Outer-mass double box

The planar two-loop double box with six perimeter lines of a common mass, a massless central rung and massless legs, which is finite in four dimensions and, up to an algebraic prefactor, equal to a single weight-four polylogarithm. Caron-Huot and Henn computed it in 2014, and it is recomputed here without using their answer.

The integral

The outer-mass double box is a planar two-loop four-point diagram: two boxes sharing a central rung, with all six propagators around the perimeter carrying a common mass $m$ and the central rung and the four external legs massless, $p_i^2=0$. It is a scalar Feynman integral of the standard form

$$I_G \;=\; \int \prod_{\ell=1}^{L} d^d l_\ell \;\prod_{e\in G}\frac{1}{q_e^2-m_e^2}\,,\qquad d=4-2\varepsilon,$$

where $l_1,\dots,l_L$ are the loop momenta, the product runs over the internal lines $e$ of the graph $G$, $q_e$ is the momentum flowing through line $e$ and $m_e$ its mass; here there are $L=2$ loops, six of the seven masses equal $m$ and the rung is massless. The integral depends on the two Mandelstam invariants through the dimensionless ratios

$$u=\frac{4m^2}{-s},\qquad v=\frac{4m^2}{-t},\qquad s=(p_1+p_2)^2,\quad t=(p_2+p_3)^2,$$

both positive in the Euclidean region $s,t\lt0$. The integral is finite, so it is evaluated directly in $d=4$ with no expansion in $\varepsilon$. Write $I_5$ for the integral with every propagator present once, in $d=4$ at $m=1$ with the measure normalized as in Caron-Huot and Henn, and $g_{10}(u,v)=-\tfrac18\,s^2t\,\sqrt{1+u}\,\sqrt{1+u+v}\;I_5$ for its normalized form at Euclidean $u,v\gt0$, keeping the labels these two functions carry in the literature cited below. The prefactor is fixed by the leading singularitythe algebraic prefactor read off the maximal cut; dividing by it leaves a pure function of uniform transcendental weight of the integral.

Caron-Huot and Henn computed this integral in 2014 by solving a differential equation for it directly in four dimensions. Besides rational functions of $u$ and $v$, the alphabet contains only three square roots, $\beta_u=\sqrt{1+u}$, $\beta_v=\sqrt{1+v}$ and $\beta_{uv}=\sqrt{1+u+v}$. The substitution $u=(1-w^2)(1-z^2)/(w-z)^2$, $v=4wz/(w-z)^2$ rationalizes all three at once and maps the Euclidean region to $0\lt z\lt w\lt1$. The twelve-letter alphabet, the maximal cuts and the symbolthe tensor of logarithms that encodes the iterated-integral structure of a polylogarithmic function; two functions with the same symbol differ at most by lower-weight functions times constants also follow from the Landau equations of the graph alone, and the symbol can be recovered by regression from high-precision numerical samples of the integral. The computation here combines the two ideas.

The alphabet is rebuilt here from a Landau singularity analysis of the diagram, which gives six rational letters ($u$, $v$, $1{+}u$, $1{+}v$, $u{+}v$, $1{+}u{+}v$) and a list of candidate algebraic letters (those that go to their inverse when the sign of one square root is flipped); a Regge-limit argument removes the candidates containing $\sqrt u$ or $\sqrt v$ and leaves exactly the twelve letters. The general weight-four symbol in the twelve letters has too many unknown coefficients to determine directly from numerical values of the integral, and three conditions first reduce the space to a basis of 161 weight-four symbols. The first is integrabilitythe condition that a tensor of logarithms $\mathcal{S}$ is the symbol of an actual function, written $d\mathcal{S}\wedge d\mathcal{S}=0$ of the symbol, the condition introduced by Goncharov, Spradlin, Vergu and Volovich in 2010. The second is a first-entry condition that lets only the $s$- and $t$-channel thresholds and pseudo-thresholds open a branch cut. The third is a definite parity under the sign flips of the three square roots, odd in $\beta_u$ and $\beta_{uv}$ and even in $\beta_v$. In the Landau bootstrap the maximal cuts select the answer in this space; here an integer-relation fita search, by PSLQ or lattice reduction, for exact rational coefficients that match high-precision numerical values of the 161 coefficients to high-precision values of the integral on a Euclidean grid selects it instead.

