Renormalized Position Space Amplitudes in a Massless

CERN-PH-TH-2015-016

Renormalized position space amplitudes in a massless QFT Ivan Todorov Theory Division, Department of Physics, CERN, CH-1211 Geneva 23, Switzerland permanent address: Institute for Nuclear Research and Nuclear Energy Tsarigradsko Chaussee 72, BG-1784 Soﬁa, Bulgaria [email protected]; [email protected]

Abstract Ultraviolet renormalization of 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 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) 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 [LZ]. Contents

1 Introduction 2

2 Euclidean space renormalization of the massless ϕ4 theory 3 2.1 Terminology and conventions ...... 3 2.2 Synopsis of earlier results ...... 4 2.3 Vacuum completion of four-point graphs ...... 8

3 Renormalized position space amplitudes 8 3.1 Primitively divergent ϕ4 graphs. Conformal anomaly . . . . . 8 3.2 Associate homogeneity law for amplitudes with subdivergences 10 3.3 Four-point ϕ4 amplitudes as conformal invariant functions of four complex variables ...... 11

4 Outlook 14

1 1 Introduction

The Epstein-Glaser [EG] position space approach to the ultraviolet (UV) renormalization of a local quantum field theory (QFT) uses, following Bogol- ubov ([BS] and references therein), an adiabatic procedure in which each cou- pling constant g is replaced by a test function g(x) 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 factor- ization condition (see [NST12] and [NST]). Integrating over internal vertices - which would correspond to taking the adiabatic limit g(x) → g(6= 0) - would not keep track of the localization of internal vertices. The aim of the present note is to demonstrate that in a conformally invariant theory, like ϕ4 in D = 4 dimensions, such an integration actually does not pose problems. In fact, Lowenstein and Zimmermann have proven long ago [LZ] (in the momentum space framework) that the UV renormalized massless ϕ4 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 [GGV] which presents a major step in solving this problem (appearing in fact at an ad- vanced 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). It provides a systematic study of all graphs up to order g4 (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 con- crete typical cases to adding to the development of the general machinery of [NST]. 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 ϕ4 theory. We recall in Sect. 2.3 Schnetz’s vacuum completion of 4-point graphs, [Sch, S13]. We elaborate on his characteriza- tion of primitively divergent ϕ4 graphs and introduce the conformal anomaly in Sect. 3.1. Sect. 3.2 surveys the associate homogeneity law for amplitudes with subdivergences. We demonstrate in Sect. 3.3 that every 4-point amplitude in the ϕ4 theory can be presented as a conformal 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).

2 2 Euclidean space renormalization of the massless ϕ4 theory

2.1 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

Figure 1: Four point graphs with one (Γa), two (Γb) and three (Γc) independent external 4-vectors. of a graph of Γ is obtained by eliminating a non-empty subset of the vertices of Γ together with the adjacent half-edges. (We use the terms ’line’ and ’edge’ interchangeably.) The 2-point vertex graph Γ1 of Fig. 2 is a subgraph of the corresponding 2-point amplitude G1 but the 4-point graph Γa of Fig. 1 is not a subgraph of Γ1 (since the two graphs have the same set of vertices). There

Figure 2: Self-energy graph without (Γ1) and with (G1) external propagators are only graphs with an even number of external lines (2n-point graphs) in the ϕ4 theory. The (euclidean) position space Feynman rules for a massless scalar theory can be summarized as follows. To each vertex i (= 1, ..., V (Γ)) of the graph

3 α Γ we associate an euclidean 4-vector xi (= xi , α = 1, 2, 3, 4). To an internal line with end points i, j corresponds a (massless) propagator

1 X4 , x = x − x , x2 = (xα)2 . (2.1) x2 ij i j ij α=1

d4x Each internal vertex x is integrated with a measure π2 (a convention that goes back at least to old work of Broadhurst and helps displaying the number theoretic content of renormalization - see the discussion after Proposition 2.1 below). Thus, the Feynman amplitude G4 corresponding to the 4-loop graph Γ4 on Fig. 3 is given by

