiral symmetry is responsible for the smallness of SM fermion masses.

Naturalness of scalar masses
Next, we study the relation between the relevant symmetry of scalar mass m φ and radiative corrections on m φ , in order to take a hint for a naturalness of m h . Unless a theory has dimensional parameters except for m φ , the classical conformal symmetry appears in the limit of m φ → 0. In the same way as the QED, the scale invariance is broken down as where δm 2 φ represents the radiative corrections on m 2 φ , β k are β functions for coupling constants a k , and O k are operators with the mass dimension 4.
In the φ 4 theory, δm 2 φ at the one-loop level is most commonly written by where λ φ is the quartic self-coupling constant of φ, and the ellipsis stands for Λ independent terms. For example, the unregularized one is given by where we rotate to the Euclidean space and carry out the integration for the angles of momentum space. The δm 2 φ of (6) is obtained by replacing ∞ by Λ 2 − m 2 φ on the final expression in (7).
As seen from (6), it is widely thought that m φ is not a natural parameter, because δm 2 φ does not vanish in the limit of m 2 φ → 0, in the appearance of the quadratic term of Λ. However, if the quadratic term is subtracted or absent from some reason, m φ can be a natural parameter. Bardeen reexamined the naturalness in the SM and pointed out that the classical scale invariance implies the naturalness of the Higgs boson mass m h [10]. The reasoning is illustrated as follows. In the SM, the scale invariance is broken as where δm 2 h represents the radiative corrections on m 2 h . The anomalous terms are quantum corrections induced from loop contributions due to particles with masses smaller than the reference energy scale. It is quite unlikely that the radiative corrections on masses affect them. Hence, the anomalous divergence of the scale current should remain in the limit of m h → 0, and δm 2 h should be proportional to not Λ 2 but m 2 h . In other words, the classical symmetries should be restored in the limits of m h → 0 and β k → 0.
In effective field theories, ambiguities can exist in the regularization procedure, and such ambiguities, in most cases, are resolved by considering symmetries realized manifestly at the low-energy scale [11]. If we had a theory with a high calculability and predictability, regularization dependent quantities would be absent. In this regard, quantities depending on the regularization method should be subtracted or eliminated, unless the subtraction induces any physical effects.
The dimensional regularization is known as a regularization procedure, that does not induce quadratic divergences for scalar masses. Using it, δm 2 φ at the one-loop level is given by where ǫ = 4−D (D is the dimension of space-time) and γ = 0.577 · · · is the Euler constant. The δm 2 φ becomes infinite in the limit of ǫ → 0, i.e., D → 4. The 2/ǫ corresponds to ln(Λ 2 /m 2 φ ), and then the quadratic divergence is absent. Fujikawa gave a scheme on the subtractive renormalization of the quadratic divergences of scalar mass [12]. In case that the subtraction of quadratic divergences induces no physical effects on the low-energy theory, such a scheme is useful to treat physical quantities including scalar masses.
Aoki and Iso studied the quadratic divergences of scalar mass from the viewpoint of the Wilsonian renormalization group, and found that they can be absorbed into a position of the critical surface, which means the subtraction of them [13].
Extensions of the SM have been proposed by adopting the classical conformal invariance as a guiding principle [14,15,16,17,18]. 1

Naturalness of Higgs boson mass
Before we study the subtraction of quadratic divergences from the viewpoint of hidden symmetry, we discuss the naturalness of Higgs boson mass. In the SM, the radiative corrections on the Higgs mass squared m 2 h at the one-loop level are given by where c h and c ′ h are functions of the SM parameters such that Here λ is the quartic self-coupling constant of Higgs boson, g ′ is the U (1) Y gauge coupling constant, y t is the top Yukawa coupling constant, and contributions from other fermions are omitted. If we face the quadratic divergences squarely, the fine tuning among parameters is necessary to explain the observed value m h 126GeV, unless Λ ≤ O(1)TeV or c h = 0 is realized. Here, the condition Λ ≤ O(1)TeV means that a new physics beyond the SM must exist around the terascale, unless nature requires the fine tuning. The condition , 2 which leads to a value m h 320GeV at the weak scale. 3 If all quadratic divergences are subtracted, δm 2 h also vanishes in the limit of m h → 0. Then, the classical conformal symmetry seems to control quantum corrections as the chiral symmetry does. If this feature holds for a higher order corrections, the classical conformal symmetry is not broken down perturbatively (although it is broken down anomalously without threatening the consistency of theory). In this way, the conformal symmetry might be responsible to the smallness of Higgs boson mass, comparing with a high-energy scale M U . Or it might be said that the regularization ambiguities can be resolved by the conformal symmetry.
