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An introduction to Lattice Gauge theories Antonio Rago University of Wuppertal CP3 -Origins Odense-17 August 2010 A. Rago (U. Wuppertal) Lattice Introduction 17 August 2010 1 / 20 Overview of the Lattice Approach LGT calculations are a non-perturbative implementation of ﬁeld theory using the Feynman path integral approach. A. Rago (U. Wuppertal) Lattice Introduction 17 August 2010 2 / 20 Overview of the Lattice Approach LGT calculations are a non-perturbative implementation of ﬁeld theory using the Feynman path integral approach. The calculations proceed exactly as if the ﬁeld theory was being solved analytically (had we the ability to do the calculations). A. Rago (U. Wuppertal) Lattice Introduction 17 August 2010 2 / 20 Overview of the Lattice Approach LGT calculations are a non-perturbative implementation of ﬁeld theory using the Feynman path integral approach. The calculations proceed exactly as if the ﬁeld theory was being solved analytically (had we the ability to do the calculations). The starting point is the partition function in Euclidean space-time Z= ¯ DAµDψDψeS where S is the gauge action, for example: 1 ¯ S= d4 x Fµν Fµν − ψM ψ . 4 and M is the Dirac operator. The fermions are represented by Grassmann variables ψ and ¯ ψ. A. Rago (U. Wuppertal) Lattice Introduction 17 August 2010 2 / 20 The ambitious plan Deﬁning a ﬁnite dimensional system The Yang-Mills action for gauge ﬁelds have to be transcribed onto a discrete space-time lattice in such a way as to preserve 1 gauge invariance 2 chiral symmetry 3 topology, 4 and a one-to-one relation between continuum and lattice ﬁelds. Discretisation Errors & Continuum limit it is necessary to quantify and remove the discretisation errors by a reliable extrapolation to a = 0. Generating background gauge ﬁelds the path integral is analogous to the Boltzmann factor in the partition function for statistical mechanics systems. It can be regarded as a probability weight for generating conﬁgurations. Expectation values & Correlation functions A. Rago (U. Wuppertal) Lattice Introduction 17 August 2010 3 / 20 Discretisation There are a number of possible ways to discretized space-time in 4 Euclidean dimensions: hypercubic, body-centered cubic, hexagonal, random lattices. Of these the simplest is the isotropic hypercubic grid with spacing: a = aS = aT and size NS × NS × NS × NT . aT aS NT NS A. Rago (U. Wuppertal) Lattice Introduction 17 August 2010 4 / 20 Discretisation The lattice transcription of the matter ﬁelds ψ(x) → ψi is straightforward. A. Rago (U. Wuppertal) Lattice Introduction 17 August 2010 5 / 20 Discretisation The lattice transcription of the matter ﬁelds ψ(x) → ψi is straightforward. The construction of gauge ﬁelds is less intuitive. In the continuum: Aµ (x) carry 4-vector Lorentz indices Wilson noticed that y ψ(y) = Pe x igAµ (x)dxµ ψ(x) This suggested that gauge ﬁelds be associated with links that connect sites on the lattice. ˆ µ U (i, i + µ) ≡ Uµ (i) = eiagAµ (i+ 2 ) A. Rago (U. Wuppertal) Lattice Introduction 17 August 2010 5 / 20 Discretisation The lattice transcription of the matter ﬁelds ψ(x) → ψi is straightforward. The construction of gauge ﬁelds is less intuitive. In the continuum: Aµ (x) carry 4-vector Lorentz indices Wilson noticed that y ψ(y) = Pe x igAµ (x)dxµ ψ(x) This suggested that gauge ﬁelds be associated with links that connect sites on the lattice. ˆ µ U (i, i + µ) ≡ Uµ (i) = eiagAµ (i+ 2 ) aT ψi aS NT Aµ NS A. Rago (U. Wuppertal) Lattice Introduction 17 August 2010 5 / 20 Symmetries The SO(4) symmetric group is broken by the discretisation and replaced with the dihedral group D3 . Translations have to be by at least one lattice unit, so the allowed