砂漠と自然性 ‐場の理論をこえて‐

砂漠と自然性

‐場の理論をこえて‐

2013年10月22日 於 中央大

川合 光

（京都大学）

Y. Hamada, K. Oda and HK: arXiv:1210.2358 (PRD),

arXiv:1305.7055 , 1308.6651

HK: Int. J. of Mod. Phys. A vol. 28, nos. 3 & 4 (2013) 1340001

PART 1 砂漠の可能性

LHC gave beautiful results

But in some sense, they indicate

“the worst scenario”.

Higgs particle was discovered, but nothing else.

Especially, no signal of the SUSY.

We need to reconsider the origin of the

fine tuning.

Suppose the underlying fundamental theory, such as string theory, has the momentum

scale m_S and the coupling constant g_S . The naturalness problem

Then, by dimensional analysis and the power

the low energy effective theory are given as

follows:

dimension 2 (Higgs mass)

dimension 4

(vacuum energy or cosmological constant)

naturalness problem (cont.’d)

dimension 0

(gauge and Higgs couplings) dimension -2 (Newton constant)

2 2 .

S N

S

G g

 m

1 2 3

2

, , ,

.

S

H S

g g g g

λ g





2 2

2 0 .

H g mS S

m  +

2 4

0 g_s m_S .

λ  ⋅ ⁻ +

( )²

( )

2 2 2 18

100 GeV 10 GeV

H S S

m   g m 

(^{2 ~ 3meV})^{4 mS4}

(

)

λ _{  }

unnatural ! →

unnatural ! !→

tree

SUSY as a solution to the naturalness problem

Bosons and fermions cancel the UV divergences:

However, SUSY must be spontaneously broken at some momentum scale M_SUSY ,

below which the cancellation does not work.

+

bosons fermions

⇒ 2 2

SUSY . mH  M

+ ⇒ λ = ^0.

⇒

2 0.

mH =

⇒ λ _ ^M_SUSY^4.

(cont’d)

Therefore, if M

Higgs mass is naturally understood,

although the cosmological constant is still a big problem.

However, no signal of new particles is observed in the LHC below 1 TeV.

We have to think about other possibilities.

Possible explanation to the naturalness problem other than SUSY

1. We do not have to mind. We should simply take the parameters as they are.

2. Anthropic principle.

The parameters should be such that we can exist.

a) In some model, the wave function of the

universe is a superposition of various worlds each of which has different low energy

effective Lagrangians:

1 2 3 .

Ψ = Ψ + Ψ + Ψ + 

We are sitting in one of them. The parameters there must be such that we exist.

Anthropic principle. (cont.’d)

b) The universe has different parameters place by place. We are sitting at one place, where

the parameters are such that we can exist.

3. The parameters are fixed by some non-

perturbative effect of quantum gravity/string theory such as Coleman’s baby universe

mechanism.

Although we do not understand the real reason, nature chooses the parameters as we observe.

Possibility of desert

It may not be right to doubt the SM in the high

energy region by the reason that it is not natural.

The right attitude would be to examine simply

whether the SM is valid to the string scale or some new physics is needed below the scale .

If it is the former case, there is a possibility for

the desert, that is, we have only the SM below the string scale.

String theory

SM m_s

Can the SM valid to the Planck/string scale?

In order to answer the question, we consider the SM Lagrangian with cutoff momentum Λ,

and estimate its bare parameters in such a way that the observed low energy parameters are recovered.

If no inconsistency arises, it means that the SM can be valid to the energy scale Λ.

+  .

The bare coupling λ^Ｂ

As usual, the bare couplings can be approximated by the running couplings at Λ in a mass

independent scheme such as MS bar.

The error can be evaluated once the cutoff scheme is specified, and is expected as small as the two-loop corrections.

( ) ( ) ( )

,

i i ijk j k

B MS MS MS

j k

λ = λ Λ + ∑ b λ Λ λ Λ

: dimensionless couplings

(gauge, Yukawa, Higg sself couplings)

i

λ B

( ) ^.

i i

B MS

λ  λ Λ

We can approximate

13

• In general, the bare mass consists of quadratically divergent part and logarithmically divergent part:

• Here we consider only the first part, or we simply assume

• is determined by an order by order

perturbative calculation in the bare couplings demanding

13

The bare mass m

1

2 2

2 2

log 2 .

phys

phys

B m b

m a   m  

= Λ +   Λ  +

2

m

2 0.

mphys =

2 0 : mphys =

1-loop

2-loop

2

1 2

1 ,

I 16

π ^Λ

 ² 1 ¹

200 .

