Unequal-mass kite

The two-loop kite self-energy with massive lines of mass 1, 1 and √2, whose sunrise sub-diagram defines an elliptic curve that is not modular. Its finite part is new at these masses and is given here as a single integral over the periods of that curve, starting from an analytically derived boundary value.

The integral

The kite is a two-loop self-energya diagram with one external momentum p flowing in and out; it corrects a particle's propagator and depends only on p squared: a single off-shell momentum $p$ enters at one vertex and leaves at the other, and the only kinematic variable is $s = p^2$. The family is

$$J(\nu_1,\ldots,\nu_5) \;=\; \int d^d k_1\, d^d k_2\; \frac{1}{D_1^{\nu_1} D_2^{\nu_2} D_3^{\nu_3} D_4^{\nu_4} D_5^{\nu_5}}\,, \qquad d = 4-2\varepsilon,$$

with loop momenta $k_1, k_2$ and the five propagators

$$D_1 = k_1^2 - 1,\quad D_2 = k_2^2,\quad D_3 = (k_1-k_2)^2 - 1,\quad D_4 = (k_1-p)^2,\quad D_5 = (k_2-p)^2 - 2.$$

The three massive lines $D_1, D_3, D_5$ are the sunrise sub-diagram of the figure, and the maximal cutput every internal line of the sub-diagram on shell simultaneously; the integral over what survives exposes the diagram's underlying geometry of that sunrise is an elliptic curve, so $J(1,1,1,1,1)$ cannot be written in multiple polylogarithms alone. The family has fourteen master integralsthe finite basis of integrals to which every integral of the family reduces through integration-by-parts identities, and the quantity computed is the finite $\varepsilon^0$ coefficient of the top-sector masterthe master integral with every propagator of the diagram present once $J(1,1,1,1,1)$, written $J^{(0)}(s)$, as a function of $s=p^2$; its $\varepsilon^{-2}$ and $\varepsilon^{-1}$ parts vanish.

Kite integrals of this kind appear in two-loop self-energies, for example the two-loop electron self-energy in QED studied by Hönemann, Tempest and Weinzierl (2018). Remiddi and Tancredi (2016) set up the differential equations and dispersion relations for the massive sunrise and the kite, and for three equal masses Adams, Bogner, Schweitzer and Weinzierl (2016) gave the kite to all orders in $\varepsilon$ in elliptic polylogarithms. In that equal-mass case the sunrise curve is the modular curve of the congruence subgroup $\Gamma_1(6)$. Adams and Weinzierl (2017) showed that the kernel multiplying the sunrise in the differential equation of the top-sector master is then itself a modular forma function on the upper half-plane with a fixed transformation law under a group of integer Möbius transformations and a convergent expansion in the nome q; such functions form small, explicitly known spaces of $\Gamma_1(6)$, so the answer is a combination of elliptic polylogarithms with constant coefficients. Those coefficients can be fixed by an integer-relation (PSLQ) fit to precise numerical values of the integral.

At squared sunrise masses $(1,1,2)$ the sunrise curve, taken in the Weierstrass form of Bogner, Müller-Stach and Weinzierl (2019), is not modular. Its $j$-invariant differs from that of the $\Gamma_1(6)$ curve at every value of $s$ (at $s=-2$ it is $10976$, against $15664.667\ldots$ at equal mass), and the base of the family is not a modular curve. The kernel is then no longer a modular form, because the period of the second kindthe quasi-period: the integral of the curve's other differential, x dx/y, around the same cycle as the period; period and quasi-period together fix the Wronskian through the Legendre relation enters it with a coefficient that is not a rational function of $s$. As a result no combination of elliptic polylogarithms with constant coefficients and $s$-independent arguments, the form that suffices at equal mass, reproduces the integral. One way out is to let the arguments of the elliptic polylogarithms move with $s$, as Broedel, Duhr, Dulat, Penante and Tancredi (2019) did in writing the kite with three distinct masses as a pure combination of elliptic polylogarithms of uniform weight three; this mass point is the $m_1=m_2$ limit of their eq. (5.6). The other is to keep both periods of the curve and their Wronskian explicit and solve the differential equation from a boundary value, as is done here. An $\varepsilon$-factorized differential equation for the kite family with five different masses is also known, from Giroux, Pokraka, Porkert and Sohnle (2024). What is new here is the finite part itself, for $s\lt0$ at these masses, obtained from an analytically derived boundary value and repeated at a second mass point, together with its continuation to the physical region; no earlier evaluation at these masses is known.

