section viii the weak interaction. 181 the weak interaction the weak interaction accounts for many...
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The Weak Interaction The WEAK interaction accounts for many decays in particle physics
Examples:
Characterized by long lifetimes and small cross-sections
eμ
τe-
μe-
νpenνμπ
ννeτννeμ
;
;
3
Two types of WEAK interaction:
CHARGED CURRENT (CC): W Bosons
NEUTRAL CURRENT (NC): Z0 Boson
The WEAK force is mediated by MASSIVE VECTOR BOSONS:
MW ~ 80 GeV
MZ ~ 91 GeV
Examples:
Weak interactions of electrons and neutrinos:
W
e
eνW e
eν
0Z
eν
eν0Z
e
e
4
Boson Self-Interactions In QCD the gluons carry “COLOUR” charge. In the WEAK interaction the W and Z0 bosons carry the WEAK
CHARGE W also carry EM charge
BOSON SELF-INTERACTIONS
W
W
0Z
W
W
W
W
W
W
W
W
0Z
0Z
W
W0Z
W
W
5
Fermi Theory
Weak interaction taken to be a “4-fermion contact interaction” No propagator Coupling strength given by the FERMI CONSTANT, GF
GF = 1.166 x 10-5 GeV-2
Decay in Fermi Theory
Use Fermi’s Golden Rule to get the transition rate
where Mif is the matrix element and (Ef) is the density of final states.
n
p
e
eν
GF
f2
fi EρM2Γ π
e-νpen
f
f dE
dNEρ
6
Density of Final States: 2-body vs. 3-body (page 54)
TWO BODY FINAL STATE:
Only consider one of the particles since the other fixed by (E,p) conservation.
THREE BODY FINAL STATE (e.g. decay):
Now necessary to consider two particles – the third is given by (E,p) conservation.
dEdΩ
2π
EdN 3
2
ee3
2e
3
22 dEdΩ
2π
EdEdΩ
2π
ENd
Relativistic (E ~ p)i.e. neglect mass of finalstate particles.
7
In nuclear decay, the energy released in the nuclear transition, E0, is shared between the electron, neutrino and the recoil kinetic energy of the nucleus:
Since the nucleus is much more massive than the electron /neutrino:
and the nuclear recoil ensures momentum conservation.
For a GIVEN electron energy Ee :
recoilνe0 TEEE
νe0 EEE
0ν dEdE
ee3
2e
ν3
2ν
ν0
dEdΩ2π
EdΩ
2π
E
dE
dN
dE
dN
8
Assuming isotropic decay distributions and integrating over ded gives:
Matrix Element
In Fermi theory, (4-point interaction)
and treat e, as free particles
e4
2e
2
e0e4
2e
2ν
e3
2e
3
2ν2
0
dE4π
EEEdE
4π
EEdE
2π
E
2π
E4π
dE
dN
3
2e
2
e02
fie
e4
2e
2
e02
fi
2π
EEEM
dE
dΓ
dE4π
EEEM2πdΓ
rdψψψψGM 3νepnFfi
rpirpie eψeψ e
rdψeψGM 3n
rpppFfi
νe
9
Typically, e and have energies ~ MeV, so << size of nucleus and
Corresponds to zero angular momentum (l = 0) states for the e and . ALLOWED TRANSITIONS
The matrix element is then given by
where the nuclear matrix element accounts for the overlap of the nuclear wave-functions.
If the n and p wave-functions are very similar, the nuclear matrix element
Mfi large and decay is favoured:
SUPERALLOWED TRANSITIONS
-210rp
1ψ,ψe
222
nuclear2F
3np
2Ffi MGrdψψGM
2
nuclearM
12 nuclearM
10
Here, assume (superallowed transition):
SARGENT RULE:
e.g. - and - decay (see later)
By studying lifetimes for nuclear beta decay, we can determine the strength of the weak interaction in Fermi theory:
12 nuclearM
3
50
2F
50
50
50
3
2F
E
0 e2e
2
e03
2F
2e
2
e03
2F
e
60π
EGΓ
4
E2
5
E
3
E
2π
GdEEEE
2π
GΓ
EEE2π
G
dE
dΓ
0
5 0Eτ
25- GeV10003.0136.1 FG
2Ffi GM
2
11
Beta-Decay Spectrum
Plot of versus is linear
2e
2
e03e
EEE2πdE
dΓ
2FG
e02ee
EEE
1
dE
dΓ
2ee E
1
dE
dΓ e0 EE
KURIE PLOT
keVEe
e33 eHeH ν 2
ee E
1
dE
dΓ
0E
12
Mass
m= 0 (neglect mass of final state particles)
End point of electron spectrum = E0
m 0 (allow for mass of final state particles)
Density of states (page 54)
me known, m small only significant effect is where Ee E0
2e
2e
2
e0
2ν2
e
2
e03
2F
eE
m1EE
m1EEE2π
G
dE
dΓ
dEdΩ
2π
EdN 3
2
dEdΩ
p
E
2π
pdN 3
2
13
eE
2ee E
1
dE
dΓ
KURIE PLOT
Most recent results (1999)Tritium decay:
me < 3 eV
If neutrinos have mass, me << me
Why so small ?
