IFUP-TH 2008/18 Weighted Power Counting And Lorentz Violating Gauge Theories. 9 0 II: Classi(cid:28) ation 0 2 n a J 8 Damiano Anselmi 2 Dipartimento di Fisi a (cid:16)Enri o Fermi(cid:17), Università di Pisa, ] Largo Ponte orvo 3, I-56127 Pisa, Italy, h t and INFN, Sezione di Pisa, Pisa, Italy - p damiano.anselmidf.unipi.it e h [ 2 v 4 7 Abstra t 4 3 . 8 We lassify the lo al, polynomial, unitary gauge theories that violate Lorentz symmetry expli itly at 0 8 highenergiesandarerenormalizablebyweightedpower ounting. Westudythestru tureofsu htheories 0 : and provethat renormalization does not generate higher time derivatives. We work out the onditions to v i renormalize verti es that are usually non-renormalizable, su h as the two s alar-two fermion intera tions X and the four fermion intera tions. A number of four dimensional examples are presented. r a 1 1 Introdu tion Lorentz symmetry is a fundamental assumption behind the Standard Model of parti le physi s. Experimental bounds on the parameters of the Lorentz violating Standard-Model extension [1℄ are often very pre ise [2℄. Nevertheless, several authors, inspired by di(cid:27)erent onsiderations, have suggestedthatLorentzsymmetryandCPT ouldbebrokenatveryhighenergies[3℄. Theproblem ofLorentzviolationhasattra tedalotofinterest, in osmology,astrophysi s,high-energyphysi s. If Lorentz symmetry were not exa t our understanding of Nature would hange onsiderably. We an imagine that the Standard Model is orre ted by Lorentz violating terms of higher Λ L dimensions, multiplied by inverse powers of a s ale , whi h an be understood as the s ale Λ L of Lorentz violation. If is su(cid:30) iently large, the orre ted model an be organized so that it agrees with all present experimental data, yet it predi ts violations of Lorentz symmetry starting ∼ Λ L from energies . If we do not assume exa t Lorentz invarian e at arbitrarily high energies, yet demand lo ality and unitarity, several theories that are not renormalizable by the usual power ounting be ome renormalizableintheframeworkofa(cid:16)weightedpower ounting(cid:17) [4℄,whi hassignsdi(cid:27)erentweights to spa e and time. The large momentum behavior of propagators is improved by quadrati terms ontaining higher spa e derivatives. The set of verti es is arranged so that no higher time derivatives are generated by renormalization, in agreement with (perturbative) unitarity. S alar and fermion theories of this type have been studied in [4, 5℄. In ref. [6℄, to whi h we refer as (cid:16)paper I(cid:17) from now on, the basi properties of Lorentz violating gauge theories have been derived. Here we give an exhaustive lassi(cid:28) ation of gauge theories, investigate their stru ture and study a number of four dimensional examples. We sear h for theories that are lo al and polynomial, free of infrared divergen es in the Feyn- man diagrams, and renormalizable by weighted power ounting. To avoid the presen e of ertain spurious subdivergen es, originated by the pe uliar form of the gauge-(cid:28)eld propagator, spa etime is split into spa e and time and other restri tions are imposed. The paper is organized as follows. In se tion 2 we review the weighted power ounting. In se tion 3 we study the stru ture of renormalizable theories ontaining gauge (cid:28)elds and matter. We work out the onditions for polynomiality and renormalizability, and