Massive neutrinos
Dirac vs. Majorana
Niels Martens
Supervisor: Dr. J.G. Messchendorp
8 okt 2010
Introduction
Introduction
Outline
• Introduction
– Helicity
– Chirality
– Parity violation in weak interactions
• Theory
– SM: massless lefthanded neutrinos
– Massive neutrinos
•
•
•
•
Dirac mass
Majorana mass
Dirac-Majorana mass terms
Possible scenarios
• Experiments
– Neutrinoless double beta decay
• Results Heidelberg-Moscow cooperation
2
Introduction
Introduction
Helicity & Chirality
• Helicity: projection of spin in
the direction of momentum
• Ill-defined when m≠0
(Lorentz transformation)
 Chirality states (eigenstates
of weak interaction):
superposition of helicity
states
3
Introduction
Introduction
Parity violation in weak interactions
• Parity operation: x  -x
• V  -V
• AA
• Goldhaber experiment (1957): measuring neutrino
helicity
• Electron capture in 152Eu
e 152Eu152Sm  e  
• Two co-linear events of opposite parity expected:
4
Introduction
Introduction
Parity violation in weak interactions
P
• Only lefthanded photons observed  only lefthanded
neutrinos
• Later experiments: only righthanded anti-neutrinos
5
Theory
Neutrinos in the Standard Model
• Fermion; spin-½
• Massless
• only lefthanded neutrinos, righthanded
anti-neutrinos
6
Theory
Neutrinos in the standard model
• Massless spin-½ particles are described by the
Dirac eqation for massless particles:

i   L  0

i    0

i   R  0
p
p
 R, L   R, L
7
Theory
Massive neutrinos – Dirac
neutrino
• Flavour oscillations  (small) neutrino mass!!
• How to incorporate this in SM/ extend SM?
(1) Dirac mass
(i     mD )  0
i    L  m R

i   R  m L
 Boost can change handedness
 coupling between two helicity states
 A single four-component spinor
8
Theory
Massive neutrinos – Dirac
neutrino
• Dirac mass term in Lagrangian
LMass  mD  mD ( L  R )( L  R )
 mD ( L R  R L )
• What other mass terms are possible?
9
Theory
Massive neutrinos – Majorana
neutrino
(2) Majorana mass
L
Mass
L
c
1
c
 mL ( L R  R L )
2
R
Mass
L
c
1
 mR ( L R  R Lc )
2
• Neutrino is chargeless, so it can be its own
antiparticle
 mM couples particle and antiparticle
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Theory
General case: Dirac-Majorana-mass
(3) Dirac-Majorana mass term
c
c
2L  mD ( L R  L )  mL L  mR L R  h.c.

c
  L,  L

 mL

 mD
c
R
mD 

mR 
c
R
 Rc 
   h.c.
 
 R
• Diagonalizing M gives two mass eigenvalues:
m1, 2

1
 (mL  mR )  (mL  mR ) 2  4mD2
2

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Theory
Different scenarios
m1, 2

1
 (mL  mR )  (mL  mR ) 2  4mD2
2
(a) mL  mR  0  m1,2  mD
 

: pure Dirac case
(Dirac field)
(b) mD  0  m1, 2  mL, R : pure Majorana case
c
c





,






1
L
R
2
R
L
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Theory
Different scenarios
(c) Seesaw model
mR  mD ; mL  0
 mD2 
mD2
 m1 
; m2  mR 1  2   mR
mR
 mR 
• Explains:
1   L  Rc , 2   Lc  R
– light mass of neutrinos
– the experimental fact that only lefthanded neutrinos couple to
the weak interaction.
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Experiments
Related experiments
• Tritium β-decay
• Flavor oscillations
• Neutrinoless double β-decay
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Experiments
Neutrinoless double β-decay
• β—decay: n  p  e   e
• Double β--decay:
( A, Z )  ( A, Z  2)  2e  2 e
• Could any nucleus be used?
No:
* M N ( A,Z )  M N ( A,Z 2)  2me  2m

e
* Single β-decay must be forbidden
 M N ( A,Z )  M N ( A,Z 1)  me
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Experiments
Neutrinoless double β-decay
• Semi-empirical mass/Weizsäcker formula:
16
Experiments
Neutrinoless double β-decay
• 35 naturally occurring isotopes which decay via
2β , all even-even
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Experiments
Neutrinoless double β-decay
-
• So how can 2β show that the neutrino is a
majorana particle?
Neutrinoless double beta decay
( A, Z )  ( A, Z  2)  2e  X
2 e
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Experiments
Neutrinoless double β-decay
• 2 necessary conditions:
– Particle-antiparticle matching
  
– Helicity matching
 m 0
Virtual neutrino line
e
M
Mass
L
c
c
1
1
c
c
 mL ( L R  R L )  mR ( R L  L R )
2
2
• If neutrinoless double β-decay occurs, the
neutrino is a massive majorana particle.
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Experiments
Neutrinoless double β-decay
• Experimental signatures:
– Two e- from same place at same time
– Daughter nucleus (Z+2,A)
– Neutrinoless case: sharp defined kinetic energy of
electrons, instead of continuous spectrum
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Experiments
Neutrinoless double β-decay
1
2
2
 M m
T1/ 2
• Theoretical uncertainty (76Ge): 1.5 < |M| < 4.6
• Half-lives
• β : from seconds to 105 y
• 2νββ: ~1020 y
• 0 νββ: > 1025 y
• mν ~ 50 meV  100 kg needed for 1 event/y
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Experiments
Neutrinoless double β-decay
• Experimental difficulties:
– Count rate: How to measure T1/2 beyond 1025 y!?
– Source strength: expensive!
– Background: Cosmic rays, 2νββ, natural
radioactive decay
– Energy resolution
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Experiments
Heidelberg-Moscow Experiment
Source strength  11,0 kg
enriched 76Ge: Source = detector
Background  find a
mountain and dig a hole
Enormous half-lives 
experiment run from 1990
till 2003 (but, stability then
becomes a problem)
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Experiments
Heidelberg-Moscow experiment
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Conclusions
• None… yet
• Since neutrinos do have mass, the SM has to
be extended.
• Theoretically, massive neutrinos can have a
Dirac and/or Majorana nature.
• Reliable 0νββ observations would prove that the
neutrino is a Majorana particle and give the
neutrino mass, but at the moment 0νββexperiments face many difficulties.
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Bibliography
• C. Giunti & C.W. Kim, Fundamentals of
neutrino physics and astrophycis, Oxford
University Press, 2007
• K. Zuber, Neutrino Physics, IOP
Publishing, 2004
• H.V. Klapdor-Kleingrothaus et al. / Physics
Letters B 586 (2004) 198–212
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