aa r X i v : . [ m a t h - ph ] F e b On the lo al Borel transform of Perturbation TheoryChristoph Kopper ∗ Centre de Physique Théorique, CNRS, UMR 7644E ole Polyte hniqueF-91128 Palaiseau, Fran eAbstra tWe prove existen e of the lo al Borel transform for the perturbative series ofmassive ϕ -theory. As ompared to previous proofs in the literature, the presentbounds are mu h sharper as regards the dependen e on external momenta, they areexpli it in the number of external legs, and they are obtained quite simply througha judi iously hosen indu tion hypothesis applied to the Wegner-Wilson-Pol hinski(cid:29)ow equations. We pay attention not to generate an astronomi ally large numeri al onstant for the inverse radius of onvergen e of the Borel transform. ∗ kopper pht.polyte hnique.fr 1 Introdu tionPerturbation theory in quantum (cid:28)eld theory is suspe ted to be divergent. The divergentbehaviour an be dire tly related to the presen e of nontrivial minima of the lassi ala tion in the omplex oupling onstant plane [Li℄, and one speaks of instanton singu-larities in onsequen e. Starting from the expansion in terms of Feynman diagrams thesingularity an also be related to the in rease of the number of Feynman diagrams athigh orders in perturbation theory. In theories like ϕ , this number grows as N ! , whereN is the order of perturbation theory. This indi ates divergent behaviour. In four dimen-sions this divergen e has never been proven however. The main obstru tion stems fromthe renormalization subtra tions whi h are required to an el short distan e singularities.They lead to the appearan e of ontributions of opposite sign in the Feynman amplitudes.A lower bound on perturbative ontributions would then require to ontrol the absen eof e(cid:30) ient sign an ellations, a task whi h has turned out to be too di(cid:30) ult up to thepresent day. Thus divergen e an only be proven in three or fewer dimensions where therenormalization problem is marginal or absent [Sp℄, [Br℄, [MR℄. In the four-dimensional ase the very need for renormalization implies the appearan e of a new (hypotheti al)sour e of divergen e of the perturbative expansion, named renormalon singularity after 'tHooft [tH℄. This type of singularity is related - in the language of Feynman graphs - to thepresen e of graphs whi h require a number of renormalization subtra tions proportionalto the order of perturbation theory. In a stri tly renormalizable theory it typi ally leadsto a orresponding power of the logarithms of the momenta (cid:29)owing through the diagram.For example for the diagram of Fig.1 we obtain an integral of the type Z d p p + m ) log N ( p + m m ) ∼ N ! , where N is the number of bubble graph insertions and p the momentum (cid:29)owing throughthe big loop. Su h a behaviour is obviously not ompatible with a onvergent perturbationexpansion.It was then proven in the seminal work of de Calan and Rivasseau [CR℄ that the twosour es of divergent behaviour do not onspire to deteriorate the situation even more.Even in the presen e of both instanton and renormalon type singularities the Borel trans-form of the perturbation expansion has a (cid:28)nite radius of onvergen e, i.e. perturbativeamplitudes at order N do not grow more rapidly than N ! . In fa t one of the main resultsof [CR℄ is that the number of graphs whi h require k ≤ N renormalization subtra tionsis bounded by ( const ) N N ! k ! so that the bound they present on their amplitudes, whi h isof the form ( const ′ ) N k ! , is su(cid:30) ient to prove lo al existen e of the Borel transform.2 pp Figure 1: A renormalon diagram in ϕ -theoryThe subje t of large orders of perturbation theory was taken up by several authors inthe sequel. The bounds were improved and generalized in the paper [FMRS℄. In [CPR℄the result was extended to massless ϕ -theory. Lo al existen e of the Borel transformfor QED was proven in the book [FHRW℄. David, Feldman and Rivasseau [DFR℄ madeessential progress in proving that the radius of onvergen e of the Borel transformedseries for the ϕ -theory is not smaller than what is expe ted from the analysis of typi alsimple graphs ontributing to the renormalon singularity as the one of Fig.1. Namelythey showed that this radius is bounded below by the inverse of the (cid:28)rst oe(cid:30) ient of the β -fun tion, as suspe ted by 't Hooft. In fa t this oe(cid:30) ient is al ulated from a sub lassof diagrams of whi h the one shown in Fig.1 is a representative. They are obtained byiteratively repla ing in all possible ways elementary verti es by the one-loop bubble graphwhi h apppears as a multiple insertion in Fig.1. The proof required a judi ious partialresummation te hnique applied to the perturbative expansion, of a similar kind as the oneemployed previously in [Ri℄ to prove the existen e (beyond perturbation theory) of planar"wrong" sign ϕ -theory. It also made use of the pre ise upper bounds on the perturbativeseries in the absen e of renormalon type diagrams established previously in [MR℄ and[MNRS℄. Finally Keller [Ke℄ (cid:28)rst proved the lo al existen e of the Borel transform inthe framweork of the Wegner-Wilson-Pol hinski (cid:29)ow equations whi h we also use in thispaper.As ompared to the previous papers our motivation and in onsequen e the results aredi(cid:27)erent. Our paper is of ourse losest in spirit to [Ke℄, whi h is the only one where thedependen e on the number of external legs is expli itly ontrolled. The paper is part of alarger program to get rigorous ontrol of the properties of the S hwinger or Green fun tionsof quantum (cid:28)eld theory with the aid of (cid:29)ow equations. A review is in [Mü℄, for re entnovel results see e.g. [KMü℄, [Ko℄. Our aim is not only to ontrol the large order behaviour3f perturbation theory in the sense of the mathemati al statement on the existen e of thelo al Borel transform. We would like to ontrol the whole set of S hwinger fun tions atthe same time as regards their large momentum behaviour. This is in fa t ne essary ifthe bounds on the S hwinger fun tions are supposed to serve as an ingredient to furtheranalysis. If for example they appear as an input in the (cid:29)ow equations, or simliarly inS hwinger-Dyson type equations, bad bounds on one side will typi ally undermine goodones on the other side ; for example bad high momentum behaviour will lead to bad highorder behaviour when losing loops and integrating over loop momenta. In the same way,sin e an n -point fun tion an be otained by merging two external lines and forming aloop in an ( n + 2) -point fun tion, bounds whi h are not su(cid:30) iently strong as regards thedependen e on n , will not be of mu h use either. We need bounds on the high momentumbehaviour whi h do not in rease faster than logarithmi ally with momentum (apart fromthe two-pont fun tion), and whi h are thus optimal for the four-point fun tion, in the sensethat they are saturated by ertain individual Feynman amplitudes. Su h bounds wereproven in [KM℄, however without ontrol on the behaviour at large orders of perturbationtheory or at large number of external legs. In the above ited papers the ontrol onthe high momentum behaviour is far from su(cid:30) ient, in [CR℄ and in [Ke℄ the radius of onvergen e of the Borel transform shrinks as an inverse power of momentum, in theother papers the result is not framed in momentum spa e but rather in distributionalsense making use of various norms, and ertainly too far from optimal to be used in theabove des ribed ontext. We note that bounds in