Threshold banana — CY₃ rung

The four-loop banana integral tuned to its particle-production threshold: the maximal-cut geometry is a Calabi–Yau threefold, one walk around the threshold rotates two of its six periods by $\pm i$, and both quarter-integer connection coefficients close in the ring of the lemniscatic constant $\Gamma(\tfrac14)$ — a constant the banana ladder had not produced before.

The integral

The four-loop banana is five massive propagators stretched between the same two vertices, with one external leg attached at each vertex:

$$I(p^2)\;=\;\int d^d l_1\, d^d l_2\, d^d l_3\, d^d l_4\;\frac{1}{\big(l_1^2-M_1\big)\big(l_2^2-M_2\big)\big(l_3^2-M_3\big)\big(l_4^2-M_4\big)\,\big((l_1+l_2+l_3+l_4-p)^2-M_5\big)}\,.$$

It depends on a single kinematic variable, $t=p^2$, the squared momentum flowing in one vertex and out the other, together with the five internal masses-squared $(M_1,\ldots,M_5)$. Its maximal cutthe integral with every internal line put on shell; it solves the homogeneous part of the diagram's differential equation and carries the diagram's geometry is the period of a Calabi–Yau threefolda complex three-dimensional Calabi–Yau manifold — one complex dimension above the banana's K3, two above the elliptic sunrise: as $t$ varies the threefold varies, and the holomorphic period $\varpi_0(t)$ — the unique power-series solution of the diagram's order-six Picard–Fuchs operatorthe linear differential operator in $t$ that annihilates the maximal-cut period, normalized to $1$ at the MUM pointthe point of maximal unipotent monodromy, the standard expansion point for Calabi–Yau periods, here at $p^2\to\infty$ — is the function on which banana analyses at this loop order are built. Everything on this page happens at the level of that period, exact in $t$ with no $\varepsilon$-expansion anywhere: the closed objects are the exact local data of $\varpi_0$ at the threshold, on the one-variable slice with the masses frozen.

Tune the masses-squared to $(1,1,1,1,16)$ — so that $\sqrt{M_5}=4=1{+}1{+}1{+}1$ — and the heavy line sits at the threshold coalescencethe kinematic tuning $\sqrt{M_{L+1}}=\sum_i\sqrt{M_i}$, where the heavy line sits exactly on the normal threshold of the others and the threshold collides with $p^2=0$. At that tuning the order-six Picard–Fuchs operator picks up a pair of quarter-integer roots in its indicial polynomialthe polynomial whose roots $\rho$ are the allowed leading powers $t^\rho$ of solutions at the singular point $t=0$:

$$3057647616\,(\rho-2)^2(\rho-1)^2(4\rho-7)(4\rho-5)\;\Longrightarrow\;\{\,1,1,2,2,\,\tfrac54,\,\tfrac74\,\}.$$

The $\tfrac54$ and $\tfrac74$ exponents give monodromythe linear map on the solution space induced by carrying the integration contour once around a singular point and back eigenvalues $e^{5\pi i/2}=+i$ and $e^{7\pi i/2}=-i$: an order-four semisimple block. Walk once around the threshold and two of the six period solutions rotate by $\pm i$; four times around brings them back. The numbers computed and given in closed form here are the two connection coefficients $c_{5/4}$ and $c_{7/4}$ with which $\varpi_0$ projects onto the fractional branches $\Phi_{5/4}=t^{5/4}(1+O(t))$ and $\Phi_{7/4}=t^{7/4}(1+O(t))$ — displayed in The result — together with the exact indicial data above, the unipotent control at the nearby non-threshold point $(1,1,1,1,9)$, and the monodromy ladder through CY₄. This is the CY₃ lift of the K3 rung solved earlier.

Why it matters

The banana tower is the standard home of Calabi–Yau geometry in the Feynman-integral literature, and the standard analyses of its periods — the all-loop treatments of arXiv:2008.10574 and arXiv:1912.06201 — are organized around the MUM point, where the monodromy is unipotent by construction: circling the singular point shifts the answer by logarithms and nothing worse. A finite particle-production threshold is a different kinematic locus. The K3 rung established that at the threshold coalescence $(1,1,1,9)$ the three-loop banana's monodromy is genuinely non-unipotent (a half-integer indicial root, an order-two eigenvalue), and with it comes a new transcendental that no logarithmic ansatz reaches. This entry establishes that the phenomenon survives the climb to CY₃, and that the kind of transcendental changes with the rung.

