QED light-by-light graphs with a kite self-energy

Three-loop light-by-light scattering in QED with the electron mass kept: one electron line of the one-loop box carries the two-loop crossed-photon self-energy (the kite topology), and the electron propagator adjoining the insertion appears either once or squared. Both integrals are new and are given as single dispersion integrals over the squared mass of the dressed line.

The integral

Light-by-light scattering, $\gamma\gamma\to\gamma\gamma$, is the interaction of four massless on-shell photons ($p_i^2=0$) through loops of virtual electrons of mass $m$. With all momenta incoming the Mandelstam invariantsthe Lorentz-invariant combinations of external momenta that a scattering amplitude can depend on are $s=(p_1+p_2)^2$, $t=(p_1+p_3)^2$ and $u=-s-t$; the two planar channels of the box are $s$ and $u$. The electron mass sets the scale, and $m^2=1$ in the numerical work. Every integral in the family of this graph has the form

$$I[\nu_1,\ldots,\nu_n](s,t;m^2)\;=\;\int d^dl_1\,d^dl_2\,d^dl_3\;\prod_{j=1}^{n}\frac{1}{D_j^{\nu_j}},\qquad d=4-2\varepsilon,$$

up to an overall normalization convention. For this graph $n=9$; writing the loop momentum of the box as $\ell=l_1$ and those of the insertion as $k_1=l_2$, $k_2=l_3$, the propagators are

$$ \begin{aligned} D_1&=(\ell+p_1)^2-m^2, & D_2&=(\ell+p_1+p_2)^2-m^2, & D_3&=(\ell+p_1+p_2+p_3)^2-m^2,\\ D_4&=\ell^2-m^2, & D_5&=k_1^2-m^2, & D_6&=k_2^2-m^2,\\ D_7&=(\ell-k_1-k_2)^2-m^2, & D_8&=(\ell-k_1)^2, & D_9&=(\ell-k_2)^2. \end{aligned} $$

Here $D_1,D_2,D_3$ are the undressed sides of the box, $D_5,\ldots,D_9$ form the crossed-photon self-energythe electron emits photon 1, emits photon 2, absorbs photon 1, absorbs photon 2: two interleaved photon arcs on one fermion line, the same five-propagator topology as the two-loop kite, and $D_4$ is the electron line on either side of the insertion, whose two segments are the stubs. Both stubs carry the momentum $\ell$, so the QED graph in the figure is $I_{\rm KP}=I[1,1,1,2,1,1,1,1,1]$, with $D_4$ squared, while the single-stub member $I_{K_1}=I[1,1,1,1,1,1,1,1,1]$ has one power of it. Both $I_{\rm KP}$ and $I_{K_1}$ are ultraviolet finite. The quantities computed are their values at $d=4$ as functions of $s$ and $t$ at Euclidean points, $s,t\lt0$ with $u$ below its two-particle threshold $4m^2$, where the integrals are real.

A self-energy insertion on one line of a loop is a two-particle-reduciblethe graph falls into two pieces when the two stub lines on either side of the insertion are cut piece of the amplitude, and such a graph is a dispersion integral over the squared mass $w$ of the dressed line: the one-loop box with that line given mass squared $w$, weighted by the spectral densitythe imaginary part of the self-energy across its cut, as a function of the invariant mass flowing through it of the insertion, as in the two-loop self-energy methods of Bauberger, Berends, Böhm and Buza (1995) and the dispersion relations of Remiddi and Tancredi (2016) for the sunrise and the kite. Sabry (1962) computed the spectral functions of the two-loop electron propagator, and the kite's master integralsa finite basis of integrals to which every integral of the family reduces by integration-by-parts identities are known as iterated integrals of modular forms for $\Gamma_1(6)$functions on the upper half-plane with a prescribed transformation law under $\Gamma_1(6)$, the subgroup of $SL(2,\mathbb{Z})$ that is the symmetry group of the equal-mass sunrise curve from Hönemann, Tempest and Weinzierl (2018); the density used here is derived instead from the differential equation of the kite. Bargiela, Chakraborty, Gambuti and Ozcelik (2026) computed the three-loop amplitudes with massless fermions and Ajjath, Chaubey and Shao (2023) the two-loop amplitudes with the fermion mass kept; at three loops with the electron mass kept, $I_{\rm KP}$ and $I_{K_1}$ do not appear to have been computed before.

