Lattice Field Theory
Karl Jansen
• Lattice: Non-perturbative aspects of eld theories Lattice Quantum Chromodynamics Fermion-Higgs Systems Spin Systems Chiral Gauge Theories Supersymmetry Quantum Gravity Algorithms and Machines General articles
Lectures, review articles
• R. Gupta Introduction to Lattice QCD, hep-lat/9807028
• C. Davies Lattice QCD, hep-ph/0205181
• M. Lüscher Advanced Lattice QCD, hep-lat/9802029 Chiral gauge theories revisited, hep-th/0102028
• A.D. Kennedy Algorithms for Dynamical Fermions, hep-lat/0607038
1 Books about Lattice Field Theory
• T. DeGrand and C. DeTar Lattice methods for Quantum Chromodynamics World Scienti c, 2006
• H.J. Rothe Lattice gauge theories: An Introduction World Sci.Lect.Notes Phys.74, 2005 • J. Smit Introduction to quantum elds on a lattice: A robust mate Cambridge Lect.Notes Phys.15, 2002
• I. Montvay and G. Münster Quantum elds on a lattice Cambridge, UK: Univ. Pr., 1994
• Yussuf Saad Iterative Methods for sparse linear systems Siam Press, 2003
2 The Lectures http://www-zeuthen.desy.de/∼kjansen/
The harmonic oscillator programme J. Gonzales Lopez, A. Nube, D. Renner http://www-zeuthen.desy.de/∼kjansen/
3 Plan of Lectures
• Introduction to the Lattice Quantum Mechanics on the lattice Schwinger model • Algorithms, Errors and Machines Foundations of simulation algorithms Hybrid Monte Carlo Algorithm (plus improvements Intermezzo: a demonstration Auto correlation time and error analysis A selection of supercomputers
• Lattice QCD Static Potential
• Lattice QCD today
4
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â ìèð åø îïàñíûõ æèâîòíûõ!
Dobrodosli na reˇsetku i opasne ˇzivotinje na njoj Kal¸c rjate sto plègma kai ta gria jhrÐa tou
5 ½ Quantum mechanical oscillator on a Euclidean time-lattice Feynman and Hibbs; Creutz and Freedman, Annals Phys.13,1981
We start with Feynman path integral in quantum mechanics
R i S Z = Dqe~ cl
• Scl classical action, e.g. quantum mechanical oscillator
R 1 2 Scl = dt 2q˙ (t) − V (q(t))
• q(t) are classical paths
6 perform an analytical continuation to imaginary (in general complex) time
t → −iτ f(t) → f(τ) with euclidean time τ the action then becomes
" # Z 1 dq(τ) 2 S → (−i)dτ − V (q(τ)) 2 d(−iτ) Z 1 = i dτ q˙2+V (q) 2
≡ iSE
SE is the euclidean action and R −1S Z = Dqe ~ E the euclidean path integral
7 the expression for the euclidean path integral
1 R − SE R 1 2 Z = Dqe ~ ,SE = dτ 2q˙ + V (q) introduce a time lattice with N = T/a lattice points
0 { a T q(τ) → q(na) , q˙ → q˙lattice boundary condition: q(N + 1) = q(0) discretization of continuum time-derivative q˙ not unique
8 Discretizing the Quantum Mechanical Oscillator Verstegen, Phys.Lett.B122,63,1983
R 2 1 2 SE = dt q˙ (t) + 2mq (t)
