Transcript document

Fromm Institute for Lifelong Learning, University of San Francisco
Modern Physics for Frommies III
A Universe of Leptons, Quarks and
Bosons; the Standard Model of
Elementary Particles
Lecture 6
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Agenda
• Administrative Matters
• Patterns and Symmetries in Nature (con.)
• Isospin and the Weak Interaction
• Color and the Strong Interaction
• Broken Symmetry
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Administrative Matters
•Full schedule of colloquia is posted on the Wiki and
should be posted in Fromm Hall. Next colloquium is in
March
•A list of popular books pertaining to Elementary Particle
Physics is posted on the Wiki. It has been updated this
week.
•There are corrections to the slides for last week’s lecture
which are posted.
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Rx for invoking a symmetry group at a vertex:
1) Identify or postulate a set of N fermions that are observed, or
expected, to act as a ‘fundamental’ multiplet. N is the dimension
of a symmetry group G, and the set of fermions is an N-plet
under G.
2) An operation from G when applied to a member of the N-plet
transmutes it into another member.
3) Every transmutation is interpreted as being due to the emission
or absorption of a field boson (aka a gauge boson).
e.g. e is a fermion that cannot be changed into something else
electromagnetically  it is a singlet under some 1-D group, call it U(1)
There is only one boson associated with the group, g.
Note that g is itself uncharged.
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How can a symmetry produce a force field?
Take a table cloth or a square piece of Al foil. Pretend that the material
extends to infinity.
Now rotate the cloth
thru some arbitrary
angle
Any piece of the cloth
appears the same as before.
The cloth is invariant under
global rotations
Special relativity does not allow the simultaneous rotation of the
entire universe.
Only local symmetry rotations are allowed, that is a symmetry
where the amount of rotation varies from event to event in space
time
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Secure the edges of the cloth. Place a finger near the center and give
an arbitrary twist
A spray of wrinkles radiates
The cloth that was
outward from the twisted area.
under the finger still
The local twist cannot be
appears the same. Local
connected smoothly with the
invariance.
undisturbed cloth at large
distances.
This, and any other, local symmetry creates a field. The wrinkles are
analogous to the field lines or lines of force as they were known in
pre-quantum days. In a Feynman diagram, the wrinkles appear as
gauge bosons.
Back to the EM field:
Gauge groups act like rotations. Simplest case is rotation
in a plane.
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What quantity (other than position looks like a rotation angle?
How about the phase ()?
Suppose that phase symmetry is a local symmetry is a general property
of Nature, which if applied locally, produces field wrinkles.
Reinterpret Feynman diagram for e- moving through space.
Direction and l → momentum vector, f → energy
Absolute  cannot be measured, only D s are observable.
Probably  universe is symmetric
under global phase changes but
special relativity doecn’t allow
application of global symmetries.
Our e- should be invariant under local
D s which produce wrinkles in
space-time. Photons carry the
wrinkles
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Gauge
twist
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Gauge
untwist
Gauge
wrinkle
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In QED
D 
1
 electron charge ) photon field ) space-time step )
Can we try to apply this to other forces?
This D is a simple number as it looks like rotation in a plane, group U(1)
and the photon is a simple beast.
The gauge symmetry of the photon is exact.
D is proportional to the field strength, Eg
As Eg → 0, D → 0
Eg2  p 2c 2  m 2c 4
For a massive boson, Eg → mc2. A massive particle cannot be
turned off “gracefully”.
Dg is a single number, it is Abelian. It carries no memory of
the way the g was created (no information about vertex
coupling).  g does not carry a charge.
Other forces might be harder and more complicated
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Recall
Rx for invoking a symmetry group at a vertex:
1) Identify or postulate a set of N fermions that are observed, or
expected, to act as a ‘fundamental’ multiplet. N is the dimension
of a symmetry group G, and the set of fermions is an N-plet
under G.
2) An operation from G when applied to a member of the N-plet
transmutes it into another member.
3) Every transmutation is interpreted as being due to the emission
or absorption of a field boson (aka a gauge boson).
Can we find sets of fermions that resemble each other very closely
in some respects? How about the proton and the neutron?
mp  mn (0.14%) and the strong forces between pp, np and nn are
practically identical
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We’ve talked about this before as strong and weak isospin
Treat n and p as different states of the nucleon.
 p I3   1 2 
 p


  or  n   
n 
 I 3    1 2 
In “quantum mechanics language” we see the nucleon amplitude, N,
as a superposition of p and n amplitudes
N  p sin

