The $\eta$ decay into 3$\pi$ in asymmetric nuclear medium

We explore how the $\eta$-$\pi^0$ mixing angle and the $\eta$ meson decay into $\pi^{+}\pi^{-}\pi^0$ and 3$\pi^{0}$ are modified in the nuclear medium on the basis of the in-medium chiral effective theory with the isospin asymmetry $\alpha$ varied, where $\alpha\equiv \delta\rho/\rho$ with $\delta \rho=\rho_n-\rho_p$ and $\rho=\rho_n+\rho_p$. We find that the larger the isospin asymmetry $\delta \rho$ and smaller the total density $\rho$, the more enhanced the mixing angle. We show that the decay width in the nuclear medium has an additional density dependence which cannot be renormalized into that of the mixing angle: The additional term originates from the vertex proportional to a low energy constant $c_1$, which only comes into play in the nuclear medium but not in the free space. It turns out that the resultant density effect on the decay widths overwhelms that coming from the isospin asymmetry, and the higher the $\rho$, the more enhanced the decay widths; the width for the $\pi^{+}\pi^{-}\pi^0$ decay is enhanced with a factor two to three at the normal density $\rho_0$ with a minor increase due to $\delta \rho$, while that for the 3$\pi^0$ decay shows only a small increase of around 10 percent even at $\rho_0$. We mention the possible relevance of the partial restoration of chiral symmetry to the unexpected density effect on the decay widths in the nuclear medium.


Introduction
The quantitative understanding of the η meson decay into three π's is one of the long standing problems in hadron physics [1][2][3][4][5][6][7][8][9].The decay process is prohibited by the G parity conservation with the isospin symmetry being taken for granted, and the electromagnetic correction is found to vanish in the leading order [1].The origin of the decay is attributed to the isospin-symmetry breaking inherent in Quantum Chromodynamics (QCD), the small current-quark mass difference between u and d quarks 1 .Due to the isospin-symmetry breaking, the observed η and π 0 do not correspond to the eigenstate of the flavor SU(3) but to their mixed state; where we denote the mass eigenstate as η and π 0 , and the flavor SU(3) eigenstate as η 8 and π 3 .The angle θ is called the η-π 0 mixing angle.The π 3 component of the η meson enables the meson to decay into 3π's, telling that the mixing angle plays an essential role in the decay.However, the analysis based on the current algebra shows a large discrepancy with the experimental result [2].Recent theoretical development has revealed the significance of the final-state interaction between pions in addition to the u and d quark mass difference for the quantitative account of the experimental data 2 .A phenomenological inclusion of the effect of the final state interaction [4] shows a good agreement with the observed decay width and it is shown that the higher order contribution of chiral perturbation theory to the decay process which contains the effect of the final state interaction is quite large [5,7,8].The modification of the hadron properties in the environment, which is characterized by temperature, baryonic density, electromagnetic field and so on, is one of the interesting topics of hadron physics [17].Furthermore, effects of the isospin-asymmetry in a nuclear medium on hadron properties are being investigated in various systems, including the pion nucleus system [18,19], the few-body system of the hyper-nuclei [20], and the equation of state of the nuclear matter [21,22]: In this paper, we show that the η decay into three pions provide yet another example for revealing the interesting effects caused by the isospin-asymmetry of nuclear medium.
As for the in-medium properties of the η meson, intensive search for η-bound states in a nucleus has been made [23], where the focus is put on the η-optical potential and/or a possible mass shift in the nuclear medium.
In this paper, we study the effect of the asymmetric nuclear medium focusing on the three-π decay of the η in the isospin-asymmetric nuclear medium, where the isospin breaking background field is present in addition to the different u and d quark masses [24].We shall show that the isospin-asymmetry in the nuclear medium increases the mixing of the η and the π 0 meson and thereby leads to an enhancement of the decay rates of the η meson to three pions.We shall also find that the (isosymmetric) total baryon density unexpectedly causes an enhancement of the decay width, which is found to be associated with the phenomenon of the partial restoration of chiral symmetry in the nuclear medium.
