ptep ptep Progress of Theoretical and Experimental Physics 2050-3911 Oxford University Press 10.1093/ptep/ptv168 ptv168 Papers Theoretical Particle Physics B30 B39 B87 Natural solution to the naturalness problem: The universe does fine-tuning HamadaYuta * KawaiHikaru * KawanaKiyoharu * Department of Physics, Kyoto University, Kyoto 606-8502, Japan E-mail: hamada@gauge.scphys.kyoto-u.ac.jp, hkawai@gauge.scphys.kyoto-u.ac.jp, kiyokawa@gauge.scphys.kyoto-u.ac.jp 122015 21122015 2015 12 123B03 2492015 30102015 © The Author(s) 2015. Published by Oxford University Press on behalf of the Physical Society of Japan. 2015 This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted reuse, distribution, and reproduction in any medium, provided the original work is properly cited. Funded by SCOAP3

We propose a new mechanism to solve the fine-tuning problem. We start from a multi-local action , where the 's are ordinary local actions. Then, the partition function of this system is given by , where represents the parameters of the system whose Hamiltonian is given by , is the radius of the universe determined by the Friedman equation, and , which is determined by , is a smooth function of . If a value of , , dominates in the integral, we can interpret that the parameters are dynamically tuned to . We show that this indeed happens in some realistic systems. In particular, we consider the strong CP problem, the multiple point criticality principle, and the cosmological constant problem. It is interesting that these different phenomena can be explained by one mechanism.

 Subject Index B30 B39 B87 SCOAP3 arxiv-id arXiv:1509.05955 Introduction and general idea

Since the discovery of the Higgs particle [1, 2], it has become more and more important to consider the fine-tuning problem [3–5]. The natural and conservative approach is to seek solutions in the context of ordinary local field theory. However, even if such theory can solve an individual problem such as the quadratic divergence of the Higgs mass, it seems difficult to answer all the problems simultaneously. Therefore, it is necessary to consider a new framework or principle beyond ordinary local field theory. There are many interesting proposals such as asymptotic safety [6], hidden duality and symmetry [7–9], classical conformality [10–15], the multiple point criticality principle (MPP) [16–33], and the maximum entropy principle [34–38].

Among them, Coleman's theory on baby universes and the multiverse [39] seems promising. Although Coleman's first work explains the smallness of the cosmological constant (CC), its validity is unclear because it is based on Euclidean gravity. Therefore, a Lorentzian version is inevitably needed to give reliable predictions. In Refs. [34–37], such improvements are actually done. They are essentially based on a multi-local action and the existence of the multiverse. The former is relatively natural because it arises if we take the topology change into account [39–42]. On the other hand, the idea of the multiverse is slightly uncommon because there is no evidence so far that we live in the multiverse. Therefore, it is meaningful to consider whether we can solve the fine-tuning problem without relying on the existence of the multiverse. The purpose of this paper is to give a new framework to solve the fine-tuning problem based on a multi-local action.

We consider the partition function of the multi-local action, where is an ordinary local action, and the 's, 's, are constants. Here, we have assumed that the universe is created at , and evolves to . Then, Eq. (1) is obtained after summing up the wormholes. See Fig. 1 for an example. Thus, by expressing as a Fourier transform we can write the partition function of the system as follows: where is the radius of the universe determined by the Friedman equation, and represent initial and final states which are independent of , and we have assumed that the universe eternally expands like our universe.1 If there is a point that strongly dominates in the integral of Eq. (4), the observer in the universe finds that the parameters are fixed at . In particular, Eq. (4) is equivalent to At first glance, it seems difficult to evaluate because it involves the total history of the universe. However, in almost all the time, the universe is sufficiently expanded, and its energy density is very close to that of the vacuum. More concretely, we have where is the space volume, is the vacuum energy density, and is a time such that the energy density of the state is sufficiently close to . By substituting Eq. (6) in Eq. (4), we obtain As we will see in the following discussion, we can find from Eq. (8) in some phenomenologically interesting systems.

Universe and baby universes. Here, we show wormholes having two legs. The left and right figures represent the partition function and Hamiltonian points of view respectively.

We can also repeat the above argument from the point of view of Coleman's baby universes as follows. As discussed in [39, 40], instead of using the multi-local action, the wormhole effect can be expressed by introducing operators and that describe the creation and annihilation of a baby universe. Namely, the action of the universe with baby universes is given by See the right figure in Fig. 1 for an example. Because the 's are conserved quantities, for each set of their eigenvalues, Eq. (9) is an ordinary local action for the universe. Therefore, we can develop the quantum theory for the total system consisting of the universe and baby universes: Here, is the complete set of the Fock space of the baby universes: and is the Hamiltonian for the universe that corresponds to the action . Then, for the initial state , the wave function at is given by Thus, by considering the limit, and multiplying a final state by Eq. (12), we actually obtain Eq. (4) where This is the derivation of Eq. (4) from the point of view of Coleman's baby universes. Note that the weight function is not important in the following discussion because we will consider the general consequences which do not depend on the detail of as long as it is a smooth function. In the following, we do not consider such a special case that or is .

