Universal Range Effects to Efimov Features Chen Ji ECT* / - PowerPoint PPT Presentation

Universal Range Effects to Efimov Features Chen Ji ECT* / INFN-TIFPA EMMI RRTF, 02.06.2016 In Collaboration with: Bijaya Acharya, Eric Braaten, Lucas Platter, Daniel Phillips

EFT for 3 identical bosons LO ( r/a ) 0 EFT Lagrangian for 3 identical bosons � � � � i∂ 0 + ∇ 2 i∂ 0 + ∇ 2 � � d − g L = ψ † ψ − d † d † ψψ + h.c + hd † dψ † ψ + · · · 4 m − ∆ √ 2 m 2 terms with more derivatives are at higher orders ( r/a ) n Non-perturbative features at LO atom-atom (dimer) scattering (tune g ) 1 ∝ = + 1 /a + ik atom-dimer scattering (tune h ) t 0 = + + t 0 + t 0 Bedaque, Hammer, van Kolck ’99

LO renormalization LO 3BF h : tune H (Λ) = Λ 2 h/ 2 mg 2 : fix one 3-body observable limit cycle: H (Λ) periodic for Λ → Λ(22 . 7) n Bedaque et al. ’00 scaling invariance → Efimov physics Efimov ’71

Universal physics at LO Universal features in three-body systems (Efimov effects) 3-body spectrum: a function of scattering length a geometric spectrum E n = (22 . 7) − 2 E n − 1 in the limit a → ∞ universal relation of recombination features a ∗ = a + / 4 . 5 = − a − / 21 . 3 i.e. a − ( n ) = 22 . 7 a − ( n − 1) Zaccanti et al. ’09

Range effects in EFT range effects on universal physics 2 k 2 + · · · 2-body observable: k cot δ 0 = − 1 a + r 3-body observables: → in r/a expansion r/a corrections (fixed a ): Hammer, Mehen ’01 (NLO) Bedaque, Rupak, Griesshammer, Hammer ’03 (N 2 LO, partial resummation) Platter, Phillips ’06 (N 2 LO, partial resummation) CJ, Phillips ’13 (N 2 LO, full perturbation) r/a corrections (variable a ): Platter, CJ, Phillips ’09 (NLO, partial resummation) CJ, Platter, Phillips ’10; ’12 (NLO, full perturbation)

N 2 LO ( r /a ) 2 range effects (fixed a ) N 2 LO corrections to atom-dimer scattering amplitude: in 2 nd order perturbation theory ( ∼ r 2 /a 2 ): N 2 LO dimer: t 0 t 0 N 2 LO 3BF: t 0 t 0 t 0 t 0 + · · · two NLO t 0 t 0 t 0 t 0 t 0 t 0 terms: + · · · t 0 t 0 t 0 N 2 LO 3-body force: H 2 ( E, Λ) = r 2 Λ 2 h 20 (Λ) + r 2 mE 3 h 22 (Λ) → one additional 3-body input is needed CJ, Phillips FBS ’13 c.f. Bedaque et al. ’03 & Platter, Phillips ’06

N 2 LO Renormalization in Helium Trimers H 2 = r 2 Λ 2 h 20 + r 2 mE t h 22 H 2 = r 2 Λ 2 h 20 100 1.5 (0) B 0 1 0 (0) B 1 (1) [ γ n B d ] (0) [ γ n B d ] (0) 0.1 B 2 0.5 -100 B n 0 B n (1) B 0 (1) B 1 -0.5 (1) -200 B 2 -1 2 3 4 5 2 3 4 5 10 10 10 10 10 10 10 10 Λ [γ] Λ [γ] a ad → LO/NLO/N 2 LO a ad → LO/NLO/N 2 LO B (1) → N 2 LO B (1) = B (1) + rB (1) + r 2 B (1) t t 0 1 2 B (0) = B (0) + rB (0) + r 2 B (0) t 0 1 2

N 2 LO three-body 3BF H 2 (Λ) = r 2 Λ 2 h 20 (Λ) + r 2 mE t h 22 (Λ) 40 0.8 20 0.6 1 / h 20 ( Λ) 1 / h 22 ( Λ) 0 0.4 -20 0.2 -40 0 1 2 3 4 5 1 2 3 4 5 10 10 10 10 10 10 10 10 10 10 Λ [γ] Λ [γ] 1 /h 22 (Λ) [partial version] 1 /h 20 (Λ)

