The masses of $D_{sj}^\ast$(2317) and $D_{sj}^\ast$(2463) in the MIT bag model

The masses of the $D_{sj}^\ast$(2317)and $D_{sj}^\ast$(2463) were found in the MIT bag model in a reasonable agreement with the experimental values. The spectrum of ${0^\pm,1^\pm}$ for $D$ and $B$ mesons is presented. The equalities between the mass splittings $(1^+ - 1^-)(c\bar{q})\approx (1^+ - 1^-)(c\bar{s})$ and $(0^+ - 0^-)(c\bar{q})\approx (0^+ - 0^-)(c\bar{s})$, where $q=u, d$ are confirmed despite of the fact that $SU(3)_L\times SU(3)_R$ chiral symmetry is broken by the non-zero strange quark mass.


I. INTRODUCTION
A new scalar meson with the cs flavor content and a mass around 2.32 GeV has been recently discovered by the BaBar collaboration [1] as a narrow resonance in the D s π 0 decay channel and confirmed by the CLEO collaboration [2]. This state aroused a particular interest because of a plain conflict with the theoretical calculations which predict the mass about 100 − 300 MeV above its experimental value [3]. This discrepancy opened a possibility for more refine explanations of the new state [4]. However, it is not clear, weather the difference between the predictions and experiments rises a real issue or it is rather a problem of the models. Besides all that, there are successful explanations of the meson spectrum based on the chiral models [5,6,7,8,9] under the assumption of the SU (3) L × SU (3) R chiral symmetry. The QCD sum rules results [10] also agree with the experiments within the theoretical uncertainty.
This short note is, as a matter of fact, a supplement to the papers devoted to the spectroscopy of hadrons which contain one heavy quark [11,12,13]. In these papers the MIT bag model [14] was adapted to heavy-light systems (Qq, Qqq were Q = c, b and q = u, d, s) and masses of mesons and barions were calculated for the lowest energy states Using the same model and the same parameters (with justified changes in the quark masses) one can find the values of 2.29 GeV for the 0 + and 2.46 for the 1 + cs states.

