Combinatorial set up Physics set up Tying the combinatorics to the physics A sample result Solving Dyson-Schwinger equations by chord diagrams Karen Yeats University of Waterloo Algebraic and combinatorial perspectives in the mathematical sciences seminar June 4, 2020 Thanks to NSERC and the CRC program.
Combinatorial set up Physics set up Tying the combinatorics to the physics A sample result Rooted chord diagrams A rooted chord diagram with n chords is a matching on { 1 , 2 , . . . , 2 n } . We can draw it on a line or on a circle. The directed intersection graph of a chord diagram has a vertex for each chord, and an edge between chords { a , b } , { c , d } ( a < b , c < d ) if a < c < b < d , i.e. the chords cross .
Combinatorial set up Physics set up Tying the combinatorics to the physics A sample result Connectivity and terminal chords A rooted chord diagram is connected if the directed intersection graph is weakly connected. A chord is terminal if it has no outgoing edges in the directed intersection graph.
Combinatorial set up Physics set up Tying the combinatorics to the physics A sample result Chord orders There are two important ways to order the chords of a chord diagram. Both extend the partial order given by the directed intersection graph. Order chords by left end-point. Order chords recursively as follows:
Combinatorial set up Physics set up Tying the combinatorics to the physics A sample result The world of perturbative quantum fi eld theory A picture of perturbative quantum fi eld theory:
Combinatorial set up Physics set up Tying the combinatorics to the physics A sample result Feynman graphs and Feynman integrals So from a combinatorial perspective, we have something like generating series for Feynman graphs weighted by the Feynman integrals. An example Feynman integral:
Combinatorial set up Physics set up Tying the combinatorics to the physics A sample result A Dyson-Schwinger equation Like how in enumerative combinatorics we get functional equations for the generating functions from decompositions of the objects, we get Dyson-Schwinger equations in quantum fi eld theory. Take an example like G ( x , L ) = 1 − x k · q � � d 4 k � k 2 G ( x , log( k 2 ))( k + q ) 2 − same � q 2 � q 2 = µ 2 which is a Dyson-Schwinger equation for a piece of Yukawa theory, solved by Broadhurst and Kreimer. Which piece?
Combinatorial set up Physics set up Tying the combinatorics to the physics A sample result Rearranging From � G ( x , L ) = 1 − x � k · q d 4 k � k 2 G ( x , log( k 2 ))( k + q ) 2 − same � q 2 � q 2 = µ 2 We can expand 1 / G ( x , log( k 2 )) as a series in x and log( k 2 ), use log � a = ∂ � a � | � =0 , recollect the series, to get G ( x , L ) = 1 − xG ( x , ∂ − ρ ) − 1 ( e − L ρ − 1) F ( ρ ) | ρ =0 where F ( ρ ) is the regularized integral for the one-loop diagram.
Combinatorial set up Physics set up Tying the combinatorics to the physics A sample result . . . we fi nd new diagrammatics Interpret G ( x , L ) = 1 − xG ( x , ∂ − ρ ) − 1 ( e − L ρ − 1) F ( ρ ) | ρ =0 as an equation in formal power series. Recursively it determines the coe ffi cients of G ( x , L ) in terms of F ( ρ ) = f 0 ρ − 1 + f 1 + f 2 ρ + · · · This works for the speci fi c F ( ρ ) of the Yukawa example, or for general F ( ρ ), also for more general Dyson-Schwinger equations x k G ( x , ∂ − ρ ) 1 − sk ( e − L ρ − 1) F k ( ρ ) | ρ =0 � G ( x , L ) = 1 − k ≥ 1 These also have expansions as sums over combinatorial objects – new diagrammatics.
Combinatorial set up Physics set up Tying the combinatorics to the physics A sample result The initial result with Marie Theorem G ( x , L ) = 1 − xG ( x , ∂ − ρ ) − 1 ( e − L ρ − 1) F ( ρ ) | ρ =0 with F ( ρ ) = f 0 ρ − 1 + f 1 + f 2 ρ + · · · is solved as a series by ( − L ) i x | C | f t k − t k − 1 · · · f t 2 − t 1 f | C | − k � � G ( x , L ) = 1 − f t 1 − i 0 i ! C i ≤ t 1 where the sum is over rooted connected chord diagrams and the t 1 < t 2 < · · · < t k are the terminal chords of the chord diagram C. Generalization with Hihn.
