From Cosmological From Cosmological ConstantConstant

to Sin Distributionto Sin Distribution

ICRR neutrino workshop Nov. 02, 2007ICRR neutrino workshop Nov. 02, 2007

Taizan Watari (U. Tokyo)Taizan Watari (U. Tokyo)

0707.344 (hep-ph) + 0707.346 (hep-ph) 0707.344 (hep-ph) + 0707.346 (hep-ph)

with L. Hall (Berkeley) and M. Salem (Tufts)with L. Hall (Berkeley) and M. Salem (Tufts)

13q

Three Issues Three Issues

Small but non-vanishing cosmological Small but non-vanishing cosmological constantconstant

Large mixing angles in neutrino oscillation Large mixing angles in neutrino oscillation What are “generations”?What are “generations”?

Can we ever learn anything profound from Can we ever learn anything profound from precise measurements in the neutrino precise measurements in the neutrino sector?sector?

Cosmological Constant Cosmological Constant ProblemProblem

Extremely difficult to explainExtremely difficult to explain

A possible solution by S. Weinberg ’87A possible solution by S. Weinberg ’87 Structures (such as galaxies): Structures (such as galaxies):

formed only for moderate formed only for moderate Cosmological Constant.Cosmological Constant.

That’s where we find ourselves. That’s where we find ourselves.

120 40 10 .Dark Energy PMr -¹ p

Key ingredients of this Key ingredients of this solutionsolution

CC of a vacuum can take almost any CC of a vacuum can take almost any value theoretically [i.e., a theory with value theoretically [i.e., a theory with multiple vacua]multiple vacua]

Such multiple vacua are realized in Such multiple vacua are realized in different parts of the universe. different parts of the universe.

just like diversity + selection in biological just like diversity + selection in biological evolution.evolution.

Any testable consequences ??Any testable consequences ??

What if other parameters What if other parameters (Yukawa)(Yukawa)

are also scanning?are also scanning? Do we naturally obtain Do we naturally obtain

hierarchical Yukawa eigenvalues,hierarchical Yukawa eigenvalues, generation structuregeneration structure in the quark in the quark

sector,sector,

but not for the lepton sector?but not for the lepton sector?

31 21 2 3

31 2

, , ,L L L

uu uQ Q Q

dd d

æ öæ ö æ ö ÷÷ ÷ çç ç ÷÷ ÷ çç ç ÷÷ ÷ çç ç÷ ÷ ÷ç ç çè ø è ø è ø: : :

A toy model generating A toy model generating statisticsstatistics

In string theory compactification, In string theory compactification,

Use Gaussian wavefunctions in Use Gaussian wavefunctions in overlap integral:overlap integral:

equally-separated hierarchically small equally-separated hierarchically small Yukawas.Yukawas.

6.( ) ( ) ( )m

IJ I m Jy d y y y yy g j yµ ò

Generation StructureGeneration Structure With random Yukawa matrix With random Yukawa matrix

elements, elements,

In our toy model, In our toy model,

Generation StructureGeneration Structure

originates from localized wavefunctions originates from localized wavefunctions of quark doublets and Higgs boson:of quark doublets and Higgs boson: No flavour symmetry, yet fine.No flavour symmetry, yet fine. No intrinsic difference between three quark No intrinsic difference between three quark

doubletsdoublets

Large mixing angles in the lepton sectorLarge mixing angles in the lepton sector non-localized wavefunctions for lepton non-localized wavefunctions for lepton

doubletsdoublets

Lepton Sector Lepton Sector PredictionsPredictions

Mixing angles without cutsMixing angles without cuts

Two large angles, Two large angles, After imposing cuts After imposing cuts

13(sin 0.18) 67%.P q < =

213(sin 3 10 ) 96%.P q -> ´ »

SummarySummary

Multiverse, motivated by the CC problemMultiverse, motivated by the CC problem Scanning Yukawa couplings: statistical Scanning Yukawa couplings: statistical

understanding of masses and mixings, understanding of masses and mixings, possibly w/o a symmetry.possibly w/o a symmetry. Generation structure: correlation between up and Generation structure: correlation between up and

down-type Yukawa matricesdown-type Yukawa matrices Localized wavefunctions of q and h are the origin Localized wavefunctions of q and h are the origin

of generations.of generations. Successful distributions for the lepton sector, Successful distributions for the lepton sector,

too,too, with very largewith very large

213(sin 3 10 ).P q -> ´

spare slidesspare slides

Family pairing structureFamily pairing structurecorrelation between the up and down Yukawa correlation between the up and down Yukawa

matricesmatrices Introduce a toy landscape on an extra Introduce a toy landscape on an extra

dimension dimension

Quarks and Higgs boson have Gaussian wave Quarks and Higgs boson have Gaussian wave functionfunction

Matrix elements are given by overlap integralMatrix elements are given by overlap integral

The common wave functions of quark The common wave functions of quark doublets doublets

and the Higgs boson introduce the correlation.and the Higgs boson introduce the correlation.