prove that higher time derivatives are absent. In se tion 4 we derive su(cid:30) ient onditions for the absen e of spurious subdivergen es. In se tion 5 we study the onditions to renormalize verti es that are usually non-renormalizable, in parti ular the two s alar-two fermion intera tions and the four fermion intera tions, and illustrate a number of four dimensional examples. In se tion 6 we onsider the most general type of Lorentz violations. Se tion 7 ontains our on lusions. In appendix A we re all the form of the gauge-(cid:28)eld propagator and thedispersion relations. In appendix B we study the renormalizability of our theories to all orders, using the Batalin-Vilkovisky formalism. 2 2 Weighted power ounting In this se tion we review the weighted power ounting riterion of refs. [4, 5℄ and a number of resultsfrompaper I.Thesimplestframework to studytheLorentz violationsisto assumethatthe d O(1,d−1) O(1,dˆ−1)×O(d¯) -dimensional Lorentz group isbroken to a residual symmetry . The d M = Rd Mˆ ×M¯ -dimensional spa etime manifold is split into the produ t of two submanifolds, dˆ Mˆ = Rdˆ a -dimensional submanifold , ontaining time and possibly some spa e oordinates, and d¯ M¯ = Rd¯ ∂ (∂ˆ,∂¯) a -dimensional spa e submanifold . The partial derivative is de omposed as , ∂ˆ ∂¯ Mˆ M¯ where and a tonthesubspa es and ,respe tively. Coordinates, momenta andspa etime indi es are de omposed similarly. We (cid:28)rst study renormalization in this simpli(cid:28)ed framework and later generalize our results to more general breakings (see se tion 6). For simpli ity, we assume separate invarian es under C, P and T throughout this paper. Consider a free s alar theory with (Eu lidean) lagrangian 1 1 L = (∂ˆϕ)2+ (∂¯nϕ)2, 2 2Λ2n−2 (2.1) free L Λ n > 1 L where is an energy s ale and is an integer . It is invariant under the weighted res aling xˆ → xˆ e−Ω, x¯ → x¯ e−Ω/n, ϕ → ϕ eΩ( /2−1), ž (2.2) = dˆ+d¯/n Λ L where ž is the (cid:16)weighted dimension(cid:17). Note that is not res aled. The intera ting theory is de(cid:28)ned as a perturbative expansion around (2.1). For the purposes of renormalization, the masses and the other quadrati terms an be treated perturbatively, sin e O [O] the ounterterms depend polynomially on them. Denote the (cid:16)weight(cid:17) of an obje t by and assign weights to oordinates, momenta and (cid:28)elds as follows: 1 1 [xˆ]= −1, [x¯]= − , [∂ˆ] = 1, [∂¯] = , [ϕ] = ž −1, n n 2 (2.3) Λ L while is weightless. The lagrangian terms of weight ž are stri tly renormalizable, those of weightssmallerthanžsuper-renormalizableandthoseofweightsgreaterthanžnon-renormalizable. The weighted power ounting riterion amounts to demand that the theory ontains no parame- ter of negative weight. The onsiderations just re alled are easily generalized to fermions, whose −1)/2 weight is (ž . A = AaTa Ta A = (Aˆ,A¯) µ µ The gauge (cid:28)eld , with anti-Hermitian, is de omposed as . The ovariant derivative D = (Dˆ,D¯) = (∂ˆ+gAˆ,∂¯+gA¯) (2.4) indu es the weight assignments 1 [gAˆ]= [Dˆ] = 1, [gA¯]= [D¯] = , n 3 g ∼ (∂ˆAˆ)2 where is the gauge oupling. On the other hand, the weight-ž kineti term gives [Aˆ]= /2−1 [g] = 2− /2 ž , so ž . The (cid:28)eld strength is split as Fˆ ≡ F , F˜ ≡ F , F¯ ≡ F . µν µˆνˆ µν µˆν¯ µν µ¯ν¯ (2.5) We (cid:28)nd 1 1 2 [Aˆ] = ž−1, [A¯]= ž−2+ , [Fˆ] = ž, [F˜]= ž−1+ , [F¯]= ž−2+ . 