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

Extracting Excited Mesons from the Finite Volume Michael D¨ oring Meson 2014, Krak´ ow, Poland, May 29-June 3, 2014

The cubic lattice Side length L , V = L 3 ( + L t ), periodic boundary conditions ! Ψ( x ) = Ψ( x + ˆ e i 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 Window of acceptable L Avoided level crossing (resonance OR threshold) increasingly difficult to measure ) s ( e c E [MeV] n a n o s e R lowest threshold -bL bound state: E(L)~M B +a e -1 ] L [M π

L¨ uscher equation Periodic boundary conditions e i L ) = exp ( i L q i ) Ψ( x ) ⇒ q i = 2 π ! Ψ( x ) = Ψ( x + ˆ L n i , n i ∈ Z , i = 1 , 2 , 3 Integrals → Sums � d 3 � q q | 2 ) → 1 q = 2 π � q | 2 ) , n ∈ Z 3 (2 π ) 3 g ( | � g ( | � � L � n , � L 3 � n L¨ uscher equation p cot δ ( p ) = − 8 π E � ˜ G ( E ) − Re G ( E ) � p : c.m. momentum E : scattering energy ˜ G − Re G : known kinematical function → one phase at one energy.

Moving frames to get more levels ( Σ ∗ → π Λ ) Operators with non-zero momentum of the center-of-mass: � P = � p 1 + � p 2 � = 0 Rummukainen, Gottlieb, NPB (1995)

Breaking of cubic symmetry through boost q ∗ boosted with P = (0 , 0 , 0) → 2 π Example: Lattice points � L (0 , 0 , 2) :

Need for an interpolation in energy ( K π 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 Periodic B.C.: Twisted B.C.: ππ , ¯ e i L ) = e i θ i Ψ( � KK → f 0 (980) . Ψ( � x + ˆ e i L ) = Ψ( � x ) Ψ( � x + ˆ x ) Three unknown transitions Periodic in 2 dim.: Periodic/antiperiodic: V ( ππ → ππ ) V ( ππ → ¯ KK ) V ( ¯ KK → ¯ KK ) θ i = 0 θ i = π /2 θ i = π 1100 E [MeV] _ θ 1 = 0 θ 1 = 0 KK 1000 900 2 2.5 3 3.5 -1 ] L [M π Twisted B.C. for the s -quark: θ 2 = 0 θ 2 = π u ( � x + ˆ e i L ) = u ( � x ) d ( � x + ˆ e i L ) = d ( � x ) e i L ) = e i θ i s ( � s ( � x + ˆ x )

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 L i [ × 10 − 3 ] and q max [MeV]. L 1 L 2 L 3 L 4 0 . 873 +0 . 017 0 . 627 +0 . 028 − 0 . 710 +0 . 022 − 3 . 5 [fixed] − 0 . 028 − 0 . 014 − 0 . 026 L 5 L 6 + L 8 L 7 q max [MeV] 2 . 937 +0 . 048 1 . 386 +0 . 026 0 . 749 +0 . 106 981 [fixed] − 0 . 094 − 0 . 050 − 0 . 074 A resonance is characterized by its (complex) pole position and residues, corresponding to resonance mass, width, and branching ratio. Table: Pole positions z 0 [MeV] and residues a − 1 [ M π ] in different channels. I , L , S : isospin, angular momentum, strangeness. I L S Resonance sheet z 0 [MeV] a − 1 [ M π ] a − 1 [ M π ] − 31 − i 19 ( ¯ 0 0 0 σ (600) 434+ i 261 KK ) − 30+ i 86 ( ππ ) pu 16 − i 79 ( ¯ 0 0 0 f 0 (980) pu 1003+ i 15 KK ) − 12+ i 4 ( ππ ) 1 / 2 0 − 1 κ (800) 815+ i 226 − 36+ i 39 ( η K ) − 30+ i 57 ( π K ) pu − 10 − i 107 ( ¯ 1 0 0 a 0 (980) pu 1019 − i 4 KK ) 21 − i 31 ( πη ) − 2+ i 0 ( ¯ 0 1 0 φ (1020) p 976+ i 0 KK ) — 1 / 2 1 − 1 K ∗ (892) 889+ i 25 − 10+ i 0 . 1 ( η K ) 14+ i 4 ( π K ) pu − 11+ i 2 ( ¯ 1 1 0 ρ (770) pu 755+ i 95 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)] 80 0 250 0 ( π K --> π K) [deg] 0 ( π K --> π K) [deg] -5 0 ( ππ --> ππ ) [deg] 60 200 -10 f 0 (980) 40 150 κ (800) -15 σ (600) 100 δ 1/2 20 δ 0 δ 3/2 -20 50 0 0 400 600 800 1000 700 800 900 1000 1100 700 800 900 1000 1100 0 200 200 1 ( π K --> π K) [deg] 0 ( ππ --> ππ ) [deg] 1 ( ππ --> ππ ) [deg] -10 150 150 -20 100 100 K*(892) ρ (770) -30 δ 2 δ 1/2 50 δ 1 50 -40 0 0 400 600 800 1000 700 800 900 1000 1100 1200 400 600 800 1000 E [MeV] E [MeV] E [MeV]

