Meson Spectroscopy and Resonances Sinéad Ryan School of Mathematics, Trinity College Dublin, Ireland Hadron 2011, 13 th June 2011
Lattice QCD Lattice - a nonperturbative, gauge-invariant regulator for QCD Nielson-Ninomiya theorem ⇒ Lattice spacing a chirally symmetric quarks missing, Quarks fields ��� ��� ��� ��� ��� ��� ��� ��� on sites ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� but can discretise quarks by ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� �� �� ��� ��� ��� ��� ��� ��� �� �� ��� ��� trading-off some symmetries. ��� ��� ��� ��� �� �� ��� ��� ��� ��� ��� ��� �� �� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� In finite volume, V = L 4 , finite d.o.f ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� Gauge fields on links ��� ��� ��� ��� ��� ��� ��� ��� and path-integral is large but finite ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� integral. Wick rotation, analytic continuation t → iτ , − i ℏ S → i ℏ S Enables importance sampling ie Monte Carlo Lose direct access to dynamical properties of the theory like decay widths.
The current landscape From 2001 [Ukawa, 2001: the “Berlin Wall”] � − 6 � m π × L 5 × a − 7 C flops ∝ m ρ to 2009 [Giusti, 2006] � − 2 � m π × L 5 × a − 7 C flops ∝ m ρ dramatic improvements in scaling with quark mass [Hasenbusch ’01, Lüscher ’03,04]. With fall in cost of CPU cores, simulations at physical quark masses possible.
The current landscape C. Hoelbling, Lattice 2010 arXiv:1102.0410 ETMC '09 (2) 6 ETMC '10 (2+1+1) MILC '10 QCDSF '10 (2) 5 0.1% QCDSF-UKQCD '10 BMWc '10 PACS-CS '09 4 RBC/UKQCD '10 0.3% L[fm] JLQCD/TWQCD '09 HSC '08 3 BGR '10 (2) 1% 2 1 100 200 300 400 500 600 700 M π [MeV] Dynamical simulations with N f = 2 or 2 + 1 Large volumes, L ≥ 3fm ⇒ O ( 1 %) on m π . Light quark masses, now close to or at m π . Lattice spacing, continuum extrapolations or
Disclaimer Not a review talk. I will discuss challenges and recent progress, showing results from the Hadron Spectrum Collaboration and others. Other lattice talks at this meeting Daniel Mohler 13/6 (17:30) James Zanotti 14/6 (10.30) Bernhard Musch 14/6 (14:30) Robert Edwards 15/6 (12.30) [Baryon Spectroscopy and Resonances]
Disclaimer Not a review talk. I will discuss challenges and recent progress, showing results from the Hadron Spectrum Collaboration and others. Other lattice talks at this meeting Daniel Mohler 13/6 (17:30) James Zanotti 14/6 (10.30) Bernhard Musch 14/6 (14:30) Robert Edwards 15/6 (12.30) [Baryon Spectroscopy and Resonances] Plan spectroscopy: methods, challenges and solutions results: light and charm meson spectroscopy resonances: challenges and possible solutions recent results for light meson resonances
Spectroscopy
Spectroscopy - making measurements 100 10 1 0.1 Energy of a (colorless) QCD state 0.01 0.001 0.0001 log (C(t) ) 1e-05 extracted from a two-point function in 1e-06 1e-07 1e-08 1e-09 Euclidean time, C ( t ) = 〈 ϕ ( t ) | ϕ † ( 0 ) 〉 . 1e-10 1e-11 1e-12 1e-13 1e-14 0 10 20 30 40 50 60 70 80 90 100 110 120 130 a t /t Inserting a complete set of states, lim t → ∞ C ( t ) = Ze − E 0 t . Observing the exponential fall of C ( t ) at large t , the energy can be measured.
Spectroscopy - making measurements 100 10 1 0.1 Energy of a (colorless) QCD state 0.01 0.001 0.0001 log (C(t) ) 1e-05 extracted from a two-point function in 1e-06 1e-07 1e-08 1e-09 Euclidean time, C ( t ) = 〈 ϕ ( t ) | ϕ † ( 0 ) 〉 . 1e-10 1e-11 1e-12 1e-13 1e-14 0 10 20 30 40 50 60 70 80 90 100 110 120 130 a t /t Inserting a complete set of states, lim t → ∞ C ( t ) = Ze − E 0 t . Observing the exponential fall of C ( t ) at large t , the energy can be measured. Excited state energies from a matrix of correlators: C ij ( t ) = 〈 ϕ i ( t ) | ϕ † j ( 0 ) 〉 . Solving a generalised eigenvalue problem C ( t 1 ) v = λC ( t 0 ) v gives lim ( t 1 − t 0 ) → ∞ λ n = e − E n ( t 1 − t 0 ) .
