Sensitivity study of $\bar{K}_1(1270)$ decay dynamics using four $D\to \bar{K}_1(1270)(\to \bar K\pi\pi)e^+\nu$ decay channels

The strange axial-vector mesons offer interesting possibilities for the study of quantum chromodynamics in the non-perturbative regime. Due to the presence of a strange quark with mass greater than the up and down quark masses, SU(3) symmetry is broken so that the $ ^3P_1 $ and $ ^1P_1 $ states mix with each other to construct the mass eigenstates, $ \bar{K}_1(1200) $ and $ \bar{K}_1(1400) $, by the mixing angle $ \theta_{\bar{K}_1} $ [1]. The mixing angle $ \theta_{\bar{K}_1} $ plays a crucial role in determining the theoretical calculations, such as the helicity form factors and branching fractions (BFs) for semileptpnic D-meson decays into strange axial-vector mesons [2−4].

Semileptonic charm decays, induced by the quark-level process $ c\to se^+\nu_e $, are predominantly mediated by pseudoscalar (K) and vector ($ K^*(892) $) mesons, i.e., contain a kaon and at most one pion in the final-state hadronic systems [5, 6]. However, semileptonic charm decays to higher-multiplicity final states are expected to proceed predominantly via the axial–kaon system [7] and are therefore strongly suppressed. The $ D\to \bar{K}\pi\pi e^+\nu_e $ decays provide a unique opportunity to study the properties and interactions of $ \bar{K}_1(1270) $ and $ \bar{K}_1(1400) $ mesons in a clean environment, without any additional hadrons in the final states. Such studies can lead to a better determination of $ \theta_{\bar{K}_1} $, as well as more precise measurements of the masses and widths of the $ \bar{K}_1 $ mesons, all of which currently carry large uncertainties [8]. Furthermore, by exploiting the measured properties of $ D\to \bar{K}_1(1270) \ell^+\nu_\ell $ and $ B\to \bar{K}_1(1270)\gamma $ decays, the photon polarization in $ b\to s\gamma $ can be determined without considerable theoretical ambiguity [9, 10]. Charge-conjugate decays are implied throughout the paper.

The BFs of $ \bar K_1(1270) $ decays to different two-body final states of $ \bar K\rho,\ \bar K^{*}_0(1430)\pi,\ \bar K^*(892)\pi,\ \bar K \omega, \bar Kf_0(1370) $ reported by the Particle Data Group (PDG) [8] are mostly based on a study of the $ K^-\pi^+\pi^- $ system conducted in a $ K^- p\to K^-\pi^-\pi^+ p $ scattering experiment in 1981 [11], combined with a recent BESIII measurement of the branching ratio $ {\cal B}(\bar K(1270)\to\bar K^*(892)\pi)/{\cal B}(\bar K(1270)\to\bar K\rho) $ in the $ D_s^+\to K^-K^+\pi^+\pi^0 $ decay [12]. All these BFs possesses large uncertainties, that lead to $ \sim $20% uncertainties on the $ \bar K_1(1270)\to \bar K \pi\pi $ BFs [13], becoming a bottleneck for precise BF measurements on any decays with $ \bar K_1(1270) $ as intermediate particles.

Although not used by the PDG for the BF averages, there are still a number of other measurements on the $ \bar K_1(1270) $ decays. Based on an amplitude analysis of the decay $ B^+\to J/\psi K^+\pi^+\pi^- $, the Belle collaboration found the BFs of $ \bar{K}_1(1270)\to \bar{K}\rho, \bar K\omega $, and $ \bar{K}^*(892)\pi $ to be generally consistent with the PDG averages within two standard deviations, while the measured BF of $ \bar{K}_1(1270)\to K_0^*(1430)\pi $ is significantly smaller [14]. Later measurements of the BF ratio $ \alpha\equiv\dfrac{{\cal{B}}(\bar{K}_1(1270)\to \bar K^*\pi)}{{\cal{B}}(\bar{K}_1(1270)\to \bar{K}\rho)} $, where $ {\cal{B}}(\bar{K}_1(1270)\to \bar K^*\pi)={\cal{B}}(\bar{K}_1(1270)\to \bar{K}^*(1430)\pi) $ $ +{\cal{B}}(\bar{K}_1(1270)\to \bar{K}^*(892)\pi) $, yield different results depending on the decay channels used [15−18], whereas they are expected to be identical under the narrow width approximation for the $ \bar{K}_1(1270) $ meson assuming $ CP $ conservation in strong decays [19].

