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We present an in-depth study of masses and decays of excited scalar and pseudoscalar states in the Extended Linear Sigma Model (eLSM). The model also contains ground-state scalar, pseudoscalar, vector and axial-vector mesons. The main objective is to study the consequences of the hypothesis that the f0(1790) resonance, observed a decade ago by the BES Collaboration and recently by LHCb, represents an excited scalar quarkonium. In addition we also analyse the possibility that the new a0(1950) resonance, observed recently by BABAR, may also be an excited scalar state. Both hypotheses receive justification in our approach although there appears to be some tension between the simultaneous interpretation of f0(1790)/a0(1950) and pseudoscalar mesons η(1295), π(1300), η(1440) and K(1460) as excited states.
One of the most important features of strong interaction is the existence of the hadron spectrum. It emerges from confinement of quarks and gluons – degrees of freedom of the underlying theory, Quantum Chromodynamics (QCD) – in regions of sufficiently low energy where the QCD coupling is known to be large [1–4]. Although the exact mechanism of hadron formation in non-perturbartive QCD is not yet fully understood, an experimental fact is a very abundant spectrum of states possessing various quantum numbers, such as for example isospin I, total spin J, parity P and charge conjugation C.
This is in particular the case for the spectrum of mesons (hadrons with integer spin) that can be found in the listings of PDG – the Particle Data Group . In the scalar channel (JP = 0+), the following states are listed in the energy region up to approximately 2 GeV:
The pseudoscalar channel (JP = 0-) is similarly well populated:
A natural expectation founded in the Quark Model (see Refs. [6, 7]; for a modern and modified version see for example Refs. [8, 9]) is that the mentioned states can effectively be described in terms of constituent quarks and antiquarks – ground-state resonances. In this context, we define ground states as those with the lowest mass for a given set of quantum numbers I, J, P and C. Such a description is particularly successful for the lightest pseudoscalar states π, K and η.
However, this cannot be the full picture as the spectra contain more states than could be described in terms of the ground-state structure. A further natural expectation is then that the spectra may additionally contain first (radial) excitations of states, i.e., those with the same quantum numbers but with higher masses. (In the spectroscopic notation, the excited scalar and pseudoscalar states correspond, respectively, to the 2 3P0 and 2 1S0 configurations.) Of course, the possibility to study such states depends crucially on the identification of the ground states themselves; in the case of the scalar mesons, this is not as clear as for the pseudoscalars. Various hypotheses have been suggested for the scalar-meson structure, including meson–meson molecules, states and glueballs, bound states of gluons – see, e.g., Refs. [10–76]. Results of these studies are at times conflicting but the general conclusion is nonetheless that the scalar ground states (as well as the glueball and the low-energy four-quark states) are well defined and positioned in the spectrum of particles up to and including the f0(1710) resonance.
The main objective of this work is then to ascertain which properties the excited scalar and pseudoscalar states possess and whether they can be identified in the physical spectrum.
Our study of the excited mesons is based on the Linear Sigma Model [77–80]. This is an effective approach to low-energy QCD – its degrees of freedom are not quarks and gluons of the underlying theory but rather meson fields with various values of I, J, P and C.
There are several advantages that the model has to offer. Firstly, it implements the symmetries of QCD as well as their breaking (see Sect. 2 for details). Secondly, it contains degrees of freedom with quantum numbers equal to those observed experimentally and in theoretical first-principles spectra (such as those of lattice QCD). This combination of symmetry-governed dynamics and states with correct quantum numbers justifies in our view the expectation that important aspects of the strong interaction are captured by the proposed model. Note that the model employed in this article is wide-ranging in that it contains the ground-state scalar, pseudoscalar, vector and axial-vector states in three flavours (u, d, s), the scalar dilaton (glueball) and the first excitations in the three-flavour scalar and pseudoscalar channels. Considering isospin multiplets as single degrees of freedom, there are 16 ground states and 8 excited states plus the scalar glueball in the model. For this reason, it can be denoted the “Extended Linear Sigma Model” (eLSM). A further advantage of eLSM is that the inclusion of degrees of freedom with a certain structure (such as states here) allows us to test the compatibility of experimentally known resonances with such structure. This is of immediate relevance for experimental hadron searches such as those planned at PANDA@FAIR .