This paper is organized as follows.In Sec. 2, we introduce the model Lagrangian and explain the calculation of the η-π 0 mixing angle and the decay amplitude of the η into three π's in the asymmetric nuclear medium in chiral effective field theory.We evaluate the ηπ 0 mixing angle in the asymmetric nuclear medium in Sec. 3. Then we discuss the decay amplitude of the η meson decay into 3π's in the isospin-asymmetric nuclear medium with numerical results in Sec. 4. A brief summary and concluding remarks are presented in Sec. 5.In Appendix, we present the explicit forms of the meson-baryon vertices derived from the chiral Lagrangian and used in the calculation in the text.

Preliminaries
To investigate the η meson decay into three π's in the nuclear medium, we apply the chiral effective field theory in the nuclear medium.A full account of the in-medium chiral perturbation may be seen in Refs.[25,26].The basic degrees of freedom are the flavor-octet pseudoscalar mesons and baryons.Then the chiral Lagrangian needed for our calculation reads [27,28] L =L (2)  ππ + L (4)  ππ + L (1)

L
(2) Here, and • • • means the trace in the flavor space.The Lagrangian which determine the interaction between the hadrons is constructed so as to be invariant under the chiral transformation of the hadron fields as The parameters f, B 0 , m i , L i , H i , g A , and c i appearing in the Lagrangian are low-energy constants (LEC's), the values of which cannot be fixed solely from the symmetry and determined phenomenologically; the values which are used in our calculation are presented in Refs.[27,29].
The relevant degrees of freedom of baryon fields in Eq. ( 9) are proton and neutron because we are interested in the medium modification by the nucleon background.We denote the nucleons in a doublet form as N = t (p, n).Although it is known [23,30] that the coupling with N * (1535) resonance contribute to the η self-energy, the incorporation of N * (1535) and other excited baryons with strangeness is beyond the scope of the present work.We shall later give a brief comment on possible modification of the results due to the coupling with N * (1535).The meson-baryon vertices are derived by expanding U with respect to the meson fields π a .The explicit forms of the vertices to be used in our calculation are presented in Appendix A.
The medium effect is contained in the nucleon propagator iG(p, k f ), where m N and k f are the nucleon mass and Fermi momentum, respectively: The first term of Eq. ( 18) is the contribution of the nucleon propagation in the free space and the second term account for the Pauli blocking effect of nuclear medium.The number density and the Fermi momentum of the nucleon are related by ρ p,n = k (p,n)3 f 3π 2 .The total baryon density ρ and asymmetric density δρ are defined by ρ = ρ n + ρ p and δρ = ρ n − ρ p , respectively.Note that δρ > 0 means that ρ n > ρ p in the present definition.The nuclear asymmetry is also defined by α = δρ/ρ.
We note that the value of the Fermi momentum k f around the normal nuclear density ρ 0 =0.17fm −3 is roughly 2m π .In our calculation, we regard k f as small as the pseudoscalar meson masses and momenta, which are the expansion parameter in the ordinary chiral perturbation theory.We call k f and the masses of the pseudoscalar mesons as the small quantities and denote generically by q in the following.Hence, our calculation is the expansion with respect to the number of the mesons or nucleon loops because these loops supply additional small quantities compared with the tree level.In addition, we regard the nucleon mass m N as a large enough quantity and neglect the ratios of the other quantities to m N .
Here, we note that the the states of the nuclear medium is treated as a Fermi gas in the leading order in the present formalism, and accordingly where the nucleon-nucleon interaction is switched off initially.
We calculate the η-3π decay width up to O(q 5 ) in the leading order of the asymmetric density δρ.The final states of the three π's can be two patterns, i.e., π 0 π + π − and three π 0 's.The decay amplitude in the free space up to O(q 4 ) is given in Refs.[5,7].

The η-π 0 propagator in the asymmetric nuclear medium
This section is devoted to the calculation of the η-π 0 propagator and the η-π 0 mixing angle in the asymmetric nuclear medium.Here, we denote η and π 0 as the mesons of the mass eigenstate and the η 8 and π 3 as the SU(3) eigenstate and the mass and SU(3) eigenstates are related by Eq. (1).
