Quantum phase transition and absence of quadratic divergence in generalized quantum field theories

In ordinary thermodynamics, around first-order phase transitions, the intensive parameters such as temperature and pressure are automatically fixed to the phase transition point when one controls the extensive parameters such as total volume and total energy. From the microscopic point of view, the extensive parameters are more fundamental than the intensive parameters. Analogously, in conventional quantum field theory (QFT), coupling constants (including masses) in the path integral correspond to intensive parameters in the partition function of the canonical formulation. Therefore, it is natural to expect that in a more fundamental formulation of QFT, coupling constants are dynamically fixed a posteriori, just as the intensive parameter in the micro-canonical formulation. Here, we demonstrate that the automatic tuning of the coupling constants is realized at a quantum-phase-transition point at zero temperature, even when the transition is of higher order, due to the Lorentzian nature of the path integral. This naturally provides a basic foundation for the multi-critical point principle. As a concrete toy model for solving the Higgs hierarchy problem, we study how the mass parameter is fixed in the $\phi^4$ theory at the one-loop level in the micro-canonical or further generalized formulation of QFT. We find that there are two critical points for the renormalized mass: zero and of the order of ultraviolet-cutoff. In the former, the Higgs mass is automatically tuned to be zero and thus its fine-tuning problem is solved. We also show that the quadratic divergence is absent in a more realistic two-scalar model that realizes the dimensional transmutation. Additionally, we explore the possibility of fixing quartic coupling in $\phi^4$ theory and find that it can be fixed to a finite value.


