Crossed light-by-light box

The non-planar three-loop light-by-light box with the electron mass kept, whose two internal photons cross along the diagonals. Its finite part is new. Because the top sector couples to the K3 surface of the three-loop banana, the finite part contains iterated integrals of the K3 period and is not a multiple polylogarithm in the eight letters of its maximal cut.

The integral

In the crossed box an electron loop of mass $m$ runs around the four sides of a square with one external photon at each corner, of momentum $p_i$ ($i=1,\ldots,4$), massless and on shell so that $p_i^2=0$, and two internal photons join opposite corners. With $n=6$ lines, the four electron sides and the two photon diagonals, the family is

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

with propagators $D_j=q_j^2-m^2$ on the electron lines and $D_j=q_j^2$ on the photon lines, where $q_j$ is the momentum flowing through line $j$. Each loop measure $d^dl_i$ is divided by $i\pi^{d/2}$. The invariants are $s=(p_1+p_2)^2$ and $t=(p_2+p_3)^2$, with $u=(p_1+p_3)^2=-s-t$ because the photons are massless. The mass is set to $m^2=1$ and no factor of $e^{\gamma_E\varepsilon}$ is removed, so Euler's constant $\gamma_E$ appears in the expansion coefficients; all values are at Euclidean kinematics, $s,t\lt0$. Twenty-two of the family's master integralsthe finite set of integrals to which every integral of a family reduces by integration-by-parts identities lie in the top sector. The quantity computed is the finite $\varepsilon^0$ coefficient, written $m_{18}^{(0)}(s,t)$, of the top-sector masterthe master integral with every propagator of the diagram present once $m_{18}=I[1,1,1,1,1,1]$ (the subscripts are the labels of the masters used in the paper), along the curve $1/s+1/t=-\tfrac38$: a one-parameter slice through the point $(s,t)=(-8,-4)$ on which $t=-8s/(3s+8)$.

In QED four photons scatter through a virtual electron loop, and the process $\gamma\gamma\to\gamma\gamma$ has been observed in ultraperipheral lead–lead collisions at the LHC (ATLAS, 2017). The three-loop QED amplitude with the electron mass kept contains this family of integrals. At two loops the master integrals with a massive fermion loop are polylogarithmic (Ajjath A H, Chaubey and Shao, 2023), and at three loops the amplitudes with massless internal lines are known (Bargiela, Chakraborty, Gambuti and Ozcelik, 2026); the three-loop family with the mass kept does not appear to have been computed before. The crossed box is also the six-line pinchthe sub-topology obtained by contracting a propagator to a point of the ten-propagator non-planar QED graph in which each corner of the box is resolved into two vertices joined by an extra electron propagator; the top-sector geometry of that ten-line graph is treated together with the QED parent graphs of the kite insertion.

The maximal cutthe integral with every propagator of the sector put on shell, which isolates the sector's intrinsic geometry of the top sector is rational, and on the cut the differential equations that the master integrals satisfy in $s$ and $t$ have logarithmic singularities only, located on the zero sets of the eight letters

$$\{\,s,\ t,\ s+t,\ s+t+4,\ s+t+16,\ s-4,\ t-4,\ 2s+t\,\}$$

where the integrals can branch, so from this cut alone one would expect multiple polylogarithms. The full family is not polylogarithmic, however, because of a sub-sector: contracting both photons leaves the four electron lines joining two vertices, the equal-mass three-loop banana at momentum squared $u$, whose maximal cut is a family of K3 surfaces with singular fibers at $u\in\{0,4,16\}$ (Bloch, Kerr and Vanhove, 2015; Pögel, Wang and Weinzierl, 2022). As a result $m_{18}^{(0)}$ mixes multiple polylogarithms with K3 periodsintegrals of the holomorphic differential form over cycles of the surface; as functions of $u$ they satisfy a linear differential equation, the Picard–Fuchs equation and is not a combination of multiple polylogarithms over the eight letters of its maximal cut (polylogarithmic alphabets in other bases of the differential equation are not excluded), because an iterated integral of the K3 period enters it with nonzero coefficient.