discr P 1 1 1 2 m 2 SE = a n a 2(1 + β)q(n + a) − βq(n) − 2(1 − β)q(n − a) + 2 q (n)
β = 1 : [q(n + a) − q(n)] /a
1 0 1 00 2 = a q(n) + aq + 2q a + · · · − q(n) =q ˙ + O(a)
β = 0 : [q(n + a) − q(n − a)] /2a
1 0 1 00 2 0 1 00 2 2 = 2a q(n) + aq + 2q a + · · · − q(n) + q a − 2q a =q ˙+O(a )
9 Eects of dierent Discretizations with q(n) = √1 P eipnaqˆ(p) , p = jπ/aN , j = 0, 1, 2, ··· ,N − 1 2 p
discr 1 P ∗ SE = 2a p qˆ (p)D(p)ˆq(p)
2 1 2 4 1 2 D(p) = 4 sin (2pa) + 4(β − 1) sin (2pa) + a m
• dierent β : obtain continuum p2 with dierent rate
• β = 0 : obtain continumm p2 behaviour for p → 0 and p → π
10 Lessons from discretizing the Oscillator
• dierent choices of lattice derivatives can lead to O(a), O(a2), O(a3), ··· lattice artefacts → very dierent rates to approach continuum limit
• there are wrong choices of lattice derivatives → lead to wrong continuum theory
11 The Transfer Matrix
Restricting ourselves to the choice qn+1−qn (T=N ) q˙ → a a
“q −qn ”2 ff −a P 1 n+1 +V (q) Q R ~ n 2 a Z = n dqne we obtain he continuum expression by sending N → ∞ while keeping T xed ⇒ a → 0
12 relation to canonical quantization:
2 1 2 1 q0−q a2q˙n + V (qn) = a2 a + V (q)a introduce coordinate and momentum operators:
pˆ|pi = p|pi;q ˆ|qi = q|qi pˆ = qˆ˙ = (qˆ0 − qˆ)/a
2 2 „ 0 « „ ˆ0 « a1 q −q +V (q)a a1 q −qˆ +V (ˆq)a e 2 a = hq0|e 2 a |qi
Z 2 = dphq0|e−pˆ a|pihp|e−V (ˆq)a|qi
Z 2 = dp hq0|pihp|qi e−p ae−V (q)a | {z 0 } eip(q−q ) (1)
13 using the Baker-Hausdor formula
1 eAeB = eA+B+2[A,B]+... we obtain for an in nitesimal change from q(t) → q0(t + a)
2 hq0|e−pˆ ae−V (ˆq)a|qi + O(a2) = hq0|e−aH|qi with Hamilton operator H
1 2 H = 2pˆ + V (ˆq)
14 pathintegral and operator picture
classical fields operator picture picture
Z 1 1 − SE − HT Dqe ~ = hqN|e ~ |q0i Z Y −1Ha = dqihqN|e ~ |qN−1i · ... i −1Ha ·hqi+1|e ~ |qii · ... −1Ha ·hq1|e ~ |q0i
15 −1Ha the operator T = e ~ is called the transfer matrix and describes in nitesimal time steps of the system the matrix elements of the transfer operator are
−1Ha hqi+1|e ~ |qii with qN = q0 ≡ q
Z = TrTN
→ partition function in statistical mechanics
16 inserting an energy eigenbasis
−1Ha X −1Ha hqi+1|e ~ |qii = hqi+1|e ~ |EihE|qii E X −1Ea = hqi+1|EihE|qiie ~ E
→ euclidean Hamilton operator
~ H = −a ln T
17 ⇒ transfer matrix has to be positive de nite
• positivity of T is su cient to guarantee that all n-point functions can be rotated back to Minkowski space (reconstruction theorem)
• positivity of T ⇔ Osterwalder-Schrader re ection positivity
• Schrödinger equation in euclidean time
1 ∂ 2∆Ψ = ∂τ Ψ
becomes analoguous to a diusion equation (with diusion constant 1 ) D = 2
18 Schwinger model: 2-dimensional Quantum Electrodynamics Schwinger 1962 the free Dirac equation (~ = c = 1)
(iσµ∂µ − m) Ψ = 0 can be derived from the Lagrange density
