Sunrise

The elliptic prototype — the diagram the Eichler-integral library was written for: closed at order ε⁰ as an Eichler integral on the Γ₁(6) modular curve — a classically known function space, recomputed here without using the published values as fitted inputs at any step and checked against independent evaluations at points kept out of the fit — with the three unequal-mass assignments closed at ε⁰ in the same space and the ε¹ correction carried across them.

The integral

The sunrise is the minimal two-loop diagram: two vertices joined by three massive propagators, with a single external momentum $p$ flowing in one side and out the other,

$$J_{111}(d,\,p^2) \;=\; \int d^d l_1\, d^d l_2\; \frac{1}{\bigl(l_1^2-m_1^2\bigr)\,\bigl(l_2^2-m_2^2\bigr)\,\bigl((l_1+l_2-p)^2-m_3^2\bigr)}\,, \qquad d = 2-2\varepsilon,$$

up to an overall normalization convention — the subscript records one power of each propagator, and the exact conventions ($J_{111} = -S_{111}$ in the notation of arXiv:1704.08895, $m^2 = \mu^2 = 1$, no $e^{\gamma\varepsilon}$ prefactor) are fixed in the expression file below. In the equal-mass case all three internal lines carry the same mass $m$, so the integral depends on one dimensionless variable, $t = p^2/m^2$ — the squared external momentum measured in units of the internal mass; the equal-mass case is closed here at order ε⁰. The same function space then closed the three unequal-mass assignments $(m_1^2, m_2^2, m_3^2) = (1,1,2),\ (1,2,3),\ (1,1,4)$ at ε⁰, and the ε¹ correction is carried across all three unequal masses.

After the loop integrations the integrand carries the square root of a quartic, so the contour has to thread around four branch points rather than collect residues at poles, and the surface that records those threadings is a torus. This is the simplest Feynman integral whose answer is elliptic.

The torus is a specific one from number theory. The integral can blow up only where $t$ hits 0, 1, 9 or infinity, and those four points are exactly the four cuspsThe points at the boundary of a modular curve where the elliptic fiber pinches to a nodal rational curve; the natural boundary data of a modular problem. of the modular curve for the congruence subgroup Γ₁(6)A specific finite-index subgroup of SL(2,ℤ); its quotient of the upper half-plane is a genus-zero curve with exactly four cusps, and it indexes elliptic curves with a marked point of order six.; the Beauville classification of rational elliptic surfaces picks that curve out uniquely from the fiber pattern (1, 2, 3, 6). The natural language for the answer is therefore modular formsHolomorphic functions on the upper half-plane that transform with a fixed weight under a congruence subgroup; at each weight they span a finite-dimensional vector space. on Γ₁(6) and their once-integrated cousins, Eichler integralsIterated integrals of modular forms along a path in the upper half-plane — the elliptic generalisation of the polylogarithms..

Why it matters

This is the elliptic prototype, the simplest diagram beyond polylogarithms and the standard test case for every elliptic method in use, with a literature to match. Laporta and Remiddi gave the equal-mass graph a full analytic treatment in 2005; Bloch and Vanhove then recast the answer in terms of the elliptic dilogarithm, tying the diagram to the number theory of its curve; Adams, Bogner and Weinzierl extended the elliptic-dilogarithm form to arbitrary internal masses in two dimensions; and Adams and Weinzierl assembled the all-orders representation as iterated integrals of modular forms on Γ₁(6) — the representation this page's result is checked against. The references below trace that arc.

Γ₁(6) is as clean as elliptic Feynman geometry gets: four cusps, a finite ring of modular forms at each weight, the whole surface pinned down by the Beauville classification. That made it the right place to build the Eichler-integral machinery once: the Eichler library was written for this integral, and every elliptic and Calabi–Yau entry further down the gallery runs on the code it left behind. Within BootLoops the sunrise is the validation anchor of the elliptic ladder — the answer was known, so the harness had to reproduce it blind, comparing only afterwards, before the same machinery could be trusted on diagrams nobody had closed.

