Transcript Weak mixing

Mixing in Weak Decays
• Charged Weak Current (exchange of Ws)
causes one member of a weak doublet to
change into the other
   e  e  


e
W
e
• Taus and muons therefore decay into the
lightest member of the doublet (their
neutrinos)
• electrons are stable as the (e,nu) doublet is
the lightest doublet. The virtual W can’t
convert to anything
“e decay”
e
e
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W
1
Mixing in Weak Decays
• In the same context, the heavier quark
doublets decay via as c,t are heavier
 c  t 
   c  s"W "; t  b"W "
 s  b 
• s and b quarks should then be stable (their
lightest baryons) as the lightest member of
their doublets
• But they aren’t……due to mixing between
the 3 generations
• for quarks the mass eigenstates are not the
same as the decay eigenstates
• b mass eigenstate: what has mb
• b decay eigenstate: what interacts with W-t
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Quark Mixing: 2 Generations
• If assume only 2 generations. Mixing matrix
 cosq C
M  
  sin q C
sin q C  Vud Vus 
  

cosq C  Vcd Vcs 
• where qC is the Cabibbo angle
• M then rotates from the mass eigenstates (d,s) to
the decay eigenstates (d’,s’) (usually deal with
mixing of charge 1/3 quarks but both mix)
 d 
d 
   M  
 s 
s
 u 
u
   M  
 c 
c
• look at weak vertices (2 identical ways)
W
c
c
Vcs
s’
u
d’
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s
c
Vcd
d
3
Charm Decay
• Charmed meson can Beta decay to lighter mesons
which have either s or d quarks
D  (cd )  K 0 ( sd )      
•
D    0 or  0 (dd )      
BF  .07
BF  .003
Vcd2 sin 2 q C .05
.003
 .043  2 

2
.07
Vcs cos q C .95
• Modulo slightly different phase space, the ratio of
these decays depends only on the different mixing.
Direct measurement of the mixing angle.
d
s

W
d
c
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
4
Kaon Decays
• Historically first place mixing observed
• decay rates depend on same phase space and spin
factors as charged pion decay
• Observed rates only 5% of what they “should” be
and Cabibbo proposed a mixing angle whose
source was unknown at the time
• This (partially) lead to a prediction that the c quark
must exist
mK  494MeV   1.2 108 sec
K  (us )   
BF  64%
K     0
BF  21%
K     0 0 or     BF  7%
K    0   or  0 e 
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BF  8%
5
3 Quark Generations
• For 3 generations need 3X3 matrix. It is unitary and
has some phases which don’t matter and can be
defined by 3 angles and 1 phase (phase gives
particle antiparticle differences….antiparticles use
M* Hermitian adjoint)
• called Cabibbo-Kobyashi-Maskawa (CKM) matrix
and was predicted by K-M before the third
generation was discovered
•
Vud Vus Vub   .97 .22 .004

 

Vcd Vcs Vcb    .22 .97 .04 
V
  .01 .04 .997
V
V
ts
tb 
 td


• Each Vij tells what factor needed for W vertex.
Shown are experimental values. No theory predicts
the amount of mixing,
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CKM Matrix Numerology
• For N generations need NXN unitary matrix.
Matrix has 2N2 terms (real and complex) and it has
N2 constraints (rows x columns = 0,1).
For 2N quarks have 2N-1 arbitrary phases
•
N=
2
3
N2
=
4
9
2N-1 =
3
4
parameters needed
1
4
• 3 generations  3 angles (Euler angles) + 1 phase
• CKM* acts on antiquarks. phase causes a small
particle-antiparticle difference. Need at least 3
generations to have CP violation/matter-dominated
Universe
CKM 
0   c2
1 0

 
 0 c1 s1   0
 0  s1 c1  s 2e i

 
0  s 2ei   c3 s3 0 
 

1
0     s 3 c3 0 
0
c2   0
0 1 
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B mesons
• B mesons contain b quarks (D mesons contain c
quarks)

