aa r X i v : . [ h e p - ph ] A ug Magnetic aspects of QCD and compact stars
Toshitaka
Tatsumi
Department of Physics, Kyoto University, Kyoto 606-8502
Magnetic properties of quark matter are discussed. The possibility of ferromagnetictransition is studied by using the one-gluon-exchange interaction. Magnetic susceptibility isevaluated within Landau Fermi liquid theory, and the important roles of the screening forthe gluon propagation are elucidated. Static screening for the longitudinal gluons improvesthe infrared singularities, while the transverse gluons receive only dynamic screening. Thelatter property gives rise to a novel non-Fermi-liquid behaviour in the magnetic susceptibility.The critical density is estimated to be around the nuclear density and the Curie temperatureseveral tens MeV. The spin density wave is also discussed at moderate densities, where chiraltransition becomes important. Pseudoscalar condensate as well as scalar one takes a spatiallynon-uniform form in a chirally invariant way. Accordingly magnetization oscillates like spindensity wave. These results should have some implications on compact star phenomena. §
1. Introduction
Nowadays there are many studies about the phase diagram of QCD on temperature-density plane. Here we’d like to explore some magnetic phases by considering thespin degrees of freedom. Phenomenologically such magnetisms should be related toobservation of compact stars, especially their magnetic evolution. The origin of thestrong magnetic field in compact stars is a log-standing problem since the first dis-covery of pulsars. Recent discovery of magnetars with huge magnetic field (10 G)seems to revive this issue. Their origin is not clear yet, while some ideas such asfossil field, dynamo scenario has been proposed. If QCD has a potential to producesuch magnetic filed, it gives a microscopic origin. Actually we studied a possibilityof spontaneous magnetization of quark matter, since microscopic nuclear-mattercalculations have shown negative results. We have shown that quark matter wouldbe ferromagnetic state due to the Bloch mechanism, in analogy with electron gas,and suggested that magnitude of the magnetic field amounts to be O (10 − G ) ifsuch state develops inside compact stars.We shall discuss a possibility of spin density wave (SDW) phase as another in-teresting magnetic aspect of QCD in relation to chiral transition. It is well knownthat restoration of chiral symmetry is important at moderate densities and manyefforts have been devoted to figure out their properties. Assuming the non-uniformcondensates of pseudoscalar as well as scalar channel, we have studied another pathof chiral transition. We have seen that such phase appears near the phase bound-ary of chiral transition, and thereby restoration of chiral symmetry is delayed tohigher densities or temperatures. We can also see that magnetization in this phaseshows an oscillating shape like SDW. This phase may then be characterized by localferromagnetism and global anti-ferromagnetism. typeset using
PTP
TEX.cls h Ver.0.9 i T. Tatsumi §
2. Chiral symmetry and spin density wave
First we see the appearance of spin density wave in relation to the chiral tran-sition. Restoration of chiral symmetry is important at moderate densities or finitetemperature and many studies including lattice gauge simulations or effective modelstudies have been done to figure out the QCD phase diagram on the temperature-density plane. Assuming non vanishing pseudosclar condensate as well as scalarone, we consider the following non-uniform configuration ∗ ) : Note that this configu- h ¯ ψψ i = ∆ cos q · r h ¯ ψiγ τ ψ i = ∆ sin q · r . (2.1) -3 -2 -1 0 1 2 3-1 0 1-1 0 1 z ψψ ψ i γ τ ψ Fig. 1. ration respects SU (2) × SU (2) chiral symmetry and the system is a charge eigenstate.It is called dual chiral density wave (DCDW) (Fig. 1). Both condensates constructthe complex order parameter, in cotrast to the usual discussion