2 february 2011modern physics iii lecture 41 modern physics for frommies iii a universe of leptons,...

2 February 2011 Modern Physics III Lecture 4 1

Modern Physics for Frommies IIIA Universe of Leptons, Quarks and

Bosons; the Standard Model of Elementary Particles

Lecture 4

Fromm Institute for Lifelong Learning, University of San Francisco


Agenda• Administrative Matters• Quantum Field Theories

• Clear up some stuff from last time• Relativistic Wave Equations• Fermions and Pauli Blocking

• Second or Field Quantization• Quantum Electrodynamics (QED)• Weak Interactions• Strong Interactions


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.


Wave Equations Revisited

The (time dependent) Schrödinger equation is an eigenvalue equation of the form: H E

Differential operator, the Hamiltonian

A number called the eigenvalue returned when is an eigenfunction

A particular solution called an eigenfunction

For a free particle (no potential energy), classically2

kinetic energy2

pE

m

Quantum mechanically, we rewrite things in terms of operators2 2

is momentum, so is free particle kinetic energy2

p i Em

where2 2 2

22 2 2x y z

Finally is the Hamiltonian (total energy) operatorH it


2 22 2 or 0

2 2i i

t m m t

Forms of the Schrödinger equation usually found in textbooks2

2 2 2 2 4Now replace the nonrelativistic with the relativistic 2

pE E p c m c

m

and write in the spirit of the Schrodinger equation2 2 2 2 2 2

2 2 2 4 2 2 2 42 2

c or c 02 2

c cm m

t m t m

Forms of the Klein -Gordon equation


Pauli Blocking, White Dwarves, Neutron Stars and Black Holes

A star uses up its nuclear fuel, Fusion stops and the star begins to collapse “under its own weight”. The endpoint of this collapse is dependent on the mass of the star.

For 1.4ChandesekarM M M

The collapse is stopped by electron degeneracy pressure resulting from the Pauli principle.

the result is a white dwarf

e- squeezed too close together different energy levels. Adding another e- to a given volume requires the energy to raise it to a higher level pressure.

5

3P

Can also look at this using the Heisenberg Uncertainty Principle.


x p Collapse x gets smaller p gets bigger

This zero point or Heisenberg speed is in addition to the usual temperature dependent speed of gas particles, and is present regardless of temperature, and contributes to the gas pressure

7 10 3Typical white dwarf: 10 10 kg / m

If M > MChandra electron degeneracy pressure is insufficient to stop collapse. 1010 kg / m3

Electron Fermi energy > n –p mass difference inverse decay

is favored over decay

ee p v n

en pv

ee p v n


1014 kg/m3 - Individual nuclei are so neutron rich that they start to fall apart

Remaining nuclei surrounded by sea of free neutronsThis is called the neutron drip phase

1016 kg/m3 - Neutron degeneracy pressure starts to become important

1018 kg/m3 - Neutron degeneracy finally halts the collapse provided that M < 3

End up with a neutron star… typical mass of 1.4 with a radius of 10km

M

M

For 3 neutron degeneracy fails and the collapse continues M M

forming a black hole


Four Fundamental Forces:Name Strength Range “Charge” Carrier

Gravity V. weak ∞ Mass (+) G

Weak Nuclear Weak 10-18 m1 Weak isospin (±) W ±, Z0

Electromagnetic Strong ∞ Charge (±)

Strong Nuclear V. Strong ∞

In principle2

Color charge (red, green, blue)

G (8 kinds)

The short ranges of the 2 nuclear forces make it impossible to deduce the Maxwell-like classical field equations. Description is available only via QFT.

Tables are turned for gravity

1. Proton radius 10-16 m

2. ∞ for bare quarks if they were ever bare. In reality, color screening redices range to nuclear scale

__________________________________________________________

Range of force ≈ Compton wavelength of carrier = h

mc


Pre 1974

Three quarks for Muster Mark!-James Joyce, Finnegans Wake (1939)

Post 1974

0 , , and

p uud ud ud uu dd

n udd uds

= 2 3 , 1 3 , 3 1/u d sq e q e q e

= 2 3

1 3

1/ 3

c

b

t

q e

q e

q e

Murray Gell-Mann1969 Nobel Prize


Ignore direction of blue arrows

Quantum Electrodynamics (QED)

Actually 2 diagrams, can propagate either way.

