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E2-96-482
M.K. Volkov*
THE PSEUDOSCALAR AND VECTOR EXCITED MESONS IN THE U (3) * U (3) CHIRAL MODEL
Submitted to «#z%epHaa $H3HKa»
♦E-mail: [email protected]
1996 t* 1. Introduction
Investigation of the radial excitations of the light mesons is of great interest in hadronic physics. So far there are the questions connected with the exper imental and theoretical descriptions of the radial excitations of pseudoscalar mesons. For instance, the ir' meson with the mass (1300±100)MeV is usually identified as the first radial excitation of the pion [1]. However, indications of a light resonance in diffractive production of 3?r-states have lead to spec ulations that the mass of the it1 may be considerably lower, at ~ 750 MeV [2]. So far there are no experimental data concerning the excited states of the kaons [1]. In this paper we attempt to give the theoretical predictions for the masses and the weak decay constants of these excited mesons. A theoretical description of radially excited pions poses some interesting challenges. The physics of normal pions is completely governed by the sponta neous breaking of chiral symmetry. A convenient way to derive the properties of soft pions is by way of an effective Lagrangian based on a non-linear realiza tion of chiral symmetry [3]. When attempting to introduce higher resonances to extend the effective Lagrangian description to higher energies, one must ensure that the introduction of new degrees of freedom does not spoil the low-energy theorems for pions, which are universal "consequences of chiral symmetry. A useful guideline in the construction of effective meson Lagrangians is the Nambu-Jona-Lasinio (NJL) model, which describes the spontaneous breaking of chiral symmetry at quark level using a four-fermion interaction [4, 5, 6]. The bosonization of this model and the derivative expansion of the result ing fermion determinant reproduce the Lagrangian of the linear sigma model, which embodies the physics of soft pions as well as higher-derivative terms. With appropriate couplings the model allows to derive also a Lagrangian for vector and axial-vector mesons. This not only gives the correct structure of the terms of the Lagrangian as required by chiral symmetry, but also quan titative predictions for the coefficients, such as /,, /%, g r, g p, etc., which are in good agreement with phenomenology. One may therefore hope that a suit able generalization of the NJL-model may provide a means for deriving an effective Lagrangian including also the excited mesons. When extending the NJL model to describe radial excitations of mesons, one has to introduce non-local (finite-range) four-fermion interactions. Many non-local generalizations of the NJL model have been proposed, using ei ther covariant euclidean [7] or instantaneous (potential-type) [8,-9] effective quark interactions. These models generally require bilocal meson fields for bosonization, which makes it difficult to perform a consistent derivative ex
l pansion leading to an effective Lagrangian. A simple alternative is the use of separable quark interactions. There are a number of advantages of work ing with such a scheme. First, separable interactions can be bosonized by introducing local meson fields, just as the usual NJL-model. One can thus derive an effective meson Lagrangian directly in terms of local fields and their derivatives. Second, separable interactions allow one to introduce a limited number of excited states and only in a given channel. An interesting method for describing excited meson states in this approximation was proposed in [10]. Furthermore, the separable interaction can be defined in Minkowski space in