Neutrino oscillation physics II
Alberto Gago
PUCP
CTEQ-FERMILAB School 2012
Lima, Perú - PUCP
Oscillation in matter
• When neutrinos go through matter they can suffer coherent
forward elastic scattering (e.g its four momentum is unchanged)
which modifies the mixing angle.
L. Wolfenstein, Phys. Rev. D 17, 2369 (1978); ibid. D 20, 2634 (1979)
S. P. Mikheyev, A. Yu Smirnov, Sov. J. Nucl. Phys. 42 (1986) 913.
Neutrino interactions
The inelastic and absorption
neutrino Interactions are
negligible . They produce a
mean free path of the order
of
 1014 cm
l ~ 
 E / GeV



Oscillation in matter
The low-energy charged current hamiltonian is given by:
HCC

ee
GF
e   (1   5 ) e e  (1   5 )e
2
We can rearrange this using Fierz identities
HCC

ee
GF
e   (1   5 )e e  (1   5 ) e
 
 

2 
Je
J
If we average out the electron current in a medium of electrons we
have
e  oe  Ne



e  e  ve



e   5e   e


electron density
velocity
spin
Since the electrons of the medium are non-relativistic, unpolarized and
isotropically distributed only the electron density term survives.
Oscillations in matter
• Then :
HCC
ee
1  5 
 e
 2GF N e ( x) e 0 

2 

Ve
Y  
 Y   matt 

Ve  2GF N e ( x)  2GF  e matt   3.82 10 14 eV  e 
3 
m
0
.
5
1
gr
/
cm
 

N


 matt  matter density
Ne
Ye 
N p  Nn
mN  nucleon mass
1
GF N n
2
matter electrically neutral
VNC  

 O 10

eV 
Ve  O 10 12 eV
Ve
13
at the solar core
at the Earth`s core
Oscillation in matter
• Since the matter potential is a time like
component:
p 2  m 2  ( E  V ) 2  E 2  2
EV  E 2  A
A
E 2  p 2  m2  A
E
2
m
p 2  m2  A  p 
2p
A

2p
Similar to the vacuum part
Oscillation in matter
• The vacuum hamiltonian:
d  1
i 
dt   2
d  e
i 
dt   
  E1
  

 0

E
  U  1
0



0   1

E2   2

 and using


0   e
U 
E2    
 1


 2


i d  e

dt   

 e


 U

 



 mk2 
 , E k  p 1 
 2 p 




 
 m12 0     e
1
p
U 
U 
2
 
2 p  0 m2    
 





vacuum hamiltonian
d  e
i 
dt   

E
  U  1
0



0   e
U 
E2    


i d  e

dt   

 
 m12
1
p
U 
 
2
p
 0
 
0   1  2 E 2GF N e

U 
2 p 
m22 
0
0    e

0    




Oscillation in matter
d   e
i
dx   




A





  1  m12
0  
1  2 2G N E 0    e 



t  x 
U
U 
F
e

   
  2E  0 m2 
2
E

2 


0
0    







 
H vacuum
Matter potential


pE
p has been absorbed since it is a global phase
Evolution equation in matter
Oscillation in matter
• From:
d  e
i 
dx   
 1   m12 0    A 0    e  1 ~   e
U 


U  
 
Mmat 
 2E   0 m2 


 

 0 0      2 E
2


 




• We have for constant density:
m


d
1
i  m
dx   2

m
 1 m


~ U m 1

U M
 2 E mat   2m
~ 2 ,m
~2 ) 

Diag ( m
1
2




 e
where 
 

Evolution equation in the diagonal basis
m



  U m 1

  2m






Oscillation in matter
• We get for a constant density:
2
2
m

m
1
2
1  A
~
m 

2
2
2
2
m

m
1
2
2
1  A
~
m2 

2
2
2
1
m  sin 2  (m
2 2
21
2
cos 2  A) 2
2
21
m  sin 2  (m
2 2
21
2
m sin 2
tan 2 m 
2
m21 cos 2  A
cos 2  A) 2
2
21
Vacuum angle
2
21
Mixing angle in matter
Oscillation in matter
• Similar form to the vacuum oscillation formula for 2:
P e  

m L
 sin 2 m sin 1.27  21 
E

2
2
  m
~2  m
~2 
m21  m
2
1
sin2 2 m 
m  sen 2
2 2
21
2
2
m21
 m21
2
 ( A  m21
cos 2 ) 2
m  sin 2
m  sin 2
2 2
21
2
2 2
21
2
2
 ( A  m21
cos 2 ) 2
For antineutri nos : A   A
Oscillation in matter
sin2 2 m 
m  sin 2
2 2
21
2
m  sin 2  ( A  m
2 2
21
2
2
21
cos 2 ) 2
2
 sin 2 2 m  1 @ resonance point A  m21
cos 2   m 

