INTRODUCTION TO NUCLEAR EFFECTIVE FIELD THEORIES

Transcription

1 INTRODUCTION TO NUCLEAR EFFECTIVE FIELD THEORIES Ubirajara U. van Kolck Institut de Physique Nucléaire d Orsay and University of Arizona c Supported in part by CNRS and US DOE 1

2 Simplicity, like everything else, must be explained. S. Weinberg

3

4 References: U. van Kolck, Effective Field Theories of Loosely Bound Nuclei, in The Euroschool on Exotic Beams, Vol. IV C. Scheidenberger and M. Pfützer (eds.), Springer, Berlin Heidelberg (014) Lect. Notes Phys. 879 (014) 13 U. van Kolck, Les Houches Lectures on Effective Field Theories for Nuclear and (some) Atomic Physics, in Proceedings of the 017 Les Houches Summer School on EFTs in Particle Physics and Cosmology, S. Davidson et al. (eds.), Oxford University Press (to appear) arxiv: (nucl-th) 4

5 O u t l i n e Effective Field Theories Introduction What is effective Summary EFT for Short-Range Forces Two-Body System Three-Body System More-Body Systems Long-range Forces 5

6 Wanted Dead or Alive FORMULATION OF NUCLEAR PHYSICS CONSISTENT WITH STANDARD MODEL (SM) OF PARTICLE PHYSICS Reward understanding emergence of complexity at the most fundamental level: nucleus made out of quarks and gluons interacting strongly (QCD), yet exhibiting many regularities use of nuclei as laboratories for physics beyond the SM Beware coupling constants not small: not an easy problem! 6

7 Not an easy problem 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 they know what the problem is. M. L. Goldberger Midwestern Conference on Theoretical Physics, Purdue University,

8 Nuclear Physics The canons of tradition I II III IV Nuclei are essentially made out of non-relativistic nucleons in two isospin states (protons and neutrons) The interaction potential is mostly two-body, but there is evidence for smaller three-body forces Isospin is a good symmetry, except for the Coulomb interaction, breaking in two-nucleon scattering lengths, and smaller effects External probes (e.g. photons) interact mainly with each nucleon, but there is evidence for smaller two-nucleon currents but WHY? 8

9 Quantum Chromodynamics On the road to infrared slavery 1 Up, down quarks are relatively light,, and thus relativistic m, 5 MeV ud The interaction is a multi-gluon, and thus a multi-quark, process e.g. 3 Isospin symmetry is not obvious: ε = m m d d m + m u u 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? 9

10 Strongly interacting particles (hadrons) (many observed states not shown!) Mesons Baryons S =1 S = 0 S = 3 S =1 QCD Scale m π Exception: pion 140 MeV/c we ll return to it! M QCD M QCD 1000 MeV/ c = 1 GeV/ c 10

11 B A Q M nucc 10 MeV m M N QCD Nuclear scales r A 13 1 fm Q M c nuc (electron scattering) Q M nuc c 100 MeV/c 11

12 Multi-scale problems H atom H p α c p = 1 + α; ; α e 1 1 m e r m e c m e cr 4π c 137 r R α c E( R) p m e R R R ( ) de R 0 dr = R = αmec Three scales mec = pc 0.5 MeV c α me = 3.6 kev p 1 E α mec = 13.6 ev m e α α (from now on, units such that = 1, c = 1 ) 1

13 However no obvious small coupling in nuclear forces. 1 PDG QCD fine-structure constant Needed: method that does not rely on small couplings M QCD EFFECTIVE FIELD THEORY 13

14 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,

15 Ingredients Relevant degrees of freedom 15

16 Seurat, La Parade (detail) 16

17 Ingredients Relevant degrees of freedom choose the coordinates that fit the problem All possible interactions 17

18 Example: Earth-moon-satellite system Rm 1.7 Mm d 384 Mm RE 6.4 Mm -body forces +3-body forces 3-body force change in resolution 18 Wikipedia

19 Ingredients Relevant degrees of freedom choose the coordinates that fit the problem All possible interactions Symmetries what is not forbidden is compulsory 19

20 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 1 δαij 1 amenable to perturbation theory 0

21 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 1

22 t Hooft 79 After scales have been identified, the remaining, dimensionless parameters are ( 1) 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

