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Meson Lagrangians of the U (3) Group in the Model with Four-Quark Interactions

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Meson Lagrangians of the U (3) Group in the Model with Four-Quark Interactions v ' /, •' .' * объединенный ИНСТИТУТ ядерных исследований дубна Е2-83-19 D.V.Kreopalov, M.K.Yolkov MESON LAGRANGIANS OF THE U(3) GROUP IN THE MODEL WITH FOUR-QUARK INTERACTIONS Submitted to "ТМФ" 1983 1. INTRODUCTION In ref.' ' it is shown that one can construct a sigma-model for the pion interaction, a Yang-Mills Lagrangian for vector meson interaction, and a Lagrangian for mixed interactions of pions with vector mesons on the basis of a simple effective Lagrangian with four-quark interactions of scalar, pseudoscalar, and vector types. The relevant composite-hadron Lagrangians are derived in a straight-forward way by using the technique of path-integrals over collective fields. It is also shown that for electromagnetic interactions of quarks with photons this approach automatically leads to the well-known vector dominance model. Our model is a further development of ideas expounded in refs/2-4'. Let us present in brief main points of this approach. All mesons are considered to be composite two-quark systems. Interaction between mesons is mediated by quark loops. The same concerns the interaction of mesons with photons. Phenomenological Lagrangians are constructed only within one-loop approximation with divergent loops. The divergences in our model are removed by the renormalization of the meson fields. These renormaliza- tions completely define the strength of meson interaction, phe­ nomenological meson vertices. Renormalization of meson fields is determined by kinetic terms arising from loop diagrams with two meson legs. All meson vertices for strong interaction can be expressed via one vertex specific for the vector dominance model.. This is the vertex g which describes the decay p -»2я (gp/Ф7 ~ 3). it turns out that the strong vertex describing the interaction of pions with each other and with quarks, g. is con­ 2 nected withgp:g =V6g and equals g /4rr = 1/2. Thus, in the sec­ tor for pseudoscalar and scalar mesons there appear arguments for the use of the usual perturbation theory even for the desc­ ription of strong interactions *• In this paper the model proposed in is generalized to the group U(3). Here we shall describe strong interactions of sca­ lar, pseudoscalar„ vector, and pseudovector meson nonets. The main point is the construction of phenomenological Lagrangians in :-?hich all interaction constants are again connected with each other uniquely and are expressed via the constant g (or g ). *The same value for the constant g is obtained in ref. 1 At the same time we discuss mass formulae for the above par­ ticles. However, in deriving mass formulae it is not possible to get such unambiguous results as for the coupling constants of meson strong interactions because of a considerably greater number of arbitrary parameters. Indeed, if coupling constants of strong interactions of mesons are completely determined by renormalizations of meson fields, the definition of meson mas­ ses includes also parameters G; of the strength of effective four-quark interactions and some other parameters (see'1/ and a further text of the present paper). Thus, we pretend to quantitative results only for the pheno- menological interaction constants of meson fields. As for the mass formulae of 36 sorts of mesons, only qualitative results can be expected. We note that the Lagrangians obtained allow a satisfactory description of numerous decays of the mesons and of many intrin­ sic characteristics of mesons. In the following section an effective Lagrangian will be derived for strong interactions of scalar, pseudoscalar, vector, and pseudovector nonets of mesons. In the third section a ge­ neralized sigma model is found. In the fourth section we dis­ cuss mass formulae for scalar and pseudoscalar mesons. In the fifth section we describe vector mesons and their interactions with scalar and pseudoscalar particles. Pseudovector particles and their interactions with other fields are discussed in sec­ tion 6. The paper is concluded with a brief discussion of the results obtained. 2. EFFECTIVE LAGRANGIANS Consider an effective quark Lagrangian corresponding to the group U(3) and describing four-quark interactions of scalar, pseudoscalar, vector, and pseudovector types 2 2 £ (q.q) = qli(9'-[m0 + A(\0 -\/2AB)]|q + —i[(qAQq) +(qiy5Aaq) ] - [ - — (qy/Qq) + (Ч)ЪУ,ЛЧ) i • We do not consider here tensor mesons though their inclu­ sion in our model is not difficult in principle. 