The η and π 0 propagator in the asymmetric nuclear medium D(p; k f ) reads where ) are the propagator of the pseudoscalar mesons in the triplet and octet state in the free space, respectively, and the Π i (k f ) is the in-medium self-energy.The Π η8π3 (k f ) is the transition amplitude of the η 8 and π 3 mesons.The meson mass squared m 2 η8 and m 2 π3 are the pole of D η8 and D π3 , respectively, and the off-diagonal term of the η-π 0 mass matrix m 2 η8π3 is equal to Π η8π3 .The η-π 0 mixing angle θ is obtained in terms of the masses; The diagrams which contribute to the η-π 0 mixing are shown in Fig. 1.We denote the contribution from the diagrams (i), (ii), and (iii) as Π Fig. 1 The diagrams contributing to the η-π 0 mixing angle in the asymmetric nuclear medium.The dashed and double-solid lines represent the meson and nucleon propagations, respectively.The white box at the vertex of diagram (iii) means the L πN -originated vertex.
The η N N and π 0 N N vertices relevant to Π η8π3 are given in Eqs.(A4) and (A5), and we have Here, the minus sign of the right hand side of the first line comes from the nucleon loop, and the tr in the first line means the trace of the gamma matrix.In our calculation, we take the η-rest frame, so the η momentum k is given by k = (m η , 0).Eliminating the contribution from the nucleon propagation in the free space, we have Changing the integration valuable from p to p ′ = p + k in the first term in the bracket, we obtain Here, the nucleon mass is treated as a large quantity, and hence the nucleon energy E N ( p) is approximated by m N and the initial η meson is at rest.Noting that the π 0 couples to proton and neutron with the opposite sign, we obtain the final form as Now, we decompose Π η8π3 that come from the terms proportional to c 1 and c 5 in the chiral Lagrangian in Eq. ( 6), respectively.The ηπ πN is given in Eq. (A8), and thus we have for Π which is reduced to With the use of the ηπ 0 N N vertex coming from the c 5 term in Eq. (A11), Π η8π3 is reduced to which tells us that the contribution from the nuclear medium is given by Incorporating the contribution in the free space and nuclear medium effect given in Eqs.(24), (25), and (27), we have for the in-medium η-π 0 mixing angle θ (ρ) where Equation (28) shows that the in-medium mixing angle depends not only on the asymmetric density δρ appearing in the second term, but also on the total baryon density ρ in the third Fig. 2 The η-π 0 mixing angle in the asymmetric nuclear medium up to O(q 5 ).The solid lines are the contour lines and plotted per 0.5 of the mixing angle normalized with the free space value.The horizontal and vertical axis are the proton density, ρ p [fm −3 ], and the neutron density, ρ n [fm −3 ].The dashed and dotted lines are the constant-ρ and vanishing-δρ lines, respectively.The lower-left and the upperright regions of the figure correspond to the small and large total baryon density ρ and the upper and lower sides of the dotted line are the neutron-and proton-rich region, respectively.The value is normalized by the mixing angle θ (0) ≃ 1.058 × 10 −2 [rad] in the free space.The mixing angle is smaller than that of free-space value in the blue region where ρ p is large, and larger in the red region where the ρ n is large.term.We see that δρ enhances the mixing angle in the neutron-rich asymmetric nuclear medium, while ρ reduces the mixing angle because the coefficient c 1 is negative.The parameter c 1 is determined so as to reproduce the low-energy πN scattering [29] and the sign reflects the nature of the low-energy πN interaction.The resultant decay width is obtained from the balance of these effects.From the assignment of the isospin, the proton and neutron density affect the decay in the same way as the u and d quark mass do.The larger density difference of the u and d quarks means the stronger violation of the isospin-symmetry, so the η-π 0 mixing angle is enhanced in the neutron-rich nuclear medium.The contour plot of the η-π 0 mixing angle in the asymmetric nuclear medium is presented in Fig. 2: In the present work, we use the following values in the numerical calculation [29]; c 1 = −0.93 ± 0.10 GeV −1 , c 5 = −0.09± 0.01 GeV −1 , and f = 93 MeV.We note that the LEC's c i have uncertainties of some 10 percent.The masses of all the hadrons are taken to be the experimental values listed in PDG [16] and thus m 2 1 = 5165.86MeV 2 .One can 8 (a) Fig. 4 The diagrams contributing to the η decay into three π's.The meanings of the lines and vertices are same as in Fig. 1.The diagrams (a), (b), and (c) are the contributions from the mixing angle and (d) to (g) give the medium effects on the η-3π decay amplitude directly.
find from this figure that the η-π 0 mixing angle is enhanced in the neutron-rich asymmetric nuclear medium, and tends to be slightly suppressed by the total baryon density.We show the nuclear asymmetry α dependence of the η-π 0 mixing angle in Fig. 3: One sees that the mixing angle is enhanced by α, and the slope of the α dependence is bigger for the higher total baryon density.In fact, the slope of the mixing angle in terms of α in the small density is given as which is proportional to ρ.
