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. All the other three groups have More specifically, the charmonium is identified with not reported the masses for both the neutral and the charged D∗(2400). 0 χ ( 3 ) ¯ In addition, the masses reported by different groups are very different c1 2 P1 . Some calculations favor a larger cc com- ∗0 from each other. As a result, the world-averaged masses for the neutral ponent over the D0 D component (see e.g. [20,21]). ∗( ) and charged D0 2400 shown in Table 1 are very different as well. The Then the question is how to explain the mass of X(3872) difference is well beyond the expectation from isospin splitting. Thus, the two averaged masses given by the Particle Data Group (PDG) [1] through the charmonium picture. are not consistent with each other. In this work, we consider both PDG masses for D∗(2400). In Sect. 2 we focus on the closeness of D∗ and ∗ 0 s0 In short, the qq¯ quark model has difficulties in describing D0 masses and discuss its implication to the scalar B sector. In Sect. 3 + + we derive two different sets of scalar B meson masses, corresponding light scalar mesons below 1 GeV,0 and 1 charmed mesons ∗( ) + ( ) to two different PDG masses for D0 2400 . and 1 charmonium-like state X 3872 . Common wisdom 123 Eur. Phys. J. C (2017) 77 :668 Page 3 of 13 668
( ) ∗( ) for the resolution of aforementioned difficulties lies in the widths of f0 500 and K0 800 , and the narrowness of coupled channel effects which will distort the quark model f0(980) and a0(980) owing to the very limited phase space calculations. available as they are near the K K threshold. ∗ In the quark potential model, the predicted masses for Ds0 In [48] we have studied near mass degeneracy of scalar ∗0 and D0 are higher than the measured ones by order 160 and charmed and bottom mesons. Qualitatively, the approximate 70 MeV,respectively [7,8]. It was first stressed and proposed mass degeneracy can be understood as a consequence of self- ∗ ( ) ∗( )0 in [22] that the low mass of Ds0 2317 (D0 2400 )arises energy effects due to strong coupled channels which will from the mixing between the 0+ cs¯ (cq¯) state and the DK push down the mass of the heavy scalar meson in the strange (Dπ) threshold (see also [23]). This conjecture was realized sector more than that in the non-strange partner. However, we in both QCD sum rule [24,25] and lattice [26–29] calcu- showed that it works in the conventional model without heavy lations. For example, when the contribution from the DK quark expansion, but not in the approach of heavy meson continuum is included in QCD sum rules, it has been shown chiral perturbation theory (HMChPT) as mass degeneracy ∗ ∗ ∗ that this effect will significantly lower the mass of the Ds0 and the physical masses of Ds0 and D0 cannot be accounted state [24]. Recent lattice calculations using cs¯, DK and D∗ K for simultaneously. Mass shifts in the strange charm sector ∗ ( ) interpolating fields show the existence of Ds0 2317 below are found to be largely overestimated. It turns out that the ( ) ∗ the DK threshold [26,27] and Ds1 2460 below the D K conventional model works better toward the understanding threshold [28].2 All these results indicate that the strong cou- of near mass degeneracy. pling of scalars with hadronic channels will play the essential Our previous work was criticized by Alhakami [49]who role of lowering their masses. followed the framework of [50] to write down the general By the same token, mass shifts of charmed and bottom expression of HMChPT and fit the unknown low-energy scalar mesons due to self-energy hadronic loops have been constants in the effective Lagrangian to the experimentally calculated in [31]. The results imply that the bare masses of measured odd- and even-parity charmed mesons. Using the scalar mesons calculated in the quark model can be reduced results from the charm sector, Alhakami predicted the spec- significantly. Mass shifts due to hadronic loops or strong cou- trum of odd- and even-parity bottom mesons. He concluded pled channels have also been studied in different frameworks that the near degeneracy of non-strange and strange scalar ∗ ( ) to explain the small mass of Ds0 2317 [32–35]. In the same B mesons is confirmed in the predictions using HMChPT. spirit, even if X(3872) is dominated by the cc¯ component, He then proceeded to criticize that we should use physical χ the mass of c1 can be shifted down due to its strong coupling masses instead of bare masses to evaluate the hadronic loop with DD¯ ∗ channels [36–40]. effects and that we have missed the contributions from axial- Both f0(980) and a0(980) have the strong couple chan- vector heavy mesons to the self-energy of scalar