Towards cohomology of renormalization: bigrading the combinatorial Hopf algebra of rooted trees

The renormalization of quantum field theory twists the antipode of a noncocommutative Hopf algebra of rooted trees, decorated by an infinite set of primitive divergences. The Hopf algebra of undecorated rooted trees, ${\cal H}_R$, generated by a single primitive divergence, solves a universal problem in Hochschild cohomology. It has two nontrivial closed Hopf subalgebras: the cocommutative subalgebra ${\cal H}_{\rm ladder}$ of pure ladder diagrams and the Connes-Moscovici noncocommutative subalgebra ${\cal H}_{\rm CM}$ of noncommutative geometry. These three Hopf algebras admit a bigrading by $n$, the number of nodes, and an index $k$ that specifies the degree of primitivity. In each case, we use iterations of the relevant coproduct to compute the dimensions of subspaces with modest values of $n$ and $k$ and infer a simple generating procedure for the remainder. The results for ${\cal H}_{\rm ladder}$ are familiar from the theory of partitions, while those for ${\cal H}_{\rm CM}$ involve novel transforms of partitions. Most beautiful is the bigrading of ${\cal H}_R$, the largest of the three. Thanks to Sloane's {\tt superseeker}, we discovered that it saturates all possible inequalities. We prove this by using the universal Hochschild-closed one-cocycle $B_+$, which plugs one set of divergences into another, and by generalizing the concept of natural growth beyond that entailed by the Connes-Moscovici case. We emphasize the yet greater challenge of handling the infinite set of decorations of realistic quantum field theory.Comment: 21 pages, LaTe

Feynman diagrams as a weight system: four-loop test of a four-term relation

At four loops there first occurs a test of the four-term relation derived by the second author in the course of investigating whether counterterms from subdivergence-free diagrams form a weight system. This test relates counterterms in a four-dimensional field theory with Yukawa and $\phi^4$ interactions, where no such relation was previously suspected. Using integration by parts, we reduce each counterterm to massless two-loop two-point integrals. The four-term relation is verified, with $= 0 - 3\zeta_3 + 6\zeta_3 - 3\zeta_3 = 0$, demonstrating non-trivial cancellation of the trefoil knot and thus supporting the emerging connection between knots and counterterms, via transcendental numbers assigned by four-dimensional field theories to chord diagrams. Restrictions to scalar couplings and renormalizable interactions are found to be necessary for the existence of a pure four-term relation. Strong indications of richer structure are given at five loops.Comment: minor changes, references updated, 10 pages, LaTe

Massive 3-loop Feynman diagrams reducible to SC* primitives of algebras of the sixth root of unity

In each of the 10 cases with propagators of unit or zero mass, the finite part of the scalar 3-loop tetrahedral vacuum diagram is reduced to 4-letter words in the 7-letter alphabet of the 1-forms $\Omega:=dz/z$ and $\omega_p:=dz/ (\lambda^{-p}-z)$, where $\lambda$ is the sixth root of unity. Three diagrams yield only $\zeta(\Omega^3\omega_0)=1/90\pi^4$. In two cases $\pi^4$ combines with the Euler-Zagier sum $\zeta(\Omega^2\omega_3\omega_0)=\sum_{m> n>0}(-1)^{m+n}/m^3n$; in three cases it combines with the square of Clausen's $Cl_2(\pi/3)=\Im \zeta(\Omega\omega_1)=\sum_{n>0}\sin(\pi n/3)/n^2$. The case with 6 masses involves no further constant; with 5 masses a Deligne-Euler-Zagier sum appears: $\Re \zeta(\Omega^2\omega_3\omega_1)= \sum_{m>n>0}(-1)^m\cos(2\pi n/3)/m^3n$. The previously unidentified term in the 3-loop rho-parameter of the standard model is merely $D_3=6\zeta(3)-6 Cl_2^2(\pi/3)-{1/24}\pi^4$. The remarkable simplicity of these results stems from two shuffle algebras: one for nested sums; the other for iterated integrals. Each diagram evaluates to 10 000 digits in seconds, because the primitive words are transformable to exponentially convergent single sums, as recently shown for $\zeta(3)$ and $\zeta(5)$, familiar in QCD. Those are SC$^*(2)$ constants, whose base of super-fast computation is 2. Mass involves the novel base-3 set SC$^*(3)$. All 10 diagrams reduce to SC$^*(3)\cup$SC$^* (2)$ constants and their products. Only the 6-mass case entails both bases.Comment: 41 pages, LaTe

A dilogarithmic 3-dimensional Ising tetrahedron

In 3 dimensions, the Ising model is in the same universality class as $\phi^4$-theory, whose massive 3-loop tetrahedral diagram, $C^{Tet}$, was of an unknown analytical nature. In contrast, all single-scale 4-dimensional tetrahedra were reduced, in hep-th/9803091, to special values of exponentially convergent polylogarithms. Combining dispersion relations with the integer-relation finder PSLQ, we find that $C^{Tet}/2^{5/2} = Cl_2(4\alpha) - Cl_2(2\alpha)$, with $Cl_2(\theta):=\sum_{n>0}\sin(n\theta)/n^2$ and $\alpha:=\arcsin\frac13$. This empirical relation has been checked at 1,000-digit precision and readily yields 50,000 digits of $C^{Tet}$, after transformation to an exponentially convergent sum, akin to those studied in math.CA/9803067. It appears that this 3-dimensional result entails a polylogarithmic ladder beginning with the classical formula for $\pi/\sqrt2$, in the manner that 4-dimensional results build on that for $\pi/\sqrt3$.Comment: 8 pages, LaTeX; Eq(25) simplified; Eqs(27,33) and refs[3,18] adde

Transcendental numbers and the topology of three-loop bubbles

We present a proof that all transcendental numbers that are needed for the calculation of the master integrals for three-loop vacuum Feynman diagrams can be obtained by calculating diagrams with an even simpler topology, the topology of spectacles.Comment: 4 pages in REVTeX, 1 PostScript figure included, submitted to Phys. Rev. Let

Evaluations of k-fold Euler/Zagier sums: a compendium of results for arbitrary k

Euler sums (also called Zagier sums) occur within the context of knot theory and quantum field theory. There are various conjectures related to these sums whose incompletion is a sign that both the mathematics and physics communities do not yet completely understand the field. Here, we assemble results for Euler/Zagier sums (also known as multidimensional zeta/harmonic sums) of arbitrary depth, including sign alternations. Many of our results were obtained empirically and are apparently new. By carefully compiling and examining a huge data base of high precision numerical evaluations, we can claim with some confidence that certain classes of results are exhaustive. While many proofs are lacking, we have sketched derivations of all results that have so far been proved.Comment: 19 pages, LaTe