A Explicit expression of each interaction term

Pion properties at finite nuclear density based on in-medium chiral perturbation theory

Abstract

The in-medium pion properties, i.e. the temporal pion decay constant , the pion mass and the wave function renormalization, in symmetric nuclear matter are calculated in an in-medium chiral perturbation theory up to the next-to-leading order of the density expansion . The chiral Lagrangian for the pion-nucleon interaction is determined in vacuum, and the low energy constants are fixed by the experimental observables. We carefully define the in-medium state of pion and find that the pion wave function renormalization plays an essential role for the in-medium pion properties. We show that the linear density correction is dominant and the next-to-leading corrections are not so large at the saturation density, while their contributions can be significant in higher densities. The main contribution of the next-to-leading order comes from the double scattering term. We also discuss whether the low energy theorems, the Gell-Mann–Oakes–Renner relation and the Glashow–Weinberg relation, are satisfied in nuclear medium beyond the linear density approximation. We also find that the wave function renormalization is enhanced as largely as 50% at the saturation density including the next-to-leading contribution and the wave function renormalization could be measured in the in-medium decay.

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D33

1 Introduction

Spontaneous breakdown of chiral symmetry (SSB) characterizes the vacuum and low-energy dynamics of Quantum ChromoDynamics (QCD) [1]. The non-vanishing chiral condensate is considered as one of the order parameters of SSB and gives a characteristic scale for hadron physics. SSB is considered to be responsible for the origin of constituent quark mass after the current quark mass is given by the Higgs condensate slightly. According to the spontaneous breakdown, the pseudoscalar mesons such as appear as the Nambu-Goldstone (NG) bosons.

Recently, in order to investigate the mechanism of the dynamical mass generation, partial restoration of the chiral symmetry in the nuclear medium has gained considerable attention. This phenomenon is incomplete restoration of chiral symmetry with sufficient reduction of the absolute value of the chiral condensate in the medium and will lead to various changes of hadron properties. Once we understand the partial restoration of chiral symmetry, we can predict other in-medium hadronic quantities through low energy theorems and vice versa.

From this point of view, vast theoretical and experimental efforts are devoted for this topic. The density dependence of the chiral condensate is evaluated in various approaches, for example, the well-known linear density approximation providing the model-independent low density theorem [2, 3], the relativistic Bruckner-Hartree-Fock theory approach[4] and systematic calculations by in-medium chiral perturbation theory beyond the linear density calculation [6, 5]. According to the model-independent linear density approximation, the leading density correction to the chiral condensate is determined by the N sigma term and with the empirical value of the sigma term it has been found that the leading correction gives enough large contribution at the normal nuclear density:

(1)

where , is the in-medium and in-vacuum condensate and is the normal nuclear density. Recently, an in-medium sum rule satisfied in any density region has been derived model-independently by current algebra method and the low energy theorems, such as the Gell–Mann–Oakes–Renner relation, the Glashow–Weinberg relation and the Weinberg–Tomozawa relation, are discussed within the linear density order [7].

To investigate the pion properties in nuclei, deeply bound pionic atoms have received much attention [8, 9, 10]. Theoretically, the binding energies and decay widths of the 1s and 2p deeply-bound pionic atom states are estimated [12, 13, 11] and in Refs. [14, 15] hadronic quantities, such as pion optical potential, have been calculated beyond the linear density. Experimentally the reduction of the chiral condensate is estimated quantitatively through reduction of the -wave isovector parameter in the -nucleus optical potential [10]. The parameter is regarded as an in-medium isovector N scattering length. These results show that the reduction of means the repulsive enhancement of the -wave -nucleon interaction in nucleus. Another examples are low energy -nucleus scattering and interaction in nuclei in the scalar-isoscalar channel. The low energy -nucleus scattering also show that s-wave -nucleus interaction are enhanced repulsively [16, 17]. According to the theoretical discussion given in Refs. [18, 19], the in-medium interaction in the scalar-isoscalar part will also has attractive enhancement thanks to the partial restoration of chiral symmetry in nuclear medium and the experimental observation of the invariant mass spectrum of the production off nuclear targets performed in Refs. [20, 21, 22] could have a hint of such a enhancement.

