Three-loop ladder

The three-loop triangle ladder with nine massless lines and off-shell external legs, a function of the ratios of their virtualities, built from classical polylogarithms of weight six. Its closed form has been known since Ussyukina and Davydychev (1993) and is recovered here exactly.

The integral

The ladder is planar, and as a scalar Feynman integral it has the form

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

with $L=3$ loop momenta $l_\ell$, where the second product runs over the nine internal lines $e$ of the graph $G$, $q_e$ is the momentum flowing through line $e$, and every mass $m_e$ is zero. With all three legs off shell the integral is $(p_3^2)^{-3}$ times the dimensionless ladder function $\Phi^{(3)}(X,Y)$ of the two ratios $X=p_1^2/p_3^2$ and $Y=p_2^2/p_3^2$, in the normalization of Ussyukina and Davydychev. In the variables $z,\bar z$ with $X=z\bar z$ and $Y=(1-z)(1-\bar z)$, $\Phi^{(3)}$ is $1/(z-\bar z)$ times a sum of differences $\mathrm{Li}_n(z)-\mathrm{Li}_n(\bar z)$ of classical polylogarithms weighted by powers of $\ln(z\bar z)$; the holomorphic blockthe function of one variable from which the two-variable ladder function is assembled, as a difference of two copies with conjugate arguments $f_3(x)$ is the corresponding one-variable sum, the polylogarithms $\mathrm{Li}_n(-x)$ weighted by the same powers of $\ln x$. Its argument is the ratio $x$, related to $X$ by

$$X=\frac{p_1^2}{p_3^2}=\frac{x}{1+x},\qquad x=\frac{p_1^2}{p_3^2-p_1^2},$$

so that $x=X/(1-X)$. As $p_2^2\to0$, $\Phi^{(3)}$ itself diverges like $\ln^3 Y$, while $f_3$ stays finite. In the small-$Y$ expansion of $(1-X)\,\Phi^{(3)}$ organized in powers of $\Lambda=\ln((1-X)/Y)$, $f_3(x)$ is $3!$ times the $\Lambda$-independent part, with the two constants left by the polylogarithm inversion subtracted, so the relation between the two functions is an algebraic identity rather than a limit. The off-shell ladder is finite in four dimensions, and the quantity computed is $f_3(x)$ at $\varepsilon=0$ on the half-line $x\gt0$, a pure functiona function of uniform transcendental weight, with no rational or algebraic prefactor left of weight $2L=6$.

Ussyukina and Davydychev evaluated the three- and four-point ladder graphs with an arbitrary number of rungs in 1993 and showed that the two series give the same functions $\Phi^{(L)}$: the $L=1$ function is the one-loop off-shell box (equivalently the off-shell triangle), and the $L=2$ function the off-shell massless double box. Broadhurst and Davydychev (2010) gave an integral representation valid for all $L$ (arXiv:1007.0237), Drummond et al. (2013) analyzed the conformal symmetry of these off-shell integrals (arXiv:1303.6909), and Derkachov, Isaev and Shumilov (2023) re-derived the series in an operator formalism built on conformal triangles (arXiv:2302.11238).

The branch points of $f_3$ lie only at $x=0$, $x=-1$ and $x=\infty$, so its lettersthe arguments of the logarithms from which the function is built as an iterated integral; a word is an ordered string of letters, one per unit of weight are $x$ and $1+x$, and the space of weight-six words in these two letters has dimension $2^6=64$, small for a three-loop integral. The factor $(p_3^2)^{-3}$ in the definition fixes the normalization of $f_3$. As a test of the fitting step on a function space where the answer is known, an integer-relation (PSLQ) fit of the 64 words to high-precision values of $f_3$ at points $x$ in $[0.01,0.71]$ recovers the four rational coefficients.

Closed form

The integer-relation fit over the weight-six words in $x$ and $1+x$ gives

$$f_3(x)=120\,\mathrm{Li}_6(-x)-60\,\ln x\,\mathrm{Li}_5(-x)+12\,\ln^2 x\,\mathrm{Li}_4(-x)-\ln^3 x\,\mathrm{Li}_3(-x),$$

where $\mathrm{Li}_n$ is the classical polylogarithm of weight $n$. The coefficients are the $L=3$ case of the all-orders Ussyukina–Davydychev formula, in which the coefficient of $\ln^k x\,\mathrm{Li}_{2L-k}(-x)$ is $c^{(L)}_k=(-1)^k\,(2L-k)!/(k!\,(L-k)!)$. As a combination of Goncharov wordsiterated integrals $G(a_1,\dots,a_6;x)$ over the kernels $dx/x$, index $0$, and $dx/(1+x)$, index $-1$, $f_3=-6\,G(0,0,0,-1,0,0;x)+6\,G(0,0,-1,0,0,0;x)$: two weight-six words with a single index $-1$ each. The two-variable function $\Phi^{(3)}(X,Y)$ is real for $X,Y\gt0$ with $4XY\gt(X+Y-1)^2$, the region where Euclidean three-point momenta with these virtualities exist.

Checks

Evaluator

Same family

The massless planar double box has the same two-letter alphabet $\{x,\,1+x\}$, with $x=t/s$ the ratio of its two Mandelstam invariants, and every coefficient of its Laurent expansion, from the $\varepsilon^{-4}$ pole to the finite part, is a combination of harmonic polylogarithms. It is the on-shell four-point integral of Smirnov (1999), with all four legs massless, whereas the $L=2$ member of the ladder series here is the double box with off-shell legs.

The paper

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

References

← back to Feynman diagrams