hep-lat/ Dec 93

Transcription

1 GLUON VERSUS MESON EXCHANGE IN HADRON-HADRON SYSTEMS ON THE LATTICE 1 H. MARKUM, K. RABITSCH, W. SAKULER Institut fur Krnphysik, Tchnisch Univrsitat Win A-100 Vinna, Austria hp-lat/ Dc 93 Th intraction of spatially xtndd havy hadrons is invstigatd in th framwork of lattic QCD with dynamical quarks. In addition to th baryon-baryon potntial rsults for th baryon-antibaryon and for th mson-mson systm ar prsntd. It is shown that th xpctd dipol forcs hav a vry short rang and that sa quarks play a minor important rol. 1 Supportd in part by "Fonds zur Fordrung dr wissnschaftlichn Forschung" undr Contract No. P7510-TEC. 1

2 QCD as a quantum ld thory has du to th uncrtainty principl a cration probability of virtual gluons and quarks. As a consqunc th nuclon-nuclon forcs ar mdiatd by gluon xchang btwn th constitunt quarks for short distancs whras for longr distancs th production of quark-antiquark pairs is th dominating mchanism [1]. Th continuous advanc of th tchniqus of lattic QCD maks th study of systms consisting of a fw valnc quarks in th prsnc of sa quarks fasibl []. This papr invstigats th inunc of both gluons and dynamical quarks on static hadron-hadron systms. Th sa quarks constitut th mson xchang as xpctd by th Yukawa thory. In our approximation of QCD th valnc quarks of a hadron ar rstrictd to ful- ll th static Dirac quation. Th gluons ar tratd as SU(3)-Maxwll lds U x and th sa quarks as Dirac lds x; x in Kogut-Susskind discrtization. Th fr nrgy F (~r 1 ; : : : ; ~r N ) of a systm of N quarks and antiquarks in a gluonic and frmionic ld is dnd as a thrmodynamical xpctation valu and xprssd by th Fynman pathintgral for th product of N Polyakov loops L(~r i ) [3] xp 1 T F (~r 1; : : : ; ~r N ) = R D[U]D[ ; ] L(~r1 ) : : : L y (~r N ) S[U; ; ] R D[U]D[ ; ] S[U; ; ] = hl(~r 1 ) : : : L y (~r N )i : (1) S(U; ; ) = S G (U)+S F (U; ; ) is th total action of th systm with tmpratur T. For th gluonic action S G w us Wilson's plaqutt action with th invrs gluon coupling = 6=g and for th frmionic action S F w mploy th Kogut-Susskind formulation with n f avors. Th tim volution of th valnc quarks is takn as th static quark propagator, th so-calld Polyakov loop L(~r) = 1=3 Tr Q N t i=1 U =(~r; t i ). W construct a quark wav function (~r) by distributing th static quark chargs in Gaussians (~r) = ( p ) 3= r L(~r) () ovr a sphr with radius R on th lattic. Sinc th rsults turn out to dpnd only wakly on th width w st () to a uniform distribution. For th dscription of

