#[1]Scholarpedia (en) [2]Scholarpedia Atom feed

Bogoliubov-Parasiuk-Hepp-Zimmermann renormalization scheme

   From Scholarpedia
   Klaus Sibold (2010), Scholarpedia, 5(5):7306. [3]doi:10.4249/scholarpedia.7306
   revision #137544 [[4]link to/cite this article]
   Jump to: [5]navigation, [6]search
   Post-publication activity
   (BUTTON)

   Curator: [7]Klaus Sibold
   Contributors:


   0.20 -

   [8]Jonathan R. Williford
   0.20 -

   [9]Benjamin Bronner
   0.20 -

   [10]Riccardo Guida
   0.20 -

   [11]Carlo Maria Becchi

   [12]John H. Lowenstein
     * [13]Dr. Klaus Sibold, Institut für Theoretische Physik Fakultät für Physik
       und Geowissenschaften Universität Leipzig

   The Bogoliubov, Parasiuk, Hepp, Zimmermann (abbreviated BPHZ) [14]renormalization
   scheme is a mathematically consistent method of rendering Feynman amplitudes
   finite while maintaining the fundamental postulates of relativistic quantum field
   theory (Lorentz invariance, unitarity, [15]causality). Technically it is based on
   the systematic subtraction of momentum space integrals. This distinguishes it
   from other methods of renormalization. For massless particles the scheme has been
   enlarged by Lowenstein and is then called BPHZL.

Contents

     * [16]1 The problem
     * [17]2 Diagrammatics
     * [18]3 Application
     * [19]4 General remarks
     * [20]5 References
     * [21]6 Further Reading
     * [22]7 See also

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 [23]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: [24]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.

   \[\tag{1} S_{\mathrm{fin,in}} = \delta_{\mathrm{fin,in}} -i(2\pi)^4\delta (\sum
   q_{\mathrm{in}} - \sum q_{\mathrm{fin}}){\mathcal{M}}_{\mathrm{fin,in}} \]

   where \(\sum q_{\mathrm{in}}\;\;\left(\sum q_{\mathrm{fin}}\right)\) are the sum
   of initial (respectively final) momenta that should be equal by momentum
   conservation. The probability density for the transition \(|{\mathrm{in}}\rangle
   \rightarrow |{\mathrm{fin}}\rangle\)

   is \(\mathcal{M}_{\mathrm{fin,in}}\mathcal{M}_{\mathrm{fin,in}}^*\ ,\) where
   \(\mathcal{M}_{\mathrm{fin,in}}\) is defined by equation ([25]1).

   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, see Figure
   [26]5.
   Figure 5: Tree diagram with the inclusion of the operator \(\partial_\mu \varphi
   \partial_\nu \varphi\ .\) This diagram contributes to the matrix element of the
   energy-momentum tensor, \(\langle 3\, \mathrm{particles}|T_{\mu\nu}|3\,
   \mathrm{particles}\rangle\ .\) (Note that the complete expression of
   \(T_{\mu\nu}\) includes \(\partial_\mu \varphi \partial_\nu \varphi\) and other
   additional terms.)

   As long as the diagrams in question have the form of trees the rules yield
   mathematically well defined expressions and maintain Lorentz [27]covariance.
   Tree-level transition amplitudes violate however unitarity (conservation of
   probabilities in physical processes), and causality which are the further
   fundamental properties which should be valid for a theory of elementary
   particles. Actually [28]loops of propagators (closed paths in the diagrams) have
   to appear, if unitarity and causality are requested: indeed
   \(\mathcal{M}_{\mathrm{fin,in}}\) appears as a loop-ordered formal series of
   diagrams. (A \(L\)-loop diagram being weighted by \(\hbar ^L\ ,\) where\( \hbar\)
   is the Planck's constant.) Loops imply however (according to the rules) that one
   has to perform non-trivial 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 \(\varphi^4\ .\)

   It is thus necessary to analyze this situation carefully and to set up modified
   rules which do respect the fundamental postulates (Lorentz covariance, unitarity,
   causality), lead to meaningful expressions which then, eventually, can be checked
   by experiment. Any such set of rules 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 with \(I\) internal lines, \(V\) vertices, \(N\)
   external lines and \(L\) 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 do not have to be considered, they serve only as a remainder for external
   momenta entering the diagram. Diagrams and subdiagrams are supposed to be
   "spanned" by their lines, the vertices attached to the lines of a diagram or
   subdiagram also belong to the diagram (resp. subdiagram). To every line of type
   \(\varphi_a\) (by now: an internal one) is associated a propagator,
   \(\Delta_{\mathrm{c}}^{(a)}\ ,\) to every vertex \(v\) a polynomial \(P_v\) in
   the momenta. Examples for non-trivial momentum dependence contributing to power
   counting are shown in Figure [29]7.
   Figure 7: (Top) Propagator associated to a fermionic line. (Bottom) Triple gluon
   vertex.
   Figure 8: Example: diagram contributing to the \(L\)-loop 4-pt function in
   \(\varphi^4\) 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
   \(k_l\)  \(l=1,...,L\) of independent loops has to be performed. In the simple
   example of Figure [30]8 this results in the expression: \[\tag{2} \int
   \prod_{l=1}^L \left( d^4k_l\frac{1}{(p-k_l)^2 -m^2}\frac{1}{k_l^2 -
   m^2}\right)\;.\]

   A degree \(d(\gamma)\ ,\) called the [31]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: \[\tag{3}
   d(\gamma) = 4 - \sum_a d_a N_a + \sum_v (d_v - 4)\]

   where:
     * \(N_a\) is the number of external lines of type \(\varphi_a\ ,\)
     * \(d_a \) is the UV-dimension of field \(\varphi_a\) and is given by \( \deg
       (\Delta_c^{(a)})=2 d_a-4\) (for example, \(d_a=1\) for a scalar boson in
       \(4\) dimensions),
     * \(d_v = \sum_a d_a n_{a,v} + \deg(P_v)\) (\(n_{a,v}\) being the number of
       fields of type \(\varphi_a\) at vertex \(v,\) and \(\deg(P_v)\) being the
       degree of the polynomial in the momenta associated to vertex \(v\)).

