TWO-QUARK BOUND STATES: MESON SPECTRA AND REGGE TRAJECTORIES
1 2 G. Ganbold • t and G.V. Efimov1
(1) Bogoliubov Laboratory of Theoretical Physics, JINR, Dubna (2) lnstitute of Physics and Technology, MAS, Ulaanbaatar 1' E-mail: ganbold@thsunl .jinr. ru
Abstract We consider simple relativistic quantum field models with the Yukawa interaction of quarks and gluons. In the presence of the analytic confinement there exist bound states of quarks and gluons at relatively weak coupling. By using a minimal set of free parameters (the quark masses and the confinement scale) and involving the mass-dependent coupling constant of QCD we satisfactorily explain the observed meson spectra and the asymptotically linear Regge trajectories.
1. Introduction
The conventional theoretical description of colorless hadrons within the QCD implies they are bound states of quarks and gluons considered under the color confinement. A realisation of the confinement is developed in [l, 2] by using an assumption that QCD vacuum is realized by the self-dual homogeneous vacuum gluon field. Accordingly, this can lead to the quark and gluon confinement as well as a necessary chiral symmetry breaking. Hereby, propagators of quarks and gluons in this field are entire analytic fonctions in the p2-complex plane, i.e. the analytic confinement takes place. On the other hand, there exists a prejudice to the idea of the analytic confinement (for example, [3]). Therefore, it seems reasonable first to consider simple quantum field models in order to investigate only qualitative effects. In our earlier paper [4] we considered simple scalar field models to clarify the "pure role" of the analytic confinement in formation of the hadron bound states. In doing so, we have demonstrated just a mathematical sketch of calculations of the mass spectrum for the "mesons" consisting of two "scalar quarks" by omitting some quantum degrees of freedom such as the spin, color and flavor. These models gave a quite reasonable sight to the underlying physical principles of the hadron formation mechanism and the spectra, and have resulted in qualitative descriptions of "mesons", their Regge trajectories and "glueballs" [5]. In this paper we present a more realistic extention of our earlier consideration by taking into account the omitted quantum characteristics associated with color, flavor and spin degrees of freedom for constituent quarks and gluons. Below we calculate the mass spectrum of pseudoscalar and vector mesons as well as glueballs.
67 2. The Model
Conventionally, consiclering this problem within QCD one deals with cornplicatecl and elaborate calculatious, because the confinement is achievecl as a result of strong interaction involving high-orcler correction:; [6]. Then, there arises a problem of correct and effective summation of these contributions. On the other liane!, the use of a QFT methocl is effective when the coupling is not Then, lower orders of a perturbative technique can result in accurate estimates in QCD) of observables. Particularly, one can effectiYely use the one-gluon cxchange approximation. \Ve consider a relativistic physics and for the haclronization processes use the Bethe Salpetcr equation. Because, when the bincling energy is not neglible the rdativistic cor rections arc considerable. Of course, in doing so, we use a minimal set of frce paramctcrs. Considcr the quark and gluon fields: w~ 1 (:x:) = W~~l~,rflavour(:r),
L(x) = ( \]!:, ·) 1 ( •a •b) \ Wii + 2
