Light-by-light box with a vacuum-polarization insertion

Three-loop light-by-light scattering in QED with the electron mass kept: one electron line of the one-loop box emits and reabsorbs a photon that itself carries an electron loop. The integral is new and is given here as an exact function of the space-time dimension, poles included, in the form of a dispersion integral over the 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 planar channels of the box are $s$ and $u$. The electron mass sets the scale, and $m^2=1$ in the numerical work. Every member of the three-loop family is an integral of 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 overall normalization. For this graph $n=8$; 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

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

Here $D_1,D_2,D_3$ are the undressed sides of the box and $D_4$ is the electron line on either side of the insertion, whose two segments are called the stubs; $D_5$ is the electron inside the self-energy, $D_6$ the internal photon on either side of the fermion loop, and $D_7,D_8$ that loop. Both stubs carry the momentum $\ell$ and both photon segments the momentum $k_1$, so the graph in the figure is $I_{\rm VP}=I[1,1,1,2,1,2,1,1]$, with $D_4$ and $D_6$ squared. The one-loop subdivergences of the vacuum polarization and of the self-energy around it give $I_{\rm VP}$ a double pole in $\varepsilon$. The quantity computed is $I_{\rm VP}$ itself at fixed $\varepsilon$ in $d=4-2\varepsilon$, as a function of $s$ and $t$ at Euclidean points ($s,t\lt0$ with $u$ below its two-particle threshold $4m^2$, where the integral is real), together with the first four coefficients of its Laurent expansion $I_{\rm VP}(\varepsilon)=\sum_{k\ge-2}I_{\rm VP}^{(k)}\varepsilon^k$.

Classical electrodynamics is linear, so two beams of light cross without interacting; in QED four photons scatter through a loop of virtual electrons. The ATLAS collaboration (2017) found evidence for $\gamma\gamma\to\gamma\gamma$ in ultraperipheral lead–lead collisions at the LHC, and ATLAS and CMS have since observed it there; the process is also a search channel for axion-like particles. At two loops the amplitudes with the electron mass kept are polylogarithmic (Ajjath, Chaubey and Shao (2023)); at three loops the massless amplitudes are known (Bargiela, Chakraborty, Gambuti and Ozcelik (2026)), and with the mass kept this graph, the simplest with a second fermion loop, does not appear to have been computed before.

A self-energy insertion on one line of a loop is a two-particle-reduciblecutting the two stub lines on either side of the insertion separates the graph into two pieces 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). The spectral functions of the two-loop electron propagator, this insertion among them, go back to Sabry (1962); here the density is derived instead from the differential equation of the self-energy and comes out exact in $d$. On the maximal cutall eight propagators placed on shell at once the graph factorizes into the polylogarithmic one-loop box times the cut self-energy, so anything beyond polylogarithms must come from the insertion. The elliptic part comes from the three-electron cut of the self-energy, which opens above $w'=9m^2$; the three-body phase-space integral is defined on the elliptic curve of the equal-mass sunrise, the modular curve of $\Gamma_1(6)$ of Bloch and Vanhove (2015) and Adams and Weinzierl (2017).

Closed form

For any $d$, $I_{\rm VP}$ is a single dispersion integral,

$$I_{\rm VP}(\varepsilon)=\frac1\pi\int_{m^2}^\infty\!\rho_{\rm VP}(w';\varepsilon)\,K_P(w';\varepsilon)\,\mathrm{d}w'.$$

Here $\Sigma_{\rm VP}(w')$ is the two-loop self-energy insertion at squared momentum $w'$ and $\rho_{\rm VP}(w';\varepsilon)=-\mathrm{Im}\,\Sigma_{\rm VP}(w'+i0;\varepsilon)$ is its spectral density. Let $K(w)$ be the one-loop box with three lines of mass $m$ and, in place of the dressed line, one line of squared mass $w$. The kernel is that box twice subtracted at $w=m^2$, $K_P(w)=\frac{K(w)-K(m^2)}{(w-m^2)^2}-\frac{K'(m^2)}{w-m^2}$, evaluated at $w=w'$; $K(m^2)$ is the box with four lines of mass $m$ and $K'(m^2)$ its derivative in the mass of one line. The two subtractions come from partial fractions between the dispersion denominator and the squared stub propagator $D_4^{\,2}$. At general $\varepsilon$ the box $K(w)$ is a two-fold Feynman-parameter integral, which reduces to dilogarithms at $\varepsilon=0$, and along $w'$ it is evaluated by transportnumerical solution of a differential equation along a path, starting from a point where the integrals are known exactly of the differential equation of the one-loop box family, starting from the boundary point $w'=5m^2$, where its eight master integralsthe finite basis of integrals to which every integral of a family reduces by integration-by-parts identities are computed directly from their Feynman-parameter representations.

