Part II: (S)RG and Low-Momentum Interactions To understand the - - PowerPoint PPT Presentation

How will we approach this problem: QCD à à NN (3N) forces à à Renormalize à à “Solve” many-body problem à à Predictions

To understand the properties of complex nuclei from first principles Renormalizing NN Interactions Basic ideas of RG Low-momentum interactions Similarity RG interactions Benefits of low cutoffs G-matrix renormalization

Part II: (S)RG and Low-Momentum Interactions

Renormalization of Meson-Exchange Potentials

Can we just make a sharp cut and see if it works?

Ok, high momentum is a pain. I wonder what would happen to low-energy observables…

Low-to-high momentum makes life difficult for low-energy nuclear theorists, so let’s get rid of it Vfilter(k0, k) ≡ 0; k, k0 > 2.2 MeV

Sharp cut

Renormalization of Meson-Exchange Potentials

Can we just make a sharp cut? Nope! Low-energy physics is not correct

·

100 200 300

Elab (MeV)

−20 20 40 60

phase shift (degrees)

1S0

AV18 phase shifts after low-pass filter

k = 2 fm

−1

Glad I didn’t bet money

• n that… I wonder what

went wrong

Renormalization of Meson-Exchange Potentials

Can we just make a sharp cut? Nope! Low-energy physics is not correct Phase shifts involve couplings of low-to-high momenta Lesson: Must ensure low-energy physics is preserved! hk|V |k0i +

Λ

X

q=0

hk|V |qihq|V |k0i ✏k0 ✏q +

1

X

q=Λ

hk|V |qihq|V |k0i ✏k0 ✏q

·

100 200 300

Elab (MeV)

−20 20 40 60

phase shift (degrees)

1S0

AV18 phase shifts after low-pass filter

k = 2 fm

−1

Glad I didn’t bet money

• n that… I wonder what

went wrong

Renormalization of Meson-Exchange Potentials

To do properly, from T-matrix equation, define low-momentum equation: Lower UV cutoff, but preserve low-energy physics!

Renormalization of Meson-Exchange Potentials

To do properly, from T-matrix equation, define low-momentum equation:

Require : d dΛ T = 0

Lower UV cutoff, but preserve low-energy physics! Leads to renormalization group equation for low-momentum interactions d dΛV Λ

low k(k0, k) = 2

π V Λ

low k(k0, Λ)T Λ(Λ, k)

1 − (k/Λ)2

Renormalization of Meson-Exchange Potentials

Run cutoff to lower values – decouples high-momentum modes Start from some initial at high cutoff Λ0 “Universality” at low momentum Λ ≈ ΛData

VNN

Renormalization of Meson-Exchange Potentials

Diagonal Off-diagonal These are all our favorite OBE NN potentials… These are all our favorite OBE NN potentials… at low momentum Universal collapse in both diagonal/off-diagonal components, most partial waves

Renormalization of Chiral EFT Potentials

These are all our favorite Chiral EFT NN potentials… These are all our favorite Chiral EFT NN potentials… at low momentum Diagonal Off-diagonal Differences remain in off-diagonal matrix elements. Why?

Renormalization of Chiral EFT Potentials

These are all our favorite Chiral EFT NN potentials… These are all our favorite Chiral EFT NN potentials… at low momentum Diagonal Off-diagonal Differences remain in off-diagonal matrix elements Sensitive to agreement for phase shifts (not all fit perfectly)

Renormalization of NN Potentials

Veff = VL + δVc.t.(Λ)

symbols: Vlow k " = 2 fm-1

Why is it mostly a shift?

Overall effect of evolving to low momentum Main effect is shift in momentum space

Renormalization of NN Potentials

Overall effect of evolving to low momentum Main effect is shift in momentum space – delta function Removes hard core (unconstrained short-range physics)! Veff = VL + δVc.t.(Λ)

symbols: Vlow k " = 2 fm-1

Why is it mostly a shift?

Improvements in Perturbation Theory

Explore improvements in symmetric infinite matter calculations Order by order in many-body perturbation theory (MBPT) No clear convergence with increasing order in bare potential

Improvements in Perturbation Theory

Explore improvements in symmetric infinite matter calculations Order by order in many-body perturbation theory (MBPT) No clear convergence with increasing order in bare potential Significant improvement with low-momentum interactions!

Improvements in Perturbation Theory

Explore improvements in symmetric infinite matter calculations Order by order in many-body perturbation theory (MBPT) No clear convergence with increasing order in bare potential Significant improvement with low-momentum interactions!

