Ice-cream cone

A two-loop elliptic integral, the textbook example in the Landau analysis of sequential discontinuities, computed here on the one-scale slice $p_{12}^2=p_{34}^2$, $p_{56}^2=-1$, at equal mass and at one set of fully distinct internal masses. The structure of its top sector in two dimensions (rank two, algebraic leading singularities, the sunrise sub-sectors entering as extensions) was worked out by Lairez and Vanhove, by Doran, Harder, Pichon-Pharabod and Vanhove, and by de la Cruz and Vanhove; what is new here is the explicit function on that slice, with high-precision values and an evaluator.

The integral

The ice-cream cone is the two-loop family

$$\mathrm{ICC}[\nu_1,\nu_2,\nu_3,\nu_4] \;=\; \int d^d k\, d^d l\; \frac{1}{D_1^{\nu_1}\, D_2^{\nu_2}\, D_3^{\nu_3}\, D_4^{\nu_4}}\,, \qquad d = 2-2\varepsilon,$$

up to an overall normalization convention, with loop momenta $k, l$ and the four massive propagators

$$D_1 = k^2 - m_1^2, \qquad D_2 = (k - P_3)^2 - m_2^2, \qquad D_3 = l^2 - m_3^2, \qquad D_4 = (k + P_1 - l)^2 - m_4^2.$$

The lines $D_1, D_2$ are the triangle "cone"; $D_3, D_4$ form the bubble sitting on its third edge, and together with a cone line they make the scoopthe three-line sunrise sub-diagram that carries the elliptic curve; named for the ball of ice cream on top of the cone: contracting a cone line collapses the diagram onto a three-line sunrise (masses $m_1, m_3, m_4$), and contracting the bubble instead leaves an ordinary one-loop bubble. The sunrise is where the elliptic curve lives. The three external momenta $P_1, P_2, P_3$ — each the total momentum of a pair of external legs, hence the invariant labels $p_{12}^2, p_{34}^2, p_{56}^2$ — sum to zero. The slice computed here keeps a single external scale $x = p_{12}^2 = p_{34}^2$ and fixes the third invariant $p_{56}^2 = -1$. Off that slice the family has since been carried along two lines: at $p_{12}^2=-2$ a nine-master connection in $p_{34}^2$ reproduces an independent evaluation to 57 digits, and at $p_{12}^2=-3$ a second such connection, seeded at one point and transported to another, reproduces an independent value there to 57 digits on all nine masters; the landing from those lines back onto the slice at $p_{34}^2=p_{12}^2$, an apparent singularity of the connection, is complete, to 47 digits at $-3$ and 48 at $-2$ against the on-slice records. The family reduces to seven master integralsthe finite basis of independent integrals to which every integral of the family reduces via integration-by-parts identities at equal mass and nineteen at generic mass, and $T(x)$ below is the top master $\mathrm{ICC}[1,1,1,1]$. Two mass configurations were closed: the equal-mass case with all four lines at $m^2 = 1$, and a generic distinct-mass case with $(m_1^2, m_2^2, m_3^2, m_4^2) = (1, 2, 3, 5)$.

The diagram earns its place in the gallery because of a fact about its singularities. A standard working assumption in Landau analysis, the strict hierarchical principle, says that a Feynman integral's leading singularities are inherited from its sub-diagrams: pinch a propagator and you should land on a singularity that was already there. The ice-cream cone violates that rule. Hannesdottir, McLeod, Schwartz and Vergu showed in arXiv:2211.07633 (JHEP 07 (2023) 236, §6.4) that its leading Landau locus carries an extra "bubble-like" branch — a configuration where all four lines go on shell while two of the Feynman parameters vanish — that no sub-diagram supplies. On the slice $p_{12}^2 = p_{34}^2 = x$, $p_{56}^2 = -1$ the cone's own leading Landau locus (HMSV Eq. 6.48, all four Feynman parameters nonzero) is the quadratic

$$L_{\rm bub} = x^2 - 10x + 5 = (x - r_+)(x - r_-), \qquad r_\pm = 5 \pm 2\sqrt{5},$$

the point where the scoop's spectral mass reaches its threshold, equivalently the Landau locus of the triangle with masses $(m, m, 2m)$. It belongs to the top sector alone and is absent from the sunrise alphabet $\{x,\, x-1,\, x-9\}$. HMSV's bubble-like singularity, the branch that violates the strict hierarchical principle, sits at $p_{56}^2 = (m_1+m_2)^2$, off this slice. At generic distinct masses the same mechanism gives the irreducible quartic $L_{\rm bub} = x^4 - 40x^3 + 348x^2 - 880x + 196$.

