Role of a four-quark and a glueball state in pion-pion and pion-nucleon scattering

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 $f_{0}(1710)$), as well as the nucleon and its chiral partner. We extend this model by the additional light scalar meson $f_{0}(500)$, predominantly a putative four-quark state. We investigate various interaction terms of the four-quark and glueball states with the other particles, some of which preserve and some of which explicitly break the $U(1)_{\textrm{A}}$ symmetry. We test our model by performing a global fit to masses and decay widths of the scalar resonances and pion-pion scattering lengths. We also discuss the influence of the scalar four-quark state and the glueball on the baryon sector by evaluating pion-nucleon scattering parameters. We find that the inclusion of $f_{0}(500)$ improves the description of pion-pion and pion-nucleon scattering lengths.


I. INTRODUCTION
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 ∼2 GeV [1]. This problem is exceptionally difficult, due to the large number of hadrons and the intrinsically strong interaction between them.
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. [2,3] solidly reproduces properties of conventional quark-antiquark and threequark states. Even after many years, it still provides a useful starting point for many considerations. Yet, the effect of mesonic quantum corrections are not taken into account [for an extension in this direction, see Ref. [4]] and various candidates for non-conventional mesons (such as glueballs, hybrids, and multiquark states) cannot be easily accounted for. On the other hand, numerical simulations of QCD on the lattice are now capable of reproducing a large part of the QCD spectrum [see e.g. Ref. [5]]. Nowadays even scattering lengths can be computed, see e.g. Ref. [6]. However, there is still a long way to go towards an exhaustive description of all properties of low-energy QCD using lattice simulations.
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 [7][8][9][10][11] [and the related quark-meson model, see Refs. [12][13][14] and refs. therein]. More recently, Dyson-Schwinger equations have been employed to calculate meson and baryon masses in an approach which starts directly from the QCD Lagrangian and respects chiral symmetry [for reviews, see Refs. [15,16]].
Another approach to describe meson properties in the low-energy domain is chiral perturbation theory (ChPT), see e.g. Refs. [17][18][19][20][21]. It is based on a non-linear realization of chiral symmetry and the primary method to study hadronic low-energy properties in a systematic and welldefined way. ChPT is originally devised to study the interactions of the (pseudo)-Goldstone bosons emerging from chiral symmetry breaking, i.e., for two quark flavors the pions. The description of other, heavier mesons becomes more difficult [22,23].
Other chiral approaches, so-called linear sigma models [see e.g. Refs. [24][25][26][27][28][29]], are based on the linear realization of chiral symmetry, hence contain hadrons and their chiral partners on the same footing. In particular, within the last ten years a chiral model, called extended Linear Sigma Model (eLSM), has been developed [27,28,30] in an attempt to include as many resonances as possible. The eLSM is based on both chiral symmetry and dilatation invariance and correctly models their respective explicit, anomalous, and spontaneous breaking mechanisms. In the mesonic sector, the eLSM contains also (axial-)vector meson degrees of freedom besides the standard (pseudo)scalar mesons. The lightest scalar glueball is included as a dilaton field in the Lagrangian. Moreover, the model was extended to include the lightest pseudoscalar glueball [31], a vector glueball, pseudovector and excited vector mesons [32], as well as excited (pseudo)scalar mesons [33]. In the low-energy limit, the eLSM correctly reduces to ChPT [34], thus showing the compatibility of these two different approaches to hadronic physics. In the baryonic sector, the eLSM was developed for two flavors in Ref. [35] and for three flavors in Ref. [36] on the basis of the mirror assignment for the chiral partner of the nucleon [37], in such a way that chirally invariant mass terms for baryons are possible.
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 [47], they could be bound states formed by a colored diquark (in the antisymmetric color-antitriplet and antisymmetric flavorantitriplet representation) and a colored anti-diquark (in the corresponding color-triplet and flavor-triplet representation) [48][49][50][51]. In this picture, the resonance f 0 (500) is a [u, d][ū,d] four-quark state. The other members of the nonet are formed similarly using also [us], [ds], and the corresponding anti-diquarks.
