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Scalar Meson Spectroscopy with a Fine Lattice CPC(HEP & NP), 2011, 35(10): 896{901 Chinese Physics C Vol. 35, No. 10, Oct., 2011 Scalar meson spectroscopy with a fine lattice FU Zi-Wen(Ff©)1;1) Carleton DeTar2;2) 1 Key Laboratory of Radiation Physics and Technology (Sichuan University), Ministry of Education; Institute of Nuclear Science and Technology, Sichuan University, Chengdu 610064, China 2 Physics Department, University of Utah, Salt Lake City, UT 84112, USA Abstract: With sufficiently light u and d quarks the isovector (a0) and isosinglet (f0) scalar meson propagators are dominated at large distances by two-meson states. In the staggered fermion formulation of lattice QCD, taste-symmetry breaking causes a proliferation of multihadron states that complicates the analysis of these channels. Of special interest is the bubble contribution, which makes a considerable contribution to these channels. Using numerical simulation we have measured the correlators for both a0 and f0 channels in the \Asqtad" improved staggered fermion formulation in a MILC fine (a = 0:09 fm) lattice ensemble. We analyze those correlators using rooted staggered chiral perturbation theory (rSχPT) and achieve chiral couplings that are well consistent with previous determinations. Key words: lattice QCD, scalar meson, bubble contribution PACS: 11.15.Ha, 12.38.Gc, 12.39.Fe DOI: 10.1088/1674-1137/35/10/002 1 Introduction fermion QCD. An explanation of the nonstandard fea- tures of the scalar correlators is provided by rSχPT The recent evident successes of lattice QCD simu- [17{19]. In that theory all pseudoscalar mesons come lations with improved staggered fermions demand an in multiplets of 16 tastes. The π and K multiplets intensive examination of the \fourth-root trick" us- are split in similar ways. For η and η0 mesons only ing fractional powers of the determinant to simulate the taste singlet η and η0 obtain physical mass. Some the correct number of quark species. This trick au- of the remaining members of the η multiplet remain tomatically performs the transition from four tastes degenerate with the pions. According to taste sym- to one taste per flavor for staggered fermions at all metry selection rules, any two mesons coupling to a orders. The procedure is known to introduce nonlo- taste-singlet a0 must have the same taste. All tastes calities and violations of unitarity at nonzero lattice are equally allowed. Among other states, the taste spacing [1]. Hence there are strong theoretical argu- singlet a0 couples to the Goldstone pion (pseudoscalar ments [2{5] that the fourth-root trick is valid. taste) and an η, also with pseudoscalar taste and of One can test the fourth-root trick numerically [6{ the same mass. This two-body state at twice the mass 12]. Alternatively, low-energy results of staggered of the Goldstone boson accounts for the anomalous fermion QCD simulations can be compared with pre- low-energy component in that channel. dictions of rSχPT [13, 14]. Since staggered chi- In our previous work we extended the analysis of ral perturbation theory becomes standard chiral per- Refs. [18, 19], examined the scalar meson correlators turbation theory in the continuum limit, agreement in full QCD on a MILC coarse (a = 0:12 fm) lat- between rooted QCD and rSχPT at nonzero lat- tice ensemble and compared their two-meson content tice spacing would suggest that lattice artifacts pro- with predictions of rSχPT [20, 21]. Since the appear- duced by the fourth-root approximation are innocu- ance of the two-meson intermediate state (i.e. bubble ous. There are two recent tests of agreement between contribution) is a consequence of the fermion deter- rooted staggered fermion QCD and rSχPT [15, 16]. minant, an analysis of this correlator gives a direct The a0 channel has been studied in staggered test of the fourth root recipe. In this work we extend Received 8 December 2010, Revised 6 January 2011 1) E-mail: [email protected] 2) E-mail: [email protected] ©2011 Chinese Physical Society and the Institute of High Energy Physics of the Chinese Academy of Sciences and the Institute of Modern Physics of the Chinese Academy of Sciences and IOP Publishing Ltd No. 10 FU Zi-Wen et al: Scalar meson spectroscopy with a fine lattice 897 the analysis of Refs. [20, 21] and carry out a quan- In the taste-pseudoscalar and taste-tensor sectors, titative comparison of the measured