Cosmological Constant Problem and Scale Invariance Taichiro Kugo - PowerPoint PPT Presentation

1 Cosmological Constant Problem and Scale Invariance Taichiro Kugo Maskawa Institute, Physics Dept., Kyoto Sangyo University December 17 – 20, 2018 KEK Theory Workshop 2018 @ KEK, Tsukuba

2 1 Cosmological Constant Problem Dark Clouds hanging over the two well-established theories Quantum Field Theory ⇐ ⇒ Einstein Gravity Theory I first explain my view point on what is actually the problem. Recently observed Dark Energy Λ 0 , looks like a small Cosmological Con- stant (CC): 10 − 29 gr / cm 3 ∼ 10 − 47 GeV 4 ≡ Λ 0 Present observed CC (1) We do not mind this tiny CC now, which will be explained after our CC problem is solved. However, we use it as the scale unit Λ 0 of our discussion in the Introduction.

3 What is the true problem? → Essential point: multiple mass scales are involved! There are several dynamical symmetry breakings and they are necessarily accompanied by Vacuum Condensation Energy (potential energy): In particular, from the success of the Standard Model, we are confident of the existence of at least TWO symmetry breakings: ( 200 GeV ) 4 ∼ 10 9 GeV 4 ∼ 10 56 Λ 0 Higgs Condensation ∼ qq ⟩ 4 / 3 ∼ ( 200 MeV ) 4 ∼ 10 − 3 GeV 4 ∼ 10 44 Λ 0 QCD Chiral Condensation ⟨ ¯ Nevertheless, these seem not contributing to the Cosmological Constant! It is a Super fine tuning problem: c : initially prepared CC ( > 0) c − 10 56 Λ 0 : should cancell, but leaving 1 part per 10 12 ; i.e., ∼ 10 44 Λ 0 c − 10 56 Λ 0 − 10 44 Λ 0 : should cancell, but leaving 1 part per 10 44 ; i.e., ∼ Λ 0 c − 10 56 Λ 0 − 10 44 Λ 0 ∼ Λ 0 : present Dark Energy

4 c = initially prepared CC 4321 , 0987654321 , 0987654321 , 0987654321 , 0987654321 × Λ 0 ∼ 10 56 Λ 0 654321 , 098765 � �� � 12 digits c + V Higgs = × Λ 0 ∼ 10 44 Λ 0 4321 , 0987654321 , 0987654321 , 0987654321 , 0987654321 � �� � 44 digits c + V Higgs + V QCD = present Dark Energy 1 × Λ 0 ∼ Λ 0 Note that the vacuum energy is almost totally cancelled at each stage of spontaneous breaking as far as the the relevant energy scale order.

5 Contents Part I: Scale Invariance is a Necessary Condition 2. (Quantum) Vacuum Energy ≃ Vacuum Condensation Energy (potential) 3. Some conclusions from the simple observation Part II: Scale Invariance is a Sufficient Condition? 4. Scale Invariance may solve the problem 4-1. Classical scale invariance 4-1. Quantum scale invariance? 5. Quantum scale-invariant renormalization 6. Some conclusions from the simple observation In this context, the use of quantum scale-invariant prescription was pro- posed by M. Shaposhnikov and D. Zenhausern, Phys. Lett. B 671 (2009)

6 162 This is actually a very good paper. I introduce you their scenario and point the problem.

7 Part I: SI is Necessary Vacuum Energy ≃ vacuum condensation energy 2 People may suspect that there are “Two” origins of Cosmological Constant (Quantum) Vacuum Energy 1 ∑ ∑ 2 ℏ ω k − ℏ E k (2) k ,s k ,s Infinite, No controle, simply discarded ↕ (Classical) Potential Energy V ( ϕ c ) : potential (3) Finite, vacuum condensation energy They are separately stored in our (or my, at least) memory, but actually, almost the same object, as we see now. We now show for the vacuum energies in the SM that quantum Vacuum Energy = Higgs Potential Energy (4)

8 Let us see this more explicitly. For that purpose, consider Simplified SM: ψ ( x ) + 1 − λ ( ) ( ) L r = ¯ iγ µ ∂ µ − yϕ ( x ) ∂ µ ϕ ( x ) ∂ µ ϕ ( x ) − m 2 ϕ 2 ( x ) 4! ϕ 4 ( x ) − hm 4 . (5) ψ 2 Effective Action (Effective Potential) is calculated prior to the vacuum choice. (i.e., calculable independently of the choice of the vacuum) Calculating Formula: L (Φ + ϕ ) = L ( ϕ ) + ∂ L ( ϕ ) ∂ϕ Φ + 1 ( ) iD − 1 2Φ F ( ϕ ) Φ + L int . (Φ; ϕ ) (6) ( i ⟨ )⟩ ∫ ∫ d 4 x L ( ϕ ) + i [ ] iD − 1 d 4 x L int (Φ; ϕ ) Γ[ ϕ ] = 2 ℏ ln Det F ( ϕ ) − i ℏ exp (7) ℏ 1PI ( i ⟨ )⟩ ∫ ∫ d 4 k V [ ϕ ] = V 0 ( ϕ ) + 1 [ ] iD − 1 d 4 x L int (Φ; ϕ ) i (2 π ) 4 ln det F ( k ; ϕ ) + i ℏ exp (8) 2 ℏ ℏ 1PI

