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<p>The gradient property of the renormalisation group (RG) is examined to four-loop order in scalar-fermion systems in <i>d</i> = 4 and <i>d</i> = 4 − <i>ε</i> dimensions. The crucial role played by the beta shift, which is a modification of the standard dim-reg beta function, is elucidated, and specific conditions that it needs to satisfy for the RG flow to be gradient are derived. Over a thousand gradient-flow conditions are found, all of which are scheme-independent and satisfied whenever the full set of results needed to check them is available. It is shown, in the framework of the <i>ε</i> = 4 − <i>d</i> expansion, that the space of conformal field theories (CFTs) is dominated by those with non-zero beta shift as the number of fields grows. Physical properties of CFTs obtained as solutions where the beta functions are not zero in the <i>ε</i> expansion are discussed.</p>

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Gradient RG flow in scalar-fermion QFTs

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2026年5月7日

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  • William H. Pannell,
  • William P. Ronayne,
  • Andreas Stergiou

摘要

The gradient property of the renormalisation group (RG) is examined to four-loop order in scalar-fermion systems in d = 4 and d = 4 − ε dimensions. The crucial role played by the beta shift, which is a modification of the standard dim-reg beta function, is elucidated, and specific conditions that it needs to satisfy for the RG flow to be gradient are derived. Over a thousand gradient-flow conditions are found, all of which are scheme-independent and satisfied whenever the full set of results needed to check them is available. It is shown, in the framework of the ε = 4 − d expansion, that the space of conformal field theories (CFTs) is dominated by those with non-zero beta shift as the number of fields grows. Physical properties of CFTs obtained as solutions where the beta functions are not zero in the ε expansion are discussed.