Problems in Lattice Gauge Fixing
We review many topics and results about numeric gauge fixing in lattice QCD.
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arXiv subjects
C. Parrinello
Publications and source records attributed to C. Parrinello.
Lattice calculation of $1/p^2$ corrections to $α_s$ and of $Λ_{\rm {QCD}}$ in the $\widetilde{MOM}$ scheme
We report on very strong evidence of the occurrence of power terms in $\as(p)$, the QCD running coupling constant in the $\widetilde{MOM}$ scheme, by analyzing non-perturbative measurements from the lattice three-gluon vertex between 2.0 and 10.0 GeV at zero flavor. While putting forward the caveat that this definition of the coupling is a gauge dependent one, the general relevance of such an occurrence is discussed. We fit $Λ_{\bar{\rm MS}}^{(n_f=0)}= 237 \pm 3 ^{+ 0}_{-10}$ MeV in perfect agreement with the result obtained by the ALPHA group with a totally different method. The power correction to $\as(p)$ is fitted to $(0.63\pm 0.03 ^{+ 0.0}_{- 0.13}) {\rm GeV}^2/p^2$.
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Asymptotic Scaling and Infrared Behavior of the Gluon Propagator
The Landau gauge gluon propagator for the pure gauge theory is evaluated on a 32^3x64 lattice with a physical volume of (3.35^3x6.7)fm^4. Comparison with two smaller lattices at different lattice spacings allows an assessment of finite volume and finite lattice spacing errors. Cuts on the data are imposed to minimize these errors. Scaling of the gluon propagator is verified between beta=6.0 and beta=6.2. The tensor structure is evaluated and found to be in good agreement with the Landau gauge form, except at very small momentum values, where some small finite volume errors persist. A number of functional forms for the momentum dependence of the propagator are investigated. The form D(q^2)=D_ir+D_uv, where D_ir(q^2) ~ (q^2+M^2)^-ηand D_uv is an infrared regulated one-loop asymptotic form, is found to provide an adequate description of the data over the entire momentum region studied - thereby bridging the gap between the infrared confinement region and the ultraviolet asymptotic region. The best estimate for the exponent ηis 3.2(+0.1/-0.2)(+0.2/-0.3), where the first set of errors represents the uncertainty associated with varying the fitting range, while the second set of errors reflects the variation arising from different choices of infrared regulator in D_uv. Fixing the form of D_uv, we find that the mass parameter M is (1020+/-100)MeV.
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The Massive Yang-Mills Model and Diffractive Scattering
We argue that the massive Yang-Mills model of Kunimasa and Goto, Slavnov, and Cornwall, in which massive gauge vector bosons are introduced in a gauge-invariant way without resorting to the Higgs mechanism, may be useful for studying diffractive scattering of strongly interacting particles. We perform in this model explicit calculations of S-matrix elements between quark states, at tree level, one loop, and two loops, and discuss issues of renormalisability and unitarity. In particular, it is shown that the S-matrix element for quark scattering is renormalisable at one-loop order and is only logarithmically non-renormalisable at two loops. The discrepancies in the ultraviolet regime between the one-loop predictions of this model and those of massless QCD are discussed in detail. Also, some of the similarities and differences between the massive Yang-Mills model and theories with a Higgs mechanism are analysed at the level of the S-matrix. Finally, we briefly discuss the high-energy behaviour of the leading-order amplitude for quark-quark elastic scattering in the diffractive region. The above analysis sets up the stage for carrying out a systematic computation of the higher order corrections to the two-gluon exchange model of the Pomeron using massive gluons inside quantum loops.
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Search for 1/p^2 Corrections to the Running QCD Coupling
We investigate the occurrence of power terms in the running QCD coupling by analysing non-perturbative measurements of alpha_s (p) at low momenta obtained from the lattice three-gluon vertex. Our exploratory study provides some evidence for power contributions to alpha_s (p) proportional to 1/p^2. Possible implications for physical observables are discussed.
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Search for Lambda^2/p^2 corrections to the QCD running coupling
We investigate the occurrence of power terms $(Lambda^2/p^2$ in the running QCD coupling by analysing non-perturbative measurements of $α_s(p)$ at quite low momenta obtained from the lattice three-gluon vertex. Our study provides some evidence for such a contribution. The phenomenological implications of such a presence are reviewed.
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The running QCD coupling in the pre-asymptotic region
We study deviations from the perturbative asymptotic behaviour in the running QCD coupling by analysing non-perturbative measurements of alpha_s (p) at low momenta as obtained from the lattice three-gluon vertex. Our exploratory study provides some evidence for power corrections to the perturbative running proportional to 1/p^2.
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The structure of the gluon propagator
The gluon propagator has been calculated for quenched QCD in the Landau gauge at beta=6.0 for volumes 16^3x48 and 32^3x64, and at beta=6.2 for volume 24^3x48$. The large volume and different lattice spacings allow us to identify and minimise finite volume and finite lattice spacing artefacts. We also study the tensor structure of the gluon propagator, confirming that it obeys the lattice Landau gauge condition.
