# Wilson loops, confinement and chiral symmetry relations

As per my limited understanding of confinement, I understood it as phenomenon between fermions of a gauge theory, such as $SU(3)$ gauge theory in the low energy limit (where the coupling constant becomes infinite at some finite energy scale $\Lambda_{Landau Pole}$ ) where the strong force directly increases with distance unlike the electromagnetic $U(1)$ gauge theory.

Now, I did not know of any precise definition of confinement which shows the above statement explicitly. Upon reading a few references, I have seen some variants of confinement and I still fail to understand the exact mathematical definition of confinement and relations between the below statements.

1. Quarks are not freely observed in nature. Hence only color singlet combinations $\bar{q}q$ and $qqq$ are allowed combinations. This is called quark confinement. And quarks are said to be deconfined when no longer exist in color-singlet states.

$q^f(x) = \begin{pmatrix}\psi^f_{red}(x)\\\psi^f_{blue}(x)\\\psi^f_{green}(x)\end{pmatrix}$

Question: Is it from experimental data that we postulate that, "Ok, let's just say quarks are confined, then lets decide what combinations of quarks are allowed if we make the hypothesis that a new quantum number color exists."

Further, is confinement a statement on color? I wouldn't like to think so, but the above definition seems to be very specific to color

2. A theory is confining if all finite-energy states are invariant under a global gauge transformation. $U(1)$ gauge theory QED is not confining, because there are finite-energy states (such as the state of a single electron) that have nonzero electric charge, and hence change by a phase under a global gauge transformation. (From Pg.494 of Srednicki)

Question: I havn't come across the energy states of QED's Hamiltonian in any of the standard textbooks of peskin, srednicki, zee etc. So I'm not sure how to check this symmetry in QED and QCD.

3. Chiral symmetry breaking or restoration indicates confinement / deconfinitement.

Question: Is there any explicit way to realise this? I know the chiral symmetry $\psi(x) \rightarrow e^{i\alpha\gamma^5}\psi(x)$ gives us a current $\partial_\mu j^\mu = 2im\bar{\psi}\gamma^5\psi$ which is conserved in the chiral limit $m \rightarrow 0$. How does this statement got to do with confinement

4. The Vacuum expectation value (VEV) of the Wilson loop can diagnose whether or not a gauge theory exhibits confinement.

Question: There is a 4 page derivation in Pg.494 of Srednicki that shows the perimeter law relates confinement to it but I'm not sure as to why in the world would we be interested take VEV of a Wilson loop. How is it of physical interest? (I know Wilson loop is an interesting object that generates gauge invariant objects, but not sure if that helps in realizing VEV in a nice way.)

Main Question: Finally, the main question was to understand the relation between each of the above definitions of confinements and which statement implies the other or equivalent to the other. If you can back your arguments with precise expressions or references with computations of the same, I would be immensely grateful!

Confinement is not a sharp, mathematically well defined, notion in a theory with massless or light fermions (in the fundamental representation). This is why the Millenium Prize Problem is stated as a mass gap problem for pure gauge theories.

In a pure gauge theory the area law for the Wilson loop provides a sharp definition of confinement. Physically, the Wilson loop is related to the potential between a heavy quark-anti-quark pair (in a pure gauge theory these are just external probes). This is the case because the propagator of a static quark is just a gauge link. The area law corresponds to a linear potential, which formalizes the physical picture of a flux tube with constant energy per unit length.

The notion that an isolated test charge has infinite energy can be formalized using the Polyakov line, a gauge link along the euclidean time direction. The idea is that $P\sim \exp(-F_QT)$, where $F_Q$ is the free energy of an isolated charge. In the confined phase $\langle P\rangle =0$, and in the deconfined phase $\langle P\rangle\neq 0$. $P$ transforms non-trivially under the center of the gauge group ($Z_N$ in the case of $SU(N)$), so deconfinemnet corresponds to broken center symmetry.

There is no direct relation between chiral symmetry breaking and confinement. Neither implies the other, although in QCD there appears to be only a single crossover phase transition where both deconfinement and chiral restoration happen.

Finally, what do we mean by confinement in a theory with light fermions? Basically, just a smooth continuation of the phenomena that are seen in the limit $m_q\to\infty$. For example, QCD exhibits a regime in which the heavy quark (say $c\bar{c}$) potential is linear. The presence of light quarks implies that as $x\to\infty$ the potential is not linear, but goes to a constant. This is called "string breaking", and corresponds to the process $c\bar{c}\to (c\bar{q})+(\bar{c}q)$. Similarly, QCD has a deconfining phase transition that evolves into a sharp transition as $m_q\to\infty$.

• Thank for this answer. Could you please provide some idea of Question 2 if you're familiar with it.
– user28174
Nov 23 '17 at 11:01
• This notion is somewhat difficult to formalize. The best known attempt is known as the Kugo-Ojima confinement criterion, academic.oup.com/ptp/article/60/6/1869/1846386 Nov 23 '17 at 13:40
• You might also mention center symmetry breaking. Nov 23 '17 at 15:02