Non-linear and weak-coupling expansion in Quantum Field Theory
aa r X i v : . [ h e p - t h ] S e p Non-linear and weak-coupling expansion inQuantum Field Theory
Vincenzo Branchina a,b, , Alberto Chiavetta a,c , Filippo Contino a,b a Department of Physics and Astronomy, University of Catania,Via Santa Sofia 64, 95123 Catania, Italy, b INFN, Sezione di Catania, Via Santa Sofia 64, 95123 Catania, Italy, c Scuola Superiore di Catania, Via Valdisavoia 9, 95123 Catania, Italy
Abstract
A formal expansion for the Green’s functions of an interacting quantum field theory ina parameter that somehow encodes its “distance” from the corresponding non-interactingone was introduced more than thirty years ago, and has been recently reconsidered inconnection with its possible application to the renormalization of non-hermitian theories.Besides this new and interesting application, this expansion has special properties alreadywhen applied to ordinary (i.e. hermitian) theories, and in order to disentangle the pecu-liarities of the expansion itself from those of non-hermitian theories, it is worth to pushfurther the investigation limiting first the analysis to ordinary theories. In the presentwork we study some aspects related to the renormalization of a scalar theory within theframework of such an expansion. Due to its peculiar properties, it turns out that at anyfinite order in the expansion parameter the theory looks as non-interacting. We showthat when diagrams of appropriate classes are resummed, this apparent drawback disap-pears and the theory recovers its interacting character. In particular we have seen thatwith a certain class of diagrams, the weak-coupling expansion results are recovered, thusestablishing a bridge between the two expansions.
Even though almost hundred years passed since the pioneeristic work of Dirac on thequantization of the electromagnetic field [1], it can hardly be said that the subject ofQuantum Field Theory (QFT) is a closed chapter of theoretical physics. This is strik-ingly evident with the appearance of divergent contributions to physical amplitudes, whosepresence prompted the introduction of renormalization [2, 3, 4, 5, 6, 7, 8, 9], that indepen-dently of the divergences themselves has proved to be a fruitful and insightful approachto QFTs [10, 11, 12, 13, 14, 15, 16, 17, 18]. However, in spite of the efforts made alongseveral decades for implementing QFTs and the related renormalization program in per-turbative [19, 20, 21, 22] and/or non-perturbative [23, 24, 25, 26, 27, 28, 29, 30] frameworks,these still remain evergreen subjects for new discoveries, sometimes even for frustration. [email protected] [email protected] fi[email protected] ~ expansion, weak-coupling expansion, truncations of Dyson-Schwinger-type,Borel-resummations, Pad´e approximants, resummation of infrared and ultraviolet renor-malons [31, 32, 33, 34], just to mention few well known approaches.By considering the case of a scalar theory with a typical polynomial interaction, al-most thirty years ago an expansion in terms of a non-linearity parameter that somehowencodes the “distance” between the linear (non-interacting) theory and the interactingone was proposed in [35, 36]. Such a formal expansion was reconsidered in a recent pa-per [37], where it was suggested that it could be the key for implementing a systematicrenormalization program for non-hermitian (in particular PT-symmetric) quantum fieldtheories.In these latter years great interest has grown around PT-symmetric theories [38, 39,40], and impressive successful applications to many different areas of quantum mechanicshave been realized [41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58,59, 60]. The extension to QFTs has also been considered [61, 62, 63, 64, 65, 66, 67,68]. However, while in the case of PT-symmetric quantum mechanics the boundaryconditions on the Schrodinger equation are imposed in complex Stokes sectors, the searchfor the Stokes wedges for the integration over infinitely many complex field variables isan insurmountable task. The great advancement that the expansion in the parameter ǫ seems to bring is that it allows to perform the functional integration along the real- φ axis(rather than in the complex- φ domain), as the functional integral that defines the theory(see Eqs. (6), (7) and (8) below) converges term-by-term in powers of ǫ [37].PT-symmetric theories, however, have their own peculiar features, so that, in order tomake progress in understanding the expansion itself, it is useful to disentangle its intrinsicproperties from those that are specific to this kind of theories . To this end, it is usefulto deepen the analysis by studying first the application of this expansion to ordinarytheories. Earlier attempts in this direction where made in [70], however a more completeanalysis, and in particular the implementation of a systematic renormalization programwithin the framework of such an expansion is still missing. An ambitious program wouldbe to study the renormalization of ordinary theories with the same degree of systematicityrealized in the context of the weak-coupling expansion. Clearly this is an enormous task,and the aim of the present work is more modest.Our goal is to investigate some aspects related to the renormalization of theories forwhich the Green’s functions are calculated within the framework of such an expansion.In order to better appreciate the peculiar properties of this approach, in this paper weconsider the simplest possible framework, by studying the case of an ordinary scalar theorywith interaction term φ ǫ ) . For ǫ = 0 the non-interacting linear theory is obtained,and this is why we said that the expansion is organized in terms of a parameter (theparameter ǫ ) that measures the distance between the interacting and the correspondingnon-interacting theory. Note that for ǫ = 1 the physically relevant φ theory is recovered.The starting point of the procedure introduced in [35, 36], and later reconsidered in [70, The extension of this analysis to PT-symmetric theories is left for future work [69]. φ ǫ ) in powers of ǫ , φ ǫ ) = φ ∞ X k =0 ǫ k k ! î log Ä φ äó k . (1)For our scopes it is important to note that each order in ǫ contains a power of thelogarithm of the scalar field, and that such logs are treated by considering an adaptationof the replica-trick [72] that is suitable to the expansion of the theory’s Green’s functions.This is obtained with the help of the formal identity: φ log k ( φ ) = lim N → d k dN k φ N . (2)The whole procedure is built around this identity [35], and aims at finding approxima-tions to the Green’s functions G n in terms of an expansion in powers of ǫ . The actualimplementation of this program, however, presents several delicate aspects that will bediscussed in detail in the paper.In previous works the calculation of the Green’s functions G n for any n was performedat first order in ǫ , while at order ǫ only G and G were calculated [71]. One of ourgoals is to push the calculation of the generic G n beyond the leading order. The generalframework and the notation will be established by repeating first the calculation of the O ( ǫ ), for which we reobtain the results of [35], and later the full O ( ǫ ) contribution willbe obtained.To this end we introduce a systematic diagrammatic expansion for the G n ’s, andsuccessively define “effective vertices” that allow to express the final results in an effectiveand simple way, even at higher orders in ǫ . However we will show that more and moredelicate issues appear when higher and higher order contributions are considered, and wewill show that this is due to a peculiar ultraviolet behavior of the G n ’s that is related tothe properties of this expansion. We will be confronted with the appearance of series forwhich the convergence is not always guaranteed, and we will need appropriate proceduresto regularize these series.The ultraviolet behaviour of the Green’s functions will be studied with the introductionof a physical momentum cut-off Λ, and the different contributions to the G n will beclassified with respect to their dependence on Λ. As an outcome of this analysis, we willfind an unexpected and in a sense astonishing behaviour with Λ. More specifically, wewill see that (barring unconventional ways to renormalize the theory that at present arenot supported and/or justified and on which we will comment in due time) as long as weconfine ourselves to a finite order in ǫ , for space-time dimensions d >
2, the G n with n > → ∞ , thus suggesting that at any finite order the theory turns out to benon-interacting.Let us apply these results to a φ theory. Although for d = 4 a final word on the inter-acting character of the theory is still missing, we certainly know that in d = 3 dimensionsthe theory is interacting. In principle we cannot exclude that the interacting characterof the theory could be recovered with the help of a non-conventional renormalization,however it seems to us that these results signal a weakness of the expansion when welimit ourselves to consider finite order approximations to the Green’s functions.Motivated by these observations, we looked for better approximations to the G n ’s ex-ploring the possibility of resumming classes of diagrams from all orders in ǫ . In particular,3esorting to the classification in terms of the physical cut-off Λ referred above, we startedby picking up the leading contributions (in terms of their dependence on Λ) from eachorder in ǫ . Interestingly we found that this resummation provides a bridge between theexpansion in ǫ and the leading order of the usual weak-coupling expansion in the couplingconstant g . Even more interestingly we found that, extending this criterion also to thenext-to-leading terms, it is possible to recover the results of the weak-coupling expansionalso at order g . Along the same lines, we have also indicated the resummations thatshould lead to higher order results in g .Besides being of great importance on their own, these results shade some light on the“strange” ultraviolet behaviour (no interaction) of the Green’s functions in the frameworkof this expansion: finite orders in ǫ seem to give a “too poor” truncation for the interactingterm (1) in the lagrangian.The rest of the paper is organized as follows. Section 2 is partially devoted to reviewingsome of the results presented in [35, 36]. After introducing the strategy to realize the ex-pansion in powers of ǫ , we will define a set of Feynman rules useful to give a diagrammaticrepresentation for the Green’s functions. We will then consider the O ( ǫ ) contributionsto the G n ’s with the help of the corresponding diagrammatic representation, and analysetheir behaviour in terms of the physical cut-off Λ. In Section 3 we will move to the O ( ǫ ),and classify the different contributions with respect to their Λ-dependence. In Section4 we will consider higher order contributions to the Green’s functions and again classifythem according to their ultraviolet behaviour. Section 5 is devoted to the resummationof the leading diagrams, while in Section 6 we will push forward the resummation to theorder g . Section 7 is for the conclusions. ǫ and leading order results The original idea presented in [35, 36] for solving a self-interacting scalar theory with inter-action of the kind φ p , with integer p , consists in considering the (Euclidean) Lagrangiandensity (in d -dimensions with ǫ > L = 12 ( ∇ φ ) + 12 m φ + 12 gµ φ Ä µ − d φ ä ǫ , (3)and putting forward a scheme to calculate the Green’s functions based on an expansionin the parameter ǫ , considered as small. In this sense, the parameter ǫ measures thedeparture from the linear theory. Since the interaction term is defined as φ ( φ ) ǫ , varying ǫ from zero to p − φ p . Note that this analytic continuation follows apath that preserves the parity symmetry, so that if for instance we consider ǫ = , theinteraction φ ( φ ) ǫ gives rise to φ | φ | rather than φ (that would have been obtainedfrom φ ( φ ) ǫ ). So doing, the different theories for different values of ǫ are uniquely andunambiguously defined.Clearly, once the Green’s functions are calculated to a given order in ǫ , the final goalis to send ǫ → p − ǫ → p − ǫ consists in writing the term( µ − d φ ) ǫ as e ǫ log ( µ − d φ ) and expanding the exponential in powers of ǫ , so that (3) isreplaced by the highly non-polynomial lagrangian: L = 12 ( ∇ φ ) + 12 m φ + 12 gµ φ ∞ X k =0 ǫ k k ! î log Ä µ − d φ äó k . (4)Successively, including the ǫ term in the “free lagrangian” L L ≡
