An emergence of local spatial parity breaking (LPB) in central heavy-ion collisions (HIC) at high energies is discussed. The QCD phenomenology of LPB in the fireball is induced by a difference between the number densities of right- and left-handed chiral fermions which is triggered by a chiral (axial) chemical potential. For the description of peculiarities of LPB, a number of QCD-inspired models are considered and confronted to certain lattice results. In particular, from the meson effective Lagrangian, it is found that the lightest states may become massless and some scalars turn out to be stable. In experimental studies, the asymmetry in production of longitudinal and transverse polarized states of ρ and ω mesons for different values of the invariant mass can serve as a characteristic indication of local spatial parity breaking which can be derived from an abnormal yield of dilepton pairs in the PHENIX, STAR and ALICE collaborations

Andrianov, Alexander A.

Andrianov, Vladimir A.

Espriu, D. (Domènec)

Acta Physica Polonica B Proceedings Supplement

Diposit Digital de la Universitat de Barcelona

Vol. 9 (2016) Acta Physica Polonica B Proceedings Supplement No 3QCD WITH CHIRAL CHEMICAL POTENTIAL:MODELS VERSUS LATTICE∗A.A. Andrianov, V.A. AndrianovSaint-Petersburg State University, 199034 St. Petersburg, RussiaD. EspriuICCUB, University of Barcelona, Martí i Franquès 1, 08028 Barcelona, Spain(Received July 19, 2016)An emergence of local spatial parity breaking (LPB) in central heavy-ion collisions (HIC) at high energies is discussed. The QCD phenomenologyof LPB in the fireball is induced by a difference between the number densi-ties of right- and left-handed chiral fermions which is triggered by a chiral(axial) chemical potential. For the description of peculiarities of LPB, anumber of QCD-inspired models are considered and confronted to certainlattice results. In particular, from the meson effective Lagrangian, it isfound that the lightest states may become massless and some scalars turnout to be stable. In experimental studies, the asymmetry in production oflongitudinal and transverse polarized states of ρ and ω mesons for differentvalues of the invariant mass can serve as a characteristic indication of localspatial parity breaking which can be derived from an abnormal yield ofdilepton pairs in the PHENIX, STAR and ALICE collaborations.DOI:10.5506/APhysPolBSupp.9.5151. Chiral imbalance in heavy-ions collisionsThe behaviour of baryonic matter under extreme conditions in HIC hasreceived a lot of attention [1,2]. New properties of QCD in the environmentwere tested in current accelerator experiments on RHIC and LHC [3]. Amedium generated in these collisions (a fireball) may serve for experimentaland theoretical studies of various phases of a matter.The dedicated experimental study of hadron correlations in non-centralheavy-ion collisions at RHIC [4] and LHC [5] revealed a signal of the separa-tion of electric charges predicted in [6] as a signature of local P- and CP-odd∗ Presented at “Excited QCD 2016”, Costa da Caparica, Lisbon, Portugal, March 6–12,2016.(515)516 A.A. Andrianov, V.A. Andrianov, D. Espriufluctuations in QCD matter. The subsequent studies [7, 8] improved thetheoretical understanding of the underlying phenomenon, the “chiral mag-netic effect” (CME) in the reactions for peripheral-ion collisions (see [9] fora review).On the contrary, a gradient density of isosinglet pseudoscalar conden-sate can be formed as a result of large, “long-lived” topological fluctuationsof gluon fields in the fireball in central collisions (see [10] for details). Todescribe various effects of hadron matter in a fireball with parity break-ing, we must introduce the axial/chiral chemical potential [10]. At finitetemperature, the transitions between the vacuum states with different topo-logical Chern–Simons numbers can be induced by a classical thermal activa-tion process, the so-called “sphaleron” [11]. In QCD matter, sphalerons areabundant [12] and induce the quark chirality non-conservation. There aresome experimental indications of an abnormal dilepton excess in the rangeof low invariant masses and rapidities, and moderate values of the trans-verse momenta (see the review in [13]), which can be thought of as a resultof LPB in the medium (the details can be found in [14]). In particular, inheavy-ion collisions at high energies, with raising temperatures and baryondensities, metastable state can appear in the fireball with a non-trivial topo-logical/axial charge T5, which is related to the gluon gauge field GiT5(t) =18pi2∫vold3x εjkl Tr(Gj∂kGl−i23GjGkGl), j, k, l = 1, 2, 3 , (1)where the integration is over the finite fireball volume. Its jump ∆T5 canbe associated with the space-time integral of the gauge-invariant Chern–Pontryagin density∆T5 = T5 (tf )− T5(0) = 116pi2tf∫0dt∫vold3xTr(GµνG˜µν). (2)It is known that the divergence of isosinglet axial quark current J5,µ =qγµγ5q is locally constrained via the relation of partial conservation of axialcurrent (PCAC) affected by the gluon anomaly∂µJ5,µ − 2im̂qqγ5q = Nf2pi2Tr(GµνG˜µν). (3)This relation allows to find the connection of a non-zero topological chargewith a non-trivial quark axial charge Qq5. Namely, integrating (3) over afinite volume of fireball, we come to the equalityddt(Qq5 − 2NfT5) ' 2i∫vold3x m̂qqγ5q , (4)QCD with Chiral Chemical Potential: Models Versus Lattice 517Qq5 =∫vold3x q†γ5q = 〈NL −NR〉 , (5)where 〈NL − NR〉 stands for the vacuum averaged difference between leftand right chiral densities of baryon number (chiral imbalance). Therefore,it follows that in the chiral limit and for a finite fireball volume, the axialquark current is conserved in the presence of non-zero topological charge.If for the lifetime of fireball and the size of hadron fireball of the order ofL = 5–10 fm, the created topological charge is non-zero, 〈∆T5〉 6= 0, thenit may be associated with a topological chemical potential µθ or an axialchemical potential µ5 [10]. Thus, we have〈∆T5〉 ' 12Nf〈Qq5〉 ⇐⇒ µ5 '12Nfµθ . (6)Thus, adding to the QCD Lagrangian the term ∆Ltop = µθ∆T5 or ∆Lq =µ5Qq5, we get the possibility of accounting for non-trivial topological fluctu-ations (“fluctons”) in the nuclear (quark) fireball.2. Effective meson theories in search of LPBFor the detection of LPB in the hadron fireball, let us consider an effectivetheory describing the electromagnetic interactions in a fireball. In this case,we keep in mind mechanisms of hadron interactions which rely on vectordominance model [15].The quark–meson interaction is described by the Lagrangian,Lint = q¯γµV µq ; Vµ ≡ −eAµQ+ 12gωωµIq+12gρρ0µλ3+1√2gφφµIs , (7)while Q = λ32 +16Iq − 13Is, gω ' gρ ≡ g ' 6 < gφ ' 7.8 and the values ofthe constants are extracted from the decays of the vector mesons. Here, Iqand Is are the unit matrices in the non-strange and strange quark sectors,and λ3 is a corresponding Gell-Mann matrix. The parity-odd contributionis given by the Chern–Simons term,LCS(k) = −14εµνρσ Tr[ζˆµ Vν(x)V ρσ(x)]= 12Tr[ζˆ jkl Vj ∂kVl], (8)which describes the mixing of photons and vector mesons under the localspatial parity breaking. We can obtain the relation ζ = Nc g2µ5/8pi2, whereNc is a number of colours, and numerically ζ ' 1.5µ5.518 A.A. Andrianov, V.A. Andrianov, D. EspriuThe analysis of the massive Chern–Simons electrodynamics [10] hasshown that in the case of an isosinglet pseudoscalar background field, thespectrum of massive vector mesons splits into three polarizations with themasses m2V,+ < m2V,L < m2V,−. The position of resonance poles for trans-verse polarizations of ρ0, ω mesons is shifted with the wave vector |~k| andalso a resonance broadening occurs that leads to an increased contributionof the dilepton production compared with the situation with resonances invacuum (for details, see [16]). Then, the question arises. Could the splittingbe measured in experiments with heavy-ion collisions?For this purpose, there should be analysed an effect of anomalous dilep-ton pair production in the range of low invariant masses and rapidities, andmoderate transverse momenta which was established in a series of experi-ments with heavy-ion collisions in recent years [13]. It is well-known thatthe angular distribution of leptons carries the information on polarizations.However, the current angular distribution studies based on full angular av-erage do not seem to detect possible parity-odd effects.In order to isolate the transverse polarizations, we perform different cutschoosing the angle θA for the analysis and study the variations of the ρ(and ω) spectral functions. A quite visible secondary peak appears in aP -odd medium (see Fig. 1)! To isolate the transverse polarizations in thespectrum, we selected different angle sectors and studied the changes inthe ρ-meson spectral function. Various experimental possibilities for itsidentification were discussed in [16].Fig. 1. ρ-spectral function depending on the dielectron invariant massM in vacuum(µ5 = 0) and in a parity-breaking medium with µ5 = 300 MeV for different rangesof the angle θA between the two outgoing leptons in the laboratory frame.Thus, a signal (a phase) with spatial parity breaking in heavy-ion col-lisions (in a fireball) can be sought in experiments “event by event” usingan excess yield of dilepton pairs and predominantly with different circularpolarizations outside the resonance region of the ρ- and ω-meson invariantmasses.QCD with Chiral Chemical Potential: Models Versus Lattice 5193. Effective scalar meson Lagrangian with axialchemical potentialLet us now consider the QCD-inspired scalar meson model with axialchemical