Rearranging Dyson-Schwinger equations

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, <disp-formula content-type="math/mathml"> \[ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="gamma 1 Superscript r Baseline left-parenthesis x right-parenthesis equals upper P Subscript r Baseline left-parenthesis x right-parenthesis minus normal s normal i normal g normal n left-parenthesis s Subscript r Baseline right-parenthesis gamma 1 Superscript r Baseline left-parenthesis x right-parenthesis squared plus left-parenthesis sigma-summation Underscript j element-of script upper R Endscripts StartAbsoluteValue s Subscript j Baseline EndAbsoluteValue gamma 1 Superscript j Baseline left-parenthesis x right-parenthesis right-parenthesis x partial-differential Subscript x Baseline gamma 1 Superscript r Baseline left-parenthesis x right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msubsup> <mml:mi> γ </mml:mi> <mml:mn>1</mml:mn> <mml:mi>r</mml:mi> </mml:msubsup> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>=</mml:mo> <mml:msub> <mml:mi>P</mml:mi> <mml:mi>r</mml:mi> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo> − </mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal">s</mml:mi> <mml:mi mathvariant="normal">i</mml:mi> <mml:mi mathvariant="normal">g</mml:mi> <mml:mi mathvariant="normal">n</mml:mi> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:msub> <mml:mi>s</mml:mi> <mml:mi>r</mml:mi> </mml:msub> <mml:mo stretchy="false">)</mml:mo> <mml:msubsup> <mml:mi> γ </mml:mi> <mml:mn>1</mml:mn> <mml:mi>r</mml:mi> </mml:msubsup> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:msup> <mml:mo stretchy="false">)</mml:mo> <mml:mn>2</mml:mn> </mml:msup> <mml:mo>+</mml:mo> <mml:mrow> <mml:mo>(</mml:mo> <mml:munder> <mml:mo> ∑ </mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>j</mml:mi> <mml:mo> ∈ </mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">R</mml:mi> </mml:mrow> </mml:mrow> </mml:munder> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> <mml:msub> <mml:mi>s</mml:mi> <mml:mi>j</mml:mi> </mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> <mml:msubsup> <mml:mi> γ </mml:mi> <mml:mn>1</mml:mn> <mml:mi>j</mml:mi> </mml:msubsup> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>)</mml:mo> </mml:mrow> <mml:mi>x</mml:mi> <mml:msub> <mml:mi mathvariant="normal"> ∂ </mml:mi> <mml:mi>x</mml:mi> </mml:msub> <mml:msubsup> <mml:mi> γ </mml:mi> <mml:mn>1</mml:mn> <mml:mi>r</mml:mi> </mml:msubsup> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\gamma _1^r(x) = P_r(x) - \mathrm {sign}(s_r)\gamma _1^r(x)^2 + \left (\sum _{j \in \mathcal {R}}|s_j|\gamma _1^j(x)\right ) x \partial _x \gamma _1^r(x)</mml:annotation> </mml:semantics> </mml:math> \] </disp-formula> where <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="r element-of script upper R"> <mml:semantics> <mml:mrow> <mml:mi>r</mml:mi> <mml:mo> ∈ </mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">R</mml:mi> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">r \in \mathcal {R}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper R"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">R</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {R}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is the set of amplitudes of the theory which need renormalization, <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="gamma 1 Superscript r"> <mml:semantics> <mml:msubsup> <mml:mi> γ </mml:mi> <mml:mn>1</mml:mn> <mml:mi>r</mml:mi> </mml:msubsup> <mml:annotation encoding="application/x-tex">\gamma _1^r</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is the anomalous dimension associated to <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="r"> <mml:semantics> <mml:mi>r</mml:mi> <mml:annotation encoding="application/x-tex">r</mml:annotation>

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