The density dependence of the η-π 0 mixing angle has an uncertainty coming from those of the LEC's, c 1 and c 5 [29].The resultant uncertainty of the mixing angle is about ten percent.

The η-3π decay width in the asymmetric nuclear medium
In this section, we estimate the η-3π decay width in the asymmetric nuclear medium.The partial width Γ at the rest frame of a particle with mass M reads Here, n is the number of the identical particles in the final state, and M the matrix element of the decay, and s = (p η − p π 0 ) 2 , t = (p η − p π + ) 2 the Mandelstam valuables.
The diagrams contributing to the η-3π decay amplitude is shown in Fig. 4. The diagrams (a), (b), and (c) affect the decay amplitude through the η-π 0 mixing angle, and the diagrams (d) to (g) give the medium effects on the decay amplitude directly.The in-medium η-π 0 mixing angle has been already calculated in Sec. 3, i.e., the contributions from the diagrams (a), (b), and (c).We evaluate the diagrams (d) to (g) in Fig. 4 and the decay width in the asymmetric nuclear medium in this section.In the first and second subsection, we calculate the decay amplitude of η into π 0 π + π − and 3π 0 , respectively, and we show the results of the numerical estimation of the decay width in the third subsection.

The η decay into π
In this subsection, we calculate the η decay amplitude into π 0 π + π − in the asymmetric nuclear medium.We show the matrix elements of η decay into π 0 π + π − which come from the diagrams (d) to (g) in Fig. 4.
Diagrams (d) and (e) do not contribute to the decay amplitude because both the ηπ 0 π + π − NN and the ηπ 0 N N vertices coming from L With the use of the vertices of η N N and π 0 π + π − N N given in Eqs.(A4) and (A13), the contribution from the diagram (f ) is written as The minus sign comes from the fermion loop.Eliminating the pure-vacuum contributions and setting A = 2p 0 − p + − p − , we have Changing the integration valuable of the first term in the bracket p into p ′ = p + k and keeping the leading order of the momentum, iM (f ) is reduced to Approximating E N ( p) to m N , we arrive at Using the energy conservation, A 0 = 3E π 0 − m η , and taking account of the opposite sign of the vertex between π and proton or neutron, we find that the leading contribution containing the density effect finally has the following form; Here, we decompose the contribution from the diagram (g) in Fig. 4 in two parts; the term proportional to c 1 and c 5 , respectively; M (g) as M (g) = M (g1) + M (g5) , where M (g1) and M (g5) are proportional to c 1 and c 5 .The ηπ 0 π + π − N N vertices proportional to c 1 and c 5 are presented in Eqs.(A21) and (A22), respectively.
With the ηπ 0 π + π − N N vertex Eq. (A21), M (g1) is given as Eliminating the vacuum part of the nucleon propagator, M (g1) is reduced to Using the η3π 0 N N vertex Eq. (A22), M (g5) is written as Omitting the terms which come from the free space propagation of nucleon, we obtain M (g5) as Thus, M (g) is written as Taking account of the contributions from the free space and the modification of the η-π 0 mixing angle, we have the matrix element of the η decay into π 0 π + π − in the asymmetric nuclear medium up to O(q 5 ) as where Here, s 0 is given as s 0 = m 2 η /3 + m 2 π , and M (4)vac η→π 0 π + π − is the meson one-loop contribution in the free space which is known to give a large contribution as mentioned in Sec. 1.The details of the form and calculation of M (4)vac η→π 0 π + π − are given in Refs.[5,7].M (f) and M (g) are given in Eqs. ( 37) and ( 42), respectively.We denote the η-π 0 mixing angle in the free space by θ (0) , which is given by setting ρ = δρ = 0 in Eq. (28).In this calculation, we have assumed that the isospin-symmetry breaking is so small that we can make the approximation that sin θ ∼ θ ∼ tan 2θ/2.

