Three-loop banana at the mass threshold

The three-loop banana self-energy with squared masses (1, 1, 1, 9), in which the heavy line sits exactly at the threshold for producing the three light particles. Its maximal cut is a K3 period with a square-root branch at that point, and the coefficient of that branch in the holomorphic period is new and is given here in closed form.

The integral

The three-loop banana is the two-point graph with four propagators joining two vertices. With one external momentum $p$ the family is

$$ I[\nu_1,\nu_2,\nu_3,\nu_4] \;=\; \int d^d l_1\, d^d l_2\, d^d l_3\; \frac{1}{\bigl(l_1^2 - M_1\bigr)^{\nu_1}\bigl(l_2^2 - M_2\bigr)^{\nu_2}\bigl(l_3^2 - M_3\bigr)^{\nu_3}\bigl((l_1+l_2+l_3-p)^2 - M_4\bigr)^{\nu_4}}\,, \qquad d = 4-2\varepsilon, $$

with loop momenta $l_1,l_2,l_3$ and the four squared masses set to $(M_1, M_2, M_3, M_4) = (1,1,1,9)$ in units of the light mass, so that the only kinematic variable is $t = p^2$. The family is written in $d=4-2\varepsilon$ dimensions; the computation is at $d=2$, where the full integral is finite and its maximal cutthe integral with all four propagators put on shell, which isolates the geometry underneath the integral is a pure periodthe integral of the surface's holomorphic two-form over a compact two-cycle; as the momentum $t$ varies it is a transcendental function of $t$ of a K3 surfacea compact complex surface with a nowhere-vanishing holomorphic two-form, the two-dimensional Calabi–Yau manifold; the cut of the $L$-loop banana is a Calabi–Yau variety of dimension $L-1$, an elliptic curve for the two-loop sunrise and a K3 surface here. The quantity computed is the connection coefficient $c_{3/2}$: the amount of the square-root branch at the threshold $t=0$ contained in the holomorphic period $\varpi_0(t)$, the solution of the surface's fourth-order Picard–Fuchs equationthe linear differential equation in $t$, with rational coefficients, that every period of the family satisfies given at large momentum by the all-loop series of Bönisch, Fischbach, Klemm, Nega and Safari in $z=-1/t$, $\varpi_0=1-12z+204z^2-\cdots$, so that $\varpi_0\to1$ as $t\to\infty$. The coefficient $c_{3/2}$ is an exact constant that does not depend on the regulator $\varepsilon$.

Bloch, Kerr and Vanhove (2015) gave the differential equation of the equal-mass three-loop banana in two dimensions and the K3 geometry behind it. Klemm, Nega and Safari (2020) and Bönisch, Fischbach, Klemm, Nega and Safari (2021) wrote the periods of the banana with any number of loops and arbitrary masses as explicit series around $1/t=0$, the large-momentum point of maximal unipotent monodromythe linear map on the space of solutions induced by carrying the integration contour once around a singular point and back, where $\varpi_0$ is accompanied by solutions with one and two powers of $\log(1/t)$. Broedel, Duhr and Matthes (2022) expressed the equal-mass three-loop integral in meromorphic modular forms, and Duhr and Maggio (2025) solved the three-loop banana with three equal masses and one different mass, the family that contains $(1,1,1,9)$, by expansions around points where the local monodromy is unipotent.

At equal masses the three-loop banana has square-root branch points at its finite singular points, with local exponents $\{0,\tfrac12,1\}$ and a semisimple monodromy, $\mathrm{diag}(1,-1,1)$ in the basis of Broedel, Duhr and Matthes; Bönisch, Duhr, Fischbach, Klemm and Nega (2022) found such square-root branch points in the equal-mass bananas of every odd loop order. None of these works expands a K3 period around a point where a square-root branch and a logarithmic block occur together, as they do at $t=0$ for the masses $(1,1,1,9)$.

