Isoscalar scattering and the /f 0 (500) resonance Ral Briceo - PowerPoint PPT Presentation

Isoscalar ππ scattering and the σ /f 0 (500) resonance Raúl Briceño rbriceno@jlab.org [ with Jozef Dudek, Robert Edwards & David Wilson] HadSpec Collaboration Lattice 2016 Southampton, UK July, 2016

Motivation scattering/spectroscopy precision tests of SM ¯ ¯ ¯ d d d π π π u u u ¯ ¯ d d ¯ b K d d d f 0 (500) / σ f 0 (500) / σ ¯ ¯ ¯ u u u π π π d d d long range nuclear forces Sample of previous lattice efforts: d d Alford & Jaffe (2000) n n u u d d Prelovsek, et al. (2010) Fu (2013) π π Wakayama, et al. (2015) Howarth & Giedt, (2015) d d n n d d Z. Bai et al. (RBC, UKQCD) (2015) u u

The experimental situation summary of various experiments

The experimental situation “ f 0 (500) / σ ” E σ = 449( 22 16 ) MeV Γ σ = 550(24) MeV Peláez (2015)

Finite vs. infinite volume spectrum Infinite volume Im[s] first Riemann sheet bound state Re[s] branch cut - where threshold scattering takes place s = E 2 cm

Finite vs. infinite volume spectrum Infinite volume Im[s] second Riemann sheet narrow resonance Re[s] Re[s] broad resonance s = E 2 s R = ( E R − i 2 Γ R ) 2 cm

Finite vs. infinite volume spectrum finite volume finite volume eigenstates no continuum of states: no cuts no sheet structure “only a discrete number of modes no resonances no asymptotic states: can exist in a finite volume” no scattering

Lüscher formalism det[ F − 1 ( E L , L ) + M ( E L )] = 0 spectrum satisfy: an exact mapping L finite volume spectrum scattering amplitude M = scattering amplitude E L = finite volume spectrum L = finite volume F = known function

Lüscher formalism spectrum satisfy: det[ F − 1 ( E L , L ) + M ( E L )] = 0 Lüscher (1986, 1991) [elastic scalar bosons] Rummukainen & Gottlieb (1995) [moving elastic scalar bosons] Kim, Sachrajda, & Sharpe /Christ, Kim & Yamazaki (2005) [QFT derivation] Bernard, Lage, Meissner & Rusetsky (2008) [N π systems] RB, Davoudi , Luu & Savage (2013) [generic spinning systems] Feng, Li, & Liu (2004) [inelastic scalar bosons] Hansen & Sharpe / RB & Davoudi (2012) [moving inelastic scalar bosons] RB (2014) / RB & Hansen (2015) [moving inelastic spinning particles]

Extracting the spectrum Two-point correlation functions: C 2 pt. X ab ( t, P ) ⌘ h 0 |O b ( t, P ) O † Z b,n Z † a,n e − E n t a (0 , P ) | 0 i = n Evaluate all Wick contraction - [distillation - Peardon, et al. (Hadron Spectrum, 2009)] e.g. π [000] π [110] 0.002 m π = 236 MeV 0 e E 0 t C ( t, 0) -0.002 0.016 0 4 8 12 16 20 24 0.012 0.008 0.004 0 -0.004 0 4 8 12 16 20 24 28 32 36

Extracting the spectrum Two-point correlation functions: C 2 pt. X ab ( t, P ) ⌘ h 0 |O b ( t, P ) O † Z b,n Z † a,n e − E n t a (0 , P ) | 0 i = n Evaluate all Wick contraction - [distillation - Peardon, et al. (Hadron Spectrum, 2009)] e.g. π [000] π [110] m π = 236 MeV close up (d) 0.002 (b) (a) 0 (c) (c) -0.002 0 4 8 12 16 20 24