Hence, the problem whether the weak scale relating the Higgs boson mass is stabilized against radiative corrections in the framework of SM (a narrow definition of the naturalness problem) 4 can be solved by the subtraction of quadratic divergences. Then, the naturalness can become a powerful guiding principle to construct an effective theory. 5 In posed [15]. As an extension including dark matter candidates, a model with a strongly interacting hidden sector, to trigger the breakdown of electroweak symmetry, has been constructed [16]. Recently, various models, to generate the weak scale and provide dark matter candidates, have been proposed [19,20,21,22]. 2 The same type of condition was derived through a tadpole diagram concerning the Higgs boson [23]. 3 Recently, it is pointed out that c h = 0 holds around M Pl and there is a possibility that the bare Higgs mass vanishes there [24]. 4 We use the terminology "the naturalness problem" in a wider sense, that is, it should be regarded as a collective term for a fine tuning problem concerning mass parameters of scalar fields such as the Higgs mass, which contains the gauge hierarchy problem. 5 Wells presented an interesting observation toward the SM from the QED using the naturalness as the guiding principle [25].
other words, symmetries such as the chiral symmetry, the gauge symmetry and the conformal symmetry become powerful tools for a realistic model-building, from the viewpoint of the effective field theory. There is a possibility that all fields, in our low-energy world, are massless at M U . At this stage, the following questions (other parts of the naturalness problem) arise. One is what induces the negative mass squared of the Higgs boson around the weak scale, starting the massless state at M U , this is, what is the origin of the weak scale. A possible solution has been proposed based on the extension of the SM with the U (1) B−L gauge symmetry and new particles around the terascale [14,17]. In particular, the TeV scale B − L model proposed in [18] has several excellent features such as the classical conformality, the flatness of Higgs potential at M U , and the predictability relating m h 126GeV.
The other is the problem whether the weak scale is stabilized against large radiative corrections due to heavy particles in the framework of field theory including a high-energy physics, e.g., a grand unified theory. This is (the technical side of) the gauge hierarchy problem [6,7]. For instance, in the presence of heavy particles with masses M I and some SM gauge quantum numbers, m 2 h generally receives large radiative corrections of O(M 2 I ) in addition to the quadratic term of Λ such that and the stability of the weak scale is threatened. Here,c h and c ′′ hI are also functions of parameters. Then, the fine tuning is indispensable for M 2 I ≫ m 2 h in the appearance of the quadratic term of M I (a part of logarithmic divergences), even if the quadratic divergences of O(Λ 2 ) are removed and unless some miraculous cancellation mechanism works among corrections of heavy particles. We will come back to this problem in subsection 3.3.

Naturalness and duality
Let us explore the possibility that the quadratic divergences are removed, in the expectation that the quadratic divergences might be artifacts of regularization procedure and the calculation scheme can be selected by the physics.
Concretely, we pursue other reasoning to suggest the subtraction of quadratic divergences, based on the conjecture that an ultimate theory does not induce any large radiative corrections for low-energy fields owing to a symmetry, and such a symmetry is hidden in the SM. 6 We present a (tricky) calculation scheme, that rules out quadratic divergences thanks to a hidden duality.
In the following, it is shown that the logarithmic corrections on a scalar mass can be picked out by specifying the duality in the effective field theory.

Basic idea
Our method is based on the following assumptions relating features of an underlying theory. 7 (a) There is an ultimate theory, which has a fundamental energy scale. We denote the scale as Λ, for simplicity. (b) The ultimate theory has a duality between the physics at a higher-energy scale (E Λ) and that at a lower-energy scale (E Λ). It consists of the following two features. (b1) The physics is invariant under a duality transformation, e.g., E → E ′ = Λ 2 /E . (b2) The physics is only described by one of the two energy regions, relating with each other by the transformation. (c) A remnant of the duality is hidden in quantities of the low-energy physics involved with Λ, e.g., radiative corrections on parameters.
To illustrate our idea, let us consider quantum corrections on a parameter a at the one-loop level given by, where p 2 is an Euclidean momentum squared for a massless virtual particle running in the loop, and f (p 2 ) is a function of p 2 .