Alternatively, the light scalar mesons could be (loosely bound) molecular states formed from, or unbound states in the scattering continuum of [52][53][54][55][56][57][58][59][60], two color-neutral mesons. The latter possibility is supported by studies where they emerge as companion poles of conventional qq seed states [4,[61][62][63][64][65] [for the dynamical generation of a 0 (980) and K * 0 (800) starting from an eLSM-inspired Lagrangian, see Refs. [66,67]]. Thus, even if the above mentioned approaches differ in the interpretation of the internal structure, all agree on a predominantly four-quark nature of the light scalars. In this respect, the findings of the eLSM are consistent with these results.
The scalar state f 0 (500) is particularly important since it is the lightest state with the quantum numbers of the vacuum [for a review, see Ref. [68]]. It is expected to be relevant both in pion-pion, pion-nucleon, as well as nucleon-nucleon scattering. However, this state was not yet included in the eLSM, although some preliminary attempts were made in studies at non-zero density [69], at nonzero temperature [70], and of neutron-proton scattering [71]. Moreover, in a comparison of the eLSM with ChPT it was recently stressed that the f 0 (500) is necessary for a proper description of the pion-pion scattering lengths [34].
The main goal of the present work is the inclusion and systematic investigation of the light four-quark state f 0 (500) within the eLSM. To this end, we consider masses, decay widths, as well pion-pion and pion-nucleon scattering lengths, where f 0 (500) plays a decisive role. At the same time, we shall also investigate the effects of the glueball/dilaton field (identified with f 0 (1710)) on these quantities. We show that the presence of both f 0 (500) and f 0 (1710) offers a satisfactory description of experimental results in the meson sector and, in the baryon sector, at least an improved description of data in comparison to models without these states. This paper is organized as follows: in Sec. II we couple a four-quark field (which is the predominant component of f 0 (500)) both to the mesonic and to the baryonic sector of the eLSM as an additional chirally invariant scalar (thus, in order to avoid double counting one should not generate it via loop contributions within this approach). For the sake of definiteness, we will use the diquark-antidiquark picture of this state in our considerations, but this is actually of secondary importance for the two-flavor version of the eLSM studied here.
In the baryonic sector, we make use of the so-called mirror assignment, first proposed in Ref. [37] and further studied in Refs. [35,[72][73][74][75][76][77][78]: here, the condensation of f 0 (500) (corresponding to a four-quark condensate) and of the dilaton f 0 (1710) (corresponding to a gluon condensate) contribute to the baryonic mass terms of the mirror model in addition to the condensation of the standard quark-antiquark meson f 0 (1370).
Then, in Sec. III we present our results for masses, decay widths, and pion-pion as well as pion-nucleon scattering lengths and volumina. Here, one observes that f 0 (500) is crucial to obtain a correct description of data for the pion-pion scattering length a 0 0 and the pionnucleon scattering length a + 0 . Moreover, a detailed study of mixing between a bare four-quark state, a bare quarkantiquark state, and a bare glueball confirms that the field f 0 (500) is mostly a four-quark state, f 0 (1370) is mostly quarkonium, and f 0 (1710) is mostly gluonic. We conclude this work with a summary and a discussion of the results in Sec. IV. Details of the calculations are deferred to the Appendix. We use natural units = c = 1; the convention for the metric tensor of flat Minkowski space-time is g µν = diag(+, −, −, −).

A. Quarkonium multiplets
For an arbitrary number of flavors we can arrange (pseudo)scalar quarkonium fields into multiplets using the current which is a matrix in flavor space and where the color indices are implicitly contracted. This is a so-called heterochiral [79] scalar because it transforms under the global chiral symmetry where U L,R are SU (N f ) L,R transformations and α is the parameter of the U (1) A transformation. For two flavors Φ can be written as where t a = τ a /2 are the generators of U (2), namely half the Pauli matrices for a = 1, 2, 3, and half the unit matrix for a = 0. Similarly, we define the right-and left-handed vector currents, These are homochiral multiplets, i.e., they transform under the chiral symmetry G fl × U (1) A as For two flavors the (axial-)vector fields are contained in the right-and left-handed meson matrices These non-exotic multiplets lead to the well-known chirally invariant Lagrangian [28,35,80] where is the dilaton potential, responsible for the breaking of dilatation symmetry (trace anomaly) [81][82][83][84]. The scalar glueball with mass m G emerges upon the shift G → G 0 + G. The scalar glueball represents a color-neutral gluonic bound state. Since it is not composed of any quarks of any flavor, it is trivially a chiral singlet.