position spa e, if optimal in the abovesense, ould serve as well as those in momentum spa e. We adressed the problem inmomentum spa e here sin e it is of more ommon use in short distan e physi s. For workwith (cid:29)ow equations in position spa e see [KMü℄.We would also like to stress the fa t that we pay mu h attention to the fa t not toprodu e astronomi al1 onstants in the lower bounds on the inverse radius of onvergen eof the Borel transformed S hwinger fun tions. The paper ould have been onsiderablyshortened without that e(cid:27)ort, and the reader will easily (cid:28)nd his shortened path throughthe paper, if he is not interested in that aspe t. The onstants obtained in the literatureare typi ally astronomi ally large ; in some restri ted sense this is even true for the optimalresult [DFR℄, sin e the bound obtained is on asymptoti ally large orders of perturbationtheory, allowing smaller orders to be very large. In a losed system of equations it is againnot possible to relax on low orders of perturbation theory without having a drawba k onhigher orders. Further onsiderable e(cid:27)ort seems ne essary if one wants to obtain a lose1an astronomi al onstant would be one of the form n where n is a large integer. Our aim is toshow that a small value of n an be a ommodated for.4o realisti value for this inverse radius. It requires more expli it al ulations in lowestorders whi h are of ourse doable sin e the (cid:29)ow equations provide an expli it al ulationals heme.Our paper is organized as follows. We (cid:28)rst present the (cid:29)ow equation framework aswe will use it in the proof. Then we olle t some elementary auxiliary bounds whi h areto be used in the proof of the subsequent proposition. This part ould be onsiderablyshortened, were it not for the above mentioned aim to avoid the appearan e of astronom-i al onstants. Then we present our results and their proof. The reader familiar withthe domain will realize that the proof is omparatively short and (hopefully) transparent.The hardest part of the work onsisted in (cid:28)nding out the pertinent indu tion hypothesis.2 The (cid:29)ow equation frameworkRenormalization theory based on the (cid:29)ow equation (FE) [WH℄ of the renormalizationgroup [Wi℄ has been exposed quite often in the literature [Po℄, [KKS℄, [Mü℄. So we will in-trodu e it rather shortly. The obje t studied is the regularized generating fun tional L Λ , Λ of onne ted (free propagator) amputated Green fun tions (CAG). The upper indi es Λ and Λ enter through the regularized propagator C Λ , Λ ( p ) = 1 p + m { e − p m − e − p m } or its Fourier transform ˆ C Λ , Λ ( x ) = Z p C Λ , Λ ( p ) e ipx , with Z p := Z R d p (2 π ) . (1)We assume ≤ Λ ≤ Λ ≤ ∞ so that the Wilson (cid:29)ow parameter Λ takes the roleof an infrared (IR) uto(cid:27)2, whereas Λ is the ultraviolet (UV) regularization. The fullpropagator is re overed for Λ = 0 and Λ → ∞ . For the "(cid:28)elds" and their Fouriertransforms we write ˆ ϕ ( x ) = R p ϕ ( p ) e ipx , δδ ˆ ϕ ( x ) = (2 π ) R p δδϕ ( p ) e − ipx . For our purposesthe (cid:28)elds ˆ ϕ ( x ) may be assumed to live in the S hwartz spa e S ( R ) . For (cid:28)nite Λ andin (cid:28)nite volume the theory an be given rigorous meaning starting from the fun tionalintegral e − ~ ( L Λ , Λ0 ( ˆ ϕ )+ I Λ , Λ0 ) = Z dµ Λ , Λ ( ˆ φ ) e − ~ L ( ˆ φ + ˆ ϕ ) . (2)2Su h a uto(cid:27) is of ourse not ne essary in a massive theory. The IR behaviour is only modi(cid:28)ed for Λ above m . 5n the r.h.s. of (2) dµ Λ , Λ ( ˆ φ ) denotes the (translation invariant) Gaussian measure with ovarian e ~ ˆ C Λ , Λ ( x ) . The fun tional L ( ˆ ϕ ) is the bare a tion in luding ounterterms,viewed as a formal power series in ~ . Its general form for symmetri ϕ theory is L Λ , Λ ( ˆ ϕ ) = g Z d x ˆ ϕ ( x ) ++ Z d x { a (Λ ) ˆ ϕ ( x ) + 12 b (Λ ) X µ =0 ( ∂ µ ˆ ϕ ) ( x ) + 14! c (Λ ) ˆ ϕ ( x ) } , (3)the parameters a (Λ ) , b (Λ ) , c (Λ ) ful(cid:28)ll a (Λ ) , c (Λ ) = O ( ~ ) , b (Λ ) = O ( ~ ) . (4)They are dire tly related to the standard mass, oupling onstant and wave fun tion ounterterms. On the l.h.s. of (2) there appears the normalization fa tor e − I Λ , Λ0 whi his due to va uum ontributions. The exponent I Λ , Λ diverges in in(cid:28)nite volume so thatwe an take the in(cid:28)nite volume limit only when it does not appear any more. We donot make the (cid:28)nite volume expli it here sin e it plays no role in the sequel. For a morethorough dis ussion see [Mü℄, [KMR℄.The FE is obtained from (2) on di(cid:27)erentiating w.r.t. Λ . It is a di(cid:27)erential equationfor the fun tional L Λ , Λ : ∂ Λ ( L Λ , Λ + I Λ , Λ ) = (5) = ~ h δδ ˆ ϕ , ( ∂ Λ ˆ C Λ , Λ ) δδ ˆ ϕ i L Λ , Λ − h δδ ˆ ϕ L Λ , Λ , ( ∂ Λ ˆ C Λ , Λ ) δδ ˆ ϕ L Λ , Λ i . By h , i we denote the standard s alar produ t in L ( R , d x ) . Changing to momentumspa e and expanding in a formal powers series w.r.t. ~ we write L Λ , Λ ( ϕ ) = ∞ X l =0 ~ l L Λ , Λ l ( ϕ ) . From L Λ , Λ l ( ϕ ) we then de(cid:28)ne the CAG of order l in momentum spa e through δ (4) ( p + . . . + p n ) L Λ , Λ n,l ( p , . . . , p n − ) = 1 n ! (2 π ) n − δ ϕ ( p ) . . . δ ϕ ( p n ) L Λ , Λ l | ϕ ≡ , (6)where we have written δ ϕ ( p ) = δ/δϕ ( p ) . The CAG are symmetri in their momentumarguments by de(cid:28)nition. Note that by our de(cid:28)nitions the free two-point fun tion is not ontained in L Λ , Λ l ( ϕ ) , sin e it is attributed to the Gaussian measure in (2). This isimportant for the set-up of the indu tive s heme, from whi h we will prove our boundsbelow. We thus de(cid:28)ne L Λ , Λ n,l ≡ for l < , n ≥ , and L Λ , Λ , ≡ . ∂ Λ ∂ w L Λ , Λ n,l ( p , . . . p n − ) = ( 2 n + 22 ) Z k ( ∂ Λ C Λ , Λ ( k )) ∂ w L Λ , Λ n +2 ,l − ( k, − k, p , . . . p n − ) (7) − X l l l,w w w w n + n = n +1 n n c { w j } " ∂ w L Λ , Λ n ,l ( p , . . . , p n − ) ( ∂ w ∂ Λ C Λ , Λ ( q )) ∂ w L Λ , Λ n ,l ( p n , . . . , p n − ) sy with q = − p − . . . − p n − = − p n = p n +1 + . . . + p n . Here we have written (7) dire tly in a form where also momentum derivatives of the CAG(6) are performed. In this paper we will restri t for simpli ity to up to 3 derivatives alltaken w.r.t. one momentum p i , sin e our aim is in the (cid:28)rst pla e to bound the S hwingerfun tions themselves, and not their derivatives 3. We use the shorthand4 notations ∂ w := Y µ =0 ( ∂∂p i,µ ) w µ with w = ( w , . . . , w ) , | w | = X µ w µ and w ! = w ! . . . w ! , c { w j } = w ! w ! w ! w ! . The symbol sy means taking the mean value over those permutations π of (1 , . . . , n ) ,for whi h π (1) < π (2) < . . . < π (2 n − and π (2 n ) < π (2 n + 1) < . . . < π (2 n ) .For the derivatives of the propagator we (cid:28)nd the following relations ∂ Λ C Λ , Λ ( p ) = − e − p m , ∂ p µ e − p m = − p µ Λ e − p m , (8) ∂ p µ ∂ p ν e − p m = h p µ p ν − δ µν i e − p m , (9) ∂ p µ ∂ p ν ∂ p ρ e − p m = h − p µ p ν p ρ + 4Λ (cid:0) δ µν p ρ + δ µρ p ν + δ νρ p µ (cid:1)i e − p m . (10)3In distributing the derivatives over the three fa tors in the se ond term on the r.h.s. with the Leibnizrule, we have ta itly assumed that the momentum p i appears among those from L Λ , Λ n ,l . If this is notthe ase one has to parametrize L Λ , Λ n ,l in terms of (say) ( p , . . . p n ) with p n = − p n +1 − . . . − p n ,to introdu e the p i -dependen e in L Λ , Λ n ,l . For an extensive systemati treatment in luding the generalsituation where derivatives w.r.t. several external momenta are present, see [GK℄. This situation, also onsidered in [KM℄, ould be analysed here too at the prize of basi ally notational