Transcript Yukawa Couplings in Landscapes

From Cosmological Constant
to Sin q13 Distribution
ICRR neutrino workshop
Nov. 02, 2007
Taizan Watari (U. Tokyo)
0707.344 (hep-ph) + 0707.346 (hep-ph)
with L. Hall (Berkeley) and M. Salem (Tufts)
Three Issues

Small but non-vanishing cosmological constant

Large mixing angles in neutrino oscillation


What are “generations”?
Can we ever learn anything profound from
precise measurements in the neutrino sector?
Cosmological Constant Problem

Extremely difficult to explain
0 ¹ r Dark Energy p 10
- 120

4
P
M .
A possible solution by S. Weinberg ’87
Structures (such as galaxies):
formed only for moderate Cosmological Constant.
 That’s where we find ourselves.

Key ingredients of this solution

CC of a vacuum can take almost any value
theoretically [i.e., a theory with multiple vacua]

Such multiple vacua are realized in different
parts of the universe.


just like diversity + selection in biological evolution.
Any testable consequences ??
What if other parameters (Yukawa)
are also scanning?

Do we naturally obtain
hierarchical Yukawa eigenvalues,
 generation structure in the quark sector,

æu1 ÷
ö
QL 1 : çç ÷
,
÷
çèd1 ÷
ø

æu2 ÷
ö
QL 2 : çç ÷
,
÷
çèd 2 ÷
ø
but not for the lepton sector?
æu3 ÷
ö
QL 3 : çç ÷
,
÷
çèd3 ÷
ø
A toy model generating statistics

In string theory compactification,
yIJ µ
òd
6
m
y y ( y ) I g j m ( y )y ( y ) J .

Use Gaussian wavefunctions in overlap integral:

equally-separated hierarchically small Yukawas.
Generation Structure

With random Yukawa matrix elements,

In our toy model,
Generation Structure

originates from localized wavefunctions of
quark doublets and Higgs boson:
No flavour symmetry, yet fine.
 No intrinsic difference between three quark doublets


Large mixing angles in the lepton sector

non-localized wavefunctions for lepton doublets
Lepton Sector Predictions

Mixing angles without cuts
Two large angles, P(sin q13 < 0.18) = 67%.
 After imposing cuts

P(sin q13 > 3´ 10- 2 ) » 96%.
Summary


Multiverse, motivated by the CC problem
Scanning Yukawa couplings: statistical understanding
of masses and mixings, possibly w/o a symmetry.
Generation structure: correlation between up and downtype Yukawa matrices
 Localized wavefunctions of q and h are the origin of
generations.


Successful distributions for the lepton sector, too,

with very large P(sin q13 > 3´ 10- 2 ).
spare slides

Family pairing structure
correlation between the up and down Yukawa matrices

Introduce a toy landscape on an extra dimension
S1 [0 : L].
Quarks and Higgs boson have Gaussian wave
é( y- a )
ù
ê
ú
function
ê
ú
- 1/ 4
2
d
ë
û
y i ( y; ai ) : (p ML) e
.
 Matrix elements are given by overlap integral

2
i
2
l ij(u ) = gM ò dy y ( y; ai ) y ( y; b j ) y ( y,0),
l kj( d ) = gM ò dy y ( y; ck ) y ( y; b j ) y ( y,0).
The common wave functions of quark doublets
and the Higgs boson introduce the correlation.
Neutrino Physics

The see-saw mechanism
Assume non-localized wavefunctions for 5 s.
 Introduce complex phases.
 Calculate the Majorana mass term of RH neutrino
by
cij = ò dy y n ( y; yi ) y n ( y; y j ) j ( y; y* ).

R ,i

R, j
SB
Neutrino masses: hierarchy of all three matrices
m1 = m
add up. Hence very hierarchical see-saw
masses.
solar , matm .

Mixing angle distributions:


Bi-large mixing possible.
CP phase distribution
P(sin q13 > 3´ 10- 2 ) » 96%.
log10 (sin q13 )

The Standard Model of particle physics has
3(gauge)+22(Yukawa)+2(Higgs)+1 parameters.

