1. Topological susceptibility at zero and finite T in SU(3) Yang-Mills theory.

The topological susceptibility χ is calculated on the lattice for the pure SU(3) gauge theory both at zero temperature and across the deconfining transition. We obtain χ ≈ (170(7)(4) MeV)⁴ at zero temperature (the first error comes from our simulation and the second error derives from the determination of the physical units) while at the transition the following figure describes our main results:

The horizontal line and the grey band are the value and error of χ at zero temperature. Data are obtained from simulations on a 32³x 8 lattice with two different operators (called 1-smear and 2-smear respectively).

The numerical value obtained for χ is in agreement with the expectation from the analysis of the U_A(1) problem.

In the Erratum Nucl. Phys. B679 397 (2004) a few trivial numerical errors are corrected. The new results and figures are qualitatively identical (the figure shown above is the corrected one). The old version of the paper together with the Erratum (appearing as an appendix to the old paper) can be found in hep-lat/9605013.

2. O(n) Symmetric λΦ⁴ Theory: The Gaussian Effective Potential Approach.

The triviality problem of the λΦ⁴ theory in 4 dimensions with N fields is studied by calculating the Gaussian Approximation of the Effective Potential of the theory. Under such an approximation two apparently interacting phases are found. The bare coupling constant λ_B runs with the cutoff Λ as λ_B∝ 1/log(Λ/Ω) where Ω is a mass parameter related to the scalar particle and is one of the variational parameters in the trial wave functional of the vacuum.

3. Scanning the topological sectors of the QCD vacuum with hybrid Monte Carlo.

4. Hybrid Monte Carlo and topological modes of full QCD.

These two are technical papers for full QCD lattice practitioners. It is shown that the updating of topological modes is particularly slow in Monte Carlo simulations of full QCD both with Wilson ( paper 3) and staggered ( paper 4) fermions. A comparison of decorrelation efficiencies is performed with other more local observables showing that many more hits of HMC are needed for the updating of topology than for the updating of the other observables.

5. α_s from the nonperturbatively renormalized lattice three gluon vertex.

The coupling constant of strong interactions at the scale μ, α_s(μ), is an extremely important quantity of QCD. The rate at which α_s(μ) runs with μ is determined by the scale Λ of QCD. This is a fundamental parameter of the standard model. In the paper this parameter is calculated for the quenched theory obtaining the value Λ_MS=0.31(5) MeV. First the 3-gluon proper Green function is extracted from a Monte Carlo simulation in the Landau gauge. Then a renormalization scheme analogous to the MOM^∼ is imposed. The matching with the MS scheme is performed by a perturbative computation in the continuum without using lattice perturbation theory. The gauge field on the lattice is defined as A_μ(x)≡ -½ i [U_μ(x) - U_μ(x)^†]_traceless in terms of the links. In the Monte Carlo simulation the 3-gluon Green function and the gluonic propagator are evaluated at a particular set of external momenta in such a way that the proper 3-gluon function can be obtained by a simple algebra. The rotational invariance of the result has been checked. The calculation of the full QCD coupling constant by P. Boucaud et al. is based on the above method.

6. The Three loop lattice free energy.

The paper contains the first analytical calculation of the third term in the free energy of pure Yang--Mills. Such a result proves to be very useful to improve the precision in the extraction of the quenched gluon condensate and to define effective schemes on the lattice. The calculation was done by using the software prepared by us and introduced in Nucl. Phys. B413 553 (1994). The result has been extended to full QCD with Wilson fermions in Phys. Lett. B426 361 (1998) (Erratum:ibid. B553 337 (2003)) where the coefficients are extracted for several fermion masses up to three loops.

7. Renormalization and topological susceptibility on the lattice: SU(2) Yang-Mills theory.

The topological susceptibility χ of a gauge theory can be calculated on the lattice by measuring the observable 〈Q_LQ_L〉 where Q_L is the lattice regularized topological charge operator. However, in general, the above expectation value does not yield the continuum value of the susceptibility but instead the combination Z² a⁴χ + M where a is the lattice spacing and Z and M are multiplicative and additive renormalization constants respectively. Unless these two renormalization constants are known accurately enough, it will not be possible to extract the sought topological susceptibility from numerical simulation data. The above expression can be justified in the context of perturbation theory by using the Product Expansion Theorem.

Regularizations of the topological charge density operator have been defined around mid 90's for which M vanishes and Z is unity.

Z and M have been determined by using perturbation theory (see for instance my paper Nucl. Phys. B413 553 (1994) where M is calculated at three loops for the naïve definition of Q_L). However the method is intrinsically ambiguous since there is little control on the higher order terms. This problem, which is common in every perturbative calculation, is particularly acute in the case of lattice regularization. In the paper Phys. Rev. D48 2284 (1993) an improvement of the method to evaluate Z and M is utilized. The idea was put forward in a seminal paper by two of the authors, Phys. Lett. B275 429 (1992). This is a non-perturbative technique which avoids the systematic errors implicit in perturbation theory calculations at few loops.

In a nutshell the renormalization constants Z and M are simulated on configurations with a well-known instanton content in such a way to be able to impose a renormalization scheme such as the MS where the topological charge in the presence of one instanton is fixed to be 1. Needless to say, the validity of this procedure lies on an (yet unproved) extension of the Operator Product Theorem beyond perturbation theory.

In Phys. Rev. D54 1044 (1996) a consistency check of the method is performed. In Phys. Rev. D74 094503 (2006) another check is shown where the calibration of the topological content of configurations is done by counting fermionic zero modes and the result compared with the usual calibration based on standard cooling. In Phys. Rev. D77 056008 (2008) a further check is done. In this case the main goal was to verify the so-called Haldane conjecture which states that the mass gap of the O(3) non-linear sigma model in 2 dimensions with a θ term shall vanish when θ=π. Only by using the values of Z calculated with the present method, we were able to recover the correct position (θ=π) where the mass gap closes.