Non-planar two-loop W-pair production

The non-planar two-loop double box for quark–antiquark annihilation into a W pair, with exact top-quark and W masses. The planar graphs were computed by He, Zhang, Han and collaborators (2024); the non-planar graph is new. Every master integral is given as an exact function of one kinematic invariant, the other two held constant, with some boundary constants fixed numerically.

The integral

Two-loop corrections to $q\bar q\to W^+W^-$ in which the $W$ bosons attach to a top-quark line depend on $m_t$, $m_W$ and two scattering invariants. The non-planar graph above is the one He et al. (2024) draw as their fig. 1d, and in their labeling the family of integrals is

$$\mathcal T_4[\nu_1,\nu_2,\nu_3,\nu_6,\nu_7,\nu_8,\nu_9] \;=\; \int \frac{d^d l_1}{i\pi^{d/2}}\, \frac{d^d l_2}{i\pi^{d/2}}\; \frac{1}{D_1^{\nu_1} D_2^{\nu_2} D_3^{\nu_3} D_6^{\nu_6} D_7^{\nu_7} D_8^{\nu_8} D_9^{\nu_9}}\,, \qquad d = 4-2\varepsilon,$$

with loop momenta $l_1, l_2$, a $+i0$ in every propagator, no factor of $e^{\gamma_E\varepsilon}$, $m_t=1$ in all numerical values below, and the seven propagators

$$D_1 = l_1^2,\quad D_2 = (l_1+p_1)^2,\quad D_3 = (l_1-p_2)^2,\quad D_6 = (l_2-p_2)^2,\quad D_7 = (l_1-l_2)^2,$$

$$D_8 = (l_1-l_2-p_4)^2 - m_t^2,\qquad D_9 = (l_2-p_2-p_3)^2 - m_t^2.$$

The quark legs are massless, $p_1^2 = p_2^2 = 0$, the $W$ legs are on shell, $p_3^2 = p_4^2 = m_W^2$, and $m_t$ in $D_8$, $D_9$ is kept exact. All four momenta are incoming, and $p_4$ in $D_8$ stands for $-p_1-p_2-p_3$. The Mandelstam invariantsthe Lorentz-invariant combinations of external momenta that a scattering amplitude can depend on are $s = (p_1+p_2)^2$ and $t = (p_2+p_3)^2$, and dividing by the one internal scale leaves three dimensionless variables,

$$x = -\frac{s}{m_t^2}, \qquad y = -\frac{t}{m_t^2}, \qquad z = -\frac{m_W^2}{m_t^2}.$$

The result is the set of Laurent coefficients in $\varepsilon$, from $\varepsilon^{-3}$ through $\varepsilon^{1}$, of all 76 master integralsthe finite basis of integrals to which every integral of the family reduces through integration-by-parts identities of the family, together with three auxiliary integrals of the top sector (the sector in which all seven propagators are present), as exact functions of $y$ on the line $(x,z) = (17/2,\,5/3)$ with $0\lt y\lt 1$. On this line $s\lt0$ and $m_W^2\lt0$, while the third invariant $u=(p_1+p_3)^2=(31/6+y)\,m_t^2$ lies above the $t\bar t$ threshold, so the line is reached from the physical scattering region by analytic continuation and 36 of the 79 functions are complex on it.

Master integrals for two-loop $q\bar q\to W^+W^-$ with exact top-quark mass dependence enter the next-to-next-to-leading-order QCD corrections to $W$-pair production at hadron colliders through loops of third-generation quarks (He et al., 2024). That work computed the master integrals of the three planar top-quark topologies, one of which already contains an elliptic sector; the non-planar topology was not treated there, and to our knowledge no earlier result for it exists. Of its 76 master integrals, 53 coincide with masters of the planar topologies and 23, those whose propagators include all four of $D_6$, $D_7$, $D_8$ and $D_9$, are new.

With $x$ and $z$ fixed, the master integrals satisfy an ordinary differential equation in $y$ whose matrix is determined exactly, by rational reconstruction from numerical integration-by-parts reductions on the line. On the line $(x,z)=(17/2,\,5/3)$ the function space, the letters and the one elliptic curve of the family are all known explicitly; the curve differs from that of the planar topologies, and the top sector adds no new transcendental functions of its own. The exact differential equation in $y$ has also been reconstructed on a second line, $(x,z)=(17/2,\,7/4)$, where it has the same structure sector by sector, but the dependence on $x$ and $z$, and with it a closed form in all three variables, remains open.

