Central-mass double box

An independent computation of the scalar top-sector master of the central-rung-mass planar double box — explicit iterated integrals in both Mandelstam variables through weight four, every constant classical — agreeing with the published result of Schwanemann and Weinzierl (topology C of arXiv:2412.07522).

The integral

The object is the two-loop planar double box with four massless external legs and one internal mass $m$ on the central rung,

$$I[\nu_1,\dots,\nu_7] \;=\; \int d^d k_1\, d^d k_2\; \frac{1}{D_1^{\nu_1} D_2^{\nu_2} D_3^{\nu_3} D_4^{\nu_4} D_5^{\nu_5} D_6^{\nu_6} D_7^{\nu_7}}\,, \qquad d = 4 - 2\varepsilon,$$

up to an overall normalization convention, with loop momenta $k_1, k_2$ and the seven propagators

$$D_1 = k_1^2,\quad D_2 = (k_1+p_1)^2,\quad D_3 = (k_1+p_1+p_2)^2,\quad D_4 = k_2^2,$$

$$D_5 = (k_2+p_1+p_2)^2,\quad D_6 = (k_2+p_1+p_2+p_3)^2,\quad D_7 = (k_1-k_2)^2 - m^2.$$

The one mass sits on $D_7$, the central rung. The external momenta all square to zero, $p_i^2 = 0$, so the kinematics reduce to the two Mandelstam invariantsthe Lorentz-invariant combinations of external momenta that a scattering amplitude can depend on $s = (p_1+p_2)^2$ and $t = (p_2+p_3)^2$; the third invariant is dependent, $s + t + u = 0$. The rung mass sets the unit, $m^2 = 1$, leaving the two ratios $s/m^2$ and $t/m^2$. The family reduces to a small basis of master integralsthe finite basis of independent integrals to which every integral of the family reduces via integration-by-parts identities, and $J$ denotes the top-sector master with all seven propagators.

The leading-singularitythe maximal-cut residue of the integrand; stripping it off as a prefactor leaves a "pure" function of uniform transcendental weight prefactor is $R = s^2(t-1)$ at $m^2 = 1$, and the closed form below is for the pure function

$$g(\varepsilon) \;=\; R\,\varepsilon^4 e^{2\varepsilon\gamma_E}\, J \;=\; \sum_w g^{(w)}\,\varepsilon^w,$$

where $\gamma_E$ is the Euler–Mascheroni constant. With the prefactor stripped, the leading term is exactly $g^{(0)} = 1$.

Why it matters

The planar double box is old ground in its simpler dressings: Smirnov solved the fully massless case analytically in 1999, and Caron-Huot and Henn gave the massive outer-mass version, with its single square root, as iterated integrals in 2014. The variant with the mass on the central rung — the $\gamma$-$Z$-$\gamma$ ladder of NNLO electroweak Møller and Bhabha scattering, equivalently the $g$-$V$-$g$ ladder of mixed QCD-electroweak four-fermion scattering — was computed by Schwanemann and Weinzierl in December 2024 (topology C of arXiv:2412.07522, SciPost Phys. 18, 172 (2025): all nineteen master integrals in multiple polylogarithms through weight four, with public code). This page gives an independent computation of the scalar top-sector master at unit internal mass, derived without their result as input, in a different (Euclidean, fibration-basis) representation; on the $s+i0$ sheet their canonical master $J^{C}_{17}$ equals $(s/m^2)^{2\varepsilon}\,g(\varepsilon)$ in the normalization above, and the two agree through weight four at Euclidean points to the working precision.

On paper a polylogarithmic two-loop box is the easy class, and this one has a seven-letter rational alphabetthe finite set of "letters" — rational or algebraic functions of the kinematics — whose iterated d log integrals span the answer, no square root and no elliptic curve. What made it hard for a value-fit was conditioning: the number of candidate weight-$w$ functions grows roughly like $k^w$ with alphabet size $k$ — order a hundred at weight 2, thousands at weight 4 — and a raw fit over an early ansatz that had dropped one of the seven letters could not be made to close by adding samples.

