Lecture 27: May 1st 2012
Reading 9.5-9.6
Homework
1) 9.5, 9.29
When Comparing to the experimental result, remember that the decay to a muon/electron
is only part of the total decay width of the tau. Is solving the problem using the
assumption muon mass is small supported by the experimental data?
2) What quarks or anti-quarks does n e not interact with via the W boson? Via the Z
boson?
3) 9.16
1) Quark generations
Consider the decay:
The Width should be (adapting pion result): G =
GF2 f K2 me2 2
mK - me2 )
(
3
8pmK
Since f p and f K involve the same type of strong interactions between quarks by isospin
symmetry we don’t expect them to be different.
Using this if we compare the ratio of the widths of
experimental value of 0.126 we find:
G ( K - ® e-n e )
and
to the
3
2
2
fK2 mp ( mK - me ) mK
=
=
¹ 0.126
2
mp
G ( p - ® e-n m ) fp2 m3 ( mp2 - me2 )
K
2
We expected the kaon decay width to be larger!
A solution to this is to have a suppression factor when decaying across generations in the
matrix element. For the standard d to u decay there is a factor of cosqc and for the
suppressed s to u decay there is a factor of sin 2 q c and qc =13.1.
It was also observed that s to d transitions only occurred at small rates. There had to be a
mechanism to suppressed them.
The solution was the introduction of weak quark eigenstates as the states that couple to
the weak force and express the strong eigenstates as superposition’s of the weak
eigenstates and viceversa.
Weak states expressed as super-positions of strong states.
d ' = d cosq c + ssin qc
s' = -d sin q c + scosq c
and strong states expressed as super-positions of weak states.
d = d ' cosq c - s' sin qc
s = d ' sinq c + s' cos qc
or represented as a rotation matrix
æ d ' ö æ cosq c sin qc öæ dö
ç '÷=ç
÷ç ÷
è s ø è-sin q c cosq c øè s ø
æ d ö æcos qc
ç ÷=ç
è s ø è sin q c
-sin q c öæ d ' ö
÷ç ÷
cosqc øè s' ø
The weak force will only relate quarks of the same weak generation
Then for weak quark decay.
For the interaction within the generation d  u:
é
ù
-igw m
g 1- g 5 u(1,d)ú
êëu (3,u)
û
2 2
é
ù
-igw m
g 1- g 5 u(1,d ' )cosqc - u(1,s' )sin qc ú
êëu (3,u)
û
2 2
(
)
(
)(
)
Then only the contribution within the weak generation contributes.
é
ù
-igw cosqc m
g 1- g 5 u(1,d' )ú
êëu (3,u)
û
2 2
(
)
The amplitude will have a factor of cos2 qc
For interactions across the generation s  u:
é
ù
-igw m
g 1- g 5 u(1,s)ú
êëu (3,u)
û
2 2
é
ù
-igw m
g 1- g 5 u(1,d ' )sinqc + u(1,s' )cosqc ú
êëu (3,u)
û
2 2
(
)
(
)(
)
Only allow interactions within the weak generation contributes
é
ù
-igw sin q c m
g 1- g 5 u(1,d' )ú
êëu (3,u)
û
2 2
The amplitude will have a factor of sin2 q c
(
)
2) CKM matrix
In fact we know that there are 6 quarks. One can write down a general rotation matrix as.
Were I’ve chosen the Wolfenstein-Parameterization which expands the sin and cos terms
of the first generation as a power series.
The portion of the values of the CKM matrix:
Transitions from the 3rd to the 1st generation, b  u transitions, are possible but small.
They also involve a complex phase! The complex phase allows for CP violation
Example: The CKM matrix elements can be factored out to the front.
Determination of the CKM elements:
Vud nuclear beta (
Vus
(
) or pion decay (
)
)or semileptonic kaon decay
(
)
Vub semiletpnic B meson decay with no kaons(strange quarks in the final state)
(
). Note these are very rare decays.
Vcd semilepton D meson decay to pions
charmed particle production by neutrinos
Vcs semileptonic D meson decay to kaons
(
,
), or more purely
(
)
Vcb From the lifetime of the B hadrons since these decays dominate decays dominate the
decay of the B.
(
)
Vtd and Vts. The top quark is observed to decay to b quarks with a 100% branching ratio
to experimental errors. Vtd and Vts are determined from Bd and Bs meson oscillation.
Vtb direct limit on how close to one determined from top decay and a more precise value
determined from unitarity of eigenstate mixing probabilities requires that
|Vtd|2+|Vts|2+|Vtb|2=1
There are other unitarity requirements of the form
åV V
ik
k
as a triangle in the complex plane.
