Download ModPhys III Lecture 5 - University of San Francisco

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 5
9 February 2011
Modern Physics III Lecture 5
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Agenda
• Administrative Matters
• Quantum Field Theories
• Second or Field Quantization
• Electromagnetic Interaction
• Weak Interaction
• Strong Interaction
• Patterns and Symmetries in Nature
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Administrative Matters
•Full schedule of colloquia is posted on the Wiki and
should be posted in Fromm Hall
•A list of popular books pertaining to Elementary Particle
Physics is posted on the Wiki. It will be updated when
appropriate.
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Exchange vector bosons form a T =1 triplet
 W T3 
W 
 1 


 0
 
W0


W
or
T

0


3

 W   
W  
 T3   1 






W0 rather than Z0 for technical
reasons involving electro-weak
unification
W+ (T3 = +1) is emitted in transitions {(T3 = +1⁄2) → (T3 = −1⁄2)}
W− boson (T3 = −1) is emitted in transitions {(T3 = −1⁄2) → (T3 =+ 1⁄2)}.
e.g. b decay
d  uW 
W   e 
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It took a great deal of effort to get a renormalizable electro-weak QFT
Sheldon Glashow
Abdus Salam
1926 - 1996
Steven Weinberg
1979 Nobel Prize
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Strong Interaction:
A little history:
1911 Rutherford discovers the nucleus
1932 Chadwick discovers the neutron
→ Nucleus is protons + neutrons
What holds the nucleus together?
Need a strong but short range force.
1947 Powell et al.
discover pion, p, in
cosmic rays
1935 Yukawa potential
g 2  ar
g 2 a  ar
VSI (r )   e or FSI   2 e
r
r
Could be mediated by a scalar boson
of mass  200 me
P+
p0
P+
Muon, m not it.
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In the 1960s both theory and experiment began
to  substructure to “elementary” particles
Partons: quarks, antiquarks and gluons
Hideki Yukawa
1907 -1981
1949 Nobel Prize
Update Yukawa’s picture
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Simple Quark Model (no color):
Baryons:
u
d
Arrows are
quark spins
u
d
Sq = 1/2
u
d
Proton
Neutron
Mesons:
u
u
d
u
d
u
or dd
p
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
p

Modern Physics III Lecture 5
p
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Color and Color Charge: Color was originally introduced to beat the
Pauli principle.
u


u

Arrows are
quark spins
Sq = 1/2
s3 2
Clearly Pauli blocked
u
u
u
Add a new number, color
u
There are also
anticolors for
the antiquarks
Mrs. Pauli’s favorite son is now happy!
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Theory of strong interactions between quarks is called quantum
chromodynamics (QCD)
Mediated by force carriers called gluons
Bound states: qq color - color  meson
3q 3 different colors  baryon
3q 3 different colors  antibaryon
Note that in all of the above the colors add up to “white”.
Color is not observed in our “usual” particles (p, n, p, K etc.).
Gluons: 8 of them carrying both color and anticolor
BLUE ANTIRED GLUON
rb
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From the 3 color states, one can form 9 bicolor color - anticolor states
RR , RB , RG
BR , BB , BG
GR , GB , GG
Removing the colorless state of the trace
reduces us to 8 combinations which
exchange color between quarks
Confinement: Why don’t we see free quarks?
Gluons themselves carry color charge (unlike the photon which
is electrically neutral). They participate in strong interactions.
These g – g interactions constrain color fields to string like
objects, flux tubes.
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Stretching the tube requires more and more energy
At some distance it is energetically more favorable to pull a qq
pair out of the vacuum than to increase the tubelength.
e p  K   0
(1-3) e  strikes p exciting the
d quark
(4) An ss pair is created
(5) quarks rearrange into color