a 3-dimensional (yet covariant) way, with form factors depending only on the part of the quark-antiquark relative momentum transverse to the meson momentum [9, 11, 12]. This is essential for a correct description of excited states, since it ensures the absence of spurious relative time excitations [13]. Finally, as we have shown [12], the form factors defining the separable inter action can be chosen so that the gap equation of the generalized NJL-model coincides with the one of the usual NJL-model, whose solution is a constant (momentum-independent) dynamical quark mass. Thus, in this approach it is possible to describe radially excited mesons above the usual NJL vacuum. Aside from the technical simplification the latter means that the separable generalization contains all the successful quantitative results of the usual NJL model. In our previous paper [12] the theoretical foundations for the choice of the pion-quark form factors in a simple extension of the NJL model to the nonlocal quark interaction were discussed. It was shown that we can choose these form factors such that the gap equation conserves the usual form and gives the solution with a constant constituent quark mass. The quark condensate also does not change after including the excited states in the model, because the tadpole connected with the excited scalar field is equal to zero (the quark loop with the one excited scalar vertex - vertex with form factor). Now we shall use these form factors for describing the first excited states of the pseudoscalar and vector meson nonets in the framework of the more realistic C7(3) * U(3) chiral model [4, 5, 6]. We shall take into account the connections of the scalar and vector coupling constants which have appeared in this model and the additional renormalization of the pseudoscalar fields connected with the pseudoscalar - axial vector transitions. For simplicity, we shall suppose that the masses of the up and down quarks are equal to each other and shall take into account only the mass difference between (up, down) and strange quarks (mu and ms). Then we have in this model the five basic parameters: mu, m8 , A3 (3-dimensional cut-off parameter), G\ and
2 G2 (the four-quark coupling constants for the scalar-pseudoscalar coupling (Gi) and for the vector - axial-vector coupling (Gg)). For the definition of these parameters we shall use the experimental values: the pion decay constant fr = 93MeV, the p-meson decay constant g p to 6.14 (|£ to 3), the pion mass M* to HOMeF, p-meson mass Mp = 770 MeV and the kaon mass M/v to 495MeV. Using these five parameters we can describe the masses of the four meson nonets (pseudoscalar, vector, scalar and axial-vector) 1 and all the meson coupling constants describing the strong interactions of the meson with each other and with the quarks. For the investigation of the excited states of the mesons it is necessary to consider the nonlocal four-quark interactions. We have shown that for description of the excited states of the pseudoscalar and vector meson nonets it is enough to use only three different form factors in the effective four- quark interactions. These form factors contain three arbitrary parameters and describe the excited states of the mesons consisting : 1) of the u and d quarks (7 ru/); 2) of the (u,d) and the strange quarks (K‘, K*')\ 3) of the strange quarks For the determination of these parameters we shall use the experimental values of the masses of the excited vector mesons p', K* and
