Nuclear physics around the unitarity limit

Nuclear physics around the unitarity limit Sebastian König Nuclear Theory Workshop TRIUMF, Vancouver, BC February 28, 2017 SK, H.W. Grießhammer, H.-W. Hammer, U. van Kolck, arxiv:1607.04623 [nucl-th] SK, arxiv:1609.03163 [nucl-th] Nuclear physics around the unitarity limit p. 1

Nuclear physics around the unitarity limit Perturbative perspectives for nuclear effective field theories Sebastian König Nuclear Theory Workshop TRIUMF, Vancouver, BC February 28, 2017 SK, H.W. Grießhammer, H.-W. Hammer, U. van Kolck, arxiv:1607.04623 [nucl-th] SK, arxiv:1609.03163 [nucl-th] Nuclear physics around the unitarity limit p. 1

Prelude hierarchy of forces (natural in EFT) many-body forces two-body off-shell tuning Various approaches depart from focusing on two-body input... JISP16 Shirokov et al., PLB 644 33 (2007) two-body only, but input from nuclei up to 16 O N2LO opt, N2LO sat Ekstöm et al., PRL 110 192502 (2013), PRC 91 051301 (2015) simultaneous fit to N N + light nuclei, saturation properties SRG-evolved 2N + N2LO 3N Simonis et al., PRC 93 (2016) predict realistic saturation properties nuclear lattice calculations Elhatisari et al.. PRL 117 132501 (2016) use input from α-α scattering Nuclear physics around the unitarity limit p. 2

Nuclear EFT overview Nuclear physics around the unitarity limit p. 3

Nuclear EFT overview Nuclear physics around the unitarity limit p. 3

Nuclear scales revisited Nuclear physics around the unitarity limit p. 4

Nuclear scales revisited unitarity limit: 1/a = 0 SK et al. JPG 43 055106 (2016) Nuclear physics around the unitarity limit p. 4

Nuclear scales revisited unitarity limit: 1/a = 0 SK et al. 1607.04623 Nuclear physics around the unitarity limit p. 4

The unitarity expansion Basic setup SK, H.W. Grießhammer, H.-W. Hammer, U. van Kolck, arxiv:1607.04623 [nucl-th] two-body physics (LECs) effective range expansion assume a s= 1 S 0,t= 3 S 1 = 1/a s,t = 0 at leading order still need LO pionless three-body force! reproduce triton energy exactly finite a, Coulomb, ranges perturbative corrections! Nuclear physics around the unitarity limit p. 5

The unitarity expansion Basic setup SK, H.W. Grießhammer, H.-W. Hammer, U. van Kolck, arxiv:1607.04623 [nucl-th] two-body physics (LECs) effective range expansion assume a s= 1 S 0,t= 3 S 1 = 1/a s,t = 0 at leading order still need LO pionless three-body force! reproduce triton energy exactly finite a, Coulomb, ranges perturbative corrections! Capture gross features at leading order, build up the rest as perturbative fine structure! shift focus away from two-body details nuclear sweet spot 1/a s,t < Q A < 1/R? note: zero-energy deuteron at LO and NLO exact SU(4) W symmetry at LO! cf. Vanasse+Phillips, FB Syst. 58 26 (2017) original cupcake by Slashme (via Wiki Commons) Nuclear physics around the unitarity limit p. 5

Helium results 3 He at 1 S 0 and full unitarity good NLO established for 1 S 0 unitarity SK, Hammer, Grießhammer, van Kolck (2015/16, 2016/17) -B [MeV] -7.80 exp. 3 He -7.78-8.00-8.28 1 S0 unitarity LO + NLO at/app, Coulomb -8.80 exp. 3 H -8.78 100 1000 10000 Λ [MeV] Nuclear physics around the unitarity limit p. 6

Helium results 3 He at 1 S 0 and full unitarity good NLO established for 1 S 0 unitarity still looks good with full unitarity...... even though E B (d) still zero at NLO SK, Hammer, Grießhammer, van Kolck (2015/16, 2016/17) -B [MeV] -7.80 exp. 3 He -7.78-8.00-8.28 1 S0 unitarity LO + NLO at/app, Coulomb full unitarity LO + NLO ad, at/app, Coulomb -8.80 exp. 3 H -8.78 100 1000 10000 Λ [MeV] Nuclear physics around the unitarity limit p. 6

