**‘Two-mass-easy’ box.** We set $p_2^2 = p_4^2 = 0$, $m_i^2 = 0$ for all $i$ and $s = (p_1 + p_2)^2$ and $t = (p_1 + p_3)^2$. We recall the result given in ref. [100],

$$ \tilde{J}_4(s, t, p_1^2, p_3^2) = \frac{2 c_\Gamma}{\epsilon^2 (st - p_1^2 p_3^2)} \left\{ (-s)^{-\epsilon} + (-t)^{-\epsilon} - (-p_1^2)^{-\epsilon} - (-p_3^2)^{-\epsilon} \right. \\ \left. + \sum_{j=0}^{3} (-1)^j \left( \frac{s+t-p_1^2-p_3^2}{\alpha_j} \right)^\epsilon {}_2F_1 \left( \epsilon, \epsilon; 1+\epsilon; \frac{st-p_1^2p_3^2}{\alpha_j} \right) \right\}, \qquad (B.43) $$

with

$$ \begin{aligned} \alpha_0 &= (p_1^2 - s)(p_1^2 - t) & \alpha_1 &= (p_1^2 - s)(s - p_3^2) \\ \alpha_2 &= (p_3^2 - s)(p_3^2 - t) & \alpha_3 &= (p_1^2 - t)(t - p_3^2). \end{aligned} $$

$$ C_{e_1,e_3} \tilde{J}_4(s, t, p_1^2, p_3^2) = - \frac{2}{(st - p_1^2 p_3^2)} \left[ \frac{1}{\epsilon} - \log \left( \frac{s(p_1^2 p_3^2 - st)}{(s - p_1^2)(s - p_3^2)} \right) \right] + \mathcal{O}(\epsilon), \qquad (B.44) $$

$C_{e_2,e_4} \tilde{J}_4(s, t, p_1^2 p_3^2)$ is obtained by symmetry.

$$ C_{e_1,e_2} \tilde{J}_4(s, t, p_1^2, p_3^2) = \frac{2}{(st - p_1^2 p_3^2)} \left[ \frac{1}{\epsilon} - \log \left( \frac{p_1^2(st - p_1^2 p_3^2)}{(p_1^2 - s)(p_1^2 - t)} \right) \right] + \mathcal{O}(\epsilon), \qquad (B.45) $$

$C_{e_3,e_4} \tilde{J}_4(s, t, p_1^2 p_3^2)$ is obtained by symmetry.

$$ C_{e_1,e_2,e_3,e_4} \tilde{J}_4(s, t, p_1^2, p_3^2) = 2 \frac{e^{\gamma_E \epsilon} \Gamma(1-\epsilon)}{\Gamma(1-2\epsilon)} \frac{(s+t-p_1^2-p_3^2)^\epsilon}{(st-p_1^2 p_3^2)^{1+\epsilon}}. \qquad (B.46) $$

$$ j_4(s, t, p_1^2, p_3^2) = \frac{2}{(st - p_1^2 p_3^2)}. \qquad (B.47) $$

**‘Two-mass-hard’ box.** We set $p_3^2 = p_4^2 = 0$, $m_i^2 = 0$ for all $i$, $s = (p_1 + p_2)^2$ and $t = (p_1 + p_3)^2 \equiv sx$. Given that the triangle with three-external masses $\tilde{J}_3(s, p_1^2, p_2^2)$ appears in the diagrammatic coaction of this diagram, it is convenient to introduce $y$ and $\bar{y}$ such that

$$ y\bar{y} = \frac{p_1^2}{s}, \qquad (1-y)(1-\bar{y}) = \frac{p_2^2}{s}. \qquad (B.48) $$

Up to order $\epsilon^0$, the expression for the two-mass-hard box can be found e.g. in ref. [101]:

$$ \tilde{J}_4(s, t, p_1^2, p_2^2) = (-s)^{-2-\epsilon} c_\Gamma \frac{(y\bar{y})^\epsilon x^{-1-2\epsilon}}{((1-y)(1-\bar{y}))^{-\epsilon}} \left[ \frac{1}{\epsilon^2} + 2\mathrm{Li}_2 \left( 1 - \frac{x}{y\bar{y}} \right) \right. \\ \left. + 2\mathrm{Li}_2 \left( 1 - \frac{x}{(1-y)(1-\bar{y})} \right) - \frac{\pi^2}{12} \right] + \mathcal{O}(\epsilon). \qquad (B.49) $$

$$ C_{e_1,e_3} \tilde{J}_4(s, t, p_1^2, p_2^2) = - \frac{s^{-2-\epsilon}}{x^{1+2\epsilon}} \left[ \frac{1}{\epsilon} + \log(y\bar{y}) + \log((1-y)(1-\bar{y})) \right] + \mathcal{O}(\epsilon) \qquad (B.50) $$
$$
C_{e2,e4} \tilde{J}_4(s, t, p_1^2, p_2^2) = -2 \frac{s^{-2-\epsilon}}{x^{1+2\epsilon}} \left[ \frac{1}{\epsilon} + \log((x - y\bar{y})(x - (1-y)(1-\bar{y}))) \right] + \mathcal{O}(\epsilon) \quad (B.51)
$$

$$
C_{e1,e2} \tilde{J}_4(s, t, p_1^2, p_2^2) = \frac{s^{-2-\epsilon}}{x^{1+2\epsilon}} \left[ \frac{1}{\epsilon} + \log \left( \frac{(1-y)(1-\bar{y})}{y\bar{y}} \right) + 2\log(x - y\bar{y}) \right] + \mathcal{O}(\epsilon) \quad (B.52)
$$

