Semileptonic $B_c \to \eta_c, J/\psi$ transitions

Using a systematic, symmetry-preserving continuum approach to the Standard Model strong-interaction bound-state problem, we deliver parameter-free predictions for all semileptonic Bc → ηc, J/ψ transition form factors on the complete domains of empirically accessible momentum transfers. Working with branching fractions calculated therefrom, the following values of the ratios for τ over μ final states are obtained: Rηc = 0.313(22) and RJ/ψ = 0.242(47). Combined with other recent results, our analysis confirms a 2σ discrepancy between the Standard Model prediction for RJ/ψ and the single available experimental result.


Introduction
The B c meson was discovered a little over twenty years ago [1]. With mass m B c = 6.2749(8) GeV [2], it lies below the threshold for BD decay; and since it is an open flavour state, electromagnetic decays are forbidden. Thus, within the Standard Model, only flavour-changing weak decays are possible. Consequently, B c has a relatively long lifetime [2]: 0.510(9) ps, (1) which is, e.g. ten-billion-times longer than that of the η c . These things make the B c an especially interesting system: it is the lightest open-flavour bound-state of the two heaviest quarks in Nature that are experimentally pliable; and lives long enough to make measurements possible. Flavour-changing B c weak decays involve one of the following transitions:b →ū,b →c, c → s, c → d. Specific entries in the Cabbibo-Kobayashi-Maskawa (CKM) matrix modulate the strengths of these transitions. Since |V cs | is the largest of the four that can be involved here, one may anticipate that B c → B s transitions dominate. This expectation is supported by contemporary calculations, e.g. Refs. [3][4][5]. Another factor is the available phase space. For instance, with η c , J/ψ final states, this is more than ten-times larger than for B; and such magnification may be sufficient to overwhelm the factor of roughly six suppression from |V cb |/|V cd |. Calculations of the branching fractions ratio bear this out, e.g. [5]: B B + c →η c + ν /B B + c →B 0 + ν ≈ 6, where is a light lepton. (This is a longstanding qualitative prediction [6,7].) Therefore, it is not surprising that the B c was discovered in decays to J/ψ final states, especially given the narrow, prominent decay width for J/ψ → + − . * Corresponding Author Email addresses: zqyao@smail.nju.edu.cn (Zhao-Qian Yao), (2) -red circle, empirical result from LHCb Collaboration [8]; blue star -our prediction; grey square -lQCD result [9,10]; and gold band -unweighted mean of central values from contemporary calculations [11][12][13][14][15][16] (Details provided below in connection with Table 5B.) Data acquired in the last decade, potentially indicating violations of lepton universality in b-quark decays [17][18][19][20][21][22][23], raise studies of the semileptonic decays of B c -mesons with groundstate charmonia final-states to a new level of importance in the search for physics outside the Standard Model paradigm. In fact, the LHCb collaboration has reported [8]: and stated that this result lies approximately two standarddeviations (2σ) above the range of central values predicted by reliable calculations within the Standard Model, as highlighted by Fig. 1. Such a discrepancy could signal violation of lepton universality in Nature's weak interactions.

Transition Form Factors: Definitions
We consider the following transition matrix elements: where µν ; the squared-momentum-transfer is t = −Q 2 ; and t M ± = (m B c ± m M ) 2 , M = η c , J/ψ. (t − is the largest accessible value of t in the identified physical decay process.) The scalar functions in Eqs. (3) are the semileptonic transition form factors, which express all effects of hadron structure on the transitions. Ensuring the absence of kinematic singularities in Eqs. (3), symmetries require With predictions for the transition form factors in hand, one can compute the associated decay branching fractions from the differential decay width for B c → Ml + ν l : whereas when M = J/ψ, After integrating Eq. (5) to obtain the required partial widths, one quotes the branching fractions, B B c →Ml ν l , with respect to the total width determined from Eq. (1).