23 h A classical example: the flat Earth light object near surface of a large body E mh g E und mgr d.o.f.: mass sym: eff m ( h,,y) = ( h) V x V eff i g Veff ( h) m gih const mg h i 0 g h = = = parameters (neglecting quantum corrections ) R M naturalness: V und ( h) = GMm + 1 g = ( 1) = ( 1) g i + 1 = i R i+ 1 mgi h E h i mgih Eund R 1 R + h h R 1 1 i GM = m R i= 0 R g h i g i ( ) + = 1 1 i g R i itself the first term in a low-energy EFT of general relativity 3

24 Going a bit deeper A short path to quantum mechanics a P= PA= + A + A + A b sum over all paths RULE b Ai exp i dt ( q( t)) a each path contributes a phase given by the classical* action Path Integral Feynman 48 ( ) A = Dq exp i dt ( q( t)) dq ( t i ) i * Not strictly true when interactions involve time derivatives (Salam + Strahdee 68) classical path ( dt q t ) δ ( ( )) = 0 4

25 q q( t i + n) q( t i + n) EFFECTIVE THEORY q( t i ) tit i M t j ( q( t )) i 1 Λ ( t i+ n t i+ n q t i ( ) 1 Q t j + 1 t j + scale of fine-structure of dynamics scale of variation of long-range dynamics coarse-graining scale (cutoff) 1 dq + ( t t ) i + dt t dq + ( t ti ) ) dt t i t i 5

26 More generally, ( ) und A = Dq exp i dt ( q) = Dq δ q f ( q) ( ) ( ) EFT Dq exp i dt ( q) Λ i ( ) ( ) dqt ( ) δ qt ( ) f qt ( ) i i i EFT d d q q = cd+ n( M, Λ) O d+ n q, d dn, = 0 dt ( ) n Naturalness c c 0 d + n d + n M e.g. 4 dq VEFT ( q ) = cq 0 + cq + dt Observables ~ expansion in Q M 6

27 All information is in the S matrix k = k k elastic scattering (for simplicity) (conservation of energy) θ : given by certain probability amplitude k the scattering amplitude T( k, θ) T ( k) P( cosθ) Tl 1 ( κ + iκ ) = 0 r i angular momentum 1. Parametrization by phase shifts:. Low-energy expansion: 3. Poles: T κ r = 0 κ 0 i k = l l= 0 bound states resonances l Heisenberg 43 (center-of-mass frame) l θ partial-wave amplitude k k ( ) ( iδ k ) T k effective range E = B< 0 E = E iγ R Legendre polynomial = 4π i exp l( ) 1 µ k ( k) = µ 1 r + k P + k + ik 1 l l 4 l+ 1 l l π k al 4 scattering length shape parameter R 7

28 characteristic external momentum ( ) Q Q Q T T ( Q)~ N( M ) F, ; c ( Λ) ν= ν i M m Λ = { } ν ν normalization min dt 0 dλ = invariance ) renormalization-group ν non-analytic, from the solution of a dynamical equation (e.g. Schrödinger eq.) low-energy constants or Wilson PCKLJS coefficients ν =ν ( d, n, power counting For k ~ m, truncate consistently with RG invariance so as to allow systematic improvement (perturbation theory): T ( ) Q Q T ν 1, = + M Λ Λ T ( ν ) dt dλ ( ν ) Q = Λ Warning: renormalization NOT optional Otherwise, just a model since dependent on regularization choice 8

29 second quantization : qt () ψ(,), rt ψ (,) rt dt 3 dt d r d, dt t r + Lorentz invariance representation of SO(3,1) 4 d x x µ EFFECTIVE FIELD THEORIES Euler + Heisenberg 36 Weinberg Wilson, early 70s 9

30 ( 4 { }) free int A= DψDψ exp i d x ( ψ) + ( ψ) = DψDψ 1 + i d x int ( ψ) + i d x int ( ψ) + exp i d x free( ψ) ( ) int momentum space = λ ψψ 4 ( ) = iλ (skip many steps ) = i p m + iε p1 p ' p ' 1 + l p p1 p l = = dl 4 i i iλ iλ ( π) ( p + l) m + iε ( p l) m + iε 4 1 but divergent from high-momentum region needs a cutoff to separate high and low momenta

