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Standard Model and the Planck Scale Dec 12 2015 at NCTS Hikaru Kawai (Kyoto Univ.) In collaboration with K. Kawana, Y. Hamada, K. Oda, and S. C. Park. Outline The observed values of the Higgs and top masses indicate that SM is valid up to the string/Planck scale and no new physics is forced to appear. In string theory, non-supersymmetric tachyon-free vacua appear more frequently than supersymmetric ones. The simplest guess is that SM is directly connected to string theory at the string scale. Only minor modifications come in between the SM and the string scale. However we need some mechanisms by which the fine tunings of the Higgs mass and the cosmological constant are performed. The naturalness problem Suppose the underlying fundamental theory, such as string theory, has the momentum scale mS and the coupling constant gS . Then, by dimensional analysis and the power counting of the couplings, the parameters of the low energy effective theory are given as follows: naturalness problem (cont.’d) 2 gS GN ∼ 2 . mS dimension -2 (Newton constant) g1 , g2 , g3 ∼ g S , dimension 0 (gauge and Higgs couplings) H ∼ 2 gS . tree dimension 2 (Higgs mass) 2 mH ∼ 0 g S mS . 2 2 unnatural ! → mH ∼ 100GeV ≪ g S mS ∼ 10 GeV 2 2 2 2 18 2 dimension 4 (vacuum energy or cosmological constant) ∼ 0 g s mS . 2 unnatural ! !→ ∼ 2 ~ 3meV 4 ≪ mS ∼ 10 GeV 4 18 4 4 SUSY as a solution to the naturalness problem Bosons and fermions cancel the radiative corrections: + bosons fermions + ⇒ mH 0. 2 2 ⇒ mH ∼ M SUSY . 2 ⇒ 0. ⇒ ∼ 4 M SUSY . However, SUSY must be spontaneously broken at some momentum scale MSUSY , below which the cancellation does not work. (cont’d) Therefore, if MSUSY is close to mH , the Higgs mass is naturally understood, although the cosmological constant is still a big problem. However, no signal of the SUSY partners is observed in the LHC below 1 TeV. It is better to think about the other possibilities. Possibility of desert The first thing we should know is whether the SM is valid to the string/Planck scale or it breaks down below the scale. If it is the former case, there is a possibility that the SM is directly connected to the Planck scale physics. Is the SM valid to the Planck/string scale? Y. Hamada, K. Oda and HK: arXiv:1210.2358 , arXiv:1305.7055 , 1308.6651 In order to answer the question, we consider the bare Lagrangian of SM with cutoff momentum Λ, . and determine the bare parameters in such a way that the observed parameters are recovered. If no inconsistency arises, it means that the SM can be valid to the energy scale Λ. Bare parameters as a function of the cutoff [1405.4781] 2-loop RGE mHiggs =125.6 GeV The MC mass should be distinguished from the QCD pole mass. The latter has a rather large error. 