mesons. nel K K . They are often viewed as K K molecules, which Motivated by the above-mentioned criticisms [49], in this accounts for their near degeneracy with 2m K . Schematically, work we shall re-examine our calculations within the frame- ( ) ∗ the self-energy K K loop diagram of a0 980 will shift its work of HMChPT. We show that the closeness of Ds0 and ∗0 mass to the physical one. In the unitarized chiral perturbation D0 masses can be achieved by taking into account the addi- ( ) ∗( ) ( ) theory, light scalar mesons f0 500 , K0 800 , f0 980 and tional contribution, which was missing in our previous work, a0(980) can be dynamically generated through their strong from axial-vector heavy mesons to the self-energy diagrams couplings with ππ, K π, K K , and K K , respectively [41– of scalar mesons by adjusting two relevant strong couplings 45]. Alternatively, it is well known that the tetraquark pic- and the renormalization scale μ appearing in the loop dia- ture originally advocated by Jaffe [46] provides a simple gram. Then we proceed to confirm that near degeneracy solution to the mass and width hierarchy problems in the observed in the charm sector will imply the similarity of ∗ ∗ light scalar meson sector. The tetraquark structure of light Bs0 and B0 masses in the B system. ≈ scalars accounts for the mass hierarchy pattern ma0(980) This work is organized as follows. In Sect. 2 we consider m ( ) > m ∗( ) > m ( ), Moreover, the S-wave 4- the self-energy corrections to scalar and axial-vector heavy f0 980 K0 800 f0 500 quark nonet can be lighter than the P-wave qq¯ nonet above mesons in HMChPT. In the literature, the self-energy loop 1 GeV due to the absence of the orbital angular momen- diagrams were sometimes evaluated in HMChPT by neglect- tum barrier and the presence of strong attraction between ing the corrections from mass splittings and residual masses the diquarks (qq)3∗ and (q¯q¯)3 [47]. The fall-apart decays to the heavy meson’s propagator. We shall demonstrate in ( ) → ππ ∗( ) → π ( ), ( ) → f0 500 , K0 800 K , and f0 980 a0 980 Sect. 3 that the calculation in this manner does not lead to K K are all OZI-superallowed. This explains the very broad the desired degeneracy in both charm and B sectors simulta- ∗ ∗ neously. The masses of Bs0 and B0 are discussed in Sect. 4 2 with focuses on the predictions based on heavy quark sym- A recent lattice calculation with N f = 2 + 1 + 1 optimal domain- ± ± ∗ metry and possible 1/m Q and QCD corrections. Section 5 wall fermions [30] yields a mass of 2317 15 5MeVforDs0 and ± ± ( ) 2463 13 9MeVforDs1 2460 . comes to our conclusions. 123 668 Page 4 of 13 Eur. Phys. J. C (2017) 77 :668
Fig. 1 Self-energy π, K, η π, K, η contributions to scalar and axial-vector heavy mesons
0+ 0−, 1+ 0+ 1+ 1−, 0+ 1+
(a) (b)
2 Mass shift of scalar and axial-vector heavy mesons the total angular momentum of the light degrees of freedom): due to hadronic loops + v/ = 1 [ ∗ γ μ − γ ], Ha Paμ Pa 5 + + 2 Self-energy hadronic loop corrections to 0 and 1 heavy 1 μ ∗ mesons have been considered in the literature [31,48–54]. Sa = (1 + v/)[P γμγ − P ], (2.5) 2 1a 5 0a Since chiral loop corrections to heavy scalar mesons have = , , = ( 0, +, +) both finite and divergent parts, it is natural to consider the with a u d s, for example, Pa D D Ds .The framework of HMChPT where the divergences and the renor- nonlinear chiral symmetry√ is realized by making use of the malization scale dependence arising from the chiral loops unitary matrix = exp(i 2φ/fπ ) with fπ = 93 MeV and induced by the lowest-order tree Lagrangian can be absorbed φ being a 3 × 3 matrix for the octet of Goldstone bosons. In into the counterterms which have the same structure as the terms of the new matrix ξ = 1/2, the axial-vector field A A = i (ξ †∂ ξ − ξ∂ ξ †) next-order tree Lagrangian. reads 2 μ μ .InEq.(2.4), the parameter The heavy meson’s propagator in HMChPT has the S is the residual mass of the S field; it measures the mass expression splitting between even- and odd-parity doublets and can be expressed in terms of the spin-averaged masses, i , (2.1) ∗ + 3M + M ∗ 2v · k − (v · k) 3MP MP P1 P0 MH ≡ , MS≡ , (2.6) 4 4 where v and k, respectively, are the velocity and the residual so that momentum of the meson defined by p = vm0+k, and (v·k) is the 1PI self-energy contribution. In general, (v · k) is S = MS− MH . (2.7) complex as its imaginary part is related to the resonance’s width. The particle’s on-shell condition is then given by There exist two corrections to the chiral Lagrangian (2.4): one from 1/m Q corrections and the other from chiral sym- v · ˜ − (v · ˜) = . 