Closed form

The result is a representation of $J^{(0)}(s)$ as a single integral in $s$ whose integrand is built from the two periods of the sunrise curve, the integrals of $dx/y$ around its two independent cycles. The curve is

$$ E_{(1,1,2)}:\quad y^2 \;=\; 4\,(x-e_1)(x-e_2)(x-e_3),\qquad k^2=\frac{e_3-e_2}{e_1-e_2},\quad Z_3=e_1-e_2, $$

where the branch points $e_1,e_2,e_3$ are algebraic functions of $s$ fixed by the four sunrise thresholds $M_a=(\pm1\pm1\pm\sqrt2)^2$, in the form given by Bogner, Müller-Stach and Weinzierl. Two of them collide at the sunrise threshold $s=6+4\sqrt2$ and at each of the pseudo-thresholds $s=6-4\sqrt2$ and $s=2$. The branch points define the elliptic modulus $k^2$ and the period normalization $Z_3$; at the boundary point $s=-2$ used below the modulus is $k^2=(2+\sqrt2)/4$. The two periods are complete elliptic integrals and their Wronskian follows from them,

$$ \psi_1(s)=\frac{2}{\sqrt{Z_3}}\,K\!\big(k^2\big),\qquad \psi_2(s)=\frac{2i}{\sqrt{Z_3}}\,K\!\big(1-k^2\big),\qquad W(s)=\psi_1\psi_2'-\psi_2\psi_1'\,, $$

with $K$ the complete elliptic integral of the first kind, primes denoting derivatives in $s$, and $W$ determined exactly by the Legendre relation $K(k^2)\,E(1-k^2) + E(k^2)\,K(1-k^2) - K(k^2)\,K(1-k^2) = \tfrac{\pi}{2}$, in which $E$ is the complete elliptic integral of the second kind. The finite part of the top-sector master is then the variation of parametersthe textbook method for an inhomogeneous linear differential equation: take the homogeneous solutions, promote their constant coefficients to functions, and fix those functions by a first-order quadrature against the source integral

$$ J^{(0)}(s)\;=\;J^{(0)}(-2)\;+\;\int_{-2}^{\,s}\frac{\psi_1(s')^{\,3}}{W(s')}\;S(s')\;\mathrm{d}s'\,, $$

where the source $S(s)$, a rational function of $s$, is assembled from the lower, polylogarithmic master integrals through the top row of the $14\times14$ matrix $A(d,s)$ in the differential equation $d\vec J/ds=A(d,s)\,\vec J$ obeyed by the vector $\vec J$ of the fourteen master integrals. The entries of that row are rational functions of $s$ and come from integration-by-parts reduction. Every denominator in $A$ factors over the four letters $\{s,\,s-1,\,s-2,\,s^2-12s+4\}$. The last letter, absent at equal mass, vanishes at the sunrise threshold and pseudo-threshold $s=6\pm4\sqrt2$. The points $s=1$ and $s=2$ are the thresholds of the one-loop bubble sub-diagrams with masses $(0,1)$ and $(0,\sqrt2)$. All singular points therefore lie off the negative real axis, and the integration path from $s=-2$ never meets a pole. In practice $J^{(0)}(s)$ is evaluated by transportnumerical solution of the differential equation along a path in $s$, starting from a point where the integrals are known exactly of the boundary value at $s=-2$ along that axis, using the exact matrix $A(d,s)$.

