EXAM 2 Results Mean 65 6 Std Dev

EXAM 2 Results Mean, = 65. 6 Std Dev, = 14. 4 30 40 50 60 70 80 90 100 Breakdown: Problem 1 Problem 2 Problem 3 a Problem 3 b Problem 4 Problem 5 20 20 20 12 16 12 14 -20 2 -20 0 -12 0 -16 0 -12 19. 4 7. 5 8. 2 7. 0 3. 2 Compilation of total scores past 4 PHYS 926 sections 200 300 400 500 600

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)

QUANTUM CHROMO-DYNAMICS Q. C. D b g bg bg b bg g rg b b bg rb r r u d p+ u u p d

But since the gluons are CHARGE CARRIERS themselves they also interact with ONE ANOTHER! interactions include: 3 -gluon vertex 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 1 In QED each vertex introduces a factor of = 137 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 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/ ~molecular distances r

Vacuum Polarization In QED the vacuum can sprout virtual e+e- wink in and out of existence but are polarized for their brief existence, partially screening the TRUE CHARGE by contributions from: e- each “bubble” is polarized e+ e- e- e+ 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+ 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 Me. V electron beams to observe! Scattering experiments with 0. 5 Me. V 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 existence but are polarized for their brief existence, partially screening the TRUE CHARGE by contributions from: The matrix element for the single loop process: e- X(p 2) is a function of p 2 in text: X(p 2)=( /3 ) ln ( | p 2 |/me 2 ) e+ e- effective = (1 + m 2 + m 3 +. . . ) e 2/ħc e+ e- e- e+ Notice: as goes up e+ 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 2 s½ (n=2, ℓ= 0, j = ½) 2 p½ (n=2, ℓ= 1, j = ½) are expected to be perfectly degenerate 1947 Lamb & Retherford found 2 s½ energy state > 2 p½ 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. ? ? q 3 q 1 ur nflav ur q 4 q 2 urur is one example. Like e+e- pair production this always screens the quarks electric charge 1 of 3 the time This bubble can happen shielding color charge nflavor × ncolor different ways. driving s up at short distances, down at large distances. Obviously only the colorless G 3, G 8 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 b b r g ncolor ways g r r g b r g ncolor ways r b b b g each ncolor ways Each of these anti-shield (drive s down at short distances, up at large distances)

b b r g ncolor 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 2 nflavor diagrams that SHIELD color 4 ncolor diagrams that ANTI-SHIELD In fact there are even more diagrams contributing to ANTI-SHIELDING. SHIELD: 2 nflavor = 12 ANTI-SHIELD: 11 ncolor = 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 “asymptotic freedom” • at large separations the coupling between color charges grow HUGE “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 s q 2, Ge. V 2/c 2

If try to separate quarks u u d d gr u u d G m 8 G m 3 d

If try to separate quarks u u d d u gr u rr d d p+ u u d d d

If try to separate quarks u u d p+ u u d d d

If try to separate quarks u u d

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

e + e – + – 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