In case that δa diverges, the infinities come from p 2 = ∞ and/or p 2 = 0, and hence it is ordinarily regularized as where µ 0 is a fictitious mass parameter.
Here, let us show that the expression (14) is necessarily obtained and the form of δa is restricted, based on the above assumptions, by specifying the duality transformation.
First, we rewrite (13) as Note that (15) is reduced to (13) in the limit of µ 2 0 → 0. Using the assumption (b), (14) is obtained, if the domain of integration [µ 2 0 , Λ 2 ] is transformed into [Λ 2 , Λ 4 /µ 2 0 ] under a remnant of duality and the following relation holds, Next, we take p 2 → p ′2 = Λ 4 /p 2 as the remnant of duality transformation. Hereafter, we refer to the remnant of duality transformation as the duality transformation or the 7 Our idea is inspired by the world-sheet modular invariance in string theory. We will comment on the world-sheet modular invariance in subsection 3.3. duality, in most cases. Then, using assumptions (b1) and (c), the the following relation is derived, From (16) and (17), the form of f (p 2 ) is restricted as f (p 2 ) = c(p 2 )/p 2 where c(p 2 ) is a function invariant under the change p 2 → Λ 4 /p 2 , e.g., c(p 2 ) = p 2 + Λ 4 /p 2 . Unless we consider effects of heavy particles with masses of O(Λ) such as threshold corrections, f (p 2 ) does not contain Λ and then δa is determined as where c −1 is a p 2 -independent quantity. Our procedure can be regarded as not a mere regularization but a recipe to obtain finite physical values, because Λ is (big but) finite and infinities are taken away by the symmetry relating integration variables, like string theory. It is also regarded as the operation to pick out parts that satisfy assumptions. In case that f (p 2 ) does not contain Λ, it is simply denoted by where Du[ * ] represents the operation, and f (p 2 ) is expanded in a series.

Radiative corrections on scalar mass
We apply our method to radiative corrections on m 2 φ . In case that the bare mass is zero, the unregularized one is given by If we demand that the duality p 2 → Λ 4 /p 2 is hidden in δm 2 φ and the physics can be described by the region below Λ, δm 2 φ turns out to be zero such that Next, we study the case with a non-zero bare mass, based on the momentum cutoff method and the proper time method.
(i) The momentum cutoff method First, we separate the original one into the quadratic and logarithmic divergent parts such that where Λ φ is a provisional cutoff parameter (Λ 2 φ ≫ m 2 φ ), which goes to the infinity in the limit of m 2 φ → 0. In case with Λ 2 φ = (Λ 4 /m 2 φ ) − m 2 φ , we find that the second term on the right-hand side in (22) is invariant under the change p 2 + m 2 φ → Λ 4 /(p 2 + m 2 φ ), but the first term is not. Furthermore, in this case, the integration from p 2 = 0 to p 2 = Λ 2 φ is divided into that from p 2 = 0 to p 2 = Λ 2 − m 2 φ (≪ Λ 2 φ ) and that from p 2 = Λ 2 − m 2 φ to p 2 = Λ 2 φ , and these integrals for the second term take a same value. Note that the duality transformation reduces to that in the massless case, in the limit of m 2 φ → 0. Here, we impose the duality relating p 2 +m 2 φ → Λ 4 /(p 2 +m 2 φ ) on quantities relevant to Λ. If the physics from p 2 = 0 to p 2 = Λ 2 −m 2 φ is same as that from p 2 = Λ 2 −m 2 φ to p 2 = Λ 2 φ and the physics is only described by one of the two regions, Λ is naturally introduced and the desired expression is obtained as Note that δm 2 φ vanishes in the limit of m 2 φ → 0.
(ii) The proper time method Using the proper time method, δm 2 φ is given as where t is a parameter called a proper time. First, we separate δm 2 φ into the quadratic and logarithmic divergent parts by expanding the exponential factor such that whereΛ φ is a provisional cutoff parameter (Λ 2 φ ≫ m 2 φ ), which goes to the infinity in the limit of m 2 φ → 0. In case withΛ 2 φ = Λ 4 /m 2 φ , we find that the second term on the righthand side in (25) is invariant under the change t → 1/(Λ 4 t ), but others are not. Furthermore, in this case, the integration from t = 1/Λ 2 φ to t = 1/m 2 φ is divided into that from t = 1/Λ 2 φ to t = 1/Λ 2 and that from t = 1/Λ 2 to t = 1/m 2 φ , and these integrals for the second term take a same value.
