Fermion Many-Body Systems II

Transcription

1 Outline Atoms Stars Answer RG Department of Physics Ohio State University June, 2004

2 Outline Outline Atoms Stars Answer RG Universality (Un)like Overview of Cold Atoms Overview of Neutron Stars Answer: How are Cold Atoms Like Neutron Stars? Renormalization Group Methods

3 Outline Atoms Stars Answer RG Universality (Un)like Nuclear and Cold Atom Many-Body Problems Lennard-Jones and nucleon-nucleon potentials O -O potential (mev) n-n potential (MeV) n-n system O -O system n-n distance (fm) O 2-O 2 distance (nm) [figure borrowed from J. Dobaczewski] Are there universal features of such many-body systems? How can we deal with hard cores in many-body systems?

4 Outline Atoms Stars Answer RG Universality (Un)like How Are Cold Atoms Like Neutron Stars? Regal et al., ultracold fermions Chandra X-Ray Observatory image of pulsar in 3C58

5 Outline Atoms Stars Answer RG Universality (Un)like How Are Cold Atoms UnLike Neutron Stars? Regal et al., ultracold fermions Chandra X-Ray Observatory image of pulsar in 3C58

6 Outline Atoms Stars Answer RG Universality (Un)like How Are Cold Atoms UnLike Neutron Stars? Regal et al., ultracold fermions Microscopic Chandra X-Ray Observatory image of pulsar in 3C58 Macroscopic

7 Outline Atoms Stars Answer RG Universality (Un)like How Are Cold Atoms UnLike Neutron Stars? Regal et al., ultracold fermions Microscopic Atoms (bosons and fermions) Chandra X-Ray Observatory image of pulsar in 3C58 Macroscopic Nucleons (protons, neutrons)

8 Outline Atoms Stars Answer RG Universality (Un)like How Are Cold Atoms UnLike Neutron Stars? Regal et al., ultracold fermions Microscopic Atoms (bosons and fermions) Temperature: K Chandra X-Ray Observatory image of pulsar in 3C58 Macroscopic Nucleons (protons, neutrons) Temperature: K

9 Outline Atoms Stars Answer RG Low T Trap/Cool Imaging Bosons and Fermions at Low Temperature Temperature of a gas of particles Distribution of (kinetic) energies Temperature as measure of average energy When are quantum effects important? [Hint: λ = h/p = h/mv] Temperature and energy states Bosons vs. fermions What atoms can you trap magnetically? If the atom has an unpaired electron it has a magnetic moment, so external magnetic fields exert forces

10 Outline Atoms Stars Answer RG Low T Trap/Cool Imaging

11 Outline Atoms Stars Answer RG Low T Trap/Cool Imaging How Do You Trap and Cool Atoms? Optical Molasses applet Transfer momentum to atoms by absorbing photons from forward direction (and re-emit randomly); use Doppler shift Magnetic trapping applet Evaporative cooling applet Removal of most energetic atoms by rf-induced evaporation In higher magnetic field = in resonance with rf or microwaves, = spin flips and atom escapes Optical trapping and cooling Use a single focused beam from a high-power ultra-stable CO 2 laser. Near focal point a potential well is formed. Slowly reduce laser intensity to allow hottest atoms to escape. Why do you need optical trapping for fermions?

12 Outline Atoms Stars Answer RG Low T Trap/Cool Imaging What Do These Experiments Look Like?

13 Outline Atoms Stars Answer RG Low T Trap/Cool Imaging Imaging the Atoms Measure low temperatures by looking at extension of cloud = larger means more energetic Size of atomic cloud found by shadow of laser absorption = atoms energy and temperature (since field known) Kinetic energy distribution by turning off trap and imaging ballistic expansion

14 Outline Atoms Stars Answer RG Low T Trap/Cool Imaging Forming a BEC The sharp peak is the signature of a BEC

15 Outline Atoms Stars Answer RG Facts Pulsars Anatomy QCD Phases The Fate of Stars Too light (< 8 solar masses) = white dwarf Too heavy (> 40 solar masses) = black hole Just right = star goes supernova, with neutron star left behind

16 Outline Atoms Stars Answer RG Facts Pulsars Anatomy QCD Phases Neutron Star vs. Bar Harbor Radius 10 km Mass 1.4 M sun Density g/cc Spin rate 38, 000 rpm Magnetic field Gauss

