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It follows that the dimensionality

DI(G0) = α(S0) + dimS0− dim[Λ(I)S0] (5.195)

is the net number of momentum factors (including those of derivative couplings associated with the vertices) of the subgraph G0 _{counting α}

j for each line j and 2ω for each integration.

If the subdiagram G0 associated to the subspace S0 consists only of internal lines of G lines then Λ(I)S0 _{= 0 and thus in this case D}

I(G0) is the superficial degree of divergence of the

subdiagram G0_.

Now it is simple to rewrite (5.118) for our special application to Feynman diagrams as αI(S) = max

S∈G0DI(G

0_{). (5.196)}

According to Lemma 3 in other words we can say that max runs over all subgraphs G0 containing just that very set E∞ of external lines j for which ~Vj is not perpendicular to S.

We can summarise this in the

Theorem 6. Asymptotic behaviour of Green’s functions: If a set E∞ of external lines

of a Feynman diagram G, which is overall convergent as well as its subgraphs according to theorem 5, goes to infinity in the sense of Weinberg’s theorem then the integral associated with G has the asymptotic behaviour O(ηαI(E∞)_{(ln η)}βI(~L) _{where α}

I(E) is the maximum of the

dimensionality (defined by (5.195)) of all subgraphs G0 of G including the external lines in E_∞ and no more external lines of G.

5.9

BPH-Renormalisation

Now we have all ingredients to prove the main theorem of perturbation theory, namely that for a theory which has only coupling constants with dimensions O(pk_{) with k ≥ 0 we can}

renormalise any diagram to any order by adding counterterms to the effective action which are of the same form as in the classical action. This means that only a finite number of parameters (wave function renormalisation constants, masses and coupling constants) have to be fixed at any order of perturbation theory to experiment. We call a theory with only renormalisable interaction terms in the Lagrangian, superficially renormalisable, because we have to show now that they are really renormalisable. The original parameters in the classical Lagrangian, the so-called bare parameters, are infinite due to the infinities of finite diagram but cannot be observed in principle. This is what we call renormalisability in the strict sense. The great advantage of such a renormalisable theory is that one needs only a finite number of parameters which have to be fixed to experiment. All other comes out of the model given by the original Lagrangian containing these parameters.

The mathematical task is thus to show that for a theory with renormalisable couplings, i.e. such with coupling constants of momentum order greater or equal 0, one can render any diagram finite by subtracting a polynomial in its external momenta which can be written as a term like that in the original Lagrangian if interpreted in position space.

Up to now we have calculated some examples by using the method of dimensional regulari- sation as an intermediate step to give the integrals a meaning and to find a systematic way to subtract the divergent parts of the integrals. We have also seen that the renormalisation

program works for our examples using this regularisation scheme. The great advantages of dimensional regularisation are that it respects most of the symmetries which are so important in quantum field theory, especially that of local gauge invariance (which is important for QED and essential for the more complicated non-abelian gauge theories which build the heart of our nowadays understanding of elementary particles within the standard model). For our present task it is not so appropriate as for the practical calculations.

We come now to the important idea of Boguliubov and Parasiuk who invented a systematic scheme to render the Feynman integrals finite by handling the integrands first without intro- ducing an intermediate step of regularisation. They use directly Weinberg’s power counting theorem, which was so hard to prove in the previous section, to subtract polynoms of the external momenta of the divergent diagrams and its divergent sub-diagrams up to the or- der given by the superficial degree of divergence. If we can show that such a scheme works then we are done with the task to show the renormalisability of superficially renormalisable theories, because then all infinities are rendered finite by subtracting counterterms in the La- grangian which are local (i.e., they are polynomials of the external momenta momenta which is true by construction) and of the same form as of the terms already present in the original Lagrangian18.

