Nucleating Quark Droplets in the Core of Magnetars

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E. and o h eeatqatte eae otenceto pro- nucleation the to results related nucleation quantities Our all relevant obtain approximation. the for thin-wall to the order in in III parameters proceed Section we potential. approxima- how effective the the shows and compute to model used effective tions the describe briefly values small tension. so surface not the for even of nucleation quantities ob- favor these is to of the seems time combination and and nucleation non-trivial a The tension from surface tained affected. the are rate radius, nucleation correlation critical the the potential, chemical length, critical chiral We the the how matter. for star show scales compact pro- time magnetized in typical and conversion the phase parameters for estimate nucleation an approximation, relevant vide thin-wall all the the extract within as we effec- and known lin- one-loop potential, also the the tive From quarks, use ex- to model. we an quark-meson coupled framework, two-flavor of model a a presence sigma As in the ear droplets field. in magnetic symmetric environment ternal chirally dense investigat- and of by cold formation magnetars the in matter ing quark of nucleation chiral matter. the magnetar of for case mechanism the is whether driving in assess them the transition to of be field each can magnetic how nucleation external an compute by these to affected of needs combination non-trivial one a quantities, conversion from phase built the are for control scales process external time an the can Since of the tension) variations parameter. to surface radius, differently the critical very and react the in potential as ingredients chemical different (such critical [12], process Ref. nucleation become of has the analysis as the However, in field. clear magnetic the of as function behavior a non-monotonic interesting Ref. an in exhibiting model [17], Nambu–Jona-Lasinio the within estimated Refs. in ingredient estimated key later a was that which [12–16]. [10] tension, Ref. surface in the shown is been has It rate. h ae sognzda olw.I eto Iwe II Section In follows. as organized is paper The homogeneous of possibility the assess we paper this In was matter quark magnetized for tension surface The Fraga ecia hs ovrini magnetized in conversion phase chiral he nfrnts ml auso h surface the of values small so not for en xrc l eeatnceto parame- nucleation relevant all extract s niomn ntepeec fan of presence the in environment nse atri antr,w investigate we magnetars, in matter k en h ulaintm em to seems time nucleation the define o il rtclrdu,creainlength correlation radius, critical tial, etv oeta ftetwo-flavor the of potential ffective † o, 2 cess are presented in Section IV. Section V presents our system under consideration. Moreover, in order to per- summary. form a semi-analytic study, some simplifying approximations are needed. The first regards the fermionic contribution to the II. EFFECTIVE THEORY effective potential. As the action is quadratic in the fermion fields, we can formally integrate over the quarks, A. General Framework so that their contribution to the effective potential is given by a determinant. However, as the quarks cou- To study the phase conversion process, we adopt the ple to σ, one is left to compute a fermionic determinant linear sigma model coupled to quarks (LSMq) [18] as our in the presence of an arbitrary background field, which effective theory description of the chiral sector of strong cannot be done in closed form, unless for systems in 1+1 interactions. The Lagrangian is given by dimensions under some special circumstances [21–23]. As customary, we consider the quark gas as a thermal bath µ = ψ¯f [iγ ∂µ g(σ + iγ5τ π)]ψf in which the long-wavelength modes of the chiral field L − · evolve, so that the calculation is performed considering 1 µ π µπ + (∂µσ∂ σ + ∂µ ∂ ) (1) a static and homogeneous background field. This pro- 2 · cedure can be further improved, e.g. via a derivative λ 2 2 2 2 (σ + π v ) + hσ . expansion [24–26]. − 4 − The contribution from the mesons to the effective The model contains a fermionic SU(2) chiral doublet, potential is also subject to simplifying approximations. First, it has been shown that the pions do not affect ap- ψf , representing the up and down constituent quarks, and four mesons – one scalar, σ, and three pseudoscalars, preciably the phase conversion process, so their dynamics π. The mesons can be grouped into a single O(4) chi- is usually discarded and the whole analysis can be done π setting π = π = 0. Second, since λ 20, quantum cor- ral field φ (σ, ). It is well know that the LSMq re- h i ≈ produces correctly≡ all the chiral low energy phenomenol- rections arising from the sigma self interaction are usually ignored, and its contribution to the effective potential is ogy of strong interactions, such as mesons masses and 2 the spontaneous and (small) chiral symmetry breaking, taken to be classical . which are present in the mesonic self-interaction potential. The model parameters are fixed accordingly [18]. Moreover, it was argued [19] that both QCD with two B. Effective potential at one loop in a magnetic flavors of massless quarks and the model we consider be- background long to the same universality class, thus exhibiting the 1 same behavior at criticality . Our aim is to study the chiral transition in a cold and Due to spontaneous symmetry breaking, the σ field ac- dense environment in the presence of an external con- quires a non-vanishing vacuum expectation value. How- stant and homogeneous magnetic field, as a very simpli- ever, for sufficiently high temperatures, the condensate fied model for the core of a magnetar. Adapting the pre- melts and chiral symmetry is approximately restored. vious setup to describe such a system is straightforward. Therefore, in this context the expectation value of the The interaction with the magnetic field is introduced via σ field plays the role of an approximate order parameter minimal coupling, i.e the derivatives acting on quarks are for the chiral transition, being exact only in the limit of traded for Dµ ∂µ + iqAµ. Following previous work, we vanishing quark (and pion) masses, which happens for use the aforementioned≡ approximations when computing h = 0. In this limit, the model becomes truly chiral, and the effective potential. the pions behave as Goldstone bosons. So, to investi- In this setup, the effective potential for two flavors of gate the phase conversion in the LSMq, one ultimately quarks with N colors in the presence of a homogeneous needs to study how the expectation value σ =σ ¯ varies c and static magnetic field B in the cold and dense limit as a function of the relevant control parameters,h i such can be written as sum of three contributions [28]: as temperature, chemical potentials and external fields. As usual in this approach, the effective potential formal- vac med ism rises as the appropriate means for the description of Veff(¯σ)= Ucl(¯σ)+ Uf (¯σ,B)+ Uf (¯σ,µ,B) . (2) phase transitions. In the spirit of effective theory descrip- tions, we will not be concerned with numerical precision, The first term is just the classical potential for σ, the but rather in obtaining qualitative information about the