$$
C_{e2,e3} \tilde{J}_4(s, t, p_1^2, p_2^2) = \frac{s^{-2-\epsilon}}{x^{1+2\epsilon}} \left[ \frac{1}{\epsilon} - \log \left( \frac{(1-y)(1-\bar{y})}{y\bar{y}} \right) + 2\log(x - (1-y)(1-\bar{y})) \right] \\
+ \mathcal{O}(\epsilon) \tag{B.53}
$$

$$
C_{e1,e2,e3} \tilde{J}_4(s, t, p_1^2, p_2^2) = -\frac{1}{s^2 x} + \mathcal{O}(\epsilon) \quad (B.54)
$$

$$
C_{e1,e2,e3,e4} \tilde{J}_4(s, t, p_1^2, p_2^2) = 2 \frac{e^{\gamma_E \epsilon} \Gamma(1-\epsilon) (x-\bar{y}(1-y))^{\epsilon} (x-y(1-\bar{y}))^{\epsilon}}{\Gamma(1-2\epsilon) s^{2+\epsilon} x^{1+2\epsilon}} \quad (B.55)
$$

$$
j_4(s, t, p_1^2, p_2^2) = \frac{2}{s^2 x}. \tag{B.56}
$$

Box with a single massive propagator. We set $p_i^2$ for all $i$, $m_2^2 = m_3^2 = m_4^2 = 0$, $s = (p_1 + p_2)^2$ and $t = (p_1 + p_3)^2$:

$$
\begin{equation}
\begin{aligned}
\tilde{J}_4(s, t, m_1^2) = \frac{(-t)^{-\epsilon}}{t(s - m_1^2)} & \left[ \frac{1}{\epsilon^2} + \frac{2}{\epsilon} \log\left(\frac{m_1^2}{m_1^2 - s}\right) - \frac{1}{2} \log^2\left(-\frac{m_1^2}{t}\right) \right. \\
& \left. + G\left(1, 0, -\frac{m_1^2}{t}\right) + 4 \log\left(\frac{s}{t}\right) G\left(-\frac{s}{t}, -\frac{m_1^2}{t}\right) - 4G\left(-\frac{s}{t}, 0, -\frac{m_1^2}{t}\right) \right. \\
& \left. + 4G\left(-\frac{s}{t}, -\frac{s}{t}, -\frac{m_1^2}{t}\right) - \frac{13\pi^2}{12} \right] + \mathcal{O}(\epsilon)
\end{aligned}
\tag{B.57}
\end{equation}
$$

$$
C_{e1} \tilde{J}_4(s, t, m_1^2) = \frac{1}{t(m_1^2 - s)} \left[ -\frac{2}{\epsilon} + 2\log(-t) + \log\left(\frac{m_1^2}{m_1^2 + t}\right) + 4\log\left(\frac{m_1^2 - s}{s}\right) \right] + \mathcal{O}(\epsilon) \quad (B.58)
$$

$$
C_{e1,e3} \tilde{J}_4(s, t, m_1^2) = \frac{2}{t(m_1^2 - s)} \left[ \frac{1}{\epsilon} + 2\log\left(\frac{s}{s - m_1^2}\right) - \log(-t) \right] + \mathcal{O}(\epsilon) \quad (B.59)
$$

$$
C_{e2,e4} \tilde{J}_4(s, t, m_1^2) = \frac{1}{t(m_1^2 - s)} \left[ \frac{1}{\epsilon} + \log \left( \frac{m_1^2(m_1^2 + t)}{t} \right) + 2 \log (m_1^2 - s) \right] + \mathcal{O}(\epsilon) \quad (B.60)
$$

$$
C_{e1,e2,e3} \tilde{J}_4(s, t, m_1^2) = \frac{1}{t(m_1^2 - s)} + \mathcal{O}(\epsilon) \quad (B.61)
$$

$$
C_{e1,e2,e3,e4} \tilde{J}_4(s, t, m_1^2) = 2 \frac{e^{\gamma_E \epsilon} \Gamma(1-\epsilon) s^{\epsilon} (s+t)^{\epsilon}}{\Gamma(1-2\epsilon) t^{1+\epsilon} (s-m_1^2)^{1+2\epsilon}} \quad (B.62)
$$

$$
j_4(s, t; m_1^2) = \frac{2}{t(s - m_1^2)}. \qquad (B.63)
$$
## C The Hopf algebras of graphs and MPLs

In this appendix we present a brief overview and references on Hopf algebras and comodules. We start by a general introduction and then focus on the special instances of Hopf algebras and comodules on cut graphs encountered in this paper. All vector spaces and algebras will always be assumed to be defined over $\mathbb{Q}$.

### C.1 Bialgebras, Hopf algebras and comodules

A bialgebra is an algebra $H$ together with two maps, the coproduct $\Delta: H \to H \otimes H$ and the counit $\varepsilon: H \to \mathbb{Q}$, satisfying the following properties:

1) Coassociativity:

$$ (\Delta \otimes \text{id})\Delta = (\text{id} \otimes \Delta)\Delta. \qquad (C.1) $$

2) $\Delta$ and $\varepsilon$ are algebra homomorphisms:

$$ \Delta(a \cdot b) = \Delta(a) \cdot \Delta(b) \quad \text{and} \quad \varepsilon(a \cdot b) = \varepsilon(a) \cdot \varepsilon(b), \quad \forall a, b \in H. \qquad (C.2) $$

3) The counit and the coproduct are related by

$$ (\varepsilon \otimes \text{id})\Delta = (\text{id} \otimes \varepsilon)\Delta = \text{id}. \qquad (C.3) $$

An element $a \in H$ is called group-like if $\Delta(a) = a \otimes a$.

