Effective Field Theory for Cold Atoms I

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1 Effective Field Theory for Cold Atoms I H.-W. Hammer Institut für Kernphysik, TU Darmstadt and Extreme Matter Institute EMMI School on Effective Field Theory across Length Scales, ICTP-SAIFR, Sao Paulo, Brazil, 2016 EFT for Cold Atoms p. 1

2 Agenda 1. EFT for Ultracold Atoms I: Effective Field Theories & Universality 2. EFT for Ultracold Atoms II: Cold Atoms & the Unitary Limit 3. EFT for Ultracold Atoms III: Weak Coupling at Finite Density 4. EFT for Ultracold Atoms IV: Few-Body Systems in the Unitary Limit 5. Beyond Ultracold Atoms: Halo Nuclei and Hadronic Molecules Literature G.P. Lepage, TASI Lectures 1989, arxiv:hep-ph/ D.B. Kaplan, arxiv:nucl-th/ E. Braaten, HWH, Phys. Rep. 428 (2006) 259 [arxiv:cond-mat/ ] EFT for Cold Atoms p. 2

3 Resolution of a Microscope Resolution of microscope is limited Diffraction effects: Light is EM wave (λ 550 nm) Resolution: d = λ A N Aperture: A N d max 0.4µm Quantum mechanics: particles have wave character matter waves λ = hc E, E m (de Broglie, 1924) QM microscope EFT for Cold Atoms p. 3

4 Hydrogen Atom Quantum bound state of e and p Energy levels (lowest order): Schrödinger-Eq. for V(r) = e 2 /r E = E 0 = m eα 2 2n 2, α = e2 4π [ Corrections beyond leading order: E = E 0 1+O(α, m ] e,...) m p corrections from EM interaction: fine structure L S α 4,... corrections from proton structure: finite proton mass m p hyperfine structure: µ p finite proton size: r p EFT for Cold Atoms p. 4

5 Dimensional Analysis Estimate effect of r p on E 0 Natural scales: a 0 = 1/(m e α) 0.5 Å Proton charge radius: G E (q 2 ) = 1+q 2 r 2 p/6 {}}{ G E(0)+... G E(0) 1/m 2 p, q 1/a 0 = ( ) me α m p Size of correction to E 0 : E Hz 30 khz Measurement of proton size in (muonic) hydrogen (m µ /m e 200) Proton radius puzzle: r p µh = ± fm r p H = ± fm (cf. Pohl et al., Ann. Rev. Nucl. Part. Sci. 63 (2013) 175 [arxiv: ]) How to construct an effective theory to calculate these corrections? EFT for Cold Atoms p. 5

6 Construction of an Effective Theory How to construct an effective theory to calculate these corrections? Need to identify: Low and high scales in the problem: M lo, M hi,... Active degrees of freedom: coordinates, particles,... Symmetries to constain interactions Power counting: organize the theory through expansion in M lo /M hi Guiding principle: Include long-range physics explicitly, parametrize short-range physics in low-energy constants Consider a classical example EFT for Cold Atoms p. 6

7 Example: Multipole Expansion Separation of scales: r r Short-range physics: expand in r r power counting Breakdown scale: r r r O ρ r φ( r) = l,m C l Y lm (Ω) r l+1 q lm {}}{ d 3 r ρ 1 ( r )Ylm (Ω )(r ) l 1 r r = l,m C l Y lm (Ω)Y lm (Ω ) rl < r l+1 > EFT for Cold Atoms p. 7

8 Example: Multipole Expansion Separation of scales: r r Short-range physics: expand in r r power counting Breakdown scale: r r ρ r O r ρ 2 φ( r) = l,m C l Y lm (Ω) r l+1 q lm {}}{ d 3 r ρ 1 ( r )Ylm (Ω )(r ) l + d 3 r ρ 2( r ) r r Long-range physics: include explicitly Use these ideas in the framework of Quantum Field Theory EFT for Cold Atoms p. 8

