From Cosmological Constant to Sin Distribution

1
From Cosmological Constantto Sin
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)

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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?

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Cosmological Constant Problem

• Extremely difficult to explain
• A possible solution by S. Weinberg 87
• Structures (such as galaxies)
  formed only for moderate Cosmological
  Constant.
• Thats where we find ourselves.

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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 ??

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What if other parameters (Yukawa)are also
scanning?

• Do we naturally obtain
• hierarchical Yukawa eigenvalues,
• generation structure in the quark sector,
• but not for the lepton sector?

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A toy model generating statistics

• In string theory compactification,
• Use Gaussian wavefunctions in overlap integral
• equally-separated hierarchically small Yukawas.

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Generation Structure

• With random Yukawa matrix elements,
• In our toy model,

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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

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Lepton Sector Predictions

• Mixing angles without cuts
• Two large angles,
• After imposing cuts

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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
  down-type Yukawa matrices
• Localized wavefunctions of q and h are the origin
  of generations.
• Successful distributions for the lepton sector,
  too,
• with very large

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spare slides
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• Family pairing structure
• correlation between the up and down Yukawa
  matrices
• Introduce a toy landscape on an extra dimension
• Quarks and Higgs boson have Gaussian wave
  function
• Matrix elements are given by overlap integral
• The common wave functions of quark
  doublets
• and the Higgs boson introduce the correlation.

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Neutrino Physics
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• The see-saw mechanism
• Assume non-localized wavefunctions for s.
• Introduce complex phases.
• Calculate the Majorana mass term of RH neutrino
  by
• Neutrino masses hierarchy of all three matrices
  add up. Hence very hierarchical see-saw masses.

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• Mixing angle distributions
• Bi-large mixing possible.
• CP phase distribution

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• 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.
• though they have a certain hierarchical pattern

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theories of flavor (very simplified)

• Flavor symmetry and its small breaking
• Predictive approach use less-than 20 independent
  parameters to derive predictions.
• Symmetry-statistics hybrid approach
• Use a symmetry to explain the hierarchical
  pattern.
• The coefficients are just random and of
  order unity.

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ex. symmetry-statistics hybrid

• an approximate U(1) symmetry broken by
• U(1) charge assignment (e.g.)

3
0
0
0
2
are random coefficients of order unity.
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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?

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randomly generated matrix elements
Hall Murayama Weiner 99 Haba Murayama 00

• Neutrino anarchy
• Generate all -related matrix elements
  independently, following a linear measure
• explaining two large mixing angles.
• Power-law landscape for the quark sector
• Generate 18 matrix elements independently,
  following
• The best fit value is

Donoghue Dutta Ross 05
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Let us examine the power-law model more closely
for the scale-invariant case
Results (eigenvalue distributions)
Hierarchy is generated from statistics for
moderately large
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mixing angle distributions
pairing
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?
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• Family pairing structure
• correlation between the up and down Yukawa
  matrices
• Introduce a toy landscape on an extra dimension
• Quarks and Higgs boson have Gaussian wave
  function
• Matrix elements are given by overlap integral
• The common wave functions of quark
  doublets
• and the Higgs boson introduce the correlation.

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inspiration

• in certain compactification of Het. string
  theory,
• Yukawa couplings originate from overlap
  integration.
• Domain wall fermion, Gaussian wavefunctions and
  torus fibration ? see next page.

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domain wall fermion and torus fibration

• 5D fermion in a scalar background
• Gaussian wavefunction at the domain wall.
• 6D on with a gauge flux F on it.
• looks like a
  scalar bg. in 5D.
• chiral fermions in eff. theory
• Generalization -fibration on a 3-fold B.

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• 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
• Distribution of Yukawa couplings (ignoring
  correlations)

scale invariant distribution
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To understand more analytically....

• FN factor distribution ?

FroggattNielsen type mass matrices
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Distribution of Observables

• 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 .
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quick summary

• hierarchy from statistics
• FroggattNielsen like Yukawa matrices
• hence family pairing structure
• FN charge assignment follows automatically.
• The scale-invariant distr. follows for
• Geometry dependence?
• How to accommodate the lepton sector?

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Geometry Dependence
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exploit the FN approximation

• FN suppression factor for q or qbar
• FN factors the largest, middle and smallest of
  three randomly chosen FN factors as above.

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compare and

• FN factors / eigenvalues / mixing
  angles

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• 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

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Neutrino Physics
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• The see-saw mechanism
• Assume non-localized wavefunctions for s.
• Introduce complex phases.
• Calculate the Majorana mass term of RH neutrino
  by
• Neutrino masses hierarchy of all three matrices
  add up. Hence very hierarchical see-saw masses.

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• Mixing angle distributions
• Bi-large mixing possible.
• CP phase distribution

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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 .
• No FN str. in RH Majorana mass term
• large hierarchy in the see-saw neutrino masses.
• Large probability for observable .

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The scale invariant distribution of Yukawa
couplings for B S1 becomes
for B T2,
for B S2.
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• 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.
• In the t Hooft solution,
  
  the instanton-center coordinates
  can be chosen freely.
• F-theory (or IIB) flux compactification can be
  used to study the scanning of complex-structure
  (vector bundle in Het) moduli.