The quest to fully understand the internal structure of hadrons, in particular the dynamics as well as the spatial and momentum distributions of their constituents - the partons -, inspired a large community of theoreticians and experimentalists to commit to the continuous development of the underlying theory aspects and to design and exploit a variety of constantly modernizing high-energy experiments all over the world.
In Quantum Chromodynamics (QCD), the theory of strong interaction, the internal structure of hadrons as explored in hard-scattering reactions is encoded in Parton Distribution Functions (PDFs). It was proven theoretically and demonstrated experimentally that the intrinsic transverse motion of partons and various spin-orbit correlations play an important role in a complete three-dimensional picture of a fast-moving hadron. In the context of the presently accepted understanding, the spin and momentum structure of the nucleon is based on a set of Transverse Momentum Dependent (TMD) PDFs that describe the distributions of longitudinal and transverse momenta of partons and their correlations with nucleon and quark polarisations. The PDFs, in particular those of mesons, serve also as an important input for the general understanding of the hadron-mass generation mechanisms through the dynamical chiral symmetry breaking of the scale invariance in QCD, the conundrum of “Emergence of Hadron Mass”. The fully multi-dimensional description of hadron structure is complemented by Generalized Parton Distributions (GPDs) that describe the spatial distributions of partons in the plane transverse to the momentum of a fast-moving hadron. Projecting out the pure transverse extension of the partons and thus integrating over the longitudinal correlations, GPDs contain information about electromagnetic form factors and thereby, in principle, also on the proton radius. Results from canonic measurements of the proton radius by scattering experiments and atomic spectroscopy using electrons and muons did not converge up to now. Presently, the experimental determination of the spatial extent of the distribution of the electric charge in the proton is one of the challenging and open aspects of modern physics, which is often referred to as “proton radius puzzle”.
The COMPASS experiment  at CERN comprises a state-of-the-art spectrometer and, optionally, a unique polarized solid-state target; operation in the M2 beamline (SPS, North area, EHN2 hall) started in 2001. Since then, COMPASS has become one of the well-recognised leaders in the field of measurements related to hadron structure and spectroscopy studies. Between 2002 and 2018, a wealth of data was collected using high energy muon and hadron beams and a variety of polarised and unpolarised targets. The COMPASS collaboration has been addressing a remarkable variety of physics topics, in particular in connection with studies related to TMD PDFs and GPDs. The results have been published in distinguished scientific journals and they triggered strong interest and attention of a large community of theorists and experimentalists all over the world. In 2021-2022, COMPASS will have its last data-taking campaign in order to complete its experimental program on nucleon spin structure studies.
North Area. Protons leaving the SPS enter a target station (bottom left), leading to 6 km of secondary beamlines for experiments in three halls. Source: https://gis.cern.ch.
In the history of studying the spin structure of the nucleon, two important quantities playing a central role in the framework of TMD PDFs are the transversity distribution hq1 and the Sivers function f⟂q1T. Their existence was conjectured by theorists more than three decades ago. First pioneering Semi-Inclusive measurements of hadrons produced in Deep Inelastic Scattering (SIDIS) were performed in the early 2000’s by the HERMES (DESY, Germany) and COMPASS experiments using transversely polarised targets; both hq1 and f⟂q1T were found to be accessible and significantly different from zero.
In a transversely polarised nucleon, the distributions of quark transverse momentum were found to be asymmetric with respect to the plane given by the directions of nucleon spin and momentum. This asymmetry is known as the Sivers effect and is embedded in the TMD PDF f⟂q1T. It has the remarkable property of being naively time-reversal odd, which implies that it should have opposite sign when measured in SIDIS on the one hand, and in the Drell-Yan process or W and Z-boson production on the other. The experimental test of this fundamental prediction, which in the TMD factorization approach is a direct consequence of QCD gauge invariance, is a major challenge in hadron physics and one of the goals of COMPASS.
