scholarly journals Generating a mass gap using Feynman diagrams in an asymptotically free theory

2018 ◽  
Vol 175 ◽  
pp. 11010 ◽  
Author(s):  
Venkitesh Ayyar ◽  
Shailesh Chandrasekharan
Keyword(s):  
Field Theory ◽  
Monte Carlo ◽  
Partition Function ◽  
Perturbation Series ◽  
Feynman Diagrams ◽  
Two Dimensional ◽  
Mass Gap ◽  
Free Theory ◽  
Lattice Field ◽  
Massless Fermions

Using the example of a two dimensional four-fermion lattice field theory, we show that Feynman diagrams can generate a mass gap in a theory with massless fermions that interact via a marginally relevant coupling. We show this by introducing an infrared cutoff that makes the perturbation series for the partition function convergent. We use a Monte Carlo approach to sample sufficiently high orders of diagrams and thus expose the presence of the mass gap.

scholarly journals Interface Roughening in Two Dimensions

2021 ◽  
Vol 182 (3) ◽  
Author(s):  
Gernot Münster ◽  
Manuel Cañizares Guerrero
Keyword(s):  
Field Theory ◽  
Monte Carlo ◽  
Capillary Wave ◽  
System Size ◽  
Two Dimensions ◽  
Two Dimensional ◽  
Wave Approximation ◽  
Statistical Field ◽  
Statistical Field Theory ◽  
Interface Roughening

AbstractRoughening of interfaces implies the divergence of the interface width w with the system size L. For two-dimensional systems the divergence of $$w^2$$ w 2 is linear in L. In the framework of a detailed capillary wave approximation and of statistical field theory we derive an expression for the asymptotic behaviour of $$w^2$$ w 2 , which differs from results in the literature. It is confirmed by Monte Carlo simulations.

A COMPACT BOSON COUPLED TO TWO-DIMENSIONAL GRAVITY

1991 ◽  
Vol 06 (15) ◽  
pp. 2743-2754 ◽  
Author(s):  
NORISUKE SAKAI ◽  
YOSHIAKI TANII
Keyword(s):  
Field Theory ◽  
Partition Function ◽  
Quantum Gravity ◽  
Matrix Models ◽  
Riemann Surfaces ◽  
Partition Functions ◽  
Two Dimensional ◽  
Dimensional Gravity ◽  
The Continuum ◽  
Continuum Field

The radius dependence of partition functions is explicitly evaluated in the continuum field theory of a compactified boson, interacting with two-dimensional quantum gravity (noncritical string) on Riemann surfaces for the first few genera. The partition function for the torus is found to be a sum of terms proportional to R and 1/R. This is in agreement with the result of a discretized version (matrix models), but is quite different from the critical string. The supersymmetric case is also explicitly evaluated.

scholarly journals ON BORWEIN’S CONJECTURES FOR PLANAR UNIFORM RANDOM WALKS

2019 ◽  
Vol 107 (3) ◽  
pp. 392-411 ◽  
Author(s):  
YAJUN ZHOU
Keyword(s):  
Field Theory ◽  
Quantum Field Theory ◽  
Random Walks ◽  
Euclidean Plane ◽  
Feynman Diagrams ◽  
Quantum Field ◽  
Two Dimensional ◽  
Third Order ◽  
The Third ◽  
Uniformly Convergent

Let $p_{n}(x)=\int _{0}^{\infty }J_{0}(xt)[J_{0}(t)]^{n}xt\,dt$ be Kluyver’s probability density for $n$-step uniform random walks in the Euclidean plane. Through connection to a similar problem in two-dimensional quantum field theory, we evaluate the third-order derivative $p_{5}^{\prime \prime \prime }(0^{+})$ in closed form, thereby giving a new proof for a conjecture of J. M. Borwein. By further analogies to Feynman diagrams in quantum field theory, we demonstrate that $p_{n}(x),0\leq x\leq 1$ admits a uniformly convergent Maclaurin expansion for all odd integers $n\geq 5$, thus settling another conjecture of Borwein.

ON A PHASE STRUCTURE OF A TWO-DIMENSIONAL (φ2)2 FIELD THEORY

1989 ◽  
Vol 04 (18) ◽  
pp. 4977-4990 ◽  
Author(s):  
G. V. EFIMOV
Keyword(s):  
Field Theory ◽  
Phase Transitions ◽  
Phase Structure ◽  
Weak Coupling ◽  
Scalar Fields ◽  
Perturbation Series ◽  
Coupling Constants ◽  
Two Dimensional ◽  
Effective Coupling ◽  
Two Phases

Two models of scalar fields with the interaction Lagrangians gφ4 and [Formula: see text] are considered in ℝ2. There are phase transitions in these models for a certain g = gc. It is shown that the spontaneous symmetry breaking takes place for g > gc. The description of the two phases for g < gc and g > gc is given. The effective coupling constants in perturbation series are less than unity for both the phases so that these models describe the systems with weak coupling. In the second model the "Goldstone" particles have nonzero masses in the phase g > gc.

scholarly journals RG-inspired machine learning for lattice field theory

2018 ◽  
Vol 175 ◽  
pp. 11025 ◽  
Author(s):  
Sam Foreman ◽  
Joel Giedt ◽  
Yannick Meurice ◽  
Judah Unmuth-Yockey
Keyword(s):  
Field Theory ◽  
Machine Learning ◽  
Monte Carlo ◽  
Renormalization Group ◽  
Coarse Graining ◽  
Large Datasets ◽  
Logarithmic Divergence ◽  
Lattice Field ◽  
Components Analysis ◽  
The Relationship

Machine learning has been a fast growing field of research in several areas dealing with large datasets. We report recent attempts to use renormalization group (RG) ideas in the context of machine learning. We examine coarse graining procedures for perceptron models designed to identify the digits of the MNIST data. We discuss the correspondence between principal components analysis (PCA) and RG flows across the transition for worm configurations of the 2D Ising model. Preliminary results regarding the logarithmic divergence of the leading PCA eigenvalue were presented at the conference. More generally, we discuss the relationship between PCA and observables in Monte Carlo simulations and the possibility of reducing the number of learning parameters in supervised learning based on RG inspired hierarchical ansatzes.

CONVERGENT WEAK COUPLING EXPANSIONS FOR LATTICE FIELD THEORIES THAT LOOK LIKE PERTURBATION SERIES

1989 ◽  
Vol 01 (01) ◽  
pp. 47-87 ◽  
Author(s):  
GERHARD MACK ◽  
ANDREAS PORDT
Keyword(s):  
Field Theory ◽  
Perturbation Theory ◽  
Weak Coupling ◽  
Free Field ◽  
Perturbation Series ◽  
Field Theories ◽  
Lattice Field Theory ◽  
Lattice Field ◽  
Local Coupling ◽  
New Form

We propose a new form of convergent weak coupling expansion for the lattice field theory. It has the advantage that it is very similar to standard (Feynman) perturbation theory. Convergence is proven for sufficiently weak local coupling, i.e. when the theory is close to a free field theory. In the proof, use of analyticity in field variables, as pioneered by Kupiainen and Gawedzki, is supplemented with techniques for handling derivatives with respect to free propagators.

scholarly journals HIGH PRECISION SIMULATION TECHNIQUES FOR LATTICE FIELD THEORY

1993 ◽  
Vol 04 (02) ◽  
pp. 451-458 ◽  
Author(s):  
Ulli Wolff
Keyword(s):  
Field Theory ◽  
Monte Carlo ◽  
High Precision ◽  
Critical Slowing Down ◽  
Field Theories ◽  
Cluster Algorithms ◽  
Slowing Down ◽  
Monte Carlo Algorithms ◽  
Simulation Techniques ◽  
Lattice Field

An overview is given over the recently developed and now widely used Monte Carlo algorithms with reduced or eliminated critical slowing down. The basic techniques are overrelaxation, cluster algorithms and multigrid methods. With these tools one is able to probe much closer than before the universal continuum behavior of field theories on the lattice. This is demonstrated by reviewing some applications.

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