Non-planar $q\bar q\to W^+W^-$

The non-planar two-loop double box for $W$-pair production with the exact top-quark mass — the one branch that the reference computation of these integrals did not treat — computed here and closed in explicit symbolic form on a one-dimensional kinematic line.

The integral

The object is the family of two-loop integrals

$$\mathcal T_4[\nu_1,\nu_2,\nu_3,\nu_6,\nu_7,\nu_8,\nu_9] \;=\; \int d^d l_1\, d^d l_2\; \frac{1}{D_1^{\nu_1} D_2^{\nu_2} D_3^{\nu_3} D_6^{\nu_6} D_7^{\nu_7} D_8^{\nu_8} D_9^{\nu_9}}\,, \qquad d = 4-2\varepsilon,$$

up to an overall normalization convention, with loop momenta $l_1, l_2$ and the seven propagators

$$D_1 = l_1^2,\quad D_2 = (l_1+p_1)^2,\quad D_3 = (l_1-p_2)^2,\quad D_6 = (l_2-p_2)^2,\quad D_7 = (l_1-l_2)^2,$$

$$D_8 = (l_1-l_2-p_4)^2 - m_t^2,\qquad D_9 = (l_2-p_2-p_3)^2 - m_t^2.$$

The two quark legs are massless, $p_1^2 = p_2^2 = 0$; the two $W$ legs are on shell, $p_3^2 = p_4^2 = m_W^2$; and the top-quark mass $m_t$ in the chain $D_8, D_9$ is kept exact rather than expanded away. Following the reference paper, all four external momenta are written as incoming, so $p_4 = -p_1-p_2-p_3$, and the Mandelstam invariantsthe Lorentz-invariant combinations of external momenta that a scattering amplitude can depend on are $s = (p_1+p_2)^2$ and $t = (p_2+p_3)^2$. Dividing by the one dimensionful scale leaves three dimensionless variables,

$$x = -\frac{s}{m_t^2}, \qquad y = -\frac{t}{m_t^2}, \qquad z = -\frac{m_W^2}{m_t^2}.$$

The graph is genuinely non-planarthe rung $D_7$ and the massive diagonal $D_9$ cross, with no way to redraw the graph flat while keeping the four external legs on the outer face, and the family reduces to 76 master integralsthe finite basis of independent integrals to which every integral of the family reduces via integration-by-parts identities. The closed form below is constructed with $x$ and $z$ held fixed and $y$ running — the choice, and why it loses nothing structural, is explained at the start of The result.

Why it matters

Two-loop $q\bar q\to W^+W^-$ with the top mass kept exact is a required ingredient for the NNLO QCD corrections to $W$-pair production — a benchmark process for the electroweak sector at the LHC, where the top-quark loops are not a small effect. arXiv:2409.08879 (W.-J. He, R.-Y. Zhang, L. Han et al., JHEP 12 (2024) 136) computed the planar top branches of this integral family and wrote, in as many words, "we focus primarily on planar topologies" — leaving the non-planar branch of their fig 1d uncomputed. This is that branch.

To our knowledge no non-planar two-mass two-loop $q\bar q\to W^+W^-$ integral has been published, so the 23 masters unique to this branch appear to be the first entries of that missing layer. "First computation" carries the same caveat: groups working on two-loop $q\bar q\to WW$ could have an unpublished non-planar computation in preparation.

What was hard

First, the graph is non-planar and carries the exact top mass, so the integrals depend on three independent scales at once — the regime where standard two-loop technology thins out; the reference paper did not treat this branch. Second, one six-master sector (sector 481) has a loop-by-loop cut that is an elliptic curve fibre by fibre over a rational surface, the curve non-CM and non-congruence; its $\varepsilon^0$ block is reducible to first order, so no named elliptic function class is invoked, and no ready-made class of functions (elliptic polylogarithms, Eichler integrals) was used for it. Third, a subtle trap: the reference paper's propagators are written for incoming momenta, and transplanting them onto outgoing-$W$ kinematics silently defines a different integral family — one with 98 masters and the wrong geometry — while every internal consistency check still passes. That wrong object was computed first, caught, and discarded; the corrected family's identity with the paper's fig 1d was then proven three independent ways (an exact match of the graph polynomials, an edge-by-edge match of the diagram, and numerical agreement with the paper's published planar results).