Prediction of levels (also for M π � = M phys . ) π 2.5 3 3.5 2 2.5 3 3.5 2 (I,L,S)=(0,0,0) 1200 (I,L,S)=( ½,0,-1) 1200 (I,L,S)=(0,1,0) (I,L,S)=(2,0,0) [ σ (600), f 0 (980)] [ φ (1020)] [ κ (800)] [ ππ repulsive] 1400 1000 1000 1400 1200 E [MeV] E [MeV] 800 800 1200 1000 600 600 800 400 400 1000 1200 (I,L,S)=(1,1,0) (I,L,S)=(1,0,0) (I,L,S)=(½,1,-1) (I,L,S)=(3/2,0,-1) [ ρ (770)] 1100 [a 0 (980)] [K*(892)] [ π K repulsive] 1400 1400 1000 1000 1200 1200 E [MeV] E [MeV] 800 900 1000 1000 600 800 800 800 700 400 600 2 2.5 3 3.5 2 2.5 3 3.5 2 2.5 3 3.5 2 2.5 3 3.5 -1 ] -1 ] -1 ] -1 ] L [M π L [M π L [M π L [M π [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 [ V 2 ≡ V LO known/fixed from f π , f K , f η ; s ≡ E 2 ] � − 1 � V 2 − V fit V fit = = a + b ( s − s 0 ) + c ( s − s 0 ) 2 + d ( s − s 0 ) 3 + · · · 4 V fit , 4 V 2 2 950 80 κ (800) κ (800) 0 , s 1 ) fit (s 900 0 ( π K --> π K) [deg] 60 0 , s 1 , s 2 ) fit 850 (s E [MeV] 800 0 , s 1 , s 2 , s 3 ) fit 40 (s 750 700 δ 1/2 20 650 600 0 2 2.5 3 3.5 4 700 800 900 1000 1100 -1 ] E [MeV] L [M π Figure: Solid line: Actual phase shift. Error Figure: Pseudo lattice-data and bands of the ( s 0 , s 1 ) , ( s 0 , s 1 , s 2 ) , and ( s 0 , s 1 , s 2 ) fit to those data with ( s 0 , s 1 , s 2 , s 3 ) fits. uncertainties (bands).

The κ (800) pole 300 0 , s 1 , s 2 , s 3 ) fit 0 , s 1 ) fit area (s area (s 0 , s 1 , s 2 ) fit, ∆ E=5MeV area (s 250 0 , s 1 , s 2 ) fit Im E [MeV] area (s 200 Actual pole position 0 , s 1 , s 2 , s 3 ) in V 4 fit Fit (s 0 , s 1 , s 2 ) in V 4 fit Fit (s 0 , s 1 ) in V 4 fit Fit (s 150 0 ) in V 4 fit Fit (s 700 750 800 850 900 950 Re E [MeV]

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 ): V ij ( E ) = a ij + b ij ( E 2 − 4 M 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 f 0 (980) pole position 1150 250 25 1100 0 ( ππ −−> ππ ) [deg] increased precision for δ 20 1050 200 Im E [MeV] E [MeV] KK 1000 15 f 0 (980) 950 150 10 900 δ 0 850 5 100 KK KK 1.6 1.8 2 2.2 2.4 2.6 0 850 900 950 1000 1050 1100 980 990 1000 1010 -1 ] L [M π E [MeV] Re E [MeV]

Mixing of partial waves in boosted multiple channels: σ (600) [M.D., E. Oset, A. Rusetsky, EPJA (2012)] ππ & ¯ KK in S -wave, ππ in D -wave. Organization in Matrices ( A + 1 ), e.g. 420 � 800 P = (2 π/ L )(0 , 0 , 1) , (2 π/ L )(1 , 1 , 1) , and (2 π/ L )(0 , 0 , 2) : 400 750  V (11) V (12)  0 S S E [MeV] 380 700 V (21) V (22) V = 0   S S V (22) 0 0 360 650 D   G R (1) ˜ 0 0 00 , 00 600 340 G R (2) ˜ G R (2) ˜ ˜ = 0 G   00 , 00 00 , 20   G R (2) ˜ G R (2) ˜ 550 0 2 3 2 2.5 20 , 00 20 , 20 -1 ] -1 ] L [M π L [M π Phase extraction: Expand and fit V S , Solid: Levels from A + V D simultaneously to different 1 . representations, as in case of Non-solid: Neglecting the D -wave. multi-channels (reduction of error).

Phase shifts from a moving frame: the σ (600) Comparison: Variation of L vs moving frames 300 The first two levels 0 , s 1 ,s 2 ,s 3 ) fit area (s for the first five boosts: 20 pts., ∆ E=10 MeV : Pseudo-data [10 MeV error] 250 Im E [MeV] P=2 π /L x ... 900 (0,0,0) (0,0,1) 800 200 (0,1,1) Actual position 0 , s 1 , s 2 , s 3 ) fit (1,1,1) Central, (s 0 , s 1 , s 2 ) fit 700 (0,0,2) Central, (s Level 2 Moving frame: 2 L’s 0 , s 1 ) fit Central, (s 150 E [MeV] 600 300 500 250 Im E [MeV] 400 Level 1 0 , s 1 ,s 2 ,s 3 ) fit area (s 11 pts., ∆ E=10 MeV 300 200 2 2.5 3 3 : 6 L’s -1 ] Vary volume L L [M π 150 300 350 400 450 500 550 Re E [MeV]

Asymmetric boxes & boosts M.D., R. Molina, GWU Lattice Group [A. Alexandru et al.] K*(892) 1 ( π K --> π K) [deg] 100 M π =138 MeV (891,48) M π =230 MeV (922,32) 50 δ 1/2 M π =305 MeV (941,16) L x = L , L y = L , L z = x L 0 x = 1 , 1 . 26 , 2 . 04 860 880 900 920 940 2 π � E [MeV] L P = (0 , 0 , 0) , (0 , 0 , 1) → Resonance not covered by eigenlevels. → Find other boosts/spatial setups.