2 See, however, Ref. [27], where the authors consider thermal me- 1 Recent lattice results seem to challenge this connection in the son fluctuations using resummations, and Ref. [12], where the chiral limit [20], although further detailed studies are still neces- authors compute the one-loop correction to the classical poten- sary. tial and treat systematically vacuum terms. 3 second gives the fermionic vacuum contribution the cold, i.e. T 0, limit to compute the effective potential, see Eq. (4).∼ When we focus on thermal nucleation, N we are ultimately comparing temperature with the height U vac = c (q B)2 ζ′ ( 1, x ) + f −2π2 f H − f of barrier separating true and false vacua, whereas when f " X (3) we consider the cold limit we compare it with the quark 2 2 x xf x chemical potential. Indeed, in our setup the tempera- f − log x + f , − 2 f 4 ture is high enough to enable thermal activation and low # enough to justify the use of the zero-temperature effective potential3. where x = M 2/(2 q B), M = gσ¯ is the quark f q | f | q Our aim is to estimate typical times scales for the nu- dynamically-generated mass, qf is the electric charge of ′ cleation process and to understand under which condi- quark species f and ζH denotes the derivative with re- spect to the first argument of the Hurwitz ζ-function. tions it is favored. In other words, which are the features Finally, the last term of Eq. (2) is the medium contribu- that can make nucleation happen effectively in magnetar tion due to the quarks (see e.g. Ref. [29]) matter, producing chirally symmetric matter in the core of such stars. As mentioned previously, a key quantity νmax seems to be the surface tension, since it is the amount of N med c 2 2 energy needed to build up a barrier separating the two Uf = 2 (2 δν0) qf B µ µ MfB + − 4π =0 − | | " − phases. In other words, the surface tension is the ener- f ν q X X getic cost to create a bubble. µ + µ2 M 2 2 − fB MfB log , −  qMfB 

  (4) A. Extracting nucleation parameters from the effective potential In this last expression we assume that both fermion species have the same chemical potential µ. In addition, 2 2 Since we are not concerned with numerical precision, MqB = Mq +2 qf B denotes the magnetic correction to the quark mass| and| ν is an integer value that labels Lan- but rather with obtaining reasonable estimates and the dau levels. The last occupied level is given by: qualitative functional behavior, it is convenient to work with approximate analytic relations by fitting the effec- 2 2 tive potential in the relevant region. This can be done µ Mf νmax = − . (5) conveniently using a quartic polynomial and imposing the 2 q B $ | f | % thin-wall limit. In the range between the critical chemical potential, µc, and the spinodal, µsp, the effective po- This effective potential exhibits a first-order phase tran- tential can be written in the following Landau-Ginzburg sition for a critical value µc(B) of the chemical potential. form [25, 31]:

4 III. SURFACE TENSION AND NUCLEATION n Veff a φ . (6) ≈ n n=0 Given the effective potential, we can proceed to the X study of the phase conversion process driven by the chiral Although this approximation is not able to reproduce the transition. The physical setup we have in mind is that three minima of Veff, the polynomial form gives a good of a collapsing star and, more specifically, the scenario quantitative description of the function in the region con- of magnetar formation. Thus, we investigate whether taining the two minima representing the symmetric and chirally symmetric matter can be nucleated as the density broken phases as well as the barrier between them. increases in the presence of a strong magnetic field. A quartic potential such as Eq. (6) can always be In our analysis, we focus on homogeneous nucleation. written in the form Dynamically, there are two ways by which nucleation can occur: thermal activation and quantum tunneling. 2 2 2 At the temperatures that correspond to the scenario at (ϕ)= α(ϕ a ) + jϕ, (7) V − hand, of the order of 10 30 MeV, and in the presence of a − barrier in the effective potential, thermal activation is by with the coefficients above defined in terms of the an as far the dominant way [10]. Once the barrier disappears, the initial state of the system is no longer in a metastable vacuum, so that spinodal decomposition takes place and the phase conversion is explosive [30]. It is important to state that there is no contradiction 3 It has been shown that thermal fluctuations and quantum vac- in considering thermal activation of bubbles and taking uum corrections compete when they are included in Veff [12]. 4 follows [25, 31]: IV. RESULTS

α = a4, (8a) 2 In this section we use the method described above 1 a2 3 a3 a2 = + , (8b) to describe quantitatively the nucleation process in the 2 −a4 8 a4 " # LSMq in the presence of a magnetic background field. 3 We compute different nucleation parameters for the for- a1 1 a2 1 a3 j = a4 + , (8c) mation of chirally symmetric droplets in a chirally asym- a4 − 2 a3 8 a4 " # metric medium for values of the external magnetic field 1 a3 that are compatible with what one expects to be relevant ϕ = φ + . (8d) 4 a4 to magnetar matter. As an initial step, we analyze how the critical chemical potential depends on B. The new potential (ϕ) reproduces the original Veff (φ) up to a shift in the zeroV of energy. We are interested in the effective potential only between µc and µsp. At µc, we will have two distinct minima of equal depth. This clearly corresponds to the choice j = 0 in Eq. (7), so A. Landau level filling and oscillations that has minima at ϕ = a and a maximum at ϕ = 0. V ± The minimum at ϕ = a and the maximum move closer When studying the critical behavior of the LSMq in the together as the chemical− potential is shifted and merge 3 presence of an external magnetic field, the first question at µsp. Thus, the spinodal requires j/αa = 8/3√3 in 3 − we should consider is how the position of the critical line Eq. (7). The parameter j/αa falls roughly linearly from is affected by B. The plot in Fig. 1 shows the behavior of 0, at µ = µ , to 8/3√3 at the spinodal. c − the critical chemical potential, µc(B), normalized by the In the thin-wall limit the explicit form of the critical critical chemical potential in the absence of the external bubble is given by [22] field, µ (0) = µ0 305 MeV. c c ≈ 1 r Rc ϕb(r, ξ, Rc)= ϕf + 1 tanh − , (9) ξ√2α − ξ where ϕf is the new false vacuum, Rc is the radius of 1.00 2 ′′ the critical bubble, and ξ = 2/m, with m (ϕf ), is a measure of the wall thickness. The thin-wall≡ V limit 0 c corresponds to ξ/R 1 [22], which can be rewritten Μ c 0.95 3 ≪ c as (3 j /8αa ) 1. Nevertheless, it was shown in [31, Μ 32], for| | the case≪ of zero density and finite temperature, that the thin-wall limit becomes very inaccurate as one approaches the spinodal. (This is actually a very general 0.90 feature of this description [30].) In this vein, the analysis presented below is to be regarded as semi-quantitative 0 5 10 15 20 25 and provides estimates, not accurate results. 2 In terms of the parameters α, a, and j defined above, eBmΠ one finds [25, 31] j FIG. 1. Critical chemical potential, µc as a function of B. ϕt,f a 2 , (10) ≈± − 8αa The solid line is just a guide for the eye. 1/2 1 ξ = 2 (11) α(3ϕ a2) From the plot it is clear that µ has a nonmonotonic " f − # c dependence on B, it oscillates and reaches a minimum in the thin-wall limit. The surface tension, Σ, is given by value for eB 10m2 . The results show clearly that the ≈ π ∞ 2 presence of a moderate external magnetic field can reduce dϕb 2 Σ dr 3 , (12) the value of µc up to 15%. ≡ 0 dr ≈ 3αξ 2 Z The small oscilations observed for eB . 4mπ are anal- and the critical radius is obtained from Rc = (2Σ/∆V ), ogous to the de Haas–van Alphen oscillations in metallic where ∆V V (φf ) V (φt) 2a j . Finally, the free crystals. They are related to the fact that, as we vary the ≡ − ≈ | | 2 energy of a critical bubble is given by Fb = (4πΣ/3)Rc, magnetic field, the degeneracy of the Landau levels and and from knowledge of Fb one can evaluate the nucleation the spacing between them are modified, so that the level rate Γ e−Fb/T . In calculating thin-wall properties, we filling varies with B. On the other hand, the behavior ∼ 2 shall use the approximate forms for φt, φf , Σ, and ∆V for eB & 4mπ is purely due to the lowest Landau level for all values of the potential parameters. filling. For a detailed discussion see Ref. [33]. 5

