Chiral symmetry restoration and strong CP violation in a strong magnetic background
aa r X i v : . [ h e p - ph ] O c t Chiral symmetry restoration and strong CP violationin a strong magnetic background
Eduardo S. Fraga ∗ Instituto de Física, Universidade Federal do Rio de Janeiro,Caixa Postal 68528, 21941-972, Rio de Janeiro, RJ , BrazilE-mail: [email protected]
Ana Júlia Mizher
Instituto de Física, Universidade Federal do Rio de Janeiro,Caixa Postal 68528, 21941-972, Rio de Janeiro, RJ , BrazilE-mail: [email protected]
Motivated by the phenomenological scenario of the chiral magnetic effect that can be possiblyfound in high-energy heavy ion collisions, we study the role of very intense magnetic fields andstrong CP violation in the phase structure of strong interactions and, more specifically, their in-fluence on the nature of the chiral transition. Direct implications for the dynamics of phase con-version and its time scales are briefly discussed. Our results can also be relevant in the case of theearly universe. ∗ Speaker. c (cid:13) Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-ShareAlikeLicence. http://pos.sissa.it/ hiral symmetry restoration and strong CP violation in a strong magnetic background
Eduardo S. Fraga
1. Introduction and motivation
Although topologically nontrivial configurations of the gauge fields allow for a CP-violatingterm in the Lagrangian of QCD [1, 2], experiments indicate that its numerical coefficient, knownas q , is vanishingly small, q . − [3, 4]. Nevertheless, in spite of the fact that spontaneousbreaking of P and CP were proved to be forbidden in the true vacuum of QCD for q = chiral magnetic effect [9].The quark-gluon plasma possibly possesses regions with nonzero winding number, Q w = B restricts the quarks (all in the lowest Landaulevel, aligned with B ) to move along its direction. For a topologically nontrivial domain with e.g. Q w = −
1, left-handed quarks are converted into right-handed ones, inducing an inversion of thedirection of momentum and, consequently, a net current and a charge difference are created alongthe direction of the magnetic field. The system is, then, P-odd.The magnetic fields involved, although short-lived for very high energies [9], will certainlylive longer for lower values of √ s , and can attain enormous intensities, above those considered formagnetars [10] and comparable only to the ones believed to be present during the early stages ofthe universe [11]. The presence of such extreme magnetic fields coupled with strong CP violationraises several theoretical questions. The first one is how the QCD phase diagram is altered by thepresence of a nonzero uniform magnetic field that plays the role of another “control parameter”.Next one can ask where are the possible metastable CP-odd states and how “stable” they are, i.e.how long their lifetimes are. These questions are directly related to possible modifications in thenature of the phase transitions of strongly interacting matter and to the relevant time scales for phaseconversion. As we have mentioned above, there is already an on-going experimental investigationthat seems to show indications of the chiral magnetic effect [12]. Furthermore, pioneer latticestudies on the chiral magnetization of the non-Abelian vacuum also seem to be able to capturethese effects [13].In what follows, we discuss the effects of a strong and constant magnetic background and ofCP violation on the chiral transition at finite temperature and vanishing chemical potential. For thispurpose, we adopt the linear sigma model coupled with two flavors of quarks as our effective fieldtheory [14], following the notation and conventions of Ref. [15]. We show that for high enoughmagnetic fields the chiral transition is no longer a crossover. Instead, it is turned into a first-ordertransition [16].To include the effects of the presence of the axial anomaly and CP violation, we add a term2 hiral symmetry restoration and strong CP violation in a strong magnetic background Eduardo S. Fraga that mimics the presence of nontrivial gauge field configurations, the ’t Hooft determinant [17].The rich vacuum structure brought about by a nonzero q term in the action has an influence onthe chiral transition, and generates a more complex picture in the analysis of the phase diagramof strong interactions [18]. Working in a mean field approximation in an extended CP-odd linearsigma model, our results can be cast in terms of condensates of the fields [19]. In this framework,three different phases occur, making the topography of extrema very rich, and allowing for theexistence of metastable minima in certain situations.
