Chiral Nuclear Effective Field Theory

Transcription

1 Chiral uclear Effective Field Theory U. van Kolck University of Arizona Supported in part by US DOE and Sloan Foundation Background by S. Hossenfelder

2 In emoriam Vijay Pandharipande See talks by Carlson, Sick 9/9/6 v. Kolck, Pion Renormalization

3 Outline Effective Field Theories Pionful (uclear) EFT on-perturbative Renormalization and Power Counting Role of the Delta Isobar Outlook See parallel session R 9/9/6 v. Kolck, Pion Renormalization

4 Wanted Dead or Alive QCD EXPLAATIO OF UCLEAR PHYSICS Reward understanding of gross features: Why is B A ~ev QCD ~GeV? How large are few-nucleon forces? Why is isospin a good symmetry? Beware coupling constants not small 9/9/6 v. Kolck, Pion Renormalization 4

5 uclear physics scales ln Q perturbative QCD His scales are His pride, Book of Job ~ GeV ~ m, m, 4 f, K QCD ρ no small coupling ~ ev ~ ev ~ f, / r, m, ℵ nuc ~ a K need a more general way to treat multi-scale problems Effective Field Theory (EFT) (expansion in ) 9/9/6 v. Kolck, Pion Renormalization 5 Q

6 E What is Effective? ( 4 L ), Z= Dφ Dφ exp i d x ( φ φ ) H L und H L Λ m = Dϕδ ϕ ( φ ) ( ) ( 4 i d xl ) EFT ϕ Dϕ exp ( ) f Λ L L renormalization-group invariance = EFT i i d= i( d, n) Z Λ = c ( m ϕ ) (, Λ) O (, ) d n local underlying symmetries 9/9/6 v. Kolck, Pion Renormalization 6

7 ν ( ) Q Q T T ( Q)~ ( ) c, i( ), i ; i F Λ = % ν Λ ν ν= ν m m T Λ = normalization ν =ν ( d, n, K) min power counting e.g. # loops L non-analytic, from loops For Q ~ m, truncate consistently with RG invariance so as to allow systematic improvement (perturbation theory): ν max + = + ( max ) T T Q ν O ν ν max max + T Q = O Λ Λ ( ) 9/9/6 v. Kolck, Pion Renormalization 7 Λ

8 uclear physics scales ln Q ~ GeV ~ m, m, 4 f, K QCD perturbative QCD ρ hadronic th with chiral symm this talk sum? ~ ev ~ f, / r, m, nuc K sum Q nuc ~ ev ℵ ~ a sum Q ℵ halo nuclei See talks by Hammer, Braaten 9/9/6 v. Kolck, Pion Renormalization 8

9 Q m QCD uclear EFT pionful EFT degrees of freedom: nucleons, pions, deltas (+ roper?, ) m Δ m ~m ( m m ~. 5 m, K) symmetries: Lorentz, P, T, chiral expansion in: Q QCD Q m Q mρ, K Q 4 f non-relativistic multipole pion loop L EFT +, m, mδ m = c{ d, p, f } { d, p, f} QCD f f QCD calculated from QCD: lattice, fitted to data d 9/9/6 v. Kolck, Pion Renormalization 9 p f f QCD = L Δ= ( Δ) Δ d + f chiral symmetry

10 L L = + + K + g A + r r + i τ ( ) + K + τσ ( ) + K 4f f 4f + h r r A + +Δ [ i ( mδ m ) + K] Δ+ K+ ( TSΔ+ Η.c. ) ( ) ( + K) f r C C () ( μ) K m f 4f () S + + ( ) T ( σ ) + r r = + τ ( ) + K ( ) + mp mn τ τ+ K m 4f f r + + b ( ) b( ) bm ib4εijkεab cσkτc( ib)( jc ) f + + K g A + r r i σ + Η.c ( ) ( + ) 4m f τ K d + + r r + τσ ( ) ( K) f L () =K ( ) E + + Form of pion interactions determined by 9/9/6 v. Kolck, Pion Renormalization chiral symmetry

11 Role of the RG ( p p' +K) Q +L f f ( ' + K) p p dl l f 4 f Q ( p p ' +K) f 4 ( ) ( l m ) f ( 4 f ) cq absorbed absorbed # Λ # Q l nλ # Q Q ( 4 ) f Q QCD O Q Λ 4 LO LO Λ c ln Λ Λ ' ( 4 ) Λ ' aturalness: f 9/9/6 v. Kolck, Pion Renormalization QCD absorbed in higher-order coefficients

12 A=, : chiral perturbation theory ν T ΔE Q c ν ν Q Q F ν QCD m nucleon Weinberg 79 Gasser + Leutwyler 84 Gasser, Sainio + Svarc 87 Jenkins + anohar 9 dense but short-ranged ν = A + L+ V Δ ν = A # loops i i # vertices of type i i min long-ranged but sparse. fm 9/9/6 v. Kolck, Pion Renormalization QCD m.4 fm

13 r r r g ( S ( q) + σ σ ) f q m f ˆ A q r τ τ + l k E = m 4 d l i V r r 4 OPE V ( ) l + k m l m iε l + k m l m iε r d l m l = spin/isospin f f r + K ( ) l k l + m C aturalness: C m 4 f absorbed f # Λ + # Q + O f m Q 4 f f for Q Q Λ 9/9/6 v. Kolck, Pion Renormalization 4 f m f RESU ev absorbed in higher-order coefficients LO OPE But other counterterms?

