Soft Collinear Effective Theory for Heavy QCD Axions

Abstract: We develop a soft-collinear effective theory (SCET) framework for heavy QCD axion, considering two low-energy realizations and taking $B\to Ka$ as a benchmark mode. In the first realization, $aG\widetilde G$ is assumed to be the only independent axion interaction at the scale $μ\sim m_b$. We show that eliminating the redundant flavor-changing derivative-gluon operator in the weak effective theory generates a new dimension-seven axion operator identified as $\mathrm{O}_{\partial ag}$. We match this operator onto SCET and derive the corresponding leading-power soft and spectator-scattering contributions to $B\to Ka$. We obtain a factorized expression for the spectator contribution in terms of perturbative hard kernels and the $B$- and $K$-meson light-cone distribution amplitudes. The spectator contribution arises at the same order in the power expansion as the soft-overlap term and amounts to approximately $25\%$ of the soft contribution. In the second realization, the Wilson coefficient of $aG\widetilde G$ is assumed to be present above the electroweak scale. Renormalization-group evolution and matching then induce a direct $b\to sa$ operator, which subsequently results in a dominant soft form-factor contribution, whereas the gluonic spectator term turns out to be numerically subleading ($\sim 6-7\%$). We thus identify the conditions under which spectator scattering becomes relevant for heavy-axion production in rare $B$-meson decays. Finally, we derive the corresponding bounds on the axion decay constant $f_a$ for both realizations and compare their phenomenological implications.

• The paper develops a leading-order SCET factorization framework for heavy QCD axion production in B → Ka decays and identifies the dimension-seven operator O∂ag as the dominant low-energy contribution from a purely gluonic axion coupling.
• Spectator scattering contributes about 25%–30% of the soft amplitude when the gluonic coupling is imposed near the b-quark scale, enhancing the branching fraction by 1.5–1.7 and strengthening bounds on fa by roughly 25%–30%.
• When the gluonic coupling exists above the electroweak scale, two-loop matching generates a direct b → sa operator that dominates, reducing the spectator correction to 6%–7% and predicting branching fractions of 2.7–17.3 × 10⁻⁶ for fa = 200 GeV.

Overview

The paper develops a soft-collinear effective theory (SCET) framework for heavy QCD axions produced in rare flavor-changing neutral-current $B$-meson decays, using $B \to Ka$ as a benchmark mode (2608.12467). The central object of study is the minimal gluonic interaction $$\mathcal{L}_{\rm ALP} \supset (C_{gg}f_a)\, a G_{\mu\nu}^A \widetilde G^{A\mu\nu}$$, with $C_{gg} = \alpha_s/8\pi$ as in KSVZ-type realizations. The authors consider two distinct ultraviolet boundary conditions: (i) the gluonic coupling is the only independent axion interaction at $\mu \sim m_b$, with no electroweak evolution or matching, and (ii) the coupling is present above the electroweak scale, where renormalization-group evolution and two-loop electroweak matching generate a direct $b \to sa$ operator. The analysis is carried out at leading order in $\alpha_s$ in the position-space formulation of SCET.

Generation of a new dimension-seven operator

A central technical observation concerns the treatment of the redundant operator $$\mathcal{O}_{Dg} = \bar s \gamma^\mu P_L T^A b (D^\nu G_{\nu\mu}^A)$$ in the weak effective theory. In the Standard Model, this operator is eliminated using the gluon equation of motion, leaving only the four-quark penguin operators $O_3$–$O_6$. In the presence of $B \to Ka$0, however, the gluon equation of motion acquires an axion-dependent term, so that eliminating $B \to Ka$1 generates a genuinely new dimension-seven interaction

$B \to Ka$2

The coefficient $B \to Ka$3 inherits a numerically enhanced factor $B \to Ka$4 from the charm-loop contribution to $B \to Ka$5, compared with $B \to Ka$6 for the chromomagnetic coefficient $B \to Ka$7. Consequently, effects descending from $B \to Ka$8 dominate over those from $B \to Ka$9, and the four-quark penguin operators contribute only at an additional loop level. One caveat is acknowledged: treating $$\mathcal{L}_{\rm ALP} \supset (C_{gg}f_a)\, a G_{\mu\nu}^A \widetilde G^{A\mu\nu}$$0 as a local Wilson coefficient holds for gluon virtualities $$\mathcal{L}_{\rm ALP} \supset (C_{gg}f_a)\, a G_{\mu\nu}^A \widetilde G^{A\mu\nu}$$1, which is exact for hard-collinear gluons but only approximate for hard ones.

