String perturbation theory

String perturbation theory expands amplitudes in powers of the String coupling by summing over worldsheet topologies. It is reliable where the local effective coupling $e^{\Phi (X)}$ is small.

Tree-level scattering of four ordered open string tachyons gives the Veneziano amplitude. Its poles factorize through intermediate open-string states.

The ordered tree-level four-tachyon open string amplitude is proportional to $\Gamma (-1-{\alpha }^{\prime }s)\Gamma (-1-{\alpha }^{\prime }t)/\Gamma (-2-{\alpha }^{\prime }(s+t))$. Its Mandelstam variables locate the mass levels, and pole residues reveal spin content.

At high energy with $x=-t/s\in (0,1)$ fixed, reflection and Stirling formulas for the Gamma function give the displayed exponential envelope, away from real-axis resonance poles or after suitable analytic continuation. It is softer than the power-law fixed-angle behavior of typical point-particle interactions and exposes the finite string length. This limit differs from the fixed-$t$ Regge limit and does not by itself prove finiteness of every loop amplitude.

In units ${\alpha }^{\prime }=1$, generic fixed $t$ gives simple poles at $s=n-1$, $n\ge 0$, with residue $-{\prod }_{j=2}^{n+1}(t+j)/n!$. The empty product is one. A residue zero removes that pole, and fixed $t$ must be away from its own channel singularities.

For identical open-string tachyons, $s+t+u=-4/{\alpha }^{\prime }$. At ${\alpha }^{\prime }s=n-1$, the ordered Veneziano amplitude residues in the $s,t$ and $s,u$ terms differ by $(-1{)}^n$, so their symmetric sum cancels odd-level $s$-channel poles at generic $t$. In particular the massless vector does not couple to this identical-scalar singlet amplitude. It remains a state of the open-string spectrum and is visible in suitable ordered or color-dependent amplitudes. Pole absence due to an external-state selection rule must not be interpreted as absence of the exchanged state from the theory.

The pole at $s=0$ has residue $-(t+2)=-(t-u)/2$ for unit-slope tachyons. It is proportional to conserved scalar-pair vector currents and linear in the scattering-angle cosine, exhibiting a massless spin-one intermediate string state.