First Lattice QCD calculation of semileptonic decays of charmed-strange baryons Ξc

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Qi-An Zhang, Jun Hua, Fei Huang, Renbo Li, Yuanyuan Li, Cai-Dian Lü, Peng Sun, Wei Sun, Wei Wang and Yi-Bo Yang. First Lattice QCD calculation of semileptonic decays of charmed-strange baryons Ξc[J]. Chinese Physics C.
Qi-An Zhang, Jun Hua, Fei Huang, Renbo Li, Yuanyuan Li, Cai-Dian Lü, Peng Sun, Wei Sun, Wei Wang and Yi-Bo Yang. First Lattice QCD calculation of semileptonic decays of charmed-strange baryons Ξc[J]. Chinese Physics C. shu
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First Lattice QCD calculation of semileptonic decays of charmed-strange baryons Ξc

    Corresponding author: Peng Sun, sunpeng@njnu.edu.cn
    Corresponding author: Wei Wang, wei.wang@sjtu.edu.cn
    Corresponding author: Yi-Bo Yang, ybyang@itp.ac.cn
  • 1. Key Laboratory for Particle Astrophysics and Cosmology (MOE), Shanghai Key Laboratory for Particle Physics and Cosmology, Tsung-Dao Lee Institute, Shanghai Jiao Tong University, Shanghai 200240, China
  • 2. INPAC, Key Laboratory for Particle Astrophysics and Cosmology (MOE), Shanghai Key Laboratory for Particle Physics and Cosmology, School of Physics and Astronomy, Shanghai Jiao Tong University, Shanghai 200240, China
  • 3. Nanjing Normal University, Nanjing, Jiangsu, 210023, China
  • 4. Institute of High Energy Physics, Chinese Academy of Sciences, Beijing 100049, China
  • 5. School of Physics, University of Chinese Academy of Sciences, Beijing 100049, China
  • 6. CAS Key Laboratory of Theoretical Physics, Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing 100190, China
  • 7. School of Fundamental Physics and Mathematical Sciences, Hangzhou Institute for Advanced Study, UCAS, Hangzhou 310024, China
  • 8. International Centre for Theoretical Physics Asia-Pacific, Beijing/Hangzhou, China

Abstract: While the standard model is the most successful theory to describe all interactions and constituents in elementary particle physics, it has been constantly examined for over four decades. Weak decays of charm quarks can be used to measure the coupling strength between quarks in different families and serve as an ideal probe for CP violation. As the lowest charm-strange baryons with three different flavors, $\Xi_c$ baryons (made of $csu$ or $csd$) have been extensively studied in experiment. In this work, we use the state-of-the-art lattice QCD techniques, and generate 2+1 clover fermion ensembles with two lattice spacings, $a=(0.108{\rm{fm}},0.080{\rm{fm}})$. We then present the first ab-initio lattice QCD calculation of $\Xi_c\to \Xi$ form factors. Our theoretical results for the $\Xi_{c}\to \Xi \ell^+\nu_{\ell}$ decay widths are consistent with and about two times more precise than the latest measurements by ALICE and Belle collaborations. Based on the latest experimental measurements, we independently obtain the quark-mixing matrix element $|V_{cs}|$, which is in good agreement with the results from other theoretical approaches.

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    I.   INTRODUCTION
    • Since the establishment in 1960s, the standard model (SM) of particle physics has achieved many remarkable successes, and has been constantly examined for over four decades. Nowadays searching for new physics (NP) beyond the SM is the primary objective in particle physics, which usually proceeds in two distinct directions. On the one side, new particles can be directly produced in high energy collisions for instance at large hadron collider. On the other side, it is greatly valuable to examine various low-energy observables with prestigious high precision that can give an indirect search for NP.

      Weak decays of heavy charm and bottom quarks provide an ideal platform to test the standard model of particle physics, especially the Cabibbo-Kobayashi-Maskawa (CKM) paradigm which describes quark mixing and CP violation. Any significant deviation from SM expectation for CKM matrix would provide definitive clues for new physics beyond SM. Most of previous analysis has concentrated on the meson sector like B and D mesons, while recently heavy baryon decays started to determine $ |V_{ub}/V_{cb}| $ from $ \Lambda_b \to p \mu^-\bar{\nu}_\mu $ and $ \Lambda_b \to \Lambda_c \mu^-\bar{\nu}_\mu $ [1], and $ |V_{cs}| $ from $ \Lambda_c\to \Lambda e^+\nu_e $ [2, 3].

