P780.02 Spring 2003 L15
Neutrino oscillations/mixing
Richard Kass
The derivation of neutrino oscillations is very similar to the derivation of “strangeness”
oscillations (Lec. 7) and B meson oscillations.
To make the derivation “simple” assume that CP is conserved, there are only 2 types of
neutrinos and both neutrinos are stable (t1 = t2 = ).
At t=0 we have an electron (ne) and muon (nm) neutrino which are both mixtures of n1 and n2.
ne(t=0) ne= n1cosq+n2sinq
nm(t=0) nm= -n1sinq+n2cosq
Since we don’t know (beforehand) how “mixed” the neutrinos are we use q to describe the
mixture. Note: for the kaon case we assumed equal amounts of K1 and K2 or q=45 degrees.
The mass eigenstates (n1 and n2) propagate through space with energy E1 and E2 according to:
n e (t )  n1 e iE1t cosq  n 2 e iE 2t sin q
n m (t )   n1 e iE1t sin q  n 2 e iE 2t cosq
We are interested in the case where the neutrinos are relativistic (E>>m) and therefore:
m2
2
2
E  p m  p
2p
Assuming the same energy (and E= p) for both neutrino components we can write:
n m (t )  e
i ( p  m12 / 2 En )t
( n1 sin q  n 2 e
itm 2 / 2 En
cosq ) with m2  m12  m22
The probability of observing a ne at x (=ct) given that a nm was produced at t=0 is:
P(nmne)=|< ne|nm(t)> |2
M&S 11.1.1
P780.02 Spring 2003 L15
Richard Kass
Neutrino Oscillations/Mixing
P(n m  n e )  sin q cosq (1  e
 itm 2 / 2 En
2
)
If we measure mass in eV, x in meters, and E in MeV we can write the above as:
1.27 xm 2
x
P (n m  n e )  sin 2q sin (
)  sin 2 2q sin 2 ( )
En

2
2

En
1.27m2
The probability of observing a nm at x given that a nm was produced at t=0 is:
P(nmnm)=|< nm|nm(t)> |2
1.27 xm 2
P (n m  n m )  1  sin 2q sin (
)
En
2
2
In order to have neutrino oscillations:
1) at least one neutrino must have mass
2) the neutrinos must mix
Since the oscillation depends on m2 the mass of the neutrinos
must be determined from “other” experiments:
n energy endpoint experiments
double b-decay experiments
P780.02 Spring 2003 L15
The SuperKamiokande Experiment
Richard Kass
Original purpose was to search for proton decay: pe+0 (baryon # violation).
Found lepton number violation instead!
Use water as target and detector medium
Need lots of protons to get neutrino interactions.
Size: Cylinder of 41.4m (Height) x 39.3m (Diameter)
Weight: 50,000 tons of pure water
Need to identify e-’s and, m’s, 0’s (use Cerenkov radiation)
Reject unwanted backgrounds (cosmic rays, natural radiation)
103m underground at the Mozumi mine
of the Kamioka Mining&Smelting Co Kamioka-cho, Japan
P780.02 Spring 2003 L15
Atmospheric Neutrinos
Richard Kass
Atmospheric neutrinos are the end
product of high energy collisions of
cosmic rays (mostly protons) with
the nuclei in our upper atmosphere.
Neutrinos are mostly the result of
pion decay (and subsequent muon
decay) but kaons also contribute to
neutrino production.
From the figure on the right we (naively)
expect for the number of muon and
electron induced interactions:
R
N (n m n m )
Nn e
2
The experiments cannot distinguish
the charge of the lepton produced in the
neutrino interaction.
N(n m n m )  N(n

