Light scalar nonets in pole-dominated QCD sum rules T. Kojo (Kyoto Univ.) D. Jido (YITP)

Light scalar nonets ~ candidates of exotic hadrons isospin: 01/210 mass (MeV): 600 ?800 ?980 width:broad narrow 2 - quark picture has the difficulty: To obtain J P = 0 + state, P-wave excitation (~500MeV) is needed. The masses exceed ~ 1 GeV. The assignment assuming the ideal mixing: wrong ordering Natural explanation for mass ordering & decay mode & width Possible strong diquark correlation (?) → mass < 1 GeV 4 -quark picture leads the favorable prescription (Jaffe, 1977)

The purpose of my talk The purpose of my talk is: to provide the information to consider the relevant constituents possible mixing with 2q & 2qG for light scalar nonets, using the QCD Sum Rules. We already know the 2q operator analysis fails to reproduce the lightness of light scalar nonets. Therefore, we perform the QSR analysis using the tetraquark operators.

2, Basics of QCD Sum Rules & typical artifacts in the application to the exotics

QCD Sum Rules (QSR) ？ hardsoft OPE smalllarge Borel window OPE bad OPE good

Procedures for estimating the physical quantities 3, Select Sth to give the good stability against the variation of M. 2, Plot physical quantities as functions of M 2 : effective mass: E E th 1, Set the Borel window for each Sth : M min < M < M max constraint for OPE convergenceconstraint for continuum suppression highest dim. / whole OPE < 10 %pole / whole spectral func. > 50 % peak like structurestability against M variation

QSR for Exotics 2. Good continuum suppression 3. small M 2 -dependence 1. Good OPE convergence Difficulty to analyze Exotics R.D.Matheus and S.Narison, hep-ph/ M 2 small ? When quark number of operator is large, realizing the conditions 1, 2 becomes extremely difficult. Indeed, in most of the previous works, OPE convergence is not good, and In the M 2 region where pole ratio is too small, the artificial stability of the physical parameters emerges! ( Even in the meson and baryon cases ) To avoid the artificial stability, we must estimate the physical values in the Borel window. pole / whole contribution in the spectral integral is less than ~20 %!

QSR artifact ~ pseudo peak artifact : outside of the Borel window dim 10 ~ 12dim 0 ~ 8 E E th dim 8 ~ 12 dim 0 ~ 6 E E th pseudo peak ! dim 0 ~ 6 dim 8 ~ 12 Spectral function

Pseudo peak artifact ~ Impact on physical quantities massresidue spectral functionpole dominance These artifacts are easily rejected by Borel window. & inclusion of higher dimension (> 6) terms.

Impact of width on physical quantities ？ example: Breit-Wigner form (pole mass = 600 MeV) effective mass: We will estimate the physical quantities considering the error from width effects. output: mass width = 400 MeV input: Breit-Wigner

3, Tetraquark operator analysis

Calculation Linear combination: θ will be chosen to achieve: Set up of the operator: up to dim12 within vacuum saturation OPE: Must be calculated to find the Borel window ! weak M – dep. weak Eth – dep. well-isolated peak-like structure (not strongly affected by background) well-separated from threshold Borel window most important for meaningful estimation

Annihilation diagrams ~ Flavor dependence The number of annihilation diagrams strongly depends on the flavor. 2q - 4q, 2qG - 4q & cyclic diquark base: pure singlet: pure octet: large small num. of annihilation diagrams mixing

Effective mass for pure singlet & octet ( in the SU(3) chiral limit ) mass: 0.7 ~ 0.85 GeV Eth: 1.0 ~ 1.3 GeV mass: 0.6 ~ 0.75 GeV Eth: 0.8 ~ 1.3 GeV May be broad, or small pole to background ratio

Effective residue for pure singlet & octet smaller than singlet residue small pole to background ratio?

σ(600) mass: 0.6 ~ 0.75 GeV Eth: 0.9 ~ 1.3 GeV f 0 (980) (preliminary) mass: 0.75 ~ 0.90 GeV Eth: 1.1 ~ 1.5 GeV Effective mass plots for σ & f 0

a 0 (980) (preliminary) a 0 - channel No stability in the Borel window in the arbitrary θ mass residue (×10 7 GeV 8 ) Results for pure octet ~ mass & residue for a 0 κ - channel shows almost same behavior.

Summary We perform the tetraquark op. analysis within the Borel window. To find the Borel window, the higher dim. calculation is inevitable to include the sufficient low energy correlation. Our analysises imply ( within our operator combinations ): σ(600) and f 0 (980) are more likely 4q rather than 2q state. ( in 2q op. case, their masses are ~ GeV) singlet channel has well-developed enhancement around E~ 0.7GeV. octet channel may be strongly affected by low energy contaminations. ( pole to background ratio may be large. ) ( pole to background ratio may be small or no pole. ) The difference between singlet and octet originates from annihilation diagrams, 4q→2q or 4q→2qG.

