#### Excited scalar and pseudoscalar mesons in the extended linear sigma model

Eur. Phys. J. C
Excited scalar and pseudoscalar mesons in the extended linear sigma model
Denis Parganlija 2
Francesco Giacosa 0 1
0 Institut für Theoretische Physik, Johann Wolfgang Goethe-Universität , Max-von-Laue-Str. 1, 60438 Frankfurt am Main , Germany
1 Institute of Physics, Jan Kochanowski University , ul. Swietokrzyska 15, 25-406 Kielce , Poland
2 Institut für Theoretische Physik, Technische Universität Wien , Wiedner Hauptstr. 8-10, 1040 Vienna , Austria
We present an in-depth study of masses and decays of excited scalar and pseudoscalar q¯q 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 q¯q states.
1 Introduction
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
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 q¯q 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 q¯q structure. A further natural expectation is
then that the spectra may additionally contain first (radial)
excitations of q¯q 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 3 P0 and 2 1 S0 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,
q¯q¯qq 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 q¯q
ground states (as well as the glueball and the low-energy
fourquark 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 q¯q 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
lowenergy 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 q¯q 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 q¯q ground states and 8 q¯q 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 q¯q 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
[
81
].
With regard to vacuum states, the model has been used
in studies of two-flavour q¯q mesons [
82
], glueballs [
83–87
],
K1 and other spin-1 mesons [
88,89
] and baryons [90]. 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 q¯q 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 q¯q 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. [
96
] 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 [
126
], QCD string
approaches [
127
] and QCD domain walls [
128
]. 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 [94].
The outline of the article is as follows. The general
structure and results obtained so far regarding ground-state q¯q
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 q¯q 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
h¯ = c = 1; the metric tensor is gμν = diag(+, −, −, −).
2 The model
2.1 General remarks
A viable effective approach to phenomena of non-pertur
bative 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 (N f )L × U (N f )R symmetry (L and R denote the
’left’ and ’right’ components and N f the number of quark
flavours; global, broken in vacuum spontaneously by the
nonvanishing chiral condensate q¯ q [
133, 134
], at the quantum
level via the axial U (1) A anomaly [135] and explicitly by
the non-vanishing quark masses); dilatation symmetry
(broken at the quantum level [
136, 137
] but valid classically in
QCD without quarks); C P T 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
N f × N f ; 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.
The eLSM Lagrangian has the following general structure:
L = Ldil. + L0 + LE
and in Sects. 2.2 and 2.3 we discuss the structure of the
Lagrangians contributing to L as well as their matter
content.
2.2 Ground-state Quarkonia and Dilaton: Lagrangian and
the matter content
This section contains a brief overview of the results obtained
so far in the Extended Linear Sigma Model that contains
N f = 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
].
In Eq. (1), Ldil implements, at the composite level, the
dilatation symmetry of QCD and its breaking [
144–149
]:
1 1 m2G
Ldil. = 2 (∂μG)2 − 4 2
G4 ln
G2
2 −
G4
4
(1)
(2)
where G represents the dilaton field and is the scale that
explicitly breaks the dilatation symmetry. Considering
fluctuations around the potential minimum G0 ≡ leads to
the emergence of a particle with J PC = 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 N f = 3 and the dilaton are collected in the
L0 contribution to Eq. (1), as in Refs. [
82, 94, 95
]:
L0 = Tr[( Dμ
)†( Dμ
)] − m20
− λ1[Tr( †
1
− 4 Tr(L 2μν + Rμ2ν )
+ Tr
G
G0
× ⎜⎜
⎝
ωN√−K2¯ρ00++ fK1¯N√10−2a10 KωS0++ fK1S10 ⎟⎟⎠
where Ti (i = 0, . . . , 8) denote the generators of U (3),
while Si represents the scalar, Pi the pseudoscalar, Viμ the
vector, Aiμ the axial-vector meson fields. (Note that we
are using the non-strange–strange basis defined as ϕN =
√13 √2 ϕ0 + ϕ8 and ϕS = √13 ϕ0 − √2 ϕ8 with ϕ ∈
(Si , Pi , Viμ, Aiμ).)
Furthermore,
Dμ
≡ ∂μ
− i g1(Lμ
−
Rμ)
is the derivative of 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
Lμν ≡ ∂μ Lν − ∂ν Lμ,
Rμν ≡ ∂μ Rν − ∂ν Rμ.
The following symmetry-breaking mechanism is
implemented:
– The spontaneous breaking of the U (3)×U (3) chiral
symmetry requires setting m20 < 0.
– The explicit breaking of the U (3)×U (3) chiral as well as
dilatation symmetries is implemented by terms
describing non-vanishing quark masses: H = diag{h N , h N , h S },
= diag{0, 0, δS } and E0 = diag{0, 0, S }.
– The U (1)A (chiral) anomaly is implemented by the
determinant term c1(det − det †)2 [
150,151
].
We also note the following important points:
– All states present in the Lagrangian (3), except for the
dilaton, possess the q¯q structure [
82,152
]. The
argument is essentially based on the large-Nc behaviour of
the model parameters and on the model construction in
terms of the underlying (constituent) quark fields. The
ground-state Lagrangian (3) contains a pseudoscalar field
assigned to the pion since it emerges from spontaneous
breaking of the (chiral) U (3) × U (3) symmetry.
Furthermore, the vector meson decaying into 2π is identified
with the rho since the latter is experimentally known to
decay into pions with a branching ratio of slightly less
Rμ =
8
than 1. Pion and rho can be safely assumed to
represent (very predominant) q¯q states and hence the large-Nc
behaviour of their mass terms has to be N 0.
Additionc
ally, the rho-pion vertex has to scale as Nc−1/2 since the
states are quarkonia. Then, as shown in Ref. [
82
], this
is sufficient to determine the large-Nc behaviour of all
ground-state model parameters and of the non-strange
and strange quark condensates. As a consequence, the
masses of all other ground states scale as Nc0 and their
decay widths scale as 1/Nc. For this reason, we identify
these degrees of freedom with q¯q states.
