INTRODUCTION TO EFFECTIVE FIELD THEORIES OF QCD

Transcription

1 INTRODUCTION TO EFFECTIVE FIELD THEORIES OF QCD Ubirajara U. van Kolck Institut de Physique Nucléaire d Orsay and University of Arizona Supported in part by CNRS, Université Paris Sud, and US DOE 1

2 Outline Effective Field Theories Introduction What is Effective Example: NRQED Summary QCD at Low Energies Towards Nuclear Structure

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, 00), nucl-th/ D.B. Kaplan, Effective field theories, Lectures at 7th Summer School in Nuclear Physics Symmetries, Seattle, WA, Jun 1995, nucl-th/

4 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! 4

5 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,

6 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 a sizable breaking in two-nucleon scattering lengths and other, smaller effects External probes (e.g. photons) interact mainly with each nucleon, but there is evidence for smaller two-nucleon currents but WHY? 6

7 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? 7

8 Strongly interacting particles (hadrons) (many observed states not shown!) Mesons Baryons S = 0 S =1 S = 3 S =1 QCD Scale Exception: pion m c M π 140 MeV/ QCD we ll return to it! M QCD 1000 MeV/ c = 1 GeV/ c 8

9 B A Q M m M c nuc N QCD 10 MeV Nuclear scales r 1 fm 13 A Q M c nuc (electron scattering) Q M c100 MeV/ c nuc 9

10 H atom H Multi-scale problems p αc p = 1 + α; ; α e 1 1 m e r m e c m e cr 4π c 137 r R αc E( R) p m e R R R ( ) de R 0 dr = R = αmec Three scales mec = pc 0.5 MeV α mec = 3.6 kev p 1 E α mec = 13.6 ev m e α α (from now on, units such that = 1, c = 1 ) 10

11 However no obvious small coupling in nuclear forces. 1 PDG, 005 QCD fine-structure constant Needed: method that does not rely on small couplings M QCD EFFECTIVE FIELD THEORY 11

12 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, 197 1

13 Ingredients Relevant degrees of freedom 13

14 Seurat, La Parade (detail) 14

15 Ingredients Relevant degrees of freedom choose the coordinates that fit the problem All possible interactions 15

16 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 16 Wikipedia

17 Ingredients Relevant degrees of freedom choose the coordinates that fit the problem All possible interactions Symmetries what is not forbidden is compulsory 17

18 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 18

19 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 19

20 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 0

21 h A classical example: the flat Earth light object near surface of a large body Emh g E und mgr d.o.f.: mass sym: eff m ( h,, ) = ( h) V xy V eff eff i ( ) { h = m const } ih = + m h+ ηh + V g g i= 0 parameters (neglecting quantum corrections ) R M naturalness: Vund ( 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 ( ) = i g R i itself the first term in a low-energy EFT of general relativity 1

22 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 classical path ( dt q t ) δ ( ( )) = 0

23 q q( t i + n) q( t i + n) EFFECTIVE THEORY q( t i ) tit i M und 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 3

24 More generally, ( ) und A = Dq exp i dt ( q) = Dqδ q f ( q) ( ) ( ) EFT Dq exp i dt ( q) Λ i ( ) ( ) dqt ( ) δ qt ( ) f qt ( ) i i i d d q q = c ( M, Λ) O q, ( ) EFT d + n d + n 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 4

25 All information is in the S matrix k = k k elastic scattering (for simplicity) (conservation of energy) k k (center-of-mass frame) θ k k θ : given by certain probability amplitude the scattering amplitude l= 0 ( k) ( θ) T( k, θ) = Tl Pl cos angular momentum partial-wave amplitude Legendre polynomial parametrized by phase shift δ l ( k ) Tl 1 ( κ + iκ ) = 0 r i κ = r κ i 0 0 bound states resonances E = B< E = E iγ R 0 R 5

