Feynman Path Integral Formulation

88 3 Gravity in 2 + ε Dimensions
( a1
λ 0 → λ 0
[1 −
ε + a ) ]
2
ε 2 G
(3.91)
with coefficients
a 1 = − 8 α
− 1)2 − 1)(β − 3)
+ 8(β + 4(β
(β − 2) 2 α(β − 2) 2
(β − 1)2
a 2 = 8
(β − 2) 2 . (3.92)
One notices that the kinematic singularities in the graviton propagator, proportional
to 1/(d −2), can combine with the one loop ultraviolet divergent part of momentum
integrals, as in
∫
1
ε
d d k 1
(2π) d k 2 ∼ 1
ε 2 , (3.93)
to give terms of order 1/ε 2 in Eq. (3.91). Generally it is better to separate the ultraviolet
divergence from the infrared one, by using for example the following regulated
integral
∫
d d k 1
(2π) d (k 2 + μ 2 ) a = 1 Γ (a − d/2)
(4π) d/2 (μ 2 ) d/2−a , (3.94)
Γ (a)
for a = 1 and μ → 0.
One can then follow the same procedure for the √ gR term. First one needs to
expand the Einstein term to quadratic order in the quantum field h μν
√ḡ
¯R = √ gR
+ √ g { 1
4
∇ ρ h μ ν∇ ρ h ν μ − 1 2 ∇ νh ν μ∇ ρ h ρμ + 1 2 Rσ ρμνh ρ σ h μν} + ...
(3.95)
where ∇ μ denotes the covariant derivative with respect to the background metric
g μν . The complete expansion was given previously in Eq. (1.94). The same expansion
then needs to be done for the gauge fixing term of Eq. (3.87) as well, and
furthermore it is again convenient to choose as a background field the flat metric
g μν = δ μν . For the one loop divergences associated with the √ gR term one then
finds
μ ε (1
16πG →
με
− b )
16πG ε G , (3.96)
with the coefficient b given by (Gastmans et al, 1978; Christensen et al, 1980)
b = 2 3
· 19 +
4(β − 1)2
(β − 2) 2 . (3.97)
Thus the one-loop radiative corrections modify the total Lagrangian to
L →−
(1 με
− b ) √gR
[ (
16πG ε G a1
+ λ0 1 −
ε + a )
2 √g
G]
. (3.98)
ε 2