Excited(ing) State Spectroscopy in Lattice QCD John Bulava PH Dept. - PowerPoint PPT Presentation

Excited(ing) State Spectroscopy in Lattice QCD John Bulava PH Dept. - TH Division CERN Aug. 30 th , 2012 GGI Workshop - New Frontiers in Lattice Gauge Theory

why study excited states? Where are the ‘missing’ reso- Mass/(MeV/c 2 ) nances? ( N ∗ ) N(I=1/2) ∆ (I=3/2) QM QM exp exp H 3,11 (2420) 2400 ◮ Compare experiment (PDG F 37 (2390) G 19 (2250) D 35 (2350) H 19 (2220) H 39 (2300) D 15 (2200) G 17 (2190) 2200 ’09) to quark model P 11 (2100) S 31 (2150) S 11 (2090) F 35 (2000) D 13 (2080) F 37 (1950) D 33 (1940) F 15 (2000) 2000 (Capstick, Roberts ’00) D 35 (1930) F 17 (1990) P 33 (1920) P 13 (1900) P 31 (1910) P 13 (1720) F 35 (1905) P 11 (1710) 1800 S 31 (1900) ◮ Too many d.o.f in the QM? D 13 (1700) P 31 (1750) F 15 (1680) D 33 (1700) D 15 (1675) S 11 (1650) S 31 (1620) 1600 P 33 (1600) Reduced by diquarks (Jaffe) S 11 (1535) D 13 (1520) P 11 (1440) 1400 ◮ Experiment is mostly P 33 (1232) 1200 → N π . States which couple 1000 weakly? N ∗ program at P 11 (939) JLab. Where are the ‘QCD Exotica’? ◮ Hybrids?: Charmonia, GLuEX, BESIII ◮ Tetraquarks?

Spectroscopy in Lattice QCD ◮ Finite a , L , and T. Calculate Euclidean n-point functions ◮ Spectral rep. of two point functions: A n e − E n t + O ( e − ET ) , C 2 pt ( t ) = � 0 |O ( t ) ¯ � O (0) | 0 � = n A n = |� 0 | ˆ O| n �| 2 What are ‘Resonances’? ◮ Def: Poles in the S-matrix on 2nd Riemannian sheet ◮ Often show up as ‘bumps’ in the cross section Maiani-Testa No Go Theorem: S-matrix elements cannot be obtained from (I.V.) Euclidean correlators (except in principle at threshold).

K¨ all´ en-Lehmann representation: � ∞ 1 d µ 2 ρ ( µ 2 ) ∆( p ) = p 2 − µ 2 + i ǫ 0 In I.V., ρ ( µ 2 ) has δ -functions, and a continuum above threshold. In F.V. ρ ( µ 2 ) is discrete above threshold. Behavior of F.V. energies below inelastic thresholds is well known (L¨ uscher ‘86). Generalizations: ◮ Moving frames: Gottlieb, Rummukainen ‘95 ◮ Multiple 2-particle channels: Liu, Feng, He ‘05 ◮ ... ◮ 2 Non-identical particles, moving frame: Leskovec, Prelovsek ‘12 General prescription to extract I.V. resonance info above 3 (or more) particle thresholds lacking.

Below threshold, F.V energy corresponds to I.V. bound state up to O ( e − m π L ). Near threshold, F.V energies are distorted. Avoided level crossing occurs. Example: taken from (Gottlieb, Rummukainen ’95) 4! ρ 4 + λ 2 4! φ 4 + g ◮ Two scalars: 4 m φ > m ρ > 2 m φ , L int = λ 1 2 ρφ 2 ◮ Spectrum from GEVP with single and multi-particle ops. ◮ Left: g = 0, Right: g = 0 . 008

The GEVP Solve (L¨ uscher, Wolff ‘90) C ( t ) v n ( t , t 0 ) = λ n ( t , t 0 ) C ( t 0 ) v n ( t , t 0 ) , C ij ( t ) = �O i ( t ) ¯ O j (0) � Method 1: ◮ Avoids diagonalization at large t ◮ Solve GEVP for t = t ∗ . Discard λ n ( t ∗ , t 0 ). ◮ Use { v n ( t ∗ , t 0 ) } to rotate C ( t ). ◮ Ensure that the off-diagonal elements remain small. Method 2: ◮ Diagonalize on each t ◮ (Blossier, et al. ‘10): If t 0 > t / 2 E eff n ( t , t 0 ) = − ∂ t log λ n ( t , t 0 ) = E n + O ( e − ( E N +1 − E n ) t )

