1. 1 Scalar and Pseudoscalar Glueballs Hai-Yang Cheng Academia Sinica, Taipei December 4, 2009 Hadron 2009, Tallahassee, Florida

2. 2 Glueball: color-singlet bound state of gluons as gluons have a self coupling Lightest glueballs in pure YM theory: J PG =0 ++ 1710  50  80 MeV J PG =2 ++ 2390  30  120 MeV J PG =0 -+ 2560±35±120 MeV Y. Chen et al. PR, D73, 014516 (2006) Glueball will mix with qq states so that a pure glueball does not exist in nature Mass of 0 -+ glueball could be drastically affected What is the effect of quark degrees of freedom on glueballs ?

3. 3 Scalar glueball Chun-Khiang Chua, Keh-Fei Liu, HYC, PR, D74, 094005 (2006)

4. 4 Scalar Mesons (J P =0 + )

5. 5 Amsler & Close (1995) claimed f 0 (1500) discovered at LEAR as an evidence for a scalar glueball because its decay to ,KK, ,  ’ is not compatible with a simple qq picture. f 0 (1500): dominant scalar glueball  1550 MeV [ Bali et al. ’93, Amsler et al. ’95] f 0 (1710):  is suppressed relative to KK  primarily ss dominated f 0 (1370): KK is suppressed relative to   dominated by nn states G ss nn M S >M G >M N M G  1500 MeV, M S -M N  200-300 MeV Amsler, Close, Kirk, Zhao, He, Li…

6. 6 Difficulties: Near degeneracy of a 0 (1450) and K 0 * (1430) cannot be explained due to the mass difference between M S and M n If f 0 (1710) is ss dominated,  [J/  f 0 (1710)] = 6  [J/  f 0 (1710)] [Close,Zhao] BES   [J/  f 0 (1710)] = (3.3  1.3)  [J/   f 0 (1710)] J/  →  gg and the two gluons couple strongly to glueball ⇒ If f 0 (1500) is primarily a glueball, J/    f 0 (1500) >> J/   f 0 (1710) BES   [J/    f 0 (1710)]  > 4  [J/   f 0 (1500)] If f 0 (1500) is composed mainly of glueball, then the ratio R=  (f 0 (1500)   )/  (f 0 (1500)   ) = 3/4 flavor blind < 3/4 for chiral suppression R expt (f 0 (1500))=4.1  0.4, R expt (f 0 (1710))=0.41+0.11 -0.17

7. 7 In our work we reply on two simple inputs for mass matrix: approximate SU(3) symmetry in scalar meson sector (> 1GeV) a 0 (1450), K 0 * (1430), D s0 * (2317), D 0 * (2308)  M S should be close to M N a feature confirmed by LQCD LQCD  K 0 * (1430)=1.41±0.12 GeV, a 0 * (1450)=1.42±0.13 GeV  near degeneracy of K 0 * (1430) and a 0 (1450) This unusual behavior is not understood and it serves as a challenge to the existing QM glueball spectrum from quenched LQCD M G before mixing should be close to 1700 MeV Mathur et al.

8. 8 uu dd ss G x: quark-antiquark annihilation y: glueball-quarkonia mixing first order approximation: exact SU(3)  M U =M D =M S =M, x=x s =x ss, y s =y y=0, f 0 (1710) is a pure glueball, f 0 (1370) is a pure SU(3) singlet with mass = M+3x ⇒ x = -33 MeV y  0, slight mixing between glueball & SU(3)-singlet qq. For |y|  x|, mass shift of f 0 (1370) & f 0 (1710) due to mixing is only  10 MeV  In SU(3) limit, M G is close to 1700 MeV a 0 octet singlet G

9. 9 Chiral suppression in scalar glueball decay If G→PP coupling is flavor blind, If f 0 (1710) is primarily a glueball, how to understand its decay to PP ? chiral suppression: A(G→qq)  m q /m G in chiral limit LQCD [Sexton, Vaccarino, Weingarten]  [Chao, He, Ma] : m q is interpreted as chiral symmetry breaking scale [Zhang, Jin]: instanton effects may lift chiral suppression Chiral suppression at hadron level should be not so strong perhaps due to nonperturbative chiral symmetry breaking and hadronization [Carlson et al; Cornwall, Soni; Chanowitz]

10. 10 Consider two different cases of chiral suppression in G→PP: (i) (ii) In absence of chiral suppression (i.e. g  =g KK =g  ), the predicted f 0 (1710) width is too small (< 1 MeV) compared to expt’l width of 137  eV  importance of chiral suppression in G  PP decay Scenario (ii) with larger chiral suppression is preferred

