In 1974, Wilson [3] showed how the lattice controller could be made such that the gauge invariance of the theory was preserved. The strongly coupled limit of lattice theory also provides a standard against which the results of the numerical simulations can be compared. However, when removing the limit value, a renormalization of the coupling constant g is necessary.
You have to set the exposed parameters (here g), i.e. adjust g as a function of the grid spacing a, so quantities of physical. The first step is to find the location of the critical points in exposed coupling space. This direction will be specified by the properties of the phases near the critical point.
For some theories, the region of convergence may extend from the high temperature range to the critical point of interest. Another problem is that the sum of the strong coupling series in the convergence region does not uniquely determine the theory everywhere.
PROPERI'IES OF LATTICE QCD
Even if the coupling constant is very large, there are very light particles such as the pion if the bare quark mass is small. However, if we follow the quark part of the theory carefully, this problem can be overcome, and we will see in Chapters 3 and 4 that the first few kinds of expansion are still tractable. These results should be compared with those from estimates of strong coupling either in .
Some of the formalisms used, such as the Euclidean lattice theory with Kogut–Susskind fermions, have a continuous chiral symmetry. In these cases, the chiral symmetry breaks dynamically and the pion appears as a Goldstone boson. experiment gives 1J3 mev.) in the current algebra relation. Let us also mention some of the properties of lattice QCD at finite temperature.
Below Tc, the free energy of an isolated quark in the fundamental representation is infinite, while above Tc, this free energy becomes finite and the quarks are released. However, there is good evidence that ground quark confinement is not a necessary condition for chiral symmetry breaking to occur [21].
1,1] PADE FIT TO DATA FROM A
KAPPA *10- 1
CHIRAL SYMMETRY BREAKING IN LATTICE GAUGE THEORIES
THE ORDER PARAME'I'ER AT STRONG COUPIJNG AND LARGE N
In two dimensions, these loops actually restore the ground state chiral invariance from Coleman's theorem. The methods used were an exact evaluation of the generating function of the lattice gauge theory in infinite coupling with arbitrary sources. We wish to find <#>m.q as mq -+0 since it is an order parameter of chiral symmetry.
One is then to take the trace about spin~ and color. we must add terms of the form. where the £:i are Wilson loops and the last trace comes from the expansion of. Set up a recursion that summarizes the paths of length L + 1 in terms of the number of paths of length L. This chapter dealt with the large N limit of the lattice measure theory under strong coupling.
Note that the background gauge field has no memory for the direction of chiral symmetry breaking. To see the role played by the center of the meter group, it is necessary to move to a much higher order. If we do not include the inner quark rings, the Green's functions of the two theories are identical up to factors of C.
As for the computation of <1/nf>A=<»•, the basic building blocks are the irreducible tree graphs. This last constraint ensures the uniqueness of the translation, which leads a plaque to the origin. This condition completely satisfies the constraints, and we have a one-to-one labeling of the graphs.
Then dress the locations of this graph with any number of irreducible tree graphs constrained to take their first step off the picture. Now find the only one such that, if we consider this the first step of the graph, you will never return to the origin with a lree. In fact, the number of these links is exactly the pendulum number of the graph, which we have determined to be one.
12] First stage of dressing the plaque: the thin lines are limited to stay on the plaque. 13] A graphic that does not appear at the first stage of the plaque application.
THE LARGE N SPECTRUM AT STRONG COUPLING
GRAPIDCAL METHOD FOR OBTAINING THE SPECTRUM
The fermionic part of the action 1f; Q1f; has a diagonal stretch in the space indices and a hopping term from site to site. Now in the strong coupling limit, the only paths that survive are those with trivial color factors, that is, the paths that include zero area. In fact, we can add up all random walks of the meson, since the ones that backtrack do not contribute to the amplitude if we calculate things correctly.
The large d limit greatly simplified the calculation of <#>. and the same is true for the meson propagator. One can calculate the graphs of the two-point function exactly for any d and determine the spectrum exactly. It defines the part of the graph that intersects the curk line at the site o.
Proceed to the next place and analogously define the part of the graph which is considered to be clothed. This re-normalizes the mass so that each trunk step is related by a factor of cx-2 )'/-LX(-/'J.L)t. For a large d limit, this restriction becomes irrelevant and we find the result of the previous section.
The other eigenvectors or low mass live in the corners of the Brillouin zone but have the same energy. Thus, one has an estimate of the hadron masses, which converges statistically with the sample size. We define mass m of the particle to be energy at zero spatial momentum E(p =0).
The spectrum of the naive fermion theory is the same as in the theory of Kogut-Susskind fermions, but with a fourfold multiplicity. In the limit of strong coupling, the quarks are always on top of each other, so the spin dependence of the force cannot have its effect. We have seen how the large N limit of lattice gauge theory SU(N) can be exactly solved by strong coupling.
We must restrict ourselves to the pion, but since the propagation of the other mesons is simply associated with shifts in momentum, there is no loss of generality. Each step on the stem will lead to a spin xspin factor when dressed and we get a geometric series for the propagator.