Figure 3: Four-loop 4-point graph Γ4

Z 1 Y4 1 d4x G (x , ..., x ) = . (2.2) 4 1 4 x2 x2 x2 x2 (x − x)2 π2 12 23 34 14 i=1 i

Note that the integral G4 is absolutely convergent for non-coinciding arguments, xi 6= xj for i 6= j. Each GΓ 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 R4(V (Γ)−1) - if its product with the volume form (of order 4(V (Γ) − 1)) has a non-positive degree of homogeneity, i.e. if κ := 2L(Γ) − 4(V (Γ) − 1) ≥ 0,L(Γ) being the number of internal lines of Γ. A divergent graph Γ is primitive if each of its proper connected subgraphs γ is convergent - i.e., if 4(V (γ) − 1) > 2L(γ).

2.2 Synopsis of earlier results In a massless QFT a Feynman amplitude G(~x) is a homogeneous function of ~x ∈ RN ; if G corresponds to a connected graph with V vertices then

4 N = 4(V −1). It is superficially divergent if G defines a homogeneous density in RN \D of non-positive degree, where D is a subvariety (diagonal) of lower dimension: G(λ~x) dN λx = λ−κG(~x) dN x , κ ≥ 0 , x ∈ RN \D (λ > 0) ; (2.3) κ is called (superficial) degree of divergence. In a scalar QFT with a propagator (2.1) a connected graph with a set L of internal lines gives rise to a Feynman amplitude that is a multiple of the product Y 1 G(~x) = . (2.4) x2 (i,j)∈L ij If G is superficially divergent (i.e. if κ = 2L − N ≥ 0 where L is the number of lines in L) then it is divergent - that is, it does not admit a homogeneous extension as a distribution on RN . (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. 5.2 of [NST].) For a primitively divergent amplitude the following proposition (Theorem 2.3 of [NST12]) serves as a definition of both a residue ResG and of a renormalized amplitude Gρ(~x). Proposition 2.1. If G(~x) (2.4) is primitively divergent then for any smooth norm ρ(~x) on RN one has 1 [ρ(~x)]²G(~x) − (Res G)(~x) = Gρ(~x) + O(²). (2.5) ² Here Res G is a distribution with support at the origin. Its calculation is reduced to the case κ = 0 of a logarithmically divergent graph by using the identity (−1)κ (Res G)(~x) = ∂ ...∂ Res (xi1 ...xiκ G)(~x) (2.6) κ! i1 iκ where summation is assumed (from 1 to N) over the repeated indices i1, ..., iκ. If G is homogeneous of degree −N then

i (Res G)(~x) = res (G) δ(~x) (for ∂i(x G) = 0) . (2.7) Here the numerical residue res G is given by an integral over the hypersurface Σρ = {~x| ρ(~x) = 1}: Z 1 XN res G = G(~x) (−1)i−1xidx1 ∧ ...dxˆ i... ∧ dxN , (2.8) πN/2 Σρ i=1

