Polylogarithms

The first rung above the elementary functions: how the logarithm grows into the dilogarithm, how iterated integrals organize the whole family, and how the symbol turns identities among these functions into linear algebra — up to the first integrals that refuse to join.

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

From the logarithm up

Every function on this ladder descends from one integral, the simplest in the calculus:

$$\log x \;=\; \int_1^x \frac{dt}{t}.$$

The integrand $dt/t$ has a pole at $t=0$, and that pole is the whole story of the logarithm: the branch cut, the $2\pi i$ ambiguity, the appearance of $\log$ in every loop integral with a massless threshold. The natural next move — integrate the logarithm against $dt/t$ — produces the dilogarithm:

$$\mathrm{Li}_2(x) \;=\; -\int_0^x \frac{dt}{t}\,\log(1-t) \;=\; \sum_{k=1}^{\infty}\frac{x^k}{k^2},$$

where the series follows by expanding $-\log(1-t)=\sum_k t^k/k$ and integrating term by term; it converges for $|x|\le 1$. At $x=1$ it gives Euler's solution of the Basel problem, $\mathrm{Li}_2(1)=\zeta(2)=\pi^2/6$. The function is old: it already appears in Leibniz's 1696 correspondence with Johann Bernoulli, and Euler studied it in the eighteenth century. The dilogarithm is still sometimes called Spence's function, after William Spence's 1809 essay on "logarithmic transcendents"; Landen found some of its earliest functional equations.

Iterating the same move builds the whole classical tower. Define

$$\mathrm{Li}_n(x) \;=\; \int_0^x \frac{dt}{t}\,\mathrm{Li}_{n-1}(t), \qquad \mathrm{Li}_1(x)=-\log(1-x), \qquad \mathrm{Li}_n(x)=\sum_{k=1}^\infty \frac{x^k}{k^n}.$$

Each integration against $dt/t$ raises the weightthe number of integrations needed to build the function from rational ones; $\log$ has weight 1, Li$_n$ has weight $n$, and $\pi$ and $\zeta(n)$ count as weight 1 and $n$ by one: $\log$ has weight one, $\mathrm{Li}_2$ weight two, $\mathrm{Li}_n$ weight $n$.

These functions satisfy relations that the series representation does not remotely suggest. The classic example is the five-term identity of the dilogarithm. Written with the Rogers dilogarithm $L(x)=\mathrm{Li}_2(x)+\tfrac12\log x\,\log(1-x)$, which strips off the elementary-logarithm dressing, it reads

$$L(x)+L(y)\;=\;L(xy)+L\!\left(\frac{x(1-y)}{1-xy}\right)+L\!\left(\frac{y(1-x)}{1-xy}\right),$$

valid for $0<x,y<1$. The five-term combination — five dilogarithms at five algebraically related arguments — vanishes. No identity of this kind exists for $\mathrm{Li}_2$ alone at a single argument. Polylogarithms carry hidden algebraic structure, and the rest of this page builds the frame that makes that structure visible and computable.

Iterated integrals

The right frame is the iterated integral, developed systematically by the mathematician Kuo-Tsai Chen in the 1960s and 1970s. Fix a collection of differential one-forms $\omega_1,\dots,\omega_n$ — for us, forms like $dt/t$ and $dt/(1-t)$ — and a path $\gamma$ from $0$ to $x$. The iterated integral of the word $\omega_1\omega_2\cdots\omega_n$ is the integral over the ordered simplex:

$$\int_0^x \omega_1\,\omega_2\cdots\omega_n \;=\; \int\limits_{0\,\le\, t_1\,\le\, t_2\,\le\,\cdots\,\le\, t_n\,\le\, x} f_1(t_1)\,dt_1\; f_2(t_2)\,dt_2\cdots f_n(t_n)\,dt_n,$$

where $\omega_i=f_i(t)\,dt$. A word is a sequence of letters; the weight of the function is the length of the word. In this notation the dilogarithm is the two-letter word

$$\mathrm{Li}_2(x)\;=\;\int_0^x \frac{dt_1}{1-t_1}\,\frac{dt_2}{t_2},$$

the innermost integration carrying the $1/(1-t)$ kernel — check it by differentiating both sides with respect to $x$.

Iterated integrals obey one universal multiplication law, the shuffle product: the product of two iterated integrals over the same path is the sum of iterated integrals over all interleavings of their letters that preserve each factor's internal order. The two-letter case is the one to verify by hand:

$$\left(\int_0^x \omega_a\right)\left(\int_0^x \omega_b\right) \;=\; \int_0^x \omega_a\,\omega_b \;+\; \int_0^x \omega_b\,\omega_a.$$

The proof is a picture. The left side is an integral over the square $\{0\le t_1\le x\}\times\{0\le t_2\le x\}$; the square is the union of the triangle $t_1\le t_2$ and the triangle $t_2\le t_1$; each triangle is one of the two iterated integrals on the right. As a concrete instance take $\omega_a=\omega_b=dt/(1-t)$, whose single-letter integral is $-\log(1-x)$. The shuffle says $\log^2(1-x) = 2\int_0^x \frac{dt_1}{1-t_1}\frac{dt_2}{1-t_2}$, and differentiating both sides confirms it in one line. For longer words the combinatorics grows — a length-$m$ word times a length-$n$ word produces $\binom{m+n}{m}$ interleavings — but the principle never changes.

