"Lattice QCD calculations of the excitedstate spectrum, and the low-energy degrees of freedom of QCD

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1 "Lattice QCD calculations of the excitedstate spectrum, and the low-energy degrees of freedom of QCD David Richards Jefferson Laboratory/Hadron Spectrum Collaboration Kyoto, 26 Feb, 205

2 Outline Spectroscopy: theory and experiment Lattice QCD Spectroscopy Recipe Book Results and insight Light-meson spectroscopy and isoscalar Decay constants Charmonium Baryons, and the search for gluons Strong decays Summary

3 Spectroscopy Classic means of determining underlying degrees of freedom in a theory. Probe the strong interaction and its underlying field theory Quantum Chromodynamics (QCD) Spectroscopy and QCD What are the key degrees of freedom describing the bound states - protons, neutrons, pions,???? How do they change as we vary the quark mass - Charmonium? What is the origin of confinement, describing 99% of observed matter? If QCD is correct and we understand it, expt. data must confront ab initio calculations What is the role of the gluon in the spectrum search for exotics

4 Meson Spectrum S2 L S Simple quark model (for neutral mesons) admits only certain values of J PC P = ( ) l+ C = ( ) l+s Exotic Mesons are those whose values of J PC are in accessible to quark model: 0 +-, -+, 2 +- Multi-quark states: Hybrids with excitations of the flux-tube Study of hybrids: revealing gluonic degrees of freedom of QCD. Glueballs: purely, or predominantly, gluonic states 4

5 Baryon Spectroscopy No baryon exotics, ie quantum numbers not accessible with simple quark model; but may be hybrids! Nucleon Spectroscopy: Quark model masses and amplitudes states classified by isospin, parity and spin. Real%parts%of%N*%pole%values%% Ours PDG%4* N*%with%3*,%4* PDG%3* N*%with%*,%2* PDG 8 5 Ours 6 Missing, because our pictures do not capture correct degrees of freedom? Do they just not couple to probes? q 3 > q 2 q> EBAC: Kamano, Nakamura, Lee, Sato - 202

6 Quantum Chromodynamics (QCD) QED Photon, γ Charged particles, e, µ, u, d, Photon is neutral α e =/37 QCD Gluons, G Quarks: u, d, s, c, b, t Gluons carry color charge Theory is non-abelian α s ~O() Infrared Slavery Lattice QCD Asymptotic Freedom Pert. Theory Gluons in spectrum! Gluons in three-jet event

7 Lattice QCD Lattice QCD enables us to undertake ab initio computations of many of the low-energy properties of QCD Continuum Euclidean space time replaced by four-dimensional lattice 24 3 x 28 for talk today ψ, ψ are Grassmann Variables Importance Sampling

8 Hierarchy of Computations Capability Computing - Gauge Generation Capacity Computing - Observable Calculation Highly regular problem, with simple boundary conditions very efficient use of massively parallel computers using data-parallel programming.

9 0 2 Mflops / $ Science per Dollar for (some) LQCD Capacity Applications Optimized LQCD Clusters A 250 node cluster, optimized to a limited number of science problems is a cost effective platform. Accelerators beat that by a factor of ten! QCDSP Cluster Performance USQCD Clusters QCDOC GPUs have proven to be especially cost effective Intel Xeon Phi is the newest accelerator technology with promise? 2007 Japanese Earth Simulator BlueGene/L BlueGene/P Vector Supercomputers trend line IBM BlueGene/Q In one form or another, massively parallel is the future!

10 Low-lying Hadron Spectrum Benchmark of LQCD Durr et al., BMW Collaboration Science 2008 Control over: Quark-mass dependence Continuum extrapolation finite-volume effects (pions, resonances)

11 Variational Method Subleading terms Excited states Construct matrix of correlators with judicious choice of operators C ij (t, 0) = V 3 X Z N i ~x,~y ho i (~x, t)o j (~y, 0)i = X N hn O i (0) 0i Delineate contributions using variational method: solve C(t)v (N) (t, t 0 )= N (t, t 0 )C(t 0 )v (N) (t, t 0 ). N (t, t 0 )! e E N (t t 0 ), Zi N Zj N 2E N e E N t Eigenvectors, with metric C(t 0 ), are orthonormal and project onto the respective states v (N 0 ) C(t 0 )v (N) = N,N 0 Z N i = p 2m N e m N t 0 /2 v (N) j C ji (t 0 ).