momenta are discrete 2πn k= n = 0, 1, ..., N Na A local gauge transformation Vi acts ψi → Vi ψi ψi → ψi Vi† ¯ ¯ † Uµ (i) → Vi Uµ (i)Vi+µ The only gauge invariant operators must be written like: ¯ ψi U (Ci,j )ψj U (Ci,i ) A. Rago (U. Wuppertal) Lattice Introduction 17 August 2010 6 / 20 Quenched Approximation In the rest of this lecture we will work in the “quenched approximation”. All the dynamics of the fermions and the related problems will be presented in next lecture (A. Patella ... good luck) A. Rago (U. Wuppertal) Lattice Introduction 17 August 2010 7 / 20 Quenched Approximation In the rest of this lecture we will work in the “quenched approximation”. All the dynamics of the fermions and the related problems will be presented in next lecture (A. Patella ... good luck) Quenched approximation: Once integrated integrated out exactly the fermionic ﬁelds 4 1 Z= DAµ det M e d x( 4 Fµν Fµν ) A. Rago (U. Wuppertal) Lattice Introduction 17 August 2010 7 / 20 Quenched Approximation In the rest of this lecture we will work in the “quenched approximation”. All the dynamics of the fermions and the related problems will be presented in next lecture (A. Patella ... good luck) Quenched approximation: Once integrated integrated out exactly the fermionic ﬁelds 4 1 Z= DAµ det M e d x( 4 Fµν Fµν ) det M is an highly non-local term A. Rago (U. Wuppertal) Lattice Introduction 17 August 2010 7 / 20 Quenched Approximation In the rest of this lecture we will work in the “quenched approximation”. All the dynamics of the fermions and the related problems will be presented in next lecture (A. Patella ... good luck) Quenched approximation: Once integrated integrated out exactly the fermionic ﬁelds 4 1 Z= DAµ det M e d x( 4 Fµν Fµν ) det M is an highly non-local term the partition function is an integral over only background gauge conﬁgurations A. Rago (U. Wuppertal) Lattice Introduction 17 August 2010 7 / 20 Quenched Approximation In the rest of this lecture we will work in the “quenched approximation”. All the dynamics of the fermions and the related problems will be presented in next lecture (A. Patella ... good luck) Quenched approximation: Once integrated integrated out exactly the fermionic ﬁelds 4 1 Z= DAµ det M e d x( 4 Fµν Fµν ) det M is an highly non-local term the partition function is an integral over only background gauge conﬁgurations One can write the action, by re-exponentiating 1 S = Sgauge + Squarks = d4 x Fµν Fµν − i log (det Mi ) 4 i where the sum is over the quark ﬂavors. We deﬁne the quenched approximation setting det M = constant A. Rago (U. Wuppertal) Lattice Introduction 17 August 2010 7 / 20 A pure gauge action The simplest loop you can draw on a lattice is a square composed by the product of four links. ˆ † ˆ † Uµν = Uµ (i)Uν (i + µ)Uµ (i + ν )Uν (i) If you make an a expansion of such a term what you will ﬁnd is: 2 a4 2 ReTr(1 − Uµν ) = Fµν F µν + O(a6 ) g 4 with Fµν = ∂µ Aν − ∂ν Aµ + g [Aµ , Aν ] with the derivative term obtained from 1 ∂x f (x) → ∆x (f (x + a) − f (x)) a A. Rago (U. Wuppertal) Lattice Introduction 17 August 2010 8 / 20 The Haar Measure The last ingredient needed to complete the formulation of LGT as a quantum ﬁeld theory via the path integral is to deﬁne the measure of integration over the gauge degrees of freedom. Unlike the continuum ﬁelds Aµ , lattice ﬁelds are SU(3) matrices with elements that are bounded in the range [0, 1]. Wilson proposed an invariant group measure, the Haar measure. f (U )dU = f (V U )dU = f (U V )dU As an example for SU(2) we have: U = x0 Id + ix · σ detU = x2 = (x0 )2 + x2 = 1 1 dU = δ(x2 − 1)d4 x π A. Rago (U. Wuppertal) Lattice Introduction 17 August 2010 9 / 20 Continuum limit and critical behaviour We just mentioned the possibility