gg → Zγ with exact top-quark mass

Two gluons produce a Z boson and a photon through a top-quark loop. The two elliptic master integrals of the non-planar two-loop graph, with exact top-quark mass and a fixed Z-to-top mass ratio, are new. Their elliptic curve is modular, and they are obtained by integrating an exact differential equation in the kinematics from closed-form boundary values.

The integral

The process is of mixed QCD–electroweak order, and the two-loop family of the drawn graph is

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

up to an overall normalization, with loop momenta $l_1, l_2$; $D_1,\ldots,D_7$ are the inverse propagators of the drawn graph, three massless and four of mass $m_t$, and $D_8$, $D_9$ are irreducible numerators. The gluons and the photon are massless and the $Z$ is on shell. 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, $t=(p_1+p_3)^2$ with $p_3$ the photon momentum (all momenta incoming), and $u=M-s-t$. The quantities computed are the two elliptic master integralsa finite basis of integrals to which every member of the family reduces by integration-by-parts identities of the six-propagator sector in canonical normalization, denoted $g_{31}$ and $g_{32}$, for the Laurent orders $\varepsilon^{-2}$ through $\varepsilon^{4}$. They are given on the line $t=-1/3$ for $-1/4\le s\le 0$ and as functions of $t$ at $s=-1/4$, at the fixed ratio $M=7/25$. The six masters of the seven-propagator top sector of the drawn graph are not computed here.

The gluon-fusion channel of $Z\gamma$ production enters the diboson rate measured by ATLAS and CMS at the LHC, and its Standard Model prediction includes two-loop diagrams in which the $Z$ and the photon couple to a top-quark loop. For a massless quark loop the two-loop helicity amplitudes were computed by Gehrmann, Tancredi and Weihs (2013). The non-planar three-point function with a massive quark loop and massless external legs, which is not expressible in multiple polylogarithms, was given by von Manteuffel and Tancredi (2017). Ahmed, Chaubey, Kaur and Maggio (2024) computed the full non-planar four-point family of that kind, the $M=0$ parent of this one. The heavy-quark-loop amplitudes for $gg\to\gamma\gamma$ and their crossings were then obtained analytically by Becchetti, Coro, Nega, Tancredi and Wagner (2025) and Coro, Nega, Tancredi and Wagner (2026). In the neighboring gluon-fusion channels $gg\to ZH$ and $gg\to ZZ$ the top-quark loop has been treated by expansion in the quark mass by Davies et al. (2025); the elliptic sectors of $gg\to Z\gamma$ at $m_Z\neq0$ appear to have had no analytic treatment. An exact-$m_t$ result at one mass ratio gives the function the amplitude contains rather than its expansion in the quark mass.

On the maximal cutevery propagator of the loop set on shell; what survives of the integrand exposes the geometry of the sector of the massive loop one integration is left, over the inverse square root of a quartic,

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

where $P$ is the remaining integration variable on the cut. The quartic defines an elliptic curve whose $j$-invariant is unchanged under $s\to M^2/s$. At $M = 0$ it is the curve of the massless-leg family, which 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 has genus 0 and three cusps with Hauptmodulthe single function that generates all modular functions of the group $x=s$ and $j=(x^2+16x+16)^3/(x(x+16))$. A nonzero $Z$ mass replaces the Hauptmodul by $x=(s-M)^2/s$, a quadratic base change within the same family, so $\tau(s)$ is still a $\Gamma_0(4)$ parameter, fixed by $(\eta(\tau)/\eta(4\tau))^8 = 256\,s/(s-M)^2$ with $\eta$ the Dedekind eta function and $\tau$ the ratio of the two periods of the curve. In the variable $x$ the group has three cusps, of widths $(1,1,4)$; pulled back to $s$ by the quadratic map they become five: $s=0$ and $s=\infty$, $s=M$, and the two threshold roots $r_{1,2}$ of $(s-M)^2+16s=0$.

Order by order in $\varepsilon$ the maximal-cut solutions for $g_{31}$, $g_{32}$ are iterated integrals whose kernels are modular forms for $\Gamma_0(4)$ (functions of $\tau$ transforming with a fixed weight under the group), and the level of the group, the integer $N$ in $\Gamma_0(N)$, fixes which constants their integrals between cusps can produce. At level four the weight-two constant is Catalan's constant$G = \sum_{n\ge 0}(-1)^n/(2n+1)^2 = 0.91596\ldots$ $G$, the value $L(\chi_{-4},2)$ of the Dirichlet $L$-function of the odd character $\chi_{-4}$ modulo 4; for the curve of the equal-mass sunrise, modular for $\Gamma_1(6)$, the analogous constant is $L(\chi_{-3},2)$, with $\chi_{-3}$ the odd character modulo 3, and the two differ already in the first decimal place. These kernels have weight two or three, and $\Gamma_0(4)$ has no cusp forms of either weight, so every modular kernel that enters is an Eisenstein series; at weight three the kernels are the two eta quotients $\eta(\tau)^4\eta(2\tau)^6\eta(4\tau)^{-4}$ and $\eta(\tau)^{-4}\eta(2\tau)^6\eta(4\tau)^4$, which span the two-dimensional space of weight-three modular forms for $\Gamma_0(4)$ with character $\chi_{-4}$.

