EXAM 2 Results
30 40 50 60 70 80 90 100
Mean, = 65.6Std Dev, = 14.4
Breakdown:
Problem 1 20 14-20 19.2 Problem 2 20 14-20 19.4Problem 3a 20 2-20 7.5 Problem 3b 12 0-12 8.2Problem 4 16 0-16 7.0Problem 5 12 0-12 3.2
200 300 400 500 600
Compilation of total scores past 4 PHYS926 sections
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:
)(2
1 bbrr )(
2
1 ggrr )(
2
1 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
kj
ijkcgii
kj
ijkcgii
GGcGG
GGcGG
interactions include:
3-gluonvertex
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
1
R• 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
A charge imbedded in a di-electric 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
Vacuum Polarization In QED the vacuum can sprout virtuale+e wink in and out of existence but are polarized for theirbrief existence, partially screening the TRUE CHARGE bycontributions from:
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 andtabulated on the insidecover of nearly every
physics text.
THAT wouldbe the fully screened“effective charge”
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 wink in and out of existence but are polarized for theirbrief existence, partially screening the TRUE CHARGE bycontributions 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 |/me2 )
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 diagrams
that result in sheilding color charge.
But ALSO (completely UNlike QED)QCD includes diagrams like:
rr
g g
bb b
each
ncolor
ways
r
g
r
g
b b ncolor ways
ncolor waysr
g
r
r
g
b
Each of theseanti-shield
(drive s downat short distances,
up at large distances)
r
g
bb
ncolor ways
for this bubbleto be formed
but
b r
b g
doesn’t shield at allin 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.
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 COLOR
Quarks are confined in color singlet packagesof 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
_
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
