Renormalization of Spontaneaously Broken SU(2) Yang-Mills Theory with Flow Equations
aa r X i v : . [ m a t h - ph ] F e b Renormalization of Spontaneaously Broken SU(2)Yang-Mills Theory with Flow Equations
Christoph Kopper ∗ Centre de Physique Th´eorique, CNRS, UMR 7644Ecole PolytechniqueF-91128 Palaiseau, FranceVolkhard F. M¨uller † Fachbereich Physik, Technische Universit¨at KaiserslauternD-67653 Kaiserslautern, Germany
Abstract
Abstract: We present a renormalizability proof for spontaneously broken SU (2)gauge theory based on Flow Equations. It is a conceptually and technically simplifiedversion of the earlier paper [KM] including some extensions. The proof of [KM] alsowas incomplete since an important assumption made implicitly in the proof of Lemma2 there is not verified. So the present paper is also a corrected version of [KM]. ∗ [email protected] † [email protected] Introduction
The differential flow equations [WH] of the renormalization group [W] offer a powerful toolfor a unified approach to the analysis of systems with infinitely many degrees of freedom.Although first conceived for an analysis of such systems beyond perturbation theory, itwas realized by Polchinski [P] that these equations also paved the way for a new elegantapproach to perturbative renormalization theory . Local gauge theories, however, presentparticular difficulties in this approach because the momentum space regulator violates gaugeinvariance. Thus dimensional renormalization is in practice the most popular scheme forrenormalizing such theories in perturbation theory. But at the same time this scheme isrestricted to Feynman graphs. It not only defies to be given rigorous meaning in path integralformulations, it does not even directly apply in a mathematical sense to perturbative Greenfunctions as a whole without splitting them into graphs. Thus, in some sense it is farthestaway from nonperturbative analysis, and it does not allow to address a number of interestingconceptual, mathematical and quantitative questions. The authors analysed spontaneouslybroken SU(2)-Yang-Mills theory with flow equations in [KM]. This analysis was simplified in[M]. In an endeavour to further simplify and clarify the analysis, which was also caused bylecturing on the subject several times, we came across an error in [KM], which reappeared in[M] by quotation. In fact Lemma 2 in [KM] cannot be proven without an assumption madeimplicitly in its proof, which did not take into account the presence of irrelevant boundaryterms in the bare action. These terms have been “forgotten” because the context of theproof had changed in the progress of our work, after the Lemma had been written. Sincewe have found quite a number of further simplifcations in the mean time, since the subjectis important in physics, and since a correction of [KM] required quite a lot of changes, evenif the line of argument stays the same, we preferred to write a self-contained modern and(hopefully !) mathematically correct version of our previous paper.The strategy of proof remains that of [KM]. The (ultraviolet) power counting part of theflow equation renormalization proof is universal and simple for all renormalizable theories.For gauge theories we have to show that gauge invariance can be restored when the cutoffs aretaken away. On the level of the Green functions (which are not gauge invariant) this meansthat we have to verify the Slavnov-Taylor identities (STI) of the theory. They then allow toargue that physical quantities such as the S-matrix are gauge-invariant [Z]. On analysing theflow equations (FE) for a gauge theory one realizes that the restoration of the STI dependson the choice of the renormalization conditions chosen and cannot be true in general. More Wilson himself remarked already in the late sixties that this should be possible, as we learned from E.Br´ezin. ~A ) . The question is then: Can we use the freedom in adjusting therenormalization conditions such that the STI are nevertheless restored in the end? To answerthis question a first observation is crucial: The violation of the STI in the regularized theorycan be expressed through Green functions carrying an operator insertion, which depends onthe regulators. FE theory for such insertions tells us that these Green functions will vanishonce the cutoffs are removed, if we achieve renormalization conditions on the noninsertedGreen functions such that the inserted ones, which are calculated from those, have vanishingrenormalization conditions for all relevant terms, i.e. up to the dimension of the insertion(which is 5 in our case). Comparing the number of relevant terms for the SU(2) theory - 37(see App.A)- and for the insertion - 53 (see App.C) -, we realize that it is not possible to makevanish 53 terms on adjusting 37 free parameters, unless there are linear interdependences.These interdependences are revealed in the analysis of the present paper. As comparedto [KM] we also include the proof of the validity of the equation of the antighost in therenormalized theory for suitable renormalization conditions.This paper is organized as follows. In Section 2 we introduce the classical action of themodel and the BRST-transformations, [BRS], [T]. In Section 3 we introduce the conceptsfrom FE theory and recall the statements on renormalizability we need. In particular weintroduce the above mentioned operator insertions. When using FE it is natural to analysethe generating functional of free propagator amputated Schwinger functions. The analysisof the STI is however technically simpler for one-particle irreducible vertex functions so thatwe introduce the generating functionals of both, together with the corresponding renormal-izability statements. In Section 4 we derive the violated Slavnov-Taylor identities (VSTI) forthe regularized theory in various forms for the bare and the renormalized functionals. TheSections 1 to 4 follow closely the line of [KM]. In Section 5 we present the new tool requiredin view of the fact that Lemma 2 of [KM] has become obsolete. Namely the generatingfunctional of the vertex functions is not only expanded w.r.t. to fields and momenta, butalso w.r.t. the mass parameters, as far as their presence indicates improvement of UV powercounting. The corresponding redefinition of relevant renormalization constants permits a complete analysis of the relevant part of the STI in terms of the renormalization conditions.We do not need any more to jump from bare to renormalized functionals and vice versa.It is then possible to show that for suitable renormalization conditions the inserted func-tional decribing the violation of the STI has no relevant part. This result together with anobvious bound on its irrelevant part at the regularization scale Λ , following directly from3he properties of regulator, permits to prove that the violation disappears for Λ → ∞ sothat the STI hold in this limit. This proof finally elucidates the fact the validity of the STIcan directly and fully be settled by analysing the (large) system of equations describing itsrelevant part at the renormalization point. This aim was not achieved in [KM].We reproduce the appendices of [KM] with slight notational changes. In Appendix Awe list all 37 relevant terms allowed by the global symmetries of SU (2)-Yang-Mills theory.In Appendix B the 7 relevant terms appearing in the inserted functionals describing theBRST-transformations are listed. In Appendix C we list the 53 equations corresponding tothe relevant contributions to the inserted functional describing the violation of the STI. Byanalysis of this system of equations we show restoration of gauge symmetry in the (properly)renormalized theory.A reader familiar with the power counting results following from the flow equations canskip the major part of Section 3. He might use it for finding some notations also used inlater Sections and to get acquainted with the mass expansion of the Schwinger functionswhich is used for the first time in this paper. It is described in the last part of Section 3.1(from (55) onwards) and in the last page of Section 3.2 (from (88) onwards). Following closely the monograph of Faddeev and Slavnov [FS], we collect some basic proper-ties of the classical Euclidean SU(2) Yang-Mills-Higgs model on four-dimensional Euclideanspace-time. The fields of the model are a triplet { A aµ } a =1 , , of real vector fields and thecomplex scalar doublet { φ α } α =1 , . The classical action has the form S inv = Z dx (cid:26) F aµν F aµν + 12 ( ∇ µ φ ) ∗ ∇ µ φ + λ ( φ ∗ φ − ρ ) (cid:27) , (1)with the field strength tensor F aµν ( x ) = ∂ µ A aν ( x ) − ∂ ν A aµ ( x ) + gǫ abc A bµ ( x ) A cν ( x ) (2)and the covariant derivative ∇ µ = ∂ µ + g i σ a A aµ ( x ) (3)acting on the SU(2)-spinor φ . The parameters g, λ, ρ are real positive, ǫ abc is totally skewsymmetric, ǫ = +1, and { σ a } a =1 , , are the standard Pauli matrices. The action (1) isinvariant under local gauge transformations of the fields12 i σ a A aµ ( x ) −→ u ( x ) 12 i σ a A aµ ( x ) u ∗ ( x ) + g − u ( x ) ∂ µ u ∗ ( x ) ,φ ( x ) −→ u ( x ) φ ( x ) , (4)4ith u : R → SU(2), smooth. The choice of a stable equilibrium point of the action (1)leads to spontaneous symmetry breaking, dealt with by reparametrizing the complex scalardoublet as φ ( x ) = B ( x ) + iB ( x ) ρ + h ( x ) − iB ( x ) ! , (5)where { B a ( x ) } a =1 , , is a real triplet and h ( x ) the real Higgs field. Moreover, in place of theparameters ρ, λ the masses m = 12 gρ, M = (8 λρ ) (6)are used. Aiming at a quantized theory, pure gauge degrees of freedom have to be eliminated.We choose the ’t Hooft gauge fixing, with α ∈ R + , S g.f. = 12 α Z dx ( ∂ µ A aµ − αmB a ) . (7)With regard to functional integration this condition is implemented by introducing anticom-muting Faddeev-Popov ghost and antighost fields { c a } a =1 , , and { ¯ c a } a =1 , , , respectively,and forming with these six independent scalar fields the additional term in the action S gh = − Z dx ¯ c a (cid:8) ( − ∂ µ ∂ µ + αm ) δ ab + 12 αgmhδ ab + 12 αgmǫ acb B c − g∂ µ ǫ acb A cµ (cid:9) c b . (8)Hence, we have the total “classical action” S BRS = S inv + S g . f . + S gh , (9)which is decomposed as S BRS = Z dx {L quad ( x ) + L int ( x ) } (10)into its quadratic part, where ∆ ≡ ∂ µ ∂ µ , L quad = 14 ( ∂ µ A aν − ∂ ν A aµ ) + 12 α ( ∂ µ A aµ ) + 12 m A aµ A aµ + 12 h ( − ∆ + M ) h + 12 B a ( − ∆ + αm ) B a − ¯ c a ( − ∆ + αm ) c a , (11)and into its interaction part L int = gǫ abc ( ∂ µ A aν ) A bµ A cν + 14 g ( ǫ abc A bµ A cν ) + 12 g (cid:8) ( ∂ µ h ) A aµ B a − hA aµ ∂ µ B a − ǫ abc A aµ ( ∂ µ B b ) B c (cid:9) + 18 gA aµ A aµ (cid:8) mh + g ( h + B a B a ) (cid:9) + 14 g M m h ( h + B a B a ) + 132 g (cid:18) Mm (cid:19) ( h + B a B a ) − αgm ¯ c a (cid:8) hδ ab + ǫ acb B c (cid:9) c b − g ǫ acb ( ∂ µ ¯ c a ) A cµ c b . (12)5nspecting the quadratic part (11) we recognize two favourable consequences of the partic-ular gauge fixing (7) : this part is diagonal in the fields (no coupling A aµ ∂ µ B a appears) andall fields are massive.As a prerequisite to state the symmetries of S BRS (10), composite classical fields are intro-duced as follows: ψ aµ ( x ) = (cid:8) ∂ µ δ ab + gǫ arb A rµ ( x ) (cid:9) c b ( x ) ,ψ ( x ) = − gB a ( x ) c a ( x ) ,ψ a ( x ) = (cid:8) ( m + 12 g h ( x )) δ ab + 12 gǫ arb B r ( x ) (cid:9) c b ( x ) , Ω a ( x ) = 12 gǫ apq c p ( x ) c q ( x ) . (13)We can then write (8) in the form S gh = − Z dx ¯ c a {− ∂ µ ψ aµ + αmψ a } . (14)The classical action S BRS (10), shows the following symmetries:i) Euclidean invariance: S BRS is an O(4)-scalar.ii) Rigid SO(3)-isosymmetry: The fields { A aµ } , { B a } , { c a } , { ¯ c a } are isovectors and h anisoscalar; S BRS is invariant under spacetime independent SO(3)-transformations.iii) BRS-invariance:The BRS-transformations of the basic fields [BRS] are defined as A aµ ( x ) −→ A aµ ( x ) − ψ aµ ( x ) ε,h ( x ) −→ h ( x ) − ψ ( x ) ε,B a ( x ) −→ B a ( x ) − ψ a ( x ) ε, (15) c a ( x ) −→ c a ( x ) − Ω a ( x ) ε, ¯ c a ( x ) −→ ¯ c a ( x ) − α (cid:0) ∂ ν A aν ( x ) − αmB a ( x ) (cid:1) ε with the composite fields (13), and ε is a Grassmann element not depending on space-time,that commutes with the fields { A aµ , h, B a } but anticommutes with the (anti-) ghosts { c a , ¯ c a } .To show the BRS-invariance of the total classical action (9) one first observes that thecomposite classical fields (13) are themselves invariant under the BRS-transformations (15).Herewith, and using (14), it follows easily that the sum S g . f . + S gh is invariant under the trans-formation (15). Finally, on S inv act only the BRS-transformations of the fields A aµ , B a , h ,6hich amounts to local gauge transformations.We observe that upon scaling the composite fields (13) entering the BRS-transformationsas well as S gh (14), by a factor of λ , the corresponding S BRS remains invariant under suchBRS-transformations.