17 Outline Atoms Stars Answer RG Facts Pulsars Anatomy QCD Phases More Neutron Star Fun Facts... Roughly a giant nucleus (mostly neutrons) bound by gravity (?) Extremely large gravitational field: A marshmallow dropped from shoulder height onto the surface of a neutron star would hit the surface with the kinetic energy of an atom bomb. In 10 billion year lifetime of galaxy, probably 10 8 to 10 9 neutron stars formed In gauss field, the binding energy of hydrogen is 160 ev Blackbody radiation peaked in X-ray (but small surface area)

18 Outline Atoms Stars Answer RG Facts Pulsars Anatomy QCD Phases Pulsating Radio Sources or Pulsars In 1967, Joceyln Bell observed sec. period radio pulses Rotating neutron stars by process of elimination 300, 000 km/s s < 10, 000 km white dwarf pulsational, rotational, orbital periods too large Synchrotron radiation from charged particles along field lines magnetic poles don t align with rotational = lighthouse effect

19 Outline Atoms Stars Answer RG Facts Pulsars Anatomy QCD Phases SAXJ : X-Ray Millisecond Pulsar Billion year history of binary system: companion spirals into massive star massive star core goes supernova = neutron star companion loses mass = accretion disk accretion = faster spin, up to few millisecond period and X-rays, which vaporize companion at end, compact rapidly rotating neutron star with pair of radio beams NASA s Rossi X-ray Timing Explorer (RXTE)

20 Neutron Stars Outline Atoms Stars Answer RG Facts Pulsars Anatomy QCD Phases Dilute neutrons at surface = neutron superfluid High density core: color superconductor from quark Cooper pairs (???)

21 Outline Atoms Stars Answer RG Facts Pulsars Anatomy QCD Phases The QCD Phase Diagram

22 Outline Atoms Stars Answer RG Review Atoms Large a 0 1-D Quick Review of Scattering P/2 + k P/2 + k sin(kr+δ) P/2 k P/2 k 0 R Relative motion in frame with P = 0: ψ(r) r e ik r + f (k, θ) eikr r where k 2 = k 2 = ME k and cos θ = ˆk ˆk Differential cross section is dσ/dω = f (k, θ) 2 Central V = partial waves: f (k, θ) = l (2l + 1)f l(k)p l (cos θ) where f l (k) = eiδ l (k) sin δ l (k) k and the S-wave phase shift is defined by = 1 k cot δ l (k) ik u 0 (r) r sin[kr + δ 0 (k)] = δ 0 (k) = kr for hard sphere r

23 Outline Atoms Stars Answer RG Review Atoms Large a 0 1-D Effective Range Expansion Total cross section: σ total = 4π k 2 (2l + 1) sin 2 δ l (k) l=0

24 Outline Atoms Stars Answer RG Review Atoms Large a 0 1-D Effective Range Expansion Total cross section: σ total = 4π k 2 (2l + 1) sin 2 δ l (k) What happens at low energy (λ = 2π/k 1/R)? l=0 k cot δ 0 (k) k 0 1 a r 0k a 0 is the scattering length and r 0 is the effective range

25 Outline Atoms Stars Answer RG Review Atoms Large a 0 1-D Near-Zero-Energy Bound States Bound-state or near-bound state at zero energy = large scattering lengths (a 0 ± )

26 Outline Atoms Stars Answer RG Review Atoms Large a 0 1-D Near-Zero-Energy Bound States Bound-state or near-bound state at zero energy = large scattering lengths (a 0 ± ) For kr 0, the total cross section is { 4πa0 2 4πa 2 σ total = σ l=0 = 1 + (ka 0 ) 2 = 0 for ka 0 1 4π k 2 for ka 0 1 (unitarity limit)

27 Outline Atoms Stars Answer RG Review Atoms Large a 0 1-D Trapped Fermion Atoms Low densities and temperatures = only a 0 enters Use Feshbach resonances to tune scattering length a 0 ± scattering length (a o ) B (gauss) Superfluidity? BCS-BEC crossover? Universal behavior?

28 Outline Atoms Stars Answer RG Review Atoms Large a 0 1-D Trapped Fermion Atoms Low densities and temperatures = only a 0 enters Use Feshbach resonances to tune scattering length a 0 ± scattering length (a o ) B (gauss) Superfluidity? BCS-BEC crossover? Universal behavior?

29 Outline Atoms Stars Answer RG Review Atoms Large a 0 1-D Trapped Fermion Atoms Low densities and temperatures = only a 0 enters Use Feshbach resonances to tune scattering length a 0 ± scattering length (a o ) B (gauss) N 0 / N B (gauss) t (ms) B (gauss) Superfluidity? BCS-BEC crossover? Universal behavior?