Feynman rules take care of this.

We can have higher order processes as well, e.g. 2 photon exchange

In principle, to calculate ee → ee in we would have to sum an infinite number of diagrams.


Saved: Each vertex contains a factor of the E.M. coupling constant or the fine structure constant.

1

137

We can do a perturbation expansion and get away with considering only the leading terms, e.g. electron gyromagnetic ratio required 72 diagrams (up to 7 vertices) and several years of calculation to match experimental accuracy. It matched to 9 decimal places.

The more vertices in a diagram, the weaker the amplitude and probability is given by (amplitude)2.


Renormalization: We’re not finished yet. There is a class of diagrams that could wreck our theory.

Compared to the lowest order diagram, this has 2 extra vertices. We would expect this amplitude to be down by 1372 10,000

More to the game than just counting vertices. other QFT rules also

The looped e+e- as well as the are virtual. The ’s energy, fixed by the kinematics of the ee → ee collision can be shared in an infinite number of ways between the looped particles.

We have to sum up all these ways of dividing the energy and find that the amplitude for this process is infinite.

What to do?


Feynman, Schwinger and Tomonaga 1965 Nobel Prize

Julian Schwinger Sinichiro Tomonaga 1918 – 1994 1906-1979

RenormalizationThink about theoretical predictions of QFT vs. physically observable experimental quantities measured in the lab.

Observed ee scattering. QM what goes on between the 2 electrons is not observable by experimenter.

Arbitrarily let's call the left hand e the projectile and the other the target


Think of the process as the exchange of a virtual and a target e “dressed” by the vacuum fluctuation loop.

When your experiment measures the e as a target you are measuring all possible manifestations of the target, bare and dressed.

The bare e is not an observable. Observable is the combination of bare and dressed. Not just the above, we can surround the target e with an arbitrarily complex mess of virtual loops connected by virtual photons.

Measure the electron charge experimentally: 191.6 10 Coulombeq e

This is bare effect of vacuum fluctuations eq

Adjust or renormalize the bare electron charge so that your calculated dressed charge matches the experimental value.


If for a proposed QFT of any of the forces, the adjustment of a finite number of parameters, e.g. the charge and mass, renders the calculation of all observables finite; then the theory is renormalizable, workable and is a possible fundamental theory of the force.

Most physicists are very satisfied with the situation. They say: 'Quantum electrodynamics is a good theory and we do not have to worry about it any more.' I must say that I am very dissatisfied with the situation, because this so-called 'good theory' does involve neglecting infinities which appear in its equations, neglecting them in an arbitrary way. This is just not sensible mathematics. Sensible mathematics involves neglecting a quantity when it is small - not neglecting it just because it is infinitely great and you do not want it!

- Paul Dirac as late as 1975

This makes some people more than a little nervous


The shell game that we play ... is technically called 'renormalization'. But no matter how clever the word, it is still what I would call a dippy process! Having to resort to such hocus-pocus has prevented us from proving that the theory of quantum electrodynamics is mathematically self-consistent. It's surprising that the theory still hasn't been proved self-consistent one way or the other by now; I suspect that renormalization is not mathematically legitimate.

-Richard Feynman 1985

But it does work and it works very well!


What about the other forces?

Weak Interaction:

Yes, in fact the weak interaction and the E.M. interaction are unified at a sufficiently high energy

decay

n p e

tCharged current

Neutral current

t


Weak Isospin:

Strong isospin, a.k.a. isoptopic spin, isobaric spin, I-spin was initially introduced by Heisenberg following the discovery of the neutron (1932)

n and p have almost the same mass and under their strong interactions are almost the same. There is a symmetry here between n and p .

Noether’s Theorem: For every symmetry exhibited by aphysical law, there is a corresponding observable quantity that is conserved.