The Lagrangian L g3,g4,g5,g6 describes (axial-)vector meson interactions, see Appendix of Ref. [35]. After chiral symmetry breaking and a shift of the axial-vector fields to eliminate bilinear mixing terms between the latter and the pions, derivatively coupled four-pion interactions emerge. Therefore, in principle the coupling constants g 3 , g 4 , g 5 , g 6 have an influence on the pion-pion scattering lengths. However, it was shown in Ref. [34] that varying the values for these coupling constants between ±100 (i.e., a range of values that is in agreement with ChPT and moreover appears to be of a natural order of magnitude) the change in the pion-pion scattering lengths is only of the order of a few percent. Therefore, we will neglect these coupling terms in the following.

B. Four-quark multiplets
There are several ways to incorporate four-quark states into a chiral model. Here, we will follow the approach of Ref. [85]. We start with the three-flavor case and then reduce it to two flavors. There are several reasons for choosing this approach: 1. Writing down all relevant terms for three flavors facilitates an extension of the current work to the case of three flavors.
2. We will be able to compare our approach with other ones, e.g. those of Refs. [86,87].
3. It is easier to see which terms are large-N c dominant, enabling us to choose only the most relevant terms.
4. For three flavors a four-quark nonet has the same chiral structure as the quarkonium nonet, while for N f = 3 the four-quark multiplet will have a different chiral structure, which makes three-flavor models somewhat special when considering four-quark states [88].
For N f = 3 the right-and left-handed diquark fields are defined as with the charge-conjugation matrix C = iγ 2 γ 0 (in the Dirac representation), where the flavor indices are in small letters and color indices in capital letters. For the sake of simplicity we will drop the color indices in the following. In the next step, we construct the right and left-handed diquark matrices where R/L , such that we obtain diquark matrices with well defined parity via the linear combinations These are composed of scalar and pseudoscalar diquarks, defined as such that (again suppressing color indices) In the following we will only be interested in scalar diquarks, assuming that the pseudoscalar ones are not relevant for low-energy hadron phenomenology. A strong attraction between two quarks in a color antitriplet (3 C ), a flavor antitriplet (3 F ) and spin-zero configuration [47] is obtained in studies based on one-gluon exchange [89], instantons [90,91], the NJL model [92], and Dyson-Schwinger equation (DSE) [93]. In these studies it is also shown that the pseudoscalar diquark turns out to be considerably heavier.
Considering only scalar diquarks we can therefore construct a scalar four-quark nonet by which explicitly reads where a 0 and K * 0 are identified with a 0 (980) and K * 0 (800), and admixtures of χ and χ s are assigned to the physical states f 0 (500) and f 0 (980). Now we are able to construct chirally invariant interaction terms that couple scalar four-quark states to scalar quarkonia: In each term there is a diquark and an anti-diquark such that the expression becomes color neutral. Furthermore, all terms are also invariant under parity, charge conjugation, and U (1) A transformations. In order to keep our effective model as simple as possible, in the following we will only consider the first term in Eq. (19), which is the leading one in the large-N c expansion, where we used Eqs. (13) and (17) and neglected all terms containing pseudoscalar diquarks. 1 For two flavors, Eq. (20) reduces to where we abbreviated g (1) χ ≡ −2g χ and only considered T 33 ≡ χ because this is the only four-quark state that exists for two flavors. This term is similar to the determinant term in Eq. (8), which models the U (1) A anomaly. Since the chiral condensate also induces a condensate of the scalar four-quark state, Eq. (20) generates a contribution to the masses of σ and π and, of the same magnitude but with opposite sign, to those of η and a 0 .