30 Outline Atoms Stars Answer RG Review Atoms Large a 0 1-D Trapped Fermion Atoms Low densities and temperatures = only a 0 enters Use Feshbach resonances to tune scattering length a 0 ± scattering length (a o ) B (gauss) Superfluidity? BCS-BEC crossover? Universal behavior?

31 Outline Atoms Stars Answer RG Review Atoms Large a 0 1-D Metaphor for a Fermion Condensate (JILA) These dancers represent our fermionic atoms. They appear to be moving independently, however, there is a subtle pairing: You can see this from the dancers eye contact and body language. These pairs act like bosons and can form a condensate.

32 Outline Atoms Stars Answer RG Review Atoms Large a 0 1-D Metaphor for Bound Molecules (JILA) To see the condensation of pairs of fermionic atoms we suddenly bring together the two atoms (or dancers here) in each of these subtle pairs. When we look at the motion of these bound pairs the condensation apparent.

33 Outline Atoms Stars Answer RG Review Atoms Large a 0 1-D What is Superfluidity? Superfluidity is a state of matter characterized by the complete absence of viscosity Superfluids in a closed loop can flow endlessly without friction Discovered by Kapitsa, Allen, and Misener in 1937

34 Outline Atoms Stars Answer RG Review Atoms Large a 0 1-D What is Superfluidity? Superfluidity is a state of matter characterized by the complete absence of viscosity Superfluids in a closed loop can flow endlessly without friction Discovered by Kapitsa, Allen, and Misener in 1937 Where is superfluidity found? 4 He bosons = BEC with interactions 3 He fermions = Cooper pairs Bose condense (as in superconductivity) Cold atomic gases Neutron stars

35 Outline Atoms Stars Answer RG Review Atoms Large a 0 1-D Superfluid Vorticity Quantized vorticity: velocity around each line determined by h/m (m is mass of one atom) Quantum phenomenon on a macroscopic scale # of vortex lines h/m: 1000 in 1 cm radius container at 1 rpm [For the experts: vortex lines in superfluid analogous to magnetic flux lines in type II superconductors.]

36 Outline Atoms Stars Answer RG Review Atoms Large a 0 1-D How Cold Must a Superfluid Be? Need coherence: large-scale quantum correlations Roughly the de Broglie wavelength must be comparable to the average distance d between particles Use λ = h/mv and mv 2 /2 3kT /2 to estimate T c h2 3mkd 2 Transition temperatures: 4 He transition at 2.17 K ( C) 3 He transition at 2.6 mk neutron star can be superfluid even at much higher temperatures because of very high density (small d)

37 Outline Atoms Stars Answer RG Review Atoms Large a 0 1-D Checking for Fermion Superfluidity John Thomas group at Duke Use a focused CO 2 laser beam to trap 6 Li atoms (fermions) into cigar-shaped cloud and apply magnetic field Switch optical trap off and on to give gas a light tap Observe length of oscillations at different temperatures normal gas: atom collisions damp oscillations superfluid: unified long-lasting oscillations Result: Longer oscillations with lower temperatures!

38 Outline Atoms Stars Answer RG Review Atoms Large a 0 1-D Trapped Fermion Atoms vs. QCD Atoms: Change magnetic field = resonant scattering

39 Outline Atoms Stars Answer RG Review Atoms Large a 0 1-D Trapped Fermion Atoms vs. QCD Atoms: Change magnetic field = resonant scattering QCD: Adjust quark mass theoretically = m π changes more Universal properties?

40 Outline Atoms Stars Answer RG Review Atoms Large a 0 1-D Large Scattering Length Problem Attractive with a 0 If R 1/k F a 0, then expect scale invariance Energy and gap are pure numbers times E FG = 3 5 kf 2 2M R ~ 1/ k F a 0

41 Outline Atoms Stars Answer RG Review Atoms Large a 0 1-D Large Scattering Length Problem Attractive with a 0 If R 1/k F a 0, then expect scale invariance Energy and gap are pure numbers times E FG = 3 5 2M EFT power counting says: sum everything with leading vertices Easy in free space = geometric sum of bubbles Many more diagrams in the medium k 2 F R ~ 1/ a 0 k F

42 Outline Atoms Stars Answer RG Review Atoms Large a 0 1-D Calculations Summing All Leading Diagrams Green s Function Monte Carlo (GMFC) [J. Carlson et al.] Solve many-body S-eqn with variational wf improved by GFMC GFMC applied to real nuclei up to A = 12 with energies to 1% Large a 0 : Convenient potential tuned to a 0

