Bogoliubov-Parasiuk-Hepp-Zimmermann renormalization scheme

• Dr. Klaus Sibold, Institut für Theoretische Physik Fakultät für Physik und Geowissenschaften Universität Leipzig

The Bogoliubov, Parasiuk, Hepp, Zimmermann renormalization scheme (abbreviated BPHZ scheme) is a mathematically consistent method of rendering Feynman diagrams finite while maintaining the fundamental postulates of relativistic quantum field theory (Lorentz invariance, unitarity, causality).

The problem

For elucidating the problem let us have a look at an intuitive representation of processes involving particles at the subatomic level. Elementary particles like electrons, quarks, photons and gluons interact with each other: in scattering processes incoming particles collide and give rise to outgoing particles, the transition from such an initial state to a final state obeying the rules of quantum mechanics. (According to the laws of quantum mechanics the states belong to a Hilbert space, which is associated to the physical system one is dealing with.) Pictorially this is described in terms of Feynman diagrams.

Figure 1: \(e^+e^-\) annihilate into a photon, photon disintegrates into a \(\mu^+\mu^-\) pair.	Figure 2: Vertex: interaction.
Figure 3: Propagator: virtual photon.	Figure 4: External lines: physical fermion and physical antifermion.

Such pictorial descriptions become quantitative by assigning to the lines, vertices and the diagram as a whole appropriate mathematical expressions, every diagram contributing quantitatively to the transition amplitude of the physical process in question. These transition amplitudes form the elements of the scattering matrix \(S\), which maps every initial state to a final state:

<math Smatrix>

S_{fin,ini} = \delta_{fin,ini} -i(2\pi)^4\delta (\sum q_{ini} - \sum q_{fin})\mathcal{M}\;, </math> where \(\sum q_{ini}\) (\(\sum q_{fin}\)) are the sum of initial (respectively final) momenta that should be equal by momentum conservation. The probability density for the transition \(|ini\rangle \rightarrow |fin\rangle\) is <review>???proportional to</review> \(\mathcal{MM^*}\), where \(\mathcal{M}\) is defined by equation (<ref>Smatrix</ref>).

By a slight change of diagrams and rules one is able to find as well the matrix elements of other operators: one just singles out one new vertex representing the operator in question. If, e.g. one is interested in matrix elements of the energy-momentum tensor, one vertex in a Feynman diagram is obtained from the expression of this tensor as a function of the fields in the theory, see figure <ref>figure_5</ref>.

Figure 5: Tree diagram with the inclusion of the operator \(\partial_\mu \phi \partial_\nu \phi\). This diagram contributes to the matrix element of the energy-momentum tensor, \(\langle 3|T_{\mu\nu}|3\rangle\). (Note that the complete expression of \(T_{\mu\nu}\) includes \(\partial_\mu \phi \partial_\nu \phi\) and other additional terms.)

As long as the diagrams in question have the form of trees the rules can be easily spelled out in such a way that the fundamental properties of a theory of elementary particles, which one wants to maintain, can indeed be realized. These are: Lorentz covariance, unitarity (conservation of probabilities in physical processes) and causality. In fact these postulates fix the rules. However as soon as closed loops of propagators appear one has to perform non-trival integrations which may just have infinity as a result. The rules, one has set up, were too naive.

Figure 6: Example: one-loop contribution to 4-pt function in \(\phi^4\).

It is thus necessary to analyze this situation carefully and to set up rules which do respect the fundamental postulates and lead to meaningful expressions which then, eventually, can be checked by experiment. Any such scheme is called a renormalization scheme. In this note we describe a specific renormalization scheme, named after its inventors: Bogoliubov, Parasiuk, Hepp, Zimmermann — abbreviated as BPHZ.

Diagrammatics

Let us look at a Feynman diagram \(\gamma\) with \(I\) internal lines, \(V\) vertices, \(N\) external lines and \(m\) closed loops. It turns out, that infinities can be traced back to diagrams which are one-particle irreducible: they are connected and stay so, if one single line is cut in the diagram. In this spirit external lines too do not have to be considered. Diagrams and subdiagrams are supposed to be spanned by their lines. To every line (by now: an internal one) is associated a propagator, to every vertex a polynomial in the momenta (see figure <ref>figure_7a</ref>).

Figure 7: (Top) Propagator associated to a fermionic line. (Bottom) Triple gluon vertex.

Figure 8: Example: diagram contributing to the m-loop 4-pt function in \(\phi^4\) theory.

A flow of momentum has to be chosen such that one has conservation of momentum at every vertex and thus for the diagram as a whole. An integration over the momenta \(k_l\)  \(l=1,...,m\) of independent closed loops has to be performed. In the simple example of figure <ref>figure_8</ref> this results in the expression:

<math integral>\int \prod_{l=1}^m \left( d^4k_l\frac{1}{(p-k_l)^2 -m^2}\frac{1}{k_l^2 - m^2}\right)\;.</math>

A degree \(d(\gamma)\), called the ultraviolet degree of divergence, is assigned to each diagram \(\gamma\) by scaling the momenta \(k_l\) in the corresponding integral by a real number \(\rho\), by considering the limit \(\rho\rightarrow \infty\) and by defining \(d(\gamma)\) as the degree of the overall power of \(\rho\) (including the contribution from the rescaling of the integration measure). \(d(\gamma)\) measures the "growth" of the integrand for large internal momenta and thus whether the integral has a chance to exist or not.

