INTRODUCTION TO CHIRAL EFFECTIVE FIELD THEORY

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

2 Outline Effective Field Theories at Low Energies and chiral symmetry Chiral EFT Nuclear bound states and reactions Example: nuclear EDMs Renormalization issues Outlook & summary

3 Reference: 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 (04) Lect. Notes Phys. 879 (04) 3 3

4 0. Chiral EFT? nuclear physics atomic physics condensed-matter physics and beyond molecular physics NRQED ? -lite +NR ( or 3 flavors) (6 flavors) (SUSY) GUT? Electroweak Th + higher-dim ops E(GeV) QED General Relativity + higher-curvature terms 0 3 Fermi Th 0 0 r(fm) Peeling the onion 4

5 d.o.f.s EFT at a few GeV= underlying theory for nuclear physics l f + l = ν f u q = d leptons: quarks: a photon: A µ gluons: symmetries: SO(3,) global, U e() gauge, SU c(3) gauge G µ und 3 ( ) = lf i + eql A mf lf Fµν F 4 f = ( ) Tr µν + q i + eqa q + gg s q G G µν ( mu + md) qq ( mu md) qτ 3q mm u d + θ qiγ5q + m + m u d µν QED + -lite higher-dimension interactions: suppressed by larger masses unnaturally small T violation (strong CP problem) θ < 0 Ql = 0 0 Q q = τ 3 = 6 e.g. G F 9 M 5 W, Z

6 Focus on strong-interacting sector: four parameters ) m = m = 0, = 0, θ = 0 u d e single, dimensionless parameter 4 4 d x d x q( i gsg) q Tr µν = / + / G G invariant under scale transformations but in ( 4 ) Z DG Dq Dq exp i d x = scale invariance anomalously broken by dimensionful regulator coupling runs ( Q ) α GeV s ( dimensional transmutation ) µν chiral limit x q G λ λ 3 x q λg PDG 6

7 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 GeV Observation: pion mass m 40 MeV M breakdown of naturalness? NO! spontaneous breaking of chiral symmetry 7

8 Why is the pion special? ( ) ( ) Tr µν = q i / + gg/ q + q i / + gg/ q G G L s L R s R u q = d invariant under +γ q γ 5 q 5 ( ) q exp i q α τ SU() SU() SO(4) L( R) L( R) L( R) LR ( ) chiral symmetry L R µν m m N σ m m N + broken by vacuum down to ( α τ) q exp i q V α = α ± α RL / V A SU() isospin V SO(3) (axial transformations broken) 8

9 Chiral Limit V chiral circle σ two isospin axis not shown qiγτq 5 qq f pion decay constant (in chiral limit) piece invariant under + ε [function of on chiral circle] EFT = µ + 4 f µ 9

10 ) m 0 m, = 0, θ = 0 u d e ( ) Tr µν = q i / + gg s / q G Gµν + ( mu + md) qq + ( mu md) qτ 3q + v.k th component of SO(4) vector 3 rd component of SO(4) = ( γ τ ) P = ( qτq, qiγ q) S qi q qq 5, 5 vector break SO(4) SO(3) U() (explicit chiral-symmetry breaking) (isospin violation) 0

11 Chiral Limit two isospin axis not shown qiγτq 5 V qq f 9 MeV pion decay constant slightly-tilted chiral circle σ piece invariant under + ε [function of ] EFT = + piece in qq direction [function of explicitly] + isospin breaking µ Q ( m + m ) u u d ( m m ) d CHIRAL SYMMETRY WEAK PION INTERACTIONS

12 3) e 0, θ = 0 Two types of interactions: soft photons explicit d.o.f. in the EFT D = ieq A µ µ q µ Fµν = µ Aν ν Aµ hard photons integrated out of EFT und = + µν e qqqγµ q D ( ) qqqγνq 34 comp of antisymmetric tensor SO(4) εijkqiγ µ γτ 5 kq qiγ µ τ jq = qiγτ µ iq 0 breaks (and SO(3) in particular) U() F µ v.k. 93 = EFT soft photons + further isospin breaking e α 4

13 4) θ 0 mm = + qi q + m θ γ + u d 5 mu d 4 th component of SO(4) vector P = ( qτq, qiγ q) 5 T violation linked to isospin violation: in EFT, combination is + mm m + m u d ( mu md) P 3 θ P 4 u d Hockings, Mereghetti + v.k., 0 5) continue with higher-order operators, e.g. T-violating quark EDM and color-edm P-violating four-quark operators De Vries, Mereghetti, Timmermans + v.k., Kaplan + Savage 96 Zhu, Maekawa, Holstein, Musolf + v.k. 0 3