The boundary value

The point $s=0$ is a regular singular point of the differential equation but not a Landau singularity of the masters, so the physical solution is the unique branch bounded there. At $s=0$ partial fractions reduce every master to two-loop vacuum integrals, products of tadpoles and vacuum sunsets, whose only transcendental constants are $\gamma_E$, $\zeta_2$, $\ln 2$ and Catalan's constant $G$. Catalan's constant comes from the vacuum sunset with masses $1$, $1$, $\sqrt2$: its finite part contains the Clausen values $\mathrm{Cl}_2(2\alpha)$ at twice the angles $\alpha$ of the triangle with those masses as sides, here right isosceles, and the combination equals $-8\,G$. The residue matrix at $s=0$ has no positive integer eigenvalue, so these vacuum values fix every Taylor coefficient of the bounded branch, and transporting that branch to $s=-2$ along the differential equation gives the boundary value $J^{(0)}(-2) = -0.8887421666\ldots$.

The boundary value is itself a period integral of the curve. In the normalization $N_{\rm kite} = 4s\,J^{(0)}(s)\big|_{s=-2} = 7.109937332\ldots$ used in the expression file it takes the form

$$ N_{\rm kite} \;=\; c_1\,\psi_1(-2) + c_2\,\psi_2(-2) \;+\; \psi_2(-2)\int_{-4}^{-2}\frac{\psi_1(t)\,R(t)}{W(t)}\,dt \;-\; \psi_1(-2)\int_{-4}^{-2}\frac{\psi_2(t)\,R(t)}{W(t)}\,dt, $$

where $t$ runs along the negative axis and $R$ is the source term for $4s\,J^{(0)}$, read from the same top row of $A$ as $S$. The lower limit $t=-4$ is an arbitrary base point on the negative axis; the constants $c_1 = 26.35941243\ldots$ and $c_2 = -35.49989517\ldots$ depend on that choice and are fixed by the bounded solution at $s=0$. An integer-relation search finds no expression for $N_{\rm kite}$ in terms of the classical weight-two constants, the Dirichlet $L$-values of conductor $4$ and $8$, or products of $\Gamma(1/4)^2$ with the periods $\psi_{1,2}(-2)$, and none should exist: at $s=-2$ the curve is $y^2 = x^3 - 756x + 7344$, which has conductor $128$, rank one and no complex multiplication, so there is no reason for its periods to reduce to values of the $\Gamma$ function.

A second mass point

The same construction goes through with the sunrise masses squared set to $(1,1,3)$, that is internal masses $(1,0,1,0,\sqrt3)$: the singular points become $s\in\{0,1,3,7\pm4\sqrt3\}$, the residue matrix at $s=0$ has the same eigenvalues, the Clausen values of the vacuum sunset are taken at the angles $(\pi/6,\pi/6,2\pi/3)$ of the $(1,1,\sqrt3)$ mass triangle in place of Catalan's constant, and the boundary point is $s=-5/3$.

The physical region

Above the thresholds the same differential equation supplies the $s+i0$ value: the solution is continued from $s=-2$ along a path in the upper half of the complex $s$-plane, past the five real singular points, and back to the real axis at the physical value of $s$, which gives the real and imaginary parts of $J^{(0)}$ there.

The complete expression, with the curve data and the boundary value to full precision, is in kite-expression.md.

Checks

Evaluator

Same family

• The equal-mass kite has all three massive lines at one mass; its sunrise curve is modular for $\Gamma_1(6)$ and the known result is a five-term combination of ordinary and elliptic polylogarithms.
• The four-mass kite with a massless rung has four different rim masses and a massless central line, so every three-line cut contains a massless line and no elliptic curve appears.
• The sunrise sub-diagram on its own, at these and two other mass sets, is the unequal-mass sunrise.
• The ice-cream cone with generic masses is another integral on an elliptic curve that is not modular, likewise obtained from its differential equation and analytically derived boundary data; the $gg\to Z\gamma$ integrals at exact top-quark mass live on a curve that is modular, for $\Gamma_0(4)$, and are solved the same way, by transport of the exact differential equation from boundary constants derived analytically at $s=0$.

The paper

• Polylogarithmic, elliptic, and Calabi–Yau Feynman integrals from a hybrid bootstrap (PDF) — Matthew D. Schwartz. Section 5.4, “The unequal-mass kite,” derives the transport representation, the boundary value and the continuation to the physical region.

References

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