1. Lecture 12: Electroweak Kaon Regeneration & Oscillation
The Mass of the W
The Massless Photon & Broken Symmetry
The Higgs
Mixing and the Weinberg Angle
The Mass of the Z
Z Decay
Useful Sections in Martin & Shaw:
Chapter 9, Chapter 10

2. So what are kaons???  that depends... who wants to know?!
Regeneration
So what are kaons???
 that depends... who wants to know?!
Ko, Ko  states of definite strangeness
K1o, K2o  states of definite CP
KSo, KLo  states of definite lifetime KL
KL + KS
strong interaction with
matter picks out Ko & Ko
which then re-mix

3. So what are kaons??? ''Regeneration"
 that depends... who wants to know?!
Ko, Ko  states of definite strangeness
K1o, K2o  states of definite CP
KSo, KLo  states of definite lifetime KL
''Regeneration"
KL + KS
strong interaction with
matter picks out Ko & Ko
which then re-mix

4. Strangeness Oscillation
Amplitudes for decaying states KSo and KLo as a function of time are
AS(t) = AS(0) exp(imSt) exp(St/2) S  ℏ/S
AL(t) = AL(0) exp(imLt) exp(Lt/2) L  ℏ/L
K1o = 1/2 ( Ko + Ko )
K2o = 1/2 ( Ko  Ko ) or
Ko = 1/2 ( K1o + K2o )
Ko = 1/2 ( K1o  K2o )
≃ 1/2 ( KSo + KLo )
≃ 1/2 ( KSo  KLo )
AK(t) = 1/2 ( AS(t) + AL(t) )
AK(t) = 1/2 ( AS(t)  AL(t) )

5. Strangeness Oscillation Intensities
Thus, if we start with a pure Ko beam at t=0, the intensity at time t will be
I(Ko) = 1/2 [AS(t) + AL(t)][AS*(t) + AL*(t)]
(setting AS(0) = AL(0) = 1)
= 1/4 {exp(St) + exp(Lt) + 2 exp[(S+L)t/2] cosmt }
and similarly,
= 1/4 {exp(St) + exp(Lt)  2 exp[(S+L)t/2] cosmt }
I(Ko) = 1/2 [AS(t) AL(t)][AS*(t) AL*(t)] Ko
where m  mLmS
= 3.49x1012 MeV
(m/m ≃ 7x1015)

6. K, W and Z
A B C D E
F G H I J
L M N O
P Q R S T
U V X Y

7. Weak Coupling & the W Mass
Recall that the ''matrix element" for scattering from a Yukawa potential is
f V o = g2/(q2+M2)
In the Fermi theory of decay, this is what essentially becomes GF
or, more precisely,
GF/2 = g2/(q2+M2) = 4W/(q2+M2)
  GF2 and the relatively small value of GF characterizes
the fact that the weak interaction is so weak
We can get this small value either by making W small or by making M large
So what if we construct things so W =  ???
 UNIFICATION !!
Assuming M ≫ q2 ,
M =  4 2  / GF
 M ~ 100 GeV
CERN, 1983
MW = 80 GeV !!
 = 1/137
GF = 105 GeV2

8. p W- u hadrons d e, ,  u e-, -, - d hadrons
Stochastic Cooling
Electron Cooling

10. Electroweak Interlude
A Brief
Theoretical Interlude
(electroweak theory... at pace!!)

11. ( )L ( )L Wo  1/2 ( )
Weak Isospin
But how can this be the ''same" force when the
W’s are charged and the photon certainly isn’t !?
Is there a way we can ''bind up" the W’s along with a neutral
exchange particle to form a ''triplet" state (i.e. like the pions) ??
Well, like with the pions, we seem to have a sort of ''Weak" Isospin
since the weak force appears to see the following left-handed doublets e e
( )L     u d
( )L c s t b
as essentially two different spin states: IW(3) =  1/2 (like p-n symmetry)
e e W+
Thus, in
the process
The W+ must carry
away +1 units of IW(3)
so let’s symbolically denote W+  
and, similarly, W  
If IW = 1 for the W’s then, similar to the o, there is also a neutral state:
Wo  1/2 ( )
(which completes the triplet)

