Weak decays of the B_c meson to charmonium and D mesons in the relativistic quark model

Worth your time if you work on Bc decays or on quark-model form factors, but read it as a model paper, not a measurement. The real contribution is in abstract:s1: they compute the full q² dependence of the Bc → ηc, J/ψ, ηc′, ψ′, D and D* form factors as overlap integrals of their relativistic wave functions, rather than fixing one kinematical point and extrapolating with a pole or Gaussian ansatz. That is a real methodological step, and the appendix gives every formula (A.1–A.18), so the algebra is checkable. The model parameters were fixed earlier from spectra and radiative decays, not from these channels, so the rates are not fitted values in disguise. Table II is the most useful page in the paper: ten calculations side by side, showing the literature disagreed by a factor of two on Bc → ηc eν and J/ψ eν. Their numbers land on the low side, agreeing with Refs. [22–25] rather than [17–20]. That documentation is a service to the field independent of whether their numbers are right. Where it is soft: there is not a single uncertainty on any number in Tables II–V. That is the real problem, and I think it is more load-bearing than the Γ(2) extrapolation worry the reader flagged. The leading overlap integral carries the full model dependence of the Bc and charmonium wave functions and of the boost and Wigner rotation in Eq. (14); without an error estimate you cannot tell whether the factor-of-two gap to Refs. [17–20] is physics or model choice. The heavy-quark-symmetry check is waved through (Sec. IV, 'easy to check') and never displayed. On the Γ(2) extrapolation to q²_max for D and D*: the authors acknowledge it, Fig. 4 does suggest the subleading pieces are small and weakly q²-dependent, and those channels are CKM-suppressed anyway, so I would weight that concern lower than the reader did. Recommendation: send it out. It is a competent, honest model calculation with a genuine improvement in how the q² dependence is obtained, and the comparison table earns its place. Ask the referees to press on error estimates and to show the heavy-quark-limit relations explicitly rather than asserting them.

The central claim has two parts: (i) form factors are obtained as overlap integrals over the whole q² range, and (ii) the resulting charmonium rates sit a factor of two below Refs. [17–20]. For (i), the leading Γ(1) contribution is exact given the wave functions, and for charmonium the Γ(2) expansion in p/m_b,c is well justified across the range. The reader's weakest_assumption — that Γ(2) is extrapolated to q²=q²_max for Bc→D,D* — is correctly located but is not load-bearing for the headline: those channels are CKM-suppressed and yield rates of order 10⁻¹⁷ GeV (Table II), and Fig. 4 already shows the subleading V^S(2), V^V(2) terms for D* are small and weakly q²-dependent. The more load-bearing vulnerability for the headline charmonium numbers is that Γ(1) inherits the full model dependence of the Bc and charmonium wave functions and of the boost prescription (Eq. 14, Wigner rotation), and the paper attaches no uncertainty to any form factor or rate. Consequently the factor-of-two spread against Refs. [17–20] cannot be attributed to a physical effect rather than to model choice. Two further unshown checks compound this: the heavy-quark-symmetry consistency is asserted ('easy to check', Sec. IV) but not displayed, and the claim that Γ(2) 'introduces only small uncertainties' (Secs. III–IV) is supported by one figure for one form factor.

The reader's verdict of CONDITIONAL is appropriate and I would not move it. The paper is a competent model calculation with a genuine methodological improvement (full q² dependence from overlap integrals rather than single-point extrapolation) and parameters fixed in prior independent work, so the circularity burden is low. I agree with the reader that the Γ(2) extrapolation for D mesons is a real soft spot and that the absence of error bars makes the factor-of-two spread against Refs. [17–20] hard to interpret. Where I differ is on which concern is most load-bearing. The reader's weakest_assumption targets the D channels, which are CKM-suppressed and contribute negligibly to the headline; the paper is also explicit about the limitation in Sec. III and Sec. IV, and Fig. 4 supplies numerical evidence that the subleading terms are small. The more load-bearing issue for the central claim is that the leading overlap integral, which sets the overall normalization of the charmonium rates, carries the full unquantified model dependence of the wave functions and the boost prescription (Eq. 14). That is what makes the factor-of-two disagreement with Refs. [17–20] uninterpretable, and it is the point a reader would most want addressed before using Table II as a reference. The unshown heavy-quark-symmetry check and the single-figure support for the Γ(2) smallness claim are secondary but worth flagging. None of this is a correctness flaw; it is a request for uncertainty quantification on a model prediction, which is exactly what the CONDITIONAL verdict already encodes. Hence UNCHANGED, with partial agreement on the location of the weakest assumption.

The full q² determination of the form factors is a genuine methodological improvement over the single-point-plus-extrapolation procedure that dominates the earlier literature, and the Appendix supplies every formula needed to reproduce the calculation. The model parameters (quark masses, A, B, ε, κ) were fixed in prior work from quarkonium spectra, radiative decays, and B→D semileptonic decays, so the Bc rates reported here are predictions rather than fitted values; that is a real strength. The factor-of-two spread against Refs. [17-20] for the CKM-favored channels is a useful, falsifiable benchmark for CDF and LHCb. The paper would be considerably stronger if the claimed reduction in uncertainty were quantified rather than asserted.