The η decay into three π 0 's
In this subsection, we give the decay amplitude for the η to three π 0 's in the asymmetric nuclear medium.
First of all, the contributions from the diagrams (d), (e), and (f ) vanish in the three π 0 's case.The diagrams (d) and (e) give no contribution in the same way as in the case of the π 0 π + π − decay.The contribution of the diagram (f ) also vanishes because the 3π 0 N N vertex is zero as is shown in Appendix A.3.Thus, the decay amplitude in the asymmetric nuclear medium is solely given by the sum of the contributions from the diagram (g).M (g1) , which is proportional to c 1 , is written as where the η3π 0 N N vertex appearing from c 1 is presented in Eq. (A18).The nuclear medium modification is evaluated to be The η3π 0 N N vertex with c 5 is given in Eq. (A22), and the term proportional to c 5 in M (g5) , reads Omitting the free-space part in the right hand side of the equation, we obtain Summing up Eqs. ( 46) and ( 48), we have Thus we have the matrix element of the η decay into 3π 0 as with Here, M (4)vac 3π 0 is the contribution from the meson one-loop and its detailed form is presented in Refs.[5,7].We note that the amplitude does not depend on the η-π 0 mixing angle in the leading order on account of the symmetry of the final state consisting of the identical three π 0 's.For this reason, the medium modification of the η decay into three π 0 's is small, which will be demonstrated in the numerical calculation given in the next subsection.

Numerical results
Using the definition of the decay width given in Eq. ( 31) and the matrix elements presented in Eqs. ( 43) and (50), we evaluate the partial width of the η decay into 3π.
Figure 5 and 6 are the contour plot of the the decay width of the η into π 0 π + π − and into three π 0 's, respectively.The widths are normalized with the each value at ρ = δρ = 0, respectively.] Fig. 6 η → 3π 0 decay width in the asymmetric nuclear medium up to O(q 5 ).The axes and the lines are the same as those in Fig. 5.The contour is plotted per 0.03 of the normalized width.The width in the free space is 298eV.
First, we discuss the η-π 0 π + π − decay width.From Figure 5, one finds that the width is large in the higher density region (upper right of the figure), and the decay width is enhanced in the proton rich region.We show the δρ dependence of the decay width with some values of fixed ρ in Fig. 7.The parabolic-shape dependence on δρ comes from the sin 2 θ dependence of the decay width: The minimum points of the decay width are not located at α = 0 due to the explicit symmetry breaking in the free space caused by the different u, d quark masses.We find that the decay width is enhanced by the total baryon density ρ.
Next, we discuss the η-3π 0 decay.Figure 6 is the contour plot of the η-3π 0 decay in the asymmetric nuclear medium and Figure 8 shows the δρ dependence of the η-3π 0 decay width with some fixed values of the total density ρ.The decay width of the η into 3π 0 shows an enhancement by the total baryon density much the same way as the the η into π 0 π + π − decay.This enhancement is caused by the third term in Eq. (50).One finds that the effect of the total baryon density ρ overwhelms that of the isospin-asymmetry δρ on the η-3π 0 decay.Nonetheless the neutron-rich medium enhances the decay width as one can see in Fig. 8.The relative smallness of the effect of the isospin-asymmetry δρ on the decay in comparison with that on the π 0 π + π − decay can be attributed to the fact that the mixing angle dependence of the decay amplitude is suppressed by the crossing symmetry.
Here, we comment on the uncertainties of the decay widths coming from those of the LEC's.The uncertainties of the η decay width into the π 0 π + π − and 3π 0 are both about ten percent.We show Figs. 9 and 10 to visualize the uncertainties for the normal nuclear densities as an example: The solid lines are the central values and the shaded areas show the uncertainties from the LEC's.Fig. 7 The nuclear asymmetry α dependence of the η decay width into π 0 π + π − with some fixed ρ.The solid, dashed, and dotted lines are the decay width at ρ = ρ 0 , ρ 0 /2, and ρ 0 /4, respectively.The decay width is normalized with the value at ρ and δρ = 0.The uncertainty of the η decay width into π 0 π + π − at the normal nuclear density.The solid line corresponds to the decay width with the LEC's at the central values.The shaded area represents the uncertainty of the width due to that of the LEC's.Others are the same as those of Fig. 7.The uncertainty of the η decay width into 3π 0 due to those of LEC's at the normal nuclear density.The line, and the shaded area are same as those of Fig. 9.
