The Proton Radius and Other Aspects of Nucleon Structure From Lattice QCD

Principal Investigator:
Jeremy Green

Theoretical Physics Department, CERN, Geneva, Switzerland

Local Project ID:

HPC Platform used:

Date published:

Protons and neutrons are basic building blocks of ordinary matter, as are electrons. Unlike electrons, they are composite particles, bound by the strong force. As a result, protons have a rich internal structure and nonzero size. There has been a decades-long experimental effort to understand properties of the proton such as its internal distributions of charge, spin, and momentum. However, these properties can also be determined from theory.

Within the Standard Model of particle physics, the strong force is described by quantum chromodynamics (QCD), the theory of quarks and gluons. In principle, all of the properties of protons and neutrons can be predicted by the handful of parameters needed to specify QCD: an overall scale, plus the masses of the quarks. However, performing calculations in QCD is challenging. The most comprehensive approach for studying particles like the proton is lattice QCD, a way of transforming the theory into a finite problem suitable for numerical calculations. Lattice QCD works by representing the three dimensions of space and one of time using a discrete lattice (or grid). Although the problem size is finite, it can still be very large: this project used lattice sizes of 24³×48, 32³×48, 48⁴, and 64⁴. This approach is well suited for calculations on supercomputers: each computing node is assigned a small chunk of the whole lattice, and network communications are used when data from outside the local chunk are needed.

In addition to the parameters of QCD, two additional parameters govern lattice QCD: the lattice spacing a and the box size L. (The number of sites in each spatial dimension is L/a.) The connection with true QCD can be made by repeating a calculation with many values of a and L, then extrapolating a to zero and L to infinity, but this brute force approach is very computationally expensive.

One interesting property is the proton’s radius, which was the subject of a disagreement between experiments a decade ago. (To be more precise, this refers to the charge radius, i.e. the average radius of the charge distribution of the proton.) Formally, the definition of the proton radius is based on a function called the electric form factor. In experiments, this function is mapped out by performing elastic scattering of electrons off protons; its value depends on the momentum transferred between the two particles. (Here “elastic” means scattering without creating new particles.) The proton radius is obtained from the slope of the electric form factor at zero momentum transfer. In lattice QCD, one of the effects of a calculation with finite box size is that only a discrete set of momenta can exist in the box. This means that the smallest available nonzero momentum transfer is generally (2π/L)², which limits the accuracy of the radius that can be obtained.

Another pair of important properties are the magnetic moments of the proton and neutron. Like the electron, both of these particles act like tiny magnets. In experiments, the magnetic moments (i.e. the strengths of the magnets) can be measured very precisely. For electrons, this leads to a very precise test of the Standard Model. On the other hand, lattice QCD calculations are required to predict the proton and neutron magnetic moments, and such extreme precision cannot be achieved. Instead, these calculations can serve as tests of lattice QCD methodology. The magnetic moments are typically determined by computing a function called the magnetic form factor at nonzero momentum transfer. One must then extrapolate to zero momentum transfer, where the magnetic form factor equals the magnetic moment.

A major part of this project focussed on investigating a new approach for directly obtaining the proton radius and magnetic moment at zero momentum transfer, without fitting or extrapolation. This approach is a variant of an earlier method developed by the same project team. A key feature of these methods is that the distortions caused by the finite box size are expected to shrink exponentially with L. It was found that the new variant method yields much smaller statistical fluctuations for the radius than the original version, making it more practical for obtaining precise results [1]. Subsequently, the dependence on the box size was numerically studied by performing calculations using two lattices that differ only in L [2]. Preliminary results indicate that for typical box sizes, the finite-L distortions may not be negligible (Figures 1 and 2).

A side project investigated a potential methodology improvement with widespread applicability. Lattice QCD calculations of proton structure all have to deal with a major nuisance: an exact proton state cannot be created. Instead, one can only create a mixture of the proton and some unwanted “excited states”, and the excited states must be filtered out [3]. This filtering can introduce considerable noise into the calculation. The side project studied a computationally inexpensive way of removing some of the excited states [4]. Unfortunately, the outcome was rather mixed: one quantity showed a substantial reduction of excited-state contributions (see Figure 3) but some others showed an increase. The results indicate that a more comprehensive and systematic approach is needed to remove excited states.


  1. [1]  N. Hasan, “A lattice QCD study of nucleon structure with physical quark masses,” Ph. D. thesis, University of Wuppertal (2019) doi:10.25926/19rj-ej28

  2. [2]  J. Green, “Electromagnetic form factors and the proton radius,” Talk presented at Advances in Lattice Gauge Theory 2019, indico.cern.ch/event/790129/contributions/3499760/

  3. [3]  N. Hasan, J. Green, S. Meinel, M. Engelhardt, S. Krieg, J. Negele, A. Pochinsky and S. Syritsyn, “Nucleon axial, scalar, and tensor charges using lattice QCD at the physical pion mass,” Phys. Rev. D 99, 114505 (2019) doi:10.1103/PhysRevD.99.114505

  4. [4]  J. R. Green, M. Engelhardt, N. Hasan, S. Krieg, S. Meinel, J. W. Negele, A. V. Pochinsky and S. N. Syritsyn, “Excited-state effects in nucleon structure on the lattice using hybrid interpolators,” Phys. Rev. D 100, 074510 (2019) doi:10.1103/PhysRevD.100.074510

Principal investigator

Jeremy Green
Theoretical Physics Department, CERN, Geneva, Switzerland

Project team members

Michael Engelhardt
New Mexico State University, Las Cruces, NM, USA

Eric Gregory, Nesreen Hasan, Stefan Krieg, Thomas Lippert
JARA-HPC, JSC, Forschungszentrum Jülich GmbH, Jülich and Bergische Universität Wuppertal, Wuppertal

Stefan Meinel
University of Arizona, Tucson, AZ, USA

John Negele, Andrew Pochinsky
Center for Theoretical Physics, MIT, Cambridge, MA, USA

Sergey Syritsyn
Department of Physics and Astronomy, Stony Brook University, Stony Brook, NY, USA and RIKEN/BNL Research Center, Brookhaven National Laboratory, Upton, NY, USA

Scientific Contact

Dr. Jeremy Green
Department of Theoretical Physics
Esplanade des Particules 1
P.O. Box, CH-1211 Geneva 23 (Switzerland)
e-mail: jeremy.green [@] cern.ch

Local project ID: chmz37

March 2021