A further reason is that all states entering the matrix
in Eq. (4) can be decomposed in terms of (constituent)
quark currents whose behaviour under chiral
transformation is such that all terms in the Lagrangian (except
for symmetry-breaking or anomalous ones) are chirally
symmetric [
152
].
Note that our excited-state Lagrangian (16) will have
exactly the same structure as the ground-state one.
Considering the above discussion, we conclude that its
degrees of freedom also have the q¯q structure.
– The number of terms entering Eq. (3) is finite under the
requirements that (i) all terms are dilatationally invariant
and hence have mass dimension equal to four, except
possibly for those that are explicitly symmetry breaking
or anomalous, and (ii) no term leads to singularities in
the potential in the limit G → 0 [
153
].
– Notwithstanding the above point, the glueball will not be
a subject of this work – hence G ≡ G0 is set
throughout this article. With regard to the ground-state mesons,
we will be relying on Ref. [
94
] since it contains the
latest results from the model without the glueball. (For the
model version with three-flavour q¯q states as well as the
scalar glueball; see Ref. [
95
].)
– There are two scalar isospin-0 fields in the Lagrangian
(3): σN ≡ n¯ n (n: u and d quarks, assumed to be
degenerate) and σS ≡ s¯s. Spontaneous breaking of the chiral
symmetry implies the existence of their respective
vacuum expectation values φN and φS. As described in Ref.
[
94
], shifting of σN,S by φN,S leads to the mixing of
spin1 and spin-0 fields. These mixing terms are removed by
suitable shifts of the spin-1 fields that have the following
general structure:
V μ → V μ + Z SwV ∂μ S,
(10)
where V μ and S, respectively, denote the spin-1 and
spin-0 fields. The new constants Z S and wV are
fielddependent and read [
94
]
w f1N = wa1 =
g1φN
ma21 w f1S =
√
2g1φS
m2f1S
i g1(φN −
where fπ and fK , respectively, denote the pion and kaon
decay constants.
The ground-state mass terms can be obtained from
Lagrangian (3); their explicit form can be found in Ref.
[
94
] 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 1. The following is of
importance here:
– Table 1 contains no statement on masses and
assignment of the isoscalar states σN and σS. The reason is
that their identification in the meson spectrum is unclear
due to both theoretical and experimental uncertainties
[
154,155
]. In Ref. [94], the preferred assignment of
σN was to f0(1370), not least due to the best-fit result
mσN = 1363 MeV. The resonance σS was assigned to
f0(1710). Note that a subsequent analysis in Ref. [
95
],
which included the scalar glueball, found the assignment
of σS to f0(1500) more preferable; f0(1710) was found
to be compatible with the glueball. These issues will be
of secondary importance here since no mixing between
excited and ground states will be considered. (We also
note that decays of the excited states into f0(1500) and
f0(1710) would be kinematically forbidden.
Excitedstate masses are discussed in Sect. 3).
– Table 1 also contains no statement on the axial-vector
kaon K1. Reference [
94
] obtained m K1 = 1282 MeV as
the best-fit result. One needs to note, however, that PDG
listings [
5
] contain two states to which our K1 resonance
could be assigned: K1(1270) and K1(1400). Both have
(11)
(12)
(13)
(14)
(15)
a significant mutual overlap [
156–174
]; analysis from
the Linear Sigma Model suggests that our K1 state has
a larger overlap with K1(1400) [
89
]. Nonetheless, we
will use m K1 = 1282 MeV for decays of excited states
involving K1 – this makes no significant difference to our
results since the decays with K1 final states are
phasespace suppressed for the mass range of excited mesons.
– The states η and η arise from mixing of ηN and ηS in
Lagrangian (3). The mixing angle is θη = −44.6◦ [
94
];
see also Refs. [
175–183
].
2.3 Excited scalars and pseudoscalars
2.3.1 Lagrangian
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:
LE = Tr[(Dμ E )†(Dμ E )] + α Tr[(Dμ E )†(Dμ )
2
G0
Tr( †E E )
+ (Dμ )†(Dμ E )] − (m0∗)2
− λ0
G
G0
2
Tr( †E
− λ1∗ Tr( †E E ) Tr( † )
− λ2∗ Tr( †E E
+
†
E )
− κ1 Tr( †E
− κ2[Tr( †E
− κ3 Tr( †E
− ξ1 Tr( †E
− κ4[Tr( †E E )]2
†
+
− ξ2 Tr( †E †E +
− ξ3 Tr( † E †E E +
− ξ4 Tr( †E E )2
E
†
+
+
+
†
†
†
†
+
†
E E
E ) Tr( † )
†)
E )]2
E ) Tr( †E E )
†
E
†)
†
E )
†E E †E )
+ Tr( †E E E1 + E †E E1)
+ c1∗[(det − det †E )2
+ (det † − det E )2] + c1∗E (det E − det
†E )2
h∗
+ 21 Tr( †E
+ h21∗E Tr( †E E ) Tr(L2μ + Rμ2)
+ h2∗ Tr( †E Lμ Lμ
†
+ Rμ E
Rμ + Rμ E Rμ)
+ h2∗E Tr[|Lμ E |2 + | E Rμ|2]
† Lμ Lμ
+
†
+
†
E ) Tr(L2μ + Rμ2)
E
Table 1 Best-fit results for
masses of ground-state mesons
and pseudoscalar decay
constants present in Eq. (3),
obtained in Ref. [
94
]. The values
in the third column will be used
in this article in order for us to
remain model-consistent. Note
that the errors in the fourth
column correspond either to the
experimental values or to 5% of
the respective central values
(whichever is larger)
+ 2h3∗ Tr(Lμ E Rμ † + Lμ
+ 2h3∗E Tr(Lμ E Rμ †E ).
