Effective Field Theory in the Harmonic Oscillator Basis

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

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

Interactions from Chiral EFT • low-energy effective field theory for the relevant degrees of freedom (π, N) based on the symmetries of QCD c 1=-0. 81 Ge. V-1 c 3=-3. 2 Ge. V-1 • long-range pion dynamics explicitly c 4= 5. 4 Ge. V-1 c. E=-0. 029 • short-range physics absorbed in contact π terms, low energy constants (LEC) fitted to experiment π LO N 2 LO N 3 LO

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 Me. V), the lightest meson beyond the pions • use regulator functions f(q) to cut off the interaction beyond Λχ

Determination of LECs • LECs determined from NN scattering phaseshifts and bound-state properties (2 H, 3 He) • these few-body calculations employ practically infinitely large bases, fully capturing all the relevant low- and highenergy physics E. Epelbaum, ar. Xiv: 1302. 3214 D. Gazit, ar. Xiv: 0812. 4444

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

Oscillations of Phase Shifts • projecting the interaction onto smaller model spaces introduces oscillations in the phase shifts • interaction appropriate to be used in many-body calculations?

Harmonic-Oscillator EFT renormalization methods HO-EFT • 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 • determine the LECs of the chiral interaction that only lives in the small space

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

Convergence in finite Oscillator Spaces • HO basis acts as an additional regulator that cuts off the interaction at ΛUV too small 10 Me. V 20 Me. V 30 Me. V ΛUV a little too small too large (resolution)

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

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

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

Convergence of Many-Body Calculations • no interaction beyond Nmax=10, so fast convergence of the Nmax=10, 12, 14 sequence is expected. . . • . . . and observed: • 40 Ca: 0. 1 Me. V • 90 Zr: 1 Me. V • 132 Sn: 6 Me. V heavy nuclei from first principles? Nmax 40 Ca 90 Zr 132 Sn 10 -402. 0 -918. 4 -1230. 0 12 -402. 4 -923. 1 -1249. 3 14 -402. 5 -924. 6 -1255. 6 ∞ -402. 5 -925. 4 -1260. 1 Egs from Coupled Cluster with Singles and Doubles (CCSD)

Improving the IR Properties k k’ 1 k’ k 1 k 2 k 3 k 4 k 5 k’ 2 k’ 3 k’ 4 k’ 5 • HO-EFT has a close connection to a Discrete Variable Representation using the discrete momentum basis • matrix elements of the original interaction at the discrete momenta are well reproduced • below ΛIR and above ΛUV the interaction may be very off

Improving the IR Properties k’ 11 k 22 k 33 k 4 k 55 k’ 22 k’ 33 k’ 44 k’ k’ 55 k’ • “sacrifice” the reproduction at the highest momentum in favor of a reproduction at selected low momentum

NLO, 1 s 0 -1 s 0 Channel, Λχ=500 Me. V k’ 1 k’ 2 k’ 3 k’ 4 k’ 5 k’ 1 k 1 k 2 k 3 k 4 k 5 k’ 2 k’ 3 k’ 4 k’ 5

NLO, 1 p 1 -1 p 1 Channel, Λχ=500 Me. V

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