It can be show that \(d(\gamma)\) can be expressed as follows:

<math d>d(\gamma) = 4 - \sum d_a N_a + \sum (d_V - 4)</math>

where:

• \(d_a \) is the UV-dimension of field \(\phi_a\) (for example, \(d_a=1\) for a scalar boson in \(4\) dimensions),
• \(N_a \) is the <review>?</review>
• \(d_V = \sum d_a n_{a,V}(\phi_a) + degr(P_V)\) (\(n_{a,V}\) being the number of fields of type \(\phi_a\) at vertex \(V\) and \(degr(P_V)\) <review>is the ????</review>).

For the example in figure <ref>figure_6</ref> one finds \(d(\gamma)= 0\), hence the diagram is (logarithmically) divergent. Examples for non-trivial momentum dependence contributing to power counting are shown in figures <ref>figure_7a</ref> and <ref>figure_8</ref>.

Since the integrands are rational functions of the momenta, one can enforce convergence by subtracting the first \(d(\gamma)\) terms of their Taylor expansion around the configuration in which all external momenta are vanishing. This ad hoc prescription can be justified by observing that on the diagrammatic level this procedure amounts in subtracting pointlike vertices carrying a polynomial of degree \(d(\gamma)\) in external momenta. Hence, if on a formal level the fundamental postulates are satisfied, they will also be maintained after this redefinition, which leads to a meaningful expression. It is important here that one works perturbatively: to a given, say, one-loop diagram one adds a point like vertex, which in the two-loop approximation appears as an additional interaction vertex of the theory. Of course, one has introduced by this procedure free parameters which must be fixed — by the so called normalization conditions. Different schemes require different values for such parameters, but after this renormalization all schemes agree in their results.

It goes hand in hand with the perturbative construction that the proper definition of the finite part of a diagram is recursive. First those divergent subdiagrams of a large diagram have to be subtracted which have the smallest loop number, then one has to consider those (sub)diagrams of which they are subdiagrams etc. The diagrams have to be ordered. As long as divergent, one-particle irreducible subdiagrams are disjoint from or properly contained in each other this is not problematic: the respective subtractions do not interfere. If however neither of these situations is realized, one says diagrams overlap, subtractions do interfere and one has to give a prescription as how to proceed.

Figure 9: 2-loop diagram for 2-pt fct. in \(\phi^4\). The diagram has the structure that follows.
Subdiagrams: \(\gamma_i \subset \gamma \quad i= 1,2,3\;;\quad\) \(\gamma_i \cap \gamma_j \ne 0 \quad i\ne j\;;\quad\) (\(\gamma\)'s overlap).
Trees: \(U_0 = \emptyset,\; U_\gamma = \{\gamma\},\; U_i = \{\gamma_i\} i=1,2,3,\; U_{i\gamma} = \{\gamma, \gamma_i\}.\; i=1,2,3\).
Forest: \(\mathcal{F}_\gamma = \bigcup U_\alpha\).

Zimmermann <review>precise citation to refs needed</review> this problem by introducing the notion of trees: families of non-overlapping, divergent, one-particle irreducible subdiagrams (renormalization parts). The subtracted integrand \(R_\gamma(p,k)\) associated with an integrand \(I_\gamma(p,k)\) is then defined as a sum over all trees of renormalization parts of the diagram \(\gamma\); the sum over all trees represents the forest of \(\gamma\) and the subtracted integrand is given by the forest formula:

<math forests>

R_\gamma (p,k)=S_\gamma \sum_{U \in\mathcal{F}\gamma}\prod_{\lambda \in U} (-t^{d(\lambda)}_{p^\lambda}S_\lambda) I_\gamma (U) \;. </math> In equation (<ref>forests</ref>),

• \(I_\gamma(U)\) is the integrand written in variables fitting to \(U\);
• \(S_\lambda\) is substitution operator, relabelling momenta appropriately
• <review>who is U?</review>
• <review>who is \(\lambda\)?</review>
• <review>who is \(t^d\)?</review>
• <review>who is \(\mathcal{F}\gamma\)?</review>

<review> Please explain more precisely notations and extend this part that is the main point of the article. It would be nice to expand a the forest formula in the case of the example in figure 9. It would be nice to show that the formulas amounts to products of (1-T) for non overlapping divergences. </review>

Using forest formula together with a specific prescription as to go around the poles in the propagators, Zimmermann was then able to prove absolute convergence of the integrals

\[\int d^4k_1...d^4k_L R_\gamma (p,k)\;.\]

Application

In fact, with this type of construction one is not only able to study diagrams contributing to the \(S-\)matrix, but also to those forming matrix elements of composite operators. One just takes those as vertices into account in the power counting formula and proceeds via the forest formula. Hence one can now derive on the fully quantized level equations of motions, can construct currents and thus check whether symmetries are realizable and can establish other relations between operators, e.g. operator product expansions.

