Structure of Mass Gap between Two Spin Multiplets
aa r X i v : . [ h e p - ph ] D ec TKU-07-2 Oct. 2007
Structure of Mass Gap between Two Spin Multiplets
Takayuki Matsuki ∗ Tokyo Kasei University, 1-18-1 Kaga, Itabashi, Tokyo 173-8602, JAPAN
Toshiyuki Morii † Graduate School of Science and Technology, Kobe University,Nada, Kobe 657-8501, JAPAN
Kazutaka Sudoh ‡ Institute of Particle and Nuclear Studies,High Energy Accelerator Research Organization,1-1 Ooho, Tsukuba, Ibaraki 305-0801, JAPAN (Dated: October 1, 2007)Studying our semirelativistic potential model and the numerical results, which succeeds in pre-dicting and reproducing recently discovered higher resonances of D , D s , B , and B s , we find a simpleexpression for the mass gap between two spin multiplets of heavy-light mesons, (0 − , − ) and (0 + , + ).The mass gap between chiral partners defined by ∆ M = M (0 + ) − M (0 − ) and/or M (1 + ) − M (1 − )is given by ∆ M = M (0 + ) − M (0 − ) = M (1 + ) − M (1 − ) ≈ Λ Q − m q in the limit of heavy quark sym-metry, and including 1 /m Q corrections, we have ∆ M ≈ Λ Q − m q + (1 . × + 4 . × · m q ) /m Q with Λ Q ≈
300 MeV, a light quark mass m q , and a heavy quark mass m Q . This equation holds bothfor D and D s heavy mesons. Our model calculations for the B and B s also follow this formula. PACS numbers: 12.39.Hg, 12.39.Pn, 12.40.Yx, 14.40.Lb, 14.40.NdKeywords: potential model; spectroscopy; heavy mesons
I. INTRODUCTION
The discovery of the narrow D sJ particles by BaBar [1] and CLEO [2] and soon confirmed by Belle [3] immediatelyreminded people an effective theory approach proposed by Nowak et al. and others [4, 5, 6, 7]. They constructedan effective Lagrangian for heavy mesons from the Nambu-Jona-Lasinio type four-fermi interactions and combined itwith 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 − , − ) and (0 + , + ), are degenerate in the limit in which the chiralsymmetry is an exact symmetry of the vacuum and the heavy quark symmetry is exactly realized. From this effectivetheory, 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 bearound ∆ M = g π f π ≈
349 MeV, where g π is the coupling constant for 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 theexperiments (around 350 MeV), people thought that underling physics may be explained by their SU (3) effectiveLagrangian [8, 9]. However, when (0 + , + ) for D meson were found by Belle and FOCUS, and later reanalyzed byCLEO, their explanation needs to be modified even though some people still study in this direction; in fact, theeffective 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 . Inother words, the forumula can be applied equally for any of these heavy meson multiplets. Thus, it is required tofind the mass gap formula, if it exists, which agrees well with the experiments and explains the physical ground of itsformula.In this paper, using our semirelativistic potential model, we first give our formula for the mass gap between chiralpartners 0 + (1 + ) and 0 − (1 − ) for any heavy meson, D , D s , B , and B s , among which the known mass gaps, i.e., theones for D and D s , agree well with the experiments although there is some ambiguities for D meson data. Next weshow how this mass gap depends on a light quark mass m q for q = u, d , and s , where we neglect the difference between ∗ E-mail: [email protected] † E-mail: [email protected] ‡ E-mail: [email protected] u and d quarks. Our formula naturally explain that the mass gap for D is larger than that for D s and predict themass 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] M X ( j P ) = m Q + E k ( m q ) + O (1 /m Q ) , (1)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 | − | k | , P = k | k | ( − | k | +1 , E k ( m q ) = E ( j P , m q ) . (2)To begin with, we study the heavy meson mass without 1 /m Q corrections so that we can see the essence of the massgap. States with the same | k | value are degenerate in a pure chiral limit and without confining scalar potential, whichis defined as m q → S ( r ) → S ( r ) is turned on and confines quarksinto heavy mesons but keeping vanishing light quark mass intact. In fact, in this limit our model gives the mass gapbetween two spin multiplets ∆ M ≈