A Hopf algebra is a bialgebra $H$ together with an involution $S: H \to H$ that satisfies

$$ m(\text{id} \otimes S)\Delta = m(S \otimes \text{id})\Delta = \varepsilon \quad \text{and} \quad S(a \cdot b) = S(b) \cdot S(a) \quad \forall a, b \in H, \qquad (C.4) $$

where $m$ denotes the multiplication in $H$. If $H$ is a Hopf algebra, then a $H$ (right-) comodule is a vector space $A$ together with a map $\rho: A \to A \otimes H$ such that

$$ (\rho \otimes \text{id})\rho = (\text{id} \otimes \Delta)\rho \quad \text{and} \quad (\text{id} \otimes \varepsilon)\rho = \text{id}. \qquad (C.5) $$

### C.2 Incidence Hopf algebras

In this section we discuss a special incarnation of a Hopf algebra, which plays an important role on this paper. More precisely, we present the incidence algebra, which is a simple combinatorial construction on partially ordered sets. The incidence algebra is defined as follows. Let $P$ be a partially ordered set, i.e., a set together with a partial order $\le$ on its elements. If $A \le B$, we define the interval $[A, B] = \{X \in P | A \le X \le B\}$, and we denote by $P_2$ the algebra generated by all intervals $[A, B]$. Then $P_2$ can be turned into a bialgebra, known as the incidence bialgebra [102, 103], with the coproduct

$$ \Delta([A, B]) = \sum_{A \le X \le B} [A, X] \otimes [X, B], \qquad (C.6) $$

and the counit

$$ \varepsilon([A, B]) = \begin{cases} 1, & \text{if } A = B, \\ 0, & \text{otherwise.} \end{cases} \qquad (C.7) $$
It is easy to see that elements of the form $[A, A]$ are grouplike, $\Delta([A, A]) = [A, A] \otimes [A, A]$.

An incidence bialgebra can be turned into a Hopf algebra by the construction of an antipode, if and only if the elements $[A, A]$ are invertible [104]. Let us define $\tilde{P}_2$ as the bialgebra obtained by adding to $P_2$ new grouplike generators $[A, A]^{-1}$, $A \in P$, which are multiplicative inverses of $[A, A]$. Then $\tilde{P}_2$ is a Hopf algebra, and the antipode is unique and involutive (i.e. $S^2 = 1$). It is given by [104]

$$ S([A, B]) = \begin{cases} [A, A]^{-1}, & \text{if } A = B, \\ \sum_{A=X_0 \le X_1 \le \dots \le X_k = B} \frac{(-1)^k}{[A, A]} \prod_{i=1}^{k} \frac{[X_{i-1}, X_i]}{[X_i, X_{i-1}]}, & \text{otherwise.} \end{cases} \quad (C.8) $$

Since $S$ is involutive, this implies $S([A, A]^{-1}) = S^2([A, A]) = [A, A]$.

### C.3 Examples of incidence Hopf algebras

**The Hopf algebra of MPLs.** A first example of an incidence Hopf algebra is the Hopf algebra $\mathcal{H}$ of MPLs (for generic values of the arguments). It is easy to see that the coproduct on MPLs in eq. (2.18) can be cast in the form of eq. (C.6) if we identify the set $P$ with the set of all words $\vec{a}$ formed from the alphabet $\{a_1, \dots, a_n\}$ (including the empty word). The partial order on $P$ is the natural ordering induced by the inclusion of words. We can then identify an interval of words $[\vec{b}, \vec{a}]$ with the function $G_{\vec{b}}(\vec{a}; z)$, and the interval $[\emptyset, \vec{a}]$ is identified with the MPL $G(\vec{a}; z)$.

**The incidence Hopf algebra of pairs of subsets.** A second example of an incidence Hopf algebra is provided by the set $\mathcal{P}(E)$ of all subsets of a set $E$. The subsets are ordered by inclusion, and the set of intervals $\mathcal{P}_2(E)$ is the set of all pairs $[A, B]$ with $A \subseteq B \subseteq E$. The incidence coproduct on $\mathcal{P}_2(E)$ takes the form

$$ \Delta([A, B]) = \sum_{A \subseteq X \subseteq B} [A, X] \otimes [X, B], \quad (C.9) $$

There are two natural families of Hopf subalgebras in $\tilde{P}_2(E)$:

1) Nested subsets containing a given subset $C$. Define $\tilde{P}_2(E, C)$ as the subalgebra generated by elements $[A, B]$ and $[A, A]^{-1}$ with $C \subseteq A \subseteq B \subseteq E$. It is easy to check that $\tilde{P}_2(E, C)$ is a Hopf subalgebra of $\tilde{P}_2(E)$.

2) Nested subsets of a minimum cardinality $k$. For $k$ a positive integer, we define $\tilde{P}_{2,k}(E)$ as the subalgebra generated by elements $[A, B]$ and $[A, A]^{-1}$ with $|A| \ge k$. Here too, $\tilde{P}_{2,k}(E)$ is a Hopf subalgebra of $\tilde{P}_2(E)$.