The shuffle product accounts for the hidden relations of the last section: products of low-weight polylogarithms and single higher-weight polylogarithms inhabit the same space of words, so identities like the five-term relation become statements of linear algebra in a finite-dimensional vector space — fix the letters, fix the weight, list the words, and every identity is a linear relation among them.

HPLs and MPLs

Quantum field theory forced physicists to standardize this machinery. In 1999 Remiddi and Vermaseren introduced the harmonic polylogarithms (HPLs): iterated integrals in the letters

$$\frac{dt}{t}, \qquad \frac{dt}{1-t}, \qquad \frac{dt}{1+t},$$

exactly the alphabetthe finite list of letters (equivalently, of $d\log$ forms) out of which a given problem's iterated integrals are built that massless and single-mass-scale two-loop integrals need. The classical $\mathrm{Li}_n$ are the HPL words with a single $1/(1-t)$ letter; generic words, like the one with letters $\frac{dt}{1-t}\frac{dt}{t}\frac{dt}{1+t}\frac{dt}{t}$, are genuinely new functions with no expression in terms of classical polylogarithms.

The general case allows the poles to sit anywhere. Goncharov's multiple polylogarithms (MPLs), formalized in the 1990s, are defined recursively:

$$G(a_1,a_2,\dots,a_n;x)\;=\;\int_0^x \frac{dt}{t-a_1}\,G(a_2,\dots,a_n;t), \qquad G(;x)=1, \qquad G(\underbrace{0,\dots,0}_n;x)=\frac{\log^n x}{n!},$$

with letters $dt/(t-a_i)$ for arbitrary constants $a_i$ — the alphabet of a given problem. HPLs are the special case $a_i\in\{0,\pm1\}$, and $\mathrm{Li}_n(x)=-G(0,\dots,0,1;x)$ with $n-1$ zeros.

Setting $x=1$ with letters drawn from $\{0,1\}$ produces the multiple zeta values,

$$\zeta(s_1,\dots,s_k)\;=\;\sum_{n_1>n_2>\cdots>n_k\ge 1}\frac{1}{n_1^{s_1}\cdots n_k^{s_k}},$$

the constants that populate the $\varepsilon$-expansions of dimensionally regularized integrals. Euler already studied the depth-two cases and proved $\zeta(2,1)=\zeta(3)$; the systematic theory took off in the early 1990s with Hoffman and Zagier. And the constants are not private to loop diagrams: the Grassmannian string integral $X(3,6)$ — the string-type branch of the integral class — evaluates in multiple zeta values, every wilder constant cancelling from the final answer (the quantum-gravity page).

The bottom of the tower is where collider computations start. The one-loop box with off-shell legs evaluates to dilogarithms and squared logarithms — weight two, the generic weight of the finite part of any one-loop integral in four dimensions. Two-loop massless integrals reach weight four: Smirnov's 1999 closed form for the massless planar double box is a tower of HPLs in the single ratio $x=t/s$, and its $\varepsilon$-expansion coefficient at order $\varepsilon^j$ has uniform weight $4+j$. The rule of thumb, provable in wide classes of cases, is that an $L$-loop integral in $d=4-2\varepsilon$ reaches weight $2L$ at its finite order. The pattern extends upward: the three-loop massless ladder tops out at weight six, and the two-loop Sudakov form factor, pushed order by order in $\varepsilon$, climbs through weight eight — all still inside the HPL world (the constants are $\zeta$ values).

The symbol

For a single dilogarithm, the series is a fine representation. For a two-loop amplitude spread over thousands of MPL words, one wants a representation in which the hidden relations are automatic. That representation is the symbol.

The starting observation: the derivative of any weight-$n$ MPL is a finite sum

$$dF \;=\; \sum_i F_i \; d\log \phi_i,$$

where each $F_i$ is an MPL of weight $n-1$ and each $\phi_i$ is a letter of the alphabet — an algebraic function of the kinematic variables. Now iterate: differentiate the $F_i$, then their coefficients, all the way down to weight zero, and record only the letters. The result is the symbol, a sum of $n$-fold tensor products of letters, defined recursively by

$$\mathcal{S}(F)\;=\;\sum_i \mathcal{S}(F_i)\otimes \phi_i.$$

For example $\mathcal{S}(\log x)=x$, $\mathcal{S}(\log x\log y)=x\otimes y+y\otimes x$, and $\mathcal{S}(\mathrm{Li}_2(x))=-(1-x)\otimes x$: the two letters record the two integration kernels $d\log(1-t)$ and $d\log t$, in the order they were applied.