Ψ(¯ x)(iσµ∂µ − m) Ψ(x)
σµ :-Pauli matrices (will also use γµ) Ψ 2-component Grassmann vector
→ represent fermionic elds (electron and anti-electron)
19 Principle of Local Gauge Invariance interaction of elds mediated by gauge degrees of freedom: photon
→ replace derivative with the covariant derivative
∂µΨ(x) → (∂µ − ig0Aµ(x))Ψ(x) ≡ DµΨ(x) with Aµ gauge potential, g0 bare coupling
R 2 Sferm = d xΨ(¯ x)[Dµ + m] Ψ(x) is locally gauge invariant under a gauge transformation Λ(x):
iΛ(x) Aµ(x) → Aµ(x) + ∂µΛ(x) , Ψ(x) → e Ψ(x) Gauge Field Self-Interaction
R 2 Sgauge = d xFµνFµν ,Fµν(x) = ∂µAν(x) − ∂νAµ(x) equations of motion: obtain classical Maxwell equations
20 Schwinger Model
Quantization via Feynman path integral
R −Sgauge−S Z = DAµDΨDΨ¯ e ferm
• existence of bound states (mass gap)
• asymptotic free (g0 → 0 for distance between charges going to zero)
• super-renormalizable
• exactly solvable for zero fermion mass (Coleman)
⇒ valuable test laboratory for QCD (simulations can be done on your desk top)
21 Going to the Lattice introduce a 2-dimensional lattice with lattice spacing a
elds Ψ(x), Ψ(¯ x) on the lattice sites x = (t, x) integers we then de ne discrete derivatives
1 ∗ 1 ∇µΨ(x) = a [Ψ(x + aµˆ) − Ψ(x)] , ∇µΨ(x) = a [Ψ(x) − Ψ(x − aµˆ)] with µˆ a unit vector in direction µ replace (symmetric derivative): 1 ∗ ∂µ → 2 ∇µ + ∇µ
22 Lattice Partition Function
2 P ¯ 1 ∗ Z = R DΨ¯ DΨe−a x Ψ(x)[2σµ(∇µ+∇µ)+m]Ψ(x) R Q R DΨ = x dΨ(x) with R dΨ(x) a usual Grassmann integral version of naive lattice Schwinger model involves only classical elds
Mathews-Salam formulae: for a n × n matrix D Z Y ¯ dΨ(¯ x)dΨ(x)e−ΨxDx,yΨy = detD x
Z ¯ Y ¯ ¯ −ΨxDx,yΨy −1 dΨ(x)dΨ(x)ΨiΨje = D ij detD x
23 a rst look at the continuum limit we introduce external sources η, η¯ and write the action as S = Ψ¯ KΨ
Z(η, η¯) = R DΨ¯ DΨe−ψD¯ Ψ+ψη¯ +¯ηΨ the 2-point function (Green's function) G is obatined by
1 ∂ ∂ G = Z(0,0) ∂η ∂η¯Z(η, η¯) η=¯η=0 we have
−1 −1 −1 −1 Z(η, η¯) = R DΨ¯ DΨe−(Ψ¯ −ηD¯ )D(Ψ−D η)+¯ηD η = det(D)eηD¯ η and obtain
−1 1 ∗ −1 G = D = 2γµ ∇µ + ∇µ + m
24 we evaluate the 2-point function in momentum space impose periodic boundary conditions
Ψ(x + Lµ) = Ψ(x) This means that
• the momenta at restricted −π/a < pµ ≤ π/a (the rst Brillouin zone)
• the momenta are quantized (⇐ eip(x+L) = eipx) pµ = 2nπ/Lµ, n = 0, 1,...,Lµ − 1 and we have
1 P ˜ ipx Ψ(x) = V p Ψpe
2 with V = L the lattice volume assuming Lµ ≡ L for each µ
25 h i 1 ∗ ˜ ipx 1 ip(x+aµ) ip(x−aµ) ˜ ˜ ipx 2γµ ∇µ + ∇µ + m Ψpe = 2aγµ e − e Ψp + mΨpe | {z } eipx(eipaµ−e−iapµ)
−1 i ˜ 1 i −1 −aγµ sin pµa+m G = a (iγµ sin pµa) + m ≡ a (ˆpµγµ + m) = pˆ2+m2 continuum limit a → 0
i −aγµ sin pµa → γµpµ and we obtain
˜ −ipµγµ+m lima→0 G = p2+m2 this is exactly the free continuum propagator!
