Ｍ ultiverse and the Naturalness Problem

Hikaru KAWAI 　

2012/ 12/ 4at Osaka University

Contents1. Low energy effective theory of quantum gravity / string theory2. Wave function of the multiverse3. Naturalness and Big Fix

I will discuss the low energy effective action of quantum gravity and the possibility of solving the naturalness problem by using it.

1. Low energy effective theory of quantum gravity / string theory

Low energy effective theory of quantum gravity / string theory is obtained by integrating out the short distance physics.

why not

...mattergauge eff RgxdS D

Because of the symmetry, it should be

Usually, action is additive.Is that all?

.

,41

,

4int

240

int0

AexdS

FxdS

SSS

.int0 SSS (Sugawara 　‘ 82 )

An observer finds only a local field theory.The coupling constants are determined by the state or the history of the universe.

eff int 0 0 int 0 int .S S S S S S S

Actually, in quantum gravity or matrix model, thereare some mechanisms that the low energy effective action becomes

eff ,

( ) ( ).

i i i j i j i j k i j ki i j i j k

Di i

S c S c S S c S S S

S d x g x O x

)(xOi: local operators

By replacing each factor to its expectation value as is usually done in the mean field approximation, we have

“multi local action”

Consider Euclidean path integral which involves the summation over topologies,

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.

Multi local action from wormhole (1) Coleman ‘88

Summing over the number of wormholes, we have

bifurcated wormholes ⇒ cubic terms, quartic terms, …

4 4

,

( ) ( ) ( ) ( ) exp .i ji j

i j

c d xd d y g x g y O x O yg S

Therefore, a wormhole contribute to the path integral can be expressed as

4 4

,

( ) ( ) ( ) ( )exp .i ji j

i j

c d x d y g x g y O xg yS Od

x

y

4 4

0 ,

4 4

,

1 ( ) ( ) ( ) ( )!

exp ( ) ( ) ( ) ( ) .

n

i ji j

N i j

i ji j

i j

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

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

Thus wormholes contribute to the path integral is

Multi local action from wormhole (2)

The effective action becomes a factorized form

.)()(

,eff

xOxgxdS

SSScSScScS

iD

i

kjikji

kjijji

ijii

ii

By introducing the Laplace transform

effexp exp .i ii

Z d S d w d S

Coupling constants are not merely constant but to be integrated.

eff 1 2 1 2exp , , , , exp ,i ii

S S S d w S

we can express the path integral as

Multi local action from wormhole (3)

)],[21],[

41(1 2

2 AAATrg

S

Various possibilities for the emergence of space-time(1) Aμ as the space-time coordinates

mutually commuting Aμ 　⇒　space-time

Multi local action from IIB matrix model (1)

(2) Aμ as non-commutative space-time non-commutative Aμ 　⇒　 NC

space-time (3) Aμ as derivatives

Asano , Tsuchiya and HK

embedding in ten matrices

,

( ) ( 1.. )0 ( 1..10)

ba b

aC a D

Aa D

bb

aa CA ,)(

If the covariant derivative on a manifold acts on the regular representation field, its action can be decomposed into D scalars.

,( ) , ( 1,.., )baC a D : the Clebsh-Gordan coefficients

vector r r rV V V V r : regular representation

: regular representation field on D-dim manifold M

Hanada, HK, Kimura

Multi local action from IIB matrix model (2)

We can calculate the low energy effective action by using the background field method,and we obtain

.)()(

,eff

xOxgxdS

SSScSScScS

iD

i

kjikji

kjijji

ijii

ii

xy xz

Multi local action from IIB matrix model (3)

y

in the Lorentzian space timeWe consider Lorentzian theory from now on.

the effective action of quantum gravity/string is given by

.)()(

,eff

xOxgxdS

SSScSScScS

iD

i

kjikji

kjijji

ijii

ii

More precisely, the path integral is given by

effexp exp .i ii

Z d i S d w d i S

the coupling constants are determined by the state

eff, , ,

2 3 .i i i j i j i jk i j ki i j i j k

S c S c S S c S S S

We have seen

Coupling constants are not merely constant but to be integrated.

the question

What physics does the multi local action describe?Are some special values of the coupling constants favored? (Naturalness problem)

In the following, as an ad hoc assumption, we postulate the probabilistic interpretation for the multiverse wave function.

