Pure SU(2) gauge theory is the simplest asymptotically free theory in four dimensions. To investi... more Pure SU(2) gauge theory is the simplest asymptotically free theory in four dimensions. To investigate Euclidean quantum gravity effects in a fundamental length scenario, we simulate 4d SU(2) lattice gauge theory on a dynamically coupled Regge skeleton. The fluctuations of the skeleton are governed by the standard Regge-Einstein action. From a small 2 • 4 3 lattice we report exploratory numerical results, limited to a region of strong gravity where the Planck mass and hadronic masses take similar orders of magnitude. We find a range of the Planck mass where stable bulk expectation values are obtained which vary smoothly with the gauge coupling, and a remnant of the QCD deconfining phase transition is located.
Proceedings of The 32nd International Symposium on Lattice Field Theory — PoS(LATTICE2014), 2015
With the emergence of the Yang-Mills gradient flow technique there is renewed interest in the iss... more With the emergence of the Yang-Mills gradient flow technique there is renewed interest in the issue of scale setting in lattice gauge theory. Here I compare for the SU(3) Wilson gauge action non-perturbative scale functions of Edwards, Heller and Klassen (EHK), Necco and Sommer (NS), both relying on Sommer's method using the quark potential, and the scale function derived by Bazavov, Berg and Velytsky (BBV) from a deconfining phase transition investigation by the Bielefeld group. It turns out that the scale functions are based on mutually inconsistent data, though the BBV scale function is consistent with the EHK data when their low β (β = 5.6) data point is removed. Besides, only the BBV scale function is consistent with three data points calculated from the gradient flow by Lüscher. In the range for which data exist the discrepancies between the scale functions are only up to ±2% of their values, but clearly visible within the statistical accuracy.
We reexamine the approach to four-dimensional Euclidean quantum gravity based on the Regge calcul... more We reexamine the approach to four-dimensional Euclidean quantum gravity based on the Regge calculus. A cutoff on the link lengths is introduced and consequently the gravitational coupling and the cosmological constant become independent parameters. We determine the zero curvature, R = 0, line in the coupling constant plane by numerical simulations. When crossing this line we find a strong, probably first order, phase transition line with indications of a second order endpoint. Beyond the endpoint the transition through the R = 0 line appears to be a crossover. Previous investigations, using the Regge or the Dynamical Triangulation approach, dealt with a limit in which the first order transition prevails.
We investigate the eigenvalue spectrum of the staggered Dirac matrix in SU(3) gauge theory and in... more We investigate the eigenvalue spectrum of the staggered Dirac matrix in SU(3) gauge theory and in full QCD as well as in quenched U(1) theory on various lattice sizes. As a measure of the fluctuation properties of the eigenvalues, we consider the nearest-neighbor spacing distribution, P (s). We further study two-color QCD at nonzero chemical potential, µ, by constructing the spacing distribution of adjacent eigenvalues in the complex plane. We find that in all regions of their phase diagrams, compact lattice gauge theories have bulk spectral correlations given by random matrix theory, which is an indication for quantum chaos. In the confinement phase, the low-lying Dirac spectrum of these quantum field theories is well described by random matrix theory, exhibiting universal behavior.
We compare the low-lying spectrum of the staggered Dirac operator in the confining phase of compa... more We compare the low-lying spectrum of the staggered Dirac operator in the confining phase of compact U(1) gauge theory on the lattice to predictions of chiral random matrix theory. The small eigenvalues contribute to the chiral condensate similar as for the SU(2) and SU(3) gauge groups. Agreement with the chiral unitary ensemble is observed below the Thouless energy, which is extracted from the data and found to scale with the lattice size according to theoretical predictions.
We present investigations of the potential between static charges from a simulation of quantum gr... more We present investigations of the potential between static charges from a simulation of quantum gravity coupled to an SU(2) gauge eld on 6 3 4 and 8 3 4 simplicial lattices. In the well-de ned phase of the gravity sector where geometrical expectation values are stable, we study the correlations of Polyakov loops and extract the corresponding potentials between a source and sink separated by a distance R. In the con ned phase, the potential has a linear form while in the decon ned phase, a screened Coulombic behavior is found. Our results indicate that quantum gravitational e ects do not destroy con nement due to non-abelian gauge elds.