43 Outline Atoms Stars Answer RG Review Atoms Large a 0 1-D Calculations Summing All Leading Diagrams Green s Function Monte Carlo (GMFC) [J. Carlson et al.] Solve many-body S-eqn with variational wf improved by GFMC GFMC applied to real nuclei up to A = 12 with energies to 1% Large a 0 : Convenient potential tuned to a 0 Lattice calculation for two spins = no fermion sign problem! Path integral evaluated on discretized space-time lattice QCD lattice calculations = best nonperturbative results Large a 0 : Proposed by J.-W. Chen and D. Kaplan In progress by QCD lattice gauge theorist [M. Wingate]

44 Outline Atoms Stars Answer RG Review Atoms Large a 0 1-D Calculations Summing All Leading Diagrams Green s Function Monte Carlo (GMFC) [J. Carlson et al.] Solve many-body S-eqn with variational wf improved by GFMC GFMC applied to real nuclei up to A = 12 with energies to 1% Large a 0 : Convenient potential tuned to a 0 Lattice calculation for two spins = no fermion sign problem! Path integral evaluated on discretized space-time lattice QCD lattice calculations = best nonperturbative results Large a 0 : Proposed by J.-W. Chen and D. Kaplan In progress by QCD lattice gauge theorist [M. Wingate] Could there be an additional EFT expansion?

45 Outline Atoms Stars Answer RG Review Atoms Large a 0 1-D GFMC Results [J. Carlson et al.] Extrapolate to large numbers of fermions 20 Pairing gap ( ) = 0.9 E FG E/E FG 10 odd N E = 0.44 N E FG even N N Energy per particle: E/N = 0.44(1)E FG

46 TABLE BCS cro Outline Atoms Stars Answer RG Review Atoms Large a 0 1-D Diffusion MC Results [G.E. Astrakharchik et al.] Square-well potential tuned to a 0 Extrapolate to large numbers of fermions Energy per particle: E/N = 0.42(1)E FG

47 Outline Atoms Stars Answer RG Review Atoms Large a 0 1-D Is There An Additional Expansion?

48 Outline Atoms Stars Answer RG Review Atoms Large a 0 1-D Is There An Additional Expansion? No parameters with dimensions. What about geometry?

49 Outline Atoms Stars Answer RG Review Atoms Large a 0 1-D Is There An Additional Expansion? No parameters with dimensions. What about geometry? J. Steele: Organize according to space-time dimension P/2 q k F 1 P/2 q k F 1 2 D/2

50 Outline Atoms Stars Answer RG Review Atoms Large a 0 1-D Is There An Additional Expansion? No parameters with dimensions. What about geometry? J. Steele: Organize according to space-time dimension P/2 q k F 1 P/2 q k F 1 2 D/2 Leading 1/D analytic expansion yields: ( E A = E FG k ) Fa 0 /9 a + O(1/D) 0 4 π 2k F a 0 9 E FG [cf. 0.44(1) E FG ]

51 Outline Atoms Stars Answer RG Review Atoms Large a 0 1-D Is There An Additional Expansion? No parameters with dimensions. What about geometry? J. Steele: Organize according to space-time dimension P/2 q k F 1 P/2 q k F 1 2 D/2 Leading 1/D analytic expansion yields: ( E A = E FG k ) Fa 0 /9 a + O(1/D) 0 4 π 2k F a 0 9 E FG [cf. 0.44(1) E FG ] Subleading??? Pairing gap???

52 Outline Atoms Stars Answer RG Review Atoms Large a 0 1-D Recap: How are Cold Atoms Like Neutron Stars? Low temperatures (on an appropriate scale) = means low energies, low resolution = quantum mechanical (wave) effects are large Only the scattering length a 0 is important if cold and dilute = universal physics when a 0 is large! Cold fermions with large scattering lengths are similar Superfluid atoms in traps Superfluid neutrons in neutron stars

53 Outline Atoms Stars Answer RG Review Atoms Large a 0 1-D Experimental Evidence for Pairing in Nuclei B(N, Z ) = [ ( ) ] 2 N Z (15.6 MeV) A A (17.2 MeV)A 2/3 (0.70 MeV) Z 2 A 1/3 + (6 MeV)[( 1) N + ( 1) Z ]/A 1/2 Odd-even staggering of binding energies