Very similar terms are obtained for the coupling to 1 We note that this term is precisely the same as the one in Eq. (16) of Ref. [87]. However, in that work the interaction term is constructed by using the four-quark matrix analogue of Φ, which transforms in the same manner as Φ except under U (1) A transformations. Although both approaches yield the same interaction term as above, the other two interaction terms of Eq. (19) are only found using our approach.
(axial-)vector quarkonia: Again, we are only interested in the leading-order term at large-N c . Neglecting the pseudoscalar diquarkD we obtain: For two flavors, This term reduces to: where we abbreviated g AV ≡ −2g AV . Interestingly, this term looks structurally similar to Eq. (21). It generates a contribution to the masses of the isoscalar (axial-)vector mesons and, of the same magnitude but with opposite sign, to those of the isovector (axial-)vector mesons.
Introducing also a kinetic and mass term for the scalar tetraquark, we find the complete two-flavor four-quark Lagrangian to be From this Lagrangian and Eq. (8) we can derive masses and decay widths as well as the pion-pion scattering lengths.

C. Masses of the scalar-isoscalar resonances
The terms in the Lagrangian which, upon condensation of σ, G, and χ, give rise to the mass matrix for the scalarisoscalar resonances are We perform a shift of the scalar-isoscalar fields by their respective vacuum expectation values, σ → ϕ + σ, G → G 0 + G, χ → χ 0 + χ. This leads to mass terms for these three fields, which can be compactly written in matrix form as where m 2 σ is given in Eq. (B4) and Note that the (13) and (31) elements of M were simplified using the condition that χ 0 is an extremum of the potential energy density. The real, symmetric mass matrix M can be diagonalized by an orthogonal transformation O, where the eigenvalues M 2 H , M 2 S , and M 2 G correspond to the (squared) masses of the physical fields f 0 (500), f 0 (1370), and f 0 (1710). These fields are linear combinations of the unphysical fields χ, σ, and G, given by 

D. Baryons
Baryons are implemented in the eLSM in the so-called mirror assignment [35,37,94]. One introduces two baryon doublets, Ψ 1 and Ψ 2 , where Ψ 1 has positive parity and Ψ 2 is its chiral partner with negative parity. In the mirror assignment these fields transform under chiral transformations as i.e., Ψ 1,R/L transforms like a vector under SU (N f ) R/L , as expected, while Ψ 2,R/L transforms in a mirror way, like a vector under SU 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 non-zero 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 χ, and/or the glueball field G, and subsequent condensation of χ and G. Both G and χ are chiral singlets as shown in App. A.
Furthermore, it is possible to introduce another interaction term that violates U (1) A symmetry but gives a contribution to the nucleon mass as well. This term is given by Such an anomalous term (which also breaks the dilatation symmetry) yields a four-point vertex that has not been considered before in this model, see also the last diagram of Fig. 2 (found in App. E).
The terms in the baryon Lagrangian which are relevant for pion-nucleon scattering are then where D µ 1/2,R = ∂ µ − ic 1/2 R µ , D µ 1/2,L = ∂ µ − ic 1/2 L µ . Upon condensation of χ, G, and σ a baryonic mass term is generated The mass term mixes Ψ 1 and Ψ 2 , so that the physical fields are obtained by a unitary transformation, where δ is the mixing parameter.

III. RESULTS
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.

A. Global fit in the meson sector
The assignment of our effective hadronic degrees of freedom is given in Tab. I. For this assignment the experimental data for the decay widths and the pion-pion scattering lengths are given in Tab. II. For the mass of the pion we use the isospin-averaged mass, m π = 138 MeV. For the mass of a 1 we choose 1277 MeV, which is slightly above the upper error band of the PDG data. The reason for this choice is the mass splitting between the isoscalar and the isovector (axial-)vector mesons generated by the coupling between the four-quark state and the (axial-)vector mesons in Eq. (24), see also App. B 2 and B 3. Since the masses of ρ, ω, and f 1 are known to very good precision, see Tab. I, we are forced to increase the theoretical value for the mass of a 1 such that the mass splitting between that state and f 1 is of the same order as that between ρ and ω. Furthermore, the physical η meson contains a considerabless admixture, which, in a pure two-flavor scenario, has to be eliminated by a rotation in the η − η sector. The result is the value m η = 755 MeV for the mass of the purely non-strange η meson [28].  [1], the decay width ΓS→ππ is taken from Ref. [95], and the scattering lengths from Ref. [96].