What can we learn from the 20 observables in
the Yukawa sector?

maybe ... not much. It does not seem that there is a
beautiful and fundamental relation that governs all
the Yukawa-related observables.
mc = 1.25GeV mb = 4.20GeV
mm = 106MeV

| Vub |= 3.96´ 10- 3
D me2 = 8.0´ 10- 5 eV2
sin 2 (2q12 ) = 0.86
though they have a certain hierarchical pattern
theories of flavor (very simplified)

Flavor symmetry and its small breaking
L=
å
a

ca
F ija
M*
l ij =
y i y jf ,
å
a
ca
F ija
M*
.
Predictive approach:
use
less-than
20
independent
a
F
parameters ca ij
to derive predictions.
M*

Symmetry-statistics hybrid approach:
Use a symmetry to explain the hierarchical pattern.
 The coefficients ca are just random and of order unity.

ex. symmetry-statistics hybrid


an approximate U(1) symmetry broken by F = eM*.
e = 1.
U(1) charge assignment (e.g.)
101 = (Q,U , E)1
ni
æF
l ij = cij ççç
èM
3
102
103
2
0
51,2,3 = ( D, L)1,2,3 h
0
n + nj
i
ö
÷
÷
÷
÷
ø
*
,
cij
are random coefficients of order unity.
l u : l c : l t » e 6 : e 4 :1,
l d : l s : l b » e 3 : e 2 :1,
Vus : Vub : Vcb » e : e 3 : e 2 .
0
pure statistic approaches

Multiverse / landscape of vacua
best solution ever of the CC problem
 supported by string theory (at least for now)


Random coefficients fit very well to this
framework.

But, how can you obtain hierarchy w/o a
symmetry?
randomly generated matrix elements

Neutrino anarchy
Hall Murayama Weiner ’99
Haba Murayama ‘00
Generate all 5 -related matrix elements independently,
following a linear measure dP(l ) µ dl ,
 explaining two large mixing angles.


Power-law landscape for the quark sector
Donoghue Dutta Ross ‘05

Generate 18 matrix elements independently, following
L = l ij y i y jf ,
dP(l ) µ l - 1+ ddl .

The best fit value is d = - 0.16.
Let us examine the power-law model more closely
for the scale-invariant case d = 0.
xi =
ln(l i / l min )
.
ln(l max / l min )
Results: (eigenvalue distributions)
dP( x1,2,3 ) » dx1dx2dx3 36x34 x22.
x1 £ x2 £ x3.
æl min
ln
Hierarchy is generated from statistics for moderately large ççè
l
ö
÷
.
÷
ø
max
mixing angle distributions
e.g.
L
2
L
1
u - d
u1L - d3L u3L - d2L pairing
VCKM
æ
1ö
÷
çç
÷
÷
ç
÷
: ç1
.
÷
çç
÷
÷
çè 1 ÷
ø
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 structure
correlation between the up and down Yukawa matrices

Introduce a toy landscape on an extra dimension
S1 [0 : L].
Quarks and Higgs boson have Gaussian wave
é( y- a )
ù
ê
ú
function
ê
ú
- 1/ 4
2
d
ë
û
y i ( y; ai ) : (p ML) e
.
 Matrix elements are given by overlap integral

2
i
2
l ij(u ) = gM ò dy y ( y; ai ) y ( y; b j ) y ( y,0),
l kj( d ) = gM ò dy y ( y; ck ) y ( y; b j ) y ( y,0).
The common wave functions of quark doublets
and the Higgs boson introduce the correlation.
inspiration

in certain compactification of Het. string theory,
W=


ò WÙ(dB -
tr (AdA - AAA)).
Yukawa couplings originate from overlap integration.
Domain wall fermion, Gaussian wavefunctions
and torus fibration  see next page.
domain wall fermion and
torus fibration

5D fermion in a scalar background


2
T
6D on
with a gauge flux F on it.



Gaussian wavefunction at the domain wall.
A6 = F56 ( x5 - x5,0 ) looks like a scalar bg. in 5D.
æ
ö
chiral fermions in eff. theory: ò ççç F56 ÷÷÷.
è 2p ø
Generalization: T 3 -fibration on a 3-fold B.

introducing “Gaussian Landscapes” (toy models)
calculate Yukawa matrix by overlap integral on a mfd B
 use Gaussian wavefunctions
 scan the center coordinates of Gaussian profiles


Results: try first for the easiest B = S .
1

Distribution of Yukawa couplings (ignoring correlations)
scale invariant
distribution
d / L = 0.08.