Why it matters

The ice-cream cone has been the textbook test case for the hierarchical principle since the principle was first written down. Landshoff, Olive and PolkinghorneP. V. Landshoff, D. I. Olive, J. C. Polkinghorne, Nuovo Cimento A 43 (1966) 444 introduced the hierarchy principle in 1966; this diagram became the textbook counterexample and was singled out in this strict form by HMSV 2022. BoylingJ. B. Boyling, Nuovo Cimento A 53 (1968) 351 gave the principle a homological formulation two years later. The modern resolution, due to HMSV, is to replace the strict form with a Pham-refined weak formdrop the requirement that all Feynman parameters of the dominating sub-diagram be nonzero: a branch point may sit on any branch of a dominating diagram's Pham locus, on-shell and Landau loop equations only: on Pham loci the principle is restored, because the offending singularity is recognised as a branch of the cone's own Pham locus rather than a lower-order singularity that switched on out of nowhere.

The strict hierarchy fails here and the Pham-refined weak form admits the extra branch, but nobody had the closed function to read either statement off directly. The new content is the cone's own Landau letterthe leading Landau locus of the full diagram, present in no sub-diagram $L_{\rm bub}$the quadratic $x^2-10x+5$ (equal mass) or quartic $x^4-40x^3+348x^2-880x+196$ (generic mass): the cone's own leading Landau locus on the slice, all four lines on shell with all four Feynman parameters nonzero. HMSV derived it in 2022 from the Landau equations alone (their Eq. 6.48). This calculation exhibits that predicted Landau singularitya value of the external kinematics at which internal propagators can go simultaneously on shell and pinch the integration contour; the places a Feynman integral can develop branch points inside the closed function itself: $L_{\rm bub}$ is the exact denominator of an integration-by-parts reduction coefficient, sits in the top sector of the differential equation and nowhere else, and appears as the last-entry symbol letter of the weight-3 answer. The top-down singularity prediction and the bottom-up function now agree.

The same demonstration runs at generic distinct masses. There the quartic $L_{\rm bub}$ again localises to the top sector only, and the scoop curve is the Adams–Bogner–Weinzierl distinct-mass sunrise quartic with no ice-cream-cone input at all. So the cone's own Landau letter is a property of the full diagram across the whole mass space, exactly as the Landau analysis said it should be.

What was hard

First, the function space: at grade zero the top sector closes on a second-order operator whose two solutions are algebraic, $y_\pm(x) = [\tfrac12(2x^2+8x-5 \pm \sqrt5\,(2x-1)\sqrt{4x+1})]^{-1/2}$, and the sunrise enters only through the source of that equation (the third-order operator of de la Cruz and Vanhove (arXiv:2401.09908) factorises as a first-order operator times this second-order one) — so the answer is not polylogarithmic but one fold of the sunrise period and its derivative against algebraic kernels, an elliptic polylogarithm of length two, and the boundary constants that anchor it match nothing in the certified constant rings, so fitting them was never an option: they had to be derived, from analytic conditions at $x = \infty$ and, independently, from an explicitly written parametric integral of the graph built with no reduction machinery at all. Second, the cone's leading Landau locus had to be exhibited exactly, not just numerically. The HMSV Landau prediction was reproduced analytically (their eq. 6.48), and then confirmed by an entirely separate route: the exact integration-by-parts reduction of the once-dotted top integral onto the top master produces the coefficient

$$\frac{(2-d)\,x + d - 3}{x^2 - 10x + 5}\,,$$

with $L_{\rm bub}$ in the denominator as a matter of exact algebra. It appears only in the top sector of the connection matrixthe matrix $A(d,x)$ in the first-order system $dM/dx = A\,M$ obeyed by the master integrals; its pole structure encodes the symbol alphabet; the scoop block, the equal-mass two-loop sunrise on $\Gamma_1(6)$the congruence subgroup of $\mathrm{SL}_2(\mathbb{Z})$ whose modular curve parametrises the equal-mass sunrise elliptic family, with cusps at $0,1,9,\infty$, never sees it, and the same localisation holds at generic mass. Third, the generic-mass case leaves the modular world entirely: its scoop curve sits on no congruence subgroup, so no ready-made catalogue of functions or constants exists for it — the construction had to run on the period pair directly.