   For the example in Figure [32]6 one finds \(d(\gamma)= 0\ ,\) hence the diagram
   is (logarithmically) divergent.

   Since, at least for massive fields, the integrand is a rational function of the
   momenta, analytic at the origin of momentum space, one can enforce convergence by
   Taylor expanding around vanishing external momenta and subtracting all terms up
   to and including degree \(d(\gamma)\) in this expansion. (The operator that
   performs a Taylor expansion in a given set of momenta \(p\) up to -and included-
   degree \(d(\gamma)\) is denoted by \(t^{d(\gamma)}_p\)). 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
   \(d(\gamma)\ .\) Indeed, formally the subtraction procedure is equivalent to
   introducing a new diagram in which the divergent subdiagram has been replaced by
   a vertex \(v\) with suitably chosen \(P_v\ ,\) known as counterterm. 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 the fact that one works perturbatively (loop expansion): e.g. the
   counterterm defined to subtract a one-loop diagram (i.e. of order \(\hbar\)) when
   inserted as an interaction vertex in diagram with four loops (i.e. of order
   \(\hbar^4\)), will give rise to a contribution of order five loops (i.e. of order
   \(\hbar^5\)).

   Of course, by this procedure one has introduced for every counterterm a free
   parameter, which must be fixed by the so called normalization conditions.
   Different schemes require different values for such parameters, but after this
   re-normalization 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. Given a [33]multiloop
   Feynman diagram, one has first to subtract the divergent subdiagrams which have
   the smallest loop number, then one has to consider larger (sub)diagrams of which
   included the previous subdiagrams etc. In a word, the diagrams have to be ordered
   in some way to be properly treated.

   As long as divergent, one-particle irreducible subdiagrams are mutually disjoint
   (with respect to their lines, irrespective of vertices), or properly contained in
   each other, this is not problematic because the respective subtractions do not
   interfere. As an example, consider Figure [34]8 with \(L=2\ .\) It is fairly
   obvious (and can be proven rigorously) that Dyson's formula for the renormalized
   Feynman diagram \(R_{\gamma} (p,k)\) (Dyson F.J., 1949), \[ R_{\gamma} (p,k) =
   S_{\gamma} \prod_{\lambda\in U} (1-t^{d(\lambda)}_{p^\lambda}S_{\lambda})
   I_{\gamma} (U), \] leads to convergence. Here \(U\) is the set \(U=\{\gamma,
   \gamma_1, \gamma_2\}\) of diagrams \(\gamma,\, \gamma_1\) and \(\gamma_2\ .\)
   Figure 9: Identification of diagrams \(\gamma\ ,\) \(\gamma_1\) and \(\gamma_2\
   .\)

   \(I_\gamma(U)\) is the integrand written in variables fitting to the set \(U.\)
   \(S_\lambda\) is a substitution operator, relabeling momenta appropriately.
   \(p^\lambda\) is the set of the external momenta of the subdiagram \(\lambda,\)
   as prepared by \(S_\lambda\ .\)

   Given a set of subdiagrams \(U=\{\gamma_1,\cdots,\gamma_n\}\ ,\) if none of the
   conditions \[\gamma_i \subseteq \gamma_j,\, \gamma_j \subseteq \gamma_i,\,
   \gamma_i \cap \gamma_j = \emptyset\] holds for any couple \((\gamma_i,\gamma_j)\)
   of elements of U , then the diagrams are said to overlap. (Note that inclusion
   and disjunction are here intended in terms of lines.) See Figure [35]10 for an
   example.
   Figure 10: 2-loop diagram for the 2-point function in \(\varphi^4\ .\) 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).
   Forests: \(U_0 = \emptyset,\; U_\gamma = \{\gamma\},\; U_i = \{\gamma_i\}
   i=1,2,3,\; U_{i\gamma} = \{\gamma, \gamma_i\}.\; i=1,2,3\ .\)
   Family of Forests: \(\mathcal{F}_\gamma = \bigcup U_\alpha\ .\)

   In general, when divergent one-particle irreducible subdiagrams overlap,
   subtractions do interfere and one has to give a prescription as how to proceed.
   Zimmermann (Zimmermann W., 1969) solved this problem by introducing the notion of
   forests, defined as families of divergent one-particle irreducible (sub)diagrams
   (known as renormalization parts) which are strongly non-overlapping, i.e. which
   are pairwise strongly-disjoint (disjoint in terms of lines and vertices) or
   included one in the other (in terms of lines).

   In order to understand Zimmermann's solution it is instructive to continue with
   the example \(L=2\) in Figure [36]8. If one multiplies out the product in the
   Dyson formula and takes into account that \[
   (1-t^{d(\gamma)}_{p^{\gamma}})t^{d(\gamma_1)}_{p^{\gamma_1}}t^{d(\gamma_2)}_{p^\g
   amma2} I_\gamma(U)=0 \] one can rewrite the result in the form \[ R_{\gamma}
   (p,k) = S_{\gamma} \sum_{U\in F_\gamma} \prod_{\lambda\in U}
   (-t^{d(\lambda)}_{p^\lambda} S_{\lambda}) I_{\gamma} (U), \] if one chooses as
   family of forests \(\mathcal{F}_\gamma\) \[ \mathcal{F}_\gamma= \{ \emptyset,
   \{\gamma\},\{\gamma_1\}, \{\gamma_2\}, \{\gamma, \gamma_1\}, \{\gamma,
   \gamma_2\}\}, \] i.e. all the sets of renormalization parts of \(\gamma\) which
   are strongly-non-overlapping. Note that, remarkably enough, the diagrams
   \(\gamma_1\) and \(\gamma_2\) are not strongly-disjoint (nor related by
   inclusion) because they have a vertex in common (and only that), hence the
   forests \(\{\gamma_1, \gamma_2\}\) and \(\{\gamma, \gamma_1, \gamma_2\}\) do not
   appear in \(\mathcal{F}_\gamma\ .\) Moreover, it is essential to define
   \(\mathcal{F}_\gamma\) such that the empty set belongs to it (one correspondingly
   sets \(I_\gamma(\emptyset)=I_\gamma (\{\gamma\})\ ;\) no \(t^d_p\) is needed).

