Four-loop equal-mass banana

The four-loop banana of five equal-mass propagators, whose maximal cut is a Calabi–Yau threefold period, computed by Bönisch, Duhr, Fischbach, Klemm and Nega (2021) and Pögel, Wang and Weinzierl (2022). Here it is rederived by a different route, with only its large-momentum limit taken from them; the five master integrals through the finite part follow from one series in $1/p^2$.

The integral

The diagram is the four-loop member of the banana family, with one external momentum $p$. At $m^2=1$ the integrals depend on $p^2$ alone, through the variable $z=-1/p^2$. The family is

$$I_{\nu_1\nu_2\nu_3\nu_4\nu_5} \;=\; \int \frac{d^d k_1\, d^d k_2\, d^d k_3\, d^d k_4}{D_1^{\nu_1}\,D_2^{\nu_2}\,D_3^{\nu_3}\,D_4^{\nu_4}\,D_5^{\nu_5}}\,, \qquad d = 2-2\varepsilon,$$

with loop momenta $k_1,\dots,k_4$, the measure $d^dk/(i\pi^{d/2})$ for each loop, and the propagators

$$D_i = k_i^2 - m^2 \quad (i=1,\dots,4), \qquad D_5 = (k_1+k_2+k_3+k_4-p)^2 - m^2 .$$

Integration-by-parts reduction leaves five master integralsthe finite basis of integrals to which every integral of the family reduces by integration-by-parts identities: the fourth power of the one-loop tadpole, $I_0=I_{11110}=\Gamma(\varepsilon)^4$, the top-sector masterthe master integral with every propagator of the diagram present once $I_1=I_{11111}$, and three further top-sector integrals $I_2, I_3, I_4$ that carry the numerators $\big((k_1-p)^2\big)^2$, $\big((k_1+k_2)^2\big)^2$ and $\big((k_1-p)^2\big)^3$. The quantities computed are the Laurent coefficients of all five masters from $\varepsilon^{-4}$ through $\varepsilon^0$ on the Euclidean axis $p^2\lt0$. The top-sector master has no poles in $\varepsilon$ in two dimensions, and its finite part is written $I^{(0)}_{11111}(p^2)$.

The equal-mass banana graphs form a sequence in the loop number, and the geometry of the maximal cutthe integral with every propagator put on its mass shell, which satisfies the homogeneous part of the differential equation and fixes the class of functions in the answer gains one complex dimension with each loop: an elliptic curve for the two-loop sunrise, a K3 surface at three loops, and at four loops a Calabi–Yau threefold from the one-parameter Hulek–Verrill family. Klemm, Nega and Safari (2019) obtained the banana periods at any loop order from Gel'fand–Kapranov–Zelevinsky hypergeometric systems, and Bönisch, Fischbach, Klemm, Nega and Safari (2020) described the analytic structure of the banana integrals at every loop order in two dimensions. Bönisch, Duhr, Fischbach, Klemm and Nega (2021) extended the construction to dimensional regularization and wrote the large-momentum limit of the equal-mass banana as an alternating sum of products of $\Gamma$ functions over hard and soft regions, and Pögel, Wang and Weinzierl (2022) brought the differential equation of the four-loop integral to $\varepsilon$-factorized form. In two dimensions the banana integrals are also moments of Bessel functions, studied by Broadhurst (2016); at Euclidean momentum the finite part of the top-sector master is $I^{(0)}_{11111}(p^2)=-16\int_0^\infty x\,J_0\big(x\sqrt{-p^2}\,\big)\,K_0(x)^5\,dx$.

Eliminating $I_2, I_3, I_4$ from the order-$\varepsilon^0$ maximal-cut block of the five-master differential equation gives the Picard–Fuchs operatorthe ordinary differential operator in the kinematic variable that annihilates the periods of the cut geometry of the cut directly from the reduction: the fourth-order operator numbered 34 in the tables of Calabi–Yau equations of Almkvist, van Enckevort, van Straten and Zudilin (2005). With $\theta=z\,d/dz$,

$$\mathcal L_4=\theta^4+z\,(35\theta^4+70\theta^3+63\theta^2+28\theta+5)+z^2(\theta+1)^2(259\theta^2+518\theta+285)+225\,z^3(\theta+1)^2(\theta+2)^2 .$$

Its point of maximal unipotent monodromy, the MUM pointthe most degenerate singular point of the family, where all four solutions collapse onto one power series and its logarithmic partners; the natural expansion point for Calabi–Yau periods, is $z=0$, that is $p^2\to\infty$, where the indicial equation is $\theta^4=0$ and the holomorphic solution is the Hulek–Verrill period $\varpi_0(z)=\sum_{n\ge0}A_n(-z)^n$ with $A_n=\sum_{k_1+\dots+k_5=n}\big(n!/(k_1!\cdots k_5!)\big)^{2}=1,\,5,\,45,\,545,\,7885,\dots$. The other three singular points, $z=-1,-\tfrac19,-\tfrac1{25}$, are the thresholds $p^2=m^2$, $(3m)^2$ and $(5m)^2$; the full five-master differential equation has in addition only the singular point $p^2=0$, and no spurious singularity.