Further, we can use these Hopf subalgebras to define certain comodules. Suppose that we consider pairs in which the smaller subset is fixed. To be precise, for $C \subseteq E$, we define $\tilde{Q}_2(E, C)$ as the subalgebra of $\tilde{P}_2(E)$ generated by elements $[C, B]$ with $C \subseteq B \subseteq E$. However, $\tilde{Q}_2(E, C)$ is not a Hopf subalgebra. Instead, we can see it as a $\tilde{P}_{2,|C|+1}(E)$
comodule. The coaction is given by the map $\rho : \tilde{\mathcal{Q}}_2(E, C) \to \tilde{\mathcal{Q}}_2(E, C) \otimes \tilde{\mathcal{P}}_{2,|C|+1}(E)$ defined by

$$ \rho([C, B]) = \sum_{\substack{C \subset X \subseteq B \\ X \neq C}} [C, X] \otimes [X, B]. \qquad (C.10) $$

**The incidence Hopf algebra of cut graphs.** It is readily apparent that the coaction on cut graphs defined in eq. (4.1) takes the form of an incidence algebra. The incidence algebra for the family of cut graphs obtained by pinches of a cut graph $(G, E_G)$ is then $\tilde{\mathcal{P}}_2(E_G)$. For a cut graph $(G, C)$, it is $\tilde{\mathcal{P}}_2(E_G, C)$. However, we introduce several modifications:

* The ordering of pairs is reversed compared to the mathematical literature such as eq. (C.6), so that in the body of this paper, the subsets appear on the right.

* As a matter of notation, we write the elements in the form $(G, C)$ rather than $[E_G, C]$, to reinforce the notion that we are acting on graphs with physical interpretations.

* We specifically exclude the empty set from the sum. (Compare the indexing in eq. (4.1) with eq. (C.6), where there is no such restriction, and no special treatment for the case $A = \emptyset$.) In this respect, our operation differs from the mathematical literature on incidence algebras. In fact, this modification means that we are working with the comodule $\tilde{\mathcal{Q}}_2(E_G, C)$ on which the Hopf algebra $\tilde{\mathcal{P}}_{2,|C|+1}(E_G)$ coacts according to eq. (C.10). In other words, we can identify the coaction $\Delta_{\text{Inc}}$ on cut graphs defined in eq. (4.1) with the coaction $\rho$ of eq. (C.10).

This incidence algebra does not involve any information about the graphical structure other than the list of edges.

## D Compactification of cut and uncut one-loop integrals

In ref. [45] it was shown that in order to study cuts of one-loop Feynman integrals, it is convenient to write them as integrals over a compact quadric in the complex projective space $\mathbb{CP}^{D+1}$ [85, 89, 105]. In this appendix we briefly summarize the main results presented there, and we compute the cut integrals necessary to completely determine the differential equations satisfied by a generic one-loop integral as discussed in Section 9.2. Throughout this appendix, we will work in Euclidean kinematics to match ref. [45].

We equip $\mathbb{CP}^{D+1}$ with the bilinear form

$$ (Z_1 Z_2) = z_1^\mu z_{2\mu} - \frac{1}{2} Z_1^+ Z_2^- - \frac{1}{2} Z_1^- Z_2^+, \qquad (D.1) $$

where $z_1^\mu z_{2\mu}$ denotes the usual Euclidean scalar product. If we work in the coordinate patch $Z^+ = 1$, then to each propagator $D_i = (k^E - q_i^E)^2 + m_i^2$ we associate the point $X_i \in \mathbb{CP}^{D+1}$ defined by

$$ X_i = \begin{bmatrix} (q_i^E)^\mu \\ (q_i^E)^2 + m_i^2 \\ 1 \end{bmatrix}, \quad 1 \le i \le n. \qquad (D.2) $$
A one-loop integral is then written as

$$I_n^D = \frac{(-1)^n e^{\gamma_E \epsilon}}{\pi^{D/2}} \int_{\Sigma} \frac{d^{D+2}Y \delta((YY))}{\mathrm{Vol}(GL(1))} \frac{[-2(X_\infty Y)]^{n-D}}{[-2(X_1 Y)] \dots [-2(X_n Y)]}, \quad (D.3)$$

where $\Sigma$ is the real quadric defined by $(YY) = 0$ (take e.g. $Y = [(k^E)^\mu, (k^E)^2, 1]^T$), and we have introduced the lightlike 'point at infinity'

$$X_{\infty} = \begin{bmatrix} 0^{\mu} \\ 1 \\ 0 \end{bmatrix}. \qquad (D.4)$$

In ref. [45], it was shown that all one-loop integrals, cut or uncut, can be written in terms of the same class of functions $Q_n^D$ defined as

$$Q_n^D(X_1, \dots, X_n, X_0) = \frac{(-1)^n e^{\gamma_E \epsilon}}{\pi^{D/2}} \int_{\Sigma} \frac{d^{D+2}Y \delta((YY))}{\mathrm{Vol}(GL(1))} \frac{[-2(X_0 Y)]^{n-D}}{[-2(X_1 Y)] \dots [-2(X_n Y)]}, \quad (D.5)$$

where the integration runs over the real quadric $(YY) = 0$. In the special case where $X_0 = X_\infty$, eq. (D.5) reduces to an ordinary one-loop integral,

$$I_n^D = Q_n^D(X_1, \dots, X_n, X_\infty). \qquad (D.6)$$

The cut integral $C_I I_n^D$, where $C$ is a subset of the propagators in eq. (D.3) has a simple interpretation in $CP^{D+1}$. Let $P_i = \{X \in CP^{D+1} : (Y X_i) = 0\}$ and $P_C = \bigcap_{i \in C} P_i$. Then the cutting operation is, loosely speaking, a projection onto the hyperplane $P_C$: the integration contour is given by $\Sigma \cap P_C$, which is still a quadric, and the points $Y$ and $X_j$ for $j \notin C$ are transformed into $Y_\perp$ and $X'_{C,j}$, defined as

$$Y_\perp = Y - \sum_{i \in C} a_i X_i, \qquad X'_{C,j} = X_j - \sum_{i \in C} a_{ij} X_i, \qquad (D.7)$$

