Transcript No Slide Title

6 of the gluon fields are independent linear combinations of
the simple gluon fields we enumerated
Gm1 = (rg + gr)/2 Gm4 = (bg + gb)/2 Gm6 = (rb + br)/2
Gm2=-i(rg - gr)/2 Gm5=-i(bg - gb)/2 Gm7=-i(rb + br)/2
linear combinations of the color/anticolor states
rb
q
rg
br
bg
gr
gb
b
r
q
Gm6 or Gm7
rg or
Gm1 = (1/2 ) (rg + gr )
Gm2 = (-i/2 )(rg - gr )
gr
or inverting
1
2
1
(
G

iG
)  rg
2
1
2
(G1 - iG 2 )  gr
The COLOR SINGLET would be 1/3 ( rr + gg + bb )
…does not seem to exist
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:
1
(rr
2
- bb )
1
2
( rr - gg )
1
2
(bb - gg )
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)
1
G 
(bb - gg )
2
3
1
G 
(2rr - bb - gg )
6
8
QUANTUM CHROMO-DYNAMICS Q.C.D
b
bg
g
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!
L
8
 F
gauge
field
im
F
i
m
i 1
m
i

im
 ( G -  G -
2g
c
jm
k
cijkG G ) 
( m G i -  G i m - 2cg cijkGmj Gk )
interactions include:
3-gluon
vertex
4-gluon
vertex
with
coupling
~g
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
1
QED interactions respect the 2 behavior of the Coulomb potential
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 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 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/
~molecular
distances
r
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-
each “bubble”
is polarized
e+
e-
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+
eTHAT would
be the fully screened
“effective charge”
e+
The corresponding “intermolecular” spacing
that’s appropriate here would be the
COMPTON WAVELENGTH of the electron

-10
C 
 2.43  10 cm
mc
(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- pairs that 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-
m  X(p2) is a function of p2
in text:
X(p2)=(/3p) ln ( | p2 |/me2 )
e+
e-
e+
e-
e+
e-
effective =
(1 + m + m2 + m3 + ...)
e2/ħc
e+
e-
e-
e+
Notice: as m goes up
e+
effective
goes up and
m goes 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.
 (0)   ( p  0)
2
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
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. ??
q3
ur
q4
nflav
q1
ur
q2
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 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
b
b
r
g
ncolor ways
g
r
b
r
g
r
g
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
2nflavor diagrams that SHIELD color
4ncolor diagrams that ANTI-SHIELD
In fact there are even more diagrams
contributing to ANTI-SHIELDING.
SHIELD: 2nflavor
= 12
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
“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 q2
s
q2, GeV2/c2
If try to separate quarks
u
u
d
d
gr
u
u
d
d
d
gr
u
u
d
Gm8
Gm3
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
p
d
u
u
d
d
d
p
u
u
d
d
d
If try to separate quarks
u
u
d
Hadrons
_
g
q
_
q
q
Hadrons
q
LEP (CERN)
Geneva
e+e–  m+m–
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