CHIRAL PHASE TRANSITIONS

Transcription

1 CHIRAL PHASE TRANSITIONS arxiv:hep-th/ v5 30 Ot 2001 M. D. Sadron and P. Żenzykowski Department of Theoretial Physis, Institute of Nulear Physis, Kraków, Poland Otober 28, 2018 Abstrat We show that melting the quark mass, the salar σ mass and the quark ondensate leads uniquely to the quark-level SU(2) linear σ model field theory. Upon thermalization, the hiral phase transition urve requires T = 2f CL 180 MeV when µ = 0, while the ritial hemial potential is µ = m q 325 MeV. Transition to the superondutive phase ours at T (SC) T (SC) < 180 MeV. = /e γ E. Coloured diquarks suggest Invited talk at the Chiral Flutuations in Hadroni Matter International Workshop, Orsay, Frane, Sept Permanent address: Physis Dept., University of Arizona, Tuson, AZ, USA sadron@physis.arizona.edu zenzyko@iblis.ifj.edu.pl 1

2 1 Introdution Aording to the now prevailing views, it is expeted that at high temperature and/or density the standard hadroni matter will undergo a phase transition to a new state. Suh prospets have motivated present experimental searhes for the signatures of the quark-gluon plasma in high-energy heavy ion ollisions. The appearane of suh state(s) may also affet properties of strongly interating matter at suh extreme onditions as those ourring inside of neutron stars. Clearly, if we are to understand the underlying phenomena better, various theoretial studies should be pursued. Temperature dependene of the properties of low density QCD has been explored fairly well. The basis for our understanding is provided by lattie field theory alulations whih, for QCD with two massive flavours, exhibit a phase transition to a hirally symmetri phase at T = 173 ± 8 MeV for improved staggered fermions [1]. Dependene on baryon harge density or hemial potential µ is muh more hypothetial, sine lattie Monte Carlo tehniques are not well suited to the ase when µ is non-zero. Nonetheless, on the basis of general arguments and of an insight gained from various models, one may onjeture the struture of the phase diagram in QCD with two massless quarks. Thus, the salient features of this QCD phase diagram in the Tµ plane are known or expeted as follows (see Fig.1 [2]). i)whentemperaturet israisedatµ = 0,thehiralQCDondensate qq melts at T 173±8 MeV or so, and the system enters into a quark-gluon plasma (QGP) phase, where the hiral ondensate is identially zero. The transition from the low temperature hadroni phase to the high temperature QGP phase is most probably of seond order. Beause the QGP and hadroni phases differ by the expetation value of qq, a (urved) line of phase transitions must begin at T = T, µ = 0 and ontinue for µ > 0 without terminating. By ontinuity for low µ, these phase transitions must be of seond order (this is indiated by a dotted line in Fig. 1). 2

3 ii) When the hemial potential µ of the system is inreased at zero temperature, one first enounters a transition from the phase of zero number density to that of nulear matter density. This transition is of first order and by ontinuity remains suh also at low T. In Fig. 1 this is indiated by a solid line. The hadroni gas at low T and small µ has number density n(t,µ) 0, and is onneted analytially to the nulear matter phase at high enough T 0. The latter is related to the binding energy of nuleons in nulear matter (T 0 10 MeV). iii) When the hemial potential and number density are further inreased at zero temperature, the interations of quarks beome weak due to(qcd) asymptoti freedom and olour interations beome sreened at shorter and shorter lengths. Thus, nonperturbative phenomena suh as hiral symmetry breaking should be absent at suffiiently high µ and a phase transition should our at some µ. Most likely, it is of first order. Sine regions to the left and right of the point µ = µ, T = 0 differ in the value of the quark ondensate, a line of first order transitions (marked solid) must start at this point and ontinue away from T = 0 without terminating. Most likely, it joins the line of seond order phase transitions emanating from µ = 0, T = T at a triritial point T 3. iv) There are arguments that in the region of low T to the right of µ, one should expet a nonvanishing value of a diquark ondensate qq whih spontaneously breaks olour symmetry. In this region, for temperatures below T (SC) (of order 100 MeV), one expets to find olour superondutivity. The nature of this phase transition to a non-superonduting phase at higher temperatures is not lear yet and, onsequently, this transition line is marked with dashes. In the following it will be shown how some of the above features as well as other aspets of the underlying theory are predited in a linear sigma model with a built-in dynamial symmetry breaking struture. 3