      The study of weak decays of charmed baryons $ \Xi_c^{+,0} $ especially $ \Xi_c\to \Xi\ell^+\nu $ decays is greatly valuable from various aspects. First of all, the combination of form factors from lattice QCD (LQCD) and experimental results for branching fractions of semileptonic decays allows an independent determination of $ |V_{cs}| $. Secondly, a comparison of theory calculation and experimental measurements provides a stringent test of theoretical models. Thirdly, compared to the isosinglet counterpart $ \Lambda_c $ whose decays have been extensively in experiment [412] and from LQCD [13, 14], the iso-doublet $ \Xi_c^{+,0} $ baryons have more versatile decay modes. The closeness of decay branching fractions for the exclusive $ \Lambda_c\to \Lambda \ell^+\nu $ and inclusive $ \Lambda_c \to X\ell^+\nu $ modes [8] shows a very different pattern with heavy bottom/charm mesons. The combined analysis of $ \Xi_c\to \Xi \ell^+\nu $ decays and $ \Lambda_c $ decays can provide a way to validate/invalidate this pattern, which is valuable to understand the underlying dynamics in baryonic transition, and test the flavor SU(3) symmetry [1517]. Moreover, decays of $ \Xi_c $ have played an important role in the study of doubly-charmed baryon $ \Xi_{cc}^{++} $ [18], precision measurement of the lifetime of $ \Xi_b^0 $ [19] and the discovery of new exotic hadron candidates $ \Omega_c $ [20].

      Since the first observation of the inclusive semileptonic decay [21], a number of different decay modes of $ \Xi_c $ have been studied in experiments [2226]. In addition to measuring branching fractions for suppressed modes, the LHCb has also searched for CP violation in $ \Xi_c^+\to pK^-\pi^+ $ [25]. Very recently, the ALICE [27] and Belle [28] collaborations have measured the branching fractions for $ \Xi_c\to \Xi\ell^+\nu $:

      $ {\cal{B}}_{\rm ALICE}(\Xi_{c}^0\to \Xi^- e^+\nu_{e}) = (2.43\pm0.25\pm0.35\pm0.72) {\text{%}}, $

      (1)

      $ {\cal{B}}_{\rm Belle}(\Xi_{c}^0\to \Xi^- e^+\nu_{e}) = (1.72\pm 0.10 \pm 0.12 \pm 0.50){\text{%}} , $

      (2)

      $ {\cal{B}}_{\rm Belle}(\Xi_{c}^0\to \Xi^- \mu^+\nu_{\mu}) = (1.71\pm 0.17 \pm 0.13 \pm 0.50){\text{%}}, $

      (3)

      where the last errors arise from the uncertainties in $ {\cal{B}}(\Xi_{c}^0\to \Xi^-\pi^+) $ [26].

      On theoretical side, the $ \Xi_{c}\to \Xi $ transition depends on six form factors which parametrize the matrix elements of vector and axial-vector currents between the $ \Xi_{c} $ and Ξ baryons. Most available theoretical analyses of these form factors are based on phenomenological models [2934], but results vary substantially depending on explicit assumptions. A first-principle calculation is extremely crucial for a precise determination of CKM matrix element, and reliable analysis of CP violation in nonleptonic decays. In this work, we use the-state-of-the-art LQCD techniques and for the first time in the literature calculate $ \Xi_{c}\to \Xi $ form factors. Predictions for semi-leptonic decay widths are also presented, based on which the $ |V_{cs}| $ is extracted. As we will show, our results greatly improve the theoretical calculations, and are more precise than the experimental measurements. These results also serve as mandatory inputs for future analysis of non-leptonic decays particularly in the factorization scheme.

    II.   LATTICE SETUP
    • This work is based on 2+1 flavor ensembles generated with tree level tadpole improved clover fermion action and tadpole improved Symanzik gauge action. One step of Stout link smearing is applied to the gauge field used by the clover action to improve the stability of the pion mass for given bare quark mass. The tadpole improvement factors for quarks and gluons are tuned to the fourth root of the plaquette using Stout link smearing and the original gauge links. We start from the ensemble s108 with bare coupling $ \beta = \dfrac{10}{g^2} = 6.20 $ and size $ 24^3\times 72 $, determine the lattice spacing using Wilson flow [35], and tune the bare coupling for the s080 ensemble with smaller lattice spacing to make their physical volume to be roughly the same. The information on the two ensembles used in this paper can be found in Tab. I.

      $\beta=\frac{10}{g^2}$$L^3\times T$a$c_{\textrm{sw}}$$\kappa_l$$m_{\pi} $$\kappa_s$$m_{\eta_s} $
      s1086.20$24^3 \times 72$0.1081.161-0.27702850.1330640
      s0806.41$32^3 \times 96$0.0801.141-0.22952950.1318650

      Table 1.  Parameters of the 2+1 flavor clover fermion ensembles used in this calculation. The $\pi$/$\eta_s$ masses and the lattice spacings are given in units of MeV, and fm, respectively.

      On these two ensembles, we use the charm quark mass $ m_c^{s108}a = 0.485 $ and $ m_c^{s080}a = 0.235 $, respectively, by requiring the corresponding $ J/\psi $ mass to have its physical value $ m_{J/\psi} = 3.096900(6) $GeV [36] within 0.3% accuracy.

