MSc defense: Andreas Forum

Speaker: Andreas Forum

Title: Symbols and Coaction of $l$-loop Sunrise Integrals


We construct a symbol and coaction operator on $l$-loop equal-mass sunrise master integrals. Our construction allows us to define a coaction through the symbol operator. The symbol operator itself is defined on a pair of forms $[\xi_i,\xi_j]$, who are elements of a unipotent matrix $T_N$. To find the unipotent matrix $T_N$, one needs to consider a unipotent Gauss-Manin type differential equation with a nilpotent connection $N$. The unipotent matrix $T_N$ is then given by considering a bar construction power series of a this connection. The equal-mass sunrise master integrals themselves do not satisfy such a unipotent Gauss-Manin differential equation. Instead we expand our basis with $l$ number of $\tau_i$'s for $0 \leq i\leq l-1$, defined as the ratio between two periods in the frobenius basis, such that the equal-mass sunrise integrals are part of a solution which satisfy the unipotent differential equation. We provide a general prescription for $T_N$ at arbitrary loop order $l$. \\
We give explicit examples of the coaction and symbols for our equal-mass integrals at $l=2,4$. We find that for $l=2$ the symbol length is two and for $l\geq 3$ the symbol length is three. We comment on the meaning of these results and compare our findings for $l=2$ with earlier works by Bönisch et al. [1803.10256]. \\
Finally we comment on further improvements and discuss how to generalise the answer to generic mass.