5 (a hat over an argument meaning, as usual, that this argument is omitted). The residue res G is independent of the (transverse to the dilation) surface N−1 Σρ since the form in the integrant is closed in the projective space P . We note that N is even, in fact, divisible by 4, so that PN−1 is orientable. The functional res G is a period according to the definition of [KZ, M-S]. Such residues are often called ”Feynman” or ”quantum” periods in the present context (see e.g. [Sch]). We shall use in what follows ”residue” and ”period” interchangeably. The convention of accompanying the 4D volume d4x by a π−2 (2π2 being the volume of the unit sphere S3 in four dimensions), reflected in the prefac- tor, goes back at least to Broadhurst [B, BK] and is adopted in [Sch, BS12]; it yields rational residues for the graph Γa of Fig. 1 and for that of Fig. 2. For graphs with three or higher number of loops `(= h1(Γ), the first Betti number of the graph) one encounters, in general, multiple zeta values of overall weight not exceeding 2` − 3 (cf. [BK, Sch, S13]). If we denote by L = L(Γ) and V = V (Γ) the numbers of internal lines and vertices of a connected graph Γ then ` = L − V + 1(= V − 1 for a connected 4-point graph in the ϕ4 theory). Examples of graphs with three and four loops are Γc on Fig. 1 and Γ4 on Fig. 3, respectively. With the above choice of the 4D volume form the only residues at three, four and five loops (in the ϕ4 theory) are integer multiples of ζ(3), ζ(5) and ζ(7), respectively. The first double zeta value, ζ(5, 3), appears at six loops (with a rational coefficient) (see the census in [Sch]). All known residues were (up to 2013) rational linear combinations of multiple zeta values [BK, Sch]. A seven loop graph was re- cently demonstrated [P, B14] to involve multiple Deligne values - i.e., values of hyperlogarithms at sixth roots of unity. The self-energy graph Γ1 on Fig. 2 is viewed as primitively (quadrati- cally) 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 2 −3 1 2 −2 divergent 4-point graph at Γa of Fig. 1 using the identity (x ) = 8 ¤(x ) where ¤ stands for the 4-dimensional Laplace operator. We have (cf. Eqs. (3.2) and (3.3) of [NST12]): µ ¶ 1 x2 ² 1 1 xα x2 ∂ lim[ − δ(x)] = ∂α [ ln ] , ∂α := , ²→0 π2(x2)2 `2 ² 2π2 (x2)2 `2 ∂xα µ ¶ 1 x2 ² 1 1 xα x2 Γ1(x) = lim[ − Box δ(x)] = ¤ ∂α[ ln ] , (2.9) ²→0 π2(x2)3 `2 8² 16π2 (x2)2 `2

6 where δ(x) is the 4-dimensional Dirac δ-function. 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, 1 λ6Γ (λx) = Γ (x) + ¤ δ(x) ln λ. (2.10) 1 1 4

The renormalized Feynman amplitude G(x1, ..., x4) 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):

12 λ G(λx1, ..., λx4) = G(x1, ..., x4) + res(G) δ(x12)δ(x23)δ(x34) ln λ. (2.11) 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 [NST], 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 the following associate homogeneity law: Xn (ln λ)j λN G(λ~x) = G(~x) + R (G)(~x) , (2.12) j j! j=1 where the distributions Rj(G) can be viewed as generalized residues: XN j−1 α Rj(G) = Res[(E + N) G(~x)] , E = x ∂α. (2.13) α=1 One proves that only the coeﬃcient to the highest power of the logarithm,

n−1 Rn(G) = res[(E + N) G(~x)]δ(~x) , (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 normaliza- tion condition consists in ﬁxing a zero of the Fourier transform of Feynman amplitudes. For instance, the Fourier transform of Γ1(x) (2.9), p2 p2 Γ˜ (p) = ln( )(µ2`2eγ = 1 for γ = −Γ0(1)) (2.15) 1 8 µ2 2 2 vanishes for p = µ (while Γ1(x) only vanishes for x → ∞).

7 2.3 Vacuum completion of four-point graphs Following Schnetz [Sch, S13] we associate to each 4-point graph Γ of the ϕ4 theory a completed vacuum graph Γ,¯ obtained from Γ by joining all four external lines in a new vertex ”at infinity”. An n-vertex 4-regular vacuum graph - having four edges incident with each vertex and no tadpole loops - gives rise to n 4-point graphs (with (n − 1) vertices each) corresponding to the n possible choices of the vertex at infinity. The introduction of such completed graphs is justified by the following result (see Sect. 2.3, Proposition 2.6 and Sect. 2.4, Theorem 2.7 of [Sch] as well as Sect. 3.1 below). Theorem 2.2. A 4-regular vacuum graph Γ¯ 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 Γ is primitively divergent iff its completion Γ¯ 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 self-energy graph Γ1 of Fig. 2 (Proposition 3.1 below). The only primitive 4-point graph with a rational period is the one loop graph Γa of Fig. 1 (with residue 1).The n loop zig-zag graph [BS12] has a residue that is a rational multiple of ζ(2n − 3), n = 3, 4, .... The first two zig-zag diagrams are the graphs Γc of Fig. 1 and Γ4 on Fig. 3. Their ¡2n−2¢ residues are n−1 ζ(2n − 3), n = 3, 4 (see [T] for an elementary derivation and further references).

