by Bhavna Prasad
Guided by Rutherford’s gold foil experiment in the early 1900’s, one has come to picture
a proton or a neutron as a tiny solid particle, something like a minuscule billiard ball,
placed at the center of an atom. In reality, dominating the maximum fraction of the visible
universe, these particles are far from simple. They are dynamical systems made of quarks
and gluons that constantly interact through the strong nuclear force. Perhaps one of the
simplest questions one can ask about them is, how big are protons and neutrons, really?
The answer comes from a quantity known as the charge radius, and today one of the most
powerful methods for computing it theoretically is a technique called lattice QCD.
The charge radius does not correspond to a simple measurement of a sphere’s radius, nor
does it represent a hard edge or physical boundary. Instead, it is roughly defined by the
slope of the Fourier transform of the electric charge distribution within the particle at zero
momentum transfer. In the case of the proton, which carries a positive electric charge, the
charge radius characterizes the spatial distribution of this positive charge. For the neutron,
which is electrically neutral overall, the negative charge radius reflects the distribution of the
negatively charged down quark, being located further from the center than the positively
charged up quark.
Physicists measure charge radii by scattering electrons off of nucleons and capturing how
the electrons are deflected, encoded in an observable known as scattering cross section. One
of the first attempts to do this was in the 1950’s. However, different experiments have
produced different results, which led to what is known as the Proton radius puzzle.
Inevitably, we want to probe this theoretically using Quantum Chromodynamics (QCD).
QCD is the theory governing the dynamics of quarks and gluons. Unlike electromagnetism,
the strong force becomes extremely strong at the energy scales relevant for nucleons. This
makes straightforward calculations impossible with standard methods. This is where lattice
QCD enters the story. Lattice QCD is a way of solving QCD using computers by turning
space and time into a grid, or lattice. One begins by replacing continuous space and time
into discrete points. Quarks live on the vertices of this grid, and gluons live on the links
connecting those points. The equations of QCD are then translated into a form suitable for
large-scale numerical simulations.
One then uses powerful supercomputers to simulate the behavior of quarks and gluons
and extract physical properties of nucleons. To access the charge radius, we study how
a nucleon responds to an electromagnetic probe, much like in scattering experiments, but
inside a computer simulation. The key steps for doing this are:
- Create a proton- or neutron-like particle on the lattice using the quark and gluon
fields. - Insert an electromagnetic current synonymous to virtual photon interaction in electron-nucleon
scattering in experiments. - Measure the electric form factors, encoding the structure of nucleons with varying
momentum transfer. - Finally, extract the charge radius related to how the electric form factor changes at very low momentum.
It may seem that this conceptually simple idea would allow for an easy implementation,
but it turns out that the actual computation involves significant effort to attain meaningful
control over the numerical uncertainties. When we move from a continuous space-time
onto a discrete grid, we break natural symmetries of the physical world. So, one tries to
mimic it as closely as possible by taking very fine grids, thereby greatly increasing the
computational effort required, which is starting to become more tractable with modern
supercomputers. Calculations are then repeated using progressively finer grids and larger
grid-size, and extrapolated to obtain the physical continuum. Despite these challenges,
steady progress over the past two decades has led to increasingly precise determinations of
nucleon charge radii.
While charge radii are an example, numerous other observables have been made available
using lattice QCD. It is thus transforming our understanding of protons and neutrons from
abstractly defined dynamical systems into numerically tractable and improvable results,
thereby continually informing our understanding of how the strong force shapes the size and
structure of matter itself.