I  I

It turns out that 2-loop contribution is small in the case of the SM.

Renormalization group equation

Initial values

G. Degrassi, S. Di Vita, J. Elias-Miro, J. R. Espinosa, G. F.

Giudice, G. Isidori and A. Strumia,

Higgs mass and vacuum stability in the Standard Model at NNLO," JHEP 1208 (2012) 098 [arXiv:1205.6497 [hep-ph]].

Bare parameters of the cutoff theory (1)

m_Higgs =126GeV m_top =172 GeV

Ytop

U(1) SU(2)

SU(3)

Higgs self coupling Higgs mass ²

log₁₀ Λ[GeV}

No inconsistency arises below the string scale.

Bare parameters of the cutoff theory (2)

Ytop

U(1) SU(2)

SU(3)

Higgs self coupling Higgs mass ²

log₁₀ Λ[GeV}

Higgs field becomes unstable.

m_Higgs =126GeV m_top =190 GeV

Bare parameters of the cutoff theory (3)

m_Higgs =126GeV m_top =150 GeV

Ytop U(1)

SU(2) SU(3)

Higgs self coupling

Higgs mass ²

log₁₀ Λ[GeV}

No inconsistency arises, but the Higgs self

coupling tends to diverge.

Bare parameters of the cutoff theory (4)

Ytop

U(1) SU(2)

SU(3)

Higgs self coupling Higgs mass ²

log₁₀ Λ[GeV}

Higgs field becomes unstable.

m_Higgs =100GeV m_top =172 GeV

Bare parameters of the cutoff theory (5)

m_Higgs =150GeV m_top =172 GeV

Ytop

U(1) SU(2)

SU(3)

Higgs self coupling

Higgs mass ²

log₁₀ Λ[GeV}

No inconsistency arises, but the Higgs self

coupling tends to diverge.

m_top =171 GeV

Standard Model Criticality Prediction:

Top mass 173 ± 5 GeV and Higgs mass 135 ± 9 GeV.

Froggatt, Nielsen(1995)

Stability of the potential

m_Higgs =125GeV

m_top =171.31 GeV

m_Higgs =127GeV

m_top =172.29 GeV

m_Higgs =129GeV

m_top =173.26 GeV

Froggatt Nielsen by the recent values

small m^t

large m^t

Cut off dependence of the bare mass

Bare Higgs mass becomes zero if mt=170GeV.

Quadratic coupling vanishes if mt=171GeV.

Both m_B² and λ vanish around the Planck scale

Λ=M^Pl

Three quantities,

become close to zero around the Planck/string scale.

Triple coincidence

( )

, ,

B _λ B

m

λ β λ

( )

, ,

B _λ B

m

λ β λ

• The SM can be valid to the string scale.

Desert is possible.

• The experimental value of the Higgs mass seems to be just on the stability bound.

Nature seems to like the marginal stability.

• The bare Higgs mass becomes close to zero at the string scale. It implies that Higgs particle comes from a massless state of tree level string.

• The Higgs self coupling and the beta function also becomes close to zero at the string scale. It indicates that the Higgs

potential becomes almost flat around the string scale, which opens the possibility that the Higgs field plays the roll of

inflaton.

• It is important to know the top mass within 1% error.

Summary of the Higgs bare parameters

PART 2 自然性

－場の理論を少しはみ出す試みー

We consider the possibility that the fine tunings result from not the conventional local field theory but something slightly beyond.

2-1. Froggatt and Nielsen

“PREdicted the Higgs Mass”

H.B.Nielsen, arXiv:1212/5716

Why do we start from the canonical ensemble?

[ ]^{dϕ exp}

(

)

∫

In the ordinary quantum theory, the path integral of the form

On the other hand in the statistical mechanics, the most fundamental concept is the micro

canonical ensemble

[ ]

(

)

Z =

∫

and the canonical ensemble is shown to be equivalent in the thermodynamic limit:

[ ]^{dϕ δ}

(

)

(

)

∫ ∫

The total energy is given first, and the temperature is determined as a result.

Example: Set of many water molecules in a cylinder with a fixed pressure.

[ ]^{dϕ δ}

(

)

(

)

∫ ∫

p

water vapor

E T

water

vapor water + vapor

T_*

T is automatically tuned to T_* for wide range of E.

T corresponds to coupling constants in field theory.

Question:

Why isn’t quantum theory defined by micro canonical like path integral?

