{
  "id": "0810.2249",
  "version": "v1",
  "published": "2008-10-13T15:17:09.000Z",
  "updated": "2008-10-13T15:17:09.000Z",
  "title": "Growth estimates for Dyson-Schwinger equations",
  "authors": [
    "Karen Yeats"
  ],
  "comment": "86 pages, the author's PhD thesis",
  "categories": [
    "math-ph",
    "math.MP"
  ],
  "abstract": "Dyson-Schwinger equations are integral equations in quantum field theory that describe the Green functions of a theory and mirror the recursive decomposition of Feynman diagrams into subdiagrams. Taken as recursive equations, the Dyson-Schwinger equations describe perturbative quantum field theory. However, they also contain non-perturbative information. Using the Hopf algebra of Feynman graphs we will follow a sequence of reductions to convert the Dyson-Schwinger equations to the following system of differential equations, \\[ \\gamma_1^r(x) = P_r(x) - \\sgn(s_r)\\gamma_1^r(x)^2 + (\\sum_{j \\in \\mathcal{R}}|s_j|\\gamma_1^j(x)) x \\partial_x \\gamma_1^r(x) \\] where $r \\in \\mathcal{R}$, $\\mathcal{R}$ is the set of amplitudes of the theory which need renormalization, $\\gamma_1^r$ is the anomalous dimension associated to $r$, $P_r(x)$ is a modified version of the function for the primitive skeletons contributing to $r$, and $x$ is the coupling constant. Next, we approach the new system of differential equations as a system of recursive equations by expanding $\\gamma_1^r(x) = \\sum_{n \\geq 1}\\gamma^r_{1,n} x^n$. We obtain the radius of convergence of $\\sum \\gamma^r_{1,n}x^n/n!$ in terms of that of $\\sum P_r(n)x^n/n!$. In particular we show that a Lipatov bound for the growth of the primitives leads to a Lipatov bound for the whole theory. Finally, we make a few observations on the new system considered as differential equations.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2008-10-13T15:17:09.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "dyson-schwinger equations",
      "growth estimates",
      "differential equations",
      "lipatov bound",
      "perturbative quantum field theory"
    ],
    "tags": [
      "dissertation"
    ],
    "publication": {
      "journal": "Ph.D. Thesis",
      "year": 2008,
      "month": "Oct"
    },
    "note": {
      "typesetting": "TeX",
      "pages": 86,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "inspire": 799250,
      "adsabs": "2008PhDT........80Y"
    }
  }
}