Kite

A two-loop self-energy with an elliptic curve inside, in three mass configurations: the known equal-mass answer recomputed blind; the unequal-mass case, where the curve turns non-modular, symbolic fitting provably fails, and the differential equation is transported from an analytically derived boundary; and the massless-rung configuration, a genus-zero family with four distinct masses whose finite part is transported from an analytic seed at $p^2 = 0$.

The integral

The kite is a two-loop self-energya diagram with one external momentum flowing in and out; as a function of that single invariant it corrects a particle's propagator: a single off-shell momentum $p$ enters at one vertex and leaves at the other, and the only kinematic variable is the squared momentum $s = p^2$ it carries. The family is

$$J[\nu_1,\ldots,\nu_5] \;=\; \int d^d k_1\, d^d k_2\; \frac{1}{D_1^{\nu_1} D_2^{\nu_2} D_3^{\nu_3} D_4^{\nu_4} D_5^{\nu_5}}\,, \qquad d = 4-2\varepsilon,$$

with loop momenta $k_1, k_2$ and, in the unequal-mass configuration that is this page's main result, the five propagators

$$D_1 = k_1^2 - 1,\quad D_2 = k_2^2,\quad D_3 = (k_1-k_2)^2 - 1,\quad D_4 = (k_1-p)^2,\quad D_5 = (k_2-p)^2 - 2.$$

Two lines are massless and three are massive, and the three massive lines $D_1, D_3, D_5$ form a three-line sunrise sub-diagram — pinch both massless lines and the kite collapses onto it. That sunrise fixes the geometry: its maximal cutput every internal line of the sub-diagram on shell simultaneously; the integral over what survives exposes the diagram's underlying geometry is an elliptic curve, so the kite lives in elliptic-function territory from the start. The object closed in every configuration is the finite $\varepsilon^0$ coefficient of the top master $J(1,1,1,1,1)$, the integral with all five propagators present once.

Three mass configurations are treated. In the equal-mass kite the three massive lines share a single mass $m$, fixed to $m^2 = 1$, and the variable is $t = p^2/m^2$; the sunrise curve is the congruence modular curve $\Gamma_1(6)$, and the closed form has been known since Adams, Bogner, Schweitzer and Weinzierl wrote it down in 2016 (arXiv:1607.01571) — a known answer, which makes it one of the blind recomputations behind this site. In the unequal-mass kite the masses become $(1, 0, 1, 0, \sqrt2)$ — the propagators displayed above. The sunrise inside now carries squared masses $(1, 1, 2)$, and the curve changes: a direct check of the j-invariantthe single complex number that classifies an elliptic curve up to isomorphism; two curves with different j are genuinely different geometries shows it matches $\Gamma_1(6)$ at no value of the external momentum. The $(1,1,2)$ family is non-modular in the precise sense that its base is not a modular curve: its $j$-invariant is not the $\Gamma_1(6)$ one at any $s$. A closed form for the kite with three distinct masses, as a pure weight-three combination of elliptic polylogarithms with punctures that move with $s$, is in Broedel, Duhr, Dulat, Penante and Tancredi (arXiv:1902.09971, Sec. 5), of which $(1,1,2)$ is the $m_1 = m_2$ limit; what is new here is the transport from an analytically derived boundary, its held-out precision, the second mass point and the Minkowski continuation. In the massless-rung configuration the five lines carry squared masses $(1, 4, 9, 2, 0)$, four distinct masses on the rim and a massless rung, treated in its own section below.

Why it matters

Validation comes first in this program: of the paper's thirty results, fifteen reproduce known integrals or calibration runs, taking no published value as a fitted input, and they calibrate the pipeline the fifteen new ones then run through under the same checks. The equal-mass kite is one of those fifteen reproductions. BootLoops derived the function space from constraints, fit the five coefficients of the answer against its own high-precision values with the published result locked away, recognized them by PSLQ, and only then opened the Adams–Bogner–Schweitzer–Weinzierl expression for comparison — which matched. The five-word closed form can be rerun at any Euclidean point with kite-equal-evaluate.py, which rebuilds the periods, the nome and the q-series from their definitions at runtime, checks the result against four independent reference values (two of them, t = −3 and t = −10, kept out of the fit) and against the digits printed in the paper, and exits nonzero on any mismatch. A pipeline that can do that has earned some trust one mass-shift later, where no literature answer exists.