The result

The ice-cream cone is closed at both equal mass ($m^2=1$) and generic distinct mass ($m^2=1,2,3,5$), at weight 3 ($\varepsilon^0$) and weight 4 ($\varepsilon^1$), as an explicit variation-of-parameters word over an algebraic kernel pair, with the period pair of the scoop's elliptic curve entering through the source.

Equal mass. Throughout, $x$ is the one scale left floating on the slice and $T(x)$ is the top master $\mathrm{ICC}[1,1,1,1]$ at $\varepsilon^0$. Eliminating the once-dotted top from the differential equation leaves a second-order operator

$$L_T = \partial_x^2 + \tilde p_1\,\partial_x + \tilde p_0, \qquad \tilde p_1 = \frac{14x^5-155x^4+520x^3-316x^2+22x-5}{(x-1)(4x+1)(x^2-10x+5)(x^2-4x+1)}, \quad \tilde p_0 = \frac{6(x^4-8x^3+24x^2-4x+3)}{(x-1)(4x+1)(x^2-10x+5)(x^2-4x+1)},$$

whose two solutions are algebraic,

$$y_\pm(x) = q_\pm(x)^{-1/2}, \qquad q_\pm = \tfrac12\big(2x^2+8x-5 \pm \sqrt5\,(2x-1)\sqrt{4x+1}\big), \qquad q_+ q_- = (x-1)^2 (x^2-10x+5).$$

The top obeys $L_T[T] = \alpha S + \beta S' + g_1 - g_t\,\mathrm{Tri}_0$ with $S$ the weight-2 scoop sunrise, $\mathrm{Tri}_0 = 4\ln\varphi/\sqrt5$ the $d=2$ bubble at $p_{56}^2$, and the exact rational kernels

$$\alpha = \frac{-8(2x^4-9x^3-15x^2-25x-15)}{3(x-9)(x-1)(4x+1)(x^2-10x+5)(x^2-4x+1)}, \quad \beta = \frac{-2(11x^5-63x^4+48x^3-340x^2+165x-45)}{3(x-9)(x-1)(4x+1)(x^2-10x+5)(x^2-4x+1)},$$

$$g_1 = \frac{8}{(x-9)(x-1)(x^2-10x+5)}, \qquad g_t = \frac{2(2x^3-27x^2+18x-1)}{(x-1)(4x+1)(x^2-10x+5)(x^2-4x+1)}.$$

The answer is the variation-of-parameters word over the algebraic pair:

$$T(x) = c_+\,y_+(x) + c_-\,y_-(x) + \int_{-1}^{x} \frac{y_-(x)\,y_+(x') - y_+(x)\,y_-(x')}{W(x')}\,\big(\alpha S + \beta S' + g_1 - g_t\,\mathrm{Tri}_0\big)(x')\,dx', \qquad W = y_+ y_-' - y_+' y_-,$$

with the closed weight-2 scoop source $S(x) = A\,\psi_1(x) + B\,\psi_2(x) + 6\,P(x)$, $A = -2.4880163439\ldots$, $B = +0.3371299225\ldots$, where $P(x)$ is a fixed particular solution of the scoop's inhomogeneous equation and the source constant $6$ is exact. Every per-word object is explicit — the kernels $\alpha, \beta, g_1, g_t$, the algebraic pair $y_\pm$, the sunrise period pair in the source, and the $d\log$ alphabetthe finite list of factors whose logarithms build the iterated integrals: the complete set of places the answer can be singular $\{x, x-1, x-9, 2x+1, 4x+1, L_{\rm bub}\}$ — and the cone's own letter $L_{\rm bub} = x^2-10x+5$ occupies the last-entry slot, exactly where the Landau analysis of the full diagram put it. For Euclidean $x < -1/4$ the two kernels are complex conjugates and $T$ is real. Written through the scoop's spectral mass $s$, the whole $\varepsilon^0$ top is the one-fold integral of logarithms

