Theory of lepton mixing - I
Andrea Romanino
CERN Theory Division
Andrea Romanino, CERN Theory Division
1
LNF Spring School – May 2003
Part I of 4:
1. (AR) Theoretical background
2. (PL) Phenomenology of neutrino oscillations
3. (AR) Theoretical ideas on ν masses and mixings
4. (PL) Future prospects for new experimental studies
Andrea Romanino, CERN Theory Division
2
LNF Spring School – May 2003
Interest of neutrino physics
• New data!
• Data interpretation (now) pretty clean
• Astrophysics (probe of SUN, SNe, HE sources,…)
• Cosmology (CMB, structure formation, nucleosynthesis,
baryogenesis,…)
• Particle physics (access Λ ~ MGUT , unification, flavor)
Andrea Romanino, CERN Theory Division
3
LNF Spring School – May 2003
Flavor and mass eigenstates
 e ,  , 
“flavor eigenstates”, paired to
charged leptons in CC:
 1 , 2 , 3
“mass eigenstates”, diagonalize
the neutrino mass matrix:
 ei  Uih h
Andrea Romanino, CERN Theory Division
U unitary (PMNS)
4
Jl

g

ei   PL ei
2
mij ei  e j  mh h h
( ei  Uih* h )
LNF Spring School – May 2003
Oscillations
 ei  Uih*  h
 ej e
iHt
e iHt  ei  Uih*e iEht  h
 ei  U jh e
iEht
Uhi†
mh2
Eh  p 
2E
P ( ei   e j )   e j e
iHt
 ei
2
In the simplest case:
 e   1 cos    2 sin 
    1 sin    2 cos 
2

m
L
2
2
P ( e    )  sin 2 sin
4E
(to be integrated over energy, position and convoluted with, cross
section, resolution, efficiency…)
Andrea Romanino, CERN Theory Division
5
LNF Spring School – May 2003
o In vacuum only
o Simplified derivation: p constant? E constant? It does not really
matter (change of variable in the wave packet integral)
o m 2 and sin 2 2 require a sensitivity to L/E dependence

o m 2L /( 4E )  1  P  m 2 sin 2
m 2L /( 4E )  1  P  sin 2 2 / 2
optimal L/E: m 2L /( 4E ) ~ 1
Andrea Romanino, CERN Theory Division
6
 L /( 4E )2
2
(P/L^2 indep. of L)
LNF Spring School – May 2003
3 ν oscillations
Standard parametrization of U:
s13e i
1
 c13


