Let us remember that the lattice gauge theory is only a regularization for the investigation of the continuum field theory. The results obtained by means of lattice caluclations, make physical sense only in the continuum limit a → 0. At the same time, the continuum renormalized parameters such as masses or running couplings, must take finite values. Hence in order to go to the continuum limit, one has to require that the couplingsβ and κdepend on the lattice spacing a(see [9, 10]).
It is necessary to know which lines in the (β, κ) plane correspond to constant physics in the limita→0. One of such lines is obtained by means of the study of
lattice renormalized masses. They can be extracted from corresponding correlators in the following way [62]:
ΓΦ(τ) =X
~ x
hΦ0Φxi ∝exp(−mΦτ), x= (~x, τ), a→0. (2.32) For example, the operator for the extraction of the pseudoscalar ’pion’ mass mπ is Φ(π)x =ψxγ5ψx.
Keeping withτ =an4the integer valuen4in the (2.32) finite, one can formulate the condition for the bare lattice parameters in the continuum limit:
amphys →0, a→0. (2.33)
where mphys = mphys(β, κ) are masses of physical particles or energies of their ground states extracted according to (2.32).
Equation (2.33) determines the critical lineκ=κc(β) where the fermion parti-cles ina units have zero masses [63] – [65]. It means that along this line known as the chiral limit line, the chiral symmetry broken by the Wilson mass term (2.14) is partially restored. On the other hand, since the powers of the 1/amphys contribute to fermion observables like (2.27) or (2.28) and the masses mphys are finite, the chiral limit line in the language of the thermodynamics must be a line of at least second order phase transition. Instead of the bare fermion massm0, one can define the naive lattice fermion massmq [63] as
amq = 1 2
1
κ − 1 κc(β)
. (2.34)
Studies of the 4-dimensional U(1) model with Wilson fermions have shown (see e.g. [6, 7], [17, 18], [63] – [70]) that such a theory has a nontrivial phase structure (Figure 2.1). It consists of at least 4 phases in the (β, κ) plane separated by different order phase transition lines. But there is a difference between the quenched approximation andNf = 2 dynamical fermions. While in the quenched case the critical line separating Coulomb and confinement phase has the same β value equal to 1.01(1) (Figure 2.1a), in the dynamical case it coincides partially with the chiral limit line (Figure 2.1b) [70]. This line κ = κc(β) connects the points κc(0) = 1/4 and κc(∞) = 1/8 at the Wilson coefficient r = 1 [63]. The deviation of κc(β) from the exact perturbative value 1/8 can be explained as an influence of the chirally noninvariant Wilson mass-like term (2.14).
0 0
(a)
confinement phase
Coulomb phase κ
0.25 0.125
1.01(1) β 8
chiral limit κ=κc(β)
0 0
weakly 1st order 1st order
(b)
confinement phase
Coulomb phase 4th
(Aoki) phase
3rd phase κ
0.25 0.125
1.01(1) β 8
2nd order
higher order transition?
1 st order
Figure 2.1: Phase structure of compact lattice QED in the quenched approx-imation (a) and with Nf = 2 dynamical fermions [70] (b).
We are interested mostly in the Coulomb phase because it describes the usual static Coulomb potential and the vanishing photon mass. It is characterized by the suppression of magnetic monopoles [17]. However, it is worth to discuss also the confinement phase having many similarities with QCD one. In this phase in quenched approximation, the static potential for charged particles is directly proportional to the distance between them [8], the corresponding gauge bosons acquire a non-zero mass and one detects a condensation of monopole-antimonopole pairs [17].
These phases are separated by the line of the first order phase transition [17], [71] – [73] (see Figure 2.1). It means that one has to search for the continuum limit points (β?, κ?) outside this line. According to the above presented arguments, these points should lie on the curveκ=κc(β) in the Coulomb phase. The precise numerical value for the β? point is unknown so far in spite of numerous efforts in this direction (see e.g. [74] – [76]). We will not touch this problem but note that it requires a very careful study of the renormalized masses and coupling constant [13] – [16].
At the same time, the investigation [77] of the confinement phase near the chiral limit and also the 4th (Aoki) phase [29] (see Figure 2.1b) is complicated because the well-known method for such purposes, the hybrid Monte Carlo algorithm [39, 40], does not work well in the case of large condition numbersζ (2.30) (see also [33, 78]).
To decrease this number, in case ofNf = 2 one can introduce the following twisted mass term [30, 31]:
hψγ5⊗τ3ψ, (2.35)
and then at the evaluation of desired observables take the limit h → 0. The investigation of the Aoki phase led to the conclusion that there the composite pseudoscalar fermion masses are equal to 0 and the combined parity-flavour sym-metry is broken [29] – [32]. But in order to better understand the properties of this phase, one should use an alternative to the hybrid Monte Carlo algorithm. And moreover, presently the studies of the lattice compact QED were done in the frame-work of the quenched approximation or for even dynamical fermion flavours [77].
It would be interesting to investigate also the dynamical models with odd fermion flavours. The problems of such investigation will be discussed later together with the consideration of the dynamical fermion algorithms.
In the following let us use the convention a = 1 for the lattice spacing. If it
is necessary, dimensions can be easily re-inserted. The lattice size will be V = Ns3×N4, with N4 ≥Ns in order to compute masses from correlators as (2.32).
Chapter 3
Gauge fixing on the lattice
3.1 Motivation
As it was already mentioned, studying the gauge invariant observables on the lattice in the framework of compact gauge models [8] does not require a gauge fixing (see [9]). Nevertheless, applying the Faddeev-Popov trick [79] to the integral (2.24) by inserting the unity:
1 = ∆FP[U] Z
[dg]δ(F[Ug]),
where ∆FP[U] is the so-called Faddeev-Popov determinant, and by integrating out the gauge transformation fieldg, we get the following expression for the averaged gauge invariant operator O (2.25):
hOi= 1 Z
Z
[dU]∆FP[U]δ(F[U])O[U]e−SG[U]detNfM[U], (3.1) similar to the continuum field theory case.
However, in the case of perturbative study of a lattice model [46], one has to introduce a gauge fixing term just as in the continuum theory. Moreover, the evaluation of such gauge invariant objects as Wilson loops is very simple e.g. in 2-dimensional gauge models when an additional gauge fixing method is employed [9].
We note that usual gauge invariant values describe either composite particles or bounded states of quantum fields e.g. mesons or glueballs [10]. At the same time, studies of gauge dependent observables like photon or fermion propagator can give us more detailed and natural information about quantum objects such as
behaviour of renormalized Green functions [14]. But the straightforward averaging of gauge dependent operators over gauge field without any gauge fixing term leads, according to group symmetry properties, to zero. For instance, in the case of fermion propagator one has:
as it follows from the covariance property (2.19) of the Wilson matrix, and where the translational invariance of the theory has been used. Hence, to consider on the lattice the gauge dependent objects, one has to use a gauge fixing procedure.
And the expression (3.1) can serve as a definition for the average value of a gauge dependent observable.
In our case, studying the ’Landau pole’ problem in the compact lattice QED requires an investigation of the renormalized coupling constant and fermion mass [15, 16]. They can be extracted in the best way from the gauge dependent photon (link) and fermion correlators by the method analogous to (2.32) (see [13, 14]).
That is why we are interested in the study of gauge dependent objects in the U(1) theory on the lattice.