Introduction to Effective Field Theories in QCD

Transcription

1 Introduction to Effective Field Theories in QCD Ubirajara U. van Kolck University of Arizona Supported in part by US DOE 6//9 v. Kolck, Intro to EFTs Background by S. Hossenfelder

2 Outline Effective Field Theories Introduction What is Effective Example: NRQED Summary QCD at Low Energies Towards Nuclear Structure 6//9 v. Kolck, Intro to EFTs

3 References: U. van Kolck, L.J. Abu-Raddad, and D.M. Cardamone, Introduction to effective field theories in QCD, in New states of matter in hadronic interactions (Proceedings of the Pan American Advanced Studies Institute, ), nucl-th/558 D.B. Kaplan, Effective field theories, Lectures at 7th Summer School in Nuclear Physics Symmetries, Seattle, WA, 8-3 Jun 995, nucl-th/ //9 v. Kolck, Intro to EFTs 3

4 There are few problems in nuclear theoretical physics which have attracted more attention than that of trying to determine the fundamental interaction between two nucleons. It is also true that scarcely ever has the world of physics owed so little to so many It is hard to believe that many of the authors are talking about the same problem or, in fact, that M. they L. Goldberger know what the problem is. Midwestern Conference on Theoretical Physics, Purdue University, 96 6//9 v. Kolck, Intro to EFTs 4

5 Nuclear Physics The canons of tradition I II III IV Nuclei are essentially made out of non-relativistic nucleons (protons and neutrons), which interact via a potential The potential is mostly two-nucleon, but there is evidence for smaller three-nucleon forces Isospin is a good symmetry of nuclear forces, except for a sizable breaking in two-nucleon scattering lengths and other, smaller effects External probes --such as photons-- interact mainly with each nucleon but there is evidence for smaller two-nucleon currents but WHY? 6//9 v. Kolck, Intro to EFTs 5

6 Quantum Chromodynamics On the road to infrared slavery Up, down quarks are relativistic, interacting via multi-gluon exchange The interaction is a multi-quark process e.g. 3 Isospin symmetry is not obvious: ε = m m d d m + m u u 3 4 External probes can interact with collection of quarks difficulty quarks and gluons not the most convenient degrees of freedom at low energies How does nuclear structure emerge from QCD? 6//9 v. Kolck, Intro to EFTs 6

7 short table gives the name, the quantum numbers (where known), and the status of baryons in the Review. Only the baryons with 3- This 4-star status are included in the main Baryon Summary Table. Due to insucient data or uncertain interpretation, the other entries in or short table are not established as baryons. The names with masses are of baryons that decay strongly. For N,, and resonances, the the wave is indicated by the symbol L I J, where L is the orbital angular momuntum (S, P, D, :::), I is the isospin, and J is the total partial + c c *** + c *** + cc PDG, 5 Baryon Summary Table angular momentum. For and resonances, the symbol is L I J. P **** (938) p (94) n P **** (3) P 33 **** (6) P 33 *** P **** (45) S **** + P **** P **** P **** ; P **** N(44) P **** (6) S 3 **** (5) D 3 **** ; P **** (53) P 3 **** N(5) D 3 **** (7) D 33 **** (6) P *** (6) * (385) P 3 **** N(535) S **** N(65) S **** N(675) D 5 **** N(68) F 5 **** (75) P 3 * (9) S 3 ** (95) F 35 **** (67) S **** (69) D 3 **** (8) S *** (9) P 3 **** (8) P *** (48) * (56) ** (58) D 3 ** (6) S ** (69) *** (8) D 3 *** (95) *** (3) *** Hadronic N(7) D 3 *** N(7) P *** N(7) P 3 **** (9) P 33 *** (93) D 35 *** (94) D 33 * (8) F 5 **** (83) D 5 **** (89) P 3 **** (66) P *** (67) D 3 **** (69) ** () * (5) ** (37) ** Scales N(9) P 3 ** (95) F 37 **** () * (5) * (75) S *** N(99) F 7 ** (77) P * () F 35 ** () F 7 * N() F 5 ** (775) D 5 **** ; **** (5) S 3 * () G 7 **** N(8) D 3 ** (84) P 3 * (5) ; *** () G 37 * () F 5 *** (3) H 39 ** (35) D 3 * N(9) S * (88) P ** (38) ; ** N() P * (35) H 9 *** (95) F 5 **** (35) D 35 * (47) ; ** N(9) G 7 **** (94) D 3 *** (39) F 37 * (585) ** **** (4) G 39 ** N() D 5 ** () S * N() H 9 **** (4) H 3 **** (3) F 7 **** c (593) + *** N(5) G 9 **** (7) F 5 * (75) I 3 3 ** c (65) + *** c (765) + * (95) K 3 5 ** N(6) I *** (8) P 3 ** N(7) K 3 ** c (88) + ** () G 7 * c (455) **** (5) *** (54) + *** c (5) *** (455) ** (86) * + c *** (6) ** (3) * (37) * *** c c (645) *** c (79) *** c (85) *** c *** M QCD MeV/ c * b *** b, ; b * **** Existence is certain, and properties are at least fairly well explored. *** Existence ranges from very likely to certain, but further conrmation is desirable and/or 6//9 v. Kolck, Intro to EFTs 7 well determined. not are etc. fractions, branching numbers, quantum ** Evidence of existence is only fair. * Evidence of existence is poor.

8 Nucleus (g.s.) H( d ) 3 H() t 3 He 4 He( α ) 5 He Nuclear Matter P J I ( MeV) E r ( fm) E A +.9 ( MeV) Nuclear Scales Q M c MeV/ c E 6//9 v. Kolck, Intro to EFTs 8 ch.6(6).755(86).959(34).676(8) ( fm ) k F Friar, 93 M M nuc nuc QCD c MeV r fm M nuc c

9 Multi-scale problems H atom r p R R p e p H = + O ; me 4π r mec mec r E ( R) ( ) e m R π R e 4 de R dr = R = αmec α e 4π c 37 Three scales mec = pc.5 MeV α mec = 3.6 kev p E α mec = 3.6 ev m e α α 6//9 v. Kolck, Intro to EFTs 9 (from now on, units such that =, c = )

10 However no obvious small coupling in nuclear forces..3 PDG, 5 QCD fine-structure constant α s (μ).. Needed: method that does not rely on small couplings μ GeV EFFECTIVE FIELD THEORY 6//9 v. Kolck, Intro to EFTs

11 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 97 6//9 v. Kolck, Intro to EFTs

12 Ingredients Relevant degrees of freedom 6//9 v. Kolck, Intro to EFTs

13 Seurat, La Parade (detail) 6//9 v. Kolck, Intro to EFTs 3

14 Ingredients Relevant degrees of freedom choose the coordinates that fit the problem All possible interactions 6//9 v. Kolck, Intro to EFTs 4

15 Example: Earth-moon-satellite system Rm.7 Mm d 384 Mm RE 6.4 Mm -body forces +3-body forces Wikipedia change in resolution 6//9 v. Kolck, Intro to EFTs 5