[ ]

( ) ⁽

⁾

[ ]

( (

) )

4 †

0

2 2 4 † 2

0

exp

exp .

Z d d x I i S

dm d i S m d x m I

φ δ φ φ φ

φ φ φ φ

= −

= − +

∫ ∫

∫ ∫ ∫

Here we see what happens if we start with

One value of dominates in the RHS:m²

( ( ) )

2 2

exp .

Z =

∫

(next two slides)

[ ]

(

)

Z =

∫

Assume that the

effective potential for S

has two minima. φ²

Veff

2

φ2 2

φ1

2 2

m < mc m² = m_c² m² > m_c²

2 2

φ _ φ2 φ₁² ≤ φ² ≤ φ₂² φ² _ φ₁²

[ ]

( (

) )

2 2 4 † 2

exp 0

Z =

∫

∫

∫

The original

effective potential.

4 †

0. d x φ φ = I

∫

2 2

1 I0 /V 2 ,

φ ≤ ≤ φ

If m² should be equal to

in order for the vacuum to be a mixture of the two phases such that

2

mc

In other words, F in

behaves as ^{Z =}

_∫

(

( )

)

2 2

2 0

(φ − I /V m)



2 2

1 0

(φ − I /V m)



m2

F

2

mc

φ2

Veff

2

φ2 2

φ1

If there is no special reason, it is natural to expect

φ2

Veff

2

φ2 2

φ1

What is the most probable value for ? φ2²

2 2

2 m_P .

φ _

Then can be naturally satisfied, because is expected to be of order

2 2

1 I0 /V 2

φ ≤ ≤ φ

0 /

I V m_P² .

The Higgs potential should have a degenerate minimum at a large value of the field.

mH = 125.6 GeV

generalization

[ ]

( ( ) ) ⁽

⁾

( )

( (

( ) ) )

4 † 2

2 2 2 4 † 2

exp

exp .

Z d d x M i S

dm w m d i S m d x M

φ ρ φ φ φ

φ φ φ φ

= −

= − −

∫ ∫

∫ ∫ ∫

Again dominates in the RHSm²  m_c²

2 2 2

1 M 2 .

φ < < φ

The micro canonical like path integral can be generalized to

is natural.

Planck scale M 

2 2

2 m

 φ

2 2

1 m

 φ m2

F

2

mc

slope ^{M 2}

( ) ( ( ( ) ) )

2 2 2 2 2

exp ,

Z =

∫

naturalness

[ ]

( ( ) ) ( (

^{( )} ) )

( )

[ ]

( (

( ) ( ) ) )

( ) ( ( ( ) ) )

( ) ( ( ( ) ( ) ) )

( ) ( ( ( ) ) )

4 † 2

2 2

2 4 † 2

2

2 4 †

2 4 †

2

2

2 2 2 2

2 2 2 2 2

2 2 2 2

2

2 2

exp

exp

exp

exp exp

R

R

R

R R

R

Z d d x M i S

dm w m

d i S m d x M

dm w m iV F m M m

dm w m iV F m

m d

M m

dm w m iV F

x

m d x m

m m

m M m

m

φ ρ φ φ φ

φ φ

φ

φ φ

φ

φ φ

−

= −

= ⋅

− −

= − −

=

− +

− − −

−

−

−

−

∫ ∫

∫

∫

∫ ∫

∫

∫

∫

∫



Even if a mass is generated by the radiative correction, it is negligible:

2-2. Coleman’s Baby Universe

Consider Euclidean path integral which involves the summation over topologies,

Coleman (‘88) an explicit mechanism to get the factorized action

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 universe.

[ ]

topology

exp .

dg −S

∑ ∫

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 insert ion of a local operator.

Summing over the number of wormholes, we have

bifurcated wormholes

⇒ cubic terms, quartic terms, …

[ ] ^{4 4} ( )

,

( ) ( ) ⁱ ( ) ^j ( ) exp .

i j i j

c d x

dg ∑ d y g x g y O x O y −S

∫ ∫

Therefore, a wormhole contribute to the path integral as

[ ] ^{4 4}

,

( ) ( ) ( ) ( )

exp _{i j} ^{i j} .

i j

c d x d y g x g y O x

g S O y

d  

 − + 



∑ ∫

∫

x y

4 4

0 ,

4 4

,

1 ( ) ( ) ( ) ( )

!

exp ( ) ( ) ( ) ( ) .

n

i j

i j

N i j

i j

i j i j

c d x d y g x g y O x O y n

c d x d y g x g y O x O y

∞

=

 

 

 

 

=  

 

∑ ∑ ∫

∑ ∫

Thus wormholes contribute to the path integral as

The effective action becomes a factorized form

. ) (

) (

eff ,

x O

x g

x d

S

S S

S c

S S c

S c

S

i D

i

k j

i k

j i

k j i j

j i

i j

i i

i i

∫

∑

∑

∑

=

+ +

+

= 

By introducing the Laplace transform

[ ]exp ( eff ) ( ) [ ]exp _{i i} .

i

Z = dφ − S = d wλ λ dφ ^ − λ S ^



∑

∫ ∫ ∫

Coupling constants are not merely constant but to be integrated.

