slides PDF - Theory Group
slides PDF - Theory Group
slides PDF - Theory Group
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ψ(n)<br />
Basics:<br />
a<br />
µ<br />
U µ (n)=e igaAµ(n)<br />
S = S G + S F Difficult to simulate --><br />
S F = ! (k)D(k,n)!(n)<br />
!O" =<br />
Lattice fermions<br />
C. Alexandrou, University of Cyprus PSI, December 18th 2007<br />
"<br />
k,n<br />
Integrate out<br />
# dUnd$ nd$ nO[U,$ ,$ ]e %S<br />
&<br />
n<br />
Z<br />
It is well known that naïve discretization of Dirac action<br />
S = d F 4 x !(x) # " ! + m%<br />
$ µ µ & !(x)<br />
' (<br />
!(x + µ)- !(x -µ)<br />
! !(x) = µ<br />
2a<br />
=<br />
90’s Det[D]=1 : quenched<br />
# dUndet[D]O[U,D %1 ]e %SG &<br />
n<br />
Z<br />
leads to 16 species<br />
A number of different discretization schemes have been developed to avoid doubling:<br />
Wilson : add a second derivative-type term --> breaks chiral symmetry for finite a<br />
Staggered: “distribute” 4-component spinor on 4 lattice sites --> still 4 times more species,<br />
take 4th root, non-locality<br />
Nielsen-Ninomiya no go theorem: impossible to have doubler-free, chirally<br />
symmetric, local, translational invariant fermion lattice action BUT…
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