of taking the limit for a → 0. It’s clear that at large distance correlator of two ﬁelds goes like φ(x1 )φ(x2 ) c ∝ e−m|x1 −x2 | The correlation length which determines the rate of the exponential decay is therefore given by: 1 ξ= am If a continuum limit with a ﬁnite physical mass exists By a suitable choice of bare parameters we can approach a limit where a → 0 while m stays ﬁnite ξ→∞ We are in presence of a critical point (second order phase transition) Our concern is SU(N) gauge theories. Such theories are characterised by having an UV ﬁxed point at g = 0. A. Rago (U. Wuppertal) Lattice Introduction 17 August 2010 10 / 20 The Wilson loop Elitzur’s theorem: In gauge theories it is impossible to ﬁnd a local quantity acting as an order parameter. A. Rago (U. Wuppertal) Lattice Introduction 17 August 2010 11 / 20 The Wilson loop Elitzur’s theorem: In gauge theories it is impossible to ﬁnd a local quantity acting as an order parameter. Average values cannot distinguishes the Coulomb phase, in which charge states can be produced, from the so called conﬁned phase. A. Rago (U. Wuppertal) Lattice Introduction 17 August 2010 11 / 20 The Wilson loop Elitzur’s theorem: In gauge theories it is impossible to ﬁnd a local quantity acting as an order parameter. Average values cannot distinguishes the Coulomb phase, in which charge states can be produced, from the so called conﬁned phase. To study conﬁnement in pure gauge theories we need a gauge invariant non local quantity A. Rago (U. Wuppertal) Lattice Introduction 17 August 2010 11 / 20 The Wilson loop Elitzur’s theorem: In gauge theories it is impossible to ﬁnd a local quantity acting as an order parameter. Average values cannot distinguishes the Coulomb phase, in which charge states can be produced, from the so called conﬁned phase. To study conﬁnement in pure gauge theories we need a gauge invariant non local quantity Consider a rectangular loop CR,T of side lengths R and T W (CR,T ) = U CR,T The static quark potential V (R) is deﬁned by means of the large T behaviour of the corresponding Wilson loop in the fundamental representation. 1 V (R) ≡ − lim log W (CR,T ) T →∞ T so that W (CR,T ) ∼ Ce−T V (R) T →∞ A. Rago (U. Wuppertal) Lattice Introduction 17 August 2010 11 / 20 Strong coupling One well-established nonperturbative method is strong coupling. A. Rago (U. Wuppertal) Lattice Introduction 17 August 2010 12 / 20 Strong coupling One well-established nonperturbative method is strong coupling. Strong coupling is tied to the lattice and cannot be derived directly for a continuum theory. A. Rago (U. Wuppertal) Lattice Introduction 17 August 2010 12 / 20 Strong coupling One well-established nonperturbative method is strong coupling. Strong coupling is tied to the lattice and cannot be derived directly for a continuum theory. Once expanded each lattice integral in power of 1/g 2 A. Rago (U. Wuppertal) Lattice Introduction 17 August 2010 12 / 20 Strong coupling One well-established nonperturbative method is strong coupling. Strong coupling is tied to the lattice and cannot be derived directly for a continuum theory. Once expanded each lattice integral in power of 1/g 2 It’s possible to perform the integrations that turns out to be only factorised group integration A. Rago (U. Wuppertal) Lattice Introduction 17 August 2010 12 / 20 Strong coupling One well-established nonperturbative method is strong coupling. Strong coupling is tied to the lattice and cannot be derived directly for a continuum theory. Once expanded each lattice integral in power of 1/g 2 It’s possible to perform the integrations that turns out to be only factorised group integration The largest contribution correspond to the minimal area bounded to the Wilson loop 2 W (CR,T ) ∼ e−RT ln g If no phase transition occurs when g 2 varies from inﬁnity to zero the gauge theory underlying is always in a conﬁnement phase Ex. SU(2) SU(3) in 4d On the other side is a phase transition exists, the phase in which we can demonstrate the existence of a conﬁning behaviour is not connected to the continuum limit. Ex. SU(N) with N> 3 in 4d A. Rago (U. Wuppertal) Lattice Introduction 17 August 2010 12 / 20 Thermodynamics of Quantum Field Theory and the Lattice In order to describe a ﬁnite temperature theory we require that one dimension (the Time) is compactiﬁed with periodic boundary conditions. 1 The compactiﬁcation length LT a = T , where T is the temperature. Periodic boundary conditions implies a new global symmetry of the action. The symmetry group is the center C of the gauge group (SU(N ) → ZN ). This symmetry can be realised as follows. UT (i) → W0 UT (i) ∀ i belonging to an hyperplane a ﬁxed time A. Rago (U. Wuppertal) Lattice Introduction 17 August 2010 13 / 20 Thermodynamics of Quantum Field Theory and the Lattice The space-like plaquettes are not aﬀected by the transformation −1 In timelike plaquette two contribution appear: W0 and W0 . Since they belong to the center they cancel each other. The Wilson action is invariant under such transformation. It is impossible to re-absorb this global twist by means of local gauge transformations. It is possible to deﬁne gauge invariant observables which are topologically non-trivial: The Polyakov loop LT P (x) ≡ Tr UT (i). i=1 A. Rago (U. Wuppertal) Lattice Introduction 17 August 2010 14 / 20 Thermodynamics of Quantum Field Theory and the Lattice The relevant feature of the Polyakov loop is that it transforms under the above discussed symmetry as follows: P (x) → W0 P (x); thus it is a natural order parameter for this symmetry. It will acquire a non zero expectation value if the center symmetry is spontaneously broken. Its expectation value is related to the free energy of a single isolated quark. Signature of deconﬁnement: phase transition which separates the regime in which the center symmetry is unbroken from the broken symmetric phase. A. Rago (U. Wuppertal) Lattice Introduction 17 August 2010 15 / 20 A partial summary What we have learned (or at least mentioned) so far Discretized the theory on a lattice A. Rago (U. Wuppertal) Lattice Introduction 17 August 2010 16 / 20 A partial summary What we have learned (or at least mentioned) so far Discretized the theory on a lattice Deﬁne the Wilson action A. Rago (U. Wuppertal) Lattice Introduction 17 August 2010 16 / 20 A partial summary What we have learned (or at least mentioned) so far Discretized the theory on a lattice Deﬁne the Wilson action Search for a continuum limit A. Rago (U. Wuppertal) Lattice Introduction 17 August 2010 16 / 20 A partial summary What we have learned (or at least mentioned) so far Discretized the theory on a lattice Deﬁne the Wilson action Search for a continuum limit Order parameter for conﬁnement at zero Temperature (Wilson Loop) A. Rago (U. Wuppertal) Lattice Introduction 17 August 2010 16 / 20 A partial summary What we have learned (or at least mentioned) so far Discretized the theory on a lattice Deﬁne the Wilson action Search for a continuum limit Order parameter for conﬁnement at zero Temperature (Wilson Loop) Order parameter for conﬁnement at non-zero Temperature (Polyakov Loop) A. Rago (U. Wuppertal) Lattice Introduction 17 August 2010 16 / 20 A partial summary What we have learned (or at least mentioned) so far Discretized the theory on a lattice Deﬁne the Wilson action Search for a continuum limit Order parameter for conﬁnement at zero Temperature (Wilson Loop) Order parameter for conﬁnement at non-zero Temperature (Polyakov