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Document 1181260
Weak Interactions
●
●
Some
weak interaction basics:
■ Weak force is responsible for β decay e.g. n → pev (1930’s).
Chapter 8 M&S
■ Interaction involves both quarks and leptons.
■ Not all quantum numbers are conserved in weak interaction:
◆ parity, charge conjugation, CP
◆ isospin
◆ flavor (strange, charm, bottom, top)
■ Weak (+EM) are “completely” described by the Standard Model.
Weak interactions has a very rich history:
■ 1930’s: Fermi’s theory described β decay.
■ 1950’s: V-A (vector-axial vector) theory:
◆ Yang & Lee describe parity violation.
◆ Feynman and Gell-Mann describe muon decay and decay of strange mesons
■ 1963: N. Cabibbo proposes “quark mixing”:
◆ “explains” why rates for decays with ΔS = 0 > ΔS = 1:
BR(K − → µ −ν µ ) = 63.5%
◆
€
BR(π − → µ −ν µ ) = 99.9%
Quarks in strong interaction are not the same as the ones in the weak interaction:
❍ weak interaction basis different than strong interaction basis:
(K 0 ,K 0 ) ⇔ (K S ,K L )
K.K. Gan
€
L10: Weak Interactions
1
Weak Interactions
●
●
Weinberg-Salam-Glashow
(Standard Model 1970’s-today):
■ unify weak and EM forces
■ predict neutral current (Z) reactions
■ gives relationship between mass of W and Z
■ predict/explain lots of other stuff, e.g. no flavor changing neutral currents
■ predict existence of Higgs (“generates” mass in Standard Model)
■ construct renormalizable gauge theory
The picture is still incomplete:
■ must input lots of parameters into the Standard Model (e.g. masses)
■ where’s the Higgs and how many are there?
■ how many generations of quarks and leptons are there?
■ how to explain the mass pattern of quarks and leptons?
◆ neutrinos have mass!
■ CP violation observed with quarks!
◆ is there CP violation with leptons?
K.K. Gan
L10: Weak Interactions
2
Weak Interactions
●
Classification
Type
Leptonic
Semileptonic
of weak interactions
Comment
leptons only
leptons and quarks
Examples
µ − → e−ve vµ
ve e− → ve e−
n → pe−ve (Δs = 0)
K + → µ + vµ (Δs = 1)
−
0 −
B → D µ vµ (Δb = 1)
●
€
●
Non - leptonic
quarks only
Λ → π − p & K + → π +π 0
Quarks and leptons are grouped into doublets (SU(2)).
■ Sometimes called families or generations.
■ For every quark doublet there is a lepton doublet:
Q( e )
2
 u  c
t 
3
   
 
− 13
d  s
b
 e   µ   τ  −1
  ν   
0
ν e   µ  ν τ 
νµ,τ
W-
e-
νµ
W-
e-
νe,τ
W-
µ-
ντ
W-
µ-
νe,µ
W-
W-
Charged
■
€
νe
Not Allowed
Allowed
■
current interaction involves the exchange of a W boson. τ-
τ-
W’s couple to leptons in the same doublet.
W coupling to leptons/quarks is a combination of vector and axial vector (V-A) terms:
Ju = uγu(1- γ5)u
◆ parity violating charged current
K.K. Gan
L10: Weak Interactions
3
Cabibbo Model
conjecture (1963):
■ quarks participating in weak interaction are a mixture of quarks that participate in the strong interaction
■ explain certain decay patterns in the weak interactions
■ originally had only to do with the d and s quarks:
d' = d cos θ + s sin θ
☞ the form of the interaction (charged current) has an extra factor for d and s quarks
d quark : J u ∝ γ u (1− γ 5 ) cos θ c
Purely leptonic decays
€
s quark : J u ∝ γ u (1− γ 5 ) sin θ c
(e.g. muon decay) do not
s
d
contain the Cabibbo factor
u 

u
W-
W-
=



sinθc
€
 d ′  d cos θ c + s sin θ c  cosθc
νµ
µ+
●
●
Cabibbo’s
u
u
The Cabibbo angle is important for determining the rate of many reactions.
€
■ The Cabibbo angle can measured using data from the following reactions:
2
BR(K + → µ + vµ ) sin 2 θ c mK 1− (mµ /mK )2 