26 but now comes this little devil, asking
what if pµ ≈ π ? with pµ = kµ + π/a : sin pµa = sin(kµ + π) = − sin kµ which means that for a → 0 we obtain
˜ −ikµγµ+m G → − k2+m2 ⇒ again we obtain the continuum 2-point funktion! the opposite sign can be interpreted such that this continuum fermion at p ≈ π/a has the opposite chirality from the one at p ≈ 0
→ nd proliferation of fermions (0, 0) (π, 0) Brillouin zones 4 fermions (0, π) (π, π)
27 The Wilson-Dirac Operator can we repair this? Wilson suggested to add a second derivative term
2 P ¯ 2 S → a x Ψ[γµ∂µ − r ∂µ +m] Ψ(x) |{z}∗ ∇µ∇µ h i−1 ˜ 1 r P → G = a (iγµ sin pµa) + a µ(1 − cos pµa) + m now, for a → 0, pµ π/a
˜ −ipµγµ+m G → p2+m2 we play the same devils trick pµ = kµ + π/a, kµ 1 r r r a(1 − cos [kµ + π/a] a) = a(1 + cos kµa) → a ˜ r G → a(1 + cos kµa) ⇒ at the other corners of the Brillouin zone the fermions become in nitely heavy and decouple in the continuum limit
28 how to implement covariant derivative? matter elds live on lattice points separated by the lattice spacing a geometrical interpretation of the gauge elds if we have a curve zµ(s) parametrized by s, 0 ≤ s ≤ 1
a phase v at y (s = 0) would be transported along this curve to x (s = 1)
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R y v(1) = U z(x, y)v(0) = ei x dzµAµ(z)v(0) U z(x, y) is called the parallel transporter
29 Gauge Invariance under a gauge transformation
Aµ(x) → Aµ(x) + ∂µΛ(x)
R y the parallel transporter ei x dzµAµ(z) transforms as
U z(x, y) → eiΛ(x)U z(x, y)e−iΛ(y) and hence for
Ψ(x) → eiΛ(x) , Ψ(¯ x) → e−iΛ(x)Ψ(¯ x) the expression
Ψ(¯ x)U z(x, y)Ψ(y) is gauge invariant
30 Lattice gauge covariant derivative on the lattice, zµ is just a line element, a so-called link that connects the points x and y = x + aµˆ :
U(x, µ) = eiaAµ(x) ∈ U(1) this suggest to take the lattice derivatives
1 ∇ Ψ(x) = [U(x, µ)Ψ(x + µ) − Ψ(x)] µ a 1 ∇∗ Ψ(x) = Ψ(x) − U −1(x − µ, µ)Ψ(x − µ) µ a then we obtain the gauge invariant expression
¯ ∗ ∗ ¯ Ψ(x) mq + γµ(∇µ + ∇µ) − r∇µ∇µ Ψ(x) ≡ Ψ(x)DWilsonΨ(x)
31 The Gauge Field Action Is there a gauge invariant correspondence to the eld strength tensor using the parallel transporter? the link variables transform as U(x, µ) → g(x)U(x, µ)g−1(x + µ) need to parallel transport around a closed loop
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32 The Lattice Gauge Action
Euclidean pure gauge action
P P N † S = µ,ν Tr 2 − Up + U x µ<ν 2 p g0 |{z}≡β
−1 −1 Up = U(x, µ)U(x + µ, ν)U (x + ν, µ)U (x, ν) where g0 is the bare gauge coupling N relates to the group N = 1 for U(1), N for SU(N), for QCD: N = 3 the action converges in the continuum limit a → 0 to the expression
N S → 2 FµνFµν, a → 0 g0 note that † † 2 − Up − Up = (1 − Up)(1 − Up) ≥ 0
33 The Mass Spectrum goal: non-perturbative computation of bound state spectrum
→ euclidean correlation functions Reconstruction theorem relates this to Minkowski space operator O(x, t) with quantum numbers of a given particle correlation function decays e−Et ,E2 = m2 + p2 ⇒ zero momemtum P O(t) = x O(x, t) correlation function
1 X −Ht hO(0)O(t)i = Z h0|O(0)e |nihn|O(0)|0i n 1 X = |h0|O(0)|ni|2e−(En−E0)t Z n
34 connected correlation function
2 −E t limt→∞ hO(0)O(t)i − |hO(0)i| ∝ e 1 vanishing of connected correlation function
→ cluster property ⇒ locality of the theory
100 Data periodic boundary conditions Fit
10 P −Ent −En(T −t) hO(0)O(t)ic = n cn e + e
1 −mt −m(T −t) C(t/a) 1 t T : hO(0)O(t)ic ∝ e + e
0.1
0.01 0 10 20 30 40 50 60
t/a
35 Hadron Spectrum in Schwinger model hadrons are bound states in QCD
• mesons pion, kaon, eta, ...