4
2
 A  m21
 sin2 2 m  sin2 2
2
 A  m21
 sin2 2 m  0 (it happens when we increase N e or the energy E )
 Fake CP violation P e    P e   even with   0 (two generations)
and we can also have a fake CPT violation since P e  e  P e  e
MSW effect
cos 2 m 
2
m21
cos 2  A
m  sin 2  ( A  m
2 2
21
2
2
21
cos 2 ) 2
Oscillation in matter
m  
Matter suppresion
Cos2=0.38
m 2  0 & 
m 2  0 & 
m 2  0 & v
m 2  0 & 
We can deduce the sign of m2
Oscillation in matter
• For varying density the evolution equation is
described by:
iU
m
m
m



d  m  1   1 m  ~
m
1
U  m  
U MmatU  m 


dx   2   2 E
 2 
  m21

m
d  1   4 E
i  m  
dx  2   d m
i
 dx
d m 
 m 
i
dx  1 
m
 21  2m 

4E 
Non – diagonal hamiltonian
Oscillation in matter
• Adiabatic regime: slow varying density
d m
m21

dx
4E
-  1m and  2m are always instantaneous eigenstates of the Hamiltonia n
- evolution of  1m and  2m are decoupled
Negligible transition s  1m  2m : P m  m  Pc  0 Pc  crossing probabilit y 
1
m21
 ( x)  4 E 
d m
dx
2
 
m 2
21
dN e
4 2 E GF sin 2 m
dx
2
 x  1 adiabatic regime
Oscillations in matter
The survival probability in the adiabatic case is:
2
xv

2
x0
  A e  i  e
 A i  e   A e  i  A i  e




 propagatio n 
i 1, 2 
i 1, 2
i Ei ( x ) dx
P e  e
production
detection
Fast oscillations -(averaged out) large source-detector distance
P e  e  cos 2  cos 2  0  sin 2  sin 2  0
1
 1  cos 2 cos 2 0 
2
Neutrino production at high Ne : cos 20  -1
production
detection
 P e  e  sin 2 
2
Oscillations in matter
• Non- adiabatic regime: fast varying density
 x   1 non - adiabatic regime  P
P e  e 

m
1
 2m
 Pc  0
2
2
2
A e  i  A i  j
 A j  e








i , j 1, 2 
production
crossing probabilit y
detection
Within an interval around
the resonance.
Oscillation in matter
• Crossing probability:
F 
 

 
exp    R F   exp    R

2
 2

 2 sin 2 
Pc 
F 
 
1  exp    R

2
 2 sin 2 
Extreme non - adiabatic limit  R  1
Density profile
Ne  x
F
1
x
1 - tan 
Ne  e
Pc  cos 2 
2
Adiabatic case  R  1
Pc  1
Oscillation in matter
• The survival probability is given by:
P e  e
1 1

    Pc  cos 2 cos 2 0
2 2

Low Energy (adiabatic)
High Energy(adiabatic)
cos 2 0  cos 2
cos 2 0  1
1
P e  e  1  sin 2 2
2
P e  e  sin 2 
Matter effects can be neglected
Matter effects are important
Oscillation in matter
How this probability looks like:
P e  e
1
P e  e  1  sin 2 2
2
P e  e  sin 2 
Keep on mind this plot!
Three neutrino scheme
The 3 framework within the experimental context:
0   ij  
U PMNS
0    2
2
m31
2
m21
 
0   c13
0 s13e i   c12 s12 0 
1 0
 

 

  0 c23 s23    0
1
0     s12 c12 0 
 0  s c    s e  i 0 c   0

0
1
23
23  
13
13  


LBL & atms
Reactor & atms
Solar & reactors
* mainly sensitive
Solar neutrinos
Solar net reaction
4 1 H 4 He  2e   2 e  26.73 MeV
CNO-chain
pp - chain
Solar neutrino problem
Objective of the first solar
neutrino experiment
To demonstrate that the
Solar Standard model
was correct
Experiment
Data/ SSM
Homestake
(e37Cl->37Ar+e)
Sage + Gallex
(e71Ga->71Ge+e)
Superkamiokande
(xe->xe)
0.340.03
0.560.04
0.460.02
Borexino
Solution to the solar neutrino problem
SNO ( D2O phase) observes:
 meas  
e
 e d  ppe
 X d   X pn