With regard to vacuum states, the model has been used in studies of two-flavour mesons , glueballs [83–87], K1 and other spin-1 mesons [88, 89] and baryons . It is, however, also suitable for studies of the QCD phase diagram [91–93]. In this article, we will build upon the results obtained in Refs. [94, 95] where ground-state resonances and the glueball were considered in vacuum. Comparing experimental masses and decay widths with the theoretical predictions for excited states, we will draw conclusions on structure of the observed states; we will also predict more than 35 decays for various scalar and pseudoscalar resonances (see Sect. 3.3).
Irrespective of the above advantages, we must note that the model used in this article also has drawbacks. There are two that appear to be of particular importance.
Firstly, some of the states that might be of relevance in the region of interest are absent. The most important example is the scalar glueball whose mass is comparable [54, 58, 61, 64, 65] to that of the excited states discussed here. The implementation of the scalar glueball is actually straightforward in our approach (see Sect. 2) but the amount of its mixing with excited states is as yet unestablished, mainly due to the unfortunate lack of experimental data (discussed in Sect. 2.3.1).
Secondly, our calculations of decay widths are performed at tree level. Consequently, unitarity corrections are not included. A systematic way to implement them is to consider mesonic loops and determine their influence on the pole positions of resonances. Substantial shift of the pole position may then improve (or spoil) the comparison to the experimental data. However, the results of Ref.  suggest that unitarity corrections are small for resonances whose ratio of decay width to mass is small as well. Since such resonances are present in this article (see Sect. 3.3.3), the corrections will not be considered here.
Excited mesons were a subject of interest already several decades ago [97, 98]; to date, they have been considered in a wide range of approaches including QCD models/chiral Lagrangians [99–104], Lattice QCD [105–110], Bethe-Salpeter equation [111–114], NJL Model and its extensions [115–125], light-cone models , QCD string approaches  and QCD domain walls . Chiral symmetry has also been suggested to become effectively restored in excited mesons [129, 130] rendering their understanding even more important. A study analogous to ours (including both scalar and pseudoscalar excitations and their various decay channels) was performed in extensions of the NJL model [117–119, 121, 122]. The conclusion was that f0(1370), f0(1710) and a0(1450) are the first radial excitations of f0(500), f0(980) and a0(980). However, this is at the expense of having very large decay widths for f0(1370), f0(1500) and f0(1710); in our case the decay widths for f0 states above 1 GeV correspond to experimental data but the resonances are identified as quarkonium ground states .
The outline of the article is as follows. The general structure and results obtained so far regarding ground-state resonances are briefly reviewed in Sects. 2.1 and 2.2. Building upon that basis, we present the Lagrangian for the excited states and discuss the relevant experimental data in Sect. 2.3. Two hypotheses are tested in Sect. 3: whether the f0(1790) and a0(1950) resonances can represent excited states; the first one is not (yet) listed by the PDG but has been observed by the BES II and LHCb Collaborations [131, 132] and is discussed in Sect. 2.3.1. We also discuss to what extent it is possible to interpret the pseudoscalar mesons η(1295), π(1300), η(1440) and K(1460) as excited states. Conclusions are presented in Sect. 4 and all interaction Lagrangians used in the model can be found in Appendix A. Our units are ħ = c = 1; the metric tensor is gμν = diag( + , - , - , - ).
A viable effective approach to phenomena of non-perturbative strong interaction must implement the symmetries present in the underlying theory, QCD. The theory itself is rich in symmetries: colour symmetry SU(3)c (local); chiral U(Nf)L × U(Nf)R symmetry (L and R denote the ’left’ and ’right’ components and Nf the number of quark flavours; global, broken in vacuum spontaneously by the non-vanishing chiral condensate [133, 134], at the quantum level via the axial U(1)A anomaly  and explicitly by the non-vanishing quark masses); dilatation symmetry (broken at the quantum level [136, 137] but valid classically in QCD without quarks); CPT symmetry (discrete; valid individually for charge conjugation C, parity transformation P and time reversal T); Z3 symmetry (discrete; pertaining to the centre elements of a special unitary matrix of dimension Nf × Nf; non-trivial only at non-zero temperatures [138–143]) – all of course in addition to the Poincaré symmetry.Terms entering the Lagrangian of an effective approach to QCD should as a matter of principle be compatible with all symmetries listed above. Our subject is QCD in vacuum. In this context, we note that the colour symmetry is automatically fulfilled since we will be working with colour-neutral degrees of freedom; the structure and number of terms entering the Lagrangian are then restricted by the chiral, CPT and dilatation symmetries.
