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Cosmological Constant Problem and ScaleInvariance

Taichiro Kugo
Maskawa Institute, Physics Dept., Kyoto Sangyo University
August 04, 2017

PPP2017 @ 基礎物理学研究所

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Cosmological Constant Problem
Dark Clouds hanging over the two well-established theories
Quantum Field Theory ⇐⇒ Einstein Gravity Theory

We know the recently observed Dark Energy Λ0, which looks like a small
Cosmological Constant (CC):Present observed CC 10−29gr/cm3 ∼ 10−47GeV4 ≡ Λ0(1)We do not mind this tiny CC, which will be explained after our CC problem
is solved. However, we use it as the scale unit Λ0 of our discussion.

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本当に問題なのは何か？

Essential point: multiple mass scales are involved!
There are several dynamical symmetry breakings and they are necessarily
accompanied by vacuum condensation energy:
In particular, we are conﬁdent from the success of the Standard Model ofthe existence of at least two symmetry breakings:
Higgs Condensation ∼QCD Chiral Condensation ⟨q̄q⟩4/3 ∼( 200 GeV )4 ∼ 109GeV4 ∼ 1056Λ0( 200 MeV )4 ∼ 10−3GeV4 ∼ 1044Λ0Nevertheless, these seem not contributing to the Cosmological Constant!
It is a Super ﬁne tuning problem:c : initially prepared CC (> 0)c − 1056Λ0 : should cancell, but leaving 1 part per 1012; i.e., ∼ 1044Λ0c − 1056Λ0 − 1044Λ0 : should cancell, but leaving 1 part per 1044; i.e., ∼ Λ0c − 1056Λ0 − 1044Λ0 ∼ Λ0 : present Dark Energy

4c = initially prepared CC
56654321,
0987654321,0987654321,0987654321,0987654321,0987654321
×Λ∼
10
Λ00|{z}
12 桁c + VHiggs
44×Λ∼
10Λ04321,0987654321,0987654321,0987654321,0987654321
0{z}|44 桁c + VHiggs + VQCD = present Dark Energy
1 × Λ0 ∼ Λ0Note that the vacuum energy is almost totally cancelled at each stage of
spontaneous breaking as far as the the relevant energy scale order.

52真空エネルギー ≃ 真空凝縮エネルギー
宇宙定数の「2 つ」の起源

Vacuum Energy in QFT:

∑1k,s2ℏωk −∑

ℏEk

(2)

k,s

Vacuum Condensation Energy:
V (ϕc) : potential(3)

They are separately stored in our (or my, at least) memory, but actually, almost the sameobject, as we see now.
場の理論のテキストの最初に現れる「真空エネルギー」は無限大。しかし、masslessの場合の無限大は、例えば Supersymmetry でボソン−フェルミオン間で相殺していると考える。
massless からのズレのエネルギー密度は、有限に計算され、それは結局は真空凝縮
エネルギー V (ϕc) などに効いている。Consider the chiral quark condensation in QCD.
For simplicity, consider NJL model as a parallel model for the realistic QCD:]G[LNJL = q̄iγ µ∂µq +(q̄q)2 + (q̄iγ5q)241→ q̄(iγ µ∂µ − σ − iγ5π)q − (σ 2 + π 2)G6The eﬀective potential V (σ, π) is a function of σ 2 + π 2 and can be computed at the π = 0section V (σ) = V (σ, π = 0):∫1 2d4 pV (σ) = σ −ln det(/p − σ)Gi(2π)4But the second term is nothing but the vacuum energy∫∑ √d4 p−ln det(/p − σ) = −ℏ p2 + σ 2 + (σ-independent const)
4i(2π)
p,s
implying that⟨q̄q⟩ condensation energy ≃ Dirac sea vacuum energy(4)

Moreover, in a Shwinger-Dyson approach to realistic QCD, the quark mass is calculated< ΛQCD, and the condensation energy is
as a function Σ(p) possessing the support only ∼computed ﬁnite.