54 Outline Atoms Stars Answer RG Review Atoms Large a 0 1-D Experimental Evidence for Pairing in Nuclei Odd-even staggering of binding energies Energy gap in spectra of deformed nuclei Low-lying 2 + states in even nuclei Deformations and moments of inertia (theory requires pairing)

55 Outline Atoms Stars Answer RG Review Atoms Large a 0 1-D 1-D Molecules and Cooper Pairs Consider Fermions in one dimension with λδ(x) attraction In the COM frame with relative coordinate x, 1 M d 2 ψ(x) dx 2 λ δ(x)ψ(x) = B 2 ψ(x) always a bound state with B 2 = Mλ 2 /4 with ψ(x) = Ae M B 2 x (e.g., solve by integrating S-eqn across x = 0) extent of bound state increases as λ 0 (cf. 3-D behavior)

56 Outline Atoms Stars Answer RG Review Atoms Large a 0 1-D 1-D Molecules and Cooper Pairs Consider Fermions in one dimension with λδ(x) attraction In the COM frame with relative coordinate x, 1 M d 2 ψ(x) dx 2 λ δ(x)ψ(x) = B 2 ψ(x) always a bound state with B 2 = Mλ 2 /4 with ψ(x) = Ae M B 2 x (e.g., solve by integrating S-eqn across x = 0) extent of bound state increases as λ 0 (cf. 3-D behavior) = at very low density, a gas of noninteracting two-fermion composite particles ( molecules )

57 Outline Atoms Stars Answer RG Review Atoms Large a 0 1-D 1-D Molecules and Cooper Pairs Consider Fermions in one dimension with λδ(x) attraction In the COM frame with relative coordinate x, 1 M d 2 ψ(x) dx 2 λ δ(x)ψ(x) = B 2 ψ(x) always a bound state with B 2 = Mλ 2 /4 with ψ(x) = Ae M B 2 x (e.g., solve by integrating S-eqn across x = 0) extent of bound state increases as λ 0 (cf. 3-D behavior) = at very low density, a gas of noninteracting two-fermion composite particles ( molecules ) What happens at higher density?

58 Outline Atoms Stars Answer RG Review Atoms Large a 0 1-D How Does the Fermi Sea Affect Bound States? Pauli blocking = Look in momentum space: (H 0 + V ) ψ = E 2 ψ and ψ(x) = k C k e ikx k ψ = C k

59 Outline Atoms Stars Answer RG Review Atoms Large a 0 1-D How Does the Fermi Sea Affect Bound States? Pauli blocking = Look in momentum space: (H 0 + V ) ψ = E 2 ψ and ψ(x) = k C k e ikx k ψ = C k Project the Schrödinger equation on k and insert a complete set of states to evaluate V : = k H 0 ψ + k k V k k ψ = E 2 k ψ or 2ɛ 0 k C k λ k C k = E 2 C k

60 Outline Atoms Stars Answer RG Review Atoms Large a 0 1-D How Does the Fermi Sea Affect Bound States? Pauli blocking = Look in momentum space: (H 0 + V ) ψ = E 2 ψ and ψ(x) = k C k e ikx k ψ = C k Project the Schrödinger equation on k and insert a complete set of states to evaluate V : = k H 0 ψ + k k V k k ψ = E 2 k ψ or 2ɛ 0 k C k λ k C k = E 2 C k Let A = k C k, solve for C k and sum over k: k C k = A = λ L A k 1 2ɛ 0 k E 2 or 1 = λ L k 1 2ɛ 0 k E 2

61 Outline Atoms Stars Answer RG Review Atoms Large a 0 1-D In free space, we can calculate the integral as L For E 2 < 0, we recover our previous result: 1 = λ dk 2π k 2 /2M + E 2 = λ M = E 2 = Mλ2 2 E 2 4 For E 2 > 0, we get scattering states with energies equal to the asymptotic kinetic energy

62 Outline Atoms Stars Answer RG Review Atoms Large a 0 1-D In free space, we can calculate the integral as L For E 2 < 0, we recover our previous result: 1 = λ dk 2π k 2 /2M + E 2 = λ M = E 2 = Mλ2 2 E 2 4 For E 2 > 0, we get scattering states with energies equal to the asymptotic kinetic energy In medium, include Pauli blocking through the integral limit: 1 = λ 2π 2 dk k 2 /2M E 2 k F 2 ɛ F=kF /2M 1 = λ M π 2 ɛ F dɛ 1 ɛ 2ɛ E 2 This equation has two relevant types of solutions, when z E 2 2ɛ F < 0 and 0 < E 2 < 2ɛ F (or 0 < z < 1)