Closed form

On the line the 79 functions, collected in a vector $M$, obey $\partial_y M = A(\varepsilon,y)\, M$ with a connection matrix $A$ whose entries are known exactly as rational functions of $y$ and $\varepsilon$. The matrix $A$ is block lower-triangular with 36 diagonal blocks, so the function class of a master is set by its own diagonal block and by the blocks feeding its inhomogeneous terms. For 29 blocks a rational change of basis (Lee, 2015) gives $\varepsilon$-factorized $d\log$ formthe connection is $\varepsilon$ times a sum of constant matrices multiplying logarithmic differentials $d\log\alpha_i(y)$ of the letters; the hallmark of polylogarithmic integrals, 6 blocks have square-root letters and admit no such change of basis, and in one, the block of the five-propagator sector $(D_1,D_6,D_7,D_8,D_9)$, the maximal cutthe integral with every propagator of the sector put on shell, which isolates the sector's intrinsic geometry is an integral over an elliptic curve. Each component of $M$ then follows, order by order in $\varepsilon$, by variation of parameters from the homogeneous solutions of its diagonal block, the already-solved blocks entering its inhomogeneous terms, and its boundary constant.

Polylogarithmic blocks

The functions whose own block and source blocks are all in $d\log$ form are multiple polylogarithmsgeneralized (Goncharov) polylogarithms $G(a_1,\dots,a_w;y)$, the iterated integrals of $dy/(y-a_i)$ that are the standard functions of multi-loop calculations with up to six indices $a$ drawn from fifteen letters:

$$a \;\in\; \Big\{\,0,\; -1,\; \pm\tfrac{5}{3},\; -\tfrac{31}{12},\; -\tfrac{31}{6},\; -\tfrac{41}{6},\; -\tfrac{25}{6},\; -\tfrac{7}{2},\; -\tfrac{73}{48},\; -\tfrac{175}{48},\; \tfrac{91}{24},\; -\tfrac{215}{24},\; r_+,\; r_-\Big\},$$

where $r_\pm = \tfrac{1}{12}\big({-31\pm\sqrt{561}}\big)$ are the roots of $18y^2+93y+50$. The solution of each such block is a matrix of these polylogarithms, expanded in powers of $\varepsilon$, acting on the vector of boundary constants. For example, one entry of the solution matrix at first order in $\varepsilon$ is

$$-\tfrac{2}{3}\,G(0;y) \;+\; \tfrac{187}{18}\,G(-1;y),$$

and the higher orders are polylogarithms with more indices in the same letters, with rational coefficients. In the six square-root blocks the iterated integrals have, besides these letters, algebraic kernels involving further quadratic polynomials in $y$, among them two with complex roots.

The top sector

At $\varepsilon=0$ (four dimensions) the diagonal block of the four top-sector functions (the top-sector master and the three auxiliary integrals) has a fundamental system of rational solutions: two are polynomial in $y$ and the other two have the quadratic $18y^2+93y+203$ as their only denominator. Write $y=\tfrac13+\tfrac{4}{15}\xi$, so that $\xi=0$ at the boundary point $y_0=1/3$, and let $P(\xi)=\xi^2+\tfrac{175}{8}\xi+\tfrac{1475}{8}$, which is $\tfrac{25}{32}(18y^2+93y+203)$. The four solutions are

$$Y_1 = \begin{pmatrix} \dfrac{\tfrac{1475}{8} - 5\xi}{P} \\[2pt] 0 \\ 0 \\[2pt] \dfrac{85\,\xi}{P} \end{pmatrix},\qquad Y_2 = \begin{pmatrix} 0 \\ 1 \\ \tfrac{4}{165}\,\xi \\ 0 \end{pmatrix},\qquad Y_3 = \begin{pmatrix} 0 \\ 0 \\ 1 \\ 0 \end{pmatrix},\qquad Y_4 = \begin{pmatrix} -\dfrac{\tfrac{15}{4}\,\xi}{P} \\[2pt] 0 \\ 0 \\[2pt] \dfrac{\tfrac{1475}{8} + \tfrac{215}{8}\xi}{P} \end{pmatrix}.$$

When one of the three auxiliary integrals is replaced by another integral of the top sector, the block takes $\varepsilon$-factorized $d\log$ form, with the two letters $18y^2+93y+50$ and $18y^2+93y+203$.