( )

(

)

exp , , , , exp _{i i} ,

i

S S S dλ w λ λ ^ λ S ^

− =  − 



∑

∫

 

we can express the path integral as

A solution to the cosmological constant problem

( ) [ ]^exp

( )

Z =

∫

∫

∫

∫

dominates irrespectively of S4

r

( )

( ( ) )

( ) ( )

2 4

exp

exp 1/ , 0

no solution, 0

d w dr r r

d wλ

Λ Λ − − + Λ

Λ Λ >

Λ  Λ <

∫ ∫

∫





r S

− 1 Λ

Λ  0 ^w( )^{Λ .}

( ) [ ]^exp ( ( ))

Z = ∫ d wλ λ ∫ dφ − S λ

including multiverse

( )

single single

0

1 exp .

!

n n

Z Z

n

∞

=

= ∑ =

n

Difficulty (1)

Problem of the Wick rotation WDW eq.

←wrong sign

“Ground state” does not make sense.

total 0

H Ψ =

total universe matter graviton

2 universe

1

2 ^a

H H H H

H p

a

= + + +

 

= −  + 





Wick rotation is not well defined. t

matter ,

H 

universe

H : size of the universe

a

matter

H is bounded from below.

universe

H is bounded from above.

The overall phase of Z_single

The multiverse partition function

( ) ( )

multi exp single .

Z ^

∫

[ ] ( )

single exp

Z =

∫

is important.

We need subtle analyses.

Difficulty (2)

Overall phase of the Partition function

2-3. Lorentzian Path integral of

the factorized action

It is natural to imagine that the low energy effective action of a theory including gravity has the same

structure as Coleman’s:

. ) (

) (

eff ,

x O

x g

x d

S

S S

S c

S S c

S c

S

i D

i

k j

i k

j i

k j i j

j i

i j

i i

i i

∫

∑

∑

∑

=

+ +

+

= 

More precisely, the path integral is given by

[ ]exp ( eff ) ( ) [ ]exp _{i i} .

i

Z =

∫

∫

∫

∑

eff

, , ,

2 3 .

i i i j i j i jk i j k

i i j i j k

S ^

∑

∑

∑

Coupling constants are not merely constant but to be integrated.

e.g. IIB matrix model

Y. Asano, A. Tsuchiya, HK

[ ]exp ( eff ) ( ) [ ]exp _{i i} .

i

Z = dφ i S = d wλ λ dφ ^i λ S ^

 ∑ 

∫ ∫ ∫

It is natural to apply this action to the multiverse.

[ ]

[ ]

1 0

1 single universe

exp 1

! exp

n i i

i n

i i

i

d i S Z

n

Z d i S

φ λ

φ λ

∞

=

  =

 

 

 

=  

∑ ∑

∫

∫ ∑

n

( )^exp( ¹ ( ))^.

d wλ λ Z λ

= ∫

2-4. Partition function of a single universe

Basic problem

Define and evaluate the partition function of a single universe:

( )

[ ]

1 exp _{i i} .

i

Z λ = dφ ^ i λ S ^



∑

∫

The path integral of a universe

( ) [ ] ( )

[ ]

(

)

( )

( )

1

1 0

0 0

exp

exp exp ˆ

ˆ

E E

Z d i S

f dadpdN i dt pa N H i f dT iTH i

f H i

f i

λ

λ λ

λ φ

δ

φ φ

∞

−∞

= =

=

= −

= −

=

=

∫

∫ ∫

∫



i f

2 3

1 1 1

ˆ ( )

H 2 p a U a

a a

λ = − −

: radius of the universe Question: a

Is there a natural choice for them?

T

If the initial and final states are given, the path integral is evaluated as follows.

( )

E E E E

φ φ _′ = δ − ^{′ 2 3 4}

( ) 1 C^matt C^rad U a = a − Λ − a − a

Initial state

For the initial state, we assume that the universe emerges with a small size ε.