Quantization of the theory by means of functional integration in the realm of (formal) powerseries is based on a Gaussian measure related to the quadratic part (11) of S BRS (10).Denoting the differential operators appearing there by D µν := ( − ∆ + m ) δ µν − − αα ∂ µ ∂ ν , ˜ D := − ∆ + M , D := − ∆ + αm , (16)we write Z dx L quad ( x ) = 12 h A aµ , D µν A aν i + 12 h h, ˜ Dh i + 12 h B a , DB a i − h ¯ c a , Dc a i . (17)To these differential operators (16) are associated the (free) propagators C µν ( x, y ) = 1(2 π ) Z dk e ik ( x − y ) C µν ( k ) , (18)and similarly in the other cases, with C µν ( k ) = 1 k + m (cid:16) δ µν − (1 − α ) k µ k ν k + αm (cid:17) , C ( k ) = 1 k + M , S ( k ) = 1 k + αm . (19)A Gaussian product measure, the covariances of which are a regularized version of the prop-agators (18), (19), forms the point of departure. We choose the cutoff function, improvingslightly the former one of [M], σ Λ ( k ) = exp (cid:16) − ( k + m )( k + αm )( k + M )( k ) Λ (cid:17) . (20)It is positive, invertible and analytic, and has the property ddk σ Λ ( k ) | k =0 = 0 (21)which will be helpful in the analysis of the relevant part of the STI later on. Employing thiscutoff function we define the regularized propagators, with UV-cutoff Λ < ∞ and a flowparameter Λ satisfying 0 ≤ Λ ≤ Λ , C Λ , Λ µν ( k ) ≡ C µν ( k ) σ Λ , Λ ( k ) := C µν ( k ) (cid:0) σ Λ ( k ) − σ Λ ( k ) (cid:1) (22)7nd similarly for C ( k ) , S ( k ). The particular choice (20) implies ∂ Λ C Λ , Λ µν ( k ) = − · ( k + αm ) δ µν − (1 − α ) k µ k ν Λ · ( k + M )( k ) Λ σ Λ ( k ) , and similarly in the other cases. Herefrom follow the bounds, using C Λ , Λ ( k ) as a collectivesymbol for the propagators considered, (cid:12)(cid:12) ∂ w ∂ Λ C Λ , Λ ( k ) (cid:12)(cid:12) ≤ ( c | w | σ ( k ) f or ≤ Λ ≤ m , Λ − −| w | P | w | ( | k | Λ ) σ Λ ( k ) f or Λ > m . ) (23)On the l.h.s. ∂ w denotes a | w |− fold partial momentum derivative (see below (39)). Moreover,the polynomials P | w | have nonnegative coefficients, which, as well as the constants c | w | ,depend on α, m, M, | w | only. Considering σ Λ ( k ), (20), as a function of (Λ , k ), it cannot beextended continuously to (0 , σ (0) := lim k → σ ( k ) = 0, and hence σ , Λ (0) = σ Λ (0) = 1.It is convenient to introduce a short collective notation for the various fields and their sources:i) We denote the bosonic fields and the corresponding sources, respectively, by ϕ τ = ( A aµ , h, B a ) , J τ = ( j aµ , s, b a ) , (24)ii) and all fields and their respective sources byΦ = ( ϕ τ , c a , ¯ c a ) , K = ( J τ , ¯ η a , η a ) . (25)The sources η a and ¯ η a are Grassmann elements and have ghost number +1 and −
1, respec-tively. In the sequel, we exclusively use left derivatives with respect to these quantities.The characteristic functional of the Gaussian product measure with the covariances ¯ hC Λ , Λ from (22), (19) is then given by Z dµ Λ , Λ (Φ) e h h Φ ,K i = e h P Λ , Λ0 ( K ) , (26)where h Φ , K i : = Z dx (cid:16) X τ ϕ τ ( x ) J τ ( x ) + ¯ c a ( x ) η a ( x ) + ¯ η a ( x ) c a ( x ) (cid:17) , (27) P Λ , Λ ( K ) = 12 h j aµ , C Λ , Λ µν j aν i + 12 h s, C Λ , Λ s i + 12 h b a , S Λ , Λ b a i − h ¯ η a , S Λ , Λ η a i . (28)Aiming at a quantized descendant of the classical theory, we consider the generating func-tional L Λ , Λ (Φ) of the connected amputated Schwinger functions (CAS) e − h ( L Λ , Λ0 (Φ)+ I Λ , Λ0 ) = Z dµ Λ , Λ (Φ ′ ) e − h L Λ0 , Λ0 (Φ ′ +Φ) , (29) L Λ , Λ (0) = 0 . (30)8he constant I Λ , Λ is the vacuum part of the theory which is proportional the volume becauseof translation invariance. It therefore requires to consider the theory at first in a finite volumeΩ ⊂ R . For details see [KMR].Since the regularization necessarily violates the local gauge symmetry, the bare functional L Λ , Λ (Φ) = Z dx L int ( x ) + L Λ , Λ c.t. (Φ) (31)in a first stage has to be chosen sufficiently general in order to allow for the restoration ofthe Slavnov-Taylor identities at the end. Therefore, we add to the interaction part (12) ofclassical origin counter terms L Λ , Λ c.t. , which a priori include all local terms of mass dimension ≤ O (4)-invariance and SO (3)-isosymmetry. There are 37 such terms, by definition all at least of order O (¯ h ). Thegeneral bare functional is presented in Appendix A.From (29) the corresponding flow equation follows upon differentiation with respect to theflow parameter Λ , ∂ Λ e − h ( L Λ , Λ0 (Φ)+ I Λ , Λ0 ) = ¯ h ˙∆ Λ , Λ e − h ( L Λ , Λ0 (Φ)+ I Λ , Λ0 ) , (32)where the r.h.s. is obtained on derivation of the Gaussian measure dµ Λ , Λ (Φ ′ ) and observingthat the integrand is a function of Φ ′ + Φ. The “dot” appearing on the functional Laplaceoperator∆ Λ , Λ = 12 (cid:10) δδA aµ , C Λ , Λ µν δδA aν (cid:11) + 12 (cid:10) δδh , C Λ , Λ δδh (cid:11) + 12 (cid:10) δδB a , S Λ , Λ δδB a (cid:11) + (cid:10) δδc a , S Λ , Λ δδ ¯ c a (cid:11) (33)denotes differentiation with respect to Λ. Hence, we arrive at the flow equation ∂ Λ (cid:0) L Λ , Λ (Φ) + I Λ , Λ (cid:1) = ¯ h (cid:16) X τ D δδϕ τ , ˙ C Λ , Λ τ δδϕ τ E + 2 D δδc a , ˙ S Λ , Λ δδ ¯ c a E(cid:17) L Λ , Λ (Φ) − X τ D δL Λ , Λ δϕ τ , ˙ C Λ , Λ τ δL Λ , Λ δϕ τ E − D δL Λ , Λ δc a , ˙ S Λ , Λ δL Λ , Λ δ ¯ c a E . (34)Since we restrict to perturbation theory, the generating functional will be considered withina formal loop expansion L Λ , Λ (Φ) = ∞ X l =0 ¯ h l L Λ , Λ l (Φ) . (35)Furthermore, decomposing into particular n -point Schwinger functions we use a multiindex n , the components of which denote the number of each source field species appearing: n = ( n A , n h , n B , n ¯ c , n c ) , | n | = n A + n h + n B + n ¯ c + n c . (36)9ecause of (12) there will not appear 1- and 2-point functions at the tree level ( l = 0). If wedo not regard the vacuum part, we can study the flow of the n -point functions in the infinitevolume limit Ω → R . Due to translation invariance, it is convenient to consider also theFourier transformed source field ˆΦ, the conventions used are Z p := Z R d p (2 π ) , Φ( x ) = Z p e ipx ˆΦ( p ) −→ δ Φ( x ) := δδ Φ( x ) = (2 π ) Z p e − ipx δ ˆΦ( p ) . (37)Given these conventions, the momentum representation of the n -point function with multi-index n , (36), at loop order l is obtained as an | n | -fold functional derivative(2 π ) | n |− δ n ˆΦ( p ) L Λ , Λ l (Φ) | Φ=0 = δ ( p + · · · + p | n | ) L Λ , Λ l,n ( p , · · · , p | n | ) . (38)For the sake of a slim appearance, the notation does not reveal how the momenta are assignedto the multiindex n , and in addition, the O (4)- and SO (3)-tensor structure remains hidden.By definition the n -point function is completely symmetric (antisymmetric) if the variablesthat belong to each of the bosonic (fermionic) species occurring are permuted. As momentumderivatives of n -point functions have to be considered, too, we also introduce the shorthandnotation w = ( w , , · · · , w n − , ) , w i,µ ∈ N , ∂ w := n − Y i =1 4 Y µ =1 (cid:16) ∂∂p i,µ (cid:17) w i,µ , | w | = X i,µ w i,µ . (39)The system of flow equations (FE) for the connected amputated Schwinger functions (CAS)then follows from (34), using (35),(38), and finally performing the momentum derivatives(39) ∂ Λ ∂ w L Λ , Λ l,n ( p , · · · , p | n | ) = X n ′ , | n ′ | = | n | +2 c n − n ′ Z k ( ∂ Λ C Λ , Λ ( k )) ∂ w L Λ , Λ l − , n ′ ( k, − k, p , · · · , p | n | )(40) − X l l l, w w w wn ,n , | n | + | n | = | n | +2 c { w i } " c n ,n ∂ w L Λ , Λ l ,n ( p , . . . , p | n |− , p ′ ) · ( ∂ w ∂ Λ C Λ , Λ ( p ′ )) ∂ w L Λ , Λ l ,n ( − p ′ , . . . , p | n | ) s,a . The field assignment of the propagators C Λ , Λ on the r.h.s. is not written, it is implicit in themultiindices n ′ , n , n related to n . In the linear term the integrated momentum k refers tothat of the fields from n ′ − n and the factor c n − n ′ has the value 1 / − p ′ = p + . . . + p | n |− . Furthermore10he subscripts s, a indicate full (anti)symmetrization according to the statistics of the variousfields, requiring the combinatorial constants c n ,n to rule out those permutations, which actsolely within a given CAS. The combinatoric coefficients c { w i } stem form the Leibniz ruleand have the values c { w i } = w ! w ! w ! w ! , where w ! = Q i,µ w i,µ ! .To end up with Schwinger functions fulfilling the Slavnov-Taylor identities (STI), wehave to consider Schwinger functions with a composite field inserted, too. Two kinds of suchinsertions have to be dealt with: local insertions implementing the BRS-variations, and aspace-time integrated insertion representing the intermediate violation of the STI.The classical composite BRS-fields (13) all have mass dimension 2 and transform as vector-isovector, scalar-isoscalar, scalar-isovector and scalar-isovector, respectively. Moreover, thefirst three have ghost number 1, whereas the last one has ghost number 2. Hence, addingcounterterms, we introduce the bare composite fields( ψ aµ ) , Λ ( x ) = R ∂ µ c a ( x ) + R g ǫ arb A rµ ( x ) c b ( x ) , (41a)( ψ ) , Λ ( x ) = − R g B a ( x ) c a ( x ) , (41b)( ψ a ) , Λ ( x ) = R m c a ( x ) + R g h ( x ) c a ( x ) + R g ǫ arb B r ( x ) c b ( x ) , (41c)(Ω a ) , Λ ( x ) = R g ǫ apq c p ( x ) c q ( x ) , (41d)keeping the notation from (13) but using it henceforth exclusively according to (41a)-(41d).We set R i = 1 + O (¯ h ) , (42)thus viewing the counterterms again as formal power series in ¯ h ; the tree order ¯ h providesthe classical terms (13). Observe that for l > ψ aµ and ψ a of (13) do require R and R , respectively, as counterterms.Moreover, it is important to note that the modified composite fields (41a)-(41d) remain invariant under the BRS-transformations (15) upon assuming the conditions R = R = R , R R = ( R ) (43)and employing the generalized composite fields (41a)-(41d) in place of the original ones, (13).To deal with Schwinger functions showing one insertion, the bare interaction (31) is modifiedadding the composite fields (41a)-(41d) coupled to corresponding sources˜ L Λ , Λ ( ξ ; Φ) := L Λ , Λ (Φ) + L Λ , Λ ( ξ ) , (44) For details see [M], eq.