Draw a tree graph, pick a root, and read off the series for the corresponding fully nested massive correlator, computed with the spectral gluing algorithm of arXiv:2607.18223. An optional second step assembles the full correlator, with all on-shell terms and every Schwinger–Keldysh component.
1. Draw the graph
Every dot is a bulk vertex (with external energy Xi and twist pi), every line a massive exchange (with internal energy Yij and mass parameter μij). Start from an example or build your own, then pick the root: the vertex whose energy is the largest one in the kinematic region you are interested in.
2. The formula
Add at least one vertex.
Here mi is the summation index attached to the line joining vertex i to its parent (the root has none). The notation used above is collected here:
The series converges for , and each leaf satisfies Y ≤ X.
How the tool got there
Orienting every line away from the root gives the incidence matrix Q (entry −1 where a line leaves a vertex, +1 where it arrives). Deleting the root row gives the reduced matrix Qr, and the on-shell condition Qrx = −(p̃(r) + m) is solved by the path matrix Pr = Qr−1, whose columns record the path from the root to each other vertex.
The on-shell values of the spectral variables are x* = −Pr(p̃(r) + m):
3. The full correlator optional
For the full correlator every vertex sits on one of the two branches of the in-in contour, which gives 2V colourings of the graph (complex-conjugate pairs are combined). A line between two vertices of the same colour carries a time-ordered propagator. Its spectral density splits into an on-shell part, which cuts the line, and an analytic part, which is glued with the algorithm above. Lines between different colours are already factorised. The tool writes out every resulting term.
Which end of a cut line carries the Boltzmann weights e±πμ, and which vertex roots each analytic piece, depends on how the energies Xi compare. Order the vertices by energy, largest first. By default the order follows the tree away from the root chosen above.
Energy ordering
Preparing terms…
The series converge in the region . In addition, the Lauricella functions need ∑j uij < 1 at every vertex.
What is computed
The first formula is the maximally-nested analytic contribution to the all-plus (fully time-ordered) master integral, obtained in Sec. 4.2 of the paper by setting the spectral parameters of the principal-value part on shell. The optional full correlator adds the on-shell terms of every time-ordered line and all Schwinger–Keldysh components. Both assume generic twists and masses; for special values of p̃i some lower parameters of the Lauricella functions can hit non-positive integers and the series has to be treated by a limit.
Choosing a different root, or a different energy ordering, gives the expansion in a different kinematic region, so it is worth trying several on the same graph.
Read the paper: All Tree-Level Massive Cosmological Correlators via Spectral Gluing