12
B  ub
B  u b   1.7  10 s
B 0  db
B 0  d b   1.6  1012 s
Bc  cb
Bc  c b   0.5  1012 s
Bs0  sb
Bs0  s b   1.5  1012 s
• B, D and  lifetimes are just long enough so their
path lengths can be detected
• use to measure B properties and identify B,D, in
ee,pp collisions. For B mesons
E
p
g
v
bgc
7 GeV 5 GeV 1.4
.71c
.3 mm
20.6
20
4.1
.97c
1.2 mm
50.3
50
10
.995c
3 mm
H  bb ,
t  bW
Z  bb ,
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B Decays
• B mesons are dominated by the decay of the b
quark. As large mass, phase space differences are
small and can get branching fractions by just
counting
x3
d c , du
D  , 

W
e  
   e 
db
B0
u c
d
s
B 0  X
1
1


B 0  all 1  1  1  3  3 9
2
b  c Vcb
.042


 100
2
2
b  u Vub
.004
2
measured in
data
5
( B  D ) Vcb mB .042  5.3 
4



1
.
7

10


( K   ) Vus 2 mK5 .222  .5 
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9
Particle-antiparticle mixing and CP
violation
• There is another type of “mixing” which is
related to quark mixing. This can lead to
observation and studies of CP violation
• consider the mesons which are neutral and
composed of different types of quarks
0
0
0
d
0
s
0
0
0
d
0
s
K (ds ) D (uc ) B (db ) B ( sb )
K (d s) D (u c) B (d b) B ( s b)
• Weak interactions can change particle into
antiparticle as charge and other quantum
numbers are the same. The “strangeness”
etc are changing through CKM mixing
d
K
0
u,c,t
s
W
K0
d
s
u, c,t
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• Depends onVij at each W vertex
• as V and V* are different due to phase,
gives particle-antiparticle difference and CP
violation (any term with t-quark especially)
• the states which decay are admixtures of the
“strong” state (a rotation). They can have
different masses and different lifetimes
K1   K 0  b K 0
K2   K 0  b K 0
• #particle vs #antiparticle will have a time
dependence. Eg. If all particle at t=0, will be
a mixture at a later time
• the phenomenology of K’s is slightly
different than B/D’s and we’ll just do K’s in
detail. Kaons rotate and give long-lived and
short-lived decays. B/D also rotate but
lifetimes are ~same.
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Neutral Kaon Semi-leptonicDecay
• Properties for “long” and “short” lived
K 0 : m ass  498MeV , mK L  mK S  3 1012 MeV
 K  1010 sec  K  5 108 sec
S
L
• Semi-leptonic (Beta) decays. Positive or
negative lepton tells if K or anti-K decayed
K 0 (ds )    (du )  e  or    
K 0 (d s)    (d u )  e  or   
• partial width is exactly the same as charged
K decay (though smaller BF for Short and
larger for Long).
4
BF  7 10 K S  b 
BF  0.3K L  b 
BF

BF

 0.7  107 sec1
 0.6 107 sec1
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Neutral Kaon Hadronic Decays
• Also decay hadronically
K 0 (ds )       or  0   0
K 0 (ds )       or  0   0
K 0 (ds )         0 or  0   0   0
K 0 (ds )         0 or  0   0   0
• Both decay to same final states which
means the mixed states K1 and K2 also
decay to these 2pi and 3pi modes. Means
initial states can mix and have interference
d
u
d
K K
0
0
d
u
s
u d
d
d
s
u
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Sidenote C+P for Pions
• Parity operator Pf(x,y,z)=f(-x,-y,-z).
Intrinsic parity for psuedoscaler mesons
(like K,pi) is -1
• Charge conjugation operator C. Changes
particle to antiparticle.
C 
C K   K
C0  0
C K0   K 0
C (C  0 )  C (  0 )  2  0    1
• Can work out eigenvalue. As C changes
charge, C=-1 for photon
C
e-
e+
=
• given its decay, pion has C= +1
0
BF
(

 gg )
0
7
  gg

4

10
BF( 0  ggg )
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Neutral Kaon Hadronic Decays
• 2 pion and 3 pion are CP eigenstates with
eigenvalue +1 for 2pi and -1 for 3pi
CP        
CP  0 0    0 0
CP    0      0
CP 3 0   3 0
• K1 and K2 also CP eigenstates
1
( K 0  K 0 )  K S CP  1
2
1