The pion-pion scattering parameters in the eLSM have been first calculated in Ref. [97], but without a scalar four-quark state and without a dynamical scalar glueball. It was found that the pion-pion scattering length m π a 0 0 is in the range of experimental data only for a small mass of the scalar-isoscalar quarkonium σ field, while m π a 2 0 agrees well with experimental data for all values of the σ mass [cf. Fig. 2 in Ref. [97]]. Here we examine how the results change if we consider a scalar four-quark state χ in addition to the quarkonium state σ. At first, we neglect the scalar glueball.
The Lagrangians (8) and (25) contain ten parameters that are of relevance for our fit: µ 2 , λ 1 , λ 2 , c, m 2 1 , h 1 +h 2 ≡ h, h 3 , g χ , g AV , m χ . The parameters λ 2 , h 3 , c, µ 2 , m 2 1 , g AV can be expressed by the physical masses of Tab. I or by the remaining model parameters, see App. B. Furthermore, λ 1 is large-N c suppressed and is therefore set to zero. Thus, only the three parameters h, g χ , m χ need to be fitted. We used the standard χ 2 procedure to fit the parameters and determine the errors (χ 2 = χ 2 (h, g χ , m χ )): The result of this fit is presented in Tab given. The parameters in the lower left box are calculated from the fitted parameters. θ is the mixing angle between χ and σ. In the right box the numerical results for the observables are given.
Next, we consider the scalar glueball as dynamical field as well. Now, M G and G 0 are additional fit parameters. Then the χ 2 function is given as The results of the fit for this scenario are given in Tab. IV. From this fit the following scalar-isoscalar mixing matrix is obtained: which corresponds to the following admixtures of the physical states: f 0 (500) : 100% χ, 0% σ, 0% G , Let us briefly discuss these results: • Our aim was to correctly reproduce the masses and decay widths of f 0 (500), f 0 (1370), and f 0 (1710). The fit agrees well with experimental data, only for the scattering length m π a 2 0 the theoretical and experimental error bands overlap just barely (the theoretical value is slightly too large). In both fits, with and without a dynamical glueball, similar parameters are obtained, which results in very similar observables.
• Although the value of g χ is very small, it is numerically not negligible. In App. B we show that g AV is proportional to the inverse of g χ . Thus, g AV would diverge if we send g χ → 0.
• Since g χ is very small, the coupling g AV between the scalar four-quark state and the (axial-)vector mesons is rather large, g AV ∼ −12 GeV in both fits.
• We also tried to identify H = f 0 (980), another possible candidate for a four-quark state, but no reasonable fit results were obtained. Our investigation clearly favors the (non-strange) scalar four-quark state to be a light and broad state.
• The value of Γ S→ππ used in our fit is somewhat problematic due to its uncertain value in the literature. In Ref. [95] a mass-dependent decay width of 325 MeV at m f0(1370) = 1309±1±15 MeV is found, while Ref. [98] quotes the value 91 MeV. If we exclude this width from our fit, we obtain Γ S→ππ ≈ 3 GeV for the case without dynamical glueball and Γ S→ππ ≈ 750 MeV for the case with dynamical glueball, while the other observables change only slightly. In order to increase the fit quality, we decided to use the larger value Γ S→ππ = 325 MeV in our fit.
• We obtain a fit of similar quality if we assign G = f 0 (1500), but at a cost of a very large dilaton condensate G 0 > 1.5 GeV. Thus, we cannot make any prediction about whether f 0 (1500) or f 0 (1710) is more likely the glueball candidate. This question has been addressed in a model similar to ours [30], where the authors found f 0 (1710) to be the scalar glueball while f 0 (1500) was found to be mostly an ss quarkonium state.
• The elements of the matrix (41) which correspond to the mixing between σ and G are somewhat larger than those of Ref. [30], most likely due to the missing strange scalar-isoscalar σ S in our two-flavor model.
