Extracting Excited Mesons from the Finite Volume Michael D oring - - PowerPoint PPT Presentation

Extracting Excited Mesons from the Finite Volume

Michael D¨

• ring

Meson 2014, Krak´

• w, Poland, May 29-June 3, 2014

The cubic lattice

Side length L, V = L3 (+Lt), periodic boundary conditions Ψ(x)

!

= Ψ(x + ˆ ei L) → finite volume effects → Infinite volume L → ∞ extrapolation Lattice spacing a → finite size effects Modern lattice calculations: a ≃ 0.07 fm → p ∼ 2.8 GeV → (much) larger than typical hadronic scales; not considered here. Unphysically large quark/hadron masses → chiral extrapolation required.

Resonances decaying on the lattice

Eigenvalues in the finite volume L [Mπ

• 1]

E [MeV] bound state: E(L)~MB+a e

• bL

R e s

• n

a n c e ( s ) lowest threshold Avoided level crossing

(resonance OR threshold)

Window of acceptable L

increasingly difficult to measure

L¨ uscher equation

Periodic boundary conditions Ψ(x)

!

= Ψ(x + ˆ ei L) = exp (i L qi) Ψ(x) ⇒ qi = 2π L ni, ni ∈ Z, i = 1, 2, 3 Integrals → Sums

• d3

q (2π)3 g(| q |2) → 1 L3

• n

g(| q|2),

• q = 2π

L n,

• n ∈ Z3

L¨ uscher equation p cot δ(p) = −8π E ˜ G(E) − Re G(E)

p: c.m. momentum E: scattering energy ˜ G − ReG: known kinematical function → one phase at one energy.

Moving frames to get more levels (Σ∗ → πΛ)

Operators with non-zero momentum of the center-of-mass: P = p1 + p2 = 0

Rummukainen, Gottlieb, NPB (1995)

Breaking of cubic symmetry through boost

Example: Lattice points q∗ boosted with P = (0, 0, 0) → 2π

L (0, 0, 2):

Need for an interpolation in energy (Kπ scattering)

More need for an interpolation in energy (coupled channels)

Twisting the boundary conditions [Bernard, Lage, Meißner, Rusetsky, JHEP (2011), M.D., Meißner, Oset, Rusetsky, EPJA (2011)]

S-wave, coupled-channels ππ, ¯ KK → f0(980). Three unknown transitions

V(ππ → ππ) V(ππ → ¯ KK) V( ¯ KK → ¯ KK)

2 2.5 3 3.5 L [Mπ

• 1]

900 1000 1100 E [MeV] θi = 0 θi = π/2 θi = π _ KK

Twisted B.C. for the s-quark: u( x + ˆ eiL) = u( x) d( x + ˆ eiL) = d( x) s ( x + ˆ eiL) = eiθis( x) Periodic B.C.: Ψ( x + ˆ eiL) = Ψ( x) Periodic in 2 dim.: θ1 = 0 θ2 = 0 Twisted B.C.: Ψ( x + ˆ eiL) = eiθiΨ( x) Periodic/antiperiodic: θ1 = 0 θ2 = π

Energy interpolation through unitarized ChPT

[M.D., Meißner, JHEP (2012)] using IAM [Oller, Oset, Pel´ aez, PRC (1999)] Unitary extension of ChPT, can be matched to ChPT order-by-order.

Table: Fitted values for the Li [×10−3] and qmax [MeV].

L1 L2 L3 L4 0.873+0.017

−0.028

0.627+0.028

−0.014

−3.5 [fixed] −0.710+0.022

−0.026

L5 L6 + L8 L7 qmax [MeV] 2.937+0.048

−0.094

1.386+0.026

−0.050

0.749+0.106

−0.074

981 [fixed] A resonance is characterized by its (complex) pole position and residues, corresponding to resonance mass, width, and branching ratio.

Table: Pole positions z0 [MeV] and residues a−1[Mπ] in different channels. I, L, S: isospin, angular momentum, strangeness.

