Bananas: multi-edge graphs and their Feynman integrals

TL;DR

The paper develops a recursive, dispersion-based framework for banana graphs $b_n$, expressing the principal-imaginary part $\Im(\Phi_R^D(b_n))$ as an iterated one-dimensional integral built from the seed $\Im(\Phi_R^D(b_2))$ and evolving to $V_n^D$ through $(n-2)$ nested integrals. It analyzes threshold and pseudo-threshold structures, revealing how monodromy and phase-space geometry underpin the integrals, including elliptic behavior in low dimensions (e.g., $b_3$) and more intricate non-elliptic features for higher $n$ (e.g., $b_4$). The work then connects these iterated-integral representations to differential equations, integration-by-parts identities, and master-integral counting, providing both a concrete computational scheme and a geometric interpretation of the master-set size $2^n-1$. Throughout, the authors illustrate how dispersion and iterative integration yield a scalable approach to compute $\Phi_R^D(b_n)$ across dimensions, while highlighting the role of thresholds, pseudo-thresholds, and the algebraic structure arising from successive square-root nestings.

Abstract

We consider multi-edge or banana graphs $b_n$ on $n$ internal edges $e_i$ with different masses $m_i$. We focus on the cut banana graphs $\Im(Φ_R(b_n))$ from which the full result $Φ_R(b_n)$ can be derived through dispersion. We give a recursive definition of $\Im(Φ_R(b_n))$ through iterated integrals. We discuss the structure of this iterated integral in detail. A discussion of accompanying differential equations, of monodromy and of a basis of master integrals is included.

Figures (7)

• Figure 1: Banana graphs $b_n$ on $|b_n|=(n-1)$ loops. We indicate momenta at internal edges $e_1,\ldots e_n$ labeling from top to bottom. We assign mass square $m_i^2$ to edge $e_i$. A positive infinitesimal imaginary part is understood in each popagator. Both vertices have an external edge with incoming momenta $k_n$ and $-k_n$. Note that edges $e_1, \ldots,e_j$, $n>j\geq 2$ constitute a banana graph $b_j$ with external momentum $k_{j}$ flowing through. It is a $(j-1)$-loop subgraph of $b_n$. In particular we have a sequence $b_2\subset b_3\subset\cdots \subset b_n$ of graphs which gives rise to an iterated integral.
• Figure 2: The bubble $b_2$. It gives rise to a function $\Phi_R^D(b_2)(k_2^2,m_1^2,m_2^2)$. We compute its imaginary part $\Im\left(\Phi_R^D(b_2)(k_2^2,m_1^2,m_2^2)\right)$ below. It starts an induction leading to the desired iterated integral for $\Im(\Phi_R^D(b_n))$. The edges $e_1,e_2$ are given in red or blue. Shrinking one of them gives a tadpole integral $\Phi_R^D(t_1)(m_1^2)$ (red) or $\Phi_R^D(t_2)(m_2^2)$ (blue).
• Figure 3: We indicate momenta and masses at internal edges from top to bottom. We now also indicate momentum $s_n^j$ for edges $e_2,\ldots, e_n$. The mass-shell conditions encountered in the computation of $V_n^D$ enforce $k_j^2=s_n^{n-j}$ for $2\leq j\leq n$. Eq.(\ref{['tensors']}) simply expresses the fact that $-2k_j\cdot k_{j+1}=(k_{j+1}-k_j)^2-k_{j+1}^2-k_j^2$ with $(k_{j+1}-k_j)^2=m_{j+1}^2$.
• Figure 4: The doubling of propagators indicated by a dot on the edge creates a problem.
• Figure 5: The graph $b_3$ and its triangular cell $C_3$. The codimension-one boundaries (sides) are given by the condition $A_i=0$, indicated in the figure by $i=0$, $i\in\{1,2,3\}$. The graph $b_3$ with two yellow leaves as external edges is put in the barycenter. All its edges are put on-shell. The cell decomposes into six sectors $m_iA_i>m_jA_j>m_kA_k$ as indicated by $i>j>k$. The lines $m_iA_i=m_jA_j$ (indicated by $i=j$) start at the midpoint $\mathrm{mid}_{i,j}:\, A_k=0,\,A_im_i=A_jm_j$ of the co-dimension one boundary $A_k=0$ and pass through the barycenter $\mathrm{bc}:\,m_1A_1=m_2A_2=m_3A_3$ towards the corner $c_k:\,A_i=A_j=0$, labeled $k$. Such corners are removed. For these three lines the three intervals $[\mathrm{mid}_{i,j},\mathrm{bc}]$ from the midpoints of the sides to the barycentre of the cell form the spine. It indicated in turquoise. The bold hashed line indicated by $2<3$ (so $m_2A_2<m_3A_3$) on the left and $2<1$ (so $m_2A_2<m_1A_1$) on the right is an example of a fibre over one (the vertical) part (on the $1=3$-line) of the spine (the turquoise line from $m_1A_1=m_3A_3,A_2=0$ to the barycentre). On the left, along the fibre the ratio $A_2/A_3<m_3/m_2$ is a constant, on the right similarly. Finally, to the two yellow leaves we assign incoming four-momenta $k_3,-k_3$ with $k_3^2=s$. The spine partitions the cell $C_3$ into three 2-cubes, boxes $\Box(j)$ with four corners for any $\Box(j)$: $\mathrm{mid}_{i,j},\mathrm{bc},\mathrm{mid}_{j,k}, c_j$. For each such box $\Box(j)$ there is a diaginal $d_j$. It is a line from a corner to the barycenter: $\mathrm{d}_j:\,]c_j,\mathrm{bc}]$ for which we have $m_iA_i=m_kA_k$. We assign to this diagonal $\mathrm{d}_j$ a graph for which edges $e_i,e_k$ are on-shell and edge $e_j$ carries a dot. Along the diagonal $\mathrm{d}_j$ we have $A_jm_j>(A_im_i=A_km_k)$.

Theorems & Definitions (17)

• Remark 1.1
• Remark 2.1
• Theorem 2.2
• proof
• Remark 2.3
• Proposition 3.1
• proof
• Proposition 3.2
• proof
• Remark 3.3