The elliptic three-loop integrals of hadronic vacuum polarization in chiral perturbation theory

Abstract

This work presents a detailed account of the Feynman integrals required for the three-loop hadronic vacuum polarization calculation performed in arXiv:2510.12885. We explain how to compute each of the three-loop integrals, and outline the mathematical framework underlying their evaluation. This culminates in a practical numerical implementation that enables fast and accurate evaluation of these integrals for arbitrary complex values of the photon virtuality.

Figures (16)

• Figure 1: The integrals $E_1$, $E_2$, and $E_3$ (left to right), shown as functions of the nome $q$ over the complex unit disk. This disk corresponds to a fractal tiling of infinitely many copies of the complex $t$-plane (see \ref{['fig:q']}), and we outline one such copy in white here. For each integral, its value at $t=0$ (limit as $q\to1$) has been subtracted to make the features more visible. The integral values are shown using domain coloring: the hue represents their phase (with red and cyan for positive and negative real values, respectively), while the contours represent their absolute value. The latter is a power of $2$ on the contours, and decreases toward the shaded side. As a reference for reading the contours, the top corner of the white outline (corresponding to $t=4$) has $0.5 < |E_1(2;4)-E_1(2;0)| < 1$, $0.25<|E_2(2;4)-E_2(2;0)|<0.5$, and $0.125<|E_3(2;4)-E_3(2;0)|< 0.25$. Note the branch cut on $q\in[-1,0]$, which corresponds to two copies of $t\in[16,\infty)$ (see \ref{['fig:q']}).
• Figure 2: Left: the mapping $\tau\mapsto t$ visualized in the complex plane, covering one period in the real direction. The complex values are represented as in \ref{['fig:elliptic']}. The $\times$-shaped contour intersections are saddle points at $t=4$ (for instance at $\tau=\pm\frac{1}{4}+\frac{i\sqrt3}{12}$) and $t=16$ (for instance at $\tau=\pm\frac{1}{2}+\frac{i\sqrt3}{6}$). Right: a sketch (to scale) of the $\mathop{\mathrm{Re}}\nolimits(\tau)>0$ half of the same plot, showing a selection of the infinitely many $\tau$-plane images of the real $t$-axis, colored according to which kinematic region they reproduce: spacelike subthreshold ($t<0$, blue), timelike subthreshold ($0<t<4$, orange), two-pion ($4<t<16$, green) and multi-pion ($16<t$, violet). The shaded area (if extended upward to infinity) contains all $t$ with $\mathop{\mathrm{Im}}\nolimits(t)>0$ exactly once, with contours indicating (bottom to top) $\mathop{\mathrm{Im}}\nolimits(t)=0.001$, $0.1$, $0.5$, and $1$; its boundary (bold) contains all real $t$ exactly once, ordered counterlockwise.
• Figure 3: Like \ref{['fig:tau']}, but showing $q\mapsto t$ over the unit disk. The lines in the sketch have been extended onto the $\mathop{\mathrm{Re}}\nolimits(\tau)<0$ part (lower half of the disk), and thereby produce the outline used in \ref{['fig:elliptic']}. Note how inside the shaded region, $q$ approaches $1$ extremely slowly as $|t|$ approaches $0$, ensuring good convergence of our sums almost everywhere.
• Figure 4: Like \ref{['fig:tau']}, but showing $t_ {}\mapsto t$ as defined in \ref{['eq:t-to-tau']}. In the sketch on the right, the shaded area is $\Omega_>$, and the two inverse images of the real $t$-line are shown with bold colored lines as in \ref{['fig:tau']}. The dotted rectangle indicates the region shown in \ref{['fig:rho24']} (bottom row) and \ref{['fig:rhoF']}. The thin gray lines are the same contours of constant $\mathop{\mathrm{Im}}\nolimits(t)$ as in \ref{['fig:tau']}, and the dashed ones indicate their images under the involutory mapping $z\mapsto (9z-9)/(z-9)$, which is how they appear in $\varpi_r$.
• Figure 5: The construction of $\rho_2(z)$ (top row) and $\rho_4(z)$ (bottom row). Left: The principal branch of $\sqrt{(z-4)(z-16)}$ and $\sqrt[4]{z^3 - 9z^2 + 3z - 3}$, respectively, visualized in the complex plane using colors and contours like in \ref{['fig:tau']}. Middle: The corrected functions $\rho_2(z)$ and $\rho_4(z)$, respectively. Right: A schematic picture of their relation. Colored lines indicate where the argument of the root [$(z-4)(z-16)$ and $z^3 - 9z^2 + 3z - 3$, respectively] is real: red for positive values and cyan for negative. Dotted lines do not cause branch cuts. Solid lines cause cuts that are removed by analytically continuing to different branches, starting on the principal branch in the region marked by $\star$ and picking up roots of unity ($-1$ and $+i$, respectively) when crossing a line in the direction indicated by the arrows. Zigzag lines are left as branch cuts also in $\rho_2(z)$ or $\rho_4(z)$. The dotted circle indicates the boundary between $\Omega_<$ and $\Omega_>$ (compare \ref{['fig:tsun']}).