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 \(|ini\rangle \rightarrow |mat|fin\rangle\) is <review>(comment: In order to avoid misunderstanding in Eq.(1) the dependence of \(\mathcal{M}\) on the initial and final states should be put into evidence)</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. <review>(comment:One has to give an idea of the origin of difficulties, I suggest the following insertion) However the simplest tree diagrams violate these rules, e.g. unitarity. Therefore one has to add further diagrams containing closed loops of propagators. After this \(\mathcal{M}\) appears as a loop ordered formal series of diagrams. </review> 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 <review>(suggested insertion)</review> \(\phi_a\) line (by now: an internal one) is associated a propagator <review>(suggested insertion)</review> \(\Delta_a\), to every vertex a polynomial<review>(suggested insertion)</review> \(P_V\) 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 \(\phi_a\) <review>(suggested insertion)) and is given by \( degr\Delta_a=2 d_a-4\) </review> (for example, \(d_a=1\) for a scalar boson in \(4\) dimensions),
• \(d_V = \sum d_a n_{a,V}(\phi_a) + degr(P_V)\) (\(n_{a,V}\) being the number of fields of type \(\phi_a\) at vertex \(V\) and \(degr(P_V)\).

For the example in <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>.

<review>(comment:The following paragraph deals with three different subjects : subtraction, counter terms and normalization conditions. It should be expanded discussing each item. I have tried to introduced some suggested insertions).</review> Since, <review> at least in the case of massive fields, </review>the integrands are rational functions of the momenta <review>analytic in the origin of momentum space</review>, 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. <review> Indeed, formally, the subtraction procedure is equivalent to introducing a new counter-diagram in which the divergent part has been replaced by a vertex with suitably chosen \( P_V \). </review> 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 <review> (recursively) </review>: 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 <review> which is usually called counter term </review>.

Of course, one has introduced by this procedure <review> for every counter term </review> a free parameter. <review> Indeed in general the divergent part of a diagram is identified up to a constant </review> 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 (Zimmermann W. (1969)) 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 < review> the tree </review> \(U\);
• \(S_\lambda\) is substitution operator, relabelling momenta appropriately

<review>(comment: It would be nice to expand a the forest formula in the case of the example in figure 6. 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

<review> (comment:This section can be understood only by experts. I have introduced below a number of suggestions giving hints about the main ideas)</review>

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, <review> verify if they are conserved</review> and thus check whether symmetries are realizable. <review> One can also </review>establish other relations between operators, e.g. operator product expansions.

<review> The technical difficulty in this analysis originates from the fact that the composite operators appearing in the field or in the current conservation equations correspond to vertices which introduce extra subtractions, that is, extra contribution to \( d(\gamma)\) in Eq.(3). There are situations in which the extra subtractions are “anisotropic” meaning that the extra contribution depend on the external legs of \(\gamma\) while in other situations the extra subtractions are constants. In both cases one has forest formulae with subtraction degrees higher than their naive dimension.

The difficulty is overcome thanks to an identity proven by Zimmermann, and thus named after him, which allows the reduction of extra subtracted composite operators to a linear combination of naively subtracted ones.</review> <review>(comment: The following paragraph should be omitted)</review> (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 simplest example is that of </review> a mass term for a scalar field \(-1/2 m^2\int \phi^2\)<review> which</review> has naive dimension 2. But one obtains also finite diagrams, if it is being assigned dimension, i.e. subtraction degree, 4. <review> We shall denote the first vertex by \(-1/2 m^2[\int \phi^2]_2\) and the second one by \(-1/2 m^2[\int \phi^2]_4\) </review> 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

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

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.

<review>(comment: Here again it is just for experts: why functionals? what is <math >\phi</math>? What is meant for classical theory? It is definitely not a matter of notations. Hence the following sentence should be modified)</review>(In order to show this when considering symmetries we need yet some more notations.) <review> The first problem one finds considering symmetries is to identify how they act on Feynman diagrams and time-ordered Green functions. Since these are infinite in number a symmetry of the theory should be translated into an infinite number of equations. This difficulty can be overcome introducing functionals. Let <math >\phi(x)</math> be a </review>test function <review>taking values in the classical field space and let the Fourier transform of \(\Gamma^{(m)}_n(x_1,...,x_n)\) denote the sum of all one-particle-irreducible diagrams having \(n\) external legs and \(m\) loops. One introduces the generating functional for 1PI Green functions through the formal series \[ \Gamma = \sum_{n=1}^\infty[\frac{1}{n!} \int dx_1...dx_n \phi (x_1) ... \phi(x_n) \sum_{m=0}^\infty \Gamma^{(m)}_n(x_1,...,x_n)] \]

In the tree approximation (no loop) the one-particle-irreducible diagrams are given by point like objects i.e. vertices and the functional \(\Gamma^{(0)}\) coincides with the classical action, the space-time integral of the Lagrangian density. Therefore, in this approximation, the invariance of the action under a field transformation \(\delta\phi\) can be translated into a functional differential equation \[ W\Gamma^{(0)} \equiv \int dx \delta \phi(x)\frac{\delta}{\delta\phi(x)}\Gamma^{(0)}=0\ , \] named Ward identity. Extending the differential equation to diagrams with loops one faces the extra subtraction problem discussed above. Extra subtractions induce further terms into the Ward identity corresponding to diagram with the insertion of the vertex \(Q(x)\). The new terms are given in the form \(\int dx[Q(x)]\cdot \Gamma\). In this way one proves a remarkable theorem (action principle) which corresponds to the general validity of the broken Ward identity

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

where the explicit form of vertex insertion \(Q\) depends on \(W\). Notice that the potential deviation from symmetry, \([\int Q]\cdot\Gamma\), starts not earlier than at one-loop order.</review>

<review>(comment The paragraphs in brackets should be omitted.)</review> (\(\Gamma \) 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 <review>(insert)</review><review>the Ward identity</review> \[W\Gamma = 0\] can be extended to all orders of perturbation theory.

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, finding 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 somewhat more involved, but leads qualitatively to they same conclusion, hence in \(\phi^4\) dilatation and special conformal symmetry are incurably 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 truly non-trivial quantum behavior 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