The BootLoops integral class

The umbrella page of the function-class ladder: what kinds of integrals BootLoops computes, what they look like on paper, and the diagnostics that tell you which rung of the ladder an integral lives on.

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The ladder: The BootLoops integral class · Polylogarithms · Elliptic · K3 · Calabi–Yau

BootLoops computes definite integrals exactly — closed forms, checked to thirty digits or more against values held out from every derivation step. This page describes the class of integrals the program handles and the single organizing fact about them: their answers sort into a short ladder of function classes, and the geometry hidden inside each integral tells you in advance which class its answer belongs to. Four sibling pages then treat the rungs one at a time: polylogarithms, elliptic functions, K3 periods, and Calabi–Yau periods.

What the integrals look like

Strip away the applications and one shape remains. An integral in the BootLoops class has an algebraic integrand — typically a product of powers of polynomials in the integration variables — over a domain cut out by polynomial equalities and inequalities, with everything the problem cares about (kinematics, hop rates, data counts) carried in the polynomials' coefficients and exponents. The simplest member has one variable and two linear forms: Euler's beta function,

$$B(a,b)\;=\;\int_0^1 x^{\,a-1}(1-x)^{\,b-1}\,dx\;=\;\frac{\Gamma(a)\,\Gamma(b)}{\Gamma(a+b)}\,.$$

Everything the program computes is this object's extended family — more variables, more polynomials, deeper geometry underneath. Four branches of the family cover the site.

Evidence integrals — statistics. A Bayesian model comparison asks for the evidence: the probability of the data averaged over everything the model's parameters could have been,

$$Z \;=\; \int_\Theta p(\mathrm{data}\mid\theta)\;p(\theta)\;d\theta\,.$$

For a mixture model — count data, model parameters $\theta$ ranging over simplices, a Dirichlet prior — the integrand is a product of powers of linear forms in $\theta$: a many-variable beta integrand. Integrals of exactly this species carry the mixture-model verdicts, the phylogenetics tree scores, and the composite-authorship probabilities.

Lattice integrals — condensed matter. The return probability of a random walker on a lopsided cubic crystal comes down to the triple integral

$$W_S(\alpha,\beta,\gamma;w)\;=\;\frac{1}{\pi^3}\int_0^\pi\!\!\int_0^\pi\!\!\int_0^\pi\frac{dk_1\,dk_2\,dk_3}{\,w-\alpha\cos k_1-\beta\cos k_2-\gamma\cos k_3\,}\,,$$

first evaluated in special cases by G. N. Watson in 1939 and central to the condensed-matter applications. Substituting $z_j=e^{ik_j}$ turns each cosine into $\tfrac12(z_j+z_j^{-1})$: the integrand becomes a rational function and the domain becomes the torus $|z_j|=1$ — the same species, in trigonometric disguise.

String-type integrals — points in a plane. Let the exponents $a$ and $b$ depend on particle momenta and the beta function becomes the Veneziano amplitude, the 1968 formula that began string theory. The Veneziano integral runs over the position of one point on a line, with three more fixed at $0$, $1$, and $\infty$; promote those four points to $n$ points in a plane and you get the Grassmannian string integrals $X(d,n)$. The first member beyond the string, $X(3,6)$, is computed exactly on the quantum-gravity page.

Feynman-type integrals — colliders, gravity, cosmology. A scalar $L$-loop Feynman integral starts life in momentum space, but Feynman parameters — one parameter $x_e$ per propagator, the standard first move of loop computation — turn it into the same parametric shape as everything above:

$$I_G \;\propto\; \Gamma\!\left(\nu - \tfrac{LD}{2}\right)\int_0^\infty \prod_e dx_e\;\delta\!\Big(1-\sum_e x_e\Big)\;\frac{\,\mathcal U(x)^{\,\nu-(L+1)D/2}}{\,\mathcal F(x)^{\,\nu-LD/2}}\,,$$

with $\nu$ the number of propagators, $D$ the spacetime dimension, and the two Symanzik polynomialstwo polynomials in the Feynman parameters, determined entirely by the graph: U from spanning trees, F from spanning two-forests plus the mass term $\mathcal U$ and $\mathcal F$ fixed by pure graph theory — $\mathcal U$ from the spanning trees of the graph, $\mathcal F$ from its two-forests plus the mass term. Products of powers of two polynomials over a simplex; the whole difficulty of multi-loop computation is compressed into the geometry of the hypersurface $\mathcal F(x)=0$. The wavefunction coefficients of inflationary cosmology are close cousins — energy integrals with rational integrands — and the black-hole-scattering integrals of gravitational-wave theory are the same Feynman species with nonrelativistic propagators.