( K 0  b K 0 )  K L CP  1
2
K1 
K2
 ( K S )  0.9 1010 s
 ( K  )  1.2 108 s
 ( K L )  5.2 108 s
• different values of matrix element if initial
and final states are the same CP eigenstate
or if they are not CP eigenstates (like K+ or
beta decays)
• if CP is conserved, K1/Ks decays to 2 pions
and K2/KL decays to 3 pions. More phase
space for 2 pions and so faster decay,
shorter lifetime.
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Decay and Interference
• From Schrodinger eq. plane wave solutions
 ( K S )  AS (t  0)e
 ( K L )  AL (0)e
(
(
S
2
ims )
eiEt /  , E  m
L
imL )
2
  e t /    1
2
assume: K s  K1; K L  K2
• the two amplitudes have to be added and
then squared. Gives interference. Example:
start with pure K0
K0 
1
1
( K L  K S )  AS (0)  AL (0) 
2
2
• Intensity is this amplitude squared

I ( K 0 )   ( K S )   ( K L ) * ( K S )   * ( K L )



1 S t
 e  e Lt  2e ( S L )t / 2 cos m t
4
m  mL  mS  105 eV
• small mass difference between the two
weak decay eigenstates
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Decay and Interference
• Do the same for anti-K

I ( K 0 )   ( K S )  ( K L ) * ( K S )  * ( K L )


1 S t
 e  e Lt  2e ( S L )t / 2 cos m t
4
• get mixing. Particle<->antiparticle varying
with time.
• At large time get equal mixture = 100% KL
• the rate at which Kanti-K depends on
1/m. You need to mix K<->antiK before
they decay to have KS and KL
m S  0.47 " K S " , " K L " decays
But
If
(m) 1   K 0  just " K 0 " decays
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
KS Regeneration
KL 
1
(K 0  K 0 )
2
• Assume pure KL beam
• strikes a target made up of particles (p,n)
• different strong interaction cross section for
K and anti-K 0
K ( d s )  n   (uds )   0
K 0  n   (uds )   0
• mix of K-antiK no longer 1:1. Example,
assume “lose” 0.5 antiK, 0.0 K. gives
(ignoring phases and so not quite right)
0
K
0
K 
 aKL  bKS 
2
a( K 0  K 0 )  b( K 0  K 0 )  a  34 , b 
1
4
• First observed by Lederman et al. measures
particle/antiparticle differences. Useful
experimental technique
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CP Violation
•
•
•
•
•
1
2
3
C changes particle to antiparticle
P operator flips space (mirror image)
T time reversal t  -t
fundamental axiom (theory?) of quantum
mechanics CPT is conserved
Weak interaction violate all 3. CP violation
is the same as T violation. Three
observations (so far) of this
Universe is mostly matter (Sakharov
1960s)
KL decay to 2 pions (Christianson, Cronin,
Fitch and Turlay, 1964)
neutral B decays
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spark chambers and so poor mass
resolution. Identify K->2 as in
forward direction
m ostly: K L         0
45  9 K L      
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CP Violation in K decays
• Ks and KL (the particles which have
different lifetimes) are NOT eigenstates of
CP. Instead K1 and K2 are
1
K1, 2 
(K 0  K 0 )
2
1
KL 
( K 2  K1 ) |  | 2.2 103
1 |  |2
KS 
1
1 |  |2
K0
( K1  K 2 )
K
0
KS
K1
KL
K2
• When KL decays, mostly it is decaying to a
CP=-1 state(3 pions) but sometimes to a
CP=+1 state (2 pions)
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CP violation in K decays
• CP is then explained by having a phase in the
mixing between K and anti-K
• other sources of CP violation (“fifth force”) are
ruled out as inconsistent with the various ways of
observing CP violation
K L    
BF  2.1103
K L   0 0
BF  9 10 4
K S     0
BF  (3) 107
ch arg e asym m etry  L  0.3 102
( K L      )  ( K L      )

(or e)
 
 
( K L     )  ( K L     )
am p( K L   0 0 )
 .9950 .0008
 
am p( K L    )
d
K
0
u,c,t
s
W
K0
d
s
u, c,t
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Indirect vs Direct CP
• Indirect CP is due to the mixing (the box diagram)
• Direct is in the decay and that the charged and
neutral modes are slightly different (different
isospin)
W
g
K L    
BF  2.1  103
K L   0 0
BF  9  104
am p( K L   0 0 )
 .9950 .0008
 
am p( K L    )
d
K
0
u,c,t
s
W
K0
d
s
u, c,t
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Fermilab proposal 617 January 1979
20+ year experiment at FNAL and CERN
wrong. small effect and very
large hadronic factors
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B’s: Mixing and CP violations
• Neutral B’s (and Ds) also mix and have CP violating
decays. These depend on CKM matrix elements (and
are better at determining them than K decays). Bs and
Ks both oscillate a few times before they decay
• different than K system as many decay channels most
of which are not CP eigenstates. Also no “L,S” as the
lifetimes of the “1,2” states are about the same
B1, 2 

