IN THE EXTENDED NAMBU JONA-LASINIO MODEL

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1 THE DECAYS ( (89), (4), (7), (65), (6), (64)) IN THE EXTENDED NAMBU JONA-LASINIO MODEL M.. Volkov,. Nurln, Bogoliubov Lbortory of Theoreticl Physics, JINR, Dubn 498, Russi. Dubn Stte University, Dubn 498, Russi E-mil: volkov@theor.jinr.ru nurln.qnt@mil.ru The ( (89), (4), (7), (65), (6), (64)) decys in the extended NJL model re clculted. The results obtined for ( (89), (4), (7)) decys re in stisfctory greement with the experimentl dt. There re no experimentl dt for other types of decy. Therefore, the obtined results should be considered s predictions for further experiments. PACS: 3.35 DxDecysoftus/.39 Fe Chirl Lgrngins

2 Introduction In the tu lepton decys, intermedite chnnels with vector nd xil-vector mesons in the ground nd first rdil-exited sttes ply very importnt role. As rule, for description of those processes vrious phenomenologicl models, which describe the meson interction t low energies, re used. One of the most effective models of this type is the extended NJL model. This model contins miniml number of rbitrry prmeters in comprison with other phenomenologicl models [-5]. In the frmework of this model the following processes were described: [6], ( ) [7], [8], (55)( (95)) [9], [], (, ) [], [] where the chnnel with intermedite vector mesons,, [4],, f ply n importnt role. On the other hnd, in the 3 [3], [5] decys the chnnel with intermedite xil-vector mesons plys the dominnt role (see [6]). As intermedite vector nd xil-vector chnnels ply n importnt role in the clcultion of the forementioned processes, no doubt, the cretion mechnism of vector or xil-vector mesons by mens of tu lepton current is very interesting. This pper is devoted to the solution of this problem.. The Lgrngin of the extended NJL model for the vector (89), (4) nd xil-vector (7), (65), (6), (64) mesons. In the extended NJL model, the qurk-meson interction Lgrngin for the vector (89), (4) nd xil-vector (7), (65), (6), (64) mesons tkes the form [,7]: ' L int( q,q,,,,,, ) q { 5( b ) where qq, ' ' ( ) b 5( ) } q () - re the qurk fields with msses m m 8MeV, m 4MeV ; (89), (4) re vector nd (7), (65), (7), (65) re xil-vector mesons, u d S, () g sin g ' f ( k )sin * * * sin

3 b g cos g ' f k cos * * * sin g sin g ' f ( k )sin sin b g cos g ' f k cos sin (3) 84.74, nd 79.8, re the mixing ngles, f k d k is the form fctor for the description of the first rdilly excited sttes nd d - is the slope prmeter ( d The coupling constnts re: g I m u,m 3 ud.784gev, s / d us.76gev ). 3, g I m,m u s / / g I m u,m u, g I m u,m 3 3 u / (4) The integrl I hs the following form: n n f Nc f (k ) 4 I m, m i 4 3 k d k m k m k (5) where 3.3 GeV is the cut-of prmeter. The mplitudes of the processes (,,,,, ) The digrms of the processes,,,,, ) re shown in Figs,. ( Figure.The digrm describing decys (, ) L u u - lepton current) ( V 3

4 Figure. The digrm describing decys (, ) (, ),( LA u 5 u - lepton current) The mplitudes of these processes tke the form: G A L V p g p p m m g e C C F 3 ( ) ( ) V us s u ( ) g L V m m m e C C 3 ( ) V us s u ( ) g 3 A L V m ( ) ( m m ) e C ( C ) A us s u g A L 6 ( ) ( ) A Vud m m u e C C (6) g where C C * sin sin sin sin R V sin sin sin sin R B, C, C * cos cos cos cos R V cos cos (7) cos cos R B where R V I mu, ms,, I m m I m m u s u s, R B fud I mu, md fud fud,,. (8) I m m I m m u d u d.6637 MeV is the Fermi constnt, us Cbbibo-obyshi-Mskw mtrix V, V ud re elements of the 3. Numericl estimtions The squre of the mplitude of the decy (89) tkes the form: 4