Spectroscopy - making measurements Lattice operators are bilinears with path-ordered products between quark and anti-quark fields; different offsets, connecting paths and spin contractions give different projections into lattice symmetry channels. Need ops with good overlap onto low-lying spectrum Good idea to smooth fields spatially before measuring: smearing
Spectroscopy - making measurements Lattice operators are bilinears with path-ordered products between quark and anti-quark fields; different offsets, connecting paths and spin contractions give different projections into lattice symmetry channels. Need ops with good overlap onto low-lying spectrum Good idea to smooth fields spatially before measuring: smearing Distillation [Hadron Spectrum Collab.] Reduce the size of space of fields (on a time-slice) preserving important features. all elements of the (reduced) quark propagator can be computed: allows for many operators, disconnected diagrams and multi-hadron operators. combined with stochastic methods to improve volume scaling.
Spectroscopy The “naive” spectroscopy of heavy and light mesons As well as control of usual lattice systematics ( a → 0, L → ∞ , m π realistic) need statistical precision at % percent level reliable spin identification heavy quark methods
Spectroscopy The “naive” spectroscopy of heavy and light mesons As well as control of usual lattice systematics ( a → 0, L → ∞ , m q ∼ m π ) need statistical precision at percent level to include multi-hadrons and study resonances reliable spin identification heavy quark methods
statistical precision at percent level “distillation” - a new approach to simulating correlators. Particularly good for spectroscopy. enables precision determination of disconnected diagrams, crucial for isoscalar spectroscopy large bases of interpolating operators now feasible, for better determination of excited states via variational method.
Spectroscopy The “naive” spectroscopy of heavy and light mesons As well as control of usual lattice systematics ( a → 0, L → ∞ , m π realistic) need statistical precision at % percent level reliable spin identification understanding symmetries and connection between lattice and continuum designing operators with overlap onto J PC of interest. heavy quark methods
Reliable spin identification Continuum: states classified by irreps ( J P ) of O ( 3 ) . The lattice breaks O ( 3 ) → O h . ( g, u ) ( g , u ) ( g , u ) ( g , u ) , E ( g , u ) , T O h has 10 irreps: { A , A , T } 1 2 1 2 Continuum spin assignment then by subduction J 0 1 2 3 4 . . . 1 1 A 1 . . . A 2 1 . . . E 1 1 . . . 1 1 1 T 1 . . . T 2 1 1 1 . . . Design good operators: start from continuum, “latticize” ( D latt for D ) continuum operators. These lattice operators subduced from J should have good overlap with states of continuum spin J . Study overlaps ( Z ).
Reliable spin identification - overlaps Hadron Spectrum Collaboration, 2010 overlaps for J −− 16 3 lattice m π ≈ 700 MeV.
Spectroscopy The “naive” spectroscopy of heavy and light mesons As well as control of usual lattice systematics ( a → 0, L → ∞ , m π realistic) need statistical precision at % percent level reliable spin identification heavy quark methods
Heavy quarks in lattice QCD O ( am Q ) errors are significant for charm and large for bottom. These sectors require particular methods. Relativistic actions Isotropic ( a s = a t ): Effective Theories needs very fine lattices. NRQCD: m c not heavy Working well for charm, enough? Good for extended to (nearly) bottomonium. bottom Fermilab: works well [arXiv:1010.3848]. but difficult to improve. Anisotropic ( a s � = a t ): Also works for reduce relevant bottomonium. [See talk temporal a t m Q errors. by Mohler] Works well for charm (see later). In general, O ( am Q ) can be controlled and methods have been shown to agree.
Results
Results: Light Isovector Spectrum Hadron Spectrum Collaboration, 2010 1.6 1.6 1.6 1.4 1.4 1.4 1.2 1.2 1.2 1.0 1.0 1.0 0.8 0.8 0.8 0.6 0.6 0.6 0.4 0.4 0.4
Results: isovector exotic summary 2.5 2.0 previous 1.5 studies quenched dynamical 1.0 0.1 0.2 0.3 0.4 0.5 0.6 Recent isovector exotics compared with older results. Note the improvement in precision.
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