Local operator
Local operator (or composite operator) refers to the definition of a local product of fields and their space-time derivatives, in an interacting perturbative quantum field theory. As quantum fields are operators valued distributions in some Hilbert space, the definition of products of fields and their space-time derivatives at the same space-time point, called a local or a composite operator, requires some care. After a reminder on the free field case (normal order, Wick theorem), the fully interacting case is discussed with emphasis on locality (normal products, Zimmermann identities, operator mixing) and illustrations in \(\Phi^4\) theory and Quantum Electrodynamics (with the electromagnetic current).
Motivation
The basic objects of quantum field theory, the quantum fields, are local operators. Although unphysical objects, they are the building blocks for any physical quantity such as the energy-momentum tensor, the conserved (or softly broken) currents,... Think for example of the electromagnetic interaction in Quantum Electrodynamics: it is expressed via the electromagnetic current \(J_\mu(x)\) interacting with the photon field \(A^\mu(x)\) through the interaction Lagrangian density \( \mathcal{L}_{int.}(x) = J_\mu(x)A^\mu(x)\,.\) (Note that Greek letters denote Lorentz indices ranging from \(0 \) to \( 3\ ,\) and that we adhere to the convention that repeated Lorentz indices \(\mu,\nu\cdots\ ,\) spin indices \(\alpha,\beta\cdots \) and "isospin" indices \(i, j\cdots\) are summed.)
Locality of the interaction means that, for instance in Q.E.D. the electromagnetic current \(J_\mu(x)\) is an operator that depends on the basic spinorial fields \(q^i_\alpha(x)\ ,\) all taken at the same point \(x\): \[J_\mu(x) = \sum_{ i}e_i [\bar{q}^i_\alpha(x) \gamma_\mu^{\alpha\beta} q^i_\beta(x)]\,.\]
This definition of the local current \(J_\mu(x)\) looks simple at the classical level but needs a precise definition in a quantum theory: indeed, as will be shortly recalled, even the definition of the products of free fields \(\bar{q}^i_\alpha(x)\) and \(q^j_\beta(x)\) at the same space-time point is not straightforward and requires normal ordering (subsection "The normal order prescription..." ); moreover, in presence of interactions, the renormalization program has to take these local operators into account (section "The interacting 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 free field case
Definition of a local operator
Let \(\Phi^i(x)\) be the set of quantum local operators describing the multiplet of free classical fields \(\phi^i(x), i= 1,2 ,... , N\,.\) A local (composite) operator \(\mathcal{O}(x)\) would be an arbitrary polynomial in the fields and their space-time derivatives (notations are those of Zimmermann (Zimmermann W., 1970)): \[\tag{1} \begin{array}{rcl} \mathcal{O}_{\{\mu\}}\{\Phi^i(x)\} & = & \Phi_{(\mu)_1}^{i_1}(x)...\Phi_{(\mu)_m}^{i_m}(x) ,\\ \Phi_{(\mu)_k}^{i_k}(x) & = & \partial_{(\mu)_k}\Phi^{i_k}(x) , \\ (\mu)_k & = & (\mu_{k_1},..., \mu_{k_{p(k)}}), \quad \mathrm{if}\ p(k) > 0, \quad (\mu)_k=\emptyset \quad \mathrm{if}\ p(k) = 0 ,\\ \partial_{(\mu)_k} & = & \partial_{\mu_{k_1}}...\partial_{\mu_{k_p}} \quad \mathrm{if}\ p(k) > 0, \quad \partial_{(\mu)_k} =\mathbf{1} \quad \mathrm{if}\ p(k) = 0 ,\\ \{\mu\} & = & ((\mu)_1...(\mu)_p), \\ \partial_{\mu_{k_\ell}}&=&{\partial}/{\partial x^{\mu_{k_\ell}}}. \end{array} \]
In any relativistic quantum field theory, canonical commutation relations imply that for space-like separations, the operators commute:
\[[\Phi^i(x), \Phi^j(y)] = [\mathcal{O}(x), \Phi^i(y)] = ... = 0 \ \mathrm{ for}\ (x-y)^2 = (x^0-y^0)^2 - \sum_{a=1}^3 (x^a-y^a)^2 < 0\,;\] this guarantees locality and causality.
Then, an obvious difficulty for the definition of a composite operator comes from the singularity in the product of local quantum fields at the same point. This results from general principles, but it is easily seen for free fields in the annihilation and creation operators formalism. In this context a free field is decomposed as
\[ \Phi^i(x) = \int d\mathbf{k}\,[a^i(k)e^{-ik.x} +a^{\dagger i}(k)e^{+ik.x}] ,\]
where \(a^{\dagger i}(k)\) (respectively \(a^i(k)\)) are the operator of creation (resp. annihilation), and where \(\mathbf{k}=(k^1,k^2,k^3)\) is a 3-vector, \(\ k^0 =\omega_k = \sqrt{\mathbf{k}^2 + m^2},\) \( d\mathbf{k}= \displaystyle \frac{1}{(2\pi)^3} \frac{d^3\mathbf{k}}{2\omega_k} \) is the Lorentz invariant measure, and \(k.x = k^0 x^0 -\sum_{a=1}^3 k^a x^a\) is the Lorentz scalar product of two quadrivectors \(x^\mu\) and \(k^\mu\ .\)
Annihilation and creation operators allow the construction of the Fock space in which the quantum operators act. By definition, the vacuum state \(\mid 0 \rangle \) is annihilated by any annihilation operator \(a^i(k)\mid 0 \rangle = 0\) (and the adjoint relation is \( \langle 0 \mid a^{\dagger i}(k) = 0\)). The one-particle state of momenta \(k\) is \(\mid i\,,\ k> = a^{\dagger i}(k)\mid 0 \rangle \ ;\) similarly N-particle states are obtained by applying to the vacuum state N creation operators.
