Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Primitive asymptotics in $φ^4$ vector theory

Paul-Hermann Balduf, Johannes Thürigen

Abstract

A longstanding conjecture in $φ^4_4$ theory is that primitive graphs dominate the beta function asymptotically at large loop order in the minimal-subtraction scheme. Here we investigate this issue by exploiting additional combinatorial structure coming from an extension to vectors with $O(N)$ symmetry. For the 0-dimensional case, we calculate the $N$-dependent generating function of primitive graphs and its asymptotics, including arbitrarily many subleading corrections. We find that the leading asymptotic growth rate becomes visible only above $\approx 25$ loops, while data at lower order is suggestive of a wrong asymptotics. Our results also yield the symmetry-factor weighted sum of 3-connected cubic graphs, and the exact asymptotics of Martin invariants. For individual Feynman graphs, we give bounds on their degree in $N$ depending on their coradical degree, and construct the primitive graphs of highest degree explicitly. We calculate the 4D primitive beta function numerically up to 17 loops, and find its behaviour to be qualitatively similar to the 0D case. The locations of zeros quickly approach their large-loop asymptotics at negative integer $N$, while the growth rate of the beta function differs from the asymptotic prediction even at 17 loops.

Primitive asymptotics in $φ^4$ vector theory

Abstract

A longstanding conjecture in theory is that primitive graphs dominate the beta function asymptotically at large loop order in the minimal-subtraction scheme. Here we investigate this issue by exploiting additional combinatorial structure coming from an extension to vectors with symmetry. For the 0-dimensional case, we calculate the -dependent generating function of primitive graphs and its asymptotics, including arbitrarily many subleading corrections. We find that the leading asymptotic growth rate becomes visible only above loops, while data at lower order is suggestive of a wrong asymptotics. Our results also yield the symmetry-factor weighted sum of 3-connected cubic graphs, and the exact asymptotics of Martin invariants. For individual Feynman graphs, we give bounds on their degree in depending on their coradical degree, and construct the primitive graphs of highest degree explicitly. We calculate the 4D primitive beta function numerically up to 17 loops, and find its behaviour to be qualitatively similar to the 0D case. The locations of zeros quickly approach their large-loop asymptotics at negative integer , while the growth rate of the beta function differs from the asymptotic prediction even at 17 loops.

Paper Structure

This paper contains 35 sections, 17 theorems, 103 equations, 26 figures, 7 tables.

Table of Contents

  1. Introduction
  2. Background and Motivation
  3. Outline and results
  4. Discussion and Outlook
  5. Setup
  6. $\textrm{O}(N)$-symmetric $\phi^4$ theory
  7. Renormalization, primitive graphs, and periods
  8. The beta function
  9. Definition and asymptotics
  10. Primitive beta function
  11. Martin invariant
  12. QFT in zero dimensions
  13. Vacuum graphs
  14. External edges, connected and 1PI graphs
  15. Counterterms, primitive graphs, Martin invariants
  16. ...and 20 more sections

Key Result

Lemma 4

A $(2p-1)$-loop multiedge graph without external edges has

Figures (26)

  • Figure 1: A 4-valent vertex allows for three non-isomorphic decompositions, each of which consists of a pair of edges. To compute a Feynman amplitude, all three decompositions need to be summed.
  • Figure 2: The fish graph (also known as double edge, 1-loop multiedge, banana, or bubble) has two vertices. In a sum over all decompositions, both vertices produce three terms according to \ref{['fig:vertex_decomposition']} which yields nine terms in total, only one of which contains a circuit. Thus, the circuit partition polynomial of this graph is $N+8$. Notice that the circuit partition polynomial depends on whether the four external edges are open or pairwise connected, compare \ref{['ex:fish_chain']}.
  • Figure 3: A 5-loop completion (left) has seven vertices. Removing one of the vertices yields a 5-loop decompletion. In this case, only two of the seven possible decompletions are non-isomorphic.
  • Figure 4: Feynman graphs representing the first three summands of $Z(\hbar, 0)$. This partition function enumerates vacuum graphs. The first line shows the graph, where the solid dot is a vertex to be decomposed according to \ref{['fig:vertex_decomposition']}. The second line is the automorphism symmetry factor and the power of $\hbar$, multiplied by the circuit partition polynomial.
  • Figure 5: Feynman graphs representing the first two summands of $\partial^2_j Z(\hbar, j)$. These are vacuum graphs with one marked edge, indicated with a white 2-valent vertex in the first row. Equivalently, upon cutting the white vertex, one obtains graphs with exactly two external edges, shown in the second row. The replacement \ref{['eta_replacement']} guarantees that there is no sum over the $\textrm{O}(N)$ index $j$ of the external $\phi_j$.
  • ...and 21 more figures

Theorems & Definitions (51)

  • Definition 1
  • Definition 2
  • Example 3
  • Lemma 4
  • Definition 5
  • Definition 6
  • Example 7
  • Lemma 8
  • Lemma 9
  • Conjecture 10
  • ...and 41 more