Chiral partner structure of light nucleons in an extended parity doublet model

We study chiral partner structure of four light nucleons, $N(939)$, $N(1440)$, $N(1535)$ and $N(1650)$ using an effective chiral model based on the parity doublet structure. In our model we introduce four chiral representations, $({\bf 1},{\bf 2})$, $({\bf 2},{\bf 1})$, $({\bf 2},{\bf 3})$ and $({\bf 3},{\bf 2})$ under ${\rm SU}(2)_{\rm L} \otimes {\rm SU}(2)_{\rm R}$ symmetry. We determine the model parameters by fitting them to available experimental values of masses, widths and the axial charge of $N(939)$ together with the axial charges of $N(1535)$ and $N(1650)$ by lattice analyses. We find five groups of solutions: In a group the chiral partner to $N(939)$ is $N(1440)$ having small chiral invariant mass. In another group, the chiral partner is a mixture of $N(1535)$ and $N(1650)$ having a large chiral invariant mass. We claim that off-diagonal elements of axial-charge matrix can be used for distinguishing these groups. We also discuss changes of masses associated with chiral symmetry restoration, which could emerge in high density matter.


I. INTRODUCTION
One of the most important features of QCD relevant to the low-energy hadron physics is the chiral symmetry and its spontanious breaking. The spontanious symmetry breaking generates mass differences between chiral partners as well as the mixing among different chiral representations. It is interesting to study the role of the chiral symmetry breaking to determine the properties and structures of baryons such as amount of the masses of baryons generated by the chiral symmetry breaking and the chiral partner of ground state nucleon.
In a hadronic model for light nucleons based on the parity doublet structure [1][2][3][4][5] the chiral partner of N (939). One of the important feature of the model is the existence of the chiral invariant mass denoted by m 0 which is not originated from the spontaneous chiral symmetry breaking. In other word, the mass splitting between N (939) and its chiral partner N (1535) denoted by m 0 , which implies that the masses of N (939) and its chiral partner tend to m 0 when the chiral symmetry is restored and their mass splitting is given by spontanious chiral symmetry braking.
Particularly, in Ref. [15], 3)] to reproduce the axial charge of nucleon as pointed in Ref. [8], and that the chiral invariant mass of the N (939) is about 800 MeV. The large value of the chiral invariant mass of N (939) seems consistent with the results by the lattice QCD analysis in Ref. [17,18], which shows that is almost constant even if temperature is increased.
In this paper, we make a general analysis using a two-flavor parity doublet model including chiral [(1, 2) ⊕ (2, 1)] and [(2, 3) ⊕ (3, 2)] representations under the chiral SU(2) L ⊗SU(2) R symmetry, with including derivative interactions to the pion fields. We will show that there exist five groups of solutions distinguishable by chiral inavarinat masses and mixing rates of nucleons. In a group of solutions, the chiral partner of N (939) is N (1440) having small chiral invariant mass of about 100MeV and N (939) is dominated by [(2, 3) ⊕ (3, 2)] representation. In another group, on the other hand, N (939) belongs dominantly [(1, 2) ⊕ (2, 1)] representation having a large chiral invariant mass, and chiral partner of N (939) is a mixture of N (1535) and N (1650). Futhermore, we give predictions of off-diagonal elements of axial-charge matrix, which could be checked in future lattice analysis. We also show changes of nucleon masses when the vacuum expectation value of σ, which is an order parameter of the spnoaneous chiral symmetry breaking, is changed. This paper is organized as follows: In section II, we construct an extended model with parity doublet structure. Section III is a main part, where we show the numerical results of fitting on the chiral invariant masses and chiral partner structure. In sections IV and V, we study off-diagonal components of axial-charges matrix and change of nucleons mass as predictions. Finally we will give a brief summary and discussions in section VI. In this section we introduce four baryon fields with parity doublet structure and construct a Lagrangian for baryons and scalar and pseudoscalar mesons based on the SU(2) L ⊗SU(2) R chiral symmetry.