ompli ation.4slightly abusive, sin e the index i is suppressed in w l ∈ N a) X ≤ l ,l ,l l l l + 1) ( l + 1) ≤ l + 1) , X ≤ l ,l ,l l l l + 1) ( l + 1) ≤ l + 1) , (11)b) X ≤ n ,n ,n n n +1 n n ≤ n , X ≤ n ,n ,n n n +1 n n ≤ n . (12)Proof : a) The inequality an be veri(cid:28)ed expli itly for l ≤ . Assuming l > we have X ≤ l ,l ,l l l l + 1) ( l + 1) = 2( l + 1) + l − X k =1 k + 1) ( l − k + 1) (13) ≤ l + 1) + Z l dx ( x + 1) ( l − x + 1) = 2( l + 1) + Z l +11 dx (cid:16) a + bxx + c − bx ( l + 2 − x ) (cid:17) , where a = 1( l + 2) , b = 2( l + 2) , c = 3( l + 2) . The integral equals then l + 2) (cid:16) − l + 1 ] + 4 l + 2 log( l + 1) (cid:17) ≤ l + 1) for l > , (14)and the bound is thus also veri(cid:28)ed for l > . The se ond statement in (11) is a dire t onsequen e of the (cid:28)rst sin e a term l +1) is subtra ted on the l.h.s.b) We may again assume n > on verifying the lowest values expli itly. The statementthen follows from the proof of a) through X ≤ n ,n ,n n n +1 n n = X ≤ n ,n ,n n n − n + 1) ( n + 1) ≤ n + sup ≤ n ≤ n − n + 1) ( n − n ) X ≤ n ,n ,n n n − n + 1) ( n + 1) n + 12( n − X ≤ n ≤ n − n + 1) ( n − n ) ≤ n + 12( n −
1) 3 n ≤ n , where we used the bound (14) on (13) in the last but se ond inequality. The se ondinequality in b) then follows dire tly from the previous al ulation.Lemma 2 :a) For integers n ≥ , n , n ≥ , l , l , λ , l , λ ≥ X l l l , n n n +1 , λ ≤ l , λ ≤ l , λ λ λ l + 1) ( l + 1) n n n ! n ! n ! λ ! λ ! λ ! ( n + l − n + l − n + l − ≤ K l + 1) n , where we may hoose K = 20 . (15)For n , n ≥ X l l l , n n n +1 , λ ≤ l , λ ≤ l , λ λ λ l + 1) ( l + 1) n n n ! n ! n ! λ ! λ ! λ ! ( n + l − n + l − n + l − ≤ K l + 1) n . (16)b) For n ≥ , n = 2 , n = n − X l l l , λ ≤ l , λ ≤ l ,λ + λ = λ l + 1) ( l + 1) n n n ! n ! n ! λ ! λ ! λ ! ( n + l − n + l − n + l − ≤ K ′ l + 1) n , where we may hoose K ′ = ( 34 ) · ≤ . . (17) ) For n ≥ , n = 1 , n = n X l l l , λ ≤ l , λ ≤ l ,λ + λ = λ l + 1) ( l + 1) n n n ! n ! n ! λ ! λ ! λ ! ( n + l − n + l − n + l − ≤ K ′′ l + 1) n , where we may hoose K ′′ = 5 . (18)9roof : a) We have n ! n ! n ! λ ! λ ! λ ! ( n + l − n + l − n + l − nn n (cid:0) n − n − (cid:1) (cid:0) λλ (cid:1) h(cid:0) n + l − n + l − (cid:1)i − . We note that (cid:0) n − n − (cid:1) (cid:0) ll (cid:1) ≤ (cid:0) n + l − n + l − (cid:1) . (19)This follows dire tly from the standard identity p X k =0 (cid:0) n − p − k (cid:1) (cid:0) lk (cid:1) = (cid:0) n + l − p (cid:1) , assuming without limitation that n − ≥ l and setting p = inf { n + l − , n + l − ( n + l ) } ≤ n + l − ≤ n − .Se ondly we show that for l = l + l X λ ≤ l , λ ≤ l , λ + λ = λ λ ! λ ! λ ! ≤ (cid:0) ll (cid:1) . (20)For the indu tive proof we assume l ≥ and without loss l ≤ l . To realize by indu tionon ≤ k ≤ l that A k := h(cid:0) ll (cid:1)i − X λ ≤ l , λ ≤ l , λ + λ = l − k ( l − k )! λ ! λ ! ≤ , we start from A = 1 . Then assuming that we have A k − ≤ for k ≥ we (cid:28)nd A k = l − ( k − l − ( k − A k − + h(cid:0) ll (cid:1)i − (cid:0) l − kl (cid:1) ≤ − l l − ( k −
1) + l l ( l − . . . ( l − ( k − l − . . . ( l − ( k − . This equals for k = 1 and an be bounded for k ≥ through − l l − ( k − (cid:0) − ( l − l − . . . ( l − ( k − l ( l − . . . ( l − ( k − (cid:1) ≤ . For l < k ≤ l it is immediate to see that A k ≤ A k − sin e the sum for A k does not ontain more nonvanishing terms than the one for A k − , and a nonvanishing term in A k an be bounded by a orresponding one in A k − : ( l − k )! λ ! λ ! ≤ ( l − ( k − λ + 1)! λ ! . X λ ≤ l , λ ≤ l , λ + λ = λ nn n ( n + l − n − λ ! ( n + l − n − λ ! ( n − λ !( n − l )! ≤ nn n . (21)Using Lemma 1 we then get X l + l = l , n + n = n +1 nn n l + 1) ( l + 1) n n ≤ l + 1) n . (22)The statements (16) and parts b) (17) and ) (18) follow from Lemma 1 and (21).Lemma 3 : For v ≤ and a i , x ∈ R the following inequality holds e − x v Y i =1 , | x + a i | ) ≤ c ( v ) v Y i =1 , | a i | ) , (23)where we may hoose c (0) = 1 , c (1) = 1 . , c (2) = 2 . , c (3) = 5 . . (24)Proof : The inequality is trivial if one allows for large onstants. Suppose v = 3 . Wemay suppose without limitation that | a | ≥ | a | ≥ | a | ≥ (if a i ≤ we may pass tothe ase v − ), and that | x | ≤ sup | a i | sin e the expression on the l.h.s. of (23) ismaximized if all a i ∈ R are parallel and anti-parallel to x . In this ase, assummingthat | a | | a | | a | ≥ (1 + | x | ) , the inequality at (cid:28)xed produ t | a | | a | | a | and at (cid:28)xed | x | , be omes most stringent if | a | , | a | = 1 + | x | . It then takes the form e − x (1 + | x | ) ≤ c (3) | a | − | x || a | with | a | > | x | . (25)If | a | | a | | a | < (1 + | x | ) , the bound is satis(cid:28)ed if we demand e − x ≤ c (3) 1(1 + | x | ) . This relation is also su(cid:30) ient for (25) to hold. The expression e − x (1 + | x | ) is maximalfor | x | = √ − and bounded by . . The ases v = 2 and v = 1 are treated analo-gously. 11emma 4 : For r ∈ N and a ≥ Z x e − | x | log r ( | x | + a ) ≤
14 log r + a + 13 ( r !) / , (26)where log + x := log(sup(1 , x )) .Proof : Again the only nontrivial point is to avoid bad numeri al onstants in the bound.Remembering the de(cid:28)nition (1), (cid:28)rst note that for r ≤ , a ≤ Z x e − | x | log r ( | x | + a ) ≤ Z x e − | x | log r Z | x |≥ e − | x | log r ( | x | + a ) ≤ (1 . r π + 18 π X n ≥ e − n / n log r (3 + n ) ≤
13 ( r !) / √ (27)on bounding the sum numeri ally ; we also used the fa t that the derivative of the inte-grand w.r.t. | x | is negative for | x | ≥ . Se ondly, for r ≤ , a > a ) − r Z x e − | x | log r ( | x | + a ) = Z x e − | x | (cid:2) | x | a )log a (cid:3) r (28) ≤ Z x e − | x | (1 + log 2log a ) r + 18 π X n ≥ e − n / n (cid:2) n )log 3 (cid:3) ≤ π (1 + log 2log a ) r + 6 . π ≤
14 + 13 ( r !) / log r a √ on bounding the sum numeri ally and on noting that the last inequality is valid taking a = 3 on the l.h.s. and a = 5 on the r.h.s., and also for a = 5 on the l.h.s. and a = e on the r.h.s. For log a ≥ the last bound an be repla ed by independently of r ≤ .Thirdly, for r > , a ≤ r Z x e − | x | log r ( | x | + a ) ≤ Z x e − | x | log r ( | x | + r ) ≤ log r r Z x e − | x | (cid:2) | x | r log r (cid:3) r ≤ log r r Z x e − | x | + | x | log r ≤ log r r e
12 log2 6 π Z ∞− e − z z dz ≤
110 log r r on majorizing for r = 6 and ompleting the square in the last but se ond integral. Then
110 log r r ≤
13 ( r !) / , noting that log r r/ ( r !) / ≤ . , the maximal value being attained for r = 15 .In the fourth pla e we have for a > r > quite similarly r a Z x e − | x | log r ( | x | + a ) = Z x e − | x | (cid:2) | x | a )log a (cid:3) r ≤ Z x e − | x | (cid:2) | x | r log r (cid:3) r ≤ . s ∈ N , a > , M > κ ≥ m > λ = l X λ =0 λ λ ! Z Mκ dκ ′ κ ′− s − log λ (sup( aκ ′ , κ ′ m )) ≤ κ − s s λ = l X λ =0 λ λ ! log λ sup( aκ , κm ) . (29)Proof : We have Z Mκ dκ ′ κ ′− s − log λ (sup( aκ ′ , κ ′ m )) ≤ κ − s s log λ + ( aκ ) + Z sup( √ ma, M )sup( κ, √ ma ) dκ ′ κ ′− s − log λ ( κ ′ m ) , and the last integral an be bounded by Z Mκ dκ ′ κ ′− s − log λ ( κ ′ m ) ≤ κ − s s λ ! λ X ν =0 log ν ( κm ) ν ! 1 s λ − ν . (30)We then (cid:28)nd λ = l X λ =0 n λ λ ! log λ + ( aκ ) + 12 λ λ X ν =0 log ν ( κm ) ν ! 