U   c23 s23 
1
  s c   s e i
c13

23 23 
13
 c12 s12 



s
c
 12 12 


1




c12c13
s12c13
s13e i

   s12c23  c12s23s13e i c12c23  s12s23s13e i s23c13

i
i
s
s

c
c
s
e

s
c

s
c
s
e
s23c13
23 12
12 23 13
 12 23 12 23 13





in terms of the physical (and observable in oscillations)
parameters
0  θ12,θ23,θ13  π/ 2, 0  δ  2π
Andrea Romanino, CERN Theory Division
7
LNF Spring School – May 2003
Standard labeling of eigenstates
2
2
uniquely defines the labeling
0  m21
 m32
2
2
m21
 0 by definition, m32 can have both signs:
3
2
Normal
m
32  0
e.g.:
2
m1  m2 « m3
(inverse
hierarchical)
m1  m2  m3
(degenerate)
2
m1  m2  m3
1
(neither)
Andrea Romanino, CERN Theory Division
e.g.:
m2  m1  m3
1
(hierarchical)
normal
2
Normal
m
32  0
inverted
m1  m2  m3
(degenerate)
3
8
LNF Spring School – May 2003
Parameters accessible to oscillations
2
m32
~ 2.5  10 3 eV 2  23 ~ 45o
(ATM, K2K)
2
m21
~ 0.7  10  4 eV 2  12 ~ 30  35o
(SUN,KamLAND)
 13  10o
(CHOOZ, Palo Verde)
2
sign ( m32
)?  ?
(superbeams? nu-facts?)
23 ~ 45o (  45o ?)
12 ~ 30  35o  45o
Guidelines for model building:
13  10o
2
2
m21
/ m32
 0.025 « 1
Andrea Romanino, CERN Theory Division
9
LNF Spring School – May 2003
3ν  2ν
Exact (cumbersome) 3ν formulae:
P ( ei   e j )  P ( e j   ei )  PCP  PCP
P ( ei   e j )  P ( e j   ei )  PCP  PCP
ji
ji
ji
2
2
2
PCP  ij  4 Re J12 S12
 4 Re J23 S23
 4 Re J31 S31
PCP  8  ij JCP S12S23S31
ji
(Shk
2
mhk
L
 sin
)
4E
ji
†
†
Jhk  U jhUhi
UikUkj
, Im(Jhk )  ji  hk JCP ,  ji  1,0
2
S12 «
Chooz:
ATM:
SUN:
2
1,
2
S23 
2
2
2
S12 « 1, S23  S13, 13 « 1 :
2
2
S23, S13
suppressed by 13 :
Andrea Romanino, CERN Theory Division
2
m23L
P ( e   e )  1  sin 213 sin
4E
2
2
2 m23L
P (    )  sin 223 sin
4E
P ( e    , ) « 1
2
S13 :
10
2
2
m21L
P ( e   e )  1  sin 212 sin
4E
2
2
LNF Spring School – May 2003
CP-violation
O(1)
?
LBL
PCP   cos 13 sin 212 sin 223 sin 213 sin  S21S32S13
Large angles (unlike in quark sector) enhance CP-violation
P ( e    )  P ( e    )
1
1
1
a 

, a ~

P ( e    )  P ( e    ) sin 213
N
sin 213
 stat. error ~ a / a ~ constant with 13
(until N  0 or subdominant effects start dominate)
Andrea Romanino, CERN Theory Division
11
LNF Spring School – May 2003
Matter effects
Incoherent scattering – typical mean free paths
(depend on flavor, “simplified” energy dependence):
 (E ) ~ 10cm 100MeV/E 2
in proto-neutron star cores
 (E ) ~ 1010 km 10MeV/E 2 in the Sun
 (E ) ~ 109 km GeV/E
in the Earth's mantle
Coherent forward scattering is enhanced by GF E 2
incoherent : dPsc / dx ~ GF2E 2n
coherent :
 E 
dPsc / dco ~ GF E ~ 10 5 

 GeV 
dco / dx ~ GF n
It affects the neutrino phases in a flavor dependent way
Andrea Romanino, CERN Theory Division
12
LNF Spring School – May 2003
In matter:
Ve

 m12





1
  univ. terms
U †   m022
H 
U
m22
2E 





m32 
m032 




Free Hamiltonian
V Ve V  2 GF ne
V V
MSW potential
(neutral matter, nν « ne )
(tree level, neutral matter, Lμ  Lτ  0)
νx
νe
e
W
νe
+ u, d,…
Z
e
e
for ν :U  U * ,V  V
Andrea Romanino, CERN Theory Division
νx
13
e
LNF Spring School – May 2003
Propagation in constant density
Oscillation formulae still hold with θ  θm , Δm 2  (Δm 2 )m , where
θm , (Δm 2 )m depend on the neutrino energy
The Earth:
Mantle
Core
m ~ 35 g / cm3
c ~ 10 15 g / cm3
Propagation in the Earth affects
•Atmospheric ν’s (only through the subdominant νe  ν μ,τ )
•Solar, SN ν’s (D/N effect)
•Terrestrial experiments (Long Baseline)
Andrea Romanino, CERN Theory Division
14
LNF Spring School – May 2003
Resonance (2ν)
2EV
 2
sin



