3. Lattice QCD Or: Using Large Computers for Fun References: [(Path - PowerPoint PPT Presentation

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 1310.1760 [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

(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

(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. �� ∂ Φ ( t ) � � 2 i S [ Φ ] = + i � − m 2 Φ 2 ( t ) d t 2 ∂ t �� ∂ Φ ( x E ) � � 2 → − 1 � + m 2 Φ 2 ( x E ) Euclideanise i t → x E = : − S E [ Φ ] d x E ∂ x E 2 − ∂ 2 � � → − 1 � + m 2 d x E Φ ( x E ) Φ ( x E ) Rewrite as 2nd derivative ∂ x 2 2 E � − 1 � → − a a 2 ( Φ n + 1 − 2 Φ n + Φ n − 1 )+ m 2 Φ 2 ∑ Φ n Discretise á la Runge-Kutta RK2 n 2 lattice sites n   − 2 a 2 + m 2 1 ... 0 0 a 2   Φ 1 1 − 2 1 a 2 + m 2 ...  0  . Convert to matrix on vector �  →∝ � a 2 a 2 � Φ T   Φ = . Φ   .  1 − 2 1  a 2 + m 2 ... 0    a 2 a 2 Φ 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

Momentum Restriction: Brillouin Zone and Miniumum Momentum � � S E [ Φ ] = a − 1 ∑ a 2 ( Φ n + 1 − 2 Φ n + Φ n − 1 )+ m 2 Φ 2 Φ n n 2 lattice sites n � d k 2 π e i kna Φ ( k ) at momentum k . Solve by Discrete Fourier Transform Φ n = → HW 1 a → 0 → m 2 + k 2 + O ( a 2 ) . − Result: Correct continuum limit for relativistic E - p relation: propagator a Resolution a cannot resolve high momenta/high-frequency oscillations. ⇒ 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 = ± 2 π L ⇒ Discretised momenta k = 0 , ± 2 π L , ± 4 π L ,..., ± π = a PHYS 6610: Graduate Nuclear and Particle Physics I, Spring 2018 H. W. Grießhammer, INS, George Washington University III.3.3

Fermion Doubling Problem � → ← � Discretise First Derivative 1 Ψ → 1 ¯ / / ¯ Ψ ∂ − ∂ Ψ n ( Ψ n + 1 − Ψ n − 1 ) “symmetric form”, no γ µ s here 2 2 a ⇒ 2 decoupled chains, S E [ k ] = S E [ π Only next-to-nearest-neighbours interact . = a − k ] identical. 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 Can Show: unavoidable with first deriva- tive ( 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 λ a ∂ 2 One Way Around: “Wilson Fermions” add ¯ ∂ x 2 Ψ to action ( λ some dimensionless parameter). 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

(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

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

A More Realistic Example – And Some People’s Fantasies [NPLQCD arXiv:1508.07583v1 [hep-lat]] Effective-mass shift ∆ E = 2 M N − M ( deuteron ) in 32 3 × 96 lattice, using lattice units. [HALQCD arXiv:1502.04182v2 [hep-lat]] Fit-error construction: At least 3 different people Eff. shift ∆ E = 4 M N − M ( 4 He ) in ( 4 . 3fm ) 3 ( 48 4 lattice), in lattice units. use different algorithms to identify plateaus indepen- dently, each providing an error estimate. Total error Quote: “Fit result with one standard deviation er- is statistical sum of all. ror band and total error including the systematic one is expressed by solid and dashed lines, re- Watch out for strong correlation of points: spectively.” same lattice data! PHYS 6610: Graduate Nuclear and Particle Physics I, Spring 2018 H. W. Grießhammer, INS, George Washington University III.3.7

(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

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

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

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

Action Density: “Pure-Glue” Vacuum Fluctuations [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.3.12

QCD Precision Spectroscopy: Quarkonia & Heavy-Light Mesons 12 η c , ϒ ′ , ϒ set scales of r b2 h b (2P) r b1 (2P) m c , m b , α s ( Q 2 0 ) r b0 [ ’’ [ (1D) d ’ [ ’ 10 r b2 b r b1 (1P) d b [ h b (1P) r b0 MESON MASS (GEV) 8 expt fix parameters postdictions B c predictions 6 B * B s s B B * 4 r c2 s ’ d ’ r c1 HFS c h c r c0 L · � � d c J/ s S D s 2 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

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

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. Problem: Light-quark admixture = ⇒ Lattice computation: Bad signal-to-noise, quark loops give huge corrections! “Quenched” computation: GlueX: Glueball search in Hall D is major no disconnected quark lines. motivation for JLab 12GeV -upgrade. Unique experimental signal difficult. [Morningstar/Peardon Phys. Rev. D 60 ( 1999! ) 034509] PHYS 6610: Graduate Nuclear and Particle Physics I, Spring 2018 H. W. Grießhammer, INS, George Washington University III.3.15

Other Observables Need to be phrased as energy-differences! Isovector magnetic form factor G p − n M ( Q 2 ) Forefront includes: – Parton Distribution Functions – QCD phase diagram – scattering: ππ , NN ,. . . – weak interactions – beyond Standard Model [EMT collaboration, arXiv 0811.0724 [hep-lat]] PHYS 6610: Graduate Nuclear and Particle Physics I, Spring 2018 H. W. Grießhammer, INS, George Washington University III.3.16