A. Model construction
The chiral representations of quarks under SU(2) L ⊗SU(2) R are written as where the 2 and 1 in above bracket express doublet and singlet, respectively. Since baryons are expressed as direct products of three quarks, we have following possibilities for the chiral representations of baryons: After the chiral symmetry is spontaneously broken down to the flavor symmetry, nucleons appear from the representations of (2, 1) ⊕ (1, 2) and (3, 2) ⊕ (2, 3). In this paper we introduce two baryon fields corresponding to these two representations: Here the subscripts l and r express the chirality: for i = 1, 2.
For clarifying the representations under the chiral symmetry, we explicitly write the superscripts of the baryon fields as , η (a,αβ) 2r (5) where a, b = 1, 2 are for SU(2) L and α, β = 1, 2 for SU(2) R . Note that the superscripts ab and αβ of η fields are symmetrized to express 3 representation: e.g. η (a,αβ) 1l = η (a,βα) 1l . The transformation properties under the parity and the charge conjugation are defined as where C = iγ 2 γ 0 and Ψ = ψ, η. The covariant derivatives for the fields are expressed as and 1r,2l , (9) where L µ and R µ are the external gauge fields introduced by gauging the chiral SU(2) L ⊗SU(2) R symmetry.
Next we introduce a 2×2 matrix field M expressing a nonet of scalar and pseudoscalar mesons made of a quark and an antiquark. The representation under SU(2) L ⊗SU(2) R of the M is The transformation properties under the parity and the charge congugation are given by The covariant derivative for M is expressed as Using the fields introduced above we construct a Lagrangian invariant under the chiral SU(2) L ⊗SU(2) R symmetry.
Let us first consider terms including only ψ 1 and ψ 2 and their Yukawa interaction to M field. In the present analysis, we include interaction terms with one M field. Then, possible terms are expressed as A part including η 1 and η 2 with M is Yukawa interaction terms connecting ψ fields to η fields are expressed as In addition to the non-derivative interactions shown above, we need to include derivative interactions. Possible interaction terms including one derivative are given by Combining the above terms together, the Lagrangian in the present analysis is given by where the mesonic part L meson is written as where V (M ) is a meson potential term. In this paper, we do not specify the form of the potential, but we assume that this potential provides the vacuum expectation value (VEV) of M as M = diag(f π /2, f π /2), where f π is the pion decay constant.

B. Mass matrix
We have constructed the Lagrangian by requiring the chiral SU(2) L ⊗SU(2) R invariance. To study the properties of nucleons, we decompose baryons in the chiral representation to irreducible representations of the flavor symmetry as In the following, for convenience we redefine the nucleon fields as: The mass terms for the redefined fields are expressed as and The mass eigenstates denoted by are obtained by diagonalizing the above mass matrix M N . We note that parities of all fields in N are even due to the redefinition given in Eq. (22). Two eigenvalues of the mass matrix M N are negative in our analysis, and we regard the parities of these state as negative.

C. One pion interactions and axial charges
The interaction terms of nucleons to one pion are given from the Lagrangian as where π = π · τ and (29) Axial-vector charge matrix is determined as where In the present analysis, we identify the mass eigenstates as

III. CHIRAL INVARIANT MASSES AND PARTNER STRUCTURE
In this section we determine the values of model parameters and study the mixing structure of relevant baryons.
As we said in the previous section, we set the VEV of σ to be the pion decay constant:  [19]. The column indicated by P = ± shows the parity of the nucleon. Unit of masses and widths is MeV. The error of m N (939) expresses the mass difference between the proton and neutron. [lat] indicates that the value is obtained by the lattice analysis in Ref. [20].
Beside this parameter, there are twelve parameters in this model: 0 , g 1 , g 2 , g 3 , g 4 , y 1 , y 2 , a 1 , a 2 , a 3 , a 4 We list values of relevant physical quantities determined from experiments and lattice analyses in Table.I. Among them, we use the following ten physical values as inputs: nucleon masses: and axial charges: In addition to the above inputs, we use the following range of g A (N (1535)) to restrict the parameters: Furthermore, we restrict the parameters by requiring all the components of axial-charge matrix on the physical base are no larger than 5.
In this analysis, we first fix the chiral invariant masses m We find that solutions are categorized into five groups as shown in Fig.1. In Group 1 indicated by purple + symbols, both chiral invariant masses are less than 100 MeV. In the range where m In Group 5 indicated by yellow , the chiral invariant mass m 0 takes large value of about 1000 MeV. In Fig. 2, we show the mixing structure of nucleons: N (939), N (1440), N (1535) and N (1650) for Group 1 to Group 5. Here the horizontal axis shows the value of axial-charge of N (1535) and the vertical axis shows the percentages of ψ 1 indicated by magenta symbols, η 1 by brown • symbols, ψ 2 by green symbols and η 2 by navy symbols. In Table II, we summarize features of mixing rates for each group. The first row in Fig. 2 shows that the dominant component of N (939) is η 1 indicated by brown • belonging to [(2, 3) ⊕ (3, 2)] representation. We note that we cannot find any solutions for g A (N (1535)) −0.1 in the Group 1. One can easily see that N (1440) is dominated by η 2 (navy ), N (1535) by ψ 1 (magenta ) belonging to [(1, 2) ⊕ (2, 1)] representation and N (1650) bv η 2 (green ). Since η 1 and η 2 are chiral partners to each other, we conclude that N (1440) dominated by η 2 is In Group 2 (the second row of Fig. 2), η 1 (brown •) belonging to [(2, 3) ⊕ (3, 2)] representation is a dominant component in the N (939) and ψ 1 (magenta ) almost occupies N (1535), similarly to Group 1. A difference between Group 1 and Group 2 appears in the rate of ψ 2 (green ) in N (1440). In Group 1, the mixing rate of ψ 2 in N (1440) is smaller than 0.1 as can be seen in the first row of Fig. 2. On the other hand, the rate of ψ 2 is larger than 0.2 and ψ 2 is included in N (1440) dominantly as shown in the second row of Fig. 2. Here the rate of η 2 component (navy ) included in N (1650) is high, but N (1440) and N (1535) include a certain amount of the η 2 component. So, it is difficult to identify the chiral partner of N (939) in Group 2.
In Group 3, Group 4 and Group 5, N (939) is composed of ψ 1 (magenta ) or η 2 (navy ) dominantly and negative parity nucleons, N (1535) and N (1650) have ψ 2 (green )) or η 2 (brown •) mainly, as can be seen in the third, fourth and fifth rows in Fig. 2. This indicates that the chiral partner of N (939) is a mixture of two negative parity nucleons in these groups, differently from Group 1 and Group 2. Table II

