In-medium Pion and Partial Restoration of Chiral Symmetry

In-medium Pion and Partial Restoration of Chiral Symmetry

Abstract

The partial restoration of chiral symmetry in nuclear medium is investigated in a model independent way by exploiting operator relations in QCD. An exact sum rule is derived for the quark condensate valid for all density. This sum rule is simplified at low density to a new relation with the in-medium quark condensate , in-medium pion decay constant and in-medium pion wave-function renormalization . Calculating at low density from the iso-scalar pion-nucleon scattering data and relating to the isovector pion-nucleus scattering length , it is concluded that the enhanced repulsion of the s-wave isovector pion-nucleus interaction observed in the deeply bound pionic atoms directly implies the reduction of the in-medium quark condensate. The knowledge of the in-medium pion mass is not necessary to reach this conclusion.

YITP-07-23,  KUNS-2142

, ,

1 Introduction

Exploring possible evidence of partial restoration of chiral symmetry in the nuclear medium is one of the most important and challenging problems in hadron physics. Experimental studies along this line have been carried out by the spectroscopy of deeply bound pionic atoms [1], by low energy pion-nucleus scatterings [2], and by the production of di-pions in hadron-nucleus and photon-nucleus reactions [3, 4, 5]. Important experimental observations are (i) a repulsive enhancement of the in-medium -nucleon interaction in the pionic atoms and pion-nucleus scattering, and (ii) an attractive enhancement of the scalar-isoscalar - interaction in nuclei. In the theoretical side, it was suggested that the reduction of the temporal part of the pion decay constant in the nuclear medium is intimately related to the phenomenon (i) [6, 7]. It was also argued that the reduction of is responsible for the phenomenon (ii) [8].

The purpose of this paper is to present theoretical basis of the direct connection between and the in-medium quark condensate (the order parameter of the chiral phase transition) only by using the current commutation relations, spectral decomposition and the linear density approximation.1 We derive an exact sum rule relating the in-medium quark condensate and hadronic matrix elements valid for all density in the chiral limit. We also derive following model-independent relations at low density: The im-medium Glashow-Weinberg (GW) relation in the chiral limit, the in-medium TomozawaWeinberg (TW) relation in the chiral limit, and the in-medium Gell-MannOakesRenner (GOR) relation away from the chiral limit. Here is the pseudo-scalar pion coupling constant, is the isovector pion-nucleus scattering amplitude at zero spacial momentum and is the isospin averaged pion mass. In particular, the in-medium GW relation, which has not been discussed in previous literatures, gives a new way to estimate from the experimental data of the pionic atoms and the pion-nucleus and pion-nucleon scatterings without recourse to the information of as will be shown in this paper.

The organization of this paper is as follows: In section 2, we derive the exact sum rule for the quark condensate valid for all density in the chiral limit. In section 3, we show that the exact sum rule leads to the in-medium GW relation and the in-medium GOR relation which are valid at low density. In section 4, we evaluate the reduction of by relating it to the isoscalar N scattering amplitude. In section 5, we derive the in-medium TW relation in the chiral limit. In section 6, utilizing all the results derived, the in-medium quark condensate is related to the observables in -nucleus and N scattering data. In section 7, we discuss how the exact sum rule is expressed off the chiral limit, based on the PCAC relation. In section 8 the relation of the present argument to the in-medium chiral perturbation theory is discussed. Section 9 is devoted to summary and concluding remarks.

2 Exact sum rule in the chiral limit at all density

Let us first discuss how the in-medium quark condensate is expressed in terms of hadronic quantities. We consider the following correlation function in isospin symmetric nuclear matter in the chiral limit:

 Πab5(q)=∫d4x eiq⋅x∂μ⟨Ω|T[Aaμ(x)ϕb5(0)]|Ω⟩, (1)

where is the pseudoscalar density with the Pauli matrix and the quark field ; denotes the axial vector current associated with the SU(2) chiral transformation. The ground state of the isospin symmetric nuclear matter denoted by in Eq.(1) is normalized as and is specified by the baryon density .

Invoking an operator relation with the conservation of the axial current in the chiral limit, is written in terms of the in-medium quark condensate as

 Πab5(0)=⟨Ω|[Qa5,ϕb5]|Ω⟩=−iδab⟨¯qq⟩∗ , (2)

where and with the scalar density defined by .

