The Radii of Baryons

Transcription

1 Jounal Heading Yea; Vol. (No.): page ange DOI: 0.592/j.xxx.xxxxxxxx.xx The Radii of Bayons Maio Evealdo de Souza Depatmento de Físia, Univesidade Fedeal de Segipe, São Cistovão, , Bazil Astat Consideing the model in whih the effetive inteation etween any two quaks of a ayon an e appoximately desied y a simple hamoni potential, and making use of the expession of the enegy otained in Catesian oodinates fo the aove mentioned model, we find a geneal expession fo the adii of ayons. We then apply the expession to some ayons and find vey onsistent values fo the adii of ayons and an expeimental onfimation fo the gound state of. Keywods Radii of ayons, ayon spetosoy.. Intodution Thee ae vaious models that deal with the adii of ayons suh as the Skyme model [], the / N expansion [2,], hial petuation theoy [4-7], lattie QCD [8,9], meson-ayon dynamis with hial symmety [0], hial onstituent quak model ( CQM) [] et. Howeve, none of these models has deived a fomula fo the adii of ayons. On the expeimental side thee ae a ouple of impotant woks that have epoted the adii of the gound states of some ayons [2,,4]. The pesent wok is an updated vesion of the pe-pint [5]. It is ased on efeene [6] whih alulated most enegy levels of ayons y means of two simple fomulas (one in Catesian oodinates and anothe one in pola ylindial oodinates) with whih we an pedit levels yet to e found. The fomulas do not apply to levels esulting fom hadoni moleules. One of the enegy levels pedited was the st exited state of (enegy of 5.9 GeV) whih has eently een found y CDF [7] with enegy equal to ( ) MeV. *Coesponding autho: maiodesouza.ufs@gmail.om Pulished online at Copyight yea Sientifi & Aademi Pulishing. All Rights Reseved 2. Deivation of an equation fo the adius of a ayon We make use of Eq. () elow fo the enegies of ayons whih was dedued in efeene [6] E h ( n ) h ( m ) h ( l ) () 2 whee n, m, k 0,,2,,4,... and h, h 2, h ae the masses of quaks. On the othe hand it is well known that the aveage potential enegy of a hamoni osillato is half of the total enegy, that is, 2 h k n whee n 0,,2,,... ut sine the quaks ae in a plane and as thee ae two spatial degees of feedom in the plane fo eah quak, we have E h ( n ) h ( m ) h ( k ) n, m, k k k2 2 k k k k 2 2 whee,2, efe to the quaks of the ayons and in whih j and q ae the two othogonal i ij iq (2) () dietions. Eq. () aove was otained onsideing thee

2 2 autho name (et al.): atile title independent osillatos. Thus we an make the assoiation h ( n ) k (4) 2 i i i And defining the adius of a ayon as we otain 2 2 (5) h ( n ) h ( m ) h ( l) 2 k k2 k (6) The onsisteny of Eq. (6) is poven with the alula- tion of fo the gound state of and its ageement with the expeimental value. has the alulated enegy.24 GeV and expeimental enegy.22 GeV. In this ase fo ( n, m, l) thee ae the possiilities (0,0,);(0,,0);(,0,0) and thus, The level n m l 2; E.55 GeV As it is shown on Tale thee ae many states of N and. Fo ( n, m, l) thee ae the possiilities (2,0,0),(0,2,0),(0,0,2) and (,,0),(,0,),(0,,). The oesponding adii ae Calulation of the adii of ayons The masses of quaks h, h 2, h wee taken fom Patile Data Goup [8] as m m 0. GeV, m 0.5 GeV, m.7 GeV, m 5 GeV, and s m 74 GeV. t.. Bayons N,, As shown aove E 0.( n m l ), and thus, as alulated aove h n m l (7) k Using the value h 0.GeV and the expeimental value otain k GeV/fm 2. Theefoe, we have fm [4] fo the poton adius, we n m l (8) in femis. We now apply this aove fomula to all the othe levels. u d The level n m l ; E.86 GeV We see on Tale states of N and. Fo ( n, m, l) thee ae the possiilities (,0,0), (0,,0), (0,0,); (0,,2), (0,2,), (,0,2), (,2,0), (2,,0), (2,0) and (,,). Fo the fist set we epesent the adius y 00, and fo the 2 nd set y 02. The oesponding adii ae ; ; Doing the same fo the othe levels, and summaizing the esults we otain Tale elow.... The level n m l ; E.24 GeV As it is shown on Tale this is the fist state of and

3 Jounal Heading Yea; Vol. (No.): page ange Tale. Radii fo levels of ayons N and aoding to n m l. ( GeV ) 0, 0, 0 0.9( N ) n m l.24( ) n m l n m l n m l n m l n m l n m l n m l n m l n m l n m l n m l n m l n m l n m l n m l n m l n m l n m l n m l n m l n m l n m l n m l n m l n m l n m l n m l n m l Bayons and As shown aove E 0.( n m 2) 0.50( l ), and thus, aoding to Eq. (6) we have Using the expeimental value fm fo the gound state of [4] and the aove value of k GeV/fm 2, we otain fom Eq. (9) k GeV/fm 2. Theefoe, the equation fo the adii of these ayons is given y n m (0) l.2.. The level n m, l 0; E.4 GeV The possiilities fo ( n, m, l ) ae (0,,0), (,0,0), and thus, Eq. (0) yields The level (0, 0,); E.62 GeV adius We have fo ( n, m, l ) the set (0,0,), and hene, the Calulating the othe levels in the same way and odeing the integes of the sets (n,m,l) fom smalle to lage, and aanging the esults on a tale, we otain Tale 2 elow... Bayons Using Eq. (6) and what was alulated aove, we have that the adii of these ayons ae desied y n () m l... The gound state (0, 0, 0) ; E. GeV h h n m l k k (9) Fom Eq. () we otain fm whih is vey lose to the expeimental value of

