In high energy particle physics, specifically in hadron-beam scattering experiments, transverse momentum distributions (TMDs) are the distributions of the hadron's quark or gluon momenta that are perpendicular to the momentum transfer between the beam and the hadron.
Specifically, they are probability distributions to find inside the hadron a parton with a transverse momentum
TMDs provide information on the confined motion of quarks and gluons inside the hadron and complement the information on the hadron structure provided by parton distribution functions (PDFs) and generalized parton distributions (GPDs).
[1] In all, TMDs and PDFs provide the information of the momentum distribution (transverse and longitudinal, respectively) of the quarks (or gluons), and the GPDs, the information on their spatial distribution.
TMDs are an extension of the concept of parton distribution functions (PDFs) and structure functions that are measured in deep inelastic scattering (DIS).
denotes the fraction of hadron longitudinal momentum carried by the parton, and identifies with the Bjorken scaling variable in the infinite energy-momentum limit.
In all, there are 16 dominant (viz leading-twist) independent TMDs, 8 for the quarks and 8 for the gluons.
In turn, the correlations provide access the dynamics of partons in the transverse plane in momentum space.
Thus, TMDs are comparable and directly complementary to the generalized parton distributions (GPDs) which describe the parton dynamics in the transverse plane in position space.
Formally, TMDs access the correlations between a parton orbital angular momentum (OAM) and the hadron/parton spin because they require wave function components with nonzero OAM.
Therefore, TMDs allow us to study the full three-dimensional dynamics of hadrons, providing more detailed information than that contained in conventional PDF.
One example of the importance of TMDs is that they provide information about the quark and gluon OAM.
Those are not directly accessible in regular DIS, but are crucial to understand the spin content of the nucleon and resolve the nucleon spin crisis.
In fact, lattice QCD calculations indicate that quark OAM is the dominant contribution to the nucleon spin.
[2] Similarly to quark TMDs, gluon TMDs allow access to the gluonic orbital angular momentum, another possibly important contribution to the nucleon spin.
They are: Our initial understanding of the short-distance nucleon structure has come from deep inelastic scattering (DIS) experiments.
This description is essentially one-dimensional: DIS provides us with the parton momentum distributions in term of the single variable x, which is interpreted in the infinite momentum limit (the Bjorken limit) as the fraction of the nucleon momentum carried by the struck partons.
This entails that to measure TMDs, we need to gather more information from the scattering process.
In DIS, only the scattered lepton is detected while the remnants of the shattered nucleon are ignored (inclusive experiment).
Semi-inclusive DIS (SIDIS), where a high momentum (i.e. leading) hadron is detected in addition of the scattered lepton, allows us to obtain the needed additional details about the scattering process kinematics.
This latter retains the information on its motion inside the nucleon, including its transverse momentum
Consequently, the structure functions entering the SIDIS cross-section or asymmetries are convolutions of a
Therefore, precise knowledge of fragmentation functions is important to extract TMDs from experimental results.
Quark TMDs measurements were pioneered at DESY by the HERMES experiment.
Quark and gluon TDM measurements are an important part of the future electron–ion collider scientific program.