Renormalization of position space amplitudes in a massless QFT

Ultraviolet renormalization of position space massless Feynman amplitudes has been shown to yield associate homogeneous distributions. Their degree is determined by the degree of divergence while their order—the highest power of logarithm in the dilation anomaly—is given by the number of (sub)divergences. In the present paper we review these results and observe that (convergent) integration over internal vertices does not alter the total degree of (superficial) ultraviolet divergence. For a conformally invariant theory internal integration is also proven to preserve the order of associate homogeneity. The renormalized 4-point amplitudes in the φ4 theory (in four space-time dimensions) are written as (non-analytic) translation invariant functions of four complex variables with calculable conformal anomaly. Our conclusion concerning the (off-shell) infrared finiteness of the ultraviolet renormalized massless φ4 theory agrees with the old result of Lowenstein and Zimmermann [23].

1. INTRODUCTION On-shell scattering amplitudes are simpler to study in momentum space, so perturbative renormalization in quantum field theory (QFT) was originally developed starting with an ultraviolet momentum cutoff. On the other hand, Stueckelberg and Bogolubov had realized early on (see [4] and references therein) that the principle of relativistic causality in position space may serve as a true basis of renormalization theory: not only counter-terms have to be local but causal factorization gives rise to a recursive procedure that allows to reduce ultraviolet (UV) renormalization to that of primitively divergent amplitudes. In fact, Fourier transform is an old example of what is now fashionable to call a duality transformation: it maps a large momentum problem into a small distance one. The position space approach was developed systematically by Epstein-Glaser [16] in the early seventies but only gained popularity much later. It does not use global Poincaré invariance and thus offers a way to develop perturbative QFT and operator product expansions on a curved background (see [17,20] and references to earlier work of Brunetti, Duetsch, Fredenhagen and of Hollands et al. cited there).
It uses, following Bogolubov et al., an adiabatic procedure in which each coupling constant g is replaced by a test function that vanishes at infinity. This forces one to treat all vertices as external and allows to set up a recursive procedure for Feynman graphs of increasing order in terms of a causal factorization condition (see [NST12] and [NST]). Integrating over internal vertices-which would correspond to taking the adiabatic limit -would not keep track of the localization of internal vertices. Here  ( 0) g x g → ≠ we demonstrate that in a conformally invariant theory, like in dimensions, such an integration actually does not pose problems. In fact, Lowenstein and Zimmermann have proven long ago [23] (in the momentum space framework) that the UV renormalized massless theory is infrared finite. Various parts of the position space problem have also been addressed by a number of authors. We refer to the recent paper [19] which presents amajor step in solving this problem (appearing in fact at an advanced stage of our own work on the subject) and has, in addition, the virtue of giving a careful account of earlier contributions (within a bibliography of 49 entries-see also [18] for a later survey). It provides a systematic study of all graphs up to order (and up to three loops with no tadpoles). We choose to follow the elementary readable style of this work, preferring the outline of the argument in concrete typical cases to adding to the development of the general machinery of [27]. We calculate in particular the dilation anomaly of a logarithmically divergent 4-point graph with an arbitrary 4-point subdivergence.
We start in Sect. 2 by reviewing earlier results on euclidean space renormalization of the massless theory: the notion of residue and renormalization of primitively divergent graphs (Sect. 2.2), and the recursion based on the causal factorization requirement allowing to renormalize arbitrary associate homogeneous amplitudes (Sect. 2.3). We recall in Sect. 2.4 Schnetz's vacuum completion of 4-point graphs, [29,30]. We elaborate on his characterization of primitively divergent graphs and introduce the conformal anomaly in Sect. 3.1. Sect. 3.2 surveys the associate homogeneity law for amplitudes with subdiver- ϕ gences. We demonstrate in Sect. 3.3 that every 4-point amplitude in the theory can be presented as a translation invariant function of four complex variables. Using this representation we exhibit the dilation anomaly of a divergent 4-point graph with any primitive 4-point subdivergence (the reduced graph-in which the 4-point subgraph is substituted by a single internal vertex-is having four loops in this generic case).

