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9. Quantum chromodynamics
1
9. QUANTUM CHROMODYNAMICS
Written October 2009 by G. Dissertori (ETH, Zurich) and G.P. Salam (LPTHE, Paris).
9.1. Basics
Quantum Chromodynamics (QCD), the gauge feld theory that describes the
strong interactions oF colored quarks and gluons, is the SU(3) component oF the
SU(3)
×
SU(2)
×
U(1) Standard Model oF Particle Physics.
The Lagrangian oF QCD is given by
L
=
X
q
¯
ψ
q,a
(
iγ
μ
∂
μ
δ
ab
−
g
s
γ
μ
t
C
ab
A
C
μ
−
m
q
δ
ab
)
ψ
q,b
−
1
4
F
A
μν
F
Aμν
,
(9
.
1)
where repeated indices are summed over. The
γ
μ
are the Dirac
γ
matrices. The
ψ
are
quarkfeld spinors For a quark oF ﬂavor
q
and mass
m
q
, with a colorindex
a
that runs
From
a
=1to
N
c
=3
,
i.e.
quarks come in three “colors.” Quarks are said to be in the
Fundamental representation oF the SU(3) color group.
The
A
C
μ
correspond to the gluon felds, with
C
running From 1 to
N
2
c
−
1=8
,
i.e.
there are eight kinds oF gluon. Gluons are said to be in the adjoint representation oF the
SU(3) color group. The
t
C
ab
correspond to eight 3
×
3 matrices and are the generators oF
the SU(3) group (cF. the section on “SU(3) isoscalar Factors and representation matrices”
in this
Review
with
t
C
ab
≡
λ
C
ab
/
2). They encode the Fact that a gluon’s interaction with
a quark rotates the quark’s color in SU(3) space. The quantity
g
s
is the QCD coupling
constant. ±inally, the feld tensor
F
A
μν
is given by
F
A
μν
=
∂
μ
A
A
ν
−
∂
ν
A
A
μ
−
g
s
f
ABC
A
B
μ
A
C
ν
[
t
A
,t
B
]=
if
ABC
t
C
,
(9
.
2)
where the
f
ABC
are the structure constants oF the SU(3) group.
Neither quarks nor gluons are observed as Free particles. Hadrons are colorsinglet (
i.e.
colorneutral) combinations oF quarks, antiquarks, and gluons.
Abinitio predictive methods For QCD include lattice gauge theory and perturbative
expansions in the coupling. The ±eynman rules oF QCD involve a quarkantiquark
gluon (
q
¯
qg
) vertex, a 3gluon vertex (both proportional to
g
s
), and a 4gluon vertex
(proportional to
g
2
s
). A Full set oF ±eynman rules is to be Found For example in ReF. 1.
UseFul coloralgebra relations include:
t
A
ab
t
A
bc
=
C
F
δ
ac
,where
C
F
≡
(
N
2
c
−
1)
/
(2
N
c
)=
4
/
3 is the colorFactor (“Casimir”) associated with gluon emission From a quark;
f
ACD
f
BCD
=
C
A
δ
AB
where
C
A
≡
N
c
= 3 is the colorFactor associated with gluon
emission From a gluon;
t
A
ab
t
B
ab
=
T
R
δ
AB
T
R
=1
/
2 is the colorFactor For a gluon to
split to a
q
¯
q
pair.
The Fundamental parameters oF QCD are the coupling
g
s
(or
α
s
=
g
2
s
4
π
) and the quark
masses
m
q
.
K. Nakamura
et al.
,JPG
37
, 075021 (2010) (http://pdg.lbl.gov)
July 30, 2010
14:57
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9. Quantum chromodynamics
This review will concentrate mainly on perturbative aspects of QCD as they
relate to collider physics. Related textbooks include Refs. 1–3. Some discussion of
nonperturbative aspects, including lattice QCD, is to be found in the reviews on “Quark
Masses” and “The CKM quarkmixing matrix” of this
Review
. LatticeQCD textbooks
and lecture notes include Refs. 4–6, while recent developments are summarized for
example in Ref. 7. For a review of some of the QCD issues in heavyion physics, see for
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This note was uploaded on 06/07/2011 for the course PHYS 4132 taught by Professor Kutter during the Spring '11 term at University of Florida.
 Spring '11
 Kutter

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