Central-mass double box

The two-loop planar double box with four massless legs and one massive propagator on the central rung, the photon-Z-photon ladder in electroweak corrections to Møller and Bhabha scattering. Its top-sector master integral, computed by Schwanemann and Weinzierl in 2024, is rederived here by a different route and written as iterated integrals in the two Mandelstam invariants through the finite part.

The integral

All four external legs are massless, $p_i^2=0$, and the only internal mass $m$ sits on the central rung $D_7$. The family is

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

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 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, $u=(p_1+p_3)^2$, is fixed by $s + t + u = 0$. The rung mass sets the unit, $m^2 = 1$, and the results are given on the Euclidean sheet $s\lt0$, $t\lt0$. The quantity computed is the top-sector masterthe master integral with every propagator of the diagram present once $J=I[1,1,1,1,1,1,1]$, presented as the pure function $g$ obtained by multiplying $J$ by $R = s^2(t-1)$, the inverse of its leading singularitythe maximal-cut residue of the integrand; dividing by it leaves a function of uniform transcendental weight,

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

where $\gamma_E$ is the Euler–Mascheroni constant, and the overall normalization of the measure is fixed by requiring $g^{(0)} = 1$. Each coefficient $g^{(w)}$ has uniform transcendental weight $w$. The result is the five coefficients $g^{(0)},\dots,g^{(4)}$, which together determine the Laurent expansion of $J$ from $\varepsilon^{-4}$ through $\varepsilon^0$.

Simpler versions of this diagram have long been known in closed form. Smirnov (1999) solved the fully massless double box, and Caron-Huot and Henn (2014) gave the version with a common mass on the six perimeter lines, whose alphabetthe finite set of rational or algebraic functions of the kinematics whose iterated logarithmic integrals span the answer carries one square root, as iterated integrals. With the mass on the central rung instead, the diagram is the $\gamma$-$Z$-$\gamma$ ladder of electroweak corrections to Møller and Bhabha scattering at next-to-next-to-leading order, and equally the gluon-$V$-gluon ladder, with $V$ a massive electroweak boson, of mixed QCD–electroweak corrections to four-fermion scattering. Schwanemann and Weinzierl (2024) computed this family, their topology C, giving all nineteen master integralsthe finite basis of integrals to which every integral of the family reduces by integration-by-parts identities in multiple polylogarithms through weight four. The scalar top-sector integral $J$ is one of three masters in the top sector; on the $s+i0$ sheet their canonical master $J^{C}_{17}$ equals $(s/m^2)^{2\varepsilon}\,g(\varepsilon)$ in the normalization above. The result here is derived from the family's own integration-by-parts reduction and from integer-relation (PSLQ) fits to high-precision numerical values, and is written in a different basis of iterated integrals, adapted to the Euclidean region.

The function space follows from the singularity structure of the family. Its canonical differential equation is rational, so no algebraic letter can occur, and the alphabet of the top-sector integral consists of the seven rational components of the family's Landau singular locus, the principal $A$-determinant of its graph polynomial, computed as in Fevola, Mizera and Telen (2024) and Correia, Giroux and Mizera (2025). The letter $s+t-st$, which a loop-by-loop cut analysis can miss, is required by the dilogarithm $\mathrm{Li}_2(s+t-st)$ in the weight-two function below. The first entries of the symbola tensor of alphabet letters that lists the order in which a polylogarithm is built up by logarithmic integrations, with the constants stripped lie in $\{s,\ 1-s,\ 1-t\}$, the physical thresholds.

