Non-planar hexa-box

A two-loop, five-point, non-planar integral family with two off-shell legs — the two-loop layer that $WW$+jet and $ZZ$+jet collider predictions still lack — solved here: thirteen new master integrals, with the eight-propagator top sector in explicit closed form through its finite part.

The integral

The two-mass non-planar hexa-box (two external legs off shell at one common mass, $p_1^2=p_3^2=m^2$; the evaluator files call the family 'one-mass-pair') is the family of two-loop five-point integrals

$$I[\nu_1,\dots,\nu_{11}] \;=\; \int d^d k_1\, d^d k_2\; \frac{D_9^{-\nu_9}\, D_{10}^{-\nu_{10}}\, D_{11}^{-\nu_{11}}}{D_1^{\nu_1}\, D_2^{\nu_2}\, D_3^{\nu_3}\, D_4^{\nu_4}\, D_5^{\nu_5}\, D_6^{\nu_6}\, D_7^{\nu_7}\, D_8^{\nu_8}}\,, \qquad d = 4-2\varepsilon,$$

up to an overall normalization convention, with loop momenta $k_1, k_2$, the eight massless propagators

$$D_1 = k_1^2,\qquad D_2 = k_2^2,\qquad D_3 = (k_1-k_2+p_4)^2,\qquad D_4 = (k_2-p_4)^2,$$

$$D_5 = (k_1+p_1+p_2+p_3+p_4)^2,\qquad D_6 = (k_1-k_2+p_1+p_2+p_3+p_4)^2,$$

$$D_7 = (k_1-k_2+p_2+p_3+p_4)^2,\qquad D_8 = (k_1-k_2+p_3+p_4)^2,$$

and the three irreducible numerators $D_9=(k_1+p_1)^2$, $D_{10}=(k_2+p_2)^2$, $D_{11}=(k_2+p_3)^2$, which appear only with $\nu_i\le 0$. The five external legs carry momenta $p_1,\dots,p_5$ with $p_1+p_2+p_3+p_4+p_5=0$: three are massless and on shell, $p_2^2=p_4^2=p_5^2=0$, and the two non-adjacent legs $p_1,p_3$ are off shell at equal virtuality $p_1^2=p_3^2=m^2$ — a $W^+W^-$ or $ZZ$ pair plus a jet. The graph is genuinely non-planarit contains a $K_{3,3}$ minor, the Kuratowski obstruction: no redrawing puts all eight lines in the plane with the five legs on the boundary.. The kinematics carries six independent scales: the five cyclic two-particle invariants $s_{12}, s_{23}, s_{34}, s_{45}, s_{15}$, where $s_{ij}=(p_i+p_j)^2$, plus the common virtuality $m^2$. Equal virtuality covers $WW$+jet and $ZZ$+jet; the unequal-mass ($WZ$+jet) family is the seven-scale generalization. Everything is computed in dimensional regularization, as a Laurent series in $\varepsilon$.

The family reduces to 98 master integralsthe finite basis of independent integrals to which every integral of the family reduces via integration-by-parts identities — here spread over 62 sectors; 76 of the masters live in known sub-topologies, leaving 22 genuinely new. in 62 sectors. The graph automorphism $\sigma$ that swaps $p_1\leftrightarrow p_3$ at equal virtuality pairs the 22 genuinely new masters down to 13 independent ones in six $\sigma$-orbits, with the 8-propagator top sector (sector 255, three masters) at the apex. The maximal cutthe integral with every propagator put on shell — it isolates the top-sector geometry: here a genus-zero curve, so the family is polylogarithmic, not elliptic. of the top sector is genus zero: the family is hard for bookkeeping reasons, not geometric ones. No kinematic slice is taken — all six scales stay independent — and the closure runs along one-dimensional paths through the six-scale space (three sampled segments, described below; the top sector on two of them), the first from the base point $(s_{12},s_{23},s_{34},s_{45},s_{15};m^2)=(-3,-5,-7,-2,-11;1)$, where the boundary data are set, to the far point $(-3,-5,-7,-11,-17;2)$, reserved for verification. Each of the ten sub-orbit masters is determined from its leading pole through order $\varepsilon^{2}$, and the three top-sector masters through the finite part $\varepsilon^{0}$. At $m^2=0$ the graph lands exactly on one of the known non-planar hexa-box topologies of the one-mass literature — the anchor for its boundary data.