Loop) An analytical method to justify conﬁnement of the continuum theory if the theory does not have any bulk PT A. Rago (U. Wuppertal) Lattice Introduction 17 August 2010 16 / 20 A partial summary What we have learned (or at least mentioned) so far Discretized the theory on a lattice Deﬁne the Wilson action Search for a continuum limit Order parameter for conﬁnement at zero Temperature (Wilson Loop) Order parameter for conﬁnement at non-zero Temperature (Polyakov Loop) An analytical method to justify conﬁnement of the continuum theory if the theory does not have any bulk PT What we miss now is to understand how to obtain results from our discretized model. A. Rago (U. Wuppertal) Lattice Introduction 17 August 2010 16 / 20 Dynamics Ergodic Hypotheses: average on a statistical ensemble time average Problem: give a dynamic to the system Fundamental request: at the equilibrium you must generate conﬁguration according to the Boltzmann distribution A. Rago (U. Wuppertal) Lattice Introduction 17 August 2010 17 / 20 Markov Chain n t t−1 Series of conﬁgurations Cm for which Cm depends only on Cn according to a probability distribution Pnm (upper index time, lower index conﬁguration index) Irreducible if for each Cj it’s possible to reach Cl l > j, or equivalently it exists a time k for which k Pjl = i1 ...in Pji1 Pi1 i2 ...Pin l = 0 for every j, k k Aperiodic if Pii = 0 for each i Ci state is said to be positive if it reappear in a ﬁnite time A. Rago (U. Wuppertal) Lattice Introduction 17 August 2010 18 / 20 Equilibrium distribution Given a {Ci } Markov chain irreducible, aperiodic and with only positive states The equilibrium distribution exists and is unique hence it doesn’t depend on the start state N lim Pij = Pi N →∞ A. Rago (U. Wuppertal) Lattice Introduction 17 August 2010 19 / 20 Equilibrium distribution Given a {Ci } Markov chain irreducible, aperiodic and with only positive states The equilibrium distribution exists and is unique hence it doesn’t depend on the start state N lim Pij = Pi N →∞ The equilibrium distribution is stationary 1 Pj = Pij Pi i A. Rago (U. Wuppertal) Lattice Introduction 17 August 2010 19 / 20 Equilibrium distribution Given a {Ci } Markov chain irreducible, aperiodic and with only positive states The equilibrium distribution exists and is unique hence it doesn’t depend on the start state N lim Pij = Pi N →∞ The equilibrium distribution is stationary 1 Pj = Pij Pi i if the variance of the reappear time is ﬁnite N 1 Pi O(Ci ) = O = lim O(Cj ) N →∞ N i j=1 A. Rago (U. Wuppertal) Lattice Introduction 17 August 2010 19 / 20 Detailed balance Montecarlo Dynamics: any Markov dynamics A. Rago (U. Wuppertal) Lattice Introduction 17 August 2010 20 / 20 Detailed balance Montecarlo Dynamics: any Markov dynamics Problem: Given an action, build a Markov dynamic A. Rago (U. Wuppertal) Lattice Introduction 17 August 2010 20 / 20 Detailed balance Montecarlo Dynamics: any Markov dynamics Problem: Given an action, build a Markov dynamic Necessary condition: not known A. Rago (U. Wuppertal) Lattice Introduction 17 August 2010 20 / 20 Detailed balance Montecarlo Dynamics: any Markov dynamics Problem: Given an action, build a Markov dynamic Necessary condition: not known Suﬃcient condition: Detailed balance e−βH(Ci ) Pij = e−βH(Cj ) Pji A. Rago (U. Wuppertal) Lattice Introduction 17 August 2010 20 / 20 Detailed balance Montecarlo Dynamics: any Markov dynamics Problem: Given an action, build a Markov dynamic Necessary condition: not known Suﬃcient condition: Detailed balance e−βH(Ci ) Pij = e−βH(Cj ) Pji and you are still free to choose Pij , decide you algorithm A. Rago (U. Wuppertal) Lattice Introduction 17 August 2010 20 / 20

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