12 ( ∇ φ ) + 12 Ä m + gµ ä φ , (5)the lagrangian L is written as L = L + L int ≡ L + ∞ X k =1 L k , (6)where L k = 12 gµ φ ǫ k k ! î log Ä µ − d φ äó k . (7)It is worth to stress that from now we will consider the theory as defined by thelagrangian (6) rather than by (3). This is however not the last step of the strategy. As wewill see other steps are needed to give a precise meaning to the expansion of the Green’sfunctions in powers of ǫ .The partition function Z and the n -points Green’s functions are given as usual [33]: Z = Z D φ e − R d d u L , (8) G n ( x , . . . , x n ) = 1 Z Z D φ φ ( x ) . . . φ ( x n ) e − R d d u L . (9)where from now on we consider even values of n for parity reasons.As mentioned above, the aim of the present section is to provide the building blocksfor the calculation of the connected Green’s functions to any given order in ǫ . To thisend, we consider e − S int = e − R d d u L int as a perturbation term, then expand it in Eqs. (8)and (9) and collect different powers of ǫ , thus getting: e − S int = 1 − Z d d u L + − Z d d u L + 12 Z d d u L Z d d w L + − Z d d u L + Z d d u L Z d d w L − Z d d u L Z d d w L Z d d z L + . . . (10)Following then the usual procedure, for the calculation of the Green’s functions (9) weinvert the order of the space and functional integrals.At this point, however, we note that all the “interaction terms” L k are of the kind φ log k ( µ − d φ ). In [37], and previously in [35, 36] in an implicit manner, it was suggested5hat each of these interactions can be rewritten in terms of powers of the field with thehelp of the formal identity µ − d φ log k ( µ − d φ ) = lim N → d k dN k µ (2 − d ) N φ N , (11)that is an adaptation of the replica trick introduced in [72]. The actual calculation of theGreen’s functions follows a procedure based on this formal replacement, that consists ofseveral steps.First of all the operators lim N → d k dN k , lim N → d k dN k , . . . have to be taken on theleft of the functional integrals (actually, we have one of these operators for each of the L k appearing in the products in (10)). Then, at a first stage the variables N i have to beconsidered as integers numbers, so that the calculation of the functional integrals is tracedback to an application of the Wick’s theorem. Subsequently, the resulting functions haveto be extended to real values of the N i ’s, and then the derivatives with respect to N i ’sand the limits N i → Z with its zeroth-order (in ǫ ) expression Z , andcalculate the functional integrals involved in the described procedure considering onlyconnected diagrams. From now on, we will refer to G n as to the connected Green’sfunctions.This whole procedure can be resumed by introducing a set of Feynman rules fromwhich we obtain the diagrams corresponding to the desired Green’s functions. To thisend we first introduce the “auxiliary” interaction lagrangian: L auxint = ∞ X k =1 λ k ( N ) φ N , (12)with λ k ( N ) ≡ gµ − d )( N − ǫ k k ! , (13)that gives rise to infinitely many 2 N -legs vertices, each related to the O ( ǫ k ) “coupling” λ k ( N ). This brings us to the definition of the vertices λ k
21 2 N ≡ − λ k ( N ) . (14)With the definitions given above the O ( ǫ m ) contribution to the n -points connectedGreen’s functions is obtained by considering first the connected diagrams that can bedrawn to that given order in ǫ with the vertices (14) (that amounts to perform theusual Wick’s contractions with the appropriate combinatorial factors), and then, beforeintegrating in the coordinates of the vertices, applying to each of these diagrams theoperator lim N → d k dN k (15)6or each λ k -vertex that appears.Naturally we should ask ourselves if with these prescriptions we recover the correctterms of the expansion in ǫ of the Green’s functions. Arguments in this sense are given in[35, 36], and for those cases for which the check could be performed the coincidence wasexplicitly shown.In the next subsection we consider the connected Green’s functions to O ( ǫ ) and drawthe corresponding Feynman diagrams according to the Feynman rules given above. O ( ǫ ) The systematic expansion of the Green’s functions in the non-linearity parameter ǫ startswith the contributions of order ǫ and ǫ , that were already calculated in [35, 36]. Themain goal of the present subsection is to derive these results in terms of Feynman diagramsobtained with the rules introduced above. We will also express the O ( ǫ ) contribution to theGreen’s functions in terms of O ( ǫ ) “effective vertices”, to be later defined. The usefulnessof these definitions will be more apparent when moving to higher order diagrams.Let us begin by noting that, since the interaction Lagrangian contains terms that areat least O ( ǫ ), the two-points function G ( x , x ) is the only connected Green’s functionthat has an order ǫ contribution, that we denote with G (0)2 ( x , x ). For the free lagrangiangiven in Eq. (5), this is nothing but the Feynman propagator (from now on M = m + gµ ) G (0)2 ( x , x ) = ∆( x − x ) = Z d d p (2 π ) d p + M e − ip ( x − x ) . (16)Moving to O ( ǫ ), we see that at this order we need to consider only the vertex λ , sothat the O ( ǫ ) contribution to the generic n -points Green’s function is: G (1) n ( x , . . . , x n ) = Z d d u lim N → ddN ® − λ ( N ) 1 Z Å Z D φ e − S φ ( x ) . . . φ ( x n ) φ ( u ) N ã C ´ (17)where C in the path integral is for connected and where, as explained before, the derivativeand the limit with respect to N will be properly defined when the analytic extension toreal values of N near N = 1 will be considered. Focusing now on the path integral inthe curly brackets, we note that for integers N < n a vanishing contribution to G (1) n isobtained (there are no connected diagrams in this case), while for N ≥ n we get − λ ( N ) Z Å Z D φ e − S φ ( x ) . . . φ ( x n ) φ ( u ) N ã C = λ x x n x N − n = − λ ( N ) [∆(0)] N − n C n ( N ) n Y i =1 ∆( x i − u ) , (18)where C n ( N ) is the combinatorial factor coming from the contractions: C n ( N ) = [2 N (2 N − . . . (2 N − n + 1)] (2 N − n − n factors related to the contractions of n of the 2 N fields of the vertex with n fields in different spacetime points, while the double factorial comes from the N − n self-loops obtained from the leftover fields. Putting together all the odd factors, we canrewrite (19) as: C n ( N ) = 2 n ï N ( N − . . . ( N − n ò (2 N − n ( N ) n (2 N − x ) m to denote the falling factorial, defined as( x ) m = x ( x − . . . ( x − m + 1) ∀ x ∈ R , m ∈ N + (21)for positive integer values of m and extended to ( x ) = 1 for the case m = 0 . From (21)we see that C n ( N ) vanishes for integers N < n . For this reason, the expression in the lastmember of (18) gives the correct result for the path integral also when N < n .This observation is crucial to the procedure. In a sense, we aim at a “definition”of the path integral in (18) even for real values of N , and in particular around N =1, where the derivative with respect to N has to be calculated. This is achieved byconsidering the analytic extension of the last member of (18), more precisely of the factor − λ ( N ) [∆(0)] N − n C n ( N ) . It is worth to stress that, while this procedure provides theanalytically extended function of real variable N even for N < n , the latter keeps memoryof the fact that the original path integral in (18) is actually performed for integer valuesof N with N ≥ n , and this is why the factor [∆(0)] N − n appears.Once this analytic extension is performed, the O ( ǫ ) contribution to the n -point Green’sfunction can be expressed in terms of an n -legs “effective vertex” defined asΠ (1) n = 121 n ≡ lim N → ddN n − λ ( N ) [∆(0)] N − n C n ( N ) o (22)where the superscript (1) indicates that Π (1) n is O ( ǫ ). With this definition: G (1) n ( x , . . . , x n ) = Π (1) n Z d d u n Y i =1 ∆( x i − u ) . (23)We are then left with the calculation of the Π (1) n ’s. Replacing in (22) the combinatorialfactor C n ( N ) in (20) and the coupling λ ( N ) in (13), we obtain:Π (1) n = lim N → ddN ï − ǫ g µ − d )( N − [∆(0)] N − n n ( N ) n (2 N − ò = − ǫ g µ ñ ô n − lim N → ddN f n ( N ) (24)where f n ( N ) = ( N ) n (2 N − î µ − d ∆(0) ó N − , (25)and is the function that need to be analytically extended. The falling factorial ( N ) n isjust a polynomial in N and as such it is an entire function of N . On the contrary, the8emifactorial (2 N − N . With the help of the relation(2 N − N − N + )Γ( ) [73] valid for integers N , we can finally define the analyticextension of f n ( N ) to real values of its variable as f n ( x ) = ( x ) n Γ( x + )Γ( ) î µ − d ∆(0) ó x − , (26)that is analytic around x = 1. For simplicity in the following we continue to indicate thereal variable as N . We are now in the position to take the derivative and the limit.Due to the presence of the factor ( N −
1) in the falling factorial, the derivative of f n ( N )with respect to N has a simple behaviour in the limit N → n = 4 , , . . . ,while the case n = 2 has to be treated separately. Let us begin with the latter. Case n = 2 . We have f ( N ) = N Γ Ä N + ä Γ Ä ä î µ − d ∆(0) ó N − , (27)so that lim N → ddN f ( N ) = log î µ − d ∆(0) ó + 1 + Γ ′ ( )Γ( ) ≡ K , (28)and the 2-points effective vertex becomesΠ (1)2 = 1 = − ǫ g µ K . (29)
Case n > . As stressed above, in these cases the falling factorial contains the factor( N − N → N −
1) is non-vanishing. Using the symbolic expression x ( m ) for the rising factorial x ( m ) = x ( x + 1) . . . ( x + m − x ) m = ( − m ( − x ) ( m ) , we can rewritethe function f n as f n ( N ) = N ( N − − n − (2 − N ) ( n − Γ Ä N + ä Γ Ä ä î µ − d ∆(0) ó N − . (30)Being 1 ( m ) = m !, we easily obtain:lim N → ddN f n ( N ) = ( − n − Å n − ã ! (31)that in turn gives for the n -legs effective vertex ( n > (1) n = 121 n = ( − n − Å n − ã ! ǫ g µ ñ ô n − . (32)We can now write the Green’s functions up to O ( ǫ ) in a convenient form.9 wo-points Green’s function G . Up to order ǫ we have G ( x , x ) = ∆( x − x ) + Π (1)2 Z d d u ∆( x − u )∆( x − u ) . (33)Going to Fourier space, and using (29) for Π (1)2 , for ‹ G ( p ) we obtain ‹ G ( p ) ≡ = + 1= 1 p + M − p + M ǫ g µ K p + M . (34)Eq. (34) has a resemblance with the corresponding 2-point Green’s function of the ordi-nary weak-coupling expansion at O ( g ). The O ( ǫ ) term is the free propagator (as thecorresponding O ( g ) term in the weak-coupling expansion), while the O ( ǫ ) term containsthe radiative corrections, here carried by the residual loop factor K of the effective vertexΠ (1)2 .Noting that the bubble diagram in Eq. (34) is the lowest 1PI self-energy diagram withinthe ǫ -expansion, we can get the first approximation to the self-energy by resumming thefollowing geometric series:= + 1 + 1 1 + . . . (35)getting ‹ G ( p ) = 1 p + M + ǫgµ K (36)from which the radiatively corrected mass turns out to be: m R = M + ǫgµ ( log î µ − d ∆(0) ó + 1 + Γ ′ ( )Γ( ) ) . (37)Few comments are in order. We know that the loop integral ∆(0), which is nothing but∆(0) = Z d d p (2 π ) d p + M , (38)is a divergent quantity when d ≥