4 4 autho name (et al.): atile title fm [6]. This esult onfims the validity of the geneal fomula Tale. Radii fo levels of ayons aoding to n m l h ( n ) h ( m ) h ( l). 2 k k2 k Using Eq. () we otain that the othe levels of have the adii summaized on Tale elow. Tale 2. Radii fo levels of ayons and aoding to n m l. ( GeV ) mnl 0, 0, n m, l , 0, n m 2, l n m 2, l n m, l n m, l n m, l , 0, n m 2, l n m 2, l n m 4, l n m 4, l n m 4, l n m, l m n, l m n, l , 0, m n 6, l m n 6, l m n 6, l m n 6, l m n 5, l m n 5, l m n 5, l ( GeV ) mnl (fm) 0, 0, , 0, n 0, m l , 0, n, m l , 0, n 0, m l n 0, m l , 0, Bayons Fom what was alulated aove, and with the use of Eq. (6), we otain that the adii ae desied y n m l. (2) We an, thus, pedit that the gound state has a adius of aout fm. The alulation of the othe levels podue Tale 4 elow. Tale 4. Pedited adii fo ayons aoding to n m l. ( GeV ) mnl 0, 0, n m l n m l n m l

5 Jounal Heading Yea; Vol. (No.): page ange 5 4. Conlusion Consideing that the effetive inteation etween any two quaks of a ayon an e appoximately desied y a simple hamoni osillato, we deive a geneal fomula that desies the enegy levels of ayons and using it we an otain a geneal fomula fo the desiption of the adii of ayons. The alulation is vey onsistent and agees vey well with the expeimental value fo the gound state of. Sine the fomula fo the adius was dedued fom the expession fo the enegy in Catesian oodinates thee is not a way at the moment of identifying the adii in tems of J and L. REFERENCES [] C. Goi, S. Boffi, and D. O. Riska, Mean squae adii of hypeons in the ound-state soliton model, Nulea Physis A, vol. 547, p. 6, 992. [2] A. J. Buhmann and R. F. Leed, Lage N, onstituent quaks, and N, hage adii, Physial Review D, vol. 62, p , [] A. J. Buhmann and R. F. Leed, Stutue of stange ayons, Physial Review D, vol. 67, p , 200. [4] S. Cheedket, V. E. Lyuvovitskij, T. Gutshe, A. Faessle, K. Pumsa-ad, and Y. Yan, Eletomagneti fom fatos of the ayon otet in the petuative hial quak model, Euopean Physis Jounal A, vol. 20, p. 7, [5] S. J. Puglia, M. J. Ramsey-Musolf, and Shi-Lin Zhu, Otet ayon hage adii, hial symmety, and deuplet intemediate states, Physial Review D, vol 6, p. 0404, 200. [6] D. Andt and B. C. Tiuzi, Chage adii of the meson and ayon otets in quenhed and patially quenhed hial petuation theoy, Physial Review D, vol. 68, p , 200. [7] L. S. Geng, J. M. Camalih, and M. J. V. Vaas, Eletomagneti stutue of the lowest-lying deuplet esonanes in ovaiant hial petuation theoy, Physial Review D, vol. 80, p , [8] P. Wang, D. B. Leinwee, A. W. Thomas, and R. D. Young, Chial extapolation of otet-ayon hage adii, Physial Review D, vol. 79, p , [9] S. Boinepalli, D. B. Leinwee, P. J. Moan, A. G. Willimas, J. M. Zanotti, and J. B. Zhang, Eletomagneti stutue of deuplet ayons in the hial egime, Physial Review D, vol. 80, p , [0] T. Sekihaa, T. Hyodo, and D. Jido, Eletomagneti Mean Squaed Radii of (405) in Meson-ayon Dynamis with Chial Symmety, Pogess of Theoetial Physis supplement, vol. 74, p. 266, [] N. Shama and H. Dahiya, Chage adii of otet and deuplet ayons, AIP Confeene Poeedings, vol. 88, issue, p. 458, 20. [2] J. J. Muphy, II, Y. M. Shin, and D. M. Skopik, Poton fom fato fom 0.5 to 0.79 fm 2, Physial Review. C, vol. 9, p. 225, 974. [] B. Povh and J. Hüfne, Geometi intepetation of hadon-poton total oss setions and a detemination of hadoni adii, Physial Review Lettes, vol. 58, p. 62, 987. [4] I. Eshih et al. (SELEX Collaoation), Measuement of the hage adius y -eleton elasti satteing, Physis Lettes B, vol. 522, p. 2, 200. [5] M. E. de Souza, Calulation of the enegy levels and sizes of ayons with a nonental hamoni potential, axiv: hep-ph/ [6] M. E. de Souza, Calulation of almost all enegy levels of ayons, Papes in Physis, vol., p. 0000, 20. [7] I. V. Goelov(CDF Collaoation), Evidene fo the ottom ayon esonane axiv: hep-ex/ *0 with the CDF II deteto, [8] J. Beinge et al. (PDG), Physial Review E, vol. 86, p. 0000, 202.