• baryons neutron, proton, Delta, .. for the computation of the hadon spectrum
construct operators with the suitable quantum numbers
compute the connected correlation function
take the large time limit of the correlation function
36 Lorentz symmetry, parity and charge conjugation rotational symmetry → hypercubic group: discrete rotations and re ections classi cation of operators: irreducible representations R (note hypercubic group is a subgroup of SO(3))
parity charge conjugation T Ψ(x, t) → γ0Ψ(−x, t) Ψ(x, t) → CΨ¯ (x, t)
T −1 Ψ(¯ x, t) → Ψ(¯ −x, t)γ0 Ψ(¯ x, t) → −Ψ (x, t)C
C charge conjugation matrix C = γ0γ2 C has to satisfy
−1 T ∗ CγµC = −γµ = −γµ
37 Contraction
• 2-point-function calculation ¯ OΓ(x) = ψΓψ(0)
hOΓ(x)OΓ(0)i =
ψ¯(x)Γψ¯(0)ψ(x)Γψ(0) (2)
= tr[ΓS(x, 0)ΓS(0, x)]
in terms of eigenvalues and eigenvectors:
X 1 X †α β †γ δ tr[ΓS(x, 0)ΓS(0, x)] = (φj (x)Γαβφi (x))(φi (0)Γγδφj(0)) λiλj λi,λj αβγδ
38 Example: pion operator OPS(x, t) = Ψ(¯ x, t)γ5Ψ(x, t) correlation function
X ¯ ¯ fP (t) ≡ hOPS(0)OPS(t)i = ψ(x, t)γ5Ψ(x, t) ψ(0, 0)γ5Ψ(0, 0) x X = Tr SF (0, 0; x, t) γ5SF (x, t; 0, 0)γ5 x | † {z } =SF (0,0;x,t)
−1 used Wick's theorem and SF = D the fermion propagator ⇒ need to compute inverse of the fermion matrix
2 |h0|P |PSi| −m t −m (T −t) a t T : fPS(t) = · e PS + e PS 2mPS | 2 {z } ≡FPS/2mPS
FPS pion decay constant
39 Eective Masses exponential deacy of correlator Γ(t) = hO(0)O(t)ic de ne an eective mass Γ(t+1) meff(t) = − ln Γ(t)
periodic boundary conditions:
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40 Results in the Schwinger model (N. Christian, K. Nagai, B. Pollakowski, K.J.) scaling variable
√ 2/3 2 z ≡ (mf β) , 1/β = a Analytical results:
Strong coupling, small fermion mass (Smilga)
√ √ 2/3 βmπ = 2.008 βm = 2.008z
Large mass (Gattringer)
√ √ 2/3 βmπ = 2.163 βm = 2.163z
41 Scaling of dierent lattice Schwingermodels
Used: Wilson, Hypercube, Twisted Mass, Overlap fermions
√ 2/3 z = ( βmf ) = 0.4
0.88 0.88 β β
√ 0.86 0.86 √ π π 2 m m • observe a scaling 0.84 0.84
0.82 0.82 • universality of continuum limit
0.80 0.80
0.78 0.78
0.76 0.76 Wilson Hypercube 0.74 Twisted mass Overlap 0.74
1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 1 1 1 β β
42 Comparison with continuum results
2.5 1.1 data theor β R Rm = m /m
√ mπ π π π
π 2 m 1 4: strong coupling 1.5 0.9 •: large mass 0 0.2 0.4 0.6 0.8 1 1 z
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43 Lattice QCD change:
• 2-d → 4-d
• gauge eld U(x, µ) ∈ U(1) → U(x, µ) ∈ SU(3)
• Pauli matrices σµ → Gellmann-matrices γµ
• spinors become 12-component complex vectors
• theory needs renormalization
44 Gauge Field Self-Interaction
R 4 Sgauge = d xTrFµνFµν
Fµν(x) = ∂µAν(x) − ∂νAµ(x) + ig0 [Aµ(x),Aν(x)]
Aµ(x) vector potential (Lie Algebra) g0 bare coupling
[Aµ(x),Aν(x)] 6= 0 → self-interaction
