Lattice Quantum Chromo Dynamics and the Art of Smearing

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1 Lattice Quantum Chromo Dynamics and the Art of Georg Engel March 25, 2009 KarlFranzensUniversität Graz Advisor: Christian B. Lang Co-workers: M. Limmer and D. Mohler 1 / 29

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3 Continuum Theory Regularization: The Lattice Lattice: Quarks and Gluons Lattice and Symmetries Lattice Gauge Lattice Fermion Chiral Symmetry 3 / 29

4 Continuum Theory Continuum Theory Regularization: The Lattice Lattice: Quarks and Gluons Lattice and Symmetries Lattice Gauge Lattice Fermion Chiral Symmetry Strongly interacting elds: Quarks (Fermions): up, down, strange, charm, bottom and top represented by the eld ψ (f) (x) Gluons (Bosons) represented by the eld A µ (x) Commonly accepted theory for strong interaction: Quantum Chromo Dynamics The generalization of Feynman's quantum mechanical Path Integral to Quantum Field Theory is called Functional Integral. Within this formalism, the action S gives the probability for each specic eld conguration via: P (ψ, ψ, A) e S(ψ,ψ,A) E.g. the expectation value of an operator O(ψ, ψ, A) is given by: O(ψ, ψ, A) = O(ψ, ψ, A) F G = 1 D [ A ] D [ ψ, ψ ] e S(ψ,ψ,A) O(ψ, ψ, A) Z 4 / 29

5 Continuum Theory II Continuum Theory Regularization: The Lattice Lattice: Quarks and Gluons Lattice and Symmetries Lattice Gauge Lattice Fermion Chiral Symmetry The (Euclidean) action S is the space-time integral of the QCD-Lagrangian: S = L[ψ, ψ, A] = N f f=1 d 4 xl ψ (f) (x) [ γ µ D µ + m (f)] ψ (f) (x) + 1 2g 2 Tr[ F µν (x)f µν (x) ] D µ (x) = µ + ia µ (x) F µν (x) = i [ D µ, D ν ] = µ A ν (x) ν A µ (x) + i [ A µ (x)a ν (x) ] Most of the known (hadronic) mass in universe is created by strong interaction. 5 / 29

6 Regularization: The Lattice Continuum Theory Regularization: The Lattice Lattice: Quarks and Gluons Lattice and Symmetries Lattice Gauge Lattice Fermion Chiral Symmetry There are several divergencies in continuum QCD, regularization is necessary. Theory cannot be veried by experiments at arbitrary high energies. One possible regularization: Introduce momentum ultraviolet-cuto, equivalent to a minimum distance, as seen by Fourier-transformation. If we continue to require local gauge symmetry, we obtain. Integral over spacetime turns into a sum, resulting nite number of integrals over elds is computed using Monte Carlo techniques. Continuum limit of the lattice theory is a possible denition of a renormalized continuum theory. Lattice spacing a 0 The lattice theory gives predictions for experiment only in the continuum limit. universality classes of operators on the lattice. Lattice quantum eld theory is very similar to classical statistical physics. 6 / 29

7 Lattice: Quarks and Gluons Continuum Theory Regularization: The Lattice Lattice: Quarks and Gluons Lattice and Symmetries Lattice Gauge Lattice Fermion Chiral Symmetry a { Quarks Gluon Quarks ψ(n), ψ(n) Gluons Links Gauge Transporter U µ (n) 7 / 29

8 Naive Lattice Fermion Continuum Theory Regularization: The Lattice Lattice: Quarks and Gluons Lattice and Symmetries Lattice Gauge Lattice Fermion Chiral Symmetry Continuum free fermion action: S F [ ψ, ψ ] = d 4 x ψ(x) (γ µ µ + m) ψ(x) Discretization of partial derivative on the lattice: µ ψ(na) = ψ ((n + ˆµ)a) ψ ((n ˆµ)a) 2a Naive lattice ansatz for free fermion action: S F [ ψ, ψ ] = a 4 n Λ ψ(n) ( 4 µ=1 + O(a 2 ) γ µ ψ(n + ˆµ) ψ(n ˆµ) 2a + mψ(n) ) wrong!! 8 / 29