That mass shift is the second part of the page, and a differential-equation-transport closure on a non-modularan elliptic curve not governed by any congruence subgroup of SL(2,ℤ), so its periods are not modular forms and admit no standard q-expansion Feynman curve — the route the generic-mass ice-cream cone and the qq̄→WW nonplanar box also take. What the kite supplies is the clean witness that transport can be provably mandatory: the equal-mass sunrise and kite were fittable in a modular basis on $\Gamma_1(6)$, where the periods are modular forms, the integration kernel is itself a q-series and so is the answer, while this configuration carries the same algebraic prefactor and yet, because its kernel is non-modular (an Eichler-integral, period-mixing obstruction), admits no such word list. The exact-top-mass gg→Zγ masters meet the same wall on a conductorthe integer N labelling the congruence subgroup Γ₀(N) on which a rational elliptic curve becomes modular; small conductor means a small, well-tabulated space of modular forms-4 curve — a blind value fit there reaches 1.43 digits against a 78-digit positive control, and the boundary is delivered as a named Eichler integral on $\Gamma_0(4)$. For the unequal-mass kite the modular dictionary is empty, and the only route to a verified answer is to carry the period data through the differential equation itself.

It also sharpened the program's advance test for whether an elliptic integral will yield to a symbolic fit. Until this diagram that test looked only at the $\varepsilon^0$ prefactor; the unequal-mass kite passes it and is still provably unfittable, because the obstruction hides one layer deeper, in the inhomogeneous integration kernel. Catching that sharpened the criterion for every elliptic diagram that followed.

What was hard

First, the geometry leaves the modular world: the base of the $(1,1,2)$ sunrise family is not a modular curve, so the fixed-puncture elliptic-polylogarithm word bases of the equal-mass case do not apply (the closed form of Broedel et al. for three distinct masses uses punctures that move with $s$). Second, the failure of symbolic fitting is provable, and the proof required looking past the standard criterion. The natural test — an algebraic $\varepsilon^0$ prefactor, which this integral has — predicts a fit; the fit nonetheless fails, flooring at 9–12 digits on points kept out of the fit while the same basis fits a manufactured test function to 72 digits, so the obstruction is mathematical, not a limitation of the fit. The reason is the integration kernel $K = \psi_1^3/W$, built from the periods of the sunrise curve and their Wronskian: for equal masses Adams and Weinzierl showed $K$ is itself a modular q-series, while for the $(1,1,2)$ masses an exact-Wronskian probe places it genuinely outside the modular span, with the second-kind period active and non-rational. That probe fits the equal-mass kernel to 116 digits with two words; the unequal-mass kernel plateaus at 19 digits however many words are added (2 through 13), with the same agreement on fitted and unfitted points at every basis size — a plateau that is neither precision-limited nor sample-limited: the kernel simply lies outside any constant-coefficient span of these functions. Third, with fitting closed off, the boundary data for the differential equation had to be derived rather than measured: the physical solution is the unique branch bounded at $s = 0$, where its limits are a closed-form two-loop vacuum ring in classical constants, and everything else follows by transporting that branch. The massless-rung configuration added a wall of its own: with a massless line the point $p^2 = 0$ is a regular singular point of the system, with exponents $-1+\varepsilon$ and $-2+2\varepsilon$ beside the analytic branch, and the closure had to be seeded there, at the vacuum values of the masters.

The result

Equal mass — the known answer, recomputed blind. Here $t = p^2/m^2$ is the single kinematic variable, $G(\ldots;t)$ are ordinary multiple polylogarithms with letters 0 and 1, and $\overline{E}$ are the Adams–Bogner–Schweitzer–Weinzierl elliptic polylogarithms, evaluated at the third root of unity $\zeta_3 = e^{2\pi i/3}$ in the nome $-q$, where $q = e^{i\pi\tau}$ and $\tau = \psi_2/\psi_1$ is the ratio of the two periods of the sunrise curve (singular fibers at the threshold $t=1$ and pseudo-threshold $t=9$). In the engine normalization ($m^2 = \mu^2 = 1$, no $e^{\gamma_E\varepsilon}$ prefactor) the $\varepsilon^0$ top master is the five-term word list

