Speaker
Description
Amplitude are geometric objects and we have ambitions to classify them. As functions of discrete indices of color, charge, helicity, ..., and spin, they are tensors on the Fock spaces of elementary particles. Linear spaces, like the Fock spaces, they have automorphisms under which a given tensor is transformed into another in the equivalent class. In this sense, equivalent classes of tensors are orbits under the action of linear automorphism groups, it is thus natural to represent each equivalent class of tensors by normal forms composed of `invariants'. E.g., matrices under congruence they classify $(0,2)$-type bilinear forms, while under similarity they classify $(1,1)$-type linear endomorphisms, two such $(1,1)$ forms are equivalent if and only if their corresponding characteristic matrices share the same set of Smith invariant factors over a polynomial PID (principal ideal domain). On the other hand, amplitude in massive theory is quite often associated with families of elliptic curves parameterized by the Mandelstam variables. Just like automorphisms of linear spaces leads to the transformation between equivalent tensors, isomorphisms of elliptic curves lead to the modular transformation of the symbol letters. Just like equivalent classes of tensors are represented by normal forms, the space of equivalent classes of elliptic curves are represented by fundamental domains of moduli curves (spaces) $\Gamma\backslash\mathbb{H}$. In this spirit, we are aimed at a classification of different scattering processes according to the underlying moduli space, deriving potential dimension formulas and function bases for the symbol letters. In this talk, I report a first step towards the land of amplitude beyond genus one -- the complete analytic computation of the 2-loop Bhabha scattering in Quantum Electrodynamics. I will start from the correspondence between modular curves $\Gamma\backslash\mathbb{H}$ and elliptic moduli space with level structures, and then use the sunrise family to show how we parameterize the punctured Riemann sphere through principal modular function, and eventually jump over to modular parameterization of elliptic K3 surface through the universal curves $\mathcal{E}_{\Gamma_1(4)\backslash\mathbb{H}}$. I will highlight the choice of a \emph{universal rational marked point} inspired by \textrm{Mordell–Weil theorem}, and show that Bhabha scattering and planar top quark production at sector 79 are described by the same moduli space of $\mathscr{M}_{1;2}[4]$ with $4$-level structures, thus should be represented by the same class of function space.
