Finite-volume Hamiltonian for coupled channel interactions in lattice QCD

Transcription

1 Finite-volume Hamiltonian for coupled channel interactions in lattice QCD Jiajun Wu Theory Group, Physics Division Argonne National Laboratory Collaborators: T.-S. Harry Lee, Ross D. Young, A. W. Thomas arxiv: IHEP, Beijing

2 Outline Introduction Hamiltonian for ππ scattering Finite-box Hamiltonian method Applications to Lattice QCD Lattice spectra from the experiment data Summary and Outlook

3 Resonance Region Introduction

4 Resonance Region QCD Introduction Experiment Data (cross section)

5 Resonance Region QCD Nonperturbative Introduction? Experiment Data (cross section)

6 Resonance Region QCD Nonperturbative Introduction? Experiment Data (cross section) One way Lattice QCD

7 Resonance Region QCD Nonperturbative Introduction? Experiment Data (cross section) One way? Lattice QCD Finite-volume & Euclidean time

8 Resonance Region QCD Nonperturbative One way Lattice QCD Introduction?? Experiment Data (cross section) Partial Wave Analysis Partial Wave S matrix (phase shift and inelasticity) Finite-volume & Euclidean time Finite-Volume energy eigenstate s spectrum

9 Resonance Region QCD Nonperturbative One way Lattice QCD Introduction?? Experiment Data (cross section) Partial Wave Analysis Partial Wave S matrix (phase shift and inelasticity) Finite-volume & Euclidean time Luscher Method Finite-Volume energy eigenstate s spectrum

10 Resonance Region QCD Nonperturbative One way Lattice QCD Introduction?? Experiment Data (cross section) Partial Wave Analysis Partial Wave S matrix (phase shift and inelasticity) Finite-volume & Euclidean time Luscher Method Finite-Volume energy eigenstate s spectrum?

11 Introduction The limitation of Luscher method. One channel case: one (E~L) one (E~( E~δ) Two channel case: Three (E ~ L 1, L 2, L 3 ) Three (E~δ 1, δ 2, η)

12 Introduction The limitation of Luscher method. One channel case: one (E~L) one (E~( E~δ) Two channel case: Three (E ~ L 1, L 2, L 3 ) Three (E~δ 1, δ 2, η)

13 Several E Several L Introduction The limitation of Luscher method. One channel case: one (E~L) one (E~( E~δ) Two channel case: Three (E ~ L 1, L 2, L 3 ) Three (E~δ 1, δ 2, η) Difficult! Lattice spectrum Several L Several E

14 Outline Introduction Hamiltonian for ππ scattering Finite-box Hamiltonian method Applications to Lattice QCD Lattice spectra from the experiment data Summary and Outlook

15 Hamiltonian for ππ scattering H = H0 + HI H = σ m σ + α( k ) m + k + m + k α( k ) i i i α α1 α α2 α α i= 1, n α σ i > bare state with mass m i α(k α )> the channels such as ππ, KK,

16 Hamiltonian for ππ scattering H = H0 + HI H = σ m σ + α( k ) m + k + m + k α( k ) i i i α α1 α α2 α α i= 1, n α σ i > bare state with mass m i α(k α )> the channels such as ππ, KK, H = gˆ + vˆ I + gˆ = α( kα) gi, α σi + σi gi, α α( kα) α αβ, i= 1, n vˆ = α( k ) v β( k ) α α, β β

17 Hamiltonian for ππ scattering Scattering Equation: (Partial Wave) V, t,, E ( k k ) ( k k ) 2 αγ, α γ γβ, γ β tαβ, ( kα, kβ, E) = Vαβ, ( kα, kβ) + kγ dkγ E m γ γ1+ k γ m γ1+ k γ + i ε

18 Hamiltonian for ππ scattering Scattering Equation: (Partial Wave) V, t,, E ( k k ) ( k k ) 2 αγ, α γ γβ, γ β tαβ, ( kα, kβ, E) = Vαβ, ( kα, kβ) + kγ dkγ E m γ γ1+ k γ m γ1+ k γ + i ε