45 The Continuum Action for Quantum Chromodynamics
R 4 R 4 S = SFerm + Sgauge = d xΨ(¯ x)[γµDµ + m] Ψ(x) + d xTrFµνFµν Holy Principles of Quantum Chromodynamics
• Gauge Invariance ↔ renormalizability
• Local Theory ↔ (micro) causality
• Lorentz Invariance ↔ relativistic theory
• Chrial Invariance ↔ spontaneous symmetry breaking ↔ spectrum of light mesons massless limit:
action invariant under chiral transformations (γ5 = γ0γ1γ2γ3, α real)
Ψ → eiαγ5Ψ , Ψ¯ → e−iαγ5Ψ¯ = Ψ¯ eiαγ5
equivalent condition: γ5D(m = 0) + D(m = 0)γ5 = 0 ⇐ very important physical consequences for the low energy behaviour of QCD
46 Chiral Symmetry
Wilson fermion action h i 4 P ¯ r P S = a p Ψ(p) mq + iγµ sin pµa + a µ(1 − cos pµa)
• gauge invariance: discussed later
• Locality: okay
• continuum limit: okay
• Lorentz Invariance: replaced by hypercubic group
iαγ −iαγ • Chiral invariance for mq = 0 , : Ψ → e 5Ψ , Ψ¯ → e 5Ψ¯ broken by mass-like term r a chiral symmetry only recovered in continuum limit (Bochicchio, Maiani, Martinelli, Rossi, Testa)
47 clash between chiral symmetry and fermion proliferation
→ Nielsen-Ninomiya theorem:
For any lattice Dirac operator D the conditions
• D is local (bounded by Ce−γ/a|x|)
2 • D˜(p) = iγµpµ + O(ap ) for p π/a
• D˜(p) is invertible for all p 6= 0
• γ5D + Dγ5 = 0 can not be simultaneously ful lled
48 Proof of Nielsen-Ninomya theorem (Friedan)
−1 µνρ chiral current jµ(p) = [6(i2π)] tr(∂νS∂ρS)(p)
0 R ∗ S(p) = θ(K(p) − p ) H = p ψ K(p)ψ
P(1) ∂µjµ = 0 chiral invariance
P(2) jµ = 0 vanishes away from spectrum of K(p) locality
P(3) jµ(p) depends only on the spectral projection of K near p
From P3: chiral charge (number of left-handed minus number of right-handed fermions) R I = dpj0(0, p) R R From P1: dpj0(0, p) = dpj0(p0, p) R From P2: dpj0(p0, p) = 0 ⇒ chiral charge vanishes ⇒ always have doublers
The theorem I = 0 simply states the fact that the Chern number is a cobordism invariant
49 let us go to outer space in extra dimensions (following D. Kaplan) also, let us start with continuum eld theory we consider a 5-dimensional theory (free fermions for the moment) with a mass defect in one extra dimension s
D5 = ∂µγµ + m0 + γ5∂s + m(s) −m; s → −∞ m(s) = +m; s → +∞
+m 0 s -m 0
50 let us try to solve the massless Dirac equation
D5(m0 = 0)ψ = 0 this can be solved by the ansatz
ipx ψ± = e Φ±(s)u± Z s 0 0 Φ±(s) = exp ± m(s )ds 0 γ5u± = ±u±
only Φ− normalizable ⇒ only one solution
• massless fermion travelling along the domain wall
• it has a de nite chirality
• bound to the domain wall with exponential fall-o with a rate |m| when going to |s| 1
51 on a (still in nite) lattice → −m; s ≤ a m(s) = 0; s = 0 +m; s ≥ a
derivatives → nite lattice dierences
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s but, there is now a second normalizable solution Φ+(s) = (−1) Φ−(s) → doubler in the extra dimension
