354,671 research outputs found
Operator product expansion algebra
We establish conceptually important properties of the operator product
expansion (OPE) in the context of perturbative, Euclidean -quantum
field theory. First, we demonstrate, generalizing earlier results and
techniques of arXiv:1105.3375, that the 3-point OPE, , usually interpreted only as an
asymptotic short distance expansion, actually converges at finite, and even
large, distances. We further show that the factorization identity 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.Comment: 53 pages, v2: typos removed, minor corrections, v3: typos removed,
minor changes in introduction, v4: Note added in proo
Operator product expansion and analyticity
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.Comment: Preprint PRA-HEP 99/04, 20 page
The Lorentz Anomaly via Operator Product Expansion
The emergence of a critical dimension is one of the most striking features of
string theory. One way to obtain it is by demanding closure of the Lorentz
algebra in the light-cone gauge quantisation, as discovered for bosonic strings
more than fourty years ago. We give a detailed derivation of this classical
result based on the operator product expansion on the Lorentzian world-sheet
The operator product expansion on the lattice
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.Comment: 7 pages, 4 postscript figures. Contribution to Lattice 2007,
Regensbur
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