String coupling ($g_s$)

The string coupling controls topology change and string splitting and joining. A constant background dilaton ${\Phi }_0$ sets $g_s=e^{{\Phi }_0}$.

For a connected oriented closed-string worldsheet of genus $g$ with $n$ external closed-string vertices, the perturbative amplitude carries $g_s^{2g-2+n}$. The exponent combines the dilaton factor $g_s^{-\chi }$ for Euler characteristic $\chi =2-2g$ with one $g_s$ from each normalized vertex.

Connected oriented closed string worldsheets of genus $g$ give the spacetime effective-action term $g_s^{2g-2}{S}_g$. Each $S_g$ has a separate low-energy derivative expansion in ${\alpha }^{\prime }$. The string coupling orders quantum string loops, while ${\alpha }^{\prime }$ orders effects of finite string size.

A constant dilaton contributes ${\Phi }_0\chi (\Sigma )$ to the Euclidean worldsheet action by the Gauss-Bonnet theorem. With string coupling $g_s=e^{{\Phi }_0}$ and Euler characteristic $\chi =2-2g$, the path-integral weight is $g_s^{2g-2}$. Canonically normalized external closed-string vertices add one power of $g_s$ each.

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.

The dilaton is a scalar closed-string background field coupled to the worldsheet Ricci scalar. Its constant mode determines the string coupling, while spacetime variation modifies the background-field equations.

In two worldsheet dimensions, ${\delta }_{\omega }(\sqrt{g}R)=-2\sqrt{g}{\Delta }_g\omega$. Thus the standard dilaton curvature coupling does not have classical Weyl invariance for a general spacetime-varying dilaton. A constant dilaton gives the topological Euler characteristic instead. In the quantum sigma model, this classical variation combines with the loop anomaly; vanishing improved sigma-model beta functions is the full background conformal condition.

With Euclidean curvature coupling $(4\pi {)}^{-1}\int \sqrt{h}\phi R$, a constant dilaton contributes ${\phi }_0\chi$ by the Gauss-Bonnet theorem. The genus-$g$ closed-string weight is $e^{-{\phi }_0\chi }=g_s^{2g-2}$ with $g_s=e^{{\phi }_0}$. If the coefficient is instead written as $\int \sqrt{h}\Phi R$, then in the same convention $\phi =4\pi \Phi$. The field normalization must be fixed before reading off the coupling.

With worldsheet signature $(-,+)$, $e^{i{S}_L}=e^{-S_E}$, and the curvature convention $R[e^{2\omega }h]=e^{-2\omega }(R[h]-2{\square }_h\omega )$, the usual negative Lorentzian kinetic action becomes positive in Euclidean signature. A positive Lorentzian term $(4\pi {)}^{-1}\int \sqrt{-h}R\Phi$ becomes a negative Euclidean curvature coupling. Therefore the conventional Euclidean dilaton with positive curvature coupling is $\phi =-\Phi$. The leading metric-dilaton Weyl condition is $R_{ab}+2{\nabla }_a{\nabla }_b\phi =0$, or $R_{ab}-2{\nabla }_a{\nabla }_b\Phi =0$ with that literal positive Lorentzian coupling. Keeping the same named field and reversing the action's curvature-coupling sign reverses the Hessian term. The continuation follows directly from $d\tau =-id{\tau }_E$ and $S_E=-i{S}_L$; the geometric scalar curvature continues without an extra arbitrary sign.