Questions: Confinement and Hadrons

Question 1

The potential between a quark and an antiquark at large separation r goes as V(r) ~ sigma r, where sigma (the string tension) is approximately 1 GeV/fm. What happens physically when you try to pull a quark-antiquark pair apart?

• A. The quarks accelerate away from each other until they escape
• B. The energy stored in the color flux tube grows until it exceeds 2 m_q, at which point a new quark-antiquark pair is created from the vacuum — you end up with two mesons instead of two free quarks
• C. The potential eventually flattens to a constant, allowing separation at sufficient energy
• D. The quarks radiate gluons that carry away the excess energy

Answer: (B)

This is string breaking. The color flux tube between the quark and antiquark stores energy proportional to its length. When the energy exceeds the rest mass of a quark-antiquark pair (about 300 MeV for light quarks), it is energetically favorable to create a new pair from the vacuum. The newly created quark binds with the original antiquark, and the newly created antiquark binds with the original quark, producing two separate mesons. You can never isolate a single quark — adding energy simply creates more hadrons. This is why high-energy collisions produce jets of hadrons rather than free quarks.

Question 2

Lattice QCD is a method for computing QCD predictions non-perturbatively by discretizing spacetime on a grid and evaluating the path integral numerically. It has successfully computed the proton mass to within a few percent of the experimental value.

• T. True
• F. False

Answer: True

Lattice QCD replaces continuous spacetime with a discrete lattice (typically with spacing a ~ 0.1 fm), which provides a natural ultraviolet cutoff. The path integral becomes a finite-dimensional integral that can be evaluated by Monte Carlo methods. The proton mass (938 MeV), computed from first principles with no free parameters except the quark masses and the QCD coupling, agrees with experiment to within about 2%. Lattice QCD has also successfully computed meson masses, the pion decay constant, and other non-perturbative quantities. It is currently the only systematic method for first-principles calculations in the confining regime of QCD.

Question 3

All observed hadrons are either mesons (quark-antiquark) or baryons (three quarks). Exotic hadrons like tetraquarks (two quarks + two antiquarks) or pentaquarks are forbidden by QCD.

• T. True
• F. False

Answer: False

QCD requires hadrons to be color-neutral, but this does not restrict them to only mesons (q q-bar) and baryons (qqq). Any color-singlet combination is allowed. Tetraquarks (qq q-bar q-bar), pentaquarks (qqqq q-bar), and glueballs (bound states of gluons with no quarks) are all consistent with QCD. Several tetraquark and pentaquark candidates have been observed at the LHC and other experiments (e.g., the X(3872), Z_c(3900), and the P_c pentaquark states discovered by LHCb in 2015 and 2019). These exotic hadrons are more difficult to produce and identify than conventional mesons and baryons, but they are real predictions of QCD.

Question 4

Explain why approximately 99% of the proton's mass comes from the energy of the gluon field and quark kinetic energy, rather than from the intrinsic masses of the quarks.

Model answer: The proton mass is approximately 938 MeV. The up and down quark masses are approximately 2-5 MeV each, totaling about 10 MeV for the three valence quarks — roughly 1% of the proton mass. The remaining 99% comes from two sources: the kinetic energy of the confined quarks (the uncertainty principle requires large momenta when quarks are confined to a region of size ~1 fm) and the energy stored in the gluon field that binds them. This is a purely relativistic and quantum effect: E = mc^2 applied to the field energy gives the proton its mass. In this sense, most of the mass of ordinary matter is 'made of' the energy of the strong force rather than the intrinsic masses of quarks.

This is confirmed by lattice QCD, which computes the proton mass from the QCD Lagrangian with light quark masses as inputs. If you set the quark masses to zero (the chiral limit), the proton mass decreases by only about 5-10%, not to zero. The strong interaction generates mass from pure energy — a dramatic manifestation of E = mc^2.