2 2 n 2 2 n 2 n (2.6) In the presen e of gauge intera tions the renormalizable theories are still those that do not ontain parameters of negative weights. To single out the super-renormalizable theories we an re(cid:28)ne this requirement, demanding that no parameters have weights smaller than some non- χ negative onstant . Indeed, if that happens Feynman diagrams are ertainly multiplied by χ oe(cid:30) ients of weights greater than or equal to , so no new ounterterms are turned on by g renormalization. Applying the re(cid:28)ned requirement to the gauge oupling we (cid:28)nd 0 ≤ χ ≤ 2− /2. ž (2.7) ≤ 4 In parti ular, we must have ž . It is onvenient to write the gauge-(cid:28)eld a tion S = ddx(L +L ) ≡ S +S , 0 Q I Q I (2.8) Z S S S Q I Q as the sum of two ontributions and : olle ts the gauge-invariant quadrati terms ≤ S I of weight ž, onstru ted with two (cid:28)eld strengths and possibly ovariant derivatives, while ≤ −χ olle ts the vertex terms of weights ž , onstru ted with at least three (cid:28)eld strengths and possibly ovariant derivatives. L Q Up to total derivatives the quadrati part of the lagrangian reads (in the Eu lidean frame- work) 1 1 L = Fˆ2 +2F η(Υ¯)F +F τ(Υ¯)F + (D F )ξ(Υ¯)(D F ) . Q 4 µν µˆν¯ µˆν¯ µ¯ν¯ µ¯ν¯ Λ2 ρˆ µ¯ν¯ ρˆ µ¯ν¯ (2.9) (cid:26) L (cid:27) Υ¯ ≡ −D¯2/Λ2 η τ ξ The proof an be found in paper I. Here L and , and are polynomials of degrees n−1 2n−2 n−2 , and , respe tively. We have expansions n−1 2j η(Υ¯) = η Υ¯i, [η ]= , n−1−i j n (2.10) i=0 X η i and similar, where are dimensionless onstants of non-negative weights. The free a tion is positive de(cid:28)nite if and only if k¯2 η > 0, η˜≡ η+ ξ > 0, τ >0, Λ2 (2.11) L 4 η τ ξ k¯2/Λ2 where now , and are fun tions of L. Furthermore, we assume η > 0, τ > 0, η˜ = η +ξ > 0, η > 0, τ > 0. 0 0 0 0 0 n−1 2n−2 (2.12) The (cid:28)rst three onditions ensure that the propagators have the best UV behaviors. The other two onditions, together with d ≥ 4, (2.13) ensurethattheFeynmandiagramsarefreeofIRdivergen esatnon-ex eptionalexternalmomenta, despite the fa t that the gauge (cid:28)elds are massless. The reason is that, under the mentioned assumptions, the IR behavior of Feynman diagrams is governed by the low-energy theory 1 L = (Fa )2+2η (Fa )2+τ (Fa )2 , IR 4 µˆνˆ n−1 µˆν¯ 2n−2 µ¯ν¯ (2.14) (cid:2) (cid:3) whi h has an ordinary power ounting. The BRST symmetry [7℄ oin ides with the usual one, g sAa =DabCb = ∂ Ca+gfabcAbCc, sCa = − fabcCbCc, µ µ µ µ 2 sC¯a=Ba, sBa = 0, sψi = −gTaCaψj, ij et ., with the weight assignments [C]= [C¯]= ž −1, [s] = 1, [B] = ž. 2 2 (2.15) We hoose the gauge-(cid:28)xing λ L = sΨ, Ψ = C¯a − Ba+Ga , Ga ≡ ∂ˆ·Aˆa+ζ(υ¯)∂¯·A¯a, gf 2 (2.16) (cid:18) (cid:19) λ υ¯ ≡ −∂¯2/Λ2 ζ where is a dimensionless, weightless onstant, L and is a polynomial of degree n−1 . Compatibly with (2.12) we assume ζ > 0, ζ > 0, ζ > 0. 0 n−1 (2.17) The total gauge-(cid:28)xed a tion is S = ddx(L +L +L ) ≡ S +S . Q I 0 gf gf (2.18) Z The propagator is reported in appendix A, together with the dispersion relations. η τ ξ ζ i i i i For the purposes of renormalization, we an treat the weightful parameters , , and , i > 0 , perturbatively, be ause the divergent parts of Feynman diagrams depend polynomially on 5 them. In this framework, the propagators we use in the high-energy analysis of the diagrams are (A.2)-(A.4) with the repla ements k¯2 n−1 k¯2 2(n−1) k¯2 n−2 k¯2 n−1 η → η , τ → τ , ξ → ξ , ζ → ζ , 0 Λ2 0 