16 Ingredients Relevant degrees of freedom choose the coordinates that fit the problem All possible interactions Symmetries what is not forbidden is compulsory 6//9 v. Kolck, Intro to EFTs 6

17 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 u v αijuv i j ij + δα uv ij ij i j no, say, uv δαij amenable to perturbation theory 6//9 v. Kolck, Intro to EFTs 7

18 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//9 v. Kolck, Intro to EFTs 8

19 t Hooft 79 After scales have been identified, the remaining, dimensionless parameters are O ( ) unless suppressed by a symmetry cow non-sphericity Occam s razor: simplest assumption, to be revised if necessary Expansion in powers of E E und fine-tuning energy of probe energy scale of underlying theory 6//9 v. Kolck, Intro to EFTs 9

20 h A classical example: the flat Earth -- light object near surface of a large body E mgh E und mgr d.o.f.: mass m sym: V ( h, x, y) = V ( h) eff eff eff i ( ) { h = m const } ih = + m h+ ηh + V g g i= parameters (neglecting quantum corrections ) R M naturalness: Veff ( h) = GMm + g = O() = O() g i + = O i R i+ mgi h E h i mgih Eund R R + h h R i GM = m R i= R ( ) 6//9 v. Kolck, Intro to EFTs g h i g i + = itself the first term in a low-energy EFT of general relativity i g R i

21 Going a bit deeper A short path to quantum mechanics a PP= = AA+ + AA+ A 3 b sum over all paths RULE b Ai exp i dtl( q( t)) a each path contributes a phase given by the classical action Path Integral ( ) = L Feynman 48 A Dqexp i dt ( q( t)) dq ( t i ) i 6//9 v. Kolck, Intro to EFTs

22 q q( t i + n) q( t i + n) EFFECTIVE THEORY q( t i ) tit i + M und t j Λ L ( q( t )) i L( q t i t i+ n t i+ n ( ) m t j + t j + scale of fine-structure of underlying curve coarse-graining scale (cutoff) dq + ( t t ) i + dq + ( t ti ) ) dt dt 6//9 v. Kolck, Intro t to EFTs t i t i scale of variation of curve

23 QM + special relativity: quantum field theory qt () ϕ( rt,) ϕ( x) 3 4 dt dt d r d x d dt x μ EFFECTIVE FIELD THEORIES 6//9 v. Kolck, Intro to EFTs 3

24 Partition function ( 4 { L }) free Lint Z= Dϕ exp i d x ( ϕ) + ( ϕ) { } L ( ) L ( ) exp ( L ( )) int int free = Dϕ + i d x ϕ + i d x ϕ + i d x ϕ momentum space L int = λ ϕ 4 4 = iλ (skip many steps ) = i p m + iε p p ' p ' + l p p p l = = dl 4 i i iλ iλ ( π ) ( p + l) m + iε ( p l) m + iε 4 6//9 v. Kolck, Intro to EFTs 4

25 E M Λ m H ( Q M ) φ > L ( Q M ) φ < What is Effective? ( ) ( 4 L ), Z= Dφ Dφ exp i d x ( φ φ ) = H L und H L Dϕδ ϕ ( φ ) ( 4 i d xl ) EFT ϕ Dϕ exp ( ) f Λ L Euler + Heisenberg 36 Weinberg Wilson, early 7s L renormalization-group invariance = EFT i i d= i( d, n) details of the underlying dynamics Z Λ = c ( m ϕ ) ( M, Λ) O (, ) d n local φh : Δ x < Q M φl : Δ x > Q M 6//9 v. Kolck, Intro to EFTs 5 underlying symmetries

26 characteristic external momentum ν ( ) Q Q T T ( Q)~ N( M ) c, i( ), i ; i M F Λ = ν Λ ν ν= ν m m T Λ = normalization ν =ν ( d, n, ) min power counting e.g. # loops L non-analytic, from loops For Q ~ m, truncate consistently with RG invariance T so as to allow systematic improvement (perturbation theory): ν + ( ν ) Q = T + O N M ( ν ) T ( ν ) Q = O T ln Λ Λ 6//9 v. Kolck, Intro to EFTs 6

27 Why is this useful? Because in general the appropriate degrees of freedom below M are not the same as above (, ) φ = φ φ H ϕ L Examples: M is mass of physical particle -- virtual exchange in coefficients i φl φ 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 c φl φ H φl ϕ ϕ ϕ c i ϕ 6//9 v. Kolck, Intro to EFTs 7

28 How can we do it? Two possibilities: know and can solve underlying theory -- c i get s in terms of parameters in ( φ, φ ) und H L know but cannot solve, 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. L Z 4 Note: proven only for scalar field with symmetry in E, but no known counterexamples Ball + Thorne 94 6//9 v. Kolck, Intro to EFTs 8

29 Bira s EFT Recipe. identify degrees of freedom and symmetries. construct most general Lagrangian 3. run the methods of field theory compute Feynman diagrams with all momenta ( regularization ) relate to observables, which should be independent of ( renormalization ) Q < Λ c ( Λ), Λ Λ i Q M O controlled expansion in () not a model form factor 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 what is not forbidden is mandatory! 6//9 v. Kolck, Intro to EFTs 9

30 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) 993 6//9 v. Kolck, Intro to EFTs 3

31 Time for a paradigm change, perhaps? 6//9 v. Kolck, Intro to EFTs 3

32 . Chiral EFT? nuclear physics atomic physics condensed-matter physics and beyond molecular physics NRQED 5 9 QCD ( or 3 flavors) QCD QED (6 flavors) Electroweak Th + higher-dim ops (SUSY)? GUT? Fermi Th General Relativity + higher-curvature terms The world as an onion E(GeV) 6 3 r(fm) 6//9 v. Kolck, Intro to EFTs 3

33 A quantum example: non-relativistic QED (NRQED) single fermion ψ of mass M, massless spin- boson A μ Lund = ψ ( id M ) ψ Fμν F 4 p Lorentz, parity, time-reversal, and U() gauge invariance = μ p i M + iε = ieγ μ p ν μ = i p η μν μν + iε interactions Dμ = μ ieaμ Fμν = μaν νaμ e = 4πα 3 perturbation theory How do E&M bound states arise? 6//9 v. Kolck, Intro to EFTs 33

34 Q M p + q i = = p + q M + iε p + + M + iε i ( p γ p γ + q + M) ( q ) ( p q) i ( p γ + M p γ + q) = p q p q + q p q q + iε i ( + γ ) p q = O ( Q) = + q + iε q = q = O ( Q) Q ± γ p = p + M = M + O P± PP ± ± = P±, PP ± = M projector onto energy states heavy-fermion formalism imt Ψ e Pψ ψ = ( P + P ) ψ = e ( Ψ +Ψ ) imt ± ± 6//9 v. Kolck, Intro to EFTs 34 ± + + Georgi 9 particles: annihilates creates antiparticles: creates annihilates