Ok, the interactions look perturbative, but something is wrong here…

Improvements in Perturbation Theory

Explore improvements in symmetric infinite matter calculations Order by order in many-body perturbation theory (MBPT) No clear convergence with increasing order in bare potential Significant improvement with low-momentum interactions! Does not saturate – what might be missing?

Ok, the interactions look perturbative, but something is wrong here…

Improvements in Perturbation Theory

No clear convergence with increasing order in bare potential Significant improvement with low-momentum interactions! Does not saturate – what might be missing?

Ok, the interactions look perturbative, but something is wrong here…

H (Λ) = T + VNN (Λ) + V3N (Λ) + V4N (Λ) + · · ·

Similarity Renormalization Group

Complementary method to decouple low from high momenta Decouples high-momentum

!0 !1 !2 k’ k

ails in lecture 2)

!0 !1 !2 k’ k

(technical

Similarity Renormalization Group Drives Hamiltonian to band-diagonal

Wegner, Glazek/Wilson (1990s)

Similarity Renormalization Group

Apply a continuous unitary transformation, parameterized by s: where differentiating (exercise) yields: where Never explicitly construct unitary transformation Instead choose generator to obtain desired behavior: Many options, e.g., Drives H(s) to band-diagonal form H = T + V → H(s) = U(s)HU †(s) ≡ T + V (s) dH(s) ds = [η(s), H(s)] η(s) ≡ dU(s) ds U †(s) η(s) = [T, H(s)]

Wegner, Glazek/Wilson (1990s)

η(s) = [G(s), H(s)]

Illustration of SRG Flow

Drive H to band-diagonal form with kinetic-energy generator: With alternate definition of flow parameter: η(s) = [T, H(s)] λ2 = 1 √s λ = 8.0 fm−1 Argonne V18

1S0

Illustration of SRG Flow

λ = 4.0 fm−1 Argonne V18

1S0

Drive H to band-diagonal form with standard choice: With alternate definition of flow parameter: η(s) = [T, H(s)] λ2 = 1 √s

Illustration of SRG Flow

λ = 3.0 fm−1 Argonne V18

1S0

Drive H to band-diagonal form with standard choice: With alternate definition of flow parameter: η(s) = [T, H(s)] λ2 = 1 √s

Illustration of SRG Flow

λ = 2.5 fm−1 Argonne V18

1S0

Drive H to band-diagonal form with standard choice: With alternate definition of flow parameter: η(s) = [T, H(s)] λ2 = 1 √s

Illustration of SRG Flow

λ = 2.0 fm−1 Argonne V18

1S0

Drive H to band-diagonal form with standard choice: With alternate definition of flow parameter: η(s) = [T, H(s)] λ2 = 1 √s

Other Generator Choices: Block Diagonal

Create block diagonal form like Vlowk? With alternate definition of flow parameter: λ2 = 1 √s G(s) = HBD = ✓ PH(s)P QH(s)Q ◆ Argonne V18

3S1

λ = 10.0 fm−1

Create block diagonal form like Vlowk? With alternate definition of flow parameter: λ2 = 1 √s G(s) = HBD = ✓ PH(s)P QH(s)Q ◆ Argonne V18

3S1

λ = 5.0 fm−1

Other Generator Choices: Block Diagonal

λ = 2.0 fm−1 Create block diagonal form like Vlowk? With alternate definition of flow parameter: λ2 = 1 √s G(s) = HBD = ✓ PH(s)P QH(s)Q ◆ Argonne V18

3S1

Other Generator Choices: Block Diagonal

SRG Renormalization of Chiral EFT Potentials

These are all our favorite Chiral EFT NN potentials… These are all our favorite Chiral EFT NN potentials… SRG evolved Exhibit similar “universal” behavior as low-momentum interactions!

Diagonal Vλ k k

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

k [fm

−1]

−2.0 −1.5 −1.0 −0.5 0.0 0.5 1.0

Vλ(k,k) [fm]

550/600 [E/G/M] 600/700 [E/G/M] 500 [E/M] 600 [E/M]

λ = 1.5 fm

−1

1S0

Off-Diagonal Vλ k 0

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

k [fm

−1]

−2.0 −1.5 −1.0 −0.5 0.0 0.5 1.0

Vλ(k,0) [fm]

550/600 [E/G/M] 600/700 [E/G/M] 500 [E/M] 600 [E/M]

λ = 1.5 fm

−1

1S0

⇒ ≈

Diagonal Vλ(k, k)

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

k [fm

−1]

−2.0 −1.5 −1.0 −0.5 0.0 0.5 1.0

Vλ(k,k) [fm]