      The extraction of $ \Xi_c\to \Xi $ form factors requires the lattice QCD calculation of both three-point correlation function (3pt) from $ \Xi_c $ to Ξ, and also the two point correlation functions (2pt) of both $ \Xi_c $ and Ξ. The 3pt with weak current $ J^\mu = V^\mu-A^\mu = \bar s \gamma^\mu(1-\gamma_5)c $ is defined by,

      $ \begin{aligned}[b] C_{3}^{V-A}(q^2, t, t_{\rm seq}) =& \int d^3\vec x d^3 \vec y e^{-i\vec p_{\Xi}\cdot \vec x} e^{-i\vec q\cdot \vec y} T_{\gamma'\gamma} \\ & \times \langle 0| \chi_{\gamma}^{\Xi}(\vec x, t_{\rm seq}) J^{\mu}(\vec y, t)\overline \chi_{\gamma'}^{\Xi_c}(\vec 0, 0)|0\rangle , \end{aligned} $

      (4)

      where $ \chi_{\gamma}^{\Xi, \Xi_c} $ is the interpolation field of Ξ and $ \Xi_c $, respectively, T is a combination of the Dirac matrix that is chosen to project out the form factor. $ C_3^{V-A}(q^2, t, t_{\rm seq}) $ is related to the bare form factor F using standard parameterization of 3pt with one excited state,

      $ \begin{aligned}[b] C_3^{V-A}(q^2, t, t_{\rm seq}) =& \frac{f_{1}f_{2}m_{1}^2m_{2}^2}{4E_1E_2}e^{-E_2(t_{\mathrm{seq}}-t)-E_1t} \\ &\times \left( F+c_1e^{-\Delta E_1t}+c_2e^{-\Delta E_2 (t_{\rm seq}-t)} \right), \end{aligned}$

      (5)

      in which $ f_{i} $, $ m_{i} $ and $ E_{i} $ are decay constants, masses and ground-state energies of $ \Xi_c $ ($ i = 1 $) and Ξ ($ i = 2 $). $ \Delta E_{i} $ describe the mass differences between the first excited states and ground states, $ c_{i} $ and the following $ d_i $ are parameters for the excited state contamination. For the 2pt with $ B = 1, 2 $,

      $ \begin{aligned}[b] C_{2}^{B}(t) =& \int d^3\vec x e^{-i \vec p_{B} \cdot \vec x} T^{\prime}_{\gamma'\gamma} \langle 0|\chi_{\gamma}^{B}(\vec x, t) \overline \chi_{\gamma'}^{B}(\vec 0,0) |0\rangle \\ =& \frac{f_B^2m_B^2}{2E_B}e^{-E_Bt}\left(1+d e^{-\Delta E_B t}\right), \end{aligned}$

      (6)

      we choose $ T^\prime $ as the identity matrix to simplify the expressions. In order to eliminate the contributions from excited-state, we can define the following ratios for different projection matrices T and current operator $ V^{\mu}/A^{\mu} $,

      $ R_{V/A}(T,\mu) = \sqrt{\frac{C_{3}^{V/A}(q^2, t, t_{\rm seq})C_{3}^{V/A}(q^2, t_{\rm seq}-t, t_{\rm seq})} {C_2^{B_1}(t_{\rm seq}) C_2^{B_2}(t_{\rm seq}) } }, $

      (7)

      where the subscript V or A corresponds to the vector or axial-vector current in the 3pt. After making use of the reduction formula, the ratios $ R_F $ for the six form factors $ F = (f_\perp, f_+, f_0, g_\perp, g_+, g_0) $ can be constructed by different combinations of $ R_{V/A}(T, \mu) $. More details can be found in Appendix B. Then we adopt the parameterization,

      $ \begin{aligned}[b] R_F =& F\Bigg(\frac{1+c_1e^{-\Delta E_1t}+c_2e^{-\Delta E_2 (t_{\rm seq}-t)}}{1+d_1e^{-\Delta E_1t_{\rm seq}}}\\&\times \frac{1+c_1e^{-\Delta E_1 (t_{\rm seq}-t)}+c_2e^{-\Delta E_2t}}{1+d_2e^{-\Delta E_2t_{\rm seq}}}\Bigg)^{1/2} \\ \simeq & F[1+ c' (e^{-\Delta E't/2}+ e^{-\Delta E'(t_{\rm seq}-t)})], \end{aligned} $

      (8)

      to eliminate excited-state contaminations and obtain the desired form factor F. It should be noted that Eq. (9) is the complete form from the parametrization of 3pt and 2pt and employed in the fit for most cases, while for the largest negative $ q^2 $ case $ \left(\vec{p}_2 = (0,0,2)\times\dfrac{2\pi}{La}\right) $, the lattice results are noisy and the contributions from effective mass gap $ \Delta E_i $ is too small to be reliably determined from fitting the ratios, thus we will take an approximation in Eq. (10) for this case. To determine the form factor F as well as the parameters $ c_i $ and $ E_i $, we fit the results of ratios $ R_F $ with different $ t_{\mathrm{seq}} $. In Appendix C, we also try to apply a joint fit for the results of ratios and 2pt, which will show consistent results.

