Multivariate Ramanujan series for the D=2 sunset#

This backend reaches the first graph-to-series milestone for a massive, unit-power, two-loop sunset in D=2. It generates exact Schwinger bracket matrices and Gamma families from the graph, resolves coalescing poles analytically, selects a justified convergence region, and sums the finite multivariate series at arbitrary precision. Evaluation performs no quadrature, QMC, numerical regulator extrapolation, or fallback to the sector backend.

The symbolic compiler is reusable for other graphs. Numerical family selection, pole resolution, and convergence proofs are currently specific to this sunset.

Use it#

The base package dependencies suffice; SciPy is needed only for optional comparisons with the Massive Euclidean multiloop graphs numerical backend.

import mpmath as mp
from ramanujan_qft import ScalarEdge, ScalarGraph, compile_sunset_rmt

graph = ScalarGraph([ScalarEdge(0, 1, 1) for _ in range(3)],
                    external_momenta={0: [10], 1: [-10]})
series = compile_sunset_rmt(graph)
result = series.evaluate(dps=30)
print(series.chart)
print(mp.nstr(result.value, 25))
print(result.converged)

momentum
0.6290167075906141495908273
True

The invariant is \(p_E^2=100\), not 10. The normalization is the positive Euclidean integral

\[ I=\int\frac{d^2k\,d^2\ell}{\pi^2} \frac{1}{(k^2+m_1^2)(\ell^2+m_2^2)((p-k-\ell)^2+m_3^2)}. \]

integrate_graph_rmt(graph, dps=30) is the compile-and-evaluate convenience function. It and series.evaluate return RMTSeriesResult, whose value is an mpmath number. truncation_bound, error_estimate, terms, shells, working_dps, chart, and convergence_ratio expose the summation diagnostics.

A dominant mass also permits zero momentum:

from ramanujan_qft import integrate_graph_rmt
vacuum = ScalarGraph([ScalarEdge(0, 1, m2) for m2 in (1, 4, 100)])
value = integrate_graph_rmt(vacuum, dps=30)
print(value.chart)
print(mp.nstr(value.value, 25))

mass
0.185949303854726222708126

Exact graph-to-bracket compilation#

For any supported ScalarGraph, schwinger_brackets obtains \(U\) and \(K=F-U\sum_i m_i^2\alpha_i\) from the graph. It expands the mass exponentials, \(\exp(-K/U)\), and the remaining power of \(U\). Each monomial contributes an exact integer exponent column. The integration weights and the power of \(U\) provide the bracket offsets. A basis is solved by exact linear algebra and converted into a formal Gamma-function summand.

For the nonzero-momentum sunset, use indices \((k,r_1,r_2,r_3,a,b,c)\) for the momentum exponential, the three mass exponentials, and the three monomials of \(U\). The brackets are

\[\begin{split} \begin{pmatrix} 1&1&0&0&1&1&0\\ 1&0&1&0&1&0&1\\ 1&0&0&1&0&1&1\\ 1&0&0&0&1&1&1 \end{pmatrix} \begin{pmatrix}k\\r_1\\r_2\\r_3\\a\\b\\c\end{pmatrix} +\begin{pmatrix}\nu_1\\\nu_2\\\nu_3\\D/2\end{pmatrix}=0. \end{split}\]

This matrix is constructed from the polynomials; it is not stored as a hard-coded matrix. In a square bound-index basis \(B\), with free indices \(n_f\), the compiler solves \(n_b=-B^{-1}(c+A_f n_f)\) and applies the Gamma/bracket rule with determinant factor \(1/|\det B|\). Free factorial indicators, monomial coefficients, \(1/\Gamma(D/2+k)\), and \(1/\prod_i\Gamma(\nu_i)\) are retained. No dependent rows are silently discarded, and determinant tests use no floating-point threshold. A nonsingular basis is not a convergence proof.

from ramanujan_qft import schwinger_brackets
import sympy as sp
eta = sp.Symbol("eta", real=True)
brackets = schwinger_brackets(graph, propagator_shifts=(eta, eta, eta))
assert brackets.matrix.shape == (4, 7)
basis = brackets.solve((0, 4, 5, 6))
assert abs(basis.determinant) == 1
# Inspect basis.values and basis.term for the exact solved indices and Gamma term.
assert len(series.gamma_terms) == 8

eta shifts the propagator powers solely to expose the removable pole collisions. It is not a dimensional regulator or an epsilon-expansion feature. The compiled object retains bracket_system, families, gamma_terms, summation_indices, regulator, and the finite coefficient_expression. The evaluator checks that the generated scale constraint matches the sunset Mellin kernel before using the optimized coefficient formula below.

The bracket rule follows the generalized RMT framework of González, Moll and Schmidt. Only candidate algebra is produced for arbitrary topologies; the current numerical compiler recognizes the sunset by connectivity, not polynomial degree.