In particular the pion decay constant is a fundamental quantity of chiral symmetry breaking. The in-medium decay constant also has been investigated in the linear density approximation [23] and recently the chiral condensate and the decay constant have been evaluated in the next-to-leading order based on chiral order counting [24]. In this paper, we discuss a general in-medium pion state and evaluate in-medium pionic quantities such as the decay constant, mass and the pseudo scalar coupling beyond linear density approximation. This paper is organized as follows. In Sec.2, we explain the general formulation of the in-medium chiral perturbation theory and discuss an expansion by Fermi momentum counting. In Sec.3, we discuss in-medium pion state and define the in-medium pionic quantities. Here we will find that the pion wave function renormalization plays an important role for the in-medium pionic quantities. In Sec.4, we evaluate the in-medium pion self energy, wave function renormalization, pion decay constant and pseudo-scalar coupling and show the numerical results of the density dependence of them up to in Fermi momentum expansion in symmetric nuclear matter and in isospin limit. We also discuss whether the in-medium low energy theorems, the Gell-Mann–Oakes–Renner relation and the Glashow–Weinberg relation, are satisfied or not. Finally we discuss the in-medium process caused by chiral anomaly. In Sec.5, we summarize our paper.

2 In-medium chiral perturbation theory

Chiral perturbation theory (CHPT) is a powerful tool describing low energy dynamics of the Nambu-Goldstone (NG) bosons and nucleons as an effective field theory of QCD [25, 26, 27, 28]. CHPT is constructed based on chiral symmetry and its spontaneous breaking and consists in systematic expansion of NG boson momentum and the quark mass. The chiral order counting scheme makes it possible to categorize Lagrangian and Feynman diagrams in terms of powers of momentum and the quark mass and to estimate magnitude of possible corrections for the amplitude, such as the current-current correlation functions. Thus, CHPT describes quantitatively the S-matrix elements of the QCD currents.

In this decade, chiral effective theory for nuclear matter has been developed [29, 14] and is applied to study nuclear matter properties, such as the nuclear matter energy density[30] and also used to study partial restoration of chiral symmetry and the in-medium changes of the pion properties, such as in-medium pion mass , decay constant [6, 24] and 1s and 2p energy levels of deeply-bound pionic atoms [11].

2.1 Basics of the formulation

For the calculations of the in-medium pion quantities, we evaluate the Green’s functions in the ground state of nuclear matter, and, in particular, our interests are hadron properties in nuclear medium as a bound state of the nucleon many body system. In quantum field theory, the transition amplitude between the ground state in the presence of the external fields is the fundamental quantity, and the generating functional is defined by the transition amplitude, which can be calculated by the path integral formalism of the quantum field theory. In the in-medium chiral perturbation theory, one prepares the ground state of nucleon Fermi gas at asymptotic time as a reference state [29]. The generating functional for the connected Green functions in nuclear matter is given as follows:

(2)
(3)

where is the chiral field parametrized by the NG boson field in the nonlinear realization of chiral symmetry, is the nucleon field, and represents the scalar, pseudo scalar, vector and axial vector external fields . The Lagrangian is described by the free pion and nucleon fields and should include, in principle, all the interaction among pions and nucleons within the Lagrangian. Since we use the Lagrangian described by the free nucleon field and the Lagrangian prescribes the nucleon interaction, the reference state can be the ground state of the free Fermi gas of nucleons defined by

(4)

where is the nucleon creation operator with momentum and the index represents the spin and isospin and is the number of the momentum states below the nucleon Fermi momentum , which is obtained by the nucleon density as , and is the 0-particle state.

The connected -point Green functions can be calculated by taking functional derivatives of with respect to the external sources :

(5)

where is the corresponding current operator to the external source . Considering the symmetry property of the current operator under the chiral rotation, one can identify the quark contents of the current operators, such as the pseudo-scalar current and the axial-vector current . If we evaluate the generating functional non-perturbatively with appropriate nucleon-nucleon interactions, we can describe nuclear matter in principle and deduce the QCD current Green functions in nuclear matter. Note that this prescription resembles the description of deuteron in terms of the free nucleons, in which one starts with the free nucleon field and with appropriate nucleon-nucleon interaction and one can find deuteron as a bound state of two nucleons by solving the Bethe-Salpeter equation non-perturbatively.