Terminology and Conventions
We call a graph n-point if it has n external (half-) lines. Thus, each of the three graphs on Fig. 1 corresponds to a 4-point vertex function (i.e. a Feynman amplitude without propagators attached to the external lines). Any subgraph of a graph of is obtained by eliminating a non-empty subset of the vertices of together with the adjacent half-edges. (We use the terms "line" and "edge" interchangeably.) The 2-point vertex graph of Fig. 2 is a subgraph of the corresponding 2-point amplitude but the 4-point graph of Fig. 1 is not a subgraph of (since the two graphs have the same set of vertices). There are only graphs with an even number of external lines (2n-point graphs) in the theory. The (euclidean) position space Feynman rules for a massless theory can be summarized as follows.
i i  Fig. 2. Self-energy graph without ( ) and with ( ) external propagators. Note that the integral is absolutely convergent for non-coinciding arguments, for . Each is a locally integrable homogeneous function for non-coinciding arguments and defines a Schwartz distribution off the large diagonal. It may diverge-i.e., not admit a homogeneous extension as a distribution on the entire space -if its product with the volume form (of order ) has a non-positive degree of homogeneity, i.e. if being the number of internal lines of . A divergent graph is primitive if each of its proper connected subgraphs is convergent-i.e., if .

Renormalization of Primitively Divergent Graphs
In a massless QFT a Feynman amplitude is a homogeneous function of ; if corresponds to a connected graph with vertices then . It is superficially divergent if defines a homogeneous density in of non-positive degree, where is a subvariety (diagonal) of lower dimension: is called (superficial) degree of divergence. In a scalar QFT with propagators a connected graph with a set of internal lines gives rise to a Feynman amplitude that is a multiple of the product where is the number of lines in ) then it is divergent-that is, it does not admit a homogeneous extension as a distribution on . (For more general spin-tensor fields whose propagator has a polynomial numerator a superficially divergent amplitude may, in fact, turn out to be convergent-see Sect.
Here is a distribution with support at the origin. Its calculation is reduced to the case of a logarithmically divergent graph by using the identity where summation is assumed (from 1 to N) over the repeated indices . If is homogeneous of degree-then Here the numerical residue is given by an integral over the hypersurface : (a hat over an argument meaning, as usual, that this argument is omitted). The residue is independent of the (transverse to the dilation) surface since the form in the integrant is closed in the projective space . We note that N is even, in fact, divisible by 4, so that is orientable. Remark 2.1. The use of a (homogeneous) norm as a regulator appears to be more flexible than dimen- TODOROV sional regularization and should be also applicable in the presence of an axial (or chiral) anomaly. The functional is a period according to the definition of [22,25]. Such residues are sometimes called Feynman or quantum periods in the present context (see e.g. [29]). We shall use in what follows "residue" and "period" interchangeably.
The convention of accompanying the 4D volume by a ( being the volume of the unit sphere in four dimensions), reflected in the prefactor, goes back at least to Broadhurst [5,7] and is adopted in [12,29]; it yields rational residues for the graph of Fig The self-energy graph on Fig. 2 is viewed as primitively (quadratically) divergent in the configuration space-while it is treated as a diagram with overlapping divergences in the traditional momentum space picture. Its renormalized contribution can be obtained from that of the logarithmically divergent 4-point graph at of Fig. 1 where δ(x) is the 4-dimensional Dirac δ-function and is a free parameter. As noted in the beginning-and illustrated in the above example-the extension of a homogeneous primitively divergent amplitude is no longer homogeneous. It satisfies instead an associate homogeneity condition which fixes the dilation anomaly; in particular, The renormalized Feynman amplitude of an arbitrary primitively divergent 4-point graph (with a single external half-line at each external vertex) is an associate homogeneous distribution (of order one): For graphs with subdivergences one first renormalizes the contributions of all primitively divergent subgraphs, then a similar procedure is applied to the resulting associate homogeneous amplitude (see [27], Sect. 4 and Appendix D.2). Remarkably, at each step one just solves a 1-dimensional problem. For a renormalized 4-point function with n (sub) divergences one then has an order associate homogeneity law: where the distributions can be viewed as generalized residues: (2.13) One proves that only the coefficient to the highest power of the logarithm, (2.14) is independent of the ambiguity of renormalization (i.e., independent, in our case, of the scale parameters-like in Eq. (2.9)). The standard normalization condition consists in fixing a zero of the Fourier transform of Feynman amplitudes. For instance, the Fourier transform of (2.9), (2.15) vanishes for (while only vanishes for ). 6 1 1

