Scattering amplitudes from lattice QCD

Transcription

1 Scattering amplitudes from lattice QCD David Wilson Old Dominion University Based on work in collaboration with J.J. Dudek, R.G. Edwards and C.E. Thomas. Jefferson lab theory center 20th October 2014.

2 Two talks Today: Introductory stuff The methods we are using Elastic scattering Next Week: Jo Dudek Resonances in coupled channel scattering from lattice QCD David Wilson Resonances in πk Scattering 2

3 Introduction Lagrangian of QCD Spectrum of hadrons L QCD = X q q i /D - m q q F µ F µ mêgev /D = µ (@ µ - iga µ ) F µ µ A A µ + g [A µ, A ] J P Coloured quark and gluon degrees of freedom. Only colourless states, no asymptotic quarks and gluons. Excited state spectrum contains many interesting open questions. Interesting effects near thresholds: tetraquarks? meson-meson bound states? f0(980) in ππ scattering and new charmonium states, eg: Z(4430). Hybrid states, containing explicit gluonic degrees of freedom. Could be seen in new experiments, like Glue-X. David Wilson Resonances in πk Scattering 3

4 Strong coupling Lagrangian of QCD g 2 4 PDG 2013 L QCD = X q q i /D - m q q F µ F µ /D = µ (@ µ - iga µ ) F µ µ A A µ + g [A µ, A ] Many interesting consequences: Confinement - no asymptotic quarks or gluons. Dynamical chiral symmetry breaking. Light physical pion ~ goldstone boson of the symmetry breaking. Cannot use perturbation theory: Non-perturbative methods needed. Several options including: Models Schwinger-Dyson+Bethe-Salpeter Effective Field Theories Lattice QCD David Wilson Resonances in πk Scattering 4

5 Path integrals The starting point is the path integral: hx b e -iht x a i = Z Dx(t)e is[x(t)] t x b x a x David Wilson Resonances in πk Scattering 5

6 Path integrals To solve numerically, consider the discretised version t hx b e -iht x a i = xti Z dx ti e is (x ti ) x b t N... t 1 x a t 0 x David Wilson Resonances in πk Scattering 6

7 Path integrals Evaluate correlation functions from the path integral: h0 O i (t)o j (0) 0i = 1 Z 0 Z D D DA O i (t)o j (0) eis [,,A] In order to deal with strong coupling: Solve the QCD path integral numerically. Integrate over gauge field configurations: Infinitely many possibilities. Store field values on a discrete set of points David Wilson Resonances in πk Scattering 7

8 Lattice QCD y a L x Use a finite spacetime volume, L 3 t. (L Roughly 2-3fm in these studies). Use a finite number of points, with separation a ~ 0.1fm. (L/a = 16, 20, 24) Quarks live on discrete points and the gluons live on the links between them. Use periodic boundary conditions: Volume becomes a torus. David Wilson Resonances in πk Scattering 8

9 Lattice QCD x t Use a finite spacetime volume, L 3 t. (L Roughly 2-3fm in these studies). Use a finite number of points, with separation a ~ 0.1fm (L/a = 16, 20, 24). Change variables to Euclidean spacetime to simplify integration. Quarks live on discrete points and the gluons live on the links between them. Use periodic boundary conditions: Volume becomes a torus. David Wilson Resonances in πk Scattering 9

10 Correlation functions Evaluate correlation functions from the path integral: C ij (t) =h0 O i (t)o j (0) 0i = 1 Z D Z D DA O i (t)o j (0) e-s [,,A] 0 Leads to the ground state energy for large t: C ij (t) = X n 1 2E n h0 O i nihn O j 0i e-e nt = Z? i Z j 2E n e -E nt The symmetries of the operators dictate which states can be extracted 5 J P = 0 - i J P = 1 - David Wilson Resonances in πk Scattering 10

11 Symmetry on the lattice The lattice has a cubic symmetry. It does not have the O(3) symmetry of continuous space. Eg: 2D QM vs Continuous rotational spatial symmetry e i! e i +i e i! e i +in /2 Only symmetric at discrete angles David Wilson Resonances in πk Scattering 11

