Symmetry Origin of Observable Nonunitary Neutrino Mixng Matrix in TeV Scale Seesaw Models

Transcription

1 Symmetry Origin of Observable Nonunitary Neutrino Mixng Matrix in TeV Scale Seesaw Models Ernest Ma Physics and Astronomy Department University of California Riverside, CA 92521, USA Symmetry Origin of Observable Nonunitary Neutrino Mixing Matrix (INT2010) back to start 1

2 Contents Seesaw Variations Zero Neutrino Mass from Cancellation Observable Nonunitary Neutrino Mixing Matrix Symmetry Origin of the Texture Hypothesis Concluding Remarks Symmetry Origin of Observable Nonunitary Neutrino Mixing Matrix (INT2010) back to start 2

3 Seesaw Variations With 1 doublet neutrino ν and 1 singlet neutrino N, their 2 2 mass matrix is the well-known ( ) 0 md M νn =, m D m N resulting in the famous seesaw formula m ν m D 2 /m N. Hence ν N mixing m D /m N m ν /m N < 10 6, for m ν < 1 ev and m N > 1 TeV. Symmetry Origin of Observable Nonunitary Neutrino Mixing Matrix (INT2010) back to start 3

4 Consider now 1 ν and 2 singlets: N 1,2. Their 3 3 mass matrix is then 0 m D 0 M νn = m D m 1 m N, 0 m N m 2 resulting in m ν m D 2 m 2 /(m N 2 m 1 m 2 ). Since the limit m 1 = 0 and m 2 = 0 corresponds to lepton number conservation (L = 1 for ν and N 2, L = 1 for N 1 ), their smallness is natural the inverse seesaw [Mohapatra/Valle(1986)]. Symmetry Origin of Observable Nonunitary Neutrino Mixing Matrix (INT2010) back to start 4

5 Here ν N 1 mixing m ν /m D, whereas ν N 2 mixing m D /m N, and they are not constrained to be the same as was in the canonical seesaw. For example, let m D 10 GeV, m N 1 TeV, m 2 10 kev, then m ν 1 ev, and ν N 1 mixing is very small, but ν N 2 mixing 10 2 is large enough to be observable. Note that the geometric mean of the two mixings is again 10 6 as before. Symmetry Origin of Observable Nonunitary Neutrino Mixing Matrix (INT2010) back to start 5

6 Linear Seesaw: [Barr(2004), Malinsky/Romao/Valle(2005)] 0 m D m D M νn = m D 0 m N m D m N 0 m ν 2m D m D /m N, which is only linear in m D. Ma(2009): For the linear seesaw to work, m D must be very small. In the limit it is zero, M νn is the same as that of the inverse seesaw in the same limit, so they must have the same origin. Symmetry Origin of Observable Nonunitary Neutrino Mixing Matrix (INT2010) back to start 6

7 To prove this, let m D /m D = tan θ, then rotate the (N 1, N 2 ) basis by θ, we get 0 m D /c 0 M νn = m D /c m N s 2 m N c 2 0 m N c 2 m N s 2 where c = cos θ, c 2 = cos 2θ, and s 2 = sin 2θ. For small θ, this is just the inverse seesaw, with m ν 2m Nm ( ) 2 D md = 2m Dm D. m D m 2 N m N Symmetry Origin of Observable Nonunitary Neutrino Mixing Matrix (INT2010) back to start 7

8 Zero Neutrino Mass from Cancellation If the inverse or linear seesaw mechanisms are invoked to get large ν N mixing for each family, we will need 2 singlet neutrinos for each doublet neutrino, thus implying a 9 9 mass matrix. Is that really necessary? Buchmuller/Wyler(1990), Pilaftsis(1992,2005), Kersten/Smirnov(2007): Consider first two families: ν 1,2 and N 1,2, with M N = diag(m 1, M 2), where ( ) ( ) a1 b M D = 1 a 1 b 2 a1 = ( b a 2 b 1 a 2 b 2 a 1 b 2 ). 2 Symmetry Origin of Observable Nonunitary Neutrino Mixing Matrix (INT2010) back to start 8