Because the above point of view is rather uncommon, let us see a simple example here: where This is a harmonic oscillator having an additional bi-local action . By using a Lagrangian multiplier , the path integral of this system can be rewritten as follows: where Here, in the last line in Eq. (16), we have changed the variable: One can see that Eq. (16) actually has the form of Eq. (4). However, is not fixed to any special value because the integrand in Eq. (16) is a regular function of in this case. On the other hand, as we will see, if we consider systems with infinite degrees of freedom, is indeed fixed to a special value in several cases.

In the following sections, we study a few examples which show that the above argument is not an armchair theory. In particular, we consider the strong CP problem, the MPP, and the cosmological constant problem (CCP). The MPP is a proposal to solve the fine-tuning problem, which claims that, when a field theory has two vacua, the parameters are fixed so that they become degenerate. In regard to the CCP, we will see that the CC is nearly fixed to zero by generalizing Eq. (4) to the Wheeler–DeWitt wave function. In order to explain the positive small CC, we will give an argument based on the existence of the multiverse.

This paper is organized as follows. In Sect. 2, we give a few important mathematical preliminaries. In Sect. 3, we discuss the strong CP problem. In Sect. 4, we derive the MPP from Eq. (4). In Sect. 5, we consider the CCP by generalizing Eq. (4) to the Wheeler–DeWitt wave function. In Sect. 6, we give a summary and discussion.

 Mathematical preliminaries

In this section, we consider the singular behavior of in the limit, which plays a crucial role in the following discussion.

If is smooth, and has a stationary point , we obtain by using the saddle point approximation. Note that the right-hand side is suppressed by the factor .

If is smooth and monotonic in the 0$]]> region, and , we have where is a step function. The proof is as follows. By multiplying a test function with finite support to , and integrating from 0 to , we obtain Thus, one can see that Eq. (20) holds in the limit. Note that the right-hand side is proportional to , which is small compared with Eq. (19).

Similarly, we can obtain the following equation for that is smooth and monotonic in each of the regions, \lambda _{0}$]]> and : Note that the right-hand side is non-zero only when is not smooth at . We will call such a point a non-analytic point.

 Strong CP problem

In the QCD Lagrangian, there exists a CP-violating topological term Experimentally, there is a strong upper bound on [5, 43], which is unnaturally small. This is the so-called “strong CP problem.” We can naturally solve this problem as follows.

From the general argument of Sect. 1, the partition function is given by where is the energy density of the vacuum (see [44] for an example), and is the volume of the space time. If we consider such a final state with a finite winding number,2 using Eq. (19), we obtain3 By substituting this into Eq. (25), the partition function becomes This shows that the partition function is strongly dominated by and worlds, as in the many-world interpretation.

 The multiple point criticality principle

In this section, we derive the MPP by assuming that the potential of a scalar field has two minima at and , where we take . Here, is one of the coupling constants of the theory. We assume that two minima become degenerate when is equal to zero, and that the signature of is chosen as 0,\nonumber\\ V(\phi_{1}(\lambda),\lambda)&>V(\phi_{2}(\lambda),\lambda)\quad \hbox{ for }\lambda<0. \end{align}]]> See Fig. 2 for an example. Then, the true vacuum expectation value and the vacuum energy density are given by 0 \end{cases},\quad \varepsilon(\lambda)= \begin{cases} V(\phi_{2}(\lambda))&\hbox{ for }\lambda<0\\ V(\phi_{1}(\lambda))&\hbox{ for }\lambda>0.\end{cases}\end{equation}]]>

Schematic behavior of . The blue line corresponds to , and the green (red) line corresponds to 0$]]> .

According to Eq. (4), the partition function is given by

We assume that and are monotonic functions of in 0$]]> and respectively, and that their derivatives are not equal at . Because, for generic , has no singularity at , we can use Eq. (22): By substituting this into Eq. (31), we obtain Thus we have derived the MPP in the context of the multi-local action.

 Generalization to Wheeler–DeWitt wave function

In this section, we consider the generalization of Eq. (4) to the Wheeler–DeWitt wave function and the fine-tuning of the physical CC . Unfortunately, it is not easy to consider the negative region because we do not know what happens after the Big Crunch. On the other hand, in the positive region 0$]]>, we can consider the Wheeler–DeWitt wave function for the entire region of the radius of the universe, and examine which value of dominates. Therefore, in the following, we take only the region 0$]]> into account in the path integral.

