Effective Field Theories for lattice QCD
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1 Effective Field Theories for lattice QCD Stephen R. Sharpe University of Washington S. Sharpe, EFT for LQCD: Lecture 1 New horizons in lattice field theory, Natal, Brazil 1
2 Outline of Lectures 1. Overview & Introduction to continuum chiral perturbation theory (ChPT) 2. Illustrative results from ChPT; SU(2) ChPT with heavy strange quark; finite volume effects from ChPT and connection to random matrix theory 3. Including discretization effects in ChPT 4. Partially quenched ChPT and applications, including a discussion of whether mu=0 is meaningful S. Sharpe, EFT for LQCD: Lecture 3 New horizons in lattice field theory, Natal, Brazil 2 /50
3 Outline of lecture 1 Why LQCD needs effective field theories (EFTs) Focus in these lectures on ChPT, Symanzik s low-energy effective theory for lattice field theories (SET) and their combination Examples of uses of ChPT in recent simulations Introduction to continuum ChPT Emphasize features & results relevant to LQCD and later lectures S. Sharpe, EFT for LQCD: Lecture 1 New horizons in lattice field theory, Natal, Brazil 3 /50
4 (or ChPT) S. Sharpe, EFT for LQCD: Lecture 1 New horizons in lattice field theory, Natal, Brazil 4 /50
5 Some continuum ChPT References S. Sharpe, EFT for LQCD: Lecture 1 New horizons in lattice field theory, Natal, Brazil 5 /50
6 ChPT for LQCD References S.R. Sharpe, Applications of Chiral Perturbation Theory to Lattice QCD, hep-lat/ (a write-up of an earlier, shorter form of these lectures) M.F.L. Golterman, Applications of chiral perturbation theory to lattice QCD, arxiv: [hep-lat] (from 2009 Les Houches summer school) S. Sharpe, EFT for LQCD: Lecture 1 New horizons in lattice field theory, Natal, Brazil 6 /50
7 Why LQCD needs EFTs Most calculations require extrapolation in light (u & d) quark masses ~ mstrange physical point ~ mlight Landscape of recent Nf=2+1 simulations [Fodor & Hoelbling, RMP 2012] S. Sharpe, EFT for LQCD: Lecture 1 New horizons in lattice field theory, Natal, Brazil 7 /50
8 Why LQCD needs EFTs ALL calculations require extrapolation in lattice spacing a N.B. Leading discretization error is proportional to a 2 with modern actions physical point Landscape of recent Nf=2+1 simulations [Fodor & Hoelbling, RMP 2012] S. Sharpe, EFT for LQCD: Lecture 1 New horizons in lattice field theory, Natal, Brazil 8 /50
9 Why LQCD needs EFTs Most calculations require extrapolation to infinite box size L Exception is work in epsilon-regime (Mπ L < 1) physical mass Landscape of recent Nf=2+1 simulations [Hoelbling 2010] S. Sharpe, EFT for LQCD: Lecture 1 New horizons in lattice field theory, Natal, Brazil 9 /50
10 Why LQCD needs EFTs LQCD continues to need extrapolations in m, a & L Also needs extrapolation from mu=md (isospin limit) to physical mu & md, and from αem=0 to αem=1/137 - Increasingly important as percent-level precision is attained (e.g. in fk/fπ, BK,...) - First calculations including isospin breaking and EM in measure (either directly or by reweighting) are underway but far from production status Chiral Perturbation Theory, incorporating discretization effects using Symanzik s effective theory, provides the necessary functional forms for these extrapolations ChPT + SET provide predictions for the small-volume epsilon-regime which can be used to determine physical parameters These are the topics I aim to introduce in these lectures Other EFTs (not discussed here) play a crucial role in lattice calculations - Heavy Quark Effective Theory in the calculation of B-meson properties - Non-Relativistic QCD for simulating heavy quarks - Pion-less (and pion-full) nuclear EFT to extend lattice calculations of multinucleon properties to a broader array of quantities S. Sharpe, EFT for LQCD: Lecture 1 New horizons in lattice field theory, Natal, Brazil 10/50
11 Why LQCD needs EFTs Widespread use of unphysical simulations ChPT (+ SET) allows one to remove unphysical effects, or (in the case of PQQCD) to see how to use the extra information to learn about physical quantities S. Sharpe, EFT for LQCD: Lecture 1 New horizons in lattice field theory, Natal, Brazil 11/50
