Quantum Chromo Dynamics (QCD), as the fundamental theory of the strong interaction predicts the existence of exotic mesons made of gluons. Observation

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1 Scalar Glueball Decay Into Pions In Eective Theory Hongying Jin and Xinmin Zhang Institute of High Energy Physics, Academia Sinica, P.O.Box 98(4), Beijing 39, China Abstract We examine the mixing between the sigma meson and the "pure" glueball eld H and study the decays of the scalar glueball candidate f (5) (a linear combination of and H) into two and four pions in eective linear sigma model. From recent experimental data on f (5) decay to pions we point out that the mixing angle is of order of.. PACS numbers 3.25.Hw 2.28.Lg jhy@hptc5.ihep.ac.cn, xmzhang@bepc3.ihep.ac.cn

2 Quantum Chromo Dynamics (QCD), as the fundamental theory of the strong interaction predicts the existence of exotic mesons made of gluons. Observations of these gluonium will provide a direct conrmation on the special feature of non-abelian gauge theory. For scalar glueballs the candidates reported in the literature include f (5) and f (75)[,2,3]. However, the non-perturbative dynamics of QCD makes it dicult to do the calculation on the glueball mass, decay width and mixing with the sigma meson. Besides Lattice QCD[8], one has considered various phenomenological models, such as the potential models[4], bag models[5], ux tube[6] models and the QCD sum rule[7] in order to study the glueball mass and its two body decays. In this paper we take an eective lagrangian approach and discuss the glueball decay into twoaswell as four-pion nal states. Particularly we will pay attention to the mixing between the glueball and the sigma meson. It is well known that non-linear sigma model describes the dynamical properties of the pions. To construct an eective lagrangian for glueball decay into pions one could simply include the glueball meson into the non-linear sigma model. Under the chiral symmetry, the glueball transforms as a singlet. For instance, interaction responsible for the decay of the scalar glueball into multi-pions is given by HTr@ () where = exp i~ ~=f and H the "pure" glueball eld. However, with mass of scalar glueball around 5 MeV or 7MeV, the momentum transfer from the decay of the glueball into pions is so large that resonances, such as sigma and rho mesons will show up and play an essential role in the processes of the glueball decay. Thus we use in this paper an eective lagrangian with sigma and rho mesons included explicitly i.e. the linear sigma model. The identication of the sigma particle is highly controversial. It is listed in the Particle Data Tables[4] as a very broad meson with mass around 4 2 MeV and full width 6 2 MeV. This particle, needed in the linear sigma model, has been argued to play an important role in nuclear physics[9] and in the study on the chiral phase transition[]. In recent years, many works are devoted to search for it in the 2

3 scattering[3]. Regarding its role in the glueball physics, BES[] and MKIII[2] data on B(J=! f (5)! 4) and B(J=! f (75)! 4), Crystal Barrel data[3] on pp! f (5)! (2; KK;2; ; 4) show f (5) decays into 4 dominantly through sigma channel(s partial wave), and also hints it strongly couples to 2. In this paper, we consider, for simplicity, the linear sigma model for two avors. As usual we introduce the eld = 2 +i~~ 2 ; (2) where is unity matrix and ~ the Pauli matrices with normalization condition Tr a b = 2 ab. Under a SU L (2) SU R (2) chiral transformation, transforms as! L y R: (3) A renormalizable lagrangian of linear sigma model is given now by L = Trf@ g [Trf y g f 2 2 ]2 ; (4) where f is the vacuum expectation value of the sigma eld and the self coupling constant. Let's now add the O ++ "pure" glueball state H on the lagrangian. Given that the H is made of gluons and singlet under the chiral symmetry, there are only two terms which give rise to the interaction among H, the sigma and pions, L H = g f H[Trf y g f 2 2 ]+g 2H 2 [Trf y g f 2 ]; (5) 2 where two free parameters, g and g 2 are introduced to describe the strength of couplings of one and two Htothe sigma or pions. For the self interaction of glueballs, the lagrangian is given by L H H = H@ H + 2 m2 H H2 + f 3 H 3 + f 4 H 4 ; (6) where m 2 H is the mass of glueball, f 3 ;f 4 the self coupling constants. After the chiral symmetry is broken by non-vanishing vacuum expectation value of the sigma eld, lagrangian in (5) generates not only the interaction for glueball decay, but also 3