5.9.1 Some examples of the method

In this section we understand the Feynman rules as rules for building the integrands rather than the integrals of the perturbation series. To define the subtraction operator we start again with the simple example of the one-loop contribution to the four point vertex, the dinosaur diagram: Γ = p1 p2 p3 p4 l + q l with q = p1+ p2 = p3+ p4. (5.197)

According the Feynman rules the integrand of the diagram Γ is I(q, l) = λ

2G(l)G(l + p) with G(k) =

1

m2_{− k}2_{− iη}. (5.198)

Power counting shows that the superficial degree of divergence is

Ds(Γ) = 0. (5.199)

This diagram does not contain any divergent sub-diagrams in the sense of Definition 2 in the previous section. Indeed breaking up any number of lines to extract sub-diagrams gives a tree level diagram which is finite. Such diagrams are called primitively divergent.

18_{In this chapter we consider theories in which all superficially renormalisable interaction terms are present}

and consistent with Poincar´e symmetry. One should note that the Poincar´e symmetry is no restriction for the renormalisation program since all integrands of Feynman integrals are tensors or contractions of tensors with γ-matrices and thus manifestly Lorentz covariant. The same holds true for the φ4–theory which we take as a simple toy model to have simple examples at hand: Here the superficially renormalisable terms ∝ φ and ∝ φ3

are missing but due to the symmetry of the Lagrangian with respect to transformations φ → −φ all Green’s functions with an odd number of external legs vanish.

5.9 _{· BPH-Renormalisation}

The idea of Boguliubov and Parasiuk is to render primitively divergent diagrams finite by subtracting the Taylor expansion around 0 with respect to the external momenta of this diagram (in our case it depends only on the one momentum q which is due to the simplicity of the diagram) up to the order given by Ds(Γ) from the integrand. The general definition of

the Taylor operator is given by tj(q)f (q1, . . . , qk) = j X n=0 1 n!q µ1 m1· · · q µn mn  ∂n ∂qµ1 m1· · · q µn mn f (q1, . . . , qk)  q1=...=qk=0 . (5.200) Herein summation over the Lorentz indices µl and the independent external momenta labels

ml is understood. For the Taylor operator with respect to the external momenta of the

diagram Γ to order Ds(Γ) we write tΓ. Application of the Taylor tγ operator belonging to

a sub-diagram γ is defined to act on the external momenta of γ only and not to affect the complementary diagram Γ\ γ. Here we introduce the notation we shall need in this and the next sections: We read the diagram not only as an expression of the Feynman integrals or integrands but also as a set of the lines and vertices it is built off. By definition the external lines are never contained in the diagram.

In our case of the dinosaur the subtraction of the overall divergence is given by RΓ= (1− tΓ)Γ =

λ

2G(l)[G(l + q)− G(l)]. (5.201) The degree of divergence is_{−1 and thus the integral over this expression exists, and the power} counting theorem tells us that the integral has the asymptotic behaviour O[(ln q)β]. Using the techniques of Feynman parameterisation (C.16) and (C.8) for ω = 2 we find the same result as we have calculated in section 6.4 with the dimensional regularisation technique. The subtraction at external momentum 0 is in this case the same as we defined as the physical scheme. Of course one can change the scheme also in the BPHZ technique by subtracting an arbitrary value so that one may define λ to be the physical coupling on another set of external momenta. We shall investigate the meaning of such changes of the renormalisation scheme in another chapter of these notes when we talk about the renormalisation group. Now consider the diagram

Γ = p1 p2 p3 p3− q = iλ 3 2 G(l1)G(l1− q)G(l2)G(l1+ l2− p3) with q = p1+ p2. (5.202)

which is not only overall divergent (its superficial degree of divergence is Ds(Γ) = 0, i.e. it is

logarithmitically divergent) but contains a divergent subdiagram γ around which we draw a box: −tγΓ = γ = IΓ\γ(−tγ)γ =− iλ3 2 G(l1)G(l1− q)G 2_(l 2) with q = p1+ p2. (5.203)

We define that a diagram Γ with one (or more disjoined) subdiagrams stand for the diagram were the boxed subgraphs are substituted by the negative of the Taylor operator up to the

order of superficial divergence with respect to the external lines of the subdiagram while the remainder Γ_{\ γ is unaffected to any manipulations}19_.