4 He trimer B (1) B (0) a ad [ γ − 1 ] r ad [ γ − 1 ] Input [ B d ] [ B d ] t t TTY potential 1.738 96.33 1.205 a ad LO 1.723 97.12 1.205 0.8352 a ad NLO 1.736 89.72 1.205 0.9049 a ad , B (1) N 2 LO 1.738 116.9 1.205 0.9132 t B (1) LO 1.738 99.37 1.178 0.8752 t B (1) NLO 1.738 89.77 1.201 0.9130 t B (1) N 2 LO , a ad 1.738 115.9 1.205 0.9135 t CJ, Phillips FBS ’13

4 He trimer B (1) B (0) a ad [ γ − 1 ] r ad [ γ − 1 ] Input [ B d ] [ B d ] t t TTY potential 1.738 96.33 1.205 a ad LO 1.723 97.12 1.205 0.8352 a ad NLO 1.736 89.72 1.205 0.9049 a ad , B (1) N 2 LO 1.738 116.9 1.205 0.9132 t B (1) LO 1.738 99.37 1.178 0.8752 t B (1) NLO 1.738 89.77 1.201 0.9130 t B (1) N 2 LO , a ad 1.738 115.9 1.205 0.9135 t CJ, Phillips FBS ’13

4 He trimer B (1) B (0) a ad [ γ − 1 ] r ad [ γ − 1 ] Input [ B d ] [ B d ] t t TTY potential 1.738 96.33 1.205 a ad LO 1.723 97.12 1.205 0.8352 a ad NLO 1.736 89.72 1.205 0.9049 a ad , B (1) N 2 LO 1.738 116.9 1.205 0.9132 t B (1) LO 1.738 99.37 1.178 0.8752 t B (1) NLO 1.738 89.77 1.201 0.9130 t B (1) N 2 LO , a ad 1.738 115.9 1.205 0.9135 t CJ, Phillips FBS ’13

4 He trimer B (1) B (0) a ad [ γ − 1 ] r ad [ γ − 1 ] Input [ B d ] [ B d ] t t TTY potential 1.738 96.33 1.205 a ad LO 1.723 97.12 1.205 0.8352 a ad NLO 1.736 89.72 1.205 0.9049 a ad , B (1) N 2 LO 1.738 116.9 1.205 0.9132 t B (1) LO 1.738 99.37 1.178 0.8752 t B (1) NLO 1.738 89.77 1.201 0.9130 t B (1) N 2 LO , a ad 1.738 115.9 1.205 0.9135 t Difference in 2 renormalization schemes (LO → NLO → N 2 LO): atom-dimer effective range r ad : 5% → 0.9% → 0.02% ground-state trimer B (0) : 2% → 0.07% → 0.9% t CJ, Phillips FBS ’13

NLO ( r/a ) range effects Calculate r/a correction to atom-dimer amplitude NLO dimer: t 0 t 0 t 0 t 0 t 0 t 0 NLO 3BF: NLO 3-body force: H 1 (Λ) = r Λ h 10 (Λ) + r/a h 11 (Λ) a fixed: h 11 is absorbed (no additional 3-body input is needed) a varies: one additional 3-body input is needed

NLO three-body 3BF H 1 (Λ) = r Λ h 10 (Λ) + r/a h 11 (Λ) 0 0 -0.5 -0.1 1 / h 10 ( Λ) -1 1 / h 11 ( Λ) -0.2 -1.5 -0.3 -0.4 -2 -0.5 -2.5 1 2 3 4 5 6 1 2 3 4 5 6 10 10 10 10 10 10 10 10 10 10 10 10 Λ / κ * Λ / κ * 1 /h 10 (Λ) 1 /h 11 (Λ)

Recombination of 7 Li Atoms NLO ⋆ Experiment †‡ LO LO -264 † 3A res a − , 0 [ a B ] -264 -244 -264 1160 † rec min a + , 1 [ a B ] 1254 1160 1160 180 ‡ Ad res a ∗ , 1 [ a B ] 281 259 210(44) † Gross et al. ’09 ‡ Machtey et al. ’12 ⋆ Ji, Phillips, Platter ’10

Universal Relations at NLO recombination features are correlated by a i,n = λ n θ i κ − 1 + ( J i + n σ ) r ; i = ∗ , + , − ; λ = 22 . 694 ∗ J i is non-universal but J i − J j is a universal number e.g. J + − J − = σ/ 2 , σ = 1 . 095 κ ∗ r and J i can be fixed by the ratio of two Efimov features e.g. fix ( a − , 1 /a − , 0 ) /λ , ( a − , 2 /a − , 1 ) /λ deviate from 1 due to range effects predict ratios of any other features ( a i,n +1 /a i,n ) /λ CJ, Braaten, Phillips, Platter, PRA 2015