$$ I_{\rm kite}(t) \;=\; \frac{1}{4t}\Big[\, -\tfrac{2\pi^2}{3}\,G(1;t) \;-\; 8\,G(0,1,1;t) \;+\; 4\,G(1,0,1;t) \;-\; 108\,\mathrm{Cl}_2\!\big(\tfrac{2\pi}{3}\big)\,\overline{E}_{1;-1}(\zeta_3;1;-q) \;-\; 108\,\overline{E}_{0,2;-2,0;2}(\zeta_3,\zeta_3;1,-1;-q) \,\Big], $$

with $1/(4t)$ the algebraic prefactor and $\mathrm{Cl}_2(2\pi/3)$ the Clausen value, the standard $\Gamma_1(6)$ cusp constant. Every coefficient is a rational times at most that one constant, and the singular points are $t = 0, 1, 9$. This is the closed form of Adams, Bogner, Schweitzer and Weinzierl, recovered blind — the published expression entered only after the fit, for comparison.

Unequal mass — the transport representation. In the fixed three-puncture word bases there is no finite constant-coefficient elliptic-polylogarithm word list that reproduces this master; the kernel-modularity probe above confirms it, and the closed form of Broedel et al. uses punctures that move with $s$ instead. The terminal form is the transport representation: the explicit rational connection, the explicit period pairthe two independent contour integrals ∮ dx/y around the a- and b-cycles of an elliptic curve; together they span every solution of the curve's second-order Picard–Fuchs equation of the (1,1,2) curvethe Feynman elliptic curve of the sunrise sub-diagram with squared masses (1,1,2), in BMSW Weierstrass form, and a single analytically derived boundary, assembled by variation of parametersthe textbook method for an inhomogeneous linear ODE: take the homogeneous solutions, promote their constant coefficients to functions, and fix those functions by a first-order quadrature against the source.

The curve is

$$ E_{(1,1,2)}:\quad y^2 \;=\; 4\,(x-e_1)(x-e_2)(x-e_3),\qquad k^2=\frac{e_3-e_2}{e_1-e_2},\quad Z_3=e_1-e_2, $$

where $e_1, e_2, e_3$ are the branch points of the curve as functions of $s$, fixed by the four sunrise thresholds $M_a=(\pm 1\pm 1\pm\sqrt2)^2$; $k^2$ is the resulting elliptic modulus and $Z_3$ the period normalization. The period pair and Wronskian are closed in complete elliptic integrals,

$$ \psi_1(s)=\frac{2}{\sqrt{Z_3}}\,K\!\big(k^2\big),\qquad \psi_2(s)=\frac{2i}{\sqrt{Z_3}}\,K\!\big(1-k^2\big),\qquad W(s)=\psi_1\psi_2'-\psi_2\psi_1'\,, $$

with $K$ and $E$ the complete elliptic integrals of the first and second kind (primes marking the complementary modulus $1-k^2$), and $W$ determined exactly by the Legendre relation $K E' + K' E - K K' = \tfrac{\pi}{2}$. The $\varepsilon^0$ top master is then the period-pair variation-of-parameters integral

$$ J^{(0)}_{\text{top}}(s)\;=\;J^{(0)}_{\text{top}}(-2)\;+\;\int_{-2}^{\,s}\frac{\psi_1(s')^{\,3}}{W(s')}\;S(s')\;\mathrm{d}s'\,, $$

where the source $S(s)$ is rational in $s$, assembled from the lower polylogarithmic masters by the 14×14 connection $A(d,s)$ of the family's fourteen master integralsthe finite basis of independent integrals to which every integral of the family reduces via integration-by-parts identities. Every denominator in $A$ factors over the four letters $\{s,\,s-1,\,s-2,\,s^2-12s+4\}$ — the last an algebraic threshold at $s = 6 \pm 4\sqrt2$ that the equal-mass case does not have — so the integration path along the negative real axis never meets a pole.

The same construction, re-run with the sunrise masses set to $(1,1,3)$, reproduces separately computed independent values at two points kept out of the construction, $s=-5/3$ and $s=-10/3$, to 128 digits on all sixteen non-rational entries per point, the six exact rational $\varepsilon^{-2}$ entries matching exactly; the closed form itself remains the $(1,1,2)$ configuration — the second mass set is a reproduction of the construction, not a second closure; the evaluator below computes this line too (--masses 1,1,3) and checks both points against the recorded values. The exact reduction table at $x = 3$ behind that run was itself checked entry by entry — all 406 coefficients of the 29-target table and all 196 entries of the 14×14 connection — against four independent numerical reductions at distinct values of $(d,s)$, every entry exact.

