EFT as the bridge between Lattice QCD and Nuclear Physics

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1 EFT as the bridge between Lattice QCD and Nuclear Physics David B. Kaplan QCHSVII Ponta Delgada, Açores, September 2006 National Institute for Nuclear Theory

2 Nuclear physics from lattice QCD? Not yet, but on the visible horizon Lattice QCD is getting Dynamical fermions Chiral symmetry Constantly improved algorithms Faster computers But still: Lattice spacing too big Volumes too small Quarks too heavy Euclidean formulation Most nuclei too big

3 Effective field theory (EFT) can make up for shortcomings of the lattice, allowing one to: Extrapolate to physical quark masses (or m q =0) Parametrize finite lattice spacing effects Parametrize finite volume effects Extract physics from cheaper fermions Determine S-matrix elements from Euclidean simulations Study complicated hadronic systems (eg, 2 baryons)

4 Outline: I. Uses of chiral perturbation theory a. Extrapolations in a, L, m q b. Fun & games with sick (but cheap) actions II. Nuclear effective theory a. The pionless theory & 2 baryons in a box b. The pionful theory, and m q extrapolation III. The EFT frontier a. hyperons b. meson-baryon interactions c. 3-body forces

5 I. Uses of chiral perturbation theory Current cost (in Tflop-years) for dynamical Wilson fermions: h #confs i h m i 1 h V i 5/4 C ost MeV 32 fm 4 Staggered fermions roughly similar; chiral fermions (domain wall, overlap) ~ 100 x more costly h a 0.08 fm i 6 (Giusti, Lattice 06) Chiral perturbation theory allows one to: Work at moderate m and extrapolate to lighter quark mass Work at moderate a and extrapolate to a -> 0 Work with light, chiral valence quarks & heavier sea quarks Work at moderate V and extrapolate to larger lattices

6 Using chiral perturbation theory to extrapolate to small m is the conventional use of the theory. But extrapolating to small a? Isn t this UV physics? 1. Symanzik action: match lattice action to continuum action with local operators suppressed by powers of a. tune away For Wilson fermions, Symanzik action includes!dim 3 chiral symmetry violating mass: a 1 ψψ!dim 5 chiral symmetry violating ops, e.g.: a ψσ µν G µν ψ!dim 6 Lorentz violating ops, e.g: a 2 3 ψd µ γ µ ψ 2. Use spurions to incorporate these lattice artifacts into the chiral Lagrangian

7 End product: a chiral Lagrangian similar to continuum version, but with simultaneous expansions in (a p) and (p/f), and spurions reflecting the smaller symmetry group of the lattice theory. Ideally: want to eliminate finite lattice space corrections; symmetry breaking terms especially. i. Use chiral fermions (domain wall, overlap) to eliminate chiral symmetry violating operators O(a -1 ), O(a), flavor violation, nasty operator mixing ii.tune bare action to reduce O(a 2 ) corrections But expensive to have dynamical chiral sea quarks!

8 One temporary solution: mixed actions Nonchiral sea quarks (staggered, Wilson) Chiral valence quarks (domain wall, overlap) Ghosts to cancel valence quark loops Some benefits of chiral symmetry...cheaper...more knobs to turn

9 m valence m s physical lattice QCD PQ χpt m sea m s

10 Extra knobs example: extraction of (2L 8 -L 5 ) Regular and exotic pion masses in O(p 4 ): M 2 π ± = 2 mb{ mb + f 2 M 2 SV = (m sea + m)b [ 1 + mb N8π 2 f 2 log B N8π 2 f 2 [ (2 m msea ) log ( 2 mb µ 2 ) + m m sea ] (2L 8 L 5 ) + N16m seab (2L f 2 6 L 4 ) } (24) ( ) 2 mb + 8(m sea + m)b (2L µ 2 f 2 8 L 5 ) + N16m seab (2L f 2 6 L 4 ) ] (25) M 2 SS = 2m sea B [ 1 + m seab N8π 2 f 2 log + 16m seab f 2 ( ) 2msea B µ 2 (2L 8 L 5 ) + N16m seab f 2 (2L 6 L 4 ) ]. (26) A Cohen, DK, A Nelson, 1999 M 2 π + M 2 SS 2M 2 SV independent of tree level & (2L 6 -L 4 ) vary m sea, m and fit to get (2L 8 -L 5 )

