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Range e�ects on E�mov physics in cold atoms
An E�ective �eld theory approach
Chen Ji ‖ TRIUMF
in collaboration with D. R. Phillips, L. Platter
INT workshop, November 7 2012
Canada's national laboratory for particle and nuclear physics
Laboratoire national canadien pour la recherche en physique nucléaire et en physique des particules
Range e�ects on E�mov physics in cold atoms 1 / 22
Universal features at low energies
Separation of scales:a� `
2-body S-wave universality:Bd = 1/Ma2
Physics at large distance is insensitiveto physics at short distance
Large-distance physics is studied in`/a expansion
E�ects from SR-dynamics can beincluded in perturbation theory
Range e�ects on E�mov physics in cold atoms 2 / 22
Universal features at low energies
Separation of scales:a� `
2-body S-wave universality:Bd = 1/Ma2
Physics at large distance is insensitiveto physics at short distance
Large-distance physics is studied in`/a expansion
E�ects from SR-dynamics can beincluded in perturbation theory
Range e�ects on E�mov physics in cold atoms 2 / 22
Large-scattering-length physics
Universal Physics exists in systems with`� a
Nuclear Physics
Few-nucleon systems (NN , NNN):i.e., `tnp ∼ 1.7 fm, atnp ∼ 5.4 fm
Halo nuclei (6He, 11Li)Esep � Ecore
Atomic Physics
Cold atomic gases (133Cs, 7Li, 39K):r0 and a varies near Feshbachresonance
4He atoms (dimer, trimer, tetramer):`vdw ∼ 7Å, a ∼ 100Å
[www.theurbn.com]
[www.anl.gov]
Range e�ects on E�mov physics in cold atoms 3 / 22
Large-scattering-length physics
Universal Physics exists in systems with`� a
Nuclear Physics
Few-nucleon systems (NN , NNN):i.e., `tnp ∼ 1.7 fm, atnp ∼ 5.4 fm
Halo nuclei (6He, 11Li)Esep � Ecore
Atomic Physics
Cold atomic gases (133Cs, 7Li, 39K):r0 and a varies near Feshbachresonance
4He atoms (dimer, trimer, tetramer):`vdw ∼ 7Å, a ∼ 100Å
720 730 740 750 760B (G)
-1000
-500
0
500
1000
Un
it o
f a
B
a
a
r0
7Li atoms [Gross et al. '09]
6He tetramer [D. Blume]
Range e�ects on E�mov physics in cold atoms 3 / 22
E�ective �eld theory
An approach to systems with a separation of scalesSystems with `� a → an EFT with contact interactions
Few-nucleon systems → pionless EFTHalo nuclei → halo EFTAtomic systems → short-range EFT
Physical quantities are expanded in powers of r0/a
Contact interactions at LO
2-body contact interaction (LO) introduce a dimer �eld
= −iC0
C0 determined by a 2-body observable
3-body contact interaction (LO)
= −iD0
D0 determined by a 3-body observable Bedaque, Hammer, van Kolck '99
Range e�ects on E�mov physics in cold atoms 4 / 22
E�ective �eld theory
An approach to systems with a separation of scalesSystems with `� a → an EFT with contact interactions
Few-nucleon systems → pionless EFTHalo nuclei → halo EFTAtomic systems → short-range EFT
Physical quantities are expanded in powers of r0/a
Contact interactions at LO
2-body contact interaction (LO) introduce a dimer �eld
= −iC0 = −i√2gC0=g
2/∆==============⇒
C0 determined by a 2-body observable
3-body contact interaction (LO)
= −iD0 = ihD0=−3hg2/∆2
=================⇒
D0 determined by a 3-body observable Bedaque, Hammer, van Kolck '99
Range e�ects on E�mov physics in cold atoms 4 / 22
EFT for 3 identical bosons
LO (r0/a)0 EFT Lagrangian for 3 identical bosons
