17.2 Off-Shell Superstring Amplitudes
349
Using the operator G (17.33), the propagator can be written generically as
= b
+
0 b
−
0
G
L 0 + ¯
L 0
δ(L
−
0 ).
(17.46)
Remark 17.5 (Propagators) NS and R states correspond, respectively, to bosonic
and fermionic fields: the operators L
+
0 and X 0 can be interpreted as the (massive)
Laplacian and Dirac operators, such that both propagators can be written as
NS ∼
1
k 2 + m 2 ,
, R ∼
i /
∂ + m
k 2 + m 2 .
(17.47)
To motivate the identification of X 0 with the Dirac operator, remember that X (z)
contains a term e φ(z) G(z) (this is the only term that contributes on-shell), where
G(z) in turn contains ψ μ ∂X μ . But, the zero-modes of ψ μ and ∂X μ correspond,
respectively, to the gamma matrix γ μ and momentum k μ when acting on a state.
The PCO zero-mode insertion inside the propagator has another virtue. It was
noted previously that states with N pic = −3/2 are infinitely degenerate since one
can apply β 0 an arbitrary number of times. These states have large negative ghost
numbers. Considering a loop amplitude, all these states would appear in the sum
over the states and lead to a divergence. The problem is present only for loops
because the ghost number is not fixed: in a tree propagator, the ghost number is
fixed and only a finite number of β 0 can be applied. But, the PCO insertion turns
these states into N pic = −1/2 states. In this picture number, one cannot create an
arbitrarily large negative ghost number since γ n
0 can only increase the ghost number.
17.2.3 Spurious Poles
A spurious pole corresponds to a singularity of the amplitude that cannot be
interpreted as the degeneration limit of Riemann surfaces. As a consequence, they
do not correspond to infrared divergences and do not have any physical meaning;
they must be avoided in order to define a consistent theory. To achieve this, the
section S g,m,n must be chosen such that it avoids all spurious poles. However, while
it is always possible to avoid these poles locally, it is not possible globally (this is
related to the results from [3]). Poles can be avoided using vertical integration: two
methods have been proposed, in the small (Sen–Witten) [28, 31] and large (Erler–
Konopka) [8] Hilbert spaces, respectively. Before describing the essence of both
approaches, we review the origin of spurious poles.
Origin
Spurious poles arise in three different ways:
• two PCOs collide;
349
Using the operator G (17.33), the propagator can be written generically as
= b
+
0 b
−
0
G
L 0 + ¯
L 0
δ(L
−
0 ).
(17.46)
Remark 17.5 (Propagators) NS and R states correspond, respectively, to bosonic
and fermionic fields: the operators L
+
0 and X 0 can be interpreted as the (massive)
Laplacian and Dirac operators, such that both propagators can be written as
NS ∼
1
k 2 + m 2 ,
, R ∼
i /
∂ + m
k 2 + m 2 .
(17.47)
To motivate the identification of X 0 with the Dirac operator, remember that X (z)
contains a term e φ(z) G(z) (this is the only term that contributes on-shell), where
G(z) in turn contains ψ μ ∂X μ . But, the zero-modes of ψ μ and ∂X μ correspond,
respectively, to the gamma matrix γ μ and momentum k μ when acting on a state.
The PCO zero-mode insertion inside the propagator has another virtue. It was
noted previously that states with N pic = −3/2 are infinitely degenerate since one
can apply β 0 an arbitrary number of times. These states have large negative ghost
numbers. Considering a loop amplitude, all these states would appear in the sum
over the states and lead to a divergence. The problem is present only for loops
because the ghost number is not fixed: in a tree propagator, the ghost number is
fixed and only a finite number of β 0 can be applied. But, the PCO insertion turns
these states into N pic = −1/2 states. In this picture number, one cannot create an
arbitrarily large negative ghost number since γ n
0 can only increase the ghost number.
17.2.3 Spurious Poles
A spurious pole corresponds to a singularity of the amplitude that cannot be
interpreted as the degeneration limit of Riemann surfaces. As a consequence, they
do not correspond to infrared divergences and do not have any physical meaning;
they must be avoided in order to define a consistent theory. To achieve this, the
section S g,m,n must be chosen such that it avoids all spurious poles. However, while
it is always possible to avoid these poles locally, it is not possible globally (this is
related to the results from [3]). Poles can be avoided using vertical integration: two
methods have been proposed, in the small (Sen–Witten) [28, 31] and large (Erler–
Konopka) [8] Hilbert spaces, respectively. Before describing the essence of both
approaches, we review the origin of spurious poles.
Origin
Spurious poles arise in three different ways:
• two PCOs collide;