12 Symmetry on the lattice vs Continuous rotational spatial symmetry e i! e i +i e i! e i +in /2 Only symmetric at discrete angles Cubic symmetry groups mix the continuum angular momentum: Irrep J P A , 4 +,... T1-1 -, 3 -,... David Wilson Resonances in πk Scattering 12

13 Operators with overall momentum Because momentum is quantised, different energies can be accessed by considering operators with an overall momentum ~p = 2 L ~n E 2 lat = E 2 cm L ~n Overall zero momentum: (0, 0, 0) (0, 0, 0), (1, 0, 0) (-1, 0, 0),... One unit: (1, 0, 0) (0, 0, 0), (1, 1, 0) (-1, 0, 0),... Useful to consider systems with ~n =(0, 0, 0), (1, 0, 0), (1, 1, 0), (1, 1, 1), (2, 0, 0),,... Less symmetry: More mixing of angular momentum! David Wilson Resonances in πk Scattering 13

14 Extracting a spectrum Getting the ground state is useful, but we want to extract the whole spectrum in a finite volume. Fitting subleading exponentials doesn t get very far: With very precise data, sometimes a second state can be found. A solution: The variational method. C ij (t)v n j = n (t)c ij (t 0 )v n j If more than one operator overlaps onto the same state represented by some eigenvector the generalised eigenvalue problem can be solved and then as many states as operators may be extracted. n(t) e -E n(t-t 0 ) v n i... a large basis of operators are needed David Wilson Resonances in πk Scattering 14

15 Operators and the variational method Use a large basis of operators C(t)v n = n (t)c(t 0 )v n n(t) e -E n(t-t 0 ) O i = O i =! D...! D i = {1, 0, 5, 0 5, i, 0 i, 5 i, [ i, j]} Use the variational method with a large correlation matrix to obtain an optimal spectrum. David Wilson Resonances in πk Scattering 15

16 Operators and the variational method t ê a t t ê a t t ê a t 1.15 e E 1t 1 (t) e E 2t 2 (t) e E 3t 3 (t) e E 4t 4 (t) 1.4 t ê a t T - 1 irrep ( JP = 1 - ) using cc operators C(t)v n = n(t) e -E n(t-t 0 ) L. Liu et al David Wilson Resonances in πk Scattering 16 n (t)c(t 0 )v n

17 Meson-meson energy levels T1 - irrep: contains states with J P =1 -, 3 -,4 -,... a t E cm q q 0.30 [111] [-1-1-1] The spectrum should also contain multiparticle states: (~p 1 ) (~p 2 ) 0.25 [110] [-1-10] No continuum of energies: allowed momentum is quantised 0.20 [100] [-100] Two meson states do not appear to overlap well onto the singleparticle operators we have used David Wilson Coupled-channel scattering from lattice QCD 17

18 Meson-meson energy levels We could construct something simple to overlap on to two-pion states q 5 q q 5 q But it s better to make use of variational method solutions: v n represents the variationally-optimal pion, so to create such a state we use: C(t)v n = n (t)c(t 0 )v n n(t) e -E n(t-t 0 ) n = X i v n i O i 0.2 q q q Dq 0.15 q DDq 0.1 q DDDq David Wilson Coupled-channel scattering from lattice QCD 18

19 Meson-meson energy levels T1 - irrep: contains states with J P =1 -, 3 -,4 -,... a t E cm q q 0.30 [111] [-1-1-1] 0.25 [110] [-1-10] 0.20 [100] [-100] 0.15 David Wilson Coupled-channel scattering from lattice QCD 19

20 Meson-meson energy levels T1 - irrep a t E cm q q q q David Wilson Coupled-channel scattering from lattice QCD 20

21 Two particles in a finite volume Simple 1-d problem Periodic B.C. s x 1 x 1 2 L 2 No interactions total energy is just the sum For a single particle: E =(~p m 2 1) 1 2 +(~p m 2 2) 1 2 ~p i 2 = 2 n L 2 Non-interacting energies in a finite volume are known from the single-particle analysis If we measure the energies on the lattice and find a difference, this shift must be due to interactions. Lüscher et al David Wilson Coupled-channel scattering from lattice QCD 21