L = ψ†(i∂0 +
∇2
2m
)ψ−d†
(i∂0 +
∇2
4m−∆
)d− g√
2
(d†ψψ + h.c
)+hd†dψ†ψ+· · ·
terms with more derivatives are at higher orders (r0/a)n
Non-perturbative features at LO
atom-atom (dimer) scattering (tune g)
= + ∝ 11/a+ik
atom-dimer scattering (tune h)
t0 = t0 t0+ + +
Bedaque, Hammer, van Kolck '99
Range e�ects on E�mov physics in cold atoms 5 / 22
LO renormalization
Without 3BF:
3-body spectrum:cuto� dependent (Λ ∼ 1/`)Platter '09
LO 3BF h:
tune H(Λ) = Λ2h/2mg2:�x one 3-body observable
limit cycle:H(Λ) periodic for Λ→ Λ(22.7)n
Bedaque et al. '00scaling invariance → E�mov physicsE�mov '71
101
102
103
Λ [B2
1/2]
101
102
103
B3[B
2]
Range e�ects on E�mov physics in cold atoms 6 / 22
Universal physics at LO
Universal features in three-body systems (E�mov e�ects)3-body spectrum: a function of scattering length a
geometric spectrumEn = (22.7)−2En−1 in the limit a→∞
universal relation of recombination featuresa∗ = a+/4.5 = −a−/21.3i.e. a−(n) = 22.7a−(n−1)
Zaccanti et al. '09Range e�ects on E�mov physics in cold atoms 7 / 22
Recombination features of cold atoms
loss rates of free atoms inultracold atomic gases
39K atoms
Zaccanti et al. '09
7Li atoms |F = 1,mF = 0〉Gross et al. '09
7Li atoms |F = 1,mF = 1〉Pollack et al. '09
Pic. credit: Ferlaino, Grimm '10
Range e�ects on E�mov physics in cold atoms 8 / 22
Beyond universality: range e�ects
range e�ects on universal physics
2-body observable: k cot δ0 = − 1a + r0
2 k2 + · · ·
3-body observables: → in r0/a expansion
Previous EFT calculations of r0/a corrections (�xed a):
Hammer, Mehen '01 (NLO)Bedaque, Rupak, Griesshammer, Hammer '03 (N2LO, partial iteration)Platter, Phillips '06 (N2LO, partial iteration)
We perform a rigorous perturbative caluclationNLO (systems with variable a)
ultracold atomic gases (r0/a varies near Feshbach resonance)3-body recombination
N2LO (systems with a �xed a)4He atoms (r0/a ∼ 0.07)4He trimer (bound state), atom-dimer phase shift (scattering state)
Range e�ects on E�mov physics in cold atoms 9 / 22
NLO (r0/a) range e�ects
Calculate r0/a correction to atom-dimer amplitudeNLO correction to atom-atom propagator (dimer):
= ir02πmg2
1/a+ik−1/a+ik
NLO correction to atom-dimer contact term (3BF):
= i2mg2
Λ2H1(a,Λ)
Contribution in 1st order perturbation theory:
NLO dimer:
NLO 3BF: t0 t0 t0t0
t0 t0
NLO 3-body force:
H1(Λ) = r0Λ h10(Λ) + r0/a h11(Λ)a �xed: h11 is absorbed (no additional 3-body input is needed)a varies: one additional 3-body input is needed
Range e�ects on E�mov physics in cold atoms 10 / 22
NLO three-body 3BF
H1(Λ) = r0Λ h10(Λ) + r0/a h11(Λ)
101
102
103
104
105
106
Λ/κ*
-2.5
-2
-1.5
-1
-0.5
0
1 /
h10(Λ
)
1/h10(Λ)
101
102
103
104
105
106
Λ/κ*
-0.5
-0.4
-0.3
-0.2
-0.1
0
1 /
h11(Λ
)
1/h11(Λ)
Range e�ects on E�mov physics in cold atoms 11 / 22
In ultracold atomic gases
a and r0 are functions of the magnetic �eld
7Li |F = 1,mF = 0 > 7Li |F = 1,mF = 1 >
Gross et al. '09
Range e�ects on E�mov physics in cold atoms 12 / 22
Recombination of 7Li Atoms
|F = 1,mF = 0〉
3-body recombination rate of 7Liatoms
NLO (r0/a) range e�ects torecombination
positions are not shifted bydeep-dimer at LO[Braaten et al. '08] 0.1 1
a [1000 aB]
0.1
1
10
K3[1
000 a
B]
LO: rn to a−
NLO: rn to a−
& a+
Gross et al. data
Experiment†‡ LO LO NLO?