4 a mixing between the glueball H and the sigma eld. The mass matrix for (H; ) is where m 2 =2f 2. m 2 = m 2 H g f 2 g f 2 To diagonalize the mass matrix, we introduce an unitary matrix V = cos sin sin cos m 2 C B C A; (7) C A : (8) Then the mass eigen-states are given by H C B A C B H C A ; (9) with = g f 2 =(m2 H m 2 ). Substituting (9) into (5) and (4), we obtain the lagrangian for H decay L H = H f m2 H 2f 2 +( m2 H+2m 2 2f 2g 2 f ) 2 +O( 2 )g: () In (), the couplings of the glueball to sigma and the pions are proportional to the mixing angle. This could be understood by the observation that in the limit of large N c, the mixing between H and is of the same order as that for H to decay into light hadrons (see gure ). Furthermore, under the assumption of vector meson dominance[], meson can be introduced by in (4) by D ig( ). We found that the direct coupling of H and vanishes at the tree level. Identifying the glueball state H with f (5), the phenomenologies of our model can be summarized as follows: i) Glueball decay into two pions: The decay width of f (5) into two pions is given by q (H! 2) = (m H ) 2 m 2 H f 4m2 : () Experimentally Crystal Barrel Group reported[3] that Br(f (5)! 2) : Br(f (5)! 4) = 4:39 :6 : 4:9 3:2 and total = 2 2MeV. Assuming 4

5 that Br(f (5)! 4) is about 5% one has that (2) =7:563MeV. Using this value, we obtain :5: (2) ii)glueball decay into four pions: the dominant decay of the glueball into four pions is through the intermediate two sigma states. In Figure 2, we plot the ratio of glueball decay into 2 to 2 as function of g 2. This measurement can be used to determine the parameter g 2. Before concluding our paper, we make two remarks: ) In the chiral limit which we work on in this paper, the mixing of glueball with sigma is a consequence of the spontaneous chiral symmetry breaking. However, when including the explicit symmetry breaking eects by the quark mass, especially when extending the lagrangian for two avors to three avors, there would be one term proportional to the explicit symmetry breaking which also generates mixing of the glueball with the sigma meson as shown in Fig. 3; 2) To make a comparison of our result from eective theory with other models, we estimate the mixing angle, as an example, by using QCD sum rule. The low energy theorem states that [] Z i dx < jto(x)oj >= d O < jo()j >; (3) where d O is the mass dimension of operator O, O is the trace of the energy momentum tensor = ( s) 4 s G G b s 8 G G ; (( s ) is Gell-Mann-Low function, b= for pure Yang-Mills QCD) which vanishes in classical level. To obtain the mixing angle, let us rstly choose operator O to be O = (uu + dd)= p 2. Assuming that the lowest-lying ++ ground state, i.e., saturates the correlation function we obtain < jo ()j > m 2 <joj >= d < jo ()j >: (4) 5

6 Similarly, taking now the operator O to be O, we have < jo()jh > m 2 H <HjOj >= d G < jo()j >: (5) In deriving (5), we have also assumed that the ground state of ++ "pure" glueball saturates the l.h.s of (5). Dening < jojh >= f H, < jo j>=f,we can roughly dene the eective eld of the "pure" glueball H as H = f H O f H <joj >:: (6) Therefore, From eqs (4-7), we get the mixing angle = 3m2 f f H < jo j >= <joj > f H : (7) 3m 2 2f m H q < joj > < jo j >: (8) Note that the mixing angle is proportional to the quark's condense. The parameter f in (8) can be estimated again by the QCD sum rule. Following[2], we have f 2 = 6 6 M 4 ( + 3 s M < jmqqj > M < j s 4 G G j > 48 3 s 8 M < jqqj > 6 )em2 =M2 ; where M is a parameter with mass dimension in Borel transformation, and M GeV. For the quark and gluon condensate, we take the standard values < jmqqj >= (:Gev) 4 ; (9) < jqqj >=< juuj >=< j ddj >= (:25Gev) 3 ; (2) < j s G G j >= (:33Gev) 4 : Taking a common value, m = 5MeV[3] (8) gives :8: (2) The mixing angle in (2) is consistent with that in (2). In conclusion, we have studied the decay of the scalar glueball into pions and its mixing with the sigma particle in eective theory. Our results show that f (5) decays into four 6

7 pions through 2 intermediate states. The can contribute to the four-pion decay mode at one-loop level with branch ratio (2) :: (22) (2) Furthermore we t our model to the recent experimental data on the decay of f (5) and obtain that the mixing of glueball with sigma is small, which is consistent with our estimation based on low energy theorem and QCD sum rule. ACKNOWLEDGMENTS We thank Zhenping Li, Qing Wang and L.S. Kisslinger for discussions. This work was supported in part by the national natural science foundation of China. 7

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10 Figure Captions Fig. (a) Illustration of glueball decay into 2 or 2; (b)illustration of glueball mixing with induced by the quark condense. Fig.2 Ratio of f (5) decay into 2 to 2 as function of g 2. In the numerical calculation, we take m = 5GeV Fig.3 Illustration of glueball mixing with induced by the quark masses.