Operator product expansion

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Guy Bonneau (2009), Scholarpedia, 4(9):8506. doi:10.4249/scholarpedia.8506 revision #91610 [link to/cite this article]
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Operator product expansion (OPE) refers to the expression of products of composite operators (known also as local operators) at short distance. Indeed, in quantum field theory these products are singular when their space-time point coincide. After an analysis in the free field case (using normal order and the Wick theorem), with examples taken from \(\Phi^4\) theory and QCD (its electromagnetic current), the fully interacting case is discussed. The Wilson-Zimmermann short distance operator product expansion is presented and some hints are given on its understanding, with particular emphasis on power counting. As an application, the high energy behaviour of the total hadronic annihilation cross section - the famous \(R\) ratio - is analysed.

Contents

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 \(e^+ e^-\) cross section, one needs to control the singular behaviour of local operator products when their application points coincide.

Indeed, let \(J_\mu (x)\) be the 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 \(\mu,\,\nu\,...\)and "isospin" indices \(i,\,j\,...\) are summed). Neglecting the electron mass, to the first order in the electromagnetic interaction, the total annihilation cross section may be written as: \[\tag{1} \begin{array}{lcl} \displaystyle\sigma_{e^+e^- \rightarrow hadrons } & = & \frac{e^4}{2(q^2)^3}L^{\mu\nu}\int d^4x \exp(iq.x)<0\mid J_\mu (x)J_\nu (0)\mid 0> \\ & = & \displaystyle\frac{16\pi^2 \alpha^2}{(q^2)^3}L^{\mu\nu} Im \left[ i \int d^4x \exp(iq.x)<0\mid T[J_\mu (x)J_\nu (0)]\mid 0> \right] \end{array}\]


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

Figure 1: Illustration of eq. (1)

As illustrated on Figure 1, the high energy behaviour of the annihilation cross section is related to the short distance expression for the vacuum polarisation matrix element\[{\rm lim}_{x \rightarrow 0} <0\mid T[J_\mu (x)J_\nu (0)]\mid 0>\,. \] 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 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, \( \mathcal{A}\) and \( \mathcal{B}\) being two local operators, Wilson ansatz was that the bilocal operator \( \mathcal{A}(x +\chi/2 )\mathcal{B}(x-\chi/2) \) may be expanded on some basis of local operators \(\mathcal{C}_i(x)\) with coefficient functions \(c_i(\chi, x)\) that may be singular when \(\chi \rightarrow 0\,,\)

\[\tag{2} \displaystyle \mathcal{A}(x +\chi /2)\mathcal{ B}(x-\chi/2) = \sum_{i=1}^{i=n} c_i(\chi, x)\mathcal{ C}_i(x ) + \mathcal{R}(x, \chi) \]


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

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

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

The free field case

To simplify the notations, most of the time the definitions and properties will be given for spinless fields. The generalization to spinors, vectors,... is straightforward, but one should not forget to replace commutation relations by anticommutation ones (etc ...) when going from bosonic to fermionic fields.

The Wick theorem and the free field Operator Product Expansion

Let \(\Phi^i(x)\) be the set of local operators describing the free multiplet of fields \(\phi^i (x),\ i= 1,\ 2... \ N\) and \(\mid \phi ^i >\) the corresponding states in the Fock space.

A local operator \(\mathcal{ O}(x)\) will be an arbitrary polynomial in the fields and their space-time derivatives (Zimmermann W., 1970): \[\tag{3} \begin{array}{rcl} :\mathcal{ O}_{\{\mu\}}\{\Phi^i(x)\}: & = & :\Phi_{(\mu)_1}^{i_1}(x)...\Phi_{(\mu)_m}^{i_m}(x): ,\\ \Phi_{(\mu)_k}^{i_k}(x) & = & \partial_{(\mu)_k}\Phi^{i_k} ,\\ (\mu)_k & = & (\mu_{k_1},..., \mu_{k_{p(k)}}),\ \ \ p(k) \geq 0 ,\\ \partial_{(\mu)_k} & = & \partial_{\mu_{k_1}}...\partial_{\mu_{k_p}} ,\ \ \partial_{(\mu)_k} = \mathbf{1}\ \mbox{if}\ p(k) = 0 ,\\ \{\mu\} & = & ((\mu)_1...(\mu)_p), \\ \end{array} \]

where the colons :: indicate the normal ordering prescription (required for the existence of the local product of fields in (3)).