(2.28). Λ , Λ ( ξ ) = Z dx { γ aµ ( x ) ψ aµ ( x ) + γ ( x ) ψ ( x ) + γ a ( x ) ψ a ( x ) + ω a ( x )Ω a ( x ) } . (45)According to the properties of these composite fields, the sources γ aµ , γ, γ a are Grassmannelements, they all have canonical dimension 2 and ghost number − ω a has canon-ical dimension 2 and ghost number − ψ τ = ( ψ aµ , ψ, ψ a ) , γ τ = ( γ aµ , γ, γ a ) , ξ = ( γ τ , ω a ) . (46)Using now (44) in place of L Λ , Λ as the bare action in the representation (29) provides thefunctional ˜ L Λ , Λ ( ξ ; Φ) , from which the generating functional of the regularized CAS with one insertion ψ ( x ) follows as L Λ , Λ γ ( x ; Φ) := δδγ ( x ) ˜ L Λ , Λ ( ξ ; Φ) | ξ =0 , (47)and similarly for the other insertions from (45). In the infinite volume limit, and performinga Fourier transform of the insertion position we obtainˆ L Λ , Λ γ ( q ; Φ) = Z dx e iqx L Λ , Λ γ ( x ; Φ) . (48)After loop expansion the n -point function with one insertion ψ is obtained as δ ( q + p + · · · + p | n | ) L Λ , Λ γ ; l,n ( q ; p , · · · , p | n | ) := (2 π ) | n |− δ n ˆΦ( p ) ˆ L Λ , Λ γ ; l ( q ; Φ) | Φ=0 , (49)and similarly as regards the other insertions.Starting from the analog of (34) for the modified generating functional ˜ L Λ , Λ ( ξ ; Φ), whichemerges from the bare action (44), and restricting to one insertion by the operation (47),leads to a linear flow equation for L Λ , Λ γ ( x ; Φ) . Proceeding then as before in the derivationof (40), yields the system of differential FE for the CAS with one insertion ψ∂ Λ ∂ w L Λ , Λ γ ; l,n ( q ; p , · · · , p | n | ) = X n ′ , | n ′ | = | n | +2 c n − n ′ Z k ( ∂ Λ C Λ , Λ ( k )) ∂ w L Λ , Λ γ ; l − , n ′ ( q ; k, − k, p , · · · , p | n | )(50) − X l l l, w w w wn ,n , | n | + | n | = | n | +2 c { w i } " c (1) n ,n ∂ w L Λ , Λ γ ; l ,n ( q ; p , · · · , p | n |− , p ′ ) · ( ∂ w ∂ Λ C Λ , Λ ( p ′ )) ∂ w L Λ , Λ l ,n ( − p ′ , · · · , p | n | ) s,a . The notation is that of (40), with − p ′ = q + p + · · · + p | n |− , however. Since ghost andantighost in (34) do not appear symmetrically, the ¯ c ( c )-derivative appears once in n ( n )12nd once in n ( n ). It is obvious that each of the other insertions (45) leads to a similarsystem of flow equations.As will turn out in Section 4, the initial regularization, necessarily violating the STI,leads to a bare space-time integrated insertion of the form L Λ , Λ (Φ) = Z dx N ( x ) , N ( x ) = Q ( x ) + Q ′ ( x ; Λ − ) . (51)The individual terms of N ( x ) involve at most five fields and have ghost number 1. Further-more, Q ( x ) is a local polynomial in the fields and their derivatives, having canonical massdimension D = 5, whereas Q ′ ( x ; Λ − ) is nonpolynomial in the field momenta but suppressedby powers of Λ − . To obtain the generating functional L Λ , Λ (Φ) with one (bare) insertion(51) we can resort to the local case, considering the bare local insertion L Λ , Λ ( ̺ ) = Z dx ̺ ( x ) N ( x ) (52)and proceed as before. Observing (47), (48) we obtain L Λ , Λ (Φ) = Z dx δδ̺ ( x ) ˜ L Λ , Λ ( ̺ ; Φ) | ̺ =0 = Z dx L Λ , Λ ̺ ( x ; Φ) = ˆ L Λ , Λ ̺ (0; Φ) . (53)Performing again a loop expansion, the CAS n -point function with one insertion (51) isobtained as δ ( p + · · · + p | n | ) L Λ , Λ l,n ( p , · · · , p | n | ) := (2 π ) | n |− δ n ˆΦ( p ) L Λ , Λ l (Φ) | Φ=0 . (54)For these CAS holds again a system of linear FE. According to the preceding treatment ofthe integrated insertion we only have to take (50) at the fixed momentum value q = 0 of theinsertion, and then replace each symbol L Λ , Λ γ ; l,n (0 ; · · · ) by the new symbol L Λ , Λ l,n ( · · · ) .Polchinski realized the flow equations (40) to open the way for a simple inductive proof ofrenormalizability. The mathematical proof was carried through in [KKS] on simplifying stillPolchinski’s argument. The FE for composite operators (50) were introduced and analysedin [KK]. For a recent presentation see [M].The analysis of the STI, however, as will be shown in Section 4, requires to trace in theperturbative expansion the effect of the super-renormalizable three-point couplings presentin the interaction. To this end we scale in the tree-level part (12) of (31) the mass parametersappearing in the three-point couplings, as well as in the BRS-insertions the part proportionalto m , see (41c), by a common factor of λ > m → λm , M → λM . (55)13 ote however that we do not scale the mass parameters which are present in the regularizedpropagators appearing in the flow equations. All CAS will then depend smoothly on λ , andwe expand them as L Λ , Λ l,n ( λ ; ~p ) = ∞ X ν =0 ( m λ ) ν L ( ν ) , Λ , Λ l,n ( ~p ) , ~p = ( p , · · · , p | n | ) , (56) L Λ , Λ γ ; l,n ( λ ; q ; ~p ) = ∞ X ν =0 ( m λ ) ν L ( ν ) , Λ , Λ γ ; l,n ( q ; ~p ) , (57)where for suitable (physically natural !) renormalization schemes the sum is finite, its sizedepending on l and n , as will be shown below. We adopt the following Renormalization scheme : Relevant terms are those which satisfy | n | + | w | + ν ≤ L Λ , Λ , | n | + | w | + ν ≤ L Λ , Λ γ ,in agreement with the bounds to be derived below.At tree level we then have ( ∂ w L ( ν ) , Λ , Λ ,n )( ~
0) = 0 , if | n | + | w | + ν < . (58)For l ≥
1, we use renormalization conditions on the relevant terms as follows: we impose( ∂ w L ( ν ) , , Λ l,n )( ~ ! = 0 , if | n | + | w | + ν < , (59)whereas if | n | + | w | + ν = 4 , on the r.h.s. a free constant r ( ν ) , l, n can be chosen.Correspondingly, in the case of an insertion, we have at the tree level( ∂ w L ( ν ) , Λ , Λ γ ; 0 ,n )(0; ~ , if | n | + | w | + ν < , (60)and employ renormalization conditions( ∂ w L ( ν ) , , Λ γ ; l,n )(0; ~ ! = 0 , if | n | + | w | + ν < , (61)but if | n | + | w | + ν = 2 , on the r.h.s. again a free constant can be chosen.Because of the expansions (56) and (57) the FE (40) and (50) have to be adjusted attributinga superscript ( ν ) to the CAS and to sum ν + ν = ν , in complete analogy to the loop index l . Using these extended FE the following bounds can be deduced, Proposition 1
Let l ∈ N and ≤ Λ ≤ Λ , then | ∂ w L ( ν ) , Λ , Λ l,n ( ~p ) | ≤ (Λ + m ) −| n |−| w |− ν P (log Λ + mm ) P ( | ~p | Λ + m ) , (62) Notice, that for l = 0 there are no CAS with | n | ≤ ∂ w L ( ν ) , Λ , Λ γ ; l,n ( q ; ~p ) | ≤ (Λ + m ) −| n |−| w |− ν P (log Λ + mm ) P ( | q, ~p | Λ + m ) . (63) In these bounds P i , i = 1 , , denote (each time they appear possibly new) polynomialswith nonnegative coefficients independent of Λ , Λ , ~p, q, m . The coefficients may depend on n, l, w, and the other free parameters of the theory α, M/m , g . These bounds are uniform in Λ . The proof is solely based on power counting for renormal-izable theories, it does not involve the symmetry structure of the Yang-Mills theory. Proof : To prove (62) one proceeds by induction as follows: ascending in N := 2 l + | n | , forgiven N ascending in l , for given N, l ascending in ν , and for given N, l, ν descending in | w | .Given n , the irrelevant cases | n | + | w | + ν > ”downwards” with initial conditions equal to zero. In contrast, the relevantones, i.e. | n | + | w | + ν ≤
4, choosing the particular momentum value ~p = 0, are integratedfrom the initial point Λ = 0 ”upwards” with initial conditions (59) and the remaining oneschosen freely, hereafter this result has to be extended to general ~p via the Taylor formula f ( ~p ) = f (0) + ~p · Z ( ~∂ f )( t~p ) dt . Descending in | w | , the integrand in the respective remainder of the Taylor extension hasalready been bounded previously. A derivative by induction provides another factor of(Λ + m ) − , which can be combined with the momentum factor of the remainder to increasethe degree of the bounding polynomial. A key to this induction is the property that in thetree order there are no CAS with | n | ≤
2. Bounding the linear term of the FE (cid:12)(cid:12)(cid:12) X n ′ , | n ′ | = | n | +2 c n − n ′ Z k ( ∂ Λ C Λ , Λ ( k )) ∂ w L ( ν ) , Λ , Λ l − , n ′ ( k, − k, ~p ) (cid:12)(cid:12)(cid:12) ≤ X n ′ , | n ′ | = | n | +2 Λ Z k ′ | Λ ∂ Λ C Λ , Λ (Λ k ′ ) | | ∂ w L ( ν ) , Λ , Λ l − , n ′ (Λ k ′ , − Λ k ′ , ~p ) |≤ Λ X n ′ , | n ′ | = | n | +2 (Λ + m ) −| n ′ |−| w |− ν P (log Λ + mm ) P ( | ~p | Λ + m ) ≤ (Λ + m ) −| n |−| w |− ν − P (log Λ + mm ) P ( | ~p | Λ + m ) , after a change of the integration variable k = Λ k ′ one uses the bounds (23) and (62) andthen performs the k ′ -integration.The proof of (63) is analogous to the proof of (62): One has to observe the inherent demar-cation between relevant and irrelevant, and to employ the bound (62) required to treat the This term generates a new loop. L ( ν ) , Λ , Λ l,n ( ~p ) ≡ , if ν > l + | n | − , L ( ν ) , Λ , Λ γ ; l,n ( q, ~p ) ≡ , if ν > l + | n | − . (64)These statements follow inductively from the FE, once they hold for the terms fixed by theboundary conditions. Note that the first of these relations can be understood in terms ofFeynman graphs as following from the upper bound on the number of trivalent vertices ata given loop-order. The second one takes into account additionally that the BRS-insertions(41c) also include one factor of m .To also prove convergence for Λ → ∞ (which a physicist would grant as a consequenceof uniformity) one has to analyse the FE, derived w.r.t. Λ , using the same inductivetechnique. It is then possible to prove [M] that | ∂ Λ ∂ w L ( ν ) , Λ , Λ l,n ( ~p ) | ≤ Λ − (Λ + m ) −| n |−| w |− ν P (log Λ m ) P ( | ~p | Λ + m ) , (65) | ∂ Λ ∂ w L ( ν ) , Λ , Λ γ ; l,n ( q ; ~p ) | ≤ Λ − (Λ + m ) −| n |−| w |− ν P (log Λ m ) P ( | q, ~p | Λ + m ) , (66)for Λ large enough. Herefrom we can infer the existence of the limits Λ → ∞ at fixedvalue of Λ . Our analysis of the Slavnov-Taylor identities (STI) and the proof of their restoration willbe based on a presentation in terms of proper vertex functions (1PI), since the extractionof relevant parts from the STI is simpler and more transparent in terms of those than interms of the CAS. To present their relation with the CAS considered so far, we introducethe shorthand notation˜ L ( ξ ; Φ) := ˜ L Λ , Λ ( ξ ; ϕ τ , c, ¯ c ) , C τ := C Λ , Λ τ , S := S Λ , Λ , (67)for the generating functional of the CAS with insertion (45) and for the regularized propa-gators. From ˜ L ( ξ ; Φ) we define the ”classical fields” Φ ≡ ( ϕ τ , c, ¯ c ) by ϕ τ ( x ) = ϕ τ ( x ) − Z dy C τ ( x − y ) δ ˜ L ( ξ ; Φ) δϕ τ ( y ) ,c a ( x ) = c a ( x ) + Z dy S ( x − y ) δ ˜ L ( ξ ; Φ) δ ¯ c a ( y ) , ¯ c a ( x ) = ¯ c a ( x ) − Z dy S ( x − y ) δ ˜ L ( ξ ; Φ) δc a ( y ) . (68)16he generating functional of the proper vertex functions ˜Γ( ξ ; Φ) ≡ ˜Γ Λ , Λ ( ξ ; ϕ τ , c, ¯ c ) is thengiven by the transform ˜Γ( ξ ; Φ) = ˜ L ( ξ ; Φ) − X τ h ϕ τ , C − τ ϕ τ i + h ¯ c, S − c i + X τ h ϕ τ C − τ ϕ τ i − h ¯ c, S − c i − h ¯ c, S − c i , (69)with Φ = Φ(Φ) on the r.h.s., according to (68). Since we are only interested in the kernelsto be derived from the generating functional Γ we may always assume the field variables tobe sufficiently regular so that the