2. In these cases, we need to regularize this divergence.In this respect we note that, with the exception of the Theory of Everything, any quantumfield theory is an effective theory. From the physical point of view this means that we canencode our ignorance on physics above a given scale Λ by introducing this scale in thetheory as a physical momentum cut-off when summing the quantum fluctuations. Havingthis in mind, we regularize ∆(0) with the help of a momentum cut-off Λ.To be more specific, it is worth at this point to focus on the d = 4 case, where∆(0) = 116 π Ç Λ − M log Λ M å + O (Λ − ) . (39)Replacing this result in (37), we obtain m R = M + ǫgµ ( log Λ µ + 1 + Γ ′ ( )Γ( ) − log(8 π ) ) + O (Λ − ) , (40)10rom which we see that the radiative correction to the mass diverges as log Λ, irrespectivelyof the value of ǫ , i.e. irrespectively of the degree of the self-interaction.Considering the λ φ theory, that corresponds to the case ǫ = 1 (with the replacement g = λ ), it is worth to remind that within the framework of the weak coupling expansionfor the radiatively corrected mass at O ( g ) we have m R = m + λ π Ç Λ − m log Λ m å + O (Λ − ) . (41)Eq. (40) is greatly intriguing. As compared with Eq. (41), it seems to suggest that withinthe ǫ -expansion the unnatural quadratically divergent correction to the scalar mass ishealed, as the correction diverges only logarithmically. If confirmed at higher orders, sucha result could be of great interest for the naturalness/hierarchy problem. In the followingwe will further investigate this question. Green’s functions G n for n > . As mentioned above, in these cases there is no order ǫ term. As for the O ( ǫ ) contribution to G n , replacing in Eq. (23) the vertex factor (32)we obtain: G (1) n ( x , . . . , x n ) = ( − n − ǫgµ ñ ô n − Å n − ã ! Z d d u n Y i =1 ∆( x i − u ) , (42)that in momentum space is: ‹ G n ( p , . . . , p n ) = ( − n − Å n − ã ! ǫgµ ñ ô n − n Y i =1 p i + M , (43)with the external momenta conservation to be imposed.It is worth to compare the result (43) with the corresponding one in weak couplingexpansion. While in the latter case for an interaction of the kind φ p there are non-vanishing Green’s functions at first order only for n ≤ p , Eq. (43) apparently shows anon-zero result at O ( ǫ ) for any value of n .However, sticking again to the d = 4 case, from (43) we see that all the amputatedGreen’s functions seem to vanish in the limit Λ → ∞ as inverse powers of the cut-off,again irrespectively of the value of ǫ : ‹ G n ( p , . . . , p n ) ∼ n − . (44)This would mean that for d = 4 all the Green’s functions starting from n = 4 (and thenall the scattering amplitudes) vanish in the limit Λ → ∞ , thus suggesting (at least atthis order in ǫ ) that the theory is non-interacting. Similar results (with different degreesof vanishing with Λ) hold for any d ≥
2. If confirmed at higher orders, this result wouldhave serious consequences for the theory. One of the scopes of the present paper is tofurther investigate this issue.Moreover Eq. (44) shows that the ‹ G n ’s are suppressed by higher inverse powers of thecut-off as n increases. The reason is intrinsic to the procedure outlined above. In fact,before proceeding to the analytic extension, the path integral in (18) was calculated byconsidering sufficiently large values of the integer N ( N > n ), and the limit N → − n that givesthe suppressed behaviour shown in (44).In passing we note that a possible way out from this apparent non-interacting characterof the theory would be to consider unconventional renormalization patterns that couldmake the scattering amplitudes finite and non-vanishing. If we again consider the d = 4case and focus our attention on the four-points Green’s function, from Eq. (43) we see thatthe only way to keep G finite (non-vanishing) is to require for the coupling constant g abehaviour with Λ such that g ∼ Λ . From Eq. (40) we then see that the radiative correction δm to the mass is no longer proportional to log Λ, but rather goes as δm ∼ Λ log Λ,so that the original hope of having a less severe naturalness problem is immediatelydisattended. Moreover we note that, even imposing such a behaviour with Λ to thecoupling constant g , from (44) we see that the Green’s functions starting from n = 6would still vanish.In order to address the two questions posed in this section, namely the degree ofdivergence of the radiative correction to the mass (hierarchy problem) and the vanishingof the Green’s functions with n >
2, we have to move to higher orders in ǫ . In the followingsection we begin with the O ( ǫ ). O ( ǫ ) In this section we push forward the analysis to second order in ǫ . Even though the two-and four-points Green’s functions have already been considered in [71], here we performthe calculations for all the G n , showing that in the general case there are delicate problemsto be taken into account.Going back to Eq. (10) we see that the O ( ǫ ) contributions to the Green’s functions,that we denote with G (2) n ( x , . . . , x n ), come from the third and fourth terms of this equa-tion. This can be read in terms of the Feynman rules for the auxiliary vertices definedin Eq. (13): for each of the Green’s function we have two O ( ǫ ) classes of diagrams, onewith a single vertex λ and one with two vertices λ . Every vertex carries a power of thecoupling constant g . Therefore, disregarding the trivial dependence on g carried by themass M in the factored out propagators, the first class of diagrams is O ( ǫ g ) while thesecond one is O ( ǫ g ). Let us analyse separately these two contributions, that we indicaterespectively with G (2 ,g ) n and G (2 ,g ) n . ǫ g and ǫ g contributions The order ǫ g . The contribution at this order comes from the vertex λ ( N ) and is: G (2 ,g ) n ( x , . . . , x n ) = Z d d u lim N → d dN ® − λ ( N ) 1 Z Å Z D φ e − S φ ( x ) . . . φ ( x n ) φ ( u ) N ã C ´
12 lim N → d dN λ x x n x N − n = lim N → d dN n − λ ( N ) [∆(0)] N − n C n ( N ) o Z d d u n Y i =1 ∆( x i − u ) . (45)As can be easily verified, the diagram in (45) is practically the same as the one in (18),with the only difference that the O ( ǫ ) vertex λ (and correspondingly the second deriva-tive operator) appears rather than the O ( ǫ ) vertex λ . As for the self-loops and thecombinatorial factor, they are exactly the same. Similarly to what we have done at O ( ǫ ),we now define the n -legs O ( ǫ ) effective vertex Π (2) n as:Π (2) n = 2 n ≡ lim N → d dN n − λ ( N ) [∆(0)] N − n C n ( N ) o . (46)Replacing in (46) the expressions for λ ( N ) and C n ( N ) in Eqs. (13) and (20), we have:Π (2) n = − ǫ g µ ñ ô n − lim N → d dN f n ( N ) , (47)where the function f n ( N ) is defined in Eq. (25) for integers N , and its analytic extensionis given in (26). Due to the presence of the factor ( N − n = 2 and n >
2. Performing the calculations, we get: n = 2 Π (2)2 = − ǫ g µ ñ K − ψ ′ Ç åô (48) n > (2) n = ( − n − Å n − ã ! ǫ gµ ñ ô n − Ä K − H n − ä (49)where K is defined in Eq. (28), while H n stands for the n -th Harmonic number. The order ǫ g . The calculation of this term is somehow more complicated than theprevious one. Due to the presence of two vertices λ , the number of different possiblediagrams grows enormously, and there are subtleties that need to be carefully treated.From (10), the contribution to G n to this order is: G (2 ,g ) n ( x , . . . , x n ) = 12 Z d d u d d w lim N → lim M → ddN ddM ( − λ ( N )) ( − λ ( M )) × Z Å Z D φ e − S φ ( x ) . . . φ ( x n ) φ ( u ) N φ ( w ) M ã C . (50)In order to list the different diagrams that can be built, we can begin by consideringthe contractions of the n “external” fields φ ( x ) , . . . , φ ( x n ) with the fields φ ( u ) in the13oint u . We can connect r , with 0 ≤ r ≤ n , of them with φ ( u ), leaving the remaining n − r fields for contractions with φ ( w ). For each of these choices, we still have the freedomto use the leftover fields φ ( u ) and φ ( w ) to link the two vertices u and w with as manylinks as we like, with the upper limit posed by the number of available fields, that in turndepends on the values of N and M . Moreover, the number of u − w links must have thesame parity of r , so that all the remaining fields at each vertex are contracted in pairs,giving rise to self-loops. For this reason, the even ( r = 2 j ) and odd ( r = 2 j + 1) caseshave to be treated separately. “Even” contribution. As for the O ( ǫ ) case, the path integral in (50) gives non van-ishing results only for sufficiently large integers N and M , namely N > j and
M > n − j ,so that connected diagrams can be drawn. In this case, indicating with G (2 ,g ) n, E the sum ofall the diagrams of this kind , we have: G (2 ,g ) n, E = 12 lim N → M → ddN ddM n/ X j =0 Min X l =1 λ u λ w x x j x j +1 x n l N − j − l M − n + j − l + n j ! − = 12 Z d d u d d w lim N → M → ddN ddM n/ X j =0 Min X l =1 î − λ ( N )∆(0) N − j − l ó î − λ ( M )∆(0) M − n + j − l ó × C n,j,l ( M, N ) j Y i =1 ∆( x i − u ) n Y h =2 j +1 ∆( x h − w ) ∆( u − w ) l + n j ! − . (51)where the coefficient C n,j,l ( M, N ) contains all the combinatorial factors, the Ä n j ä permuta-tions corresponds to permutations of external points in the diagram, and the upper limitof the l -sum is Min ≡ min ( N − j, M − n + j ) . Rigorously speaking, Eq. (51) takes aprecise mathematical meaning only for real values of the variables N and M , with thethe functions of N and M in (51) being analytic around ( N = 1 , M = 1). This meansthat we have to search for the analytic extensions of these functions. This is what weare going to do, but before reaching that point let us continue to consider N and M asintegers, and stick on the above mentioned conditions N > j and
M > n − j .We now take out from the limits and the derivatives in (51) those factors that do notdepend on N and M , thus getting: G (2 ,g ) n, E = 12 Z d d u d d w n/ X j =0 j Y i =1 ∆( x i − u ) n Y h =2 j +1 ∆( x h − w ) lim N → M → ddN ddM Min X l =1 C n,j,l ( M, N ) × î − λ ( N )∆(0) N − j − l ó î − λ ( M )∆(0) M − n + j − l ó ∆( u − w ) l + n j ! − . , (52) As explained above, the two vertices λ are connected with an even number 2 l of lines. As we startwith l = 1, the minimal number of internal lines is 2, while the maximal number is 2 Min. In (51) this isindicated by connecting these two vertices with two continuous lines and two dashed ones that representthe sum over l . M and N .Let us consider first the combinatorial factor C n,j,l ( M, N ). It is: C n,j,l ( M, N ) = 2 N (2 N − . . . (2 N − j − l + 1) (2 N − j − l − × M (2 M − . . . (2 M − n + 2 j − l + 1) (2 M − n + 2 j − l − l )!= 1(2 l )! C j +2 l ( N ) C n − j +2 l ( M ) . (53)The last line of (53) is obtained by noting that each of the two factors in the second memberis of the same form of the combinatorial coefficients C n in Eq. (19). Such a factorizationis quite remarkable, and can be understood by looking at the diagram in (51). Actually,the combinatorial factor C n,j,l ( M, N ) takes into account:1. the contractions of 2 j + 2 l (out of 2 N ) φ ( u ) fields with fields at other space-timepoints: this is the definition of C j +2 l ( N );2. the contractions of n − j + 2 l (out of 2 M ) φ ( w ) fields with fields at other space-timepoints: this is the definition of C n − j +2 l ( M );3. the fact that the 2 l contractions made between fields φ ( u ) and φ ( w ) are indistin-guishable: the factor l )! cures the over-counting of these permutations.Thanks to this factorization, the sum over l can then be written as: Min X l =1 î − λ ( N ) C j +2 l ( N )∆(0) N − j − l ó î − λ ( M ) C n − j +2 l ( M )∆(0) M − n + j − l ó ∆( u − w ) l (2 l )! . (54)Replacing in (54) the expression for λ given in (13), together with the combinatorialfactors in (20), we have: Min X l =1 " − ǫgµ ñ ô j + l − f j +2 l ( N ) − ǫgµ ñ ô n − j + l − f n − j +2 l ( M ) ∆( u − w ) l (2 l )! , (55)where we used the functions f m defined in (25) as: f m ( N ) = ( N ) m (2 N − î µ − d ∆(0) ó N − . (56)From (56) we see that, as the only dependence on the parameter m is contained in thefalling factorial ( N ) m , using the property( x ) a + b = ( x ) a ( x − a ) b ∀ a, b ∈ Z with a, b ≥ , ∀ x ∈ R , (57)we can split the two falling factorials in f j +2 l ( N ) and f n − j +2 l ( M ) as( N ) j + l = ( N ) j ( N − j ) l ; ( M ) n − j + l = ( M ) n − j Å M − n j ã l