35 Lund =Ψ id Ψ Ψ iγ D Ψ +Ψ i D Ψ Ψ i + M Ψ ( γ ) ( D ) other, heavy d.o.f.s 4 F F μν μν L anomalous magnetic moment =O() ( 4 L ) und, δ ( ) Z = DA DΨ DΨ exp i d x ( Ψ Ψ, A) DΨ Ψ Ψ ( 4 L ) EFT = DA DΨ exp i d x ( Ψ, A) e jk μν =ΨiD Ψ+ ΨD Ψ+ Ψσ Ψ ε F + Fμν F + M M 4 EFT i ijk non-relativistic expansion eκ jk + ΨσiΨ εijk F + M Pauli term complete square, do Gaussian integral most general Lag with Ψ, A invariant under U() gauge, parity, time-reversal, and Lorentz transformations 6//9 v. Kolck, Intro to EFTs 35

36 L EFT p p p p a ( ) b = F F μν F ( F ) M F μν M F μν μν + μν + μν + ν μ = = i p η μν + iε p 3 p 4 ρ σ ( ( ) ) e μναβ +Ψiv DΨ + Ψ v D D Ψ + ( + κ) ΨvαSβΨ ε Fμν + M M v (, ) i μ = iev v p + ( p ( v p) ) + + iε μ σ M S, ν μ e { ν α β = i( p + p' ) + ( + κ) ε v S q } e μ μναβ = i ( η μν v μ v ν ) M M q μ () () γ γ + ΨΨΨΨ+ Ψ M M S Ψ Ψ S Ψ+ i = + M ν μ ( γ γ S S) p p Euler + Heisenberg 36 i = M { a ημρηνσ p p3p p4 b[ ]} 6//9 () () v. Kolck, Intro to EFTs 36 etc.

37 Various processes at low energies: e.g. T γγ = + light-by-light scattering no explicit fermion-antifermion pair creation! T γψ = Compton scattering Thompson limit no change in heavy-fermion number! 6//9 v. Kolck, Intro to EFTs 37

38 Back to atomic bound states: the NRQED perspective ( p ', p ') ( p ', p ') T ψψ = ( p, p ) ( p, p) p p p' = O Q p' CoM frame ( ) Q = O M higher powers of Q M ie ie ie e = = p p iε Q ( p p' ) + iε ( p p ' ) ( ') + iε ( p p' ) 6//9 v. Kolck, Intro to EFTs 38 ( ) V r = e 4π r

39 ( p + p + l l, ) ( l, ) l ( p + p ' + l ' l, ) l = e 4 4 dl ( π ) 4 ( l + p) ( l p ') l + p + iε l + p' + iε M M l + l + iε l l + iε ( p p ' ) ( p p' ) 4 3 Q M 3 4 = ie 4 3 dl 3 ( π ) ( l + p) ( l p ') l p + iε l p' + iε M M 4 l l + iε l iε ( ) p p' ( p p' + ) Q Q + 3 e 4 Q 3 α ( ) 4π QQQ Q 4π Q e just as expected 6//9 v. Kolck, Intro to EFTs 39

40 ( p + p + l 4 l, 3 ) Q ( l, ) l Q ( p ( p + l ) M l, Q ) l 3 4 = e 4 = ie + 4 dl 4 ( π ) ( l + p) ( l + p) l + p + iε l + p + iε M M 3 4 p p + l + iε l l + iε 4 ( ' l ) ( p p ' + ) 3 dl ( π ) + ( ) M p + M ( ) ( l + p) p p ' p p' + l + iε M ( l + p) p iε l + p l i 3 4 Q M e e M e + α α + 4π Q Q Q 4π Q Q Q ε 6//9 v. Kolck, Intro to EFTs 4 infrared enhancement!

41 dl Q M Q dl + + time-ordered perturbation theory Q 6//9 v. Kolck, Intro to EFTs 4

42 () T ψψ = + + bound state at e M e + O α + Q Q Q M O α Q Q αm Q E α M M () V ψψ () = V () ψψ + + = + () V ψψ V ψψ () T ψψ () V ψψ () V ψψ = e = O Q Coulomb potential Lippmann-Schwinger eq. = Schroedinger eq. pˆ () () () () + V ψψ ψ = E ψ M 6//9 v. Kolck, Intro to EFTs 4 known results

43 But more: () V ψψ = α e = O 4π Q () T ψψ () V ψψ () T ψψ () T ψψ () = V ψψ () T ψψ () V ψψ () V ψψ () T ψψ E = E + ψ Vψψ () () () () () 6//9 v. Kolck, Intro to EFTs 43 ψ α 4π E () = O

44 () V ψψ = + + Q e = O M Q () T ψψ = () () () () ( ) E = E + ψ Vψψ ψ + Q M () = O E piece μ μ ψ δ ψ = μ μ ψ ( ) ( ) ( ) ( ) 3 ()* (3) () () dr r r r magnetic interaction 6//9 v. Kolck, Intro to EFTs 44

45 N O T E (3) T ψψ starting at, sufficiently many derivatives appear at vertices that Λ loops bring positive powers of, which need to be compensated by γ () i ( Λ) and higher-order counterterms (3) V ψψ = α Q e = O 4π M Q α M ln Λ () i γ = O ( α ) Etc. 6//9 v. Kolck, Intro to EFTs 45

46 Example: g factor for electron bound in H-like atoms electron Larmor frequency measured known trapped-ion ion charge cyclotron frequency ion mass measured electron mass ψ ψ ( κ ) g = + () ψ = + + () ψ Pachucki, Jentschura + Yerokhin 4 u = m C( gs) 6//9 v. Kolck, Intro to EFTs 46 Most precise determination of electron mass (expt)(th)

47 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//9 v. Kolck, Intro to EFTs 47

48 Introduction to Effective Field Theories in QCD U. van Kolck University of Arizona Supported in part by US DOE 6//9 v. Kolck, Intro to EFTs 48

49 Outline Effective Field Theories QCD at Low Energies QCD and Chiral Symmetry Chiral Nuclear EFT Renormalization of Pion Exchange Summary Towards Nuclear Structure 6//9 v. Kolck, Intro to EFTs 49

50 References: S. Weinberg, Phenomenological Lagrangians, Physica A96:37,979 S. Weinberg, Effective chiral Lagrangians for nucleon-pion interactions and nuclear forces, Nucl.Phys.B363:3-8,99 S.R. Beane, P.F. Bedaque, L. Childress, A. Kryjevski, J. McGuire, and U. van Kolck, Singular potentials and limit cycles, Phys.Rev.A64:43,, quant-ph/73 A. Nogga, R.G.E. Timmermans, and U. van Kolck, Renormalization of one-pion exchange and power counting, Phys.Rev.C7:546,5, nucl-th/565 6//9 v. Kolck, Intro to EFTs 5

51 . Chiral EFT? nuclear physics atomic physics condensed-matter physics and beyond molecular physics NRQED 5 9 QCD ( or 3 flavors) QCD QED (6 flavors) Electroweak Th + higher-dim ops (SUSY)? GUT? Fermi Th General Relativity + higher-curvature terms The world as an onion E(GeV) 6 3 r(fm) 6//9 v. Kolck, Intro to EFTs 5

