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. 30th, 2012

GGI Workshop - New Frontiers in Lattice Gauge Theory

why study excited states?

Where are the ‘missing’ reso- nances? (N∗)

◮ Compare experiment (PDG

’09) to quark model (Capstick, Roberts ’00)

◮ Too many d.o.f in the QM?

Reduced by diquarks (Jaffe)

◮ Experiment is mostly

→ Nπ. States which couple weakly? N∗ program at JLab.

1000 1200 1400 1600 1800 2000 2200 2400 P11(939) P11(1440) D13(1520) S11(1535) S11(1650) D15(1675) F15(1680) D13(1700) P11(1710) P13(1720) P13(1900) F17(1990) F15(2000) D13(2080) S11(2090) P11(2100) G17(2190) D15(2200) H19(2220) G19(2250) P33(1232) P33(1600) S31(1620) D33(1700) P31(1750) S31(1900) F35(1905) P31(1910) P33(1920) D35(1930) D33(1940) F37(1950) F35(2000) S31(2150) H39(2300) D35(2350) F37(2390) H3,11(2420)

Mass/(MeV/c2)

N(I=1/2) ∆(I=3/2) exp exp QM QM

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:

C 2pt(t) = 0|O(t) ¯ O(0)|0 =

• n

Ane−Ent + O(e−ET), An = |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

• btained from (I.V.) Euclidean correlators (except in principle at

threshold).

K¨ all´ en-Lehmann representation: ∆(p) = ∞ dµ2 ρ(µ2) 1 p2 − µ2 + iǫ In I.V., ρ(µ2) has δ-functions, and a continuum above threshold. In F.V. ρ(µ2) 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−mπL). Near threshold, F.V energies are distorted. Avoided level crossing

• ccurs.

Example: taken from (Gottlieb, Rummukainen ’95)

◮ Two scalars: 4mφ > mρ > 2mφ, Lint = λ1 4! ρ4 + λ2 4! φ4 + g 2ρφ2 ◮ 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)vn(t, t0) = λn(t, t0)C(t0)vn(t, t0), Cij(t) = Oi(t) ¯ Oj(0) Method 1:

◮ Avoids diagonalization at large t ◮ Solve GEVP for t = t∗. Discard λn(t∗, t0). ◮ Use {vn(t∗, t0)} to rotate C(t). ◮ Ensure that the off-diagonal elements remain small.

Method 2:

◮ Diagonalize on each t ◮ (Blossier, et al. ‘10): If t0 > t/2

E eff

n (t, t0) = −∂tlogλn(t, t0) = En + O(e−(EN+1−En)t)

In practice, weakly coupled low-lying states are problematic Example: C 2pt

ij

(t) =

m ψimψ∗ jme−Emt ◮ Energies: r0Em = m, m = 1..20 ◮ A 3 × 3 GEVP, ψmi, i = 1..3 chosen empirically ◮ Solve for E eff n (t, t0 = t/2)

t/r 1 2 3 (t)

eff

E r 0.990 0.995 1.000 1.005 1.010 1.015 1.020 1.025 Level 1 t/r 1 2 3 (t)

eff

E r 1.8 1.9 2.0 2.1 2.2 2.3 2.4 2.5 Level 2 t/r 1 2 3 (t)

eff

E r 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 Level 3

Reduce ψi1, i = 1..3 by 100:

t/r 1 2 (t)

eff

E r 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 Level 1 t/r 1 2 (t)

eff

E r 1.0 1.5 2.0 2.5 3.0 3.5 4.0 Level 2 t/r 1 2 (t)

eff

E r 2 4 6 8 10 12 Level 3

Simple ops aren’t enough

Using the GEVP, can access the ψim = limt→∞ ψeff (t) Test case (JB, Donnellan, Sommer, ‘11): PS static-light mesons, Nf = 0, as = at ∼ 0.09fm, L ∼ 1.5fm, mq ∼ ms

• Ops. made from different smearings. Columns: m = 1..5, Rows:

r/r0 = 0.0, 0.36, 0.62, 1.13

Spatially Extended operators: (Basak, et al. ’05), (Foley, et al. ‘07), (Dudek, et al. ‘08)

✍✌ ✎☞ ✉✉ ✉ single- site ♠ ✉ ✉ ✉ singly- displaced ❤ ✉ ✉ ✉ doubly- displaced-I ❤ ✉ ✉ ✉ doubly- displaced-L ❡ ✉ ✉ ✉ triply- displaced-T ❡✈

single- site

❡ ✈

singly- displaced

❡ ✈

doubly- displaced-L

❡ ✈

triply- displaced-U

❡ ✈ ✓ ✓

triply- 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 {Oi}

t0 tf t0 tf

Needed even for isoscalar single hadrons

tf t0 tf t0

Distillation

Exact smeared-smeared or point-smeared all-to-all propagators (Peardon, et al ‘09) SQ−1S = viKijv†

j

S =

Nev

• i

v†

i vi, ∆vi = λivi

Smearing controlled by λmax. Requires Ninv ∼ Nev, but Nev ∼ V .