Λ2 0 Λ2 0 Λ2 (cid:18) L(cid:19) (cid:18) L(cid:19) (cid:18) L(cid:19) (cid:18) L(cid:19) everyothertermbeingtreatedasavertex. Intermediatemasses anbeaddedtothedenominators, to avoid IR problems, and removed immediately after al ulating the divergent parts. P (pˆ,p¯) pˆ p¯ k k k,n Were all that isaweighted polynomial in and , ofdegree , where isamultiple 1/n P (ξnpˆ,ξp¯) kn ξ k,n of , if is a polynomial of degree in . A propagator is regular if it is the ratio P (kˆ,k¯) r P′ (kˆ,k¯) (2.19) 2s r 2s r s of two weighted polynomials of degrees and , where and are integers, su h that the P′ (kˆ,k) denominator 2s is non-negative (in the Eu lidean framework), non-vanishing when either kˆ 6= 0 k 6= 0 or and has the form P′(kˆ,k¯) = ωˆ(kˆ2)s +ω¯(k¯2)ns+··· , s (2.20) ωˆ > 0 ω¯ > 0 (kˆ2)j−m(k¯2)mn j < s 0 ≤ m ≤ j with , , where the dots olle t the terms with , , j = s 0 < m < s and , . The regularity onditions ensure that the derivatives with respe t to kˆ k¯ ω¯ 6= 0 kˆ improve the large- behavior (be ause ), besides the large- and overall ones, and the k¯ kˆ ωˆ 6= 0 k¯ derivatives with respe t to improve the large- behavior (be ause ), besides the large- kˆ k¯ k¯ and overall ones. For this reason, the -subdivergen es are lo al in and the -subdivergen es kˆ kˆ k are lo al in . The -subintegrals and the -subintegrals, whi h annot behave worse than the kˆ k - -integrals, are automati ally ured by the ounterterms that subtra t the overall divergen es kˆ k of the - -integrals. Su h ounterterms are, for example, the (cid:28)rst terms of the (cid:16)weighted Taylor expansion(cid:17) around vanishing external momenta [4℄. A propagator that does not satisfy (2.19) an generate spurious ultraviolet subdivergen es in kˆ k Feynman diagrams when tends to in(cid:28)nity at (cid:28)xed, or vi eversa. The gauge and ghost prop- agators (A.2), (A.3) are regular at non-ex eptional momenta, be ause the positivity onditions (2.11) and (2.17) ensure that the denominators are positive-de(cid:28)nite in the Eu lidean framework. hA¯A¯i Moreover, the onditions (2.12) ensure that all su h propagators but satisfy (2.19)-(2.20) hA¯A¯i k kˆ in the Feynman gauge (A.4). Instead, is regular when tends to in(cid:28)nity at (cid:28)xed, but kˆ k ∼ 1/kˆ2 not when tends to in(cid:28)nity at (cid:28)xed, where it behaves like . To ensure that no spurious kˆ subdivergen e is generated by the -subintegrals, a more areful analysis must be performed, to whi h we devote se tion 4. The result is that the su(cid:30) ient onditions to ensure the absen e of spurious subdivergen es in lude dˆ= 1, d = n= , even odd (2.21) 6 plus other restri tions stated at the end of se tion 4. In parti ular, spa etime is split into spa e and time. In se tion 6 we prove that, be ause of the spurious subdivergen es, more general type dˆ> 1 of Lorentz violations ( ) are disfavored. The absen e of spurious subdivergen es ensures the lo ality of ounterterms. Consider a G r diagram equipped with the subtra tions that take are of its diverging proper subdiagrams. G pˆ p¯ r i i Di(cid:27)erentiating a su(cid:30) ient number of times with respe t to any omponents , of the p i external momenta , we an arbitrarily redu e the overall degree of divergen e and eventually produ e a onvergent integral. Therefore, overall divergen es are polynomial in all omponents of the external momenta. 