 KLT orthogonality

ID: klt-orthogonality

KLT orthogonality states that vectors built from different solutions of the scattering equations are orthogonal under the field-theory KLT bilinear form. For generic massless kinematics, label the

(n - 3)!

solutions by

i

a_{1}, \dots, a_{n}

PT (a_{1}, \dots, a_{n}) = \frac{1}{σ _{a_{1} a_{2}} σ _{a_{2} a_{3}} \dots σ _{a_{n} a_{1}}}, σ_{ab} = σ_{a} - σ_{b} .

Let

V_{i} (α)

evaluate this factor at solution

i

in ordering

(1, α, n - 1, n)

and

U_{j} (β)

in ordering

(1, β, n, n - 1)

. If

S [α ∣ β]

is the KLT momentum kernel, then

\sum_{α, β} V_{i} (α) S [α ∣ β] U_{j} (β) = 0 (i \neq = j),

with nonzero diagonal pairings generically.^[ref]

The pairing is bilinear, without complex conjugation; it is not an ordinary positive Hermitian inner product. At four points there is only one solution, so the off-diagonal statement is vacuous. At five points two solutions yield two vanishing cross-pairings. The proof relates the pairing to a generalized scattering-equation Jacobian whose rank drops from

n - 3

n - 4

for distinct solutions, forcing the relevant minor to vanish. This eliminates mixed-solution terms when combining gauge-theory amplitudes into gravitational amplitudes.^[ref]

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