For n and p this is the strong isospin, I = ½. In analogy to ordinary spin, the two form a doublet in isospin space

3

3

1 2 or 1

2

p

n

Ip

n I

n and p are symmetric under the strong interaction. E.M. interaction breaks this symmetry.


Similarly, the pions form an I = 1 triplet:

0

3

03

3

1

or 0

1

I

I

I

Weak isospin is a quantum number relating to the weak interaction and parallels the idea of isospin under the strong interaction.

Use T rather than I to avoid confusion with strong isospin

Fermions with negative chirality (also called left-handed fermions) have T = 1⁄2 and can be grouped into doublets with T3 = ±½ that behave the same way under the weak interactions.

3

3

1 2 1

2

uct

dsb

T

T

quarks 3

3

1 21

2

e

e

T

T

leptons


Aside: Helicity and ChiralitySpin quantization axis is direction of motion

S

p

S

p


Exchange vector bosons form a T =1 triplet

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)}.

0

3

03

3

1

or 0

1

W

W

W

TW

W T

W T

W0 rather than Z0 for technical reasons involving electro-weak unification

d uW

W e

e.g. decay


It took a great deal of effort to get a renormalizable electro-weak QFT

Sheldon Glashow Abdus Salam Steven Weinberg1926 - 1996

1979 Nobel Prize

2 February 2011 Modern Physics III Lecture 4 2426 January 2011 Modern Physics III Lecture 2 24

Strong Interaction:

1911 Rutherford discovers the nucleus

1932 Chadwick discovers the neutron→ Nucleus is protons + neutrons

A little history:

What holds the nucleus together?

Need a strong but short range force.

2 2

SI SI 2( ) or ar arg g a

V r e F er r

1935 Yukawa potential

P+

P+

Could be mediated by a scalar boson of mass 200 me

Muon, not it.

1947 Powell et al. discover pion, in cosmic rays


Hideki Yukawa1907 -1981

1949 Nobel Prize

In the 1960s both theory and experiment began to substructure to “elementary” particles

Partons: quarks, antiquarks and gluons

Update Yukawa’s picture


Simple Quark Model (no color):

u

u

d

Arrows are quark spins

Sq = 1/2u

d

d

Baryons:

Proton Neutron

Mesons:

u d d uu u

or dd


u d

Scratch


Color and Color Charge: Color was originally introduced to beat the Pauli principle.

u

u

u

3 2s

Arrows are quark spins

Sq = 1/2 Clearly Pauli blocked

u

u

uAdd a new number, color

Mrs. Pauli’s favorite son is now happy!

There are also anticolors for the antiquarks


Theory of strong interactions between quarks is called quantum chromodynamics (QCD)

Mediated by force carriers called gluons

Bound states: color - color meson

3 3 different colors baryon

3 3 different colors antibaryon

qq

q

q

Note that in all of the above the colors add up to white. Color is not observed in our “usual” particles (p, n, , K etc.).

Gluons: 8 of them carrying both color and anticolor

rb

Actually 2

rb br


From the 3 color states, one can form 9 bicolor color - anticolor states

,

,

,

,

,

,

B

B BB

G

G

G G

RR R R

B

B

R

R 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.


Stretching the tube requires more and more energy

At some distance it is energetically more favorable to pull a pair out of the vacuum than to increase the tubelength.

qq

0e p K

0

(4) An pair is

(5)

(1-3) strikes e

quarks rearrange

xciting

into co

cre

the

qua

lor

r

singlets, an

a

d

k

ted

e p

d

s

K

s


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, EM, coupling constant increases with energy

What happens in QCD?


In QCD the virtual gluons emitted by quarks not only create pairs but also more gluons with all the allowed bicolor signatures

qq

In QCD the virtual quark-antiquark pairstend to screen the color charge. However, QCD has an additionalwrinkle: 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

http://en.wikipedia.org/wiki/Color_charge

http://en.wikipedia.org/wiki/Asymptotic_freedom


H. David Politzer Frank Wilczek David Gross

Nobel Prize 2004

2 february 2011modern physics iii lecture 41 modern physics for frommies iii a universe of leptons,...

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elementary particle

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black hole slide

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neutron stars

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