Hadronic physics from the lattice

Hadronic physics from the lattice Chris Michael c.michael@liv.ac.uk University of Liverpool Hadronic physics from the lattice p.1/24

Hadronic Structure - Introduction What are hadrons made of? Is a meson made of qq, q qqq, meson-meson,..? A state that can decay strongly (resonance) necessarily has a meson-meson component - is this important? Are unstable particles different from stable ones? What is the nature of light-light scalar mesons? Where is the glueball? Are there hybrid mesons,...? Lattice QCD is a first-principles method of attack. But we use unphysical (too heavy) quark masses, we have Euclidean time, we have cubic rotational symmetry only,... What can we learn? Hadronic physics from the lattice p.2/24

Lattice QCD - Introduction Measure C(t) = Vacuum Meson(0)Meson(t) Vacuum with operator "Meson" made out of lattice quark and gluon fields creating a meson of required quantum numbers. With Euclidean Time: C(t) = c 1 e m1t + c 2 e m2t +... So ground state (lightest m i ) dominates. Using variational methods it is possible to extract the next state (first excited state) sometimes. A 2-body continuum is tricky - but in a finite spatial box it is discrete - see later. Does an unstable particle behave differently? Hadronic physics from the lattice p.3/24

Hadronic Decays - Introduction Relatively few hadronic states are stable (under QCD with degenerate u and d quarks). Mesons: Stable: π K η D D s B B s B c Ds B Bs D s (0 + ) B s (0 + ) Γ < 1 MeV: η D ψ(1s) ψ(2s) χ 1 χ 2 Υ(1S) Υ(2S) Υ(3S) Γ < 10 MeV: ω φ χ 0 X(3872) Γ > 10 MeV: ρ f 0 a 0 h 1 b 1 a 1 f 2 f 1 a 2, etc., inc η c. Masses of unstable states defined as 90 0 phase shift still seem to fit patterns well: ρ(776); ω(783) are close in mass despite having widths of 150; 8 MeV resp. (1232); Σ(1385); Ξ(1530); Ω(1672) are roughly equally spaced in mass despite having widths of 120; 37; 9; 0 MeV resp. mass defined as real part of pole fits less well. eg. (1232) mass 22 MeV less CM66 Hadronic physics from the lattice p.4/24

Decays in Euclidean Time on a Lattice NO GO. At large spatial volume, two-body continuum masks resonance state. Extraction of spectral function from correlator C(t) is ill-posed unless a model is made. (Since the low energy continuum dominates at large t). CM89, Maiani Testa 90 GO. For finite spatial volume (L 3 ), two-body continuum is discrete and Lüscher showed how to use the small energy shifts with L of these two-body levels to extract elastic scattering phase shifts. The phase shifts then determine the resonance mass and width. The ρ appears as a distortion of the π n π n energy levels where q = 2πn/L. Lüscher 91 Hadronic physics from the lattice p.5/24

Lattice evaluation of ρ ππ As a check of this approach to unstable particles on a lattice, the coupling of ρ to ππ has been determined from first principles McNeile CM 02 Method is to arrange ρ and ππ state (with definite relative momentum) to be approximately degenerate in energy on a lattice. Then several independent methods allow to determine the transtion amplitude x. Hadronic physics from the lattice p.6/24

Lattice evaluation: ρ ππ Determine coupling constant from lattice (where decay does not proceed) and compare with experiment: method m val m sea ḡ Lattice xt s s 1.40 +47 23 Lattice ρ shift s s 1.56 +21 13 φ K K s u, d 1.5 K Kπ u, d/s u, d 1.44 ρ ππ u, d u, d 1.39 Note lattice has heavier sea quarks than experiment Hadronic physics from the lattice p.7/24

Lattice evaluation of A BC (i) Consider off-diagonal correlator: A B C 0 -X -0 0 t T (ii)t is not observed on a lattice - so sum over t. T t=0 e m(a)t xe m(bc)(t t) ==> xte m(a)t if m(a) m(bc) (iii) so plot (normalised) transition from lattice versus T : slope is x, which can be related to continuum coupling g 2. (iv) like measuring the wave-function overlap - but with no model for the wave-functions. Hadronic physics from the lattice p.8/24

