Chiral perturbation theory vs. Linear Sigma Model in a chiral imbalance medium

We compare Chiral Perturbation Theory (ChPT) and the Linear Sigma Model (LSM) as realizations of low energy QCD for light mesons in a chirally imbalanced medium. The relations between the low-energy constants of the Chiral Lagrangian and the corresponding constants of the Linear Sigma Model are established as well as the expressions for the decay constant of the $\pi$-meson in the medium and the mass of the $a_0$. In the large $N_c$ count taken from QCD the correspondence of ChPT and LSM is remarkably good and give a solid ground for search of chiral imbalance manifestation in pion physics. A possible experimental detection of chiral imbalance tracks (and therefore a phase with Local Parity Breaking) in the charged pion decays inside the fireball is outlined.


Introduction
The possible generation of a phase with Local Parity Breaking (LPB) in nuclear matter at extreme conditions such as those reached in heavy ion collisions (HIC) at RHIC and LHC [4] has been examined recently [1,2,3]. It has been suggested in [1] that at high temperatures an isosinglet pseudoscalar background could arise due to large-scale topological charge fluctuations (studied recently in lattice QCD simulations [5]). These considerations led eventually to the observation of the so-called Chiral Magnetic Effect (CME) [1] in the STAR and PHENIX experiments at RHIC [6]. The effect should be most visible for noncentral HIC where large angular momenta induce large magnetic fields contributing to the chiral charge separation. For central collisions it was proposed in [2] that the presence of a phase where parity was spontaneously broken could be a rather generic feature of QCD at finite density. Local parity breaking can be induced by difference between the densities of the right-and left-handed chiral fermion fields (Chiral Imbalance).
In the hadron phase we shall assume that as a consequence of topological charge fluctuations, the environment in central HIC generates a pseudoscalar background growing approximately linearly in time. This background is associated with a constant axial vector whose zero component is identified with a chiral chemical potential. In such an environment one could search for possible manifestation of LPB in electromagnetic and lepton probes. In particular, in [7] it was shown that a good part of the excess of dileptons produced in central heavy-ion collisions [8] might be a consequence of LPB due to the generation of a pseudoscalar isosinglet condensate whose precise magnitude and time variation depends on the dynamics of the HIC. This paper is mostly concerned with the possibility of identifying LPB in the hadron phase of QCD in HIC. Such a medium would be simulated by a chiral chemical potential µ 5 . Adding to the QCD Lagrangian the term ∆L q = µ 5 q † γ 5 q ≡ µ 5 ρ 5 , we allow for non-trivial topological fluctuations [7] in the nuclear (quark) fireball, which are ultimately related to fluctuations of gluon fields. The transition of the quark-gluon medium characteristics to a hadron matter reckons on the quark-hadron continuity [9] after hadronization of quark-gluon plasma. The behavior of various spectral characteristics for light scalar and pseudoscalar (σ, π a , a a 0 ) -mesons by means of a QCD-motivated σ-model Lagrangian was recently derived for SU L (2) × SU R (2) flavor symmetry including an isosinglet chiral chemical potential [10]. The structural constants of the σ-model Lagrangian were taken as input parameters suitable to describe the light meson properties in a chiral medium. In this way there is no reliable predictability in the determination of the hadron system response on chiral imbalance, and reaching quantitative predictions requires a phenomenologically justified hadron dynamics. To increase predictability, we extend the vacuum Chiral Lagrangians [11,12] with phenomenological low-energy structural constants taking into account the chiral medium in the fireball with a chiral imbalance. It is shown that σ-model parametrization of [10] fits well the pion phenomenology at low energies as derived from ChPT.
It is also described how pions modify their dynamics in decays in a chiral medium, in particular, charged pions stop decaying into muons and neutrinos for a large enough chiral chemical potential.

Chiral Lagrangian with chiral chemical potential
The Chiral Lagrangian for pions describing their mass spectra and decays in the fireball with a chiral imbalance can be implemented with the help of softly broken chiral symmetry in QCD transmitted to hadron media, properly constructed covariant derivative where we skipped the electromagnetic field. The axial chemical potentialis introduced as a constant time component of an isosinglet axial-vector field.
In the framework of large number of colors N c [12] the SU(3) Chiral Lagrangian in the strong interaction sector contains the following dim=2 operators [12], where < ... > denotes the trace in flavor space, j µ ≡ U † ∂ µ U, the chiral field U = exp(iπ/F 0 ), the bare pion decay constant F 0 ≃ 92MeV , χ(x) = 2B 0 s(x) and M 2 π = 2B 0mu,d , the treelevel neutral pion mass. The constant B 0 is related to the chiral quark condensate <qq > as F 2 0 B 0 = <qq >. Taking now the covariant derivative in (1) it yields Herein we have used the identity for U ∈ SU(n),< j µ >= 0. In the large N c approach the dim=4 operators [12] in the Chiral Lagrangian are given by where L 0 ,L 3 , L 5 are bare low energy constants. For SU(3) and SU(2) < j µ >= 0 and there is the identity whereas for SU (2) there is one more identity Applying these identities one finds the four-derivative Gasser-Leutwyler (GL) operators for the SU(3) chiral Lagrangian with For SU(2) one has a further reduction of the dim=4 lagrangian, with normalization so that We stress that this chain of transformations is valid only if < j µ >= 0. The response of the Chiral Lagrangian on chiral imbalance is derived with the help of the covariant derivative (1) applied to the Lagrangian (4), We notice that this result is drastically different from what one could obtain from the final Lagrangian (9). This is because the identities (5) and (6) are violated if < j µ > = 0. The above modifications change differently the coefficients in the dispersion law in energy p 0 and three-momentum | p| for the mass shell as well as modify the mass term for pions (all together it gives the inverse propagator of pions), In the leading order of large N c expansion (neglecting the RG logarithm contribution) the empirical values of the SU(2) GL constants [11] are They can be obtained also if they are normalized at the RG scale µ ≃ M π ≃ 140MeV, log m π /µ ≃ 0. Thus in the pion rest frame i.e. the pion decay constant is growing and its mass is decreasing in the chiral media.