B. Nucleation parameters quark chemical potential for diﬀerent values of magnetic ﬁeld. It is interesting to notice that, as a consequence of

Oscillations are not only seen in the behavior of µc(B). the critical chemical potential oscillation, the metastable In fact, as the following plots show, all the nucleation region shifts when the magnetic field varies: first to lower parameters have a non-trivial oscillatory dependence on values of µ and then in the opposite direction. the magnetic field. As mentioned in the previous section, the correlation length, ξ, provides a measure of the thickness of the bubble wall. The thin-wall approximation relies on the as- 20 sumption that ξ/R 1 or, equivalently, that the free c ≪ HaL eB=0 energy difference between both vacua is small compared 15 to the barrier between them. In Fig. 3 we plot this quan- 2 HbL eB=5mΠ

L tity as a function of quark chemical potential. As one 2 fm HcL eB=10m should expect, this assumption is reasonable far from the

H Π

c 10 HcL HdL HbL HeL HaL

R spinodal, in the vicinity of the critical line. Nevertheless, H L 2 d eB=15mΠ in the spirit of providing estimates and the qualitative 2 5 HeL eB=20mΠ behavior, we apply the thin-wall limit in the whole range of chemical potentials between µc and µsp.

0 0.90 0.95 1.00 1.05 50 0 Μ Μc HaL eB=0 40 HeL L 2 2 HbL eB=5mΠ fm

FIG. 2. Critical radius of chirally symmetric droplets as a 30 HcL eB=10m2 function of the quark chemical potential for diﬀerent values HdL Π

MeV 2

eB H of . 20 HdL eB=15mΠ S HcL 2 HeL eB=20mΠ Recall that whenever a bubble is formed, its interior 10 HaL tends do lower the free energy of the system, since the HbL field within sits on the true vacuum. On the other hand, 0 the surface of the bubble tends to increase it, as discussed 0.90 0.95 1.00 1.05 1.10 1.15 previously. The critical bubble is the one whose energetic 0 ΜΜc gain due to the volume exactly compensates the cost of the surface. Thus, to minimize the energy, any bubble smaller than the critical will shrink and the ones that FIG. 4. Surface tension as a function of quark chemical po- are bigger will expand. Therefore, the radius of the crit- tential for different values of eB. ical bubble, or critical radius, sets the threshold between suppressed and favored bubbles. Finally, in Fig. 4 we present the results for the surface tension as a function of quark chemical potential for dif- 3.0 ferent magnetic fields. This plot shows clearly that for HaL eB=0 2 B . 5mπ the presence of an external magnetic field can 2.5 2 actually reduce the energetic cost to build up the bubble HbL eB=5mΠ wall, which would in principle favor nucleation in this 2.0 HcL eB=10m2 Π scenario. However, the behavior of µc as a function of c