The boundary value $J^{(0)}_{\text{top}}(-2) = -0.8887421666\ldots$ is derived, not measured: it is the $s=-2$ value of the unique branch of the connection bounded at the regular singular point $s=0$, whose $s=0$ limits are the closed-form two-loop vacuum ring — every constant classical: $\gamma_E$, $\zeta_2$, $\ln 2$, and Catalan, the last entering through $\xi(1,1,2)=-8\,\text{Catalan}$ — carried to $s=-2$ along the differential equation itself. Independent numerical evaluations enter the construction nowhere; they serve only as after-the-fact checks. Beyond the two exact facts $M_{10}^{(\varepsilon^{-2})}=(s+2)(s+6)$ and $M_{9}^{(\varepsilon^{-2})}=3$ from lower sectors, the boundary entries are transcendentals with no PSLQ relation to named constants at sane height across the rings we searched — consistent with no closed form.

The boundary itself has an explicit period-pair representation. Writing $N_{\rm kite} = 4s\,J^{(0)}_{\text{top}}(s)\big|_{s=-2} = 7.109937332\ldots$, variation of parameters on the curve's own period pair gives

$$ N_{\rm kite} \;=\; c_1\,\psi_1(-2) + c_2\,\psi_2(-2) \;+\; \psi_2(-2)\int_{-4}^{-2}\frac{\psi_1(t)\,R(t)}{W(t)}\,dt \;-\; \psi_1(-2)\int_{-4}^{-2}\frac{\psi_2(t)\,R(t)}{W(t)}\,dt, $$

with $R(t)$ the explicit rational source from the top row of the connection and the two constants derived analytically from the $s=0$ regularity — $c_1 = 26.35941243\ldots$, $c_2 = -35.49989517\ldots$, outputs of the derivation, not fitted inputs. A PSLQ search for the boundary across five rings — period-lattice bilinears; Dirichlet L-values at the bad prime 2 (Catalan, $L(\chi_{\pm 8},2)$); periods times logarithms of the alphabet letters at $s=-2$; classical weight-two constants ($\pi^2$, $\log^2 2$, $\mathrm{Cl}_2(\pi/4)$) times periods; and the lemniscatic period scale $\Gamma(1/4)^2$ — returns no relation at bounded height (bases up to 15 elements, 80–100 digits), with three positive controls recovered on the same rings, and the arithmetic explains why: the $s=-2$ fiber is the rank-one, conductor-128 curve $y^2 = x^3 - 756x + 7344$, non-CM, so no named closed form is expected even though the construction that produces the number uses only $\gamma_E$, $\zeta_2$, $\ln 2$ and Catalan through convergent series.

The massless-rung configuration

The third configuration moves the same five-line graph to squared masses $(1,4,9,2,0)$: masses $1$ and $2$ on the left rim, $3$ and $\sqrt2$ on the right rim, and a massless rung joining the two internal vertices. The object computed is the finite $\varepsilon^0$ coefficient of the top master $\mathrm{TOP}(t) = I(1,1,1,1,1)$ as a function of $t = p^2$. Both three-line cuts pass through the massless rung, $(1,0,\sqrt2)$ and $(2,0,3)$, and a sunrise with a massless line has a genus-zero maximal cutput every internal line of the sub-diagram on shell simultaneously; the integral over what survives exposes the diagram's underlying geometry: the Källén polynomial of the pair that contains the massless line is a perfect square, and what remains under the root is a quadratic. So this family carries no elliptic curve at all. The triple $(1,4,9)$ that an earlier version of this page called a threshold-tuned sunrise is the left bubble times the tadpole of mass $3$, and the Landau points $t = 4, 16, 36$ of a $(1,2,3)$ sunrise occur nowhere in the connection. The transcendental content is organised by seven rational letters, two of them quadratic over $\mathbb Q$, and by the four square roots $\sqrt{(t-1)(t-9)}$, $\sqrt{(t-1)(t-25)}$, $\sqrt{t^2-6t+1}$, $\sqrt{t^2-22t+49}$ of the two bubbles and the two cuts; whether the top master is a multiple polylogarithm over these roots is not decided here. What the row delivers is the transport itself: an exact twenty-two-master connection seeded analytically at $p^2 = 0$ with no AMFlow input, held out at seven points to 81--110 digits and reproduced at a second mass configuration.