In our case there are no further subdivergent diagrams. Thus we have ¯

RΓ= (1− tγ)IΓ =

iλ3

2 G(l1)G(l2)G(l1− q)[G(l1+ l2− p) − G(l2)]. (5.204) Power counting shows that this expression is O(l₁−5) and thus the integration over l1 is con-

vergent while the result of this integration is of order O[l₁−4(ln l1)β]. Thus the integral over l1

is divergent. Here we have introduced the definition of ¯RΓ which is the integrand according

to the diagram rules for Γ with all subdivergences removed.

Now it is simple to remove also the overall divergence, namely by using the rule for primitively divergent integrals. This is sufficient to render the whole diagram finite because by definition

¯

RΓ does not contain any subdivergence. Thus in our case the integrand of the completely

renormalised diagram reads

RΓ= (1− tΓ) ¯RΓ= (1− tΓ)(1− tγ)IΓ=

= G(l1)G(l2){G(l1− q)[G(l1+ l2+ p3)− G(l2)]− G(l1)[G(l1+ l2)− G(l2)]}.

(5.205) We see that nested divergences are no problem for the BPH-formalism. Two diagrams γ1 and

γ2 are called nested if γ1⊂ γ2 or γ2 ⊂ γ120.

Now the power counting theorem discussed at length at sections 6.7 and 6.8 shows that the so renormalised integrals are convergent and up to logarithms behave for large external momenta asymptotically as given by superficial power counting. Of course the BPH-subtractions of the subdivergences and overall divergences lead to an integrand which all have a superficial degree of divergence less than 0 and according to the theorem are thus perfectly finite.

The only thing we have to do is to give a general prescription how to render a general diagram finite by this formalism which task is completed in the next section by giving the explicit solution of the recursive BPH-subtraction scheme which is called Zimmermann’s forest formula.

5.9.2 The general BPH-formalism

For sake of completeness we start anew with defining the classes of subdiagrams of a given diagram relevant for the BPH-formalism. Let Γ be an arbitrary Feynman diagram. Then two subdiagrams γ1 and γ2 are called

• disjoined if the do not share any common internal line or vertex. The shorthand notation for this case is γ1∩ γ2=∅.

• nested if the one diagram, e.g. γ1, is a subdiagram of the other, e.g. γ2: γ1 ⊆ γ2

19_{According to the diagram rules one can always write I}

Γ = IΓ\(γ1∪...∪γk)Iγ1· · · Iγk were γ1, . . . , γk are

arbitrary mutually disjoined 1PI subdiagrams of Γ. Since γ1, . . . , γkare mutually disjoined tγj (which means

that any two diagrams have neither a line nor a vertex in common) acts only on Iγj while all Iγj0 for j 6= j

0

can be taken out of the operator tγj

20_γ

1⊂ γ2means that all lines and vertices of γ1are also contained in γ2and γ2contains lines and/or vertices

5.9 _{· BPH-Renormalisation}

• overlapping if they are neither disjoined nor nested: γ1◦ γ2.

Diagrams and subdiagrams are called renormalisation parts if they are superficially divergent, i.e., d(γ)_{≥ 0. Suppose we have renormalised all subdivergences of Γ by applying the above} described subtraction method (called R-operation) to any divergent subdiagram (including all nested subdivergences of the subdiagrams themselves). The corresponding integrand is named ¯RΓ. Due to Weinberg’s power counting theorem then the integrand is either finite

(which is the case if Γ is not a renormalisation part) or it can be rendered finite by the R- operation applied by subtracting the Taylor expansion with respect to the external momenta of Γ if it is a renormalisation part. Thus this last step can be written compactly as

RΓ= (1− tΓ) ¯RΓ. (5.206)

Here and in the following we define tΓ as the Taylor expansion with respect to the external momenta around an arbitrary appropriate renormalisation point21 up to the order given by superficial degree of divergence of the considered (sub)diagram. If the diagram is not superficially divergent, i.e. has a superficial degree of divergence d(γ) < 0 than tγ is simply

the operator which maps the integrand to 0. The ¯Rγ themselves are defined recursively by

¯ Rγ= Iγ+ X {γ1,...,γc} I_{γ\{γ1,...,γc}} c Y τ =1 Oγτ (5.207) with Oγ=−tγR¯γ. (5.208)

The sum has to be taken over all sets_{γ1, . . . , γc} mutually disjoined subdiagrams of γ which

are different from γ.

