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

Part II: (S)RG and Low-Momentum Interactions 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 How will we approach this problem: QCD à à NN (3N) forces à à Renormalize à à “Solve” many-body problem à à Predictions

Renormalization of Meson-Exchange Potentials 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 Can we just make a sharp cut and see if it works? Sharp cut V filter ( k 0 , k ) ≡ 0; k, k 0 > 2 . 2 MeV

Renormalization of Meson-Exchange Potentials Can we just make a sharp cut? · Nope! Low-energy physics is not correct 1 S 0 − 1 k = 2 fm 60 phase shift (degrees) AV18 phase shifts Glad I didn’t bet money 40 on that… I wonder what went wrong 20 0 after low-pass filter − 20 0 100 200 300 E lab (MeV)

Renormalization of Meson-Exchange Potentials Can we just make a sharp cut? · Nope! Low-energy physics is not correct 1 S 0 − 1 k = 2 fm 60 phase shift (degrees) AV18 phase shifts Glad I didn’t bet money 40 on that… I wonder what went wrong 20 0 after low-pass filter − 20 0 100 200 300 E lab (MeV) Phase shifts involve couplings of low-to-high momenta Λ 1 h k | V | q ih q | V | k 0 i h k | V | q ih q | V | k 0 i X X h k | V | k 0 i + + ✏ k 0 � ✏ q ✏ k 0 � ✏ q q =0 q = Λ Lesson: Must ensure low-energy physics is preserved!

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 d Λ T = 0 Lower UV cutoff, but preserve low-energy physics! Leads to renormalization group equation for low-momentum interactions V Λ low k ( k 0 , Λ ) T Λ ( Λ , k ) d low k ( k 0 , k ) = 2 d Λ V Λ 1 − ( k/ Λ ) 2 π

Renormalization of Meson-Exchange Potentials Run cutoff to lower values – decouples high-momentum modes Start from some initial V NN at high cutoff Λ 0 Λ ≈ Λ Data “Universality” at low momentum

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

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

Renormalization of NN Potentials symbols: V low k " = 2 fm -1 Why is it mostly a shift? V e ff = V L + δ V c . t . ( Λ ) Overall effect of evolving to low momentum Main effect is shift in momentum space

Renormalization of NN Potentials symbols: V low k " = 2 fm -1 Why is it mostly a shift? V e ff = V L + δ V c . t . ( Λ ) Overall effect of evolving to low momentum Main effect is shift in momentum space – delta function Removes hard core (unconstrained short-range physics)!

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) Ok, the interactions look perturbative, but something is wrong here… 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) Ok, the interactions look perturbative, but something is wrong here… No clear convergence with increasing order in bare potential Significant improvement with low-momentum interactions! Does not saturate – what might be missing?

Improvements in Perturbation Theory H ( Λ ) = T + V NN ( Λ ) + V 3N ( Λ ) + V 4N ( Λ ) + · · · Ok, the interactions look perturbative, but something is wrong here… No clear convergence with increasing order in bare potential Significant improvement with low-momentum interactions! Does not saturate – what might be missing?

Similarity Renormalization Group Wegner, Glazek/Wilson (1990s) Complementary method to decouple low from high momenta ails in lecture 2) (technical k ’ k ’ k k ! 2 ! 1 ! 2 ! 1 ! 0 ! 0 Similarity Renormalization Group Decouples high-momentum Drives Hamiltonian to band-diagonal

Similarity Renormalization Group Wegner, Glazek/Wilson (1990s) Apply a continuous unitary transformation, parameterized by s: H = T + V → H ( s ) = U ( s ) HU † ( s ) ≡ T + V ( s ) where differentiating (exercise) yields: η ( s ) ≡ d U ( s ) d H ( s ) U † ( s ) where = [ η ( s ) , H ( s )] d s d s Never explicitly construct unitary transformation Instead choose generator to obtain desired behavior : η ( s ) = [ G ( s ) , H ( s )] Many options, e.g., Drives H(s) to band-diagonal form η ( s ) = [ T, H ( s )]

Illustration of SRG Flow Drive H to band-diagonal form with kinetic-energy generator: η ( s ) = [ T, H ( s )] λ 2 = 1 With alternate definition of flow parameter: √ s 1 S 0 Argonne V 18 λ = 8 . 0 fm − 1