Matching onto SCET and the spectator-scattering contribution

The operator $$\mathcal{L}_{\rm ALP} \supset (C_{gg}f_a)\, a G_{\mu\nu}^A \widetilde G^{A\mu\nu}$$2 is matched onto SCET$$\mathcal{L}_{\rm ALP} \supset (C_{gg}f_a)\, a G_{\mu\nu}^A \widetilde G^{A\mu\nu}$$3 using the standard machinery: collinear Wilson lines $$\mathcal{L}_{\rm ALP} \supset (C_{gg}f_a)\, a G_{\mu\nu}^A \widetilde G^{A\mu\nu}$$4 to build gauge-invariant gluonic and quark building blocks, BPS field redefinitions to decouple soft-collinear interactions, and heavy-quark effective theory matching for the $$\mathcal{L}_{\rm ALP} \supset (C_{gg}f_a)\, a G_{\mu\nu}^A \widetilde G^{A\mu\nu}$$5 field. The leading-power SCET operator is

$$\mathcal{L}_{\rm ALP} \supset (C_{gg}f_a)\, a G_{\mu\nu}^A \widetilde G^{A\mu\nu}$$6

The momentum scaling of the external legs fixes the internal gluon to be $$\mathcal{L}_{\rm ALP} \supset (C_{gg}f_a)\, a G_{\mu\nu}^A \widetilde G^{A\mu\nu}$$7-hard-collinear. A notable point is the treatment of the axion momentum: two-body kinematics with $$\mathcal{L}_{\rm ALP} \supset (C_{gg}f_a)\, a G_{\mu\nu}^A \widetilde G^{A\mu\nu}$$8 gives $$\mathcal{L}_{\rm ALP} \supset (C_{gg}f_a)\, a G_{\mu\nu}^A \widetilde G^{A\mu\nu}$$9, so the axion is assigned a non-relativistic scaling $C_{gg} = \alpha_s/8\pi$0; the paper shows this assignment is consistent even in the limit $C_{gg} = \alpha_s/8\pi$1 where the axion would become $C_{gg} = \alpha_s/8\pi$2-collinear, and the final results are insensitive to this choice. The axion mass range probed is roughly $C_{gg} = \alpha_s/8\pi$3 GeV.

The spectator-scattering amplitude is computed from the time-ordered product of the SCET Lagrangian terms, with the hard-collinear gluon attaching to the soft $C_{gg} = \alpha_s/8\pi$4-meson spectator quark. Inserting the leading-twist kaon and $C_{gg} = \alpha_s/8\pi$5-meson light-cone projectors and evaluating the Dirac trace (only the antisymmetric, parity-odd part survives against the external $C_{gg} = \alpha_s/8\pi$6), the amplitude factorizes into perturbative hard kernels convoluted with $C_{gg} = \alpha_s/8\pi$7 and $C_{gg} = \alpha_s/8\pi$8. Remarkably, the dependence on the kaon momentum fraction $C_{gg} = \alpha_s/8\pi$9 cancels entirely in the hard kernel, and the result collapses to a compact form governed by the inverse moment $\mu \sim m_b$0:

$\mu \sim m_b$1

The authors also demonstrate that a competing spectator topology with $\mu \sim m_b$2-hard-collinear gluons is kinematically forbidden, since it would require a fine-tuned cancellation in manifest power counting and leads to an inconsistent scaling for the second internal gluon.

Soft-overlap contributions

For the soft contribution, the one-loop QCD amplitude with insertion of $\mu \sim m_b$3 and a hard gluon of virtuality $\mu \sim m_b$4 is matched onto the SCET current $\mu \sim m_b$5, whose matrix element is parameterized by the soft form factor $\mu \sim m_b$6, with $\mu \sim m_b$7 from the light-cone quark model. The matching coefficient is

$\mu \sim m_b$8

An analogous calculation is performed for the chromomagnetic operator $\mu \sim m_b$9 combined with $b \to sa$0, which generates both a spectator amplitude $b \to sa$1 and a soft-overlap contribution with a coefficient containing logarithms of $b \to sa$2. The $b \to sa$3-induced spectator amplitude is found to be about 15% of the $b \to sa$4 spectator amplitude, consistent with the hierarchy $b \to sa$5.

The electroweak-generated direct operator

In the second realization, $b \to sa$6 is present above $b \to sa$7, and two-loop electroweak matching generates the direct flavor-changing operator $b \to sa$8 with a Wilson coefficient of order $b \to sa$9, whose explicit form involves polylogarithmic functions of $\alpha_s$0 for $\alpha_s$1. Matching this operator onto SCET yields both the leading $\alpha_s$2-type current (soft overlap) and the subleading $\alpha_s$3-type currents (spectator scattering), the latter contributing

$\alpha_s$4

The soft-overlap matrix element reproduces the conventional LCSR-based form-factor description, and the spectator result is cross-checked against the known form-factor expressions of Beneke and Feldmann.