15o that: f j +2 l ( N ) = f j ( N ) ( N − j ) l ; f n − j +2 l ( M ) = f n − j ( M ) Å M − n j ã l . (58)Replacing Eqs. (58) in (55), we can take out from the sum the l -independent terms, thusgetting: " − ǫgµ ñ ô j − f j ( N ) − ǫgµ ñ ô n − j − f n − j ( M ) × Min X l =1 ñ ô l ∆( u − w ) l (2 l )! ( N − j ) l Å M − n j ã l . (59)Using the identity (2 l )! = (2 l )!!(2 l − l l ! 2 l Ä ä ( l ) , and turning the falling factorialsto rising factorials through the relation ( x ) m = ( − m ( − x ) ( m ) , Eq. (59) becomes: " − ǫgµ ñ ô j − f j ( N ) − ǫgµ ñ ô n − j − f n − j ( M ) × Min X l =1 ( j − N ) ( l ) Ä n − j − M ä ( l ) Ä ä ( l ) l ! ñ ∆( u − w )∆(0) ô l . (60)As Min= min Ä N − j, M − n + j ä , and we are still considering N and M as integers,the sum over l can be written as Min X l =1 ( j − N ) ( l ) Ä n − j − M ä ( l ) Ä ä ( l ) l ! ñ ∆( u − w )∆(0) ô l = F j − N, n − j − M ; 12 ; ñ ∆( u − w )∆(0) ô ! − , (61)where the function F ( a, b ; c ; z ) is the Gaussian hypergeometric function defined by F ( a, b ; c ; z ) = ∞ X l =0 ( a ) ( l ) ( b ) ( l ) ( c ) ( l ) z l l ! ∀ z ∈ C with | z | < , (62)which in (61) is truncated to a polynomial in z = h ∆( u − w )∆(0) i , as the sum over l runs up toMin due to the vanishing of the rising factorials in (61) for l > M in . In passing we notethat in this case the condition | z | < " − ǫgµ ñ ô j − f j ( N ) − ǫgµ ñ ô n − j − f n − j ( M ) × " F j − N, n − j − M ; 12 ; ñ ∆( u − w )∆(0) ô ! − . (63)We got the expression we were looking for. Although Eq. (63) was obtained for N and M integers such that N > j and
M > n − j , we would like to use this expression to get16n analytic extension of the “even contribution” to the path integral in (50) to real valuesof N and M close to N = 1 and M = 1.To this end we begin by noting that, the functions f j ( N ) and f n − j ( M ) respectivelyvanish for N < j and
M < n − j while for N = j and/or M = n − j the factor F − f m ( N ) defined in (56) can be easily extended toreal values of N , as already shown in Section 2 (see Eq.(26)).Let us consider next the extension of the function F to real N and M . This extensionhas to be done for any couple of points ( u, w ) as these are the variables on which we haveto integrate (see Eq.(52)). For a generic couple ( u, w ) such that u = w the condition | ∆( u − w )∆(0) | < u = w we have | ∆( u − w )∆(0) | = 1. Let usbegin by considering the case u = w , leaving aside for the moment the case u = w .Under this condition ( u = w ), we can analytically extend F in (63) to real values of N and M with the help of the definition (62). Moreover, as for any fixed couple of values( z, c ) the function F ( a, b ; c ; z ) is an entire function of a and b , the derivatives and thelimits with respect to N and M can be performed term by term in the series.Inserting in (63) the left-hand side of (61), with Min sent to infinity, appropriatelyrecombining the rising factorials of (61) with the functions f j ( N ) and f n − j ( M ) makinguse of Eqs. (58), and then considering the derivatives and the limits of (63) we get: ∞ X l =1 lim N → ddN " − ǫgµ ñ ô j + l − f j +2 l ( N ) × lim M → ddM " − ǫgµ ñ ô n − j + l − f n − j +2 l ( M ) ∆( u − w ) l (2 l )! . (64)Next we have to consider the case u = w , that corresponds to z = | ∆( u − w )∆(0) | = 1. Thiscase is more delicate as z = 1 lies on the border of the convergence circle of the series(62). However the convergence of this power series is guaranteed even for z = 1 underthe further condition Re ( c − a − b ) > , (65)that in our case becomes N + M + 12 − n > . (66)Once again we note that, as long as N > j and
M > n − j this condition is fulfilled.However we need to consider the analytic extension around the point ( N = 1 , M = 1),and in this region the condition (66) is not fulfilled starting from n = 6. As a consequence,for n ≥ u = w .However we have to stress that, even though for n = 2 and n = 4 the series (64)converges in the whole domain of space-time integration, this does not mean that the finalexpression after integration is convergent. Actually we have checked (even numerically)that there are two distinct possible sources of divergences: (i) the series involved in theanalytic extension could diverge in the ultraviolet regime u → w (that is the case weencountered for n > u → w , the whole17xpression could be ultraviolet divergent once the space-time integration is performed.This is due to the fact that the resummed function, although everywhere finite, couldbe non-integrable or to the absence of dominated convergence that make it impossible toperform the integration by series.These are genuinely novel features of the O ( ǫ ) contributions to the Green’s functions.At the order ǫ considered in the previous section, such a problem did not show up as, inthat case, the functions to be analytically extended were factorized out of the space-timeintegral. Here, instead, we had to cope with analytic extensions of functions involvedin space-time integrals that are related to the connection of two vertices at differentpoints, and such analytic extension gave rise to hypergeometric infinite series that bringdivergences in the ultraviolet regime.These divergences pose delicate problems and deserve further investigation. One wayto take care of them is through the introduction of a cut-off L max in the power series,and this will allow to treat together the cases u = w and u = w . Once the infinite upperlimit is replaced with L max , Eq.(64) can be extended to all the space-time domain ofintegration. We have not further studied this problem, but we are rather assuming thatthis is a suitable regularization and this is the implementation we will follow below.Under this assumption, we can finally write for G (2 ,g ) n, E G (2 ,g ) n, E = 12 n/ X j =0 L max X l =1 lim N → ddN " − ǫgµ ñ ô j + l − f j +2 l ( N ) × lim M → ddM " − ǫgµ ñ ô n − j + l − f n − j +2 l ( M ) l )! × Z d d u d d w j Y i =1 ∆( x i − u ) n Y h =2 j +1 ∆( x h − w ) ∆( u − w ) l + n j ! − . . (67)Eq.(67) is a very welcome result as each of the factorized limits in the sum over l isnothing but an effective vertex as given in (24). For G (2 ,g ) n, E we finally have the elegantand compact result: G (2 ,g ) n, E = 12 n/ X j =0 L max X l =1 Π (1)2 j +2 l Π (1) n − j +2 l l )! × Z d d u d d w j Y i =1 ∆( x i − u ) n Y h =2 j +1 ∆( x h − w ) ∆( u − w ) l + n j ! − . . (68)With the help of the diagrammatic representation of the effective vertices Π (1) n givenin (32), we can write (68) as: 18 (2 ,g ) n, E = 12 n X j =0 L max X l =1 u w x x j x j +1 x n l + n j ! − . (69)Note that the factor l )! of (68) is recovered once the indistinguishability of the 2 l linesconnecting the effective vertices Π (1) in u and w is taken into account. It is worth tonote that l )! is the only genuine factorial term that is left in G (2 ,g ) n, E , the other factorialterms appearing in the intermediate steps of our derivation have been replaced by analyticcontinuations and then absorbed in the definition of the Π (1) ’s.Finally we observe that, by keeping the points u and w attached to the vertices, weare considering them as two distinguishable points. Actually, as u and w are interchange-able, every pair of diagrams corresponding to j = i and j = n/ − i yields to the samecontribution because they are equal under the exchange u ⇐⇒ w . Therefore, if we onlyconsider distinct diagrams under the exchange of vertices, we can cancel the prefactor in the right-hand side of (69). However, for generic values of n , it is not always true that n is an integer number, and this is why we have to keep the factor in our expression.Finally when j = n/ “Odd” contribution. By considering now the case where the two vertices are joinedwith an odd number of lines, we have : G (2 ,g ) n, O = 12 lim N → M → ddN ddM n/ − X j =0 Min X l =0 λ u λ w x x j +1 x j +2 x n l + 1 N − j − l − M − n + j − l + Ç n j +1 å − . (70)Performing similar steps to those done for the even case, the contribution G (2 ,g ) n, O to theGreen’s functions G (2 ,g ) n turns out to be: G (2 ,g ) n, O = 12 n − X j =0 L max X l =0 Π (1)2 j +2 l +2 Π (1) n − j +2 l l + 1)! ×× Z d d u d d w j +1 Y i =1 ∆( x i − u ) n Y h =2 j +2 ∆( x h − w ) ∆( u − w ) l +1 + n j +1 ! − . As compared to the diagrams appearing in the “even” contribution (Eq. (51)), in (70) we have drawnonly one continuous internal line, as in this case we have an odd number, namely 2 l + 1, of internal lines,starting from l = 0.
19 12 n − X j =0 L max X l =0 u w x x j +1 x j +2 x n l + 1 + n j +1 ! − , (71)where L max is introduced for the same reasons encountered in the even case.From Eqs. (69) and (71) we see that the O ( ǫ ) diagrams coming from two vertices λ are easily expressed in terms of the effective vertices Π (1) n of O ( ǫ ) . This allows to writeall the O ( ǫ g ) contributions to the n -points Green’s functions without referring to theoriginal Feynman rules given in Section 2. What we need to do, instead, is to draw thediagrams with two effective vertices of the type Π (1) , according to the following simplerules:1. Draw all the different diagrams with n external points and two “effective vertices”Π (1) that fulfil the following conditions:(a) r (with 0 ≤ r ≤ n ) of the external points have to be attached at one of thetwo vertices, with the remaining points are attached to the other one;(b) the two vertices have to be linked by a number of internal lines that is limitedby the cut-off L max , and this number must have the same parity of r (we haveat our disposal only vertices with an even number of legs);2. Attach to each vertex with 2 i legs the effective vertex function Π (1)2 i ;3. Associate a propagator to each line, integrating over the internal spacetime points;4. Associate the combinatorial factor m ! if the two vertices are connected m times ;5. For any r with 0 ≤ r ≤ n −
1, consider the Ä nr ä permutations of external points,each corresponding to a different channel. For r = n/ Ä nn/ ä . Let us apply the above results to the two-points Green’s function G .Collecting all the O ( ǫ ) contributions we have: As we are considering only distinct diagrams, the factor in (69) and (71) is cancelled by an exchangefactor 2 for the interchange of the vertices u ⇐⇒ w , as explained in the text above. (2)2 ( x , x ) = 2 x x + L max X l ≥ x x l + 1 + L max X l ≥ x x l (72)where 2 x x = Π (2)2 Z d d u ∆( x − u )∆( x − u ) (73)1 1 x x l + 1 = Π (1)2+2 l Π (1)2+2 l (2 l + 1)! Z d d u d d w ∆( x − u )∆( u − w ) l +1 ∆( x − w )(74)11 x x l = Π (1)2+2 l Π (1)2 l (2 l )! Z d d u d d w ∆( x − u )∆( x − u )∆( u − w ) l (75)Going to momentum space, the O ( ǫ ) contribution to ‹ G ( p ) is then: ‹ G (2)2 ( p ) = 2 + 1 1+ X l ≥ l + 1 + X l ≥ l (76)where 2 = 1 p + M Π (2)2 p + M (77)1 1 = 1 p + M Π (1)2 p + M Π (1)2 p + M (78)1 1 l + 1 = 1 p + M ñ Π (1)2+2 l Π (1)2+2 l I l +1 ( p )(2 l + 1)! ô p + M ( l ≥
1) (79)11 l = 1 p + M ñ Π (1)2+2 l Π (1)2 l I l (0)(2 l )! ô p + M ( l ≥