      The vector and axial-vector $ c\to s $ currents on the lattice suffer from finite renormalization. Such a renormalization can be defined by the ratio of the conserved-like-vector-current $ V_{\rm c} $ and the local current V in the hadron matrix element,

      $ \begin{aligned}[b] R^{q_1\to q_2}_V(t) =& \frac{\langle M_1(T/2)\sum_{\vec{x}}V^{q_1\to q_2}_{\rm c}(\vec{x},t)M_2(0)\rangle}{\langle M_1(T/2)\sum_{\vec{x}}V^{q_1\to q_2}(\vec{x},t)M_2(0)\rangle}\\ =& Z^{q_1\to q_2}_V+{\cal{O}}(e^{-T/4 \Delta E}), \end{aligned} $

      (9)

      where $ M_{1,2} $ are interpolating operators for pseudoscalar mesons and $ \Delta E $ is the mass gap between the ground state and first exited state. For the $ c\to s $ current, one can use either the combination $ (M_1,M_2) = (\eta_s, D_s) $, or the geometric average of those of the $ s\to s $ current and $ c \to c $ current using $ (M_1,M_2) = (\eta_s, \eta_s) $ and $ (\eta_c,\eta_c) $, respectively. We illustrate the $ Z_V $ in Fig. 1, in which the crosses and dots correspond to $ R^{c\to s}_V(t) $ and $ \sqrt{R^{c\to c}_V(t)R^{s\to s}_V(t)} $, respectively. Constant fits can describe the data at medium large $ t\sim T/4 $ well, and the difference between two definitions becomes smaller for the finer s080 ensemble (upper yellow data), and both of them are also closer to one compared to the values for the coarser s108 ensemble (lower blue data). Thus the differences between the two strategies arise from discretization effects. In the following discussion, we will use $ Z^{c\to s}_V $ to obtain the central values of the final result, then repeat the analysis with $ \sqrt{Z^{c\to c}_VZ^{s\to s}_V} $ and treat the differences as a systematic uncertainty. Due to the chiral symmetry breaking of the clover fermion action, the renormalization factor in the axial-vector current is not exactly the same as the vector one. Thus we use the off-shell quark matrix elements to define $ Z_A $ as,

      Figure 1.  (color online) Lattice results for $R_V^{c\to s}$ and $\sqrt{R_V^{c\to c}R_V^{s\to s}}$. The bands correspond to the ground state contributions $Z_V^{c\to s}$ and $\sqrt{Z_V^{c\to c}Z_V^{s\to s}}$ on the s080 and s108 ensembles, respectively.

      $ Z^{c\to s}_A\equiv Z^{c\to s}_V\sqrt{ \frac{\mathrm{Tr}[\langle c|V^{\mu}|c\rangle\gamma^{\mu}\gamma_5]}{\mathrm{Tr}[\langle c|A^{\mu}|c\rangle\gamma^{\mu}]} \frac{\mathrm{Tr}[\langle s|V^{\mu}|s\rangle\gamma^{\mu}\gamma_5]}{\mathrm{Tr}[\langle s|A^{\mu}|s\rangle\gamma^{\mu}]}}, $

      (10)

      with multiple off-shell quark momenta $ p^2 $. With $ a^2p^2 $ extrapolation and three values of $ p^2 $ in the range of $ a^2p^2\in[4,8] $, we obtain $ {Z_A}/{Z_V} $ = 1.010231(69) and 1.020296(68) on s108 and s080, respectively.

    III.   NUMERICAL RESULTS
    • Choosing different reference time slices, we preform 48$ \times $393 measurements on the s108 ensemble, and 72$ \times $436 measurements on the s080 ensemble. The lattice results for the ratios $ R_{f_\perp} $ with $ \vec{p}_{\Xi} = (0,0,1)\times \dfrac{2\pi}{La} $$ \left(\dfrac{2\pi}{La}\simeq 0.48{\rm GeV}\right) $ are shown in Fig. 2. The $ \chi^2/d.o.f $ are below/close to 1 for most fits of 400 bootstrap samples, which indicates a good fit quantity, and the colored bands in the left panel of Fig. 2 predicted by the fit agree with the data points well. To further validate the results, we calculate the differential summed ratio [37],

      Figure 2.  (color online) Lattice results for the $f_\perp(\Xi_c\to \Xi)$ form factor on the s080 ensemble with $\vec{p}_{\Xi}=(0,0,1)\times \frac{2\pi}{La}$, in the source-sink separation range $[12a, 20a]$. The left panel shows a two-state fit with the excited state contamination using the parametrization defined in Eq. (9), and the right panel gives the differential summed ratio. The ground-state matrix element (the grey band) obtained from the two-state fit agree with the differential summed ratio well when $t_{\rm seq}>14$.

      $ \tilde{R}(t_{\rm seq}) \equiv \frac{SR(t_{\rm seq})-SR(t_{\rm seq}-\Delta t)}{\Delta t}, $

      (11)

      and show the results in the right panel of Fig. 2, where $ SR(t_{\rm seq})\equiv\sum_{t_c<t<t_{\rm seq}-t_c}R_F(t, t_{\rm seq}) $, $ t_c = 3a $ is the requirement used in the fits to suppress higher excited states contributions. One can see that $ \tilde{R}(t_{\rm seq}) $ agrees with the grey band from the two-state fit well when $ t_{\rm seq}>14a $.