The elliptic sector

The maximal cut of the five-propagator sector localizes onto the curve, in the $(z_4,w)$ plane,

$$w^2 \;=\; g_1(z_4)\, G_2(z_4),$$

with $z_4, z_5$ the two Baikov variablesintegration variables in a representation of the integral in which the propagators themselves become coordinates left on the cut, $w$ the square root that the cut produces, $g_1$ and $G_2$ the Baikov polynomials of the two loops ($G_2$ here is a Baikov polynomial, unrelated to the polylogarithms $G(a_1,\dots;y)$ above), and, explicitly,

$$g_1 = -\tfrac{1}{4}\left(y^2 - 2yz_5 + 2y + 4z\,z_4 + z_5^2 - 2z_5 + 1\right),$$

$$\begin{aligned} G_2 = \tfrac{1}{16}\Big(&x^2y^2 + 2x^2y z_4 + 2x^2 y + x^2 z_4^2 - 2x^2 z_4 + x^2 + 2xy^2 z_5 - 4xyz\,z_4 - 2xyz\,z_5 + 2xy\,z_4 z_5 - 4xy\,z_4 \\ &+ 2xy\,z_5 + 4xz^2 z_4 - 4xz\,z_4^2 + 2xz\,z_4 z_5 + 4xz\,z_4 - 2xz\,z_5 + y^2 z_5^2 - 2yz\,z_5^2 + z^2 z_5^2\Big). \end{aligned}$$

The product is a cubic in $z_4$, so the geometry is an elliptic curve fibered over the $z_5$ line (a rational elliptic surface with two nodal and five two-component singular fibers, $2I_1+5I_2$ in Kodaira's notation, Euler number 12) whose $j$-invariant varies with $z_5$. Each curve carries a rational 2-torsion point, so the curve together with that point defines a point of the modular curve $X_0(2)$, and $j$ is a rational function of the Hauptmodulthe single rational coordinate that parametrizes a genus-zero modular curve $h$ of $X_0(2)$:

$$j \;=\; \frac{(h+256)^3}{h^2}, \qquad h = \frac{6787441}{213725}\,,$$

where the value of $h$ is that of the fiber $z_5=7/3$ at the boundary point $y=1/3$. The curve is not of complex-multiplication type, and it is not parametrized by modular forms of any congruence subgroup. On the line, however, the order-$\varepsilon^0$ block of this sector is reducible: its differential operator in $y$ factorizes into six first-order operators with rational coefficients, so that two of its six homogeneous solutions are rational functions of $y$ and the other four are iterated integrals of algebraic functions; the block is then solved by variation of parameters without elliptic special functions. The local exponents of the full $\varepsilon$-dependent block lie in $\mathbb{Z}+\mathbb{Z}\varepsilon$ except at the zeros of $Q_1 = 225y^2+5370y+19801$ and $Q_2 = 900y^2-12180y-7751$, where the exponent is $-3/2$ for every value of the dimension; hence no $\varepsilon$-factorized form with rational letters exists for this block on the line, although one may exist over the function field extended by $\sqrt{Q_1}$ and $\sqrt{Q_2}$ (Frellesvig, 2021).

Boundary constants

Every function has a finite limit as $y\to0$, and these 79 limits can be taken as the integration constants. Thirty-nine are known exactly: 23 are limits of the planar polylogarithms of He et al. continued to the line, 11 are $\Gamma$-function and hypergeometric values of vacuum and bubble integrals, and 5 are given by convergent integral representations. The other 40, among them every constant of the square-root blocks and of the elliptic sector, are fixed instead by the values of their functions at $y_0=1/3$. Those values are obtained, to any requested precision, by transportnumerical solution of a differential equation along a path, starting from a point where the integrals are known exactly in an auxiliary mass given to every propagator, from large mass, where $\Gamma$-function vacuum integrals fix the solution, down to zero. For 24 of the 40 the leading Laurent coefficient is rational or a combination of $\ln2$ and $\ln3$; no closed form is known for the other 16 leading coefficients.

The complete expression, with the alphabet, the sector bases, the polylogarithmic words, the elliptic-sector data and the boundary constants, is in the expression file inside the bundle under Evaluator below.

Checks

Evaluator

Same family

gg → Zγ with exact top-quark mass is the other two-loop four-point computation with an exact internal top-quark mass; its elliptic curve is modular for $\Gamma_0(4)$ and its boundary constants are known in closed form, whereas the curve of this family is not attached to any congruence subgroup and its elliptic-sector constants are fixed numerically. The elliptic masters of gg → H are a known two-loop result with an exact top-quark loop, rederived independently. The non-planar hexa-box is the other new non-planar collider family, five-point and polylogarithmic.

The papers

• Polylogarithmic, elliptic, and Calabi–Yau Feynman integrals from a hybrid bootstrap (PDF) — Matthew D. Schwartz.
• Automated computation of Feynman integrals with BootLoops (PDF) — Matthew D. Schwartz. A preliminary version of the paper.

References

← back to Feynman diagrams