2 k Re k 0 (2.2) metry breaking. The 1/m Q corrections are given by [58,59] The physical mass reads 1 H ¯ μν L /m = λ Tr[Haσ Haσμν] 1 Q 2m 2 Q ˜ 1 ˜ S ¯ μν m = m + v · k = m + Re(v · k). (2.3) −λ Tr[S σ S σμν] , (2.8) 0 0 2 2 a a Consider the self-energy diagrams depicted in Fig. 1 for with scalar and axial-vector heavy mesons. We will evaluate the H 1 2 2 S 1 2 2 λ = ( ∗ − ), λ = ( − ∗ ), 2 MP MP 2 MP MP (2.9) loop diagrams in the framework of HMChPT in which the 4 4 1 0 low-energy dynamics of hadrons is described by the formal- where λ (λ ) is the mass splitting between spin partners, ism in which heavy quark symmetry and chiral symmetry are H S namely, P∗ and P (P and P∗) of the pseudoscalar (scalar) synthesized [55–57]. The relevant Lagrangian is [58] 1 0 doublet. We will not write down the explicit expressions for L = Tr H¯ (iv · D) H +Tr S¯ ((iv · D) −δ )S chiral symmetry breaking terms and the interested reader is b ba a b ba ba S a ¯ μ referred to [50]. The masses of the heavy mesons can be +gTr H γμγ A H b 5 ba a expressed as ¯ μ +hTr Sbγμγ5A Ha + h.c. ba H ¯ μ 3 λ +g Tr S γμγ A S + h.c. , (2.4) = − 2 + , b 5 ba a MPa M0 a 2 m Q ∗ where H denotes the odd-parity spin doublet (P, P ) and S 1 λH ∗ ∗ = + 2 + , ( , ) = / MPa M0 a the even-parity spin doublet P0 P1 with j 1 2(j being 2 m Q 123 Eur. Phys. J. C (2017) 77 :668 Page 5 of 13 668
3 λS 2 ∗ 2 ˜ 2g i MP = M0 + S − + a, (ω ) =− 0a D K D 2 2 m Q 1 1 fπ 2 λS 1 2 ˜ d4q q2−(v · q)2 M = M0 + S + + a, (2.10) P1a × 2 m Q (2π)4 (q2−m2+i)(v · k− − 1 M −˜ +i) S 4 S u ˜ 2g2 i where a and a denote the residual mass contributions =− 2 to odd- and even-parity mesons, respectively. Note that fπ 2 λH /m ≈ 1 (M ∗ − M ) ≡ 1 M and λS/m ≈ 2 Q 2 P P 2 P 2 Q d4q q2 − (v · q)2 1 ( − ∗ ) ≡ 1 × , 2 MP MP 2 MS in the heavy quark limit. The ( π)4 ( 2 − 2 + )(v · + ω + ) 1 0 ∗ ∗ 2 q m i q i (v) (v) (v) (v) D1 propagators for Pa , Pa , P0a and P1a read (2.15) i 3 , where ω = v·k+M ∗ −M −(M −M )+ M − D Ds D D D 4 D 2(v · k + 3 M − ) + i 1 0 1 1 4 P a u, and use of Eq. (2.10) has been made. Likewise, the full −i(gμν − vμvν) D∗ propagator reads , (2.11) 0 1 2(v · k − MP − a) + i 4 3 ˜ i 2(v · k − S + MS − u) 4 and 3 1 − Dπ (ωD) + Dη(ωD) + D K (ωD ) i , 2 6 s s 3 ˜ − 2(v · k − S + MS − a) + i 1 4 +3 (ω ) + 1 (ω ) + (ω ) . −i(gμν − vμvν) D π D D η D D K D , (2.12) 2 1 1 6 1 1 s1 s1 (v · − − 1 − ˜ ) + 2 k S 4 MS a i (2.16) ˜ ˜ respectively. Since many parameters such as S, MS, s and u in ∗ Eqs. (2.14) and (2.16) are unknown, we are not able to deter- Consider the hadronic loop contribution to Ds0 in Fig. 1(a) with the intermediate states D0 and K +. The self-energy loop mine the mass shifts from the above equations. Assuming integral is that the bare mass M is the one obtained in the quark model, then from Eq. (2.10)wehave 2 (ω ) = 2h i 3 ˜ DK D S − MS + s = MD∗ − M0 f 2 2 s0 π 4 4 (v · )2 3 d q q = M ∗ − M − M + . (2.17) × Ds0 D D u (2π)4 (q2−m2 +i)(v · k+ 3 M − +i) 4 K 4 D u 2 With = 2h i f 2 2 π 4 2 3 d q (v · q) F(v · k)D∗ ≡ 2(v · k − MD∗ + MD + MD − u) × , (2.13) s0 s0 4 (2π)4 (q2 − m2 + i)(v · q + ω + i) K D − (ω ) + 2 (ω ) Re 2 DK D Ds η Ds 3 where m is the mass of the Goldstone boson. The residual 2 + (ω ) + (ω ) , momentum k of the heavy meson in the loop is given by 2 D K D D η D 1 1 3 s1 s1 k = p +q −vM = q +k +v(M ∗ − M ), and ω = v · D Ds0 D D 3 (v · ) ∗ ≡ (v · − M ∗ + + − ) ∗ 3 F k D 2 k D MD MD u k+MD −MD + MD −u. The calligraphic symbol has 0 0 s0 4 ∗ 4 been used to denote the bare mass. The full Ds0 propagator becomes
i (2.14) (v · − + 3 − ˜ ) −[ (ω ) + 2 (ω ) + (ω ) + 2 (ω )] 2 k S MS s 2 DK D Ds η Ds 2 η 4 3 D1 K D1 3 Ds1 Ds1 after taking into account the contributions from the chan- 0 +, + 0, +η 0 +, + 0, +η nels D K D K Ds and D1 K D1 K Ds1 .In Eq. (2.14), 123 668 Page 6 of 13 Eur. Phys. J. C (2017) 77 :668
3 1 Note that the function F(−m/ω) can be recast into the form −Re Dπ (ωD) + Dη(ωD) 2 6 + (ω ) m 1 Ds K Ds F − = G(ω, m) ω ω (2.24) +3 (ω ) D π D 2 1 1 with 1 ⎧√ + (ω ) + (ω ) , − ω D η D D K D (2.18) ⎪ ω2 − m2 [cosh 1( ) − iπ],ω>m, 6 1 1 s1 s1 ⎨√ m (ω, )= 2 − ω2 −1(− ω ), ω2 < 2 ∗ G m m cos m the on-shell conditions read F(v · k˜) ∗ = 0forD and ⎩⎪ √ m Ds0 s0 2 2 −1 ω ∗ − ω − m cosh (− ), ω < −m. F(v · k˜) ∗ = 0forD . The physical masses are then given m D0 0 by (2.25) ˜ 3 ˜ The parameter appearing in Eqs. (2.21) and (2.22)is MD∗ = M0 + v · k = MD + MD − u + v · k. (s)0 4 an arbitrary renormalization scale. In the dimensional regu- 2 (2.19) larization approach, the common factor − γE + ln4π + 1 with = 4 − n can be lumped into the logarithmic term Since is of order 1 MeV [60], it can be neglected in u ln(2/m2). In conventional practice, one often chooses practical calculations. Note that in the above equation, one ∼ χ , the chiral symmetry breaking scale of order 1 GeV, should not replace M0 by the bare mass MD∗ . Indeed, in (s)0 to get numerical estimates of chiral loop effects. However, the absence of chiral loop corrections, M ∗ = M + v · k˜ = Ds0 0 as pointed out in [48], contrary to the common wisdom, the M + − 3 M + ˜ = M ∗ , as it should. 