Here, we impose the duality relating t → 1/(Λ 4 t ) on quantities relevant to Λ. If the physics from t = 1/Λ 2 φ to t = 1/Λ 2 is same as that from t = 1/Λ 2 to t = 1/m 2 φ and the physics is only described by one of the two regions, Λ is naturally introduced and the desired expression is obtained as In a similar way as the momentum cutoff method, δm 2 φ vanishes in the massless limit.
The region around p 2 = Λ 2 φ or t = 1/Λ 2 φ corresponds to the ultra-violet (UV) one, and that around p 2 = 0 or t = 1/m 2 φ corresponds to the infra-red (IR) one. Hence, the symmetry relating p 2 + m 2 φ → Λ 4 /(p 2 + m 2 φ ) or t → 1/(Λ 4 t ) might suggest that Λ has a physical meaning as a fundamental scale and there is an equivalence between the physics at the UV region and that at the IR one, in an ultimate theory.

Different choice
It is important to examine the applicable scope of our scheme. Here, we point out that the result depends on the choice of duality transformation, by giving an example.
Based on the proper time method, δm 2 φ is rewritten as where τ 2 = Λ 2 t . Let us make the complex parameter τ = τ 1 + i τ 2 play the role of the modular parameter in string theory. The world-sheet modular transformation is given by where a, b, c and d are integers, and the transformation is generated by the compositions of two types of transformations τ → τ + 1 and τ → −1/τ. If we require the invariance under the transformation (28) and assume that the physics is only described by an independent region, which is not connected with by the transformation, the following expression is obtained, where F stands for the fundamental region defined by The value of (29) is different from that of (26). The difference of values comes from that of the invariant measures, i.e., the invariant measure for τ → −1/τ is d 2 τ/τ 2 2 , but that for t → 1/(Λ 4 t ) is d t /t up to a sign factor. Note that both τ → −1/τ and t → 1/(Λ 4 t ) connect the UV region to the IR one, and τ → −1/τ reduces to τ 2 → 1/τ 2 , which corresponds to t → 1/(Λ 4 t ), in case with τ 1 = 0.
In this way, we find that the value of δm 2 φ depends on the form of duality transformation and need to specify it in order to obtain a physically meaningful value. We expect that the form of duality transformation is determined by matching the counterpart in the ultimate theory.
We add a comment on radiative corrections in string theory. From the world-sheet modular invariance for the closed string, δm 2 φ (radiative corrections of the scalar mass including contributions from innumerable string states) should be given by where G(τ) is a world-sheet modular invariant function, i.e., G(τ) = G(τ + 1) and G(τ) = G(−1/τ). In cases that SUSY holds exactly, G(τ) vanishes, and then δm 2 φ = 0. Even if SUSY is broken down, there is a possibility that G(τ) vanishes in conspiracy with infinite towers of massive particles, as suggested in Ref. [26].
In string theory, the world-sheet modular invariance is deeply connected to the consistency of theory, and radiative corrections should be given in the world-sheet modular invariant form for the closed string. On the other hand, in the effective field theory, a corresponding symmetry stays in the background if it exists at all, and the consistency of theory would not be necessarily threatened if it is overlooked. Hence, radiative corrections in the effective theory are not generally given in the duality invariant form, and the projection to pick out the invariant parts would be required.
Finally, we give a conjecture on a solution for the one side of the naturalness problem whether the weak scale is stabilized against radiative corrections, taking string theory as a candidate of the ultimate theory. In string theory, the world-sheet conformal invariance induces the massless string states, and the world-sheet modular invariance guarantees the finiteness of physical quantities. We conjecture that, owing to some powerful symmetry (such as SUSY) in addition to the world-sheet modular invariance, the masslessness of scalar particles would be protected against quantum corrections, and the above-mentioned problem would not be caused. This could be understood in the framework of low-energy effective field theory, as follows. We assume that the theory is described by only massless string states, effects of massive string states are introduced as non-renormalizable interactions among massless particles, and they do not cause (the technical side of) the gauge hierarchy problem. In the field theory limit, if τ reduces to τ 2 with τ 1 = 0, the duality transformation τ → −1/τ reduces to τ 2 → 1/τ 2 , which corresponds to t → 1/(Λ 4 t ). Then, massless scalar fields do not receive any radiative corrections on their masses, as seen from (26). This matches the conjecture based on string theory. For the case with massive light scalar fields, a more careful consideration is needed and it is beyond the scope of this paper, because it is deeply related to the other side of the naturalness problem what is the origin of the weak scale or the Higgs mass.
on the assumptions relating features of the ultimate theory. The assumptions are summarized as follows. Beyond the SM, there is an ultimate theory with a fundamental scale Λ and a duality between the physics at the UV region beyond Λ and that at the IR region, and a remnant of the duality is hidden in the lower-energy theory. We have shown that the logarithmic corrections can be picked out by specifying the duality transformation. Because the logarithmic corrections are compatible with a specific duality, it is expected that the subtraction of quadratic divergences could be justified in the ultimate theory.
