On equations for the multi-quark bound states in Nambu Jona-Lasinio model R.G. Jafarov

Transcription

1 On equations for the multi-quark bound states in Nambu Jona-Lasinio model R.G. Jafarov Institute for Physical Problems Baku tate University Baku, Azerbaijan

2 INTRODUCTION Multi-particle equations are a traditional basis for a quantum-field description of bound states in particle physics. A well-known eample of these equations is the two-particle Bethe-alpeter equation (BE) for the two-particle amplitude and for the twoparticle bound state: Bethe H. and alpeter E.E.: Phys.Rev. 8 (95) 309 The multi-particle (three or more particle) generalization of BE have been also studied. A straightforward generalization of twoparticle BE has been intensively studied in sities-seventies of last century. A best eposition of these studies can be found in the work of Huang and Weldon: Huang K. And Weldon H.A.: Phys.Rev D (975) 57 An essential imperfection of the original Bethe-alpeter approach to multi-particle equations was a full disconnection of the approach with the field-theoretical equations for Green functions (which are known as chwinger-dyson equations).

3 This imperfection has been eliminated by Dahmen and Jona- Lasinio, which had included the BE to the field-theoretical Lagrangian formalism with the consideration of functional Legendre transformation with respect to bilocal source of fields: Dahmen H.D. and Jona-Lasinio G.: Nuovo Cim. 5A (967) 807 Then this approach has been generalized for multi-particle equations with consideration of Legendre transformation with respect to multi-local sources: Rochev V.E.: Theor.Mat.Fiz. 5 (98) However, these theoretical constructions had not solved the principial dynamical problem of quantitative description of real bound states ( nucleons, mesons etc.). A solution of BE-type equations has been founded as a very complicated mathematical tool even for simple dynamical model. There is a main reason of a comparatively small popularity of the method of multi-particle BE- type equations among the theorists.

4 Much more popular approach to the problem of hadronization in QCD is based on the t Hooft s conjecture that QCD can be regarded as an effective theory of mesons and glueballs: t Hooft G.: Nucl. Phys. B74 (974) 46 ubsequently, it was shown by Witten that the baryons could be viewed as the solitons of the meson theory: Witten E.: Nucl. Phys. B60 (979) 57 Further development of these ideas has been successful and has leaded to the prediction of pentaquark states in baryon spectrum: Diakonov D., Petrov V. and Polyakov M.: Z.Phys, A359 (997) 305. (It should be noted that the idea of pentaquarks was put forward for the first time in: Zeldovich Ya. B. and akharov A.D.: Yadernaya Fizika, 4 (966) 395.) Nevertheless, the investigations of multi-quark equations are of significant interest due to the much less model assumptions in this approach in comparison with the chiral-soliton models.

5 The solutions of multi-quark equations will provide us almost ehaustive information about the structure of hadrons. There is the basic motivation of present work. We shall investigate Nambu - Jona-Lasinio (NJL) model with quark content which is one of the most successful effective models of QCD in the non-perturbative region. ( for review see: Vogl U. and Weise W.: Prog. Part. and Nucl. Phys. 7 (99) 95 Klevansky.P.: Rev. Mod. Phys. 64 (99) 649 ) In overwhelming majority of the investigations, the NJL model has been considered in the mean-field approimation or in the leading order of /n-epansion. However, a number of perspective physical applications of NJL model is connected with multi-quark functions ( for eample: meson decays, pion-pion scattering, baryons, pentaquarks etc.). These multi-quark functions arise in higher orders of the mean-field epansion (MFE) for NJL model.

6 In present report we review some preliminary results of investigation of higher orders of MFE for NJL model. We have used an iteration scheme of solution of chwinger-dyson equation with fermion bilocal source, which has been developed in work: Rochev V.E.: J. Phys. A: Math. Gen. 33 (000) 7379 We have considered equations for Green functions of NJL model in MFE up to third order. The leading approimation and the first order of MFE maintain equations for the quark propagator and the two-particle function and also the first-order correction to the quark propagator. A consideration of these equations is the usual field of investigations of NJL model. The second order of MFE maintains the equations for four-particle and three-particle functions, and the third order maintains the equations for siparticle and five-particle functions. Here we discuss first results of investigation of the second-order equations for four-particle and three-particle Green functions.