1 2 I1 = 2 16 [Hamada, Oda, HK ,1210.2538, 1308.6651] Higgs self coupling λ λ takes a small value at high energy scale. [1405.4781] If Mt=171GeV, λ=βλ=0 at 1017-18GeV. [Hamada, Oda, HK,1210.2538, 1308.6651] Bare mass If Mt=170GeV, the bare mass becomes 0 at MP. [Hamada, Oda, HK,1210.2538] Triple coincidence Three quantities, B , B , mB become close to zero around the Planck/string scale. This means that the Higgs potential becomes zero around the Planck/string scale. V 𝑚𝑠 𝜙 Such situation was predicted by Froggatt and Nielsen ’95. Multiple Point Criticality Principle (MPP) Review of MPP In the ordinary quantum theory, we start with the path integral of the form of canonical ensemble: d exp S . On the other hand in the statistical mechanics, the most fundamental concept is the micro canonical ensemble Z d H E , and the canonical ensemble follows in the thermodynamic limit: d H E d exp H / T . Important point: In the micro canonical ensemble we control extensive variables, while in the canonical ensemble we control intensive variables. Example: Consider a given number of water molecules put in a box with a given volume. In the micro canonical ensemble the energy is controlled. p E L vapor S G 𝑬𝒄 vapor water T If 𝑬 < 𝑬𝒄 , the system is automatically on the critical line. The coexisting phase is obtained without fine tuning. Intensive variables are tuned “miraculously”. Let’s imagine that quantum theory is defined by a path integral of the micro canonical ensemble type d S C . It is equivalent to the ordinary quantum mechanics, if the space-time volume is sufficiently large. If C is in some region, the coupling constants are automatically tuned in such a way that two vacua degenerates and coexist. In other words, it is natural to have degenerate vacua. V 𝑚𝑠 𝜙 Working hypotheses It is not clear how much we should trust, but let’s assume MPP as a working hypothesis. MPP: Coupling constants are tuned so that the vacuum is on the boundary of various phases. We can also imagine another working hypothesis. Maximum Entropy Principle (MEP): Coupling constants are tuned so that the entropy of the universe becomes maximum. Probably many authors have made similar statements, and there would be many possible arguments. For example, suppose that we pic up a universe randomly from the multiverse. Then the most probable universe is expected to be the one that has the maximum entropy. (T. Okada and HK) Higgs inflation Question: Can the Higgs field play the roll of inflaton? Hamada, Oda, HK: arXiv:1308.6651 Hamada, Oda, Park and HK: arXiv: 1403.5043 Higgs inflation (1) modest approach We trust the effective potential only below the string scale, and try to make bounds on the parameters. Hamada, Oda and HK: arXiv: 1308.6651 (2) optimistic