Closed form

The pole part of $m_{18}$ does not depend on the kinematics: the $\varepsilon^{-3}$ and $\varepsilon^{-2}$ coefficients vanish and the $\varepsilon^{-1}$ coefficient is exactly $2\zeta_3$, the pole coefficient of the tetrahedral vacuum integral $V_{4N}$ of Broadhurst (1999), to which $m_{18}$ reduces as $m^2\to\infty$, equivalently at $s=t=0$. The boundary value there is

$$m_{18}^{(0)}(0,0)=\bar V_{4N}-6\gamma_E\zeta_3,\qquad \bar V_{4N}=6\zeta_3-22\zeta_4-8\zeta_2\log^2 2+\tfrac{4}{3}\log^4 2+32\,\mathrm{Li}_4(\tfrac12),$$

where $\bar V_{4N}$ is the finite part of $V_{4N}$, $\zeta_n$ are zeta values and $\mathrm{Li}_4$ is the weight-four polylogarithm. The last four terms of $\bar V_{4N}$ combine into $-14\zeta_4-16\,U_{3,1}$, where $U_{3,1}=\sum_{j\gt k\gt 0}(-1)^{j+k}/(j^3 k)$ is Broadhurst's irreducible alternating double sum, and the $32\,\mathrm{Li}_4(\tfrac12)$ term comes entirely from the K3 part. The integration-by-parts identities also give the exact scaling relation $(s\partial_s+t\partial_t)\,m_{18}^{(0)}=-2\Delta$, with $\Delta=m_{34}^{(0)}+m_{35}^{(0)}+3\zeta_3$, where $m_{34}=I[1,1,2,1,1,1]$ and $m_{35}=I[1,1,1,2,1,1]$ are the top-sector masters with one electron propagator squared, exchanged with each other under the crossing $s\leftrightarrow t$: $m_{35}(s,t)=m_{34}(t,s)$. Integrating this relation along a ray from the origin, where $\Delta$ vanishes and $m_{18}^{(0)}$ equals the boundary value above, gives $m_{18}^{(0)}$ at any $(s,t)$ once $\Delta$ is known along the ray.

In $s$ the vector $\vec m$ of master integrals satisfies $\partial_s\vec m=A_s\vec m$ with a matrix $A_s$ of rational functions, the connection in $s$. The K3 periods enter $m_{18}$ through a single entry of $A_s$, the rational function of $s$ and $t$ that couples $m_{18}$ to the banana master $m_{13}=I[1,1,1,1,0,0]$ (both diagonals contracted). At $d=4$ and fixed $t$ its partial fractions in $s$ have poles only at $s=0$ and at the banana letters $u=0,4,16$; the exact entry is written out, with the rest of the connection data, in the expression file inside the bundle under Evaluator. The K3 part of $m_{18}$ is accordingly the one-fold integral $\int_\Gamma K(u',t)\,m_{13}^{(0)}(u')\,du'$ with $K(u',t)=-A_s[m_{18}\to m_{13}]$ taken at general $t$ and at $s=-u'-t$, along a contour $\Gamma$ in $u'$. When the equation is integrated in $s$ at fixed $t$, the pole at $s=0$ generates $\int^{u}m_{13}(u')\,du'/(u'+t)$, and the K3 part therefore depends on $s$ and $t$ separately rather than through $u=-s-t$ alone. The holomorphic K3 period that enters $m_{13}^{(0)}$ is the Domb series $\varpi_0(u)=\sum_n \mathrm{Domb}_n\,(u/64)^n$, where the $\mathrm{Domb}_n$ are the Domb numbers (OEIS A002895).

All the masters are obtained by transportnumerical solution of the differential equation along a path, starting from a point where the integrals are known exactly of the differential equation, order by order in $\varepsilon$, along the curve $1/s+1/t=-\tfrac38$ from boundary data known in closed form in the large-mass limit, where every master reduces to vacuum integrals. At the point $(s,t)=(-8,-4)$ of the curve, where $u=12$,

$$m_{18}^{(0)}(-8,-4)=-10.42013800\ldots\,.$$

Along the curve, $m_{18}^{(0)}$ has a word form: it can be written as a combination of words, iterated integralsa repeated integral in which each kernel, or letter, is integrated against the result of the previous ones; letters in a given order form a word in thirteen letters, several of them modular formsholomorphic functions on the upper half-plane that transform in a fixed way under a group of integer matrices, here $\Gamma_1(6)$, the group of the equal-mass sunrise and banana pulled back from the banana K3, with exact rational coefficients fixed by the differential equation and the boundary data. As a function of $s$ on the curve it splits into a K3 part and a rational part,

