Feynman Path Integral Formulation

80 3 Gravity in 2 + ε Dimensions
∫ Λ
1
2 (N − 1) d d k
(2π) d ln(k2 + m 2 ) , (3.52)
which makes the evaluation of the trace straightforward. As should be clear from
Eq. (3.49), the parameter m can be interpreted as the mass of the φ field. The functional
integral in Eq. (3.50) can then be evaluated by the saddle point method. It is
easy to see from Eq. (3.51) that the saddle point conditions are
with the function Ω d (m) given by the integral
σ 2 = 1 − (N − 1)Ω d (m)T ,
m 2 σ = 0 (3.53)
∫ Λ d d k 1
Ω d =
(2π) d k 2 + m 2 . (3.54)
The latter can be evaluated in terms of a hypergeometric function,
1 Λ d [
Ω d =
2 d−1 π d/2 Γ (d/2) m 2 d 2 F 1 1, d 2 ;1+ d 2 ]
2 ; −Λ m 2
, (3.55)
but here one only really needs it in the large cutoff limit, m ≪ Λ, in which case one
finds the more tractable expression
Ω d (m) − Ω d (0)=m 2 [c 1 m d−4 + c 2 Λ d−4 + O(m 2 Λ d−6 )] , (3.56)
with c 1 and c 2 some d-dependent coefficients.
From Eq. (3.53) one notices that at weak coupling and for d > 2 a non-vanishing
σ-field expectation value implies that m, the mass of the π field, is zero. If one sets
(N − 1)Ω d (0)=1/T c , one can then write the first expression in Eq. (3.53) as
σ(T )=±[1 − T/T c ] 1/2 , (3.57)
which shows that T c is the critical coupling at which the order parameter σ vanishes.
Above T c the order parameter σ vanishes, and m(T ) is obtained, from Eq. (3.53),
by the solution of the nonlinear gap equation
1
T =(N − 1) ∫ Λ
d d k 1
(2π) d k 2 + m 2 . (3.58)
Using the definition of the critical coupling T c , one can now write, for 2 < d < 4,
for the common mass of the σ and π fields
m(T )
( 1
∼ − 1 ) 1/(d−2)
, (3.59)
m≪Λ T c T