The transversity PDF, which is a chiral-odd quantity, describes the distribution of transversely polarised quarks in a transversely polarised nucleon. Measuring the transversity PDF for both u and d valence quark flavours allows the determination of the tensor quark charges δu and δd. Their difference represents the isovector nucleon charge that is a fundamental property of the nucleon. Together with the vector and axial charges, it characterises the nucleon as a whole. The tensor charges are being calculated with steadily increasing accuracy by continuum and lattice approaches to QCD. Recently, the connection between the isovector nucleon charge and possible novel tensor interactions at the TeV scale in neutron and nuclear β-decays and its potential contribution to the neutron electric dipole moment have also been investigated, and constraints on new physics beyond the standard model have been derived.
Since the transversity PDF is chiral-odd, it can be accessed only in processes where it couples to another chiral-odd quantity, e.g. in SIDIS measurements with transversely polarised nucleon targets. Whereas a reasonably large statistics on SIDIS data taken with proton targets was collected by the HERMES and COMPASS experiments, the information available from measurements with neutron and deuteron targets is rather scarce. A measurement with a 3He target was performed at Jefferson Lab at much lower energy and with limited statistics. Deuteron data were collected in the early years of COMPASS running with a small-acceptance spectrometer. Hence the statistical uncertainties of the deuteron transverse spin asymmetries are considerably larger than those of the corresponding proton asymmetries. Correspondingly, the d-quark PDFs are determined considerably worse than the u-quark ones, which is shown in the left panel of Fig. 1 for the transversity functions, i.e. for xhu1 and xhd1.
In order to better determine the d-quark transversity and other TMD PDFs and to provide valuable input for the determination of the isovector nucleon charge gT, COMPASS requested one more year of data taking with a muon beam and a transversely polarized deuteron target , more specifically a 6LiD target with a very favourable ratio between polarisable deuterons and overall nucleons. The start is scheduled for 2021 and the required statistics is expected to be acquired by 2022. After 150 days of data taking, a combined analysis of all the existing proton data and the new deuteron data should allow the extraction of the transversity PDFs with the accuracy illustrated in the right panel of Fig.1. Integrating numerically the u-quark and d-quark transversities in the Bjorken-x range 0.008<x<0.21, the new deuteron data will considerably improve the precision in the d-quark tensor charge, the uncertainty goes from 0.108 to 0.040. For the u-quark, the accuracy is improved from 0.032 to 0.019. The final uncertainty for the isovector nucleon charge is expected to be ±0.044, which is an improvement of a factor of two compared to existing data.
Fig.1: The curves show the transversity distributions xhu1(red) and xhd1(black) as a function of Bjorken-x, with 68% and 90% confidence bands obtained using replicas for the present (left) and projected (right) uncertainties of the COMPASS deuteron data. The points show the values extracted from the measured asymmetries.
Altogether, with one additional year of data taking on a deuteron target and using the existing proton data, an accurate flavour separation for the transversity and Sivers PDFs will be possible. These measurements will stay unique for many years to come. In combination with the data collected in 2007 and 2010 on transversely polarised protons, the new deuteron data will constitute the legacy of COMPASS on the transverse spin nucleon structure.
A new era to investigate fundamental questions connected to the origin of the visible mass in the universe will commence soon, again using the unique M2 beam line at CERN. The COMPASS++/AMBER collaboration is about to start a new generation of experiments, with the flagship goal to investigate the emergence of hadron mass. In January 2019, a Letter of Intent (LoI)  was submitted to the SPSC in order to establish a "New QCD facility at the M2 beam line of the CERN SPS". Such an unrivalled project will allow for a great variety of measurements able to address fundamental issues of strong interactions in the medium and long-term future. The proposed measurements cover a wide range in Q2 and hence in the distance scale for probing the hadron. At lowest values of Q2, or equivalently large distances, AMBER will determine the proton charge radius through elastic muon-proton scattering. At intermediate Q2, hadron spectroscopy studies are envisaged using dedicated meson beams. At high Q2, i.e. small distances, the structure of mesons and baryons will be studied via the Drell-Yan process and charmonia production. The LoI describes the envisaged physics goals, as well as sensitivity reach and competitiveness for such a future general-purpose fixed-target installation at CERN. The list of foreseen measurements is displayed in the following table.