3 Renormalized position space amplitudes

3.1 Primitively divergent ϕ4 graphs. Conformal anomaly We ﬁrst note that the primitively divergent vacuum graphs of the ϕ4 theory have either two or four external legs. The only 4-regular vacuum graph with ¯ three vertices is the completion Γa of Γa (Fig. 1). Calling a vacuum graph simple if it contains at most one edge joining any two of its vertices, one can ¯ prove that Γa is the only non-simple completed primitive graph. Proposition 3.1. The only primitively divergent 2-point Feynman amplitude corresponds to the graph Γ1 of Fig. 2.

8 Proof. Cutting off an external vertex of a given 2-point graph Γ we obtain a 4-point graph that is the trivial single vertex graph for Γ = Γ1. 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 ϕ4 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 κ = 2L − 4(V − 1) = 0. Proposition 3.3. The period of a completed primitive graph Γ¯ is equal to the residue of each 4-point graph Γ = Γ¯ − v (obtained from Γ¯ by cutting off an arbitrary vertex v). The resulting common period can be evaluated from Γ¯ by choosing arbitrarily three vertices {0, e (s.t. e2 = 1), ∞}, setting all propagators corresponding to edges of the type (xi, ∞) equal to 1 and integrating over the remaining n − 2 vertices of Γ (n = V (Γ)): Z nY−1 d4x P er(Γ)¯ ≡ res(Γ) = Γ(e, x , ..., x , 0) i . (3.1) 2 n−1 π2 i=2 Sketch of proof. For a given choice of the vertex at infinity (3.1) follows from (2.7). The independence of the choice of the point at infinity follows xi from conformal invariance; the conformal inversion Ir : xi → 2 , i = 2, ..., n, xi exchanges the (arbitrarily chosen) xn = 0 and ∞ while the integral remains invariant since 2 2 4 1 xi xj 4 d x Ir : 2 → 2 , d x → 2 4 . (3.2) xij xij (x ) The conformal invariance is broken in a controlable way by renormalization. For a special conformal transformation x + cx2 dx2 g x = , (dg x)2 = , ω(c, x) = 1 + 2cx + c2x2 (3.3) c ω(c, x) c ω(c, x)2 one obtains the conformal anomaly by substituting λ in (2.11) by 1 for ω(c,xi) any i ∈ (1, 2, 3, 4). The δ-function ensures that the result is independent of the choice of i. The cocycle condition that implements the group law is satisfied because of the identity

ω(c1 + c2, x) = ω(c1, x)ω(c2, gc1 x) . (3.4)

9 3.2 Associate homogeneity law for amplitudes with subdivergences The study of graphs with a 2-point subdivergence requires the computation of the dressed propagator G1 of Fig. 2:

Z 4 Z 4 d x d y Γ1(x − y) G1(x12) = 2 2 2 2 . (3.5) π π (x1 − x) (y − x2)

An intelligent way to compute G1 consists in using (2.12) to ﬁrst derive its dilation law: 2 ln λ λ G1(λx12) = G1(x12) − 2 2 , (3.6) π x12 where we have used the fact that the integrand in (3.5) involves the Green function of the 4D Laplacian: 1 ¤1 2 2 = −δ(x1 − x) . (3.7) 4π (x1 − x)

The general form of G1(x) satisfying (3.6) is:

ln( `2 ) G (x) = x2 . (3.8) 1 2π2x2 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 St(x) corresponding to the ”stye graph” displayed on Fig. 4:

Figure 4: The ”stye”: a 4-point graph with a 2-point subdivergence

G (x) St(x) = 1 for x 6= 0 . (3.9) x2

10 The extension of the distribution St to the entire R4 is again reduced to an 1-dimensional problem by integrating the corresponding density with respect to the angles: Z 3 3 d ω 2 ` r dr St(rω) 2 = ln dr. S3 π r r Its general associate homogeneous extension, the renormalized stye 1-form R(St) is:

Z 3 3 ext d ω R(St) := r dr St (rω) 2 = S3 π `0 rl0 ` r d(ln( ) ln( ) = d[(ln )2 − (ln )2]θ(r) (3.10) r `2 + `0 ` where `0 > 0 is another scale and θ(r) is the Heaviside step function. (For `0 = ` we recover the extension given by Proposition A.1 of [NST].) The associate homogeneity law for R(St) reads: r R(St)(λr) = R(St)(r) − 2d(ln ) ln λ − δ(r)dr(ln λ)2 . (3.11) ` + We see that the term with (ln λ)2 is indeed independent of the ambiguity (`), ` while the coeﬃcient to ln λ is 2d(ln r )+ and thus depends on `. We shall consider the case of 4-point subdivergences within our treatment of 4-point functions in the ϕ4 theory in Sect. 3.3 below.

3.3 Four-point ϕ4 amplitudes as conformal invariant functions of four complex variables

Every four points, x1, ..., x4, 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 xi by a complex number zi so that 2 2 xij = |zij| = (zi − zj)(¯zi − z¯j). (3.12) To make the correspondence between 4-vectors x and complex numbers z explicit we fix a unit vector e and let n be a variable unit vector parametrizing a 2-sphere orthogonal to e. Then any euclidean 4-vector x can be written (in spherical coordinates) in the form:

x = r(cosρ e + sinρ n) , e2 = 1 = n2 , en = 0 , r ≥ 0 , 0 ≤ ρ ≤ π . (3.13)

11 The 4D volume element is written in these coordinates as Z d4x = r3drsin2ρ dρ d2n , d2n = 4π . (3.14) S2 We associate with the vector x (3.13) a complex number z such that: z = reiρ → x2(= r2) = zz¯ , (x − e)2 = |1 − z|2 = (1 − z)(1 − z¯) (3.15)

Z 4 2 Z d x 2 d z 4 2 2 = |z − z¯| , δ(x)d x = δ(z)d z . (3.16) n∈S2 π π S2 As massless Feynman amplitudes give rise to well deﬁned functions for non-coinciding arguments the ϕ4 4-point amplitudes are conformally covari- ant for such arguments. For a graph with four distinct external vertices the amplitude (integrated over the internal vertices) has scale dimension 12 (in mass or inverse length units) and can be written in the form: g(u, v) F (z) Q Q G(x1, ..., x4) = 2 = 2 (3.17) i

2 2 2 2 x12x34 x14x23 2 z12z34 u = 2 2 = zz¯ , v = 2 2 = |1 − z| , z = . (3.18) x13x24 x13x24 z13z24

We shall outline how one can compute the amplitude G4 corresponding to the graph Γ4 of Fig. 3 in terms of the variables z (see [S13, T]). We shall then use the integral in (2.2), Z Y4 1 d4x I(x , ..., x ) := x2 x2 x2 x2 G (x , ..., x ) = , (3.19) 1 4 12 23 34 14 4 1 4 (x − x)2 π2 i=1 i to calculate the dilation anomaly of a 4-point graph with a primitive 4-point subdivergence. According to (3.17) and (3.19) this amounts to evaluating the 2 2 conformal invariant amplitude F (z) = x13x24I(x1, ..., x4). There are two ways to do it: one may either expand the original x-space integrand in Gegenbauer polynomials [CKT] (after sending x1 to inﬁnity) or use the theory of single- valued multiple polylogarithms [B04, S13] (for a pedagogical derivation and more references - see [T]). The result is:

1−z 2Li2(z) − 2Li2(¯z) + ln zz¯ln 4iD(z) F (z) = 1−z¯ = (3.20) z − z¯ z − z¯

12 where D(z) is the Bloch-Wigner single valued dilogarithm [Bl, Z]. The real valued function µ ¶ ˜ z12z34 D(z1, z2, z3, z4) = D (3.21) z13z24 is known since Lobachevsky to give the volume of the (oriented) ideal tetrahe- dron with vertices z1, ..., z4 on the absolute (horosphere) of the 3-dimensional hyperbolic space [M]. 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 unde odd permutations of the arguments z1, ..., z4. The complete 4-point integral I can be written in the form Z d2z Y4 I = I(z , ..., z ) = |z − z¯|2 |z − z|−2 = 1 4 π i i=1 ˜ 4iD(z) 4iD(z1, ..., z4) = 2 2 = , (3.22) |z13| |z24| (z − z¯) z12z34z¯13z¯24 − z13z24z¯12z¯34 where z is given by (3.18). The identity z14z23 = z13z24 − z12z34 implies that the denominator in (3.22) 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 [D] 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 [B04] of single-valued multipolylogarithms labeled by words of the 2-letter alphabet {0, 1}. We shall just need the weight two functions Pab, a, b ∈ {0, 1} defined as single-valued solutions of the differential equation

P (z) ∂ P (z) = a ,P (z) = ln zz¯ , P (z) = ln |1 − z|2 (3.23) z ab z − b 0 1

(satisfying the initial condition Pab(0) = 0 for (ab) 6= (00), and P00(z) = (ln zz¯)2 2 ). Their sum satisﬁes the shuﬄe relation P01 + P10 = P0P1 while

P01(z) − P10(z) = 4iD(z) . (3.24)

13 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 z12z34, is small:

−2 I(z1, ..., z4) ∼ |z13z24| ∂z(P01 − P10)|z∼0 = −2 z13z24 2 = |z13z24| (ln | | + 2) for z12z34 → 0 . (3.25) z12z34 Thus I 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 C3. Let S(y1, ..., y4) be a renormalized primitively divergent 4-point amplitude that appears as a subdivergence in Z Y4 d4y G (x , ..., x ) = S(y , ..., y ) i . (3.26) S 1 4 1 4 π2(x − y )2 i=1 i i The dilation law for S,

λ12S(λ~y) = S(~y) + res(S) δ(~y) ln λ (3.27) implies that the dilation anomaly of GS for non-coinciding arguments is

12 λ GS(λx1, ..., λx4) − GS(x1, ..., x4) = G4(x1, ..., x4) res(S) ln λ , (3.28)

2 where G4 is given by (2.2). It follows that the coeﬃcient res2(GS) to (ln λ) which is independent of the renormalization ambiguity is given by the product of residues:

res2(GS) = res(G4) res(S)(res(G4) = 20 ζ(5)) . (3.29)

(As noted in Sect. 2.3 G4 is the amplitude of the second of the ”zig-zag graphs”, whose residues, conjectured by Broadhurst and Kreimer, have been evaluated in [BS12]; res(G4) has been also calculated using higher depth zeta values in [T].)

4 Outlook

Quantum ﬁeld theory which once signaled, according to Freeman Dyson [D72], a divorce between mathematics and physics, now seems to be the best

14 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. [BEK, BlK, BKr, BS14]). 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 Hörmander[H]). Here we complete this study in the case of the conformally invariant (at least at the classical level) euclidean ϕ4 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. [W12]). Acknowledgments. The work started with [NST12, NST] and contin- ued with the present paper was initiated by Raymond Stora whom I thank for sharing his insight with me and for a critical reading of the manuscript. The importance of extending the Epstein-Glaser renormalization program to amplitudes with integrated over internal vertices was stressed to me by Detlev Buchholz. Discussions with Nikolay Nikolov at an early stage of this work are also gratefully acknowledged. I thank the Theory Division of the Department of Physics of CERN for hospitality during the course of this work. The work of the author has been supported in part by Grant DFNI T02/6 of the Bulgarian National Science Foundation.

15 References

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