Back up slide

0 ¯ ¯ (1) 1¯ + (1) 0 ++ (0) 0 + ¯ (1) 1 + ¯(1) π(137) 0 + (1/2) ρ(770) a 1 (1230) J PG (I) M (MeV) a 2 (1320) 2 + ¯(1) vector, axial vector, tensor Scalar meson (L=0, S=1) (L=1, S=0) (L=1, S=1) QQ singlet – octet mixing valence: (QQ)(QQ)(QQ)-(QQ) GG or σ(600) f 0 (980) a 0 (980) K 0 * (800) f 0 (1370) f 0 (1500) a 0 (1450) K 0 * (1430) f 0 (1710) 2q?(L=1, S=1)

Pole dominance ~ importance of higher dim. terms Pole / Whole spectral function (σ- case ) Only after dim. 8 terms contributes, M max becomes large. +dim.8

2-quark Annihilation diagrams 4-quark 3-loop, α suppression 4-loop, α suppression no suppression factor, few loops few loops, but equal to zero Annihilation diagrams have more importance in 4q op. case than in 2q op. case. split singlet & octet Annihilation diagrams increase in higher dim. terms. important especially in low energy region.

4q-2q or 4q-2qG mixing dim 10, 12dim 8dim 6dim 0 ~ 4 E ~ 1 GeV 2qG mixing 2q mixing ~ 2 GeV singlet octet qualitative behavior of This 2qG mixing is turned out to be essential for the large correlation in low energy ~ 1 GeV.

4q-2q or 4q-2qG mixing essential for low energy enhancement can be interpreted as diquark-diquark correlation ? contributes mainly 1~ 2 GeV enhancement can be interpreted as 2q component above 1GeV ?

OPE diagrams ( for massless limit )

Theoretical suggestions 1: mass (MeV): 600 ?800 ?980 width(MeV): ~ 400 ? 50 ~ 300 ~500 ? Natural explanation for mass ordering & decay mode & width Possible strong diquark correlation (?) → mass < 1 GeV 4 -quark picture leads the favorable prescription. (due to strong chromo-magnetic interaction) 1, Jaffe: ( MIT bag model ) PRD15, (1976) 267

Theoretical suggestions 2 & 3: 2, Weinstein & Isgur: ( 4-particle Shrodinger eq. ) PRL48, (1982) 659 qqqq bound states normally do not exist. a0(980), f0(980) → loosely KK bound states. T. Barnes (estimate a0, f0 → 2γ width) PLB165, 434 (1985) a0 (2q) → 2γ : width ~ 1.6 keV ~ 8 ×exp. width a0 (KK) → 2γ: width ~ 0.6 keV ~ 3 ×exp. width f0 (KK) → 2γ: width ~ exp. width 3, Narison: ( phenomelogy with QCD sum rules cal. ) PRD73, (2006) σ, f0 → strong 2q – glueball mixing. a0 → 2q, not 4q κ → 2q ( strong interference with nonresonant background) ( a0 → 2γwidth is 1/1000 small in 4q case) ( σ, f0 → 2π width is too small in 4q case) ( but all cal. of width in QSR is suspicious)

Experimental results: 1, Exp. at Fermi lab. ( E791 Collaboration ) PRL12, (2002) PRL86, 770(2001) PRL12, 765(2001) σ ( mass = 478±17 MeV width = 324±21 MeV ) ( no evidence for σ) κ ( mass = 797±62 MeV width = 410±130 MeV ) c s s s W+ u d c d d d u d

M (MeV) Scalar meson σ(600) f 0 (980) a 0 (980) K 0 * (800) f 0 (1370) f 0 (1500) a 0 (1450) K 0 * (1430) f 0 (1710) Kentucky group a0(1450), K0(1430) → 2q σ(600) → 4q overlap fermion (χ-symmetry) volume dep. UK QCD group Nf=2 sea quark (partially quenched) a0(980) → reproduced within 2q? Scalar collaboration dynamical fermion (including glueball mixing) disconnected diagram dominate (σ case) light σ Lattice: No KK, using (ud) picture

Dispersion relation, OPE, quark-hadron duality QCD side, … sum of local operators Operator Product Expansion information of QCD vacuum （ OPE ） hard soft qq Hadronic side spectral function ？ simple parametrization

Constraint for M Borel trans. Information of low energy we want to know OPE bad OPE good smalllarge Borel window Within the Borel window, we represent mass & residue as functions of the unphysical expansion parameter M ( & physical value Sth )., physical parameter should not depend on M

QSR artifact ~ pseudo peak artifact E E th dim 8 ~ 12 dim 0 ~ 6 E E th pseudo peak ! dim 0 ~ 6 dim 8 ~ 12 : outside of the Borel window Spectral function