Proven series regions#

Write \(Q=\sqrt{p_E^2}\) and use positive masses \(m_i=\sqrt{m_i^2}\).

Chart	Required strict inequality	Squared expansion ratios
`momentum`	\(Q>m_1+m_2+m_3\)	\(z_i=m_i^2/Q^2\)
`mass`	\(M>m_a+m_b+Q\), with \(M\) the largest mass	\((z_a,z_b,w)=(m_a^2/M^2,m_b^2/M^2,Q^2/M^2)\)

chart="auto" selects a covered region; an explicit chart may also be requested. These are sufficient domains, not a claim of maximal convergence. The equal-mass vacuum, the intervening region, and the boundaries are refused with RMTDomainError, even when the integral itself is finite. For example, the known vacuum value \(4\operatorname{Cl}_2(\pi/3)/\sqrt3\) is an independent benchmark for the existing numerical backend, not a value supplied by these RMT charts.

The separate Equal-mass sunset: vacuum and Euclidean continuation API now covers equal-mass vacuum and finite Euclidean momenta by a differential-equation Taylor series. An RMT call never silently switches to this method. Unequal-mass gaps remain open.

Domain decisions use rational lower and upper enclosures of sums of square roots, constructed using integer square roots. A positive gap is required; there is no floating-point tolerance that turns a boundary into an interior point. Inconclusive enclosures are refused. Very small exact gaps also increase the required working precision.

The momentum expansion is the coalesced form of a Lauricella \(F_C^{(3)}\) representation. The general sunset series and these convergence conditions are discussed by Ananthanarayan, Friot and Ghosh. Here is a separate outline connecting the generated brackets to that expansion. For general powers, set \(A=\sum_i\nu_i\) and \(Z=\sum_i z_i\). In a common initial Mellin-Barnes strip the Schwinger integral gives

\[ \frac{(Q^2)^{2-A}}{\prod_i\Gamma(\nu_i)} \int_{\cal C}\frac{d^3z}{(2\pi i)^3} \frac{\Gamma(A+Z-2)}{\Gamma(3-A-Z)} \prod_i\Gamma(-z_i)\Gamma(1-\nu_i-z_i) \left(\frac{m_i^2}{Q^2}\right)^{z_i}. \]

The two right-pole choices for each mass generate eight families, exactly the choices between a mass index and the complementary \(U\) index in the bracket matrix. The standard Lauricella contour closure applies when \(\sum_i m_i/Q<1\). Analytic continuation in the powers then permits \(\nu_i\to1\) inside this region. One of the eight regulated families vanishes identically: the all-\(U\)-index family contains \(1/\Gamma(-N)=0\) for \(N=n_1+n_2+n_3\ge0\), independently of dimension and propagator powers. The sum of the remaining regulated families has a finite limit at \(D=2\), \(\nu_i=1\). The resulting series has the absolute majorant given below.

For the mass chart, there are four small-mass pole families. It can also be justified by the coordinate-space identity \(I=4\int_0^\infty rJ_0(Qr)\prod_iK_0(m_i r)\,dr\): expand the two lighter \(K_0\) factors and \(J_0\), retaining \(K_0(Mr)\). The absolute expanded integrand is bounded at infinity by a polynomial/logarithmic factor times \(e^{-(M-m_a-m_b-Q)r}\); it is integrable at zero. Thus termwise integration is justified in the stated strict domain.

Pole collisions and finite coefficients#

Let \(N=n_1+n_2+n_3\), \(H_n=\sum_{j=1}^n1/j\) and \(H_n^{(2)}=\sum_{j=1}^n1/j^2\), with both zero at \(n=0\). Define

\[ W_n=\frac{\Gamma(N+1)^2}{\prod_i\Gamma(n_i+1)^2}. \]

In the momentum chart, \(b_i=H_{n_i}-H_N-\tfrac12\log z_i\) and

\[ I=\frac4{Q^2}\sum_{n_1,n_2,n_3\ge0}(-1)^N W_n\prod_i z_i^{n_i} \left(\sum_{i<j}b_i b_j-\frac32H_N^{(2)}\right). \]

In the mass chart, the first two indices describe the lighter masses and the third the momentum. With \(b_a=H_{n_1}-H_N-\tfrac12\log z_a\) and \(b_b=H_{n_2}-H_N-\tfrac12\log z_b\),

\[ I=\frac4{M^2}\sum_{n_1,n_2,n_3\ge0}(-1)^{n_3}W_n z_a^{n_1}z_b^{n_2}w^{n_3} \left(b_a b_b+\frac{\zeta(2)-H_N^{(2)}}2\right). \]

At \(Q=0\), only \(n_3=0\) remains, so evaluation is a double series. These are finite limits of the generated Gamma families, not fitted expressions. series.coefficient((n1,n2,n3), dps=...) evaluates an individual summand.