Rμ †E )
Fit (MeV)
and we also set E1 = diag{0, 0, SE }.
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 σNE and σSE 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
1 There is a subtle point pertaining to the condensation of excited states
in σ -type models: as discussed in Ref. [
184
], 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. [
111
].)
Observable
Model ground state assigned to
Experiment (MeV)
are 8 states in Eq. (17): σNE , σSE , a0E and K0 E (scalar) and ηNE ,
ηE , π E and K E (pseudoscalar); the experimental
informa
S
tion on states with these quantum numbers is at times limited
or their identification is unclear:
– Seven states are listed by the PDG in the scalar
isosinglet (I J PC = 00++) channel in the energy region up
to 2 GeV: f0(500)/σ , f0(980), f0(1370), f0(1500),
f0(1710), f0(2020) and f0(2100). The last two are
termed unestablished [
5
]; the others have been subject of
various studies in the last decades [
10,11,15–52,82,94
].
As mentioned in the Introduction, the general conclusion
is that the states up to and including f0(1710) are
compatible with having ground-state q¯q or q¯q¯qq structure; the
presence of the scalar glueball is also expected [
42,53–
72,83,95
]. However, none of these states is considered as
the first radial excitation of the scalar isosinglet q¯q state.
A decade ago, a new resonance named f0(1790) was
observed by the BES II Collaboration in the π π final
states produced in J / radiative decays [131]; there had
been evidence for this state in the earlier data of MARK
III [
185
] and BES [
186
]. Recently, LHCb has confirmed
this finding in a study of Bs → J / π π decays [
132
].
Since, as indicated, the spectrum of ground-state scalar
quarkonia appears to be contained in the already
established resonances, we will work here with the hypothesis
that f0(1790) is the first excitation of the n¯ n ground state
(≡ σNE ). The assignment is further motivated by the
predominant coupling of f0(1790) to pions [
131
].
The data of Ref. [
131
] will be used as follows: m f0(1790) =
(1790 ± 35) MeV and f0(1790)→ππ = (270 ± 45)
MeV, with both errors made symmetric and given as
arithmetic means of those published by BES II.
Additionally, Ref. [
131
] also reports the branching ratios
J / → φ f0(1790) → φπ π = (6.2 ± 1.4) · 10−4 and
J / → φ f0(1790) → φ K K = (1.6 ± 0.8) · 10−4.
Using f0(1790)→ππ = (270 ± 45) MeV and the
quotient of the mentioned branching ratios we estimate
f0(1790)→K K = (70±40) MeV. These data will become
necessary in Sects. 3.2 and 3.3. We note, however, already
at this point that the large uncertainties in f0(1790)
decays – a direct consequence of uncertainties in the
J / branching ratios amounting to ∼23% and 50% –
will lead to ambiguities in prediction of some decays
(see Sect. 3.3.1). These are nonetheless the most
comprehensive data available at the moment, and more data
would obviously be of great importance.
The assignment of our excited isoscalar s¯s state σSE will
be discussed as a consequence of the model [particularly
in the context of f0(2020) and f0(2100)].
– Two resonances are denoted as established by the PDG
in the I J PC = 10++ channel: a0(980) and a0(1450)
[
5
]. Various interpretations of these two states in terms
of ground-state q¯q or q¯q¯qq structures or meson–meson
molecules have been proposed [
20,23,24,26,28,30–32,
36–41,43,49,52,73,74,76
].
Recently, the BABAR Collaboration [
187
] has claimed
the observation of a new resonance denoted a0(1950) in
the process γ γ → ηc(1S) → K¯ K π with significance
up to 4.2 σ . There was earlier evidence for this state in
the Crystal Barrel data [
188,189
]; see also Refs. [
190,
191
]. We will discuss the possible interpretation of this
resonance in terms of the first I J PC = 10++excitation
as a result of our calculations.
– Two resonances are candidates for the ground-state q¯q
resonance in the scalar-kaon channel (with alternative
interpretations – just as in the case of the a0
resonances – in terms of q¯q¯qq structures or meson–meson
molecules): K0 (800)/κ and K0 (1430); controversy still
surrounds the first of these states [
11,20,26,28,30–
32,34,35,37,39,49,74–76
].
A possibility is that K0 (1950), the highest-lying
resonance in this channel, represents the first excitation,
although the state is (currently) unestablished [
5
]. This
will be discussed as a result of our calculations later on.
– The pseudoscalar isosinglet (I J PC = 00−+) channel has
six known resonances in the energy region below 2 GeV
according to the PDG [
5
]: η, η (958), η(1295), η(1405),
η(1475) and η(1760).
Not all of them are without controversy: for example,
the observation of η(1405) and η(1475) as two different
states was reported by E769 [
192
], E852 [
193
], MARK
III [
194
], DM2 [
195
] and OBELIX [
196,197
], while they
were claimed to represent a single state named η(1440)
by the Crystal Ball [198] and BES [
199,200
]
Collaborations. It is important to note that a clear
identification of pseudoscalar resonance(s) in the energy region
between 1.4 GeV and 1.5 GeV depends strongly on a
proper consideration, among other, of the K K threshold
opening (m K + m K = 1385 MeV) and of the existence
of the I J PC = 01++ state f1(1420) whose partial wave
is known to influence the pseudoscalar one in
experimental analyses (see, e.g., Ref. [193]). A comprehensive
study of BES II data in Ref. [
201
], which included an
energy-dependent Breit–Wigner amplitude as well as a
dispersive correction to the Breit–Wigner denominator
(made necessary by the proximity to the K K
threshold), has observed only a marginal increase in fit quality
when two pseudoscalars are considered. In line with this,
our study will assume the existence of η(1440) to which
our ηSE state will be assigned. We will use mη(1440) =
(1432 ± 10) MeV and η(1440)→K K = (26 ± 3) MeV
[
199,200
] in Sects. 3.2 and 3.3.2; the error in the decay
width is our estimate. We emphasise, however, that our
results are stable up to a 3% change when η(1475) is
considered instead of η(1440).2
Our state ηNE will be assigned to η(1295) in order to test
the hypothesis whether an excited pseudoscalar
isosinglet at 1.3 GeV can be accommodated in eLSM (and
notwithstanding the experimental concerns raised in Ref.