Extracting the spectrum Two-point correlation functions: C 2 pt. X ab ( t, P ) ⌘ h 0 |O b ( t, P ) O † Z b,n Z † a,n e − E n t a (0 , P ) | 0 i = n Evaluate all Wick contraction - [distillation - Peardon, et al. (Hadron Spectrum, 2009)] Use a large basis of operators with the same quantum numbers ‘Diagonalize’ correlation function variationally ~ PL e.g. ~ d = 2 ⇡ = [110] a t E cm … m π = 391 MeV 0 . 25 L/a s = 24 ηη | thr . 0 . 20 KK | thr . ηη KK ππ 0 . 15 ¯ ψ Γ ψ ¯ ψ Γ ψ all KK mm ππ ηη 0 . 1

Extracting the spectrum Two-point correlation functions: C 2 pt. X ab ( t, P ) ⌘ h 0 |O b ( t, P ) O † Z b,n Z † a,n e − E n t a (0 , P ) | 0 i = n Evaluate all Wick contraction - [distillation - Peardon, et al. (Hadron Spectrum, 2009)] Use a large basis of operators with the same quantum numbers ‘Diagonalize’ correlation function variationally ~ PL e.g. ~ correct spectrum d = 2 ⇡ = [110] a t E cm … m π = 391 MeV 0 . 25 L/a s = 24 ηη | thr . 0 . 20 KK | thr . ηη KK ππ 0 . 15 ¯ ψ Γ ψ ¯ ψ Γ ψ all KK mm ππ ηη 0 . 1

Finite volume spectra m π =391 MeV m π =236 MeV 40

Finite volume spectra 1000 1100 900 1000 800 900 700 800 600 700 500 600 24 32 40 24 32 40 24 32 40 24 32 40 24 32 40 16 20 24 16 20 24 16 20 24 16 20 24 16 20 24 m π =391 MeV m π =236 MeV det[ F − 1 ( E L , L ) + M ( E L )] = 0 Spectrum satisfies: One channel, ignoring partial wave mixing: cot δ 0 ( E cm ) + cot φ ( P, L ) = 0 Use a various parametrizations e.g. 40 M − 1 = K − 1 + I, Im( I ) = − ρ [unitarity] g 2 K = s 0 − s + c

Scattering amplitude vs m π 1 0.5 0 -0.5 -1 -0.06 -0.03 0 0.03 0.06 0.09 0.12 HadSpec Collaboration

Scattering amplitude vs m π 1 [scattering lengths] -1 0.5 0 -0.5 -1 -0.06 -0.03 0 0.03 0.06 0.09 0.12 HadSpec Collaboration

Scattering amplitude vs m π 1 0.5 0 [bound state] -0.5 -1 -0.06 -0.03 0 0.03 0.06 0.09 0.12 HadSpec Collaboration

Scattering amplitude vs m π 180 150 120 90 60 30 0 0.01 0.03 0.05 0.07 0.09 0.11 0.13 HadSpec Collaboration

Scattering amplitude vs m π 180 150 120 [no narrow resonance] 90 60 30 0 0.01 0.03 0.05 0.07 0.09 0.11 0.13 HadSpec Collaboration

The σ /f 0 (500) vs m π s 0 = ( E σ − i 2 Γ σ ) 2 , g 2 σππ = lim s → s 0 ( s 0 − s ) t ( s ) 0 300 500 700 900 -100 2 Γ σ / MeV − Im √ s 0 = 1 -200 -300 800 600 400 200 0 150 200 250 300 350 400

The σ /f 0 (500) vs m π s 0 = ( E σ − i 2 Γ σ ) 2 , g 2 σππ = lim s → s 0 ( s 0 − s ) t ( s ) 0 300 500 700 900 -100 2 Γ σ / MeV − Im √ s 0 = 1 -200 -300 800 600 400 200 0 150 200 250 300 350 400

The σ /f 0 (500) vs m π s 0 = ( E σ − i 2 Γ σ ) 2 , g 2 σππ = lim s → s 0 ( s 0 − s ) t ( s ) 0 300 500 700 900 -100 2 Γ σ / MeV − Im √ s 0 = 1 -200 -300 800 600 400 200 0 150 200 250 300 350 400