Here, we discuss the origin of the enhancement of the decay widths with the total baryon density.It is found that the dominant density dependence for both the two decays comes from the term which is proportional to the low energy constant c 1 in Eq. ( 43) and in Eq. (50) for the π 0 π + π − and 3π 0 decay, respectively.The parameter c 1 is related to σ πN by c 1 = − σπN 4m 2 π because we assume that all the loop corrections of the nucleon mass are renormalized into LEC [31].Substituting this relation into the second term of Eq. ( 43), one finds that the prefactor of the second term reads It is noteworthy here that the coefficient σ πN ρ/m 2 π f 2 is nothing but the quantity that represents the reduction rate of the quark condensate or chiral order parameter in the nuclear medium with the linear density approximation [32,33]; where qq ρ=0 and qq ρ are the quark condensates at ρ = 0 and non-zero, respectively.Thus, one should be able to rewrite the decay amplitude given in Eq. (43) ( Eq. ( 50)) for the π 0 π + π − (3π 0 ) decay in terms of the reduction of the chiral order parameter δ qq .Here, we rewrite the M η→π 0 π + π − Eq. ( 43) in terms of the renormalized pion decay constant f * ; where From the first to second line, we have regarded σ πN ρ/m 2 π f 2 as a small quantity, and thus the ρ dependence is absorbed into f * 2 .As one can see in Eq. ( 53), the total baryon density dependence of the decay amplitude of the η → π 0 π + π − process can be renormalized into the density dependence of the pion decay constant3 .Thus, one may say that the enhancement of the 3π decay width originates from the chiral restoration in the nuclear medium, although more detailed analysis is necessary to establish the relevance of the partial restoration of the chiral symmetry in the enhancement of the 3π decay of the η at finite baryon density.Nevertheless, it is worth mentioning that a similar mechanism is responsible for an enhancement of the ππ cross section near the 2π threshold in the σ meson channel in nuclear matter by Jido, Hatsuda, and one of the present authors [35], where it is found that the reduction of the chiral condensate implying the partial restoration of chiral symmetry in nuclear matter is responsible for this enhancement.Furthermore, they clarified that a 4π-nucleon vertex shown in Fig. 5 in Ref. [35], which has the same structure diagram (g) in Fig. 4 in the present article, is responsible for the enhancement.

Brief summary and concluding remarks
In this paper, we have studied the η-π 0 mixing angle defined in Eq.( 1) and the η decay into 3π in the nuclear medium with its isospin-asymmetry varied on the basis of the inmedium chiral effective theory, where the Fermi momentum k f as well as the pseudoscalar meson masses and momenta are treated as the small expansion parameters.We have found that both the quantities are significantly modified in the nuclear medium by the asymmetry δρ = ρ n − ρ p and the total baryon densities ρ = ρ n + ρ p , although the densities affect both the quantities in different manners.Whereas the mixing angle increases along with the asymmetric density δρ, but decreases along with the increase of the total baryon density.In terms of the α dependence of the mixing angle, the mixing angle increases along with the α, the slope of which is greater for the larger total baryon density.The increase of the mixing angle means that the physical η meson has more component of the π 3 , the isospin eigen state of the neutral pion.It has turned out the total and asymmetric densities tend to increase the decay width of the η into π 0 π + π − in an additive way, although the total density dependence overwhelms that of the isospin: For example, the width is enhanced with a factor 2 to 3 at ρ = ρ 0 depending on the isospin asymmetry α.
The enhancement of the width due to the isospin asymmetry is traced back to the increase of the π 3 component in the physical η state.
The η-3π 0 decay width in the nuclear medium is enhanced by the total baryon density ρ as in the case of η → π 0 π + π − decay, although the enhancement is relatively smaller than the latter case.This is because the mixing angle dependence of the η-3π 0 decay amplitude is suppressed by the symmetrization of the three π 0 's in the final state.The neutron-rich asymmetric medium slightly enhances the decay width in the η-3π 0 case.