$$T(x) = 2\int_4^\infty \frac{ds}{\sqrt{s(s-4)}}\;\frac{2(c-2s)\,B(s;x) + (2c-5)\,\mathrm{Tri}_0}{2c^2-10s}, \qquad c = s+1-x, \qquad B(s;x) = \int_0^1 \frac{da}{a + s(1-a) - x\,a(1-a)},$$

which reproduces the stored 330-digit $T(-1)$ to its full length in seconds (the same form at masses $(1,2,3,5)$ reaches 209 of the 210 stored digits); its $\mathrm{Tri}_0$ term is elementary (residues at the cone's Landau points $s_\pm(x)$) and the rest is a length-two elliptic polylogarithm on the scoop curve with third-kind punctures at $s_\pm(x)$. The same construction one order deeper closes the weight-4 coefficient $T^{(1)}(x)$. The two boundary values

$$T(-1) = 0.6098423912\ldots, \qquad T^{(1)}(-1) = -0.6775187338\ldots$$

fix $c_+, c_-$ at each weight (full-precision strings in the downloads), and both are derived, not fit, by two independent routes that agree. The first is analytic matching at $x=\infty$: no half-integer power $x^{-3/2}$ can survive the expansion of a Feynman integral, and matching onto the hard bubble sets the constant log slot $\tau_0 = 0$ — the differential equation then reproduces the bubble physics on its own, a resonance forcing the log coefficient $\tau_1 = 8\ln\varphi/\sqrt5$ with no further input, $\varphi$ the golden ratio. The second is a self-contained, explicitly written convergent parametric integral of the graph, with no reduction machinery anywhere in its construction. No finiteness assumption is imposed. The explicit constants of the derivation chain live in the golden-ratio dilogarithm ring built on $\mathrm{Tri}_0 = 4\ln\varphi/\sqrt5$ and

$$J = \int_0^1 \frac{\ln(1+a-a^2)}{1+a-a^2}\,da = \frac{2}{\sqrt5}\Big(\ln 5\,\ln\varphi + \mathrm{Li}_2\big(\tfrac{5-\sqrt5}{10}\big) - \mathrm{Li}_2\big(\tfrac{5+\sqrt5}{10}\big)\Big),$$

the bubble's weight-3 constant closing as $2\gamma_E\,\mathrm{Tri}_0 + J$ and its weight-4 companion as $-\bigl((2\gamma_E^2+\zeta_2)\,\mathrm{Tri}_0 + 2\gamma_E J + \tfrac12 K\bigr)$, with $K = \int_0^1 \frac{\ln^2(1+a-a^2)}{1+a-a^2}\,da$. The boundary constants $c_\pm$ themselves match nothing in the certified weight-3 constant ring under integer-relation search: the value at $x=-1$ of a length-two elliptic polylogarithm with punctures at the cone's Landau points, integer-relation-negative against the searched rings at the stated heights, with no closed form claimed for them.

Generic distinct mass, $m^2 = (1,2,3,5)$. At generic mass the top sector is again rank two with algebraic (Liouvillian) solutions (Lairez–Vanhove; Doran–Harder–Pichon-Pharabod–Vanhove), the $(1,3,5)$ scoop and the $(2,3,5)$ sub-sunrise entering through the source; our 19-master elimination there is third-order and we have not reduced it to the second-order form, so the generic result is delivered as the transported connection with its derived boundary data. The scoop curve is

$$Q_{\rm sc} = x^4 - 36x^3 + 302x^2 - 564x + 121 = \prod_\pm\bigl(x - (\pm 1 \pm \sqrt{3} \pm \sqrt{5})^2\bigr).$$

The top master's self-coupling and the scoop→top coupling in the 19-master differential equation are the explicit rationals

$$A_{\rm top} = \frac{-270x^4 + 2706x^3 - 8658x^2 + 10676x - 4004}{(4x+1)\,(x^4 - 40x^3 + 348x^2 - 880x + 196)},$$

$$c_S = \frac{-192x^9 + 8550x^8 - 118439x^7 + 691024x^6 - 1762968x^5 + 1499645x^4 + 880795x^3 - 1000411x^2 - 61114x + 58410}{(2x+1)(4x+1)\,(x^4 - 40x^3 + 348x^2 - 880x + 196)\,(x^4 - 36x^3 + 302x^2 - 564x + 121)},$$