14 Nuclear physics scales ln Q ~ GeV ~00 MeV perturbative M ~ m, m, f, N ρ 4 hadronic th with chiral sym M f r m nuc ~, / NN,, His scales are His pride, Book of Job this lecture brute force (lattice),? Q M ~30 MeV ℵ ~ ann Q M nuc another time no small coupling expansion in 4

15 Qm M Chiral EFT d.o.f.s: nucleons, pions, deltas, Roper? ( ) + + p N = + = i ( ) n = 0 3 symmetries: Lorentz, P, T, chiral ( m mn ~ m, m N mn ~3 m) ++ + p = N = 0 n Non-linear realization of chiral symmetry Weinberg 68 Callan, Coleman, Wess + Zumino 69 (chiral) covariant derivatives pion chiral invariants D fermions see for example Weinberg s Quantum Theory of Fields, vol µ µ + f 4 f i µ µ τ Eµ E D f µ µ + S4's, P3 's, F34's (( u d) ) m = m + m M m u m + md = M 5

16 Open parentheses on parametrizations ϕ = 4 f hiral circle φ ϕ 4 f = ϕ i i ϕ i f f + isospin rotation φ φ+ iαv φ ϕ ϕ iα V ϕ D D + iα D µ µ V µ rotates 4 stereographic projection axial rotation φ φ αaϕ 4 ϕ ϕ + α φ 4 4 A αa f f 4f α nonlinear! D D α A D f µ µ µ rotates with α V i αa f analogous for µ Weinberg 68 isospin symmetry of covariant objects ensures chiral invariance 6 A

17 Σ τ ϕ4 + iτφ = f exp i fσ f sigma-model parametrization exponential parametrization Callan, Coleman, Wess + Zumino 69 Choice of fields does not affect observables: all parametrizations are equivalent i i Σ exp αl τ Σexp αr τ invariants constructed with Tr e.g. Σ µ Σ= iτ D e.g. ( µ ) µ Tr Σ µ Σ = 8 D µ Dµ less intuitive but easier to generalize to SU(3) L SU(3) R 7

18 chiral-breaking terms --- an example: common quark mass stereographic projection ( γ ) S = qi τq qq 5, exponential parametrization mu + m d qq qlm( x) qr qrm ( x) ql chiral symmetry broken explicitly just as in S + 4 th component of SO(4) vector = +, + 4f f 4f f M( x) LM( x) R spurion field 4 = + + f f 8 4 pion mass term Tr Σ + Σ invariant if Close parentheses M M mu + m d = + f pion mass term 8

19 Schematically, p f n D,, mr m N m ψψ + EFT = c{ npf,, } { np,, f } M MQ CD f f M f M calculated from : lattice, fitted to data = () isospin conserving NDA: naïve α dimensional = ε, isospin breaking 4 analysis ( ) = f f n+ p+ d + 0 = 0 chiral index chiral symmetry 9

20 = g A + + N i 0 τ ( 0) + N + N τσ N ( ) + 4f f 4f + h A + + [ i 0 ( m mn ) + ] + + ( N TS + Η.c. ) ( ) ( + ) f C N N C N (0) ( µ ) m f 4f () S + + ( ) T ( σ N ) + = N + τ ( ) + ( ) + mp mn τ3 3 τ+ N mn 4f f + + N b ( 0 ) b3( ) bm ib4εijkεabcστ k c ( ib)( jc ) N f + + g A + in σ N + Η.c. ( 0 ) ( + ) 4mN f τ h A + i TS N + ( ) 4mN f Η.c. ( 0) + d N + NN + + τσ N ( ) ( + ) f Form of pion interactions E () = + ( N N) 3 determined by chiral symmetry 0

21 A= 0, : chiral perturbation theory T E Q c ν ν ν Q Q F ν M m ν = A+ L+ V ν = A i i i min nucleon Weinberg 79 Gasser + Leutwyler 84 Gasser, Sainio + Švarč 87 Bernard, Kaiser + Meißner 90 Jenkins + Manohar 9 dense but short-ranged Q M # loops expansion in Q Q Q m m 4 f # vertices of type i N ρ, non-relativistic multipole pion loop long-ranged but sparse M 0.3 fm m.4 fm