52 QCD Game Plan lattice EFT necessary to extrapolate to large small r m π NCSM, Few-nucleon systems? lattice, Many-nucleon systems want model independence Infinite-nucleon system 6//9 v. Kolck, Intro to EFTs 5

53 d.o.f.s L EFT at a few GeV= underlying theory for nuclear physics 3 ( ) = l i + eca m l Fμν F 4 und f f f f = ( ) Tr μν + q i + eqa + gs G q G G μν ( mu + md) qq ( mu md) qτ 3q mm u d + θ qiγ5q + m + m u d l f + l = ν f u q = d leptons: quarks: a photon: A μ gluons: symmetries: SO(3,) global, U () gauge, SU (3) gauge em 6//9 v. Kolck, Intro to EFTs 53 μν c QED + QCD higher-dimension interactions: suppressed by larger masses unnaturally small T violation (strong CP problem) G μ C = 3 + 3τ 3 Q= = 3 6 G F θ < e.g. M 9 W, Z

54 Focus on strong-interacting sector: four parameters ) m = m =, =, θ = u d e 4 4 d x QCD d x q( i gsg) q Tr μν L = /+ / G G but in single, dimensionless parameter invariant under scale transformations ( 4 ) QCD = L Z DG Dq Dq exp i d x scale invariance anomalously broken by dimensionful regulator coupling runs ( Q ) α GeV s ( dimensional transmutation ) α s (μ).3.. μν chiral limit x q G λ λ 3 x q λg Λ 6//9 v. Kolck, Intro to EFTs 54 μ GeV

55 Non-perturbative physics at Q GeV Assumption : confinement only colorless states ( hadrons ) are asymptotic Observation: (almost) all hadron masses > GeV Assumption : naturalness masses are determined by characteristic scale M QCD GeV Observation: pion mass m 4 MeV π M QCD breakdown of naturalness? NO! spontaneous breaking of chiral symmetry 6//9 v. Kolck, Intro to EFTs 55

56 L Why is the pion special? ( ) ( ) Tr μν = q i /+ g G/ q + q i /+ g G/ q G G QCD L s L R s R u q = d +γ q γ 5 q 5 μν invariant under q exp i q ( α τ ) L( R) L( R) L( R) chiral symmetry SU () SU () SO(4) L R m m N σ m m π N + broken by vacuum down to q exp( iα τ ) q isospin SU () SO(3) L+ R 6//9 v. Kolck, Intro to EFTs 56

57 Chiral Limit V QCD two isospin axis not shown qq chiral circle π σ qiγ τ q 5 f π pion decay constant (in chiral limit) L piece invariant under π π+ ε [function of on chiral circle] EFT = π μ π + 4 f μ π π 6//9 v. Kolck, Intro to EFTs 57

58 ) m m, =, θ = u d e L QCD ( ) Tr μν = q i /+ gsg/ q G Gμν + ( mu + md) qq + ( mu md) qτ 3q + v.k th component of SO(4) vector 3 rd component of SO(4) S = qiγτq qq P = qτ q, qiγ q ( ) 5, ( ) 5 vector SO(4) SO(3) break U () (explicit chiral-symmetry breaking) (isospin violation) 6//9 v. Kolck, Intro to EFTs 58

59 Chiral V QCD Limit slightly-tilted chiral circle two isospin axis not shown qiγ τ q 5 qq f π 9 MeV pion decay constant π σ L piece invariant under π π+ ε [function of ] EFT = π + piece in qq direction [function of π explicitly] + isospin breaking μ Q ( m + m ) u u d ( m m ) d CHIRAL SYMMETRY WEAK PION INTERACTIONS 6//9 v. Kolck, Intro to EFTs 59

60 3) e, θ = Two types of interactions: soft photons explicit d.o.f. in the EFT D = iqa e μ μ μ Fμν = μaν νaμ hard photons integrated out of EFT ( ) μν Lund = e qqγ μq D qqγνq + 34 comp of antisymmetric tensor SO (4) F μ εijkqiγ μγ5τk q qiγ μτ jq = qiγτ μ iq breaks (and SO (3) in particular) U () v.k. 93 L = EFT soft photons + further isospin breaking e α 4π 6//9 v. Kolck, Intro to EFTs 6

61 4) θ L und mm u d = + θ qiγ5q + mu + md 4 th component of SO(4) vector P = ( qτ q, qiγ q) 5 T violation linked to isospin violation: in EFT, combination is u d + mm m + m u d ( m m ) P 3 θ P 4 5) continue with higher-order operators, e.g. T-violating quark EDM and color-edm P-violating four-quark operators u d Hockings, Mereghetti + v.k., in preparation De Vries, Mereghetti, Timmermans + v.k., in progress Kaplan + Savage 96 Zhu, Maekawa, Holstein, Musolf + v.k. 6//9 v. Kolck, Intro to EFTs 6

62 Nuclear physics scales ln Q perturbative QCD His scales are His pride, Book of Job ~ GeV ~ MeV ~3 MeV M ~ m, m, 4π f, QCD N M ~ f, / r, m, ℵ nuc ~ ann ρ π hadronic th with chiral symm π NN π brute force (lattice),? this talk Q M QCD Q M nuc no small coupling halo nuclei expansion in 6//9 v. Kolck, Intro to EFTs 6

63 Q mπ M QCD Nuclear EFT ( m Δ m ~ m π ) d.o.f.s: nucleons, pions, deltas N + π ( π + π ) p N = + π = π i n ( π π ) = π 3 π symmetries: Lorentz, P, T, chiral pionful EFT Δ Δ Δ Δ ++ + Δ= Non-linear realization of chiral symmetry Weinberg 68 Callan, Coleman, Wess + Zumino 69 (chiral) covariant derivatives chiral invariants pion fermions D μ μπ π + f 4 f π π D it E ( ) μ μ μ E π D + S4's, P3's, F34's (( u d) QCD ) mπ = O m + m M 6//9 v. Kolck, μ Intro to EFTs μ 63 fπ m u + m d m π = O M QCD

64 Schematically, L EFT p f n + D, D, mδ m N mπ π ψψ c{ n, p, f } { npf,, } M QCD MQ CD f π fπ M QCD = f π M QCD calculated from QCD: lattice, fitted to data = O () α = O ε, 4π isospin conserving isospin breaking (NDA: naïve dimensional analysis) ( Δ) = f f L Δ n+ p+ d + Δ= chiral index chiral symmetry 6//9 v. Kolck, Intro to EFTs 64

