Eur. Phys. J. C (2017) 77:668 DOI 10.1140/epjc/s10052-017-5252-4 Regular Article - Theoretical Physics Masses of scalar and axial-vector B mesons revisited Hai-Yang Cheng1, Fu-Sheng Yu2,a 1 Institute of Physics, Academia Sinica, Taipei 115, Taiwan, Republic of China 2 School of Nuclear Science and Technology, Lanzhou University, Lanzhou 730000, People’s Republic of China Received: 2 August 2017 / Accepted: 22 September 2017 / Published online: 7 October 2017 © The Author(s) 2017. This article is an open access publication Abstract The SU(3) quark model encounters a great chal- 1 Introduction lenge in describing even-parity mesons. Specifically, the qq¯ quark model has difficulties in understanding the light Although the SU(3) quark model has been applied success- scalar mesons below 1 GeV, scalar and axial-vector charmed fully to describe the properties of hadrons such as pseu- mesons and 1+ charmonium-like state X(3872). A common doscalar and vector mesons, octet and decuplet baryons, it wisdom for the resolution of these difficulties lies on the often encounters a great challenge in understanding even- coupled channel effects which will distort the quark model parity mesons, especially scalar ones. Take vector mesons as calculations. In this work, we focus on the near mass degen- an example and consider the octet vector ones: ρ,ω, K ∗,φ. ∗ ∗0 eracy of scalar charmed mesons, Ds0 and D0 , and its impli- Since the constituent strange quark is heavier than the up or cations. Within the framework of heavy meson chiral pertur- down quark by 150 MeV, one will expect the mass hierarchy bation theory, we show that near degeneracy can be quali- pattern mφ > m K ∗ > mρ ∼ mω, which is borne out by tatively understood as a consequence of self-energy effects experiment. However, this quark model picture faces great due to strong coupled channels. Quantitatively, the closeness challenges in describing the even-parity meson sector: ∗ ∗0 of Ds0 and D0 masses can be implemented by adjusting two relevant strong couplings and the renormalization scale • Many scalar mesons with masses lower than 2 GeV appearing in the loop diagram. Then this in turn implies the have been observed and they can be classified into two mass similarity of B∗ and B∗0 mesons. The P∗ P inter- s0 0 0 1 nonets: one nonet with mass below or close to 1 GeV, action with the Goldstone boson is crucial for understand- such as f (500) (or σ ), K ∗(800) (or κ), f (980) and ing the phenomenon of near degeneracy. Based on heavy 0 0 0 a (980) and the other nonet with mass above 1 GeV quark symmetry in conjunction with corrections from QCD 0 such as K ∗(1430), a (1450) and two isosinglet scalar and 1/m effects, we obtain the masses of B∗ and B 0 0 Q (s)0 (s)1 mesons. Of course, the two nonets cannot be both low- mesons, for example, MB∗ = (5715 ± 1) MeV + δS, s0 lying 3 P qq¯ states simultaneously. If the light scalar = ( ± ) + δ δ / 0 MB 5763 1 MeV S with S being 1 m Q s1 nonet is identified with the P-wave qq¯ states, one will corrections. We find that the predicted mass difference of encounter two major difficulties: first, why are a (980) 48 MeV between B and B∗ is larger than that of 20– 0 s1 s0 and f (980) degenerate in their masses? In the qq¯ model, 30 MeV inferred from the relativistic quark models, whereas 0 the latter is dominated by the ss¯ component, whereas the difference of 15 MeV between the central values of MB s1 the former cannot have the ss¯ content since it is an and MB is much smaller than the quark model expectation 1 I = 1 state. One will expect the mass hierarchy pattern of 60–100 MeV.Experimentally, it is important to have a pre- ∗ m f (980) > m K ∗(800) > ma (980) ∼ m f (500). However, cise mass measurement of D mesons, especially the neutral 0 0 0 0 0 this pattern is not seen by experiment. In contrast, it is one, to see if the non-strange scalar charmed meson is heav- ≈ > ∗ > ma0(980) m f0(980) m K (800) m f0(500) experimen- ier than the strange partner as suggested by the recent LHCb (0 ) ∗( ) ∗± tally. Second, why are f0 500 and K0 800 so broad measurement of the D0 . compared to the narrow widths of a0(980) and f0(980) even though they are all in the same nonet? • ∗( ) In the scalar meson sector above 1 GeV, K0 1430 with mass 1425 ± 50 MeV [1] is almost degenerate in masses with a0(1450), which has a mass of 1474 ± 19 MeV [1] a e-mail: [email protected] despite having one strange quark for the former. 