The hadronic contributions to can be read off from the spectral representation:

 Πab5(q)=−iqμ∫∞0dω′(σ+μ(ω′,→q)ω−ω′+iϵ−σ−μ(ω′,→q)ω+ω′−iϵ) , (3)

where with the energy measured from the nuclear matter ground state. The spectral functions give the strength of the hadronic excitations with the energy and momentum and are expressed in terms of the matrix elements of and :

 σ+μ(ω,→q) = i∑ℓ⟨Ω|Aaμ|Ωℓ⟩⟨Ωℓ|ϕb5|Ω⟩δ→q,→pℓδ(εℓ−ω)2εℓ , (4) σ−μ(ω,→q) = i∑ℓ⟨Ω|ϕb5|Ωℓ⟩⟨Ωℓ|Aaμ|Ω⟩δ−→q,→pℓδ(εℓ−ω)2εℓ, (5)

where are the eigenstates of the QCD Hamiltonian normalized as with the eigenvalue measured from the ground state and the spatial volume . In the infinite volume limit, the summation should be replaced as with the density of states .

Since the states and have isospin 0 and 1, respectively, we put the isospin label explicitly to the latter state as for later convenience. In the soft limit , we have zero modes denoted by with the property as , and the non-zero modes denoted by with the property at .2 Leaving the discussion on the physical content of these states to the next section, we focus on the symmetry aspect in the nuclear medium.

With the vector characterizing Lorentz frame of the nuclear matter, the matrix elements of and for the states and are generically written as

 ⟨Ωbℓ(k)|ϕa5(x)|Ω⟩ = δabG∗1/2ℓeik⋅x, (6) ⟨Ω|Aaμ(x)|Ωbℓ(k)⟩ = iδab[nμ(n⋅k)N∗ℓ+kμF∗ℓ]e−ik⋅x. (7)

Here the space-time independent constants, (pseudo-scalar coupling constant), and and (axial-vector coupling constants) are the functions of and . The factor in front of is introduced to make and have the same dimension. Because of the conservation law of the axial current in the chiral limit, and are not independent: . Taking the rest frame of the nuclear matter where , we have

 ε2ℓ(N∗ℓ+F∗ℓ)−→k2F∗ℓ=0, (8)

with . According to this relation, the axial-vector coupling of the non-zero modes should vanish in the limit .

The correlator (3) can be calculated with the matrix elements (6) and (7): For example, the spectral function defined in (5) is evaluated to be

 iδab∑ℓωεℓN∗ℓ+(ωεℓ−→q2)F∗ℓ2εℓ(ω−εℓ+iϵ)G∗1/2ℓ −−−→qμ→0   i2δab∑α(N∗α+F∗α)G∗1/2α, (9)

where we have used Eq.(8) together with the fact that can have finite values only for the zero-modes ().

Note that the result is independent of how to take the soft limit, . The contribution from is found to be the complex conjugate of . Thus the hadronic contribution to the correlator (1) in the soft limit becomes

 Πab5(0)=iδab∑αRe[(N∗α+F∗α)G∗1/2α]. (10)

where the summation is taken for all the zero modes.

Equating Eqs.(2) and (10), we finally obtain an exact sum rule in the chiral limit which is valid for all densities,

 ∑αRe[(N∗α+F∗α)G∗1/2α]=−⟨¯qq⟩∗ . (11)

At zero density, only the pion contributes to the sum rule and Eq. (11) reduces to the well-known Glashow-Weinberg relation, [10]. On the other hand, at finite density, the left hand side of Eq. (11) may receive various contributions not only from the in-medium pion but also from other collective excitations in nuclear matter, so that full dynamical treatment of the many body system would be necessary. However, the discussion becomes much simpler when the linear density approximation is adopted as will be shown in the next section.

Before closing this section, let us consider a zero mode with a linear dispersion relation for small momenta. In this case, the matrix element (7) can be written in a simpler form. Let us introduce the temporal and spatial “decay constants” as

 ⟨Ω|Aa0(0)|Ωbα(k)⟩=iδabεαFtα , (12) ⟨Ω|Aai(0)|Ωbα(k)⟩=iδabkiFsα . (13)

Then, by using the dispersion relation together with Eq. (8) and , we have

 Fsα=F∗α,   Ftα=N∗α+F∗α=Fsα/v2α. (14)

Here it should be noticed that the relation for the pionic mode at finite temperature is derived in [11, 12].

