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<p>In this article, we initiate the study of operator product expansions (OPEs) for the sine-Gordon model. For simplicity, we focus on the model below the first threshold of collapse (<InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\beta &lt;4\pi \)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>β</mi> <mo>&lt;</mo> <mn>4</mn> <mi>π</mi> </mrow> </math></EquationSource> </InlineEquation>) and on the singular terms in OPEs of derivative-type fields <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\partial \varphi \)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>∂</mi> <mi>φ</mi> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\bar{\partial }\varphi \)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mover accent="true"> <mrow> <mi>∂</mi> </mrow> <mrow> <mo stretchy="false">¯</mo> </mrow> </mover> <mi>φ</mi> </mrow> </math></EquationSource> </InlineEquation>. We prove that compared to corresponding free field OPEs, the sine-Gordon OPEs develop logarithmic singularities and generate Wick ordered exponentials. Our approach for proving the OPEs relies heavily on Onsager-type inequalities and associated moment bounds for GFF correlation functions involving Wick ordered exponentials of the free field.</p>

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Operator Product Expansions of Derivative Fields in the Sine-Gordon Model

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  • 发表日期：

    2026年3月9日

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  • Alex Karrila,
  • Tuomas Virtanen,
  • Christian Webb

摘要

In this article, we initiate the study of operator product expansions (OPEs) for the sine-Gordon model. For simplicity, we focus on the model below the first threshold of collapse ( \(\beta <4\pi \) β < 4 π ) and on the singular terms in OPEs of derivative-type fields \(\partial \varphi \) ∂ φ and \(\bar{\partial }\varphi \) ∂ ¯ φ . We prove that compared to corresponding free field OPEs, the sine-Gordon OPEs develop logarithmic singularities and generate Wick ordered exponentials. Our approach for proving the OPEs relies heavily on Onsager-type inequalities and associated moment bounds for GFF correlation functions involving Wick ordered exponentials of the free field.