PHYS 6610: Graduate Nuclear and Particle Physics I, Spring Institute for Nuclear Studies The George Washington University.

Transcription

1 PHYS 6610: Graduate Nuclear and Particle Physics I H. W. Grießhammer INS Institute for Nuclear Studies The George Washington University Institute for Nuclear Studies Spring 2018 III. Descriptions 3. Lattice QCD Or: Using Large Computers for Fun References: [(Path Integral: Ryd 5; Sakurai: Modern QM 2.5); CL 10.5; PDG 18; Wagner arxiv [hep-lat]; Alexandru, Lee, Freeman, Lujan, Guo;... ] PHYS 6610: Graduate Nuclear and Particle Physics I, Spring 2018 H. W. Grießhammer, INS, George Washington University III.3.0

2 (a) Motivation of the Path Integral [Ryd 5; Sakurai: Mod. QM 2.5] Historic Note [Ryd Chap. 5] (b) Path Integrals on a Computer PHYS 6610: Graduate Nuclear and Particle Physics I, Spring 2018 H. W. Grießhammer, INS, George Washington University III.3.1

3 (c) Free Fields on the Lattice Points Fields: x(t) Φ(x µ ) For A Massive Real Scalar Field Consider 1-Dimensional Case: only time direction, nothing else generalisation straightforward. [ is[φ] = + i ( Φ(t) ) 2 dt m 2 Φ (t)] 2 2 t [ Euclideanise it x E 1 ( Φ(xE ) 2 ) dx E +m 2 Φ 2 (x E )] =: S E [Φ] 2 x E Rewrite as 2nd derivative 1 dx E Φ(x E ) [ 2 2 xe 2 + m ]Φ(x 2 E ) Discretise á la Runge-Kutta RK2 a [ ] 1 2 Φ n lattice sites n a 2 (Φ n+1 2Φ n + Φ n 1 ) + m 2 Φ 2 n 2 + m Φ 1 a 2 a 2 Convert to matrix on vector Φ =. Φ T m a 2 a 2 a m Φ a Φ 2 a 2 a 2 N This is a Linear Chain of Coupled Harmonic Oscillators: Dislocation Φ n at point n by spring with constant 1 a 2, nearest-neighbour interactions, m provides additional drag. PHYS 6610: Graduate Nuclear and Particle Physics I, Spring 2018 H. W. Grießhammer, INS, George Washington University III.3.2

4 Momentum Restriction: Brillouin Zone and Miniumum Momentum S E [Φ] = a 2 lattice sites n Solve by Discrete Fourier Transform Φ n = [ ] 1 Φ n a 2 (Φ n+1 2Φ n + Φ n 1 ) + m 2 Φ 2 n Result: Correct continuum limit for relativistic E-p relation: dk 2π eikna Φ(k) at momentum k. Resolution a cannot resolve high momenta/high-frequency oscillations. 1 a 0 m 2 + k 2 + O(a 2 ). propagator a HW = Useful momenta must be inside Brillouin Zone π a k π a. black: k < π a ; red: k > π a In finite lattice volume, there is also a smallest nonzero momentum. Example hypercube with Periodic Boundary Condition Φ n = Φ n+n : k min = ± 2π L = Discretised momenta k = 0,± 2π L,±4π L,...,±π a PHYS 6610: Graduate Nuclear and Particle Physics I, Spring 2018 H. W. Grießhammer, INS, George Washington University III.3.3

5 Fermion Doubling Problem Discretise First Derivative 1 2 Ψ[ ] Ψ 1 2a Ψ n (Ψ n+1 Ψ n 1 ) symmetric form, no γ µ s here Only next-to-nearest-neighbours interact. = 2 decoupled chains, S E [k] = S E [ π k] identical. a In particular, S E [k 0] = S E [k π a ]: Fermions at border of Brillouin zone contribute as much as fermions at rest! Fermion Doubling Problem One Way Around: Wilson Fermions add Ψλa 2 One Can Show: unavoidable with first derivative (Nielsen-Ninomiya No-Go Theorem). Even Worse: doubling in each dimension = 2 4 = 16 zero-energy fermions in d = 4, instead of the 1 we want. Ψ to action (λ some dimensionless parameter). x2 Such a bosonic RK-2 term breaks degeneracy but vanishes for a 0. There are other remedies ( staggered,... ). All remedies carry a hefty computational prize. PHYS 6610: Graduate Nuclear and Particle Physics I, Spring 2018 H. W. Grießhammer, INS, George Washington University III.3.4

