Combinatorics of Feynman diagrams and algebraic lattice structure in QFT

Transcription

1 Combinatorics of Feynman diagrams and algebraic lattice structure in QFT Michael Borinsky 1 Humboldt-University Berlin Departments of Physics and Mathematics - Alexander von Humboldt Group of Dirk Kreimer Resurgence, Physics and Numbers, borinsky@physik.hu-berlin.de M. Borinsky (HU Berlin) Combinatorics of Feynman diagrams and algebraic lattice structure in QFT 1

2 Table of Contents 1 Introduction Motivation Zero dimensional Quantum Field Theory The Hopf algebra of Feynman diagrams 2 Lattice structure of Feynman diagrams From diagram counting to the Hopf algebra The incidence Hopf algebra Lattice structure in physical QFTs 3 Applications Density of primitive diagrams Connection to irreducible partitions 4 Conclusions M. Borinsky (HU Berlin) Combinatorics of Feynman diagrams and algebraic lattice structure in QFT 2

3 Motivation Why study the combinatorics of Feynman diagrams? (Divergent) perturbation expansions in QFTs dominated by number of diagrams. Number of generators of the Hopf/Lie algebra of Feynman diagrams. M. Borinsky (HU Berlin) Combinatorics of Feynman diagrams and algebraic lattice structure in QFT 3

4 Zero dimensional Quantum Field Theory What is the simplest way to analyze the combinatorics of Feynman diagrams? Zero dimensional QFTs! Extensively studied 2. Idea: Replace the path integral by an ordinary integration. 2 Cvitanović, Lautrup, and Pearson M. Borinsky (HU Berlin) Combinatorics of Feynman diagrams and algebraic lattice structure in QFT 4

5 Zero dimensional Quantum Field Theory For ϕ k -theory in zero dimensions: dϕ Z ϕ k (j, λ) := e ϕ2 2 2π R +λ ϕk k! +jϕ, where λ counts the number of vertices and j the number of external edges. This integral is meant to be calculated perturbatively, i.e. by termwise integration: Z ϕ k (j, λ) := { ( ) 1 dϕ e ϕ2 λϕ k n } 2 (jϕ) m. n!m! n,m 0 2π k! R M. Borinsky (HU Berlin) Combinatorics of Feynman diagrams and algebraic lattice structure in QFT 5

6 Diagrammatically: Z ϕ 3 = Analogous for Yukawa, QED, quenched QED, QCD,... M. Borinsky (HU Berlin) Combinatorics of Feynman diagrams and algebraic lattice structure in QFT 6

7 Zero dimensional Quantum Field Theory Use the exponential formula to obtain the connected diagrams: Diagrammatically: W (j, λ) = log(z(j, λ)). W ϕ 3 = Calculate the classical field ϕ c (j, λ) := W j, Shift source variable j j + j 0 such that ϕ c (j ) vanishes at j = 0. M. Borinsky (HU Berlin) Combinatorics of Feynman diagrams and algebraic lattice structure in QFT 7

8 Perform a Legendre transformation to obtain the effective action: Γ(ϕ c, λ) := W j ϕ c, Γ is the generating function for all 1PI Feynman diagrams. Diagrammatically: Γ ϕ 3 = In the following: Parametrize Γ with instead of λ to count loops instead of vertices Γ(ϕ c, ). M. Borinsky (HU Berlin) Combinatorics of Feynman diagrams and algebraic lattice structure in QFT 8

9 The Hopf algebra structure of 1PI Feynman diagrams Starting point for the Hopf algebra of Feynman diagrams: H fg is the Q-algebra generated by all mutually non-isomorphic 1PI diagrams. With Disjoint union as multiplication, a unit u, a counit ɛ and the coproduct encapsulating the BPHZ-algorithm: : T H fg H fg Γ γ Γ γ= γ i with each γ i 1PI and sup.div. γ Γ/γ H fg becomes a Hopf algebra. H fg is equipped with a grading given by the loop number. H fg = L 0 H fg(l) M. Borinsky (HU Berlin) Combinatorics of Feynman diagrams and algebraic lattice structure in QFT 9

10 Example Take all 1PI sub-diagrams of a graph: { ( ) P =,,,,,,,, },,, M. Borinsky (HU Berlin) Combinatorics of Feynman diagrams and algebraic lattice structure in QFT 10

11 Example Keep only the superficially divergent ones: ( ) P s.d. = {,,,,,,,, },,, M. Borinsky (HU Berlin) Combinatorics of Feynman diagrams and algebraic lattice structure in QFT 11

12 Example = γ {,,,, } = I + I + γ /γ = }{{} M. Borinsky (HU Berlin) Combinatorics of Feynman diagrams and algebraic lattice structure in QFT 12

