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Neutrino states in oscillation
experiments
– are they pure or mixd?
Pheno 07,
May, 07-09, 2007, Madison , Wisconsin
Marek Zralek, Univ. of Silesia
1. INTRODUCTION
Common approach to oscillation phenomena
D
FLUX
DETECTOR
P
Calculated for
massless
neutrinos
1) For production and detection cross section - massless
neutrino
2) Factoryzation
3) Transition probability
In vacuum or in matter
Where flavour states are given by
Z.Maki,M.Nakagawa,S. Sakata,
Prog.Theor.Phys. 28(1962)870
How we can convince that it is correct?
C.Giunti, C.W.Kim, J.A.Lee,U.W.Lee,Phys. Rev,
D48(1993) 4310.
W.Grimus,P.Stockinger, Phys.Rev.D54 (1996) 3414.
Full Quantum field theoretical treatment
Neutrino propagate over macroscopic distance (sometimes
astronomical)  it is unnatural to consider them as virtual
Quantum-field-theoretical model of neutrino oscillation in
which the propagating neutrino is described by a wave packet
state determined by the production process
C.Giunti JHEP 0211(2002)017
For the process:
It was proposed:
And finally the neutrino states are given by:
This approach is not fully correct:
1) Particles which take part in the production and detection processes have
spins, we don’t known what to do with them,
2) All time the neutrino state is pure quantum mechanical states, even for non
relativistic neutrinos,
3) We don’t know how to incorporate physics beyond tha SM.
We propose to use density matrix approach, then
1. We know what to do with any properties of accompanied particles,
2. We can check, when neutrino state is pure, and when it is mixed,
3. Any New Physic (NP) in neutrino interaction can be easy considered,
4. In very natural way we are able to take into account neutrino space
localization (wave packet approach) ,
5. We exactly know, when the formula for neutrino transition factorize,
6. For relativistic neutrino and their SM Left-Handed interaction, we
reproduce the standard formulae
2. DENSITY MATRIX FOR PRODUCED NEUTRINOS
We consider production neutrino process:
  e,  ,
i  1, 2,3
For each particle (without neutrino) we introduce wave packet (given by
experimental condition):
In momentum representation:
Final results don’t depend on the shape of wave pockets - we use Gauss
distribution.
In coordinate space:
We calculate:
Let us assume the effective Hamiltonian:
Then:
First we integrate over particle momenta:
Or in the other way:
First we integrate over particle momenta, using:
We obtain:
If we introduce:
We can integrate over d4x:
=
Where
p=p
  
Let us assume now that initial particles are not polarized, we can
define density matrix for final neutrino:
Normalization condition:
The amplitudes we calculate for general interaction:
l
PL PR
i
l
PL PR
W+
A
i
H+
B
A
B
We calculate the amplitude in the CM frame
Everything have to be transformed to the laboratory frame
Lorentz transformation ()
Helicity states feel Lorentz transformation:
p


p


z
Helicity and Wigner rotation
For Wigner rotation:
For rotation of helicity states:
Wigner rotation and rotation for helicity states
near threshold
Wigner rotation
Rotation for helicity states
a
a
1
1
0.8
0.8
b
0.6
0.6
0.4
0.4
0.2
b
b
0.2
2 10
-6
4 10
-6
6 10
-6
8 10
-6
0.00001
2 10
-6
4 10
E [MeV]
500, m
-6
6 10
-6
8 10
-6
0.00001
E [MeV]
1 eV,
3
6 10
5 10
4 10
  500


b
-7
12
-7

b
0.00006
23
 
24
-7
0.00004
3 10
2 10
1 10
-7
-7
0.00002
-7
1
2
3
4
1
5
2
3
4
5
4
5
  50
Neutrino energy [MeV]
L
Neutrino energy [MeV]
5000
L
20
4000
15
3000
10
2000
5
1000
1
2
3
4
5
CM
1
2
3
CM
6 10
-7
5 10
-7
4 10
-7
3 10
-7
2 10
-7
1 10
-7


  50
12
1
2
3
4

23
 
24
5
Neutrino energy [MeV]
Neutrino energy [MeV]
L
L
CM
CM
Standard Model,
Mass hierarchy
1  Tr ( 2 )
m1  0, m 2  0.009 eV, m 3  0.05 eV
me  0.51099892 MeV,
md  5 MeV,
m u  2.25 MeV,
 p  10 eV,
Threshould = 25 MeV 2 .
1  Tr ( )
2
With right-handed currents,
 R  0.01
U  U
L
R
All couplings
 L   R = L  0.01;
R  0.02
UL  UR  VL = VR
Dependence
on the mass hierarchy
eV
=0[eV]
eV
=1[eV]
Dependence
on the scattering angle
3. NEUTRINO PROPAGATION AND DETECTION
The statistical operator in the detector place, after time T:
and density matrix:
Let us assume, now that neutrinos are detected in the process:
The transition cross section for flavour  neutrino detection:
To allow full and all wave pockets to pass:
Number of flavour  neutrinos in
the detector:
S
Total number of
neutrinos = N
d
Final cross sections:
1
p
1
32 s
2 sA 1
1, 1
1
AL
3
L
L
ULi ULk
AL
L
L
V Li ULk
AL
L
L
L
L
ULi V Lk
V Li V Lk
1 i; 1 k ;
1
p
1
32 s
2 sA 1
1, 1
1
AR
3
R
URi URk
R
AR
R
R
V Ri URk
AR
1, 1
1
R
R
R
R
R U i
V Ri V Rk
V Rk
1 i; 1 k ;
1, 1
p
32 s
D1 Re
1
2 sA 1
L
R
URi V Lk
1 i; 1 k
D1 Re
L R
V Ri ULk
1 i; 1 k
D2 Re
L R
URi ULk
1 i; 1 k
Normalized elements of the production density matrix:
1,i ;
1,k
1
Nor
A
2
L
1
2
L
A
2
1,i ;
ULi ULk
L R L
A
2
R
2
R
A
L
V Ri V Rk
L R R
R
L
R
ULi V Rk
R
L
V Ri ULk
;
1,k
1
Nor
A
2
R
1
2
2
R
A
URi URk
R L L
A
L
2
L
2
L
A
V Li V Lk
R L R
R
URi V Lk
V Li URk
;
1, 1
AL
L
L
ULi ULk
AL
==
L
L
VLi ULk
L
L
ULi VLk
AL
L
L
VLi VLk
x
A
2
L
2
L
UL i UL k
A
2
R
2
R
VR i VR k
1
2
e
A
I
L
R
L
m2 m2
i
k
2E
A
L
L
R
R
R
U L i U L k U L i UL k e
L
i,k
1
R
I
2 m2
mi
k
2E
UL i VR k
L
3
P
L
L
VR i UL k
4. CONCLUSIONS
1. States are not pure near threshold, pure states appear for
relativistic neutrinos and charge current left-handed
production and detection mechanism,
2. For searching a physics beyond the SM, neutrino production
and detection states are not necessary pure,
3. States are mixed, if right-handed (RH), scalar-LH-RH or
pseudoscalar-RH – LH interactions are present,
4. Wigner rotation for helicity neutrino states are completely
negligible in practice,
5. Only for relativistic neutrinos produced and detected by the LH
mechanism the oscillation rates factorize.