both carrying the cone's own quartic $L_{\rm bub} = x^4 - 40x^3 + 348x^2 - 880x + 196$ in their denominators; $L_{\rm bub}$ divides denominators only in the top sector of the system and is absent from both scoop blocks. The boundary value $T(-1) = 0.1348970572\ldots$ fixes $c_+, c_-$, by the same $x=\infty$ matching with the hard-bubble slot $\tau_0 = \ln 15 \cdot \mathrm{Tri}_0^{\rm gen}$, $\mathrm{Tri}_0^{\rm gen} = \ln(1+\sqrt2)/\sqrt2$. The $(1,3,5)$ curve sits on no congruence subgroup, so there is no reference ring of known constants to compare against; $c_+$ and $c_-$ are given as convergent period integrals, with no closed form claimed. At generic mass the downloadable script replaces 21 of the 46 stored boundary reference values by derived closed forms and written-out integrals and recomputes the other 25 at runtime on its default run — seven exact rationals, eight elementary closed forms, eight resolved at the boundary (six by four derived resonance slots at the infinity cusp, two by a coupled two-pass solve), and two recomputed directly at the reference precision — so the stored strings serve only as checks, which the recomputed values match to a floor of 109 digits, limited by the length of the stored strings. The stored evaluations are now carried to 210 digits, and the served evaluator checks its transported value at $x=-2$ against an independent evaluation there to 118 digits (the row's gate) and its recomputed $T(-1)$ against the stored string to 133. The derivation itself has since been carried through at arbitrary rational masses in a second script, icc-genmass-evaluate.py (the evaluator above unchanged), the reference masses recovered as a control; at a second mass set never used in the construction, $m^2=(1,3,2,7)$, the derived boundary value at $x=-2$, recorded before any independent number existed at these masses, agrees with an AMFlow evaluation at two working precisions (agreeing with itself to at least 84 digits on all nineteen masters) to at least 86 digits — the independent pair's own floor, so the count is that evaluation's precision rather than a disagreement; on the served script the value reproduces the AMFlow number to 98 digits at working precision 108. A third mass set, $m^2=(1,\tfrac52,\tfrac12,3)$, is checked the same way: an AMFlow evaluation at two working precisions, agreeing with itself to 85 digits on all nineteen masters, reproduces the derived value at $x=-2$ to 51 digits, the evaluator's own floor at 60 working digits.

There is no compact closed form for the boundary constants, and none is claimed: the deliverable is the explicit word above — exact rational kernels, the period pair, and derived (equal mass) or stored-and-documented (generic mass) boundary constants — together with a script that recomputes the whole chain from scratch at any point on the slice.

Everything on this page was validated against independent evaluations at kinematic points never used in the construction, to at least 45 digits; the equal-mass evaluator at its default precision agrees with the independent value at $x=-2$, never used in the construction, to 134 digits (the row gate; the script's default run prints 134.9 d), and at working precision 260 against the full 330-digit record to 260 digits (icc-evaluate.py --config eqmass --dps 260, minutes-class on one core). MANIFEST.sha256 lists every file of the bundle with its checksum.

Off the symmetric slice $p_{12}^2 = p_{34}^2$ on which the closed forms above live, the cone's nine-master system has also been transported along $p_{34}^2$ on two lines, $p_{12}^2 = -2$ and $p_{12}^2 = -3$, from a seed point to a target point with the exact rational connection of the off-slice basis; at the targets the twelve $p_{34}^2$-dependent master-and-order pairs on each line agree with independent evaluations to 57 digits — and at order $\varepsilon^1$, transported from seeds computed one order deeper, to 47 digits on both lines (47.56 and 47.41; the independent evaluations at the two seeds agree with their higher-precision twins to 48.11 and 48.02) — and the six pairs of the three $p_{34}^2$-independent masters, carried unchanged along the line, agree with the on-slice records and the closed-form tadpole and sunrise values to 57 digits at worst; the seed value at $(-2,-3,-1)$ that the first line starts from agrees between two independent evaluations at different working precisions to 57 digits. The step from those lines back onto the slice is now taken: at the point $p_{34}^2 = p_{12}^2$, where the nine-master basis degenerates to the seven of the slice, the connection has an apparent singularity (integer exponents, no logarithms), and the mean of the transported solution on a small circle around it lands the six slice-facing masters onto the slice — at $-3$ to 47 digits and at $-2$ to 48 digits against the on-slice records (measured 47.92 and 48.08); the seventh slice master is not in the off-slice basis and is not landed. icc-offslice-evaluate.py runs both lines and, with --land, the landing; land_fixed_eps.py beside it performs the landing and re-measures the agreement.