The ρ dependent part of the decay amplitudes of the η to 3π given in in Eq. (43) and Eq.(50) are both proportional the low energy constant c 1 , which is in turn related with the π-N sigma term σ πN .Thus, we find that the effect of the total baryon density can be rewritten in terms of the change of the quark condensate in the nuclear medium.This suggests that the enhancement of the decay rate may result from the partial restoration of chiral symmetry in the nuclear medium.Further analyses are, however, necessary to elucidate the underlying mechanism of the enhancement and its possible relationship with the chiral restoration in the nuclear medium.
It may be possible to observe the modification with the nucleus target experiment with the large isospin asymmetry (N − Z)/A at the facilities, for example, SPring-8 and J-PARC in Japan or FAIR, COSY, and MAMI in Germany.
In our calculation, the decay widths in the free space fairly reproduce the experimental data: The values are roughly 60% in the η → π 0 π + π − case and 70% in the η → 3π 0 case compared with the experimental data [16].To reproduce the experimental data in the free space more precisely, the higher order terms in the chiral perturbation should be included [7].Some of them may be resummed into the final state interaction of π's, more precisely, the s-wave π + π − correlations, which may make the σ resonance as mentioned in Sec. 1.
The nuclear medium would affect the spectral properties of the σ mode through the chiral restoration [17,[35][36][37][38]), and thus a full account of the medium effect on the σ mode including its possible softening further effects of the nuclear medium on may have a significant impact on the η decay in the nuclear medium.
Our calculations do not take the effect of the N * (1535) resonance into account as mentioned in Sec. 3. It is known [23,30] that the coupling with resonance modifies the in-medium selfenergy or the optical potential of the η: It is suggested that the η-N interaction is attractive, and the attraction might lead to a reduction of the η mass, say of 50 MeV order, in the nuclear medium.Furthermore, there arises an induced η-π 0 coupling through the N * -nucleon hole excitations in the nuclear medium.The formula of the η-π 0 mixing angle in Eq. ( 1) tells us that both the reduction of the η mass and the additional η-π 0 coupling would cause an additional ρ dependence but not the δρ dependence of the mixing angle and hence an enhancement of the π 0 π + π − decay width.On the other hand, the 3π 0 decay is independent of the mixing angle as shown in Eq. ( 50), so the effect of the resonance on the 3π 0 decay width should be small.We hope that we can report on a quantitative analyses of such an additional density dependence coming from the coupling with N * of the mixing angle and the decay width of the η in near future.
Recently, possible effects of the external magnetic field on hadron properties are a focus of intensive studies; the relevant physical quantities include mass spectra of light hadrons [39,40] and psudoscalar-vector mixing rate in heavy quarkonia as well as their mass shifts [41,42]).We note that a strong magnetic field can be also a possible source of the isospin-symmetry breaking due to the difference of the electromagnetic charges of the u and d quarks, and thus the external magnetic field may change the hadron properties as we have shown in the present work.This subject is left as a future study.
A.4.2.The vertex from c 1 term.The ηπ 0 π + π − N N vertex from the c 1 term is given as

Fig. 3
Fig.3The nuclear asymmetry α dependence of the η-π 0 mixing angle in the nuclear medium.The horizontal and vertical axes represent the nuclear asymmetry and the ηπ 0 mixing angle normalized by the value at ρ = δρ = 0.The solid, dashed, and dotted lines are the lines with ρ = ρ 0 , ρ 0 /2, and ρ 0 /4, respectively.

3 ]Fig. 5 η
Fig.5η → π 0 π + π − decay width in asymmetric nuclear medium up to O(q 5 ).The horizontal and the vertical axes represent ρ p and ρ n , respectively.The solid line represents the contour of the decay width.The dashed and dotted lines mean the constantρ and the vanishing-δρ lines, respectively.The width is normalized by the value at ρ = δρ = 0.The contour is plotted per 0.3 of the normalized width.The width in the free space is 163eV.
Fig. 10The uncertainty of the η decay width into 3π 0 due to those of LEC's at the normal nuclear density.The line, and the shaded area are same as those of Fig.9.