VOLUME 78, NUMBER 1 PHYSICAL REVIEW LETTERS 6JANUARY 1997
Two-Loop Renormalization in Coulomb Gauge QED
Gregory S. Adkins and Plamen M. Mitrikov
Franklin and Marshall College, Lancaster, Pennsylvania 17604
Richard N. Fell
Physics Department, Brandeis University, Waltham, Massachusetts 02254
(Received 9 October 1996)
We study the renormalization of Coulomb gauge QED at two-loop order. A convenient formula is
given for the evaluation of the noncovariant Feynman integrals that occur in Coulomb gauge QED. We
calculate the two-loop Coulomb gauge vertex renormalization constant Z
1
, and discuss the absence of
infrared divergences in this gauge. [S0031-9007(96)02071-6]
PACS numbers: 12.20.Ds, 11.10.Gh
Coulomb gauge has been used in quantum electrody-
namics since its foundation. The Coulomb gauge photon
propagator
D
mn
k
√
1
k
2
0
0
1
k
2 d
ij
2
ˆ
k
i
ˆ
k
j
!
(1)
contains the most direct description of the Coulomb
interaction between charges and the magnetic interaction
between currents of any gauge. However, with the advent
of covariant methods and the renormalization program,
Coulomb gauge has fallen out of favor for many purposes.
Its noncovariant structure complicates the evaluation of
Feynman integrals and the associated algebra. Covariant
gauges such as the Feynman gauge have been used almost
exclusively in the calculation of scattering processes and
magnetic moments.
Bound state problems, however, are another matter.
Here the dominant physics is the Coulombic binding. The
inherent superiority of Coulomb gauge in this regime has
been widely recognized, and Coulomb gauge has been in
continual use, albeit seldom in problems where ultraviolet
divergences appear, and never in problems with two-loop
ultraviolet divergences.
Anticipating the need to use Coulomb gauge in bound
state problems with multiloop ultraviolet divergences, we
have worked out a simple formula for use in evaluating
noncovariant Feynman integrals, and have calculated the
two-loop vertex renormalization constant Z
1
in Coulomb
gauge. We show that Z
1
is free of infrared divergences in
this gauge.
Noncovariant Feynman integrals can be performed in
essentially the same way as covariant ones. Consider, for
example, the integral
I
Z
d
n
i p
n2
1
2A
21
1 2 ? p 1 M
2
a
, (2)
where n is the dimension of spacetime and A
21
is a
diagonal matrix with A
21
ij
2d
ij
and A
21
00
. 0. An
infinitesimal negative imaginary part is implicit in the
mass term. By completing the square and defining
0
through
0
1 Ap with A
21
A g diag1, 2d
ij
,
one finds that
I
Z
d
n
0
i p
n2
1
2
0
A
21
0
1D
a
, (3)
where D pAp 1 M
2
. Taking
0
A
12
00
, one has
I
p
jdetAj
Z
d
n
00
i p
n2
1
2
002
1D
a
. (4)
The remaining integral is covariant, and is done in the
usual way via Wick rotation. The result is
I
p
jdetAj
1
Ga
Ga2 n2
D
a2n2
. (5)
It is not much harder to work out a more general in-
tegral formula for an integrand having several denomina-
tors and a numerator which is a polynomial in . We use
Feynman parameters to combine denominators. The com-
plete formula is
Z
d
n
i p
n2
T
2A
21
1
1 2 ? p
1
1 M
2
1
a
1
··· 2A
21
N
1 2 ? p
N
1 M
2
N
a
N
Z
d
N
x
x
a
1
21
1
··· x
a
N
21
N
Ga
1
··· Ga
N
p
jdetAj
(
T
0
Ga2 n2
D
a2n2
2
1
2
T
1
Ga2 n2 2 1
D
a2n221
1
1
4
T
2
Ga2 n2 2 2
D
a2n222
1 ···
)
.
(6)
0031-9007 96 78(1) 9(4)$10.00 © 1996 The American Physical Society 9