Effective Field Theory and. the Nuclear Many-Body Problem

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1 Effective Field Theory and the Nuclear Many-Body Problem Thomas Schaefer North Carolina State University 1

2 Schematic Phase Diagram of Dense Matter T nuclear matter µ e neutron matter? quark matter µ 2

3 Nuclear Effective Field Theory Low Energy Nucleons: Nucleons are point particles Interactions are local Long range part: pions Advantages: Systematically improvable Symmetries manifest (Chiral, gauge,...) Connection to lattice QCD 3

4 Effective Field Theory Effective field theory for pointlike, non-relativistic neutrons L eff = ψ ( i M ) ψ C 0 2 (ψ ψ) 2 + C [ 2 (ψψ) (ψ 2 ] ψ) + h.c Simplifications: neutrons only, no pions (very low energy) Match to effective range expansion p cot δ 0 = 1 a Λ2 n ( ) p 2 n+1 r n Λ 2 Coupling constants C 0 = 4πa M C 2 = 4πa2 M r 2 4

5 Few Body Physics: Successes NN scattering: N 3 LO potentials External currents: np dγ etc. Three body systems: Efimov effect, Phillips line Four body physics,... 5

6 The Nuclear Matter Problem is Hard: Traditional View NN Potential has a very strong hard core 3-body forces, isobars, relativity,... important Saturation density too small 6

7 The Nuclear Matter Problem is Hard: EFT View NN Potential has a very strong hard core Short distance behavior not relevant 3-body forces, isobars, relativity,... important 3-body: Yes; Isobars, relativity: Absorbed in counterterms Saturation density too small Yes: NN system and nuclear matter (?) are fine tuned 7

8 Toy Problem (Neutron Matter) Limiting case ( Bertsch problem) (k F a) (k F r) 0 a r k 1 F No Expansion Parameters! Universal properties [E F = k 2 F /(2m), n f = (2mµ) 3/2 /(3π 2 )] (E/A) T =0 = ξ(e (0) /A) = ξ 3 5 E F T =0 = ζe F [T c = ζ T F ] P (T, µ) = 2 5 µn f (µ)f(t/t F ) 8

9 Perfect Liquids sqgp (T=180 MeV) Neutron Matter (T=1 MeV) Trapped Atoms (T=1 nev) 9

10 Universality What do these systems have in common? dilute: rρ 1/3 1 a r k 1 F strongly correlated: aρ 1/3 1 Feshbach Resonance in 6 Li Neutron Matter 10

11 You have to work with the scattering length (and range parameters) you have, not with the scattering length you want! Donald H. Rumsfeld (SecDef, retired) 11

12 Warmup: Low Density Expansion Finite density: L L µψ ψ Modified propagator G 0 (k) αβ = δ αβ ( Perturbative expansion θ(k k F ) k 0 k 2 /2M + iɛ + θ(k F k) ) k 0 k 2 /2M iɛ E A = k2 F 2M [ ( 2 3π (k F a) π 2 (11 2 log(2))(k F a) 2 ) ]

13 Low Density Expansion: Higher orders Effective range corrections Logarithmic terms E A = k2 F 2M 1 10π (k F a) 2 (k F r) E A = k2 F 2M (g 1)(g 2) 16 27π 3 (4π 3 3)(k F a) 4 log(k F a) related to log divergence in 3 3 scattering amplitude local counterterm D(ψ ψ) 3 exists if g 3 13

14 Nonperturbative Methods Lattice Field Theory Other numerical methods: GFMC, VMC,... Expansion in number of species (large N) Expansion in dimensionality (large d, ɛ = 4 d) 14

15 Lattice Field Theory P (MeV/fm 3 ) fc f 1 f 2 b 1 b 2 s 1 s 2 FP ρ/ρ N Lee, Schaefer (2004) pure neutron matter, T = 4 MeV 15

16 Large N approximation(s) Large N gives mean field dynamics. What mean field? Determined by symmetries of the interaction SU(2N) symmetric interaction L = C 0 (ψ f ψ f ) 2 ρ = 1 N ψ ψ N N(C 0 N) Sp(2N) symmetric interaction (J = (σ 2 )... (σ 2 )) L = C 0 ψf J fg ψ g 2 Φ = 1 N ψ f J fg ψ g N N(C 0 N) 16