[203]). We will use the PDG value mη(1295) = (1294±4)
MeV for determination of mass parameters in Sect. 3.2.
The PDG also reports total
η(1295) = (55 ± 5) MeV; the
relative contributions of η(1295) decay channels are
uncertain. Nonetheless, we will use total
η(1295) in Sect. 3.3.2.
– Two states have the quantum number of a pion excitation:
π(1300) and π(1800), with the latter being a candidate
for a non-q¯q state [
5
]. The remaining π(1300) resonance
may in principle be an excited q¯q isotriplet; however, due
to the experimental uncertainties reported by the PDG
[mπ(1300) = (1300±100) MeV but merely an interval for
π(1300) = (200 − 600) MeV] this will only be discussed
as a possible result of our model.
– Two states are candidates for the excited kaon: K (1460)
and K (1830). Since other excited states of our model
have been assigned to resonances with energies 1.4
GeV, we will study the possibility that our I J P = 21 0−
state corresponds to K (1460). This will, however, only be
discussed as a possible result of the model since the
experimental data on this state is very limited: m K (1460) ∼
1460 MeV; K (1460) ∼ 260 MeV [
5
].
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
res2 The η(1405) resonance would then be a candidate for the pseudoscalar
glueball [
202
].
onances 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 2. The data are used in Sect. 3.
2.3.2 Parameters
The following parameters are present in Eq. (16):
g1E , α, m0∗, λ0, λ1∗,2, κ1,2,3,4,
ξ1,2,3,4, SE , c1∗, c1∗E , h1∗,2,3, h1∗,E2,3.
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:
– All large-Nc suppressed parameters are set to zero since
their influence on the general phenomenology is expected
to be small and the current experimental uncertainties do
not permit their determination. Hence the parameters λ1∗,
h1∗ and κ1,2,3,4 are discarded.
– The parameter c1∗ is set to zero since it contains a term
∼ (det )2, which would influence ground-state mass
terms after condensation of σN and σS . Such
introduction of an additional parameter is not necessary since, as
demonstrated in Ref. [
94
], the ground states are very well
described by Lagrangian (3).
– As a first approximation, we will discard all parameters
that lead to particle mixing and study whether the
assignments described in Table 2 are compatible with
experiment. Hence we discard the parameters α, λ0 and ξ1; note
that mixing is also induced by κ1,2 and c1∗ but these have
already been discarded for reasons stated above.3
3 However, there would be no mixing of pseudoscalar isosinglets ηE
N
and ηSE 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).
I J P
00+
00−
00−
00+
10+
10−
Assignment
f0(1790)
η(1295)
η(1440)
Possible overlap with
f0(2020)/ f0(2100) to
be discussed as a
model consequence
Possible overlap with
a0(1950) to be
discussed as a model
consequence
Possible overlap with
π(1300) to be
discussed as a model
consequence
Possible overlap with
K0 (1950) to be
discussed as a model
consequence
Possible overlap with
K (1460) to be
discussed as a model
consequence
We use
– Parameters that lead to decays with two or more excited
final states are not of relevance for us: all states in the
model have masses between ∼ 1 GeV and ∼ 2 GeV and
hence such decays are kinematically forbidden.
(Parameters λ2∗ and ξ2 that contribute to mass terms are obviously
relevant and excepted from this criterion.) Hence we can
discard ξ3,4, c1∗E and h1∗,E2,3.
Note that the above criteria are not mutually exclusive:
some parameters may be set to zero on several grounds, such
as for example κ1.
Consequently we are left with the following undetermined
parameters:
g1E , m0∗, λ2∗, ξ2, SE , h2∗,3.
The number of parameters that we will actually use is even
smaller, as we discuss in Sects. 2.3.3 and 2.3.4.
This is obvious after substituting the strange condensate φS
by the non-strange condensate φN via Eq. (15). The modified
mass terms then read
2.3.3 Mass terms
The following mass terms are obtained for the excited states
present in the model:
ξ2.
Mass terms for all eight excited states can hence be described
in terms of only three parameters from Eq. (16): C1∗, C2∗ and
m2σNE = (m0∗)2 +
ma20E = (m0∗)2 +
λ2∗ + ξ2 2
2 φN ,
λ2∗ + ξ2 2
2 φN ,
m2π E = m2NE = (m0∗)2 +
η
m2ηSE = (m0∗)2 − 2 SE + λ2∗ − ξ2 φS2
m2σSE = (m0∗)2 − 2 SE + λ2∗ + ξ2 φS2,
E λ2∗ φ2N
m2K E = (m0∗)2 − S + 4
λ2∗ − ξ2 2
2 φN ,
λ2∗ φ2,
− √ξ22 φN φS + 2 S
m2
E λ2∗ φ2N
K E = (m0∗)2 − S + 4
0
λ2∗ φ2.
+ √ξ22 φN φS + 2 S
(20)
(21)
(22)
(23)
(24)
(25)
(26)
(27)
(28)
(29)
The mass terms (21)–(27) contain the same linear
combination of m0∗ and λ2∗:
λ2∗ φ2N ,
C1∗ = (m0∗)2 + 2
and the mass terms (24)–(27) contain the same linear
combination of λ2∗ and SE :
E
C2∗ = λ2∗ Z K fK (Z K fK − φN ) − S .
(30)
(31)
(32)
(33)
(34)
(35)
(36)
(37)
2.3.4 Decay widths
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
|k| 2
A→BC = I 8π m2A |MA→BC | ,
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 [
96
].