1 The mass formulae for the axial-vector mesons and , especially, for the scalar mesons give only qualitative results (20 — 30% accuracy). 2Remind, one more additional parameter , connected with the gluon anomaly, was used in the usual NJL model, when we described the ground states of the 17 and rf mesons (1/(1) problem) [5]. The problem of the radial excitations of the light mesons, including the 17 and 17 ', in the framework of the potential model was discussed in works [14].
3 2. U(S) * U(3) chiral Lagrangian with the excited meson states
In the usual U(3) * U(3) NJL model a local (current-current) effective quark interaction is used L[q,q] = j d4xq(x) (i$ - m°) q(x) + Lint , (1)
Tint = j dAx^{jas{x)jas{x)+ jaP{x)jaP{x))
- y+ (2) where m° is the current quark mass matrix. We suppose that « m®. is,p,v ,,A(:r) denote, respectively, the scalar, pseudoscalar, vector and axial- vector currents of the quark field ([/(3)-flavor),
3s(x) = g(^)A°g(x), jp(x) = 9(2)175 A*g(x), = ?(x)7^A°9(x), jJ'V) = 9(x)7 57 flAa^(x). (3)
Here Aa are the Gell-Mann matrices, 0 < a < 8. The model can be bosonized in the standard way by representing, the 4-fermion interaction as a Gaussian functional integral over scalar, pseudoscalar,vector and axial-vector meson fields [4, 5, 6]. The effective meson Lagrangian, which is obtained by inte gration over the quark fields, is expressed in terms of local meson fields. By expanding the quark determinant in derivatives of the local meson fields one then derives the chiral meson Lagrangian. The Lagrangian (2) describes only ground-state mesons. To include excited states, one has to introduce effective quark interactions with a finite range. In general, such interactions require bilocal meson fields for bosonization [7, 9]. A possibility to avoid this complication is the use of a separable interaction, which is still of current-current form, eq.(2), but allows for non-local vertices (form factors) in the definition of the quark currents, eqs.(3),
A»t = / ^ Z [ir
-----7T +.?A,i(X)-?A1i(a0] (4)
d4x2 q(xi)Fs i(x\xuX2)q(x2), (5)
4 3p,Ax) = J j d4x2 q(xi)Fpi(x] xux2)q(x2)y (6)
i^(x) = J (^xi J d4x2q(xi)Fyf(x]xuX2)q(x2), (7)
= J d4x{ J d4x2q(x])F^(x;xl,x2)q(x2). (8)
Here, F^'f(x; X\, x2), 1 = 1,...TV, denote a set of non-local scalar, pseu doscalar, vector and axial-vector quark vertices (in general momentum and spin-dependent), which will be specified below. Upon bosonization we obtain £bos( + j d4x (v?(x)F2Ax’, *1^2) + +V;a,%T)F£f(x; * 1, X2) + Aai'tt{x)Fa/‘{x\ 3:1. x2))]g(.r2) - / ^ E + /T'w)] (9) i= 1 This Lagrangian describes a system of local meson fields, a®(.r), 0?(x), V'-a,/'(.r), A“’p (x), t = 1,... TV, which interact with the quarks through non-local ver tices. These fields are not yet to be associated with physical particles , which will be obtained after determining the vacuum and diagonalizing the effective meson Lagrangian. In order to describe the first radial excitations of mesons (N = 2), we take the form factors in the form (see [12] ) C"2(k) = A*/.(k),; c;2(k) = «TsA«/.(k), F“£(k) = Y-A"/,,(k),; F3;5(k) = 7 57"A"/«(k). (10) /„(k) = c„(l +d„k2). (11) We consider here the form factors in the momentum space and in the rest frame of the mesons (PmeJ(>n = 0. k and P are the relative and total momentum of the quark-antiquark pair.). For the ground states of the mesons the functions Z?(k) = 1. After integrating over the quark fields in eq.(9), one obtains the effective Lagrangian of the (T],cr2, 5 ^r(aa + + al + 4>i) + 2(f2^a + A2a + V* + A2) 2G -z'M Tr logjz# - m° + (^ + i75 °a - a- < °'a >0 (14) and redefine the quark masses ma = mj— < a'a> . (15) Then eq. (13) can be rewritten in the form of the usual gap equation m, = mj + 8Gim,7i(m, ), (i = u, d, s) (16) where r d4ki 1 = -iN. (17) a3 (2tt)4 (m? — k2)n and m, are the constituent quark masses. In order to obtain the correct coefficients of kinetic terms of the mesons in the one-quark-loop approximation, we have to make the renormalization of the meson fields in eq. (12) <7 .