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Lecture 3 – neutrino oscillations and hadrons
● Review of leptons● Neutrino oscillations● Introduction to hadrons
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The lepton family
Lepton (antilepton)
Charge (e)
Mass (GeV)
e- (e+) -1 (+1) 0.0005e,(e) 0 0(+) -1 (+1) 0.105 0 0(+) -1 (+1) 1.8 0 0
Spin 1/2
6 leptons + antileptons. Divided into 3 families/flavours/multipletes: electron,muon tau
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- -
Lepton interactions
1; 1; 0
1; 1; 0
ElectromagneticIn:
Out: e
e
L L L
L L L
1; 0; 0
1; 0; 0
In:
Out: e
e
L L L
L L L
e- e-e-
W-
+
Charged leptons interact via the em and weak forces.Neutrinos only interact via the weak force.
Lepton number is always conserved at a vertex and in the whole process.
As for all forces: Charge conservation and energy-momentum conservation for incoming and outgoing particles.
Charge is conserved at a vertex though energy can appear to beviolated when dealing with ”internal lines” (lecture 1).
2; 0; 0
2; 0; 0
Weak: neutral currentIn:
Out: e
e
L L L
L L L
2; 0; 0
2; 0; 0
Weak; Neutral currentIn:
Out: e
e
L L L
L L L
1; 1; 0
1; 1; 0
Weak: charged currentIn:
Out: e
e
L L L
L L L
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Neutrino masses3 3Use decays eg
Mass and momenta of all particles except neutrino known and used to infer unobserved neutrino mass.
eH He e
Lepton Mass (MeV)e < 0.0003 <0.2 <18
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Neutrino oscillations
1 2 3 1 2 3 1 2 3
1 2
, ,
,
Hypothesis: are eigenstates of flavour (lepton number) but not "mass eigenstates".
+ ; ; (3.1)
are mass eigenstates with definite energy:
e
e
i
a b c d e f g h k
2 2
0
, , ....
(3.2)
(1.1) ; Implicit notation: ; are co-efficients which must be measured.
Remarks: (1) It is nonsense to ask "what is the mass of, eg ?" Any measurement wou
iiE ti
i i i
e
t e
E p m a b c
1 2 3
1 2 3
.
, ,
ld return a value of , or .(2) The weak force produces a flavour eigenstate, eg
That particle then propagate through space as a mixed state of , each with different energies. If
m m m
m
1 2 3 , and have different masses, this allows interference effects and theconversion of neutrino species, eg (next slides).
can then be measured in a dector via a weak interaction.Lepton nu
m m
mber is violated. The oscillation length determines whether or not this would be observed in a laboratory experiment.
Strategy:(1) Neutrino oscillation theoretical framework(2) How neutrinos are produced and measured.(3) Interpretation and final results.
Whole process violates: LL
Conserves L
Conserves L
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Neutrino oscillation formalism
sin cos cos sin
sin cos cos sin
,
Simplicity - consider two neutrino eigenstates, and eg and
and (3.3) Obs: .
and (3.4)
- mass eigens
e
i j
i j i j
i j
0 0
.
0 1 ; 0 0 0 sin 0 cos
sin cos
tates with definite energy: ; (3.2)
is a mixing angle - must be measured.Particle starts off as a
;
ji
i
iE tiE ti i j j
i j
iE ti j
t e t e
t e t e
22 2
2 22
cos sin sin cos
sin 2sin cos 1 14
sin 2 sin 22 2cos 4sin4 4 2
=
Probabili
j
ji
j i j ij ji i
iE t
iE tiE ti j
i E E t i E E tiE t iE tiE t iE t
j ij i
t t t e e
t e e e e e e
E E tE E t
2
sin 2 sin2
ty of : (3.5)
Neutrino oscillations!!
j iE E tP
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22 2 2 2
2
2 2
2
2 2 2 2
2
1
12 2
2 2
sin 2 sin
(highly relativistic particle)
For a flavour neutrino consisting of two neutrino mass states and i j
j i j ij i
j
mE p m pp
m mE p pp p
m m m mE E
p E
m mP
22
2 22 2
0 2 20
4
4sin 2 sin sin 2 sin4
Alternatively if neutrino travels a distance (nu)
= (3.6) Oscillation length
i
j iij
j i
tE
L ct t
m m L EP L LE L m m
2 2 2 0
0
(3.7)
For oscillations - a mass difference and
a non-zero mixing angle are needed. j im m m
Real(i)
probability prob. prob.
