Model building:

1
Model building The simplest neutrino mass
matrix
see Harrison and Scott Phys. Lett. B594, 324
(2004), hep-ph/0402006, Phys. Lett. B557, 76
(2003). Basic idea use the experimental data
and guess some underlying symmetries. Based on
them find values or ranges for the so far unknown
parameters. Counting parameters 3 mixing
angles 1 CP
phase 2 mass
differences 1
absolute mass scale
------------------------
7 altogether
2

• A simple example
• Let the mass matrix for two flavors be of the
  form
• (in flavor basis)
• 0 e
• M m , where e ltlt 1
• e 1
• This matrix has two eigevalues, m2 m and m1
  e2m.
• The mixing angle is q e (m1/m2)1/2.
• Thus the mixing angle is related to the ratio
• of masses. This works for quarks where the
  Cabibbo
• angle qC (md/ms)1/2

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• What is known empirically?
• a) Two mass differences Dm212 8 x 10-5 eV2
• Dm322 2 x
  10-3 eV2
• their ratio, Dm212/ Dm322 0.04 is a small
  number.
• b) Two mixing angles, q23 450, q12 350 are
  large and
• reasonably well determined. The third mixing
  angle,
• q13 is only constrained from above, sinq13 lt
  0.17.
• Perhaps sinq13 is another small parameter.
• c) Nothing is known about the CP phase d.
• Yet we would like to know q13 and d, since CP
  violation
• effects are proportional to sin2q12 sin2q23
  sin2q13 sind,
• i.e. CP violation is unobservable if q13 or d
  vanish.

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• The mixing matrix as of now looks like this
• n1 n2
  n3
• e 0.84 0.54
  0.0(0.17)
• U m -0.38(-0.48) 0.60(0.54) -0.71
• t -0.38(-0.28) 0.60(0.66) 0.71
• Here the first entry is for q13 0 and the
  (second)
• for q13 0.17 i.e. the maximum allowed value.
• (The possible deviation of q23 from 450 is
  neglected,
• also, the CP phase d is assumed to vanish.)
• Note that the second column n2 looks like a
  constant
• made of 1/ 3 0.58, i.e. as if n2 is maximally
  mixed.
• The m and t lines are almost identical suggesting
  another
• symmetry.

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• In fact, this matrix resembles the tri-bimaximal
  matrix
• e (2/3)1/2 (1/3)1/2
  0
• U m -(1/6)1/2 (1/3)1/2
  -(1/2)1/2
• t -(1/6)1/2
  (1/3)1/2 (1/2)1/2
• Consider groups Sn of permutation of of n
  elements, and in
• particular the chain S1 S2 S3. One has three
  Class operators
• 1 0 0
  1 0 0
• C(1) I 0 1 0 C(2) P(mt) 0
  0 1
• 0 0 1
  0 1 0
• 
  1 1 1
• and C(3) P(em) P(mt) P(te) 1 1
  1 i.e. democratic
• 
  1 1 1

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• The most general hermitian class operator is
• 
  stu u u
• Mn2 sC(1) tC(2) uC(3) u
  su tu
• 
  u tu su
• With real s,t,u. The eigenvalues are
• m12 st, m22 s t 3u, m32 s t.
• When Mn2 is diagonalized, one arrives at the
  tri-bimaximal matrix.

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• Now consider a more general approach, with
  matrices
• I, P(em), P(mt) , P(te), and
• 0 1 0
  0 0 1
• P(e,m,t) 0 0 1 and P(t,m,e)
  1 0 0
• 1 0 0
  0 1 0
• The most general Mn2 matrix is
• Mn2 aI b P(e,m,t) b P(t,m,e) x P(mt)
  y P(te) z P(em),
• where a,x,y,z are real and b is complex.
• This is a general representation of S3 group.
• One can now express the eigenvalues and
  eigenvectors in terms of
• these parameters. Parameter a only affects
  absolute masses, no
• mass square differences or eigenvectors. Also,
  since the P are
• not independent, we can add a real constant to a
  and b, provided
• we subtract the same from x,y,z.
• The n2 eigenvector is automatically maximally
  mixed.

