3-BODY QUANTIZATION CONDITION IN UNITARY FORMALISM [Eur.Phys.J. A53 - PowerPoint PPT Presentation

3-BODY QUANTIZATION CONDITION IN UNITARY FORMALISM [Eur.Phys.J. A53 (2017) no.9] [Eur.Phys.J. A53 (2017) no.12] [Phys.Rev. D97 (2018) no.11] [arXiv:1807.04746] M a x i m M a i T h e G e o r g e W a s h i n g t o n U n i v e r s i t y

● Many unsolved questions of QCD involve 3-body channels ● Roper-puzzle & π channel π N ↔ π channels ↔ spectroscopy spin-exotics ● a ( 1 2 6 0 ) ρ / π σ π π 1 π π a 1 ● X etc.. ( 3 8 7 2 ) π π ρ σ π π 2 Ma x i m Ma i ( G WU )

● Many unsolved questions of QCD involve 3-body channels ● Roper-puzzle & π channel π N ↔ π channels ↔ spectroscopy spin-exotics ● a ( 1 2 6 0 ) ρ / π σ π π 1 π π a 1 ● X etc.. ( 3 8 7 2 ) π π ρ σ π π ● Best theoretical tool: Lattice QCD → some (preliminary) studies: ● π π N & a ( 1 2 6 0 ) Lang et al.(2014)Lang et al.(2016) 1 ● π ρ I = 2 [I=2,πρ] Woss et al. (2017) ● more is under way... 3 Ma x i m Ma i ( G WU )

● Many unsolved questions of QCD involve 3-body channels ● Roper-puzzle & π channel π N ↔ π channels ↔ spectroscopy spin-exotics ● a ( 1 2 6 0 ) ρ / π σ π π 1 π π a 1 ● X etc.. ( 3 8 7 2 ) π π ρ σ π π ● Best theoretical tool: Lattice QCD → some (preliminary) studies: ● π π N & a ( 1 2 6 0 ) Lang et al.(2014)Lang et al.(2016) 1 ● π ρ I = 2 [I=2,πρ] Woss et al. (2017) ● more is under way... ● However, Lattice spectrum is discretized → m spectrum a p p i n g t o i n fi n i t e v o l u m e this talk: Q U A N T I Z A T I O N C O N D I T I O N F O R 3 - B O D Y S Y S T E M S 4 Ma x i m Ma i ( G WU )

2-body case Lüscher(1986) ● o n e - t o - o n e m a p p i n g ● Various extensions: multi-channels, spin, ... Gottlieb,Rummukainen,Feng,Meißner, Li,Liu,Doring,Briceno,Rusetsky,Bernard… 5 Ma x i m Ma i ( G WU )

2-body case Lüscher(1986) ● o n e - t o - o n e m a p p i n g ● Various extensions: multi-channels, spin, ... Gottlieb,Rummukainen,Feng,Meißner, Li,Liu,Doring,Briceno,Rusetsky,Bernard… 3-body case ● p no o r e s u m a b l y n e - t o - o n e m a p p i n g → complex kinematics (8 variables) → sub-channel dynamics ● important theoretical developments and p numerical investigation i l o t Sharpe,Hansen,Briceno,Hammer,Rusetsky,Polejaeva,Griesshammer,Davoudi,Guo… MM/Doring(2017) Pang/Hammer/Rusetsky/Wu(2017) Hansen/Briceno/Sharpe(2018) Doring/Hammer/MM/Pang/Rusetsky/Wu (2018) ● First data driven study of the volume spectrum → ( π ) and ( π ) systems + + + + + π π π → comparison with Lattice QCD results MM/Doring (2018) > this talk < 6 Ma x i m Ma i ( G WU )

UNITARY ISOBAR INF.-VOL. AMPLITUDE Eur.Phys.J. A53 MM et al. (2017) 1) T is a sum of a dis/connected parts 7 Ma x i m Ma i ( G WU )

UNITARY ISOBAR INF.-VOL. AMPLITUDE Eur.Phys.J. A53 MM et al. (2017) 1) T is a sum of a dis/connected parts 2) Disconnected part = spectator + tower of “ i ” s o b a r s ➢ f u n c t i o n s w i t h c o r r e c t r i g h t - h a n d - s i n g u l a r i t i e s f o r e a c h Q N τ ( M ) i n v ➢ c o u p l i n g t o a s y m p t o t i c s t a t e s : c u t - f r e e - f u n c t i o n v ( q , p ) 8 Ma x i m Ma i ( G WU )

UNITARY ISOBAR INF.-VOL. AMPLITUDE Eur.Phys.J. A53 MM et al. (2017) 1) T is a sum of a dis/connected parts 2) Disconnected part = spectator + tower of “ i ” s o b a r s ➢ f u n c t i o n s w i t h c o r r e c t r i g h t - h a n d - s i n g u l a r i t i e s f o r e a c h Q N τ ( M ) i n v ➢ c o u p l i n g t o a s y m p t o t i c s t a t e s : c u t - f r e e - f u n c t i o n v ( q , p ) 3) Connected part = general 4d BSE-like equation w.r.t kernel B ( p , q ; s ) 9 Ma x i m Ma i ( G WU )