6 (d) QCD on the Lattice (e) Heavy-Quark Potential in the Strong Coupling Limit (f) Very Rough Outline of Lattice Computations (g) A Few Selected Problems in Lattice QCD PHYS 6610: Graduate Nuclear and Particle Physics I, Spring 2018 H. W. Grießhammer, INS, George Washington University III.3.5

7 Temporal Correlation Function Example [Wagner] Everything (Masses, input m q ) is given in units one dimension-ful quantity: lattice spacing a. lim t E B meson( t E) e H t E B meson(t E = 0) exp t E M B meson PHYS 6610: Graduate Nuclear and Particle Physics I, Spring 2018 H. W. Grießhammer, INS, George Washington University III.3.6

8 A More Realistic Example And Some People s Fantasies [NPLQCD arxiv: v1 [hep-lat]] Effective-mass shift E = 2M N M(deuteron) in lattice, using lattice units. Fit-error construction: At least 3 different people use different algorithms to identify plateaus independently, each providing an error estimate. Total error is statistical sum of all. Watch out for strong correlation of points: same lattice data! [HALQCD arxiv: v2 [hep-lat]] Eff. shift E = 4M N M( 4 He) in (4.3fm) 3 (48 4 lattice), in lattice units. Quote: Fit result with one standard deviation error band and total error including the systematic one is expressed by solid and dashed lines, respectively. PHYS 6610: Graduate Nuclear and Particle Physics I, Spring 2018 H. W. Grießhammer, INS, George Washington University III.3.7

9 (h) Very Few (Even More Selected) Lattice Results Extrapolation to Physical Masses Use known low-energy Nuclear Physics (Chiral EFT) to cut down on computational cost. Not just a linear extrapolation! [Duerr et al. Science (2008)] PHYS 6610: Graduate Nuclear and Particle Physics I, Spring 2018 H. W. Grießhammer, INS, George Washington University III.3.8

10 Static Potential between Infinitely Heavy Quarks (Quarkonium) Infinitely heavy = no recoil = no retardation or colour radiation = Potential makes sense. Appears quite linear. [Kenway UKQCD 1999] PHYS 6610: Graduate Nuclear and Particle Physics I, Spring 2018 H. W. Grießhammer, INS, George Washington University III.3.9

11 Energy Density: Flux Tube for a Heavy Meson [Leinweber et al. 2003, click here for homepage] PHYS 6610: Graduate Nuclear and Particle Physics I, Spring 2018 H. W. Grießhammer, INS, George Washington University III.3.10

12 Action Density: Flux Tube for a Heavy Baryon [Leinweber et al. 2003, click here for homepage] PHYS 6610: Graduate Nuclear and Particle Physics I, Spring 2018 H. W. Grießhammer, INS, George Washington University III.3.11

13 [Leinweber et al. 2003, Action Density: Pure-Glue Vacuum Fluctuations click here for homepage] PHYS 6610: Graduate Nuclear and Particle Physics I, Spring 2018 H. W. Grießhammer, INS, George Washington University III.3.12

14 QCD Precision Spectroscopy: Quarkonia & Heavy-Light Mesons d b d b [ [ [ h b (2P) h b (1P) r b2 r b1 (2P) b0 r b2 r b1 (1P) b0 [(1D) η c,ϒ,ϒ set scales of m c, m b, α s (Q 2 0 ) MESON MASS (GEV) 8 6 B c B s B B * s B * expt fix parameters postdictions predictions 4 d c s r c2 r h c1 c rc0 HFS d c J/s L S 2 D s D [PDG 2013 Fig. 14.8] PHYS 6610: Graduate Nuclear and Particle Physics I, Spring 2018 H. W. Grießhammer, INS, George Washington University III.3.13

15 QCD Spectroscopy: Systems With Light Quarks Approaching physical pion masses, good accuracy. [Duerr et al. Science (2008), from PDG 2013 Fig. 14.7] PHYS 6610: Graduate Nuclear and Particle Physics I, Spring 2018 H. W. Grießhammer, INS, George Washington University III.3.14