Introduction and Outline
By the discovery of the Higgs boson at LHC, it has been confirmed that the electroweak symmetry breaking is triggered by the Higgs mechanism.However, the question of why there is a huge hierarchy between the electroweak scale 10 2 GeV and the Planck scale 10 18 GeV is not unveiled yet.In the Standard Model (SM), the electroweak scale is merely obtained by fine-tuning the mass parameter of the Higgs potential, which makes the Higgs boson mass sensitive to ultraviolet (UV) scales such as the grand unification or Planck scale.Although supersymmetry has been discussed as one of the most promising new physics scenarios beyond the SM because of the stabilization of the Higgs boson mass to be the electroweak scale, it does not explain the smallness of the electroweak scale, nor is found near the expected TeV scale.In order to confront this fine-tuning problem, more radical and fundamental approaches beyond ordinary quantum field theory (QFT) appears to be required [1,2,3,4,5,6,7,8,9,10,11,12,13,14].
Nature might already give us some important clues to tackle this problem.Let us recall a basic notion of statistical mechanics: there are several different formulations depending on which parameters to use as control parameters, and they are equivalent in the thermodynamic limit V → ∞.In particular, the most fundamental formulation is obtained with the micro-canonical ensemble where all the extensive parameters, e.g., energy E, volume V , number of particles N are chosen as control parameters, while intensive parameters, e.g., temperature T , pressure p, and chemical potential µ, are determined as functions of these extensive parameters in the thermodynamic limit.Here, the important point is that this correspondence is not injective when a system undergoes phase transitions.For example, in a first-order phase transition, the temperature stays at the critical temperature T cri (E) until the system releases or absorbs all the latent heat, which means that the critical point spans the finite region of E in the micro-canonical formulation. 1 More generally, with a finite probability, intensive parameters are fixed at the point where the extensive quantities become discontinuous.
In this paper, we explore a correspondence in QFT similar to that observed in statistical mechanics.Our objective is to address the fine-tuning problem by determining parameters in the canonical partition function of QFT from the micro-canonical partition function, thereby eliminating the need for tuning.In essence, the fine-tuning problem is resolved when a finite region in the parameter space of micro-canonical (or further generalized) QFT corresponds to a specific point in the canonical-QFT parameter space that appears fine-tuned.We also acknowledge various attempts to construct micro-canonical QFT, as discussed in previous studies [1,2,3,11,12,13,14] and the references therein.
The concept of generalized QFT is both fascinating and intriguing; however, the actual finetuning of couplings remains unclear, necessitating further research.As a first step, we investigate the free scalar theory to explore how the bare mass-squared parameter m 2 B is fixed in the generalized QFT.We calculate the generalized partition function and find two critical points in the large volume limit: m 2 B = 0 and O(Λ 2 ), where Λ denotes the cut-off scale.In particular, while the latter corresponds to a saddle point of the vacuum energy and depends on the regularization schemes in general, the former corresponds to its discontinuity and does not depend on the regularization schemes.In this regard, one might find m 2 B = 0 physically more preferable, meaning that massless theory is naturally realized in the micro-canonical (or further generalized) picture.This can be also interpreted as a theoretical explanation of the origin of the so-called classical scale invariance or classical conformality in the literature [15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30], which is also the basic assumption behind the Coleman-Weinberg mechanism [31].
We proceed to examine the φ 4 theory at the one-loop level and generalize it to a large-N model.In both cases, the mass term receives UV divergent corrections δm 2 UV ∼ Λ 2 , prompting us to investigate at which value the renormalized mass m 2 := m 2 B + δm 2 UV is fixed in the generalized QFT.A key distinction from the above free theory lies in the vacuum transition at m 2 = 0; For m 2 > 0, we observe a trivial vacuum expectation value (VEV) φ = 0, while it becomes nonzero for m 2 < 0. This behavior corresponds to the discontinuity of the (second) derivative of the vacuum energy at m 2 = 0, which remains a critical point in both cases.Although our analysis is limited to the one-loop level or the large N limit, we anticipate that the conclusions will remain unaffected by higher order corrections as long as the theory is in the perturbative region.
In our next non-trivial example, we examine a two-real-scalar model at the one-loop level.This model has been extensively studied in a phenomenological context [32,33,34,35,36,37] because it can realize the Coleman-Weinberg mechanism [31] under the assumption of classical conformality, i.e., m 2 = 0.However, in the generalized QFT, this is not an assumption but a point of discussion for determining where and how renormalized masses are fixed.To streamline our analysis, we assume Z 2 × Z 2 symmetry and focus on the coupling space where one real scalar φ can develop a nonzero VEV φ = 0 due to the radiative potential.We discover that the classical conformal point m 2 φ = m 2 S = 0 is not a critical point in the generalized partition function.Instead, we identify another critical point with m 2 φ = 0 and m 2 S = 0, which corresponds to a quantum first-order phase transition point.From a phenomenological perspective, this critical point may serve as an alternative possibility for a dimensional transmutation mechanism [36,37] compared to the conventional classical conformal point; see also Ref. [35] for other possible critical points without Z 2 symmetry.
The organization of this paper is as follows: In Sec. 2, we introduce the generalized QFT and explain how the fine-tuning of coupling constants can be automatically achieved within this framework.We also review the standard discussion on the equivalence between different ensembles in statistical mechanics to clarify the underlying concept.In Sec. 3, we investigate the free scalar theory in the context of generalized QFT, focusing on how the bare mass-squared parameter is fixed in the large volume limit.In Sec. 4, we delve into a more non-trivial example by considering the φ 4 theory, discussing how the renormalized mass-squared is fixed at the one-loop level, as well as in the large-N model.In Sec. 5, we explore the possibility of automatically tuning the quartic coupling constant.In Sec. 6, we analyze the two-real-scalar model within a simple parameter space where only one scalar can develop a VEV.Finally, in Sec. 7, we summarize our result.
Throughout the paper, we work in natural units = c = 1 and employ the metric convention 2 Generalized Quantum Field Theory In this section, we briefly review the basics of statistical mechanics and introduce a generalized partition function in the micro-canonical formulation of QFT, and illustrate how the tuning of the coupling constants becomes possible.See e.g.Ref. [3] and Appendix D in Ref. [38] for other reviews.