The canonical commutation relations for the creation/annihilation operators read \[\tag{2} [a^i(k), a^{\dagger j}(k')] = 2\omega_k \delta^{ij}(2\pi)^3 \delta^3 (\mathbf{k-k'})\ , \quad [a^i(k), a^{j}(k')] = 0\ , \quad[a^{\dagger i}(k), a^{\dagger j}(k')] = 0\ , \]
or, in terms of free fields,
\[\tag{3} [\Phi^i(x),\Phi^j(y)] = i \delta^{ij}\Delta(x-y)\,\mathbf{1} ,\]
where \( \mathbf{1}\) is the identity operator (which is local according to the previously given definition of locality) and \(\Delta(x-y)\) is a c-number valued distribution, singular when \( y = x\ :\)
\[\tag{4} \displaystyle \Delta(x-y) =\frac{1}{i} \int d\mathbf{k}[e^{-ik.(x-y)} - e^{+ik.(x-y)}] \quad\Rightarrow\quad \partial_{x^{0}}\Delta(x-y)\mid_{(x^{0} = y^{0})} = - \delta^3(\mathbf{x} - \mathbf{y}) . \]
The commutation relations (3) imply that the fields are operator valued distributions; moreover, the singular behaviour (4) causes difficulties for the definition of composite operators. For example, if one computes the Hamiltonian operator for this free bosonic field, one immediately obtains: \[ H = \frac{1}{2} \int d\mathbf{k}\;\omega_k\;[a^i(k)a^{\dagger i}(k) +a^{\dagger i}(k)a^i(k)]\] whose vacuum expectation value, \(\langle 0\mid H\mid 0 \rangle \ ,\) is infinite!
The normal order prescription for the product of free field operators
The problem of having divergent vacuum expectation values is usually solved by the normal ordering prescription: in any product of creation and annihilation operators, the creation operators are moved ((anti-)commuted) to the left place. Then, denoting the normal order of any local operator \(A\) by \(:A:\ ,\) one now gets:
\[ H = \ :\frac{1}{2}\int d\mathbf{k}\;\omega_k\;[a^i(k)a^{\dagger i}(k) +a^{\dagger i}(k)a^i(k)]:\ =\int d\mathbf{k}\;\omega_k\; a^{\dagger i}(k)a^i(k) \]
whose vacuum expectation value is now finite... and vanishing (as it should) !
Another example is: \[ :\Phi^i (x)\Phi^j(y):\ =\ :\int d\mathbf{k} d\mathbf{k'}[a^i(k)e^{-ik. x} +a^{\dagger i}(k)e^{+ik. x}] [a^j(k')e^{-ik'. y} +a^{\dagger j}(k')e^{+ik'. y}]:\ :\] \[= \int d\mathbf{k} d\mathbf{k'}[a^i(k)a^j(k') e^{-ik. x} e^{-ik'. y} + a^{\dagger j}(k') a^i(k) e^{-ik. x} e^{ik'. y} + a^{\dagger i}(k) a^j(k') e^{ik. x} e^{-ik'. y} + a^{\dagger i}(k) a^{\dagger j}(k')e^{ik. x} e^{ik'. y} ] \ :\]
\[=\ :\Phi^j(y)\Phi^i(x): \;.\] (The result uses the symmetry \((x,k,i)\leftrightarrow(y,k',j) \) of the intermediate expression.)
The normal ordered product \(:\Phi^i (x)\Phi^j(y):\) is then continuous when \(y\rightarrow x\ :\) this limit defines unambiguously the free field composite operator \(:\Phi^2(x):\,.\) (Note that we define \(\Phi^2(x)=\Phi^i(x)\Phi^i(x)\ .\))
An important remark at this point is that, 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 a weak sense, i.e. in terms of the limits of all possible correlation functions with insertion of the operator (i.e. all vacuum expectation values of the form
\(\langle 0\mid \mathcal{O}(y)\;\prod_\ell \Phi^{i_\ell}(x_\ell) \mid 0\rangle\ ,\) for any number of elementary fields \(\Phi^i(x)\)).