At squared masses $(1,1,1,9)$ the pseudo-threshold $t=(\sqrt{M_4}-\sqrt{M_1}-\sqrt{M_2}-\sqrt{M_3})^2$, where the heavy line can turn into the three light ones, coincides with $t=0$ (the threshold, for short). The Picard–Fuchs operator is singular at the thresholds and pseudo-thresholds $(\sum_i\pm\sqrt{M_i})^2=0,\,4,\,16,\,36$, at the further pair $t=11\pm\sqrt{105}$, and at infinity. Its indicial equation at $t=0$, in the local exponent $\rho$, is

$$ 18432\,(\rho-1)^3\,(2\rho-3) \;=\; 0\,, $$

so the local exponents are $\rho\in\{1,1,1,\tfrac32\}$. The three unit exponents form one unipotent Jordan block of rank three. The exponent $\tfrac32$ gives a monodromy eigenvalue $e^{3\pi i}=-1$: carried once around the threshold, one of the four local solutions comes back with its sign flipped, as one circuit of a Möbius band reverses orientation, and a second circuit undoes the flip, so this part of the monodromy has order two. The local structure differs from the equal-mass case: the eigenvalue $-1$ occurs together with a rank-three unipotent block rather than inside a semisimple monodromy, so a circuit of this point both shifts the periods by logarithms and reverses a sign. It also differs from the generic-mass case, for instance $(1,1,4,9)$, where the local monodromy at $t=0$ is unipotent and a circuit only adds logarithms. A point with these local exponents, $\{1,1,1,\tfrac32\}$, is called a point of mixed type. A double-cover (Baikov) model of the cut suggests that the K3 fiber at $t=0$ is a non-semistable degeneration whose monodromy has a finite part of order two; this geometric label is supported by the period monodromy but not proved. The two-loop counterpart is the sunrise with squared masses $(1,1,4)$, whose elliptic curve degenerates at $t=0$ with a monodromy of order four.

The series for the periods around $1/t=0$ do not fix $c_{3/2}$, which is a connection constant between the point at infinity and the threshold. It is computed here by numerical transportnumerical solution of the differential equation along a path in $t$, starting from a point where the solution is known exactly of the differential equation from large momentum to $t=0$. The maximal-cut differential operator of $I[1,1,1,1]$ at $d=2$, obtained from the integration-by-parts reductionthe reduction of every integral of the family to a finite set of master integrals by linear identities among them of the family at symbolic $d$, equals the Picard–Fuchs operator of $\varpi_0$ up to a unit shift of the Euler operator $\theta_z=z\,d/dz$, $z=-1/t$, so the fractional exponent at $t=0$ is a property of the Feynman family's own differential equation.

Closed form

Near the threshold the holomorphic period decomposes in the local Frobenius basis as

$$ \varpi_0(t) \;=\; A_0\,\Phi^{(0)}(t) + A_1\,\Phi^{(1)}(t) + A_2\,\Phi^{(2)}(t) \;+\; c_{3/2}\,\Phi_{3/2}(t), $$

where $\Phi^{(0)}, \Phi^{(1)}, \Phi^{(2)}$ are the three unipotent Frobenius solutions at $t=0$, power series carrying zero, one and two logarithms, and $\Phi_{3/2}$ is the square-root branch, a Frobenius series in $t$ with rational coefficients, normalized to leading coefficient one:

$$ \Phi_{3/2}(t) \;=\; t^{3/2}\Bigl(1 + \tfrac{19}{216}\,t + \tfrac{181}{17280}\,t^2 + \cdots\Bigr). $$

The branch of $t^{3/2}$ is $\arg t=-\pi$ as reached from the Euclidean axis; the other branch reverses the sign of $c_{3/2}$. With the local solutions written in the variable $e^{-i\pi}p^2$ of the Bessel representation below, which is positive for Euclidean momenta and in which the odd-order coefficients of $\Phi_{3/2}$ change sign relative to the display above, the unipotent coefficients are $A_1=-\sqrt3/(24\pi)=\tfrac32\,c_{3/2}$ and $A_2=0$, both of which follow analytically from that representation, and $A_0=\sqrt3\,\log 24/(12\pi)$, which is equivalent to an evaluation of a Bessel moment that has not been proved and holds numerically to high precision. The coefficient of the square-root branch is

$$ c_{3/2} \;=\; -\frac{\sqrt{3}}{36\,\pi} \;=\; -0.0153146915\ldots\,, \qquad 432\,\pi^2\,c_{3/2}^{\,2} \;=\; 1\,, $$

obtained by applying to $\varpi_0$ the spectral projectorthe polynomial in the matrix $M_0$ that is the identity on one chosen eigenspace and annihilates every other generalized eigenspace onto the $-1$ eigenspace of the threshold monodromy matrix $M_0$, the $4\times4$ matrix by which the four local solutions transform when continued once around $t=0$:

$$ P_{-1} \;=\; -\tfrac{1}{8}\,(M_0 - \mathbb{1})^3\,, \qquad P_{-1}\!\cdot\!\varpi_0 \;=\; c_{3/2}\,\Phi_{3/2}\,. $$

