I Introduction

September, 2017

Masses of Scalar and Axial-Vector Mesons Revisited

Hai-Yang Cheng, Fu-Sheng Yu

Taipei, Taiwan 115, Republic of China

School of Nuclear Science and Technology, Lanzhou University

Lanzhou 730000, People’s Republic of China

Abstract

The SU(3) quark model encounters a great challenge in describing even-parity mesons. Specifically, the quark model has difficulties in understanding the light scalar mesons below 1 GeV, scalar and axial-vector charmed mesons and charmonium-like state . A common wisdom for the resolution of these difficulties lies on the coupled channel effects which will distort the quark model calculations. In this work, we focus on the near mass degeneracy of scalar charmed mesons, and , and its implications. Within the framework of heavy meson chiral perturbation theory, we show that near degeneracy can be qualitatively understood as a consequence of self-energy effects due to strong coupled channels. Quantitatively, the closeness of and masses can be implemented by adjusting two relevant strong couplings and the renormalization scale appearing in the loop diagram. Then this in turn implies the mass similarity of and mesons. The interaction with the Goldstone boson is crucial for understanding the phenomenon of near degeneracy. Based on heavy quark symmetry in conjunction with corrections from QCD and effects, we obtain the masses of and mesons, for example, , with being corrections. We find that the predicted mass difference of 48 MeV between and is larger than that of MeV inferred from the relativistic quark models, whereas the difference of 15 MeV between the central values of and is much smaller than the quark model expectation of MeV. Experimentally, it is important to have a precise mass measurement of mesons, especially the neutral one, to see if the non-strange scalar charmed meson is heavier than the strange partner as suggested by the recent LHCb measurement of the .

## I Introduction

Although the SU(3) quark model has been applied successfully to describe the properties of hadrons such as pseudoscalar and vector mesons, octet and decuplet baryons, it often encounters a great challenge in understanding even-parity mesons, especially scalar ones. Take vector mesons as an example and consider the octet vector ones: . Since the constituent strange quark is heavier than up or down quark by 150 MeV, one will expect the mass hierarchy pattern which is borne out by experiment. However, this quark model picture faces great challenges in describing the even-parity meson sector:

• Many scalar mesons with masses lower than 2 GeV have been observed and they can be classified into two nonets: one nonet with mass below or close to 1 GeV, such as (or ), (or ), and and the other nonet with mass above 1 GeV such as , and two isosinglet scalar mesons. Of course, the two nonets cannot be both low-lying states simultaneously. If the light scalar nonet is identified with the P-wave states, one will encounter two major difficulties: First, why are and degenerate in their masses? In the model, the latter is dominated by the component, whereas the former cannot have the content since it is an state. One will expect the mass hierarchy pattern : . However, this pattern is not seen by experiment. In contrast, it is experimentally. Second, why are and so broad compared to the narrow widths of and even though they are all in the same nonet?

• In the scalar meson sector above 1 GeV, with mass MeV (1) is almost degenerate in masses with which has a mass of MeV (1) despite having one strange quark for the former.

• In the even-parity charmed meson sector, we compare the experimentally measured masses and widths with what are expected from the quark model (see Table 1). There are some prominent features from this comparison: (i) The measured masses of and are substantially smaller than the quark model predictions. (ii) The physical mass is below the threshold, while is below . This means that both of them are quite narrow, in sharp contrast to the quark model expectation of large widths for them. (iii) and are almost equal in their masses, while is heavier than even though the latter contains a strange quark. 1 (iv) The masses of and predicted by the quark model are consistent with experiment. These four observations lead to the conclusion that and charmed mesons have very unusual behavior not anticipated from the quark model.

• The first particle, namely , observed by Belle in 2003 in decays (8), has the quantum numbers (9). cannot be a pure charmonium as it cannot be identified as with a mass 3511 MeV (1) or with the predicted mass of order 3950 MeV (10). Moreover, a pure charmonium for cannot explain the large isospin violation observed in decays. The extreme proximity of to the threshold suggests a loosely bound molecule state for . On the other hand, cannot be a pure molecular state either for the following reasons: (i) It cannot explain the prompt production of in high energy collisions (11); (12). (ii) The ratio is predicted to be much less than unity in the molecular scenario, while it was measured to be and by Belle (13). (iii) For the ratio , the molecular model leads to a very small value of order (16); (14); (15) while the charmonium model predicts to be order of unity. The LHCb measurement yields (17). Hence, cannot be a pure molecular state.

The above discussions suggest that is most likely an admixture of the -wave molecule and the -wave charmonium as first advocated in (11)

 |X(3872)⟩ = c1|c¯c⟩P−wave+c2|D0¯¯¯¯¯D∗0⟩S−wave+c3|D+D∗−⟩S−wave+⋯. (1)

More specifically, the charmonium is identified with . Some calculations favor a larger component over the component (see e.g. (18); (19)). Then the question is how to explain the mass of through the charmonium picture.