The closure anchors the transport at the threshold itself. The family reduces to twenty-two master integrals with an exact rational connection $\partial_t M = A(\varepsilon,t)\,M$, linear in $\varepsilon$, singular only on $t \in \{0,\, 1,\, 9,\, 25,\, \tfrac{85}{4},\, 3\pm2\sqrt2,\, 11\pm6\sqrt2\}$ — the $d\log$ letters $\{t,\, t-1,\, t-9,\, t-25,\, 85-4t,\, t^2-6t+1,\, t^2-22t+49\}$ — and pole-free along $t<0$. At the threshold-vacuum point $p^2 = 0$ the residue matrix has eigenvalue spectrum $\{0\,(\times 8),\ -1+\varepsilon\,(\times 13),\ -2+2\varepsilon\,(\times 1)\}$; every non-analytic branch is unbounded, so boundedness at $t = 0$ fixes every homogeneous constant to zero analytically — the same mechanism as the $s=0$ regularity boundary of the unequal-mass configuration above. The seed is classical: partial-fractioning the degenerate propagator pairs collapses all twenty-two masters to two-loop vacuum integrals — the tadpole and a one-massless-line vacuum sunset derived from scratch by Mellin–Barnes plus a reflection collapse — and the analytic branch is then carried along the pole-free Euclidean axis. The top component of that seed is also a one-fold Bessel integral, $\mathrm{TOP}(0) = -\tfrac{8}{21}\int_0^\infty \tfrac{dx}{x}\,[K_1(x) - 2K_1(2x)]\,[\sqrt2\,K_1(\sqrt2\,x) - 3K_1(3x)] = -0.2910157568\ldots$, which reproduces the classical seed to 49 digits.

The evaluator kite-threshold-evaluate.py rebuilds the twenty-two-master transport from the $p^2 = 0$ seed at any requested precision and evaluates the top master at any Euclidean point; the exact connection it reads is kite-threshold-connection.json. Its --AB and --AB-ring options, kept for the record, print the coordinates $A$, $B$ of the top master in the period frame of a $(1,2,3)$ sunrise curve at $t_0 = -2$ and the integer-relation scans run on them; that curve is not a sub-topology of this family, so those coordinates are a frame choice and not periods of the integral.

The same construction, re-run with the outer mass moved to squared masses $(1,4,9,3,0)$, reproduces separately computed independent values at two points kept out of the construction, $t=-2$ and $t=-7/3$, to 98 digits on all fifteen non-rational entries per point, the seven exact rational $\varepsilon^{-2}$ entries matching exactly — the transported values were on record before the independent numbers at these masses existed; the closed form itself remains the $(1,4,9,2,0)$ configuration. The evaluator below computes this configuration too (--masses 1,4,9,3,0) and checks both points against the recorded values. The exact reduction table at $m_3^2 = 3$ behind that run was itself checked entry by entry — all 968 coefficients and all 484 connection entries — against four independent numerical reductions at distinct values of $(d,s)$, every entry exact.

Everything on this page was validated against independent evaluations at points never used in the construction, to at least 60 digits; the equal-mass configuration additionally reproduces the published Adams–Bogner–Schweitzer–Weinzierl closed form.

Downloads: kite-expression.md · kite-evaluate.py · kite-boundary.py · kite-equal-evaluate.py (--point t with $t>1$ now evaluates the physical region, continuing the served hauptmodul around the threshold on two independent routes and checking against the Minkowski reference values at $t=4$ and $16$ to 67 digits and the spectral density at $t=12$, $50$, $100$; the cusps $t=1$ and $t=9$ are refused by name) · kite-threshold-evaluate.py · kite-threshold-connection.json · kite_exact_parse.py · MANIFEST.sha256 (every file of the bundle with its checksum)

The evaluators read the exact connection through kite_exact_parse.py, in exact rational arithmetic on plain Python with the mpmath and python-flint libraries; no computer-algebra system is imported at runtime. The parsed coefficient lists, kite-connection-A14-graded.json for the masses $(1,1,2)$ and kite-connection-A14-x3-graded.json for $(1,1,3)$, sit beside the scripts and are checked against the connection files each time a script starts.

References

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