Renormalization group methods in nuclear few- and many-body problems

Renormalization group methods in nuclear few- and many-body problems Lecture 1 S.K. Bogner (NSCL/MSU) 2011 National Nuclear Physics Summer School University of North Carolina at Chapel Hill

Useful readings for these lectures 2

Lecture 1 outline Objective: Give an overview of how renormalization group methods can be used to simplify microscopic few- and many-body calculations in low energy nuclear physics. Technical details will be revisited in lectures 2 and 3. 3

Frontiers in low E nuclear theory 6

What are the relevant degrees of freedom? Nucleonic matter (our domain in nuclear structure) 7

no 1-size fits all method Density functional theory covers the most ground, but is the most phenomenological 8

Bottom-up approach to nuclear structure DFT Shell model Ab-initio RG EFT QCD 9

Nuclear Interactions 10

Choosing the right DOF: The effective NN interaction How to get it? Ideally, from lattice QCD effective field theory + phase shifts (or phenomenological meson-exchange models) 11

NN central potential VC(r) for mπ = 530 MeV from lattice QCD 12

NN Scattering review 13

Low energy limit: Effective range expansion 14

Phenomenological NN Models short- and mid-range tuned (~ 20 parameters) to phase shifts and deuteron pole 15

Phenomenological NN Potential Models (CD-Bonn, Argonne v18, Reid93, Nijmeigen I and II,...) all share one pion exchange (OPE) at long distances model-dependent mid-range attraction and short-distance repulsion fit ~ 6000 NN data with χ 2 /dof ~ 1 many ab-initio successes in light nuclei but difficult to estimate theoretical errors and range of applicability no obvious connection to QCD not obvious how to define fully consistent 3NF s and operators (e.g., meson-exchange currents) hard to work with in most many-body methods chiral EFT (lecture 2) addresses these shortcomings 16

Renormalization Group Methods The method in its most general form can I think be understood as a way to arrange in various theories that the degrees of freedom that you re talking about are the relevant ones for the problem at hand. -S. Weinberg 17

Why is textbook nuclear physics is so hard? Repulsive core & strong tensor force => low and high k modes strongly coupled by the interaction (reminder: typical k ~ 1 fm -1 in nuclei) V l=0 (k, k )= d 3 rj 0 (kr) V (r) j 0 (k r ) 18

Why is textbook nuclear physics is so hard? Repulsive core & strong tensor force => low and high k modes strongly coupled by the interaction (reminder: typical k ~ 1 fm -1 in nuclei) Complications: strong correlations, non-perturbative, poorly convergent basis expansions,... 19

Many short wavelengths => Large matrices to diagonalize 20

Why large Λ s are painful Suppose we want to compute the ground state E of a nucleus with mass number A by brute force diagonalization. Assume the interaction has a cutoff Λ. Exercise: Estimate how the size of the s.p. basis scales with Λ. Given this, estimate the size of the Hamiltonian matrix for 16 O. Hints: 1) The basis must be sufficiently extended in space to capture the size of the nucleus (R ~ 1.2A 1/3 fm). 2) The basis must be sufficiently extended in momentum to capture the size of the cutoff Λ in the Hamiltonian. 3) Use a phase space argument to get # of sp states 22

Why large Λ s are painful Suppose we want to compute the ground state E of a nucleus with mass number A by brute force diagonalization. Assume the interaction has a cutoff Λ. Exercise: Estimate how the size of the s.p. basis scales with Λ. Given this, estimate the size of the Hamiltonian matrix for 16 O. Answer: # of s.p. states D ~ Λ 3 A Dim(H) = # of A-body Slater determinants = D!/(D-A)!/A! e.g., for Λ = 4.0 fm -1 Dim(H) ~ 10 14 23

Why large Λ s are painful Suppose we want to compute the ground state E of a nucleus with mass number A by brute force diagonalization. Assume the interaction has a cutoff Λ. Moral: Easiest way to extend the reach of ab-initio to heavier nuclei is to use lower resolutions (Λ) physical scales kf and mπ ~ 1 fm -1... 24

Arguments for using low-resolution interactions SKB et al.,arxiv:0912.3688 NN models share same long-distance physics (Vπ) Phase shifts to Elab ~ 350 MeV (krel ~ 2.1 fm -1 ); beyond this, totally model-dependent Most H(Λ) have Λ >> Λdata ~ 2.1 fm -1 kf ~ 1.35 fm -1, mπ ~ 0.7 fm -1 Why work so hard to treat high k modes that are unconstrained by NN data? 25

Low-pass filter on fourier transform of a 2d-image Much less information needed BUT Long-wavelength info preserved 26