3A res a−(-) [aB] -264† -264 -244 -264
rec min a+ [aB] 1160† 1254 1160 1160
Ad res a∗ [aB] 180‡ 281 259 210(44)
† Gross et al. '09 ‡ Machtey et al. '12 ? Ji, Phillips, Platter '10
Range e�ects on E�mov physics in cold atoms 13 / 22
Recombination of 7Li Atoms
|F = 1,mF = 0〉
3-body recombination rate of 7Liatoms
NLO (r0/a) range e�ects torecombination
positions are not shifted bydeep-dimer at LO[Braaten et al. '08] 0.1 1
a [1000 aB]
0.1
1
10
K3[1
000 a
B]
LO: rn to a−
NLO: rn to a−
& a+
Gross et al. data
Experiment†‡ LO LO NLO?
3A res a−(-) [aB] -264† -264 -244 -264
rec min a+ [aB] 1160† 1254 1160 1160
Ad res a∗ [aB] 180‡ 281 259 210(44)
† Gross et al. '09 ‡ Machtey et al. '12 ? Ji, Phillips, Platter '10
Range e�ects on E�mov physics in cold atoms 13 / 22
Recombination of 7Li Atoms
|F = 1,mF = 0〉
3-body recombination rate of 7Liatoms
NLO (r0/a) range e�ects torecombination
positions are not shifted bydeep-dimer at LO[Braaten et al. '08] 0.1 1
a [1000 aB]
0.1
1
10
K3[1
000 a
B]
LO: rn to a−
NLO: rn to a−
& a+
Gross et al. data
Experiment†‡ LO LO NLO?
3A res a−(-) [aB] -264† -264 -244 -264
rec min a+ [aB] 1160† 1254 1160 1160
Ad res a∗ [aB] 180‡ 281 259 210(44)
† Gross et al. '09 ‡ Machtey et al. '12 ? Ji, Phillips, Platter '10
Range e�ects on E�mov physics in cold atoms 13 / 22
Recombination of 7Li Atoms
|F = 1,mF = 0〉
3-body recombination rate of 7Liatoms
NLO (r0/a) range e�ects torecombination
positions are not shifted bydeep-dimer at LO[Braaten et al. '08] 0.1 1
a [1000 aB]
0.1
1
10
K3[1
000 a
B]
LO: rn to a−
NLO: rn to a−
& a+
Gross et al. data
Experiment†‡ LO LO NLO?