9 One neutrino mass is zero because the determinant of M D is zero by construction. If we now also impose the arbitrary condition b 2 1/M 1 + b 2 2/M 2 = 0, then the other neutrino mass is zero as well. However, ν N mixing may be large, because a i b j need not be small. In other words, zero mixing is now not the limit of zero neutrino mass as in the previous seesaw formulas. If small deviations from this texture are present, small neutrino masses will appear, but the large ν N mixing will remain. This appears to be fine tuning, but a lot of phenomenological studies have been done in this context. Symmetry Origin of Observable Nonunitary Neutrino Mixing Matrix (INT2010) back to start 9

10 Observable Nonunitary Neutrino Mixing Matrix The addition of heavy singlet fermions N i to the Standard Model induces both the well-known dimension-five operator Λ 1 f αβ (L α Φ)(L β Φ) [mass mixing] and the less-known dimension-six operator Λ 2 f αβ (Φ Lα )i µ γ µ (L β Φ) [kinetic mixing]. Symmetry Origin of Observable Nonunitary Neutrino Mixing Matrix (INT2010) back to start 10

11 Both operators will probe m D /m N, which is of course too small in the canonical seesaw, but if the inverse seesaw or linear seesaw or the texture hypothesis is used, then they will provide useful phenomenological constraints. Antusch/Biggio/Fernandez-Martinez/Gavela/Lopez- Pavon(2006), Antusch/Baumann/Fernandez-Martinez(2009): Let the neutrino mixing matrix be (1 + η)u, then η < Symmetry Origin of Observable Nonunitary Neutrino Mixing Matrix (INT2010) back to start 11

12 Ohlsson/Popa/Zhang(2010): For the texture hypothesis, i.e. a 1 M D = a 2 ( b 1 b 2 b 3 ) a 3 with the condition b 2 1/M 1 + b 2 2/M 2 + b 2 3/M 3 = 0, the parameters η eτ and η µτ are correlated: η eτ η µτ < 1. Symmetry Origin of Observable Nonunitary Neutrino Mixing Matrix (INT2010) back to start 12

13 Symmetry Origin of the Texture Hypothesis He/Ma(2009): To understand the mechanism and symmetry of the texture hypothesis, change the neutrino basis to M νn = 0 0 m m 2 m 1 0 M 1 M 3 0 m 2 M 3 M 2. Symmetry Origin of Observable Nonunitary Neutrino Mixing Matrix (INT2010) back to start 13

14 If M D of the texture hypothesis is now rotated on the left with tan 1 (a 1 /a 2 ) and on the right with tan 1 (b 1 /b 2 ), then m 1 = 0 automatically. Furthermore, M 1 = cos 2 θ R M 1 + sin 2 θ R M 2 = (b 2 1/M 1 + b 2 2/M 2)M 1M 2/(b b 2 2) = 0 as well. Hence ν 1 and ν 2 = (M 3 ν 2 m 2 N 1 )/ M m2 2 are massless, the latter showing also how the large mixing occurs between ν 2 and N 1, i.e. through the inverse seesaw mechanism. However, lepton number conservation would not only forbid M 1 but also M 2, which is arbitrary here. Where is the symmetry which does this? Symmetry Origin of Observable Nonunitary Neutrino Mixing Matrix (INT2010) back to start 14

