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					    (Pseudo)-Dirac neutrinos and
            leptogenesis
                               Steven Abel
                         s.a.abel@durham.ac.uk


SAA, A Dedes and K Tamvakis, Phys Rev D 71 (2005) 033003; SAA and V Page,
                  JHEP 0605:024,2006: hep-ph/0601149




                                                          (Pseudo)-Dirac neutrinos and leptogenesis – p.
Outline

1. Options for neutrino masses
2. Natural neutrino masses in supergravity
3. A model for small Dirac masses
4. Neutrinogenesis: the idea
5. Affleck-Dine: the idea
6. Affleck-Dine neutrinogenesis
7. The baryon number



                                             (Pseudo)-Dirac neutrinos and leptogenesis – p.
Options for ν-mass?
General neutrino mass matrix (one flavour)
                      1 ¯c       1
            −Lmass   = χL Mν χL + χ L Mν χL
                                   ¯
                      2          2
with                      
                      ν
              χ=              ¯      ¯ ¯
                            , χc = −(ν c , ν )
                      νc
                                       
                               mL mD
                  Mν =                 
                               mT MN
                                D




                                                  (Pseudo)-Dirac neutrinos and leptogenesis – p.
Options for ν-mass?
a) Dirac
                                     
                            0    mD
                  Mν =               
                            mT
                             D   0

Two degenerate eigenvalues; ±mD . Maximal mixing
(θ = π ) between active and sterile neutrino but no
      4
oscillations




                                             (Pseudo)-Dirac neutrinos and leptogenesis – p.
Options for ν-mass?
b) Pseudo-Dirac
                                        
                            mL      mD
                  Mν =                  
                            mT
                             D      MN

Two eigenvalues; mD ± (mL + MN ). Maximal mixing
(tan θ = M2mD L ) oscillations between active and sterile
          N +m
neutrino even with one generation.




                                                (Pseudo)-Dirac neutrinos and leptogenesis – p.
Options for ν-mass?
c) See-Saw
                                            
                             mL         mD
                  Mν =                      
                             mT
                              D         MN

Sterile state ν c is almost mass eigenstate.
Negligible mixing (tan θ =    2mD
                             MN +mL
                                    )   and hence negligible
active ↔sterile.




                                                   (Pseudo)-Dirac neutrinos and leptogenesis – p.
Can Dirac neutrinos be natural ?
Consider supergravity: K, W where
                                  ¯
e.g. W ⊃ QHu U c + QHd Dc and K ⊃ ΦΦ leads to ∂φ∂φ∗ .
Instead of W ⊃ λLHu N c assume
                           ∗
                 LHu N c LHd N c
              K⊃        +        + h.c.
                  M       M
Assume supersymmetry breaking is communicated by
gravity with m3/2 ∼ 1TeV. Then Hu ≈ mtop and M ≈ MP
gives
                   m3/2
             mν ≈       ( Hu + Hd ) ∼ 10−4 eV
                    M
Close to measured value 0.05 eV!
                                           (Pseudo)-Dirac neutrinos and leptogenesis – p.
Supergravity calculation
Most general K, W are (si ≡hidden fields, y α ≡visible
“matter” fields)


             W (s, y) = W (h) (s) + W (o) (s, y)
      K(s, s∗ , y, y ∗ ) = K (h) (s, s∗ ) + K (o) (s, s∗ , y, y ∗ )




                                                          (Pseudo)-Dirac neutrinos and leptogenesis – p.
Supergravity calculation
Visible matter Lagrangian
                         ¯
                         β µ
          L = iKαβ χ σ ∂µ χα − (mαβ χα χβ + h.c.)
                 ¯ ¯ ¯


where the crucial terms are ...
                   (o)          ¯   (o)             ¯   (o)    (h)
      2mαβ                        (o)
             = Wαβ − K γ δ Kαβ δ Wγ − K ij Kαβ¯ Wi
                               ¯              j
                                      ¯       (o)         ¯   (o)         (h)
                                            (o)
                                −K γ i Kαβ¯Wγ − K iδ Kαβ δ Wi
                                          i              ¯
                          (o)             ¯   (o)         ¯   (o)        (h)
                                        (o)
             + m3/2 (Kαβ − K γ δ Kαβ δ Kγ − K ij Kαβ¯ Ki
                                     ¯              j
                                      ¯       (o)        ¯    (o)      (h)
                                            (o)
                                −K γ i Kαβ¯Kγ − K iδ Kαβ δ Ki )
                                          i              ¯




