gg → Zγ

Two gluons fuse to a Z boson and a photon through a top-quark loop. The two-loop elliptic master integrals with the top mass kept exact had no analytic treatment on record; they are solved here at one mass ratio by identifying the curve's modular group, deriving every boundary constant analytically at the $s=0$ cusp, and transporting the exact differential equation; the elliptic endpoint is a numerical march to 137 digits, not a classical number.

The integral

Two gluons fuse to a Z boson and a photon through a closed quark loop, at two loops in mixed QCD–electroweak coupling. The object is the family of two-loop integrals

$$I[\nu_1,\ldots,\nu_9] \;=\; \int d^d l_1\, d^d l_2\; \frac{1}{D_1^{\nu_1} D_2^{\nu_2} \cdots D_9^{\nu_9}}\,, \qquad d = 4-2\epsilon,$$

up to an overall normalization convention, with loop momenta $l_1, l_2$ and the nine $D_j$ the inverse propagators and irreducible numerators of the two-loop non-planar four-point topology whose parent family ($M=0$) was published with ancillary data in arXiv:2402.07311 — one loop of massless lines and one loop carrying the top quark, with the Z-boson mass switched on. The deformed exact-$m_t$ family had no closed form on record.

The external legs are the two incoming gluons and the outgoing photon, all massless, plus the Z boson. In units $m_t^2 = 1$ the kinematics is

$$s = (p_1+p_2)^2, \qquad s + t + u = M, \qquad M = \frac{m_Z^2}{m_t^2} = \frac{7}{25},$$

where $p_1, p_2$ are the incoming gluon momenta, $t$ is the squared momentum transfer between an incoming gluon and an outgoing leg, and the third invariant $u$ is fixed by momentum conservation. Everything is computed in $d = 4 - 2\epsilon$ dimensions, through the full Laurent expansion from $\epsilon^{-2}$ to $\epsilon^{4}$.

The family reduces by IBP identitiesintegration-by-parts identities: total-derivative relations that reduce every integral of a family to a finite basis of master integrals to master integrals, and the hardest sectors (E125 and E127, eight master integrals) are elliptic. E125 holds the canonical pair $g_{31} = R_a$, the holomorphic-period master, and $g_{32} = R_b$, the quasi-period master; E127 holds six punctured companions. On the maximal cutset every propagator of the loop on shell; what survives of the integrand exposes the geometry of the sector of the top loop the sector lands on a single elliptic curve, the deformed quartic

$$y^2 = P\,(P+s-M)\,(P^2 + (s-M)P - 4s),$$

where $P$ is the one integration variable left on the cut and $y$ is the square root of the quartic in $P$ that defines the curve; at $M = 0$ this collapses to the parent curve $y^2 = P\,(P+s)\,(P^2 + sP - 4s)$. The curve is modular for $\Gamma_0(4)$the index-6 subgroup of $\mathrm{SL}_2(\mathbb{Z})$ of matrices with lower-left entry divisible by 4; its modular curve is genus 0 with three cusps, with parent cusps $\{0,-16,\infty\}$; the $Z$ mass acts as the quadratic base change $x=(s-M)^2/s$ of the parent's $x=s$ within the same family, $j=(x^2+16x+16)^3/(x(x+16))$, equivalently $(\eta(\tau)/\eta(4\tau))^8 = 256\,s/(s-M)^2$, and the deformation splits the singular locus into five cusps.

Conductorthe level $N$ of the congruence subgroup $\Gamma_0(N)$ on which the curve becomes modular; it dictates which Dirichlet $L$-values can appear as boundary constants 4 fixes the arithmetic. The weight-2 boundary generator native to a $\Gamma_0(4)$ curve is Catalan's constant$G = \sum_{n\ge 0}(-1)^n/(2n+1)^2 = 0.91596\ldots$, equal to the Dirichlet $L$-value $L(\chi_{-4},2)$. The equal-mass sunrise that opened the elliptic side of this project lives on conductor 3, where the analogous constant is $L(\chi_{-3},2)$; the two differ already in the first decimal place.