14 A > : resummed chiral perturbation theory Weinberg 9, 9 Ordonez + v.k. 9 V T infrared enhancement! T = V + + V = V + V m Q ΔE A-nucleon irreducible A-nucleon reducible Schematically, T imq (4 m ) V + V V + K 4 4 m V iq ( ) Q bound state at 4 i m V iℵ 9/9/6 v. Kolck, Pion Renormalization 4

15 Ordonez + v.k. 9 v.k. 94 V = For parity violation See talk by aekawa V = higher powers of Q etc. more nucleons 9/9/6 v. Kolck, Pion Renormalization 5

16 Issue: power counting (relative sizes) = O f = O f Q ( 4 f )? f QCD 4 O m? QCD 4 Q O m? QCD O m ( 4 ) Etc. 9/9/6 v. Kolck, Pion Renormalization 6

17 aïve Dimensional Analysis (DA) 4 f f f m QCD QCD Weinberg 9, 9, 9 Ordonez + v.k. 9 v.k. 94 Ordonez, Ray + v.k LO: S-wave contacts + OPE (non-perturbative pions) LO: P-wave contacts + TPE + forces via delta + (PUT) sublos also iterated in Lippman-Schwinger eq. ℵ f f f B ev m 4 similar to phenomenological potential models: at LO, [ V ] [ AV8] [ ] short range+ OPE short range+ OPE V T c term T' long range + non-local terms 9/9/6 v. Kolck, Pion Renormalization 7

18 e.g., Fujita-iyazawa pot v.k. 94 Friar, Hueber + v.k. 99 Coon + Han V = + + one unknown parameter two unknown parameters + Tucson-elbourne pot with T pot a a' = a m c c c' = See talk by Robilotta t ( qq r r, ') a bq q ' c q q ' d q q δ r r r r = r r r αβ εαβγτ γσ ' + αβ K ( ) ( ) cf. Brazil pot 9/9/6 v. Kolck, Pion Renormalization 8

19 ote: OT your usual potential! Ordonez + v.k. 9 (cf. Stony Brook TPE) σ + ω e.g., chiral v.d. Waals force Kaiser, Brockmann + Weise 97 V C (r) [ev] for m r σ + ω Isoscalar Central Potential Rentmeester et al., ijmegen PSA of 95 pp data long-range pot # bc models with σ, ω, might be misleading 9/9/6 v. Kolck, Pion Renormalization 9 r [fm] χ min OPE 6. OPE + TPE ( lo) OPE + TPE ( nlo) 94.5 ijm parameters found consistent with data! at least as good!

20 any successes of Weinberg s counting, e.g., See talks by achleidt, eissner At LO, fit to phase shifts comparable to those of realistic phenomenological potentials With LO and LO potentials, good description of observables and 4 binding energy levels of p-shell nuclei Gueorguiev, avratil, Ormand + Vary 5 Entem + achleidt Epelbaum, Gloeckle + eissner 4 Epelbaum et al. Binding Energy (ev) Exp: () Thy: -64.* * Convergence study not completed o-core Shell odel 9/9/6 v. Kolck, Pion Renormalization

21 Is Weinberg s power counting consistent? BUT o! g ˆ A m S() r τ τ + K e f 4 ( mr ) r r r r S () rˆ = σ rˆσ rˆ σ σ attractive in some channels m r singular potential not enough contact interactions for RG invariance even at LO! 9/9/6 v. Kolck, Pion Renormalization

22 9/9/6 v. Kolck, Pion Renormalization

23 Renormalization of the r n potential V() r R Λ r OPE: C r () ( Λ) δ Λ ( r) V ( R) Θ r R λ f ( r r ) mr ( ) n r r m= m r = m λ = m 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,, R T so that ( k r ) O R = s R r 9/9/6 v. Kolck, Pion Renormalization

24 n = u R< r = K ( n r ) const.+ V ( ) ( R) = + ln + O F k + λ R 8mR mr R R * Beane, Bedaque, Savage + v.k. R r ( ) ( λ, r ), R = λ ln + O R ( R ) R R integer determined by low-energy data * V 6 k = ( ev ) ( ) R (fm) R V R approaches fixed point exact solution, fit to scatt length analytical form, fit to scatt length neglecting (ln R) 9/9/6 v. Kolck, Pion Renormalization 4 R term

