Transcript Elementary Particle Physics

Lecture 13 – the weak force
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When is it weak and when is it strong ?
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Weak interactions of quarks
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
Cabibbo angle

CKM matrix
Handedness in charged current interactions.
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When is a force weak or strong ?
We use unnecessarily confusing terminology for the forces
(weak,strong..) which arose due to the historical development of
particle physics. If we were to start to study fundamental forces
today we'd probably give them different names.
A basic question....
Q) A process takes place through force A, eg a scattering between
two particles at a fixed centre-of-mass energy. The cross section was found
to be m uch lower than the cross section for scattering via force B between
two particles of the similar masses at the same energy. Can we say that force
A is always weaker than force B ?
A) No. The strength of a force depends on many factors. Sometimes a force
acts in a powerful way, other times its rather puny.
In particular, for very high energy collisions, the weak force isn't weak at all.
This is one the main messages of this lecture!
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Q) What determines the size of the cross section for a process ?
A) Take simple case a particle of mass m scatters off a fixed potential:
2
d
m2
Born approximation:
 2 M  q  (11.01)
d  4
d
We can think of
depending on:
d
(1) Phase space, mass m of particle.
(2) Amplitude: M
2
Pi , mass=m
Pf , mass=m
gX
Yukawa potential with born approximation/single
X,|Q|
particle X exchange (lecture 2):
 g X2
4 X
M Q   2
=
(2.32)
2
2
2
Q  MX Q  MX
2
gX
We can regard the amplitude depending on (i) 4-momentum transfer Q 2
(ii) "charge"/coupling g X ,  X (iii) rest mass of the exchanged particle: M X
Is a force strong or weak ? That depends on the above factors!
Obs ! From lecture 12: the "charge"/coupling varies with Q 2 though for the
electromagnetic and weakreactions we're considering this variation is slight
and we can consider the couplings to be constant. Also, we took a simple
non-relativistic scattering as an example - many processes are more complicated
though our example is a useful generic guide to understanding the strengths of
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forces.
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When the weak force is weak
Compare weak and electromagnetic decays in region: Q 2  M W2
Compare main decays of particles  0 ,  0 . Roughly the same mass difference
between initial and final states
100 GeV
0  p   
0  
Decay rate: 
;  0  1010 s ; weak
;   0  1018 s ; electromagnetic
amplitude  phase space 
2
Assume same phase space for each decay and that each channel is the only
channel.
M em
M weak
2
2
1010

6 1018
108
tiny 4-momentum exchange: Q
0.15 GeV (pion mass)
 decay is weak in relation with respect to other forces.
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When the weak force become stronger
Compare electromagnetic and weak forces for Q 2  M W2 .
Consider the following deep-inelastic scattering processes at HERA collider:
(a) e   p  e   X and (b) e   p   e  X .
e-
e-
e
e-
,Z0
W-
p
p
(a) proceeds via em and weak (b) proceed s via weak only.
Very rough order of magnitude estimate:
d
(a)
2
2
Amplitude
for
(a)
dQ

2
d
(b) Amplitude for (b)
2
dQ
2 2
g
Q2
2
W
g
Q 2  M W2
2
(13.01)
For Q 2  M W2 expect the two processes
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Electromagnetic and weak force strength
As expected, for Q 2  M W2 the cross
sections for electromagnetic and weak
processes are of similar strength.
Data from the HERA electron-proton
collider.
e
Evidence for electroweak unification
(next lecture).
MW2
6400 GeV2
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The weak force in detail
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Charged currents
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Basic processes for quarks and leptons
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Lepton-quark symmetry
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Cabibbo angle
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CKM matrix
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V-A theory
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Basic charged current processes involving leptons
Weak coupling at a vertex: gW
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Quark interactions via the weak force.
Consider, eg, decay p  n  e    e .
Coupling of the W to the u - d quarks= gud
Is gud  gW ?
To answer this, need to consider all possible quark couplings.
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Cabibbo mixing
Charged current weak interactions take place between lepton doublet members:
  
 e 
  ,   (consider first 2 generations for simplicity)
 e 
  
u 
c 
 d '   cos C sin C   d 
uarks doublets are:   ,   where    
   (13.02)
d
'
s
'
s
'

sin

cos

 
 
  
C
C  s 
d '  d cos C  s sin C s '  d sin C  s cos C states seen by the weak force
d , s, c, u are the physical quark states.
Obs! implicit Dirac notation: d ', d , s... are d ' , d , s ...
 the quark states seen by the weak force are mixed states of the physical quarks.
C =Cabibbo angle and must be determined from experiment.
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Lepton quark symmetry
c
s’
c
s
c
d
gcd
s '  d sin C  s cosC
gcs  gW cosC
gcd  gW sin C
Charged current interactions lead to cross generation interactions:
u  d, s ;
c  s, d
Formally: gW  coupling of W to charged lepton+neutrino;
gud  g cs  gW cos C =coupling of u  d , c  s (13.03)
gus  g cd  gW sin C = coupling of u  s, c  d (13.04)
 Same weak interaction for leptons in a doublet as for quarks in a doublet if
mixed states d ' and s ' are considered.
 lepton-quark symmetry.
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Determination of the Cabibbo angle
Compare decay rates: K        and         .
K-
gud
  K       
       


s
u
gus2
 2  tan 2 C
gud
gus
(13.05)
 tan C  0.232  0.002  C  13.1o  0.1o (13.06)
 other decays.
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Cabibbo–Kobayashi–Maskawa Matrix
Consider mixing for three generations: CKM matrix U .
 d '
 d   Vud Vus Vub   d 
 