Why it matters

$gg \to Z\gamma$ is a measured LHC process. The gluon-fusion channel feeds the diboson rate that ATLAS and CMS record, and the two-loop top-mass corrections sit inside the NLO prediction those measurements are compared against. The two-loop helicity amplitudes with massless quark loops have been known since Gehrmann, Tancredi and Weihs (arXiv:1302.2630); the top-quark loop has so far been reached through mass expansions (as in the neighbouring gluon-fusion diboson channels, arXiv:2509.07072). An exact-$m_t$ solution at one mass ratio replaces a large-mass expansion with the function the physical amplitude actually contains.

Earlier boundary code supplied only the conductor-3 constant $L(\chi_{-3},2)$ of the equal-mass sunrise, while the conductor-4 ring generated by Catalan's constant was the one this curve required; building the $\Gamma_0(4)$ ring explicitly, verified to at least 130 digits with its generator confirmed to 149, corrected that live error in the boundary code.

What was hard

The geometry itself was a trap: the curve is modular for $\Gamma_0(4)$, conductor 4, rather than the conductor 3 of the equal-mass sunrise, the only conductor the prior boundary code supplied. The deformed $j$-invariant satisfies $j(M^2/s) = j(s)$ exactly, and the resulting torsion-free genus-zero quotient has index 6 and cusp widths $(1,1,4)$. The massless parent ($M=0$) is the $x=s$ member of the same $\Gamma_0(4)$ family, the $Z$ mass acting as the quadratic base change $x=(s-M)^2/s$ that splits the parent's three cusps into five; the modular group dictates the entire boundary arithmetic.

The natural first attack is the one that closed the equal-mass sunrise elsewhere in this gallery: build the modular word basis, sample the masters numerically, and fit exact rational coefficients. It fails here, and the failure is structural, not numerical. Two fitting attempts — a blind weight-graded value fit of the elliptic master, then a cross-validation over a seven-dimensional basis of second-kind subtractions (Weierstrass $\eta/\omega$, Kronecker–Zagier) — both failed against evaluation data withheld from the fits, reaching at best 1.43 digits (against a 78-digit positive control on the same machinery) and 0.15 digits respectively, proving the machinery sound and the object at fault. The obstruction: the quasi-period master $g_{32}$ is inhomogeneous over nine lower polylogarithmic subsectors that switch on at order $\epsilon^0$ and depend on $t$, while any second-kind subtraction, $\eta/\omega = -(\pi^2/3)\,E_2(\tau)/\omega_0^2$, depends only on $s$; no $s$-only subtraction can remove a $t$-carried tail, so no fit of the $s$-only elliptic boundary could ever reproduce it. The numerics measure the wall; the $t$-dependence argument proves it. The decision rule BootLoops now uses for when to abandon fitting values and solve the differential equation instead was calibrated on this diagram.

With the fit ruled out, the differential-equation route closed the problem: an exact $\epsilon$-forma basis rotation after which the differential equation reads $dM = \epsilon\,A\,M$ with $A$ independent of the dimensional regulator $\epsilon$; the system then integrates order by order in $\epsilon$ as iterated integrals connection in both kinematic directions, whose flatness on the $2\times 2$ elliptic block — the compatibility of the $s$- and $t$-derivatives on the elliptic rows of the connection — holds exactly; 25 cusp boundary constants derived analytically, by Frobenius expansion at the five $s$-cusps together with the $M \to 0$ degeneration onto the published parent family, so that no fitted value enters; and certified transport carrying them to any interior point.

The result

The exact-top-mass two-loop $gg \to Z\gamma$ elliptic masters — the E125 pair $g_{31} = R_a$, $g_{32} = R_b$, plus their six punctured E127 companions — live on the single deformed $\Gamma_0(4)$ curve displayed above. No compact one-line closed form in $s$ and $t$ exists for these masters, and none is claimed: the $t$-dependent part is not a pure-modular object, so the solution has three explicit pieces — the $\Gamma_0(4)$ structure of the homogeneous block in $s$, with the elliptic endpoint from an analytic $s$-march, a set of analytically derived boundary constants, and exact polynomials in $t$.