• We find in both fits very similar values for the pion-pion scattering lengths, indicating that the scalar glueball is actually not important for pionpion 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., m π → 0, as required by low-energy theorems.
• To further underline the importance of a light scalar-isoscalar resonance we can take the limit g χ → 0 and g AV → 0 to turn off the interactions of the four-quark state. Performing a fit in this limit leads to m π a 0 0 = 0.156 and m π a 2 0 = −0.044, i.e., results comparable to those of Ref. [97] for large σ masses. This result is obtained for both cases, with and without a dynamical scalar glueball.

B. Pion-nucleon scattering parameters
Some of the parameters of the baryon Lagrangian have been already determined in Ref. [35] and are reported in Tab. V. In Ref. [35] the isospin-even and isospin-odd scattering lengths have been calculated in a model without a scalar four-quark state and a scalar glueball. The authors found m π a (−) 0 = 0.0834 ± 0.0087 for the isospin-odd scattering length, which is in surprisingly good agreement with the experimental value, see the first entry of the last column in Tab. VI. However, there were several errors in Eq. (19) Parameter Value Experiment of Ref. [35]. The correct formula is given in App. E, and the correct value is the first entry of the second column in Tab. VI. This value is now outside the experimental error band. The isospin-even scattering length m π a (+) 0 has also been calculated in Ref. [35]. There was also a sign error in Eq. (18) of Ref. [35], which changes the behavior of m π a (+) 0 as a function of the parameter m 1 as shown in Fig. 2 of Ref. [35]. The correct formula is also given in App. E. We do not show the corrected graphs; in the first entry of the second column in Tab. VII we simply list the correspondig value for m 1 = 643 MeV obtained from the global fit of Ref. [28]. The theoretical value is about a factor of 7 larger than the experimental value.
In addition, here we also calculate isospin-even and isospin-odd scattering volumes and effective range parameters within the set-up of Ref. [35]. The corresponding values are shown in the second to fourth rows in Tabs. VI and VII. The isospin-odd scattering volumes and the range parameter deviate by factors of 0.5 to 3 from the experimental values, the isospin-even scattering volumes by factors of 0.4 to 2, while the isospin-even range parameter is about a factor 7 too small. Note that all scattering parameters have also theoretical errors originating from the uncertainties of the χ 2 fit in determining the parameters. However, we omitted the errors because they are smaller than 5% and therefore not a reliable measure of uncertainty.
The theoretical values in Tabs. VI and VII were computed without a scalar four-quark state or scalar glueball, i.e., they are just a correction and extension of the results of Ref. [35]. While the isospin-odd scattering parameters are influenced neither by a scalar four-quark state nor by a scalar glueball, and thus cannot be further improved within our model, we can still study the question whether the introduction of these states can at least improve the description of the isospin-even scattering parameters. The explicit calculations of the pion-nucleon scattering amplitudes and the isospin-even scattering parameters are deferred to App. E. Compared to the model of Ref. [35] we have three new couplings, a, b, and c N , the values of which are only constrained by the linear combination (36), which should have the correct value of m 0 in order to reproduce the mass of the nucleon and its chiral partner. In a first step we consider only one parameter at a time, while setting the other ones to zero. In this way we distinguish three cases: C: In Tab. VIII we consider case A, where only the scalar four-quark state contributes to the explicit mass term. In this case the scalar glueball contributes only indirectly to the pion-nucleon scattering parameters via the G − σ and G − χ mixing. Due to the small value of the fourquark condensate found in the fit of the meson sector the coupling between the scalar four-quark state and the nucleons must be extremely large in order to obtain m 0 = 462 MeV. This leads to scattering parameters that are off by several orders of magnitude.