3A res a−(-) [aB] -264† -264 -244 -264
rec min a+ [aB] 1160† 1254 1160 1160
Ad res a∗ [aB] 180‡ 281 259 210(44)
† Gross et al. '09 ‡ Machtey et al. '12 ? Ji, Phillips, Platter '10
Range e�ects on E�mov physics in cold atoms 13 / 22
Recombination of 7Li Atoms
|F = 1,mF = 1〉7Li(mF = 1) recombination[Pollack et al. '09]
data disagree with universality bya factor of 2
EFT analysis at NLO agrees withuniversality
10-1
100
101
a [1000 aB]
10-2
10-1
100
101
102
K3
[1
00
0 a
B]
LO: rn to a−
1
NLO: rn to a−
1 & a
−
2
Experiment LO LO NLO[Pollack et al. '09] [Ji et al. '10]
3A res a−(-) [aB] -298 -298 -278 -298
3A res a− [aB] -6301 -6763 -6301 -6301rec min a+ [aB] 2676 1415 1319 1348(151)Ad res a∗ [aB] 608 317 295 356(55)
Range e�ects on E�mov physics in cold atoms 14 / 22
Recombination of 7Li Atoms
|F = 1,mF = 1〉7Li(mF = 1) recombination[Pollack et al. '09]
data disagree with universality bya factor of 2
EFT analysis at NLO agrees withuniversality
10-1
100
101
a [1000 aB]
10-2
10-1
100
101
102
K3
[1
00
0 a
B]
LO: rn to a−
1
NLO: rn to a−
1 & a
−
2
Experiment LO LO NLO[Pollack et al. '09] [Ji et al. '10]
3A res a−(-) [aB] -298 -298 -278 -298
3A res a− [aB] -6301 -6763 -6301 -6301rec min a+ [aB] 2676 1415 1319 1348(151)Ad res a∗ [aB] 608 317 295 356(55)
Range e�ects on E�mov physics in cold atoms 14 / 22
Recombination of 7Li Atoms
|F = 1,mF = 1〉7Li(mF = 1) recombination[Pollack et al. '09]
data disagree with universality bya factor of 2
EFT analysis at NLO agrees withuniversality
10-1
100
101
a [1000 aB]
10-2
10-1
100
101
102
K3
[1
00
0 a
B]
LO: rn to a−
1
NLO: rn to a−
1 & a
−
2
Experiment LO LO NLO[Pollack et al. '09] [Ji et al. '10]
3A res a−(-) [aB] -298 -298 -278 -298
3A res a− [aB] -6301 -6763 -6301 -6301rec min a+ [aB] 2676 1415 1319 1348(151)Ad res a∗ [aB] 608 317 295 356(55)
Range e�ects on E�mov physics in cold atoms 14 / 22
Recombination of 7Li Atoms
|F = 1,mF = 1〉7Li(mF = 1) recombination[Pollack et al. '09]
data disagree with universality bya factor of 2
EFT analysis at NLO agrees withuniversality
10-1
100
101
a [1000 aB]
10-2
10-1
100
101
102
K3
[1
00
0 a
B]
LO: rn to a−
1
NLO: rn to a−
1 & a
−
2
Experiment LO LO NLO[Pollack et al. '09] [Ji et al. '10]
3A res a−(-) [aB] -298 -298 -278 -298
3A res a− [aB] -6301 -6763 -6301 -6301rec min a+ [aB] 2676 1415 1319 1348(151)Ad res a∗ [aB] 608 317 295 356(55)
Range e�ects on E�mov physics in cold atoms 14 / 22
Universal relations at NLO
1a = γ − 1
2r0γ2
NLO corrections
γ0 = − 0.210γ(−)− + r0(a0) · I(−)