'")iJ _ ,-,..., etjJ .a ""!('. __ 'C (I IL <.>fJ ·-· (l y) l,ij' } pv - ·- 01, \Ve guess that the matrix elements of hadron processcs at large distance (in the con finement region) are in fact integrated characteristics of quark (gluon) propagators and their vertices. Their tiny detailecl behavuors may be not so important, but important is to take into account the correct symrnetry foatures. Our aim is to find the most plain forms of these propagators which keep the essen tial properties and result in a qualitative and semi-quantative description of the hadron spectra. For tltis purpose, we consider the following propagators: A2 DllV(") = 0 511v -- -x2J\2/4 ab :i, ab ( 41f) 2 e S~s(P) = 5ii 2, [ip + m(l + /5w(m))]"a · · rn- 1' where A - the confinement energy scale, m - the quark mass and 1 1 - ·u 'U 2 1 1 - 'U ,,~-° 1 }·-! wlm) = jdu (-) _, d'U -- ' 1 +·u l-'l.l2 { (l+'U) 1-·u,2 () ! These are entire analytic functions and represent reasonable approximations of the exact quark and gluon propagators. Consicler the partition fonction and take an explicit integration over ,P-variable: z - JJ o\flow c-.CF[W,'1'] JO]-.C;,,t[W,'1L, // o\flol]J e-LF[W,,vJ-.C2[W.'vJ-.C:i['ÏL,'v]-.Cc+ ... 68 Then, the lowest bound states of quarks and gluons are expressed as follows. Mesons: 2 L2 = !}__ ô e-CB[] ( 'Î'fw) D ( 'Î'fw) 2 J Glue balls: 2 r _ 3(3h) o Baryons: J:,3 = g:h Jô e-CB[) ('Î'rw) D ('Î'rw) D ('Î'rw) D 3. Two-Quark Bound States Omitting elaborous details of intermediate calculations, we write shortly only important steps of our approach for de.scribing two-quark (mesonic) stable bound states. i. Allocation of one-gluon exchange between quark currents 2 Lz = ~ jj dx1dx2 'L,(ifo(x1)'YµtaqJi(x1)D';!:,(xi,x2)(ijh(x2)1.,tbqh(x2)). fih ii. Introduction of the centre-of-masses system: mh mh X1 =x+f.iy, Xz = x-Çzy, 6 = mfi +mh' 6= mh +mh iii. Fierz transformation for colorless bilocal quark currents i('Yµ,)a1/31i('Yµ,)a2.62 = -1 · s - ! . V+ 0. T +!.A- 1. p 2 2 oµµ' o""' = ! . s - ! . V+ ! . T - ! . A - ! . p 4 4 4 4 4 L2 = ~ jj dxdy L, JJJih(x,y)JJhf,(x,y). Jfih iv. Orthonormal basis {UQ(x)} indiced by quantum numbers Q = {nll":lm}: JJJih(x, y)= JD(y)(iJJi (x + f.iy)f Jqh(x - Çzy)) = L, JJQJih(x)UQ(y), Q v. Diagonalization of L2 on colorless quark currents: g2 Lz = 2 L JdxJjJ-(x)JN(x), N = JQfdz, JJQ(x) = iiJi (x)ViQ(B)qh(x). N 69 vi. Gaussian representation by using auxialiary meson fields: =li DENDBf.r vii. Integration over q, if and introduction of an effective action 1 2 SN [B l 2(BN) +TI· + g(BNVv·)S). viii. H adron'izatfon A nsatz to identify BN(x) as meson fields with N = { J Q f f}. I ( ., ) SN [13 l = -2 BJ,r + Tr + g(BNVv)S), 'Ir= Tr/I'r,. ix. Partition fonction for mesons m1d extraction of the full quaclrntic (kinetic) tenu: ZN j DEN ln[l+9(BNVN)8] JDEN C- x. The diagonalization of the quadratic form is equivalent to solution of the Bethe Salpeter equation (in the ladcler approximatiou) on the orthonormal system UN: 2 g2Tr(VNSVN,S) = (UNI1pUN·) = >w(-p )o;J•Ôqq• xi. The mass spectrum and vertice-fuuctions are defined from: 1 = >w(l'vl}r), V:v(D) = r, j dy Uq(y) ;; xii. Below we investigate only two-quark bourn:! statcs for mcsons and \liill uot consicler "''"''·"'"S intcraetion betwcen mesons is dcscribed by functional 3.1. Bethe-Salpeter Equation Mesons 2 The of g I11,(J:, y) is to the solution CJf t.he Bethe- Salpetcr equatiou: jfb:dy UJq(x) 'y) (u) = 5QC!' kcrnel y) is real and symmctric that allows 011e to use a variational Clwose a nonnalizecl trial wavc fonction (for sirmllicitv consicler oximation for (:r) i.Jliri•} a) = C1 ') Ci=J\-~[..i..-l î '------,, :[ 1b: /wi{l.}(:r, cl!-'= 1, (1•}.I where c > 0 - variational parnmcter; - four-dimensional orthogonal polynomials. 70 3.2. Mass Equation Variational solution (on the mass shell p2 = - M?) for the Bethe-Salpeter equation for mesons reads: 1 = niax L {{ dy1clY2 wQ(y1) g2ITp(Y1, Y2) Wq(Y2) >..q(M1) (1) Q {lt} .!/ 1 _ . 4 . {M,2(µÎ+µ~)-(m.î+m~)} 2 - 0'. 8 2 2 exp 37r m 1 m2 2 + 1 2 / 2 / d 1 { M,2(p.1 - p.o) } . max [b(l b/2)] + (-1) dAl exp A - O A= 1 + 2b, rn; = n11) A, l'vfz = lvlrneson/ A p = { 1, n, x={+1,-1} for { P - Psev.doscalar, V - V ector} 3.3. Mass-dependent Coupling Constant Generally, the coupling constant a., = g2 /47r is a free parameter of our mode! and it can be found as a good fitting parameter for known meson masses. However, it is interesting to consider Cts in dependence on meson masses, for example, by using known results for the QCD running coupling [7, 8]. For this purpose we have tested varions versions of such mass dependendence (Fig.l): 2 1 1 (30 (M ) [QCD asymptotics] as(M) = as(Mo) + 47r ln MJ ' 47r 1 ( 1 ) [A.Nesterenko(2002)] as(M) = f3o