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Is the wilson plaue always positive in the lattice simulation?

  1. Mar 22, 2012 #1
    More precisely, my question is, Is the wilson loop used to calculate the heavy quark potential always positive in the lattice simulation?

    (i) as we are usually told, wilson loops of the following form are related with
    the heavy quark potentials
    ..........................................
    _________ <WC_______.........
    |...............................|../\ ...
    |WD................../\......|...|....
    |...............................|...r....
    | \/...................WB....|...|....
    |_____WA>__________|..\/....
    <------------- t -------------->..
    ..........................................

    <W>=<W_{r,t}> ~ e^{-V( r )*t}
    so we expect <W> is always positive(or negative,
    any way, it should not change signs we we change t and r)

    (ii) but by definition

    <W>= <tr[e^{iAa ta} * e^{i Ab tb} * e^{-iAc tc} * e^{-i Ad td}]>

    = <tr[WA * WB * WC * WD]>

    since in numerical simulations, the link variables WA, WB, WC, WD et al
    are random unitary matrix, we cannot expect the trace of their products
    is positive.

    I encounter this problem because in one of my little exercises to write
    lattice QCD codes, I got results just as I explained in (ii), so I cannot extract
    the heavy quark potentials, because some times I get W(r,t)>0, some times
    W(r,t)<0. Especially worse is, I encounter the case
    W(r,t)>0
    W(r,t+a)<0
    So the proportion W(r,t+a)/W(r,t) ~ e^{-V( r )*a} < 0, this is really a disappointing
    outcome.

    What key points did I ignored ? Who can tell me or give me some
    hints? If you have comments or remarks, please do not hesitate to reply this
    thread or write me email, dfzeng2000@hotmail.com
     
    Last edited: Mar 22, 2012
  2. jcsd
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