Statistical mechanics
To clarify the idea, let us briefly recall statistical mechanics.When control parameters are temperature T and volume V , the system is described by the canonical partition function where E n denotes the energy eigenvalue and F V (T ) = −T log Z V (T ) is the Helmhortz free energy.
On the other hand, we can also consider the micro-canonical formulation where E is a control parameter instead of T .The corresponding partition function, the number of states, is given by where S V (E) denotes the entropy and ∆E denotes a sufficiently small energy interval whose effects on observables vanish in the thermodynamic limit V → ∞.In the following, we omit the subscript V for simplicity.The equivalence between the canonical and micro-canonical formulations can be shown as where s = S/V (ε = E/V ) represents the entropy (energy) density.The integration is dominated by the saddle point in the thermodynamic limit V → ∞ as

Partition function Thermodynamic function
Table 1: Relation between canonical and micro-canonical formulations in statistical mechanics where C = ∂E/∂T is the specific heat and Eqs. ( 5) and ( 6) are nothing but the Legendre transformation between the free energy and the entropy in thermodynamics.It is also straightforward to check the equivalence of the ensemble averages of a general observable x.The canonical ensemble average is given by where x n = E n |x|E n .This can be written as where is the micro-canonical ensemble average.Because the integrand in Eq. ( 9) has a strong peak at E = E * as in Eq. ( 4), we obtain in the thermodynamic limit V → ∞.In particular, during the first-order phase transition, T is equal to the critical temperature T c (E), but E varies between the two phases.This means that the coexisting phases are described in the finite parameter region of the micro-canonical scheme.In this sense, a fine-tuning to a first-order phase transition point T = T c (E) is automatically realized as long as E is the finite region.The correspondence is summarized in Table 1.

Micro-canonical formulation of QFT
Now let us return to QFT.Conventionally we start from "canonical" partition function with the bare coupling constants { λ } := { λ j } j=0,1,2,... : where O j [φ] (j = 0, 1, 2, . . . ) denotes a spacetime integral of a local operator such as 4 , etc.After the renormalization procedure, we can calculate physical observables finitely as functions of (renormalized) couplings.However, the problem is that there is no principle to pick up specific values of { λ } theoretically, and this is the very origin of the fine-tuning problems.Here comes the analogy between QFT and statistical mechanics into play: What if we start from the "micro-canonical" picture in QFT?
The number of states Ω(E) in statistical mechanics can naturally be promoted to the partition function in QFT as where we write { A } := { A j } j=0,1,2,... and each A j is an "extensive parameter" corresponding to O j [φ], which is proportional to the spacetime volume V d .By using the Fourier transform of the delta function, Eq. ( 13) can be written as Now, one can see that we have an ensemble average of various coupling constants and their weight is proportional to the canonical partition function (12).

Further generalized QFT
Further, we can generalize [3] the product of delta functions in Eq. ( 13) to an arbitrary weight function where ω({ λ }) is the Fourier transform of W ({ x }): In this generalization, we assume that W ({ x }) can depend on extensive parameters solely through does not depend on any extensive parameters.

Parameter Partition function Generating function
If there exists a strong peak { λ * } of Z({ λ }) in the infinite volume limit V → ∞, the generalized QFT is equivalent to the ordinary canonical QFT whose (bare) coupling constants are fixed at {λ * }.In other words, the fine-tuning of coupling constants is automatically realized in the generalized partition function.We will see that only the behavior of the canonical partition function Z({ λ }) matters for the realization of the Higgs fine-tuning, regardless of whether it is in the micro-canonical QFT (13) or in the generalized QFT (15).See Table 2 for the summary of naive correspondence.In the following sections, we will verify this fine-tuning mechanism for the (bare) mass term m 2 B in scalar field theory.

Fixing mass in free scalar theory
We study the free scalar theory in the generalized QFT and show how the bare mass parameter is fixed.