52 solution ⇒ add Wilson term also in extra dimension
∗ ∗ D5 = ∂µγµ + m0 + γ5∂s + m(s) − ∇µ∇µ − ∇s∇s this kills all the doublers and we are left with a single chiral fermion on the lattice
on a nite lattice the extra dimension has an extent Ns
and we have to impose some boundary conditions
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53 we can also choose open boundary conditions in the extra dimension (Furmann, Shamir) ⇒ chiral zero modes appear as surface modes (reminiscent of Shockley modes in solid state physics numerically solving the Dirac equation
(a) periodic boundary conditions
(b) open boundary conditions (≈ Schrödinger functional)
54 gauging the 5-dimensional Dirac operator: gauge only the 4-dimensional part
∗ ∗ ∗ D5 = DW − m0 + γ5∂s + m(s) − ∇s∇s ≡ DµγµDµDµ − m0 + γ5∂s + m(s) − ∇s∇s ⇒ our 5-dimensional lattice action becomes P ¯ SDW = x,y,s,s0 ψ Dx,yδs,s0 + Ds,s0δx,y Ψ
1 X D = (1 + γ )U(x, µ)δ + (1 − γ )U †(y, µ)δ + (m − 4)δ x,y 2 µ x+µ,y µ x−µ,y 0 x,y µ P δ 0 − mP δ 0 − δ 0, s = 1 + 2,s − Ns,s 1,s Ds,s0 = P+δs+1,s0 + P−δs−1,s0 − δs,s0, 2 ≤ s ≤ Ns − 1 0 0 0 P−δNs−1,s − mP+δ1,s − δNs,s , s = Ns projectors P± = (1 ± γ5)/2 • m is the domain wall mass → determines the rate of exponential decay in the extra dimension
• m0 is the quark mass → has to be tuned to zero to give exactly chiral fermions
55 if we de ne (Neuberger, Kikukawa, Noguchi) K ≡ 1 ± 1γ asM M = D − m ± 2 2 52+asM W 0 then the domain wall operator can be written as an eective 4-dimensional operator Ns Ns K+ −K− aDNs = 1 + γ5 Ns Ns K+ +K− in nite Ns limit ⇒ 4-dimensional operator
aD ≡ limNs→∞ aDNs = 1 + γ5 sign (K+ − K−) , which is written as
aD = 1 − √ A ,A = − asM . A†A 2+asM anti-commutation relation for D
γ5D + Dγ5 = 2aDγ5D
56 D is a (practical !) example of an operator that satis es the celebrated Ginsparg-Wilson relation
γ5D + Dγ5 = 2aDγ5D
−1 −1 ⇒ Dxy γ5 + γ5Dxy = 2aγ5δx,y
−1 D anti-commutes with γ5 at all non-zero distances → only mild (i.e. local) violation of chiral symmetry Ginsparg and Wilson arrived at this expression already in the early days of lattice gauge theories from a completely dierent path ⇐ block spinning from the continuum local (Hernandez, K.J., Lüscher) solution: overlap operator Dov (Neuberger)
† −1/2 Dov = 1 − A(A A) with A = 1 + s − Dw s a tunable parameter, 0 < s < 1
57 Moreover: Ginsparg-Wilson relation implies an exact lattice chiral symmetry (Lüscher): for any operator D which satis es the Ginsparg-Wilson relation, the action
S = ψDψ¯ is invariant under the transformations
1 δψ = γ5(1 − 2aD)ψ ¯ ¯ 1 δψ = ψ(1 − 2aD)γ5
⇒ have a notion of chiral symmetry on the lattice
1 γ5 → γ5(1 − 2aD) the lattice operator D enjoys many properties of the continuum operator:
ZA = ZV = 1, anomaly, index theorem, ...
