Function Spaces, Symbols, and Coaction Patterns

Many dimensionally regulated loop integrals—and the amplitudes assembled from them—occupy much smaller function spaces than dimensional analysis alone would allow. In polylogarithmic sectors, iterated integrals are organized by a symbol alphabet, transcendental weight, integrability, entry conditions, and discontinuities. Coaction constructions can relate functions and cuts in specified integral classes. None of these statements is universal: a symbol loses constants and branch data, and elliptic or more general periods require a larger language.

Required background. Differential Equations for Master Integrals supplies the master-integral system and boundary conditions.

Helpful background. Asymptotic Scales, Remainders, Uniformity, and Optimal Truncation supplies controlled limit expansions used to fix boundary data.

Iterated integrals and alphabets

Given dimensionless, nonvanishing functions $\phi_a(x)$ called letters, an iterated integral is built recursively from logarithmic one-forms $\mathrm d\log\phi_a$. Dimensionful quantities may appear only through ratios; multiplying a letter by a nonzero constant does not change its logarithmic differential. In a kinematic patch and basis where an $\epsilon$-form exists, a canonical differential equation for a vector of master integrals has the form

$$\mathrm d\boldsymbol I(x,\epsilon) =\epsilon\sum_a A_a\,\mathrm d\log\phi_a(x) \boldsymbol I(x,\epsilon),$$

with constant matrices $A_a$. Expanding in $\epsilon$ produces iterated integrals whose entries are drawn from the alphabet

$$\mathcal A=\{\phi_1,\ldots,\phi_N\}.$$

The existence of this form is an additional property of the chosen integral family and basis, not a theorem for arbitrary master-integral systems. Henn’s original proposal derives the iterated-integral solution once the $\epsilon$-factorized form is reached and illustrates it for planar massless two-loop $2\to2$ master integrals Henn 2013, pp. 251601-1–251601-4.

Zeros and poles of the letters are candidate singular loci. They need not all be physical: spurious letters can cancel in the complete amplitude, and algebraically related letters can describe the same locus. The alphabet also depends on variables and algebraic extensions.

For example,

$$\mathrm d\operatorname{Li}_2(z) =-\log(1-z)\,\mathrm d\log z,$$

so its symbol is

$$\mathcal S(\operatorname{Li}_2 z) =-(1-z)\otimes z.$$

The rightmost entry records the final differential in this convention. Authors sometimes reverse tensor order, so the convention must be declared before comparing symbols.

Symbol, weight, and integrability

For a pure weight-$w$ iterated integral $F$, the symbol is a rank-$w$ tensor

$$\mathcal S(F) =\sum_{a_1,\ldots,a_w} c_{a_1\cdots a_w} \phi_{a_1}\otimes\cdots\otimes\phi_{a_w}.$$

Not every tensor integrates to a function. Integrability requires, at each adjacent slot,

$$\sum_{a_1,\ldots,a_w} c_{a_1\cdots a_w} \mathrm d\log\phi_{a_k}\wedge \mathrm d\log\phi_{a_{k+1}} \,T_k(a_1,\ldots,a_w)=0,$$

where $T_k$ is the ordered tensor product of all letters except the two tested slots $k$ and $k+1$. The condition is imposed for every $k=1,\ldots,w-1$. Linear equations implementing it can reduce a symbol ansatz dramatically.

Transcendental weight counts the length of an iterated integral in a pure basis: logarithms have weight one, $\operatorname{Li}_n$ weight $n$, and products add weights. Dimensional regularization can mix weights through rational prefactors and the $\epsilon$ expansion. Uniform weight is a property of selected integrals and theories, not a universal law of amplitudes.

The symbol definition modulo multiplicative constants, the polylogarithm example above, and the resulting beyond-the-symbol ambiguity are exhibited in a two-loop six-point planar $\mathcal N=4$ super-Yang–Mills application by Goncharov et al. 2010, pp. 151605-2–151605-3. That example does not by itself establish the same function space for other theories, multiplicities, or loop orders.

Physical entry conditions

The first entry is tied to possible branch discontinuities. If an amplitude can develop a discontinuity only when a physical invariant $s_I$ crosses a threshold, a first-entry condition restricts the first slot to functions of the allowed $s_I$. Subsequent entries encode iterated discontinuities and differential information.

This condition is necessary but not sufficient for a physical amplitude. Ordinary Steinmann relations forbid double discontinuities in overlapping channels and therefore constrain the first two symbol entries when the symbol/discontinuity dictionary applies. Their concrete use in the planar six-gluon $\mathcal N=4$ super-Yang–Mills hexagon-function space is demonstrated by Caron-Huot et al. 2016, pp. 241601-1–241601-4. “Extended Steinmann” restrictions on later adjacent slots are additional patterns established only in particular amplitude families. Final-entry restrictions may follow from supersymmetry or differential equations in special theories. Every such rule has a domain; it should be derived or cited rather than promoted to a universal alphabet law.

Discontinuities, generalized cuts, and differential equations provide complementary information:

$$\text{cuts}\longleftrightarrow\text{discontinuities}, \qquad \mathrm dF\longleftrightarrow\text{last entries}, \qquad \text{boundary values}\longleftrightarrow\text{symbol kernel}.$$

A viable bootstrap combines all three.