0 S 4 S s Ds0 renormalization scale has to be larger than the chiral sym- For the self-energy of the axial-vector meson, we con- metry breaking scale of order 1 GeV in order to satisfy the sider Ds1 as an illustration which receives contributions from on-shell conditions. In general, there exist two solutions for ∗0 +, ∗+ 0, ∗+η ∗0 +, ∗+ 0, ∗+η v · ˜ v · D K D K Ds and D0 K D0 K Ds0 inter- k due to two intercepts of the curve with the k axis. v · ˜ mediate states (see Fig. 1b). The full Ds1 propagator reads We shall consider the smaller solution for k as the other
−i(gμν − vμvν) , (2.20) 1 ˜ 2 2 2(v · k − S − MS − s) −[2 D∗ K (ωD∗ ) + D∗η(ωD∗ ) + 2 ∗ (ω ∗ ) + ∗ η(ω ∗ )] 4 3 s s D0 K D0 3 Ds0 Ds0
ω ∗ = v · + − ∗ − 1 ω = where D k MD MD MD and ∗ solution will yield too large masses. It could be that a higher- s1 4 D0 3 v · k + M − M ∗ − (M ∗ − MD) + MD. order heavy quark expansion needs to be taken into account Ds1 D0 D0 4 The loop integrals in Eqs. (2.13) and (2.15) obey the to justify the use of ∼ χ . expressions [59–61] In our previous study [48] we argued that near mass degen- eracy and the physical masses of D∗ and D∗ cannot be 2 ω s0 0 (ω) = 2h accounted for simultaneously in the approach of HMChPT. 2 2 fπ 32π In this work we show that near mass degeneracy can be imple- 2 m mented by taking into account the additional contributions × (m2 − 2ω2)ln − 2ω2 + 4ω2 F − m2 ω from axial-vector heavy mesons to the self-energy diagram Fig. 1a of scalar mesons. Since ωP ∼ MP∗ − MP +···, (2.21) 0 ω ∼ M ∗ − M − (M − MP ) +··· and M ∗ > MP , P P0 P1 P1 P0 and [49,50] 1 M ∗ < M , we find numerically that (ω ) contributes P0 P1 2 ω destructively to the mass shifts. Moreover, the self-energy of (ω) = 2g ∗ ∗ D0 (Ds0) is sensitive (insensitive) to the coupling g . There- f 2 32π 2 π fore, we can adjust the couplings h, g and the renormaliza- 2 ∗ ∗ 2 2 10 2 2 2 2 m μ × (3m −2ω )ln − ω +4m +4(ω − m )F − , tion scale to get a mass degeneracy of Ds0 and D0 .We m2 3 ω take the quark model of Godfrey et al. [7] as an illustration; (2.22) ∗ it predicts the bare masses of 2480 and 2400 MeV for Ds0 and D∗, respectively. We find that h = 0.51, g = 0.25 respectively, with 0 ⎧ and = 1.27 GeV will lead to MD∗ = 2222 MeV √ √ s0 2− ∗ ⎨ x 1 ( + 2 − ), | |≥ , and MD = 2225 MeV. By fixing h and and varying 1 √x ln x x 1 x 1 0 F = ( ∗ , ∗ ) = ( , ) 1−x2 π −1 x the coupling g , we obtain MD MD 2338 2241 , x ⎩− − tan √ , |x|≤1. s0 0 x 2 1−x2 (2277, 2235) and (2222, 2225) MeV for g = 0, 0.15 and (2.23) 0.25, respectively. This explains why we could not obtain 123 Eur. Phys. J. C (2017) 77 :668 Page 7 of 13 668
δ ≡ −M ∗ ∗0 Table 2 Mass shifts ( M M ) of heavy scalar mesons calculated focused on the closeness of the masses of Ds0 and D0 .The in HMChPT. The renormalization scale is taken to be = 1.3 (1.2) GeV ∗± ∗0 measured mass of D0 by LHCb is greater than D0 and for scalar D (B) mesons. Bare masses are taken from [7]. All masses ∗ Ds0 by an amount of the order of 30 MeV. More precise and widths are given in MeV and only the central values are listed here ∗0 measurement of the D0 mass is certainly needed. If it turns Meson Bare mass M δMM ∗0 ∗ out that the mass of D0 is larger than that of Ds0, then we ∗0 ∗ ∗ − need a larger |g | to render D heavier than D . (iv) There Ds0 2480 162 2318 0 s0 ∗ − exist other contributions which are not considered in Fig. 1, D0 2400 79 2321 ∗ 0 0 B∗ 5831 −136 5694 for example, the intermediate states D K , Dsη , ··· for s0 ∗ ∗ − the self-energy diagram of D . (v) The calculated masses B0 5756 55 5701 s0 − are sensitive to the choice of the renormalization scale .Of Ds1 2550 98 2452 − course, physics should be independent of the renormaliza- D1 2460 41 2419 − tion scale. However, this issue cannot be properly addressed Bs1 5857 85 5772 B 5777 −34 5743 in the phenomenological model discussed here. In view of 1 this, we should not rely on the chiral loop calculations to make quantitative predictions on the scalar meson masses. For this purpose, we will turn to heavy quark symmetry in ∗ ∗ near degeneracy of Ds0 and D0 in our previous work with Sect. 4. g = 0. Hence, although the coupling g, which characterizes ∗ the interaction of P0 P1 with the Goldstone boson, is small, it does play a crucial role for achieving near degeneracy. The 3 Comparison with other work in HMChPT absolute mass is not an issue as we can scale up the unknown mass M0 by an amount of 96 MeV for scalar and axial-vector + + ∗ Self-energy hadronic loop