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Lattice Methods in Field Theory Contents Lattice Methods in Field Theory Contents 1. Motivation 2. Basics—Euclidean quantisation 3. Lattice gauge fields Jonathan Flynn 4. Lattice fermions University of Southampton 5. Lattice QCD 6. Numerical simulations BUSSTEPP 2002 University of Glasgow 1 2 Why lattice QCD? evaluate non-perturbative strong interaction effects in 1 Motivation • physical amplitudes using large scale numerical simulations: observables found directly from QCD lagrangian 1.1 Theoretical long-distance QCD effects in weak processes are • The lattice regularisation of quantum field theories frequently the dominant source of uncertainty in extracting fundamental quantities from experiment is the only known nonperturbative regularisation • admits controllable, quantitative nonperturbative Example: K –K mixing and BK • calculations provides insight into how QFT's work and enables • study of unsolved problems in QFT's s W d 0 t 0 1.2 Applications of lattice field theories K t K QED: `triviality', fixed point structure, . d s • W Higgs sector of the SM: bounds on Higgs mass, Since mt ;m L can do perturbative analysis at high W QCD • baryogenesis, . scales where QCD is weak and run by renormalisation group down to low scales. Left with: Quantum gravity • 2 n SUSY 0 mt dγn (1 γ5)s dγ (1 γ5)s 0 1 K C αs(µ); − − K m + O • h j m2 m4 j i m6 QCD: hadron spectrum, strong interaction effects in W W W • weak decays, confinement, chiral symmetry breaking, C( ): calculable perturbative coefficient (as long as • · exotics, finite T and/or density, fundamental µ=LQCD not too small) parameters ( , quark masses) αs evaluate matrix element on a lattice with µ 1=a (a is • lattice spacing) ∼ match lattice result to continuum at scale µ 1=a • ∼ 3 4 CKM matrix and unitarity triangle Unitarity triangle Vud Vus Vub V V ∗ +V V ∗ +V V ∗ = 0 Vcd Vcs Vcb = ud ub cd cb t d t b Vt d Vt s Vt b ! 1 2 3 1 =2 A ( i ) C K M Dms & Dmd λ λ λ ρ η f i t t e r Dm − 2 −2 4 d λ 1 λ =2 Aλ + O(λ ) 0.8 − − Aλ3(1 ρ i η) Aλ2 1 ! − − − 0.6 h Unitarity: VudVu∗b +VcdVc∗b +Vt dVt∗b = 0 0.4 3 7 VudVu∗b = Aλ (ρ + i η) + O(λ ) |e | |Vub/Vcb| 3 7 0.2 K VcdVc∗b = Aλ + O(λ ) sin 2b − WA 3 7 0 Vt dVt∗b = Aλ (1 ρ i η) + O(λ ) − − -1 -0.5 0 0.5 1 where ρ = ρ(1 λ2=2) and η = η(1 λ2=2). r − − Measurement V other Constraint CKM × b u V 2 (ρ;η) ub 2 2 ! ρ + η |e | b c V K ! cb α 2 2 2 2 DM V fB BB f (mt ) (1 ρ) + η d j t d j d d − 2 2 ρ + i η 1 ρ i η DM V fB BB − − d t d d d (1 )2 + 2 2 ρ η DMs V f B − t s Bs Bs ε f (A; η;ρ;B ) ∝ η(1 ρ) γ β K K − (CKMfitter Spring 2002: H Hocker¨ et al, hep-ph/0104062; (0;0) (1;0) http://ckmfitter.in2p3.fr/) 5 6 2 Basics: Euclidean quantisation Lattice embedded in d-dimensional Euclidean spacetime Tat . with sin2β from BaBar and Belle, Standard Model is in good shape. Errors in the nonperturbative parameters are now the limiting factor in more precise testing to look for effects from New Physics. Las There is also a rich upcoming experimental programme in the next few years which will need or test lattice results: as B-factories: constraining unitarity triangle, rare decays • Tevatron Run II: DM , DG , b-hadron lifetimes, . • Bs Bs CLEOc: leptonic and semileptonic D decays, masses of at • quarkonia, hybrids, glueballs as ;a lattice spacings LHC: . t • Las length in spatial dimension(s) Tat length in temporal dimension Matter fields live on lattice sites x. Example: scalar field x = nas ; n = 0;:::;L 1 φ(x) with j − x = ma ; m =;:::;T 1 0 t − 7 8 2.1 Lattice as a regulator Fourier transform of a lattice scalar field in one dimension: x = na, n = 0;:::;L 1, with periodic boundary conditions: − L 1 − ipna φ˜(p) = a ∑ e− φ(x) 2.2 Euclidean quantisation on the lattice n=0 φ(x + La) = φ(x) Path integral well-defined in Euclidean space discretisation implies Wick • Minkowski Euclidean φ˜(p) periodic with period 2π=a rotation • momenta lie in first Brillouin zone • π π iε prescription avoids poles < p − a ≤ a Procedure have introduced a momentum cutoff; • π 1. Continuum classical Euclidean field theory L = a 2. Discretisation lattice action −! spatial periodicity implies momentum p quantised in • 3. Quantisation functional integral units of 2π=La −! gauge invariance and gauge fields, fermions: later • Lattice provides both UV and IR cutoffs. Ultimately want infinite volume (L;T ¥) and continuum (a 0) limits. Most effort devoted to!continuum limit. ! 