In that free field case, the operator product expansion simply results from the Wick theorem which writes:

\[ \begin{array}{rcl} \displaystyle :\mathcal{O}_{\{\mu\}}\{\Phi^a(x)\}:\ :\mathcal{O}_{\{\nu\}}\{\Phi^b(y)\}: & = & :\mathcal{O}_{\{\mu\}}\{\Phi^a(x)\}\mathcal{O}_{\{\nu\}}\{\Phi^b(y)\}:\ + \\ & + & \displaystyle \sum_{1-contraction\ [(x,i)(y,j)]} \Delta_+^{ij}(x-y) :\mathcal{O}_{\{\mu\}}\{\Phi^a(x)\}\mathcal{O}_{\{\nu\}}\{\Phi^b(y)\}: + \\ & + & \displaystyle \sum_{2-contractions\ [(x,i)(y,j)],[(x,i')(y,j')]}\Delta_+^{ij}(x-y) \Delta_+^{i'j'}(x-y) :\mathcal{O}_{\{\mu\}}\{\Phi^a(x)\}\mathcal{O}_{\{\nu\}}\{\Phi^b(y)\}: +\\ & + & ...\\ \end{array} \]

where:

  • the "contraction \([(x,i)(y,j)]\)" means the suppression of the fields \(\Phi^i(x)\) and \(\Phi^j(y)\) in the chain \( \mathcal{ O}_{\{\mu\}}\{\Phi^a(x)\}\mathcal{ O}_{\{\nu\}}\{\Phi^b(y)\}\,,\)
  • the bilocal normal ordered operators \( :\mathcal{ A}(x)\mathcal{ B}(y):\) are well defined when \(y \rightarrow x\,.\) As a consequence, they can be Taylor expanded in \(y\) around \(x\,,\) so giving the looked-for expansion on local operators \(:\mathcal{ C}_i(x):\,,\)
  • \( \Delta_\pm^{ij}(x-y) = \delta^{ij}\int d\mathbf{ k}[e^{\mp ik.(x-y)}]\) diverges when \( y \rightarrow x ,\ d\mathbf{ k}\) being the Lorentz invariant measure \(\displaystyle = \frac{1}{(2\pi)^3} \frac{d^3\mathbf{ k}}{2\omega_k}\,, \ k_0 = \omega_k = \sqrt{\mathbf{ k}^2 + m^2}\,.\) For further use, note that \(\Delta_\pm^{ij}(x-y)\) may be rewritten as\[\displaystyle \delta^{ij}\int \frac{d^4k}{(2\pi)^4}\theta(k_0)2\pi \delta(k^2-m^2) e^{\mp ik.(x-y)}\] and, in the limit of a vanishing mass, one gets:
\[\tag{4} \Delta_\pm^{ij}(x-y) \simeq \frac{-\delta^{ij}}{4\pi^2}\frac{1}{(x-y)^2 \mp i\epsilon (x^0-y^0)}\,,\]
  • the Wick theorem is usually written for time-ordered products where the \(\Delta_+^{ij}(x-y)\) are replaced by the Feynman propagator \(\displaystyle i\Delta_F^{ij}(x-y) = \delta^{ij}\int \frac{d^4k}{(2\pi)^4}\frac{e^{ -ik.(x-y)}}{k^2-m^2 + i\epsilon}\ .\) Note that, in the limit of a vanishing mass one gets:
\[\tag{5} i\Delta_F^{ij}(x-y) \simeq \frac{-\delta^{ij}}{4\pi^2}\frac{1}{(x-y)^2 - i\epsilon }\,.\]


Examples for a spin zero multiplet \(\Phi^i(x)\)

Let us illustrate this on the simplest case \[\Phi^i(x+\chi/2) \Phi^j(x-\chi/2) = :\Phi^i(x+\chi/2) \Phi^j(x-\chi/2): +\ \Delta_+^{ij}(\chi) \mathbf{ 1} \]

What would the limit of the left hand side be when \(\chi \rightarrow 0 \ ?\) Let us make some preliminary remarks:

  • here and in the sequel, when speaking of the limit of a local operator \(\mathcal{ O}(y)\) when \(y \rightarrow x,\) this should be understood in the weak sense, i.e. as limits of all Green functions with an insertion of the local operator and any number of external fields ;
  • a difficulty immediately appears: the above mentioned limit a priori depends on the direction of \(\chi.\) In the sequel, besides a scaling positive real parameter \(\rho \,,\) a 4-vector \(\eta^\mu\) is introduced\[\chi^{\mu} = \rho\eta^{\mu} .\]

