Three well-known definitions of R-operation in the BPHZ formalism are presented. The equivalence between the Zimmermann’s forest formula and the factorized version of R-operation is proved

Węgrzyn, Paweł

Acta Physica Polonica B

English

Jagiellonian Univeristy Repository

Vol. B20 (1989) ACTA PHYSlCA POLONICA No 10ON THE EQUIVALENCE OF DIFFERENT DEFINITIONS OF R-OPERATION** This work was partially supported by Polish Govemment Research Grants CPBP 01.03. ♦♦ Address: Instytut Fizyki, Uniwersytet Jagielloński, Reymonta 4, 30-059 Kraków, Poland.(859)By P. WęgrzynInstitute of Physics, Jagellonian University, Cracow**(Receiυed April 7, 1989)Three well-known definitions of R-operation in the BPHZ formalism are presented. The equivalence between the Zimmermann,s forest formula and the factorized version of R-operation is proved.PACS numbers: ll.10.Gh1. IntroductionThe renormalization theory provides a generał framework for carrying out pertur- bation calculations of QFT in a well-defined and unique way. The most generał scheme for extracting finite parts from divergent Feynman amplitudes is the BPHZ formalism. This well-known procedurę was given and elaborated by Bogoliubov, Parasiuk, Hepp, Zimmermann and Zavyalov [1-6]. In this approach, the renormalized amplitudę is ob- tained by using sonie suitable set of subtractions, known as R-operation, which removes the ultraviolet infinities from the Feynman amplitudę while preserving unitarity, Lorentz invariance and causality (locality). The essence of this method is that if we choose a re­normalization point in the space of the extemal momenta and we expand the Feynman amplitudę in a Taylor series about this point, the divergences appear only in the leading co- efficients of the Taylor expansion. Furthermore, the R-operation is a solution to Bogo- liubov,s recursive relation (see [4]), what implies'that the subtracted terms can be directly related to Lagrangian counter terms and, hence, in the case of renormalizable theories, we obtain the interpretation in terms of field, coupling constant and mass renormalization.A convenient method for executing the R-operation, adopted throughout this paper, is to perform subtractions on the parametric function in the integrand of Feynman am­plitudę in the absence ot regularization. While this procedurę introduces considerable simplifications it also deals with integrands of non-existing integrals (the original theory deals with complete Feynman,s integrals and the integrands in themselves have no phys- 860ical meaning). For this reasón careful treatments of' renormalization always use some sort of regularization. Nevertheless, the familiar renormalization prescriptions, which involve the analytical, dimensional or Pauli-Villars regularization, give the same results as the scheme based on the idea of performing suitable subtractions on the integrand of the Feynman amplitudę (see [6, 7]). From this fact follows the equivalence of these quoted regularization methods, motivated by the concept that the finał results should not depend on a choice of regularization. However, one could choose such a “pathological” regula­rization that the commutativity of the subtraction procedurę with the integrations would be violated. Considering such regularization as improper [7], we assume that the R-ope- ration can be applied either to the whole integral or to the integrand. Choosing the latter way, we can dispense with regularization.In Section 2, we wish to piesent three known definitions of the R-operation, intro- duced by Appelquist [8], Zimmermann [3, 5] and Bergere and Zuber [9, 10], The most popular and useful is the Zimmermann formula incorporating the sum over all forests consisting of divergent subdiagrams (renormalization parts). Sometimes