Introduction to Effective Field Theories in QCD Ubirajara U. van - PowerPoint PPT Presentation

Multi-scale problems ⎡ ⎤ ⎛ ⎞ ⎛ ⎞ 2 2 2 2 � p e p = − + 1 ; ⎢ O ⎥ ⎜ ⎟ ⎜ ⎟ H π 2 2 2 2 2 2 4 ⎢ ⎥ ⎝ ⎠ ⎝ ⎠ m r m c m c r ⎣ ⎦ e e e ∼ H r R ⎛ ⎞ 2 2 � e ( ) − ∼ atom ⎜ ⎟ � E R 2 π 2 4 ∼ ⎝ ⎠ m R R p e R ( ) 2 1 � dE R e α ≡ ≅ = = � 1 0 R π α 4 � 137 c dR m c e m c = 2 0.5 MeV α e Three α = 2 ∼ 3.6 keV p c m c e α scales 2 1 p − α = 2 2 ∼ ∼ 13.6 eV E m c 2 2 e m e (from now on, units such that ) = = 6/2/2009 v. Kolck, Intro to EFTs 9 � 1, 1 c

PDG, 2005 However… no obvious small coupling in nuclear forces. 0.3 QCD α s ( µ ) “fine-structure” 0.2 constant 0.1 0 2 Needed: method that does not 1 10 10 µ GeV rely on small couplings EFFECTIVE FIELD THEORY 6/2/2009 v. Kolck, Intro to EFTs 10

I do not believe that scientific progress is always best advanced by keeping an altogether open mind. It is often necessary to forget one’s doubts and to follow the consequences of one’s assumptions wherever they may lead ---the great thing is not to be free of theoretical prejudices, but to have the right theoretical prejudices. And always, the test of any theoretical preconception is in where it leads. S. Weinberg, The First Three Minutes 1972 6/2/2009 v. Kolck, Intro to EFTs 11

Ingredients � Relevant degrees of freedom 6/2/2009 v. Kolck, Intro to EFTs 12

Seurat, La Parade (detail) 6/2/2009 v. Kolck, Intro to EFTs 13

Ingredients � Relevant degrees of freedom choose the coordinates that fit the problem � All possible interactions 6/2/2009 v. Kolck, Intro to EFTs 14

Example: Earth-moon-satellite system R � 1.7 Mm R � 6.4 Mm d � 384 Mm m E Wikipedia 2-body forces � 2+3-body forces change in resolution 6/2/2009 v. Kolck, Intro to EFTs 15

Ingredients � Relevant degrees of freedom choose the coordinates that fit the problem � All possible interactions what is not forbidden is compulsory � Symmetries 6/2/2009 v. Kolck, Intro to EFTs 16

A farmer is having trouble with a cow whose milk has gone sour. He asks three scientists—a biologist, a chemist, and a physicist—to help him. The biologist figures the cow must be sick or have some kind of infection, but none of the antibiotics he gives the cow work. Then, the chemist supposes that there must be a chemical imbalance affecting the production of milk, but none of the solutions he proposes do any good either. Finally, the physicist comes in and says, “First, we assume a spherical cow…” � � ∑ + ∑ α → ⋅ δα u v u v u v ij i j ij i j ij ij no, say, δα � 1 u v 1 2 ij amenable to perturbation theory 6/2/2009 v. Kolck, Intro to EFTs 17

Ingredients � Relevant degrees of freedom choose the coordinates that fit the problem � All possible interactions what is not forbidden is compulsory � Symmetries not everything is allowed � Naturalness 6/2/2009 v. Kolck, Intro to EFTs 18

‘t Hooft ‘79 After scales have been identified, the remaining, dimensionless parameters are ( ) 1 O cow unless suppressed by a symmetry non-sphericity… Occam’s razor: simplest assumption, to be revised if necessary fine-tuning energy of probe E Expansion in powers of energy scale of E und underlying theory 6/2/2009 v. Kolck, Intro to EFTs 19

A classical example: the flat Earth -- light object near surface of a large body d.o.f.: mass m ≡ ∼ � E m h g E m gR ( ) ( ) = sym: , , und h V h x y V h eff eff ∞ { } (neglecting ∑ ( ) = = + + η 2 + const … i V h m g h m g h h quantum eff i = 0 corrections…) i parameters + 1 ⎛ ⎞ i g m g h E h ( ) ( ) = + = × = × naturalness: O 1 1 1 O O ⎜ ⎟ i g + 1 i ⎝ ⎠ i i R m g h E R i und R − 1 − i ∞ 1 ⎛ ⎞ ⎛ ⎞ 1 i g G M ( ) ∑ ( ) = − + = − = 1 i ⎜ ⎟ ⎜ ⎟ V h G M m m h g 1 + 2 eff i ⎝ ⎠ ⎝ ⎠ i R h R R R = 0 i ≡ � h R g M itself the first term in a low-energy EFT of general relativity… 6/2/2009 v. Kolck, Intro to EFTs 20

Going a bit deeper… A short path to quantum mechanics 2 2 = + = + + P A A P A A A a 1 2 1 2 3 sum over all paths b Path Integral Feynman ‘48 ⎛ ⎞ b ( ) ∫ ∝ exp ( ( )) ⎜ L ⎟ A i dt q t = ∫ ∫ exp ( ( )) L A Dq i dt q t i RULE ⎝ ⎠ a each path contributes a phase ∏∫ ( ) dq t given by the classical action i i 6/2/2009 v. Kolck, Intro to EFTs 21

q ( ) q t + 2 i n ( ) q t + i n EFFECTIVE THEORY ( ) q t i � t t + t + t + t scale of fine-structure 1 2 i i i n i n of underlying curve 1 M und scale of variation 1 m of curve coarse-graining scale t t + t + 1 j j 2 j (cutoff) 1 Λ 2 1 ( ) ( ) dq d q ( ) → L ( ) + − + … + − 2 L ( ) ( ) q t q t t t t t i i 2 i i 2 dt dt 6/2/2009 v. Kolck, Intro to EFTs 22 t t i i

QM + special relativity: quantum field theory � → ϕ ≡ ϕ ( ) ( , ) ( ) q t r t x → ≡ 3 4 dt dt d r d x ∂ d → ∂ x μ dt EFFECTIVE FIELD THEORIES 6/2/2009 v. Kolck, Intro to EFTs 23

Partition function ( ) { } ∫ ∫ = ϕ ϕ + ϕ 4 exp ( ) ( ) L L Z D i d x free int { } ( ) 2 ⎡ ⎤ ∫ ∫ ∫ ∫ = ϕ + ϕ + ϕ + ϕ 4 4 4 1 ( ) ( ) … exp ( ) L L L D i d x i d x i d x ⎣ ⎦ int int free momentum space (skip many steps…) λ ϕ i = = 4 = i λ L − + i ε int 2 2 4 p m ' ' p p 1 2 4 d l i i ∫ = λ λ i i π + − + ε − − + ε 4 2 2 2 2 (2 ) ( ) ( ) p l m i p l m i − 1 2 + p l p l 2 1 = … p p 1 2 6/2/2009 v. Kolck, Intro to EFTs 24