Transition Form Factors: Matrix Elements
At leading-order (rainbow-ladder, RL) in the most widely used CSM truncation [41,42], which has been employed to unify, inter alia, all semileptonic pseudoscalar-to-pseudoscalar transitions involving π, K, D (s) initial states [40,43], the matrix elements in Eqs. (3) take the following form: (9) where N c = 3 and the trace is over spinor indices. There are three types of matrix-valued functions in Eq. (9). The simplest are the propagators for the dressed-quarks involved in the transition process: S f (s), f = c, b; then there are the Bethe-Salpeter amplitudes for the mesons involved: Γ M ; and, finally, the dressed b → c weak transition vertex: W bc µ . Each of these quantities can be computed once the kernel of the RL Bethe-Salpeter equation is specified. Importantly, Eq. (9) preserves the identities in Eqs. (4), both algebraically and numerically.
With a realistic kernel, RL truncation provides a sound description of systems wherein [30]: (i) orbital angular momentum does not play a large role and (ii) the non-Abelian anomaly can be neglected. In such cases, corrections to the truncation largely cancel amongst themselves. η c , J/ψ, B c are amongst the systems for which these conditions hold. Herein, we use the RL kernel detailed in Refs. [31][32][33]: }/s, m t = 0.5 GeV. Following standard practice, in solving all integral equations [44], we use a mass-independent momentum-subtraction renormalisation scheme, fixing each renormalisation constant in the chiral limit, with renormalisation scale ζ = 19 GeV=: ζ 19 . Table 1: Static properties of mesons evaluated using bound-state equations defined by the kernel specified in Eqs. (10), (11). Normalisation: the empirical value of the pion's leptonic decay constant is f π ≈ 0.092 GeV. Empirical values (expt.), where available, drawn from Ref. [2]; and lattice-QCD (lQCD) results for leptonic decay constants from Refs. [51][52][53]. The mean absolute relative error between our predictions and empirical results is 3.6%. (All dimensioned quantities in GeV.) The elaboration of Eqs. (10), (11) and their connection with QCD are described in Refs. [31,32,34]. Here, we simply reiterate some points. (i) The interaction is consistent with that found through studies of QCD's gauge sector, capitalising on the fact that the gluon propagator is a bounded, smooth function of spacelike momenta, which achieves its maximum value on this domain at s = 0 [45][46][47]; and the dressed gluon-quark vertex does not possess any structure which can qualitatively alter these features [48]. (ii) It is specified in Landau gauge because, inter alia, this gauge is a fixed point of the renormalisation group and ensures that sensitivity to the form of the gluon-quark vertex is minimal, thus providing the conditions for which RL truncation is most accurate. (iii) The interaction preserves the one-loop renormalisation group behaviour of QCD; hence, e.g. the quark mass-functions produced are independent of the renormalisation point. (iv) On s (2m t ) 2 , Eq. (11) defines a two-parameter Ansatz, the details of which determine whether such corollaries of emergent hadron mass as confinement and dynamical chiral symmetry breaking are realised in solutions of the bound-state equations [49,50].
The analyses in Ref. [33] determined that one can unify the properties of a diverse range of systems using ω = 0.8 GeV, ς 3 = Dω = (0.6 GeV) 3 and we use these values hereafter. An additional feature of Eq. (11) is that with a given value of ς, results for observable quantities are practically insensitive to variations ω → ω(1 ± 0.1); so, there is no issue of fine tuning.
We now illustrate the qualities of the framework by computing an array of heavy pseudoscalar meson static properties, i.e. their masses and leptonic decay constants. Solving the gap and Bethe-Salpeter equations (see, e.g. Ref. [39] and Ref.
one obtains the results in Table 1. The mean absolute relative error between our predictions and empirical values is 3.6%. For later use, we note that r b:c =m b /m c = 4.32 and, equivalently, The masses in Eq. (12) These quantities are analogous to the "running masses" often quoted in connection with heavy quarks and our predictions are within 3% of those listed elsewhere [2]. It is worth remarking on some important physical aspects of the weak transition vertex, W cb µ . Pseudoscalar-to-pseudoscalar transitions only involve the vector part, which possesses poles = 0. Pseudoscalar-to-vector transitions also involve the axial-vector part. This has poles at Q 2 + m 2 B c ,B c1 = 0. The presence of these poles is a prerequisite for any valid analysis of B c → η c , J/ψ semileptonic transitions. They are manifest in our treatment.