Partition function of free scalar theory
We consider the free scalar theory in the d-dimensional spacetime: Here, we introduce the bare mass term according to the generalized QFT, while leaving the kinetic term (17) as is, for simplicity.Then the generalized partition function is defined by where ω(m 2 B ) is the Fourier transform of W (x) and Z(m 2 B ) denotes the ordinary canonical partition function We assume that ω(m 2 B ) is an ordinary smooth function.We can now perform the path-integral as where = η µν ∂ µ ∂ ν , a = A/V d , iε is the Feynman's prescription, and with p µ E being the Euclidean momentum.This is apparently UV divergent and we need a regularization.

Mass-squared in free scalar theory
To discuss how the mass is tuned, we employ cut-off and dimensional regularizations.

Cut-off regularization
First, let us study the cut-off regularization.As a function of m 2 B , the integrand in Eq. ( 23) has a branch cut at and the integration line −∞ < m 2 B < +∞ is located below this branch cut as shown in the left panel in Fig. 1.Then, Eq. ( 24) can be evaluated as where S d−1 denotes the area of a d − 1 dimensional sphere.We see that the function F (m 2 B ) contains the imaginary part for m 2 B < 0, which gives a large suppression in the partition function when V d is large.Qualitatively, the contribution from On the other hand, there is no such suppression for m 2 B > 0, and the integrand is a rapidly oscillating function of m 2 B .In this case, the saddle point can exist at the point determined by When d = 4, this equation becomes where f (ξ) is a monotonic function satisfying f (0) = 1 and f (∞) = 0; see the right panel in Fig. 1.Thus, there can be a saddle point when a/Λ 2 ≤ (32π 2 ) −1 , while the exponent,

Dimensional regularization
Let us also calculate the partition function in the dimensional regularization.We define = 4 − d.
In this case, the free energy is calculated as where µ is the renormalization scale and c MS contains both the finite and divergent terms.For m 2 B ≤ 0, we again have the imaginary part from the logarithmic term, and the partition function is highly suppressed.On the other hand, for m 2 B > 0, the saddle point is determined by which always has a solution unlike the cut-off case.If we identify µe c MS as the cut-off scale, the location of the saddle points is in both of the regularizations.

Mass tuning in free scalar theory
As we have seen, the existence of a saddle point depends on the regularization scheme.But in any case, m 2 B = 0 holds over a wide range of the parameter space.First, if there is no saddle point, the m 2 B integration is dominated by the boundary m 2 B = 0 by the mathematical formula (88) for any smooth weight function ω(m 2 B ).On the other hand, if there is a saddle point, am ) is monotonic, and the boundary m 2 B = 0 is always dominant due to the mathematical formula (88).In conclusion, m 2 B = 0 appears to be a unique critical point that is physically reasonable in the generalized partition function (23).

Equivalence in large volume limit
Finally, let us confirm the equivalence between generalized QFT and canonical QFT in the large volume limit.When A = 0, the first derivative of ) is continuous and nonzero at m 2 B = 0 while the second derivative is discontinuous.Then, we can use the mathematical formula (92) and Ω(A) is evaluated as lim where N is an unimportant numerical factor.That is, lim We apparently see the equivalence in the large volume limit.
We can also check the equivalence of correlation functions.We introduce a source term as As long as J(x) is a finite supported function, i.e., when it has no volume dependence, the factor G(J; m 2 B ) does not have exponentially large volume dependence, and the m 2 B integral in Eq. ( 37) is dominated by the critical point of Z(m 2 B ). Namely, we have lim Then, by taking the functional derivatives with respect to J(x), we obtain which corresponds to Eq. ( 11) in statistical mechanics.
4 Fixing mass in φ 4 theory In this section, we analyze the φ 4 theory in the scope of generalized QFT.In particular, we study how the mass term is fixed by studying its critical point at the one-loop level, and also in the large-N extended model.Our investigation reveals that the renormalized mass settles at zero in the one-loop analysis as well as in the large-N model, due to the discontinuity of the derivative of the vacuum energy.
In this section, we assume that the quartic coupling is prefixed at a certain value, as in the ordinary QFT.The possibility of its tuning will be discussed in the next section.