A single property unites all four branches, whatever their origin. Each one, viewed as a function $f(t)$ of a parameter $t$ (a momentum ratio, a hop rate, a hyperparameter), satisfies a linear differential equation of finite order with rational-function coefficients:

$$c_r(t)\,\frac{d^r f}{dt^r} + c_{r-1}(t)\,\frac{d^{r-1} f}{dt^{r-1}} + \cdots + c_0(t)\, f \;=\; 0\,,\qquad c_k \in \mathbb{Q}(t).$$

Functions with this property are called holonomicalso "D-finite": satisfying a finite-order linear ODE with rational coefficients in each of its variables; sums, products, and integrals of holonomic functions are again holonomic, and integrals of the shared shape above are holonomic in every parameter — for the Feynman branch this follows from the integration-by-parts identities described below. BootLoops works with the equation, never the integrand. A holonomic function is pinned down by finitely many data: its equation plus a few boundary constants. Everything BootLoops does — classifying an integral, fixing its function space, transporting values to arbitrary precision — runs on the differential equation.

The ladder

As loops, masses, and kinematic scales accumulate, the answers to these integrals climb a ladder of function classes: rational and logarithmic functions, then polylogarithms, then elliptic functions, then K3 periodsintegrals of algebraic functions over the closed loops and surfaces of a geometric object — defined from scratch later on this page, then Calabi–Yau periods. Every branch of the family populates the ladder — the evidence and string integrals live low on it, the Watson integral sits on the K3 rung — but the cleanest illustrations are Feynman graphs, and each rung below has a canonical one.

Rational/logarithmic. The one-loop massless bubble — two propagators, one external momentum — evaluates to a single logarithm:

$$B(p^2) \;=\; \frac{1}{\varepsilon} \;-\; \ln\!\frac{-p^2}{\mu^2} \;+\; 2 \;+\; O(\varepsilon)\,,$$

in $\overline{\rm MS}$-style normalization. One scale produces one logarithm; this is the ladder's bottom rung.

Polylogarithmic. The one-loop massless box — four propagators, two invariants $s$ and $t$ — is the canonical example one rung up. Its closed form is

$$I_{\rm box} \;=\; \frac{c_\Gamma}{st}\left\{\frac{2}{\varepsilon^2}\Big[(-s)^{-\varepsilon}+(-t)^{-\varepsilon}\Big] \;-\; \ln^2\frac{s}{t} \;-\; \pi^2\right\} + O(\varepsilon)\,, \qquad c_\Gamma=\frac{\Gamma(1+\varepsilon)\Gamma^2(1-\varepsilon)}{\Gamma(1-2\varepsilon)}\,,$$

and at higher weight the functions that appear are the multiple polylogarithmsiterated integrals of logarithmic differentials $d\log f(x)$, generalizing $\ln$ and $\mathrm{Li}_n$; each satisfies a linear ODE with rational coefficients $G(a_1,\dots,a_n;x)$, built by integrating one logarithmic kernel at a time. Most integrals a collider physicist meets live on this rung; the polylogarithms page covers it.

Elliptic. The two-loop equal-mass sunrise — three massive lines between two vertices — is the classic first integral that polylogarithms cannot express. It has resisted polylogarithmic evaluation since Sabry's calculation in 1962:

$$J_{111} \;=\; \int d^d l_1\, d^d l_2\; \frac{1}{\bigl(l_1^2-m^2\bigr)\bigl(l_2^2-m^2\bigr)\bigl((l_1+l_2-p)^2-m^2\bigr)}\,,\qquad d=2-2\varepsilon.$$

Its answer needs the periods of an elliptic curve — a torus. The sunrise page works this example in full.

K3. The three-loop equal-mass banana — four massive lines between the same two vertices — climbs one more rung:

$$I_{1111} \;=\; \int \frac{d^d l_1\, d^d l_2\, d^d l_3}{\big(l_1^2-m^2\big)\big(l_2^2-m^2\big)\big(l_3^2-m^2\big)\big((l_1+l_2+l_3-p)^2-m^2\big)}\,,\qquad d=2-2\varepsilon.$$

Its geometry is a K3 surface, a two-complex-dimensional space that plays the role the torus played one rung down. The banana page treats it.