In Tabs. IX and X we consider the cases B and C, where m 0 originates either exclusively from the gluon condensate or from the anomaly contribution, respectively. In these cases, the scalar four-quark state contributes to pion-nucleon scattering only via the χ − σ and χ − G coupling. The results are rather similar for both cases, and the numerical values rather close to those of the model of Ref. [35], cf. Tab. VII. This is expected because on the one hand the glueball is rather heavy and thus cannot substantially influence the scattering parameters in case B. On the other hand, the additional anomalous contribution ∼ c N ϕ 6 to the σN N * coupling is about a factor of 2 smaller in magnitude than the Yukawa coupling (ĝ 1 e δ −ĝ 2 e −δ )/4, and thus case C is, as far as pionnucleon scattering is concerned, essentially identical to the model of Ref. [35]. Case A is markedly different from cases B and C because of the presence of the light fourquark field f 0 (500), which has a substantial impact on pion-nucleon scattering parameters.
We now perform a simultaneous χ 2 fit for all three parameters a, b, and c N , respecting the constraint (36). Using the constraint that b is positive yields the results of Tab. XI. It should also be noted that we find basically the same results if we set either b = 0 or c N = 0. It is important, however, that a is non-vanishing. Although the agreement between the theoretically calculated scattering parameters and experimental data is now in general better, the values still deviate by factors of 1.5 to 3 (and the range parameter has even the wrong sign).

IV. SUMMARY AND DISCUSSION
In this paper, we studied the influence of the light four-quark state f 0 (500) and the scalar glueball on pionpion and pion-nucleon scattering in the framework of the eLSM for N f = 2 flavors. We first performed a χ 2 fit to properties in the mesonic sector. We found a physically acceptable minimum for which the resonance f 0 (500) is (almost) exclusively a four-quark state, f 0 (1370) predominantly a light quark-antiquark state, and f 0 (1710) predominantly a gluonic state. The masses and the decay widths of these resonances as well as pion-pion scattering lengths can be correctly described: in particular a 0 0 is strongly dependent on the additional attraction on account of f 0 (500), and the presence of the latter is necessary for a good description of the data. The resonance f 0 (1710) is quite heavy, hence it does not affect the results in a substantial way, but its presence is nevertheless important to stabilize the fit (smaller errors, reasonable upper limit for the gluon condensate G 0 ). Another notable result is the observation that the (axial-)vectors turn out to interact quite strongly with the scalar fourquark state (g AV /G 0 ∼ −30), while the coupling to the (pseudo)scalar quarkonia is rather small (g χ /G 0 ∼ 0).
Then we studied the role of f 0 (500) and f 0 (1710) in the baryonic sector of the eLSM. The nucleon and its chiral partner were incorporated into the model in the so-called mirror assignment and theoretical expressions for the pion-nucleon scattering parameters, namely the isospin-even and isospin-odd scattering lengths, scattering volumes, and effective range parameters were derived. First, we presented results without the scalar four-quark and glueball state, correcting and extending results of Ref. [35]. Our results are found to be in the correct order of magnitude compared to experimental data but there is still room for improvement. Therefore, in a first attempt to improve the results, we considered the effects of both a four-quark state and a glueball state and, in addition, also those of an anomalous term. We found that the scalar four-quark state f 0 (500) has the greatest impact on the values of the scattering parameters. However, physically reasonable results are only obtained when two parameters are fitted simultaneously: the coupling of the four-quark state to the nucleon in combination with either the glueball-nucleon coupling or the coupling to the U (1) A -violating interaction term.
A straightforward extension of this work would be to consider the three-flavor case. Two additional resonances appear in the scalar-isoscalar sector: the strangeantistrange quarkonium (predominantly f 0 (1500)) and the four-quark state f 0 (980) (a kaon-kaon state in the molecular picture, a [u, s][ū,s] + [d, s][d,s] state in the tetraquark picture). Moreover, also the quarkonium states a 0 (1450) and K * 0 (1430) and the four-quark states a 0 (980) and K * 0 (800) would enter this extended scenario. In this context, the difference between different internal structures of the four-quark states (meson-meson or diquark-diquark) would also become visible in terms of different Clebsch-Gordon coefficients.
In the baryonic sector, an interesting future work would be to continue the nonzero-density study of Ref. [69]. Due to the recent discovery of gravitational waves emitted by neutron-star binary mergers, it is expected that the equation of state of nuclear matter at high density can be more precisely determined in the future.
of the State Administration of Foreign Experts Affairs of China.