300 MeV as follows;∆ M = E (1 + , − E (1 − ,
0) = E (0 + , − E (0 − ,
0) = 295 . D, and D s , = 309 . B, and B s , (3)This gap is mainly due to gluon fields which confines quarks into heavy mesons. It is interesting that obtained valuesare close to Λ QCD ≈
300 MeV. Next, turning on a light quark mass which explicitly breaks a chiral symmetry, wehave SU (3) flavor breaking pattern of the mass levels, i.e., mass of D becomes different from that of D s with the samevalue of j P . Since we assume m u = m d , there still remains SU (2) iso-spin symmetry. Note that even after chiralsymmetry 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 from0 to 0.2 GeV, we have calculated the m q dependence of ∆ M and have obtained Fig. 1, in which ∆ M is linearlydecreasing with m q . From Fig. 1, we find that the mass gap between two spin multiplets for a heavy meson X canbe written as ∆ M = M X (0 + ) − M X (0 − ) = M X (1 + ) − M X (1 − ) = g Λ Q − g m q , (4)Λ Q = 300 MeV , (cid:26) g = 0 . , g = 1 . , for D/D s g = 1 . , g = 1 . , for B/B s , (5)where the values of g , and g are estimated by fitting the optimal line with Fig. 1. Since both g and g are veryclose to 1, we conclude that the mass gap is essentially given by∆ M = Λ Q − m q (6)Though the physical ground of this result is out of scope at present, Eq. (6) is serious, since it is very different fromthe 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 correctionscan be given by Eq. (1) with an eigenvalue E k being given by the following eigenvalue equation. m q + S + V − ∂ r + kr ∂ r + kr − m q − S + V ! u k ( r ) v k ( r ) ! = E k u k ( r ) v k ( r ) ! . (7) m q [GeV] ∆ M ( m q ) [ G e V ] charmbottom m u m s FIG. 1: Plot of the mass gap between two spin multiplets. Light quark mass dependence is given. The horizontal axis is lightquark mass m q and the vertical axis is the mass gap ∆ M .TABLE I: Optimal values of parameters.Parameters α cs α bs a (GeV − ) b (GeV)0.261 ± ± ± ± m u,d (GeV) m s (GeV) m c (GeV) m b (GeV)0.0112 ± ± ± ± χ /d.o.f18 8 107.55 Using this equation, the mass gap between k = +1 and k = −
1, which are corresponding to the spin multiplets(0 − , − ) and (0 + , + ), respectively, when they are degenerate, is re-expressed as∆ M = M (0 + ) − M − (0 − ) = M (1 + ) − M − (1 − )= Z d x πr ( Φ † ( r ) m q + S + V − ∂ r + r ∂ r + r − m q − S + V ! Φ ( r ) − Φ †− ( r ) m q + S + V − ∂ r − r ∂ r − r − m q − S + V ! Φ − ( r ) ) = Z dr h Φ † ( r ) K Φ ( r ) − Φ †− ( r ) K − Φ − ( r ) i + m q Z dr h Φ † ( r ) β Φ ( r ) − Φ †− ( r ) β Φ − ( r ) i . (8)From this equation we can see that the mass gap linearly depends on m q . Here the radial wave function Φ k ( r ) andthe massless free kinetic term K k with the quantum number k are given byΦ k ( r ) = u k ( r ) v k ( r ) ! , K k = S ( r ) + V ( r ) − ∂ r + kr ∂ r + kr − S ( r ) + V ( r ) ! . (9)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) canbe explained or we can intuitively understand this result in our formulation as follows. The quantum numbers k = − TABLE II: Degenerate masses of model calculations and their mass gap between 0 + (1 + ) and 0 − (1 − ) for n = 1. M ( D ) M ( D s ) M ( B ) M ( B s )0 − / − + / + + (1 + ) − − (1 − ) 283 195 293 204 and k = 1 correspond to ℓ = 0 and ℓ = 1 respectively, where ℓ is the angular momentum of a light antiquark relativeto a heavy quark as can be seen from Table I of Ref.