                                                               (Pseudo)-Dirac neutrinos and leptogenesis – p.
Supergravity calculation

                i¯
                 j   (o)   (h)           (o)
        2mαβ ⊃ K Kαβ¯Wi
                    j
                                 + m3/2 Kαβ




                                               (Pseudo)-Dirac neutrinos and leptogenesis – p. 1
A model
How do we ensure that the Dirac mass terms are not in W ?
Need a symmetry that treats K and W differently
-R-symmetry

      Q            Uc       Dc       L         Ec       Nc              Hu              Hd
 2 − r d − rh   rd + 2rh     rd   1 + rh    1 − 2rh     −1            −rh                rh


W   ⊃ YU QHu U c + YD QHd Dc + YE LHd E c
                                                            ∗
K ⊃ c1 (s, s∗ ) Hu Hd + c2 (s, s∗ ) LHu N c + c3 (s, s∗ ) LHd N c + h.c.



                                                       (Pseudo)-Dirac neutrinos and leptogenesis – p. 1
A model
find Dirac mass terms

       mD = v sin β[ m3/2 (c2 − c1 c3 ) − F s (c2 + c3 cot β),s ]
        ν


If mD = (0.04 − 0.05)eV and c2,3
     ν                                      M −1 and
 (c2,3 ),s M −2

                     mD M 2
                       ν
       Fs                         = (1.6 − 2.8) × 10−13 M 2
               v sin β(1 + cot β)
         √
If F =
   s
             3m3/2 MP then 100 < m3/2 < 104 GeV gives

                4 × 1016 GeV < M < 5 × 1017 GeV

                                                       (Pseudo)-Dirac neutrinos and leptogenesis – p. 1
A model
Relaxing lepton number conservation also get
Pseudo-Dirac neutrinos from

                   W ⊃ Y4 (LHu )2
                   K ⊃ c4 W (h) (N c )2

Note that ms ∼ F s /M ∼ 10 TeV is high




                                               (Pseudo)-Dirac neutrinos and leptogenesis – p. 1
Neutrinogenesis: the idea
If both B and L conserved in K, W how can we get nett B?
Use sphalerons (Dick, Lindner, Ratz, Wright); e.g drift and
decay of heavy φ

                    ¯                     ¯
           φ → νL + νR → −α(B + L) + νL + νR

 • νR is inert because mD is small and holds lepton
                        ν
    number
 • sphalerons change B + L but B − L = 0

 • CP violation in φ decays



                                              (Pseudo)-Dirac neutrinos and leptogenesis – p. 1
Affleck Dine: the idea

                       CP violating lump
                                                    |φ|∼Μ (?)


        Field ends with B nonzero




                                           (Pseudo)-Dirac neutrinos and leptogenesis – p. 1
Affleck Dine neutrinogenesis
Consider superpotential of effective global SUSY theory

W = YU QHu U c +YD QHd Dc +YE LHd E c +Yν LHu N c +µHu Hd

Yν ≈ 10−12 is an incredibly small coupling.




                                              (Pseudo)-Dirac neutrinos and leptogenesis – p. 1
Affleck Dine neutrinogenesis
D − f lat directions correspond to scalars LHu and N c :
                                  
                              1  φ 
                      L = √
                               2    0
                                  
                              1  0 
                     Hu = √
                               2    φ
                   Nc = ν
                        ˜
                        ¯




                                              (Pseudo)-Dirac neutrinos and leptogenesis – p. 1
Affleck Dine neutrinogenesis
Potential
                   V = VSB + VHubble + VF

V                                         ¯
    = (m2 − cφ H 2 )|φ|2 + (m2 − cν H 2 )|ν |2
        φ                    ν            ˜

                           ¯           |Yν |2 2 2
        + (Yν (A + cA H)φ2 ν + h.c.) +
                           ˜                 |φ | + |Yν |2 |ν φ|2
                                                            ˜
                                                            ¯
                                         4




                                                  (Pseudo)-Dirac neutrinos and leptogenesis – p. 1
Affleck Dine neutrinogenesis
Minimum at
                        cφ H
        |φmin |
                        2 |Yν |
                    
                          −cφ      |A|
                    
                        2cν −cφ   |Yν |
                                        ,   cA H   |A|
              ˜
              ¯
             |ν |         −cφ      H
                    
                        2cν −cφ   |Yν |
                                        ,   cA H   |A|




                                                    (Pseudo)-Dirac neutrinos and leptogenesis – p. 1
Affleck Dine neutrinogenesis
Dynamics of nLR = nL − nR

                       nν = n L + n R

                         i ˙∗     ˙
                  nL   =   (φ φ − φφ∗ )
                         2
                  nR         ˙ ∗¯ ¯¯
                                   ˙
                             ¯ ν − ν ν ∗)
                             ˜
                       = −i(ν ˜ ˜ ˜

Need to solve eqns of motion, e.g.

