Ice-cream cone with generic masses

The two-loop ice-cream cone with four different internal masses (squared masses 1, 2, 3, 5) and three off-shell legs, on a one-parameter slice. Its finite part in two dimensions is new. It depends on the non-modular elliptic curve of the sunrise sub-diagram and is computed here from the differential equation of the family and boundary values derived analytically.

The integral

The ice-cream cone is a one-loop triangle with a one-loop bubble inserted on one of its three edges: four massive internal lines and three off-shell external momenta $P_1, P_2, P_3$, each the total momentum of a pair of external legs, with $P_1^2=p_{34}^2$, $P_2^2=p_{12}^2$ and $P_3^2=p_{56}^2$. The family is

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

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 $D_1, D_2$ form the cone and $D_3, D_4$ the bubble on the remaining edge. The bubble together with the cone line $D_1$ is the scoopthe three-line sunrise sub-diagram with masses $m_1$, $m_3$, $m_4$ that carries the elliptic curve, named for the ball of ice cream on top of the cone: contracting $D_2$ collapses the diagram onto this sunrise, with masses $(m_1, m_3, m_4)$. Contracting $D_1$ instead gives the sub-sunrise with masses $(m_2, m_3, m_4)$. Here the squared masses are $(m_1^2, m_2^2, m_3^2, m_4^2) = (1, 2, 3, 5)$, and the kinematics is the symmetric slice with one external scale: $x = p_{12}^2 = p_{34}^2$ at fixed $p_{56}^2 = -1$. For real $x\lt-\tfrac14$ the Källén function$\lambda(a,b,c)=a^2+b^2+c^2-2ab-2bc-2ca$, the triangle function of two-body kinematics of the three external invariants, $1+4x$, is negative, so the kinematics is Euclidean and the integral converges absolutely. The family has 19 master integralsthe finite basis of integrals to which every integral of the family reduces via integration-by-parts identities at this mass point. 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 $I[1,1,1,1]$ as a function of $x$.

The ice-cream cone is the two-loop member of the ice cone family of Duhr, Klemm, Nega and Tancredi (2023) and the standard counterexample to the strict hierarchical principle of Landshoff, Olive and Polkinghorne (1966), by which the leading singularities of a diagram would all be inherited from its sub-diagrams: its leading Landau locus has a branch that no sub-diagram supplies. In two dimensions the maximal cut of the top sector satisfies a second-order equation with algebraic solutions, and the elliptic curves of the sunrise sub-sectors enter only through the inhomogeneous terms; this structure was established by Lairez and Vanhove (2023), by Doran, Harder, Pichon-Pharabod and Vanhove (2024) and by de la Cruz and Vanhove (2024). No earlier evaluation of the function itself is known.

At squared masses $(1,2,3,5)$ two elliptic curves enter the top integral through its sunrise sub-sectors. The scoop curve is that of the sunrise with squared masses $(1,3,5)$, an instance of the arbitrary-mass sunrise solved by Adams, Bogner and Weinzierl (2014), with threshold letter

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

whose roots are the normal threshold and the three pseudo-thresholds of that sunrise. This family of curves is not modular: its monodromy group is not a congruence subgroupa subgroup of $\mathrm{SL}_2(\mathbb{Z})$ defined by congruence conditions on the matrix entries; when the periods of a family of curves mix only by such matrices, the solution can be written in modular forms, and the curve at $x=-1$, the point where the boundary values are fixed, has $j$-invariant $470596/225$, which is not an algebraic integer, so the curve has no complex multiplication either. There is therefore no basis of modular forms to write the answer in, unlike the equal-mass case on $\Gamma_1(6)$, and the periods of the curve enter the boundary values directly, as complete elliptic integrals. The sub-sunrise with squared masses $(2,3,5)$ defines a second, distinct curve, with the threshold quartic $x^4-40x^3+352x^2-960x+576=\prod_\pm\bigl(x-(\sqrt2\pm\sqrt3\pm\sqrt5)^2\bigr)$. The cone also has a leading Landau singularity of its own on the slice: the leading singularity of the one-loop triangle with internal masses $(m_1, m_2, m_3+m_4)$, in which the bubble is pinched at its two-particle threshold. At equal mass this is the quadratic $x^2-10x+5$; at these masses the same mechanism gives the irreducible quartic $L_{\rm bub} = x^4 - 40x^3 + 348x^2 - 880x + 196$, which collects that locus together with its conjugate from the bubble's pseudo-threshold.