73Quantum Gravity is irrelevant
CC problem is to be considered in Einstein Gravity theory.
Einstein gravity is a unique Low Energy Eﬀective Theory(5)

Just like Chiral Lagrangian()
L = fπ2 tr ∂µU †∂ µU
U = exp(iπ/fπ ),π = π a(x)T a< fπ , i.e., in the lowest (second) oris a unique Eﬀective Theory in the low energy region E ∼der in the derivative. We know that the fundamental theory describing the strong interaction
is QCD. But, whatever the dynamical theory is beyond E > fπ , the sysytem is described by
the the Nambu-Goldstone (NG) bosons π based on the coset SU (3)L × SU (3)R /SU (3)V ,and the dynamics is uniquely described by this non-linear sigam model. The non-linearly
realized chiral symmetry uniquely determines the dynamics of the NG bosons, self-coupling
and coupling to other matters in the low energy regime. Moreover, even the quantum
correction in this system can be computed by this Lagrangian in the sense of Weinberg.

8In exactly the same manner, the general coordinate (GC) invariance uniquely determine
the Lagrangian in the lowest (second) order in the derivative; that is, it is the EinsteinHilbert action. In this analogy, it is worth noticing
Graviton is a NG tensor boson corresponding to GL(4) → SO(3,1)
Nakanishi-Ojima (1979)
So the Einstein-Hilbert action is exactly analogous to the chiral Lagrangian, and MPl is thecounterpart of the pion decay constant fπ :
∫{}
4 √
422µν
Seﬀ = d x −g c0MPl + c1MPlR + c2R + c3Rµν R + · · ·gµν = ηµν + hµν /MPl
The CC term (with no derivatives) is consistent with GC invariance and its natural scale is4O(MPl).
Below the Planck energy scale MPl, the dynamics is uniquely described by the E-H action
plus interaction terms with matter ﬁelds. The quantum gravity is quite irrelevant to any
problem in much lower energy region than Planck sacale, E ≪ MPl, in particular, to theCC problem associated with the spontaneous breaking of Electro-weak symmetry and chiralsymmetry.

94Scale Invariance solves the problem!

Our world is almost scale invariant: that is, the standard model Lagrangian is scale
invariant except for the Higgs mass term!
If the Higgs mass term comes from the spontaneous breaking of scale invariance at higher
energy scale physics, the total system can be really be scale invariant.

4.1 Classical Scale Invariance
Suppose that our world has no dimensionful parameters.
Suppose that the eﬀective potential V of the total system looks likeV (ϕ) = V0(Φ) + V1(Φ, h) + V2(Φ, h, φ)↓↓↓M ≫µ≫mand it is scale invariant. Then, classically, it satisﬁes the scale invariance relation :∑ ∂ϕi i V (ϕ) = 4V (ϕ),∂ϕi
⟨ i⟩so that the vacuum energy vanishes at any stationary point ϕ = ϕi0:V (ϕ0) = 0.

(6)

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Important point is that this holds at every stages of spontaneous symmetry breaking.
In the above potential V , we can retain only V0(Φ) when discussing the physics at scale
M , since φ and ϕ are expected to get VEVs of order µ or lower. Then the scale invarianceguarantees V0(Φ0) = 0.
If we discuss the next stage spontaneous breaking at energy scale µ, we should takeV0(Φ) + V1(Φ, h), and can conclude V0(Φ′0) + V1(Φ′0, h0) = 0.
Similarly, at scale m, we have the potential V0(Φ) + V1(Φ, h) + V2(Φ, h, φ), and canconclude V0(Φ′′0 ) + V1(Φ′′0 , h′0) + V2(Φ′′0 , h′0, ϕφ0) = 0.
This miracle is realized since the scale invariance holds at each energy scale of spontaneous
symmetry breaking.
For the help of understanding, we now write a toy model of potentials.
1V0(Φ) = λ0(Φ21 − ε0Φ20)2,2in terms of two real scalars Φ0, Φ1, to realize a VEV√⟨Φ0⟩ = M and ⟨Φ1⟩ = ε0M.