and such that $(Y_\perp X_i) = (X'_{C,j} X_i) = 0$ for $i \in C$. In particular, while $X_\infty$ is lightlike, $X'_{C,\infty}$ might not be. It is then easy to see that, taking $C = \{1, \dots, c\}$,

$$C_C I_n^D = \frac{(-1)^n (2\pi i)^{\lfloor c/2 \rfloor} e^{\gamma_E \epsilon}}{(-2)^c \pi^{D/2} \sqrt{(-1)^c Y_C}} \int_{\tilde{S}_C} \frac{d^{D-c+2} Y_\perp \delta((Y_\perp Y_\perp))}{\mathrm{Vol}(GL(1))} \frac{[-2(X'_{C,\infty} Y_\perp)]^{(n-c)-(D-c)}}{[-2(X'_{C,c+1} Y_\perp)] \dots [-2(X'_{C,n} Y_\perp)]} \\ = \frac{2^{-c} (2\pi i)^{\lfloor c/2 \rfloor}}{\pi^{c/2} \sqrt{(-1)^c Y_C}} Q_{n-c}^{D-c}(X'_{C,c+1}, \dots, X'_{C,n}, X'_{C,\infty}). \qquad (D.8)$$

Here, $Y_C$ is the modified Cayley determinant. Both $Y_C$ and the Gram determinant $\mathrm{Gram}_C$ have simple expressions in terms of the $X_i$:

$$\det(X_i X_j)_{i,j \in C} = \begin{cases} (-1)^c Y_C, & \text{if } \infty \notin C, \\ \frac{(-1)^{c-1}}{4} \mathrm{Gram}_{C \setminus \{\infty\}}, & \text{if } \infty \in C. \end{cases} \qquad (D.9)$$
It is often useful to write the scalar products $(X'_{C,i}X'_{C,j})$ in terms of the $X_i$:

$$ (X'_{C,i} X'_{C,j}) = \frac{1}{(-1)^c Y_C} \det \begin{pmatrix} (X_1 X_1) \dots (X_c X_1) & (X_i X_1) \\ \vdots & \vdots \\ (X_1 X_c) \dots (X_c X_c) & (X_i X_c) \\ (X_1 X_j) \dots (X_c X_j) & (X_i X_j) \end{pmatrix}. \quad (D.10) $$

The functions $Q_n^D$ can easily be written as parametric integrals after combining denominators with Feynman parameters [105]. In particular, in the case where none of the $X_i$ are lightlike, which is the case relevant for cuts of a generic one-loop integral, we find

$$ \begin{aligned} Q_n^D(X_1, \dots, X_n, X_0) = & \\ = & \frac{(-1)^n e^{\gamma_E \epsilon} \Gamma(D/2)}{\Gamma(D-n)} \int [db] \left( \frac{b_0}{\sqrt{(X_0 X_0)}} \right)^{D-n-1} \left( -\sum_{i=0}^n b_i^2 - 2 \sum_{\substack{i,j=0 \\ i<j}}^n b_i b_j u_{ij} \right)^{-D/2}, \end{aligned} \quad (D.11) $$

where

$$ u_{ij} = \frac{(X_i X_j)}{\sqrt{(X_i X_i)(X_j X_j)}}, \qquad \int [db] = \left( \prod_{i=0}^{n} \int_{0}^{\infty} \frac{db_i}{\sqrt{(X_i X_i)}} \right) \delta(1-h(b)), \quad (D.12) $$

with $h(b) = \sum_{i=0}^n h_i b_i$ such that the $h_i \ge 0$ are not all zero (see e.g. ref. [8]). The Y-integration in eq. (D.5) is performed after Feynman parametrization using

$$ I(X) = \int_{\Sigma} \frac{d^{D+2}Y \delta((YY))}{\mathrm{Vol}(GL(1))} [-2(YX)]^{-D} = \frac{\pi^{D/2} \Gamma(D/2)}{\Gamma(D)} [-(XX)]^{-D/2}. \quad (D.13) $$

Note that eq. (D.11) is only valid for $D \ne n$, because otherwise the integrand has a pole at $b_0 = 0$ and a problematic factor of $\Gamma(D-n)$. The case $D = n$, however, is important, because it covers precisely the DCI integrals considered in Section 7. In this case, we see from eq. (D.5) that $Q_n^n$ does not depend on $X_0$ and it is straightforward to repeat the derivation of eq. (D.11) for $D = n$. We find

$$ Q_n^n(X_1, \dots, X_n) = (-1)^n e^{\gamma_E \epsilon} \Gamma(n/2) \int [db] \left( -\sum_{i=1}^n b_i^2 - 2 \sum_{\substack{i,j=1 \\ i<j}}^n b_i b_j u_{ij} \right)^{-n/2}, \quad (D.14) $$

where the integration measure $[db]$ does not involve the variable $b_0$. Equation (D.14) describes an integral over a projective simplex of an integrand with singularities along a quadric. Integrals of this type have been considered by Goncharov in ref. [87].

