PPROPAGATOR-CANCELLING SCALAR FIELDS
PAUL-HERMANN BALDUF
Abstract.
We examine a large class of scalar quantum field theories where vertices are able tocancel adjacent propagators. These theories are obtained as diffeomorphisms of the field variableof a free field. Their connected correlations functions can be computed efficiently with an algebraicprocedure dubbed connected perspective in earlier work. Specifically,(1) We compare the Feynman rules in momentum-space to their position-space counterparts,they agree. This is a a-posteriori justification for the Feynman rules of the connected perspective.(2) We extend the connected perspective to also incorporate counterterm vertices.(3) We examine a specific choice of diffeomorphism that assumes the vertices at different valenceproportional to each other. We find that these relations lead to various simplifications.(4) We perturbatively compute the connected 2-point function to all orders for an arbitrarydiffeomorphism of a massless field. We thereby give a systematic perturbative derivation of theexponential superpropagator known from the literature.(5) We compute a number of one- and two-loop counterterms. We find that for the specificdiffeomorphism, the one-loop counterterms arise from the bare vertices by a non-linear redefinitionof the momentum.(6) We show that every diffeomorphism fulfils a set of infinitely many Slavnov-Taylor-like iden-tities which express diffeomorphism-invariance of the S-matrix at loop-level.Finally, we comment on surprising similarities between the structure of propagator cancellingscalar theories and gauge theories . Introduction
Motivation.
This paper concerns propagator-cancelling scalar theories, that is scalar quan-tum field theories where vertex Feynman rules are able to cancel adjacent propagators. We restrictourselves to those theories which can be obtained from a free field theory by a non-linear redefinitionof the field variable. The implications of the latter restriction depend on the precise form of thepropagator. If the propagator is quadratic in momenta and massless, then all propagator-cancellingtheories fall in this class, see theorem 1.4. In the following, the word ”diffeomorphism” will alwaysrefer to a transformation of the field variable, not to be confused with spacetime diffeomorphisms.There are several reasons to study propagator-cancelling theories and especially diffeomorphisms:(1) Non-linear field redefinitions do not alter the S -matrix. This is frequently used to simplifythe Lagrangian of a quantum field theory, e.g. when treating gauge invariance [46, ch. 6.3]or non-local interactions [16]. The invariance of the S -matrix has been demonstrated inthe path integral formalism [53, 1] and also using graph theory [65, 42, 43, 2, 50, 49]. Thebehaviour of correlation functions of the field diffeomorphism has never been addressed indetail to the author’s knowledge apart from the statement that they vanish onshell.(2) It is well known that linear shifts in the field variable of an interacting field alter the type ofinteraction,e.g. [56]. Remarkably, they do not destroy renormalizability of a theory, whichmakes it possible to formulate theories with spontaneous symmetry breaking. One can askwhether a similar mechanism is at work in non-linear field transformations. Date : February 9, 2021.The author thanks Dirk Kreimer for helpful discussion as well as Daniel Reiche and Alexandra Gl¨uck forproofreading. a r X i v : . [ m a t h - ph ] F e b
3) There are two different connections with quantum Einstein gravity. Firstly, in gravitythe renormalization Hopf algebra equals the core Hopf algebra [36]. This is also true fora propagator-cancelling theory, which qualifies the latter as a model for the algebraic be-haviour of gravity. Secondly, although gravity is not a diffeomorphism of a free field [3], non-linear redefinitions of the field variable have been considered already decades ago [30]. Bynow, several different choices have been proposed, all of which lead to a non-renormalizablequantum theory. A better general understanding of field diffeomorphisms might clarifywhat can and what cannot be altered by them, apart from mere onshell invariance.(4) Historically, field redefinitions were prominently used to treat non-polynomial interactionterms, see section 1.2, but these computations often relied on ad-hoc prescriptions. Theyconcentrated on the 2-point-function, leaving open the question how they relate to themodern formalism of perturbation theory.1.2.
Historical background.
Perturbative renormalization of quantum fields is possible if thebare Lagrangian contains only coupling constants with non-negative mass dimensions [61, 27].Since this classification is based on order-by-order perturbation theory, it is conceivable that non-renormalizable interactions still give finite results either by a non-perturbative treatment [52], byanalytic properties of the interaction term [5], or by non-trivial cancellation effects of higher orderterms in the perturbation expansion [33, 19, 45]. An often studied example of a non-renormalizableinteraction are theories of Liouville-type [48] L = 12 ∂ µ φ∂ µ φ − exp( gφ ) . (1.1)In two dimensions, Liouville theory is solved by mapping its solutions to modes of a free fieldvia B¨acklund transformation [14] which enjoyed significant attention in the 1980s [11, 20] for itsconnection to string theory [58]. In four dimensions, it has been treated, e.g., in [54, 22].A suitable field diffeomorphism, i.e. a global redefinition φ ( x ) = ∞ (cid:88) j =0 a j ρ j +1 ( x ) , a = 1 , (1.2)of the field variable φ in terms of another field ρ by constant coefficients a j ∈ C , can possiblyturn a theory with non-polynomial interaction term into a theory with polynomial interaction, butnon-standard propagator [72]. This motivated the study of such non-standard propagators, wherethe exponential superpropagator G ( x ) := exp (cid:0) − g G F ( x ) (cid:1) , G F ( x ) a suitable free propagator(1.3)is the most prominent example. Contrary to ordinary propagators, it is not a tempered distributionin position space [31] and defining its Fourier transform to momentum space requires additionalassumptions or constraints. This has produced a number of results, e.g. [52, 47, 5], which differ byfinite terms δ k in each order in p . With the conventions of [9], G ( p ) = ig π p + iπ g ∞ (cid:88) k =0 − k g k ( p ) k (cid:16) ln (cid:16) g p (cid:17) + δ k (cid:17) Γ( k + 1)Γ( k + 2)Γ( k + 3) . (1.4)1.3. Notation and conventions.
Let ˆ s be the differential operator defining a free field theoryvia its Lagrangian L = 12 φ ˆ sφ. (1.5) he Fourier transform s p of the field differential operator, s p · e ipx := ˆ se ipx , (1.6)is called the offshell variable of the theory. A momentum p is said to be onshell if s p = 0. Example 1.1 (Standard scalar theories) . The most common free scalar theory has the Lagrangian L = − φ∂ µ ∂ µ φ − m φ , field differential operator ˆ s = − ∂ µ ∂ µ − m and offshell variable s p = p − m . The corresponding massless theory is obtained by setting m = 0, i.e. L = − φ∂ µ ∂ µ φ, ˆ s = − ∂ µ ∂ µ , s p = p . In the following, we will make reference to a specific free Lagrangian of type eq. (1.5) by itscorresponding offshell variable. Addition of indices is understood as acting on the momenta, i.e. ifthere are numbered momenta p , p , . . . then s := s p + p which in general is not equal s + s .If an edge e of a Feynman graph carries momentum p e then we similarly write s e ≡ s p e for thecorresponding offshell variable.Many statements of the present paper are purely combinatoric in nature and can be formulatedin any dimension of spacetime. For the sake of clarity, in all explicit calculations we use dimensionalregularization [8, 66], working in D = 4 − (cid:15) dimensions. In this way, a Feynman graph evaluates toa Laurent series in the regularization parameter (cid:15) . The part of this series with negative exponentsof (cid:15) diverges in the physical limit D →
4. it represents the divergent part of the graph underconsideration. For a graph Γ, we define R [Γ] to be the projection of the amplitude onto itsdivergent part, Γ = ∞ (cid:88) k = − n (cid:15) k c k ⇒ R [Γ] := − (cid:88) k = − n (cid:15) k c k . (1.7)The propagator G F ( z ) of a theory is defined as the Green function of the corresponding fielddifferential operator eq. (1.5), ˆ sG F ( z ) = iδ ( z ). In momentum space it is given by G F ( p ) = is p + i , (1.8)where, in the following, the causal prescription + i e in a Feynman graphin momentum space thus contributes to the amplitude with a factor is e . The 4-dimensional Fouriertransforms of the propagator is p for a massless field reads, e.g. [29, p.30], s = p : G F ( x ) = (cid:90) d D k (2 π ) D ik e − ikx = i (2 π ) x (cid:0) (cid:15) (cid:0) γ E + ln( πx ) (cid:1) + O ( (cid:15) ) (cid:1) . (1.9)We will often encounter symmetric sums over all permutations of certain terms, therefore weintroduce a shorthand notation: Definition 1.2.
The expression (cid:104) k (cid:105) f ( x , x , . . . , x n ) denotes the sum over all k different permuta-tions of arguments in the function f ( x , . . . , x n ). Example 1.3 (4-valent tree amplitude) . There are six different ways to choose two out of fouroffshell variables but external momentum conservation of a 4-valent graph identifies them pair-wise ike in s = s . Consequently there are only three actually different ways of building suchoffshell variables: (cid:104) (cid:105) s = 1 s + 1 s + 1 s Field diffeomorphisms.
Diffeomorphisms of free fields have been studied in detail recently[42, 43, 2, 50]. In this section we list some central results. If a diffeomorphism eq. (1.2) φ ( x ) = (cid:80) ∞ j =1 a j − ρ j ( x ) is applied to a free Lagrangian eq. (1.5), one obtains a theory with an infinite setof n -valent interaction vertices with Feynman rules [2] iv n = i n − (cid:88) k =1 a n − k − a k − ( n − k )! k ! (cid:88) P ∈ Q ( n,k ) s P (1.10)where s P is the offshell variable eq. (1.6) of a set of momenta P and Q ( n,k ) is the set of all possibilitiesto choose k out of n external edges without distinguishing the order. Especially, for k = 1 and k = n −
1, the summands are proportional to the offshell variables adjacent to the vertex, iv n = i ( n − a n − ( s + . . . + s n ) + O ( s i + j ) . (1.11)For s = p − m , the sum can be computed explicitly to produce [43] iv n = if n · ( s + s + . . . + s n ) + ig n m , (1.12) where f n = B n − , (2! a , a , . . . ) + B n − , (2! a , a , . . . ) ,g n = 12 n ( n − n − (cid:88) k =0 a n − k − a k ( n − k − k. Here, B n,k ( x , . . . ) are the incomplete Bell polynomials, see e.g. [17]. If additionally m = 0, suchthat s p = p , then by momentum-conservation all sums of partitions of momenta can be rewrittenin terms of the external momenta and iv n = i ( n − · ( s + s + . . . + s n ) · n − (cid:88) k =0 a n − − k a k ( n − k − k + 1) . (1.13)For scalar fields where the propagator is of quadratic order in momentum, so s p = p or s p = p + m , essentially all propagator-cancelling theories are diffeomorphisms of a free field: Theorem 1.4.
Consider a scalar field theory ρ with propagator quadratic in momentum. (1) If ρ has interaction vertices iv n = k n · ( p + . . . + p n ) + r n ∀ n > where k n , r n ∈ R , then ρ is a unique diffeomorphism of a field φ such that the vertices of φ are independent ofmomenta, iv (cid:48) n = ir (cid:48) n where r (cid:48) n ∈ R . (2) If ρ is additionally a massless field and has interaction vertices iv n = k n · ( p + . . . + p n ) ,then it is a diffeomorphism of a free massless scalar field φ . (3) There is no diffeomorphism between two power-counting renormalizable theories.Proof.
For the first two cases, the most general interaction vertex which can be obtained from thecorresponding φ can be brought into the required form, see [2]. For (2), see eq. (1.13) and notethat by tuning all parameters a n , the proportionality constants of each vertex can be adjustedindependently to match any given k n .(3) follows from the fact that a theory can only be power-counting renormalizable if no vertex isproportional to squared momenta. By (1), there is a unique diffeomorphism to produce this form.The resulting theory might or might not be renormalizable, depending on valence n of the verticeswhere the remaining coefficients are r (cid:48) n (cid:54) = 0. (cid:3) .5. Propagator cancellation and tree sums.
The vertices eq. (1.10) are capable of cancellinga propagator is e where e is an edge adjacent to the vertex by means of s e · is e = i . This means thata vertex iv n residing in a graph Γ will generally change the topology of Γ which poses a challengeto computations as well as combinatorial arguments in perturbation theory.It turns out that the vertex Feynman rules fulfil - for any choice of the diffeomorphism parameters { a i } i - an infinite set of identities, as is seen most clearly in the form eq. (1.10): The momentum-dependent factors s P are in one to one correspondence with the possibilities to construct the n -valentvertex by joining two vertices of lower valence with an internal propagator. Since each vertex andpropagator comes with an imaginary factor i , a graph with two joined vertices carries an overallminus sign compared to a single vertex at the same position. In this way, the contributions of s P to a n -point-vertex cancel out the contributions of the same s P arising from two joined vertices oflower valence. All that remains from iv n are those terms where s P = s e is the offshell variable ofan external edge of iv n and we have (cid:88) v j (cid:63)v k = v n iv j is P iv j (cid:12)(cid:12)(cid:12) Terms not proportional to an external s e = − iv n (cid:12)(cid:12)(cid:12) Terms not proportional to an external s e . (1.14)The product v j (cid:63) v k means summation over all ways to choose the valences j, k and also all possiblepermutations of external edges. On the right hand side the restriction implies that both vertices v j and v k cancel the intermediate propagator is P such that an overall factor s P arises, which, if P is a non-trivial partition, is not proportional to any external s e . From eq. (1.14) it follows that inthe connected tree-level n -point amplitude, the only remaining contributions are proportional tothe external offshell variables s e , that is, this amplitude has the form − ib n ( s + . . . + s n ) =: iV n (1.15)where b n is a c-number independent of kinematics. Definition 1.5 (Tree sum b n ) . The tree sums b n for n ≥ ρ with a total of n external edges, where n − s e = 0 for these edges e ) and the last external edge is offshell. The propagator is e of this offshell edge e is included in b n . Finally, b := 1.It can be shown [43, 2] that, regardless of the concrete form of s p , b n +2 = n (cid:88) k =1 ( n + k )! n ! B n,k ( − a , − a , . . . , − n ! a n )(1.16)where B n,k are the incomplete Bell polynomials. Moreover, the same b n also are the coefficients ofthe inverse diffeomorphism of eq. (1.2), ρ ( x ) = ∞ (cid:88) n =1 b n +1 n ! φ n ( x ) . (1.17)This implies a n = 1( n + 1)! n (cid:88) k =1 B n + k,k (0 , − b , − b , − b , . . . ) . (1.18)1.6. The connected perspective.
The fact that the tree sums definition 1.5 are mere numberswithout any remaining internal propagators motivates to use these tree sums as metavertices incomputing connected correlation functions. This approach is dubbed connected perspective , asopposed to the ordinary perspective with vertex Feynman rules eq. (1.10), and works according tothe following Feynman rules: heorem 1.6 (Feynman rules of the connected perspective of a free field diffeomorphism) . Assuming vanishing of tadpoles, the n -point connected amplitude is obtained by summing over allgraphs Γ such that (1) Each internal edge e ∈ Γ contributes a propagator factor is e . (2) Γ is built from ( k > -valent metavertices with amplitude iV k = − ib k ( s + s + . . . + s k ) .Keeping a summand s e in this amplitude amounts to cancelling the adjacent edge e . (3) The metavertices do not cancel internal edges of Γ . (4) There are no internal metavertices.
Point (4) is redundant and kept for clarity. An internal metavertex is one that is not adjacentto any external edge of the graph.These Feynman rules are slightly unconventional, but they have the advantage that no morecancelling of edges occurs, as opposed to eq. (1.12). The so-constructed graphs Γ coincide withthe topologies which eventually are left if the Feynman amplitudes are constructed naively fromvertices v n . For further explanations and examples see [2].If tadpole graphs vanish then all - tree-level and loop-level - connected correlation functions ofthe diffeomorphism field ρ differ from the respective ones of φ only by terms proportional to someoffshell variable s e for an external edge e . The onshell connected correlation functions, that is, theelements of the S -matrix, are unaltered by a diffeomorphism.Note that the construction of connected tree-level amplitudes with theorem 1.6 reminds of onshellmethods from S -matrix theory. For example, the tree-level amplitude with precisely two externallegs offshell has the general form (cid:80) iV j · is e · iV k where the internal edge e crucially is not cancelled.That means, from the perspective of the metavertices V j,k , this edge appears “onshell” in the sensethat the terms ∝ s e vanish in V j,k . In other words, the connected tree-level amplitude has polesonly arising from internal propagators as asserted in BCFW relations [12, 13]. However, in thepresent case the individual constituents iV j are not completely onshell because then they wouldvanish identically. But at least they behave like onshell amplitudes with regard to the internaledges.1.7. Content.