(7)This M is totally spontaneous and we suppose it be Planck mass giving the Newton coupling

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constant via the scale invariant Einstein-Hilbert term∫{}
4 √22µνSeﬀ = d x −g c1Φ0 R + c2R + c3Rµν R + · · ·gµν = ηµν + hµν /MPl
If GUT stage exists, ε0 may be a constant as small as 10−4 and then Φ1 gives the scalarﬁeld breaking GUT symmetry.
V1(Φ, h) causes the electroweak breaking:)
1 ( †2 2V1(Φ, h) = λ1 h h − ε1Φ1 ,2with very small parameter ε1 ≃ 10−24. This reproduces the Higgs potential when h is theHiggs doublet ﬁeld and ε1Φ21 term is replaced by the VEV ε1ε0M 2 = µ2/λ1.
V2(Φ, h, φ) causes the chiral symmetry breaking, SU(2)L×SU(2)R → SU(2)V . Using the2 × 2 matrix scalar ﬁeld φ = σ + iτ · π, we may similarly write the potential()2V2(Φ, h, φ) = 14 λ2 tr(φ†φ) − ε2Φ21 + Vbreak(Φ, h, φ)with another small parameter ε2. The ﬁrst term reproduces the linear σ-model potential
invariant under the chiral SU(2)L×SU(2)R transformation φ → gLφgR when ε2Φ21 is replaced by the VEV ε2ε0M 2 = m2/λ2. The last term Vbreak stands for the chiral symmetry
breaking term which is caused by the explicit quark mass terms appearing as the result of12tiny Yukawa couplings of u, d quarks, yu, yd, to the Higgs doublet h; e.g.,( ())
12†∗Vbreak(Φ, h, φ) = ε2Φ1 tr φ yuϵh ydh + h.c.
24.2 Quantum Mechanically
However, we have neglected the scale invariance anomaly in quantum ﬁeld theory. Actually, if we take account of the renormalization point µ, we have the RGE()∑∑∂∂∂µ+βa(λ)+γi(λ)ϕi
V (ϕ) = 0∂µ∂λ∂ϕaiai
and the dimension counting identity()∑∂∂+ϕiV (ϕ) = 4V (ϕ)µ∂µ∂ϕii
From these we obtain(∑i(1 − γi(λ))ϕi∂−∂ϕi∑aβa(λ)∂∂λa)
V (ϕ) = 4V (ϕ)

This is the correct equation in place of the above naive one:
∑ ∂ϕiV (ϕ) = 4V (ϕ)∂ϕii13This shows the anomalous dimension γi(λ) is not the problem.
βa(λ) terms may be problematic:∑∂−→
4 V (ϕ0) = −βa(λ)
V (ϕ0)∂λaaSo, an obvious possibility is that all the coupling constants go to the Infrared Fixed Points:βa(λIR) = 0. But,

What does this equation really imply?
We argue that the potential value V (ϕ0) at the stationary point ϕ = ϕ0,dV= 0,dϕ ϕ=ϕ0is zero at any µ, even before reaching the IR limit µ = 0.
The potential value V (ϕ0) = V0(λ; µ2) at stationary points satisﬁes the RGE:()∑∂∂µβa(λ)+V0(λ; µ2) = 0∂µ∂λaa(The ﬁrst term µ∂/∂µ may be replaced by 4 since V0 = µ4v(λ).)

14The solution is given byV0(λ; µ2) = V (λ̄(t); µ2e4t),
where t = ln µ,dλ̄a(t)= βa(λ̄(t)) with λ̄a(t = 0) = λa.
dt
Or, writing V0(λ; µ2) = µ4v(λ), we must havev(λ) = e4tv(λ̄(t))→v(λ̄(t)) = e−4tv(λ)

Taking the IR limit t → −∞ (µ → 0) givesv(λIR) = e+∞v(λ)
If v(λIR) is ﬁnite, then we havev(λ) = 0→V0(λ; µ2) = 0

That is, provided that IR ﬁxed point λIR, as well as the theory on top of that point, exist,
then, the vanishing property of the potential value at stationary point is not injured by theanomaly!

155Example Calculation in λϕ4 theoryOne-loop RGE-improved tree potential:
V (ϕ, λ; µ2) =λ4!

