PI: Roger Horsley

The Gross-Llewellyn Smith sum rule from lattice QCD

The Gross–Llewellyn Smith (GLS) sum rule is one of the cleanest and most elegant results in particle physics. It shows that neutrinos can be used to ‘count’ the number of quarks inside a proton. In these experiments, high-energy neutrinos are fired at protons or neutrons. As they interact only through the weak force, they usually pass straight through matter. When they do interact, however, they hit individual quarks inside the proton. The GLS sum rule states that if you add up all neutrino–proton scattering events in the right way, the result is essentially the number 3. Why 3? Because a proton contains three valence quarks – two u quarks and one d quark. In other words, neutrino scattering directly reveals the proton’s quark content. By comparing neutrino and antineutrino scattering, the experiments isolate a specific part of the proton’s internal structure that depends only on the number of quarks, not on complicated details of how they interact. In the real world, valence quarks are surrounded by gluons and short-lived quark–antiquark pairs. These quantum effects slightly reduce the result from exactly 3. The size of this reduction is predicted by quantum chromodynamics (QCD), the theory of the strong force. In the figure, we compare our results (coloured points) [1] to the experimental points (black stars) [2]. We find good agreement. The horizontal dotted line shows 3, the number of valence quarks. The black line shows the expected perturbative QCD result, [3]. There is a clear reduction from 3 due to QCD effects.

The simulations and analysis were partially performed on the DiRAC-3 Extreme Scaling service (Edinburgh) and Data Intensive service (Cambridge), as part of the project ‘Hadron structure: the parton model and beyond’ (PPTP349).

[1] K. U. Can, J. A. Crawford, R. Horsley, P. E. L. Rakow, T. G. Schar, G. Schierholz,

H. Stüben, R. D. Young, J. M. Zanotti [QCDSF], Phys. Rev. D 111 (2025) 11, [arXiv:2502.19704].

[2] J. H. Kim, et al., Phys. Rev. Lett. 81 (1998) 3595, [arXiv:hep-ex/9808015].

[3] S. A. Larin, J. A. M. Vermaseren, Phys. Lett. B 259 (1991) 345.