I L S Resonance sheet z0 [MeV] a−1 [Mπ] a−1 [Mπ] σ(600) pu 434+i 261 −31−i 19 ( ¯ KK) −30+i 86 (ππ) f0(980) pu 1003+i 15 16−i 79 ( ¯ KK) −12+i 4 (ππ) 1/2 −1 κ(800) pu 815+i 226 −36+i 39 (ηK) −30+i 57 (πK) 1 a0(980) pu 1019−i 4 −10−i 107 ( ¯ KK) 21−i 31 (πη) 1 φ(1020) p 976+i 0 −2+i 0 ( ¯ KK) — 1/2 1 −1 K ∗(892) pu 889+i 25 −10+i 0.1 (ηK) 14+i 4 (πK) 1 1 ρ(770) pu 755+i 95 −11+i 2 ( ¯ KK) 33+i 17 (ππ)

Fit to meson-meson PW data using unitary ChPT with NLO terms

[M.D., Meißner, JHEP (2012)] using IAM [Oller, Oset, Pel´ aez, PRC (1999)]

400 600 800 1000 50 100 150 200 250 δ0

0 (ππ --> ππ) [deg]

σ(600) f0(980) 700 800 900 1000 1100 20 40 60 80 δ1/2

0 (πK --> πK) [deg]

κ(800) 700 800 900 1000 1100

• 20
• 15
• 10
• 5

δ3/2

0 (πK --> πK) [deg]

400 600 800 1000 E [MeV]

• 40
• 30
• 20
• 10

δ2

0 (ππ --> ππ) [deg]

700 800 900 1000 1100 1200 E [MeV] 50 100 150 200 δ1/2

1 (πK --> πK) [deg]

K*(892) 400 600 800 1000 E [MeV] 50 100 150 200 δ1

1 (ππ --> ππ) [deg]

ρ(770)

Prediction of levels (also for Mπ = M phys.

π

)

2 2.5 3 3.5 400 600 800 1000 1200 E [MeV] 2 2.5 3 3.5 800 1000 1200 1400 2 2.5 3 3.5 L [Mπ

• 1]

600 800 1000 1200 1400 E [MeV] 2 2.5 3 3.5 L [Mπ

• 1]

800 1000 1200 1400 (I,L,S)=(0,0,0) (I,L,S)=( ½,0,-1) (I,L,S)=(1,0,0) (I,L,S)=(3/2,0,-1) [σ(600), f0(980)] [κ(800)] [a0(980)] [πK repulsive] 400 600 800 1000 1200 E [MeV] 1000 1200 1400 2 2.5 3 3.5 L [Mπ

• 1]

700 800 900 1000 1100 E [MeV] 2 2.5 3 3.5 L [Mπ

• 1]

400 600 800 1000 1200 (I,L,S)=(2,0,0) (I,L,S)=(0,1,0) (I,L,S)=(½,1,-1) (I,L,S)=(1,1,0) [ππ repulsive] [φ(1020)] [K*(892)] [ρ(770)]

[M.D., Meißner, JHEP (2012) Loops in t- and u-channel (1-loop calculation): [Albaladejo, Rios, Oller, Roca,

arXiv: 1307.5169; Albaladejo, Oller, Oset, Rios, Roca, JHEP (2013)]

Reconstruction of the κ(800) stabilized by ChPT

Fit potential [V2 ≡ VLO known/fixed from fπ, fK, fη; s ≡ E2] V fit =

• V2 − V fit

4

V 2

2

−1

, V fit

4

= a + b(s − s0) + c(s − s0)2 + d(s − s0)3 + · · ·

2 2.5 3 3.5 4 L [Mπ

• 1]

600 650 700 750 800 850 900 950 E [MeV]

κ(800)

Figure: Pseudo lattice-data and (s0, s1, s2) fit to those data with uncertainties (bands).

700 800 900 1000 1100 E [MeV] 20 40 60 80 δ1/2

0 (πK --> πK) [deg]

κ(800)

(s

0, s 1, s 2, s 3 ) fit

(s

0, s 1 ) fit

(s

0, s 1, s 2 ) fit

Figure: Solid line: Actual phase shift. Error bands of the (s0, s1), (s0, s1, s2), and (s0, s1, s2, s3) fits.

The κ(800) pole

700 750 800 850 900 950 Re E [MeV] 150 200 250 300 Im E [MeV] Actual pole position Fit (s

0, s 1, s 2, s 3 ) in V4 fit

Fit (s

0, s 1, s 2 ) in V4 fit

Fit (s

0, s 1 ) in V4 fit

Fit (s

0 ) in V4 fit

area (s

0, s 1, s 2, s 3 ) fit

area (s

0, s 1, s 2 ) fit

area (s

0, s 1 ) fit

area (s

0, s 1, s 2 ) fit, ∆E=5MeV

Coupled-channel systems with thresholds

[M.D., Meißner, Oset/Rusetsky, EPJA 47 (2011)]