Helium results 3 He at 1 S 0 and full unitarity good NLO established for 1 S 0 unitarity still looks good with full unitarity...... even though E B (d) still zero at NLO SK, Hammer, Grießhammer, van Kolck (2015/16, 2016/17) 4 He (zero-range, no Coulomb) -B [MeV] -7.80 exp. 3 He -7.78-8.00-8.28 1 S0 unitarity LO + NLO at/app, Coulomb full unitarity LO + NLO ad, at/app, Coulomb -8.80 exp. 3 H -8.78 100 1000 10000 Λ [MeV] -25.0-27.5 exp. 4 He -Bα [MeV] -30.0-32.5-35.0-37.5 unitarity LO, lmax y 0 unitarity LO, jmax y 3/2 500 750 1000 1250 1500 1750 2000 2250 SK, Hammer, Grießhammer, van Kolck (2016/17) Λ [MeV] cf. also Platter (2004) Nuclear physics around the unitarity limit p. 6

Helium results 3 He at 1 S 0 and full unitarity good NLO established for 1 S 0 unitarity still looks good with full unitarity...... even though E B (d) still zero at NLO SK, Hammer, Grießhammer, van Kolck (2015/16, 2016/17) 4 He (zero-range, no Coulomb) -B [MeV] -7.80 exp. 3 He -7.78-8.00-8.28 1 S0 unitarity LO + NLO at/app, Coulomb full unitarity LO + NLO ad, at/app, Coulomb -8.80 exp. 3 H -8.78 100 1000 10000 Λ [MeV] -25.0-27.5 exp. 4 He -Bα [MeV] -30.0-32.5-35.0 LO with physical ad,t, lmax y 0 LO with physical ad,t, jmax y 3/2-37.5 unitarity LO, lmax y 0 unitarity LO, jmax y 3/2 500 750 1000 1250 1500 1750 2000 2250 SK, Hammer, Grießhammer, van Kolck (2016/17) Λ [MeV] cf. also Platter (2004) Nuclear physics around the unitarity limit p. 6

Helium results 3 He at 1 S 0 and full unitarity good NLO established for 1 S 0 unitarity still looks good with full unitarity...... even though E B (d) still zero at NLO SK, Hammer, Grießhammer, van Kolck (2015/16, 2016/17) 4 He (zero-range, no Coulomb) -B [MeV] -7.80 exp. 3 He -7.78-8.00-8.28 1 S0 unitarity LO + NLO at/app, Coulomb full unitarity LO + NLO ad, at/app, Coulomb -8.80 exp. 3 H -8.78 100 1000 10000 Λ [MeV] -25.0 -Bα [MeV] -27.5-30.0-32.5-35.0 LO with physical ad,t, lmax y 0 LO with physical ad,t, jmax y 3/2 unitarity LO + NLO ad,t, jmax y 3/2 unitarity LO + NLO ad,t, lmax y 0 exp. 4 He -37.5 unitarity LO, lmax y 0 unitarity LO, jmax y 3/2 500 750 1000 1250 1500 1750 2000 2250 SK, Hammer, Grießhammer, van Kolck (2016/17) Λ [MeV] cf. also Platter (2004) Nuclear physics around the unitarity limit p. 6

Some details binding energies at LO: find zeros of det(1 K(E)), K(E) = Faddeev(-Yakubowsky) kernel NLO energy shift: E = Ψ V (1) Ψ, Ψ = LO wavefunction Ψ = (1 P 34 PP 34 )(1 + P) ψ A + (1 + P)(1 + P) ψ B wavefunction convergence slower than eigenvalue convergence! need more mesh points and partial-wave components... Energy balance sample calculation with physical scattering lengths at LO: Λ / MeV 800 1000 1200 1400 E kin / MeV 113.67 140.58 168.44 197.09 E pot / MeV -139.77-167.41-195.76-224.62 E kin and E pot not observable sum converges as cutoff is increased, individual values do not! Nuclear physics around the unitarity limit p. 7

Helium results 3 He at 1 S 0 and full unitarity good NLO established for 1 S 0 unitarity still looks good with full unitarity...... even though E B (d) still zero at NLO SK, Hammer, Grießhammer, van Kolck (2015/16, 2016/17) 4 He (zero-range, no Coulomb) -B [MeV] -7.80 exp. 3 He -7.78-8.00-8.28 1 S0 unitarity LO + NLO at/app, Coulomb full unitarity LO + NLO ad, at/app, Coulomb -8.80 exp. 3 H -8.78 100 1000 10000 Λ [MeV] -25.0 -Bα [MeV] -27.5-30.0-32.5-35.0 LO with physical ad,t, lmax y 0 LO with physical ad,t, jmax y 3/2 unitarity LO + NLO ad,t, jmax y 3/2 unitarity LO + NLO ad,t, lmax y 0 exp. 4 He -37.5 unitarity LO, lmax y 0 unitarity LO, jmax y 3/2 500 750 1000 1250 1500 1750 2000 2250 SK, Hammer, Grießhammer, van Kolck (2016/17) Λ [MeV] cf. also Platter (2004) Nuclear physics around the unitarity limit p. 8