[14]. An excited state with ℓ = 1 ( k = +1) is more relativisticcompared with the one with ℓ = 0 ( k = − v ( r ) becomes larger than v − ( r ) sincethey are normalized as ( u k ) + ( v k ) = 1. Hence ( u ) − ( v ) = Φ † ( r ) β Φ ( r ) becomes smaller than Φ †− ( r ) β Φ − ( r ).Thus the coefficient of m q becomes negative. As a matter of fact, linear m q dependence of ∆ M is not yet definitesince radial wave functions u k and v k are also dependent on m q . However, looking at Eq. (4) or Fig. 1 which are thenumerical calculation of our model, we can say that implicit dependence on m q of these wave functions is numericallysmall. Thus the above physical and intuitive interpretation of linear m q dependence of ∆ M 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 heavymesons occurs differently from what people in [4, 5, 6] originally considered. Let us briefly explain the mechanismthat these authors considered as a generation of the mass gap, which is due to the paper [5]. The Lagrangian for thechiral multiplets, which couples to the heavy quark sector, can be written as follows. L chiral = ¯ ψ ( i/∂ − m q ) ψ − g ¯ ψ L Σ ψ R − g ¯ ψ R Σ † ψ L −
12 Λ Tr (cid:0) Σ † Σ (cid:1) , (10)where ψ is the chiral quark field with three flavors and Σ is the 3 × ψ T = ( u, d, s ) , Σ = 12 σI + iπ a λ a , (11)When this Lagrangian is combined with the effective theory for heavy hadrons, the effective mass of a constituentquark is given by h σ i + m q . Then the mass gap is given by∆ M = g π ( h σ i + m q ) . (12)where g π is the Yukawa coupling constant between the heavy meson and a chiral multiplet and is taken to be g π = 3 . h σ i = f π . This expression is obtained in the heavy quark symmetric limit and should be compared withour Eq. (6). Instead of minus sign for the term m q that we obtained, the authors of [5] obtained plus sign as shownin the above equation. The same result is obtained even if we use the nonlinear Σ model [8]. IV. /m Q CORRECTIONS
Next let us study the case when 1 /m Q corrections to the mass gap are taken into account. Part of the results isgiven in [15]. In Table III, we give our numerical results in the cases of n = 1 and n = 2 (radial excitations). Valuesin brackets are taken from the experiments. Our values seem to agree with the experimental ones though the fit isnot as good as the case for the absolute values of heavy meson masses. We assume the form of the mass gap with the1 /m Q corrections as follows. ∆ M = ∆ M + c + d · m q m Q . (13)Using Eq.(4) for D and D s mesons, i.e. ∆ M = g Λ Q − g m q = 295 . − . m q , we obtain the values of theparameters c and d for D/D s mesons given in Table III, which are given by c = 1 . × MeV , d = 4 . × MeV . (14) TABLE III: Model calculations of the mass gap. Values in brackets are taken from the experiments. Units are MeV.Mass gap ( n = 1) ∆ M ( D ) ∆ M ( D s ) ∆ M ( B ) ∆ M ( B s )0 + − −
414 (441) 358 (348) 322 2391 + − −
410 (419) 357 (348) 320 242( n = 2) ∆ M ( D ) ∆ M ( D s ) ∆ M ( B ) ∆ M ( B s )0 + − −
308 274 206 1601 + − −
350 327 216 171TABLE IV:
D/D s meson mass spectra for both the calculated and experimentally observed ones. Units are MeV. s +1 L J ( J P ) M calc ( D ) M obs ( D ) M calc ( D s ) M obs ( D s ) S (0 − ) 1869 1867 1967 1969 S (1 − ) 2011 2008 2110 2112 P (0 + ) 2283 2308 2325 2317” P ”(1 + ) 2421 2427 2467 2460TABLE V: B/B s meson mass spectra for both the calculated and experimentally observed ones. Units are MeV. s +1 L J ( J P ) M calc ( B ) M obs ( B ) M calc ( B s ) M obs ( B s ) S (0 − ) 5270 5279 5378 5369 S (1 − ) 5329 5325 5440 − P (0 + ) 5592 − − ” P ”(1 + ) 5649 − − The term c/m Q lifts the constant g Λ Q about 100 MeV and the term d/m Q gives deviation from -1 to the coefficientfor 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. B (0 + ) − B (0 − ) ≈ B (1 + ) − B (1 − ) ≈ , B s (0 + ) − B s (0 − ) ≈ B s (1 + ) − B s (1 − ) ≈