3 Stru ture of renormalizable theories In this se tion we investigate renormalizable and super-renormalizable theories in detail. We study the onditions for renormalizability and polynomiality, and investigate the time-derivative stru ture. In se tion 4we study thespurious subdivergen es, while se tion 5 isdevoted to expli it examples, mainly four dimensional. ≥ χ χ We know that the theories ontain only parameters of weights , where satis(cid:28)es (2.7). λ i i Call the oupling multiplying the -th vertex belonging to the physi al se tor and denote the n n ≥ 3 [λ ] ≥ χ i i i number of its external legs by . Clearly, and . By polynomiality, the number of χ ≡ min [λ ] i i physi al verti es is (cid:28)nite, so we an take . De(cid:28)ne [λ ] i κ≡ min . i ni−2 Sin e the gauge oupling multiplies three-leg verti es, we have [λ ] ≥ (n −2)κ ∀i, 0 ≤ κ≤ 2− ž, i i and 2 (3.1) χ > 0 κ > 0 g¯ κ λ = λ¯ g¯ni−2 i i and if and only if . Introdu e a oupling of weight and write . Then [λ¯ ] ≥ 0 1/α¯ α¯ = g¯2 i (3.1) ensures . The theory an be reformulated in the (cid:16) form(cid:17) ( ), namely as 1 L = L¯ (g¯A,g¯ϕ,g¯ψ,g¯C¯,g¯C,λ¯), 1/α¯ r α¯ (3.2) ϕ ψ where and are matter (cid:28)elds (s alars and fermions, respe tively) and the redu ed lagrangian L¯ g¯ λ¯ g = g¯ρ r depends polynomially on and the 's. The gauge oupling an be parametrized as , ρ λ¯ where has a non-negative weight and is in luded in the set of the 's. A generi vertex of (3.2) has the stru ture λ¯ g¯ni−2∂ˆk∂¯mAˆpA¯qC¯rCrϕsψ¯tψt, i (3.3) n = p+q+2r+s+2t p,q,r,k,m,s t i where and and are integers. Formula (3.3) and analogous expressions in this paper are meant (cid:16)symboli ally(cid:17), whi h means that we pay attention to the 7 (cid:28)eld- and derivative- ontents of the verti es, but not where the derivatives a t and how Lorentz, gauge and other indi es are ontra ted. L Every ounterterm generated by (3.2) (cid:28)ts into the stru ture (3.2). Indeed, onsider a -loop E I v i i diagram with external legs, internal legs and verti es of type . The leg- ounting gives n v = E +2I = E +2(L+V −1) i i i , so the diagram is multiplied by a produ t of ouplings P g¯Pi(ni−2)vi λ¯vi = α¯Lg¯E−2 λ¯vi. i i (3.4) i i Y Y g¯E−2 Weseethata fa torizes, asexpe ted. Moreover, ea h looporder arriesanadditional weight 2κ of at least . κ = 2− /2 g¯= g 1/α When ž we an take , whi h gives the theories onsidered in paper I. They have a lagrangian of the form 1 L = L (gA,gϕ,gψ,gC¯,gC,λ). 1/α α r (3.5) The lass (3.2) is mu h ri her than the lass (3.5), yet it is does not over the most general ase. To move a step forward towards the most general lass of theories, it is useful to show how to 1/α¯ gauge s alar-fermion theories. Express the matter theory in form, namely 1 L = L¯ (g¯ϕ,g¯ψ,λ¯ ). sf sf matter α¯ (3.6) [g] ≥ [g¯] g = g¯ρ [ρ] ≥ 0 We assume that and write , with . In this way, the gauge intera tions an ρ be swit hed o(cid:27) letting tend to zero. Covariantize the derivatives ontained in (3.6) and add the 1/α pure gauge theory, plus extra terms allowed by the weighted power ounting. We obtain a mixed theory of the form 1 1 1 L = L (gA,gC¯,gC,λ )+ L¯ (g¯ϕ,g¯ψ,λ¯ )+ ∆L(gA,gC¯,gC,g¯ϕ,g¯ψ,λ). g g sf sf α α¯ α¯ (3.7) ∆L L¯ sf Here ontains both the terms ne essary to ovariantize and the mentioned extra terms. G E L g¯ ρ Consider a diagram with external legs and loops. Using the - parametrization and G α¯Lg¯E−2 repeating the argument