This section contains a brief overview of the results obtained so far in the Extended Linear Sigma Model that contains Nf = 3 scalar, pseudoscalar, vector and axial-vector quarkonia and the scalar glueball. The discussion is included for convenience of the reader and in order to set the basis for the incorporation of the excited quarkonia (Sect. 2.3). All details can be found in Refs. [94, 95].
where G represents the dilaton field and Λ is the scale that explicitly breaks the dilatation symmetry. Considering fluctuations around the potential minimum G0 ≡ Λ leads to the emergence of a particle with JPC = 0++ – the scalar glueball [83, 95].
Terms that (i) are compatible in their structure with the chiral, dilatation and CPT symmetries of QCD and (ii) contain ground-state scalar, pseudoscalar, vector and axial-vector quarkonia with Nf = 3 and the dilaton are collected in the ℒ0 contribution to Eq. (1), as in Refs. [82, 94, 95]:
In Eq. (3), the matrices Φ, Lμ, and Rμ represent the scalar and vector nonets:
where Ti (i = 0, …, 8) denote the generators of U(3), while Si represents the scalar, Pi the pseudoscalar, the vector, the axial-vector meson fields. (Note that we are using the non-strange–strange basis defined as and with .)
is the derivative of Φ transforming covariantly with regard to the U(3)L × U(3)R symmetry group; the left-handed and right-handed field strength tensors Lμν and Rμν are, respectively, defined as
The following symmetry-breaking mechanism is implemented:
We also note the following important points:
The ground-state mass terms can be obtained from Lagrangian (3); their explicit form can be found in Ref.  where a comprehensive fit of the experimentally known meson masses was performed. Fit results that will be used in this article are collected in Table Table1.1. The following is of importance here:
With the foundations laid in the previous section, the most general Lagrangian for the excited scalar and pseudoscalar quarkonia with terms up to order four in the naive scaling can be constructed as follows:
The covariant derivative DμΦE is defined analogously to Eq. (7):
and we also set E1 = diag.
Spontaneous symmetry breaking in the Lagrangian for the excited (pseudo)scalars will be implemented only by means of condensation of ground-state quarkonia σN and σS, i.e., as a first approximation, we assume that their excited counterparts and do not condense.1 As a consequence, there is no need to shift spin-1 fields or renormalise the excited pseudoscalars as described in Eqs. (10)–(11).
We now turn to the assignment of the excited states. Considering isospin multiplets as single degrees of freedom, there are 8 states in Eq. (17): , , and (scalar) and , , πE and KE (pseudoscalar); the experimental information on states with these quantum numbers is at times limited or their identification is unclear:
As indicated in the above points, with regard to the use of the above data for parameter determination we exclude as input all states for which there are only scarce/unestablished data and, additionally, those for which the PDG cites only intervals for mass/decay width (since the latter lead to weak parameter constraints). Then we are left with only three resonances whose experimental data shall be used: f0(1790), η(1295) and η(1440). For clarity, we collect the assignment of the model states (where possible), and also the data that we will use, in Table Table2.2. The data are used in Sect. 3.
The following parameters are present in Eq. (16):
The number of parameters relevant for masses and decays of the excited states is significantly smaller as apparent once the following selection criteria are applied:
Note that the above criteria are not mutually exclusive: some parameters may be set to zero on several grounds, such as for example κ1.
The following mass terms are obtained for the excited states present in the model:
This is obvious after substituting the strange condensate ϕS by the non-strange condensate ϕN via Eq. (15). The modified mass terms then read
Mass terms for all eight excited states can hence be described in terms of only three parameters from Eq. (16): , and ξ2.
Our objective is to perform a tree-level calculation of all kinematically allowed two- and three-body decays for all excited states present in the model. The corresponding interaction Lagrangians are presented in Appendix A. As we will see, there are more than 35 decays that can be determined in this way but all of them can be calculated using only a few formulae.
