Equal-mass ice-cream cone

The two-loop ice-cream cone, a triangle with a bubble on one edge, at equal masses on a one-scale slice. Vanhove and collaborators showed that in two dimensions its homogeneous solutions are algebraic and the sunrise sub-diagram's elliptic curve enters only through the inhomogeneous term. Its finite part and the next term in the dimensional regulator are new on this slice.

The integral

The ice-cream cone is the two-loop family

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

up to overall normalization, 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 cone and $D_3, D_4$ form the bubble on its third edge, the scoopthe two-line bubble: the ball of ice cream sitting on the cone. Contracting the cone line $m_2$ collapses the diagram onto a sunrise with masses $(m_1, m_3, m_4)$, the scoop sunrise; contracting $m_1$ instead gives $(m_2,m_3,m_4)$, and at equal mass the two coincide. The three external momenta $P_1, P_2, P_3$ sum to zero, and their squares are the invariants labeled in the figure, $P_1^2=p_{34}^2$, $P_2^2=p_{12}^2$, $P_3^2=p_{56}^2$; $P_3$ enters at the vertex where the two cone lines meet and $P_1$ at the vertex where the line $m_1$ meets the bubble. The slice computed here keeps one external scale $x = p_{12}^2 = p_{34}^2$, fixes $p_{56}^2 = -1$, and sets all four internal masses equal, $m^2=1$, so that every invariant is measured in units of the internal mass. On this slice the family has seven master integralsa finite set of integrals to which every integral of the family reduces by integration-by-parts identities. The quantity computed is $T(x)$, the order-$\varepsilon^0$ coefficient of the top-sector masterthe master integral with every propagator of the diagram present once $[1,1,1,1]$, which has weighttranscendental weight: the number of iterated integrations (logarithms, periods) the function is built from three, together with its order-$\varepsilon^1$ coefficient $T^{(1)}(x)$ of weight four.

Landshoff, Olive and Polkinghorne stated the hierarchical principle for the singularities of Feynman integrals in 1966 and Boyling gave it a homological form in 1968; in its strict form, every singularity an integral inherits from a sub-diagram is a leading singularity of that sub-diagram, with all its Feynman parameters nonzero. The ice-cream cone is the standard counterexample to the strict form: its Landau locus has a bubble-like branch on which two of the four Feynman parameters vanish, and the weak form of the principle drops the requirement of nonzero Feynman parameters. The slice $p_{56}^2=-1$ misses that branch, which sits at $p_{56}^2=(m_1+m_2)^2$, but crosses the cone's own leading Landau locus restricted to the slice, with all four lines on shell and all four Feynman parameters nonzero:

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

the locus on which the invariant mass flowing through the scoop reaches its two-particle threshold, equivalently the Landau locus of the one-loop triangle with the same external momenta and internal masses $(m, m, 2m)$. This singularity is not a singularity of any sub-diagram, and $L_{\rm bub}$ is not among the letters $\{x,\, x-1,\, x-9\}$ of the sunrise $d\log$ alphabetthe finite list of factors whose logarithmic derivatives build the iterated integrals: the complete set of places the answer can be singular. With lightlike legs in two dimensions the same graph is the two-loop member of the ice cone family of Duhr, Klemm, Nega and Tancredi, where it is algebraic rather than elliptic.

The structure of the top sector in $d=2$ was established by Lairez and Vanhove (2023), by Doran, Harder, Pichon-Pharabod and Vanhove (2024) and by de la Cruz and Vanhove (2024): the maximal cut of the top sector has rank two with algebraic leading singularities, and the elliptic curves of the sunrise sub-sectors enter only through the inhomogeneous terms. The third-order operator of de la Cruz and Vanhove for the top integral factorizes as a first-order operator times a second-order one with algebraic solutions, so $T$ is one integration of the scoop sunrise and its derivative against algebraic kernels: a length-two elliptic polylogarithm on the scoop curve, and not a multiple polylogarithm. At equal mass that curve is the modular curve of $\Gamma_1(6)$ with cusps at $x\in\{0,1,9\}$, the curve of the equal-mass sunrise. The letter $L_{\rm bub}$ first appears in the integration-by-parts reductionthe linear relations among integrals of one family that follow from integrating a total derivative; they reduce every integral to the master integrals: the top-sector integral with one propagator squared (the dotted top integral) reduces onto $[1,1,1,1]$ and sub-sector integrals, and the coefficient of $[1,1,1,1]$ is

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

In the differential equation $L_{\rm bub}$ appears in the denominators of the top-sector rows only.