potential [17] for a description of some properties of low-energymesons in medium and how it manifests itself in LPB. First of all, the axialchemical potential is introduced and treated as a constant time componentof an isosinglet axial-vector field in the non-strange sector. For the light-est isotriplet pseudoscalar pi and scalar a0 states, the piece of the effectiveLagrangian that is bilinear in fields looks as followsL = 12(∂a0)2 + 12(∂pi)2 − 12m21a20 − 12m22pi2 − 4µ5a0p˙i . (9)Evidently, new eigenstates p˜i arise from the mixture of scalars and pseu-doscalars, then for big 3-momentum, p˜i becomes massless and further ontachyonic [17]. For the same 3-momentum, the quasi-scalar state σ looksstable.4. Chemical potentials in the NJL modelIn [18], we incorporated both a vector and an axial chemical potentialsin the Nambu–Jona-Lasinio (NJL) model with the purpose of unravellingthe landscape of different stable phases of the theory. It turns out thatthe inclusion of µ5 changes radically the phase structure of the model andshows that µ is not a key player in ushering a thermodynamically stablephase where parity is violated in the NJL model, but µ5 is. It leads to anon-trivial dependence of the scalar condensate (see Fig. 2) in the chirallybroken phase.Fig. 2. Evolution of the constituent quark mass M : left panel — depending on µfor different values of the axial chemical potential µ5; right panel — depending onµ5 for different values of the chemical potential µ.520 A.A. Andrianov, V.A. Andrianov, D. EspriuCritical temperature with axial potential switched on was computed onlattice [19] and a fairly well correspondence with the results of [18] was estab-lished. The main result is that with an increasing chiral chemical potential,the dynamical mass and critical temperature raises up.5. Conclusions and outlookIn this paper, we described a possibility of local spatial parity breakingemerging in a dense hot baryon medium (fireball) in heavy-ion collisions athigh energies. We stress that LPB is not forbidden by any physical prin-ciple in QCD at a finite temperature/density. We suggested a generalizedLagrangian of the vector meson dominance model in the presence of theChern–Simons interaction. It turns out that the spectrum of massive vectormesons splits into three components with different polarizations and withdifferent effective masses m2V,+ < m2V,L < m2V,−, and a resonance broadeningoccurs that leads to an increase of the spectral contribution to the dileptonproduction as compared with the vacuum state. The proposed mechanismfor generating local spatial parity breaking helps to explain qualitativelyand quantitatively the anomalous yield of dilepton pairs in the CERES,PHENIX, STAR, NA60, and ALICE experiments.This work has been supported through grants FPA2013-46570, 2014-SGR-104 and Consolider CPAN. Funding was also partially provided bythe Spanish MINECO under project MDM-2014-0369 of ICCUB (Unidadde Excelencia ‘Maria de Maeztu’). A.A. and V.A. were supported by grantRFBR project 16-02-00348 and SPbSU project 11.41.593.2016.REFERENCES[1] P. Jacobs et al., arXiv:0705.1930 [nucl-ex].[2] J.-P. Blaizot et al., Nucl. Phys. A 873, 68 (2012).[3] A. Andronic et al., Nucl. Phys. A 837, 65 (2010).[4] B. Abelev et al., Phys. Rev. C 81, 054908 (2010).[5] B. Abelev et al. [ALICE Collaboration], Phys. Rev. Lett. 110, 012301 (2013).[6] D.E. Kharzeev, Phys. Lett. B 633, 260 (2006).[7] D.E. Kharzeev, L.D. McLerran, H.J. Warringa, Nucl. Phys. A 803, 227(2008).[8] K. Fukushima, D.E. Kharzeev, H.J. Warringa, Phys. Rev. D 78, 074033(2008).[9] D.E. Kharzeev, Prog. Part. Nucl. Phys. 75, 133 (2014).QCD with Chiral Chemical Potential: Models Versus Lattice 521[10] A.A. Andrianov, V.A. Andrianov, D. Espriu, X. Planells, Phys. Lett. B 710,230 (2012).[11] F.R. Klinkhamer, N.S. Manton, Phys. Rev. D 30, 2212 (1984); V.A. Kuzmin,V.A. Rubakov, M.E. Shaposhnikov, Phys. Lett. B 155, 36 (1985).[12] L.D. McLerran, E. Mottola, M.E. Shaposhnikov, Phys. Rev. D 43, 2027(1991).[13] I. Tserruya, Landolt–Börnstein 23, 176 (2010).[14] A.A. Andrianov, V.A. Andrianov, D. Espriu, X. Planells, Theor. Math.Phys. 170, 17 (2012); A.A. Andrianov, V.A. Andrianov, Theor. Math. Phys.185, 1370 (2015).[15] J.J. Sakurai, Ann. Phys. 11, 1 (1960).[16] A.A. Andrianov, V.A. Andrianov, D. Espriu, X. Planells, Phys. Rev. D 90,034024 (2014).[17] A.A. Andrianov, D. Espriu, X. Planells, Eur. Phys. J. C 73, 2294 (2013).[18] A.A. Andrianov, D. Espriu, X. Planells, Eur. Phys. J. C 74, 2776 (2014).[19] V.V. Braguta et al., Phys. 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QCD with chiral chemical potential: Models versus lattice

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QCD with chiral chemical potential: Models versus lattice

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