D ln l : ( L / d )2 /11.
To understand more analytically....
2
2
é
ù
a
+
b
a
b
1/ 4
i
j
i
j
(u )
ú.
 l ij » g (4Md / 9p ) exp êê2
ú
3d
êë
ú
û
(e
- ai2 / 3 d 2
- b 2j / 3 d 2
)´ (e
2
2
æda db ö
p
dr
3
p
d
d (ln l )
÷
ç
=
»
.
 dP(l ) » çç 2 ÷
2
2
÷
è L ø
L
L
2
2
2
r := a + b ,
2
r
- ln l :
2


æd ö
dP(l )
= 3p çç ÷
.
÷
÷
ç
èLø
d ln l
FN factor distribution 
3d
)´ ( e
- aib j / 3 d 2
)
=1.
Froggatt—Nielsen type
mass matrices
2.
Distribution of Observables
d / L = 0.08.

Three Yukawa eigenvalues (the same for u and d sectors)

Three mixing angles
family pairing
The family pairing originates from the localized wave functions of
. H.
qL &
quick summary







hierarchy from statistics
Froggatt—Nielsen like Yukawa matrices
hence family pairing structure
FN charge assignment follows automatically.
The scale-invariant distr. follows for B = S1.
Geometry dependence?
How to accommodate the lepton sector?
Geometry Dependence
exploit the FN approximation

FN suppression factor for q or qbar:


2
ær ö
e= e
;
y º ln e µ - çç ÷
:
÷
÷
çèd ø
dP
d 2 dV (r 2 )
= f ( y) µ
.
2
dy
V dr
- r 2 /3d 2
FN factors: the largest, middle and smallest of three
randomly chosen FN factors as above.
dP( y1, y2 , y3 )
= 3! f ( y1 ) f ( y2 ) f ( y3 )
dy1dy2dy3
Q( y3 - y2 )Q( y3 - y1 )Q ( y2 - y1 ).
compare

FN factors:
B= T
2
/ eigenvalues /
and S
2
mixing angles

dP
d 2 dV (r 2 )
= f ( y) µ
,
2
dy
V dr
The original
carrying info. of
geometry B, is integrated once or twice in
obtaining distribution fcns of observables.
details tend to be smeared out.
 power/polynomial fcns of log of masses / angles in
Gaussian landscapes.
 broad width (weak predictability)


Dimension dependence: FN factor distribution
S
2
S3
Neutrino Physics

The see-saw mechanism
Assume non-localized wavefunctions for 5 s.
 Introduce complex phases.
 Calculate the Majorana mass term of RH neutrino
by
cij = ò dy y n ( y; yi ) y n ( y; y j ) j ( y; y* ).

R ,i

R, j
SB
Neutrino masses: hierarchy of all three matrices
m1 = m
add up. Hence very hierarchical see-saw
masses.
solar , matm .

Mixing angle distributions:


Bi-large mixing possible.
CP phase distribution
P(sin q13 > 3´ 10- 2 ) » 96%.
log10 (sin q13 )
In Gaussian Landscapes,





Family structure from overlap of localized
wavefunctions.
FN structure with hierarchy w/o flavor sym.
Broad width distributions.
Non-localized wavefunctions for 5 .
No FN str. in RH Majorana mass term
large hierarchy in the see-saw neutrino masses.
 Large probability for observable | U e 3 | .

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
should come from scanning vector-bdle moduli.
Instanton (gauge field on 4-mfd not 6-mfd) moduli space
is known better.
æ
ö
r
÷
ç
÷
1+ å
,
 In the t Hooft solution, A = - h ¶ ln ç
÷
ç
÷
(y- y ) ø
ur è
the instanton-center coordinates y can be chosen freely.
I
2
a
a
m
mn
I
n
2
I

I
F-theory (or IIB) flux compactification can be used to
study the scanning of complex-structure (vector bundle
in Het) moduli.