15 Let ν 1,2, N 1,2 have L = 1, 1, 3, 1. Add the usual Higgs doublet (φ + 1, φ0 1) with L = 0 and the Higgs singlet χ 2 with L = 2. Then m 2 comes from φ 0 1, M 2 from χ 2, and M 3 from χ 2. m 1 = 0 at tree level, because there is no Higgs doublet with L = 4, and M 1 = 0 at tree level, because there is no Higgs singlet with L = ±6. In one loop, M 1 will be induced, thus giving ν 2 an inverse seesaw mass = M 1 m 2 2/M 2 3. Once ν 2 is massive, ν 1 also gets a two-loop radiative mass from the exchange of two W s [Babu/Ma(1988)]. Symmetry Origin of Observable Nonunitary Neutrino Mixing Matrix (INT2010) back to start 15

16 χ 2 χ 2 N 1 N 2 N 1 Figure 1: One-loop generation of M 1. Symmetry Origin of Observable Nonunitary Neutrino Mixing Matrix (INT2010) back to start 16

17 W ν 2,3 l ν 1 ν 1 l ν 2,3 W Figure 1: Two-W generation of neutrino mass. Symmetry Origin of Observable Nonunitary Neutrino Mixing Matrix (INT2010) back to start 17

18 With small m 1 and M 1, the 4 4 neutrino mass matrix M νn is reduced to the 2 2 ( ) m1 M ν 2 M 2 /M3 2 m 1 m 2 /M 3 m 1 m 2 /M 3 M 1 m 2 2/M3 2. Since M 2 M 3 in this hypothesis, the (1,1) entry is a canonical seesaw, whereas the (2,2) entry is an inverse seesaw and the (1,2) or (2,1) entry is a linear seesaw. In this basis, only ν 2 has possible large mixing with N. Similarly, if 3 families are considered with 3 N, only one linear combination of the 3 ν may mix significantly with N. Symmetry Origin of Observable Nonunitary Neutrino Mixing Matrix (INT2010) back to start 18

19 For three families, let m m 2 0 M νn = m 3 m M 1 M 4 M 5 0 m 2 0 M 4 M 2 M m 3 M 5 M 6 M 3. The texture hypothesis is equivalent to m 1 = m 2 = 0 and M 1 = M 4 = 0. Symmetry Origin of Observable Nonunitary Neutrino Mixing Matrix (INT2010) back to start 19

20 To enforce this pattern at tree level, use L = 1 for ν 1,2,3 as usual, but L = 3, 2, 1 for N 1,2,3. Add one Higgs doublet with L = 0 and three Higgs singlets with L = 2, 3, 4. Then M 1 = M 4 = 0, because there is no Higgs singlet with L = ±6 or L = ±5, but will become nonzero in one loop, whereas m 1 = m 2 = 0 to all orders. To obtain nonzero m 1,2, a second Higgs doublet with L = 4, 1 may be added. ν 1,2 is then massive in one loop, and ν 2,1 becomes massive in two loops. Symmetry Origin of Observable Nonunitary Neutrino Mixing Matrix (INT2010) back to start 20

21 Lepton number has been used as a global U(1) symmetry, but a discrete version (e.g. Z 7 ) also works. The global U(1) may also be gauged, using either U(1) B L or U(1) χ from E 6 where Q χ = 5(B L) 4Y. Many possible signatures of these extensions may be looked for at the LHC. For example, with U(1) χ, it is possible to produce N in pairs through q q Z N N, if kinematically allowed. The final states to be analyzed are l ± l W ± W and l ± l ± W W. Without new physics, N is not likely to be observable. [Ibarra/Molinaro/Petcov(2010).] Symmetry Origin of Observable Nonunitary Neutrino Mixing Matrix (INT2010) back to start 21

22 Concluding Remarks TeV scale seesaw models may allow ν N mixing to be large enough, so that the nonunitarity of the neutrino mixing matrix, i.e. (1 + η)u, may become observable. In the texture hypothesis, special choices of lepton symmetry are required to maintain the assumed pattern, in which case extra Higgs particles must appear. The phenomenological constraints on η eτ and η µτ are of order Neutrino oscillations are not much affected in such models. Symmetry Origin of Observable Nonunitary Neutrino Mixing Matrix (INT2010) back to start 22