Resonance properties from finite volume energy spectrum

Transcription

1 Resonance properties from finite volume energy spectrum Akaki Rusetsky Helmholtz-Institut für Strahlen- und Kernphysik Abteilung Theorie, Universität Bonn, Germany NPB 788 (2008) 1 JHEP 0808 (2008) 024 arxiv: In collaboration with: V. Bernard, D. Hoja, M. Lage, U.-G. Meißner K. Polejaeva MAMI and beyond, Mainz, 1 April 2009 A. Rusetsky, MAMI and beyond, Mainz, 1 April 2009 p.1

2 Plan Introduction Research program: Quark mass dependence of the resonance characteristics Matrix elements magnetic moments, etc Generalization to the inelastic case Summary, outlook A. Rusetsky, MAMI and beyond, Mainz, 1 April 2009 p.2

3 Introduction Lattice calculations are always done at... L Finite lattice spacing a 0 Finite lattice volume L 3 T (Euclidean space) (Mostly) at unphysical quark masses m q m phys q T a Resonances do not correspond to the isolated energy levels Resonance properties can be extracted, studying the behavior of the lattice spectrum in a finite volume (Lüscher) Effective field theory can be used to perform various extrapolations A. Rusetsky, MAMI and beyond, Mainz, 1 April 2009 p.3

4 Unstable particles: Lüscher s approach M. Lüscher, lectures given at Les Houches (1988); NPB 364 (1991) 237 U. Wiese, NPB (Proc. Suppl.) 9 (1989) 609 T. DeGrand, PRD 43 (1991) 2296 S.R. Beane, P.F. Bedaque, A. Parreño and M.J. Savage, PLB 585 (2004) 106; NPA 747 (2005) 55 C.h. Kim, C.T. Sachrajda and S.R. Sharpe, NPB 727 (2005) 218 N.H. Christ, C.h. Kim and T. Yamazaki, PRD 72 (2005) T L M π L 1, a 0, m q : below threshold value The energy spectrum is real Momenta are small: p 2π/L M π Finite-volume corrections to the energy levels are only power-suppressed in L Studying the dependence of the energy levels on L gives the scattering phase in the infinite volume Resonances A. Rusetsky, MAMI and beyond, Mainz, 1 April 2009 p.4

5 Extracting the resonance Lüscher s formula for the P 33 phase shift (mixing neglected): δ(p) = Z 00 (1; q 2 ) = π 3/2 q πn + arctan Z 00 (1; q 2 ), 1 1 lim 4π s 1 (n 2 q 2 ) s n Z 3 q = pl 2π Note: The rotation symmetry is broken down to cubic symmetry: no complete diagonalization mixing of the partial waves Partial diagonalization can be achieved by a special choice of the basis Including nucleon spin in Lüscher s formula: V. Bernard, M. Lage, U.-G. Meißner and AR, JHEP 0808 (2008) 024 A. Rusetsky, MAMI and beyond, Mainz, 1 April 2009 p.5

6 Quark mass dependence in a finite volume m q m phys q Finite-volume corrections to the energy spectrum should be calculated for unphysical quark masses Use effective field theory in a finite volume At M π L, the expressions should agree with Lüscher s formula Calculations at O(ǫ 3 ), O(ǫ 4 ) in Small Scale Expansion (SSE): V. Bernard, U.-G. Meißner and AR, Nucl. Phys. B 788 (2008) 1 V. Bernard, D. Hoja, U.-G. Meißner and AR, arxiv: Establish the convergence of the chiral expansion Establish a reliable procedure to extract the width from the data at unphysical quark masses A. Rusetsky, MAMI and beyond, Mainz, 1 April 2009 p.6

7 Baryon self-energy at O(ǫ 4 ) V. Bernard, D. Hoja, U.-G. Meißner and AR, arxiv: ^a,e^ 1 1 c A c A ^c 1 ^a 1 b,b 3 6 c A Infinite volume (B = N, ): m B = m B + y 1 (Mπ 2 M π) 2 + y 2 (Mπ 3 M π) 3 + y 3 (Mπ 4 M π) 4 ( + y 4 Mπ 4 ln M ) π m M M π 2 π B m B Z F 2 (Φ B(m N, m, M π ) Φ B ( m N, m, M π )) Z = c 2 A + 2( m m N )(b 3 + b 6 ) A. Rusetsky, MAMI and beyond, Mainz, 1 April 2009 p.7

8 Finite-volume correction to the -mass at M π L 1 m E = Z F 2 ((E + m N) 2 M 2 π) q2 6E 3 1 4π 3/2 EL Z 00(1; q 2 ) q = L 2π λ 1/2 (E 2, m 2 N, M2 π) 2E, q 2 > 0 Z = Z + 2(b 3 + b 6 )c A (E m ) + O(ǫ 2 ) Lüscher s formula can be reproduced with tan δ(p) = Z p 3 (E + m N ) 2 Mπ 2 F 2 48πE 2 m E c A, b 3, b 6 c A, b 3, b 6 A. Rusetsky, MAMI and beyond, Mainz, 1 April 2009 p.8

9 Finite-volume corrections applied m N, m [GeV] M π [GeV] C. Alexandrou et al. [ETM Coll.], PRD 78 (2008) ĉ 1 = 1.6 GeV 1 â 1 = 1.7 GeV 1 ê 1 = 1.4 GeV 3 ê 1 = 6.4 GeV 3 L (fm) M π M N E ++, E +,0 δe ++, δe +, ± ± ± ± ± ± ± ± A. Rusetsky, MAMI and beyond, Mainz, 1 April 2009 p.9