Closed form

The top sector has two master integrals, $I[1,1,1,1]$ and one with an irreducible numerator. Eliminating the second from their coupled system, whose homogeneous solutions are algebraic, gives a third-order operator for $T$, which has not been reduced here to a second-order form with an explicit pair of algebraic kernels as at equal mass; its singular set consists of $x=0$, the roots of the two sunrise quartics and of $L_{\rm bub}$, and the roots of the quadratic $3x^2-18x+11$. $T$ is given here through the exact differential equation of the family, with its connection matrixthe matrix $A(d,x)$ in the first-order system $dM/dx = A\,M$ obeyed by the vector $M$ of master integrals; its poles are the letters of the alphabet rational in $x$ and in the masses, solved from a boundary vector derived at $x=-1$. In that system the self-coupling $A_{\rm top}$ of the top master and the coefficient $c_S$ that couples the scoop sunrise master $S$ to it are the rational functions

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

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

Both carry the cone's quartic $L_{\rm bub}$ in the denominator; the linear factors $2x+1$ and $4x+1$ beside it are apparent singularities of the basis and its normalization, not of $T$. In the full system $L_{\rm bub}$ appears only in the top-sector rows and not in either sunrise block, as the Landau analysis of the full diagram requires. Values of $T$ at Euclidean $x\lt-\tfrac12$, below the zero of $2x+1$, 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 masters from $x=-1$.

The Feynman-parameter representation of the top integral given for the equal-mass ice-cream cone carries over to these masses: the bubble acts as a single line of squared mass $s$, integrated over $s$ from $(\sqrt3+\sqrt5)^2$ upward against $2/\sqrt{\lambda(s,3,5)}$ with $\lambda$ the Källén function, and the remaining loop is a triangle with lines of squared mass $1$, $2$ and $s$. In two dimensions that triangle is a sum of three logarithms, so for $x\lt-\tfrac14$ the function $T(x)$ is also a convergent one-fold integral of logarithms over the bubble mass $s$, which needs no boundary value.

The boundary value

The boundary vector at $x=-1$, the coefficients of orders $\varepsilon^{-2}$ to $\varepsilon^{0}$ of the masters, is derived analytically. The masters that do not depend on $x$ are elementary; among them is the two-dimensional bubble at $p_{56}^2=-1$ with the cone masses $m_1, m_2$, equal to $\ln(1+\sqrt2)/\sqrt2$. The sunrise masters are convergent one-fold integrals over the same bubble mass $s$, and the periods of the scoop curve at $x=-1$ are complete elliptic integrals with the rational Weierstrass roots $(\tfrac{11}{3},\tfrac23,-\tfrac{13}{3})$. The remaining components follow from the behavior at large $x$. In the variable $u=1/x$, expansion by regions allows only integer powers of $u$ times logarithms; this excludes the local solution of the top sector at $u=0$ whose exponent is half-integer. Matching to the hard bubble at large momentum then fixes the constant term of the $u^1$ solution to $\ln 15\cdot\ln(1+\sqrt2)/\sqrt2$, where the $\ln 15$ comes from $m_3^2 m_4^2 = 15$. Four sub-sector masters have resonant indicial exponents at $u=0$; their boundary constants lie in $\mathbb{Q}[\gamma_E,\ln2,\ln3,\ln5,\zeta_2]$. Integrating the connection from $u=0$ to $x=-1$ with these conditions gives the boundary value $T(-1) = 0.1348970572\ldots$. An integer-relation (PSLQ) search finds no expression for $T(-1)$ in terms of the periods of the curve or of the $L$-values and regulators of $\mathbb{Q}(\sqrt3,\sqrt5)$, the natural constants in the absence of modular forms, so the two integration constants of the top sector are left as convergent period integrals.

The same derivation applies at any rational squared masses $(1, m_2^2, m_3^2, m_4^2)$; two further mass points, $(1,3,2,7)$ and $(1,\tfrac52,\tfrac12,3)$, are computed this way as checks. The order-$\varepsilon^1$ coefficient at squared masses $(1,2,3,5)$ follows from the same conditions at large $x$ one order deeper, by the construction described under order $\varepsilon^1$ of the equal-mass ice-cream cone.

The complete expression, with the connection data at both mass configurations and the boundary values at $x=-1$, is in the expression file.

Checks

Evaluator

Same family

The equal-mass ice-cream cone has all four lines at one mass $m$; its scoop curve is the modular curve of $\Gamma_1(6)$ and its top-sector operator has an explicit pair of algebraic kernels, so the top master and its order-$\varepsilon^1$ term have closed forms. The unequal-mass sunrise is the scoop topology on its own, the two-point function with unequal masses, at the squared-mass sets $(1,1,2)$, $(1,2,3)$ and $(1,1,4)$, where it is a sum of elliptic dilogarithms. The unequal-mass kite is another two-loop integral on a sunrise curve that is not modular, solved the same way from its differential equation and a boundary value in closed form.

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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