3 baryons w tak th product of thr quark wav functions and for msons corrlatd wav functions of quark and antiquark. Bcaus th static quarks carry no spin and avor all baryons and msons ar dgnrat. Th total wav function of th hadron-hadron systm is constructd to b a product of Gauss functions with two cntrs sparatd by som distanc d. To comput th path intgral (1) w prform Mont Carlo itrations for QCD without dynamical frmions and 1000 itrations in th prsnc of frmions. W tak a spac-tim lattic of siz with priodic boundary conditions. In th pur gluonic cas w choos = 5:6 which corrsponds via th rnormalization group quation to a lattic constant of a 0:5 fm and to a spatial xtnsion in z dirction of about fm. For full QCD at = 5: ( = :9) w st th numbr of avors n f = 3 (n f = ) and th dynamical quark mass m = 0:1. This choic lads to comparabl lattic constants all lying in th connmnt rgim. W calculat th total nrgy of th hadron-hadron systm according to (1) and subtract twic th nrgy of a singl hadron. In Fig. 1a w show th rlativ potntial nrgy of th two-baryon systm with baryons of radius for varying distancs d btwn th two clustrs. W nd an attractiv potntial with an ctiv rang of about R. Th rason for th attractiv forcs btwn th two baryons is th Coulomb plus linar typ of potntial btwn quarks which tris to attract th two thr-quark clustrs. Whn th two baryons bcom sparatd thir colors ar saturatd and thy volv to isolatd color singlts which do not intract. At last on th lattic no long rang forcs can b rsolvd. Taking dynamical frmions into account w obtain no drastic ct. This is also obsrvd in mass spctrum calculations whr sa quarks giv ris to only a 10 pr cnt ct. In Fig. 1b w display th two-baryon systm with baryon radius R =. W nd attractiv intractions only in th ovrlap rgion and practically no avor dpndnc. W now turn to th baryon-antibaryon systm in Fig. 1c and again s an attractiv intraction in th ovrlap rgim. Th dirnc btwn th baryon-baryon and baryonantibaryon potntials might b intrprtd qualitativly by th fact that th baryon- 3

4 antibaryon systm can build thr msons which is nrgtically favord. Switching on dynamical frmions w obsrv practically no ct. Incrasing th radius of th hadron sphr to R = w rcogniz in Fig. 1d mor or lss th sam bhavior as for. Th static mson-mson systm is dpictd in Figs. 1 {f for radii and R =, rspctivly, and bhavs similar to th prvious systms. To conclud, this study dals with a truncatd Dirac quation rstricting th valnc quarks to static color chargs. Th construction of th quark orbitals lads to baryons of rasonabl siz but taks no kintic nrgy into account. It turns out that th gluon xchang dominats and sa quark cts ar small. Th aim would b to valuat th S matrix for scattring of two nuclons consisting of thr quarks ach. This rquirs th calculation of a four-point hadron-grn-function bing quivalnt to a corrlation function of six quark propagators in th QCD path-intgral. Thr has bn som progrss in computing mson scattring-amplituds from th nit volum dpndnc of th bound-stat nrgis []. A mthod to xtract ctiv potntials from th two-hadron spctrum was proposd rcntly [5]. Rfrncs [1] K. Holind, Phys. Rp. 68 (1981) 11. [] H. Markum, M. Minhart, G. Edr, M. Fabr and H. Lb, Phys. Rv. D31 (1985) 09; K. Rabitsch, H. Markum and W. Sakulr, Phys. Ltt. B, in print. [3] L. D. McLrran and B. Svtitsky, Phys. Rv. D (1981) 50. [] S. R. Sharp, R. Gupta and G. W. Kilcup, Nucl. Phys. B383 (199) 309. [5] H. R. Fibig, Talk at th 3 rd Workshop on Lattic Fild Thory (Vinna, 1993).

5 (a) Baryon{Baryon n f = 3; = 5: n f = ; = : Distanc d[a] (b) -0. Baryon{Baryon R = Distanc d[a] n f = 3; = 5: n f = ; = : (c) Baryon{Antibaryon n f = 3; = 5: n f = ; = : (d) Baryon{Antibaryon R = n f = 3; = 5: n f = ; = : Distanc d[a] Distanc d[a] 0. () (f) Mson{Mson n f = 3; = 5: n f = ; = :9-0. Mson{Mson R = n f = 3; = 5: n f = ; = : Distanc d[a] Distanc d[a] Fig. 1: Baryon-baryon potntials (a,b) as a function of cntr of mass distanc d for pur gluon xchang and with sa quarks for baryons with radii and R = in lattic units. Th intraction is attractiv and taks plac mainly in th ovrlap rgion. Th baryon-antibaryon potntials (c,d) turn out to b dpr and th mson-mson potntials (,f) ar shallowr. Dynamical quarks hav no pronouncd ct. Error bars ar in th siz of th symbols. 5