UNITARY ISOBAR INF.-VOL. AMPLITUDE Eur.Phys.J. A53 MM et al. (2017) 1) T is a sum of a dis/connected parts 2) Disconnected part = spectator + tower of “ i ” s o b a r s ➢ f u n c t i o n s w i t h c o r r e c t r i g h t - h a n d - s i n g u l a r i t i e s f o r e a c h Q N τ ( M ) i n v ➢ c o u p l i n g t o a s y m p t o t i c s t a t e s : c u t - f r e e - f u n c t i o n v ( q , p ) 3) Connected part = general 4d BSE-like equation w.r.t kernel B ( p , q ; s ) 4) 2- and 3-body unitarity constrains B , τ → relativistic 3d integral-equation v C B = v → useful for phenomenological applications v v + + +... - 1 τ = → unknowns: v , C , m 1 / m 0 0 10 Ma x i m Ma i ( G WU )

3-BODY QUANTIZATION CONDITION Eur.Phys.J. A53 MM/Doring(2017) ● Power-law fjnite-volume efgects ↔ on-shell confjgurations in T ↔ I ↔ Unitarity is crucial m T ● Replace integrals by sums: { E * | T - 1 ( E * ) = 0 } = { E n . E i g e n v a l u e s i n a b o x } ⚠ B is NOT regular → projection to irreps essential some useful techniques: Doring/Hammer/MM/… (2018) 11 Ma x i m Ma i ( G WU )

3-BODY QUANTIZATION CONDITION Eur.Phys.J. A53 MM/Doring(2017) ● Power-law fjnite-volume efgects ↔ on-shell confjgurations in T ↔ I ↔ Unitarity is crucial m T ● Replace integrals by sums: { E * | T - 1 ( E * ) = 0 } = { E n . E i g e n v a l u e s i n a b o x } ⚠ B is NOT regular → projection to irreps essential some useful techniques: Doring/Hammer/MM/… (2018) ➢ Final result in terms of shells s (/) and basis vector index u (/) W – total energy ϑ – multiplicity L – lattice size E s – 1p. energy 12 Ma x i m Ma i ( G WU )

3-BODY QUANTIZATION CONDITION Eur.Phys.J. A53 MM/Doring(2017) ● Power-law fjnite-volume efgects ↔ on-shell confjgurations in T ↔ I ↔ Unitarity is crucial m T ● Replace integrals by sums: { E * | T - 1 ( E * ) = 0 } = { E n . E i g e n v a l u e s i n a b o x } ⚠ B is NOT regular → projection to irreps essential some useful techniques: Doring/Hammer/MM/… (2018) ➢ Final result in terms of shells s (/) and basis vector index u (/) W – total energy ϑ – multiplicity L – lattice size E s – 1p. energy ● Possible work-fmow: 1)Fix & to 2-body channel (Lattice or Exp. data) + + +... v = - 1 τ = in to 3-body data (Lattice or Exp. data) 2)Fix C C B = 13 Ma x i m Ma i ( G WU )

PHYSICAL APPLICATION arXiv:1807.04746 MM/Doring(2018) ● Interesting system: π + π + π + ➢ LatticeQCD results for ground level available for π + π + π + π + π + & Detmold et al.(2008) ➢ Repulsive channel → Q : d o e s t h e “ i s o b a r ” p i c t u r e h o l d ? ➢ L → B = 2 . 5 f m , m = 2 9 1 / 3 5 2 / 4 9 1 / 5 9 1 M e V o n u s Q : c h i r a l e x t r a p o l a t i o n i n 3 b o d y s y s t e m ? π 14 Ma x i m Ma i ( G WU )

PHYSICAL APPLICATION arXiv:1807.04746 MM/Doring(2018) ● Interesting system: π + π + π + ➢ LatticeQCD results for ground level available for π + π + π + π + π + & Detmold et al.(2008) ➢ Repulsive channel → Q : d o e s t h e “ i s o b a r ” p i c t u r e h o l d ? ➢ L → B = 2 . 5 f m , m = 2 9 1 / 3 5 2 / 4 9 1 / 5 9 1 M e V o n u s Q : c h i r a l e x t r a p o l a t i o n i n 3 b o d y s y s t e m ? π I. 2-body subchannel: ➢ one-channel problem: π -system in S-wave, I=2 π ➢ 2-body amplitude consistent with 3-body one 15 Ma x i m Ma i ( G WU )

PHYSICAL APPLICATION arXiv:1807.04746 MM/Doring(2018) ● Interesting system: π + π + π + ➢ LatticeQCD results for ground level available for π + π + π + π + π + & Detmold et al.(2008) ➢ Repulsive channel → Q : d o e s t h e “ i s o b a r ” p i c t u r e h o l d ? ➢ L → B = 2 . 5 f m , m = 2 9 1 / 3 5 2 / 4 9 1 / 5 9 1 M e V o n u s Q : c h i r a l e x t r a p o l a t i o n i n 3 b o d y s y s t e m ? π I. 2-body subchannel: ➢ one-channel problem: π -system in S-wave, I=2 π ➢ 2-body amplitude consistent with 3-body one 0 1) Fix λ to exp. data , M 0 -20 ☹ incoorrect m behavior! π -40 δ [°] -60 ChPT @ NLO K-mat @ LO -80 IAM Isobar: λ=const. -100 Isobar: IAM 400 600 800 1000 σ [MeV] 16 Ma x i m Ma i ( G WU )