Coupled-channel $\Lambda_{c}K^{+}-pD_{s}$ Interaction in Flavor $ \textrm{SU}\left(3\right) $ Limit of Lattice QCD

We study $S$-wave interactions in the $I\left(J^{p}\right)=1/2\left(1/2^{-}\right)$ $\Lambda_{c}K^{+}-pD_{s}$ system on the basis of the coupled-channel HAL QCD method. The potentials which are faithful to QCD S-matrix below the $ pD^{*} $ threshold are extracted from Nambu-Bethe-Salpeter wave functions on the lattice in Flavor $ \textrm{SU}\left(3\right) $ Limit. For the simulation, we employ $ 3 $-flavor full QCD gauge configurations on a $\left(1.93 \:\textrm{fm} \right)^{3}$ volume at $m_{\pi}\simeq 872$ MeV. %\textcolor{red}{For the charm quark, the relativistic heavy quark action is employed to treat its dynamics on the lattice}. We present our results of the S-wave coupled-channel potentials for the $\Lambda_{c}K^{+}-pD_{s}$ system in the $1/2\left(1/2^{-}\right)$ state as well as scattering observables obtained from the extracted potential matrix. We observe that the coupling between $\Lambda_{c}K^{+}$ and $pD_{s}$ channels is weak. The phase shifts and scattering length obtained from the extracted potential matrix show that the $\Lambda_{c}K^{+}$ interaction is attractive at low energy and stronger than the $pD_{s}$ interaction though no bound state at $m_{\pi}\geq872$ MeV.


I. INTRODUCTION
The study of interactions between a charmed meson and a nucleon (N) is an active field to investigate the property of the charmed hadronic matters.The interaction between a baryon and a meson due to the residual strong force (nuclear force) can form bound, resonance or molecular states.
Currently, the observation of new XYZ states [1] with hidden charm and/or beauty [2], also the charmed tetraquarks [3][4][5], and further pentaquark states [6], make the studies of systems contain charm hadrons like D and D * more desirable.Very recently, for the first time, the ALICE Collaboration measures the interaction between nucleons and charm hadrons through the femtoscopic analysis, i.e., the two-particle momentum correlations of pD − pair in high-multiplicity pp collisions [7].
Furthermore, it has been known that the presence of heavy hadrons (those contain charm and bottom quarks) as impurities in a nuclear or quark matter leads to occurrence of the Kondo effect as well (in the case of quark matter it is known as QCD Kondo effect) [8][9][10], this effect was originally observed in metal including impurities in the context of condensedmatter physics [11,12].The Kondo effect changes the thermodynamic and transport properties of the quark/nuclear matter by converting perturbative interaction at high energy scale to non-perturbative one at low energy scale in medium.To be more specific, the Kondo effect due to existence of Σ c (Σ * c ) baryons [13,14] and Ds D mesons [15,16] as heavy impurities in nuclear matter are calculated while the spin and/or isospin exchange provides the non-Abelian interaction correspond to the Kondo effect.It is also demanded to consider the charge-conjugate state D s and D mesons as well.In such case, it is necessary to consider additional new channels such as D s N → KΛ c [15], which is tackled here.
Therefore, a determination of the scattering parameters of systems involving charm hadron are essential.Since the nature of these interactions is due to non-perturbative QCD, the best tool to study them is lattice QCD simulations that is based on the first-principles calculations of QCD.Today this is possible by the modern high performance computing facilities together progress in theoretical method like HAL QCD [17] and non-relativistic effective field theories [1,18].They have equipped us to explore new forms of matter.
Motivating by the above discussions, it is desirable to determine the coupled channel D s : Strange D meson : (cs) and p is proton (uud)), by lattice QCD simulations.In the recent years, an approach to investigate hadron interactions in lattice QCD has been proposed by the HAL QCD Collaboration [19,20].One of the advantages of the HAL QCD method is that it can be extended straightforwardly to the case of inelastic scatterings.The extended method, namely coupled-channel HAL QCD method [21], has been applied to the hyperon-baryon(hyperon) systems [22,23], charmed baryon-(charmed) baryon systems [24,25], charmed tetraquarks states [4,5,26], resonance states [27] and meson-baryon bound states [28,29].The calculated scattering amplitude from obtained potentials can be used to compare the scattering observables with experimental data [4].
Here, we consider the inelastic regions for the pD s scattering by using coupled-channel HAL QCD method.In particular, we focus on the S-wave Λ c K + − pD s and calculate the scattering observables from obtained potentials in the infinite volume, such as phase shifts for Λ c K + and pD s systems and the inelasticity of the scatterings.This paper is organized as follows.In section II, we review the coupled-channel approach to the baryon-meson interactions by the HAL QCD method in lattice QCD.In section III, the numerical setup on the lattice and definitions of baryon and meson operators are summarized.We present our results on the coupled-channel potential for the Λ c K + − pD s system in the S-wave J p = 1/2 − state, the phase shift and scattering length by solving Schrödinger equation with the extracted potential in section IV.And finally, section VII is devoted to summary and conclusion.