13 Tool to calculate finite amplitudes: The group of characters, G Hfg A. Consists of algebra morphisms H fg A. With A a unital algebra. Product of φ, ψ G Hfg A The unit is u A ɛ H fg. is defined as, φ ψ = m A (φ ψ). The inverse can be expressed using the antipode S on H fg, m (S id) = u ɛ: φ 1 = φ S M. Borinsky (HU Berlin) Combinatorics of Feynman diagrams and algebraic lattice structure in QFT 13

14 φ denotes the character which maps a Feynman diagram H fg to its amplitude. This amplitude is infinite. We are interested in the renormalized amplitude given by, φ R := S φ R φ. S φ R is the twisted antipode defined as S φ R := R φ S For a multiplicative renormalization scheme. M. Borinsky (HU Berlin) Combinatorics of Feynman diagrams and algebraic lattice structure in QFT 14

15 From diagram counting to the Hopf algebra Use the simple Feynman rules of 0-dimensional QFT: φ : H fg Q[[ ]], γ γ. For a 1PI diagram γ and γ its loop number. Connect to the path integral formulation, e.g. for ϕ 3 -theory: 2 Γ ϕ 2 c = φ(x ) 3 Γ ϕ 3 c = φ(x ) where X := 1 γ 1PI res γ= γ Aut γ X := 1 + γ 1PI res γ= γ Aut γ. M. Borinsky (HU Berlin) Combinatorics of Feynman diagrams and algebraic lattice structure in QFT 15

16 Use the toy renormalization scheme: R = id S φ R = S φ = φ S This amounts to renormalization of zero dimensional QFT. Using this the counterterms or Z-factors can be obtained, Z = S φ (X ) Z = S φ (X ). M. Borinsky (HU Berlin) Combinatorics of Feynman diagrams and algebraic lattice structure in QFT 16

17 Explicitly: Using the combinatorial form of Dyson s equation 3 X r = L 0 Q 2L X r X r L with the invariant charge Q := X (X ) 3 2. Z r = 1 φ(x r )[ (S φ (Q)[ ]) 2 ] r {, } 3 Kreimer M. Borinsky (HU Berlin) Combinatorics of Feynman diagrams and algebraic lattice structure in QFT 17

18 The generating function Z ( ) counts primitive diagrams 4. But why is this the case? Does it work for general QFTs? Study S φ. 4 Cvitanović, Lautrup, and Pearson M. Borinsky (HU Berlin) Combinatorics of Feynman diagrams and algebraic lattice structure in QFT 18

19 The incidence Hopf algebra of posets H P is the Q-algebra generated by all mutually non-isomorphic partially ordered sets (posets). With a unique smallest element ˆ0 and a unique largest element ˆ1. With Cartesian product as multiplication of two posets P 1, P 2 : and the coproduct 5 P 1 P 2 = {(s, t) : s P 1 and t P 2 } with (s, t) (s, t ) iff s s and t t : H P H P H P, P x P[ˆ0, x] [x, ˆ1], where [x, y] is the interval, the subset {z P : x z y}. 5 Schmitt M. Borinsky (HU Berlin) Combinatorics of Feynman diagrams and algebraic lattice structure in QFT 19

20 Lemma 1 6 There is a Hopf algebra morphism, ξ : H fg H P, γ P s.d. (γ), mapping a 1PI diagram to its poset of divergent subdiagrams, ordered by inclusion, is a Hopf algebra morphism. For example γ Prim(H fg ) : ξ(γ) = γ, 6 Borinsky (in preparation). ( ) or ξ = γ µ 1 µ 2. M. Borinsky (HU Berlin) Combinatorics of Feynman diagrams and algebraic lattice structure in QFT 20

21 The map ξ is compatible with the Hopf algebra structure: ξ m H fg = m H P (ξ ξ) (ξ ξ) H fg = H P ξ Example: = + + ξ : = + + M. Borinsky (HU Berlin) Combinatorics of Feynman diagrams and algebraic lattice structure in QFT 21

22 The antipode is also compatible: ξ S H fg = S H P ξ For a subspace H fg(l) H fg, ξ the toy Feynman rules φ act as a characteristic function on H P : φ(x) = L φ ξ(x) x H fg(l) where φ : H P Q, P 1. M. Borinsky (HU Berlin) Combinatorics of Feynman diagrams and algebraic lattice structure in QFT 22

23 Eventually, we want to calculate S φ = φ S H fg. Can be obtained in H P for elements in H fg(l) H fg : S φ = L φ ξ S H fg = L φ S H P ξ µ := φ S H P is a well studied object, called the möbius function of a poset 7. Especially interesting are möbius functions on lattices. 7 Schmitt 1994; Stanley M. Borinsky (HU Berlin) Combinatorics of Feynman diagrams and algebraic lattice structure in QFT 23