1) (80)21nd I r ( k ) indicates the multi-loop integral I r ( k ) = Z d d p (2 π ) d . . . d d p r − (2 π ) d p + M . . . p r − + M k − P r − i =1 p i ) + M (81)where r ≥ k is the total momentumcarried in the loop.It is worth to note that the second term in the right-hand side of (76) (i.e. thediagram in (78)) would correspond to the l = 0 contribution in the sum that appears inthe third term. We wrote it separately as it is a reducible diagram, actually the doubleiteration of the 1PI diagram previously found at O ( ǫ ) (see Eq.(34)). Finally we observethat in Eqs. (77)-(80) the amputated diagrams are obtained once the propagators p + M are removed from their extreme left and right sides. Collecting as before all the O ( ǫ ) contributions we get: G (2)4 ( x , . . . , x ) = x x x x + L max X l ≥ x x x x l + L max X l ≥ x x x x l + 1 + 3 perm. + L max X l ≥ x x x x l + 2 perm. (82)Going to momentum space, and writing directly the amputated diagrams we have:22 G (2)4 , amp ( p , . . . , p ) = + + 3 perm. + L max X l ≥ l + 1 + 3 perm. + L max X l ≥ l + L max X l ≥ l + 2 perm. (83)where = Π (2)4 (84) = Π (1)2 p + M Π (1)4 (85) l + 1 = Π (1)2+2 l Π (1)4+2 l I l ( p ) (86) l = Π (1)4+2 l Π (1)2 l I l (0) (87) l = Π (1)2+2 l Π (1)2+2 l I l ( q ) (88)where the momentum p carried in the propagator in (85) and in the loop integral in (86)is one of the external momenta p , p , p or p , depending on the specific permutationthat we are considering, while the momentum q in (88) is p + p , p + p or p + p ,depending on the channel s , t or u considered.23s above we observe that the third term in the right-hand side of (83) (i.e. the diagramin (85)) would correspond to the l = 0 contribution in the sum that appears in the fourthterm of the same equation. As before, we have not included it in that sum as it is areducible diagram, where one of the external lines (propagators) is corrected by the O ( ǫ )bubble (effective vertex) Π (1)2 (see Eq.(29)). d = 4 In order to get a better insight on the cut-off dependence of the radiative corrections tothe tree level propagator G (0)2 ( x , x ) = ∆( x − x ), it is worth to consider the familiar d = 4 case. The four-points Green’s function G will be also considered, and the resultsof this section will be useful for our later purposes.First of all we note that in the contribution of order O ( ǫ g ), that is in the diagram (77),the dependence on the cut-off is contained in the effective vertex Π (2)2 given in Eq. (48),whose leading divergence, due to the factor log(∆(0)) present in K (defined in (28)), is:Π (2)2 ≡ − ǫ g µ ñ K − ψ ′ Ç åô ∼ (log Λ) . (89)Moving to O ( ǫ g ), we note that the reducible diagram in (78) clearly diverges as(log Λ) , as it is the square of the 1PI O ( ǫ ) diagram that, as we know, diverges as log Λ1 1 ∼ (log Λ) . (90)Concerning all the other contributions, the dependence on Λ is contained in the twoeffective vertices and in the loop integrals. For the Π (1) ’s we know the exact cut-offdependence, that is given by the ∆(0) factors:Π (1)2 ≡ − ǫ g µ K ∼ log(Λ) (91)Π (1) n ≡ n = ( − n − n − Å n − ã ! ǫ g µ ∆(0) n − ∼ n − ( n ≥ . (92)As for the loop integrals, although we do not know their exact regularized expressions,we can calculate their superficial degree of divergence, thus getting: I ( k ) ∼ log(Λ) (93) I r ( k ) ∼ Λ r − ( r >
2) (94)Collecting the above results, the cut-off dependence of the amputated diagrams cor-responding to the second line of (76), that are the only O ( ǫ g ) 1PI diagrams, turns outto be 241 ∼ (log(Λ)) Λ (95)11 l ∼ ∀ l ≥ l + 1 ∼ ∀ l ≥ O ( g ) diagram (89). The only other O ( ǫ ) diagram that is of the sameorder in Λ is the 1P reducible diagram (90). We note that the degree of divergence of the O ( ǫ ) contribution is higher than the one of O ( ǫ ): (log(Λ)) as compared to log(Λ).These observations will be useful when we will later consider resummations of dia-grams.Similar considerations can be made for the G n ’s with higher n . In particular weconcentrate on the four-point Green’s function G . First of all we notice that in thecontribution of order O ( ǫ g ), that is the diagram (84), the dependence on the cut-off iscontained in the effective vertex Π (2)4 , and we haveΠ (2)4 ≡ = − ǫ gµ K ∆(0) ∼ log(Λ)Λ . (98)Moving to the O ( ǫ g ), the amputated 1P reducible diagram (85) has the cut-offdependence ∼ log(Λ)Λ , (99)being the product of the two subdiagrams Π (1)2 and Π (1)4 (see Eqs. (91) and (92)).Concerning all the other contributions, the dependence on Λ is contained in the twoeffective vertices and in the loop integrals. Making use of Eqs. (91)-(94), their cut-offdependence turns out to be 25 l + 1 ∼ ∀ l ≥ ∼ (log(Λ)) Λ (101) l ∼ ∀ l ≥ ∼ log(Λ)Λ (103) l ∼ ∀ l ≥ O ( g ) diagram (98). As compared to the O ( ǫ ) result (seeEq. (44) for n = 4), the cut-off dependence of this O ( ǫ ) contribution to the radiativecorrection to G is enhanced (less vanishing) with respect to the lower order result: log ΛΛ as compared to .As stressed above for G , these observations will be crucial for our later developments,where we will be interested in resummations of selected classes of diagrams at each orderin ǫ . To this end we need to gain some knowledge on the higher orders contributions tothe G n . The next Section is devoted to derive some general results in this direction. The next step of the expansion in ǫ consists in considering the O ( ǫ ) contribution. Start-ing from this order, however, the procedure outlined in the previous sections presentsincreasing difficulties, and at present it is not clear to us how to overcome these problemsfor the most general case. Nevertheless, a restricted but important subclass of diagramscan still be treated by resorting to the same techniques previously introduced.Let us shed some light on this point by considering the different cases that we canface when we move to the generic O ( ǫ k ). In principle one should start the calculation26rom the Feynman rules derived in Section 1 and perform all the steps of the replica trickprocedure described in that Section. According to those rules, the O ( ǫ k ) contributionsto the G n ’s are obtained by taking all the combinations of vertices λ i such that the totaldegree in ǫ of the resulting diagrams is ǫ k . Three cases have to be distinguished:1. Single-vertex diagram. This diagram is obtained when we take the vertex λ k . It ispractically the same as those appearing in Eqs. (18) and (45), with the difference thatthe elementary vertex is λ k rather than λ or λ , and that we have the k-derivativewith respect to N rather than the first or second derivative. As a consequence, thisdiagram is easily given in terms of the O ( ǫ k ) n -legs effective vertex:Π ( k ) n = k n ≡ lim N → d k dN k n − λ k ( N ) [∆(0)] N − n C n ( N ) o = − ǫ k k ! g µ ñ ô n − lim N → d k dN k f n ( N ) . (105)Diagrams of this kind are handled with the same techniques described before.2. 2-vertices diagrams. These diagrams are obtained by taking a vertex λ i and a vertex λ k − i (with 1 ≤ i ≤ k − λ i and λ k − i (ratherthan twice λ ), and that the derivatives with respect to N and M are of degree i and k − i (rather than first order derivatives in both cases). In this case we perform theanalytic extensions of the appropriate functions similarly to what we have done forthe O ( ǫ g ). Naturally, we are assuming again that the series defining our Green’sfunctions are truncated at a given cut-off L max , and under this assumption thesediagrams are given in terms of the O ( ǫ i ) and O ( ǫ k − i ) effective vertices. Referringagain to the (similar) Feynman diagrams in x-space given in Eqs. (69) and (71),these contributions are written as i k − i x x j x j +1 x n l i k − i x x j +1 x j +2 x n l + 1 (106)3. 3 or more vertices. These diagrams are more complicated. The only point we canclearly show is that the combinatorial coefficients are factorized as before. How-ever, the analytic extensions of the series are much more involved, and we have notbeen able to perform this step in all its generality. Actually, while for the two-vertex diagrams the sum over the number of links between the two vertices almoststraightforwardly gives rise to the hypergeometric function F , in these latter caseswe have m vertices, with m >
2. As a consequence, there are Ä m ä sums over thenumber of links among these vertices, and it is much more difficult to gain con-trol on the analytic extensions of the corresponding series, mainly in connection27ith their convergence properties. More precisely we have checked that, when itis possible to define these analytic extensions, they should be again given in termsof hypergeometric functions, even though of a higher number of variables (multiplehypergeometric functions). Clearly in these cases the problem with the convergenceof the series is more severe than before, and the possibility of defining the analyticextension of these series again resorts on the assumption that they can be consis-tently truncated. If this is doable, the derivatives and the limits with respect to thevariables N i can be done term by term, and the corresponding contributions to theGreen’s functions are given again in terms of the effective vertices Π ( i ) n .Once all the contributions to the Green’s functions are written in terms of the effectivevertices Π’s, it is possible to analyse their dependence on the momentum cut-off Λ similarlyto what we have previously done for the first and second order contributions.Moreover we note that, when the systematic renormalization program of the theory ateach order in ǫ will be undertaken, the possibility of writing all the contributions in termsof effective vertices would make it possible to apply to the expansion in ǫ the results ofthe BPHZ theorem concerning the subleading divergences. This is however beyond thescopes of the present work and is left for future investigations.As usual we are interested in the amputated 1PI diagrams, with total momentumconservation already factored out. Moreover, in the diagrams it is always understood thatat each effective vertex an even number of legs is attached, even though for notationalsimplicity we do not indicate it explicitly.The leading behaviour of the 1-vertex diagram contributing to the G n at O ( ǫ k ) is: k n = Π ( k ) n ∼ log k − ΛΛ n − (107)where we adopt the convention that for n = 2, Λ has to be read as log Λ in the numerator.This convention will be also used for all the diagrams considered below.From (107) we see that for n > k , at any finite orderwe get a vanishing contribution to G n , as no power of log Λ can underdo the suppressiondue to the inverse power of Λ.Concerning the diagrams with two effective vertices, we have i k − i x x a x a +1 x n r = Π ( i ) a + r Π ( k − i ) n − a + r I r ( p ) r ! ∼ (log Λ) i − Λ a + r − (log Λ) k − i − Λ n − a + r − Λ r − (108)where r ≥ introduced above has to be applied.28oving now to the case of three effective vertices, the diagrams to be considered are: i i i x x a y y b z z c r st (109)where a + b + c = n and i + i + i = k . Moreover, as we are taking into account only 1PIdiagrams, we must have either r = 0 and s, t ≥ r, s, t, ≥ r = s = t =1 (see Eq. (110) below) has a convergent triangleloop, so that the leading behaviour in terms of the cut-off is: i i i x x a y y b z z c ∼ (log Λ) i − Λ a (log Λ) i − Λ b (log Λ) i − Λ c (110)In all the other cases, the superficial degree of divergence of the loop integrals is non-negative, more specifically D = 4( r + s + t − − r + s + t ) = 2( r + s + t − ≥
0. The(superficial) cut-off dependence of the diagram is then (for r + s + t ≥ i i i x x a y y b z z c r st ∼ (log Λ) i − Λ a + r + t − (log Λ) i − Λ b + r + s − (log Λ) i − Λ c + s + t − Λ r + s + t − (111)To summarize, leaving aside the subleading dependence on log Λ, the leading behaviourwith the cut-off of the 1PI diagrams is given by: • ∼ Λ − n • ∼ Λ − n • ∼ Λ − n (Eq.(110)) or Λ − − n (Eq.(111))29e can now extend the previous analysis to a generic number of effective vertices V . Indicating as usual with N I the number of internal lines, and referring to a genericdimension d , the superficial degree of divergence D coming from the loop integrals is: D = d ( N I − V + 1) − N I , (112)where the integral is (superficially) convergent when D < V vertices with a total numberof legs n + 2 N I . Therefore, referring to (107) with generic d , that givesΠ ( k ) n ∼ (log Λ) k − Λ ( d − )( n − ∼ Λ − ( d − )( n − (113)we have V Y i =1 Π k i a i ∼ V Y i =1 (log Λ) k i − Λ ( d − )( a i − ∼ ( d − )( n +2 N I − V ) = 1Λ[ ( d − ) n +( d − N I − ( d − V ) ] , (114)where in the last members of (113) and (114) the subleading logarithms are neglected.We have to distinguish the two cases D ≥ D <
0. In the first case, the cut-offdependence of the diagram with n external lines, V vertices and N I internal lines, comesfrom both the effective vertices (114) and the loop integrals (112). In d dimensions it is:Λ − [ d − n +2 V − d ] ( d =4 → Λ − n − V ) (115)and it is clear that the dominant (less suppressed) diagram is the one with V = 1, i.e.(113).In the second case, when D <