approach We trust the potential including the string scale. We assume that nature does fine tunings if they are necessary. We introduce a non-minimal coupling. Hamada, Oda, Park and HK, arXiv: 1403.5043 Cook. Krauss, Lawrence , Long and Sabharwal, arXiv: 1403.4971 Bezrukov and Shaposhnikov, arXiv: 1403.6078 Higgs potential V 4 4 mH = 125.6 GeV 𝜑[GeV] Modest approach We trust the above effective potential only below some scale Λ . Above Λ it is not described by SM, but we need string theory. string SM 𝚲 Necessary conditions At present we do not have enough knowledge above Λ. But there is a possibility that the Higgs potential in the stringy region has a proper shape, and the inflation occurs in this region. In order for this scenario to work, the necessary conditions are d VSM 0 for , d VSM V*. where 3 AS r* 4 65 4 V* r* M P 1.3 10 GeV . 2 0.11 2 mH 125.6 GeV r 0.1 𝑟 ∼ 0.01 is possible if the cutoff scale is ∼ 1017 GeV . Optimistic approach We assume that the Higgs potential can be trusted up to the inflection point region. The naïve guess is that inflation occurs if the parameters are tuned such that the inflection point almost becomes a saddle point. 𝜑[GeV] However, this inflection point can not produce a realistic inflation: sufficient 𝑁∗ in this region ⇒ 𝜑∗ ∼ saddle point ⇒ small 𝜀 ⇒ too large 𝐴𝑆 ∼ 𝑉∗ /𝜀 𝜑[GeV] Non minimal Higgs inflation with flat potential We introduce a non-minimal coupling R 2 as the original Bezrukov-Shaposhnikov’s. Then a realistic Higgs inflation is possible. In the Einstein frame the effective potential becomes V h , h 𝜑[GeV] h 1 h / MP 2 2 . 𝝃 need not be very large r* 4 V* 1.3 10 GeV . 0.11 65 𝑉 ∼ 𝜆 𝜑ℎ 𝜑ℎ4 ∼ 𝜆 𝜑ℎ 𝑀𝑃4 /𝜉 2 . What is new here is that 𝜆 𝜑 becomes small around the string scale: 𝜆 𝜑ℎ ∗ ∼ 10−6 . This allows rather small values for 𝜉 ( 𝜉 ∼ 10) . realistic values of 𝒓 and 𝒏𝑺 𝜇𝑚𝑖𝑛 𝑐 = 𝜉 𝑀𝑃 𝜆′ 𝜇𝑚𝑖𝑛 = 0 For example, 𝑚𝐻 = 125.6GeV ⇒ 𝜇𝑚𝑖𝑛 ∼ 5.5 × 1017 GeV Hamada, Oda, Park, HK: arXiv:1408.4864 What we have seen so far. • All the coupling constants are small at the string scale. It may suggest that our vacuum is close to one of the perturbative vacua of string theory. • Higgs potential becomes almost flat around the string scale. −6 4 −6 4 V ∼ 10 𝜑 ∼ 10 𝑚𝑠 • This suggests that the Higgs potential is zero in the string tree level and is generated by the 1-loop correction, because the string 1-loop correction is proportional to 𝐶𝑙𝑜𝑜𝑝 = 𝑆9 2 2𝜋 10 ∼ 10−7 . • Higgs inflation is possible if one of the parameters is tuned so that the potential nearly has a saddle point. • Such fine tuning might be understood by MEP. inflation ⇒ large entropy Can we say something beyond 𝑀𝑃 by string theory? Higgs potential in string theory Higgs potential for large field values can be estimated, and turns out to be compatible with