Closed form

Write $L=\log z$, remove the Euler–Mascheroni constant by setting $M_j=e^{4\gamma_E\varepsilon}I_j$ for $j=0,\dots,4$, and rescale them by powers of $z$ into $\widehat M=(M_0,\,z^{-1}M_1,\,z\,M_2,\,z\,M_3,\,z^{2}M_4)$, each entry a Laurent polynomial in $\varepsilon$ from $\varepsilon^{-4}$ to $\varepsilon^{0}$. The rescaled masters obey a first-order system $\theta\,\widehat M=C(z,\varepsilon)\,\widehat M$, in which the entries of $C$ are polynomials in $z$ with integer coefficients over the common denominator $20(1+z)(1+9z)(1+25z)$. The matrix $C(z,\varepsilon)$ is holomorphic at the MUM point, so $z=0$ is a regular singular point of the system; its residue matrix $C_0=C(0,\varepsilon)$, acting on the twenty-five Laurent coefficients of the five masters taken together, is nilpotent of index 8. All five masters through order $\varepsilon^0$ are therefore one log-Frobenius series, $\widehat M(z)=F(z)$ with

$$F(z) \;=\; \sum_{n\ge0} z^n \sum_{j=0}^{7} F_{n,j}\, L^j, \qquad F_0(L) = e^{C_0 L}\, v,$$

where $F_0(L)=\sum_j F_{0,j}L^j$ is the $n=0$ term and every coefficient vector $F_{n,j}$ with $n\ge1$ follows from the boundary vector $v$ by a recursion in $n$ whose coefficients are the integer coefficients of $C(z,\varepsilon)$. The logarithms stop at $L^7$ because $C_0^8=0$.

The boundary vector collects, master by master, the coefficients of $\varepsilon^{-4},\dots,\varepsilon^{0}$ at $n=0$, and lies entirely in the ring of zeta values $\zeta_2,\zeta_3,\zeta_4$:

$$ \begin{aligned} v_{M_0}&=\big(1,\,0,\,2\zeta_2,\,-\tfrac{4}{3}\zeta_3,\,6\zeta_4\big),\qquad v_{M_1}=(0,\,0,\,0,\,0,\,0),\\ v_{M_2}&=\big(-1,\,-2,\,-2-2\zeta_2,\,-2-4\zeta_2+\tfrac{4}{3}\zeta_3,\,-50-4\zeta_2+\tfrac{104}{3}\zeta_3-6\zeta_4\big),\\ v_{M_3}&=\big(-\tfrac{3}{2},\,-1,\,-3\zeta_2,\,-2\zeta_2+2\zeta_3,\,-60+\tfrac{52}{3}\zeta_3-9\zeta_4\big),\\ v_{M_4}&=\big(1,\,3,\,\tfrac{9}{2}+2\zeta_2,\,\tfrac{21}{4}+6\zeta_2-\tfrac{4}{3}\zeta_3,\,\tfrac{453}{8}+9\zeta_2-52\zeta_3+6\zeta_4\big). \end{aligned} $$

The tadpole block $v_{M_0}$ is the expansion of $\Gamma(1+\varepsilon)^4e^{4\gamma_E\varepsilon}$, since $M_0=\Gamma(\varepsilon)^4e^{4\gamma_E\varepsilon}$. The top-sector master is finite in two dimensions and has no free boundary constant, $v_{M_1}=0$: its coefficients at $n=0$ are generated from the tadpole block and the constants of $M_2,M_3,M_4$ by powers of $C_0$. Three conditions on the general solution fix the vector: the tadpole block, the vanishing of $v_{M_1}$ in all five orders, and agreement of the $n=0$ logarithmic terms with the large-momentum limit of Bönisch–Duhr–Fischbach–Klemm–Nega and Pögel–Wang–Weinzierl, the one input taken from earlier work. Because the logarithms at $n=0$ come from powers of $C_0$ acting on $v$, these conditions are a linear system with rational coefficients and right-hand sides in the zeta ring, and the vector above is its unique solution in exact arithmetic.

For the top-sector master alone, the system at order $\varepsilon^0$ amounts to an inhomogeneous fourth-order equation with a boundary condition at the MUM point. The operator $\mathcal L_4$ applied to $(-p^2)\,I^{(0)}_{11111}$, regarded as a function of $z$, gives the constant $-5!=-120$, and at $z=0$ the coefficients of $L^0,\dots,L^4$ in $(-p^2)\,I^{(0)}_{11111}$ are $(0,\,80\zeta_3,\,0,\,0,\,-5)$. In this normalization neither $\zeta_2$ nor $\zeta_4$ appears.

The series in $z$ converges near the MUM point, inside the nearest threshold $|z|\lt\tfrac1{25}$. Any Euclidean $p^2\lt0$ is reached by transportnumerical solution of the differential equation along a path in the kinematic variable, starting from a point where the integrals are known exactly along the positive $z$ axis, which meets no singular point. Removing the factor $e^{4\gamma_E\varepsilon}$ order by order in $\varepsilon$ then gives the $I_j$. The full matrix $C(z,\varepsilon)$, together with $v$, is tabulated in the evaluation script banana-4loop-evaluate.py.

Checks

Evaluator

Same family

• The three-loop equal-mass banana has four lines of one mass; its maximal cut is a period of a K3 surface, the symmetric square of the sunrise elliptic curve, and its finite part is a two-term formula on the disk around $p^2=0$.
• The four-loop banana at the mass threshold takes squared masses $(1,1,1,1,16)$, where the pseudo-threshold coincides with $p^2=0$ and the monodromy of the Calabi–Yau period there acquires a part of finite order; the new results are the threshold coefficients, which involve $\Gamma(\tfrac14)$.

The paper

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

References

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