      To access the $ q^2 $ distribution, we employ the z-expansion parametrization of form factors that arises from analyticity and unitarity [38]

      $ f(q^2) = \frac{1}{1-q^2/(m_{\rm pole}^{f})^2} \sum\limits_{n = 0}^{n_{\rm max}} (c_{n}^f+ {d^f_{n}a^2}) [z(q^2)]^n, $

      (12)

      where

      $ z(q^2) = \frac{\sqrt{t_+-q^2} - \sqrt{t_+-t_0}}{\sqrt{t_+-q^2} + \sqrt{t_+-t_0}}. $

      (13)

      $ t_0 = q_{\rm max}^2 = (m_{\Xi_c}-m_{\Xi})^2 $ and $ t_+ = (m_D+m_K)^2 $, and $ d^f_{n} $ describes the discretization error of each z-expansion parameter $ c_{n}^f $. The pole masses in the form factor are $ m_{\rm pole}^{f_+,f_\perp} = 2.112 $ GeV, $ m_{\rm pole}^{f_0} = 2.318 $ GeV, $ m_{\rm pole}^{g_+,g_\perp} = 2.460 $ GeV, and $ m_{\rm pole}^{g_0} = 1.968 $ GeV. We collect the fitted parameters in Tab. II, and show the $ q^2 $ dependent form factor in the continuum limit (by eliminating the $ d^f_{n} $ terms) from the fit and also the lattice results at given $ q^2 $ in Fig. 3. As shown in the figure, our form factor results for the s108 ensemble and the s080 ensemble show only small discretization effects.

      $c_0$$c_1$$c_2$
      $f_{\perp}$1.51±0.09−1.88±1.211.71±0.49
      $f_{0}$0.64±0.09−1.83±1.220.56±0.51
      $f_+$0.77±0.07−4.09±1.180.35±0.49
      $g_{\perp}$0.56±0.07−0.35±1.260.15±0.29
      $g_{0}$0.63±0.07−1.37±1.360.15±0.29
      $g_+$0.56±0.080.00±1.380.14±0.29

      Table 2.  Results for the z-expansion parameters describing the form factors with statistical errors.

      Figure 3.  (color online) The $q^2$ distribution for the $\Xi_c\to \Xi$ form factors. The z expansion approach has been used to fit the lattice data. An extrapolation to the continuum limit has been made, and the shaded regions correspond to the final results with $a\to 0$.

      In Fig. 4, we use the above form factors to predict differential decay widths (in units of $ {\rm ps}^{-1} {\rm GeV}^{-2} $) for $ \Xi_{c}^0\to \Xi^- \ell^+\nu $ divided by $ |V_{cs}|^2 $ as a function of $ q^2 $. Results for $ \Xi_{c}^+\to \Xi^0 \ell^+\nu $ are also similar. Using the lifetime from PDG: $ \tau({\Xi^0_c}) = (1.53\pm0.06)\times 10^{-13} $s and $ \tau({\Xi^-_c}) = (4.56\pm $$ 0.05)\times 10^{-13} $s, and $ |V_{cs}| = 0.97320\pm0.00011 $ [36], one can obtain the decay branching fractions:

      Figure 4.  (color online) Predictions for the differential decay widths for the $\Xi_{c}^0\to \Xi^- e^+\nu_e$ and $\Xi_{c}^0\to \Xi^- \mu^+\nu_\mu$, divided by $|V_{cs}|^2$ in units of ${\rm ps}^{-1} {\rm GeV}^{-2}$.

      $ \begin{aligned}[b] &{\cal{B}}(\Xi_{c}^0\to \Xi^- e^+\nu_e) = 2.38(0.30)_{\mathrm{stat.}}(0.32)_{\mathrm{ext.}}(0.07)_{\mathrm{ren.}}{\text{%}} ,\\& {\cal{B}}(\Xi_{c}^0\to \Xi^- \mu^+\nu_\mu) = 2.29(0.29)_{\mathrm{stat.}}(0.30)_{\mathrm{ext.}}(0.06)_{\mathrm{ren.}}{\text{%}},\\& {\cal{B}}(\Xi_{c}^+\to \Xi^0 e^+\nu_e) = 7.18(0.90)_{\mathrm{stat.}}(0.96)_{\mathrm{ext.}}(0.20)_{\mathrm{ren.}}{\text{%}} ,\\& {\cal{B}}(\Xi_{c}^+\to \Xi^0 \mu^+\nu_\mu) = 6.91(0.87)_{\mathrm{stat.}}(0.91)_{\mathrm{ext.}}(0.19)_{\mathrm{ren.}}{\text{%}} . \end{aligned} $

      (14)

      The first errors come from statistical fluctuations, while the second and third ones are systematic uncertainties arising from the differences between continuum-extrapolated results and the ones using the s080 ensemble, and the differences between the results using $ Z_{V/A}^{c\to s} $ or $ \sqrt{Z_{V/A}^{c\to c}Z_{V/A}^{s\to s}} $ in the renormalization, respectively. Our predictions for branching fractions are consistent with model results in Ref. [29], but smaller than the ones in Ref. [30, 33]. Compared to the previous theoretical results which have typically 30% ~ 50% parametric uncertainties and uncontrollable systematic uncertainties, our results have greatly improved the theoretical predictions. Our calculation also indicates sizable SU(3) symmetry breaking effects compared to $ \Lambda_c\to \Lambda\ell^+\nu $ decays [3, 13].

