One-loop cut construction

7.1 One-loop cut construction

The maximal $N = 8$ supergravity amplitudes can be obtained by applying the KLT equations to express them in terms of maximally supersymmetric $N = 4$ gauge theory amplitudes. For $N = 8$ supergravity, each of the states of the multiplet factorizes into a tensor product of $N = 4$ super-Yang–Mills states, as illustrated in Eq. (9

). Applying the KLT equation (10

) to the product of tree amplitudes appearing in the $s12$ channel two-particle cuts yields:

∑ M tree(− L ,1,2,L ) × M tree(− L ,3,4,L ) N= 8 4 1 3 4 3 1 states ∑ = − s2 Atree(− L1, 1,2,L3) × Atree(− L3, 3,4,L1) 12N =4 4 4 ∑ states × Atr4ee(L3,1,2,− L1 ) × Atr4ee(L1,3,4,− L3 ), (54 ) N=4 states

where the sum on the left-hand side runs over all 256 states in the $N = 8$ supergravity multiplet. On the right-hand side the two sums run over the 16 states (ignoring color degrees of freedom) of the $N = 4$ super-Yang–Mills multiplet: a gluon, four Weyl fermions and six real scalars.

The $N = 4$ super-Yang–Mills tree amplitudes turn out to have a particularly simple sewing formula [29 ],

∑ Atr4ee(− L1, 1,2,L3) × Atr4ee(− L3, 3,4,L1) N=4 states = − is s Atree(1, 2,3,4)-----1---- -----1----, (55 ) 12 23 4 (L1 − k1 )2 (L3 − k3)2

which holds in any dimension (though some care is required to maintain the total number of physical states at their four-dimensional values so as to preserve the supersymmetric cancellations). The simplicity of this result is due to the high degree of supersymmetry.

Using the gauge theory result (55 ), it is a simple matter to evaluate Eq. (54 ). This yields:

∑ M tree(− L ,1,2,L ) × M tree(− L ,3,4,L ) N =8 4 1 3 4 3 1 states [ ] = is s s M tree(1, 2,3,4) ----1-----+ ----1----- 12 23 13 4 (L1 − k1)2 (L1 − k2)2 [ ] -----1---- ----1----- × (L − k )2 + (L − k )2 . (56 ) 3 3 3 4

The sewing equations for the $s23$ and $s13$ kinematic channels are similar to that of the $s12$ channel.

Applying Eq. (56 ) at one loop to each of the three kinematic channels yields the one-loop four graviton amplitude of $N = 8$ supergravity,

1loop ( κ)4 ℳ 4 (1,2,3,4) = − i -- s12s23s13 M t4ree(1,2,3,4) ( 2 ) × ℐ14 loop(s12,s23) + ℐ14loop(s12,s13) + ℐ14loop(s23,s13), (57 )

in agreement with previous results [69

]. The gravitational coupling $κ$ has been reinserted into this expression. The scalar integrals are defined in Eq. (51

), inserting $𝒫 = 1$ . This is a standard integral appearing in massless field theories; the explicit value of this integral may be found in many articles, including Refs. [69 , 27 ]. This result actually holds for any of the states of $N = 8$ supergravity, not just external gravitons. It is also completely equivalent to the result one obtains with covariant Feynman diagrams including Fadeev–Popov [59 ] ghosts and using regularization by dimensional reduction [122 ]. The simplicity of this result is due to the high degree of supersymmetry. A generic one-loop four-point gravity amplitude can have up to eight powers of loop momenta in the numerator of the integrand; the supersymmetry cancellations have reduced it to no powers.