Structure of Mass Gap between Two Spin Multiplets

Studying our semirelativistic potential model and the numerical results, which succeeds in predicting and reproducing recently discovered higher resonances of $D$, $D_s$, $B$, and $B_s$, we find a simple expression for the mass gap between two spin multiplets of heavy-light mesons, $(0^-,1^-)$ and $(0^+,1^+)$. The mass gap between chiral partners defined by $\Delta M=M(0^+)-M(0^-)$ and/or $M(1^+)-M(1^-)$ is given by $\Delta M=M(0^+)-M(0^-)=M(1^+)-M(1^-)\approx \Lambda_{\rm Q}-m_q$ in the limit of heavy quark symmetry, and including $1/m_Q$ corrections, we have $\Delta M\approx \Lambda_{\rm Q}-m_q+(1.28\times 10^{5}+4.26\times 10^{2}\cdot m_q)/m_Q$ with $\Lambda_{\rm Q}\approx 300$ MeV, a light quark mass $m_q$, and a heavy quark mass $m_Q$. This equation holds both for $D$ and $D_s$ heavy mesons. Our model calculations for the $B$ and $B_s$ also follow this formula.


I. INTRODUCTION
The discovery of the narrow D sJ particles by BaBar [1] and CLEO [2] and soon confirmed by Belle [3] immediately reminded people an effective theory approach proposed by Nowak et al. and others [4,5,6,7]. They constructed an effective Lagrangian for heavy mesons from the Nambu-Jona-Lasinio type four-fermi interactions and combined it with the chiral multiplets so that the mass of heavy mesons can be related to the Higgs scalars of chiral Lagrangian, and they found that two spin multiplets, j P = (0 − , 1 − ) and (0 + , 1 + ), are degenerate in the limit in which the chiral symmetry is an exact symmetry of the vacuum and the heavy quark symmetry is exactly realized. From this effective theory, they derived the Goldberger-Treiman relation for the mass gap between chiral partners 0 + (1 + ) and 0 − (1 − ) instead of the heavy meson mass itself and predicted the mass gap between chiral partners of heavy mesons to be around ∆M = g π f π ≈ 349 MeV, where g π is the coupling constant for 0 + → 0 − + π and f π is the pion decay constant.
Finding that the mass gap between chiral partners 0 + (1 + ) and 0 − (1 − ) in the case of D s agrees well with the experiments (around 350 MeV), people thought that underling physics may be explained by their SU (3) effective Lagrangian [8,9]. However, when (0 + , 1 + ) for D meson were found by Belle and FOCUS, and later reanalyzed by CLEO, their explanation needs to be modified even though some people still study in this direction; in fact, the effective Lagrangian approach [8] predicts about 94 MeV smaller mass gap for D mesons than that for D s mesons, while the experimental mass gap for D mesons is about 70 ∼ 80 MeV larger than that for D s mesons [10]. Furthermore, what they originally predicted could not be identified as any of heavy meson multiplets for D, D s , B, and B s . In other words, the forumula can be applied equally for any of these heavy meson multiplets. Thus, it is required to find the mass gap formula, if it exists, which agrees well with the experiments and explains the physical ground of its formula.
In this paper, using our semirelativistic potential model, we first give our formula for the mass gap between chiral partners 0 + (1 + ) and 0 − (1 − ) for any heavy meson, D, D s , B, and B s , among which the known mass gaps, i.e., the ones for D and D s , agree well with the experiments although there is some ambiguities for D meson data. Next we show how this mass gap depends on a light quark mass m q for q = u, d, and s, where we neglect the difference between u and d quarks. Our formula naturally explain that the mass gap for D is larger than that for D s and predict the mass gaps for B and B s .