application of the inverted regularized propagators makessense. By functional derivation we deduce the relations δ ˜Γ( ξ ; Φ) δϕ τ ( x ) = Z dy C − τ ( x − y ) ϕ τ ( y ) ,δ ˜Γ( ξ ; Φ) δc a ( x ) = Z dy S − ( x − y ) ¯ c a ( y ) , δ ˜Γ( ξ ; Φ) δ ¯ c a ( x ) = − Z dy S − ( x − y ) c a ( y ) , (70)forming the inverse of the relations (68). Moreover, acting on the ”classical fields” (68) withthe respective inverse propagators C − τ and S − , and then using (70), provides the crucialrelations between the generating functionals ˜ L ( ξ ; Φ) and ˜Γ( ξ ; Φ)(2 π ) − C − τ ( p ) ϕ τ ( − p ) = δ ˜Γ( ξ ; Φ) δϕ τ ( p ) − δ ˜ L ( ξ ; Φ) δϕ τ ( p ) , (2 π ) − S − ( p ) c a ( − p ) = − δ ˜Γ( ξ ; Φ) δ ¯ c a ( p ) + δ ˜ L ( ξ ; Φ) δ ¯ c a ( p ) , (71)(2 π ) − S − ( p ) ¯ c a ( − p ) = δ ˜Γ( ξ ; Φ) δc a ( p ) − δ ˜ L ( ξ ; Φ) δc a ( p ) , written in terms of Fourier transformed fields. Functional derivation of (69) with respect tothe source γ ( x ) at fixed Φ leads to δ ˜Γ( ξ ; Φ) δγ ( x ) (cid:12)(cid:12)(cid:12) ξ =0 = δ ˜ L ( ξ ; Φ) δγ ( x ) (cid:12)(cid:12)(cid:12) ξ =0 , (72)and to analogous equations as regards the other sources γ aµ , γ a , ω a .Restricting again to perturbation theory we consider the proper vertex functions whichcorrespond to the various types of CAS dealt with up to now. Hence, we define propervertex functions without insertion, with one local insertion as in (47), (48), and with aglobal one as in (53), keeping the same notations. Since by definition ˜Γ( ξ ; Φ) has no vacuum This transform corresponds to the familiar Legendre transform of the connected (non-amputated)Schwinger functions. Λ , Λ l , Γ Λ , Λ γ ; l , Γ Λ , Λ l we obtain the corresponding n -point proper vertexfunctions of loop order l in analogy with (38), (49), (54),(2 π ) | n |− δ n Φ( p ) Γ Λ , Λ l (Φ) | Φ ≡ = δ ( p + · · · + p | n | ) Γ Λ , Λ l,n ( p , · · · , p | n | ) , (73)(2 π ) | n |− δ n Φ( p ) Γ Λ , Λ γ ; l ( q ; Φ) | Φ ≡ = δ ( q + p + · · · + p | n | ) Γ Λ , Λ γ ; l,n ( q ; p , · · · , p | n | ) , (74)(2 π ) | n |− δ n Φ( p ) Γ Λ , Λ l (Φ) | Φ ≡ = δ ( p + · · · + p | n | ) Γ Λ , Λ l,n ( p , · · · , p | n | ) . (75)The FE for the ˜ L -functional implies a corresponding flow equation for the proper vertexfunctional ˜Γ. Performing the Λ-derivative of the transform (69) and observing that theclassical fields Φ , (68), themselves depend on Λ due to (70), eventually yields( ∂ Λ ˜Γ)( ξ ; Φ) = ∂ Λ ˜ L ( ξ ; Φ) − X τ h ϕ τ , ∂ Λ C − τ ϕ τ i + h ¯ c , ∂ Λ S − c i (76)+ X τ h ϕ τ , ∂ Λ C − τ ϕ τ i − h ¯ c , ∂ Λ S − c i − h ¯ c , ∂ Λ S − c i , where ( ∂ Λ ˜Γ) denotes the derivative of the functional ˜Γ itself. Inserting now the flow equa-tion for ˜ L ( ξ ; Φ) which has the same form as (34), and eliminating in its bilinear terms thefunctionals δδ Φ ˜ L using the equations (68), provides the flow equation of the vertex functional( ∂ Λ ˜Γ)( ξ ; Φ) + ( ∂ Λ ˜ I )( ξ ) − X τ h ϕ τ , ∂ Λ C − τ ϕ τ i + h ¯ c , ∂ Λ S − c i = ¯ h ˙∆ Λ , Λ ˜ L ( ξ ; Φ) , (77)where one should remember the dependence on the parameters Λ , Λ from (67) and thedefinition (33). At this stage the fields Φ can be considered as autonomous (test) functionsof the functional ˜Γ , not depending on Λ. On the l.h.s. the second term is the vacuum part,since ˜Γ( ξ ; 0) = 0, and the subsequent terms subtract the (regularized) two-point tree orderfrom ( ∂ Λ ˜Γ)( ξ ; Φ) . The resulting functional still has to be expressed in terms of proper vertexfunctions. Performing a loop expansion and functional derivatives w.r.t. the fields we obtainfrom (77) for | n | ≥ δ n Φ | Φ ≡ : ( ∂ Λ ˜Γ l )( ξ ; Φ) = ˙∆ Λ , Λ ˜ L l − ( ξ ; Φ) , l ≥ . (78)Since the vacuum part has disappeared we can now pass to the infinite volume limit. On theright hand side the functional ˜ L l − ( ξ ; Φ) is first acted upon by two particular Φ-derivativesfrom the functional Laplace operator, then followed by an n -fold functional derivative with again to be viewed on finite volume before passing to correlation functions L ( ξ ; Φ) via the chain rule together with (70), and hereafterconsider the outcome within a loop expansion, δ ( p + q ) δ l, (2 π ) C Φ , Φ ′ ( p ) = δ ˜Γ l ( ξ ; Φ) δ Φ( p ) δ Φ ′ ( q ) − (2 π ) X Φ ′′ l l l Z k δ ˜ L l ( ξ ; Φ) δ Φ( p ) δ Φ ′′ ( k ) C Φ ′′ Φ ′′′ ( k ) δ ˜Γ l ( ξ ; Φ) δ Φ ′′′ ( − k ) δ Φ ′ ( q ) . (79)This identity forms the point of departure to relate successively n -point functions of the L -and the Γ- functional. We have to deal with it in the case without insertion, setting ξ ≡ L -functionalwith and without insertion appear, δ Γ Λ , Λ γ ; l ( q ; Φ) δ Φ( p ) δ Φ ′ ( p ′ ) = (2 π ) X Φ ′′ l l l (cid:16) Z k δ L Λ , Λ γ ; l ( q ; Φ) δ Φ( p ) δ Φ ′′ ( k ) C Λ , Λ Φ ′′ Φ ′′′ ( k ) δ Γ Λ , Λ l (Φ) δ Φ ′′′ ( − k ) δ Φ ′ ( p ′ )+ Z k δ L Λ , Λ l (Φ) δ Φ( p ) δ Φ ′′ ( k ) C Λ , Λ Φ ′′ Φ ′′′ ( k ) δ Γ Λ , Λ γ ; l ( q ; Φ) δ Φ ′′′ ( − k ) δ Φ ′ ( p ′ ) (cid:17) . (80)Taking (80) at momentum q = 0 and replacing the subscript γ by the subscript 1 providesthe relation in the case of the integrated insertion.From (79) without insertion, considered at loop order l = 0 and at Φ = Φ ≡
0, follows inthe first step, because of the key property L Λ , Λ ,n ( k, − k ) ≡
0, if | n | = 2,1 = C Λ , Λ Φ , Φ ′ ( p ) Γ Λ , Λ , n ( p, − p ) , n ˆ= (Φ , Φ ′ ) . (81)Before returning to the flow equation we note, that in order to obtain from (79) with ξ ≡ n -point functions of the L - and the Γ- functional,we have to act upon these equations repeatedly by Φ - derivation, to be performed on the L -functional via the chain rule. The chain rule derivatives δ Φ /δ Φ can be read from (70). Inparticular, on account of the propagators C Λ , Λ ( k ) vanishing at Λ = Λ and observing (81),one realizes, ascending with | n | ,Γ Λ , Λ l,n ( p , · · · , p | n | ) = L Λ , Λ l,n ( p , · · · , p | n | ) , ( l, | n | ) = (0 , , (82) Here Φ ′′′ is determined by Φ ′′ , cf. (17). Λ , Λ l,n ( p , · · · , p | n | ) = L Λ , Λ l,n ( p , · · · , p | n | ) , (83)and similarly in the case of the local insertions.We now return to the FE (78) and first treat the case without insertion, thus we set there ξ ≡
0. Performing in addition the momentum derivatives (39) we obtain the system, for | n | ≥ , ( l, | n | ) = (0 , ∂ Λ ∂ w Γ Λ , Λ l,n ( p , · · · , p | n | ) = 12 ′ X | n ′ | = | n | +2 Z k ( ∂ Λ C Λ , Λ ( k )) ∂ w L Λ , Λ l − , n ′ ( k, − k ; p , · · · , p | n | ) . (84)The summation extends on the various propagators as stated in (79), not distinguishedhere notationally, the corresponding pair of fields together with n determine n ′ . Moreover,the momentum derivative ∂ w concerns the momenta p , · · · , p | n | of the configuration n . Togenerate the functions on the r.h.s. of (84) we have to act on (78), after setting ξ ≡
0, with δ n Φ | Φ ≡ , and these derivatives are directly applied on the L - functional. Hence the functions L Λ , Λ l,n in (84), differing from the CAS L Λ , Λ l,n . The vanishing 2-point CAS in the tree order,together with its correspondence (81) then allow to express inductively the functions L Λ , Λ l,n on the r.h.s. of (84) in terms of proper vertex functions, ascending in l , and for fixed l ascending in | n | . The r.h.s of (84) then emerges in the form L Λ , Λ l − , n ′ ( k, − k ; p , · · · , p | n | ) = Γ Λ , Λ l − , n ′ ( k, − k, p , · · · , p | n | ) + · · · , (85)where the dots represent chains Γ C Γ and higher iterations, formed of proper vertex func-tions Γ Λ , Λ l ′ , n ′′ with ( l ′ , n ′′ ) prior to ( l − , n ′ ), joined via (free) propagators.In the case of one local insertion the equation (78) has to be derived with respect to thesource at zero source, cf. (47),(48). Performing again the momentum derivation leads to thethe system of flow equations for proper vertex functions with one local insertion, | n | ≥ ∂ Λ ∂ w Γ Λ , Λ γ ; l,n ( q ; p , · · · , p | n | ) = 12 ′ X | n ′ | = | n | +2 Z k ( ∂ Λ C Λ , Λ ( k )) ∂ w L Λ , Λ γ ; l − ,n ′ ( q ; k, − k ; p , · · · , p | n | ) , (86)The r.h.s. of (86) is now obtained in complete analogy to the case without insertion, ther.h.s. is now extracted inductively from (80) in place of (79). By this operation, both the L -functions with and without insertion appear. Proceeding inductively as before, and usingthe already determined L -functions without insertion, provides the function on the r.h.s. ofthe system (86), as L Λ , Λ γ ; l − , n ′ ( q ; k, − k ; p , · · · , p | n | ) = Γ Λ , Λ γ ; l − , n ′ ( q ; k, − k, p , · · · , p | n | ) + · · · , (87)20here the dots again represent a sum of chains, each of which contains exactly one insertedfactor Γ Λ , Λ γ ; l ′′ ,n ′′ , which has already been determined previously in the inductive procedure.Finally, in the case of an integrated insertion, we obtain the system (86) at the particularmomentum value q ≡ Λ , Λ l,n ( λ ; ~p ) = ∞ X ν =0 ( mλ ) ν Γ ( ν ) , Λ , Λ l,n ( ~p ) , ~p = ( p , · · · , p | n | ) , (88)Γ Λ , Λ γ ; l,n ( λ ; q ; ~p ) = ∞ X ν =0 ( mλ ) ν Γ ( ν ) , Λ , Λ γ ; l,n ( q ; ~p ) . (89)We first consider the tree level l = 0. In the case of (88) the scaling (55) of the interactionresults in ( ∂ w Γ ( ν ) , , Λ ,n )( ~ , | n | = 3 , | w | + ν = 1 . (90)Whereas there is no | n | = 1 content, the 2-point functions are fixed by the regularizedpropagators (81) (the masses of which are not scaled). The vertex functions with insertion(89) satisfy ( ∂ w Γ ( ν ) , , Λ γ ; 0 ,n )(0; ~ , | n | + | w | + ν < . (91)Owing to the expansions (88) and (89), in both FE (84) and (86) a superscript ( ν ) has tobe attached to the respective n -point function on the l.h.s. and on the n ′ -point functionspresent on the r.h.s. We then use the same inductive scheme which leads to the bounds(62),(63) on the CAS and may deduce renormalizability of the proper vertex functions. Forthe relevant terms the choice of the renormalization conditions is as follows , l ≥ ∂ w Γ ( ν ) , , Λ l, n )( ~ ! = 0 , if | n | + | w | + ν < , (92)but if | n | + | w | + ν = 4 , a nonvanishing constant can be chosen on the r.h.s.,whereas in the case of an insertion( ∂ w Γ ( ν ) , , Λ γ ; l, n )(0; ~ ! = 0 , if | n | + | w | + ν < , (93)but if | n | + | w | + ν = 2 , again a nonvanishing constant on the r.h.s. may be imposed.Proceeding inductively as indicated we obtain the bounds: Proposition 2 | ∂ w Γ ( ν ) , Λ , Λ l,n ( ~p ) | ≤ (Λ + m ) −| n |−| w |− ν P (log Λ + mm ) P ( | ~p | Λ + m ) , ( l, | n | ) = (0 , , (94)21 ∂ w Γ ( ν ) , Λ , Λ γ ; l,n ( q ; ~p ) | ≤ (Λ + m ) −| n |−| w |− ν P (log Λ + mm ) P ( | q, ~p | Λ + m ) , (95) The notations are those from (62),(63).