63 Outline Atoms Stars Answer RG Review Atoms Large a 0 1-D Graphical Solution (Plot LHS vs. RHS) π 0 < z < 1 2 z 1/2 If 2πk F < 2, there is a solution M λ with z = E 2 /2ɛ F < 0 = like molecules z < z 1/2

64 Outline Atoms Stars Answer RG Review Atoms Large a 0 1-D Graphical Solution (Plot LHS vs. RHS) π 0 < z < 1 z < 0 2 z 1/ z 1/2 If 2πk F < 2, there is a solution M λ with z = E 2 /2ɛ F < 0 = like molecules If 2πk F > 2, there is a M λ solution with 0 < z < 1 = like Cooper pairs

65 Outline Atoms Stars Answer RG Review Atoms Large a 0 1-D Graphical Solution (Plot LHS vs. RHS) π 0 < z < 1 z < 0 2 z 1/ z 1/2 We can define through E 2 = 2ɛ F If 2πk F < 2, there is a solution M λ with z = E 2 /2ɛ F < 0 = like molecules If 2πk F > 2, there is a M λ solution with 0 < z < 1 = like Cooper pairs As z 1 (weak coupling), expand to find = 8ɛ F e 2πk F/M λ So the gap has an essential singularity in the coupling constant

66 Outline Atoms Stars Answer RG Review Atoms Large a 0 1-D Graphical Solution (Plot LHS vs. RHS) π 0 < z < 1 z < 0 2 z 1/ z 1/2 We can define through E 2 = 2ɛ F If 2πk F < 2, there is a solution M λ with z = E 2 /2ɛ F < 0 = like molecules If 2πk F > 2, there is a M λ solution with 0 < z < 1 = like Cooper pairs As z 1 (weak coupling), expand to find = 8ɛ F e 2πk F/M λ So the gap has an essential singularity in the coupling constant In 3-D weak-coupling BCS, ɛ F e π/2kf a

67 Outline Atoms Stars Answer RG Review Atoms Large a 0 1-D Compare 3-D Lennard-Jones and 1-D δ Function In 3-D, low-density corresponds to weak coupling (kinetic energy dominates) In 1-D, high-density corresponds to weak coupling (kinetic energy dominates)

68 Outline Atoms Stars Answer RG Laws Vlowk Many-Body 3-Body EFT Recap Famous Laws of Theoretical Physics I Steven Weinberg enumerated these laws while discussing the renormalization group.

69 Outline Atoms Stars Answer RG Laws Vlowk Many-Body 3-Body EFT Recap Famous Laws of Theoretical Physics I Steven Weinberg enumerated these laws while discussing the renormalization group. Weinberg s Three Laws of Progress in Theoretical Physics 1. Conservation of Information: You will get nowhere by churning equations. 2. Do not trust arguments based on the lowest order of perturbation theory 3. You may use any degrees of freedom you like to describe a physical system, but if you use the wrong ones, you ll be sorry!

70 Outline Atoms Stars Answer RG Laws Vlowk Many-Body 3-Body EFT Recap Principles of Effective Low-Energy Theories

71 Outline Atoms Stars Answer RG Laws Vlowk Many-Body 3-Body EFT Recap Principles of Effective Low-Energy Theories If system is probed at low energies, fine details not resolved

72 Outline Atoms Stars Answer RG Laws Vlowk Many-Body 3-Body EFT Recap Principles of Effective Low-Energy Theories If system is probed at low energies, fine details not resolved use low-energy variables for low-energy processes short-distance structure can be replaced by something simpler without distorting low-energy observables

73 Outline Atoms Stars Answer RG Laws Vlowk Many-Body 3-Body EFT Recap NN Potential and Scattering Phase Shifts n-n potential (MeV) n-n system O -O system n-n distance (fm) low 0 energies 0.4 = 0.8 large a O 2-O 2 distance (nm) zero crossing = hard core phase shift (degrees) np 1 S 0 (PWA93) E lab (MeV)

74 V NN (k,k) [fm] Outline Atoms Stars Answer RG Laws Vlowk Many-Body 3-Body EFT Recap NN Potential and Scattering Phase Shifts Now look at NN potentials in momentum space: s-wave k V k Paris Bonn A Argonne v18 Idaho A CD Bonn Nijmegen94 II Nijmegen94 I k [fm -1 ] phase shift (degrees) np 1 S 0 (PWA93) E lab (MeV) Many different potentials with same phase shifts (χ 2 /dof 1)

