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The operator product expansion on the lattice  [PDF]
W. Bietenholz,N. Cundy,M. G?ckeler,R. Horsley,H. Perlt,D. Pleiter,P. E. L. Rakow,C. J. Roberts,G. Schierholz,A. Schiller,J. M. Zanotti
Physics , 2007,
Abstract: We investigate the Operator Product Expansion (OPE) on the lattice by directly measuring the product (where J is the vector current) and comparing it with the expectation values of bilinear operators. This will determine the Wilson coefficients in the OPE from lattice data, and so give an alternative to the conventional methods of renormalising lattice structure function calculations. It could also give us access to higher twist quantities such as the longitudinal structure function F_L = F_2 - 2 x F_1. We use overlap fermions because of their improved chiral properties, which reduces the number of possible operator mixing coefficients.
Operator product expansion and analyticity  [PDF]
Jan Fischer,Ivo Vrkoc
Physics , 1999, DOI: 10.1142/S0217751X9900227X
Abstract: We discuss the current use of the operator product expansion in QCD calculations. Treating the OPE as an expansion in inverse powers of an energy-squared variable (with possible exponential terms added), approximating the vacuum expectation value of the operator product by several terms and assuming a bound on the remainder along the euclidean region, we observe how the bound varies with increasing deflection from the euclidean ray down to the cut (Minkowski region). We argue that the assumption that the remainder is constant for all angles in the cut complex plane is not justified. Making specific assumptions on the properties of the expanded function, we obtain bounds on the remainder in explicit form and show that they are very sensitive both to the deflection angle and to the class of functions chosen. The results obtained are discussed in connetcion with calculations of the coupling constant \alpha_{s} from the \tau decay.
Operator product expansion and confinement  [PDF]
V. Shevchenko,Yu. Simonov
Physics , 2001, DOI: 10.1103/PhysRevD.65.074029
Abstract: Operator product expansion technique is analyzed in abelian and nonabelian nonsupersymmetric field theoretical models with confinement. Special attention is paid to the regimes where nonzero virtuality of vacuum fields is felt by external currents. It is stressed that despite physics of confinement is sometimes considered as being caused by "soft" vacuum gluon fields, it can exhibit pronounced "hard" effects in OPE both in coordinate and momentum spaces.
Operator product expansion algebra  [PDF]
Jan Holland,Stefan Hollands
Mathematics , 2012, DOI: 10.1063/1.4810780
Abstract: We establish conceptually important properties of the operator product expansion (OPE) in the context of perturbative, Euclidean $\varphi^{4}$-quantum field theory. First, we demonstrate, generalizing earlier results and techniques of arXiv:1105.3375, that the 3-point OPE, $< O_{A_1} O_{A_2} O_{A_3} > = \sum_{C} \cal{C}_{A_1 A_2 A_3}^{C} $, usually interpreted only as an asymptotic short distance expansion, actually converges at finite, and even large, distances. We further show that the factorization identity $\cal{C}_{A_1 A_2 A_3}^{B}=\sum_{C}\cal{C}_{A_1 A_2}^{C} \cal{C}_{C A_3}^{B}$ is satisfied for suitable configurations of the spacetime arguments. Again, the infinite sum is shown to be convergent. Our proofs rely on explicit bounds on the remainders of these expansions, obtained using refined versions, mostly due to Kopper et al., of the renormalization group flow equation method. These bounds also establish that each OPE coefficient is a real analytic function in the spacetime arguments for non-coinciding points. Our results hold for arbitrary but finite loop orders. They lend support to proposals for a general axiomatic framework of quantum field theory, based on such `consistency conditions' and akin to vertex operator algebras, wherein the OPE is promoted to the defining structure of the theory.
Associativity of the operator product expansion  [PDF]
Jan Holland,Stefan Hollands
Physics , 2015,
Abstract: We consider a recursive scheme for defining the coefficients in the operator product expansion (OPE) of an arbitrary number of composite operators in the context of perturbative, Euclidean quantum field theory in four dimensions. Our iterative scheme is consistent with previous definitions of OPE coefficients via the flow equation method, or methods based on Feynman diagrams. It allows us to prove that a strong version of the "associativity" condition holds for the OPE to arbitrary orders in perturbation theory. Such a condition was previously proposed in an axiomatic setting in [1] and has interesting conceptual consequences: 1) One can characterise perturbations of quantum field theories abstractly in a sort of "Hochschild-like" cohomology setting, 2) one can prove a "coherence theorem" analogous to that in an ordinary algebra: The OPE coefficients for a product of two composite operators uniquely determine those for $n$ composite operators. We concretely prove our main results for the Euclidean $\varphi^4_4$ quantum field theory, covering also the massless case. Our methods are rather general, however, and would also apply to other, more involved, theories such as Yang-Mills theories.
Operator product expansion at finite temperature  [PDF]
S. Mallik
Physics , 1997, DOI: 10.1016/S0370-2693(97)01335-X
Abstract: We extend an earlier, configuration space method to find the Wilson coefficients of operators appearing in the short distance expansion of thermal correlation functions of different quark bilinears. Considering all the different correlation functions, there arise, up to dimension four, two new operators, in addition to the two appearing already in the vacuum correlation functions. They would contribute substantially to the QCD sum rules, when the temperature is not too low.
3-Cocycles and the Operator Product Expansion  [PDF]
Javier P. Muniain,Jose Wudka
Physics , 1997, DOI: 10.1103/PhysRevD.55.5341
Abstract: Anomalous contributions to the Jacobi identity of chromo-electric fields and non-Abelian vector currents are calculated using a non-perturbative approach that combines operator product expansion and a generalization of Bjorken-Johnson-Low limit. The failure of the Jacobi identity and the associated 3-cocycles are discussed.
On the Operator Product Expansion in Noncommutative Quantum Field Theory  [PDF]
Frederic Zamora
Physics , 2000, DOI: 10.1088/1126-6708/2000/05/002
Abstract: Motivated by the mixing of UV and IR effects, we test the OPE formula in noncommutative field theory. First we look at the renormalization of local composite operators, identifying some of their characteristic IR/UV singularities. Then we find that the product of two fields in general cannot be described by a series expansion of single local operator insertions.
Breakdown of the operator product expansion in the 't Hooft model
Mondejar, Jorge;Pineda, Antonio
High Energy Physics - Phenomenology , 2008, DOI: 10.1103/PhysRevLett.101.152002
Abstract: We consider deep inelastic scattering in the 't Hooft model. Being solvable, this model allows us to directly compute the moments associated with the cross section at next-to-leading order in the 1/Q^2 expansion. We perform the same computation using the operator product expansion. We find that all the terms match in both computations except for one in the hadronic side, which is proportional to a non-local operator. The basics of the result suggest that a similar phenomenon may occur in four dimensions in the large N_c limit.
Breakdown of the operator product expansion in the 't Hooft model  [PDF]
Jorge Mondejar,Antonio Pineda
Physics , 2008, DOI: 10.1103/PhysRevLett.101.152002
Abstract: We consider deep inelastic scattering in the 't Hooft model. Being solvable, this model allows us to directly compute the moments associated with the cross section at next-to-leading order in the 1/Q^2 expansion. We perform the same computation using the operator product expansion. We find that all the terms match in both computations except for one in the hadronic side, which is proportional to a non-local operator. The basics of the result suggest that a similar phenomenon may occur in four dimensions in the large N_c limit.
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