On the K3 the only fractional indicial gap was $\tfrac12$, so $\Gamma(\tfrac12)=\sqrt\pi$ supplied the one new constant and $c_{3/2}$ lived in the rational ring $\mathbb{Q}(\sqrt3,\pi^{-1})$. On the CY₃ the gaps are $\{\tfrac14,\tfrac12,\tfrac34\}$, and the Barnes-type connection coefficientthe constant relating two local Frobenius solutions across a fractional indicial gap; built from products and ratios of $\Gamma$ at the exponent differences pulls in $\Gamma(\tfrac14)$ — the lemniscatic constant$\Gamma(\tfrac14)=3.6256\ldots$, the period of the lemniscate $r^2=\cos2\theta$ and the first "named" $\Gamma$-value not expressible in $\pi$ and square roots. Here $\Gamma(\tfrac14)$ enters the banana ladder as a local connection coefficient of the holomorphic period itself rather than as a boundary value.

The ladder prediction behind this page was falsifiable, and it nearly got falsified in a useful way. The threshold monodromy order runs $4,2,4,2,\ldots$ up the tower (period two in $L-1$, where $L$ is the loop order), fixed by exact indicial data through CY₄. A naive single-weight reading had forecast a unipotent collapse at the CY₄ rung; the exact order-eight indicial polynomial there, computed on a two-mass slice of the period, refuted the forecast, carrying a lone half-integer $\tfrac32$ exactly as the corrected counting demands. Beyond CY₄ the ladder, and with it the persistence of the non-unipotency, is conjectural.

It also sharpens the function-space lesson. The two non-unipotent branches $\Phi_{5/4}$, $\Phi_{7/4}$ cannot be matched by any combination of products of holomorphic CY₃ periods — the coefficients have to be carried to the threshold along the differential equation itself — and the closed forms below show why: the answer lives in a ring no expansion around the MUM point spans.

What was hard

Near the threshold the transported period is dominated by the unipotent logarithm tower, whose components are roughly $10^4$ times larger than the fractional admixture, so a direct linear solve fishes the answer out of subtraction noise. The K3 rung had hit the same wall, and its cure — spectral-projector extractionbuild the idempotent onto a chosen eigenspace as a polynomial in the local monodromy matrix $M_0$, so the extraction is exact algebra rather than an ill-conditioned linear solve — lifts cleanly. Here $M_0$ is the $6\times 6$ matrix by which one circuit of $t=0$ acts on the six-dimensional solution space, and $\mathbb{1}$ is the identity; with eigenvalues $\{1,1,1,1,+i,-i\}$ the rank-four unipotent block is killed by $(M_0-\mathbb{1})^4$, and the two semisimple branches are split by a single linear factor each:

$$P_{+i}=\tfrac{i}{8}(M_0-\mathbb{1})^4(M_0+i\mathbb{1}),\qquad P_{-i}=-\tfrac{i}{8}(M_0-\mathbb{1})^4(M_0-i\mathbb{1}),$$

so $P_{\pm i}\cdot\varpi_0=c_{5/4,\,7/4}\,\Phi_{5/4,\,7/4}$ with no $6{\times}6$ solve, and with every component of the transported state required to return the same coefficient as a running consistency check.

Second, no external evaluation of this integral was available to compare against: a direct numerical computation of the unequal-mass four-loop banana at this point exceeded the available hardware. The validation was therefore built from two transport routes engineered to share nothing but the Picard–Fuchs operator — different language, arithmetic, coordinate chart, series seed, and monodromy contour — with the operator itself reconstructed exactly over the rationals (its residual on the maximal-cut period series is identically zero), the exact product identity below as an internal cross-check, and the known K3 value $c_{3/2}=-\sqrt3/(36\pi)$ reproduced as a positive control on the same code path.

The family has since been run as a Feynman family as well: an integration-by-parts reduction of the $(1,1,1,1,16)$ family at symbolic dimension (nine master integrals, seven in the top sector) gives a maximal-cut operator whose $d=2$ form, of order six, is the Picard–Fuchs operator of the period exactly, up to a shift of the Euler operator by one unit. The connection coefficients recomputed from that operator share their first 45 digits (for each of $c_{5/4}$ and $c_{7/4}$) with the recorded values at the two-precision floor, 75 at the higher precision alone. The physical master is finite at $d=2$ at two AMFlow points, its two goals at the regular point agreeing to 62 digits on every coefficient, and its fractional threshold coefficients at $d=2$ (the $5/4$ and $7/4$ branches) are bounded by $10^{-23}$ with the log tower absent — to this precision the fractional monodromy is confined to the maximal cut. The two transport routes above remain each other's check.

The result

Throughout, $t=p^2$ is the single kinematic invariant of the slice, with the masses fixed at $(M_1,\ldots,M_5)=(1,1,1,1,16)$, and $\varpi_0(t)$ is the holomorphic period of the maximal-cut Calabi–Yau threefold, normalized to $1$ at the MUM point and transported from there to the threshold along the differential equation. Near $t=0$ the six-dimensional solution space splits into four integer-exponent Frobenius solutionsthe canonical local solutions at a regular singular point: one power series per indicial root, with logarithms where roots repeat $\Phi^{(j)}$ (exponents $\{1,1,2,2\}$, carrying the logarithms) and the two quarter-integer branches, so the transported period decomposes as

$$\varpi_0(t) \;=\; \sum_{j=0}^{3} A_j\,\Phi^{(j)}(t) \;+\; c_{5/4}\,\Phi_{5/4}(t) \;+\; c_{7/4}\,\Phi_{7/4}(t), \qquad \Phi_\alpha(t)=t^{\alpha}\bigl(1+O(t)\bigr),$$

with $A_j$ the coefficients on the integer tower and $c_{5/4}$, $c_{7/4}$ the two connection coefficients that carry the new transcendental content.

