Light-by-light box with a kite insertion

The three-loop light-by-light box in which one massive electron line carries the two-loop kite self-energy, with the electron segments flanking the insertion removed. Its value is new. Because the diagram separates into two pieces when the dressed line is cut, that value is a dispersion integral of the one-loop box against the spectral density of the kite.

The integral

Light-by-light scattering is the process $\gamma\gamma\to\gamma\gamma$: four photons with momenta $p_1,\ldots,p_4$, all massless and on shell ($p_i^2=0$), interacting through a loop of virtual electrons of mass $m$. At three loops with the electron mass kept, every integral of the light-by-light family 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, with propagators $D_j=q_j^2-m^2$ on electron lines and $D_j=q_j^2$ on internal photon lines, where each $q_j$ is the combination of the loop momenta $l_1,l_2,l_3$ and the external momenta flowing through line $j$. 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 $u=(p_2+p_3)^2=-s-t$ (the photons are massless); the planar channels of the box are $s$ and $u$; and $m^2$, set to $1$, fixes the overall scale. The kite-insertion integral has $n=8$ lines with every index $\nu_j$ equal to one: the three undressed sides of the box are electron lines, and the kite that replaces the fourth side has three electron lines of mass $m$ and two massless photons. The integral is ultraviolet finite, and the quantity computed is its value in four dimensions, written $I(s,t,m^2)$, for $s,t\lt0$ with $u=-s-t$ below the two-electron threshold $4m^2$, where it is real. The kite self-energy depends on a single further variable, $w$, the squared invariant mass flowing through the dressed line.

Since the ATLAS collaboration reported evidence for $\gamma\gamma\to\gamma\gamma$ in ultraperipheral lead–lead collisions in 2017, the QED prediction has been pushed to higher orders: at two loops the master integralsthe finite set of integrals to which every integral of a family reduces by integration-by-parts identities with the electron mass kept are polylogarithmic (Ajjath, Chaubey and Shao, 2023), and at three loops the amplitudes are known for a massless electron (Bargiela, Chakraborty, Gambuti and Ozcelik, 2026). With the mass kept at three loops, no master integral of the family appears to have been computed before.

Because the kitethe two-loop self-energy shaped as a rhombus with a diagonal: here three electron lines of mass $m$ and two massless photons, the crossed-photon electron self-energy, which contains the equal-mass sunrise as a sub-diagram joins two corners of the box directly, the graph is not itself a QED Feynman diagram. It is the QED diagram with the same self-energy after the two electron segments flanking the insertion are shrunk to points. With $m^2=1$ the two-particle threshold in $w$ is at $w=1$ and the three-particle threshold at $w=9$. On the maximal cutthe integral with every propagator put on shell, which isolates the geometry attached to the diagram the integral factorizes into the one-loop massive box, which is polylogarithmic, times the cut self-energy. Above $w=9$ the three-particle cut of the kite is a phase-space integral over the elliptic curve of the equal-mass sunrise, the curve with modular group $\Gamma_1(6)$the level-six congruence subgroup of the integer two-by-two matrices of determinant one; the ratio of the two periods of the equal-mass sunrise curve transforms under it; Adams and Weinzierl introduced iterated integrals of the modular forms of this group to describe the sunrise and the kite in 2017. The singular points $w\in\{0,1,9,\infty\}$ of that curve are the only possible singular points of the kite's spectral density $\rho(w)$.

Closed form

Cutting the dressed line separates the diagram into the one-loop box and the kite, so the integral is a dispersion integral over $w$:

$$I(s,t,m^2)=\frac1\pi\int_{m^2}^\infty\!\rho(w)\,\mathrm{Box}_1(s,t;w)\,\mathrm{d}w,\qquad \rho(w)=-\,\mathrm{Im}\,J_{\rm top}(w+i0),$$

where $\mathrm{Box}_1(s,t;w)$, the one-loop box with three lines of mass $m$ and one line of squared mass $w$, is a closed-form combination of logarithms and dilogarithms, and $\rho$, the spectral density of the kite, is the negative of the imaginary part of its top-sector masterthe master integral with every propagator of the diagram present once $J_{\rm top}$ just above the cut, at $w+i0$. Below the three-particle threshold the density is polylogarithmic of weight two ($m^2=1$),

$$\rho(w)=-\frac{\pi}{w}\Big[2\ln(w-1)\ln w+3\,\mathrm{Li}_2(1-w)\Big],\qquad m^2\le w\le 9m^2,$$

and above it the three-particle cut of the equal-mass sunrise inside the kite adds a term that is a single integral over the periodsthe two integrals of the holomorphic differential, one around each cycle of the elliptic curve, as functions of $w$ of the sunrise curve. The density satisfies the kite's differential equation in $w$ (the discontinuity of a solution is again a solution), and regularity at the two-particle threshold $w=1$ fixes the coefficient $-2\pi$ of the threshold term $(w-1)\ln(w-1)$ and with it every constant in the closed form. At large $w$,

$$\rho(w)=\frac{4\pi\ln w+6\pi}{w^2}+\frac{8\pi\ln w+\tfrac{\pi}{2}}{w^3}+\cdots$$

and the density obeys the sum rule $\frac1\pi\int_{m^2}^\infty\rho\,\mathrm{d}w=6\zeta_3$, which follows from the large-momentum limit of the kite self-energy.

The complete expression, with the conventions of the kite master integrals and the derivation of the boundary vector at $w=5$, is in the expression file.

Checks

Evaluator

Same family

• The light-by-light box with a sunrise insertion dresses the same box line with the three parallel electron lines of the equal-mass sunrise instead of the kite. Shrinking the kite's two photons gives it, the smallest elliptic member of the family.
• The QED light-by-light graphs with a kite self-energy keep the electron segments flanking the kite, so that every vertex joins two electrons and a photon.
• The light-by-light box with a vacuum-polarization insertion dresses the line with a one-loop self-energy whose photon carries an electron loop of its own, giving two fermion loops.
• The crossed light-by-light box puts the two internal photons on the diagonals; its top-sector integral mixes multiple polylogarithms with the periods of the K3 surface of the three-loop banana, which appears when both photons are contracted.

The inserted self-energy on its own, three electron lines of mass $m$ and two massless photons as a two-point function of $w$, is the equal-mass kite.

The paper

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

References

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