What was hard

Three things, none of them integration. First, the function space had to be found before anything could be fitted, because the answer lives outside every polylogarithmic catalogue. The geometry hands it over: the singular set {0, 1, 9, ∞} read off the Landau equations is exactly the four cusps of Γ₁(6), fixing the polylogarithmic letters to $\{t,\ t-1,\ t-9\}$ — the soft point and the two physical thresholds — and the elliptic data to two objects: the holomorphic periodThe integral of the holomorphic one-form dx/y around a closed cycle of the elliptic curve; the basic elliptic transcendental, playing the role 2πi plays for logarithms. ψ₁/π of the curve as overall prefactor, and the depth-two Eichler integral $I(1,f_3)$ of the weight-three modular formA function on the curve's parameter space with a fixed transformation law; for Γ₁(6) every form below weight four is built from Eisenstein series, the first cusp form appearing only at weight four. $f_3$ of Γ₁(6), with the cusp structure supplying the one constant that can appear at this order, $L(\chi_{-3},2)$. Integrability and the first-entry condition — only the soft point and the physical thresholds may open a branch cut — prune the word ansatz; the surviving coefficients are over-determined, fitted by PSLQAn integer-relation algorithm: feed it high-precision decimals and it returns the small-integer linear combination that vanishes, if one exists. on a grid of Euclidean points with one point kept out of the fit, every coefficient required to come back an exact rational.

Second, no public software could evaluate $I(1, f_3;\, q)$, the depth-two iterated integral of a weight-three modular form that carries the answer, so a library had to be written: Eichler, built for this diagram, which has since grown the Dirichlet-L dictionary, the period transport and the Calabi–Yau routines the later diagrams needed.

Third, the period frame. The naive maximal-cut period from the Abel–Jacobi parametrization sits near a cusp and gives a spuriously small Eichler integral, and in that frame the fit stalled at zero digits; the correct dressing period is the Adams–Weinzierl Feynman-curve period $\hat\psi_1 = (4/\pi)\,\bigl[(t-1)^3(t-9)\bigr]^{-1/4}K(k^2)$, and once that frame is used the whole structure $J = -\hat\psi_1\bigl(B_0 + I(1,f_3)\bigr)$ falls out, with the boundary constant $B_0 = \tfrac{3}{2}\sqrt{3}\,L(\chi_{-3},2)$ — a Dirichlet L-valueThe value at an integer point of an L-series built from a periodic arithmetic character; the number-theoretic constants that replace ζ(2) and ζ(3) on an elliptic curve. the cusp structure of Γ₁(6) supplies — matched to its independent reference to 70 digits.

The result

Throughout, $t = p^2/m^2$ is the kinematic variable defined above and $J^{(0)}$ is the ε⁰ coefficient of the master integral $J_{111}$.

At leading order in ε the sunrise is the elliptic period times a boundary constant plus an Eichler integral on Γ₁(6):

$$ J^{(0)}(t) \;=\; -\,\frac{\psi_1(t)}{\pi}\,\Bigl(\,\tfrac{3}{2}\sqrt{3}\,L(\chi_{-3},2) \;+\; I(1,f_3;\,q_C)\Bigr)\,, $$

with the three pieces given explicitly by

$$ \frac{\psi_1}{\pi} \,=\, \frac{4}{\pi}\,\bigl[(t-1)^3(t-9)\bigr]^{-1/4}\,K(k^2)\,, \qquad q_C \,=\, -\,e^{\,i\pi\,\psi_2/\psi_1}\,, \qquad I(1,f_3;\,q_C) \,=\, \sum_{n\ge 1}\frac{a_n}{n^{2}}\,q_C^{\,n}\,. $$

