Ginsparg-Wilson Fermions and the Chiral Gross-Neveu Model
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1 Ginsparg-Wilson Fermions and the DESY Zeuthen 14th September 2004 Ginsparg-Wilson Fermions and the
2 QCD predictions Perturbative QCD only applicable at high energy ( 1 GeV) At low energies (100 MeV - 1 GeV) QCD is nonperturbative Nonperturbative methods are needed to verify that the QCD Lagrangian describes physics of hadrons at low energy Lattice QCD Formulation which is mathematically well defined at the nonperturbative level Provides momentum cutoff 1/a Systematical error due to discretization continuum limit Quantitative results obtained using numerical simulations Ginsparg-Wilson Fermions and the
3 1 A No-Go Theorem Naive, Wilson, (Staggered) Fermions 2 Symmetries, Renormalization Ginsparg-Wilson Fermions and the
4 A No-Go Theorem Naive, Wilson, (Staggered) Fermions Euclidean action of N free massless fermions In the continuum S f = dx d ψ(x) D ψ(x) D = γ µ µ anti-commutes with γ 5 thus S f is invariant under On the lattice SU(N) V SU(N) A U(1) V U(1) A S f = a d x ψ(x) D ψ(x), x = a(n 0,..., n d ), n µ Z Dψ(x) = y D(x, y)ψ(y), D e ixp u = D(p) e ixp u, p π/a Ginsparg-Wilson Fermions and the
5 A No-Go Theorem Naive, Wilson, (Staggered) Fermions Theorem (Nielsen-Ninomiya no-go) Following can not hold simultaneously: 1 D(p) is an analytic periodic function of the momenta pµ with period 2π/a 2 D(p) = iγµ p µ + O(ap 2 ) for p π/a 3 D(p) is invertible at all nonzero momenta 4 D anti-commutes with γ 5 Or equivalently 1 D(x, y) is local 2 theory describes Dirac particle 3 no doublers 4 action has chiral symmetry Ginsparg-Wilson Fermions and the
6 A No-Go Theorem Naive, Wilson, (Staggered) Fermions Naive fermions D = 1 2 γ µ( µ + µ ) With lattice forward and backward difference operators µ ψ(x) = {ψ(x + aˆµ) ψ(x)}/a µψ(x) = {ψ(x) ψ(x aˆµ)}/a Fourier transform D(p) = (1/a) µ γ µ sin(ap µ ) has in four dimensions 16 zeros in the Brillouin zone Fermion doubling Ginsparg-Wilson Fermions and the
7 A No-Go Theorem Naive, Wilson, (Staggered) Fermions Wilson fermions Adding a dimension d + 1 operator to the action, which modifies the Dirac operator removes the unwanted zeros D W = 1 2 {γ µ( µ + µ ) a µ µ} D W (p) = (1/a) µ {γ µ sin(ap µ ) + 2 a sin2 (ap µ /2)} giving a mass 1/a to the doublers, but breaks chiral symmetry explicitly effects renormalization of bare parameters: m q = 0 not protected Ginsparg-Wilson Fermions and the
8 A No-Go Theorem Naive, Wilson, (Staggered) Fermions Ginsparg-Wilson relation γ 5 D + D γ 5 = a Dγ 5 D chiral symmetry partly preserved, i.e. the fermion propagator anti-commutes with γ 5 at nonzero distances D(x, y) 1 γ 5 + γ 5 D(x, y) 1 = a γ 5 δ xy extended chiral symmetry solutions exist that are in accordance with points (1)-(3) of the no-go theroem Ginsparg-Wilson Fermions and the
9 A No-Go Theorem Naive, Wilson, (Staggered) Fermions Possible solution D = 1 ( 1 A(A A) 1/2), A = 1 a D W a derivates from D W only through lattice corrections of order a Fourier transform a D(p) = 1 (1 12 ) ( a2ˆp 2 iaγ µ pµ ) 1/2 2 a4 ˆp µˆp 2 ν 2 µ>ν where p µ = (1/a) sin(ap µ ) and ˆp µ = (2/a) sin(ap µ /2) Conditions (1), (2) and (3) are fulfilled, but chiral symmetry modified... Ginsparg-Wilson Fermions and the
10 A No-Go Theorem Naive, Wilson, (Staggered) Fermions Continuous symmetry - Lattice chiral symmetry Consider infinitesimal variations ψ ψ + ɛ δψ, ψ ψ + ɛ δψ where Then δψ = γ 5 ( ad ) ψ δψ = ψ δ(ψdψ) = ψdγ 5 ( ad ) ψ + ψ = ψ (Dγ 5 + γ 5 D adγ 5 D) ψ = 0 (1 12 ad ) γ 5 (1 12 ad ) γ 5 Dψ Ginsparg-Wilson Fermions and the