12 Challenges To appreciate difficulty of extracting excited states, need to understand signal-to-noise ratio in two-point functions. Consider correlation function: C(t) = 0 O(t)O(0) 0 Then the fluctuations behave as e Et DeGrand, Hecht, PRD46 (992) 2 (t) 0 O(t)O(0) 2 0 C(t) 2 e 2m t Signal-to-noise ratio degrades with increasing E - Solution: anisotropic lattice with lattice spacing at < as For heavy quarks m at << Cubic symmetry of lattices M 2 M E M T2 a 2

13 Glueball Spectroscopy - I Triple-gluon vertex - Pure Yang-Mills spectrum. Predicts existence of bound states. Morningstar, Peardon 97,99 Observe emergence of degeneracies

14 Spectroscopy with Quarks Anisotropic lattices - to precisely resolve energies Variational method - with sufficient operator basis to delineate states Identification of spin - Many Values of Lattice Spacing? Anisotropic fermion action Edwards, Joo, Lin, PRD78 (2008) 8 < S G [U] = Nc g : X x,s>s 0 S F [U,, ] = X x (x) ũ t ( 2 " 2 apple 5 3u 4 P ss 0 s g + f ũ t ũ 2 s 2u 6 s R ss 0 ũ t ˆm 0 + Ŵt + X f X s s Ŵ s + X x,s ts ˆFts + f ũ 3 s apple X 4 3u 2 su 2 t s<s 0 ss 0 ˆFss 0 P st 2u 4 su 2 R st t #) (x). Two anisotropy parameters to tune, in gauge and fermion sectors 9 = ; =3.5 g = 0 f = 0 / Dispersion Relation a s ' 0.2 fm a t ' fm

15 Anisotropic Clover Generation - I Tuning performed for three-flavor theory Challenge: setting scale and strange-quark mass Lattice coupling fixed Proportional to ms to LO ChPT Omega Express physics in (dimensionless) (l,s) coordinates H-W Lin et al (Hadron Spectrum Collaboration), PRD79, (2009 ) Proportional to ml to LO ChPT

16 Anisotropic Clover II Low-lying spectrum: agrees with experiment to 0% m 400 MeV No chiral extrapolations - resonances

17 Variational Method: Meson Operators Aim: interpolating operators of definite (continuum) JM: O JM Starting point Introduce circular basis: h0 O JM J 0,M 0 i = Z J J,J ( x, 0 M,M 0 t) Di D j... ( x, t)! D m= = i p 2!D x! D m=0 = i! D z! D m=+ = i! D y pi!d 2 x + i D! y. Straighforward to project to definite spin - for example J = 0,, 2 ( D [] J= )J,M = X,m ;,m 2 J, M! m D m2. m,m 2 Use projection formula to find subduction under irrep. of cubic group - operators are closed under rotation! Irrep, Row Irrep of R in Λ Action of R 7

18 Correlation functions: Distillation Use the new distillation method. Observe Eigenvectors of Laplacian Truncate sum at sufficient i to capture relevant physics modes we use 64: set weights f to be unity Meson correlation function Includes displacements Decompose using distillation operator as M. Peardon et al., PRD80, (2009) Perambulators Momentum Projection at Source and Sink!