1 [0 : ].S L

( )2

2( )

1/ 4 2( ; ) .iy a

di iy a ML ey p

é ù-ê ú- ê ú- ë û:

( ) ( ; ) ( ; ) ( ,0),uij i jgM dy y a y b yl y y y= ò( ) ( ; ) ( ; ) ( ,0).dkj k jgM dy y c y b yl y y y= ò

Neutrino PhysicsNeutrino Physics

The see-saw mechanismThe see-saw mechanism

Assume non-localized wavefunctions for s. Assume non-localized wavefunctions for s.

Introduce complex phases. Introduce complex phases. Calculate the Majorana mass term of RH Calculate the Majorana mass term of RH

neutrino byneutrino by

Neutrino masses: hierarchy of all three Neutrino masses: hierarchy of all three matrices add up. Hence very hierarchical matrices add up. Hence very hierarchical see-saw masses. see-saw masses.

, ,*( ; ) ( ; ) ( ; ).

R i R jij i j

SBc dy y y y y y y

n ny y j=ò

5

1 solar atm, .m m m=

Mixing angle distributions:Mixing angle distributions:

Bi-large mixing possible.Bi-large mixing possible. CP phase distributionCP phase distribution 10 13log (sin )q

213(sin 3 10 ) 96%.P q -> ´ »

The Standard Model of particle physics The Standard Model of particle physics has 3(gauge)+22(Yukawa)+2(Higgs)+1 has 3(gauge)+22(Yukawa)+2(Higgs)+1 parameters.parameters.

What can we learn from the 20 What can we learn from the 20 observables in the Yukawa sector?observables in the Yukawa sector? maybe ... not much. It does not seem that maybe ... not much. It does not seem that

there is a beautiful and fundamental there is a beautiful and fundamental relationrelation that governs all the Yukawa-related that governs all the Yukawa-related observables. observables.

though they have a certain hierarchical though they have a certain hierarchical patternpattern

1.25GeVcm = 4.20GeVbm =3| | 3.96 10ubV-= ´

106MeVmm=2 5 28.0 10 eVm -D = ´e

212sin (2 ) 0.86q =

theories of flavor (very theories of flavor (very

simplified)simplified) Flavor symmetry and its small breakingFlavor symmetry and its small breaking

Predictive approach: use less-than 20 Predictive approach: use less-than 20 independent parameters to derive independent parameters to derive predictions.predictions.

Symmetry-statistics hybrid approach:Symmetry-statistics hybrid approach: Use a symmetry to explain the hierarchical Use a symmetry to explain the hierarchical patternpattern.. The The coefficients are just randomcoefficients are just random and of order and of order

unity.unity.

*

,aij

ia ja

L cM

y y fF

=å*

.aij

ij aa

cM

lF

=å

*

aij

ac MF

ac

ex. symmetry-statistics ex. symmetry-statistics hybridhybrid

an approximate U(1) symmetry an approximate U(1) symmetry broken bybroken by

U(1) charge assignment (e.g.) U(1) charge assignment (e.g.)

*.MeF =1.e=

1 110 ( , , )Q U E= 210 310 1,2,3 1,2,35 ( , )D L= h

in 3 2 0 00

*

,i jn n

ij ijc Ml

+æ öF ÷ç ÷= ç ÷ç ÷çè ø ijcare random coefficients of order unity.6 4

3 2

3 2

: : : :1,

: : : :1,

: : : : .

u c t

d s b

us ub cbV V V

l l l e e

l l l e e

e e e

»

»

»

pure statistic approachespure statistic approaches

Multiverse / landscape of vacuaMultiverse / landscape of vacua best solution ever of the CC problembest solution ever of the CC problem supported by string theory (at least for supported by string theory (at least for

now)now) Random coefficients fit very well to Random coefficients fit very well to

this framework.this framework.

But, how can you obtain hierarchy But, how can you obtain hierarchy w/o a symmetry?w/o a symmetry?

randomly generated matrix randomly generated matrix elementselements

Neutrino anarchyNeutrino anarchy Generate all -related matrix elements Generate all -related matrix elements

independently, following a linear measureindependently, following a linear measure explaining two large mixing angles.explaining two large mixing angles.