3 In-medium Glashow-Weinberg relation at low density

The exact sum rule in the chiral limit, Eq. (11), is reduced to a simple form at low density. The expectation value of an arbitrary operator in nuclear matter up to the linear term in density reads

 ⟨Ω|O|Ω⟩≃⟨0|O|0⟩+ρ⟨N|O|N⟩. (15)

For in Eq.(1), the first term in the right hand side of Eq.(15) is dominated by the massless pion in the chiral limit. The second term corresponds to the forward scattering amplitude of the external fields ( and ) with the nucleon multiplied by .

As shown in Fig.1, there are four different contributions to the forward amplitude; (a) the term with double pole where both and couple directly to the pion, (b,c) the term with single pole where either or couples directly to the pion, and (d) the term without pole where the external fields couple directly to the nucleon.

It is easy to see that the -wave coupling of the pion or the external fields to the nucleon gives vanishing contribution to the forward amplitude in (a-d). On the other hand, the -wave coupling in (a,b,c) can leave finite contributions even in the soft limit due to the pion pole(s). In particular, particle-hole excitations do not contribute to the sum rule in the linear-density approximation. This leads to the conclusion that only the pionic mode with possible medium modification contributes to the sum rule in the previous section at low density:

 FtπG∗1/2π=−⟨¯qq⟩∗. (16)

Note that Eq.(16) is only valid if both sides are expanded up to .

Taking the ratio of Eq.(16) and its counterpart at zero density, we find the scaling law:

 (FtπFπ)Z∗1/2π= ⟨¯qq⟩∗⟨¯qq⟩ , (17)

with the in-medium wave function renormalization . As will be shown in the next section, in-medium change of from 1 can be evaluated from the isosinglet pion-nucleon scattering amplitude, while is related to the pion-nucleus isovector scattering lengths through the in-medium TW relation. Therefore, Eq.(17) gives a direct link between the in-medium modification of the quark condensate and that of the pion decay constant3.

Alternative relation between and is obtained by taking the matrix element of the PCAC relation
slightly away from the chiral limit. With the matrix elments given in Eqs. (6) and (12) with , we have , where and are the values in the chiral and soft limit. Since the in-medium pion mass is given by , we have . Combining this with Eq.(11), we obtain

 (Ftπ)2m∗2π=−2mq⟨¯qq⟩∗, (18) (FtπFπ)2(m∗πmπ)2= ⟨¯qq⟩∗⟨¯qq⟩. (19)

Eq.(18) is the in-medium generalization of the Gell-Mann–Oakes–Renner relation [14] and was derived before in the Nambu–Jona-Lasinio model [15] and in chiral perturbation theory [16, 11]. Theoretically, Eq.(17) is equivalent to Eq.(19). Experimentally, the information on is necessary to check the latter sum rule, which is relatively difficult.

4 In-medium wave-function renormalization

Let us consider the relation between the pion wave function renormalization constant and the isospin singlet pion-nucleon scattering amplitude. For this purpose, we introduce the off-shell N amplitude near the pion pole through the operator , as is done in [17],

 TabπN(ν,νB;mπ)=δab T(+)+12[τa,τb] T(−) ≡iGπq2q′2∫d4xeiq⋅x⟨N(P′)|T[ϕa5(x)ϕb5(0)]|N(P)⟩ (20)

with the in-coming (out-going) pion momentum () and the kinematical variables defined as and . In the forward limit with , the scattering amplitude is a function solely of . Thus, the isospin singlet amplitude for small is written as

 T(+)(ω;mπ)≃α+βω2 (21)

Thanks to the special off-shell extrapolation given in Eq.(20), the coefficients and can be evaluated as follows [17]: (i) At the off-shell Weinberg point, we have with the sigma term  [18]. (ii) At the on-shell threshold, we have with the scattering length  [19]. Combining these, we obtain

 β≃σπNF2πm2π+(1+mπmN)4πaπNm2π. (22)

Numerically, the first term in the right hand side of Eq.(22) dominates over the second term and we find .

