Resolving the low mass puzzle of $\Lambda_c(2940)^+$

For the long standing low mass puzzle of $\Lambda_c(2940)^+$, we propose an unquenched picture. Our calculation explicitly shows that the mass of the $\Lambda_c(2P,3/2^-)$ state can be lowered down to be consistent with the experimental data of $\Lambda_c(2940)^+$ by introducing the $D^*N$ channel contribution. Additionally, we give a semi-quantitative analysis to illustrate why the $\Lambda_c(2940)^+$ state has a narrow width. It means that the low mass puzzle of $\Lambda_c(2940)^+$ can be solved. What is more important is that we predict a mass inversion relation for the $2P$ $\Lambda_{c}^+$ states, i.e., the $\Lambda_c(2P,1/2^-)$ state is higher than the $\Lambda_c(2P,3/2^-)$, which is totally different from the result of conventional quenched quark model. It provides a criterion to test such an unquenched scenario for $\Lambda_c(2940)^+$. We expect the future experimental progress from the LHCb and Belle II.

The Λ c (2940) + was first observed in the D 0 p mass spectrum by the BaBar Collaboration [8], and later, confirmed by Belle in decay mode Σ c (2455)π [9]. More importantly, in 2017, the LHCb further measured Λ c (2940) + as a P-wave state with J P = 3/2 − [10]. Now, PDG listed the mass and decay width of Λ c (2940) + as M = 2939.6 +1.3 −1.5 MeV and Γ = 20 +6 −5 MeV [1]. There exists low mass puzzle for the Λ c (2940) + , which results in the difficulty to arrange Λ c (2940) + under the framework of charmed baryon [5,6,8,[10][11][12][13]. Due to this reason, the exotic hadronic molecular configuration to Λ c (2940) + was proposed [14]. However, we need to exhaust different possibili-ties under the conventional framework before confirming the existence of exotic state. Along this line, it is obvious that we are not satisfied with the present solution to the low mass puzzle of Λ c (2940) + when looking back on the present research status of Λ c (2940) + . New idea must emerge for solving this low mass puzzle.  [11,15]. The D s1 (2460) ± (J P = 1 + ) is close to the D * K threshold and nearly 90 MeV below the quenched one, and it is also a heavy-light meson as D * s0 (2317) ± and thus not shown. For Λ c (2940) + , LHCb suggested that the most likely spin-parity J P is 3/2 − [10]. Thus, we adopted J P = 3/2 − assignment to Λ c (2940) + in our discussion of Λ c (2940) + in this work.
When checking the whole observed hadrons, we notice four states Λ(1405) 0 , D * s0 (2317) ± , D s1 (2460) ± , and X(3872), which have been established in experiments. If further comparing Λ c (2940) + with these four states, we find the similarities: 1) there exists low mass problem, i.e., they are are about 100 MeV lower than the corresponding theoretical results from ordinary (quenched) quark model calculation; 2) they are close to some s-wave channel thresholds as shown in the figure; 3) especially they have P-wave quantum number. In Fig. 1, we illustrate these common features.
In fact, we may draw inspiration from the research progress around Λ(1405) 0 , D * s0 (2317) ± , D s1 (2460) ± , and X(3872). For understanding low mass problems existing in these states, the unquenched quark model by including coupled-channel effect was developed, which was applied to explain why the masses of the corresponding physical states can be lowered down to be consistent with experimental data (see Refs. [16][17][18][19][20] for example). Due to the similarities illustrated above, we naturally conjecture whether unquenched picture can happen for discussed Λ c (2940) + .
In this work, we construct an unquenched picture to test such a scenario. Our calculated results explicitly show that the low mass puzzle of Λ c (2940) + can be solved, which provides a unique view point to decode the nature of Λ c (2940) + without including so called exotic state assignment to Λ c (2940) + . Success of solving the low mass puzzle of Λ c (2940) + makes that Λ c (2940) + becomes the first typical example affected by the unquenched effect in the heavy baryon family. What is more important is that group of Λ c (2940) + with Λ(1405) 0 , D * s0 (2317) ± , D s1 (2460) ± , and X(3872) constructs a complete chain. Until now, the unquenched effect can be seen in different types of hadronic system (from light baryon to charmedstrange meson containing heavy-light quarks, cc doubleheavy meson system, and to charmed baryon with heavy-light quarks), which is a fantastic phenomenon.