65 L L π π = g A + π + N i τ ( π π) + N + N τσ N ( π) + 4fπ fπ 4fπ + h A + +Δ [ i ( mδ mn ) + ] Δ+ + ( N TSΔ+ Η.c. ) ( π) ( + ) fπ C N N C N () ( μπ) m π fπ 4fπ () S + + ( ) T ( σ N ) + = N + τ ( π π) + ( ) + mp mn τ3 π 3π τ+ N mn 4fπ fπ + + N b ( ) b3( ) bm ib4εijkεab cσkτc( iπb)( jπc ) N f π π π + + π g A + in σ N + Η.c ( ) ( + ) 4mN f τ π π d N NN τσ N ( π) ( ) f L () = π ( ) 3 E N N + + Form of pion interactions determined by 6//9 v. Kolck, Intro to EFTs 65 chiral symmetry

66 A=, : chiral perturbation theory c ν T ΔE Q ν ν Q Q F ν M QCD mπ ν = A + L+ V Δ ν = A i i i min nucleon Weinberg 79 Gasser + Leutwyler 84 Gasser, Sainio + Svarc 87 Jenkins + Manohar 9 dense but short-ranged Q M # loops expansion in Q m Q m N # vertices of type i, non-relativistic multipole long-ranged but sparse M.3 fm ρ 6//9 QCD v. Kolck, Intro to EFTs 66 Q 4π fπ pion loop QCD m π.4 fm

67 Analogous to NRQED Weinberg 79 Gasser + Leutwyler 84 T ππ = current algebra Weinberg 66 Gasser, Sainio + Svarc 87 Jenkins + Manohar 9 T π N = Etc. N.B. For 3 Q E ( mδ mn ) < O M QCD a resummation is necessary Phillips + Pascalutsa Long + v.k., in preparation 6//9 v. Kolck, Intro to EFTs 67

68 A > : resummed chiral perturbation theory Weinberg 9, 9 V A-nucleon irreducible V m Q N ΔE infrared enhancement! l A-nucleon reducible E e.g. V 4 d l i V V ( π ) 4 l + k m N l mn iε l + k mn l mn iε V 3 d l m = V N mnq V + O V 3 ( π ) l k 4π k = m N Q instead of 6//9 v. Kolck, Intro to EFTs (4 π ) 68

69 i 3 ( qˆ + σ σ ) g A q S( ) τ τ fπ q + mπ fπ S tensor force ( qˆ) = 3σ qˆσ qˆ σ σ 3 Q Q Q Q 4 π f π ( 4π ) Q ( 4π ) π f Q Q Q f Q = O M QCD 3 Q Q Q Q Q 4 fπ π mδ m Q Q Q 4 f mδ m ( ) ( π ) π 4 N N π Q m Q Q m Q Q f Q Q Q f f f 3 N N 4 π 4π 4π fπ π π μπ π 6//9 v. Kolck, Intro to EFTs μ 69 π ( ) = O f ( ) = O for Q μπ

70 T () = + + bound state at Q μπ Q E μ π m N M QCD Q + O + μ O μπ fπ π fπ Q V () = + () V + = V + V () () 4π fπ M nuc = μπ fπ f m T V () () N π Nuclear scale arises naturally from chiral symmetry () V = +? Is PE all there is in leading order? That is, are observables cutoff independent with PE alone? 6//9 v. Kolck, Intro to EFTs 7

71 Issue: relative importance of pion exchange and short-range interactions g ( S ( q) + σ σ ) i f 3 N ˆ A q 4π τ τ π q + mπ m μπ g (3) () A τ τ δ () r fπ V r mπ mπ r = + e 4π r S = S = (3) m A π m r m π π m r π τ ˆ τ δ π πr πr πr πr g V () r = () r e e S () r f ( m ) m 3 much more singular --and complicated!-- than ie e e V ( r) = 4π r ( p p' ) iε Q S j j j+ j j( j+ ) j 6 j+ j+ j j( j+ ) j + j + 6 j+ j+ 6//9 v. Kolck, Intro to EFTs 7

72 Assume contact interactions are driven by short-range physics, and scale with M QCD according to naïve dimensional analysis (W power counting) Weinberg 9, 9 Ordonez + v.k. 9 Ordonez, Ray + v.k. 96 Entem + Machleidt 3 Epelbaum, Gloeckle + Meissner 4... C ( σ σ ) ( 3) C σ + σ 4 4 4π 4π μ μ () () m N m N 4π (3) V() r = δ () r m N μ 4π V r = δ r (3) () () m N μ S = S = μ μ i π C ( i ) in LO 4π m μ N π Q M QCD in NLO (terms linear in Q M QCD break P, T ) etc. 6//9 v. Kolck, Intro to EFTs 7

73 V N = Ordonez + v.k. 9 v.k V 3N = higher powers of Q Etc. more nucleons 6//9 v. Kolck, Intro to EFTs 73

74 6//9 v. Kolck, Intro to EFTs 74 QCD Q f M π O f π O 3 3 QCD Q f M π O 4 4 QCD Q f M π O LO -body 3-body 4-body NLO QCD Q M f π O NNLO NNNLO (parity violating) ETC.

75 Hierarchies many-body forces V V V A canon emerges! N 3N 4N Weinberg 9, 9 Similar explanation for isospin-breaking forces V V V IS IV CSB J J J N N 3N external currents v.k. 93 Rho 9 6//9 v. Kolck, Intro to EFTs 75

76 Ordonez + v.k. 9 v.k. 94 V N = similar to phenomenological potential models, e.g. AV8 (OPE)^ + non-local terms Stoks, Wiringa + Pieper 94 6//9 v. Kolck, Intro to EFTs 76

77 6//9 v. Kolck, Intro to EFTs 77

78 But: NOT your usual potential! Ordonez + v.k. 9 σ + ω e.g., (cf. Stony Brook TPE) chiral v.d. Waals force 6 for m r π Rentmeester et al., 3 Kaiser, Brockmann + Weise 97 Nijmegen PSA of 95 pp data V C (r) [MeV] π σ + ω Isoscalar Central Potential r [fm] Similar results in other channels, e.g. spin-orbit force! long-range pot #bc models with σ, ω, might be misleading 6//9 v. Kolck, Intro to EFTs 78 χ min OPE 3 6. OPE + TPE ( lo) OPE + TPE ( nlo) Nijm parameters found consistent with πn data! at least as good!

79 Fujita + Miyazawa 58 v.k. 94 Friar, Hueber + v.k. 99 Coon + Han V 3N = two unknown parameters + Coon et al. 78 Tucson-Melbourne pot with a a m c, ' q ' q = δ ' ε τ σ q q ' + αβ ( t ( qq )) a bq c N ( q ) d 3 π αβ αβγ γ π c TM potential 6//9 v. Kolck, Intro to EFTs 79

80 Many successes of Weinberg s counting, e.g., To N3LO (w/o deltas), fit to NN phase shifts comparable to those of realistic phenomenological potentials Ordonez, Ray + v.k. 96 Epelbaum, Gloeckle + Meissner Entem + Machleidt 3 Entem + Machleidt 3 VPI PSA NNNLO NNLO Nijmegen PSA NLO 6//9 v. Kolck, Intro to EFTs 8

81 With N3LO N and NLO 3N potentials (w/o deltas), good description of 3N observables and 4N binding energy levels of p-shell nuclei Epelbaum et al. Gueorguiev, Navratil, Nogga, Ormand + Vary 7 6//9 v. Kolck, Intro to EFTs 8