0, the cut-off dependence is given uniquely by (114).For any fixed value of V , the condition D < V ≤ N I < dd − ( V − V = 2, the condition D <
V >
2, among the diagrams with
D <
0, the leading one (i.e.the one less suppressed by powers of Λ) is the diagram with the minimal possible valueof N I , that is N I = V . It goes as:Λ − ( d − ) n ( d =4 → Λ − n ) (116)Comparing (116) with (113), again we see that the dominant contribution comes from(113), that is from the diagram with only one effective vertex.In the present section, extending the analysis previously applied to the O ( ǫ ), we havebeen able to identify the leading contributions to the G n at any order in ǫ . However, toour dissatisfaction, we have seen that at any finite order in ǫ the radiative correctionsgive a too strong suppression factor with the physical cut-off Λ so that, disregardingpossible unconventional renormalization prescriptions (flows) on which we have arleadycommented, it seems that all the physical amplitude vanish, thus resulting in a non-interacting theory.In our opinion this is due to the fact that truncating the expansion to a finite order in ǫ we get a too mild impact of the interaction term on the Green’s functions. However wehave seen that, increasing the power of ǫ , the suppression factor that causes the vanishing30f the G n becomes milder and milder, thus rising the hope that an infinite resummationof different orders in ǫ could rescue the theory.With this in mind, in the next section we will focus our attention on the leading order(in Λ) contributions at each order in ǫ and attempt to a resummation of these diagrams.The way for this analysis has been paved in this section. We now proceed to the resummation of the leading contributions to the Green’s functions G n , and note that we have to treat separately (as it was for the previous sections) the n = 2 and n > G In the previous sections we have seen that the O ( ǫ ) contribution to ‹ G is:1 (117)that diverges as log ∆(0). At order ǫ there are two leading diagrams,2 1 1 (118)both diverging as log ∆(0), while for the order ǫ the leading divergences are containedin the three diagrams:3 2 1 1 1 1 (119)all of them diverging as log ∆(0). Similarly for the higher orders.Our goal is to proceed to the resummation of the infinite “triangle” of diagrams whosefirst instances are given in Eqs. (117)-(119). This can be done by resumming first the 1PIdiagrams in the left side of the triangle, and then considering the geometric series arisingfrom their iteration, that means resumming all the diagrams in the triangle.Considering then the amputated 1PI bubbles k , i.e. the effective verticesΠ ( k )2 , we obtain the resummed two-legs vertex function (i.e. proper self-energy) that wename Γ :Γ ≡ ∞ X k =1 k = ∞ X k =1 Π ( k )2 = ∞ X k =1 lim N → d k dN k ¶ − λ k ( N ) [∆(0)] N − C ( N ) © = − g µ ∞ X k =1 ǫ k k ! lim N → d k dN k f ( N ) = − g µ [ f (1 + ǫ ) − f (1)]= − g µ " ( ǫ + 1) Γ( ǫ + )Γ( ) î µ − d ∆(0) ó ǫ − , (120)31here the resummation in the second line of (120) is possible due to the analyticity onthe positive real axis of the function f ( x ) in (27), and the final result is given in the lastmember.It is worth to stress that the above result is obtained for generic real positive valuesof ǫ . At the same time, what is physically interesting at the end of the calculation is toconsider integer values of ǫ that correspond to relevant interacting theories as φ ( ǫ = 1), φ ( ǫ = 2), . . . . In these latter cases, being Γ( ǫ + )Γ( ) = 2 − ǫ (2 ǫ + 1)!!, Eq.(120) becomes:Γ ǫ ∈ Z + = gµ − ( ǫ + 1)(2 ǫ + 1)!! gµ − d ) ǫ ∆(0) ǫ . (121)Few comments are in order. Let us start by recalling that the lagrangian defining thetheory (see Eq.(3)) is: L = 12 ( ∇ φ ) + 12 m φ + 12 gµ − d ) ǫ φ ǫ = 12 ( ∇ φ ) + 12 Ä m + gµ ä φ + ñ gµ − d ) ǫ φ ǫ − gµ φ ô , (122)where the second line is written according to the splitting in free and interaction La-grangians as given in Eqs.(5)-(6). Considering then as L the first two terms of the secondline in Eq.(122), and as L int the terms in the square brackets, we find the central resultof this subsection, namely that Eq.(121) is nothing but the proper self-energy obtainedwithin the framework of the weak coupling expansion. Indeed, at the first perturbativeorder in g we obtain the two diagrams:= g µ (123) ǫ = − gµ − d ) ǫ (2 ǫ + 2) (2 ǫ + 1)!! ∆(0) ǫ (124)whose sum gives the proper self energy in (121). Note that the diagram in (123) comesfrom the term − gµ φ in L int and that the mass term in the loop integral ∆(0) is M = m + gµ . We would like to stress that would we have used the splitting in L and L int according to the first line of (122), in (121) the term gµ would be absent, and thepropagator ∆(0) would contain the mass m . This is simply related to the freedom ofmoving φ terms from the free to the interaction lagrangian.We are now ready to proceed to the resummation of the whole “triangle” of diagramsin Eqs. (117)-(119). Iterating the proper self-energy insertion (120), we obtain for the fullpropagator the approximation (for integer values of ǫ (120) coincides with (121)):= 1 p + M " gµ − ǫ ( ǫ + 1) Γ( ǫ + )Γ( ) gµ − d ) ǫ ∆(0) ǫ p + M + " gµ − ǫ ( ǫ + 1) Γ( ǫ + )Γ( ) gµ − d ) ǫ ∆(0) ǫ p + M ! + · · · = 1 p + M − ï gµ − ǫ ( ǫ + 1) Γ( ǫ + )Γ( ) gµ − d ) ǫ ∆(0) ǫ ò (125)32rom which the renormalized mass turns out to be: m R = M − gµ + 2 ǫ ( ǫ + 1) Γ( ǫ + )Γ( ) gµ − d ) ǫ ∆(0) ǫ = m + 2 ǫ ( ǫ + 1) Γ( ǫ + )Γ( ) gµ − d ) ǫ ∆(0) ǫ (126)Eq.(126) is one of the central results of the present work and we now explain this pointmaking a certain number of comments.First of all we stress that the above resummation, containing iterated insertions of(123), provides the term − gµ that, combined with M , restores m as the tree level massterm.We note however that the mass in the loop integral ∆(0) is still M = m + gµ .As we will see in the next section, the reason is that in order to get even in ∆(0) thesame shift M → m we have to go further in the systematic approximation of the properself-energy. More specifically, we will see that the resummation of subleading diagramswill provide, among others, iterated insertions of (123) inside the loops, thus progressivelyshifting the mass M towards m also in the radiative corrections.We are now ready to appreciate the importance of (126). To this end, we comparethe result for m R at O ( ǫ ) (see Eq.(37) in Section 2) with the result obtained after resum-mation in (126). In the first case, the radiative correction to the mass is proportional to ǫ log(∆(0)), which for d ≥ ǫ , while after resummation the radiative correction goes as Λ d − ǫ .A better insight on this result is gained if we consider integer values ǫ = 1 , , . . . thatcorrespond to interacting theories φ , φ , . . . . For the sake of definiteness let us specify tothe case ǫ = 1 and d = 4, i.e. to the 4-dimensional φ theory. In this case the renormalizedmass turns out to be: m R = m + 6 g ∆(0) , (127)where the second term is due to the tadpole diagram in (124) with only one loop ( ǫ = 1).Using then Eq.(39) for ∆(0), we get m R = m + 3 g π Ç Λ − M log Λ M å (128)that coincides with Eq.(41) obtained in the framework of the weak-coupling expansion,with the only difference that in the logarithmic divergence m is replaced by M . Theexplanation for this difference is given above (it is cured when higher order approximationsare considered).In a sense Eq. (128) is as a deceptive result. In fact at O ( ǫ ) we found that the radiativecorrection to the mass goes as log(Λ) (Eq.(40)) and this gave rise to the hope that withinthe ǫ -expansion the hierarchy problem could be enormously alleviated. However Eq. (128)shows that the resummation of the leading diagrams at any order in ǫ restores the weak-coupling result, i.e. a quadratically divergent correction.We may ask ourselves, how is it possible that such a resummation switches a loga-rithmic to a quadratic divergence. The answer is contained in Eq. (120), whose leading33ehaviour is given by (note that here ǫ = 1 and ∆(0) ∼ Λ ): X k k ∼ X k ǫ k k ! log[∆(0)] k ∼ [∆(0)] ǫ (129)This result sounds like a warning and apparently confirms the suspicion that we haveexpressed above, namely that when we consider the expansion of the lagrangian (122) inpowers of ǫ and limit ourselves to a finite order in ǫ , this could result in grasping a toopoor truncation of the physical interaction. In the following we will further investigatethis crucial point. G n for n > As we already noted in Section 4, the diagrams with one effective vertex have a leadingbehaviour with Λ, but there are also reducible diagrams that have the same superficialcut-off dependence. The contribution of the latters, however, is already taken into accountwhen dealing with Green’s functions with a smaller number of points n . For this reason, wefocus on the resummation of the effective vertices only, that are 1PI diagrams. Amputatingthe external propagators and factoring out the δ -momentum conservation, we get theresummed n -legs vertex function that we name Γ n :Γ n = ∞ X k =1 k n = ∞ X k =1 Π ( k ) n = ∞ X k =1 lim N → d k dN k ¶ − λ k ( N ) [∆(0)] N − C n ( N ) © = − g µ ñ ô n − ∞ X k =1 ǫ k k ! lim N → d k dN k f n ( N )= − g µ ñ ô n − [ f n (1 + ǫ ) − f n (1)] (130)where, again, the last step can be done due to the analyticity of the function f n ( x ) onthe positive real axis. Referring to Eq.(26), we see that f n (1) = 0 ∀ n > n = ∞ X k =1 Π ( k ) n = − g µ ñ ô n − ( ( ǫ + 1) n Γ( ǫ + )Γ( ) î µ − d ∆(0) ó ǫ ) . (131)As for the case of G , this result is obtained for generic positive real values of ǫ . Movingto the physically relevant cases, i.e. to integer values of ǫ , Eq.(131) becomes:Γ n ǫ ∈ Z + = − g µ − d ) ǫ n − ( ǫ + 1) n (2 ǫ + 1)!!∆(0) ǫ +1 − n (132)This is the main result of the present subsection. As can be immediately checked,Eq.(132) is nothing but the O ( g ) contribution to the n -points Green’s functions within34he framework of the weak-coupling expansion: n n − ǫ = − gµ − d ) ǫ (2 ǫ + 2) n (2 ǫ + 1 − n )!! ∆(0) ǫ +1 − n . (133)From the lagrangian (122) we see that the mass term in the loops is M = m + gµ . Inpassing we note that considering also loops with iterated insertions of the crossed diagram(123) we would obtain the shift M → m (see comments below Eq. (126)).Referring to (130) we stress that the resummation (131) gives only one term as f n (1) =0, and this reflects the fact that the term − gµ φ does not give connected diagrams for n >