Euler + Heisenberg ’36 Weinberg ’67 … ’79 What is Effective? Wilson, early 70s E … ( ) φ > H Q M ( ) ∫ ∫ ∫ = φ φ φ φ 4 exp ( ) L Z D D i d x M , H L und H L Λ ( ) ∫ × ϕ δ ϕ − φ ( ) D f ( ) φ < Λ Q M L L ( ) ∫ ∫ m = ϕ ϕ 4 exp ( ) L D i d x EFT ∞ ( ) = ∑ ∑ Λ ∂ ϕ ( , ) ( , ) d L n c M O m EFT i i = 0 ( , ) d i d n 1 1 φ Δ < ∼ : x H Q M ∂ renormalization-group local Z ∂Λ = 0 1 1 invariance φ Δ > : ∼ x L Q M details of the 6/2/2009 v. Kolck, Intro to EFTs 25 underlying symmetries underlying dynamics

characteristic external momentum ν Λ ∞ ⎡ ⎤ ⎛ ⎞ Q Q ∑ ∑ � ∞ = Λ ( ) ( ) ~ ( ) ( ) ; ⎜ ⎟ T T Q N M c F ⎢ ⎥ ν ν , , i i ⎣ ⎦ ⎝ ⎠ M m m ν ν = i min ∂ normalization T non-analytic, ∂Λ = 0 from loops ν = ν ( , , … ) “power counting” d n e.g. # loops L For Q ~ m , truncate consistently with RG invariance so as to allow systematic improvement (perturbation theory): ⎛ ⎞ ν + 1 ν ∂ ⎛ ⎞ ( ) ⎛ ⎞ Q T Q ν = + = ν ( ) ( ) ⎜ ⎟ O O ⎜ ⎟ ⎜ ⎟ T T N T ⎜ ⎟ ∂ Λ Λ ⎝ ⎠ ln ⎝ ⎠ M ⎝ ⎠ 6/2/2009 v. Kolck, Intro to EFTs 26

Why is this useful? ϕ Because in general the appropriate degrees of freedom below M ( ) are not the same as above φ = φ φ , H L Examples: ϕ ϕ φ φ L L M is mass of physical particle -- � c φ i virtual exchange in coefficients ϕ ϕ c φ φ H i L L (Appelquist-Carazzone decoupling theorem) M is scale associated with breaking of continuous symmetry -- � appearance of massless Goldstone bosons or gauge-boson mass (Goldstone’s theorem, Higgs mechanism) M is scale of confinement -- rearrangement of whole spectrum � M is radius of Fermi surface -- BCS behavior � 6/2/2009 v. Kolck, Intro to EFTs 27

How can we do it? Two possibilities: � know and can solve underlying theory -- ( ) get ‘s in terms of parameters in φ φ L , c i und H L � know but cannot solve, or or do not know, underlying theory -- use Weinberg’s “theorem”: Weinberg ‘79 “The quantum field theory generated by the most general Lagrangian with some assumed symmetries will produce the most general S matrix incorporating quantum mechanics, Lorentz invariance, unitarity, cluster decomposition and those symmetries, with no further physical content.” Note: proven only for scalar field with symmetry in , Ball + Thorne ‘94 Z E 2 4 but no known counterexamples 6/2/2009 v. Kolck, Intro to EFTs 28

Bira’s EFT Recipe 1. identify degrees of freedom and symmetries what is not forbidden 2. construct most general Lagrangian is mandatory! 3. run the methods of field theory Q < Λ • compute Feynman diagrams with all momenta (“regularization”) ( ) , c Λ Λ Λ • relate to observables, which should be independent of i (“renormalization”) not a model form factor Q ( ) controlled expansion in M × O 1 “naturalness”: what else? unless suppressed by symmetry… contrast to models, which have fewer, but ad hoc, interactions; useful in the identification of relevant degrees of freedom and symmetries, but plagued with uncontrolled errors 6/2/2009 v. Kolck, Intro to EFTs 29

A significant change in physicists’ attitude towards what should be taken as a guiding principle in theory construction is taking place in recent years in the context of the development of EFT. For many years (…) renormalizability has been taken as a necessary requirement. Now, considering the fact that experiments can probe only a limited range of energies, it seems natural to take EFT as a general framework for analyzing experimental results. T.Y. Cao, in Renormalization, From Lorentz to Landau (and Beyond), L.M. Brown (ed) 1993 6/2/2009 v. Kolck, Intro to EFTs 30

Time for a paradigm change, perhaps? 6/2/2009 v. Kolck, Intro to EFTs 31

condensed-matter physics and beyond molecular atomic physics nuclear physics Chiral EFT 0.1 physics NRQED ? 1 The world QCD (2 or 3 flavors) as an onion 2 10 QED QCD (6 flavors) Fermi Th Electroweak Th 15 10 + higher-dim ops (SUSY) 19 GUT? 10 General Relativity + higher-curvature terms ? 10 − 10 − 10 − 10 − 1 20 16 (fm) 3 1 r (GeV) E 6/2/2009 v. Kolck, Intro to EFTs 32

A quantum example: non-relativistic QED (NRQED) single fermion ψ of mass M , massless spin-1 boson A μ Lorentz, parity, time-reversal, and U(1) gauge invariance = ∂ − D ieA μ μ μ = ∂ − ∂ F A A μν μ ν ν μ 1 ( ) μν = ψ − ψ − L iD M F F μν und 4 ν − η i i = p μν = p − + i ε + ε p M 2 p i μ μ 1 perturbation ∝ = πα = γ interactions 4 ∼ e ie theory μ 3 How do E&M bound states arise? 6/2/2009 v. Kolck, Intro to EFTs 33

� Q M � � ( ) 0 γ 0 − ⋅ γ + + i p p q M i = = � � ( ) + − + ε 2 ( ) 2 + − + − + ε p q M i 0 0 2 p q p q M i + p q � � ( ) γ + − ⋅ γ + 0 0 i p M p q = � � � + − ⋅ − + ε 0 0 02 2 2 2 p q q p q q i q p ( ) + γ 0 1 i � � = + ( ) … = ∼ O p q Q + ε 0 2 q i � ( ) 0 = = O q q Q ± γ ⎛ ⎞ 0 1 2 � Q = = ≡ , 0 0 = 2 + 2 = + P P P P P O ⎜ ⎟ P p p M M ± ± ± ± ∓ ± 2 ⎝ ⎠ M ± projector onto energy states Georgi ’90 “heavy-fermion ( ) ( ) Ψ ≡ P ψ ψ = + ψ = − Ψ + Ψ i M t i M t e P P e formalism” ± ± + − + − particles: annihilates creates antiparticles: creates annihilates 6/2/2009 v. Kolck, Intro to EFTs 34

� � � ( � ) 1 ( ) = Ψ Ψ − Ψ γ ⋅ Ψ + Ψ γ ⋅ Ψ − Ψ + Ψ − F F μν 2 L iD i D i D i D M + + − + + − − − μν 0 0 und 4 + other, heavy d.o.f.s ( ) ( ) ∫ ∫ ∫ ∫ ∫ = Ψ Ψ 4 Ψ Ψ × Ψ δ Ψ−Ψ exp ( , ) L Z DA D D i d x A D + − + − + , und ( ) complete square, ∫ ∫ ∫ = Ψ Ψ 4 exp ( , ) L DA D i d x A do Gaussian integral EFT � 1 1 e = Ψ Ψ + Ψ Ψ + Ψ σ Ψ ε + − μν + 2 … … L jk i D D F F F μν 0 EFT i ijk 2 2 4 M M non-relativistic expansion Pauli term κ e + Ψ σ Ψ ε + … jk F i ijk 2 M most general Lag with Ψ, A anomalous magnetic moment invariant under U(1) gauge, parity, time-reversal, =O(1) and Lorentz transformations 6/2/2009 v. Kolck, Intro to EFTs 35

