# Is the wilson plaue always positive in the lattice simulation?

1. Mar 22, 2012

### sufive

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