240 MeV , (15)which should be compared with our model calculations, 321 and 241 MeV, in Table III. Thus the linear dependence ofthe 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 < . 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, onefinds that mass differences between D and D s becomes smaller compared with those of the 0 − and 1 − states. Thiscan be seen from Table IV and was first discussed in Ref.[16] by Dmitraˇsinovi´c. He claimed that considering D sJ asa four-quark state, one can regard this phenomena as flavor SU (3) recovery. However, in our interpretation, this isnot 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 − / − state to the 0 + / + state, the mass of D s meson is elevated by about 100 MeV smaller than that of 0 − / − 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 whenthe transition from 0 − / − to 0 + / + occurs, and their values remain to be m u ( d ) = 11 . m s = 92 . Mass Gap of Heavy Baryons –
When we apply our formula to the heavy-light baryons which include two heavyquarks, ( ccs ), ( ccu ), ( bcs ), ( bcu ), ( bbs ), and ( bbu ), mass gaps between two pairs of baryons, like ( ccs ) and ( ccu ), willbe given by Eq. (6) in the heavy quark symmetric limit and by Eq. (13) with 1 /m Q corrections where we have toreplace m Q with m Q + m Q . Here the isospin symmetry is respected since in our model m u = m d . This speculation m q [GeV] ∆ M ( m q ) [ G e V ] + - 0 - (charm)1 + - 1 - + - 0 - (bottom)1 + - 1 - FIG. 2: Plot of the mass gap between two spin multiplets. Light quark mass dependence is given. The horizontal axis is lightquark mass m q and the vertical axis is the mass gap ∆ M . 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]. [1] BaBar Collaboration, B. Aubert et al., Phys. Rev. Lett. , 242001 (2003).[2] CLEO Collaboration, D. Besson et al., Phys. Rev. D , 032002 (2003);[3] Belle Collaboration, P. Krokovny et al. , Phys. Rev. Lett. , 262002 (2003); Y. Mikami et al. , Phys. Rev. Lett. , 012002(2004).[4] M. A. Nowak, M. Rho, and I. Zahed, Phys. Rev. D , 4370 (1993).[5] W. A. Bardeen and C. T. Hill, Phys. Rev. D , 409 (1994).[6] D. Ebert, T. Feldmann, R. Friedrich and H. Reinhardt, Nucl. Phys. B , 619 (1995); D. Ebert, T. Feldmann, and H.Reinhardt, Phys. Lett. B , 154 (1996).[7] A. Deandrea, N. Di Bartolomeo, R. Gatto, G. Nardulli, and A. D. Polosa, Phys. Rev. D , 034004 (1998).[8] W. A. Bardeen, E. J. Eichten, and C. T. Hill, Phys. Rev. D , 054024 (2003).[9] M. Harada, M. Rho, and C. Sasaki, Phys. Rev. D , 074002 (2004).[10] Belle Collaboration, K. Abe et al. , Phys. Rev. D , 112002 (2004).[11] T. Matsuki and T. Morii, Phys. Rev. D , 5646 (1997); See also T. Matsuki, Mod. Phys. Lett. A , 257 (1996), T.Matsuki and T. Morii, Aust. J. Phys. , 163 (1997), T. Matsuki, T. Morii, and K. Seo, Trends in Applied Spectroscopy, , 127 (2002); T. Matsuki, T. Morii, and K. Sudoh, AIP Conference Proceedings, Vol. 814, p.533 (2005), a talk given by T.Matsuki at the ”XI International Conference on Hadron Spectroscopy” (Hadron05), held at Rio De Janeiro, Brazil 21-26Aug. 2005; hep-ph/0510269 (2005).[12] T. Matsuki, K. Mawatari, T. Morii, and K. Sudoh, hep-ph/0408326.[13] T. Matsuki, K. Mawatari, T. Morii, and K. Sudoh, Phys. Lett. B , 329 (2005).[14] T. Matsuki, T. Morii, and K. Sudoh, Prog. Theor. Phys. , 1077 (2007), ; hep-ph/0605019.[15] T. Matsuki, T. Morii, and K. Sudoh, Eur. Phys. J. C , 701 (2007), a talk given by T. Matsuki at the IV-th InternationalConference on Quarks and Nuclear Physics (QNP06), held at Madrid, Spain 5-10 June 2006; hep-ph/0610186.[16] V. Dmitraˇsinovi´c, Phys. Rev. Lett. , 162002 (2005).[17] M. Savage and M. B. Wise, Phys. Lett. B , 177 (1990).[18] T. Ito, T. Morii, and M. Tanimoto, Z. Phys. C59