XJ E2-98-1O1. M.K.Volkov, M.Nagy*, V.L.Yudichev. SCALAR MESONS IN THE NAMBU JONA-LASINIO MODEL WITH THE 'thooft INTERACTION

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1 XJ OBbEflMHEHHbIM MHCTMTYT I ii i; n MCCJIEflOBAHMM fly6ha E2-98-1O1 M.K.Volkov, M.Nagy*, V.L.Yudichev SCALAR MESONS IN THE NAMBU JONA-LASINIO MODEL WITH THE 'thooft INTERACTION Submitted to «Acta Physica Slovaca» Permanent address: Institute of Physics, Slovak Academy of Sciences, Bratislava, Slovakia 1998

2 1. Introduction During the last years noticeable progress has been achieved in both experimental and theoretical investigations of scalar mesons. The low-mass sigma meson appeared in the last Review of Particle Properties (1996) [1]. A new theoretical analysis of experimental data on the low-mass sigma meson has been completed in the papers [2, 3]. Scalar mesons in the NJL model with the 't Hooft interaction were investigated in [4, 5]. In the papers [5] the coupling of the qq states to the two-pion continuum and a model of confinement were also included into consideration. Here we continue these investigations in the framework of the standard NJL model with the 't Hooft interaction, following the papers [4]-[6]. Our work is organized as follows. In Sec. 2, we describe the characteristics of our NJL model with the 't Hooft interaction. We define the main parameters of this model - the cut-off parameter A and the constituent u-quark mass rn u using the experimental values of the pion decay coupling constant 1^=93 MeV, the p- meson coupling constant g p = 6.14 ( ^ w 3), describing the decay p > 2TT, and the relation g p v6g a. In Sec. 3, we describe the masses of the isovector and strange mesons. In Sec. 4, we calculate the quark-loop contributions to the masses of the?/, 7/', a and f 0 mesons. In Sec. 5, the numerical estimations of the quark-loop contributions to the meson masses are carried out. In Sec. 6 the strong decay widths of the scalar mesons are estimated. Sec. 7 contains discussion and the conclusion. 2. The NJL model with the 't Hooft interaction The U(3)xU(3) version of the NJL model supplemented by the 't Hooft interaction takes the form L = q(id-m o )q+- r 8 -K {det[f(l + ls)q) + det[q(l - ls)q]} (1) where A,- (i=l,...,8) are the Gell-Mann matrices and A 0 = i/fl, with 1 being the unit matrix; m is a current quark mass matrix with diagonal elements m, m", m (ra «m ). The Lagrangian (1) can be rewritten in the form (see [6]) L = q(id - m )q + \ ^ where Ti = \i (i = l,...,7), T, = A, = ) (2) 1

3 r 9 = A, = (-A o = G± 4Km s h{m s ), = G[ ±] = G± AKm u h(m u ), = G, G > = ±4\/2Km u I 1 {m u ). (3) Here m u and m s are the constituent quark masses and 7 "( m ') = (2TT) 4 J dek (k* + m?) n (4) Here we have used the Euclidean space and the cut-off parameter A. Let us define the parameters m u and A, following the papers [7, 8, 9]. We shall use the four equations 1) the Goldberger-Treiman relation 2) the relation between g p and g a q q [7]-[10] 3) the relation between g aqq and g Tqq, which was obtained by taking into account the -K a.\ transitions (see [8, 9]) (5) (6) where M ai is the a r meson mass; 4) the expression for g aqg through a logarithmically divergent integral [7, 8] (7) u)]->. (8) From equations (5)-(7) it is possible to express the constituent u-quark mass through the observable values F T = 93 MeV, g p «6.14 and M ai = 1230 MeV 12 1, m u = 280 MeV. (9) Then, from (6) and (8) we define A = 1250 MeV. (10)