   Going back to the general case, it turns out that this observation is decisive
   for the generalization of the subtraction prescription to all diagrams, i.e.  the
   sum over all families of strongly-non-overlapping, divergent
   one-particle-irreducible (sub)diagrams of a given diagram \(\gamma\) is the right
   notion to lead to convergence in the general case, even when \(\gamma\) contains
   overlapping renormalization parts.

   The subtracted integrand \(R_\gamma(p,k)\) associated with an integrand
   \(I_\gamma(p,k)\) is then defined as the sum over all possible forests of
   strongly-non-overlapping renormalization parts of the diagram \(\gamma\)
   (including the empty set with no subtraction), as given by the forest formula: \[
   R_\gamma (p,k)=S_\gamma \sum_{U\in F_\gamma}\prod_{\lambda\in U}
   (-t^{d(\lambda)}_{p^\lambda}S_\lambda) I_\gamma (U). \]

   Using the 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 \(\int d^4k_1...d^4k_m R_\gamma (p,k)\ .\)

   The absolute convergence originates from a very elegant treatment of the
   \(\varepsilon\) in the propagator: Zimmermann replaced the standard
   \(i\varepsilon\) by \[ \Delta_{\mathrm{c}}(p)= \frac{i}{p^2 - m^2 +
   i\varepsilon({\mathbf{p}}^2+m^2)} \] and showed that this definition leads to a
   Euclidean majorant and minorant for the Minkowski propagator. The respective
   inequalities read \[ \frac{1}{\sqrt{1+\varepsilon^2}}\frac{1}{k_0^2+{\mathbf
   {k}}^2+m^2} \leq \frac{1}{|k_0^2-{\mathbf{k}}^2-m^2 +i\varepsilon({\mathbf
   {k}}^2+m^2)|} \leq \frac{1}
   {\sqrt{1+\frac{4}{\varepsilon^2}}}\frac{1}{k_0^2+{\mathbf {k}}^2+m^2}. \] Hence
   one avoids the problems of conditional convergence appearing in the conventional
   formulation. Lorentz covariance is recovered in the limit of vanishing
   \(\varepsilon\ .\)

   Let us illustrate these remarks in the simplest possible example, the diagram for
   \(L=1\) in Figure [37]8.

   Since the degree of divergence is \(d(\gamma)=0\) one has to subtract from the
   integrand \(I_\gamma(p,k)\) just its value at \(p=0\) and obtains for the desired
   integral (up to numerical overall factors) \[ \int d^4k\, R_{\gamma} (p,k) = \int
   d^4k \left(\frac{1}{(p-k)^2 - m^2
   +i\varepsilon(({\mathbf{p}}-{\mathbf{k}})^2+m^2)} \frac{1}{k^2 - m^2
   +i\varepsilon({\mathbf {k}}^2+m^2)} - \frac{1}{k^2 - m^2
   +i\varepsilon({\mathbf{k}}^2+m^2)} \frac{1}{k^2 - m^2
   +i\varepsilon({\mathbf{k}}^2+m^2)}\right) \] This integral clearly converges
   absolutely since \[ \int d^4k|R_\gamma(p,k)|\leq \int d^4k
   \frac{1}{1+\frac{4}{\varepsilon^2}} \left|\frac{-p^2+2pk}{((p-k)^2_E + m^2)(k_E^2
   + m^2)^2}\right| \] does so by Euclidean power counting.

   The existence of the limit \(\varepsilon \rightarrow 0\) is difficult to see in
   this momentum space form of the integral. One can however verify it by going over
   to another parametrization (Feynman parameters). It turns out that the integral
   approaches a Lorentz covariant distribution.

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
   [38]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 relations between composite operators on the fully quantized level. A very
   important example is provided by [39]operator product expansions. Another one is
   constituted by equations of motions and currents. One can now verify if the
   latter are conserved and thus check whether symmetries are realizable on the
   quantum level.

   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 contributions to \(d(\gamma)\) in Eq. ([40]3). There are situations in
   which the extra subtractions are "anisotropic" meaning that the extra
   contributions depend on the external legs of \(\gamma\) and not just on their
   dimensionally weighted sum, while in other situations the extra subtractions are
   constants. In both cases one has forest formulae with subtraction degrees higher
   than their naive dimension.

   This 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. The simplest
   example is that of a mass term for a scalar field \(m^2\int \varphi^2\) which has
   naive dimension 2. But one obtains also finite diagrams, if it is being assigned
   dimension, i.e. subtraction degree, 4. We shall denote the first vertex by
   \(m^2[\int \varphi^2]_2\) and the second one by \(m^2[\int \varphi^2]_4\ .\) 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 \(\varphi^4\) interaction it reads

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

   [41]Figure 10.png

   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 we consider in some
   detail how symmetries can be implemented in quantum field theory using the BPHZ
   renormalization scheme.