2. Effective theory for the chiral transition in a strong magnetic background
Modifications in the vacuum of CP-symmetric QCD by the presence of a magnetic field havebeen investigated previously within different frameworks, mainly using effective models [20, 21,22, 23, 24, 25, 26, 27], especially the NJL model [28], and chiral perturbation theory [29, 30, 31],but also resorting to the quark model [32] and certain limits of QCD [33]. Most treatments havebeen concerned with vacuum modifications by the magnetic field, though medium effects wereconsidered in a few cases, as e.g. in the study of the stability of quark droplets under the influenceof a magnetic field at finite density and zero temperature, with nontrivial effects on the order of thechiral transition [34]. More recently, magnetic effects on the dynamical quark mass [35] and onthe thermal quark-hadron transition [36], as well as magnetized chiral condensates in a holographicdescription of chiral symmetry breaking [37], were also considered. Recent applications of quarkmatter under strong magnetic fields to the physics of magnetars using the NJL model can be foundin [38].To investigate the effects of a strong magnetic background on the nature and dynamics of thechiral phase transition at finite temperature, T , and vanishing chemical potential, we adopt thelinear sigma model coupled to two favors of quarks. Following the notation of Ref. [15], we havethe lagrangian L = y f (cid:2) i g m ¶ m − g ( s + i g ~ t · ~ p ) (cid:3) y f + ( ¶ m s¶ m s + ¶ m ~ p¶ m ~ p ) − V ( s ,~ p ) , (2.1)where V ( s ,~ p ) = l ( s + ~ p − v ) − h s is the self-interaction potential for the mesons, exhibitingboth spontaneous and explicit breaking of chiral symmetry. The N f = y f represent the up and down constituent-quark fields y = ( u , d ) . The scalar field s plays therole of an approximate order parameter for the chiral transition, being an exact order parameter formassless quarks and pions. The latter are represented by the pseudoscalar field ~ p = ( p , p + , p − ) ,and it is common to group together these meson fields into an O ( ) chiral field f = ( s ,~ p ) . In whatfollows, we implement a simple mean-field treatment with the customary simplifying assumptions,where quarks constitute a thermalized fluid that provides a background in which the long wave-length modes of the chiral condensate evolve. At T =
0, the model reproduces results from chiralperturbation theory for the broken phase vacuum. In this phase, quark degrees of freedom areabsent (excited only for T > s field is heavy, M s ∼
600 MeV, and treated classically.On the other hand, pions are light, and fluctuations in p + and p − couple to the magnetic field, B ,as will be discussed below, whereas fluctuations in p give a B -independent contribution that weignore, for simplicity. For T >
0, quarks are relevant (fast) degrees of freedom and chiral sym-metry is approximately restored in the plasma for high enough T . In this case, we incorporate3 hiral symmetry restoration and strong CP violation in a strong magnetic background Eduardo S. Fraga quark thermal fluctuations in the effective potential for s , i.e. we integrate over quarks to oneloop. Pions become rapidly heavy only after T c and their fluctuations can, in principle, matter sincethey couple to B . The parameters of the lagrangian are chosen such that the effective model re-produces correctly the phenomenology of QCD at low energies and in the vacuum, in the absenceof a magnetic field. Standard integration over the fermionic degrees of freedom to one loop, us-ing a classical approximation for the chiral field, gives the effective potential in the s direction V e f f = V ( f ) + V q ( f ) , where V q represents the thermal contribution from the quarks that acquire aneffective mass M ( s ) = g | s | . The net effect of the term V q is correcting the potential for the chiralfield, approximately restoring chiral symmetry for a critical temperature T c ∼
150 MeV [15]. s (MeV) -4-3-2-10 V e ff ( s , T , B = m p ) / T T = 120 MeVT = 140 MeVT = 160 MeVT = 180 MeV
Figure 1:
Evidence for a first-order chiral tran-sition in the effective potential for eB = m p . s (MeV) -0.9-0.89-0.88-0.87-0.86-0.85 V e ff ( s , T , B = m p ) / T T = 180 MeV
Figure 2:
Zoom of the barrier for eB = m p . Assuming that the system is now in the presence of a strong magnetic background that is con-stant and homogeneous, one can compute the modified effective potential following the procedureoutlined in Ref. [16]. In what follows, we simply sketch some of the main results. For definiteness,let us take the direction of the magnetic field as the z -direction, B = B ˆz . The effective potentialcan be generalized to this case by a simple redefinition of the dispersion relations of the fields inthe presence of B , using the minimal coupling shift in the gradient and the field equations of mo-tion. For this purpose, it is convenient to choose the gauge such that A m = ( A , A ) = ( , − By , , ) .Decomposing the fields into their Fourier modes, one arrives at eigenvalue equations which havethe same form as the Schrödinger equation for a harmonic oscillator potential, whose eigenmodescorrespond to the well-known Landau levels. The latter provide the new dispersion relations p n = p z + m + ( n + ) | q | B , p n = p z + m + ( n + − s ) | q | B , (2.2)for scalars and fermions, respectively, n being an integer, q the electric charge, and s the sign ofthe spin. Integrals over four momenta and thermal sum-integrals are modified accordingly, yieldingsums over the Landau levels.In our effective model, the vacuum piece of the potential will be modified by the