Besides solving the low mass puzzle of Λ c (2940) + , we predict a mass inversion for the 2P Λ + c states, i.e., the mass of 2P 1/2 − Λ + c state is expected to be larger than that of Λ c (2940) + under the unquenched picture. We should emphasize that there must exist such mass inversion relation for the 2P Λ + c states if the unquenched effect plays an important role to Λ c (2940) +1 . Since the predicted 2P 1/2 − Λ + c state is still missing, searching for this missing state becomes a crucial point to test the unquenched scenario for Λ c (2940). It will be a good opportunity for the future experimental study at LHCb and Belle II. An unquenched picture for Λ c (2940) + :-Before introducing the unquenched picture for Λ c (2940) + , we firstly mention what is the low mass puzzle of Λ c (2940) + . According to the calculations of quenched quark models [5,6,11,12] and Regge trajectory analysis [13], the mass of Λ c (2P) state with J P = 3/2 − , which is tentatively named as Λ c (2P, 3/2 − ) for convenience of later discussion, should be in the range 3000 ∼ 3040 MeV, which is 60 ∼ 100 MeV larger than the measured resonance parameter of Λ c (2940) + . This phenomenon results in the confusion for its nature in past years. Making comparison with the D * N threshold, we notice that Λ c (2P, 3/2 − ) may couple with this D * N channel via s-wave interaction. In fact, the situation of Λ c (2P, 3/2 − ) is similar to that of several typical states like Λ(1405) 0 , D * s0 (2317) ± , D s1 (2460) ± , and X(3872), where the masses of corresponding bare states are larger than the corresponding observed values and there exist s-wave couplings between the typical states with the concrete thresholds.
In our calculation, we only select the D * N channel contribution to the discussed Λ c (2P, 3/2 − ). This treatment is due to suggestion by Geiger and Isgur in Ref. [21,22]. Usu-ally, all possible hadronic channels coupled with a bare state should be included. But, this consideration makes the calculation become impractical [21], where the trivial and nontrivial coupled channel cases should be distinguished by different treatments [21,22]. Isgur indicated that the long-range coupled channel effects from the nonperturbative quark loops can be absorbed by the string tension while the qq pair creation at short distances just changes the running coupling constant α s [22]. Thus, in most cases, the trivial coupled channel effect can be renormalized in the parameters α s and b in the quenched quark model. It also naturally explains why most of the observed meson and baryon states can be depicted in the quenched quark models [11,15]. Isgur further pointed out that the nontrivial coupled channel effect can not be treated as the adiabatic approximation when a resonance state strongly couples to nearby s-wave threshold [22]. It is the cases of Λ(1405), D * s (2317) ± , D s (2460) ± , and X(3872) states since they couple strongly with the nearby NK, DK, D * K, andD * D thresholds, respectively. In this work, the Λ c (2940) + has been verified as the first known heavy baryon state which should be considered the nontrivial coupled channel effect seriously.
For reflecting the contribution from the D * N channel, we need to write out the so called full Hamiltonian of the physical Λ c (2P, 3/2 − ) [18,[23][24][25][26][27][28] whereĤ 0 depicts the discrete mass spectrum of the bare charmed baryon, which has expression Here, the parameters α c , b, and C represent the strength of the color Coulomb potential, the strength of linear confinement, and a mass-renormalized constant, respectively. The spin-dependent interactions, V spin i j , contain the spin-spin contact hyperfine interaction, the tensor interaction, the spin-orbit interaction of color-magnetic piece, and the Thomas precession term (see Refs. [11,15] for more details). The color factor F i · F j is taken as −2/3 for the baryon system (the meson system, F i · F j = −4/3). In our calculation, the masses of u/d and c quarks are taken as 0.370 GeV and 1.88 GeV, respectively. For the baryons, the parameters α s , b, and σ (where the σ is a parameter in contact term, and one can refer from [29]) are taken as 0.554, 0.120 GeV 2 , and 1.60 GeV, respectively. In our work, we also reproduce the masses of light and charmed mesons since their wave functions shall be used in our unquenched calculation. The values of α s , b, and σ for these meson systems are taken as 0.561, 0.142 GeV 2 , and 1.08 GeV, respectively. Finally, the constant C are fixed as C udc = −0.630 GeV, C N = −0.746 GeV, C π = −0.655 GeV, and C D = −0.700 GeV for the different hadron systems.
