Flavor and CP symmetries in the standard model effective field theory

The <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mi>C</a:mi><a:mi>P</a:mi></a:math> properties of effective operators are closely related to the spacetime and internal symmetries, such as the flavor symmetry, in the standard model effective field theory (SMEFT). In this work, we utilize the flavor symmetry to organize and reduce numbers of independent Wilson coefficients of the SMEFT operators. We classify the dimension-6 and dimension-8 baryon/lepton-number-conserving operators based on their <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:mi>C</c:mi><c:mi>P</c:mi></c:math> properties. The <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"><e:mi>U</e:mi><e:mo stretchy="false">(</e:mo><e:mn>1</e:mn><e:msup><e:mo stretchy="false">)</e:mo><e:mn>4</e:mn></e:msup></e:math> rephasing symmetry is applied to distinguish <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"><i:mi>C</i:mi><i:mi>P</i:mi></i:math>-violating and <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"><k:mi>C</k:mi><k:mi>P</k:mi></k:math>-odd operators, which leads to reduction of the independent <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"><m:mi>C</m:mi><m:mi>P</m:mi></m:math>-violating phases. The <o:math xmlns:o="http://www.w3.org/1998/Math/MathML" display="inline"><o:mi>U</o:mi><o:mo stretchy="false">(</o:mo><o:mn>3</o:mn><o:msup><o:mo stretchy="false">)</o:mo><o:mn>5</o:mn></o:msup></o:math> and <s:math xmlns:s="http://www.w3.org/1998/Math/MathML" display="inline"><s:mi>U</s:mi><s:mo stretchy="false">(</s:mo><s:mn>2</s:mn><s:msup><s:mo stretchy="false">)</s:mo><s:mn>5</s:mn></s:msup></s:math> flavor symmetries let us classify the <w:math xmlns:w="http://www.w3.org/1998/Math/MathML" display="inline"><w:mi>C</w:mi><w:mi>P</w:mi></w:math>-violating phases as the flavor invariants, which can be enumerated by the Hilbert series and obtained explicitly by the Young tensor method. Then after introducing the minimal flavor violation (MFV) hypothesis, we present the flavor structures of the dimension-6 and dimension-8 SMEFT operators under the MFV hypothesis utilizing the spurion method.