Approximate degeneracy of heavy-light mesons with the same $L$

Careful observation of the experimental spectra of heavy-light mesons tells us that heavy-light mesons with the same angular momentum $L$ are almost degenerate. The estimate is given how much this degeneracy is broken in our relativistic potential model, and it is analytically shown that expectation values of a commutator between the lowest order Hamiltonian and ${\vec L}^{~2}$ are of the order of $1/m_Q$ with a heavy quark mass $m_Q$. It turns out that nonrelativistic approximation of heavy quark system has a rotational symmetry and hence degeneracy among states with the same $L$. This feature can be tested by measuring higher orbitally and radially excited heavy-light meson spectra for $D/D_s/B/B_s$ in LHCb and forthcoming BelleII.


I. INTRODUCTION
Ever since the discovery of X(3872), D s0 (2317), and D s1 (2460) in 2003, there have been many more XYZ as well as higher radially and orbitally excited particles found at Belle, BESII, BESIII, BaBar, and LHCb [1]. There are a couple of problems for these particles. One is that most of them appear at thresholds and hence there may be kinematical explanations possible. Another point is that some of them should be multiquark states because they cannot be explained as higher excited states of ordinary quarkonium due to the charged states.
When focusing on higher orbital excitations of the heavylight system, we see some tendency of their spectroscopy which has not yet been explained by heavy quark symmetry. The problem is described as follows. Even though the angular momentum L is not a good quantum number in the heavy quark system, it seems that masses of states with the same L are close to each other even for the heavy-light system.
To explain this approximate degeneracy among heavy-light mesons with the same L observed in experiments, we need to show, at least analytically or numerically, how small matrix elements of this resultant difference operator are. One of the powerful quark models is the relativized Godfrey-Isgur (GI) model [2,3] in which their lowest order Hamiltonian commutes with L even in their relativized formulation. Hence, there is no wonder within their formulation why the masses with the same L are close to each other. However, when calculating commutator of the lowest order Hamiltonian and L in our relativistic potential model [4,5], we obtain nonvanishing result. Difference between the GI and our models is in that we cast a light quark into a four-component Dirac spinor which causes non-vanishing commutator as seen below while the GI treats it a two-component spinor. * Electronic address: matsuki@tokyo -kasei.ac.jp † Electronic address: lvqifang@ihep.ac.cn ‡ Electronic address: dongyb@ihep.ac.cn § Electronic address: morii@kobe-u.ac.jp In the past decades, the heavy-light meson families have become a rich structure as seen in PDG [1]. Even though it does not take into account the heavy quark symmetry, the GI model [2,3] has been successful in reproducing and predicting low lying hadrons and heavy-light mesons except for D sJ . This model respects angular momentum conservation at the lowest order so that states with the same angular momentum L are degenerate without spin-orbit interactions.
Let us look at numerical results of models only for D mesons which include a heavy quark c and compare them with each other and with experimental data in Table I. A model in the second column [2,6,7] is the GI model itself and a model in the seventh column [10,11] is a nonrelativistic potential model including a one-loop computation of the heavyquark interaction. Those in the third column [12,13] use the Bethe-Salpeter formulation to expand the system in terms of 1/m Q , while ours in the sixth column [4] uses the Foldy-Wouthouysen-Tani transformation to obtain the equation of motion for a Qq bound system and is essentially the same formulation as that of Ref. [12]. Hence the following arguments given in Sect. II can be derived from Refs. [12,13], too. Finally Ref. [14] uses a quasipotential approach whose details are given in their paper. Similar tables for D s /B/B s mesons can be easily obtained and they give tendency similar to Table  I. Because we would like to extract and show the essence of our claim, we omit them in this article. It is not amazing to see that states with the same L of the GI model have similar mass values for states with the same L because it respects L. However, it is surprising that even models respecting heavy quark symmetry produce the results similar to the GI model, which can be seen from Table I.
States in Table I are assigned definite values of 2S +1 L J in the first column. Even though our relativistic wave function is not an egenstate of L in our formulation [4], we can still assign 2S +1 L J to each state in the nonrelativistic limit.
In the last two columns of Table I MeV between multiples (0 − , 1 − ) with L = 0 and (0 + , 1 + ) with L = 1, 49 MeV between (0 + , 1 + ) and (1 + , 2 + ) with the same L = 1, 320 MeV between (1 + , 2 + ) with L = 1 and (1 − , 2 − ) with L = 2, etc. We can see that mass differences within a spin doublet and between doublets with the same L are very small compared with a mass gap between different multiplets with different L, which is nearly equal to the value of the QCD Λ QCD ∼ 300 MeV 1 [1] for n f = 4.