As a consequence, the singularity in the product of two free fields may be extracted: \[\tag{5} \Phi^i(x)\Phi^j(y) = :\Phi^i(x) \Phi^j(y): + \langle 0\mid\Phi^i(x)\Phi^j(y)\mid0 \rangle \mathbf{1} =\ :\Phi^i(x)\Phi^j(y): +\ \Delta_+^{ij}(x-y) \mathbf{1} \;, \]
where \[\tag{6} \Delta_+^{ij}(x-y) = \delta^{ij}\int d\mathbf{k}[e^{-ik.(x-y)}]= \delta^{ij}\int \frac{d^4k}{(2\pi)^4}\theta(k_0)2\pi \delta(k^2-m^2) e^{-ik.(x-y)} \]
is singular when \(y \rightarrow x\ .\) In the limit of a vanishing mass one gets: \[\tag{7} \Delta_+^{ij}(x-y) \simeq \frac{-\delta^{ij}}{4\pi^2}\frac{1}{(x-y)^2-i\epsilon (x^0-y^0)}\,. \]
Equation (5) is the simplest application of the Wick theorem, more frequently written for time-ordered products of fields: in this case the \(\Delta_+^{ij}(x-y)\) are replaced by the Feynman propagator \(i\Delta_F^{ij}(x-y)\ .\)
The product of composite operators in a free field theory: extension of the Wick theorem
The considerations of the previous section can be generalized to the product of any local (normal ordered) operator, each of the form \( :\mathcal{O}_{\{\mu\}}\{\Phi^i(x)\}: = :\Phi_{(\mu)_1}^{i_1}(x)...\Phi_{(\mu)_m}^{i_m}(x) : \,.\)
The product of general (normal ordered) local operators a priori depends on their order and will behave in a singular way when their application points coincide. The Wick theorem allows the reduction of such a product: \[ \begin{array}{rcl} :\mathcal{O}_{\{\mu\}}\{\Phi^a(x)\}: :\mathcal{O}_{\{\nu\}}\{\Phi^b(y)\}: & = & :\mathcal{O}_{\{\mu\}}\{\Phi^a(x)\}\mathcal{O}_{\{\nu\}}\{\Phi^b(y)\}: \\ &&\;+\displaystyle \sum_{1-contraction\ [(x,i)(y,j)]}\Delta_+^{ij}(x-y) :\mathcal{O}_{\{\mu\}}\{\Phi^a(x)\}\mathcal{O}_{\{\nu\}}\{\Phi^b(y)\}:\\ &&\;+ \displaystyle \sum_{2-contractions\ [(x,i)(y,j)],[(x,i')(y,j')]}\Delta_+^{ij}(x-y) \Delta_+^{i'j'}(x-y) :\mathcal{O}_{\{\mu\}}\{\Phi^a(x)\}\mathcal{O}_{\{\nu\}}\{\Phi^b(y)\}:\\ &&\;+ ... \end{array} \] where contraction [(x,i)(y,j)] means the suppression of the fields \(\Phi^i(x)\) and \(\Phi^j(y)\) in the subsequent chain \(:\mathcal{O}_{\{\mu\}}\{\Phi^a(x)\}\mathcal{O}_{\{\nu\}}\{\Phi^b(y)\}:\ .\) (As previously said, the Wick theorem is more frequently written for time-ordered products: in this case the \(\Delta_+^{ij}(x-y)\) are replaced by the Feynman propagator \(i\Delta_F^{ij}(x-y)\ .\))
For example, one gets:
\[ :\Phi^2(x): : \Phi^2(y): = :\Phi^2(x)\Phi^2(y): +\ 4 \Delta_+^{ij}(x-y) :\Phi^i(x) \Phi^j(y): +\ 2\Delta_+^{ij}(x-y) \Delta_+^{ij}(x-y)\;, \] and, taking into account the vanishing of the vacuum expectation value of any normal ordered product, \[\tag{8} \langle 0\mid:\Phi^2(x): : \Phi^2(y):\mid0 \rangle \ =\ 2 \Delta_+^{ij}(x-y)\Delta_+^{ij}(x-y)\;. \]
Note that the vacuum expectation value in equation (8) diverges when \(y\rightarrow x\ ,\) on the contrary of \( \langle 0\mid :[\Phi^2(x)]^2:\mid 0 \rangle \) which vanishes!
The canonical dimension of a local operator
To each free field operator is assigned a canonical dimension \(d\) (also called mass dimension or classical dimension or - shortly - dimension) and to each space-time derivative and mass parameter a dimension 1. The dimension \(d\) is given by the condition that, as it should, the free part of the Lagrangian density \(\mathcal{L}_0(\Phi^i(x),\,m)\) has canonical dimension 4, that is\( \mathcal{L}_{0}(\lambda^d\Phi^i(\lambda x),\,\lambda m)) = \lambda^4\mathcal{L}_{0}(\Phi^i(x),\,m)\,.\) For instance, it turns out that the canonical dimension of a spin zero free field is 1 and that of a spin 1/2 is 3/2.
Then, it can be checked that, for each \( n\,, \) there is only a finite number of ( linearly independent) local operators of dimension \(\le n\ .\) Hence, a finite basis can always be constructed: for instance, a basis for \(0(N)\) scalar operators constructed in terms of \(\Phi^i(x)\ ,\) and having dimension not greater than 4, is \( \{\mathbf{1}\,, :\Phi^2(x):\ =\ :\Phi^i(x).\Phi^i(x):\,,\ :[\Phi^2(x)]^2:\,,\ :\partial_\mu \Phi^i(x).\partial^\mu\Phi^i(x):\,, \ :\Phi^i(x).\partial_\mu\partial^\mu \Phi^i(x): \} \,. \)
As it will appear later on, the renormalization of a given operator of dimension \(d\) can only mix this operator with operators of dimension less or equal to \(d\), so this tool of power counting will be useful in the sequel.