To obtain a relation between and , we now consider the correlation of in symmetric nuclear matter in the chiral limit expanded up to linear in density according to Eqs.(15), (20) and (21);

 Dab(q) = ∫d4xeiq⋅x⟨Ω|T[ϕa5(x)ϕb5(0)]|Ω⟩ (23) −−→q=0 iδabGπ[1ω2−1ω2T(+)(ω;0)ρ1ω2], (24) = iδabGπ(1−βρ)ω2=iδabG∗πω2. (25)

Then, we obtain

 Z∗1/2π≡(G∗πGπ)1/2=1−γρρ0, (26)

with . Notice that the reduction of in the nuclear medium given in Eq.(26) stems solely from the -wave pion-nucleon interaction.

5 In-medium Tomozawa-Weinberg relation

In-medium pion properties are conventionally expressed in terms of the pion-nucleus optical potential. For instance, the -wave potential for is parametrized as [6]

 2mπUs=−4π[1+mπmN] (b∗0(ρ)ρ−b∗1(ρ)δρ), (27) =−T(+)∗(ω=mπ;mπ)ρ−T(−)∗(ω=mπ;mπ)δρ, (28)

with the isoscalar density and the isovector density . The parameters and represent the pion-nucleus scattering lengths in isoscalar and isovector channels, respectively. are the isoscalar and isovector pion-nucleus scattering amplitude at zero spatial momentum. The value of extracted from the pionic atom and -nucleus scattering data is larger than that in the vacuum, which indicates an enhanced repulsion from nuclei [1, 2]. This extra repulsion expressed by is interpreted as originating from the in-medium reduction of through the in-medium generalization of the Tomozawa-Weinberg relation [6, 7].

Let us derive a relation between and in a way parallel to the derivation of in the previous section using current commutation relations. We consider the following axial vector correlator in a slightly asymmetric nuclear matter in the chiral limit:

 Πabν(q)=∫d4x eiq⋅x∂μ⟨Ω′|T[Aaμ(x)Abν(0)]|Ω′⟩ . (29)

Then using the current conservation and the relation satisfied in the chiral limit, we obtain the sum rule

 Πab0(0)=iϵab3⟨Ω′|V30|Ω′⟩≃iϵab312δρ. (30)

On the other hand, Eq.(29) at the chiral and soft limits is saturated with linear in isovector density as

 Π120(q)−−→q=0iω[ωFtπω2⋅T(−)∗(ω;0)δρ⋅ωFtπω2], (31)

Comparing Eqs. (30) and (31), we find

 T(−)∗(ω;0)≃ω2(Ftπ)2. (32)

This is an in-medium generalization of the Tomozawa-Weinberg relation [20, 17] and was obtained before in Ref.[6] with chiral perturbation approach.

Using the definitions Eqs.(27,28) together with Eq.(32), we obtain a formula relating the in-medium change of the isovector scattering length and the in-medium change of the pion decay constant in the chiral limit:

 b1b∗1=(FtπFπ)2. (33)

6 In-medium quark condensate

Now, inserting Eqs.(26) and (33) into Eq.(16), we arrive at one of the central results in this work,

 ⟨¯qq⟩∗⟨¯qq⟩≃(b1b∗1)1/2(1−γρρ0), (34)

which directly relates the in-medium quark condensate with the observables related to the pion in nuclei. The deeply bound pionic atom data suggest the repulsive enhancement of  [1]. The N scattering data tell that . Thus, Eq.(34) implies that these experimental facts give a direct evidence of the reduction of the quark condensate in nuclear medium. Quantitatively, the experimental value of is obtained as at the effective density in deeply bound pionic atoms [1]. With this value and estimated in Sec. 4 together with the linear density approximation, we find for the ratio of the quark condensates . We also evaluate this ratio with obtained in elastic -nucleus scatterings [2] assuming the effective density . The result is . These numbers are consistent with that given by the formula expressed with the pion-nucleon -term (see footnote 17.) For further quantitative argument, one has to take into account explicit chiral symmetry breaking effects and higher density contributions.

Here we emphasize that the renormalization of the pion field is inevitable in describing the pion dynamics when partial restoration of chiral symmetry occurs. This is understood clearly in the context of the chiral effective theory. The chiral effective theory is based on a consistent decomposition of the field variables on the chiral manifold with the original symmetry to radial () and angular (pionic, i.e., Nambu-Goldstone) ones. For dynamics in the vacuum, the relevant degree of freedom is the massless angular mode expressed by the dimensionless chiral field . Since the pion field has the dimension of energy and the order parameter of the dynamical symmetry breaking provides the only relevant energy scale in the chiral limit, the pion field should be normalized by the quark condensate. Therefore, when partial restoration of the chiral symmetry takes place, the pion field is necessarily renormalized according to the reduction of the quark condensate [8]; see also Sec.8.

