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Exploring the Spectrum of QCD Using a Space-Time Lattice

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Exploring the Spectrum of QCD Using a Space-Time Lattice ExploringExploring thethe spectrumspectrum ofof QCDQCD usingusing aa spacespace--timetime latticelattice Colin Morningstar (Carnegie Mellon University) New Theoretical Tools for Nucleon Resonance Analysis Argonne National Laboratory August 31, 2005 August 31, 2005 Exploring spectrum (C. Morningstar) 1 OutlineOutline z spectroscopy is a powerful tool for distilling key degrees of freedom z calculating spectrum of QCD Æ introduction of space-time lattice spectrum determination requires extraction of excited-state energies discuss how to extract excited-state energies from Monte Carlo estimates of correlation functions in Euclidean lattice field theory z applications: Yang-Mills glueballs heavy-quark hybrid mesons baryon and meson spectrum (work in progress) August 31, 2005 Exploring spectrum (C. Morningstar) 2 MonteMonte CarloCarlo methodmethod withwith spacespace--timetime latticelattice z introduction of space-time lattice allows Monte Carlo evaluation of path integrals needed to extract spectrum from QCD Lagrangian LQCD Lagrangian of hadron spectrum, QCD structure, transitions z tool to search for better ways of calculating in gauge theories what dominates the path integrals? (instantons, center vortices,…) construction of effective field theory of glue? (strings,…) August 31, 2005 Exploring spectrum (C. Morningstar) 3 EnergiesEnergies fromfrom correlationcorrelation functionsfunctions z stationary state energies can be extracted from asymptotic decay rate of temporal correlations of the fields (in the imaginary time formalism) Ht −Ht z evolution in Heisenberg picture φ ( t ) = e φ ( 0 ) e ( H = Hamiltonian) z spectral representation of a simple correlation function assume transfer matrix, ignore temporal boundary conditions focus only on one time ordering insert complete set of 0 φφ(te) (0) 0 = ∑ 0 Htφ(0) e−Ht nnφ(0) 0 energy eigenstates n (discrete and continuous) 2 −−()EEnn00t −−()EEt ==∑∑neφ(0) 0 Ane nn z extract A 1 and E 1 − E 0 as t → ∞ (assuming 0 φ ( 0 ) 0 = 0 and 1 φ ( 0 ) 0 ≠ 0) August 31, 2005 Exploring spectrum (C. Morningstar) 4 EffectiveEffective massmass ⎛ C(t) ⎞ z the “effective mass” is given by meff (t) = ln⎜ ⎟ ⎝ C(t +1) ⎠ z notice that (take E 0 = 0 ) ⎛ A e−E1t + A e−E2t +⎞ lim m (t) = ln⎜ 1 2 ⎟ → ln e−E1 = E t→∞ eff ⎜ −E1(t+1) ⎟ 1 ⎝ A1e + ⎠ z the effective mass tends to the actual mass (energy) asymptotically z effective mass plot is convenient visual tool to see signal extraction seen as a plateau z plateau sets in quickly for good operator z excited-state contamination before plateau August 31, 2005 Exploring spectrum (C. Morningstar) 5 ReducingReducing contaminationcontamination z statistical noise generally increases with temporal separation t z effective masses associated with correlation functions of simple local fields do not reach a plateau before noise swamps the signal need better operators better operators have reduced couplings with higher-lying contaminating states z recipe for making better operators crucial to construct operators using smeared fields – link variable smearing – quark field smearing spatially extended operators use large set of operators (variational coefficients) August 31, 2005 Exploring spectrum (C. Morningstar) 6 PrincipalPrincipal correlatorscorrelators z extracting excited-state energies described in C. Michael, NPB 259, 58 (1985) Luscher and Wolff, NPB 339, 222 (1990) z can be viewed as exploiting the variational method + z for a given N × N correlator matrix Ct αβ () = 0 O α ( t ) O β ( 0 ) 0 one defines the N principal correlators λα (t,t0 ) as the eigenvalues of −1/ 2 −1/ 2 C(t0 ) C(t) C(t0 ) where t 0 (the time defining the “metric”) is small −(t−t0 )Eα −t∆Eα z can show that limt→∞ λα (t,t0 ) = e (1+ e ) eff ⎛ λα (t,t0 ) ⎞ z N principal effective masses defined by m α ( t ) = ln ⎜ ⎟ ⎝ λα (t +1,t0 ) ⎠ now tend (plateau) to the N lowest-lying stationary-state energies August 31, 2005 Exploring spectrum (C. Morningstar) 7 PrincipalPrincipal effectiveeffective massesmasses z just need to perform single-exponential fit to each principal correlator to extract spectrum! can again use sum of two-exponentials to minimize sensitivity to t mi n z note that principal effective masses (as functions of time) can cross, approach asymptotic behavior from below z final results are independent of t 0 , but choosing larger values of this reference time can introduce larger errors August 31, 2005 Exploring spectrum (C. Morningstar) 8 UnstableUnstable particlesparticles (resonances)(resonances) z our computations done in a periodic box momenta quantized discrete energy spectrum of stationary states Æ single hadron, 2 hadron, … z scattering phase shifts Æ resonance