   Clearly we have to understand how symmetry transformations act on Feynman
   diagrams and thereafter on the different types of Green functions which can be
   expressed as sums of diagrams. Since there are infinitely many, say, time ordered
   Green functions a symmetry of the theory should be translated into an infinite
   number of equations. A convenient tool for treating them at once are functionals
   generating the desired Green functions upon differentiation with respect to
   suitably chosen set of auxiliary functions. Let now \(\phi(x)\) denote a test
   function with values in the classical field space and let the Fourier transform
   \(\Gamma_n^{(L)}(x_1,...,x_n)\) denote the sum of all one-particle-irreducible
   diagrams having \(n\) external legs and \(L\) closed loops. Then one introduces
   the generating functional for 1PI Green functions through the formal series \[
   \Gamma = \sum_{n=1}^\infty\left[\frac{1}{n!} \int dx_1...dx_n \phi (x_1) ...
   \phi(x_n) \sum_{L=0}^\infty \Gamma^{(L)}_n(x_1,...,x_n)\right]. \] In the tree
   approximation (no loop) the one-particle-irreducible Green functions are given by
   pointlike objects, i.e. "vertices" and the functional \(\Gamma^{(0)}\) can be
   identified with the classical action, the spacetime integral of the
   [42]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:

   \[ W\Gamma^{(0)} \equiv \int \delta \phi\frac{\delta}{\delta\phi}\Gamma^{(0)}=0,
   \] named Ward identity, \(W\) being the Ward identity operator.

   Extending the differential equation to diagrams with closed loops one faces the
   extra subtraction problem discussed above. Extra subtractions induce further
   terms into the Ward identity corresponding to diagrams with the insertion of an
   additional vertex \(Q(x)\ ,\) more precisely (and this a non-trivial statement)
   as a normal product \(\int dx [Q(x)]\cdot \Gamma\ .\) This is the content of a
   remarkable theorem (action principle) which corresponds to the general validity
   of the broken Ward identity \[ W\Gamma = \left[\int dx \, Q(x)\right]\cdot\Gamma.
   \] Here the explicit form of the insertion \(Q\) and its subtraction degree
   depend on \(W\ .\) Notice that the potential deviation from symmetry, \([\int
   Q]\cdot \Gamma\ ,\) is at least of one-loop order if we started from an invariant
   classical action.

   The most interesting question is now, whether a Ward identity:

   \[ W\Gamma=0 \] holds to all orders of [43]perturbation theory.

   Linear symmetry transformations in massive theories can be extended naively to
   all loop orders, if the classical action is invariant. 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 \(u,v\ .\)

   Does this breaking disappear for vanishing mass? In order to answer this question
   appropriately one has to enlarge the BPHZ subtraction scheme, since momentum
   subtractions at vanishing external momenta would lead to spurious infrared
   divergences. One proceeds by introducing an auxiliary mass term \[ \Gamma_M =
   -\frac{1}{2}M^2(s-1)^2 \int dx \phi^2, \] where the variables \(s\) and \(s-1\)
   participate in the subtractions like external momenta of a diagram. Ultraviolet
   subtractions are performed at \(s=0\ ,\) hence do not introduce infrared
   divergences, subsequent infrared subtractions, namely subtractions with respect
   to \(s-1\) re-install the correct infrared behavior, in particular the pole at
   \(s=1\) of the propagator, i.e. lead to the massless theory (Lowenstein J. and
   Zimmermann W. (1975); Lowenstein J. (1976)). We shall name this enlarged scheme
   BPHZL. Now one has to treat symmetries analogously to the massive case. And one
   arrives at the analogous conclusion: in the \(\varphi^4\) theory dilatation and
   special conformal symmetry are incurably violated: one says, they are anomalous.

   In the systematic study of symmetries (non-linear, internal, local gauge
   symmetry, [44]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 behaviour of a quantum field theory.

General remarks

   What are the great successes of the BPHZ renormalization scheme? The first
   certainly was the confirmation of Wilson's hypothesis on operator product
   expansions in perturbation theory based on Zimmermann's normal products. This
   provided the basis of confidence for the rich application of Wilson's ideas in
   particle physics.

   The second is the treatment of symmetries. Once Zimmermann and Lowenstein had
   enlarged the subtraction scheme (Lowenstein J. and Zimmermann W. (1975);
   Lowenstein J. (1976)) as to treat successfully and with full mathematical rigor
   theories containing massless particles the road was open to quantize
   [45]non-abelian gauge theories, in particular non-vectorlike (i.e. chiral, cf.
   [46][1]) ones, i.e. theories containing left- or right-handed fields only.

   Basing the required analysis solely on power counting and the action principle
   Becchi, Rouet and Stora were able to quantize non-abelian gauge theories and in
   particular to give a clear cut criterion under which conditions those were
   physical, namely maintaining the axioms: here unitarity is the crucial issue.
   After having translated broken [47]gauge transformations into the language of a
   symmetry with anti-commuting parameters, thereafter called [48]BRS
   transformations, they showed that the breaking of this symmetry leads to
   violation of unitarity. Absence of this breaking is assured to all orders of
   perturbation theory if the respective anomaly coefficient in the one-loop
   approximation vanishes. This restricts the admissible representations of fermions
   in the model.

   Interestingly enough, general \(N=1\) supersymmetric non-abelian gauge theories
   belong also to this wide class of chiral theories, hence indeed are also prime
   candidates to be quantized using BPHZL. This has been done. (Piguet O. and Sibold
   K. (1986))

   It is remarkable that only as late as 1998 the first all order renormalization of
   a simplified version of the electroweak standard model has been achieved (Kraus
   E. (1998)). And it has been based on this scheme.

   The successful quantization of all of these non-abelian gauge theories is thus at
   the very basis of today particle physics, the success being attributable to the
   BPHZL renormalization scheme.

