PHASE STRUCTURE OF THREE- AND FOUR-DIMENSIONAL ϕ4 FIELD THEORY

1992 ◽  
Vol 07 (19) ◽  
pp. 4539-4558 ◽  
Author(s):  
G.V. EFIMOV ◽  
S.N. NEDELKO
Keyword(s):  
Field Theory ◽  
Renormalization Group ◽  
Strong Coupling ◽  
Phase Structure ◽  
Space Time ◽  
Strong Coupling Regime ◽  
Canonical Transformations ◽  
Coupling Constant ◽  
Large Coupling ◽  
Large Coupling Constant

The strong coupling regime of gϕ4 theory in space-time Rd for d=3, 4 is investigated by the methods of canonical transformations and the renormalization group. It is shown that the model describes a system symmetric under transformation ϕ→−ϕ both at small and large coupling constant g. Comparison with the case d=2 shows a crucial influence of the renormalization structure of the theory on its phase structure.

PHASE STRUCTURE OF THE THREE-DIMENSIONAL $g (\vec {\bf \phi}^2)^2$ FIELD THEORY

1992 ◽  
Vol 07 (05) ◽  
pp. 987-1006 ◽  
Author(s):  
G. V. EFIMOV ◽  
S. N. NEDELKO
Keyword(s):  
Phase Structure ◽  
Field Model ◽  
Three Dimensional ◽  
Two Dimensions ◽  
Symmetric System ◽  
Canonical Transformations ◽  
Coupling Constant ◽  
Large Coupling ◽  
Large Coupling Constant ◽  
Invariant Interaction

The phase structure of the three-dimensional superrenormalizable scalar-field model with the O (N)-invariant interaction g(ϕ2)2 is considered. By the method of canonical transformations, it is shown that this model describes the O (N)-symmetric system at both the small and the large coupling constant g. At the same time, phase transitions accompanied by symmetry rearrangement are possible at intermediate values of g. Comparison with the case of two dimensions shows a crucial influence of higher-order renormalization on the phase structure of the model.

scholarly journals Preparation and Spectroscopic Characterization of trans-Fe(CO)3L1L2 (L1, L2 = Phosphine or Phosphite)

1989 ◽  
Vol 44 (6) ◽  
pp. 641-646 ◽  
Author(s):  
Hidenari Inoue ◽  
Takeshi Kuroiwa ◽  
Tsuneo Shirai ◽  
Ekkehard Fluck
Keyword(s):  
Spectroscopic Characterization ◽  
Coupling Constant ◽  
Positive Slope ◽  
Phosphorus Ligands ◽  
Large Coupling ◽  
Iron Carbonyls ◽  
Trigonal Bipyramidal ◽  
Large Coupling Constant ◽  
Back Donation

The 57Fe Mößbauer spectra of mixed ligand complexes of the type trans-Fe(CO)3L1L2 (L1 = triphenylphosphine or triphenylphosphite and L2 = phosphine or phosphite) show a quadrupolesplitting doublet typical of the disubstituted iron carbonyls in trigonal bipyramidal symmetry. The inverse linear dependence of the isomer shifts on the CO stretching frequencies is interpreted on the basis of the strengthening triple-bond nature of the carbonyl ligands with increasing iron-to-phosphorus π-back donation. A linear correlation, with a positive slope, between the isomer shifts and the quadrupole splittings has revealed that the phosphorus-to-iron σ-donation is offset by the iron-to-phosphorus π-back donation. A correlation between the coordination shifts and the isomer shifts demonstrates that the iron-to-phosphorus π-back donation plays an important role in the Fe-P bond. The relatively large coupling constant of 2J(P,P) reflects a strong interaction between trans-phosphorus ligands through the P-Fe-P bond

STRONG-COUPLING REGIME IN $g\phi^4_2$ THEORY

2003 ◽  
Vol 18 (09) ◽  
pp. 1637-1656
Author(s):  
G. V. EFIMOV ◽  
G. GANBOLD
Keyword(s):  
Strong Coupling ◽  
Order Approximation ◽  
Main Idea ◽  
Strong Coupling Regime ◽  
Perturbation Scheme ◽  
Coupling Constant ◽  
Coupling Behavior ◽  
Gaussian Transform ◽  
Bare Coupling Constant ◽  
Higher Order Corrections

The problem of the strong-coupling regime is considered in the scalar superrenormalizable field theory [Formula: see text]. By using the Gaussian transform, we have found an optimal representation within which the exact strong-coupling behavior of the free energy is already obtained in the leading-order approximation. Within this representation, the interaction becomes slower as the bare coupling constant grows, so the higher-order corrections can be systematically estimated by using a modified perturbation scheme. The next-to-leading terms give rise in insignificant corrections for finite coupling. The regularization procedure regroups the initial counterterms so that the divergencies are exactly removed in final expressions. The main idea is demonstrated in the simplest examples of a plain quartic integral and the anharmonic oscillator.

scholarly journals Exact four-point function and OPE for an interacting quantum field theory with space/time anisotropic scale invariance

2021 ◽  
Vol 2021 (10) ◽  
Author(s):  
Hidehiko Shimada ◽  
Hirohiko Shimada
Keyword(s):  
Field Theory ◽  
Quantum Field Theory ◽  
Scale Invariance ◽  
Asymptotic Series ◽  
Space Time ◽  
Quantum Field ◽  
Coupling Constant ◽  
Point Function ◽  
Calogero Model ◽  
Primary Operator