What the symbol forgets

Constants such as $\pi^2$ or $\zeta_n$ have zero symbol, and products containing them can lie partly in the symbol kernel. The symbol also does not select a branch, an $i0$ prescription, or a base point for iterated integration. Thus

$$\mathcal S(F)=\mathcal S(G)$$

implies only that $F-G$ is invisible to the chosen symbol map, not that the functions are equal.

To reconstruct an amplitude, fix:

• beyond-the-symbol constants and lower-weight products;
• a Euclidean or other base region;
• analytic-continuation paths and $i0$ prescriptions;
• boundary values at regular points or controlled limits;
• rational and algebraic prefactors.

Numerical comparison in one region cannot establish the correct monodromy elsewhere.

Coaction-like organization

Multiple polylogarithms possess a Hopf-algebra coproduct, and some Feynman-integral families exhibit relations in which functions, discontinuities, and cuts organize into paired tensor factors. For dimensionally regulated one-loop scalar integrals, Abreu and collaborators propose a diagrammatic coaction whose first entries are pinched graphs and whose second entries are cut graphs. The central graph-to-function identity is explicitly a conjecture, supported there by nontrivial one-loop examples rather than proved for arbitrary integrals Abreu et al. 2017, § 4.3, pp. 20–21; § 6, pp. 25–34.

In amplitude practice, “coaction principle” can mean the empirical closure of a chosen space under appropriate coproduct components, discontinuities, or derivatives. This is strong ansatz evidence, not automatically a theorem for every multiloop integral or quantum field theory. State the function class and precise coaction being used.

Elliptic and more general sectors

The two-loop sunrise integral with nonzero internal masses already contains an elliptic curve. In two spacetime dimensions it is ultraviolet finite; its second-order homogeneous differential equation has elliptic periods, and the arbitrary-mass answer is written as a period times elliptic dilogarithms rather than ordinary multiple polylogarithms Adams, Bogner, and Weinzierl 2014, §§ 3, 5, 8–9, pp. 4–11. This example is an obstruction to an ordinary rational-letter multiple-polylogarithm ansatz, not a claim that every two-loop integral is elliptic.

In elliptic sectors one may use elliptic multiple polylogarithms, modular forms, periods, and kernels adapted to the curve. The ordinary symbol can still describe polylogarithmic subpieces, but it cannot encode the full answer. More complicated geometries can lead beyond elliptic functions.

The frontier evidence table therefore distinguishes established polylogarithmic symbol calculus from a conjectural one-loop diagrammatic coaction and an explicit elliptic obstruction. This classification was checked against the cited primary papers on 9 August 2026; it does not assert a complete function classification for general amplitudes.

A bootstrap workflow

For a polylogarithmic candidate amplitude:

1. derive or infer the alphabet from differential equations and singular loci;
2. impose integrability on a weight-graded symbol ansatz;
3. apply physical first-entry, symmetry, crossing, and known adjacent-entry constraints;
4. match generalized cuts, discontinuities, and differential equations;
5. integrate the symbol in a declared function basis;
6. fix beyond-the-symbol terms from boundary values and limits;
7. continue with an explicit branch prescription and compare independent numerical points;
8. test whether elliptic periods appear before claiming completeness of the polylogarithmic ansatz.

Common pitfalls

Equating a symbol with a function. Constants, branch choices, and boundary values are missing. Fix them independently.

Calling every singular letter physical. Variable choices and intermediate integrals can introduce spurious letters. Check the complete amplitude’s discontinuities.

Assuming uniform maximal weight universally. Masses, rational prefactors, matter content, and non-supersymmetric sectors can mix weights.

Forcing an elliptic integral into a polylogarithmic alphabet. Analyze the maximal cut or differential equation’s homogeneous solutions before choosing the function space.

Check your understanding

Differentiate $\operatorname{Li}_2(z)$ to recover the symbol $-(1-z)\otimes z$ in the page’s tensor-order convention. Then add $\pi^2/6$ and explain why the symbol is unchanged although the functions differ.

Where to continue

• Differential Equations for Master Integrals constructs the systems from which alphabets and boundary data arise.
• Generalized Unitarity and Integrand Reduction supplies discontinuity and cut constraints.
• Color–Kinematics Duality contains the shared evidence table.

References

• Abreu, Samuel, Ruth Britto, Claude Duhr, and Einan Gardi. “Diagrammatic Hopf Algebra of Cut Feynman Integrals: The One-Loop Case.” Journal of High Energy Physics 12 (2017): 090. DOI. Open preprint.
• Adams, Luise, Christian Bogner, and Stefan Weinzierl. “The Two-Loop Sunrise Graph in Two Space-Time Dimensions with Arbitrary Masses in Terms of Elliptic Dilogarithms.” Journal of Mathematical Physics 55 (2014): 102301. DOI. Open preprint.
• Caron-Huot, Simon, Lance J. Dixon, Andrew McLeod, and Matt von Hippel. “Bootstrapping a Five-Loop Amplitude Using Steinmann Relations.” Physical Review Letters 117 (2016): 241601. DOI. Open preprint.
• Goncharov, Alexander B., Marcus Spradlin, Cristian Vergu, and Anastasia Volovich. “Classical Polylogarithms for Amplitudes and Wilson Loops.” Physical Review Letters 105 (2010): 151605. DOI. Open preprint.
• Henn, Johannes M. “Multiloop Integrals in Dimensional Regularization Made Simple.” Physical Review Letters 110 (2013): 251601. DOI. Open preprint.