12 Why ChPT needs LQCD! ChPT contains many a priori unknown parameters Low-energy constants or LECs Some can be determined by matching with experiment, but many cannot LQCD provides additional dials that are unavailable in the real world Quark masses can be tuned---making a feature out of a bug Allows determination of LECs that are difficult or impossible to obtain otherwise ChPT can then be used to extrapolate, approximately, from the simple processes that LQCD can calculate, e.g. K ππ, to those that it cannot, e.g. K 3π LQCD can test how well ChPT converges in a much more controlled setting than experiment S. Sharpe, EFT for LQCD: Lecture 1 New horizons in lattice field theory, Natal, Brazil 12/50
13 Outline of today s lecture Why LQCD needs effective field theories (EFTs) Focus in these lectures on ChPT, Symanzik s low-energy effective theory for lattice field theories (SET) and their combination Examples of uses of ChPT in recent simulations Introduction to continuum ChPT Emphasize features & results relevant to LQCD and later lectures S. Sharpe, EFT for LQCD: Lecture 1 New horizons in lattice field theory, Natal, Brazil 13/50
14 Examples of ChPT in use today Chiral-continuum extrapolation of fb using improved Wilson fermions - Left: fit to form using heavy-meson ChPT + a 2 corrections. Curvature from chiral logs - Right: fit to linear ansatz + a 2 (used to estimate extrapolation error) - a = 0.045, & fm Bahr et al. [ALPHA collaboration], arxiv: S. Sharpe, EFT for LQCD: Lecture 1 New horizons in lattice field theory, Natal, Brazil 14/50
15 Examples of ChPT in use today Chiral-continuum extrapolation of D K form factor using improved staggered fermions and staggered ChPT (SChPT) - Left: using SU(3) SChPT; Right: using SU(2) SChPT - mlight / ms ranges from 0.4 (blue) to 0.1 (red) [compared to ~0.04 for physical case] - Lattice spacings range from fm (triangles) to 0.12 fm (squares) Bailey et al. [FNAL-MILC collaboration], arxiv: S. Sharpe, EFT for LQCD: Lecture 1 New horizons in lattice field theory, Natal, Brazil 15/50
16 Examples of ChPT in use today Chiral-continuum extrapolation of pion decay constant and mass using improved staggered fermions and staggered ChPT (SChPT) - Includes partially quenched data (mvalence < msea) - curvature due to NLO chiral logs (which include discretization corrections--- taste breaking --- essential for good fits) - fits include continuum NNLO chiral logs fπ mπ 2 /m mlight mlight Bernard et al. [MILC collaboration], arxiv: S. Sharpe, EFT for LQCD: Lecture 1 New horizons in lattice field theory, Natal, Brazil 16/50
17 Examples of ChPT in use today Distribution of small eigenvalues of the Hermitian Wilson-Dirac operator in small (1.5 fm) box compared to Wilson ChPT in the epsilon-regime - Left: fit to top. charge 0 configs (determines leading order LECs in ChPT) - Right: corresponding prediction for top. charge 1 configs (note asymmetry) fπ mπ 2 /m Damgaard et al., arxiv: S. Sharpe, EFT for LQCD: Lecture 1 New horizons in lattice field theory, Natal, Brazil 17/50
18 Outline of today s lecture Why LQCD needs effective field theories (EFTs) Focus in these lectures on ChPT, Symanzik s low-energy effective theory for lattice field theories (SET) and their combination Examples of uses of ChPT in recent simulations Introduction to continuum ChPT Emphasize features & results relevant to LQCD and later lectures S. Sharpe, EFT for LQCD: Lecture 1 New horizons in lattice field theory, Natal, Brazil 18/50
19 Essence of EFTs Consider only low energy degrees of freedom - Integrate out high energy or massive degrees of freedom - Requires a separation of scales Use symmetries alone to constrain terms in effective Lagrangian - Systematically order terms using an appropriate power counting in (low scale)/(high scale) S. Sharpe, EFT for LQCD: Lecture 1 New horizons in lattice field theory, Natal, Brazil 19/50
20 Essence of EFTs Consider only low energy degrees of freedom - Integrate out high energy or massive degrees of freedom - Requires a separation of scales Use symmetries alone to constrain terms in effective Lagrangian - Systematically order terms using an appropriate power counting in (low scale)/(high scale) Examples: p QCD 1/a Analogous to Fermi theory of weak interactions p S. Sharpe, EFT for LQCD: Lecture 1 New horizons in lattice field theory, Natal, Brazil 20/50