10 Determining the width Γ = Z p3 cm 6πF 2 π ( m + m N ) 2 M 2 π 4 m 2 Consistency condition: m = E (L 1 ) + δe (L 1 ; c A, b 3, b 6 ) = E (L 2 ) + δe (L 2 ; c A, b 3, b 6 ) = Determine c A, b 3, b 6, width Fixing Z = 2.14 (physical value) b 3 + b GeV 1 (natural size) At present, however, the large error bars on the data below threshold preclude one from a meaningful extraction of the decay width! A. Rusetsky, MAMI and beyond, Mainz, 1 April 2009 p.10

11 Extracting bound-state pole (preliminary, 1+1 dimensions) How are the mass and the width of a resonance defined? M L 1 Non-relativistic effective Lagrangian: L = Φ (i t 2W( ))2W( )Φ + C 0 Φ Φ ΦΦ + derivatives No particle creation/annihilation Scattering amplitude in a finite volume: T = = ( ) 1 1 4p s cot δ(p) JL 0 (p) J L 0 (p) = 1 2 s 1 L k 1 k 2 p 2 = 1 4p s cot pl 2 The energy spectrum: 2δ(p) + pl = 2πn (Lüscher) A. Rusetsky, MAMI and beyond, Mainz, 1 April 2009 p.11

12 Second Riemann sheet z Assume effective-range expansion: 1 4p s cot δ(p) = A 0 + A 1 p 2 + A 2 p 4 + fix A 0, A 1, A 2 from the lattice data π L 0100 Resonance position on the second Riemann sheet: A 0 + A 1 (p ) 2 + A 2 (p ) 4 + = 1 4p s Analytic continuation of Lüscher s eq.: cot πq i, as p p q i and L = 2πq p as p p A. Rusetsky, MAMI and beyond, Mainz, 1 April 2009 p.12

13 Implications: an example Extract magnetic moment from the energy shift in the external magnetic field B (see, e.g. C. Aubin et al, arxiv: ) m(b) = m µ z B + O(B 2 ) Can one apply Lüscher s formalism to obtain the magnetic moments of resonances? Volume-dependence of the three-point function? Infinite-volume limit, as p p, q i Vertex: on-mass-shell external momenta, infinite volume A. Rusetsky, MAMI and beyond, Mainz, 1 April 2009 p.13

14 Lüscher s formalism in the inelastic case KN scattering with I = 0: M. Lage, U.-G. Meißner and AR, in preparation Two channels: KN = 1, Σπ = 2 Λ(1405) resonance between two thresholds Extract properties of the Λ(1405) Extract KN scattering length (complex) Lippmann-Schwinger equation in the infinite volume: T 11 = H 11 + H 11 iq 1 T 11 + H 12 iq 2 T 21 T 21 = H 21 + H 21 iq 1 T 11 + H 22 iq 2 T 21 H ij determine the position of the Λ(1405) pole Scattering length: a 11 = H 11 (s) + iq 2(s)(H 12 (s)) 2 1 iq 2 (s)h 22 (s), at s = (m N + M K ) 2 A. Rusetsky, MAMI and beyond, Mainz, 1 April 2009 p.14

15 Scattering length in lattice QCD Lüscher s formula: Measure E = E(L), the ground-state energy of the two-particle system in a finite box of a size L At large L, E a L 3 { 1 + c 1 a L + c 2 c 1, c 2 are known numbers a 2 } L 2 + O(L 6 ) Extract scattering length a (used to measure ππ, πk, KN scattering lengths in the past) Can be applied, however, only to the elastic case, real scattering length A. Rusetsky, MAMI and beyond, Mainz, 1 April 2009 p.15

16 NR EFT: generalization to a finite volume d 3 k (2π) 3 ( ) 1 L 3 ( ), k = 2πn L, k n Z3 Lüscher s zeta function: : iq 2 πl Z 00 (1; k 2 ), k = Lq 2π Lippmann-Schwinger equation in a finite volume: T 11 = H 11 + H 11 Z 00 (1; k 2 1)T 11 + H 12 Z 00 (1; k 2 2)T 21 T 21 = H 21 + H 21 Z 00 (1; k 2 1)T 11 + H 22 Z 00 (1; k 2 2)T 21 E n = E n (H 11, H 12, H 22 ; L), replace H ij (s) H ij (s t ) A. Rusetsky, MAMI and beyond, Mainz, 1 April 2009 p.16

17 The structure of the energy levels n=3 n= n=2 KN threshold Λ (1405) E n [GeV] n= M π L Fit the ground state and the first excited level to H 11, H 12, H 22 Determine the scattering length and the parameters of Λ (1405) A. Rusetsky, MAMI and beyond, Mainz, 1 April 2009 p.17

18 Summary, outlook Effective field theory with explicit is used to calculate quark mass dependence of energy levels in a finite volume: Large finite-volume corrections emerge below threshold Lüscher s approach has been applied to study of the static properties of resonances (magnetic moments, etc.) on the lattice: In 1+1 dimensions, external field method allows one to extract the on-mass-shell vertex function in the infinite volume The case with 3+1 dimensions: in progress Matrix elements, involving resonance states: in progress Lüscher s approach is generalized to the case of multi-channel scattering: The approach can be used to extract the KN scattering lengths and the parameters of the Λ(1405) resonance A. Rusetsky, MAMI and beyond, Mainz, 1 April 2009 p.18