it is desirable, from both a formal and calculational point of view, to use a factorized version of the R-operation being the single product of all individual subtraction operators. Such a sub­traction formula was proposed by Appelquist [8]. However, adopting the factorized ver- sion of R-operation one should take care of the peculiarities c nnected with the over- lapping of divergent subdiagrams (this problem is not solved in Appelquist,s work). To keep the mathematical correctness one must generalizc slightly the parametric κ-represen- tation for subtractions [11], Moreover, the equivalence between the factorized and the recursive renormalization formulae was proved in [8] only for the case of renormalizable theories. Bergere and Zuber [9] defined the R-operation as the product of subtraction operators over all possible, whether divergent or not, subdiagrams of the Feynman dia­gram. This formulation is almost independent of the topological structure of the diagram. Nevertheless, they proved it is equivalent to Zimmermann,s R-operation (up to a finite renormalization for theories with spins).In Section 3, we ρrove the equivalence between the Appelquist factorized definition of the R-operation and the Zimmermann forest formula. For the sake of simplicity, we limit ourselves to the spinless theory with no derivative couplings. Referring to a spe- cial class of Feynman diagrams, namely T-diagrams, this equivalence was established in [11], In this paper, we prove it without limiting to any special topology of divergent diagrams.The notation, terminology and mathematical concepts applied in this paper are ex- plained in Aρpendix.2. R-operationTo present different definitions of the R-operation, we begin with introducing some background to describe the subtractions procedurę.We consider an arbitrary proper Feynman diagram appearing in the Z>-dimensional scalar field theory with no derivative couplings, that contains N vertices and L intemal 861lines. The Feynman amplitudę corresponding to this diagram can be presented in the standard parametric integral representation (omitting a numerical factor) (see [4-6])where k = (kt, ...,kN) denotes the set of external momenta, α = (a1, ...,aŁ), da = da.1...daL. The functions appearing in (1) are defined aswhere the sum runs over all trees in Γ,(2)(3)where the sum runs over all two-trees in Γ (each of the two-tree naturally separates the vertices into two disjoint non-empty sets F1 and K2).Let us denote by φ this part of the integrand which reflects the behaviour in the ultra- violet region (related to the lower limit of the parametric integration), namely* = (4)A family of all divergent subdiagrams (renormalization parts) included in the dia­gram Γ we denote by 3?r= {yι,>,2> ∙∙∙.y∣ι}∙ (5)To realize the subtraction in external momenta for any divergeht subdiagram y„ we introduce the κ-ρarametrization (see [4, 6, 9-11]). The paramćtrized function φ(κ) is obtained from φ by dilatation of all a; ∈ yr by κr (for any 1 ≤ r ≤ R) and multiplΓcation Rby the factor J~J k^dp^ω'r=lwhere pr denotes the number of loops in the subdiagram yr.The subtraction oρeration Ojr, which truncates the Taylor expansion about the ex- ternal momenta corresponding to the subdiagram yτ, may be regarded as the truncation (1)(6)862of the Taylor expansion with respect to the parameter κr where y denotesφ or any expression of the form Oγil...Oyikφ. The auxiliary number Ωr is large enough to make y and its derivatives non-singular at all ρoints κr ∈ [0, 1 ]. Al- though the definition of the subtraction operation (7) involves pretty arbitr'ary parameter Ωr, the result is always the same no matter which value is chosen for Ωr. NevertHeless, Ωr cannot be fixed because its smallest possible value is always determined