Section 2 gives an interpretation of field diffeomorphisms in position space andthereby a posteriori motivates the peculiar Feynman rules of the connected perspective, theorem 1.6.In section 3 we derive the connected 2-point-function in momentum space. In section 4, we extendthe connected perspective to also incorporate counterterm vertices. We compute several of thesemeta-counterterms and show how to use them to remove subdivergences. Section 5 concerns aclass of diffeomorphisms, where b n = λ n − , called exponential diffeomorphism. We compute theexplicit functional form of all such diffeomorphisms in position- and momentum space as well as the2-point-functions. Further, we demonstrate that for this class of diffeomorphisms, all connectedamplitudes can be computed from 2-point functions. In section 6, we extract 1PI countertermsfrom the previously computed meta-counterterms. In particular, we show that for the exponentialdiffeomorphism, the one-loop counterterms are structurally equal to the bare vertices iv n and thatthis is not the case at two-loop order. Finally, we derive the 1PI counterterm of the 2-point-functionto all orders in perturbation theory. In section 7, we show that diffeomorphism-invariance of the S -matrix implies infinitely many Slavnov-Taylor identities between the 1PI counterterms. In section 8we explore the structural similarities between a field diffeomorphism and a nonabelian gauge theory.We argue that the recursion relations which define the connected tree-level amplitudes in the scalarcase are the equivalent of Berends-Giele relations in QCD. . Correlation functions in position space
Structure of correlation functions.
The transformed field ρ ( x ) is related to the underlyingfree field φ via the inverse diffeomorphism eq. (1.17), ρ ( x ) = ∞ (cid:88) n =1 b n +1 n ! φ n ( x ) . (2.1)This allows for a straightforward computation of the correlation functions of ρ in position space interms of the correlation functions of φ by expanding each field operator ρ ( x ) according to eq. (2.1).The 1-point function is just the expectation value of ρ ( x ). Wick’s Theorem [75] resolves (cid:104) φ n ( x ) (cid:105) into a sum of all possible products of position space propagators eq. (1.9). Since there is onlyone spacetime point, they are G F (0), which corresponds to a position-space tadpole graph and isindependent of momenta. We assume that all tadpole-graphs vanish, G F (0) ! = 0 . (2.2)Consequently the 1-point-function vanishes as well (cid:104) ρ ( x ) (cid:105) = 0.For the 2-point function, the Wick expansion reads (cid:104) ρ ( x ) ρ ( y ) (cid:105) = ∞ (cid:88) t =1 ∞ (cid:88) t =1 b t +1 b t +1 t ! t ! (cid:10) φ t ( x ) φ t ( y ) (cid:11) . The right hand side are correlation functions of a free field φ . By Wick’s theorem and using eq. (2.2),the terms contributing to the 2-point function consist of an arbitrary number of edges between thetwo spacetime points x, y . Especially, as they do not involve any other vertex than these two, theycan be interpreted as Feynman diagrams on two external vertices, see graph A in fig. 1. There are t ! different Wick contractions for each summand and (cid:104) ρ ( x ) ρ ( y ) (cid:105) = ∞ (cid:88) t =1 b t +1 t ! t ! t ! G t F ( x − y ) = ∞ (cid:88) t =1 b t +1 t ! G tF ( x − y ) . (2.3)Conversely, this position-space representation of the 2-point-function allows to compute masslessmultiedges via Fourier transform as done in [7].The 3-point function is sketched in graph B in fig. 1 and can be written as (cid:104) ρ ( x ) ρ ( y ) ρ ( z ) (cid:105) = ∞ (cid:88) t =1 ∞ (cid:88) t =1 ∞ (cid:88) t =1 b t +1 b t +1 b t +1 t ! t ! t ! (cid:10) φ t ( x ) φ t ( y ) φ t ( z ) (cid:11) . Let l , l , l be the number of propagators between the points, then t = l + l , t = l + l , t = l + l ⇔ l = 12 ( t + t − t ) , l = 12 ( t + t − t ) , l = 12 ( t + t − t ) . We can rewrite the sums over t i in terms of l i where l i ≥ t j ≥
1. Afterworking out the combinatoric prefacotrs, the non-tadpole part of the 3-point function reads (cid:104) ρ ( x ) ρ ( y ) ρ ( z ) (cid:105) = (cid:88) lj ∈ N l + l + l ≥ b t +1 b t +1 b t +1 l ! l ! l ! G l F ( x − y ) l G F ( y − z ) G l F ( x − z ) . This structure continues in all n -point functions, see e.g. C in fig. 1. For the 4-point-function, therelations between t j and l j read t = l + l + l , t = l + l + l , t = l + l + l , t = l + l + l . : φ t ( x ) φ t ( y ) B : φ t ( x ) φ t ( z ) φ t ( y ) l l l C : φ t ( x ) φ t ( v ) φ t ( z ) φ t ( y ) l l l l l l Figure 1.
Contributions to connected correlation functions in position space. Eachdot represents a factor of φ . Graphs where dots of the same monomial φ j (i.e. samespacetime-point) are connected, are excluded due to eq. (2.2). A : 2-point function, B : 3-point function, C : 4-point function.Summing over all permutations, one then obtains (cid:104) ρ ( x ) ρ ( y ) ρ ( z ) ρ ( v ) (cid:105) = (cid:88) l ,...,l ∈ N t j ≥ ∀ j b t +1 b t +1 b t +1 b t +1 l ! l ! l ! l ! l ! l ! G l F ( x − v ) G l F ( x − y ) · · · G l F ( x − z ) . (2.4)The requirement t j ≥ l = 1 , l = 1 and all other l j zero is a valid contribution to eq. (2.4),yet it is not a connected correlation function since it is actually a product of two 2-point-functions. Theorem 2.1.
If tadpoles vanish, then the complete n -point amplitude in position space is (cid:104) ρ ( x ) · · · ρ ( x n ) (cid:105) = (cid:88) l ,...,lk ∈ N t j ≥ ∀ j b t +1 · · · b t n +1 l ! · · · l k ! G l F ( x − x ) G l F ( x − x ) · · · G l k F ( x n − − x n ) where k = n ( n − is the number of ways to form pairs x i − x j . t j are sums of n − indices l i labellingthe n − edges incident to a vertex j in a completely connected graph on n vertices. Especially,each l i appears in precisely two distinct t j and each pair { t i , t j } shares precisely one l i .Proof. By Wick’s theorem, the complete amplitude is a sum of all possible contractions. It remainsto show that the combinatorial factors take the claimed form.Consider a completely connected graph on n vertices and choose an arbitrary labeling of its n ( n − =: k edges with non-negative integers { l i } . Each vertex j is incident to precisely n − t j . Each edge connects two vertices, so eachedge label l i contributes to precisely two distinct t j . Also, each pair of vertices { i, j } is connectedby precisely one edge, hence the corresponding { t i , t j } have precisely one edge label l i in common.The diffeomorphism eq. (2.1) implies t j ≥ j , hence the summation over { l i } mustensure these conditions but is otherwise unconstrained and l i = 0 is allowed. Now fix some j andexamine the vertex j . It has connections to ( n −
1) remaining vertices with, say, multiplicities l , . . . , l n − . There are (cid:0) t j l (cid:1) ways to choose the l strands leading to the first neighbor vertex, what emains are t j − l unused points at j . So there are (cid:0) t j − l l (cid:1) choices to connect to the second neighborvertex and so on. The same holds true for any other vertex, so there arises an overall symmetryfactor of (cid:18) t l (cid:19)(cid:18) t − l l (cid:19) · · · (cid:18) l n − l n − (cid:19) · (cid:18) t l (cid:19)(cid:18) t − l l n (cid:19) · · · (cid:18) l n − l n − (cid:19) · · · = t ! · · · t n ! l ! l ! l ! l ! · · · l k ! l k ! . This factor counts the possibilities to permute how vertices are connected, but not the additionalpossibilities to permute the l i internal lines of such a connection, which induces another factor l i !for each connection. Finally, each vertex j comes with a factor b tj +1 t j ! and hence the overall factorof the summands is as claimed.Each index l j counts the number of Wick contractions between the corresponding pair { i, j } ofvertices. This translates to a factor G l j F ( x i − x j ) in the summand. (cid:3) Note that the combinatoric prefactor is excactly what one would expect for Feynman graphswhere a j -valent vertex has the amplitude b j j ! . These are the ordinary Feynman amplitudes inposition space. They do not involve integrals since integration would only be necessary for unde-terminded inner points, not for loops as in momentum-space.Note further that by b t j depending on ( n −
1) of the indices l j , the k sums in theorem 2.1 arenot independent from each other as long as the coefficients { b t j } are unspecified.2.2. Interpretation.
The position-space correlation functions as computed from Wick’s theoremin section 2.1 allow for a very transparent interpretation of the momentum-space Feynman rules ofthe connected perspective theorem 1.6. Namely, the fact that the metavertices cancel an adjacentpropagator in momentum space is equivalent to this vertex being an external (i.e. at the position ofan argument of the n -point function), not an inner (i.e. at an unspecified position to be integratedover) vertex in position space. To see this, consider first the n -valent metavertex iV n in momentumspace, including its adjacent propagators. This is supposed to be the tree-level contribution to theconnected n -point amplitude. We set s (cid:54) = 0 and s = . . . = s n = 0. (cid:104) ρ ( p ) · · · ρ ( p n ) (cid:105) = − ib n s i n s · · · s n = b n n (cid:89) j =2 is j . (2.5)Now consider the summand in the connected n -point amplitude in position space which is propor-tional to b n . By theorem 2.1, it is the summand where all b t j +1 except one are 1 and consequently t j = 1 . The sum over { l j } then collapses to n terms, namely, each l j is either 0 or 1 such that thesum of the l j is n −
1. The term corresponding to eq. (2.5) is the one where t = n − (cid:104) ρ ( x ) · · · ρ ( x n ) (cid:105) = b n n (cid:89) j =2 G F ( x j − x ) . (2.6)Such terms can be matched one by one to corresponding terms in eq. (2.5): The term with position-space propagators originating from one position x j corresponds to the term where the momentum-space propagator is j is cancelled, as can be seen by an explicit Fourier transform of eq. (2.6), usingthe momentum-space propagators eq. (1.9): (cid:104) ρ ( p ) · · · ρ ( p n ) (cid:105) = n (cid:89) k =1 (cid:90) d x k b n n (cid:89) j =2 G F ( x j − x ) n (cid:89) i =1 e ip i x k = b n δ ( p + p + . . . + p n ) n (cid:89) j =2 is j . he result equals eq. (2.5) up to an overall delta function, which is not written in the momentumspace functions by convention. The relation between both terms is represented graphically in thefirst row of fig. 2. position-space: ∝ b φ ( x ) φ ( x ) φ ( x ) φ ( x ) φ ( x ) momentum-space: p p p p p ∝ b b φ ( x ) φ ( x ) φ ( x ) φ ( x ) φ ( x ) p p p p p ∝ b b b φ ( x ) φ ( x ) φ ( x ) φ ( x ) φ ( x ) p p p p p Figure 2.
Correspondence between momentum-space and position-space Feynmanrules for selected graphs of the 5-point function. If a vertex cancels an externalpropagator in momentum space, it is turned from an internal to an external vertexin position space. A momentum p i is the Fourier transform of a position x i . Theedge which is cancelled by a metavertex is marked with an arrowhead.Their counterpart in position-space makes it completely obvious why the momentum-space Feyn-man rules in the connected perspective theorem 1.6 allow for maximal n metavertices in a n -pointamplitude: There are not more than n points in position-space which could possibly be turned intometavertices. This argument can even be applied to loop graphs in momentum-space as shown inthe bottom row in fig. 2. If a momentum-space amplitude contains loops, it involves integrals over nternal momenta. All integrations in momentum-space are leftovers of the Fourier transform, theirultraviolet divergence corresponds to two of the external coordinates approacing each other.3. The 2-point function in momentum space
Using the Feynman rules of the connected perspective theorem 1.6, the connected amplitudewith two external momenta is supported on Feynman graphs with up to two metavertices, bothof which are external. Excluding Tadpoles, the only remaining graph topology is that of l -loopmultiedges M ( l ) where the vertices are ( l + 2)-valent metavertices iV l +2 = − ib l +2 · s cancelling theexternal propagator is , see fig. 3. The two external propagators are not included. G ( s ) = − is + ∞ (cid:88) l =1 ( − ib l +2 s ) M ( l ) ( s )( l + 1)! =: − is (cid:16) G fin2 ( s ) + G div2 ( s ) (cid:17) + O ( (cid:15) ) . (3.1)Here we have separated the singular part of G ( s ) where R is defined in eq. (1.7): G div2 ( s ) := is R [ G ( s )] = − is ∞ (cid:88) l =1 b l +2 M ( l )div ( s )( l + 1)! . (3.2) G ( s ) = + + + . . . = − is + ( − ib s ) M (1) ( s ) + ( − ib s ) M (2) ( s ) + . . . Figure 3.
The amputated connected two-point-amplitude G in momentum-spacewith external momentum s := p in the connected perspective. Only multiedgescontribute. The two external propagators are not included. Both metaverticescancel the external edges as indicated with arrowheads. Example 3.1 (Massless theory) . In all explicit examples, we consider the massless theory s p = p in D = 4 − (cid:15) dimensions. Using lemma A.1, G ( s ) = − is − ∞ (cid:88) l =1 b l +2 ( l + 1)! ( − is ) l +1 (4 π ) l ( l !) (cid:18) (cid:15) − l + 1) H l − lγ E − l ln s + l ln(2 π ) (cid:19) + O ( (cid:15) ) . The two constituents are G fin2 ( s ) = − ∞ (cid:88) l =1 b l +2 ( l + 1)! ( − is ) l (4 π ) l ( l !) ((2 l + 1) H l − − lγ E + l ln(4 π ) − l ln s ) , (3.3) G div2 ( s ) = − ∞ (cid:88) l =1 b l +2 ( l + 1)! ( − is ) l (4 π ) l ( l !) (cid:15) . (3.4)As announced in section 2.2, the divergent part G div2 corresponds to the coincidence of points inposition space, i.e. x = 0, as can be verified by compuing the Fourier-transform of the summandsin eq. (3.2), using e.g.[34, pp. 155, 163], which produces terms ∝ ( i (cid:3) ) l δ ( x ).The following comments are in order: • Even if the multiedges M ( l ) are technically 1PI graphs, G ( s ) is indeed the amputated con-nected , not the 1PI 2-point-function. The latter requires systematic use of 1PI countertermsand will be computed in section 6.3. In the onshell limit s → G ( s ) reproduces the free two-point function. The S -matrix isunaltered as claimed in [2]. • One of G fin2 and G div2 can be chosen freely by picking suitable coefficients b n . If one of thosefunctions is fixed, the other one is, too. Especially, to have G ≡ − is for all s (i.e. afree 2-point function), one has to set all b j = 0 and consequently all n -point-functions arefree, such a theory is a free theory altogether. This is in accordance with the Jost-Schroer-Federbush-Johnson-Theorem [32, 24, 57]. • One could be tempted to choose b n ∝ √ (cid:15) such that G div2 ( s ) becomes a regular function, butthis choice does not remove the divergences of higher n -point functions. It seems impossibleto render the theory finite by choosing an “infinitesimal” diffeomorphism.At least in the massless case, some higher correlation functions can in principle be computedbut the explicit results are not of immediate interest for the following sections. For illustration,we consider the 3-point-function in appendix B. Here, we merely note that starting from two looporder, the 3-point and all higher correlation functions contain non-lokal divergences such as (cid:15) ln s in eq. (B.2). This indicates that, to remove these subdivergences, we have to extend the connectedperspective in a way to also include counterterm vertices, which will be done in the following section.4. Divergences in the connected perspective
In this chapter, we extend the formalism of the connected perspective (theorem 1.6) to in-corporate even counterterms, eventually giving rise to finite amplitudes. To this end, we define“meta-counterterms” C k which share the combinatorial properties of metavertices: they absorb allpossible internal cancellations and appear in graphs without changing the graph topology.It turns out that the metacounterterms can be classified by three numbers j, k, l correspondingto the graphs they arise from, namely C ( l ) n,k cancels the superficial divergence of graphs with • n external legs • k ≤ n external legs offshell (which implies precisely k metavertices) • l loops.For fixed n , the l -grading also implies an overall order in momentum. To impose k ≤ n , we definie C ( l ) n,k>n := 0 ∀ n, k, l since there are no graphs and consequently no divergences with k > n , i.e. more metaverticesthan external legs. All counterterms are in the minimal subtraction (MS) scheme in dimensionalregularization, but remember that we demand vanishing of tadpoles. In massive theories, wheretadpoles do not vanish automatically, they produce an additional contribution to the countertermswhich we do not include explicitly.In the connected perspective, by theorem 1.6 a metavertex must not cancel internal edges . Thesame restriction applies to meta-counterterms with the only difference that a meta-countertermcan cancel more than one of its adjacent edges. Theorem 4.1.