Renormalization of Associate Homogeneous Distributions;
Causal Factorization In order to treat the general case of a graph with subdivergences we shall define ultraviolet renormalization by induction with respect to the number of vertices. Assume that all contributions of diagrams with less than points are renormalized. If then is an arbitrary connected n-point graph its renormalized contribution should satisfy the following inductive causal factorization requirement.
Let the index set of be split into any two non-empty non-intersecting subsets Let for . Let further and be the renormalized distributions associated with the subgraphs whose vertices belong to the subsets and , respectively. We demand that for each such splitting our euclidean distribution , defined on all partial diagonals, exhibits the factorization property: (2.16) where are factors (of type (2.1)) in the rational function which are understood as multiplicators on . Remark 2.2. In the Lorentzian signature case one demands that the points indexed by the set precede those of and uses Wightman functions instead of in the counterpart of (2.16)-thus justifying the term causal (see Sect. 2.2 of [27]).
We shall add to this basic physical requirement two more mathematical conventions (MC) which will substantially restrict the notion of renormalization used in this paper.
(MC1) Renormalization maps rational homogeneous functions onto associate homogeneous distributions of the same degree of homogeneity; it extends associate homogeneous distributions defined off the small diagonal to associate homogeneous distributions of the same degree (but possibly of higher order) defined everywhere on . (MC2) The renormalization map commutes with multiplication by polynomials. If we extend the class of our distributions by allowing multiplication with smooth functions of no more than polynomial growth (in the domain of definition of the corresponding functionals), then this requirement will imply commutativity of the renormalization map with such multipliers.
The induction is based on the following.  . This completes the proof of our statement.
The first step in implementing the above inductive procedure consists in the renormalizatioin of primitively divergent graphs surveyed in Sect. 2.2. Schnetz's notion of completion of 4-point graphs, reviewed in the next subsection, offers a general picture of primitively divergent graphs in the in the -theory.

Proposition 2.3. A 4-regular vacuum graph with at least three vertices is said to be completed primitive if the only way to split it by a four edge cut is by splitting off one vertex. A 4-point Feynman amplitude corresponding to a connected 4-regular graph is primitively divergent iff its completion is completed primitive. All 4-point graphs with the same primitive completion have the same residue.
There are infinitely many primitive 4-point graphs while there is a single primitive 2-point graph: the selfenergy graph of Fig. 2 (Proposition 3.1 below). The only primitive 4-point graph with a rational period is the one loop graph of Fig. 1 (with residue 1).The n loop zig-zag graph [12] has a residue that is a rational multiple of . The first two zig-zag diagrams are the graphs of Fig. 1 and on Fig. 3.
Their residues are (see [31] for an elementary derivation and further references).

Conformal Anomaly
We first note that the primitively divergent vacuum graphs of the theory have either two or four external legs. The only 4-regular vacuum graph with three vertices is the completion of ( Fig. 1). Calling a vacuum graph simple if it contains at most one edge joining any two of its vertices, one can prove that is the only non-simple completed primitive graph. Proof. Cutting off an external vertex of a given 2-point graph we obtain a 4-point graph that is the trivial single vertex graph for . The Proposition then follows from the following simple fact about 4-point graphs.

Lemma 3.2. Each non-trivial connected 4-point graph of the theory is either primitive logarithmically divergent or contains a subdivergence.
The Lemma follows from the fact that for a connected 4-point 4-regular graph the number of internal lines is L = 2(V -1) and hence the superficial degree of divergence is .
The conformal invariance is broken in a controlable way by renormalization. For a special conformal transformation (3.3) one obtains the conformal anomaly by substituting in (2.11) by for any . The -function ensures that the result is independent of the choice of . The cocycle condition that implements the group law is satisfied because of the identity (3.4)

Associate Homogeneity Law for Amplitudes with Subdivergences
The study of graphs with a 2-point subdivergence requires the computation of the dressed propagator of Fig. 2: An intelligent way to compute consists in using (2.12) to first derive its dilation law: where we have used the fact that the integrand in (3.5) involves the Green function of the 4D Laplacian: The general form of satisfying (3.6) is: We observe that (convergent) integration over internal vertices preserves the order of associate homogeneity of the integrand. The power of the logarithm only increases if one encounters another ultraviolet divergence (typically in an UV divergent graph with a subdivergence) as illustrated by the amplitude corresponding to the "stye graph" displayed on Fig. 4: The extension of the distribution to the entire is again reduced to an 1-dimensional problem by integrating the corresponding density with respect to the angles: Its general associate homogeneous extension, the renormalized stye 1-form is: where is another scale and is the Heaviside step function. (For we recover the extension given by Proposition A.1 of [27].) The associate homogeneity law for reads: We see that the term with is indeed independent of the ambiguity ( ), while the coefficient to is and thus depends on .
We shall consider the case of 4-point subdivergences within our treatment of 4-point functions in the theory in the next subsection.