Need for an interpolation in energy (→ Unitarized ChPT,. . . ) Expand a two-channel transition V in energy (i, j: ππ, ¯ KK): Vij(E) = aij + bij(E2 − 4M 2

K)

Include model-independently known LO contribution in a, b. Or even NLO contributions (7 LECs: more fit parameters). lattice data & fit extracted phase shift f0(980) pole position

1.6 1.8 2 2.2 2.4 2.6 L [Mπ

• 1]

850 900 950 1000 1050 1100 1150 E [MeV]

KK increased precision for δ

850 900 950 1000 1050 1100 E [MeV] 100 150 200 250

δ0

0 (ππ −−> ππ) [deg]

KK

980 990 1000 1010

Re E [MeV]

5 10 15 20 25

Im E [MeV]

KK f0(980)

Mixing of partial waves in boosted multiple channels: σ(600)

[M.D., E. Oset, A. Rusetsky, EPJA (2012)] 2 3 L [Mπ

• 1]

340 360 380 400 420 E [MeV] 2 2.5 L [Mπ

• 1]

550 600 650 700 750 800

Solid: Levels from A+

1 .

Non-solid: Neglecting the D-wave. ππ & ¯ KK in S-wave, ππ in D-wave. Organization in Matrices (A+

1 ), e.g.

• P = (2π/L)(0, 0, 1), (2π/L)(1, 1, 1),

and (2π/L)(0, 0, 2): V =

 

V (11)

S

V (12)

S

V (21)

S

V (22)

S

V (22)

D

 

˜ G =

  

˜ GR (1)

00,00

˜ GR (2)

00,00

˜ GR (2)

00,20

˜ GR (2)

20,00

˜ GR (2)

20,20

  

Phase extraction: Expand and fit VS, VD simultaneously to different representations, as in case of multi-channels (reduction of error).

Phase shifts from a moving frame: the σ(600)

Comparison: Variation of L vs moving frames

The first two levels for the first five boosts:

2 2.5 3 L [Mπ

• 1]

300 400 500 600 700 800 900 E [MeV] (0,0,0) (0,0,1) (0,1,1) (1,1,1) (0,0,2) Level 1 Level 2 P=2π/L x ... : Pseudo-data [10 MeV error]

150 200 250 300 Im E [MeV] Actual position Central, (s

0, s 1, s 2, s 3) fit

Central, (s

0, s 1, s 2) fit

Central, (s

0, s 1) fit

20 pts., ∆E=10 MeV area (s

0, s 1,s 2,s 3) fit

Moving frame: 2 L’s

300 350 400 450 500 550 Re E [MeV] 150 200 250 300 Im E [MeV]

area (s

0, s 1,s 2,s 3) fit

Vary volume L

3: 6 L’s

11 pts., ∆E=10 MeV

Asymmetric boxes & boosts

M.D., R. Molina, GWU Lattice Group [A. Alexandru et al.]

Lx = L, Ly = L, Lz = x L x = 1, 1.26, 2.04

L 2π

P = (0, 0, 0), (0, 0, 1)

860 880 900 920 940 E [MeV] 50 100 δ1/2

1 (πK --> πK) [deg]

K*(892)

Mπ=138 MeV Mπ=230 MeV Mπ=305 MeV (891,48) (922,32) (941,16)

→ Resonance not covered by eigenlevels. → Find other boosts/spatial setups.

Three-particle intermediate states

J,M ℓ,S,m π N π J,M π σ, ρ σ, ρ π Δ Δ ℓ,S,m π Consistent 3-body cuts σ, ρ π π N

πN scattering: Known large inelasticities ππN [π∆, σN, ρN,. . . ] ππ/πN boosted subsystems. Is it enough to include (boosted) 2-particle subsystems in the propagator? No. Three-body s-channel dynamics requires particle exchange transitions. ⇒ Three-body unitarity

[Aaron, Almado, Young, PR 174 (1968) 2022, Aitchison, Brehm, PLB 84 (1979) 349, PRD 25 (1982) 3069; Hansen, Sharpe, Davoudi, Brice˜

• no. . . ]

Perspectives

Rapid progress in the actual ab-initio calculations of resonances/phase shifts: ρ(770), a0(980), K ∗(s, p, d), N(1535), N(1650), ∆(1232), . . . . Close to the physical point, finite volume effects dominate the spectrum. Use finite volume effects in your favor: L¨ uscher & extensions (coupled channels, moving frames, twisted boundary conditions,. . . ) Energy interpolation needed in many aspects —Unitarized ChPT & coupled-channel approaches can provide a framework.