1 2 3 4 5 6 7 8 9 10

r/as

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Ψ(r)

N=64 N=32 N=8

10 20 30 40 50

n

0.1 0.2 0.3 0.4 0.5 0.6

λn

12

3 lattice

16

3 lattice

Calculating the eigenpairs

Solution of Nt 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( ˜ ∆ + λC) λL − λC , A = Tn(B) Cost is dominated by global re-orthog. of the Krlov space. Cost ∼ N2

ev ∗ Nitr ∗ V ∼ V 4

Largest test: L = 3.8fm, Nev = 384, still a tiny fraction of the total cost. Dominant cost is Dirac matrix inv. Inv.cost ∼ Nev ∗ V ∼ V 2

Distillation results

Results on small volumes (< 2.4 − 2.9fm):

◮ N, ∆, and Ω baryons: (JB, et al. ‘10) ◮ ππ-scattering: (Dudek, et al. ‘12) ◮ Dπ-scattering: (Mohler, et al. ‘12) ◮ Kπ-scattering: (Lang, et al. ‘12) ◮ ρ and a1 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): as = at = 0.12fm, mπ = 266MeV, Ls = 2fm, D∗

0 channel

(JP = 0+)

4 6 8 10 12 14

t

1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8

aE

qq: O1,3 ; Dπ: Ο 5,6 4 6 8 10 12 14

t

just qq: O1,3,4 4 6 8 10 12 14

t

1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8

just Dπ: Ο5,6

Left: single D∗

0 ops and Dπ ops

Middle: just single D∗

0 ops

Right: just Dπ ops

Stochastic LapH

Introduce noise in the subspace (Morningstar, et al. ‘11) η(r)

aα (x, t) = ρ(r) αi (t)vi at(x)

SQ−1S = Er(ψη†), ψ(r) = SQ−1η(r) Dilute in (α, i, t)-space: η[d] = P[d]η, ψ[d] = SQ−1η[d], SQ−1S =

• d

Er(ψ[d]η[d]†)

0.2 0.4 0.6 0.8

ND

−½

10 20 30 40 50

σ/σgn

LapH noise lattice noise

C(t=5) triply-displaced-T nucleon

0.2 0.4

ND

−½

5

σ/σgn

(TF,SF,LI8) (TF,S1,LI16) (TF,S1,LI12) (TF,S1,LI8)

20

3 lattice

16

3 lattice

C(t=5) triply-displaced-T nucleon

Correlator construction is simple and efficient!

◮ For connected lines: Ninv/cfg. ∼ 32 × Nt0 ◮ For disconnected lines: Ninv/cfg. ∼ 32 × 16 × (0.03fm/at)

Example: 4 connected lines, Nt0 = 1, Ninv/cfg = 128 Ωijk(t0) = aabc

αβγ

• x0

eip·x0η[i]

aα(x0)η[j] bβ(x0)η[k] bγ(x0)

Σijk(t, t0) = aabc

αβγ

• x

eip·xψ[i]

aαt0(x)ψ[j] bβt0(x)ψ[k] bγt0(x)

Aij(t, t0) =

• x

ψ[i]†

t0 (x)A0ψ[j] t0 (x),

ωij(t, t0) =

• x

eip·xψ[i]†

t0 (x)Γψ[j] t0 (x)

ρij(t0) =

• x0

eip·x0η[i]†(x0)Γη[j](x0) Correlation functions: Cπ(t − t0) = ωijρij, Cfπ(t − t0) = Aijρij, CN(t − t0) = ΣijkΩ∗

ijk,

C I=2

ππ (t − t0) = ωij ρjk ωkℓ ρℓi − C 2 π

Stochastic LapH and quark-disc. diagrams

Exact all-to-all is ‘wasteful’(Wong, et al. ‘10):

◮ HSC Lattice: Nf = 2 + 1, as = .12fm = 3.5at,

mπ = 400MeV, Ls = 1.9fm

◮ Left: ‘Box’ diagram for ππ, Right: Disc. contribution to scalar ◮ For the scalar:

◮ Distillation: Ninv/cfg. = 16384 ◮ Stochastic LapH: Ninv/cfg. = 1024

2 4 6 8 10 12 14 16 18 20 22 24

τ/at

20 40 60 80 100

C(τ)

[F,F,F]t0-t[F,F,F]t-t [F,F,I8]t0-t[I16,F,I8]t-t

2 4 6 8 10 12 14 16

τ/at

1 2 3

C(τ)

[F,F,F] [I16,F,I8]

Scalar I = 0 channel (JB, D. Lenkner, et al. ‘11):

◮ HSC Lattice: Nf = 2 + 1, as = .12fm = 3.5at,

mπ = 400MeV, Ls = 1.9fm

◮ 5x5 GEVP: 2 single meson ops., 2 ππ ops., 1 glueball op. ◮ Results of a preliminary diagonalization

t

5 10 15 20

)

• 1

t

m (a

0.2 0.4 0.6 0.8

Level 0

t

5 10 15 20 0.2 0.4 0.6 0.8

Level 1

t

5 10 15 20 0.2 0.4 0.6 0.8

Level 2

t

5 10 15 20 0.2 0.4 0.6 0.8

Level 3

Conclusions

◮ All-to-all propagators can be stochastically estimated

efficiently

◮ Effort ∼ V with Ninv fixed, Ninv/cfg ∼ 1200 is sufficient ◮ Useful for ‘ordinary’ C(t), bare current insertions.

◮ First finite box spectrum calculations are underway:

a 0.9 − 0.12fm, mπ > 260 − 300MeV, L 2.5 − 3fm

◮ In principle, δ(k) below > 3 hadron thresholds can be

• extracted. At mπ ∼ 300MeV, this covers interesting physics!

◮ σ, f0(980) ◮ Roper resonance ◮ Λ(1405)

◮ To Do:

◮ Phase shifts above inelastic thresholds, similar issues to

experiment

◮ smaller a, mπ, larger L ◮ Interaction with perturbative probes?