=flX> 4>a = fM'„ v? = g°vvr, a; = g°vAY, (18) where C dz/44 kj = [4/2(mi,mj)]_», = / — /19) J aA,(2ir)3 v/71 " 4(m? - k2)(mj - k1) 9v V6g" (20) After taking into account the pseudoscalar - axial-vector transitions ( 9$ — 2Sul (21) 6 where Zr — \ ^ « 0.7 for pions. (Mai = 1.23GeV is the mass of the axial-vector a\ meson, [1], mu = 280 MeV [5] ). We shall assume that all Za « Zv « 0.7. After these renormalizations the part of the Lagrangian (12) describing the ground states of mesons takes the form 1 ' ,a2„2 , „a2j.2\ , n9ya2 ' iNc Tr log iQ - m° + (g>„ + + “(t/.K'' + 7s7#A")) A” (22) For simplicity we omitted here the index r of the meson fields. From the Lagrangian (22) in the one-loop approximation the following expressions for the meson masses are obtained [5] 1 ^ 8/i(m u) sil = (23) |G, G\ n%u 4 Z/2(mu,mu)’ Mk — g\ —----4(/i(mu) -f Ii(ms)) + Z l(ms - mu)2, vi 1 (24) 9k = 4Z/2(mti,ms)’ mi = -h- = (25) iGi 8g2h(m„,Tn„)' (26) M\. = p h{mu, m8) Now let us fix our basic parameters. For that we shall use the five experimental values [4, 5]: 1) The pion decay constant /, = 93MeV . 2) The p-meson decay constant g p « 6.14. Then from the Goldberger-Treimann identity we obtain mu — fug* (28) and from eqs. (20) and (21) we get 9p f*9p 9r = mu — = 280MeV. (29) V6Z' Vez 7 From eqs. (19) and (20) we can obtain (see [15]) /2(mu,mu) = A3 = l.OSGey. (30) lg p 3) Mw « 140MeV. The eq. (23) gives G, = 3.48GeF~ 2 (see [15]). 4) Mp = 770 MeV. The eq. (25) gives G2 = 16GeV~ 2. 5) Mk ~ 495AfeV. The eq. (24) gives ms = 460MeV. After that the masses of r),rf and K*, M\.. = M\ . + 6m,m,, (31) . = Ml. + 4m,m,. (32) We can calculate the values of all the coupling constants, describing the strong interactions of the scalar, pseudoscalar, vector and axial-vector mesons with each other and with the quarks, and describe all the main decays of these mesons (see [5]). 3. The effective Lagrangian for the ground and excited states of the pions and kaons To describe the first excited states of the all meson nonets, it is enough to use only three different form factors /Q(k) (see eq. (11)) /uu(k) = cutl(l T duuk ), f«Jk) = c„,(l + d„k2), fss(k) = c„(l + d„k2). (33) Following our work [12] we can fix the parameters duu, du8 and dsa by using •the conditions /i/uu(mu) = 0, l{u‘(mu) + l{u,(m8) = 0, l{"(ms) = 0, (34) where (35) The eqs. (34) allows us to conserve the gap equations in the form usual for the NJL model (see eqs. (16)), because the tadpoles with the excited 3To calculate the masses of the 17 and rf mesons, it is necessary to take into account the gluon anomaly [5]. 8 scalar external fields do not contribute to the quark condensates and to the constituent quark masses. Using eqs. (34) we obtain for all da close values duu = -1.78 GeV~\ dus = -1.75 GeV~2, dss = -1.72 GeV^. (36) Now let us consider the free part of the Lagrangian (12). For the pseu doscalar meson we obtain A,2,( «J=1a—0 Here = (*?) 2 + 2^+^r, (ti? + = 2 A? AT. (*!?'+ (e!)1 = 2KfAf, W?)2 = W)2, ( 4>)1 arid ® arc the components of the r] and rf mesons. The quadratic form K?j(P), eq.(37), is obtained as K$(P) = (39) A'“ (P) = - ;tr [ 1 1 - i N, ./A;, (2%)^ ^ + 2"^ - 77l 1 ' $ ~\tf ~ mq> f\ = 1, /2“ = /a(k). (40) m. mu (a = 0,..., 7); m® = ms; m°, = mu (a = 0, ...,3); m“, = ms (a = 4,...,8). (41) mu and ms arc the constituent quark masses (mu «m Mf = (Zf-'P " 4(/(/“(»»;) + /(%",")] + A2»„(|l=1. .. (45) U| 9 Here, //“ and I^a denote the usual loop integrals arising in the momentum expansion of the NJL quark determinant, but now with zero, one or two factors /a(k), eqs.