*
jk
= 0, which can be represented
3) Neutral weak interactions: Z boson
In general we will have two types diagrams. Annihilations of fermion anti-fermion pairs
to a Z or scattering of fermion by a Z. Scattering interactions involving identical pairs of
particles or annihilations of fermion anti-fermion pairs to the same type of pair and will
involve multiple diagrams and similar complexities to the similar interactions for the
electromagnetic force. Further complicating things there are interactions that can take
place via a W or a Z or a Z and a photon.
-igZ m
g ( cV - c A g 5 )
2
-i( gmn - qm qn / M Z2 )
Weak Z vertex factor:
Weak Z propagator:
q 2 - M Z2
For small q this propagator reduces to:
igmn
2gZ2
gz2
and
the
constants
or
,
G
=
N
M Z2
8M Z2
4M Z2
Neutral current interactions will not change the flavor of quarks. The CKM matrix
Enforces this.
s  d: Naively this would happen at tree level via the Z and is known as a flavor
changing neutral current. FCNC.
Lets abbreviate with “Z” for the vertex factor.
[u(3, d)Zu(1, s)]
é(u(1, d ' )cosq - u(1, s' )sinq ) Z (u(1, d ' )sinq + u(1, s' )cosq )ù
c
c
c
c û
ë
Only allow interactions within the generation
é cosq sinq u(1, d ' )Zu(1, d ' ) - cosq sinq u(1, s' )Zu(1, s' ) ù
c
c
c
c
ë
û
(
)
However, the in generation interaction should have the same strength for both types of
weak quark eigenstates so this term is 0.
You can still have the s  d process
at higher order.
The kaon interaction would involve a d  u  s set of transitions in a box diagram with
an internal fermion propagator for the u line
m
é
ùæ i(g qm + m) öé
ù
-igw m
-ig
5
÷êu (2,s) w g m (1- g 5 ) u(u)ú
g (1- g ) u(1,d)úçç 2
êëu (u)
2
ûè q - m ÷øë
û
2 2
2 2
æ -i(g m q + m) ö
gW2
m
m
5
'
÷ u (2,d' )g m (1- g 5 ) u(u)
cosq c sin qc
u (u)g (1- g ) u(1,d ) çç
2
2
÷
8
è q -m
ø
[
]
[
]
However, despite the fact that there are four weak vertices involved the calculation gives
too large a rate. However, if there were another type of quark possible in the loop
besides the u quark – the c quark, a second generation version of the u, then the c quark
would couple as:
æ -i(g m q + m) ö
gW2
m
m
5
'
÷ u (2,s' )g m (1- g 5 ) u(c)
-cosq c sin qc
u (c)g (1- g ) u(1,s ) çç
2
2
÷
8
è q -m
ø
[
]
[
]
and cancel out the other terms, except in that the integrals over the internal momentum of
the u and c quarks are slightly different allowing for a small rate. In fact we see that 10
in 1 billion neutral kaon decays go to two muons. The combination of no tree level Z
contribution and the loop level contribution being suppressed in known as the GIM
mechanism.
In addition if you consider the interaction
, if you rotate this into the flavor
eigenstates you get
. The CKM matrix elements don’t
matter for the neutral current Z boson interaction.
For the charged weak W boson interactions we had experimental evidence that the
current was a pure V-A, cV = -cA interaction from maximal parity violation in nuclear
decays and the fact that the neutrino was purely right handed. This was not a given fact
until we performed the measurements. For the neutral weak Z boson interactions the
measurements indicate that not only do these interactions not have cV = -cA , but they are
different for each type of quark and lepton, u, d, e-, .
u: cV =
1 4 2
1
- sin qW ,c A =
2 3
2
1 2
1
d: cV = - + sin2 qW ,c A = 2 3
2
1
1
e-: cV = - + 2sin2 qW ,c A = 2
2
1
1
: cV = ,c A =
2
2
The GSW model of electroweak unification makes predictions for all these parameters.
For instance for the primary coupling constants and masses.
gw =
ge
ge
,gz =
, MW = M z cosq w,
sinq w
sin q w cosq w
and therefore GN =
q w =28.7
2gz2
2 cos2 qW gz2
2 cos2 qW gW2
2gW2
=
=
=
= GF
8M Z2
8M W2
8M W2 cos2 qW
8M W2
The relationship between the constants via the Weinberg mixing angle indicates some of
the physics. More on this later.