1
( B0  B 0 )
2
m
m/hbar
KS
0.9  1010 sec 4  1012 Mev,0.5  1010  / s 0.5
Bd
1.5  1012 sec 3  1010 Mev,0.5  1012  / s 0.8
Bs
1.5  1012 sec 1.5  108 Mev,19  1012  / s 30
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Glashow-Weinberg-Salam Model
• EM and weak forces mix…or just EW
force. Before mixing Bosons are massless:
• Group Boson Coupling Quantum No.
SU(2)L W1,2,3 g
T weak isospin
U(1)
B
g’ Y leptonic hypercharge
T3  Y / 2  Q (elec.ch arge)
• Interaction Lagrangian is
 
YB

Lint  gT  W  g
2
• convert to physical fields. Neutrals mix
(B,W3Z, photon). W,Z acquire mass.
Force photon mass=0. Higgs Boson
introduced to break mass symmetry (A is
same field as in EM….4 vector)
A  sin qW W 3  cosqW B
Z  cosqW W 3  sin qW B
g
qW  weak m ixing angle tanqW 
g 27
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Higgs Boson
• breaking electroweak symmetry gives:
massive W+Z Bosons
mass=0 photon
1 or more scalar particles (Higgs)
minimal SUSY, 2 charged and 3 neutral
• Higgs couples to mass and simplistically
decays to the most massive available
particles
• “easy” to produce in conjunction with heavy
objects (helps to discover??)
 ( HZ )   ( HW )
 20% mH  100GeV
 ( H ) production
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Standard Model Higgs Boson
• Branching fraction
depends on mass
• Use ZH,WH for
m<135 GeV
• Use WW for m>
135 GeV
• Current limits use
1-2 fb-1
• D0: 12 Higgs decay
channels + 20
analyses combined
WH  l   bb
ZH  l  l  bb
ZH   bb
H  WW  l l
D0+CDF
Limits 1.4-8
times SM
(2007)
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• Look at EM and 2 weak “currents”
LEM
LEM
LEM
 g sin qW T3W  g  cosqW
3
YB
2
Y
g  cosqW  g sin qW  e
 g sin qW (T3  ) A
2
T3  Y / 2  Q

 g sin qW A Q  eA J em
• charged current. Compare to mu/beta decay
(have measured weak force, eg. weak
mixing only “new” free parameter)

Lcc  g TW  TW

e / sin qW 37.3GeV
g2
GF
g


M



W
2
5/ 4
5/ 4
8M W
sin qW
2
2
GF 2
GF
• weak neutral current
LNC
g
g
2


Z (T3  sin qW Q ) 
Z  J NC
cosqW
cosqW
g 2 / cos2 qW GF
MW

 MZ 
2
8M Z
cosqW
2
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W and Z couplings
 e 
1
  T3   Q  1,0
2
 e L
e R T3  0 Q  1
• EW model has
left-handed doublets
right handed singlets
[  T  0 Q  0]
• W couplings to left-handed component and
always essentially the same
R
3
gW   T 3   12 g doublet;  0 sin glet
• Z to left-handed doublet
Z to right-handed singlet
g L " g z " (T 3  Q sin 2 qW )
g R " g z " (Q sin 2 qW )
e
Z
[T 3  Q sin 2 qW ]
sin qW cosqW
• redefine as Vector and Axial parts of V-A
Z
e
cV  g L  g R  T 3  2Q sin 2 qW
cA  g L  g R  T 3
e
c c
2
V
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2
A
31
Z decays/vertices
ee
 e e
uu
dd
  
  
cc
ss
  
  
Z
T3
Q
cV
cA
2
cA
 cV2

1
2

1
0
 .0 8

1
2
1
2
.2 6
1
2
1

2
.5 0

bb
1
2
2

3

.1 9

1
2
1

3

 .3 4
1
1

2
2
.2 9
.3 7
Color factor of 3 for quarks
cV  g L  g R  T 3  2Q sin 2 qW
cA  g L  g R  T 3
sin 2 qW  0.21
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Z Branching Fraction
• Can use couplings to get branching modes
• PDG measured values in ()
•
Z  ee
.26