7 Mean-field epansion in bilocal-source formalism We consider NJL model with the Lagrangian ˆ g L = ψi ψ + + [( ) ( ) ] a ψψ ψiγ τ ψ 5. A generating functional of Green functions ( vacuum epectation values of T-products of fields) can be represented as the functional integral with bilocal source: G ( η) = D( ψ, ψ ) epi dl ddyψ ( y) η( y, ) ψ ( ) Here η ( y, ) [ ] is the bilocal source of the quark field..

8 The n-th functional derivative of G over source (n-point) Green function: δη δ n G ( y, ) δη( y ) n, n η η=0 = is the n-particle { ψ ( ) ψ ( y ) ψ ( ) ( )} n ψ y 0 0T n y n y n.

9 Jafarov R.G. and Rochev V.E.: Centr. Eur. J. Phys. (004) 367 Translational invariance of the functional measure gives us the functional-differential chwinger-dyson equation for the generating functional G: δ ( y) G + i ) δη δg ( y, ) + + δ ig δη tr δg γ 5 tr γ δg ( y, ) δη(, ) δη( y, ) δη(, ) δ 5 = = d η (, ) δg δη ( y, ). We shall solve this equation employing the method of work:

10 A leading approimation is the functional The leading approimation generates the linear iteration scheme: Functional G G ( ) ( ) ( 0) 0 = eptr η ( 0 ) + G( ) + G ( ) +, = G n G ( n ) = P( n) G( 0 ), where ( ) is a polynomial of n-th order over the bilocal source : P n P ( n) ( n) G = ( n) n n η + n η ( n)! ( n )! ( n) n η + ( n) ( n) η + η.

11 The unique connected Green function of the leading ( 0 ) approimation is the quark propagator. Other connected Green functions appear in the following iteration steps. The quark propagator in the chiral limit is p ˆ = m p ( 0 ) ( ) ( ) where mis the dynamical quark mass, which is a solution of gap equation: = 8ign c dp p 4 ( ) π m The iteration-scheme equations give us the equations for firstorder two-particle function and for propagator (R.G. Jafarov and V.E. Rochev: Centr. Eur. J. of Phys. (004) 367.) :

12 () = + ( ) Two-quark function equation () = i ( ) + i () The equation of quark propagator

13 Here the graphical notation are used: = i ( 0) = τ τ ig γ γ 5 5 ( k ) n ( k ) = n

14 To describe the solution of the first-order equation for two-particle function and for future purposes we introduce the composite meson propagators by following way: a) Let us define scalar-scalar function () ( ) tr ψψ ( ) ψψ ( ) σ From the equation for two-particle function we obtain (in momentum space) σ ig ( p) = ( iδ ( p) ) Here we define the following function, which we call meson propagator: Z ( ) ( p) p = ( ) ( p ) Δ σ 4m p I 0 4m where Z ( p) = and I ( p) = 0 dq [ m ( p q) ]( m q ). I 0 σ ~ σ

15 b) Pseudoscalar-pseudoscalar function is defined as a b a b ab ( () τ τ ) tr γ γ π 5 ψγ ψ ( ) ψγ ψ ( ) τ τ From the equation for two-particle function we obtain (in momentum space) ab δ Δ π π ig ab ab ( p) = ( i ( p) ) Here we define the pion propagator. Δ ab π ( p ) = δ ab Z p π ( p ), ( p) Z = π I I 0 ( 0) ( ). p 0

16 econd order equations econd-order generating functional is G = Tr η 4 3 4! 3! 0 ( ) ( ) 4 ( ) 3 ( ) ( ) ( ( η ) + Tr( η ) + Tr( η ) + Tr( ) G ). The equations for four-quark and three-quark functions are ( ) 4 3 ( ) = + ( ) 4 Four particle functions equation

17 ( ) 3 i = ( ) i () + + ( ) ( ) 4 3 Three particle functions equation i

18 The equation for the four-quark function has a simple eact solution which is the product of first-order two-quark functions: ( ) ( ) 4 = 3 ( ) As it seen from this solution, the ππ scattering in NJL model is suppressed, i.e. in the second order of MFE this scattering is absent, and it can be arise in the third order only.