Why it matters

$VV$+jet at NNLO is on the community wishlist (Les Houches 2023: NNLO QCD desired, currently unknown), and the two-loop double-virtual integrals are the missing layer. The literature surrounds this family on three sides without touching it: the complete planar two-mass five-point set exists (Abreu–Chicherin–Sotnikov–Zoia, JHEP 10 (2024) 167), one planar two-mass penta-box family was bootstrapped at the differential-equation level (Jiang–Liu–Xu–Yang), and the non-planar hexa-boxes are known with exactly one off-shell leg (Abreu–Ita–Page–Tschernow; also via the simplified differential-equation approach, and completed into the one-mass pentagon functions in PRL 132, 141601). The one-mass papers position themselves as ingredients for single-boson+jets; the two-mass non-planar layer that $WW$+jet needs has, as far as a fresh June-2026 literature sweep can tell, no published computation. The 13 masters closed here are that layer's first non-planar entries — including the top sector, through its finite part.

The novelty claim carries a date stamp: "no published computation" means first-public as of June 2026. The groups behind the one-mass non-planar and planar two-mass sets are visibly working toward $VV$+jets, so a computation could exist in preparation.

What was hard

First, the width: six independent scales and no kinematic slice. A from-scratch analysis of the family's singularity structure is out of reach at this width, so the working alphabet was assembled as a union, closed under the $p_1\leftrightarrow p_3$ symmetry, of the alphabets of the neighbouring known families — 374 letters from the one-mass non-planar hexa-box (344 rational and 30 algebraic), 49 from the massless pentagon, 2 from the two-mass four-point — plus 12 genuinely new two-mass letters (9 rational, 3 algebraic), 437 in all (404 rational, 33 algebraic). One-variable slice differential equations confirmed the coverage: every singularity the system actually has is a specialization of a letter in the union. Counting came before computing: a symbol-level cascade over this alphabet (integrability, first-entry, last-entry and parity constraints) cut the free constants to a handful per weight before anything expensive was launched, and a singularity-completeness check run after the fact certified every first-type singularity of the top sector as a letter already present in the alphabet.

Second, the $\varepsilon$-structure. All thirteen masters are closed by one mechanism, exact-rational transport along the path; what varies is the $\varepsilon$-structure met on the way, and a survey of the 31 diagonal blocks of the path system finds constant rational rescalings $\varepsilon$-factorizing at most 17 of them. Orbit by orbit, two become $\varepsilon$-linear after one rational rescaling (a sunrise $\Gamma$-ratio); two carry a spurious $\varepsilon$-pole that moves with the kinematics and had to be expanded around layer by layer (the resulting $\varepsilon$-factorised forms are local to the expansion point; nothing verified below relies on them); sector 223 provably admits no global diagonal $\varepsilon$-rescaling at all — spurious poles injected by the reduction algorithm itself grow in degree with the $\varepsilon$-layer — and was closed by multi-layer Taylor transport instead. The top sector rides on the full 98-master system: its rational path connection reaches degree 83, and a fresh reduction of the family at a new kinematic point costs hours, so both the sampling and the independent verification were expensive. Its closure came from a variation-of-parameters layer recursion on the exact rational path differential equation, with the deeper layers' denominators built from validated lower-layer factors rather than fit blind. One sampled point sat on a degenerate slice where the reduction returns a wrong-but-internally-consistent answer; an off-curve consistency check caught it and now screens every sample.