1 s λ − ν o ≤ λ = l X λ =0 log λ ( aκ )2 λ λ ! + 2 λ = l X λ =0 log λ ( κm )2 λ λ ! ≤ λ = l X λ =0 λ λ ! log λ sup( aκ , κm ) . Lemma 6 :Here and in the following we set κ = Λ + m .a) e − m ≤ K κ , where K = 6 . , (31) p e − p ≤ κ e , | p | e − p ≤ κ √ e . (32)b) For | w | ≤ : | ∂ w e − p e − m | ≤ K ( | w | ) κ − [sup( κ, | p | )] −| w | . (33)with K (0) = K , K (1) = K e = 4 . , K (2) = 77 . , K (3) = 37 . | ∂ w e − p e − m | ≤ K ′ ( | w | ) κ − [sup( κ, | p | )] −| w | . (34)13ith K ′ (0) = K , K ′ (1) = K e = 9 . , K ′ (2) = 135 , K ′ (3) = 407 . ) For ≤ τ ≤ and p ( τ ) = − τ p − p − p : | p | e − p sup( κ, η (4)1 , ( τ p , p , p , p ( τ ))) ≤ e − / , | p | e − p sup( κ, η (4)1 , ( τ p , p , p , p ( τ ))) ≤ e , (35)where η is de(cid:28)ned below (43), | p | e − p sup( τ | p | , κ ) ≤ √ e , p e − p sup( τ | p | , κ ) ≤ κ e , | p | e − p sup( τ | p | , κ ) ≤ κ ( 3 e ) / . (36)Proof : a) The bound (31) follows from e − m ≤ κ sup x ≥ (1 + x ) e − x , (37)and the fun tion of x is maximized for x = √ − . To prove (32) note p e − p ≤ κ sup x n x e − x o = 2 κ e , | p | e − p ≤ κ sup x ≥ n x e − x o = κ √ e . b) The bounds are proven similarly as in a). For w = 0 the result follows from a).For | w | = 1 , , we use (8), (9),(10). We may suppose that the axes have been hosensu h that p is parallel to one of them. For | w | = 1 we then (cid:28)nd | κ ∂ w e − p e − m |≤ inf n | p | sup x { x e − x } sup y ≥ { (1 + y ) e − y } , κ sup x ≥ { x e − x } sup y ≥ { (1 + y ) e − y } o . For | w | = 2 we obtain | κ ∂ w e − p e − m |≤ inf n | p | sup x {| x − x | e − x } sup y ≥ { (1+ y ) e − y , κ sup x ≥ {| x − | e − x sup y ≥ { (1+ y ) e − y } o . For | w | = 3 we get | κ ∂ w e − p e − m |≤ inf n | p | sup x {| − x + 32 x | e − x } sup y ≥ { (1 + y ) e − y } , κ sup x ≥ {| − x + 32 x | e − x } sup y ≥ { (1 + y ) e − y } o . x and y and taking the maximal onstant inea h of the three expressions gives the numeri al onstants of (34).The bounds (33) follow on repla ing e − x → e − x in maximizing the previous expressions. ) The (cid:28)rst bound (35) follows from κ, η (4)1 , ( τ p , p , p , p ( τ ))) | p | e − p ≤ | p | κ e − p ≤ | p | Λ e − p ≤ e − / and the se ond bound follows analogously.The bounds (36) are obtained by the same reasoning.Lemma 7:a) Z Λ0 d Λ ′ Λ ′− e − m / Λ ′ κ ′ log λ ( κ ′ m ) ≤ K log λ +1 (cid:0) κm (cid:1) λ + 1 with K = K . , (38)b) Z Λ0 d Λ ′ Λ ′− e − m / Λ ′ κ ′ log λ ( κ ′ m ) ≤ K ′ log λ +1 (cid:0) κm (cid:1) λ + 1 with K ′ = 14 . . (39)Proof: The integrals are bounded through Z κ/m dxx ( xx − s e − x − log λ x ≤ sup y ≥ (cid:16) (1 + y ) s e − y (cid:17) log λ +1 ( κ/m ) λ + 1 , where s ∈ { , } . The sup leads to the numeri al onstants.Lemma 8: For λ ∈ [0 , and x, y ∈ R d , if | x + y | ≥ | x | then | λx + y | ≥ λ | x | .Proof: | λx + y | ≥ | x + y | − | (1 − λ ) x | ≥ | x | − (1 − λ ) | x | = λ | x | .4 Sharp bounds on S hwinger fun tionsWith the aid of the FE (7) it is possible to establish a parti ularly simple indu tive proofof the renormalizability of ϕ theory. Renormalizability in fa t appears as a onsequen eof the following bounds [KKS℄, [Mü℄ on the fun tions L Λ , Λ n.l :Boundedness | ∂ w L Λ , Λ n,l ( ~p ) | ≤ κ − n −| w | P (log κm ) P ( | ~p | κ ) , (40)15onvergen e | ∂ Λ ∂ w L Λ , Λ n,l ( ~p ) | ≤ κ − n −| w | P (log Λ m ) P ( | ~p | κ ) . (41)The P i denote polynomials with nonnegative oe(cid:30) ients, whi h depend on l, n, | w | , butnot on ~p, Λ , κ = Λ + m, Λ . The statement (41) implies renormalizability, sin e it provesthe limits lim Λ →∞ , Λ → L Λ , Λ ( ~p ) to exist to all loop orders l . But the statement (40) hasto be obtained (cid:28)rst to prove (41).The standard indu tive s heme whi h is used to prove these bounds, and whi h wewill also employ in the proof of the subsequent proposition, goes up in n + l and for given n + l des ends in n , and for given n, l des ends in | w | . The r.h.s. of the FE is thenprior the l.h.s. in the indu tive order, and the bounds an thus be veri(cid:28)ed for suitableboundary onditions on integrating the r.h.s. of the FE over Λ , using the bounds of theproposition. Terms with n + | w | ≥ are integrated downwards from Λ to Λ , sin e forthose terms we have the boundary onditions at Λ = Λ following from (3) ∂ w L Λ , Λ n,l ( p , . . . p n − ) = 0 for n + | w | ≥ , whereas the terms with n + | w | ≤ at the renormalization point - whi h we hooseat zero momentum for simpli ity - are integrated upwards from to Λ , sin e they are(cid:28)xed at Λ = 0 by renormalization onditions, whi h de(cid:28)ne the relevant parameters ofthe theory. We will hoose for simpli ity L , Λ ,l (0 , ,
0) = δ l, g , L , Λ ,l (0) = 0 , ∂ p L , Λ ,l (0) = 0 , (42)though more general hoi es ould be a ommodated for without any problems5.Our new result ombines the sharp bounds on the high momentum behaviour from[KM℄ with good ontrol on the onstants hidden in the symbols P in (40), (41) .In the Theorem and the Proposition we use the following notations and assumptions :We denote by ( p , . . . , p n ) a set of external momenta with p + . . . + p n = 0 , and wede(cid:28)ne ~p = ( p , . . . , p n − ) , | ~p | = sup ≤ i ≤ n | p i | . Furthermore η (2 n ) i,j ( p , . . . , p n ) := inf n | p i + X k ∈ J p k | / J ⊂ (cid:0) { , ..., n } − { i, j } (cid:1)o . (43)Thus η (2 n ) i,j is the modulus of the smallest subsum of external momenta ontaining p i butnot p j . We assume ≤ Λ ≤ Λ , and we write κ = Λ + m .5It would amount to absorb the new onstants in the respe tive lower bounds on K in part B of theproof. 16ur main result an then be stated as follows :Theorem :There exists a onstant ˜ K > su h that |L Λ , Λ n,l ( ~p ) | ≤ κ − n ˜ K l + n − n ! ( n + l )! λ = l X λ =0 log λ (cid:0) sup( | ~p | κ , κm ) (cid:1) λ λ ! for n > , (44) |L Λ , Λ ,l ( p ) | ≤ sup( | p | , κ ) ˜ K l ( l + 1) l ! λ = l − X λ =0 log λ (cid:0) sup( | p | κ , κm ) (cid:1) λ λ ! , l ≥ . (45)The Theorem follows from the subsequent Proposition. In the Proposition the bounds arepresented in a form su h that they an serve at the same time as an indu tion hypothesisfor the statements to be proven. We then have to in lude also bounds on momentumderivatives of the S hwinger fun tions in order to have a omplete indu tive s heme.Proposition :We assume | w | ≤ , where the derivatives are taken w.r.t. some momentum p i . Further-more j ∈ { , . . . , n } / { i } . There exists a onstant K > su h that for n > | ∂ w L Λ , Λ n,l ( ~p ) | ≤ κ − n K l + n − ( l + 1) n ! n ( n + l − (cid:0) sup( κ, η (2 n ) i,j ) (cid:1) | w | λ = l X λ =0 log λ (cid:0) sup( | ~p | κ , κm ) (cid:1) λ λ ! . (46)For n = 4 , | w | ≥ | ∂ w L Λ , Λ ,l ( ~p ) | ≤ K l − / ( l + 1) (1 + l )! 1 (cid:0) sup( κ, η (4) i,j ) (cid:1) | w | λ = l − X λ =0 log λ (cid:0) sup( | ~p | κ , κm ) (cid:1) λ λ ! (cid:1) . (47)For n = 4 , | w | = 0 |L Λ , Λ ,l ( ~p ) | ≤ K l ( l + 1) (1 + l )! λ = l − X λ =0 log λ (cid:0) sup( | ~p | κ , κm ) (cid:1) λ λ ! (cid:16) (cid:0) sup( | ~p | κ , κm ) (cid:1)(cid:17) , (48) |L Λ , Λ ,l (0 , p , p ) | ≤ K l ( l + 1) (1 + l )! λ = l X λ =0 log λ (cid:0) sup( | ~p | κ , κm ) (cid:1) λ λ ! . (49)For n = 2 , | w | = 3 | ∂ w L Λ , Λ ,l ( p ) | ≤ sup( | p | , κ ) − K l − − ( l + 1) l ! λ = l − X λ =0 log λ (cid:0) sup( | p | κ , κm ) (cid:1) λ λ ! . (50)17or n = 2 , ≤ | w | ≤ , l ≥ | ∂ w L Λ , Λ ,l ( p ) | ≤ sup( | p | , κ ) −| w | K l − ( l + 1) l ! λ = l − X λ =0 log λ (cid:0) sup( | p | κ , κm ) (cid:1) λ λ ! (cid:16) | p | κ , κm )) (cid:17) . (51)For n = 2 , | w | ∈ { , } , l ≥ | ∂ w L Λ , Λ ,l (0) | ≤ κ −| w | K l − ( l + 1) l ! λ = l − X λ =0 log λ (cid:0) κm (cid:1) λ λ ! . (52)Remarks :Note that j in (46) - (47) is otherwise arbitrary apart from the ondition j = i , so thatthe bound arrived at will be in fa t | ∂ w L Λ , Λ n,l ( ~p ) | ≤ κ − n K l + n − ( l + 1) n ! n inf j, ≤ j ≤ n (cid:0) sup( κ, η (2 n ) i,j ) (cid:1) | w | ( n + l − λ = l X λ =0 log λ (cid:0) sup( | ~p | κ , κm ) (cid:1) λ ! . We will hoose j = 2 n in the proof. This means that the momentum p n will beeliminated on both sides of the FE.Sin e the elementary vertex has a weight g , a perturbative S hwinger fun tion L n,l arries a fa tor ( g ) l + n − For simpli ity of notation we repla e this fa tor by one in thesubsequent proof. So the (cid:28)nal numeri al bound on the S hwinger fun tions in terms ofthe onstant K , see (76) below, should be multiplied by this fa tor.Proof :The above des ribed indu tive s heme starts from the onstant L Λ , Λ , at loop order 0.From this term, irrelevant tree level terms with n > are produ ed by the se ond termon the r.h.s. of the FE. For those terms the Proposition is veri(cid:28)ed from a simpli(cid:28)ed versionof part A) II) of the proof, where all sums over loops are suppressed. Note also that thetwo-point fun tion for l = 1 is given by the momentum independent tadpole whi h isbounded by κ . We will subsequently assume that l ≥ for simpli ity of notation.A) Irrelevant terms with n + | w | ≥ :I) The (cid:28)rst term on the r.h.s. of the FEa) n > :Integrating the FE (7) w.r.t. the (cid:29)ow parameter κ ′ from κ to Λ + m gives the followingbound for the (cid:28)rst term on the r.h.s. of the FE - denoting Λ ′ = κ ′ − m and, as a shorthand,18 ~p | n +2 = sup( | ~p | , | k | , | − k | ) = sup( | p | , . . . , | p n | , | k | ) , η (2 n +2) i, n = η (2 n +2) i, n ( ~p, k, − k ) : (2 n + 1)(2 n + 2)2 Z Λ + mκ dκ ′ Z k ′ e − k m ′ κ ′ − (2 n +2) K l + n − l ( n + 1)! ( n + 1) × ( n + l − (cid:0) sup( κ ′ , η (2 n +2) i, n ) (cid:1) | w | λ = l − X λ =0 log λ (sup( | ~p | n +2 κ ′ , κ ′ m ))2 λ λ ! ≤ ( nn + 1 ) (2 n + 1) K l + n − l n ! n ( n + l − λ = l − X λ =0 λ λ ! (53) × K Z Λ + mκ dκ ′ κ ′ − n −| w | Z k κ ′ (cid:0) sup(1 , η (2 n +2) i, n κ ′ ) (cid:1) | w | e − k ′ log λ (cid:0) sup( | ~p | n +2 κ ′ , κ ′ m ) (cid:1) . We used Lemma 6, (31). We bound the momentum integral as follows, setting x = kκ ′ : Z x (cid:0) sup(1 , η (2 n +2) i, n κ ′ ) (cid:1) | w | e − x log λ (cid:0) sup( | ~p | n +2 κ ′ , κ ′ m ) (cid:1) ≤ sup x n e − x (cid:0) sup(1 , η (2 n +2) i, n κ ′ ) (cid:1) | w | o Z x e − x log λ (cid:0) sup( | ~p | n +2 κ ′ , κ ′ m ) (cid:1) . (54)The (cid:28)rst term is bounded6 with the aid of Lemma 3, (23), as sup x n e − x (cid:0) sup(1 , η (2 n +2) i, n κ ′ ) (cid:1) | w | o ≤ c ( | w | ) 1 (cid:0) sup(1 , η (2 n ) i, n κ ′ ) (cid:1) | w | . To bound the integral in (54), we note that sup( | ~p | n +2 κ ′ , κ ′ m ) ≤ sup( | ~p | κ ′ + | k | κ ′ , κ ′ m ) so that the integral an be bounded using Z x e − x log λ (sup( | x | + a, b )) ≤ Z x e − x log λ ( | x | + a ) + Z x e − x log λ b (55)with a = | ~p | κ ′ and b = κ ′ m . We have Z x e − x log λ b = 14 π log λ b . (56)6by the de(cid:28)nition of η (43) we have η (2 n +2) i, n ∈ {| q | , | q ± k |} , if η (2 n ) i, n = | q | .19sing Lemma 4, (26) and π + ≤ , we an then bound the integral from (54) by Z x e − x log λ (sup( | ~p | n +2 κ ′ , κ ′ m )) ≤ K (cid:16) log λ (sup( | ~p | κ ′ , κ ′ m )) + [ λ !] / (cid:17) , (57)where K = 13 . (58)With these results (53) an now be bounded by ( nn + 1 ) (2 n + 1) K l + n − l n ! n ( n + l − K K c ( | w | ) (cid:0) sup(1 , η (2 n ) i, n κ ) (cid:1) | w | (59) × λ = l − X λ =0 Z Λ + mκ dκ ′ κ ′ − n −| w | λ λ ! (cid:16) log λ (sup( | ~p | κ ′ , κ ′ m )) + [ λ !] / (cid:17) . Using Lemma 5, (29) we (cid:28)nd - writing s = 2 n + | w | − - λ = l − X λ =0 λ λ ! Z Λ + mκ dκ ′ κ ′− s − (cid:16) log λ (sup( | ~p | n κ ′ , κ ′ m )) + [ λ !] / (cid:17) ≤ κ − s s n λ = l − X λ =0 λ λ ! log λ sup( | ~p | κ , κm ) + 2 o ≤ κ − s s λ = l − X λ =0 λ λ ! log λ sup( | ~p | κ , κm ) . Using this bounds in (59), the (cid:28)rst term on the r.h.s. of the FE then satis(cid:28)es the indu tionhypothesis (46)7, κ − n K l + n − ( l + 1) n ! n ( n + l − (cid:0) sup( κ, η (2 n ) i, n ) (cid:1) | w | λ = l − X λ =0 λ λ ! log λ sup( | ~p | κ , κm ) , on imposing the lower bound on KK − ( nn + 1 ) (2 n + 1) ( l + 1) l K K c ( | w | ) 5(2 n + | w | − ≤ . (60)b) n = 4 , | w | ≥ :The only hange w.r.t. part a) is that we have to verify the bound with an addditionalfa tor of K − / appearing in (47). We therefore arrive at the bound K − ( 23 ) l + 1) l K K c ( | w | ) 5 | w | ≤ . (61)7we may note that for this term the sum extends up to l − only20) n = 2 , | w | = 3 :Due to the momentum derivatives the orresponding ontribution for l = 1 vanishes.Using the indu tion hypothesis on | ∂ w L Λ , Λ ,l − ( ~p ) | for l ≥ as in (53) we obtain in loseanalogy with A) I) a) and b) the following bound ( 12 ) K l − − ( l + 1) κ sup( | p | , κ ) l ! λ = l − X λ =0 log λ (cid:0) sup( | p | κ , κm ) (cid:1) λ λ ! in agreement with (50), on imposing the lower bound K −
38 ( l + 1) l K K c (3) 5 ≤ . (62)II) The se ond term on the r.h.s. of the FEa) n > :We sum over all ontributions without taking into a ount the fa t that some of themare suppressed by supplementary fra tional powers of K . Some additional pre autionis required in the presen e of relevant terms, i.e. underived four-point fun tions, andtwo-point fun tions derived at most twi e. These fun tions are de omposed as L ,l ( p , p , p ) = L ,l (0 , p , p ) + p ,µ Z dτ ∂ ,µ L ,l ( τ p , p , p ) . (63)For the two-point fun tion we may suppose without limitation that p = ( p , , , . Wethen write p instead of p , ∂ instead ∂∂p and interpolate ∂ L ,l ( p ) = ∂ L ,l (0) + p Z dτ ∂ L ,l ( τ p ) , (64) ∂ L ,l ( p ) = p ∂ L ,l (0) + p Z dτ (1 − τ ) ∂ L ,l ( τ p ) , (65) L ,l ( p ) = L ,l (0) + 12 p ∂ L ,l (0) + p Z dτ (1 − τ ) ∂ L ,l ( τ p ) . (66)In ase of the four-point fun tion we use the bound from (49) for the (cid:28)rst term of thede omposition, and the bound from (47) for the se ond term. Here the interpolatedmomentum p will be (without loss of generality) supposed to be the momentum q of thepropagator linking the two terms on the r.h.s. of the FE. We then will use the bound (35)to get rid of the momentum fa tor produ ed through interpolation. Thus we an avoid21sing (48) whi h would not reprodu e a bound mat hing with our indu tion hypothesis.For the two-point fun tion we similarly use either the bounds (52) at zero momentum, or(50), together with (36) and (31), for the interpolated term.These de ompositions lead to additional fa tors in the bounds. So as not to produ e toolengthy expressions we will (cid:28)rst write the bounds only for the ontributions where theadditonal fa tors are not present and add the modi(cid:28) ations ne essitated by those termsafterwards (see after (74)).A se ond point has to be lari(cid:28)ed (whi h is treated in a fully expli it though notationallymore omplex way in [GK℄). When deriving both sides of the (cid:29)ow equation w.r.t. themomentum p i , there may arise two situations for the se ond term on the r.h.s. : eitherthe two momenta p i and p n appear both as external momenta of only one term L n i ,l i ,or ea h of them appears in a di(cid:27)erent L n i ,l i . In the (cid:28)rst ase the derivatives only applyto the term where they both appear, and not to the se ond one whi h is independent of p i , nor to the propagator linking the two terms. In the se ond ase also the other termand the linking propagator depend on p i via the momentum q of the propagator whi his a subsum of momenta ontaining p i . Applying then the indu tion hypothesis to bothterms we get a produ t of η -terms whi h an be bounded by a single one : (cid:0) sup( κ, η (2 n ) i, n ) (cid:1) | w | (cid:0) sup( κ, η (2 n ) i, n ) (cid:1) | w | ≤ (cid:0) sup( κ, η (2 n ) i, n ) (cid:1) | w | + | w | , (67)sin e one veri(cid:28)es that the set of momenta over whi h the inf is taken in η in the termson the l.h.s. of (67) is ontained in the one on the r.h.s. of (67). Here η (2 n ) i, n has beenintrodu ed as in (43) for the momentum set { p , . . . , p n − , q } , where q = − p − p − . . . − p n − , and we understand (without introdu ing new notation) that η (2 n ) i, n has beenintrodu ed as in (43) for the momentum set { q, p n , . . . , p n } where q takes the role of p i .The reasoning remains the same, if permutations of these momentum sets are onsidered,whi h still leave p i and p n in di(cid:27)erent sets.Integrating the indu tive bound on the se ond term on the r.h.s. of the FE from κ to Λ + m then gives us the following bound - where we also understand that the sup w.r.t.the previously mentioned permutations has been taken for the momentum attributions Z Λ + mκ dκ ′ κ ′ − (2 n +2) K l + n − X l + l = l , w + w + w = w,n + n = n +1 c { w i } n ( l + 1) n ! n n ( l + 1) n ! n × (cid:0) sup( κ ′ , η (2 n ) i, n ) (cid:1) | w | ( n + l − λ = l X λ =0 log λ (cid:0) sup( | ~p | κ ′ , κ ′ m ) (cid:1) λ λ ! 2Λ ′ | ∂ w e − q m ′ | (cid:0) sup( κ ′ , η (2 n ) i,j ) (cid:1) | w | ( n + l − λ = l X λ =0 log λ (cid:0) sup( | ~p | κ ′ , κ ′ m ) (cid:1) )2 λ λ ! . We use (67) to bound the previous expression by X l + l = l , n + n = n +1 , λ ≤ l , λ ≤ l l + 1) ( l + 1) n n n ! n ! n ! ( λ + λ )! λ ! λ ! ( n + l − n + l − n + l − × K l + n − ( n + l − n ! Z Λ + mκ dκ ′ κ ′ − n log λ + λ (cid:0) sup( | ~p | κ ′ , κ ′ m ) (cid:1) λ + λ ( λ + λ )! × X w + w + w = w c { w i } ′ | ∂ w e − q m ′ | (cid:0) sup( κ ′ , η (2 n ) i, n ) (cid:1) | w | + | w | . Using Lemma 2, (15) and Lemma 6, (33), and the fa t that sup( | q | , κ ′ ) −| w | (cid:0) sup( κ ′ , η (2 n ) i, n ) (cid:1) | w | + | w | ≤ (cid:0) sup( κ ′ , η (2 n ) i, n ) (cid:1) | w | we then arrive at the bound K l + 1) n K l + n − n ! ( n + l − Z Λ + mκ dκ ′ κ ′ − n −| w | X ≤ λ ≤ l log λ (cid:0) sup( | ~p | κ ′ , κ ′ m ) (cid:1) λ λ ! × X w i c { w i } K ( | w | ) (cid:0) sup(1 , η (2 n ) i, n κ ′ ) (cid:1) | w | . (68)Using also Lemma 5 we verify the bound (46) κ − n K l + n − l + 1) n n ! ( n + l − X ≤ λ ≤ l log λ (cid:0) sup( | ~p | κ , κm ) (cid:1) λ λ ! 1 (cid:0) sup( κ, η (2 n ) i,j ) (cid:1) | w | , on imposing the lower bound on KK − · K n n + | w | − K X w i c { w i } K ( | w | ) ≤ , n > . (69)b) n = 4 , | w | ≥ :We obtain in the same way, using Lemma 2 ) K − / K K ′′ X { w i } c { w i } K ( | w | ) ≤ . (70)23) n = 2 , | w | = 3 :For the two-point fun tion we obtain K − / K K ′′ X { w i } c { w i } K ( | w | ) ≤ . (71)Taking both ontributions from the r.h.s. of the FE together, the lower bounds on K be ome for n > K (cid:16) K ( nn + 1 ) c ( | w | ) (2 n + 1) ( l + 1) (2 n + | w | − l + 6 n n + | w | − K X { w i } c { w i } K ( | w | ) (cid:17) ≤ K , (72)and for n = 2 resp. n = 1 K (cid:16) · K c ( | w | ) l + 1) | w | l + 6 · K ′′ | w | X { w i } c { w i } K ( | w | ) (cid:17) ≤ K , (73) K (cid:16) · K c (3) ( l + 1) l K − + 6 K ′′ X { w i } c { w i } K ( | w | ) (cid:17) ≤ K . (74)We now ome ba k to the modi(cid:28) ations required be ause of the de ompositions (63),(64), (65), (66). We introud e the shorthands P { w i } c { w i } K ( | w | ) ≡ ˜ K ( w ) ≡ ˜ K and P { w i } c { w i } K ′ ( | w | ) ≡ ˜ K ′ ( w ) ≡ ˜ K ′ . In order not to in(cid:29)ate too mu h the values of the onstants we distinguish di(cid:27)erent ases. In ea h ase we have to repla e the fa tors K ˜ K from (69) resp. K ′′ ˜ K from (70) and from (71) by the following ones:i) n > : K K + 2 K ′ ˜ K + 2 K ′ √ e K / ˜ K ′ + 2 K ′′ ˜ K + K ′′ ( 1 √ e + 12 2 e + 1 K / ) ˜ K ′ , ii) n = 3 : K K + K ′ ˜ K + K ′ ( 2 √ e K / + 2 e K / ) ˜ K ′ + 2 K ′′ ˜ K + 2 K ′′ ( 1 √ e + 12 2 e + 1 K / ) ˜ K ′ , iii) n = 2 : K ′′ ˜ K + 2 K ′′ √ e K / K ′ , iv) n = 1 : K ′′ ˜ K + K ′′ ( 1 √ e + 12 2 e + 1 K / ) ˜ K ′ . K by K / if no two- or four-point fun tions appear by Lemma2, (16). In the other ases we use Lemma 2, (17) or(18), and we use the de ompositionswhi h then give rise to a sum of ontributions. Fa tors of appear if there exist two ontributions of the required type. To bound the individual terms from the de ompositionwe also have to use Lemma 6 ), sin e there appear momentum dependent fa tors inthe interpolation formulas whi h have to be bounded with the aid of the regularizingexponential. The terms multiplied by ˜ K thus arise from the boundary terms, thosemultiplied by ˜ K ′ from interpolated ones where the bounds (34) instead of (33) have tobe used sin e the regularizing exponential has to be split up and used for bounding twotypes of momentum fa tors. In the ases n = 2 and n = 1 there appear one four- andon two-point fun tion resp. two two-point fun tions on the r.h.s. of the FE. Only one ofthese fa tors has to be de omposed however, sin e in the (cid:28)nal bound we an tolerate onefa tor of (1 + log (cid:0) sup( | ~p | κ , κm ) (cid:1) a ording to the indu tion hypotheses for these two ases,see (48), (51).The (cid:28)nal lower bound on K whi h also turns out to be the most stringent one in theend, stems from the ase n = 3 . It is thus the following one n K ( 34 ) c ( | w | ) 7 ( l + 1) l + 18 h K K + K ′ ˜ K + K ′ ( 2 √ e K / + 2 e K / ) ˜ K ′ + 2 K ′′ ˜ K + 2 K ′′ ( 1 √ e + 12 2 e + 1 K / ) ˜ K ′ io K | w | ≤ K . (75)The numeri al lower bound on K dedu ed from (75) in the worst ase | w | = 3 is K ≥ . · . (76)One ould ertainly gain several orders of magnitude by more arefully bounding individ-ual spe ial ases (see above for one point). The basi sour e of the (still) large numeri al onstant is in the fa t that we have to re onstru t the relevant terms from their derivatives.B) Relevant terms with n + | w | ≤ :a) n = 4 , | w | = 0 :We (cid:28)rst look at L Λ , Λ ,l ( ~ whi h is de omposed as L Λ , Λ ,l ( ~
0) = L , Λ ,l ( ~
0) + Z Λ0 d Λ ′ ∂ Λ ′ L Λ ′ , Λ ,l ( ~ , (77)25here the (cid:28)rst term is vanishes for l ≥ , see (42). For the se ond term we obtain byindu tion from the (cid:28)rst term on the r.h.s. of the FE the bound ( 62 ) Z Λ+ mm dκ ′ Z k ′ e − k m ′ κ ′− K l − l · (1 + l )! λ = l − X λ =0 log λ (sup( | k | κ ′ , κ ′ m ))2 λ λ ! ≤ K K ( 62 ) 12 · K l − l (1+ l )! λ = l − X λ =0 λ λ ! Z Λ+ mm dκ ′ κ ′− (cid:16) log λ ( κ ′ m ) + ( λ !) / (cid:17) , (78)where we used again (31) and (57), remembering that | ~p | = | k | in the present ase. Wehave Z κm dκ ′ κ ′ (cid:16) log λ ( κ ′ m ) + [ λ !] / (cid:17) = log λ +1 ( κm ) λ + 1 + log( κm ) [ λ !] / , (79) λ = l − X λ =0 (cid:16) log λ +1 ( κm )2 λ ( λ + 1)! + log( κm ) 12 λ λ ! / (cid:17) ≤ inf n λ = l X λ =1 log λ ( κm )2 λ λ ! , λ = l − X λ =0 log λ ( κm )2 λ λ ! (1+log κm ) o . (80)Using the (cid:28)rst of these bounds in (78), the (cid:28)rst term on the r.h.s. of the FE is boundedin agreement with the indu tion hypothesis by K l ( l + 1) (1 + l )! λ = l X λ =0 log λ ( κm )2 λ λ ! , (81)assuming the lower bound on KK − K K ( 62 ) 2 · ( l + 1) l ≤ . (82)In the ontribution from the se ond term on the r.h.s. of the FE we have one ontributionwith n = 2 and one ontribution with n = 1 or vi e versa. Integrating the FE (7) w.r.t.the (cid:29)ow parameter at vanishing momentum gives the indu tive bound, using (49), (52)and Lemma 2 ) · Z Λ+ mm dκ ′ ′ e − m ′ κ ′ K l − X l + l = l , l ≥ (1 + l )!( l + 1) l !( l + 1) λ = l X λ =0 log λ ( κ ′ m )2 λ λ ! λ = l − X λ =0 log λ ( κ ′ m )2 λ λ ! ≤ K l − K ′′ ( l + 1) (1 + l )! Z Λ0 d Λ ′ Λ ′ e − m ′ κ ′ λ = l − X λ =0 log λ ( κ ′ m )2 λ λ ! . K ′′ K ≤ K .