1ϕ4123λ2 λϕ1−ln
32π 2µ2λOr, denoting 4πϕ = φ, 12 λϕ2 = αφ2, 32π
2 = α,αφ4192π V =1 − 3α ln(αφ2/µ2)2

図 1: y = αx2 /(1 − 3α ln(αx)), α = 3/100, µ = 1

16停留点 φ20 ：∂V
αφ2=(2(1 − 3α ln(·)) + 3α) = 0∂φ2(1 − 3α ln(·))211→ φ20 = 0 or ln(αφ20/µ2) = +( 2 3α)
2µ11→ φ20 = 0 or φ20 =exp+→ ∞e∞
α2 3αas α → 0+(8)停留値：

αφ402V0(φ0) = 0 or V0(φ0) == − φ40 → −∞2e∞−(3α/2)
3as α → 0+dλ= β(λ) leads todt()
111t=−b1 λ̄(t) λIn this case of β(λ) = b1λ2, the RGE[ ()]141v(λ̄(t)) = v(λ)e−4t = v(λ) exp−b1 λ̄(t) λBut t → −∞ で λ̄(t) → λIR = 0 だが、v(0) は free theory だから、それが発散するのはおかしい。ゆえに、v(λ) = 0.

17In case of non-trivial IR ﬁxed point, β(λ) ≃ b(λ − λIR) with b > 0,()1
λIR − λ̄(t)t = lnb
λIR − λ)4/b(λIR − λ→ v(λ̄(t)) = v(λ)e−4t = v(λ)λIR − λ̄(t)t → −∞ で λ̄(t) → λIR−(9)186Gauge Hierarchy上では)2()21 (V2 = λ2 h†h − ε1Φ21や V2 ⊃ 14 λ2 tr(φ† φ) − ε2Φ212で、単に小さな parameter ε1 ≃ 10−24, ε2 ≃ 10−30 を仮定することによって gauge hierarchy を「実現」した。
しかし、例えば chiral symmetry breaking scale ε2 の値は、GUT を仮定すれば、SU(3)gauge coupling g3 が、ε0M からくり込み群発展で running coupling ḡ3(µ) が O(1) になる
スケール µ ≃ ΛQCD で chiral symmetry が起こる、ε1ε0M ≃ ΛQCD 、ということで決まっている。Planck/GUT scale M と QCD scale ΛQCD との関係は、スケール M でのゲージ結合定数 g3 がどういう値をとっていたかによって決まる。
Electroweak breaking scale を決める ε1 も、例えば Technicolor 等の下のレベルのゲージ相互作用があれば、同様に決まるだろう。
Bardeen が昔から指摘しているように、Higgs scalar の mass term は、Higgs が elementary の場合でも、sacale invariance があれば 4 次の相互作用項が起源なので、2 次発散
ではなく log 発散であり、ε1 がゼロであれば Φ1 との相互作用も切れるので、ε1 のくりこみは対数補正でかつ ε1 に比例している。同様な指摘は、Shaposhnikov-Zenhausern によってもなされている。
W. A. Bardeen, “On naturalness in the standard model,” FERMILAB-CONF-95-391-T.
M. Shaposhnikov and D. Zenhausern, Phys. Lett. B 671 (2009) 162

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宇宙定数問題の解決に scale invariance が本質的な役割を果たすというアイデアは、

E. Rabinovici, B. Saering and W. A. Bardeen, Phys. Rev. D 36 (1987) 562.
M. Shaposhnikov and D. Zenhausern,ibidC. Wetterich, Nucl. Phys. B 302 (1988) 668.にあるが、前 2 者は quantum にも exact scale invariance を必要としている。どれも、multi-step spontaneous breaking との関連は言っていない。
また後者 2 者は、共に gravity の asymptotic safety を要求しているが、宇宙項問題の本
質には Planck scale 以上の UV の振る舞いは無関係のはず。

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これから
1. ∃Dilaton は何か？ → Higgs ?
2. Dilaton 質量は？ → anomaly が効く!
3. 現在の宇宙定数 Λ0 の値はどう説明？
4. インフレーションはどう起こるのか？
5. Running cosmological constant? : Loop eﬀects of matter and graviton ﬁelds below thePlanck scale.
6. Thermal eﬀects.
7. Scale invariant Beyond Standard Model の構成。
8. Scale Invariant Einstein Gravity: Local scale invariance will be meaningless.