What was hard

A naive weight-graded fit stalls at weight 2. Weight 1 falls immediately to a fit against high-precision samples, with exact integer coefficients. At weight 2 the first ansatz alphabet, taken from a loop-by-loop maximal-cut scan, had dropped one of the seven letters — $s+t-st$, which the dilogarithm $\mathrm{Li}_2(s+t-st)$ below needs — and a raw fit over it was ill-conditioned: a scan in the number of sample points leaves a flat residual that does not shrink with precision, and adding samples does not close it. A candidate eighth letter, $s^2-t$, proposed by an early maximal-cut scan, lowered the residual without closing the fit, and the two square roots the loop-by-loop scan had returned, $r_1 = \sqrt{s(s+4)}$ and $r_2 = \sqrt{s(s-8)}$, changed nothing; neither the roots nor $s^2-t$ appear in any word of the final tables, and the alphabet is the seven rational faces of the principal $A$-determinant. Over those seven letters, with symbol integrability and the physical first-entry set $\{s,\ 1-s,\ 1-t\}$ added as constraints, the direct fit closes weights 0–2 as the 8-term formula below, checked to 108 digits at kinematic points never used in the fit. At weight 3 a raw fit is span-deficient — a classical-polylogarithm product ansatz matches its own samples to $2\times10^{-5}$ and fresh points to only three digits — so the higher weights go through symbol constraints and a second route.

Differential-equation transportintegrating the integral's first-order linear system dM/dt = A(ε,t)·M from a single high-precision boundary point to any other point in kinematic space sidesteps the wall, because its cost scales with the number of master integrals — sixteen on the transport slice — and is blind to how many letters the answer uses. BootLoops built the $t$-derivative system directly from the integration-by-parts reduction, took a single high-precision boundary value at $t = -\tfrac13$, and integrated the system along $t$; a second transport between two other points, which never touches the boundary, gives an independent non-circular check. One trap: with four massless legs $s + t + u = 0$, so the $t$-derivative operator must impose $\partial u/\partial t = -1$, and the natural-looking choice $\partial u/\partial t = 0$ over-constrains the operator and corrupts the raised-index masters; with $-1$ the integrand identity matches an independent symbolic differentiation exactly.

Transport gives the function as a certified system plus a boundary value, and was at first the only handle on weights 3 and 4. The explicit two-variable words at weights 3 and 4 below come from a constrained symbol fit with no new reduction: alphabet closure, first entry in $\{s,\ 1-s,\ 1-t\}$ and word-level matching to the $s=-1$ slice are imposed as exact linear constraints before any numerics (cutting the 281-column weight-3 ansatz to 42 free parameters), and least squares on certified auxiliary-mass-flow samples then fixes every coefficient as an exact rational (at weight 4, 44 additional sample points, used as fit data only). The transport survives as the independent cross-check. The trade-off this diagram exposed — a raw direct fit stalls on conditioning long before transport does — set the routing for every later polylogarithmic target in BootLoops.

The result

Every Laurent order of the top-sector master from $\varepsilon^{-4}$ through $\varepsilon^0$ — transcendental weights 0 through 4 of the pure function $g$ — is an explicit iterated integral in both variables, over the seven-letter rational alphabet

$$\{\,s,\ t,\ s+t,\ 1+s,\ 1-s,\ 1-t,\ s+t-st\,\} \qquad (m^2 = 1).$$

The geometry is genus zero — no elliptic curve, no Calabi–Yau, and no square root: the family's canonical differential equation is rational in these seven letters, and the symbola tensor of alphabet letters recording the sequence of logarithmic differentiations of a polylogarithm — the combinatorial skeleton of the function with constants stripped of the top master uses no other letter through weight four, in contrast with the outer-mass double box of Caron-Huot and Henn, which carries a square root. Every constant is classical ($\zeta_2$, $\zeta_3$, $\zeta_4$ and logarithms), with no new period.

Weights 0–2, explicit in both variables. With $m^2 = 1$ and the prefactor $R = s^2(t-1)$ stripped, the pure function through weight 2 is, in classical-polylog notation — $\log$ is the ordinary logarithm and $\mathrm{Li}_n$ the classical polylogarithm, $\mathrm{Li}_n(z) = \sum_{k\ge 1} z^k/k^n$, with $\mathrm{Li}_2(z) = -\int_0^z \log(1-w)\,dw/w$ the dilogarithm:

$$ \begin{aligned} g^{(0)} &= 1,\\[2pt] g^{(1)} &= -2\log(-s) \;-\; 4\log(1-t),\\[2pt] g^{(2)} &= 2\log^2(-s) \;-\; 2\log(-s)\,\log(1+s) \;+\; 8\log(-s)\,\log(1-t) \;+\; 8\log^2(1-t)\\ &\quad -\,2\,\mathrm{Li}_2(-s) \;+\; 10\,\mathrm{Li}_2(t) \;+\; 2\,\mathrm{Li}_2(s+t-st) \;-\; \tfrac{1}{6}\pi^2. \end{aligned} $$

The same answer can be read at the symbol level. Writing $(a, b)$ for the tensor entry $d\log a \otimes d\log b$ — the record of which letter is differentiated first and which second — the weight-2 symbol is

$$ \mathcal{S}[g^{(2)}] = 4(s,s) + 8(s,1{-}t) + 8(1{-}t,s) + 16(1{-}t,1{-}t) - 10(1{-}t,t) - 2(s,1{+}s) - 2(1{-}t,s{+}t{-}st) - 2(1{-}s,s{+}t{-}st). $$

Six of the seven letters appear at weight 2; $s+t$ enters only at weight 3 and above. All coefficients are integers of height at most 16.