Numerical results and phenomenology

The key quantitative findings are the spectator-to-soft ratios in the two scenarios:

Scenario	Source	Ratio $\alpha_s$5
Gluonic coupling at $\alpha_s$6	$\alpha_s$7	$\alpha_s$8–$\alpha_s$9
Coupling above $$\mathcal{O}_{Dg} = \bar s \gamma^\mu P_L T^A b (D^\nu G_{\nu\mu}^A)$$0 (KSVZ)	$$\mathcal{O}_{Dg} = \bar s \gamma^\mu P_L T^A b (D^\nu G_{\nu\mu}^A)$$1	$$\mathcal{O}_{Dg} = \bar s \gamma^\mu P_L T^A b (D^\nu G_{\nu\mu}^A)$$2–$$\mathcal{O}_{Dg} = \bar s \gamma^\mu P_L T^A b (D^\nu G_{\nu\mu}^A)$$3
Chromomagnetic operator	$$\mathcal{O}_{Dg} = \bar s \gamma^\mu P_L T^A b (D^\nu G_{\nu\mu}^A)$$4	$$\mathcal{O}_{Dg} = \bar s \gamma^\mu P_L T^A b (D^\nu G_{\nu\mu}^A)$$5–$$\mathcal{O}_{Dg} = \bar s \gamma^\mu P_L T^A b (D^\nu G_{\nu\mu}^A)$$6

In the low-energy gluonic scenario, the spectator term is of the same power counting as the soft term and enhances the branching fraction by a factor of $$\mathcal{O}_{Dg} = \bar s \gamma^\mu P_L T^A b (D^\nu G_{\nu\mu}^A)$$7–$$\mathcal{O}_{Dg} = \bar s \gamma^\mu P_L T^A b (D^\nu G_{\nu\mu}^A)$$8 for constructive interference, strengthening the inferred bound on $$\mathcal{O}_{Dg} = \bar s \gamma^\mu P_L T^A b (D^\nu G_{\nu\mu}^A)$$9 by roughly 25%–30%; the relative sign must be retained, since destructive interference would instead weaken the limit. The predicted branching fraction is

$O_3$0

which probes only $O_3$1 because the short-distance coefficient carries the flavor-changing suppression, but remains competitive with hadronic-ALP bounds from pion and kaon decays. A zero-background Belle II search with $O_3$2 would reach only about $O_3$3 events at 90% CL in this scenario.

In the electroweak scenario, the direct operator dominates, giving $O_3$4 for UV scales of $O_3$5 TeV, with the gluonic spectator term reduced to a $O_3$6–$O_3$7 correction. The paper also compares the SCET matrix elements with the conventional LCSR $O_3$8 approach: the spectator contribution makes the SCET result larger than the LCSR prediction (still overlapping within the combined $O_3$9–$O_6$0 uncertainties dominated by $O_6$1–$O_6$2 GeV) and reverses its mass dependence from increasing to decreasing in $O_6$3.

The appendix on ALP–meson mixing notes that in Case 1, where $O_6$4 GeV, the interference term from $O_6$5–$O_6$6 mixing can reach $O_6$7 of the direct contribution away from resonance, a potentially relevant effect whose phase structure is left unexamined.

Limitations and open questions

The analysis is restricted to leading order in $O_6$8; the renormalization-group evolution of the axion-involved SCET operators is discussed only qualitatively, with resummation effects estimated at 10%–20% on individual amplitudes and partial cancellation in $O_6$9, since the leading cusp logarithms cancel in the ratio. A complete resummed analysis is explicitly deferred. The minimal gluonic Lagrangian is, by the authors' own admission, not a complete bottom-up EFT basis—derivative quark-current couplings and electroweak gauge-boson operators are required for renormalizability order by order, and their omission is a deliberate isolation of the $B \to Ka$00 hadronic consequences. The treatment of $B \to Ka$01 as a local Wilson coefficient is approximate for hard gluon virtualities above $B \to Ka$02. For $B \to Ka$03, the gluon pair at the vertex could become hard-collinear, generating genuinely new SCET$B \to Ka$04 structures outside the mass window considered. Finally, the phase-dependent interference between direct production and $B \to Ka$05 mixing in Case 1 remains an open phenomenological question.

Conclusion

This paper establishes a systematic SCET factorization for heavy-axion production in $B \to Ka$06, identifying the dimension-seven operator $B \to Ka$07 as the dominant low-energy consequence of a purely gluonic axion coupling below the electroweak scale, and quantifying the leading-power spectator-scattering contribution that is invisible in conventional form-factor treatments. The main physical conclusion is that the relative importance of spectator scattering versus soft overlap is controlled by the ultraviolet boundary condition: it is a $B \to Ka$08–$B \to Ka$09 effect when the gluonic coupling is imposed at $B \to Ka$10, but only $B \to Ka$11–$B \to Ka$12 when electroweak evolution generates a direct $B \to Ka$13 operator. This delineates the conditions under which precision analyses of rare $B \to Ka$14 decays must go beyond the total form factor $B \to Ka$15.