IV. AXIAL CHARGES
In the previous section, we discussed the mixing structure of nucleons together with their chiral invariant masses. In this section, we study axial charges in detail.
We define transition axial charge as off-diagonal elements of following axial-charge matrix on physical base.
where In the present model, the following relation the diagonal axial charges is satiefied: Now we use axial charge:g A (N 1 ) = g A (N (939)) = 1.272 and g A (N 4 ) = g A (N (1650)) = 0.55 as input. So this relation is We plot this relation in Fig.3. This plot shows that, when the axial charge of N 1535) is in the range consistent with the lattice analysis, the axial charge of N (1440) is negative. In Fig. 4 we plot predicted values of g A (N 2 N 3 ) and g A (N 2 N 4 ). This shows that |g A (N 2 N 3 )| of Group 1 and Group 2 is always larger than 2 and 0.5, respectively. On the other hand, |g A (N 2 N 4 )| of Group 1 is smaller than 2 and that of Group 2 is above 1. We see that Group 1 is able to be distinguished from other groups. Predicted values of g A (N 3 N 4 ) and g A (N 1 N 3 ) are plotted in Fig. 5. We note that |g A (N 1 N 3 )| belonging to Group 1 and Group 2 are larger than 1, and that |g A (N 3 N 4 )| of Group 2 and Group 5 lies between 1 and 3 and that of Group 1 is above 1. In particular, |g A (N 3 N 4 )| ∼ 2 in Group 5. We plot the values of g A (N 1 N 2 ) and g A (N 1 N 4 ) in Fig. 6. From Fig.6, we can find that |g A (N 1 N 2 )| belonging to Group 1 (purple + symbols) and Group 2 (blue symbols) are no larger than 1, while that belonging to Group 3 (light green × symbols) and Group 4 light blue × + symbols are below 1.5. |g A (N 1 N 4 )| belonging to Group 3 (light green × symbols) are larger than 0.5.
We summarize typical predicted values of transition axial-charges and the range of transition axial-charges in Table. IV.

V. CHANGE OF NUCLEON MASSES
In this section we study the change of nucleon masses when the VEV of σ changed . We plot the dependences group (m  of nucleon masses on the value of the VEV for Groups 1-4 in Fig. 7 and those for Group 5 in Fig. 8 for some choices of the chiral invariant masses, m   Figure 7 shows that nucleons masses are decreased as σ 0 is decreased in Group 1, 2, 3 and 4, if chiral invariant masses are smaller than mass of N (939). We note that, for some parameter choices in Group 4, mass of ground state nucleon is increased as σ 0 is decreased. In the case of Group 5 shown in Fig. 8, the value of m shows that the mass of the ground state is stable for σ 0 > 60 MeV and it decreased towards m (2) 0 as σ 0 is decreased from 60 MeV. On the other hand, the right panel [(1210MeV, 1050MeV)] shows that all the masses are stable against the change or σ 0 .
Since σ 0 is an order parameter of chiral symmetry, Figs. 7 and 8 show that nucleon masses are degenerated to chiral invariant masses when thechiral symmetry is restored in e.g., high temperature and/or density.  2)], whose chiral invariant mass is about 1000 MeV, and the chiral partner of N (939) is a mixture of negative parity nucleons.
We gave predictions off-diagonal elements of axialcharge matrix called transition axial-charges, which shows that some groups can be excluded when some of them are determined by e.g., lattice analysis in future. We also study the change of nucleon masses when the VEV of σ, σ 0 , is changed. In Groups 1-4, all nucleons masses are decreased with decreasing σ 0 if two chiral invariant masses are smaller than mass of N (939) as shown in Fig. 7. In Group 5, on the other hand, the behavior depends on the value of m (2) 0 : For small m (2) 0 the mass of N (939) is stable for σ 0 > 60 MeV and it decreases toward m (2) 0 , while for large m (2) 0 it is stable for all σ 0 . This seems consistent with the lattice analysis in Ref. [17,18], which shows that, with increasing temperature, the mass of the positive parity nucleon mass is stable, while that of the negative parity nucleon mass decreased.