The result

The family is polylogarithmic: the maximal cutthe integral with every propagator put on shell — it isolates the top-sector geometry; genus zero here, so the answer is polylogarithmic, not elliptic of the top sector is a genus-zero curve, so every master is an iterated integral of d-log formslogarithmic differential forms $d\log(\text{letter})$ — the building blocks whose iterated integrals give the polylogarithms of multi-loop calculations in the six scales. The alphabet those forms are drawn from is the structural half of the answer: a 437-letter union assembled from the neighbouring known families (404 rational letters and 33 algebraic ones carrying $\sqrt{\mathrm{Gram}_5}$, the square root of the Gram determinant of the external momenta), extended by two genuinely new sets found here. Writing $P_{abcd}=\mathrm{tr}_4(p_a p_b p_c p_d)$ for the Dirac trace of four external momenta, the 15 new parity-odd letters are the ratios

$$\frac{P_{abcd}-\sqrt{\mathrm{Gram}_5}}{P_{abcd}+\sqrt{\mathrm{Gram}_5}}\,,$$

each with $P_{abcd}^2-\mathrm{Gram}_5$ factoring over the rational alphabet and each reducing at $m^2\to 0$ to a known massless odd pentagon letter; one further even letter identified in the six scales and absent from the union is

$$X_8 \;=\; -\,\frac{4\,\mathrm{Gram}_3(p_3,p_4,p_5)}{s_{45}}\,,$$

with $\mathrm{Gram}_3$ the Gram determinant of the three momenta listed; five more irreducible degree-2 path factors are recorded as candidates.

One choice was made up front: the explicit solution is constructed along kinematic paths through the six-scale space rather than over the whole space at once. The path construction pins down the function space and the form of the answer — the alphabet above, which weights appear, the one-fold radical structure below. What follows are the closed forms along the path, and the result is a path closure: the masters are evaluable along sampled kinematic segments, not yet as a full six-scale function. The segments are three: the first varies $s_{45}$, $s_{15}$, $m^2$ at fixed $s_{12}$, $s_{23}$, $s_{34}$ and carries all 98 masters; the second varies all six scales, including complex-$t$ detours around the path's poles, and carries the five subsector orbits only; the third varies $s_{12}$, $s_{23}$, $s_{34}$ at fixed $s_{45}$, $s_{15}$, $m^2$ and carries all 98 masters, its transport equation reconstructed as exact rational functions of the path parameter from samples at 176 nodes and, at three fixed half-integer values of $d$, reproducing an independent evaluation at the segment's end point to at least 61 digits on all 98 masters (the integer values of $d$ are excluded because the masters have poles there).

The top sector, explicit along the path. Writing $t\in[0,1]$ for the path variable — the line in six-scale space from the base point $(-3,-5,-7,-2,-11;1)$ to the verification point $(-3,-5,-7,-11,-17;2)$, run on a detour into the complex plane around the path's poles — each Laurent coefficient of the three top-sector masters is a finite one-fold sum

$$n_{i,k}(t) \;=\; \sum C\,\sqrt{\mathrm{rad}(t)}\;R(t)\,G_{\text{word}}(t)\,,$$

where $n_{i,k}$ is the coefficient of $\varepsilon^{k}$ in top master $i$, each $C$ is an exact constant, each $R(t)$ is an exact rational kernel, each $G_{\text{word}}(t)$ is an iterated integral labelled by a word in the path letters, and the radical is the quartic

$$\mathrm{rad}(t) \;=\; Q_4(t) \;=\; 3721\,t^4+6724\,t^3+11824\,t^2+16400\,t+7040\,,$$

whose square root the top block's homogeneous solutions carry, with indicial eigenvalues $(\tfrac12,\tfrac12,0)$ and $Q_4(1)=45709$ at the endpoint. Along this straight-line parametrization, on which the square root of the Gram determinant restricts to the square root of an irreducible quartic in $t$ (a genus-one radical), no polylogarithmic form over rational or algebraic letters of $t$ exists in the raw or the rotated basis (a kernel-rank certificate); this is a property of the parametrization, not of the integrals, and the one-fold representation is the delivery form on the line.