Reference [29] propose a method to derive the in-medium chiral Lagrangian in a systematic expansion in terms of the chiral counting and the Fermi sea insertion. In this method, the Fermi momentum is regarded as a small parameter as the chiral order. Performing the integral in terms of the nucleon field by using the Gauss integral formula for the bilinear form of the nucleon interaction, one obtains the generating functional characterized by double expansion of Fermi sea insertions and chiral orders:

(6)

where denotes Fourier transformation of the spacial variables except for , is the nucleon energy for the momentum and is the nonlocal vertex defined by the in-vacuum quantities as , where the interactions and the free nucleon propagator are given by the in-vacuum chiral perturbation theory. The isodoublet matrix restricts the momentum integral for the nucleon momenta up to the Fermi momentum:

(7)

The nonlocal vertex is expanded in terms of the bilinear local vertex as

(8)

and the vertex is also expanded in terms of the chiral order. The in-medium pion Lagrangian can be obtained by the generating functional (6) as .

For the calculation of the amplitude, one uses the free nucleon propagators for the chiral expansion of the nonlocal vertex function while the Fermi sea nucleon term in the Fermi sea insertion among the nonlocal vertices . It has been shown explicitly in Ref. [31] that this expansion scheme is consistent with the conventional relativistic many body theory in the sense that one can use directly the in-medium nucleon propagator, that is the Fermi gas propagator,

(9)
(10)

in the calculation. Here, the free propagator and the medium part of the Fermi gas propagator are denoted by , with the isospin index . The medium part represents Fermi sea effect. We can calculate the connected functions using usual perturbative expansions with the in-medium nucleon propagator. In this way, one can deal with density contributions from nuclear medium by perturbative expansion of nuclear Fermi momentum.

2.2 Density expansion

In the in-medium CHPT, we can classify the current Green functions in terms of the order of the small parameters in the expansion of the pion momentum, the quark mass and the Fermi momentum in the similar way to the in-vacuum CHPT. The chiral order of a specific diagram is counted as [29]

(11)
(12)

where is the number of pion loops, is the number of the pion propagators, is the chiral dimension from the pion chiral Lagrangian, is the chiral dimension of the nonlocal in-medium vertex with Fermi sea insertions and is the chiral dimension of the vertex. In this chiral counting, we count Fermi gas propagator as like free nucleon propagator, so that the counting rule is the same as in-vacuum chiral perturbation theory.

In this work, we focus on the Fermi momentum dependence of the Green functions and count only by Fermi momentum orders. We assume that the in-vacuum loop effects are renormalized into counter terms in the chiral Lagrangian and we use the in-vacuum physical values to fix the low energy constants (LECs). For example, we translate LEC appearing in into the physical sigma term :

(13)

since the term gives the leading order of the sigma term in the chiral perturbation theory and once one calculates the higher orders for the sigma term, they should also contribute to the in-medium quantities in the same manner as . An empirical value of the sigma term is [32]. and a recent analysis based on relativistic formulation of chiral perturbation theory suggests [33, 34]. Another example is the isoscalar scattering length . In tree order, is given with LECs from as

(14)

The scattering length is used to determine the combination of the LECs . Recent calculations based on CHPT give at better than the 95 confidence level [35]. In this way, we take the in-vacuum physical values to determine LECs and perform systematic calculations for density effects of the pionic observables based on the counting of Fermi momentum orders.

As we have discussed in Ref. [31], beyond the order of in the density expansion, the dynamics alone cannot predict the in-medium quantities, because we encounter divergence in loop calculations. These divergence can be removed, once we introduce counter terms expressed by contact interactions. The interactions are to be determined by the in-vacuum dynamics.

2.3 Chiral Lagrangian and parametrization of chiral field

We use the following chiral Lagrangian in this work. The chiral Lagrangian for the meson sector is given with the chiral field parametrized by the pion field as

(15)

with the covariant derivative for the chiral field

(16)

given by the vector and axial vector external fields and counted as , and field

(17)

given by the scalar and pseudoscalar fields and counted as . The scalar field is replaced by the quark mass matrix in calculation. This Lagrangian is counted as .

In this paper, in order to make the perturbative calculation simple, as we have done in the previous paper [31], we use the following parametrization of the chiral field proposed by [37, 38]:

(18)

where satisfies

(19)

The chiral Lagrangian for the nucleon sector is as follows:

(20)

where is the chiral interaction for the nucleon in the bilinear form. The chiral interaction can be expanded in terms of the chiral order as and is counted as . The explicit form of the leading term reads

(21)

with the vector current

(22)

and the axial current

(23)

where the field is defined by a square root of the chiral field : . The expression of the next leading term is given as

(24)

with , and the covariant derivative for the nucleon field . Here we have omitted irrelevant terms in the present work for the in-medium pion properties in symmetry nuclear matter.