One way to derive the limits uses the logarithmic \(K_0\) series. The relevant Mellin kernels are \(\Gamma(N+1)^2\) for the retained \(K_0\) factor and \(\Gamma(N+1)/\Gamma(-N)\) for \(J_0\). The Bessel Mellin transforms and modified Bessel Mellin transform give these kernels. Reflection writes the second as \(-\sin(\pi N)\Gamma(N+1)^2/\pi\). Taking the log-coefficient derivatives before setting \(N\) to an integer yields the finite harmonic expressions above. The momentum derivation uses the continued Mellin kernel: integrating the expanded growing powers against \(J_0\) as ordinary convergent integrals would be invalid. Its integral identity is justified by the Mellin-Barnes chart, not by that invalid interchange.

Tail bounds and numerical checks#

For either chart let \(r\) denote its three nonnegative squared ratios and \(\rho=(\sqrt{r_1}+\sqrt{r_2}+\sqrt{r_3})^2<1\). The multinomial theorem gives

\[ \sum_{|n|=N}W_n r^n\le\rho^N. \]

If \(L\) is the largest \(-\tfrac12\log r_i\) among the mass variables, the logarithmic bracket has absolute value at most \(c(L+H_N)^2+d\), where \((c,d)=(3,\pi^2/4)\) for momentum and \((1,\pi^2/12)\) for mass. Using \(H_{N+k}\le H_N+k/(N+1)\), the omitted tail after shell \(N\) is bounded by

\[ \frac4{S}\rho^{N+1}\left[ \frac{cC^2+d}{1-\rho} +\frac{2cCa}{(1-\rho)^2} +\frac{ca^2(1+\rho)}{(1-\rho)^3} \right],\quad C=L+H_N,\quad a=\frac1{N+1}, \]

with \(S=Q^2\) or \(M^2\). The implementation uses a conservative rational upper bound on \(\rho\), slightly inflated against arithmetic rounding. It sums complete total-degree shells and uses this tail bound for stopping; a short run of small terms is never treated as convergence.

The mathematical truncation bound is evaluated with mpmath guard digits. Estimated arithmetic error, agreement at two working precisions, and final rounding contribute to error_estimate. This is not a rigorous interval arithmetic certificate. Work is bounded by max_shells, max_terms, and max_dps; an unmet target raises RMTSeriesConvergenceError. A partial result is attached when available. Close to the boundary the series may be impractical.

Measured default-budget coverage#

These measurements use max_shells=128, max_terms=250000 and the default precision cap. “Pass” means the returned value met the requested relative accuracy and its reported error against a separate 45/60-digit parameter reference. “Budget” means a typed RMTSeriesConvergenceError, not a divergent integral or an inaccurate accepted result. All rows are inside a justified chart. Mass inputs below are squared masses; \(Q\) is Euclidean momentum.

\((m_1^2,m_2^2,m_3^2)\)	\(Q\)	Chart	\(\rho\)	10 digits	20 digits	30 digits
\((1,1,1)\)	10	momentum	0.09	Pass	Pass	Pass
\((1,1,1)\)	6	momentum	0.25	Pass	Pass	Pass
\((1,1,1)\)	\(30/7\)	momentum	0.49	Pass	Pass	Pass
\((1,1,1)\)	4	momentum	0.5625	Pass	Pass	Budget
\((1,1,4)\)	5	momentum	0.64	Pass	Budget	Budget
\((1,1,16)\)	1	mass	0.5625	Pass	Pass	Budget

Reproduce with python tools/audit_sunset_budgets.py; raw outcomes, source hashes, shell counts and timings are in benchmarks/data/sunset_budgets.json. This finite grid is not a universal cutoff in \(\rho\): logarithms, cancellation, scale, precision and zero ratios also affect cost. With three nonzero ratios, shell \(N\) has \((N+1)(N+2)/2\) terms. The default term cap can therefore stop a run before the shell cap; the measured refusals stopped after 113 shells and 246,905 terms per precision pass.

Increase the budgets explicitly for an interior point, reduce the requested digits when appropriate, or use the separately named equal-mass Taylor backend. For unequal masses, integrate_graph offers a float64 statistical estimate within its supported domain. An RMT call does not silently select a fallback.

Validation and remaining work#

Tests compare against independently generated 40/55-digit parameter quadratures, a live nested quadrature, and the existing Hepp-sector backend. They also test the regulated Gamma-family limits, exact exponent algebra, truncation bounds, equal and unequal masses, both charts, mass rescaling, permutations, and strict boundary refusal. The reference generator is tools/generate_sunset_rmt_references.py; it imports no package evaluator. The numerical pole-limit calculation is used only in tests.

Further RMT extensions need justified continuation between the charts, particularly for unequal masses. Equal-mass vacuum and low Euclidean momentum are already covered by the separate Taylor method. Other topologies need their own family selection, collision resolution and convergence proof. Non-unit powers, epsilon expansions, massless graphs, tensors and physical-region continuation are not implemented by this API.