Charged current
Neutral current
 meas     
e
Measurement of the solar neutrino
flux compatible with the SSM
NC
SNO
 1.01  0.12
SSM
This confirms that neutrinos suffer a
flavour conversion
 meas   
e
1

6.36 
xe → xe
Solar neutrinos
Borexino
vacuum dominated
Do you remember this probability plot?
..MSW transition
Matter dominated
N e  N e c132
By the way the survival probability in 3

2
2
P3  s134  c134 Pmatt
m21
, sin 2 212
e
e

Solar neutrinos
Reactor-experiments:  e   e disappearance
L
180 km
 e flux
KamLAND

E
4 MeV
53 reactors
Atmospheric neutrinos
R
N (    )
N ( e  e )
2
Atmospheric neutrinos
The atmospheric neutrino anomaly
was found trying to understand the
background involved in nucleon
decay searches
Then the Super-Kamiokande experiment
came into the game and…
Atmospheric neutrinos
….observed in 1998 neutrino oscillations
No oscillations hypothesis
Long-Baseline experiments(LBL)
Disappearance experiments     
    
The MINOS experiment
The K2K experiment
L
735 km

Ev
3 GeV
L
250 km

Ev
1.3 GeV
Long-Baseline experiments(LBL)
MINOS experiment
K2K experiment
No oscillations
hep-ex/0606032
No oscillations
R. Nichol -Neutrino 2012
Why we believe in neutrino oscillation
due to mass?
Oscillation maxima

atmospheri c sector : m , sin  23
2
31
2


2
solar sector : m21
, sin 2 12
Oscillation pattern depends on L/E (not a minor detail
in the confirmation of oscillation due to mass)

Searches for 13 -LBL
T2K
   e
L
295 km

E
0.6 GeV
T. Nakaya – Neutrino 2012
Results of 13
T2K
MINOS
Now is not consistent
with sin 2 213  0 @ 90% C.L
R. Nichol
Neutrino 2012
T. Nakaya Neutrino 2012
10 electron - neutrino candidates
060
sin 2 213  0.10400..045
@  0 with 3.2  signal significance
This term explains the
periodic behaviour in 
P   e
2 leading
P
term

2
2
 2

2
2
2  m31
2
2  m21
 s23 sin 213 sin 
L   c23 sin 212 sin 
L 
 4E 
 4E 
2
2
2

  m21


m31
2  m31

J
cos  
L   
L   sin 
L 

4E   4E 
c13 sin 213 sin 212 sin 2 23

 4E 
Search for 13
• Reactors : Source of  e
e  p  e  n

L
1000  m 
~


E
3  MeV 
valid for
P e  e
2
 m31

 1  sin 213 sin 
L 
 4E 
2
2
prompt signal E prompt  E e  0.782 MeV
2013
coincidence
Similar detection concept
in KamLAND
France
Korea
China
E. Lisi
Results of 13 - Reno
4.9 signal significance
FD-8% deficit
ND-1.8% deficit
sin 2 213  0.113  0.013 stat  0.019syst  13  9.82o
Soo-Bong Kim – Neutrino 2012
Only rates
Results of 13 - Double Chooz
Rates + Shape
sin 2 213  0.109  0.030 stat  0.025syst
 13  9.63o
depletion
M. Ishitsuka – Neutrino 2012
Results of 13 - Daya Bay
No oscillation
>8 from null hyp.
deficit
D. Dwyer- Neutrino 2012
sin 2 213  0.089  0.010 stat  0.005syst  13  8.68o
Only rates
Global analysis-3
Normal
Hierarchy
Degeneracy in  23
Inverted
Hierarchy
SuperK atmospheri c data
favoured  23 not maximal
 
 23 not maximal  0, 
 4
Fogli et. Al. Neutrino 2012
Global analysis-3
Global analysis-3
2
5
m21
 7.54 00..26
eV 2
22  10
2 .6 %
18
sin 2 12  0.307 00..16
3.0%
2
m31
 2.4300..106 10 3 eV 2
5 .4 %
25
sin 2 13  2.4100..25
 10-2
10.0%
24
sin2 23  0.386 00..21
14.0%