Prediction of levels & Chiral extrapolation → find suitable lattice setups to cover resonance region with eigenstates. provide maximal precision of extracted data. Analysis of lattice data.

Thank you to the Organizers! Thank you for slides: R. Brice˜ no, G. Engel, C. Lang, B. Menadue, M. Petschlies, A. Rusetsky, G. Schierholz, M. Wagner, D. Wilson.

Fit to meson-meson data using unitary ChPT with NLO terms

[M.D., Meißner, JHEP (2012)] using IAM [Oller, Oset, Pel´ aez, PRC (1999)] Unitary extension of ChPT, can be matched to ChPT order-by-order.

Table: Fitted values for the Li [×10−3] and qmax [MeV].

L1 L2 L3 L4 0.873+0.017

−0.028

0.627+0.028

−0.014

−3.5 [fixed] −0.710+0.022

−0.026

L5 L6 + L8 L7 qmax [MeV] 2.937+0.048

−0.094

1.386+0.026

−0.050

0.749+0.106

−0.074

981 [fixed] A resonance is characterized by its (complex) pole position and residues, corresponding to resonance mass, width, and branching ratio.

Table: Pole positions z0 [MeV] and residues a−1[Mπ] in different channels. I, L, S: isospin, angular momentum, strangeness.

I L S Resonance sheet z0 [MeV] a−1 [Mπ] a−1 [Mπ] σ(600) pu 434+i 261 −31−i 19 ( ¯ KK) −30+i 86 (ππ) f0(980) pu 1003+i 15 16−i 79 ( ¯ KK) −12+i 4 (ππ) 1/2 −1 κ(800) pu 815+i 226 −36+i 39 (ηK) −30+i 57 (πK) 1 a0(980) pu 1019−i 4 −10−i 107 ( ¯ KK) 21−i 31 (πη) 1 φ(1020) p 976+i 0 −2+i 0 ( ¯ KK) — 1/2 1 −1 K ∗(892) pu 889+i 25 −10+i 0.1 (ηK) 14+i 4 (πK) 1 1 ρ(770) pu 755+i 95 −11+i 2 ( ¯ KK) 33+i 17 (ππ)

The a0(980)

[Wagner, Daldrop, Abdel-Rehim, Urbach et. al. [ETMC], JHEP (2013) & new results]

500 1000 1500 2000 2500 3000 3500 2 4 6 8 10 12 14 meffective in MeV t/a mol(K+K), tetra(g5), 2-particle(K+K), 2-particle(ss+pi)

0.2 0.4 0.6 0.8 1 1 2 3 4 5 Meff

GEVP(t)

t State 0 State 1 2*mK

Mπ ∼ 300 MeV, no singly disconnected diagrams. Operators: ¯ KK molecular, diquark-antidiquark, meson-meson. Two low-lying states, large overlap with meson-meson. Mπ ∼ 300 MeV, singly disconnected diagrams included. Operators: q¯ q, ¯ KK molecular. Again, two low lying states, no information on additional state.

The Λ(1405)

[M. D./Haidenbauer/Meißner/Rusetsky, EPJA 47 (2011)]

(Non-factorizing/off-shell) Lippman-Schwinger equation in the finite volume,

T(P)(q′′, q′) = V(q′′, q′) + 2π2 L3

∞

• i=0

ϑ(P)(i) V(q′′, qi) T(P)(qi, q′) √s − Ea(qi) − Eb(qi) , qi = 2π L √ i .

2 2.5 3 3.5 L [Mπ

• 1]

1300 1400 1500 1600 1700 E [MeV] from UCHIPT (Oset/Ramos, NPA (1997)) from Had-Exch. (Mueller,..., NPA (1990)) πΣ _ KN Λ(1670) Λ(1405)

Access to sub- ¯ KN-threshold dynamics: Discrepancies of lowest levels: levels sensitive to different Λ(1405) dynamics. One- or two-pole structure:

Will NOT lead to additional level. but shifted threshold levels.