corrections to 0 and 1 heavy heavy mesons, i.e. P0a and P1a in Eq. (2.10), so that we have mesons have been discussed in the literature within the frame- MD∗ = 2318 MeV and MD∗ = 2321 MeV. s0 0 work of HMChPT [31,48–54]. Guo, Krewald and Meißner Using the same set of values for the parameters h and g (GKM) [31] considered three different models for calcu- but = 1.18 GeV for even-parity B mesons, we see from ∗ ∗ lations. Models I and III correspond to non-derivative and Table 2 that the closeness of Bs0 and B0 masses also holds derivative couplings of the scalar meson with two pseu- in the scalar B sector. The predicted masses for axial-vector doscalar mesons, while Model II is based on HMChPT. How- charmed mesons M = 2452 MeV and M = 2419 MeV Ds1 D1 ever, our HMChPT results are different from GKM. This can are in agreement with the respective measured masses (see be traced back to the loop integral (v · k), which has the . ± . ± Table 1): 2459 5 0 6 MeV and 2427 36 MeV. following expression in [31]: The coupling constant h at tree level can be extracted from ∗0, ∗± 0 2h2 m2 2 the measured widths of D0 D0 and D 1. We find the val- (ω) = −2ω ln − 2ω + 4G(ω, m) = . ± . ∗( )0 . ± . 2 π 2 2 ues h 0 60 0 07 from D0 2400 ,0514 0 017 from fπ 32 m ∗ ± ( ) . ± . ( )0 2 2 2 D0 2400 and 0 79 0 17 from D1 2430 .Itisobvi- 2h m m ∗ ± = − ω − ω + ωF − , ous that D (2400) gives the best determination of h as its 2 2 2 ln 2 2 4 0 fπ 32π m ω mass and width measured recently by LHCb [2,3] are more (3.1) ∗0 0 = . accurate than D0 and D 1. Our result of h 0 508 agrees well with the LHCb experiment. As for the coupling g, its where use of Eq. (2.24) has been made. Comparing with magnitude and even the sign relative to the coupling g in Eq. (2.21), we see that the m2 coefficient in Eq. (3.1) should Eq. (2.4) are unknown. A recent lattice calculation yields be replaced by ω2 and a term (m2/ω)ln(2/m2) in [···]is h = 0.84(3)(2) and g =−0.122(8)(6) [62]. missing. Contrary to the claim made by GKM, the self-energy Several remarks are in order. (i) The chiral loop calcula- contribution should not vanish in the chiral limit. Some other tions presented here are meant to demonstrate that the mass detailed comparisons are referred to [48]. Contributions from difference between strange heavy scalar mesons and their axial-vector heavy mesons to the self-energy of scalar mesons non-strange partners can be substantially reduced by self- were also not considered in this work. energy contributions. Near mass degeneracy in the scalar Mehen and Springer [50] have systematically studied chi- charm sector will imply the same phenomenon in the scalar B ral loop corrections to the masses of scalar and axial-vector system. (ii) For the bare masses of even-parity heavy mesons, heavy mesons. For the propagators of heavy mesons, they we have used the relativistic quark model of [7] as an illus- only keep the i/(2v · k) terms and neglect all mass splittings ˜ tration. If we use the quark model predictions from [8], near such as S, MP,S and residual masses such as a and a degeneracy can be implemented by employing h = 0.52 and in Eqs. (2.11) and (2.12). Then they set v · k = 0inside g = 0.28 and similar given before. (iii) So far we have the loop integral. As a consequence, ω = Mext − Mint,the 123 668 Page 8 of 13 Eur. Phys. J. C (2017) 77 :668
/( v · ) ∗ ∼ Table 3 Same as Table 2 except that the propagator i 2 k is used Bs0 is 8 MeV.However, the renormalization scale is chosen for all heavy meson states and v · k is set to be zero inside the loop to be = 317 MeV in [49], which is too small according to integral calculation the spirit of ChPT. Meson Bare mass M δMM
∗ − Ds0 2480 117 2363 D∗ 2400 −44 2356 ∗ 0 4 Masses of B(s)0 and B(s)1 mesons ∗ − Bs0 5831 63 5768 B∗ 5756 −27 5729 ∗ 0 The states B(s)0 and B(s)1 are yet to be observed. The predic- − Ds1 2550 48 2502 tions of their masses in the literature are collected in Table 4. − D1 2460 22 2438 They can be classified into the following categories: (i) rela- − Bs1 5857 40 5817 tivistic quark potential model [7,8,63–67], (ii) nonrelativistic − B1 5777 9 5768 quark model [68,69], (iii) heavy meson chiral perturbation theory [48,70–72], (iv) unitarized chiral perturbation theory [73–78], (v) lattice QCD [79–82], (vi) potential model with mass difference between external and internal heavy meson one loop corrections [64,83,84], (vii) QCD sum rules [85], states. If we follow this prescription and repeat the calcula- and (viii) others, such as the nonlinear chiral SU(3) model tions presented in the last section, we will obtain different [86,87], the semi-relativistic quark potential model [88], the results. The physical masses of the scalar charmed mesons MIT bag