A simple generalization of the low-energy theorem for the effective Higgs-gluon-gluon coupling for the case of simultaneous decoupling of several heavy quarks

We extend in an extremely simple and straightforward way the well-known low-energy theorem for an effective Higgs-like scalar-gluon-gluon coupling [1] (as well as a similar one for for the effective coupling of the Higgs-like field to the light scalar quark currents) in QCD including arbitrary number of heavy quarks in addition to the light ones. The application of the generalized low-energy theorem allows the extraction of the four-loop effective Higgs-gluon-gluon coupling valid for extensions of the Standard Model with additional heavy quarks from the 3-loop results of [2] for the decoupling constant of $α_s$.

💡 Research Summary

The paper presents a concise yet powerful generalization of the low‑energy theorem (LET) that relates the effective Higgs‑gluon‑gluon coupling to the decoupling of heavy quarks in QCD. While the classic LET was originally derived for a single heavy quark (the top) and later extended to two‑loop order, the author shows that the underlying derivation does not rely on the number of heavy flavours. By carefully examining the dependence of the decoupling constants ζ_α (for the strong coupling) and ζ_m (for the quark masses) on the heavy‑quark masses, the author rewrites the original LET expressions (C₁ = Σ_h m_h ∂{m_h} ln ζ_α, C₂ = 1 + Σ_h m_h ∂{m_h} ln ζ_m) in a form that is manifestly renormalization‑group (RG) invariant for an arbitrary set of heavy quarks.

The key technical advance is the introduction of RG‑improved LETs (eqs. 2.6 and 2.8). By applying the RG operator d/dμ² to the logarithmic relations between the full and effective couplings, the author derives expressions in which the derivative with respect to the coupling a = α_s/π appears only once, reducing the required loop order of the decoupling constants by one. Consequently, an L‑loop result for ζ_α or ζ_m suffices to obtain the (L + 1)‑loop coefficient functions C₁ and C₂. This insight dramatically simplifies the computation of higher‑order corrections.

To demonstrate the utility of the formalism, the paper first reproduces the known three‑loop result for C₁ in a theory with several heavy quarks of different masses. Starting from the well‑known two‑loop decoupling constant ζ_α, the author performs a straightforward substitution of the heavy‑quark logarithms L_{μh} = ln(μ²/m_h²) and the overall factor T_F → n_h T_F, arriving at a compact expression for C₁ at three loops (eq. 3.4‑3.5). This result matches the much more involved direct diagrammatic calculation performed in earlier works, confirming the correctness of the RG‑improved LET.

The most striking application is the extraction of the four‑loop coefficient C₁ for an arbitrary number of heavy quarks. The three‑loop decoupling constant ζ_α, computed in the literature for simultaneous decoupling of bottom and charm quarks, is expressed as ζ_α = 1 + d₁ a + d₂ a² + d₃ a³. Inserting this into the RG‑improved LET (2.6) and using the known four‑loop QCD β‑function and quark‑mass anomalous dimension, the author derives a compact analytic formula for C₁ up to a⁴ (eq. 4.3). The coefficients ˜C₁,₁ … ˜C₁,₄ are given solely in terms of the β‑function coefficients (β_i, β′i) and the decoupling constants d_i. As an explicit example, the paper evaluates C₁⁽⁴⁾ for n_f = 6, n_l = 4 (t and b heavy, u,d,s,c light), expanding in the limit m_b ≪ m_t and discarding power‑suppressed terms. The resulting expression contains only linear logarithms L{μt}, L_{μb}, their cubes, and a ζ(3) term, confirming the earlier observation that higher‑order polylogarithms cancel in the final C₁.

The conclusion emphasizes that the generalized LET and its RG‑improved form provide a systematic, low‑effort pathway to obtain high‑order effective Higgs‑gluon couplings in extensions of the Standard Model featuring multiple heavy quarks (e.g., fourth‑generation models, vector‑like quarks, or multi‑Higgs scenarios). The method also opens the door to resummation of large logarithms via standard RG techniques, a task the author earmarks for future work. Overall, the paper delivers a clear conceptual advance—showing that high‑loop effective couplings can be built from lower‑loop decoupling data—while delivering concrete new results at four loops, thereby enhancing the precision toolkit available for Higgs phenomenology.