Light-by-light box with a sunrise insertion

A three-loop contribution to photon–photon scattering through a massive electron loop: the one-loop box with the two-loop equal-mass sunrise inserted on one internal line, all six internal lines carrying the electron mass. The integral is new, and its only elliptic curve is that of the equal-mass sunrise sub-diagram.

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 loops of virtual electrons of mass $m$. In this graph one of the four lines of the one-loop electron box carries the two-loop equal-mass sunrise. The graph belongs to the family

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

where the measure is $d^dl/(i\pi^{d/2})$ per loop with no factor $e^{\varepsilon\gamma_E}$, and the $n=6$ propagators are $D_j=q_j^2-m^2$, one for each electron line, $q_j$ being the combination of the loop momenta $l_1,l_2,l_3$ and the external momenta flowing through line $j$. The Mandelstam invariantsthe Lorentz-invariant combinations of external momenta that a scattering amplitude can depend on are $s=(p_1+p_2)^2$ and $t=(p_2+p_3)^2$; the third, $u=(p_1+p_3)^2$, satisfies $s+t+u=0$ because the photons are massless. Throughout, $m^2=1$. Two quantities are computed. The first is the top-sector masterthe master integral with every propagator of the diagram present once $I[1,1,1,1,1,1]\,(s,t;m^2)$ through order $\varepsilon^0$ for $s,t\lt0$, where it is real. The second is the sunrise insertion on which it depends: the two-loop equal-mass sunrise in the same dimension and normalization, as a function of the virtuality $w=k^2$, where $k$ is the momentum through the line that carries the insertion (the dressed line). The Laurent series of the insertion begins at order $\varepsilon^{-2}$. The elliptic sectors are analyzed in an auxiliary three-point family in which $k$ flows through a propagator of mass $m$ and no propagator carries $(k+p_1+p_2)^2$, so that its integrals do not depend on $s$.

The ATLAS collaboration (2017) reported evidence for light-by-light scattering in ultraperipheral lead–lead collisions at the LHC. The two-loop amplitudes with the fermion mass kept are polylogarithmic and were computed by Ajjath, Chaubey and Shao (2023); at three loops Bargiela, Chakraborty, Gambuti and Ozcelik (2026) computed the amplitudes with massless internal fermions. With the electron mass kept, this three-loop graph appears to be new.

A three-loop light-by-light graph with one electron loop has two internal photons. When both attach to the same side of the box with their endpoints interleaved (emit, emit, absorb, absorb), that side is the two-loop kite self-energy, and contracting the kite's two photon propagators leaves the equal-mass sunrise on the dressed line: three parallel electron propagators between two vertices. Bloch and Vanhove (2015) wrote the sunrise integral on its elliptic curve as an elliptic dilogarithm, and Adams and Weinzierl (2017) wrote it to all orders in $\varepsilon$ as iterated integrals of modular formsholomorphic functions on the upper half-plane that transform simply under a subgroup of the integer two-by-two matrices, here the congruence subgroup $\Gamma_{1}(6)$ for $\Gamma_{1}(6)$. Inside the box the curve is fibered over the virtuality $w$ of the dressed line, with singular fibers at $w\in\{0,1,9,\infty\}$ of Kodaira typesthe classification of the degenerate fibers of an elliptic fibration; the fiber $I_n$ is a cycle of $n$ components $I_1$, $I_2$, $I_3$ and $I_6$; the other three lines of the box add no further curve, so the function space is that of the equal-mass sunrise continued to higher orders in $\varepsilon$. On the maximal cut of the auxiliary three-point family the dressed line is on shell, $w=m^2$, a singular fiber where the curve degenerates, so the elliptic functions are taken from the sunrise insertion itself, with the dressed line kept off shell at general $w$. Because the graph falls into two pieces when the three sunrise lines are cut, the box is a dispersion integral in $w$ over that cut, the construction Remiddi and Tancredi (2016) used for the sunrise and the kite.

Closed form

Of the ten master integralsthe finite basis of integrals to which every integral of the family reduces by integration-by-parts identities of the auxiliary three-point family, the three polylogarithmic ones are products of one-loop integrals. The seven elliptic ones are all built from one elliptic function of the virtuality of the dressed line, $\mathcal{I}_{\rm sun}(w)$: one of them is the equal-mass sunrise taken in $2-2\varepsilon$ dimensions, whose order-$\varepsilon^0$ term is $\mathcal{I}_{\rm sun}(w)$ up to a constant factor, and the other six contain that master under a one-fold integral over $w$. The sunrise insertion in $d=4-2\varepsilon$ is a combination of the same master, its derivative in $w$ and the square of the one-loop tadpole with coefficients rational in $w$ and $\varepsilon$ (the dimension-shift relations), so its order-$\varepsilon^0$ coefficient, the second quantity defined above, contains $\mathcal{I}_{\rm sun}(w)$ but is not equal to it. The function is

$$\mathcal{I}_{\rm sun}(w)=\psi_1(w)\big[\mathcal{E}[f_3](\tau(w))+c_1\big]+c_2\,\psi_2(w),\qquad (c_1,c_2)=\big(B^{(0)},0\big),\qquad \psi_1=\tfrac{2\pi}{\sqrt3}\,\varpi_0.$$