19 Verte σππ The eistence of above eact solution for the four-quark function gives us a possibility to obtain a closed equation for the threequark function. As a first step in an investigation of this rather complicated equation we shall solve a problem of definition of σππ verte with composite sigma-meson and pions. Let us introduce a function: W a τ b τ ab ( ) ( ) tr γ γ σππ ψψ ( ) ψγ ψ ( ) ψγ ψ ( ) 5 5 a τ b τ and define:

20 a) scalar verte (here, is the triangle diagram). b) pseudoscalar verte (here ). ( ) ( ) ( ) ( ) ( ) () 0 d v in V c Δ = σ σ tr ( ) ( ) ( ) ( ) ( ) v s = tr ( ) ( ) ( ) ( ) ( ) ( ) d v in V ab P c b a ab Δ = π π τ γ τ γ tr ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) v P = γ γ tr

21 With these definitions we obtain from the second-order equations ab for W σππ, which can be easy solved in the momentum space. ab The connected part of W σππ is an amplitude of decay σ π. It has a following form: n i ab con c aa ab [ W ( pp p )] = Δ ( p) [ v ( pp p ) + v ( pp p )] Δ ( p ) Δ ( p ) σππ or, graphically: σ P P π π γ 5 π p c W σππ = p σ p γ 5 π

22 Third order The third-order generating functional is G 6! ( 3 ) ( 3) 6 ( 3) 5 ( 3) 4 [ η] = Tr( η ) + Tr( η ) + Tr( η ) 6 5! 5 4! 4 + ( 3) 3 ( 3) + Tr( η ) + Tr η 3 3! The equation for si-quark functions is ( 3) ( 0 ) ( η ) + Tr( ) G ( 3) 6 5 ( ) 4 ( 3) 6 = +

23 And equation for five-quark function is finded as following ( 3) 5 ( ) 4 = ( ) 3 i ( ) 6 6 ( 3) 5 The equations for the si-quark function and for the five-quark function in our iteration scheme are new, and the equations for four-quark function ( 3) ( 3) 4, three-quark function, two-quark ( 3 3) function and quark propagator ( 3) have the same form as the second-order equations ecept of the inhomogeneous term, which contains the si-quark function and the five-quark function.

24 I Conclusion IN conclusion we list some possible physical application of the multi-quark equations.. Vector meson decays (in generalized NJL model).. Nucleons as the bound states in three-quark function (?) ( ) 3

25 3. Pentaquarks as the bound states in five-quark function (??) ( 3) 5 4. ππ scattering in four-quark function of third order. ( 3) 4

26 Appendi c W σππ W con σππ ( pp p ) = iδ ( p) [ v ( pp p ) + v ( pp p )] Δ ( p ) Δ ( p ) σ P Lets result of calculation of the triangular diagram. p ~ ( 0 ( ) ) ( 0 ( ) ) ( pp p dqtr p + q γ ( p + q) γ ) ( q). P. [ ] 0 v P 5 5 = Here - 4-momenta of sigma-meson, а p, p - 4-momenta of pions. v P = m I p + pp I p, p [ ( ) ( ) ( )] 4 p 0 3 π π

27 here I 0 is two loop integral, which we calculated in different regularizations (in dimensional-analytical regularization and in 4-cutoff regularization ), and I 3 ( p, p ) The converging integral, which on a mass shell is. dq~ ( m ( p + q) )( m q ) m ( p + q) = ( pp ) = ( p p ) p = 4m, p = p = 0, = m I ms 3 = 56m Finally we have the result for decay width A = 8g σ ππ g g σqq πqq 3 ν p ( ), Γ σ ππ m 700MeV σ