Massless planar double box

The planar two-loop double box with seven massless propagators and four massless on-shell legs, expanded in the dimensional regulator from the leading pole through the finite part. Smirnov gave its closed form in 1999 as harmonic polylogarithms of a single ratio of invariants. It is recomputed here without using that answer.

The integral

The double box is a scalar Feynman integral of the 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 the product runs over the internal lines $e$ of the graph $G$, the $l_\ell$ are the $L$ loop momenta, $q_e$ is the momentum flowing through line $e$ and $m_e$ its mass; here $L=2$ and every $m_e$ vanishes. With no internal mass to set a scale, the integral depends on the two Mandelstam invariants, apart from an overall power of $s$, only through the dimensionless ratio

$$x=\frac{t}{s},\qquad s=(p_1+p_2)^2,\quad t=(p_2+p_3)^2.$$

The quantity computed is the Laurent expansion in $\varepsilon$ of the normalized integral $I$, obtained from $I_G$ by dividing out the loop-measure factor $(i\pi^{d/2})^2$, the powers $(-s)^{-2\varepsilon}$ and $e^{-2\gamma_E\varepsilon}$ (with $\gamma_E$ Euler's constant) and fixing the overall normalization so that the coefficient of $\varepsilon^{-4}$ is the leading singularitythe rational prefactor of the integral, computed by putting every propagator on shell; dividing by it leaves a function of uniform transcendental weight $1/(-2s^2t)$. The expansion runs from order $\varepsilon^{-4}$ through the finite part, as a function of $x$ in the Euclidean region $s,t\lt0$, where $x\gt0$ and $I$ is real.

Smirnov evaluated this integral analytically in 1999 as an expansion in $\varepsilon$ whose coefficients are harmonic polylogarithmsiterated integrals of the kernels $dx/x$, $dx/(1+x)$ and $dx/(1-x)$; only the first two occur here of $x$, the first analytic result for a massless on-shell double box. The non-planar double box of Tausk followed the same year. The integral has since become a standard example for differential equations in canonical form. The family has eight master integralsa finite set of integrals of the family in terms of which every other member is a rational linear combination, by integration-by-parts identities; in the canonical basis of Henn (2013) the differential equation in $x$ has logarithmic singularities only at $x=0$ and $x=-1$, and the coefficient of $\varepsilon^j$ is a pure functionan iterated integral of uniform transcendental weight with constant rational coefficients, no rational or algebraic functions of the kinematics multiplying it of uniform transcendental weight $4+j$.

Both features, the two singular points and the uniform weight, follow from the graph alone, before any coefficient is computed. The maximal cutthe integral with every propagator put on shell, which exposes the algebraic prefactor and the geometry is rational, so the leading singularity is the rational prefactor divided out above and no elliptic curve or higher period appears. Dividing it out leaves functions built from the two letters $x$ and $1+x$, which vanish at $x=0$ (where $t=0$) and at $x=-1$ (where the third Mandelstam invariant $u=-s-t$ vanishes). Three conditions then fix which iterated integrals (words in the two letters) may appear, stated in terms of the symbolthe tensor of letters that encodes the iterated-integral structure of a polylogarithmic function; integrability is the condition that such a tensor comes from an actual function as reviewed by Duhr: the first letter of every word must be a letter whose vanishing is a Landau singularity of the graph (the first-entry condition, which both letters satisfy here); the symbol must be integrable; and discontinuities may occur only in the $s$ and $t$ channels, with none in $u$. Together with their products with zeta values of complementary weight, the words that satisfy these conditions span an ansatz of dimension $(1,2,5,11,23)$ at weights $0$ through $4$, so twenty-three unknown rational coefficients at weight four. Fitting that ansatz to high-precision numerical values of the integral at Euclidean points with an integer-relation algorithm (PSLQ) determines every coefficient as an exact rational number, weight by weight.

Closed form

Every order of the expansion is the leading singularity times a rational combination of harmonic polylogarithms of $x$ and their products with zeta values. Smirnov's function $K(x,\varepsilon)=\sum_{j\ge-4}\varepsilon^{j}K^{(j)}(x)$ is related to $I$ by $I=\frac{1}{-2s^2t}\cdot\frac{K(x,\varepsilon)}{-4}$, and through the finite part

$$ \begin{aligned} K^{(-4)}&=-4,\qquad K^{(-3)}=5\,G(0;x),\qquad K^{(-2)}=-4\,G(0,0;x)+15\,\zeta_2,\\ K^{(-1)}&=-4\,G(0,0,0;x)+4\,G(-1,0,0;x)-33\,\zeta_2\,G(0;x)+12\,\zeta_2\,G(-1;x) +\tfrac{65}{3}\,\zeta_3,\\ K^{(0)}&=32\,G(0,0,0,0;x)-16\,G(-1,0,0,0;x)-20\,G(0,-1,0,0;x)+4\,G(-1,-1,0,0;x)\\ &\quad+72\,\zeta_2\,G(0,0;x)-20\,\zeta_2\,G(-1,0;x)-60\,\zeta_2\,G(0,-1;x)+12\,\zeta_2\,G(-1,-1;x) \\ &\quad-\tfrac{88}{3}\,\zeta_3\,G(0;x)+4\,\zeta_3\,G(-1;x)+87\,\zeta_4 \end{aligned} $$

where $G(a_1,\dots,a_n;x)$ is the Goncharov iterated integral with indices $a_i=0$ and $a_i=-1$ standing for the integration kernels $dx/x$ and $dx/(1+x)$; with these two indices only, the $G$ functions are the harmonic polylogarithms of Smirnov's answer. In the weight count a word of length $n$ times $\zeta_k$ carries weight $n+k$.

Checks

Evaluator

Same family

• The outer-mass double box has the same graph with a common mass on the six perimeter lines and a massless rung, and is finite in four dimensions: a single pure function of weight four whose alphabet contains three square roots.
• The central-mass double box puts the mass on the shared rung instead and keeps the perimeter massless; its alphabet is seven rational letters in $s$ and $t$, with no square root.
• The three-loop ladder has the same two-letter alphabet $\{x,\,1+x\}$ one loop higher: a three-point ladder with nine massless lines and two off-shell legs, whose finite part has weight six.

The paper

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

References

← back to Feynman diagrams