The elliptic curve enters only through the insertion. On the maximal cutthe integral with every propagator put on shell, which isolates the intrinsic geometry of the graph the integral factorizes into the one-loop massive box, which is polylogarithmic, times the cut of the self-energy, and above $w=9m^2$ the three-particle cut of the kite is a phase-space integral on the elliptic curve of the equal-mass sunrise (Adams and Weinzierl, 2017), the $\Gamma_1(6)$ family with singular fibers at $w\in\{0,m^2,9m^2,\infty\}$. Below that threshold the density is a weight-two combination of logarithms and dilogarithms.

A second ten-line QED graph of three-loop light-by-light scattering, the non-planar fermion octagon whose two internal photons join opposite sides of the loop and cross, has not been computed; the crossed light-by-light box is its pinch with the four corner lines contracted. The octagon's maximal cut nevertheless fixes its geometry: a genus-one curve carrying thirty master integrals on the cut, whose modulus (the $j$-invariant) varies with $s$ and $t$. The curve of the graph computed here is inherited from the sunrise inside the self-energy, but the octagon's curve belongs to its own topology, so the massive three-loop $\gamma\gamma\to\gamma\gamma$ amplitude is not polylogarithmic even apart from its self-energy insertions.

Closed form

With the stubs kept, partial fractions in $\ell^2$ between the dispersion denominator $\ell^2-w$ and the stub propagator $\ell^2-m^2$ replace the one-loop box by a subtracted kernel and leave the density unchanged:

$$I_{\rm KP}=\frac1\pi\int_{m^2}^\infty\!\rho(w)\,K_P(w)\,\mathrm{d}w,\qquad K_P(w)=\frac{K(w)-K(m^2)}{(w-m^2)^2}-\frac{K'(m^2)}{w-m^2},$$

where the kernel $K(w)$ is the scalar one-loop box of Passarino and Veltman (1979) with three lines of mass $m$ and the dressed line of mass squared $w$, a closed-form combination of dilogarithms; $K(m^2)$ and $K'(m^2)$ are its value and its first derivative in $w$ at $w=m^2$, both taken from the closed form. The density $\rho(w)=-\,\mathrm{Im}\,J_{\rm top}(w+i0)$ is the spectral density of the kite, the discontinuity of its top-sector masterthe master integral with every propagator of the self-energy present once $J_{\rm top}$ across the cut. The squared stub would place a double pole at the lower endpoint $w=m^2$ of the dispersion integral; the two subtractions remove it, and $K_P$ is regular there. The single-stub member $I_{K_1}$ is the same integral with $K_1(w)=[K(w)-K(m^2)]/(w-m^2)$ in place of $K_P$. With the bare kernel $K(w)$ and the same density the integral is the box with a kite insertion, the no-stub member in the Checks table. The identity is general: any graph with a self-energy on one line of a loop reduces in this way to divided differences of the bare-line kernel integrated against the unchanged density of the insertion. At the reference point $(s,t,m^2)=(-1,-\tfrac13,1)$ the two kernel constants are $K(m^2)=0.1780502267\ldots$ and $K'(m^2)=-0.08182371131\ldots$.

The density is fixed by exact data alone. It is the discontinuity of the differential system of the nine kite master integrals in $w$, and regularity at the two-particle threshold $w=m^2$, where it behaves as $-2\pi\,(w-m^2)\ln(w-m^2)$, is the only boundary condition. Below the three-particle threshold the density is the weight-two combination of logarithms and dilogarithms given on the kite-insertion page, and above it the three-particle cut of the equal-mass sunrise inside the kite adds a single integral over that cut. The evaluation script uses both forms: inside the dispersion integral it generates the density as the power-series solution in $w$ of the kite system, started at $w=5m^2$ from values obtained by solving that system outward from $w=0$, where the master integrals have closed forms, and at fixed $w$, as in the density rows of the Checks table, it evaluates the closed form and compares the two. All dependence on $s$ and $t$ sits in the kernel $K$; an iterated-integral form in $s$ and $t$ themselves was not constructed. The complete expression, with the conventions, the density below and above the three-particle threshold and the kernel constants, is in the expression file inside the bundle under Evaluator.

Checks

Evaluator

Same family

• The light-by-light box with a kite insertion is this graph with both stubs removed, which makes the two box corners at the insertion four-valent, so it is not itself a QED diagram.
• The box with a sunrise insertion replaces a whole side of the box by the three massive lines of the equal-mass sunrise and is the simplest light-by-light graph in which the elliptic curve appears.
• The vacuum-polarization insertion dresses the line with the self-energy whose photon carries a second electron loop, and is given exactly in $d$ with its double pole in $\varepsilon$.
• The crossed light-by-light box has the two internal photons joining opposite corners; its maximal cut alone is polylogarithmic, but contracting both photons leaves the three-loop banana, whose K3 surface enters the finite top-sector master through an iterated integral of the K3 period.

The paper

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

References

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