4 2 Chiral phase transition The word hiral means handedness in Greek. Defining left- and right- handed harges 2Q i ± = Q i ±Q i 5, the SU(2)urrent (harge) algebra ofthe1960sredues to [Q i ±,Qj ±] = iǫ ijk Q k ±, [Qi ±,Qj ] = 0. We onsider hiral low energy field theories of the linear σ model (LσM) [3, 4], four-fermion [5] and infrared QCD forms. When a field theory involving fermions (quarks) is thermalized [6], the fermion propagator is replaed by i(p/+m) p 2 m 2δ(p2 m 2 )(p/+m). (1) 2 e p 0/T +1 We shall show in many ways that the hiral symmetry restoration temperature T ( melting onstituent quarks, salar σ mesons, or the quark ondensate) is for 2 flavours in the hiral limit (CL) Here the sale f CL T = 2f CL 180 MeV. (2) is fixed from experiment [7] to be 90 MeV. This T sale is ompatible with omputer lattie results [1] T = 173±8 MeV. i) First we melt the SU(2) quark tadpole graph [8]: 0 m q (T ) = m q + 8N g 2 m q m 2 σ ( ) T 2 J (0), (3) with quark-meson oupling g, olour number N and J + (0) = 0 xdx(e x + 1) 1 = 2 /12. Canelling out the m q sale, Eq(3) gives (assuming N = 3) T 2 = m2 σ g 2. (4) ii) Next we melt the salar σ mass with oupling λφ 4 σ/4, to find [9] 0 m 2 σ(t ) = m 2 σ 6λ with J (0) = 0 xdx(e x 1) 1 = 2 /6, resulting in ( ) T 2 J 2 2 (0), (5) T 2 = 2m2 σ /λ. (6) 4

5 Taking the LσM tree-level relation λ = m 2 σ/2f 2 we note that the m σ mass sale divides out of Eq.(6) and reovers [9] Eq.(2) T = 2f. Also, the ratio of Eq.(4) to Eq.(6) gives λ = 2g 2, (7) independent ofthem σ andt sales. Moreover, Eq.(7)holds attreeandatloop level as we shall show in Se.3. Then with T 2 = 4f 2 from Eqs(2,6), melting the quark tadpole in Eq.(4) in turn requires g 2 T 2 = 4f2 g2 = m 2 σ, (8) whih implies the LσM [3, 4] quark-level Goldberger-Treiman relation (GTR) f g = m q and the famous NJL relation [5] m σ = 2m q. We shall return to this ombined LσM-NJL sheme [3, 4] in Set.3. iii) Finally we melt the quark ondensate qq(t ) in QCD: 0 qq(t ) mq = qq +2 4N m q ( T ) J + (0), (9) in analogy with Eq.(3). Here the fator of 2 is due to thermalizing both q and q in qq (as opposed to the flavour fator of 2 in Eq.(3) due to u and d quarks). At a 1 GeV sale, the QCD ondensate is known to have the value qq 1 GeV (250 MeV) 3 for QCD oupling [10] α s 0.5. But on the quark mass shell (where m q M N /3 315 MeV) this ondensate runs down to [11] qq mq = 3m 3 q /2 (215 MeV) 3. (10) In between, this ondensate (10) freezes out at [12] α s 0.75 near the MeV σ mass. In any ase, applying Eq.(10) to the quark ondensate melting ondition Eq.(9) gives for J + (0) = 2 /12, T 2 = 3m2 q /2 or T = 2[m q /(2/ 3)]. (11) Thus Eq.(11) again requires T = 2f CL, but only if the meson-quark QCD oupling is frozen out at the value 5