Abstract We identify a nontrivial yet tractable quantum field theory model with space/time anisotropic scale invariance, for which one can exactly compute certain four-point correlation functions and their decompositions via the operator-product expansion(OPE). The model is the Calogero model, non-relativistic particles interacting with a pair potential $$ \frac{g}{{\left|x-y\right|}^2} $$ g x − y 2 in one dimension, considered as a quantum field theory in one space and one time dimension via the second quantisation. This model has the anisotropic scale symmetry with the anisotropy exponent z = 2. The symmetry is also enhanced to the Schrödinger symmetry. The model has one coupling constant g and thus provides an example of a fixed line in the renormalisation group flow of anisotropic theories.We exactly compute a nontrivial four-point function of the fundamental fields of the theory. We decompose the four-point function via OPE in two different ways, thereby explicitly verifying the associativity of OPE for the first time for an interacting quantum field theory with anisotropic scale invariance. From the decompositions, one can read off the OPE coefficients and the scaling dimensions of the operators appearing in the intermediate channels. One of the decompositions is given by a convergent series, and only one primary operator and its descendants appear in the OPE. The scaling dimension of the primary operator we computed depends on the coupling constant. The dimension correctly reproduces the value expected from the well-known spectrum of the Calogero model combined with the so-called state-operator map which is valid for theories with the Schrödinger symmetry. The other decomposition is given by an asymptotic series. The asymptotic series comes with exponentially small correction terms, which also have a natural interpretation in terms of OPE.

scholarly journals Four-pomeron vertex

2021 ◽  
Vol 81 (12) ◽  
Author(s):  
M. A. Braun
Keyword(s):  
Perturbative Qcd ◽  
Variational Approach ◽  
Equations Of Motion ◽  
Iterative Solution ◽  
Infrared Region ◽  
The Other ◽  
Coupling Constant ◽  
Large Coupling ◽  
Large Coupling Constant

AbstractThe four-pomeron vertex is studied in the perturbative QCD. Its dominating terms of the leading (zeroth and first) orders in the coupling constant and subdominant in the number of colors are constructed. The vertex consists of two terms, one with a derivative in rapidity $$\partial _y$$ ∂ y and the other with the BFKL interaction between pomerons. The corresponding part of the action and equations of motion are found. The iterative solution of the latter is possible only for rapidities smaller than 2 and quite large coupling constant $$\alpha _s$$ α s , of the order or greater than unity, when the quadruple pomeron interaction is relatively small. Also iteration of the part with $$\partial _y$$ ∂ y is unstable in the infrared region and compels to introduce an infrared cut. The variational approach with simple trying functions allows to find the minimum of the action at $$\alpha _s$$ α s of the order 0.2 and rapidities up to 25. Numerical estimates for O–O collisions show that actually the influence of the quadruple pomeron interaction turns out to be rather small.

scholarly journals Nonlocal Scalar Quantum Field Theory—Functional Integration, Basis Functions Representation and Strong Coupling Expansion

Particles ◽  
2019 ◽  
Vol 2 (3) ◽  
pp. 385-410 ◽  
Author(s):  
Matthew Bernard ◽  
Vladislav A. Guskov ◽  
Mikhail G. Ivanov ◽  
Alexey E. Kalugin ◽  
Stanislav L. Ogarkov
Keyword(s):  
Field Theory ◽  
Quantum Field Theory ◽  
Strong Coupling ◽  
Functional Integration ◽  
Strong Coupling Expansion ◽  
Basis Functions ◽  
Model Parameters ◽  
Quantum Field ◽  
Primary Field ◽  
Coupling Constant

Nonlocal quantum field theory (QFT) of one-component scalar field φ in D-dimensional Euclidean spacetime is considered. The generating functional (GF) of complete Green functions Z as a functional of external source j, coupling constant g and spatial measure d μ is studied. An expression for GF Z in terms of the abstract integral over the primary field φ is given. An expression for GF Z in terms of integrals over the primary field and separable Hilbert space (HS) is obtained by means of a separable expansion of the free theory inverse propagator L ^ over the separable HS basis. The classification of functional integration measures D φ is formulated, according to which trivial and two nontrivial versions of GF Z are obtained. Nontrivial versions of GF Z are expressed in terms of 1-norm and 0-norm, respectively. In the 1-norm case in terms of the original symbol for the product integral, the definition for the functional integration measure D φ over the primary field is suggested. In the 0-norm case, the definition and the meaning of 0-norm are given in terms of the replica-functional Taylor series. The definition of the 0-norm generator Ψ is suggested. Simple cases of sharp and smooth generators are considered. An alternative derivation of GF Z in terms of 0-norm is also given. All these definitions allow to calculate corresponding functional integrals over φ in quadratures. Expressions for GF Z in terms of integrals over the separable HS, aka the basis functions representation, with new integrands are obtained. For polynomial theories φ 2 n , n = 2 , 3 , 4 , … , and for the nonpolynomial theory sinh 4 φ , integrals over the separable HS in terms of a power series over the inverse coupling constant 1 / g for both norms (1-norm and 0-norm) are calculated. Thus, the strong coupling expansion in all theories considered is given. “Phase transitions” and critical values of model parameters are found numerically. A generalization of the theory to the case of the uncountable integral over HS is formulated—GF Z for an arbitrary QFT and the strong coupling expansion for the theory φ 4 are derived. Finally a comparison of two GFs Z , one on the continuous lattice of functions and one obtained using the Parseval–Plancherel identity, is given.

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