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 someunderlying symmetries. Based on them find valuesor ranges for the so far unknown parameters.

Counting parameters: 3 mixing angles 1 CP phase 2 mass differences 1 absolute mass scale ------------------------ 7 altogether

A simple example:Let the mass matrix for two flavors be of the form(in flavor basis): 0 M = m , where << 1 1

This matrix has two eigevalues, m2= m and m1 = m.

The mixing angle is m1/m2)1/2.

Thus the mixing angle is related to the ratioof masses. This works for quarks where the Cabibboangle C ~ (md/ms)1/2

What is known empirically?

a) Two mass differences: m212 ~ 8 x 10-5 eV2

m322 ~ 2 x 10-3 eV2

their ratio, m212/ m32

2 ~ 0.04 is a small number.

b) Two mixing angles, are large and reasonably well determined. The third mixing angle, is only constrained from above, sin < 0.17.

Perhaps sin is another small parameter.

c) Nothing is known about the CP phase .

Yet we would like to know and , since CP violation

effects are proportional to sin2sin2sin2sini.e. CP violation is unobservable if or vanish.

The mixing matrix as of now looks like this:

e 0.84 0.54 0.0(0.17)U = -0.38(-0.48) 0.60(0.54) -0.71 -0.38(-0.28) 0.60(0.66) 0.71

Here the first entry is for = 0 and the (second)

for = 0.17 i.e. the maximum allowed value.

(The possible deviation of from 450 is neglected,

also, the CP phase is assumed to vanish.)Note that the second column looks like a constant

made of 1/ 3 = 0.58, i.e. as if is maximally mixed.

The and lines are almost identical suggesting anothersymmetry.

In fact, this matrix resembles the tri-bimaximal matrix

e (2/3)1/2 (1/3)1/2 0

U = -(1/6)1/2 (1/3)1/2 -(1/2)1/2

-(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() = 0 0 1

0 0 1 0 1 0

1 1 1

and C(3) = P(e) + P() + P(e) = 1 1 1 i.e. democratic

1 1 1

The most general hermitian class operator is

s+t+u u uM

= sC(1) + tC(2) + uC(3) = u s+u t+u

u t+u s+u

With real s,t,u. The eigenvalues are:

m12 = s+t, m2

2 = s + t + 3u, m32 = s – t.

When M is diagonalized, one arrives at the tri-bimaximal

matrix.

Now consider a more general approach, with matrices I, P(e), P() , P(e), and 0 1 0 0 0 1P(e, = 0 0 1 and P(e) = 1 0 0 1 0 0 0 1 0

The most general M matrix is

MabP(e, + b* P(e) + x P() + y P(e) + z P(e),

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 ofthese parameters. Parameter a only affects absolute masses, nomass square differences or eigenvectors. Also, since the P arenot independent, we can add a real constant to a and b, providedwe subtract the same from x,y,z.The eigenvector is automatically maximally mixed.

The matrix Mis the most general 3x3 matrix that commutes

with the democracy operator D,

[M, D ] = 0. Hence the eigenvectors are eigenstates of D,

with and corresponding to D = 0, and to D = 1.

By ignoring a, and putting Re(b) = 0, we can rewrite M as

x z y 0 1 -1M

= z y x + i Im(b) -1 0 1

y x z 1 -1 0

We have thus separated Minto the real and imaginary parts

and reduced the number of parameters to four.

This is a consequence of the democracy invariance or in otherwords, the requirement that is maximally mixed.

Further simplification is achieved by requiring permutation

symmetry in accord with the empirical evidence.This is achieved by setting y = z.(In the standard parametrization this corresponds to sin2 and | sin | = 1 )

The mixing angle is not constrained,

|Ue3| = sin = (2/3)1/2 sintan 2 = (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 ~ -m2atm/2, y = z = m2

atm/3(m2sol/ m2

atm – sin),

d = (3)1/2Im(b) = m2atm/2 sin 2sin2 = 3/2 sin2 <<1

Now, since m2sol/ m2

atm << 1, and sin< 1 from experiment,

we see that y << x.What happens if we require that z = y = 0?

To have that, we must have m2sol/ m2

atm = sin, i.e.

we have found a relation between the mass differencesand the unknown mixing angle or

This then yields a testable prediction

|Ue3| = sin(2 m2sol/ 3m2

atm)1/2 ~ 0.13 +- 0.03

This is the “simplest mixing matrix”. The large parameters, and are fixed by the assumed symmetries. The overall

scale is m2atm/2 = (d2 + x2)1/2, and there is one small

parameterd/(3)1/2x ~ 0.2. It is not clear why this is so, but one needsto explain only one small parameter and the assumed

symmetries.

Note that the predicted is near the empirical upper limit

and thus relatively easy to check.Unrelated “prediction” by Ramond hep-ph/0401001relates the mixing angles to quark masses, but givesa similarly optimistic value sin~ (ms/2mb)1/2 ~ 0.12

However, there are many other model builders who predictmuch smaller value of sin

The discussion shows the important role of the .

How can one measure this quantity?

is related to -> e or e -> oscillations

(e -> is clearly impossible or very difficult).

One possibility is to use nuclear reactor and tryto improve substantially the results of the CHOOZand Palo Verde experiments.

The other possibility is to use the long baseline beam from an accelerator and look for the

electron appearance.

Thus, choosing the distance L such that sin atm ~ 1,

the accuracy with which the e flux can be determined

corresponds to the accuracy of sin2determination.

The present limit corresponds to ~4% (but ata distance where sin atm < 1). At a

substantially better accuracy, the systematicerrors would dominate and a second “monitor”detector is needed.

There are plans to perform such measurementsin France (“DoubleCHOOZ”), US (Diablo Canyon andBraidwood), a China (Daya Bay).

Another possibility is to use an accelerator beam

with E ~ 1 GeV and L ~ 1000km and look for the

e 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

m2atm.

Also, effects of the CP phase must be considered.There is a large number of “parameter

degeneracies.

A more complete formula with the effects of the CP phaseand the lowest order matter effects included is:

Again, there is a large number of proposals.Clearly, knowing sin2 from a reactor experiment

would help in reducing the degeneracies.

Altogether, determination of sin2and is clearly

the next big issue and will keep people busy forthe next decade.