Equal-mass sunset: vacuum and Euclidean continuation

integrate_sunset_taylor complements the multivariate Multivariate Ramanujan series for the D=2 sunset backend. It evaluates the unit-power, equal-mass D=2 sunset with a transformed Taylor series, a Clausen boundary value, and a differential-equation recurrence. It uses no quadrature. This is a separate analytic method; it does not generate new RMT Gamma families or silently change the method of an RMT call.

import mpmath as mp
from ramanujan_qft import ScalarEdge, ScalarGraph, integrate_sunset_taylor
g = ScalarGraph([ScalarEdge(0, 1, 1)]*3)
result = integrate_sunset_taylor(g, dps=30)
print(mp.nstr(result.value, 25))

2.343907238689458890601562

The graph must have exactly equal positive squared masses, three parallel edges, unit powers and D=2. Finite Euclidean external momenta are supported subject to max_terms and max_dps. Unequal masses are explicitly refused. compile_sunset_taylor(graph) returns an inspectable SunsetTaylorSeries; evaluate(dps=30, max_terms=512, max_dps=2000) returns SunsetTaylorResult. Unmet work/precision targets raise SunsetTaylorConvergenceError, with a partial result when available. The RMT API retains its own strict domains.

Derivation and normalization

Use the positive Euclidean loop measure \(d^2k\,d^2\ell/\pi^2\). For equal masses, write \(s=p_E^2/m^2\) and \(I=m^{-2}f(-s)\). On the unit simplex,

\[f(-s)=\int_\Delta\frac{dx\,dy}{U+sxyz},\qquad U=xy+xz+yz,\quad z=1-x-y.\]

Since \(U/(xyz)=1/x+1/y+1/z\ge9\), set \(v=xyz/U\in[0,1/9]\), \(r=s/(9+s)\in[0,1)\), and obtain

\[f(-s)=(1-r)\sum_{n=0}^\infty b_n r^n,\qquad b_n=\int_\Delta\frac{(1-9v)^n}{U}\,dx\,dy.\]

The coefficients obey \(0\le b_n\le b_0\) and decrease with \(n\). The integrable dominating function \(1/U\) proves the interchange and convergence for every finite \(s\ge0\). After K coefficients, the tail obeys

\[0\le I-I_K\le \frac{b_0}{m^2}r^K.\]

The known boundary value is \(b_0=4\operatorname{Cl}_2(\pi/3)/\sqrt3\). The equal-mass Picard-Fuchs equation in \(t=-s\) is

\[t(t-1)(t-9)f''+(3t^2-20t+9)f'+(t-3)f=-6.\]

The equation and normalization follow Müller-Stach, Weinzierl and Zayadeh; the boundary value and series methods are treated by Laporta and Remiddi. Their D=2 loop measure gives a value smaller by a factor of 16; multiplying their \(\sqrt3\operatorname{Cl}_2(\pi/3)/12\) by 16 gives our \(b_0\). Substitute \(t=-9r/(1-r)\) and \(f=(1-r)G(r)\). The transformed equation is

\[r(1-r)(8r+1)G''+(1+14r-24r^2)G'+(2-8r)G=\frac6{1-r}.\]

Matching coefficients gives, with \(b_{-1}=0\),

\[b_{n+1}=\frac{6-(7n^2+7n+2)b_n+8n^2b_{n-1}}{(n+1)^2}.\]

This recurrence computes the integral moments above, starting from the Clausen boundary value. The apparent singularity at \(r=-1/8\) in the ODE does not limit this particular solution; the integral argument establishes its power series on \(|r|<1\).

Arithmetic stability and validation

Forward recurrence excites an unwanted solution growing approximately as \((-8)^n\). Requested decimal precision alone is insufficient. The evaluator allocates at least one guard digit per coefficient, propagates an arithmetic error estimate through the recurrence, and compares two working precisions. The analytic tail and final rounding are included in error_estimate. This remains a numerical diagnostic, not an interval-arithmetic certificate.

The term count follows the mathematical lower bound \(I\ge b_0(1-r)/m^2\) and does not depend on potentially corrupted partial sums. Large Euclidean momentum makes \(r\) close to one and may exhaust the budget; the high-momentum RMT chart is then usually more practical. There is no automatic fallback.

Tests cover the vacuum Clausen value, independent one-dimensional parameter quadrature after analytic elimination of one parameter, exact common scaling, the RMT boundary \(p_E=3m\), overlap with the RMT chart at \(p_E=6m\), partial-tail bounds, precision exhaustion, and forbidden numerical fallbacks.