MPP. Hamada, Oda , HK: arXiv: 1501.04455 Higgs potential in string theory To be concrete we take heterotic string, and consider a 4 dimensional perturbative vacuum that is nonsupersymmetric and tachyon-free. Because the Higgs particle is a real scalar and massless at the tree level, its emission vertex is written as 𝑉 𝑘 = 𝑂(𝑧, 𝑧)𝑒 𝑖𝑘⋅𝑋 , where 𝑂(𝑧, 𝑧) is a real (1,1) operator. The Higgs potential for the field value 𝜙 can be obtained by evaluating the partition function for the world sheet action . flat potential in tree level ⇔ exactly marginal A typical exactly marginal operator is given as 𝑂 𝑧, 𝑧 = 𝑂 where 𝑂 1,0 and 𝑂 1,0 𝑧 ×𝑂 0,1 are (1,0) and (0,1) operators. 0,1 𝑧 , Then by using bosonization 𝑂 1,0 𝑧 = 𝜕𝑌(𝑧) , 𝑂 0,1 𝑧 = 𝜕𝑍 𝑧 , we can evaluate the partition function. It is not clear whether this is completely general or not, but we consider this form. Examples of factorized 𝑶(𝒛, 𝒛) ★ Higgs comes from gauge-Higgs unification. Emission vertex of gauge field 𝐴𝜇𝑎 is 𝜕𝑧 𝑋𝜇 𝑗 𝑎 𝑧 𝑒 𝑖𝑘⋅𝑋 . Then the emission vertex of 𝑎 𝐴5 is 𝜕𝑧 𝑋 5 𝑗 𝑎 𝑧 𝑒 𝑖𝑘⋅𝑋 . ★ Generic cases in the Fermionic construction. ★ Higgs comes from an untwisted sector in the orbifold construction. This is rather generic. Simplest example - radion potentialAs the simplest example we consider the case 𝑂 𝑧, 𝑧 = 𝜕𝑧 𝑋 5 𝜕𝑧 𝑋 5 , where 𝑋 5 is a compactified dimension with the compactification radius 𝐿. Then the world sheet action to consider is 𝑆 = 𝑆0 + 𝜙 𝑆0 = 𝑑2 𝑧 𝑂(𝑧, 𝑧) = 𝑑2 𝑧(1 + 𝜙)𝜕𝑧 𝑋 5 𝜕𝑧 𝑋 5 + ⋯ . 𝑑2 𝑧𝜕𝑧 𝑋 5 𝜕𝑧 𝑋 5 This comes back to the original form 𝑆0 by the redefinition 𝑋′ 5 = 1 + 𝜙 𝑋5. But the compactification radius is changed from 𝐿 to 𝐿′= 1 + 𝜙 𝐿. The radion potential for large field can be evaluated by considering the dimensional reduction of 5D gravity to 4D: 𝐶: 5D cosmological constant 𝑆𝑒𝑓𝑓 = 5 𝑔MN 𝑑 5 𝑥 −𝑔 𝑔𝜇𝜈 = 0 5 (ℛ 5 − 𝒞) + ⋯ . 0 ′ , 𝐿 = ′2 𝐿 1 + 𝜙𝐿 . ⇒ 𝜇𝜈 𝑆𝑒𝑓𝑓 = 1𝑔 ′ ′ 𝑑 𝑥 −𝑔𝐿′(ℛ − 𝜕 𝐿 𝜕 𝐿 − 𝒞) + ⋯ . 𝜇 𝜈 2 𝐿′2 4 Converting it to the Einstein frame 𝑔𝜇𝜈 = ′ 𝐸 𝐿 𝑔𝜇𝜈 , we have 𝑆𝑒𝑓𝑓 = 𝐿′= 1 + 𝜙(𝑥) 𝐿 𝐸𝜇𝜈 1 𝑔 𝒞 4 𝐸 ′ ′ 𝐸 𝑑 𝑥 −𝑔 (ℛ − 𝜕 𝐿 𝜕 𝐿 − ) + ⋯ . 𝜇 𝜈 2 𝐿′2 𝐿′ The Einstein frame potential is runaway for large radius if cosmological constant in 5D is positive. This is true for all orders. It follows only from the low energy effective action in 5D. General 𝟏, 𝟎 × 𝟎, 𝟏 case Background for 𝜕𝑧 𝑌𝜕𝑧 𝑍 corresponds to the Lorentz boost for the momentum lattice for Y and Z. 3cases: runaway, periodic, chaotic In the runaway case, a new space dimension is generated and opened up from the internal degree of freedom. In string theory there is no essential difference between internal degrees of freedom and the spacetime coordinates. 𝑉 4D 5D 𝜙 A toy model We consider 9D toy model instead of 4D theory. 