In short, the quark model has difficulties in describing light scalar mesons below 1 GeV, and charmed mesons and charmonium-like state . A common wisdom for the resolution of aforementioned difficulties lies on the coupled channel effects which will distort the quark model calculations.

In the quark potential model, the predicted masses for and are higher than the measured ones by order 160 MeV and 70 MeV, respectively (6); (7). It was first stressed and proposed in (20) that the low mass of () arises from the mixing between the () state and the () threshold (see also (21)). This conjecture was realized in both QCD sum rule (22); (23) and lattice (24); (25); (26) calculations. For example, when the contribution from the continuum is included in QCD sum rules, it has been shown that this effect will significantly lower the mass of the state (22). Recent lattice calculations using , and interpolating fields show the existence of below the threshold (24) and below the threshold (25). 2 All these results indicate that the strong coupling of scalars with hadronic channels will play an essential role of lowering their masses.

By the same token, mass shifts of charmed and bottom scalar mesons due to self-energy hadronic loops have been calculated in (28). The results imply that the bare masses of scalar mesons calculated in the quark model can be reduced significantly. Mass shifts due to hadronic loops or strong coupled channels have also been studied in different frameworks to explain the small mass of (30); (29); (31); (32). In the same spirit, even if is dominated by the component, the mass of can be shifted down due to its strong coupling with channels (33); (34); (35).

Both and have the strong couple channel . They are often viewed as molecules, which accounts for their near degeneracy with . Schematically, the self-energy loop diagram of will shift its mass to the physical one. In the unitarized chiral perturbation theory, light scalar mesons , , and can be dynamically generated through their strong couplings with , , , and , respectively (36). Alternatively, it is well known that the tetraquark picture originally advocated by Jaffe (37) provides a simple solution to the mass and width hierarchy problems in the light scalar meson sector. The tetraquark structure of light scalars accounts for the mass hierarchy pattern , Moreover, the -wave 4-quark nonet can be lighter than the -wave nonet above 1 GeV due to the absence of the orbital angular momentum barrier and the presence of strong attraction between the diquarks and (38). The fall-apart decays , , and are all OZI-superallowed. This explains the very broad widths of and , and the narrowness of and owing to the very limited phase space available as they are near the threshold.

In (39) we have studied near mass degeneracy of scalar charmed and bottom mesons. Qualitatively, the approximate mass degeneracy can be understood as a consequence of self-energy effects due to strong coupled channels which will push down the mass of the heavy scalar meson in the strange sector more than that in the non-strange partner. However, we showed that it works in the conventional model without heavy quark expansion, but not in the approach of heavy meson chiral perturbation theory (HMChPT) as mass degeneracy and the physical masses of and cannot be accounted for simultaneously. Mass shifts in the strange charm sector are found to be largely overestimated. It turns out that the conventional model works better toward the understanding of near mass degeneracy.

Our previous work was criticized by Alhakami (40) who followed the framework of (41) to write down the general expression of HMChPT and fit the unknown low-energy constants in the effective Lagrangian to the experimentally measured odd- and even-parity charmed mesons. Using the results from the charm sector, Alhakami predicted the spectrum of odd- and even-parity bottom mesons. He concluded that the near degeneracy of nonstrange and strange scalar mesons is confirmed in the predictions using HMChPT. He then proceeded to criticize that we should use physical masses instead of bare masses to evaluate the hardronic loop effects and that we have missed the contributions from axial-vector heavy mesons to the self-energy of scalar mesons.

Motivated by the above-mentioned criticisms (40), in this work we shall re-examine our calculations within the framework of HMChPT. We show that the closeness of and masses can be achieved by taking into account the additional contribution, which was missing in our previous work, from axial-vector heavy mesons to the self-energy diagrams of scalar mesons by adjusting two relevant strong couplings and the renormalization scale appearing in the loop diagram. Then we proceed to confirm that near degeneracy observed in the charm sector will imply the similarity of and masses in the system.

This work is organized as follows. In Sec. II we consider the self-energy corrections to scalar and axial-vector heavy mesons in HMChPT. In the literature, the self-energy loop diagrams were sometimes evaluated in HMChPT by neglecting the corrections from mass splittings and residual masses to the heavy meson’s propagator. We shall demonstrate in Sec. III that the calculation in this manner does not lead to the desired degeneracy in both charm and sectors simultaneously. The masses of and are discussed in Sec. IV with focus on the predictions based on heavy quark symmetry and possible and QCD corrections. Sec. V comes to our conclusions.