Z  all 3 * .26  3 * .50  2 * 3 * .29  3 * 3 * .37
.26

 .036 (.034)
7.3
Z   3 * .50

 0.21 (.20)
Z  all
7.3
Z  bb 3 * .37

 0.15 (.15)
Z  all
7.3
Z  qq 3 * 3 * .37  2 * 3 * .29

 0.70 (.70)
Z  all
7.3
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Neutrino Physics
• Three “active” neutrino flavors (from Z width
measurements). Mass limit from beta decay
m e  3 eV
mex2  104 eV 2 x   or 
m   0.2 MeV
m2x  103 eV 2 x   (or inactive)
m  18 MeV
• Probably have non-zero masses as they oscillate
(right-handed neutrinos? messes up electroweak)
• Only have weak interactions and can be either
charged or neutral currents
e
W
charge
e
e
e
 en  e p
n   p
 e e   e  e
n e
n,p,e  e p   e p
neutral
e
p
Z
n,p,e
P461 - particles III
 ee   ee
 e  e
34
Neutrino Cross Sections
• Use Fermi Golden Rule
Rate 
2
| M |2  phase space

• M (matrix element) has weak interaction
physics…W, Z exchange ~ constant at
modest neutrino energies. Same G factor as
beta decay
1
1
2
G
g

2
2 8M W
q M
2

2
W
M
2
W
E  M W
• cross section depends on phase space and
spin terms. Look at phase space first for
charged current. Momentum conservation
integrates out one particle
 e e  e e (CC )
phase space  pe2 dpe p2 dp
pcm  pe  p   
4G 2
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
2
pcm
35
Neutrino Cross Sections II
• Look in center-of-momentum frame
s  M 2  Etot2  ptot2


pe   p  ptot  0 Etot  E  Ee  2 p
 s  (2 p) 2
4G 2 p 2 G 2 s




• s is an invariant and can also determine in
the lab frame
ptot  p  E
Etot  E  me
s  E2  2me E  me2  p2  2me E
G 2 2m E


• cross section grows with phase space (either
neutrino energy or target mass)
 (p) mp

 2000
 (e) me
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Neutral Currents
• The detection of some reactions proved that
neutral current (and the Z) exist
   e    e
   p    p
• the cross section depends on the different
couplings at each vertex and measure the
weak mixing angle
e
16 4
sin qW )

3
G 2 me E 1 4 2
16

(  sin qW  sin 4 qW )

3 3
3
 
 e
G 2 me E
(1  4 sin 2 qW 
• about 40% of the charged current cross
section. due to Z-e-e coupling compared to
W-e-nu coupling
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Neutrino Oscillations
• Different eigenstates for weak and mass
weak : e ,  , 1, 2 , 3 : mass
• can mix with a CKM-like 3x3 matrix with
(probably) different angles and phases then
quarks. The neutrino lifetime is ~infinite
and so mix due to having mass and mass
differences (like KL and KS)
• example. Assume just 2 generations (1
angle)
    1 cosq  2 sin q
 e  1 sin q  2 cosq
• assume that at t=0 100% muon-type
  (t  0)  1  e (t  0)  0
 1 (t  0)  cosq  2 (t  0)  sin q
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Neutrino Oscillations II
• Can now look at the time evolution
• from the Scrod. Eq. And assuming that the
energy is much larger than the mass
1, 2 (t )  1, 2 (0)e
mi2
Ei  p 
2p
iE1, 2t
  c 1
• probability of e/mu type vs time (or length
L the neutrino has traveled) is then
2
  (t )  cos q e
2
iE1t
 sin q e
2
iE2t 2
2
4

m
Lc
 1  sin 2 2q sin 2
4 Ec
• where we now put back in the missing
constants and used 2 trig identities
LE
t
c p
2 sin q cosq  sin 2q
( E2  E1 )t
cos(E2  E1 )t  1  2 sin
2
2
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39
Neutrino Oscillations III
• Oscillation depends on mixing angle and
mass difference (but need non-zero mass or
no time propagation)
2
4

