Bogoliubov-Parasiuk-Hepp-Zimmermann renormalization scheme
| Klaus Sibold (2010), Scholarpedia, 5(5):7306. | doi:10.4249/scholarpedia.7306 | revision #137544 [link to/cite this article] |
The Bogoliubov, Parasiuk, Hepp, Zimmermann (abbreviated BPHZ) renormalization 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).
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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. 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. 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.
\[ S_{fin,ini} = \delta_{fin,ini} -i(2\pi)^4\delta (\sum q_{ini} - \sum q_{fin})\mathcal{M} \]
\(\mathcal{MM^*}\): probability density for \(|ini\rangle \rightarrow |fin\rangle\), momentum conservation: sum over initial, resp. final momenta
By a slight change of diagrams and rules one is able to find eventually the matrix elements of other operators as well: one just singles out one vertex as 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 provided by this tensor as a function of the fields in the theory.
figure 5 \(\partial_\mu \phi \partial_\nu \phi\) in a tree diagram
contribution of part of the energy-momentum-tensor to a \(\langle 3|T_{\mu\nu}|3\rangle\) matrix element
As long as the diagrams in question have the form of trees the rules can be easily spelled out in such a way that 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, 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 one, 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. 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. Over the momenta \(k_l\) \(l=1,...,m\) of independent closed loops an integration has to be performed.
Examples for non-trivial momentum dependence contributing to power counting:
figure 7 triple gluon vertex
figure 8
\(\int \prod_{l=1}^m d^4k_l\frac{1}{(p-k_l)^2 -m^2}\frac{1}{k_l^2 - m^2}\)
example: m-loop 4-pt fct in \(\phi^4\)
Scaling the momenta \(k_l\) by a real number \(\rho\) and considering the limit \(\rho\rightarrow \infty\) to every integrand including the integration measure a degree, \(d(\gamma)\), is assigned, the power of \(\rho\), which is called the ultraviolet degree of divergence. It measures the growth of the integrand for large internal momenta and thus whether the integral has a chance to exist or not.
\(d(\gamma) = 4 - \sum d_a N_a + \sum (d_V - 4)\)
\(d_a =\) UV-dimension of field \(\phi_a\)
\(d_V = \sum d_a n_{a,V}(\phi_a) + degr(P_V)\)
\(n_{a,V}=\) number of fields of type \(\phi_a\) at vertex \(V\)
For the example in figure 6 one finds \(d(\gamma)= 0\), hence the diagram is (logarithmically) divergent.
Since the integrands are rational functions of the momenta one can enforce convergence by Taylor expansion around vanishing external momenta and subtracting the first \(d(\gamma)\) terms. This ad hoc prescription can be justified by observing that on the diagrammatic level this amounts to subtract pointlike vertices carrying a polynomial in external momenta of degree \(d(\gamma)\). 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 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 \(\phi^4\)
subdiagrams
\(\gamma_i \subset \gamma \quad 1= 1,2,3\)
\(\gamma_i \cap \gamma_j \ne 0 \quad i\ne j\)
\(\gamma\)'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 solved 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.
\[ R_\gamma (p,k)=S_\gamma \sum_{U \in\mathcal{F}\gamma}\prod_{\lambda \in U} (-t^{d(\lambda)}_{p^\lambda}S_\lambda) I_\gamma (U) \] (\(I_\gamma(U)\): integrand written in variables fitting to \(U\); \(S_\lambda\): substitution operator, relabelling momenta appropriately)
Together with a specific prescription as to go around the poles in the propagators he 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. 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
\[m^2[\int \phi^2]_2 = m^2[\int \phi^2]_4 +u[\int \partial\phi\partial\phi]_4 + v[\int\phi^4]_4 \]
figure 10
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
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.
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.