In practice, weakly coupled low-lying states are problematic Example: C 2 pt m ψ im ψ ∗ jm e − E m t ( t ) = � ij ◮ Energies: r 0 E m = m , m = 1 .. 20 ◮ A 3 × 3 GEVP, ψ mi , i = 1 .. 3 chosen empirically ◮ Solve for E eff n ( t , t 0 = t / 2) 1.025 6.5 (t) (t) (t) 2.5 Level 1 Level 2 eff eff eff 6.0 1.020 Level 3 E E 2.4 E 0 1.015 0 0 5.5 r r r 2.3 5.0 1.010 2.2 4.5 1.005 2.1 4.0 1.000 2.0 3.5 0.995 1.9 3.0 0.990 2.5 1.8 0 1 2 3 0 1 2 3 0 1 2 3 t/r t/r t/r 0 0 0 Reduce ψ i 1 , i = 1 .. 3 by 100: 2.2 12 (t) (t) (t) 4.0 Level 2 2.0 Level 3 eff eff eff 10 E E E 3.5 1.8 0 0 0 r r r 8 3.0 1.6 Level 1 2.5 1.4 6 2.0 1.2 4 1.5 1.0 1.0 2 0.8 0 1 2 0 1 2 0 1 2 t/r t/r t/r 0 0 0

Simple ops aren’t enough Using the GEVP, can access the ψ im = lim t →∞ ψ eff ( t ) Test case (JB, Donnellan, Sommer, ‘11): PS static-light mesons, N f = 0, a s = a t ∼ 0 . 09 fm , L ∼ 1 . 5 fm , m q ∼ m s Ops. made from different smearings. Columns: m = 1 .. 5, Rows: r / r 0 = 0 . 0 , 0 . 36 , 0 . 62 , 1 . 13

Spatially Extended operators: (Basak, et al. ’05), (Foley, et al. ‘07), (Dudek, et al. ‘08) ✉ ✉ ❡ ✉ ✎☞ ✉ ✉ ♠ ✉✉ ✉ ✉ ✉ ✉ ❤ ✉ ❤ ✉ ✉ ✉ ✍✌ single- singly- doubly- doubly- triply- site displaced displaced-I displaced-L displaced-T ❡ ❡ ✈ ✓ ✓ ❡✈ ❡ ✈ ❡ ✈ ✈ single- singly- doubly- triply- triply- site displaced displaced-L displaced-U displaced-O The Goal: For each symmetry channel pick a maximum energy. Include an op. for each (known) state below that energy. J=0, I=0 (Isoscalar-scalar channel): ◮ single σ -meson operators ◮ single glueball operators ◮ I = 0 two pion operators, moving and at rest ◮ ¯ K − K operators, moving and at rest ◮ ...

all-to-all Needed to include multi-hadrons in {O i } t f t f t 0 t 0 Needed even for isoscalar single hadrons t f t 0 t f t 0

Distillation Exact smeared-smeared or point-smeared all-to-all propagators (Peardon, et al ‘09) SQ − 1 S = v i K ij v † j N ev � v † S = i v i , ∆ v i = λ i v i i Smearing controlled by λ max . Requires N inv ∼ N ev , but N ev ∼ V . 0.6 1 3 lattice 12 N=64 0.5 0.9 N=32 N=8 0.8 0.7 0.4 λ n 0.6 Ψ (r) 0.5 0.3 0.4 3 lattice 16 0.3 0.2 0.2 0.1 0.1 0 10 20 30 40 50 0 0 1 2 3 4 5 6 7 8 9 10 r/a s n

Calculating the eigenpairs Solution of N t 3d eigenproblems of a hermitian operator. Use a Krylov-Spectral Restarted Lanczos (KRSL) algorithm (Wu, Simon ‘00) Chebyshev acceleration is very helpful B = 1 + 2( ˜ ∆ + λ C ) , A = T n ( B ) λ L − λ C Cost is dominated by global re-orthog. of the Krlov space. Cost ∼ N 2 ev ∗ N itr ∗ V ∼ V 4 Largest test: L = 3 . 8 fm , N ev = 384, still a tiny fraction of the total cost. Dominant cost is Dirac matrix inv. Inv . cost ∼ N ev ∗ V ∼ V 2

Distillation results Results on small volumes ( < 2 . 4 − 2 . 9 fm ): ◮ N , ∆ , and Ω baryons: (JB, et al. ‘10) ◮ ππ -scattering: (Dudek, et al. ‘12) ◮ D π -scattering: (Mohler, et al. ‘12) ◮ K π -scattering: (Lang, et al. ‘12) ◮ ρ and a 1 meson decay: (Prelovsek, et al. ‘11) ◮ I = 0 mesons: (Dudek, et al. ‘11) ◮ Charmonium: (Liu, et al. ‘11) ◮ Hybrid Baryons: (Dudek, Edwards ‘12)