In the connected perspective of a free field diffeomorphism, all amplitudes can bemade finite if meta-counterterms are included according to the following rules: (1)
Proceed according to the BPHZ renormalization prescription, recursively replacing divergentsubgraphs γ ⊂ Γ by their corresponding meta-counterterm which subtracts the divergence.Finally, remove the superficial divergence. (2) The meta-counterterm C n,k is inserted in place of a graph on k metavertices, it cancels k out of its n adjacent edges simultaneously. (3) No internal edge of a graph must be cancelled, neither by a metavertex nor by a meta-counterterm. There are neither internal metavertices nor internal meta-counterterms.
Like in theorem 1.6, Point (4) is implied by the first three and kept for clarity
Proof.
For the general procedure to render Feynman integrals finite, we can ignore the physicalinterpretation of the connected perspective and just apply the well-known procedure of BPHZrenormalization [6, 28, 76, 74, 39, 37]. That is, in an l -loop amplitude one first has to subtract allsubdivergences which arise from graphs with less than l loops and finally the superficial divergence,which is guaranteed to be local.To be shown is the compatibility of this procedure with the combinatorial restrictions of theconnected perspective, i.e. that there is an 1:1 correspondence between divergent subgraphs andmeta-counterterms which are inserted as stated in the theorem. ⇒ : If γ ⊂ Γ is a divergent subgraph with n legs and k metavertices, then all k metaverticesare external in Γ and γ is adjacent to at least k external legs of Γ. At the same position a meta-counterterm C n,k can be inserted since at least k external edges are available to be cancelled. ⇐ : A meta-counterterm C n,k is, amongst others, a sum over all different ways to assign the k cancellations to its n ≥ k legs. If a leg e is cancelled, then it is adjacent to a metavertex ofthe graph whose divergence contributes to C n,k . Consequently, when inserted into a graph Γ, ameta-counterterm is a sum of counterterms, some of which would cancel internal edges of Γ. Bythe theorem, we must not include the terms which cancel internal edges. This means that onlythose terms in C n,k remain which are divergences of allowed graphs in the connected perspective.Finally, if Γ is a divergent graph with n external edges and k metavertices and superficial degreeof divergence greater than zero, then the amplitude of Γ contains some new positive power ofmomenta s j which did not arise from metavertices. One might think that this factor invalidatesthe above discussion since it is in principle arbitrary which momentum is chosen as a scale variable,thus introducing an arbitrary edge cancellation.This ist not so, because in fact s can not be choosen freely amongst the external momenta of γ . It can only depend on the total momenta entering the graph γ at metavertices. Let V be ametavertex adjacent to j > s , . . . , s j . Then s can only depend on s ... + j , not onany of the individual momenta. But a factor of s ... + j does not cancel any adjacent propagator,therefore it does not influence the above discussion. Now let V be a metavertex adjacent to only asingle external leg s . Then it is possible to choose s = ∝ s and s cancels the adjacent edge. Butthat edge is cancelled anyway since V needs to cancel one of its external legs by theorem 1.6. Soagain, the overall momentum scale can not introduce any new cancellations. (cid:3) Example 4.2. (3-point-function) The requirement of not cancelling internal edges automaticallyselects the correct parts of the meta-counterterms. Consider the three-loop graph Γ shown infig. 4. It has a quadratic subdivergence γ ⊂ Γ . This subdivergence is removed by the countertermgraph ˜Γ where a meta-counterterm C (2)4 , is inserted into the cograph Γ γ . On the other hand, thegraph Γ amounts to a different orientation of γ in the same cograph. However, Γ is not presentin the connected perspective since it has an internal metavertex. This restriction is automaticallyrespected by the meta-counterterm C (2)4 , : When cancelling two edges, only those graphs contributeto C (2)4 , where said edges are incident to two distinct metavertices, see fig. 5. If we label the edgesof C (2)4 , as 1 , , ,
4, then the graphs shown in fig. 5 are ∝ s or permutations thereof, but not ∝ s j where j ∈ { , , , } . The only cancellations stem from the metavertices. Lemma 4.3.
In the graphs contributing to the connected n -point amplitude, all possible subdiver-gences can be removed with meta-counterterms C m,k where k < n .Proof. Follows from theorems 1.6 and 4.1: The connected n -point amplitude is supported on graphsΓ with at most n metavertices. A subdivergence is given by a subgraph γ with k vertices where = 12 γ Γ = does notexist γ ˜Γ = (2) Figure 4.
Three-loop contributions to the 3-point function. They are two differ-ent ways to insert the divergent subgraph γ , only Γ contributes in the connectedperspective. The meta-counterterm is drawn as a crossed out metavertex. (2) = C (2)4 , = −R + + does notexist + . . . Figure 5.
Relevant part of C (2)4 , . Indicated by arrows, the rightmost graph involvesa metavertex which cancels two adjacent edges simultaneously, and another onewhich cancels no adjacent edge. By theorem 1.6, such metavertices do not exist.Consequently, this graph does not contribute to C (2)4 , . k ≤ n . If k = n , then the cograph Γ γ contains just a single metavertex and is a tadpole whichis assumed to vanish. Hence k < n . The superficial divergence of a graph with k metavertices issubtracted in C m,k where m ≥ k is the number of legs. If γ is primitive, this finishes the proof,otherwise proceed inductively, using meta-counterterms C m ,k where k < k . (cid:3) k=0 and k=1 legs offshell. If all external legs are onshell, i.e. k = 0, then the amplitudesof the connected perspective vanish, consequently there is no divergence and C ( l ) n, = 0 ∀ n, l ⇒ C n, = 0 . (4.1)If only one external leg is offshell, the amplitude is supported on graphs with a single metavertex.Such graphs are tadpoles and we assume them to vanish. We therefore have C ( l ) n, = 0 ∀ n, l ⇒ C n, = 0 . (4.2)Graphically, these two identities are shown in fig. 6.... = 0 ... = 0 Figure 6.
The meta-counterterms C n,k for connected amplitudes vanish identicallyif k = 0 or k = 1, i.e. zero or one external leg is offshell.4.2. n=2 legs. The two-point-function n = 2 is supported on l -loop multiedge graphs M ( l ) ( s ).Since tadpoles are assumed to vanish, these graphs have no subdivergences. Correspondingly, o other meta-counterterms C n,k> are necessary as asserted by lemma 4.3. The l -loop meta-counterterm for the 2-point-function is just the divergent part of − M ( l ) , C ( l )2 , ( s ) = − ( − ib l +2 s ) M ( l )div ( p )( l + 1)! = b l +2 s M ( l )div ( p )( l + 1)! , (4.3)and the all-order counterterm consequently is C , ( s ) := ∞ (cid:88) l =1 C ( l )2 , = − ( − is ) ∞ (cid:88) l =1 b l +2 ( l + 1)! M ( l )div ( p ) . Example 4.4 (Massless theory) . In the massless theory in D = 4 − (cid:15) dimensions, the divergencesof multiedges are given by eq. (A.3) and therefore C , ( s ) = + ∞ (cid:88) l =1 b l +2 ( l + 1)! ( − is ) l +1 (4 π ) l ( l !) (cid:15) . This of course coincides with − ( − is ) G div2 ( s ) from eq. (3.4). Adding this counterterm, the connectedtwo-point-function eq. (3.1) is finite: G R ( p ) := G ( p ) + C , ( p ) = − is (cid:16) − G fin2 ( s ) (cid:17) . (4.4)4.3. k=2 legs offshell. A meta-counterterm with k = 2 of its legs offshell represents the superficialdivergence of a graph on 2 metavertices i.e. a multiedge. ( l ) = −R ( l ) + ( l ) Figure 7.
Meta-counterterm C ( l )3 , according to eq. (4.5). For the indicated orien-tation of cancelled edges, only two graphs contribute.For graphs on k = 2 metavertices, but with n > n = 3 external edges, one of the metavertices is adjacent to oneexternal edge and the other one to the remaining two, see fig. 7, and there are three ways to choosewhich two edges are offshell. The l -loop meta-counterterm reads C ( l )3 , = b l +2 b l +3 ( l + 1)! (cid:16) s ( s + s ) M ( l )div ( s ) + s ( s + s ) M ( l )div ( s ) + s ( s + s ) M ( l )div ( s ) (cid:17) . (4.5) ( l ) = −R ( l ) + ( l ) + ( l ) + ( l ) Figure 8.
Meta-counterterm C ( l )4 , according to eq. (4.6) for one of six orientations.With n = 4 external edges and k = 2 metavertices, two different configurations of multiedgesare possible: Either each metavertex is adjacent to two external edges or one of them to three andone to only one external edge. In the former case, the multiedge depends on a sum offshell variable i + j . There are six possibilities to choose two out of four edges offshell, each of them contributesfour graphs as shown in fig. 8; the sum can be written as C ( l )4 , = (cid:104) (cid:105) · b l +2 b l +4 s ( s + s + s ) M ( l )div ( s )( l + 1)! + (cid:104) (cid:105) · b l +3 ( s + s )( s + s ) M ( l )div ( s )( l + 1)! . (4.6)As expected, C ( l )4 , again cancels only two out of its four external edges.Computing the higher valent meta-counterterms, this pattern continues: C ( l )5 , = (cid:104) (cid:105) · b l +2 b l +5 s ( s + s + s + s ) M ( l )div ( s )( l + 1)! + (cid:104) (cid:105) · b l +3 b l +4 ( s + s )( s + s + s ) M ( l )div ( s )( l + 1)!(4.7) C ( l )6 , = (cid:104) (cid:105) · b l +2 b l +6 s ( s + . . . + s ) M ( l )div ( s )( l + 1)! + (cid:104) (cid:105) · b l +3 b l +5 ( s + s )( s + . . . + s ) M ( l )div ( s )( l + 1)!+ (cid:104) (cid:105) · b l +4 ( s + s + s )( s + s + s ) M ( l )div ( s )( l + 1)! . The meta-counterterm C ( l ) n, is a symmetric sum of terms, each of which corresponds to a way topartition n into two nonempty disjoint sets. These partitions are elegantly given by Bell polynomials B n, ( x , x , . . . ) = n − (cid:88) j =1 (cid:18) nj (cid:19) x j x n − j . (4.8)Compare e.g. eq. (4.7) to the Bell polynomials B , ( b l +2 , b l +3 , b l +4 , . . . ) = 5 b l +2 b l +5 + 10 b l +3 b l +4 B , ( b l +2 , b l +3 , b l +4 , . . . ) = 6 b l +2 b l +6 + 15 b l +3 b l +5 + 10 b l +4 Lemma 4.5.
The l -loop meta-counterterm with n edges, two of which are cancelled, is C ( l ) n, = 12 n − (cid:88) j =1 (cid:104) K j (cid:105) b l +1+ j b l + n +1 − j ( s + . . . + s j )( s j +1 + . . . + s n ) M ( l )div ( s ... + j )( l + 1)! where K j = (cid:0) nj (cid:1) .Proof. C ( l ) n, is given by the divergences of multiedge graphs M ( l ) . A multiedge graph amounts to apartition of the n external edges into precisely two nonempty disjoint sets, each of which containsthe edges connected to one of the two vertices in M ( l ) . We sum over symmetric permutations,therefore it is sufficient to store the cardinality j of one of the two sets and fix this set to be { s , . . . , s j } . The other set contains all remaining variables, which are to be connected to thesecond vertex. For one fixed permutation of the external edges, we have to sum over all ways tocancel one edge of each of the sets, this produces a factor ( s + . . . + s j ) · ( s j +1 + . . . + s n ).The momentum flowing through the multiedge is the sum of either set, we choose s ... + j . Themultiedge comes with a symmetry factor and we have to sum over j . The vertex-factors ( − i ) produce one minus sign which is cancelled because the meta-counterterm is minus the divergenceof the graph.The number of possible permutations is given by the Bell polynomial B n,k with values eq. (4.8).Note that the factor therein already accounts for the possibility to exchange both sets. (cid:3) xample 4.6 (One loop) . Assume that M (1)div is independent of momenta, this is true for examplein D = 4 − (cid:15) dimensions for quadratic propagators. Then the explicit prefactors in lemma 4.5constitute the only momentum-dependence. The product ( s + . . . + s j )( s j +1 + . . . + s n ) contains j · ( n − j ) summands. There are K j = (cid:0) nj (cid:1) such terms and the sum is symmetric. The elementarysymmetric quadratic polynomial E ( s , . . . , s n ) has n ( n − factors, therefore (cid:104) K j (cid:105) ( s + . . . + s j )( s j +1 + . . . + s n ) = (cid:0) nj (cid:1) j ( n − j ) n ( n − E ( s , . . . , s n ) = 2 (cid:18) n − j − (cid:19) E ( s , . . . , s n )and C (1) n, = E ( s , . . . , s n ) M (1)div n − (cid:88) j =1 (cid:18) n − j − (cid:19) b j +2 b n − j +2 . k=3 legs offshell. Meta-counterterms with k = 3 represent the superficial divergence oftriangle graphs where the sides are multiedges. These multiedges constitute subdivergences whichhave to be subtracted with suitable C ( l )2 ,k meta-counterterms. The latter are known from lemma 4.5and thanks to lemma 4.3 they are sufficent to eliminate all possible subdivergences. The removalof a subdivergence for k = 3 has already been illustrated in example 4.2.To clarify the procedure, we compute the counterterm of the massless connected 2-loop 3-pointamplitude from appendix B. In the connected perspective, three different topologies contribute.We will ignore in the following the multiedge graph M (2) , which produces C (2)3 , as indicated in fig. 7and was computed already in eq. (4.5). The two remaining graphs are shown in fig. 9.Γ A = s s Γ B = s s Figure 9.
The two topologies of two-loop graphs contributing to the connectedthree-vertex correlation function where all three legs are cancelled. Each graph hasthree different permutations s → s → s , they are not indicated.If s = p and D = 4 − (cid:15) , the graphs Γ A , Γ B are given by eq. (B.3) and example A.2, where wesum over all orientations:Γ A = is s s b b π ) (cid:18) (cid:15) + 1 (cid:15) (15 − γ E + 6 ln(4 π ) − s + ln s + ln s )) (cid:19) + finite terms(4.9)Γ B = (cid:104) (cid:105) · ib b s s s · M (1) ( s )2! M (1) ( s )2!= ib b s s s π ) (cid:18) (cid:15) + 1 (cid:15) (12 − γ E + 6 ln(4 π ) − s + ln s + ln s )) (cid:19) + finite terms . The topologies Γ A and Γ B involve divergent subgraphs. To eliminate the subdivergences, we needthe counterterm graph ˜Γ from fig. 10. It contains the meta-counterterm of the 1-loop multiedge, C (1)4 , from eq. (4.6), which is to be inserted into another 1-loop multiedge where it must not cancel Γ = (1) ˜Γ = C (2)3 , = (2) Figure 10.
Counterterm graphs to absorb the divergences of the graphs shownin fig. 9. ˜Γ absorbs the subdivergences of both Γ A and Γ B while ˜Γ cancels thesuperficial divergences of Γ A + Γ B .internal edges. Let s , s be the external edges as indicated in fig. 9, then the appropriate amplitudeof the meta-counterterm is C (1)4 , (cid:12)(cid:12)(cid:12) s = s =0 = 2 b b s s M (1)div
2! + 2 b s s M (1)div . Summing over all three orientations, the counterterm graph has the amplitude˜Γ = − i (cid:0) b b + b b (cid:1) s s s M (1)div (cid:16) M (1)fin ( s ) + M (1)fin ( s ) + M (1)fin ( s ) (cid:17) − i (cid:0) b b + b b (cid:1) s s s (cid:16) M (1)div (cid:17) + finite terms . If s = p and D = 4 − (cid:15) , this is˜Γ = − i (cid:0) b b + b b (cid:1) s s s π ) (cid:15) (6 − γ E + 3 ln(4 π ) − ln s − ln s − ln s )(4.10) − i (cid:0) b b + b b (cid:1) s s s π ) (cid:15) + finite terms . The remaining graph ˜Γ from fig. 10 absorbs the overall divergence, C (2)3 , = −R (cid:16) Γ A + Γ B + ˜Γ (cid:17) . (4.11)In the massless case, using eqs. (4.9) and (4.10), C (2)3 , = is s s π ) (cid:18)(cid:0) b b + b b (cid:1) (cid:15) − b b (cid:15) (cid:19) . (4.12) 5. Exponential diffeomorphisms
So far, the set of parameters { a j } j ∈ N in eq. (1.2) or equivalently { b j } j ∈ N in eq. (1.17) has beenarbitrary. In the present section, we choose u ∈ N fixed and arbitrary and require b n = n = 2 λ n − ∃ k ∈ N : uk = n −
20 else . (5.1)We call this class of theories exponential diffeomorphisms .Note that the choice eq. (5.1) implies that not only the n -valent metavertex eq. (1.15), butactually all summands of the connected n -point function are proportional to λ n − . More precisely,the connected tree-level n -point function has the form G tl n = − iλ n − ( E ( s , . . . , s n ) + R n )(5.2)where E ( s , . . . , s n ) = s + . . . + s n is the elementary symmetric polynomial of order one and R n is a rational function symmetric in { s , . . . , s , . . . , s , . . . } of overall order one in s . .1. Inverse field diffeomorphism.