Computational Scheme and Results
The integration in Eq. (9) samples the appearing functions on a material domain of their complex-valued arguments. So long as the masses of the initial and final state mesons are similar, i.e. the ratio of the current-masses of the quarks involved, r Q 1 :Q 2 , does not differ too much from unity, the integral can readily be evaluated using simple numerical techniques because t − and, hence, the maximum momentum of the recoiling meson, remains modest. However, at some value of r Q 1 :Q 2 =: r f , t − becomes so large that singularities associated with the analytic structure of the dressed-quark propagators [54,55] move into the complex-s 2 integration domain and straightforward numerical techniques fail.
This sort of problem was solved in Ref. [56] by using perturbation theory integral representations (PTIRs) [57] for each matrix-valued function in the integrand defining the associated matrix element. However, constructing accurate PTIRs is time consuming; and here the challenge is compounded because the complete set of integrands involves 46 distinct scalar functions. Like Ref. [40], we therefore adopt a different approach.
(I) -We consider the semileptonic transitions of a fictitious cQ pseudoscalar meson: B cQ → η cc , J/ψ cc . All relevant Schwinger functions and, subsequently, the transition form factors are computed as a function ofm Q as it is increased from the point r Q  (III) -Having completed these exercises, we combine the outcomes to produce our final results.
It is worth remarking that the SPM is founded on interpolation via continued fractions [66,67]. It is typically augmented today by statistical sampling. The approach avoids any assumptions on the function used for the representation of input and captures both local and global features of that source. This latter aspect underpins the reliability of subsequent extrapolations. The SPM can accurately reconstitute a complex-valued function within a radius of convergence determined by that one of the function's branch points which lies nearest the real domain from which the sample points are drawn. The statistical aspect ensures that one has a genuine estimate of the uncertainty associated with any extrapolation.
To elucidate further, we first compute the value of a given   (18) inspecting the C(N, M) combinatorial possibilities for the M element subset and eliminating those functions which fail to satisfy a simple physical constraint; namely, we insist that each interpolation be smooth on the domain of required current-quark masses. For all quantities considered, this constraint yields n I ≈ 100, 000 acceptable interpolations. Our prediction for X is then obtained by extrapolating each of the associated n I physical SPM interpolants to the target current-mass and reporting as the result that value which sits at the centre of the band within which 68% of the interpolants lie. This 1σ band is quoted as the uncertainty in the result. The reliability of our SPM procedure is readily illustrated. The meson masses in Table 1 were computed directly via the Bethe-Salpeter equation using the masses in Eq. (12). One may equally compute masses using the procedures described in (I) and (II) above. Using (I) on r η QQ f ≤ r Q :b ≤ 1, we find m B c = 6.259(1) GeV; and employing (II) on 1 ≤ r Q :c ≤ r η c f , m B c = 6.281(6) GeV. Hence, the final SPM result is m B c = 6.270(4) GeV, (14) matching the directly computed value in Table 1. Repeating this exercise using the limiting current-masses in the J/ψ channel, the SPM result is 6.267(8) GeV, again agreeing with Table 1.
On the physical domain associated with any value of r Q 1 :Q 2 , each of the transition form factors can accurately be interpolated using the following function: where α 1,2,3 and m are functions of r Q :b or r Q :c . It is the coefficients α 1,2,3 for which we develop SPM interpolations. The results are listed in Table 2.

