Kiefer C. Quantum gravity

54 COVARIANT APPROACHES TO QUANTUM GRAVITY
the four-momentum. After a rather long calculation, Bjerrum-Bohr et al. (2003a)
find (restoring c)
V (r) =− Gm (
1m 2
1+3 G(m 1 + m 2 )
r
rc 2 + 41 )
G
10π r 2 c 3 . (2.100)
All terms are fully determined by the non-analytic parts of the one-loop amplitude;
the parameters connected with the higher curvature terms in the action
contribute only to the analytic parts. It is for this reason that an unambiguous
result can be obtained. Note that (2.100) corresponds to an effective gravitational
constant G eff (r) >G.
Although arising from a one-loop amplitude, the first correction term is in fact
an effect of classical GR. It can be obtained from the Einstein–Infeld–Hoffmann
equations, in which none of the two bodies is treated as a test body. Interestingly,
such a term had already been derived from quantum-gravitational considerations
by Iwasaki (1971).
The second correction is proportional to and is of genuine quantum-gravitational
origin. The sign in front of this term indicates that the strength of the
gravitational interaction is increased as compared to the pure Newtonian potential.
21 The result (2.100) demonstrates that a definite prediction from quantum
gravity is in principle possible. Unfortunately, the correction term, being of the
order (l P /r) 2 ≪ 1, is not measurable in laboratory experiments: Taking for r the
Bohr radius, the correction is of the order of 10 −49 . We remark that similar techniques
are applied successfully in low-energy QCD (in the limit of pion masses
m π → 0) and known under the term ‘chiral perturbation theory’ (which is also a
‘non-renormalizable theory’ with a dimensionful coupling constant); see, for example,
Gasser and Leutwyler (1984). Quantum corrections to the Schwarzschild
and Kerr metrics are calculated along these lines in Bjerrum-Bohr et al. (2003b).
The second example is graviton–graviton scattering. This is the simplest lowenergy
process in quantum gravity. It was originally calculated in tree level by
DeWitt (1967c). For the scattering of a graviton with helicity +2 with a graviton
with helicity −2, for example, he found for the cross-section in the centre-of-mass
frame, the expression
dσ
dΩ =4G2 E 2 cos12 θ/2
sin 4 θ/2 , (2.101)
where E is the centre-of-mass energy (and similar results for other combinations
of helicity). One recognizes in the denominator of (2.101) the term well known
from Rutherford scattering. DeWitt (1967c) also considered other processes such
as gravitational bremsstrahlung.
One-loop calculations can also be carried out. In the background-field method,
the quantum fields f µν occur only in internal lines; external lines contain only
the background field ḡ µν . It was already mentioned that this makes the whole
21 There had been some disagreement about the exact number in (2.100); see the discussion
in Bjerrum-Bohr et al. (2003a).