In that weak sense, the normal ordered product \(:\Phi^i(x+\chi/2) \Phi^j(x-\chi/2):\) is continuous and infinitely differentiable in \(\chi\) in a vicinity of \(0\,;\) then: \[\tag{6} \Phi^i(x+\chi/2) \Phi^j(x-\chi/2) = \Delta_+^{ij}(\chi)\mathbf{1}\ + :\Phi^i(x) \Phi^j(x):\ + \chi^\mu :[\partial_\mu\Phi^i(x)\Phi^j(x)-\Phi^i(x)\partial_\mu\Phi^j(x)]/2: + R^{ij}(x, \rho, \eta ) . \]


A trivial, but important, remark is the increasing canonical dimension of the local operators that appear in the expansion and, simultaneously, the less and less singular character of their coefficient function in the limit \(\chi\rightarrow 0\,.\)

For further use, one can give another illustration of this short distance expansion on a particular (now a time-ordered) Green function (still for free fields), the external legs being amputated (\({\tilde\Phi}^a\) represents the Fourier transform of the field \(\Phi^a(x) \)).

\[\tag{7} \begin{array}{rcl} <0\mid T[ \Phi^i(x+\chi/2) \Phi^j(x-\chi/2) \tilde{\Phi}^a(p_a) \tilde{\Phi}^b(p_b) ]\mid 0>\mid^{amputated,\,Free} & = & i\Delta_F(\chi)\delta^{ij}i\delta^{ab}(p_a^2 - m^2 )\delta^4(p_a- p_b ) + \\ + \left[ \delta^{ia} e^{ip_a.(x+\chi/2)}\delta^{jb} e^{ip_b.(x-\chi/2)} + [(a, p_a) \leftrightarrow (b, p_b)]\right] & = & - \Delta_F(\chi)\delta^{ij}\delta^{ab}(p_a^2 - m^2 )\delta^4(p_a- p_b ) +\\ + (\delta^{ia} \delta^{jb} +\delta^{ja}\delta^{ib})e^{i(p_a+ p_b).x} & + & i \rho\eta^\mu e^{i(p_a+ p_b).x} (p_a - p_b)_\mu (\delta^{ia}\delta^{jb}-\delta^{ja}\delta^{ib})/2 + \mathcal{ O}(\rho^3) . \end{array}\]

This gives an expansion similar to the one in eq. (6): \[\tag{8} T\left[\Phi^i(x+\chi/2) \Phi^j(x-\chi/2)\right]\ =\ i\Delta_F^{ij}(\chi)\mathbf{1} +\ :\Phi^i(x) \Phi^j(x):\ + \chi^\mu :[\partial_\mu\Phi^i(x)\Phi^j(x)-\Phi^i(x)\partial_\mu\Phi^j(x)]/2: + R'^{ij}(x, \rho, \eta ). \]

Note the change in the singular behaviour of the OPE between eqns. (6) and (8) due to the specific, inherent singularity of a time ordered product.

Thanks to the Wick theorem, this immediately generalizes to the product of two local operators \(\mathcal{ O}(x).\) Consider for instance:

\[\tag{9} :\Phi^2(x+\chi/2): :\Phi^2(x-\chi/2):\ =\ :\Phi^2(x+\chi/2)\Phi^2(x-\chi/2):\ +\ 4 \Delta_+^{ij}(\chi) :\Phi^i(x+\chi/2)\Phi^j(x-\chi/2): +\ 2 \Delta_+^{ij}(\chi)\Delta_+^{ij}(\chi)\mathbf{1} \]

and here also use Taylor expansion inside the normal products \(:\Phi^i(x+\chi/2)\Phi^j(x-\chi/2): \) and \( :\Phi^2(x+\chi/2)\Phi^2(x-\chi/2):\) to get:

\[\tag{10} \begin{array}{rcl} :\Phi^2(x+\chi/2): :\Phi^2(x-\chi/2): & = & :[ \Phi^2(x)]^2: +\ 2 \Delta_+^{ij}(\rho\eta)\Delta_+^{ij}(\rho\eta)\mathbf{1} +\ 4 \Delta_+^{ij}(\rho\eta) :\Phi^i(x)\Phi^j(x): +\\ & + & 2 [\rho\eta^\mu \Delta_+^{ij}(\rho\eta)] :[(\partial_\mu\Phi^i(x))\Phi^j(x) - \Phi^i(x)(\partial_\mu\Phi^j(x))]: + \\ & + & \frac{1}{2}[\rho^2\eta^\mu \eta^\nu \Delta_+^{ij}(\rho\eta)] :[(\partial_\mu\partial_\nu\Phi^i(x))\Phi^j(x) + \Phi^i(x) (\partial_\mu\partial_\nu\Phi^j(x)) - 2 \partial_\mu\Phi^i(x)\partial_\nu\Phi^j(x)]:\ + \\ & + & R(x, \rho, \eta ) \\ \end{array}\]


where (see eq (Figure 1)) all coefficient functions are singular in the limit \(\rho \rightarrow 0\,,\) and the rest \(R(x, \rho, \eta )\) vanishes in that limit.