II. ANALYTICAL ANALYSIS
Using the heavy quark symmetry, the lowest order Hamiltonian in our relativistic potential model [4,5] is given by whose commutation relation with L = r × p is given by On the other hand, we have the following commutation relation, with a light quark spin Σ q /2. Adding Eqs. (2) and (3), we obtain conservation of j ℓ = L + Σ q /2 of light-quark degrees of freedom as expected, H 0 , j ℓ = 0. Because matrices related to a heavy quark are not included in H 0 , a heavy quark spin Σ Q /2 also commutes with H 0 , H 0 , Σ Q /2 = 0, which means a total angular momentum J = L + Σ q /2 + Σ Q /2 also conserves, We would like to estimate the expectation value of [H 0 , L 2 ] whose explicit form is given by There is a lemma that if we calculate the expectation value, The actual wave function includes both positive-and negative-energy states, Ψ ± ℓ in regard to a heavy quark, where ℓ = {k, j, m} with a total angular momentum j and its z-component m. Here the quantum number k is related to the angular momentum of a light quark j ℓ and the parity P for a heavy-light meson as [5,21], Wave functions are defined as [5], In the case of k = −1 ( j P ℓ = (1/2) − ), we obtain the following results up to the first order of 1/m Q , [5,22], where we give J P in the parentheses on the l.h.s. and all the constants, c k,k ′ 1± , are of the order of 1/m Q . On the r.h.s there appear a wave function with a negative-energy component of a heavy quark, Ψ − , together with a positive energy one, Ψ + . After some calculations, we obtain the matrix elements, where σ i 's are Pauli matrices, p i is a momentum operator, and f n (r, p) is defined in Eq. (4). When estimating ψ ℓ (0 − )|M|ψ ℓ (0 − ) , there is no surviving term up to the first order in 1/m Q . This is because Ψ + −1 |M|Ψ + −1 vanishes due to the lemma even though we have Eq. (11) and cross terms of Ψ + ℓ and Ψ − ℓ ′ vanish because of Eq. (10). Hence, the surviving term starts from the order of (1/m Q ) 2 . When estimating ψ ℓ (1 − )|M|ψ ℓ (1 − ) and taking into account the above estimate and c −1,1 1− ∼ 1/m Q , there remain cross terms in k, Ψ + −1 |M|Ψ + 2 with k quantum numbers in subindices and its conjugate, which are of the order of 1/m Q and hence it is suppressed for large m Q . The similar arguments for other higher states give the same conclusion and the expectation value of a matrix element for a higher state is all the same order of magnitude, i.e., at most 1/m Q .
In order to obtain a complete symmetry, we just need to neglect a lower component radial wave function v k (r) which makes Eq. (11) vanish. Neglecting v k (r) in Eq. (7), we obtain a nonrelativistic wave function in the heavy quark system and a little calculation shows us that this is an eigenfunction of L 2 as, L 2 y k jm = k(k + 1)y k jm = L(L + 1)y k jm , The D meson masses in MeV from different quark models and experimental data. Models of ZVR [12], DE [13], EFG [14], and MMS [4] respect heavy-quark symmetry.
gaps between different spin multiplets are nearly equal to Λ QCD ∼ 300 MeV, which coincides with the observation of heavy-light mesons. Future measurement of higher orbitally and/or radially excited states and their masses by LHCb and forthcoming BelleII is waited for to test our observation.