   More generally speaking this scheme is a perfect tool for studying structural
   relations: current algebras, including the algebra of the energy-momentum tensor
   and superconformal algebras; theories with vanishing \(\beta\)-functions (often
   called "finite" theories); similarly one can rigorously formulate topological
   field theories and extract the relevant information. The precise definition of
   anomalies and their interrelations is at its core, since one can obtain them
   constructively. In this sense the BPHZL scheme is still effectively used and has
   not been superseded by any other scheme.

   In recent years another aspect has become of interest: the algebraic structure
   which is behind Feynman rules on the one hand, and behind the forest formula on
   the other. In its simplest form the forest formula carries a [49]Hopf algebra
   structure and becomes as such an element of a rich mathematical theory. Like for
   a similar algebraization in pure mathematics it is to be expected that this
   process also reaches now renormalization theory and produces new, unexpected
   results which could not be found in the concrete realizations. Hopefully they
   lead to new physical insight.

   Having spoken so much on one scheme one should perhaps put it in the context of
   other renormalization schemes and try to contrast it with those.

   First of all one has to recall that all renormalization schemes are equivalent
   upon finite renormalizations. This statement is the content of a theorem due to
   K. Hepp who has given an axiomatic characterization of what a renormalization
   scheme is. Roughly speaking it is any set of prescriptions which is
   mathematically consistent and tells one, how to obtain, say Green functions or
   operators (like the scattering operator) satisfying Lorentz covariance, unitarity
   and causality. (This may or may not be realized via Feynman diagrams!) After this
   theorem the use of any specific scheme is a matter of practice but not a matter
   of principle.

   So, BPHZL does not maintain BRS invariance (even in vector-like models) hence is
   not a very practical tool for explicit calculations in such theories. Here, for
   instance dimensional regularization and subsequent renormalization is the most
   practical scheme because it is naively compatible with this type of gauge
   invariance. Dimensional renormalization is however at least as cumbersome as
   BPHZL in chiral models, since there is no naive treatment of \(\gamma_5\ ,\) the
   latter being a genuine object of four-dimensional spacetime. If one wants to see
   how a renormalization scheme constructively exhausts the axioms (in particular
   unitarity and causality) one will use the Epstein-Glaser method because there the
   prescriptions of how to construct the \(S\)-operator are directly based on these
   principles. Similarly, if one wants to maintain causality one might stick to
   analytic renormalization. For very concrete models one will set up combinations
   of these schemes in order to facilitate explicit computations.

   Looking back at about forty years it seems that as far as structural relations
   are concerned and their use in physically relevant models BPHZL is the leading
   scheme, just because it is constructive and does not only signal, for instance
   the breakdown of a symmetry, but at the same time explicitly exhibit how the
   symmetry is broken. It is however to be repeated: this is a matter of practice
   and not of principle.

References

     * Dyson, Freeman J. (1949) 'The S-matrix in [50]quantum electrodynamics' Phys.
       Rev. 75: 1736. [51]doi:10.1103/PhysRev.75.1736.

     * Kraus, Elisabeth (1998) 'Renormalization of the electroweak standard model to
       all orders' Annals of Physics 262: 155. [52]doi:10.1006/aphy.1997.5746.

     * Lowenstein, John and Wolfhart Zimmermann (1975) 'The Power Counting theorem
       for Feynman Integrals with Massless Propagators.' Communications in
       Mathematical Physics 44: 73. [53]doi:10.1007/BF01609059.

     * Lowenstein, John (1976) 'Convergence Theorems for Renormalized Feynman
       Intgrals with Zero-mass Propagators.' Communications in Mathematical Physics
       47: 53. [54]doi:10.1007/BF01609353.

     * Piguet, Olivier and Klaus Sibold (1986) Renormalized Supersymmetry. Boston:
       Birkhäuser. [55]doi:10.1007/978-1-4684-7326-1.

     * Zimmermann, Wolfhart (1968) 'The Power Counting Theorem for Minkowski
       Metric.' Communications in Mathematical Physics 11: 1.
       [56]doi:10.1007/BF01654298.

     * Zimmermann, Wolfhart (1969) 'Convergence of Bogoliubov's Method of
       Renormalization in Momentum Space.' Communications in Mathematical Physics
       15: 208. [57]doi:10.1007/BF01645676.

   Internal references
     * Jean Zinn-Justin and Riccardo Guida (2008) [58]Gauge invariance.
       [59]Scholarpedia, 3(12):8287. [60]doi:10.4249/scholarpedia.8287.

     * Gerard 't Hooft (2008) [61]Gauge theories. Scholarpedia, 3(12):7443.
       [62]doi:10.4249/scholarpedia.7443.

     * Guy Bonneau (2009) [63]Local operator. Scholarpedia, 4(9):9669.
       [64]doi:10.4249/scholarpedia.9669.

     * Vladimir Alexandrovich Smirnov (2009) [65]Multiloop Feynman integrals.
       Scholarpedia, 4(6):8507. [66]doi:10.4249/scholarpedia.8507.

     * Guy Bonneau (2009) [67]Operator product expansion. Scholarpedia, 4(9):8506.
       [68]doi:10.4249/scholarpedia.8506.

Further Reading

     * Bogoliubov, Nikolai N. and Dimitri V. Shirkov (1959) Introduction to the
       theory of quantized fields. Wiley-Interscience

     * Collins, John (1984) Renormalization. Cambridge
       [69]doi:10.1017/CBO9780511622656.005.

     * DeWitt, Cecile and Raymond Stora (eds.) (1971) Statistical mechanics and
       quantum field theory. Gordon and Breach (in particular pp. 429-500)

     * Itzykson, Claude and Jean-Bernard Zuber (1980) Quantum field theory.
       McGraw-Hill, Inc.

     * Kugo, Taichiro (1997) Eichtheorie. Springer (German)
       [70]doi:10.1007/978-3-642-59128-0.