Short distance behaviour of the product of two electromagnetic currents

As another example, let us consider the value of the hadronic electromagnetic tensor, still in a free field approximation of Quantum Chromodynamics\[ <0\mid [J_\mu (x)J_\nu (y)]\mid 0> \] with \(\displaystyle J_\mu(x) = \sum_{quarks\ i}[e_i\bar{q}^i(x)\gamma_\mu q^i(x)] .\) Taking into account the vanishing of the vacuum expectation value of any normal ordered operator, this dimension 6 quantity may be expressed (in the limit of vanishing quark masses) as (adapting eq. (9) to this fermionic case)\[\tag{11} \begin{array}{rcl} <0\mid J_\mu (x)J_\nu (y) \mid 0> \mid^{Free}& \simeq & -\displaystyle\sum_{i,j} e_i e_j Tr [\gamma_\mu (i\gamma_\rho\partial^\rho_x\Delta_+^{ij}(x-y))\gamma_\nu (i \gamma_\sigma\partial_y^\sigma\Delta_-^{ji}(y-x)) ]\\ & \simeq & - \displaystyle 4[\sum_i e_i^2 ][2g_{\mu\rho}g_{\nu\sigma} -g_{\mu\nu}g_{\rho\sigma}] (\frac{-1}{4\pi^2})^2 \partial^\rho_x\left[\frac{1}{(x-y)^2 -i\epsilon(x^0-y^0)}\right] \partial^\sigma_y\left[\frac{1}{(x-y)^2 +i\epsilon(y^0-x^0)}\right]\\ & = & \displaystyle \left(\sum_i\frac{e_i^2}{12\pi^4}\right) [\partial_\mu\partial_\nu - g_{\mu\nu} \Box][\frac{1}{\chi^2 - i\epsilon \chi^0}]^2 \ \ {\rm where}\ \chi = x-y\,.\\ \end{array}\]


Note that gauge invariance is recovered.

The Fourier transform of the distribution \([\frac{1}{\chi^2 - i\epsilon \chi^0}]^2\) being given by \( 2\pi^3\theta(q_0)\theta(q^2)\) (where \( \theta\) is the step function), the hadronic cross section (1) is readily found\[\displaystyle \sigma_{e^+e^- \rightarrow hadrons } \simeq \frac{4\pi\alpha^2}{3q^2}\sum_{i} e_i^2 \,.\]

To summarize, in that free case, the product of the currents is expressed by a dimension 6 singularity times the identity operator and the ratio \(R_0\) of the hadronic to muonic annihilation cross sections in the limit of vanishing fermion masses is readily obtained: \[ \displaystyle \frac{\sigma_{e^+e^- \rightarrow hadrons }}{\sigma_{e^+e^- \rightarrow \mu^+ \mu^-}} = R_{0} = \sum_{quarks\ i} e_i^2\,.\]

In the next section, the non-trivial OPE will be used to discuss the modifications of this free-field approximation due to interactions.

The interacting case in quantum field theory

As soon as fields are interacting ones, everything becomes more complicated: for example, in the Feynman graph approach, loops come into the game and at the same moment the need for regularization and renormalization, with definite rules to ensure all-loop order renormalizability.

Local operators in quantum field theory

The precise definition of a quantum local operator has been given (in the BPHZ framework ) by Zimmermann (Zimmermann W. (1970)) through the normal product prescription (not to be confused with the normal order prescription).

Let \(d\) be the canonical dimension of the composite classical operator \(\mathcal{O}(x)\,.\) The normal product \(N_\delta [\mathcal{O}(x)] ,\) where \(\delta \ge d\,,\) is defined through Green functions with an \(\mathcal{O}(x)\)insertion and any number of external fields, by adding a rule for the subtraction of subgraphs that contain the vertex associated to the inserted operator: for such subgraph (say with \(n\) "external" lines), the power of the Taylor series in "external" momenta to be subtracted is \(\delta - n\) and not \(4-n\) (in the dimensional renormalization scheme, as shown by Breitenlohner and Maison (1977), the subtraction is automatically a minimal one , i.e. \(\delta =d\)).