The result

The integrals depend on the three kinematic variables $x$, $y$, $z$. One choice was made up front: fix $x$ and $z$ at the generic values $(x,z) = (17/2,\,5/3)$ and work with $y$ running — a one-dimensional slice of the kinematic space. The point of the restriction was to pin down the function space on that line and understand the form of the answer there before anything else: which classes of functions appear, the letters the line realizes (fifteen in its polylogarithmic words; the family's full singularity alphabet has 36), the elliptic curve, which masters are genuinely new. Everything that follows — the exact connection, the symbolic forms, the boundary constants and the standalone evaluator — is delivered on that line and nowhere else. The line lies in the region $x,y,z>0$ of the reference paper's numerical checkpoint: $s<0$, $m_W^2<0$ and $u=(31/6+y)\,m_t^2$ above the $t\bar t$ threshold, a continuation region rather than the physical scattering region; 36 of the 79 components are complex there, and the evaluator's reference comparisons are of real parts (the imaginary parts, transported from the node bank by the same solve, agree with the complex reference values at worst 42 and 42 digits at the two reference points; python3 eval_row35.py --gate-im repeats that comparison against the imaginary parts of the reference values shipped in row35_reference_im.json). The closed forms below are the forms on this slice, and the answer as a single formula in all three variables was not computed: the extension to full three-variable kinematics is staged but was not attempted, and it remains open. The differential-equation data behind this construction are derivatives in $y$ alone, to which 11 of the family's 36 letters are invisible and which allow no integrability check across $(x,y,z)$, so no three-variable canonical-form statement can be made from them. At one test point off the line, $(x,y,z)=(8,1/3,3/2)$, the line's rotations to canonical form were carried over block by block on 28 diagonal blocks: the 15 whose rotation is a product of letters agree exactly in rational arithmetic, the other 13 do not lift from one line — a statement about one point, not a two-variable form. A $d\log$ ansatz on the full 36-letter alphabet is the route sized for the extension.

On this line, the 76 masters (plus three auxiliary integrals carried alongside) obey a first-order system $\partial_y M = A(\varepsilon,y)\, M$ whose connection $A$ is now known exactly, in block-triangular form with 36 diagonal blocks: 29 blocks solved in logarithms of rational and algebraic functions ($d\log$ formthe derivative of the solution is a rational combination of logarithmic differentials $d\log(y-a)$ — the hallmark of polylogarithmic integrals; 28 in the Laporta basis, sector 487 once its basis is exchanged for a cut-reduced one), 6 blocks with square-root letters, and one block, sector 481, whose loop-by-loop cut is fibrewise elliptic. Every component of the solution is constructed symbolically from this data. The three ingredients:

The polylogarithmic part. Most of the tower is explicit GPLgeneralized (Goncharov) polylogarithms $G(a_1,\dots,a_w;y)$ — the standard iterated-integral functions of multi-loop calculations words, of depth up to six, over fifteen letters:

$$a \;\in\; \Big\{\,0,\; -1,\; \pm\tfrac{5}{3},\; -\tfrac{31}{12},\; -\tfrac{31}{6},\; -\tfrac{41}{6},\; -\tfrac{25}{6},\; -\tfrac{7}{2},\; -\tfrac{73}{48},\; -\tfrac{175}{48},\; \tfrac{91}{24},\; -\tfrac{215}{24},\; r_+,\; r_-\Big\},$$

where $r_\pm = \tfrac{1}{12}\big({-31\pm\sqrt{561}}\big)$ are the roots of $18y^2+93y+50$. A typical entry of the first $\varepsilon$-layer of the main tower reads

$$-\tfrac{2}{3}\,G(0;y) \;+\; \tfrac{187}{18}\,G(-1;y),$$

and the deeper layers are words in the same letters with exact rational coefficients.

The simplest block, written out in full. Sector 487 (four masters) is solved by exact rational functions. With $y = \tfrac13 + \tfrac{4}{15}t$ and $D(t) = t^2 + \tfrac{175}{8}t + \tfrac{1475}{8}$, a fundamental system of solutions is

$$Y_1 = \begin{pmatrix} \dfrac{\tfrac{1475}{8} - 5t}{D} \\[2pt] 0 \\ 0 \\[2pt] \dfrac{85\,t}{D} \end{pmatrix},\qquad Y_2 = \begin{pmatrix} 0 \\ 1 \\ \tfrac{4}{165}\,t \\ 0 \end{pmatrix},\qquad Y_3 = \begin{pmatrix} 0 \\ 0 \\ 1 \\ 0 \end{pmatrix},\qquad Y_4 = \begin{pmatrix} -\dfrac{\tfrac{15}{4}\,t}{D} \\[2pt] 0 \\ 0 \\[2pt] \dfrac{\tfrac{1475}{8} + \tfrac{215}{8}t}{D} \end{pmatrix},$$

verified by an exact zero-remainder substitution into the system. The other rational and radical blocks have the same character with more terms.

