How to Find Velocity in Quantum Chromodynamics: A Comprehensive Guide

Quantum Chromodynamics (QCD) is the fundamental theory that describes the strong interaction between quarks and gluons, the fundamental particles that make up hadrons like protons and neutrons. To determine the velocity of these partons (quarks and gluons) within a hadron, physicists rely on the concept of deep inelastic scattering (DIS), where a high-energy lepton (such as an electron or muon) collides with a hadron, revealing the internal structure of the hadron.

Understanding Deep Inelastic Scattering (DIS)

In a DIS experiment, the key measurable quantities are the four-momenta of the incoming and outgoing particles. These can be used to calculate the following kinematic variables:

1. Squared four-momentum transfer (Q^2): This represents the amount of momentum transferred from the lepton to the hadron during the collision.
2. Energy transfer (ν): This is the amount of energy transferred from the lepton to the hadron.
3. Bjorken scaling variable (x): This variable is defined as x = Q^2 / (2mν), where m is the mass of the hadron. The Bjorken x can be interpreted as the fraction of the hadron’s momentum carried by the parton that the photon interacts with.

Inferring Parton Velocity from DIS Kinematics

The velocity of the partons inside the hadron can be inferred from the kinematic variables of the DIS process, particularly the Bjorken scaling variable x. This is because in the Bjorken limit, where Q^2 and ν are much larger than the hadron mass, the structure functions of the hadron exhibit scaling behavior, suggesting that the photon is scattering off point-like constituents (quarks) with no intrinsic scale.

The relationship between the Bjorken x and the parton velocity v is given by:

x = Q^2 / (2mν) = p_parton / p_hadron

where p_parton is the momentum of the parton and p_hadron is the momentum of the hadron. Rearranging this equation, we can express the parton velocity as:

v = p_parton / p_hadron = x

This means that the Bjorken x variable can be directly interpreted as the fraction of the hadron’s momentum carried by the parton, which is also equal to the parton’s velocity in the Bjorken limit.

Measuring Parton Velocity in QCD Experiments

To measure the velocity distribution of partons inside a hadron, experiments typically measure the cross-section for DIS processes as a function of the kinematic variables Q^2, ν, and x. By fitting the measured cross-section to theoretical predictions based on QCD, researchers can extract information about the velocity distribution of partons within the hadron.

The general procedure for measuring parton velocity in QCD experiments is as follows:

1. Perform DIS Experiment: Collide a high-energy lepton (e.g., electron or muon) with a hadron (e.g., proton or neutron) and measure the four-momenta of the incoming and outgoing particles.
2. Calculate Kinematic Variables: Use the measured four-momenta to calculate the squared four-momentum transfer Q^2, the energy transfer ν, and the Bjorken scaling variable x.
3. Measure DIS Cross-section: Measure the cross-section for the DIS process as a function of the kinematic variables Q^2, ν, and x.
4. Fit to QCD Predictions: Compare the measured cross-section to theoretical predictions based on QCD. By fitting the data to the QCD model, researchers can extract information about the velocity distribution of partons inside the hadron.

The velocity distribution of partons within a hadron is not a directly measurable quantity but rather an inferred quantity based on the kinematics of the DIS process and the theoretical predictions of QCD.

Theoretical Considerations and Challenges

The interpretation of the parton velocity in QCD is not straightforward and requires careful consideration of several theoretical aspects:

1. Parton Model and Scaling: The Bjorken scaling behavior of the structure functions in the DIS process suggests that the photon is scattering off point-like constituents (quarks) with no intrinsic scale. This allows the parton velocity to be directly related to the Bjorken x variable.
2. Confinement and Hadronization: Quarks and gluons are confined within hadrons, and the process of hadronization, where partons form observable hadrons, can complicate the interpretation of parton velocities.
3. Non-perturbative QCD Effects: At low momentum transfers, the strong coupling constant becomes large, and non-perturbative QCD effects, such as the formation of bound states and the dynamics of confinement, become important, making the interpretation of parton velocities more challenging.
4. Relativistic Effects: Partons inside hadrons can have relativistic velocities, and the interpretation of their velocities must account for relativistic kinematics.

Numerical Examples and Data Points

To illustrate the measurement of parton velocities in QCD, let’s consider a few numerical examples:

1. Proton Structure Function Measurements: Experiments at the HERA collider have measured the proton structure function F2(x,Q^2) as a function of Bjorken x and Q^2. These measurements have allowed researchers to extract the velocity distribution of quarks and gluons inside the proton. For example, at Q^2 = 10 GeV^2, the quark velocity distribution peaks around x ≈ 0.1, indicating that the most probable quark velocity is about 10% of the proton’s velocity.

2. Neutron Structure Function Measurements: Experiments have also measured the structure functions of neutrons, which provide information about the velocity distribution of partons inside neutrons. Comparisons between proton and neutron structure functions can reveal differences in the parton velocity distributions, reflecting the different internal structures of these hadrons.

3. Parton Distribution Functions (PDFs): Theoretical models and global analyses of DIS data have led to the determination of parton distribution functions (PDFs), which describe the probability distributions of finding partons with a given momentum fraction x inside a hadron. These PDFs can be used to infer the velocity distribution of partons within the hadron.

By combining the experimental measurements of DIS cross-sections with the theoretical predictions of QCD, physicists can gain valuable insights into the velocity distribution of partons inside hadrons, which is crucial for understanding the internal structure and dynamics of these fundamental particles.

Conclusion

Determining the velocity of partons (quarks and gluons) inside hadrons is a crucial aspect of understanding the strong interaction described by Quantum Chromodynamics (QCD). By utilizing the concept of deep inelastic scattering (DIS) and the associated kinematic variables, researchers can infer the velocity distribution of partons within a hadron. This process involves measuring the DIS cross-section, fitting the data to QCD predictions, and extracting information about the parton velocity distribution. While the interpretation of parton velocities in QCD is not straightforward and requires careful consideration of theoretical aspects, the continued advancement of experimental techniques and theoretical understanding has led to significant progress in this field of research.

References

1. Quantum chromodynamics, CERN Yellow Report, 2004-001, p1-47, https://cds.cern.ch/record/726186
2. New perspectives in quantum chromodynamics, IAEA NCL Collection, 25019746, https://inis.iaea.org/search/search.aspx?orig_q=RN:25019746
3. Quantum Chromodynamics, American Physical Society Division of Particles and Fields Long Range Planning Working Group on Quantum Chromodynamics QCD, https://www.aps.org/units/dpf/upload/qcd.pdf
4. EXCLUSIVE PROCESSES IN QUANTUM CHROMODYNAMICS*, SLAC-PUB-4947, https://www.slac.stanford.edu/cgi-wrap/getdoc/slac-pub-4947.pdf
5. Quantum Chromodynamics – Research Explorer, University of Manchester, https://www.research.manchester.ac.uk/portal/en/researchers/frank-close(d4d4d4d4-d4d4-d4d4-d4d4-d4d4d4d4d4d4).html