As an immediate consequence, and as it happens in the free field case, the vacuum expectation value of any normal product vanishes: it is by construction a normal ordered normal product !

Moreover, due to operator mixing through renormalization, the definition of the normal product \(N_\delta [\mathcal{ O}(x)]\) requires the simultaneous definition of the complete set of operators of canonical dimension \(\le \delta\,.\) Of course, this set is limited by global symmetries. The subtraction algorithm, whatever it may be, involves normalization conditions to define precisely this set: to say it in other words, it means that an equivalent number of parameters has to be introduced (along with masses, couplings, ..) into the game to completely define the theory.

An important remark is the absence of renormalization for operators corresponding to conserved currents (or equivalently, the vanishing of the anomalous dimension of the corresponding operator) and, more generally, the reduction in the number of above mentioned extra parameters in the presence of symmetries (Ward identities).

The OPE in interacting perturbative quantum field theory

But the (weak) limit of the product of two normal products \(N_d [\mathcal{O}(x+\chi/2)]N_{d'} [\mathcal{O}'(x-\chi/2)]\) when \( \chi \rightarrow 0\) will be singular, contrarily to the expected result\( N_{d +d' }[\mathcal{O}(x)\mathcal{O}'(x)].\) This analysis is the aim of the short distance expansion, which after Wilson writes: \[\tag{12} \displaystyle N_d [\mathcal{O}(x+\chi/2)]N_{d'} [\mathcal{O}'(x-\chi/2)] = N_{d+d'}[\mathcal{O}(x)\mathcal{O}'(x)] + D_0(\chi)\mathbf{1} + \sum_{i=1}^p D_i(\chi)N_{d_i} [\mathcal{A}_i(x)] + \mathcal{R}(x,\chi), \]


where:

  • the set \(\{ \mathbf{1},\,N_{d_i} [\mathcal{ A}_i]\}\) form a basis of local operators with increasing canonical dimension (minimally subtracted ones thanks to the Zimmermann identities) ;
  • \( p \geq p_0 \) where \( p_0\) is the number of local operators with canonical dimension \(d_i \leq d + d'\,;\) for \( i = 1, ..., p_0\,,\) the \(D_i(\chi)\) are singular functions ;
  • at the classical level, the behaviour of the coefficient functions \(D_i(\chi)\) when \(\chi^{\mu}\ = \rho\eta^{\mu} \rightarrow 0 \ ,\) the vector \(\eta\) being fixed, is \(\rho^{d_i - (d+d')}\ ;\) renormalization effects will modify this result and the scaling properties of these coefficients are given by the Callan-Symanzik equation, with in particular operator mixing (an interesting case is asymptotically free theories, where deviations from canonical scaling are limited to logarithms ) ;
  • the rest \( \mathcal{ R}(x,\chi)\) is a bilocal operator, regular in the weak limit sense \(\rho \rightarrow 0 \ .\) At the classical level, it behaves as \(\rho^{(d_p +1- (d+d' ))} \ ,\) and here also, renormalization effects will modify this result.

Note that a similar OPE holds for time-ordered products of local operators, with different coefficient functions: \[\tag{13} \displaystyle T\left[N_d [\mathcal{O}(x+\chi/2)]N_{d'} [\mathcal{O}'(x-\chi/2)]\right] = N_{d+d'}[\mathcal{O}(x)\mathcal{O}'(x)] + E_0(\chi)\mathbf{1} + \sum_{i=1}^p E_i(\chi)N_{d_i} [\mathcal{A}_i(x)] + \mathcal{R}'(x, \chi), \]


In the case where \(\mathcal{O}(x)\) is a scalar field, a complete proof of eq.(13)- including the independence with respect to the direction \(\eta_\mu\) - may be found in (Zimmermann W., 1973). Hereafter, some hints are presented.

Illustration of the short distance expansion in interacting quantum field theory

Let us first consider a simple case.