Combinatorial set up Physics set up Tying the combinatorics to the physics A sample result Questions this asks for combinatorics Chord diagrams are longstanding objects investigated by combinatorialists. However The terminal chords are a new parameter. What can we say about the statistics of the gaps between terminal chords? Some results with Courtiel. The recurrences that appear reminded N. Zeilberger of recurrences he saw in lambda calculus and led to a new bijection with bridgeless maps (with Courtiel and Zeilberger). Invariance under choices? Bijective connections between di ff erent classes of chord diagrams which arise? (ask Ali Mahmoud and Lukas Nabergall) Can we categorify? (ask Lucia Rotheray)
Combinatorial set up Physics set up Tying the combinatorics to the physics A sample result Questions this asks for physics Feynman diagrams give combinatorially indexed expansions, but each gives something complicated. This is an expansion in combinatorial objects where each object contributes something very simple. The chord diagram expansion is particularly nice for looking at the leading log expansion, next to leading log expansion etc. (with Courtiel) What does it tell us about which terms dominate the perturbative expansion? What does it tell us about the renormalization group equation? How generally does it apply (some work in progress by Lukas Nabergall)? Could it truly be an alternate diagrammatics for perturbative quantum fi eld theory?
Combinatorial set up Physics set up Tying the combinatorics to the physics A sample result Questions for everyone But why? Why chord diagrams? Is there a natural way to go directly from Feynman diagrams to chord diagrams, at least in the Yukawa case. Kurusch and I are working on this. Can the proof of the chord diagram expansion be made more combinatorial – yes, work in progress by Lukas Nabergall.
Combinatorial set up Physics set up Tying the combinatorics to the physics A sample result Next-to k leading log expansions Let’s discuss one result in more detail, see arXiv:906.05139 (with Courtiel) First recall ( − L ) i x | C | f t k − t k − 1 · · · f t 2 − t 1 f | C | − k � � G ( x , L ) = 1 − f t 1 − i 0 i ! C i ≤ t 1 Note, G ( x , L ) is triangular. The degree of L is at most the degree of x in each term.
Combinatorial set up Physics set up Tying the combinatorics to the physics A sample result Leading log part; next-to-leading log part The leading log part is The next-to-leading log part is
Combinatorial set up Physics set up Tying the combinatorics to the physics A sample result After that After that something di ff erent happens. We had ( − L ) i x | C | f t k − t k − 1 · · · f t 2 − t 1 f | C | − k � � G ( x , L ) = 1 − f t 1 − i 0 i ! C i ≤ t 1
Combinatorial set up Physics set up Tying the combinatorics to the physics A sample result More general chord diagram expansions But for the results of arXiv:906.05139 we need more general G ( x , L ). Now our chord diagrams have weighted chords. There is a parameter s indicating the insertion growth rate. They are ω -marked: the intervals covered by c contain d ( c ) s − 2 marks.
Combinatorial set up Physics set up Tying the combinatorics to the physics A sample result Algorithms for the expansions The analogous expansion in these objects solves the generalized Dyson-Schwinger equation (Hihn,Y rephrased by Courtiel). There is an automatable procedure to calculate any particular next-to k leading log expansion. Julien implemented it in Maple. It extends the ideas we saw in the simpler case – identify which families of chord diagrams contribute.
Combinatorial set up Physics set up Tying the combinatorics to the physics A sample result Dichotomy There is a dichotomy between s = 1 and s ≥ 2. Theorem (Courtiel) For s ≥ 2 The dominant types in the next-to k leading log expansion are those with k 1 adjacent terminal chords of weight 1 , fi rst terminal chord of weight 1 , k 2 non-terminal chords of weight 2 , and all other chords non-terminal of weight 1 (k 1 + k 2 = k). The number of these diagrams grows like ( s − 1) k 1 � k , k 1 � ( n ) k n − 1 s − 1 s n − k − 1 . Γ (1 − 1 log s ) k ! The leading terms in the expansion involve only a 1 , 0 , a 2 , 0 and a 1 , 1 .
Combinatorial set up Physics set up Tying the combinatorics to the physics A sample result Theorem (continued) For s = 1 The dominant types in the next-to k leading log expansion are those with only one terminal chord of weight 2 , k non-terminal chords of weight 2 , and all other chords non-terminal of weight 1 . The number of these diagrams grows like 1 ( k − 1)! log( n ) k − 1 n − 2 . The leading terms in the expansion involve only a 1 , 0 and a 2 , 0 .
Combinatorial set up Physics set up Tying the combinatorics to the physics A sample result Extra space
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