Exotic mesons States with exotic quantum numbers J PC = 1 +, 0 +, 2 +, 0 are definitely not described by the quark model (that predicts C = ( 1) L+S, P = ( 1) L+1 ). Hybrid exotic mesons with qq plus excited glue. Multiquark operators (qqqq). Typical meson creation operators used in lattice QCD calculations of hybrid meson spectroscopy. 1 ψγ j ψ 1 ǫ ijk ψγ 5 F ij ψ 1 + ψγ j F jk ψ 1 + ψγ j γ 5 ψ ψγ 5 ψ where F is the QCD field strength tensor. No mention of potentials or wave functions for Hybrid mesons No mention of flux tubes Hadronic physics from the lattice p.9/24

1 + masses. Summary of lattice results from the MILC collaboration, (hep-lat/0209097). Lattice mostly still predicts lightest 1 + state to be at 1.9(2) GeV. ( σ 0.5 GeV). Current results are (strictly) upper bounds (if decays are ignored), but for light quarks the lattice result at 1.9(2) GeV is not in great agreement with experiment 1.6 GeV. Hadronic physics from the lattice p.10/24

Two recent quenched calculations Some recent lattice QCD calculations quote results close to 1600 MeV. Quenched is not really QCD. The Adelaide group (hep-lat/0509106) computed m 1 + = 1.74(24) GeV from a quenched lattice QCD calculation. The lattice parameters were a 0.13 fm and L 2.6 fm. Lightest pseudoscalar mass 320 MeV. Cook and Fiebig extracted the masses of the 1 + mesons from a quenched lattice QCD calculation using the maximum entropy method. Lattice parameters a t 0.07 fm and L 1.7 fm. They claim to see masses of 1 + at 1400, 1600, and 1900 MeV. The extrapolations of the 1 + mass to the physical quark mass needs physics of decays included. Hadronic physics from the lattice p.11/24

Hybrid meson decay S-wave decay of J PC = 1 + spin-exotic hybrid meson to πb 1. Illustration M(hybrid)=1.6 GeV L=3 fm N f = 2. 2 π + b 1 q = 2π/L Energy 1.5 π + b 1 q = 0 H L = 3fm 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 m q /m s Hadronic physics from the lattice p.12/24

S-wave meson decay: test Test S-wave decays from the lattice: b 1 πω (using method to be used for Hybrid meson decay). McNeile CM 06 Hadronic physics from the lattice p.13/24

Hybrid meson decay: results From lattices with N f = 2 flavours of sea-quark, a hybrid meson state at 2.2(2) GeV with a partial width to πb 1 of 400(120) MeV and to πf 1 of 90(60) MeV. McNeile CM 06 maybe smaller widths in limit m q physical value Hadronic physics from the lattice p.14/24

Flavour singlet mesons OZI-rule etc: QCD evaluation of disconnected quark diagram for 0 + and 1 flavour singlet mesons (η and ω). connected 0 disconnected 0 0 1.2 1 0.8 0.04 0.03 ETMC r 0 /a = 6.61 L = 32a ETMC r 0 /a = 5.22 L = 32a ETMC r 0 /a = 5.22 L = 24a mη GeV 0.6 0.4 ETMC r 0 /a = 6.61 UKQCD r 0 /a = 5.32 ETMC r 0 /a = 5.22 L = 32 ETMC r 0 /a = 5.22 L = 24 UKQCD r 0 /a = 5.04 CP-PACS r 0 /a = 4.49 DWF r 0 /a = 4.28 mω mρ GeV 0.02 0.01 0.2 0 0 0.1 0.2 0.3 0.4 0 0 0.1 0.2 0.3 0.4 m 2 π GeV2 m 2 P GeV2 Hadronic physics from the lattice p.15/24

Scalar Mesons uū + d d, s s, glueball, and meson-meson components are possible for flavour-singlet scalars decays to ππ or ηπ are allowed in dynamical lattice studies. 0 ++ Glueball ππ. Sexton Vaccarino Weingarten 96(Quenched) Glueball mixing with q q meson: hadronic transition Weingarten Lee 98,00(Quenched); McNeile CM 01 ππ is seen in f 0 correlator ETMC 07. Full study needed but disconnected diagram for f 0 ππ is very noisy: start on flavour non-singlets... Q T Hadronic physics from the lattice p.16/24