At $\varepsilon^{-4}$ the forms collapse to exact rationals,

$$m_0[\varepsilon^{-4}] = -\tfrac{571}{1079925}, \qquad m_1[\varepsilon^{-4}] = \tfrac{46591}{8639400}, \qquad m_2[\varepsilon^{-4}] = -\tfrac{854177}{8639400},$$

where $m_0,m_1,m_2$ are the three top-sector masters; along the path these three coefficients are in fact rational functions of $t$, with common denominator $m^2(-s_{15})\,s_{45}^2\,s_{35}\,s_{14}=(t+1)(6t+11)(9t+2)^2(10t+7)(16t+9)$ evaluated on the path and cubic, quartic and quintic integer numerators (written out in hexabox-expression.md, Section 2a), reducing to the values above at $t=1$. In the $\varepsilon$-factorized endpoint rotation $\hat g = (\sqrt{45709}\,g_0,\ \sqrt{45709}\,g_1,\ g_2)$ — with $(g_0,g_1,g_2)$ the rotation of the three masters into a uniform-weight basis and $\sqrt{45709}$ exactly the path radical $\sqrt{Q_4(1)}$ — the leading rationals are

$$\hat g_0[\varepsilon^{-4}] = \tfrac{123164227}{3427200}, \qquad \hat g_1[\varepsilon^{-4}] = \tfrac{203357363}{3427200}, \qquad \hat g_2[\varepsilon^{-4}] = -\tfrac{940511}{1713600},$$

with denominator $3427200 = 2^7\, 3^2\, 5^2\, 7\, 17$, alphabet-smooth primes only (in the master normalization the leader carries the radical explicitly, $\hat g_0[\varepsilon^{-4}] = \tfrac{123164227}{3427200}\sqrt{45709}$). The $\varepsilon^{-3}$ imaginary parts are exact $\pi$-rationals,

$$\frac{\mathrm{Im}\,\hat g_0[\varepsilon^{-3}]}{\pi} = \tfrac{33402719852867}{1005262876800}, \qquad \frac{\mathrm{Im}\,\hat g_1[\varepsilon^{-3}]}{\pi} = \tfrac{75672918098083}{1005262876800}, \qquad \frac{\mathrm{Im}\,\hat g_2[\varepsilon^{-3}]}{\pi} = -\tfrac{375001639771}{502631438400},$$

with denominator $1005262876800 = 2^7\, 3^4\, 5^2\, 7\, 13\, 17\, 23\, 109$ — again alphabet primes only. These six constants — three rational $\varepsilon^{-4}$ leaders and three $\pi$-rational $\varepsilon^{-3}$ imaginary parts — are the complete named boundary set. At $\varepsilon^{-2}$ through $\varepsilon^{0}$ the same construction gives layered iterated-integral closed forms with exact kernels, each layer's certified closed functions feeding the next as sources.

The massless boundary. At $m^2=0$ the sector-219 master $m_{219}$ collapses onto an explicit combination of the published massless non-planar pentagon basis, with integer coefficients over $\sqrt{\Delta}$:

$$\varepsilon^4 e^{2\varepsilon\gamma_E}\, m_{219}\Big|_{m^2=0} \;=\; \frac{1}{\sqrt{\Delta}}\Big({-}6\,I_{55} - 3\,I_{3} - 3\,I_{18} + 3\,I_{32} - 3\,I_{44} + 3\,I_{51} - 3\,I_{52}\Big),$$

where $\gamma_E$ is the Euler–Mascheroni constant, the $I_k$ are elements of the d-log basis of arXiv:1809.06240 in that paper's numbering, and $\Delta$ is the massless five-point Gram determinant, with $\mathrm{Gram}_5|_{m^2=0}=\Delta$ exactly. One orbit (sector 231) vanishes identically at $m^2=0$ and gets no constraint from this limit; its normalization comes instead from the $\varepsilon$-factorization requirement and its $\sigma$-partner.