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• The matrix Mn2 is the most general 3x3 matrix
  that commutes
• with the democracy operator D,
• Mn2, D 0. Hence the eigenvectors are
  eigenstates of D,
• with n1 and n3 corresponding to D 0, and n2 to
  D 1.
• By ignoring a, and putting Re(b) 0, we can
  rewrite Mn2 as
• x z y 0
  1 -1
• Mn2 z y x i Im(b) -1 0 1
• y x z 1
  -1 0
• We have thus separated Mn2 into the real and
  imaginary parts
• and reduced the number of parameters to four.
• This is a consequence of the democracy invariance
  or in other
• words, the requirement that n2 is maximally
  mixed.

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• Further simplification is achieved by requiring m
  - t permutation
• symmetry in accord with the empirical evidence.
• This is achieved by setting y z.
• (In the standard parametrization this corresponds
  to
• sin2 q23 1 , and sin d 1 )
• The mixing angle q13 is not constrained,
• Ue3 sin q13 (2/3)1/2 sinc , tan 2c
  (3)1/2 Im(b)/(x-y),
• We are now left with three parameters, x, y,
  Im(b).
• We can express them in terms of observables
• x -Dm2atm/2, y z Dm2atm/3(Dm2sol/ Dm2atm
  sin2c),
• d (3)1/2Im(b) Dm2atm/2 sin 2c, sin2 c 3/2
  sin2q13 ltlt1

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• Now, since Dm2sol/ Dm2atm ltlt 1, and sinc lt 1
  from experiment,
• we see that y ltlt x.
• What happens if we require that z y 0?
• To have that, we must have Dm2sol/ Dm2atm
  sin2c , i.e.
• we have found a relation between the mass
  differences
• and the unknown mixing angle c or q13.
• This then yields a testable prediction
• Ue3 sinq13 (2 Dm2sol/ 3Dm2atm)1/2 0.13 -
  0.03
• This is the simplest mixing matrix. The large
  parameters
• q23,q12 and d are fixed by the assumed
  symmetries. The overall
• scale is Dm2atm/2 (d2 x2)1/2, and there is
  one small parameter
• d/(3)1/2x 0.2. It is not clear why this is so,
  but one needs
• to explain only one small parameter and the
  assumed symmetries.

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• Note that the predicted q13 is near the
  empirical upper limit
• and thus relatively easy to check.
• Unrelated prediction by Ramond hep-ph/0401001
• relates the mixing angles to quark masses, but
  gives
• a similarly optimistic value
• sinq13 (ms/2mb)1/2 0.12
• However, there are many other model builders who
  predict
• much smaller value of sinq13.

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• The discussion shows the important role of the
  q13.
• How can one measure this quantity?
• q13 is related to m -gt e or e -gt m oscillations
• (e -gt t is clearly impossible or very difficult).
• One possibility is to use nuclear reactor and try
• to improve substantially the results of the CHOOZ
• and Palo Verde experiments.
• The other possibility is to use the long baseline
• nm beam from an accelerator and look for the
• electron appearance.

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• Thus, choosing the distance L such that sin Datm
  1,
• the accuracy with which the ne flux can be
  determined
• corresponds to the accuracy of sin22q13
  determination.
• The present limit corresponds to 4 (but at
• a distance where sin Datm lt 1). At a
• substantially better accuracy, the systematic
• errors would dominate and a second monitor
• detector is needed.
• There are plans to perform such measurements
• in France (DoubleCHOOZ), US (Diablo Canyon and
• Braidwood), a China (Daya Bay).

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• Another possibility is to use an accelerator nm
  beam
• with En 1 GeV and L 1000km and look for the
• ne appearance.
• Such a beam will go necessarily through a large
  amount
• of matter and thus matter effects must be
  included.
• Note that matter effects depend on the sign of
  Dm2atm.
• Also, effects of the CP phase d must be
  considered.
• There is a large number of parameter
  degeneracies.

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A more complete formula with the effects of
the CP phase d and the lowest order matter
effects included is
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• Again, there is a large number of proposals.
• Clearly, knowing sin22q13 from a reactor
  experiment
• would help in reducing the degeneracies.
• Altogether, determination of sin22q13 and d is
  clearly
• the next big issue and will keep people busy for
• the next decade.