JHEP11(2018)094
formula can still be used to compute the discontinuity across such branch points; how-
ever in this case the on-shell states cannot collectively be regarded as an intermediate
state in the sum −i T
†
|nihn|T .
2. In computing −i T
†
T we need to reverse the signs of i in the propagators of the
amplitude to the right of the cut so that it represents a matrix element of T
†
. This
does not follow from Cutkosky’s formula for discontinuity.
While for any specific graph one can do a more detailed analysis taking into account all
these effects, the general proof of unitarity based on this approach becomes cumbersome.
These problems were overcome in [7, 8] where a different proof of unitarity was given based
on the ‘largest time equation’. An alternative proof was given in [9] based on the old
fashioned perturbation theory. However unlike Cutkosky’s original analysis, which did not
depend on the detailed structure of the vertices as long as they do not introduce additional
singularities at finite momentum, the analysis of [7–9] requires working in coordinate space
where at least for the time coordinate the propagator and vertices are expressed in the
position space instead of the momentum space. Unfortunately for string field theory,
for which the vertices are exponentials of quadratic functions of momenta, there is no
convenient representation of the vertices in the coordinate space. Therefore the analysis
of [7–9] do not apply.
This difficulty was overcome in [11] that expressed T − T
†
as a sum over cuts diagrams
by working directly in momentum space. Furthermore the part of the contribution to the
right of the cut was shown to be hermitian conjugated, representing a contribution to T
†
.
This method is well-suited for string field theory and other non-local theories, but also for
ordinary quantum field theories with local vertices, giving an alternative proof of unitarity.
In the next two sections we shall illustrate how this works for the box diagram and tree
diagrams in φ
4
theory.
3 Unitarity of the box diagram
Usually in φ
4
theory with interaction term −(λ/4!)
R
d
D
x φ
4
one takes the propagator of
momentum k to be −i/(k
2
+ m
2
− i) and the vertex to be −iλ/4!. Furthermore in the
computation of the T-matrix we have an overall factor of i. However, as in [11], we shall
use a slightly different but equivalent convention where for computation of the T-matrix
we use the following rules:
1. The propagator of momentum k is given by 1/(−k
2
− m
2
+ i).
2. The vertex is given by λ/4!.
3. For each loop integral we have a factor of i.
4. If the diagram has n
c
disconnected components then we have a factor of i
1−n
c
.
We shall drop the overall momentum conserving delta function (2π)
D
δ
(D)
(
P
i
p
i
), asso-
ciated with each connected component, from the expressions for the amplitudes. The
– 3 –
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