0 γ(−)−
2
γ∗ = − 0.939γ(−)− + r0(a∗) · I(−)
∗ γ(−)−
2
Linear correlation at NLO
I(−)∗ = −0.309 + 7.17 I(−)
0
Universal relation at NLO
γ∗ = − 0.939γ(−)− − 0.309 r0(a∗)γ
(−)−
2+ 7.17 r0(a∗)
r0(a0)
(γ0 + 0.210γ
(−)−
)γ∗ = 4.47γ0 − 7.02 r0(a∗)γ20 + 0.566 r0(a∗)
r0(a(−)− )
(γ(−)− + 4.76γ0
)
Ji, Phillips, Platter '12
Range e�ects on E�mov physics in cold atoms 15 / 22
Universal relations at NLO
1a = γ − 1
2r0γ2
NLO corrections
γ0 = − 0.210γ(−)− + r0(a0) · I(−)
0 γ(−)−
2
γ∗ = − 0.939γ(−)− + r0(a∗) · I(−)
∗ γ(−)−
2
Linear correlation at NLO
I(−)∗ = −0.309 + 7.17 I(−)
0
Universal relation at NLO
γ∗ = − 0.939γ(−)− − 0.309 r0(a∗)γ
(−)−
2+ 7.17 r0(a∗)
r0(a0)
(γ0 + 0.210γ
(−)−
)γ∗ = 4.47γ0 − 7.02 r0(a∗)γ20 + 0.566 r0(a∗)
r0(a(−)− )
(γ(−)− + 4.76γ0
)
Ji, Phillips, Platter '12
Range e�ects on E�mov physics in cold atoms 15 / 22
Universal relations at NLO
1a = γ − 1
2r0γ2
NLO corrections
γ0 = − 0.210γ(−)− + r0(a0) · I(−)
0 γ(−)−
2
γ∗ = − 0.939γ(−)− + r0(a∗) · I(−)
∗ γ(−)−
2
Linear correlation at NLO
I(−)∗ = −0.309 + 7.17 I(−)
0
Universal relation at NLO
γ∗ = − 0.939γ(−)− − 0.309 r0(a∗)γ
(−)−
2+ 7.17 r0(a∗)
r0(a0)
(γ0 + 0.210γ
(−)−
)γ∗ = 4.47γ0 − 7.02 r0(a∗)γ20 + 0.566 r0(a∗)
r0(a(−)− )
(γ(−)− + 4.76γ0
)
Ji, Phillips, Platter '12
Range e�ects on E�mov physics in cold atoms 15 / 22
Universal relations at NLO
1a = γ − 1
2r0γ2
NLO corrections
γ0 = − 0.210γ(−)− + r0(a0) · I(−)
0 γ(−)−
2
γ∗ = − 0.939γ(−)− + r0(a∗) · I(−)
∗ γ(−)−
2
Linear correlation at NLO
I(−)∗ = −0.309 + 7.17 I(−)
0
Universal relation at NLO
γ∗ = − 0.939γ(−)− − 0.309 r0(a∗)γ
(−)−
2+ 7.17 r0(a∗)
r0(a0)
(γ0 + 0.210γ
(−)−
)
γ∗ = 4.47γ0 − 7.02 r0(a∗)γ20 + 0.566 r0(a∗)
r0(a(−)− )
(γ(−)− + 4.76γ0
)
Ji, Phillips, Platter '12
Range e�ects on E�mov physics in cold atoms 15 / 22
Universal relations at NLO
1a = γ − 1
2r0γ2
NLO corrections
γ0 = − 0.210γ(−)− + r0(a0) · I(−)
0 γ(−)−
2
γ∗ = − 0.939γ(−)− + r0(a∗) · I(−)
∗ γ(−)−
2
Linear correlation at NLO
I(−)∗ = −0.309 + 7.17 I(−)
0
Universal relation at NLO
γ∗ = − 0.939γ(−)− − 0.309 r0(a∗)γ
(−)−
2+ 7.17 r0(a∗)
r0(a0)
(γ0 + 0.210γ
(−)−
)γ∗ = 4.47γ0 − 7.02 r0(a∗)γ20 + 0.566 r0(a∗)
r0(a(−)− )
(γ(−)− + 4.76γ0
)Ji, Phillips, Platter '12
Range e�ects on E�mov physics in cold atoms 15 / 22
N2LO (r0/a)2 range e�ects
N2LO corrections to atom-dimer scattering amplitude:in 2nd order perturbation theory (∼ r20/a2):
N2LO dimer:
N2LO 3BF:
two NLOterms:
t0 t0
t0 t0 t0 t0
t0 t0 t0 t0 t0t0 + · · ·
t0t0 t0 + · · ·
N2LO 3-body force:H2(E,Λ) = r20Λ2 h20(Λ) + r20mE3 h22(Λ)
