Is the wilson plaue always positive in the lattice simulation?

[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
hints? If you have comments or remarks, please do not hesitate to reply this
thread or write me email, dfzeng2000@hotmail.com

 

[eljose79]

Thank you for your question regarding the use of Wilson loops to calculate the heavy quark potential in lattice simulations. This is a very interesting and complex topic, and I would be happy to provide some insight as a scientist in this field.

First of all, your understanding of the relationship between Wilson loops and the heavy quark potential is correct. The Wilson loop is a gauge-invariant quantity that can be used to extract information about the potential between two heavy quarks. In lattice QCD simulations, the Wilson loop is typically calculated as an expectation value of a product of link variables along a closed path, as shown in the diagram you provided.

Now, onto your main question - is the Wilson loop always positive in lattice simulations? The short answer is no. The Wilson loop can take on both positive and negative values, depending on the specific configuration of the link variables. This is because the link variables are random unitary matrices, as you mentioned, and their product can result in either a positive or negative value. Therefore, it is not surprising that you have encountered cases where W(r,t)>0 and W(r,t)<0.

However, this does not mean that the heavy quark potential is not calculable from the Wilson loop. In fact, the heavy quark potential can still be extracted from the Wilson loop by taking the logarithm of the expectation value and using the relation <W> ~ e^{-V(r)*t}. This is because the potential is related to the slope of the logarithm of the Wilson loop as a function of the separation distance r. So even if the Wilson loop is negative, the logarithm will still be negative and the potential can be extracted.

In addition, it is important to note that in lattice simulations, the Wilson loop is averaged over many configurations of the link variables. This means that the overall result will be a combination of positive and negative values, resulting in an average that is close to zero. Therefore, it is crucial to perform a statistical analysis of the data to extract the potential accurately.

In conclusion, it is not uncommon to encounter negative Wilson loops in lattice simulations, but this does not mean that the heavy quark potential cannot be extracted. It is important to take the logarithm of the Wilson loop and perform a statistical analysis to obtain accurate results. I hope this helps to clarify your understanding. If you have any further questions or concerns, please do not hesitate to reach out. Best of luck with your lattice Q