9D string obtained by compactifying one direction (the ninth direction) of the10D 𝑺𝑶 𝟏𝟔 × 𝑺𝑶(𝟏𝟔) non-supersymmetric heterotic string on S1 with radius 𝑟. The ninth component of the 10D gauge field becomes a scalar, whose emission vertex is Example of GHU 𝑎 9 𝑎 𝑖𝑘⋅𝑋 𝑉 = 𝜕𝑧 𝑋 𝑗 𝑧 𝑒 , where 𝑗 𝑎 (𝑧) is the 𝑆𝑂 16 × 𝑆𝑂(16) current. We then introduce a background 𝐴 for one component 𝑎. One-loop potential in Jordan frame periodic in A increasing in r One-loop potential in Einstein frame Runaway for large r True for all orders We have seen that we have a runaway vacuum in addition to our vacuum. Eternal inflation at domain wall Domain wall between two vacua: ✴ For a given random initial condition. If the curvature at the maximum is of order one in the Planck unit, the domain wall is wide enough that ✴ DW supports eternal inflation. ✴ A solution to horizon problem. Eternal inflation at false vacuum is also possible. Higgs inflation works. Need another inflaton. Solution to CC problem If we assume MPP, our vacuum should degenerate with the runaway vacuum, which has vanishing CC. So the CC of our vacuum should be zero. 𝑉 𝜙 This might be too good. CC of our universe is not exactly zero ~ 2.2 𝑚𝑒𝑉 . 4 Quantum fluctuation of action In general a system follows quantum mechanics. In the classical limit, it is described by the least action principle. However the value of the action is not completely determined because of the quantum fluctuation. Classical action is defined only within this error. The fluctuation is evaluated as 2 (Δ𝑆) ∼ 4 𝑑 𝑥ℒ 𝑥 4 𝑑 𝑦ℒ 𝑦 (𝜕𝜙)2 ∼𝑉× 4 𝑀𝑃 (𝜕𝜙)2 . 𝑉: space-time volume The classical Lagrangian density is determined up to an ambiguity Δℒ ∼ 2 𝑀𝑃 𝑉 . If 𝑉 = ∞, Δℒ = 0. For a de Sitter space with Hubble parameter 𝐻, −4 it is natural to take 𝑉 ∼ 𝐻 . So we have Δℒ ∼ 2 𝑀𝑃 ⋅𝐻 . 2 In particular the CC is determined up to the 2 2 4 error ∼ 𝑀𝑃 ⋅ 𝐻 ∼ 𝑚𝑒𝑉 . Inflation as the string scale physics Question: If everything is in string scale, can the inflation occur? It is possible if some number of fine tunings are allowed. Hamada, Kawana, HK: arXiv: 1507.03106 In terms of dimensionless fields, the low energy effective action of string theory is typically 𝜑 can be any field that becomes S= 𝑚𝑠8 𝑔𝑠2 + = = 𝑑10 𝑥 𝑔 𝐴 𝜑 𝑅 𝑚𝑠8 𝑔𝑠2 𝑑10 𝑥 𝑔𝐵(𝜑) 𝜕𝜑 𝑚𝑠8 𝑉6 2 𝑔𝑠 + scalar in 4D. We assume that the tree-level potential is zero. + 𝐶𝑙𝑜𝑜𝑝 𝑚𝑆10 𝑑10 𝑥 𝑔𝑉(𝜑) + ⋯ 4 𝐶𝑙𝑜𝑜𝑝 = 𝑑 𝑥 𝑔𝐴(𝜑)𝑅 𝑚𝑠8 𝑉6 2 𝑔𝑠 𝑀𝑃2 2 𝑑 4 𝑥 𝑔𝐵(𝜑) 𝜕𝜑 ∼ 10−7 in 10D 4 + 𝐶𝑙𝑜𝑜𝑝 𝑚10 𝑉 𝑑 𝑥 𝑔𝑉(𝜑) + ⋯ 𝑠 6 2 𝑚 𝑠 2 𝑀𝑃 = 2 (𝑚𝑠6 𝑉6 ) 𝑔𝑠 4 𝑑 𝑥 𝑔𝐴(𝜑)𝑅 +𝑀𝑃2 𝑑 4 𝑥 𝑔𝐵(𝜑) 𝜕𝜑 2 𝑆9 2 2𝜋 10 2 + 𝐶𝑙𝑜𝑜𝑝 𝑔𝑠2 𝑀𝑃2 𝑚𝑠2 𝑑 4 𝑥 𝑔𝑉(𝜑) + ⋯ In the Einstein frame we have S= 𝑀𝑃2 𝑑 4 𝑥 𝑔 𝑅 +𝑀𝑃2 𝑑 4 𝑥 𝑔𝐶(𝜑) 𝜕𝜑 2 + 𝐶𝑙𝑜𝑜𝑝 𝑔𝑠2 𝑀𝑃2 𝑚𝑠2 𝑑 4 𝑥 𝑔𝑈(𝜑) + ⋯ In terms of the canonical-like field this becomes S= 2 𝑀𝑃 𝑑4 𝑥 𝑔 𝑅 +𝑀𝑃2 𝑑 4 𝑥 𝑔 𝜕𝜒 2 + 𝐶𝑙𝑜𝑜𝑝 𝑔𝑠2 𝑀𝑃2 𝑚𝑠2 𝑑 4 𝑥 𝑔 𝑊(𝜒) + ⋯ If we do a fine tuning we have a saddle point for 𝑾 𝝌 : W 𝜒 ~𝑂(1) S= 𝑀𝑃2 𝑑4 𝑥 𝑔 𝑅 +𝑀𝑃2 4 𝑑 𝑥 𝑔 𝜕𝜒 2 + 𝐶𝑙𝑜𝑜𝑝 𝑔𝑠2 𝑀𝑃2 𝑚𝑠2 4 𝑑 𝑥 𝑔 𝑊(𝜒) + ⋯ 𝐶𝑙𝑜𝑜𝑝 ∼ 10−7 Higher derivatives in the effective action can be consistently neglected because 𝑯~ 𝑪𝒍𝒐𝒐𝒑 𝒈𝒔 𝒎𝒔 ≪ 𝒎𝒔 . If we assume the value of the potential matches with that of the SM, we have 𝑪𝒍𝒐𝒐𝒑 𝒈𝟐𝒔 𝑴𝟐𝑷 𝒎𝟐𝒔 ∼ 𝟏𝟎𝟔𝟒 𝐆𝐞𝐕 𝟒 . 