Then the generalization of Eq. (4) to the Wheeler–DeWitt wave function is given by where is the Hamiltonian for the radius of the universe with the bare CC [36], represents the other degrees of freedom, that is, the matter and radiation including gravitons, and () is an initial (a final) state of the universe. (We have assumed that the initial universe has the radius .) The integration over comes from the path integral for the lapse function. In the following discussion, we assume where represents the large radius of the universe. Before considering the CCP, we first re-derive the results of the previous sections from Eq. (34).

Fixing other parameters than the cosmological constant

Equation (34) differs from Eq. (4) in that it contains the integration over the time . However, the results in the previous sections can also be obtained from Eq. (34) because, for a fixed value of , the integral is dominated by at which the radius of the universe becomes . More concretely, we have where we have omitted the integrations over and for simplicity. Here, satisfies the following Friedman equation and boundary condition: where is the volume of the space. We can understand Eq. (35) within the Born–Oppenheimer approximation [45] by assuming that the expansion rate is very small compared with the energy scale of the other degrees of freedom. See Appendix A for the details.

Using Eq. (35), we can re-derive the results of the previous sections. For a sufficiently large value of , we can replace the Hamiltonian by the vacuum energy , as in Eq. (8). Therefore, the right-hand side of Eq. (35) becomes where is a time such that the energy density of the state is sufficiently closed to that of the vacuum. Then, Eq. (34) can be written as Thus, by repeating the same argument, we can re-derive the results of the previous sections.

 Solving the cosmological constant problem

In order to fix , it is sufficient to consider the effective action for the radius because only the vacuum energy is relevant: where the effective Hamiltonian is Here, stands for the energy density of the matter and radiation including gravitons. One can see that plays the role of a potential of the radius of the universe. By inserting the complete set into Eq. (39), we obtain where is the zero-energy eigenstate, and represents the principal value. Here, we have used the following identity:4 The second term in Eq. (42) comes from the fact that we have chosen as the beginning of the universe. However, because the universe is well described classically by the Friedman equation, we expect that only the small region around is relevant in this integral. In Appendix B, we actually check this by using the WKB (Wentzel–Kramers–Brillouin) approximation. As a result, Eq. (42) can be approximated by Let us now evaluate by using the WKB approximation. The Wheeler–DeWitt wave function is given by [36] where is the classical momentum. For simplicity, we consider a matter-dominated universe, where is the total energy of the matter. Then, we can evaluate the exponent in Eq. (45): where is the radius of the universe at the time when the matter-dominated era starts. is a smooth and monotonic function of , which satisfies Thus, by substituting Eq. (47) into Eq. (45), and using Eq. (20), we obtain where is the zero-energy eigenstate with . By substituting this into Eq. (44), we have This indicates that the whole universe is described by the Wheeler–DeWitt wave function with . However, this result is inconsistent with cosmological observation [46], which supports a small but non-zero . In the next section, we discuss a possibility to explain the discrepancy by assuming the multiverse.

 Summary and discussion

We have proposed a new mechanism to solve the fine-tuning problem of the universe: Assuming a multi-local action, we have obtained the partition function Eq. (4) or its generalization to the Wheeler–DeWitt wave function Eq. (34). We have found that, in some phenomenologically interesting cases, there is a special point that strongly dominates in the partition function. This fact can be understood as the dynamical fine-tuning of the parameters. In particular, we have solved the strong CP problem and the CCP, and also derived the MPP. To obtain these results, we have assumed that the final state of the universe is fixed to a generic state. Although the justification of this assumption remains an open problem, it is interesting and remarkable that the different phenomena of field theory are explained by a unique mechanism.

Finally, let us discuss the possibility of obtaining a small but nonzero CC. As we will see in the following, we can obtain a fluctuation of the CC that is consistent with the observed value if we assume (i) the existence of the multiverse, and (ii) that only the region within the horizon is relevant to examine the partition function. Taking wormhole effects into consideration [34–37], we obtain the generalized partition function of the multiverse instead of Eq. (4): where is a coupling constant, and is the partition function of a single universe. We have assumed that all the universes are copies of our universe. Equation (51) indicates a new possibility to fix the parameter : it is fixed to the saddle point of even if itself does not have a strong peak as . can be expanded as From the path integral expression, can be expressed as the expectation value of the local action : Therefore, the fluctuation of around is given by Typically, because where we have assumed that the cut-off scale is , and that by dimensional analysis. Thus, one can see that is of order . This leads to the fluctuation of the vacuum energy density , which is typically where we have replaced with . Thus, if is , this fluctuation is consistent with the observed value. Although this conclusion is based on a few nontrivial assumptions, it is interesting that such a small CC can be obtained by maximizing the partition function as a function of the parameter.