5 m m m A G V m m m C (9) 3 F us ( ) s u g m The squres of the mplitudes of other decys hve n nlogous form. The decy width for the process (89) tkes the form: 3 3 v k A d p d p Г( ) π δ p p p m () E π E π v k τ τ v k where the phse volume: d p m m () E π E π v 3 3 v d pk 4 4 π δ p 3 3 v pk pτ 8m k The theoreticl vlues nd experimentl dt obtined for tu decy re given in the tble. Decys Theor NJL [MeV] Epx PDG [MeV][9] (89) (.7.5) (4) (7).97,7 3 (65) 6.78 (6) 3. (64).437 3,4 3,78 (.6.44) In this tble, there re no theoreticl nd experimentl dt for the decy widths ( 77), (45), becuse this informtion is contined in the [8]. The decys ( (89), (4)) were lso considered in tht pper. However, lrge errors were mde there. Therefore, in this pper, we gin tke into considertion these decys in the frmework extended NJL model. 4. Conclusion The experimentl dt of the considered here processes for tu lepton decys into the ground stte nd first-rdilly exited stte re vilble only for the decys ( (89), (4)). Hence, we nlyse the obtined theoreticl results only for the ( (89), (4)) decys. The experiment shows tht the (4) decy width is 8 time less thn the (89) decy width. In our model, this rtio is given by the dependence of the mplitude on the mixing ngles of the ground nd excited sttes (4 5

6 times). Further greement with the experimentl dt is chieved due to the structure of the mplitude nd describe in the phse volume. In describing the trnsition of the lepton current into the vector meson mny uthor use the model of vector dominnce [-4]. It is interesting to note tht in our cse the use of the model of vector dominnce leds to the sme results s it ws obtined by mens of MVD. At the sme time, in more complicted decy processes of tu leptons, when vector mesons re intermedite sttes nd decy into more complex finl products, it is ppers tht multipliction of the pp (,, ) term by the top tends to be zero. This tkes plce for the processes. For decys such s ( ( (3)), ( ),, ( )) we cnnot neglect the pp term. Therefore, our grdient invrint form of the mplitude for describing the trnsition of the lepton current into vector (xil-vector) meson, in most cses, coincides with MVD. Acknowledgements We re grteful to A.B. Arbuzov nd A.A. Pivovrov for useful discussions. References. M..Volkov, C.Weiss, Phys. Rev. D56 (997).. M..Volkov, Phys. Atom. Nucl. 6 (997) M..Volkov, D. Ebert, M. Ngy, Int. J. Mod. Phys. A3 (998) M.. Volkоv, V. Yudichev, Phys. Prt. Nucl. 3 () Volkov M.., Rdzhbov A.E. The Nmbu Jon-Lsinio Model nd Its Development // Phys. Usp. 6. V. 49. P Volkov. M., ostunin. D. G, PEPAN Lett. (3) Volkov. M., ostunin D. G, Phys. Rev. D86 () Volkov. M., Arbuzov.A. B, ostunin.d. G, Phys. Rev. D86 () Volkov. M., Arbuzov.A. B, ostunin. D. G, Phys. Rev. C89 (4) 5. Volkov. M., Pivovrov.A. A, Mod.Phys.Lett A 3, 6543 (6).. Volkov.M., Pivovrov.A. A, JETP Lett. 3, 63 (6) [Pism Zh.Eksp.Teor.Fiz. 3, 697 (6)].. Volkov. M.., Pivovrov. A.A, Mod.Phys.Lett. A 3, 6538 (6). 3. Ivnov.Y. P, Osipov.A. A, Volkov.M., Z. Phys. C 49, 563 (99). 4. Ivnov.Y. P, Osipov. A. A, Volkov. M., Phys. Lett. B4, 498 (99). 5. Vishnev A. V., Volkov M.., ostunin D.G.The Decy τ fπντin the Nmbu-Jon-Lsinio Model // Eur. Phys. J. A. 4. V. 5. P М.К. Волков, А.Б. Арбузов.//УФН, DOI:.3367/UFNr Vishnev A. V., Volkov M.. Rdilly Excited Axil-Vector Mesons in the extended Nmbu- Jon- Lsinio model // Int. J.Mod.Phys. A. 4. V. 9. P Ahmdov.A. I, linovsky.yu. L, Volkov.M., Int.J.Mod.Phys. A3 (5) 6, Olive.A. et l. Prticle Dt Group Collbortion. Chin. Phys. C. 4. V. 38. Okun L. B. Leptons nd Qurks // 989. Moscow. Nuk.. Lopez Cstro. G nd Flcon.D. A, Phys. Rev. D54 (996) 44. Tllp.A nd Lopez Cstro.G, Phys. Rev. D77 (8) 3 3. Guo. Z.-H., Phys. Rev. D78 (8) Gomez Dumm. D nd Roig.P, Phys. Rev. D86 () 769 6