                   ¨ + 3H φ + ∂V = 0
                   φ      ˙
                              ∂φ∗

                                            (Pseudo)-Dirac neutrinos and leptogenesis – p. 2
Affleck Dine neutrinogenesis
Gives

          nν + 3Hnν = 0
          ˙
                                 ¯
        nLR + 3HnLR = 4Im(Yν Aφ2 ν )
        ˙                        ˜




                                       (Pseudo)-Dirac neutrinos and leptogenesis – p. 2
Affleck Dine neutrinogenesis
Thermal history
                                                        log nLR
                                                             24
                                                             22
                                                             20
                                                             18
                                                             16
                                                             14
                                                             12
                                                             10
                                                              8
                                                              6
                                                              4
                                                              2           T Tew
                               H 100GeV                                   at 14
                                                                                   log t tR
                         -6      -4     -2                   T TR     2        4
                                                             H 1GeV


                                                                                               10
                                                                                        4·10
                                                                                               10
                                          2·10
                                                 12                                     3·10
                                                                                               10
                                                 12
                                                                                        2·10
                                          1·10                                                 10
                                                                                        1·10
              10    10    10    10   10      9           9                 7       7      7            7       7      7
         -3·10     -2·10
             -2.5·10 -1.5·10
                           -1·10 -5·10                5·10            -6·10 -4·10 -2·10 10          2·10 4·10 6·10
                                        12                                         -1·10
                                  -1·10                                                        10
                                                                                       -2·10
                                                 12                                            10
                                     -2·10                                             -3·10
                                                                                               10
                                                                                       -4·10




                                                                                                           (Pseudo)-Dirac neutrinos and leptogenesis – p. 2
Affleck Dine neutrinogenesis
 • H   m3/2 : inflaton oscillation R ∼ t2/3 , matter
   dominated chaotic motion:

               R3 H 2 φ2 = const ⇒ nLR ∼ const

 • H(TR ) < H < m3/2 : matter dominated cyclic motion:


                R3 m2 φ2 = const ⇒ nLR ∼ t−1
                    φ

 • H < H(TR ): radiation dominated cyclic motion:

                    3                               −3
                R       m2 φ2
                         φ      = const ⇒ nLR ∼ t    2




                                                         (Pseudo)-Dirac neutrinos and leptogenesis – p. 2
Affleck Dine neutrinogenesis
Constraints:

 • Gravitino bound: TR < 109 GeV. If TR = 109 GeV then
    H(TR ) ∼ T 2 /MP ∼ 1GeV.
            ¯
 • Decay of ν oscillations after Tew : τdecay ∼ 4π/(Yν2 mν )
            ˜                                            ˜
    gives Tdecay ∼ 100MeV




                                                (Pseudo)-Dirac neutrinos and leptogenesis – p. 2
Final baryon number
Until reheating
                       ρnLR     m2 |A/Yν |2
                                 3/2
                            ∼
                        ρI          m2 MP
                                     3/2
                                         2


After reheating use mφ,ν nLR = ρnLR scales like matter and
                       ¯
                       ˜
entropy

    nLR   |A/Yν |2 TR
        ≈     2
     s      MP mφ
                                2           2
                  −9      A         10−12        TR      100GeV
          = 10
                       100GeV        Yν         1T eV      mφ


                                                        (Pseudo)-Dirac neutrinos and leptogenesis – p. 2
Final baryon number
Finally when sphalerons are active
                      8
               B = L = nR T > Tew
                      23
                      44
               B = L=     nR T < Tew
                      137




                                       (Pseudo)-Dirac neutrinos and leptogenesis – p. 2
Summary
 • Supergravity provides attractive possibilities for
   (Pseudo)-Dirac neutrinos.
 • mν ≈ 0.05eV consistent with
   4 × 1016 GeV < M < 5 × 1017 GeV
 • Affleck-Dine works remarkably well with D-flat
   directions (LHu and N c ). Find nB /s ∼ 10−9 if
   TR ∼ 1TeV
 • Baryon number related to DM →Mν ≈ 1 GeV
                                 ˜

 • Pseudo-Dirac neutrinos can be accomodated.



                                               (Pseudo)-Dirac neutrinos and leptogenesis – p. 2

				
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