of chiral transitionwith uniform scalar condensate; we shall see that the amplitude ∆ gives an effectivemass for quarks and the phase degree of freedom θ = q · r gives rise to a mag-netic property. Similar ideas of chiral density wave have been proposed as relevantphases in the large N c limit, where only non-uniform scalar condensate has beenconsidered.Some results have been presented in Figs. 3,4 by using NJL model as an effectivemodel of QCD at moderate densities. Different from the usual result denoted by thethin dotted curve, DCDW appears around the critical density with finite momentum q (Fig. 1). Chiral restoration is then delayed by the appearance of DCDW. Themagnitude of q is O (2 k F ), which suggests that nesting effect of the Fermi surfaceshould be responsible to appearance of DCDW. Actually we can show that thecorrelation function C ( q ) = lim ω → F.T. h ¯ ψiγ τ ψ ( x ) ¯ ψiγ τ ψ (0) i diverges at finite q with O (2 k F ) at the critical density. Accordingly magnetization of quark matter exhibits h ¯ ψΣ z ψ i = M cos( q · r ) , (2.2)which implies there develops SDW in the DCDW phase.It should be interesting to see a similarity to pion condensation in hadronicmatter, where nucleons take a anti-ferromagnetic spin ordering in the presence ofclassical pion field. §
3. Ferromagnetism and magnetic susceptibility
In the first study about ferromagnetic instability in QCD we calculated theenergy of the spin polarized quark matter. Since quark matter is color neutral as ∗ ) We only consider the chiral limit in the following. agnetic aspects of QCD µ / Λ M / Λ , q / Λ Order Parameters at T = M/ Λ q/ Λ M/ Λ (q=0) µ c1 µ c2 Fig. 2. Wave number q and the dynamical mass M = 2 G∆ are plotted as functions of thechemical potential at T = 0. Solid (dotted)line for M with (without) the density wave,and dashed line for q . µ / Λ T / Λ χ SB χ SB with DCDW
DCDW
Fig. 3. Phase diagram of chiral transition onthe temperature-density plane. DCDW ap-pears at relatively low temperature. a whole, the Fock exchange energy gives the leading-order contribution. Using therelation, h λ i i ab h λ i i ba = 1 / − / (2 N c ) δ ab , we can see that it repulsively works for anyquark pair. Then two particles with the same spin can avoid the repulsive interactiondue to the Pauli principle to favor ferromagnetic order. On the other hand the kineticenergy is totally increased. So when the energy gain in their interaction exceeds theincrease of the kinetic energy, we can expect a ferromagnetic instability. This is theBloch mechanism. A calculation has been done by using the one-gluon-exchangeinteraction to find a weakly first-order phase transition around the nuclear density.To get more insight into the magnetic properties of quark matter we have recentlystudied magnetic susceptibility within Fermi-liquid theory ∗ ) . By applying tinyand uniform magnetic field B we examine the magnetization h M i of quark matter toevaluate magnetic susceptibility, χ M = ∂ h M i /∂B | N,T,B =0 , which can be expressedin terms of the Landau-Migdal parameters: χ M = (cid:16) g D µ q (cid:17) / (cid:18) π N c k F E F − f s + ¯ f a (cid:19) , (3.1)where f s , ¯ f a are spin-independent and -dependent Landau-Migdal parameters, re-spectively.3.1. Screening effects for gluons
Landau-Migdal parameters usually includes infrared (IR) divergences in gaugetheories QCD/QED, so that it is essential to take into account the screening ef-fect to improve them. The HDL resummation can be achieved by using the quarkpolarization operator; longitudinal gluons are statically screened in terms of the De-bye mass, while transverse gluons are only dynamically screened due to the Landaudamping. Thus Debye screening surely improves the IR divergence for longitudinalgluons, while there still remains the IR divergences coming from transverse gluons. ∗ ) We assume here the second order phase transition, but we shall find the similar critical densityto the one in ref..