19 Distillation - II Meson correlation functions N 3 Baryon correlation functions N 4 Stochastic sampling of eigenvectors - stochastic LaPH Morningstar et al, Phys.Rev.D83:4505,20 Alternative idea: simpler orthonormal basis for the smearing function L X i (x) i (y) = (x y), X i (x) j (x) = ij x i Colour-wave basis i(x) =e ipx s,s 0 c,c 0 Z.Brown, K.Orginos, arxiv:20.953

20 Identification of Spin Hadspec collab. (dudek et al), , PRD82, C ij = dim( ) X 0 O [J] i( ) O[J] j( ) 0 Operators know their parentage Exploit to determine spins

21 Isovector Meson Spectrum - I Dudek et al, PRL 03:26200 (2009) Isovector spectrum with quantum numbers reliably identified Nf = 3 theory - three mass-degenerate strange quarks { Exotic

22 Isovector Meson Spectrum - II Nf = 2+, mπ = 397 MeV States with Exotic Quantum Numbers previous studies quenched dynamical Dudek, Edwards, DGR, Thomas, arxiv:

23 Interpretation of Meson Spectrum Z N i = p 2m N e m N t 0 /2 v (N) j C ji (t 0 ). D [2] J= Vanishes for unit gauge field In each Lattice Irrep, state dominated by operators of particular J

24 rd excited state is dominantly hybrid? with some look at the overlaps nd excited state is dominantly with some st excited state is dominantly with some hybrid? Anti-commutator of covariant derivative: vanishes for unit gauge! ground state is dominantly Use lattice QCD to build phenomenology of bound states Dudek, arxiv:06.555

25 Isoscalar Meson Spectrum negative parity positive parity Dudek et al, arxiv: , arxiv: exotics isoscalar isovector YM glueball Diagonalize in 2x2 flavor space C = C +2D 2 D s 2 D s C ss + D ss J. Dudek et al., PRD73, 502 Spin-identified single-particle spectrum: states of spin as high as four. Hidden flavor mixing angles extracted - except 0 -+, ++ near ideal mixing First determination of exotic isoscalar states: comparable in mass to isovector 25

26 Charmonium Operator construction follows light-quark Liuming Liu et al, arxiv: Ignore annihilation contributions Charm quark mass set from ηc with scale set using Ω D s D s Exotics atm 0.65 DD Volume-dependence small quote results at larger volume 26

27 Charmonium - II D-wave Ds Ds Appearance of multiplets from n 2s+ LJ quark potential model M-Mhc HMeVL P-wave DD 500 S-wave Hybrid supermultiplet Z 6 4,3 Hyb 8p NR, r NR < D J=,3 Hyb J M-Mhc HMeVL D s D s DD HT 2 L 2 -+ HEL

28 f X = m 2 X Pseudoscalar Decay Constants Expectation from WT identity Compute in LQCD m q h0 Xi e.g. Chang, Roberts, Tandy, arxiv: f N 0,N 0 McNeile and Michael, hep-lat/

29 Pseudoscalar Decay - II Axial-vector current mixes on lattice E. Mastropas, DGR, PRD(204) apple A I 4 =(+ma t m ) A U 4 4 ( )a t@ 4 P C A4,N (t) = X h0 A 4 (~x, t) N (~y, 0) 0i V!e m N t m N f N 3 ~x,~y where N = p 2m N e m N t 0 /2 v (N) i O i 29

30 Look at overlaps with different classes of operators a0xd3_j3_j0 J0_A axd3_j3_j J0_A bxd_j J0_A bxd3_j30_j J0_A bxd3_j32_j J0_A bxd3_j32_j3 J4_A pion_2xd0_j0 J0_A pion_2xd2_j0 J0_A pionxd0_j0 J0_A pionxd2_j0 J0_A rho_2xd2_j J0_A rhoxd2_j J0_A (2) 0.274(3) 0.367(3) 0.42() 0.499(5) Strong suppression for second HYBRID state 30

31 Excited Baryon Spectrum - I Construct basis of 3-quark interpolating operators in the continuum: 7 J= N M Flavor x Spin x Orbital Subduce to lattice irreps: M D[2] L=2,S 2.0 O [J] n,r = X M S J,M n,r O[J,M] : = G g/u,h g/u,g 2g/u.8.6 H u R.G.Edwards et al., arxiv: lattices m = 524, 444 and 396 MeV Observe remarkable realization of rotational symmetry at hadronic scale: reliably determine spins up to 7/2, for the first time in a lattice calculation Continuum antecedents 3