Helium results 3 He at 1 S 0 and full unitarity good NLO established for 1 S 0 unitarity still looks good with full unitarity...... even though E B (d) still zero at NLO SK, Hammer, Grießhammer, van Kolck (2015/16, 2016/17) 4 He (zero-range, no Coulomb) -B [MeV] -7.80 exp. 3 He -7.78-8.00-8.28 1 S0 unitarity LO + NLO at/app, Coulomb full unitarity LO + NLO ad, at/app, Coulomb -8.80 exp. 3 H -8.78 100 1000 10000 Λ [MeV] -Bα [MeV] -25.0-27.5-30.0-32.5-35.0-37.5 LO with physical ad,t, lmax y 0 LO with physical ad,t, jmax y 3/2 unitarity LO + NLO ad,t, jmax y 3/2 unitarity LO + NLO ad,t, lmax y 0 unitarity LO, lmax y 0 unitarity LO, jmax y 3/2 exp. 4 He 500 750 1000 1250 1500 1750 2000 2250 Λ [MeV] 4 He resonance state 0.3 MeV above 3 He + p threshold just below threshold at unitarity LO boson calculations with nuclear scales shift by about 0.2 0.5 MeV SK, Hammer, Grießhammer, van Kolck (2016/17) cf. also Platter (2004) Nuclear physics around the unitarity limit p. 8

Few-nucleon correlations EB( 3 H) (MeV) 11 10 9 8 7 exp. EB( 3 H) exp. 2 an d LO a phys. t = 23.7 fm LO at = LO at = + NLO a phys. t LO a phys. t + NLO rt LO at = + NLO a phys. t, rt 8.2 8.1 8.0 0.6 0.65 0.7 6-0.4 0 0.4 0.8 1.2 1.6 2 2 a n d (fm) Phillips line = f( + ) ( 1 S 0 unitarity only ) Nuclear physics around the unitarity limit p. 9

Few-nucleon correlations EB( 3 H) (MeV) 11 10 9 8 7 exp. EB( 3 H) exp. 2 an d LO a phys. t = 23.7 fm LO at = LO at = + NLO a phys. t LO a phys. t + NLO rt LO at = + NLO a phys. t, rt 8.2 8.1 8.0 0.6 0.65 0.7 6-0.4 0 0.4 0.8 1.2 1.6 2 2 a n d (fm) Phillips line = f( + ) ( 1 S 0 unitarity only ) Tjon line unitarity LO, jmax = 3/2 40 30 = f( ) exp. Bα [MeV] 70 60 50 20 4 6 8 10 12 BT [MeV] Nuclear physics around the unitarity limit p. 9

Few-nucleon correlations EB( 3 H) (MeV) 11 10 9 8 7 exp. EB( 3 H) exp. 2 an d LO a phys. t = 23.7 fm LO at = LO at = + NLO a phys. t LO a phys. t + NLO rt LO at = + NLO a phys. t, rt 8.2 8.1 8.0 0.6 0.65 0.7 6-0.4 0 0.4 0.8 1.2 1.6 2 2 a n d (fm) Phillips line = f( + ) ( 1 S 0 unitarity only ) Tjon line unitarity LO, jmax = 3/2 LO with physical ad,t, jmax = 3/2 40 30 = f( ) exp. Bα [MeV] 70 60 50 20 4 6 8 10 12 BT [MeV] Nuclear physics around the unitarity limit p. 9

Few-nucleon correlations EB( 3 H) (MeV) 11 10 9 8 7 exp. EB( 3 H) exp. 2 an d LO a phys. t = 23.7 fm LO at = LO at = + NLO a phys. t LO a phys. t + NLO rt LO at = + NLO a phys. t, rt 8.2 8.1 8.0 0.6 0.65 0.7 6-0.4 0 0.4 0.8 1.2 1.6 2 2 a n d (fm) Phillips line = f( + ) ( 1 S 0 unitarity only ) Tjon line unitarity LO, jmax = 3/2 LO with physical ad,t, jmax = 3/2 unitarity LO + NLO ad,t, jmax = 3/2 40 30 = f( ) exp. Bα [MeV] 70 60 50 20 4 6 8 10 12 BT [MeV] Nuclear physics around the unitarity limit p. 9

Unitarity expansion(s) at second order Various contributions at N 2 LO... SK, 1609.03163 [nucl-th] 1 quadratic scattering-length corrections 2 two-photon exchange 3 quadratic range corrections 4 isospin-breaking effective ranges: r pp r np 5 mixed Coulomb and range corrections! e.g., log. divergence, new pd counterterm! Nuclear physics around the unitarity limit p. 10

Unitarity expansion(s) at second order Various contributions at N 2 LO... SK, 1609.03163 [nucl-th] 1 quadratic scattering-length corrections 2 two-photon exchange 3 quadratic range corrections 4 isospin-breaking effective ranges: r pp r np 5 mixed Coulomb and range corrections! e.g., log. divergence, new pd counterterm! Energy shift from T-matrix B 2 = (E + B 0 ) 2 T (2) (E; k, p) + B 1 (E + B 0 )T (1) (E; k, p) lim E B 0 B (0) (k) B (0) (p) cf. Ji+Phillips (2013), Vanasse (2013) Nuclear physics around the unitarity limit p. 10