by the actual structure of the argument y. The following lemma will be useful for usLemma 1: If a family of divergent subdiagrams γh, ..., γik forms a tree family (see Appen- dix), then the productM7ι^M7ι^ ••• ^7lkΦdoes not depend on the order of subtraction operators M7l and all auxiliary parameters Ωl may be fixed to be zero. The proof is given in Ref. [11],We have now set up enough mathematical concepts to describe the R-operation. Let us start with the standard Zimmermann,s definition [3]*z = 1+ Σ {∏(-Mγr)}, (8)where the sum runs over the set of all non-empty forests.Since any forest is a tree family, from Lemma 1 we see that working with Zimmer- mann,s formula for the R-operation (8) we are allowed to set all auxiliary parameters Ωr in the κ-representation for subtractions equal to zero. Moreover, any product of sub­traction operators Mγ appearing in the (8) does not depend on the order of its factors.A precursor of Zimmermann,s formula is the natural definition being the product of all subtraction operatorsRp≡OnOyι ... O7R = FI or (9)This definition was first proposed by Appelquist [8], He proved that this expression is welLdefined, namely it does not depend on the order of application of the subtraction operators O7r over φ, moreover, he demonstrated that this definition is equivalent to Bogo- liubov,s renormalization formula in the renormalizable scalar theory case. In Section 3, we will ρrove the equivalence of (8) and (9) for any would-be scalar theory (the result can be simply generalized to theories with spins and derivative couplings). Let us remark that the product (9) is independent of the order of its factors, but the subtraction opera­tors do not commute in generał. Moreover, the auxiliary parameters Ωr are no longer(7)(7a)(7b)863fixed to be zeros and their choice depends on the order of applications of the subtraction operators.Bergere and Zuber [9] put forward the third possible formula for the subtraction proceduręRbz ≡ ∏ o1, (10)zerwhere the product is taken over all possible subdiagrams of Γ (a subdiagram is defined here as a set lines).This definition is completely independent of the topology of the Feynman diagram except for the total number of internal lines. The independence of the order of subtraction operators and the equivalence with the Zimmermann forest formula are proved in Ref. [9]. .The above equivalent form of the R-operation provides us with remarkable insight nto the mechanism of removing the ultraviolet divergences.3. Demonstration of the equivalence of Zimmermann's and factorized formulaeIn this section, we intend to give a generał proof of the equivalence of the R-operation definitions (8) and (9).Since the definition of the R-operation is independent of the order of the subtraction operators, we are allowed to order it in such a way that Olr stands to the left of <λ,ι if yr ⊂ ys. We are intending to establish recursively the following identity∏ = ∏ OJ1+ Σ ∏(-Λf7i)]≠∙ (U)⅛=1 fc=lwhere the sum runs over the non-empty forests included in the set= {?r+ 1» ^lr÷l∙> •••» '/«}' (12)The first step is the trivial check that the identity (11) is satisfied for r = R-l. Next, assuming that the identity (11) is satisfied for order r we shall prove it for order (r-l). Let us consider the right hand side of (11)(∏<W,[1 + Σ ∏(-ΛQ]≠. (13)t = 1 JrcΛr+ I ⅛∙ΛSome subdiagrams belonging to the set ó^r+ł may overlap with the subdiagram γr. Let {ε1,ε2, ...,εm} denote the subset of ∙3fr+1 composed of all elements overlapping with yr. Using the fact that the subtraction operators commute when the corresponding pro­duct is associated with non-overlapping subdiagrams (see Lemma 1), the exρression (13) can be written in the form(∏1 [1+ Σ ∏ (-M7i)+M,r ⅛ AkMc,tk = 1 Jr ∈Jfr iβJsr k ≡ 1-M1r∑AuMtlMll + ... +(-l)m+1Mrr∕iu∙ mΛftlMl2...