II. SEMIRELATIVISTIC QUARK POTENTIAL MODEL AND STRUCTURE OF MASS GAP
Mass for the heavy meson X with the spin and parity, j P , is expressed in our formulation as [11] where the quantum number k is related to the total angular momentum j and the parity P for a heavy meson as [12] j = |k| − 1 or |k|, P = k |k| To begin with, we study the heavy meson mass without 1/m Q corrections so that we can see the essence of the mass gap. States with the same |k| value are degenerate in a pure chiral limit and without confining scalar potential, which is defined as m q → 0 and S(r) → 0 [13]. We consider the scenario that a chiral symmetry breaking and a confinement take place in two steps. First the degeneracy is broken due to gluon fields when S(r) is turned on and confines quarks into heavy mesons but keeping vanishing light quark mass intact. In fact, in this limit our model gives the mass gap between two spin multiplets ∆M ≈ 300 MeV as follows; This gap is mainly due to gluon fields which confines quarks into heavy mesons. It is interesting that obtained values are close to Λ QCD ≈ 300 MeV. Next, turning on a light quark mass which explicitly breaks a chiral symmetry, we have SU (3) flavor breaking pattern of the mass levels, i.e., mass of D becomes different from that of D s with the same value of j P . Since we assume m u = m d , there still remains SU (2) iso-spin symmetry. Note that even after chiral symmetry is broken, there is still degeneracy between members of a spin multiplet due to the heavy quark symmetry, i.e., SU (2) f × SU (2) spin symmetry, with SU (2) f rotational flavor symmetry and SU (2) spin rotational spin symmetry. By using the optimal values of parameters in Ref. [14], which is listed in Table I, degenerate masses without 1/m Q corrections for D, D s and B, B s mesons are calculated and presented in Table II. Furthermore, by changing m q from 0 to 0.2 GeV, we have calculated the m q dependence of ∆M 0 and have obtained Fig. 1, in which ∆M 0 is linearly decreasing with m q . From Fig. 1, we find that the mass gap between two spin multiplets for a heavy meson X can be written as where the values of g 0 , and g 1 are estimated by fitting the optimal line with Fig. 1. Since both g 0 and g 1 are very close to 1, we conclude that the mass gap is essentially given by Though the physical ground of this result is out of scope at present, Eq. (6) is serious, since it is very different from the one of an effective theory approach as mentioned later. This result is exact when O (1/m Q ) terms are neglected. As we will see later, since 1/m Q corrections are nearly equal to each other for two spin doublets, the above equation (6) between two spin multiplets holds approximately even with 1/m Q corrections.
Let us see how the mass gap can be written in our formulation [11]. Heavy meson mass without 1/m Q corrections can be given by Eq. (1) with an eigenvalue E k 0 being given by the following eigenvalue equation.   Using this equation, the mass gap between k = +1 and k = −1, which are corresponding to the spin multiplets (0 − , 1 − ) and (0 + , 1 + ), respectively, when they are degenerate, is re-expressed as From this equation we can see that the mass gap linearly depends on m q . Here the radial wave function Φ k (r) and the massless free kinetic term K k with the quantum number k are given by Numerically the coefficient of m q becomes negative, while the first term in Eq. (8) is approximately given by 300 MeV, which is nearly equal to the scale parameter of QCD, Λ QCD . That a coefficient of m q becomes negative in Eq. (8) can be explained or we can intuitively understand this result in our formulation as follows. The quantum numbers k = −1  and k = 1 correspond to ℓ = 0 and ℓ = 1 respectively, where ℓ is the angular momentum of a light antiquark relative to a heavy quark as can be seen from Table I of Ref. [14]. An excited state with ℓ = 1 (k = +1) is more relativistic compared with the one with ℓ = 0 (k = −1), which means a lower component v 1 (r) becomes larger than v −1 (r) since they are normalized as (u k ) 2 + (v k ) 2 = 1. Hence (u 1 ) 2 − (v 1 ) 2 = Φ † 1 (r)βΦ 1 (r) becomes smaller than Φ † −1 (r)βΦ −1 (r). Thus the coefficient of m q becomes negative. As a matter of fact, linear m q dependence of ∆M is not yet definite since radial wave functions u k and v k are also dependent on m q . However, looking at Eq. (4) or Fig. 1 which are the numerical calculation of our model, we can say that implicit dependence on m q of these wave functions is numerically small. Thus the above physical and intuitive interpretation of linear m q dependence of ∆M 0 is correct.