Illustration of SRG Flow Drive H to band-diagonal form with standard choice: η ( s ) = [ T, H ( s )] λ 2 = 1 With alternate definition of flow parameter: √ s 1 S 0 Argonne V 18 λ = 4 . 0 fm − 1

Illustration of SRG Flow Drive H to band-diagonal form with standard choice: η ( s ) = [ T, H ( s )] λ 2 = 1 With alternate definition of flow parameter: √ s 1 S 0 Argonne V 18 λ = 3 . 0 fm − 1

Illustration of SRG Flow Drive H to band-diagonal form with standard choice: η ( s ) = [ T, H ( s )] λ 2 = 1 With alternate definition of flow parameter: √ s 1 S 0 Argonne V 18 λ = 2 . 5 fm − 1

Illustration of SRG Flow Drive H to band-diagonal form with standard choice: η ( s ) = [ T, H ( s )] λ 2 = 1 With alternate definition of flow parameter: √ s 1 S 0 Argonne V 18 λ = 2 . 0 fm − 1

Other Generator Choices: Block Diagonal Create block diagonal form like V lowk ? ✓ ◆ PH ( s ) P 0 G ( s ) = H BD = 0 QH ( s ) Q With alternate definition of flow parameter: λ 2 = 1 √ s 3 S 1 λ = 10 . 0 fm − 1 Argonne V 18

Other Generator Choices: Block Diagonal Create block diagonal form like V lowk ? ✓ ◆ PH ( s ) P 0 G ( s ) = H BD = 0 QH ( s ) Q With alternate definition of flow parameter: λ 2 = 1 √ s 3 S 1 λ = 5 . 0 fm − 1 Argonne V 18

Other Generator Choices: Block Diagonal Create block diagonal form like V lowk ? ✓ ◆ PH ( s ) P 0 G ( s ) = H BD = 0 QH ( s ) Q With alternate definition of flow parameter: λ 2 = 1 √ s 3 S 1 λ = 2 . 0 fm − 1 Argonne V 18

SRG Renormalization of Chiral EFT Potentials ⇒ ≈ Diagonal V λ ( k , k ) Off-Diagonal V λ ( k , 0 ) 1.0 1.0 − 1 − 1 λ = 5.0 fm λ = 5.0 fm 0.5 1 S 0 0.5 1 S 0 These are all our 0.0 0.0 V λ (k,k) [fm] V λ (k,0) [fm] favorite Chiral EFT − 0.5 − 0.5 NN potentials… − 1.0 − 1.0 550/600 [E/G/M] 550/600 [E/G/M] 600/700 [E/G/M] 600/700 [E/G/M] − 1.5 − 1.5 500 [E/M] 500 [E/M] 600 [E/M] 600 [E/M] − 2.0 − 2.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 − 1 ] − 1 ] Diagonal V λ k k Off-Diagonal V λ k 0 k [fm k [fm 1.0 1.0 − 1 − 1 λ = 1.5 fm λ = 1.5 fm These are all our 0.5 1 S 0 0.5 1 S 0 favorite Chiral EFT 0.0 0.0 V λ (k,k) [fm] V λ (k,0) [fm] NN potentials… − 0.5 − 0.5 SRG evolved − 1.0 − 1.0 550/600 [E/G/M] 550/600 [E/G/M] 600/700 [E/G/M] 600/700 [E/G/M] − 1.5 − 1.5 500 [E/M] 500 [E/M] 600 [E/M] 600 [E/M] − 2.0 − 2.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 − 1 ] − 1 ] k [fm k [fm Exhibit similar “universal” behavior as low-momentum interactions!

Renormalization of Nuclear Interactions H ( Λ ) = T + V NN ( Λ ) + V 3N ( Λ ) + V 4N ( Λ ) + · · · Evolve momentum resolution scale of chiral interactions from initial Λ χ Remove coupling to high momenta, low-energy physics unchanged Bogner, Kuo, Schwenk, Furnstahl AV AV 18 18 Universal at low-momentum N 3 LO LO V low k ( Λ ): lower cutoffs advantageous for nuclear structure calculations