7 The sum rule beyond the chiral limit

Let us generalize the in-medium sum rule Eq.(11) in the chiral limit to the case with a finite quark mass . From the PCAC relation , Eq.(2) is easily generalized to

 Πab5(0)−2mqDab(0)=−iδab⟨¯qq⟩∗ (35)

where is the correlation function of the pseudoscalar density given in Eq.(23).

Also, the definition of the coupling constants, Eqs.(6) and (7), together with the PCAC relation lead to the generalization of Eq.(8),

 ε2ℓ(N∗ℓ+F∗ℓ)−→k2F∗ℓ=2mqG∗1/2ℓ. (36)

The hadronic matrix elements in the left hand side of Eq.(35) can be evaluated as before and we find the sum rule away from the chiral limit,

 ∑ℓRe[(N∗ℓ+F∗ℓ)G∗1/2ℓ]=−⟨¯qq⟩∗, (37)

where the summation is taken over all states coupled to and , and the matrix elements are evaluated at . Modes may be classified into would-be zero modes () and would-be non-zero modes (). Then, eq.(36) shows that, in the soft limit, for would-be zero-modes, while for would-be non-zero modes. Namely, the quark mass correction to in the rhs of Eq.(37) receives both effects. To evaluate them, detailed hadronic model calculations such as given in Ref. [21] are needed.

8 Relation to the chiral effective theory

So far, we have derived the in-medium sum rule Eq.(11) based only on general operator relations in QCD without assuming any hadronic descriptions. The sum rule can be used for checking consistency of theoretical models with the fundamental symmetry of QCD, and also for experimental confirmations of partial restoration of chiral symmetry in nuclear medium, once the matrix elements of the currents are experimentally extracted, as discussed in section 6.

Let us demonstrate here how the in-medium GW relation Eq.(16) is expressed in terms of the low energy constants in chiral perturbation theory. An effective Lagrangian for the pion in nuclear medium is obtained in the mean field approximation of the nucleon field [16]:

 Leff = (f24+c32ρ)Tr[DμUDμU†]+~c22ρTr[D0UD0U†] (38) +(f24+c1ρ)Tr[U†χ+χ†U],

where is the isoscalar density , are the low energy constants with and is the tree-level pion decay constant in the vacuum in the chiral limit. We also introduce the chiral vector currents, and , in the covariant derivative and the scalar and pseudoscalar external fields, and , in with a normalization constant . The chiral field is given by

 U=exp[iπa⋅τaf∗]. (39)

The represents the normalization of the pion field and is left as a parameter to be fixed so as to normalize the pion kinetic term properly [8]. Expanding the effective Lagrangian (38) in terms of the pion field, we find the normalization in the linear-density approximation to be

 f∗=f[1+ρf2(c3+~c2)]. (40)

The axial current and the pseudoscalar density in the effective Lagrangian are evaluated through the formulas and . Evaluating the matrix elements of the axial current and the pseudoscalar density for the in-medium one-pion state in the tree approximation, we obtain

 Ftπ=f2+(2c3+2~c2)ρf∗,   G∗1/2π=2B0f2+4c1ρf∗. (41)

Thus, combining Eq.(41) together with the pion field normalization (40), we obtain

 FtπG∗1/2π=−2B0(f2+4c1ρ), (42)

in the linear density approximation. This is exactly the same formula as Eq.(16) for the medium modification of the quark condensate, since .

To evaluate the sum rule beyond the linear density approximation, one has to calculate the matrix elements with taking into account of dynamical properties of the pion and nucleons in nuclear medium. For this purpose, the chiral effective theory is one of the powerful tool which gives systematic description in terms of density and chiral symmetry breaking off soft limit.

9 Summary

In this paper, we have derived various in-medium sum rules in a model-independent way by exploiting operator relations and chiral symmetry in QCD. We have found an exact sum rule Eq.(11) valid for any density in the chiral limit. Also, a new scaling law Eq.(17) as a generalization of the Glashow-Weinberg relation is deduced from the exact sum rule at low density, in which the in-medium quark condensate, the in-medium pion decay constant and the in-medium wave-function renormalization of the pion field are related. With the information of the N scattering data and the in-medium Tomozawa-Weinberg relation Eq.(33), the quark condensate is related directly to the isovector -nucleus scattering length as shown in Eq.(34).