Below $w'=9m^2$ only the two-body cut contributes, and there the imaginary part of the self-energy, and with it the density, is elementary; in units $m^2=1$, used in the displayed formulas from here on,

$$\frac{1}{\pi}\,\mathrm{Im}\,\Sigma_{\rm VP}(w';\varepsilon)=\frac{\Gamma(1+\varepsilon)\Gamma(1-\varepsilon)}{\varepsilon\,\Gamma(2-2\varepsilon)}\Big[{-}\tfrac{(1-2\varepsilon)(w'+1)}{w'-1}+\tfrac{\varepsilon(w'-1)}{6}\Big]\frac{(w'-1)^{-2\varepsilon}}{w'^{\,1-\varepsilon}}.$$

This form follows from the differential equation of the self-energy in $w'$, whose integration constants are fixed analytically by regularity at the two-particle threshold $w'=m^2$. Above the three-body threshold $w'=9m^2$ the density acquires a second piece, a single integral over the three-particle phase space:

$$\rho_{3b}(w')=\pi\,c_1^2\,w'^{\,\varepsilon-1}\!\int_4^{(\sqrt{w'}-1)^2}\!\!dq\,(q-4)^{\frac12-\varepsilon}q^{-\frac52}\big[((\sqrt{w'}-1)^2-q)((\sqrt{w'}+1)^2-q)\big]^{\frac12-\varepsilon},\qquad c_1=\tfrac{\Gamma(1-\varepsilon)}{\Gamma(2-2\varepsilon)},$$

where $q$ is the squared invariant mass of the electron pair produced by the vacuum-polarization loop. The branch points $\{0,4,(\sqrt{w'}\mp1)^2\}$ of the integrand are the roots of the quartic that defines the equal-mass sunrise curve, so at $\varepsilon=0$ the integral $\rho_{3b}$, the elliptic part of the density, is a periodthe integral of an algebraic function around a cycle on the curve of that curve. The pole parts of the self-energy are closed forms, with $\gamma_E$ the Euler–Mascheroni constant:

$$\Sigma_{\rm VP}^{(-2)}(w')=-\frac{1}{w'-1},\qquad \Sigma_{\rm VP}^{(-1)}(w')=-\frac{2\log(1-w')}{1-w'}-\frac{\log(1-w')}{w'}+\frac{2\gamma_E}{w'-1}.$$

The pole coefficients of $I_{\rm VP}$ follow from these together with the Taylor coefficients of the kernel at $w'=m^2$, which are two-fold moment integrals over the Feynman parameters and carry the dependence on $(s,t)$: the double pole comes from the threshold region $w'\to m^2$, where the non-integer power $(w'-m^2)^{-2\varepsilon}$ of the two-body density multiplies the double pole $1/(w'-m^2)^2$ of the squared stub propagator. At the reference point $(s,t,m^2)=(-1,-1/3,1)$ the Laurent coefficients are $I_{\rm VP}^{(-2)}=-0.04540615882\ldots$, $I_{\rm VP}^{(-1)}=-0.01534480520\ldots$, $I_{\rm VP}^{(0)}=-0.1466053426\ldots$ and $I_{\rm VP}^{(1)}=0.0397158224\ldots$. The finite coefficient $I_{\rm VP}^{(0)}$ is a single convergent integral over $w'$, obtained by expanding every factor of the dispersion integrand in $\varepsilon$ and subtracting the threshold behavior at $w'=m^2$ analytically.

The complete expression, with the Feynman-parameter form of the kernel and the pole coefficients to full precision, is in the expression file inside the bundle under Evaluator.

Checks

Evaluator

Same family

• The light-by-light 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 graph of this family in which the $\Gamma_1(6)$ curve appears.
• The box with a kite insertion dresses the line with the two-loop kite self-energy and pinches both stubs; that graph has one fermion loop instead of two and is finite at $d=4$.
• The QED graphs with a kite self-energy keep one or both stubs beside the kite insertion, so that the stub propagator enters once or squared; both graphs are finite at $d=4$.
• The crossed light-by-light box has the two internal photons joining opposite corners of the fermion loop; its most complicated integrals are iterated integrals built on the periods of a K3 surface, the one attached to the three-loop banana graph.

The paper

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

References

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