Towards Solving the Proton Spin Puzzle by Lattice QCD

Principal Investigator:
Karl Jansen

Deutsches Elektronen-Synchrotron/DESY, Zeuthen (Germany)

Local Project ID:

HPC Platform used:

Date published:

In the project Lattice QCD simulations were carried out to compute the individual contributions of quarks and gluons to the proton spin which the value ½ in nature. The result confirms the experimental data which has been collected during the past 30 years and which indicates that only a small fraction of the proton spin is carried by the intrinsic spin of the quarks.

The Origin of the Proton Spin Puzzle

Protons and neutrons, also called nucleons, are the building blocks of the nuclei of atoms. Together with the much lighter electrons, they make up basically all of the stable matter surrounding us on earth. The intrinsic angular momentum of nucleons (and electrons) as quantum mechanical particles is exactly 1/2 in units of ħ and gives rise to the magnetic properties of many materials. Understanding the internal structure of the nucleons has posed an enormous experimental and theoretical challenge since more than half a century.

The symmetries and regular patterns in the properties of the nucleons and the zoo of further hadrons observed in high-energy physics experiments have led to the picture that nucleons are bound states of three constituents, called quarks. Their binding force must be strong at large distances in order to explain "confinement", i.e. the fact that quarks can not be directly observed in experiments but remain confined inside hadrons.

However, somewhat surprisingly, deep inelastic scattering experiments with a polarized proton target by the European Muon Collaboration (EMC) at CERN (and subsequent experiments, e.g. at SLAC and DESY) in the late 1980’s found that only a small fraction of the proton spin can be attributed to the spin of the individual quarks within the proton. This astonishing result was often called the ”proton spin puzzle'' because it is in contrast to the theoretical expectation from constituent quark models.

In these models, the nucleon is considered as a bound state of just three massive quarks. The mass of such constituent quarks is large (about a third of the nucleon mass) because it effectively accounts for part of the binding energy from these strong interactions. In this picture, even when relativistic effects are taken into account, the quark models predict that a large fraction of the nucleon spin is carried by the quarks, and thus can not explain the experimental results.

The Proton in QCD

The situation has become different, but not less challenging in the framework of Quantum Chromodynamics (QCD). This is the relativistic quantum field theory which describes the strongly interacting sector of the Standard Model of particle physics. The strong interactions are mediated by massless gauge fields, called gluons. The quark fields of QCD have a relatively small mass of only a few percent of the proton mass. These QCD quarks are related in a very complicated dynamical way to the physical nucleon states - and to the constituent quarks of the naive quark models.

In the fully dynamical framework of QCD also gluons and the orbital angular momentum of the quarks can contribute to the total nucleon spin. Thus one obtains the following sum rule

where Sq is the contribution from the intrinsic quark spin, Lq is the quark orbital angular momentum, and Jg is the gluon total angular momentum.The quark contributions also include effects of sea quarks (i.e. virtual quark-antiquark pairs) and are summed over all quark flavors q=u, d, s, c, ... (not only the valence quarks u and d of the nucleon). Some care is also needed to properly define Lq and Jg such that all three contributions in the above decomposition are gauge invariant. Moreover, due to non-trivial renormalization effects, each term on the l.h.s. of (1) may become dependent on the momentum transfer, Q2, in the scattering process, while the r.h.s. is of course independent of Q2.

The quantitative test of the sum rule (1) from experimental data or theory predictions poses a huge challenge: Disentangling quark and gluon contributions requires deep inelastic scattering experiments where different particles, in particular leptons (as in the EMC experiment) and also protons (as in recent experiments at RHIC) are used to probe the polarized protons within a fixed target or a second beam. Then, the measured ”structure functions'' need to be integrated (and usually extrapolated) over suitable kinematical regions to extract the individual contributions in (1).

Also on the theory side, the computation of these quantities is extremely difficult, because the formation of the proton as a complicated bound state of the quarks and gluons in QCD involves highly non-linear effects. These cannot be computed perturbatively, i.e. by an expansion in the coupling strength of QCD, because this coupling becomes large at the energy scales relevant for the hadronic bound states. Only numerical simulations within lattice QCD, where the theory is formulated on a discrete and finite space-time lattice, allow for a non-perturbative treatment of QCD in the low-energy regime.

The Proton Spin from Lattice QCD

In the project, large scale and extensive Lattice QCD simulations have been carried out with four dynamical quarks with degenerate and approximately physical mass values for u and d quarks and non-perturbatively improved twisted-mass action to reduce discretization effects. In Lattice QCD, the individual contributions to the proton spin according to (1) can be obtained from matrix elements of suitable renormalized local operators Oibetween proton states. To extract these matrix elements one computes ratios of correlation functions between two interpolating fields that create and annihilate proton states at large Euclidian time separation and with or without insertion of Oi. Different methods to compute these ratios have been used to verify that the time separations are sufficiently large to suppress contributions from excited nucleon states.

Correlation functions where the operator couples only to gluon fields or to quarks that are not directly connected to the proton fields, so called disconnected diagrams, have an inherently bad signal-to-noise ratio. This difficulty has now been overcome by using novel algorithms and by exploiting special properties of twisted-mass fermions.


The results for the individual contributions to the proton spin, Sq + Lq and Jg according to (1), computed at Q2 = 0, are summarized in Fig 2. Since these values have been obtained from simulations at a single lattice spacing of about 0.07 fm, no continuum extrapolation has been performed yet. Within the statistical and systematic uncertainties, the sum of all contributions nicely adds up to ½ as expected from (1).

Adding up only the intrinsic spin contributions, Sq, from all quarks, one finds that only about 20 % of the total proton spins arises from the spin of the quarks, in accordance with the experimental data. The remaining proton spin is due to the gluons and the angular momentum of the quarks (which both are absent e.g. in simple quark model predictions). This result is in line with phenomenological analyses and an important step towards resolving the proton spin puzzle, which originates from the simplified picture of a proton being made up effectively only of constituent quarks without orbital angular momentum and without explicitly including gluons.

In addition, the contributions of gluons and (valence and sea) quarks to the linear momentum of the proton have been computed and are found to consistently add up to the overall linear momentum of the proton.


1) A. Adare et al., (PHENIX Collaboration), Inclusive double- helicity asymmetries in neutral-pion and eta-meson production in p+p collisions at √s= 200 GeV, Phys. Rev. D90 (2014) no. 1, 012007

2) C. Adolph et al. (COMPASS Collaboration), The spin structure function g1p of the proton and a test of the Bjorken sum rule, Phys. Rev. Lett. B 753 (2016) 18

3) C. Alexandrou et al., Nucleon Spin and Momentum Decomposition Using Lattice QCD Simulations, Phys. Rev. Lett. 119 (2017) 142002

4) A. Abdel-Rehim et al., Disconnected quark loop contributions to nucleon observables in lattice QCD, Phys. Rev. D89 (2014) no.3, 034501

Scientific Contact:

Dr. Karl Jansen
Deutsches Elektronen-Synchrotron (DESY)
Platanenallee 6, D-15738 Zeuthen/Germany
e-mail: Karl.Jansen [@] desy.de

JSC Project ID: hch02

April 2019