The perturbative deuteron Use efficient method to calculate T-matrix in perturbation theory! Vanasse, PRC 88 044001 (2013) T-matrix perturbation theory for C 0 = C (0) 0 + C (1) 0 + C (2) 0 + at NLO, the deuteron remains at zero energy...... but it moves to κ (1) = 1/a t at N 2 LO expansion in momentum, not energy B 0 = (κ(0) ) 2 M N, B 1 = 2κ(0) κ (1) M N, B 2 = (κ(1) ) 2 M N, κ (0) 0 Nuclear physics around the unitarity limit p. 11

The perturbative deuteron Use efficient method to calculate T-matrix in perturbation theory! Vanasse, PRC 88 044001 (2013) T-matrix perturbation theory for C 0 = C (0) 0 + C (1) 0 + C (2) 0 + at NLO, the deuteron remains at zero energy...... but it moves to κ (1) = 1/a t at N 2 LO expansion in momentum, not energy B 0 = (κ(0) ) 2 M N, B 1 = 2κ(0) κ (1) M N, B 2 = (κ(1) ) 2 M N, κ (0) 0 Nuclear physics around the unitarity limit p. 11

The perturbative deuteron Use efficient method to calculate T-matrix in perturbation theory! Vanasse, PRC 88 044001 (2013) T-matrix perturbation theory for C 0 = C (0) 0 + C (1) 0 + C (2) 0 + at NLO, the deuteron remains at zero energy...... but it moves to κ (1) = 1/a t at N 2 LO expansion in momentum, not energy B 0 = (κ(0) ) 2 M N, B 1 = 2κ(0) κ (1) M N, B 2 = (κ(1) ) 2 M N, κ (0) 0 Nuclear physics around the unitarity limit p. 11

The perturbative deuteron Use efficient method to calculate T-matrix in perturbation theory! Vanasse, PRC 88 044001 (2013) T-matrix perturbation theory for C 0 = C (0) 0 + C (1) 0 + C (2) 0 + at NLO, the deuteron remains at zero energy...... but it moves to κ (1) = 1/a t at N 2 LO expansion in momentum, not energy B 0 = (κ(0) ) 2 M N, B 1 = 2κ(0) κ (1) M N, B 2 = (κ(1) ) 2 M N, κ (0) 0 Nuclear physics around the unitarity limit p. 11

More 3 He results SK, arxiv:1609.03163 [nucl-th] with range corrections, there is a new pd three-body force at N 2 LO...... but the convergence of the unitarity expansions can be checked for the zero-range case! Nuclear physics around the unitarity limit p. 12

More 3 He results SK, arxiv:1609.03163 [nucl-th] with range corrections, there is a new pd three-body force at N 2 LO...... but the convergence of the unitarity expansions can be checked for the zero-range case! -7.50-7.55 -. (MeV) -7.60-7.65 f1ll 1nitarity LO + NLO 1 S0 1nitarity LO + NLO -7.70 exp. 3 He 1000 10000 Λ (MeV) Nuclear physics around the unitarity limit p. 12

More 3 He results SK, arxiv:1609.03163 [nucl-th] with range corrections, there is a new pd three-body force at N 2 LO...... but the convergence of the unitarity expansions can be checked for the zero-range case! -7.50-7.55 -. (MeV) -7.60-7.65-7.70 exp. 3 He full unitarity LO + NLO 1 S0 unitarity LO + NLO full unitarity LO + NLO + N 2 LO 1 S0 unitarity LO + NLO + N 2 LO 1000 10000 Λ (MeV) good convergence of half- and full-unitarity expansions! Nuclear physics around the unitarity limit p. 12

Perturbative p-d phase shifts At intermediate energies, Coulomb is perturbative for pp/pd scattering! SK et al. (2015); SK (2016/17) 0 3 Kievsky et al. Arvieux η 1/3 for k 20 MeV δ(k) (deg) Perturbative subtracted phase shifts 10 13 20 23 30 LO + NLO + N 2 LO 10 20 30 40 30 k (MeV) δ(k) δ full (k) δ c (k) = δ (0) full (k) δ c (0) (k) + δ (1) full (k) δ(1) c (k) + δ (2) full (k) δ(2) c (k) + cf. also SK, Hammer (2014) Nuclear physics around the unitarity limit p. 13

Summary and outlook Nuclear physics around the unitarity limit p. 14

Summary and outlook Nuclear physics around the unitarity limit p. 14

Summary and outlook Nuclear physics around the unitarity limit p. 14