123 668 Page 2 of 13 Eur. Phys. J. C (2017) 77 :668 Table 1 Measured masses and J P Meson Mass (MeV) (MeV) Mass (MeV) widths of even-parity charmed mesons. The four p-wave + ∗( )0 ± ± 0 D0 2400 2318 29 267 40 2340–2410 Large charmed meson states are ∗ ± ∗, , ∗ D (2400) 2351 ± 7 230 ± 17 2340–2410 Large denoted by D0 D1 D1 and D2 , 0 respectively. In the heavy quark + ( )0 ± ± +107 ± 1 D1 2430 2427 26 25 384−75 74 2470–2530 Large limit, D has j = 1/2andD 1 1 1+ D (2420)0 2420.8 ± 0.531.7 ± 2.5 2417–2434 Small has j = 3/2with j being the 1 ± total angular momentum of the D1(2420) 2432.2 ± 2.425± 6 2417–2434 Small light degrees of freedom. The + ∗( )0 . ± . ± . 2 D2 2460 2460 57 0 15 47 7 1 3 2460–2467 Small data are taken from the Particle ∗ ± D (2460) 2465.4 ± 1.346.7 ± 1.2 2460–2467 Small Data Group [1]. The last two 2 + ∗ ( )± . ± . < columns are the predictions 0 Ds0 2317 2317 7 0 6 3.8 2400–2510 Large + ( )± . ± . < from the quark model [7,8]. 1 Ds1 2460 2459 5 0 6 3.5 2528–2536 Large “Large” means a broad width of + ± 1 Ds1(2536) 2535.10 ± 0.06 0.92 ± 0.05 2543–2605 Small order 100 MeV, while “small” + ∗ ( )± . ± . ± . implies a narrow width of order 2 Ds2 2573 2569 1 0 8169 0 8 2569–2581 Small 10 MeV 3 3 • In the even-parity charmed meson sector, we compare the χc1(1 P1) with a mass 3511 MeV [1]orχc1(2 P1) with experimentally measured masses and widths with what the predicted mass of order 3950 MeV [11]. Moreover, a are expected from the quark model (see Table 1). There pure charmonium for X(3872) cannot explain the large are some prominent features from this comparison: (i) the isospin violation observed in X(3872) → J/ψω, J/ψρ ∗, , ∗ ( ) measured masses of D0 D1 Ds0 and Ds1 are substan- decays. The extreme proximity of X 3872 to the thresh- tially smaller than the quark model predictions. (ii) The old suggests a loosely bound molecule state D0 D¯ ∗0 ∗ ( ) ( ) physical Ds0 mass is below the DK threshold, while Ds1 for X 3872 . On the other hand, X 3872 cannot be a is below DK∗. This means that both of them are quite nar- pure DD¯ ∗ molecular state either for the following rea- row, in sharp contrast to the quark model expectation of sons: (i) It cannot explain the prompt production of ∗( )0 ∗ ( ) ( ) large widths for them. (iii) D0 2400 and Ds0 2317 are X 3872 in high-energy collisions [12,13]. (ii) The ratio ∗( )± ≡ ( 0 → 0 ( ))/ ( − → − ( )) almost equal in their masses, while D0 2400 is heavier R1 B K X 3872 B K X 3872 ∗ 1 than Ds0 even though the latter contains a strange quark. is predicted to be much less than unity in the molecular , ∗, ∗ . ± . ± . (iv) The masses of D1 D2 Ds1 and Ds2 predicted by scenario, while it was measured to be 0 50 0 30 0 05 the quark model are consistent with experiment. These and 1.26 ± 0.65 ± 0.06 by Belle [14,15]. (iii) For the + + four observations lead to the conclusion that 0 and 1 ratio R2 ≡ (X(3872) → ψ(2S)γ )/ (X(3872) → charmed mesons have very unusual behavior not antici- J/ψ(1S)γ ), the molecular model leads to a very small pated from the quark model. value of order 3 × 10−3 [16–18], while the charmonium • The first XYZ particle, namely X(3872), observed by model predicts R2 to be of order of unity. The LHCb mea- ± ± + − Belle in 2003 in B → K + (J/ψπ π ) decays [9], surement yields R2 = 2.46 ± 0.64 ± 0.29 [19]. Hence, has the quantum numbers J PC = 1++ [10]. X(3872) X(3872) cannot be a pure DD¯ ∗ molecular state. The cannot be a pure charmonium as it cannot be identified as above discussions suggest that X(3872) is most likely an admixture of the S-wave DD¯ ∗ molecule and the P- wave charmonium as first advocated in [12], 1 Strictly speaking, the masses of the neutral and charged states of ∗( ) D0 2400 are not consistently determined due mainly to its broadness. 0 ∗0 The mass of the charged one is primarily from the LHCb measurements |X(3872)=c1|cc¯P-wave + c2|D D S-wave [2,3], while the neutral one is from BaBar [4], Belle [5]andFOCUS + ∗− [6]. It is worthwhile to notice that only FOCUS has measured both the +c3|D D S-wave +··· . (1.1) ∗( ) neutral and the charged D0 2400 , and their masses are quite similar, with a small difference of a few MeV.