1 ( ) ( ) a b 2 2 � μν μν μν = − + + + … L F F F F F F μν μν μν 4 4 EFT 4 M M Euler + Heisenberg ‘36 ρ ν ν − η i p { μν p [ ] } = 3 i p ⎡ ⎤ 2 = η η ⋅ ⋅ + + … … a p p p p b + ε 2 ⎣ ⎦ p p p i μρ νσ 1 3 2 4 4 4 1 μ M μ σ ( ) 1 e ( ) ( ) 2 μναβ +Ψ ⋅ Ψ + Ψ ⋅ − 2 Ψ + + κ Ψ Ψ ε + … 1 iv D v D D v S F α β μν 2 M M � ( ) v ≡ 1,0 μ i = � p = ( ) 1 σ iev μ ⎛ ⎞ ( ) 2 ⋅ + − ⋅ + + ε 2 … ≡ ⎜ v p p v p i 0, 2 ⎟ S 2 M ⎝ ⎠ ν p ′ μ 2 { } ( ) e e ( ) ( ) = η − = + + + κ ε ν α β ' 2 1 i v v i p p v S q μν μ ν μναβ μ 2 M M μ p q γ γ (0) (1) + ΨΨ ΨΨ + Ψ Ψ ⋅Ψ Ψ + … 0 0 S S 2 2 M M ( ) i etc. = γ (0) + γ (1) ⋅ 6/2/2009 v. Kolck, Intro to EFTs 36 S S 0 0 1 2 2 M

Various processes at low energies: e.g. = + … T light-by-light scattering γγ no explicit fermion-antifermion pair creation! + … = + + Compton scattering T γψ no change in Thompson limit heavy-fermion number! 6/2/2009 v. Kolck, Intro to EFTs 37

Back to atomic bound states: the NRQED perspective � − � ( ) ( ) 0 0 ' , ' ' , ' p p p p = + + + … T ψψ � − � ( ) ( ) 0 , 0 , CoM p p p p frame + + … + � � ( ) = ∼ ' O p p Q Q ⎛ ⎞ 2 higher powers of Q 0 0 = ∼ ' O ⎜ ⎟ p p M ⎝ ⎠ M − 2 2 2 2 ie ie ie e = = � ∼ � � � � ( ) ( ) ( ) 2 2 2 ( ) 2 − + ε 2 − − ε ' ' 0 − 0 − − + ε Q ' ' p p i p p i p p p p i 2 e ( ) → = V r π 4 r 6/2/2009 v. Kolck, Intro to EFTs 38

2 ( 1 ( 4 + 0 0 , 1 1 0 + 0 ' , d l p l p l ∫ = 4 � � e � � � � ( ) ) � ) 4 + − � 2 2 π ( ) ( ' ) 2 l p l p + − + ' + − + ε + − + ε 0 0 0 0 p l ' p l l p i l p i 2 2 � M M ( ) 3 0 , 4 l l 1 1 � � ( ) � � ( ) 2 2 02 − 2 + ε 0 − 0 + 0 − − + + ε l l i ' ' p p l p p l i 0 l 3 1 1 d l ∫ = 4 � � � � ie ( ) � � 3 + − 2 2 π ( ) ( ' ) 2 l p l p − 0 + − ε − 0 + − ε ' l p i l p i 2 2 3 4 M M 1 2 4 3 1 1 4 � � � ( ) ( ) � � 2 2 − ε 2 − − − − + + ε 0 0 l i ' ' p p l p p l i 2 Q M + 3 … Q Q α 3 2 1 1 1 1 Q e 4 ∼ ∼ e just as expected… ( ) π 2 2 2 π 4 4 Q Q Q Q Q 6/2/2009 v. Kolck, Intro to EFTs 39

2 1 4 ( 1 1 ( d l + ∫ 0 0 , 0 − 0 , = 4 � � p l p l � � e ( ) 4 + + 2 2 π ( ) ( ) � 2 � ) l p l p � ) � 0 + 0 − + ε − 0 + 0 − + ε + − + l p i l p i ( ) p l p l 2 2 M M � ( ) 3 4 1 1 0 , l l � � ( ) � � ( ) 2 − + ε 2 02 2 − + − − + + ε 0 0 0 l l i ' ' p p l p p l i 0 l 3 1 1 d l ∫ = 4 � ie � � � ( ) 3 + 2 2 π 2 ( ) � ⎛ ⎞ 2 + l p 2 ( ) − + − ε l p 0 2 0 − − 2 + ε p i ⎜ ⎟ p l i M 2 ⎝ M ⎠ 3 4 1 4 3 1 � 2 � 2 � ⎛ ⎞ + 2 ( ) ( ) � � 2 l p 0 − 0 − − − + + ε 2 2 ' ' ⎜ ⎟ p p p p l i Q M 2 ⎝ M ⎠ + 3 4 … Q Q � 1 ⎛ ⎞ 3 α 2 2 1 1 Q M e e M 4 + α + ∼ ∼ … ⎜ ⎟ e π 2 2 2 π 2 2 4 4 ⎝ ⎠ Q Q Q Q Q Q infrared enhancement! 6/2/2009 v. Kolck, Intro to EFTs 40

∫ 0 dl → + + … + + 1 1 ∝ ∝ 2 Q M Q ∫ 0 dl “time-ordered + → + … perturbation theory” 1 ∝ Q 6/2/2009 v. Kolck, Intro to EFTs 41

(0) = + + … T ψψ bound state at α ∼ ⎧ ⎫ Q M ⎛ ⎞ 2 2 1 e M e + α + ∼ … ∼ 1 O ⎨ ⎬ ⎜ ⎟ 2 ⎛ ⎞ 2 2 Q ⎝ ⎠ Q ⎩ Q ⎭ Q M − α 2 M ∼ ∼ − α E 1 O ⎜ ⎟ M ⎝ ⎠ Q (0) (0) V T ψψ ψψ (0) = + + … = (0) + V V ψψ ψψ (0) (0) V V ψψ ψψ Lippmann-Schwinger eq. ⎛ ⎞ = Schroedinger eq. 2 Coulomb e = (0) = ฀ O ⎜ ⎟ V ⎛ ⎞ 2 ψψ ˆ potential 2 p ⎝ ⎠ Q + (0) ψ (0) = ( 0 ) ψ (0 ) ⎜ ⎟ V E ψψ 2 M ⎝ ⎠ known results… 6/2/2009 v. Kolck, Intro to EFTs 42

But more : ⎛ ⎞ α 2 e = + + … + … + = (1) ฀ O ⎜ ⎟ V ψψ π 2 4 ⎝ ⎠ Q (0) T ψψ (1) (0) V T ψψ ψψ (1) (1) = + + + T (1) V V ψψ ψψ ψψ (0) (1) T V ψψ ψψ (0) T ψψ = + ψ ψ (1 ) (0) (0) (1) (0) E E V ψψ α ⎛ ⎞ = (0) ฀ O ⎜ ⎟ E π ⎝ 4 ⎠ 6/2/2009 v. Kolck, Intro to EFTs 43

⎛ ⎞ 2 2 Q e = + … (2) = + ฀ O ⎜ ⎟ V ψψ 2 2 ⎝ ⎠ M Q (2) = … T ψψ = + ψ ψ + … (2 ) (1) (0) (2) ( 0 ) E E V ⎛ ⎞ 2 ψψ Q = (0) ฀ O ⎜ ⎟ E 2 ⎝ ⎠ M � � � � � � � � ( ) ( ) ( ) ( ) 2 ∫ ∝ μ ⋅ μ 3 ψ (0)* δ (3) ψ (0) = μ ⋅ μ ψ (0) piece 0 d r r r r 1 2 1 2 magnetic interaction 6/2/2009 v. Kolck, Intro to EFTs 44

N (3) starting at , sufficiently many derivatives appear at vertices that T ψψ O Λ loops bring positive powers of , which need to be compensated by T ( ) γ Λ ( ) and higher-order “counterterms” i E 0 ⎛ ⎞ α 2 2 = Q e (3) + = + … + V ฀ O ⎜ ⎟ ψψ π 2 2 4 ⎝ ⎠ M Q α 2 ( ) γ ( ) = O α 2 ∝ Λ ↔ ln i ฀ 0 2 M Etc. 6/2/2009 v. Kolck, Intro to EFTs 45