Power-law landscape for the quark sectorPower-law landscape for the quark sector Generate 18 matrix elements independently, Generate 18 matrix elements independently,

following following

The best fit value is The best fit value is

,iij jL l y y f= 1( ) .dP ddl l l- +µ

Hall Murayama Weiner ’99Haba Murayama ‘00

Donoghue Dutta Ross ‘05

5( ) ,dP dl lµ

0.16.d=-

Let us examine the power-law model more closely for the scale-invariant case

4 21,2,3 1 2 3 3 2( ) 36 .dP x dx dx dx x x»

Results: (eigenvalue distributions)

min

max min

ln( / ).

ln( / )i

ixl ll l

=

1 2 3.x x x£ £

0.d=

Hierarchy is generated from statistics for moderately large min

maxln .l

læ ö÷ç ÷ç ÷è ø

2 1L Lu d-

1 3L Lu d- 3 2

L Lu d- pairing1

1 .

1CKMV

æ ö÷ç ÷ç ÷ç ÷ç ÷ç ÷ç ÷÷çè ø

:

mixing angle distributions

e.g.

Family pairing structure is not obtained.

Who determines the scale-invariant (box shaped) distribution?

How can both quark and lepton sectors be accommodate within a single framework?

Family pairing structureFamily pairing structurecorrelation between the up and down Yukawa correlation between the up and down Yukawa

matricesmatrices Introduce a toy landscape on an extra Introduce a toy landscape on an extra

dimension dimension

Quarks and Higgs boson have Gaussian wave Quarks and Higgs boson have Gaussian wave functionfunction

Matrix elements are given by overlap integralMatrix elements are given by overlap integral

The common wave functions of quark The common wave functions of quark doublets doublets

and the Higgs boson introduce the correlation.and the Higgs boson introduce the correlation.

1 [0 : ].S L

( )2

2( )

1/ 4 2( ; ) .iy a

di iy a ML ey p

é ù-ê ú- ê ú- ë û:

( ) ( ; ) ( ; ) ( ,0),uij i jgM dy y a y b yl y y y= ò( ) ( ; ) ( ; ) ( ,0).dkj k jgM dy y c y b yl y y y= ò

inspirationinspiration

in certain compactification of Het. in certain compactification of Het. string theory,string theory,

Yukawa couplings originate from Yukawa couplings originate from overlap integration.overlap integration.

Domain wall fermion, Gaussian Domain wall fermion, Gaussian wavefunctions and torus fibration wavefunctions and torus fibration see next page.see next page.

( )( )tr .W dB AdA AAA= WÙ - -ò

domain wall fermion and domain wall fermion and torus fibrationtorus fibration

5D fermion in a scalar background5D fermion in a scalar background

Gaussian wavefunction at the domain wall.Gaussian wavefunction at the domain wall. 6D on with a gauge flux F on it.6D on with a gauge flux F on it.

looks like a scalar bg. in looks like a scalar bg. in 5D.5D.

chiral fermions in eff. theory:chiral fermions in eff. theory:

Generalization: -fibration on a 3-fold B. Generalization: -fibration on a 3-fold B.

2T

6 56 5 5,0( )A F x x= -56 .

2

F

p

æ ö÷ç ÷ç ÷çè øò

3T

introducing “Gaussian Landscapes” (toy introducing “Gaussian Landscapes” (toy models)models) calculate Yukawa matrix by overlap integral on a calculate Yukawa matrix by overlap integral on a

mfd Bmfd B use Gaussian wavefunctionsuse Gaussian wavefunctions scan the center coordinates of Gaussian profilesscan the center coordinates of Gaussian profiles

Results: try first for the easiestResults: try first for the easiest Distribution of Yukawa couplings (ignoring Distribution of Yukawa couplings (ignoring

correlations)correlations)

1.B S=

/ 0.08.d L=

2ln ( / ) /11.L dlD :

scale invariant distribution

To understand more To understand more analytically....analytically....

FN factor distribution FN factor distribution

( )2 2

1/ 4( )2

4 / 9 exp .3

i j i juij

a b a bg Md

dl p

é ù+ -ê ú» -ê úê úë û

( ) ( ) ( )2 2 22 2 / 3 /3/3 1 .j i jib d a b da de e e- -- ´ ´ =

2 2

2 2 2

3 (ln )( ) .

da db dr d ddP

L L L

p p ll

æ ö÷ç» = »÷ç ÷çè ø2 2 2: ,r a b= +

2

2ln .3

rd

l- :2

( )3 .

ln

dP d

d L

lp

l

æö÷ç= ÷ç ÷çè ø

Froggatt—Nielsen type mass matrices

Distribution of Distribution of ObservablesObservables

Three Yukawa eigenvalues Three Yukawa eigenvalues (the same for u (the same for u and d sectors)and d sectors)

Three mixing angles Three mixing angles family family pairingpairing

The family pairing originates from the localized wave functions of .& .Lq H

/ 0.08.d L=

quick summaryquick summary

hierarchy from statisticshierarchy from statistics Froggatt—Nielsen like Yukawa matricesFroggatt—Nielsen like Yukawa matrices hence family pairing structurehence family pairing structure FN charge assignment follows FN charge assignment follows

automatically.automatically. The scale-invariant distr. follows for The scale-invariant distr. follows for Geometry dependence?Geometry dependence? How to accommodate the lepton sector?How to accommodate the lepton sector?

