Abstract:We compute the three-loop scattering amplitude of four gravitons in N = 8 supergravity. Our results are analytic formulae for a Laurent expansion of the amplitude in the regulator of dimensional regularisation. The coefficients of this series are closed formulae in terms of well-established harmonic poly-logarithms. Our results display a remarkable degree of simplicity and represent an important stepping stone in the exploration of the structure of scattering amplitudes. In particular, we observe that to this … Show more
“…a quantity that diverges as 2 ǫ for small ǫ. This expression is valid for general values of ǫ up to the subleading level in the Regge limit and we checked that in this regime it reproduces the data of [52] where the one-loop result is written explicitly up to O(ǫ 4 ).…”
Section: Exponentiation At the First Subleading Eikonalmentioning
confidence: 65%
“…At this order the amplitudes depend on the details of the theory and we focused on the case of N = 8 supergravity. 16 By using the explicit results of [52] we compared the two exponentiations at all loops for the first four terms in the ǫ → 0 expansion. As discussed in section 5, there is an impressive agreement between the eikonal prediction (2.15) and the explicit results of [52] that satisfy perfectly the IR exponentation in momentum space (2.4).…”
Section: Discussionmentioning
confidence: 99%
“…16 By using the explicit results of [52] we compared the two exponentiations at all loops for the first four terms in the ǫ → 0 expansion. As discussed in section 5, there is an impressive agreement between the eikonal prediction (2.15) and the explicit results of [52] that satisfy perfectly the IR exponentation in momentum space (2.4). However there is a mismatch for one term appearing at the lowest power of 1/ǫ and the lowest power of log(q 2 ) accessible with the current data.…”
High-energy massless gravitational scattering in N = 8 supergravity was recently analyzed at leading level in the deflection angle, uncovering an interesting connection between exponentiation of infrared divergences in momentum space and the eikonal exponentiation in impact parameter space. Here we extend that analysis to the first non trivial sub-leading level in the deflection angle which, for massless external particles, implies going to two loops, i.e. to third post-Minkowskian (3PM) order. As in the case of the leading eikonal, we see that the factorisation of the momentum space amplitude into the exponential of the one-loop result times a finite remainder hides some basic simplicity of the impact parameter formulation. For the conservative part of the process, the explicit outcome is infrared (IR) finite, shows no logarithmic enhancement, and agrees with an old claim in pure Einstein gravity, while the dissipative part is IR divergent and should be regularized, as usual, by including soft gravitational bremsstrahlung. Finally, using recent threeloop results, we test the expectation that eikonal formulation accounts for the exponentiation of the lower-loop results in the momentum space amplitude. This passes a number of highly non-trivial tests, but appears to fail for the dissipative part of the process at all loop orders and sufficiently subleading order in ǫ, hinting at some lack of commutativity of the relevant infrared limits for each exponentiation.
“…a quantity that diverges as 2 ǫ for small ǫ. This expression is valid for general values of ǫ up to the subleading level in the Regge limit and we checked that in this regime it reproduces the data of [52] where the one-loop result is written explicitly up to O(ǫ 4 ).…”
Section: Exponentiation At the First Subleading Eikonalmentioning
confidence: 65%
“…At this order the amplitudes depend on the details of the theory and we focused on the case of N = 8 supergravity. 16 By using the explicit results of [52] we compared the two exponentiations at all loops for the first four terms in the ǫ → 0 expansion. As discussed in section 5, there is an impressive agreement between the eikonal prediction (2.15) and the explicit results of [52] that satisfy perfectly the IR exponentation in momentum space (2.4).…”
Section: Discussionmentioning
confidence: 99%
“…16 By using the explicit results of [52] we compared the two exponentiations at all loops for the first four terms in the ǫ → 0 expansion. As discussed in section 5, there is an impressive agreement between the eikonal prediction (2.15) and the explicit results of [52] that satisfy perfectly the IR exponentation in momentum space (2.4). However there is a mismatch for one term appearing at the lowest power of 1/ǫ and the lowest power of log(q 2 ) accessible with the current data.…”
High-energy massless gravitational scattering in N = 8 supergravity was recently analyzed at leading level in the deflection angle, uncovering an interesting connection between exponentiation of infrared divergences in momentum space and the eikonal exponentiation in impact parameter space. Here we extend that analysis to the first non trivial sub-leading level in the deflection angle which, for massless external particles, implies going to two loops, i.e. to third post-Minkowskian (3PM) order. As in the case of the leading eikonal, we see that the factorisation of the momentum space amplitude into the exponential of the one-loop result times a finite remainder hides some basic simplicity of the impact parameter formulation. For the conservative part of the process, the explicit outcome is infrared (IR) finite, shows no logarithmic enhancement, and agrees with an old claim in pure Einstein gravity, while the dissipative part is IR divergent and should be regularized, as usual, by including soft gravitational bremsstrahlung. Finally, using recent threeloop results, we test the expectation that eikonal formulation accounts for the exponentiation of the lower-loop results in the momentum space amplitude. This passes a number of highly non-trivial tests, but appears to fail for the dissipative part of the process at all loop orders and sufficiently subleading order in ǫ, hinting at some lack of commutativity of the relevant infrared limits for each exponentiation.
“…Analytic continuation into another scattering region may be performed for example by following the steps discussed in ref. [57,58]. Similarly, permutations of external legs of our master integrals can be obtained using the methods detailed in refs.…”
Section: Classification Of the D Log Integrals According To Their Infmentioning
confidence: 99%
“…Similarly, permutations of external legs of our master integrals can be obtained using the methods detailed in refs. [57,58]. We checked that the permutations of our master integrals satisfy the permuted systems of differential equations.…”
Section: Classification Of the D Log Integrals According To Their Infmentioning
We compute all master integrals for massless three-loop four-particle scattering amplitudes required for processes like di-jet or di-photon production at the LHC. We present our result in terms of a Laurent expansion of the integrals in the dimensional regulator up to 8 th power, with coefficients expressed in terms of harmonic polylogarithms. As a basis of master integrals we choose integrals with integrands that only have logarithmic poles -called d log forms. This choice greatly facilitates the subsequent computation via the method of differential equations. We detail how this basis is obtained via an improved algorithm originally developed by one of the authors. We provide a public implementation of this algorithm. We explain how the algorithm is naturally applied in the context of unitarity. In addition, we classify our d log forms according to their soft and collinear properties.
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