Improvement when adding multi-hadron ops. (Prelovsek, et al. ‘12): a s = a t = 0 . 12 fm , m π = 266 MeV , L s = 2 fm , D ∗ 0 channel ( J P = 0 + ) qq: O 1,3 ; D π: Ο 5,6 just qq: O 1,3,4 just D π: Ο 5,6 1.8 1.8 1.7 1.7 1.6 1.6 1.5 1.5 aE 1.4 1.4 1.3 1.3 1.2 1.2 1.1 1.1 1 1 4 6 8 10 12 14 4 6 8 10 12 14 4 6 8 10 12 14 t t t Left: single D ∗ 0 ops and D π ops Middle: just single D ∗ 0 ops Right: just D π ops

Stochastic LapH Introduce noise in the subspace (Morningstar, et al. ‘11) η ( r ) a α ( x , t ) = ρ ( r ) α i ( t ) v i at ( x ) SQ − 1 S = E r ( ψη † ) , ψ ( r ) = SQ − 1 η ( r ) Dilute in ( α, i , t )-space: η [ d ] = P [ d ] η, ψ [ d ] = SQ − 1 η [ d ] , SQ − 1 S � E r ( ψ [ d ] η [ d ] † ) = d 5 50 C(t=5) triply-displaced-T nucleon (TF,S1,LI8) C(t=5) triply-displaced-T nucleon 40 3 lattice 20 LapH noise (TF,S1,LI12) 3 lattice 16 lattice noise 30 σ/σ gn (TF,S1,LI16) σ/σ gn (TF,SF,LI8) 20 10 0 0 0 0.2 0.4 0 0.2 0.4 0.6 0.8 − ½ − ½ N D N D

Correlator construction is simple and efficient! ◮ For connected lines: N inv / cfg . ∼ 32 × N t 0 ◮ For disconnected lines: N inv / cfg . ∼ 32 × 16 × (0 . 03 fm / a t ) Example: 4 connected lines, N t 0 = 1, N inv / cfg = 128 e i p · x 0 η [ i ] a α ( x 0 ) η [ j ] b β ( x 0 ) η [ k ] Ω ijk ( t 0 ) = a abc � b γ ( x 0 ) αβγ x 0 e i p · x ψ [ i ] a α t 0 ( x ) ψ [ j ] b β t 0 ( x ) ψ [ k ] Σ ijk ( t , t 0 ) = a abc � b γ t 0 ( x ) αβγ x ψ [ i ] † t 0 ( x ) A 0 ψ [ j ] e i p · x ψ [ i ] † t 0 ( x )Γ ψ [ j ] � � A ij ( t , t 0 ) = t 0 ( x ) , ω ij ( t , t 0 ) = t 0 ( x ) x x � e i p · x 0 η [ i ] † ( x 0 )Γ η [ j ] ( x 0 ) ρ ij ( t 0 ) = x 0 Correlation functions: C π ( t − t 0 ) = ω ij ρ ij , C f π ( t − t 0 ) = A ij ρ ij , C N ( t − t 0 ) = Σ ijk Ω ∗ ijk , C I =2 ππ ( t − t 0 ) = ω ij ρ jk ω k ℓ ρ ℓ i − C 2 π

Stochastic LapH and quark-disc. diagrams Exact all-to-all is ‘wasteful’(Wong, et al. ‘10): ◮ HSC Lattice: N f = 2 + 1, a s = . 12 fm = 3 . 5 a t , m π = 400 MeV , L s = 1 . 9 fm ◮ Left: ‘Box’ diagram for ππ , Right: Disc. contribution to scalar ◮ For the scalar: ◮ Distillation: N inv / cfg . = 16384 ◮ Stochastic LapH: N inv / cfg . = 1024 3 100 [F,F,F] [F,F,F] t 0 -t [F,F,F] t-t [I16,F,I8] [F,F,I8] t 0 -t [I16,F,I8] t-t 80 2 60 C( τ ) C( τ ) 40 1 20 0 0 2 4 6 8 10 12 14 16 18 20 22 24 0 2 4 6 8 10 12 14 16 τ /a t τ /a t

Scalar I = 0 channel (JB, D. Lenkner, et al. ‘11): ◮ HSC Lattice: N f = 2 + 1, a s = . 12 fm = 3 . 5 a t , m π = 400 MeV , L s = 1 . 9 fm ◮ 5x5 GEVP: 2 single meson ops., 2 ππ ops., 1 glueball op. ◮ Results of a preliminary diagonalization 0.8 0.8 0.8 0.8 ) -1 Level 0 Level 1 Level 2 Level 3 t m (a 0.6 0.6 0.6 0.6 0.4 0.4 0.4 0.4 0.2 0.2 0.2 0.2 0 0 0 0 0 5 10 15 20 0 5 10 15 20 0 5 10 15 20 0 5 10 15 20 t t t t