11 Chiral Lagrangians also in principle let one compute leading finite volume dependence, and extrapolate to infinite volume Gasser, Leutwyler Pion is lightest particle...most sensitive to boundary conditions in a box. Can compute leading volume dependence of physical quantities; not clear yet whether this will be a practical alternative to simulating at larger volumes.

12 Example #1: calculation of g A LHPC collaboration (Edwards et al.), hep-lat/ , Phys. Rev. Lett Experiment g A m! (GeV ) Mixed action (Staggered fermion sea, domain wall valence); solid line, shaded region = PQchiPT fit extrapolated to infinite volume, small quark mass.

13 Example #2: calculation of f K /f π NPLQCD collaboration (Beane, Bedaque, Orginos, Savage) hep-lat/ Use PQchiPT, mixed action (staggered sea, domain wall valence quarks) Compute to one loop, O(p 4 ), extract GL coefficient L 5 Result: f K /f π = ± Compare: f K /f π exp = ±.012 f K /f π MILC = ±.004 ±.013 Staggered fermions, not mixed action

14 II. Nuclear effective theory Nuclear = more than one baryon! The starting point: can lattice QCD be used to compute baryon-baryon scattering lengths (NN, NY, YY)? vs Minkowski Euclid Lattice gives Euclidean correlation functions, not S-matrix elements...but by measuring energy levels of two particles in a box, one can deduce the phase shifts. (Lüscher)

15 Why can t Euclidean correlation functions give phase shifts? Correlation functions computed with interpolating operators...can always add to these operators terms which vanish on-shell in Minkowski space...gives unknown Z factors Can compute ratios reliably; Z s drop out. For example, can compute energies, by seeing how correlation functions fall off with Euclidean time. 2-particle phase shifts can be extracted from spatial volume dependence of the 2-particle energies

16 Use EFT to study 2 baryons in box to extract phase shifts Low energy NN interactions: integrate out the pion. Left only with NN contact interactions. NN vertex in the EFT(/π) : = C 0 (µ) + C 2 (µ)p For NN scattering, equivalent to the effective range expansion; When external currents are included -> a powerful theory

17 Example: best determination of theoretical errors in SNO experiment for neutrino-deuteron break-up & pp fusion cross-sections by fitting L 1A Axial isovector 2-nucleon current: A a NN L 1,A t s a 3 S 1, 1 S 0 di-nucleon operators L 1,A accounts for 5% discrepancies between different potential model calculations.

18 Feynman NN scattering amplitude in the EFT(/π) : ia = = 1 1 A = ( 4π M ) 1 p cot δ(p) ip, = p (E H 0 ) p = ME 1 E H + iɛ (E H 0) p Energy eigenvalues given by zeros of Re[1/A] Energies in a box: L-dependent shift related to infinite volume phase shift

19 Eigenvalue equation in box: 0 = Re [ (ia) 1] = Re 1 Box = Re 1 E = 0 L = + E = 0 L = Box = M 4π [p cot δ] L=,p= E i M + ( n physical, infinite volume phase shift 1 (nπ/l) 2 /M E i finite, computable function of L, E i d 3 k 1 (2π) 3 k 2 /M )

20 Arrive at: Infinite volume phase shift p cot δ(p) = 1 πl S ( p 2 L 2 4π 2 ), p 2 = ME n Energy eigenvalue in box S(η) = lim Λ Λ j 1 j 2 η 4πΛ Integer triplets Measure E n! in Euclidean box -> phase shift δ(p n ) Lüscher; Beane, Bedaque, Parreño, Savage