Mass-squared at one-loop level
We first introduce a bare action without the bare mass term: where we have also included the bare cosmological constant term for later convenience.For simplicity, we have imposed the Z 2 symmetry φ → −φ, which leaves only the quadratic and quartic terms in the renormalizable potential.
Our starting point is the generalized partition function Ω(A) which can be expressed in terms of the canonical partition function Z(m 2 B ) by the same procedure as in Sec.3.1: where The canonical partition function Z can be obtained from the ordinary effective action Γ[φ] via Z = e i min φ Γ [φ] .At the one-loop level, we obtain where 2 φ 2 .Now the one-loop effective potential is given by denote the renormalized parameters.We can regard these renormalized couplings as our parameters instead of the bare couplings (m 2 B , λ B ).As a consistency check, let us consider the free theory limit: λ B = λ R = 0.In this case, there is no difference between the bare mass and the renormalized mass, and the effective potential is exactly given by which has a trivial minimum at φ = 0 and the corresponding vacuum energy is which is nothing but Eq. ( 32).Let us come back to the interacting scalar theory.We assume λ R > 0 to ensure the stability of the effective potential.As usual, we can obtain the RG improved effective potential by choosing the renormalization scale µ appropriately.For m 2 R ≥ 0, the VEV is trivial v = 0, and we take µ = m R e −3/4 , which will give a simple expression of the vacuum energy (as the first line in Eq. (50) below).On the other hand, for m 2 R < 0, the field φ develops a nonzero VEV, which is determined by By choosing the renormalization scale at µ 2 = M 2 φ (v)e −1 , the VEV is given by Now the vacuum energy as a function of m 2 R is given by which shows that the second derivative of V min (m 2 R ) is discontinuous at m 2 R = 0 as Note that the first derivative of ) is continuous and nonzero at m 2 R = 0 as long as A = 0, which means that we can use the mathematical formula Eq. ( 92).Then, the partition function is evaluated as lim which leads to lim where s is the field that may acquire a VEV.The Lagrangian now becomes We can introduce an auxiliary scalar field c in the Lagrangian such that the partition function does not change after performing the path-integral over c: By performing the path-integral of π i , the partition function is given as where λB := λ B N/6.The variation of c gives while the variation of s gives As long as we focus on the ground state, we can omit s.For a given value of λB , the parameter space of m 2 B is divided into two regions as shown in Fig. 2, which correspond to the broken phase (blue) and the unbroken phase (uncolored) respectively.Now let us discuss the behavior of the partition function in each region.

Broken phase
When s = 0, we have m 2 B + c = 0. Eq. (62) then becomes which indicates the parameter space of broken phase.This is shown by the blue region in Fig. 2. We represent the critical value of m 2 B as −m 2 Λ .For d = 4, it is For the parameters obeying Eq. ( 64), the canonical partition function Eq. ( 61) becomes very simple in the large N analysis.In fact, One can see that there is a saddle point at m 2 B = −2 λB A/V d if it is smaller than the critical value −m 2 Λ .Otherwise, the exponent in Eq. ( 67) is a monotonic function for m 2 B ≤ −m 2 Λ .In particular, the second derivative is given by

Unbroken phase
When s = 0, the system is in the unbroken phase.In this case, c is determined by the gap equation This equation has a solution only for which is consistent with Eq. ( 64).In this case, the canonical partition function becomes where c depends on m 2 B via the gap equation ( 69).The exponent in Eq. ( 71) is a monotonically decreasing function of m 2 B and the second derivative with respect to 1 + λB By comparing Eq. ( 68) and Eq. ( 72), one can see that the second derivative is discontinuous at the critical point m 2 B = −m 2 Λ .Thus, we conclude that m 2 B is fixed at −m 2 Λ by the mathematical formula (92) (as long as A = 0) at which the renormalized mass m 2 R = m 2 B + c is zero.