Example: g factor for electron bound in H-like atoms ( ) = + κ 2 1 g electron known Larmor frequency ion mass ψ ψ (0) measured + = + … measured ψ ψ (0) electron trapped-ion ion charge mass cyclotron frequency Pachucki, Jentschura + Yerokhin ‘04 ⎛ ⎞ m = 12 ( ) C gs ⎜ ⎟ u 12 ⎝ ⎠ 6/2/2009 v. Kolck, Intro to EFTs 46 Most precise determination of electron mass (expt)(th)

Summary Nuclear systems involve multiple scales but no obvious small coupling constant EFT is a general framework to deal with a multi-scale problem using the small ratio of scales as an expansion parameter Applied to low-energy QED, EFT reproduces well-known facts and also provides a systematic expansion for the potential -- NRQED is in fact the framework used in state-of-the-art QED bound-state calculations Stay tuned: next, how we can make nuclear physics as systematic as QED 6/2/2009 v. Kolck, Intro to EFTs 47

Introduction to Effective Field Theories in QCD U. van Kolck University of Arizona Supported in part by US DOE 6/2/2009 v. Kolck, Intro to EFTs 48

Outline � Effective Field Theories � QCD at Low Energies QCD and Chiral Symmetry Chiral Nuclear EFT Renormalization of Pion Exchange Summary � Towards Nuclear Structure 6/2/2009 v. Kolck, Intro to EFTs 49

References: S. Weinberg, Phenomenological Lagrangians, Physica A96:327,1979 S. Weinberg, Effective chiral Lagrangians for nucleon-pion interactions and nuclear forces, Nucl.Phys.B363:3-18,1991 S.R. Beane, P.F. Bedaque, L. Childress, A. Kryjevski, J. McGuire, and U. van Kolck, Singular potentials and limit cycles, Phys.Rev.A64:042103,2001, quant-ph/0010073 A. Nogga, R.G.E. Timmermans, and U. van Kolck, Renormalization of one-pion exchange and power counting, Phys.Rev.C72:054006,2005, nucl-th/0506005 6/2/2009 v. Kolck, Intro to EFTs 50

condensed-matter physics and beyond molecular atomic physics nuclear physics Chiral EFT 0.1 physics NRQED ? 1 The world QCD (2 or 3 flavors) as an onion 2 10 QED QCD (6 flavors) Fermi Th Electroweak Th 15 10 + higher-dim ops (SUSY) 19 GUT? 10 General Relativity + higher-curvature terms ? 10 − 10 − 10 − 10 − 20 16 (fm) 3 1 1 r (GeV) E 6/2/2009 v. Kolck, Intro to EFTs 51

Game Plan QCD lattice large r necessary to extrapolate to EFT small m π NCSM,… ? want model independence Few-nucleon lattice,… systems Many-nucleon systems Infinite-nucleon system 6/2/2009 v. Kolck, Intro to EFTs 52

EFT at a few GeV= underlying theory for nuclear physics ⎛ ⎞ ⎛ + ⎞ u l =⎜ ⎟ =⎜ d.o.f.s leptons: quarks: ⎟ photon: A μ gluons: q l a G μ f ν ⎝ ⎠ d ⎝ ⎠ f symmetries: (3,1) global, (1) gauge, (3) gauge SO U SU em c ⎛ ⎞ 1 0 3 1 ( ) ∑ =⎜ μν ⎟ = ∂ + − − C L l i e CA m l F F 0 0 ⎝ ⎠ μν und f f f 4 = 1 f ⎛ ⎞ + τ 0 1 3 23 QED = = 3 1 Tr ⎜ ⎟ Q ( ) ⎡ ⎤ + ∂ + + − μν 0 6 − ⎝ 13 ⎠ q i e QA g G q G G + ⎣ ⎦ μν s 2 QCD 1 1 ( ) ( ) τ − + − − m m qq m m q q 3 u d u d 2 2 e.g. higher-dimension interactions: m m + θ γ + … G ∝ suppressed by larger masses 2 u d 1 qi q M + 5 , F W Z m m u d unnaturally small T violation θ < − 9 10 (strong CP problem) � 6/2/2009 v. Kolck, Intro to EFTs 53

Focus on strong-interacting sector: four parameters = = = θ = 1) 0, 0, 0 “chiral limit” m m e u d single, dimensionless parameter ⎧ ⎫ 1 Tr ( ) ∫ ∫ = ∂ / + / − μν 4 4 L ⎨ ⎬ d x d x q i g G q G G λ − → 1 μν x x QCD s ⎩ 2 ⎭ → λ 32 q q invariant under scale transformations but in → λ G G ( ) = ∫ ∫ ∫ ∫ 4 exp L Z DG Dq Dq i d x QCD Λ scale invariance 0.3 “anomalously broken” by dimensionful regulator α s ( µ ) coupling runs 0.2 ( ) α ∼ 1GeV ∼ 1 s Q 0.1 (“dimensional transmutation”) 6/2/2009 v. Kolck, Intro to EFTs 54 0 2 1 10 10 µ GeV

Non-perturbative physics at Q ∼ 1 GeV Assumption 1: confinement only colorless states (“hadrons”) are asymptotic Observation: (almost) all hadron masses > 1 GeV � Assumption 2: naturalness masses are determined by characteristic scale ∼ 1 GeV M QCD Observation: pion mass π � 140 MeV � m M QCD breakdown of naturalness? NO! “spontaneous breaking” of chiral symmetry 6/2/2009 v. Kolck, Intro to EFTs 55

Why is the pion special? 1 Tr ( ) ( ) μν = ∂ / + / + ∂ / + / − L q i g G q q i g G q G G μν QCD L s L R s R 2 + γ 1 ⎛ ⎞ − γ 1 u 5 = d ⎜ ⎟ 5 q q q ⎜ ⎟ 2 2 ⎝ ⎠ chiral symmetry invariant under � � ( ) → α ⋅ τ × exp (2) (2) ∼ (4) q i q SU SU SO ( ) ( ) ( ) L R L R L R L R broken by vacuum down to isospin � m m σ π � � ( ) → α τ ⋅ � (2) + ∼ (3) exp m m SU SO q i q L R N N − + 6/2/2009 v. Kolck, Intro to EFTs 56

Chiral V QCD Limit chiral circle σ two isospin axis qq not shown π γ τ f π qi q 5 1 pion decay constant (in chiral limit) ∂ π on chiral circle] EFT = → + piece invariant under [function of L π π ε μ ⎛ ⎞ 2 π − + ∂ 1 … π ⎜ ⎟ μ 2 4 f ⎝ ⎠ π 6/2/2009 v. Kolck, Intro to EFTs 57

≠ ≠ = θ = 2) 0 , 0, 0 m m e u d 1 Tr ( ) μν = ∂ / + / − L q i g G q G G μν QCD s 2 v.K. ’93 1 1 + + + − τ + … ( ) ( ) m m q q m m q q 3 u d u d 2 2 4 th component of 3 rd component of (4) vector (4) vector SO SO � � ( ) ( ) = γ τ = τ γ , , S qi q qq P q q qi q 5 5 break → → (4) (3) (1) SO SO U (explicit chiral-symmetry breaking) (isospin violation) 6/2/2009 v. Kolck, Intro to EFTs 58

Chiral V QCD Limit slightly-tilted chiral circle σ two isospin axis qq not shown π f π ≅ 92 MeV γ τ qi q pion decay constant 5 1 ∝ EFT = ∂ π → + ] piece invariant under L π π ε [function of Q μ ∝ + π ( ) explicitly] + piece in direction [function of m m qq u d ∝ − + isospin breaking ( ) m m u d CHIRAL SYMMETRY WEAK PION INTERACTIONS 6/2/2009 v. Kolck, Intro to EFTs 59