21 Important observation by Beane, Bedaque, Parreño, Savage: Need L>> effective range, not L>> scattering length! One can study a real deuteron (or nn pair) in a fm box; smaller box for unphysically heavy pion

22 How to compute the phase shift at unphysical quark mass, and extrapolate to the physical value? Need to consider the nuclear effective theory with pions

23 Nuclear effective theory, with dynamical pions (A story of confusion & controversy) Why? Irrelevant 4-fermion interaction gives rise to large scattering lengths -> strongly coupled theory, large anomalous dimensions, power counting not obvious; Tensor force: 1/r 3 long-range force gives essential singularities in wavefunction at origin -> not amenable to perturbative renormalization.

24 Weinberg s prescription: LO! Construct NN potential in chiral expansion = ! Solve Schrödinger eq Kaplan/Savage/Wise (KSW) prescription:! Weinberg s prescription fails: counterterms occur at higher order than divergences; infinite number of counterterms needed in tensor channel! Better: sum up leading contact interaction to all orders; insert pions, chiral corrections perturbatively = LO NLO

25 KSW expansion is wrong in triplet channel Many people showed: summed up tensor interaction only requires one counterterm nonperturbatively, even though diagramatically, it appears to need an infinite number; Mehen & Stewart showed KSW perturbative expansion tensor force does not converge numerically Weinberg expansion is also wrong Some KSW objections are valid - counterterms at wrong order; Nogga, Timmermans & van Kolck showed that a counterterm is required at LO in any partial wave with attractive tensor interaction Also: ordering of the Weinberg expansion known to be incorrect in the 3-body system (Bedaque et al.)

26 ! [deg] [deg] Nogga, Timmermans, van Kolck, nucl-th/ P " [fm -1 ] 3 P2! [deg] " [fm -1 ] 3 P0 3 D3 Cutoff dependence of phase shifts in attractive triplet channels at laboratory energies of 10 MeV (solid line) line), and 100 MeV (dotted line). Weinberg s scheme: Cutoff dependence from 1-pion exchange, without counterterm (two different momenta) 3 P0 1 [fm4 ] ! [deg] Cutoff independence of phase shift with one UV subtraction 10 (4 different momenta) National Institute for 0 Nuclear Theory 5 3

27 Best nuclear EFT expansion seems to be BBSvK (Beane, Bedaque, Savage, van Kolck, nucl-th/ ): -> Expand about m q =0 world Looks like KSW in spin singlet channel; Weinberg in repulsive spin triplet channels; Nogga, Timmermans & van Kolck in attractive spin-triplet channels...

28 Application to a lattice calculation: quark mass dependence of NN phase shifts. Pioneering effort: NPLQCD hep-lat/ Precision results will require ~ 10 Tflop-years

29 Step 1: Two baryons III. The frontier NN: Compare with data to understand systematics of lattice calculation Learn about apparent fine-tuning & quark mass dependence 2-body currents NY, YY Input for theory of hypernuclei Input for theory of hyperonic atoms Role of hyperons in neutron stars

30 Multi-hadron physics from lattice QCD, continued Step 2: Baryon-meson (Coupled channels, quark annihilation diagrams) Nπ/ /N Resonance properties Currents N K/Σπ/Λπ/ΞK/Λ(1405) Strangeness in neutron stars, kaon condensation

31 Step 3?: 3-baryons Number of quark contractions grows factorially! Theorists should study 3-baryons in a box, figure out how to extract spectrum, 3-body force Valuable input for ab initio nuclear structure computations, up to A ~ 12, such as 3α 12 C Lacking experimental data in I=3/2 channel

32 Conclusions Lattice QCD is effective field theory Smart EFT calculations can accelerate the convergence between lattice and reality EFT absolutely necessary to make lattice nuclear physics progress Lots of interesting problems available to work on, both analytical and numerical.

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