The sibling crossing. The same hexa-box with the two massive legs on adjacent corners — the crossing that carries a two-mass threshold — is not solved. What exists for it is one value at one point: its top sector's three masters, with two independent reduction routes agreeing on every one of the fifteen coefficients and two working precisions agreeing to 56 digits at $\varepsilon^0$.

Scope. The five sub-orbits below the top sector are certified transport representations rather than symbolic words: an exactly reconstructed rational path connection, a layer-by-layer transport recipe, and boundary data (for sector 223 the downloadable evaluator reads the transported values from archive rather than re-deriving them live — the one remaining final-form gap among the orbits). Every kernel of the top-sector recursion is an exact rational function of the path parameter: the three top-row kernels whose degree exceeds the univariate reconstruction ceiling, carried as numeric fits in earlier bundles, enter as the exact layers of the factored-route object (the fits agree with them to at least 68 digits along the path), so the attainable precision of the $\varepsilon^{-2}..\varepsilon^{0}$ coefficients is set by the 70-digit boundary seeds alone (two neighbouring kernel families in the same row have since been reconstructed as exact bivariate rationals and certified, and the four $m\ge5$ top-row kernels with their higher layers now enter as exact objects reconstructed on 101 path nodes). The closed form's own $\varepsilon^{1}$ and $\varepsilon^{2}$ layers are established: with the $\varepsilon^{3,4}$ boundary seeds in hand and those four kernels (with their $m=7,8,9$ layers) and the $m=8,9$ layers of the nine couplings to the $\varepsilon^{-4}$ master entered as exact objects, each reproducing path nodes its fit never saw, the recursion reproduces the two oracle records to 40 and 35 digits at $\varepsilon^{1}$ and $\varepsilon^{2}$ (57 against the goal-60 record, the two-precision pair 71–75, its $\varepsilon^{-4}..\varepsilon^{0}$ control at 44); with those objects entered as zero it read 0 and 0 digits. The $\varepsilon^{-3}$ real parts are a structured negative, not closed: the boundary injection carries rationals of height around $10^{12}$ into every logarithm coefficient, so naming them by integer-relation methods would need far more precision than is available; the imaginary side closes because it collapses to a single rational. The quartic $Q_4$, on the other hand, is identified: it is the five-point Gram determinant along the path, $Q_4(t)=\mathrm{Gram}_5(s(t))$ identically in $t$, so $Q_4(0)=7040$ and $Q_4(1)=45709$ are the values of $\mathrm{Gram}_5$ at the base point and the verification point. On the path it is the second-type (Gram) Landau locus of the eight-propagator sector, and a direct Euler-characteristic-drop test confirms it ($\chi_{\rm reg}$ drops from 101 to 90 on the $Q_4=0$ locus); an earlier negative was a candidate-generation gap of the finite-field letter reconstruction, and the earlier label caveat is withdrawn.

Boundary data. The 447 tagged boundary constants belong to the derivation run's evaluator (eval_row24.py, the evaluator of the paper's row 24), not to the downloadable evaluator above, which consumes the sector-219 seeds (13 rows, 50 digits) and the sector-255 seed (70 digits) and never reads them. Their provenance is an offline recompute of the transport chain: the chain's generator (chain_transport.py, sha-pinned) is on the record beside its landings; the evaluator's recompute mode reassembles all 447 strings from those landings identically, and all 447 regenerated from definition at 90 and 120 working digits reproduce the stored strings digit for digit (the imaginary parts of the real rows are structural zeros, named and not counted). The retired interim values survive only as a cross-check that evaluator re-verifies — refusing to run on any mismatch — at every load. Ten of the 447 are, in addition, determined by a residue-constraint system: regularity of the solution at the five detour points of the transport path (apparent singular points of the kernels) is imposed as residue conditions and the resulting null-space problem solved numerically, which fixes exactly two tagged towers of five constants each ($k=0,\dots,4$); these ten are numerical values without closed forms. Beyond the six constants named exactly above, the other 457 endpoint constants of the closure are certified numbers (about 60 digits, none named) — a multi-point integer-relation search over the expected constant ring returned zero matches out of 457. That census (six named plus 457 unnamed) is organized by a different tag set from the 447 tagged constants of the transport-chain recompute described above. The downloadable evaluator re-derives the entire top-sector closure from the exact rational path differential equation on every call, with nothing archived between runs (its sector-219 recompute is seeded at the fit point, as the script itself discloses).