31 Two possibilities: EFFECTIVE FIELD THEORIES Euler + Heisenberg 36 Weinberg Wilson, early 70s know and can solve underlying theory -- get ci s in terms of parameters in und know but cannot solve, or do not know, underlying theory -- invoke Weinberg s folk 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, but no known counterexamples Z E4 Ball + Thorne 94 31

32 Bira s EFT Recipe 1. 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 controlled expansion in ( 1) what is not forbidden is mandatory! 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 3

33 Nonrenormalizable theories are as renormalizable as renormalizable theories. S. Weinberg 33

34 A modern view of renormalization 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.),

35 Time for a paradigm change, perhaps? 35

36 Halo EFT Pionless EFT Chiral EFT? nuclear physics atomic physics condensed-matter physics and beyond molecular physics NRQED QCD-lite+NRQCD ( or 3 flavors) QCD (6 flavors) (SUSY) GUT?? E(GeV) 0 QED Electroweak Th + higher-dim ops General Relativity + higher-curvature terms Fermi Th The world as an onion r(fm) 36

37 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 EFT is not just any field theory with effective degrees of freedom Symmetries and renormalization essential for model independence Stay tuned: next, how we can describe low-energy nuclear physics systematically 37

38 INTRODUCTION TO NUCLEAR EFFECTIVE FIELD THEORIES c U. van Kolck Institut de Physique Nucléaire d Orsay and University of Arizona Supported in part by CNRS and US DOE 38

39 O u t l i n e Effective Field Theories EFT for Short-Range Forces Scales Degrees of freedom Symmetries Action Two-Body System Three-Body System More-Body Systems Long-range Forces 39

40 > 1990 : Very Brief History of Nuclear EFTs Weinberg generalizes Chiral Perturbation Theory to nuclei --- Chiral EFT. He identifies infrared enhancements in multi-nucleon amplitudes and suggests: i) the potential be calculated in the usual ChPT expansion; ii) amplitudes be calculated from the exact solution of the Schrödinger equation with truncated potentials > 199 : Others carry out this procedure obtaining good description of few-body data with only limited cutoff variation > 1996 : Weinberg s prescription shown to fail to produce renormalizable amplitudes; problem ignored by most nuclear physicists > 1998: Pionless EFT developed to resolve these issues; it additionally finds great success for light nuclei > 1998: Pionless EFT extended with perturbative pions and shown to work more or less as well as Pionless EFT > 005: Weinberg s power counting modified so that nonperturbative pions today: produce renormalizable amplitudes Phenomenological problems with Weinberg s prescription mounting in the case of heavier nuclei 40

41 B 3 = 8.48 MeV 5 B ev = ( 13 1 A ) = κ + η + liquid-drop formula v. Weizsäcker 36 41

42 Finite-range two-body potential R V () r r λ f( r R) = + µ R ( r R) n nucleons: pion exchange µ = mn 940 MeV n = 3, R= 1 mπ 1.4 fm λ = mπ MNN 1 f ( r R) = exp( r R) 4 He atoms: two-photon exchange µ = m 4 He n = 6, R = lvdw 5.4 A λ = 1 f ( r R) = 1 4

43 A measure of (inverse) size: the binding momentum Q mb A A A B Q A 1 = µ B right position of two-body pole A 1 i= 1 Qi m all particles contribute equally to energy nucleons: 4 He atoms: unitarity limit ( ) 1 Q R 3 ( Q R) 1 3 ( Q R) 1 ( Q R) 1 10 ( Q ) 1 R 3 B 1 µ B R ψ ( r R) µ Br r λ f( r R) V () r = + µ R ( r R) n e r 43

44 1 µ B ψ ( r R) r e µ Br r EFT B R V () r λ f( r R) = + µ R ( r R) n B 1 µ B ψ () r V() r C e r µ B r r δ () r 4π r = + derivatives of the delta function 0 multipole expansion 44

45 Halo EFT Pionless EFT Chiral EFT? nuclear physics atomic physics condensed-matter physics and beyond molecular physics NRQED QCD-lite+NRQCD ( or 3 flavors) QCD (6 flavors) (SUSY) GUT?? E(GeV) 0 QED Electroweak Th + higher-dim ops General Relativity + higher-curvature terms Fermi Th The world as an onion r(fm) 45

46 On blackboard 46

47 INTRODUCTION TO NUCLEAR EFFECTIVE FIELD THEORIES c U. van Kolck Institut de Physique Nucléaire d Orsay and University of Arizona Supported in part by CNRS and US DOE 47