In conclusion, the theory of the multi-local action is attractive and promising in solving the fine-tuning problem of the universe.

 Funding

Open Access funding: SCOAP3.

 Acknowledgments

This work is supported by Grants-in-Aid for Japan Society for the Promotion of Science (JSPS) Fellows No. 251107 (YH) and No. 271771 (KK).

 Appendix A. The Born–Oppenheimer approximation <p>In this appendix, we give a justification of Eq. (<xref rid="ptv168M35" ref-type="disp-formula">35</xref>) based on the Born–Oppenheimer approximation. Because the time scale of <inline-formula><tex-math notation="LaTeX" id="ImEquation176"><![CDATA[$a$]]></tex-math></inline-formula> is much longer than that of the matter and radiation, the total wave function can be obtained as follows.</p> <p><bold><italic>Step1</italic></bold>: We first solve the Schrödinger equation for the matter and radiation assuming that <inline-formula><tex-math notation="LaTeX" id="ImEquation177"><![CDATA[$a_{cl}(t)$]]></tex-math></inline-formula> is a slowly changing c-number function of <inline-formula><tex-math notation="LaTeX" id="ImEquation178"><![CDATA[$t$]]></tex-math></inline-formula>: <disp-formula id="ptv168M58"><label>(A1)</label><tex-math notation="LaTeX" id="DmEquation58"><![CDATA[\begin{equation} -i\frac{\partial}{\partial t}|\psi_{MR}(t)\rangle=\hat{H}_{MR}(a_{cl}(t))|\psi_{MR}(t)\rangle.\end{equation}]]></tex-math> </disp-formula> Then, the expectation value of the energy as a function of <inline-formula><tex-math notation="LaTeX" id="ImEquation179"><![CDATA[$a$]]></tex-math></inline-formula> is given: <disp-formula id="ptv168M59"><label>(A2)</label><tex-math notation="LaTeX" id="DmEquation59"><![CDATA[\begin{equation} E_{MR}(a,t):=\langle \psi_{MR}(t)|\hat{H}_{MR}(a)|\psi_{MR}(t)\rangle.\end{equation}]]></tex-math> </disp-formula> <bold><italic>Step2</italic></bold>: By using <inline-formula><tex-math notation="LaTeX" id="ImEquation180"><![CDATA[$E_{MR}(a,t)$]]></tex-math></inline-formula>, we solve the Schrödinger equation for <inline-formula><tex-math notation="LaTeX" id="ImEquation181"><![CDATA[$a$]]></tex-math></inline-formula>: <disp-formula id="ptv168M60"><label>(A3)</label><tex-math notation="LaTeX" id="DmEquation60"><![CDATA[\begin{equation} -i\frac{\partial}{\partial t}|\psi_{r}(t)\rangle=\left(\hat{H}_{G}+E_{MR}(\hat{a})\right)|\psi_{r}(t)\rangle. \end{equation}]]></tex-math> </disp-formula> Then, we identify <inline-formula><tex-math notation="LaTeX" id="ImEquation182"><![CDATA[$a_{cl}(t)$]]></tex-math></inline-formula> with the expectation value of the radius, <disp-formula id="ptv168M61"><label>(A4)</label><tex-math notation="LaTeX" id="DmEquation61"><![CDATA[\begin{equation} a_{cl}(t)=\langle \psi_{r}(t)|\hat{a}|\psi_{r}(t)\rangle. \end{equation}]]></tex-math> </disp-formula> By solving Eqs. (63)–(66) in a self-consistent manner, we obtain <disp-formula id="ptv168M62"><label>(A5)</label><tex-math notation="LaTeX" id="DmEquation62"><![CDATA[\begin{align} &\langle a|\otimes\langle f_{MR}|e^{-i\left(\hat{H}_{G}(\Lambda_{B})+\hat{H}_{MR}(\overrightarrow{\lambda};\hat{a})\right)t}|\epsilon\rangle\otimes |i_{MR}\rangle \nonumber\\ &\quad \sim\langle a|\psi_{r}(t)\rangle \langle f_{MR}|T\left\{\exp\left(-i\int_{0}^{t}dt'\hat{H}_{MR}(a_{cl}(t'))\right)\right\}|i_{MR}\rangle. \end{align}]]></tex-math> </disp-formula> Then, within the Born–Oppenheimer approximation, the Wheeler–DeWitt wave function is given by</p> <p><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="ptv168M63"/></p> <p>Here, because <inline-formula><tex-math notation="LaTeX" id="ImEquation183"><![CDATA[$\langle a|\psi _{r}(t)\rangle $]]></tex-math></inline-formula> has a peak at <inline-formula><tex-math notation="LaTeX" id="ImEquation184"><![CDATA[$a_{cl}(t)$]]></tex-math></inline-formula> as a function of <inline-formula><tex-math notation="LaTeX" id="ImEquation185"><![CDATA[$a$]]></tex-math></inline-formula>, the <inline-formula><tex-math notation="LaTeX" id="ImEquation186"><![CDATA[$t$]]></tex-math></inline-formula> integral of Eq. (A6) is dominated by <inline-formula><tex-math notation="LaTeX" id="ImEquation187"><![CDATA[$T_{a}$]]></tex-math></inline-formula> such that <disp-formula id="ptv168M64"><label>(A7)</label><tex-math notation="LaTeX" id="DmEquation64"><![CDATA[\begin{equation} a_{cl}(T_{a})\sim a. \end{equation}]]></tex-math> </disp-formula> Thus, we obtain Eq. (<xref rid="ptv168M35" ref-type="disp-formula">35</xref>).</p> </sec> </app> <app><title>Appendix B. Evaluation of the principal value <p>In this appendix, we evaluate <disp-formula id="ptv168M65"><label>(B1)</label><tex-math notation="LaTeX" id="DmEquation65"><![CDATA[\begin{equation} PV\int_{-\infty}^{+\infty}\frac{dE}{E}\langle a|E;\Lambda\rangle\langle E;\Lambda|\epsilon\rangle \end{equation}]]></tex-math> </disp-formula> by using the WKB approximation. he WKB solution of <inline-formula><tex-math notation="LaTeX" id="ImEquation188"><![CDATA[$\langle a|E;\Lambda \rangle $]]></tex-math></inline-formula> is given by <disp-formula id="ptv168M66"><label>(B2)</label><tex-math notation="LaTeX" id="DmEquation66"><![CDATA[\begin{equation} \langle a|E;\Lambda\rangle=M_{pl}\sqrt{\frac{a}{p_{cl}}}\exp\left(i\int^{a} da'p_{cl}(a')\right), \end{equation}]]></tex-math> </disp-formula> where <disp-formula id="ptv168M67"><label>(B3)</label><tex-math notation="LaTeX" id="DmEquation67"><![CDATA[\begin{equation} p_{cl}(a)=M_{pl}a^{2}\sqrt{2\left(\frac{\rho(a)}{6}-\frac{E}{a^{3}}\right)}:=M_{pl}a^{2}\sqrt{\frac{\tilde{\rho}(a)}{3}}. \end{equation}]]></tex-math> </disp-formula> Then, for a sufficiently large value of <inline-formula><tex-math notation="LaTeX" id="ImEquation189"><![CDATA[$a$]]></tex-math></inline-formula>, we have <disp-formula id="ptv168M68"><label>(B4)</label><tex-math notation="LaTeX" id="DmEquation68"><![CDATA[\begin{align} \int_{a_{M}}^{a} da p_{cl}(a) &=\frac{M_{pl}}{3^{\frac{3}{2}}}\left(a^{3}\sqrt{\tilde{\rho}(a)}-a_{M}^{3}\sqrt{\tilde{\rho}(a_{M})}+\frac{M-E}{\sqrt{\Lambda}}\log\left[\frac{a^{\frac{3}{2}}(\Lambda+\sqrt{\Lambda\tilde{\rho}(a)})}{a_{M}^{\frac{3}{2}}(\Lambda+\sqrt{\Lambda\tilde{\rho}(a_{M})})}\right]\right)\nonumber\\ &:=\frac{M_{pl}a^{3}}{3^{\frac{3}{2}}}g(\Lambda,E,a) \underset{a\gg a_{M}}{\simeq}\frac{M_{pl}a^{3}}{3^{\frac{3}{2}}}\sqrt{\tilde{\rho}(a)}. \end{align}]]></tex-math></disp-formula></p> <p>By substituting Eq. (<xref rid="ptv168M66" ref-type="disp-formula">B2</xref>) and Eq. (<xref rid="ptv168M68" ref-type="disp-formula">B4</xref>) into Eq. (<xref rid="ptv168M65" ref-type="disp-formula">B1</xref>), we obtain <disp-formula id="ptv168M69"><label>(B5)</label><tex-math notation="LaTeX" id="DmEquation69"><![CDATA[\begin{equation} PV\int_{-\infty}^{\infty}\frac{dE}{E} M_{pl}\sqrt{\frac{a}{p_{cl}}}\exp\left(i\frac{M_{pl}a^{3}}{3^{\frac{3}{2}}}\sqrt{\tilde{\rho}(a)}\right)\langle E;\Lambda|\epsilon\rangle. \end{equation}]]></tex-math> </disp-formula> By expanding the exponent around <inline-formula><tex-math notation="LaTeX" id="ImEquation190"><![CDATA[$E=0$]]></tex-math></inline-formula>, we have <disp-formula id="ptv168M70"><label>(B6)</label><tex-math notation="LaTeX" id="DmEquation70"><![CDATA[\begin{align} &M_{pl}\sqrt{\frac{a}{p_{cl}}}\exp\left(i\frac{M_{pl}a^{3}}{3^{\frac{3}{2}}}\sqrt{\rho(a)}-i\frac{M_{pl}E}{3^{\frac{3}{2}}\sqrt{\rho(a)}}+{\mathcal{O}}(E^{2})\right)\nonumber\\ &\qquad=\langle a|0;\Lambda\rangle\exp\left(-i\frac{M_{pl}E}{3^{\frac{3}{2}}\sqrt{\rho(a)}}+{\mathcal{O}}(E^{2})\right). \end{align}]]></tex-math> </disp-formula> Therefore, only the region <disp-formula id="ptv168M71"><label>(B7)</label><tex-math notation="LaTeX" id="DmEquation71"><![CDATA[\begin{equation} |E|\lesssim\frac{\sqrt{\rho(a)}}{M_{pl}}\sim\frac{\sqrt{\Lambda}}{M_{pl}} \end{equation}]]></tex-math> </disp-formula> contributes to the integral.<sup><xref ref-type="fn" rid="fn5">5</xref></sup> Therefore, it is self-consistent to show <inline-formula><tex-math notation="LaTeX" id="ImEquation191"><![CDATA[$\Lambda =0$]]></tex-math></inline-formula> by using only the zero-energy eigenstate <inline-formula><tex-math notation="LaTeX" id="ImEquation192"><![CDATA[$|0\rangle $]]></tex-math></inline-formula>.</p> </sec> </app> <app><title>Appendix C. Symmetry enhancement <p>In this appendix, we study a mechanism by which symmetry is enhanced. In particular, we consider a scalar field <inline-formula><tex-math notation="LaTeX" id="ImEquation193"><![CDATA[$\phi $]]></tex-math></inline-formula> with the effective potential <disp-formula id="ptv168M72"><label>(C1)</label><tex-math notation="LaTeX" id="DmEquation72"><![CDATA[\begin{equation} V(\phi)=a_{1}\phi+\frac{a_{2}}{2}\phi^{2}+a_{3}\phi^{3}+\frac{a_{4}}{4}\phi^{4}, \end{equation}]]></tex-math> </disp-formula> where <inline-formula><tex-math notation="LaTeX" id="ImEquation194"><![CDATA[$a_{4}$]]></tex-math></inline-formula> is fixed to a positive value so that the system is bounded below. We can eliminate <inline-formula><tex-math notation="LaTeX" id="ImEquation195"><![CDATA[$a_{3}\phi ^{3}$]]></tex-math></inline-formula> by shifting the field, <inline-formula><tex-math notation="LaTeX" id="ImEquation196"><![CDATA[$\phi \rightarrow \phi +\phi _{0}$]]></tex-math></inline-formula>, and the potential becomes <disp-formula id="ptv168M73"><label>(C2)</label><tex-math notation="LaTeX" id="DmEquation73"><![CDATA[\begin{equation} V(\phi)=a_{1}\phi+\frac{a_{2}}{2}\phi^{2}+\frac{a_{4}}{4}\phi^{4}. \end{equation}]]></tex-math> </disp-formula> In Fig. <xref ref-type="fig" rid="ptv168FC1">C.1</xref>, we show the typical shapes of <inline-formula><tex-math notation="LaTeX" id="ImEquation197"><![CDATA[$V(\phi )$]]></tex-math></inline-formula>. In the following discussion, we fix <inline-formula><tex-math notation="LaTeX" id="ImEquation198"><![CDATA[$a_{2}$]]></tex-math></inline-formula> and vary <inline-formula><tex-math notation="LaTeX" id="ImEquation199"><![CDATA[$a_{1}$]]></tex-math></inline-formula>. We denote the negative (positive) vacuum expectation value by <inline-formula><tex-math notation="LaTeX" id="ImEquation200"><![CDATA[$\phi _{1}(a_{1})$]]></tex-math></inline-formula> <inline-formula><tex-math notation="LaTeX" id="ImEquation201"><![CDATA[$(\phi _{2}(a_{1}))$]]></tex-math></inline-formula>. According to the general argument in Sect. 1, the partition function is <disp-formula id="ptv168M74"><label>(C3)</label><tex-math notation="LaTeX" id="DmEquation74"><![CDATA[\begin{equation} Z= \int da_{1}f(a_{1})\exp\left(-i\varepsilon(a_{1})V_{4}\right)\langle f|\psi\big(t^{*};a_{ 1}\big)\rangle, \end{equation}]]></tex-math> </disp-formula> where <inline-formula><tex-math notation="LaTeX" id="ImEquation202"><![CDATA[$\varepsilon (a_{1})$]]></tex-math></inline-formula> is the vacuum energy density of this system, which is given by <disp-formula id="ptv168M75"><label>(C4)</label><tex-math notation="LaTeX" id="DmEquation75"><![CDATA[\begin{equation} \varepsilon(a_{ 