The QCD Quark Propagator in Coulomb Gauge and - Institut für Physik

52
10 3
m Bound State
10 2
10 1
SD
AD
ρ
10 0
10 -4 10 -3 10 -2
µ IR
Figure 5.4: The masses of the ρ as well as the scalar (SD) and axial-vector (AD) diquarks
as functions of the infrared regulator µ IR . The mass of the π is identically zero for all
values of µ IR and therefore not shown in the graph. All quantities are given in appropriate
units of √ σ c .
states from the physical spectrum. We have illustrated these effects in figure 5.4 for values
of 10 −4 ≤ µ IR ≤ 10 −2 .
Possessing amplitudes we are in the position to compute charge radii. To this end we
need to know the electromagnetic form factors for low momenta. The pion electromagnetic
form factor F π (q 2 ) (analogously for the diquarks) is defined in terms of the electromagnetic
vertex [GMW84, GMW83]
〈π(p ′ )|J µ (0)|π(p)〉 = ieQ (p µ + p ′ µ )F π(q 2 ) , (5.13)
where J µ the electromagnetic current, Q = 0, ±1 is the pion charge and q = p ′ − p. The
form factor is normalised via
F π (0) = 1 , (5.14)
which is the same as demanding the charge of the π + to be one. The mean square charge
radius is then given as
〈r 2 〉 = 6 ∂F π
∂q 2 (q2 = 0) . (5.15)
In order to calculate the pion form factor at non-zero momentum transfer we need to know
the wave function of a moving pion. The integral equations (5.11) however give only the
Bethe-Salpeter wave function for a pion at rest. In a covariant gauge a wave function
expanded in covariants does not need any boost. However in our approach we have to
boost the wave function. Since we are only interested in low momenta, a Galilean boost
is sufficient.