Calabi–Yau. The four-loop equal-mass banana — five massive lines — sits on a Calabi–Yau threefold, three complex dimensions, and each further loop of the banana family climbs one complex dimension higher. The threshold CY₃ page works a four-loop example.

Where does the geometry come from? From the maximal cut: replace every propagator $1/(q_e^2-m_e^2)$ by a delta function $\delta(q_e^2-m_e^2)$ — put every internal line on shell — and do the localized integrals. Whatever integral is left over solves the homogeneous part of the integral's differential equation, and it always takes the same form: an algebraic differential form integrated over a closed cycle of a geometric object. For the one-loop box that object is trivial — the cut localizes completely, to a point, and the leftover is an algebraic function, which is why the box is polylogarithmic. For the sunrise, the cut integrand carries the square root of a quartic polynomial, $1/\sqrt{q(x)}$; the surface on which that square root is single-valued is a torus, and the leftover integral wraps its cycles. For the three-loop banana the cut is a two-dimensional integral over a K3 surface; for the four-loop banana, a threefold. The intermediate case — a Riemann sphere with marked points, where the cut leaves only simple poles — is the geometric home of the polylogarithmic rung.

These leftover cut integrals are periods. A period is the integral of an algebraic differential form over a closed cycle of an algebraic variety — a space cut out by polynomial equations. The first example is the basic contour integral of complex analysis:

$$2\pi i \;=\; \oint_{|z|=1} \frac{dz}{z}\,.$$

The form $dz/z$ is algebraic; the cycle winds once around the puncture at $z=0$; the output is a transcendental number. Every logarithm and every $\zeta$-value arises this same way from cycles on spheres with marked points. The second example is the complete elliptic integral that gives the exact period of a pendulum:

$$K(k)\;=\;\int_0^1 \frac{dx}{\sqrt{(1-x^2)(1-k^2x^2)}}\,,\qquad \oint_\gamma \frac{dx}{y} \;=\; 4\,K(k) \quad\text{on the curve } y^2=(1-x^2)(1-k^2x^2).$$

The curve $y^2 = $ quartic is an elliptic curve; the cycle $\gamma$ encircles two of its four branch points; and $K$ is, literally, a period of the torus — the analogue, one rung up, of $2\pi i$. K3 surfaces and Calabi–Yau threefolds each carry a distinguished volume form, and its integrals over their cycles are the periods that four- and five-line banana graphs deliver as answers. Kontsevich and Zagier made "period" a precise, countable class of numbers in their 2001 essay; for our purposes the geometric picture above is all you need. In one sentence: the function class of a Feynman-type integral is the period class of its maximal-cut geometry — a point, a sphere with marked points, an elliptic curve, a K3 surface, a Calabi–Yau threefold.

How you know

In practice you do not stare at the integrand and guess. A sequence of diagnostics, each computable, sorts an integral onto its rung. Here they are in the order a practitioner applies them.

1. The order of the differential equation. At one loop, the many integrals of a computation famously collapse onto a handful of scalar ones. Integration-by-parts (IBP) identities — the observation of Chetyrkin and Tkachov (1981) that $\int d^dl\; \partial_\mu(\text{anything}) = 0$ in dimensional regularization — extend that reduction to all loops: every integral of a family collapses onto finitely many master integralsthe finite basis of independent integrals to which IBP identities reduce an entire family, and differentiating the masters with respect to a kinematic variable $t$ lands back in the family, so the masters obey a first-order linear system

$$\frac{d}{dt}\,\vec I(t,\varepsilon) \;=\; A(t,\varepsilon)\,\vec I(t,\varepsilon)\,,$$

with $A$ a matrix of rational functions. The finiteness of the master basis is a theorem of Smirnov and Petukhov (2010). Eliminating all but one component turns the block for a given sector into a single scalar equation whose order equals the number of masters in that block. That order is the first diagnostic: order one means rational/logarithmic; order two or more means the ladder is in play. The Watson integral's proved order — five, one above the ceiling any product of elliptic integrals can reach — settled that no elliptic closed form exists for the anisotropic walk.