The interacting case in perturbative quantum field theory
As soon as fields interact, everything complicates. In the perturbative approximation of interacting quantum field theories, perturbative calculations of Green functions (vacuum expectation values of a time-ordered product of fields at different space-time points) are expressed, order by order, by Feynman amplitudes which are sums of integrals pictorially described by the so-called Feynman graphs. Now, when Feynman graphs with loops come into the game, it happens that the corresponding integrals may diverge for large values of the loop momenta (the integration variables). These divergences are called ultraviolet (UV) divergences. An ultraviolet superficial degree of divergence can be defined to control if the integral is convergent (degree negative) or divergent (degree non-negative). Then, to consistently get rid of divergences, one needs a regularization and a renormalization scheme, with definite rules to ensure all-loop order finiteness - as well as Lorentz invariance and unitarity.
Simple regularization schemes are the one which uses a momentum cut-off at some scale \(\Lambda\) or the popular dimensional regularization method, based on analytic continuation of the Feynman integrand to a complex space-time dimension \(D = 4 - \epsilon\ .\)
For one-loop contributions to some Green function, the renormalization scheme is easy, because in this case the divergent part may be simply subtracted ( by "divergent part" one means the part of the integral that diverges when the cut-off \(\Lambda\to\infty\) or when \(\epsilon\to 0\)). This divergent part is not uniquely defined as an arbitrary finite (in the limit of an infinite cut-off) quantity can always be added to it. Then, to uniquely fix the finite part of the Feynman amplitude, a normalization condition is required.
The chosen set of subtraction rules and normalization conditions defines what is called a renormalization scheme. Examples of renormalization schemes are the BPHZ's one (with normalization conditions at vanishing external momenta) or minimal dimensional renormalization where these normalization conditions are "implicit" ones (one often speaks of an intermediate renormalization).
Of course, for multiloop Feynman integrals, the analysis is much more involved as sub-integrals may be themselves divergent: so a recursive procedure should be defined. The first step is to consider the proper (or 1-particle irreducible) Feynman amplitudes which correspond to the graphs that cannot be broken into two parts cutting a single line. Among those, the primitively divergent graphs, i.e. the graphs whose superficial degree of divergence is \(\geq 0\), obviously require subtractions: but, as sub-graphs may also be divergent, even for a superficially convergent diagram, one has first to apply the subtraction procedure to these sub-diagrams before extracting the possible divergent part of the whole diagram (when the diagram is primitively divergent). The subtraction of such sub-divergences is technically difficult, especially when there are so-called overlapping sub-divergences.
In the same manner, as the Green functions with insertion of any quantum local operator will offer new primitive divergences, new subtractions and normalization conditions will be necessary to define such composite operator at the perturbative quantum level. This will be detailed in the subsection "Definition of normal products and operator-mixing".
The role of locality in perturbative quantum field theory
One more time, locality plays an essential role: it may be proved the existence of consistent subtraction algorithms. A subtraction algorithm is consistent if, once applied up to order (N-1), the divergent parts of the Feynman amplitudes at order N are polynomial in masses and external momenta, the degree of the polynomial being given by the superficial degree of divergence of the Feynman amplitude. This "local" character of the divergences means that their subtraction may be interpreted as resulting from the addition to the effective action of new properly chosen terms (known as counterterms) that are local polynomials in the fields and their derivatives, and whose coefficients depend on the regularization parameter (cut-off \(\Lambda\ ,\) dimensional parameter \(\epsilon\ .\) ..). For any renormalizable theory, these counterterms, having the same form as the terms in the classical Lagrangian density, simply redefine (in a regularization dependent manner) the fields and parameters of the theory. When counterterms absent in the classical Lagrangian density - in particular of canonical dimension > 4 - appear, the theory is non-renormalizable.
Of course, each of the required normalization conditions defines a parameter of the theory (masses, couplings, wave functions ...). Some of these parameters may be unphysical ones: for example, the parameters associated with field renormalizations or with gauge fixing play no role when one computes a physical quantity (as soon as the calculation is done in a consistent way!).
Moreover, global and local symmetries (e.g. parity, gauge invariance .. ) relate Green functions through Ward identities and then reduce the number of the free parameters of a theory. Think for example of the current conservation in Quantum Electrodynamics: although the 2-photon Green function is quadratically divergent, gauge invariance and parity limit the number of normalization conditions (or parameters) to one (think of the renormalization constant \(Z_3\)); in the same way, the vertex function renormalization requires one normalization condition (that defines the physical electron charge). Moreover, the Ward identity relates this vertex renormalization to the photon, electron and electric charge renormalizations: another way of saying this is that, due to its conservation, the electromagnetic current composite (local) operator \(J^\mu(x)\) requires no extra renormalization when fields and charge renormalizations are properly done.
Definition of normal products and operator-mixing
In an interacting quantum field theory, the precise definition of a local operator has been given by Zimmermann in the BPHZ framework, through the normal product prescription which extends to all orders of perturbation theory the notion of the normal order prescription for free fields discussed in the subsection "The normal order prescription..." (Zimmermann W., 1970).
Let \(d_\mathcal{O}\) be the canonical dimension of the composite classical operator \(\mathcal{O}(x)\ .\) The normal product \(N_\delta[\mathcal{O}(x)]\ ,\) where \(\delta \ge d_\mathcal{O},\) is defined through its Green functions (vacuum expectation values of the time-ordered product of an arbitrary number of elementary fields and a single \(\mathcal{O}(x)\) insertion), by adding a rule for the subtraction of subgraphs that contain the vertex associated to the inserted operator: for each 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. \(\delta = d_\mathcal{O}\ .\))
It follows from the prescription above that the vacuum expectation value of any normal product vanishes, as it does in the free field case: \[\langle 0\mid T[N_\delta[\mathcal{O}(x)]]\mid 0\rangle=0\;.\] Hence the normal product is by construction a normal ordered normal product!