2
 n cos
Pure p

/ 2
2
Pure n
 is called the mixing angle
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 p  1 2 
Mixing angle: Consider the p, n doublet,    

n

1
2
  

Superposition allows us to write any quantum
state as a linear combination of other states
0°
  a1 1  a2 2
   1 cos  2 sin 
Thus the symmetry behaves like a rotation with a
mixing angle .
Isospin mixing angle
To maintain normalzation and to ensure that
superposition of the combination state with itself
yields the combination itself, we choose to write
the combination as
60°
120°
180°
A symmetry acting on an N-plet behaves like a
rotation in some abstract N-space
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We don’t observe any other particles resembling p and n as closely
as they resemble each other.  N has precisely 2 states: it is a
doublet under the isospin group.
 I-spin group is 2-D  the group is non- Abelian. Recall rotations
of dice and Rubik’s cube.
(?) internal degree of
Abelian: A  B  B  A
freedom of gauge boson.
Non-Abelian: A  B  C   ? )
A, B, C symmetry rotations
(?) is associated with the charge of the gauge boson.
Yang-Mills Fields:
Ignore Dq and Dm so isospin symmetry is exact.
Again relativity allows only local symmetry.
Global would be unobservable anyway
For EM: e →(twist) → e + g
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For the nulcleon: 4 possible transformations
(1) p → (twist) → p + b1 Could be caused by
same boson
(2)
n
→
(twist)
→
n
+
b
2
b for boson
(3) p → (twist) → n + b3
(4) n → (twist) → p + b4
3 different field quanta: general property for N > 1
N-plet with local symmetry  (N2 – 1) gauge bosons
Yang-Mills field or special unitary group of 2 dimensions, SU(2)
Non-Abelian means order of twists counts  memory of vertex
interaction  Y-M boson can carry isospin charge.
n, p have Dq  Some Y-M bosons carry electric charge as well
Y-M bosons can interact amongst themselves
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A RED HERRING:
Set of 3 candidates: the pions, an I-spin triplet
p
0
n
p
n
0
  
 0
 
  


n

p
p
Or by inverting the
time order of the 

n
p

n
Nucleon state changes
 has right mass for Compton wave length to match range of
nuclear forces
lC 
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2mc
 10 15 m
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Problems: Recall, exact local U(1)  massless, spin 1 (vector) boson
[the table cloth is actually an analogy to 4-D space time so the simplest
possible field quantum is spin 1]
We assumed exact local SU(2) symmetry   should be a massless
vector boson.
Instead, m   140 MeV and m 0  135 MeV
Unlike g,  is not stable,
   1016 sec and    108 sec
0

Long enough for nuclear force by factor of 107 but why the 0, 
difference
Answer:  is not truly “elementary”, it is composed of a qq pair
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The Color Field:
If the  is composite of 2 components, one spin up and one spin
down
s  s1  s2  0
1
2
Composite => decay no longer unexpected
Still have   vs. 0 difference to explain
Suppose  composite => baryons are also composite
What is the fermion multiplet for baryons? Why can’t we see
the constituents?
Turns out N = 3 is simplest possible case and luckily it appears
to work.
Basic triplet of fermions are of course the quarks and the
symmetry group is SU(3)
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EM, U(1): 1 charge (-) [thanks Ben] 1 anticharge (+)
Su(3):
3 charges (R, B, G) 3 charges  R, B, G )
(N2 -1) = 8 massless vector bosons (gluons)
Non-Abelian => gluons interact amongst themselves => gluon
emission can change quark color just like SU(2) bosons shange
isospin charge.
We have never seen color directly =>
Gluons are exchanged in such a way as to paint the world white
Simplest ways to make white (color singlets):
qcolor qcolor
qR qB qG
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mesons
baryons
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Now the  exchange nuclear force diagram looks like
time
If we stop here we can only make one baryon, qR + qB + qG
Even at low energies we have 2 (p,n). How do we make more baryons?
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We need to add an additional degree of freedom or quantum number,
call it flavor.
p charged
 u and d have different electrical charges
n neutral
Baryons being a white combination of 3 quarks  quark
electrical charges are integral multiples of 1/3.
Suspicion: Does ‘whiteness’ and integral electric sharge
being the same  a deeper unity between color and electric
charge?
What hadrons can we expect to make? Depends on the number
of quark flavors. We know of 6: u, d, s, c, b and t.
There are arguments that these 6 are all the quarks there are.
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The pion revisited:
0
uu or dd
uu  gg Need 2 g s to conserve momentum
dd  gg
  10-16 sec EM time scale.
Nuclear time scale is  10-23 sec .
What about ?


u
d
u
d
ud  gg and ud  gg
We need a small amplitude for
processes like
u  d and d  u
(small so that   10-8 sec)
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Weak Decay:
  need small amplitudes u  d and d  u , then uu or dd  2g
 Small mixing angle, i.e, u acts like d small part of the time   long
What symmetry?
SU(2) if no mixing at all, like n, p doublet.
W  
 0
Apply as local gauge symmetry → triplet of gauge bosons  W 
W  