3) ≠ θ = 0, 0 e Two types of interactions: = ∂ − D i QA e μ μ μ � “soft” photons – explicit d.o.f. in the EFT = ∂ − ∂ F A A μν μ ν ν μ � “hard” photons – “integrated out” of EFT v.K. ’93 ( ) μν = − γ ∂ γ + 2 2 L … … e qQ q D qQ q μ ν und ε γ γ τ γ τ ⎛ ⎞ qi q qi q 34 comp of μ μ 5 = ⎜ ijk k j ⎟ F μ − γ τ antisymmetric tensor 0 qi q ⎝ ⎠ μ i breaks → (4) (3) (1) (and in particular) SO SO U EFT = L ∝ soft photons e α ∝ + further isospin breaking 4 π 6/2/2009 v. Kolck, Intro to EFTs 60

4) θ ≠ 0 m m = + θ γ + … … L u d qi q 5 + un d m m u d � ( ) 4 th component of = τ γ (4) vector , SO P q q qi q 5 T violation linked to isospin violation: in EFT, combination is 1 m m ( ) − − + θ u d m m P P 3 4 + 2 u d m m u d Hockings, Mereghetti + v.K., in preparation 5) continue with higher-order operators, De Vries, Mereghetti, e.g. T-violating quark EDM and color-EDM Timmermans + v.K., in progress … P-violating four-quark operators Kaplan + Savage ’96 Zhu, Maekawa, Holstein, Musolf + v.K. ’02 6/2/2009 v. Kolck, Intro to EFTs 61

Nuclear physics scales “His scales are His pride”, Book of Job ln Q perturbative QCD brute force π ~ , , 4 , … ~1 GeV M m m f (lattice), …? ρ π QCD N hadronic th this talk with chiral symm Q M ~100 MeV π … ~ , 1/ , , M f r m QCD π nuc NN ℵ ~30 MeV ~ 1 a Q M NN nuc halo nuclei no small coupling expansion in 6/2/2009 v. Kolck, Intro to EFTs 62

Nuclear EFT pionful EFT ∼ � Q m M π QCD Δ − • d.o.f.s: nucleons, pions, deltas ( ~ 2 ) m m m π N ⎛ ++ ⎞ ( ) Δ ⎛ ⎞ + − π + π 2 ⎜ ⎟ π ⎛ ⎞ ⎜ ⎟ + ⎛ ⎞ Δ 1 p ⎜ ⎟ ⎜ ⎟ ( ) ⎜ ⎟ Δ = ⎜ = ⎜ + − = π = − π − π ⎟ π 2 N i ⎟ ⎜ ⎟ Δ 0 2 ⎜ ⎟ ⎝ ⎠ n ⎜ ⎟ ⎜ ⎟ π ⎜ ⎟ π ⎜ ⎟ ⎝ ⎠ 0 − Δ 3 ⎝ ⎠ ⎝ ⎠ • symmetries: Lorentz, P, T, chiral Weinberg ’68 Non-linear realization of chiral symmetry Callan, Coleman, Wess + Zumino ‘69 chiral invariants + 's, 's, 's S P F 4 3 34 ∂ π ⎛ ⎞ ⎛ ⎞ 2 π ( ) μ ( ) (chiral) ≡ − + pion 1 … = O + ⎜ ⎟ ⎜ ⎟ 2 D m m m M μ 2 π 4 u d QCD ⎝ ⎠ ⎝ ⎠ f f covariant π π ⎛ ⎞ ( ) 2 m ≡ ∂ − ⋅ fermions derivatives D + = π ⎜ ⎟ O i t E m m ⎜ ⎟ μ μ μ u d M ⎝ ⎠ QC D π ≡ × E D μ μ 6/2/2009 v. Kolck, Intro to EFTs 63 f π

Schematically, p f n ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ 2 2 − ψ ψ + 2 2 , D , π m m m ∑ D = Δ π 2 2 ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ L N c f M ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ π { , , } 2 2 2 EFT n p f QCD M M f f M ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ { , , } π π n p f QCD Q C D QCD = O (1) isospin conserving (NDA: naïve α ⎛ ⎞ calculated from QCD: lattice, … dimensional = ε isospin breaking , 4 O ⎜ ⎟ analysis) fitted to data π ⎝ ⎠ ∞ = ∑ L f f Δ ( ) Δ ≡ + + − ≡ + − ≥ 2 2 0 n p d 2 2 Δ= 0 chiral symmetry “chiral index” 6/2/2009 v. Kolck, Intro to EFTs 64

⎛ ⎞ ⎛ ⎞ 2 2 1 π 1 π … = ∂ − + − − + (0) 2 2 2 L ( π ) 1 … π 1 … ⎜ ⎟ ⎜ ⎟ m μ π 2 2 2 2 2 4 ⎝ ⎠ ⎝ ⎠ f f π π … � ⎡ ⎤ ⎛ ⎞ 2 1 � π g + + ∂ − ⋅ × ∂ + + + σ ⋅⋅ ∇ − + τ π π … τ π … ( ) ( ) 1 ⎜ ⎟ ⎢ ⎥ A N i N N N 0 0 2 2 4 2 4 ⎣ ⎦ ⎝ ⎠ f f f π π π � � … ( ) h [ ] ( ) + Δ + ∂ − − + Δ + + + Δ + ⋅ ⋅ ∇ + … … T π … ( ) Η .c. ( ) 1 A i m m N S Δ 0 N 2 f π � ( ) ( ) 2 2 − + − + σ C N N C N N S T ⎡ ⎤ 2 � � ⎛ ⎞ ⎛ ⎞ 1 1 1 1 + ⎢ ⎥ (1) = ∇ + ⋅ × ∇ + + − τ − π ⋅ + L τ ( π π ) … ( ) π τ … ⎜ ⎟ ⎜ ⎟ N m m N 3 3 2 2 2 4 2 p n 2 ⎢ ⎥ ⎝ ⎠ ⎝ ⎠ m f f ⎣ ⎦ π π … N � 1 ⎡ ⎤ + + ∂ 2 − ∇ 2 − 2 2 + ε ε σ τ ∂ π ∂ π + ( π ) ( π ) 2 π ( )( ) … N b b b m ib N ⎣ ⎦ π 2 0 3 1 4 2 ijk ab c k c i b j c f π … � � g ( ) ⎡ ⎤ − + σ ⋅ ∇ + ⋅ ∂ + Η .c ( ) 1 … τ π A iN N ⎣ ⎦ 0 4 m f π N � … � d N ( ) + + + σ ⋅ ⋅ ∇ + τ ( π ) 1 … N N N f π ( ) 3 + − E N N (2) = … Form of pion interactions L determined by 6/2/2009 v. Kolck, Intro to EFTs 65 chiral symmetry

Weinberg ’79 A= 0, 1: chiral perturbation theory Gasser + Leutwyler ’84 … Gasser, Sainio + Svarc ’87 Jenkins + Manohar ’91 … 1 1 T ∼ Δ E Q nucleon nucleon ν ⎛ ⎞ ⎛ ⎞ dense but Q Q ∑ ⎜ ⎟ ∼ ⎜ ⎟ c F ⎜ ⎟ short-ranged ν ν ⎝ ⎠ M m ⎝ ⎠ ν π QCD ∑ ν = − + + Δ ≥ ν = − 2 2 2 A L V A min i i i # loops # vertices of type i ≈ 1 0.3 fm M QCD long-ranged expansion in but sparse ⎧ non-relativistic Q m N ⎪ Q m π ≅ 1 1.4 fm ∼ … multipole , ⎨ Q m ρ M ⎪ 6/2/2009 v. Kolck, Intro to EFTs 66 π 4 QCD pion loop ⎩ Q f π