82 γ d dγ measured: Illinois 94, SAL, Lund 3 extracted nucleon polarizabilities: Beane, Malheiro, McGovern, Phillips + v.k. 4 Many reactions: γ d pp pn dπ ppπ pp pnπ + pp dπ + dπ threshold amplitude predicted: Beane, Bernard, Lee, Meissner + v.k. 97 confirmed: SAL 98, Mainz measured: IUCF 9-, TRIUMF 9-, Uppsala 95- S waves sensitive to high orders: Miller, Riska + v.k. 96 P waves converge, fix 3BF LEC: Hanhart, Miller + v.k. CSB asymmetry sign predicted: Miller, Niskanen + v.k. confirmed: TRIUMF 3 dd απ measured: IUCF 3 mechanisms surveyed: Fonseca, Gardestig, Hanhart, Horowitz, Miller, Niskanen, Nogga +v.k PARITY, TIME-REVERSAL VIOLATION, etc. 6//9 v. Kolck, Intro to EFTs 8

83 BUT Is Weinberg s power counting consistent? No! 3 g ˆ A m S() r π τ τ + e 3 fπ 4 π ( mπr ) attractive in some channels m r π singular potential not enough contact interactions for renormalization-group invariance even at LO 6//9 v. Kolck, Intro to EFTs 83

84 6//9 v. Kolck, Intro to EFTs 84

85 Renormalization of the r n potential C (3) ( Λ) δ Λ ( r) V ( R) θ V() r r R R Λ r r r λ f ( r r ) mr ( ) n OPE: m= mn r = mπ λ = mπ μπ f( r r) = exp( r r) s wave matching ψ n ( r R r ) u () n r r ( λ ) V V = F r R mr cot mr n,, so that T R ( k r ) = O T s s ln R r 6//9 v. Kolck, Intro to EFTs 85

86 n Beane, Bedaque, Childress, Kryjevski, McGuire + v.k. Two regular solutions that oscillate! if no counterterm, will depend on cutoff R model dependence 4 λ λ un( r r ) = cos δ + n + n n ( r r ) ( n )( r r ) determined by low-energy data n n R λ Fn ( λ, r, R) = λ tan + δ n n + 4 r ( n )( R r ) u(r) 4 exact 3rd 4 H = mr V (R) r 3 st nd exact vs perturbation th //9 v. Kolck, Intro to EFTs 86 limit-cycle-like behavior R

87 Same is true in all channels were attractive singular potential is iterated λ r ll ( + ) + 3 r r Beane, Bedaque, Savage + v.k. Nogga, Timmermans + v.k. 5 Pavon Valderrama + Ruiz-Arriola, 6 Birse, 6, 7 Long + v.k., [ ll ( + ) ] ( λr ) 3 ll ( + ) r λr r 3 r l = 3λ ( ll+ ) r r but rl r for ll ( + ) λ M singular potential only needs to be iterated in a few waves, where counterterms are needed OPE: 3M QCD ll ( + ) < 5 μ π l < 6//9 v. Kolck, Intro to EFTs 87

88 certain counterterms that in Weinberg s counting were assumed suppressed by powers of are in fact suppressed by powers of Q M Q QCD 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//9 v. Kolck, Intro to EFTs 88

89 S Examples Other singlet channels δ 6 4 δ 6 Beane, Bedaque, Savage + v.k. Λ = 985 MeV LO EFT Λ = 49 MeV Λ =4 MeV Nijmegen PSA 3 P (MeV) NLO EFT δ [deg] δ [deg] P F3 Nogga, Timmermans + v.k. 5 (cf. Birse + McGovern 4) D G T L [MeV] 5 5 T L [MeV] 3 LO EFT ( Λ = fm ) Nijmegen PSA 6//9 v. Kolck, Intro to EFTs 89 P (MeV)

90 δ [deg] Attractive triplet channels LO EFT m π χ-limit Nijmegen PSA P (MeV) 3 S Beane, Bedaque, Savage + v.k. Repulsive triplet channels Λ = 3 P - 3 F δ [deg] δ [deg] T L [MeV] LO EFT ( fm ) Nijmegen PSA Nogga, Timmermans + v.k. 5 (cf. Pavon Valderrama + Ruiz-Arriola, 6) 6//9 5 5 v. Kolck, Intro to EFTs T [MeV] T [MeV] 3 P 3 F ε 3 P 5 5 T L [MeV]

91 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//9 v. Kolck, Intro to EFTs 9

92 Introduction to Effective Field Theories in QCD U. van Kolck University of Arizona Supported in part by US DOE 6//9 v. Kolck, Intro to EFTs 9

93 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//9 v. Kolck, Intro to EFTs 93

94 References: U. van Kolck, Effective field theory of short-range forces, Nucl.Phys.A645:73-3,999, nucl-th/9887 P.F. Bedaque, H.-W. Hammer, and U. van Kolck, The three-boson system with short-range interactions, Nucl.Phys.A646: ,999, nucl-th/9846 I. Stetcu, B.R. Barrett, and U. van Kolck, No-core shell model in an effective-field-theory framework, Phys.Lett.B(to appear),7, nucl-th/693 P.F. Bedaque, H.-W. Hammer, and U. van Kolck, Narrow resonances in effective field theory, Phys.Lett.B569:59-67,3, nucl-th/347 6//9 v. Kolck, Intro to EFTs 94

95 Nuclear physics scales ln Q perturbative QCD His scales are His pride, Book of Job ~ GeV M QCD ~ mn, m ρ, 4π fπ, brute force (lattice),? hadronic th with chiral symm Chiral EFT ~ MeV M nuc ~ fπ, / rnn, mπ, Q M QCD ~3 MeV ℵ ~ a NN Q M nuc no small coupling this talk expansion in 6//9 v. Kolck, Intro to EFTs 95

96 Lots of interesting nuclear physics at instead of E MeV E 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//9 v. Kolck, Intro to EFTs 96

97 - pionful EFT an overkill at lower energies! e.g. NN s channel: s cf. Bethe + Peierls 35 channel: (real) bound state = deuteron (virtual) bound state ℵ ~ m 45 MeV< N B d m π ℵ ~ m 8 MeV N B * d m π ℵ 4.5 fm multipole expansion of meson cloud: contact interactions among local nucleon fields 6//9 v. Kolck, Intro to EFTs 97

98 Q ℵ M nuc pionless EFT d.o.f.: nucleons symmetries: Lorentz, P, T L EFT = N i + N + C N N N N mn N N C 3 N N N N 8mN C N N N N + omitting spin, isospin 6//9 v. Kolck, Intro to EFTs 98

99 ~ ic ( Λ) l p etc. 4 dl C ( Λ) 4 ( π ) l + p l + p l + p + iε l + p m N m N 3 k dl = im N C ( Λ) m 3 N ( ) ( ) + iε ( π ) l k iε Λ Λ m N = i C ( Λ) dl k dl π + l + k+ iε l k iε k k = im N C ( Λ) Λ + i + O π 4π 4πΛ ~ ic ( Λ) k non- absorbed in absorbed in C ( Λ) analytic C ( Λ) in E 6//9 v. Kolck, Intro to EFTs 99 ic Q Q ( Λ) I ( Λ) M M l

