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EXAM 2 Results

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EXAM 2 Results

Mean, = 65.6

Std Dev, = 14.4

30 40 50 60 70 80 90 100

Breakdown:

Problem 1 20 14-20 19.2

Problem 2 20 14-20 19.4

Problem 3a 20 2-20 7.5

Problem 3b 12 0-12 8.2

Problem 4 16 0-16 7.0

Problem 5 12 0-12 3.2

Compilation of total scores past 4 PHYS926 sections

200 300 400 500 600

The remaining octet states involve G3 and G8

which do not change color.

We need 2 states ORHTOGONAL to the

sterile singlet state. The possibilities are:

and obviously only 2 are actually independent.

We need to find two that are also orthogonal

to each other, the convention is to use

(see again how 3 and 8 were defined)

QUANTUMCHROMO-DYNAMICSQ.C.D

b

b

bg

bg

g

g

bg

rg

b

b

bg

rb

r

r

u

u

d

u

d

p+

p

But since the gluons are

CHARGE CARRIERS themselves

they also interact with

ONE ANOTHER!

interactions include:

with

coupling

~g

3-gluon

vertex

4-gluon

vertex

with

coupling

~g2

This means all STRONG processes

are much more complicated

with many more Feynman diagrams contributing:

Besides the “tree-level”

and familiar “2nd-order” processes:

we also have the likes of:

and

QED interactions respect the behavior of the Coulomb potential

- infinite reach involves smallest
- energy-momentum transfers
- close single boson exchanges involve potentially
- large energy-momentum transfers

But something MUCH different happens with abelian theories

Most distant reaching individual branches

still involve the smallest momentum carriers

The field lines are better represented

(qualitatively) by color flux tubes:

Since the exchanged gluons are attracted to one another

the field is even more “confined” than an electric dipole!

Further complications

1

137

In QED each vertex introduces a factor of =

to all calculations involving the

process.

That factor is so small, we need only deal with

a limited number of vertices (“higher order”

diagrams can often be neglected.

Contributing sums CONVERGE.

Calculations in the theory are

PERTURBATIVE.

But judging by the force between 2 protons:

s > 137 ~ 1

With so many complicated, higher order diagrams

HOW CAN ANYTHING BE CALCULATED?

CHARGE IN A DI-ELECTRIC MEDIUM

A charge imbedded in a di-electric can polarize

the surrounding molecules into dipoles

A halo of opposite charge

partially cancels Q’s field.

Q

qeff =

Q

dielectric constant

but once within intermolecular distances

you will observe the FULL charge

Q

Q/

r

~molecular

distances

Vacuum Polarization In QED the vacuum can sprout virtual

e+e-wink in and out of existencebut are polarized for their

brief existence, partially screening the TRUE CHARGE by

contributions from:

e-

e+

each “bubble”

is polarized

The TRUE or BARE

charge on an electron

is NOT what’s measured

by e&M experiments and

tabulated on the inside

cover of nearly every

physics text.

e-

e+

e-

e+

e-

e+

e-

e+

e-

e+

THAT would

be the fully screened

“effective charge”

The corresponding “intermolecular” spacing

that’s appropriate here would be the

COMPTON WAVELENGTH of the electron

(related to the spread of the electron’s own wavefunction)

To get within THAT distance of another electron

requires MeV electron beams to observe!

Scattering experiments with 0.5 MeV electron beams

(v = c/10)

show the nominal electron charge requires a

6×10-6 = 0.0006% correction

Vacuum Polarization In QED the vacuum can sprout virtual

e+e-wink in and out of existencebut are polarized for their

brief existence, partially screening the TRUE CHARGE by

contributions from:

The matrix element for

the single loop process:

m X(p2)is a function of p2

in text:

X(p2)=(/3) ln( | p2 |/me2 )

e-

e+

e-

e+

e-

e+

effective=

(1 + m + m2 + m3 + ...)

e2/ħc

e-

e+

e-

e+

e-

e+

Notice:as mgoes upaeffectivegoes up and

mgoes up as p2 goes up.

Thus higher momentum virtual particles

have a higher probability

of creating these dipole pairs

…and higher momentum virtual particles

are “felt” by (exchanged between)

only the closest of interacting charges.

is the charge as seen “far” from the source, e

The true charge is HIGHER.

The Lamb Shift

Relativistic corrections insufficient

to explain hyperfine structure

2p½(n=2, ℓ= 1, j = ½)

2s½(n=2, ℓ= 0, j = ½)

are expected to be perfectly degenerate

1947Lamb & Retherfordfound

2s½energy state > 2p½state

- Bethe’s explanation:
- Coulomb’s law inadequate
- The field is quantized (into photons!)
- and spontaneously produces e+e- pairs near
- the nucleus…partially screening its charge
- Corrects the magnetic dipole moment
- of both electron and proton!

What happens in Q.C.D. ??

q3

q4

ur

Like e+e- pair production

this always screens

the quarks electric charge

nflav

q1

q2

ur

ururis one example.

This bubble can happen

nflavor× ncolordifferent ways.

1

3

of the time

shielding

color charge

driving s up

at short distances,

down at large distances.

Obviously only the colorless G3, G8 exchanges

can mediate this particular interaction

This makes 2 × nflavor diagrams

that result in sheilding color charge.

But ALSO (completely UNlike QED)

QCD includes diagrams like:

r

r

b

b

ncolor ways

g

g

Each of these

anti-shield

(drive s down

at short distances,

up at large distances)

r

b

r

r

ncolor ways

g

g

r

each

ncolor

ways

r

b

b

b

g

g

r

b

b

g

ncolor ways

for this bubble

to be formed

but

br

bg

doesn’t shieldat all

in fact brings the color charges

right up closer the to target

enhances the sources color charge

In short order we just found

2nflavor diagrams that SHIELD color

4ncolor diagrams that ANTI-SHIELD

In fact there are even more diagrams

contributing to ANTI-SHIELDING.

= 12

SHIELD: 2nflavor

ANTI-SHIELD:11ncolor

= 33

QCD coupling DECREASES at short distances!!

- 2 important consequences:
- at high energy collisions between hadrons
- s 0
- for impacts that probe small distances
- quarks are essentially free
- at large separations the coupling between
- color charges grow HUGE

“asymptotic freedom”

“confinement”

All final states (even quark composites)

carry no net color charge!

Naturally occurring stable “particles” cannot carry COLOR

Quarks are confined in color singlet packages

of 2 (mesons) color/anticolor

and 3 (baryons) all 3colors

Variation of the QCD coupling parameter s with q2

s

q2, GeV2/c2

If try to separate quarks

u

u d

d

gr

u

u

d

d

Gm3

d

gr

u

u

d

d

Gm8

If try to separate quarks

u

u d

d

u

gr

u

d

rr

d

p+

u

d

d

u

d

If try to separate quarks

u

u d

p+

u

d

d

u

d

p+

u

d

d

u

d

If try to separate quarks

u u d

Hadrons

_

g

q

_

q

q

q

LEP (CERN)

Geneva

Hadrons

e+e– +–e+e– qqe+e– qqg

OPAL Experiment

_

e+e- q q g 3 jets

JADE detector at PETRA e+e- collider

(DESY, Hamburg, Germany)

2-jet event