4 3. The masses of the isovector and strange mesons Now let us first consider bosonization of the diagonal parts of the Lagrangian (2) including the isovector and strange mesons. In Sec. 4, we complete the bosonization of the last part of the Lagrangian (2) containing the nondiagonal terms. After renormalization of the meson fields we obtain [7, 8] -ITX In I 1 - -~^ X)(^i7sA,-^ + g a, A,-a,-) 1, (11) where <j>i and cr,- are the pseudoscalar and scalar fields, respectively, TT 2 = w + 27T+7T-, A' 2 = K K + K+K-, a 2 0 = af + 2a+ao, A' 2 = A 0 'A 0 * + A'* + A o - G* = G + AKm,h(m s ), GK = G* - 4A'(7n s /i(m s ) - m u /i(m u )), G ao = G*-8Km,I 1 {m s ), (12) GK> =G n - AK{m s I l {m s ) + J7I U /,(JJI U )), = [4/ 2 (ro u )]-\ ^. = [4/ 2 (m u, m. s )}~\ r,, K f. 4, 0(A 2 ~A: 2 ) h{m u,m 3 ) = - ^ j ^* (Jfc2+m2)(Jfc2 + m j ) = Ujf^l ) f^l > ) (13) 2 - ml), g K = Z)l 2 g K., Z««Z K * Then, in the one-loop approximation, the following expressions for the meson masses are obtained [8] - 4[/,(m tt ) + JiK)] + Z(m, - m u ) 2, (14) The first relation is used to define the parameter G^. For the experimental values M x o = 135 MeV we have G, = 4.92 GeV" 2. (15) 3

5 4. The masses of the rj, r)', a and / 0 mesons The nondiagonal part of the Lagrangian (2) has the form AL = i = (16) ^ ), (a = u, s) (^ = u, s) where Gl+> = G T - SKmMm.), Gi"> = G,, ^,/ik), (17) After bosonization we obtain & L = ^ 7 /"U ) a /?'//3 ^~ CTa '-' >ap Cr P~ -i Trln Jl [g ga t78aai? o + g ga A,g,]} (19) where G s -Gus J </. = ffas«, ^ = [4/ 2 A (m s )]- 1/2, g Vu =g^u, g n. = Z 1/2 g a,. (21) From the Lagrangian (19), in the one-loop approximation, the following expressions for the mass terms are obtained vhere -2{mlal + m]ol) = -\ (22) (23)

6 l = 9l, (^(T 5 )- 1-8J,(m.)) + 4m*, (24) After diagonalization of the Lagrangian (22) we find masses of the pseudoscalar and scalar mesons r), TJ', a and /o Let us define the mixing angle for the pseudoscalar mesons TJ $ = 7] cos 6 +?/' sin, (25) ] (26) TJ U = -j; sin ^ + v' cos ^, 9 = (27) where #<> w 35.3 is the ideal mixing angle (ctg 0 O = \/2) and 6 is the singlet-octet mixing angle For the scalar mesons we use the relations ' uu (28) a u = a cos <j> + /o sin ^, where ^ is the singlet-octet mixing angle and <T S = IT sin ^ + /o cos ^, if, = $ 0 - cf> (29) 5. Numerical estimations of quark-loop contributions to meson masses Using for the parameters m a and K the values m s = 425 MeV, K = 13.3 GeV" 5 (31) we obtain the following estimations for the masses of the pseudoscalar and scalar mesons M* = 135 MeV, M K = 495 MeV, M n = 520 MeV, Af,. = 1000 MeV. (32)

7 M a = 550 MeV, M h = 1130 MeV, M ao = 810 MeV, M K. = 960 MeV, (33) K Note that the pion and kaon masses are the input parameters in our model. The experimental data are [1] M^o = ± MeV, M T ± = MeV, M K+ = ± MeV, M K o = ± MeV, M v = ± 0.19 MeV, M n, = ± 0.14 MeV, 6 «-20 [11]. (34) M, o ( ) = MeV, M /o(98o) = 980 ± 10 MeV, M ao = ± 0.9 MeV, M K; = 1429 ± 6 MeV. (35) Comparing the theoretical results with the experimental data we can see that we have obtained satisfactory results for the pseudoscalar mesons and for the octetsinglet mixing angle of the (rjrj 1 ) mesons. However, for the scalar mesons we have got the masses smaller than the experimental data (except for the /o meson). 6. Strong decays of the scalar mesons. Now, let us show what the information concerning the strong decays of the scalar and pseudoscalar mesons we can obtain from the Lagrangian (11) and (19). These Lagrangians allow us to get the following expressions for the scalar-pseudoscalar meson vertices, describing the corresponding strong decays of the scalar mesons 2ml Z^2 (36) With these vertices one obtains the following decay widths r ff _ «700 MeV, T fo _ n «20 MeV, r ao _^ «130 MeV, rjf.^, w 330 MeV. {il)