The hyperbolic function of order u of the r th kind is definedas [69] H u,r ( x ) := ∞ (cid:88) k =0 x uk + r Γ ( uk + 1 + r ) . Lemma 5.1.
If the connected n -point functions ib n of a field ρ ( x ) fulfil eq. (5.1) for a fixed u ∈ N ,then ρ is related to a free field φ ( x ) by λρ ( x ) = H u, ( λφ ( x )) = λφ · (cid:40) F (1; 2 | λφ ( x )) , u = 1 F u − (cid:16) {} ; (cid:8) u , u , . . . , u − u , u +1 u (cid:9) (cid:12)(cid:12)(cid:12) (cid:16) λφ ( x ) u (cid:17) u (cid:17) u ≥ . Proof.
If a propagator-cancelling field has metavertex amplitudes b n then it is related to a free field φ by the diffeomorphism eq. (1.17) ρ ( x ) = ∞ (cid:88) n =1 b n +1 n ! φ n ( x ) = φ ∞ (cid:88) k =0 ( λφ ) ku ( uk + 1)! = φ ∞ (cid:88) k =0 ( λφ ) ku Γ( uk + 2) . For u = 1, the series is ∞ (cid:88) k =0 ( λφ ) k Γ( k + 2) = ∞ (cid:88) k =0 Γ( s )Γ( k + 2) Γ( k + 1)Γ(1) ( λφ ) k k ! = F (1; 2 | λφ ) . In the general case, use Gauss’ product formula for Gamma functions [25, Sec. 26] twice: ∞ (cid:88) k =0 uk + 2) ( λφ ) ku = ∞ (cid:88) k =0 (2 π ) u − Γ (cid:0) uku (cid:1) · · · Γ (cid:0) u +1+ uku (cid:1) u uk + ( λφ ) uk = ∞ (cid:88) k =0 (2 π ) u − Γ (cid:0) u + k (cid:1) · · · Γ (1 + k ) Γ (cid:0) u +1 u + k (cid:1) u Γ (cid:0) u (cid:1) · · · Γ (cid:0) u +1 u (cid:1) u (2 π ) u − Γ(2) ( λφ ) uk u uk . (cid:3) We note in passing that the diffeomorphisms given by eq. (5.1) fulfild u d φ u ρ ( x ) = λ u · ρ ( x )and are therefore sums of terms ρ ∝ e q i λφ where q i are u th roots of unity. This fact motivates thename exponential diffeomorphisms . Example 5.2.
For small u , the hypergeometric functions evaluate to u = 1 : ρ = λ − (cid:16) e λφ − (cid:17) u = 2 : ρ = (2 λ ) − (cid:16) e λφ − e − λφ (cid:17) = λ − sinh ( λφ ) u = 3 : ρ = (3 λ ) − (cid:18) e λφ + ( − e − ( − λφ + ( − e ( − λφ (cid:19) = (3 λ ) − e − λφ (cid:18) e λφ + 2 sin (cid:18)
16 (3 √ λφ − π ) (cid:19)(cid:19) u = 4 : ρ = (2 λ ) − (sin( λφ ) + sinh( λφ )) . .2. Field diffeomorphism.
Using eq. (1.18), the diffeomorphism coefficients a n can be computedin principle, but there seems to be no easy explicit formula. One obtains u = 1 : a n = ( − n λ n n + 1 u > a n = (cid:40) ( − k λ n ( n +1)! · α k , n = k · u { α k } k ∈ N have recently been interpreted in terms of Whitney numbers[23]. Example 5.3.
The case u = 1 amounts to α n = n !. Other examples of { α k } are u = 2 : { , , , , , . . . } [63, A001818] u = 3 : { , , , , , . . . } [63, A292750] u = 4 : { , , , , . . . } u = 5 : { , , , , . . . } For u = 1 ,
2, the function φ ( ρ ) can be obtained by inverting the function ρ ( φ ) from example 5.2: u = 1 : φ = λ − ln(1 + λρ )(5.3) u = 2 : φ = λ − asinh( λρ ) = λ − ln( (cid:112) λρ ) + λρ ) . It is instructive to write down the Lagrangian density for these diffeomorphisms in the case s = p − m , namely u = 1 : L = − ∂ µ φ∂ µ φ − m φ = −
12 1(1 + λρ ) ∂ µ ρ∂ µ ρ − m λ ln (1 + λρ )(5.4) u = 2 : L = −
12 11 + ( λρ ) ∂ µ ρ∂ µ ρ − m λ sinh ( λφ ) . In both cases, the kinetic term is multiplied by the inverse squared of the field. The mass termon the other hand becomes an unorthodox transcendental interaction term. Setting m = 0 anddefining a field (cid:37) := 1 + λρ , eq. (5.4) takes the form u = 1 : L = − λ · (cid:37) ∂ µ (cid:37)∂ µ (cid:37) (5.5)which vagely reminds of the Einstein-Hilbert-Lagrangian. Using eq. (1.13) and s j = p j , the vertexFeynman rules of the Lagrangian eq. (5.5), in terms of the field ρ = λ − ( (cid:37) − iv n = i ( − n λ n − ( n − · (cid:0) p + p + . . . + p n (cid:1) . Two-point-function in position space.Lemma 5.4.
If the field ρ ( x ) fulfils eq. (5.1) for a fixed u ∈ N , then its connected full two-pointfunction in position space is G ( z ) = 1 λ H u, ( λ G F ( z )) where G F ( z ) is the Greens function of the original field differential operator, i.e. ˆ s z G F ( z ) = iδ ( z ) and H u, can be expressed in F u − like in lemma 5.1. roof. The non-tadpole part of the 2-point function is eq. (2.3), using eq. (5.1), it reads (cid:104) ρ ( x ) ρ ( y ) (cid:105) = ∞ (cid:88) t =1 (cid:0) λ t − (cid:1) δ t − uk t ! G tF ( x − y ) = G F ( x − y ) ∞ (cid:88) k =0 (cid:0) λ G F ( x − y ) (cid:1) uk Γ( uk + 2) . (5.6)This is up to a different argument the same series as in the proof of lemma 5.1. (cid:3) Example 5.5 (Massless field) . Consider the standard, massless theory s = p with propagatoreq. (1.9). Like in example 5.2, one obtains u = 1 : G ( z ) = λ − (cid:18) exp (cid:18) i Γ(1 − (cid:15) ) λ ( z ) − (cid:15) π − (cid:15) (cid:19) − (cid:19) = λ − (cid:18) e i λ z π )2 − (cid:19) + O ( (cid:15) )(5.7) u = 2 : G ( z ) = λ − sinh (cid:18) Γ(1 − (cid:15) ) λ ( z ) − (cid:15) π − (cid:15) (cid:19) . In stark contrast to the free propagator G F (eq. (1.9)) or the perturbative 2-point function ofany renormalizable theory, these functions have an essential singularity at z = 0. But apart fromthat, they are finite in the limit (cid:15) →
0. Especially, the case u = 1 amounts to the exponentialsuperpropagator eq. (1.3).5.4. Higher correlation functions in position space.
By theorem 2.1, the n -point function inposition space for a theory fulfilling eq. (5.1) with u = 1 is (cid:104) ρ ( x ) · · · ρ ( x n ) (cid:105) = (cid:88) l ,...,lk ∈ N t j ≥ ∀ j λ t − · · · λ t n − l ! · · · l k ! G l F ( x − x ) G l F ( x − x ) · · · G l k F ( x n − − x n ) . Here, l j ∈ N represent the number of edges between a pair of points and t j ∈ N the number ofedges at one point j ∈ { , . . . , n } . Hence each l i contributes to precisely two t j and t + . . . + t n =2( l + . . . + l k ) and (cid:104) ρ ( x ) · · · ρ ( x n ) (cid:105) = 1 λ n (cid:88) l ,...,lk ∈ N t j ≥ ∀ j λ l · · · λ l k l ! · · · l k ! G l F ( x − x ) G l F ( x − x ) · · · G l k F ( x n − − x n ) . (5.8)Recognizing the 2-point function G ( z ) := ∞ (cid:88) l =0 λ l l ! G lF ( z ) , we see that eq. (5.8) is the sum of all ways to connect the n points by those 2-point functions.This gives us yet another interpretation of the recursion relations eq. (5.1): They are the uniquechoice of parameters b n for which the position-space correlation functions factor into products ofposition-space superpropagators.5.5. Two-point function in momentum space.
Assuming recursion relations eq. (5.1), theconnected amputated 2-point function eq. (3.1) for s p = p in D = 4 − (cid:15) dimensions specialises to G fin2 ( s ) = ∞ (cid:88) k =1 (cid:0) − iπ λ s (cid:1) uk Γ( uk + 2)Γ ( uk + 1) ((2 uk + 1) H uk − − ukγ E + uk ln(2 π ) − uk ln s )(5.9) G div2 ( s ) = ∞ (cid:88) k =1 (cid:0) − iλ s (cid:1) uk (4 π ) uk Γ( uk + 2)Γ ( uk + 1) . (5.10) estricting further to u = 1, the finite part G fin2 ( s ) = ∞ (cid:88) k =1 (cid:0) − iπ λ s (cid:1) k Γ( k + 2)Γ ( k + 1) ((2 k + 1) H k − − kγ E + k ln(2 π ) − k ln s )(5.11)almost coincides with the exponential superpropagator eq. (1.4). The difference of our formulaeq. (5.11) to the various historic results is a finite constant δ k for every order s k . In a renormalizabletheory, such differences are expected when using different renormalization prescriptions, comparee.g. [15]. In the present, non-renormalizable case, the historic results assumed different non-standard conditions to fix a unique Fourier-transform, our own result is in MS and assumes thevanishing of tadpoles. Equations (5.7) and (5.11) show that the exponential superpropagator canindeed be obtained within rigorous perturbation theory both in position space and in momentumspace. Lemma 5.6. In D = 4 − (cid:15) dimensions with s p = p , if a theory fulfils eq. (5.1) then the divergentpart of its connected amputated 2-point function is G div2 ( s ) = (cid:40) F (cid:0)(cid:8) ; 1 , , (cid:12)(cid:12) − iπ λ s (cid:9)(cid:1) − u = 1 F u − (cid:16) {} ; u , u , u , u , u , . . . , u − u , u − u , u − u , , , u +1 u (cid:12)(cid:12)(cid:12) (cid:16) − iλ s (4 π ) u (cid:17) u (cid:17) − u > . Proof.
Analogous to lemma 5.1. UseΓ ( uk + 2) Γ ( uk + 1) = Γ (cid:0) u + k (cid:1) · · · Γ (cid:0) u − u (cid:1) Γ (cid:0) u +1 k (cid:1) Γ (cid:0) u + k (cid:1) · · · Γ (cid:0) u − u + k (cid:1) Γ (cid:0) u (cid:1) · · · Γ (cid:0) u − u (cid:1) Γ (cid:0) u +1 u (cid:1) Γ (cid:0) u (cid:1) · · · Γ (cid:0) u − u (cid:1) Γ ( k + 1) u uk to rewrite eq. (5.10) as a hypergeometric function. (cid:3) One loop divergences.
Consider massless one-loop graphs, not including external propaga-tors. In D = 4 − (cid:15) dimensions, the only divergent one-loop graph is the one-loop multiedge M ( l ) which contributes to all n -point functions, see fig. 11. Similar to eq. (5.2), G (1) n ∝ λ n . G (1)4 = + + + convergent G (1)5 = + + ++ + + convergent Figure 11.
Divergent contribution to the 4 − and 5-point amplitudes. Permuta-tions of external edges are not indicated. xample 5.7 (5-point function) . Assuming eq. (5.1) for u = 1, the graphs in fig. 11 evaluate to G (1)4 = − λ (cid:18) (cid:104) (cid:105) s ( s + s + s ) M (1) ( s ) + (cid:104) (cid:105) s s s ( s + s ) M (1) ( s )++ (cid:104) (cid:105) ( s + s )( s + s ) M (1) ( s ) (cid:17) + convergent graphs G (1)5 = − λ (cid:18) (cid:104) (cid:105) s ( s + s + s + s ) M (1) ( s ) + (cid:104) (cid:105) s s s ( s + s + s ) M (1) ( s )+ (cid:104) (cid:105) ( s + s )( s + s + s ) M (1) ( s ) + (cid:104) (cid:105) ( s + s ) 1 s s s s M (1) ( s )+ (cid:104) (cid:105) s s s s s ( s + s ) M (1) ( s ) + (cid:104) (cid:105) ( s + s ) s s ( s + s ) M (1) ( s ) (cid:19) + c . We are interested only in the leading terms of G (1) n , which do not involve uncancelled propagators,because all other terms are products of G (1) j with j < n . The divergence of the leading terms iscancelled by the meta-counterterm C (1) n, . Lemma 5.8. In D = 4 − (cid:15) dimensions and for s p = p and assuming the relations eq. (5.1) for u = 1 , the one-loop meta-counterterm with n edges cancels precisely two of its edges and reads C (1) n ≡ C (1) n, = − λ n n − (4 π ) (cid:15) E (cid:0) p , . . . , p n (cid:1) .E ( s , s , . . . , s n ) is the elementary symmetric polynomial of order two in n variables.Proof. Follows from lemma 4.5 since at one-loop level, the only divergent graph is the one-loop-multiedge, or, equivalently, C (1) n,k> = 0. The multiedge has a symmetry factor and, as computedin example A.2, its divergent part is momentum-independent, M (1)div ( s P j ) = − (4 π ) − (cid:15) − . Thereforewe insert eq. (5.1) into example 4.6: C (1) n, = − E ( s , . . . , s n ) 12 (4 π ) − (cid:15) − n − (cid:88) j =1 (cid:18) n − j − (cid:19) λ j λ n − j = − E ( s , . . . , s n ) 12(4 π )2 (cid:15) (1 + 1) n − . (cid:3) Higher correlation functions in momentum space.
We have seen in section 5.4 thatif eq. (5.1) is fulfilled with u = 1, then the n -point-amplitudes factor into products of all-order2-point-functions in position space. A similar statement holds in momentum space. Theorem 5.9 (Feynman rules of exponential diffeomorphism) . Assume the coefficients b n fulfileq. (5.1) for u = 1 . Then the connected ( n > -point amplitude G n is given by the followingconnected graphs: (1) The vertices are metavertices V n = − iλ n − ( s + . . . + s n ) for any valence n ≥ . (2) Edges between metavertices are given by the function G ( s ) . (3) G ( s ) is the full, amputated 2-point-function, the sum of all multiedges eq. (5.11) . (4) In G n> , there is at most one edge directly between any two metavertices. (5) Every Metavertex cancels precisely one of the n external edges. (6) Every external edge which is not cancelled by a metavertex is dressed by G ( s ) .Proof. Consequence of theorem 1.6 by identification of the connected 2-point amplitude with a fullpropagator. The only point to be shown is that the recursion relations eq. (5.1) are sufficient toidentify each internal multiedge with the corresponding term in G . o see this, let there be two metavertices V n , V n and between them j propagators, this objectis ∝ λ n + n − . There are n − j resp. n − j edges left adjacent to the metavertices which are notbetween them. Now, the contribution to G which has j internal edges is the ( j − λ j − due to its two vertices V j +1 ∝ λ j − . The two vertices cancel thetwo outer edges of the multiedge, hence these propagators do not contribute to G . To make upfor the missing external edges, one needs to connect to G two metavertices V n − j +1 and V n − j +1 respectively. These produce factors λ n − j − and λ n − j − . In total, the j -edge term of C togetherwith the two vertices is ∝ λ j − n − j − n − j − = λ n + n − . This is the same factor as if we didnot insert C and instead used j internal edges.Phrased differently, connecting C to a vertex V n amounts to splitting V n into two parts V n · V n − n +2 and cancelling the intermediate propagator. But in both cases, the amplitude is λ n − byeq. (5.1). (cid:3) Observe that to compute a n -point function by theorem 5.9, assuming G is known, only finitelymany integrals remain to be solved. The graph with highest loop number is the completely con-nected graph on n vertices, it has n ( n − internal edges and hence ( n − n + 2) loops. Example 5.10 (Lowest amplitudes) . Following theorem 5.9, the lowest n -point amplitudes areshown in fig. 12. If one knows G , one can compute G with only one integration and G withthree integrations. Since all metavertices are external, the graphs constructed from theorem 5.9 allcarry symmetry factor 1. G = G = (cid:104) (cid:105) + G = (cid:104) (cid:105) + (cid:104) (cid:105) + (cid:104) (cid:105) + (cid:104) (cid:105) + (cid:104) (cid:105) + Figure 12.