Short distance behaviour of the product of two scalar fields

As in the free field case, consider the simplest example where eq.(8) writes:

\[\tag{14} T\left[\Phi^i(x+\chi/2) \Phi^j(x-\chi/2)\right]= \ i\Delta_F^{ij}(\chi)\mathbf{1} + :\Phi^i(x) \Phi^j(x): + \rho\eta^\mu :[\partial_\mu\Phi^i(x)\Phi^j(x)-\Phi^i(x)\partial_\mu\Phi^j(x)]/2: + R^{ij}(x, \rho, \eta ) . \]


Recall that OPE holds in the weak limit sense: so, to simplify the graphical analysis of the Green functions with their subtraction algorithm, consider the vacuum expectation value \(A\) of a time-ordered product, the external legs being amputated (see eq.(7)): \[ A = <0 \mid T [ \Phi^i(x+\chi/2) \Phi^j(x-\chi/2) \tilde{ \Phi}^a(p_a) \tilde{ \Phi}^b(p_b)]\mid 0>\mid^{amp.} \] to be compared to: \[ B = <0\mid T [N_2[\Phi^i(x)\Phi^j(x)] \tilde{\Phi}^a(p_a)\tilde{\Phi}^b(p_b)]\mid 0>\mid^{amp.} .\]

Up to first order in the coupling constant \(\lambda\ ,\) with \(q = (p_a +p_b)\ ,\) and the interaction Lagrangian density being \(\frac{\lambda}{8} (\Phi^2)^2\ ,\) the Green function \(A\) writes (Figure 2):

Figure 2: First order expression for Green function A

\[A = i\Delta_F^{ij}(\chi)[i(p_a^2 -m^2)\delta^{ab}\delta^4(p_a -p_b)] + e^{iq.x}\left[\delta^{ia}\delta^{jb}e^{i(p_a -p_b).\chi/2} + \delta^{ja} \delta^{ib}e^{-i(p_a -p_b).\chi/2}\right] + \] \[ -i\lambda e^{iq.x}(\delta^{ij}\delta^{ab} +\delta^{ia}\delta^{jb} +\delta^{ja}\delta^{ib}) \int \frac{d^4 k}{(2\pi)^4} \frac{e ^{ik.\chi}}{((k+q/2)^2 - m^2)((k-q/2)^2 - m^2)} \,.\]

At the same order, a subtraction being done according to the \(N_2\) normal product prescription, \(B\) writes (Figure 3):

Figure 3: First order expression for Green function B

\[B = e^{iq.x}[\delta^{ia}\delta^{jb} + \delta^{ja} \delta^{ib}] -i\lambda e^{iq.x}(\delta^{ij}\delta^{ab} +\delta^{ia}\delta^{jb} +\delta^{ja}\delta^{ib})\displaystyle\int \frac{d^4 k}{(2\pi)^4} \left[ \frac{1}{((k+q/2)^2 - m^2)((k-q/2)^2 - m^2)}- \frac{1}{(k^2 - m^2)^2}\right]\,.\]

Due to the appearance of a term in \(\delta^{ij}\delta^{ab}\ ,\) one immediately sees that Green functions of new operators enter into the game: let

\[B' = <0\mid T [N_2[\delta^{ij}\Phi^m(x)\Phi^m(x)] \tilde{\Phi}^a(p_a)\tilde{\Phi}^b(p_b)]\mid 0>\mid^{amp.} = 2 \delta^{ij}\delta^{ab}e^{iq.x} + ...\,,\]

then, still up to first order in \(\lambda\,,\) one obtains: \[\tag{15} A = i\Delta_F^{ij}(\chi)\delta^{ab}[i(p_a^2 -m^2)\delta^4(p_a -p_b)] + B + C(\chi)[B +B'/2] + \tilde{R}^{ij}(\chi, x ;p_a, p_b) \]


where:

  • \(C(\chi)\) is a distribution, singular when \(\chi \rightarrow 0\ :\)

\[\displaystyle C(\chi) = -i\lambda \int \frac{d^4k}{(2\pi)^4} \frac{e^{ik.\chi}}{(k^2 - m^2 +i\epsilon)^2} = -\frac{\lambda}{16\pi^2} [ \ln(-m^2 \chi^2/4) + 2\gamma] + \mathcal{O}(\chi^2\ln(-m^2 \chi^2))\,;\]

  • \(\tilde{R}^{ij}(\chi, x ;p_a, p_b)\) continuously vanishes for \(\chi\rightarrow 0\ :\)

\[\tag{16} \begin{array}{rcl} \tilde{R}^{ij}(\chi, x ;p_a, p_b) & = & e^{iq.x}\left[\delta^{ia}\delta^{jb}(e^{i(p_a -p_b).\chi/2}-1) + \delta^{ja} \delta^{ib}(e^{-i(p_a -p_b).\chi/2}-1)\right] - i\lambda e^{iq.x}(\delta^{ij}\delta^{ab} +\delta^{ia}\delta^{jb} +\delta^{ja}\delta^{ib}) \times \\ & \times & \displaystyle \int \frac{d^4k}{(2\pi)^4}(e ^{ik.\chi} -1)\left[\frac{1}{(k+q/2)^2 - m^2)(k-q/2)^2 - m^2)} - \frac{1}{(k^2 - m^2)^2}\right] \end{array}\]