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Modelling the gluon propagator
Scaling of the Landau gauge gluon propagator calculated at beta=6.0 and at beta=6.2 is demonstrated. A variety of functional forms for the gluon propagator calculated on a large (32^3x64) lattice at beta=6.0 are investigated.
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Gluon Propagator in the Infrared Region
The gluon propagator is calculated in quenched QCD for two different lattice sizes (16^3x48 and 32^3x64) at beta=6.0. The volume dependence of the propagator in Landau gauge is studied. The smaller lattice is instrumental in revealing finite volume and anisotropic lattice artefacts. Methods for minimising these artefacts are developed and applied to the larger lattice data. New structure seen in the infrared region survives these conservative cuts to the lattice data. This structure serves to rule out a number of models that have appeared in the literature. A fit to a simple analytical form capturing the momentum dependence of the nonperturbative gluon propagator is also reported.
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Status of alpha_s determinations from the non-perturbatively renormalised three-gluon vertex
We demonstrate the feasibility of computing alpha_s from the lattice three-gluon vertex in the Landau gauge. Data from 16^4 and 24^4 quenched lattices at beta=6.0 are presented. Our main result is that 2-loop asymptotic scaling is observed for momenta in the range 1.8-2.3 Gev, where lattice artifacts appear to be under control.
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Alpha_s from the non-perturbatively renormalised lattice three-gluon vertex
We compute the running QCD coupling on the lattice by evaluating two-point and three-point off-shell gluon Green's functions in a fixed gauge and imposing non-perturbative renormalisation conditions on them. Our exploratory study is performed in the quenched approximation at beta=6.0 on 16^4 and 24^4 lattices. We show that, for momenta in the range 1.8-2.3 Gev, our coupling runs according to the two-loop asymptotic formula, allowing a precise determination of the corresponding Lambda parameter. The role of lattice artifacts and finite-volume effects is carefully analysed and these appear to be under control in the momentum range of interest. Our renormalisation procedure corresponds to a momentum subtraction scheme in continuum field theory, and therefore lattice perturbation theory is not needed in order to match our results to the MSbar scheme, thus eliminating a major source of uncertainty in the determination of alpha_MSbar. Our method can be applied directly to the unquenched case.
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Soft Pomeron Physics on the Lattice
We discuss strategies for using lattice QCD to investigate some topics in strong interaction phenomenology which are usually related to soft pomeron exchange. These include hadronic cross-sections at high energies and diffractive scattering at HERA. Some numerical results are presented in the framework of the Landshoff-Nachtmann pomeron model.
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Lattice Study of the Pomeron
We investigate the phenomenology of the Landshoff-Nachtmann two-gluon-exchange model of the Pomeron using gluon propagators computed in the Landau gauge within quenched lattice QCD simulations. As the propagators have been evaluated entirely from QCD first principles, our results provide a consistency check of the model. Finally, we report a proposal to investigate the structure of the Pomeron-quark vertex, using a colour-singlet Pomeron source constructed from the gauge fields.
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Soft Covariant Gauges on the Lattice
We present an exploratory study of a one-parameter family of covariant, non-perturbative lattice gauge-fixing conditions, that can be implemented through a simple Monte Carlo algorithm. We demonstrate that at the numerical level the procedure is feasible, and as a first application we examine the gauge dependence of the gluon propagator.
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The Landshoff-Nachtmann Pomeron on the Lattice
We investigate the Landshoff-Nachtmann two-gluon-exchange model of the Pomeron using gluon propagators computed in the Landau gauge within quenched lattice QCD calculations. We first determine an effective gluon-quark coupling by constraining the Pomeron-quark coupling to its phenomenological value $β_0 = 2\, \gev^{-1}$. We then provide predictions for a variety of diffractive processes. As the propagators have been evaluated entirely from QCD first principles (although in the quenched approximation), our results provide a consistency check of the Landshoff-Nachtmann model. We address the issue of the possible gauge-dependence of our results, which will be the object of a future study.
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The Strong Coupling Constant from the Lattice 3-Gluon Vertex
We compute the QCD running coupling on the lattice as defined from the 3-gluon vertex. We present the results of an exploratory study at $β=6.0$ on a $16^4$ lattice, which show that for momenta larger than 2 \Gev, the coupling runs according to the 2-loop asymptotic formula, allowing a precise determination of the $Λ$ parameter. Our renormalization procedure corresponds to a momentum subtraction scheme in the continuum and, most remarkably, one does not need lattice perturbation theory to match the results to MSbar scheme. We obtain $α^{\MSB}(M_Z) = 0.115 \pm 0.003 \pm 0.008$.
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The SU(3) running coupling from lattice gluons
We provide numerical results for the running coupling in $SU(3)$ Yang-Mills theory as determined from an analysis of lattice two and three-point gluon correlation functions. The coupling is evaluated directly, from first principles, by defining suitable renormalisation constants from the lattice triple gluon vertex and gluon propagator. For momenta larger than 2 GeV, the coupling is found to run according to the 2-loop asymptotic formula. The influence of lattice artifacts on the results appears negligible within the precision of our measurements, although further work on this point is in progress.
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