singlets, K and 
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Color Screening and Running Coupling Constants:
In QED we had some trouble with
loop diagrams.
We fixed this up by invoking the dressing or screening of the bare
electron
Vacuum polarization or charge screening
reduces the observed charge.
As we increase probe energy (probe shorter
distances) the observed charge grows
The EM, aEM, coupling constant increases with energy
What happens in QCD?
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In QCD the virtual gluons emitted by quarks not only create qq
pairs but also more gluons with all the allowed bicolor
signatures
In QCD the virtual quark-antiquark pairstend to screen
the color charge. However, QCD has an additional wrinkle:
its force-carrying particles, the gluons, themselves carry color
charge, and in a different manner. Each gluon carries both a
color charge and an anti-color magnetic moment. The net
effect of polarization of virtual gluons in the vacuum is not to
screen the field, but to augment it and affect its color. This is
sometimes called antiscreening. Getting closer to a quark
diminishes the antiscreening effect of the surrounding virtual
gluons, so the contribution of this effect would be to weaken
the effective charge with decreasing distance.
1 Fermi = 1 fm = 1x10-15 m
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Asymptotic freedom
H. David Politzer
Frank Wilczek
David Gross
Nobel Prize 2004
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Patterns and Symmetries in Nature
Our normal space E3, Physics doesn’t change when moved. It
is invariant under translation and rotation.
Symmetry groups
Group: A set of elements closed under a prescription relating the
elements. e.g. a and b in the group  a  b in the group
 is associative but not necessarily commutative
 A unit element 1 
1  a  a 1  a
Each element has an inverse a-1  aa-1 = 1
Examples: The integers form a group under addition, 1 + 3 =4.
Hours on a clock. A finite number of elements (12).
1 = 12 because h +12 = h
If h = 1, h-1 = 11 because 1 + 11 =12 = 1
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In Physics, groups are associated with manipulations, e.g translation
and rotation
Translation group: Carry from here to there
Rotation group: Rotate so much about a specified axis.
Things that remain the same when manipulated by an element are
said to be symmetrical.
Examples: A sphere under all rotations.
A cube under 90° rotations about 3 specific axes.
A shoe is not symmetric but a pair is.
Approximate symmetries: Still useful. The Earth is not a perfect
sphere but can often be treated as one.
Some groups can be shown to be equivalent, e.g. group of
rotations of 30° is equivalent to the clock number group above.
Representation  specific instance of a group, e.g. the clock
numbers are a representation of the 2-D 30°rotation group
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NOW RESTRICT OURSELVES TO FINITE GROUPS
The elements of a represntation are classified into multiplets
e.g. A sphere is a singlet under rotation
A square in a plane is a quartet
An equilateral triangle in a plane is a triplet
A cube about 3 orthogonal axes is an octet
Rotations in a plane commute or are Abelian.
All 1-D groups are Abelian
Finite 3-D rotations are non-Abelian, see Rubik’s Cube
Invariants: Symmetries leave something unchanged.
e.g. translation maintains the distance between points, r2 = x2 + y2.
 a conserved quantity, p. Rotations preserve distances and angles
 conserved quantity, L
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Symmetries in Relativity:
The frame independence of c  a “super distance” in 4-D spacetime
which is invariant under a corresponding “super rotation”. This is the
4-length or interval s 2  x 2  y 2  z 2
The super rotations are of course the Lorentz trnsforms a.k.a. the
Lorentz group.
Quantizing Spin: Magnetic fields with their “vector product behavior”
arise from the spin of the exchanged g. The occurrence of both
repulsive and attractive forces also requires consideration of the
exchanged spin (the ball throwing analogy breaks doen here).
Recall the wave justification of Bohr’s quantization
rule: y must be a standing wave around the
equator.
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2p
2p r  n n  1, 2, 3,  and  =
mv
mvr  n
Angular momentum
We are dealing here with the rotation of the particle itself so we
have the spin angular momentum
S n
It doesn’t matter that we have used a finite radius for the path of this phase
measurement; This is merely an accounting device for the phase, a pure number
independent of physical size.
Now suppose we measure the phase for multiple loops around the equator
rotation
marker
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Phase Wheel
1 revolution = 2p radians = 360°
Corresponds to 1 complete wavelength
n=1
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Our requirement is just that the path closes after an integral number n
of revolutions of the wheel.
This could easily well happen going around the equator twice
Suppose n is even, say n = 4
Clearly mark 3 coincides with 1 and 2 with 4
n=4
1
3
2
4
Superposition of two indistinguishable
paths each with n/2 wavelengths
n even  n/2 = m is also an integer
We end up with a class of particles with spin
quantized according to
S m
just as before
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n=3
Now let n be an odd integer, say 3
3
The pair wise coincidence is
replaced by an anticoincidence
Each mark on the 2nd loop is
midway between 2marks on
the 1st.
1
2
The 2 loops are not the same. We obtain a class of particles with an
equatorial circumference of n/2 with quantized spin
n
S
2
n odd
This process can be generalized to more looping of the phase
wheel.The above 2 classes of particles are the only 2 that appear and
are the bosons and fermions respectively.
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Boson: Phase wrapping is a simple circle. A boson turned 360° is
indistinguishable from its original
Space is rotationally symmetric so no surprise that 360° turns a
particle into itself
Fermion: Phase wrapping consists of 2 different loops. A fermion
turned 360° does not turn into itself. Instead it requires a turn of 720°.
So, what state do we obtain with a 360° rotation?
360°does not return the same amplitude but we really believe
space to be rotationally symmetric.
Only way out appears to be y → -y for a fermion turned through 360°
Remember, we don’t observe amplitudes (y), only probabilities (P).
Py
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2
 y
2
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What happens if we superpose 2 indistinguishable particles.
Distinguishable  different quantum numbers, momenta or positions
2 particles in a 1-D box. In analogy to going to a higher D box
n1n2 ( x1 , x2 )  y n1 ( x1 )y n2 ( x2 )
Particles are identical so we actually have to write
1
y n1 ( x1 )y n2 ( x2 ) y n2 ( x1 )y n1 ( x2 ) 
 n1n2 ( x1 , x2 ) 
2
    bosons 
 