H 
m 2
 sin  cos 


sin  cos   m 2
 2E  universal terms
cos2  
Resonant enhancement of the mixing angle :
2EV
Δm 2
 cos 2θ  (sin 2θ )m  1, (Δm 2 )m  sin 2θ Δm 2
Θ = 0.06
ν , Δm 2  0
Θ = 0.6
Also: (Δm 2 )m  Δm 2
•Resonance width = tan 2θ (sin2 2θ  1 / 2)
•θ  45o  resonance only if V  Δm 2  0
2
•SUN: V  0, Δm21
 0  res. only if θ12  45o
2
) » 1   e   2 m
note also: (2EV ) /( m21
sin 2θ
(sin 2θ )m
Andrea Romanino, CERN Theory Division
15
LNF Spring School – May 2003
Resonances and neutrino beams
• Enhance P (νe  ν μ ) or P (νe  ν μ ) through θ13 resonance . Need
- E ~ E res ~ 10 GeV
- L ~ λ0 / 4  π /(2V ) ~ 2500 km (independently of E )
2
• θ13 «1  the resonance is - in the ν channel (V  0) if Δm32
0
2
- in the ν channel (V  0) if Δm32
0
2
• opportunity of measuring sign(Δm32
)
Andrea Romanino, CERN Theory Division
16
LNF Spring School – May 2003
Propagation in varying density (2 ν)
H (t )  H free VMSW (t )
time-dependent hamiltonian
Adiabatic evolution: no ν1  ν 2 transitions
adiabaticity condition (2 ν’s):
d m ( m2 )m
«
dx
2E
Adiabatic resonance crossing  large flavor swap even for small θ
ν2
Θ = 0.06
ν1
2EV » m 2
V 0
 e   2 m
 2   e sin     cos 
P ( e    )  cos2 
(the adiabatic approximation must
break at small θ )
resonance layer
Andrea Romanino, CERN Theory Division
17
LNF Spring School – May 2003
Level crossing
dθm (Δm2 )m
• The adiabatic approximation
is worst at resonance
«
dt
2E
Δm 2
sin 2 2θ
• Adiabatic condition at the resonance: γ 
»1

2E (V /V )res cos 2θ
• If γ ~
 1 but γ » 1 at production and detection,
P (ν1  ν2 )  Pc  e πγ / 2
(LZ)
2
2
• E.g.: SN neutrinos (Δm32
 0) or antineutrinos (Δm32
 0)
for θ13  10 3
Andrea Romanino, CERN Theory Division
18
LNF Spring School – May 2003
Solar neutrinos
Evolve adiabatically in the Sun, “resonance” crossed by8B neutrinos
8
B
pp
Θ = 0.6
P (νe  νe )
pp
νB
Be
sin2 2θ12
P 1
2
P  sin2 θ
Smirnov
Andrea Romanino, CERN Theory Division
19
2EV
2
Δm21
LNF Spring School – May 2003
Supernova neutrinos
• Probe of core-collapse supernova physics
• Sensitivity to neutrino parameters (limited by the uncertainties on
the source)
• Constraint on exotic neutrino mixing
M ~ 1.5Msun
R ~ 8000 km
ρ ~ 10 9 g / cm3
R ~ 30 km
E rel
T ~ 0.7 MeV
0.01% photons
ρ ~ 3  1014 g / cm3