8 These values are in qualitative agreement with the experimental data TfZ TT ~ ( ) MeV, I ^ ~ (30-78) MeV,?l x o\. ~ (50-100) MeV, r^_,^ w (263 ±26 ±21) MeV. 7. Conclusion Our calculations have shown that the 't Hooft interaction allows us to describe the masses of pseudoscalar meson's masses and their singlet-octet mixing angle in satisfactory agreement with the experiment 2. For the scalar meson masses we have also obtained results more close to experimental data than in the NJL model without the 't Hooft interaction. However, the masses of the a 0 and especially the Kg mesons are noticeably less then the experimental ones. In order to get satisfactory result for the /o meson it is necessary to take into account the mixing of the f 0 and a mesons with the glueball state (see [12]). The problem with masses of a 0 and KQ mesons could be solved in the framework of the four-quark (? 2? 2 ) MIT-bag model [13] or the KK molecule model [14]. However, it is interesting to note that in spite of a very rough description of the scalar meson's masses our model allows us to obtain a qualitatively true picture of the strong decay widths of the scalar mesons. Acknowledgments This work has been supported in part by the grant from RFFI No and the Heisenberg-Landau program, References [1] Review of Particle Properties, Phys. Rev. D54 (1996) 1. [2] S. Ishida et al. - Progr. Theor. Phys. 95 (1996) 745. [3] M. Svec, A. de Lesquen, L. van Rossum - Phys. Rev. D42 (1992) 949; hepph/ [4] V. Dmitrasinovic - Phys. Rev. C53 (1996) In [7]-[8] the pseudoscalar T; and 17' meson's masses have been described by moans of introducing an additional isoscalar quadratic term into the meson Lagrangian, connected with the gluon anomaly. There were obtained results very close to our work.

9 [5] L. S. Celenza, C. M. Shakin, J. Szweda - Int. J. Mod. Phys. E2 (1993) 437; L. S. Celenza, Xiang-Dong, C. M. Shakin - Phys. Rev. C56 (1997) [6] S. P. Klevansky - Rev. Mod. Phys. 64 (1992) 649. [7] M. K. Volkov, D. Ebert - Sov. J. Nucl. Phys. 36 (1982) 736; Z. Phys. C16 (1983) 205; M. K. Volkov - Ann. Phys. 157 (1984) 282. [8] M. K. Volkov - Sov. J. Part, and Nuclei 17 (1986) 186. [9] D. Ebert, H. Reinhardt, M. K. Volkov - Progr. Part. Nucl. Phys. 35 (1994) 1. [10] H. Kikkawa - Progr. Theor. Phys. 56 (1976) 974. [11] F. J. Oilman and R. Kauffman, Phys. Rev. D 36 (1987) [12] K. Kusaka, M. K. Volkov, W. Weise - Phys. Lett. B302 (1993) 145. [13] R. L. Jarre, Phys. Rev. D15 (1977) 267, D15 (1977) 281. [14] J. Weinstein and N. Isgur, Phys. Rev. D41 (1990) Received by Publishing Department on April 21, 1998.

10 Волков М.К., Надь М., Юдичев В.Л. Скалярные мезоны в модели Намбу Иона-Лазинио с взаимодействием 'т Хофта Е Мы вычисляем спектр масс нонета скалярных мезонов в модели Намбу Иона-Лазинио (НИЛ-модели), которая включает взаимодействие 'т Хофта. Взаимодействие 'т Хофта увеличивает массы скалярных мезонов, однако такое увеличение оказывается недостаточным для того, чтобы объяснить спектр масс скалярных мезонов в целом. Смешивание с глюболом могло бы улучшить ситуацию для а- и/ 0 -мезонов. Для описания масс a Q - и К ^-мезонов необходимо привлекать другие модели. Описаны ширины сильных распадов скалярных мезонов. Работа выполнена в Лаборатории теоретической физики им. Н.Н.Боголюбова ОИЯИ. Препринт Объединенного института ядерных исследований. Дубна, 1998 Volkov M.K., Nagy M., Yudichev V.L. Scalar Mesons in the Nambu Jona-Lasinio Model with the 4 Hooft Interaction E We calculate the mass spectra of the pseudoscalar and scalar meson nonets in the Nambu Jona-Lasinio model with the 't Hooft interaction. We obtain satisfactory result for the pseudoscalar mesons. For the scalar mesons, the 4 Hooft interaction somewhat increases the values of the masses. However, it is not sufficient to explain the whole scalar mass spectrum. The situation could be improved for the a and / Q mesons through mixing with the glueball state. For the description of the masses of a Q and К 0 mesons, it is necessary to involve the other models. The strong decay widths of the scalar mesons are described. The investigation has been performed at the Bogoliubov Laboratory of Theoretical Physics, JINR. Preprint of the Joint Institute for Nuclear Research. Dubna, 1998

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