[Paper Review] Pseudoscalar decay constants at large N_c

Roland Kaiser, H. Leutwyler|ArXiv.org|Jun 10, 1998

Quantum Chromodynamics and Particle Interactions40 citations

TL;DR

This paper calculates pseudoscalar decay constants in the large-N_c limit of QCD using an extended chiral effective field theory that includes the η′ as a Goldstone boson. It shows that the singlet axial current's anomalous dimension induces scale dependence in effective couplings, and derives a one-loop correction to the η–η′ mixing angle difference, predicting a reduction from 16° to ~14° when higher-order corrections are included.

ABSTRACT

In the large N_c limit, the variables required to analyze the low energy structure of QCD in the framework of an effective field theory necessarily include the degrees of freedom of the eta'. We evaluate the decay constants of the pseudoscalar nonet to one loop within this extended framework and show that, as a consequence of the anomalous dimension of the singlet axial current, some of the effective coupling constants depend on the running scale of QCD. The calculation relies on a simultaneous expansion in powers of momenta, quark masses and 1/N_c. Talk given at the Workshop on Nonperturbative Methods in Quantum Field Theory, NITP/CSSM, University of Adelaide, Australia, Feb. 1998.

Motivation & Objective

• To extend chiral effective field theory to include the η′ as a Goldstone boson in the large-N_c limit of QCD.
• To analyze the decay constants of the pseudoscalar nonet using a simultaneous expansion in momenta, quark masses, and 1/N_c.
• To investigate how the anomalous dimension of the singlet axial current affects the running scale dependence of effective coupling constants.
• To compute one-loop corrections to the η–η′ mixing pattern and refine the low-energy theorem for the mixing angle difference.

Proposed method

• Formulates an effective Lagrangian that includes the η′ as a dynamical Goldstone mode in the large-N_c limit.
• Uses a simultaneous expansion in powers of momenta, quark masses, and 1/N_c to compute one-loop contributions to decay constants.
• Introduces two distinct mixing angles, θ₈ and θ₀, for the octet and singlet axial currents, respectively, to describe η–η′ mixing.
• Derives scale-invariant expressions for decay constants and mixing angles, with F₀ dependent on the running scale due to the anomalous dimension.
• Applies the low-energy theorem relating the mixing angle difference to the coupling constant L_A, which is scale-invariant due to cancellation of running effects.
• Computes tree-level results for the mixing angle difference using the effective Lagrangian, incorporating corrections from L₄, L₅, and L₁₈ couplings.

Experimental results

Research questions

• RQ1 How do the pseudoscalar decay constants of the nonet behave in the large-N_c limit when the η′ is treated as a Goldstone boson?
• RQ2 What is the impact of the U(1) anomaly on the running scale dependence of effective couplings in the effective Lagrangian?
• RQ3 How do one-loop corrections modify the leading-order prediction for the difference between the η–η′ mixing angles?
• RQ4 To what extent do higher-order corrections reduce the mixing angle difference predicted by the leading-order low-energy theorem?
• RQ5 Can the scale-invariant combination L_A be used to correlate the mixing angle difference with the magnitude of F₀?

Key findings

• The decay constant F₈ is predicted to be 1.34 times F_π, with no scale dependence, in agreement with chiral perturbation theory.
• The singlet decay constant F₀ depends on the running scale of QCD due to the anomalous dimension of the singlet axial current.
• The combination L_A = (2L₅^r + 3L₁₈^r)/√(1+Λ₁) is scale-invariant, ensuring that physical observables like the mixing angle difference remain scale-independent.
• The mixing angle difference sin(θ₀−θ₈) is predicted to be approximately 14° when including higher-order corrections, compared to 16° in the leading-order formula.
• The result is sensitive to the value of L_A, which is estimated to be dominated by L₅^r in the large-N_c limit, yielding F̄₀ ≈ F_π.
• The tree-level calculation of the mixing angle difference yields sin(θ₀−θ₈) = 8√2(M_K²−M_π²)L_A / (3F₈F̄₀), providing a scale-invariant, model-independent expression.