T. Tatsumi At T = 0 these divergences cancel each other in (3.1) to give a meaningful result.We shall see an interesting effect caused by the dynamic screening in section 3.3.3.2. Magnetic transition at T = 0The magnetic susceptibility is given in Fig. 4 at T = 0. We can see that mag- -40-20 0 20 40 0 0.5 1 1.5 2 OGE incl. Debye scr. with NF=3incl. Debye scr. with NF=1 kF [1/fm] χ Fig. 4. Magnetic susceptibility at T = 0. The solid curveshows the result with the simple OGE without screening,while the dashed and dash-dotted ones shows the screen-ing effect with N f = 1 (only s quark)and N f = 2 + 1( u, d, s quarks), respectively. -1.4-1.2-1-0.8-0.6-0.4-0.2 0 0.2 0.4 0.6 0.8 0 0.5 1 1.5 2 2.5 3 κ κ l n ( / κ ) NF =1 NF =3 Fig. 5. Flavor dependence of thecontribution of the screeningeffects. netic susceptibility diverges around the nuclear density and quark matter is in theferromagnetic phase below the critical density. When the screening effects are takeninto account, the curve is shifted in two ways, depending on the number of flavors;it shifted to lower densities for N f = 1, while to higher densities for N f = 3. Thusthe screening effects favors the ferromagnetic phase for N f = 3, different from thecase of N f = 1. This is in contrast with the usual argument for electron gas, wherethe correlation effect is always disfavors the magnetic transition. Such behaviorcan be seen by looking at the contribution of the screening effects to χ M (Fig. 5),which reads ∆χ − M ∝ κ ln(2 /κ ) , (3.2)with κ = m D / k F , where the Debye screening mass can be written as m D = P flavors g / π k F,i E F,i . Thus the screening effect in quark matter is qualitativelydifferent from electron gas.3.3.
Finite temperature effects and non-Fermi-liquid behavior
Our framework can be easily extended to finite temperature case.
We, here-after, consider the low temperature case (
T /µ ≪ Σ + ( ω ) ≃ Re Σ + ( µ ) − C f g v F π ( ω − µ ) ln m D | ω − µ | + ∆ reg ( ω − µ ) (3.3)near the Fermi surface, with v F being the Fermi velocity and C f = ( N c − / N c . The second term appears due to the transverse gluons and logarithmically diverges agnetic aspects of QCD marginal
Fermi liquid.
The temperature dependent term in χ M is finally given by δχ − M = χ − h π k F (cid:18) E F − m + m E F (cid:19) T + C f g v F k F E F (cid:0) k F + k F m + m (cid:1) T ln (cid:16) m D T (cid:17) i + O ( g T ) . (3.4)We can see that there appears T ln T term beside the usual T one. This is a novelnon-Fermi-liquid effect. It should be interesting to compare this result with othertypical non-Fermi-liquid effect in the specific heat or the gap function in colorsuperconductivity.
Moreover, it is to be noted that the spin fluctuation effectgives T ln T term as a leading-order contribution in electron gas. Finally the magnetic phase digram is presented on the temperature-density plane(Fig. 5), where we can see the Curie temperature of several tens MeV. T [ M e V ] kF [1/fm] Fig. 6. Magnetic phase diagram in the density-temperature plane. The solid, dashed, dash-dotted,dotted curves show the results for the full expression, the one without the T ln T term, withoutthe κ ln κ term, and without the T ln T and κ ln κ terms. The open (filled) circle indicates theCurie temperature at k F = 1 . .
6) fm − while the squares show those when we disregard the T ln T dependence. §
4. Summary and concluding remarks
We have discussed some magnetic aspects of QCD on the temperature-densityplane.First we have demonstrated appearance of DCDW near the phase boundary ofchirak transition. In recent papers Nickel discussed the appearance of the real kinkcrystal (RKC).
The tricritical point is then Lifshitz point in this case. This is aninteresting possibility, but more studies are needed to elucidate the relation betweenRKC and DCDW phases, while he cocluded that RKC is more favored than DCDWphase.We have studied the static magnetic susceptibility of quark matter by utilizingFermi-liquid theory to see ferromagnetic transition, where the screening effects forgluon propagators become very important; the static screening by the Debye massgives g ln g term at T = 0, while it produces a novel non-Fermi liquid effect by T. Tatsumi the T ln T term. For a more realistic study, however, some non-perturbative effectsare taken into account at moderate densities. Moreover, the restoration of chiralsymmetry is also important there.Observational signals of the magnetic phases may be found out in thermal evo-lutions of compact stars, besides the direct evidence of magnetic evolution. Nambu-Goldstone bosons, as a consequence of spontaneous symmetry breaking in the mag-netic phases, may contribute to thermodynamical quantities to modify their thermalevolutions. Acknowledgements
This work was partially supported by the Grant-in-Aid for the Global COEProgram “The Next Generation of Physics, Spun from Universality and Emergence”from the Ministry of Education, Culture, Sports, Science and Technology (MEXT)of Japan and the Grant-in-Aid for Scientific Research (C) (16540246, 20540267).
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