Transport of the large-momentum series to a small circle about $t=0$, and then once around that circle, gives the $4\times4$ monodromy matrix $M_0$. Solving the transported period directly against the four local solutions is ill-conditioned: the square-root admixture is roughly $10^4$ times smaller than the dominant unipotent period, and a direct solve loses those four orders of magnitude in precision. The projector avoids this loss: because $M_0-\mathbb{1}$ is nilpotent of index three on the unipotent block, $(M_0-\mathbb{1})^3$ annihilates all three unipotent solutions, logarithms included, and acts as $(-2)^3=-8$ on the square-root branch, so a single polynomial in $M_0$ isolates $c_{3/2}\Phi_{3/2}$ by exact algebra. The same construction extracts any non-unipotent component of a period: the spectral idempotent onto an eigenvalue is a polynomial in $M_0$ fixed by its minimal polynomial. An integer-relation (PSLQ) fit to the projected value gives the closed form above.

The coefficient $c_{3/2}$ also follows analytically from the position-space Bessel representation of the period, an integral over a radial variable $y$ of the product $J_0(y)^3J_0(3y)$ of Bessel functions, one factor $J_0(\sqrt{M_i}\,y)$ for each line, against a Hankel-function kernel $H_0^{(1)}$. The non-oscillatory tail of that product begins $-\tfrac{\sqrt{3}}{18\pi^2}\,y^{-3}$: the $1/\sqrt{3}$ is $(M_1M_2M_3M_4)^{-1/4}$ and the $1/\pi^2$ comes from the four Bessel asymptotics. Integrating this term against the kernel is a Mellin transform of $H_0^{(1)}$, which multiplies it by $2^{-3}\Gamma(-\tfrac12)^2=\pi/2$ and turns it into $-\tfrac{\sqrt{3}}{36\pi}\,t^{3/2}$ exactly. The next tail term reproduces the Frobenius coefficient $\tfrac{19}{216}$ of $\Phi_{3/2}$. The closed form contains no special value of the K3 surface's $L$-function and none of its complex-multiplication periods (the Chowla–Selberg products of $\Gamma(a/15)$ with integer $a$): $c_{3/2}$ lies in the ring generated by $\sqrt3$ and $1/\pi$.

The full integral

At $d=2$ and Euclidean momenta the full integral $I[1,1,1,1]$ is finite, and to the accuracy computed it is analytic at the threshold: it has no $t^{3/2}$ component and no logarithm at $t=0$, so the square-root branch and the non-unipotent monodromy belong to the maximal cut alone. It satisfies the maximal-cut differential equation with an inhomogeneous term from the subsector integrals; near $z=-1/t=0$ it is a particular solution of that equation plus a combination $\sum_{i=0}^{3}a_i\,\Pi_i$ of the Frobenius solutions of the maximal-cut operator at $z=0$: $\Pi_0=z\varpi_0$, its partners $\Pi_1$ and $\Pi_2$ with one and two powers of $\log z$, and the second power-series solution $\Pi_3=z^2(1+\cdots)$, with

$$ a_0 = 19.2329104\ldots,\qquad a_1 = 0,\qquad a_2 = -13.1833474\ldots,\qquad a_3 = -38.5332202\ldots $$

in one fixed normalization of the measure and of the basis, which also fixes the particular solution. Details are in Polylogarithmic, elliptic, and Calabi–Yau Feynman integrals from a hybrid bootstrap.

The complete expression, with the integer relations and the high-precision value of $c_{3/2}$, is in the expression file inside the bundle under Evaluator.

Checks

Evaluator

Same family

The four-loop banana at the mass threshold, squared masses $(1,1,1,1,16)$, has a Calabi–Yau threefold as its cut, a threshold monodromy of order four, and two fractional-branch coefficients in the ring of the lemniscatic constant $\Gamma(\tfrac14)$. The three-loop equal-mass banana is the same graph with four lines of one mass, where the K3 period is modular and the finite part of the integral is given in closed form around $p^2=0$, on the disk $\lvert p^2\rvert\lt 4m^2$ with $m$ the common mass.

The papers

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

References

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