Equal-mass kite

The two-loop kite self-energy with three lines of mass $m$ and two massless lines. The three massive lines form the equal-mass sunrise, the sub-diagram whose maximal cut is the elliptic curve of the integral. The kite's finite part, computed in elliptic polylogarithms by Adams, Bogner, Schweitzer and Weinzierl in 2016, is recomputed here without using the published answer.

The integral

The kite is a two-loop self-energya diagram with a single external momentum p flowing in and out; it is a correction to a particle's propagator and depends only on p²: a single off-shell momentum $p$ enters at one vertex and leaves at the other. Three of the five lines carry the same mass $m$ and the other two are massless, so the only kinematic variable is $t=p^2/m^2$, and $m^2$ is set to $1$. 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

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

Contracting the two massless lines $D_2$ and $D_4$ gives the equal-mass sunrise on the massive lines $D_1$, $D_3$, $D_5$, and that sub-diagram fixes the class of functions: its maximal cutput every internal line of the sub-diagram on shell simultaneously; the integral over what survives exposes the diagram's underlying geometry defines an elliptic curve. The curve degenerates at the two thresholds $t=1$ and $t=9$ ($p^2=m^2$ and $p^2=9m^2$), and together with $t=0$ and $t\to\infty$ these are the four singular points of the integral. The quantity computed, written $J^{(0)}(t)$, is the finite $\varepsilon^0$ coefficient of the top-sector masterthe master integral with every propagator of the diagram present once $J(1,1,1,1,1)$ divided by $(i\pi^{d/2})^2$, one factor per loop, with no factor $e^{\gamma_E\varepsilon}$. The integral is finite at $\varepsilon=0$.

The kite enters the two-loop electron self-energy in QED, whose spectral functions Sabry computed in 1962. Remiddi and Tancredi (arXiv:1602.01481) derived differential equations and dispersion relations for the massive sunrise and the kite in 2016, and in the same year Adams, Bogner, Schweitzer and Weinzierl (arXiv:1607.01571) gave the equal-mass kite to all orders in $\varepsilon$ in terms of elliptic polylogarithms. These are iterated integrals on the sunrise curve, written as power series in the nomethe expansion variable of functions on an elliptic curve, built from the ratio of its two periods; power series in it converge away from the singular points $q=e^{i\pi\tau}$, where $\tau=\psi_2/\psi_1$ is the ratio of the two periodstwo integrals of the holomorphic differential, one around each cycle of the elliptic curve; every solution of the curve's Picard–Fuchs differential equation is a combination of them of the curve. Besides $q$ they depend on auxiliary arguments, which for the sunrise and the kite are sixth roots of unity (here $e^{2\pi i/3}$ and $\pm1$). The curve of the equal-mass sunrise is modular, as Bloch and Vanhove (arXiv:1309.5865) showed: carrying its periods around the singular points mixes them only by matrices of the subgroup $\Gamma_1(6)$ of $SL(2,\mathbb{Z})$, so $t$ is a modular function of $\tau$ for that group and the four singular points of the integral lie at the four cusps of $\Gamma_1(6)$. Adams and Weinzierl (arXiv:1704.08895) then wrote the whole sunrise and kite family as iterated integrals of modular formsfunctions of τ that transform with a fixed weight under a congruence subgroup of SL(2,ℤ) and have convergent expansions in the nome for $\Gamma_1(6)$, and Hönemann, Tempest and Weinzierl (arXiv:1811.09308) recomputed the full two-loop electron self-energy with these functions in 2018.

Beyond the functions of the sunrise curve, the two massless lines bring in only logarithmic singularities at $t=0$ and $t=1$, the letters $t$ and $1-t$ of ordinary polylogarithms. The kite satisfies a first-order differential equation in $t$ with the sunrise as its source (Remiddi and Tancredi), so its finite part is obtained by integrating sunrise-type functions against a kernel, $\psi_1^3/W$ with $W=\psi_1\psi_2'-\psi_2\psi_1'$ the Wronskian of the periods. That kernel is itself a modular form for $\Gamma_1(6)$ with a $q$-series expansion (Adams and Weinzierl), so each integration of the differential equation stays within the elliptic polylogarithms of the curve and brings in only constant coefficients. Dividing out an algebraic prefactor $1/(4t)$ then leaves a quantity of transcendental weight at most three (at most three iterated integrations), and only five such functions can occur: three harmonic polylogarithms with letters $\{0,1\}$ and two elliptic polylogarithms of $\Gamma_1(6)$ at the third root of unity. An integer-relation (PSLQ) fit to values of the integral at a few points below threshold gives the five coefficients as exact numbers, and the result is the expression of Adams et al.

Closed form

With $G(a_1,\ldots,a_w;t)$ the multiple polylogarithms with letters $0$ and $1$ (harmonic polylogarithms), and $\overline{E}$ the elliptic polylogarithms of Adams et al., taken at the cube root of unity $\varrho=e^{2\pi i/3}$ and at nome $-q$, the finite part of the top-sector master is

$$ J^{(0)}(t) \;=\; \frac{1}{4t}\Big[\, -\tfrac{2\pi^2}{3}\,G(1;t) \;-\; 8\,G(0,1,1;t) \;+\; 4\,G(1,0,1;t) \;-\; 108\,\mathrm{Cl}_2\!\big(\tfrac{2\pi}{3}\big)\,\overline{E}_{1;-1}(\varrho;1;-q) \;-\; 108\,\overline{E}_{0,2;-2,0;2}(\varrho,\varrho;1,-1;-q) \,\Big], $$

where $\mathrm{Cl}_2(2\pi/3)$ is the Clausen function at $2\pi/3$. Apart from it and $\pi^2$, every coefficient is a rational number. The expression holds for real $t\lt1$ with $t\neq0$. Here $\overline{E}_{n;m}(x;y;q)$ is the double $q$-series of Adams et al. with integer weights $n,m$ on its two summation indices, and $\overline{E}_{0,2;-2,0;2}$ is its depth-two iterate. The two $\overline{E}$ functions are power series in the nome and are summed numerically once $q$ is computed from the periods of the sunrise curve at the given $t$. In this notation the kernel $\psi_1^3/W$ of the differential equation is $-6\,\overline{E}_{0;-2}(\varrho;-1;-q)$.

The physical region

For $t\gt1$ the nome is continued along $t\to t+i0$ through the upper half of the complex $t$-plane, and the harmonic polylogarithms take their $t+i0$ boundary values. The continued expression is complex, and its imaginary part is minus the spectral density of the kite. The closed form holds on either side of the thresholds $t=1$ and $t=9$ but not at the thresholds themselves, where the sunrise curve degenerates.

Checks

Evaluator

Same family

The unequal-mass kite moves one of the three masses to $\sqrt2$: the sunrise curve is then no longer modular, and the finite part is given as a single integral over the periods of that curve from an analytically derived boundary value, a representation that is new. In the four-mass kite with a massless rung the four rim lines carry four different masses and the rung is massless; both sunrise sub-diagrams of a kite contain the rung, so neither defines an elliptic curve and the integral is polylogarithmic. The equal-mass sunrise is the three massive lines on their own, the integral whose curve, periods and nome the kite inherits. Inserting the kite in place of one line of the one-loop photon box gives the light-by-light box with a kite insertion, where its spectral density enters a dispersion integral.

The paper

• Polylogarithmic, elliptic, and Calabi–Yau Feynman integrals from a hybrid bootstrap (PDF) — Matthew D. Schwartz. Section 5.2, “The equal-mass kite,” gives the function space of the kite from the sunrise curve, the result with the fit of its coefficients, sample values, and the continuation to the physical region.

References

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