550/600 [E/G/M] 600/700 [E/G/M] 500 [E/M] 600 [E/M]

λ = 5.0 fm

−1

1S0

Off-Diagonal Vλ(k, 0)

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

k [fm

−1]

−2.0 −1.5 −1.0 −0.5 0.0 0.5 1.0

Vλ(k,0) [fm]

550/600 [E/G/M] 600/700 [E/G/M] 500 [E/M] 600 [E/M]

λ = 5.0 fm

−1

1S0

Renormalization of Nuclear Interactions

AV AV18

18

N3LO LO

Vlow k(Λ): lower cutoffs advantageous for nuclear structure calculations

Evolve momentum resolution scale of chiral interactions from initial Remove coupling to high momenta, low-energy physics unchanged

Λχ

Bogner, Kuo, Schwenk, Furnstahl

Universal at low-momentum

H (Λ) = T + VNN (Λ) + V3N (Λ) + V4N (Λ) + · · ·

Smooth vs. Sharp Cutoffs

AV AV18

18

N3LO LO

Similar but not exact same results – will be differences in calculations

Can have sharp as well as smooth cutoffs Remove coupling to high momenta, low-energy physics unchanged

Bogner, Kuo, Schwenk, Furnstahl

AV AV18

18

H (Λ) = T + VNN (Λ) + V3N (Λ) + V4N (Λ) + · · ·

SRG-Evolution of Different Initial Potentials

EFT1 EFT1

Lots of pretty pictures, but how does it actually help?

SRG evolution of two different chiral EFT potentials

EFT2 EFT2

H (Λ) = T + VNN (Λ) + V3N (Λ) + V4N (Λ) + · · ·

Revisit Low-Pass Filter Idea

What’s the difference now?

Ok, high momentum is a pain. I wonder what would happen to low-energy observables…

Low-to-high momentum makes life difficult for low-energy nuclear theorists

Vfilter(k0, k) ≡ 0; k, k0 > 2.2 MeV

AV1 V18 SR SRG

Revisit Low-Pass Filter Idea

Low-energy observables were preserved – now sharp cut makes sense!

Ok, high momentum is a pain. I wonder what would happen to low-energy observables…

Low-to-high momentum makes life difficult for low-energy nuclear theorists

Vfilter(k0, k) ≡ 0; k, k0 > 2.2 MeV

500 1000

• 100
• 50

50 AV18 AV18 [kmax = 2.2 fm-1] Vs [kmax = 2.2 fm-1] 500 1000

• 50

50 100 500 1000

• 30
• 20
• 10

500 1000

• 40
• 20

20 500 1000

Elab [MeV]

• 10
• 5

5

phase shift [deg]

1S0 3S1 3P0 3F3 3D1

Benefits of Lower Cutoffs

Often work in HO basis – does this make a difference there? Removes coupling from low-to-high harmonic oscillator states Expect to speed convergence in HO basis Explicitly see why this causes problems later!

5 10 15 20

• 5

5 10

< 0 | Vlow k | n > [MeV]

1S0 !=2.0 fm

• 1

1S0 !=5.0 fm

• 1

5 10 15 20 25

n

• 15
• 10
• 5

5 10 15

3S1 !=2.0 fm

• 1

3SD1 !=2.0 fm

• 1

3S1 !=5.0 fm

• 1

3SD1 !=5.0 fm

• 1

Benefits of Lower Cutoffs

Exactly what happens in no-core shell model calculations Probably equally helpful in normal shell-model calculations? Come back to this later…

Benefits of Lower Cutoffs

Use cutoff dependence to assess missing physics: return to Tjon line Varying cutoff moves along line Still never reaches experiment Lesson: Variation in physical observables with cutoff indicates missing physics Tool, not a parameter!

ing

Benefits of Lower Cutoffs

Triton binding energy - again clearly improved convergence behavior Clear dependence on cutoff – more than one, look closely… What is the source(s)?

Benefits of Lower Cutoffs

Triton binding energy - again clearly improved convergence behavior Clear dependence on cutoff – more than one, look closely… What is the source(s)? 1) SRG cutoff dependence

Benefits of Lower Cutoffs

Triton binding energy - again clearly improved convergence behavior Clear dependence on cutoff – more than one, look closely… What is the source(s)? 1) SRG cutoff dependence 2) Initial cutoff dependence Something missing in each case!