Real(i)Real()
L
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Neutrino Interactions in Matter
Probability of interaction ~ 10-5 / km water at 100 TeV energy
100 billion neutrinos pass through your thumbnail each second but only 1-2 will interact in your body during your lifetime.
1
Beam of neutrinos entering and propagating through matter:Probability density function for distances between successive collisions:
(3.8)
interaction length mean distance between intera
x
dx e dx
ctions (also called mean free path)
I0x)
interaction
x
0
103
106
109
1 103 106
E (TeV)N
e utr i
no in
tera
c tio
n le
n gth
i In
wa t
er/k
m
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relics from Big Bang
natural radioactive decay in Earthnuclear reactors
explosions of supernova
remnants of supernova
nuclear reactions in Sun
interactions of cosmic rays in atmosphere
Active Galactic Nuclei
Neutrinos arriving at Earth – not for the exam Fl
ux/e
nerg
y (c
m-2s-1
MeV
-1)
Neutrino energy (eV)
Relevant for this lecture
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Atmospheric neutrinos and Super Kamiokande
Boat
,
Cosmic rays interact in atmosphere
to produce pions , (this lecture)
(2 muon neutrinos, 1 electron neutrino)Neutrinos enter Super-K detector.Underground in Japanese Alps.
ee
, ,
Neutrinos measured in Super-Ks water Cerenkov detector.
Eg (X anything)
electrons and muons distinguished by Cerenkov light - measured inphotomultipliers. Energy and dir
e p e X p X p X
ection measured.Expt: 50,000 tons of water, 1200 photomultipliers
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Super-K: oscillations2
1.3
flux of muon-neutrinosExpect flux of electron-neutrinos
Observed Additional observationsFlux of electron-neutrinos from below=Flux of electron-neutrinos from above
electron neutrinos haven't
R
R
oscillated.Flux of muon-neutrinos from below< Flux of muon-neutrinos from above
muon neutrinos have oscillated to tau neutrinos:
Super-K
15 km
13000 km
From above
From below
no oscillations
P(
(Dat
a/Pr
edic
tion
for o
scill
atio
ns)
oscillations Earth
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Atmospheric neutrino results
2 3 2
sin cos cos sin
45 3 10
Evidence for muon neutrino turning into a tau neutrinos.Super-K data + constraints from other experiments:
and (3.9)
eV (3.atm i atm j atm i atm j
oatm atmm
10)Note: approximate with two flavour and two mass state neutrinos.
E=1 GeV
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relics from Big Bang
natural radioactive decay in Earthnuclear reactors
explosions of supernova
remnants of supernova
nuclear reactions in Sun
interactions of cosmic rays in atmosphere
Active Galactic Nuclei
Neutrinos arriving at Earth – not for the exam Fl
ux/e
nerg
y (c
m-2s-1
MeV
-1)
Neutrino energy (eV)
Relevant for this lecture
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Neutrinos in the sun – not for the exam
● Fusion process leading to
● ke measured flux on earth can be predicted
Solar neutrino energies
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Solar neutrino problem
37 37
Ray Davis et al. 1968Vat of 600 tons chlorine (cleaning fluid) inHomestead gold mine in South Dakota.Depth reduces background from cosmic rays
Looked for Cl Ar Collected 33 argon atoms. (flu
e e
shed out with He and their radioactive decays counted)
Prediction of standard solar model 100.
Solar neutrino problem: 1/3 of expected rate of solar neutrinos
Hypothesis: neutrino oscillations respons
, .
ible: ;
Unable to check since experiment insensitive e e
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SNO
2 ( )
( ) ( )
( ) , ,
Sudbury Neutrino Observatory (SNO)Ontario, underground water Cerenkov device.