fermions
  

For fermions:  nn ( x, x)  0
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Truly identical fermions cannot exist.
Forces fermions to aggregate in finite sized lumps when piled together
Prevents fermions from building up large scale coherent fields
when exchanged as virtual particles
Identify fermions with matter
Bosons have no objection to being identical
Bosons aggregate together in the lowest allowed energy state
without building up finite sized lumps.
Responsible for phenomena such as superconductivity and
superfluidty.
Identify bosons with forces
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Fermion
Boson?
Demo with string
Have to change the topology of the phase path.
Unless you are allowed to go
outside the plane you cannot
make these transformations
Unless you can step out of space into another dimension, you can
never change fermion to boson or vice–versa.
It can be shown formally that, unless the universe has extra
freedom beyond space-time, spin can only change in whole units
of ħ.
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Polarization and Gauge Invariance:
A particle with non-zero-spin can have a certain orientation in space
called the polarization.
Pick a fixed, but otherwise arbitrary, direction (call it the z-axis)
ms = +1/2
ms = +1
s = 1/2
ms = -1/2
s=1
For a given value of s, there are 2s + 1
values for ms ranging from s to –s in
steps of 1.
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ms = 0
ms = -1
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Examples of picking a z-axis:
Apply a magnetic or electric field
Direction of motion
The line drawn in the sand by Col. Travis
etc.
The choice of a direction is called fixing a gauge and the fact that the
Physics is independent of this choice is called gauge invariance
Zero mass particles: e,g, g
Lorentz symmetry  s 2  x 2  c 2t 2 is invariant
so is m2c 4  p 2c 2  E 2
Let m → 0, so we get E = pc, and apply E = hf and p = h/
pc = hf = (h/)c or c =f a wave propagating with
velocity c
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Now, let’s fix the gauge so that the z-axis is in the direction of motion.
This is called the radiation gauge or helicity frame.
c
ms = -1
ms = 0
ms = +1
A particle with zero rest mass can only have 2 degrees of spin
freedom. Similarly the handedness of a massless particle cannot be
reversed.
c
ms = -1/2
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We have seen that the laws of relativity and the law of spin and
statistics are consequences of the Lorentz invariance of space time.
Can we extend the use of symmetries to Multiplets forces, i.e. the way
in which fermions and bosons couple at a vertex
Multiplets: Sets of things that have something in common.
Singlet  an object that is not changed under a symmetry operation
e.g. a circle is a singlet under rotation, a square is a singlet under
rotations of 90°
Doublet  a pair of objects that transform into each other.
e.g. p,n are a doublet under 180° rotations in strong isospin
space.
etc..
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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.
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