52
K. P. Satheesh
dV e f f
d M 2 =
m
2
R
λ R
d
2 V e f f
d(M 2 ) 2 =
1
3λ R
which can be written as
m
2
R
λ R
=
m
2
λ
+
1
2
I 2
1
λ R
=
1
λ
+
1
2
I 2
Giving the effective potential and effective mass which are finite.
V e f f = −
M
4
2λ R
+
M
2
2
˜
φ
2
+
m
2
R
λ R
M
2
+
M
4
64π 2
ln
M
2
2 −
1
2
M
2
= m
2
R +
λ R
2
˜
φ
2
+
λ R
3π 2 M
2 ln
M
2
2
It is interesting to note that in quantum mechanics the variational approximation
using Gaussian state produces very accurate results. But in field theory the results
are not extremely accurate. Using other wave functionals like coherent state wave
functional may produce more accurate results which is to be verified.
6 Effective Action for φ 6 Model
The procedure developed above for φ
4 model can be extended to φ
6 model. The
coupling effects will show up only at three-loop levels and we write the expansions
up to three loops.
=
dt
x
ˆ
˙ ˆ
φ −
1
2
(∇ ˆ
φ)
2
− V ( ˆ
φ) +
x,y
˙
G − 2
x,y,z
G
−
x
1
8
G
−1
(x, x, t) −
1
2
∇
2
x G(x, y, t)| x=y +
1
2
V
(2)
( ˆ
φ)G(x, x, t)
−
1
8
V
(4)
( ˆ
φ)
x
G(x, x, t)
2
−
1
16
V
(6)
( ˆ
φ)
x
G
3
(x, x, t)
.
(29)