Depending on the final states, the interaction Lagrangians
presented in Appendix A can have one of the following
general structures:
– For a decay of the form S → P1 P2, where S is a scalar
and P1 and P2 are pseudoscalar particles, the generic
structure of the interaction Lagrangian is
LS P1 P2 = DS P1 P2 S P1 P2 + ES P1 P2 S∂μ P1∂μ P2
+ FS P1 P2 ∂μ S∂μ P1 P2,
where DS P1 P2 , ES P1 P2 and FS P1 P2 are combinations
of (some of the) parameters entering Lagrangian (16).
According to Eq. (37), the decay width reads in this case
|k|
S→P1 P2 = I 8π m2 |DS P1 P2 − ES P1 P2 K1 · K2
S
+ FS P1 P2 K · K1|2,
where K , K1 and K2 are respectively 4-momenta of S,
P1 and P2.
– For a decay of the form S → V P, where V is a vector
and P is a pseudoscalar particle, the generic structure of
the interaction Lagrangian is
LSV P = DSV P SVμ∂μ P,
where DSV P is a combination of (some of the) parameters
entering Lagrangian (16). The decay width reads in this
case
×
S→V P = I 8π|km| 2 DS2V P
S
(m2S − m2V − m2P )2
4m2V
− m2P .
– For a decay of the form S → V1V2, where V1 and V2 are
vector particles, the generic structure of the interaction
Lagrangian is
LSV1V2 = DSV1V2 SV1μV2μ,
where DSV1V2 is a combination of (some of the)
parameters entering Lagrangian (16). Then the decay width reads
(38)
(39)
(40)
(41)
(42)
|k| 2
S→V1V2 = I 4π m2 DSV1V2
S
(m2S − m2V1 − m2V2 )2
×
8m2V1 m2V2
As is evident from Appendix A, the most general
interaction Lagrangian for 3-body decays of the form S → S1 S2 S3
is
LSS1 S2 S3 = DSS1 S2 S3 S S1 S2 S3 + ESS1 S2 S3 S(∂μ S1∂μ S2)S3
+ (analogous terms with derivative couplings
among final states only). (44)
The ensuing formula for the decay width reads
1
S→S1 S2 S3 = I 32(2π )3m3S (mS1 +mS2 )2
(m23)max .
(mS−mS3 )2
× dm212
(m23)min .
dm223
MS→S1 S2 S3 2 (45)
where m212 = (K S1 + K S2 )2, m223 = (K S2 + K S3 )2 and
(E2∗)2 − m2S2 +
(E3∗)2 − m2S3
, (46)
(E2∗)2 − m2S2 −
(E3∗)2 − m2S3
, (47)
2
2
(m23)min . = (E2∗ + E ∗)2
3
(m23)max . = (E2∗ + E ∗)2
3
−
−
with
E2∗ =
m212 − m2S1 + m2S2 , E3∗ =
2m12
m2S − m212 − m2S3
2m12
. (48)
As is evident from Appendix A, our decay widths depend
on the following parameters: g1E , λ2∗, ξ2 and h2∗,3. 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, h2∗,3, can be calculated from decays with ground states
in the outgoing channels – we will discuss this in Sect. 3.3.
3 Masses and decays of the excited states: results and consequences
3.1 Parameter determination: general remarks
Combining parameter discussion at the end of Sects. 2.3.3
and 2.3.4, the final conclusion is that the following
parameters need to be determined:
C1∗, C2∗, ξ2, h2∗
and
h∗
3
(49)
with C1∗ and C2∗ parameter combinations defined in Eqs. (28)
and (29).
As is evident from mass terms (30)–(36) and Appendix
A, C1∗ and C2∗ 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, h2∗ and h3∗ appear only
in decay widths (with no excited final states). Hence our
parameters are divided in two sets, one determined by masses
(C1∗, C2∗ and ξ2) and another determined by decays (h2 and
h3).
Parameter determination will ensue by means of a χ 2 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 χ 2 fit is that error
calculation for parameters and observables is then
straightforward.
The general structure of the fit function χ 2 fit is as follows:
χ 2( p1, . . . , pm ) =
n
i=1
Oith.( p1, . . . , pm ) − Oexp. 2
i
Oexp.
i
for a set of n (theoretical) observables Oth. determined by
i
m ≤ n parameters p j . 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 Oiexp. and Oiexp.. Parameter errors pi are
calculated as the square roots of the diagonal elements of the
inverse Hessian matrix obtained from χ 2( p j ). Theoretical
errors Oi for each observable Oi are calculated by
diagonalising the Hesse matrix via a special orthogonal matrix M
M H M t ≡ diag{eigenvalues of H }
and rotating parameters pi such that
q = M ( p − pmin .)
Table 3 Masses of the excited
states present in the model.
Masses marked with an asterisk
are used as input. There is mass
degeneracy of σNE and aE
0
because we have discarded
large-Nc suppressed parameters
in our excited-state Lagrangian
(16) – see Sect. 2.3.2. The
degeneracy of ηNE and π E is a
feature of the model
Model state
σ E
N
ηNE
ηSE
σSE
aE
0
K0 E
π E
K E
(50)
(51)
(52)
I J P
00+
00−
00−
00+
10+
21 0+
10−
where p contains all parameters and pmin . realises the
minimum of χ 2( p1, . . . , pm ). Then we can determine Oi via
Oi =
n
j=1
∂ Oi (q1, . . . qm )
∂q j
at fit value of Oi
2
q j
(53)
(see also Chapter 39 of the Particle Data Book [
5
]).
3.2 Masses of the excited states
Following the discussion of the experimental data on excited
states in Sect. 2.3.1 and particle assignment in Table 2, we
use the following masses for the χ 2 fit of Eq. (50): mσNE ≡
m f0(1790) = (1790±35) MeV, mηNE ≡ mη(1295) = (1294±4)
MeV and mηSE ≡ mη(1440) = (1432 ± 10) MeV. Results for
C1∗, C2∗ and ξ2 are
With these parameters, the general discussion from Sect. 3.1
allows us to immediately predict the masses of σSE , a0E , K0 E ,
π E and K E . They are presented in Table 3.
