Lecture 12: Electroweak

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
Regeneration
So what are kaons???
? that depends... who wants to know?!
K1o, K2o ? states of definite CP
KSo, KLo ? states of definite lifetime
3
Regeneration
So what are kaons???
? that depends... who wants to know?!
K1o, K2o ? states of definite CP
KSo, KLo ? states of definite lifetime
4
Strangeness Oscillation
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 ?
?/?S
AL(t) AL(0) exp(?imLt) exp(??Lt/2) ?L ?
?/?L
or
?Ko? 1/?2 ( ?K1o? ?K2o? )
? 1/?2 ( ?KSo? ?KLo? )
? 1/?2 ( ?KSo? ? ?KLo? )
AK?(t) 1/?2 ( AS(t) AL(t) )
5
Strangeness Oscillation Intensities
Thus, if we start with a pure Ko beam at t0, the
intensity at time t will be
(setting AS(0) AL(0) 1)
I(Ko) 1/2 AS(t) AL(t)AS(t) AL(t)
1/4 exp(??St) exp(??Lt) 2
exp?(?S?L)t/2 cos?mt
and similarly,
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/(q2M2)
In the Fermi theory of ??decay, this is what
essentially becomes GF or, more precisely,
GF/?2 g2/(q2M2) 4??W/(q2M2)
? ? 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
? UNIFICATION !!
So what if we construct things so ?W ? ???
Assuming M q2 ,
? 1/137 GF 10?5 GeV?2
8
p p
9
(No Transcript)
10
Electroweak Interlude
A Brief Theoretical Interlude
(electroweak theory... at pace!!)
11
Weak Isospin
But how can this be the ''same" force when the
Ws are charged and the photon certainly isnt
!?
Is there a way we can ''bind up" the Ws 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
as essentially two different spin states IW(3)
? 1/2 (like p-n symmetry)
Thus, in the process
The W must carry away 1 units of IW(3)
so lets symbolically denote W ? ???
and, similarly, W? ? ???
If IW 1 for the Ws then, similar to the ?o,
there is also a neutral state
Wo ? 1/?2 ( ??????????)
(which completes the triplet)
12
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 wed 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
Ws 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")
13
Mixing the Photon Z
Further suppose that this field is blind to weak
isospin and, thus, allows for its 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
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)
where the right-most terms represent the weak
isospin - violating terms
Assume couplings to Ws are all the same (GW) but
coupling to ? may be different (GG )
MW2 GW2
For the W? the mass would then simply be given by
(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
15
Massless Photon / Massive Z
From the second of these
Substituting into the first
M4 ? M2 GG2 M2 GW2 ? GW2 GG2 GW2 GG2
M2 ( M2 ? GG2 ? GW2) 0
? M2 0 or M2 GW2 GG2
Note also that MZ?????W
16
Weinberg Angle Z Mass
We can parameterize the ? as a mixture of Wo and
? as follows
? ? ? sin?W ? Wo cos?W
M2 ? M2 (? sin?W ? Wo cos?W) 0
Thus, applying M2
0 ( GG2?? GW GG?W? ) sin?W ?? (GW2 Wo ? GW
GG ??? cos?W
GW GG?sin?W ?? GW2 cos?W 0
Coefficient of Wo ?
Coefficient of ? ?
GG2?sin?W ? GW GG cos?W 0
MZ?2/MW2 (GW2 GG2)/GW2 1/cos2?W
is satisfied separately for each generation
17
Neutral Current Event
Neutral Current Event (Gargamelle
Bubble Chamber, CERN, 1973)
??
??
p
18
Z Discovery
From comparing neutral and charged current rates
? sin2?W 0.226
MW 80 GeV
19
Flavour-changing neutral currents
While were here...
So, consider the coupling to the Z0

Probability ? product of wave functions
Flavour-Changing Neutral Currents ? never seen!
20
GIM mechanism
Postulate 2 doublets
(Glashow, Iliopolis Maiaini GIM mechanism)

21
Resonant Cross Section
(recall ? ?/?)
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 ?/2 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
And the cross section will be proportional to the
square of the propagator
so, roughly, ?/2 ? M?
In fact, a full relativistic treatment yields
24
The Z Resonance
Thus, for the production of Z0 near resonance
and the subsequent decay to some final state
''X"
since ???????e?e?? can be related by
time-reversal to ??e?e??????
Peak of resonance ? MZ
Height of resonance ? product of branching ratios
25
Z Decay Generation Limit
Results
???????hadrons??? 1.741 ? 0.006 GeV
MZ 91.188 ? 0.002 GeV ?Z 2.495 ? 0.003 GeV
???????l ??l ????????? 0.0838 ? 0.0003 GeV
? 2.495 !!
1.741 (3 x 0.0838) 1.9924
So whats left ???
26
End To Generation Game
An End To The Generation Game ???
(not necessarily a bad thing!)