1,654 research outputs found
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 and , where is the sixth root of unity. Three diagrams
yield only . In two cases combines
with the Euler-Zagier sum ; in three cases it combines with the square of Clausen's
. The case
with 6 masses involves no further constant; with 5 masses a
Deligne-Euler-Zagier sum appears: . The previously unidentified term in the
3-loop rho-parameter of the standard model is merely . 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 and , familiar in QCD. Those are
SC constants, whose base of super-fast computation is 2. Mass involves
the novel base-3 set SC. All 10 diagrams reduce to SCSC 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
-theory, whose massive 3-loop tetrahedral diagram, , 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 , with and
. This empirical relation has been checked at
1,000-digit precision and readily yields 50,000 digits of , 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 ,
in the manner that 4-dimensional results build on that for .Comment: 8 pages, LaTeX; Eq(25) simplified; Eqs(27,33) and refs[3,18] adde
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
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 , 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
Renormalization automated by Hopf algebra
It was recently shown that the renormalization of quantum field theory is
organized by the Hopf algebra of decorated rooted trees, whose coproduct
identifies the divergences requiring subtraction and whose antipode achieves
this. We automate this process in a few lines of recursive symbolic code, which
deliver a finite renormalized expression for any Feynman diagram. We thus
verify a representation of the operator product expansion, which generalizes
Chen's lemma for iterated integrals. The subset of diagrams whose forest
structure entails a unique primitive subdivergence provides a representation of
the Hopf algebra of undecorated rooted trees. Our undecorated Hopf
algebra program is designed to process the 24,213,878 BPHZ contributions to the
renormalization of 7,813 diagrams, with up to 12 loops. We consider 10 models,
each in 9 renormalization schemes. The two simplest models reveal a notable
feature of the subalgebra of Connes and Moscovici, corresponding to the
commutative part of the Hopf algebra of the diffeomorphism group:
it assigns to Feynman diagrams those weights which remove zeta values from the
counterterms of the minimal subtraction scheme. We devise a fast algorithm for
these weights, whose squares are summed with a permutation factor, to give
rational counterterms.Comment: 22 pages, latex, epsf for figure
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, , generated by a single primitive divergence, solves a universal problem
in Hochschild cohomology. It has two nontrivial closed Hopf subalgebras: the
cocommutative subalgebra of pure ladder diagrams and
the Connes-Moscovici noncocommutative subalgebra of
noncommutative geometry. These three Hopf algebras admit a bigrading by ,
the number of nodes, and an index 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 and and infer a simple
generating procedure for the remainder. The results for
are familiar from the theory of partitions, while those for
involve novel transforms of partitions. Most beautiful is the bigrading of
, 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 , 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
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