As usual in quantum field theory, normalization conditions are necessary to define precisely the interacting (renormalized) composite operators. These conditions concern the proper (or 1-particle irreducible) Feynman amplitudes (i.e. corresponding to some sum of proper Feynman graphs). The BPHZ normalization conditions (zero momentum or "minimal" ones) for a normal product \(N_\delta[\mathcal{O}_k(0)]\) are: \[\tag{9} \tau^{(\delta-n)}_{(p_{\ell})} \langle 0\mid T\left[N_\delta[\mathcal{O}_k(0)] \prod_{\ell=1}^n\tilde{\Phi}(p_\ell)\right]\mid 0 \rangle\mid^{\mathrm{Proper}} = \langle 0 \mid T\left[N_\delta[\mathcal{O}_k(0)]\prod_{\ell=1}^n\tilde{\Phi}(p_\ell)\right]\mid 0 \rangle\mid^{\mathrm{Proper,\,Trivial}} , \]
where:
- \(\tilde{\Phi}(p)\) represents the Fourier transform of the field \(\Phi(x)\ ,\)
- proper graph has been defined above,
- trivial means computed at the tree-level (diagrams with no loops), where \(N_\delta[\mathcal{O}_k(0)] = :\mathcal{O}_k(0):\ ,\)
- when \((\delta -n)\ge 0 \ ,\) the Taylor operator \(\tau^{(\delta -n)}_{(p_{\ell})}\) creates a polynomial of degree \(\delta -n\) in the external momenta \({p_{\ell}}\ ;\) otherwise, when \(\delta < n\ ,\) it vanishes.
The right hand side in (9) requires that new parameters are defined in the theory.
Moreover, and as will be illustrated in the next subsection, one cannot discuss the full Green functions of a single operator \(N_\delta[\mathcal{O}(x)]\) but one should introduce a whole basis of operators of canonical dimension \(\le d_{\mathcal{O}} \ ;\) it will be explained that, given a set \(\{ :\mathcal{O}_k(x):\}\ ,\) a basis of free field composite operators of canonical dimension \( d_k \le d_{\mathcal{O}}\ ,\) the normal products \(N_\delta[\mathcal{O}_k(x)]\) offer a basis for renormalized composite operators of canonical dimension \(\le d_{\mathcal{O}}\,.\)
Example in scalar \(\Phi^4\) theory
Consider for instance, in massive \(\Phi^4 \) theory, the normal ordered operator \(:\Phi^6(x):\,.\) Let us show that \(12 \) normalization conditions and the simultaneous definition of \(11\) other local operators will be necessary to construct the normal product \(N_6[\Phi^6]\ !\)
Let us introduce into the classical Lagrangian density, an external source \(\chi(x)\) for the local operator \(\frac{1}{6!}:\Phi^6(x):\) \[\mathcal{L}_{sources} = :\chi(x)\frac{1}{6!}\Phi^6(x):\,.\]
We will explain the announced operator mixing in the Feynman graph approach.
Using power counting to evaluate the superficial degree of divergence of diagrams, it follows that the lowest-order contributions to the superficially divergent Green functions with one insertion of the local operator \(:\frac{1}{6!}\Phi^6(x):\) come form the graphs shown in figures 1-4 below.
Using the above mentioned property of locality of the divergent parts at loop order N when all necessary subtractions up to order (N-1) are done, one successively follows the steps below.
STEP 1: The superficial degree of divergence of \[ \langle 0\mid T[N_6[\frac{1}{6!}\Phi^6(0)] \prod_{\ell=1}^6 \tilde{\Phi}(p_\ell)]\mid 0 \rangle\mid^{_\mathrm{Proper}}\] is zero: then this Green function, subtracted for all its subdivergences, has a \(\{p_\ell\}\)-independent divergence \(a_1\ ;\) this requires addition of a counterterm of the form \[\tag{10} -\chi(x)\left[a_1 \frac{1}{6!}\Phi^6 \right]\ .\]
The lowest order contribution is a 1-loop graph (see Figure 1). At this order there is no necessity of subgraph subtraction and the \(a_1\) divergence is readily obtained. Then, implementing the normalization condition eq.(9) for \(\delta =6, n=6\ ,\) the normal product of the operator is readily defined up to the one-loop order.