The main tool in these investigations is an identity, proven by Zimmermann and thus named after him. It is based on the observation that composite operators may appear in the power counting formula and then in the forest formula with subtraction degrees higher than their naive dimension. <review> The relation among the naive dimension of a composite operator and the degree of subtraction is not clear at this stage.</review> E.g. a mass term for a scalar field \(-1/2 m^2\int \phi^2\) has naive dimension 2. But one obtains also finite diagrams, if it is being assigned dimension, i.e. subtraction degree, 4. Of course the integrals obtained for the two prescriptions will, in general, be different. The Zimmermann identity now states that their difference can be expressed in terms of vertices with dimension (and power counting degree) 4.

In the example of one scalar field with \(\phi^4\) interaction it reads

\[m^2[\int \phi^2]_2 = m^2[\int \phi^2]_4 +u[\int \partial\phi\partial\phi]_4 + v[\int\phi^4]_4 \]

<review>The notation [O]_2 []_4 has not been defined</review>

The Zimmermann coefficients \(u,v\) appearing here are at least of order one-loop. This is obvious, because in the trivial order — no loops, pointlike vertices — the two objects agree, since there are no subtractions to be performed. This innocently looking identity is actually one of the most fundamental relations in quantum field theory. In order to show this when considering symmmetries we need yet some more notations.

\[ \Gamma = \sum_{n=1}^\infty[\frac{1}{n!} \int dx_1...dx_n \phi (x_1) ... \phi(x_n) \sum_{m=0}^\infty \Gamma^{(m)}_n(x_1,...,x_n)] \]

\(\Gamma \) denotes the generating functional for 1PI Green functions, where \(\Gamma^{(m)}_n(x_1,...,x_n)\) is its contribution with \(m\) loops. The Fourier transform of the latter is just the sum of all one-particle-irreducible diagrams having \(n\) external legs and \(m\) loops. Analogously, the functional with one vertex insertion \(Q(x)\) is denoted by \([Q(x)]\cdot \Gamma\). The tree approximation (no loop) is given by point like objects i.e. vertices (!), which can be identified with the classical theory. A field transformation \(\delta\phi\) can now be implemented as a differential operator \(W\) acting on \(\Gamma\)\[W\Gamma \equiv \int \delta \phi\frac{\delta}{\delta\phi}\Gamma\], named Ward identity operator. It is a remarkable theorem (action principle) that

\[W\Gamma = [\int dx Q(x)]\cdot\Gamma\]

where the explicit form of vertex insertion \(Q\) depends on \(W\).

On the classical level a symmetry of the action can be expressed as

\[W\Gamma^{(0)}=0\]

hence for those \(\Gamma\) the potential deviation from symmetry, \([\int Q]\cdot\Gamma\), starts not earlier than at one-loop order. The most interesting question is now, whether such an identity can be extended to all orders of perturbation theory:

\[W\Gamma = 0\]

Linear symmetry transformations in massive theories can be extended naively to all loop orders, if the classical action is invariant. Notable examples are translations and Lorentz transformations. Dilatations and special conformal transformations, however, do not leave invariant the mass term. Then one has to use the Zimmermann identity, finds that these symmetries are broken in one-loop (and subsequently in all higher orders) and that the breaking can be expressed in terms of the coefficients \(u,v\). For massless theories an analogous treatment is somewht more involved, but leads qualitatively to they same conclusion, hence in \(\phi^4\) dilatation and special conformal symmetry are uncurably violated: one says, they are anomalous.

In the systematic study of symmetries (internal, local gauge symmetry, supersymmetry) it always turned out that with the help of the respective Zimmermann identities one could decide whether the symmetries were anomalous or not and one was able to give an explicit expression for the breaking in terms of the Zimmermann coefficients. This points to the universal character of this identity. Even outside of perturbation theory it is such an identity which governs the truely non-trivial quantum behaviour of a quantum field theory.

References

• Lowenstein, John and Wolfhart Zimmermann (1975) 'The Power Counting theorem for Feynman Integrals with Massless Propagators.' Communications in Mathematical Physics 44: 73.

• Lowenstein, John (1976) 'Convergence Theorems for Renormalized Feynman Intgrals with Zero-mass Propagators.' Communications in Mathematical Physics 47: 53.

• Piguet, Olivier and Klaus Sibold (1986) Renormalized Supersymmetry. Boston: Birkhäuser.

• Zimmermann, Wolfhart (1968) 'The Power Counting Theorem for Minkowski Metric.' Communications in Mathematical Physics 11: 1.

• Zimmermann, Wolfhart (1969) 'Convergence of Bogoliubov's Method of Renormalization in Momentum Space.' Communications in Mathematical Physics 15: 208.

See also

Algebraic renormalization, Composite operator, Gauge theories, Multiloop Feynman integrals, Operator product expansion, Renormalization

Sponsored by: Dr. Riccardo Guida, Institut de Physique Théorique, CEA & CNRS, Gif-sur-Yvette, France

Reviewed by: Dr. John H. Lowenstein, New York University

Reviewed by: Anonymous

Accepted on: 2010-05-11 09:42:32 GMT