Case 1: Price of Low Cutoffs = Induced Forces

Life Lesson: no free lunch – not even at Summer Schools, apparently L Consider Hamiltonian with only two-body forces: And Simply expand with creation/annihilation operators: H = T + VNN η(s) = [T, H(s)] dH(s) ds = [η(s), H(s)] = [[T, T + V (s)] , T + V (s)]

Case 1: Price of Low Cutoffs = Induced Forces

Life Lesson: no free lunch – not even at Summer Schools, apparently L Consider Hamiltonian with only two-body forces: And Simply expand with creation/annihilation operators: Three-body terms will appear even when initial 3-body forces absent Call these induced 3N forces (3N-ind) H = T + VNN η(s) = [T, H(s)] dH(s) ds = [η(s), H(s)] = [[T, T + V (s)] , T + V (s)]

dV (s) ds = hhX a†a, X a†a†aa i , X a†a†aa i = · · · + X a†a†a†aaa + · · ·

Induced 3N Forces

Effect of including 3N-ind? Exactly initial up to neglected 4N-ind NN-only clear cutoff dependencs

1 2 3 4 5 6 7 10 20

λ [fm

−1]

−8.6 −8.4 −8.2 −8.0 −7.8 −7.6 −7.4

Ground-State Energy [MeV]

NN-only

3H

Expt.

N

3LO (500 MeV)

1 2 3 4 5 10 20

λ [fm

−1]

−29 −28 −27 −26 −25 −24

Ground-State Energy [MeV]

NN-only

4He

N

3LO (500 MeV)

Expt.

VNN

Induced 3N Forces

Effect of including 3N-ind? Exactly initial up to neglected 4N-ind NN-only clear cutoff dependencs 3N-induced – dramatic reduction in cutoff dependence! Lesson: SRG cutoff variation a sign of neglected induced forces

1 2 3 4 5 6 7 10 20

λ [fm

−1]

−8.6 −8.4 −8.2 −8.0 −7.8 −7.6 −7.4

Ground-State Energy [MeV]

NN-only NN + NNN-induced

3H

Expt.

N

3LO (500 MeV)

1 2 3 4 5 10 20

λ [fm

−1]

−29 −28 −27 −26 −25 −24

Ground-State Energy [MeV]

NN-only NN+NNN-induced

4He

N

3LO (500 MeV)

Expt.

VNN

1 2 3 4 5 6 7 10 20

λ [fm

−1]

−8.6 −8.4 −8.2 −8.0 −7.8 −7.6 −7.4

Ground-State Energy [MeV]

NN-only NN + NNN-induced

3H

Expt.

N

3LO (500 MeV)

1 2 3 4 5 10 20

λ [fm

−1]

−29 −28 −27 −26 −25 −24

Ground-State Energy [MeV]

NN-only NN+NNN-induced

4He

N

3LO (500 MeV)

Expt.

Induced 3N Forces

Effect of including 3N-ind? Exactly initial up to neglected 4N-ind NN-only clear cutoff dependencs 3N-induced – dramatic reduction in cutoff dependence! Lesson: SRG cutoff variation a sign of neglected induced forces Still far from experiment and remaining (minor) cutoff dependence!

VNN

Aside: G-matrix Renormalization

Standard method for softening interaction in nuclear structure for decades: Infinite summation of ladder diagrams Need two model spaces: 1) M space in which we will want to calculate (excitations allowed in M) 2) Large space Q in which particle excitations are allowed To avoid double counting, can’t overlap – matrix elements depend on M

Gijkl(ω) = Vijkl + X

mn∈Q

Vijmn Q ω − εm − εn Gmnkl(ω)

Aside: G-matrix Renormalization

Standard method for softening interaction in nuclear structure for decades: Iterative procedure Dependence on arbitrary starting energy!

G-matrix Renormalization

Standard method for softening interaction in nuclear structure for decades:

What happens as we keep increasing M?

Gijkl(ω) = Vijkl + X

mn∈Q

Vijmn Q ω − εm − εn Gmnkl(ω)

G-matrix Renormalization

Results of G-matrix renormalization vs. SRG

AV1 V18 N3LO LO

Removes some diagonal high-momentum components Still large low-to-high coupling in both interactions No indication of universality Clear difference compared with SRG-evolved interactions!

G-m G-mat G-m G-mat SR SRG SR SRG+ G+ G-m G-mat SR SRG SR SRG+ G+ G-m G-mat

Summary

Low-momentum interactions can be constructed from any VNN via RG Low-to-high momentum coupling not desirable in low-energy nuclear physics Evolve to low-momentum while preserving low-energy physics Universality attained near cutoff of data Low-momentum cutoffs remove low-to-high harmonic oscillator couplings Cutoff variation assesses missing physics interaction level: tool not a parameter