48 O u t l i n e Effective Field Theories EFT for Short-Range Forces Two-Body System Amplitude Connection to ERE, potentials Potentials and fine tuning Non-trivial fixed point, scale invariance Three-Body System More-Body Systems Long-range Forces 48

49 On blackboard 49

50 LO N LO 1 S 0 Nucleon-nucleon phase shifts Pionless EFT N 4 LO N LO resum Nijmegen PSA fit predict (0) C 00 (1) C 0 B d = B = d a r 1 1 Chen, Rupak + Savage 99 3 S 1 = 5.4 fm (exp) = 1.75 fm (exp) 1.91 MeV (NLO). MeV (exp) fit predict (0) C 01 (1) C 1 a r 0 0 = = 0.0 fm (exp).78 fm (exp) B = 0.09 MeV (NLO) d * N LO NLO LO Nijmegen PSA

51 INTRODUCTION TO NUCLEAR EFFECTIVE FIELD THEORIES c U. van Kolck Institut de Physique Nucléaire d Orsay and University of Arizona Supported in part by CNRS and US DOE 51

52 Effective Field Theories O u t l i n e EFT for Short-Range Forces Two-Body System Three-Body System Auxiliary Field Amplitude Bound states and correlations Limit cycle, discrete scale invariance More-Body Systems Long-range Forces 5

53 53 Issue: at what order a three-body force? 0 (4 ) ( ) m i D π Λ First, LO two-body force: 0 4 Q C m m π Q m m C Q π π lo mq m C Q M π π if there is a shallow two-body state need to resum LO two-body force hi D M NDA: but with fine tuning?

54 Auxiliary field Kaplan 97 v.k. 99 T = full propagator of auxiliary dimeron d with mass m removed by choice of heavy field LO LEC NLO LEC + + d = ψ i + + ψ + d + σ i + + d t m t 4m π d ψψ+ ψψ d h0 d dψψ+ m 54

55 i i π m i π m 4 π i dl m ( π ) 0 0 ( l + p) 0 0 ( l + p) l + p + iε l + p + iε mn mn i k = = θ 1Λ + ik + θ 1 + Λ LO = + + ( ) 1 1 = = i + θ1λ + ik + M hi = 1 a R NLO ( ) i ik = = + σ + θ R 1 m k Λ m σ R = mr renormalization simpler than before because momentum energy 55

56 k 4 +, p B E m k m E, p LO T = k, m p E = km B, p 4 3k B 4m 4π ~ Q 3 Q 4π 1 4π ~ ~ Q 4π Q Q Mlo = + T 3 T Λ 3 dl TK 3 OPE 3 = KOPE + λ ( ) 3 π D 0 Skorniakov--Ter-Martirosian 57 56

57 Only half-off-shell amplitude enters: T E = T ( p,) l υ (, l p) Λ 3 dl 3 3 3( p, p) = υ3( p, p) λ 3 π 0 1 a l υ ( p, p) = 3 p m ( ) me 8π me p p p p k bosons S = 0 λ =1 nucleons S = 3 S =1 λ = 1 two equations that decouple in the UV λ =1 & λ = 1 57

58 Simplest case: S wave Λ λ 3,0 υ3,0 3,0(, k) = υ 3,0(, k) π 0 1 a 3l Λ λ υ3,0 p l T3, 0 = υ 3,0( p, k) dl l π 0 1 a l T ( p,) l (, l k) T p p dl l + 4 me (,) (, l k) υ k me = p k p + k + p k me 4π 3,0(, ) ln p + k p k p p k > 1 a Λ 4λ dl p + p l + l ( p k) = ln T,0( l k) 3π p p pl + l As before, first look at UV properties: T me 3,0 3 T ( p k) p + 8λ sin( sπ 6) 3s cos( sπ ) ( s 1) Ansatz = 1 Danilov 63 3,0 58

59 Danilov 63 nucleons S = 3 λ = 1 s.17 ( x) T ( p a,0) Nd,3/ Bedaque + v.k. 97 Bedaque, Hammer + v.k. 98 different regulators = p a T Nd, 3/( pa ~ 1, 0) Λ 0 TNd, ( p a 1, 0) 3/ 3.17 p 59