Connected amplitudes of a theory fulfilling eq. (5.1) with u = 1. Aprefactor (cid:104) j (cid:105) indicates the presence of in total j graphs of the corresponding topologywhere external edges are permuted.6. Knowing the meta-counterterms C ( l ) n,k , which cancel the divergences of connected amplitudes, wecan reconstruct the amputated 1PI counterterms which we call c ( l ) n,k . The indices n, l, k have thesame meaning as for the meta-counterterms in section 4. Eventually, the sum c ( l ) n := (cid:80) nk =0 c ( l ) n,k epresents the l -loop counterterm to the 1PI n -point function, i.e. this is the object which normallyis called counterterm in a local quantum field theory.6.1. The 1PI one-loop graph of the 2-point-function coin-cides with the connected 2-point function eq. (4.3), therefore C (1)2 = C (1)2 , = b s M (1)div c (1)2 . (6.1)For the 3-point function, the connected 3-point divergence is the product of the 1PI 3-pointdivergence c (1)3 and three adjacent connected 2-point divergences. To one-loop order, this productcan contain only one divergent term in total, either c (1)3 or one of the propagator corrections,consequently C (1)3 = c (1)3 + (cid:104) (cid:105) iv is c (1)2 ( s ) . (6.2)First consider the case where all external legs are onshell, i.e. C (1)3 , . Then the meta-counterterms C (1)3 and C (1)2 vanish due to eq. (4.1) and eq. (6.2) simplifies to0 = c (1)3 , + 0 . (6.3)Now let one of the legs be offshell. The connected counterterm C (1)3 , vanishes due to eq. (4.2) butone of the terms C (1)2 in eq. (6.2) survives, so C (1)3 , = 0 = c (1)3 , + (cid:104) (cid:105) b ( − is ) is c (1)2 , ( s ) . (6.4)This implies that c (1)3 , = − (cid:104) (cid:105) b c (1)2 , ( s ) = − b (cid:0) s + s + s (cid:1) M (1)div . (6.5)For the higher n -point functions these arguments become increasingly cumbersome; they are muchmore transparent if carried out in a graphical manner, see fig. 13.eq. (6.1): (1) = (1) = b s M (1)div eq. (6.4): (1) (cid:124) (cid:123)(cid:122) (cid:125) =0 = (1) + (1) + (1) (cid:124)(cid:123)(cid:122)(cid:125) = + (1) (cid:124)(cid:123)(cid:122)(cid:125) = eq. (6.5): (1) = − (1) = − b · b s M (1)div Figure 13.
Graphical representation for the computation of c (1)3 , . The perpendic-ular line indicates an external edge which must not be cancelled by the adjacentvertex. or c (1)3 , , the meta-counterterm C (1)3 , does not vanish, see eq. (4.5). The construction of c (1)3 , isshown in fig. 14, it yields c (1)3 , = C (1)3 , − (cid:104) (cid:105) ( − ib s ) is c (1)2 , ( s ) − (cid:104) (cid:105) ( − ib s ) is c (1)2 , ( s )= (cid:104) (cid:105) b (cid:0) b − b (cid:1) s s M (1)div b (cid:0) b − b (cid:1) M (1)div · E ( s , s , s ) . (6.6)Note that we assumed that M (1)div be independent of momenta only to simplify notation, the resultwould be ∝ b ( b − b ) regardless. −R + = (1) = (1) + (1) + (1) Figure 14.
Graphical notation for the computation of c (1)3 , . The meta-countertermhas been taken from fig. 7.Finally, C (1)3 , = 0 and there is no divergent connected graph that cancels three external edges atone loop, therefore c (1)3 , = 0 . (6.7) (1) = 0 = (1) + (cid:104) (cid:105) (1) + (cid:104) (cid:105) (1) Figure 15.
Construction of the onshell 4-point meta-counterterm from metaver-tices 1PI counterterms.Figure 15 indicates the contributions to the onshell 4-point meta-counterterm C (1)4 . Usingeq. (6.5), the last two summands partially cancel and what remains is c (1)4 , = −(cid:104) (cid:105) c (1)3 , is ( − ib s ) = − b (cid:0) s + s + s (cid:1) M (1)div (cid:15) . With one external leg offshell, the meta-counterterm fig. 6 still vanishes and we obtain fig. 16.Equation (6.5) implies that the second and third graph cancel and therefore c (1)4 , = (cid:104) (cid:105) (cid:0) b − b (cid:1) b M (1)div s − (cid:104) (cid:105) b (cid:0) b − b (cid:1) M (1)div ( s + s + s ) s . The meta-counterterms C ( l ) n are always symmetric polynomials in the external momenta. Con-versely, the 1PI counterterms can also contain inner offshell variables s i + j + ... . This is completelyanalogous to the 1PI vertices iv n eq. (1.12) which, unlike the metavertices iV n , must contain suchinternal offshell variables in order to fulfil eq. (1.14).By power-counting, the one-loop 1PI counterterm c (1) n is proportional to two powers of offshellvariables, hence, five different dependencies are possible: Square of an external offshell variable s j , square of an internal one s i + j + ... , two different external ones s i s j , two different internal ones s i + j + ... s k + l + ... or a mixture of both types s j s k + l + ... . In every case, symmetric sums are understood. = 0 = (1) + (cid:104) (cid:105) (1) + (cid:104) (cid:105) (1) + (cid:104) (cid:105) (1)(1) = 0 = (1) + (cid:104) (cid:105) (1) + (cid:104) (cid:105) (1) Figure 16.
Lemma 6.1.
The summand in c (1) n , which is proportional to a square of an external offshell variable,is +( n − a n − b M (1)div (cid:0) s + . . . + s n (cid:1) , where a n is given by eq. (1.18) .Proof. This can be proved by rigorously constructing C (1) n, from trees involving precisely one 1PIcounterterm c (1) j of valence j ≤ n and arbitrary many tree sums b k . However, an inductive proof isshorter. The 3-valent term eq. (6.5) has the required form.Assume the statement holds for any valence j < n . Consider the divergence of the n -valentconnected amplitude, where one external edge e is double-cancelled. It is given by the followingcontributions, shown in fig. 17:(1) A sum over all trees where a counterterm c (1) j, double-cancels e . There is a suitable numberof tree sums b k connected to c (1) j, .(2) A sum over all trees where a counterterm c (1)2 is inserted into e , thus double-cancelling it.(3) A single vertex iv n connected to a counterterm c (1)2 in the edge e . To produce an overallfactor s e , the vertex iv n must cancel e , therefore only the part ∝ s e is relevant.(4) The counterterm c (1) n, double-cancelling e .The first two cases cancel each other because of the induction hypothesis. On the other hand, theoverall sum is zero since it is C (1) n, = 0 by eq. (4.2). Therefore,0 = iv n (cid:12)(cid:12)(cid:12) summand ∝ s e is e c (1)2 ( s e ) + c (1) n, (cid:12)(cid:12)(cid:12) summand ∝ s e . Using eq. (1.11) and eq. (6.1) and summing over all n orientations produces the statement. (cid:3) Lemma 6.2.
In the 1PI counterterm c (1) n , the contributions proportional to s p , the square of theoffshell variable of some partition p of the external momenta, is b M (1)div n − (cid:88) k =2 (cid:88) p ∈ Q ( n,k ) k ! a k − a n − k − ( n − k )! s p where Q ( n,k ) denotes the set of all possibilities to choose k out of n external legs. valence n = 0 = (cid:88) (1) valence < n + (1) + (1) valence n + (1) Figure 17.
Proof of lemma 6.1. The bracket vanishes by induction hypothesis.
Proof.
Analogous to the proof of lemma 6.1. Setting all external legs onshell, the sum of allconnected graph divergences vanishes by eq. (4.1), C (1) n, = 0 .The tree point counterterm is c (1)3 , = 0 since there are no internal edges at all. Assume now thatfor all j < n , the counterterms c (1) j, are chosen such that no factor s e remains in the sum of allconnected trees with j external legs. Following fig. 18, at n external legs, there are four structuresin the sum of connected trees, each of which involves precisely one counterterm c (1) :(1) Trees T , where at least one vertex is connected through an edge e which is not triple-cancelled. Then, triple-cancellation can only occur in the original tree T , but it has valence < n and is, when summed over all trees, free of triple-cancellation by induction hypothesis.All trees with more than two vertices fall in this category by powercounting.(2) A single counterterm, where one vertex is added through a new edge e which is triple-cancelled. There can be only one such edge e by power-counting and it represents a partitionof the external legs into precisely two sets, one on each side.(3) Two single vertices cancelling the edge e and a counterterm c (1)2 in that same edge e .(4) A new counterterm c (1) n, .Since (1) vanishes by induction hypothesis, the last three contributions have to cancel. We knowfrom lemma 6.1 that cases (2) and (3) add up to zero, only that in the present construction, case(3) has two interchangeable vertices and hence an overall factor . Consequently, (3) = − · (2)and (2) + (3) = (2).The counterterm c (1) n, must be chosen to be the sum of all ways to partition the n external legsinto two sets where the first set has k − k -valent counterterm c (1) k, andthe second set has n − k + 1 elements connected to a vertex iv n − k +2 . The set of all such partitionsis denoted Q ( n,k − . For the vertex iv n − k +2 , the only relevant summand ist the one cancelling theedge e from eq. (1.11), for c (1) k, the only relevant one is the part double-cancelling e from lemma 6.1,because all other contributions do not produce a triple-cancelled edge e . c (1) n, = − · n − (cid:88) k =3 c (1) k, is e iv n − k +2 = − n − (cid:88) k =3 (cid:88) p ∈ Q ( n,k − ( k − a k − b M (1)div · s e · a n − k ( n − k + 1)!Note that this proof is similar to the derivation of b n from iv j in [2], but backwards. (cid:3) Lemma 6.3. If b n = λ n − then c (1) does not contain any summands which are proportional to s e · s f , where e (cid:54) = f can be external or internal offshell variables.Proof. The Lemma concerns in principle three different contributions: valence n = 0 = (cid:88) (1) e no triple cancellation+ n − (cid:88) k =3 (cid:88) Q ( n,k ) (1) e valence k + (1) Figure 18.
Proof of lemma 6.2. The first sum involves a triple cancellation in thesubtree left of e . This subtree has less than n external legs and vanishes by inductionhypothesis.(1) Summands proportional to two distinct external edges,(2) Summands proportional to one internal edge and one external edge,(3) Summands proportional to two distinct internal edges.Assume case (1) is shown to vanish and consider case (2). Let e be the internal edge in question,there is a factor ∝ s e in the counterterm, that means, e is double-cancelled. There are three graphswhich can give rise to such contributions, each of them consist of the edge e and one vertex on eachof its ends: First, a vertex c ( n ) j, which double-cancels e , connected to some b k which does not cancel e . Second, a b j which cancels e , then a 2-point-counterterm c (1)2 , in e and another b k which doesnot cancel e . And third, some b j which cancels e , connected to c (1) n − n +2 , which cancels e once. Thefirst two of these add up to zero due to fig. 17. The third one is proportional to c (1) n − j +2 , which isassumed to vanish. Hence, case (2) does not contribute.Similarly, case (3) does not occur if one assumes that case (1) vanishes by the repetition of theabove argument for two distinct internal edges.It remains to show that case (1) vanishes. Use induction. If we assume that c (1) j, = 0 for all2 < j < n then at n legs there are only three topologies which cancel two distinct external edges:(1) The counterterm c (1) n, ,(2) A treesum − ib n with one counterterm c (1)2 , ,(3) Two treesums − ib j , − ib n − j +2 connected by a counterterm c (1)2 , .The latter two topologies amount to a sum over all partitions of the n external edges into twononempty distinct sets. Let j be the cardinality of one of these sets, then the two treesumscontribute b j +1 · b n − j +1 , which is also valid for j = 1 or j = n −
1. The counterterm c (1)2 , is knownfrom eq. (6.1). Finally, cancelling one external edge at each of the tree sums produces a factor( s + . . . + s j ) · ( s j +1 + . . . + s n ), where we have assumed that { s , . . . , s j } are the j edges connectedto the first tree sum. In summary, the latter two topologies produce terms of the form (cid:104) K j (cid:105) b j +1 b n − j +1 b ( s + . . . + s j )( s j +1 + . . . + s n ) M (1)div (cid:104) K j (cid:105) denotes the number of possible ways to choose j out of n momenta. We have to sumover j from 1 to n − to account for the interchangeability of thetwo vertices. ut this is precisely the same construction which defines the one-loop meta-counterterm C (1) n, inlemma 4.5. The summands match one to one, the only difference is the prefactors, we obtain c (1) n, = C (1) n, − n − (cid:88) j =1 (cid:104) K j (cid:105) b j +1 b n − j +1 b ( s + . . . + s j )( s j +1 + . . . + s n ) M (1)div
2= 12 n − (cid:88) j =1 (cid:104) K j (cid:105) (cid:0) b j +2 b n − j +2 − b j +1 b n − j +1 b (cid:1) ( s + . . . + s j )( s j +1 + . . . + s n ) M (1)div . For every j , the prefactor vanishes if b j +2 = b j +1 b . This is eq. (5.1). (cid:3) Theorem 6.4. If b n = λ n − then the one-loop counterterm c (1) n has the same structure as the 1PIvertex v n eq. (1.10) , c (1) n = v n (cid:12)(cid:12)(cid:12) s e → M (1)div2 λ s e . Proof.
Add the contributions of lemmas 6.1 and 6.2 to obtain c (1) n = ( n − a n − b M (1)div (cid:0) s + . . . + s n (cid:1) + b M (1)div n − (cid:88) k =2 (cid:88) p ∈ Q ( n,k ) a n − k − a k − ( n − k )! k ! s p = 12 n − (cid:88) k =1 (cid:88) p ∈ Q ( n,k ) a n − k − a k − ( n − k )! k ! s p b M (1)div . This equals the vertex eq. (1.10) up to the stated replacement. Given b n = λ n − , lemma 6.3 assertsthe absence of other terms. (cid:3) Theorem 6.4 asserts that if we set is R := is − ( is ) b M (1)div , (6.8)then iv Rn := 12 n − (cid:88) k =1 a n − k − a k − ( n − k )! k ! (cid:88) p ∈ Q ( n,k ) is Rp = iv n + c (1) n is a “renormalized” vertex in the sense that using iv Rn in place of iv n , all one-loop divergences areremoved from the theory. The “renormalization” eq. (6.8) is a divergent non-linear rescaling of aquantity, much like the rescaling g R = g + O ( g ) in conventional renormalization, only that it isnot a rescaling of a coupling parameter, but, in a certain sense, a non-linear rescaling of spacetime.6.2. The meta-counterterm of the connected 2-point-functionis C (2)2 , = − b s M (2)div ( s ) from eq. (4.3). On the other hand, it can be built from 1PI countertermsas shown in fig. 19. After working out the cancellations due to eq. (6.5), what remains is c (2)2 , ( s ) = C (2)2 , − c (1)2 , is c (1)2 , = b s M (2)div ( s )6 − ib s (cid:32) M (1)div ( s )2 (cid:33) . (6.9) Example 6.5 (Massless theory) . Using example A.2, in the massless theory the counterterm reads c (2)2 , = b s is π ) (cid:15) − ib s (cid:18) − π ) (cid:15) (cid:19) = is π ) (cid:18) b (cid:15) − b (cid:15) (cid:19) . = (2) + (1) (1) + (cid:104) (cid:105) (1) + (cid:104) (cid:105) (1) = (cid:104) (cid:105) (1) C (1)3 , = 0 Figure 19.