Moreover, we can also obtain the bounds (65) - (66) in the case of proper vertex functionsderived w.r.t. Λ . To examine the violation of the STI produced by the UV cutoff Λ we depart from thegenerating functional of the regularized Schwinger functions at the physical value Λ = 0 ofthe flow parameter, Z , Λ ( K ) = Z dµ , Λ (Φ) e − h L Λ0 , Λ0 (Φ)+ h h Φ ,K i . (96)The Gaussian measure dµ , Λ (Φ) corresponds to the quadratic form h Q , Λ (Φ), cf. (26), Q , Λ (Φ) = 12 h A aµ , (cid:0) C , Λ (cid:1) − µν A aν i + 12 h h, ( C , Λ ) − h i + 12 h B a , ( S , Λ ) − B a i − h ¯ c a , ( S , Λ ) − c a i . (97)We define regularized BRS-variations (15),(41a)-(41d) of the fields by δ BRS ϕ τ ( x ) = − ( σ , Λ ψ τ )( x ) ε,δ BRS c a ( x ) = − ( σ , Λ Ω a )( x ) ε, (98) δ BRS ¯ c a ( x ) = − (cid:0) σ , Λ ( 1 α ∂ ν A aν − mB a ) (cid:1) ( x ) ε . The BRS-variation of the Gaussian measure has the form dµ , Λ (Φ) dµ , Λ (Φ) (cid:16) − h δ BRS Q , Λ (Φ) (cid:17) , (99)and inspecting (97) we observe that the factor σ , Λ of the variations (98) just cancels itsinverse entering the inverted propagators. Hence, the BRS-variation of the Gaussian measurehas mass dimension D = 5. Requiring the regularized generating functional Z , Λ ( K ), (96),to be invariant under the BRS-variations (98) of the integration variables, provides the violated Slavnov-Taylor identities (VSTI)0 ! = Z dµ , Λ (Φ) e − h L Λ0 , Λ0 (Φ)+ h h Φ ,K i (cid:16) δ BRS h Φ , K i − δ BRS ( Q , Λ + L Λ , Λ ) (cid:17) . (100) Again one should stay in finite volume as long as the vacuum part is involved. Z , Λ ( K, ξ ) := Z dµ , Λ (Φ) e − h ˜ L Λ0 , Λ0 ( ξ ;Φ)+ h h Φ ,K i , (101)and introduce a regularized BRS-operator D Λ = X τ (cid:10) J τ , σ , Λ δδγ τ (cid:11) + (cid:10) ¯ η a , σ , Λ δδω a (cid:11) + (cid:10) α ∂ ν δδj aν − m δδb a , σ , Λ η a (cid:11) . (102)ii) The BRS-variations of the bare action and of the Gaussian measure L Λ , Λ ε : = − δ BRS (cid:16) Q , Λ + L Λ , Λ (cid:17) = Z dx N ( x ) ε (103)form a space-time integrated insertion with ghost number 1. The variation of L Λ , Λ , however,keeps the regularizing factor σ , Λ of (98), thus the integrand N ( x ) is no longer a polynomialin the fields and their derivatives. We can initially treat the integrand N ( x ) as a localinsertion with a source ρ ( x ), cf. (52). Introducing the corresponding bare action ˜ L Λ , Λ ( ρ ; Φ)similarly to (44), we define the functional ˜ Z , Λ ( K, ρ ) in analogy to (101).In terms of these modified Z -functionals the VSTI (100) can now be written D Λ ˜ Z , Λ ( K, ξ ) | ξ =0 = Z dx δδ̺ ( x ) ˜ Z , Λ ( K, ρ ) | ρ =0 . (104)The modified Z -functional (101) is related to the corresponding generating functional ofmodified CAS by ˜ Z , Λ ( K, ξ ) = e h P , Λ0 ( K ) e − h (˜ L , Λ0 ( ξ ; ϕ τ , c, ¯ c )+ I , Λ0 ) , (105)and analogously in case of ˜ Z , Λ ( K, ρ ). Furthermore, the variables of the Z - and the L -functional satisfy ϕ τ ( x ) = Z dy C , Λ τ ( x − y ) J τ ( y ) ,c a ( x ) = − Z dy S , Λ ( x − y ) η a ( y ) , ¯ c a ( x ) = − Z dy S , Λ ( x − y ) ¯ η a ( y ) . (106) Abusing notation we let the variables ρ and ξ , respectively, denote different functions. The vacuum part I , Λ is the same as in the case without insertion, since the latter has nonzero ghostnumber Z , Λ ( K, ρ ), we derive, using the defi-nitions (47), (53) and denoting the differential operators (16) by D τ in accord with ϕ τ , the violated Slavnov-Taylor identities of the CAS: (cid:10) c a , D (cid:0) α ∂ ν A aν − mB a (cid:1)(cid:11) − (cid:10) c a , σ , Λ (cid:0) ∂ ν δL , Λ δA aν − m δL , Λ δB a (cid:1)(cid:11) + X τ (cid:10) ϕ τ , D τ L , Λ γ τ (cid:11) − (cid:10) ¯ c a , DL , Λ ω a (cid:11) = L , Λ . (107)Starting from the relations (72) between the generating functionals of the vertex- andSchwinger-functions we can convert (107) at the (physical) value Λ = 0 into the violatedSlavnov-Taylor identities for proper vertex functions , on substituting there the fields Φ ac-cording due to (70), and employing (71), (72), X τ D δ Γ , Λ δϕ τ , σ , Λ Γ , Λ γ τ E − D δ Γ , Λ δc a , σ , Λ Γ , Λ ω a E − D α ∂ ν A aν − mB a , σ , Λ δ Γ , Λ δ ¯ c a E = Γ , Λ ( ϕ τ , c a , ¯ c a ) , (108)with Γ , Λ ( ϕ τ , c a , ¯ c a ) = L , Λ ( ϕ τ , c a , ¯ c a ) . (109)In the analysis of the STI it will turn out that we need the form of their explicit violation“on the bare side”, Γ Λ , Λ (Φ) , too. From the definition (103) we directly determine the barefunctional L Λ , Λ (Φ) , using (44) and (45), L Λ , Λ (Φ) = h c a , D ( 1 α ∂ ν A aν − mB a ) i + X τ h ϕ τ , D τ L Λ , Λ γ τ i − h ¯ c a , DL Λ , Λ ω i (110) − D δL Λ , Λ δ ¯ c a , σ , Λ ( 1 α ∂ ν A aν − mB a ) E + X τ D δL Λ , Λ δϕ τ , σ , Λ L Λ , Λ γ τ E − D δL Λ , Λ δc a , σ , Λ L Λ , Λ ω E . The functional L Λ , Λ (Φ) generates n -point functions with 2 ≤ | n | ≤
5. Moreover, we ob-serve that only the terms emerging from the BRS-variation of the bare interaction L Λ , Λ have mass dimension greater than D = 5 , because of the cutoff function σ , Λ ( k ) (cf.remark after (99)). Given the functional L Λ , Λ , its n -point functions coincide with those ofthe functional Γ Λ , Λ , due to the identity (83).24 Restoration of the Slavnov-Taylor Identities
To restore the STI, it is in particular necessary to make vanish the relevant part of theviolating functional Γ , Λ . It will then turn out that this is also sufficient in the limitΛ → ∞ . Namely the irrelevant contributions to this functional at the bare scale Γ Λ , Λ ,which stem from the regulating function σ , Λ , are sufficiently bounded in terms of inversepowers of Λ so that we may apply Proposition 3 providing the bound (119).The freedom we dispose of to achieve this task is the freedom of choosing the renormal-ization conditions for the relevant terms appearing in the functionals Γ , Λ l,n and Γ , Λ γ ; l,n . Oninspection of the VSTI (108) one realizes that there is an obstacle on this way of proceeding :Since the insertion defining the functional Γ Λ , Λ is of dimension 5, we have to apply up to5 field- and momentum-derivatives on (108) in order to exhaust all relevant terms. We firstnotice that momentum derivatives of the cutoff function σ , Λ ( k ) = σ Λ ( k ) do not con-tribute to the relevant terms looked for, cf. (21). Hence, in the terms generated from (108)by these field- or momentum-derivatives there apply d (field or momentum)-derivatives tothe factors of the form δ Γ /δϕ in (108), and d (field or momentum)-derivatives apply to thefactors of the form Γ γ , ∂A a , or mB a , where d + d ≤
5. If d ≥ , Λ γ ; l,n , they generate irrelevant contributions, since the insertions in Γ , Λ γ ; l,n are of dimension 2. In our earlier paper [KM] such contributions to the VSTI were denotedby ”irr” in its Appendix C. They hampered the analysis of the relevant part of the VSTI atthe renormalization scale in our previous efforts since they cannot be controlled explicitly interms of the renormalization conditions. The only way out can be that the relevant termsfrom Γ , Λ l,n multiplying these irrelevant terms can always be made to vanish so as to avoidthe a priori unknown irrelevant terms to appear. One then realizes however that there arecontributions in Γ , Λ l,n , present already at the tree level l = 0 , which do not satisfy thiscriterion, namely the nonvanishing super-renormalizable three-point couplings, as well as themass term of the 2-point functions (see Appendix A).We present the following solution to this problem : The functionals Γ , Λ γ ; l,n and Γ , Λ l,n are expanded at zero momentum not only w.r.t. the fields and the momenta but also w.r.t.to the number of super-renormalizable vertices, or otherwise stated w.r.t. to the number ofmass parameters appearing in these couplings, see Section 3.2, (88) and (89). The degreeof divergence then diminishes with this number, in fact the corresponding bounds (94) and(95) show that the presence of an explicit mass term produces a gain in power counting by This property is at the origin of our particular choice of the cutoff function. L Λ , Λ and Γ Λ , Λ inherited from themass scaling (55), L Λ , Λ l,n ( λ ; ~p ) = ∞ X ν =0 ( mλ ) ν L ( ν ) , Λ , Λ l,n ( ~p ) , ~p = ( p , · · · , p | n | ) , (111)Γ Λ , Λ l,n ( λ ; ~p ) = ∞ X ν =0 ( mλ ) ν Γ ( ν ) , Λ , Λ l,n ( ~p ) . (112)Since we aim at a consistent mass expansion of the VSTI, (108), we first observe, that we alsohave to perform the mass scaling (55) of the BRS-variation α ( ∂ ν A aν ( x ) − αmB a ( x )) of theantighost appearing, cf.(15), in accord with our treatment of the BRS-insertions. We thenwant to determine via (108) the relevant part of the functional Γ , Λ , given by the values( ∂ w Γ ( ν ) , , Λ l,n )( ~ , | n | + | w | + ν ≤ l ∈ N ,( ∂ w Γ ( ν ) , , Λ l, n )( ~ ! = 0 , if | n | + | w | + ν < , (113)irrelevant contributions from the functionals Γ ( ν ) , , Λ γ τ , Γ ( ν ) , , Λ ω then are annihilated by mul-tiplication and only contributions of these functionals with | n | + | w | + ν ≤ l ≥ | n | = 3 ,(90).Here, we remind the reader that we do not apply the mass expansion to the free propagator,but only to the boundary terms appearing in the FE. Now the inverted free propagatorsΓ , Λ , n , | n | = 2 , appear in (108) as boundary terms at Λ = 0 for the functions Γ Λ , Λ l,n , andthey are then mass expanded, (55), thus satisfying (113), too. Therefore it is important toremember that the FE and the VSTI are derived before mass expanding. Afterwards weconsistently apply the mass expansion to all boundary terms and make the correspondingstatement on the bounds for the vertex functions which is verified inductively.The renormalization conditions (92) imposed on (a subset of) the relevant terms of thevertex functions imply zero renormalization conditions for the leading contributions to all26he two-point functions : δm ν ) = 0 , Σ ¯ cc ( ν ) (0) = 0 , Σ BB ( ν ) (0) = 0 , Σ hh ( ν ) (0) = 0 for ν ≤ , (114)and also Σ AB ( ν ) (0) = 0 for ν = 0 ; κ ( ν ) = 0 for ν ≤ . (115)Here we use the notations of App. A. The respective relevant parts of the inserted func-tionals Γ γ are collected in App. B. The restricted set of renormalization conditions (93) isautomatically satisfied, even in the nonvoid case with | n | = 1, n ≡ c a : Γ , Λ γ a ; n (0; ~ m R , (116)due to the explicit factor of m to be scaled according to (55).The functionals L Λ , Λ (Φ) , Γ Λ , Λ (Φ) serve to control the violation of the STI. They containirrelevant boundary terms at Λ = Λ , in contrast to the functionals without insertion orwith a BRS-insertion. These boundary terms are due to the presence of the factors σ , Λ ,cf. the remarks after (110). They are proportional to σ , Λ ( p ) − O (( p ) / Λ ) , as followsfrom (20), since the terms proportional to σ , Λ (0) = 1 are relevant.We first assert the bound on the bare functional Γ Λ , Λ , valid for l ∈ N , | ∂ w Γ ( ν ) , Λ , Λ l,n ( ~p ) | ≤ (Λ + m ) −| n |−| w |− ν (cid:16) log Λ m (cid:17) r P ( | ~p | Λ ) , (117)and trivially satisfied, unless 2 ≤ | n | ≤