A direct fit of numerical values to a basis of polylogarithms is ill-conditioned for a seven-letter alphabet, because the number of candidate functions of weight $w$ grows like $k^w$ for $k$ letters. Imposing the alphabet, the integrability of the symbol and the first-entry condition as exact linear constraints, and only then fitting, determines every coefficient as an exact rational. At weights three and four, products of classical polylogarithms no longer span the answer, and the two-variable fit is done instead in a basis of Goncharov polylogarithms$n$-fold iterated integrals of the forms $du/(u-c_i)$ ending at the argument; the general multiple polylogarithms of weight $n$. On the slice $s=-1$ the function is also obtained by transportnumerical solution of the differential equation along a path in $t$, starting from a point where the integrals are known to high precision of the differential equation in $t$ from a single boundary value at $t=-\tfrac13$; the cost of the transport is set by the sixteen master integrals that couple on the slice, not by the size of the function space. Matching the two-variable fit to the resulting slice expressions is one more exact constraint.

Closed form

Through weight four, the pure function $g$ is an explicit iterated integral in $s$ and $t$ over the seven-letter rational alphabet

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

The constants in the two-variable form are $\zeta_2=\pi^2/6$, $\zeta_3$ and $\zeta_4=\pi^4/90$, with $\zeta_n=\sum_{k\ge1}k^{-n}$; on the slice $s=-1$ powers of $\log 2$ and $\mathrm{Li}_4(1/2)$ also appear.

Weights 0–2 in both variables

Writing $\mathrm{Li}_n(z) = \sum_{k\ge 1} z^k/k^n$ for the classical polylogarithm, so that $\mathrm{Li}_2(z) = -\int_0^z \log(1-u)\,du/u$ is the dilogarithm, the pure function through weight two is

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

The symbol of $g^{(2)}$, writing $(a, b)$ for $d\log a \otimes d\log b$, 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 two; $s+t$ enters only at weight three and above.

Weights 3–4 at $s=-1$

On the slice $s=-1$ the alphabet in $t$ collapses to the three letters $\{t,\ 1-t,\ 2t-1\}$, and the letter $1-s$ freezes to the constant $2$, which is why powers of $\log 2$ appear among the coefficients. In the variable $x=-t$ the three letters vanish at $x=0$, $-1$ and $-1/2$. Write $G[a_1,\dots,a_n]$, with each $a_i$ one of the letters $t$, $1-t$, $2t-1$, for the Goncharov polylogarithm $G(c_1,\dots,c_n;\,x)$ at $x=-t$ whose point $c_i$ is the zero of the letter $a_i$. In this notation the weight-three function is the 15-term 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 four on the slice has the same structure with 44 terms: weight-four polylogarithms with last letter $1-t$, lower-weight ones multiplying $\log 2$, $\pi^2$, $\log^2 2$, $\zeta_3$ and $\log^3 2$, and a constant that contains $\mathrm{Li}_4(1/2)$.

Weights 3–4 in both variables

Off the slice, weights three and four are written in a fibration basis: products of Goncharov polylogarithms $G(\vec a;\, x = -t)$ with indices from $\{0,\, -1,\, s,\, s/(1-s)\}$, the points at which the letters $t$, $1-t$, $s+t$ and $s+t-st$ vanish in $x$, times $G(\vec b;\, y = -s)$ with indices from $\{0,\, 1,\, -1\}$, the zeros of $s$, $1+s$ and $1-s$ in $y$, times powers of $\log x$ and $\log y$ and the constants $\zeta_2$, $\zeta_3$, $\zeta_4$. In this basis $g^{(3)}$ is a list of 39 terms and $g^{(4)}$ a $t$-dependent list of 123 terms plus a remainder of 29 that do not depend on $t$, each an exact rational number multiplying one basis element. The two lists restricted to $s=-1$ reduce to the slice formulas above.

The complete expression, with the weight-four slice terms, the two-variable lists and the conventions, is in the expression file inside the bundle under Evaluator.

Checks

Evaluator

Same family

• The massless planar double box is the same seven-propagator graph with every line massless, the case Smirnov solved in 1999.
• The outer-mass double box puts a common mass on the six perimeter lines and leaves the rung massless, which brings a square root into the alphabet.

The paper

• Polylogarithmic, elliptic, and Calabi–Yau Feynman integrals from a hybrid bootstrap (PDF) — Matthew D. Schwartz.

References

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