model [89], the mixture of conventional P-wave read quark–antiquark states with four-quark components [90], the chiral quark–pion Lagrangian with strong coupled channels M ∗ = M ∗ + (v · k) ∗ , M ∗ = M ∗ + (v · k) ∗ , Ds0 Ds0 Ds0 D0 D0 D0 [91], heavy quark symmetry and the assumption of flavor (3.2) independence of mass differences between 0+ and 0− states in [93]. with (see Eq. (2.18)) It is clear from Table 4 that the mass difference between 1 2 strange and non-strange scalar B mesons predicted by the (v · k)D∗ = Re 2 DK(ωD) + D η(ωD ) s0 2 3 s s relativistic quark models is of order 60–110 MeV, and that ∗ 2 Bs0 is above the BK threshold. As shown in Sect. 2,the + (ω ) + (ω ) , ∗ ∗ 2 D K D D η D 1 1 3 s1 s1 closeness of the B and B masses is expected in view of 0 s0 1 3 the near degeneracy observed in the charm sector. (v · k)D∗ = Re Dπ (ωD) Since this section based on heavy quark symmetry was 0 2 2 already discussed in detail in our previous work [48], we 1 + Dη(ωD) + D K (ωD ) shall recapitulate the main points and update the numerical 6 s s ∗ results as the masses of charged and neutral D0 under the 2 1 + (ω ) + (ω ) current measurement are somewhat different. D π D D η D 3 1 1 6 1 1 λS In the heavy quark limit, the two parameters S and 2 + (ω ) , defined in Eqs. (2.7) and (2.9), respectively, are indepen- D (3.3) Ds K s1 1 dent of the heavy quark flavor, where S measures the spin- averaged mass splitting between the scalar doublet (P, P∗) and ωD = M ∗ −MD, ω = M ∗ −M , ··· etc. The 1 0 D(s)0 D1 D(s)0 D1 ∗ and the pseudoscalar doublet (P , P) and λS is the mass results of calculations are summarized in Table 3.Itisclear ∗ that while near degeneracy is achieved for scalar charmed splitting between spin partners, namely P1 and P0 ,ofthe mesons, it is not the case for scalar B mesons. scalar doublet. From the data listed in Table 1,weareledto Note that in the work of Mehen and Springer [50] and (cu¯) = ± ,λS(cu¯) = ( ± )2 Alhakami [49], the bare mass M in the mass relation M = S 426 34 MeV 2 360 95 MeV M+ is expressed in terms of the unknown low-energy con- (4.1) stants in the effective chiral Lagrangian which are fitted to the measured masses of odd- and even-parity charmed mesons. for the scalar cu¯ meson, = M + 1 In our work, we have M 2 Re (see Eq. (2.3)) with M being inferred from the quark model. We use full prop- ( ¯) ( ¯) cd = 434 ± 29 MeV,λS cd = (302 ± 75 MeV)2 agators to evaluate the mass shifts. Based on the HMChPT S 2 framework of Mehen and Springer [50], Alhakami [49] has (4.2) shown that the non-strange and strange scalar B mesons are ∗ ¯ nearly degenerate and the mass difference between B0 and for the scalar cd meson and 123 Eur. Phys. J. C (2017) 77 :668 Page 9 of 13 668
∗ Table 4 Predicted masses (in MeV) of B(s)0 and B(s)1 mesons in the literature ∗ ∗ B0 B1 Bs0 Bs1 Relativistic quark model Di Pierro et al. [8] 5706 5742 5804 5842 Godfrey et al. (2016) [63] 5720 5738 5805 5822 Lakhina et al. [64] 5730 5752 5776 5803 Liu et al. [65] 5749 5782 5815 5843 Ebertetal. [66] 5749 5774 5833 5865 Sun et al. [67] 5756 5779 5830 5858 Godfrey et al. (1991) [7] 5756 5777 5831 5857 Nonrelativistic quark model Lu et al. [68] 5683 5729 5756 5801 Ortega et al. [69] 5741 5858 HMChPT Bardeen et al. [70] 5627 ± 35 5674 ± 35 5718 ± 35 5765 ± 35 Colangelo et al. [71,72] 5708.2 ± 22.5 5753.3 ± 31.1 5706.6 ± 1.2 5765.6 ± 1.2
Cheng et al. [48] 5715 ± 22 + δS 5752 ± 31 + δS 5715 ± 1 + δS 5763 ± 1 + δS Unitarized ChPT Torres-Rincon et al. [73] 5530 5579 5748 5799 Guo et al. [74,75] Two poles 5725 ± 39 5778 ± 7 Cleven et al. [76] 5625 ± 45 5671 ± 45 Altenbuchinger et al. [77] 5726 ± 28 5778 ± 26 Cleven et al. [78] 5696 ± 40 5742 ± 40 Lattice Gregory et al. (HPQCD) [79] 5752 ± 30 5806 ± 30 Koponen et al. (UKQCD) [80,81] 5760 ± 9 5807 ± 9 Lang et al. [82] 5713 ± 22 5750 ± 26 Chiral loop corrections I. W. Lee et al. [83] 5637 5673 5634 5672 Albaladejo et al. [84] 5709 ± 8 5755 ± 8 QCD sum rules Z. G. Wang [85] 5720 ± 50 5740 ± 50 5700 ± 60 5760 ± 60 Others Kolomeitsev et al. [86,87] 5526 5590 5643 5690 Matsuki et al. [88] 5592 5649 5617 5682 Orsland et al. [89] 5592 5671 5667 5737 Vijande et al. [90] 5615 5679 5713 Badalian et al. [91] 5675 ± 20 5725 ± 20 5710 ± 15 5730 ± 15 Lähde et al. [92] 5678 5686 5781 5795 Dmitrašinovi´c[93] 5718 ± 25 5732 ± 25 5719 ± 25 5765 ± 25
(cs¯) = . ± . ,λS(cs¯) = ( . ± . )2 S 347 7 0 6MeV 2 411 5 1 7MeV the experimental difficulty in identifying the broad states. The λS (4.3) light quark flavor dependence of S and 2 shown in Eqs. (4.1)–(4.3) indicates SU(3) breaking effects. for the cs¯ meson, where we have assumed that the charged Note that the parameter S can be decomposed in terms of ± 0 + − D1 has the same mass as D1 . The large errors associated the mass differences between 0 and 0 states and between λS + − with S and 2 in the non-strange scalar meson sector reflect 1 and 1 states defined by