19 Hamiltonian for ππ scattering Scattering Equation: (Partial Wave) V, t,, E ( k k ) ( k k ) 2 αγ, α γ γβ, γ β tαβ, ( kα, kβ, E) = Vαβ, ( kα, kβ) + kγ dkγ E m γ γ1+ k γ m γ1+ k γ + i ε

20 Hamiltonian for ππ scattering Scattering Equation: (Partial Wave) V, t,, E ( k k ) ( k k ) 2 αγ, α γ γβ, γ β tαβ, ( kα, kβ, E) = Vαβ, ( kα, kβ) + kγ dkγ E m γ γ1+ k γ m γ1+ k γ + i ε g 1 g * i, α i, β E mi v α, β

21 Hamiltonian for ππ scattering Observations & t martix S = 1 i2 ρ t,, E ρ η e α, β α α, β 0α 0β β α = 2 iδ α α α1 0α α1 0α α, α ( k k ) π k m + k m + k = S E ρ

22 Hamiltonian for ππ scattering Observations & t martix S = 1 i2 ρ t,, E ρ η e α, β α α, β 0α 0β β α = 2 iδ α α α1 0α α1 0α α, α ( k k ) π k m + k m + k = S E ρ g 1 g v α, β * i, α i, β E mi

23 Hamiltonian for ππ scattering g 1 g i * i, α, β E mi gi, α ( k ) α = g i, α π + 1 c k 2 (1 ( α α ) ) v α, β v G ( k, k )= 1 1 αβ, αβ, α β mπ (1 + ( dαk α) ) (1 + ( dβkβ) )

24 Hamiltonian for ππ scattering g 1 g i * i, α, β E mi gi, α ( k ) α = g i, α π + 1 c k 2 (1 ( α α ) ) v α, β v G ( k, k )= 1 1 αβ, αβ, α β mπ (1 + ( dαk α) ) (1 + ( dβkβ) )

25 Hamiltonian for ππ scattering

26 Hamiltonian for ππ scattering One channel case (1b-1c): only ππ, fit up to 0.9 GeV Include 5 parameters

27 Hamiltonian for ππ scattering One channel case (1b-1c): only ππ, fit up to 0.9 GeV Include 5 parameters Two channel case (1b-2c): ππ ΚΚ, fit up to 1.2 GeV Include 10 parameters

28 Outline Introduction Hamiltonian approach for ππ scattering Finite-box Hamiltonian method Applications to Lattice QCD Lattice spectra from the experiment data Summary and Outlook

29 Finite-box Hamiltonian method H ψ = E ψ Det[ H + H EI]=0 0 I 2 k π = n n L 3 Z

30 Finite-box Hamiltonian method H ψ = Det[ H + H EI]=0 0 E ψ I Eigenvalue Energy 2 k π = n n L 3 Z Lattice Size

31 Finite-box Hamiltonian method H 0 H ψ = Det[ H + H EI]=0 0 E ψ One channel case (1b-1c): m k0 + mπ = k1 + mπ fin C3( n) 2π ππ ( n) ππ ( n) g k = g k 4π L I Eigenvalue Energy 3 fin C3( n) C3( m) 2π ππ, ππ n m ππ, ππ n m v ( k, k ) = v ( k, k ) 4π 4π L H I 2 k π = n n L 3 Z Lattice Size fin fin 0 g ( k0) g ( k1)... ππ ππ fin fin fin gππ ( k0 ) vππ, ππ ( k0, k0 ) vππ, ππ ( k0, k1)... fin fin fin g ( k1 ) v, ( k1, k0 ) v, ( k1, k1)... ππ ππ ππ ππ ππ =

32 Finite-box Hamiltonian method H 0 H ψ = Det[ H + H EI]=0 0 E ψ One channel case (1b-1c): m k0 + mπ = k1 + mπ fin C3( n) 2π ππ ( n) ππ ( n) g k = g k 4π L I Eigenvalue Energy 3 fin C3( n) C3( m) 2π ππ, ππ n m ππ, ππ n m v ( k, k ) = v ( k, k ) 4π 4π L H I 2 k π = n n L 3 Z Lattice Size fin fin 0 g ( k0) g ( k1)... ππ ππ fin fin fin gππ ( k0 ) vππ, ππ ( k0, k0 ) vππ, ππ ( k0, k1)... fin fin fin g ( k1 ) v, ( k1, k0 ) v, ( k1, k1)... ππ ππ ππ ππ ππ = C ( n) 3 n The number of 2 when = C (1) = 6 C (2) = 12 C (3) = 8 n n 3 3 3