6 g = 2/ 3 = (12) This infrared QCD relation Eq.(12) was earlier stressed in ref.[13]. Its numerial value an be tested by the GTR ratio of CL masses g = (M N /3)/f CL 315/ Moreover the QCD effetive α s version of Eq.(12) is [11] α eff s = C 2F α s (m σ ) = (4/3)(/4) = /3, (13) whih also follows from Eq.(2) or g 2 /4 = /3, the latter being the LσM version as we shall see in Se.3. Of ourse α eff s = /3 1 is where the oupling is large and the quarks ondense. Another theoretial interpretation of Eq.(12) is as a Z = 0 ompositeness ondition for the LσM [14]. This justifies that an elementary salar σ(600) is only slightly less in mass than the bound state qq vetor ρ(769) i.e. the UV hiral utoff must be about 750 MeV separating elementary partiles from bound states (in the SU(2) LσM field theory). 3 Dynamially generated SU(2) LσM at T = 0 Given the above thermal analysis whih indiretly finds the T-independent relations f CL 90 MeV, f g = m q 325 MeV, λ = 2g 2, g = 2/ 3, m σ = 2m q 650 MeV, we first note that the latter σ mass is presumably f 0 ( ) as listed in the 1996,1998, 2000 PDG tables [15]. Aordingly, we now turn our attention to the SU(2) LσM field theory at T = 0. The original interating part of the SU(2) LσM lagrangian is [3] L int LσM = g ψ(σ +iγ 5 τ )ψ +g σ(σ ) λ(σ ) 2 /4, (14) g = m q /f, g = m 2 σ /2f = λf. (15) Refs. [3] work only in tree order in a spontaneous symmetry breaking (SSB) ontext, and do not speify g, g, λ exept that they obey the hiral relation 6

7 Eqs.(15). However, the dynamial symmetry breaking (DSB) version in loop order invoking the nonperturbative Nambu-type gap equations δf = f and δm q = m q was worked out in refs. [4] giving g = 2/ 3, g = 2gm q, λ = 8 2 / These gap equations are respetively (for d 4 p d 4 p/(2) 4 ) 1 = i4n g 2 (p 2 m 2 q) 2 d 4 p (16) 1 = 8iN g 2 /( m 2 σ ) (p 2 m 2 ) 1 d 4 p, (17) where the log-divergent gap Eq.(16) (LDGE) follows from the quark loop version of 0 A 3 µ 0 = if q µ and also is valid in the usual NJL model. Note that the UV utoff in the LDGE Eq.(16) is Λ 2.3m q 750 MeV, onsistently distinguishing the elementary (140), σ(650) from the bound states ρ(770), ω(780), a 1 (1260) as antiipated in Se.2 [4]. The quadrati divergent Eq.(17) is solved by using the utoff-independent dim.reg. lemma [4] for 2l = 4: [ d 4 m 2 ] q p (p 2 m 2 q) 1 = lim im2l 2 q 2 p 2 m 2 q (4) [Γ(2 l)+γ(1 l)] = im2 q 2 (4) (18) 2 beause Γ(2 l)+γ(1 l)] 1 as l 2 due to the gamma funtion identity Γ(z + 1) = zγ(z). Alternatively Eq.(18) follows by starting with the partial fration identity m 2 (p 2 m 2 ) 2 1 p 2 m 2 = 1 p 2 [ m 4 ] (p 2 m 2 ) 1, (19) 2 and integrating over d 4 p while negleting the massless tadpole d 4 p/p 2 = 0 (as is also done in dim. reg., analyti, zeta funtion, and Pauli-Villars regularizations [4]). Then the right-hand side of Eq.(18) immediately follows, so Eq.(18) is more general than dim. reg. Combining Eq.(18) with Eq.(17) and using the LDGE Eq.(16) gives m 2 σ = 2m2 q (1+g2 N /4 2 ), (20) whih when ombined with Eq.(12) implies m 2 σ/2m 2 q = 1+1 = 2, the famous NJL relation. Note that 1+1=2 is not a deliate partial anellation. Instead the loop order LσM neatly extends the tree order LσM relations Eqs.(15) to m σ = 2m q, when g = 2/ N. (21) 7

