The quenched approximation

Monte Carlo simulations are simplified drastically, if the fermion determinant detDm is set to a constant. In expectation values this constant cancels as the same constant occurs in the normalization factor Z. To understand the implica-tion of this approximaimplica-tion it is useful to note that the determinant is real. Using D=γ5D⁺γ5 we get

detD_m = det(γ₅D_mγ₅) = detD_m⁺ = (detD_m)^∗ (1.79) As the QCD Lagrangian is diagonal in the quark flavor, each flavor adds an additional factor of detDm. So for mass degenerate quarks the contribution of this determinant is (detD)^Nf with N_f the number of flavors.

If Nf is odd, the determinant can fluctuate in sign. It cannot be included in the weight of the Monte-Carlo as this is to be interpreted as a probability distribution. This is the major obstacle of simulations with oddN_f.

At the moment dynamical computations are performed with an even number of degenerate quark flavors. Then the determinant can be included in the weight.

If the series of gauge configurations has been generated including the determi-nant, the evaluation of the observables is exactly the same as in the quenched case. Quenching merely amounts to changing the weights with which configura-tions contribute. However, the numerical effort to generate these ’unquenched’

configurations is several orders of magnitudes larger than in the quenched case.

This has several reasons.

• The computation of the determinant of such a large matrix is nontrivial.

• The updating step of the Metropolis algorithm is local in the quenched approximation. We have to compare the weight of the old and the new

configuration. As the gauge action is a sum of local terms the weight is a product of local terms

e^−PpSp[U]=Y

p

e^−Sp[U] . (1.80)

If we now offer a new link variable Uµ(x) only very few of the factors change and the new weight can be computed very quickly. For the fermion determinant this is not possible since it connects the link variables in a non-local way.

In physical terms, the quenched approximation amounts to the limit of infinite masses for the sea quarks; no virtual quark–anti-quark pairs can be generated.

This is plausible, as the determinant ofDmis dominated by the diagonal elements if m → ∞. However, in many simulations this approximation has given good results. Furthermore, the sea quark masses of current simulations are quite large, e.g. of the order of 50 MeV and their contribution is, thus, quite small. However their effect will certainly get larger with smaller quark masses. In general the error introduced by the quenched approximation is estimated to be of the order of 10%. . .20%.

Chapter 2

Chiral symmetry breaking

As discussed in the introduction, chiral symmetry breaking is a key feature of the theory of strong interactions. Whereas in early theories it had to be introduced by hand, it emerges quite naturally in QCD. Even though the Lagrangian is chirally invariant, the full interacting theory is not. This allows the creation of a chiral condensate hψψ¯ i, which should vanish in a chirally invariant theory. Via the Banks-Casher relation the emergence of the chiral condensate is linked to the density of eigenvalues ρ(λ) of the Dirac operator at λ = 0. A promising explanation of its creation is given by the interaction of instantons, i.e., classical solutions of the equations of motion for the gauge fields. This is the subject of Sec. 2.1. It is interesting to search for traces of instantons in the corresponding, low lying eigenmodes of the Dirac operator. In particular, we consider two of the major properties of instantons. The first is that they are localized objects, the second is their definite chirality. In the course of Section 2.2 we first examine observables which quantify the localization of the eigenmodes. Later on we look into observables which connect localization and chirality.

In the second part of this chapter, Sec.2.3, the behavior of these observables is analyzed for increasing temperatureT. It is generally believed that for growing T a phase transition occurs in QCD from the chirally broken phase to a phase where chiral symmetry is restored. Lattice studies further indicate that this coin-cides with the deconfinement phase transition. The purpose of large experimental programs at RHIC and LHC is to collect information about these phase transi-tions. Our aim is to study localization and chirality in the two phases. This can be used to sharpen the ideas from models based on the classical solutions called calorons in the finite temperature region. Finally, a summary of the findings of this section is given in Sec. 2.4.

2.1 The mechanism of chiral symmetry break-ing

We are going to study chiral symmetry breaking by looking at the eigenvectors and eigenvalues of the Dirac operator, especially for eigenvalues close to the origin. There we define the density of eigenvalues ρ(λ) such that ρ(λ)∆λ gives the number of eigenvalues in the interval (λ, λ+ ∆λ]. The zero modes themselves do not contribute to this density. Spontaneous breaking of chiral symmetry is manifest in the emergence of a non-zero chiral condensate hψψ¯ i. The Banks-Casher relation [BC80] states that the density of eigenvalues near the origin λ= 0 is proportional to the chiral condensate.

hψψ¯ i=−π

V ρ(0) (2.1)

HereV is the volume of the finite box in which the theory is put. This truncation is necessary to define a finite density of eigenvalues. Butρ(0)/V should become constant as V → ∞.