5. Because of the identity (83) we can establish thecorresponding bound on L Λ , Λ and making use of (110). We employ the previous bounds on ∂ w L ( ν ) , Λ , Λ l,n , (62), and on ∂ w L ( ν ) , Λ , Λ l,n , (63) , at the value Λ = Λ . For σ , Λ ( k ) = σ Λ ( k )we use the bounds | ∂ w σ Λ ( k ) | ≤ Λ −| w | P | w | (cid:0) | k | Λ (cid:1) , which are an easy consequence of (20), the polynomials P | w | having nonnegative coefficientsnot depending on k . With these ingredients we prove (117).The bound on the functional Γ Λ , Λ (119) does not follow from the choice of standard renor-malization conditions for insertions. We rather assume its relevant part at the physical valueΛ = 0 of the flow parameter to vanish, l ∈ N ,( ∂ w Γ ( ν ) , , Λ l,n )( ~ , | n | + | w | + ν ≤ . (118)In Section 5.3 we will be able to verify these conditions from the VSTI (108), choosing forthe functionals entering the l.h.s. suitable renormalization conditions within the class (92),(93) considered. Assuming (118), we want to show that the corresponding irrelevant part27anishes upon shifting the UV- cutoff to infinity: Proposition 3
Given (118), then for l ∈ N , | n | ≥ and ≤ Λ ≤ Λ , | ∂ w Γ ( ν ) , Λ , Λ l,n ( ~p ) | ≤ (Λ + m ) −| n |−| w |− ν (cid:16) log Λ m (cid:17) r P ( | ~p | Λ + m ) . (119) with a positive integer r depending on n, l, w , and a polynomial P as in (62),(63).Proof : We first notice, that the bound (119) at Λ = Λ agrees with the bound (117) ,and at Λ < Λ majorizes this bound, if | n | + | w | + ν >
5. The functions ∂ w Γ ( ν ) , Λ , Λ l,n with flow parameter 0 ≤ Λ ≤ Λ are bounded integrating inductively the FE (86), adaptedto an integrated insertion and to the λ -expansion, however, as stated. We proceed in theinductive order as in the proof of the Proposition 1, but observing that the relevant termsof the functional treated here satisfy | n | + | w | + ν ≤ . If | n | = 2, the boundary valueeven vanishes and thus the function itself, satisfying (119) trivially. Proceeding, for given n in the irrelevant cases | n | + | w | + ν > | n | + | w | + ν ≤ ∂ w Γ ( ν ) , Λ , Λ
1; 0 ,n ( ~ | w | , the integrand in the respective remainderof the Taylor extension has already been bounded before, providing the bound for generalvalue ~p . Hence, the assertion is established in the tree order.Proceeding for l > L -functions appearing on the r.h.s. ofthe FE (86) have to be determined within this inductive process via (80), as expoundedin presenting the FE and supplemented in the text after (87), leading to the Proposition2. Therefore, to bound the r.h.s. one also needs the bound (94) on the vertex functionswithout insertions, to be dealt with independently before. As a result the bound deducedon | ∂ w L ( ν ) , Λ , Λ l − , n ′ | essentially coincides with the bound on | ∂ w Γ ( ν ) , Λ , Λ l − , n ′ | , cf. (87), i.e. has thesame form and power behaviour of Λ + m . This bound allows to estimate the r.h.s. of theFE and hereafter the integrations ”downwards” with initial conditions (117), and ”upwards”with initial conditions (118), of the irrelevant and relevant cases, respectively. Extendingfinally the relevant cases via the Taylor formula to general ~p completes the proof.Thus, given the condition (118), the bound (119) implies that the Slavnov-Taylor-Identitiesare restored in the limit Λ → ∞ . .2 Equation of motion of the anti-ghost Renormalization theory for nonabelian gauge theories in gauge invariant renormalizationschemes is generally based on the STI, complemented by the equation of motion of theantighost [Z], [FS]. In our scheme we rather start from a derivation of this equation fromthe functional integral. In Section 5.3 we will then show that this equation is satisfiedfor renormalization conditions compatible with the STI if in addition the renormalizationcondition for the longitudinal part of the gauge field propagator is fixed uniquely to vanish atzero momentum.The field equation follows from the representation (29). After functional derivation of(29) with respect to ¯ c a ( x ) we reexpress the r.h.s. as δL Λ , Λ (Φ) δ ¯ c a ( x ) e − h ( L Λ , Λ0 (Φ)+ I Λ , Λ0 ) = δδζ a ( x ) Z dµ Λ , Λ (Φ ′ ) e − h (cid:0) L Λ0 , Λ0 (Φ ′ +Φ)+ L Λ0 , Λ0 ( ζ ; Φ ′ +Φ) (cid:1)(cid:12)(cid:12)(cid:12) ζ =0 on extending the original bare interaction L Λ , Λ (Φ) by the insertion L Λ , Λ ( ζ ; Φ) = Z dx ζ a ( x ) δL Λ , Λ (Φ) δ ¯ c a ( x ) . (120)The source ζ a ( x ) is a Grassmann element carrying ghost number −
1. Treating now the r.h.s.analogously as in (44) - (47), we obtain the field equation of the antighost δL Λ , Λ (Φ) δ ¯ c a ( x ) = L Λ , Λ ζ a ( x ; Φ) , (121)employing the notation introduced there. On the r.h.s. appears the generating functional ofthe CAS with one local insertion corresponding to (120). The classical BRS-invariant action(9) satisfies the classical field equation δ/δ ¯ c a ( x ) S BRS = ∂ µ ψ aµ ( x ) − αmψ a ( x ), observing (14).The aim is to show that the relation following from the classical action at the tree level forthe physical value Λ = 0 of the flow parameter δL , Λ (Φ) δ ¯ c a ( x ) = ∂ µ L , Λ γ aµ ( x ; Φ) | mod − αmL , Λ γ a ( x ; Φ) | mod , (122)still holds in the renormalized theory. The label ”mod” is to signal that we have to replacein the bare insertions (41a)-(41d) R i → ˜ R i = O (¯ h ) for i = 1 , π ) δ Γ , Λ (Φ) δ ¯ c a ( q ) = − q + αm σ , Λ ( q ) c a ( − q ) − iq µ Γ , Λ γ aµ ( q ; Φ) | mod − αm Γ , Λ γ a ( q ; Φ) | mod . (123)29he first term on the r.h.s. is the tree level 2-point function. Restricting (123) to its relevantpart, σ , Λ ( q ) is replaced by σ , Λ (0) = 1 due to (21), the first term then provides the treeorder of R and R excluded in the insertions as indicated by the label mod , cf. (122).The proof of (123) or equivalently (122) consists in two steps of the same nature as thoseemployed in the previous section. We may consider the (regularized) inserted functionalΓ Λ , Λ c a ( q ; Φ) := (2 π ) δ Γ Λ , Λ (Φ) δ ¯ c a ( q ) + q + αm σ , Λ ( q ) c a ( − q )+ iq µ Γ Λ , Λ γ aµ ( q ; Φ) | mod + αm Γ Λ , Λ γ a ( q ; Φ) | mod . (124)In the mass expansion scheme it corresponds to an operator insertion of dimension 3, wherewe take into account also the momentum and mass factors in front of the last three terms.Since the flow equations for inserted functionals are linear, the new functional obeys again alinear flow equation obtained from those for the functionals on the r.h.s. by superposition.Note that the second term on the r.h.s., being a tree level contribution, does not flow.If we can choose renormalization conditions such that all relevant contributions to Γ Λ , Λ c a ( q ; Φ)vanish, we can prove by induction on the linear flow equation (the solution of which is uniquefor specified boundary conditions) that Γ Λ , Λ c a ( q ; Φ) ≡ , since such terms only appear in thefirst two terms on the r.h.s. at the tree level and cancel exactly. So the situation is simplerthan that of the functional Γ analysed in the previous section.At the end of the next section it is shown explicitly that the relevant contributions to(124) can be made to vanish for suitable renormalization conditions so that the equation ofmotion for the antighost (123) or (122) holds at the quantum level . We now require the relevant part of the functional Γ , Λ to vanish in accord with the VSTI(108). This requirement amounts to satisfy the 53 equations presented in the Appendix C. Itis satisfied in the tree order. Noticing that the normalization constants of the BRS-insertionsbehave as R i = 1 + O (¯ h ) , i = 1 , · · ·
7, we first analyse the equations IX to XXIX , but takealready into account the equations
V II d , V III c , the latter ones providing r hBA = r ¯ ccA = 0 . (125)In proceeding we use conditions determined before, if needed.From XIV b , XIV e , XV b , XXIII directly follow r AA ¯ cc = r AA ¯ cc = r BB ¯ cc = r AABB = 0 , (126)30nd than, from XIV a + c , XV II b , XV III c , XXV III, XXIX , r AAAA = r hh ¯ cc = r ¯ cc ¯ cc = r hB ¯ cc = r BB ¯ cc = 0 . (127) XV I a , XV III a , and XV a combined with XV I b , respectively, require R = R = R , R R = ( R ) . (128) XIV c : 2 F AAAA R = − F AAA gR (129) XI : F ¯ ccB (1) R = − F ¯ cch (1) R . (130)From X , XX , XIX , IX follow for the self-coupling of the scalar field8 F BBBB R = F BBh (1) gR , (131)4 F BBhh R = F BBh (1) gR , (132)8 F hhhh R R = F BBh (1) g ( R ) , (133) F hhh (1) R = F BBh (1) R , (134)and from XV I b , XV II a , XXI , XIII for the scalar-vector coupling2 F BBA R = − F hBA R , (135)4 F AAhh R = F hBA gR , (136)4 F AABB R = F hBA gR , (137) F AAh (1) R = F hBA R . (138)One easily verifies that the remaining equations of IX to XXIX are satisfied due to theseconditions (125)-(138).At this stage, all those relevant couplings with | n | = 3 , | n | = 3 : (129), (131)-(133), (136),(137). Inaddition, there are 4 conditions relating couplings with | n | = 3 : (130), (134), (135) and(138). Moreover, the normalization constants of the BRS-insertions are required to satisfythe three conditions (128).There are still 18 − I to V III to be considered. They contain therelevant parameters of Γ , Λ with | n | = 1 , , F hhh , together with the normalizationconstants of the BRS-insertions. Since 2 of these parameters have been fixed before, (125),there remain 26 to be dealt with. ( F hhh will then be determined by (134).) These parametersin addition have to obey the conditions derived before: We first observe that the condition31138) is identical to equation V I b . There remain the 5 conditions to be satisfied: 3 conditions(128), together with (130), (135). All these conditions generate 4 linear relations among theequations still to be considered: denoting by { X } the content of the bracket {· · · } appearingin equation X , we find [M, (4.94-97)]0 = α − { V III b } + gR { I b } + R (cid:0) { III a } + { III b } (cid:1) , (139)0 = gR { II b } − { V III b } + R { IV b } − R { V } , (140)0 = R { IV a } − R (cid:0) { V I a } − { V I b } (cid:1) , (141)0 = R { V } − R { V II c } . (142)Hence, the 26 parameters in question are constrained by 16 + 5 − renormalization conditions we then fix κ (3) = 0 and letΣ trans , Σ long , Σ AB (1) , ˙Σ ¯ cc , ˙Σ BB , F AAA , F
