Effective Lagrangians and Field Theory on the lattice

Method of the derivation of Chiral <strong>Effective</strong>Theories from Lattice QCD• Why QCD on the lattice. Lattice QCD as regularizationscheme.• Analytical integration on the lattice – how it’s possible?• First Step – Strong coupling regime• Lattice <strong>and</strong> space-time symmetries: lattice artifacts• Solution: R<strong>and</strong>om Lattice!• Restoration of space-time symmetries on the R<strong>and</strong>omLattice <strong>and</strong> zero-order chiral effective theory• Derivation of Chiral <strong>Effective</strong> Theories from Lattice QCD – way isfree!

Strong-coupling <strong>and</strong> gluon correlation length

Strong-coupling <strong>and</strong> gluon correlation lengthSpacing a > 0.2 fm

Strong coupling regime = Link integrals dominationwhereZ( β → ) [ DG][ Dψ][ Dψ]00= ∫− Se ψ(+)x, x, x, x,= ∑+S Tr A G A Gψμ μ μ μx,μAA=ψPψx , μ x , μ μ x=ψPx , μ x μ x , μ−+ψ± 1Pμ= r ±μ2( γ )

Lattice QCDStrong coupling regimeβ→0Gross-Brezeot trickIntegration on gluon fieldPreliminary fermion-field dependenteffective actionIntegration on fermion field<strong>Effective</strong> action on Chiral <strong>Field</strong>

Bosonic matrix M:Mαβ( x)is a metrix in the spin <strong>and</strong> flavor spacerepresenting an effective bosonic fieldM ( x) = M exp{ iS( x) + iP( x) γ + iV ( x) γ + iA ( x) γ γ + iT ( x) σ }αβ 0 5 μ μ μ μ 5 μν μνSU(2) Chiral scenario means the neglecting of contribution fromScalar, Vector, Axial-vector <strong>and</strong> Tensor mesons. Just onlyPseudoscalers are taking into account!M ( x) = M exp{ iP( x) γ }αβ0 5Stationary point of action

Interaction on fermion degrees of freedom:stationary-point expansion(S. Myint, C. Rebbi hep-lat/9401009, hep-lat/9401010)W ( λ )S U Tr xeff∞ ( k )0k( ) = −∑∑ ⎡ ( λv( ) λ0)k = 1 k ! ⎣ −x,v⎤⎦[ ]Tr ( λ ( x) − λ ) =−2 λ Tr( α)v0 02 2 2 2Tr ⎡⎣( λv( x) − λ0) ⎤⎦ = 2 λ0Tr( α ) −4 λ0Tr( α)3 3 3 3 2Tr ⎡⎣( λv( x) − λ0) ⎤⎦ = − 2 λ0Tr( α ) + 6 λ0Tr( α )α = a ∇ U ∇ U + O( a )2 +4v vU = exp( iφτ/ f π)4 4 4 4 3 3 2Tr ⎡⎣( λv( x) − λ0) ⎤⎦ = 2 λ0Tr( α ) − 8 λ0Tr( α ) + 4 λ0Tr( α )Tr ⎡( λ ( x) λ ) ⎤ 2 λ Tr( α ) 10 λ Tr( α )5 5 5 5 4⎣ v−0 ⎦ =−0+0+ …………

Integration on hyper-cubical lattice: О(4) symmetriesviolation!Lattice basis = orthogonal vectorsν =i, j, k,t

Integration on hyper-cubical lattice: О(4) symmetriesviolation!Lattice basis = orthogonal vectorsν =i, j, k,t∇ U∇ U = ∇U∇U+ +1) v v i i[ ]Tr ( λ ( x) − λ ) =−2 λ Tr( α)v0 0∼ Tr( ∂U∂U + )iiU = exp( iφτ/ f π)2 2 2 2Tr ⎡⎣( λv( x) − λ0) ⎤⎦ = 2 λ0Tr( α ) −4 λ0Tr( α)3 3 3 3 2Tr ⎡⎣( λv( x) − λ0) ⎤⎦ = − 2 λ0Tr( α ) + 6 λ0Tr( α )∼+ +Tr( ∂ U ∂ U ∂ U ∂ U )i i i i4 4 4 4 3 3 2Tr ⎡⎣( λv( x) − λ0) ⎤⎦ = 2 λ0Tr( α ) − 8 λ0Tr( α ) + 4 λ0Tr( α )Tr ⎡( λ ( x) λ ) ⎤ 2 λ Tr( α ) 10 λ Tr( α )5 5 5 5 4⎣ v−0 ⎦ =−0+0+ …………

We have a problem with reproducing of space-timesymmetry!Cause: Violation of space-time symmetry to discrete groupSequence: Lattice Artifacts!Possible solutions: Try to find more symmetrical lattice!More symmetrical lattice in 4-dim:Body Centered Hyper Cubical Lattice