9 Weinbergs Conjecture Quantum Field Theory is only" way to satisfy quantum mechanics, Lorentz invariance, and cluster decomposition to calculate the most general S-matrix consistent with given symmetries for any theory below some scale simply use the most general effective Lagrangian L eff consistent with these principles in terms of the appropriate asymptotic states (Weinberg, 1979) Need power counting scheme to organize the infinitely many terms in L eff = predictive power EFT for Cold Atoms p. 9

10 Construction of an QFT 1. Construct L respecting symmetries (e.g. gauge invariance of QED) ψ ψ = e iα(x) ψ, A µ A µ = A µ 1 e µα µ (x) 2. Retain renormalizable interactions (D 4): [ψ] = 3/2, [A] = 1 keep ψγ µ ψa µ, }{{} but drop ψσ µν ψf µν, }{{} (F µν F µν ) 2 }{{} D=4 D=5 D=8 = L = ψ(i/ e/a m)ψ 1 4 Fµν F µν 3. Calculate Feynman diagrams: Fix parameters from data predictions Example: µ e = eg e S e 2m e, g e = 2 [ 1+ e2 8π 2 +O(e 4 ) O(e) O(e 3 ) ] = + + EFT for Cold Atoms p. 10

11 Renormalizability Renormalization: method to make sense of infinities in Quantum Field Theories Some observables are sensitive to short-distances match to experiment and predict other observables Classical renormalizability: redefinition of a finite number of terms sufficient to all orders (QED: e,m e,z 3,Z 2 ) EFT: new structures appear at every order Renormalizable theories have been very successful = Standard Model (SM) Non-renormalizable interactions (D > 4) are suppressed at low energies = factors 1/M D 4 hi Modern understanding: SM is also low-energy EFT, physics beyond SM can be included through non-renormalizable interactions EFT for Cold Atoms p. 11

12 Construction of an EFT Construct most general L eff respecting underlying symmetries Exploit separation of scales: E E c Non-renormalizable theory, but only finite number of operators in L contribute at each order Low-energy constants (LEC) Symmetries and light dof s must be known E heavy dof E c light dof Work at low energies: E E c Fix LEC s from matching (to experiment, other theory,...) Calculate observables in expansion in E/E c = limited range of applicability EFT for Cold Atoms p. 12

13 Light-By-Light Scattering Classic Example (Euler, Heisenberg, 1936) Photon energy ω, electron mass m e Separation of scales: ω m e theory simplifies Invariants: F µν F µν, (F µν F µν ) 2, (F Fµν µν ) 2,... L QED [ψ, ψ,a µ ] L eff [A µ ] L eff = 1 2 ( E 2 B 2 )+ Energy expansion: e 4 360π 2 m 4 e }{{} LEC (ω/m e ) 2n Cross section: σ(ω) = 1 16π π [( E 2 B 2 ) 2 +7( E B) 2 ] e 8 m 2 e ω 6 m 6 e +O(ω 8 ) Bounds from high-power lasers (Della Valle et al., PRD 90 (2014) ) EFT for Cold Atoms p. 13

14 Fermi Theory Weak decays mediated by W ± bosons (M W 80 GeV) energy release in neutron β-decay 1 MeV [n pe ν e ] energy release in kaon decays 300 MeV [K πeν] 1 8sin 2 θ W MW 2 q2 e 2 q2 M2 W e 2 8M 2 W sin2 θ W {1+ q2 M 2 W } +... = G F = Fermi Theory of Weak Interaction EFT for Cold Atoms p. 14

15 Structure of EFTs Energy momentum expansion (derivatives acting on fields) Dimensional analysis (breakdown scale Λ) derivatives powers of typical momentum q one derivative: q/λ vertex w/ N derivatives: N (q/λ) N terms with more derivatives are suppressed if q Λ Loops: additional powers of momenta from vertices and loop momenta = loops are generally suppressed compared to trees (exceptions!) (q/λ) N (q/λ) 2N d 4 pf(p,q) } {{ } (q/λ) M EFT for Cold Atoms p. 15