9 10 Step 2: Discretisation Introduce a hypercubic lattice LE with at = as = a. x0 x1;2;3 Step 1 L = x aZ4 = 0;:::;T 1; = 0;:::;L 1 E 2 a − a − Euclidean fields φ(x) obtained formally from analytic 3 continuation L T lattice sites • 0 finite volume t ix ; φ(x;t ) φ(x) • ! − ! finite number d.o.f. Action: • Lattice action: 4 1 2 SE [φ] = d x (∂m φ) +V (φ) 2 4 1 Z SE [φ] = a ∑ ∇m φ(x)∇m φ(x) +V (φ) where µ = 0;1;2;3 and x L 2 2 E 1 λ with forward and backward lattice derivatives V (φ) = m2φ 2 + φ 4 2 4! 1 ∇m φ(x) φ(x+aµˆ) φ(x) ≡ a − Minkowski Euclidean ! 1 Lorentz symmetry O(4) symmetry ∇∗m φ(x) φ(x) φ(x aµˆ) ≡ a − − t 2 x2 invariant (x 0)2 + x2 invariant − + + Lattice Laplacian: + 0 − 1 0 1 + 3 − (x) ( ∗ ) (x) B C B + C Dφ ∑ ∇m ∇m φ B C B C ≡ m=0 @ − A @ A 1 3 = (x+a ˆ) + (x a ˆ) 2 (x) 2 ∑ φ µ φ µ φ a m=0 − − 11 12 Step 3: Quantisation—functional integral Lattice action for a free scalar field: SE [f] ZE D[φ]e− 4 1 ≡ SE [φ] = a ∑ φ(x)Dφ(x) +V (φ) Z x L − 2 2 E D[φ] is the measure, eg: D[φ] = ∏ dφ(x) x L Remarks 2 E finite number of integrations Discretisation is not unique. Can use different • • ( ) definitions for ∇m∗ and/or V (φ) as long as they become Correlation functions the same in the naive continuum limit, a 0. ! 1 S [f] φ(x ) φ(xn) D[φ]φ(x ) φ(xn)e− E Universality: discretisations fall into classes, each h 1 ··· i ≡ Z 1 ··· ∗ member of which has the same continuum limit E Z is shorthand for 0 T 0 , time-ordered vacuum Improvement: optimise choice of lattice action for • h·i h j · j i ∗ a faster approach to the continuum limit expectation value well-defined if S [φ] > 0 O(4) (eventually Lorentz symmetry) is not preserved. • E • particle spectrum implicitly determined by correlation Have cubic symmetry instead; recover O(4) symmetry • as a 0. functions ! analytically continue to Minkowski space and get • S-matrix elements (= physics) via LSZ 13 14 Lattice propagator Relation between W [J] and K : 2.3 Generating functional 1 1 1 W [J] SE [f] (J;f) 2 (J;K − J) F e = ∏ dφ(x)e− e = e Scalar product on space of fields φ over LE : ZE x L Z 2 E 4 (φ1;φ2) = a ∑ φ1(x)φ2(x) Diagonalise K through Fourier transform: x L 2 E 4 ip y 1 ip x J˜(p) = a e− · J(y); J(x) = e · J˜(p) ∑ 4 3 ∑ Action for free scalar field: y L a L T p L 2 E 2 E∗ 1 2 is the dual lattice (or set of momentum points in the SE [φ] = (φ;K φ); K = ∇∗m ∇m +m LE∗ 2 − Brillouin zone): K is a linear operator on F . 0 2π 1;2;3 2π Let J(x) be an external field (source) on L , J F , and LE∗ = p p = n0; p = n1;2;3; E 2 Ta La define the generating functional W [J] through, n = 0;:::;T 1; n = 0;:::;L 1 eW [J] e(J;f) 0 − 1;2;3 − ≡ h i 1 S [f] (J;f) = ∏ dφ(x)e− E e ZE x L Z 2 E 2π=a aT Correlation functions found by differentiating w.r.t. J(x): ∂ eW [J] = a4 φ(x)e(J;f) ∂J(x) h i 2 aL ∂ 2π=a eW [J] = (a4)2 φ(x )φ(x ) ∂J(x )∂J(x ) h 1 2 i 1 2 J=0 aL 2π=a LE LE∗ 15 16 Propagator (continued) Find: 1 4 (K − J)(x) = a G(x y)J(y) Remarks ∑ − y LE 2 As a 0 (and L;T ¥), G(x y) becomes the G(x y) is the Green function for K : • ! ! − − Euclidean Feynman propagator: ip (x y) 1 e · − 4 ip (x y) G(x y) = a 0 d p e · − 4 3 ∑ 2 2 G(x y) ! − a L T pˆ +m 4 2 2 p L∗ − −! (2π) p +m 2 E Z with using pˆ2 = p2 + O(a2). 3 2 2 apm pˆ = ∑ pˆmpˆm ; pˆm = sin Particle masses defined through poles of the m=0 a 2 • ˆ2 2 1 propagator, here poles of (p +m )− , which is periodic in each component of p with period 2π=a.
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