The trivial, but important remark is the occurrence in the right hand side of eq.(15) of a basis of dimension 2 operators with the right covariance properties \[N_2[\Phi^i(x)\Phi^j(x)]\] and \(N_2[\delta^{ij}\Phi^m(x)\Phi^m(x)]\,.\)

So, from eq.(15), one gets the OPE: \[\tag{17} T[\Phi^i(x+\chi/2) \Phi^j(x-\chi/2)]= N_2[\Phi^i(x)\Phi^j(x)] + i\Delta_F^{ij}(\chi)\mathbf{1} + C(\chi) \left\{ N_2[\Phi^i(x)\Phi^j(x)] + N_2[\delta^{ij}\Phi^m(x)\Phi^m(x)/2] \right\} + R^{ij}(x,\chi) \]


where the Green functions of the bilocal operator \(R^{ij}(x,\chi)\) are continuously vanishing when \(\chi \rightarrow 0\,.\)

Moreover, one could also extract from the regular bilocal operator \(R^{ij}(x, \chi)\) another local part, odd in \(\chi\,.\) Consider the Green function of a new dimension 3 operator, antisymmetric in \((i,\,j)\ ,\) as in eq.(6)\[N_3\left[ \partial_\mu\Phi^i(x)\Phi^j(x)-\Phi^i(x)\partial_\mu\Phi^j(x)\right]\,.\] Then the first term in eq.(16) may be rewritten as:

\[\tag{18} \begin{array}{rcl} & \ & \chi^\mu <0\mid T [N_3\left[(\partial_\mu\Phi^i(x)\Phi^j(x)-\Phi^i(x)\partial_\mu\Phi^j(x))/2\right] \tilde{\Phi}^a(p_a)\tilde{\Phi}^b(p_b)]\mid 0>\mid^{amp.} +\\ & + & e^{iq.x} \left\{ \delta^{ia}\delta^{jb}[e^{i(p_a -p_b).\chi/2}-1 -i(p_a -p_b).\chi/2) ] + \delta^{ja} \delta^{ib}[e^{-i(p_a -p_b).\chi/2}-1 +i(p_a -p_b).\chi/2)]\right\} \end{array}\]

(note that this new operator does not contribute to first order in \(\lambda\)). So, from eqs.(17), (18), one gets a more precise operator product expansion\[\tag{19} \begin{array}{rcl} T[\Phi^i(x+\chi/2) \Phi^j(x-\chi/2)] & = & N_2[\Phi^i(x)\Phi^j(x)] + i\Delta_F^{ij}(\chi)\mathbf{1} + C(\chi)N_2[\Phi^i(x)\Phi^j(x)] + C(\chi)N_2[\delta^{ij}\Phi^m(x)\Phi^m(x)/2] + \\ & + & \chi^\mu N_3[ (\partial_\mu\Phi^i(x)\Phi^j(x) - \Phi^i(x)\partial_\mu\Phi^j(x))/2] + R'^{ij}(x, \chi)\,, \end{array}\]

where the Green functions of the bilocal operator \(R'^{ij}(x,\chi)\) now vanishes as \( \chi^2\) when \( \chi\rightarrow 0\,.\) Note that the new dimension 3 operator \(N_3[ (\partial_\mu\Phi^i(x)\Phi^j(x) - \Phi^i(x)\partial_\mu\Phi^j(x))/2]\) is the unique dimension 3 operator compatible with symmetry of the left hand side under the exchange \((i \leftrightarrow j\,, \chi \leftrightarrow -\chi)\ :\) as announced, the OPE (19) involves a complete basis of operators of increasing canonical dimension ( \(\le 3\) in this example), restricted by symmetry arguments. So, this \(\mathcal{O}(\lambda)\) calculation exemplifies the Wilson Operator Product Expansion as proven by Zimmermann (Zimmermann W. (1973)). Notice that here, contrarily to Zimmermann's analysis where \(\Phi(x)\) is taken as a scalar field, an 0(N) vector field \(\Phi^i(x)\) was used: then no term linear in the quantum field may appear in the corresponding expansions eqs.(19),(20): \[\tag{20} \begin{array}{rcl} T[\Phi^i(x+\chi/2) \Phi^j(x-\chi/2)] & = & N_2[\Phi^i(x)\Phi^j(x)] + E_0(\chi)\delta^{ij}\mathbf{1} + E_0^2(\chi) N_2[\Phi^i(x)\Phi^j(x)] + E_1^2(\chi)N_2[\delta^{ij}\Phi^m(x)\Phi^m(x)/2] + \\ & + & E^{\mu 3}_0(\chi) N_3[ (\partial_\mu\Phi^i(x)\Phi^j(x) - \Phi^i(x)\partial_\mu\Phi^j(x))/2] + R^{ij}(x, \chi) \end{array}\]


Of course, any higher order analysis would involve the same basis of composite operators, but the expressions obtained in eq.(19) for the coefficient functions \(E_i^{\alpha}(\chi)\) would be modified !