75 Outline Atoms Stars Answer RG Laws Vlowk Many-Body 3-Body EFT Recap NN Scattering in the COM Frame k +k k +k T 3 2 T k +k = V B k +k + q Λ B χ 2 /dof 1 bare potential V B Probes intermediate states to q Λ B = 25 fm 1 =. 5 GeV V B V NN (k,k) [fm] Model dependent: q 3 fm k [fm -1 ] Paris Bonn A Argonne v18 Idaho A CD Bonn Nijmegen94 II Nijmegen94 I

76 Outline Atoms Stars Answer RG Laws Vlowk Many-Body 3-Body EFT Recap Low-Momentum NN Potential Bogner, Kuo, Schwenk k +k k +k T 3 T = V Λ + q Λ 2 V Λ k +k k +k Require dt dλ = 0 = renormalization group equation for V Λ Run from Λ B = 25 fm 1 to Λ = 2 fm 1 E lab. = 350 MeV V NN (k,k) [fm] Paris Bonn A Argonne v18 Idaho A CD Bonn Nijmegen94 II Nijmegen94 I k [fm -1 ]

77 Outline Atoms Stars Answer RG Laws Vlowk Many-Body 3-Body EFT Recap Low-Momentum NN Potential Bogner, Kuo, Schwenk k +k k +k T 3 T = V Λ + q Λ 2 V Λ k +k k +k Require dt dλ = 0 = renormalization group equation for V Λ Run from Λ B = 25 fm 1 to Λ = 2 fm 1 E lab. = 350 MeV V NN (k,k) [fm] Paris Bonn A Argonne v18 Idaho A CD Bonn Nijmegen94 II Nijmegen94 I k [fm -1 ]

78 Outline Atoms Stars Answer RG Laws Vlowk Many-Body 3-Body EFT Recap Low-Momentum NN Potential Bogner, Kuo, Schwenk k +k k +k T 3 T = V Λ + q Λ 2 V Λ k +k k +k Require dt dλ = 0 = renormalization group equation for V Λ Run from Λ B = 25 fm 1 to Λ = 2 fm 1 E lab. = 350 MeV V NN (k,k) [fm] Paris Bonn A Argonne v18 Idaho A CD Bonn Nijmegen94 II Nijmegen94 I k [fm -1 ]

79 Outline Atoms Stars Answer RG Laws Vlowk Many-Body 3-Body EFT Recap Low-Momentum NN Potential Bogner, Kuo, Schwenk k +k k +k T 3 T = V Λ + q Λ 2 k +k k +k V Λ Require dt dλ = 0 = renormalization group equation for V Λ Run from Λ B = 25 fm 1 to Λ = 2 fm 1. E lab = 350 MeV Same long distance physics = collapse! V NN (k,k) [fm] Paris Bonn A Argonne v18 Idaho A CD Bonn Nijmegen94 II Nijmegen94 I k [fm -1 ]

80 Outline Atoms Stars Answer RG Laws Vlowk Many-Body 3-Body EFT Recap Low-Energy Observables Unchanged Phase shift δ vs. E Lab [MeV] 0 1 S P S1 P0 5 P D1 1 ε P1 PSA93

81 Outline Atoms Stars Answer RG Laws Vlowk Many-Body 3-Body EFT Recap Low-Energy Observables Unchanged Phase shift δ vs. E Lab [MeV] 0 1 S P S1 P0 5 P D1 1 ε P1 PSA93 CD Bonn

82 Outline Atoms Stars Answer RG Laws Vlowk Many-Body 3-Body EFT Recap Low-Energy Observables Unchanged Phase shift δ vs. E Lab [MeV] 0 1 S P S1 P0 5 P D1 1 ε P1 PSA93 CD Bonn V Λ=2 fm -1

83 Outline Atoms Stars Answer RG Laws Vlowk Many-Body 3-Body EFT Recap Low-Momentum Pot l in the Many-Body Problem Removing hard core = simpler many-body starting point for neutron matter [Schwenk, Friman, Brown]

84 Outline Atoms Stars Answer RG Laws Vlowk Many-Body 3-Body EFT Recap Low-Momentum Pot l in the Many-Body Problem Removing hard core = simpler many-body starting point for neutron matter [Schwenk, Friman, Brown] Akmal et al. AV18 (1998) BHF: Bao et al. (1994) V low k Hartree-Fock Akmal et al. AV18+UIX+boost (1998) E/A [MeV] ρ [fm -3 ] Simple Hartree-Fock (circles) matches best calculations!