      The ratio of branching fractions is predicted as:

      $ \begin{aligned}[b] R_{\mu/e} = &\frac{{\cal{B}}(\Xi_{c}^0 \to \Xi^- \mu^+\nu_\mu)}{{\cal{B}}(\Xi_{c}^0\to \Xi^- e^+\nu_e)} = \frac{{\cal{B}}(\Xi_{c}^+ \to \Xi^0 \mu^+\nu_\mu)}{{\cal{B}}(\Xi_{c}^+\to \Xi^0 e^+\nu_e)} \\ =& 0.962(0.003)_{\mathrm{stat.}}(0.002)_{\mathrm{syst.}}, \end{aligned} $

      (15)

      where most uncertainties from form factors have cancelled to a large extent. The deviation from unity arises from the mass differences between muon and electron. This result is consistent with and much more precise than the Belle measurement: $ R_{\mu/e} = 1.00\pm0.11\pm0.09 $ [28], which indicates that effects not covered by our lattice calculation are probably less significant at our level of precision.

      Our results for branching fractions are consistent with and about two times more precise than the measurements by ALICE and Belle collaborations as shown in Eq. (1-3). Using the ALICE measurement [27], we give a theoretical constraint of $ |V_{cs}| $:

      $ |V_{cs}| = 0.983(0.060)_{\mathrm{stat.}}(0.065)_{\mathrm{syst.}}(0.167)_{\mathrm{exp.}}, $

      (16)

      where the first two uncertainties are statistical and systematic uncertainties of the theoretical results, and the last ones are dominant and come from experimental data. Using the Belle result [28], we also have:

      $ |V_{cs}| = 0.834(0.051)_{\mathrm{stat.}}(0.056)_{\mathrm{syst.}}(0.127)_{\mathrm{exp.}}, $

      (17)

      which is obtained by combing $ \Xi_{c}^0\to \Xi^- e^+\nu_e $ and $ \Xi_{c}^0\to \Xi^- \mu^+\nu_\mu $. Using the individual channel, we have $ |V_{cs}|_{(\ell = e)} = 0.830(0.051)_{\mathrm{stat.}}(0.055)_{\mathrm{syst.}}(0.128)_{\mathrm{exp.}} $ and $ |V_{cs}|_{(\ell = \mu)} = 0.846(0.052)_{\mathrm{stat.}}(0.056)_{\mathrm{syst.}}(0.135)_{\mathrm{exp.}} $. Both results of $ |V_{cs}| $ from ALICE and Belle data are consistent with the global fit [36] within 1-$ \sigma $.

      It is necessary to point out that the largest errors in the extracted results for $ |V_{cs}| $ are from experimental data on $ {\cal{B}}(\Xi_c^0\to \Xi^-\pi^+) $ [26]. This can be improved by more precise measurements at LHCb, Belle-II, BESIII and other experiments in future. It should also be noted that as a conservative estimate, we have included systematic uncertainties (about 6%). In the continuum extrapolation, the statistical uncertainties in the two lattice ensembles are added and the final uncertainties are also about 6%.

    IV.   CONCLUSIONS
    • In this work, we have generated 2+1 flavor ensembles with tree level tadpole improved clover fermion action and tadpole improved Symanzik gauge action. One step of Stout link smearing is applied to the gauge field used by the clover action. We then present the first lattice QCD calculation of the form factors governing the $ \Xi_{c}\to \Xi \ell^+\nu_{\ell} $ with two sets of newly generated ensemble. The continuum limits are taken based on calculations at two lattice spacings. Using the CKM matrix element $ |V_{cs}| $ from PDG and the $ \Xi_{c} $ lifetimes, we predict the branching fractions $ {\cal{B}}(\Xi_{c}^0\to \Xi^- e^+ \nu_{e}) = 2.38(0.30)_{\mathrm{stat.}}(0.32)_{\mathrm{syst.}} $%, $ {\cal{B}}(\Xi_{c}^0\to \Xi^- \mu^+ \nu_{\mu}) = 2.29(0.29)_{\mathrm{stat.}}(0.31)_{\mathrm{syst.}} $%, $ {\cal{B}}(\Xi_{c}^+\to \Xi^0 e^+ \nu_{e}) = 7.18(0.90)_{\mathrm{stat.}}(0.98)_{\mathrm{syst.}} $%, and $ {\cal{B}}(\Xi_{c}^+\to \Xi^0 \mu^+ \nu_{\mu}) = 6.91(0.87)_{\mathrm{stat.}}(0.93)_{\mathrm{syst.}} $%. Our results have greatly improved previous theoretical calculations, and are consistent with and about two times more precise than the measurements by ALICE and Belle collaborations. Our calculation also indicates sizable SU(3) symmetry breaking effects compared to $ \Lambda_c\to \Lambda\ell^+\nu $ decays. These results also serve as mandatory inputs for the analysis of non-leptonic decays in the factorization scheme. Using the measured branching fraction from two experiments together with our lattice results, we obtain the theoretical constraint of the CKM matrix element: $ |V_{cs}| = 0.983(0.060)_{\mathrm{stat.}}(0.065)_{\mathrm{syst.}}(0.167)_{\mathrm{exp.}} $ and $ 0.834(0.051)_{\mathrm{stat.}}(0.056)_{\mathrm{syst.}}(0.127)_{\mathrm{exp.}} $, where the errors come from the theoretical and experimental uncertainties, respectively.