ln(M2/Q2) 1 - (M2/Q2) ' 47r ( 1 1 ) [I.Solovtsev, D.V.Shirkov(2003)] as(M) = f3o ln(M2/Q2) - (M2/Q2) _ 1 ' where f3o = 11-2 N11av/3 and, a 8 (1777MeV) = 0.34 is known from T-lepton decay data. . ... - .. - ,,,. -- ...... Figure 1: Some mass-dependent versions of a 8 71 Note, this mass dependence allows one to explain the mass splitting effect between T<(140) and 77(547) mesons when the quark content is the same for both of them (Fig.2). Besicles, the mass-clepenclent formula due to [7] is found more acceptable for our consicl eration with q = 600 MeV. 3.4. Ground State: l = 0 Obtainecl results of our estimations for l = 0 are shown in Tables 1 and 2, and Figure 2. Figure 2: Solutions of the mass equation for T<, 77 and 7/ mesons. Note, there appears a mass splitting effect between T<(140) and 77(547) mesons when the quark content is the same for both of thern (Fig.2). This is possible only for relatively light quarks mu= md = 219MeV and the reason is a relevant combination of the decreasing fonction a.s(lvl) and an increasing fonction standing behind it in (1). But this does not appear for heavier quarb (e.g., ms = 249MeV and the unique solution is the meson r/(958). lJ:~~-• -MT~Pc: o-lM, T<(140) 140 D(1870) 1975 77(547) 547 Ds(1970) 1969 77c(2979) 2979 B(5279) 5333 77&(9300) 9300 Bs(5370) 5330 K( 495) 495 Bc(6400 ± 400) 6087 Table 1: Estimated masses for the pseudoscalar mesons. We see (Table 1) that our mode! with a set of pararneters: {A= 600AieV, m,. = md = 219MeV, ms= 249MeV, me= 844MeV, mb = 4303MeV} leads to quantitatively well description of both light and heavy meson masses. 72 ~~=1--1 M~11jPC=1-I Mp Il p(770) 770 K*(892) 892 w(782) 782 D*(2010) 2034 Table 2: Estimated masses for the vector mesons For the vector mesons, the optimal set of parameters is found: {A= 600MeV, m,, = md = 203MeV, m,. = 226MeV, me= 887MeV, mb = 4387MeV}. 3.5. Orbital Excitations: Regge Trajectories Correct description of the mesonic orbital excitations can serve as an additional testing ground for our mode!. In fact, the orbital excitations take place in larger distances and therefore, should be less sensitive to the tiny short-range details of the chosen propagators. The experimental data and our estimates for the p-meson and K-meson families are: p(770) - a2(1320) - p3 (1640) - a4 (2040), M1(770) -M2(1367) -M3 (1702) - M4 (2005), and K*(892) - K;(1430) - K;(I780) - K;(2045) M1(892) -M2(1487) -M3 (1791) -M4 (2064) -M5 (2317) -M6 (2551) -M7 (2770) correspondingly (see Fig.3). 2500 3000 2000 2000 1500 1000 1000 500 / 0~'"'7 3 4 :> Figure 3: Regge trajectories of p- and K-meson families As is expected, our mode! describes well the Regge trajectories of mesons. The most sensitive parameter A governs the slope of the Regge trajectories and its optimal value A = 600Me V is used in previous section. 73 4. Glueballs The experimental status of gluebalb is not clear up to 11ow, although they are very interesting objects from the theoretical point of view. Indeed, it is quite intriguing how can massless gluons be merged into massive object? It is evident that the structure of QCD vacuum plays the main role in formation of glueballs, any direct guess about its explicit structure is avoided. Omitting details, we just write clown the polarization kernel for the gg bound state with quantum numbers J1'C = o++: Il(p2) = 6(3h)2ma/dUdr7 e-c(ç2+7J2) D(E,) D(71)Jdze-ipzD (z + ~) D (z ~) jdE, e-2cç2 D(E,) Taking into account: A2 -A2ç2/4 c + 1/8 } - . e ' K, max = 1 ?12? D(E,) = (47r)2 { (c + :~/8)(c + 3/8 - 1/48/(c + 3/8)) ·- -··· we write the mass equation and obtain the glueball mass: 1287f 1 IT(-M8)=0, Mc=A 2 ln 27ÀK, · Substituting the coupling constant and A we obtain: ') !i- À = - = 0.33, A= 600 MeV => Afc = 1614 MeV 4. 7f This result may be compared with other predictions of the glueball mass: 1730 ± 130 MeV M.Pcarclon [1999] 1740 ± 150 MeV M.Teper [1999] 1580 MeV Y.A.Simonov [2000] 1530 ± 200 MeV H.Forkel (2001] References (1] H. Lcutwylcr, Phys. Lett. 96B 154 (1980) Nue!. Phys. 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