Chiral properties of the nucleon interpolating current and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>θ</mml:mi></mml:math>-dependent observables

We revisit the chiral properties of nucleon interpolating currents, and show that of the two leading order currents <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:msub><a:mi>j</a:mi><a:mn>1</a:mn></a:msub></a:math> and <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:msub><c:mi>j</c:mi><c:mn>2</c:mn></c:msub></c:math>, only two linear combinations <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"><e:msub><e:mi>j</e:mi><e:mn>1</e:mn></e:msub><e:mo>±</e:mo><e:msub><e:mi>j</e:mi><e:mn>2</e:mn></e:msub></e:math> transform covariantly under the anomalous <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"><g:mi>U</g:mi><g:mo stretchy="false">(</g:mo><g:mn>1</g:mn><g:msub><g:mo stretchy="false">)</g:mo><g:mi>A</g:mi></g:msub></g:math> symmetry. As a result, calculations of quantities which vanish by symmetry in the chiral limit may produce unphysical results if carried out with different linear combinations of the currents. This includes observables such as electric dipole moments, induced by the quantum chromodynamics (QCD) parameter <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"><k:mi>θ</k:mi></k:math>, and the <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"><m:mi>θ</m:mi></m:math>-dependence of the nucleon mass. For completeness, we also exhibit the leading order results for nucleon electric dipole moments (<o:math xmlns:o="http://www.w3.org/1998/Math/MathML" display="inline"><o:msub><o:mi>d</o:mi><o:mrow><o:mi>n</o:mi><o:mo>,</o:mo><o:mi>p</o:mi></o:mrow></o:msub></o:math>) induced by <q:math xmlns:q="http://www.w3.org/1998/Math/MathML" display="inline"><q:mi>θ</q:mi></q:math>, and the nucleon magnetic moments (<s:math xmlns:s="http://www.w3.org/1998/Math/MathML" display="inline"><s:msub><s:mi>μ</s:mi><s:mrow><s:mi>n</s:mi><s:mo>,</s:mo><s:mi>p</s:mi></s:mrow></s:msub></s:math>), when calculated using QCD sum rules for both the covariant choices of the nucleon interpolating current. The results in each channel, conveniently expressed as the ratios, <u:math xmlns:u="http://www.w3.org/1998/Math/MathML" display="inline"><u:msub><u:mi>d</u:mi><u:mrow><u:mi>n</u:mi><u:mo>,</u:mo><u:mi>p</u:mi></u:mrow></u:msub><u:mo>/</u:mo><u:msub><u:mi>μ</u:mi><u:mrow><u:mi>n</u:mi><u:mo>,</u:mo><u:mi>p</u:mi></u:mrow></u:msub></u:math>, are numerically consistent, and reflect the required physical dependence on <w:math xmlns:w="http://www.w3.org/1998/Math/MathML" display="inline"><w:mi>θ</w:mi></w:math>. Published by the American Physical Society 2024