Here $\psi_1(w)$ and $\psi_2(w)$ are the two periods of the sunrise curve at virtuality $w$, a basis of the solutions annihilated by the Picard–Fuchs operatorthe linear differential operator in the kinematic variable that annihilates the periods of the curve of the equal-mass sunrise,

$$L_{\rm sun}=t(t-1)(t-9)\,\frac{d^2}{dt^2}+(3t^2-20t+9)\,\frac{d}{dt}+(t-3),\qquad t=w/m^2,\qquad \varpi_0(0)=1,$$

in which $t=w/m^2$ is the customary sunrise variable, not the Mandelstam invariant of the box (with $m^2=1$ it equals $w$), and $\varpi_0$ is the power-series solution regular at $w=0$. The function $\tau(w)$ is the modular parameter of the curve, the ratio of the two periods; $\mathcal{E}[f_3]$ is the Eichler integralan iterated integral of a modular form against powers of the modular parameter; the elliptic replacement for a polylogarithm of the weight-three Eisenstein form $f_3$ on $\Gamma_{1}(6)$; and $c_1$, $c_2$ are boundary constants, of which only $c_1=B^{(0)}$ is nonzero. It is fixed at the cusp $w=0$ and is a value of the Clausen function $\mathrm{Cl}_2$,

$$B^{(0)}=2\,\mathrm{Cl}_2(\pi/3),$$

numerically $B^{(0)}=2.029883212\ldots$. At higher orders in $\varepsilon$ the boundary constant becomes a series $B^{(0)}+\varepsilon B^{(1)}+\varepsilon^2B^{(2)}+\cdots$ fixed at the same cusp. Its next two coefficients, which enter the orders $\varepsilon^1$ and $\varepsilon^2$ of the same master, lie in the ring of sixth-root-of-unity constants of Broadhurst (1999) extended by the log-sine integrals $\mathrm{Ls}_n^{(k)}(\theta)=-\int_0^\theta x^k\log^{n-1-k}\lvert 2\sin\tfrac{x}{2}\rvert\,dx$, with $\mathrm{Ls}_n\equiv\mathrm{Ls}_n^{(0)}$:

$$B^{(1)}=\tfrac{1}{6}\pi^3-2\log 3\,\mathrm{Cl}_2(\pi/3)+3\,\mathrm{Ls}_3(2\pi/3),\qquad B^{(2)}=-2\pi\zeta_3-\tfrac{1}{6}\pi^3\log 3+\log^2 3\,\mathrm{Cl}_2(\pi/3)+2\,\mathrm{Ls}_4(2\pi/3)-3\log 3\,\mathrm{Ls}_3(2\pi/3),$$

and $\mathrm{Ls}_3(2\pi/3)$ is the only constant outside Broadhurst's ring that enters at weight three; both relations were found by integer-relation (PSLQ) searches in the extended ring.

The six elliptic master integrals that contain $\mathcal{I}_{\rm sun}$ under an integral over $w$ have no boundary constants of their own. The box itself, for $s,t\lt0$ and through order $\varepsilon^0$, is a dispersion integral over the three-particle cut of the sunrise,

$$I[1,1,1,1,1,1](s,t;m^2)=\int_{9m^2}^{\infty}dw\;\rho_{\rm sun}(w)\,\big[\mathcal{B}(s,t;w)\big]_{\rm sub}+U(s,t),$$

where $\rho_{\rm sun}(w)$ is the three-particle cut density of the sunrise, which starts at $w=9m^2$; $\mathcal{B}(s,t;w)$ is the one-loop box with three lines of mass $m$ and a fourth line of squared mass $w$, a dilogarithmic function of $s$, $t$ and $w$; the bracket $[\;]_{\rm sub}$ denotes that kernel with the large-$w$ behavior subtracted exactly in $\varepsilon$; and $U(s,t)$ adds the subtraction back as generalized one-loop triangle integrals, which carry the two poles in $\varepsilon$. After the subtraction the integral over $w$ converges.

The expression file inside the bundle under Evaluator gives the closed form of $\mathcal{I}_{\rm sun}$, the Picard–Fuchs operator and the normalization of $\psi_1$, with numerical values of the two periods at a reference virtuality, of the Eichler integral at six Euclidean virtualities and of $B^{(0)}$ to full precision; the closed forms of $B^{(1)}$ and $B^{(2)}$ and the dispersion formula for the box are those of the paper, section 5.7.

Checks

Evaluator

Same family

• The box with a kite insertion: the dressed line carries the two-loop kite self-energy, three massive lines and two photons, and the same sunrise curve enters through the kite's spectral density.
• The QED graphs with a kite self-energy: the kite insertion with the two flanking electron propagators kept, as the graph appears in the amplitude; shrinking those propagators and then the kite's two photons gives the sunrise insertion.
• The box with a vacuum-polarization insertion: another electron loop sits as a bubble on an internal photon, and that integral is obtained exactly in $d$, poles included.
• The crossed box: the two internal photons join opposite corners of the box; its top sector is polylogarithmic on the maximal cut, but contracting both photons leaves the three-loop banana, and the finite part contains iterated integrals of the banana's K3 period rather than functions on an elliptic curve.

The paper

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

References

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