32 Excited Baryon Spectrum - II Broad features of SU(6)xO(3) symmetry. Counting of states consistent with NR quark model. Inconsistent with quark-diquark picture or parity doubling. [56,0 + ] [70, - ] [56,0 + ] [70, - ] [70, 0 + ], [56, 2 + ], [70, 2 + ], [20, + ] N /2+ sector: need for complete basis to faithfully extract states 32

33 Roper Resonance Kamleh et al., arxiv: Look at Radial wave function 2S state 33

34 Hybrid Baryon Spectrum Original analysis ignore hybrid operators of form D [2] l=,m Dudek, Edwards, arxiv:

35 Putting it Together Common mechanism in meson and baryon hybrids: chromomagnetic field with Eg GeV Subtract ρ Subtract N 35

36 Flavor Structure One derivative Two derivative 36

37 C ij = dim( ) X 0 O [J] i( ) O[J] j( ) 0 Block-diagonal in spin-flavor 37

38 Examine Flavor structure of baryons constructed from u, d s quarks. Can identify predominant flavor for each state: Yellow (0F), Blue (8F), Beige (F). SU(6) x O(3) Counting Presence of hybrids characteristic across all +ve parity channels: BOLD Outline R. Edwards et al., Phys. Rev. D87 (203)

39 Recent extension to doubly-charmed baryons Padmanath et al., arxiv:

40 The elephant in the room States unstable under strong interactions Meson spectrum on two volumes: dashed lines denote expected (noninteracting) multi-particle energies. Allowed two-particle contributions governed by cubic symmetry of volume Calculation is incomplete.

41 Momentum-dependent I = 2 ππ Phase Shift Dudek et al., Phys Rev D83, (20) Luescher: energy levels at finite volume phase shift at corresponding k O, ( p ) = X Operator basis m Total momentum zero - pion momentum ±p 0.40 S,m, X ˆp Y m (ˆp) O (p)o ( p)

42 Momentum-dependent I = 2 ππ Phase Shift Luescher: energy levels at finite volume phase shift at corresponding k i Dudek et al., Phys Rev D83, det h e 2i (k) U k L 2 = (20) Dudek, Edwards, Thomas, arxiv: Matrix in l lattice irrep Moving ππ system far more momenta below inelastic threshold Optimized single-pion interpolating operators more precise determination of energies

43 Energy Levels for Scattering States Slide: J. Dudek

44 Resonant I = ππ Phase Shift Feng, Renner, Jansen, PRD83, PACS-CS, PRD84, Alexandru et al Lang et al., PRD84, Dudek, Edwards, Thomas, Phys. Rev. D 87, (203) Extend to inelastic channels: Guo et al, Briceno et al., 44

45 det h First - and Successful - inelastic ij JJ 0 + i i t (J) ij (E cm) i P JJ 0 + im ~ JJ (p 0 i L) =0 Parametrized as phase shift + inelasticity t ii = ( e2i i ) 2i i,t ij = p 2 e i( i + j ) 2 p i j Dudek, Edwards, Thomas, Wilson, PRL, PRD 45

46 Pole positions in complex plane

47 Summary Spectroscopy of excited states affords an excellent theatre in which to study QCD in low-energy regime. Determining the quantum numbers and the study of the single-hadron states a solved problem Lattice calculations used to construct new phenomenology of QCD Quark-model like spectrum, common mechanism for gluonic excitations in mesons and baryons. LOW ENERGY GLUONIC DOF Prediction - there are exotics in a range accessible to the 2 GeV Upgrade of Jefferson Lab! Next step for lattice QCD: Calculations at closer-to-physical pion masses - isotropic lattices Baryons a challenge. Properties - radiative transitions, form factors. Theoretical work! Hansen and Briceno

48 Variational Method + Distillation Single distilled correlator Fit to C(t) =Ae m 0t + Be m0 t and plot C(t)/e m 0t Fit to 0 (t, t 0 )=( A)e m 0(t t 0 ) + Ae m0 (t t 0 ) and plot C(t)/e m 0(t t 0 ) Reduced contribution of excited states isotropic lattice a 2 ' 0.75 fm, SU(3)

49 And for Rho