III. INTERPRETATION DUE TO CHIRAL EFFECTIVE THEORY
The above result suggests that the physical ground of chiral symmetry breakdown or generation of mass for heavy mesons occurs differently from what people in [4,5,6] originally considered. Let us briefly explain the mechanism that these authors considered as a generation of the mass gap, which is due to the paper [5]. The Lagrangian for the chiral multiplets, which couples to the heavy quark sector, can be written as follows.
where ψ is the chiral quark field with three flavors and Σ is the 3 × 3 complex auxiliary field which are given by When this Lagrangian is combined with the effective theory for heavy hadrons, the effective mass of a constituent quark is given by σ + m q . Then the mass gap is given by where g π is the Yukawa coupling constant between the heavy meson and a chiral multiplet and is taken to be g π = 3.73 in [8], and σ = f π . This expression is obtained in the heavy quark symmetric limit and should be compared with our Eq. (6). Instead of minus sign for the term m q that we obtained, the authors of [5] obtained plus sign as shown in the above equation. The same result is obtained even if we use the nonlinear Σ model [8].
IV. 1/mQ CORRECTIONS Next let us study the case when 1/m Q corrections to the mass gap are taken into account. Part of the results is given in [15]. In Table III, we give our numerical results in the cases of n = 1 and n = 2 (radial excitations). Values in brackets are taken from the experiments. Our values seem to agree with the experimental ones though the fit is not as good as the case for the absolute values of heavy meson masses. We assume the form of the mass gap with the 1/m Q corrections as follows.
Using Eq.(4) for D and D s mesons, i.e. ∆M 0 = g 0 Λ Q − g 1 m q = 295.1 − 1.080m q , we obtain the values of the parameters c and d for D/D s mesons given in Table III, which are given by The term c/m Q lifts the constant g 0 Λ Q about 100 MeV and the term d/m Q gives deviation from -1 to the coefficient for m q in the case of D/D s . Applying this formula, Eq. (13), to the case for B/B s with m Q = m b , we obtain the mass gap as follows.
which should be compared with our model calculations, 321 and 241 MeV, in Table III. Thus the linear dependence of the mass gap on m q is also supported in the case where the 1/m Q corrections are taken into account. The calculated m q dependence of ∆M with 1/m Q corrections is presented in Fig. 2, for 0 < m q < 0.2GeV. Valuse in Table III are  calculated using those in Tables IV and V. V. MISCELLANEOUS PHENOMENA Global Flavor SU (3) Recovery -Looking at the mass levels of 0 + and 1 + states for the D and D s mesons, one finds that mass differences between D and D s becomes smaller compared with those of the 0 − and 1 − states. This can be seen from Table IV and was first discussed in Ref. [16] by Dmitrašinović. He claimed that considering D sJ as a four-quark state, one can regard this phenomena as flavor SU (3) recovery. However, in our interpretation, this is not so as we have seen that this is caused by the mass gap dependency on a light quark mass, m q , as shown in Fig.  1. That is, when the mass of D meson is elevated largely from the 0 − /1 − state to the 0 + /1 + state, the mass of D s meson is elevated by about 100 MeV smaller than that of 0 − /1 − as one can see from Fig. 1. In our interpretation, the SU (3) is not recovered since the light quark masses of m u = m d and m s do not change their magnitudes when the transition from 0 − /1 − to 0 + /1 + occurs, and their values remain to be m u(d) = 11.2 MeV and m s = 92.9 MeV, respectively, as presented in Table I.
Mass Gap of Heavy Baryons -When we apply our formula to the heavy-light baryons which include two heavy quarks, (ccs), (ccu), (bcs), (bcu), (bbs), and (bbu), mass gaps between two pairs of baryons, like (ccs) and (ccu), will be given by Eq. (6) in the heavy quark symmetric limit and by Eq. is legitimized since QQ pair can be considered to be 3 * expression in the color SU (3) space so that the baryon like QQq can be regarded as a heavy-light meson and our arguments expanded in this paper can be applied [17,18].