Integration on hyper-cubical lattice with center element asstep on the way to general solutionBasis vectors of the hyper-cubical lattice with center elementν ± = ±( e e ) / 2ij i j

Integration on hyper-cubical lattice with center element asstep on the way to general solutionBasis vectors of the hyper-cubical lattice with center elementν ± = ±( e e ) / 2ij i j[( λ ( x) − λ )]+ +∇ TrvU∇ vU = ∇iU∇ iUv 0∼ Tr( ∂iU ∂iU + )Tr ⎡⎣( λv( x) − λ0)2⎤⎦∼+ + + + + +Tr( ∂ U ∂ U ∂ U ∂ U + ∂ U ∂ U ∂ U ∂ U + ∂ U ∂ U ∂ U ∂ U )i i j j i j i j i j J i3 3 3 3 2Tr ⎡⎣( λv( x) − λ0) ⎤⎦ = − 2 λ0Tr( α ) + 6 λ0Tr( α )Non- invarianceagain!!

Unfortunately, Body Centered Hyper CubicalLattice is not what we seek…We need More symmetrical Lattice!Unfortunately, there are no more symmetrical Lattice in 4-dim thanBCHC!No more REGULAR Lattice…OK, no more regular lattice!Let us to consider R<strong>and</strong>om Lattice!

R<strong>and</strong>om Lattice: historical remark• Georgy Voronoi • Boris Delaunay1868-1909 г 1890-1980 г

R<strong>and</strong>om Lattice: step 1Let us consider R<strong>and</strong>om distribution of centers

R<strong>and</strong>om Lattice: step 2

R<strong>and</strong>om Lattice: step 3

R<strong>and</strong>om Lattice: step 4

.R<strong>and</strong>om Lattice: step 5

R<strong>and</strong>om Lattice: step 6

R<strong>and</strong>om Lattice: step 7

R<strong>and</strong>om Lattice averaging = restoration of О(D=4)invariance……………..

CFL formalism for r<strong>and</strong>om lattice(Christ, Friеdberd, Lee Nucl.Phys. B202 (1982) 89; B210 (1982) 310; 337)1. Any element or object on R<strong>and</strong>om lattice correspondto same element or object of Dual R<strong>and</strong>om lattice2. Any element or object on R<strong>and</strong>om lattice has statisticalweighsCenter iD-dim value of dual Vonoi cellLink iPlaquettejki jD-1 – dim value of the plane of theVoronoi cellD-2 - dim value ofcorrespondent dualobject

Fermion on R<strong>and</strong>om LatticeLattice action for fermion on the lattice1(μ μψ γ λ ψ )S T r l ijψ= ∑ λ = S /ij2 x , μν γvijijlij

Restoration of О(4) symmetry on R<strong>and</strong>om LatticeТ(CFL)⎛⎜⎝∑μν μ ν 2μ ν=λij ij ijL l lij⎞⎟⎠[ Ω ]=δ( δ δ δ δ δ δ )= c+ +μνρσ μ ν ρσ μρ νσ μσ ρνL2Lci iN ⎜ ∏ δ δperm utationi1i2 … i2N l k=⎛⎝∑⎞⎟⎠c N=N12 ( N + 1)!

Restoration of О(4) symmetry on R<strong>and</strong>om Lattice⎛⎜⎝∑νabνν⎞⎟⎠[ Ω ]=abii⎛⎜⎝∑ν⎞ 1aνb C d abc d ab cd ab c dν ν ν ⎟ = + +⎠ 6[ Ω]( )i i j j i j i j i j j i

<strong>Effective</strong> theory for chiral field in strong couplingregimeR<strong>and</strong>om lattices ensemble averaging => restoration ofspace symmetry on zero step of perturbation!!!2Fπ +L= Tr( ∂iU∂ iU) +21+ CTr ( ∂U∂U ∂ U∂ U +∂U∂ U ∂U∂ U +∂U∂ U ∂ U∂ U ) + C L + + + + + + +4 i i j j i j i j i j J i6 2 6F πp < F ∼ 100MeVπ

Shock-wave solutions of Chiral Born-Infeld <strong>Theory</strong>ChBI( ( ))1 1μ2βL =−f β Tr − − L L μ2 2 2πf π→ 0β → 0L ϕ ϕ2 2H= 1− 1 −( t− x)LW=−f2π4( ) μTr L L μ+Lμ= U ∂μUU = exp( iφτ/ f π)

Conclusions:• The method of derivation of chiral effective theories from Lattice QCD wasconsidered.• Basis of the method – conception of the r<strong>and</strong>om lattice ensemble averaging• Zero-order perturbation theory in strong coupling regime was considered.Restoration of O(4) space symmetry was studied.• This method could be used for any order of perturbation, for any plaquetcotributions.