Numerical methods are used to investigate some of the non-perturbative properties of lattice QCD. Observable quantities are obtained from ensemble averages in the limit of the grid spacing going to zero. In Chapter 2, we apply these methods to the calculation of the QCD potential between a pair of static quarks.
We have therefore concentrated here on improving the efficiency of Monte Carlo methods for this problem. We then describe a new approach to Monte Carlo simulations which exploits the fact that the canonical and microcanonical ensembles are equivalent in the thermodynamic limit. Note in particular that, unlike the continuum case, the lattice formulation has the advantage that it does not require a gauge-fixing term in the action.
We have now connected our formalism to the real world because this object is known to be the Euclidean form of the path integral defining the pure SU(N) gauge theory in the continuum.
The continuum limit
It would in principle be possible to generate a finite sequence of configurations evenly over the entire space of configurations, and then the ensemble average of the quantity O(C)e-/35(Cl. One chooses the form of the transition probability W so that the configurations in this sequence, which is called a Markov chain, will be asymptotically distributed with the probability distribution Eq. 1.3.3). It can be shown [ 11) that a guaranteed way to produce a Markov chain satisfying the property Eq. 1.3.3), independent of the initial configuration, is to choose a transition probability W that satisfies the condition of detailed balance, namely.
The location of the gauging action then guarantees that this W depends only on the link values in the plaques containing the link ij. Again we note in passing that the inclusion of dynamic fcrmions leads to a non-local factor in the action that destroys this useful computational feature of the pure gauge action. The grid must therefore be 'swept' by the algorithm many times before one can be confident that one has adequately traversed the configuration space of the problem.
The advantage of the heat bath scheme over the Metropolis algorithm is that the U;;· does not depend on the old U;; not, and in fact can be in any part of the group manifold.
Heat baths and pseudo heat baths
First, we should be able to apply the spirit of the SU(2) heat bath algorithm to generate an expression similar to Eq. To see how the algorithm works, consider applying the S 1 subgroup to an old connection value U. Nevertheless, the spirit of the SU(2) improvement is retained and the method is much more efficient than naive algorithms that use the full SU. (3) measure.
We say approximately because from time to time it is necessary for each of the nodes to. On Hypercube, each node can be given responsibility for a small subsection of the grid. The hypercube's nodes accept code that has been compiled from the high level.
The double arrows indicate the content and direction of the information to be sent between nodes.
Chapter 2: The SU(3) Heavy Quark Potential
Static quark potential
- Variance reduction- the PPR trick
- Scaling and the quark self-energy
- Results
- GlueiJall masses on the lattice
Continuing in the language of statistical physics, we can write the expectation value of this Wilson loop W(C) in terms of the free energy, . They observed that the product of these lines is most sensitive to the values of the link variables in the lines themselves. We applied the trick to just the parts of the loop contours in the long (time) dimension.
This strongly exponential behavior of the Wilson ioops shows that the system we measure is indeed almost entirely in the ground state. The reason for the early onset of the 2-Ioop result at g ~I is not known (and indeed there appear to be some deviations from this asymptotic scaling in the Monte Carlo results). We are now in a position to look at the asymptotic scaling properties of the potential.
This own energy contributes to the Wilson loops, which depends on the length of the quark lines. This means that the force between the quarks, i.e. the gradient of the potential function, will not contain the term self-energy (this must obviously be true since the force between quarks is an actual measurable quantity). 2.1, plotted as a function of the number of grid distances in the time direction Tla, measured at the coupling /3=6.4.
However, at these distances the statistical noise in the ensemble averages is of the same order of magnitude as the signal itself. Due to the nature of the observable glueball, slight modifications to the trick are required in the case of the glueball. The method was used to calculate the o++ mass for the SU(3) gauge group.
Contamination of the signal by unwanted conditions in the first cut gives false values for the optimal coefficients c;. However, since the actual observable consists of the correlation function between 2 smears separated in the time direction, a more straightforward improvement proceeds as follows. Finally, the table shows the most important feature of the method, which is that in the N =50 case we were able to obtain a reliable signal for the correlation functions.
3.1) Various A-loop operators; used in the construct of the glue ball wave function (Sec. Eq. 3.2.3)) for the variational Monte Carlo method.
Chapter 4: TlH' Microcanonical Renormalisat.ion Group
Critical phenomena and the renormalisation group
This means that the ensemble averages of the blocked configurations will automatically be the correct estimates for the partition function with the new action (of course, it is not yet known what the values of {3; specifying the new action' are). This frees us from having to fix a gauge, as it means the transformation is invariant. Now, the entropy O'(S) of the lattice system at an action S is defined to be the logarithm of the number of states available to the system with that action.
Thus, the distribution of the demon will go like this. where C is the specific heat of the system per unit volume V, respectively. 4.3.6) can then be neglected and the inverse temperature can easily be derived from the ensemble average of the demon energy. Some of these corrections also occur in the zero-demon energy limit, where energy is directly transferred from one part of the lattice to another.
BrieOy, the algorithm proceeds by looking at each connection variable in the grid in turn and trying to randomly change the value of the variable. In the case of an Ising model simulation, for example, the grid/daemon configuration at a given time completely determines the next grid/daemon configuration in the sequence. In the case of the Ising model, it is even possible to compare both simulations with analytical results [I 0].
It is important to understand these discrepancies between the two ensembles, especially since we are switching between the two during the calculation. The coefficient of the second term is, again, essentially C, the specific heat per unit volume. Further, our volumes are such that this error is negligible compared to statistical calculation errors.
We decided to look at the coupling renormalization flow for the SU(2) group when MCRG [20] is applied to it. Second, it is important to see how sensitive the renormaliscd coupling values are to the number of couplings measured.
Results
In the case of QCD, there is only one eigenvector of the matrix T with an eigenvalue >.> 1. The new link ij is obtained by averaging this product with that of the 5 other 3-link paths from site i to the website j. Projection of the renorm~lliscd SU(2) action on the (~r.BA .80, J space for various st:ut ing actions.
The 13tticc rcgularization has been very successful in enabling us to calculate many of the important quantities in gauge field theories. The calculation of the hadron spectrum, and the question of how QCD confines quarks within hadrons, are examples of such problems. In this way, we will be able to trace the behavior of the theory from the fairly well-understood region of small distances, to the collective behavior that occurs over large distances and is responsible for confinement.
This means the use of very weak bare couplings, where Monte Carlo methods are severely hampered by the critical slowing problem already mentioned in the body of the dissertation: that is, the extremely slow traversal of configuration space near the critical point. The use of these improved actions will be crucial in the calculation of hadron and glueball spectra. First, consider an observable that consists of the trace of the product of two terms U and Y that lie in a straight line.
Starting with a legal configuration with link values U 0 and V0, the PPR trick consists of applying a heat bath independently to each of the 2 links. Since each of these terms is generated independently by the procedure, the total transition probability is just the product of the 2 distributions (A 1.4) and (A 1.5). The environment E belonging to the U-link is now a function of the value of the V-link, and vice versa.
We conclude that the trick is not applicable to parts of an observable where a link in the observable is part of the environment of another link in the observable. In Section 2 of Chapter 3, we described how the nlass of the sticky ball can be obtained by maximizing an expression of the form (sec Eq.
