Four-mass kite with a massless rung

A two-loop kite self-energy with four different masses on the rim, (1, 2, 3, √2), and a massless line joining the two internal vertices. Unlike the massive kites, none of its cuts defines an elliptic curve. Its finite part is new at these masses and follows from the differential equation in the external momentum $p$, starting at $p=0$.

The integral

The kite is a two-loop self-energya diagram with one external momentum flowing in and out; as a function of that single invariant it corrects a particle's propagator: a single off-shell momentum $p$ enters at one vertex and leaves at the other, and the only kinematic variable is the squared momentum $s = p^2$. The family is

$$J(\nu_1,\ldots,\nu_5) \;=\; \int d^d k_1\, d^d k_2\; \frac{1}{D_1^{\nu_1} D_2^{\nu_2} D_3^{\nu_3} D_4^{\nu_4} D_5^{\nu_5}}\,, \qquad d = 4-2\varepsilon,$$

with loop momenta $k_1, k_2$ and the five propagators

$$D_1=k_1^2-1,\quad D_2=(k_1-p)^2-4,\quad D_3=k_2^2-9,\quad D_4=(k_2-p)^2-2,\quad D_5=(k_1-k_2)^2.$$

Here $D_1, D_2$ form the left half of the rim, $D_3, D_4$ the right half, and $D_5$ is the rung, so the squared masses are $(1,4,9,2,0)$. The quantity computed is the finite $\varepsilon^0$ coefficient of the top-sector masterthe master integral with every propagator of the diagram present once $J(1,1,1,1,1)$, written $J^{(0)}(s)$, as a function of $s$ for spacelike momentum, $s\lt0$.

Kites with massive lines are standard examples of two-point integrals that evaluate to elliptic functions. Numerical and dispersive treatments of massive two-loop self-energies, the kite among them, go back to Bauberger, Berends, Böhm and Buza (1995, arXiv:hep-ph/9409388). Remiddi and Tancredi (2016, arXiv:1602.01481) wrote the coupled differential equations for the massive sunrise and the kite. Adams, Bogner, Schweitzer and Weinzierl (2016, arXiv:1607.01571) then gave the equal-mass kite to all orders in $\varepsilon$ in elliptic polylogarithms, Broedel, Duhr, Dulat, Penante and Tancredi (2019, arXiv:1902.09971) the kite with three different masses, and Giroux, Pokraka, Porkert and Sohnle (2024, arXiv:2401.14307) a kite whose two sunrise cuts define two different elliptic curves. In those kites the elliptic curve comes from the maximal cutput every internal line of the sub-diagram on shell simultaneously; the integral over what survives exposes the diagram's underlying geometry of a sunrise sub-diagram with three massive lines.

Here the massless line is the rung rather than a rim line, and both three-line cuts of the graph contain it: $\{D_1,D_4,D_5\}$ with squared masses $(1,2,0)$ and $\{D_2,D_3,D_5\}$ with $(4,9,0)$. Each three-line cut is a sunrise sub-diagram. The maximal cut of a sunrise with masses $m_1,m_2,m_3$ is an integral $\int dz\,/\sqrt{\lambda(z,m_1^2,m_2^2)\,\lambda(z,s,m_3^2)}$ over the squared invariant mass $z$ of the first two lines, with $\lambda(a,b,c)=a^2+b^2+c^2-2ab-2bc-2ca$ the Källén function, and when all three lines are massive the quartic under the square root makes the cut the period of an elliptic curve. If one of the first two lines is massless, say $m_2=0$, the factor $\lambda(z,m_1^2,0)=(z-m_1^2)^2$ is a perfect square, so only a quadratic in $z$ remains under the root and the cut has genus zero. The other three-line sub-sectors that contain the rung do not depend on $s$: they are vacuum sunsets with one massless line, or vanish. Without $D_5$ the two loops decouple, so that the sector $\{D_1,D_2,D_3\}$, for example, is the one-loop bubble with masses $1,2$ times a tadpole of mass $3$. No subtopology therefore defines an elliptic curve, and the whole family is polylogarithmic.

Closed form

Integration-by-parts reduction leaves a vector $M$ of 22 master integralsthe finite basis of integrals to which every integral of the family reduces through integration-by-parts identities, which satisfies $\partial_s M = A(\varepsilon,s)\,M$ with a connection matrix $A$ that is rational in $s$, linear in $\varepsilon$, and singular only at