22 Analogous to NRQED Weinberg 79 Gasser + Leutwyler 84 T = current algebra Weinberg 66 Gasser, Sainio + Švarč 87 Bernard, Kaiser + Meißner 90 Jenkins + Manohar 9 T N = etc. N.B. For 3 Q E ( m mn ) < M ( ) ( ) N N : resummation necessary Phillips + Pascalutsa 0 Long + v.k. 08 E m m = Q : Roper important Long + v.k. 09

23 NDA d d d arbitrary diagram 4 Q (4 ) Λ (4 ) d + d d momenta ~ cutoff = short-range physics 4 d Q Q + d d ( Λ) c ( Λ) d c ( Λ) c ( Λ ) Q + d c d c ( Λ) fermions more loops, vertices other interactions naïve dimensional analysis (NDA) (perturbative renormalization) d d number of fields in operator c dimension of operator i d = d = d c d naturalness c N (4 ) red = c D 4 i M c red i = d (4 ) ( Λ) d Λ ( M ) (4 ) M d Georgi + Manohar 86 (( ) # ) red g reduced underlying theory parameter insertions reduced coupling

24 Example = + ( mu + md ) qq + M + + = (4 ) 3 4 red ( mu md) = ( m ) u m d mu + m M d EFT m = + 4 red m ( m ) M = = (4 ) m M ( red ) ( ) u + d = mu + md red ( ) = ( ) ( ) m m m m M 4

25 A > : resummed chiral perturbation theory Weinberg 90, 9 V A-nucleon irreducible V m Q N E infrared enhancement! A-nucleon reducible 0 l E V V k = m N e.g. 4 d l i V V + iε + m iε ( ) l k m N l mn l k mn l N m N 3 d l = V V + 3 ( ) l k mnq V 4 instead of Q (4 ) 5

26 i 3 ( qˆ + σ σ ) g A q S( ) τ τ f q + m f S tensor force ( qˆ) = 3σ qˆσ qˆ σ σ 3 Q Q Q Q 4 ( ) ( ) f 4 QQQ Q f 4 f Q = M 3 Q Q Q Q Q 4 f m m QQ Q f 4 f m m ( ) ( ) 4 N N Q m Q Q m Q Q f Q Q Q f f f M 3 N N f M NN NN ( ) = = for Q M NN ( ) 6

27 T (0) = + + bound state at Q M NN Q M E m M N NN Q + + f MNN f Q M NN V (0) = + (0) V + = V + V (0) (0) 4 f Mnuc = MNN f f m T V (0) (0) N Nuclear scale arises naturally from chiral symmetry (0) V = +? Is PE all there is in leading order? That is, are observables cutoff independent with PE alone? 7

28 Issue: relative importance of pion exchange and short-range interactions g ( ( ˆ) A q S q + σ σ ) 4 i τ τ f q + m 3 mn M NN g (3) () A τ τ δ () r f V r m m r = + e 4 r S = 0 S = (3) m A m r m m r τ ˆ τ δ r r r r g V () r = () r e e S () r f ( m ) m 3 much more singular --and complicated!-- than ie 4α p p' iε Q V ( r) ( ) α = r S j j j+ j j( j+ ) j 0 6 j+ j+ j 0 0 j( j+ ) j + j j+ j+ 8

29 Assume contact interactions are driven by heavier dofs, and scale with M according to naïve dimensional analysis (W power counting) Weinberg 90, 9 Ordóñez + v.k. 9 Ordóñez, Ray + v.k. 96 Entem + Machleidt 03 Epelbaum, Glöckle + Meißner (0) ( ) () ( 3) C σ σ 0 C σ + σ m m M (0) M () N N () i M M NN C () i 0 in LO 4 (3) V() r = δ () r (0) m N M 4 V r (3) () = δ () r () m N M S = 0 S = 4 m M N NN Q M in NNLO (NLO terms, linear in Q M, break P, T ) etc. 9

30 V N = Ordóñez + v.k. 9 v.k V 3N = etc. + + more nucleons higher powers of Q 30

31 -body 3-body 4-body LO in German NLO f Q M f (parity violating) NNLO Q M f NNNLO Q M f 3 3 NNNNLO Q M f 4 4 ETC. 3