     * Velo, Giorgio and Arthur S. Wightman (eds.) (1975) Renormalization theory. D.
       Reidel, Dordrecht (in particular pp. 95-160)

See also

   [71]Algebraic renormalization, [72]BRST Symmetry, [73]Composite operator,
   [74]Dimensional Renormalization, [75]Gauge theories, [76]Multiloop Feynman
   integrals, [77]Operator product expansion, [78]Renormalization ,
   [79]Supersymmetry
   Sponsored by: [80]Dr. Riccardo Guida, Institut de Physique Théorique, CEA & CNRS,
   Gif-sur-Yvette, France
   [81]Reviewed by: [82]Dr. John H. Lowenstein, New York University
   [83]Reviewed by: [84]Anonymous
   Accepted on: [85]2010-05-11 09:42:32 GMT
   Retrieved from
   "[86]http://www.scholarpedia.org/w/index.php?title=Bogoliubov-Parasiuk-Hepp-Zimme
   rmann_renormalization_scheme&oldid=137544"
   [87]Categories:
     * [88]Quantum and Statistical Field Theory
     * [89]Quantum Field Theory (Foundations)

Personal tools

     * [90]Log in

Namespaces

     * [91]Page
     * [92]Discussion

Variants

Views

     * [93]Read
     * [94]View source
     * [95]View history

Actions

Search

   ____________________ (BUTTON) Search

Navigation

     * [96]Main page
     * [97]About
     * [98]Propose a new article
     * [99]Instructions for Authors
     * [100]Random article
     * [101]FAQs
     * [102]Help

Focal areas

     * [103]Astrophysics
     * [104]Celestial mechanics
     * [105]Computational neuroscience
     * [106]Computational intelligence
     * [107]Dynamical systems
     * [108]Physics
     * [109]Touch
     * [110]More topics

Activity

     * [111]Recently published articles
     * [112]Recently sponsored articles
     * [113]Recent changes
     * [114]All articles
     * [115]List all Curators
     * [116]List all users
     * [117]Scholarpedia Journal

Tools

     * [118]What links here
     * [119]Related changes
     * [120]Special pages
     * [121]Printable version
     * [122]Permanent link

     * [123][twitter.png?303]
     * [124][gplus-16.png]
     * [125][facebook.png?303]
     * [126][linkedin.png?303]

     * [127]Powered by MediaWiki [128]Powered by MathJax [129]Creative Commons
       License

     * This page was last modified on 17 November 2013, at 05:16.
     * This page has been accessed 79,628 times.
     * "Bogoliubov-Parasiuk-Hepp-Zimmermann renormalization scheme" by [130]Klaus
       Sibold is licensed under a [131]Creative Commons
       Attribution-NonCommercial-ShareAlike 3.0 Unported License. Permissions beyond
       the scope of this license are described in the [132]Terms of Use

