gg → H

The elliptic core of two-loop Higgs production via gluon fusion: the two transcendental functions $G_1$, $G_2$ of the canonical basis of arXiv:2501.14435, re-evaluated from the paper's defining integrals and compared with auxiliary-mass flow through the paper's own rotation. A validation: the integrals and their defining forms are the source paper's; what is added is the evaluation, the values of record, and a form of $G_1$, $G_2$ in complete elliptic integrals.

The integral

This is gluon-fusion Higgs production at two loops, with the strong and the electroweak interaction both in play: a closed top-quark loop of mass $m_t$ is bridged by an internal electroweak-boson line of mass $M$ ($m_H$, $m_W$, $m_Z$, or a Goldstone), kept generic throughout — that massive rung supplies the second loop, and it is what puts the elliptic sectors on the $(m_t,m_t,M)$ sunrise curve. Two on-shell gluons with incoming momenta $p_1$ and $p_2$ enter and an off-shell Higgs exits, so the one kinematic invariant is

$$s = (p_1+p_2)^2,$$

the squared total momentum of the gluon pair. Everything is measured in units of the top mass ($m_t = 1$, so $s$ and $M^2$ are dimensionless), and the integrals live in dimensional regularization with $\varepsilon = (d-4)/2$.

Marzucca, McLeod, and Nega (arXiv:2501.14435, PRD 2025) reduced the relevant integral families 2 and 5 to canonical $\varepsilon$-forma basis of master integrals in which the differential equations carry the regulator $\varepsilon$ only as an overall prefactor, so they can be integrated order by order in $\varepsilon$ differential equations and isolated the two transcendental functions that carry the elliptic content of these families' canonical basis, $G_1(s)$ and $G_2(s)$, defined by the paper as one-fold iterated integrals over elliptic kernels (their Eqs. 3.17–3.18; equivalently the split form of Eqs. 3.21–3.22; the paper's ancillary-file rotation to the canonical basis carries $G_2$ at one quarter of the printed Eq. 3.18, which matters for the numerical check below):

$$G_1 \;=\; \tfrac{8}{3}(M^2-4)(M^2-1)\,G(k_{1E}) \;-\; \tfrac{2}{9}(M^2-1)\,G(k_{3E}), \qquad G_2 \;=\; 2\sqrt{M^2}\sqrt{4-M^2}\;G(k_{8E}),$$

where $G(k)$ denotes the paper's one-fold iterated integral over the elliptic kernel $k$, with $k_{1E}=\varpi_0/(M^2+3s-4)^2$, $k_{3E}=3\varpi_0/(M^2+3s-4)$, and $k_{8E}$ the kernel of the paper's Eq. 3.42. Here $\varpi_0(s)$ is the holomorphic period of the genus-one curve of the $(m_t, m_t, M)$ two-mass sunrisethe three-line self-energy diagram, here with two lines at the top mass and one at $M$, whose on-shell surface traces out the elliptic curve — written out explicitly in the result below. The curve's Picard–Fuchs operatorthe differential operator in $s$ that annihilates the periods of the elliptic curve; its order counts the independent periods in play has order two, and a single moving puncturea marked point on the torus whose position drifts with the kinematics $s$; it carries the residue data of the integral $z_p(s)$ sits over the extra root $s_{p1} = (4-M^2)/3$, where the kernels of both masters develop their double pole.

The paper defines $G_1, G_2$ by these integrals and evaluates them; what it leaves to future work is the elliptic-multiple-polylogarithm form of the master integrals themselves ("we believe it should be possible to express these integrals in terms of elliptic multiple polylogarithms … we have left this possibility to future work"), which this page does not attempt. The same curve class appears in the unequal-mass sunrise validation elsewhere in this gallery.

Why it matters

Gluon fusion is the dominant Higgs production mode at the LHC, and the source paper's introduction lays out the state of play: the two-loop mixed QCD–electroweak corrections involving light virtual quarks were computed over twenty years ago, while the contributions with a massive virtual top quark had been evaluated only numerically or as expansions around kinematic limits. arXiv:2501.14435 changed that by computing all the two-loop master integrals with full analytic dependence on the top, Higgs, W, and Z masses — presented as iterated integrals over bespoke elliptic kernels, with the elliptic-polylogarithm form explicitly deferred.

This page is a validation. The functions $G_1, G_2$ were defined and computed in arXiv:2501.14435 (as one-fold integrals over the holomorphic period, cross-checked in the paper against AMFlow), so — like the equal- and unequal-mass sunrise validations elsewhere in this gallery — the objects themselves are already in the literature. What is added here is the evaluation on the stated branch of the period, the values of record at ten Euclidean points, a comparison with auxiliary-mass flow through the paper's own canonical rotation at one further point, and a form of $G_1, G_2$ in complete elliptic integrals of the first and third kind. The curve, the differential equations, the kernels and the lower limit are all arXiv:2501.14435's.

What was hard

The elliptic masters defeat the direct route. Fitting their values against any finite dictionary of known constants and functions fails for a structural reason: the natural seed for such a fit carries the ratio of two elliptic periods, which varies continuously with $s$, so no finite list of algebraic coefficients can represent it — and enlarging the candidate alphabet, even severalfold, moves nothing. The same wall had already stopped the direct fit for $gg\to Z\gamma$; the way past it is the differential equation itself. In the frame where the elliptic one-form pulls back to the period $\varpi_0$ itself, the period rides inside the integration kernel and the matching coefficients stay purely algebraic. At the cusp $s=0$ the elliptic curve pinches, the masters reduce to regular tadpoles, and $G_1(0)=G_2(0)=0$ exactly, with no undetermined constant. Integrating the paper's kernels from the cusp is the paper's own definition of $G_1, G_2$; this page evaluates it.