Contributions to the two-loop divergence of the connected 2-point-function. Counterterms in the internal edges are not considered since they give riseto tadpole graphs. The last two graphs together vanish thanks to eq. (6.4)For the 3-point-function at two loops, if all legs are onshell, every graph vanishes and c (2)3 , = 0 . (6.10)If one leg is offshell, we obtain an analogue of eq. (6.4) which is shown in fig. 20. Two of the fourgraphs add to zero thanks to eq. (6.4). C (2)3 , = 0 = c (2)3 , + (cid:104) (cid:105) c (1)3 , is c (1)2 , ( s ) + (cid:104) (cid:105) ( − ib s ) is c (1)2 , ( s ) is c (1)2 , ( s ) + (cid:104) (cid:105) ( − ib s ) is c (2)2 , ( s ) ⇒ c (2)3 , = − (cid:104) (cid:105) b c (2)2 , ( s ) . (6.11)We find that c (2)3 , behaves in just the same way with respect to c (2)2 , as c (1)3 , does with respect to c (1)2 , in eq. (6.5). (2) = 0 = (2) + (1) (1) + (1) (1) + (2) = (1) (1) C (1)3 , = 0 Figure 20.
Contributions to the two-loop meta-counterterterm C (2)3 , in terms of 1PI counterterms.With two legs offshell, the vanishing of C (1)3 , and C (1)4 , eliminates all one-loop graphs with insertedcounterterms similar to the mechanism in fig. 19. What remains are the tree graphs shown in fig. 21.Inserting C ( l )3 , from eq. (4.5), one finds c (2)3 , = C (2)3 , − (cid:104) (cid:105) C (1)3 , (cid:12)(cid:12) s =0 is c (1)2 , ( s ) − (cid:104) (cid:105) ( − ib s ) is c (2)2 , ( s )= (cid:104) (cid:105) (cid:0) b b − b b (cid:1) s ( s + s ) M (2)div ( s )6 − (cid:104) (cid:105) b b is ( s + s ) (cid:32) M (1)div (cid:33) . (6.12)Compared to the one-loop case eq. (6.6), there is a structurally different contribution which stemsfrom the braced graphs in fig. 21. = C (2)3 , = (2) + (1) (1) + (1) (1) + (1)(1) + (2) = (1) (1) Figure 21.
Contributions to the two-loop meta-counterterterm C (2)3 , in terms of 1PIcounterterms. All contributions of one-loop-graphs cancel since C (1) n, = 0 eq. (4.2).The brace indicates fig. 14.Finally, allowing all three legs to be offshell, C (2)3 , is the meta-counterterm computed in eq. (4.11).This allows to reconstruct the 1PI-counterterm where we use eqs. (6.1) and (6.6) c (2)3 , = C (2)3 , − (cid:104) (cid:105) c (1)2 , ( s ) is ( − ib s ) is c (1)2 , ( s ) − (cid:104) (cid:105) c (1)2 , ( s ) is c (1)3 , (cid:12)(cid:12) s =0 = C (2)3 , − ib (cid:0) b − b (cid:1) s s s (cid:32) M (1)div (cid:33) . (6.13) Example 6.6 (Massless theory) . For the massless theory, C (2)3 , was computed in eq. (4.12) and onehas, inserting example A.2 c (2)3 , = is s s π ) (cid:18)(cid:0) b b + b b + b − b b (cid:1) (cid:15) − b b (cid:15) (cid:19) . Example 6.7 (Exponential diffeomorphism) . Assuming a massless theory and eq. (5.1), b n = λ n − ,the two-loop counterterms eqs. (6.9) and (6.11) to (6.13) simplify to c (2)2 , = is π ) λ (cid:18) (cid:15) − (cid:15) (cid:19) c (2)3 , = i (cid:0) s + s + s (cid:1) π ) λ (cid:18) (cid:15) − (cid:15) (cid:19) c (2)3 , = − i (cid:0) s s + s s + s s + s s + s s + s s (cid:1) π ) λ (cid:15) c (2)3 , = 3 is s s π ) λ (cid:18) (cid:15) − (cid:15) (cid:19) . This shows that, contrary to the one-loop-counterterms in theorem 6.4, the two-loop-countertermscan not be generated by a simple rescaling like eq. (6.8) of the momenta in the bare vertices. Thiswas to be expected because already in the connected perspective, the dunce’s cap graph Γ A in fig. 9clearly does not represent a propagator correction. Without this graph c (2)3 , = 0 but still c (2)3 , (cid:54) = 0.6.3. As opposed to G ( s ) from eq. (3.1), the 1PI Function G ( s )contains divergences logarithmic in the momentum. Removing them requires the systematic use of1PI counterterms. We have seen in fig. 19 that at 2 loops, c (2)2 can be computed from C alone anddoes not involve meta-counterterms of higher valence. This is true to all loop orders. emma 6.8. Let C ( t ) := ∞ (cid:88) l =1 C ( l )2 t l = ∞ (cid:88) l =1 b l +2 s M ( l )div ( s )( l + 1)! t l be the ordinary generating functions of meta-counterterms of the 2-point-function, then c ( l ) = − is [ t l ] C ( t ) C ( t ) − is Proof.
The stated form of C ( l )2 is eq. (4.3). Use induction to prove the lemma. At one loop, c (1)2 = C (1)2 by eq. (6.1). Consider loop order l and assume all c ( j ) for j < l are local.Then the connected l -loop 2-point-function is given by the sum over all chains of k ( j ) where 1 ≤ k ≤ l and Γ ( j ) is supposed to be the sum of all j -loop 1PI graphs. Following theusual BPHZ renormalization procedure, for each Γ ( j ) with j < l the graph can be replaced by itscorresponding 1PI counterterm c ( j ) . Thereby, we remove all divergences except one: In the casethat no c ( j ) are inserted at all but only the graphs Γ ( j ) themselves, we know by theorem 1.6 thatthese graphs sum up to M ( l ) . Therefore, the summand which arises if all the c ( j ) are inserted hasno graph to cancel. It remains as an additional divergent term which needs to be absorbed by theonly undetermined quantity, the counterterm c ( l ) .If the chain consists of k factors, the so-produced divergence amounts to all possible partitionsof l loops into k counterterms, each connected by a propagator is , in total k ! n ! (cid:18) is (cid:19) k − B n,k (cid:16) c (1) , c (2) , c (3) , . . . (cid:17) . The factorials are present because we do not distinguish between the factors, e.g. c (1) is c (1) is c (1) appears only once, not 3! times. Finally, we have to sum over all k where k = 1 represents thenew, undetermined counterterm. The result is minus the overall divergence of the connected l -loopfunction, but we know this to be C ( l )2 where no subdivergences occur. So C ( l )2 = n (cid:88) k =1 k ! n ! (cid:18) is (cid:19) k − B n,k (cid:16) c (1) , c (2) , c (3) , . . . (cid:17) . This is Faa di Brunos formula for ordinary generating functions. Define F ( t ) := ∞ (cid:88) k =1 (cid:18) is (cid:19) k − t k = − is t − is − t , c ( t ) := ∞ (cid:88) l =1 c ( l )2 t l , then C ( l ) is the l -th coefficient of a power series C ( t ) = F ( c ( t )). Inversion gives the lemma. (cid:3) Example 6.9 (Massless theory) . In the massless theory, by lemma A.1, C ( t ) = + 1 (cid:15) ∞ (cid:88) l =1 b l +2 ( − is ) l +1 (4 π ) l ( l !) ( l + 1)! t l . As a peculiar example, let b n = 0 for all n >
3, then C ( t ) = − s b π ) (cid:15) t and the l -loop counterterm is c ( l ) = − is (cid:18) − isb π ) (cid:15) (cid:19) l . f one uses analytic continuation of the sum, then the all-order counterterm is regular in the limit (cid:15) → − is : c = ∞ (cid:88) l =1 c ( l )2 = is isb π ) (cid:15) + isb → is. Apart from amusement, taking the limit (cid:15) → Z , which diverges in perturbationtheory but is assumed to fulfil 0 < | Z | < Ward-Slavnov-Taylor-Identities
In section 6, we computed several 1PI counterterms and found that many of them are related. Seee.g. eqs. (6.5) and (6.11) or the fact that the all order counterterm c of the 2-point-function doesnot require knowledge of vertex-counterterms of higher valence. All these effects can be summarizedby the following Slavnov-Taylor-like identities. Theorem 7.1.
Let c ( l ) n = (cid:80) nk =0 c ( l ) n,k be the l -loop n -valent 1PI counterterms and let Γ := − is − ∞ (cid:88) l =1 c ( l )2 ( s ) , Γ n ≥ := iv n + ∞ (cid:88) l =1 c ( l ) n , and assume that tadpoles vanish, then for n ≥ (cid:20) Γ n ( − is ) (cid:21) only s offshell = iv n (cid:12)(cid:12)(cid:12) only s offshell , (2) (cid:88) Γ j (cid:63) Γ k =Γ n (cid:20) Γ j Γ k (cid:21) onshell = − Γ n (cid:12)(cid:12)(cid:12) onshell , where the product (cid:63) implies j + k = n + 2 and a sum over all orientations of the graphs.Proof. First note that 1Γ = is − ic = is ∞ (cid:88) r =0 (cid:18) is c (cid:19) r is the non-amputated chain of all 1PI 2-point counterterms. Consequently, ( − is ) is the samechain where the outermost propagator is removed.For any n ≥
3, the connected n -point correlation function vanishes if not more than one externaledge is offshell due to theorem 1.6. Consequently, its divergent part vanishes and C n, = C n, = 0,see eqs. (4.1) and (4.2). It suffices to consider connected graphs where all internal edges are cancelledsince the remaining graphs are products of the former type.First prove (1). Use induction on n . For n = 2, (1) becomes ( − is ) = iv which is true. For n = 3, since c vanishes onshell, the connected graph where only s is offshell is Γ ( − is ) whereΓ is the counterterm of edge s . But C , = 0 and hence only the regular term survives of thissum, which is iv as claimed in (1). Now assume (1) holds for j < n . Then, in the sum of allconnected graphs, all divergent contributions cancel where s is adjacent to a j -valent counterterm,either directly or via a string of propagator counterterms. The only non-vanishing terms are thosewhere a n -valent counterterm is involved. But again, the sum over all divergent terms has to vanishand the only remaining term is iv n . This proves (1). or (2), the case n = 2 reads Γ | onshell = − Γ | onshell which is true since Γ | onshell = 0 by lemma 6.8.The same holds for n = 3 since, by eq. (4.1), Γ | onshell = 0.Assume (2) holds for j, k < n . The onshell connected amplitude can only be proportional topowers of internal momenta s e . If there is only one such internal momentum, corresponding to oneinternal edge e , then all terms proportional to s e arise from Γ j ( e ) Γ k . Since these terms are notpresent in the end result, we know Γ n must absorb them. If there is more than one edge, pick oneand call it e . Then, there are two subtrees T j , T k , each of which has valence < n and only oneexternal edge offshell, namely e . But by eq. (4.2), such trees do not contain divergent terms. Infact, as a consequence of (1), such trees do not even contain powers of internal momenta since theyare made from tree-level vertices iv k and such trees evaluate to b j by eq. (1.16). Therefore, theonly relevant contribution stems from trees with exactly one internal multi-cancelled edge, whichproves (2). (cid:3) The compatibility of theorem 7.1 with locality is expressed by the fact that such identificationsbetween different n -point-functions represent Hopf ideals in the core Hopf algebra [41, 59]. Notealso that eq. (1.14) is the tree-level version of statement (2). Technically, only statement (1) oftheorem 7.1 requires the vanishing of tadpoles, i.e. C n, = 0, whereas (2) holds regardless.Statement (1) can be rewritten in the formΓ n (cid:12)(cid:12)(cid:12) only s offshell = (cid:20) iv n is Γ ( s ) (cid:21) only s offshell , which implies c ( l ) n (cid:12)(cid:12)(cid:12) only s offshell = (cid:2) v n s − c ( s ) (cid:3) only s offshell . This means, lemma 6.1 holds to all ordersin perturbation theory. Inserting (1) into (2) produces − Γ n (cid:12)(cid:12)(cid:12) onshell = (cid:88) Γ j (cid:63) Γ k =Γ n (cid:20) iv n is Γ k (cid:21) onshell = (cid:88) Γ j (cid:63) Γ k =Γ n (cid:20) iv n is Γ ( s ) is iv k (cid:21) onshell and thereby also lemma 6.2 holds to all orders. It is lemma 6.3 which fails at higher than one-looporder: To all orders, the parts of the counterterms, which have the same momentum dependenceas the vertices iv n , can be obtained by replacing − is → Γ ( s ). But starting from two-loop order,there are additional kinematic form factors in the counterterms which are not obtained in this way.These terms can not be constructed inductively from the 2-point-counterterms.8. Analogy to gauge theory amplitudes
In this last section we argue that there are surprising similarities between a scalar field diffeo-morphism and gauge theories.The most striking one is the presence of the Slavnov-Taylor-like identities theorem 7.1. Inquantum gauge theories, the various divergent amplitudes and especially their local countertermsare related to each other by the Ward identity [73] in QED respectively Slavnonv-Taylor identities[67, 62, 68] in QCD. These identities guarantee gauge invariance for the quantized theory and implythat all divergent correlation functions can be rendered finite by a redefinition of the same couplingconstant, see for example [26]. Relations similar to theorem 7.1 were also proposed for quantumEinstein gravity [36, 38].In our case, the invariance of the S -matrix under global field diffeomorphisms takes the role oflocal gauge invariance. Conceptually, the identities theorem 7.1 play the same role as in gaugetheories: They guarantee that the counterterms of loop-level amplitudes are compatible with theinvariance, which in our case means that they vanish in the onshell limit s e →
0. As a by-product,they significantly reduce the number of independent counterterms. For the n -valent counterterm,only the summand c n,n is truly independent, all c n,k for k < n are determined by lower valence ounterterms c k . For example, in eqs. (6.9) and (6.11) to (6.13), the only independent contributionto the 3-valent counterterm is c , , whereas c , , c , and c , are determined by c . In the connectedperspective, the corresponding statement is lemma 4.3.The analogy between a scalar field diffeomorphism and QCD becomes visible in the case s p = p and a = − g , a n> = 0 of the diffeomorphism eq. (1.2). Then, using ∂ µ ρ = 2 ρ∂ µ ρ , the Lagrangianof ρ is L ρ = 12 ( − ∂ µ ρ + g ρ∂ µ ρ ) ( − ∂ µ ρ + g ρ∂ µ ρ )(8.1)which is reminiscent of the Yang-Mills-Lagrangian of QCD, L QCD = − (cid:16) ∂ µ A aν − ∂ ν A aµ + g f abc A bµ A cν (cid:17) (cid:16) ∂ µ A aν − ∂ ν A aµ + g f abc A bµ A cν (cid:17) . (8.2)The scalar field is not a Lorentz vector, so there necessarily are differences in the tensor structurebetween eq. (8.1) and eq. (8.2), and the latter also needs gauge-fixing. But still eq. (8.1) is perhapsthe closest one can get to the Yang-Mills-Lagrangian by only using scalar fields.In QCD, the maximum helicity violating (MHV) amplitudes are those where precisely two outof n external onshell gluons have a different helicity than the rest. To leading order in N of thegauge group SU ( N ), their matrix element is given by the Parke-Taylor-Formula [55] (cid:12)(cid:12) M MHV (1 − , − , + , . . . ) (cid:12)(cid:12) = g n − n − N n − ( N − n ( p · p ) (cid:88) P p · p )( p · p ) · · · ( p n · p )(8.3)where P ranges over all permutations of 1 , . . . , n . The validity of eq. (8.3) is a consequence ofBerends-Giele relations [4] in QCD. The BG relations are a recursive construction of n -gluon cur-rents ˆ J xξ (1 , , . . . , n ), which are the connected ( n + 1)-point gluon functions where precisely one legis offshell. This is almost verbatim the definition of b n +1 in definition 1.5. Consequently, the proofof eq. (1.16) in [43] is strikingly similar to the derivation of BG relations in [4]. In this sense, theconnected amplitudes iV n ∼ ib n in eq. (1.15) are a scalar analogue of the Parke-Taylor-amplitudesin QCD. In the scalar case eq. (8.1), the tree-level matrix element with one external edge offshellis the square of eq. (1.15), | V n | = | b n | n (cid:88) i =1 s i = g n − ((2 n − n (cid:88) i =1 ( p i · p i ) . (8.4)In terms of kinematics, the tree sums b n are simpler than the gluon currents ˆ J xξ since there is noremainder at all of the internal edges. Interestingly, the b n resp. | V n | can be computed easily evenif eq. (8.1) contains more than quadratic terms in each factor. This might be interesting for morecomplicated gauge theories such as quantum gravity. Example 8.1 (Exponential diffeomorphism) . In section 5, we studied in detail the choice a n = ( − n g n n +1 and, in eq. (5.5), found it reminiscent of Einstein gravity . It coincides at leading orderwith the choice a = − g in eq. (8.1) and the Lagrangian can be written in the form L ρ = 12 (cid:0) − ∂ µ ρ + g ρ∂ µ ρ − g ρ ∂ µ ρ + g ρ ∂ µ ρ ∓ . . . (cid:1) . Since b n = g n − , it has the leading order matrix element | V n | = g n − n (cid:88) i =1 ( p i · p i ) . Compared to eq. (8.4), this particular choice of diffeomorphism eliminates all combinatoric prefac-tors. e do not claim that the QCD Slavnov-Taylor Relations or the Parke-Taylor-Formula can bederived from the scalar case or vice versa, but it is interesting to observe that similar structuresexist in both cases. This implies that their presence is not exclusive to a massless spin-1 fieldand its Lorentz representations, but follows rather generically from the algebraic structure of theLagrangian being a square of an expression which is only quadratic in the fields. In this spiritit appears not surprising that the graphs of the standard model gauge theories can be generatedalgebraically from the graphs of a cubic scalar theory [40, 44, 60].9. Conclusion
We have completed a survey of propagator cancelling scalar quantum field theories. Throughout,we have restricted ourselves to those theories which are obtained by a global diffeomorphism of thefield variable of a free scalar quantum field. This subclass exhausts all massless scalar fields withquadratic propagator (theorem 1.4). The key results are:(1) We have demonstrated in section 2.2 that the Feynman rules of a field diffeomorphism inmomentum space, given by the connected perspective theorem 1.6, are in accordance withthe expected behaviour of diffeomorphisms in position space.(2) We have derived the all-orders 2-point function G for an arbitrary diffeomorphism of a freefield with propagator i/p in section 3.(3) We have extended the connected perspective to include also counterterm-metavertices andhave computed several of them in section 4.(4) In section 6, we have reconstructed 1PI-counterterms at one- and two-loop level from themeta-counterterms. Further, we showed that the 1PI counterterm of the 2-point-functioncan be computed without knowledge of higher valence counterterms (lemma 6.8).(5) In section 5, we have examined in detail those theories where the connected amplitudes areproportional to each other, b n ∼ λ n − , with the following results:(a) The explicit field diffeomorphisms giving rise to all such theories are given by lemma 5.1.For the simplest cases the transformed Lagrangian eq. (5.4) amounts to introducingthe inverse of the field variable into the kinetic term and is reminiscent of the Einstein-Hilbert-Lagrangian.(b) The all-order 2-point-function G is the only amplitude which involves infinitely manygraphs. The connected n -point loop-level amplitudes are then given by finitely manygraphs on up to n vertices, connected by G , both in momentum space (theorem 5.9)and position space (section 5.4).(c) We computed explicitly the position-space 2-point function (lemma 5.4) and its momentum-space divergence for massless fields in 4 − (cid:15) dimensions (lemma 5.6). For the simplestinstance of recursion relations, u = 1 in eq. (5.1), it turns out to be the exponentialsuperpropagator studied extensively in the early years of quantum field theory. Ourresult agrees with the literature up to a finite function which amounts to differentrenormalization conditions. Thereby we verified that the historic result is consistentwith systematic momentum-space perturbation theory.(d) We showed in theorem 6.4 that if u = 1 then all one-loop counterterms c (1) n coincidewith the tree-level vertices iv n if all offshell variables are scaled simultaneously. Thisis obvious in the connected perspective, see e.g. fig. 11 where each of the graphs is in1:1 correspondence with the replacement of precisely one propagator by a multiedge.(e) In example 6.7, we found that at two loops, this scaling alone is not sufficient to renderthe theory finite. This is again plausible from the connected perspective as the graphΓ A in fig. 9 is not a propagator correction.