     * [133]Privacy policy
     * [134]About Scholarpedia
     * [135]Disclaimers

References

   Visible links:
   1. http://www.scholarpedia.org/w/opensearch_desc.php
   2. http://www.scholarpedia.org/w/index.php?title=Special:RecentChanges&feed=atom
   3. http://dx.doi.org/10.4249/scholarpedia.7306
   4. http://www.scholarpedia.org/w/index.php?title=Bogoliubov-Parasiuk-Hepp-Zimmermann_renormalization_scheme&action=cite&rev=137544
   5. http://www.scholarpedia.org/article/Bogoliubov-Parasiuk-Hepp-Zimmermann_renormalization_scheme#mw-head
   6. http://www.scholarpedia.org/article/Bogoliubov-Parasiuk-Hepp-Zimmermann_renormalization_scheme#p-search
   7. http://www.scholarpedia.org/article/User:Klaus_Sibold
   8. http://www.scholarpedia.org/article/User:Jonathan_R._Williford
   9. http://www.scholarpedia.org/article/User:Benjamin_Bronner
  10. http://www.scholarpedia.org/article/User:Riccardo_Guida
  11. http://www.scholarpedia.org/article/User:Carlo_Maria_Becchi
  12. http://www.scholarpedia.org/article/User:John_H._Lowenstein
  13. http://www.scholarpedia.org/article/User:Klaus_Sibold
  14. http://www.scholarpedia.org/article/Renormalization
  15. http://www.scholarpedia.org/article/Causality
  16. http://www.scholarpedia.org/article/Bogoliubov-Parasiuk-Hepp-Zimmermann_renormalization_scheme#The_problem
  17. http://www.scholarpedia.org/article/Bogoliubov-Parasiuk-Hepp-Zimmermann_renormalization_scheme#Diagrammatics
  18. http://www.scholarpedia.org/article/Bogoliubov-Parasiuk-Hepp-Zimmermann_renormalization_scheme#Application
  19. http://www.scholarpedia.org/article/Bogoliubov-Parasiuk-Hepp-Zimmermann_renormalization_scheme#General_remarks
  20. http://www.scholarpedia.org/article/Bogoliubov-Parasiuk-Hepp-Zimmermann_renormalization_scheme#References
  21. http://www.scholarpedia.org/article/Bogoliubov-Parasiuk-Hepp-Zimmermann_renormalization_scheme#Further_Reading
  22. http://www.scholarpedia.org/article/Bogoliubov-Parasiuk-Hepp-Zimmermann_renormalization_scheme#See_also
  23. http://www.scholarpedia.org/article/Gluon
  24. http://www.scholarpedia.org/article/Vertex
  25. http://www.scholarpedia.org/article/Bogoliubov-Parasiuk-Hepp-Zimmermann_renormalization_scheme#Eq-1
  26. http://www.scholarpedia.org/article/Bogoliubov-Parasiuk-Hepp-Zimmermann_renormalization_scheme#fig:Figure_5.png
  27. http://www.scholarpedia.org/article/Covariance
  28. http://www.scholarpedia.org/article/Multiloop_Feynman_integrals
  29. http://www.scholarpedia.org/article/Bogoliubov-Parasiuk-Hepp-Zimmermann_renormalization_scheme#fig:Figure_7a-bis.png
  30. http://www.scholarpedia.org/article/Bogoliubov-Parasiuk-Hepp-Zimmermann_renormalization_scheme#fig:Figure_8bis.png
  31. http://www.scholarpedia.org/article/Multiloop_Feynman_integrals
  32. http://www.scholarpedia.org/article/Bogoliubov-Parasiuk-Hepp-Zimmermann_renormalization_scheme#fig:Figure_6.png
  33. http://www.scholarpedia.org/article/Multiloop_Feynman_integrals
  34. http://www.scholarpedia.org/article/Bogoliubov-Parasiuk-Hepp-Zimmermann_renormalization_scheme#fig:Figure_8bis.png
  35. http://www.scholarpedia.org/article/Bogoliubov-Parasiuk-Hepp-Zimmermann_renormalization_scheme#fig:Figure_9.png
  36. http://www.scholarpedia.org/article/Bogoliubov-Parasiuk-Hepp-Zimmermann_renormalization_scheme#fig:Figure_8bis.png
  37. http://www.scholarpedia.org/article/Bogoliubov-Parasiuk-Hepp-Zimmermann_renormalization_scheme#fig:Figure_8bis.png
  38. http://www.scholarpedia.org/article/Local_operator
  39. http://www.scholarpedia.org/article/Operator_product_expansion
  40. http://www.scholarpedia.org/article/Bogoliubov-Parasiuk-Hepp-Zimmermann_renormalization_scheme#Eq-3
  41. http://www.scholarpedia.org/article/File:Figure_10.png
  42. http://www.scholarpedia.org/article/Lagrangian
  43. http://www.scholarpedia.org/article/Perturbation_theory
  44. http://www.scholarpedia.org/article/Supersymmetry
  45. http://www.scholarpedia.org/article/Gauge_theories
  46. http://en.wikipedia.org/wiki/Chirality_(physics)
  47. http://www.scholarpedia.org/article/Gauge_invariance
  48. http://www.scholarpedia.org/article/BRST_symmetry
  49. http://www.scholarpedia.org/article/Hopf_algebra_and_quantum_fields
  50. http://www.scholarpedia.org/article/Quantum_electrodynamics
  51. http://dx.doi.org/10.1103/PhysRev.75.1736
  52. http://dx.doi.org/10.1006/aphy.1997.5746
  53. http://dx.doi.org/10.1007/BF01609059
  54. http://dx.doi.org/10.1007/BF01609353
  55. http://dx.doi.org/10.1007/978-1-4684-7326-1
  56. http://dx.doi.org/10.1007/BF01654298
  57. http://dx.doi.org/10.1007/BF01645676
  58. http://www.scholarpedia.org/article/Gauge_invariance
  59. http://www.scholarpedia.org/article/Scholarpedia
  60. http://dx.doi.org/10.4249/scholarpedia.8287
  61. http://www.scholarpedia.org/article/Gauge_theories
  62. http://dx.doi.org/10.4249/scholarpedia.7443
  63. http://www.scholarpedia.org/article/Local_operator
  64. http://dx.doi.org/10.4249/scholarpedia.9669
  65. http://www.scholarpedia.org/article/Multiloop_Feynman_integrals
  66. http://dx.doi.org/10.4249/scholarpedia.8507
  67. http://www.scholarpedia.org/article/Operator_product_expansion
  68. http://dx.doi.org/10.4249/scholarpedia.8506
  69. http://dx.doi.org/10.1017/CBO9780511622656.005
  70. http://dx.doi.org/10.1007/978-3-642-59128-0
  71. http://www.scholarpedia.org/article/Algebraic_renormalization
  72. http://www.scholarpedia.org/article/BRST_Symmetry
  73. http://www.scholarpedia.org/article/Composite_operator
  74. http://www.scholarpedia.org/w/index.php?title=Dimensional_Renormalization&action=edit&redlink=1
  75. http://www.scholarpedia.org/article/Gauge_theories
  76. http://www.scholarpedia.org/article/Multiloop_Feynman_integrals
  77. http://www.scholarpedia.org/article/Operator_product_expansion
  78. http://www.scholarpedia.org/article/Renormalization
  79. http://www.scholarpedia.org/article/Supersymmetry
  80. http://www.scholarpedia.org/article/User:Riccardo_Guida
  81. http://www.scholarpedia.org/w/index.php?title=Bogoliubov-Parasiuk-Hepp-Zimmermann_renormalization_scheme&oldid=64239
  82. http://www.scholarpedia.org/article/User:John_H._Lowenstein
  83. http://www.scholarpedia.org/w/index.php?title=Bogoliubov-Parasiuk-Hepp-Zimmermann_renormalization_scheme&oldid=75568
  84. http://www.scholarpedia.org/article/User:Anonymous
  85. http://www.scholarpedia.org/w/index.php?title=Bogoliubov-Parasiuk-Hepp-Zimmermann_renormalization_scheme&oldid=76132
  86. http://www.scholarpedia.org/w/index.php?title=Bogoliubov-Parasiuk-Hepp-Zimmermann_renormalization_scheme&oldid=137544
  87. http://www.scholarpedia.org/article/Special:Categories
  88. http://www.scholarpedia.org/article/Category:Quantum_and_Statistical_Field_Theory
  89. http://www.scholarpedia.org/article/Category:Quantum_Field_Theory_(Foundations)
  90. http://www.scholarpedia.org/w/index.php?title=Special:UserLogin&returnto=Bogoliubov-Parasiuk-Hepp-Zimmermann+renormalization+scheme
  91. http://www.scholarpedia.org/article/Bogoliubov-Parasiuk-Hepp-Zimmermann_renormalization_scheme
  92. http://www.scholarpedia.org/article/Talk:Bogoliubov-Parasiuk-Hepp-Zimmermann_renormalization_scheme
  93. http://www.scholarpedia.org/article/Bogoliubov-Parasiuk-Hepp-Zimmermann_renormalization_scheme
  94. http://www.scholarpedia.org/w/index.php?title=Bogoliubov-Parasiuk-Hepp-Zimmermann_renormalization_scheme&action=edit
  95. http://www.scholarpedia.org/w/index.php?title=Bogoliubov-Parasiuk-Hepp-Zimmermann_renormalization_scheme&action=history
  96. http://www.scholarpedia.org/article/Main_Page
  97. http://www.scholarpedia.org/article/Scholarpedia:About
  98. http://www.scholarpedia.org/article/Special:ProposeArticle
  99. http://www.scholarpedia.org/article/Scholarpedia:Instructions_for_Authors
 100. http://www.scholarpedia.org/article/Special:Random
 101. http://www.scholarpedia.org/article/Help:Frequently_Asked_Questions
 102. http://www.scholarpedia.org/article/Scholarpedia:Help
 103. http://www.scholarpedia.org/article/Encyclopedia:Astrophysics
 104. http://www.scholarpedia.org/article/Encyclopedia:Celestial_Mechanics
 105. http://www.scholarpedia.org/article/Encyclopedia:Computational_neuroscience
 106. http://www.scholarpedia.org/article/Encyclopedia:Computational_intelligence
 107. http://www.scholarpedia.org/article/Encyclopedia:Dynamical_systems
 108. http://www.scholarpedia.org/article/Encyclopedia:Physics
 109. http://www.scholarpedia.org/article/Encyclopedia:Touch
 110. http://www.scholarpedia.org/article/Scholarpedia:Topics
 111. http://www.scholarpedia.org/article/Special:RecentlyPublished
 112. http://www.scholarpedia.org/article/Special:RecentlySponsored
 113. http://www.scholarpedia.org/article/Special:RecentChanges
 114. http://www.scholarpedia.org/article/Special:AllPages
 115. http://www.scholarpedia.org/article/Special:ListCurators
 116. http://www.scholarpedia.org/article/Special:ListUsers
 117. http://www.scholarpedia.org/article/Special:Journal
 118. http://www.scholarpedia.org/article/Special:WhatLinksHere/Bogoliubov-Parasiuk-Hepp-Zimmermann_renormalization_scheme
 119. http://www.scholarpedia.org/article/Special:RecentChangesLinked/Bogoliubov-Parasiuk-Hepp-Zimmermann_renormalization_scheme
 120. http://www.scholarpedia.org/article/Special:SpecialPages
 121. http://www.scholarpedia.org/w/index.php?title=Bogoliubov-Parasiuk-Hepp-Zimmermann_renormalization_scheme&printable=yes
 122. http://www.scholarpedia.org/w/index.php?title=Bogoliubov-Parasiuk-Hepp-Zimmermann_renormalization_scheme&oldid=137544
 123. https://twitter.com/scholarpedia
 124. https://plus.google.com/112873162496270574424
 125. http://www.facebook.com/Scholarpedia
 126. http://www.linkedin.com/groups/Scholarpedia-4647975/about
 127. http://www.mediawiki.org/
 128. http://www.mathjax.org/
 129. http://creativecommons.org/licenses/by-nc-sa/3.0/deed.en_US
 130. http://www.scholarpedia.org/article/Bogoliubov-Parasiuk-Hepp-Zimmermann_renormalization_scheme
 131. http://creativecommons.org/licenses/by-nc-sa/3.0/deed.en_US
 132. http://www.scholarpedia.org/article/Scholarpedia:Terms_of_use
 133. http://www.scholarpedia.org/article/Scholarpedia:Privacy_policy
 134. http://www.scholarpedia.org/article/Scholarpedia:About
 135. http://www.scholarpedia.org/article/Scholarpedia:General_disclaimer