m
Lc
  (t )  1  sin 2 2q sin 2
4 Ec
2
 e (t )  1    (t )
2
2
• so some muon-type neutrinos are converted
to electron type. Rate depends on neutrino
energy and distance neutrino travels L/E
• go to 3 neutrino types and will have terms
with more than one mixing angle. Plus
neutrinos can oscillate into either of the
other two (or to a fourth “sterile” type of
neutrino which has different couplings to
the W/Z than the known 3 types)
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40
Neutrino Oscillations IV
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41
Neutrino Oscillations V
With three generations of neutrinos the change of one
neutrino type into another depends on many terms
You can understand the terms by measuring at different
energies and lengths
There is another effect (interactions in matter) which
we will skip that comes into play
Oscillations can also violate CP – be different if
neutrino or antineutrino beam
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42
Detecting Neutrino Oscillations
• Disappearance: flux reduction larger L/E
• Solar Neutrinos. Measure rate for both
electron neutrinos and all neutrinos (using
neutral current). Low energies (few MeV)
cause experimental thresholds for some
techniques. Compare to solar models.
rate( e  n  e  p)
Rate( e, ,  pn  e, ,  p  n)
• Atmospheric neutrinos. Measure rate as a
function of energy and length (from angle)
   
  e e 
# e 1
 production
#  2
• also electron or muon neutrinos produced at
reactors or accelerators. Compare flux near
production to far away L/E >> 1
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43
Neutrinos from Sum
• from Particle Data Group
p  p2H  e  
8
7
B8Be  e  
Be  e7Li  
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Detecting Neutrino Oscillations
• Appearance: start with one flavor detect
another
• Ideal. Tag nu production by detecting the
lepton. Then detect neutrino interaction.
Poor rates (considered pi/K beams and
muon storage rings)
• Real. Tau neutrino very difficult to detect
sources of pure electon neutrinos (reactors)
are below muon/tau threshold
•  use mostly muon neutrino beam
e
K   e e
 0.003

   
• can measure neutrino energy in detector (if
above 1 GeV. Below hurt by Fermi gas
effects). Can usually separate electron from
muon events with a very good ~100%
active detector
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45
Nova detector will be mostly liquid scintillator (like
BNL neutrino experiment of the 1980s). Greater than
80$%active.
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46
High Priority Items in Particle Physics
•
•
•
•
•
•
•
•
Quark Mixing and CP violation
Neutrino Mixing and maybe CP violation
are Quark and Neutrino mixing related?
Source of Electro-Weak symmetry
breaking (Higgs?)
Precision measurements of current
parameters (top,W,Z mass)(g-2)
what is dark matter? dark energy?
Searches for New Phenomena –
Supersymmetry, Extra Dimensions,
Leptoquarks, new quarks/leptons/bosons,
compositeness, why spin ½ vs spin 1
some NP can explain other questions
(source of CP, dark matter, etc)
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Extra Dimensions
• Possible solution to the Hierarchy Problem
MH ~ 100 GeV
MGUT/Planck ~ 1016-1019
GeV
• model of Arkani-Hamed, Dimopoulos and Dvali
 gravity propagates to n extra spatial dimensions
 gives massive stable Kaluza-Klein gravitons
GKK
 effective Plank scale MPl related to fundamental
Plank scale in (n+4) dim MD
• also model of Randall-Sundrum
 1 extra dim but large curvature
SIGNAL (real graviton)
MPl2~RnMDn+2
- high ET single photon +
missing ET
- monojet + missing ET
SIGNAL (virtual graviton)
- high mass pair: ee,,gg
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Supersymmetry
• add superpartner to
quarks, leptons, and
bosons
• Solves the Hierarchy
Problem
• lightest
supersymmetric
particle (LSP)
candidate for dark
matter
• Unification of the
gauge couplings
R-Parity:
if conserved: LSP is stable, SUSY
particles produced in pairs
not conserved: may generate 
masses/mixing
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SUSY:
Trileptons
• number of possible
decay chains
• Very clean mode
- 3 isolated leptons
- MET from  or c0
• low BF (< 0.5 pb)
• leptons can be soft
and depend on m
0
~
c  LSP
1
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