Here ψ₁ is the holomorphic period of the sunrise elliptic curve, with $K$ the complete elliptic integral of the first kind and modulus $k^2 = (e_3-e_2)/(e_1-e_2)$ built from the three roots $e_1, e_2, e_3$ of the sunrise quartic $y(y-4)\bigl(y^2-2y(t+1)+(t-1)^2\bigr)$; ψ₂ is the second, independent period, so $q_C$ is the Γ₁(6) nome, negative real for Euclidean $t<0$ and tied to the kinematic variable by the hauptmodul $t(q) = 9q\prod_{n\ge1}(1-q^{6n})^8(1-q^n)^4(1-q^{2n})^{-8}(1-q^{3n})^{-4}$; and $a_n$ are the Fourier coefficients of the weight-three modular form

$$ f_3 \;=\; 36\sqrt{3}\,\bigl(e_1^3 - e_1^2 e_2 - 4\,e_1 e_2^2 + 4\,e_2^3\bigr)\,, \qquad e_1(q) \;=\; \tfrac{1}{6} + \sum_{m\ge 1}\Bigl(\sum_{d\mid m}\chi_{-3}(d)\Bigr) q^m\,, \qquad e_2(q) \;=\; e_1(q^2)\,. $$

The boundary constant is $\tfrac{3}{2}\sqrt{3}\,L(\chi_{-3},2) = 2.0298832128\ldots$, with $L(\chi_{-3},2)$ the Dirichlet L-value of the quadratic character of conductor 3.

The ε⁰ line is the visible tip. The full blind result runs through ε⁴ — transcendental weight six — in the same ring:

$$ e^{2\gamma\varepsilon} J_{111}(2-2\varepsilon,\, t) \;=\; \tfrac{2}{\sqrt3}\; u(q)\; e^{-\varepsilon \tilde w(q)} \sum_{j\ge0} \varepsilon^j H_j(q)\,, $$

where γ is the Euler–Mascheroni constant in the standard normalizing prefactor, $q$ is the nome introduced above, $u = (\sqrt3/2)\,\hat\psi_1$ is the normalized holomorphic period with $u(0)=1$, $\tilde w$ is the exact exponential rotation that ε-factorizes the Picard–Fuchs operator, and each $H_j$ is an exact ℚ(√3) combination of iterated integrals of the five kernels {one, k2h, k3, k4E, c4} — with $k3 = -f_3/(3\sqrt3)$ in the normalization above and $c4 = \eta(\tau)^2\eta(2\tau)^2\eta(3\tau)^2\eta(6\tau)^2$ the weight-4 cusp form 6.4.a.a — in words of length up to six: 2, 4, 12, 25 and 55 words at the successive weights. The boundary constants close exactly through ε², with $r_3 = e^{2\pi i/3}$:

$$ B_0 = -\tfrac{3}{2}\sqrt{3}\,L(\chi_{-3},2)\,, \qquad B_1 = -6\,\operatorname{Im}\operatorname{Li}_3(1-r_3) \;-\; \tfrac{1}{2}\,\pi\log^2 3 \;-\; \tfrac{5}{54}\,\pi^3\,, $$

$$ B_2 = 12\,\operatorname{Im}\operatorname{Li}_4(1-r_3) \;+\; \tfrac{3}{2}\sqrt{3}\,L(\chi_{-3},4) \;-\; \tfrac{1}{4}\sqrt{3}\,\pi^2 L(\chi_{-3},2) \;+\; \tfrac{5}{54}\,\pi^3\log 3 \;+\; \tfrac{1}{6}\,\pi\log^3 3\,. $$

The two deepest constants, B₃ and B₄, remain unnamed: the searched dictionary of standard constants was insufficient at those weights, and no closed form for them is claimed. They are nonetheless exact, computable numbers — the ε³ and ε⁴ coefficients of the equal-mass vacuum integral,

$$ B_j \;=\; -\tfrac{\sqrt{3}}{2}\,\bigl[\varepsilon^j\bigr]\Bigl(e^{2\gamma\varepsilon}\, S_{\rm vac}(\varepsilon)\Bigr)\,, \qquad S_{\rm vac}(\varepsilon) \;=\; \frac{2^{\,2-\varepsilon}}{\Gamma(1-\varepsilon)}\int_0^\infty x^{1+\varepsilon}\,K_\varepsilon(x)^3\,dx\,, $$

with $K_\varepsilon$ the modified Bessel function — a definition sunrise-B34.py evaluates from scratch to any precision.