16 QCD Spectroscopy: Glueballs Colour-neutral bound states of glue are unique signal of Non-Abelian Gauge Theories. Glueballs: Any state dominated by glue. In particular when glue dictates quantum numbers. Discovery would allow direct test of QCD way beyond Constituent Quark model etc. Quenched computation: no disconnected quark lines. Problem: Light-quark admixture = Lattice computation: Bad signal-to-noise, quark loops give huge corrections! GlueX: Glueball search in Hall D is major motivation for JLab 12GeV-upgrade. Unique experimental signal difficult. [Morningstar/Peardon Phys. Rev. D60 (1999!) ] PHYS 6610: Graduate Nuclear and Particle Physics I, Spring 2018 H. W. Grießhammer, INS, George Washington University III.3.15

17 Other Observables Need to be phrased as energy-differences! Isovector magnetic form factor G p n M (Q2 ) Forefront includes: Parton Distribution Functions QCD phase diagram scattering: ππ, NN,... weak interactions beyond Standard Model [EMT collaboration, arxiv [hep-lat]] PHYS 6610: Graduate Nuclear and Particle Physics I, Spring 2018 H. W. Grießhammer, INS, George Washington University III.3.16

18 Hadron Polarisabilities: GW Leads Connecting Data & QCDGW focus Needs to be phrased as energy-difference: E = 2πα (N) E1 E π E π + π + π+ π + _ Neither Approach Uses The Other To Fit! [lattice: Lujan/Alexandru/Freeman/Lee arxiv: [hep-lat]; chiral extrapolation: hgrie/mcgovern/phillips arxiv: [nucl-th]; Downie/Feldman take data at HIγS, MAMI,... ] PHYS 6610: Graduate Nuclear and Particle Physics I, Spring 2018 H. W. Grießhammer, INS, George Washington University III.3.17

19 Phase Shifts Via Energy-Shift: Lüscher s Method (30 sec) Problem: Lattice QCD gave up time-dependence by rotation to Euclidean time. Lüscher 1991 boom since ca Solution: can still feel interactions in finite volume (cf. t-independent scattering theory) e.g. ππ scattering: compute E = E 2m π = 2 k n 2 + m 2 π 2m π = get k n, insert into Lüscher s formula k n cotδ(k n ) = 1 j Λ πl lim Λ j 1 j 2 ( k nl 2π ) 2 4πΛ (with error bars!) degrees 3 S L 24, P 0 L 32, P 0 L 24, P 1 L 32, P 1 Levinson's Theorem Experimental k m Π NN scattering at m π = 805MeV [NPLQCD PRC88 (2013) ] Valid for: below first inelasticity, L interaction range r 0 1 m π but can have scatt. length a L! Many extensions available and being worked on: 3-body, box with different lengths, coupled channels,... [Döring/Mai/..., hg... ] PHYS 6610: Graduate Nuclear and Particle Physics I, Spring 2018 H. W. Grießhammer, INS, George Washington University III.3.18

20 Phase Shifts Computed Via Energy-Shift: Tiny Effect GW focus: Alexandru, Döring,... ππ phase shifts identify ρ resonance; unphysical m π = 316MeV > m phys π = 140MeV. PHYS 6610: Graduate Nuclear and Particle Physics I, Spring 2018 H. W. Grießhammer, INS, George Washington University III.3.19

21 Alternative Worlds: Lightest Nuclei at Higher Pion Masses NPLQCD HALQCD Merger of EFT and lattice has started exploring how few-nucleon systems emerge from QCD. [J. Kirscher arxiv (got his PhD in GW s EFT group)] Surprisingly little change in few-nucleon systems but nn becomes bound when m π increased! PHYS 6610: Graduate Nuclear and Particle Physics I, Spring 2018 H. W. Grießhammer, INS, George Washington University III.3.20

22 Next: 4. Weak Interactions Familiarise yourself with: [phenomenology: PRSZR 10, 11, 12, 18.6; Per theory: Ryd 8.3-5; CL 11, 12; Per 7, 8, 5.4; most up-to-date: PDG 10, 12, 14 and reviews inside listings] PHYS 6610: Graduate Nuclear and Particle Physics I, Spring 2018 H. W. Grießhammer, INS, George Washington University III.3.21