The $B_c$ meson is the only bound state of two heavy quarks of different flavour, and its weak decay into charmonium is the channel experiments can actually reconstruct. The paper sets out to compute the form factors controlling these transitions at every value of momentum transfer, using the wave functions of a relativistic quark model rather than evaluating them at a single kinematic point and extrapolating with a pole or Gaussian ansatz. The authors argue this is necessary because the recoil momentum of the final meson is large over almost the whole allowed range, comparable to the heavy-quark masses, so a nonrelativistic treatment with a fitted $q^2$ shape would be unreliable. If the construction holds, the resulting semileptonic and nonleptonic rates carry no extrapolation ansatz, and the paper reports rates for the CKM-favoured channels roughly a factor of two below QCD sum rule and several quark-model estimates. The authors also give branching fractions for the corresponding two-body nonleptonic modes in the factorization approximation, which together with the semileptonic modes account for about 2.4 percent of the $B_c$ total width in their accounting.

What carries the argument. The machinery is the quasipotential approach to a bound quark-antiquark state: a Schrödinger-type equation whose kernel is a one-gluon-exchange plus linear-confining potential, with the confining part split into vector and scalar pieces by a mixing parameter $ε$ and a Pauli term controlled by $κ$. The weak current matrix element is the double integral of the two meson wave functions with a two-particle vertex; the final meson's wave function is boosted out of its rest frame by a Wigner rotation, which is what makes the recoil dependence relativistic rather than kinematic. The load-bearing step is the treatment of the interaction vertex: expanding it so that the quasipotential equation can absorb one of the momentum integrations turns the whole matrix element into ordinary overlap integrals, which is precisely what allows the $q^2$ dependence to come out in closed form. The model's parameters ($m_b=4.88$ GeV, $m_c=1.55$ GeV, $A=0.18$ GeV$^2$, $B=-0.16$ GeV, $ε=-1$, $κ=-1$) are the ones fixed earlier from quarkonium spectra and radiative decays, so the decay calculation introduces no new fitting.

Core claim. The central claim is that the weak decay form factors of $B_c$ into charmonium and $D$ mesons can be written as explicit overlap integrals of meson wave functions valid over the entire accessible range of $q^2$, with no interpolation between endpoints. The lowest-order vertex contribution is computed exactly, since a delta function removes one of the two momentum integrations; the subleading contribution, which involves the quark-antiquark interaction, is expanded in inverse powers of the heavy-quark masses with the recoil momentum kept unexpanded, and the heavy-quark energies approximated by their values at the recoil momentum so that the quasipotential equation can again be used to do one integration. For $B_c$ to charmonium the expansion is argued to be justified throughout; for $B_c$ to $D$ and $D^*$, where the active quark becomes light, it is justified only at large recoil and is extrapolated to the zero-recoil endpoint. The numerical output is a set of form factors at $q^2=0$ and $q^2=q^2_{max}$ (for $B_c$ to $J/ψ$, $f_+(0)=0.47$, $V(0)=0.49$, $A_1(0)=0.50$, rising to $1.07$, $1.34$, $0.88$ at maximum $q^2$), full $q^2$ curves, semileptonic rates, and factorized nonleptonic rates.

The premise it rests on. The subleading interaction contribution, derived only where the final meson's recoil momentum greatly exceeds the internal quark momentum, is carried over to the whole kinematic range, including the zero-recoil endpoint for the $D$ and $D^*$ channels, on the assertion that it is small and weakly $q^2$-dependent, an uncertainty the paper states but does not quantify.

The $q^2$ dependence of every form factor is predicted as a curve rather than a two-parameter ansatz, so the shapes of the differential rates in Figs. 11-16 are genuine predictions that can be checked bin by bin, not just the integrated rates.

For the CKM-favoured channels the predicted rates sit near $5.9\times10^{-15}$ GeV for $B_c\toη_c eν$ and $17.7\times10^{-15}$ GeV for $B_c\to J/ψ eν$, about half the QCD sum rule values, giving a sharp numerical disagreement that experiment can arbitrate.

The polarization content is specified: $Γ_L/Γ_T = 0.78$ for $B_c\to J/ψ eν$, $1.85$ for $ψ'$, and $0.53$ for $D^*$, which is a second, shape-level discriminator among the competing calculations.

With $τ_{B_c}=0.46$ ps and $a_1=1.14$, the branching fractions come out as $0.42\%$ for $B_c\toη_c eν$, $1.23\%$ for $B_c\to J/ψ eν$, and between $0.005\%$ and $0.21\%$ for the eight $B_c^+\to F^0 M^+$ nonleptonic modes listed.

Because the same wave functions reproduce the charmonium and $B_c$ spectra, the decay predictions are tied to spectroscopy: a shift in the wave functions large enough to move the rates would also move the mass levels.

The paper's disagreement with sum-rule and some quark-model results is largest exactly where the relativistic treatment matters most, in the $η_c$ and $η'_c$ channels; a lattice determination of $f_+(q^2)$ for $B_c\toη_c$ at two or three values of $q^2$ would separate 'this model underestimates' from 'the earlier calculations overestimate'.

The claim that the subleading vertex contributes little is testable inside the model itself: the ratio of the $Γ^{(2)}$ piece to the leading piece is plotted only for $V$ in $B_c\to D^*$, and reporting the same ratio for $f_+$ and $A_0$ near $q^2_{max}$ would convert the qualitative 'small uncertainties' into a number.

If the form-factor shapes hold, the ratio of $B_c\toψ'$ to $B_c\to J/ψ$ rates is a direct probe of the radial wave function's node, so a measurement of that ratio constrains the $2S$ charmonium wave function independently of the production mechanism.

The factorized nonleptonic rates scale as $a_1^2$ throughout, so the tables can be re-read for any $a_1$; the interesting residual is the ratio of $ρ$ to $π$ modes, which is fixed by decay constants alone and would test the factorization assumption rather than the form factors.

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