100 ( R) mn Λ C ( Λ) C( Λ) C C( Λ) C ( Λ) + = π mn Λ + C ( Λ) π ( R) m C( Λ) C C( Λ) N C( Λ) + 4πΛ Naïve dimensional analysis C C ( R) 4π m M N C ( R) 4π m M ( R) 4π ( R) N ( R) C C M M mnmnucm 4π M nuc etc. m N nuc M M nuc 6//9 v. Kolck, Intro to EFTs

101 But in this case: C ( R) N 4π m Q C 4 4ππ m Mℵ mn N ( R ) 44 ππ Q m m ℵ Mℵ M N N for MQ ℵM if nuc etc. C Q ( R) 44 ππ Q mmm ℵ M Mℵ N N nuc M M if nuc no b.s. at Q < M nuc, no good: just perturbation theory need one fine-tuning: M ℵ M nuc assume no other, e.g. still, etc. M M 6//9 v. Kolck, Intro to EFTs

102 T () = + + v.k Kaplan, Savage + Wise 98 Gegelia 98 { ( ) } i 4π i = ic CI + CI + = = + I m 4π Λ + + ik k ℵ C ( Λ) π 4π i k k = + O, mn ℵ+ ik Λ ℵΛ N mnc 4π = m ℵ C ( R) N k + O Λ cf. effective range expansion s wave scattering length a = ℵ bound state k = iℵ E = ℵ m N 6//9 v. Kolck, Intro to EFTs

103 () V = = () () T etc. T NN s wave V 4π m M V T () () ~ nuc N nuc T V () () M Q + + ℵ+ iq ℵ+ iq ν = ν = scattering length effective range a ~ℵ ~ nuc r M p, other waves T V T () () () Q () = O T M nuc Q 6//9 v. Kolck, Intro to EFTs 3

104 Example: square well r c T α θ mr R V () r = r ( R) cot α ( R) α + k + k + ir k kr NN ( k) = i e i zero-energy poles when α (n + ) π a = R R = R tanα α R aα 3a etc. α ( kr) cot α ( kr) + + ir k generic α = O() a r c 6//9 v. Kolck, Intro to EFTs 4 R R a αc ℵ R α = { + } αc ℵ r = R { } fine-tuning αα c + R αα R R

105 rψ ( r) N e r a E V() r e N ma α mr α mr c R a = 6//9 v. Kolck, Intro to EFTs 5 ℵ In the quantum world, one can have a b.s. with size much larger than the range of the force r provided there is fine-tuning

106 unitarity limit a (fm) - triplet scattering length 4 6 m π ( MeV ) Pion-mass dependence a 3 S Large deuteron size because m ℵ π ( ) m π M QCD m π ( M ) QCD mπ mπ M nuc m 5 5 6//9 v. Kolck, Intro to EFTs 6 π 4 B d 3 ( MeV ) Lattice QCD: quenched EFT: (incomplete) NLO Deuteron binding energy Fukugita et al. 95 cf. Beane, Bedaque, Orginos + Savage 6 Beane, Bedaque, Savage + v.k. m π ( MeV )

107 MIT group webpage Cf. trapped fermions Feshbach resonance unitarity limit a quark masses analog to magnetic field: close to critical values (( ) ) u + d M QCD ( MeV) m m m π = O 6//9 v. Kolck, Intro to EFTs 7

108 Alternative: auxiliary field Kaplan 97 v.k. 99 L EFT + + g = N i + N + T ( Δ) T + T NN + N N T mn + N N + T i + T σ 3 8mN 4m sign N integrate out auxiliary field: same Lag as before with g Δ~ ℵ, ~, π m 4 N g C = Δ, T NN = = + + 6//9 v. Kolck, Intro to EFTs 8

109 δ 9 45 fitted a r predicted = ν = 3 = ν = k (MeV) 8 s s δ. fm (exp).78 fm (exp) B d * =.9 MeV ( ν = ) ν = + ν = + Nijmegen PSA 3 6//9 v. Kolck, Intro to EFTs fitted a r predicted B d ν = + Chen, Rupak + Savage 99 = 5.4 fm (exp) =.75 fm (exp) =.9 MeV ( ν = ) B =. MeV (exp) d ν = ν = k (MeV) Nijmegen PSA

110 T Nd = + + ~ g Q m N 3 Q g g Q ~ ~ 4π Q mn Q mn ℵ = + T Nd T Nd Λ 3 dl KONE TNd = KONE + λ ( ) 3 π D 6//9 v. Kolck, Intro to EFTs

111 L EFT s 3 λ T Nd = + D N N N N N N + naïve dimensional analysis λ predicted c 3 3 4π D k cotδ [fm ] a3 = 6.33±. fm ( ν = + ) a3 = 6.35 ±. fm (exp) ~ 3 mn Mnuc k [fm ] 6//9 v. Kolck, Intro to EFTs 4π no three-body force up to ν = + 3 p ℵ p TNd p~ ℵ Λ QED-like precision! ( ν = + ) v.oers + Seagrave 67 Dilg et al. 7 3-body interaction Bedaque + v.k. 97 Bedaque, Hammer + v.k. 98 ν = + ν =

112 s λ > c λ 3 3 4π ℵ p TNd A s + Λ D ~ p cos ln δ Nd p~ 4π M mn ℵ nuc ( ν = ) T ℵ Λ Bedaque, Hammer + v.k. 99 unless Hammer + Mehen Bedaque et al. 3 6//9 v. Kolck, Intro to EFTs

113 T Nd = ~ g Q m N 4 ~ π ℵ T Nd = T Nd T Nd Λ 3 dl KONE TNd = KONE + λ ( ) 3 π D Λ 3 dl KTBF TNd + KTBF + λ ( ) 3 π D 6//9 v. Kolck, Intro to EFTs 3

114 s fitted a = predicted B t B t λ > c λ (limit cycle!).65 fm (exp) = 8.3 MeV ( ν = ) = 8.48 MeV (expt) 3 3 4π ℵ p TNd A s + Λ D ~ p cos ln δ Nd p~ 4π M mn ℵ nuc ( ν = ) k cotδ [fm ] T ℵ Λ v.oers + Seagrave 67 Kievsky et al. 96 Dilg et al. 7 Bedaque, Hammer + v.k. 99 unless Hammer + Mehen Bedaque et al. 3 ν = ν =..4.6 k [fm ] 6//9 v. Kolck, Intro to EFTs 4

115 Phillips line Bedaque, Hammer + v.k. 99 potential models Λ varying * ( ν = ) 6//9 v. Kolck, Intro to EFTs 5

116 + four-body bound state can be addressed similarly no four-body force at ν = Hammer, Meissner + Platter 4 35 Tjon line B α [MeV] 3 5 potential models exp varying Λ * ( ν = ) 7,5 8 8,5 9 B t [MeV] 6//9 v. Kolck, Intro to EFTs 6

