6 of the gluon fields are independent linear combinations ofthe simple gluon fields we enumerated

G1 = (rg + gr)/2 G

4 = (bg + gb)/2 G6 = (rb + br)/2

G2=-i(rg gr)/2 G

5=-i(bg gb)/2 G7=-i(rb + br)/2

The COLOR SINGLET would be 1/3 ( rr + gg + bb )

…does not seem to exist

linear combinations of the color/anticolor statesrb rg br bg gr gb

q b r

q

G6 or G

7

G1 = (1/2 ) (rg + gr )

G2 = (-i/2 )(rg gr ) or inverting

rgiGG

griGG

)(

)(21

21

212

1

rg or

gr

The remaining octet states involve G3 and G8

which do not change color.

We need 2 states ORHTOGONAL to thesterile singlet state. The possibilities are:

)(21 bbrr

)(21 ggrr

)(21 ggbb

and obviously only 2 are actually independent.

We need to find two that are also orthogonalto each other, the convention is to use

(see again how 3 and 8 were defined)

)(2

13 ggbbG

)2(6

18 ggbbrrG

+u d

rb

bg

bg

r r

b b

b b

g g

u d u

p

bg

bg

rg

QUANTUM CHROMO-DYNAMICS Q.C.D

But since the gluons are CHARGE CARRIERS themselves

they also interact withONE ANOTHER!

8

1i

ii FFfieldgauge

L

)(

)(2

2

kjijkc

gii

kjijkc

gii

GGcGG

GGcGG

interactions include:

3gluonvertex

withcoupling

~g

4-gluonvertex

withcoupling

~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 potential2

1R• 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 branchesstill 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

In QED each vertex introduces a factor of =to all calculations involving the

1137

process.

That factor is so small, we need only deal witha 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 diagramsHOW CAN ANYTHING BE CALCULATED?

CHARGE IN A DI-ELECTRIC MEDIUM

QQ

A charge imbedded in a dielectric can polarizethe surrounding molecules into dipoles

A halo of opposite chargepartially cancels Q’s field.

qeff = Q

dielectric constant

but once within intermolecular distancesyou will observe the FULL charge

Q

Q/

~moleculardistances

r

e

e+

e

e+

e

e+

e

e+

e

e+

e

e+

each “bubble”is polarized

The TRUE or BAREcharge on an electron

is NOT what’s measuredby E&M experiments and

tabulated on the insidecover of nearly every

physics text.

THAT wouldbe the fully screened“effective charge”

Vacuum Polarization In QED the vacuum can sprout virtuale+e pairs that wink in and out of existence but are polarized for their brief existence, partially screening the TRUE CHARGE by contributions from:

The corresponding “intermolecular” spacingthat’s appropriate here would be the

COMPTON WAVELENGTH of the electron

cmmcC

101043.2

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

To get within THAT distance of another electronrequires MeV electron beams to observe!

Scattering experiments with 0.5 MeV electron beams(v = c/10)

show the nominal electron charge requires a6×10-6 = 0.0006% correction

Vacuum Polarization In QED the vacuum can sprout virtuale+e pairs that wink in and out of existence but are polarized for their brief existence, partially screening the TRUE CHARGE by contributions from:

e

e+

e

e+

e

e+

e

e+

e

e+

e

e+

The matrix element forthe single loop process:

X(p2) is a function of p2

in text: X(p2)=(/3) ln ( | p2 |/me

2 )

effective =

(1 + + 2 + 3 + ...)

e2/ħc

Notice: as goes up effective goes up and

goes up as p2 goes up.

Thus higher momentum virtual particleshave a higher probability

of creating these dipole pairs

…and higher momentum virtual particlesare “felt” by (exchanged between)

only the closest of interacting charges.

)0()0( 2 pis the charge as seen “far” from the source, e

The true charge is HIGHER.

The Lamb ShiftRelativistic corrections insufficient

to explain hyperfine structure

2s½ (n=2, ℓ= 0, j = ½) 2p½ (n=2, ℓ= 1, j = ½)

are expected to be perfectly degenerate

1947 Lamb & Retherford found 2s½ energy state > 2p½ state

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. ??

ur

q1 q2

q3 q4

ur

Like e+e pair productionthis always screens

the quarks electric charge

of the time shielding

color charge

13

urur is one example. This bubble can happen

nflavor × ncolor different ways.driving s up

at short distances, down at large distances.

nflav

Obviously only the colorless G3, G8 exchangescan mediate this particular interaction

This makes 2 × nflavor diagramsthat result in sheilding color charge.

But ALSO (completely UNlike QED)QCD includes diagrams like:

rr

g gbb b

eachncolor

ways

rg

r

g

b b ncolor ways

ncolor waysrg

r

rg

b

Each of theseanti-shield

(drive s downat short distances,

up at large distances)

r

g

bb

ncolor waysfor this bubbleto be formed

but

b r

b g

doesn’t shield at allin fact brings the color charges

right up closer the to targetenhances 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.

SHIELD: 2nflavor

ANTI-SHIELD: 11ncolor

= 12

= 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 COLORQuarks are confined in color singlet packages

of 2 (mesons) color/anticolorand 3 (baryons) all 3 colors

Variation of the QCD coupling parameter s with q2

q2, GeV2/c2

s

If try to separate quarks

u d

u

gr

u

d

d d

u

gr

u

d

d d

u

G8

G3

If try to separate quarks

u d

u

gr

u

d

d d

u

rr

u

u

d

d d

If try to separate quarks

u d

u

u

u

d

d d

u

u

d

d d

If try to separate quarks

u u d

q

q

_

Hadrons

qq

_

Hadrons

g

LEP (CERN)Geneva

ee++ee– – ++––

ee++ee– – qqqq

ee++ee– – qqgqqgOPAL Experiment

e+e q q g 3 jets_

JADE detector at PETRA e+e collider(DESY, Hamburg, Germany)

2-jet event