ΛffJ≠.k<l(14)864The “coefficients” A are defined by-^⅛l⅛∙∙∙*n ~ Σ Π(-mtΛ (15)^∈J^klfc2∙∙∙lmwhere ^tl⅛2..,kn is the set of forests SF which obey the following conditions& C^r+ι∖{ει,ε2, ... εm} (15a)& U {εfcl. ¾, ..∙, εjkπ} is a forest. (15b)Let us consider some term in (14) corresponding to subdiagrams overlapping with γr(∏ Oγk)MγAklk2...knMtkMtk2 ... Mεkφ. (16)fc= 1To achieve our recursive proof one have to convince that each term like (16) vanishes. We pick among εkl, εkl, ...,εkn the maximal subdiagrams. A subdiagram is said to be maximal if it is not contained in any other subdiagram εk,. For ρure convenience we num- ber the maximal subdiagrams overlaρping with γr as εtl,εt2, ...,εkf (fsζn). Obviously, the expression Ak,k2.kn is identically equal to zero unless the corresponding maximal subdiagrams are mutually disjoint. For what follows, we introduce the following defi- nitions αz≡y,U%1∪⅛U...∪⅛, j = l,2,...,∕. (17)Keeping in mind that the subdiagrams εki are overlapping with γr and mutually disjoint, one observes that εjfcl overlaps with γr,ε⅛1+1 overlaps with αi, i = 1, 2, 1. (18)Let us make some further remarks. First, we see that if some forest 3' satisfies the con­ditions (15ab) then the forest & u {αγ∙} does it as well. As a result we are allowed to re- write expression (15) as Λli2→n = [ Σ ∏(-Myi)]Oα,, (19)∙^⅛∕*kik2∙"knwhere the asterisk means that the sum is taken over the forests which do not contain the subdiagram αf. Notę that the above relation remains valid if the subdiagram af is not di- vergent, that is it has a negative index, because in this case the subtraction operator re- duces to unityOxf = 1 if ωx,<0. (20)For current purposes, we evaluate the following identity1 = (Oβl+Mβl)(Oβ2+Mβ2) ... (Oxf.t+Maf.1)= OXf+MXiOX2 + MXiMX2OX}+ ,,. +MxιMx2Mxi ... Mxf.i. (21)865Putting all this together, one can rewrite the term (16) as(∏ Oyk)M,J ∑ ∏(-Λfyi)Mkn...‰fc=l ^'∈J*'klk2∙∙∙kn  ⅛∙3s^× [O.,+M.iO,2+M.1M.A3÷ ... +MxιMx2Ma3 ... Mv.,-]Oafφ. (22)To complete our proof we verify that each component of (22) vanishes, namely(∏ Oγι)MγX ∑ ∏ (-Myi)]Λfekn ... Mεkι*= 1 ^'e.τ∕*klk2∙∙∙kn  ie>xM,1M,1∙M.,.1W = ρ for g=l,2, ...,/. (23)We know that the subdiagrams εk. (i = 1, ...,∕) overlaρ with α9.1, so wechoose afamily of njutually disjoint divergent subdiagrams γll, γh,..., γiι, which are included in εkg r> cx9_j and their indices satisfy the following inequalityωll+ω∣2+ ...+ωlι≥ωn, (24)where ωn denotes the index of the subdiagram εkg∩ ag-l. Because the subdiagram αg is the sum of εkg and α9~1, one can check the following identity (for details see references [11] or [9])(∏ Ovι)M M^iOxgφ = O. (25)i=l aThe generalization of the above identity to the case (23) is trivial. Because we are not intending to repeat the highly technical proof of (25) given in Ref. [11], pages 17-21, we shall content ourselves with mentioning the most important steps and stress the differ- ences connected with the generalization to the case (23). It suffices to show that after the (κll, ..., κiv)-parametrization the expression of the formMεkg ... Mεk Mεk MaιMaι ... M^iOlgMlfφ (26)is a polynomial with respect to the parameters κll, ...,κleLemma 2: In the diagram Γ there exists a tree Tl such that∏"√T1) = P⅛1 + c<εki ∩ αi-1) ~ 1> (27a>wαi(T1) = pa., i = 1, 2, ..., g,∕, (27b)wγ,i(T1) = pγ,ι, i = 1, 2, .... s, (27c)where C(...) denotes the number of connected components. The other notations are ex- plained in Appendix.Proof: First we build trees in the subdiagrams εk. n αi.1 (for i = 2,3, ...,g). We can do it because these subdiagrams are mutually disjoint. Then, this set of lines is extended in such a way that we create a tree in the subdiagram αj. This possibility follows from the 866fact that each subdiagram εkιr> αi~1 is included in the α1 (see relation (17)). The relation (17) implies also that α1 ⊂ α2 ⊂ ... ⊂ aβ ⊂ az ⊂ Γ. (28)It means that it is possible to enlarge the tree in α1 to be a tree in α2, then to a tree in α3 and so on. Finally, we obtain a tree T1 in the whole diagram Γ and this tree, by construc- tion, satisfies conditions (27abc).Let us denote the collection of alltrees (resρ. two-trees) obeying (27abc) by ^^1 (resp. ^r2). The set .