The modular spine ($s$-direction). The homogeneous (maximal-cut) block of the E125 pair has the structure of the cusp-regularised Eichler integralthe primitive of a modular form: integrating a weight-$k$ form once in $\tau$ gives a function that transforms under the modular group with an additive polynomial correction rather than covariantly of the weight-2 Eisenstein kernel of $\Gamma_0(4)$,

$$ \mathcal{E}\bigl[B_{2,4}\bigr](s) \;=\; \int_{i\infty}^{\tau(s)}\! B_{2,4}(\tau')\,d\tau'\,\Bigr|_{\rm reg}\,, \qquad B_{2,4}(\tau) \;=\; -\tfrac{1}{24}\bigl[E_2(\tau) - 4\,E_2(4\tau)\bigr]\,, $$

where $\tau(s)$ is the modular coordinate of the curve — the ratio of its two periods at the point $s$ — $E_2$ is the weight-2 Eisenstein series, and "reg" denotes the cusp regularisation of the logarithmically divergent integral down from $i\infty$. The homogeneous solution of the period-pair block is

$$ g_{\rm spine}(s) \;=\; s^{-2\epsilon}\!\left(\frac{s-r_2}{s-r_1}\right)^{\!\frac{10\sqrt{93}}{93}\epsilon}\,\Bigl[\,c_1 + c_2\,\tau(s) + \mathcal{E}\bigl[B_{2,4}\bigr](s)\,\Bigr]\,, \qquad 625\,r_{1,2}^2 + 9650\,r_{1,2} + 49 = 0\,, $$

with $r_1, r_2$ the two elliptic-threshold cusps (the exact algebraic roots of the quadratic shown), $c_1, c_2$ two of the 25 analytically derived cusp boundary constants, and the algebraic power prefactor the exact diagonal of the canonical basis rotation — nothing in it is fitted. This is the structure of the homogeneous block: which component of which master equals $g_{\rm spine}$ is not exhibited here, and the elliptic endpoint of the masters comes from the analytic $s$-march from the $s=0$ cusp boundary ($K_{12}$, a 137-digit string).

The boundary constants. All 25 cusp boundary conditions are proven analytic — Frobenius expansion at the five $s$-cusps together with the $M \to 0$ degeneration onto the published massless parent — and no fitted or auxiliary-mass-flow-derived constant remains in the assembled masters; the one deep constant with no closed form, $K_{12}$ of the genuinely elliptic sector, is carried as a 137-digit analytic-transport digit string, with nothing else depending on it. The corner masters close as $\Gamma^2/{}_3F_2/{}_2F_1$ series in $\epsilon$, computed live by the downloadable script, and the period-polynomial constants come from the conductor-4 dictionary

$$ a_0\bigl(B_{2,4}\bigr) = \tfrac{1}{8}\,, \qquad L(\chi_{-4},2) = G\,, \qquad L(\chi_{-4},1) = \tfrac{\pi}{4}\,, \qquad a_0\bigl(E_2^{(4)}\bigr)\big|_{\infty,\,0,\,\frac12} = \bigl\{-3,\ \tfrac34,\ 0\bigr\}\,, $$

the constant terms of the two Eisenstein kernels at the three cusps together with the weight-1 and weight-2 Dirichlet $L$-values of the conductor-4 character $\chi_{-4}$. Catalan's constant $G$ enters only as the ring generator $L(\chi_{-4},2)$: an early identification of a boundary constant with $G$ itself was refuted and withdrawn, and no boundary constant here is claimed equal to $G$.