II. HAL METHOD FOR COUPLED CHANNEL
Here, we describe briefly the coupled-channel HAL QCD method [21], which will be applied to the Λ c K + − pD s system.A key quantity in the HAL QCD method is the equaltime Nambu-Bethe-Salpeter (NBS) wave function which encodes information of scattering amplitude in its asymptotic behaviour.In the center-of-mass frame, the NBS wave function of baryon-meson at Euclidean time t with the total energy W is defined by where the index C denote the flavor channel (C = Λ c K + , pD s ), and B C (φ C ) is the local interpolating operator for the baryon(meson) C i with its renormalization factor Z C i .In the case of C = Λ c K + , for instance, C 1 = Λ c and C 2 = K + .The |W stands for a QCD asymptotic in-state at the total energy of W . From the NBS wave functions, we define the energy independent non-local potentials through the following coupled-channel Schrödinger equation, where By definition, the non-local potential U ( r, r ′ ) is faithful to the QCD S-matrix unless new channel opens.In order to handle the non-locality of the potentials, we introduce the derivative expansion U ( r, r ′ ) = (V LO ( r) + V N LO ( r) + ...) δ ( r, r ′ ) , where the N n LO term is of O ∇ n .The leading-order potential matrix is extracted by using the NBS wave functions as where, In lattice QCD, the NBS wave functions can be extracted from the baryon-meson fourpoint correlation function given by with constant A C ′ n = W n J C ′ (t 0 ) 0 , where J C ′ (t 0 ) stands for the source operator for C ′ which creates baryon-meson states.The ellipses denote inelastic contributions coming from channels above C and C ′ .In fact, for large time the four-point correlation function in Eq (4) is dominated by the ground state NBS wave function.Practically, however, the accurate determination of potentials have some difficulties, first, it is to figure out the ground state domination, since t−t 0 can not be taken large enough due to statistical noises of the baryonmeson four-point correlation function [30,31], and second, in order to solve Eq. (3), we need not only the ground state of NBS wave functions but also the first excited state of NBS wave functions, which are difficult to isolate each other due to the same reasons as former.The improved method to overcome these practical difficulties, has been proposed in Ref. [32] in the case of the single channel and extended to the coupled-channel case in Refs.[20,21].

Let us consider the normalized baryon-meson four-point correlation function
this satisfies for a moderately large t − t 0 where inelastic contributions from channels other than Λ c K + and pD s can be neglected (Of course Eq. ( 6) is not exact).δ C and ∆ C ′ C in Eq. ( 6) are defined by We can then extract the potential matrix at the leading-order of the derivative expansion as h.s of Eq. ( 6)).

III. NUMERICAL LATTICE SETUP
We applied the 3-flavor full QCD gauge configurations that are generated by the CP-PACS and JLQCD Collaborations [33] with the renormalization-group-improved gauge action and We use 700 gauge configurations, and the quark propagators are calculated for the wall source at t 0 .The periodic boundary conditions are imposed in the three spacial directions, while Dirichlet boundary conditions are taken for the temporal direction at t = 16 + t 0 .In order to increase the statistics, beside averaging over forward/backward propagations, the wall source is placed at 32 different place of t 0 on each configuration.In this work, the jackknife method is used to estimate statistical error with a bin size of 20 configurations.
For the local interpolating operators in Eq. ( 4), we use following form for proton and Λ c as And for mesons, where x = ( x, t), and i, j, k are color indices.C is the charge conjugation matrix defined by C = γ 2 γ 4 , and q = u, d, s, c stands for quark operators for up-, down-, strange-and charmquarks, respectively.Flavor structures of a p, Λ c , K + and D s are given by IV. NUMERICAL RESULTS

A. Effective mass and renormalization factor
We show the effective mass plots of the temporal two-point correlators of Λ c , D s , p and K + in Fig. 1 both for the point-sink and wall-source (point-wall) and for the wall-sink and wall-source (wall-wall).To obtain the mass of hadron, m H and the overlap parameters a P W and a W W for the point-wall and wall-wall correlators, we perform single exponential fit analysis of the point-wall and the wall-wall temporal two-point correlators by employing the functional form the Z H factor for a hadron is defined by The Z H can be calculated numerically through fitting the hadron correlators with exponential function.