3.3 Decays of the excited states
3.3.1 Hypothesis: f0(1790) is an excited q¯q state
We have concluded in Sect. 3.1 that only two parameters are
of relevance for all decays predictable in the model: h2∗ and
h3∗. They can be determined from the data on the f0(1790)
resonance discussed in Sect. 2.3.1: f0(1790)→ππ = (270 ±
forming the χ 2 fif0t(1d7e9s0)c→ribKeKd i=n S(7e0ct±.34.10)wMeeoVbta[i1n31th]e.
Pfoerl-45) MeV and
lowing parameter values:
h2∗ = 67 ± 63, h3∗ = 79 ± 63.
(55)
Large uncertainties for parameters are a consequence of
propagation of the large errors for f0(1790)→π π and
particularly for f0(1790)→K K . As described in Sect. 2.3.1,
f0(1790)→K K was obtained as our estimate relying upon
J / branching ratios reported by BES II [
131
] 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 4.
1961 ± 38
21 0−
The consequences of f0(1790) input data are then as
follows:
• The excited states are generally rather narrow with the
exception of f0(1790) and η(1440) whose full decay
widths, considering the errors, are, respectively, between
∼300 and ∼500 MeV and up to ∼400 MeV. The result for
f0(1790) is congruent with the data published by LHCb
[
132
]; the large interval for the η(1440) width is a
consequence of parameter uncertainties, induced by
ambiguities in the experimental input data.
• The excited pion and kaon states are also very
susceptible to parameter uncertainties that lead to extremely
large errors for the π E and K E decay widths [O(1
GeV)]. A definitive statement on these states is
therefore not possible. Contrarily, in the case of η(1295),
the three decay widths accessible to our model (for
ηNE → ηπ π + η π π + π K K ) amount to (7 ± 3) MeV
and hence contribute very little to the overall decay width
total
η(1295) = (55 ± 5) MeV.
• Analogously to the above point, parameter uncertainties
also lead to extremely large width intervals for the decays
of scalars into vectors. These decays are therefore omitted
from Table 4, except for the large-Nc suppressed decays
σSE → ρρ and σSE → ωω.
• Notwithstanding the above two points, we are able to
predict more than 35 decay widths for all states in our
model except π E and K E . The overall correspondence of
the model states to the experimental (unconfirmed) ones
is generally rather good, although we note that our scalar
s¯s state appears to be too narrow to fully accommodate
either of the f0(2020) and f0(2100) states. The mass of
our isotriplet state a0E is also somewhat smaller than that
of a0(1950) – we will come back to this point in Sect.
3.3.3.
ηNE
ηSE
π E
10−
21 0−
3.3.2 Hypothesis: η(1295) and η(1440) are excited q¯ q
states
As indicated above, results presented in Table 4 do not allow
us to make a definitive statement on all excited pseudoscalars.
However, the situation changes if the parameters h2∗ and h∗
3
are determined with the help of the η(1295) and η(1440)
decay widths.
Using ηNE →ηπ π +η π π +π K K = (55 ± 5) MeV [
5
] and
η(1440)→K K = 26 ± 3 MeV (from Ref. [
199
]; our estimate
for the error) we obtain
h2∗ = 70 ± 2,
h3∗ = 35 ± 3.
(56)
The parameters (56) are strongly constrained and there is a
very good correspondence of the pseudoscalar decays to the
Table 5 Decays and masses for the case where η(1295) and η(1440)
are enforced as excited q¯q states. Widths marked with an asterisk were
used as input. Pseudoscalar observables compare fine with experiment
Model state
I J P
00−
00−
experimental data in this case (see Table 5). Nonetheless,
there is a drawback: all scalar states become unobservable
due to very broad decays into vectors. Thus comparison of
Tables 4 and 5 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 q¯ q states.
A possible theoretical reason is that pseudoscalars above
1 GeV may have non-q¯ q 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 q¯ q and q¯ q¯ qq
structures. In addition, the flux-tube model of Ref. [202] and
a mixing formalism based on the Ward identity in Ref.
[
209
] 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
but the scalars are unobservable due to extremely broad decays into
vector mesons
–
As in Table 3
See Appendix A
Calculated via Eqs.
(39), (41), (43),
(45) and Eq. (53)
Unobservable due to extremely
large decays into vectors
[O(1 GeV)]
55 ± 5*
26 ± 3*
3 ± 0
Suppressed
29 ± 3
Theory
m MeV
2000
1800
1600
1400
1200
1000
IJP
m MeV
2000
1800
1600
1400
1200
1000
Experiment
00
Fig. 1 Masses of excited q¯q states with isospin I , total spin J and
parity P from the Extended Linear Sigma Model (left) and masses from the
experimental data (right). Area thickness corresponds to mass
uncertainties on both panels. The lower 00+(≡ σNE ), both 00−(≡ ηNE and ηS )
E
as well as the 10+(≡ a0E ) states from the left panel were used as input.
Lightly shaded areas correspond to experimentally as yet unestablished
states. Table 6 contains the experimental assignment of the states on the
left panel and a brief overview of their dynamics
require implementation of mixing scenarios in this
channel.4
Note, however, that the results of Table 5 depend on the
assumption that the total decay width of η(1295) is
saturated by the three decay channels accessible to our model
(ηπ π , η π π and K π π ). The level of justification for this
assumption is currently uncertain [
5
]. Consequently we will
not explore this scenario further.
3.3.3 Is a0(1950) of the BABAR Collaboration an excited
qq state?
¯
Encouraging results obtained in Sect. 3.3.1, where f0(1790)
was assumed to be an excited q¯ q 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
a0(1950) = (271 ± 40) MeV [
187
].