Try a naive low-pass filter on V: V filter (k,k) 0 k, k > 2.2 fm 1 Now calculate low E observables (e.g., NN scattering) and see what happens... 27

Try a naive low-pass filter on V: δ(e) totally wrong 28

Try a naive low-pass filter on V: δ(e) totally wrong 29

Why did the low-pass filter fail? Low and high k are coupled by quantum fluctuations (virtual states) Λ k V k k V q q V k k V q q V k + + ɛ k ɛ q ɛ k ɛ q q=0 q=λ Can t simply drop high q without changing low k observables 30

2 Types of Renormalization Group Transformations k (technical details in lecture 2) k k k Λ 2 Λ 1 Λ 0 Vlow k integrate-out high k states preserves observables for k < Λ λ 0 λ 1 λ 2 Similarity RG eliminate far off-diagonal coupling preserves all observables Very similar consequences despite differences in appearance! 31

Integrating out high-momentum modes ( V low k ) Λ Λ data 32

The Similarity Renormalization Group Wegner, Glazek and Wilson Unitary transformation on an initial H = T + V H λ = U(λ)HU (λ) T + V λ λ = continuous flow parameter Differentiating with respect to λ: dh λ dλ =[η(λ),h λ] with η(λ) du(λ) dλ U (λ) Engineer η to do different things as λ => 0 η(λ) = [G λ,h λ ] G λ = T H λ driven towards diagonal in k space G λ = PH λ P + QH λ Q H λ driven to block diagonal. 33

SRG evolved NN interactions with η = [T,H] λ = 10.0 fm -1 34

SRG evolved NN interactions with η = [T,H] λ = 3.0 fm -1 35

SRG evolved NN interactions with η = [T,H] λ = 2.0 fm -1 36

Our low-pass filter now works. If you do things right, (i.e., RG/SRG transformations) problematic high-k modes can be eliminated! λ = 2.0 fm -1 37

Initially very different-looking chiral EFT potentials at N 3 LO... 38

Low-momentum universality like for V lowk Note, however, the model-dependent modes at high k along the diagonal 39

Some Consequences 40

0.25 0.2 Simplifications from lowering Λ 3 S1 deuteron probability density 1.2 1 ψ(r) 2 [fm -3 ] 0.15 0.1 0.05 Argonne v 18 Λ = 4.0 fm -1 Λ = 3.0 fm -1 Λ = 2.0 fm -1 0 0 2 4 6 8 r [fm] g(r) 0.8 0.6 0.4 0.2 weaker short-range correlations, more effective variational calcs., efficient basis expansions (SM, coupled cluster, etc.), more perturbative pair-distribution g(r) kf = 1.35 fm -1 Λ = 10.0 fm -1 (NN only) Λ = 3.0 fm -1 (NN only) Λ = 3.0 fm -1 Λ = 1.9 fm -1 Fermi gas 0 0 1 2 3 4 r [fm -1 ] 41

Weinberg Eigenvalue Analysis of Convergence Born series: T (E) =V + V 1 E H 0 V + V 1 E H 0 V 1 E H 0 V + = V + V 1 E H V If bound state Eb, series must diverge at E = Eb where (H 0 + V ) b = E b b = V b =(E b H 0 ) b For any E, generalize to find the eigenvalue of the kernel (Weinberg, 1962) 1 E b H 0 V b = b = Acting with T(E) on any Γν gives 1 E H 0 V Γ ν = η ν Γ ν T (E) Γ ν = V Γ ν ( 1+η ν + η 2 ν + ) = series diverges at E if any η ν (E) 1 42

1 E H 0 V Γ ν = η ν Γ ν (H 0 + V η ν ) Γ ν = E Γ ν 45

Weinberg Eigenvalue Analysis of Convergence Born series: T (E) =V + V 1 E H 0 V + V 1 E H 0 V 1 E H 0 V + For fixed E, find (complex) eigenvalues ην(e) [Weinberg, 1962] 1 V Γ ν = η ν Γ ν = T (E) Γ ν = V Γ ν ( 1+η ν + ην 2 + ) E H 0 = series diverges at E if any η ν (E) 1 47

Evolving to low resolution increases perturbativeness repulsive eigenvalues NOTE: 1) For real E 0, Im(η) = 0 2) repulsive eigenvalue for η 0 and visa versa (why?) 48

Evolving to low resolution increases perturbativeness repulsive eigenvalues NOTE: Any idea why in the 3S1-3D1 channel η doesn t decay as dramatically as 1S0? (HINT: typical q scale of the hard core ~ 5-7 fm -1 in both channels.) 49

Weinberg eigenvalues at finite density 50