magneticfield through the coupling of the field to charged pions. To one loop, and in the limit of high B ,4 hiral symmetry restoration and strong CP violation in a strong magnetic background Eduardo S. Fraga eB >> m p , one obtains (ignoring contributions independent of the condensates) [16] V V p + + V V p − = − m p eB p log 2 . (2.3)Thermal corrections are provided by pions and quarks. However, the pion thermal contributionas well as part of the quark thermal contribution are exponentially suppressed for high magneticfields, as has been shown in Ref. [16]. The only part of the quark thermal piece that contributes is V Tq = − N c eBT p (cid:20) Z + ¥ − ¥ dx ln (cid:16) + e − √ x + M q / T (cid:17)(cid:21) , (2.4)where N c = eB ∼ m p at different values of the temperature toillustrate the phenomenon of chiral symmetry restoration via a first-order transition. For RHIC topenergies one expects eB ∼ − m p [9]. For lower values of the field, the barrier is smaller. In Fig.2, we show a zoom of the effective potential for eB ∼ m p for a temperature slightly below thecritical one. This figure highlights the presence of a first-order barrier in the effective potential. Fora magnetic field of the magnitude that could possibly be found in non-central high-energy heavyion collisions, one moves from a crossover scenario to that of a weak first-order chiral transition,with a critical temperature ∼
30% higher [16]. Although the barrier in this case is tiny, the intensityof supercooling of the system is expected to be rather large due to the smallness of nucleation rateswhen compared to the expansion time scales for the heavy ion scenario. Therefore, even a smallbarrier can keep part of the system in the false vacuum until the spinodal instability is reachedand the system is abruptly torn apart. Nevertheless, since the magnetic field falls off very rapidlyat RHIC top energies, we expect that even for lower values of √ s only the early-time dynamicsshould be affected.As caveats, first we note that although non-central heavy ion collisions might show features ofa first-order transition when contrasted to central collisions, in this comparison finite-size effectsbecome important and have to be taken into account [39]. Second, since the magnetic field variesvery rapidly in time, it can induce strong electric fields that could play a relevant role via theSchwinger mechanism.
3. CP-odd linear sigma model
To describe the chiral phase structure of strong interactions including CP-odd effects, we adoptan effective model that reproduces the symmetries of QCD at low energy scales, and has the appro-priate degrees of freedom at each scale: the CP-odd linear sigma model coupled with two flavorsof quarks. The chiral mesonic sector is built including all Lorentz invariant terms allowed bysymmetry and renormalizability. Following Refs. [17, 40], one can write L c =
12 Tr ( ¶ m f † ¶ m f ) + a ( f † f ) − l [ Tr ( f † f )] − l [( f † f ) ] hiral symmetry restoration and strong CP violation in a strong magnetic background Eduardo S. Fraga + c [ e i q det ( f ) + e − i q det ( f † )] + Tr [ h ( f + f † )] . (3.1)The potential in the Lagrangian above displays both spontaneous and explicit symmetry breaking,the latter being implemented by the term ∼ h . The strength of CP violation is contained in the ’tHooft determinant term, which encodes the Levi-Civita structure of the axial anomaly and dependson the value of the parameter q .Expressing the chiral field f as f = √ ( s + i h ) + √ ( ~ a + i ~ p ) · ~ t , (3.2)where ~ t are the generators of SU ( ) , the Pauli matrices, the potential takes the following form(substituting the parameter h by H ≡ √ h ): V c = − a ( s + ~ p + h + ~ a ) − c q ( s + ~ p − h − ~ a ) + c sin q ( sh − ~ p · ~ a ) − H s + ( l + l )( s + h + ~ p + ~ a ) + l ( s ~ a + h ~ p + ~ p × ~ a ) , (3.3)where the parameters a , c , H , l and l are fixed by vacuum properties of the mesons [19]. Quarksare coupled to the chiral fields in the same fashion as before.Following a mean field analysis for the condensates h s i and h h i and assuming that the re-maining condensates vanish, we can compute the effective potential, which is a function of thecondensates above and of the CP violation coefficient q , and fix all the free parameters [19]. For q = q from zero to p , the minima of the effective potential rotate from the s directionalmost to the h direction, the rotation being complete only for massless quarks. - -
10 0 10 20 30 - Σ Η Figure 3:
Contour plot of the effective potentialfor q = p and T =
125 MeV. Numerical valuesare in MeV. - -
10 0 10 20 30 - Σ Η Figure 4:
Contour plot of the effective potentialfor q = p and T =
128 MeV. Numerical valuesare in MeV. hiral symmetry restoration and strong CP violation in a strong magnetic background Eduardo S. Fraga
In Figs. 3 and 4 we show contour plots of the effective potential in the case of q = p . Increas-ing the temperature, the minima move towards the center, indicating chiral symmetry restoration.However, there is a clear barrier between the global minimum and the new minimum that becomesthe true global minimum at high temperature (at h = q =
0, thissignals a first-order transition, and the possibility of metastable CP-odd states. Moreover, since thecritical temperatures for the melting of s and h condensates are different, three different phases areallowed in systems with q between zero and p : one in which both condensates are present, anotherwhere the h condensate vanishes, and a phase where both condensates vanish [19]. This can alsobe illustrated by the behavior of each condensate as a function of the temperature, as shown in Fig.5.