The symbol earns its keep twice over. First, identities: shuffle products become literal shuffles of tensor factors, and $d\log$ of a product splits additively — $\mathcal{S}$ turns every polylogarithm identity into an identity of tensors of letters, checkable by expanding logarithms of products into sums. The five-term relation, opaque at the level of functions, is a short mechanical computation at the level of symbols. Second, bootstrapping: run the logic backwards. If the physics of a problem dictates its alphabet — and for Feynman integrals the singularity structure does exactly that — then the space of candidate answers at weight $n$ is the finite-dimensional span of $n$-fold words in that alphabet. Two linear conditions carve it down: integrability, the requirement $\sum_i dF_i\wedge d\log\phi_i=0$ that a candidate tensor actually be the symbol of a function, and the first-entry condition, which restricts the first letter to physical thresholds because the first entry controls where branch cuts open. What survives is an ansatz small enough to fit, and fitting coefficients against a handful of constraints or high-precision values has produced multi-loop amplitudes that direct integration cannot touch — this is the engine of the modern amplitude bootstrap.

The symbol also forgets things, by construction. Any term proportional to a transcendental constant times a lower-weight function — $\zeta_2\log x$, $i\pi\,\mathrm{Li}_2(x)$, $\zeta_3$ itself — has vanishing symbol, because constants have zero derivative. Reconstructing a function from its symbol therefore leaves the "$i\pi$ and zeta pieces" undetermined, to be fixed by analytic input at special points; the finer coproduct structure, which keeps partial derivative information at every stage, recovers much of what the plain symbol drops. The calculus grew out of Goncharov's work on the algebraic structure of MPLs; it entered physics in 2010, when Goncharov, Spradlin, Vergu, and Volovich used it to reduce a famously long two-loop expression to a few lines of classical polylogarithms, and Duhr and collaborators then built it out into the working coproduct toolkit amplitudes practitioners use today.

When polylogarithms end

Everything above rests on one structural assumption: the derivative closes on $d\log$'s of a finite alphabet of algebraic letters. That is the polylogarithmic condition, and it is a property an integral of the BootLoops class either has or lacks. For the Feynman branch the diagnostic sits in the maximal cutthe integral with every propagator replaced by a delta function — the residue of the graph, which solves the same differential equation with trivial boundary data and so exposes the underlying geometry. When the maximal cut is an algebraic function of the kinematics, the differential equations can be brought to $d\log$ form, and the answer lives in the MPL class; every integral discussed so far on this page is of this kind.

The first integral of the Feynman branch to fall outside — and the classic first failure anywhere on the ladder — is the sunrise: two vertices joined by three massive lines. Its maximal cut is an integral of the form

$$\oint \frac{dz}{\sqrt{Q(z)}},$$

with $Q$ a quartic polynomial whose roots move with the kinematics — a complete elliptic integral, with the elliptic curve $y^2=Q(z)$ underneath. No change of variables rationalizes it, no finite alphabet of $d\log$'s reproduces its derivative, and the massive sunrise has resisted polylogarithmic evaluation since Sabry's calculation in 1962. Add more rungs between the two vertices and the geometry deepens: the three-loop equal-mass banana graph sees a K3 surface — one complex dimension above the elliptic curve — and the $L$-loop banana a Calabi–Yau $(L-1)$-fold, one more dimension with every loop. The ladder of function classes continues past the polylogarithms, and the next rung — what replaces the alphabet when the geometry is an elliptic curve — is the subject of the next page.

Where this shows up in BootLoops

Polylogarithms are the calibration class of the BootLoops program: every step has a mature tool and every answer an independent check. The polylogarithmic warm-ups page runs five classic MPL integrals blind, exactly along the arc of this page: the massless planar double box comes out as a uniform-weight HPL tower on the two-letter alphabet $\{x,\,1+x\}$ with per-weight ansatz dimensions $(1,2,5,11,23)$ at weights 0–4; the three-loop ladder is a weight-six tower of classical polylogarithms with the four coefficients $\{120,-60,12,-1\}$; the two-loop Sudakov ladder is in exact closed form, a pure zeta-value tower of uniform weight through weight twelve, from the family's own reduction onto three Gamma-function masters. Agreement with independent evaluations reaches at least 39 digits, and at least 60 digits against the published closed forms for the fully massless graphs. Those five sit among the fifteen recomputed known integrals on the validated integrals list, and the full thirty-integral portfolio — fifteen known and fifteen, to the author's knowledge, new — is the results paper Polylogarithmic, elliptic, and Calabi–Yau Feynman integrals from a hybrid bootstrap. The statistics branch sits at this ladder's rational floor, computed to the same exact standard: mixture evidences close in factorials and harmonic numbers (mixture models), and tree scores are ratios of explicit whole numbers (phylogenetics).

Each bootstrap step on this page corresponds to a tool there: Landau Alphabet fixes the letter alphabet from the graph's singularities, Ansatzer imposes symbol integrability and the first-entry condition to carve the word space, PSLQ pins the rational coefficients, and Arb with GPLEval evaluates the MPLs to arbitrary precision for the held-out checks. The method paper is Automated computation of Feynman integrals with BootLoops; the story of what happens when the alphabet runs out — the sunrise and the banana tower — is told in Bootstrapping elliptic and Calabi–Yau Feynman integrals. The full paper list is at papers.