(33), in the numerator (see (35) and below ) = -iNc (2)r)4 Lfc2)' <46) The evaluation of these integrals with a 3-momentum cutoff is described e.g. in ref. [15]. The integral over ko is taken by contour integration, and the remaining 3-dimensional integral is regularized by the cutoff. Only the di vergent. parts are kept; all finite parts are dropped. We point out that the momentum expansion of the quark loop integrals, eq.(40), is an essential part of this approach. The NJL-model is understood here as a model only for the lowest coefficients of the momentum expansion of the quark loop, not its full momentum dependence (singularities etc.). Z is the additional renormaliza tion of the ground pseudoscalar meson states taking into account the a —> Aa transitions (see eq.(21)). After the renormalization of the meson fields c = yw# (47) the part of the Lagrangian (37), describing the pions and kaons, takes the form ti2) = l [(P2 - Ml) tt? + 2r„P2 + (P2 - Ml) 4] , (48) 42) = \[{P2 - Ml - A2) /if? + (P2 - Ml - A2) K\ + 2Fk(P2 - A2) K,K2\. (49) Here 7« ° r„ = 4 (50) VzfZ: 1 m: M; = (4Z/2(mu,mu)) l[— - 8/i(m u)] = 4ZmJj(ra„ra,)’ M.L = (4/2//(m«.m«)) ~ 8i/ Z(mu)], (51) 1 M = (4Z/2(m„,m,))-,|— - 4(J,(m.) + /,(«.,))] + (Z"‘ - 1)A2 C»1 m° _|_ m° m m. + (Z-1 - 1)A2, 4Z/2(m„,m,) Ml, = (4/2/Z(m„,m„)) '[— - i{l(f[m„) + /f/(m«))]. (52) 10 After the transformations of the meson fields +-^=(yfl + Tasin 6 a - y/l - Ta cosQa) - ra sm60 + v/TTr: cosOa) 42> = 1(P2 - M,2) X2 + ±(P2 - M2) tt'2, (54) L™ = i(P2 - M%) K2 + 1(P2 - M£,) jf'2. (55) Here M(^) = 2(rrTf)[<+< (-,+) \/(M2 - M?,)2 + (2M„M,,r,)2], (56) Mi(K = 2(1 - r|.'j + + 2^2(1 - r*) (-,+) - Af^)2 + (2MKlMK,VKn (57) and K, - tan 28a — \ / ^ — 1 (58) K + Mki In the chiral limit we obtain: Af„, = 0, M,a ^ 0 (see eqs. (51)) and Af2 = < + 0«), (59) M2 = + °«)' (60) Thus, in the chiral limit the effective Lagrangian eq.(48) describes a massless Goldstone pion, 7r, and a massive particle, n'. We obtained similar results for the kaons. For the weak decay constants of the pions and kaons we obtain (see [12]) /, = mu^/2ZI2{mu, mu) (sin9wy/l + T* + cos8xy/l - F*), = muy/2Zl2(muymu) (sin9x\/1 ~Tr - cos8wy/1 + F,), (61) li fK = mu + ms y/Zl2{rnu,ms) (sinOKy/l + IV + cosOKy/1 - F%), v2 fK, = + («n0* 0 -IV - coa^Vl + H,). (62) v 2 In the chiral limit we have 1 + ra 1 -ra sm#a = cos6 a = (63) and /, = =*, /K = (^i), /„ = 0, fK, = 0. (64) 9* 2,9 k Here we used eqs.(19) and (21). Therefore, in the chiral limit we obtain the Goldberger-Treimann identities for the coupling constants g* and qk- The matrix elements of the divergences of the axial currents between meson states and the vacuum equal (PCAC relations) (Old-a; I/) = m2/^“\ (65) (OldMy 6) = m2,/*,#"6. (66) Then from eqs. (59) and (64) we can see that these axial currents are conserved in the chiral limit, because their divergences equal zero, according to the low- energy theorems. 4. The effective Lagrangian for the ground and excited states of the vector mesons The free part of the effective Lagrangian (12) describing the ground and ex cited states of the vector mesons has the form Lm(V) = -5 E E vr(P)RT(p)vi°(p), (67) i,j=l o=0 where E VT = K)2 + (pY? + 2 pYpY, (v>) 2 + (Vf? = 2 o=0 (v^f + (v;7 ")2 = 2K''h,k?’‘, (v;8 ")2 = «)2 (68) and KiT(p ) = - 6f/1' 12 d4fc l — i Nc tr 77"/; / (2tt)4 J a3 ft + ~ mq ' £ ~ \tf ~ mq> ft = 1, /2° = /a(k). (69) To order P2, one obtains R^a = - P^P" - g fW(M^)\ R%a = W^[P2g^ - PiLPu - Rfia = J%r = fW" - - ^A2g'“'<5a6|1=4..7 j. (70) Here W? = ^2, ^ = f = (71) (MH" = (^G2)-' + ^A^|6=,T, (72) + -A^|6=/, .7. (73) After renormalization of the meson fields V/“r = sfwf vr . (74) we obtain the Lagrangians 42) = -Mts'"^2 - e-p- - cfuMl)rtp\ + 2T„(g> “’P7 - P>‘PnM + (g'^P2 - P^P" - (2) = -3Ka^P2 - P'P" - + 2r*( r (2) = -^[(g"*' P2 - P"P" - g""(-A2 + Mj.))A'*' ‘A-; + 2IV (g""P2 - P''P" - g""^A2)A;"A;" + ( M » *2 _ v " 8C2/2(m„, m„)’ A 1 8Gi/'j(m„, m,) ’ = 3 «« = 3 Ml = *' 8G.Mm„m,)’ « 8G 2/2"(m»,”'»)’ Af2 ^ A/2 ^ (78) 13 /2/a(m,-, rrij) r„., = (79) \fmmi, mJ)/2//a(mt, m}) After trasformations of the vpctor meson fields, similar to eqs. (53) for the pseudoscalar mesons, the Lagrangians (75,76,77) take the diagonal form 42.’y. = - j [( +(g'"'P2 - FT" - M$.)V'»‘V‘“'], (80) where Va and Va are the physical ground and excited states vector mesons 1 K + K (- +) \J( Ml -M2)2 + (2M„MP2r,P i _ r2 = M w,w? (81) 1 K + Mj (-,+) + (2Af^,Af^r*) 2l , (82) i-4 4>i mJ. + mJ. + 3a2(i - r2,.) — 1 p2 1 - 1 K. (-,+) ^(M2. - M2.)2 + (2MJf;WK;r^.): (83) To describe the excited states of the vector mesons, we shall use the same form factors fa like in the case of the pseudoscalar mesons. Therefore, we can fix the parameters cuu, cu, and c88 using the experimental values of the masses of the vector meson excited states and then make the predictions for the masses of the excited states of the pseudoscalar mesons and vice versa. We shall here the first version. 5. Numerical estimations We can now estimate numerically the masses of the pseudoscalar and vector mesons and the weak decay constants /,, /,/, fx and /#, in our model. Because the masses formulae and others equations ( for instance, Goldber- ger Treimann identity and st) on) have new forms in the NJL model with the excited states of mesons as compared with the usual NJL model, where the excited states of mesons were ignored, the values of basic parameters of this model (mu, ms, A3, C?i, G-i) can change. However, we see that one can use the former values of the parameters A3 = 1.03 GeV and G\ — 3.48 GeV~2, because eqs. (23) and (30) change only slightly after including the excited states of mesons. For the quark masses we shall use the values mu = 285 MeV and m, = 470 MeV, which are also very close to the former values. For the coupling constant G? the new value G2 = 13.1 GeV"2 will be used, which more noticeably differs from the former value G2 = 16 GeV~2 (see section 2). It is a consequence of the fact that the mass MPl noticeably differs from the physical mass of the ground state p - Mp (see eqs. (78) and (81)). Using these basic parameters and the internal form factor parameter duu = — 1.78 GeV~2 (see eq. (36)) and choosing the external form factor parameter cuu = 1.45, one finds Mp = 770 MeV, Mp = 1.5 GeV, Mr = 137 MeV, Mr' = 1.3 GeV. (84) F, = 0.647, Tp = 0.54. (85) r, = \ZZFt ( see eqs. (50) and (79)). The experimental values are equal to Mepxp = 769.9 ± 0.8 MeV, Mtxp = 1465 ± 25 MeV, Mtxp = 139.57 MeV, Mexp = 134.98 MeV, M*?p = 1300 ± 100 MeV. . (86) From eq. (61), one obtains /, = 93 MeV, /T< = 0.86 MeV, k. 1 / M \2 (87) U Tl -1 KMj Using the internal form factor parameter dua = —1.75 GeV 2 (see eq. (36)) and choosing the external form factor parameter cus = 1.51, one finds M-. = 880 MeV, MK,< = 1450 MeV, Mk = 495 MeV, MK> = 1455 MeV, (88) VK = 0.56, rK. = 0.466 = \/ZrK. (89) The experimental values are equal to MeKxp = 891.59 ±0.24 MeV, Mexp = 1412 ± 12 MeV, Mexp = 493.677 ± 0.016 MeV, M%0P = 497.672 ± 0.031 MeV, Mexp = 1460 MeV(7). (90) From the eq. (62), one gets fK = 1.16/, = 108 MeV, fn' = 11 MeV. (91) 15 And for the (f> and The experimental values are equal to M”p = 1019.413 ± 0.008 MeV, M** v = 1680 ± 50 MeV. (93) We can use the parameters dsa and css for the calculations of the masses of the r) and if meson excited states. However, to calculate the masses of the ground and excited states of the eta-mesons it is necessary to take into account the gluon anomaly ( see [5]) and the mixing of the rj, rf, f) and rf states. Since it is a very complicated problem, we will consider this task in future in a separate paper. 