6) In theorem 7.1 we showed that the counterterms of the diffeomorphed field fulfil a set ofequations which are analogous to Slavnov-Taylor-identities in QCD. These relations reflectthe fact that the theory is invariant under diffeomorphisms at quantum level. They dramat-ically reduce the number of independent counterterms but still infinitely many independentcounterterms remain.(7) We argued that propagator cancelling scalar fields show remarkable similarities to gaugetheories and that at least three different sets of recursion relations, which have becomeindispensable in the modern understanding of perturbative quantum field theory, make anappearance:(a) Slavnov-Taylor-identities between all counterterms are realized by theorem 7.1.(b) Berends-Giele relations for the gluonic current ˆ J xξ have an analogue in the recursiveconstruction of tree sums b n as given in [43], see section 8.(c) BCFW relations are respected by the connected perspective theorem 1.6 at least attree-level, see comment in section 1.6.The similarities between scalar field diffeomorphisms and QCD might help to understand thebehaviour of more complicated gauge theories such as quantum Einstein gravity. There, contraryto eq. (8.2), higher than quadratic terms are present in the two factors in the Lagrangian, andsuch behaviour can easily be included in the scalar case, see example 8.1. Further, in the scalarcase there is an alternative proof for the recursions eq. (1.16) using generating functions [50]. Thisraises the question if a similar construction is possible in gauge theories as well.Given the above similarities, propagator cancelling scalar fields can also serve as a pedagogicalmodel to illustrate how a certain structure of the Lagrangian translates to certain behaviour ofcorrelation functions, without obfuscating this correspondence by the extensive tensor notation ofhigher spin fields.Our discussion of Slavnov-Taylor identities for the propagator cancelling field in section 7 hasremained sketchy. To make it more precise, one would need to decompose the counterterms c ( l ) n into kinematic form factors more systematically than we did. This is especially necessary for thecase of quantum gravity, where an analogue set of identities must hold to ensure gauge invarianceat quantum level. Appendix A. Massless multiedge graphs By M ( l ) ( s ) we denote the l -loop multiedge graph without vertex- and symmetry-factors. In thecase of massless fields, m = 0, all Feynman amplitudes M ( l ) are known to evaluate to Gammafunctions. To see this, consider the massless one-loop multiedge with propagator powers α, β andset s := p , it has the amplitude (e.g. [71, 51, 18]) I ( α, β ; s ) := (cid:90) d D k (2 π ) D k ) α (( k + p ) ) β = s D − α − β (4 π ) D Γ (cid:0) α + β − D (cid:1) Γ (cid:0) D − α (cid:1) Γ (cid:0) D − β (cid:1) Γ( α )Γ( β )Γ ( D − α − β ) . Since this amplitude again is a monomial in s with power D − α − β , it can be inserted recursivelyto produce multiedges with more than one loop and arbitrary propagator powers. If all propagatorshave the same power ( k ) − α , the l -loop multiedge graph has the Feynman amplitude M ( l ) = s l ( D − α ) − α (4 π ) l D (cid:32) Γ (cid:0) D − α (cid:1) Γ( α ) (cid:33) l +1 Γ (cid:0) α − l (cid:0) D − α (cid:1)(cid:1) Γ (cid:0) ( l + 1) (cid:0) D − α (cid:1)(cid:1) (A.1)These graphs need a symmetry factor l +1)! for the exchange of l + 1 equivalent edges which is notincluded in eq. (A.1). emma A.1. Let H n = (cid:80) nj =1 j − be the n th harmonic number and define s := p to be the externalmomentum squared. Then in D = 4 − (cid:15) dimensions, the massless l -loop multiedge with propagators i ( k ) − , not including the symmetry factor, has the Feynman amplitude M ( l ) ( s ) = − ( − is ) l − (4 π ) l ( l !) (cid:18) (cid:15) + (2 l + 1) H l − l (ln(4 π ) − γ E ) − l ln s (cid:19) + O ( (cid:15) ) . Proof.
Expand eq. (A.1) in (cid:15) , setting α = 1. With D = 4 − (cid:15) we have M ( l ) ( s ) = s l − − l(cid:15) (4 π ) l (2 − (cid:15) ) (Γ (1 − (cid:15) )) l +1 Γ ( − l + 1 + l(cid:15) )Γ ( l + 1 − ( l + 1) (cid:15) ) . (A.2)The only singular factor for (cid:15) → − l + 1 + l(cid:15) ) = ( − l − l ! (cid:18) (cid:15) + lψ ( l ) + O ( (cid:15) ) (cid:19) . Here, ψ ( l ) is the digamma function, with integer argument l > § ψ ( l ) = l − (cid:88) k =1 k − γ E = H l − − γ E where γ E is Euler’s constant. All other factors in eq. (A.2) are regular for (cid:15) →
0, consequentlytheir O ( (cid:15) ) coefficients need to be included to produce an overall O ( (cid:15) ) result. These are1Γ ( l + 1 − ( l + 1) (cid:15) ) = 1Γ( l + 1) + (cid:15) ( l + 1) ψ ( l + 1)Γ( l + 1) + O (cid:0) (cid:15) (cid:1) = 1 l ! (cid:0) (cid:15) ( l + 1) ( H l − γ E ) + O (cid:0) (cid:15) (cid:1)(cid:1) , (Γ(1 − (cid:15) )) l +1 = 1 + (cid:15) ( l + 1) γ E + O (cid:0) (cid:15) (cid:1) , s l − − l(cid:15) = s l − (cid:0) − (cid:15)l ln s + O (cid:0) (cid:15) (cid:1)(cid:1) , (4 π ) − l + l(cid:15) = (4 π ) − l (cid:0) (cid:15)l ln(4 π ) + O (cid:0) (cid:15) (cid:1)(cid:1) . Finally, use H l = H l − + l − and include a factor i l +1 for l + 1 internal propagators. (cid:3) Note that the presence of momentum-independent terms such as l ln(4 π ) depends on the choiceof integration measure and varies in the literature.Especially, given D − α = 1 − (cid:15) (as is fulfilled for the conventional propagator s = p in D = 4 − (cid:15) dimensions), all massless multiedge graphs have a simple singularity M ( l )div = − ( − is ) l − (4 π ) l ( l !) (cid:15) (A.3)with no logarithmic momentum-dependence. This reflects the fact that any subgraph would giverise to a massless tadpole cograph with vanishing amplitude. In this sense, massless multiedges areprimitive despite containing power-counting divergent subgraphs. Example A.2 (Multiedges) . We will use frequently the one- and two-loop multiedges which read M (1) ( s ) = − π ) (cid:18) (cid:15) + 2 − γ E + ln(4 π ) − ln s (cid:19) ,M (2) ( s ) = is π ) (cid:18) (cid:15) + 112 − γ E + 2 ln(4 π ) − s (cid:19) . ppendix B. Massless connected 3-point function
The amputated connected 3-point function in the connected perspective is given by four differentgraph topologies:(1) A single vertex G , (1) = − ib ( s + s + s )(2) One Multiedge M ( l ) where one end is adjacent to two external edges, the other to one,(3) Two multiedges joined at one external vertex,(4) Triangle-shaped graphs where the three internal edges themselves are multiedges.The first type is trivial, the second one comes in three different orientations with regard to theexternal momenta. For each orientation, there is a sum over all l -loop multiedges, G , (2) = − ∞ (cid:88) l =1 b l +2 b l +3 ( l + 1)! (cid:16) ( s + s ) s M ( l ) ( s ) + ( s + s ) s M ( l ) ( s ) + ( s + s ) s M ( l ) ( s ) (cid:17) . Observe that the three series do not coincide with the two-point function eq. (3.1): The latter hascoefficients ∝ b l +2 whereas here they are b l +2 b l +3 . Since all b n are independent, there need not beany specific relation between the two series.Using lemma A.1, G (2)3 for the massless theory s = p reads G , (2) = (cid:104) (cid:105) ∞ (cid:88) l =1 ( − i ) l +1 b l +2 b l +3 (4 π ) l ( l + 1)!( l !) ( s + s ) s l (cid:18) (cid:15) + (2 l + 1) H l − − lγ E + l ln(4 π ) − l ln s (cid:19) (B.1)where (cid:104) (cid:105) by definition 1.2 denotes the symmetric sum over three cyclic permutations of { s , s , s } .Similarly, the third topology is a product of two multiedges G , (3) = (cid:104) (cid:105) is s s ∞ (cid:88) l =1 ∞ (cid:88) l =1 b l +2 b l +2 b l + l +3 M ( l ) ( s )( l + 1)! M ( l ) ( s )( l + 1)! . (B.2)The fourth contribution to the 3-point-function for the massless theory is given by triangle graphswhere the propagators are multiedges on n edges, M ( n − .Let T ( n ,n ,n ) be the triangle graph wherethe internal edge opposed to the external momentum s j is a multiedge M ( n j − . Like for M ( l ) inlemma A.1, we do not include the symmetry factor into T ( n ,n ,n ) , it is n ! n ! n ! . Then, the fourthcontribution to the connected 3-point amplitude is the series G , (4) = ∞ (cid:88) n =0 ∞ (cid:88) n =0 ∞ (cid:88) n =0 b n + n +1 b n + n +1 b n + n +1 n ! n ! n ! T ( n ,n ,n ) ( s , s , s ) . In a massless theory, the multiedge graph M ( l ) ∝ s l ( D − α ) − α (eq. (A.1)) amounts to a propagatorwith the power − l (cid:0) D − α (cid:1) + α . This means that a triangle graph, where the edges are replaced withmultiedes, reduces up to prefactors to a simple massless triangle graph where the edges carry saidnon-integer propagator power. The latter graph evaluates to Appel’s hypergeometric F functions[10, 64]. At this point we just quote well-known results for the lowest orders. The one-loop orderis given in [18, Eq. (2.11)], it is convergent in 4 dimensions. At two-loop order, the correspondinggraph is called the massless dunce’s cap (Γ A in fig. 9), again without the symmetry factor it reads[70] T (0 , , ( s , s , s ) = 12(4 π ) (cid:18) (cid:15) + (5 − γ E − s + 2 ln(4 π )) 1 (cid:15) + finite terms (cid:19) . (B.3)If two of the external edges are onshell then T ( n ,n ,n ) can be computed similarly to the multiedgegraph and evaluates to a product of Gamma functions. eferences [1] Karyn M. Apfeldorf, Horacio E. Camblong, and Carlos R. Ordonez. “Field Redefinition In-variance in Quantum Field Theory”. In: Modern Physics Letters A issn : 0217-7323, 1793-6632. doi : . arXiv: hep-th/0003287 . url : http://arxiv.org/abs/hep-th/0003287 (visited on 01/20/2019).[2] Paul-Hermann Balduf. “Perturbation Theory of Transformed Quantum Fields”. In: Mathe-matical Physics, Analysis and Geometry issn : 1572-9656. doi : . url : https://doi.org/10.1007/s11040- 020- 09357- z (visited on 09/17/2020).[3] F. A. Berends, G. J. H. Burgers, and H. van Dam. “On Spin Three Self Interactions”. In: Zeitschrift f¨ur Physik C Particles and Fields issn : 1431-5858. doi : . url : https://link.springer.com/article/10.1007/BF01410362 (visited on 07/15/2020).[4] F. A. Berends and W. T. Giele. “Recursive Calculations for Processes with n Gluons”. In: Nuclear Physics B issn : 0550-3213. doi : . url : (visited on 01/24/2021).[5] R. J. Blomer and F. Constantinescu. “On the Zero-Mass Superpropagator”. In: NuclearPhysics B issn : 0550-3213. doi : . url : (visited on 04/27/2020).[6] N. N. Bogoliubow and O. S. Parasiuk. “Ueber die Multiplikation der Kausalfunktionen in derQuantentheorie der Felder”. In: Acta Mathematica
97 (1957), pp. 227–266. issn : 0001-5962,1871-2509. doi : . url : https://projecteuclid.org/euclid.acta/1485892235 (visited on 07/18/2019).[7] C. G. Bollini and J. J. Giambiagi. “Dimensional Regularization in Configuration Space”. In: Physical Review D issn : 0556-2821, 1089-4918. doi : . url : https://link.aps.org/doi/10.1103/PhysRevD.53.5761 (visited on 04/26/2020).[8] C. G. Bollini and J. J. Giambiagi. “Dimensional Renormalization : The Number of Dimensionsas a Regularizing Parameter”. In: Il Nuovo Cimento B (1971-1996) issn : 1826-9877. doi : . url : https://doi.org/10.1007/BF02895558 (visited on 04/25/2020).[9] C. G. Bollini and J. J. Giambiagi. “On the Exponential Superpropagator”. In: Journal ofMathematical Physics issn : 0022-2488. doi : . url : https://aip.scitation.org/doi/abs/10.1063/1.1666489 (visited on04/26/2020).[10] E E Boos and A. I. Davydychev. “Method for calculating vertex-type Feynman integrals”. In: Vestnik Moskovskogo Universiteta, Seriya 3. Fizika, Astronomiya issn :ISSN 0579-9392. url : http://inis.iaea.org/Search/search.aspx?orig_q=RN:19014691 (visited on 04/12/2020).[11] Eric Braaten, Thomas Curtright, and Charles Thorn. “An Exact Operator Solution of theQuantum Liouville Field Theory”. In: Annals of Physics issn : 0003-4916. doi : . url : (visited on 03/12/2020).[12] Ruth Britto, Freddy Cachazo, and Bo Feng. “New Recursion Relations for Tree Amplitudesof Gluons”. In: Nuclear Physics B issn : 05503213. doi : . arXiv: hep-th/0412308 . url : http://arxiv.org/abs/hep-th/0412308 (visited on 05/29/2020).[13] Ruth Britto et al. “Direct Proof Of Tree-Level Recursion Relation In Yang-Mills Theory”.In: Physical Review Letters issn : 0031-9007, 1079-7114. doi : . arXiv: hep-th/0501052 . url : http://arxiv.org/abs/hep-th/0501052 (visited on 09/03/2019).[14] Albert Victor B¨acklund. “Zur Theorie Der Partiellen Differentialgleichung Erster Ordnung”.In: Mathematische Annalen
17 (1880), p. 285. url : http://resolver.sub.uni-goettingen.de/purl?GDZPPN002245795 .[15] William Celmaster and Richard J. Gonsalves. “Renormalization-Prescription Dependence ofthe Quantum-Chromodynamic Coupling Constant”. In: Physical Review D doi : . url : https://link.aps.org/doi/10.1103/PhysRevD.20.1420 (visited on 07/31/2019).[16] Ivan V. Chebotarev et al. “S-Matrix of Nonlocal Scalar Quantum Field Theory in BasisFunctions Representation”. In: Particles doi :
10 . 3390 /particles2010009 . url : (visited on 04/28/2020).[17] Louis Comtet. Advanced Combinatorics . Dordrecht, Holland: D. Reidel Publishing Company,1974. isbn : 978-94-010-2198-2.[18] A. I. Davydychev. “Recursive Algorithm for Evaluating Vertex-Type Feynman Integrals”. In:
Journal of Physics A: Mathematical and General issn :0305-4470. doi : . url : https://doi.org/10.1088%2F0305-4470%2F25%2F21%2F017 (visited on 04/12/2020).[19] Bryce S. DeWitt. “Gravity: A Universal Regulator?” In: Physical Review Letters doi : . url : https://link.aps.org/doi/10.1103/PhysRevLett.13.114 (visited on 07/05/2019).[20] E. D’Hoker and R. Jackiw. “Classical and Quantal Liouville Field Theory”. In: PhysicalReview D doi :
10 . 1103 / PhysRevD . 26 . 3517 . url : https://link.aps.org/doi/10.1103/PhysRevD.26.3517 (visited on 12/18/2019).[21] DLMF: NIST Digital Library of Mathematical Functions . url : https://dlmf.nist.gov/ (visited on 04/05/2020).[22] G. V. Efimov. “Formulation of a Scalar Quantum Field Theory with an Essentially Non-Linear”. In: Nuclear Physics issn : 0029-5582. doi :
10 .1016/0029-5582(65)90211-7 . url : (visited on 04/25/2020).[23] John Engbers, David Galvin, and Clifford Smyth. Restricted Stirling and Lah Number Matri-ces and Their Inverses . Dec. 28, 2017. arXiv: . url : http://arxiv.org/abs/1610.05803 (visited on 01/09/2021).[24] Paul G. Federbush and Kenneth A. Johnson. “Uniqueness Property of the Twofold VacuumExpectation Value”. In: Physical Review doi : . url : https://link.aps.org/doi/10.1103/PhysRev.120.1926 (visitedon 01/03/2020).[25] Carl Friedrich Gauss. “Disquisitiones Generales circa Seriem Infinitam”. In: CommentationesSocietatis Regiae Scientiarum Gottingensis, Recentiores classis mathematicae (1813), pp. 227–270. url : https://gdz.sub.uni-goettingen.de/id/PPN35283028X_0002_2NS .[26] J. A. Gracey, H. Kissler, and D. Kreimer. On the Self-Consistency of off-Shell Slavnov-TaylorIdentities in QCD . June 19, 2019. arXiv: . url : http://arxiv.org/abs/1906.07996 (visited on 07/11/2019).