2, that in other words means that in this case there is no diagram corresponding tothe crossed diagram of (123).A striking result is that with the resummation (131) the G n with n > ǫ + 2 vanish(due to the vanishing of the falling factorial). This is in agreement with the well-knownfact that, at first order in the weak-coupling expansion, for n > ǫ + 2 we cannot drawconnected diagrams as those in (133).Last but (certainly) not least we note that, similarly to what has already been seenfor the two-points Green’s function, having the resummation restored the weak-couplingresult, the behaviour of the G n with respect to their dependence from the momentum cut-off Λ is drastically changed. Indeed Eq. (131) shows that the resummation of the leadingdiagrams generates powers of ∆(0) in the numerator so to compensate the suppressingpowers of ∆(0) in the denominator found at any finite order in ǫ : ∞ X k =1 k n ∼ n − X k =1 ǫ k k ! log[∆(0)] k − ∼ [∆(0)] ǫ − n +1 . (134)Few comments are in order. Eq. (134) shows that suitable resummations of diagramsallow to recover the interacting character of the theory. In particular, having chosen toresum the specific class of diagrams that have a leading behaviour with Λ, we recovered theresults of the weak-coupling expansion. More precisely, as a by-product of our analysis,we have established the bridge between the expansion in the non-linearity parameter ǫ and the weak-coupling expansion at O ( g ).As already hinted, it seems to us that from these and previous results we can inferthat, before proceeding to a systematic program for the renormalization of the theory, itis necessary to consider appropriate resummations of diagrams (not necessarily the oneconsidered in the present section). In fact, as we have already seen in a specific exampleat the end of section 2, the attempt of renormalizing the theory order-by-order in the ǫ -expansion seems to give rise to a “too weird” behaviour of the parameters that definethe theory. Just to mention an example and sticking on the d = 4 case, we remind that,to get a finite G at order ǫ , we had to give to the coupling constant g a dependence onΛ that goes as g ∼ Λ , that in turn gives for the radiative correction δm to the mass thecut-off dependence δm ∼ Λ log Λ (see comments below Eq. (44))35n the contrary, the resummation performed in this section “cures” the problem withthe vanishing of the cross sections, restoring the divergence structure of the weak-couplingexpansion, actually the weak-coupling results tout-court.Motivated by these results, in the following we push further the analysis on the linkbetween the two expansions. To this end, in the next section we will perform higher orderresummations, considering in particular diagrams with two effective vertices Π i . In the previous section we have considered the resummation of diagrams with one effectivevertex. We want to move now a step further in the approximation of the Green’s functions,by considering the resummation of diagrams with two effective vertices. As explainedin Section 4, the latters contain next-to-leading contributions, and even though thereare other diagrams with the same subleading behaviour, these are the only O ( g ) ones.Therefore, for the purposes of our comparison between the expansion in ǫ and the weak-coupling expansion, these are the only diagrams that we have to take into account.First of all we have to remind that, when dealing with the analytic continuation ofcontributions coming from diagrams with two auxiliary vertices, the l -series that sums overthe links between the two vertices brings an ultraviolet divergence in the loop integral forall the Green’s function with n >
4. In Sections 3 and 4 we regularized these expressionswith the help of a finite upper limit L max . So doing, all these l -series contributions areeasily written as a sum of diagrams with two Π effective vertices.At a generic order ǫ k (with k ≥ n -points Green’sfunction of the diagrams with two vertices, that we indicate with G ( k,g ) n , is given by thesum of all the diagrams of the kind given in Eq. (106), where we distinguish the two classesof “even” and “odd” diagrams with respect to the parity of the number of links betweenthe two vertices. Working on the two classes separately, we have (with obvious notations) G ( k,g ) n = G ( k,g ) n, E + G ( k,g ) n, O where: G ( k,g ) n, E = 12 k − X α =1 n X j =0 L max X l =1 αu k − α w x x j x j +1 x n l + n j ! − = 12 k − X α =1 n/ X j =0 L max X l =1 Π ( α )2 j +2 l Π ( k − α ) n − j +2 l l )! × Z d d u d d w j Y i =1 ∆( x i − u ) n Y h =2 j +1 ∆( x h − w ) ∆( u − w ) l + n j ! − . (135)36 ( k,g ) n, O = 12 k − X α =1 n − X j =0 L max X l =0 αu k − α w x x j +1 x j +2 x n l + 1 + n j +1 ! − = 12 k − X α =1 n − X j =0 L max X l =0 Π ( α )2 j +2 l +2 Π ( k − α ) n − j +2 l l + 1)! ×× Z d d u d d w j +1 Y i =1 ∆( x i − u ) n Y h =2 j +2 ∆( x h − w ) ∆( u − w ) l +1 + n j +1 ! − . . (136)Our goal is to evaluate the correction to the Green’s functions given by the sum ofthese contributions coming from all orders in ǫ , i.e. to calculate the sum of the series G ( g ) n = P ∞ k =2 G ( k,g ) n . The even and odd terms can be resummed separately, as long asthese series converges.Starting from the even case we have: G ( g ) n, E = 12 ∞ X k =2 k − X α =1 n/ X j =0 L max X l =1 Π ( α )2 j +2 l Π ( k − α ) n − j +2 l l )! " I l ( x , ... , x j ; x j +1 , ... , x n ) + n j ! − . (137)where we indicated with I l ( x , . . . , x j ; x j +1 , . . . , x n ) the integral of the propagators writ-ten in the last line of Eq. (135). Due to the presence of the cut-off L max , the general termof the double series in k and α is a sum of a finite number of terms (i.e. the sum over j and l ) so that, as long as the double series converges for each of these terms, it can besplitted as follows: G ( g ) n, E = 12 n/ X j =0 L max X l =1 ( ∞ X k =2 k − X α =1 Π ( α )2 j +2 l Π ( k − α ) n − j +2 l ) l )! " I l ( x , ... , x j ; x j +1 , ... , x n ) + n j ! − . . (138)At this point we need to evaluate the double series ∞ X k =2 k − X α =1 Π ( α )2 j +2 l Π ( k − α ) n − j +2 l . (139)It is easy to see that this double series is nothing but the Cauchy product: ∞ X k =2 k − X α =1 Π ( α )2 j +2 l Π ( k − α ) n − j +2 l = ∞ X α =1 Π ( α )2 j +2 l ! · Ñ ∞ X β =1 Π ( β ) n − j +2 l é (140)where β = k − α . Therefore, as long as the two series are convergent and at least one ofthem is absolutely convergent, we can safely calculate the sum in the l.h.s. of Eq.(140)37s the product of the two series of Π’s. We already evaluated these series in the previoussection, getting the resummed vertex functions that we denoted with Γ’s (see Eqs. (120)and (130)-(131)).We note that, as these series are nothing but the Taylor expansions around the point x = 1 of functions that are analytic in the the complex half-plane Re ( x ) > − (due to thepresence of the factor Γ( x + )), they have radius of convergence , so that the validity ofEq. (140) is guaranteed for ǫ < (as in this case the absolute convergence is guaranteed).The possibility of extending this result to larger values of ǫ is an interesting question thatwe have not pursued.Eq. (140) is crucial for this section. In fact, due to the splitting of the double series,the resummation of the subleading class of diagrams with two effective vertices is reducedto the resummations of the Π’s at each vertex of the diagrams. This in turn means thatthe result will be expressible in terms of diagrams composed of resummed Γ vertices, and(as we will see in a moment) this is ultimately the reason why the resummation of thisclass of diagrams will yield to the O ( g ) weak-coupling result.Thanks to Eq.(140) we can write the final result for the resummation of all the “even”diagrams with two effective vertices as: G ( g ) n, E = 12 n/ X j =0 L max X l =1 ∞ X α =1 Π ( α )2 j +2 l ! Ñ ∞ X β =1 Π ( β ) n − j +2 l é l )! × " I l ( x , . . . , x j ; x j +1 , . . . , x n ) + n j ! − . = 12 n/ X j =0 L max X l =1 Γ j +2 l Γ n − j +2 l (2 l )! " I l ( x , ... , x j ; x j +1 , ... , x n ) + n j ! − . (141)Following the same steps, an analogous result is obtained for the odd contribution: G ( g ) n, O = 12 n − X j =0 L max X l =0 ∞ X α =1 Π ( α )2 j +2 l +2 ! Ñ ∞ X β =1 Π ( β ) n − j +2 l é l + 1)! × " I l +1 ( x , . . . , x j +1 ; x j +2 , . . . , x n ) + n j + 1 ! − . = 12 n − X j =0 L max X l =0 Γ j +2 l +2 Γ n − j +2 l (2 l + 1)! " I l +1 ( x , ... , x j +1 ; x j +2 , ... , x n ) + n j + 1 ! − . (142)The final result is clearly obtained once we replace in both (141) and (142) the re-summed vertices Γ derived in the previous section, that for the reader’s convenience wereport here for the physically relevant cases of integer values of ǫ :Γ n = ∞ X k =1 Π ( k ) n ǫ ∈ Z + = gµ − ( ǫ + 1)(2 ǫ + 1)!! gµ − d ) ǫ ∆(0) ǫ for n = 2 − g µ − d ) ǫ n − ( ǫ + 1) n (2 ǫ + 1)!!∆(0) ǫ +1 − n for n ≥ ǫ the falling factorial ( ǫ +1) n vanishes for n > ǫ + 2. Therefore, once the resummation is performed then, the number38f links between the vertices is limited by this restriction, so that the cut-off L max doesnot play any role for integer values of ǫ . As we will see in moment, such a result is crucialto have the coincidence between the sum of (141) and (142) and the O ( g ) weak-couplingresult, as in the weak coupling expansion the number of legs at each vertex is naturallylimited by the specific interaction term.To better appreciate the results in (141) and (142), we now focus on the two- andfour-points Green’s functions, considering integer values of ǫ .Starting from the two-points function and replacing (143) in (141) and (142), we get: G ( g )2 , E = L max X l =1 Γ l Γ l (2 l )! I l ( x , x ; )= î gµ − ( ǫ + 1)(2 ǫ + 1)!! gµ − d ) ǫ ∆(0) ǫ ó î − gµ − d ) ǫ ǫ + 1) ǫ (2 ǫ + 1)!!∆(0) ǫ − ó × Z d d u d d w ∆( x − u )∆( x − u )∆( u − w ) + ǫ X l =2 î − gµ − d ) ǫ l − ( ǫ + 1) l (2 ǫ + 1)!!∆(0) ǫ +1 − l ó î − gµ − d ) ǫ l ( ǫ + 1) l +1 (2 ǫ + 1)!!∆(0) ǫ − l ó × l )! Z d d u d d w ∆( x − u )∆( x − u )∆( u − w ) l (144)and G ( g )2 , O = L max X l =0 (Γ l ) (2 l + 1)! I l +1 ( x ; x )= î gµ − ( ǫ + 1)(2 ǫ + 1)!! gµ − d ) ǫ ∆(0) ǫ ó Z d d u d d w ∆( x − u )∆( u − w )∆( x − w )+ ǫ X l =1 î − gµ − d ) ǫ l ( ǫ + 1) l +1 (2 ǫ + 1)!!∆(0) ǫ − l ó l + 1)! × Z d d u d d w ∆( x − u )∆( u − w ) l +1 ∆( x − w ) . (145)It is easy to verify that the sum of (144) and (145) is nothing but the result that wewould have obtained at the second order of the weak-coupling expansion for the two-points Green’s function of the φ ǫ theory given by the Lagrangian (122).For the sake of concreteness, let us show this coincidence for the particular case ǫ = 1,i.e. for the φ theory. In this case, starting from the splitting in free and interactionterms of the Lagrangian given in Eq. (122), the O ( g ) correction to the two-points Green’sfunction is given by the following diagrams:39 x = − g µ − d Z d d u d d w ∆( x − u )∆( x − u )∆( u − w ) (146) x x = 36 Ä gµ − d ä ∆(0) Z d d u d d w ∆( x − u )∆( x − u )∆( u − w ) (147) x x = Ä gµ ä Z d d u d d w ∆( x − u )∆( u − w )∆( x − w ) (148) x x = − g µ − d ∆(0) Z d d u d d w ∆( x − u )∆( u − w )∆( x − w )(149) x x = 36 Ä gµ − d ä ∆(0) Z d d u d d w ∆( x − u )∆( u − w )∆( x − w )(150) x x = 24 Ä gµ − d ä Z d d u d d w ∆( x − u )∆( u − w ) ∆( x − w ) (151)where the first two diagrams coincide with the two terms corresponding to l = 1 inEq. (144) (which is the only possible value of l for ǫ = 1), while the three followingdiagrams coincide with the three possible terms corresponding to l = 0 in Eq. (145), andthe last one corresponds to the l = 1 odd term. Let us finally note that the above diagramsare the usual second order Feynman diagrams for G of a φ theory, with the addition ofdiagrams that contain the insertion of two-legs vertices due to the splitting of the massterm. Moreover, the third, fourth and fifth diagrams are reducible (and in fact they arealready considered in the geometrical series of the previous section related to the radiativecorrection to the mass) while the first, second and sixth diagrams are 1PI, and genuinelyprovide the O ( g ) contribution to the proper self-energy.Moving to the 4-points Green’s function, the resummation gives:40 ( g )4 , E = L max X l =1 Γ l Γ l (2 l )! I l ( x , . . . , x ; ) + L max X l =1 (Γ l ) (2 l )! [ I l ( x , x ; x , x ) + 2 perm . ]= î gµ − ( ǫ + 1)(2 ǫ + 1)!! gµ − d ) ǫ ∆(0) ǫ ó î − g µ − d ) ǫ ǫ + 1) ǫ ( ǫ − ǫ + 1)!!∆(0) ǫ − ó × Z d d u d d w ∆( x − u ) . . . ∆( x − u )∆( u − w ) + ǫ − X l =2 î − g µ − d ) ǫ l − ( ǫ + 1) l (2 ǫ + 1)!!∆(0) ǫ +1 − l ó î − g µ − d ) ǫ l +1 ( ǫ + 1) l +2 (2 ǫ + 1)!!∆(0) ǫ − − l ó × l )! Z d d u d d w ∆( x − u ) . . . ∆( x − u )∆( u − w ) l + ǫ X l =1 î − g µ − d ) ǫ l ( ǫ + 1) l +1 (2 ǫ + 1)!!∆(0) ǫ − l ó × l )! ï Z d d u d d w ∆( x − u )∆( x − u )∆( u − w ) l ∆( x − w )∆( x − w ) + 2 perm . ò (152) G ( g )4 , O = L max X l =0 Γ l Γ l (2 l + 1)! [ I l +1 ( x ; x , x , x ) + 3 perm . ]= î gµ − ( ǫ + 1)(2 ǫ + 1)!! gµ − d ) ǫ ∆(0) ǫ ó î − g µ − d ) ǫ ǫ + 1) ǫ (2 ǫ + 1)!!∆(0) ǫ − ó × ï Z d d u d d w ∆( x − u )∆( u − w )∆( x − w ) . . . ∆( x − w ) + 3 perm . ò + ǫ − X l =1 î − g µ − d ) ǫ l ( ǫ + 1) l +1 (2 ǫ + 1)!!∆(0) ǫ − l ó î − g µ − d ) ǫ l +1 ( ǫ + 1) l +2 (2 ǫ + 1)!!∆(0) ǫ − l − ó × l + 1)! ï Z d d u d d w ∆( x − u )∆( u − w ) l +1 ∆( x − w ) . . . ∆( x − w ) + 3 perm . ò (153)Again, it is easy to check that the sum of (152) and (153) coincides with the O ( g )correction to the 4-points Green’s function in the weak-coupling expansion of the φ ǫ theory described by the Lagrangian (122). As before, specifying to the case ǫ = 1, weobtain: x x x x = 72 Ä gµ − d ä Z d d u d d w ∆( x − u )∆( x − u )∆( u − w ) ∆( x − w )∆( x − w ) (154)41 x x x = 72( gµ − d ) ∆(0) Z d d u d d w ∆( x − u )∆( u − w )∆( x − w )∆( x − w )∆( x − w ) (155) x x x x = 12 g µ − d Z d d u d d w ∆( x − u )∆( u − w )∆( x − w )∆( x − w )∆( x − w ) (156)where the first diagram corresponds to the l = 1 term of the sum in the last line ofEq.