This is the desired general prescription of the finite part of a Feynman diagram. Indeed if one takes any subdiagram of Γ by construction of ¯RΓ it has a superficial degree of divergence

less then 0 and thus is finite according to Weinberg’s theorem. By definition further also the overall superficial degree of divergence of the expression RΓ is also less than 0 and thus RΓ is

finite, again by applying Weinberg’s theorem.

Further we have shown that we had to remove only polynomial expressions in p with the order given by the superficial degree of divergence δ(γ) for any subdiagram γ of Γ. According to (5.99) in the case of φ4-theory for d = 4 only the diagrams with E ≤ 4 external legs are superficially divergent and the degree of divergence is Ds= 4− E. Further since only an even

number of fields appears in the Lagrangian also only diagrams with an even number of legs are different from 0. Thus we need to renormalise only the diagrams with 0, 2 and 4 legs. The diagrams with 0 legs are contributions to the vacuum energy which can be subtracted without changing any physical predictions. We do not need to consider them here. The diagrams with 2 legs are of divergence degree 2 and thus only polynomials of order 2 in the external momen- tum appear as counterterms. Further in our case of a scalar theory all vertex functions and

21_{In the case that massless particles are present one has to use a space like momentum otherwise one can}

use p = 0 for this so called intermediate renormalisation point. It is called intermediate because of course one can easily choice another renormalisation point and do a finite renormalisation provided. In the practical applications we shall use dimensional regularisation where this procedure is much simplified, especially infrared divergences are cured very naturally with it!

the self-energy are scalars which means that the overall counterterms for self-energy diagrams have the form δΣ = δZp2 _{− δm}2_{. We can reinterpret these counterterms as contributions}

to the effective action which looks the same as those from the original Lagrangian δZ is an infinite contribution to the bare field-normalisation and δm2 to the bare mass parameter. The remaining divergences are those of four-point vertex functions which are logarithmiti- cally divergent and have local counter terms δΓ(4) _{= δλ which are contributions to the bare}

coupling constant. This means that any divergent 1PI diagram part can be made finite by infinite contributions to the bare Lagrangian which has the same form as the finite tree-level Lagrangian we started with. The only difference is that the renormalised finite parameters of the theory, which are fixed at certain physical processes and are thus parameters carrying physical information, are substituted with infinite bare parameters which are not observable and thus the infinities are not important.

The only problem which is remaining in this description is that of the case of massless particles. In addition to the so far considered UV-divergences, i.e divergences arising from the integra- tion up to infinite momenta in the loop integrals, such theories contain also IR-divergences arising from the very soft momenta in the loop integrals. As we shall see below in the case of φ4-theory they can be renormalised by making use of a different renormalisation scheme, like the above introduced MS (minimal subtraction scheme). Then it will also become clear that we have to introduce the scale parameter µ as explained in the paragraph after eq. (5.85). In the φ4-theory this means we have to substitute λµ2 for the coupling constant instead of λ, and all counterterms will depend on µ, which means that especially the renormalised coupling constant will depend on the choice of µ although the bare coupling constant is independent of this momentum scale. For the massless case which contains no dimensionful bare parameters this means that we introduce the parameter µ with the dimension of an energy and we have to adjust the renormalised coupling at a certain value of µ which fixes the momentum scale of the physical process used to fix the coupling. As (5.93) shows if the renormalised coupling λ is small at a certain scale µ it might become large if we look at the same process at a momentum much different in scale from µ. Then the use of perturbation theory is unjustified since the effective coupling becomes large. We shall come back to this issue when we use the scale independence of the bare couplings to derive equations for the “running of the coupling” with µ, the so called renormalisation group equations which make it possible to resum leading logarithmic contributions to the renormalised coupling constant which can improve the range of applicability of perturbation theory.