58 Locality write A†A as
† 1 P A A = 1 + 4 µ6=ν {Bµν + Cµν + Dµν}
4 ∗ ∗ Bµν = a ∇µ∇µ∇ν∇ν
1 2 ∗ ∗ Cµν = 2iσµνa ∇µ + ∇µ, ∇ν + ∇ν
2 ∗ ∗ Dµν = −γµa ∇µ + ∇µ, ∇ν − ∇ν relation to plaquette
2 a [∇µ, ∇ν] ψ(x) = = {U(x, µ)U(x + aµ) − U(x, ν)U(x + ν, µ)} ψ(x + aµ + aν)
= U(x, µ)U(x + aµ)(1 − U †(x, µ)U †(x + aµ)U(x + ν, µ)U(x, ν)) ψ
= (1 − Up)U(x, ν)U(x + ν, µ)ψ(x + aµ + aν)
59 Locality
2 ⇒ ka [∇µ, ∇ν] ψ(x)k ≤ 1 − Up therefore kCµνk ≤ 2(1 − Up) kDµνk ≤ 4(1 − Up) furthermore
4 ∗ ∗ 3 ∗ ∗ Bµν = a ∇µ∇ν∇µ∇ν −a ∇µ [∇µ, ∇ν − ∇ν] | {z } | {z } >0 ≤4(1−Up)
† in total: kA Ak ≥ 1 − 30(1 − UP )
† ⇒ if 1 − UP ) ≤ = 1/30 then A A bounded from below → A†A also bounded from above
0 < u ≤ A†A < v
60 Locality
Legendre expansion
2 −1/2 P∞ k (1 − 2tz + t ) = k=0 t Pk(z)
† z = (u + v − 2A A)/(v − u) , kzk ≤ 1 , ⇒ kPk(z)k ≤ 1 introduce parameter θ: coshθ = (v + u)/(v − u) with t = e−θ we nd
† −1/2 P∞ k 1/2 (A A) = κ k=0 t Pk(z); κ = {4t/(v − u)} consider kernel:
† −1/2 P (A A) ψ(x) = y G(x, y)ψ(y)
P∞ k G(x, y) = κ k=0 t Gk(z) kGkk ≤ 1
61 Locality introduce taxi driver distance
P kx − yktaxi = µ |xµ − yµ|
Gk(x, y) = 0 for all k < kx − yktaxi/2a
† ⇐ Pk is polynomial in A A and A has only nearest neighbour interaction therefor sum:
4 κ a kG(x, y)k ≤ 1−t exp {−θkx − yktaxi/2a} for a → 0: θ/a → ∞ ⇒ point like localization in the continuum limit relies on plaquette bound (1 − UP ) ≤ 1/30
62 what about realistic gauge eld con gurations?
→ resort to numerical simulations
→ study decay properties of
ψ(x) = A(A†A)−1/2η(x) with a source vector 1 if x = y and α = 1, η (x) = α 0 otherwise compute f(r) = max kψ(x)k kx − yk = r with kx − yk the taxi driver distance see whether there is an eect of s
63 expectation value of f(r)
open symbols: s = 0
lled symbols: s = 0.4 (β = 6.0, 6.2) s = 0.2
(β = 6.4)
→ strong s-dependence
→ curve at β = 6.4, s = 0.2 practically matches the curve for free fermions
64 Summary of lecture I
Feynman's path integral in euclidean metric and the tansfer matrix formulation
R −S N −1Ha Z = Dqe E = TrT , T = e ~
Application to Quantum Chromodynamics as our theory of the strong interaction Free Wilson fermion lattice action
4 P ¯ ∗ S = a x Ψ γµ∇µ − r∇µ∇µ + m Ψ(x)
• correct continuum limit • breaks chiral symmetry explicitly consequence of Nielsen-Ninomiya theorem recovered in continuum limit • Ginsparg Wilson relation: → exact lattice chiral symmetry
γ5D + Dγ5 = 2aDγ5D → solved by overlap operator
65 The Lectures http://www-zeuthen.desy.de/∼kjansen/
The harmonic oscillator programme J. Gonzales Lopez, A. Nube, D. Renner http://www-zeuthen.desy.de/∼kjansen/
66