Fixing quartic coupling in φ 4 theory
We briefly comment on the possibility of fixing the quartic coupling.As well as the mass term, we can also consider the generalized partition function for the quartic coupling where we take the renormalized couplings as the integration parameters instead of the bare couplings for simplicity. 4Note that A λ is another extensive parameter proportional to the spacetime volume V .For simplicity, let us restrict the parameter space to λ R ≥ 0 to guarantee the stability of the system.At the one-loop level of the φ 4 theory, the vacuum energy (50) vanishes at the critical point of the mass-squared m 2 R = 0 for any λ R ≥ 0, which means that the above partition function becomes As long as A λ = 0, the exponent is a linear function of λ R , and this integration is strongly dominated by λ R = 0 by the formula Eq. (88) in the large volume limit.Thus, the free scalar theory seems to be realized in the generalized QFT in the φ 4 theory at least at one-loop level.As long as the renormalized mass m 2 R is fixed at zero and we focus on the cut-off independent part of the vacuum energy, this conclusion will not be significantly altered by the higher-loop effects because the cut-off independent part should be proportional to (m 2 R ) 2 by dimensional analysis and they vanish at the critical point m 2 R = 0. On the other hand, non-trivial saddle point can appear if we also include the cut-off dependent parts.For example, at the three-loop level, we have the following contributions to the vacuum energy: where c i > 0 are just constants containing the loop suppression factors.Thus, one can see that there exists a saddle point at This is a very interesting possibility but its physical meaning is subtle as in the mass parameter case.More detailed studies are left for future investigations.See also Appendix B for the possibility of fixing λ R in the large N limit.

Dimensional transmutation in two-real-scalar model
As a next non-trivial example, we study a two-real-scalar model [32,33,34,35,36,37] at the one-loop level.We show that an automatic tuning of the mass-squared parameter is realized so that the dimensional transmutation is successfully achieved.
For simplicity, we again impose the Z 2 × Z 2 symmetry, which leads to the following Lagrangian To simplify the discussion further, we focus on the parameter space λ φ λ S ∼ 1 and λ φS > 0, which means that the φ direction is almost flat for m 2 φ ∼ 0. φ and S play the roles of the scalar and gauge fields in the original Coleman-Weinberg mechanism.On the other hand, the S direction is always dominated by the tree-level potential, that is, its VEV is well determined by m 2 S S 2 + λ S 4! S 4 .Under these conditions, we will show that there is a critical point at m 2 S = 0 and m 2 φ = 0 which corresponds to a quantum first-order phase transition point.Although the conventional classical conformal point m 2 S = m 2 φ = 0 is not realized in the current formulation, our result serves as a concrete way of dimensional transmutation without assuming an artificial symmetry such as classical conformality.Now let us discuss the details.The one-loop effective potential in the MS scheme is given by where with Here, all the coupling constants are the renormalized ones.As usual, we can take the renormalization scale at the point µ = M where λ φ vanishes.In the following, we first examine how m 2 φ is fixed for a given value of m 2 S and then discuss the fixing of m 2 S .First, let us consider the parameter space m 2 S ≥ 0. In this case, S = 0 as long as the tree-level potential dominates in the S direction.The effective potential for φ is then In Fig. 3, we show the plots of V eff (0, φ) where the different colors correspond to the different values of m 2 φ .In particular, one can see that there exists a quantum first-order phase transition at meaning that φ achieves a nonzero VEV, φ = v φ , for m 2 φ < m 2 c .See Appendix C for the derivation of Eq. (82).Correspondingly, the VEV and the vacuum energy are given by As in the simple φ 4 theory, the additional contribution in the second line gives the discontinuity of the second derivative of the vacuum energy at m 2 φ = m 2 c , and this point is dominant in the generalized partition function.Second, we turn to the case m 2 S < 0. In this case, S already has a VEV at S 2 = −6m 2 S /λ S at tree level, and there exists the tree-level vacuum energy −3(m 2 S ) 2 /(2λ S ) as in the simple φ 4 theory discussed in the previous sections.As for the φ potential, nonzero VEV of S just changes the effective mass in Eq. (80) as Note that M 2 φ is just a constant because of λ φ = 0. Thus, we can repeat the same discussion as above and conclude that m 2 φ is still fixed at the critical point m 2 φ = m 2 c .Finally, at such a critical point of m 2 φ , the vacuum energy as a function of m 2 S is given by for m S < 0 .
As in the simple φ 4 theory, m 2 S is fixed at zero because the second derivative of this equation is discontinuous at m 2 S = 0.The conventional Coleman-Weinberg point m 2 S = m 2 φ = 0 is not realized in the current formation because such a point m 2 S = m 2 φ = 0 corresponds to the degeneracy of false vacua as discussed in Ref. [35].We leave the discussion on the criticality of general extrema for future investigation.