eE
2ee E
1
dE
dΓ
4
14
1
2
e
2e
2
e0
2ν
e0e
2e
Em1
EEm1EE
dE
dΓ
E
1
Experimentalresolution
14
Neutrino Scattering in Fermi Theory (Inverse Decay).
where Ee is the energy of the e in the centre-of-mass system and is the energy in the centre-of-mass system.
In the laboratory frame: (see page 28)
’s only interact WEAKLY have very small interaction cross-sections Here WEAK implies that you need approximately 50 light-years of water to stop a 1 MeV neutrino !
However, as E the cross-section can become very large. Violates maximum allowed value by conservation of probability at (UNITARITY LIMIT).
Fermi theory breaks down at high energies.
epnνe
n p
eeν
GF
π
sGσ
dΩ2π
EG2π
dE
dNM2πdσ
2F
3
2e2
F
2
fi
s
nνm2Es
243- cm10MeVin~ νEσ
GeV740s
Appendix F
15
Weak Charged Current: W Boson Fermi theory breaks down at high energy True interaction described by exchange of CHARGED W BOSONS Fermi theory is the low energy EFFECTIVE theory of the
WEAK interaction.
Decay:
e Scattering:
2W
2 mq
n
p
e
eν
GF
udd
udu
e
eν
n p
W
eν
GF
e
νμ
e
2W
2 Mq
1
W
Wg
Wg
eν
νμ
16
Compare WEAK and QED interactions:
CHARGED CURRENT WEAK INTERACTION At low energies, , propagator
i.e. appears as POINT-LIKE interaction of Fermi theory. Massive propagator short range
Exchanged boson carries electromagnetic charge.
FLAVOUR CHANGING – ONLY WEAK interaction changes flavour PARITY VIOLATING – ONLY WEAK interaction can violate parity
conservation.
e
2q
1γ
emg
emg μ
e
μ
egem
4
e2
e
2W
2 Mq
1
W
Wg
Wg
eν
νμ
WEAK QED
2W
2 Mq 2W
2W
2 M
1
Mq
1
fm002.0~RangeGeV4.80W
W M
1M
4π
gα
2W
W
17
Compare Fermi theory c.f. massive propagator
For compare matrix elements:
GF is small because mW is large.
The precise relationship is:
The numerical factors are partly of historical origin (see Perkins 4th ed., page 210).
The intrinsic strength of the WEAK interaction is GREATER than that of the electromagnetic interaction. At low energies (low q2), it appears weak due to the massive propagator.
-μ
μν
e
eν
GF
e
eν
W
-μ μνWg
Wg
2W
2 Mq F2W
2W G
M
g
2
G
8M
g F2W
2W
25 GeV10166.1andGeV4.80 FW GM
30
1
4π
gα0.65g
2W
WW and137
1
4π
eα
2
(see later to why different to )
FG
18
Neutrino Scattering with a Massive W Boson
Replace contact interaction by massive boson exchange diagram:
with where is the scattering angle.
(similar to pages 57 & 97)
Integrate to give:
Total cross-section now well behaved at high energies.
e
2W
2 Mq
1
W
Wg
Wg
eν
νμ
22W
2
4W
2Mq
g
32π
1
dq
dσ
22Esinq
2W
2W
2F
2W
2F
Msπ
MGσ
Msπ
sGσ
Appendix G
19
Parity Violation in Beta Decay
Parity violation was first observed in the decay of 60Co nuclei (C.S.Wu et. al. Phys. Rev. 105 (1957) 1413)
Align 60Co nuclei with field and look at direction of emission of electrons
Under parity:
If PARITY is CONSERVED, expect equal numbers of electrons parallel and antiparallel to
B e-
e-
B
P̂
e6060 νeNiCo
B
μμ;LprLpp;rr
B
J=5 J=4
20
Most electrons emitted opposite to direction of field
PARITY VIOLATION in DECAY
As 60Co heats up, thermal excitation randomises the spins
Polarized Unpolarized
T = 0.01 K
21
Origin of Parity Violation
SPIN and HELICITY
Consider a free particle of constant momentum, .
Total angular momentum, , is ALWAYS conserved. The orbital angular momentum, , is perpendicular to The spin angular momentum, , can be in any direction relative to The value of spin along is always CONSTANT.
p
SLJ
1hp
s
1hp
sp
psh
prL p
p
S
pS
“RIGHT-HANDED” “LEFT-HANDED”
Define the sign of the component of spin along the direction ofmotion as the
HELICITY
22
The WEAK interaction distinguishes between LEFT and RIGHT-HANDED states.