Scalar Mesons (a 0 ) Explore flavour non-singlet light scalars (a 0 ) Quenched has ghost effects in a 0 ηπ so use N F = 2 Measure b 1 - a 0 mass difference on a lattice: McNiele CM 06; ETMC 08 M(b 1 ) = 1235 MeV implies a 0 is near 1000 MeV r0(m(b1) m(a0)) 1 500 MeV ETMC r 0 = 6.61 ETMC r 0 = 5.22 L = 32 ETMC r 0 = 5.22 L = 24 UKQCD r 0 = 5.04 UKQCD r 0 = 4.75 DWF r 0 = 4.28 CP-PACS r 0 = 3.01 CP-PACS r 0 = 2.65 Expt 0 0 1 2 (r 0 m π ) 2 strange 3 4 Hadronic physics from the lattice p.17/24

Scalar Mesons - decays a 0 πη 8 ; a 0 KK; K 0 Kπ. Coupling similar to phenomenological estimates for a 0 (980) less clear what K 0 is appropriate. Hadronic physics from the lattice p.18/24

Do decays matter? The above analysis shows that q q states mix with two-body states with the same quantum numbers. In the real world, the two-body states are a continuum and nearby states have a predominant influence, especially for S-wave thresholds. For bound states there is an influence of nearby two-body states (eg Nπ on N) which mix to reduce the mass - this is the province of low energy effective theories, especially Chiral perturbation theory. For unstable states (resonances) the influence of the two-body continuum is less simple. For quenched QCD, however, where these two-body states are not coupled (or have the wrong sign as in a 0 ηπ); then the unstable states will be distorted. For instance the ρ will be too heavy since it is not repelled by the heavier ππ states. (with DF we saw the ρ mass decrease) Hadronic physics from the lattice p.19/24

Molecular states? Can lattice QCD provide evidence about possible molecular states: hadrons made predominantly of two hadrons? The prototype is the deuteron: n p bound by π exchange. There are states close to two-body thresholds: f 0 (980); a 0 (980) K K D s (0 + ) D(0 )K B s (0 + ) B(0 )K X(3872) D D Λ(1405) KN N(1535) ηn Some of these cases have been studied for 40 years Isopsin breaking is enhanced by mass splittings in thresholds (eg K 0 K 0 compared to K + K is 8 MeV higher) Hadronic physics from the lattice p.20/24

Molecular states? State near threshold <-> attractive interaction but chicken or egg? Vary m(q 1 ), m(q 2 ): does mass of state track two-body threshold? Explore wavefunction: is it long ranged? Explore coupling of state to two-body channel Molecule (rather than qq q q) needs spatial separation to preserve hadronic constituents. Only π exchange has long range. Examples: deuteron, also BB bound states.cm PP 99 Mesons with one heavy quark are an ideal laboratory for this study. Hadronic physics from the lattice p.21/24

Conclusions Lattice can address hadronic structure: form factors (eg. charge wavefunctions) can be evaluated decay transitions (and mixing transitions) can be evaluated LEC s of Chiral PT approach can be determined hadronic matrix elements are needed to interpret experiment (eg f B relates B meson to b quark, etc..) Hadronic physics involves unstable states. Lattice techniques are being developed to study these. Evidence that spin exotic hybrid meson is around 2 GeV and WIDE Evidence that a 0 (980) is basically a q q state. There is a lot to be learnt beyond mass spectra. Hadronic physics from the lattice p.22/24

Is the lattice 1 + a non-exotic 4 +? Issue recently raised by Dudek et al. (arxiv:0707.4162 [hep-lat]) in charmonium. Symmetries of lattice are subgroup of continuum group, so care is required in construction of operators with J PC. J Lattice Rep 0 A 1 1 T 1 2 E T 2 3 A 2 T 1 T 2 4 A 1 E T 1 T 2 So a lattice T 1 state can have J=1, 3, or 4. In principle this can be resolved by looking for the other degenerate states (in different reps) needed if J > minimum. details of reps see UKQCD hep-lat/9605025. Hadronic physics from the lattice p.23/24

Lattice and spin identification Meyer and Teper studied the J=4,6 glueballs (hep-ph/0409183) in quenched QCD to study Regge trajectories. Used continuum like interpolating operators and range of lattice spacings. Able to cleanly extract states of large J. 7 4D SU(3) glueballs 6 5 4 J 3 2 PC = ++ PC = -+ 1 PC = +- PC = -- 0 0 2 4 6 8 10 12 14 16 18 20 t = M 2 / 2πσ Current results for the masses of 1 + states are upper bounds (if decays are ignored), but for light quarks the pressing issue is that the lattice result at 1.9(2) GeV is not in great agreement with experiment 1.6 GeV. Hadronic physics from the lattice p.24/24