B. Time dependence
First, we investigate the time dependence of the diagonal potentials.The time interval t − t 0 = 8 ± 1 is selected to suppress contribution from higher excited states at smaller The Z H is defined in Eq. ( 14) and the Z p is the Z-factor of proton.t and simultaneously to avoid large statistical errors at larger t [22].Fig. 2 shows and V pDs pDs at three values of t − t 0 = 7, 8, 9. Based on the time-dependent HAL QCD method [32], within statistical errors, no significant t − t 0 dependence is observed for these potentials showing that t − t 0 = 8 is large enough to suppress inelastic contributions and that higher-order contributions in the derivative expansion are negligible.for diagonal elements of the potential matrix V ΛcK + ΛcK + and V pDs pDs , and the attraction at short distance (r < 0.5) fm in the Λ c K + channel is stronger than in the pD s channel.These (diagonal and off-diagonal) potentials, almost vanish at r > 1 fm.

D. Hermiticity
Hermiticity of the potential matrix is a sufficient condition for the probability conservation, though it is not a necessary condition.Furthermore, the Hermiticity of the potential matrix is not automatically guaranteed in the definition of the coupled-channel potential matrix in Eq. ( 2).From Fig. 3 it is apparent that the Hermiticity is largely broken, i.e. the off-diagonal parts of the potential (panel (b) and (c) in Fig. 3) matrix are different.This may indicate that it could be better to include the pD * s channel into our calculation.To treat this broken Hermiticity, we have done further analysis.Solving the coupled-channel Schrödinger equation with non-Hermitian potential is problematic because the unitarity of the S-matrix is not guaranteed.Therefore, it is common to apply Hermitian potential in phenomenological studies in nuclear physics.Since this broken Hermiticity is happened at short distances, the low energy scattering observables does not suffer from difference between

V pDs
ΛcK + and V ΛcK + pDs .In order to check this the coupled channel potential is made Hermitian by taking one of them for the off-diagonal part or their average, In Appendix A, it is shown clearly that the scattering phase shifts of Λ c K and pD s for the above 3 Hermitian potential cases are almost same within the statistical errors.Thus hereafter we consider the V pDs ΛcK + in our calculations.
V. ANALYTIC FORMS OF Λ c K + AND pD s POTENTIALS In order to use the LQCD potential in phenomenological investigations, it is useful to fit them with some functions.Accordingly for the diagonal (D) Λ c K + − Λ c K + V ΛcK + ΛcK + , pD s − pD s V pDs pDs potentials as presented in Fig. 3, the following fit function V D (r) is considered [37,38], And similarly for the off-diagonal(OD) V pDs ΛcK + as presented in Fig. 4, the following analytic form V OD (r) is selected [39], FIG.4: Hermitian (average) potential, V pDs ΛcK + in Eq. ( 15) and its fit function at t − t 0 = 8.
TABLE II: Fitted parameters in Eq. ( 16) for V ΛcK + ΛcK + with statistical errors using the data at r < 1.7 fm.α i , β i and ρ are given in units of [MeV] , fm −2 and fm −2 , respectively.

Gauss-1
Gauss- where Y is the Yukawa function with a form factor, In the Eq. ( 16), the Y form, at medium and long range distances is motivated by the picture of two-pion exchange.The Gauss form factors in the Eq. ( 16) and ( 17) describe the short range part of the potentials.Note that the pion mass m π are fixed to be the measured values on the lattice about 872 MeV.
The results of fit and corresponding parameters are given in TABLE III: Fitted parameters in Eq. ( 16) for V pDs pDs with the statistical errors using the data at r < 1.7 fm.Units are the same as those in Table II.TABLE IV: Fitted parameters in Eq. ( 15) for Hermitian V pDs ΛcK + potential with the statistical errors using the data at r < 1.7 fm.Units are the same as those in Table II.