17 Large N approximations SU(2N): Hartree + ring diagrams (x = Nk F a/π) E A = k2 F 2M [( x N R(x) +... ] ) ( ) Furnstahl & Hammer (2002) Sp(2N): BCS + fluctuations Ω N = d 3 p (2π) 3 { } ɛ 2 p + Φ 2 ɛ p mφ2 p 2 + O(1/N) ξ = /N +... = (N = 1) 17 Sachdev (2006)

18 Large d Limit In medium scattering strongly restricted by phase space P 2 + k P k 2 Find limit in which ladders are leading order λ [ ΩdC0k d 2 F M d(2π) d ] λ = const (d ) ξ = O(1/d) Steele (1999), Schaefer et al (2003) 18

19 Pairing in the Large d Limit BCS gap equation = C 0 d d p 2 (2π) d ɛ 2 p + 2 = Solution = 2e γ E F d exp ( 1 ) dλ = + = 0.375E F Pairing energy (subleading in 1/d) O(1) + O ( d 1) E A = d 4 E F ( E F ) 2 1 d. 19

20 Upper and lower critical dimension Zero energy bound state for arbitrary d ψ (r) + d 1 r ψ (r) = 0 (r > r 0 ) d=2: Arbitrarily weak attractive potential has a bound state ξ(d=2) = 1 d=4: Bound state wave function ψ 1/r d 2. Pairs do not overlap ξ(d=4) = 0 Conclude ξ(d=3) 1/2? Try expansion around d = 4 or d = 2? Nussinov & Nussinov (2004) 20

21 Epsilon Expansion EFT version: Compute scattering amplitude (d = 4 ɛ) ig id ig T = 1 Γ ( 1 d 2 ) ( m ) d/2 ( p 0 + ɛ ) 1 d/2 p 8π 2 ɛ 4π 2 m 2 i p 0 + ɛ p 2 + iδ g 2 8π2 ɛ m 2 D(p 0, p) = i p 0 + ɛ p 2 + iδ Weakly interacting bosons and fermions 21

22 Epsilon Expansion Effective lagrangian for atoms Ψ = (ψ, ψ ) and dimers φ L = Ψ ( i 0 + σ 3 2 2m ) Ψ + µψ σ 3 Ψ + Ψ σ + Ψφ + h.c. Perturbative expansion: φ = φ 0 + gϕ. Free part L 0 = Ψ [ 2 i 0 + δµ + σ 3 2m + φ 0(σ + + σ ) ]Ψ+ϕ ( i m Interacting part (g 2, µ = O(ɛ)) L I = g ( Ψ σ + Ψϕ + h.c ) + µψ σ 3 Ψ ϕ ( i m ) ϕ. Nishida & Son (2006) ) ϕ. 22

23 Epsilon Expansion Consistency conditions + = O(ɛ) Also: tadpoles cancel Effective potential O(1) O(1) O(ɛ) ξ = 1 2 ɛ3/ ɛ5/2 ln ɛ ξ(ɛ=1) = ɛ 5/ Problem: Higher order corrections large ( 100 %)! 23

24 Near two dimensions Scattering amplitude near d=2 ( ɛ = d 2) A(p 0, p) = i 2π m ɛ + O( ɛ2 ) Effective potential (similar to (k F a) expansion) g 2 = 2π ɛ m ξ = 1 ɛ + O( ɛ 2 ) O(1) O( ɛ) O( ɛ 2 ) Superfluid gap (BCS + screening correction) = 0 ( ɛ = 1) = 2µ ( e exp 1 ɛ ) 24

25 Combine expansions near d=2 and d= ξ d Conclude ξ = ( ) Arnold et al. (2006) other appl.: Kryjevski,Rupak,Schaefer (2006) 25

26 Few Body Physics Few body physics perturbative despite large scattering length + O(ɛ) O(ɛ 2 ) T ad T aa = 1 ɛ a ad /a = 1.11 (exact 1.18) + O(ɛ 2 ) O(ɛ 3 ) nd scattering (no range terms) T dd T aa = 1 2 ɛ 0.172ɛ a dd /a = 0.66 (exact 0.60) a nd (s=3/2) 4.78 fm a exp nd = 6.35 ± 0.02 fm Also: a DD (s=2) 3.15 fm Rupak (2006) 26

27 Outlook Several systemtic approaches available None of them is perfect, emphasize different aspects Can be combined in interesting ways Real nuclear matter More perturbative. Problem becomes easier? 1/a, range corrections have been studied Explicit pions, three body clusters,... Real nuclei Density functionals (LDA, KS,...) 27