Summary
In this paper, we have investigated the Higgs hierarchy problem in the generalized QFT.We first studied the free scalar theory and found that there are two critical points in the large volume where g(x) is a smooth function and x 0 is a saddle point, g (x 0 ) = 0.The proof of this equation is textbook level and we will not repeat it here.The second formula is lim where g(x) is now a real, smooth and monotonic function for x ≥ 0 and satisfies g (0) = 0.The proof is as follows.By multiplying a test function f (x) with a finite support and integrating from 0 to ∞, we have which confirms Eq. (88).Note that the contribution from x = ∞ is zero because we have assumed that f (x) has a finite support.More generally, when g(x) is a monotonic and smooth function for both sides x > 0 and x < 0, but its derivative g (x) is discontinuous at x = 0, we have lim Note that when g (x) is also continuous at x = 0, the right-hand side vanishes and higher order terms dominate.In general, when the derivatives of g(x) are continuous and nonzero up to (n − 1)th order but the n-th derivative of g(x) is discontinuous, we have lim where c is a coefficient determined by g (0), g (0),

B Fixing quartic coupling in large N model
In this Appendix, we study a possibility of fixing the quartic coupling by taking the N −1 corrections.
In the following, the mass term m where δs = s − s cl , δc = c − c cl and • • • represents the higher order terms.Note that the first term is the leading vacuum energy contribution and it is proportional to λB .Thus, this can be absorbed into the definition of A λ .In the following, we represent where µ is some (IR) scale.We can check that λ(µ) corresponds to the usual renormalized quartic coupling because Eq. ( 94) is nothing but the solution of RGE in the large N limit: In Eq. ( 93), δc part can be rewritten as Now we can perform the one-loop integration of δc as where F = V 4 Λ 4 /(32π 2 ).Note that we have singular points at λ(µ) = 0, ±∞.
Now the λ B integration in the micro-canonical partition function is given by where G(λ(µ)) := λ(µ) One can check that there exist two saddle points: where We can see that N λ ± is real and positive if we take the following large N limit.
More detailed analysis is necessary to verify the validity of this limit because we now have additional parameter A λ that is absent in the usual large N analysis.

C Details of two-real-scalar model
By putting φ2 = (m 2 S + λ φS φ 2 /2)/M 2 e 3/2 , we can rewrite Eq. (81) as See Fig. 3 for the explicit plots of Ṽφ ( φ).As a consistency check of S = 0 for m 2 S ≥ 0, let us also check the positivity of the effective mass of S at S = 0:

2 B 2 BΛ 2 . 2 BΛ 2 ,
− F (m 2 B ) is monotonic below the saddle point m Thus, when the weight function ω(m 2 B ) has a finite support for m the free energy am 2 B − F (m 2 B

Figure 2 :
Figure 2: Left: Parameter space in the large N limit.

2 2 E ) 2
is fixed at the critical point m 2 = m 2 B + c cl = 0 and we focus on d = 4. Around the large N solution (c, s) = (c cl , s cl ) studied in Section 4.2, the effective action in Eq. δc 2 + • • • ,