1})=\begin{cases} V(\phi_{1}(a_{1}))&\quad (\hbox{for } a_{1}>0),\\ V(\phi_{2}(a_{1}))&\quad (\hbox{for } a_{1}<0). \end{cases} \end{equation}]]></tex-math> </disp-formula> As a result, <inline-formula><tex-math notation="LaTeX" id="ImEquation203"><![CDATA[$a_{1}=0$]]></tex-math></inline-formula> is apparently the non-analytic point of the vacuum energy <inline-formula><tex-math notation="LaTeX" id="ImEquation204"><![CDATA[$\varepsilon (a_{1})$]]></tex-math></inline-formula>. See Fig. <xref ref-type="fig" rid="ptv168FC2">C.2</xref> for an example. Thus, by using Eq. (<xref rid="ptv168M22" ref-type="disp-formula">22</xref>), we obtain <disp-formula id="ptv168M76"><label>(C5)</label><tex-math notation="LaTeX" id="DmEquation76"><![CDATA[\begin{equation} e^{-i\varepsilon(a_{1})V_{4}}\sim-\frac{ie^{-i\varepsilon(0)V_{4}}}{V_{4}}\times\left[\left(\frac{V(\phi_{1})}{da_{1}}\right)^{-1}\Biggl|_{a_{1}=0+}-\left(\frac{V(\phi_{2})}{da_{1}}\right)^{-1}\Biggl|_{a_{1}=0-}\right]\delta(a_{1}). \end{equation}]]></tex-math> </disp-formula> By substituting this into Eq. (<xref rid="ptv168M74" ref-type="disp-formula">C3</xref>), we have <disp-formula id="ptv168M77"><label>(C6)</label><tex-math notation="LaTeX" id="DmEquation77"><![CDATA[\begin{equation} Z\sim\frac{1}{V_{4}}e^{-i\varepsilon(0)V_{4}}\langle f|\psi\big(t^{*};0\big)\rangle. \end{equation}]]></tex-math> </disp-formula> This indicates that the symmetric effective potential is favored by the multi-local action. <fig id="ptv168FC1"><label>Fig. C.1.</label> <caption><p>Typical shapes of <inline-formula><tex-math notation="LaTeX" id="ImEquation205"><![CDATA[$V(\phi )$]]></tex-math></inline-formula>. The left (right) panel shows the <inline-formula><tex-math notation="LaTeX" id="ImEquation206"><![CDATA[$a_{2}>0$]]></tex-math></inline-formula> <inline-formula><tex-math notation="LaTeX" id="ImEquation207"><![CDATA[$(<0)$]]></tex-math></inline-formula> case.</p></caption> <graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="ptv16803"/> </fig> <fig id="ptv168FC2"><label>Fig. C.2.</label> <caption><p>The minimum of the potential as a function of <inline-formula><tex-math notation="LaTeX" id="ImEquation208"><![CDATA[$a_{1}$]]></tex-math></inline-formula>.</p></caption> <graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="ptv16804"/> </fig></p> </sec> </app> </app-group> <fn-group> <fn id="fn1"><label>1</label><p>Here, for simplicity, we have assumed that <inline-formula><tex-math notation="LaTeX" id="ImEquation209"><![CDATA[$a_{cl}(t)$]]></tex-math></inline-formula> is already given. See Sect. <xref ref-type="sec" rid="s5">5</xref> for a more concrete argument.</p></fn> <fn id="fn2"><label>2</label><p>If <inline-formula><tex-math notation="LaTeX" id="ImEquation210"><![CDATA[$|f\rangle $]]></tex-math></inline-formula> is the <inline-formula><tex-math notation="LaTeX" id="ImEquation211"><![CDATA[$n$]]></tex-math></inline-formula> vacuum <inline-formula><tex-math notation="LaTeX" id="ImEquation212"><![CDATA[$|n\rangle $]]></tex-math></inline-formula> with a large value of <inline-formula><tex-math notation="LaTeX" id="ImEquation213"><![CDATA[$n\sim \Lambda _{{\rm QCD}}^{4}V_{4}$]]></tex-math></inline-formula>, the exponent of the integrand in Eq. (<xref rid="ptv168M25" ref-type="disp-formula">25</xref>) becomes stationary at the point where <inline-formula><tex-math notation="LaTeX" id="ImEquation214"><![CDATA[$\sin \theta \sim n/\left (\Lambda _{{\rm QCD}}^{4}V_{4}\right )$]]></tex-math></inline-formula>.