Closed form

Eliminating the dotted top integral from the differential equation leaves a second-order operator acting on $T$,

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

whose two homogeneous solutions are algebraic,

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

Besides the factors $(x-1)$ and $x^2-10x+5$ of $q_+q_-$, the denominators of $\tilde p_0$ and $\tilde p_1$ contain $4x+1$, where $L_T$ has local exponents $\{0,\tfrac12\}$, and $x^2-4x+1$, whose zeros $x=2\pm\sqrt3$ carry exponents $\{0,2\}$ and are not letters of the $d\log$ alphabet. The top integral obeys $L_T[T] = \alpha S + \beta S' + g_1 - g_t\,\mathrm{Tri}_0$, where $S(x)$ is the order-$\varepsilon^0$ scoop sunrise $[1,0,1,1]$, $\mathrm{Tri}_0 = 4\ln\varphi/\sqrt5$ is the $d=2$ bubble at $p_{56}^2=-1$ with $\varphi$ the golden ratio, $g_1$ and $g_t$ are the rational functions

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

and $\alpha$, $\beta$ are rational functions with denominator $3(x-9)(x-1)(4x+1)(x^2-10x+5)(x^2-4x+1)$ whose exact form is in the expression file inside the bundle under Evaluator.

Variation of parameters over the algebraic pair then gives the result,

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

with $W$ the Wronskian of the pair and $c_\pm$ two constants fixed by $T(-1)$ and $T'(-1)$. The scoop source is $S(x) = A\,\psi_1(x) + B\,\psi_2(x) + 6\,P(x)$. Here $\psi_1, \psi_2$ are the periodsthe two basic integrals of the holomorphic differential around the cycles of the elliptic curve; as functions of $x$ they solve a second-order linear differential equation, the curve's Picard–Fuchs equation of the scoop curve: $\psi_1$ is the period holomorphic at the cusp $x=0$, and $\psi_2$ is the second solution of the curve's Picard–Fuchs equation, fixed by $(\psi_2,\psi_2')=(0,1)$ at $x=-1$. The function $P(x)$ is the particular solution of that equation with unit source that vanishes together with its derivative at $x=-1$. Since $\psi_2$ and $P$ vanish at the boundary point, $A = S(-1)/\psi_1(-1) = -2.488016343\ldots$ and $B = S'(-1)-A\,\psi_1'(-1) = +0.3371299225\ldots$. The $d\log$ alphabet is $\{x,\, x-1,\, x-9,\, 2x+1,\, 4x+1,\, L_{\rm bub}\}$. The cusp letters $x$, $x-1$ and $x-9$ enter through the period $\psi_1$ in the source. The letters $2x+1$ and $4x+1$ come from the external Gram determinants; their zeros $x=-1/2$ and $x=-1/4$ are apparent (removable) singular points of the $d$-dimensional system, and $4x+1$ also appears under the square root in $y_\pm$. The cone's own letter $L_{\rm bub} = x^2-10x+5$ appears only in the outermost integration, through the branch points $r_\pm$ of $y_\pm$, where $L_T$ has local exponents $\{-\tfrac12,0\}$. For Euclidean $x\lt-1/4$ the two solutions $y_\pm$ are complex conjugates of each other, as are $c_\pm$, and $T$ is real.

Replacing the bubble by a single line of squared mass $s$ and integrating over $s$ from its threshold turns the diagram into a one-loop triangle with squared masses $(1,1,s)$, and $T$ becomes a one-fold integral of logarithms,

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

with $\mathrm{Bub}(s;x)$ the $d=2$ bubble of squared masses $(1,s)$ at momentum squared $x$. The zeros $s_\pm(x)$ of $2c^2-10s$ are the Landau points of this triangle, and $L_{\rm bub}(x)=0$ is the condition that one of them reaches the endpoint $s=4$. The $\mathrm{Tri}_0$ term integrates in closed form by residues at $s_\pm(x)$; the remainder is a length-two elliptic polylogarithm on the scoop curve whose kernels have simple poles (differentials of the third kind) at $s_\pm(x)$. This representation needs no boundary constant and converges for real $x$ below the threshold $x=9$.

The boundary values

The boundary values at weights three and four are

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

Both are derived by analytic matching at $x=\infty$. In the variable $u=1/x$ the operator $L_T$ has indicial exponents $\{1,\tfrac32\}$. The large-$x$ expansion of a Feynman integral contains only integer powers of $u$ times logarithms, so the $u^{3/2}$ branch is absent; matching onto the region where the bubble carries the large momentum sets the constant of the $u^1$ branch to zero at equal mass. The exponent $1$ is resonant with the source, so the equation itself fixes the coefficient of $u\ln u$ to be $2\,\mathrm{Tri}_0 = 8\ln\varphi/\sqrt5$ with no further input. The constants of the matching lie in the ring of logarithms and dilogarithms of the golden ratio built on $\mathrm{Tri}_0$ and

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

In the normalization of the master basis, which carries a factor $\Gamma(\varepsilon)\Gamma(1+\varepsilon)$, the bubble master formed by the two cone lines at $p_{56}^2=-1$ begins at order $\varepsilon^{-1}$ with coefficient $-\mathrm{Tri}_0$; its order-$\varepsilon^0$ and order-$\varepsilon^1$ coefficients, which enter the weight-three and weight-four derivations, are $2\gamma_E\,\mathrm{Tri}_0 + J$ and $-\bigl((2\gamma_E^2+\zeta_2)\,\mathrm{Tri}_0 + 2\gamma_E J + \tfrac12 K\bigr)$, where $K = \int_0^1 \frac{\ln^2(1+a-a^2)}{1+a-a^2}\,da$. Both boundary values are values of elliptic polylogarithms with punctures at the cone's Landau points; integer-relation (PSLQ) searches against the ring of $\Gamma_1(6)$ cusp values extended by $\mathbb{Q}(\sqrt5)$ find no relation, and no closed form for them is known.