The sector-481 block. The one block with an elliptic maximal-cut curve is sector 481. The curve itself is elliptic, with non-constant $j$; the sector's $\varepsilon^0$ differential-equation block, by contrast, is reducible to first order — on both lines and along two mass-scaling rays it factors into a chain of six first-order operators, with two exact rational solutions and four iterated integrals of radicals, and the certified basis of six homogeneous solutions used below lies in that chain's solution space (qqww-t4-sec481-chain.py re-runs the factorisation on the shipped connection). The block is regular singular at every singular point, and at generic fixed dimension an exact rational gauge transformation brings it to Fuchsian form; its local exponents lie in $\mathbb{Z}+\mathbb{Z}\varepsilon$ at every letter except the two conic letters of the chain, $Q_1 = 225y^2+5370y+19801$ and $Q_2 = 900y^2-12180y-7751$ on the line, where the exponent is $-3/2$ independently of the dimension — so no $\varepsilon$-factorised form with rational letters exists on this line, while one over the function field extended by the two conic square roots is not excluded. The maximal cutthe integral with every propagator put on shell — it isolates the sector's intrinsic geometry of its integrals localizes onto the curve

$$w^2 \;=\; g_1(z_4)\, G_2(z_4),$$

with $z_4, z_5$ the two Baikov variablesintegration variables in a representation where the propagators themselves become coordinates of the cut and, explicitly,

$$g_1 = -\tfrac{1}{4}\left(y^2 - 2yz_5 + 2y + 4z\,z_4 + z_5^2 - 2z_5 + 1\right),$$

$$G_2 = \tfrac{1}{16}\Big(x^2y^2 + 2x^2y z_4 + 2x^2 y + x^2 z_4^2 - 2x^2 z_4 + x^2 + 2xy^2 z_5 - 4xyz\,z_4 - 2xyz\,z_5 + 2xy\,z_4 z_5 - 4xy\,z_4$$

$$\qquad\qquad + 2xy\,z_5 + 4xz^2 z_4 - 4xz\,z_4^2 + 2xz\,z_4 z_5 + 4xz\,z_4 - 2xz\,z_5 + y^2 z_5^2 - 2yz\,z_5^2 + z^2 z_5^2\Big).$$

This is a cubic in $z_4$ — an elliptic curve for each value of $z_5$, so the geometry is an elliptic curve fibered over the $z_5$ line, and it is distinct from the curve of the paper's planar branches. It carries a rational 2-torsion point, which places its $j$-invariant on the classical modular curve $X_0(2)$:

$$j \;=\; \frac{(t+256)^3}{t^2}, \qquad t = \frac{6787441}{213725}\ \text{ at the anchor point},$$

but the curve itself is non-CM and non-congruence — outside the standard catalogues — and the surface it fibres is rational (Kodaira fibres $2I_1+5I_2$, Euler number 12), so this block is solved by an order-by-order variation of parameters on a certified basis of six independent solutions of its homogeneous system, rather than by any named class of special functions.

Scope. The boundary data for all 79 components now comes from a transport table with no auxiliary-mass-flow (AMFlow) input anywhere on its value path — 79 constants, one per component, re-certified at every load, real and imaginary parts both — 23 of them limits of the published planar polylogarithms, 11 derived closed forms, 5 derived written integrals, and 40 (the radical blocks and the sector-481 block among them) defined by a written transport integral in an auxiliary-mass variable, recomputable at any requested precision; per-row closed forms for those 40 are not claimed. A standalone evaluator, eval_row35.py, transports the exact connection at any requested precision and evaluates any single master in isolation. It runs on plain Python with the mpmath and python-flint libraries and reads three data files placed beside it: row35_data.json (the exact connection and the independent reference values), row35_radical_feed.json (the closed forms of three radical blocks), and the boundary node bank row35_nodes.tar.gz, unpacked beside the script; the two engine modules vopclose_vendored.py and epsfan_vendored.py travel with it. The boundary-regeneration chain downloads with the bundle as row35_chain.tar.gz (1.8 MB): tar xzf row35_chain.tar.gz beside the script unpacks row35_chain/, 763 members (the two drivers, their Python dependencies, the exact boundary tables, the reduction tables, the fixed-dimension oracle and the recorded reports), each listed with its sha256 in the chain's own manifest. The files the bundle does not host are listed in CONTAINERS.md by name, sha256 and size and are available on request until a Zenodo-class home exists: the two closed-form containers of sectors 421/453 (about a gigabyte together; placed beside the script they are checked and used automatically, and six masters then fold from closed form instead of riding the transport) and the regenerated node pair. With row35_chain/ unpacked beside it, eval_row35.py --boundary-recompute verifies every member against that manifest and then runs the from-definition chain — the exact boundary tables seed the transport in the auxiliary-mass variable and the arrival values are compared with the recorded ones (--boundary-recompute 60 --gate-bc-only, about a minute, reproduces the recorded 39–40 digits on all six boundary systems; the default form re-solves the chain at $\varepsilon=1/1013$ against the fixed-dimension oracle, and --regenerate-node re-solves any bank node from the definition at a chosen precision) — and it refuses by name if the directory is absent or any member is missing or altered; the value path stays the certified bank, whose 40-digit fan-claim cap lifts once the containers are distributable and a deeper bank is regenerated from the chain. Of the 76 masters, 53 map exactly onto the published planar results through a symbolically verified dictionary; the 23 masters of the non-planar tower are new.