85 Outline Atoms Stars Answer RG Laws Vlowk Many-Body 3-Body EFT Recap Three-Body Forces are Inevitable! What if we have three nucleons interacting? Successive two-body scatterings with short-lived high-energy intermediate states unresolved = must be absorbed into three-body force Binding Energy (MeV) Triton Binding Energy AV18 CD Bonn Experiment Λ (fm -1 ) [Bogner, Nogga, Schwenk]

86 Outline Atoms Stars Answer RG Laws Vlowk Many-Body 3-Body EFT Recap Three-Body Forces are Inevitable! What if we have three nucleons interacting? Successive two-body scatterings with short-lived high-energy intermediate states unresolved = must be absorbed into three-body force Binding Energy (MeV) Triton Binding Energy AV18 CD Bonn Experiment How do we organize three-body forces? Λ (fm -1 ) [Bogner, Nogga, Schwenk]

87 Outline Atoms Stars Answer RG Laws Vlowk Many-Body 3-Body EFT Recap Renormalization and Short-Distance Physics 3 V B = C0 2 q > Λ V B q > Λ intermediate states = replace with contact term: C 0 δ 3 (x x ) L eft = (ψ ψ) 2 + V NN (k,k) [fm] Paris Bonn A Argonne v18 Idaho A CD Bonn Nijmegen94 II Nijmegen94 I k [fm -1 ]

88 Outline Atoms Stars Answer RG Laws Vlowk Many-Body 3-Body EFT Recap (Nuclear) Many-Body Physics: Old vs. New

89 Outline Atoms Stars Answer RG Laws Vlowk Many-Body 3-Body EFT Recap (Nuclear) Many-Body Physics: Old vs. New One Hamiltonian for all problems and energy/length scales Infinite # of low-energy potentials; different resolutions = different dof s and Hamiltonians

90 Outline Atoms Stars Answer RG Laws Vlowk Many-Body 3-Body EFT Recap (Nuclear) Many-Body Physics: Old vs. New One Hamiltonian for all problems and energy/length scales Find the best potential Infinite # of low-energy potentials; different resolutions = different dof s and Hamiltonians There is no best potential = use a convenient one!

91 Outline Atoms Stars Answer RG Laws Vlowk Many-Body 3-Body EFT Recap (Nuclear) Many-Body Physics: Old vs. New One Hamiltonian for all problems and energy/length scales Find the best potential Two-body data may be sufficient; many-body forces as last resort Infinite # of low-energy potentials; different resolutions = different dof s and Hamiltonians There is no best potential = use a convenient one! Many-body data needed and many-body forces inevitable

92 Outline Atoms Stars Answer RG Laws Vlowk Many-Body 3-Body EFT Recap (Nuclear) Many-Body Physics: Old vs. New One Hamiltonian for all problems and energy/length scales Find the best potential Two-body data may be sufficient; many-body forces as last resort Avoid divergences Infinite # of low-energy potentials; different resolutions = different dof s and Hamiltonians There is no best potential = use a convenient one! Many-body data needed and many-body forces inevitable Exploit divergences

93 Outline Atoms Stars Answer RG Laws Vlowk Many-Body 3-Body EFT Recap (Nuclear) Many-Body Physics: Old vs. New One Hamiltonian for all problems and energy/length scales Find the best potential Two-body data may be sufficient; many-body forces as last resort Avoid divergences Choose diagrams by art Infinite # of low-energy potentials; different resolutions = different dof s and Hamiltonians There is no best potential = use a convenient one! Many-body data needed and many-body forces inevitable Exploit divergences Power counting determines diagrams and truncation error

94 Outline Atoms Stars Answer RG Laws Vlowk Many-Body 3-Body EFT Recap Quantum Chromodynamics and Nuclear Physics Quark Mass (m q ) Dependence of Observables S. Beane and M. Savage, Nucl. Phys. A717 (2003) 91. Crude (naive?) estimate of nonrelativistic bound state: E B p2 typical M f 2 π M 10 MeV E deuteron

95 Outline Atoms Stars Answer RG Laws Vlowk Many-Body 3-Body EFT Recap Is the Fine-Tuning Accidental? E. Epelbaum, U. Meißner, and W. Glöckle, Nucl. Phys. A714 (2003) B D [MeV] M π [MeV] Footnote from S. Weinberg paper [Nucl. Phys. B363 (1991) 3]: E. Witten says that large N c arguments... suggest that the very small binding energy of the deuteron is a fortuitous cancellation of potential and kinetic energies for the physical value N c = 3.