33 Finite-box Hamiltonian method 1b-1c: Det[ H + H EI]=0 0 I Spectrum from the Hamiltonian

34 Finite-box Hamiltonian method 1b-1c: Det[ H + H EI]=0 0 I Spectrum from the Hamiltonian Phase shift from the Hamiltonian T matrix Phase shift

35 Finite-box Hamiltonian method 1b-1c: Det[ H + H EI]=0 0 I Spectrum from the Hamiltonian Phase shift from the Hamiltonian T matrix Phase shift CONSISTENT OR NOT??

36 1b-1c: Finite-box Hamiltonian method T matrix Phase shift Luscher Method 2 2 δ ( k) = φ( q)mod π kl 2 E /4 mπ L q = = 3 2π 2π 2 1 qπ φ( q) = tan Z00(1; q ) Z (1; q ) = 3 4π n Z n q

37 1b-1c: Finite-box Hamiltonian method T matrix Phase shift 2 2 δ ( k) = φ( q)mod π kl 2 E /4 mπ L q = = 3 2π 2π 2 1 qπ φ( q) = tan Z00(1; q ) Z (1; q ) = 3 4π n Z n q

38 Finite-box Hamiltonian method 1b-2c: Det[ H + H EI]=0 0 I H H I 0 m k0 + mπ k0 + mk 0 0 = k1 + mπ k1 + mk fin fin fin fin 0 gππ ( k0) gkk ( k0) gππ ( k1) gkk ( k1) fin fin fin fin fin gππ ( k0 ) vππ, ππ ( k0, k0 ) vππ, KK ( k0, k0 ) vππ, ππ ( k0, k1 ) vππ, KK ( k0, k1) fin fin fin fin fin gkk ( k0 ) vkk, ππ ( k0, k0 ) vkk, KK ( k0, k0 ) vkk, ππ ( k0, k1 ) vkk, KK ( k0, k1) = fin fin fin fin fin gππ ( k1 ) vππ, ππ ( k1, k0 ) vππ, KK ( k1, k0 ) vππ, ππ ( k1, k1 ) vππ, KK ( k1, k1) fin fin fin fin fin gkk ( k1 ) vkk, ππ ( k1, k0 ) vkk, KK ( k1, k0 ) vkk, ππ ( k0, k1 ) vkk, KK ( k0, k1)

39 1b-2c: Finite-box Hamiltonian method

40 Finite-box Hamiltonian method 1b-2c: Det [ H + H E 0 I I]=0 L 1, L 2, L 3 E δ ( E), δ ( E), η( E) ππ KK ( ππ L L δ KK ππ E δ E KK ) ( L L E E ) 0= cos Δ ( ) +Δ ( ) ( ) ( ) ηcos Δ ( ) Δ ( ) δ ( ) + δ ( ) ππ KK ππ KK Δ ( L) = tan α qαπ 2 Z00(1, qα )

41 Finite-box Hamiltonian method 1 channel and 2 channel Finite-box Hamiltonian method Spectrum ( L~E ) Hamiltonian t matrix observations (δ 1, δ 2, η )

42 Finite-box Hamiltonian method 1 channel and 2 channel Finite-box Hamiltonian method Spectrum ( L~E ) Luscher Method One channel δ ( k) = φ( q)mod π φ( q) = tan 1 qπ Z (1; q ) Two channels ( ππ L L δ KK ππ E δ E KK ) ( L L E E ) 0= cos Δ ( ) +Δ ( ) ( ) ( ) ηcos Δ ( ) Δ ( ) δ ( ) + δ ( ) ππ KK ππ KK Hamiltonian t matrix observations (δ 1, δ 2, η )

43 Finite-box Hamiltonian method 1 channel and 2 channel Finite-box Hamiltonian method Spectrum ( L~E ) Luscher Method 1. Our approach is correct! 2. 1 channel and 2 channel is almost the same One channel δ ( k) = φ( q)mod π φ( q) = tan 1 qπ Z (1; q ) Two channels ( ππ L L δ KK ππ E δ E KK ) ( L L E E ) 0= cos Δ ( ) +Δ ( ) ( ) ( ) ηcos Δ ( ) Δ ( ) δ ( ) + δ ( ) ππ KK ππ KK Hamiltonian t matrix observations (δ 1, δ 2, η )