123 668 Page 10 of 13 Eur. Phys. J. C (2017) 77 :668
(cs¯) (cq¯) ≡ M ∗ − M ,≡ M ∗ − M , and the masses (in MeV) 0 Ds0 Ds 0 D0 D (cs¯) (cq¯) ≡ M − MD∗ ,≡ M − MD∗ . (4.4) M( ∗)0 = 5711 ± 49, M( )0 = 5748 ± 39, 1 Ds1 s 1 D1 B0 B1 M(B∗)± = 5728 ± 38, M( )± = 5754 ± 32, We have 0 B1 MB∗ = 5715 ± 1, M = 5763 ± 1. (4.12) ( ¯) 1 ( ¯) ( ¯) (cq¯) 1 (cq¯) (cq¯) s0 Bs1 cs = 3 cs + cs , = 3 + . S 4 1 0 S 4 1 0 Comparing with Eq. (4.9) we see that QCD corrections will (4.5) ∗ ∗ mainly enhance the masses of B0 and Bs0 by an amount of afewMeV. If we follow [93] assuming the heavy flavor independence of ( ) ( ) We next proceed to the corrections to b = c .In 0 and 1, namely, S S heavy quark effective theory, it follows that (see [48]for ( ¯) ( ¯) ( ¯) ( ¯) bs = cs ,bq = cq , details) 0 0 0 0 (bs¯) = (cs¯),(bq¯) = (cq¯), 1 1 1 1 (4.6) (b) (c) (c) S H 1 1 = + δS ≡ + (λ − λ ) − . S S S 1 1 2m 2m we will obtain (in MeV) c b (4.13) ∗ = ± , = ± , M(B )0 5733 29 M(B )0 5745 36 λH 0 1 While the parameter 1 is calculable, the other parameter, M(B∗)± = 5765 ± 6, M( )± = 5741 ± 36, λS / 0 B1 1 , is unknown. In [48]the1 m Q correction is estimated to M ∗ = 5716.2 ± 0.6, M = 5762.7 ± 1.8. (4.7) be Bs0 Bs1 ∗0 0 δS ∼ O(−35) MeV, (4.14) Our results for B0 and B1 masses are slightly different from those shown in the last row of Table 4. We see that the central to be compared with the estimate of −50 ± 25 MeV in [50]. ∗0 ∗ values of B0 and Bs0 are not very close. It appears that the As pointed out in [48], so long as δS is not large in (b) (c) assumption of heavy quark flavor independence of 0 and / = magnitude, the 1 m Q correction to the relation S S 1 is probably too strong. Hence, we shall follow [71,72] [see Eq. (4.13)], to a very good approximation, amounts to ∗ assuming the heavy quark flavor independence of S and lowering the masses of B( ) and B( ) by an equal amount λS s 0 s 1 2 , of |δS|:
(bq¯) (cq¯) (bs¯) (cs¯) ∗ = ± + δ , = ± + δ , = , = M(B )0 5711 49 MeV S M(B )0 5748 39 MeV S S S S S 0 1 ( ¯) ( ¯) ( ¯) ( ¯) S bq S cq S bs S cs ∗ ± = ± + δ , = ± + δ , λ = λ ,λ = λ . M(B ) 5728 38 MeV S M(B )± 5754 32 MeV S 2 2 2 2 (4.8) 0 1
M ∗ = 5715 ± 1 MeV + δS, M = 5763 ± 1 MeV + δS. In this case, we find the results (in MeV) Bs0 Bs1 (4.15) ∗ = ± , = ± , M(B )0 5705 52 M(B )0 5751 40 0 1 ∗ ∗ ∗ ± = ± , = ± , Evidently, the mass degeneracy between B and B shown M(B ) 5724 41 M(B )± 5755 34 s0 0 0 1 in Eq. (4.9) implied by heavy quark symmetry is not spoiled ∗ = ± , = ± . MB 5707 1 MB 5766 1 (4.9) s0 s1 by 1/m Q and QCD corrections. Predictions along this approach were first obtained in [71, Several remarks are in order. (i) There exists an empirical 72]. Evidently, mass degeneracy in the scalar D sector will mass relation repeat itself in the B system through heavy quark symmetry. − ∗ ∗ − MD MD MD MD Scrutiny of Table 4 shows that this degeneracy is respected s1 s0 M ∗ − M ,(141.8, 141.4, 143.8) MeV (4.16) only in few of the predictions such as [48,49,71,72,83,93]. Ds Ds Though the relations in Eq. (4.8) are valid in the heavy for charmed mesons. Therefore, one will expect the mass quark limit, they do receive 1/m Q and QCD corrections. difference relation ( ¯) ( ¯) The leading QCD correction to the relation λS bq = λS cq 2 2 − ∗ ∗ − ∗ − MB MB MB MB MB MBs (4.17) is given by [94] s1 s0 s 9/25 also valid in the B sector. The mass difference between B ( ¯) ( ¯) α (m ) s1 λS bq = λS cq s b , and B∗ in Eq. (4.15) is independent of δ ; its numerical 2 2 α ( ) (4.10) s0 S s mc value 48 ±1 MeV indeed respects Eq. (4.17). It is interesting and a similar relation with the replacement of q¯ by s¯. We then that this mass relation is not obeyed by any of the existing obtain relativistic quark models which predict a smaller mass dif- − ∗ ∼ ( ¯) ( ¯) ference MB MB 20–30 MeV (see Table 4). (ii) In λS bu = ( ± )2,λS bd = ( ± )2, s1 s0 2 326 86 MeV 2 273 68 MeV − ∗ > the charm sector we have the hierarchy MD MD S(bs¯) s1 s0 λ = (372.6 ± 1.5MeV)2, (4.11) M − M ∗ . It is natural to generalize this to the B sector: 2 D1 D0 123 Eur. Phys. J. C (2017) 77 :668 Page 11 of 13 668