This paper deals with a situation of some importance for the analysis of experimental data via Ne... more This paper deals with a situation of some importance for the analysis of experimental data via Neural Network (NN) or similar devices: Let N data be given, such that N = N s + N b , where N s is the number of signals, N b the number of background events, both unknown. Assume that a NN has been trained, such that it will tag signals with efficiency F s , (0 < F s < 1) and background data with F b , (0 < F b < 1). Applying the NN yields N Y tagged events. We demonstrate that the knowledge of N Y is sufficient to calculate confidence bounds for the signal likelihood, which have the same statistical interpretation as the Clopper-Pearson bounds for the well-studied case of direct signal observation. Subsequently, we discuss rigorous bounds for the a-posteriori distribution function of the signal probability, as well as for the (closely related) likelihood that there are N s signals in the data. We compare them with results obtained by starting off with a maximum entropy type assumption for the a-priori likelihood that there are N s signals in the data and applying the Bayesian theorem. Difficulties are encountered with the latter method.
We investigate the spectrum of the staggered Dirac operator in 4d quenched U(1) lattice gauge the... more We investigate the spectrum of the staggered Dirac operator in 4d quenched U(1) lattice gauge theory and its relationship to random matrix theory. In the confined as well as in the Coulomb phase the nearest-neighbor spacing distribution of the unfolded eigenvalues is well described by the chiral unitary ensemble. The same is true for the distribution of the smallest eigenvalue and the microscopic spectral density in the confined phase. The physical origin of the chiral condensate in this phase deserves further study.
It is noted that, in contrast to widespread believes, free fields do not only allow for global, b... more It is noted that, in contrast to widespread believes, free fields do not only allow for global, but also for local gauge invariance.
Proceedings of XXIIIrd International Symposium on Lattice Field Theory — PoS(LAT2005), 2005
It is illustrated for 4D SU(2) lattice gauge theory that sampling with a biased Metropolis scheme... more It is illustrated for 4D SU(2) lattice gauge theory that sampling with a biased Metropolis scheme is essentially equivalent to using the heatbath algorithm. Only, the biased Metropolis method can also be applied when an efficient heatbath algorithm does not exist. Other cases for which the use of the biased Metropolis-heatbath algorithm is beneficial are briefly summarized.
Proceedings of The XXVIII International Symposium on Lattice Field Theory — PoS(Lattice 2010), 2011
As long as a Higgs boson is not observed, the design of alternatives for electroweak symmetry bre... more As long as a Higgs boson is not observed, the design of alternatives for electroweak symmetry breaking remains of interest. The question addressed here is whether there are possibly dynamical mechanisms, which deconfine SU(2) at zero temperature and generate a massive vector boson triplet. Results for a model with joint local U(2) transformations of SU(2) and U(1) vector fields are presented in a limit, which does not involve any unobserved fields.
Proceedings of The XXVII International Symposium on Lattice Field Theory — PoS(LAT2009), 2010
Deconfined regions in relativistic heavy ion collisions are limited to small volumes surrounded b... more Deconfined regions in relativistic heavy ion collisions are limited to small volumes surrounded by a confined exterior. Here the geometry of a double layered torus is discussed, which allows for different temperatures in its two layers. This geometry enables one to approach the QCD continuum limit for small deconfined volumes with confined exteriors in a more realistic fashion than by using periodic boundary conditions. Preliminary data from a study for pure SU(3) lattice gauge theory support a substantial increase in a pseudo transition temperature.
We investigate the approach of pure SU(2) lattice gauge theory with the Wilson action to its cont... more We investigate the approach of pure SU(2) lattice gauge theory with the Wilson action to its continuum limit using the deconfining transition, Lüscher's gradient flow [1], and the cooling flow [2, 3] to set the scale. Of those, the cooling flow turns out to be computationally most efficient. We explore systematic errors due to use of three different energy observables and two distinct reference values for the flow time, the latter obtained by matching initial scaling behavior of some energy observables to that of the deconfining transition. Another important source of systematic errors are distinct fitting forms for the approach to the continuum limit. Besides relying in the conventional way on ratios of masses, we elaborate on a form introduced by Allton [4], which incorporates asymptotic scaling behavior. Ultimately we find that, though still small, our systematic errors are considerably larger than our statistical errors.