Assuming that f0(1790) is an excited q¯ q 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 σNE and σSE , Eqs. (30)
and (34), can be modified by reintroduction of the large-Nc
suppressed parameter κ2 and now read
m2σ E = C1∗ +
N
ξ2
2 + 2κ2
2
φN ,
(57)
4 A 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 q¯q state in Ref. [102].
m2σ E = C1∗ + 2C2∗ +
S
ξ22 + 2κ2 (φN − 2 Z K f K )2.
(58)
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 a0E corresponds
exactly to that of a0(1950). We obtain
Note that a non-vanishing value of κ2 introduces mixing of
σNE and σSE 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. 1
and for the decays in Table 6. The values of ma E , mσ E and
m K E have changed in comparison to Table 4 i0nduciSng an
incr0eased phase space. For this reason, the decay widths of
the corresponding resonances have changed as well. All other
results from Table 4 have remained the same and are again
included for clarity and convenience of the reader.
The consequences are as follows:
• The decay width of a0E is now a0E = (280 ± 90) MeV;
it overlaps fully with a0(1950) = (271 ± 40) MeV
measured by BABAR. Hence, if a0(1950) is confirmed in
future measurements, it will represent a very good
candidate for the excited isotriplet n¯ n state.
IJP
(59)
Table 6 Final results: decays and masses of the excited q¯q states. Widths marked as “suppressed” depend only on large-Nc suppressed parameters
that have been set to zero. Masses/widths marked with (*) are used as input; others are predictions
Model state I J P Mass (MeV) Decay
Width (MeV) Note
σ E
N
a0E
ηNE
ηSE
σSE
00+
1790 ± 35*
10+
• The mass of σSE is between those of f0(2020) and
f0(2100). Judging by the quantum numbers, either of
these resonances could represent a (predominant) s¯s
state; an option is also that the excited s¯s state with
I J PC = 00++ has not yet been observed in this energy
region. However, one must also remember the
possibility that q¯q–glueball mixing (neglected here) may change
masses as well as decay patterns. The decay width of σSE
is rather narrow (up to 110 MeV) but this may change if
mixing effects happen to be large.
• The mass of K0 E is qualitatively (within ∼ 100 MeV)
congruent with that of K0 (1950); the widths overlap
within 1 σ . Hence, if K0 (1950) is confirmed in future
measurements, it will represent a very good candidate
for the excited scalar kaon.
• Conclusions for all other states remain as in Sect. 3.3.1.
4 Conclusion
We have studied masses and decays of excited scalar and
pseudoscalar q¯q 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 n¯ n state. This assignment
was motivated by the observation in BES [
131
] and LHCb
[
132
] data that the resonance couples mostly to pions and
by the theoretical statement that the n¯ n ground state is
contained in the physical spectrum below f0(1790).
Furthermore, the assumption was also tested that the a0(1950)
K0 E → K π
K0 E → K1π
K E
0 → a1(1260)K
K0 E → ηK
K E
0 → f 1(1285)K
K0 E → K1η
K0 E → K0 (1430)π π
Total
–
–
Candidate state: K0 (1950);
m K0 (1950) = (1945 ± 22) MeV and
ReKq0 u(1i9r5e0s)c=on(fi2r0m1a±tio9n0[)5M]eV.
Width badly defined due to large
errors of the experimental input data
Width badly defined due to large
errors of the experimental input data
resonance, whose discovery was recently claimed by the
BABAR Collaboration [
187
], 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 6.
In essence: the f0(1790) resonance emerges as the
broadest excited q¯q state in the scalar channel with f0(1790)
= (405 ± 96) MeV; a0(1950), if confirmed, represents a
very good candidate for the excited q¯q state; K0 (1950), if
confirmed, represents a very good candidate for the excited
scalar kaon.
Our excited isoscalar s¯s 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 q¯q 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
ηπ π , η π π and π K K 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 ηto(t1a2l95) ≡ η(1295)→ηππ
+η ππ+π K K and η(1440)→K K exactly as in the data of PDG
[
5
] and BES [
199
]. Then π(1300) and K (1460) are quite well
described as excited q¯ q states – but the scalars are
unobservably broad (see Table 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 q¯ q 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 ηπ π , η π π
and π K K decays. It could therefore only be explored further
when (very much needed) new experimental data arrives –
from BABAR, BES, LHCb or PANDA [
81
] and NICA [
212
].
Acknowledgements 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.
Open Access This article is distributed under the terms of the Creative
Commons Attribution 4.0 International License (http://creativecomm
ons.org/licenses/by/4.0/), which permits unrestricted use, distribution,
and reproduction in any medium, provided you give appropriate credit
to the original author(s) and the source, provide a link to the Creative
Commons license, and indicate if changes were made.
Funded by SCOAP3.
Appendix A: Interaction Lagrangians
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 σ E
N
The Lagrangian reads
LσNE = 2
(h2 − h3)wa21 Zπ2 φN σNE (∂μηN )2 + (∂μπ )2
1
+ 2
1
+ 2
×
√
√
h2φN −
2h3φS w2K1 Z 2K σNE
× ∂μ K¯ 0∂μ K 0 + ∂μ K −∂μ K +
+ (h2 − h3)wa1 Zπ φN σNE
h2φN −
2h3φS wK1 Z K σNE
K¯ 10μ∂μ K 0 + K1−μ∂μ K + + h.c.
1 μ 2
+ 2 (h2 + h3)φN σNE (ωN ) + (ρμ)2
1 √
+ 2 h2φN + 2h3φS σNE
K¯ μ0 K μ0 + Kμ− K μ+
− ξ2 Zπ φN σNE π E · π − g1E wa1 Zπ σNE ∂μπ E · ∂μπ
1
+ 2 (h2 − h3)wa21 Zπ2 σNE σN (∂μπ )2.
(A.1)
f1μN ∂μηN + a1μ · ∂μπ
Note: the decay σ E
N
pressed.