If the quadratic divergences of scalar fields are artifacts of regularization procedure, the problem whether the weak scale is stabilized against radiative corrections in the framework of SM can be solved by the subtraction of quadratic divergences. Note that, even if the quadratic divergences are eliminated, the physics beyond the SM can induce the gauge hierarchy problem, i.e., the effective field theory becomes unnatural, because a fine tuning is required to obtain the weak scale and/or to stabilize the weak scale, if there is a high-energy physics relevant to the SM. The sources of large radiative corrections, which can ruin the stability of the weak scale, are logarithmic divergences due to heavy particles. There is a possibility that the SM (or the extension of the SM with new particles around the terascale and without new concepts such as SUSY, compositeness and extra dimensions) holds until Λ and an ultimate theory protects masses of low-energy fields against large quantum corrections by some mechanism and/or symmetry. This is a background of our previous work [27].
Finally, we discuss the applicable scope of our method. The issue is whether our calculation scheme is applicable to other systems and ordinary results are obtained or not. We anticipate that it is applicable to calculate logarithmic corrections including Λ on quantities.
We have applied our method to the radiative corrections on vacuum energy density, and shown that the logarithmic corrections can be picked out, in the appendix A. Under the assumption that the QED holds until Λ in the broken phase of electroweak symmetry, we also have applied it to the self-energy of electron, and obtained the well-known result, in the appendix B.
Because our procedure contains a provisional cutoff parameter depending on a mass, it looks like an artifact or a temporary expedient to pick out specific corrections. It is important to examine whether it is applicable to radiative corrections with higher loops by introducing several proper times and more complex models including several fields.
Even if our scheme has a limit of application or the hidden duality is a product of fantasy, our expectation would survive that the calculation scheme can be selected by the physics, and radiative corrections can be constrained by a remnant of symmetries in an ultimate theory.

A Radiative corrections on vacuum energy density
We apply our procedure to the radiative corrections on vacuum energy density δΛ V . For the scalar field φ, δΛ V at the one-loop level is commonly written as and it contains infinities. We carry out the same procedure as that for scalar masses. First, we separate δΛ V into the quartic, quadratic and logarithmic divergent parts by expanding the exponential factor such that whereΛ φ is a provisional cutoff parameter (Λ 2 φ ≫ m 2 φ ), which goes to the infinity in the limit of m 2 φ → 0. In case withΛ 2 φ = Λ 4 /m 2 φ , we find that the third term in the right-hand side of (33) is invariant under the change t → 1/(Λ 4 t ), but others are not.
Here, we impose the duality relating t → 1/(Λ 4 t ) on quantities relevant to Λ. If the physics from t = 1/Λ 2 φ to t = 1/Λ 2 is same as that from t = 1/Λ 2 to t = 1/m 2 φ and the physics is only described by one of the two regions, we obtain the relation In this way, the quartic and quadratic divergences in δΛ V are eliminated by requiring that the effective theory should have a hidden symmetry on the proper time. Note that δΛ V also vanishes, in the massless limit m φ = 0. Because the subtraction of the quartic and quadratic divergences in δΛ V induces the effect that the cosmological constant shifts, a more careful consideration is required to justify our procedure. 8

B Self-energy of electron
The self-energy of electron with the momentum p at the one-loop level is given by [29] identify z 1 ξ as a proper time t , the transformation is the same form t → 1/(Λ 4 t ) as that in case of the scalar mass m φ . Here, we impose the duality relating ξ → 1/(z 2 1 Λ 4 ξ) on quantities relevant to Λ. If the physics from ξ =m 2 /z 2 1 Λ 4 to ξ = 1/(z 1 Λ 2 ) is same as that from ξ = 1/(z 1 Λ 2 ) to ξ = 1/m 2 and the physics is only described by one of the two regions, the desired expression is obtained as where we take the limit of µ 2 γ → 0.