→ one additional 3-body input is needed (even when a is �xed)
c.f. Bedaque et al. '03 & Platter, Phillips '06
Range e�ects on E�mov physics in cold atoms 16 / 22
N2LO renormalization
H2 = r20Λ2 h20
102
103
104
105
Λ [γ]
-1
-0.5
0
0.5
1
1.5
Bn(1
) [γ
n B
d]
B0
(1)
B1
(1)
B2
(1)
aad → LO/NLO/N2LO
B(1)t = B
(1)0 + r0B
(1)1 + r20B
(1)2
H2 = r20Λ2 h20 + r20mEt h22
102
103
104
105
Λ [γ]
-200
-100
0
100
Bn(0
) [γ
n B
d]
B0
(0)
B1
(0)
0.1B2
(0)
aad → LO/NLO/N2LO
B(1)t → N2LO
B(0)t = B
(0)0 + r0B
(0)1 + r20B
(0)2
Range e�ects on E�mov physics in cold atoms 17 / 22
NLO three-body 3BF
H2(Λ) = r20Λ2 h20(Λ) + r20mEt h22(Λ)
101
102
103
104
105
Λ [γ]
-40
-20
0
20
40
1 /
h20(Λ
)
1/h20(Λ)
101
102
103
104
105
Λ [γ]
0
0.2
0.4
0.6
0.8
1 /
h22(Λ
)
1/h22(Λ) [partial version]
Range e�ects on E�mov physics in cold atoms 18 / 22
4He trimer
Input B(1)t [Bd] B
(0)t [Bd] aad [γ−1] rad [γ−1]
TTY potential 1.738 96.33 1.205
aad LO 1.723 97.12 1.205 0.8352aad NLO 1.736 89.72 1.205 0.9049
aad, B(1)t N2LO 1.738 116.9 1.205 0.9132
B(1)t LO 1.738 99.37 1.178 0.8752
B(1)t NLO 1.738 89.77 1.201 0.9130
B(1)t , aad N2LO 1.738 115.9 1.205 0.9135
Di�erence in 2 renormalization schemes (LO→NLO→N2LO):
atom-dimer e�ective range rad: 5% → 0.9% → 0.02%ground-state trimer B
(0)t : 2% → 0.07% → 0.9%
Compare with TTY:
∆B(0)t ∼20%r20B
(0)t ∼ 0.5
EFT expansion needs to be corrected for deep trimer with Uvdw [Gao '98]
Range e�ects on E�mov physics in cold atoms 19 / 22
4He trimer
Input B(1)t [Bd] B
(0)t [Bd] aad [γ−1] rad [γ−1]
TTY potential 1.738 96.33 1.205
aad LO 1.723 97.12 1.205 0.8352aad NLO 1.736 89.72 1.205 0.9049
aad, B(1)t N2LO 1.738 116.9 1.205 0.9132
B(1)t LO 1.738 99.37 1.178 0.8752
B(1)t NLO 1.738 89.77 1.201 0.9130
B(1)t , aad N2LO 1.738 115.9 1.205 0.9135
Di�erence in 2 renormalization schemes (LO→NLO→N2LO):
atom-dimer e�ective range rad: 5% → 0.9% → 0.02%ground-state trimer B
(0)t : 2% → 0.07% → 0.9%
Compare with TTY:
∆B(0)t ∼20%r20B
(0)t ∼ 0.5
EFT expansion needs to be corrected for deep trimer with Uvdw [Gao '98]
Range e�ects on E�mov physics in cold atoms 19 / 22
4He trimer
Input B(1)t [Bd] B
(0)t [Bd] aad [γ−1] rad [γ−1]
TTY potential 1.738 96.33 1.205
aad LO 1.723 97.12 1.205 0.8352aad NLO 1.736 89.72 1.205 0.9049
aad, B(1)t N2LO 1.738 116.9 1.205 0.9132
B(1)t LO 1.738 99.37 1.178 0.8752
B(1)t NLO 1.738 89.77 1.201 0.9130
B(1)t , aad N2LO 1.738 115.9 1.205 0.9135
Di�erence in 2 renormalization schemes (LO→NLO→N2LO):
atom-dimer e�ective range rad: 5% → 0.9% → 0.02%ground-state trimer B