Analogous to NRQED… Weinberg ’79 Gasser + Leutwyler ’84 … = + + + T + + … ππ current Weinberg ’66 … Gasser, Sainio + Svarc ’87 algebra Jenkins + Manohar ’91 … = + + + + + … T π N ⎛ ⎞ 3 Q ( ) − − < N.B. For ⎜ ⎟ a resummation is necessary � O E m m ⎜ ⎟ Δ 2 N M ⎝ ⎠ Phillips + Pascalutsa ’02 Etc. QCD Long + v.K. , in preparation 6/2/2009 v. Kolck, Intro to EFTs 67

A > 2: resummed chiral perturbation theory Weinberg ‘90, ‘91 A-nucleon irreducible V infrared infrared 1 m 0 ∼ N l V enhancement! enhancement! Δ 2 E Q 1 A-nucleon reducible 2 e.g. V 2 1 4 1 1 d l ∫ � i V V π + − − ε − + − − ε 4 0 2 2 0 2 2 (2 ) l k m l m i l k m l m i N N N N V 3 ⎛ ⎞ m d l m Q ∫ = + 2 … ∼ O N N ⎜ ⎟ V V V π − π 3 2 2 2 (2 ) ⎝ 4 ⎠ l k k = E 2 m Q instead of N π 2 (4 ) 6/2/2009 v. Kolck, Intro to EFTs 68

� � � ( ) 2 ⎛ ⎞ ⎛ ⎞ + σ ⋅ σ ˆ ( ) 2 1 S q g q ⋅ 12 1 2 ∼ τ τ ∼ ⎜ A ⎟ ⎜ ⎟ � i 1 2 + 2 2 2 2 3 ⎝ ⎠ ⎝ ⎠ tensor force f q m f π π π � � � � = σ ⋅ σ ⋅ − σ ⋅ σ ˆ ˆ ˆ ( ) 3 S q q q 12 1 2 1 2 3 2 2 2 1 1 1 1 Q Q Q Q ∼ ∼ ( ) ( ) 4 2 2 2 2 π 4 π f Q Q Q Q f π 4 f π π ⎛ ⎞ 2 Q = ⎜ ⎟ O ⎜ ⎟ 2 M ⎝ ⎠ QCD 3 2 2 2 1 1 1 1 Q Q Q Q Q ∼ ∼ ( ) ( ) − − 4 2 2 2 2 π 4 π f m m Q Q Q m m f 4 f π Δ Δ π N N π ( ) = O 1 3 2 2 1 1 1 Q m Q Q m Q Q ∼ ∼ ∼ N N π π μ 4 2 2 2 2 2 4 4 f Q Q Q f f f f π π π π π π ( ) μ for Q = O 1 ∼ 1 ≡ π μ 6/2/2009 v. Kolck, Intro to EFTs 69 π

bound state at μ 2 2 Q μ − π ∼ ∼ ∼ = + + … Q E (0) π T m M N QCD ⎧ ⎫ π ⎛ ⎞ 4 ⎪ ⎪ 1 1 1 f Q = μ ≈ π + + ∼ ∼ 1 O … ∼ ⎨ ⎬ ⎜ ⎟ M f f π π π nuc μ 2 2 ⎛ ⎞ ⎪ ⎪ m ⎝ ⎠ f f ⎩ ⎭ Q π π π − N 1 O ⎜ ⎟ Nuclear scale μ ⎝ ⎠ π arises naturally from (0) V (0) T chiral symmetry = + + … = (0) + (0) V V (0) (0) V V Is 1PE all there is in leading order? ? = (0) + V That is, are observables cutoff independent with 1PE alone? 6/2/2009 v. Kolck, Intro to EFTs 70

Issue: relative importance of pion exchange and short-range interactions � � � ( ) 2 ⎛ ⎞ ⎛ ⎞ + σ ⋅ σ ˆ 2 ( ) π 4 S q g q 12 1 2 ⋅ ∼ τ τ ∼ ⎜ A ⎟ ⎜ ⎟ � i 1 2 + μ 2 2 2 3 ⎝ ⎠ ⎝ ⎠ f q m m π π π N 2 ⎛ ⎞ ⎛ ⎞ 2 � S = m g 0 − = ⋅ − δ (3) + π ( ) τ τ ( ) m r ⎜ A ⎟ ⎜ ⎟ π V r r e S = 1 1 2 π 2 4 ⎝ ⎠ ⎝ ⎠ f r π 2 ⎫ ⎧ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ 2 2 ⎪ ⎪ 1 1 1 1 g m m − − = ⋅ δ − + + + (3) π π ˆ ( ) τ τ ( ) ( ) ⎨ m r m r ⎬ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ A V r r e π e π S r 1 2 1 2 π π 2 2 3 4 4 ( ) 3 ⎪ ⎪ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ f ⎩ r r m r m r ⎭ π π π − + 1 1 S j j j much more singular --and complicated!-- than 12 + − ( 1) 1 j j j − − 1 2 0 6 j 2 2 + + 2 1 2 1 2 ie e j j e ( ) ∼ ∼ → = � � V r ( ) 2 2 0 2 0 π − − ε 4 ' j Q r p p i + + ( 1) 2 j j j + − 1 6 0 2 j + + 2 1 2 1 j j 6/2/2009 v. Kolck, Intro to EFTs 71

Weinberg ’90, ‘91 Ordonez + v.K. ’92 Assume contact interactions are driven by Ordonez, Ray + v.K. ‘96 short-range physics, and scale with M … QCD according to naïve dimensional analysis Entem + Machleidt ’03… Epelbaum, Gloeckle + Meissner ’04 (W power counting) ... π 4 � � � � � S = 0 = δ (3) ( ) ( ) σ ⋅ σ + σ ⋅ σ − ( 1) ( 3) V r r μ − (0) (1) ∼ 1 2 1 2 C C m 0 0 0 N 4 4 π 4 � S = 1 = δ π π (3) ( ) ( ) 4 4 V r r ≡ ≡ μ m μ μ 1 N m m 0 1 N N μ μ ∼ ( ) in LO i C π 0 i π (terms linear in 2 4 Q in NLO ∼ μ 2 break P, T ) m M Q M π N QCD QCD etc. 6/2/2009 v. Kolck, Intro to EFTs 72

Ordonez + v.K. ’92 v.K. ’94 … = + + + + + … + … V 2 N + + + … + = + + … + … V 3 N + + … higher powers of Q Etc. more nucleons 6/2/2009 v. Kolck, Intro to EFTs 73

… 3-body 2-body 4-body ⎛ ⎞ 1 LO O ⎜ ⎟ 2 ⎝ ⎠ f π ⎛ ⎞ 1 Q ⎜ ⎟ O (parity violating) ⎜ ⎟ 2 f π M ⎝ ⎠ QCD ⎛ ⎞ 2 1 Q NLO ⎜ ⎟ O ⎜ ⎟ 2 2 f M ⎝ ⎠ π QCD ⎛ ⎞ 3 1 Q NNLO ⎜ ⎟ O ⎜ ⎟ 2 3 f M ⎝ ⎠ π QCD ⎛ ⎞ 4 1 Q NNNLO ⎜ ⎟ O ⎜ ⎟ 2 4 f M ⎝ ⎠ π QCD ETC. 6/2/2009 v. Kolck, Intro to EFTs 74

Hierarchies many-body forces A canon emerges! � � �… Weinberg ’90, ‘91 V V V 2 3 4 N N N isospin-breaking forces � � v.K. ’93 V V V Similar explanation for IS IV CSB � � �… Rho ’92 J J J 1 2 3 N N N external currents 6/2/2009 v. Kolck, Intro to EFTs 75