,

: probability amplitude of a universe emerging.

i µ a ε matter

µ

= = ⊗ 

a = ε

Evolution of the universe

Λ～curvature

～energy density

with

S³ topology

( ) ( )

(

)

0 1 0

, 1 sin ,

,

z

E a da p a

a p a

φ λ λ α

= ^ − λ ∫ ′ ′ +

( )^{, 2 4 ( ).}

p a λ = − a U a

2 3 4

( ) 1 C^matt C^rad U a = a − Λ − a − a WKB solution

( )

S = ∫ d x4 −g R − Λ + matter z

Λ < Λcr Λ = Λ_cr Λ < Λ^cr

( )

U a U a( ) U a( )

* a

case of closed universe

For the final state, we have two possibilities.

finite

The universe is closed.

We assume the final state is

. f = µ^′ a = ⊗ε matter

The partition function

( )

( )

( )

1

2 0

ˆ

E .

Z f H i

const

λ δ λ

φ ₌ ε

=



Λ < Λcr

case of open universe

∞

The universe is open.

It is not clear how to define the path integral for the universe:

lim .

IR

IR IR

f a c a a matter

= →∞ ⊗ 

a

( ) [ ] ( )

1 exp .

Z λ ⁼ ∫ dφ i S

As an ad hoc assumption we consider

Λ > Λcr

Case of open universe (cont’d)

( ) ( ) ( )

( )

( )

*

1 0 0

3 *

4 0

3 *

4 0

1 sin

1 sin .

IR E IR E

IR IR E

IR

IR E

Z c a a

c a a

a

c a

λ µ φ φ ε

µ α φ ε

µ α φ ε

= =

=

=

=

Λ + ′ Λ

Λ + ′ Λ





Then the partition function becomes

mat rad

2 3 4

( ) 1 C C

U a = a − Λ − a − a

( ) ( )

(

)

0 1 0

1 sin ^z

E a da p a

a p a

φ ₌ α

− ∫ ′ ′ +



The result does not depend on except for the phase which should come from the classical action.

( )^{, 2 4 ( )}

p a λ = − a U a

aIR

Under the ad hoc assumption, we have the partition function for a universe

∞

finite

1

( )

Z λ

for Λ < Λcr const of order 1 ,

for Λ > Λcr 4

(

)

1 sin _IR .

const a Λ + α^′

Λ

Then the integration for the multiverse partition

( )^exp ( ¹ ( ))^.

Z = ∫ d wλ λ Z λ

has a large peak at , which means that the cosmological constant at the late stages of the universe almost vanishes.

λ

( )^λ _cr

Λ  Λ

2-5. Naturalness and Big Fix

Big Fix

Then the multiverse partition function is given by

cr 1/ Crad

Λ 

BIG FIX

The low energy couplings are determined in such a way that the entropy at the late stages of the universe is maximized.

For simplicity we assume the topology of the space and that all matters decay to radiation at the late stages.

rad

2 4

( ) 1 C

U a = a − Λ − a

( ) ( ( ))

(

)

1

4 rad 4

exp

exp const 1 exp const .

cr

Z d w Z

C

λ λ λ

λ

=

 

 

 Λ 

 

∫

 

S3

Λ = Λcr

( ) U a

Examples of the Big Fix (1)

However, some of the couplings can be determined without knowing the details of the cosmological evolution.

rad ( )

C λ

If the cosmological evolution is completely understood, we can calculate theoretically, and all of the renormalized couplings are in principle determined.

case 1. Symmetry ^{example θQCD}

1. It becomes important only after the QCD phase transition.

2. Hadron mass spectrum is invariant under θ_QCD → −θ_QCD.

⇒ ^Crad is minimum or maximum at at least locally. ^θ_QCD ^{= 0}

θQCD

Crad

Nielsen, Ninomiya

At present, we do not have enough knowledge about the very early and late stages of the universe, especially the origin of inflation, dark energy and dark matter.

case 2. End point example Higgs coupling λ_H

1. Some (renormalized) couplings are bounded.

2. can be monotonic in them. C_rad

⇒ is maximized at the end point. C_rad

λH

Crad

λmin

A scenario for . λ_H

Fix v_h to the observed value and vary λ_H .

assuming the leptogenesis

⇒ sphaleron process ⇒ baryon number

λH _

⇒ radiation from baryon decay







⇒ Higgs mass is at its lower bound.

Examples of the Big Fix (2)