8 In fat the σ mass omputed from bubble plus tadpole graphs gives [4] [ m 2 σ = 16iN g 2 d 4 m 2 ] q p (p 2 m 1 2 q )2 (p 2 m 2 q ) = N g 2 m 2 q (22) 2 using the dim. reg. lemma Eq.(18). Then applying Eq.(12) redues Eq.(22) to the NJL relation Eq.(21). Further, the LDGE Eq.(16) nonperturbatively shrinks the u, d quark triangle for g σ and the quark box for g σσ, g to [4] g σ = 8ig 3 N m q (p 2 m 2 q ) 2 d 4 p = 2gm q = g (23) λ box = 8iN g 4 (p 2 m 2 q ) 2 d 4 p = 2g 2 = g /f = λ tree. (24) Then g σ = g when the GTR and also the NJL relation are valid. Also note that λ box = λ tree = 2g 2 in Eq.(24) was one of our major onlusions of Se.2 in Eq.(7). Ineffet, we have employed thegeneral olour number N when fittingphysial data, but on oasion (Eqs.(10-27)) we have used N = 3 (phenomenologially based on the 0 2γ deay rate). Also a theoretial basis for N = 3 is due to B. W. Lee s null tadpole ondition [16]. Speifially the true (DSB) vanishing vauum [not the false (SSB) nonvanishing vauum] is haraterized by the null sum of SU(2) LσM quark and meson tadpoles [16]. This leads to the CL equation [4] σ = 0 = i8n gm q (p 2 m 2 q ) 1 d 4 p+3ig (p 2 m 2 σ ) 1 d 4 p. (25) Using the LσM relations g = m q /f, g = m 2 σ /2f, multiplying through by f and invoking dimensional analysis to replae the respetive quadrati divergent integrals by m 2 q, m2 σ we see that Eq.(25) requires [4] N (2m q ) 4 = 3m 4 σ, (26) where the fator of 3 is due to σ σ σ ombinatoris. But sine we know this LσM-NJL sheme demands 2m q = m σ (as in Eqs.(21,23), this null tadpole ondition Eq.(26) demands [4] N = 3. There are alternative shemes [17] that suggest N, but they are not grounded on our SU(2) LσM-NJL ouplings whih require N = 3 when m σ = 2m q in Eq.(26). 8

9 4 Colour superondutivity Returning to the thermalization version (of the LσM) in Se.2, we extend this analysis to inlude a thermal hemial potential in order to omment on the µ µ region of Fig. 1. Following the first referene in ref.[9] we use exponential statistial mehanis fators [e (E±µ)/T + 1] 1 for fermions. Then, melting the quark ondensate gives a generalization of Eq.(11): T 2 + 3µ2 2 = 3m2 q 2. (27) Dividing Eq.(27) above by 3m 2 q /2 andusing Eq.(11) to formthe GTR,Eq.(27) beomes T 2 (2f ) 2 + µ2 m 2 q = 1, (28) a (hiral) ellipse in the T µ plane. From Eq.(28) by putting µ = 0 or T = 0 one an read off both the ritial temperature for the ase of a vanishing hemial potential (yielding T = 2f 180 MeV), or ritial hemial potential for a vanishing temperature (leading to µ = m q 325 MeV). Stated another way, we reall that the low temperature (T 2 o K) ondensed matter BCS equation [18] an be expressed analytially as 2 /T = 2e γ E , (29) where the Euler onstant is γ E = A link to partile physis is replaing the ondensed matter energy gap by the (onstituent) quark mass m q 325 MeV. Then the above BCS equation beomes for T = 2f : 2m q 2f = g = (30) (The numerial near-agreement between and is no aident. This is beause both equations require E p: Eq.(29) due to very low energy aoustial (as opposed to optial phonon energy p 2 /2m) phonons and Eq.(30) due to E 2 = p 2 +m 2 with m = 0 in the hiral limit). 9

10 The aoustial ee Cooper pair is now replaed by a QCD qq diquark in the superondutive region of Fig. 1, ie. for low T and µ µ. In general, the QCD energy gap is model dependent with superondutivity temperature T (SC) /(e γ E ) (31) ColoureddiquarkssuggestT (SC) < 180MeV [2]. Thisisoftheorderof10 12 o K. Reall that low temperature superondutivity ours at roughly 2 o K. 5 Conlusion In Se.2 we developed a hiral phase transition temperature T = 2f CL 180 MeV by independently melting the onstituent quark mass, the salar σ mass and the quark ondensate. In Se.3 we noted that the above thermalization proedure leads uniquely to the SU(2) quark-level LσM field theory at T = 0. Then in Se.4 we extended this piture (as shown in Fig. 1) to finite hemial potential. With hindsight our approah to a thermal hiral field theory based on the quark-level LσM is to first work at T = 0 as summarized in Se.3 and then proeed on to T = 2f CL as disussed in Se.2. The reverse approah to a realisti low energy theory has also been studied [19]. Speifially one starts at a hiral restoration temperature T (with m q (T ) = 0) involving only hiral bosons and σ [19] at T = T 200 MeV. Then only a λ 20 sale an generate a LσM field theory [20]. In our sheme, g 2/ and λ = 2g 2 = 8 2 /3 26.3, so it should not be surprising that T = 2f CL is near 200 MeV while λ = 8 2 /3 is near MeV We propose that the above quark-level SU(2) LσM field theory generates a self-onsistent thermal hiral phase transition with T 180 MeV when µ = 0, and µ 325 MeV when T = 0, and the superondutivity temperature is T (SC) e γ E / for a QCD qq energy gap. Coloured diquarks require T (SC) < 180 MeV [2], as Fig. 1 suggests. 10