The $t$-direction, in closed form. At fixed $s = -\tfrac14$ the certified fourteen-master differential equation in $t$ collapses: eleven of the fourteen masters — including $g_{31} = R_a$ — are exactly $t$-independent (their rows of the connection vanish identically; the $\Gamma_0(4)$ curve is an $s$-only object), and the remaining three close as exact polynomials in $t$,

$$ m_8(t) \;=\; c_8\,(200t-53) \;+\; \tfrac{1}{2}K_5 + \tfrac{1}{8}K_6 + \tfrac{1}{2}K_7\,, \qquad m_{11}(t) \;=\; c_{11}\bigl[(448\epsilon-236)\,t-53\bigr] + C_{11}(\epsilon)\,, $$

$$ g_{32}(t) \;=\; c_{13}\,\Bigl[\epsilon\,(200t-53)^2 - \bigl(60000\,t^2 - 31800\,t + 2809\bigr)\Bigr] + B_0(\epsilon)\,, \qquad g_{31} \;=\; K_{12}\,. $$

Here $c_8$, $c_{11}$, $C_{11}$, $B_0$ are exact rational functions of $\epsilon$ provided as integer tables with the download, $c_{13}$ is solved exactly at runtime from the endpoint condition, and $K_5, K_6, K_7, K_{12}$ are boundary constants of the derivation. Polynomiality is a theorem, not an observation: $t$ enters the family only through a single numerator power, so every candidate transcendental letter cancels identically — the apparent singularity at $t = -\tfrac{53}{236}$ seen in an earlier representation was a property of that representation only, and the masters are entire in $t$.

Downloads: zgamma-expression.md · zgamma-evaluate.py · zgamma-derived.json · zgamma-cone-At.json · zgamma-cusp-seeds.json · vendor_row20_grid/ (the 25 recorded off-line $(s,t)$ endpoints, the five $t$-cones and the transport worker behind zgamma-evaluate.py --point s,t) · vendor_row20_p12/ (a second transport at 60 working digits at each of the 25 grid points beside the 45-digit one: the two agree to at least $32$ digits at every point, an independent numerical evaluation at $(s,t)=(-\tfrac15,-\tfrac12)$ confirms $33$, and a short driver recomputes every figure from the recorded values in seconds) · vendor_row20_a2a/ (independent evaluations at two precision goals, $40$ and $60$ digits, at four more $(s,t)$ grid points with the certificate at each: over the five points so checked the goal-$60$ evaluation reproduces the transported fourteen masters to $47$ digits or better, its goal-$40$ twin agrees with it to $44$ or better, and the transport's own two-precision pair certifies $32$ at worst; a driver recomputes every certificate from the recorded endpoints and evaluations) · MANIFEST.sha256 (every file of the bundle with its checksum). The script reads all four data files; the last is an earlier seed table that it reproduces as a check rather than uses.

The branch structure, explicitly. The $\mathrm{Re}\,\tau = \pm\tfrac{1}{2}$ branch jumps that defeat any direct fit are, in the named form, the explicit period-polynomial cocyclethe additive shift an Eichler integral picks up under the modular group, here $\tau \to \tau+1$; it is a polynomial in $\tau$ with period coefficients, fixed by the constant term of the kernel

$$ \mathcal{E}\bigl[B_{2,4}\bigr](\tau+n) \;-\; \mathcal{E}\bigl[B_{2,4}\bigr](\tau) \;=\; 2\pi i\,n\,a_0 \;=\; n\cdot\frac{i\pi}{4}\,, \qquad n \in \mathbb{Z}\,. $$

At weight 3 the kernels are the two eta-quotients $\eta(\tau)^4\eta(2\tau)^6\eta(4\tau)^{-4}$ and $\eta(\tau)^{-4}\eta(2\tau)^6\eta(4\tau)^4$, with $\eta$ Dedekind's eta function, spanning $M_3(\Gamma_0(4),\chi_{-4})$; the space contains no cusp form, and $S_k(\Gamma_0(4))$ is empty through all the weights that enter, so the entire elliptic word list is Eisenstein.