Short distance behaviour of the product of two electromagnetic currents (in an asymptotically free theory)

Taking the asymptotic freedom of Quantum Chromodynamics for granted, the short distance expansion (13) of the canonical dimension 6 quantity \(<0\mid T[J_\mu (x)J_\nu (0)] \mid0>\) will be dominated by the dimension 6 leading singularity \( E_0^{\mu\nu}(x)\) where the singular function \(E_0^{\mu\nu}(x)\) may be expressed as: \[E_0^{\mu\nu}(x) = g^{\mu\nu}\frac{a}{(x^2 -i\epsilon )^3} + x^\mu x^\nu\frac{b}{(x^2 -i\epsilon )^4}\,.\]

Notice the \(i\epsilon\) terms in the denominators: the difference with the \(i\epsilon \chi^0\) of eq.(11) originates - as explained in subsection "The Wick theorem..." - in the T product that replaces the ordinary product (so that the Feynman propagator \(i\Delta_F\) replaces \(\Delta_+\)).

The quantities \(a\) and \(b\) have dimension zero: as said before, the electromagnetic current being conserved and the theory being asymptotically free, the classical power-like behaviour still holds at the quantum level up to logarithms \(\ln \mid m^2 x^2\mid\) given by the Callan-Symanzik \(\beta\) function.

Moreover, due to current conservation, \(a\) and \(b\) are related: \[E_0^{\mu\nu}(x) =c\frac{2 x^\mu x^\nu -x^2 g^{\mu\nu}}{(x^2 -i\epsilon )^4}\,,\] and the quantity \(c\) is \( \frac{R}{\pi^4}\ ,\) where \(R\) is the ratio of the hadronic to muonic annihilation cross sections (still neglecting the fermion masses)(Figure 4):

\[ \displaystyle R (q^2) = \frac{\sigma_{e^+e^- \rightarrow hadrons }}{\sigma_{e^+e^- \rightarrow \mu^+ \mu^-}} \simeq R_{\infty}[1 +\frac{3g^2(\lambda)T_f}{16\pi^2} + ..] , \]
Figure 4: Illustration of the smooth behaviour of \(R(q^2)\) at high energy (unit \(\sqrt{q^2}\) in GeV ). Taken from Particle data Group)

where

  • \(\displaystyle R_{\infty} = \sum_{quarks\ i}e_i^2\,,\)
  • \(g(\lambda)\) is the running coupling constant in QCD: using the well known expression for the Callan-Symanzyk \(\beta\) function, \(\displaystyle g^2(\lambda) \simeq \frac{48\pi^2}{(11C - 4T_f)\ln(\lambda^2)}\,, \ C\ {\rm and}\ T_f \) characterizing the quadratic Casimir operators in the representation of the fields.

Of course, the free case results of subsection "Short distance behaviour of the product of two electromagnetic currents" are recovered in the absence of QCD interactions.

References

  • P. Breitenlohner and D. Maison (1977), Commun. Math. Phys. 52, 11.
  • K. Wilson (1964), unpublished Cornell Report and Phys. Rev. 179 (1969) 1499.
  • W. Zimmermann (1970), Local operator products and renormalization in quantum field theory, in 1970 Brandeis Lectures, vol. 1, p.395, eds. S. Deser et al. (M.I.T. Press, Cambridge, 1970).
  • W. Zimmermann (1973), Annals of Phys. 77, 570.

Internal references

  • Jean Zinn-Justin and Riccardo Guida (2008) Gauge invariance. Scholarpedia, 3(12):8287.


Further reading

  • R. J. Crewther (1975), Asymptotic behaviour in quantum field theory, in Cargèse 1975, p. 345, eds. M. Levy et al. (Plenum Press, New York, 1976)

See also

Asymptotic freedom, BPHZ renormalization scheme, Callan-Symanzik equation, Dimensional renormalization, Gauge theory, Light-cone operator product expansion, Local operator, Multiloop Feynman integrals, Quantum chromodynamics, Quantum electrodynamics, Ward identities.

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