100 120 140 160 180
T(MeV) s> - q = 0 < h> - q = 0 - q = p /4|< h>| - q = p /4 - q = p /2|< h>| - q = p /2< s> - q = p |< h>| - q = p Figure 5:
Absolute value of the condensates (inMeV) as functions of the temperature. Full linesdenote the s condensate and dotted lines the h condensate. < h> -6-4-202 V ( < h > , T , B = m p , q = p ) / T T = 120 MeVT = 140 MeVT = 160 MeVT = 180 MeV
Figure 6:
Effective potential normalized by thetemperature for q = p and B = m p in the di-rection of the h condensate (in MeV). Following the steps of the previous section, we can incorporate the effects from a strong mag-netic background on top of the CP-odd linear sigma model [19]. One then finds that the criticaltemperature is raised, as well as the barrier along the h direction. Effects on the s field are anal-ogous to the ones obtained in the CP-conserving linear sigma model. In Fig. 6, we illustrate thisphenomenon for a magnetic field of eB = m p and q = p .Therefore, for nonzero q metastable CP-violating states appear quite naturally in the CP-oddlinear sigma model. (However, they were not found in an extension of the NJL model [18]. For adiscussion, see Ref. [41].) In fact, larger values of q tend to produce a first-order chiral transitionand might lead to the formation of domains (bubbles) in the plasma that exhibit CP violation [19].This reinforces the scenario of possible metastable CP-odd states in QCD that are so relevant forthe chiral magnetic effect [9]. This behavior is enhanced by the presence of a strong magnetic field,so that both effects seem to push in the same direction [16, 19].
4. Final remarks
Strong magnetic fields can modify the nature of the chiral and the deconfining transitions,opening new possibilities in the study of the phase diagram of QCD, introducing a new control7 hiral symmetry restoration and strong CP violation in a strong magnetic background
Eduardo S. Fraga parameter, besides temperature and baryon chemical potential, in the study of the thermodynamicsof strong interactions. These high magnetic fields are also essential in the context of high-energyheavy ion collisions in generating a measurable charge asymmetry to determine the presence ofCP-odd domains created by sphaleron transitions. Several relevant questions that are still not fullycovered can be raised. How strong can one make these magnetic fields in current and future exper-iments? How long lived could they be, and how uniform, for different values of √ s ? Only the firsttheoretical estimates [9, 42] and preliminary lattice simulations [13] were performed, providingencouraging results.On the one hand, due to the presence of a strong magnetic background, non-central heavyion collisions might show features of a first-order transition when contrasted to central collisions.On the other hand, finite-size effects are sizable for non-central collisions [39], so that an accuratecentrality dependence study seems to be necessary. This demands a thin binning of centrality andcontrol of finite-size effects, a very difficult (but necessary) task for experimentalists in the processof data analysis, especially in the scheduled Beam Energy Scan program at RHIC-BNL.Finally, one still needs to perform dynamical investigations within the chiral magnetic effectscenario to determine the relevant time scales and verify whether effects from the CP-odd domainssurvive long enough to produce measurable signatures. Acknowledgments
We thank D. Boer, J. Boomsma, M. Chernodub, K. Fukushima, M. Stephanov and A. Zhitnit-sky for fruitful discussions. E.S.F. is specially grateful to D. Kharzeev for valuable discussions andfor his kind hospitality in the Nuclear Theory Group at BNL. This work was partially supported byCAPES, CNPq, FAPERJ and FUJB/UFRJ.
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