Heavy water Sensitive to:
( )
Reaction (a) ins
X X e
X X X e
D O D deuteron
a D p n b D e p p
c e e
ensitive to neutrino flavour (lepton universality, lecture 2)- reaction rate agrees with predictions - nothing wrong with prediction of total number of neutrinosfrom the sun!
Reaction (b) measured at 1
, .
/3 rate of theoretical prediction(solar neutrino problem again).
Hypothesis: oscillated to
Check hypothesis with reaction (c) . Rate is sensitive to neutrinoflavour (question) and consistent
e
, with oscillations, e e e
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Results
sin cos cos sin
Electron neutrino oscillates into muon and/or tau neutrino.Solar neutrino + other experiments:
and (3.11)
arbitary combination of and , treat as e sol i sol j x sol i sol j
X
2 235 0.0001
one neutrino
(3.12)osol solm eV
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Generalise for 3 neutrinos
1
2
3
12 13 12 13 13
"
cos cos sin cos sin
MNS" matrix (3.13)
e
ie
U U
e
12 23 12 23 13 12 23 12 23 13 23 13
12 23 12 23 13 12 23 12 23 13 23 13
sin cos cos sin sin cos cos - sin sin sin sin cos
sin sin - cos cos sin - cos sin sin cos sin cos cos
i i
i i
e e
e e
1
2
3
12 23 13
12 23 13
(
, ,
35 45 10
Maki Nakagawa Sakata NMS) matrix
Mixing defined by three angles and one phase factor .Global fits to many experiments:
o o osol atm
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0.003
0.0001
Mass2 (eV2)
2 2 2 2 2 2 2 2 2 2 2 231 3 1 32 3 2 21 2 1 31 32 21
2 2 2 2 2 232 210.003 0.0001
Three neutrinos mass states three mass splittings (not independent)
(3.14)
eV eV (3.atm sol
m m m m m m m m m m m m
m m m m
0.0005 , 0.1 2
15)
Two possibilities: (a)normal (similar mass hierarchy to charged lepton:
GeV GeV GeV
(b) inverted.em m m
Flavour composition of mass state.
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Question
.
Using Feynman diagrams explain why the reactions
and are suppressed
with respect to e e
e e e e
e e
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Question
With the Minos experiment a beam of muon-neutrinos of energy 1-5 GeV (assume a flat, top-hat distribution of energy) is fired 750km from the Fermilab laboratory in Chicago to the SOUDAN mine in Minnes
2 23 2
2 2 2 23 2
- .
sin 2 0.9 - 0.003 .
ota.
The purpose of the experiment is to measure
Sketch the expected energy spectrum of muon neutrinos at the SOUDAN
mine if and eV
m m
m m
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Question
0 2
22
0 2
0 2
1.27
.
4 , .
Show that the oscillation length may be written where
is expressed in km, in and in
nu: : restore units with factors
Working in MKS-SI we expect an
ELm
eVL m E GeVc
EL cm
2
2 2 22
2 21 2 2 2 1 1
22
31 2 2 2 1 1
3 2
4
4
expression: with dimension
;
=
Now decide on the units: start with SI (m,s,J,kg.
a b
E Lm c
E M L T m M
M L TE M L T M L T c L Tm M
EM L T M L T L T Lc m
2219 19
10 10 2 2 22 22 28 8
34 10
2 41938 228
1.6 10 1.6 101 1.6 10 ( ) ( ) 1.6 10 ( )3 10 3 10
4 1.05 10 1.6 104
1.6 103 103 10
.)
eV eV GeV ; 1 kg J GeV ; J E E m kg mc c
EcELm c
m
2 2 2 2
787 0.7871.27
m= km= kmE E Em m m
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Summary of neutrino oscillations
● Neutrinos produced in the weak force are flavour eigenstates but not mass eigenstates.
● Neutrino oscillations between flavours occur as a consequence of non-zero mixing angles and a non-zero difference between the mass states
● Experiments on atmospheric and solar neutrinos demonstrate neutrino oscillations.