2. A dlog form exists → polylogarithmic. If a change of basis brings the system to Henn's canonical form (2013),

$$d\,\vec I \;=\; \varepsilon \left(\sum_i A_i\; d\!\log \alpha_i(t)\right) \vec I\,,$$

with constant matrices $A_i$ and algebraic functions $\alpha_i$ — the symbol alphabetthe finite list of algebraic functions whose dlogs build the answer as iterated integrals — then the answer is polylogarithmic at every order in $\varepsilon$: iterated integrals of the $d\log\alpha_i$, one letter at a time. Existence of a finite alphabet is the operational definition of the second rung.

3. An irreducible second-order piece → elliptic; compute the j-invariant. When no dlog form exists, factor the scalar operator. An irreducible second-order factor whose monodromythe linear transformation the solution vector undergoes when the variable travels once around a singular point and returns is that of a family of tori signals an elliptic curve, and the curve itself is sitting in the maximal cut: the quartic under the square root, $y^2 = q(x;t)$. Bringing the curve to Weierstrass form $y^2 = 4x^3 - g_2 x - g_3$ gives its j-invariant

$$j \;=\; 1728\,\frac{g_2^3}{g_2^3-27 g_3^2}\,,$$

the one number that identifies an elliptic curve up to isomorphism. The j-invariant is the curve's fingerprint: for the equal-mass sunrise it traces out the modular curve of $\Gamma_1(6)$a classical one-parameter family of tori; the subscript records extra structure the sunrise curve carries, and matters here only as a fingerprint as $t$ varies. And when the internal masses are instead tuned to a normal threshold, $m_3=m_1+m_2$, the fiber at $p^2=0$ degenerates onto the complex-multiplication point $j=1728$, with finite monodromy of order four — the observation from which the geometry paper builds its threshold-degeneration results.

4. A third-order piece that is a symmetric square → K3. A third-order operator $L_3$ is the symmetric square of a second-order operator $L_2$ when its three-dimensional solution space consists of the pairwise products of the two solutions of $L_2$:

$$\text{Sol}(L_3) \;=\; \{\,\psi_1^2,\;\psi_1\psi_2,\;\psi_2^2\,\}\,,\qquad \{\psi_1,\psi_2\}=\text{Sol}(L_2)\,.$$

Concretely: if $\psi_1$ and $\psi_2$ solve a second-order equation, their squares and product automatically solve a particular third-order one, and you can test whether a given $L_3$ is of that form by exact algebra. This is the working criterion for the K3 rung: the three-loop banana's third-order operator is the symmetric square of the sunrise's, its holomorphic period is literally the sunrise period squared, and the Taylor coefficients of that period are the integer sequence $1, 4, 28, 256, 2716, \ldots$ (the Domb numbers, OEIS A002895). This works because the banana's K3 is special: being the symmetric square of the sunrise curve, it has Picard rank 19, leaving only a rank-3 transcendental lattice — which is why a third-order equation suffices. A K3 family of lower Picard rank would need a higher-order equation, so the test certifies the banana family, not K3 surfaces in general.

5. A self-dual fourth-order operator → Calabi–Yau threefold. The fourth-order operators of the CY rung are singled out by a duality condition: the operator is conjugate to its own formal adjoint,

$$L_4^{\dagger} \;=\; \alpha^{-1}\, L_4\, \alpha \quad \text{for some algebraic function } \alpha(t)\,,$$

which forces the four solutions to pair up under a monodromy-invariant symplectic form — exactly the structure the periods of a Calabi–Yau threefold possess, where the pairing is the intersection of cycles. Operators passing this test (together with integrality conditions on their series solutions) are the "Calabi–Yau operators" of the Almkvist–van Straten–Zudilin classification program, and the four-loop equal-mass banana's operator is one of them.

What BootLoops does with each class

The diagnosis is the expensive step; everything after it is uniform across the ladder. The geometry of the maximal cut fixes the function space before any number is fitted — an alphabet of logarithms on the polylogarithmic rung, iterated integrals over the period pair of the curve on the elliptic rung, and the corresponding period structures on the K3 and Calabi–Yau rungs. The bootstrap then cuts that space down with analytic constraints (which singularities may appear, how the function behaves at thresholds and at infinity) and pins the last free coefficients with integer-relation fits at high precision. The boundary constants lie in the number ring the geometry's arithmetic dictates — $\zeta$-values on the lower rungs, elliptic periods at special points and their higher analogues above. Every result is then checked against independent high-precision values held out from every step of the derivation, to thirty digits or more. The four rung pages walk through what this looks like in each class: polylogarithms, elliptic, K3, Calabi–Yau.

Where this shows up in BootLoops

Further reading