The result

The two transcendental functions $G_1, G_2$ of families 2 and 5 — defined in arXiv:2501.14435 (its Eqs. 3.17–3.18) and re-evaluated here — are each a single integral of an algebraic prefactor against the holomorphic period $\varpi_0(s)$ of the $(m_t,m_t,M)$ sunrise curve, starting from an exact zero at the cusp, at order $\varepsilon^0$ and with full dependence on both $s$ and $M^2$:

$$G_i(s) \;=\; \int_0^s K_i(s')\,\varpi_0(s')\,\mathrm{d}s', \qquad G_i(0)=0,$$

with kernels algebraic over $\mathbb{Q}[\sqrt{\ }]$ in the roots $r_1=\sqrt{-s}$, $r_2=\sqrt{4-s}$, $r_3=\sqrt{M^2}$, $r_4=\sqrt{4-M^2}$:

$$K_1(s) \;=\; \frac{2\,(M^2-1)(M^2-s-4)}{(M^2+3s-4)^2},$$

$$K_2(s) \;=\; \frac{2\,r_1 r_2 r_3 r_4\,\bigl(M^4+3M^2s^2-13M^2s-4M^2-3s^3+16s\bigr)}{(s-4)^2\,s\,(M^2+3s-4)^2}.$$

In elliptic-polylogarithm terms, $K_i\,\varpi_0\,\mathrm{d}s$ is the Kronecker first-kind one-form $\omega_1$, evaluated at the moving puncture $z_p(s)$ — the image on the torus of the extra root $s_{p1}=(4-M^2)/3$, where both kernels carry their double pole — in the normalization where it pulls back to $\varpi_0(s)$. All transcendental content sits in the single kernel $\varpi_0$; the coefficients $K_1, K_2$ carry none.

Downloads: ggH-expression.md · ggH-evaluate.py · REF_VALUES_cured_20260909T083451Z.json (the values of record at ten Euclidean points, two working precisions each) · MANIFEST.sha256 (every file of the bundle with its checksum) · CHANGES.md · the previous bundle, byte for byte, under superseded_f216bb567e95b1fc/

The elliptic kernel, explicitly. The period $\varpi_0$ is an exact transcription of the paper's Eq. 3.15:

$$\varpi_0(s) \;=\; \frac{2i}{\pi}\cdot \frac{\sqrt{-M^2(M^2-4)}\;K(z_{\mathrm{arg}})}{\pi\,\sqrt{s-M^2}\;\bigl[(s-(M+2)^2)\,(s-(M-2)^2)\bigr]^{1/4}},$$

$$z_{\mathrm{arg}} \;=\; \frac{1}{2} \;+\; \frac{M^4 - 2(s+2)M^2 + (s-4)s}{2\,(M^2-s)\,\sqrt{s-(M+2)^2}\,\sqrt{s-(M-2)^2}},$$

where $K$ is the complete elliptic integral of the first kind (parameter convention $m = k^2$). The paper's split representation displayed in "The integral", with the period evaluated by the arithmetic–geometric mean instead of the complete elliptic integral, gives the same values; both routes share the period, so this is a check of the assembly, not of the branch.

Scope. The object is the order-$\varepsilon^0$ term, with full dependence on both $s$ and $M^2$. The elliptic curve, the differential equations, the kernels and the defining integrals are all in arXiv:2501.14435; what is new here is the evaluation, the values of record, and the period form: $G_1$ and $G_2$ are algebraic combinations of the first- and third-kind periods of the sunrise Baikov quartic (the third-kind poles over $x=\infty$ for $G_1$ and over the locus $D_3=0$ for $G_2$), an identity checked here to 59 digits at four Euclidean points with $(s-M^2)^2<4M^2$. One adjacent question — the precise $\eta$-dressing of the second-kind row in the S-dual frame — is not addressed by this computation and does not affect $G_1$ or $G_2$. The downloadable evaluator covers the Euclidean region ($s<0$, $0<M^2<4$ in top-mass units); continuation above threshold needs an $i\varepsilon$ path deformation it does not implement.

The values of record are the evaluator's own runs at ten Euclidean points, stored in the data file linked above at working precisions 60 and 90 with the agreement between the two depths recorded per point (81 digits at worst over the ten points); a separate transcription of Eqs. 3.17–3.18 integrated by quadrature agrees with them to 61 digits, a check of the transcription rather than of the period, which both share. The evaluator checks its output against the stored strings on every run and reports how far its values agree between one working precision and twice that precision. The one comparison that does not pass through the period is at $(s,M^2)=(-\tfrac74,\tfrac35)$: the thirty Laporta master integrals of family 2, evaluated by auxiliary-mass flow at two precision goals and pushed through the source paper's rotation to its canonical basis, fix $G_1$ to 57 digits and, one layer deeper through the weight-two identity of that basis (where the rotation carries one quarter of Eq. 3.18), $G_2$ to 48 digits; --point -7/4 3/5 runs both comparisons. The evaluator covers $s<0$ and $0<M^2<4$ (units $m_t=1$). --sheet +i0 is refused at this revision: the continuation above threshold and its stored strings were built on the previous branch of the period, so the physical region is not among the checked outputs. $M^2>4$ is not implemented.

References

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