(152) (that is the only non-vanishing “even” term for ǫ =1), while the second and thirddiagrams correspond to the two terms in the first line of Eq.(153) (for ǫ = 1 all the otherodd terms vanish). Similarly to the previous case, the diagrams (155) and (156) are oneparticle reducible and are related to the external propagator correction, while the firstdiagram is the genuine O ( g ) correction to G .Although in the above lines we have explicitly considered the results of the resum-mations in the cases of G and G and shown that they coincide with those of the O ( g )weak-coupling expansion for the φ theory (i.e. ǫ = 1), these results are of a more generalvalidity: this coincidence holds for all the Green’s functions and for all the integer valuesof ǫ for which (140) is fulfilled. This result is actually contained in the general Eqs. (141)and (142), where the terms appearing in the weak coupling expansion, including the com-binatorial factors, although differently organized, are all present in the Γ’s resummedvertices and in the integrals contained in the square brackets.Pushing the analysis of Section 5, in the present section we showed how the bridgebetween the expansion in the non-linearity parameter ǫ and the weak coupling expansionis realized at O ( g ): it is obtained by resumming the diagrams with two effective vertices.In particular we have seen that, as compared to the O ( g ) case, the peculiarity of the ǫ -expansion brings some technical difficulties that are in part related to the necessity ofintroducing the cut-off L max when considering generic values of ǫ . Interestingly, when wemove to integer values of ǫ , the cut-off L max does not play any longer a role, and we easilyobtain the weak coupling results. Moreover, a delicate point of this analysis concerns theconvergence and/or absolute convergence of the series involved in the calculation, moreprecisely the Cauchy product of (140). In particular, a word of caution has to be saidfor values of ǫ > /
2, where the splitting for the double series is not guaranteed (seecomments below Eq. (140)), while for ǫ < / φ theory is obtained for ǫ = 1.Before moving to the conclusions, we would like to observe that, following the sameline of reasoning of this section, we expect that the bridge between the two expansionsactually extends also to higher orders in g . In particular, sticking on the assumption thatthe contributions to the G n coming from diagrams with three or more vertices can still be42egularized with the help of L max cut-offs, the resummation of diagrams with a genericnumber of vertices m should yield to the results of the weak-coupling expansion at order g m . Renormalization and Renormalization Group techniques are crucial tools for extractingmeaningful result from interacting quantum field theories. In a recent paper [37] it wassuggested that a previously introduced formal expansion in a parameter that measuresthe “distance” between the free and the interacting theory [35], i.e. the parameter ǫ introduced in Section 2, could be the key for a systematic renormalization program ofnon-hermitian PT-symmetric theories.Motivated by this suggestion, in the present paper we took a step forward in theanalysis of such an expansion. However, in order to avoid the complications related tothe peculiar structure of non-hermitian theories, we have performed the analysis for ordi-nary interacting scalar theories, leaving the investigation of non-hermitian PT-symmetrictheories is left for future work [69].In Section 2 we started by reviewing the O ( ǫ ) results [35, 36], and then moved to theanalysis of the renormalization properties of the theory. In particular, considering thetwo-points Green’s function, we noted that, as compared to the weak-coupling expansionat O ( g ), the radiative correction to the mass at O ( ǫ ) is significantly milder. Implementingthe regularization with a physical cut-off Λ, and considering d > O ( ǫ ) correction goes as log(Λ), irrespectively of the power of the interactingterm (i.e. irrespectively of ǫ ), while for instance for a φ theory in d = 4 dimensions the O ( g ) radiative correction to the mass goes as Λ . The comparison between these tworesults seemed to suggest that within the expansion in ǫ the hierarchy problem could beenormously alleviated. However we have shown (see Section 5) that the resummation ofthe leading diagrams of each order in ǫ restores the weak-coupling result, that is a radiativecorrection that goes as Λ , thus presenting (at least at this order of approximation) thesame unnaturalness problem already encountered within the weak-coupling expansion.As for the higher order Green’s functions ( n ≥ d ≥ O ( ǫ ) all the G n (and as a consequence all the scattering amplitudes)behave as inverse powers (increasing with n ) of the cut-off, i.e. vanish with Λ → ∞ , thusimplying that at this order in ǫ the theory is non-interacting. However, similarly to thecase of G , in Section 5 we showed that the resummation of the leading diagrams fromall orders in ǫ restores the weak-coupling results, with the known dependence on Λ of the G n ’s.The systematic analysis of the order ǫ was performed in Section 3. We showedthat this calculation presents several delicate aspects mainly connected with the analyticextension of results related to diagrams that contain two vertices. In fact, while in the O ( ǫ ) calculation the functions that needed to be extended were factored out of space-timeintegrals, for the O ( ǫ ) case we have to consider analytic extensions of functions that arestill included in space-time integrals. More specifically, the analytic extension of sumsover links between two vertices gives rise to hypergeometric functions that could bringultraviolet divergences for the Green’s functions. Clearly this is a delicate problem thatdeserves further investigations, beyond the scope of the present work. In our analysis,43e took care of these divergences by introducing a numerical cut-off L max as upper limitof the power series, so rendering all the expressions finite and expressible as a sum ofdiagrams with two effective vertices.However we have shown (Section 6) that a very interesting result is obtained whenthe resummation of these two-vertex diagrams for integer value of ǫ is considered. In thiscase, due to the vanishing of the resummed Γ vertices, the series related to the number l of internal lines are automatically truncated starting from a certain value of l , thusrendering the presence of L max harmless.We then studied the dependence on the cut-off Λ of the Green’s functions at O ( ǫ )(Section 3) and then extended this analysis to higher orders in ǫ (Section 4). In this lattercase diagrams with an increasing number of vertices appear, and the problem (alreadyencountered at order ǫ ) of the divergences that arise from the analytic extensions of theseries becomes more severe. We again assumed that a suitable regularization can be madeby truncating the series with the help of numerical cut-offs L max . So doing, we were ableto express all these contributions in terms of sums of diagrams built with effective verticesΠ’s (see Eq. (105)). It was then possible to study the dependence of the Green’s functionson the physical cut-off Λ.The results of Sections 3 and 4 together showed that when considering higher andhigher orders in ǫ the two-points Green’s function G receives contributions that go ashigher and higher powers of log(Λ), and similarly the other Green’s functions G n ( n ≥ ǫ theystill vanish in the Λ → ∞ limit (see Eq. (107)).In our opinion the vanishing of the higher order Green’s functions at any finite order in ǫ can cast doubts on the possibility of realizing the renormalization of the theory withinthe framework of this expansion. This is certainly a very delicate issue that deservesfurther investigation.An attempt to implement a renormalization program in this context has been donein [70, 71], where a φ ǫ theory in d = 3 , O ( ǫ ), it seems to lackof systematicity when moving to higher orders in ǫ . More specifically, the authors beginby considering the O ( ǫ ) contribution and, in order to keep the 4-points Green’s functionfinite when Λ → ∞ , they introduce an unusual multiplicative renormalization . Whensubsequently the order ǫ contributions are added, the authors consider together the O ( ǫ )and O ( ǫ ) terms, looking for a renormalization that can make G finite (non-vanishing).However the different orders in ǫ have not the same ultraviolet behavior, and it turnsout that in order to realize this program it is necessary to make finite the terms of thehighest order, i.e. the O ( ǫ ) terms. This in turn results in the vanishing of the lower ordercontributions, i.e. the O ( ǫ ) terms, when Λ → ∞ . We have checked that this behaviorpersists also when higher orders in ǫ are considered, as it turns out that the highest ordercontributions are always the dominant ones in terms of Λ. We believe that, in order toovercame this mismatch of terms, an order-by-order renormalization procedure needs tobe implemented, and in our opinion this is a necessary ingredient for an expansion in a“small” parameter. Such a program has not yet been undertaken and is left for futurework. Naturally this is not the last word on the possibility of finding a systematic wayto renormalize the theory within the framework of the expansion in ǫ , and alternative In [70, 71] an x-space regularization, with a ∼ , is used. ǫ that could provide better approximations to the Green’s functions. Having found thatthe diagrams with a single effective vertex Π are leading with respect to their dependenceon the physical cut-off Λ, we first considered the resummation of these diagrams. So doingwe got for the Green’s functions the weak-coupling results at order g . This is one of thecentral results of the present paper. The resummation that we have performed provides aconnection between the expansion in the non-linearity parameter ǫ and the weak-couplingexpansion.Motivated by this finding, we tried to put forward a selection criterion for the diagramsrelated to the expansion in the parameter ǫ in order to check whether this connection couldbe extended to higher orders in g . With this in mind, as a second step we considered theresummation of all those diagrams with two effective vertices Π. Interestingly it turnedout that these diagrams are next-to-leading with respect to the previous ones .Compared to the resummation that leads to the O ( g ) result, in this case we faced theadditional difficulty of having to resum double series, and for that we needed to assurethe convergence and absolute convergence of each of the involved series. In particular,we found that when ǫ < / g weak-coupling results. This is of great interest, as for instance thephysically relevant φ theory lies within the convergence radius ǫ = 3 / g , indicating which class ofdiagrams should be resummed in order to get the weak-coupling results.In our opinion the results summarized above suggest that the “mild” behaviour withthe physical ultraviolet cut-off Λ that the Green’s functions present at any order in ǫ ,that would point towards a non-interacting theory at any order in ǫ , is actually due to a“too poor” truncation of the interaction term in the lagrangian (4), that is intrinsicallyrooted in this peculiar expansion. Our results indicate that a possible way to find suitableapproximations of the theory within the context of this expansion, and in particular todig out the interacting character of the theory, could be found in realizing appropriate re-summations of diagrams from all orders in ǫ . 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