Koba-Nielsen factor

The Koba-Nielsen factor is the product of powers of pairwise worldsheet insertion-point distances produced by contractions of string vertex operators.

Under $z\mapsto (az+b)/(cz+d)$ with $ad-bc=1$, the coordinate measure contributes $|c{z}_i+d{|}^{-4}$ at insertion $i$. The Koba-Nielsen factor contributes $|c{z}_i+d{|}^{-{\alpha }^{\prime }{\sum }_{j\ne i}{p}_i\cdot p_j}$. Four-momentum conservation makes this exponent ${\alpha }^{\prime }{p}_i^2$, so the factors cancel precisely on the displayed tachyon mass shell. The correlator alone is covariant; its product with all the coordinate measures is invariant.

The Virasoro–Shapiro amplitude is the crossing-symmetric tree-level four-point amplitude of closed-string tachyons. Its worldsheet integral is a complex beta integral, and its poles display the closed-string spectrum.

For $t\simeq -as$, $u\simeq -(1-a)s$ with $0<a<1$, the smooth high-energy envelope satisfies $\log |A|\simeq -2s[-a\log a-(1-a)\log (1-a)]$. The Gamma function asymptotics cancel the leading $s\log s$ terms, leaving exponential suppression. Exact real-axis resonance poles must be excluded or treated by analytic continuation.

For $a=-1-{\alpha }^{\prime }s/4$, $b=-1-{\alpha }^{\prime }t/4$ and $a+b+c=1$, the amplitude has generic simple poles at $a=-n$. The gamma function residue and recurrence giveThe residue polynomial has degree $2n$, consistent with closed-string exchange up to spin $2n$. When $a=-n+{\delta }_a$ and $b=-m+{\delta }_b$, the reciprocal factor $1/\Gamma (a+b)$ removes a putative double pole: the leading ratio is ${\left(\genfrac{}{}{0ex}{}{n+m}{n}\right)}^2(1/{\delta }_a+1/{\delta }_b)$. Thus channel singularities must be counted using the kinematic constraint, not by multiplying independent numerator poles.

The Cachazo-He-Yuan formalism expresses tree-level scattering amplitudes as integrals localized on scattering equations.

In the CHY formalism, the scattering equations relate massless external four-momenta $p_i$ to distinct marked points $z_i$ on the Riemann sphere:All momenta are taken outgoing; incoming particles have the opposite signed four-momentum. The dot product uses the spacetime metric. ^[ref]

Pairing terms with interchanged indices and using momentum conservation givesFor generic kinematic data, only $n-3$ equations are independent. Under a Möbius transformation $z\mapsto (az+b)/(cz+d)$ with $ad-bc=1$, nullness and momentum conservation give $E_i\mapsto (c{z}_i+d{)}^2{E}_i$. Thus three marked points can be fixed by this projective symmetry.

The equations also arise by making the Koba-Nielsen factor's logarithm stationary in the massless high-energy limit. The massless conditions are essential to the displayed projective-covariance argument; massive generalizations require a modified system.

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 $i$. Define the Parke-Taylor factor for a cyclic ordering $a_1,\dots ,a_n$ byLet $V_i(\alpha )$ evaluate this factor at solution $i$ in ordering $(1,\alpha ,n-1,n)$ and $U_j(\beta )$ in ordering $(1,\beta ,n,n-1)$. If $S[\alpha |\beta ]$ is the KLT momentum kernel, thenwith 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 $n-3$ to $n-4$ for distinct solutions, forcing the relevant minor to vanish. This eliminates mixed-solution terms when combining gauge-theory amplitudes into gravitational amplitudes.^[ref]