27] W. Heisenberg. “Zur Theorie der Schauer in der H¨ohenstrahlung”. In:
Zeitschrift f¨ur Physik issn : 0044-3328. doi : . url : https://doi.org/10.1007/BF01349603 (visited on 04/27/2020).[28] Klaus Hepp. “Proof of the Bogoliubov-Parasiuk Theorem on Renormalization”. In: Commu-nications in Mathematical Physics issn : 1432-0916. doi : . url : https://doi.org/10.1007/BF01773358 (visited on 07/18/2019).[29] Kerson Huang. Quantum Field Theory: From Operators to Path Integrals . New York: Wiley,1998. 426 pp. isbn : 978-0-471-14120-4.[30] C. J. Isham, Abdus Salam, and J. Strathdee. “Is Quantum Gravity Ambiguity-Free?” In:
Physics Letters B issn : 0370-2693. doi : . url : (visited on 04/27/2020).[31] Arthur M. Jaffe. “Form Factors at Large Momentum Transfer”. In: Physical Review Letters doi :
10 . 1103 / PhysRevLett . 17 . 661 . url : https ://link.aps.org/doi/10.1103/PhysRevLett.17.661 (visited on 04/27/2020).[32] R. Jost. “Properties of Wightman Functions”. In: Lectures on Field Theory and the Many-Body Problem Edited by E. R. Caianiello . New York: Academic Press, 1961.[33] Susumu Kamefuchi and Hiroomi Umezawa. “On the Structure of the Interaction of the El-ementary Pasticles, IV. On the Interaction of the Second Kind”. In:
Progress of TheoreticalPhysics issn : 0033-068X. doi : . url : https://academic.oup.com/ptp/article/9/5/529/1813843 (visited on 04/27/2020).[34] Ram P. Kanwal. Generalized Functions, Theory and Applications . 3rd ed. Springer, 2004.[35] Lutz Klaczynski.
Haag’s Theorem in Renormalised Quantum Field Theories . Feb. 1, 2016.arXiv: . url : http : / / arxiv . org / abs / 1602 .00662 (visited on 11/14/2018).[36] Dirk Kreimer. “A Remark on Quantum Gravity”. In: Annals of Physics issn : 00034916. doi :
10 . 1016 / j . aop . 2007 . 06 . 005 . arXiv: . url : http://arxiv.org/abs/0705.3897 (visited on 02/22/2019).[37] Dirk Kreimer. “Combinatorics of (Perturbative) Quantum Field Theory”. In: Physics Reports .Renormalization Group Theory in the New Millennium. IV 363.4 (June 1, 2002), pp. 387–424. issn : 0370-1573. doi : . url : (visited on 04/12/2020).[38] Dirk Kreimer. “Not so Non-Renormalizable Gravity”. In: Quantum Field Theory: CompetitiveModels . Ed. by Bertfried Fauser, J¨urgen Tolksdorf, and Eberhard Zeidler. Basel: Birkh¨auser,2009, pp. 155–162. isbn : 978-3-7643-8736-5. doi : . url : https://doi.org/10.1007/978-3-7643-8736-5_9 (visited on 05/01/2020).[39] Dirk Kreimer. “On the Hopf Algebra Structure of Perturbative Quantum Field Theories”. In: Adv. Theor. Math. Phys. doi : . arXiv: q-alg/9707029 [q-alg] .[40] Dirk Kreimer, Matthias Sars, and Walter D. van Suijlekom. “Quantization of Gauge Fields,Graph Polynomials and Graph Homology”. In: Annals of Physics
336 (Sept. 1, 2013), pp. 180–222. issn : 0003-4916. doi : . url : (visited on 09/17/2019).[41] Dirk Kreimer and Walter D. van Suijlekom. “Recursive Relations in the Core Hopf Algebra”.In: Nuclear Physics B issn : 05503213. doi :
10 . 1016 / j .nuclphysb . 2009 . 04 . 025 . arXiv: . url : http : / / arxiv . org / abs / 0903 . 2849 (visited on 07/18/2019).
42] Dirk Kreimer and Andrea Velenich. “Field Diffeomorphisms and the Algebraic Structure ofPerturbative Expansions”. In:
Lett. Math. Phys.
103 (2013), pp. 171–181. doi :
10 . 1007 /s11005-012-0589-y . arXiv: .[43] Dirk Kreimer and Karen Yeats. “Diffeomorphisms of Quantum Fields”. In:
MathematicalPhysics, Analysis and Geometry
20, 16 (June 2017), p. 16. doi : . arXiv: .[44] Dirk Kreimer and Karen Yeats. “Properties of the Corolla Polynomial of a 3-Regular Graph”.In: The Electronic Journal of Combinatorics (Feb. 25, 2013), P41–P41. issn : 1077-8926. doi : . url : (visited on 02/08/2021).[45] G. Lazarides, A. A. Patani, and Q. Shafi. “High-Energy Behavior in a Model NonpolynomialLagrangian Field Theory”. In: Physical Review D doi : . url : https://link.aps.org/doi/10.1103/PhysRevD.6.2780 (visited on 04/28/2020).[46] Benjamin W Lee. “Gauge Theories”. In: Methods in Field Theory, Les Houches Session 1975 .Ed. by Roger Balian and Jean Zinn-Justin. Les Houches Session 28. North Holland / WorldScientific, 1981, pp. 79–140. isbn : 978-9971-83-078-7. url : https://doi.org/10.1142/0004 .[47] H. Lehmann and K. Pohlmeyer. “On the Superpropagator of Fields with Exponential Cou-pling”. In: Communications in Mathematical Physics issn : 0010-3616, 1432-0916. url : https://projecteuclid.org/euclid.cmp/1103857157 (visited on04/26/2020).[48] J Liouville. “Sur l’equation Aux Differences Partielles D2logdu Dv2a2=0”. In: Journal demathematiques pures et appliquees 1re serie
18 (1853), pp. 71–72.[49] Ali Assem Mahmoud.
On the Enumerative Structures in Quantum Field Theory . Aug. 26,2020. arXiv: . url : http ://arxiv.org/abs/2008.11661 (visited on 09/29/2020).[50] Ali Assem Mahmoud and Karen Yeats. Diffeomorphisms of Scalar Quantum Fields via Gen-erating Functions . Sept. 6, 2020. arXiv: . url : http://arxiv.org/abs/2007.12341 (visited on 01/18/2021).[51] M. S Milgram and H. C Lee. “Tables of Divergent Feynman Integrals in the Axial and Light-Cone Gauges”. In: Journal of Computational Physics issn :0021-9991. doi : . url : (visited on 06/11/2019).[52] Susumu Okubo. “Note on the Second Kind Interaction”. In: Progress of Theoretical Physics issn : 0033-068X. doi :
10 . 1143 / PTP . 11 . 80 . url : https ://academic.oup.com/ptp/article/11/1/80/1852890 (visited on 04/27/2020).[53] Minoru Omote. “Point Canonical Transformations and the Path Integral”. In: Nuclear PhysicsB issn : 0550-3213. doi : . url : .[54] E. P. Osipov. “Feynman Integral for Exponential Interaction in Four-Dimensional Space-Time. I”. In: Theoretical and Mathematical Physics issn :1573-9333. doi :
10 . 1007 / BF01019297 . url : https : / / doi . org / 10 . 1007 / BF01019297 (visited on 04/25/2020).[55] Stephen J. Parke and T. R. Taylor. “Amplitude for $ n $ -Gluon Scattering”. In: Physical ReviewLetters doi :
10 . 1103 / PhysRevLett . 56 . 2459 . url : https://link.aps.org/doi/10.1103/PhysRevLett.56.2459 (visited on 01/26/2021).[56] Michael E. Peskin and Daniel V. Schroeder. An Introduction to Quantum Field Theory . 1995.842 pp. url : https://cds.cern.ch/record/257493 (visited on 01/11/2020).
57] K. Pohlmeyer. “The Jost-Schroer Theorem for Zero-Mass Fields”. In:
Communications inMathematical Physics issn : 1432-0916. doi : . url : https://doi.org/10.1007/BF01661574 .[58] A. M. Polyakov. “Quantum Geometry of Bosonic Strings”. In: Physics Letters B issn : 0370-2693. doi :
10 . 1016 / 0370 - 2693(81 ) 90743 - 7 . url : (visitedon 04/27/2020).[59] David Prinz. Gauge Symmetries and Renormalization . Dec. 31, 2019. arXiv: . url : http : / / arxiv . org / abs / 2001 . 00104 (visited on01/06/2020).[60] David Prinz. “The Corolla Polynomial for Spontaneously Broken Gauge Theories”. In: Math-ematical Physics, Analysis and Geometry issn : 1572-9656. doi : . url : https://doi.org/10.1007/s11040- 016- 9222- 0 (visited on 02/08/2021).[61] Shoichi Sakata, Hiroomi Umezawa, and Susumu Kamefuchi. “On the Structure of the Inter-action of the Elementary Particles, I. The Renormalizability of the Interactions”. In: Progressof Theoretical Physics issn : 0033-068X. doi : . url : https://academic.oup.com/ptp/article/7/4/377/1870973 (visited on04/27/2020).[62] A. A. Slavnov. “Ward Identities in Gauge Theories”. In: Theoretical and Mathematical Physics issn : 1573-9333. doi : . url : https://doi.org/10.1007/BF01090719 (visited on 08/20/2019).[63] N. J. A. Sloane (editor). The On-Line Encyclopedia of Integer Sequences . 2018. url : https://oeis.org (visited on 01/20/2021).[64] A. T. Suzuki, E. S. Santos, and A. G. M. Schmidt. “Massless and Massive One-Loop Three-Point Functions in Negative Dimensional Approach”. In: The European Physical Journal C issn : 1434-6044, 1434-6052. doi : . url : http://dx.doi.org/10.1140/epjc/s2002-01035-0 (visited on 04/07/2020).[65] G. ’t Hooft and M. Veltman. “Diagrammar”. Geneva, Sept. 3, 1973. url : http://cds.cern.ch/record/186259/files/CERN-73-09.pdf?version=1 .[66] G. ’t Hooft and M. Veltman. “Regularization and Renormalization of Gauge Fields”. In: Nu-clear Physics B issn : 0550-3213. doi :
10 . 1016 / 0550 -3213(72 ) 90279 - 9 . url : (visited on 04/24/2020).[67] J. C. Taylor. “Ward Identities and Charge Renormalization of the Yang-Mills Field”. In: Nu-clear Physics B issn : 0550-3213. doi :
10 . 1016 / 0550 -3213(71 ) 90297 - 5 . url : (visited on 08/20/2019).[68] G. ’tHooft. “Renormalization of Massless Yang-Mills Fields”. In: Nuclear Physics B issn : 0550-3213. doi : . url : (visited on07/31/2019).[69] Abraham Ungar. “Generalized Hyperbolic Functions”. In: The American Mathematical Monthly issn : 0002-9890. doi : . JSTOR: .[70] N. I. Ussyukina and A. I. Davydychev. “New Results for Two-Loop off-Shell Three-PointDiagrams”. In: Physics Letters B issn : 03702693. doi : . arXiv: hep-ph/9402223 . url : http://arxiv.org/abs/hep-ph/9402223 (visited on 09/19/2019).
71] N. I. Usyukina. “On a Representation for the Three-Point Function”. In:
Theoretical andMathematical Physics issn : 1573-9333. doi :
10 . 1007 /BF01037795 . url : https://doi.org/10.1007/BF01037795 (visited on 04/25/2020).[72] M. K. Volkov. “Quantum Field Model with Unrenormalizable Interaction”. In: Communi-cations in Mathematical Physics issn : 0010-3616, 1432-0916. url : https://projecteuclid.org/euclid.cmp/1103840467 (visited on 04/26/2020).[73] J. C. Ward. “An Identity in Quantum Electrodynamics”. In: Physical Review doi : . url : https://link.aps.org/doi/10.1103/PhysRev.78.182 (visited on 08/20/2019).[74] Steven Weinberg. “High-Energy Behavior in Quantum Field Theory”. In: Physical Review issn : 0031-899X. doi : . url : https://link.aps.org/doi/10.1103/PhysRev.118.838 (visited on 03/15/2019).[75] G. C. Wick. “The Evaluation of the Collision Matrix”. In: Phys. Rev. doi : . url : https://link.aps.org/doi/10.1103/PhysRev.80.268 .[76] W. Zimmermann. “Convergence of Bogoliubovs Method of Renormalization in MomentumSpace”. In: Communications in Mathematical Physics issn :1432-0916. doi :
10 . 1007 / BF01645676 . url : https : / / doi . org / 10 . 1007 / BF01645676 (visited on 04/12/2020).(visited on 04/12/2020).