INTRODUCTION TO EFFECTIVE FIELD THEORIES OF QCD

Transcription

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

2 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

3 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.B653:358-36,7, 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 3

4 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 this lecture no small coupling expansion in 4

5 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 5

6 - pionful EFT an overkill at lower energies! cf. Bethe + Peierls 35 e.g. NN s channel : s 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

7 unitarity limit a triplet scattering length Pion-mass dependence Lattice QCD: Fukugita et al. 95 quenched cf. Beane, Bedaque, Orginos + Savage 6 Beane, Bedaque, Savage + v.k. EFT: (incomplete) NLO ( ) m π M QCD Large deuteron size because m π m π ( M ) QCD Deuteron binding energy ℵ mπ mπ M nuc m π 7

8 Cf. trapped fermions Feshbach resonance unitarity limit a MIT group webpage quark masses analog to magnetic field: close to critical values (( ) ) u + d M QCD ( MeV) m m m π = contact EFT can, and has been, used for atomic systems with large scatt lengths: universality! 8

9 Example: square well T α θ mr R V () r = r ( R) cot α ( R) α + k + k + ir k ikr ( k) = i e α ( R) cot α ( R) + k + k ir k r zero-energy poles when α (n + ) π a c = R R = R tanα α R aα 3a etc. generic α = () a r R R a αc R α = { + } αc ℵ R r = fine-tuning αα c { } + R ααc ℵ R R 9

10 rψ ( r) N e r a e N E V() r ma α mr α mr c R a = ℵ In the quantum world, one can have a b.s. with size much larger than the range of the force provided there is fine-tuning r

11 Q ℵ M nuc pionless EFT d.o.f.: nucleons symmetries: Lorentz, P, T EFT = N i + N + C N N N N mn N N C 3 N NN N 8mN C N N N N + omitting spin, isospin

12 ~ 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 = i m N C ( Λ) m 3 N ( ) ( ) + iε ( π ) l k iε Λ Λ i mn ( ) C dl dl = Λ k π + l + k+ iε l k iε k k = im N C ( Λ) Λ + i + π 4π 4πΛ ~ ic ( Λ) k non- absorbed in absorbed in C ( Λ) analytic C ( Λ) in E ic Q Q ( Λ) I ( Λ) M M l

13 ( R) mnλ C ( Λ) C( Λ) C C( Λ) C ( Λ) + = π mnλ + 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 mn MnucM 4π M nuc etc. m N nuc M M nuc 3

14 But in this case: C ( R) N 4π m Q C 4 4ππ m Mℵ mn N ( R) 44 ππ m Q M Nm M Nℵℵ for MQ ℵM if nuc etc. C Q ( R) 44 ππ Q mmmℵ M Mℵ N N nuc M M if nuc no b.s. at need one fine-tuning: Q < M nuc M ℵ M nuc M M assume no other, e.g. still, no good: just perturbation theory, etc. 4

15 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 = +, mn ℵ+ ik Λ ℵΛ N mnc 4π = m ℵ C ( R) N k + Λ cf. effective range expansion s wave scattering length a = bound state k = iℵ ℵ E = ℵ m N 5

16 () V = T () = () () T V V T () () T V () () V () Q () = T M nuc etc. T () T NN s wave 4π m M ~ nuc N nuc M Q + + ℵ+ iq ℵ+ iq ν = ν = scattering length effective range a ~ℵ ~ nuc r M p, other waves 6

17 Alternative: auxiliary field Kaplan 97 v.k. 99 EFT + + g = N i + N + T ( ) T + T NN + N N T mn N N + σt i 3 + T + 8mN 4mN sig n integrate out auxiliary field: same Lag as before with g ~ ℵ, ~, π m 4 N g C =, T NN = = + + 7

18 ν = ν = ν = + ν = + Nijmegen PSA fitted a r predicted Chen, Rupak + Savage 99 = 5.4 fm (exp) =.75 fm (exp) Bd =.9 MeV ( ν = ) B =. MeV (exp) d fitted a r predicted = = s s. fm (exp).78 fm (exp) B =.9 MeV ( ν = ) d * ν = + ν = ν = Nijmegen PSA 8

19 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 9

20 renormalization: quartet s wave λ λ c 3 3 4π ( x) T Nd ( 3 ) ( pa,) Skornyakov + Ter-Martirosian 6 Bedaque + v.k. 97 Bedaque, Hammer + v.k. 98 different regulators = pa T ℵ Nd ( 3 ) ( p ~,) Λ 3 T ( ) Nd p ( ℵ,) p

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

22 s λ λ > c 3 3 4π pℵ p TNd A cos s ln + δ Λ D ~ 4π M mn ℵ nuc ( ν = ) TNd p~ ℵ Λ Bedaque, Hammer + v.k. 99 Hammer + Mehen unless Bedaque et al. 3