60 nucleons S = 3 postdict Bedaque + v.k. 97 Bedaque, Hammer + v.k. 98 v.oers + Seagrave 67 NLO Dilg et al. 71 LO a Nd,3/ = fm = 6.33 ± 0.10 fm a Nd,3/ = 6.35 ± 0.0 fm (exp) QED-like precision! 60

61 Danilov 63 bosons 0 S = λ =1 s is0 = ± s Bedaque, Hammer + v.k different cutoffs T ( pa ~ 1, 0) ψ d Λ 0 T ψ d p ( p a 1, 0) cos s0 ln + δ Λ 61

62 Thomas collapse Thomas 35 Kirscher, Barnea, Gazit, Pederiva + v.k. 15 6

63 T 3 = π ~ Q ~ 4π M lo T 3 = T 3 T Λ 3 dl TK 3 OPE 3 = KOPE + λ ( ) 3 π D 0 Λ 3 dl TK 3 TBF + KTBF + λ ( ) 3 π D 0 Bedaque, Hammer + v.k

64 Repeat with υ ( p, p) υ ( p, p) + h ( Λ) υ ( p, k) υ ( p, k) + h ( Λ) 3,0 3,0 0 h 0 ( ) 8 H ( Λ Λ π ) Λ T p k > 1 a UV properties: Λ 4 ( ) ln ( ) ( l k) 3π p p p l + l Λ dl p + pl+ l pl 3,0 p k = H T Λ 3, 0 only important for p Λ T p ( p a,0) cos Λ s ln + δ ( H ( Λ) ) Λ 1 3,0 0 δ ( H( Λ) ) s ln ( Λ Λ ) 0 physical, dimensionful parameter 64

65 Bedaque, Hammer + v.k Fitting to a3 different cutoffs T ( pa ~ 1, 0) ψ d Λ Renormalized! p p 0 Tψ Td ( p( a 1, 0) 1, 0) cos cos s0 ln + δ d p ψ a s0 ln ΛΛ* 65

66 Thomas collapse Fitting to B : Λ 3 Thomas 35 outside EFT physical triton Kirscher, Barnea, Gazit, Pederiva + v.k

67 Correlations Phillips line nucleon-deuteron scattering length potential models LO varying Λ * (Bedaque, Hammer + v.k. 00) (Phillips 68) expt triton binding energy

68 nucleons S = 1 Bedaque, Hammer + v.k Hammer + Mehen 01 Bedaque et al. 03 v.oers + Seagrave 67 Kievsky et al. 96 NLO LO Dilg et al. 71 fit a Nd,1/ = 0.65 fm (exp) postdict B t = MeV = 8.8 ± 0.5 MeV = 8.48 MeV (expt) B t 68

69 Studying UV properties in more detail: Bedaque, Hammer + v.k Fitting to B : Λ 3 H ( Λ) Λ D ( Λ) mc ( Λ) 0 0 ( Λ Λ + s0 ) ( Λ Λ s ) sin ln( ) arctan(1 ) sin ln( ) arctan(1 ) 0 periodicity Λ Λe π n s 0 RG limit cycle! bosons fermions with more than two states D 0 1 M 4 lo LO not just the effective-range expansion!

70 A = Discrete Scale Invariance Unitarity limit Mehen, Stewart + Wise 00 König, Grießhammer, Hammer + v.k. 16 König 16 v.k. 17 regulator-dependent number C 0 ( Λ) 1 = θ Λ 1 (0) mb = 0 scale invariance x t m α x α t m Λ α 1 Λ ψ α 3 ψ S S (0) (0) EFT EFT (includes Wigner spin-isospin symmetry) SU (4) W Mehen, Stewart + Wise 00

71 A = 3 D ( Λ) 1 Λ 1 ( s0 Λ Λ + s0 ) ( s0 Λ Λ s0 ) sin ln( ) arctan (0) sin ln( ) arctan dimensionful parameter (dimensional limit cycle! transmutation) discrete scale invariance x t m α x α n n t m Λ 1 α n ψ α Λ 3 n ψ S S (0) (0) EFT EFT α exp( ) (.7) n n = nπ s = n =,, 1, 0,1,, 0 ground state Efimov states mb = k exp( nπ s ) (0) 3n 0 k = Λ Efimov 71 ln ln β, modπ s... 0 β (includes Wigner spin-isospin symmetry) SU (4) W < M hi Braaten + Hammer 06 Bedaque, Hammer + v.k. 00