## Ii Mass shift of scalar and axial-vector heavy mesons due to hadronic loops

Self-energy hadronic loop corrections to and heavy mesons have been considered in the literature (42); (43); (44); (41); (40); (45); (28); (39). Since chiral loop corrections to heavy scalar mesons have both finite and divergent parts, it is natural to consider the framework of HMChPT where the divergences and the renormalization scale dependence arising from the chiral loops induced by the lowest-order tree Lagrangian can be absorbed into the counterterms which have the same structure as the next-order tree Lagrangian.

The heavy meson’s propagator in HMChPT has the expression

 i2v⋅k−Π(v⋅k), (2)

where and , respectively, are the velocity and the residual momentum of the meson defined by , and is the 1PI self-energy contribution. In general, is complex as its imaginary part is related to the resonance’s width. The particle’s on-shell condition is then given by

 2v⋅~k−ReΠ(v⋅~k)=0. (3)

 m=m0+v⋅~k=m0+12ReΠ(v⋅~k). (4)

Consider the self-energy diagrams depicted in Fig. 1 for scalar and axial-vector heavy mesons. We will evaluate the loop diagrams in the framework of HMChPT in which the low energy dynamics of hadrons is described by the formalism in which heavy quark symmetry and chiral symmetry are synthesized (46); (47); (48). The relevant Lagrangian is (49)

 L = Tr[¯Hb(iv⋅D)baHa]+Tr[¯Sb((iv⋅D)ba−δbaΔS)Sa] (5) + gTr[¯Hbγμγ5AμbaHa]+hTr[¯Sbγμγ5AμbaHa+h.c.]+g′Tr[¯Sbγμγ5AμbaSa+h.c.],

where denotes the odd-parity spin doublet and the even-parity spin doublet with ( being the total angular momentum of the light degrees of freedom):

 Ha=1+v/2[P∗aμγμ−Paγ5],Sa=12(1+v/)[P′μ1aγμγ5−P∗0a], (6)

with , for example, . The nonlinear chiral symmetry is realized by making use of the unitary matrix with MeV and being a matrix for the octet of Goldstone bosons. In terms of the new matrix , the axial vector field reads . In Eq. (5), the parameter is the residual mass of the field; it measures the mass splitting between even- and odd-parity doublets and can be expressed in terms of the spin-averaged masses

 ⟨MH⟩≡3MP∗+MP4,⟨MS⟩≡3MP′1+MP∗04, (7)

so that

 ΔS=⟨MS⟩−⟨MH⟩. (8)

There exist two corrections to the chiral Lagrangian (5): one from corrections and the other from chiral symmetry breaking. The corrections are given by (49); (50)

 L1/mQ=12mQ{λH2Tr[¯HaσμνHaσμν]−λS2Tr[¯SaσμνSaσμν]}, (9)

with

 λH2=14(M2P∗−M2P),λS2=14(M2P′1−M2P∗0), (10)

where () is the mass splitting between spin partners, namely, and ( and ) of the pseudoscalar (scalar) doublet. We will not write down the explicit expressions for chiral symmetry breaking terms and the interested reader is referred to (41). The masses of heavy mesons can be expressed as

 MPa = M0−32λH2mQ+Δa,MP∗a=M0+12λH2mQ+Δa, MP∗0a = M0+ΔS−32λS2mQ+~Δa, MP′1a=M0+ΔS+12λS2mQ+~Δa, (11)

where and denote the residual mass contributions to odd- and even-parity mesons, respectively. Note that and in the heavy quark limit. The propagators for , , and read

 i2(v⋅k+34ΔMP−Δa)+iϵ,−i(gμν−vμvν)2(v⋅k−14ΔMP−Δa)+iϵ, (12)

and

 i2(v⋅k−ΔS+34ΔMS−~Δa)+iϵ,−i(gμν−vμvν)2(v⋅k−ΔS−14ΔMS−~Δa)+iϵ, (13)

respectively.

Consider the hadronic loop contribution to in Fig. 1(a) with the intermediate states and . The self-energy loop integral is

 ΠDK(ωD) = (2h2f2π)i2∫d4q(2π)4(v⋅q)2(q2−m2K+iϵ)(v⋅k′+34ΔMD−Δu+iϵ) (14) = (2h2f2π)i2∫d4q(2π)4(v⋅q)2(q2−m2K+iϵ)(v⋅q+ωD+iϵ),

where is the mass of the Goldstone boson. The residual momentum of the heavy meson in the loop is given by , and . The calligraphic symbol has been used to denote the bare mass. The full propagator becomes

 i2(v⋅k−ΔS+34ΔMS−~Δs)−[2ΠDK(ωD)+23ΠDsη(ωDs)+2Π′D′1K(ω′D′1)+23Π′D′s1η(ω′D′s1)]

