Fifteen Eighty Four

The quantum world is governed by a large number of different energy or length scales, as clearly seen in the hydrogen atom, where an essentially point like electron is bound to a proton. The proton size itself is 10^-15 m, while the spatial extension of the hydrogen atom is about 10^-10 m. So at leading order, all one needs to know to calculate the energy spectrum of the hydrogen atom is the electron mass m_e and the fine-structure constant α, that governs the strength of the electromagnetic interaction between the electron and the proton.

But this result is not exact, there are further corrections from the electromagnetic interaction as well as from the proton structure, through the reduced mass of the 2-body system as well as the proton size. All these corrections are given in powers of (m_eα)² and m_e/m_{p ,} with m_p the proton mass, and can be  systematically calculated in an Effective Field Theory (EFT).

This method can be extended to systems and reactions, where exact solutions like in the case of the hydrogen atom are not known. The strong interactions are the main playground for such EFTs as they are incredibly difficult to solve, even with today’s biggest supercomputers. The basic idea of such an EFT is borrowed from quantum field theory, namely an expansion in tree graphs, one-loop diagrams, two-loop diagrams and so on. However, in contrast to the commonly used expansion in a small coupling constant, here one expands in small momenta and/or particle masses, divided by some large scale Λ, at which the EFT is no longer applicable. To make this work, a scale separation between the low-energy modes and the high-energy modes ~ Λ is mandatory.

Equally important are the symmetries and their realization for the system under consideration, as this severely limits the possible operators in the EFT. A simple example is parity invariance, which requires that a momentum operator p appears only in even powers like p², p⁴, … in the EFT. Finally, one has to work with the relevant degrees of freedom, which for the strong interactions at low energies are pions and nucleons, not the fundamental fields, the quarks and gluons.

In our book, we discuss in detail the foundations of such EFTs of the strong interactions, with an emphasis on the sector of the light quarks: up, down and strange. Selected examples are worked out in detail that show how this powerful machinery works. We believe that this book fills a gap in the available literature and would welcome comments by our readers.

Title : Effective Field Theories

ISBN: 9781108476980

Authors: Ulf-G Meißner and Akaki Rusetsky