m N m X )
 N(n

m N m Y )
The efficiency for detecting muons is usually very different than the efficiency for
detecting electrons so the measured R is not 2.
P780.02 Spring 2003 L15
Richard Kass
Atmospheric Neutrino Oscillation Results from SuperK
Measure the number of ne and nm interactions in SuperK as a function of neutrino
path length in the earth’s atmosphere.
n
Neutrinos are produced by
cosmic ray interactions in
earths atmosphere.
superK
earth
Phys. Rev. Lett. 81 (1998) 1562-1567
atmosphere
n
The nm‘s are
“disappearing”!
2002 Nobel Prize
M. Koshiba
Phys. Rev. Lett. 86(2001)5656-5660
P780.02 Spring 2003 L15
Richard Kass
Atmospheric Neutrino Oscillation Results from SuperK
SuperK does not actually see an “oscillation”.
For example use the solution with:
m2 = 2.2x10-3 eV2 assume <En>=103 MeV
osc = (/1.27)(<En>/m2) = (/1.27)(103/2.2x10-3) = 1.1x106 m (620miles)
SuperK sees too few muon neutrinos.
The number of expected muon neutrino interactions is calculated using a detailed
simulation of the detector and takes into account detection efficiency as a function of
energy and angle (atmospheric path length and detector path length).
Scenario #1: No oscillations (or equal muon and electron neutrino oscillations nenm)
number of muon and electron neutrino interactions independent of L/E.
Scenario #2: muon neutrino oscillates into electron neutrino (nmne)
excess number of electron neutrino interactions Vs. L/E
depletion of muon neutrino interactions Vs. L/E
Scenario #3: muon neutrino oscillates into tau neutrino (nmnt)
SuperK has low detection efficiency for nt interactions
constant number of electron neutrino interactions Vs. L/E
depletion of muon neutrino interactions Vs. L/E
Scenario #4: muon neutrino oscillates into a neutrino (nmnS) that doesn’t interact
Scenario #5: Combination of 3&4 or something else??
P780.02 Spring 2003 L15
The Solar Neutrino Problem
The sun only produces electron neutrinos (ne)!
Richard Kass
M&S 11.1.2
Since 1968 R.Davis and collaborators have been measuring the cross section of:
ne + 37Cl e- + 37Ar
Their measured rate is significantly lower than what is expected from the
“standard solar model”
SNU=standard solar unit
Measured:
2.550.170.18 SNU
SNU=1 capture/s/1036 target atoms
Calculated:
7.32.3 SNU
Data from the
Homestake Gold
Mine (South Dakota)
2002 Nobel Prize
R. Davis
There is a long list of other experiments have verified this “problem”.
Too few neutrinos from the sun!
P780.02 Spring 2003 L15
Richard Kass
The Solar Neutrino Energy Spectrum
Homestake:
Chlorine
ne + 37Cl e- + 37Ar
SAGE/GALLEX:
Gallium
ne + 71Ga e- + 71Ge
Figure by J. Bahcall
SuperK:
nX + e- nX enmt + e- 1/6(ne e-)
P780.02 Spring 2003 L15
The Solar Neutrino Problem
Richard Kass
P780.02 Spring 2003 L15
The SNO Detector
Located in a mine in Sudbury Canada
Uses “Heavy” water (D2O)
Detects Cerenkov light like SuperK
Richard Kass
SNO=Sudbury Neutrino Observatory
Nucl. Inst. and Meth. A449, p172 (2000)
P780.02 Spring 2003 L15
Richard Kass
Why Use “Heavy” Water?
Charged Current interaction (CC): ne + d  e- + p + p (ne + n  e- + p )
Deuterium has neutrons!
Only electron neutrinos can cause this reaction
Neutral Current Interactions (NC): nemt + d  nemt+ n + p
D2O has twice as many nucleons as H2O
Neutrons are captured by
all neutrino flavors contribute equally
deuterium and produce
energy threshold for NC reaction is 2.2 MeV
6.25 MeV g
Elastic Scattering interactions (ES): nemt + e-  nemt + emostly electron neutrinos (NC and CC)
SNO measures several quantities (fCC, fNC, fES) and from
them calculates the flux of muon and tau neutrinos
(fm+ftf):  f
CC
ne
f NC  fn e  fn m  fnt
f ES  fn e  0.154(fn m  fnt )
SuperK only
has protons!
The quantities can
be compared with the
standard solar model.
They also measure the total 8B solar neutrino flux
into NC events and compare it with the prediction of the SSM.
P780.02 Spring 2003 L15
Results from SNO
Richard Kass
+0.44 +0.46
neutral current results: Fssm = 5.05 +1.01
F
=
5.09
sno
-0.81
-0.43 -0.43
Best fit to data gives:
Flux of 8B solar neutrinos
6
 2 1
F mt  3.41  0.45  00..45
48 10 cm s
Fmt=0 if no oscillations.
“SSM”=Standard Solar Model
Strong evidence for Neutrino Flavor Mixing at 5.3s (5.5s if include SuperK).
Total active neutrino flux agrees with standard solar model predictions.
Believe that the mixing occurs in the sun (“MSW effect”)
P780.02 Spring 2003 L15
Richard Kass
The Mikheyev Smirnov Wolfenstein Effect
Neutrino oscillations can be enhanced by traveling through matter.
Origin of enhancement is very similar to a “birefringent” medium where different
polarizations of light have different indexes of refraction. When polarized light passes
through a birefringent medium the relative phase of each polarization component
evolves differently and the plane of polarization rotates.
The neutrino “index of refraction” depends on its scattering amplitude with matter:
sun is made of protons, neutrons, electrons up/down quarks, electrons
All neutrinos can interact through neutral currents equally.
Only electron neutrino can interact through CC scattering: ne+ e-  ne + e-
The “refractive index” seen by electron neutrinos is different than the one seen
by muon and tau neutrinos.
The MSW effect gives for the probability of an electron neutrino produced at t=0
to be detected as a muon neutrino:
The MSW effect is
sin 2 2q
xW
P(n e  n m )  sin 2q m sin (
)
osc
2
2
sin 2 2q m 
W2
W 2  sin 2 2q  ( D  cos2 2q ) 2
2E
D  2GF N e n2
m
very similar to
“K-short regeneration”
M&S 10.2.4
Here Ne is the electron density.
For travel through vacuum Ne=0 and the MSW result reduces to our previous result.
P780.02 Spring 2003 L15
The MSW Effect
Richard Kass
There are only a few allowed regions in (q, m2) space that are compatible with
MSW effect:
LMA= Large Mixing Angle region favored.
SNO Day and Night
Energy Spectra Alone
Combining All Experimental
and Solar Model information
From A. Hamer, APS Talk, 4/2002