The weak interaction couples preferentially to
LEFT-HANDED PARTICLES
and
RIGHT-HANDED ANTIPARTICLES
In the ultra-relativistic (massless) limit, the coupling to RIGHT-HANDED particles vanishes.
i.e. even if RIGHT-HANDED ’s exist – they are unobservable !
60Co experiment:
ps
ps
B60Co 60Ni
J=5 J=4
eν
e
p
ep
23
Parity Violation
The WEAK interaction treats LH and RH states differently and therefore can violate PARITY (i.e. the interaction Hamiltonian does not commute with ).
PARITY is ALWAYS conserved in the STRONG/EM interactions
Example
PARITY CONSERVED PARITY VIOLATED
Branching fraction = 32% Branching fraction < 0.1%
P̂
u
u
u
u
uu
η
00u
u 0
21 LL000
P
000
11πPπPπP1P
0000Jπππη
0LL1;P 2!
u
u
u
u
ddη
LP
1πPπP1P
000Jππη
0L1;P
24
PARITY is USUALLY violated in the WEAK interaction
but NOT ALWAYS !
Example
PARITY VIOLATED PARITY CONSERVED
Branching fraction ~ 21% Branching fraction ~ 6%
s
u
u
u
ddK
d
W u
s
u
u
uK
0
d
W u
LP
1πPπP1P000JππK
0
0
0L1;P
21 LL-
P
11πPπPπP1P0000JπππK
0LL1;P 2!
25
The Weak CC Lepton Vertex
All weak charged current lepton interactions can be described by the W boson propagator and the weak vertex:
W Bosons only “couple” to the lepton and neutrino within the SAME generation
e.g. no coupling
Universal coupling constant gW
STANDARD MODELWEAK CCLEPTON VERTEX
W τ,μ,e
τμe ν,ν,ν
Wg
4π
gα
2W
W +antiparticles
τμe ντ
νμ
νe
μνWe
26
Examples:
e
eν
W
-μ μν
ν,ν,ν μe
W
τ,μ,e-
udd
udu
e
eν
n p
W
bc
cc
cB ψJ
ν
W μ
e-νpen
τ-
μ-
e- ντ,νμ,νeW
μeννeμ
μν
W
τ ν
μ
τμννμτ
νμJB -c
27
Decay
Muons are fundamental leptons (m~ 206 me).
Electromagnetic decay is NOT observed; the EM interaction does not change flavour. Only the WEAK CC interaction changes flavour. Muons decay weakly:
As can use FERMI theory to calculate decay width (analogous to decay).
γeμ
μeννeμ
-μ
μν
e
eν
GF
e
eν
W
-μ μνWg
Wg
2W
2 Mm
28
FERMI theory gives decay width proportional to (Sargent rule).
However, more complicated phase space integration (previously neglected kinetic energy of recoiling nucleus) gives
Muon mass and lifetime known with high precision.
Use muon decay to fix strength of WEAK interaction GF
GF is one of the best determined fundamental quantities in particle physics.
5m
5μ3
2F
μμ m
192π
G
τ
1Γ
s1000004.019703.2 6
25 GeV1000002.016632.1 FG
29
Universality of Weak CouplingCan compare GF measured from decay with that from decay.
From muon decay measure:
From decay measure:
Ratio
Conclude that the strength of the weak interaction is ALMOST the same for leptons as for quarks. We will come back to the origin of this difference
e
eν
W
-μ μνWg
Wg
udd
udu
e
eν
n p
W
25 GeV1000002.016632.1 FG
25- GeV10003.0136.1 FG
003.0974.0
F
F
G
G
Ccos
30
DecayThe mass is relatively large
and as
there are a number of possible decay modes.
Examples
Tau branching fractions:
GeV0003.0777.1 τm
,m,m,mm ρπμτ
μν
W
τ ν
μ
eν
W
τ ν
eu
W
τ ν
d
%)2.07.64(%)1.03.17(%)1.08.17(
hadronsτνντννeτ
-τ
-τe
-
31
Lepton UniversalityTest whether all leptons have the same WEAK coupling from measurements of the decay rates and branching fractions.
Compare
If universal strength of WEAK interaction, expect
are all measured precisely
Predict
Measure
SAME WEAK CC COUPLING FOR AND
e
eν
W
-μ μνWg
Wg
eν
W
τ ν
e
Wg
Wg
5μ3
2F
eμμ
m192π
GΓ
τ
1
%1.08.17 e)B(τ
53
2F
e m192π
G
0.178
1Γ
e)B(τ
1
τ
1
5
5μ
μ m
m
τ
178.0
μμ τmm ,, s1000004.019703.2
MeV3.00.1777MeV658.105
6
μ
μ
τmm
s1001.091.2 13 τ
s1001.091.2 13 τ
32
Also compare
IF same couplings expect:
(the small difference is due to the slight reduction in phase space due to the non-negligible muon mass).