26 ν ( ) Q Q Q T = T ( Q)~ N( M ) c ν, i( Λ) Fν, i ; ν= ν i M m Λ T Λ = 0 characteristic external momentum normalization ν =ν ( d, n, ) min power counting non-analytic, from the solution of a dynamical equation (e.g. Schrödinger eq.) For k ~ m, truncate consistently with RG invariance so as to allow systematic improvement (perturbation theory): T Q M Q Λ ( ν ) = T 1 +, Λ T ( ν ) ( ν ) T Q Λ = Λ 6

27 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 7

28 ( 4 { }) free int A= DψDψ exp i d x ( ψ) + ( ψ) { } 1 ( ) ( ) exp ( ( ) ) int int free = DψDψ + i d x ψ + i d x ψ + i d x ψ momentum space int = p1 λ ψψ 4 ( ) p ' p ' 1 + l p p1 p l = = = iλ (skip many steps ) dl 4 i i 4 1 = i + p m iε iλ iλ ( π) ( p + l) m + iε ( p l) m + iε but divergent from high-momentum region needs a cutoff to separate high and low momenta N.B. This does NOT require relativity, as we ll see explicitly in lecture 8

29 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 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 9

30 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 30

31 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.),

32 Time for a paradigm change, perhaps? 3

33 0.1 1 Chiral EFT? nuclear physics atomic physics condensed-matter physics and beyond molecular physics NRQED ? QCD ( or 3 flavors) QCD (6 flavors) (SUSY) GUT? Electroweak Th + higher-dim ops E(GeV) QED General Relativity + higher-curvature terms 10 3 Fermi Th r(fm) The world as an onion 33

34 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 1 und = ψ ( id M ) ψ Fµν F 4 p = µ p i M + iε = ie γ µ p ν µ = µν i p η µν + iε Dµ = µ ieaµ Fµν = µ Aν ν Aµ interactions e = 4πα perturbation theory 1 3 How do E&M bound states arise? 34

35 Q M p+ q i = = p + q M + iε p + + M + iε 0 0 i ( p γ p γ + q + M) 0 0 ( q ) ( p q) 0 0 i ( p γ + M p γ + q) = p q p q + q p q q + iε 0 i ( 1+ γ ) p q = ( Q) = + 0 q + iε 0 q = q = ( Q) 0 0 Q 1± γ p = p + M = M + P± PP ± ± = P±, PP ± = 0 M projector onto energy states ± heavy-fermion formalism Georgi 90 imt Ψ e Pψ ψ = ( P + P ) ψ = e ( Ψ +Ψ ) imt ± ± + + particles: annihilates creates antiparticles: creates annihilates 35

36 1 und =Ψ id0 iγ D iγ D ( id0 M ) F F µν + Ψ+ Ψ Ψ + +Ψ+ Ψ Ψ + Ψ µν 4 + other, heavy d.o.f.s anomalous magnetic moment =O(1) ( 4 ) und, δ ( ) Z = DA DΨ DΨ exp i d x ( Ψ Ψ, A) DΨ Ψ Ψ ( 4 ) EFT = DA DΨ exp i d x ( Ψ, A) 1 e jk 1 µν =ΨiD Ψ+ ΨD Ψ+ Ψσ Ψ ε F + Fµν F + M M 4 EFT 0 i ijk non-relativistic expansion eκ jk + ΨσiΨ εijk F + M Pauli term complete square, do Gaussian integral most general Lag with Y, A invariant under U(1) gauge, parity, time-reversal, and Lorentz transformations 36

37 EFT p p p p 1 a ( ) b = F F + F + ( F ) + 4 ν µ = = i p η µν + iε µν 4 4 M F µν M F µν µν µν µν p 3 p 4 ρ σ 1 ( ( ) ) e µναβ +Ψiv DΨ+ Ψ v D D Ψ+ ( 1+ κ) Ψvα Sβ Ψ ε Fµν + M M v ( 1, 0) i µ = ie v 1 v p + ( ( ) ) µ σ S 0, p v p + + i ε M ν µ e { i ( p p' ) 1 ( ) ν α β = κ ε S q } e µ µναβv = i ( ηµν vµ vν ) M M q µ (0) (1) γ0 γ0 + ΨΨ ΨΨ + Ψ M M S Ψ Ψ S Ψ + i = + M ν µ (0) (1) ( γ0 γ0 S1 S) p p 1 Euler + Heisenberg 36 i = M { a ηµρηνσ p1 pp 3 p4 b[ ]} etc. 37