STEP 2: Starting from 2-loop order, a new Green function \[ \langle 0\mid T[N_6[\frac{1}{6!}\Phi^6(0)] \prod_{\ell=1}^4 \tilde{\Phi}(p_\ell)]\mid 0 \rangle\mid^{\mathrm{Proper}}\] diverges ( Figure 2). This Green function has a superficial degree of divergence equal to 2. As a consequence, after subtracting its subdivergences, a quadratic, \(\{p_\ell\}\) dependent, divergence arises: \[ m^2 a_2 + [\sum p_\ell^2]a_3 + [\sum p_\ell.p_{\ell'}]a_4\,.\] This divergence requires addition of counterterms of the form \[\tag{11} -\chi(x)\left [ a_2 m^2\frac{1}{4!}\Phi^4 - a_3 \frac{1}{3!} \Phi^3 \Box \Phi - a_4 \frac{1}{2!\,2!}\Phi^2\partial_\nu \Phi\partial^\nu \Phi \right](x)\;.\]
The lowest order contribution comes from the 2-loop graph shown in Figure 2. This graph has 1-loop divergent subgraphs of the form discussed in step 1 (see Figure 1). Hence the previously computed counterterm (10) is used to compute the 2-loop contributions to the three quantities \(a_2, a_3\) and \( a_4.\) This indicates that the 2-loop order definition of the normal product of our local operator will require its mixing with the three operators in eq. (11) (with regularization dependent coefficients \(a_\ell , \ell= 2, 3, 4\)) in such a way that eq.(9) holds true for \(\delta =6, n=4\ .\) Of course the coefficient \(a_1\) will be modified at 2-loop order, and precisely defined, thanks to eq.(9) for \(\delta =6, n=6\ .\)
STEP 3: Starting from 3-loop order, a new Green function \[\langle 0\mid T[N_6[\frac{1}{6!}\Phi^6(0)] \prod_{\ell=1}^2 \tilde{\Phi}(p_\ell)]\mid 0 \rangle\mid^{\mathrm{Proper}}\] diverges ( Figure 3). This Green function, has a superficial degree of divergence equal to 4 and, as a consequence, when subtracted for its subdivergences, it has a quartic, \(\{p_\ell\}\)-dependent, divergence: \[( m^2)^2 a_5 + m^2 [\sum p_\ell^2]a_6 + m^2 [ p_1.p_2]a_7 + [\sum (p_\ell^2)^2]a_8 + [(p_1.p_2) \sum (p_\ell^2)]a_9 +[p_1^2 p_2^2]a_{10} +[(p_1.p_2)^2]a_{11}\,;\] this requires addition of counterterms of the form \[-\chi(x)\left[a_5 ( m^2)^2 \frac{1}{2!}\Phi^2 - m^2 a_6\Phi\Box\Phi - m^2 a_7[\frac{1}{2!}\partial_\nu \Phi\partial^\nu \Phi] + a_8\Phi\Box^2\Phi + a_9 \partial_\nu \Phi\partial^\nu \Box\Phi +a_{10}\frac{1}{2!}\Box \Phi\Box\Phi +a_{11}[\frac{1}{2!}\partial_\nu\partial_\mu \Phi\partial^\nu \partial^\mu \Phi]\right](x)\ .\] Here again, previously computed counterterms are used to take care of the subdivergences. This indicates that the 3-loop order definition of the normal product of our local operator will require its mixing with the 3+7 = 10 other operators previously introduced (with regularization dependent coefficients \(a_\ell , \ell= 2, 3,..., 11\)), in such a way that eq.(9) holds true for \(\delta =6, n=2\ .\) Of course the coefficients \(a_1, a_2, a_3\) and \(a_4\) will be modified to 3-loop order, and precisely defined thanks to eq.(9) for \(\delta =6, n=6, 4\ .\)
STEP 4: Finally, the Green function \[ \langle 0\mid T[N_6[\frac{1}{6!}\Phi^6(0)] \mid 0 \rangle\mid^{\mathrm{Proper}}\] has a superficial degree of divergence of 6. Nevertheless this Green function does not depend on any momentum, hence its divergence is \(\{p_\ell\}\)-independent: \[( m^2)^3 a_{12}\,,\] and requires addition of a new counterterm of the form \[-\chi(x)a_{12} ( m^2)^3\mathbf{1}\;.\] The first non-trivial contribution to this counterterm arises at five loops, as shown in Figure 4. Here again, previously computed counterterms are used to take care of the subdivergences. This indicates that the higher loop order definition of the normal product of our local operator will require its mixing with the 3+7 +1 = 11 above mentioned ones (with regularization dependant coefficients \(a_\ell , \ell= 2, 3,..., 12\)) in such a way that eq.(9) also holds true for \(\delta =6, n=0\ .\) Again, the coefficients \(a_1, a_2..., a_{11}\) will be modified, and precisely defined thanks to eq.(9) for \(\delta =6, n=6, 4, 2\ .\)
CONCLUSIONS: The previous analysis proves the announced results that the full quantum definition of a finite composite operator \(N_6[\frac{1}{6!}\Phi^6(x)]\) requires 12 normalization conditions and operator mixing. Indeed:
- I) Thanks to eq.(9) for \(\delta =6, n=6, 4, 2, 0\,,\) 12 normalization conditions are necessary to define order by order in the loop expansion the operator \(N_6[\frac{1}{6!}\Phi^6(0)]\), in such a way that its insertion into any Green function is finite. As illustrated above, this renormalized operator appears as a linear combination of 12 operators with \(n\le6\) fields and regularization dependent coefficients :
\[\tag{12} \begin{array}{lcl} N_6[\frac{1}{6!}\Phi^6(x)]& = &(1- a_1) \frac{1}{6!}\Phi^6(x) \\ &&\, -\left[ a_2 m^2\frac{1}{4!}\Phi^4 - a_3 \frac{1}{3!} \Phi^3 \Box \Phi - a_4 \frac{1}{2!\,2!}\Phi^2\partial_\nu \Phi\partial^\nu \Phi \right](x) \\ &&\,- \left[a_5 ( m^2)^2 \frac{1}{2!}\Phi^2 - m^2 a_6\Phi\Box\Phi - m^2 a_7(\frac{1}{2!}\partial_\nu \Phi\partial^\nu \Phi) + a_8\Phi\Box^2\Phi + a_9 \partial_\nu \Phi\partial^\nu \Box\Phi +a_{10}\frac{1}{2!}\Box \Phi\Box\Phi +a_{11}(\frac{1}{2!}\partial_\nu\partial_\mu \Phi\partial^\nu \partial^\mu \Phi)\right](x) \\ &&\, -( m^2)^3 a_{12}\mathbf{1} \\ &\ = &\displaystyle\sum_{\ell = 1}^{12}Z_{1\ell}\mathcal{O}_\ell(x). \end{array} \]
Note that if the number of fields \(n\) that occurs in the normalization equation (9) goes from 6 to 0, it takes only even values due to the \(\Phi\leftrightarrow -\Phi\) invariance of the theory.
