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  M = m, where  << 1  1 This matrix has two eigevalues, m 2 = m and m 1 =   m. The mixing angle is  m 1 /m 2 ) 1/2. Thus the mixing angle is related to the ratio of masses. This works for quarks where the Cabibbo angle  C ~ (m d /m s ) 1/2

3. What is known empirically? a) Two mass differences:  m 21 2 ~ 8 x 10 -5 eV 2  m 32 2 ~ 2 x 10 -3 eV 2 their ratio,  m 21 2 /  m 32 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 sin2   sin2   sin2   sin  i.e. CP violation is unobservable if   or  vanish.

4. 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 45 0 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 another symmetry.

5. 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 S n of permutation of of n elements, and in particular the chain S 1 – S 2 – S 3. 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

6. The most general hermitian class operator is s+t+u u u M  = sC(1) + tC(2) + uC(3) = u s+u t+u u t+u s+u With real s,t,u. The eigenvalues are: m 1 2 = s+t, m 2 2 = s + t + 3u, m 3 2 = s – t. When M  is diagonalized, one arrives at the tri-bimaximal matrix.

7. Now consider a more general approach, with matrices I, P(e  ), P(  ), P(  e), and 0 1 0 0 0 1 P(e,  = 0 0 1 and P(  e) = 1 0 0 1 0 0 0 1 0 The most general M  matrix is M   a  b  P(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 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  eigenvector is automatically maximally mixed.

8. The matrix M   is 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 -1 M  = z y x + i Im(b) -1 0 1 y x z 1 -1 0 We have thus separated M   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  is maximally mixed.

9. 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 sin 2    and | sin  | = 1 ) The mixing angle    is not constrained, |U e3 | = sin    = (2/3) 1/2 sin  tan 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 ~ -  m 2 atm /2, y = z =  m 2 atm /3(  m 2 sol /  m 2 atm – sin   ), d = (3) 1/2 Im(b) =  m 2 atm /2 sin 2  sin 2  = 3/2 sin 2   <<1

10. Now, since  m 2 sol /  m 2 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  m 2 sol /  m 2 atm = sin  , i.e. we have found a relation between the mass differences and the unknown mixing angle  or    This then yields a testable prediction |U e3 | = sin    (2  m 2 sol / 3  m 2 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  m 2 atm /2 = (d 2 + x 2 ) 1/2, and there is one small parameter d/(3) 1/2 x ~ 0.2. It is not clear why this is so, but one needs to explain only one small parameter and the assumed symmetries.

11. Note that the predicted    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 sin   ~ (m s /2m b ) 1/2 ~ 0.12 However, there are many other model builders who predict much smaller value of sin  

12. 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 try to improve substantially the results of the CHOOZ and Palo Verde experiments. The other possibility is to use the long baseline  beam from an accelerator and look for the electron appearance.

14. 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 sin 2   determination. The present limit corresponds to ~4% (but at a distance where sin  atm < 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).

15. 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  m 2 atm. Also, effects of the CP phase  must be considered. There is a large number of “parameter degeneracies.

16. A more complete formula with the effects of the CP phase  and the lowest order matter effects included is:

18. Again, there is a large number of proposals. Clearly, knowing sin 2   from a reactor experiment would help in reducing the degeneracies. Altogether, determination of sin 2   and  is clearly the next big issue and will keep people busy for the next decade.