9 1-loop effective potential in the Simplified SM Use dimensional regularization for doing Mass-Independent (MI) renormalization V ( ϕ, m 2 ) = 1 2 m 2 ϕ 2 + λ 4! ϕ 4 + hm 4 + V 1-loop + δV (1) counterterms ∫ ∫ d 4 k d 4 p V 1-loop = 1 i (2 π ) 4 ln( − k 2 + m 2 + 1 i (2 π ) 4 ln( − p 2 + y 2 ϕ 2 2 λϕ 2 ) − 2 ) 2 � �� � � �� � = M 2 ψ ( ϕ ) = M 2 ϕ ( ϕ ) counterterms = D (1) 4! λϕ 4 + 1 2( E (1) m 2 + ( δm 2 ) (1) ) ϕ 2 + ( F (1) m 4 + G (1) m 2 + H (1) ) δV (1) dropping the 1 / ¯ ε parts in MS renormalization scheme, we find λ 2 3 1 16 π 2 y 4 1 4! Γ (4 , 0) D (1) λ = : ε − ϕ 4 16 π 2 2 ¯ ε ¯ λ 1 E (1) m 2 = ( δm 2 ) (1) = 0 Γ (2 , 0) εm 2 , : (9) ϕ 2 32 π 2 ¯ 1 1 Γ (0 , 0) : F (1) m 4 = G (1) m 2 = 0 , H (1) = 0 εm 4 , (10) 32 π 2 ¯ since ∫ i (2 π ) 4 ln( − k 2 + M 2 ) = M 4 d 4 k + ln M 2 1 ( − 1 µ 2 − 3 ) . (11) 64 π 2 2 ε ¯ 2 � �� � Coleman-Weinberg potential

10 So we get finite well-known renormalized 1-loop effective potential: V ( ϕ, m 2 ) = 1 2 m 2 ϕ 2 + λ 4! ϕ 4 + hm 4 +( m 2 + 1 ln m 2 + 1 ( ) ( ) 2 λϕ 2 ) 2 2 λϕ 2 − 4( yϕ ) 4 ln y 2 ϕ 2 − 3 µ 2 − 3 (12) 64 π 2 µ 2 64 π 2 2 2 Note that the general 1-loop contributions are given by ∫ d 4 k F ln ( M 2 ) ≡ 1 ∑ i (2 π ) 4 ln( − k 2 + M 2 ) ± n i F ln ( M 2 V 1-loop ( ϕ ) = i ( ϕ )) , (13) 2 i But, this shows it’s nothing but (quantum) Vacuum Energies: Zero-point osc. for boson and Dirac’s sea negative energies. Indeed, we can evaluate the LHS as follows: ∫ M 2 ∫ d 4 k F ln ( M 2 ) − F ln (0) = 1 dm 2 ∂ i (2 π ) 4 ln( − k 2 + m 2 − iε ) ∂m 2 2 0 ∫ M 2 ∫ d 4 k = 1 1 dm 2 − k 2 + m 2 − iε i (2 π ) 4 2 0 ∫ M 2 ∫ d 3 k = 1 1 dm 2 √ k 2 + m 2 (2 π ) 3 2 2 0 ( ℏ ) ∫ d 3 k √ √ k 2 + M 2 − ℏ k 2 = (14) (2 π ℏ ) 3 2 2

11 Note also that F ln (0) in the massless case vanishes in the dimensional regularization. If you apply the dimensional formula to the last expression, you can also recover the original RHS result. 3 Conclusions from these simple observation We have shown that the equivalence between the (quantum) vacuum energies and (‘clas- sical’) Higgs potential energy. From this simple observation, we can draw very interesting and important conclusions: As far as the matter fields and gauge fields are concerned, whose mass comes solely from the Higgs condensation ⟨ ϕ ⟩ , Their vacuum energies are calculable and finite quantities in terms of the renormalized λ and m 2 parameters! Note that this is because that their divergences are proportional to ϕ 4 and m 2 ϕ 2 . (At 1-loop, only ϕ 4 divergences appear.) However, the Higgs itself is an exception! The divergences of the Higgs vacuum energy are not only m 2 ϕ 2 and ϕ 4 but also the zero-point function proprtional to m 4 . In order to

12 cancel that part, we have to prepare the counterterm: m hm 4 = (1 + F ) hm 4 h 0 m 4 0 = Z h Z 2 1 1 F (1) h = ε. (15) 64 π 2 ¯ And the renormalized CC term hm 4 is a Free Parameter. Then, there is no chance to explain CC. Thus, for the calculability of CC, we need m 2 = 0, or No dimensionful parameters in the theory ⇒ (Classical) Scale-Invariance

13 Part II: Scale Invariance is a Sufficient Condition? 4 Scale Invariance may solve 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: λ ( h † h − m 2 ) 2 ( h † h − ε Φ 2 ) 2 . → (16) where Φ may be a field which appear also in front of Einstein-Hilbert term: ∫ d 4 x √− g Φ 2 R (17)

14 4.1 Classical Scale Invariance Suppose that our world has no dimensionful parameters. Suppose that the effective potential V of the total system looks like V ( ϕ ) = V 0 (Φ) + V 1 (Φ , h ) + V 2 (Φ , h, φ ) ↓ ↓ ↓ M ≫ µ ≫ m and it is scale invariant. Then, classically, it satisfies the scale invariance relation : ϕ i ∂ ∑ ∂ϕ i V ( ϕ ) = 4 V ( ϕ ) , (18) i ⟨ ϕ i ⟩ = ϕ i so that the vacuum energy vanishes at any stationary point 0 : V ( ϕ 0 ) = 0 . Important point is that this holds at every stages of spontaneous symmetry breaking. This miracle is realized since the scale invariance holds at each energy scale of spontaneous symmetry breaking.