## D.1 Explicit results for some cut integrals

In this appendix, we present results for the cut integrals needed to obtain the differential equations of one-loop integrals. The maximal cuts are trivial and given by eq. (D.13), so we will simply quote the result below. For the non-maximal cuts, we will need the functions $Q_n^D$ for specific values of $D$ and $n$. For $C_C \tilde{J}_G$ with $n_G$ even, $D = n_G - 2\epsilon$:
• for $n_C = n_G - 1$:

$$Q_1^{1-2\epsilon}(X_1, X_0) = -\frac{\sqrt{\pi} i^{2\epsilon-1}}{\sqrt{(X_1 X_1)}} [1 + 2\epsilon \log(1 + u_{10}) + \epsilon \log(X_0 X_0)] + \mathcal{O}(\epsilon^2). \quad (\text{D.15})$$

• for $n_C = n_G - 2$:

$$Q_2^{2-2\epsilon}(X_1, X_2, X_0) = \frac{i^{2\epsilon-2}}{2\sqrt{(X_1 X_1)(X_2 X_2)}} \frac{1}{\sqrt{u_{12}^2 - 1}} \log \left( \frac{u_{12} + \sqrt{u_{12}^2 - 1}}{u_{12} - \sqrt{u_{12}^2 - 1}} \right) + \mathcal{O}(\epsilon). \quad (\text{D.16})$$

For $\mathcal{C}_C J_G$ with $n_G$ odd, $D = n_G + 1 - 2\epsilon$:

• for $n_C = n_G - 1$:

$$Q_1^{2-2\epsilon}(X_1, X_0) = \frac{i^{2\epsilon-2}}{2\sqrt{(X_0 X_0)(X_1 X_1)}} \frac{1}{\sqrt{u_{10}^2 - 1}} \log \left( \frac{u_{10} + \sqrt{u_{10}^2 - 1}}{u_{10} - \sqrt{u_{10}^2 - 1}} \right) + \mathcal{O}(\epsilon). \quad (\text{D.17})$$

• for $n_C = n_G - 2$:

$$Q_2^{3-2\epsilon}(X_1, X_2, X_0) = \frac{\sqrt{\pi} i^{2\epsilon-3}}{2\sqrt{(X_0 X_0)(X_1 X_1)(X_2 X_2)}} \frac{1}{\Delta(u_{10}, u_{20}, u_{12})} \log \left( \frac{1 + u_{10} + u_{20} + u_{12} + \Delta(u_{10}, u_{20}, u_{12})}{1 + u_{10} + u_{20} + u_{12} - \Delta(u_{10}, u_{20}, u_{12})} \right) + \mathcal{O}(\epsilon). \quad (\text{D.18})$$

where we have defined the function $\Delta(a, b, c) = \sqrt{a^2 + b^2 + c^2 - 2abc - 1}$.

These expressions appear in eq. (D.8) evaluated at specific points $X'_{C,i}$, in combinations that can be written in terms of Gram and Cayley determinants. We will explain how this is done through an example, the next-to-maximal cut of an integral with an even number of propagators. The remaining cases can be treated in an analogous way. For our example, we set $C = [n-1]$ without loss of generality. We must evaluate eq. (D.15) at $X_1 = X'_{C,n}$ and $X_0 = X'_{C,\infty}$. From eqs. (D.9) and (D.10), it is easy to find that

$$(X'_{C,n} X'_{C,n}) = -\frac{Y_{[n]}}{Y_{[n-1]}}, \quad (X'_{C,\infty} X'_{C,\infty}) = \frac{\text{Gram}_{[n-1]}}{4 Y_{[n-1]}}. \quad (\text{D.19})$$

To evaluate $u_{n\infty}(C)$, i.e. $u_{10}$ for $X_1 = X'_{C,n}$ and $X_0 = X'_{C,\infty}$, we first note that

$$\det \begin{pmatrix} (X'_{C,n} X'_{C,n}) & (X'_{C,n} X'_{C,\infty}) \\ (X'_{C,n} X'_{C,\infty}) & (X'_{C,\infty} X'_{C,\infty}) \end{pmatrix} = \frac{\det(X_i X_j)_{i,j \in C \infty}}{(-1)^c Y_C}. \quad (\text{D.20})$$

This is just eq. (7.15) of ref. [45], whose proof was presented there. The determinant on the left-hand side is easily written in terms of $u_{n\infty}(C)$, $(X'_{C,n} X'_{C,n})$ and $(X'_{C,\infty} X'_{C,\infty})$ to finally get

$$u_{n\infty}^2(C) = 1 - \frac{\text{Gram}_{[n]} Y_{[n-1]}}{Y_{[n-1]} \text{Gram}_{[n-1]}}. \quad (\text{D.21})$$
The same procedure allows one to determine all the required $u_{ij}$. The function $\Delta$ appearing in eq. (D.18) is obtained from the determinant of the three-dimensional matrix involving the scalar products of the points $X'_{[n-2],n-1}$, $X'_{[n-2],n}$ and $X'_{[n-2],\infty}$, assuming propagators $n-1$ and $n$ are uncut.

We now combine all the results listed above to give the explicit expressions for the cuts required for full determination of the differential equations of one-loop integrals, as discussed in Section 9.2. For diagrams with an even number of propagators, we find the following results.

* Maximal cut:

$$ C_{EG} \tilde{J}_G = \frac{2^{1-n/2} i^{n/2}}{\sqrt{Y_{EG}}} \left[ 1 + \epsilon \log \left( \frac{\text{Gram}_{EG}}{4 Y_{EG}} \right) \right] + \mathcal{O}(\epsilon^2). \quad (D.22) $$

* Next-to-maximal cut:

$$ C_{EG\setminus e} \tilde{J}_G = - \frac{2^{-n/2} i^{n/2}}{\sqrt{Y_{EG}}} \left[ 1 + 2\epsilon \log \left( 1 + \sqrt{\frac{Y_{EG} \text{Gram}_{EG\setminus e} - \text{Gram}_{EG} Y_{EG\setminus e}}{Y_{EG} \text{Gram}_{EG\setminus e}}} \right) + \epsilon \log \left( \frac{\text{Gram}_{EG\setminus e}}{4 Y_{EG\setminus e}} \right) \right] + \mathcal{O}(\epsilon^2). \quad (D.23) $$

* Next-to-next-to-maximal cut:

$$ \begin{aligned} C_{EG\setminus\{e,f\}} \tilde{J}_G &= \frac{2^{-n/2} i^{n/2}}{\sqrt{Y_{EG}}} \log \left( \frac{\sqrt{Y_{EG\setminus e} Y_{EG\setminus f} - Y_{EG} Y_{EG\setminus\{e,f\}}} + \sqrt{-Y_{EG} Y_{EG\setminus\{e,f\}}}}{\sqrt{Y_{EG\setminus e} Y_{EG\setminus f} - Y_{EG} Y_{EG\setminus\{e,f\}}} - \sqrt{-Y_{EG} Y_{EG\setminus\{e,f\}}}} \right) + \mathcal{O}(\epsilon) \end{aligned} \quad (D.24) $$

For diagrams with an odd number of propagators, we find the following results.