    ACKNOWLEDGMENT
    • We thank Andreas Schäfer for valuable discussions, Y.B. Yin, J. Zhu and T. Cheng for pointing out the ALICE result in Ref. [27], C.P. Shen and Y.B. Li for the correspondence on the Belle measurement [28], and W.B. Qian, Y.H. Xie, H.B. Li, B.Q. Ke and X.R. Lyu for noticing us the experimental studies of $ \Xi_c $ decays at LHCb and BESIII. We greatly thank Prof. En-Ke Wang and Nu Xu for their support when the gauge configurations are generated on the cluster supported by Southern Nuclear Science Computing Center (SNSC) and also HPC Cluster of ITP-CAS. The LQCD calculations were performed using the Chroma software suite [39] and QUDA [4042] through HIP programming model [43]. The numerical calculation is supported by Chinese Academy of Science CAS Strategic Priority Research Program of Chinese Academy of Sciences, Grant No. XDC01040100. The setup for numerical simulations was conducted on the π 2.0 cluster supported by the Center for High Performance Computing at Shanghai Jiao Tong University.

    APPENDIX A: GENERATION OF CONFIGURATIONS AND TUNING THE QUARK MASSES
    • While lattice QCD is an ab-initio approach that can handle strong interactions for hadron physics, the precision in the calculation is emphatically limited to various systematic uncertainties. A major systematic uncertainty resides in the ensembles. In this work we have generated two sets of 2+1 flavors clover fermion ensembles with the lattice spacing $ a = 0.108 $fm and $ 0.08 $fm. The sea quark masses are demarcated by pseudoscalar mesons ($ \pi $ for the light quark pair $ \bar{q}q $ and $ \eta_s $ for $ \bar{s}s $ case). For example, the left panel of Fig.A1 illustrates the determination of pion mass through the two-point function on s108 and s080, respectively. The fit results are given as $ m_{\pi}^{s108} = $$ 284.5(1.5) $MeV with $ \chi^2/d.o.f = 0.80 $ and $ m_{\pi}^{s080} = $$ 295.0(8) $MeV with $ \chi^2/d.o.f = 0.82 $. For the case of charmed baryons, it is reasonable to assume that the contributions from sea charm and heavier quarks are neglected. The valence charm quark mass is determined by tuning the $ J/\psi $ mass, shown as the right panel of Fig.A1, where the disconnected diagrams are not considered. The fitted results are $ m_{J/\psi}^{s108} = 3.09657(32) $GeV and $ m_{J/\psi}^{s080} = $$ 3.08979(31) $GeV respectively, which are consistent with the physical $ J/\psi $ mass with in 0.3% accuracy.

      Figure A1.  (color online) Effective mass plots compare with 2-state fitted plateaus for pion (left panel) and $J/\psi$ (right panel) on different ensembles, in which the t-range of fitted plateaus indicate the range of data selected for the fit, and the dashed line in right panel denotes the physical $J/\psi$ mass.

    APPENDIX B: COMBINATION OF RATIOS OF CORRELATION FUNCTIONS
    • As in Eq.(5), one can define the projected ratios of correlation functions:

      $\tag{B1} \begin{aligned}[b]& R_1\equiv \frac{R_V(I, z)+R_V(\gamma^0, z)}{2}, \quad R_2\equiv R_V(\gamma^z, 0), \\& R_3\equiv R_V(\gamma_5\gamma^x, y), \quad R_4\equiv \frac{R_A(\gamma_5, z)+R_A(\gamma_5\gamma^0, z)}{2}, \; \\& R_5\equiv R_A(\gamma_5\gamma^z, 0), \quad R_6\equiv R_A(\gamma^x, y). \end{aligned} $

      We can construct the combined ratio $ R_F $ of the six form factors $ F = (f_\perp, f_+, f_0, g_\perp, g_+, g_0) $ as

      $\tag{B2} R_{f_\perp} = \frac{R_3}{4m_1N_{z}\hat{p}}, \quad R_{g_\perp} = \frac{R_6}{4m_1N_{z}\hat{p}}, $