Scope, stated plainly. The closed $t$-polynomials cover the fixed slice $s = -\tfrac14$, the slice on which the fourteen-master system was certified; the $s$-direction of the full system is transported by the same certified connection but is not itself reduced to a closed form in the downloads, and sector E127 is not covered on this slice. Four of the five deep boundary constants — $K_6$, $K_7$, $K_9$, $K_{10}$ in the equations above — are closed in exact $\Gamma$-ratio-ladder form with terminating $_3F_2$ inner sums, verified above 142 digits against the fully independent transport march at two working precisions; the fifth, $K_{12}$, sits in the genuinely elliptic sector, where no closed form is expected, and is delivered as an analytic-transport digit string of 137 verified digits with nothing else depending on it; the evaluator reads recorded endpoint strings only for $K_{12}$ and the three $t=-\tfrac13$ endpoint values. Off both lines, at 25 Euclidean $(s,t)$ points — $s\in\{-\tfrac{1}{20},-\tfrac{1}{10},-\tfrac{3}{20},-\tfrac15,-\tfrac{9}{40}\}$ and $t\in\{-\tfrac12,-\tfrac23,-\tfrac56,-1,-\tfrac54\}$, all at $M=7/25$ — the analytic $s$-march from the cusp boundary at $t=-\tfrac13$ followed by a $t$-transport of the fourteen-master $t$-cone built at each $s$ reproduces itself between 45 and 60 working digits to a worst case of 32 digits per point (the lower member's own precision class; the cone closes at every $s$, and the same builder reproduces the certified $t$-cone at $s=-\tfrac14$ exactly); at five of these points, $(s,t)=(-\tfrac15,-\tfrac12)$, $(-\tfrac{1}{10},-\tfrac12)$, $(-\tfrac{1}{20},-\tfrac54)$, $(-\tfrac{9}{40},-\tfrac54)$ and $(-\tfrac{3}{20},-\tfrac56)$, an independent evaluation reproduces the transported fourteen masters to 47 digits or better (the transport's own two-precision pair certifies 32 at worst over the five). zgamma-evaluate.py --point s,t runs that transport at any of the five $s$ and gates it against the 25 recorded endpoints.

The seven-line parent's own top sector (sector 127) is not solved here; its integrals are checked by transport between four independently evaluated points on the line $s=-1/4$ — six legs, each agreeing to at least 37 digits at 45 working digits and stable to at least 38 digits between 45 and 60 working digits (the two longest legs reach 53 and 55 digits against the independent evaluations at 60 working digits) — and no sector-127 value free of those independent evaluations is claimed. Those six legs are reproducible from the download: zgamma-s127-evaluate.py transports the 72-master sector-127 system along any of the six legs at the record's precisions with the transport code and reference data beside it, gates the six sector-127 masters at the target order by order, and states the record's digit for that leg; the default (the shortest leg at 30 digits) is minutes-class on one core, the longer legs hours-class, and every run resumes from its last checkpoint.

Validation against independent evaluations at points never used in the construction is piecewise: 126–139 digits for the sector-125 masters (of which the independent evaluation's own two runs certify 109–112), at least 37 digits on the six sector-127 transport legs, and 47 digits or better at the five off-line points that carry an independent evaluation (32 certified by the transport's own two-precision pair).

At the check point $(s,t)=(-\tfrac14,-\tfrac13)$, never used in the construction, the derived $s$-cone endpoint reproduces the independent evaluations to 136 digits across all 22 masters, 137 on the two elliptic masters and 139–141 on the six corner masters. Those counts are agreement with the deeper of the two independent runs at that point; the two runs agree with each other to 109–112 digits per order, so 109 is the count the independent evaluation itself certifies, and everything above it is agreement with the printed digits of the deeper run.

Runtime: the default run evaluates the four closed-form constants $K_6, K_7, K_9, K_{10}$ at 140 working digits and reads the recorded endpoint strings only to check them — minutes-class on one core, and the precision at which the run prints the full digit count the independent evaluations allow; --dps 160, the working precision of the derivation, is an hour-class run on a busy machine, and --k-strings restores the earlier minutes-class path that serves the recorded strings. The heavy mode --boundary-recompute recomputes the boundary at production precision and is hours-class per leg; it needs the archived cusp-boundary solver modules, which are not part of the download, and without them it stops at once with a message naming what is missing (the default run does not use them). The run prints its own wall; the walls on the record were measured on a box under heavy load and are recorded with the paper.

References

← back to the gallery