Three-loop equal-mass banana

The three-loop two-point integral with four internal lines of equal mass, whose maximal cut is a period of a K3 surface. Bönisch, Fischbach, Klemm, Nega and Safari (2020) and Pögel, Wang and Weinzierl (2022) computed it, and it is recomputed here without using the published answer. Its finite part is a two-term formula with a single transcendental constant.

The integral

The graph has four lines of the same mass $m$ joining two vertices and a single external momentum $p$, so it depends on the one dimensionless ratio $t = p^2/m^2$ (here $m^2 = 1$). The family is the set of three-loop two-point integrals

$$I_{\nu_1\nu_2\nu_3\nu_4} \;=\; \int \frac{d^d l_1\, d^d l_2\, d^d l_3}{\big(l_1^2-m^2\big)^{\nu_1}\,\big(l_2^2-m^2\big)^{\nu_2}\,\big(l_3^2-m^2\big)^{\nu_3}\,\big((l_1+l_2+l_3-p)^2-m^2\big)^{\nu_4}}\,, \qquad d = 2-2\varepsilon,$$

with loop momenta $l_1, l_2, l_3$ and $\varepsilon$ the dimensional regulator around two dimensions, in which $I_{1111}$ is finite. The measure is normalized as $d^d l_i/(i\pi^{d/2})$ per loop and no factor $e^{\gamma_E\varepsilon}$ is included ($\gamma_E$ is Euler's constant); in this normalization $I_{1111}$ at $p^2=0$ in exactly two dimensions equals $7\zeta_3$, with $\zeta_3$ Apéry's constant. Integration-by-parts reduction collapses the whole family onto four master integralsthe finite set of integrals to which every integral of the family reduces by integration-by-parts identities. The quantity computed is the order-$\varepsilon^0$ coefficient $I_{1111}^{(\varepsilon^0)}(t)$ of the top-sector masterthe master integral with every propagator of the diagram present once $I_{1111}$, first on the disk $|t| \lt 4$, which extends to the pseudo-threshold $p^2=(2m)^2$, and then on the rest of the real axis by continuation (the physical region).

At three loops the maximal cut, the integral with all four propagators put on shell, is not an elliptic integral as it is for the two-loop sunrise: in two dimensions it is the holomorphic period of a one-parameter family of K3 surfacesa K3 surface is a compact complex surface with trivial canonical bundle, the Calabi–Yau manifold of complex dimension two; its periods are integrals of the holomorphic two-form over two-cycles. The K3 surface here is the symmetric squarethe geometry whose periods are the pairwise products of the periods of an underlying curve of the elliptic curve of the equal-mass sunrise, a K3 surface of Picard rank 19, so it has the same congruence group $\Gamma_{1}(6)$ as the sunrise and its periods are products of pairs of sunrise periods. Because it is a symmetric square, the Picard–Fuchs operator (the differential operator in $t$ that annihilates the periods of the family) is the symmetric square of the second-order sunrise operator and has order three, and the holomorphic period $\varpi_0$ is the square of the sunrise period; its Taylor coefficients in the variable $x=t/64$ are the Domb numbers (OEIS A002895). The regular singular points $t = 0, 4, 16, \infty$ are the point of maximal unipotent monodromythe singular point where all three solutions of the Picard–Fuchs equation degenerate into one power series plus its products with $\log t$ and $\log^2 t$; the natural expansion point for Calabi–Yau periods (the MUM point), the two-particle pseudo-threshold at $(2m)^2$, the four-particle normal threshold at $(4m)^2$ and the large-momentum point.

Bloch, Kerr and Vanhove (2015) studied the integral in two dimensions as a higher normal function and derived its third-order inhomogeneous Picard–Fuchs equation; its values at special momenta are Bessel moments evaluated by Bailey, Borwein, Broadhurst and Glasser (2008) and by Broadhurst (2016). Bönisch, Fischbach, Klemm, Nega and Safari (2020) described the analytic structure of the banana integrals at every loop order, and Pögel, Wang and Weinzierl (2022) brought the three-loop equal-mass integral to $\varepsilon$-forma basis in which the connection matrix of the differential equation is proportional to $\varepsilon$, so that each order in $\varepsilon$ is a finite iterated integral over the entries (the letters) of the matrix with meromorphic modular formsfunctions of the period ratio $\tau$ that transform with a fixed weight under a congruence subgroup of SL(2,ℤ), here $\Gamma_1(6)$, and have convergent power-series expansions in $q=e^{2\pi i\tau}$, so the answer is known. The computation here takes one input from Pögel, Wang and Weinzierl, the leading coefficient in $\varepsilon$ of the sub-diagram with one of the four lines contracted (the sub-banana), which enters the differential equation of the top sector as an inhomogeneous term.

Because the answer is a K3 period times iterated integrals of modular forms, it lies outside any list of polylogarithmic or elliptic candidate functions, and the function space is read off the differential equation. With the top master normalized by the period, the four-master system takes $\varepsilon$-form over the six-letter $\Gamma_{1}(6)$ modular alphabet $\{1, f_{2a}, f_{2b}, f_{4a}, f_{4b}, f_6\}$ of modular weights $(0, 2, 2, 4, 4, 6)$. Each order in $\varepsilon$ of the top master is then a finite sum of iterated integrals with these letters as kernels, with integer coefficients, and the only transcendental constants are boundary values at the MUM point.