⅛r2 may be empty. Considering definitions (6) and (7), one can convince oneself that the expression (26) is of the formB f A*\ 1291where K denotes some positive number, B denotes some polynomial with respect to a andD*  = ∑ (∏ «,), (29a)T1e∙T1 ⅛T1Λ*=  ∑ {(∏ ≡¾)(∑ ki)2}. (29b)T2ε^^2 ltT2 iεVιIt is straightforward to deduce from the above and (27c) that the expression (26) is a poly­nomial with respect to the (κjl,..., κlt), so due to applications of the subtraction opera- tors Oyll, which are included in the first term of (23), this polynomial is cancelled. This proves the identity (23).The author is indebted to Dr. S. Brzezowski for helpful discussions and for a crit- ical reading of the manuscript.APPENDIXIn this Appendix we explain the mathematics and notation involved in Sections 2-3. A subdiagram is said to be fuli if any two vertices in this subdiagram are joined by all the lines which already joined them in the original diagram.A subdiagram is said to be one-particle irreducible (1 PI) of it is connected and each its intemal linę belongs to at least one loop.A subdiagram is said to be divergent if it is fuli, one-particle irreducible and its index is non-negative.The index ω, of the subdiagram γf is defined byco, = Dpr-2lr,where pr and /, are the numbers of loops and internal lines respectively. The number of loops can be calculated frompr = lr-nr+C(yr),where nr is the number of vertices and C(y,) denotes the number of connected components.867We also use the notation wr(T1)to cóunt the number of lines which belong to the subdiagram γr but not to the given tree T1. Two divergent subdiagrams γ1 and γ2 are overlapping if γ1 ∩ γ2 ≠ 0 and neither γ1 ⊂ γ2 nor γ2 ⊂ γ1.A forest is a subfamily of non-overlapρing divergent subdiagrams.A family of divergent subdiagrams is called a tree family if there exist a tree T1 of the diagram Γ such that for each of the divergent subdiagrams belonging to this family the intersection with Tl is a tree of this subdiagram.A Feynman diagram Γ is called a T-diagram if its family of divergent subdiagrams is a tree family.A sum of divergent subdiagram γl and γ2 is said to be a fuli subdiagram composed of all vertices belonging to γ1 and γ2. This sum may contain such lines which belong neither to γl nor to γ2.REFERENCES[1] N. N. Bogoliubov, O. S. Tarasiuk, Acta Mathematica 97, 227 (1957).[2] K. Hepp, Commun. Math. Phys. 2, 301 (1966).[3] W. Zimmermann, Ann. Phys. (NY) 77, 536 (1973).[4] N. N. Bogoliubov, D. V. Shirkov, Vυedenie v teoriyu kvantovannykh polei, Nauka, Moskva 1976 (in Russian).[5] Renormalization Theory: Proceedings of the International School at the ,'Ettore Majorana” Centre, 1975 — Dordrecht, Holland: G. Velo & A. S. Wightman, Reidel Publ. Comp. (1976).[6] O. I. Zavialov, Perenormirovannye diagrammy Feinmana, Nauka, Moskva 1979 (in Russian).[7] S. Brzezowski, Acta Phys. Pol. B16, 799 (1985).[8] T. Appelquist, Ann. Phys. (NY) 54, 27 (1969).[9] M. C. Bergere, J. B. Zuber, Commun. Math. Phys. 35, 113 (1974).[10] M. C. Bergere, Y.-M. Lam, Journal of Math. Phys. 17, 1546 (1976).[11] P. Węgrzyn, Acta Phys. Pol. B20, 13 (1989).

On the equivalence of different definitions of R-operation

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On the equivalence of different definitions of R-operation

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