The closed forms. Both quarter-integer connection coefficients of the CY₃ threshold banana at masses-squared $(1,1,1,1,16)$ close in the lemniscatic ring of $\Gamma(\tfrac14)$ and $\pi$ — a ring no Frobenius-at-MUM logarithmic ansatz spans:

$$c_{5/4}\;=\;-\frac{1}{\sqrt{2\pi}\,\Gamma(\tfrac14)^2}\;=\;-\frac{\Gamma(\tfrac34)^2}{2\sqrt2\,\pi^{5/2}},\qquad c_{7/4}\;=\;-\frac{5\,\Gamma(\tfrac14)^2}{384\sqrt2\,\pi^{5/2}},$$

$$c_{5/4}\,c_{7/4}\;=\;\frac{5}{768\,\pi^3}\quad\text{exactly}.$$

So far as we can tell these are the first $\Gamma(\tfrac14)$-ring values closed anywhere in the banana ladder. Every other reported special value of the family sits in a different $\Gamma$-ring: the equianharmonic $\Gamma(\tfrac13)$ of the three-Bessel moment at the third singular value (Bailey–Borwein–Broadhurst–Glasser 2008) — the equal-mass two-loop sunrise; the $\Gamma(a/15)$ ring of the level-15 CM K3 motive behind the equal-mass three-loop banana — an integral this work independently reproduces, though the banana's own deeper-layer datum has no recorded numerical value and has not been matched against that ring; and the critical $L$-values of the weight-2/weight-4 Hecke eigenforms at the attractor points of this Hulek–Verrill CY₃ (Candelas–de la Ossa–Elmi–van Straten) — none lemniscatic. The complete machine-readable form and full-precision reference values live in the downloads: cy3-banana-expression.md · cy3-banana-evaluate.py. Beyond evaluating the closed forms, the evaluator carries two derivation tiers: --derive decomposes the transported period onto the full six-dimensional Frobenius basis at the threshold by one direct linear solve in 1400-bit ball arithmetic — enough precision that the subtraction noise is affordable — recovering $c_{5/4}$ and $c_{7/4}$ to over 400 digits against the closed forms with both logarithmic coefficients vanishing (about half a minute); --hankel derives $c_{5/4}$ and $c_{7/4}$ analytically from the non-oscillatory Hankel-kernel tail of the Bessel representation and checks the Bessel-moment identities behind the unipotent data, among them $\int_0^\infty y\,J_0(y)^4 J_0(4y)\,dy=0$ (about three minutes). Both tiers read only the modules served beside it — direct_linear_extract.py, threshold_hankel_tail.py, calpha_rings.py and fixtures/, the same files the K3 rung serves — and the bundle carries a MANIFEST.sha256 and a CHANGES.md.

The two equivalent forms of $c_{5/4}$ are related by the reflection identity $\Gamma(\tfrac14)\Gamma(\tfrac34)=\pi\sqrt2$, and the same identity makes the product $c_{5/4}\,c_{7/4}$ a pure rational over $\pi^3$ — the Wronskian normalisation of the conjugate $\{+i,-i\}$ pair, with $768=12\cdot(\sum_i\sqrt{M_i})^2$ keyed to the threshold scale. The $\Gamma(\tfrac14)$ appearance follows the local indicial differences $\{\tfrac14,\tfrac12,\tfrac34\}$: a Barnes-type connection across those gaps is built from $\Gamma(\tfrac14)$, $\Gamma(\tfrac34)$ and $\Gamma(\tfrac12)=\sqrt\pi$, and the K3's simpler $\sqrt3/\pi$ ring was the special case where the only gap was $\tfrac12$. Under the order-four monodromy each $c_\alpha$ picks up a factor of $\pm i$ per circuit of the threshold, so the convention-independent statement is about $|c_\alpha|$; the signs above are in the lower-branch convention $\arg t=-\pi$ used throughout.

Two exact structural facts frame the closed forms. At the nearby non-threshold point $(1,1,1,1,9)$ the indicial exponents are $\{1,1,2,2,3,3\}$ — unipotent, exactly as the MUM-based expectation demands — so the order-four rotation is a property of the threshold tuning alone. And the order-four block sits inside the exact $4,2,4,2,\ldots$ ladder through CY₄ described above: the phenomenon is structural, not an accident of this rung.

Everything on this page was validated against independent evaluations never used in the construction, to at least 71 digits.

References

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