P780.02 Spring 2003 L10
Richard Kass
Weak Interactions
Some Weak Interaction basics
Weak force is responsible for b decay e.g. n → pev (1930’s)
interaction involves both quarks and leptons
not all quantum numbers are conserved in weak interaction:
parity, charge conjugation, CP
isospin
flavor (strangeness, bottomness, charm)
Weak (+EM) are “completely” described by the Standard Model
Chapter 8 M&S
Weak interactions has a very rich history
1930’s: Fermi’s theory described b 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
1960’s: Cabibbo Theory
N. Cabibbo proposes “quark mixing” (1963)
"explains" why rates for decays with DS =0 > DS =1
BR( K      )  63.5%
Chapter 8.2.3 M&S
BR(      )  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
previous example: ( K o , K o )  ( K s , K L )
P780.02 Spring 2003 L10
Weak Interactions
Richard Kass
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
existence of Higgs (“generates”mass in Standard Model)
Renormalizable Gauge Theory
But 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 ?
mass pattern of quarks and leptons ?
neutrinos have mass!
CP violation observed with quarks!
is there CP violation with leptons?
P780.02 Spring 2003 L10
Weak Interactions
Classification of Weak Interactions
Type
Comment
Leptonic
involves only leptons
Semileptonic leptons and quarks
Non-Leptonic involves only quarks
Richard Kass
Examples
muon decay (  evv)
ee-  eeneutron decay (Ds=0)
NOT
K+  + (Ds=1)
Allowed
B  Do   (Db=1)  Allowed
e
,
L  -p & K+ +o
WW-
Some details of Weak Interactions
quarks and leptons are grouped into doublets (SU(2))
(sometimes called families or generations)
For every quark doublet there is a lepton doublet
Q  2 / 3|e|
 u  c   t 
 d   s  b
Q  1/ 3|e|
Q  |e|
 e       
 
  e        
Q 0
Charged Current Interactions (exchange of a W boson)
W’s couple to leptons in the same doublet
The W coupling to leptons/quarks is a combination
of vector and axial vector terms:
Ju  ugu(1- g5 )u (parity violating charged
e
ee,
W-

W-
e,
W-
-
W-
Cabibbo Model
P780.02 Spring 2003 L10
Richard Kass
Cabibbo’s conjecture was that the quarks that participate in the weak interaction are a mixture of
the quarks that participate in the strong interaction.
This mixing was originally postulated by Cabibbo (1963) to explain certain decay patterns in the
weak interactions and originally had only to do with the d and s quarks.
d’ = d cosq + s sinq
Thus the form of the interaction (charged current) has an extra factor for d and s quarks
d quark: Ju a gu(1- g5 )cosqc
s quark: Ju a gu(1- g5 )sinqc
d
u
 u  


 d   d cos q c  s sin qc 
cosqc
s
W-
sinqc
WPurely leptonic decays
(e.g. muon decay) do not
contain the Cabibbo factor:
u
u
The Cabibbo angle is important for determining the rate of many
reactions:

2
The+ Cabibbo
angle
can measured 2using
data from the following
+
2
+ reactions:
BR(K   v)
mk
= sin q c
BR(+  + v)
cos 2 q c m 
1- (m / m k )
1- (m / m ) 2
cosqcor sinqc
From the above branching ratio’s we find:
qc= 0.27 radians
We can check the above by measuring the rates
u
for:K    oe  e     o e  e Find: qc= 0.25 radians
W+
d, s
P780.02 Spring 2003 L10
Cabibbo’s Model
Extensions to the Cabibbo Model:
Cabibbo’s model could easily be extended to 4 quarks:
c
u

 u  
  c 




 d   d cos q c  s sin qc   s   s cosq c  d sin q c 
 d    cosq
 
 s    sin q c
Richard Kass
sin q c  d 
 
cosq c  s 
Adding a fourth quark actually solved a long standing puzzle in weak interactions, the
“absence” (i.e. very small BR) of decays involving a “flavor” (e.g. strangeness) changing
neutral current:
0
 
9
BR( K    ) 7  10
8


10
0.64
BR( K      )
However, Cabibbo’s model could NOT incorporate CP violation and by 1977 there was evidence
for 5 quarks!
M&S section 8.3.1
The CKM model:
Cabibbo’s name
In 1972 (2 years before discovery of charm!) Kobayashi and Maskawa
was added to
extended Cabibbo’s idea to six quarks:
make “CKM”
6 quarks (3 generations or families)
3x3 matrix that mixes the weak quarks and the strong quarks (instead of 2x2)
The matrix is unitary 3 angles (generalized Cabibbo angles), 1 phase (instead of 1 parameter)
The phase allows for CP violation
Just like qc had to be determined from experiment, the matrix elements of the CKM matrix must
also be obtained from experiment.
P780.02 Spring 2003 L10
The GIM Mechanism
Richard Kass
In 1969-70 Glashow, Iliopoulos, and Maiani (GIM) proposed a solution to the
BR( K 0      ) 7  10 9
to the K0 + - rate puzzle.