44 Outline Introduction Hamiltonian for ππ scattering Finite-box Hamiltonian method Applications to Lattice QCD Lattice spectra from the experiment data Summary and Outlook

45 Applications to Lattice QCD If there are some Lattice data of spectrum, how can we change them to the observations? By our method: Fitting By Luscher method: Solving Equation Can (1 channel ) or Can not (2 or multi-channel )

46 Applications to Lattice QCD If there are some Lattice data of spectrum, how can we change them to the observations? By our method: Fitting By Luscher method: Solving Equation Can (1 channel ) or Can not (2 or multi-channel ) Fitting bring one problem: would the form of the g and v influence the last result or not? Check this problem: we will produce some Lattice spectrum data by the 1b-1c and 1b-2c models, then we will use different form of potential to fit these data, then using fitted parameters to compute the observations to check the dependence of the form of interaction.

47 Applications to Lattice QCD A g ( k ) = g 1 (1 ( ) ) i, α i, α α 2 π + cα kα G v ( k, k )= αβ, αβ, α β mπ (1 + ( dαkα) ) (1 + ( dβkβ) ) B g ( k ) = g i, α i, α α 2 2 π (1 + ( cα kα ) ) G v ( k, k )= (1 ( ) ) (1 ( ) ) αβ, α, β α β mπ + dαkα + dβkβ 2 gi, α ( cα kα ) C gi, α( kα) = e π Gα, β ( dα kα ) vαβ, ( kα, kβ )= e e 2 m π 2 ( d k ) 2 β β 4

48 Applications to Lattice QCD One channel case: A g ( k ) = g 1 (1 ( ) ) i, α i, α α 2 π + cα kα G v ( k, k )= αβ, αβ, α β mπ (1 + ( dαkα) ) (1 + ( dβkβ) ) B g ( k ) = g i, α i, α α 2 2 π (1 + ( cα kα ) ) G v ( k, k )= (1 ( ) ) (1 ( ) ) αβ, α, β α β mπ + dαkα + dβkβ 2 gi, α ( cα kα ) C gi, α( kα) = e π Gα, β ( dα kα ) vαβ, ( kα, kβ )= e e 2 m π 2 ( d k ) 2 β β 4

49 Applications to Lattice QCD Two channels case: FITTING COMPUTING

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51 Applications to Lattice QCD Two channels case: By 16 or 24 points on the two different L, Luscher method can tell us NOTHING, but our approach can give a good description of observations.

52 Summary Applications to Lattice QCD Fitting approach with our Hamiltonian method: 1. It is valid in the energy region where the spectrum data are fitted. 2. It is valid not only for one-channel case, but also for twochannels case, then we believe it would be also valid for multi-channel case. 3. It is independent of the form of the Hamiltonian.

53 Outline Introduction Hamiltonian for ππ scattering Finite-box Hamiltonian method Applications to Lattice QCD Lattice spectra from the experiment data Summary and Outlook

54 Lattice spectra from the experiment data Spectra Observations Observations Spectra

55 Lattice spectra from the experiment data Spectra Observations Observations Spectra

56 Outline Introduction Hamiltonian for ππ scattering Finite-box Hamiltonian method Applications to Lattice QCD Lattice spectra from the experiment data Summary and Outlook

57 Summary We apply the finite-volume Hamiltonian method to the ππ - KK scattering. The finite-volume Hamiltonian method is as accurate as the approach based on Luscher method in both the one-channel and two-channel cases. The finite-box Hamiltonian method can give correct prediction of scattering observables in the energy region where the spectrum data are fitted, independent of the form of the Hamiltonian. In the two channel cases, this Hamiltonian method need much less LQCD efforts than Luscher method.

58 Outlook Our approach is only in the S-wave S wave and Center Mass system (C. M.). It is the simplest system. P-wave Consider the Spin and angular momentum interaction C. M. boost system High eigenvalue of energy & Three body case & Electromagnetic form factor

59 Thank you very much