∗ − ∗ > − ∗ 0 MB MB MB MB . Indeed, the mass splittings of D0 can be implemented by adjusting the strong cou- s1 s0 1 0 the j = 1 doublet in strange and non-strange sectors satisfy plings h and g and the renormalization scale .This 2 ∗ ∗0 the relation (see [48]) in turn implies the mass similarity of Bs0 and B0 .The ∗ P0 P1 interaction with the Goldstone boson characterized − ∗ S(bs¯) − ∗ S(cs¯) MB MB λ MD MD λ s1 s0 = 2 , s1 s0 = 2 . by the strong coupling g is crucial for understanding the ( ¯) ( ¯) M − M ∗ λS bq M − M ∗ λS cq B B0 D D0 phenomenon of near degeneracy. 1 2 1 2 • (4.18) The self-energy loop diagram has been evaluated in the literature by neglecting mass splittings and residual The relation M − M ∗ > M − M ∗ is not respected Bs1 Bs0 B1 B0 masses in the heavy meson’s propagator. If we follow this by some of the models listed in Table 4. (iii) The differ- prescription, we find near degeneracy in the scalar charm ence 15 MeV between the central values of M and M Bs1 B1 sector but not in the corresponding B system. ∗ inferred from Eq. (4.15) is much smaller than the quark • The masses of B( ) and B( ) mesons (i.e. Eq. (4.15)) 3 s 0 s 1 model expectation of 60–100 MeV (cf. Table 4). Just can be deduced from the charm spectroscopy and heavy as the scalar B mesons, the mass splitting is reduced by quark symmetry with corrections from QCD and 1/m Q the self-energy effects due to strong coupled channels. (iv) effects, for example, M ∗ = (5715 ± 1) MeV + δ , Bs0 S The current world-averaged masses of neutral and charged = ( ± ) + δ δ / MB 5763 1 MeV S with S being 1 m Q ∗( ) s1 D0 2400 mesons are not the same; they differ by an amount corrections. We found that the mass difference of 48 MeV of 33 ± 29 MeV and the charged one is better measured. On ∗ ∼ between Bs1 and Bs0 is larger than the result of 20 the experimental side, it is thus important to have a more 30 MeV predicted by the quark models, whereas the dif- precise mass measurement of the scalar charmed meson. If ference of 15 MeV between the central values of MB ∗0 s1 it turns out the D0 mass is also of order 2350 MeV, we find and M is much smaller than the quark model expecta- ∗ ∗ B1 from Eq. (4.15) that the B0 mass is larger than that of Bs0 by tion of 60–100 MeV. an order of 13 MeV. • The current world-averaged masses of neutral and charged ∗ ∗ Since the masses of Bs0 and Bs1 are below the BK and D (2400) mesons are not the same; they differ by an ∗ 0 B K thresholds, respectively, their widths are expected to amount of 33±29 MeV and the charged one is more well be very narrow with the isospin-violating strong decays into measured. Experimentally, it is thus important to have a π 0 ∗π 0 Bs and Bs . Experimentally, it will be even more diffi- more precise mass measurement of the scalar charmed ∗ cult to identify the non-strange B and B mesons owing to ∗0 0 1 meson, especially the neutral one. If it turns out the D0 their broad widths. ∗ mass is of order 2350 MeV,this means D0 is heavier than ∗ Ds0 even though the latter contains a strange quark. In ∗ this case, we find that the B0 mass is larger than that of 5 Conclusions ∗ Bs0 by an order of 13 MeV.
The qq¯ quark model encounters a great challenge in describ- Acknowledgements We wish to thank Ting-Wai Chiu and Keh-Fei ing even-parity mesons. Specifically, it has difficulties in Liu for insightful discussions. This research was supported in part by understanding the light scalar mesons below 1 GeV, scalar the Ministry of Science and Technology of R.O.C. under the Grant No. and axial-vector charmed mesons and 1+ charmonium-like 104-2112-M-001-022 and by the National Science Foundation of China state X(3872). Many studies indicate that quark model under the Grant No. 11347027. results are distorted by the interaction of the even-parity Open Access This article is distributed under the terms of the Creative meson with strong coupled channels. In this work, we focus Commons Attribution 4.0 International License (http://creativecomm ∗ ons.org/licenses/by/4.0/), which permits unrestricted use, distribution, on the near mass degeneracy of scalar charmed mesons, Ds0 and D∗0, and its implications to the B sector. and reproduction in any medium, provided you give appropriate credit 0 to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. • We work in the framework of HMChPT to evaluate the Funded by SCOAP3. ∗ self-energy diagrams for P0 and P1 heavy mesons. The approximate mass degeneracy can be qualitatively under- stood as a consequence of self-energy hadronic loop cor- References rections which will push down the mass of the heavy scalar meson in the strange sector more than that in the 1. C. Patrignani et al., Particle Data Group, Chin. Phys. C 40, 100001 non-strange one. Quantitatively, the closeness of D∗ and (2016) s0 2. R. Aaij et al., LHCb Collaboration, Phys. Rev. D 92, 012012 (2015). arXiv:1505.01505 [hep-ex] 3 The splitting between strange and non-strange axial-vector bottom 3. R. Aaij et al., Phys. Rev. D 92, 032002 (2015). arXiv:1505.01710 mesons is of order 19 MeV in the HHChPT approach of [49]. [hep-ex] 123 668 Page 12 of 13 Eur. Phys. J. C (2017) 77 :668
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