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Status of light scalar mesons as non-ordinary mesons

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Status of light scalar mesons as non-ordinary mesons

J.R. Peláez

Departamento de F́ısica Teórica II. Universidad Complutense.28040 Madrid-SPAIN

E-mail: jrpelaez@fis.ucm.es

Abstract. In this talk I briefly review the status of the f0(500) and f0(980) together with
the other light scalar resonances, as well as the emerging picture of a non-ordinary light meson
multiplet, paying particular attention to unitarized Chiral Perturbation Theory, large Nc, semi-
local duality and Regge theory arguments.

1. Introduction and present status

The lightest scalar meson, recently renamed f0(500), but traditionally known as the σ meson,
has been a longstanding puzzle in our understanding of strong interactions. This resonance
is very relevant both for Nuclear and Particle Physics. For the former because, on a first
approximation, it is largely responsible for nucleon-nucleon attraction, and for the latter due
to its role in spontaneous chiral symmetry breaking and the identification of glueballs - two
fundamental features of QCD. Moreover, there is an ongoing worldwide experimental program
on strong interactions (CERN, China, Germany, Japan & USA), even looking for New Physics,
where many results depend crucially on final state interactions of pions, producing scalar mesons
in general and the f0(500) in particular. Despite this relevance, the σ properties, nature, and
even its very existence have been controversial for decades, since it was first proposed almost
60 years ago. As an example of this debate, the σ (with different names) appeared listed in the
early editions of the Particle Data Tables (PDT) [1], disappeared for quite some time of those
lists, and made it back to the tables in 2002. Surprisingly, despite being quoted with a huge
mass uncertainty from 400 to 1200 MeV and a huge uncertainty in the width, it was nevertheless
considered a “well established state”.

Finally, and after more than five decades, the controversy on the f0(500) mass and width
seems to be settled. This has been rather well known to specialists for quite some time
[2, 4, 5, 6, 3, 7, 8, 9, 10] but remained less well known to the Particle and Nuclear Physics
community at large. Actually, this situation has been finally acknowledged with a major revision
of its properties in the 2012 edition of the PDT[1], where, after more than a decade of quoting
such a disproportionately huge mass uncertainty, it has been reduced by a factor of more than
five ( and the width uncertainty by more than two), settling the case for the so-called “light
sigma” scenario, and opening the door for an even more radical revision in the future, in terms
of current dispersive analyses (see the “Note on light scalars” in the PDT).

Of course, the evidence that supports this recent recognition has been accumulating over
many years both from the experimental and theoretical side, using a wide variety of techniques,
which not only have allowed for precise and model independent determinations of the mass and
width of this meson, but which have advanced our understanding of its non-ordinary features.

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In addition, in the 2012 PDT edition, the f0(980) has also suffered a minor revision, by
changing the central value of its mass from 980 to 990 MeV whereas its mass uncertainty has
doubled to 20 MeV. Other possible multiplet partners of the f0(500), as the a0(980) and K(800)
have not suffered any changes in their quoted parameters in the last PDT edition. Note that
the K(800) is still considered as ”not well established” despite its pole has already been well
established within a rigorous dispersive approach [36]. The K(800) is therefore in a situation
somewhat similar to that of the f0(500) a few years ago: Its existence is widely accepted by the
“scalar-community” but it just lacks recognition in the PDT.

Historically, the first proposal of a scalar isoscalar field was made in the 50’s [11] and it was
soon realized by J. Schwinger that expectation values of such a scalar singlet field, which he
called σ [12], could be of relevance for the breaking of symmetries. Actually it soon became
relevant for simple models of the attractive part of the nucleon-nucleon interaction and for
the spontaneous chiral symmetry breaking of QCD. For the latter one should mention the
traditional Linear Sigma Model [13], but also the more modern and systematic approach of
Chiral Perturbation Theory [15], including its unitarization [16, 17, 4, 5, 6, 18], which is needed
to generate resonances. Apart from the questioning of their existence, light scalars have faced the
controversy on their classification in multiplets [19, 2, 9, 10] leading to the suggestion that they
may have a predominantly non-ordinary quark-antiquark nature [19, 20], which has received a
recent strong support based on the Nc dependence [21, 22, 23, 24, 25], unusual Regge behavior
[26], etc... On the experimental side, the clarification of these controversial issues was hindered by
the difficult observation from nucleon-nucleon interactions and the large systematic uncertainties
in traditional pion-nucleon experiments used to extract pion-pion scattering [27]. However, more
recent ”production” experiments, with different systematics from scattering processes, have been
very relevant in the recent clarification in the PDT (like [28]) although these also may have
problems of their own (use of Breit-Wigner parametrizations, isobar models, etc...). Finally, the
results that have partly driven the major σ revision in the PDT have been the latest NA48/2
data on [30] K → ππℓν experiments (some older can be found on [29]) together with the most
recent dispersive approaches that have provided a rigorous treatment of this resonance.

2. The f0(500) parameters

As just commented, one of the main difficulties to present a convincing case for the existence of a
σ resonance pole around 500 MeV and not higher —the so-called “light sigma” scenario— were
due to: i) the poor data on ππ scattering [27], ii) its large width and iii) the fact that the σ does
not present the familiar peak and shape of a Breit-Wigner resonance. A description in terms of
poles in the complex plane is more appropriate, although, being so wide, its associated pole is
deep in the complex plane. This demands a careful, rigorous and model independent analytic
extension after the amplitudes that describe the data have been constrained with dispersion
relations. The precise analyses of ππ scattering with dispersion relations [38, 33, 40] and their
later use to find the associated poles [33, 34, 35, 36] have driven the major revision in the PDT
[1], which now reads 400−550 MeV for the mass and 400−700 MeV for the width, as defined from
the associated pole. Actually, the PDT mass and with new estimates are ”conservative”, and
the PDT also provides a “more radical point of view”, relying only on such model independent
dispersive approaches, yielding a mass of 446± 6 MeV and a width of 552± 10 MeV.