2 𝑚 𝑠 2 𝑀𝑃 = 2 (𝑚𝑠6 𝑉6 ) 𝑔𝑠 Then we obtain 𝒈𝒔 𝒎𝒔 ∼ 𝟏𝟎𝟏𝟔.𝟓 GeV , 𝒎𝟔𝒔 𝑽𝒔 = 𝒈𝟒𝒔 𝑴𝟐𝑷 ~ 𝟐 𝟐 𝒈𝑺 𝒎𝑺 𝒈𝟒𝒔 ⋅ 𝟏𝟎𝟑 ~𝟏. not bad Inflation at higher criticality If one does not dislike the fine tunings, we can tune such that 𝑊 = 𝑊0 𝜒 1− 1− 𝜒𝑐 𝑛+1 . In terms of the canonical field 𝝋 = 𝑴𝑷 𝝌 , the potential becomes 𝜑 2 2 2 𝑉 𝜑 = 𝐶𝑙𝑜𝑜𝑝 𝑔𝑠 𝑀𝑃 𝑚𝑆 𝑊 . 𝑀𝑃 From this we obtain 𝜖= 1 2 𝑛! 1 𝜒𝑐𝑛+1 𝑛−1 𝑛 𝑁𝑛 𝑛+1 ! 2 𝑛−1 ∼ ⇒ 𝑛𝑆 = 1 − 6𝜖 + 2𝜂 ∼ 1 − 𝑛 𝑛𝑆 2 3 0.933 0.95 1 𝑁 2𝑛 𝑛−1 2𝑛 𝑛−1 𝑁 , 𝜂=− 𝑛 𝑛−1 𝑁 . . 4 5 6 ⇒𝑛≥4 0.955 0.958 0.96 favored. The potential is smoothly connected to the SM potential: 𝑉𝑆𝑀 ∼ 10−6 𝜑4 . Example: 𝒏 = 𝟓, 𝝌𝒄 = 𝟏 −7 𝜖 = 2.3 × 10 , 𝑉𝑐 4 𝑀𝑃 𝑉𝑐 𝑀𝑃4 = 1.2 × 10−13 . Predictions on the Higgs portal DM MPP imposes constraints on the Higgs portal DM. Hamada, Oda, HK: arXiv: 1404.6141 Higgs portal dark matter We consider 𝒁𝟐 invariant Higgs portal DM: 𝟏 𝟏 𝟐 𝟐 𝝆 𝟒 𝜿 𝟐 † 𝟐 ℒ = ℒ𝑺𝑴 + 𝝏𝝁 𝑺 − 𝒎𝑺 𝑺 − 𝑺 − 𝑺 𝑯 𝑯 𝟐 𝟐 𝟒! 𝟐 mHiggs =125GeV mtop =172.892 GeV κ ρ The effect of 𝑺 on 𝝀 is opposite to that of top quark. Top Yukawa lowers 𝝀 for smaller values of 𝝁, while 𝜿 increases 𝝀 almost uniformly in 𝝁. 𝝀 Suppose that at some scale 𝚲 we have: 𝝀 𝚲 = 𝟎, 𝜷 𝝀 𝚲 = 𝟎. If top mass is increased, 𝝀 becomes negative at some scale. Then we can recover that 𝝀 𝚲′ = 𝟎, 𝜷 𝝀 𝚲′ = 𝟎 by increasing 𝜿. MPP gives a relation between 𝒚𝒕 and 𝜿. The scale of the tangent point decrease. increasing Top Yukawa increasing 𝜿 𝝁 From the natural abundance, the DM mass is related to 𝜿: 𝜿 𝒎𝑫𝑴 ∼ 𝟑𝟑𝟎𝐆𝐞𝐕 × 𝟎. 𝟏 Naturalness and Big Fix Even if CC problem is solved, we still have to explain the magnitude of the weak scale. In terms of the ordinary field theory it involves a fine tuning, but it may become natural if one consider modification or correction to the field theory. There are several attempts to slightly extend the framework of the local field theory in order to explain such fine tunings. • multiple point criticality principle Froggatt, Nielsen, Takanishi • baby universe and maximal entropy principle Coleman Okada, Hamada, Kawana, HK • classical conformality Meissner, Nicolai, Foot, Kobakhidze, McDonald, Volkas Iso, Okada, Orikasa • asymptotic safety Shaposhnikov, Wetterich Summary • It is possible that SM is directly connected to the Planck scale. We may have only a minor modification of SM below the string scale. • We still need some mechanisms to explain the smallness of the CC and Higgs mass. For example, we can consider MPP or MEP. • We can justify this picture by examining