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<p>The metric and potential associated with the gradient property of renormalisation group flow in multiscalar models in <i>d</i> = 4 <i>− ε</i> dimensions are studied. The metric is identified with the Zamolodchikov metric of nearly marginal operators on the sphere. An explicit form for the associated Ricci scalar in <i>d</i> = 4 <i>− ε</i> is derived, which shows that the space of multiscalar field theories is curved. The potential is identified with a quantity <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13130_2025_27197_Article_IEq1.gif" Format="GIF" Height="22" Rendition="HTML" Resolution="72" Type="Linedraw" Width="16" /> </InlineMediaObject> <EquationSource Format="MATHML"><math display="inline"> <mover accent="true"> <mi>F</mi> <mo stretchy="true">~</mo> </mover> </math></EquationSource> <EquationSource Format="TEX">\( \overset{\sim }{F} \)</EquationSource> </InlineEquation> that was previously proposed as a weakly monotonic function interpolating between the <i>a</i>-theorem in four dimensions and the <i>F</i>-theorem in three dimensions. This implies that the <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13130_2025_27197_Article_IEq1.gif" Format="GIF" Height="22" Rendition="HTML" Resolution="72" Type="Linedraw" Width="16" /> </InlineMediaObject> <EquationSource Format="MATHML"><math display="inline"> <mover accent="true"> <mi>F</mi> <mo stretchy="true">~</mo> </mover> </math></EquationSource> <EquationSource Format="TEX">\( \overset{\sim }{F} \)</EquationSource> </InlineEquation>-theorem can be extended perturbatively to a theorem about gradient flow in <i>d</i> = 4 <i>− ε</i>.</p>

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Gradient flows and the curvature of theory space

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2025年9月16日

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  • William H. Pannell,
  • Andreas Stergiou

摘要

The metric and potential associated with the gradient property of renormalisation group flow in multiscalar models in d = 4 − ε dimensions are studied. The metric is identified with the Zamolodchikov metric of nearly marginal operators on the sphere. An explicit form for the associated Ricci scalar in d = 4 − ε is derived, which shows that the space of multiscalar field theories is curved. The potential is identified with a quantity F ~ \( \overset{\sim }{F} \) that was previously proposed as a weakly monotonic function interpolating between the a-theorem in four dimensions and the F-theorem in three dimensions. This implies that the F ~ \( \overset{\sim }{F} \) -theorem can be extended perturbatively to a theorem about gradient flow in d = 4 − ε.