Everything on this page was validated against independent evaluations at kinematic points never used in the construction, to at least 30 digits; the fifteen top-sector coefficients from $\varepsilon^{-4}$ through $\varepsilon^{0}$ match an independent evaluation run at a 60-digit precision goal to 66–68 digits, that evaluation's own digit floor.

Downloads: the full forms and exact kernels are in hexabox-expression.md; the live evaluator and its data are hexabox-evaluate.py · hexabox-data.json · hexabox_vop.py · hexabox_vop_lib.py · hexabox-vop-data.json.gz. The evaluator covers both sectors: run it with --sec both (or --sec 255 for the top sector alone); without the flag it evaluates sector 219 only. A second kinematic point, $(s_{12},s_{23},s_{34},s_{45},s_{15},m^2)=(-5,-7,-11,-13,-19,3)$, is on the record as a pair of independent evaluations at two working precisions: the three top-sector masters through $\varepsilon^0$ agree component by component to at least 56 digits (27 significant components; three that vanish are named, not counted). The six-scale function itself is not established, so the evaluator does not compute this point; hexabox-evaluate.py --record-point P2 re-runs the record's comparison on the two shipped evaluation files in record_P2/ and prints what the record is and is not. The bundle now carries the $\varepsilon^{1},\varepsilon^{2}$ succession: hexabox-evaluate.py --kmax 5 runs the top-sector recursion to $\varepsilon^{2}$ on the succeeded data bundle and gates the $\varepsilon^{1}$ and $\varepsilon^{2}$ orders against the served 60-digit strings and the two vendored oracle records at a bar of 30 digits (minutes), reproducing the gate of record — 40 and 35 digits, the $\varepsilon^{-4}..\varepsilon^{0}$ control at 44 — with --dps-double for the two-precision pair and --mutate-eps12 as its control; a bundle short of the reach is refused by name. The objects of that gate — the eleven exact kernel candidates, the two oracle records, the gate receipt and the support audit — are in vendor_row24_eps12/ (VENDOR_MANIFEST.sha256), each pinned in the evaluator. The two-precision boundary of the whole family at the verification point — all 98 masters evaluated at precision goals 30 and 60 at every order $\varepsilon^{-4}..\varepsilon^{4}$, the strings verbatim with the matched digits per component (at worst 48 digits through $\varepsilon^{0}$, 42 through $\varepsilon^{2}$, 32 over all orders) — is in vendor_row24_gatepoint/ (VENDOR_MANIFEST.sha256); hexabox-evaluate.py --gatepoint recomputes every component's digits from the two strings and checks those floors against the record (seconds), with --mutate-gatepoint as its control. The three top-row kernels' exact layers, which the data bundle now carries in place of the earlier numeric fits, are documented in vendor_row24_kernels/, and the 447 sector-255 boundary constants regenerated from definition at 90 and 120 working digits in vendor_row24_seeds/; hexabox-evaluate.py --boundary-regenerate runs the top-sector recursion on the regenerated seeds instead — they agree at the two precisions to at least 78 digits, and the kept seed cells outside their coverage that still carry the bundle's 70-digit strings hold the input floor at 69 digits — and prints the same oracle comparison as the default run (minutes). The third segment's transport equation — six exact fixed-$d$ slices, their end-point comparisons (at least 61 digits on all 98 masters), the generic-$d$ connection, the instruments and a seconds-long self-check — is in vendor_row24_pathC/; its 176 node samples are available on request. MANIFEST.sha256 lists every file of the bundle with its checksum and CHANGES.md records the bundle's changes.

References

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