Our formulas should be useful to check the consistency among in-medium pion properties at low density from the point of view of chiral symmetry, and to extract the in-medium quark condensate. Generalization of our approach to the pion-pion interaction inside a nuclear medium following the idea of [8] is one of the interesting future problems to be examined and may provide us with a further insight into the in-medium pions and possible partial restoration of chiral symmetry in nuclei. It is also an interesting and important task to estimate contributions beyond the linear density approximation  [21, 22, 23].

Acknowledgments

D.J. thanks W. Weise and S. Hirenzaki for their continuous encouragements and helpful discussions. This work was partially supported by a Grant-in-Aid for Scientific Research by Monbu-Kagakusyo of Japan (Nos.17540250, 18042001, 18540253, 20028004, 20540265, 20540273) and was done under Yukawa International Program for Quark-Hadron Sciences.

Footnotes

1. A preliminary account of this work has been reported in [9].
2. Strictly speaking, we should introduce a small quark mass and take the limit after to make the whole procedure well-defined.
3. There is also a well-known low-energy theorem at low density which relates the in-medium modification of the quark condensate to the pion-nucleon -term [13]; with MeV.

References

1. K. Suzuki et al., Phys. Rev. Lett. 92 (2004) 072302; P. Kienle and T. Yamazaki, Prog. Part. Nucl. Phys. 52 (2004) 85.
2. E. Friedman et al., Phys. Rev. Lett. 93 (2004) 122302; Phys. Rev. C 72 (2005) 034609.
3. F. Bonutti et al. [CHAOS collaboration], Phys. Rev. Lett. 77 (1996) 603; Nucl. Phys. A 677 (2000) 213; P. Camerini et al. [CHAOS collaboration], Nucl. Phys. A 735 (2004) 89.
4. A. Starostin et al. [Crystal Ball Coll.], Phys. Rev. Lett. 85 (2000) 5539; Phys. Rev. C 66 (2002) 055205.
5. J. G. Messchendorp et al., Phys. Rev. Lett. 89 (2002) 222302.
6. E. E. Kolomeitsev, N. Kaiser, and W. Weise, Phys. Rev. Lett. 90 (2003) 092501.
7. W. Weise, arXiv:nucl-th/0507058.
8. D. Jido, T. Hatsuda, and T. Kunihiro, Phys. Rev. D63 (2001) 011901.
9. D. Jido, T. Hatsuda and T. Kunihiro, Prog. Theor. Phys. Suppl. 168 (2007) 478 [arXiv:0706.0258 [nucl-th]].
10. S. L. Glashow and S. Weinberg, Phys. Rev. Lett. 20 (1968) 224.
11. R. D. Pisarski and M. Tytgat, Phys. Rev. D 54 (1996) 2989.
12. M. Harada, Y. Kim, M. Rho and C. Sasaki, Nucl. Phys. A 727 (2003) 437; Nucl. Phys. A 730 (2004) 379.
13. E. G. Drukarev and E. M. Levin, Prog. Part. Nucl. Phys. 27, 77 (1991).
14. M. Gell-Mann, R. J. Oakes and B. Renner, Phys. Rev. 175 (1968) 2195.
15. T. Hatsuda and T. Kunihiro, Prog. Theor. Phys. 74 (1985) 765.
16. V. Thorsson and A. Wirzba, Nucl. Phys. A589 (1995) 633.
17. S. Weinberg, Phys. Rev. Lett. 17, 616 (1966).
18. M. E. Sainio, PiN Newslett. 16 (2002) 138 [arXiv:hep-ph/0110413].
19. H. C. Schroder et al., Phys. Lett. B469 (1999) 25.
20. Y. Tomozawa, Nuovo Cim. 46A (1966) 707.
21. N. Kaiser, P. de Homont and W. Weise, Phys. Rev. C 77 (2008) 025204.
22. U. G. Meissner, J. A. Oller and A. Wirzba, Annals Phys. 297 (2002) 27.
23. M. Doring and E. Oset, Phys. Rev. C 77 (2008) 024602.
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