TY - JOUR
T1 - The soft-collinear bootstrap
T2 - N = 4 Yang-Mills amplitudes at six- and seven-loops
AU - Bourjaily, J. L.
AU - DiRe, A.
AU - Shaikh, A.
AU - Spradlin, M.
AU - Volovich, A.
PY - 2012
Y1 - 2012
N2 - Infrared divergences in scattering amplitudes arise when a loop momentum ℓ becomes collinear with a massless external momentum p. In gauge theories, it is known that the L-loop logarithm of a planar amplitude has much softer infrared singularities than the L-loop amplitude itself. We argue that planar amplitudes in N = 4 super-Yang-Mills theory enjoy softer than expected behavior as ℓ ∥ p already at the level of the integrand. Moreover, we conjecture that the four-point integrand can be uniquely determined, to any looporder, by imposing the correct soft-behavior of the logarithm together with dual conformal invariance and dihedral symmetry. We use these simple criteria to determine explicit formulae for the four-point integrand through seven-loops, finding perfect agreement with previously known results through five-loops. As an input to this calculation, we enumerate all four-point dual conformally invariant (DCI) integrands through seven-loops, an analysis which is aided by several graph-theoretic theorems we prove about general DCI integrands at arbitrary loop-order. The six- and seven-loop amplitudes receive non-zero contributions from 229 and 1873 individual DCI diagrams respectively. PDF and Mathematica files with all of our results are provided at http://goo.gl/qIKe8.
AB - Infrared divergences in scattering amplitudes arise when a loop momentum ℓ becomes collinear with a massless external momentum p. In gauge theories, it is known that the L-loop logarithm of a planar amplitude has much softer infrared singularities than the L-loop amplitude itself. We argue that planar amplitudes in N = 4 super-Yang-Mills theory enjoy softer than expected behavior as ℓ ∥ p already at the level of the integrand. Moreover, we conjecture that the four-point integrand can be uniquely determined, to any looporder, by imposing the correct soft-behavior of the logarithm together with dual conformal invariance and dihedral symmetry. We use these simple criteria to determine explicit formulae for the four-point integrand through seven-loops, finding perfect agreement with previously known results through five-loops. As an input to this calculation, we enumerate all four-point dual conformally invariant (DCI) integrands through seven-loops, an analysis which is aided by several graph-theoretic theorems we prove about general DCI integrands at arbitrary loop-order. The six- and seven-loop amplitudes receive non-zero contributions from 229 and 1873 individual DCI diagrams respectively. PDF and Mathematica files with all of our results are provided at http://goo.gl/qIKe8.
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U2 - 10.1007/JHEP03(2012)032
DO - 10.1007/JHEP03(2012)032
M3 - Article
AN - SCOPUS:84859616967
SN - 1126-6708
VL - 2012
JO - Journal of High Energy Physics
JF - Journal of High Energy Physics
IS - 3
M1 - 032
ER -