Appendix A.2: Lagrangian for σSE
→ ηS ηS (∼ κ1, h1) is large-Nc
supwK1 Z K σSE
σ E
S
K¯ μ0 K μ0 + Kμ− K μ+ .
(A.2)
+
+
×
+
Note: the decays σSE → π π (∼ κ1, h1), σSE → ηN ηN
(∼ κ1, h1), σSE → ρρ (∼ h1), σSE → ωN ωN (∼ h1), σSE →
a1π (∼ h1), σSE → f1N ηN (∼ h1), σSE → π E π (∼ κ2),
σSE → ηNE ηN (∼ κ2) and σSE → σS π π (∼ κ1, h1) are
largeNc suppressed.
Appendix A.4: Lagrangian for K0E
The Lagrangian reads (only K00E included; decays of other
K0E components follow from isospin symmetry):
+ 2 √i2(h2 +2h3)wa1w∗K Zπ2 ZKS K00E
× π+∂μK0−∂μπ0 −π0∂μK0−∂μπ+ . (A.4)
× ∂μK¯0∂μηN −∂μK¯0∂μπ0 +√2∂μK−∂μπ+
LK0E = 41 h2 φN +√2φS −2h3φN wa1wK1ZπZK K00E Onlythree-bodydecaysintopseudoscalarsarekinematically
allowed for this particle:
Appendix A.5: Lagrangian for ηNE
1
LηNE = 2(h2 − h3)wa21Zπ3 ηNEηN(∂μπ)2
+ (h2 − h3)wa21Zπ3 ηNE ∂μηN∂μπ · π
1
− 4(h2 − 2h3)wa1wK1ZπZ2K ηNE
× K¯0∂μK0∂μπ0 − √2K¯0∂μK+∂μπ−
− K−∂μK+∂μπ0 − √2K−∂μK0∂μπ+ + h.c.
1
− 2h2w2K1ZπZ2K ηNE
× π0∂μK¯0∂μK0 − π0∂μK−∂μK+
−√2π−∂μK+∂μK¯0 + h.c. .
(A.5)
(A.6)
×wf1SwK1ZηS ZK K00E∂μK¯0∂μηS
+ 41 h2 φN +√2φS −2h3φN wK1ZK K00E
× f1μN∂μK¯0 −a1μ0∂μK¯0 +√2a1μ+∂μK−
+ 41 h2 φN +√2φS −2h3φN wa1Zπ K00E
× K¯10μ∂μηN − K¯10μ∂μπ0 +√2K1−μ∂μπ+
i
+ 4(h2 −2h3)wa1w∗K Zπ2 ZKS K00Eπ0∂μK¯00∂μπ0 LπE = −ih3wa1ZπφN π0E ρμ−∂μπ+ −ρμ+∂μπ−
Appendix A.6: Lagrangian for ηSE
The Lagrangian reads
i
LηSE = −√2h3wK1ZKφN ηSE ∂μK¯0K μ0 +∂μK−K μ+ + h.c.
−√2K¯0∂μK+∂μπ−
+ (h2 −h3)wa21Zπ3 ηNE ∂μηN∂μπ ·π
− 2√12h2wa1wK1ZπZ2K ηSE K¯0∂μK0∂μπ0
−K−∂μK+∂μπ0 −√2K−∂μK0∂μπ+ + h.c.
1
+ √2h3w2K1ZπZ2K ηSE
× π0∂μK¯0∂μK0 +π0∂μK−∂μK+
+√2π−∂μK+∂μK¯0 + h.c. .
Note: the decay ηSE → ηSππ (∼ κ1, h1) is large-Nc
suppressed.
Appendix A.7: Lagrangian for πE
The Lagrangian reads (only π0E included; decays of π±E
follow from isospin symmetry):
+ 41 h2 −2h3 wa1wK1ZπZ2K π0E∂μπ0
× K¯0∂μK0 + K−∂μK+ + h.c.
1
+ 2 h2w2K1 Zπ Z 2K π 0E π 0 ∂μ K¯ 0∂μ K 0 + ∂μ K −∂μ K +
h2 + 2h3 wa1 wK1 Zπ Z 2K π 0E
The Lagrangian reads (only K 0E included; decays of other
K E components follow from isospin symmetry):
wK1 Z K K 0E
ωN μ∂ μ K¯ 0 − ρμ0∂ μ K¯ 0 +
2ρμ+∂ μ K −
wa1 Zπ K 0E
K¯ μ0∂ μηN − K¯ μ0∂ μπ 0 +
2Kμ−∂ μπ +
× w f1S ZηS K 0E K¯ μ0∂ μηS
1
− 2
√
K¯ 0∂μηN ∂ μπ 0
2K −∂μηN ∂ μπ +
(h2 − 2h3)wa1 wK1 Zπ2 Z K K 0E
π 0∂μηN ∂ μ K¯ 0
2π +∂μηN ∂ μ K − + ηN ∂μπ 0∂ μ K¯ 0
2ηN ∂μπ +∂ μ K −
1
+ √ h3wa1 w f1S Zπ Z K ZηS K 0E
2
1
− √ h2wK1 w f1S Zπ Z K ZηS K 0E
2 2
π 0∂μηS ∂ μ K¯ 0 −
2π +∂μηS ∂ μ K −
1
− √ h2wa1 wK1 Zπ Z K ZηS K 0E
2 2
√
√
ηS ∂μπ 0∂ μ K¯ 0 −
2ηS ∂μπ +∂ μ K −
h2wa21 Zπ2 Z K K 0E K¯ 0(∂μπ )2
× wa1 wK1 Zπ2 Z K K 0E π +∂μ K¯ 0∂ μπ −
1
+ √ (h2 + 2h3)wa1 wK1 Zπ2 Z K K 0E
2 2
×
π +∂μ K −∂ μπ 0 − π 0∂μ K −∂ μπ + .
(A.8)
Page 22 of 25
Page 24 of 25
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