3 In-medium properties of pion

3.1 Pion mass and wave function renormalization

The pion propagation in medium can be calculated by the two-point function of the pseudoscalar density:

(25)

Around the in-medium pion pole the two-point function can be written in terms of the in-medium quantities, such as the in-medium pion mass , velocity and pseudoscalar coupling as

(26)

where does not include any singularity at the pion pole. In this way, with , the in-medium pion mass is defined by the pole position of the two point function and the coupling of pion to the pseudoscalar density in medium is defined by the square root of the residue of the two-point function at the pion pole.

The two-point function can be calculated using the in-medium chiral perturbation theory given in the previous section. The calculation will be done in terms of the in-vacuum quantities like

(27)

where is the in-vacuum pion mass, is the pion self-energy in medium and is the vertex correction of the pion coupling to the pseudoscalar density. The vertex correction does not contain the pion pole and it is calculated by considering one-particle irreducible diagrams.

Expanding the self-energy in Eq. (27) around and ,

we write the two-point function in the following way

(28)

with

(29)
(30)
(31)

Comparing Eqs. (26) and (28), we obtain, at the pion pole,

(32)

Using the in-medium chiral perturbation theory we can calculate the self-energy and , we obtain the in-medium pion properties with the above equations.

3.2 In-medium state

The in-medium pion propagator can be also written by the pion field operator as

(33)
(34)

Comparing Eqs. (28) and (34), for the calculation of the pion pole we regard

(35)

The pion operator creates one pion in medium with the mass and the wave function normalization , satisfying

(36)

where we have introduced the one-pion state with a momentum in the nuclear medium by denoting .

The in-medium coupling constant defined in Eq. (26) as the residue of the pion propagator induced by the pseudoscalar density may be also written as the following matrix element:

(37)

Relation (32) can be understood by Eq. (37) with the reduction formula:

where we have understood , and we have used Eq. (35) in the second equality and Eq. (28) in the third equality.

3.3 In-medium pion decay constants

We define the in-medium decay constant in analogy of the in-vacuum decay constant as a matrix element of the axial vector current :

(38)

where we have introduced a vector characterizing the medium rest frame with and there are two form factors, and , in the presence of nuclear matter. These form factors, actually, should be functions of and according to Lorentz covariance. The pion decay constants are obtained at the mass shell point with and . We define the temporal and spatial components of the decay constant by taking as

(39)
(40)

The decay constants are obtained by

(41)
(42)

These decay constants can be calculated in the in-medium chiral perturbation theory. Making good use of the reduction formula in momentum space again, we write down the matrix element in terms of the one-particle irreducible vertex correction and the wave function renormalization:

where we mean and with is the vertex correction of the decay constant, which can be calculated by one-particle irreducible diagrams in the chiral perturbation theory.

4 Results

In this section, we investigate explicitly the in-medium pion properties with the in-medium chiral perturbation theory up to the order of in the density expansion. In last section, we find that the in-medium pion decay constant can be calculated by the pion wave function renormalization and the one-particle irreducible vertex correction . The wave function renormalization is evaluated by taking derivative of the pion self-energy with respect to the energy squared. We calculate the pion self energy first. With the self energy, we evaluate next the in-medium pion mass and the wave function renormalization, and show their density dependence. Calculating the one-particle irreducible correction for the pion decay constant, we evaluate the density dependence of the decay constant with the wave function renormalization. We also check whether the low energy relations are satisfied also in the nuclear medium. Finally we discuss the decay rate in nuclear medium. For simplicity, we concentrate on the calculation under the unpolarized symmetric nuclear matter, where the matter has spin 0 and isospin 0 with the equal proton and neutron densities, .

In the following, we write down the Feynman diagrams for each quantity and classify them according to the density dependence. As we have already discussed in the previous section, we presume that the in-vacuum LECs are fixed by the experimental observables. The vertex and mass corrections from the quantum loops of pion and nucleon are renormalized into the physical quantities. On this understanding, we replace LECs to the observed quantities and use the experimental values. In the present work the relevant replacements are the followings:

(43)
(44)
(45)

where the left hand sides are given by LECs, while the right hand sides are written in terms of the observed quantities in vacuum. In these expression, with the isoscalar scattering length . In the replacement higher order contributions in the chiral counting are already involved. In this sense, we loose strict counting of the chiral order. Taking this scheme, we do not have to calculate the loop integrals which contain only the in-vacuum propagators, because they are supposed to be already counted inside the experimental value. This fact reduces the number of the relevant diagrams which we should calculate.