* Maximal cut:

$$ C_{EG} \tilde{J}_G = \frac{2^{(1-n)/2} i^{(n-1)/2}}{\sqrt{\text{Gram}_{EG}}} \left[ 1 + \epsilon \log \left( \frac{\text{Gram}_{EG}}{Y_{EG}} \right) \right] + \mathcal{O}(\epsilon^2). \quad (D.25) $$

* Next-to-maximal cut:

$$ C_{EG\setminus e} \tilde{J}_G = \frac{2^{(1-n)/2} i^{(n-1)/2}}{\sqrt{\text{Gram}_{EG}}} \log \left( \frac{\sqrt{Y_{EG} \text{Gram}_{EG\setminus e} - \text{Gram}_{EG} Y_{EG\setminus e}} - \sqrt{-\text{Gram}_{EG} Y_{EG\setminus e}}}{\sqrt{Y_{EG} \text{Gram}_{EG\setminus e} - \text{Gram}_{EG} Y_{EG\setminus e}} + \sqrt{-\text{Gram}_{EG} Y_{EG\setminus e}}} \right) + \mathcal{O}(\epsilon) \quad (D.26) $$

* Next-to-next-to-maximal cut:

$$ C_{EG\setminus\{e,f\}} \tilde{J}_G = \frac{2^{(1-n)/2} i^{(n-1)/2}}{\sqrt{\text{Gram}_{EG}}} \log \left( \frac{a_1 + a_2 + a_3 + a_4 + a_5}{a_1 + a_2 + a_3 + a_4 - a_5} \right) \quad (D.27) $$
where

$$
\begin{align*}
a_1 &= \sqrt{\mathrm{Gram}_{E_G \setminus \{e,f\}} Y_{E_G \setminus e} Y_{E_G \setminus f}}, \\
a_2 &= \sqrt{\mathrm{Gram}_{E_G \setminus \{e,f\}} (Y_{E_G \setminus e} Y_{E_G \setminus f} - Y_{E_G} Y_{E_G \setminus \{e,f\}})}, \\
a_3 &= \sqrt{Y_{E_G \setminus f} (Y_{E_G \setminus e} \mathrm{Gram}_{E_G \setminus \{e,f\}} - \mathrm{Gram}_{E_G \setminus e} Y_{E_G \setminus \{e,f\}})}, \tag{D.28} \\
a_4 &= \sqrt{Y_{E_G \setminus e} (Y_{E_G \setminus f} \mathrm{Gram}_{E_G \setminus \{e,f\}} - \mathrm{Gram}_{E_G \setminus f} Y_{E_G \setminus \{e,f\}})}, \\
a_5 &= \sqrt{-\mathrm{Gram}_{E_G} Y_{E_G \setminus \{e,f\}}^2}.
\end{align*}
$$

Let us make some comments on the explicit results listed above. First, we stress that they only apply when the cuts are finite. Divergent cuts can be computed through a similar procedure, but they are less interesting for us in the context of Section 9.2, as they correspond to cuts of known integrals. Second, we recall that the cuts defined in ref. [45] are only determined up to an overall signs and modulo $i\pi$. Here, we have fixed the overall signs such that relations like eqs. (2.7) and (2.8) coming from the study of the homology groups of one-loop integrals are satisfied [42, 43]. In particular, we note that the maximal and next-to-maximal cuts of a diagram with even number of propagators, eqs. (D.22) and (D.23), are related by $-1/2$, and that eqs. (D.24) and (D.26) are related by sending one point to infinity (e.g., the one associated with the edge $f$ in (D.24)).

## D.2 Sending a point to infinity

We now make precise the operation of 'sending a point to infinity' [82, 86], which is particularly simple to explain in the compactified picture discussed above. We define $K_G$ as

$$ \tilde{K}_G \equiv \lim_{\epsilon \to 0} \tilde{J}_G, \qquad (D.29) $$

and assume completely generic kinematics so that the limit exists (the only divergent integral is the tadpole, but this is a trivial case that can be dealt with separately). Our discussion extends to integrals with some scales set to zero, as long as they are finite in the limit $\epsilon \to 0$. In the following, we assume that $G$ is a graph with an even number of internal edges, and $G'$ is the same graph after contracting one edge $e$, so that $E_G = E_{G'} \cup \{e\}$ and $n_G = n_{G'} + 1$. The operation of sending a point to infinity relates $\tilde{K}_G$ and $\tilde{K}_{G'}$.

