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Effective Field Theory in the Harmonic Oscillator Basis

Explore the use of an effective field theory in the harmonic oscillator basis to study nuclear interactions and their properties. Learn about the Chiral EFT method and its limitations, determination of low-energy constants, convergence in finite oscillator spaces, reproduction of phase shifts, and improvement of infrared properties.

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Effective Field Theory in the Harmonic Oscillator Basis

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  1. Effective Field Theory in the Harmonic Oscillator Basis with Thomas Papenbrock Andreas Ekström Gaute Hagen Aaina Bansal Kyle Wendt Sven Binder

  2. Nuclear Interaction • nuclear interaction not fundamental • analogous to van der Waals interaction between neutral atoms • induced via mutual polarization of quark and gluon distributions • QCD is nonperturbative at low energies, we cannot get a nuclear potential directly from QCD

  3. Interactions from Chiral EFT LO • low-energy effective field theory for the relevant degrees of freedom (π,N) based on the symmetries of QCD NLO c1=-0.81 GeV-1 c3=-3.2 GeV-1 c4= 5.4 GeV-1 cE=-0.029 • long-range pion dynamics explicitly N2LO π • short-range physics absorbed in contact terms, low energy constants (LEC) fitted to experiment π N3LO

  4. Chiral Regulator Function • χEFT is only valid up to some momentum scale Λχ, beyond which the interaction needs to be cut off • Λχ chosen below the mass of the ρ meson (770 MeV), the lightest meson beyond the pions • use regulator functions f(q) to cut off the interaction beyond Λχ

  5. Determination of LECs • LECs determined from NN scattering phaseshifts and bound-state properties (2H, 3H, 3He) • these few-body calculationsemploy practically infinitelylarge bases, fully capturing all the relevant low- and high-energy physics D. Gazit, arXiv:0812.4444 E. Epelbaum, arXiv:1302.3214

  6. Harmonic Oscillator Basis • many-body calculations are usually performed in the harmonic-oscillator (HO) basis • Nmax : maximum excitation • ω : oscillator frequency UV cutoff ΛUV N=3 N=3 N=2 captured physics N=2 N=1 Energy N=1 N=0 • beyond the lightest nuclei, we need unmanageably large basis sizes for converged results N=0 IR cutoff Nmax = 3 Nmax = 3 NCSM ground state

  7. Oscillations of Phase Shifts • projecting the interaction onto smaller model spaces introducesoscillations in the phase shifts • interaction appropriateto be used in many-bodycalculations?

  8. Harmonic-Oscillator EFT renormalization methods • rather than trying to squeeze information of an interaction originally defined in a large space into a small space that is accessible to our many-body method ... • ... define the interaction in the small space right from the outset HO-EFT • determine the LECs of the chiral interaction that only lives in the small space

  9. Convergence in finite Oscillator Spaces p basis parameters: • nucleus needs to fit into phase space: • interaction needs to be captured: x phase space covered by Oscillator basis with (Nmax , ω) or (ΛUV , L) pirated from T. Papenbrock

  10. Convergence in finite Oscillator Spaces • HO basis acts as an additional regulator that cuts off the interaction at ΛUV ΛUV a little too small ΛUV too small too large (resolution)

  11. Reproduction of NLOsim • use χEFT operator structure at NLO(11 LECs to determine) • try to reproduce another chiral interaction at NLO (NLOsim), i.e., fit LECs to phase shifts and deuteron properties of NLOsim • Nmax = 10, 80 NLOsim 80 HOEFT Nmax = 80 Nmax 10 HOEFT Nmax = 10 10 80 Nmax

  12. Reproduction of NLOsim • change in LECs < 10 % • very good reproduction of phase shifts(and deuteron properties)

  13. Fit to Realistic Phase Shifts • reasonable reproduction of np phase shifts and deuteron properties (NLO quality)

  14. Convergence of Many-Body Calculations • no interaction beyond Nmax=10, so fast convergence of the Nmax=10,12,14 sequence is expected ... • ... and observed: • 40Ca: 0.1 MeV • 90Zr: 1 MeV • 132Sn: 6 MeV Egs from Coupled Cluster with Singles and Doubles (CCSD) heavy nuclei from first principles?

  15. Improving the IR Properties • HO-EFT has a close connectionto a Discrete Variable Representationusing the discrete momentum basis k’5 k’4 k’1 k’2 k’3 k1 k2 • matrix elements of the original interaction at the discrete momenta are well reproduced k3 k k4 k’ • below ΛIRand above ΛUVthe interaction may be very off k5

  16. Improving the IR Properties k’5 k’5 k’5 k’5 k’5 k’4 k’4 k’4 k’4 k’4 k’1 k’1 k’1 k’1 k’1 k’2 k’2 k’2 k’2 k’2 k’3 k’3 k’3 k’3 k’3 • “sacrifice” the reproduction at the highest momentum in favor of a reproduction at selected low momentum k1 k1 k1 k1 k1 k2 k2 k2 k2 k2 k3 k3 k3 k3 k3 k4 k4 k4 k4 k4 k5 k5 k5 k5 k5

  17. NLO, 1s0-1s0 Channel, Λχ=500 MeV k’5 k’5 k’4 k’4 k’1 k’1 k’2 k’2 k’3 k’3 k1 k1 k2 k2 k3 k3 k4 k4 k5 k5

  18. NLO, 1p1-1p1 Channel, Λχ=500 MeV

  19. Removing the Chiral Regulator • the HO basis acts like a regulator • we would like to remove the chiral regulator f(q) • works best for IR-fixed interactions

  20. Removing the Chiral Regulator 1s0-1s0 Channel HOEFT NNLO Λχ = ∞ original NNLOΛχ = 500 MeV IR fix no IR fix

  21. Removing the Chiral Regulator 1p1-1p1 Channel original NNLOΛχ = 500 MeV HOEFT NNLO Λχ = ∞, IR-Fix

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