masses, widths (in principle) deduced from finite-box spectrum B. DeWitt, PR 103, 1565 (1956) (sphere) M. Luscher, NPB364, 237 (1991) (cube) z more modest goal: “ferret” out resonances from scattering states must differentiate resonances from multi-hadron states avoided level crossings, different volume dependences know masses of decay products Æ placement and pattern of multi-particle states known resonances show up as extra states with little volume dependence August 31, 2005 Exploring spectrum (C. Morningstar) 9 ResonanceResonance inin aa toytoy modelmodel (I)(I) z O(4) non-linear σ model (Zimmerman et al, NPB(PS) 30, 879 (1993)) 44 42 Sx=−2(κµ∑∑Φaa)Φ (x+ ˆ)+J∑Φ (x),∑Φa(x)=1 xxµ==11a August 31, 2005 Exploring spectrum (C. Morningstar) 10 ResonanceResonance inin aa toytoy modelmodel (II)(II) z coupled scalar fields: (Rummukainen and Gottlieb, NPB450, 397 (1995)) 22 Sd=∂1 42xφ +mφλ2+φ4+∂ρ+m2ρλ2+ρ4+gρφ2 2 ∫ (()µπ ( µ) πρ ) g = 0 g = 0.008 g = 0.021 August 31, 2005 Exploring spectrum (C. Morningstar) 11 YangYang--MillsMills SU(3)SU(3) GlueballGlueball SpectrumSpectrum z pure-glue mass spectrum known C. Morningstar and M. Peardon, Phys. Rev. D 60, 034509 (1999) still needs some “polishing” improve scalar states z “experimental” results in simpler world (no quarks) to help build models of gluons z mass ratios predicted, overall scale is not z mass gap with $1 million bounty (Clay mathematics institute) z glueball structure constituent gluons vs flux loops? −1 PC r0 = 410(20) MeV, states labeled by J August 31, 2005 Exploring spectrum (C. Morningstar) 12 GlueballsGlueballs (bag(bag model)model) z qualitative agreement with bag model constituent gluons are TE or Carlson, Hansson, Peterson, PRD27, 1556 (1983); J. Kuti (private communication) TM modes in spherical cavity Hartree modes with residual perturbative interactions center-of-mass correction 1983 1993 light baryon static-quark spectroscopy potential α s 1.0 0.5 1 B 4 230 MeV 280 MeV z recent calculation using another constituent gluon model shows qualitative agreement Szczepaniak, Swanson, PLB577, 61 (2003) August 31, 2005 Exploring spectrum (C. Morningstar) 13 GlueballsGlueballs (flux(flux tubetube model)model) z disagreement with one particular string model Isgur, Paton, PRD31, 2910 (1985) z future comparisons: with more sophisticated string models (soliton knots) AdS theories, duality August 31, 2005 Exploring spectrum (C. Morningstar) 14 InclusionInclusion ofof quarkquark loopsloops z scalar glueball results 2002 quark masses near strange z still exploratory z difficult to get adequate statistics z light quarks problematic z mixing, resonances no correlation matrices SESAM: PRD62, 054503 (2000) UKQCD: PRD65, 014508 (2002) HEMCGC: PRD44, 2090 (1991) August 31, 2005 Exploring spectrum (C. Morningstar) 15 UnquenchedUnquenched massesmasses z unquenched analysis (Hart, Teper, PRD65, 034502 (2002)) 1 z Wilson gauge, clover fermion action N f = 2, a ≈ 0.1fm, mq ≥ 2 ms z tensor glueball mass same as pure-gauge z scalar mass suppression: 0.85 of pure-gauge not finite volume effect independent of quark mass! Æ lattice artifact (another “curve ball”) most likely explanation: fermion action adds “adjoint piece” z quarkonium states ignored August 31, 2005 Exploring spectrum (C. Morningstar) 16 ExcitationsExcitations ofof staticstatic quarkquark potentialpotential z gluon field in presence of static quark-antiquark pair can be excited z classification of states: (notation from molecular physics) magnitude of glue spin projected onto molecular axis Λ = 0,1,2,... = Σ,Π,∆,... charge conjugation + parity about midpoint η = g (even) several higher levels not shown = u (odd) + Π g , ∆ g, ∆u , Σu , chirality (reflections in plane ∗ − Φu ,Πu ,Σ g ,... containing axis) Σ+ , Σ− Π,∆,…doubly degenerate (Λ doubling) Juge, Kuti, Morningstar, PRL 90, 161601 (2003) August 31, 2005 Exploring spectrum (C. Morningstar) 17 InitialInitial remarksremarks z viewpoint adopted: the nature of the confining gluons is best revealed in its excitation spectrum z robust feature of any bosonic string description: Nπ / R gaps for large quark-antiquark separations z details of underlying string description encoded in the fine structure z study different gauge groups, dimensionalities z several lattice spacings, finite volume checks z very large number of fits to principal correlators Æ web page interface set up to facilitate scrutinizing/presenting the results August 31, 2005 Exploring spectrum (C. Morningstar) 18 StringString spectrumspectrum z spectrum expected for a non-interacting bosonic string at large R standing waves: m = 1 , 2 , 3 , … with circular polarization ± occupation numbers: nm+ , nm− energies E E = E0 + Nπ / R string quantum number N ∞ N = ()n + n spin projection ∑ m+ m− Λ m=1 ∞ CP ηCP Λ = ∑()nm+ − nm− m=1 N ηCP = (−1) August 31, 2005 Exploring
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