From eq. (D.6), it follows that

$$ \tilde{K}_G = \frac{1}{\pi^{n_G/2}} \int_{\Sigma} \frac{d^{n_G+2}Y \delta((YY))}{\mathrm{Vol}(GL(1))} \frac{1}{[-2(X_1Y)] \dots [-2(X_{n_G}Y)]} \qquad (D.30) $$

For $G'$, we have

$$ \tilde{K}_{G'} = - \frac{1}{\pi^{n_G/2}} \int_{\Sigma} \frac{d^{n_G+2}Y \delta((YY))}{\mathrm{Vol}(GL(1))} \frac{1}{[-2(X_1Y)] \dots [-2(X_{n'_{G}-1}Y)][-2(X_{\infty}Y)]} \qquad (D.31) $$

We thus see that, up to a sign, $\tilde{K}_{G'}$ is obtained from $\tilde{K}_G$ by taking the point $X_e$ associated with an edge $e$ of $G$ to be $X_\infty$. Without loss of generality, we can choose this point to be
$X_{n_G}$, so that

$$ \tilde{K}'_G(X_1, \dots, X_{n'_G}) = -\tilde{K}_G(X_1, \dots, X_{n_G-1}, X_{n_G} = X_\infty). \qquad (D.32) $$

In this paper, we find it convenient to work with integrals normalized to the leading term in the $\epsilon$-expansion of their maximal cuts, as defined in eq. (2.10). Because of this normalization, the relation (D.32) is modified. For $G$, the normalized integral is:

$$ K_G = \frac{\sqrt{Y_{E_G}}}{2^{1-n_G/2} i^{n_G/2}} \tilde{K}_G. \qquad (D.33) $$

For $G'$, it is:

$$ K'_{G'} = \frac{\sqrt{\text{Gram}_{E'_{G'}}}}{2^{1-n_{G'}/2} i^{n_{G'}/2-1}} \tilde{K}'_{G'}. \qquad (D.34) $$

To obtain the equivalent of eq. (D.32) for the normalized integrals, we must determine how the normalization factor in $K_G$ transforms as we take the point $X_{n_G}$ to infinity. We use eq. (D.9) to write the modified Cayley determinant as a determinant of the points $X_i$ and then send $X_{n_G}$ to infinity:

$$ \frac{\sqrt{\det(X_i X_j)_{i,j \in E_G}}}{2^{1-n_G/2} i^{n_G/2}} \xrightarrow[X_{n_G} \to X_\infty]{} \frac{\sqrt{\det(X_i X_j)_{i,j \in E'_{G'} \cup \infty}}}{2^{1-n_G/2} i^{n_G/2}} = \frac{\sqrt{\text{Gram}_{E'_{G'}}}}{2^{2-n_G/2} i^{n_G/2-1}}, \qquad (D.35) $$

where in the last equality we again used eq. (D.9) to relate determinants involving the point at infinity with the usual Gram determinant. We thus find that the relation between the normalized integrals is

$$ K'_{G'}(X_1, \dots, X_{n'_G}) = -2K_G(X_1, \dots, X_{n_G-1}, X_{n_G} = X_\infty). \qquad (D.36) $$

This relation extends trivially to cut integrals, with the restriction that the point that is taken to infinity cannot correspond to a cut edge.

We note that the factor of 2 in the above relation is related to the deformation parameter $1/2$ appearing in the definition of $\hat{J}_G$, in terms of which we write our diagrammatic coaction, see eqs. (4.16) and (4.14). As an illustration of this observation, we recover the differential equation for a finite integral with an odd number of propagators at $\epsilon = 0$ by sending a point to infinity in eq. (9.13), i.e., from the differential equation of a finite integral with an even number of propagators. The same differential equation was obtained directly from the diagrammatic coaction in eq. (9.14). We start from eq. (9.13) and single out the terms containing edge $e$ in the sum over subsets of edges:

$$ dK_G = \sum_{\substack{X \subset E'_{G'} \\ n_X = n_{G'} - 1}} K_{GX} dC_X K_G + \sum_{\substack{X \subset E'_{G'} \\ n_X = n_{G'} - 2}} K_{GX_e} dC_{X_e} K_G, \qquad (D.37) $$

where $E_G = E_{G'} \cup \{e\}$ and $X_e \equiv X \cup \{e\}$. The cuts in the second term on the right-hand side involve the edge $e$, but using eq. (7.3) they can be rewritten in terms of cuts not involving $e$,

$$ C_{X_e} K_G = -2 C_X K_G - \sum_{e' \in E'_{G'} \setminus X} C_{X_e'} K_G. \qquad (D.38) $$
We can now rewrite eq. (D.37) in such a way that the edge $e$ is never cut in any of the contributions in the right-hand side:

$$dK_G = \sum_{\substack{X \subset E_{G'} \\ n_X = n_{G'} - 1}} K_{G_X} dC_X K_G + \sum_{\substack{X \subset E_{G'} \\ n_X = n_{G'} - 2}} K_{G_{X_e}} \left( -2 dC_X K_G - \sum_{e' \in E_{G'} \setminus X} dC_{X_e'} K_G \right). \quad (D.39)$$

We can now use eq. (D.36) and take the point corresponding to edge $e$ to infinity to obtain

$$dK'_{G'} = \sum_{\substack{X \subset E_{G'} \\ n_X = n_{G'} - 1}} K'_{G_X} dC_X K_{G'} + \sum_{\substack{X \subset E_{G'} \\ n_X = n_{G'} - 2}} K'_{G_X} \left( dC_X K_{G'} + \frac{1}{2} \sum_{e \in E_{G'} \setminus X} dC_{X_e} K_{G'} \right), \quad (D.40)$$

which reproduces eq. (9.14). While in eq. (9.14) the factor of 1/2 appeared directly from the diagrammatic coaction, in the derivation above it appeared because of the relation (D.36). For the operation of sending a point to infinity to be consistent with the diagrammatic coaction, the deformation parameter must thus be consistent with the coefficient appearing in eq. (D.36).

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