Weights 3–4 on the slice $s = -1$. On this slice the $t$-alphabet collapses to the three letters $\{t,\ 1-t,\ 2t-1\}$ and the words stay short enough to print. Writing $G[a_1,\dots,a_n]$ for the Goncharov polylogarithm $G(w_1,\dots,w_n; -t)$ whose letters $\{t, 1-t, 2t-1\}$ correspond to the points $\{0, -1, -1/2\}$ in the argument $x = -t$, $\log 2$ for the $s = -1$ image of the letter $1-s$, and $\zeta_3 = \sum_n 1/n^3$ for Apéry's constant, the weight-3 function is the 15-word combination

$$ \begin{aligned} g^{(3)} = \;&-8\,G[t,t,1{-}t] + 40\,G[t,1{-}t,1{-}t] - 12\,G[t,2t{-}1,1{-}t] + 40\,G[1{-}t,t,1{-}t] - 64\,G[1{-}t,1{-}t,1{-}t]\\ &+ 4\,G[1{-}t,2t{-}1,1{-}t] - 16\,G[2t{-}1,t,1{-}t] + 8\,G[2t{-}1,1{-}t,1{-}t] + 8\,G[2t{-}1,2t{-}1,1{-}t]\\ &+ \log 2\,\bigl(-12\,G[t,2t{-}1] + 4\,G[1{-}t,2t{-}1] + 8\,G[2t{-}1,2t{-}1]\bigr)\\ &+ \tfrac{5}{3}\pi^2\,G[1{-}t] + \bigl(-\tfrac{2}{3}\pi^2 + 4\log^2 2\bigr)\,G[2t{-}1] - \tfrac{50}{3}\,\zeta_3. \end{aligned} $$

Weight 4 is a combination of the same kind: 26 weight-4 words with last letter $1-t$, 9 weight-3 words $\times \log 2$, lower-weight words $\times \{\pi^2, \log^2 2\}$ (9 coefficients) or $\times$ a weight-3 constant, and a constant term; the full list, with every coefficient and the constant, is in the expression file linked below.

Weights 3–4 in both variables. Off the slice, weights 3 and 4 are explicit closed forms in both $s$ and $t$: weight 3 as 39 columns and weight 4 as a 123-column $t$-sector plus a 29-term pure-$y$ block, each column an exact rational number multiplying a product of Goncharov words $G(\vec a;\, x = -t)$ over the $x$-letters $\{0,\, -1,\, s,\, s/(1-s)\}$ (the tables carry a fifth $x$-letter, $-s^2$, that no column uses) and $G(\vec b;\, y = -s)$ over the $y$-letters $\{0,\, 1,\, -1\}$, times powers of $\log x$ and $\log y$. The constants are only the classical $\zeta_2$, $\zeta_3$, $\zeta_4$. The complete machine-readable tables live in the JSON file below, and the downloadable script evaluates the full closed form live at any Euclidean point ($s < 0$, $t < 0$).

Downloads: c3-dbox-expression.md · c3-dbox-evaluate.py · c3-dbox-2var.json · c3-dbox-identities.py (two checks in exact arithmetic at run time: the restriction of the 123-column weight-4 table to the slice $s=-1$, which reproduces the slice words printed above; and a one-parameter rational parametrisation of the two square roots $\sqrt{s(s+4)}$ and $\sqrt{s(s-8)}$ returned by an early maximal-cut scan, an identity about those roots alone, which belong to no master of the family and enter nothing on this page) · MANIFEST.sha256 (every file of the bundle with its checksum)

Everything on this page was validated against independent evaluations at points never used in the construction: the weight ≤ 2 form to 108 digits on eight kinematic loci (four never used in any fit), and the weight-3 and weight-4 forms to 109–112 and 109–111 digits on twenty such points. A twenty-first such point has since been added: one goal-120 auxiliary-mass-flow solve of the top-sector master at a never-used point takes 1:42 on four cores of a shared machine with the current toolkit (the June 2026 runs' goal-120 solves took 23–38 minutes each on eight threads, ten jobs at once), and the closed form reproduces it at every printed digit of its 130-digit evaluation (the reference value at that point is itself certified to 109 digits by a pair of solves at target precisions of 120 and 100 digits); c3-dbox-evaluate.py --point -2 --s -4 --dps 130 runs that check, and --gate-full all twenty-one.

References

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