6 of the gluon fields are independent linear combinations of the simple gluon fields we enumerated G  1 = (rg + gr)/  2 G  4 = (bg + gb)/  2 G  6 = (rb + br)/  2 G  2 =-i(rg  gr)/  2 G  5 =-i(bg  gb)/  2 G  7 =-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 states rb rg br bg gr gb q b r q G  6 or G  7 G  1 = ( 1/  2 ) (rg + gr ) G  2 = (-i/  2 )(rg  gr ) or inverting rg or gr

The remaining octet states involve G 3 and G 8 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)

++ ud rbrb bgbg bgbg r r b b b b g g udu p bgbg bgbg rgrg QUANTUM CHROMO-DYNAMICS Q.C.D

But since the gluons are CHARGE CARRIERS themselves they also interact with ONE ANOTHER! interactions include: 3  gluon vertex with coupling ~g 4-gluon vertex with coupling ~g 2

This means all STRONG processes are much more complicated with many more Feynman diagrams contributing: Besides the “tree-level” and familiar “2 nd -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 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 Q 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 eff = QQ dielectric constant but once within intermolecular distances you will observe the FULL charge Q Q/  ~molecular distances r 

e  e+ e  e+ e  e+ e  e+ e  e+ 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. THAT would be the fully screened “effective charge” Vacuum Polarization In QED the vacuum can sprout virtual e + 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” 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 = % correction

Vacuum Polarization In QED the vacuum can sprout virtual e + 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 for the single loop process:  X(p 2 ) is a function of p 2 in text: X(p 2 )=(  /3  ) ln ( | p 2 | / m e 2 )  effective =  (1 +  +  2 +  )  e 2 /ħc Notice: as  goes up  effective goes up and  goes up as p 2 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 2s ½ (n=2, ℓ = 0, j = ½) 2p ½ (n=2, ℓ = 1, j = ½) are expected to be perfectly degenerate 1947 Lamb & Retherford found 2s ½ energy state > 2p ½ 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. ?? urur q1q1 q2q2 q3q3 q4q4 urur Like e + e  pair production this always screens the quarks electric charge of the time shielding color charge 1313 u r u r is one example. This bubble can happen n flavor × n color different ways. driving  s up at short distances, down at large distances. n flav Obviously only the colorless G 3, G 8 exchanges can mediate this particular interaction This makes 2 × n flavor diagrams that result in sheilding color charge.

But ALSO (completely UNlike QED) QCD includes diagrams like: r r g g b b b each n color ways r g r g b n color ways r g r r g b Each of these anti-shield (drive  s down at short distances, up at large distances)

r g bbbb n color ways for this bubble to be formed but b r b g doesn’t shield at all in fact brings the color charges right up closer the to target enhances the sources color charge

In short order we just found 2n flavor diagrams that SHIELD color 4n color diagrams that ANTI-SHIELD In fact there are even more diagrams contributing to ANTI-SHIELDING. SHIELD: 2n flavor ANTI-SHIELD: 11n color = 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 COLOR Quarks are confined in color singlet packages of 2 (mesons) color/anticolor and 3 (baryons) all 3 colors

Variation of the QCD coupling parameter  s with q 2 q 2, GeV 2 /c 2 ss

If try to separate quarks u d u grgr u d d d u grgr u d d d u G8G8 G3G3

If try to separate quarks u d u grgr u d d d u rr u u d dddd 

If try to separate quarks u d u u u d dddd  u u d dddd

If try to separate quarks u u d

q q _ Hadrons q q _ g LEP (CERN) Geneva

e + e –   +  – e + e –  qq e + e –  qqg OPAL Experiment

e + e   q q g  3 jets _ JADE detector at PETRA e + e  collider (DESY, Hamburg, Germany)

2-jet event