- II) as 12 counterterms are required, and as they involve 11 other composite operators, these ones should be defined simultaneously, from the very beginning, with 11 other external sources \(\chi_\ell(x)\). With a similar analysis as done above, these 11 other dimension \(\leq 6\) scalar quantum local operators \(N_6[\mathcal{O}_k(x)]\) would also be defined recursively in the loop expansion by 11 normalization equations similar to eq.(9).
To summarize, the source terms in the effective Lagrangian density will be expressed as: \[ \mathcal{L}_{sources} = \displaystyle \sum_{k,\,\ell\ =1}^{12} \chi_k(x)Z_{k\ell}\mathcal{O}_\ell(x)\,,\] where the \(Z_{k\ell}\) are regularization dependent (divergent) and then normalization dependent, loop ordered formal series: should one modify the normalization conditions, i.e. the right hand sides of the 12 equations (9), these \(Z_{k\ell}\) would be modified by finite amounts.
This means that the renormalization of the whole set \(\mathcal{O}_k(x)\,, k=1,..12\) will be given by a \(12 \times 12 \) matrix. Note that \(Z_{k\ell} = 0 \ {\rm if}\ d_k < d_\ell\) where \(d_k\) is the canonical dimension of the operator \(\mathcal{O}_k(x)\,.\) Correspondingly, the renormalization group equation for Green functions with insertion of any combination of these local operators (at different space time points!) will involve an anomalous dimension \(12 \times 12\) matrix \(\gamma_{k\ell}\) reflecting the operator mixing. This matrix, along with some \(\beta\) functions related to coupling constant renormalization, dictates how the scaling properties of these composite operator are modified in the interacting theory.
Note that in our example, we have supposed the invariance of the interacting theory under the transformation \(\Phi(x)\rightarrow-\Phi(x)\ :\) if not true, other local operators would be involved into the mixing.
This example illustrates the announced general result: given a set \(\{ :\mathcal{O}_k(x):\}\ ,\) a basis of free field composite operators of canonical dimension \( d_k \le \delta\), the normal products \(N_\delta[\mathcal{O}_k(x)]\) offer a basis for renormalized composite operators of canonical dimension \(\le \delta\,.\)
Example of the electromagnetic current
As announced above, in the presence of local invariances, some Ward identities relate the Green functions: as a consequence, the corresponding renormalizations will not be all independent. In particular, the operator corresponding to a conserved current has no renormalization at all (its anomalous dimension vanishes).
Consider for example the electromagnetic current in Quantum Electrodynamics: \[J^{\mu}(x) = N_3[e \bar{\psi}(x)\gamma_{\mu} \psi(x)]\,.\] Its canonical dimension being 3, it may mix with all the other operators with canonical dimension less or equal to \( 3\ :\) \[\tag{13} N_3[A^\nu \partial_\nu A_\mu]\,, N_3[A^\nu\partial_\mu A_\nu]\]
and \(N_3[A^2\,A_\mu]\,.\)
Indeed, the same analysis as above would show that the primitively proper divergent graphs with one insertion of the current operator may a priori have either \( 2\) external fermionic lines, or \( 2\) photon lines or \( 3\) photon lines.
However, due to gauge invariance and charge conjugation invariance, the only divergent Green function with one insertion of the current operator is \[\langle 0\mid T[J^{\mu}(x) \tilde{\psi}(p)\tilde{\bar{\psi}}(p')]\mid0 \rangle \,.\] To say it in other words, mixing must preserve gauge invariance: hence the current operator, which is gauge invariant, cannot mix with the other operators listed in (13) that are not gauge-invariant (and cannot give rise to a gauge-invariant operator). It follows that the renormalization of the current is multiplicative. Of course, the renormalization of the other \(d \leq 3\ ,\) non gauge invariant, operators will still require some mixing among them (and a mixing matrix \(Z_{k\ell}\) compatible with charge conjugation invariance).
Moreover, thanks again to gauge invariance (i.e. to the conservation of the electromagnetic current), as soon as the fermion field, the photon field and the electric charge are properly renormalized, the Green function \[\langle 0\mid T[J^{\mu}(x) \tilde{\psi}(p)\tilde{\bar{\psi}}(p')]\mid 0 \rangle\] is finite: no new renormalization is required for the current operator. In other words, the anomalous dimension of the current operator vanishes.
