Correlators of heavy-light quark currents in HQET: Perturbative contribution up to 4 loops and beyond

The perturbative contribution to the correlator of two HQET heavy-light currents expanded in light-quark masses up to quadratic terms is calculated up to 4 loops. The leading large-$β_0$ limit is also considered, so that terms with the highest degrees of $n_f$ are calculated to all orders in $α_s$. Borel images of coefficient functions in this limit contain renormalon poles. Naive nonabelianization works surprisingly poorly for the coefficient functions considered here.

💡 Research Summary

The paper presents a comprehensive perturbative study of the two‑point correlator of heavy‑light currents in Heavy‑Quark Effective Theory (HQET). The authors expand the correlator in light‑quark masses up to quadratic order, thereby including the operators 1, m, m² and the sum of light‑flavour mass‑squares ∑ m_i². Using modern multiloop techniques—qgraf for diagram generation, FORM (with the color package) for Dirac and colour algebra, LiteRed for integration‑by‑parts reduction, and known four‑loop master integrals—they compute the Wilson coefficients C_O(τ) for these operators through four loops. Gauge‑parameter independence is verified up to three loops exactly and up to linear order in ξ at four loops, providing a stringent check of the calculation.

The results for the coordinate‑space coefficients are given in compact analytic form and also numerically for QCD with n_f = 4. For example, the leading coefficient C₁(τ) (dimension‑zero operator) behaves as τ⁻³ multiplied by a series 1 + 7.05316 α_s/π + 10.1485 (α_s/π)² + 125.943 (α_s/π)³ + …, while the mass‑linear coefficient C_m(τ) scales as τ⁻² with a similar series, and the quadratic‑mass coefficient C_{m²}(τ) scales as τ⁻¹ with alternating signs. The dimension‑two operator ∑ m_i² receives a new three‑loop contribution, extending the earlier two‑loop results.

In momentum space the spectral densities ρ_P(ω) are derived from the same Wilson coefficients. The authors show that all ε‑poles cancel after renormalisation, confirming the consistency of the calculation. The renormalisation‑group (RG) structure of the coefficients is displayed explicitly, with anomalous dimensions γ_n expressed through the heavy‑quark current anomalous dimension γ_j and the light‑quark mass anomalous dimension γ_m.

The second major part of the work investigates the large‑β₀ limit, i.e. the leading‑n_f approximation obtained by taking n_f → −∞ while keeping β₀ = 11 − 2 n_f/3 large. In this limit the authors resum the terms with the highest power of n_f to all orders in α_s. They derive compact expressions for the bare coefficient functions C₀^{mn}(τ) in terms of two‑loop integrals F_n(ε,u) and then perform a Borel transform. The Borel images S_n(u) exhibit infrared renormalon poles at u = ½, 1, etc., signalling factorial growth of perturbative coefficients at high orders. The anomalous dimensions in the large‑β₀ limit are shown to be γ_n,0 = −2 C_F F_n(0,0) with F_n(0,0)=3(n+1). The authors also provide explicit series for the resummed coefficients \hat C_n up to eight loops, displaying the characteristic ζ‑values (ζ₃, ζ₅, ζ₇) and π‑powers that appear in high‑order QCD perturbation theory.

A striking observation is that the “naive non‑abelianization” (NNA) prescription—replacing C_A by β₀ in lower‑order results—fails badly for the heavy‑light correlator. The NNA estimates deviate from the exact four‑loop coefficients by 30 % or more, especially in terms proportional to C_F T_F n_f ζ₅. This demonstrates that the renormalon structure of heavy‑light currents is more intricate than a simple β₀ substitution would suggest, involving non‑trivial colour‑factor interplay and higher ζ‑values.

Finally, the authors discuss the choice of renormalisation scale. Setting μ equal to the natural HQET scale μ_τ = 2 e^{−γ_E} τ (or μ_ω = 2 ω) eliminates logarithmic terms L_τ, L_ω from the expressions, leading to particularly simple forms for the coefficients. This scale choice also clarifies how the RG evolution of the coefficients compensates the explicit logarithms, an important point for practical applications such as QCD sum‑rule analyses and lattice‑QCD matching.

In summary, the paper delivers (i) the first complete four‑loop calculation of heavy‑light current correlators with quadratic mass dependence, (ii) an all‑order large‑β₀ resummation that reveals the renormalon pole structure, and (iii) a critical assessment of the naive non‑abelianization approach. These results provide essential input for precision determinations of heavy‑light meson properties, improve the theoretical control of HQET sum rules, and offer a benchmark for future high‑order perturbative studies in QCD.