The $i\varepsilon$-Prescription for String Amplitudes and Regularized Modular Integrals

Abstract

We study integrals appearing in one-loop amplitudes in string theory, and in particular their analytic continuation based on a string theoretic analog of the $i\varepsilon$-prescription of quantum field theory. For various zero- and two-point one-loop amplitudes of both open and closed strings, we prove that this analytic continuation is equivalent to a regularization using generalized exponential integrals. Our approach provides exact expressions in terms of the degeneracies at each mass level. For one-loop amplitudes with boundaries, our result takes the form of a linear combination of three partition functions at different temperatures depending on a variable $T_0$, yet their sum is independent of this variable. The imaginary part of the amplitudes can be read off in closed form, while the real part is amenable to numerical evaluation. While the expressions are rather different, we demonstrate agreement of our approach with the contour put forward by Eberhardt-Mizera (2023) following the Hardy-Ramanujan-Rademacher circle method.

Figures (11)

• Figure 1: An example of a worldsheet with insertions in closed string theory. The long tube is parametrized by $\frak{t}_E$. For large $\frak{t}_E=T_0\gg 0$, the tube is Wick-rotated to Lorentzian signature.
• Figure 2: On the $\mathfrak{t}$-plane, the integration contour runs along the real axis to a large value $T_0$. The integration contour then continues in the purely imaginary direction $T_{0}+i\mathfrak{t}_L$ due to the transformation to Lorentzian signature.
• Figure 3: On the $\mathfrak{q}$-plane, the integration contour runs in the decreasing direction along the real axis up to a small value $\mathfrak{q}_{0}=e^{-T_{0}}$. Due to the transformation to Lorentzian signature, the integration contour continues as an infinite spiral around the singular point $\mathfrak{q}=0$ at fixed radius $\mathfrak{q}_{0}$. (The inward spiral is only to visualize that the contour encircles the origin infinitely many times.)
• Figure 4: The complexification of the upper-half-plane, $\mathbb{H}^{\mathbb{C}}\simeq\mathbb{H}\times\tilde{\mathbb{H}}$. The left half displays the upper-half-plane $\mathbb{H}$ parametrized by $\tau=x+iy$. The fundamental domain $\mathcal{F}$ is displayed in blue. The right half displays the lower-half-plane $\tilde{\mathbb{H}}$ and fundamental domain $\tilde{\mathcal{F}}$ parametrized by $\tilde{\tau}=\tilde{x}-i\tilde{y}$. The complex structure of $\tilde{ \mathbb{H}}$ and $\tilde{\mathcal{F}}$ is opposite to that of $\mathbb{H}$ and $\mathcal{F}$.
• Figure 5: Left panel: The domain $\mathcal{F}_1$ and the semi-infinite strip $S_1$ (red). Right panel: The semi-infinite strip $S_Y$ (red) illustrates the fundamental domain $R_{\infty}$ ranges up to $y=i\infty$ singularity. The compact rectangle $R_{Y}$ (yellow) and finite keyhole $\mathcal{F}_{1}$ (blue) regions together illustrates the cut-off fundamental domain $\mathcal{F}_{Y}$, on which the modular integral will be finite.