The σ /f 0 (500) vs m π s 0 = ( E σ − i 2 Γ σ ) 2 , g 2 σππ = lim s → s 0 ( s 0 − s ) t ( s ) 0 300 500 700 900 -100 2 Γ σ / MeV − Im √ s 0 = 1 -200 -300 1/2 (MeV) Re s σ 400 500 600 700 800 900 1000 1100 1200 -100 PDG estimate 1996-2010 poles in RPP 2010 poles in RPP 1996 -200 1/2 (MeV) -300 Im s σ -400 Historical perspective -500 J. R. Peláez (2015) Review of Particle Physics (RPP)

The σ /f 0 (500) vs m π s 0 = ( E σ − i 2 Γ σ ) 2 , g 2 σππ = lim s → s 0 ( s 0 − s ) t ( s ) 0 300 500 700 900 -100 -200 -300 m π ~350 MeV U χ PT - Nebreda & Peláez (2015)

The σ /f 0 (500) vs m π s 0 = ( E σ − i 2 Γ σ ) 2 , g 2 σππ = lim s → s 0 ( s 0 − s ) t ( s ) U χ PT - Nebreda & Peláez (2015) 800 600 400 200 0 150 200 250 300 350 400

Outlook ππ -KK / f 0 (980) dispersive analysis chiral extrapolation Elastic form factors of composite particles

Outlook ππ -KK / f 0 (980) dispersive analysis chiral extrapolation 0 Elastic form factors of composite particles 300 500 700 900 -100 -200 -300

Outlook ππ -KK / f 0 (980) dispersive analysis chiral extrapolation, more quark masses(?) Elastic form factors of composite particles m π = 140 MeV δ 1 / � E cm / MeV E ? ⇡⇡ / MeV Bolton, RB & Wilson Phys.Lett. B757 (2016) 50-56.

Outlook ππ -KK / f 0 (980) dispersive analysis chiral extrapolation, more quark masses(?) Elastic form factors of composite particles first implementation: πγ *-to- ππ / πγ *-to- ρ formalism understood: 6.0 RB, Hansen - Phys.Rev. D94 (2016) no.1, 013008. RB, Hansen - Phys.Rev. D92 (2015) no.7, 074509. 4.0 RB, Hansen, Walker-Loud - Phys.Rev. D91 (2015) no.3, 034501. Bernard, D. Hoja, U. G. Meissner, and A. Rusetsky (2012) 2.0 0 2.0 2.1 2.2 2.3 2.4 2.5 100 50 0 2.0 2.1 2.2 2.3 2.4 2.5 RB, Dudek, Edwards, Thomas, Shultz, Wilson - Phys.Rev. D93 (2016) 114508. RB, Dudek, Edwards, Thomas, Shultz, Wilson - Phys.Rev.Lett. 115 (2015) 242001

Take-home message 180 1 150 0.5 120 0 90 60 -0.5 30 -1 0 0.01 0.03 0.05 0.07 0.09 0.11 0.13 -0.06 -0.03 0 0.03 0.06 0.09 0.12 0 300 500 700 900 -100 -200 -300 800 Wilson Dudek Edwards 600 400 HadSpec 200 Collaboration arXiv:1607.05900 [hep-ph] 0 150 200 250 300 350 400

HadSpec talks Resonance in coupled channels- David Wilson, Monday 10:30 Searches for charmed tetraquarks- Gavin Cheung, Monday 13:55 Radiative transitions in charmonium - Cian O'Hara, Monday 17:25 Optimised operators and distillation - Antoni Woss, Tuesday 14:40 a 0 resonance in πη , KK - Jozef Dudek, Tuesday 15:50 Charmed meson spectroscopy - David Tims, Thursday 14:20 D π , D η and D s K scattering - Graham Moir, Thursday 15:00 DK scattering - Christopher Thomas, Thursday 15:20 Charmed-bottom mesons - Nilmani Mathur, Friday 15:40

Scattering amplitude vs m π 180 [Levinson’s theorem] 150 120 90 60 30 0 0.01 0.03 0.05 0.07 0.09 0.11 0.13 HadSpec Collaboration