23 λ λ > c 3 3 4π renormalization: doublet s wave -I different cutoffs Danilov 63 Bedaque, Hammer + v.k. 99 T ℵ Nd ( ) ( p ~,) Λ ( ) T ( p,) cos s ln δ Nd ℵ p + Λ s.4 3

24 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 4

25 λ λ 3 3 4π renormalization: doublet s wave -I > c different cutoffs Danilov 63 Bedaque, Hammer + v.k. 99 T ℵ Nd ( ) ( p ~,) Λ ( ) p TNd ( p ℵ,) cos s ln + δ Λ * s.4 5

26 s λ λ > c (limit cycle!) 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 v.oers + Seagrave 67 Kievsky et al. 96 Bedaque, Hammer + v.k. 99 unless Hammer + Mehen Bedaque et al. 3 ν = fitted a = predicted B t B t.65 fm (exp) = 8.3 MeV ( ν = ) = 8.48 MeV (expt) Dilg et al. 7 ν = 6

27 λ λ 3 3 4π renormalization: doublet s wave -I > c different cutoffs Danilov 63 Bedaque, Hammer + v.k. 99 T ℵ Nd ( ) ( p ~,) Λ ( ) p TNd ( p ℵ,) cos s ln + δ Λ * s.4 7

28 renormalization: doublet s wave -II Bedaque, Hammer + v.k. 99 D 4π ~ ( ) m M H Λ ℵ nuc N H ( Λ) = ( Λ Λ + s ) ( Λ Λ s ) sin ln( ) arctan( ) sin ln( ) arctan( ) Limit cycle! 8

29 Phillips line Bedaque, Hammer + v.k. 99 potential models Λ varying * ( ν = ) 9

30 + four-body bound state can be addressed similarly no four-body force at ν = Hammer, Meissner + Platter 4 Tjon line potential models exp Λ ( ν = ) varying * 3

31 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? As A grows, given computational power limits number of accessible one-nucleon states IR cutoff in addition to UV cutoff λ Q Λ Q 3

32 Finite Volume Lattice Box Harmonic Oscillator L = Na N π ml π ml N max mb mb b = mω nuclear matter finite nuclei Stetcu, Barrett + v.k. 7 Mueller, Seki, Koonin + v.k. 99 Rotureau, Stetcu, Barrett + v.k. few nucleons Lee et al 5 few atoms Stetcu, Barrett + v.k. 9 Rotureau, Stetcu, Barrett + 3 v.k. Rotureau et al.

33 Two possible approaches Lattice EFT Harmonic EFT Use input EFT infinite-volume potential, Λ ; minimize regulator mismatch with Λ Λ Define EFT directly within finite volume λ, Λ ; fit parameters to binding energies or to E given by 3 Emb Γ n < N 4 me cotδ ( E me cotδ ( E) = 4πN ) = π L n mel b Emb n Γ 4π 4 cf. Fukuda + Newton 54 Lee et al 5 Luescher 9 Both being actively pursued ( ) Barrett, Vary + Zhang 93 No-Core Shell Model ( ) 33 Busch et al. 98

34 - many-body systems get complicated rapidly + (continuing) focus on simpler halo/cluster nuclei one or more loosely-bound nucleons around one or more cores e.g. 4 He 5 He 6 He B B α α * ℵ m N particle separation energy p 3/ E n 8 MeV 8 MeV resonance at ~ MeV s bound state at E n ~ MeV E m E M p N E = B B α α * α c n c core excitation energy MeV ℵ (esp. near driplines) n p p n M c 34 n

35 35

36 Q ~ ℵ M c halo EFT degrees of freedom: nucleons, cores symmetries: Lorentz, P, T Q M expansion in: non-relativistic c Q mn, Q = Q m π, multipole simplest formulation: auxiliary fields for core + nucleon states m c e.g. 4 He scalar fieldϕ 4 He + N s p p spin - field s spin -/ field T spin -3/ field T 3 36

37 Nα Bertulani, Hammer + v.k. PSA, Arndt et al. 73 LO LO NLO NLO scatt length only ER.8 MeV Γ( ).55 MeV E R NNNLO LO, NLO, NNLO 37

38 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 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 in! a M ℵ ( k ) a = E& M C.8 fm 3 ~ c c r M fine-tuning of r =.3fm in 38

39 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 3 H = = b.s. b.s. What next ( He + n + n) ( ) C = b.s. He + He + He ( p + n + n) Bertulani, Higa + v.k., in progress Kong + Ravndal 99 Rotureau + v.k., + in progress Bedaque, Hammer + v.k. 99 reactions: e.g. 7 p + Be cf. p + n d 8 B + γ + γ Higa + Rupak, in progress Chen et al. 39

40 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? 4