The Small-N Series in the Zero-Dimensional O(N) Model: Constructive Expansions and Transseries

Abstract We consider the zero-dimensional quartic O ( N ) vector model and present a complete study of the partition function Z ( g , N ) and its logarithm, the free energy W ( g , N ), seen as functions of the coupling g on a Riemann surface. We are, in particular, interested in the study of the transseries expansions of these quantities. The point of this paper is to recover such results using constructive field theory techniques with the aim to use them in the future for a rigorous analysis of resurgence in genuine quantum field theoretical models in higher dimensions. Using constructive field theory techniques, we prove that both Z ( g , N ) and W ( g , N ) are Borel summable functions along all the rays in the cut complex plane $$\mathbb {C}_{\pi } =\mathbb {C}{\setminus } \mathbb {R}_-$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>C</mml:mi> <mml:mi>π</mml:mi> </mml:msub> <mml:mo>=</mml:mo> <mml:mi>C</mml:mi> <mml:mo>\</mml:mo> <mml:msub> <mml:mi>R</mml:mi> <mml:mo>-</mml:mo> </mml:msub> </mml:mrow> </mml:math> . We recover the transseries expansion of Z ( g , N ) using the intermediate field representation. We furthermore study the small- N expansions of Z ( g , N ) and W ( g , N ). For any $$g=|g| e^{\imath \varphi }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>g</mml:mi> <mml:mo>=</mml:mo> <mml:mrow> <mml:mo>|</mml:mo> <mml:mi>g</mml:mi> <mml:mo>|</mml:mo> </mml:mrow> <mml:msup> <mml:mi>e</mml:mi> <mml:mrow> <mml:mi>ı</mml:mi> <mml:mi>φ</mml:mi> </mml:mrow> </mml:msup> </mml:mrow> </mml:math> on the sector of the Riemann surface with $$|\varphi |&lt;3\pi /2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>|</mml:mo> <mml:mi>φ</mml:mi> <mml:mo>|</mml:mo> <mml:mo>&lt;</mml:mo> <mml:mn>3</mml:mn> <mml:mi>π</mml:mi> <mml:mo>/</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> , the small- N expansion of Z ( g , N ) has infinite radius of convergence in N , while the expansion of W ( g , N ) has a finite radius of convergence in N for g in a subdomain of the same sector. The Taylor coefficients of these expansions, $$Z_n(g)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>Z</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>g</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> and $$W_n(g)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>W</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>g</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> , exhibit analytic properties similar to Z ( g , N ) and W ( g , N ) and have transseries expansions. The transseries expansion of $$Z_n(g)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>Z</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>g</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> is readily accessible: much like Z ( g , N ), for any n , $$Z_n(g)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>Z</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>g</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> has a zero- and a one-instanton contribution. The transseries of $$W_n(g)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>W</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>g</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> is obtained using Möbius inversion, and summing these transseries yields the transseries expansion of W ( g , N ). The transseries of $$W_n(g)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>W</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>g</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> and W ( g , N ) are markedly different: while W ( g , N ) displays contributions from arbitrarily many multi-instantons, $$W_n(g)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>W</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>g</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> exhibits contributions of only up to n -instanton sectors.

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