117 Summary: Expansion parameter Q M nuc ℵ M nuc r a LO: two two-nucleon + one three-nucleon interactions C, C, D () () NLO: two more two-nucleon interactions Etc. ~ larger nuclei? No-Core Shell Model! Barrett, Vary + Zhang 93 6//9 v. Kolck, Intro to EFTs 7

118 Crucial issue For a given computational power, as number of nucleons grows, number of one-nucleon states gets more limited A-body problem: shell model excluded space Q= P model space N max n+ l N nl, max P = nl nl ω What are the effective interactions in the model space? 6//9 v. Kolck, Intro to EFTs 8

119 The traditional no-core shell model approach Barrett, Vary + Zhang 93 start with god-given (can be non-local!) potential, and run the RG in a harmonic-oscillator basis e.g., chiral pot from last lecture O PO P = PO P + PHQ QO P + eff a a a a E QHQ = O + O + + O + + O ' ' ' ' a a+ a+ b A arbitrary truncation ( cluster approximation ) Feshbach projection convergence: a+ b A for fixed P P for fixed a+ b issues: systematic truncation error, consistent currents, etc. EFT addresses just these issues! 6//9 v. Kolck, Intro to EFTs 9

120 EFT + NCSM start with EFT in restricted space; fit parameters in few-nucleon systems for various and N ω ; and predict larger nuclei ω Stetcu, Barrett +v.k., 6 Stetcu, Barrett, Vary + v.k., 7 Vary + v.k., in progress max cutoffs IR UV λ = m N ω N ( ) Λ = m N + ω max 3 strategies: determine parameters from light-nuclei binding energies scattering phase shifts 6//9 v. Kolck, Intro to EFTs

121 single particle A 4 φ + m ωr A 3 ( m ) Nωr Y ( ) l l N ˆ nl( s) j = Nnl r Ln exp l r χ s j : relative coordinates Basis : Slater-determinant basis ξ ξ ψ ξ ξ φ ξ φ ξ (, ) = A ( ) ' ' '( ) nlj n l j code `a la Navratil, Kamuntavicius + Barrett (reduced dimensions, but difficult antisymmetrization) JI (,, ) r r r 3 φ φ φ ( r) ( r) ( r) ( ) ( ) ( ) ( r ) ( r ) ( r ) nl j n l j nl j 33 3 ψ r r r = φ r φ r φ r 3 nl j n l j nl j 33 3 φ φ φ nl j 3 n l j 3 nl j code: REDSTICK Navratil + Ormand 3 6//9 v. Kolck, Intro to EFTs

122 LO Pionless EFT: ingredients matrix elements of -, 3-body delta-functions, e.g. n! n! n l δ r n l L L Γ ( n 3 ) ( 3 ) + Γ n + ( ) ( ), = ( ), = n n EFT PC effectively justifies (modified) cluster approximation parameters ( Λ), ( Λ), ( Λ) C C D () () fitted to d, t, α ground-state binding energies 6//9 v. Kolck, Intro to EFTs

123 LO ( ν = ) Stetcu, Barrett +v.k., 6 Nmax 6 α + β ω + γ( ω) fits First + [MeV] First + [MeV] preliminary 3 4 Λ [MeV] Λ He Exp hω [MeV] hw = MeV hw = MeV hw = 3 MeV hw = 4 MeV hw = 5 MeV hw = 6 MeV a( ω ) + fits b( ω) Λ works within ~%! 6//9 v. Kolck, Intro to EFTs 3

124 LO Nmax 8 ( ν = ) E g.s. [MeV] Li hω= MeV hω= MeV hω=3 MeV hω=4 MeV hω=5 MeV Λ [MeV] Stetcu, Barrett +v.k., 6 a( ω ) + fits b( ω) Λ α + β ω + γ( ω) fits E g.s [MeV] preliminary hω [MeV] Bgs Λ= MeV Λ= MeV Λ=3 MeV Λ=4 MeV Λ=5 MeV MeV (exp) 6//9 v. Kolck, Intro to EFTs 4 works within ~3%

125 - many-body systems get complicated rapidly + (continuing) focus on simpler halo/cluster nuclei one or more loosely-bound nucleons around one or more cores ℵ m N E m E M p N c c (esp. near driplines) e.g. 4 He 5 He 6 He particle separation energy B B α α * p 3/ E n 8 MeV 8 MeV resonance at ~ MeV s bound state at E n ~ MeV E = B B α α * α core excitation energy MeV M 6//9 v. Kolck, Intro to EFTs c 5 n ℵ n p p n n

126 6//9 v. Kolck, Intro to EFTs 6

127 Q ~ ℵ M c halo EFT degrees of freedom: nucleons, cores symmetries: Lorentz, P, T expansion in: non-relativistic multipole simplest formulation: auxiliary fields for core + nucleon states e.g. 4 4 He scalar fieldϕ He + N Q M c s p p Q mn, Q = Q m π, 3 + spin - field s spin -/ field T + spin -3/ field T 6//9 v. Kolck, Intro to EFTs 7 m c 3

128 Nα Bertulani, Hammer + v.k. PSA, Arndt et al. 73 LO LO δ + [degrees] 3 NLO δ + [degrees] 8 4 NLO scatt length only E N [MeV] E N [MeV] ER.8 MeV Γ( ).55 MeV E R //9 v. Kolck, Intro to EFTs 8 E N [MeV] δ [degrees] 4 3 NNNLO LO, NLO, NNLO

129 δ [degrees] 8 5 LO NLO Afzal et.al. Higa, Hammer + v.k. 8 αα Bohr radius Extra fitting parameters P C Z em = P + 3 5k C 3.6 fm none k αα μ 9 3 E LAB [MeV] E = 9.7 ±.3 kev R Γ( E R ) = 5.57 ±.5 ev fitted with a and r = r 3k C More fine-tuning!!! fine-tuning of ( k ) a = O.8fm r =.3fm 6//9 E& M v. Kolck, Intro to EFTs 9 C in! a M ℵ 3 ~ c c r M fine-tuning of in

130 Coulomb interaction in higher waves: e.g. 4 4 p + He He + p cf. p + p p + p three-body bound states: e.g. ) 6 4 cf. ) He= b.s. 3 H = b.s. What next ( He+ n + n) ( ) C = b.s. He + He + He ( p + n + n) Bertulani, Higa + v.k., in preparation Kong + Ravndal 99 Hammer + v.k., in progress Bedaque, Hammer + v.k. 99 reactions: e.g. 7 p + Be cf. p + n d 8 B+ γ + γ Chen et al. 6//9 v. Kolck, Intro to EFTs 3

131 Conclusion EFT the framework to describe nuclei within the SM is consistent with symmetries incorporates hadronic physics has controlled expansion many successes so far, but still much to do grow to larger nuclei! new, systematic approach to physics near d r i p lines? 6//9 v. Kolck, Intro to EFTs 3