25 δ 6 Λ = 985 ev Λ = 49 ev S δ 6 including V fitted to eff range 4 ijmegen PSA Λ =4 ev 4 P (ev) P (ev) () () (k + ) g A Λ* C ( ) + md ( ) = + m ln + m Λ Λ f Λ ( ) Λ Λ O Λ expand around the chiral limit (perturbative pions) or promote () D to leading order (due to infrared enhancement) First breach of W pc 9/9/6 v. Kolck, Pion Renormalization 5

26 u n n F n 4 λ λ ( r r ) = cos + n + δ n n K ( r r ) ( n )( r r ) n n R λ λ, = λ tan + δ + K 4 ( r, R) Beane, Bedaque, Childress, Kryjevski, cguire + v.k. ( n )( R r ) determined by low-energy data n n r = mr V rd 6 u(r) 4 exact H 4 (R) 5 4 st nd exact vs perturbation th r 9/9/6 v. Kolck, Pion Renormalization 6 limit-cycle-like behavior R

27 S D δ 5 analogous, but for coupled channels δ - Frederico, Timoteo + Tomio 99 Beane, Bedaque, Savage + v.k. Ruiz-Arriola + Valderrama 5 P (ev) - ε P (ev) ijmegen PSA Λ = 48 ev Λ = 985 ev Λ =97 ev Log( Δ ) -5 EFT ijm PSA in ev Log( P ) 4 6 R =.5 fm R =.45 fm P (ev) () C sufficient in leading order!? 9/9/6 v. Kolck, Pion Renormalization 7 - slope ~

28 Pion-mass dependence (fm) - a S Triplet scattering length Lattice QCD: Fukugita et al. 95 quenched 4 6 m ( ev ) EFT: (incomplete) LO 4 B d ( ev ) Deuteron binding energy Beane, Bedaque, Savage + v.k. Beane + Savage Epelbaum, Gloeckle + eissner 5 5 9/9/6 v. Kolck, Pion Renormalization 8 m ( ev )

29 But what about higher partial waves? Attractive-tensor channels: ogga, Timmermans + v.k. 5 Problems! E (ev) 5 incorrect renormalization δ [deg] δ [deg]. 5. P D cutoff dependence 8 -. D P Λ [fm - ] Λ [fm - ] 9/9/6 v. Kolck, Pion Renormalization 9

30 one undetermined phase in each channel δ [deg] δ [deg] e.g., promote counterterms V l=, j= = c pp P F c [fm 4 ] limit-cycle-like behavior Λ [fm - ] P δ [deg] Λ [fm - ] 9/9/ v. Kolck, Pion Renormalization 5 5 T L [ev] T L [ev] ε P LO EFT ( Λ = fm ) ijmegen PSA cutoff independence P E (ev) 5 9

31 4 LO EFT ( Λ = fm ) δ [deg] D D W pc ( Λ = 8fm ) ijmegen PSA δ [deg] G ε T L [ev] 5 5 T L [ev] Promising 9/9/6 v. Kolck, Pion Renormalization

32 ( l ) l f centrifugal barrier + short-range interactions stronger than in Weinberg s pc for attractive tensor channels where ogga, Timmermans + v.k. 5 l < few ( P, P F, D?! ) c.f. Birse 5 + sublos in perturbation theory Second breach of W pc on the other hand: triton - -4 correct renormalization LO EFT ( j 4) E [ev] QCD indeed Λ [fm - ] 9/9/6 v. Kolck, Pion Renormalization but leading-order value not so great because of kinetic-potential energy cancellation!?

33 Can one integrate out the delta with small error? o! Pandharipande, Phillips + v.k. 5 EFT folklore: in nuclei, Q E m Δ m can integrate out delta m with small error + O Q m = O ( mδ m ) m m But b h A = 4d = f m Δ m ( mδ m ) ( m ) h 4 A b = d = f mδ m 4 m m % error 9/9/6 v. Kolck, Pion Renormalization

34 scattering: E Q m Δ m + + O Q = O ( mδ m ) m Δ m Q m while m m + Δ Q m extrapolation E m involved at threshold relatively large error from the scattering fit leaks into force m Δ m m best strategy is not to integrate out delta (cf. Ordonez, Ray + v.k. 96 Gerstendoerfer, Kaiser + Weise 98) 9/9/6 v. Kolck, Pion Renormalization 4

35 Conclusions and Outlook EFT allows a systematic, model-independent, unified description of strong interactions at low energies Power counting complicated by nonperturbative nature of nuclei Weinberg s power counting not entirely correct due to failure of DA in attractive-tensor channels where pions have to be iterated ew power counting has been formulated: more counterterms at each order relative to Weinberg s; expect even better description of observables Consistency and convergence of new power counting yet to be checked: LO calculation of two-nucleon other few-nucleon systems Finally a consistent and efficient power counting in the pionful EFT?! 9/9/6 v. Kolck, Pion Renormalization 5