11 A No-Go Theorem Naive, Wilson, (Staggered) Fermions Summary - Ginsparg-Wilson relation Ginsparg-Wilson relation allows for chiral fermions on the lattice Physically significant solutions can be found Tests and comparison to established methods like Wilson fermions are certainly needed Lattice QCD in 4 dimensions to expensive Two dimensional fermion theories are an accessible alternative Ginsparg-Wilson Fermions and the
12 Fermions in two dimensions Symmetries, Renormalization Four fermion interactions are renormalizable ψγ 1 ψψγ 2 ψ where Γ i = Γ S i Γ F i are matrices contracting the spin and flavour indices Spin sector ψψ, ψγ 5 ψ, ψγ µ ψ, ψγ µ γ 5 ψ Flavor sector: abelian and non-abelian part ψ ψ, ψ λ a ψ where λ a are the N 2 1 generators of SU(N) Ginsparg-Wilson Fermions and the
13 Symmetries, Renormalization Symmetries Spin and flavour indices connected via Fierz transformations not all possible terms are independent, e.g. (ψ ψ) 2 (ψ γ 5 ψ) 2 FT = 1 N (ψ γ µ ψ) Chiral symmetry U(N) flavor symmetry ψ e α Aγ 5 ψ, ψ ψ e αγ 5 (ψ γ µ λ a ψ) 2 ψ U ψ, ψ ψ U +, U + U = 1 a Ginsparg-Wilson Fermions and the
14 Symmetries, Renormalization Chiral Gross-Neveu model - Euclidean action S f = [ dx 2 ψ γ µ µ ψ 1 2 g S 2 { (ψ ψ) 2 (ψ γ 5 ψ) 2} 1 2 g V 2 (ψ γ µ ψ) 2] or equivalently [ S f = dx 2 ψ γ µ µ ψ g 2 (ψ γ µ λ a ψ) g 2 (ψ γ µ ψ) 2] Related through FT: g 2 = 1 2 g S 2, g 2 = gv 2 1 N g S 2 abelian coupling g 2 : vanishing β-function up to two loops non-abelian coupling g 2 : asymptotically free Ginsparg-Wilson Fermions and the
15 Symmetries, Renormalization Chiral Gross-Neveu model - Ginsparg-Wilson relation Are the four fermion interaction terms invariant under the lattice chiral transformations induced by the Ginsparg-Wilson relation? No! But rewriting e.g. (ψ γ µ ψ) 2 (ψ γ µ (1 (1/2)aD) ψ) 2 the whole action becomes invariant. Ginsparg-Wilson Fermions and the
16 Ginsparg-Wilson relation allows for chiral fermions on the lattice Physically significant solutions can be found Application in lattice QCD is desirable, but not in the reach of present computer power Tests and comparison to established methods are needed Chiral Gross-Neveu model in two dimension is physically interesting and accessible Ginsparg-Wilson Fermions and the
17 Euclidean action of a free massless fermion In the continuum S f = dx d ψ(x) D ψ(x) D = γ µ µ anti-commutes with γ 5 thus S f is invariant under SU(N) V SU(N) A U(1) V U(1) A ( ψ 1 + a ( ψ ψ 1 a ) λ a 2 (iαa V + αa A γ 5) + iα V + α A γ 5 ψ λ a 2 (iαa V αa A γ 5) + iα V + α A γ 5 ) Ginsparg-Wilson Fermions and the
18 Staggered fermions Flavour symmetry is (partly) given up to keep a part of the chiral symmetry Spin-diagonalization allows only one spinor component per lattice site For d = 4 this means a reduction of the degrees of freedom by a factor of four Remaining (from the doublers) degrees of freedom are combined to describe 4 flavours in d = 4 in the continuum limit Ginsparg-Wilson Fermions and the
19 Ginsparg-Wilson relation RG transformation γ 5 D + D γ 5 = a Dγ 5 D Relation first appeared in 1982 in a paper of Ginsparg and Wilson Block-spin renormalization in lattice QCD Considered some sort of remnant chiral symmetry Domain wall fermions Fermions in a dimensonal domain wall world reduce to four dimensional chiral theory at low energies However, chiral symmetry is realized through the Ginsparg-Wilson relation Ginsparg-Wilson Fermions and the
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