A second line. The connection's exact dependence on $y$ was reconstructed along a second line as well, $(x,z)=(17/2,7/4)$, from 91 Kira reductions, 89 of them in the fit: 1220 of the 1240 entries close directly and the remaining twenty, all in masters 75–78, close by the same known-denominator fit that closed them on the first line. At the two reductions kept out of the fit, $y=296/101$ and $y=306/103$ at $d=7/3$, every one of the 1240 entries reproduces the Kira reduction exactly, in rational arithmetic. This is an exactness check on that line, not a reconstruction of the full two-variable function, which remains open. The first line has the same check: the two Kira reductions behind its 1240/1240 verdict, at $y=29/211$ and $y=17/139$ with $d=890/223$, ship in row35_L1/; python3 eval_row35.py --line L1 --verify-exact reproduces both entry by entry, 1240/1240, in rational arithmetic. The evaluator repeats the comparison from the shipped data with python3 eval_row35.py --line L2 --verify-exact, reading the second-line connection connection_L2.json, the two reductions withheld_y296o101.json and withheld_y306o103.json, and the point list points.json placed in a folder row35_L2/ beside it; the value tiers run on the first line only. The canonical-basis question is settled line by line as well: of the 36 diagonal blocks of the line connection, 29 admit a rational rotation to an $\varepsilon$-factorised dlog form on the first line and 29 on the second (28 in the Laporta basis of this count and sector 487 in the basis with $[1,1,1,-1,-1,1,1,1,1]$ in place of $[1,1,1,-2,0,1,1,1,1]$: with an exact cut reduction on each line and a rational rotation, the same block is an $\varepsilon$-factorised dlog form, regular at $y=\infty$ on both lines, with the two letters $18y^2+93y+50$ and $18y^2+93y+203$ on the first line and $16y^2+80y+49$ and $16y^2+80y+185$ on the second), the same six sectors stopping on both (two Moser-irreducible, four with an $\varepsilon^0$ obstruction), and the off-diagonal completion of the twelve-row tower stops on both lines in the same class — at an $\varepsilon$-dependence of the residual connection that no constant or Magnus regrading removes. That is a statement about each line, not about the two-variable function, which remains open. python3 eval_row35.py --line L1 --canonical-blocks (and --line L2) re-derives that census in exact rational arithmetic from the shipped block data, row35_L1/blocks_epsform.json and row35_L2/blocks_epsform.json, and checks the line-to-line gauge in canonical_gauge_L1_L2.json.

Everything above was validated against independent evaluations at kinematic points never used in the construction: every one of the 79 components agrees to at least 37 digits (for 21 of the 79 the line connection vanishes identically, so at those the comparison re-tests the boundary constant itself; the transport is exercised on the other 58), and where this branch overlaps the published planar results, the paper's own functions are reproduced to 47 digits. The agreement grows with working precision exactly as an exact form must.

Downloads: the closed forms, the alphabet, the sector bases, and the sector-481 curve data with the derivation of its $X_0(2)$ anchor are written out in qqww-t4-expression.md; the boundary constants of all 79 components are digit strings in qqww-t4-boundary.json, and the closed-form search on the 40 transport-defined rows — leading Laurent coefficient an exact rational on 22 of them and an exact weight-one logarithm form on 2 more, no integer relation found for the other sixteen against the family's twelve $\Gamma$-vacuum and radical constants, per-row closed forms not claimed — is recorded verbatim in qqww-t4-ring-pass.json. The second-line data (connection, the two check-point reductions, the point list, and lines.json recording both lines) are the five files of row35_L2/ linked in the paragraph above; the written Mellin–Barnes representations of the corner rows 69, 42 and 43 — row 69's evaluated independently at two points (10.27 and 10.28 digits against the stored boundary values; 14.04 in the deepest run, at one working precision), row 42's an exhibit that reproduces the stored value to eight digits and is not established as a claim, row 43's written with its contour prescription but not evaluated and carrying no digit — together with the objects they are read from, are under vendor_row35_mb_corner/ (eval_row35.py --mb-corner-provenance prints them by object); MANIFEST.sha256 lists every shipped file, and CHANGES.md records every change to the bundle since it was first served; the first line's two reductions are withheld_y29o211.json and withheld_y17o139.json. The full account, with the per-class verification table, is Sec. 5.9 of Polylogarithmic, elliptic, and Calabi–Yau Feynman integrals from a hybrid bootstrap (PDF).

References

← back to the gallery