the nuclear manybody problem lecture 2
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The nuclear manybody problemLecture 2
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Renormalization of the nucleonnucleon force.
● Aim: Construct the best nucleonnucleon force applicable for light and medium mass nuclei. ● High momentum modes of the nucleonnucleon force makes manybody wavefunction expansions converge slowly. ● Construct effective interactions where high momentum modes are integrated out.
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Chiral Perturbation Theory.
“If you want more accuracy, you have touse more theory (more orders)”
Effective Lagrangian obeys QCD symmetries (spin, isospin, chiral symmetry breaking)
Lagrangian infinite sum of Feynman diagrams.
Expand in O(Q/ΛQCD)
Weinberg, Ordonez, Ray, van Kolck
NN amplitude uniquely determined by twoclasses of contributions: contact terms and pion exchange diagrams.
24 paramters (rather than 40 from meson theory) to describe 2400 data points with
χdof2≈1
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“Resolution dependent” Sources for Nonperturbative Physics
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Principles of low energy effective theories
● Highprecision potentials contain shortrange (high momentum) physics that is not constrained by phase shifts.
● Is it necessary to know the NN interaction at short distances to understand long wavelength physics?
● Introduce momentum cutoff and integrate out high momentum modes such that lowmomentum observables are unchanged (Renormalization group transformation).
● Resulting lowmomentum potential Vlowk .● Recall: Fermi momentum at saturation density kF =1.4 fm1.
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2Types of Renormalization Group Transformations
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Similarity Renormalization Group applied to nuclear structure
The evolution (flow) of the Hamiltonian to block diagonal form:
Differentiating with respect to the flow parameter s we get:
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Effective operators in the SRG
From S. Bogner
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Integrating out highmomentum modes by similarity transformations.
Define a model space P and a complement Qspace given by a cutoffin momentum space
The Hamiltonian can then be written in Block form as
The LeeSuzuki method finds a similarity transformation that bringsthe Hamiltonian to the block structure
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Low momentum potential Vlowk
S. K. Bogner, T. T. S. Kuo, and A. Schwenk, Phys. Rep. 386 (2003) 1
Different highprecision potentials
Universal lowmomentum potential
Properties of Vlowk :
●No hard core
●Nonlocal
●HartreeFock already yields bound nuclei.
kFermi = 1.4 fm1
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Deuteron wave function
Deuteron wave function for different cutoffs
Deuteron RMS radii for different cutoffs
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Light nuclei with Vlowk
A. Nogga, S. K. Bogner, and A. Schwenk, Phys.Rev. C70 (2004) 061002
As cutoff is varied, motion along Tjon line.
Addition of dependent threenucleon force yields agreement with experiment.
●Threenucleon force necessary.
●There is no “best” potential & TNF. Choose the most convenient.
Q: Can we understand this more systematically?
A: Yes, resort to a modelindependent approach via effective field theory (EFT).
Vlowk filters out lowenergy physics, while EFT starts from lowenergy modes.
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Convergence in light systems with vlowk
Convergence of triton binding energy for different cutoffs used in the renormalization.
● Converges fast with smaller● Binding energy depends on●Must include threebody force!
S Bogner et al. nuclth/0701013
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Convergence in light systems with vlowk
1. Running the interaction to lower cutoff increases the convergence2. Different cutoff L gives different converged ground state energies,and consequently different threenucleon forces.
Question:
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What happens in medium sized nuclei ?
Converged CoupledCluster ground state energies for 40Ca and 56Ni using Vsrg with = 2.5fm1
Several hundreds of MeV's overbinding ! Threenucleon force must be largely repulsive !Is threenucleon forces enough ??
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3NF from Chiral EFT
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Coupled Cluster results for He4 with 3NF
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Different contributions to E(CCSD) from 3NF in He4
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What happens in nuclear matter calculations ?
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Summary on lowmomentum interactions
● The hard core of the nucleonnucleon interaction makes manybody approaches difficult. Nonperturbative.
● By use of renormalization group theory or similarity transformations, high momentum modes can be integrated out, while preserving all twobody observables.
● However this procedure induces manybody forces since we remove degrees of freedom for the manybody system.
● Will effective threebody forces be sufficient to overcome the large overbinding seen in medium size nuclei ?
● Is there a systematic way of generating higher body forces as in the Chiral EFT approach ?
● Want a theory which minimizes the effect of manybody forces.