9 Naive Lattice Fermion Continuum Theory Regularization: The Lattice Lattice: Quarks and Gluons Lattice and Symmetries Lattice Gauge Lattice Fermion Chiral Symmetry Continuum free fermion action: S F [ ψ, ψ ] = d 4 x ψ(x) (γ µ µ + m) ψ(x) Discretization of partial derivative on the lattice: µ ψ(na) = ψ ((n + ˆµ)a) ψ ((n ˆµ)a) 2a Naive lattice ansatz for free fermion action: S F [ ψ, ψ ] = a 4 n Λ ψ(n) ( 4 µ=1 + O(a 2 ) γ µ ψ(n + ˆµ) ψ(n ˆµ) 2a + mψ(n) ) wrong!! 8 / 29

10 Gauge Invariance Continuum Theory Regularization: The Lattice Lattice: Quarks and Gluons Lattice and Symmetries Lattice Gauge Lattice Fermion Chiral Symmetry is required to be obey all symmetries, also local gauge invariance: Be Ω(n) SU(N c ), then: ψ (n) = Ω(n)ψ(n) ψ (n) = ψ(n)ω(n) ψ (n)ψ (n + ˆµ) = ψ(n)ω(n) Ω(n + ˆµ)ψ(n + ˆµ) The last expression is not gauge invariant. Introduce link variables U µ (n) with appropriate transformation law: U µ(n) = Ω(n)U µ (n)ω(n + ˆµ) ψ (n)u µ(n)ψ (n + ˆµ) = ψ(n)ω(n) Ω(n)U µ (n)ω(n + ˆµ) Ω(n + ˆµ)ψ(n + ˆµ) = ψ(n)u µ (n)ψ(n + ˆµ) These U µ (n) are the fundamental gluonic variables on the lattice. 9 / 29

11 Lattice and Symmetries Continuum Theory Regularization: The Lattice Lattice: Quarks and Gluons Lattice and Symmetries Lattice Gauge Lattice Fermion Chiral Symmetry Local gauge symmetry: Explicitly obeyed. Translational symmetry: Broken to discrete symmetry, but nicely restored in continuum limit. Rotational symmetry: Similar to translational symmetry. Finite number of irreducible representations (quantum numbers) instead of spin, but correct states are obtained in the continuum limit. Chiral symmetry: Explicitly broken if doublers are removed. Restoration possible but expensive. 10 / 29

12 Lattice Gauge Continuum Theory Regularization: The Lattice Lattice: Quarks and Gluons Lattice and Symmetries Lattice Gauge Lattice Fermion Chiral Symmetry Need gauge invariant object: trace over closed loop of gauge links Smallest possible closed loop: Plaquette U µν (n) = U µ (n)u ν (n + ˆµ)U µ (n + ˆµ + ˆν)U ν (n + ˆν) Wilson gauge action S g n = U µ (n)u ν (n + ˆµ)U µ (n + ˆν) U ν (n) µ<ν Re Tr [ 1 U µν (n) ] Improvement by taking into account larger Wilson loops. All in the same universality class, i.e. converge to Tr [ F µν (x)f µν (x) ] in the continuum limit, but improvement reduces the discretization errors. 11 / 29

13 Lattice Gauge II Continuum Theory Regularization: The Lattice Lattice: Quarks and Gluons Lattice and Symmetries Lattice Gauge Lattice Fermion Chiral Symmetry Wilson gauge action: Plaquettes 12 / 29

14 Lattice Gauge II Continuum Theory Regularization: The Lattice Lattice: Quarks and Gluons Lattice and Symmetries Lattice Gauge Lattice Fermion Chiral Symmetry Improved gauge action: Also larger loops 13 / 29