To go away from the renormalization point we pro eed as in [KM℄. In fa t, we will dis-tinguish four di(cid:27)erent situations as regards the momentum on(cid:28)gurations. The boundsestablished in part A) for the ase n = 4 , | w | = 1 are in terms of the fun tions η (4) i,j from(43). Assuming (without loss of generality) | p | ≥ | p | , | p | , | p | , we realize that η (4) i, is always given by a sum of at most two momenta from the set { p , p , p } . It is then obvious that the subsequent ases ii) and iv) over all possiblesituations. The ases i) and iii) orrespond to ex eptional on(cid:28)gurations for whi h thebound has to be established before pro eeding to the general ones. The four ases arei) { p , p , p } = { , q , v } ii) { p , p , p } su h that inf i η (4) i, = inf i | p i | iii) { p , p , p } = { p , − p , v } iv) { p , p , p } su h that inf i η (4) i, = inf j = k | p j + p k | .i) To prove the proposition in this ase, i. e. (49), we bound |L Λ , Λ ,l (0 , q, v ) | ≤|L Λ , Λ ,l (0 , , | + X µ Z dτ (cid:16) | q µ ∂ q µ L Λ , Λ ,l (0 , τ q, τ v ) | + | v µ ∂ v µ L Λ , Λ ,l (0 , τ q, τ v ) | (cid:17) . The se ond term is bounded using the indu tion hypothesis: K l − ( l + 1) X i =2 , | p i | Z dτ κ, η (4) i, ( τ ) ) (1 + l )! λ = l − X λ =0 λ λ ! log λ (cid:0) sup( | ~p τ | κ , κm ) (cid:1) . (83)We have written η ( τ ) for the η -parameter in terms of the s aled variables p τ = τ q , p τ = τ v and ~p τ for the momentum set (0 , p τ , p τ ) . Using Lemma 8 we (cid:28)nd η (4)2 , ( τ ) = τ | q | , η (4)3 , ( τ ) = τ | v | , and we thus obtain the following bound for (83) - apart from the prefa tor | q | (cid:16)Z inf(1 , κ | q | )0 dτκ + Z , κ | q | ) dττ | q | (cid:17) log λ (cid:0) sup( | ~p τ | κ , κm ) (cid:1) + (cid:18) q → v (cid:19) . (84)If | q | ≥ κ we (cid:28)nd Z κ | q | dττ log λ (cid:0) sup( | ~p τ | κ , κm ) (cid:1) ≤ Z κ | ~p | dττ log λ (cid:0) τ | ~p | κ (cid:1) ≤ Z | ~p | κ dxx log λ x = log λ +1 ( | ~p | κ ) λ + 1 | v | ≥ κ . We thus obtain a bound for (84) λ (cid:0) sup( | ~p | κ , κm ) (cid:1) + 2 log λ +1 (cid:0) sup(1 , | ~p | κ ) (cid:1) λ + 1 (85)whi h allows to bound (83) by K l − ( l + 1) (1 + l )! λ = l X λ =0 λ λ ! log λ (cid:0) sup( | ~p | κ , κm ) (cid:1) . (86)Using this bound together with the previous one on L Λ , Λ ,l (0 , , we verify the indu tionhypothesis on L Λ , Λ ,l (0 , q, v ) (49) under the ondition K − (cid:16) K K ( 62 ) 2 · ( l + 1) l + 16 K ′′ K (cid:17) + 6 K − / ≤ . (87)ii) We assume without loss of generality inf i η (4) i, = | p | . We use again an integratedTaylor formula along the integration path ( p τ , p τ , p τ ) = ( τ p , p , p + (1 − τ ) p ) . ByLemma 8 we (cid:28)nd η (4)1 , ( τ ) = | p τ | = τ | p | , η (4)3 , ( τ ) ≥ τ | p | . The boundary term for τ = 0 is bounded in i). For the se ond term we bound | X µ Z dτ (cid:16) p ,µ (cid:0) ∂ p ,µ − ∂ p ,µ (cid:1) L ( p τ , p τ , p τ ) (cid:17) |≤ K l − (1 + l )!( l + 1) λ = l − X λ =0 | p | λ λ ! Z dτ (cid:0) κ, η (4)1 , ( τ ) ) + 1sup( κ, η (4)3 , ( τ ) ) (cid:1) log λ (cid:0) sup( | ~p τ | κ , κm ) (cid:1) ≤ K l − (1 + l )!( l + 1) λ = l − X λ =0 | p | λ λ ! (cid:18)Z inf(1 , κ | p | )0 dτκ + Z , κ | p | ) dττ | p | (cid:19) log λ (cid:0) sup( | ~p τ | κ , κm ) (cid:1) ≤ K l − (1 + l )!( l + 1) λ = l − X λ =0 λ λ ! (cid:2) (cid:0) sup(1 , | p | κ ) (cid:1)(cid:3) log λ (cid:0) sup( | ~p τ | κ , κm ) (cid:1) ≤ K l − ( l + 1) (1 + l )! λ = l − X λ =0 λ λ ! log λ (cid:0) sup( | ~p | κ , κm ) (cid:1) (cid:16) (cid:0) sup( κm , | ~p | κ ) (cid:1)(cid:17) . (88)Adding the terms from i) to this term gives the lower bound on KK − (cid:16) K K ( 62 ) 2 · ( l + 1) l + 16 K ′′ K (cid:17) + 4 K − / ≤ . (89)Here we used the fa t that we may bound the term from i) also by (88) instead of (86)if we only want to verify the weaker form of the indu tion hypothesis valid for general28omenta. At the same time we have repla ed a fa tor of 6 appearing in (82) by a fa torof 2, sin e in the general ase we may use the se ond bound in (80).iii) We hoose the integration path ( p τ , p τ , p τ ) = ( τ p, − p, v ) . Here we assume withoutrestri tion that | v | ≤ | v − (1 − τ ) p | , otherwise we ould inter hange the role of v and p = − v . The boundary term leads again ba k to i). The integral R dτ is ut into fourpie es - where the on(cid:28)guration κ < | p | gives the largest ontribution : Z = Z inf(1 / , κ | p | )0 + Z / / , κ | p | ) + Z sup(1 / , − κ | p | )1 / + Z / , − κ | p | ) . They are bounded in analogy with ii) using η (4)1 , ( τ ) = τ | p | for τ ≤ / , η (4)1 , ( τ ) =(1 − τ ) | p | for τ ≥ / , relations established with the aid of Lemma 8. We get the bound K l − ( l + 1) (1 + l )! λ = | l − | X λ =0 λ λ ! log λ (cid:0) sup( | ~p | κ , κm ) (cid:1) (cid:16) (cid:0) sup(1 , | ~p | κ ) (cid:1)(cid:17) (90)so that veri(cid:28) ation of (48) requires again the lower bound (89) on K .iv) We assume without loss inf i η (4) i, = | p + p | and integrate along ( p τ , p τ , p τ ) =( p , − p + τ ( p + p ) , p ) . The boundary term has been bounded in iii). Using Lemma8 we (cid:28)nd inf η (4)2 , ( τ ) = τ | p + p | , and the integration term is then bounded through | X µ Z dτ (cid:16) ( p ,µ + p ,µ ) ∂ p ,µ L Λ , Λ ( p τ , p τ , p τ ) (cid:17) | ≤ K l − (1 + l )!