21 Essence of EFTs Consider only low energy degrees of freedom - Integrate out high energy or massive degrees of freedom - Requires a separation of scales Use symmetries alone to constrain terms in effective Lagrangian - Systematically order terms using an appropriate power counting in (low scale)/(high scale) Examples: m m,m N 1 GeV Conceptually more complicated than SET, since no perturbative treatment S. Sharpe, EFT for LQCD: Lecture 1 New horizons in lattice field theory, Natal, Brazil 21/50
22 General recipe for an EFT S. Sharpe, EFT for LQCD: Lecture 1 New horizons in lattice field theory, Natal, Brazil 22/50
23 Chiral perturbation theory I will consider ChPT for pseudo-goldstone bosons (PGBs): pions, kaons and eta I will only touch briefly on extension to heavy meson or baryon sources, in the context of SU(2) ChPT with heavy kaons (next lecture) The story goes back to Nambu s realization that the lightness of pions compared to other mesons and baryons can be understood if they are PGBs of chiral symmetry, which brings us to... S. Sharpe, EFT for LQCD: Lecture 1 New horizons in lattice field theory, Natal, Brazil 23/50
24 Chiral symmetry in QCD S. Sharpe, EFT for LQCD: Lecture 1 New horizons in lattice field theory, Natal, Brazil 24/50
25 Explicit breaking of Chiral symmetry I.E. Can one use SU(3) ChPT or is one forced to use only SU(2) ChPT? S. Sharpe, EFT for LQCD: Lecture 1 New horizons in lattice field theory, Natal, Brazil 25/50
26 Spontaneous breaking of Chiral symmetry S. Sharpe, EFT for LQCD: Lecture 1 New horizons in lattice field theory, Natal, Brazil 26/50
27 Symmetry breaking if M=0 This (massless QCD) is the theory around which chiral perturbation theory is developed S. Sharpe, EFT for LQCD: Lecture 1 New horizons in lattice field theory, Natal, Brazil 27/50
28 Turning on a (small) mass matrix M VEV of mass-term in Lagrangian leads to a potential for Ω If M diagonal and positive (conventional choice) minimizing V gives Since VEV breaks SU(3)A while SU(3)V is unbroken Pions, kaons and eta become massive pseudo-goldstone bosons (PGBs), since symmetry is explicitly broken A twisted mass leads to equivalent physics (useful for twisted-mass fermions ) S. Sharpe, EFT for LQCD: Lecture 1 New horizons in lattice field theory, Natal, Brazil 28/50
29 Constructing the EFT S. Sharpe, EFT for LQCD: Lecture 1 New horizons in lattice field theory, Natal, Brazil 29/50
30 Representing Goldstone Bosons Or: Integrate out ρ(x) by hand, considering only correlators with external θ(x) S. Sharpe, EFT for LQCD: Lecture 1 New horizons in lattice field theory, Natal, Brazil 30/50
31 Exponential parametrization of GBs S. Sharpe, EFT for LQCD: Lecture 1 New horizons in lattice field theory, Natal, Brazil 31/50
32 Building blocks for ChPT S. Sharpe, EFT for LQCD: Lecture 1 New horizons in lattice field theory, Natal, Brazil 32/50
33 Putting the blocks together Remember the rule: construct most general local Lagrangian consistent with symmetries (SU(3)L, SU(3)R, Lorentz/Euclidean, C, P, T) No derivatives: no terms since One derivative: no terms by Lorentz invariance Two derivatives: 1. No derivatives and one mass insertion: 2. Four derivatives: 3. [tr(l µ L µ )] Wess-Zumino-Witten term involving ε-tensor---gives U(1)A anomaly mediated processes, e.g. π0 γγ. Not discussed here. Two derivatives & one mass term (2 terms) Two mass terms (3 terms) =1 tr(l µ L µ )= tr(@ µ )=tr(r µ R µ ) tr(m )+tr( M ) tr(l µ L )tr(l µ L ) tr(l µ L µ L L ) tr(l µ L L µ L ) [not independent in SU(2)] [not independent in SU(2) or SU(3)] S. Sharpe, EFT for LQCD: Lecture 1 New horizons in lattice field theory, Natal, Brazil 33/50
