364
18 Momentum-Space SFT
The first property is responsible for the soft UV behaviour of string theory in
Euclidean signature, while the second prevents from performing the Wick rotation
(indeed, the pole at infinity implies that the arcs closing the contour contribute).
The g-loop n-point amputated Green functions are sums of Feynman diagrams,
each of the form:
F g,n (p 1 , . . . , p n ) ∼
dT
s
d
D s e
−G rs (T ) ) r · s −2H ri (T ) ) r ·p i −F ij (T ) p i ·p j
×
a
1
k 2
a + m 2
a
P(p i , , r ; T ),
(18.6)
where {p i } are the external momenta, { r } the loop momenta and {k i } the internal
momenta, with the latter given by a linear combination of the others. Moreover, T
denotes the dependence in the moduli parameters of all the internal vertices, and
P is a polynomial in (p i , , r ). The matrix G rs is positive definite, which implies
that:
• integrations over spatial loop momenta r converge;
• integrations over loop energies 0
r diverge.
As a consequence, the Feynman diagrams in Lorentzian signature are ill-defined:
we will explain in the next section how to fix this problem.
18.2 Generalized Wick Rotation
We have seen that loop integrals in Lorentzian signature are divergent because of the
large energy behaviour of the interactions. But, this is not different from the usual
QFT, where the loop integrals are also ill-defined in Lorentzian signature. Indeed,
poles of the propagators sit on the real axis and also give divergent loop integrals
(note that the same problem arises also here). In that case, the strategy is to define
the Feynman diagrams in Euclidean space and to perform a Wick rotation: the latter
matches the expressions in Lorentzian signature up to the iε-prescription. The goal
of the latter is to move slightly the poles away from the real axis.
Example 18.1: Scalar Field
Consider a scalar field of mass m with a quartic interaction. The 1-loop 4point Feynman diagram is given in Fig. 18.1. The external momenta are p i ,
i = 1, . . . , 4. There are one-loop momentum and two internal momenta k 1 =
and k 2 = p − , where p = p 1 + p 2 . The poles in the loop energy 0 are located
at
p ± = ±
2 + m 2 ,
q ± = p
0
±
(p − ) 2 + m 2 .
(18.7)
18 Momentum-Space SFT
The first property is responsible for the soft UV behaviour of string theory in
Euclidean signature, while the second prevents from performing the Wick rotation
(indeed, the pole at infinity implies that the arcs closing the contour contribute).
The g-loop n-point amputated Green functions are sums of Feynman diagrams,
each of the form:
F g,n (p 1 , . . . , p n ) ∼
dT
s
d
D s e
−G rs (T ) ) r · s −2H ri (T ) ) r ·p i −F ij (T ) p i ·p j
×
a
1
k 2
a + m 2
a
P(p i , , r ; T ),
(18.6)
where {p i } are the external momenta, { r } the loop momenta and {k i } the internal
momenta, with the latter given by a linear combination of the others. Moreover, T
denotes the dependence in the moduli parameters of all the internal vertices, and
P is a polynomial in (p i , , r ). The matrix G rs is positive definite, which implies
that:
• integrations over spatial loop momenta r converge;
• integrations over loop energies 0
r diverge.
As a consequence, the Feynman diagrams in Lorentzian signature are ill-defined:
we will explain in the next section how to fix this problem.
18.2 Generalized Wick Rotation
We have seen that loop integrals in Lorentzian signature are divergent because of the
large energy behaviour of the interactions. But, this is not different from the usual
QFT, where the loop integrals are also ill-defined in Lorentzian signature. Indeed,
poles of the propagators sit on the real axis and also give divergent loop integrals
(note that the same problem arises also here). In that case, the strategy is to define
the Feynman diagrams in Euclidean space and to perform a Wick rotation: the latter
matches the expressions in Lorentzian signature up to the iε-prescription. The goal
of the latter is to move slightly the poles away from the real axis.
Example 18.1: Scalar Field
Consider a scalar field of mass m with a quartic interaction. The 1-loop 4point Feynman diagram is given in Fig. 18.1. The external momenta are p i ,
i = 1, . . . , 4. There are one-loop momentum and two internal momenta k 1 =
and k 2 = p − , where p = p 1 + p 2 . The poles in the loop energy 0 are located
at
p ± = ±
2 + m 2 ,
q ± = p
0
±
(p − ) 2 + m 2 .
(18.7)