The BESIII collaboration, through performing separate studies of the four hadronic systems $ K^-\pi^+\pi^- $ [20], $ K^-\pi^+\pi^0 $ [21], $ K_S^0\pi^+\pi^- $ and $ K_S^0\pi^-\pi^0 $ [22], reported the first observations of semielectronic D-meson decays involving a $ \bar{K}_1(1270) $ and measured their BFs based on the assumed $ \bar{K}_1(1270) $ decays. In addition, quite recently, an amplitude analysis of the $ D^0\to K^-\pi^+\pi^-e^+\nu_e $ and $ D^+\to K^-\pi^0\pi^-e^+\nu_e $ decays has been performed [23] with the larger $ \psi(3770) $ dataset corresponding to an integrated luminosity of $ 20.3\; \text{fb}^{-1} $. The measured BFs are summarized in Table 1. In light of these measurements, in this work, a model-independent method is proposed to determine the BFs of $ \bar{K}_1(1270) $ decays through a simultaneous analysis of signal yields from the four decay modes $ D^0\to K^-\pi^+\pi^-e^+\nu_e $, $ D^+\to K^-\pi^+\pi^0e^+\nu_e $, $ D^0\to K^0_S\pi^-\pi^0e^+\nu_e $, and $ D^+\to K^0_S\pi^+\pi^-e^+\nu_e $. With this method, the feasibility of measuring BFs of $ {\cal{B}}(\bar{K}_1(1270)\to \bar K^*\pi) $, $ {\cal{B}}(\bar{K}_1(1270)\to \bar{K}\rho) $ and $ {\cal{B}}(\bar K_1(1270)\to \bar K\pi\pi) $, based on the current 20.3 $ {\rm{fb}}^{-1} $ $ \psi(3770) $ data sample from BESIII [24], is explored. The projected precisions on the BFs are also evaluated using pseudo-experiments.

Table 1 lists the experimentally measured values of the BF $ {\cal{B}}(D\to \bar{K}_1(1270)e^+\nu_e) $, which depend on the assumed decay BFs of the $ \bar{K}_1(1270) $. In these measurements [20−23], the BF $ {\cal{B}}(D \to \bar K\pi\pi e^+\nu_e) $ is expressed as the product of $ {\cal{B}}(D \to \bar{K}_1(1270) e^+\nu_e) $ and $ {\cal{B}}(\bar{K}_1(1270) \to \bar K\pi\pi) $ [25], where $ {\cal{B}}(\bar{K}_1(1270)\to \bar K\pi\pi) $ denotes the total BF for $ \bar{K}_1(1270) $ decays to $ K\pi\pi $ final states:

where $ f_i $ is the square of the Clebsch–Gordan coefficient corresponding to the $ i^{\text{th}} $ decay mode of $ \bar{K}_1(1270) $: $ \bar{K}^*(1430)\pi $, $ \; \bar{K}^*(892)\pi $, $ \; \bar{K}\rho $, $ \; \bar K\omega $. The last decay mode is neglected hereafter due to the smallness of the product $ {\cal B}(\bar{K}_1(1270)\to \bar K\omega)\times{\cal B}(\omega\to\pi^+\pi^-) $. Regarding the completeness of the $ \bar K_1(1270) $ decays, we define the sum $ {\cal{B}}_{\text{3body}} $ as

This is determined by $ {\cal{B}}(\bar{K}_1(1270) \to \bar K\omega) $ [8].

A transition variable, which is directly related to the BFs for different signal reconstruction modes and can be determined experimentally in a straightforward manner, is defined as:

Substituting Eq. (3) into the expression for α and defining the ratio $ \delta_\alpha\equiv\dfrac{{\cal{B}}(\bar{K}_1(1270)\to \bar{K}f_0)}{{\cal{B}}(\bar{K}_1(1270)\to \bar{K}\rho)} $, the BF ratio α can be expressed as:

The BFs for $ \bar{K}_1(1270) $ decays can then be expressed as:

Since $ \delta_\alpha $ depends on $ {\cal{B}}(\bar{K}_1(1270)\to \bar{K}\rho) $, it is necessary to remove this dependence to make the measurement model-independent. To better quantify the associated uncertainty, we introduce the shorthand $ r_{f_0}=\frac{{\cal{B}}(\bar{K}_1(1270)\to \bar{K}f_0)}{{\cal{B}}_{\text{3body}}-{\cal{B}}(\bar{K}_1(1270)\to \bar{K}f_0)}=(3.5\pm2.3) $% [8] into Eq. (4):

This leads to an updated expression for $ \delta_\alpha $:

By eliminating α in favor of $ \delta_\alpha $, the BFs $ {\cal{B}}(K^-_1(1270)\rightarrow K^{-} \pi^{+} \pi^{-}) $ and $ {\cal{B}}(\bar{K}^0_1(1270)\rightarrow K^{-} \pi^{+} \pi^0) $ in Eqs. (5-6) can be expressed as:

