Chiral condensate in nuclear matter beyond linear density using chiral Ward identity

In low-energy Quantum ChromoDynamics, spontaneous breaking of chiral symmetry is one of the most important phenomenon because this is responsible for the generation of the constituent quark mass. We focus on partial restoration of chiral symmetry in a finite density environment such as inside of atomic nucleus. Partial restoration of chiral symmetry can be observed by investigating the modification of hadron properties in nuclear matter. To examine partial restoration of chiral symmetry in nucleus experimentally, binding energy and width of 1s state of deeply bound pionic atom is measured precisely [1]. This experiment suggests that the chiral condensate which is an order parameter of the chiral symmetry breaking is reduced by about 30 % at the nuclear density. In our work [2], we analyze density corrections of the chiral condensate up to NLO order using the chiral Ward identity [3] and an in-medium chiral perturbation theory [4,5]. The in-medium chiral condensate is calculated by a correlation function of the axial current and pseudoscalar density in nuclear matter as a consequence of the chiral Ward identity. The correlation function is evaluated using the chiral perturbation theory with the hadronic quantities of pion and nucleon dynamics. We assume that all of the in-vacuum interaction vertices are known and in-vacuum loop corrections are supposed to be done by using the experimental values of the couplings in the calculation of the chiral condensate. This procedure leads to a density (fermi momentum) expansion of the chiral condensate. Based on this density expansion approach, we analyze diagrammatic structure of the current Green function which gives density effects to the condensate. This analysis shows that medium effects to the chiral condensate beyond linear density come from density corrections to πN sigma term due to interactions between pions and nuclear matter.


Introduction
Chiral symmetry breaking (χSSB) SU(N f ) L ⊗ SU(N f ) R → SU(N f ) V is an important phenomenon, which characterizes low-energy Quantum ChromoDynamics(QCD) and Hadron physics, and the nonvanishing chiral condensate qq as an order parameter of χSSB generates hadron masses.
Recently, partial restoration of chiral symmetry in the nuclear medium has gained considerable attention.It means reduction of the absolute value of the condensate in the nuclear medium and leads to various changes of hadron properties according to the reduction, for example, repulsive enhancement of s-wave π N interaction, the mass difference between the ρ and a 1 mesons and so on.Vast theoretical and experimental effort is devoted in this region [1][2][3].In the theoretical side, Ref. [1] discusses a systematic and model-independent derivation of in-medium sum rules and shows some relations between chiral condensate and experimental observables.In experimental side, Ref. [3] has extracted quantitatively the s-wave isovector scattering length of π and nucleus by observing the deeply-bound 1s energy level and decay width of pionic Sn atoms and derived the reduction of the pion decay constant.The reduction of the chiral condensate has been extracted through the Glashow-Weinberg relation and the Gell-Mann-Oakes-Renner relation: Here, qq * , qq 0 are the in-medium and in-vacuum chiral condensates, respectively, and ρ 0 is the normal nuclear density.This estimation suggests that the chiral symmetry breaking is partially restored to about 65% in the center of nucleus.Once we know the density dependence of the condensate qualitatively, we can get possibilities to predict other in-medium hadronic quantities through low energy theorems.Therefore, it is important to evaluate the in-medium condensate quantitatively.

Chiral Ward identity
In order to calculate the density dependence of the chiral condensate, we make use of the chiral Ward identity in the following correlation function: where A a µ (x) denotes the axial vector current associated with the SU(2) chiral transformation, P b (0) stands for the pseudo-scalar current, and |Ω is the nuclear matter ground state normalized as Ω|Ω = 1 and characterized by the proton and neutron density,ρ p and ρ n .Hereafter we write the expectation value Ω|O|Ω as O * for operator O. Differentiating the T -product and the axial current in the soft limit q µ → 0 and using the current algebra [Q a 5 , P b (x)] = −iδ ab qq with the axial transformation generator Q a 5 and the PCAC relation ∂ µ A a µ = −m q P a , we find the in-medium chiral condensate given by the Green functions in the soft limit: where D ab (0) ≡ lim

