DSpace at VNU: The CJT effective action approach applied to the SU(3) generalized NJL model

DSpace at VNU: The CJT effective action approach applied to the SU(3) generalized NJL model tài liệu, giáo án, bài giảng...

VNU JOURNAL OF SCIENCE, Mathematics - Physics T.XXII, Nq - 2006 T H E C J T E F F E C T I V E A C T IO N A P P R O A C H A P P L I E D T O T H E S U (3 ) G E N E R A L IZ E D N JL M O D E L P h a n H ong Lien, Nguyen N hu X uan Institute of Military Engineering Do T hi H ong Hai Ha not University of Mining and Geology Abstract: The sư(3) generalized Nambu - Jona - Lasinio (NJL) model is considered by means of the Cornwall - Jackiw - Tomboulis (CJT) effective action This method provides a very general framework for investigating many important non- pertubative effects: quark condensate and mesons, thermodynamical quantities at finite tempera­ ture It is shown that the mixing of flavors of u, d and s quarks exists in this formalism The gap equations, which are directly obtained from the effective potential involved the quark condensate IN T R O D U C T IO N Description of quark matter within the framework of non - pertubative theory turns out to be more crucial for relativistic quantum theoretical study of condensable matter At low energy (about lGeV) the non - pertubative effects concern with the confinement of quarks and the dynamical breaking of chiral symmetry In this respect, many authors constructed different symmetry conserving approximation schemes: the mean - field approximation [l]-[2], the "Ộ - derivable” method [3], the (second) random phase approximation (RPA) [4], an expansion in powers of the inverse number of colors [5], the one - loop approximation of the effective action [6] Ill these works, it is worth to mention that the CJT effective action method [7], which obviously includes the Schwinger - Dyson (SD) equation approach [8], may hopefully provide a promised approximation beyond two - loop calculations Its priority is expressed by the fact that the vacuum expectation values of field operators and propagators are treated on the same footing; therefore it takes into account all the possible correlation effects In addition to the preceding trend, one has made great attempts to investigate the role of chiral symmetry in condensate matter [1 ], [10] The Nambu - Jona - Lasinio (NJL) model originally was a model contained nucleons [9] Nowadays this model is used to study the properties of quarks instead of nucleons The SU(2) version of NJL has been applied by many authors to study the restoration of chiral symmetry at critical temperature and nonzero density [1], [11] However, the recent consideration [12], [13] indicate that the strange quark matter could be the absolute ground state of matter This leads to the SU(3) version of NJL model, which includes in addition to up and down quarks also strange quarks Our main aim is to present in detail the CJT effective action approach, which is applied to study systematically the SU(3) generalized NJL model In this connection, it is possible to consider our work as being complementary to [1 ] This paper is organized as follows In section 2, the chiral symmetry in SU(3) version of NJL model and CJT effective action formalism are presented Section is devoted to loop expansion of effective potential Hence SD equations and gap equations are directly derived In section the CJT effective potential is evaluated at T 7^ The conclusion and discussion are given in section P h a n H o n g L ien, N g u y e n N h u X u a n , D o T h i H o n g H 2 FO R M A LISM 1.1 C hiral sym m etry Let us consider the SU(3) generalized NJL model whose Lagrangian reads £ n (j7MaM- m ,)$ + ^ Y , [(*A“* ) + (* H 5Aatf )2] a=0 + GDdetf [$(1+ 75)*] 4-G odetf [$(1 - 75 )$] where 'I'(x) are quark fields (2.1) = u,d,s with three colors (Nc — 3) and three flavors (Nf = 3),Aa(a = -7- 8) axe the Gell-Mann matrices with A0 = \ J \ There are two chiral invariant coupling constant Gs and Go of four and six fermions interaction The current quark mass m q = diag(mUỉmci,ms) = ^ m a\ a (2 2) a=0,3,8 explicit breaks chiral symmetry, and the six - point vertex $ ( - )® = $ (2.3) leads to the mixing between singlet, octet and triplet The chiral transformation is defined by VE' (2.4a) A° * — W+((3)U+(a) = ỹexp - i (a 4-^75) (2.4b) Ý -V U(a)Us{0)V = exp i (' a - /375) where ^L(R) — u{a)^L{R) = exp “ a y J ^L(fi) (2-5) is the SUl (3) ® SU r (3) transformation in the three flavors, and the transformation -> Us{0)'& = exp f t y ^ / q y j 'ĩ' (2.6) is the Q5 transformation, which leads to the anomalous divergence of the flavor singlet axial current [1 ] (2 J) ômJ m= - N f G DIm fd e í/ ỉ> )+ im '7 5í ' (2.8) T h e C J T E ffectiv e A c tio n A p p ro a c h A p p lied to T h e S U (3) G e n e liz e d N J L M o d el Under SU i{3) ® SU r {3) transformation, the operators follows are transformed as $ - U(a)ư+(f3) (2.9a) s>+ -» U{P)+U+(a) (2.9b) It is evident that det/ $ and det/ $+ are invariant due det u = 1, but two last terms in (2 ) breaks the [/ ( ) symmetry q, _♦ e- " ( * ) ^ ; ^ _> q,ei6(:*) (2.10) The composite operators are defined by Sa = - ^ ^ A a^ ; mz P a = - - ^ * i j 5\ a* m2 (2 1 ) with Gs = g l /m The action corresponding to (2.1) now takes the form / [ * ¥ ] = I dz j * ( * ) ( i ^ - m 9) t f ( x ) - i m ] T ( s + p ) j ~ (2 ) A (Sa —2*75Aapa) ^ ■+■Godetf [*][£)*][£>S][DP] exp i: | y d x ị L + ^ * + f j ( x ) { x ) + ®(*)jj(s) + J sa {x) s a(x) + j ; { x ) V a{x) + ị j d x d y [ * ( x ) K ( x , y ) * ( y ) + S a(x)KZb( x , y ) S b(y) + P a( x ) K ? ( x , y ) P b( y ) ] \ (2.13) where f dx = Jq dr Ị dx, the summation here over repeat variable (indices) is assumed and external sources are time - independent; f dxty o^ = N is the number operator for u, d and s quarks The integration has to be performed over antiperiodic Grassman fields * ( 0,f ) = -* (/? ,£ ) and periodic bosonic fields B(0,£) = B (/?,£) P h a n H o n g L ien, N g u y e n N h u X u a n , D o T h i H o n g H where B = ($ ,S a, P a) The propagators of quarks, scalars and pseudoscalax meson axe determined from 52Wr Srj(x)ỏĩ](y) S2 Wr ỗ2W f = Dab(x,y): G(x,y); v ' an ỐJsa(x)ÓJ?(y) ' an ỖJ$(x)5J$(y) &ab(x,y) (2.14) We defined the mean values of field operators as follows SW p s T](x) SWft 6rj(x) (ý(x)) = ( * ( * ) ) = ¥>(*) (2.15) u s k “ ( S“(I)) = s - (l) {fw = ( P “, A transform of Wp ,S ,P ,G ,A ĩA ^ ^/3 Vì Vĩ JSi Jpi K ) V i V i J s i J p i K , K g , is defined as the double Legendre K p j dx ộ =diag(ộu,s SJ Sa = ( |S„|0) = (2 20) Pa = N r O > W * )] ~ ậ j h T r [Dab^ Dba^ + A af>(P)A ỉ»(fc) Two terms on first line of (3.2) correspond to the mean field approximation, three next terms are just one - loop approximation, and the last terms in expresses the non - perturbative interaction at two - loop and higher approximation The configuration of meson fields is determined from d V CJT dSa 9s ÍÍ d *p _ 9s X Tr [ơ(p)] m2 J (27r)‘ (3.3) It’s just the scalar density, which is invariant under Lorentz transformation (3.4) Substituting (3.2) into (2.22), we arrived at the SD equations G ~ \k ) = Gõ1k *]-£(*:) (3.5) D-b\ k ) = D - 1ab[ l - G s U s (k)] (3.6) ^ W (3.7) = A - l b[ l - G ps U%k)] P h a n H ong Lien, N g u y e n N h u X u a n , D o T h i H ong H a i where £(/c), IIs (k) and n£fe(/c) are, respectively, the self - energy of quarks, scalar and pseudoscalar mesons í É T r r ( k , p ) G( p ) r ( Pip + k)G(p + k) Z(k) = J