Closed form

In four dimensions the normalized integral is the leading singularity times one element of the constrained basis:

$$g_{10} = -\tfrac18\,s^2 t\,\beta_u\beta_{uv}\, I_5 = \mathrm{LS}\cdot G_{52},\qquad \mathrm{LS}=\tfrac18\sqrt{s(s-4m^2)}\,\sqrt{st\,(st-4m^2(s+t))},$$

where $\mathrm{LS}$ is the leading singularity written in $s$, $t$ and $m$. The fit assigns a nonzero coefficient to exactly one basis element, $G_{52}$, and that coefficient is $1$ (the subscripts on $G$, and on the letters $L_6$ and $L_9$ below, follow the numbering in section 4.5 of Polylogarithmic, elliptic, and Calabi–Yau Feynman integrals from a hybrid bootstrap). The symbol of $G_{52}$, written $\mathcal S(G_{52})$, coincides term by term with the symbol published by Caron-Huot and Henn and later rederived from the Landau equations. It has eighteen terms over the twelve-letter alphabet

$$\Big\{\,u,\ v,\ 1{+}u,\ 1{+}v,\ u{+}v,\ 1{+}u{+}v,\ \tfrac{\beta_u-1}{\beta_u+1},\ \tfrac{\beta_v-1}{\beta_v+1},\ \tfrac{\beta_{uv}-1}{\beta_{uv}+1},\ \tfrac{\beta_{uv}-\beta_u}{\beta_{uv}+\beta_u},\ \tfrac{\beta_{uv}-\beta_v}{\beta_{uv}+\beta_v},\ \tfrac{\beta_{uv}-\beta_u\beta_v}{\beta_{uv}+\beta_u\beta_v}\,\Big\},$$

where $\beta_v$ enters only through three of the six algebraic letters, and each letter is fixed up to a constant factor, which drops out of the symbol. Writing $L_6=(\beta_u-1)/(\beta_u+1)$ and $L_9=(\beta_{uv}-\beta_u)/(\beta_{uv}+\beta_u)$, the symbol begins

$$\mathcal S(G_{52}) = -L_6{\otimes}u{\otimes}L_6{\otimes}L_9 + L_6{\otimes}(1{+}u){\otimes}L_6{\otimes}L_9 - L_6{\otimes}u{\otimes}L_9{\otimes}L_6 + L_6{\otimes}(1{+}u){\otimes}L_9{\otimes}L_6 + \cdots$$

(fourteen more terms); the full eighteen-term symbol is written out in section 4.5 of the same paper.

The function $G_{52}$ follows from its symbol by iterated integration in $w$ and $z$ from the line $w=z$, which is the limit $u,v\to\infty$ where $g_{10}$ vanishes. Every letter allowed in the first entry equals one on that line, so the integration converges and introduces no constant beyond the symbol. On the physical sheet $g_{10}$ is singular only at the normal thresholds $s=4m^2$ and $t=4m^2$ and as $s,t\to\infty$; on other sheets it is also singular at $s=0$, $t=0$, $s+t=0$ and on the pseudo-threshold $st=4m^2(s+t)$.

Checks

Evaluator

Same family

The massless planar double box is the same graph with all seven lines massless. It depends on the single ratio $x=t/s$, and its Laurent expansion in $\varepsilon$, poles and finite part, is a sum of harmonic polylogarithms of $x$. The central-mass double box is the same graph with the mass $m$ on the central rung alone and all six perimeter lines massless.

The papers

• Polylogarithmic, elliptic, and Calabi–Yau Feynman integrals from a hybrid bootstrap (PDF) — Matthew D. Schwartz.
• Automated computation of Feynman integrals with BootLoops (PDF) — Matthew D. Schwartz. A preliminary version of the paper.

References

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