View source for Operator product expansion

==Motivation==

In quantum field theory, for many applications one needs to compute products of some operators: think for example of the current algebra.
Even if [[local operators]] are well defined quantities (at least in perturbative quantum field theory), one knows that their product ''a priori'' depends on their order.

Moreover, in some applications such as high energy limit for total hadronic <math>e^+ e^-</math> cross section, one needs to control the singular behaviour of local operator products when their application points coincide. 

Indeed, let <math>J_\mu (x)</math> be the [[quantum electrodynamics|electromagnetic hadronic current]] (note that Greek letters denote Lorentz indices ranging from 0 to 3, and that we adhere to the convention that repeated Lorentz indices <math>\mu,\,\nu\,...</math>and "isospin" indices <math>i,\,j\,...</math> are summed). Neglecting the electron mass, to the first order in the electromagnetic interaction, the total annihilation  cross section may be written as:
:<math>\label{annihil}
\begin{array}{lcl}
\displaystyle\sigma_{e^+e^- \rightarrow hadrons } &  = &  \frac{e^4}{2(q^2)^3}L^{\mu\nu}\int d^4x \exp(iq.x)<0\mid J_\mu (x)J_\nu (0)\mid 0> \\
& = &  \displaystyle\frac{16\pi^2 \alpha^2}{(q^2)^3}L^{\mu\nu} Im \left[ i \int d^4x \exp(iq.x)<0\mid T[J_\mu (x)J_\nu (0)]\mid 0> \right]
\end{array}</math>


with <math> L^{\mu\nu}=[g^{\mu\nu}p_+.p_- -p_+^\mu p_+^\nu - p_+^\mu p_-^\nu ]</math> and <math>q^\mu = (p_+ + p_-)^\mu = q^0\delta_0^\mu</math> in the center of mass system.

[[Image:Operator Product figureII1.jpg|frame|center|fig1|Illustration of eq. \eqref{annihil}]]   

As illustrated on <figref>Operator_Product_figureII1.jpg</figref>, the high energy behaviour of the annihilation cross section is related to the short distance expression for the vacuum polarisation matrix element:  <math>{\rm lim}_{x \rightarrow 0} <0\mid T[J_\mu (x)J_\nu (0)]\mid 0>\,. </math>  Then one needs an expression for ''the short distance behaviour of the vacuum expectation value of the time-ordered product of two electromagnetic currents'' (see [[#Short distance behaviour of the product of two electromagnetic currents|subsection "Short distance behaviour of the product of two electromagnetic currents"]]).

The aim of the Operator Product Expansion introduced by Wilson (OPE in the following) (Wilson, 1964) is the precise analysis of that limit, at the level of operators, even for ordinary products of local operators. 

Roughly, <math> \mathcal{A}</math> and <math> \mathcal{B}</math> being two local operators, Wilson ansatz was that the bilocal operator <math> \mathcal{A}(x +\chi/2 )\mathcal{B}(x-\chi/2) </math> may be expanded on some basis of local operators <math>\mathcal{C}_i(x)</math> with coefficient functions <math>c_i(\chi, x)</math> that may be singular when <math>\chi \rightarrow 0\,,</math>

:<math>\label{OPE0}
\displaystyle  \mathcal{A}(x +\chi /2)\mathcal{ B}(x-\chi/2)  = \sum_{i=1}^{i=n} c_i(\chi, x)\mathcal{ C}_i(x ) + \mathcal{R}(x, \chi) 
</math>


in such a way that any  Green function with an insertion of: 
:<math> \displaystyle \left[\mathcal{A}(x+\chi/2)\mathcal{B}(x-\chi/2) - \sum_{i=1}^{i=n} c_i(\chi, x)\mathcal{C}_i(x )\right]/c_n(\chi, x)</math> goes to zero with <math>\chi </math> (''weak limit'' of eq.\eqref{OPE0}).

In the translation invariant case <math>c_i(\chi, x)</math> does not depend on <math> x. </math>

Note that the OPE discussed here is a '' short distance expansion '' as <math>\chi \rightarrow 0\,.</math> 
As a matter of facts, in the Minkowski case, most applications (deep inelastic scattering, ...) concern the light-cone operator product expansion when <math>\chi^2 \rightarrow 0.</math>