VI. PHASE SHIFT AND SCATTERING LENGTH
The physical observables such as coupled-channel scattering phase shifts of Λ c K + − pD s interactions can be calculated.Therefore, we solve the coupled-channel Schrödinger equation with the fitted potentials (given in the previous section) in the infinite volume and extract Smatrix from the asymptotic behaviour of the wave functions.Here the convention introduced by Stapp et al. [40] is used for the definition of phase shifts in the case of coupled channel system as     where δ is so-called the bar phase shift and ǭ is the mixing angle.
At first, since it is desired to see how large the coupled channel effect, we have compared the single channel Λ c K + potential and a diagonal part of the coupled channel potential, i.e., Panel (a) and panel (b) in Fig. 6, shows resultant S-wave scattering phase shifts for Λ c K + and pD s channels as a function of the baryon-meson center-of-mass energy, respectively.The Λ c K phase-shift shows a weak attraction.Moreover, just above the Λ c K threshold the Swave phase shift shows a rapid variation, a feature indicative of a nearby pole.However, at the opening of the pD s threshold we do observe a noticeable "kink" in the Λ c K phase shift suggesting a non-zero coupling between the two channels.The coupled Λ c K, pD s system, showing an enhancement (steadily increasing) in the Λ c K phase shift.The non-zero coupling between channels is further demonstrated in Fig. 6, panel (c), which shows a clear deviation of the inelasticities, η from unity.The inelasticity indicates that the transition between the Λ c K + and pD s is weak and two channels are almost independent each other.This observation might be understood from the fact that the large mass splitting between the Λ c K + and pD s .
As mentioned above, because the two channels are almost independent each other due to the small mixing, they can be regarded as two single channels, therefore, from the lowenergy part of Λ c K + (pD s ) phase shifts in Fig. 6 (a) ((b)) the scattering length (a 0 ) and the effective range (r eff ) are derived by using the S-wave effective range expansion (ERE) up to the next-to-leading order (NLO), The results are where the central values and the statistical errors are evaluated at t − t 0 = 8, while the systematic errors are evaluated by the difference between the central values at t − t 0 and those at t − t 0 = 7 and 9.
In addition, because the electric charge of each hadron in the Λ c K + − pD s system is +e we have Coulomb repulsion.It is probably too early to consider Coulomb interaction since presents the relevant phase shifts in the single and coupled channel at t − t 0 = 8.The mild difference between single and coupled channel phase shift is due to the effect of coupled channel.The Λ c K + phase shift are plotted against the energies from the Λ c K + threshold.
In the case of single channel Λ c K + potential the corresponding fit parameters are given by Table .V in Appendix B.
the pion mass in our simulation is as heavy as 872 MeV, therefor for future lattice QCD simulation at physical point (with nearly physical quark mass, i.e., m π ≃ 146 MeV and m K ≃ 525MeV) [41], it is necessary to include Coulomb interaction.

VII. SUMMARY AND CONCLUSION
We have investigated the S-wave Λ c K + interaction using the Λ c K + −pD s coupled channel potentials obtained by the extension of the HAL QCD method in Flavor SU (3) Limit of Lattice QCD.Results of the potential matrix show that the diagonal elements, Λ c K + and pD s are both strongly attractive, while, Λ c K + has a much deeper attractive pocket than pD s .We have also observed weak off-diagonal elements of the potential matrix.They are not Hermitician, which is not guaranteed from the definition.The phase shifts and inelasticity extracted by solving Schrödinger equation in the infinite volume with the obtained potentials show that the Λ c K + channel does not have the two-body bound state at m π ≥ 872 MeV.
the non-perturbatively O (a)-improved Wilson quark action (c SW = 1.7610) at β = 6/g 2 = 1.83 (corresponding lattice spacing in the physical unit, a = 0.1209 fm[34]) on an L 3 × T = 16 3 × 32 lattice, that corresponds to (1.93) 3 × 3.87 fm in the physical unit.These configurations are available through Japan Lattice Data Grid (JLDG)[35].The hopping parameters of the configuration set correspond to the flavor SU (3) symmetric point are κ u,d = κ s = 0.13710.In the case of the charm quark, also the non-perturbatively O (a)improved Wilson quark action (c SW = 1.7610) is used.The charm quark propagators are calculated at κ c = 0.12240 in (partial) quenched QCD[36].

FIG. 1 :
FIG. 1: Effective mass plot of Λ c , D s , p and K + for (a) wall-wall (WW) and point-wall (PW), in addition, to have better comparison in the plot (b) only point-wall correlators are shown.

FIG. 5 :
FIG. 5: Panel (a) shows a comparison between the single channel Λ c K + potential (blue square) and a diagonal part of the coupled channel potential (red circle), and panel (b)

FIG. 7 :
FIG. 7: The scattering phase shifts of S-wave (a) Λ c K + , (b) pD s and (c) the inelasticity parameter correspond to the phase shifts calculated with non-Hermitian (blue), V pDs ΛcK +

TABLE I :
Hadron masses in unit of MeV.Statistical errors are shown in the parentheses.
Table II for V ΛcK + ΛcK + , Table III for V pDs pDs and Table IV for V pDs ΛcK + at three different values t − t 0 = 7, 8, 9.

TABLE V :
Fitted parameters in Eq. (16) for single channel Λ c K + potential with statistical errors using the data at r < 1.7 fm.α i , β i and ρ are given in units of [MeV] , fm −2 and fm −2 , respectively.