15 Lattice Fermion Continuum Theory Regularization: The Lattice Lattice: Quarks and Gluons Lattice and Symmetries Lattice Gauge Lattice Fermion Chiral Symmetry Fermionic action S F = a 4 f ψ(f) (n) D (f) (n, m) ψ (f) (m) Naive fermion action D (f) (n, m) = m (f) δ n,m + 1 2a ±4 µ=±1 Unwanted doublers: Got 16 instead of 1 fermion. Wilson Dirac matrix D W D (f) W (m (n, m) = (f) + 4 ) δ n,m 1 a 2a ±4 µ=±1 γ µ U µ (n) δ n+ˆµ,m ( 1 γµ ) Uµ (n) δ n+ˆµ,m Wilson term is a discretization of the Laplace operator, shifting the mass of the doublers to innity in the continuum limit. Only the physical pole, no doublers. Breaks chiral symmetry explicitly. 14 / 29

16 Chiral Symmetry Continuum Theory Regularization: The Lattice Lattice: Quarks and Gluons Lattice and Symmetries Lattice Gauge Lattice Fermion Chiral Symmetry Continuum version of chiral symmetry: Massless action invariant under avor dirac transformations (T i a generator of SU(N f )) ψ = e iαγ 5T i ψ, ψ = ψe iαγ 5T i Holds if and only if the Dirac operator satises D γ 5 + γ 5 D = 0 NoGo - theorem on the lattice, but not for the whole universality class. Lattice: Massless action invariant under ψ = e iαγ 5T i(1 a 2 D ) ψ, ψ = ψe iαt i(1 a 2 D )γ 5 Ginsparg-Wilson relation: D γ 5 + γ 5 D = a D γ 5 D 15 / 29

17 - Introduction - Applications - String Tension I - Wilson Loop Hadron Spectroscopy Quark Results - Nucleon Ground State 16 / 29

18 Link - What is it? - Introduction - Applications - String Tension I - Wilson Loop Hadron Spectroscopy Quark Results - Nucleon Ground State What is it on the very microscopic scale? What is it on the macroscopic scale? What are possible applications? Is it legal, i.e. does it preserve the expectation values of the original elds? Has to be checked in each application separately. 17 / 29

19 The vacuum (= the ground state) is made of zero-point oscillations of the field - Introduction on top of classical field configurations A class (x, t) : - Introduction - Applications - String Tension I - Wilson Loop Hadron Spectroscopy Quark Results - Nucleon Ground State Computer simulations of the vacuum in pure gauge theory. Lhs : action density, (a) before and (c) after smearing left: action density, before a right: so-called topological c Rhs : topological charge density, (b) before and (d) after smearing Computer simulations of the Yang Mills vacuum [J.Negele et al.] T op : Full conguration, dominated by quantum uctuations. Top: Bottom snapshot : of the full smooths configuration, out most dominated of the uctuations, by zero-point yielding oscillations. Bottom: approximately classical kills thecongurations. zero-point oscillations but reveals classical configurat gluon field, here: instantons and anti-instantons. 18 / 29

20 - Introduction II - Introduction - Applications - String Tension I - Wilson Loop Hadron Spectroscopy Quark Results - Nucleon Ground State Link smearing is a kind of averaging of the original links with links in the neighborhood. The original ones are called thin, the smeared ones fat links. The procedure has to be dened such that transformation laws and locality are ensured. does not change the scale. 19 / 29