BBh (1) , R (143)be chosen freely. These parameters correspond to the number of wave function renormal-izations (including one for the BRS sector) and coupling constant renormalizations of thetheory. Thus, there are 26 − R ( I b ) , R ( II b ) , R ( III b ) → R , R , R due to (128) ,F ¯ ccA ( III a ) , F BBA ( V ) → F hBA due to (135) ,F AAh (1) ( V I b ) , F ¯ ccB (1) ( IV a ) → F ¯ cch (1) due to (130) , Σ ¯ cc (2) ( V III a ) , Σ BB (2) ( II a ) , δm ( I a ) , Σ hh (2) ( V II a ) , ˙Σ hh ( V II b + c ) . (144)Now all parameters are determined, without using the equations IV b , V I a , V II c , V III b . These equations, however, are satisfied because of the relations (139)-(142). Finally, therelevant couplings with | n | = 4, as well as F hhh (1) , then are explicitely given by (129),(131)-(134), (136) and (137).We have not yet implemented the field equation of the antighost (123). Performing themass scaling as before and then extracting the local content | n | + | w | + ν ≤ ¯ cc = R , (145) α + Σ ¯ cc (2) = αR , (146) F ¯ ccA = gR , (147) F ¯ ccB (1) = α gR , (148) F ¯ cch (1) = − α gR . (149)Fixing now the hitherto free renormalization constant Σ long at the particular value Σ long = 0 ,we claim these relations to be satisfied: (145) and (147) follow at once from I b and III a + b ,respectively; (148) follows from 2 { IV a } − { IV b } , due to (147) and (128); and herefrom follow(149) due to (130), and (146) because of V III a , thus establishing the claim.Given these additional relations (145)-(149) we can adjust the procedure (144) choosing nowa reduced set of free renormalization conditions (143) in which Σ long is excluded. Proceedingsimilarly as before we find I b : Σ long = 0 , II a : Σ BB (2) = 0 , (150) III b : gR = − F AAA ¯ cc trans −→ R , R , R due to (128) , (151) II b : R = 1 + ˙Σ ¯ cc BB (cid:16) AB (1) (cid:17) , (152) I a : 1 + δm = 11 + ˙Σ BB (cid:16) AB (1) (cid:17) , (153) V : 2 F BBA = F AAA BB trans −→ F hBA −→ F AAh (1) due to (135) , (138) , (154) V II a : (cid:16) Mm (cid:17) + Σ hh (2) = 4 g F BBh (1) R R , (155) V II b + c : 1 + ˙Σ hh = (1 + ˙Σ BB ) R R . (156)Resuming the following task has been achieved: we first treated the functional Γ , Λ and its ancillary functionals Γ , Λ γ τ , Γ , Λ ω with a BRS-insertion, disregarding the STI. Thereappear 37 + 7 relevant parameters. Fixing among these parameters a priori κ = 0 (notadpoles) and Σ long = 0 (due to the field equation of the antighost), and regarding the set(143) without Σ long , as renormalization constants to be chosen freely , we can uniquely de-termine the remaining relevant parameters upon requiring the relevant part of the functional33 , Λ to vanish, (118), on account of the VSTI (108). Finally, since the relevant part of thefunctional Γ , Λ vanishes , due to Proposition 3, (119), its irrelevant part vanishes in thelimit Λ → ∞ , too. Thus perturbatively the functional Γ , ∞ and its ancillary funtionalsΓ , ∞ γ τ , Γ , ∞ ω are finite and satisfy the STI, i.e. equation (108) for Λ → ∞ with the r.h.s.vanishing. Acknowledgement :Both authors have been lecturing at ESI, Vienna, about the subject of this paper ; theensuing discussions were important for its genesis; hospitality of ESI is therefore gratefullyacknowledged.
Appendix A
The bare functional L Λ , Λ and the relevant part of the generating functional Γ , Λ for theproper vertex functions have the same general form. We present the latter and give the treeorder explicitly. At the end we state the modification to obtain the bare functional L Λ , Λ .Writing Γ , Λ ( A, h, B, ¯ c, c ) = X | n | =1 Γ | n | + Γ ( | n | > , where | n | counts the number of fields, we extracted the relevant part, i.e. its local field contentwith mass dimension not greater than four. Moreover, in the sequel we do not underline thefield variables though all arguments in the Γ- functional should appear underlined, of course.1) One-point function Γ = κ ˆ h (0) .
2) Two-point functionsΓ = Z p n A aµ ( p ) A aν ( − p )Γ AAµν ( p ) + 12 h ( p ) h ( − p )Γ hh ( p ) + 12 B a ( p ) B a ( − p )Γ BB ( p ) − ¯ c a ( p ) c a ( − p )Γ ¯ cc ( p ) + A aµ ( p ) B a ( − p )Γ ABµ ( p ) o , Γ AAµν ( p ) = δ µν ( m + δm ) + ( p δ µν − p µ p ν )(1 + Σ trans ( p )) + 1 α p µ p ν (1 + Σ long ( p )) , Γ hh ( p ) = p + M + Σ hh ( p ) , Γ BB ( p ) = p + αm + Σ BB ( p ) , Γ ¯ cc ( p ) = p + αm + Σ ¯ cc ( p ) , Γ ABµ ( p ) = ip µ Σ AB ( p ) . δm , Σ trans (0) , Σ long (0) , Σ hh (0) , ˙Σ hh (0) , Σ BB (0) , ˙Σ BB (0) , Σ ¯ cc (0) , ˙Σ ¯ cc (0) , Σ AB (0) , where the notation ˙Σ(0) ≡ ( ∂ p Σ)(0) has been used. We note that because of the regular-ization, the inverse of the regularized propagators (22) actually appears as the tree order( l = 0) of the 2-point functions. Due to the property (21), however, the regularizing factor( σ , Λ ( p )) − does not contribute to the relevant part.3) Three-point functionsWe only present the relevant part explicitly. A relevant parameter vanishing in the tree orderis denoted by r ∈ O (¯ h ), otherwise it is denoted by F . Moreover, we indicate an irrelevantpart by a symbol O n , n ∈ N , reminding that this part vanishes like an n -th power of amomentum when all momenta tend to zero homogeneously.Γ = Z p Z q (cid:8) ǫ rst A rµ ( p ) A sν ( q ) A tλ ( − p − q )Γ AAAµνλ ( p, q )+ A rµ ( p ) A rν ( q ) h ( − p − q )Γ AAhµν ( p, q )+ ǫ rst B r ( p ) B s ( q ) A tµ ( − p − q )Γ BBAµ ( p, q )+ h ( p ) B r ( q ) A rµ ( − p − q )Γ hBAµ ( p, q ) + ǫ rst ¯ c r ( p ) c s ( q ) A tµ ( − p − q )Γ ¯ ccAµ ( p, q )+ B r ( p ) B r ( q ) h ( − p − q )Γ BBh ( p, q ) + h ( p ) h ( q ) h ( − p − q )Γ hhh ( p, q )+ ¯ c r ( p ) c r ( q ) h ( − p − q )Γ ¯ cch ( p, q ) + ǫ rst ¯ c r ( p ) c s ( q ) B t ( − p − q )Γ ¯ ccB ( p, q ) (cid:9) , Γ AAAµνλ ( p, q ) = δ µν i ( p − q ) λ F AAA + O , F AAA = − g + r AAA , Γ AAhµν ( p, q ) = δ µν F AAh + O , F AAh = mg + r AAh , Γ BBAµ ( p, q ) = i ( p − q ) µ F BBA + O , F BBA = − g + r BBA , Γ hBAµ ( p, q ) = i ( p − q ) µ F hBA F hBA = g + r hBA , + i ( p + q ) µ r hBA + O , Γ ¯ ccAµ ( p, q ) = ip µ F ¯ ccA + iq µ r ¯ ccA + O , F ¯ ccA = g + r ¯ ccA , Γ BBh ( p, q ) = F BBh + O , F BBh = g M m + r BBh , Γ hhh ( p, q ) = F hhh + O , F hhh = g M m + r hhh , Γ ¯ cch ( p, q ) = F ¯ cch + O , F ¯ cch = − αgm + r ¯ cch , Γ ¯ ccB ( p, q ) = F ¯ ccB + O , F ¯ ccB = αgm + r ¯ ccB . The 3-point functions
AAB and
BBB have no relevant local content.4) Four-point functions 35efining as before parameters r and F , thenΓ | rel = Z k Z p Z q (cid:8) ǫ abc ǫ ars A bµ ( k ) A cν ( p ) A rµ ( q ) A sν ( − k − p − q ) F AAAA + A rµ ( k ) A rµ ( p ) A sν ( q ) A sν ( − k − p − q ) r AAAA + A aµ ( k ) A bµ ( p )¯ c r ( q ) c s ( − k − p − q )( δ ab δ rs r AA ¯ cc + δ ar δ bs r AA ¯ cc )+ A aµ ( k ) A bµ ( p ) B r ( q ) B s ( − k − p − q )( δ ab δ rs F AABB + δ ar δ bs r AABB )+ B a ( k ) B b ( p )¯ c r ( q ) c s ( − k − p − q )( δ ab δ rs r BB ¯ cc + δ ar δ bs r BB ¯ cc )+ h ( k ) h ( p ) h ( q ) h ( − k − p − q ) F hhhh + B r ( k ) B r ( p ) h ( q ) h ( − k − p − q ) F BBhh + B r ( k ) B r ( p ) B s ( q ) B s ( − k − p − q ) F BBBB + A rµ ( k ) A rµ ( p ) h ( q ) h ( − k − p − q ) F AAhh + h ( k ) h ( p )¯ c r ( q ) c r ( − k − p − q ) r hh ¯ cc +¯ c a ( k ) c a ( p )¯ c r ( q ) c r ( − k − p − q ) r ¯ cc ¯ cc + ǫ rst h ( k ) B r ( p )¯ c s ( q ) c t ( − k − p − q ) r hB ¯ cc (cid:9) ,F AAAA = g + r AAAA , F AABB = g + r AABB ,F hhhh = g (cid:0) Mm (cid:1) + r hhhh , F BBhh = g (cid:0) Mm (cid:1) + r BBhh ,F BBBB = g (cid:0) Mm (cid:1) + r BBBB , F
AAhh = g + r AAhh . Hence, Γ , Λ in total involves 1 + 10 + 11 + 15 = 37 relevant parameters.We now obtain the form of the bare functional L Λ , Λ , together with its order l = 0 explicitlygiven, upon deleting in the two-point functions the contributions of the order l = 0, i.e.keeping there only the 10 parameters which appear in the various self-energies. Appendix B
Analysing the STI, vertex functions (72) with one operator insertion, generated by the BRS-variations, have to be considered, too. These insertions have mass dimension D = 2. Weremind the notation (47) and (48) of the corresponding Fourier-transform, presenting the36espective relevant part of these four vertex functions with one insertion,ˆΓ , Λ γ aµ ( q, Φ) | rel = − iq µ c a ( − q ) R + ǫ arb Z k A rµ ( k ) c b ( − q − k ) gR , ˆΓ , Λ γ ( q ; Φ) | rel = − g Z k B r ( k ) c r ( − q − k ) R , ˆΓ , Λ γ a ( q ; Φ) | rel = mc a ( − q ) R + Z k h ( k ) c a ( − q − k ) 12 gR + ǫ arb Z k B r ( k ) c b ( − q − k ) 12 gR , ˆΓ , Λ ω a ( q ; Φ) | rel = ǫ ars Z k c r ( k ) c s ( − q − k ) 12 gR . There appear 7 relevant parameters R i = 1 + r i , r i = O (¯ h ) , i = 1 , ..., . All the other two-point functions, and the higher ones, of course, are of irrelevant type.