Three-Body Force (3BF) at NLO LO 3BF: RG limit cycle → discrete scaling symmetry λ n NLO 3BF: RG range-modification → discrete scaling breaking nσ 3BF up to NLO can be divided into 4 different contributions � r � � + ˜ η H ′ H (Λ) = H 0 (Λ / Λ ∗ ) � + h 10 (Λ / Λ ∗ )Λ r � + 0 (Λ / Λ ∗ ) ln(Λ /µ ) h 11 (Λ / Λ ∗ ) a � �� � �� � �� � �� � log periodic linear divergence log divergence log periodic with H ′ 0 = (Λ d/d Λ) H 0

Three-Body Force (3BF) at NLO LO 3BF: RG limit cycle → discrete scaling symmetry λ n NLO 3BF: RG range-modification → discrete scaling breaking nσ 3BF up to NLO can be divided into 4 different contributions � r � � + ˜ η H ′ H (Λ) = H 0 (Λ / Λ ∗ ) � + h 10 (Λ / Λ ∗ )Λ r � + 0 (Λ / Λ ∗ ) ln(Λ /µ ) h 11 (Λ / Λ ∗ ) a � �� � �� � �� � �� � log periodic linear divergence log divergence log periodic with H ′ 0 = (Λ d/d Λ) H 0

Three-Body Force (3BF) at NLO LO 3BF: RG limit cycle → discrete scaling symmetry λ n NLO 3BF: RG range-modification → discrete scaling breaking nσ 3BF up to NLO can be divided into 4 different contributions � r � � + ˜ η H ′ H (Λ) = H 0 (Λ / Λ ∗ ) � + h 10 (Λ / Λ ∗ )Λ r � + 0 (Λ / Λ ∗ ) ln(Λ /µ ) h 11 (Λ / Λ ∗ ) a � �� � �� � �� � �� � log periodic linear divergence log divergence log periodic with H ′ 0 = (Λ d/d Λ) H 0 Rewrite 3BF H 0 [ln(Λ / Λ ∗ )] + r aη H ′ H (Λ) = 0 [ln(Λ / Λ ∗ )] ln(Λ / Λ ∗ ) = H 0 [(1 + ηr/a ) ln(Λ / Λ ∗ )] � ln(Λ / Λ ∗ ) 1+ ηr/a � = H 0

Three-Body Force (3BF) at NLO LO 3BF: RG limit cycle → discrete scaling symmetry λ n NLO 3BF: RG range-modification → discrete scaling breaking nσ 3BF up to NLO can be divided into 4 different contributions � r � � + ˜ η H ′ H (Λ) = H 0 (Λ / Λ ∗ ) � + h 10 (Λ / Λ ∗ )Λ r � + 0 (Λ / Λ ∗ ) ln(Λ /µ ) h 11 (Λ / Λ ∗ ) a � �� � �� � �� � �� � log periodic linear divergence log divergence log periodic with H ′ 0 = (Λ d/d Λ) H 0 Rewrite 3BF H 0 [ln(Λ / Λ ∗ )] + r aη H ′ H (Λ) = 0 [ln(Λ / Λ ∗ )] ln(Λ / Λ ∗ ) = H 0 [(1 + ηr/a ) ln(Λ / Λ ∗ )] � ln(Λ / Λ ∗ ) 1+ ηr/a � = H 0 Running 3B parameter at NLO: κ ∗ ( Q, a ) = ( Q/κ ∗ ) − ηr/a κ ∗

RG Improvement insert running parameter into LO universal relation leads to Renormalization-Group Improved universal relations a i,n = λ n θ i ( λ n | θ i | ) ηrκ ∗ / ( λ n θ i ) κ − 1 + ˜ J i r ∗

RG Improvement insert running parameter into LO universal relation leads to Renormalization-Group Improved universal relations a i,n = λ n θ i ( λ n | θ i | ) ηrκ ∗ / ( λ n θ i ) κ − 1 + ˜ J i r ∗ expand a i,n up to linear-in- r correction a i,n = λ n θ i κ − 1 c.f. + ( J i + n σ ) r ∗ requires σ = ηπ/s 0