Then we could have a vertex like u  d  W and the pion  
can decay as    W   g
Observed:       99.99%  1
   e e
1.2 104
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(helicity suppressed)
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g
l
d
l+
   e eg
Again branching ratio  1.2 x 10-4
W+
d
u
____________________________________________________
  
 e 
W   e  e
New SU(2) doublets   and    verticies like 


W




 e
 
  
Actually there is a 3rd    verticies like W     
  
Quark-Lepton Families:
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Broken and Hidden Symmetries:
Claim: Weak interactions (WI) are a force due to local SU(2) symmetry
 e       u 
Pro: Suitable fermion multiplets,   ,   ,   , etc.
 
 e       d 
Suitable gauge bosons
Circumstantial evidence – other 3 known forces arise
from local symmetries.
EM
U(1)
Color SU(3)
Gravity Lorentz group (not a quantum field theory)
Con: We were deceived before when trying to apply isospin to SI
--------Weak bosons are not massless.
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We can make a massless particle disappear gracefully as follows
E 2  p 2c 2  m2c 4 consequence of Lorentz symmetry
m=0
As p  0, E  0 and p 
h
l
 0 as l  
For a massive particle, we have a minimum energy mc2. The
particle does nor disappear gracefully.
A massive particle always has energy, even when motionless
Lorentz symmetry and exact local gauge symmetry can only
apply simultaneously in the case where the gauge bosons are
massless.
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Exact Lorentz invariance => no preferred direction in space-time.
But, don’t we provide a preferred direction when we “fix the gauge”?
A choice must be made or we can’t calculate anything.,
e.g. We must specify North on a blueprint or the builder
can’t build the house where we want it
Once we fix the gauge , we have specified a preferred vector
and violated Lorentz symmetry ‘though the back door”.
This is an apparent Lorentz violation, the exact symmetry still
exists, hidden by gauge fixing.
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Another approach: Our old friend the photon
m = 0 => EM range is infinite
OK, we can see light that has travelled billions of light years.
This, of course is in a vacuum. Suppose we make the light travel
through a material, in particular a conductor. The light is quickly
attenuated and becomes indetectable. Its range is very short.
I  I 0e  r r0
Reminiscent of the Yukawa potential
The g acts as if it has acquired a mass
The cumulative screening of the conduction electros acts to make
the g appear massive. An “effective mass”
This effective mass depends on the photon’s environment.
Now suppose the whole universe is a conductor. Then, the photon
would always appear to behave as though it had a mass. We would
never know that the photon is, in truth, massless.
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Can we introduce something into the vacuum to explain the observed
masses of the W and Z bosons?
A field which interacts with the W and Z but not with g or the
gluons, i.e. this field must interact via the week isospin charge.
Regardless of whether the masses are real or effective
(generated by screening) we still have troubles with our gauge
theories.
Hidden Symmetry:
Suppose several hundred cold, naked, teen aged boys are presented
with a choice of four different styles of free clothing.
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The situation is symmetric with respect to the choice of styles and
none of the boys, fearing uncoolness, dares to pick a style, They
just stand around and freeze.
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Finally, one brave soul, on the edge of hypothermia, grabs a particular
style and puts it on. His buddys quickly follow suit.
The symmetry has been broken
Repeat the exercise with a different group of boys. A different
brave soul might very well grab a different style triggering a
different final result..
The overall symmetry of the choice of clothing styles was not
broken by the capricious selection of one style but was hidden by
the choice.
Is there a way to introduce the masses of the W and Z bosons in
such a way that the mass terms reflect a hidden rather than
broken symmetry
There are additional compensating terms that allow this to be done
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Recall that we want to patch up the wrinkling caused by our local
twist at large distances. We must use all degrees of freedom to do
this.
Massless particle: Moves at c, transverse polarization is forbidden,
helicity = + 1
Massive particle: Velocity < c, transverse polarization allowed, has 2
more degrees of freedom. Transverse polarization ( helicity = 0) has 2
possible spin orientations at right angles.
We can make a massive gauge boson from a massless one if we
can borrow 2 degrees of freedom from somewhere
We want to arrange things so that with every gauge boson we
produce 2 scalar (spin = 0) particles.
(massless spin-1) + 2x(spin-0) = (massive spin-1)
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Scalar Higgs doublet
 upper  Interact with W and Z and each other



 lower 
Pick one of the 2, say lower, and let it pervade the cosmos as a
sea of virtual fluctuations. We can’t see this uniform presense
just as we can’t see EM vacuum fluctuations
Hypothesize that this pervasive background of lower exerts a
“drag” force on anything it interacts with, giving mass to the W
and Z
We have hidden the symmetry of the doublet by picking lower
over upper. The symmetry appears to be broken.
Actually, the symmetry is not broken, it’s just that upper
Isn/t present everywhere the way lower is.
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