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We consider the two-flavor version of the extended linear sigma model (eLSM), which contains (pseudo)scalar and (axial-)vector quark-antiquark mesons, a scalar glueball [predominantly corresponding to

A major task in low-energy hadron physics is the unified description of masses, decays, and scattering properties (including scattering lengths, phase shifts, etc.) of all light hadrons (both mesons and baryons) below

Since quantum chormodynamics (QCD), the fundamental theory of the strong interaction, cannot be directly solved in the low-energy domain, various methods were developed to describe mesons and baryons. The relativistic quark model of Refs.

Another line of research has been the development of effective chiral approaches. Some make use of quark degrees of freedom, such as the famous Nambu–Jona-Lasinio (NJL) model

Another approach to describe meson properties in the low-energy domain is chiral perturbation theory (ChPT); see, e.g., Refs.

Other chiral approaches, so-called linear sigma models (see, e.g., Refs.

In general, the eLSM offers a satisfactory description of hadronic properties below 2 GeV (see Ref.

There is nowadays consensus that these resonances are most likely four-quark states. This still leaves different possibilities for the internal structure of these states: following the original proposal by Jaffe

Alternatively, the light scalar mesons could be (loosely bound) molecular states formed from, or unbound states in the scattering continuum of

The scalar state

The main goal of the present work is the inclusion and systematic investigation of the light four-quark state

This paper is organized as follows: in Sec.

In the baryonic sector, we make use of the so-called mirror assignment, first proposed in Ref.

Then, in Sec.

For an arbitrary number of flavors we can arrange (pseudo)scalar quarkonium fields into multiplets using the current

The Lagrangian

There are several ways to incorporate four-quark states into a chiral model. Here, we will follow the approach of Ref.

Writing down all relevant terms for three flavors facilitates an extension of the current work to the case of three flavors.

We will be able to compare our approach with other ones, e.g., those of Refs.

It is easier to see which terms are large-

For three flavors a four-quark nonet has the same chiral structure as the quarkonium nonet, while for

Here, we use a diquark-antidiquark picture as a concrete framework to construct the multiplet of light scalars and to couple them to conventional mesons. However, it is also possible to construct the same terms in the meson-meson molecular picture

For

Considering only scalar diquarks, we can therefore construct a scalar four-quark nonet by
^{1}

We note that this term is precisely the same as the one in Eq. (16) of Ref.

For two flavors, Eq.

Very similar terms are obtained for the coupling to (axial-) vector quarkonia:

Introducing also a kinetic and mass term for the scalar tetraquark, we find the complete two-flavor four-quark Lagrangian to be

The terms in the Lagrangian which, upon condensation of

Baryons are implemented in the eLSM in the so-called mirror assignment

The mirror assignment allows for the existence of a new chirally invariant mass term, which contributes to the baryon masses in a different manner than the chiral condensate. Thus, baryons can have nonzero masses even when the chiral condensate vanishes. Demanding dilatation invariance, the new mass term must necessarily arise from coupling the baryons to the four-quark field

Furthermore, it is possible to introduce another interaction term that violates

The terms in the baryon Lagrangian which are relevant for pion-nucleon scattering are then

Upon condensation of

We recall that the quantity

In this section we first perform a global fit of the parameters in the meson sector. Here we consider two different scenarios: first we neglect the scalar glueball and investigate the mixing of the scalar four-quark with the quarkonium state only. Then, we present the results for the full three-scalar mixing problem, which includes the scalar glueball, the four-quark, and the quarkonium state. This allows us to estimate the importance of the scalar glueball for the calculation of the decay widths of the scalar-isoscalars and the pion-pion scattering lengths. Subsequently, we will take the results from the global fit of the meson sector and calculate pion-nucleon scattering parameters.

The assignment of our effective hadronic degrees of freedom is given in Table

The masses of the fields as given by the PDG

The decay widths

Furthermore, the physical

The pion-pion scattering parameters in the eLSM were first calculated in Ref.

The Lagrangians

In the upper left box the fitted parameters are given. The parameters in the lower left box are calculated from the fitted parameters.

Next, we consider the scalar glueball as dynamical field as well. Now,

The result of the fit where the glueball is included.

From this fit the following scalar-isoscalar mixing matrix is obtained,

Our aim was to correctly reproduce the masses and decay widths of

The parameter determining the mixing of the four-quark state with the quarkonium state is

Although the value of

Since

We also tried to identify

The value of

We obtain a fit of similar quality if we assign

The elements of the matrix

We find in both fits very similar values for the pion-pion scattering lengths, indicating that the scalar glueball is actually not important for pion-pion scattering, which is not too surprising because of its large mass.

We checked that the pion-pion scattering lengths vanish in the chiral limit, i.e.,

To further underline the importance of a light scalar-isoscalar resonance we can take the limit

Some of the parameters of the baryon Lagrangian have been already determined in Ref.

Parameters determined by a fit of

In Ref.

Isospin-odd scattering parameters.

As experimental inputs for the scattering lengths, we use the results of the analysis of Refs.

The isospin-even scattering length

Isospin-even scattering parameters for

In addition, here we also calculate isospin-even and isospin-odd scattering volumes and effective range parameters within the setup of Ref.

The theoretical values in Tables

In Table

Results with

In Tables

Results with

Results with

We now perform a simultaneous

Best fit where

In this paper, we studied the influence of the light four-quark state

Then we studied the role of

For instance, the inclusion of the

Another straightforward extension of this work would be to consider the three-flavor case. Two additional resonances appear in the scalar-isoscalar sector: the strange-antistrange quarkonium [predominantly

In the baryonic sector, an interesting future work would be to continue the nonzero-density study of Ref.

The authors thank the

We used ^{®} for the numerical evaluation and the fits performed in this work. The notebooks can be found on GitHub

As in the meson sector, we construct possible tetraquark-baryon interaction terms for

After spontaneous symmetry breaking, the field

The masses of the resonances are calculated from the second partial derivative of the potential density [which is obtained from Eqs.

The vacuum expectation value of

The shift

We are interested in the decay of the scalars

The scattering amplitude is calculated from the tree-level amplitudes in Fig.

The dashed lines correspond to the pion, the wavy line to the

The scattering amplitude in the

On the other hand, the scattering amplitude for isospin

After the introduction of the scalar four-quark field, the term

From Eqs.

For the pion-pion scattering amplitude, we simply need to add the corresponding expression for the scalar glueball exchange to Eq.

The Feynman diagrams which contribute to the pion-nucleon scattering amplitude are shown in Fig.

Pion-nucleon scattering diagrams at tree level. The solid line represents the nucleon, the double line its chiral partner, the dashed line the pion, the double-dashed the scalar-isoscalars

The partial amplitudes

The correct result for the partial amplitudes for the model of Ref.

Errors in Eq. (19) of Ref.

In order to obtain the expressions for the scattering parameters for our model with scalar four-quark state and dynamical scalar glueball we need to modify the pion-nucleon scattering amplitudes by replacing the last term of