2 Here q = (u, d,s) are quark fields. It is assumed that they have colour indices over which the summation runs, XQ are Gell-Mann matrices, 0^a< 8 , A0 = V |- I , Тг[Ла,Х^ = 2SaJg ,mo is the bare mass of u and d quarks (m{j=m^=m0 , rag = ro0+V6A). Following ref. we consider a generating functional and introduce boson fields and then we can write down W(n. Ч) = i/dqdqexpli[f (q,q)+i7q+^q]l = N (2) tn = rr fdcjdq П dtr d0 dV dAaexpH[£'(Q.q.<'.4i', V.A) + 7}q+7,qH, N a=0 where n £'(q.q>,tf,v.A) =qiia-[m0+A(\0-V2xe)] + xa^ +iy6Aa0a + (3) ln + ^V„+y.X A Jq M - -i-(a +<£*) + -—(V^+A*). (V =V v ). а а 'Ъ a a OQ a a OQ a a' a a p In formula (2) we may integrate over quark fields-, as a re­ sult, we arrive at an expression in the exponential, which al­ lows us to construct an effective potential for meson fields. Before doing this, we make some transformations for a-fields in order to obtain a Lagrangian f* in the form more convenient for further calculations. Mass terms ini?' can be absorbed by fields oJn and ajn, as a result, the part of £' containing quark fields acquires the completely chiral-symmetric form (massless quarks). However, it turns out that in the effective meson Lagrangian the fields aQ and ст8 will have nonzero vacuum expectation values. To make them zero requires additional shifts in those fields by parame­ ters a and b (spontaneous breaking of chiral symmetry). As a re­ sult, we arrive at the following change of variables in £': S a а П Л V m n 0~ = о - - f o- ^8-b=^ + V2A, (4) where parameters a and b are defined by the condition <а0>0 =<а8>0=0. (5) In terms of new fields the Lagrangian £' assumes the form 3 £' = q f id - M + A0a0 + A^ + ... + kf^ + А8ст~ + ^УъФа + VQ + y5AQ)l q - 2 (6) _4_((7in + ^2)+ ^_(v2+ A«). 2Q, « ^« 2G2 в а where mu M =| mu , mu = =—, m8 = — (7) m8 J V3 V3 -a\0-b\8=-M —mnI + JL (AQ - v^A8 ) . It is seen that as a result of the spontaneous symmetry break­ ing in the effective meson Lagrangian (that will be obtained below) the bare masses of quarks m^'^and ml8^.n (1) are changed to new masses mu and mB. Subsequent calculations show that the bare masses m0 and m(?). coincide in magnitude with the quark current masses whereas the values of new masses mu and ms are close to the values of masses of quarks of composite particles m For (constituent quarks, u~"T'" convenience of further cal­ culations one more transformation can be made for a-fields, after which all the propagators in quark loops will have equal masses: CT °a = o0 + -=., f8= 8 -b . (8) v2 Now the Lagrangian £' assumes the form £'=qlia-muI+Aa[aa + iy6<Aa+Va+y5Aa]lq - (9) 1 2 ln c : f_in . ,2 л . 1 ,„2 , A 2 ^ .(a +0«)+_l_(V + A' ). 2Q1 a ra 2G^ Q Q, Integrating over quark fields in (2) we arrive at W(4, n) = ~ Г П d<r d0 dV dA exp|i[£'>, ф, V, A) - N a a a a a QQJ 1 -4«i3-mu + Ae[ae +iy5^a+VQ +УБ А^Г ,] I . where £"(<*,<£, V, A) is the effective meson Lagrangian 2 2 £'4a,0,V.A)=-^(a*%<^+ ^(Va +A a) - 2G 2G, (H) -iTrlnll + + +V + A ]1 У"а ^Л a y6 e - id -mu 3. GENERALIZED SIGMA-MODEL Let us take now the part of Lagrangian (II) containing only scalar and pseudoscalar fields. All meson fields in (11) are connected with each other via quark loops. We consider here only divergent quark loops. Divergent integrals may be of two types: quadratically divergent Ix and logarithmically diver­ gent Ig: d4k d*k I / I« .-i- f (12) (2тг)4 m--k 2 ' (&rV (m2-k8)8 The integral It enters into the definition of masses of mesons and of masses of bare (current) quarks; whereas the integral I2, into expressions for coupling constants of mesons in the effec­ tive Lagrangian. Owing to this integral all meson vertices can be expressed in terms of one vertex describing the decay p ->2.n. After these preliminary remarks we proceed to derive the effective Lagrangian for interactions of scalar and pseudosca­ lar mesons. On the whole there are met four types of divergent diagrams (see Fig.l). ч ' V / a) b) c) d) Fig.l Their contributions to the Lagrangian £" are, respectively, a) -4V6 Iimu<70 . 2 2 2 b) 212[(^oa) +(d^af] +41^+0*) -8I2mu aa , c) 4IBmuTrC(aff + фф)] , 4 4 В 2 d) -1аТг[Ы +(0) + 4Ы (£) -2офо$], (13) where a=\a, rф~=\ ф .After the change of variables a a* a a *o ="o + ^fm« (14) the contribution of all this group of diagrams can be written down in a simple form 2 2 2 ЩШ^) +%Фа)*] +4(I1+m uI8Xa; +0 - (15) 4 А 2 2 -I2Tr[(ff') + (ф) + 4(<7') (£) -2о'фРф]. Returning to fields with zero vacuum expectation values (O'Q^OQ - a , <78 = ffg-b)and renormalizing the fields so as to obtain a correct coefficient of the kinetic terms =gff R ф Ф 41 Ц ~°a a ' а** ?' в=< в>~ - (16) we arrive finally at the following Lagrangian for the interac­ tion of scalar and pseudoscalar mesons'1' R R R 2 R 2 ^ * )=|[(^ ) +(aM* > ] + 8 R 8 R 2 R 2 R 2 + (_LL + m .)[(a„ -A) + (a --£-) + 2 (a ) +(* ) ] - u I2 ° g • g 1 ' " (17) 7 8 R 2 2 R 2 2 - -r-Ugo*+ Л + у 2 - mu 0- a) + <ga8 -V2 A -b) + g X(a l ) + g a ^] - 2QX 0 2 1 R 2 8 B 2 _ jiTr|[(J -м.) +tfVi -№ >" - w.)i i - 4 g R g -iTrlnll+ —- g(o+iy ф)\' .
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