Ordonez + v.K. ’92 v.K. ’94 … = + + + + + … + … V 2 N + + + … similar to phenomenological potential models, e.g. AV18 – (OPE)^2 + non-local terms Stoks, Wiringa + Pieper ‘94 6/2/2009 v. Kolck, Intro to EFTs 76

6/2/2009 v. Kolck, Intro to EFTs 77

But: NOT your usual potential! Ordonez + v.K. ’92 (cf. Stony Brook TPE) σ + ω e.g. , + + … + 1 for chiral v.d. Waals force π → 2 ∼ 0 Rentmeester et al. ’01, ‘03 m 6 r Nijmegen PSA of 1951 pp data Kaiser, Brockmann + Weise ’97 0 χ 2 long-range pot #bc min OPE 31 2026.2 -1 2 π σ + ω + OPE TPE ( ) 28 1984.7 lo V C (r) [MeV] + -2 OPE TPE ( ) 23 1934.5 nlo Nijm78 19 1968.7 -3 at least Isoscalar Central Potential parameters found as good! consistent with π N data! -4 2.0 2.5 3.0 3.5 4.0 r [fm] models with σ , ω , … Similar results in other channels, 6/2/2009 v. Kolck, Intro to EFTs 78 might be misleading… e.g. spin-orbit force!

v.K. ’94 Friar, Hueber + v.K. ‘99 Coon + Han ’99 Fujita + Miyazawa ‘58 ... = V 3 N two unknown + + + parameters + … Tucson-Melbourne pot with TM’ → − 2 2 Coon et al. ‘78 a a m c π potential → 0 c � � � � � � � � � ( ) ( ) ( ) ⎡ ⎤ = δ + ⋅ + 2 + 2 − ε τ σ ⋅ × + , ' ' ' ' … t N q q a b q q c q q d q q ⎣ ⎦ π αβ αβγ γ 3 αβ 6/2/2009 v. Kolck, Intro to EFTs 79

Ordonez, Ray + v.K. ’96 Many successes of Weinberg’s counting, e.g. , … Epelbaum, Gloeckle + Meissner ’02 � To N3LO (w/o deltas), fit to NN Entem + Machleidt ’03 phase shifts comparable to those of … “realistic” phenomenological potentials Entem + Machleidt ’03 VPI PSA Nijmegen NNNLO PSA NNLO NLO 6/2/2009 v. Kolck, Intro to EFTs 80

� With N3LO 2N and N2LO 3N potentials (w/o deltas), good description of • 3N observables and 4N binding energy • levels of p-shell nuclei Epelbaum et al. ’02 Gueorguiev, Navratil, Nogga, Ormand + Vary ’07 6/2/2009 v. Kolck, Intro to EFTs 81

measured: Illinois ‘94, SAL ‘00, Lund ‘03 γ → γ d d extracted nucleon polarizabilities: Beane, Malheiro, McGovern, Phillips + v.K. ‘04 threshold amplitude predicted: Beane, Bernard, Lee, Meissner γ → π 0 d d + v.K. ’97 confirmed: SAL ’98, Mainz ‘01 Many reactions: measured: IUCF ’90-…, TRIUMF ’91-…, Uppsala ’95-… → pp π 0 pp S waves sensitive to high orders: Miller, Riska + v.K. ‘96 pn π + → pp P waves converge, fix 3BF LEC: Hanhart, Miller + v.K. ‘00 d π + → pp CSB asymmetry sign predicted: Miller, Niskanen + v.K. ’00 → d π 0 pn confirmed: TRIUMF ‘03 measured: IUCF ’03 → απ 0 dd mechanisms surveyed: Fonseca, Gardestig, Hanhart, Horowitz, Miller, Niskanen, Nogga +v.K. ’04 ’06 + PARITY, TIME-REVERSAL VIOLATION , etc. 6/2/2009 v. Kolck, Intro to EFTs 82

BUT Is Weinberg’s power counting consistent? No! attractive in some channels 2 ⎛ ⎞ ⎧ ⎫ 3 ˆ ( ) m g S r − ⋅ + π ∼ τ τ 12 … ⎨ ⎬ m r ⎜ ⎟ A π e 1 2 π 3 2 4 ( ) ⎝ ⎠ ⎩ ⎭ f m r π π singular potential not enough contact interactions for renormalization-group invariance even at LO 6/2/2009 v. Kolck, Intro to EFTs 83

6/2/2009 v. Kolck, Intro to EFTs 84

Renormalization of the potential 1 n r 1 Λ ≡ ( ) R V r OPE: r = ⎧ 2 m m � N ⎪ Λ δ (3) λ ( ) ( ) ( ) = C r f r r 1 ⎪ r m Λ 0 0 π 0 ⎨ 2 ⎛ ⎞ 2 n λ = μ ( ) r m r r r m ⎪ → θ − ( ) 1 0 π π ⎜ ⎟ 0 V R 0 ⎪ ⎝ ⎠ = − R ( ) exp( ) ⎩ f r r r r 0 0 ( ) u r ( ) s wave ψ ≡ ∼ � n r R r 0 n r matching so that ⎛ ⎞ ∂ T R ( ) ( ) − − = λ = 2 2 2 cot 2 , , ∼ 1 O ⎜ ⎟ s m R V m R V F r R k r T 0 0 0 0 n ∂ ln s ⎝ ⎠ R r 0 6/2/2009 v. Kolck, Intro to EFTs 85

n ≥ 2 Beane, Bedaque, Childress, Kryjevski, McGuire + v.K. ’02 Two regular solutions that oscillate! if no counterterm, will depend on cutoff R model dependence 1 − ⎛ ⎞ ⎛ ⎞ λ λ 4 = ⎜ + δ ⎟ ( � ) cos + … ⎜ ⎟ u r r determined by ⎜ ⎟ ( )( ) 0 − 2 1 n n ( ) n − n ⎝ ⎠ 2 1 r r n r r ⎝ ⎠ 0 low-energy data 0 − ⎛ ⎞ 1 2 n ⎛ ⎞ λ n R ( ) ⎜ ⎟ λ = − λ + δ + , , tan … ⎜ ⎟ F r R ⎜ ( )( ) ⎟ − 0 2 1 n 4 n − n ⎝ ⎠ 2 1 r n R r ⎝ ⎠ 0 0 3rd u � r � 6 � R � H 4 � = − 2 2 5 exact m R V 0 4 r 3 1st 2 2nd 1 exact vs perturbation th 0.03 0.04 0.05 0.06 0.07 0.08 0.09 R 6/2/2009 v. Kolck, Intro to EFTs 86 limit-cycle-like behavior

Same is true in all channels were Beane, Bedaque, Savage + v.K. ‘02 Nogga, Timmermans + v.K. ’05 attractive singular potential is iterated Pavon Valderrama + Ruiz-Arriola, ’06 Birse, ’06, ’07 λ + ( 1) r l l − + 0 Long + v.K., ‘07 3 2 r r [ ] 3 + ( 1) 4 l l + ( 1) l l ( ) 2 27 λ r 0 2 r r λ λ 3 r = − 0 r r 1 0 + l 3 2 ( 1) l l but ∼ � for + λ r ( 1) � r r l l 0 l M singular potential only needs to be iterated in a few waves, where counterterms are needed 3 M + < QCD ( 1) ∼ 5 l l OPE: μ � 2 π l < 2 � 6/2/2009 v. Kolck, Intro to EFTs 87

certain counterterms that in Weinberg’s counting Q were assumed suppressed by powers of M QCD Q are in fact suppressed by powers of lf π short-range physics more important than assumed by Weinberg’s; most qualitative conclusions unchanged, but quantitative results need improvement ACTIVE RESEARCH AREA 6/2/2009 v. Kolck, Intro to EFTs 88