6. Summary and conclusions Let us discuss the obtained results. In the.first case where we considered the mesons consisting of the up and down quarks (p, w, %) we used the five experimental values: the masses Mp, Mpt, Mw and the decay constants /, and g p in order to fix the parameters mu, A3, C?i, G2 and cutl. It allows us to predict the mass Mr> = 1.3 GeV and the weak decay constant /,» = 0.86 MeV. 4 Our prediction for the mass of 7 r' is consistent with the modern experimental data [1]. However, the mass of the first radial excited state of pion has an interesting history. Three years ago the new experimental information about excited states in the few-GeV region, e.g on the tt' meson, was obtained at IHEP (Protvino). Indications of the light resonance in diffractive production of 3?r- states have lead to speculations that the mass of the ir' may be considerably lower at « 750 MeV [2]. Our calculations showed that the first radial excited state of the p-meson corresponds to the first radial excited state of the pion with the mass 1.3 GeV. Therefore, the excited pion state with the mass 0.75 GeV is forbidden. Using the experimental values of the masses M$ and My we fix the param eters ms = 470 MeV and csa = 2. We could make some predictions in this quark sector, if we consider the eta-meson states. However, this task will be solved in our subsequent paper. A few predictions one can make for the strange mesons. Indeed, using only one experimental value of the mass K* mason in order to fix parameter cus = 1.5, we have calculated the masses of the K*,K, K* mesons and the weak decay constants fa and fhJ. We would like to emphasize that the excited 4We also can calculate all the strong meson coupling constants. 16 state of the ))scu dose alar kaon K' is not well determined experimentally at present (see [l]). Therefore, our prediction only points out the place where it is necessary to look for this stater* In conclusion, we would like to note that the pseudoscalar and vector me son masses are unambiguously connected with each other in the NJL-model. Indeed, in the usual NJL-model we can use the masses of the pseudoscalar mesons for fixing the model parameters and after that predict the masses of the vector mesons and vice versa I see (5, 16]). The same situation occurs for their excited states. Using the masses of the excited states of the vector mesons we can predict the masses of the excited states of the pseudoscalar mesons and vice versa. We have considered here the simplest extension of the NJL- model with polynomial meson -quark form factors and have shown that this model can be useful for describing the excited states of mesons. We have used assumption that, the quark-current form factors for the scalar, pseudoscalar, vector and axial- vector currents with the same flavours are equal to each other and the obtained results have supported this assumption. The author would like to thank Drs. S.B.Gerasimov and C.Weiss for the fruitful discussions and Prof.A.Di-Giacomo for the kind hospitality in Istituto Nazionale di Fisica Nucleate ( Pisa), where the main part of this work was ful filled. This work was supported by the Russian Foundation for Fundamental Research I Grant N 96.01.01223). References [lj Review of Particle Properties, Phys. Rev. D 54 (1996) 1. [2] Yu.I. Ivanshin et al., JINR preprint El-93 155 11993). [3] C. G. Callan, S. Coleman, J. Wess and B. Zumino, Phys. Rev. 177 (1969) 2247. [4] D. Ebert and M.K. Volkov, Z. Phys. C 16 (1983) 205; M.K. Volkov, Ann. Phys. (N.Y.) 157 (1984) 282. [5] M.K. Volkov, Sov. J. Part. Nucl. 17 (1986) 186. [6] D. Ebert and H. Reinhardt, Nucl. Phys. B 271 (1986) 188. f7] CD. Roberts, R.T. Cahill and J. Pra-schifka, Ann. Phys. (N.Y.) 188 (1988) 20. 