The generic formula for the decay width of particle A into particles B and C reads
where k is the three-momentum of one of the final states in the rest frame of A and ℳ is the decay amplitude (i.e., a transition matrix element). ℐ is a symmetry factor emerging from the isospin symmetry – it is determined by the number of sub-channels for a given set of final states (e.g., ℐ = 2 if B and C both correspond to kaons). Usual symmetry factors are included if the final states are identical. As we will see in Sect. 3.3, decay widths obtained in the model are generally much smaller than resonance masses; for this reason, we do not expect large unitarisation effects .
Depending on the final states, the interaction Lagrangians presented in Appendix A can have one of the following general structures:
As is evident from Appendix A, the most general interaction Lagrangian for 3-body decays of the form S → S1S2S3 is
The ensuing formula for the decay width reads
where , and
As is evident from Appendix A, our decay widths depend on the following parameters: , , ξ2 and . The first three appear only in decays with an excited final state; since such decays are experimentally unknown, it is not possible to determine these parameters (and ξ2 can be determined from the mass terms in any case; see Sect. 2.3.3). The remaining two, , can be calculated from decays with ground states in the outgoing channels – we will discuss this in Sect. 3.3.
As is evident from mass terms (30)–(36) and Appendix A, and influence only masses; ξ2 appears in decays with one excited final state and in mass terms. Since, as indicated at the end of Sect. 2.3.4, decays with excited final states are experimentall unknown, ξ2 can only be determined from the masses. Contrarily, and appear only in decay widths (with no excited final states). Hence our parameters are divided in two sets, one determined by masses (, and ξ2) and another determined by decays ( and ).
Parameter determination will ensue by means of a χ2 fit. Scarcity of experimental data compels us to have an equal number of parameters and experimental data entering the fit; although in that case the equation systems can also be solved exactly, an advantage of the χ2 fit is that error calculation for parameters and observables is then straightforward.
The general structure of the fit function χ2 fit is as follows:
for a set of n (theoretical) observables determined by m ≤ n parameters pj. In our case, m = n = 3 for masses and m = n = 2 for decay widths. Central values and errors on the experimental side are, respectively, denoted and . Parameter errors Δpi are calculated as the square roots of the diagonal elements of the inverse Hessian matrix obtained from χ2(pj). Theoretical errors ΔOi for each observable Oi are calculated by diagonalising the Hesse matrix via a special orthogonal matrix M
and rotating parameters pi such that
where p contains all parameters and pmin. realises the minimum of χ2(p1, …, pm). Then we can determine ΔOi via
(see also Chapter 39 of the Particle Data Book ).
Following the discussion of the experimental data on excited states in Sect. 2.3.1 and particle assignment in Table Table2,2, we use the following masses for the χ2 fit of Eq. (50): MeV, MeV and MeV. Results for , and ξ2 are
We have concluded in Sect. 3.1 that only two parameters are of relevance for all decays predictable in the model: and . They can be determined from the data on the f0(1790) resonance discussed in Sect. 2.3.1: Γf0(1790)→ππ = (270 ± 45) MeV and Γf0(1790)→KK = (70 ± 40) MeV . Performing the χ2 fit described in Sect. 3.1 we obtain the following parameter values:
Large uncertainties for parameters are a consequence of propagation of the large errors for Γf0(1790)→ππ and particularly for Γf0(1790)→KK. As described in Sect. 2.3.1, Γf0(1790)→KK was obtained as our estimate relying upon J/Ψ branching ratios reported by BES II  that themselves had uncertainties between ∼ 23 and 50%. We emphasise, however, that such uncertainties do not necessarily have to translate into large errors for the observables. The reason is that error calculation involves derivatives at central values of parameters [see Eq. (53)]; small values of derivatives may then compensate the large parameter uncertainties. This is indeed what we observe for most decays.
There is a large number of decays that can be calculated using the interaction Lagrangians in Appendix A, parameter values in Eq. (55), formulae for decay widths in Eqs. (39), (41), (43) and (45) as well as Eq. (53) for the errors of observables. All results are presented in Table Table44.