Transition Form Factors: Predictions and Comparisons
Our predictions for the B c → η c semileptonic transition form factors are given by Eq. (15) combined with the appropriate masses in Tables 1, 3 and coefficients listed in Table 2. The maximum recoil (t = 0) value of each form factor is listed in Table 4, compared with recent continuum and lattice estimates. Aspects of the information in Table 4 are depicted in Fig. 2 (14).) The points in both panels are preliminary lQCD results from Ref. [26]. results reported in Ref. [26] is 10(3)%, with the lQCD values lying uniformly below our curves. No further information on B c → η c is currently available from lQCD. Here, therefore, the interpolations we provide for our calculated transition form factors can be valuable in analysing future experimental data on the related transitions.
Working with our predictions and using Eqs. (5) -(8) evaluated with empirical lepton and meson masses, we obtain the B c → η c branching fractions reported in Table 5A. Our results match well with other contemporary estimates.
Similarly, our predictions for the B c → J/ψ semileptonic transition form factors are given by Eq. (15) combined with the appropriate masses in Tables 1, 3 and coefficients listed in Table 2. The maximum recoil (t = 0) value of each form factor is listed in Table 4, compared with recent continuum and lattice estimates. Once again, as highlighted by Fig. 2, different approaches produce a range of t = 0 form factor values; but there is no significant tension, with all values falling within 15% of their respective means.
Our B c → J/ψ transition form factors are depicted in Figs. 4. Comparing with lQCD results [10], despite minor qualitative differences, most notably concerning V B c →J/ψ (t) in Fig. 4B, there is semi-quantitative agreement. The interpolations we provide for our calculated transition form factors could be used to reduce a dominant systematic error in the extraction of R J/ψ  (8) and empirical lepton and meson masses: (A) -B c → η c ; and (B) -B c → J/ψ. Two uncertainties are listed with our results: first -1σ SPM uncertainty; second -from error on |V cb |. Column 3 reports the ratio of the first two columns: |V cb | cancels. Comparisons are provided with other analyses: quark model (QM) [11,12]; phenomenology (ph) [13]; sum rules (SR) [14] modelling based on perturbative QCD (mpQCD) [15]; Salpeter equation (iBS) [16]; and lQCD [9,10]. (No lQCD results are available for inclusion in Panel A.) As additional context, we list an unweighted average value for each quantity, evaluated with our prediction excluded (mean-e) and included (mean-i). Branching fractions are to be multiplied by 10 −3 .  (2) from experiment [8], paving the way to improved precision and a more stringent test of the Standard Model. Using Eqs. (5) -(8) evaluated with empirical lepton and meson masses and our predictions in Figs. 4, we obtain the B c → J/ψ branching fractions reported in Table 5B. Our results accord well with other contemporary estimates.

Conclusions and Perspectives
We employed a systematic, symmetry-preserving approach to the continuum strong-interaction bound-state problem in the Standard Model to calculate the semileptonic B c → η c , J/ψ transition form factors on the entire physical kinematic domain. The framework [Sec. 3] has been used successfully to unify the properties of mesons and baryons with 0 − 3 heavy-quarks; and from this foundation, we arrived at an array of parameterfree predictions, including the branching fractions B B c →η c l ν l , B B c →J/ψl ν l , l = µ, τ [Sec. 5].
A key result of our analysis is highlighted by Fig. 1. Namely, contemporary Standard Model calculations of the ratio R J/ψ in Eq. (2) are in agreement. Combined via the mean of their central values, they produce R J/ψ = 0.253 (16), which is approximately 2σ below the empirical result reported in Ref. [8]. If subsequent, precision experiments do not lead to a substantially lower central value, then one may conclude that lepton flavour universality is violated in semileptonic B c → J/ψ decays. However, the precision of existing empirical information is insufficient to support such a claim. Moreover, a compelling case could only be compiled by including information on semileptonic B c → η c decays. We predict R η c = 0.313 (22); and the mean obtained from modern continuum analyses is 0.31(4) [Table 5A].
Natural extensions of this work include kindred analyses of b → c transitions in the semileptonic decays of B (s) mesons with D ( * ) (s) mesons in the final-state. Existing surveys of Standard Model theory estimates of the "R" ratios associated with these additional processes yield values similar to those discussed herein, with the result for the pseudoscalar-meson finalstate being ∼ 15% greater than that for the vector-meson finalstate [69,70]. Augmenting such analyses via the parameterfree unification of the results obtained herein with predictions for these other "R" ratios should serve to increase confidence in Standard Model predictions and strengthen any case for or against lepton flavour universality in Nature. Furthermore, one could expand the coverage of our study to include a wider range of measurable quantities [4,11], providing additional benchmarks for Standard Model tests in B c decays.