Order $\varepsilon^1$

At the next order, $T^{(1)}$ and the order-$\varepsilon^1$ coefficient of the dotted top integral satisfy the same homogeneous equations as $T$ and the dotted top integral do at order $\varepsilon^0$. The source in $L_T[T^{(1)}]$ is built from the order-$\varepsilon^1$ coefficients of the sub-sector masters and the order-$\varepsilon^0$ top pair, weighted by the first two orders in $\varepsilon$ of the connection matrix of the seven masters, the matrix of coefficients in their differential equation in $x$. One more integration against the kernels $y_\pm$ gives $T^{(1)}(x)$ as a length-three iterated integral of weight four on the alphabet $\{x,\, x-1,\, x-9,\, 2x+1,\, 4x+1,\, x-r_+,\, x-r_-\}$ with the periods $\psi_1, \psi_2$ in the sources: a sum of 88 such iterated integrals with coefficients in $\mathbb{Q}(\sqrt5)$. Its one new constant is $T^{(1)}(-1)$ above, derived from the same two conditions at large $x$, with the order-$\varepsilon^1$ bubble coefficient and the order-$\varepsilon^1$ boundary constants $(-\pi^2,-18\zeta_3)$ of the scoop sunrise masters as inputs. The construction covers the generic mass point as well.

The physical region

For $x\lt-1/2$, numerical values of $T$ and $T^{(1)}$ on the Euclidean axis follow by transportnumerical solution of the differential equation along a path in $x$, starting from a point where the integrals are known exactly of the seven masters from $x=-1$. The zero of the letter $2x+1$ at $x=-1/2$ is an apparent singularity of the system, and on the band $-1/2\le x\lt-1/4$ the order-$\varepsilon^0$ value is given by the one-fold integral above, which agrees there with the connection continued around $x=-1/2$; the interval $-1/4\le x\le 0$ is not covered. For $x\gt0$ the masters are continued from $x=-1$ with the prescription $x\to x+i0$ along a path above the real singular points, which gives all seven, with their imaginary parts, at any point of the physical region.

The two-variable family

The closed form above holds on the slice only. Away from the symmetric slice $p_{12}^2 = p_{34}^2$ the equal-mass family at $p_{56}^2=-1$ has nine masters. Collected in a vector $\vec g$, the nine masters obey $\partial\vec g/\partial p_{12}^2=\Omega_{12}\,\vec g$ and $\partial\vec g/\partial p_{34}^2=\Omega_{34}\,\vec g$ with $9\times9$ matrices of rational functions, obtained from an integration-by-parts reduction at symbolic $(p_{12}^2,p_{34}^2)$; the one-form $\Omega=\Omega_{12}\,dp_{12}^2+\Omega_{34}\,dp_{34}^2$ is integrable: $d\Omega = \Omega\wedge\Omega$ holds identically. With the nine masters evaluated once by auxiliary mass flow at $(p_{12}^2,p_{34}^2)=(-3,-4)$ as boundary values, transport gives them through order $\varepsilon^1$ at any Euclidean point on either side of the diagonal $p_{12}^2=p_{34}^2$. On the diagonal the nine-master basis degenerates to the seven of the slice and the connection has an apparent singularity with integer exponents and no logarithms; averaging the transported solution on a small circle around the diagonal recovers six of the seven slice masters, the seventh not being in the off-slice basis.

The complete expression, with the homogeneous solutions, the source terms, the one-fold form and the boundary values to full precision at both orders, is in the expression file inside the bundle under Evaluator below.

Checks

Evaluator

Same family

• The ice-cream cone with generic masses has squared masses $(1,2,3,5)$ on the four lines: its scoop curve is not modular, the cone's own letter becomes a quartic, and the top-sector operator produced by the elimination is of third order.
• The equal-mass sunrise is the scoop sunrise on its own: the three-line two-point integral left when a cone line is contracted, whose period pair on the $\Gamma_1(6)$ curve supplies the source $S$ here.
• The equal-mass kite is a second two-loop integral built on the same $\Gamma_1(6)$ sunrise curve; there the integration kernel is itself a modular form, whereas for the cone the kernels are algebraic and the curve enters through the source.

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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