The observed ratio is consistent with the prediction.
SAME WEAK CC COUPLING FOR e, AND
LEPTON UNIVERSALITY
eν
W
τ ν
e
Wg
Wg
μν
W
τ ν
μ
Wg
Wg
9726.0
τe-
τμ-
ννeτB
ννμτB
005.0974.0
τe-
τμ-
ννeτB
ννμτB
33
Weak Interactions of QuarksIn the Standard Model, the leptonic weak couplings take place within a particular generation:
Natural to expect same pattern for QUARKS, i.e.
Unfortunately, not that simple !!
Example:
The decay suggests a coupling
eνW
-e
μνW
μ
νW
τ
uW
d
cW
s
tW
b
μνμK suW
u W
μs
μν
K
34
Cabibbo Mixing AngleFour-Flavour Quark Mixing The states which take part in the WEAK interaction are ORTHOGONAL combinations of the states of definite flavour (d, s) For 4 flavours, {d, u, s and c}, the mixing can be described by a single parameter
CABIBBO ANGLE (from experiment)
Weak Eigenstates Flavour Eigenstates
Couplings become:
oC 13
s
d
cossin
sincos
s
d
CC
CC
uW
d
uW
d
uW
sCcosWg CsinWg
CC sincos sd
cW
s
cW
d
cW
sCcosWg Csin Wg
CC sincos ds
+
+
35
Example: Nuclear decay
Recall
strength of ud coupling Hence, expect It works,
Cabibbo Favoured
Cabibbo Suppressed
udd
udu
e
eν
n p
W
25 GeV1000002.016632.1 FG
25- GeV10003.0136.1 FG
CW cosg C
222cosM
FG
Ccos FF GG
o13cos16632.1136.1 oC 13
uW
dCcosWg
uW
sCsinWg
cW
sCcosWg
cW
dCsin Wg
C2cos 2
M
C2sin 2
M
36
Example:
coupling Cabibbo suppressed
Example:
Expect
Measure
is DOUBLY Cabibbo suppressed
μνμK
su
C2sin 2
M
u
W μs
μν
K
CsinWg Wg
πKD,πKD 00
cu
su0D K
d
W u
CcosWg
CcosWg
cu
du0D π
s
W uK
CsinWg-
CsinWg
0028.0cos
sin
C4
C4
πKD
πKD0
0
0008.00038.0
πKD0
37
CKM MatrixCabibbo-Kobayashi-Maskawa Matrix
Extend to 3 generations
Weak Eigenstates Flavour Eigenstates
Giving couplings
uW
d
cW
s
tW
b
b
s
d
V
b
s
d
CKM
uW
dudWVg
uW
susWVg
uW
bubWVg
1λλ
λ1λ
λλ1
105.001.0
05.0cossin
01.0sincos
23
22
λ
32
λ
CC
CC2
2
tbtstd
cbcscd
ubusud
CKM
VVV
VVV
VVV
V
Csin
38
The Weak CC Quark VertexAll weak charged current quark interactions can be described by the W boson propagator and the weak vertex:
W bosons CHANGE quark flavour W likes to couple to quarks in the SAME generation, but quark state mixing means that CROSS-GENERATION coupling can occur.
W-Lepton coupling constant gW
W-Quark coupling constant gW VCKM
STANDARD MODELWEAK CCQUARK VERTEX
+antiparticles
W
qqWVg
4π
gα
2W
W
tc,u,q
bs,d,q
39
Example:
gw Vud, Vcs, Vtb O(1)gw Vcd, Vus O()gw Vcb, Vts O()
t d, b u O()very small
(Log scale)
KKπ
νμDB-
μs0s
μ
bs
cs0
sBsD
μνW
cbWVg
Wg
sD
u
ss
cs
dW
csWVg
udWVg π s
ss
us -K
uW
usWVg
usWVg K
C4sin 4
W
2gM
Csin
C4cos 4
W
2gM C
4sin 4W
2gM
40
Summary
WEAK INTERACTION (CHARGED CURRENT) Weak force mediated by massive W bosons
Weak force intrinsically stronger than EM interaction
Universal coupling to quarks and leptons Quarks take part in the interaction as mixtures of the flavour
eigenstates Parity can be VIOLATED due to the HELICITY structure of the
interaction Strength of the weak interaction given by
from muon decay.
25 GeV1000002.016632.1 FG
GeV038.0423.80 WM
137
1α
30
1α emW cf
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