e.g. the inflation, DM and precise measurements of the top and Higgs masses. • We may start thinking about Planck scale physics seriously. Our universe might close to one of the perturbative vacuum. Appendix A mechanism to have MEP Consider Euclidean path integral which involves the summation over topologies, Coleman (‘88) not consistent dg exp S . topology Then there should be a wormhole-like configuration in which a thin tube connects two points on the universe. Here, the two points may belong to either the same universe or the different universes. If we see such configuration from the side of the large universe(s), it looks like two small punctures. But the effect of a small puncture is equivalent to an insertion of a local operator. Therefore, a wormhole contribute to the path integral as dg c d ij 4 x x d y g ( x ) g ( y ) O ( x ) O ( y ) exp S . 4 i j i, j Summing over the number of wormholes, we have 1 4 4 i j ci j d x d y g ( x ) g ( y ) O ( x ) O ( y ) N 0 n ! i , j exp ci j d 4 x d 4 y g ( x ) g ( y ) O i ( x ) O j ( y ) i, j n . Thus wormholes contribute to the path integral as 4 4 i j ci j d x d y g ( x ) g ( y ) O ( x ) O ( y ) . dg exp S i, j bifurcated wormholes ⇒ cubic terms, quartic terms, … y The effective action becomes a factorized multilocal form: Seff c i Si ci j Si S j ci j k Si S j Sk , i Si d ij D i jk x g ( x )Oi ( x ) . By introducing the Laplace transform exp Seff S1 , S2 , d w 1 , 2 , exp i Si , i we can express the path integral as Z d exp Seff d w d exp i Si . i Coupling constants are not merely constant but to be integrated. A solution to the cosmological constant problem Z d w dg exp g R g . S r S 4 r 1 2 4 d w dr exp r r exp 1/ , 0 d w no solution, 0 𝚲~𝟎 dominates irrespectively of w . Euclidean action is not bounded from below. Difficulty Action is not bounded from below. Problem of the Wick rotation WDW eq. H total 0 H total H universe H graviton-matter H universe 1 2 pa 2a ←wrong sign a : size of the universe “Ground state” does not exists. Wick rotation is not well defined. H universe H graviton-matter is bounded from below. H universe is bounded from above. H matter , This is not a merely technical problem, but it is the reason why the universe can evolve from t Problem: Consider the Minkovski quantum theory given by the following factorized multilocal action: Seff c i i Si Si ci j Si S j ci j k Si S j Sk , ij i jk D d x g ( x ) O ( x ) . i T. Okada, HK: arXiv:1110.2303, 1104.1764 HK: Int. J. of Mod. Phys. A vol. 28, nos. 3 & 4 (2013) 1340001 The path integral is given by Z d exp i Seff d w d exp i i Si . i Coupling constants are not merely constants but to be integrated. superposition of various states corresponding to various↓couplings Coupling constants that give the maximum entropy dominate.