Examples 1 S Other singlet channels 0 Nogga, Timmermans + v.K. ’05 Beane, Bedaque, Savage + v.K. ‘02 (cf. Birse + McGovern ’04) Λ = 985 MeV δ LO EFT 0 8 60 Λ = 492 MeV 1 P 1 -5 6 Λ = 140 MeV δ [deg] 1 D 2 -10 40 4 -15 Nijmegen 2 -20 20 PSA -25 0 0 1 100 200 300 1 F 3 0.8 -1 | P | (MeV) δ [deg] 0.6 δ -2 0.4 60 NLO EFT -3 1 G 4 0.2 -4 0 40 0 50 100 150 200 0 50 100 150 200 T L [MeV] T L [MeV] 20 LO EFT − Λ = 1 ( 20fm ) Nijmegen PSA 100 200 300 6/2/2009 v. Kolck, Intro to EFTs 89 | P | (MeV)

Attractive triplet channels 12 30 10 3 P 0 25 8 20 δ [deg] 6 15 LO EFT 3 4 S χ -limit 1 150 10 3 P 2 2 δ 5 0 100 -2 0 0 m 1.4 π ε 2 50 Nijmegen -1 1.2 PSA 1.0 -2 δ [deg] 0.8 -3 100 200 0.6 -4 3 F 2 | P | (MeV) 0.4 -5 0.2 Beane, Bedaque, Savage + v.K. ‘02 0.0 -6 0 50 100 150 200 0 50 100 150 200 T L [MeV] T L [MeV] Repulsive triplet channels LO EFT 0 0 − Λ = 1 ( 20fm ) 3 P 1 -5 3 F 3 -1 Nijmegen PSA δ [deg] -10 -2 -15 Nogga, Timmermans + v.K. ’05 -3 (cf. Pavon Valderrama + Ruiz-Arriola, ’06) -20 -25 -4 6/2/2009 v. Kolck, Intro to EFTs 90 0 50 100 150 200 0 50 100 150 200 T L [MeV] T L [MeV]

Summary A low-energy EFT of QCD has been constructed and used to describe nuclear systems Chiral symmetry plays an important role, in particular setting the scale for nuclear bound states Nuclear physics canons emerge from chiral potential A new power counting has been formulated: more counterterms at each order relative to Weinberg’s; expect even better description of observables Stay tuned: next, how to extend EFT to larger systems 6/2/2009 v. Kolck, Intro to EFTs 91

Introduction to Effective Field Theories in QCD U. van Kolck University of Arizona Supported in part by US DOE 6/2/2009 v. Kolck, Intro to EFTs 92

Outline Effective Field Theories � QCD at Low Energies � � Towards Nuclear Structure Contact Nuclear EFT Few-Body Systems No-Core Shell Model Halo/Cluster EFT Conclusions and Outlook 6/2/2009 v. Kolck, Intro to EFTs 93

References: U. van Kolck, Effective field theory of short-range forces, Nucl.Phys.A645:273-302,1999, nucl-th/9808007 P.F. Bedaque, H.-W. Hammer, and U. van Kolck, The three-boson system with short-range interactions, Nucl.Phys.A646:444-466,1999, nucl-th/9811046 I. Stetcu, B.R. Barrett, and U. van Kolck, No-core shell model in an effective-field-theory framework, Phys.Lett.B(to appear),2007, nucl-th/0609023 P.F. Bedaque, H.-W. Hammer, and U. van Kolck, Narrow resonances in effective field theory, Phys.Lett.B569:159-167,2003, nucl-th/0304007 6/2/2009 v. Kolck, Intro to EFTs 94

Nuclear physics scales “His scales are His pride”, Book of Job ln Q perturbative QCD brute force π f ~ , , 4 , … ~1 GeV M m m (lattice), …? ρ π QCD N hadronic th Chiral EFT with chiral symm Q M ~ , 1 / , , … ~100 MeV M f r m π π QCD nuc NN ℵ ~30 MeV ~ 1 a Q M NN nuc this talk no small coupling expansion in 6/2/2009 v. Kolck, Intro to EFTs 95

Lots of interesting nuclear physics at E ∼ 1 MeV instead of E ∼ 10 MeV within a few MeV of thresholds: � many energy levels and resonances (cluster structures) � most reactions of astrophysical interest show universal features, i.e. to a very good approximation are independent of details of the short-range dynamics bonus: same techniques can be used for dilute atomic/molecular systems 6/2/2009 v. Kolck, Intro to EFTs 96

cf. Bethe + Peierls ‘35 - pionful EFT an overkill at lower energies! channel : channel : e.g. NN s s 0 1 (real) bound state = deuteron (virtual) bound state ℵ ≅ ℵ ≅ < 0 ~ 8 MeV � 1 ~ 45 MeV m B m m N B m π π * N d d ℵ ≅ 1 4.5 fm 1 multipole expansion of meson cloud: contact interactions among local nucleon fields 6/2/2009 v. Kolck, Intro to EFTs 97

ℵ pionless EFT ∼ � Q M nuc • d.o.f.: nucleons • symmetries: Lorentz, P, T ⎛ ⎞ ∇ 2 + + + = ∂ + + L ⎜ ⎟ N i N C N N N N 0 0 EFT 2 ⎝ ⎠ m omitting N spin, isospin ∇ 4 + + + + + ∇ 2 N N C N N N N 2 3 8 m N � � ′ + + + ∇ ⋅ ∇ + … C N N N N 2 6/2/2009 v. Kolck, Intro to EFTs 98

0 l iC Λ ~ ( ) 0 2 1 2 4 1 1 d l ∫ Λ 2 � � ∼ ( ) 1 C � � ( ) 0 4 + + 2 2 π ( ) ( ) 2 l p l p + − + ε − + − + ε 0 0 0 0 l p i l p i 2 Q M 2 2 m m N N 3 1 2 d l k ∫ l = − Λ ≡ 2 0 � ( ) p i m C ( ) 0 2 3 − − ε N 2 2 π m 2 l k i N 2 1 2 ⎧ Λ Λ ⎫ ⎪ ⎪ 1 1 m ∫ ∫ = − Λ + 2 2 ( ) ⎨ ⎬ N i C dl k dl l π 0 + + ε − − ε 2 2 ⎪ ⎪ ⎩ k i l k i ⎭ 1 2 0 0 Q M ⎧ ⎫ ⎛ ⎞ ⎪ 2 ⎪ 1 k k = − Λ Λ + + ≡ − Λ Λ 2 2 ( ) 2 ( ) ( ) ⎨ O ⎬ ⎜ ⎟ i m C i iC I 0 π π π Λ ⎠ 0 0 2 N 4 4 ⎪ ⎪ ⎝ ⎩ ⎭ non- absorbed in absorbed in Λ 2 ~ ( ) iC k 2 analytic C Λ 0 ( ) C Λ 2 ( ) in E etc. 6/2/2009 v. Kolck, Intro to EFTs 99

Λ Λ ⎧ ⎫ ( ) m C Λ → ( ) ≡ Λ − Λ + = ( ) ( ) 1 ( ) … 0 R ⎨ ⎬ N C C C C Λ 0 0 0 0 π 2 2 ⎩ ⎭ m + Λ 1 ( ) N C π 0 2 2 m Λ → ≡ Λ − Λ + … ( ) 2 ( ) ( ) ( ) R N C C C C 2 2 2 0 π Λ 4 … Naïve dimensional analysis π π 4 4 ≡ ( ) ( ) ∼ R R ∼ C C M M 0 0 0 nuc m M m M 0 N N nuc π 4 m ≡ ( ) ( ) ( )2 ∼ R R R N ∼ C C C M M 2 2 0 π 2 4 2 0 m M M M 2 N n uc nuc etc. 6/2/2009 v. Kolck, Intro to EFTs 100