correlators and in which there is no mixing of the isosinglet states, the 0 the predictions of rSχPT at a fine (a = 0:09 fm) lat- masses of the ηb and ηb by definition are tice ensemble. Despite the considerable complexity 2 2 2 2 M = M ; M 0 = M : (5) of channels with dozens of spectral components, the η;b Ub η ;b Sb chiral theory models the correlators precisely in terms In Table 1 we list the masses of the resulting of only a small number of chiral couplings, which we taste multiplets in lattice units for the lattice ensem- may determine through fits to the data [20, 21]. ble used in the present study with taste-breaking pa- rameters δA and δV determined in Refs. [15, 17] as measured or inferred from the measured masses and 2 Pseudoscalar meson taste multiplets 0 splittings. The mass of the ηI depends on m0. In Refs. [20, 21] we give a brief review of the rooted Table 1. Mass spectrum of pseudoscalar meson staggered chiral perturbation theory with particular for the MILC fine (a = 0:09 fm) lattice en- focus on the tree-level pseudoscalar mass spectrum, semble β = 7:09, amud = 0:0062, ams = 0:031. and achieve the rooted version of the theory through π 0 the replicated theory [22]. b a b aKb aηb aηb The tree-level masses of the pseudoscalar mesons P 0.1479 0.2532 0.1479 0.3273 are [13, 21] A 0.1646 0.2633 0.1530 0.3325 T 0.1743 0.2695 0.1743 0.3400 2 2 M 0 = B0(mf +mf0 )+a ∆b; (1) ffb V 0.1820 0.2745 0.1780 0.3430 where ff0 are two flavors which make up, b = 1; ;16 I 0.1926 0.2816 0.3064 | 2 ··· are the taste, B0 = mπ=2mq is the coupling constant of the point scalar current to the pseudoscalar field, 2 and the term of a ∆b comes from taste-symmetry 3 Scalar correlators from SχPT breaking. In this work we investigate degenerate u and d In Refs. [20, 21], using the language of the replica quarks (mu = md), so it will be convenient to intro- trick [13, 24] and through matching the point-to-point duce the notation scalar correlators in chiral low energy effective theory 2 2 and staggered fermion QCD, we rederive the \bub- MUb = 2B0mu +a ∆b; ble" contribution to the a0 channel of Ref. [19], and 2 2 MSb = 2B0ms +a ∆b; (2) then extend the result to the f0 channel. Here we just review some results required for this work. 2 2 MKb = B0(mu +ms)+a ∆b: 3.1 Isovector a0 correlator The isosinglet states (η and η0) are modified both by the taste-singlet anomaly and by the two- For simplicity we consider the ud¯ flavor state for trace (quark-line hairpin) taste-vector and taste- the a0 correlator. Only the quark-line-connected con- axial-vector operators [17, 23]. When the anomaly tribution appears in the QCD correlator. The times- parameter m0 is large, we obtain the usual result lice correlator for the a0 meson is given by, ip·x ¯ 2 1 2 2 2 2 C(p;t) = e d(x;t)u(x;t)u¯(0;0)d(0;0) ; (6) M = M + M ; Mη0;I = (m ): (3) h i η;I UI SI 0 x 3 3 O X In the taste-axial-vector sector we have where p is the chosen momentum. There exist many P + 1 3 multihadron states with J = 0 and I = 1 which can M 2 = M 2 +M 2 + δ Z ; ηA 2 UA SA 4 A − A propagate between the source and the sink. Of spe- cial interest in multihadron states is the intermediate 2 1 2 2 3 state with two pseudoscalars P1 P2 which we refer to M 0 = M +M + δ +Z ; (4) η A 2 UA SA 4 A A as the bubble contribution (B) [19]. If the masses of δ 9 P1 and P2 are small, the bubble contribution B gives Z2 = (M 2 M 2 )2 A (M 2 M 2 )+ δ2 ; A SA − UA − 2 SA − UA 16 A a considerable contribution to the a0 correlator, and it should be included in the fit of the lattice correlator where δV is the hairpin coupling of a pair of taste- in Eq. (6), that is, vector mesons, δA is the hairpin coupling of a pair of −ma0 t taste-axial mesons [17]. C(p;t) = Ae +Ba0 (p;t); (7) 898 Chinese Physics C (HEP & NP) Vol. 35 where for notational simplicity we omit the contribu- and for f hairpin(t), we just require V A in f hairpin(t). A ! V tions from the excited a0 meson; the oscillating terms correspond to a particle with opposite parity, and 3.2 Isosinglet f0 correlator other high-order multihadron intermediate states. We use the isosinglet operator for the f0 cor- The B (p;t) is in detail denoted in Appendix B of a0 relator. Both quark-line-connected and quark-line- Ref. [20]. For 1+1+1 theory, if we rearrange the terms disconnected contributions appear in the lattice QCD of the time Fourier transform of bubble correlator in correlator.
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