II. MASS CALCULATION
The mass of the bag can be written in the form [15]: where E Q(q) is the energy of the heavy(light) quark, E bag = 4 3 πBR 3 is the volume energy of the spherical bag of the radius R, E 0 = Z/R is the Casimir energy and the last two terms describe the energies of the color-magnetic and color-electric interactions between quarks. Let us first notice that for the heavy-light systems the center of mass of the bag is well defined. It coincides with the heavy quark position and the corrections due to the c.m. motion can be neglected. The energy of the heavy quark is approximated by the formula E Q = m Q + p 2 light /2m Q , where m Q is the mass of the heavy quark and the second term is the non-relativistic kinetic energy. The total momentum of the bag in the c.m. frame should be zero thus the values of the heavy and light quark momenta are equal. Let us notice that the non-relativistic kinetic energy term in E Q was neglected in the previous calculations [12,13]. This is justified if one wishes to study the lowest energy states. However for exited states, the contribution of the kinetic energy of the heavy quark is more important and should be included. The energy of the color-magnetic interactions between the heavy and light quarks is equal to [12,13] where λ = {1, 2} for barions and mesons respectively, A is calculated from the light quark wave-function and depends on its mass. The constant is inspired by the QCD running coupling formula [13,16]. The spin-spin interaction term σ Q σ q = {−3, 1} for scalar and vector mesons respectively. There are two sources which contribute to the color-electric energy: the interactions between quarks and their self-interactions. For mesons with one heavy quark it can be written as [12]: where C can be calculated and depends on the mass of the light quark. The second term in the parenthesis comes from the self-interaction of the heavy quark. The heavy quark self-interaction is divergent and the infinite term is omitted in (4) [12]. The radius of the bag is determined by minimizing the formula E ′ = E bag + E 0 + E q with respect to R [19].  Table 1.
For the scalar and pseudovector mesons, where the light quark is in the 1P 1/2 state, the light quark energy is E q = 3.81/R for the u, d quarks and E s = m 2 s + (3.91) 2 /R 2 for the strange quark. The radii are R = 5.09 GeV −1 for the non-strange and R s = 5.08 GeV −1 for the strange mesons. The color-magnetic interaction parameter is A(m q = 0) = 0.87 and A(m s ) = 0.84 whereas the color-electric parameter takes values C(m q = 0) = 0.88 and C(m s ) = 0.79. The masses of the 0 + , 1 + states are given in the Table 2.
The mass splitting relations ∆M 1 + 1 − = ∆M 0 + 0 − [5,6] (where ∆M 1 + 1 − = M 1 + − M 1 − and so on) and ∆M 1 + 0 + = ∆M 1 − 0 − [8,9] should hold in the chiral limit. Indeed the experiment finds [2] ∆M 1 + 1 − − ∆M 0 + 0 − = 1.2 ± 2.1 MeV and ∆M 1 + 0 + − ∆M 1 − 0 − = 1.6 ± 2.1 MeV for the cs mesons. The MIT bag model fails to reproduce the mass splitting equalities at this level of accuracy. This can be expected because the MIT bag model does not handle the chiral symmetry in the appropriate way. In spite of that the model predicts that the following relations hold (at the level of 10 − 20 MeV) independently of the flavor of the light quark: where q = u, d. These relations follow from the SU (3) L × SU (3) R . However this symmetry is broken by the non-zero, strange quark mass and it is not clear why it should work so well. Let us stress that from the relations (5) only the last one is confirmed by the experiments at the accuracy of 1 − 2 MeV. Other relations are the assumptions based on the SU (3) L × SU (3) R chiral symmetry or follow from the model calculations.
Finally, to be clear, let us notice that the considered MIT bag model joins several features of works [12,13]. The parameters B, Z as well as the description of the coupling α s comes from the paper [13]. The treatment of the infinity in the electric heavy quark interaction follows paper [12]. The heavy quark kinetic term was also added. The quark masses are adjusted anew to accommodate the changes in the model. The result of the previous paragraph, at the face value, can be regarded as a success of the MIT bag model. The description of the meson spectrum(similar for barions [11,12,13]) at the level of 4-5 per cent is already at the edge of the reliability of the model.
However the more careful scrutiny leads to additional questions. Let us first notice that the quark masses are not treated on an equal footing. The m u , m d , m s are current masses whereas m c , m b are constituent masses. This follows from the treatment of the heavy quark self-interaction in the color-electric energy. The infinite term is absorbed to the mass of the heavy quark. Secondly, the coupling constant at the energy scale 1/R = 0.25 − 0.1 GeV takes values α s = 1.0 − 1.2. This is certainly not a small parameter. Indeed one can try to fit a new set of the bag model parameters with current quark masses and a small coupling constant. First, the infinite term could be cut-off by the heavy quark Compton wave-length 1/m Q . Secondly, the scale of the interaction between the light and heavy quarks may be rather set by the m Q or geometrical mean m Q E q . If done, then the coupling constant can be reduced to the value of α s ∼ 0.3 − 0.4 and the heavy quarks get their current masses. But in this situation, the mass of D * sj (2317) immediately falls in the range 2.44 − 2.55 GeV. The simultaneous requirement of the current heavy quark masses and a small coupling constant can not be reconciled with a good fit to the whole meson spectrum [20].
Despite of these drawbacks the MIT bag model is capable to predict the 0 ± , 1 ± , D mesons spectrum with the accuracy of few per cent. One can also find that the mass splitting relations (5) are independent of the flavor of the light quark. These relations follow directly from the chiral symmetry SU (3) L × SU (3) R . However this symmetry is broken by the strange quark mass and it is not obvious why the relations (5) should hold.
In the further work one can try to include the light quark in the 1P 3/2 state. However the calculation of the 1P 3/2 state is more involved and require reconsideration of the model assumptions (e.g. sphericity of the bag) thus demanding much more extensive elaboration then this short note assumed.