Elliptic master integrals for gg → H

Two elliptic functions enter the two-loop mixed QCD–electroweak corrections to gluon-fusion Higgs production, with an electroweak boson exchanged across the top-quark loop. Marzucca, McLeod and Nega (2025) defined and computed them as single integrals over the period of a two-mass sunrise curve; this known result is re-evaluated here from the same integrals on a definite branch of the period.

The integral

The integrals belong to the two-loop mixed QCD–electroweak corrections to Higgs production by gluon fusion: a top-quark loop (mass $m_t$) is bridged by an internal electroweak-boson line of mass $M$, where $M$ stands for $m_H$, $m_W$ or $m_Z$ and is kept generic. The boson line forms the second loop, and the sectors that contain the $(m_t,m_t,M)$ sunrise subgraph are the elliptic ones. Two on-shell gluons with momenta $p_1$ and $p_2$ enter and an off-shell Higgs boson leaves, so the one kinematic invariant is

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

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

Marzucca, McLeod and Nega (arXiv:2501.14435, 2025) reduced two of the integral families of these corrections (families 2 and 5 in their numbering) to master integralsa finite set of integrals to which every integral of the family reduces by integration-by-parts identities obeying canonical $\varepsilon$-formdifferential equations that carry the regulator $\varepsilon$ only as an overall prefactor, so they can be integrated order by order in $\varepsilon$ differential equations. The elliptic content of the canonical basis is carried by two transcendental functions, $G_1(s)$ and $G_2(s)$, defined as one-fold iterated integrals over elliptic kernels (their Eqs. 3.17–3.18). Split into integrals of the individual kernels (the split form, their Eqs. 3.21–3.22), they read

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

where $G(k)$ denotes their one-fold iterated integral over the kernel $k$. All three kernels are built from the holomorphic period $\varpi_0(s)$ 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-quark mass and one at $M$, whose on-shell surface traces out the elliptic curve: $k_{1E}=\varpi_0/(M^2+3s-4)^2$, $k_{3E}=3\varpi_0/(M^2+3s-4)$ and $k_{8E}=K_2\,\varpi_0/(2r_3r_4)$ (their Eq. 3.42), where $r_3=\sqrt{M^2}$ and $r_4=\sqrt{4-M^2}$ are the two roots in the prefactor of $G_2$. The period $\varpi_0$, the kernel $K_2(s)$ and the combination $K_1(s)$ that multiplies $\varpi_0$ in $G_1$ are given under Closed form. The quantities computed are $G_1(s)$ and $G_2(s)$ for real $s\lt0$, with full dependence on $s$ and $M^2$; they enter the master integrals at order $\varepsilon^0$.

Gluon fusion is the dominant production mode of the Higgs boson at the Large Hadron Collider. As Marzucca, McLeod and Nega review, the two-loop mixed QCD–electroweak corrections with light virtual quarks were computed by Aglietti, Bonciani, Degrassi and Vicini (2004), while the contributions with a massive virtual top quark had been available only as an expansion around kinematic limits, by Degrassi and Maltoni (2004), or numerically, by Actis, Passarino, Sturm and Uccirati (2008). Marzucca et al. themselves computed all the master integrals of these corrections with full analytic dependence on the top, Higgs, $W$ and $Z$ masses; the elliptic sectors are given as iterated integrals over elliptic kernels. They expect that the master integrals can be expressed in terms of elliptic multiple polylogarithms and leave that form to future work; the recomputation here keeps their iterated-integral form. The curve, the differential equations, the kernels and the boundary point used here are all theirs.