The consequences of f0(1790) input data are then as follows:
As indicated above, results presented in Table Table44 do not allow us to make a definitive statement on all excited pseudoscalars. However, the situation changes if the parameters and are determined with the help of the η(1295) and η(1440) decay widths.
The parameters (56) are strongly constrained and there is a very good correspondence of the pseudoscalar decays to the experimental data in this case (see Table Table5).5). Nonetheless, there is a drawback: all scalar states become unobservable due to very broad decays into vectors. Thus comparison of Tables 4 and and55 suggests that there is tension between the simultaneous interpretation of η(1295), π(1300), η(1440) and K(1460) as well as the scalars as excited states. A possible theoretical reason is that pseudoscalars above 1 GeV may have non- admixture. Indeed sigma-model studies in Refs. [37, 41, 43, 204–208] have concluded that excited pseudoscalars with masses between 1 GeV and 1.5 GeV represent a mixture of and structures. In addition, the flux-tube model of Ref.  and a mixing formalism based on the Ward identity in Ref.  lead to the conclusion that the pseudoscalar channel around 1.4 GeV is influenced by a glueball contribution. Hence a more complete description of these states would require implementation of mixing scenarios in this channel.4
Note, however, that the results of Table Table55 depend on the assumption that the total decay width of η(1295) is saturated by the three decay channels accessible to our model (ηππ, η′ππ and Kππ). The level of justification for this assumption is currently uncertain . Consequently we will not explore this scenario further.
Encouraging results obtained in Sect. 3.3.1, where f0(1790) was assumed to be an excited state, can be used as a motivation to explore them further. As discussed in Sect. 2.3.1, data analysis published recently by the BABAR Collaboration has found evidence of an isotriplet state a0(1950) with mass ma0(1950) = (1931 ± 26) MeV and decay width Γa0(1950) = (271 ± 40) MeV .
Assuming that f0(1790) is an excited state (as already done in Sect. 3.3.1), we can implement ma0(1950) obtained by BABAR as a large-Nc suppressed effect in our model as follows. Mass terms for excited states and , Eqs. (30) and (34), can be modified by reintroduction of the large-Nc suppressed parameter κ2 and now read
The other mass terms [Eqs. (31)–(33), (35) and (36)] remain exactly the same; κ2 does not influence any decay widths. We can now repeat the calculations described in Sect. 3.2 with the addition that the mass of our state corresponds exactly to that of a0(1950). We obtain
Note that a non-vanishing value of κ2 introduces mixing of and in our Lagrangian (16). Its effect is, however, vanishingly small since the mixing angle is ∼ 11∘.
Using the mass parameters (59) and the decay parameters (55) we can repeat the calculations of Sect. 3.3.1. Then our final results for the mass spectrum are presented in Fig. Fig.11 and for the decays in Table Table6.6. The values of , and have changed in comparison to Table Table44 inducing an increased phase space. For this reason, the decay widths of the corresponding resonances have changed as well. All other results from Table Table44 have remained the same and are again included for clarity and convenience of the reader.
The consequences are as follows:
We have studied masses and decays of excited scalar and pseudoscalar states (q = u, d, s quarks) in the Extended Linear Sigma Model (eLSM) that, in addition, contains ground-state scalar, pseudoscalar, vector and axial-vector mesons.
Our main objective was to study the assumption that the f0(1790) resonance is an excited state. This assignment was motivated by the observation in BES  and LHCb  data that the resonance couples mostly to pions and by the theoretical statement that the ground state is contained in the physical spectrum below f0(1790). Furthermore, the assumption was also tested that the a0(1950) resonance, whose discovery was recently claimed by the BABAR Collaboration , represents the isotriplet partner of f0(1790).
Using the mass, 2π and 2K decay widths of f0(1790), the mass of a0(1950) and the masses of the pseudoscalar isosinglets η(1295) and η(1440) our model predicts more than 35 decays for all excited states except for the excited pion and kaon (where extremely large uncertainties are present due to experimental ambiguities). All numbers are collected in Table Table66.
In essence: the f0(1790) resonance emerges as the broadest excited state in the scalar channel with Γf0(1790) = (405 ± 96) MeV; a0(1950), if confirmed, represents a very good candidate for the excited state; , if confirmed, represents a very good candidate for the excited scalar kaon.