that leads to (3.4) we (cid:28)nd that is multiplied by , so it agrees L G g with the stru ture (3.7). On the other hand, every vertex of used to onstru t provides at A C¯ C G least two internal legs. Therefore, every external -, - and -leg of is multiplied by at least ρ one power of . This proves that the stru ture (3.7) is renormalizable. The theory is polynomial [g¯ϕ],[g¯ψ] > 0 > 2−2κ > 1 −2κ if , namely ž if s alar (cid:28)elds are present, ž if the matter se tor ontains only fermions. Now we are ready to introdu e the most general lass of theories, where di(cid:27)erent (cid:28)elds an g¯ g¯ i = 1,2,3, i 1 arry di(cid:27)erent 's. Call , the ones of ve tors, fermions and s alars, respe tively . g¯ 1 Amoregeneralsituation wheredi(cid:27)erentsubsetsof(cid:28)eldswiththesamespinhavedi(cid:27)erent 'sisalsopossible. This generalization is straightforward and left to thereader. 8 g¯ g γ¯ k = 1,2,3 1 k As in (3.2), needs not oin ide with . Call , , the oupling of minimum weight g¯ g¯ k 6=i,j g¯ g¯ i j i between and , where . Call the oupling of minimum weight among the 's. De(cid:28)ne α¯ = g¯2 a¯ = γ¯2 i i, i i. The lagrangian has the weight stru ture 1 1 1 1 L= L (g¯ A)+ L (g¯ ψ)+ L (g¯ ϕ)+ L (g¯ A,g¯ ψ) 1 1 2 2 3 3 12 1 2 α¯ α¯ α¯ a¯ 1 2 3 3 1 1 1 + L (g¯ A,g¯ ϕ)+ L (g¯ ψ,g¯ ϕ)+ L (g¯ A,g¯ ψ,g¯ ϕ). 13 1 3 23 2 3 123 1 2 3 a¯ a¯ α¯ (3.8) 2 1 A λ In we olle tively in lude also ghosts and antighosts. Any other parameters ontained in g¯ i (3.8) must have non-negative weights. The -fa tors appearing in formula (3.8) are mere tools to g¯ ψ g¯ψ 2 i keep tra k of the weight stru ture. For example, instead of we an have any , as long as [g¯] ≥ [g¯ ] 1/α¯ 1/a¯ 1/α¯ i 2 i i . Similarly, the denominators , and are devi es that lower the weights of appropriate amounts. L Every on the right-hand side of (3.8) must be polynomial in the (cid:28)elds and parameters. Moreover, we assume [g] ≥ [g¯ ], [gg¯ ] ≥ [g¯2], [gg¯ ] ≥ [g¯2]. 1 1 2 1 3 (3.9) These inequalities ensure that (3.8) is ompatible with the ovariant stru ture. Indeed, be ause of (3.9), the verti es generated by ovariant derivatives are multiplied by fa tors of weights not smaller than the ones appearing in (3.8), so they an (cid:28)t into one of the stru tures (3.8). Observe [g] ≥ [g¯] i i that (3.9) implies for every . Again, it iseasy toprove that thestru ture (3.8)ispreserved by renormalization. Assume, for [g¯ ]≥ [g¯ ] ≥ [g¯ ] 1 2 3 example, that (the other ases an be treated symmetri ally, be ause (3.9) plays g¯ = ρσg¯ g¯ = σg¯ g¯ = g¯ [ρ] ≥ 0 [σ] ≥ 0 1 2 3 no role here) and write , , , with , . In the parametrization g¯ ρ σ g¯ - - the -powers in front of ounterterms an be ounted as in (3.4). Moreover, verti es ontain σ A ψ ϕ a fa tor for every - and -leg, save two legs in -independent verti es. Sin e at least two legs σ A of every vertex enter the diagrams, ounterterms ontain at least a fa tor for every external - ψ ρ A and -leg. A similar argument applies to -fa tors andexternal -legs. Thus, every diagram with L ≥ 1 E E A E ψ A ψ loops, external legs, external -legs and external -legs is multiplied at least by α¯Lg¯E−2ρEAσEA+Eψ a fa tor and therefore (cid:28)ts into the stru ture (3.8). This argument proves also that the one-loop ounterterms generated by (3.8) have the weight stru ture ∆ L(g¯ A,g¯ ψ,g¯ ϕ), 1 1 2 3 (3.10) L α¯L−1 while at loops there is an additional fa tor of . Simpli(cid:28)ed versions of our theories an be obtained dropping verti es and quadrati terms of (3.8) that are not ontained in (3.10), be ause renormalization is unable to generate them ba k. The quadrati terms that annot be dropped are those that ontrol the behavior of propagators. Of ourse, the simpli(cid:28)ed model must also ontain the verti es related to su h quadrati terms by ovariantization. 