21 - Applications - Introduction - Applications - String Tension I - Wilson Loop Hadron Spectroscopy Quark Results - Nucleon Ground State Measurement tool (e.g. for instantons): Smooth gauge eld necessary for denition of e.g. the topological charge density. O( ) steps of smearing in order to lter topological properties of the gauge eld. Improvement of the action in the Monte Carlo process itself: Eective contributions from e.g. larger Wilson loops in the fat gauge action. Fat action is in the same universality class as thin action, showing reduces discretization eects. Chirality may be improved and the number of exceptional congurations reduced because some of the smallest eigenmodes of the Dirac operator are removed. Overlap operator: fat links in the kernel provide improvement of convergence. Improvement of the observables in measurement, e.g. in Hadron Spectroscopy: Filtering the UV-uctuations of the observable means reduced scaling violations and thus less noise in many cases. 20 / 29

22 - Validity - Introduction - Applications - String Tension I - Wilson Loop Hadron Spectroscopy Quark Results - Nucleon Ground State Legal? As measurement tool for topological objects: still under criticism, but ltering is necessary to extract topological quantities that can be interpreted in terms of continuum objects. What about the Boltzmann weight in Monte Carlo? View it as: O(S(U)) = (OS)(U) Is there a signature of too much smearing? Short-range physics is destroyed, e.g. Coulomb sector of static quark potential. Ending up with noise or a trivial gauge eld, depending on parameters. Criterion: Signal of observables. Continuum limit: Number of contributing links stays nite, thus the fat link converges to a local link. 21 / 29

23 - String Tension I - Introduction - Applications - String Tension I - Wilson Loop Hadron Spectroscopy Quark Results - Nucleon Ground State Static quark potential. Lhs : Thin link observable. Rhs : Fat link observable, reduced noise in long-range behavior, but missing Coulomb potential for short-range physics. 22 / 29

24 - Wilson Loop - Introduction - Applications - String Tension I - Wilson Loop Hadron Spectroscopy Quark Results - Nucleon Ground State APE HYP EXP Wilson: c1=0, c2=0; parameter set 2 0 #=1 #=3 #=7 #= Wilson: c1=0, c2=0; parameter set 5 0 #=1 #=3 #=7 #=15 Lhs : Wilson loops W 1 1, W 2 2, W 4 4 versus the number of smearing steps, using small smearing parameters. Many smearing steps seem favorable, but too many will result in an trivial at gauge eld. Rhs : Using large smearing parameters, the result deteriorates after a few steps. Application of a high number of steps lead to a very noisy gauge eld. APE HYP EXP 23 / 29

25 Hadron Spectroscopy - Introduction - Applications - String Tension I - Wilson Loop Hadron Spectroscopy Quark Results - Nucleon Ground State Hadron masses are extracted from the exponential behavior of correlators of operators with correct quantum numbers. O(n t )Ō(0) = 1 D [ U ] D [ ψ, ψ ] Z e S(ψ,ψ,U) O(n t ) O(0) = Ae an te ( 1 + O(e ant E ) ) Examples: Pion and Nucleon interpolator O π = u smear γ 5 d smear O N = ɛ abc u a,smear ( u T b,smear Cγ 5 d c,smear d T b,smear Cγ 5 u c,smear ) with C the charge conjugation operator. 24 / 29

26 Quark - Introduction - Applications - String Tension I - Wilson Loop Hadron Spectroscopy Quark Results - Nucleon Ground State Point sources possible, but smeared sources show better overlap with physical states and also allow for a larger basis in the variational method. We use 3 types of smeared sources (sinks): narrow, wide, derivative Narrow or wide Derivative / 29

27 Results - Nucleon Ground State Effective masses - a M N Nucleon ground state a M N = (23) Time Nucleon ground state Time Nucleon ground state a M N = (18) Time 26 / 29

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29 Lattice: Theory is regularized via UV-cuto in a gauge covariant manner. Link variables are gauge transporters connecting the sites. Universality classes of operators, in particular considering gauge and fermion action. : Averaging the links in a local and gauge covariant way. Reduces UV-uctutations of the gauge eld. May improve chiral properties of action and reduce further discretization errors. Improves the signal-to-noise ratio of many observables. 28 / 29

30 Thank You!! 29 / 29