The connection in both variables at once, $A_{p_{12}^2}$ and $A_{p_{34}^2}$ as exact rational $9\times 9$ matrices (integrable: $dA = A\wedge A$ vanishes identically; each line's connection is its restriction), has since been computed from a Kira reduction at symbolic $(p_{12}^2, p_{34}^2)$ and used to transport the nine masters from the $(-3,-4)$ seed to three points never solved before — $(-3,-15/4)$ along the $p_{12}^2=-3$ line, $(-5/2,-15/4)$ off both lines, and the mirror point $(-15/4,-5/2)$ on a complex detour around the diagonal $p_{12}^2=p_{34}^2$ — where the $\varepsilon^{-2}\ldots\varepsilon^{1}$ coefficients agree with independent evaluations to 64, 64 and 48 digits at worst (measured 64.40, 64.36 and 48.12, transported from the goal-60 seed; the third point's only independent value is a goal-40 one, whose depth is the limit there; the goal-40 seed gave 48, 47 and 47; the $\varepsilon^0$ rows to 64, 64 and 54). icc-evaluate.py --offslice X Y runs that transport to any point of the Euclidean region reachable by straight segments from the seed (a point whose route would cross a census curve is refused by name), with the connection, engine and reference values beside it under vendor_rows1517_offslice/xy/. On the slice, --land carries the seeds along the real axis to the point $x=-1/2$ and lands the value there as the mean on a small circle around it (129.5 digits at working precision 140 against the record under vendor_row15_landing/), and --band evaluates the explicit parametric integral for the top master on $-1/2\le x<-1/4$, where it agrees with the connection continued around $2x+1=0$ to 50 digits at the points probed; the interval $-1/4\le x\le 0$ is not served by any transport arm.

Downloads: icc-expression.md (the full construction, both mass configurations, with the full-precision boundary strings) · icc-evaluate.py (recomputes the answer from scratch, deriving the equal-mass boundary constants at runtime, and re-measures every agreement; the default run reproduces the equal-mass result at thirty digits in about two minutes on one processor core, and --full recomputes both mass configurations at full precision, which takes tens of minutes; the equal-mass slice is covered on the Euclidean axis $x<-\tfrac12$ by transport and in the Minkowski region $x>0$ by the $i0^+$ continuation of the same connection — --minkowski 12 transports along a complex detour above every real singular point and lands at $x=12$ above every threshold, where it reproduces an independent physical-region evaluation to 57 digits over all fourteen components including the imaginary parts (the transport's own pair agrees to 59); the threshold points themselves are refused by name; --onefold X instead evaluates the equal-mass $\varepsilon^0$ top master directly as the one-fold integral of logarithms above, at any real $x<9$ and in seconds, reproducing the stored 330-digit $T(-1)$ to its full length at working precision 345 in about half a minute; --config generic does the same at masses $(1,2,3,5)$) · icc-genmass-evaluate.py (the same derivation at any rational masses, with the $(1,3,2,7)$, $(1,2,3,5)$ and $(1,\tfrac52,\tfrac12,3)$ reference values shipped beside it — the equal-mass evaluator names it when asked for other masses, and each script checks that the other beside it is the one it was served with; the default run reproduces the $(1,3,2,7)$ value at thirty digits in about half a minute, and --dps 60 reaches the 51-digit agreement in about a minute and a half; --masses 1,5/2,1/2,3 --dps 60 does the same for the third set; past 100 working digits, where no boundary value is shipped and the script derives it in place, the order of that derivation now grows with the requested precision so the script's own consistency checks keep pace, a --dps 130 run taking about 36 minutes) · icc-offslice-evaluate.py (the two lines off the symmetric slice with their connections, seeds and reference values in vendor_rows1517_offslice/; the default run transports both lines at 60 digits in about two and a half minutes; --kmax 1 carries the transport one order further, to $\varepsilon^1$, from seeds computed one order deeper, and checks it against independent values at the targets, about four minutes per line)

References

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