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The present study illuminates the relation between null cusped Wilson loopsand their corresponding amplitudes. We find that, compared to the case with noself-crossing, the one loop expectation value of a self-intersecting Wilsonloop develops an additional 1/\epsilon singularity associated to theintersection. Interestingly, the same 1/\epsilon pole exists in the finite partof the one loop amplitude, appearing in the BDS conjecture, at thecorresponding kinematic limit. At two loops, we explore the behaviour of theremainder function R, encoding the deviation of the amplitude from the BDSconjecture. By analysing the renormalisation group equations for the Wilsonloop with a simple self-crossing, we argue that, when approaching theconfiguration with a self-crossing (u_2 \to 1, u_1\approx u_3), R diverges inthe imaginary direction like R ~ i \pi \log^3(1-u_2). This behaviour can beattributed to the non-trivial analytic continuation needed when passing fromthe Euclidean to the physical region and suggests that R has a branch cut inthe negative u_2 axis when the two other cross ratios are approximately equal(u_1 \approx u_3).

Null Wilson loops with a self-crossing and the Wilson loop/amplitude conjecture