$$m_{18}^{(0)}(s)\;=\;\underbrace{x_{17}(\varpi,\varpi';s)\;+\;\int_{-4}^{s}r_{17}(x)\,\mathrm dN_5(x)}_{\text{K3 sector}}\;+\;\underbrace{W_{\rm rat}(s)}_{\text{rational sector}},$$

where $\varpi(x)$ is the holomorphic period of the banana K3 (the holomorphic solution of its Picard–Fuchs equation, proportional to $\varpi_0$), written as a function of the curve parameter $x$ through $u=-x-t(x)$ with $t(x)=-8x/(3x+8)$, and $\varpi'=\mathrm d\varpi/\mathrm dx$; in $x_{17}$ both are taken at $x=s$. Here $x_{17}$ is a polynomial in $\varpi$, $\varpi'$ and rational functions of $s$; $r_{17}$ is the ratio of a degree-30 polynomial to a degree-26 polynomial; and $N_5$ is a new letter built from the period,

$$N_5(s)=\int_{-4}^{s}\Bigl[A(x)\,\varpi(x)+B(x)\,\frac{\varpi'(x)^2}{\varpi(x)}\Bigr]\mathrm dx,$$

with the rational functions

$$A=\frac{640}{13\cdot 3^{17}}\,\frac{(x+\tfrac{16}{3})(x^{2}+8x+\tfrac{64}{3})}{x\,(x+\tfrac83)^{2}},\qquad B=\frac{80}{13\cdot 3^{17}}\,\frac{x\,(x^{4}+20x^{3}+\tfrac{352}{3}x^{2}+\tfrac{1024}{3}x+\tfrac{4096}{9})}{(x+\tfrac83)(x+\tfrac{16}{3})}.$$

The rational part $W_{\rm rat}$ is a sum of twelve words, each with coefficient one, in the seven rational letters to which the alphabet reduces on the curve,

$$\{\,s,\;3s+8,\;3s^{2}+12s+32,\;3s^{2}+48s+128,\;s-4,\;5s+8,\;3s+4\,\}.$$

The inner integrations of each word run through a coupled block of twenty-four top-sector integrals and are evaluated numerically. The general solution of that block branches at one further point of the curve; the coefficients of the branched words in $m_{18}^{(0)}$ vanish, so the physical combination is single valued there.

The differential equations are written in a basis of forty-one integrals: the thirty-nine masters and two further top-sector integrals that reduce to them. In this basis, before any $\varepsilon$-dependent change of basis, the pattern of which integrals couple to which at order $\varepsilon^0$ in the exact two-variable connection $(A_s,A_t)$ defines a dependency graph; its strongly connected components form a directed acyclic graph of depth four, whose top component is a block of twenty-four integrals (the twenty-two top-sector masters and the two further integrals of the basis) coupled irreducibly at order $\varepsilon^0$. The differential equation of that block is rational in $\varepsilon$ but not linear in it. In the two-variable system, and in this basis, the block's equations are a coupled Volterra system; their solution would terminate as an iterated integral of finite length only after an $\varepsilon$-factorizing change of basis of the block, and none has been found. No algebraic change of basis brings the full connection to $\varepsilon$-factorized logarithmic form, since its residue matrix at $s=0$ has eigenvalues that are roots of the irreducible cubic $11\lambda^3-175\lambda^2-326\lambda+760$. The two-variable matrices $(A_s,A_t)$ are given as machine-checkable data with a short checker script; forty of their entries remain undetermined, and the coupling structure just described is established without them.

Four boundary constants of the word form are rational combinations of $1$, $\pi^2$ and $\zeta_3$; the boundary constants of the five basis integrals that couple directly to the banana masters satisfy no integer relation with zeta values through weight eight and enter the word form as numbers fixed by the transport. The complete word form is set out in the expression file inside the bundle under Evaluator, which names the data file holding the thirteen letters and the words with their exact coefficients.

Checks

Evaluator

Same family

The other four three-loop light-by-light integrals carry a self-energy insertion on one electron line of the one-loop box, and their geometry is the elliptic curve of the equal-mass sunrise rather than a K3 surface: the sunrise insertion has the bare two-loop sunrise on the line, the kite insertion the full crossed-photon self-energy, the QED parent graphs the same self-energy with its two flanking electron propagators restored, and the vacuum-polarization insertion a second electron loop on the internal photon of the self-energy. The three-loop equal-mass banana, whose maximal cut is the K3 surface that enters the crossed box, is computed on its own as a two-point function.

The paper

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

References

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