In-medium chiral perturbation theory
In order to calculate the in-medium chiral condensate, we calculate the correlation functions appearing in the right hand side of Eq. ( 3) in an in-medium chiral perturbation theory developed in Ref. [4][5][6].In Ref. [4], the generating functional of the in-medium SU(2) CHPT with external sources is derived.
Starting with the SU(2) chiral Lagrangian having nucleon bilinear terms, we consider the generating functional Z defined by a transition from the in-state to the out-state: where J = (s, p, v, a) represents the scalar, pseudo-scalar, vector and axial-vector sources, respectively and η, η † are external sources for the nucleon field.The states Ω out |, |Ω in are the out and in states defined by the nuclear Fermi gas in which nucleons are occupied up to the Fermi momentum k F : with the nucleon creation operator a † (p n ).In the path integral representation the nucleon bilinear interactions are integrated out by the Gauss integral formula.Setting the external sources η, η † to be zero at the final stage, we obtain the in-medium CHPT generating functional with pion-nucleon dynamics.It is shown that the obtained generating functional is characterized by double expansion of Fermi sea insertion and chiral counting.Similar to the in-vacuum CHPT, in which one counts the pion momentum p π and mass m π as O(p), we have a counting rule of the chiral power order [5].This counting scheme allows us to count Fermi momentum k F as O(p) because Fermi momentum k F ≈ 270MeV ≈ 2m π at the normal nuclear density ρ 0 .Using the in-medium CHPT, we can perform the order counting for density corrections systematically.We take πN chiral Lagrangian as interactions between pions and nucleons in the Fermi gas, and assume that the in-vacuum interactions are fixed by in-vacuum pion-nucleon dynamics.In other words, new parameters characterizing nuclear matter is not necessary up to a certain order.Describing the Green functions as follows: with the QCD current operator Ôi corresponding to the external source J i , we can evaluate the low energy QCD Green functions in the nuclear matter by perturbative calculations in terms of the Fermi momentum k F .In our study, we use the renormalization idea and focus on density counting by using the observed values for the couplings in the chiral Lagrangian.

Density dependence of chiral condensate
We calculate the Green functions Π ab 5 (0), D ab (0) to evaluate the in-medium chiral condensate based on the chiral power counting scheme.Π ab 5 (0) vanishes in soft limit because axial current is coupled with pion with derivative interaction but pion is not zero-mode.Therefore we consider D ab (0) in the following.In fig. 1, we list up the Feynman diagrams of D ab (0) in the symmetric nuclear matter.The Feynman diagrams in (a) are the leading corrections in the chiral counting scheme but vanishes in the soft limit, and diagram (b) is the leading order density corrections O(ρ) and reproduces the result in the linear density approximation obtained in Ref. [7], which is known as the linear density correction proportional to the πN sigma term.Diagram (c) is the next to leading order(NLO) corrections with O(ρ 4/3 ).They are the density corrections to the πP vertex.In other words, these are density corrections to the πN sigma term.We calculate the in-medium chiral condensate within NLO density corrections and obtain the analytic expression Here, a = m π /2k F .In fig.2, the density dependence of the condensate is plotted as a function of the nuclear density normalized by ρ 0 .The green, blue and red lines are the chiral condensates obtained by the linear density approximation in chiral limit, by the linear density approximation off the chiral limit and by the NLO corrections off the chiral limit.We find that the NLO density correction amounts to 5% at the 08010-p.3chiral condensate ratio ρ / ρ 0 up to NLO linear in chiral limit linear Fig. 2. The density dependence of chiral condensate in symmetric nuclear matter.The green, blue and red lines are the chiral condensates obtained by the linear density approximation in chiral limit, by the linear density approximation off the chiral limit and by the NLO corrections off the chiral limit.
normal nuclear density.Therefore the linear density approximation will be good in low density region up to around ρ 0 .When we go to further higher density, the NLO correction becomes 10% at 2ρ 0 .Diagram (d) shows fig. 1 is possible higher order corrections beyond NLO.We find that these are 1 and 2 pion-exchange effects in the Fermi gas.Higher order calculation has UV divergences from pion-loops, so that one needs renormalization of these vertices.This means that one needs not only in-vacuum πN couplings but also the NN contact terms obtained the NN dynamics.Recently, as a step in this direction, a non-perturbative chiral effective theory has been developed to improve the NN correlation by including NN contact terms using a resummation method [8].Moreover, in Ref. [9] ∆(1232) resonance contributions have been evaluated and it has been found that ∆ resonance effects are not small.

Fig. 1 .
Fig. 1.Feynman diagrams for the of the in-medium chiral condensate.solid, thick and dashed lines represent the free nucleon, the nucleon propagation in the Fermi sea, and pion, respectively.The filled and open circles are the leading and subleading order πN vertices.