Datasets and notebooks for "Precision asymptotics of string amplitudes"

Data and computational notebooks (Mathematica) for the preprint ``Precision asymptotics of string amplitudes’’ by M. M. Baccianti, L. Eberhardt and S. Mizera. The submission contains numerical data of the one-loop four-point scattering of gravitons in type II string theory.

The respository contains the following notebooks and data:

For X \in \{ImA, ImDResA, ReDResA\}, the file code/X.nb produces the data stored in data/X that computes the one-loop four-graviton string amplitude in type II string theory as described in the paper. The data data/ReA was computed using the C++ program StringQMC available as an ancillary file in https://arxiv.org/abs/2507.22105.

Let us describe the data formats:

- data/ImA contains files of the format poq_s0-smax.txt where smax indicate the maximum value of smax and p/q is the fixed ratio -t/s. As described in the paper, it is convenient for the numerical analysis to have data for rational ratios. These files contain a list whose first entry is the value of s and the second entry is the value of the imaginary part of the amplitude.

- data/ImDResA contains files of the format simn.m where n is a number giving the value of s. They contain the decay width of the s-th mass level, which is a polynomial in t of degree 2s-2. Each decay width is further subdivided into the contributions from different internal mass-levels mD and mU. For mD<mU, the contribution is doubled because it equals the contribution with mD and mU exchanged. Each entry in the list gives the value of mD, mU and the corresponding contribution to the decay width. To get the total decay width, one can call Total[list[[All,3]]], where list is the list in the file.

- data/ReA contains the output of the program StringQMC, where the file names follow a similar naming scheme as for the files in ImA, except that it also specifies the range of c that is employed for the Rademacher procedure. The output is described in the files; it consists of a list of the form {s, theta, c, Re[AClosed[s,theta,c]], Im[AClosed[s,theta,c]], Re[AClosedError[s,theta,c]], Im[AClosedError[s,theta,c]]}.

- data/ReDResA contains files of the form sren.m where n is a number giving the value of s. These files contain the data of the massshifts (both real and imaginary parts) as computed by the Rademacher procedure. The cutoff used for the Rademacher procedure is uniformly cmax=10. The entries in the file are of the form {c,DRes[A[s,theta,c]]]}, as a polynomial in t. The precision used for the computation is 100+5s. Such a high precision is necessary since there are a lot of numerical cancellations in the computation.

The folder toy integral contains the notebook and integral necessary for the toy example that is described in Section 3 of the paper, in particular to reproduce the plot of figure 5 of the paper.

Finally, the folder multiplicity matching contains a notebook that extracts the multiplicities from the numerical data. This notebook also contains the implementation of the saddle point evaluation of the amplitude around a given saddle to subleading order, as described in Section 2.6 of the paper.