12.  Higgs Mechanism 1/2 ( )
The Higgs
There is, however, another orthogonal state:
1/2 ( )
If we ascribe this to the photon, then perhaps we might expect
to see weak ''neutral currents" associated with the exchange of
a Wo with a similar mass to the W
so we’d have a nice
''single package" which describes EM and weak forces!
Hold on... any simple symmetry is obviously very badly broken
 the photon is massless and the W’s are certainly not!
The photon is also blind to weak isospin and also couples to
right-handed leptons & quarks as well
Assume the symmetry was initially perfect and all states were massless
Then postulate that there exists some overall (non-zero) ''field" which
couples to particles and gives them additional virtual loop diagrams :
(kind of like an ''aether" which produces a sort of ''drag")
but in the limit of zero momentum
transfer (rest mass), so represent as
 Higgs Mechanism

13. Further suppose that this field is blind to weak isospin
Mixing: the Photon & Z
Further suppose that this field is blind to weak isospin
and, thus, allows for it’s violation.
This would allow the neutral weak isospin states to mix
 like with the mesons (the W are charged and cannot mix)
We will call the ''pure," unmixed states Wo and 
And we will call the physical, mixed states Zo and 

14. M2 W = W + W M2 W = W + W +  M2  =  +   + Wo
Masses and Couplings
Think about mathematically introducing this Higgs coupling
by applying some ''mass-squared" operator to the initial states
(since mass always enters as the square in the propagator)
M2 W = W W
GW GW
M2 W = W W 
GW GW
GW GG
M2  =    Wo
GG GG
GG GW
where the right-most terms represent the weak isospin - violating terms
Assume couplings to W’s are all the same (GW) but coupling to  may be different (GG )
For the W the mass would then simply be given by
MW2 = GW2
(where G2 contains the coupling plus a few other factors)
For the latter 2 equations, we can think of M2 as an operator
which yields the mass-squared, M2 , for the coupled state:
M2 Wo = GW2 Wo + GW GG 
M2  = GG2  + GW GG W

15. Massless Photon / Massive Z
 = Wo
GW GG
(M2-GG2)
From the second of these:
M2 Wo = GW2 Wo Wo
GW2 GG2
(M2-GG2)
Substituting into the first:
M4  M2 GG2 = M2 GW2  GW2 GG2 + GW2 GG2
M2 ( M2  GG2  GW2) = 0
 M2 = or M2 = GW2 + GG2
Thus, associate M2 = 0 and MZ2 = GW2 + GG2
Note also that MZW

16. Ql + 3Qq = 0  MZ = MW/cosW
Weinberg Angle & Z Mass
We can parameterize the  as a mixture of Wo and  as follows:
W  ''Weinberg Angle"
   sinW  Wo cosW
Thus, applying M2 :
M2  = M2 ( sinW  Wo cosW) = 0
0 = ( GG2 + GW GGW ) sinW  (GW2 Wo  GW GG  cosW
Coefficient of Wo 
GW GGsinW  GW2 cosW = 0
Coefficient of  
GG2sinW  GW GG cosW = 0
tan W = GG / GW
''unification condition"
Ql + 3Qq = 0
''anomaly condition"
(leptons) (quarks)
MZ2/MW2 = (GW2 + GG2)/GW2 = 1/cos2W
 MZ = MW/cosW
is satisfied separately
for each generation

17. Neutral Current Event   p (Gargamelle Bubble Chamber, CERN, 1973)

18.  sin2W = 0.226 MW = 80 GeV  MZ = 91 GeV (predicted)
Z Discovery
Z  e+ e
From comparing neutral
and charged current rates
 sin2W = 0.226
MW = 80 GeV
 MZ = 91 GeV
(predicted)
MZ = 91 GeV (observed!!)