Appendix C
As a consequence of the expansion in the mass parameters the conditions following from thefact that the relevant part of the functional Γ should vanishΓ ( A, h, B, ¯ c, c ) | dim ≤ != 0 . can be reordered according to the value of ν which appears. We get contributions for0 ≤ ν ≤ ν in the various relevant couplings is indicated as a superscript inparentheses if ν > m in front ofeach STI. The power of m indicates the value of ν in the corresponding contribution to Γ .Two fieldsI) δ A aµ ( q ) δ c r ( k ) Γ | a) 0 ! = m q µ n − (1 + δm ) R + P AB (1) R + 1 + α P ¯ cc (2) o ,b) 0 ! = q q µ n − α (1 + P long ) R + α (1 + ˙ P ¯ cc ) o .II) δ B a ( q ) δ c r ( k ) Γ | a) 0 ! = m n ( α + P BB (2) ) R − ( α + P ¯ cc (2) ) − g κ (3) R o .37) 0 ! = m q n − P AB (1) R + (1 + ˙ P BB ) R − (1 + ˙ P ¯ cc ) o .Three fieldsIII) δ A rµ ( p ) δ A sν ( q ) δ c t ( k ) Γ | a) 0 ! = ( p µ p ν − q µ q ν ) n − F AAA R − α ( F ¯ ccA − r ¯ ccA )+ h α (1+ P long ) − (1+ P trans ) i gR o , b) 0 ! = ( p − q ) δ µν (cid:8) F AAA R + (1 + P trans ) gR (cid:9) ,IV) δ A rµ ( p ) δ B s ( q ) δ c t ( k ) Γ | a) 0 ! = m p µ n F BBA R + g P AB (1) R + α F ¯ ccB, (1) − r ¯ ccA o ,b) 0 ! = m q µ n g P AB (1) R + 4 F BBA R + ( F ¯ ccA − r ¯ ccA ) o ,V) δ B r ( p ) δ B s ( q ) δ c t ( k ) Γ | ! = ( p − q ) n R F BBA + (1 + ˙ P BB ) g R o ,VI) δ A rµ ( p ) δ h ( q ) δ c t ( k ) Γ | a) 0 ! = m p µ n − R F AAh (1) + R ( F hBA − r hBA ) + P AB (1) 12 gR − α F ¯ cch (1) o ,b) 0 ! = m q µ (cid:8) − R F AAh (1) + 2 R F hBA (cid:9) ,VII) δ h ( p ) δ B s ( q ) δ c t ( k ) Γ | a) 0 ! = m n ( M m + P hh (2) )( − gR ) + 2 F BBh (1) R + F ¯ cch (1) + ( α + P BB (2) ) gR o ,b) 0 ! = p n F hBA R − (1 + ˙ P hh ) gR o ,c) 0 ! = q n − F hBA R + (1 + ˙ P BB ) gR o ,d) 0 ! = k (cid:8) r hBA R (cid:9) ,VIII) δ c t ( q ) δ c s ( p ) δ ¯ c r ( k ) Γ | a) 0 ! = m n F ¯ ccB (1) R − ( α + P ¯ cc (2) ) gR o ,b) 0 ! = k n F ¯ ccA R − r ¯ ccA R − (1 + ˙ P ¯ cc ) gR o ,c) 0 ! = ( p + q ) n r ¯ ccA R o . 38our fieldsIX) δ h ( p ) δ h ( q ) δ B ( k ) δ c ( l ) Γ | ! = m (cid:8) F hhh, (1) ( − gR ) + 4 F BBhh R + 2 F BBh, (1) gR + 2 r hh ¯ cc (cid:9) .X) δ B ( k ) δ B ( p ) δ B ( q ) δ c ( l ) Γ | ! = m (cid:8) − F BBh, (1) gR + 8 F BBBB R + (cid:0) r BB ¯ cc + r BB ¯ cc (cid:1)(cid:9) .XI) δ h ( l ) δ ¯ c ( k ) δ c ( p ) δ c ( q ) Γ | ! = m (cid:8) r hB ¯ cc R + F ¯ ccB (1) gR + F ¯ cch, (1) gR (cid:9) .XII) δ c ( k ) δ ¯ c ( l ) δ c ( p ) δ B ( q ) Γ | ! = m (cid:8) F ¯ cch (1) ( − gR ) + (2 r BB ¯ cc − r BB ¯ cc ) R + F ¯ ccB (1) ( gR − gR ) + 2 r ¯ cc ¯ cc (cid:9) .XIII) δ A µ ( k ) δ A ν ( p ) δ B ( q ) δ c ( l ) Γ | ! = 2 r AABB R + r AA ¯ cc .XIII) δ A µ ( k ) δ A ν ( p ) δ B ( q ) δ c ( l ) Γ | ! = m (cid:8) − F AAh (1) gR + 4 F AABB R + 2 r AA ¯ cc (cid:9) .XIV) δ A µ ( p ) δ A ν ( q ) δ A ρ ( k ) δ c ( l ) Γ | a) 0 ! = 2 δ µν l ρ n F AAAA + r AAAA ) R + 2 F AAA gR + α r AA ¯ cc o ,b) 0 ! = δ µν ( p ρ + q ρ ) (cid:8) α r AA ¯ cc (cid:9) ,c) 0 ! = ( δ µρ l ν + δ νρ l µ ) (cid:8) − F AAAA R − F AAA gR (cid:9) ,d) 0 ! = ( δ µρ p ν + δ νρ q µ ) { } ,e) 0 ! = ( δ µρ q ν + δ νρ p µ ) (cid:8) − α r AA ¯ cc (cid:9) .XV) δ B ( p ) δ B ( q ) δ A µ ( k ) δ c ( l ) Γ | a) 0 ! = l µ n F AABB R + 2 F BBA gR o ,b) 0 ! = k µ (cid:8) r BB ¯ cc (cid:9) , 39V) δ B ( p ) δ B ( q ) δ A µ ( k ) δ c ( l ) Γ | a) 0 ! = p µ (cid:8) − r AABB R + 2 F BBA gR + F hBA gR (cid:9) ,b) 0 ! = q µ (cid:8) − r AABB R − F BBA gR + 2 F BBA gR (cid:9) ,c) 0 ! = k µ n − r AABB R + F hBA gR + r hBA gR + F BBA gR − α r BB ¯ cc o ,XVI) δ h ( p ) δ A µ ( k ) δ B ( q ) δ c ( l ) Γ | a) 0 ! = p µ (cid:8) F hBA g ( R − R ) − r hBA gR (cid:9) ,b) 0 ! = q µ (cid:8) F hBA gR − r hBA gR + 2 F BBA gR (cid:9) ,c) 0 ! = k µ n F hBA gR − r hBA gR + F BBA gR − α r hB ¯ cc o ,XVII) δ h ( p ) δ h ( q ) δ A µ ( k ) δ c ( l ) Γ | a) 0 ! = l µ (cid:8) F AAhh R − F hBA gR (cid:9) ,b) 0 ! = k µ (cid:8) r hBA gR + α r hh ¯ cc (cid:9) .XVIII) δ A µ ( k ) δ c ( p ) δ c ( q ) δ ¯ c ( l ) Γ | a) 0 ! = l µ (cid:8) F ¯ ccA g ( R − R ) + α r ¯ cc ¯ cc (cid:9) ,b) 0 ! = p µ n r AA ¯ cc R + r ¯ ccA g ( R − R ) + α r ¯ cc ¯ cc o ,c) 0 ! = q µ n − r AA ¯ cc R − r ¯ ccA gR + α r ¯ cc ¯ cc o .Five fieldsXIX) δ h ( p ) δ h ( q ) δ h ( k ) δ B ( l ) δ c ( l ′ ) Γ | ! = − F hhhh R + F hhBB R .XX) δ h ( p ) δ B ( q ) δ B ( k ) δ B ( l ) δ c ( l ′ ) Γ | ! = − F BBhh R + 2 F BBBB R .XXI) δ A µ ( k ) δ A ν ( p ) δ h ( k ) δ B ( l ) δ c ( l ′ ) Γ | ! = − F AAhh R + F AABB R . 40XII) δ A µ ( k ) δ B ( p ) δ c ( l ′ ) δ A ν ( q ) δ B ( l ) Γ | ! = r AABB ( R − R ).XXIII) δ A µ ( k ) δ B ( q ) δ A ν ( p ) δ c ( l ′ ) δ h ( l ) Γ | ! = r AABB R .XXIV) δ A µ ( k ) δ A ν ( p ) δ ¯ c ( q ) δ c ( l ) δ c ( l ′ ) Γ | ! = r AA ¯ cc R + r AA ¯ cc R .XXV) δ A µ ( k ) δ ¯ c ( q ) δ A ν ( p ) δ c ( l ) δ c ( l ′ ) Γ | ! = r AA ¯ cc (3 R − R ).XXVI) δ B ( p ) δ B ( q ) δ ¯ c ( k ) δ c ( l ) δ c ( l ′ ) Γ | ! = r BB ¯ cc ( R − R ) − r BB ¯ cc R .XXVII) δ B ( p ) δ ¯ c ( k ) δ B ( q ) δ c ( l ) δ c ( l ′ ) Γ | ! = − r hB ¯ cc R + r BB ¯ cc (3 R − R ).XXVIII) δ h ( p ) δ h ( q ) δ ¯ c ( k ) δ c ( l ) δ c ( l ′ ) Γ | ! = r hB ¯ cc R + r hh ¯ cc R .XXIX) δ h ( p ) δ B ( q ) δ c ( l ) δ ¯ c ( k ) δ c ( l ′ ) Γ | ! = 2 r hh ¯ cc R − r BB ¯ cc R + r BB ¯ cc R + r hB ¯ cc ( − R + 2 R ).41 eferences [BRS] C. 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