28
 1.0800..31
Precision era
13  8.9o  23  38.4o 12  33.6o   194.4o
arxiv: 1205.5254 G.L. Fogli, E. Lisi, A, Marrone, D. Montanino, A. Palazzo, A. M. Rotunno
LSND anomaly
LSND anomaly (muon decay at rest)
P   e  0.245  0.067  0.045%
 e appearance with 3.8 signalsignificance

2
2
2
mLSND
 0.2 eV2  matm
 msolar

 we require (at least) one additional neutrino
L
m
2
2
order of
1
mLSND
( BF )  1.2 eV
E
MeV
L
~ 0 .4
E
LSND anomaly
There are various experimental results
that constrained the LSND signal :
LSND
Allowed region
negative results
Hints for sterile neutrinos

Reactor
anomaly :
New estimation of  e flux produced by beta decay from the
fission products of 235U, 238U, 239Pu and 241Pu
2
Reactor  anomaly
U e4  0
Mention et al 1101.2755
ROLD  0.976  0.024

R
 0.943  0.023
NEW

the predicted mean flux of  e increase is 3% higher
obs
R
N e
Npred
e
m 2  1 eV 2 can explain depletion for L ~ 10  100 m 


L
~1
E


2
2
 m31

 m41

2
2
2
2
2
P e  e  1  4 1  U e3  U e 4 U e3 sin 2 
L   1  U e 4 U e 4 sin 2 
L 
 4E 
 4E 
Hints for sterile neutrinos
• MiniBooNE:
Low energy excess : E  475 MeV
     beam energy E  800 MeV
detector @ L  500 m
L
E

MB
500 m
L
~
800 MeV
E
LSND
   e
   e
tension
3.0 signal significance
C. Polly -Neutrino 2012
2.5 signal significance
Hints for sterile neutrino
E  475 MeV
E  475 MeV
MiniBooNE
neutrino vs antineutrino data
MiniBooNE vs LSND
antineutrino
C. Polly -Neutrino 2012
Sterile neutrino schemes
3+1
 U e1 U e 2 U e 3 U e 4 


U
U
U
U

2
3
4 
U   1
U 1 U 2 U 3 U 4 


U

 s1 U s 2 U s 3 U s 4 
3+2
C
  
R
The sterile does not feel the
SM interactions
s
 s  R
C
PR R  0 PL R   R
C
C
C
PR ( L ) 
1   5 
2
Sterile neutrino 3+1
• Short Baseline experiment oscillation
probability formula:
2
 m32
 4 E  0
L
2
2
  (1)  m ~  (1 eV )   2
E
 m21  0
 4 E
P(  )
Only one oscillation
frequency is present
 sin 2
()
 1  sin 2
2
   
SBL
P(  )
2
 m41
L

sin 
 4E 
()
SBL
2
   
   
2
2
 m41
L


sin 
 4E 
2
 ,   e ,  , , s
where :
two neutrino system
sin2 2  4 U  4 U  4
2

2
sin2 2  4 U  4 1  U  4
2
2

Sterile neutrino 3+1
• In particular for :
sin 2 ee  4 U e 4
2
2

related to P e  e
2
Solar neutrino oscillations require large U e1  U e 2
sin 2   4U  4
2

2
2
 1  U
2
e4
1
related to P   
2
2
atmospheri c neutrino oscillations require large U 1  U  2  U  3
2
 1  U
2
4
1
1 2
 sin 2 e  sin 2 ee sin 2 2 
4
signal in appearance expt      signal in a disappeara nce expt 
2

e

   & e   e 
Sterile neutrino 3+1-global analysis
T. Schwetz -Neutrino 2012
excluded
excluded
disappeara nce expts :     
appearance expts :
   e
&  e  e
2
m41
 0.92 eV 2 , sin 2 2 e  0.0026
2
U e 4  0.022, U  4  0.030,
2
Consistency between appearance vs disappearance data P=10-5
Sterile neutrino 3+2 analysis
There is also tension in 3+2 between
disappearance and appearance bounds.
Giunti, Laveder, 1109.4033
Conclusions
• We are in a precision era of the
measurements in the PMNS matrix.
• Mass hierarchy is still unknown.
• Some tendencies in the value of CP violation.
• Sterile neutrinos ?
• Dirac or Majorana