Questions: Running Coupling Constants

Question 1

The fine structure constant alpha is often quoted as 1/137. But alpha measured at the Z boson mass (91 GeV) is approximately 1/128. How can a 'constant' change with energy?

• A. The measurements are wrong — alpha must be a true constant
• B. Virtual particle-antiparticle pairs screen the bare charge at long distances; at higher energies (shorter distances), you probe inside the screening cloud and see a larger effective charge
• C. The Z boson modifies the electromagnetic interaction
• D. Lorentz contraction at high energies compresses the charge distribution

Answer: (B)

The vacuum acts as a dielectric medium due to virtual electron-positron pairs (and other charged particle pairs at higher energies). These pairs partially screen the bare electric charge at long distances, giving the measured value of approximately 1/137 at low energies. At higher energies (shorter distances), you resolve the charge inside the polarization cloud and measure a larger effective coupling. This is quantified by the QED beta function: beta(alpha) = 2alpha^2/(3pi) > 0, meaning alpha increases with energy. The running is slow (logarithmic), which is why alpha changes from 1/137 to only 1/128 over five orders of magnitude in energy.

Question 2

The beta function beta(g) = mu dg/dmu describes how the coupling g changes with the energy scale mu. A theory with beta(g) = 0 is called a conformal field theory. What is special about it?

• A. It has no particles
• B. It is scale-invariant — physics looks the same at all energy scales, and there is no running of the coupling constant
• C. It is non-renormalizable
• D. It has an infinite number of coupling constants

Answer: (B)

If beta(g) = 0, the coupling does not run, and the theory has no intrinsic energy scale (it is scale-invariant). Such theories are conformal field theories (CFTs) and play a central role in the study of critical phenomena (phase transitions) and in string theory (the AdS/CFT correspondence relates certain CFTs to gravity in anti-de Sitter space). A theory whose beta function has a zero at some coupling g* flows to that fixed point — the coupling asymptotically approaches g* at the corresponding energy scale. QCD's asymptotic freedom means it approaches the free-field fixed point (g* = 0) at high energies.

Question 3

In QED, the coupling increases at higher energies, while in QCD, it decreases. This means QED perturbation theory becomes less reliable at very high energies, while QCD perturbation theory becomes more reliable.

• T. True
• F. False

Answer: True

Perturbation theory converges when the coupling is small. In QED, alpha grows with energy (beta > 0), so perturbation theory is excellent at low energies but eventually breaks down at astronomically high energies (the Landau pole, around 10^{286} eV). In QCD, the strong coupling alpha_s decreases with energy (beta < 0, asymptotic freedom), so perturbative QCD works well for hard processes at high energies (like deep inelastic scattering at GeV scales) but fails at low energies (around Lambda_QCD ~ 200 MeV), where alpha_s ~ 1 and confinement occurs. The running of couplings determines the domain of validity of perturbation theory.

Question 4

Explain how the running of the three Standard Model gauge couplings provides evidence for (or against) grand unification.

Model answer: The three gauge couplings of the Standard Model — alpha_1 (hypercharge, U(1)), alpha_2 (weak, SU(2)), and alpha_3 (strong, SU(3)) — run differently with energy. alpha_1 increases, alpha_2 decreases slowly, and alpha_3 decreases rapidly. Extrapolating the measured low-energy values to higher energies using the renormalization group equations, the three couplings nearly (but not exactly) converge to a single value at around 10^{15} GeV. This near-convergence suggests grand unification — a single gauge group that breaks into the Standard Model groups at lower energies. The inexact convergence in the Standard Model is often cited as motivation for supersymmetry, which modifies the beta functions and achieves precise unification at approximately 10^{16} GeV.

The running of couplings is not merely a theoretical curiosity — it provides a window into physics at energy scales far beyond what accelerators can reach. The fact that three independent couplings, measured at 100 GeV, nearly converge when extrapolated over 13 orders of magnitude is either a remarkable coincidence or a deep clue about the structure of nature.