$$s \in \{0,\, 1,\, 9,\, 25,\, \tfrac{85}{4},\, 3\pm2\sqrt2,\, 11\pm6\sqrt2\},$$

the zeros of the seven lettersthe polynomials in $s$ whose logarithmic derivatives appear as integration kernels; together they form the alphabet of the alphabet. None of these points lies on the negative real axis. The two quadratic letters split over $\mathbb Q(\sqrt2)$: their zeros $3\pm2\sqrt2$ are the threshold and pseudo-threshold of the $\{D_1,D_4,D_5\}$ cut (masses $1,\sqrt2,0$), and $11\pm6\sqrt2$ those of the $\{D_3,D_4\}$ bubble (masses $3,\sqrt2$). Among the zeros of the linear letters, $s=9$ and $s=25$ are the thresholds of the $\{D_1,D_2\}$ bubble (masses $1,2$) and of the $\{D_2,D_3,D_5\}$ cut (masses $2,3,0$), $s=1$ is their common pseudo-threshold, and $s=85/4$ is a Landau singularitya kinematic point at which several internal lines of the diagram can go on shell at once, where the integral fails to be analytic of the kite itself. Besides the seven rational letters, four square roots appear: $\sqrt{(s-1)(s-9)}$ and $\sqrt{s^2-22s+49}$ from the two bubbles, and $\sqrt{s^2-6s+1}$ and $\sqrt{(s-1)(s-25)}$ from the two three-line cuts. The master $J^{(0)}(s)$ is then an iterated integral with kernels rational in the seven letters, the four square roots being the algebraic letters of its function space, and its boundary constants are the vacuum integrals at $s=0$. Whether it can be written as multiple polylogarithms with these four roots as arguments is not decided here.

The boundary constants are fixed at $s=0$ without numerical input. That point is a regular singular pointa point where the differential equation has at most a simple pole, so every local solution behaves like a power of the distance to it, possibly times logarithms of the system, and the residue matrix of $A$ there has the eigenvalue spectrum $\{0\,(\times 8),\ -1+\varepsilon\,(\times 13),\ -2+2\varepsilon\,(\times 1)\}$. Every local solution that is not analytic at $s=0$ is therefore unbounded. The integrals themselves are regular at $p=0$, where they are vacuum integrals, so the coefficients of all the singular solutions vanish and the solution is fixed completely by the values of the 22 masters at $s=0$. Those values are known. At $p=0$ the two lines of each half of the rim carry the same loop momentum, so partial fractions such as $\frac{1}{D_1D_2}\big|_{p=0}=\frac13\big(\frac1{D_2}-\frac1{D_1}\big)$ reduce all 22 masters to two-loop vacuum integrals (Broadhurst, Fleischer and Tarasov 1993, arXiv:hep-ph/9304303). These are products of one-loop tadpoles and the vacuum sunset (the sunrise at zero external momentum) with one massless line, and their $\varepsilon$ expansion, obtained from a Mellin–Barnes representation, contains only logarithms and dilogarithms of the mass ratios. The boundary value of the top-sector master, $J^{(0)}(0)$, is also a single integral over modified Bessel functions $K_1$,

$$J^{(0)}(0) = -\tfrac{8}{21}\int_0^\infty \tfrac{dx}{x}\,[K_1(x) - 2K_1(2x)]\,[\sqrt2\,K_1(\sqrt2\,x) - 3K_1(3x)] = -0.2910157568\ldots.$$

This is a combination of vacuum sunsets with one massless line written in coordinate space, where a propagator of mass $m$ at separation $x$ is proportional to $m\,K_1(mx)/x$ and a massless one to $1/x^2$. The partial fractions at $p=0$ replace each half of the rim by the difference of its two massive propagators, which gives the two brackets. From $s=0$ the analytic branch is continued by transportnumerical solution of the differential equation along a path in $s$, starting from a point where the integrals are known exactly along the negative real axis to any spacelike $s$.

A second mass point

Nothing in this construction is special to the value $2$ of the fourth squared mass. At squared masses $(1,4,9,3,0)$ the reduction again gives 22 masters, the singular points that move are $85/4\to297/8$, $3\pm2\sqrt2\to(1\pm\sqrt3)^2$ and $11\pm6\sqrt2\to(3\pm\sqrt3)^2$, and the boundary argument at $s=0$ is unchanged.

Checks

Evaluator

Same family

The equal-mass kite carries one mass $m$ on three lines and has its two massless lines on opposite sides of the rim, so that the three massive lines form a sunrise whose maximal cut is an elliptic curve with periods given by modular forms, the case solved by Adams, Bogner, Schweitzer and Weinzierl. The unequal-mass kite moves one of those three masses to $\sqrt2$; its sunrise curve is then no longer modular, and its finite part at those masses is new.

The paper

• Polylogarithmic, elliptic, and Calabi–Yau Feynman integrals from a hybrid bootstrap (PDF) — Matthew D. Schwartz. Section 4.7, “The four-mass kite with a massless rung,” gives the letters and square roots of the family, the iterated-integral form of the top-sector master, the boundary value at $p^2=0$ with the closed form of the vacuum sunset with one massless line, and the comparison at both mass points.

References

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