In Eqs. 9 and 10, the parameter β is the sole free parameter in the formulation, whereas $ {\cal{B}}_{\text{3body }} $ and $ r_{f_0} $ depend on external inputs, $ {\cal B}(\bar K_1(1270)\to \bar K \omega) $ and $ {\cal B}(\bar K_1(1270)\to \bar K f_0(1370)) $, from the PDG. The value of β can be determined directly from experimental data by fitting the corresponding signal yields. Once β is obtained, all other physical observables—including the BFs and related quantities—can be derived from it, as they are explicit functions of β. This framework thus provides a consistent and model-independent approach in which all derived parameters are fully constrained by the experimentally determined value of β.

The BESIII collaboration has individually measured $ D^0\to K^-\pi^+\pi^-e^+\nu_e $, $ D^+\to K^-\pi^0\pi^-e^+\nu_e $, $ D^0\to K^0_S\pi^0\pi^-e^+\nu_e $ and $ D^+\to K^0_S\pi^+\pi^-e^+\nu_e $ decays [20−22], with the double-tag method [26, 27] and the $ \psi(3770) $ dataset corresponding to an integrated luminosity of $ 2.93\; \text{fb}^{-1} $. A combined analysis of the $ D^0\to K^-\pi^+\pi^-e^+\nu_e $ and $ D^+\to K^-\pi^0\pi^-e^+\nu_e $ decays has been performed [23] with the larger $ \psi(3770) $ dataset corresponding to an integrated luminosity of $ 20.3\; \text{fb}^{-1} $. In this section, a sensitivity study is performed by simultaneously fitting across the four decay modes, to determine the BFs of $ \bar{K}_1(1270) $ decays, and the BF ratio α, at the same time.

One-dimensional pseudo-datasets of $ M_{\rm miss}^2 $ are generated for the decay modes of $ D^0\to K^-\pi^+\pi^-e^+\nu_e $ and $ D^{0,+}\to K_S^0\pi^-\pi^{0,+}e^+\nu_e $. Here $ M_{\rm miss}^2 $ is the missing mass square $ M_{\rm miss}^2 \equiv E^2_{\rm miss}/c^4-|\vec p_{\rm miss}|^2/c^2 $, with $ E_{\rm miss} $ and $ \vec p_{\rm miss} $ being the total energy and momentum of all missing particles in the event, respectively. For the decay mode of $ D^+\to K^-\pi^+\pi^0e^+\nu_e $, as the distribution of $ U_{\rm miss}\equiv E_{\rm miss}-|\vec p_{\rm miss}|c $ was used instead for signal yield extraction in Ref [20], the signal and background shapes of $ M_{\rm miss}^2 $ from the mode of $ D^{0}\to K_S^0\pi^-\pi^{0}e^+\nu_e $ are used as approximations. The expected signal yields of the decays $ D^0\to K^-\pi^+\pi^-e^+\nu_e $, $ D^+\to K^-\pi^0\pi^-e^+\nu_e $, $ D^0\to K^0_S\pi^0\pi^-e^+\nu_e $ and $ D^+\to K^0_S\pi^+\pi^-e^+\nu_e $, based on the $ 20.3\; \text{fb}^{-1} $ $ \psi(3770) $ data, are estimated with

where $ N(D\bar D) $ denotes the total number of produced $ D\bar{D} $ pairs [24], $ {\cal{B}}_i^{\rm ST} $ is the BF of the $ i^{\rm th} $ tag mode, and $ \varepsilon_{i}^{\rm DT} $ is the double-tag efficiency. The sum runs over the same tag modes as in Refs. [20−22], and the values of $ \varepsilon_{i}^{\rm DT} $ are assumed to be identical to those in Refs. [20−22].

The background events are generated using background probability density functions previously determined from Monte Carlo (MC) simulations in Refs. [21, 22]. The estimated yields of the combinatorial and $ D\to \bar K\pi\pi\pi $ peaking backgrounds are scaled by a factor of seven to account for the smaller datasets used in Refs. [20−22].

To extract the parameters of interest, a simultaneous unbinned maximum-likelihood fit is performed on the four pseudo-datasets. The probability density functions that model the signal and background components are adopted from Refs. [21, 22], where they were determined from MC simulations. During the fit, the signal and combinatorial background yields are allowed to float, while the peaking-background yields are fixed to their generated values.