32 Hierarchies many-body forces V V V A canon emerges! N 3N 4N Weinberg 90, 9 Similar explanation for isospin-breaking forces V V V IS IV CSB N N 3N v.k. 93 J J J Rho 9 external currents other canons emerge! 3

33 Ordóñez + v.k. 9 v.k. 94 V N = similar to phenomenological potential models, e.g. AV8 (OPE) + non-local terms Stoks, Wiringa + Pieper 94 33

34 34

35 But: NOT your usual potential! Ordóñez + v.k. 9 (cf. Stony Brook TPE) σ + ω e.g., chiral v.d. Waals force 6 for m 0 r Kaiser, Brockmann + Weise 97 Rentmeester et al. 0, 03 Nijmegen PSA of,95 pp data Similar results in other channels, e.g. spin-orbit force! long-range pot # bc χ min OPE OPE + TPE ( lo) OPE + TPE ( nlo) Nijm parameters found consistent with N data! at least as good! models with s, w, 35 might be misleading

36 V 3N = Fujita + Miyazawa 58 v.k. 94 Friar, Hüber + v.k. 99 Coon + Han 99 Epelbaum, Glöckle + Meißner two unknown parameters + Coon et al. 78 Tucson-Melbourne pot with a a m c 0, ' q ' q = δ ' ε τ σ q q ' + αβ ( t ( qq )) a bq c N ( q ) d 3 αβ αβγ γ c TM potential 36

37 Many successes of Weinberg s counting, e.g., To NNNNLO (w/o deltas), fit to NN phase shifts comparable to those of realistic phenomenological potentials Ordóñez, Ray + v.k. 96 Epelbaum, Glöckle + Meißner 0 Entem + Machleidt 03 Entem + Machleidt 03 VPI PSA NNNNLO NNNLO Nijmegen PSA NNLO 37

38 With NNNNLO N and NNNLO 3N potentials (w/o deltas), good description of 3N observables and 4N binding energy levels of p-shell nuclei Epelbaum et al. 0 Gueorguiev et al

39 Many reactions γd dγ measured: Illinois 94, SAL 00, Lund 03 extracted nucleon polarizabilities: Beane, Malheiro, McGovern, Phillips + v.k. 04 γ d 0 d threshold amplitude predicted: Beane, Bernard, Lee, Meißner confirmed: SAL 98, Mainz 0 + v.k. 97 pn 0 d CSB asymmetry sign predicted: Miller, Niskanen + v.k. 00 confirmed: TRIUMF 03 dd 0 α estimated: Gårdestig, Miller + v.k. discovered: IUCF 03 + many more, including PARITY, TIME-REVERSAL VIOLATION, etc. 39

40 Q M /? unknown physics T MSM ϕ, M ZW, 00 GeV Example: T violation SM = L ( u u R d d R) M q σ µν G g u g d µν ϕ + ϕ + T single Higgs: ϕ = i u ij εϕ dj g ud, = () M m, m, f, N ρ 4 GeV Mnuc f, rnn, m, 0 0 MV e (0) () = q( cq + cq τ ) σ G µν µν q+ quark ( i) g m cq = color-edm f M T 3 = N g + N + [ τ g ] χeft 0 3 f g m M = g 0, f M T/ 40

41 Nucleon EDM 0 J T = + + Crewther et al. 79 Hockings + v.k. 05 Narison 08 Ottnad et al. 0 De Vries et al. Light-nucleus EDM De Vries et al. 0 3 Bsaisou et al. 3 5 Dekens et al. 4 ψ T ψ T ψ T = J T 0 J T V T ψ T ψ T V T 0 J T ψ T De Vries et al. 0 J T J 0 T = + + Maekawa et al. V T = + + 4

42 (0) () (0) () ( ) ( ) q q ( q q ) cg C C D µ D8 µ a a + ε3ij qτγ i qqτjγµ γ5q+ ε3ij qτγ i λ qqτjγµ γ5λ q+ 4 4 m µν µν = + ε θ qiγ5q q c + c τ3 σµν G q q d + d τ3 σµν q F 8 ( γ5 τ γ5τ ) ( λ γλ 5 τλ γ5τλ ) abc a bνρ c µ a a a a f GµνG Gρ qq qi q q q qi q q q qi q q q qi q Neutron EDM d n can be produced by any source, but De Vries et al. 0 3 Bsaisou et al. 3 5 Dekens et al. 4 different chiral properties of sources different magnitudes for nuclear EDMs H H 3 He 3 H p n q term qcedm d d ( ) ( ) d d d h t d d h n n ( ) ( M m ) ( M m ) qedm ( ) ( ) ( M m ) gcedm, PS4QO LR4QO ( ) ( ) ( ) ( ) d d ( ) ( ) ( ) ( ) ( ) ( ) ( M m ) ( M m ) Farley et al specific relations storage-ring measurements (CERN?) could teach us about sources!