=
cosθc or sinθc
BR(π + → µ + vµ ) cos2 θ c mπ 1− (mµ /mπ )2 
From the above branching ratio’s we find:
u
θc = 0.27 radians
−
o −
−
o −
We can check the above by measuring the rates for K → π e ν e and π → π e ν e :
θc = 0.25 radians
◆
€
■
K.K. Gan
L10: Weak Interactions
€
W+
d, s
4
Extension to Cabibbo Model
model could easily be extended to 4 quarks:
u  u  

u
→
=
    

 d   d ′  d cos θ c + s sin θ c 
c  c  

c
→
=
    

 s   s′  s cos θ c − d sin θ c 
 d ′  cos θ c sin θ c  d 
 =
 
 s′   − sin θ c cos θ c  s 
● Adding a fourth quark actually solved a long standing puzzle in weak interactions:
■ “absence” (i.e. very small BR) of decays with a “flavor” (e.g. strangeness) changing neutral current:
BR(K 0 → µ + µ − ) 7 ×10 −9
−8
=
≈
10
€
M&S section 8.2.1
0.64
BR(K + → µ + vµ )
● Cabibbo’s model could not incorporate CP violation
Cabibbo’s name
■ 1977: there was evidence for 5 quarks!
was added to
● CKM model:
make “CKM”
€ ■ Kobayashi and Maskawa extended Cabibbo’s idea to six quarks in 1972.
◆ This is two years before discovery of charm!
◆ 2x2 → 3x3 matrix that mixes the weak quarks and the strong quarks
◆ The matrix is unitary.
☞ 1 parameter (Cabibbo angle) → 3 angles (generalized Cabibbo angles) + 1 phase
☞ The phase allows for CP violation.
❍ Just like θc, the matrix elements of the CKM matrix must also be obtained from experiment.
●
Cabibbo’s
K.K. Gan
L10: Weak Interactions
5
GIM Mechanism
●
In
■
■
€
1969-70, Glashow, Iliopoulos, and Maiani (GIM) proposed a solution to the K0 → µ+µ- rate puzzle:
BR(K 0 → µ + µ − ) 7 ×10 −9
=
≈ 10 −8
+
+
0.64
BR(K → µ vµ )
The branching fraction for K0 → µ+µ- was expected to be small.
The first order diagram is forbidden as this is not a allowed W coupling.
µ+
µ+
µ-
νµ
K+
allowed
W+
?
K0
forbidden
u
■
■
€
d
s
s
The 2nd order diagram (“box”) was calculated:
amplitude ∝ sinθccosθc
◆ Found to give a rate higher than the experimental measurement!
GIM proposed that a 4th quark existed and its coupling to the s and d quark was:
s' = s cos θ − d sin θ
◆ The new quark would produce a second “box” diagram with
amplitude ∝ -sinθccosθc
◆ These two diagrams almost cancel each other out.
❍ The amount of cancellation depends on the mass of the new quark.
❍ A quark mass of ≈ 1.5 GeV is necessary to get good agreement with the experimental data.
❍ First “evidence” for Charm quark!
K.K. Gan
L10: Weak Interactions
6
CKM Matrix
●
The
■
CKM matrix can be written in many forms:
In terms of three angles and phase:
 d ′ 
c12 c13
s12 c13

 
iδ13
iδ13
′
s
=
−s
c
−
c
s
s
e
c
c
−
s
s
s
e

12
23
12
23
13
12
23
12
23
13
 
 b′   s s − c c s eiδ13 −c s − s c s eiδ13
   12 23 12 23 13
12 23
12 23 13
■
■
€
€
This matrix is not unique,
many other 3x3 forms in
the literature. This one is
from PDG2000.
The four real parameters are δ, θ12, θ23, and θ13.
◆ s = sin, c = cos, and the numbers refer to the quark generations, e.g. s12 = sinθ12.
In terms of coupling to charge 2/3 quarks:
d ′ Vud Vus Vub d 
  