(0)t : 2% → 0.07% → 0.9%
Compare with TTY:
∆B(0)t ∼20%r20B
(0)t ∼ 0.5
EFT expansion needs to be corrected for deep trimer with Uvdw [Gao '98]
Range e�ects on E�mov physics in cold atoms 19 / 22
4He trimer
Input B(1)t [Bd] B
(0)t [Bd] aad [γ−1] rad [γ−1]
TTY potential 1.738 96.33 1.205
aad LO 1.723 97.12 1.205 0.8352aad NLO 1.736 89.72 1.205 0.9049
aad, B(1)t N2LO 1.738 116.9 1.205 0.9132
B(1)t LO 1.738 99.37 1.178 0.8752
B(1)t NLO 1.738 89.77 1.201 0.9130
B(1)t , aad N2LO 1.738 115.9 1.205 0.9135
Di�erence in 2 renormalization schemes (LO→NLO→N2LO):atom-dimer e�ective range rad: 5% → 0.9% → 0.02%ground-state trimer B
(0)t : 2% → 0.07% → 0.9%
Compare with TTY:
∆B(0)t ∼20%r20B
(0)t ∼ 0.5
EFT expansion needs to be corrected for deep trimer with Uvdw [Gao '98]
Range e�ects on E�mov physics in cold atoms 19 / 22
4He trimer
Input B(1)t [Bd] B
(0)t [Bd] aad [γ−1] rad [γ−1]
TTY potential 1.738 96.33 1.205
aad LO 1.723 97.12 1.205 0.8352aad NLO 1.736 89.72 1.205 0.9049
aad, B(1)t N2LO 1.738 116.9 1.205 0.9132
B(1)t LO 1.738 99.37 1.178 0.8752
B(1)t NLO 1.738 89.77 1.201 0.9130
B(1)t , aad N2LO 1.738 115.9 1.205 0.9135
Di�erence in 2 renormalization schemes (LO→NLO→N2LO):atom-dimer e�ective range rad: 5% → 0.9% → 0.02%ground-state trimer B
(0)t : 2% → 0.07% → 0.9%
Compare with TTY:∆B
(0)t ∼20%r20B
(0)t ∼ 0.5
EFT expansion needs to be corrected for deep trimer with Uvdw [Gao '98]Range e�ects on E�mov physics in cold atoms 19 / 22
He-4 trimer phase shift at N2LO
aad → LO/NLO/N2LO
B(1)t → N2LO
0 0.2 0.4 0.6 0.8 1k [γ]
-0.8
-0.7
-0.6
-0.5
-0.4
kcotδ
ad
[γ]
LO (aad
)
NLO (aad
)
N2LO (a
ad, B
t
(1))
B(1)t → LO/NLO/N2LO
aad → N2LO
0 0.2 0.4 0.6 0.8 1k [γ]
-0.8
-0.7
-0.6
-0.5
-0.4
kcotδ
ad
[γ]
LO (Bt
(1))
NLO (Bt
(1))
N2LO (B
t
(1), aad
)
Range e�ects on E�mov physics in cold atoms 20 / 22
Di�erence btw 2 renormalization
LO → NLO→ N2LO
0 0.2 0.4 0.6 0.8 1k [γ]
10-9
10-8
10-7
10-6
10-5
10-4
10-3
10-2
10-1
100
|∆ k
co
tδad| [γ
]
Range e�ects on E�mov physics in cold atoms 21 / 22
Conclusion
We studied e�ective-range corrections on E�mov physics inperturbation theory
NLO for varying a:
recombination in cold atomsH1(Λ) = r0Λ h10(Λ) + r0/a h11(Λ)one additional 3-body input is needed for NLO renormalization
N2LO for �xed a:
He-4 trimerH2(E,Λ) = r20Λ2 h20(Λ) + r20mE3 h22(Λ)one additional 3-body data is needed for N2LO renormalization
Range corrections are also important in nuclear physics
Range e�ects on E�mov physics in cold atoms 22 / 22