Linear Sigma Model for light pions and scalar mesons in the presence of chiral imbalance
Let us compare these constants with those ones estimated from the Linear Sigma Model built in [13,14]. The Sigma Model was build with realization of SU(2) chiral symmetry to describe pions and isosinglet and isotriplet scalar mesons. Its Lagrangian reads where H = ξ Σ ξ is an operator for meson fields, N c is a number of colours, m is an average mass of current u, d quarks, M is a "tachyonic" mass generating the spontaneous breaking of chiral symmetry, B 0 , c, λ 1 , λ 2 are real constants. The matrix Σ includes the singlet scalar meson σ, its vacuum average v and the isotriplet of scalar mesons a 0 0 , a − 0 , a + 0 , the details see in [13,14]. The covariant derivative of H including the chiral chemical potential µ 5 is defined in (1). The operator ξ realizes a nonlinear representation (see (2)) of the chiral group SU(2) L × SU(2) R , namely, ξ 2 = U.
The diagonal masses for scalar and pseudoscalar mesons read From spectral characteristics of scalar mesons in vacuum one fixes the Lagrangian parameters, λ 1 = 16.4850, λ 2 = −13.1313, c = −4.46874 × 10 4 MeV 2 , B 0 = 1.61594 × 10 5 MeV 2 [13]. The change of pion coupling constant F 0 is determined by potential parameters as compared to the ChPT definition, It is a quite satisfactory correspondence. Analogously, in the rest frame using the pion mass correction, m 2 π (µ 5 )F 2 π (µ 5 ) ≃ 2m q B 0 F π (µ 5 ) it is easy to find the estimation for wherefrom one can also guess the relation 4(l 1 + l 2 ) ∼ l 4 following from the LSM. For moving mesons with | p| = 0 and the CP breaking mixing of scalar and pseudoscalar mesons the effective masses m 2 ef f ∓ take the form, For small µ 2 5 , m 2 π ≪ m 2 a ≃ 1GeV 2 one can approximate the dependence on the wave vector p Comparing with (12) one establishes the relationship of isotriplet scalar mass and GL constants which reproduces the PDG value [15] with a remarkable precision. The predicted distortion of the mass shell condition can be detected in decays of charged pions when the effective pion mass approaches to muon mass. Let us find the threshold value for the π + → µ + ν decay. If a charged pion was generated in chiral medium its mass is lower than in the vacuum and the condition for its decay follows from (12), where we have used the relations 4(l 1 + l 2 ) ≃ l 4 and (21). The decay channel is closed for | p| 2 ≃ 0 if µ 5 ≃ 160MeV . It must be detected as a substantial decrease of muon flow originated from pion decays in fireball.

Conclusions and outlook
1. For light mesons in the chiral imbalance medium we compare the Chiral Perturbation Theory (ChPT) and the Linear Sigma Model (LSM) as realizations of low energy QCD. The relations between the low-energy constants of the Chiral Lagrangian and the corresponding constants of the Linear Sigma Model are established as well as the expressions for the decay constant of the pion in the medium and the mass of the a 0 meson are found. The resulting dispersion law for pions in the medium allows us reveal the threshold of decay of a charged pion into a muon and neutrino which can be suppressed with increasing of chiral chemical potential. The low energy QCD correspondence of ChPT and LSM in the large N c limit is remarkably good and give a solid ground for search of chiral imbalance manifestation in pion physics at HIC. It can be also shown [13], that at higher energies exotic decays of scalar mesons arise due to mixing of π and a 0 meson states in the presence of chiral imbalance. 2. A manifestation for LPB in the presence of chiral imbalance in the sector of ρ and ω vector mesons [2,3] can also happen and in this case the Chern-Simons interaction plays the main role. It turns out [2] that the spectrum of massive vector mesons splits into three components with different polarizations ±, long having different effective masses m V,+ < m V,long < m V,− .
3.We draw also attention to the recent proposal to measure the photon polarization asymmetry in πγ scattering [16,13] as a way to detect LPB due to chiral imbalance. It happens in the ChPT including electromagnetic fields due to the Wess-Zumino-Witten operators. The independent check of our estimates could be done by lattice computation (cf. [17]) . 4. Thus a possible experimental detection of chiral imbalance in medium (and therefore a phase with LPB)) in the charged pion decays and vector mesons inside of the fireball can be realized.