Closed form

The general solution of the differential equation at order $\varepsilon^0$ has four terms: the holomorphic period, the two logarithmic solutions at the MUM point, and one particular solution through which the sub-banana enters the top sector. The answer is the two-term formula

$$I_{1111}^{(\varepsilon^0)}(t) \;=\; 7\zeta_3\,\varpi_0(t)\;-\;\mathrm{Part}_{\rm reg}(t),$$

valid on the full disk $|t| \lt 4$; here $\varpi_0$ is the holomorphic K3 period and $\mathrm{Part}_{\rm reg}$ the holomorphic particular solution driven by the sub-banana. An integer-relation (PSLQ) fit of numerical values of $I_{1111}^{(\varepsilon^0)}$ at points inside the disk to these four solutions returns the coefficients $(7\zeta_3,\ 0,\ 0,\ -1)$ exactly: the two logarithmic coefficients vanish, as they must because $I_{1111}$ is analytic at $p^2=0$. The coefficient of $\varpi_0$ is thus the value of the integral at $p^2=0$, the four-Bessel moment that Bailey, Borwein, Broadhurst and Glasser evaluated as $7\zeta_3$.

Both terms of the formula, $\varpi_0$ and $\mathrm{Part}_{\rm reg}$, are explicit power series in $x=t/64$. The holomorphic K3 period is the Domb series

$$\varpi_0(x) = \sum_{n\ge 0} D_n\,x^n = 1 + 4x + 28x^2 + 256x^3 + 2716x^4 + \cdots$$

with integer coefficients fixed by the three-term recursion $(n+1)^3 D_{n+1} = 2(2n+1)(5n^2+5n+2)\,D_n - 64\,n^3 D_{n-1}$, $D_0 = 1$. The second term, $\mathrm{Part}_{\rm reg}$, is the unique holomorphic particular solution of the same order-three Picard–Fuchs operator driven by the sub-banana source $24/(t^2(t-4)(t-16))$:

$$\mathrm{Part}_{\rm reg}(t) = \sum_{k\ge 1} b_k\,x^k, \qquad b_1 = 24, \qquad n^3 b_n = 2(2n-1)(5n^2-5n+2)\,b_{n-1} - 64\,(n-1)^3 b_{n-2},$$

so the series begins $0,\ 24,\ 216,\ \tfrac{18944}{9},\ \tfrac{204440}{9},\ \dots$. Taken separately, the two series converge only for $|t|\lt4$, up to the pseudo-threshold; written as a single Taylor series in $x$ the combination converges up to the threshold $t=16$, because $I_{1111}$ is analytic for $|t|\lt16$. In the normalization of Pögel, Wang and Weinzierl, whose canonical integral is $\varepsilon^3(\pi^2/\varpi_0)\,I_{1111}$, the corresponding constant at the MUM point is $\tfrac{4}{3}\zeta_3$.

The physical region

Beyond $|t|=4$ the order-$\varepsilon^0$ coefficient is continued along a path in the upper half of the complex $t$-plane, $t\to t+i0$, by transportnumerical solution of the differential equation along a path in $t$, starting from a point where the integrals are known exactly of the four-master system, starting from a point inside $|t|\lt4$ where the two-term formula and the same recursion at the MUM point give all four master integrals exactly. Between the pseudo-threshold $t=4$ and the threshold $t=16$ the continued function is real, and above $t=16$ it is complex.

Order $\varepsilon^1$

At order $\varepsilon^0$ only the letters $1$ and $f_{4a}$ appear in the iterated integrals; at order $\varepsilon^1$, $f_{2a}$ and $f_{2b}$ enter as well ($f_{4b}$ first enters at $\varepsilon^2$ and $f_6$ at $\varepsilon^3$). Because every master integral is analytic at $p^2=0$, the logarithmic solutions drop out at this order too, and $I_{1111}^{(\varepsilon^1)}(t) = \alpha_1\,\varpi_0(t) + \mathrm{Part}^{(1)}(t)$, with $\mathrm{Part}^{(1)}$ the holomorphic particular solution of the same recursion at this order and $\alpha_1$ a single boundary constant, the order-$\varepsilon^1$ coefficient of the banana at $p^2=0$. That constant is a Bessel moment evaluated by quadrature, $\alpha_1 = -53.95234968\ldots$. An integer-relation search over the ring generated by zeta values, $\ln 2$ and $\gamma_E$, also extended by $L(f_3,2)$, finds no closed form for $\alpha_1$. Here $f_3$ is the weight-three, level-15 modular form attached to the fiber of the equal-mass sunrise family with complex multiplication by $\mathbb{Q}(\sqrt{-15})$, and its critical value has the Chowla–Selberg form $L(f_3,2) = \Gamma(\tfrac{1}{15})\Gamma(\tfrac{2}{15})\Gamma(\tfrac{4}{15})\Gamma(\tfrac{8}{15})\big/(120\sqrt{3}\,\pi)$. At order $\varepsilon^2$ and beyond the constants at $p^2=0$ are known only numerically, as Bessel moments from the one-dimensional representation of the four propagators, which is exact in $d$, and the higher orders away from $p^2=0$ are obtained by transport of the differential system; the arithmetic nature of $\alpha_1$ and of these higher-order constants is open.

The complete expression, with both recursions, the conventions and the boundary and arithmetic data, is in the expression file inside the bundle under Evaluator.

Checks

Evaluator

Same family

The four-loop equal-mass banana adds a fifth line of the same mass; its maximal cut is a period of a Calabi–Yau threefold with a Picard–Fuchs operator of order four. The three-loop banana at the mass threshold keeps three lines at mass $m$ and puts the fourth at $3m$, the production threshold of the other three, where the monodromy of the K3 period around the threshold is no longer unipotent and a new connection coefficient appears. One loop down, the maximal cut of the equal-mass sunrise is a period of the $\Gamma_{1}(6)$ elliptic curve whose symmetric square is this K3 surface, and the three-loop banana reappears as a subsector of the crossed light-by-light box.

The paper

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

References

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