The a0(980) in a multi-channel environment

M.D., Meißner, Oset, Rusetsky, EPJA (2011); see also Lage, Meißner, Rusetsky, PLB (2009)

940 960 980 1000 E [MeV] 1 2 3 4 5 6 7 2 2.5 3 L [Mπ

• 1]

900 950 1000 1050 904 MeV qmax = 1094 Mev 630 MeV 1094 MeV 904 MeV qmax = 630 MeV |t22|

2 I=1 / 10 3

E [MeV]

Mixing of partial waves

Example: S- and P-waves

Infinite volume limit: Rotational symmetry

• d3

q (2π)3 g(| q|) Yℓm(θ, φ)Y ∗

ℓ′m′(θ, φ) ∼ δℓℓ′δmm′.

Mixing of partial waves

Example: S- and P-waves

Infinite volume limit: Rotational symmetry

• d3

q (2π)3 g(| q|) Yℓm(θ, φ)Y ∗

ℓ′m′(θ, φ) ∼ δℓℓ′δmm′.

Wigner-Eckart theorem:

Mixing of partial waves

Example: S- and P-waves

Finite volume: Rotational symmetry → Cubic symmetry 1 L3

• n

g(| q|) Yℓm(θ, φ)Y ∗

ℓ′m′(θ, φ) ∼ Aℓℓ′mm′.

S − G-wave mixing, but S − P waves still orthogonal:

Mixing of partial waves

Example: S- and P-waves

Finite volume & boost: Cubic symmetry → subgroups of cubic symmetry 1 L3

• n

g(| q|) Yℓm(θ, φ)Y ∗

ℓ′m′(θ, φ) ∼ Aℓℓ′mm′.

For boost P = 2π

L (0,1,1):

Mixing of partial waves

Example: S- and P-waves

Finite volume & boost: Cubic symmetry → subgroups of cubic symmetry 1 L3

• n

g(| q|) Yℓm(θ, φ)Y ∗

ℓ′m′(θ, φ) ∼ Aℓℓ′mm′.

More complicated boosts:

Disentanglement of partial waves

[M.D., Meißner, Oset, Rusetsky, EPJA (2012)]

Example: S- and P-waves for the κ(800)/K∗(892) system

Knowledge of P-wave (from separate analysis of lattice data) allows to disentangle the S-wave: p cot δS = −8π E p cot δP ˆ GSS − 8πE( ˆ G2

SP − ˆ

GSS ˆ GPP) p cot δP + 8π E ˆ GPP

640 660 680 E [MeV] 1 2 3 4 5 6 7 8 δ1/2

0 (πK --> πK) [deg]

700 750 800 850 900 E [MeV] 20 40 60 80 L=6.6 Mπ

• 1

L=1.9 Mπ

• 1

L=1.9 Mπ

• 1

P = 2π/L (0,1,1) L=6.6 Mπ

• 1

P = 2π/L (0,1,1) Level 1 Level 2

δS ≡ δ0

1/2(πK → πK)

Red solid: Actual S-wave phase shift. Dash-dotted: Reconstructed S-wave phase shift, PW-mixing ignored. Dashed: Reconstructed S-wave phase shift, PW-mixing disentangled. small p-wave: Level shift ∆E ≃ − 6πESδP L3pω1ω2

Rummukainen, Gottlieb, NPB (1995); Kim, Sachrajda, Sharpe, NPB (2005); Davoudi, Savage, PRD (2011), Z. Fu, PRD (2012); Leskovec, Prelovsek, PRD (2012); Dudek, Edwards, Thomas, PRD (2012); Hansen, Sharpe, PRD 86 (2012); Brice˜ no, Davoudi, arXiv:1204.1110; G¨

• ckeler,

Horsley, Lage, Meißner, Rakow, Rusetsky, Schierholz, Zanotti, PRD (2012)

Three particles in a finite volume

• r

r

R R L L

2 particles 3 particles r1

• In case of 2 particles: r ≫ R, when particles are near the walls

In case of 3 particles: it may happen that r ≫ R, r1 ≃ R, when the particles are near the walls The problem with the disconnected contributions: is the finite-volume spectrum in the 3-particle case determined solely through the on-shell scattering matrix? Despite the presence of the disconnected contributions, the energy spectrum of the 3-particle system in a finite box is still determined by the

• n-shell scattering matrix elements in the infinite volume

[Polejaeva, Rusetski, EPJA (2012)]

Three particles: Threshold openings in the complex plane

Existence shown model-independently in [S. Ceci, M.D., C. Hanhart et. al., PRC 84 (2011)]

(z = E)

(z = E)

Kπ scattering in I = 3/2

• D. Wilson [HadSpec], Lattice 2013.

∆ resonance from the transfer matrix method [ETMC, arXiv:1305.6081]