34 Putting the blocks together Remember the rule: construct most general local Lagrangian consistent with symmetries (SU(3)L, SU(3)R, Lorentz/Euclidean, C, P, T) No derivatives: no terms since One derivative: no terms by Lorentz invariance Two derivatives: 1. No derivatives and one mass insertion: 2. Four derivatives: 3. [tr(l µ L µ )] Wess-Zumino-Witten term involving ε-tensor---gives U(1)A anomaly mediated processes, e.g. π0 γγ. Not discussed here. Two derivatives & one mass term (2 terms) Two mass terms (3 terms) =1 tr(l µ L µ )= tr(@ µ )=tr(r µ R µ ) tr(m )+tr( M ) tr(l µ L )tr(l µ L ) tr(l µ L µ L L ) tr(l µ L L µ L ) [not independent in SU(2)] [not independent in SU(2) or SU(3)] S. Sharpe, EFT for LQCD: Lecture 1 New horizons in lattice field theory, Natal, Brazil 34/50
35 Putting the blocks together Remember the rule: construct most general local Lagrangian consistent with symmetries (SU(3)L, SU(3)R, Lorentz/Euclidean, C, P, T) No derivatives: no terms since One derivative: no terms by Lorentz invariance Two derivatives: 1. No derivatives and one mass insertion: 2. Four derivatives: 3. [tr(l µ L µ )] Wess-Zumino-Witten term involving ε-tensor---gives U(1)A anomaly mediated processes, e.g. π0 γγ. Not discussed here. Two derivatives & one mass term (2 terms) Two mass terms (3 terms) =1 tr(l µ L µ )= tr(@ µ )=tr(r µ R µ ) tr(m )+tr( M ) tr(l µ L )tr(l µ L ) tr(l µ L µ L L ) tr(l µ L L µ L ) [not independent in SU(2)] [not independent in SU(2) or SU(3)] S. Sharpe, EFT for LQCD: Lecture 1 New horizons in lattice field theory, Natal, Brazil 35/50
36 Putting the blocks together Remember the rule: construct most general local Lagrangian consistent with symmetries (SU(3)L, SU(3)R, Lorentz/Euclidean, C, P, T) No derivatives: no terms since One derivative: no terms by Lorentz invariance Two derivatives: 1. No derivatives and one mass insertion: 2. Four derivatives: 3. [tr(l µ L µ )] Wess-Zumino-Witten term involving ε-tensor---gives U(1)A anomaly mediated processes, e.g. π0 γγ. Not discussed here. Two derivatives & one mass term (2 terms) Two mass terms (3 terms) =1 tr(l µ L µ )= tr(@ µ )=tr(r µ R µ ) tr(m )+tr( M ) tr(l µ L )tr(l µ L ) tr(l µ L µ L L ) tr(l µ L L µ L ) [not independent in SU(2)] [not independent in SU(2) or SU(3)] S. Sharpe, EFT for LQCD: Lecture 1 New horizons in lattice field theory, Natal, Brazil 36/50
37 Leading order chiral Lagrangian S. Sharpe, EFT for LQCD: Lecture 1 New horizons in lattice field theory, Natal, Brazil 37/50
38 Aside on vacuum structure S. Sharpe, EFT for LQCD: Lecture 1 New horizons in lattice field theory, Natal, Brazil 38/50
39 Leading order (P)GB properties S. Sharpe, EFT for LQCD: Lecture 1 New horizons in lattice field theory, Natal, Brazil 39/50
40 LO ChPT predictions for PGB masses [Gell-Mann Okubo relation] S. Sharpe, EFT for LQCD: Lecture 1 New horizons in lattice field theory, Natal, Brazil 40/50
41 Lessons for lattice (1) + + S. Sharpe, EFT for LQCD: Lecture 1 New horizons in lattice field theory, Natal, Brazil 41/50
42 Lessons for lattice (2) + - Recent progress using reweighting of isospin-symmetric configurations [Ishikawa et al., arxiv: & Aoki et al., arxiv: ] Much recent progress in reducing errors in disconnected diagrams using all-to-all techniques S. Sharpe, EFT for LQCD: Lecture 1 New horizons in lattice field theory, Natal, Brazil 42/50
43 Power-counting in ChPT (M=0) f 2 f 2 S. Sharpe, EFT for LQCD: Lecture 1 New horizons in lattice field theory, Natal, Brazil 43/50
44 Power-counting in ChPT (M=0) ~ Determined by unitarity (2 particle cut) S. Sharpe, EFT for LQCD: Lecture 1 New horizons in lattice field theory, Natal, Brazil 44/50
45 True expansion parameter? S. Sharpe, EFT for LQCD: Lecture 1 New horizons in lattice field theory, Natal, Brazil 45/50
46 Lessons for lattice (3) + - In disrepute(?) S. Sharpe, EFT for LQCD: Lecture 1 New horizons in lattice field theory, Natal, Brazil 46/50
47 Technical aside: adding sources S. Sharpe, EFT for LQCD: Lecture 1 New horizons in lattice field theory, Natal, Brazil 47/50
48 Technical aside: adding sources S. Sharpe, EFT for LQCD: Lecture 1 New horizons in lattice field theory, Natal, Brazil 48/50
49 LO chiral Lagrangian including sources S. Sharpe, EFT for LQCD: Lecture 1 New horizons in lattice field theory, Natal, Brazil 49/50
50 NLO chiral Lagrangian L µ `µ + i[`µ, ` ] S. Sharpe, EFT for LQCD: Lecture 1 New horizons in lattice field theory, Natal, Brazil 50/50
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