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Lattice Gauge Theory Lattice Gauge Theory Hartmut Wittig Oxford University EPS High Energy Physics 99 Tamp ere, Finland 20 July 1999 Intro duction Lattice QCD provides a non-p erturb ati ve framework to compute relations between SM parameters and exp erimental quantities from rst principles Formulate QCD on a euclidean space-time lattice 3 with spacing a and volume L T 1 a gauge invariant UV Z S S 1 G F ][d ] e h i = Z [dU ][d ! allows for sto chastic evaluation of h i Ideally want to use lattice QCD as a phenomenologi ca l to ol, e.g. M f s Z 7! m + m 2 GeV m + u s K However, realistic simulation s of lattice QCD are hard... 1 1 Lattice artefacts cuto e ects lat cont p h i = h i + O a 1 a 1 4 GeV , a 0:2 0:05 fm ! need to extrap olate to continuum limit: a ! 0 ! cho ose b etter discretisati ons: avoid small p. 2 Inclusion of dynamical quark e ects: Z Y S [U ] lat G Z = [dU ] e det D + m f f lat detD + m =1: Quenched Approximation f ! neglect quark lo ops in the evaluation of h i. cheap exp ensive 2 Scale ambiguity in the quenched approximation: p Q [MeV ] 1 ;::: ; Q = f ;m ;m ; a [MeV ]= N aQ 3 Restrictions on quark masses: 1 a m L q ! cannot simulate u; d and b quarks directly ! need to control extrap olation s in m q 4 Chiral symmetry breaking: Nielsen & Ninomiya 1979: exact chiral symmetry cannot b e realised at non-zero lattice spacing ! imp ossibl e to separate chiral and continuum limits 3 Outline: I. The lattice Dirac op erator { Improved actions { Exact chiral symmetry on the lattice II. Simulations with dynamical quarks { Light hadron sp ectrum, quenched & unquenched III. Light quark masses { Non-p erturba tive renormalisat ion { Recent estimates IV. Glueballs & heavy hybrids V. Omissions VI. Summary 4 I. The lattice Dirac op erator Massless free fermions: X 4 S = a x D x y y F x;y a D x y is lo cal 2 e b D p=i p + O ap e c D p is invertible for p 6=0 d D + D =0 5 5 ! Nielsen & Ninomiya: a{d do not hold simultaneous ly ! either left with doublers or chiral symmetry broken explicitly Staggered Kogut-Susskind fermions Wilson fermions 1 1 ar r + r r D = W 2 2 - degeneracy fully lifted; chiral symmetry broken - leading cuto e ects of order a p =1 5 O a improved Wilson fermions Can one nd discretisation for Wilson fermions with reduced cuto e ects? Symanzik improvement: Remove leading lattice artefacts by adding a counterterm to the Wilson- Dirac op erator Sheikholeslami-Wohlert,::: ia c F x D = D + m + sw SW W 0 4 Fix c g through suitable improvement sw 0 condition 2 ! leading lattice artefacts are O a in sp ectral quantities For n =0 and n =2, c has b een determined f f sw non-p erturba ti vely M. L uscher et al. 1997; R. Edwards et al. 1997; Jansen & Sommer 1998. 6 Scaling of m for m =m = 0:7 T. Klassen V PS V 1998 Non-p erturbati vel y improved Wilson action: 2 scaling b ehaviour is consistent with residual a artefacts Can push Symanzik improvement to higher orders Alford, Klassen, Lepage 1998 Chiral symmetry remains broken 7 Exact chiral symmetry on the lattice Chiral symmetry breaking may be tolerated in many applica tion s of QCD Chiral symmetry is crucial for non-p erturb ati ve formulation of - EW theory - SUSY Prop osals to circumvent the NN Theorem: { Domain Wall Fermions Kaplan 1992; Shamir, Furman 1993 { Overlap Formalism Narayanan, Neub erger, 1993{98 { \Perfect Actions" Hasenfratz, Niedermayer, :: :, 1993{98 { Ginsparg-Wil son relation Ginsparg & Wilson 1982, L uscher 1998{99 See also: T. Blum, Lattice 98 M. Luscher Lattice 99 H. Neub erger 8 Domain Wall Fermions Shamir & Furman th Intro duce discrete extra 5 dimension : 6 x x 1 N s x x 1 N 4 k s L ;D x; y x x; s s ? - ? N ;D 0 s ss DWF k ? 