5The close theoretics.! estimations of the A#,- and Mr ■ masses were obtained in the papers based on potential models [14], 17 [8] A. Lc Yaouanc, L. Oliver, O. Pene and J.-C. Raynal, Phys. Rev. D 29 (1984) 1233; A. Le Yaouanc et al, Phys. Rev. D 31 (1985) 137. [9] V.N. Pervushin et al., Fortschr. Phys. 38 (1990) 333; Yu.L. Kalinovsky et al, Few-Body Systems 10 (1991) 87. [10] A.A. Andrianov and V.A. Andrianov, Int. J. Mod. Phys. A 8 (1993) 1981; Nucl. Phys. B (Proc. Suppl.) 39 B, C (1993) 257. [11] Yu.L. Kalinovsky, L. Kaschluhn and V.N. Pervushin, Phys. Lett. B 231 (1989) 288. [12] M.K. Volkov and Ch. Weiss, Bochum Univ. preprint RUB-TPII-12/96 (1996); hep ph/9608347; To be publeshed in Phys. Rev. D. [13] R.P. Feynman. M. Kislinger and F. Ravndal, Phys. Rev. D 3 (1971) 2706. [14] S B. Gerasimov and A.B. Govorkov, Z. Phys. C 29 (1985) 61; A.B. Gov orkov, Yad. Fiz. 55 (1992) 1035; Yad. Fiz. 48 (1988) 237; S.B. Gerasimov and A.B. Govorkov, JINR preprint P2-86-758, (1986). [15] D. Ebert, Yu.L. Kalinovsky, L. Miinchow and M.K. Volkov, Int. J. Mod. Phys. A 8 (1993) 1295. [16] M.K. Volkov. A.N. Ivanov, Theor. Math. Phys. 69 (1986) 1066. Received by Publishing Department on December 24, 1996. 18 The Publishing Department of the Joint Institute for Nuclear Research offers you to .acquire the following books: Index Title E7-93-274 Proceedings of the International School-Seminar on Heavy Ion Physics. 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Bojikob M.K. ’ E2-96-482 IIceBflocKanBpHbie h BeKjopHbie BosdyxjteHHbie mc30hm B KHpaJlbHOH MOflCJIH U (3) * U (3) riocTpoeH KHpanbHbiH U (3) * U (3) cMMMCTpHMHbiH narpamKHaH, coaepxamHH noMHMO oSbiMHbix MC30HHbix noneti Taxxe hx nepBbie paaHajibHbie B036y*meHH$L JIarpaHXHaH nojiyncH 6o30HH3aiweti KBapKOBOH mo^cjih Haw6y — Hona-JIasHHUO C Cenapa6eJlbHbIMH HCJIOKaJIbHblMH B3aHMO/ieHCTBM$lMH, OHHCblBaeMblMH 4>OpM(j)aK- TOpaMH, CObTBCTCTByiomMMH 3-MepHblM BOJlHOBblM C^yHKllHBM flJlH OCHOBHbIX H B03- 6>^KaeHHbix coctohhhh. CnoHTaHnoe HapyweHHc KHpa/ibHOH cmmmctphh ynpasjia- erca ypaBHCHMCM utejiH moacjih HHJI. HepBbie paaHajibHbie BoaGyxACHHH nceaflo- CK3JiapHbIX H BCKTOpHblX MC30HHblX HOHCTOB OTlHCblBaiOTCH C nOMOlllblO TpCX pa3J!HMHblX 4)OpM(})aKTOpOB, COflepXCaiUHX TpH npOH3BOJlbHbIX HapaMCTpa. BblMHC- JICHbl Maccbl 3THX MC30HHMX COCTOJIHHH M K0HCT3HTbl HX CJiaSblX paCnaflOB / , h4- Pa6ora BbinojiHena b JlaGoparopHH TeopCTHHecKOH (|)H3hkh HM.H.H.Eoromo6o- Ba OH5IM. npenpHHT OGiejiHHeHHoro HHCTHTyra wtepHbix HccneaoBaHHH. fly6Ha, 1996 Volkov M.K. E2-96-482 The Pseudoscalar and Vector Excited Mesons in the U (3) * U (3) Chiral Model * ' A chiral U (3) * U (3) Lagrangian containing, besides the usual meson fields, their first radial excitations is considered. The Lagrangian is derived by bosonization of the Nambu — Jona-Lasinio quark model with separable non-local interactions, with form factors corresponding to 3-dimensional ground and excited state wave functions. The spontaneous breaking of chiral symmetry is governed by the NJL gap equation. The first radial excitations of the pseudoscalar and vector meson nonets are described with the help of three different form factors containing three arbitrary parameters. The masses of these meson states and the weak decay constants / , and fK, are calculated. The investigation* has been performed at the Bogoliubov Laboratory of Theoretical Physics, JINR. Preprint of the Joint Institute for Nuclear Research. Dubna, 1996 Mbkct T.E.HoneKo rioanMcaHO b neHarb 27.01.97 opMaT 60 x 90/16. OtJjceTHafl nenaTb. I/m.-hsa-jihctob 2,17 THpaac 460. 3bkb3 49673. Uchb 2604 p. # kfcaaTejibCKHH otacji 06i>ejiHHeHHoro HHCTHTyra aaepHbix HccneflOBaHHH Ay6na Mockobckoh o6jibcth