V. APPENDICES
We used Mathematica R for the numerical evaluation and the fits performed in this work. The notebooks can be found on github: https://github.com/Phillip2/eLSM Appendix A: Tetraquark-baryon interaction terms As in the meson sector, we construct possible tetraquark-baryon interaction terms for N f = 3 and then reduce them to N f = 2 in order to verify that the effective interaction of Eq. (32) can be derived within our approach. To this end, we use the formalism of Ref. [36], where four baryonic multiplets, N 1 , N 2 , M 1 , and M 2 were constructed using the quark-diquark picture for baryons, cf. Tab. III of Ref. [99] for the transformation properties under chiral symmetry, parity, and charge conjugation. We find the following terms where scalar tetraquarks couple to baryons: which reduces to Eq. (32) for two flavors using the same approach as in Sec. II B. We show this explicitly for one term: From the first to the second line we used the definition of the baryon fields with definite parity and chargeconjugation properties as defined in Ref. [99]. The N f = 2 limit in the last line is obtained by setting a = b = 3 and by reducing After spontaneous symmetry breaking, the field σ is shifted by its vacuum expectation value, i.e., σ → ϕ + σ. This yields mixing terms between axial-vector and pseudoscalar mesons, e.g. ∼ a µ 1 · ∂ µ π. Such terms can be eliminated by a redefinition of the fields, where with m 2 a1 from Eq. (B10), and

Masses
The masses of the resonances are calculated from the second partial derivative of the potential density (which is obtained from Eqs. (8) and (25)) with respect to the corresponding fields:

Parameters
The vacuum expectation value of σ, ϕ = Zf π , is determined from the axial current. The vacuum expectation value of the scalar four-quark state is determined by the minimum of the potential density: We can use the masses, whose values are given by experimental data, to fix some of the model parameters where we used Eq. (B12) in the last step of the last line. Note that g AV and g χ are inversely proportional to each other, i.e., a small value of g χ requires a large value of g AV and vice versa.
The scattering amplitude is calculated from the treelevel amplitudes in Fig. 1. The pion-pion interaction consists of three parts, which can be extracted from the eLSM Lagrangian (8). From this we obtain the scattering amplitude M ππ (s, t, u) = iδ ab δ cd A(s, t, u) + iδ ac δ bd A(t, u, s) where The coefficients A σ , B σ , and C σ are given in Eqs. (C7)-(C9), while A ρ = g 1 Z 2 m 2 ρ /m 2 a1 and B ρ = g 2 Z 2 w 2 . The scattering amplitude in the I = 0 channel is given by the relation [100] T 0 (s, t, u) = 3A(s, t, u) + A(t, u, s) + A(u, s, t) , (C21) from which the isospin-zero scattering length is computed as On the other hand, the scattering amplitude for isospin I = 2 is given by and the s-wave scattering length is extracted as After the introduction of the scalar four-quark field, the term in Eq. (C20) is replaced by The new coefficients are given as (C32) Thus, at threshold (s ≡ 4m 2 π , t = 0, u = 0), we obtain the scattering lengths m π a 0 0 = 1 32π 12(g 2 1 − h 3 )Z 4 w 2 m 2 π − 10 λ 1 + The Feynman diagrams which contribute to the pionnucleon scattering amplitude are shown in Fig. 2. The isospin-even and isospin-odd scattering lengths, scattering volumes, and effective range parameters can be cal-culated using [101]: The partial amplitudes A (±) and B (±) can be extracted by rewriting the scattering amplitude into the form T ab = A (+) + ( / q 1 + / q 2 ) 2 B (+) δ ab were a and b are the isospin-indices of the inital and final pion states. The correct result for the partial amplitudes for the model of Ref. [35] reads: A (+) = 4 g πN g ∂πN + g 2 ∂πN m N − g N * π g N * ∂π − 2g 2 N * ∂π (m N * − m N ) + 2g N N ππ + −g 2 N * π (m N + m N * ) + (m 2 N − m 2 N * ) 2 g N * π g N * ∂π + g 2 (E23)