Four-Point ϕ 4 Amplitudes as Conformal Invariant Functions of Four Complex Variables
Every four points, , can be confined by a conformal transformation to a 2-plane (for instance by sending a point to infinity and using translation invariance). Then we can represent each point by a complex number so that (3.12) To make the correspondence between 4-vectors and complex numbers explicit we fix a unit vector and let be a variable unit vector parametrizing a 2-sphere orthogonal to . Then any euclidean 4-vector can be written (in spherical coordinates) in the form: (3.13) The 4D volume element is written in these coordinates as (3.14) x z e n e x 2 2 (cos sin ) 1 0 00 x r e n e n en r = ρ + ρ , = = , = , ≥ , ≤ ρ ≤ π. We shall outline how one can compute the amplitude corresponding to the graph of Fig. 3 in terms of the variables (see [30,31]). We shall then use the integral in (2.2), (3.20) to calculate the dilation anomaly of a 4-point graph with a primitive 4-point subdivergence. According to (3.17) and (3.18) this amounts to evaluating the conformal invariant amplitude . There are two ways to do it: one may either expand the original -space integrand in Gegenbauer polynomials [13] (after sending to infinity) or use the theory of single-valued multiple polylogarithms [9,30] (for a  pedagogical derivation and more references-see [31]). The result is: where D(z) is the Bloch-Wigner single-valued dilogarithm [1,33]. The real valued function (3.22) is known since Lobachevsky to give the volume of the (oriented) ideal tetrahedron with vertices on the absolute (horosphere) of the 3-dimensional hyperbolic space [24]. It has the symmetry of a dimensionless fermionic 4-point function (in a logarithmic 2D conformal field theory). It is symmetric under even permutations and changes sign under odd permutations of the arguments . The complete 4-point integral can be written in the form (3.23) where is given by (3.19). The identity z 14 z 23 = implies that the denominator in (3.23) has the same symmetry properties as the numerator.
The (Hopf-)algebraic structure of hyperlogarithms can be used to reduce the study of their singularities (and monodromy) to the known properties of ordinary logarithms (see [22] for a physicist oriented review with applications and references to the original mathematical papers). To display the (integrable) singularities of the integral (3.22) it will be enough to expand the Bloch-Wigner dilogarithm in terms of the recursively defined Brown's basis [9] of single-valued multipolylogarithms labeled by words of the 2-letter alphabet . We shall just need the weight two functions defined as single-valued solutions of the differential equation The symmetry of the ratio (3.22) allows to determine its behaviour for various pairs of coinciding arguments by just considering the limit in which one of them, say , is small: Thus only has logarithmic singularities; it follows that any finite power of the function (3.22) is locally integrable and hence defines a distribution in the whole space .
Let be a renormalized primitively divergent 4-point amplitude that appears as a subdivergence in (3.27) The dilation law for , (3.28) implies that the dilation anomaly of for non-coinciding arguments is (3.29) where is given by (2.2). It follows that the coefficient to which is independent of the renormalization ambiguity is given by the product of residues: (3.30) (As noted in Sect. 2.4, is the amplitude of the second of the "zig-zag graphs", whose residues, conjectured by Broadhurst and Kreimer, have been evaluated in [12]; has been also calculated using higher depth zeta values in [31].) 4. OUTLOOK Quantum field theory which once signaled, according to Freeman Dyson [15], a divorce between mathematics and physics, now seems to be the best common playground of the two sciences. Not only did renormalization theory, which was viewed as a liability, become respectable in the Epstein-Glaser approach, but the key role, which the notion of residue and the applications of the (Hopf) algebra of hyperlogarithms play in it, relates it to current work in algebraic geometry and number theory (see e.g. [2,3,8,11]).
The consideration of a massless scalar QFT simplifies the treatment of renormalization by reducing it to the study of renormalization of (associate) homogeneous distributions (and extending the analysis of ( ) res G Hörmander [21]). Here we complete this study in the case of the conformally invariant (at least at the classical level) euclidean theory by including integration over internal vertices.
There seems to exist a parallel between the study of massless QFT and neglecting friction by the founding fathers of classical mechanics starting with Galileo. Such an idealization made it easier to find the simple basic laws of mechanics. Taking subsequently the corrections due to friction into account just added minor technical details to the general picture. We feel that at least as far as UV renormalization is concerned, the role of residues (that also appear in the renormalization group beta function) and their relation to modern study of periods in number theory, taking masses into account will not change substantially the overall picture and can be advantageously postponed to a later stage (hadronic masses appearing, without having been put in, as a result of the strong interaction-cf. [32]).