22 Two particles in a finite volume 1 x In simple QM: Interactions lead to phase shift δ on the wavefunction (x) e ±ipx Periodic boundary conditions for interacting particles. 2 (0) = x=l pl 2 + (p) = 0 p = 2 n L - 2 L (p) Discrete spectrum of allowed energies directly connected to the phase shift. If we measure the energies on the lattice and find a difference, this shift must be due to interactions. David Wilson Coupled-channel scattering from lattice QCD 22

23 Two particles in a finite volume 1 x 2 In 3+1 dimensions, this leads to a simple relation between the finite volume energy and the S-wave scattering length: k cot = 1 a rk2 + O(k 4 ) = 1 L X 1 2 ~n2z 3 ~n 2 - ~k L/(2 ) David Wilson Coupled-channel scattering from lattice QCD 23

24 Finite volume spectra a E 0.30 Weak interactions Lêa Small, +ve scattering length (weakly attractive) David Wilson Coupled-channel scattering from lattice QCD 24

25 Weakly repulsive scattering from QCD !, I = 2 Dudek, Edwards and Thomas David Wilson Coupled-channel scattering from lattice QCD 25

26 Weakly repulsive scattering from QCD !, I = 2 David Wilson Coupled-channel scattering from lattice QCD 26

27 Resonances t = 1 (E 2 ) E (E 2 ) m 2 R - E2 - ie (E 2 ) k 3 cm cot 1 = 6 g 2 E m2 - E 2 (s) = g2 R 6 k 3 cm s D a E cm David Wilson Coupled-channel scattering from lattice QCD 27

28 Finite volume spectra with a resonance a E 0.30 Weak interactions Lêa David Wilson Coupled-channel scattering from lattice QCD 28

29 Finite volume spectra with a resonance a E 0.30 Narrow resonance Lêa David Wilson Coupled-channel scattering from lattice QCD 29

30 Finite volume spectra with a resonance a E 0.30 Narrow resonance Lêa David Wilson Coupled-channel scattering from lattice QCD 30

31 Extracting the ρ resonance Several volumes: L=16, 20, 24. Operators in several moving frames, upto n=(2,0,0). Anisotropic lattices: temporal spacing 3.5 times finer for better energy resolution. Combination of single particle and meson-meson operators. mπ=391 MeV David Wilson Coupled-channel scattering from lattice QCD 31

32 Finite volume spectra in I=1 J= David Wilson Coupled-channel scattering from lattice QCD 32

33 A resonance from QCD π π π 1 (s) π s 1 2 (s) m 2 R - s - is 1 2 (s) (s) = g2 R 6 k 3 cm E 2 cm J. J. Dudek, R. G. Edwards and C. E. Thomas Phys. Rev. D 87, David Wilson Coupled-channel scattering from lattice QCD 33

34 Extensions Coupled-channel scattering, eg: see Jo s talk next week. K! K, K! K Also! KK,! KK Nucleons, eg: N! N Form factors, matrix elements, eg:! N! N David Wilson Coupled-channel scattering from lattice QCD 34

35 Summary Using finite volume formalism of Lüscher and others, it is possible to translate finite volume energy levels into scattering amplitudes. Scattering information, including resonances, can be obtained using lattice QCD. A large basis of operators makes it possible, through the variational method, to obtain excited states with the same quantum numbers. In order to extract energy levels that can mostly be attributed to meson-meson states we found it necessary to construct operators. Many exciting opportunities for future extensions David Wilson Coupled-channel scattering from lattice QCD 35

36 Backup slides David Wilson Coupled-channel scattering from lattice QCD 36

37 Principal correlators e E nt n(t) David Wilson Coupled-channel scattering from lattice QCD 37

38 Relative operator overlaps Z i = hn O i 0i David Wilson Coupled-channel scattering from lattice QCD 38

39 local & local only only David Wilson Coupled-channel scattering from lattice QCD 39