To minimize systematic effects from common sources such as luminosity, tagging, and tracking efficiencies, the parameter β is expressed in terms of ratios of signal yields:

where $ {\cal{N}} $ denotes the efficiency-corrected signal yield for each decay mode. Assuming $ \beta_{D^0}=\beta_{D^+} $, the average value of β is extracted from a simultaneous fit to the pseudo-datasets for the four decay modes. The remaining observables are then determined using Eqs. 9 and 10. The one-dimensional fit projections onto the $ M^2_{\rm miss} $ distributions for the four decays are shown in Fig. 1 and the fit results are summarized in Table 2.

Figure 1.  (color online) Simultaneous fit to the $ M_{\rm miss}^2 $ distributions of the pseudo-datasets. Points with error bars show the pseudo-data; red dashed lines represent the signal shapes, green dashed lines the combinatorial background shapes, and brown dashed lines the peaking backgrounds from $ D\to \bar K\pi\pi\pi $ decays.

A total of 2000 pseudo-experiments are performed to assess potential biases introduced by the fit model. The resulting distribution of the pulls, defined as $ \frac{\alpha_{\rm fit}-\alpha_{\rm nominal}}{\sigma_{\rm fit}} $, where $ \alpha_{\rm fit} $ and $ \sigma_{\rm fit} $ denote, respectively, the fitted α central value and its uncertainty in each pseudo-experiment, is shown in Fig. 2 and is consistent with a normal distribution, indicating that the fit model is unbiased in determining α.

Figure 2.  (color online) The α pull distribution is defined as the difference between the reconstructed value $ \alpha' $ and the input value $ \alpha_{\rm nominal} $, normalized by the estimated uncertainty $ \sigma_{\alpha} $.

Regarding potential sources of systematic uncertainty in the measurement, the double-tag method ensures that most tag-side uncertainties cancel. The uncertainties associated with the tracking and particle-identification efficiencies for $ e^+ $ and charged pions largely cancel in the ratios defined in Eqs. 12 and 13. The uncertainties on the $ \pi^0 $ and $ K_S^0 $ reconstruction efficiencies are 1% [13, 28]. This systematic uncertainty is evaluated by applying Gaussian constraints to the efficiency parameters during the fit, yielding a relative uncertainty of 3.8%. Similarly, the uncertainty originating from the assumed input branching fractions ($ {\cal{B}}_{\rm 3\; body} $ and $ r_{f_0} $) is estimated by applying Gaussian constraints to these input parameters, contributing an additional 2.7%. Adding these independent sources in quadrature results in a conservative total estimate of 5%.

Compared with BESIII's amplitude analysis based on the 20.3 $ {\rm{fb}}^{-1} $ $ \psi(3770) $ dataset [23], the expected statistical uncertainties on α and on the BFs for $ \bar{K}_1(1270)\to \bar{K}\rho,\bar K^*\pi $ in this work are larger, because this analysis does not exploit the full kinematic information of the $ D\to \bar K\pi\pi e^+ \nu_e $ decay (e.g., angular and $ q^2 $ distributions). Nevertheless, when systematic uncertainties are taken into account, this method achieves a significantly improved precision for $ {\cal B}(\bar{K}_1(1270)\to \bar{K}\rho,\bar K^*\pi) $. With this method, the expected precision on the BFs for $ \bar{K}_1(1270)\to K^-\pi^{+,0}\pi^- $ is comparable to the BESIII results, while the input uncertainties on $ {\cal B}(D\to \bar{K}_1(1270) e^+\nu_e) $ are considerably reduced.

In this work, a sensitivity study is performed to evaluate the feasibility of measuring the absolute branching fraction (BF) $ {\cal{B}}(\bar{K}_1(1270)\to \bar K\pi\pi) $ and the ratio $ \alpha={\cal{B}}(\bar{K}_1(1270)\to \bar{K}^*\pi)/{\cal{B}}(\bar{K}_1(1270)\to \bar{K}\rho) $. A model-independent approach to studying $ \bar{K}_1(1270) $ decays is proposed, in which signal yields are simultaneously extracted from the four $ K\pi\pi $ final states through a combined fit. The study demonstrates that a systematic uncertainty of approximately 5% can be achieved with the current $ \psi(3770) $ data sample (20.3 $ {\rm{fb}}^{-1} $) from the BESIII experiment, providing a significant improvement over previous results [23].

By not relying on specific signal decay models, the combined analysis yields substantially lower systematic uncertainties for $ {\cal{B}}(\bar{K}_1(1270)\to \bar K^*\pi) $, $ {\cal{B}}(\bar{K}_1(1270)\to \bar{K}\rho) $, and their ratio α, while providing a robust, model-independent validation of existing amplitude analysis results. Furthermore, these results lay the groundwork for high-precision probes of axial-vector meson structure and decay dynamics and will become increasingly advantageous with larger datasets from the Super Tau-Charm Factory, where the statistical uncertainties are expected to be reduced by at least one order of magnitude [29, 30].