1.B S=

Geometry DependenceGeometry Dependence

exploit the FN exploit the FN approximationapproximation

FN suppression factor for q or qbar:FN suppression factor for q or qbar:

FN factors: the largest, middle and FN factors: the largest, middle and smallest of three randomly chosen FN smallest of three randomly chosen FN factors as above. factors as above.

2 2/ 3 ;r dee -=

2 2

2

( )( ) .

dP d dV rf y

dy V dr= µ

2

ln :r

yd

eæö÷çº µ - ÷ç ÷çè ø

1 2 31 2 3

1 2 3

3 2 3 1 2 1

( , , )3! ( ) ( ) ( )

( ) ( ) ( ).

dP y y yf y f y f y

dy dy dy

y y y y y y

=

Q - Q - Q -

compare and compare and

FN factors: / eigenvalues / FN factors: / eigenvalues / mixing anglesmixing angles

2B T= 2S

The original The original carrying info. of geometry B, is carrying info. of geometry B, is integrated once or twice in obtaining integrated once or twice in obtaining distribution fcns of observables.distribution fcns of observables. details tend to be smeared out.details tend to be smeared out. power/polynomial fcns of log of power/polynomial fcns of log of

masses / angles in Gaussian landscapes. masses / angles in Gaussian landscapes. broad width (weak predictability)broad width (weak predictability)

Dimension dependence: FN factor Dimension dependence: FN factor distributiondistribution

2 2

2

( )( ) ,

dP d dV rf y

dy V dr= µ

2S

3S

Neutrino PhysicsNeutrino Physics

The see-saw mechanismThe see-saw mechanism

Assume non-localized wavefunctions for s. Assume non-localized wavefunctions for s.

Introduce complex phases. Introduce complex phases. Calculate the Majorana mass term of RH Calculate the Majorana mass term of RH

neutrino byneutrino by

Neutrino masses: hierarchy of all three Neutrino masses: hierarchy of all three matrices add up. Hence very hierarchical matrices add up. Hence very hierarchical see-saw masses. see-saw masses.

, ,*( ; ) ( ; ) ( ; ).

R i R jij i j

SBc dy y y y y y y

n ny y j=ò

5

1 solar atm, .m m m=

Mixing angle distributions:Mixing angle distributions:

Bi-large mixing possible.Bi-large mixing possible. CP phase distributionCP phase distribution 10 13log (sin )q

213(sin 3 10 ) 96%.P q -> ´ »

In Gaussian Landscapes,In Gaussian Landscapes,

Family structure from overlap of Family structure from overlap of localized wavefunctions. localized wavefunctions.

FN structure with hierarchy w/o flavor FN structure with hierarchy w/o flavor sym.sym.

Broad width distributions.Broad width distributions. Non-localized wavefunctions for .Non-localized wavefunctions for . No FN str. in RH Majorana mass termNo FN str. in RH Majorana mass term

large hierarchy in the see-saw neutrino large hierarchy in the see-saw neutrino masses.masses.

Large probability for observable .Large probability for observable .

5

3| |eU

The scale invariant distribution of Yukawa couplings for B = S^1 becomes

for B = T^2,

for B = S^2.

Scanning of the center coordinates Scanning of the center coordinates

should come from scanning vector-bdle should come from scanning vector-bdle moduli.moduli.

Instanton Instanton (gauge field on 4-mfd not 6-mfd) (gauge field on 4-mfd not 6-mfd)

moduli space is known better. moduli space is known better. In the t Hooft solution, In the t Hooft solution,

the instanton-center the instanton-center

coordinates can be chosen freely.coordinates can be chosen freely.

F-theory (or IIB) flux compactification can be F-theory (or IIB) flux compactification can be

used to study the scanning of complex-used to study the scanning of complex-

structure (vector bundle in Het) moduli. structure (vector bundle in Het) moduli.

2

2ln 1 ,

( )

aa Im nmn

I I

Ay y

rh

æ ö÷ç ÷=- ¶ +ç ÷ç ÷ç -è øå

Iyur