43 Chiral EFT has been recognized as the basis for nuclear physics. Now it is the favorite input for the blossoming ab initio methods that are revolutionizing nuclear structure/reaction physics. 43

44 BUT Is Weinberg s power counting consistent? No! attractive in some channels 3 g ˆ A m S() r τ τ + e 3 f 4 ( mr ) m r singular potential not enough contact interactions for renormalization-group invariance even at LO 44

45 Nogga, Timmermans + v.k. 05 Pavón Valderrama + Ruiz-Arriola 06 Problems! Attractive-tensor channels: E (MeV) incorrect renormalization 45

46 46

47 Renormalization of the r n potential V() r R Λ r OPE: C (3) 0( Λ) δ Λ ( r) V 0 ( R) θ r R λ f ( r r0 ) mr ( ) n 0 r r0 m = mn r0 = m λ = m M NN f( r r0) = exp( r r0) s wave matching ψ n ( r R r ) 0 u () n r r ( λ ) V V = F r R mr 0 cot mr 0 n, 0, R T s T so that ( k r ) R = s 0 R r0 47

48 Beane, Bedaque, Childress, Kryjevski, McGuire + v.k. 0 Two regular solutions that oscillate! if no counterterm, will depend on cutoff model dependence determined by low-energy data 3rd 4 exact st nd exact vs perturbation th limit-cycle-like behavior 48

49 Same is true in all channels where attractive singular potential is iterated λ r0 ll ( + ) + 3 r r 4 7 [ ll ( + ) ] ( λr ) 0 3 ll ( + ) r λr r 0 3 r l = 3λ ll ( + ) r 0 r but rl r0 for ll ( + ) λ M singular potential only needs to be iterated in a few waves, where counterterms are needed 49

50 Perturbative pions Kaplan, Savage + Wise 98 λ = m M NN NNLO LO Fleming, Mehen + Stewart 0 Nijmegen PSA NNLO + h.o. counter NLO M NN f indeed Partly perturbative pions ll 3M ( + ) > 5 M NN l > Beane, Bedaque, Savage + v.k. 0 Nogga, Timmermans + v.k. 05 Pavón Valderrama + Ruiz-Arriola 06 Birse, Long + v.k. 07 Pavón Valderrama 0 Long + Yang + subleading orders: in perturbation theory, as in NRQED 50

51 c Vl=, j= 0 = pp 3 ( ) Add counterterms Nogga, Timmermans + v.k. 05 (cf. Pavón Valderrama + Ruiz-Arriola, 06) Nijmegen PSA LO c Vl j p p ( ) d =, = = 3 0 E (MeV) W pc ( Λ = 8fm ) LO Nijmegen PSA 5

52 certain counterterms that in Weinberg s counting were assumed suppressed by powers of are in fact suppressed by powers of Q M Q lf short-range physics more important than assumed by Weinberg s; most qualitative conclusions unchanged, but quantitative results need improvement ACTIVE RESEARCH AREA 5

53 e.g. new PC Fits to data Pavón Valderrama 0 Long + Yang bands (not error estimates): coordinate-space cutoff variation fm cyan: NNLO in Weinberg s scheme Pavon Valderrama 0 53

54 Summary A low-energy EFT of 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 54

55 M ~ m, m, 4 f, N ρ M NN ~ Q ~ GeV The Nuclear EFT Landscape f ~00 MeV + non-pert pions + pert pions eventually Chiral EFT real nuclei phys m =40 MeV Outlook p?? lattice nuclei now nonrelativistic nucleons Pionless EFT Halo EFT Deformation EFT m

56 Conclusion EFT is a general framework for theory construction same method across scales model independent controlled expansion EFT is (very slowly) becoming the paradigm in nuclear physics encodes (and, more generally, B/SM) incorporates hadronic physics generates nuclear structure The nuclear EFT frontier: many bodies & lattice interplay with ab initio methods new EFTs