72 Braaten + Hammer 03 Science 348 (015) He atoms nucleons ( mb ) 14 3 Λ Borromean 3 1+ = two-body threshold ( mb Λ ) ( aλ ) 14 1 decreasing two-body attraction ( Λ ) 1 s ign( a ) a bound 3 state virtual 1+ state Vaghani, Rupak, Higa + v.k. 18 triton & helion

73 INTRODUCTION TO NUCLEAR EFFECTIVE FIELD THEORIES c U. van Kolck Institut de Physique Nucléaire d Orsay and University of Arizona Supported in part by CNRS and US DOE 73

74 O u t l i n e Effective Field Theories EFT for Short-Range Forces Two-Body System Three-Body System More-Body Systems Four bodies Beyond Long-range Forces 74

75 A = 4 LO No 4BF Unitary bosons Hammer + Platter 07 ground state first excited state Efimov descendants

76 Correlations nucleons Tjon line potential models (Tjon 74) exp Λ varying * (Hammer, Meißner + Platter 04) LO 76

77 Nucleons at unitarity König, Grießhammer, Hammer + v.k. 16 König 17 v.k. 17 unitarity LO varying Λ* incomplete unitarity NLO additional expansion in ( ) Qa 1 Nucleons perturbatively close to unitarity

78 Bosons NLO (+3) Bazak, Eliyahu + v.k. 16 Bazak, Kirscher, König, Pavón-Valderrama, Barnea + v.k He atoms LO PCKLJS NLO (+3+4) 4BF! No more-body forces up to NLO LMM LMM 78

79 Summary: Expansion parameter Q M hi LO: no-derivative two- and three-body interactions C 0 0 Around unitarity: a SINGLE physical parameter Λ*, D NLO: two-derivative two- and no-derivative four-body interactions C, E 0 Etc. 79

80 discrete scale invariance Unitary bosons doubling Hammer + Platter 06 von Stecher Gattobigio, Kievsky + Viviani 11 1 N N + 1 Efimov state Efimov descendants single scale LO Ground states B (0) N ( Λ ) N = κ N B 3 ( Λ ) 3 universal numbers κ κ3 1 κ κ N 5? Towers upon towers!

81 (Variational and Diffusion Monte Carlo) Gandolfi, Carlson, Vitiello + vk 17 LO Λ Q Λ3 Λ = saturation! cf. Piatecki + Krauth 14 Comparin + Krauth 16 κ 3 N N ( N ) Bazak, Eliyahu + v.k. 16 ( ) κn = κ 1+ ηn + N κ = 90 ± 10 η = 1.7 ±

82 A 4 Nucleons around unitarity discrete scale invariance doubling? single scale Ground states κ (0) ( ) ( ) 0 B A Λ B3 Λ κ 3 1 = κ LO A A 3 κ4 3.5 same as for bosons κ should grow slower A 5? than bosons same saturation mechanism as for bosons?

83 Larger nuclei? Adapt ab initio many-body methods 4 He excited state + 6 Li at LO with no-core shell model Stetcu, Barrett + v.k O at LO with auxiliary-field diffusion Monte Carlo Contessi et al 17 4 He at LO + resummed NLO with resonating group and stochastic variational Kirscher et al. 09; Lensky, Birse + Wallet 15 4 He, 16 O, 40 Ca at LO + resummed NLO Bansal et al many applications to exotic hadronic states, cold atoms, 83

84 B 3 = 8.48 MeV 5 B ev = ( 13 1 A ) = κ + η + liquid-drop formula v. Weizsäcker 36 84

85 INTRODUCTION TO NUCLEAR EFFECTIVE FIELD THEORIES c U. van Kolck Institut de Physique Nucléaire d Orsay and University of Arizona Supported in part by CNRS and US DOE 85

86 O u t l i n Effective Field Theories EFT for Short-Range Forces Two-Body System Three-Body System More-Body Systems Long-range Forces e 86

87 Next time 87

88 Conclusion Systems near unitarity can be described model-independently by Pionless EFT properties given by essentially one parameter Λ details obtained in perturbation theory cf. parallel results from a model perspective: Kievsky et al. How far can we go for nuclei? more nucleons nuclear matter higher orders Gezerlis et al., in progress Lyu et al., in progress Bazak et al., in progress