after taking into account the contributions from the channels and . In Eq. (II),

 Π′D′1K(ω′D′1) = −(2g′2f2π)i2∫d4q(2π)4q2−(v⋅q)2(q2−m2+iϵ)(v⋅k′−ΔS−14ΔMS−~Δu+iϵ) (16) = −(2g′2f2π)i2∫d4q(2π)4q2−(v⋅q)2(q2−m2+iϵ)(v⋅q+ω′D′1+iϵ),

where , and use of Eq. (II) has been made. Likewise, the full propagator reads

 i[2(v⋅k−ΔS+34ΔMS−~Δu)−[32ΠDπ(ωD)+16ΠDη(ωD)+ΠDsK(ωDs) +32Π′D′1π(ω′D′1)+16Π′D′1η(ω′D′1)+Π′D′s1K(ω′D′s1)]]−1. (17)

Since many parameters such as , , and in Eqs. (II) and (II) are unknown, we are not able to determine mass shifts from above equations. Assuming that the bare mass is the one obtained in the quark model, then from Eq. (II) we have

 ΔS−34ΔMS+~Δs=MD∗s0−M0=MD∗s0−MD−34ΔMD+Δu. (18)

With

 F(v⋅k)D∗s0 ≡ 2(v⋅k−MD∗s0+MD+34ΔMD−Δu)−Re[2ΠDK(ωD)+23ΠDsη(ωDs) +2Π′D′1K(ω′D′1)+23Π′D′s1η(ω′D′s1)], F(v⋅k)D∗0 ≡ 2(v⋅k−MD∗0+MD+34ΔMD−Δu)−Re[32ΠDπ(ωD)+16ΠDη(ωD)+ΠDsK(ωDs) (19) +32Π′D′1π(ω′D′1)+16Π′D′1η(ω′D′1)+Π′D′s1K(ω′D′s1)],

the on-shell conditions read for and for . The physical masses are then given by

 MD∗(s)0=M0+v⋅~k=MD+34ΔMD−Δu+v⋅~k. (20)

Since is of order 1 MeV (51), it can be neglected in practical calculations. Note that in the above equation, one should not replace by the bare mass . Indeed, in the absence of chiral loop corrections, , as it should be.

For the self-energy of the axial-vector meson, we consider as an illustration which receives contributions from and intermediate states (see Fig. 1(b)). The full propagator reads

 −i(gμν−vμvν)2(v⋅k−ΔS−14ΔMS−~Δs)−[2ΠD∗K(ωD∗)+23ΠD∗sη(ωD∗s)+2Π′D∗0K(ω′D∗0)+23Π′D∗s0η(ω′D∗s0)],

where and .

The loop integrals in Eqs. (14) and (16) have the expressions (52); (50); (51)

 Π(ω) = (2h2f2π)ω32π2[(m2−2ω2)lnΛ2m2−2ω2+4ω2F(−mω)] (22)

and (41); (40)

 Π′(ω) = (2g′2f2π)ω32π2[(3m2−2ω2)lnΛ2m2−103ω2+4m2+4(ω2−m2)F(−mω)], (23)

respectively, with

 F(1x)=⎧⎪ ⎪⎨⎪ ⎪⎩√x2−1xln(x+√x2−1),|x|≥1−√1−x2x[π2−tan−1(x√1−x2)],|x|≤1\,. (24)

Note that the function can be recast to the form

 F(−mω)=1ωG(ω,m) (25)

with

 G(ω,m)=⎧⎪⎨⎪⎩√ω2−m2[cosh−1(ωm)−iπ],ω>m√m2−ω2cos−1(−ωm),ω2

The parameter appearing in Eqs. (22) and (23) is an arbitrary renormalization scale. In the dimensional regularization approach, the common factor with can be lumped into the logarithmic term . In the conventional practice, it is often to choose , the chiral symmetry breaking scale of order 1 GeV, to get numerical estimates of chiral loop effects. However, as pointed out in (39), contrary to the common wisdom, the renormalization scale has to be larger than the chiral symmetry breaking scale of order 1 GeV in order to satisfy the on-shell conditions. In general, there exist two solutions for due to two intercepts of the curve with the axis. We shall consider the smaller solution for as the other solution will yield too large masses. It could be that higher-order heavy quark expansion needs to be taken into account to justify the use of .

In our previous study (39) we argued that near mass degeneracy and the physical masses of and cannot be accounted for simultaneously in the approach of HMChPT. In this work we show that near mass degeneracy can be implemented by taking into account the additional contributions from axial-vector heavy mesons to the self-energy diagram Fig. 1(a) of scalar mesons. Since , and , , we find numerically that contributes destructively to the mass shifts. Moreover, the self-energy of