With the HamiltonianĤ 0 and parameters presented above, the Gaussian Expansion Method [30] is adopted to solve the Schrödinger equations. The bare mass of Λ c (2P, 3/2 − ) is obtained to be 3012 MeV, which is about 70 MeV larger than the mass of the discussed Λ c (2940) + . In fact, basing on this HamiltonianĤ 0 the masses of Λ c (1S , 1/2 + ), Λ c (2S , 1/2 + ), Λ c (1P, 1/2 − ), Λ c (1P, 3/2 − ), Λ c (1D, 3/2 + ), and Λ c (1D, 5/2 + ) can be given as 2287, 2779, 2595, 2617, 2856, and 2863 (in units of MeV), which may correspond to these wellestablished Λ c (2286) + , Λ c (2765) + , Λ c (2595) + , Λ c (2625) + , Λ c (2860) + , and Λ c (2880) + , respectively. This fact shows that HamiltonianĤ 0 works well to reproduce most of charmed baryons.   In Eq. (1), the HamiltonianĤ I describes the interaction between the bare state and the D * N channel, which is responsible for the dress of the bare state. In this work, we emploŷ H I = g d 3 xψ(x)ψ(x) inspired by the quark-pair-creation (QPC) model. In the non-relativistic limit, thisĤ I can be replaced by [6] where the ω, φ, χ and Y m 1 are the color, flavour, spin, and orbit functions of the quark pair, respectively. The b † µ and d † ν are quark and antiquark creation operators, respectively. The dimensionless parameter γ describes the strength of the quarkantiquark pair created from the vacuum, which is fixed as 9.45 by the total decay width of Σ c (2520) [1]. When the D * N channel effect is taken into account, the physical state Λ c (2P, 3/2 − ), which contains a significant continuum component of D * N other than the udc component, can be represented as Here, the c 0 denotes the probability amplitude of the udc core in Λ c (2940) + , and the χ(q) is the wave function of the |D * N channel. Finally, the physical mass M phy for Λ c (2P, 3/2 − ) affected by the D * N channel can be obtained from the following equations for M where the M 0 = 3012 MeV is the bare mass of Λ c (2P, 3/2 − ), which is already mentioned above. The transition amplitude M Λ bare c (2P,3/2 − )→D * N (q) can be calculated by the QPC model, i.e., M Λ bare (1) . c 0 can be determined by following expression where the M phy is the solution of Eqs. (3)-(4) for M.
With the above preparation of quantitatively constructing unquenched model, we illustrate how to extract the physical mass of Λ c (2P, 3/2 − ). As shown in the left diagram of Fig. 2, we plot the functions M − M 0 and ∆M(M) dependent on M. A vivid cusplike behavior near the D * N threshold is exhibited. The intersection of two curves corresponds to the physical mass M phy of Λ c (2P, 3/2 − ). Our calculation explicitly reveals how the D * N channel contribution lowers the bare mass 3012 MeV to the physical mass 2937 MeV, which is consistent with experimental data of the observed Λ c (2940) + . Thus, the low mass puzzle of Λ c (2940) + can be solved well in this unquenched picture.
When checking the resonance parameters of the observed Λ c (2940) + , we notice that Λ c (2940) + is a narrow state with width 20 MeV [1,[8][9][10], which can be also understood by a simple analysis mentioned below. There were theoretical calculations for the strong decay behavior of the Λ c (2P, 3/2 − ) state in the quenched quark model, by which the total width of Λ c (2P, 3/2 − ) is predicted at about 380 MeV. Similar theoretical results were also obtained in Refs. [31,32]. It is obvious that these calculations are not consistent with experimental measurements of Λ c (2940) + as a narrow state. In this work,