38 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! 38

39 Back to atomic bound states: the NRQED perspective 0 ( p', p 0 ') ( p', p' ) T ψψ = ( p 0, p ) CoM ( p 0, p) p p p' = Q p' 0 0 frame ( ) Q = M higher powers of Q M ie ie ie 4πα = = 0 + p p iε Q ( p p' ) iε ( 0 ' ) ( p p' ) + iε ( p p' ) α V ( r) = r 39

40 ( p 0 + l 0, ( p+ l ) ( l 0, ) l p + l p' + l 0 0 ', ) 0 l = e 4 4 dl ( π ) ( l + p) 0 0 ( l p ') l + p + iε l + p' + iε M M l + l + iε l l + iε ( p p ' ) ( p p' ) Q M 3 4 = ie 4 3 dl ( π ) 0 ( l + p) 0 ( l p ') l p + iε l p' + iε M M l l + iε l iε ( 0 0 p p' ) ( p p' + ) Q Q + 3 e 4 Q 3 ( 4π ) α 4πα QQQ Q 4π Q 1 just as expected 40

41 ( p 0 + l 0, ( p+ l 4 3 ) Q ( l 0, ) l 1 Q p ( p+ l ) 0 l 0, M Q ) 0 l 3 4 = e 4 = ie + 4 dl ( π ) 0 0 ( l + p) 0 0 ( l + p) l + p + iε l + p + iε M M p p + l + iε l l + iε 4 3 dl ( ' l ) ( p p ' + ) 1 1 ( π ) ( ) M M 1 ( ) 0 0 ( l + p) p p ' p p' + l + iε M ( l + p) p iε 0 l + p + p l + i 3 Q M 1 1 α 4πα 4πα M 1 + α + 4π Q Q Q 4π Q Q Q 4π ( 4πα ) ε infrared enhancement! 41

42 dl Q 1 M = πδ ( l ) 0 l 0 l l + iε l + + M M i ε 1 Q 0 dl + + time-ordered perturbation theory 1 Q 4

43 (0) T ψψ = + + bound state at e M e α + Q Q Q M 1 α Q Q αm Q E α M M (0) V ψψ (0) = V (0) ψψ + + = + (0) V ψψ V ψψ (0) T ψψ (0) V ψψ (0) V ψψ = e = Q Coulomb potential Lippmann-Schwinger eq. = Schrödinger eq. pˆ (0) (0) (0) (0) + V ψψ ψ = E ψ M known results 43

44 But more: (1) V ψψ α 4πα = = 4π Q (0) T ψψ (1) V ψψ (0) T ψψ (1) T ψψ (1) = V ψψ (0) T ψψ (1) V ψψ (1) V ψψ (0) T ψψ E = E + ψ Vψψ ψ (1) (0) (0) (1) (0) α 4π E (0) = 44

45 () V ψψ = + + Q 4πα = M Q () T ψψ = () (1) (0) () ( 0) E = E + ψ Vψψ ψ + Q M (0) = E piece µ µ ψ δ ψ = µ µ ψ ( ) ( ) ( ) ( ) 3 (0)* (3) (0) (0) 1 dr r r r 1 0 magnetic interaction 45

46 N O T E starting at (3) T ψψ, sufficiently many derivatives appear at vertices so that loops bring positive powers of compensated by γ () i 0 ( Λ) Λ, which need to be and higher-order counterterms (3) T ψψ = etc. α Q 4πα = 4π M Q α M ln Λ α M ( i) γ ( ln Λ + constant) 0 renormalization to be determined by matching to QED (and/or from data) 46

47 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 = 1+ (0) ψ = + + (0) ψ Pachucki, Jentschura + Yerokhin 04 u = m1 C( gs) 1 Most precise determination of electron mass 47 (expt)(th)

48 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, and thus for the scattering amplitude --- 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 48