3.6 Supersymmetry in Three-Dimensional Space-Time and Noncommutativity
41
Fig. 3.4 The simplest
tadpole (super)graph
[x
m
, x
n
] = i
mn
,
(3.152)
where
mn is a constant matrix, can essentially improve the renormalization properties of the field theories. The most adequate formulation allowing to implement
these relations within the framework of the quantum field theory [51] is based on the
replacement of the usual product of the fields by their Moyal product defined as
φ 1 (x) ∗ . . . φ N (x) =
=
N
l=1
d
d k i
(2π) d (2π)
d e
i(k 1 +...+k N )x ˜
φ 1 (k 1 ) . . . ˜
φ N (k N ) exp(i
i< j≤N
k i ∧ k j ),
(3.153)
where k i ∧ k j =
mn k im k jn .
However, it turns to be that the famous UV/IR mixing effect [30] implies in the
partial conversion of the ultraviolet divergences to the infrared singularities. Indeed,
if one considers the d-dimensional nonsupersymmetric noncommutative φ
4 model,
the following simplest tadpole contribution to the two-point function of the φ field,
given by Fig. 3.4, will arise.
The contribution of this graph in d space-time dimensions is (see e.g. [52, 53])
S 2 ( p) =
λ
6
d
d k
(2π) d
2 + cos(2k ∧ p)
k 2 + m 2
φ(− p)φ( p) = −
λ
3
[
− d/2)
(4π) d/2 (m 2 ) 1−d/2 +
+
1
2(2π) d/2 (m
2
)
d/2−1 K d/2−1 (
m 2 ˜
p 2 )
(
m 2 ˜
p 2 ) d/2−1
],
(3.154)
where ˜
p
m
=
mn p n . We see that the first term of the expression (3.154) is similar to
the common UV divergent term different from that one arising in the commutative
case only by an overall numerical coefficient (this difference, from a formal viewpoint, can be explained by the fact that, when the noncommutativity is introduced,
some of contractions of the fields continue to be planar ones, i.e. do not acquire
the phase factor), while the second one is singular, and, since K n (x)| x→0 ∝
1
x n , we
arrive at the infrared singularity of the order d − 2 in the external momentum p m ,
in particular, at d = 3 we obtain the linear singularity, and at d = 4—the quadratic
one. A remarkable fact is that this singularity arises in the massive case as well as in
the massless one where the typical momentum integral looks like
41
Fig. 3.4 The simplest
tadpole (super)graph
[x
m
, x
n
] = i
mn
,
(3.152)
where
mn is a constant matrix, can essentially improve the renormalization properties of the field theories. The most adequate formulation allowing to implement
these relations within the framework of the quantum field theory [51] is based on the
replacement of the usual product of the fields by their Moyal product defined as
φ 1 (x) ∗ . . . φ N (x) =
=
N
l=1
d
d k i
(2π) d (2π)
d e
i(k 1 +...+k N )x ˜
φ 1 (k 1 ) . . . ˜
φ N (k N ) exp(i
i< j≤N
k i ∧ k j ),
(3.153)
where k i ∧ k j =
mn k im k jn .
However, it turns to be that the famous UV/IR mixing effect [30] implies in the
partial conversion of the ultraviolet divergences to the infrared singularities. Indeed,
if one considers the d-dimensional nonsupersymmetric noncommutative φ
4 model,
the following simplest tadpole contribution to the two-point function of the φ field,
given by Fig. 3.4, will arise.
The contribution of this graph in d space-time dimensions is (see e.g. [52, 53])
S 2 ( p) =
λ
6
d
d k
(2π) d
2 + cos(2k ∧ p)
k 2 + m 2
φ(− p)φ( p) = −
λ
3
[
− d/2)
(4π) d/2 (m 2 ) 1−d/2 +
+
1
2(2π) d/2 (m
2
)
d/2−1 K d/2−1 (
m 2 ˜
p 2 )
(
m 2 ˜
p 2 ) d/2−1
],
(3.154)
where ˜
p
m
=
mn p n . We see that the first term of the expression (3.154) is similar to
the common UV divergent term different from that one arising in the commutative
case only by an overall numerical coefficient (this difference, from a formal viewpoint, can be explained by the fact that, when the noncommutativity is introduced,
some of contractions of the fields continue to be planar ones, i.e. do not acquire
the phase factor), while the second one is singular, and, since K n (x)| x→0 ∝
1
x n , we
arrive at the infrared singularity of the order d − 2 in the external momentum p m ,
in particular, at d = 3 we obtain the linear singularity, and at d = 4—the quadratic
one. A remarkable fact is that this singularity arises in the massive case as well as in
the massless one where the typical momentum integral looks like