In confining large N theories with a θ angle such as four-dimensional SU(N) pure Yang-Mills theory, there are multiple metastable vacua and it makes sense to consider the parameter region of “large θ of order N” despite the fact that θ is a 2π-periodic parameter. We investigate this parameter region in the two-dimensional <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13130_2025_26307_Article_IEq1.gif" Format="GIF" Height="18" Rendition="HTML" Resolution="72" Type="Linedraw" Width="53" /> </InlineMediaObject> <EquationSource Format="MATHML"><math display="inline"> <msup> <mi mathvariant="double-struck">CP</mi> <mrow> <mi>N</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> </math></EquationSource> <EquationSource Format="TEX">$ {\mathbbm{CP}}^{N-1} $</EquationSource> </InlineEquation> model by computing the partition function on T 2. When θ/N is of order <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13130_2025_26307_Article_IEq3.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="49" /> </InlineMediaObject> <EquationSource Format="MATHML"><math display="inline"> <mi mathvariant="script">O</mi> <mfenced close=")" open="("> <mn>0.1</mn> </mfenced> </math></EquationSource> <EquationSource Format="TEX">$ \mathcal{O}(0.1) $</EquationSource> </InlineEquation> or less, we get perfectly sensible results for the vacuum energies and decay rates of metastable vacua. However, when θ/N is of order <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13130_2025_26307_Article_IEq4.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="36" /> </InlineMediaObject> <EquationSource Format="MATHML"><math display="inline"> <mi mathvariant="script">O</mi> <mfenced close=")" open="("> <mn>1</mn> </mfenced> </math></EquationSource> <EquationSource Format="TEX">$ \mathcal{O}(1) $</EquationSource> </InlineEquation>, we encounter a problem about saddle points that would give larger contributions to the partition function than the true vacuum. We discuss why it might not be straightforward to resolve this problem.

Large θ angle in two-dimensional large N $ {\mathbbm{CP}}^{N-1} $ model

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2025年5月28日

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Tsubasa Sugeno,
Takahiro Yokokura,
Kazuya Yonekura

摘要

In confining large N theories with a θ angle such as four-dimensional SU(N) pure Yang-Mills theory, there are multiple metastable vacua and it makes sense to consider the parameter region of “large θ of order N” despite the fact that θ is a 2π-periodic parameter. We investigate this parameter region in the two-dimensional ${CP}^{N - 1}$ $ {\mathbbm{CP}}^{N-1} $ model by computing the partition function on T ². When θ/N is of order $O (0.1)$ $ \mathcal{O}(0.1) $ or less, we get perfectly sensible results for the vacuum energies and decay rates of metastable vacua. However, when θ/N is of order $O (1)$ $ \mathcal{O}(1) $ , we encounter a problem about saddle points that would give larger contributions to the partition function than the true vacuum. We discuss why it might not be straightforward to resolve this problem.

Large θ angle in two-dimensional large N \( {\mathbbm{CP}}^{N-1} \) model

摘要