19. Flavour-changing neutral currents
While we’re here...
pre-ABBA weak doublet = u
d cosqC + s sinqC
( ) d´ =
( )
So, consider the coupling to the Z0 : Z0
(d cosqC + s sinqC) Z0 u +
Probability ∝ product of wave functions:
uu + (dd cos2qC + ss sin2qC) (sd + ds ) sinqC cosqC
DS = 0
DS = 1
“Flavour-Changing Neutral Currents”  never seen!

20. ( ) ( ) ( ) ( ) & + + Postulate 2 doublets:
GIM mechanism
Postulate 2 doublets:
(Glashow, Iliopolis & Maiaini: “GIM” mechanism) u
d cosqC + s sinqC
( ) d´ =
( ) c
s cosqC - d sinqC
( ) s´ =
( )
& Z0 u
(d cosqC + s sinqC) + Z0 c
(s cosqC - d sinqC) +
uu + cc + (dd+ss)cos2qC + (ss+dd) sin2qC) + (sd + ds - sd - sd) sinqC cosqC
DS = 0
DS = 1

21. Resonant Cross Section
Blam !
formation ''rate"
of initial state
(recall  = ℏ/)
( )
W =
dP
dN 0
prob for decay to
particular final state
given the total number
of available states
Transition
Rate =  0
( )
dP
dE
dE
dN
dP f
dE  (E-E0)2 + 2/4 = dN dE
( ) 1
V q2 dq d
(2) dE
( ) = 1
W = vB  / V
But recall that
V q2
2 v
( ) 1
 =
 0 f
q2 (E-E0)2 + 2/4

22. Relativistic Treatment
But this is non-relativistic!
From considering scattering from a Yukawa potential
(which followed from the relativistic Klein-Gordon equation)
we found the ''propagator"
1/(q2 + M2)
So consider the diagram:
Under a fully relativistic treatment, q is the 4-momentum transfer
and, if we sit in the rest frame of the intermediate state,
q2 = p2  E2 = E2
Also note that, for a decaying state, the intermediate mass takes
on an imaginary component M  M  i / since
 ~ exp(iE0t) = exp(iMt)
 exp{i(Mi/2)}t = exp(iMt) exp(t/2)

23. Relativistic Breit-Wigner
Thus, the propagator goes like 1
(Mi/2)2  E2 1
M2 /4  iM  E2 = 1
M2  iM  E2
(in the limit ≪ M) ≃
And the cross section will be proportional to the square of the propagator :
( )( ) 1
M2  iM  E2
M2  iM  E2
 ~ 1
(E2  M2)2 + M2 =
 =
 0 f
q2 (E-E0)2 + 2/4
compare
so, roughly, /2  M
and we’d expect
something like
 ~
 M2 0 f
E2 (E2-M2)2 + M22
 =
 M2 0 f
E2 (E2  M2)2 + M22 CM
In fact, a full relativistic
treatment yields

24. [ ]  (e+e  X) = MZ2 ee X
The Z Resonance
Thus, for the production of Z0 near resonance
and the subsequent decay to some final state ''X" :
 (e+e  X) =
MZ ee X
E (E2  MZ2)2 + MZ22 CM
[ ]
since ee can be related by time-reversal to ee
Peak of resonance  MZ
Height of resonance  product of branching ratios
ee X

BreeBrX=

25. Z Decay: Generation Limit
Results:
MZ =  GeV
Z =  GeV
hadrons=  GeV
l l =  GeV
(3 x ) =
≠ !!
(limit for light, ''active" neutrinos)
So what’s left ???
''Invisible modes"
Neutrinos !!

26. An End To The Generation Game ???
End To Generation Game
An End To The Generation Game ???
(not necessarily a bad thing!)