11 Aknowledgments M. D. S. appreiates hospitality of the Institute of Nulear Physis in Kraków. Referenes [1] F. Karsh, Quark Matter Conf. 2001; Stony Brook, Jan 2001, hepph/ ,; F. Karsh et al., Nul. Phys. B605 (2001) 579. [2] See for example: M. A. Halasz et al., Phys. Rev. D58 (1998) ; V. Miransky et al., Phys. Rev. D62 (2000) [3] M. Gell-Mann and M. Levy, Nuovo Cimento 16 (1960) 705; V. de Alfaro, S. Fubini, G. Furlan and C. Rosetti, Currents in Hadron Physis (North Holland 1973) Chap. 5. [4] R. Delbourgo and M. D. Sadron, Mod. Phys. Lett. A10 (1995) 251; R. Delbourgo, A. Rawlinson and M. D. Sadron, ibid. A13 (1998) 1893; M. D. Sadron, Kyoto Sigma Workshop, June 2000, KEK proeedings , hep-ph/ [5] Y. Nambu and G. Jona-Lasinio, Phys. Rev. 122 (1961) 345. [6] L. Dolan, R. Jakiw, Phys. Rev. D9 (1974) 3320; J. I. Kapusta, Finite- Temperature Field Theory (Cambridge University Press, U.K. 1989). [7] S. A. Coon and M.D.Sadron, Phys. Rev. C23 (1981) 1150 and M. D. Sadron, Reports on Prog. Phys. 44 (1981) 213 find 1 f CL /f = m 2 /82 f , so f 93 MeV from data f CL 90 MeV. [8] N. Bili, J. Cleymans, M. D. Sadron, Int. J. Mod. Phys. A10 (1995) [9] D. Bailin, J. Cleymans, and M. D. Sadron, Phys. Rev. D31 (1985) 164; J. Cleymans, A. Koi, and M. D. Sadron, ibid. D39 (1989) 323; also see review by T. Hatsuda, Nul. Phys. A544 (1992)

12 [10] A. de Rujula, H. Georgi, and S. Glashow, Phys. Rev. D12 (1975) 147. [11] L. R. Babukhadia, V. Elias and M. D. Sadron, J. Phys. G23 (1997) [12] A. C. Mattingly and P. M. Stevenson, Phys. Rev. Lett. 69 (1992) [13] V. Elias and M. D. Sadron, Phys. Rev. Lett. 53 (1984) 1129; also see M. D. Sadron, Ann. Phys. (N.Y.) 148 (1983) 257. [14] A. Salam, Nuovo Cimento 25 (1962) 224; S. Weinberg, Phys. Rev. 130 (1963) 776; M. D. Sadron, ibid. D57 (1998) [15] Partile Data Group, D. E. Groom et al., Eur. Phys. Journ. C15 (2000) 1. [16] B. W. Lee, Chiral Dynamis (Gordon and Breah 1972) p.12. [17] K. Akama, Phys. Rev. Lett. 76 (1996) 184 and J. Zinn-Justin, Nul. Phys. B367 (1991) 105 onsider instead a four-quark U(1) L U(1) R simple NJL model finding λ = g 2 and m σ 2m q but only when N. This does not orrespond to the SU(2) quark-level LσM requiring λ = 2g 2 and m σ = 2m q when N = 3. [18] J. Bardeen, L. Cooper, and J. Shrieffer, Phys. Rev. 106 (1957) 162; also see B. A. Green and M. D. Sadron, Physia B (2001) in press. [19] K. Rajagopal and F. Wilzek, Nul. Phys. B404 (1993) 577; M. Asakawa, Z. Huang and X. N. Wang, Phys. Rev. Lett. 74 (1995) 3126; J. Randrup, ibid. 77 (1996) [20] L. P. Csernai and I.N.Mishustin, Phys. Rev. Lett. 74 (1995)

13 Fig. 1. Shemati phase diagram T [MeV] QGP T 3 Hadroni Matter SC T 0 Nulear Matter µ µ 13