2. Wave function of the multiverse

the multiverse

Multiverse that includes indefinite number of large universes appears naturally in quantum gravity / string theory.

matrix model

・

bb

aC

,)(

bb

aC

,)(

Each block represents a large universe.

・

quantum gravity

Okada, HK

Wave Function of the multiverseThe multiverse sate is a superposition of N-verses.

multi0

;NN

d w

1 1, , ; ; ;N N Nz z z z

; ;

multiN

d w

exp i ii

Z d w d i S

Probabilistic interpretationWe assume a kind of probabilistic interpretation:

multi0

;NN

d w

1 1, , ; ; ;N N Nz z z z

1 1 1, , N N Nz z dz z z dz the probability of finding N universes with size

2 2

1 1, , ;N N Nw z z dz dz d

and the coupling constants .d

represents

Probability distribution of

1 1, , ; ; ;N N Nz z z z

2 211

0

2 2

2

, , ;!

exp ;

exp ( )

NN N

N

dz dzP z z wN

dz z w

w

2; (the life time of a universe)dz z

is chosen in such a way that is maximized,

irrespectively of

.w

Wave function of a universe (1)We assume that the universe emerges with a small size ε 　　with probability amplitude .

1

0(0) , (1)

0

, exp

exp

z z z

z dzdpdN i dt pz NH

dT z i H T

z

212

H z p z

: volume of the universez

is a superposition of the universe with various age,

z

T

+ z

T

zT + +…

T T dT

gives the probability of finding auniverse of age .

2 2z dz dT

1 exp( )

zz i dz p z

z p z

212

HH z p z zpp

2 1( )( )

1

dz z dzzp z

dzz

dT

Wave function of a universe (2)

infrared cutoffWe introduce an infrared cutoff for the size of universes.

IRz

ceases to exist

bounces back

or

2 2 2 life time of the universez dz dT

∞finite

Wave function of a universe (3)

We have seenthe coupling constants are chosen in such a way that the lifetime of the universe becomes maximum.

What values of the coupling constants make the lifetime maximum?

the question

3. Naturalness and Big Fix

Cosmological constant

Λ ～ curvature ～ energy density

(extremely small)

What value of Λ maximizes the life time of the universe?

0

For simplicity, we assume the S3

topology.

The cosmological constant in the far future is predicted to be very small.

IRz

0 cr cr

1/ 2cr radC

2 / 3cr mattC

2 / 3 4 / 3

1( ) matt radC CU zz z z

The other couplings (Big Fix) (1)

22expP w

The exponent is divergent, and regulated by the IR cutoff :

24

rad0cr

1, log .IR

R

z

Idz z C zz

cr rad1/ C

2,dz z

BIG FIX

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

assuming all matters decay to radiation

The other couplings (Big Fix) (2)

However, some of the couplings can be determined withoutknowing the detail of the cosmological evolution.

radC Quantities like are almost determined by the low energy physics.

⇒ If the low energy physics and the cosmological evolution is understood, we can calculate theoretically, and all of the renormalized couplings are in principle determined by maximizing it.case 1. Symmetry QCDexample

1. It becomes important only after the QCD phase transition.2. The effective action at QCD scale is invariant under QCD QCD.

⇒ 　　 is minimum or maximum at at least locally.

QCD 0 radCQCD

radC

Nielsen, Ninomiya

radC

The other couplings (Big Fix) (3)case 2. End point Hexample Higgs

coupling1. Some couplings are bounded. 2. The entropy of the universe can be monotonic in them.

⇒ The coupling is fixed to the end point.

H

radC

min

A scenario for .

HFix vh to the observed value and vary

.Hassuming the leptogenesis

⇒ sphaleron process ⇒ baryon number

H

⇒ radiation from baryon decay

⇒ Higgs mass is at stability bound, the stability bound.