9 Polynomiality Now we derive the onditions to have polynomiality. Consider (cid:28)rst the physi al 1/α¯ 1/a¯ 1/α¯ i i (i.e. non gauge-(cid:28)xing) se tors of the lagrangian (3.8). Apart from the fa tors , and , g¯ F g¯ ψ g¯ ϕ 1 2 3 they depend only on theprodu ts , , , and their ovariant derivatives, sopolynomiality is ensured when these obje ts have positive weights. Let us fo us for the moment on the gauge d¯> 1 [g¯ F¯] > 0 1 se tor. From (2.6) we see that if the most meaningful ondition is . If instead d¯= 1 [g¯ F˜] > 0 F¯ ≡ 0 1 the most meaningful ondition is , be ause . However, be ause of (2.21) and (2.13) we have to on entrate on the former ase. We on lude that pure gauge theories are polynomial in the physi al se tor if and only if 4 4− −2κ < , 1 n ž [g¯]= κ i i having written . In the presen e of s alars and fermions we must have 4 4− −2κ < , 1−2κ < , 2−2κ < 1 2 3 n ž ž ž. (3.11) n ≥ 2 g¯ Aˆ 1 Observe that (3.11) and ensure that the weight of is stri tly positive. Thus the Aˆ C¯ C theory is ertainly polynomial in . For the same reason, it is polynomial also in and . On g¯ A¯ [g¯ A¯] > −1/n 1 1 the other hand, the weight of an be negative, be ause (3.11) ensures only . This means that, in prin iple, the gauge-(cid:28)xing se tor an be non-polynomial. Now we show that ifthetree-level gauge(cid:28)xingis(2.16), thenthetheory ispolynomial alsointhegauge-(cid:28)xing se tor. Note that is some ases (see appendix B) the gauge-(cid:28)xing se tor does not preserve the simple form (2.16), but an a quire new types of verti es by renormalization. We need to prove that beyond the tree level, in both the physi al and gauge-(cid:28)xing se tors, A¯ the (cid:28)eld appears only in the ombinations gA¯, g¯ ∂ˆA¯, g¯ gAˆA¯, g¯ ∂¯A¯, g¯ gA¯A¯. 1 1 1 1 (3.12) 1/α¯ 1/a¯ 1/α¯ i i First observe that at the tree level this statement is true up to the fa tors , and A¯ i (Ga)2 appearingin(3.8). Indeed, appearsonlyinthefollowinglo ations: )in ,whi h ontributes ii D¯ iii only to the propagator; ) inside the ovariant derivative (also in the ghost a tion); ) inside ii A¯ g iii the (cid:28)eld strength. In ase ) is multiplied by and gives the (cid:28)rst term of (3.12). In ase ) g¯ g¯ F˜ 1 1 the (cid:28)eld strength arries an extra fa tor : gives the se ond and third terms of (3.12), while g¯ F¯ 1 gives the forth and (cid:28)fth terms. L G A¯ Next, onsider an -loop Feynman diagram and assume that the -stru ture of the renor- L−1 malized a tion is (3.12) up to the order in luded, with the tree-level aveat just mentioned. 1/α¯ 1/a¯ 1/α¯ G i i The fa tors , and of (3.8) are simpli(cid:28)ed by the internal legs of , whi h are at least A¯ G A¯ two for every vertex. Consider the -external legs of . In the (cid:28)rst ase of (3.12) the -leg is g g¯ ∂ˆ 1 a ompanied by a fa tor and in the se ond ase by a and a derivative a ting on it. In the g g¯ Aˆ 1 third ase it arries a fa tor (the being left for the -leg, in ase it is external), in the forth 10