R 2 HdL eB=15m the magnetic ﬁeld already gives a hint that the situation

Ξ 1.5 Π is not so straightforward. H L 2 1.0 e eB=20mΠ HcL HbL HdL HaL HeL 0.5 C. Estimating typical time scales 0.0 0.90 0.95 1.00 1.05 To obtain an estimate of the typical time scales in- ΜΜ0 volved in the nucleation of chirally symmetric matter in c a cold and dense medium under the influence of an external magnetic field, we need first an estimate of the FIG. 3. Ratio between the correlation length ξ and the critical nucleation rate per unit volume, which can be written as 4 −Fb/Tf radius as a function of quark chemical potential for different Γ T e , where Fb is the free energy of the crit- ∼ f values of eB. ical bubble and the pre-factor just gives an upper limit with the correct dimensions [30]. Here, we take Tf = 30 In Fig. 2, we show this quantity as a function of the MeV as a typical temperature for protostars. In doing 6 so we are neglecting the temperature dependence of the of a magnetar, in particular the likelihood of nucleating critical-bubble free energy or, as we discussed before, us- approximately chiral quark droplets. Using the one-loop ing the cold and dense effective potential since the differ- effective potential we computed all relevant nucleation ence scales justify this procedure. parameters within the thin-wall approximation and ob- In Fig. 5 we show the results for Γ as a function of the tained an estimate for the typical time scales. Our find- chemical potential for the same values of magnetic field ings indicate that nucleation may be present in the phe- adopted before. Again a nontrivial oscillation with the nomenological interesting range of magnetic fields. Of magnetic field can be detected. course, one has also to simulate in detail the evolution of the density profile of the protostar to make any stronger 1 assertion. The results obtained for the surface tension and nu- 10-4 cleation time are very interesting, showing that many

L HaL eB=0 different effects sum up in a nontrivial fashion yielding a 3 10-8 small nucleation time for cases whose surface tension are fm 2 HbL eB=5m Π not so small. Specifically, the B dependence of µc and -12 2 the fact that the difference between the free energy of MeV 10 HcL eB=10mΠ H

G the vacua increases faster for higher values of magnetic 2 HcL HdL HbL HeL HaL HdL eB=15mΠ 10-16 ﬁeld can combine in such a way that cases with a higher 2 HeL eB=20mΠ surface tension could have a smaller critical radius, ultimately favoring the nucleation picture. Therefore, for 10-20 0.90 0.95 1.00 1.05 1.10 magnetars it is not enough to consider the behavior (and value) of the surface tension to address the competition ΜΜ0 c between relevant time scales.

1030 FIG. 5. Nucleation rate as a function of the quark chemical potential for diﬀerent values of eB. 1020 HaL eB=0 1010 2 In order to estimate the typical time scales for the HbL eB=5mΠ phase conversion process, i.e. the formation of chiral L 2 s

H 1 HcL HdL HbL HeL HaL HcL eB=10mΠ quark matter in the core of magnetars, we follow Ref. Τ H L eB=15m2 [10] and define the nucleation time as being the time it 10-10 d Π takes for the nucleation of a single critical bubble inside 2 3 HeL eB=20mΠ a volume of 1 km , which is typical of the core of a proto- 10-20 neutron star, i.e.: 10-30 1 1 0.90 0.95 1.00 1.05 1.10 τ 3 . (13) 0 ≡ 1km Γ ΜΜc Fig. 6 exhibits this quantity as function of the chemical potential for different values of eB. The relevant time FIG. 6. Nucleation time as a function of quark chemical po- scale to compare is the time interval the system takes tential for different values of eB. from the critical chemical potential to the spinodal dur- ing the star collapse. Implicitly, in the expression above Despite its content of quarks and mesons, the linear we are using an approximation of constant density and sigma model provides essentially a chiral description, i.e. temperature over the core, which should give a good es- it does not contain essential ingredients to describe nu- timate as the density profile in this region of the star clear matter, such as the saturation density and the bind- is quite flat [1]. The plot shows that moderate mag- ing energy. Nevertheless, this analysis has unveiled how 2 netic fields, B . 20 mπ, can actually favor nucleation, the process of Landau level filling affects the nucleation as a given nucleation time is achieved for lower values of parameters in a nontrivial way, bringing new forms of chemical potential. competition between them and affecting qualitatively the dynamics of quark matter formation in compact stars.

V. SUMMARY AND FINAL REMARKS ACKNOWLEDGMENTS In this paper we have used the LSMq minimally coupled to an external classical magnetic ﬁeld in a cold and The authors would like to thank A. Schmitt for dis- dense environment as a simple model to describe criti- cussion. This work was partially supported by CAPES, cal properties of strongly interacting matter in the core CNPq and FAPERJ. 7

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