      $\tag{B3} \begin{aligned}[b] R_{f_0} =& -\frac{\left(E_{2}-m_{1}\right)\left(m_{1}^{2}-m_{2}^{2}\right)+\left(E_{2}+m_{1}\right) q^{2}}{8 m_{1}^{2}\left(m_{1}-m_{2}\right)\left(E_{2}+m_{2}\right) N_{z} \hat{p}} R_{1} \\ & +\frac{m_{1}^{2}-m_{2}^{2}+q^{2}}{8 m_{1}^{2}\left(m_{1}-m_{2}\right) N_{z} \hat{p}} R_{2} \\&+\frac{2 m_{1} E_{2}-m_{1}^{2}-m_{2}^{2}+q^{2}}{8 m_{1}^{2}\left(m_{2}+E_{2}\right) N_{z} \hat{p}} R_{3}, \end{aligned} $

      $\tag{B4}\begin{aligned}[b] R_{f_+} =& -\frac{\left(E_{2}-m_{1}\right)\left[\left(m_{1}+m_{2}\right)^{2}-q^{2}\right]}{8 m_{1}^{2}\left(E_{2}+m_{2}\right)\left(m_{1}+m_{2}\right) N_{z} \hat{p}} R_{1} \\ & +\frac{\left(m_{1}+m_{2}\right)^{2}-q^{2}}{8 m_{1}^{2}\left(m_{1}+m_{2}\right) N_{z} \hat{p}} R_{2} \\&+\frac{2 m_{1} E_{2}-m_{1}^{2}-m_{2}^{2}+q^{2}}{8 m_{1}^{2}\left(m_{2}+E_{2}\right) N_{z} \hat{p}} R_{3}, \end{aligned}$

      $\tag{B5} \begin{aligned}[b]R_{g_0} = & \frac{\left(E_{2}+m_{1}\right)\left(m_{1}^{2}-m_{2}^{2}\right)+\left(E_{2}-m_{1}\right) q^{2}}{8 m_{1}^{2}\left(m_{1}+m_{2}\right)\left(E_{2}-m_{2}\right) N_{z} \hat{p}} R_{4} \\& -\frac{m_{1}^{2}-m_{2}^{2}+q^{2}}{8 m_{1}^{2}\left(m_{1}+m_{2}\right) N_{z} \hat{p}} R_{5} \\ & + \Bigg[\frac{2 m_{1}\left(E_{2}-m_{2}\right)-m_{1}^{2}+m_{2}^{2}}{8 m_{1}^{2}\left(E_{2}-m_{2}\right) N_{z} \hat{p}} \\&+\frac{\left(m_{1}-m_{2}\right) q^{2}}{8 m_{1}^{2}\left(E_{2}-m_{2}\right)\left(m_{1}+m_{2}\right) N_{z} \hat{p}} \Bigg] R_{6}, \end{aligned} $

      $\tag{B6} \begin{aligned}[b]R_{g_+} = & \frac{\left(E_{2}+m_{1}\right)\left[\left(m_{1}-m_{2}\right)^{2}-q^{2}\right]}{8 m_{1}^{2}\left(E_{2}-m_{2}\right)\left(m_{1}-m_{2}\right) N_{z} \hat{p}} R_{4} \\& -\frac{\left(m_{1}-m_{2}\right)^{2}-q^{2}}{8 m_{1}^{2}\left(m_{1}-m_{2}\right) N_{z} \hat{p}} R_{5} \\ & + \Bigg[\frac{2 m_{1}\left(E_{2}-m_{2}\right)-m_{1}^{2}+m_{2}^{2}}{8 m_{1}^{2}\left(E_{2}-m_{2}\right) N_{z} \hat{p}} \\&+\frac{\left(m_{1}+m_{2}\right) q^{2}}{8 m_{1}^{2}\left(m_{1}-m_{2}\right)\left(E_{2}-m_{2}\right) N_{z} \hat{p}} \Bigg] R_6, \end{aligned} $

      where $ m_1 $ is the mass of $ \Xi_c $, and $ m_2,\; E_2 $ are the mass and energy of Ξ, respectively, $ \hat{p} = \dfrac{2\pi}{La}\simeq0.48 $ GeV is the unit momentum for both s108 and s080, and $ \vec{p}_{\Xi} = (N_x, N_y, N_z)\times\hat{p} $.

    APPENDIX C: COMPARISON OF FIT RESULTS WITH DIFFERENT PARAMETRIZATION FORMS
    • In this part, we compare the fit results of form factors $ f_{\perp} $ from the ratios $ R_{f_{\perp}} $ with different $ t_{\mathrm{seq}} $, as well as from a joint fit applying for the ratios and 2pt. Fig.C1 shows the comparison of fitted results with different strategies. Compared with the first fit strategy in Fig.(C1a), the joint fit in the right panel gives consistent fit result with a smaller error, and accordingly gives a stronger constraint of $ \Delta E_i $ than the ratios in joint fit. To give a conservative estimate, we will adopt the first fit strategy in our main text.

      Figure C1.  (color online) Compare the fit results of $f_\perp$ from the ratio $R_{f_{\perp}}(t,t_{\mathrm{seq}})$ with Eq.(9) (a), and from a joint fit together with the ratio $R_{f_{\perp}}(t,t_{\mathrm{seq}})$ as well as 2pt $C_2^{(1,2)}(t)$ (b) on the s080 ensemble with $\vec{p}_{\Xi}=(0,0,1)\times \frac{2\pi}{La}$.

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