The elliptic curve is the one found by Adams, Bogner and Weinzierl (2013) on the maximal cut of the sunrise with arbitrary masses, here with masses $(m_t,m_t,M)$. Its Picard–Fuchs operatorthe differential operator in $s$ that annihilates the periods of the elliptic curve; its order is the number of distinct periods has order two, and for generic $M$ the curve is not modularsaid of a family of elliptic curves whose periods, as functions of $s$, are modular forms of a congruence subgroup such as $\Gamma_1(6)$; integrals on such a curve become iterated integrals of modular forms, so $G_1$ and $G_2$ are not iterated integrals of modular forms. In the language of elliptic polylogarithms of Broedel, Duhr, Dulat and Tancredi (2017) and of the pure elliptic functions of Broedel, Duhr, Dulat, Penante and Tancredi (2019), the integration kernels of the differential equations (the letters) of families 2 and 5 are Kronecker–Eisenstein forms on the torus of the maximal cut, with punctures at torsion points associated with the top-quark and boson propagators. The functions $G_1$ and $G_2$ need one further moving puncturea marked point on the torus whose position drifts with the kinematics $s$, carrying the residue data of the integral $z_p(s)$, which is not a torsion point and which appears on the $s$-line as the additional singular point $s_{p1}=(4-M^2)/3$, a double pole of both kernels. Neither $G_1$ nor $G_2$ is given here as a constant-coefficient combination of iterated integrals of these letters, and the normalizations obstruct such a form: the one-forms of Broedel et al. are normalized by a different period $\psi_1$ of the curve, whereas $G_1$ and $G_2$ are single integrals against $\varpi_0$ with algebraic coefficients, and the ratio $\psi_1/\varpi_0$ is not constant in $s$. Both functions are fixed instead by their differential equation in $s$ together with the boundary point $s=0$, the cusp where the elliptic curve pinches: there the master integrals reduce to tadpoles and $G_1(0)=G_2(0)=0$ exactly.

Closed form

Each of $G_1$ and $G_2$ is a single integral of an algebraic prefactor against the holomorphic period $\varpi_0(s)$ of the $(m_t,m_t,M)$ sunrise curve, with lower limit at the cusp $s=0$:

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

with $K_1$ rational in $s$ and $M^2$ and $K_2$ rational up to the square roots $r_1=\sqrt{-s}$, $r_2=\sqrt{4-s}$, $r_3=\sqrt{M^2}$, $r_4=\sqrt{4-M^2}$:

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

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

All the transcendental content sits in $\varpi_0(s)\,\mathrm{d}s$, which in the elliptic-polylogarithm language is the pullback to the $s$-line of the Kronecker one-form of the first kind at the moving puncture $z_p(s)$.

The period is Eq. 3.15 of Marzucca et al.,

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

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

where $K(z)=\int_0^{1}\mathrm{d}x\,/\sqrt{(1-x^2)(1-z\,x^2)}$ is the complete elliptic integral of the first kind as a function of the parameter $z$ (the square of the modulus). The square root in the argument of $K$ is taken as the principal root of the product $(s-(M+2)^2)(s-(M-2)^2)$, positive for $s\lt0$, so that $z_{\mathrm{arg}}\to0$ and $\varpi_0\to1/\pi$ as $s\to0^-$. This is the branch of the period that is regular at the cusp, the one on which the boundary condition $G_i(0)=0$ rests; taking the roots of the two factors separately gives the other, logarithmically divergent period of the same curve. The representation holds in the Euclidean region $s\lt0$, $0\lt M^2\lt4$ (below the $t\bar t$ threshold), where both $G_i$ are real; the physical region $s\gt0$ requires the continuation $s\to s+i0$ of the kernels and of $\varpi_0$ and is not evaluated here.

The complete expression, with the period, its branch and the split form, is in the expression file inside the bundle under Evaluator.

Checks

Evaluator

Same family

In $gg\to Z\gamma$ with exact top-quark mass the period likewise enters multiplied by a coefficient that depends on the kinematics; the elliptic sectors of that four-point process also come from a top-quark loop, but its curve is modular. The $(m_t,m_t,M)$ curve itself is a special case of the unequal-mass sunrise curve.

The paper

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

References

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