( l + 1) λ = l − X λ =0 | p + p | λ λ ! (cid:18)Z inf(1 , κ | p p | )0 dτκ + Z , κ | p p | ) dττ | p + p | (cid:19) log λ (sup( | ~p τ | κ , κm )) whi h gives as before a bound K l − ( l + 1) (1 + l )! λ = l − X λ =0 λ λ ! log λ (cid:0) sup( | ~p | κ , κm ) (cid:1) (cid:16) (cid:0) sup(1 , | ~p | κ ) (cid:1)(cid:17) (91)so that taking into a ount the boundary term from iii) we (cid:28)nally require K − (cid:16) K K ( 62 ) 2 · ( l + 1) l + 16 K ′′ K (cid:17) + 5 K − / ≤ (92)to be in agreement with indu tion. 29) n = 2 :We again use the simpli(cid:28)ed notation (64) to (66). We will assume that l ≥ . We pro eedin des ending order of | w | starting fromb1) | w | = 2 : ∂ L ,l ( p ) = ∂ L ,l (0) + p Z dτ ∂ L ,l ( τ p ) . (93)We (cid:28)rst look at ∂ L Λ , Λ ,l (0) whi h is de omposed as ∂ L Λ , Λ ,l (0) = ∂ L , Λ ,l (0) + Z Λ0 d Λ ′ ∂ Λ ′ ∂ L Λ ′ , Λ ,l (0) , the se ond term being obtained from the r.h.s. of the FE, and the (cid:28)rst vanishing by (42).The (cid:28)rst term on the r.h.s. of the FE then gives the bound ( 42 ) Z Λ+ mm dκ ′ Z k ′ e − k m ′ κ ′− K l − − l l ! λ = l − X λ =0 log λ (sup( | k | κ ′ , κ ′ m ))2 λ λ ! ≤ K K K l − − l l ! λ = l − X λ =0 λ λ ! Z Λ+ mm dκ ′ κ ′− (cid:16) log λ ( κ ′ m ) + ( λ !) / (cid:17) , (94)where we used again (57) and (31), remembering that | ~p | = k in the present ase. Using(79) and (80) (with l → l − ) the (cid:28)rst term on the r.h.s. of the FE is then bounded inagreement with the indu tion hypothesis by K l − ( l + 1) l ! λ = l − X λ =0 log λ ( κm )2 λ λ ! under the assumption K − / K K · l + 1) l ≤ . (95)This ontribution has to be added to the one from the se ond term on the r.h.s. of theFE. We have only ontributions with n = 1 and n = 1 . The two momentum derivativeshave to apply both to the propagator or both to a fun tion L ,l ; all other ontributionsvanish at zero momentum. For the ontribution of the (cid:28)rst kind integration of the FE (7)gives the bound Z Λ0 d Λ ′ Λ ′ e − m ′ κ ′ K l − X l + l = l , l ,l ≥ l !( l + 1) l !( l + 1) λ = l − X λ =0 log λ ( κ ′ m )2 λ λ ! λ = l − X λ =0 log λ ( κ ′ m ))2 λ λ ! K l − K ′′ ( l + 1) Z Λ0 d Λ ′ Λ ′ e − m ′ κ ′ λ = l − X λ =0 log λ ( κ ′ m )2 λ λ ! , (96)where we used (9)and Lemma 2 ). Using also Lemma 7 we obtain the bound K l − K ′′ ( l + 1) K ′ λ = l − X λ =1 log λ ( κm )2 λ λ ! . (97)For the ontribution of the se ond kind integration of the FE gives in the same way thebound (again using Lemma 2 ) and Lemma 7) Z Λ0 d Λ ′ Λ ′ e − m ′ κ ′ K l − X l + l = l , l ,l ≥ l !( l + 1) l !( l + 1) λ = l − X λ =0 log λ ( κ ′ m )2 λ λ ! λ = l − X λ =0 log λ ( κ ′ m )2 λ λ ! ≤ K l − K ′′ ( l + 1) Z Λ0 d Λ ′ Λ ′ e − m ′ κ ′ λ = l − X λ =0 log λ ( κ ′ m )2 λ λ ! ≤ K l − K ′′ ( l + 1) K λ = l − X λ =1 log λ ( κm )2 λ λ ! . (98)The sum of this bound and the bounds (95), (97) is ompatible with the indu tion hy-pothesis (52) under the ondition K − / K K
36 ( l + 1) l + 8 K − (2 K ′′ K ′ + K ′′ K ) ≤ . (99)The se ond term in (93) is bounded with the aid of the indu tion hypothesis | p | Z dτ ∂ L ,l ( τ p ) | ≤ | p | Z dτ sup( τ | p | , κ ) K l − − ( l + 1) l ! λ = l − X λ =0 λ λ ! log λ (cid:0) sup( | τ p | κ , κm ) (cid:1) . Assuming that | p | > κ and also that | p | m > κ , whi h is the most deli ate ase (in theother ases some of the 3 ontributions in (100) below are absent) we ut up the integral Z dτ = (cid:16)Z κp + Z κ pmκp + Z κ pm (cid:17) dτ and (cid:28)nd | p | (cid:16)Z κp + Z κ pmκp + Z κ pm (cid:17) dτ log λ (cid:0) sup( | τ p | κ , κm ) (cid:1) sup( τ | p | , κ ) ≤ log λ ( κm ) + log λ +1 ( κm ) + log λ +1 ( pκ ) λ + 1 (100)so that we obtain the bound K l − − ( l + 1) l ! λ = l − X λ =0 λ λ ! log λ ( κm )) (cid:16) | p | κ , κm )) (cid:17) . K is obtained on adding the bound (99) stemming from theboundary term at zero momentum and this one K − / + K − / K K
12 ( l + 1) l + 8 K − (2 K ′′ K ′ + K ′′ K ) ≤ . (101)In the se ond term we again repla ed a fa tor of 6 by a fa tor of 2 as in (89).b2) | w | = 1 :In this ase we write ∂ L ,l ( p ) = ∂ L ,l (0) + p ∂ L ,l (0) + p Z dτ (1 − τ ) ∂ L ,l ( τ p ) . (102)Due to Eu lidean symmetry the (cid:28)rst term on the r.h.s. vanishes. The bound on these ond term has been al ulated in the previous se tion. The last term is bounded as inthe previous al ulation by p, κ ) K l − − ( l + 1) l ! λ = l − X λ =0 λ λ ! log λ ( κm )) (cid:16) | p | κ , κm )) (cid:17) . so that we get again the lower bound (101) on K .b3) | w | = 0 :We (cid:28)rst look at L Λ , Λ ,l (0) whi h is written as L Λ , Λ ,l (0) = L , Λ ,l (0) + Z Λ0 d Λ ′ ∂ Λ ′ L Λ ′ , Λ ,l (0) . (103)From the (cid:28)rst term on the r.h.s. of the FE, where we use the bound (49) sin e two of theexternal momenta in L Λ ′ , Λ ,l − (0 , , k, − k ) vanish, we obtain using again (31) and (57) ( 42 ) Z κm dκ ′ Z k ′ e − k m ′ K l − l l ! λ = l − X λ =0 log λ (sup( | k | κ ′ , κ ′ m ))2 λ λ ! ≤ K K K l − l l ! λ = l − X λ =0 λ λ ! Z κm dκ ′ κ ′ (cid:16) log λ ( κ ′ m ) + ( λ !) / (cid:17) ≤ K K K l − l l ! λ = l − X λ =0 λ λ ! κ (cid:16) log λ ( κm ) + ( λ !) / (cid:17) (104) ≤ K K K l − l l ! λ = l − X λ =0 λ λ ! κ λ ( κm ) . (105)32his is ompatible with the indu tion hypothesis (52) if K ≥ ( l + 1) l K K . (106)Integrating the se ond term on the r.h.s. of the FE we obtain the bound Z Λ0 d Λ ′ Λ ′ e − m ′ κ ′ K l − X l + l = l , l ,l ≥ l !( l + 1) l !( l + 1) l − X λ =0 log λ ( κ ′ m )2 λ λ ! l − X λ =0 log λ ( κ ′ m ))2 λ λ ! ≤ K l − K ′′ ( l + 1) Z Λ0 d Λ ′ Λ ′ e − m ′ κ ′ λ = l − X λ =0 log λ ( κ ′ m )2 λ λ ! ≤ κ K l − ( l + 1) l ! λ = l − X λ =1 log λ ( κm )2 λ λ ! using again Lemma 2 ) and Lemma 7 and imposing the ondition K ′′ K ≤ K . (107)To go away from zero momentum we write similarly as in (102) L ,l ( p ) = L ,l (0) + 12 p ∂ L ,l (0) + p Z dτ (1 − τ ) ∂ L ,l ( τ p ) (108)and pro eed in the same way as in the previous se tion, see (95), (96), (98), (100).Indu tive veri(cid:28) ation of (51) gives similarly as in (101) the lower bound on KK − + K − K K l + 1) l + K − (cid:16)
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