It is shown that whenever the multiplicative normalization of a fitting function is not known, le... more It is shown that whenever the multiplicative normalization of a fitting function is not known, least square fitting by χ 2 minimization can be performed with one parameter less than usual by converting the normalization parameter into a function of the remaining parameters and the data. Erratum: Jochen Heitger and Johannes Voss of Münster University found a typo in two subroutines which are used for the 4-parameter fit example given in subsection III D of this paper. In each of the subroutines, subg power2n.f and suby power2n.f, a(2) has to be replaced by a(1) in the expression for the derivative dyda(1). Elimination of this error improves the performance of the fitting method considerable. In the following subsection III D is rewritten to reflect use of the correct subroutines and the error prone routines have been replaced in the package FITM1.tgz, which is still available on the Web as described. Other parts of the paper are not affected.
Metropolis simulations of all-atom models of peptides (i.e. small proteins) are considered. Inspi... more Metropolis simulations of all-atom models of peptides (i.e. small proteins) are considered. Inspired by the funnel picture of Bryngelson and Wolyness, a transformation of the updating probabilities of the dihedral angles is defined, which uses probability densities from a higher temperature to improve the algorithmic performance at a lower temperature. The method is suitable for canonical as well as for generalized ensemble simulations. A simple approximation to the full transformation is tested at room temperature for Met-Enkephalin in vacuum. Integrated autocorrelation times are found to be reduced by factors close to two and a similar improvement due to generalized ensemble methods enters multiplicatively.
We consider 4d compact lattice QED in the quenched approximation. First, we briefly summarize the... more We consider 4d compact lattice QED in the quenched approximation. First, we briefly summarize the spectrum of the staggered Dirac operator and its connection with random matrix theory. Afterwards we present results for the low-lying eigenmodes of the Neuberger overlap-Dirac operator. In the strong coupling phase we find exact zero-modes. Subsequently we discuss possibly related topological excitations of the U(1) lattice gauge theory.
For a free scalar boson field and for U(1) gauge theory finite volume (infrared) and other correc... more For a free scalar boson field and for U(1) gauge theory finite volume (infrared) and other corrections to the energy-momentum dispersion in the lattice regularization are investigated calculating energy eigenstates from the fall off behavior of two-point correlation functions. For small lattices the squared dispersion energy defined by
Pure SU(2) gauge theory is the simplest asymptotically free theory in four dimensions. To investi... more Pure SU(2) gauge theory is the simplest asymptotically free theory in four dimensions. To investigate Euclidean quantum gravity effects in a fundamental length scenario, we simulate 4d SU(2) lattice gauge theory on a dynamically coupled Regge skeleton. The fluctuations of the skeleton are governed by the standard Regge-Einstein action. From a small 2 • 4 3 lattice we report exploratory numerical results, limited to a region of strong gravity where the Planck mass and hadronic masses take similar orders of magnitude. We find a range of the Planck mass where stable bulk expectation values are obtained which vary smoothly with the gauge coupling, and a remnant of the QCD deconfining phase transition is located.
Proceedings of The 32nd International Symposium on Lattice Field Theory — PoS(LATTICE2014), 2015
With the emergence of the Yang-Mills gradient flow technique there is renewed interest in the iss... more With the emergence of the Yang-Mills gradient flow technique there is renewed interest in the issue of scale setting in lattice gauge theory. Here I compare for the SU(3) Wilson gauge action non-perturbative scale functions of Edwards, Heller and Klassen (EHK), Necco and Sommer (NS), both relying on Sommer's method using the quark potential, and the scale function derived by Bazavov, Berg and Velytsky (BBV) from a deconfining phase transition investigation by the Bielefeld group. It turns out that the scale functions are based on mutually inconsistent data, though the BBV scale function is consistent with the EHK data when their low β (β = 5.6) data point is removed. Besides, only the BBV scale function is consistent with three data points calculated from the gradient flow by Lüscher. In the range for which data exist the discrepancies between the scale functions are only up to ±2% of their values, but clearly visible within the statistical accuracy.
We reexamine the approach to four-dimensional Euclidean quantum gravity based on the Regge calcul... more We reexamine the approach to four-dimensional Euclidean quantum gravity based on the Regge calculus. A cutoff on the link lengths is introduced and consequently the gravitational coupling and the cosmological constant become independent parameters. We determine the zero curvature, R = 0, line in the coupling constant plane by numerical simulations. When crossing this line we find a strong, probably first order, phase transition line with indications of a second order endpoint. Beyond the endpoint the transition through the R = 0 line appears to be a crossover. Previous investigations, using the Regge or the Dynamical Triangulation approach, dealt with a limit in which the first order transition prevails.
We investigate the eigenvalue spectrum of the staggered Dirac matrix in SU(3) gauge theory and in... more We investigate the eigenvalue spectrum of the staggered Dirac matrix in SU(3) gauge theory and in full QCD as well as in quenched U(1) theory on various lattice sizes. As a measure of the fluctuation properties of the eigenvalues, we consider the nearest-neighbor spacing distribution, P (s). We further study two-color QCD at nonzero chemical potential, µ, by constructing the spacing distribution of adjacent eigenvalues in the complex plane. We find that in all regions of their phase diagrams, compact lattice gauge theories have bulk spectral correlations given by random matrix theory, which is an indication for quantum chaos. In the confinement phase, the low-lying Dirac spectrum of these quantum field theories is well described by random matrix theory, exhibiting universal behavior.
We compare the low-lying spectrum of the staggered Dirac operator in the confining phase of compa... more We compare the low-lying spectrum of the staggered Dirac operator in the confining phase of compact U(1) gauge theory on the lattice to predictions of chiral random matrix theory. The small eigenvalues contribute to the chiral condensate similar as for the SU(2) and SU(3) gauge groups. Agreement with the chiral unitary ensemble is observed below the Thouless energy, which is extracted from the data and found to scale with the lattice size according to theoretical predictions.
We present investigations of the potential between static charges from a simulation of quantum gr... more We present investigations of the potential between static charges from a simulation of quantum gravity coupled to an SU(2) gauge eld on 6 3 4 and 8 3 4 simplicial lattices. In the well-de ned phase of the gravity sector where geometrical expectation values are stable, we study the correlations of Polyakov loops and extract the corresponding potentials between a source and sink separated by a distance R. In the con ned phase, the potential has a linear form while in the decon ned phase, a screened Coulombic behavior is found. Our results indicate that quantum gravitational e ects do not destroy con nement due to non-abelian gauge elds.
This paper deals with a situation of some importance for the analysis of experimental data via Ne... more This paper deals with a situation of some importance for the analysis of experimental data via Neural Network (NN) or similar devices: Let N data be given, such that N = N s + N b , where N s is the number of signals, N b the number of background events, both unknown. Assume that a NN has been trained, such that it will tag signals with efficiency F s , (0 < F s < 1) and background data with F b , (0 < F b < 1). Applying the NN yields N Y tagged events. We demonstrate that the knowledge of N Y is sufficient to calculate confidence bounds for the signal likelihood, which have the same statistical interpretation as the Clopper-Pearson bounds for the well-studied case of direct signal observation. Subsequently, we discuss rigorous bounds for the a-posteriori distribution function of the signal probability, as well as for the (closely related) likelihood that there are N s signals in the data. We compare them with results obtained by starting off with a maximum entropy type assumption for the a-priori likelihood that there are N s signals in the data and applying the Bayesian theorem. Difficulties are encountered with the latter method.
We investigate the spectrum of the staggered Dirac operator in 4d quenched U(1) lattice gauge the... more We investigate the spectrum of the staggered Dirac operator in 4d quenched U(1) lattice gauge theory and its relationship to random matrix theory. In the confined as well as in the Coulomb phase the nearest-neighbor spacing distribution of the unfolded eigenvalues is well described by the chiral unitary ensemble. The same is true for the distribution of the smallest eigenvalue and the microscopic spectral density in the confined phase. The physical origin of the chiral condensate in this phase deserves further study.
It is noted that, in contrast to widespread believes, free fields do not only allow for global, b... more It is noted that, in contrast to widespread believes, free fields do not only allow for global, but also for local gauge invariance.
Proceedings of XXIIIrd International Symposium on Lattice Field Theory — PoS(LAT2005), 2005
It is illustrated for 4D SU(2) lattice gauge theory that sampling with a biased Metropolis scheme... more It is illustrated for 4D SU(2) lattice gauge theory that sampling with a biased Metropolis scheme is essentially equivalent to using the heatbath algorithm. Only, the biased Metropolis method can also be applied when an efficient heatbath algorithm does not exist. Other cases for which the use of the biased Metropolis-heatbath algorithm is beneficial are briefly summarized.
Proceedings of The XXVIII International Symposium on Lattice Field Theory — PoS(Lattice 2010), 2011
As long as a Higgs boson is not observed, the design of alternatives for electroweak symmetry bre... more As long as a Higgs boson is not observed, the design of alternatives for electroweak symmetry breaking remains of interest. The question addressed here is whether there are possibly dynamical mechanisms, which deconfine SU(2) at zero temperature and generate a massive vector boson triplet. Results for a model with joint local U(2) transformations of SU(2) and U(1) vector fields are presented in a limit, which does not involve any unobserved fields.
Proceedings of The XXVII International Symposium on Lattice Field Theory — PoS(LAT2009), 2010
Deconfined regions in relativistic heavy ion collisions are limited to small volumes surrounded b... more Deconfined regions in relativistic heavy ion collisions are limited to small volumes surrounded by a confined exterior. Here the geometry of a double layered torus is discussed, which allows for different temperatures in its two layers. This geometry enables one to approach the QCD continuum limit for small deconfined volumes with confined exteriors in a more realistic fashion than by using periodic boundary conditions. Preliminary data from a study for pure SU(3) lattice gauge theory support a substantial increase in a pseudo transition temperature.
We investigate the approach of pure SU(2) lattice gauge theory with the Wilson action to its cont... more We investigate the approach of pure SU(2) lattice gauge theory with the Wilson action to its continuum limit using the deconfining transition, Lüscher's gradient flow [1], and the cooling flow [2, 3] to set the scale. Of those, the cooling flow turns out to be computationally most efficient. We explore systematic errors due to use of three different energy observables and two distinct reference values for the flow time, the latter obtained by matching initial scaling behavior of some energy observables to that of the deconfining transition. Another important source of systematic errors are distinct fitting forms for the approach to the continuum limit. Besides relying in the conventional way on ratios of masses, we elaborate on a form introduced by Allton [4], which incorporates asymptotic scaling behavior. Ultimately we find that, though still small, our systematic errors are considerably larger than our statistical errors.
It is shown that whenever the multiplicative normalization of a fitting function is not known, le... more It is shown that whenever the multiplicative normalization of a fitting function is not known, least square fitting by χ 2 minimization can be performed with one parameter less than usual by converting the normalization parameter into a function of the remaining parameters and the data. Erratum: Jochen Heitger and Johannes Voss of Münster University found a typo in two subroutines which are used for the 4-parameter fit example given in subsection III D of this paper. In each of the subroutines, subg power2n.f and suby power2n.f, a(2) has to be replaced by a(1) in the expression for the derivative dyda(1). Elimination of this error improves the performance of the fitting method considerable. In the following subsection III D is rewritten to reflect use of the correct subroutines and the error prone routines have been replaced in the package FITM1.tgz, which is still available on the Web as described. Other parts of the paper are not affected.
Metropolis simulations of all-atom models of peptides (i.e. small proteins) are considered. Inspi... more Metropolis simulations of all-atom models of peptides (i.e. small proteins) are considered. Inspired by the funnel picture of Bryngelson and Wolyness, a transformation of the updating probabilities of the dihedral angles is defined, which uses probability densities from a higher temperature to improve the algorithmic performance at a lower temperature. The method is suitable for canonical as well as for generalized ensemble simulations. A simple approximation to the full transformation is tested at room temperature for Met-Enkephalin in vacuum. Integrated autocorrelation times are found to be reduced by factors close to two and a similar improvement due to generalized ensemble methods enters multiplicatively.
We consider 4d compact lattice QED in the quenched approximation. First, we briefly summarize the... more We consider 4d compact lattice QED in the quenched approximation. First, we briefly summarize the spectrum of the staggered Dirac operator and its connection with random matrix theory. Afterwards we present results for the low-lying eigenmodes of the Neuberger overlap-Dirac operator. In the strong coupling phase we find exact zero-modes. Subsequently we discuss possibly related topological excitations of the U(1) lattice gauge theory.
For a free scalar boson field and for U(1) gauge theory finite volume (infrared) and other correc... more For a free scalar boson field and for U(1) gauge theory finite volume (infrared) and other corrections to the energy-momentum dispersion in the lattice regularization are investigated calculating energy eigenstates from the fall off behavior of two-point correlation functions. For small lattices the squared dispersion energy defined by
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Papers by Bernd Berg