Algebraic Interplay between Renormalization and Monodromy

Abstract

We investigate combinatorial and algebraic aspects of the interplay between renormalization and monodromies for Feynman amplitudes. We clarify how extraction of subgraphs from a Feynman graph interacts with putting edges onshell or with contracting them to obtain reduced graphs. Graph by graph this leads to a study of cointeracting bialgebras. One bialgebra comes from extraction of subgraphs and hence is needed for renormalization. The other bialgebra is an incidence bialgebra for edges put either onor offshell. It is hence related to the monodromies of the multivalued function to which a renormalized graph evaluates. Summing over infinite series of graphs, consequences for Green functions are derived using combinatorial Dyson-Schwinger equations. Contents 2.1. Graphs 2.2. Cuts 2.3. Sub-and co-graphs 3. Hopf algebras 3.1. The core Hopf algebra H core 3.2. Quotient Hopf algebras 3.3. The Hopf algebra H pC 3.4. Pre-Cutkosky Necklaces 3.5. Extension to a core coproduct for pairs (Γ, F ) 3.6. Counting spanning trees 4. Coactions 4.1. H pC → H core ⊗ H pC 4.2. Cointeracting bialgebras 4.3. Sector Decomposition 4.4. Two more coactions 5. DSEs in core and quotient Hopf algebras 5.1. Dyson-Schwinger and cut Dyson-Schwinger setup 5.2. Assembly maps vs Hochschild 1-cocycles 5.3. Core, no cuts 5.4. Core, with cuts 5.5. The coaction on Green functions 5.6. Pairs (Γ, F ) or H C 1

Pathlike Co/Bialgebras and their Antipodes with Applications to Bi- and Hopf Algebras Appearing in Topology, Number Theory and Physics

“…This switches the coproduct to its opposite. These two observations should generalize to the categorical setting, and when applied to graphs may explain some of the results of [46]. A last aspect is functoriality of the construction, see [25, Section 1.7], which is likely related the structure of cointeracting coalgebras [23,46] in the present context.…”

“…These two observations should generalize to the categorical setting, and when applied to graphs may explain some of the results of [46]. A last aspect is functoriality of the construction, see [25, Section 1.7], which is likely related the structure of cointeracting coalgebras [23,46] in the present context. Furthermore, we identified the skew primitive elements (as indecomposables) as being of central importance.…”

“…The characters need not be graded, but what is common is that they map the additive degree to a multiplicative one as in n Þ Ñ q n , where q can be of degree 0, which can also be achieved by shifting degrees or inverting elements in the target. One typical such q would be 1 2πi , see [46], or factors of ζ m p2q in the framework of Brown, see [7].…”

Bananas: multi-edge graphs and their Feynman integrals

“…Hence g(s) indeed factorizes as desired. 2 This relates to co-actions and cointeracting bialgebras [28,29] and will be discussed elsewhere.…”

“…is the projection onto the pole part and the sum is over the three cyclic permutations of i, j, k. This MS-renormalization results Φ D M S (b n ) can be related to Φ D R (b n ) if so desired. See also the discussion with regards to MS and tadpoles in [28].…”