The in-medium quantities which we are going to calculate should be evaluated at the pion pole. The in-vacuum chiral order can be counted by the pion mass. Therefore, in the perturbative expansion, the term of which the leading chiral counting in vacuum is has dependence. The factor comes from the Fermi motion correction of the nucleon when one calculates the Fermi sea loop integral. The order of the density expansion is counted as , while the order of the small parameter is given by . If one takes the density contribution up to , it is enough to consider the diagrams with .

The details of the loop calculation is summarized in appendix B.

4.1 In-medium pion self energy

Figure 1: Feynman diagrams for the pion self energy up to . (a) the leading order of the density corrections . (b) the next-to-leading order correction . In these diagrams, the dashed lines denote the pion lines, the doubled and thick lines are the nucleon lines for Fermi gas propagator and the medium part of the Fermi gas propagator , the filled and unfilled circles are the leading and the next-to-leading order vertices from the chiral Lagrangian respectively.

We show the Feynman diagrams contributing the in-medium self energy with the density corrections up to in Fig. 1. In these diagrams, the dashed doubled and thick lines are pion propagator, the Fermi gas propagator and the nucleon propagation in the Fermi gas , and the filled and unfilled circles are the leading and next-to-leading order vertices from the chiral Lagrangian, respectively. In the following calculations, we fix the external momentum as .

The diagrams for the leading order of the density expansion are given in Fig. 1(a). The self energy coming from the left diagram in Fig.1 (a) can be calculated as

(46)

where is the Fermi gas propagator given in Eq. (10), the amplitude can be obtained by the interaction Lagrangian given in Eq. (87) of appendix A.

Next, we calculate the right diagram in Fig.1(a):

(47)

The next-to-leading order contribution coming from the left diagram in Fig. 1(b) is given as

(48)

with the pion propagator , the symmetric factor and defined as

Finally we obtain

(49)

with function defined by

(50)

The self energies coming from the double scattering corrections [36] given by the middle and right diagrams of Fig. 1(b), and , have the two vertices from the leading Weinberg-Tomozawa interaction and the next-to-leading interaction with LECs , respectively. These are evaluated as

(51)
(52)

with and function defined by

(53)

Summing up all the contributions, we obtain the the self energy up to as

(55)

with .

4.2 In-medium pion mass

The in-medium pion mass is obtained by the summation of the in-vacuum mass and the self energy evaluated at the pion on the mass shell . This brings us a self-consistent equation:

(56)

Nevertheless, because the in-medium correction starts with the linear density and we are evaluating the pion mass up to , the density correction on the mass in the argument of the self energy gives higher orders in the density expansion of the in-medium pion mass. Thus, we are allowed to evaluate the self-energy at the in-vacuum on-shell for the present purpose. We evaluate the in-medium pion mass up to as

(57)

The LECs, , , , , have been determined by the in-vacuum physical quantities, the pion mass , the sigma term and factored scattering length . The linear density correction of the pion mass stems from the scattering length. Since the isosinglet scattering length is known to be a small number compared to the inverse pion mass, , the leading correction is as small as 5% at the saturation density .

In Fig. 2, we show the density dependence of the in-medium pion mass as a function of the density normalized by the normal nuclear density . In the figure, the dotted line shows the result up to the leading linear density and the solid line is for the result containing the next-to-leading order. We take the following values of the in-vacuum quantities; , , , and  [32]. One can see from Fig. 2 that the NLO correction is not small. The main contribution comes from the double scattering terms. The density correction of the pion mass at twice or three times the normal nuclear density becomes about 15 to 20%. Since the in-medium CHPT which is a low energy effective theory it would be not applicable in higher density region, nevertheless we expect that this theory would be applicable up to 3 where Fermi momentum corresponds to about 400 MeV.

Figure 2: Density dependence of the in-medium pion mass normalized by the in-vacuum pion mass, , in symmetric nuclear matter. The dotted line shows the result up to the leading linear density correction, while the solid line is the result containing the next-to-leading order correction.

4.3 In-medium wave function renormalization

Next, the wave function renormalization is obtained by evaluating the derivative of the self energy with respect to at . Again since the difference between the in-medium and in-vacuum pion masses in the self energy is counted as the higher order of the density expansion, we evaluate the derivative of the self energy at for the present purpose: