DIMENSIONAL REDUCTION AND THE NON-TRIVIALITY OF λϕ4 IN FOUR DIMENSIONS AT HIGH TEMPERATURE

1993 ◽  
Vol 08 (19) ◽  
pp. 1779-1793 ◽  
Author(s):  
DENJOE O’CONNOR ◽  
C.R. STEPHENS ◽  
F. FREIRE
Keyword(s):  
High Temperature ◽  
Finite Temperature ◽  
Degrees Of Freedom ◽  
Three Dimensional ◽  
Renormalization Group Equation ◽  
Group Equation ◽  
Four Dimensions ◽  
Anomalous Dimensions ◽  
Infinite Temperature ◽  
Second Order Phase Transition

λϕ4 theory in four dimensions is shown perturbatively to have a non-trivial fixed point at finite temperature, the relevant anomalous dimensions at the second order phase transition being the three-dimensional ones. We emphasize the importance of having renormalization schemes and a renormalization group equation that can explicitly take into account the fact that the degrees of freedom of a theory may be qualitatively different at different scales. By applying such considerations to finite temperature λϕ4 where the low temperature degrees of freedom are effectively four-dimensional and the high temperature ones three-dimensional we are able to follow perturbatively the theory from zero to infinite temperature.

scholarly journals EFFECTIVE AVERAGE ACTION IN STATISTICAL PHYSICS AND QUANTUM FIELD THEORY

2001 ◽  
Vol 16 (11) ◽  
pp. 1951-1982 ◽  
Author(s):  
CHRISTOF WETTERICH
Keyword(s):  
Field Theory ◽  
Statistical Physics ◽  
Thermal Fluctuations ◽  
Renormalization Group Equation ◽  
Group Equation ◽  
Quantum Field ◽  
Four Dimensions ◽  
Average Action ◽  
Exact Renormalization Group ◽  
Unified Description

An exact renormalization group equation describes the dependence of the free energy on an infrared cutoff for the quantum or thermal fluctuations. It interpolates between the microphysical laws and the complex macroscopic phenomena. We present a simple unified description of critical phenomena for O(N)-symmetric scalar models in two, three or four dimensions, including essential scaling for the Kosterlitz-Thouless transition.

scholarly journals THE COMPOSITE OPERATOR (CJT) FORMALISM IN THE (λφ4 + ηφ6)D=3 MODEL AT FINITE TEMPERATURE

2000 ◽  
Vol 15 (37) ◽  
pp. 2235-2244 ◽  
Author(s):  
G. N. J. AÑAÑOS ◽  
N. F. SVAITER
Keyword(s):  
High Temperature ◽  
Finite Temperature ◽  
Tricritical Point ◽  
Three Dimensional ◽  
Composite Operator ◽  
Large Set ◽  
Thermal Coupling ◽  
Coupling Constant ◽  
Feynman Graphs ◽  
Cjt Formalism

We discuss the three-dimensional λφ4+ηφ6 theory in the context of the 1/N expansion at finite temperature. We use the method of the composite operator (CJT) for summing a large set of Feynman graphs. We analyze the behavior of the thermal square mass and the thermal coupling constant in the low and high temperature limits. The existence of the tricritical point at some temperature is found using this method.

RG IMPROVEMENT OF THE EFFECTIVE POTENTIAL AT FINITE TEMPERATURE

1996 ◽  
Vol 11 (28) ◽  
pp. 2259-2269 ◽  
Author(s):  
HISAO NAKKAGAWA ◽  
HIROSHI YOKOTA
Keyword(s):  
Renormalization Group ◽  
Effective Potential ◽  
Finite Temperature ◽  
Renormalization Group Equation ◽  
Effective Procedure ◽  
Coefficient Function ◽  
Group Equation ◽  
Leading Order ◽  
Effective Potentials ◽  
The One

We present a simple and effective procedure to improve the finite temperature effective potential so as to satisfy the renormalization group equation (RGE). With the L-loop knowledge of the effective potential and of the RGE coefficient function, this procedure carries out a systematic resummation of large-T as well as large-log terms up to the Lth-to-leading order, giving an improved effective potential which satisfies the RGE and is exact up to the Lth-to-leading T and log terms. Applications to the one- and two-loop effective potentials are explicitly performed.

scholarly journals Incidences with Curves in $\mathbb{R}^d$

10.37236/5840 ◽  
2016 ◽  
Vol 23 (4) ◽  
Author(s):  
Micha Sharir ◽  
Adam Sheffer ◽  
Noam Solomon
Keyword(s):  
Recent Result ◽  
Degrees Of Freedom ◽  
Algebraic Curves ◽  
Three Dimensional ◽  
Three Dimensions ◽  
Bounded Degree ◽  
Four Dimensions ◽  
Mild Conditions ◽  
Point Line ◽  
Special Case

We prove that the number of incidences between $m$ points and $n$ bounded-degree curves with $k$ degrees of freedom in ${\mathbb R}^d$ is\[ O\left(m^{\frac{k}{dk-d+1}+\varepsilon}n^{\frac{dk-d}{dk-d+1}}+ \sum_{j=2}^{d-1} m^{\frac{k}{jk-j+1}+\varepsilon}n^{\frac{d(j-1)(k-1)}{(d-1)(jk-j+1)}} q_j^{\frac{(d-j)(k-1)}{(d-1)(jk-j+1)}}+m+n\right),\]for any $\varepsilon>0$, where the constant of proportionality depends on $k, \varepsilon$ and $d$, provided that no $j$-dimensional surface of degree $\le c_j(k,d,\varepsilon)$, a constant parameter depending on $k$, $d$, $j$, and $\varepsilon$, contains more than $q_j$ input curves, and that the $q_j$'s satisfy certain mild conditions. This bound generalizes the well-known planar incidence bound of Pach and Sharir to $\mathbb{R}^d$. It generalizes a recent result of Sharir and Solomon concerning point-line incidences in four dimensions (where d=4 and k=2), and partly generalizes a recent result of Guth (as well as the earlier bound of Guth and Katz) in three dimensions (Guth's three-dimensional bound has a better dependency on $q_2$). It also improves a recent d-dimensional general incidence bound by Fox, Pach, Sheffer, Suk, and Zahl, in the special case of incidences with algebraic curves. Our results are also related to recent works by Dvir and Gopi and by Hablicsek and Scherr concerning rich lines in high-dimensional spaces. Our bound is not known to be tight in most cases.

Evolution of Coupling Constant in Hot QCD

1997 ◽  
Vol 06 (01) ◽  
pp. 45-64
Author(s):  
M. Chaichian ◽  
M. Hayashi
Keyword(s):  
Finite Temperature ◽  
Lorentz Invariance ◽  
Background Field ◽  
Field Method ◽  
Renormalization Group Equation ◽  
Group Equation ◽  
Coupling Constant ◽  
Time Formalism ◽  
Background Field Method ◽  
The One

The evolution of QCD coupling constant at finite temperature is considered by making use of the finite temperature renormalization group equation up to the one-loop order in the background field method with the Feynman gauge and the imaginary time formalism. The results are compared with the ones obtained in the literature. We point out, in particular, the origin of the discrepancies between different calculations, such as the choice of gauge, the breakdown of Lorentz invariance, imaginary versus real time formalism and the applicability of the Ward identities at finite temperature.

The finite temperature renormalization group equation in λφ4 theory

1986 ◽  
Vol 167 (4) ◽  
pp. 406-410 ◽  
Author(s):  
Yasushi Fujimoto ◽  
Kazuo Ideura ◽  
Yoshimasa Nakano ◽  
Hiroshi Yoneyama
Keyword(s):  
Renormalization Group ◽  
Finite Temperature ◽  
Renormalization Group Equation ◽  
Group Equation

G-theory: The generator of M-theory and supersymmetry

2018 ◽  
Vol 33 (10n11) ◽  
pp. 1850058
Author(s):  
Alireza Sepehri ◽  
Richard Pincak
Keyword(s):  
Degrees Of Freedom ◽  
Three Dimensional ◽  
Negative Energy ◽  
Gauge Fields ◽  
Tensor Fields ◽  
Physical Reason ◽  
Four Dimensions ◽  
M Theory ◽  
Nonlinear Gravity

In string theory with ten dimensions, all Dp-branes are constructed from D0-branes whose action has two-dimensional brackets of Lie 2-algebra. Also, in M-theory, with 11 dimensions, all Mp-branes are built from M0-branes whose action contains three-dimensional brackets of Lie 3-algebra. In these theories, the reason for difference between bosons and fermions is unclear and especially in M-theory there is not any stable object like stable M3-branes on which our universe would be formed on it and for this reason it cannot help us to explain cosmological events. For this reason, we construct G-theory with M dimensions whose branes are formed from G0-branes with N-dimensional brackets. In this theory, we assume that at the beginning there is nothing. Then, two energies, which differ in their signs only, emerge and produce 2M degrees of freedom. Each two degrees of freedom create a new dimension and then M dimensions emerge. M-N of these degrees of freedom are removed by symmetrically compacting half of M-N dimensions to produce Lie-N-algebra. In fact, each dimension produces a degree of freedom. Consequently, by compacting M-N dimensions from M dimensions, N dimensions and N degrees of freedom is emerged. These N degrees of freedoms produce Lie-N-algebra. During this compactification, some dimensions take extra i and are different from other dimensions, which are known as time coordinates. By this compactification, two types of branes, Gp and anti-Gp-branes, are produced and rank of tensor fields which live on them changes from zero to dimension of brane. The number of time coordinates, which are produced by negative energy in anti-Gp-branes, is more sensible to number of times in Gp-branes. These branes are compactified anti-symmetrically and then fermionic superpartners of bosonic fields emerge and supersymmetry is born. Some of gauge fields play the role of graviton and gravitino and produce the supergravity. The question may arise that what is the physical reason which shows that this theory is true. We shown that G-theory can be reduced to other theories like nonlinear gravity theories in four dimensions. Also, this theory, can explain the physical properties of fermions and bosons. On the other hand, this theory explains the origin of supersymmetry. For this reason, we can prove that this theory is true. By reducing the dimension of algebra to three and dimension of world to 11 and dimension of brane to four, G-theory is reduced to F(R)-gravity.

scholarly journals EVOLUTION OF TRANSVERSE-DISTANCE DEPENDENT PARTON DENSITIES AT LARGE-XB AND GEOMETRY OF THE LOOP SPACE

2014 ◽  
Vol 25 ◽  
pp. 1460006 ◽  
Author(s):  
IGOR O. CHEREDNIKOV ◽  
TOM MERTENS ◽  
PIETER TAELS ◽  
FREDERIK F. VAN DER VEKEN
Keyword(s):  
Loop Space ◽  
Wilson Loop ◽  
Equations Of Motion ◽  
Three Dimensional ◽  
Renormalization Group Equation ◽  
Distribution Functions ◽  
Group Equation ◽  
Parton Distribution Functions ◽  
Factorization Scheme ◽  
Transverse Distance

We discuss possible applications of the equations of motion in the generalized Wilson loop space to the phenomenology of the three-dimensional parton distribution functions in the large-xB approximation. This regime is relevant for future experimental programs to be launched at the (approved) Jefferson Lab 12 GeV upgrade and the (planned) Electron-Ion Collider. We show that the geometrical evolution of the Wilson loops corresponds to the combined rapidity and renormalization-group equation of the transverse-distance dependent parton densities in the large-xB factorization scheme.

scholarly journals MASSLESS THREE-DIMENSIONAL QUANTUM ELECTRODYNAMICS AND THIRRING MODEL CONSTRAINED BY LARGE FLAVOR NUMBER

2006 ◽  
Vol 21 (18) ◽  
pp. 1451-1462
Author(s):  
A. R. FAZIO
Keyword(s):  
Finite Temperature ◽  
Quantum Electrodynamics ◽  
Degrees Of Freedom ◽  
Three Dimensional ◽  
Thirring Model ◽  
Asymptotic Freedom ◽  
Coupling Constant ◽  
Running Coupling ◽  
Running Coupling Constant ◽  
Massless Quantum

We explicitly prove that in three-dimensional massless quantum electrodynamics at finite temperature, zero density and large number of flavors, the number of infrared degrees of freedom is never larger than the corresponding number of ultraviolet. Such a result, strongly dependent on the asymptotic freedom of the theory, is reversed in three-dimensional Thirring model due to the positive derivative of its running coupling constant.

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