</p></fn> <fn id="fn3"><label>3</label><p>If <inline-formula><tex-math notation="LaTeX" id="ImEquation215"><![CDATA[$k$]]></tex-math></inline-formula> is finite in Eq. (<xref rid="ptv168M19" ref-type="disp-formula">19</xref>), we have a function of width <inline-formula><tex-math notation="LaTeX" id="ImEquation216"><![CDATA[$1/\sqrt {kg''(\lambda )}$]]></tex-math></inline-formula> instead of the delta function. Then, the width <inline-formula><tex-math notation="LaTeX" id="ImEquation217"><![CDATA[$\Delta \theta $]]></tex-math></inline-formula> is given by <inline-formula><tex-math notation="LaTeX" id="ImEquation218"><![CDATA[$\sqrt {1/\left (V_{4}\Lambda _{QCD}^{4}\right )}\sim 10^{-80}/\left (\sqrt {V_{4}H_{0}^{4}}\right )$]]></tex-math></inline-formula>, where <inline-formula><tex-math notation="LaTeX" id="ImEquation219"><![CDATA[$H_{0}$]]></tex-math></inline-formula> the present Hubble constant.</p></fn> <fn id="fn4"><label>4</label><p>The derivation is as follows: We can neglect <inline-formula><tex-math notation="LaTeX" id="ImEquation220"><![CDATA[$e^{-iEt}$]]></tex-math></inline-formula> by introducing the adiabatic factor <inline-formula><tex-math notation="LaTeX" id="ImEquation221"><![CDATA[$E\rightarrow E-i\epsilon $]]></tex-math></inline-formula>. Thus, by multiplying a smooth test function <inline-formula><tex-math notation="LaTeX" id="ImEquation222"><![CDATA[$F(E)$]]></tex-math></inline-formula> with finite support to the left-hand side, and integrating over <inline-formula><tex-math notation="LaTeX" id="ImEquation223"><![CDATA[$E$]]></tex-math></inline-formula>, we have <inline-formula><tex-math notation="LaTeX" id="ImEquation224"><![CDATA[$ \int _{-\infty }^{\infty }dE\frac {-1}{-i\left (E-i\epsilon \right )}F(E)=\pi F(i\epsilon )-PV\int _{-\infty }^{\infty }dE\frac {F(E)}{E-i\epsilon }. $]]></tex-math></inline-formula> Therefore, in the <inline-formula><tex-math notation="LaTeX" id="ImEquation225"><![CDATA[$\epsilon \rightarrow 0$]]></tex-math></inline-formula> limit, we obtain Eq. (<xref rid="ptv168M43" ref-type="disp-formula">43</xref>). Here, note that, if we also include the negative region in the time integral, we obtain the delta function <inline-formula><tex-math notation="LaTeX" id="ImEquation226"><![CDATA[$2\pi \delta (E)$]]></tex-math></inline-formula> instead of Eq. (<xref rid="ptv168M43" ref-type="disp-formula">43</xref>). This leads to the ordinary Wheeler–DeWitt state <inline-formula><tex-math notation="LaTeX" id="ImEquation227"><![CDATA[$|0;\Lambda \rangle $]]></tex-math></inline-formula>.</p></fn> <fn id="fn5"><label>5</label><p>More concretely, by substituting Eq. (<xref rid="ptv168M70" ref-type="disp-formula">B6</xref>) into Eq. (<xref rid="ptv168M65" ref-type="disp-formula">B1</xref>), and neglecting the <inline-formula><tex-math notation="LaTeX" id="ImEquation228"><![CDATA[$E$]]></tex-math></inline-formula> dependence of <inline-formula><tex-math notation="LaTeX" id="ImEquation229"><![CDATA[$\langle E;\Lambda |\epsilon \rangle $]]></tex-math></inline-formula>, we obtain <inline-formula><tex-math notation="LaTeX" id="ImEquation230"><![CDATA[$ \langle a|0;\Lambda \rangle PV\int _{-\infty }^{a^{3}\rho (a)}\frac {dE}{E}\exp \left (-i\frac {M_{pl}E}{3^{\frac {3}{2}}\sqrt {\rho (a)}}\right )=\langle a|0;\Lambda \rangle PV\int _{-\infty }^{k}\frac {dx}{x}e^{-ix}\underset {k\rightarrow +\infty }{\rightarrow }-i\pi \langle a|0;\Lambda \rangle , $]]></tex-math></inline-formula> where <inline-formula><tex-math notation="LaTeX" id="ImEquation231"><![CDATA[$k:=M_{pl}a^{3}\sqrt {\rho (a)}/3^{\frac {3}{2}}$]]></tex-math></inline-formula>.</p></fn> </fn-group> <ref-list> <title>References Aad G. et al. 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