JHEP05(2018)003
x
6
x
3
x
1
x
5
x
2
x
4
p
1
p
6
p
2
p
5
p
3
p
4
x
1
x
4
x
6
x
9
x
10
x
5
x
2
x
8
x
3
x
7
p
10
p
4
p
9
p
5
p
1
p
8
p
2
p
7
p
3
p
6
Figure 4. Two-loop diagrams: double box topology in momentum variables (integration over
momenta flowing in loops) and its dual double cross topology in region momentum variables (in-
tegration over position of the vertices — filled blobs). (A): all inflowing momenta (depicted by
double lines) are off shell, p
2
i
= x
2
i i+1
6= 0 at i = 1, . . . , 6. (B): amplitude diagram for scattering of
massless particles, i.e. the inflowing momenta (depicted by loosely dotted gray lines) are light-like
p
2
i
= x
2
i i+1
= 0 at i = 1, . . . , 10. These constraints are imposed by means of delta functions δ(x
2
i i+1
)
depicted by dashed black lines.
Figure 5. In this picture, a bi-scalar scattering amplitude of massless scalars is represented in the
dual graph depicted by dotted lines. It has doubled (w.r.t. the previous figure 2) external legs at the
convex corners of the boundary, and no legs at the concave corners. The corresponding additional
momentum variables at convex corners should satisfy the momentum conservation condition at each
dual vertex. We obtain the admissible bi-scalar amplitude by identifying the points of the boundary
of the original graph which end in the same square (i.e., the same site of the original lattice. Such
points are surrounded by ellipses in the above figure. The masslessness is ensured by additional
factors δ(p
2
j
) multiplying the external legs of the dual graph.
– 6 –
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