Table: List of measurements proposed in the COMPASS++/AMBER Letter of Intent.
In May 2019, a proposal  was submitted describing the first series of measurements. In this phase-1 of AMBER, the existing muon beam and conventional hadron beams of CERN's M2 beam line will be used.
The AMBER phase-1 proposal was approved by the CERN Research Board on December 2nd 2020. Data taking will start in 2022, right after the completion of the COMPASS data-taking campaign. The first AMBER measurement will be the determination of the spatial extent of the distribution of the electric charge in the proton in elastic muon-proton scattering in order to contribute a new and systematically largely independent result to help solving the “proton radius puzzle”.
Concerning the long-term future, the AMBER collaboration is going to propose measurements requiring kaon and antiproton beams of high energy and high intensity. This can be accomplished by an upgrade of the M2 beam line installing a radio-frequency (RF) separation stage. Such beams will allow for unique measurements that cannot be performed elsewhere. The proposed RF upgrade of the M2 beamline is presently under study at CERN EN-EA. The full AMBER project is expected to cover at least the next 15 years. As it continues to attract physicists world-wide, the physics scope of the facility will remain open for future exciting ideas, using either the RF-separated hadron beams or the already today unique muon beam.
The Theory Initiative "Perceiving the Emergence of Hadron Mass through AMBER@CERN" was launched in December 2019 as a series of tele-workshops. The 4th workshop in December 2020 saw 118 registered participants, 5 invited and 22 contributed talks. These fruitful and stimulating meetings will be continued; the 5th workshop is scheduled for March 22-26, 2021.
The proposal for phase-2, covering measurements after LS3, will be submitted in the course of 2021. Its flagship mission will be to shed light onto the conundrum of the emergence of hadron mass. Even fifty years after the discovery of quarks, science is only just beginning to grasp how quantum chromodynamics molds the most elementary hadrons: pions, neutrons, protons, etc.; and it is far from understanding how QCD produces nuclei. However, enormous progress is nowadays being made by theory. Results obtained using novel QCD lattice algorithms, which are steadily approaching realistic descriptions of hadron matter based on the physical pion mass, are beginning to agree with those from recent QCD continuum analyses. Predictions are being made that may also allow modern facilities to experimentally address the fundamental issue of the emergence of hadron mass. The Higgs mechanism, critical in so many areas of Standard Model physics, here only plays a minor role. Instead, it is the mass-scale characteristic of hadron matter that sets the scale for almost all visible mass in the Universe.
The role of emergent mass is strikingly expressed in the properties of the Nambu-Goldstone (NG) modes of the Standard Model. Their internal structure is complex; and that structure provides the clearest window onto the emergence of mass. In the absence of Higgs couplings into QCD, all properties of π and K-mesons are identical; but at realistic Higgs couplings, measurable properties of the π and K are windows onto the “Emergence of Hadron Mass” and its modulation by the Higgs boson. Stated differently, the properties of π and K mesons provide clear and direct access to both of the Standard Model's two mass-generating mechanisms and also the interference between them. Numerous predictions have already been made, relating, e.g. to parton distribution functions in pion and kaon, their similarities and differences. It appears reasonable to expect that continuing developments in theory will lead to more predictions in the near and medium-term future. Hence very soon new era experiments capable of validating contemporary and anticipated predictions are expected to be of very high priority. Eventually, an entire chapter of the Standard Model, whose writing began with Yukawa more than eighty years ago, can be completed and closed with elucidation of the structural details of the Standard Model's only NG modes, whose existence and properties are critical to the formation of everything from nucleons, to nuclei, and on to neutron stars. Also the proton charge radius of about 1fm has its origin in the strongly interacting substructure and thus is intimately linked to the emergence of the proton mass. Evidently, no claim to have understood the Standard Model is supportable until an explanation is provided for the emergence and structure of NG modes.
Contributions by: F. Bradamante, O. Denisov, J. Friedrich, N. d’Hose, A. Martin, W.-D. Nowak, C. Roberts, F. Tessarotto.