 
  
s
'

U
s

V
V
V
 
   cd cs cb   s  (13.07)
b' 
 b  V V V   b 
 
   td ts
tb   


g
g  V gW
In general U has complex elements. Magnitude of elements:
0.97419  0.00022
U  0.2256  0.0010
0.00874  0.0003
0.2257  0.0010
0.97334  0.00023
0.047  0.001
0.00359  0.00016
0.0415  0.001
0.999133  0.00004
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
 (13.08)


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More on the CKM Matrix
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The CKM matrix is generalisation of Cabibbo’s mixing for the
light quarks by Kobayashi and Maskawa
When generalising CP violation was introduced/discovered
as a property of weak decays.
Kobayashi and Maskawa won the 2008 Nobel prize for this
work.
Many particle physicists (probably Cabibbo included) were
surprised at Cabibbo’s exclusion.
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Decay of W
Can rotate earlier diagrams.
W can decay into a quark and an
antiquark or lepton and an antilepton.

Typical branching ratio in high energy limit
of fermion momenta >> masses.

2
B W  hadrons  
3
1


B W    
  e,  , 
3

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Spin dependence of charged current interactions
Spin-dependence in charged current interactions is a complex subject but there
is one interesting and straightforward result.
Generally for massive fermions in the ultra-relativistic limit (   c), the
W  couples preferentially to left-handed fermions and right-handed
anti-fermions. Spin-dependence arise from so-called vector-axial
(V-A) interactions.
This is in keeping with the observation that only left-handed neutrinos
and right-handed antineutrinos (highly relativistic, almost masless
particles!) have ever been observed.
Consider decay of charged pion:   
 same handedness/helicity of


 at rest. Pion has spin 0.
and  .

s


s

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V-A interactions
s
m   139 MeV ; m  106 MeV ; me  0.511 MeV
m
m  whereas me  m 
 w/o spin-dependent suppression electronic
s
s
decay would be favoured from kinematic arguments.
Kinetic energy available for

and  
ss
Q value: Q  m   m  m  34 MeV (13.09)
Relativistic electron  suppression of electronic mode.
Observed:
    e    e 
         
 1.230  0.004  104 (13.12)
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Top quark non-discovery
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1984 CERN
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UA1 experiment
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pp (630 GeV cm energy)
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Something they would
rather forget
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Top quark discovery
quark pdf q(x)
Anti-quark pdf q(x)
Tevatron pp collider at
2 TeV centre-of-mass energy.
World's highest energy collider.
Only collider capable of producing top quarks.
Production mechanism via pair production.
Measured by CDF and D0 experiments in 1995.
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Top quark lifetime
Consider qq pair produced in, eg, e  e collisions.
uu , dd , ss , cc , bb quarks form hadrons which can then decay.
Typical hadronisation time ?
At least the time taken for a gluon to move across 1fm ( hadron size)
  had
1015
24


3

10
s (13.13)
8
3 10
The top quark is a very special quark.
mt  174 GeV  t  4 1025 s.
 top quark decays before it has a chance to become bound. The
top is the closest we'll get to a bare quark which can be studied.
(For simplicity, diagram is a hypothetical e e reaction, in reality tops
have been produced in pp collisions - these have the requisite energies.
The lifetime argum ent is still the same however, regardless of how the
top is produced produced).
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Selected top quark decays at the Tevatron
Messy events:
Tops dominantly decay t  b  W
(Vtb 1  Vts  Vtd )
t  t  b W   b W 
Jets arising from b-quarks
 identify B-decays after 10 12 s.
(Silicon detectors - later lecture).
W  can decay to leptons and jets .
Suppress non-top backgrounds
which give similar signals.
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A top quark pair production ”event”
Event observed at the D0 experiment.
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Top quark discovery
top
background
Following background suppression selections.
Clear top signal.
Current mass average mt  174  3.3 GeV
(about as heavy as a gold nucleus).
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Summary
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The weak force was studied

The weak force is of comparable strength with the
electromagnetic force for interactions with |Q|>MW

Charged current interactions take place in quark doublets of
mixed physical states.

Cabibbo theory and the CKM matrix formalise mixing.

At relativistic energies, the W couples preferentially to lefthanded fermions and right-handed antifermions (V-A theory).
Next lecture – neutral currents, electroweak unification
and the Higgs.
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