The Bc mesons contain one bottom quark and one charm quark, making them useful test cases for non-perturbative strong-interaction physics. Their decay constants describe how strongly a meson couples to a quark current and cannot be calculated from perturbation theory alone. The authors investigated the pseudoscalar Bc, scalar Bc0*, vector Bc* and axial-vector Bc1 states. Four constants were defined with vector or axial-vector currents, while two new determinations used tensor or axial-tensor currents for Bc* and Bc1. The tensor quantities do not contribute to Standard Model decays, but they can probe tensor interactions proposed in extensions of the model.

The calculation was performed with QCD sum rules, a framework that matches a hadronic representation of quark-current correlators with an operator product expansion for the vacuum. Correlators were Borel transformed to suppress contributions from excited states and continuum thresholds. Their hadronic sides were represented by the lowest Bc state in each relevant channel plus continuum and higher states, while their perturbative sides included a leading-power contribution through next-to-leading order in the strong coupling αs and condensate corrections through dimension six. The dimension-four term was represented by the two-gluon condensate. Dimension-six contributions included a three-gluon condensate and a light-quark four-quark condensate.

One of the study’s main technical advances was the derivation of the dimension-four and dimension-six Wilson coefficients for tensor and axial-tensor currents. Previous work had supplied corresponding expressions for vector currents, and the authors independently checked those results. They also derived new analytical expressions valid when the charm and bottom quark masses differ, rather than assuming their equality. The tensor-current condensate terms were obtained through background-field expansions of the heavy-quark propagators, with separate calculations implemented using Mathematica packages for Dirac traces and loop integrals as an internal cross-check. Machine-readable expressions and the notebook used for the tensor-condensate calculation were supplied as ancillary files.

Effective continuum thresholds were not fixed by hand. For each Borel and renormalisation-scale combination, the researchers used daughter, or mass, sum rules to find the threshold that made the sum-rule mass most closely approach the physical meson mass. They then extracted each decay constant from the ground-state pole contribution. The final analysis used a Borel-parameter interval from 6 to 9 GeV², where the results were comparatively stable, and varied the renormalisation scale broadly from 3 to 9 GeV. The scale was evolved with four-loop renormalisation-group running. This wider-than-usual scale range was intended to avoid underestimating uncertainty from missing higher-order terms in αs.

The numerical scan comprised 49 Borel-parameter points and 19 renormalisation-scale points, producing 931 combinations for each of the six sum rules. Two quality filters were applied. The pole contribution had to exceed 50%, and the relative deviation between the physical and sum-rule masses had to remain below 25%. These cuts removed unphysical imaginary values and suppressed condensate and next-to-leading-order contributions where the sum rules were less reliable. Almost every sampled point passed for four of the channels. In the two Bc1 sum rules, roughly 60% of the points passed. The resulting continuum thresholds were close to, but generally above, the corresponding meson mass-squared values: 52 ± 4 GeV² for the scalar vector current, 46 ± 4 GeV² for the vector vector current, 49 ± 3 GeV² for the tensor current, 49 ± 4 GeV² for the pseudoscalar axial current, 56 ± 4 GeV² for the axial-vector axial current and 54 ± 4 GeV² for the axial-tensor current.

The extracted constants were reported as the midpoint of the full range found across the correlated scan, with the uncertainty set to half that range. Because the continuum threshold was redetermined at every Borel and scale point, the analysis did not separate the variation sources into artificial independent components. The dimension-four condensate contribution was at least one order of magnitude smaller than the perturbative contribution, while dimension-six terms were at least another order of magnitude smaller than the dimension-four corrections. Within the dimension-six terms, the four-quark condensate was more important than the three-gluon condensate. As an additional check, the authors repeated the extraction with subtracted dispersion relations rather than the Borel transform and obtained compatible results.

The refined (axial-)vector predictions were consistent with earlier QCD sum-rule determinations. They were also compatible with the published lattice-QCD results for the Bc and Bc* mesons, although the authors stressed that available lattice determinations showed substantial mutual spread: an ETM result in conference proceedings and a more recent CLQCD preprint differed, while only the earlier HPQCD calculation cited in the paper had been published in a peer-reviewed journal. The sum-rule uncertainties were larger than those achieved by some lattice calculations. Much of that difference was attributed to the deliberately conservative renormalisation-scale treatment, whereas earlier sum-rule analyses had fixed the scale or explored only a narrow range. The new tensor constants for Bc* and Bc1 had no available lattice-QCD or sum-rule counterparts for direct comparison.

Beyond the constants themselves, the study quantified how much each lowest Bc state contributes to the unitarity bounds that constrain b-to-c form factors. Using the lower ends of the calculated constant ranges and omitting excited one-particle states produced conservative estimates. The Bc0*, Bc, Bc*, Bc1, tensor-Bc* and axial-tensor-Bc1 states account, respectively, for about 9.40%, 18.9%, 17.5%, 7.14%, 24.9% and 8.04% of the corresponding operator-product-expansion bounds at zero momentum transfer. The tensor Bc* contribution was the largest. Such non-negligible saturation can narrow the allowed normalisation and shape of b-to-c form factors used in B̄-to-D(*)ℓν̄ decays and related transitions. The authors did not quantify the resulting improvement in any specific form factor, because that would require a dedicated analysis of input precision, correlations and kinematic coverage.

The result is therefore best understood as updated theoretical infrastructure rather than a new anomaly. It combines a broader uncertainty assessment for established vector-current constants with the first sum-rule determinations for two tensor-current quantities. The authors also note an important caveat: in a modified parametrisation of the dispersive bounds, the influence of the one-particle contributions is significantly smaller. Whether these updated constants ultimately yield tighter phenomenology therefore depends on how the constraints are implemented. Even so, the analysis gives a more complete set of non-perturbative inputs for studying exclusive b-to-c decays within the Standard Model and for investigating hypothetical tensor interactions in Bc and related systems.

Subject of Research: QCD sum-rule determination of Bc-meson decay constants and their use in b-to-c form-factor unitarity bounds

Article Title: Decay constants of the \(B_c\) mesons with vector and tensor currents

Article References: Costa i Reina, E., Gubernari, N., & Wüthrich, Z. (2026). Decay constants of the $$B_c$$ mesons with vector and tensor currents. The European Physical Journal C, 86(10), Article 1204. https://doi.org/10.1140/epjc/s10052-026-16381-w

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DOI: 10.1140/epjc/s10052-026-16381-w

Keywords: Bc mesons, decay constants, QCD sum rules, tensor currents, b-to-c transitions, form factors, unitarity bounds, semileptonic decays, quark condensates, Decay, constants, mesons