 108
BR( K      )
0.64
The branching fraction for K0 + - was expected to be small as the first order
diagram is forbidden (no allowed W coupling).

+

+
K+
allowed
W+
??0
K0
forbidden
u
d
s
s
nd
The 2 order diagram (“box”) was calculated & was found
to give a rate higher than the experimental measurement!
amplitude  sinqccosqc
GIM proposed that a 4th quark existed and its coupling to the s and d quark was:
s’ = scosq - dsinq
The new quark would produce a second “box” diagram with
amplitude  sinqccosqc
These two diagrams almost cancel each other out.
The amount of cancellation depends on the mass of the new quark
A quark mass of 1.5GeV is necessary to get good agreement with the exp. data.
First “evidence” for Charm quark!
P780.02 Spring 2003 L10
Richard Kass
CKM Matrix
The CKM matrix can be written in many forms:
1) In terms of three angles and phase:
c12c13
 d   
 
i 13
 s   s12c23  c12 s23s13e
 b   s s  c c s ei13
   12 23 12 23 13
s12c13
c12c23  s12 s23s13ei13
 c12 s23  s12c23s13ei13
s13e i13  d 
 
s23c13  s 

c23c13  b 
This matrix is not unique,
many other 3X3 forms in
the literature. This one is
from PDG2000.
The four real parameters are , q12, q23, and q13.
Here s=sin, c=cos, and the numbers refer to the quark generations, e.g. s12=sinq12.
2) In terms of coupling to charge 2/3 quarks (best for illustrating physics!)
d'
V ud V us V ub
d
s' =
V cd V cs V cb
s
b'
V td V ts V tb
b
3) In terms of the sine of the Cabibbo angle (q12).
This representation uses the fact that s12>>s23>>s13.
2

 d    1   / 2
 

1  2 / 2
 s   
 b   A3 (1    i )  A2
  
“Wolfenstein” representaton
A3 (   i )  d 
 
2
A
 s 
 
1
 b 
Here =sinq12, and A, ,  are all real and approximately one.
This representation is very good for relating CP violation to specific decay rates.
P780.02 Spring 2003 L10
CKM Matrix
Richard Kass
The magnitudes of the CKM elements, from experiment are (PDG2000):
Vud

 Vcd
V
 td
Vus Vub   0.9742  0.9757
 
Vcs Vcb   0.219  0.225

Vts Vtb   (0.4  1.4)  10 2
0.219  0.226
0.9734  0.9749
(3.5  4.3)  10 2
(2  5)  103 

2
(3.7  4.3)  10 

0.9990  0.9993 
There are several interesting patterns here:
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
D0 K-+ over D0 -+ (exp. find 3.8% vs 0.15%)
b  c over b  u
B0 D-+ over B0 -+ (exp. find 3x10-3 vs 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.
But as precision of experiments increases, we might see deviations from Unitarity.
P780.02 Spring 2003 L10
Measuring the CKM Matrix
Richard Kass
No one knows how to calculate the values of the CKM matrix.
Experimentally, the cleanest way to measure the CKM elements is by using
interactions or decays involving leptons.
 CKM factors are only present at one vertex in decays with leptons.
Vud: neutron decay:
npev
duev
Vus: kaon decay:
K0 +e-ve
suev
Vbu: B-meson decay:
B- ( or +)e-ve buev
Vbc: B-meson decay:
B- D0e-ve
bcev
Vcs: charm decay:
D0 K-e+ve
csev
Vcd: neutrino interactions: d -c
dc
“Spectator” Model decay of D0 K-e+ve
e,
W
c
D0
Vcs
For massless neutrinos
the lepton “CKM”
matrixis diagonal
e,

s
K-
Amplitude Vcs
Decay rate Vcs2
u
u
Called a “spectator” diagram because only one quark participates
in the decay, the other “stands around and watches”.