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• W4283273708 abstract "We review recent progress on operator mixing in the light of the theory of canonical forms for linear systems of differential equations and, in particular, of the Poincaré-Dulac theorem. We show that the matrix A(g)=-frac{gamma(g)}{beta(g)}=frac{gamma_0}{beta_0}frac{1}{g}+cdots <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML display=inline><mml:mrow><mml:mi>A</mml:mi><mml:mo stretchy=false form=prefix>(</mml:mo><mml:mi>g</mml:mi><mml:mo stretchy=false form=postfix>)</mml:mo><mml:mo>=</mml:mo><mml:mo>−</mml:mo><mml:mfrac><mml:mrow><mml:mi>γ</mml:mi><mml:mo stretchy=false form=prefix>(</mml:mo><mml:mi>g</mml:mi><mml:mo stretchy=false form=postfix>)</mml:mo></mml:mrow><mml:mrow><mml:mi>β</mml:mi><mml:mo stretchy=false form=prefix>(</mml:mo><mml:mi>g</mml:mi><mml:mo stretchy=false form=postfix>)</mml:mo></mml:mrow></mml:mfrac><mml:mo>=</mml:mo><mml:mfrac><mml:msub><mml:mi>γ</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:msub><mml:mi>β</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:mfrac><mml:mfrac><mml:mn>1</mml:mn><mml:mi>g</mml:mi></mml:mfrac><mml:mo>+</mml:mo><mml:mi>⋯</mml:mi></mml:mrow></mml:math> determines which different cases of operator mixing can occur, and we review their classification. We derive a sufficient condition for A(g) <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML display=inline><mml:mrow><mml:mi>A</mml:mi><mml:mo stretchy=false form=prefix>(</mml:mo><mml:mi>g</mml:mi><mml:mo stretchy=false form=postfix>)</mml:mo></mml:mrow></mml:math> to be set in the one-loop exact form A(g) = frac{gamma_0}{beta_0}frac{1}{g} <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML display=inline><mml:mrow><mml:mi>A</mml:mi><mml:mo stretchy=false form=prefix>(</mml:mo><mml:mi>g</mml:mi><mml:mo stretchy=false form=postfix>)</mml:mo><mml:mo>=</mml:mo><mml:mfrac><mml:msub><mml:mi>γ</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:msub><mml:mi>β</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:mfrac><mml:mfrac><mml:mn>1</mml:mn><mml:mi>g</mml:mi></mml:mfrac></mml:mrow></mml:math> . Finally, we discuss the consequences of the unitarity requirement in massless QCD-like theories, and we demonstrate that gamma_0 <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML display=inline><mml:msub><mml:mi>γ</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:math> is always diagonalizable if the theory is conformal invariant and unitary in its free limit at g =0 <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML display=inline><mml:mrow><mml:mi>g</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:math> ." @default.
• W4283273708 created "2022-06-23" @default.
• W4283273708 creator A5035905199 @default.
• W4283273708 date "2022-06-21" @default.
• W4283273708 modified "2023-09-30" @default.
• W4283273708 title "A differential-geometry approach to operator mixing in massless QCD-like theories and Poincaré-Dulac theorem" @default.
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• W4283273708 doi "https://doi.org/10.21468/scipostphysproc.7.032" @default.
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