Closed form

On the maximal cut the two elliptic masters $g_{31}$, $g_{32}$ satisfy a homogeneous two-by-two system, the maximal-cut block, whose solutions are expressed through the cusp-regularized 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-two 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 $E_2$ is the weight-two Eisenstein series and "reg" denotes the regularization of the logarithmically divergent integral at the cusp $i\infty$. The general solution of the maximal-cut block is

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

Here $r_1$, $r_2$ are the two threshold cusps and $c_1$, $c_2$ constants; the algebraic prefactor is the exponential of $\varepsilon\!\int\!\mathrm{tr}\,A_1\,ds$, with $A_1$ the order-$\varepsilon$ part of the connection of the maximal-cut block in $s$. The full masters are not of this form with constant $c_1$ and $c_2$: nine polylogarithmic subsectors enter the equation for $g_{32}$ at order $\varepsilon^0$ with $t$-dependent coefficients. Nor can a multiple of the quasi-period (the period of the second kind) absorb the $t$-dependent terms: its ratio to the period, $\eta/\omega = -(\pi^2/3)\,E_2(\tau)/\omega_0^2$ (with $\eta$ here the quasi-period, not the Dedekind function), depends on $s$ alone. The pair is therefore not a constant-coefficient combination of iterated integrals on the $\Gamma_0(4)$ curve, as the equal-mass sunrise is on its curve. It is obtained by integrating the 22 equations of the six-propagator sector and its subsectors in $s$ on the line $t=-1/3$, with an exact rational connection $A_s(\varepsilon,s)$, starting from analytic boundary values at the cusp $s=0$ and reaching $s=-1/4$ by transportnumerical solution of the differential equation along a path in $s$, starting from a point where the integrals are known exactly.

Boundary constants

At the cusp $s=0$ each of the 22 masters behaves as $a(s)+b(s)\,(-s)^{-\varepsilon}$ with $a$ and $b$ analytic, and all 22 boundary values are known in closed form: each is a sum of $\Gamma$ functions and ${}_3F_2$ values at rational arguments, a Beta-function moment (for the coefficients $b(0)$), or zero. Of the fourteen masters of the system at $s=-1/4$, eleven do not depend on $t$ and so equal their values at the corner $(s,t)=(-1/4,-1/3)$ where the two lines meet; these corner masters, numbered $K_0,\ldots,K_7$, $K_9$, $K_{10}$ and $K_{12}$ by their position in the fourteen-master basis, include the six two-loop vertex integrals $K_0,\ldots,K_5$ free of $p_1$, evaluated through Mellin–Barnes representations as $\Gamma$ functions and ${}_2F_1$, ${}_3F_2$ values. $K_6$, $K_7$, $K_9$ and $K_{10}$ have closed forms of the same $\Gamma$-function and ${}_3F_2$ type. $K_{12}$, the value of the elliptic master $g_{31}$ at $s=-1/4$, is fixed by transporting the 22 masters along $t=-1/3$ and is known only numerically; no closed form in classical constants is expected for it. The period data of the Eichler integral are those of level four,

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

namely the constant term of $B_{2,4}$, the weight-two and weight-one Dirichlet $L$-values of the character $\chi_{-4}$, and the constant terms of the level-four weight-two Eisenstein series $E_2^{(4)}$ (proportional to $B_{2,4}$) at the cusps $\infty$, $0$ and $\tfrac12$. Catalan's constant enters only through $L(\chi_{-4},2)$ in these period data; an integer-relation (PSLQ) search finds no relation between $G$ and the boundary constants of the pair. Under $\tau\to\tau+n$, that is across $\mathrm{Re}\,\tau=\pm\tfrac12$, the Eichler integral shifts by its period-polynomial cocyclethe additive shift an Eichler integral picks up under the modular group; a polynomial in $\tau$ with period coefficients, fixed by the constant term of the kernel; the masters, as functions of $s$, are continued across these lines by adding the shift:

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

Polynomials in $t$

In the six-propagator system at $s=-1/4$ no propagator carries $p_1$, so $t$ enters only through numerator powers $(l_1-p_1)^{2n}$ and each master is a polynomial in $t$ of degree $n$. Only three of the fourteen depend on $t$: the masters at positions 8 and 11 of the basis (in the numbering used for the $K_i$), written $m_8$ and $m_{11}$, and $g_{32}$ at position 13; $g_{31}$ does not, since the $\Gamma_0(4)$ curve depends on $s$ only. The differential equation for these three is solved exactly in $\varepsilon$:

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

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

Here $K_5$, $K_6$, $K_7$ and $K_{12}$ are corner masters of the system, constant in $t$, $C_{11}(\varepsilon)$ and $B_0(\varepsilon)$ are $\mathbb{Q}(\varepsilon)$-linear combinations of the corner masters, and $c_8$, $c_{11}$, $c_{13}$ are the three integration constants, fixed at $t=-1/3$ by the transported values there. Every logarithm that the general solution of the equation in $t$ could produce, $\log(53-200t)$ among them, cancels identically, so the masters are entire in $t$.

The complete expression, with the boundary constants, the level-four constants and the polynomials in $t$, is in zgamma-expression.md.

Checks

Evaluator

Same family

Two other processes computed with the top-quark mass exact involve elliptic master integrals: Higgs production in gluon fusion, whose two-loop mixed QCD–electroweak corrections contain the curve of a sunrise with two lines of mass $m_t$ and one electroweak mass, and non-planar two-loop W-pair production, in which a four-point graph for $q\bar q\to W^+W^-$ has two internal lines of mass $m_t$ and two propagators that cross.

The paper

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

References

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