Procedures for estimating the physical quantities 3, Select Sth to give the best stability against the variation of M. 2, Plot the physical quantities as functions of M 2. If these quantities heavily depend on M 2 in the Borel window, 1-pole + continuum approximation is bad.. We must consider another possibilities: 2 or 3 poles, smooth function for the scattering states and so on. 1, Set the Borel window for each Sth : M min < M < M max constraint for OPE convergenceconstraint for continuum suppression highest dim. / whole OPE < 10 %pole / whole spectral func. > 50 %

750~ ~ Example when QSR is workable: -meson case Borel window ( up to dim. 6 ) Note for physical importance of higher dim. terms of OPE: Only after including dim.6 terms of OPE (including low energy correlation), stability emerges in the Borel window. Dim.6 terms are responsible for the ρ － A1 mass splitting. (Without dim.6 terms, OPE forρand A1 give the same result.)

QSR for Exotics 2. Good continuum suppression 3. small M 2 -dependence 1. Good OPE convergence Difficulty to analyze Exotics R.D.Matheus and S.Narison, hep-ph/ M 2 small ? When quark number of operator is large, realizing the conditions 1, 2 becomes extremely difficult. Indeed, in most of the previous works, OPE convergence is not good, and In the M 2 region where pole ratio is too small, the artificial stability of the physical parameters emerges! ( Even in the meson and baryon cases ) To avoid the artificial stability, we must estimate the physical values in the Borel window. pole / whole contribution in the spectral integral is less than ~20 %!

QSR artifact ~ pseudo peak artifact E E th E pseudo peak ! dim 0 ~ 6 dim 8 ~ 12 dim 0 ~ 6 : outside of the Borel window dim 10 ~ 12dim 0 ~ 8

Pseudo peak artifact ~ examples massresidue spectral functionpole dominance

Calculation Linear combination: θ will be chosen to give the best stability in the Borel window. Set up of the operator: up to dim12 within vacuum saturation OPE: u, d-quark is treated in massless limit → x- rep. calculation s-quark mass is kept finite → p- rep. calculation regulate mass ×divergence terms resummation of the strange quark mass treatment of current quark mass: Must be calculated to find the Borel window !

diquark base: singletoctet Classification of nonets mass ordering: ideal mixing assumption:

Annihilation diagrams ~ Flavor dependence The number of annihilation diagrams strongly dependent on the flavor. & cyclic diquark base: pure singlet: pure octet: large small num. of annihilation diagrams

1, Sufficiently wide Borel window 2, Weak M dependence 3, Weak threshold dependence 4, The sufficient strength of the effective residue Criterions on selection of operators most important, well-satisfied for almost all θ necessary to avoid the contaminations below Eth necessary to avoid the contaminations from regions between “pole” and Eth necessary to avoid the truncated OPE error ( for stong low energy correlation, pole isolation )

Singlet Octet Global analysis ~ θdependence Singlet → better in Borel stability, larger residue Except some θ region, behavior is similar. θ θ residuemass

Effective mass plot ~ θ fixed to 7π/8 mass: 0.7 ~ 0.8 GeV Eth: 1.0 ~ 1.3 GeV mass: 0.6 ~ 0.75 GeV Eth: 0.8 ~ 1.3 GeV May be broad, or small pole to background ratio

Effective residue plot smaller than singlet residue

σ(600) mass: 0.6 ~ 0.75 GeV Eth: 0.9 ~ 1.3 GeV f 0 (980) (preliminary) mass: 0.75 ~ 0.90 GeV Eth: 1.1 ~ 1.5 GeV Effective mass plots for σ & f 0

a 0 (980) (preliminary) a 0 - channel No stability in the Borel window in the arbitrary θ mass residue (×10 7 GeV 8 ) Results for pure octet ~ mass & residue for a 0 κ - channel shows almost same behavior.

Experimental results: 1, Exp. at Fermi lab. ( E791 Collaboration ) PRL12, (2002) Dalitz decay of D meson PRL86, 770(2001) PRL12, 765(2001)

Summary We perform the tetraquark operator analysis within the Borel window. The states including the SU(3) singlet component: σ(600): f 0 (980): 0.60 ~ 0.75 GeV 0.75 ~ 0.90 GeV (preliminary) The states including only the SU(3) octet component: a 0 (980):no stability κ(800): no stability Real world SU(3) chiral symmetric world singlet: octet: 0.70 ~ 0.85 GeV (stability not good) The difference comes from self-annihilation processes (diagrams). Much stronger low energy correlation than 2-quark case → Borel window is easily found. Some important effects associated with strange quark mass & hadronic threshold seem to be underestimated. 0.6 ~ 0.75 GeV

Resummation of current quark mass EE effective mass shifts to high energy side. cut

Mass × divergence cut for spectral integral regulation for integral of Feynman parameter example: ( dim.8 )

a) b) c)d)

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