The dispersive results are also responsible for the minor change in the f0(980) parameters
and, when applied to πK scattering, for the most reliable K(800) determination [36]. I have
no space to discuss them here in detail but can only refer to the appropriate references on Roy
equations [39, 38, 33, 36], GKPY equations [40], Forward Dispersion relations [32, 41] and some
pedagogical introductions [31, 32].

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Journal of Physics: Conference Series 562 (2014) 012012 doi:10.1088/1742-6596/562/1/012012

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3. Chiral Symmetry and the f0(500)
Within Particle Physics, part of the interest on the f0(500) is due to its role in the spontaneous
chiral symmetry breaking of QCD. Actually, the traditional name “sigma”, given by J.Schwinger
[12], was later adopted by the classic ‘Linear Sigma Model”(LSM) formulation of such a breaking
[13]. The LSM has played a very important role in our phenomenological and more intuitive
description of low energy hadronic interactions, which has its success and shortcomings, leading
to multiple variations, i.e. [3]. Similarly, a relatively light scalar-isoscalar meson coupling
strongly to ππ, and therefore very wide, also seems to be generated in a Nambu-Jona-Lasinio
(NJL) model or its modifications within Hadron Physics [14]. Still, both the LSM and NJL
are simple models that, despite capturing many relevant features, do not provide a systematic
description of low energy hadron Physics and a clear connection to QCD, and usually require
some ad-hoc modifications. Nevertheless, a systematic analysis exists under the form of the
more modern Chiral Perturbation Theory (ChPT), which is the low energy effective Theory of
QCD. Within standard ChPT [15], only Goldstone modes appear explicitly in the Lagrangian,
and the dynamics is encoded in a set of ”low energy parameters”, whose values are saturated
[37] by heavier resonances, but not by the sigma, which is another hint of its non-ordinary
nature. ChPT is limited to low energies and resonances are not present in its Lagrangian,
but can be generated when combined with some dispersive techniques. This approach, less
systematic, known as unitarized Chiral Perturbation Theory [16, 17], has been very successful
in the description of light scalars and is particularly simple in the case of the f0(500) [4].
These unitarization techniques allow to study the quark mass dependence of the σ [42], also
recently studied on the lattice despite the difficulties in providing results for physical quark
masses [43]. Other models in the literature deal with the f0(500) together with the other light
scalars, like extended linear sigma models [3], non-linear models [9, 10], as well as the simple but
very successful Chiral Unitary Approach [5, 6], which is tightly connected to Unitarized Chiral
Perturbation Theory described before. The lesson to learn from all them is that even very simple
descriptions of low energy data, when implementing some basic chiral symmetry constraints and
using a decent analytic extrapolation, yield a f0(500) pole in fair agreement with the precise
and model independent dispersive determinations commented in the previous section.

4. The spectroscopy and nature of light scalars

These are the f0(500) features that are still under some discussion, although the situation has
also clarified considerably over the last years. The first problem with spectroscopy is how light
scalars are assigned to U(3) multiplets. There is a very strong support [19, 2, 3, 5, 6, 8] for
the sigma to belong to a nonet with the other lightest scalars, namely the f0(980), a0(980)
and K(800). Still, this assignment is obscured by the possible different mixing schemes [45, 10]
within states of the multiplet (or even other multiplets) as well as by the controversial status of
some scalar mesons, like the K(800) or the f0(1370).

Somewhat more controversial is the composition of these resonances in terms of quarks and
gluons. It was long ago suggested that the ordinary identification of mesons as qq̄ states did not
match well the light scalars due to their inverted hierarchy [19] as well as their large widths and
couplings [44]. Of course, attention has to be paid to what is meant by “composition”, since the
intuitive Foch space decomposition is not always well defined. Most interpretations are made
within some kind of ”constituent models”. However there are recent attempts to tackle this
problem from QCD via the 1/Nc behavior of the f0(500) pole generated from unitarized ChPT
[22, 8, 23], or, more recently, also on the lattice [49]. Unitarized ChPT supports a predominantly
non-qq̄ nature for the f0(500) providing a very strong support for the f0(500) meson, since it
depends on Nc in a rather different way than predicted from QCD for qq̄ mesons [46]. This
effect was also observed in ππ scattering by explicit introduction of resonant states [21, 24].
Furthermore, the same conclusion can be obtained without using a particular model, but just

4th International Hadron Physics Conference (TROIA’14) IOP Publishing
Journal of Physics: Conference Series 562 (2014) 012012 doi:10.1088/1742-6596/562/1/012012

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from data, the pole positions and the general behavior of qq̄ states in QCD [25]. Moreover,
larger Nc extrapolations [50] suggest that, even if it is not predominantly an ordinary meson, it
might contain some small admixture of a heavier qq̄ state [23]. Note this is the 1/Nc expansion
around Nc = 3 [47], although some results are also available for the ”large Nc limit” [48].

Finally, since the f0(500) has not been commonly included in the ordinary classifications of
hadrons into linear Regge trajectories with a universal slope [51], its Regge trajectory has been
shown not to follow a straight line and that its slope is of a very different order of magnitude
than ordinary mesons [26], once again supporting the non-ordinary nature of this state.

5. Acknowledgements

I thank the organizers of Troia14 for their kind hospitality and for creating such a nice scientific
environment. Work supported by the Spanish Research contract FPA2011-27853-C02-02 and
the EU-Research Infrastructure Integrating Activity “Study of Strongly Interacting Matter”
(acronym HadronPhysics2, Grant Agreement n. 227431, 7th Framework Programme).

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