24 Lattices A lattice L is a poset with with a unique greatest and a unique smallest element, ˆ1, ˆ0 and two additional binary operations: The join of two elements x, y L: x y := unique smallest element z, z x and z y and the meet, x y := unique greatest element z, z x and z y. M. Borinsky (HU Berlin) Combinatorics of Feynman diagrams and algebraic lattice structure in QFT 24

25 More examples In ϕ 4 -theory in 4 dimensions: ξ ( ) = This poset is a lattice without a grading. In ϕ 6 -theory in 3 dimensions: ξ ( ) = This poset is not a lattice. M. Borinsky (HU Berlin) Combinatorics of Feynman diagrams and algebraic lattice structure in QFT 25

26 Theorem 1 8 In all renormalizable QFTs with vertex valency 4, ξ maps Feynman diagrams to lattices. Join,, is defined as the union of two subdiagrams. Meet,, is obtained by dualisation. Physical QFTs carry a lattice structure. Encodes the overlapping structure of the divergences. Remark: Diagrams with only logarithmic subdivergences map to distributive lattices 9. 8 Borinsky (in preparation). 9 Berghoff M. Borinsky (HU Berlin) Combinatorics of Feynman diagrams and algebraic lattice structure in QFT 26

27 Why does S φ (X ) count primitive diagrams? Theorem 2 10 In a theory with only three-valent vertices, the lattice ξ(γ v ) has always one coatom for a vertex diagram γ v. In such a theory, the lattice ξ(γ p ) has always two coatoms for a propagator diagram γ p. 10 Borinsky (in preparation). M. Borinsky (HU Berlin) Combinatorics of Feynman diagrams and algebraic lattice structure in QFT 27

28 Using Rota s Crosscut Theorem 11 : S φ = µ ( ) = 0 and S φ = 1 S φ (X ) = 1 φ P Prim(H fg )(X ) S φ (X ) = ) (1 2 φ P Prim(H fg )(X ), where P Prim(H fg ) projects to the primitive elements of H fg. 11 Stanley M. Borinsky (HU Berlin) Combinatorics of Feynman diagrams and algebraic lattice structure in QFT 28

29 Applications Back to zero dimensional QFT: Z ϕ 3(j, ) := R dϕ e 2π 1 ( ϕ2 2 + ϕ3 3! +jϕ ). Can be renormalized by introducing Z factors and shifting the source j j + j 0 : Zϕ ren (j, ) := 3 R dϕ e 2π 1 ( Z ) ( ) ϕ2 ϕ3 +Z ( ) 2 3! +j ϕ+j 0 ( )ϕ, Using a contour integration on a specific order in j, zk,n ren = 1 d k ren 2πi 1+n Z j j k ϕ 3 =0, coefficients can be extracted. M. Borinsky (HU Berlin) Combinatorics of Feynman diagrams and algebraic lattice structure in QFT 29

30 Asymptotically all diagrams are connected and 1PI 12. Therefore the probabilities of a random Feynman diagram to be primitive can be obtained. 12 Wright M. Borinsky (HU Berlin) Combinatorics of Feynman diagrams and algebraic lattice structure in QFT 30

31 zk,n ren = 1 2πi d k ren 1+n Z j j k ϕ 3 =0, Using the saddle-point expansion the asymptotic behaviour of zk,n ren can be analyzed. For instance for the diagrams in ϕ 3 -theory: with z 3,n = z3,n ren lim = e 10 3 n z 3,n ( ) n+1 (6n+5)!! = 3! n (2n+1)!(3!) 2n+1 2πe e(3!) + O(n 1 ). 2 M. Borinsky (HU Berlin) Combinatorics of Feynman diagrams and algebraic lattice structure in QFT 31

32 Similar we obtain for Yukawa fermion scalar vertex: lim n QED fermion photon vertex: lim n z ren n z n = e 7 2 z ren n z n = e 5 2 Quench-QED fermion photon vertex: lim n z ren n z n = e 2 M. Borinsky (HU Berlin) Combinatorics of Feynman diagrams and algebraic lattice structure in QFT 32

33 (Quenched) QED primitives can be enumerated by other methods but asymptotics are more difficult. 13. In this case there is a similarity to irreducible partitions 14. One result in this direction 15, agrees with the asymptotic calculation for Quench QED. Connection between primitive diagrams and irreducible partitions? 13 Broadhurst 1999; Molinari and Manini Beissinger Kleitman M. Borinsky (HU Berlin) Combinatorics of Feynman diagrams and algebraic lattice structure in QFT 33

34 Conclusions The lattice structure of Quantum Field Theories is useful to analyze combinatorics of the counterterms. I.e. to quantify the divergence stemming from the explosion of diagrams. Could be used for estimates for the asymptotic behaviour of Green s functions, β functions, etc. Explicit results on the number of primitive elements can be obtained in cases with only 3-valent vertices. M. Borinsky (HU Berlin) Combinatorics of Feynman diagrams and algebraic lattice structure in QFT 34