16 Universality Universality: Physical systems with different short-distance behavior exhibit identical behavior at large distances EFT for Cold Atoms p. 16

17 Universality Universality: Physical systems with different short-distance behavior exhibit identical behavior at large distances Condensed matter systems near critical point P P c B M > 0 M < 0 T c T T c T ρ liq/gas (T) ρ c ±A(T c T) β M 0 (T) A (T c T) β liquid-gas system Ferromagnet (one easy axis) Universality class determines critical exponents: β = EFT for Cold Atoms p. 16

18 Universality and Pointilism Painting at the limit of resolution of the human eye G. Seurat, A Sunday on La Grande Jatte EFT for Cold Atoms p. 17

19 Universality and Pointilism Painting at the limit of resolution of the human eye G. Seurat, A Sunday on La Grande Jatte EFT for Cold Atoms p. 17

20 Renormalization Group RG: Important concept in many areas of physics (Gell-Mann, Low, Kadanoff, Wilson,..) Critical phenomena, evolution of coupling constants, Fermi liquid theory,... L point g = (g 1,g 2,...) in space of coupling constants L = n g n O n RG transformation integrate out high-energy modes change of resolution scale couplings flow generate EFT RG flow expressed by differential equation for g: Λ d dλ g = β(g) EFT for Cold Atoms p. 18

21 Fixed Points RG fixed point: β(g ) = 0 Λ d dλ g = 0 Universal properties Scale invariance power law behavior Linearise: Λ d dλ g = B (g g ) = g g = Λ B c Critical exponent: B EFT for Cold Atoms p. 19

22 Fixed Points RG fixed point: β(g ) = 0 Λ d dλ g = 0 Universal properties Scale invariance power law behavior Linearise: Λ d dλ g = B (g g ) = g g = Λ B c Critical exponent: B System looks the same at different length scales M. Frame, Yale EFT for Cold Atoms p. 19

23 Fixed Points RG fixed point: β(g ) = 0 Λ d dλ g = 0 Universal properties Scale invariance power law behavior Linearise: Λ d dλ g = B (g g ) = g g = Λ B c Critical exponent: B System looks the same at different length scales M. Frame, Yale EFT for Cold Atoms p. 19

24 Limit Cycles RG limit cycle: (Wilson, 1971) Family L of Lagrangians: g(λ) = g(µ n Λ) L (θ) parametrized by angle θ lnλ [ Complex critical exponent: g = Re Λ 2πi/lnµ] = cos(2πlnλ/lnµ) Discrete scale invariance Log-periodic behavior EFT for Cold Atoms p. 20

25 Limit Cycles RG limit cycle: (Wilson, 1971) Family L of Lagrangians: g(λ) = g(µ n Λ) L (θ) parametrized by angle θ lnλ [ Complex critical exponent: g = Re Λ 2πi/lnµ] = cos(2πlnλ/lnµ) Discrete scale invariance Log-periodic behavior Russian Dolls EFT for Cold Atoms p. 20

26 Low-Energy Universality Renormalization Group: Systems with very different fundamental interactions can behave similarly at low energies = Universal properties Nuclear Physics: Nucleons and pions... Atomic Physics: Born-Oppenheimer plus van der Waals At sufficiently low energy: short-range interactions V ~ C 6 r 6 r Contact interactions p 2... EFT for Cold Atoms p. 21

27 Summary Short-distance physics can not be resolved at low energies Effective (Field) Theory: general method to exploit separation of scales = power counting in M lo /M hi Renormalizable and non-renormalizable field theories Universality: systems with different short-distance physics can behave identically at long distances = Renormalization group Fixed points Limit cycles scale invariance discrete scale invariance EFT for Cold Atoms p. 22

28 Additional Transparencies EFT for Cold Atoms p. 23