11. 11 M S -M N  25 MeV is consistent with LQCD result  near degeneracy of a 0 (1450), K 0 * (1430), f 0 (1500) Because nn content is more copious than ss in f 0 (1710),  (J/  f 0 (1710)) = 4.1  ( J/   f 0 (1710)) versus 3.3  1.3 (expt)  (J/   f 0 (1710)) >>  (J/  f 0 (1500)) in good agreement with expt.  [J/  →  f 0 (1710)]   [J/  →  f 0 (1500)] : primarily a glueball : tends to be an SU(3) octet : near SU(3) singlet + glueball content (  13%) M N =1474 MeV, M S =1498 MeV, M G =1666 MeV, M G >M S >M N Amseler-Close-Kirk Our scenario for scalar glueball and its mixing with qq is consistent with all the existing data

12. 12 Pseudoscalar glueball Hsiang-nan Li, Keh-Fei Liu, HYC Phys. Rev. D 79, 014024 (2009)

13. 13 1980: J/  + resonance (1.44 GeV) was seen by MarK II and identified as E(1420) [now known as f 1 (1420)] first discovered at CERN in 1963. Renamed the ¶(1440) by Crystal Ball & Mark II in 1982 1981:  (1440) was proposed to be a pseudoscalar glueball by Donoghue, Johnson; Chanowitz; Lacaze, Navelet,… 2005: BES found a resonance X(1835) in J/   +  -  ’ 2006: X(1835) as an 0 -+ glueball by Kochelev, Min; Li; He, Li, Liu, Ma It is commonly believed that  (1440) now known as  (1405) is a leading candidate of G 1.Br[J/  (1405)]  O(10 -3 ) larger than J/  (1295),  (2225)  KK ,  :  (1475) in KK  was seen, but  (1405) in  was not Expt’l review: Masoni, Cicalo, Usai, J. Phys. G32, 293 (2006)

14. 14 Pseudoscalar glueball interpretation of  (1405) is also supported by the flux-tube model (Faddeev et al.), but not favored by most of other calculations: LQCD ⇒  2.6 GeV [Chen et al; Morningstar, Peardon] QCDSR ⇒ m G > 1.8 GeV QM ⇒ 2.62 GeV > m G > 2.22 GeV All yield m(0 ++ ) < m(0 -+ )

15. 15  -  ’-G mixing Extend Feldmann-Kroll-Stech (FKS) mechanism of  -  ’ mixing to include G. Consider flavor basis  q =(uu+dd)/√2,  s =ss. In absence of U(1) anomaly,  q &  s mix only through OZI-violating effects where  =  + 54.7 o,  G is the mixing angle between G and  1 (  8 is assumed not to mix with glueball). Mixing matrix depends only on  and  G

16. 16 Applying EOM: Six equations for many unknowns. We need to reply on large N c rules (‘t Hooft)

17. 17 To leading order in 1/N c, keep f q,s, neglect f q,s g, f s q, f q s keep m qq 2  m  2, m ss 2  2m K 2 -m  2, neglect other mass terms KLOE analysis of   ’ ,  has been criticized by Escribano, Nadal; Thomas New KLOE analysis ⇒  = (40.4±0.6) o,  G =(20 ±3) 0 for f s /f q =1.352  0.007 This result for m G is stable against OZI corrections m G =(1.4±0.1) GeV

18. 18 Is this compatible with LQCD which implies m G > 2GeV ? Lattice results are still quenched so far. It has been noticed that m G in full QCD with dynamic fermions is substantially lower than that in quenched approximation [Cabadadze (1998)] Contrary to the mainstream, we conjecture that pseudoscalar glueball is lighter than the scalar one due to dynamic fermion or axial anomaly. It is urgent to have a full lattice QCD calculation quenched unquenched

19. 19 supported by a recent analysis based on chiral Lagrangian with instanton effects, a solution for U A (1) problem [Song He, Mei Huang, Qi-Shu Yan, arXiv:0903.5032]  Has a connection with FKS formulism generalized to 3x3 case  Consider two scenarios for scalar & pseudoscalar glueball mass difference,  m g =m(0 ++ )-m(0 -+ ), by fixing m(0 ++ ) 1.m(0 ++ )=660 MeV,  m g ~ - (0.1-0.3) GeV: no such a candidate 2.m(0 ++ )=1710 MeV,  m g ~ (0.2-1.0) GeV: possible candidates are  (1295),  (1405),  (1475)

20. 20 Conclusions We use two simple & robust results to constrain mixing matrix of f 0 (1370), f 0 (1500) and f 0 (1710): (i) empiric SU(3) symmetry in scalar meson sector > 1 GeV, (ii) scalar glueball mass  1700 MeV Exact SU(3) ⇒ f 0 (1500) is an SU(3) octet, f 0 (1370) is an SU(3) singlet with small mixing with glueball. This feature remains to be true even when SU(3) breaking is considered Analysis of  -  ’-G mixing yields m G  1.4±0.1 GeV, suggesting that  (1405) is a leading candidate of pseudoscalar glueball. A full lattice QCD calculation is urgent.