The Structure of the Proton

Large nuclear physics facilities in several places around the world, like the Large Hadron Collider (LHC) at Geneva in Switzerland, provide important insight on the structure of the proton and neutron. While current experiments continue to collect valuable data there is only so far, such experiments can go toward unlocking the inner workings of the building blocks of matter. A new experiment, the Electron-Ion Collider, is currently under construction and will become a novel tool for exploring the inner microcosm of protons and neutrons dominated by gluons.

Besides the experimental research at such large-scale collider and detector facilities, also great theoretical effort is needed to unlock the secrets of the nature of the strong force. Within the framework of lattice QuantumChromoDynamics (QCD) the structure of nucleons (= protons or neutrons) can be investigated from first principles utilizing large computational resources. Using this method, we perform large-scale computer simulations in order to extract important quantities, like e.g. specific matrix elements which are not accessible experimentally. Such quantities can be related, for example, to so-called Beyond-the-Standard-Model (BSM) physics or certain distributions of partons inside a nucleon. While the Standard Model is based on three fundamental forces and gives an (almost too) perfect description of the nature that we can test in experiments, it poses several puzzles which, in conclusion, leads to the idea of the existence of BSM physics. To test this idea experimentally a good understanding of the structure the nucleon is essential and input from lattice QCD calculations is needed, in particular very precise data is needed. Note that precise information on nucleon structure is limited due to its nonperturbative nature which makes numerical simulations necessary.

The main challenge of lattice QCD simulations in nowadays is to control all systematic uncertainties in order to obtain reasonably small errors on the calculated quantities which then can be matched up with results from experiments. Over the last 10 years we have performed a substantial number of simulations [1] within a large collaboration [2], where we have accumulated data in total of more than 1 Peta Byte so far. This allows us by now to obtain control over relevant systematics, which is however still a very challenging task. For example, simulations can only be performed at finite volumes and unphysical simulation parameters. In addition, such calculations come with a statistical error. Therefore, an extrapolation to infinite volumes and physical parameters at sufficiently small statistical errors is mandatory, and that are not the only systematic effects that need to be controlled [3].

At the end of such an analysis we are left with the quantities that finally provide information on nucleon structure, i.e. how the quark and gluon constituents account for the physical properties such as the momentum and the spin of protons and neutrons, or how the nucleon responds, for example, to electromagnetic or weak probes. Other quantities are (unpolarised) parton distribution functions (PDFs) and scalar and tensor nucleon matrix elements, which give the nucleon couplings associated with BSM interactions, and are relevant for low energy precision beta decay experiments or experiments measuring the neutron electric dipole moment. Quantities like helicity and transversity PDFs contain information about the longitudinal and transverse spin of quarks and gluons.

While such quantities connected to nucleon structure calculated from our simulations are not so easy to interpret for non-experts, there is however one important outcome of our simulations that is straightforward to imagine, namely the mass of the nucleon. In principle, the mass of the nucleon can be predicted from our simulations, however, due to the complexity of the theory, the error for this quantity from the computation is still significant, in particular in view of its extremely precise experimental determination. One should note however that for other, less known, particles this is not the case and lattice QCD simulations can provide essential information in these cases. Nevertheless, the computation of the mass of the nucleon from lattice QCD simulations is still of great interest, because it serves as an important benchmark. The calculated mass from any simulation has to reproduce its experimental value, within errors of course, and any discrepancy would indicate that the systematics are not under control. In Fig. 1 (figure taken from Ref. [4]) we show our result for the nucleon mass (along with other particles) in comparison to the experimental values and calculations from other groups. We find great agreement for our calculated nucleon mass with respect to its experimental value at a level of 0.8 percent (and even better for the other particles shown in the plot). This is currently the best calculation for such particles world-wide within the rigor approach that we apply here.
 

Image: 
Comparison of the calculated mass of the nucleon N (as well as the lambda, sigma, xi, and omega particle) from our simulations (RQCD 22, blue circles [4]) to results from other collaborations (BMW 08 [5], ETM 14 [6], χQCD 18 [7], PNDME 19 [8], Fermilab 19 [9] , Miller et al. 22 [10], Mainz 22 [11]). The corresponding vertical lines represent the experimental values.

References:
[1] M. Bruno et. al., JHEP 02 (2015) 043
[2] Status of CLS configurations for N_f=2+1 flavors. https://www-zeuthen.desy.de/alpha/public-cls-nf21/
[3] RQCD and ALPHA Collaborations, S. Kuberski et al., D and D$_{s}$ decay constants in Nf = 2 + 1 QCD with Wilson fermions, JHEP 07 (2024), 090
[4] RQCD Collaboration, G. S. Bali, S. Collins, P. Georg, D. Jenkins, P. Korcyl, A. Schäfer, E. E. Scholz, J. Simeth, W. Söldner and S. Weishäupl, JHEP 05 (2023) 035 
[5] BMW collaboration, S. Dürr et al., Ab-initio determination of light hadron masses, Science 322 (2008) 1224 
[6] ETM collaboration, C. Alexandrou, V. Drach, K. Jansen, C. Kallidonis and G. Koutsou, Baryon spectrum with Nf = 2 + 1 + 1 twisted mass fermions, Phys. Rev. D 90 (2014) 074501
[7] Y.-B. Yang, J. Liang, Y.-J. Bi, Y. Chen, T. Draper, K.-F. Liu et al., Proton mass decomposition from the QCD energy momentum tensor, Phys. Rev. Lett. 121 (2018) 212001
[8] PNDME collaboration, Y.-C. Jang, R. Gupta, H.-W. Lin, B. Yoon and T. Bhattacharya, Nucleon electromagnetic form factors in the continuum limit from (2 + 1 + 1)-flavor lattice QCD, Phys. Rev. D 101 (2020) 014507
[9] Y. Lin, A. S. Meyer, C. Hughes, A. S. Kronfeld, J. N. Simone and A. Strelchenko, Nucleon mass with highly improved staggered quark
[10] N. Miller et al., The hyperon spectrum from Lattice QCD, PoS LATTICE2021 (2022) 448
[11] K. Ottnad, D. Djukanovic, H. B. Meyer, G. von Hippel and H. Wittig, Mass and isovector matrix elements of the nucleon at zero-momentum transfer, PoS LATTICE2022 (2023) 117