Higgs 125 5GeVm without SUSY

The other couplings (Big Fix) (4)mnon-trivial example QCD coupling or

proton massWe assume that dark matters decay faster than protons,.

If the curvature term balances with matter (protons)before the proton decay, the universe bounce back when the protons decay.The earlier the protons decay, the less Crad remains. Crad is maximized if the curvature term balances with the energy density when the protons decay.

matt rad2 3 4

1 C CU aa a a

a

U

*a

The other couplings (Big Fix) (5)(cont’d)

The curvature term balances with the energy density when the protons decay.

2 3* *

1 , .BGM M N m

a a

1

2 3*a GM

: total baryon number: proton mass: proton life time

BNm

⇒ GM 4

4 5 2,GUT B

P

m N mGMg m m

2 4

64

P GUT

B

m mmg N

⇒

2 4105

4 6 10P GUTB

m mNg m

in our universe

Cf.

378 1010 protons in 10 ly

⇒ 9 10present 10 10 lya Reasonable

?

4. Summary

SummaryIn the quantum gravity or string theory, the low energy effective action is not a simple local one but the multi-local one.

The multiverse naturally appears, and it becomes a superposition of states with various values of the coupling constants.

The coupling constants are fixed is such a way that the lifetime of the universe is maximized.For example the cosmological constant in the far future is predicted to be very small: .

Some scenario other than the probabilistic interpretation.Comparison of different dimensional space-time?Generalization to the landscape?

Future problems

The Higgs mass is predicted at its lower bound.

Appendix A. Partition function

effexp exp .i ii

Z d i S d w d i S

We consider the partition function of the multiverse:

10

1 single universe

1exp!

exp

ni i

i n

i ii

d i S Zn

Z d i S

n

1exp .d w Z

the question

How to evaluate the partition function for a single universe:

1 exp .i ii

Z d i S

The path integral for a universe

1

1

0

0 0

exp

exp

ˆexp

ˆ

E E

Z d i S

f dzdpdN i dt pz NH i

f dT iTH i

f H i

f i

i

f

21ˆ2

H z p z matter

: volume of the universezQuestion: Is there a natural choice for them?

T

If the initial and final states are given, the path integral is well defined.

E E E E

Initial state

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

,: probability amplitude of a universe emerging.

i z matter

z

Final state: case 1 For the final state, we have two possibilities.

finite

The universe shrinks back.We assume the final state is

.f z matter

The partition function

1

2

0

ˆ

.E

Z f H i

const

Final state: case 2 (1)

∞The universe keeps expanding. It is not clear how to define the path integral for the universe:

for larg, e . IRIRf c z matt r ze

0 0

1| sin ,,

z

Ez dz p zz p z

4S d x gR

z

T

f z matter

1 expZ d i S

If we take , the path integral includes the history of the universe for the corresponding time . TTherefore as a trial we take

Final state: case 2 (2)

*1 0 0

*04

1 sin .

E IR E

IR EIR

Z c z

c zz

Then the partition function becomes

mat rad2 / 3 4 / 3

1( ) C CU zz z z

IRc z

0 0

1, sin ,,

z

E z dz p zz p z

If , the result does not depend on except for the phase which should be there in the classical limit:

, 2 ( )p z U z

IRz

*1 0 4

1 sin .E IRZ z

IR IRf z z matter ad hoc assumption

Under this ad hoc assumption, we have

the partition function for a universe

∞

finite 1 .Z

for cr of order 1 ,const

for cr 4

1 sin .IRconst z

Then the integration for the multiverse partition

1exp .Z d w Z has an infinite peak at , which again indicates that the cosmological constant at the late stages of the universe vanishes.

0

Big Fix

Then the multiverse partition function is given by

cr rad1/C

BIG FIX

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.

0 cr0 cr cr

rad2 / 3 4 / 3

1( ) CU zz z

1

4rad4

exp

1exp const exp const .cr

Z d w Z

C

3S