The same function space closes the three unequal-mass members $(m_1^2,m_2^2,m_3^2)=(1,1,2),\ (1,2,3),\ (1,1,4)$ at ε⁰ by one uniform per-puncture formula in depth-two Kronecker elliptic polylogarithms, on tori carrying three marked points — all distinct for $(1,2,3)$, two coincident for $(1,1,2)$ and $(1,1,4)$; the three fitted coefficients came back the exact rationals $[-1,-\tfrac13,\tfrac83]$. The ε¹ correction closes for all three masses with no fitted parameter: it is the depth-three elliptic-polylogarithm polynomial of Bogner, Müller-Stach and Weinzierl (101 words over a 14-letter Γ₁(6) kernel alphabet, with ζ(3) entering its boundary constant) combined with the ε⁰ master through an exact frame constant. That 14-letter alphabet is not closed from scratch here — the soft-point constraint alone leaves 2926 of its 3151 words free, and the integrability pruning is not carried out. A fifth mass set, $(1,1,3)$, never fitted, is predicted by the same per-puncture formula and reproduces fresh independent evaluations at $t=-3$ and $t=-6$ to 160 and 159 digits (their lower-precision twins to 110 and 109); sunrise-genmass-evaluate.py evaluates the closed form at any rational masses and runs that check. The complete expression, conventions included, is in sunrise-expression.md, and sunrise-evaluate.py rebuilds the ε⁰ closed form from scratch — q-series, hauptmodul inversion and Eichler sum — at any point and precision. Each unequal-mass member has its own evaluator: sunrise-row09-evaluate.py for $(1,1,2)$, sunrise-row10-evaluate.py for $(1,2,3)$ and sunrise-row11-evaluate.py for $(1,1,4)$ evaluate the ε⁰ closed form on the shared layer sunrise_empl.py and compare it with independent reference values — for $(1,1,2)$ two of them, $t=-3$ and $t=-7$, were kept out of the fit and are checked separately from the four fit points, agreeing to at least 159 digits (and, above threshold, the same closed form continued to $t+i0$ reproduces independent evaluations at $t=16$ and $t=20$ — fresh points, predicted before they were computed — to at least 71 digits by the reference pair, more than 100 against the 60-digit reference, and at $t=12$ to 100 digits; --point 16 runs that check); for $(1,2,3)$ two, $t=-3$ and $t=-9$, neither in the fit, agreeing to 159 and 159 digits; for $(1,1,4)$ three, none of them in the fit — $t=-3$, checked against a value computed to a goal of 280 digits and agreeing to 289, and $t=-9$ and $t=-11$, agreeing to 159 each (two further reference points from the fit-era verification grid, $t=-5/2$ and $t=-7$, agree to at least 159) — and, for $(1,1,2)$ and $(1,1,4)$, with the digits printed in the paper; sunrise-row12-evaluate.py checks the ε¹ correction across all three mass sets against nine independent reference values, three per mass set, none of them held out — the closure has no free parameter, so every point is a prediction — agreeing to 288–308 digits at a working precision of 300 (the floor at each point is its reference's own goal), and against the paper's printed $E^{(1)}$ — the polynomial's 101 coefficients were also re-fit from the reference grid alone, every coefficient free, returning the published values exactly with four held-out points predicted to 104 digits or better.

Downloads (unequal masses): sunrise-row09-evaluate.py · sunrise-row10-evaluate.py · sunrise-row11-evaluate.py · sunrise-row12-evaluate.py · sunrise-genmass-evaluate.py · sunrise_empl.py · the continued layer sunrise_empl_cont.py · frame_ode.py · data: m113 · the row-12 re-fit · row09 · row10 · row11 · row12 · MANIFEST.sha256

Downloads: sunrise-expression.md · sunrise-evaluate.py (its stored reference values now include $t=-9$, a 128-digit truncation of an independent evaluation, so --point -9 checks a third point to 127 digits) · sunrise-B34.py

The equal-mass ε⁰ closed form reproduces the Adams–Weinzierl Γ₁(6) result of arXiv:1704.08895 and agrees with independent evaluations to 139–140 digits in a leave-one-out fit over five Euclidean points — 140 at the never-fit point $t=-3$ — with an independent Eichler-integral reference to 47 digits, and on its boundary constant to 70.

References

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