   Hidden links:
 137. http://www.scholarpedia.org/article/File:Figure_1.png
 138. http://www.scholarpedia.org/article/File:Figure_1.png
 139. http://www.scholarpedia.org/article/File:Figure_2.png
 140. http://www.scholarpedia.org/article/File:Figure_2.png
 141. http://www.scholarpedia.org/article/File:Figure_3.png
 142. http://www.scholarpedia.org/article/File:Figure_3.png
 143. http://www.scholarpedia.org/article/File:Figure_4.png
 144. http://www.scholarpedia.org/article/File:Figure_4.png
 145. http://www.scholarpedia.org/article/File:Figure_5.png
 146. http://www.scholarpedia.org/article/File:Figure_5.png
 147. http://www.scholarpedia.org/article/File:Figure_6.png
 148. http://www.scholarpedia.org/article/File:Figure_6.png
 149. http://www.scholarpedia.org/article/File:Figure_7a-bis.png
 150. http://www.scholarpedia.org/article/File:Figure_7a-bis.png
 151. http://www.scholarpedia.org/article/File:Figure_8bis.png
 152. http://www.scholarpedia.org/article/File:Figure_8bis.png
 153. http://www.scholarpedia.org/article/File:Figure_11.png
 154. http://www.scholarpedia.org/article/File:Figure_11.png
 155. http://www.scholarpedia.org/article/File:Figure_9.png
 156. http://www.scholarpedia.org/article/File:Figure_9.png
 157. http://www.scholarpedia.org/article/Bogoliubov-Parasiuk-Hepp-Zimmermann_renormalization_scheme
 158. http://www.scholarpedia.org/article/Bogoliubov-Parasiuk-Hepp-Zimmermann_renormalization_scheme
 159. http://www.scholarpedia.org/article/Main_Page