Our excited isoscalar state has a mass of (2038 ± 24) MeV, placed between the masses of the nearby f0(2020) and f0(2100) resonances; also, its width is relatively small ( ≤ 110 MeV). We conclude that, although any of these resonances may in principle represent a state, the introduction of mixing effects (particularly with a glueball state) may be necessary to further elucidate their structure.
Our results also imply a quite small contribution of the ηππ, η′ππ and πKK decays to the overall width of η(1295). For η(1440), the decay width is compatible with any value up to ∼ 400 MeV (ambiguities due to uncertainty in experimental input data).
It is also possible to implement and Γη(1440)→K⋆K exactly as in the data of PDG  and BES . Then π(1300) and K(1460) are quite well described as excited states – but the scalars are unobservably broad (see Table Table5).5). Hence, in this case, there appears to be tension between the simultaneous description of η(1295), π(1300), η(1440) and K(1460) and their scalar counterparts as excited states. This scenario is, however, marred by experimental uncertainties: for example, it is not at all clear if the width of η(1295) is indeed saturated by the ηππ, η′ππ and πKK decays. It could therefore only be explored further when (very much needed) new experimental data arrives – from BABAR, BES, LHCb or PANDA  and NICA .
We are grateful to D. Bugg, C. Fischer and A. Rebhan for extensive discussions. The collaboration with Stephan Hübsch within a Project Work at TU Wien is also gratefully acknowledged. The work of D. P. is supported by the Austrian Science Fund FWF, Project No. P26366. The work of F. G. is supported by the Polish National Science Centre NCN through the OPUS project nr. 2015/17/B/ST2/01625.
Here we collect all interaction Lagrangians that are used for calculations of decay widths throughout this article. Vertices for large-Nc suppressed decays are not included but briefly discussed after each Lagrangian in which they appear.
Appendix A.1: Lagrangian for
The Lagrangian reads
Note: the decay ( ∼ κ1, ) is large-Nc suppressed.
Appendix A.2: Lagrangian for
The Lagrangian reads
Note: the decays ( ∼ κ1, ), ( ∼ κ1, ), (), (), (), (), ( ∼ κ2), ( ∼ κ2) and ( ∼ κ1, ) are large-Nc suppressed.
Appendix A.3: Lagrangian for
The Lagrangian reads (only included; decays of follow from isospin symmetry):
Appendix A.4: Lagrangian for
The Lagrangian reads (only included; decays of other components follow from isospin symmetry):
Appendix A.5: Lagrangian for
Only three-body decays into pseudoscalars are kinematically allowed for this particle:
Appendix A.6: Lagrangian for
The Lagrangian reads
Note: the decay ( ∼ κ1, ) is large-Nc suppressed.
Appendix A.7: Lagrangian for πE
The Lagrangian reads (only π0E included; decays of π±E follow from isospin symmetry):
Appendix A.8: Lagrangian for KE
The Lagrangian reads (only K0E included; decays of other KE components follow from isospin symmetry):
1There is a subtle point pertaining to the condensation of excited states in σ-type models: as discussed in Ref. , it can be in agreement with QCD constraints but may also, depending on parameter choice, spontaneously break parity in vacuum. Study of a model with condensation of the excited states would go beyond the current work. (It would additionally imply that the excited pseudoscalars also represent Goldstone bosons of QCD which is disputed in, e.g., Ref. .)
2The η(1405) resonance would then be a candidate for the pseudoscalar glueball .
3However, there would be no mixing of pseudoscalar isosinglets and in the model even if all discarded parameters were considered. The reason is that there is no condensation of excited scalar states in Lagrangian (16).
4A similar mixing scenario may (as a matter of principle) also exist in the case of the scalars discussed here. However, the amount of theoretical studies is significantly smaller here: for example, a glueball contribution to f0(1790) has been discussed in Refs. [210, 211] while – just as in our study – the same resonance was found to be compatible with an excited state in Ref. .
Denis Parganlija, Email: ta.ca.neiwut.pti.peh@psined, http://hep.itp.tuwien.ac.at/~denisp/
Francesco Giacosa, Email: lp.ude.kju@asocaigf, http://www.ujk.edu.pl/strony/Francesco.Giacosa/