Degree of Transcendentality and Feynman Diagrams

Question

Physicists computing multiloop Feynman diagrams have introduced various techniques and conjectures that involve the notion of Degree of Transcendentality (DoT). From what I understand one defines

1) $DoT(r)=0$, r rational

2) $DoT(\pi^k)=k$, $k \in {\mathbb N}$,

3) $DoT(\zeta(k))=k$,

4) $DoT( a \cdot b)= DoT(a)+DoT(b)$

One then proves for example that the $\ell$-loop contribution to a certain scaling function in $N=4$ Supersymmetric gauge theory consists of a sum of terms all of which have DoT equal to $2 \ell-2$.

This can't be rigorous mathematically, since it is not even known that $\zeta(2n+1)$ is transcendental, but is there some circle of ideas, or conjecture in mathematics that if true would give a precise definition to DoT?

Answer 1

Your transcendentality reminds me about the Institute of Algebraic Meditation at Höör (Sweden). To be honest, your definition corresponds to what is known as the weight of a (multiple) zeta value (see Michael Hoffman's http://www.usna.edu/Users/math/meh/mult.html, especially the references on MZVs). These indeed occur in the computation of Feynman's diagrams. As for conjectures related to the transcendental number theory tag, a belief is that $\pi$ and odd zeta values $\zeta(3)$, $\zeta(5)$, etc, are algebraically independent over the rationals.