0 D x; y =D x; y + x y D 0 0 ss ss ss k D x; y : Wilson-Dirac op erator with mass M . ? D : contains Dirac mass m. 0 ss For m =0; N !1: s - no fermion doublers - chiral mo des are trapp ed in 4-dim. domain walls at either end In practice: work at nite N s ! decouplin g of chiral mo des not exact { terms which break chiral symmetry are exp onential ly suppressed 9 Pion mass: 2 N s am = C am + ae Blum, Soni, Wingate, hep-lat/9902016, N =10 s ! Can realise almost exact chiral symmetry at non- zero a at the exp ense of simulating 5-dim. theory { N 10 30 may b e sucient? s 10 Ginsparg-Wi l so n Relation Replace condition D + D =0 by the weaker 5 5 relation Ginsparg & Wilson 1982 D + D = aD D 5 5 5 If D satis es GWR, then this implies an exact global symmetry L uscher 1998 1 1 aD ; aD = 1 = 1 5 5 2 2 ! \lattice chiral symmetry" ! Flavour-singlet case: Ward identitie s have correct anomaly Construction of gauge theories with lo cal chiral symmetry Ab eli an and non-Ab elia n relies on GWR L uscher 1998{99 Overlap Formalism Narayanan, Neub erger Reinterpretation of the domain wall prop osal; Overlap op erator Neub erger 1998: 1=2 y 1 D = 1 A A A ; A =1 aD N W a D : massless Wilson-Dirac op erator W ! satis es GW-relation 11 Witten's SU2 anomaly on the lattice Witten 1982: SU2 gauge theory coupled to a single left-handed fermion is mathematic all y inconsistent: Z Z S [A] D G Z = [dA]e ][d ] e [d Weyl Z 1=2 S [A] G = [dA] det D [A] e There is a non-trivial gauge transformation U x such that 1=2 U 1=2 det D [A] = det D [A ] Exp ectation value w.r.t. Z is indetermi na te: h i =\0/0" D hermitian: eigenvalues are real and come in pairs, U Vary gauge eld smo othly from A to A : U A t=1 t A + tA ; t 2 [0; 1] Atiyah-Singer: numb er of eigenvalues that b ecome negative as t is varied from 0 to 1 is odd. 12 Compute the eigenvalues of the overlap op erator D in a lattice simulation, as A is smo othly N U O. Bar, Lattice 99 turned into A D : eigenvalues come in complex conjugate pairs; N exp ect level crossing for Im 0.2 0.1 0 -0.1 -0.2 0.4 0.5 0.6 Crossing observed for lowest eigenvalue ! Numerical pro of of Witten's SU2 anomaly Only p ossible if lattice Dirac op erator has the correct chiral prop erties 13 II. Simulations with dynamical quarks Quenched light hadron sp ectrum deviates from exp eriment CP-PACS Collab oration, hep-lat/9904012: Light Hadron Spectrum in Quenched QCD final results from CP−PACS 1.8 1.6 Ξ Ω 1.4 Σ Λ Ξ* φ 1.2 N Σ* m (GeV) K* 1.0 ∆ 0.8 experiment K K input φ input 0.6 0.4 m m to o small by 10 16 4 6 K K m =m =1:143 0:033 Exp.: 1:22; 2:5 N Quenched QCD describ es the light hadron sp ectrum at the level of 10. 14 Can sea quark e ects account for the deviation of the quenched sp ectrum from exp eriment? Oa Improvement even more imp ortant: { exp ensive to simulate very small lattice spacings { need to separate lattice artefacts from sea quark e ects Recent simulatio ns with n =2 f Collab oration GFlops Gluons Quarks CU/BNL/Riken US 250 Wilson DWF CP-PACS J 300 Iwasaki SW, tad UKQCD UK 28 Wilson SW, n.p. SESAM/TL D/I 14 Wilson Wilson > MILC US 7:3 LW staggered ! identify two light avours with physical u; d quarks 15 Observables in full QCD with n =2 avours: f Example: meson propagator: val m sea ; ; 5 j 5 j m Sea quarks are still relatively heavy: sea val m = m : m m 0:67 0:86 UKQCD PS = ; =0:169 0:6 0:8 CP-PACS m m V sea Study the dep endence of observables on m ; sea Extrap olate in m to physical m =m Incorrect n for Kaon physics: f val sea ! cho ose m >m val sea ;m = m and identify m = m u;d s ! \partially quenched" 16 Sea quark e ects in the light hadron sp ectrum: Study the continuum limit of hadron masses in \partially quenched" QCD CP-PACS, T. Kaneko, Lattice 99 K* and φ masses, K input RC (combined fits) 1.05 χ Φ quenched PW (q PT fits) 1.00 0.95 m [GeV] K* 0.90 0.85 0.80 0.0 0.2 0.4 0.6 0.8 1.0 1.2 a [GeV−1] Mesons: discrepancy with exp erimental sp ectrum diminishe d when sea quarks are \switched on" Quantify sea quark e ects in the continuum limit Can remainin g di erences b e blamed on - incorrect n =2 for strange hadrons? f - continuum extrap olation s? - extrap olation s in the quark masses? 17 III.
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