~ Eb ~ 3  10 53 erg  1% kinetic energy
T ~ 30 MeV
99% neutrinos

λ ~ 10 cm  tdiff
Andrea Romanino, CERN Theory Division
20
3r 2
~
~ 10 sec
λ
LNF Spring School – May 2003
Future Sne
Detector
SK
SNO
LVD
KamLAND
ν events
(from 10kpc)
~ 8000
~ 800
~ 400
~ 330
mν  1 eV (0ν 2β )  time spectrum undistorted
@ Neutrinosp here :
Eνe ~ 11 MeV  Eνe ~ 16 MeV  Eν x ~ 25 MeV
2
@ Earth : the energy spectra depend on θ13, sign (Δm31
)
e.g. : NH & θ13  0.05  Φ(νe )  Φ0 (ν μ ,τ )
2
Δm31
resonance crossed by ν 's (ν 's) if NI (HI);
Pc  0 (Pc  1) if θ13  0.05 (θ13  0.001)
2
(Δm21
resonance is always adiabatic)
2
(Δm31
 2E (Vμ Vτ ) resonance plays a role if Φ(ν μ )  Φ(ντ ))
Present claims plagued by low statistics
Andrea Romanino, CERN Theory Division
21
LNF Spring School – May 2003
Constraints on exotic scenarios
Energy loss argument:
dε loss
 1019 erg/s/g
dt
Constrains invisible escape channels:
axions, KK gravitons, exotic ν mixing
e.g. : sin2 2θs  108 for large Δm 2
Andrea Romanino, CERN Theory Division
22
LNF Spring School – May 2003
0ν2β decay
•Signals L-violation
•Probes the Majorana nature of neutrinos
•Allows to access parameters unaccessible to oscillations:
- absolute mass scale
- Majorana phases
Andrea Romanino, CERN Theory Division
23
LNF Spring School – May 2003
Reminder: Dirac vs Majorana
A massive charged fermion (e) necessarily involves
2 Weyl spinors and 2 particles (e  , e  ):
m ee c  h.c.,
q (e )  1, q (e c )  1
Dirac mass term
 c
Dirac spinor : ψ   e   m ee c  h.c.  m ψ ψ
e 
A massive neutral fermion (ν) only requires
1 Weyl spinor and corresponds to 1 particle (ν  ν ):
m
νν  h.c., q (ν )  0 Majorana mass term
2
(breaks any charge associated with ν)
m
m
ν
Majorana spinor : ψ     νν  h.c.  ψ ψ
2
2
ν 
Andrea Romanino, CERN Theory Division
24
(ψ ψT C )
LNF Spring School – May 2003
Dirac (m 
= 0):
ν L 0  ν   O (m / E ) ν 
ν L 0  ν   O (m / E ) ν 
 0):
Majorana (m =
ν L 0  ν   O (m / E ) ν 
ν L 0  ν   O (m / E ) ν 
Created by ν , can be annihilated by ν
In oscillations, once the O(m/E) terms have been neglected
- the elicity does not play a role
- there is no L-violation
- oscillation formulae are identical for Dirac and Majorana ν’s
Andrea Romanino, CERN Theory Division
25
LNF Spring School – May 2003
Physical parameters
Quarks
md di di
i
c
 mui ui uic
Leptons
g
 Vij uiW d j
2
mei ei eic
g
mi
 νi νi 
Uij eiW ν j
2
2
 e iφ1
  e iφ1
  e1iφ1


  Standard  

iφ2
iφ2
iα



V ,U  
e
e
e
param.

 
 

e iβ 
e iφ3  
e iφ3  

 
 

9
=
3
+
unphysical
Andrea Romanino, CERN Theory Division
3+1
26
+
2
physical in U
LNF Spring School – May 2003
2
Δm21
me ,μ ,τ
2
| Δm32
|
2
sign( Δm32
)
θ12 , θ23 , θ13 , δ
Andrea Romanino, CERN Theory Division
27
mlightest
α
β
LNF Spring School – May 2003
0ν2β decay
(A, Z )  (A, Z  2)  2 e  ; e.g. :
Γ | mee |2 Q
76
G 76Se  2 e 
2
2
2
2
2 2iα
2 2iβ
mee  Ueh
mh  c13
(m1c12
 m2s12
e )  m3s13
e
β decay
> endpoint
>
d
u

e 3
3

;
e.g.
:
H

He

e
 νe
e
X >
dN

dEν
mee
>
>
(A, Z )  (A, Z
ν
 1)  e  eν
|Ueh |2νeΓ(mh2 ) e Γ((m †m )ee )
>
d2
u
2 >2 2
2
2
>
(m m )ee |Ueh | mh  c13 (m1 c12  m22s12
)  m32s13
†
Andrea Romanino, CERN Theory Division
2
28
LNF Spring School – May 2003
(m †m )ee  (2.2 eV)2
| mee | O (1)  0.4 eV
(Mainz, Troitsk)
(Heidelberg - Moscow)
 (0.3 eV)2
(Katrin)
 O (1)  0.01 eV
(Genius)
Feruglio Strumia Vissani
Andrea Romanino, CERN Theory Division
29
LNF Spring School – May 2003
?
Andrea Romanino, CERN Theory Division
Neutrino mass
matrix
30
Measurements
LNF Spring School – May 2003
Some references
Neutrino oscillations: M.C. Gonzalez-Garcia, Y. Nir hepph/0202058
Matter effects (classic): T.K. Kuo , James Pantaleone
Rev.Mod.Phys.61:937,1989
Supernova neutrinos: Stars as Laboratories for
Fundamental Physics, University of Chicago Press
(astrophysics); Lunardini Smirnov, various papers
(particle physics)
0ν2β decay: F. Feruglio, A. Strumia, F. Vissani, hepph/0201291
General: Neutrino Physics, Ed. By K. Winter, Second
Edition
Ask for more…
Andrea Romanino, CERN Theory Division
31
LNF Spring School – May 2003