 
′
s
=
V
V
V
cd
cs
sb
  
 s 
b′ Vtd Vts Vtb b 
◆ Best for illustrating physics!
In terms of the sine of the Cabibbo angle (θ12).
 d ′  1− λ2 /2
λ
Aλ3 (ρ − iη )  d 
 
  
2
2
′
s
=
−
λ
1−
λ
/2
A
λ
 s
  
“Wolfenstein” representaton
 b′   Aλ3 (1− ρ − iη ) −Aλ2



1
  
 b
◆ This representation uses the fact that s12 >> s23 >> s13.
◆ λ = sinθ12, and A, ρ, η are all real and approximately one.
◆ This representation is very good for relating CP violation to specific decay rates.
◆
€
s13e−iδ13   d 
 
s23c13   s 
c23c13   b 
K.K. Gan
L10: Weak Interactions
7
CKM Matrix
●
The
magnitudes of the measured CKM elements (PDG2000):
Vud Vus Vub   0.9742 − 0.9757
0.219 − 0.226
(2 − 5) ×10 −3 

 
−2 
0.9734 − 0.9749 (3.7 − 4.3) ×10 
 Vcd Vcs Vcb  =  0.219 − 0.225
V

−2
(3.5 − 4.3) ×10 −2 0.9990 − 0.9993 
 td Vts Vtb   (0.4 −1.4) ×10
■
€
There are several interesting patterns:
1)  The CKM matrix is almost diagonal (off diagonal elements are small).
2)  The further away from a family, the smaller the matrix element (e.g. Vub << Vud).
3)  Using 1) and 2), we see that certain decay chains are preferred:
❍ c → s over c → d:
BR(D0 → K-π+) ~ 3.8%
BR(D0 → π-π+) ~ 0.15%
❍ b → c over b → u:
BR(B0 → D-π+) ~ 3x10-3
BR(B0 → π-π+) ~ 1x10-5
4)  Since the matrix is supposed to be unitary there are lots of constraints among the matrix elements:
*
*
Vud
Vud + Vcd
Vcd + Vtd* Vtd = 1
*
*
Vub
Vud + Vcb
Vcd + Vtb* Vtd = 0
❍
❍
So far experimental results are consistent with expectations from a unitary matrix.
As precision of experiments increases, we might see deviations from unitarity.
€
K.K. Gan
L10: Weak Interactions
8
Measuring the CKM Matrix
●
No
■
one knows how to calculate the values of the CKM matrix.
Cleanest way to measure the CKM elements is to use interactions or decays involving leptons.
◆ CKM factors are only present at one vertex in decays with leptons.
Vud
neutron decay
n → pe−ve
d → uev
Vus
kaon decay
K 0 → π + e−ve
s → uev
Vbu
B - meson decay
B − → (ρ + or π + )e−ve b → uev
Vbc
B - meson decay
B − → D0 e−ve
b → cev
Vcs
charm decay
D0 → K −e+ ve
c → sev
Vcd neutrino interaction
vµ d → µ −c
d →c
€
“Spectator” model decay of D0 → K-e+ve
Called a “spectator” diagram because
only one quark participates in the decay,
the other “stands around and watches”. W
c
D0
u
K.K. Gan
Vcs
e,µ
For massless neutrinos
the lepton “CKM”
matrix is diagonal
ve,vµ
Decay rate ∝ |Vcs|2
s
Ku
L10: Weak Interactions
9
Fly UP