The Zimmermann identities
In a previous subsection, we mentioned the notion of over-subtracted normal product, i.e. of a normal product \(N_\delta[\mathcal{O}(x)]\,,\) with \(\delta\) greater than the canonical dimension \(d_\mathcal{O} \) of the operator \(\mathcal{O}(x)\ .\)
It has been shown by Zimmermann (1970), that an over-subtracted operator \(N_\delta[\mathcal{O}(x)]\) (with \(\delta> d_\mathcal{O} \)) may be expressed in terms of a basis of minimally subtracted operators: \[\tag{14} N_\delta[\mathcal{O}(x)] = N_{d_\mathcal{O}}[\mathcal{O}(x)] + \sum_\ell c_\ell N_{d_{\mathcal{A}_\ell}}[\mathcal{A}_\ell(x)]\;, \]
where the family of local operators \([\mathcal{O}(x),\mathcal{A}_\ell(x)] \) is a basis of local operators of canonical dimension less or equal to \(\delta\) (possibly restricted by symmetry arguments: parity, charge conjugation, Lorentz covariance, ...) and the coefficients \(c_\ell,\) vanishing with the Planck constant \(\hbar,\) may be expressed as definite Green functions. Expressions like the one shown in (14) are known as Zimmermann identities.
Zimmermann identities allow the rewriting of the basis (mentioned in the last sentence of subsection Example in scalar \(\Phi^4\) theory) \(N_\delta[\mathcal{O}_k(x)]\) of composite operators of dimension \(d_k \le \delta \) as a basis of minimally subtracted ones \(N_{d_k}[ \mathcal{O}_k(x)]\,.\)
Zimmermann identities (14) have been adapted in (Bonneau G., 1980) to the dimensional renormalization scheme, where composite operators are always minimally subtracted. In this case, the introduction of a factor \((4-D)\ ,\) where \(D\) is the complex regularizing dimension, in front of an operator \(\mathcal{O}(x),\) gives rise to a similar expansion: \[\tag{15} N[(4-D)\mathcal{O}(x)]= c_0 N[\mathcal{O}(x)] + \sum_\ell c_\ell N[\mathcal{A}_\ell(x)] \]
where the family of local operators \([\mathcal{O}(x),\mathcal{A}_\ell(x)] \) is now a basis of local operators of dimension \(\le d_\mathcal{O}\ ,\) and the coefficients \(c_0, c_\ell,\) vanishing with the Planck constant \(\hbar,\) may be expressed as residues of the simple poles in the variable \((D-4)\) of properly subtracted (for all their subdivergences) definite proper Green functions.
Such operators, that vanish at the tree level when the regularizing dimension goes to 4, but contribute in the fully interacting situation, are called evanescent operators. Apart from \(N[(4-D)\mathcal{O}(x)]\ ,\) examples are linked to the non-anticommutation of Dirac matrices \(\gamma_\mu\) and \(\gamma^5\) or with the loss of algebraic properties of spinors in supersymmetric theories when the regularizing dimension \( D \neq 4\) (consider for instance the evanescent operator \( N[\bar{\Psi}\{\gamma^5,\gamma_\mu\}_+\Psi(x)]\) involved into the Ward identity for the axial current).
These identities (eq.(15) also hold for such evanescent operator, with the understanding that the basis of operators in the right hand side should be extended to all evanescent and non evanescent operators of canonical dimension \(\le d_\mathcal{O}\) (again possibly restricted by symmetry arguments: parity, charge conjugation, Lorentz covariance, ...)
These Zimmermann identities are useful to discuss anomalies in renormalized Ward identities in BPHZ renormalization scheme (eq.(14)) or in dimensional renormalization scheme (eq.(15), Bonneau G., 1980).
The product of composite operators in interacting quantum field theory
In the short-distance limit, \(y \rightarrow x\ ,\) and as it was in the free field case (section : The product of composite operators ...) the product of two well-defined normal products \(N_\delta[\mathcal{O}(x)]\) and \(N_{\delta'}[\mathcal{O}'(y)]\) is a priori different from the normal product \(N_{\delta + \delta'}[\mathcal{O}(x)\mathcal{O}'(x)]\) (as usual, the limit being in a weak sense), and, moreover it is generally singular!
More specifically, it happens that the renormalization of Green functions involving the product of two or more composite operators may require (in addition to the mixing of operators) new type of subtractions known as contact terms.
The short distance behavior (\(y \rightarrow x\)) of the product of composite operators can be characterized via the so-called operator product expansion. Alternatively, the behavior of the product of composite operators when \((x-y)^2\rightarrow 0 \) (in the Minkowsky metric) is studied via the so-called light-cone expansion.
References
- 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).
- P. Breitenlohner and D. Maison (1977), Commun. Math. Phys. 52, 11.
- Bonneau G. (1980), Nucl. Phys. B167, 261 and B171, 477.
Internal references
- Jean Zinn-Justin and Riccardo Guida (2008) Gauge invariance. Scholarpedia, 3(12):8287.
- Vladimir Alexandrovich Smirnov (2009) Multiloop Feynman integrals. Scholarpedia, 4(6):8507.
- Guy Bonneau (2009) Operator product expansion. Scholarpedia, 4(9):8506.
Further reading
- C. Itzykson and J.B. Zuber Quantum field theory, McGraw-Hill Inc. 1980 (in particular Chap 3.1, 4.2 and 8.2)
See Also
Bogoliubov-Parasiuk-Hepp-Zimmermann renormalization scheme, operator product expansion, Renormalization, Multiloop Feynman integrals, light-cone expansion.