Test of the vector portal with dark fermions in the charge-exchange reactions in the NA64 experiment at CERN SPS

We consider an experiment to search for dark sector particles in invisible (or semi-invisible) decays of neutral mesons <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:msup><a:mi>M</a:mi><a:mn>0</a:mn></a:msup><a:mo>=</a:mo><a:msup><a:mi>π</a:mi><a:mn>0</a:mn></a:msup></a:math>, <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:mi>η</c:mi></c:math>, <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"><e:msup><e:mi>η</e:mi><e:mo>′</e:mo></e:msup></e:math>, <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"><g:mi>ω</g:mi></g:math>, <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"><i:msub><i:mi>f</i:mi><i:mn>2</i:mn></i:msub><i:mo stretchy="false">(</i:mo><i:mn>1270</i:mn><i:mo stretchy="false">)</i:mo></i:math>, which produced the charge-exchange reactions <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"><m:mrow><m:msup><m:mrow><m:mi>π</m:mi></m:mrow><m:mrow><m:mo>−</m:mo></m:mrow></m:msup><m:mo>+</m:mo><m:mo stretchy="false">(</m:mo><m:mi>A</m:mi><m:mo>,</m:mo><m:mi>Z</m:mi><m:mo stretchy="false">)</m:mo><m:mo stretchy="false">→</m:mo><m:msup><m:mrow><m:mi>M</m:mi></m:mrow><m:mrow><m:mn>0</m:mn></m:mrow></m:msup><m:mo>+</m:mo><m:mo stretchy="false">(</m:mo><m:mi>A</m:mi><m:mo>,</m:mo><m:mi>Z</m:mi><m:mo>−</m:mo><m:mn>1</m:mn><m:mo stretchy="false">)</m:mo><m:mo>;</m:mo><m:msup><m:mrow><m:mi>M</m:mi></m:mrow><m:mrow><m:mn>0</m:mn></m:mrow></m:msup><m:mo stretchy="false">→</m:mo></m:mrow></m:math> of high-energy pions (or kaons) on a nuclei target in the NA64 experiment at the CERN SPS. This reaction chain would lead to a striking signature of the signal event—the complete disappearance of the beam energy in the setup []. Using data obtained from the measurements at IHEP (Protvino) and Fermilab (Batavia) we show that the integral cross sections <u:math xmlns:u="http://www.w3.org/1998/Math/MathML" display="inline"><u:mi>σ</u:mi></u:math> for the production of the <w:math xmlns:w="http://www.w3.org/1998/Math/MathML" display="inline"><w:mrow><w:msup><w:mrow><w:mi>M</w:mi></w:mrow><w:mrow><w:mn>0</w:mn></w:mrow></w:msup><w:mtext> </w:mtext><w:mtext> </w:mtext><w:mi mathvariant="normal">s</w:mi></w:mrow></w:math> slightly deviate from the phenomenological formula <z:math xmlns:z="http://www.w3.org/1998/Math/MathML" display="inline"><z:mi>σ</z:mi><z:mo>∼</z:mo><z:msup><z:mi>Z</z:mi><z:mrow><z:mn>2</z:mn><z:mo>/</z:mo><z:mn>3</z:mn></z:mrow></z:msup></z:math>, where <bb:math xmlns:bb="http://www.w3.org/1998/Math/MathML" display="inline"><bb:mi>Z</bb:mi></bb:math> is the nuclei charge. In particular, we present the formulas for the differential and integral cross sections that explicitly depend on the Mandelstam and <db:math xmlns:db="http://www.w3.org/1998/Math/MathML" display="inline"><db:mi>Z</db:mi></db:math> variables. Derived formulas are used to predict the <fb:math xmlns:fb="http://www.w3.org/1998/Math/MathML" display="inline"><fb:msup><fb:mi>M</fb:mi><fb:mn>0</fb:mn></fb:msup></fb:math> yield as a function of beam energy for several target nuclei, and to estimate the projection sensitivity for the proposed search in the dark vector portal model. Sensitivity to different decay modes of <hb:math xmlns:hb="http://www.w3.org/1998/Math/MathML" display="inline"><hb:mrow><hb:msup><hb:mrow><hb:mi>M</hb:mi></hb:mrow><hb:mrow><hb:mn>0</hb:mn></hb:mrow></hb:msup><hb:mi mathvariant="normal">s</hb:mi></hb:mrow></hb:math> is explored. Published by the American Physical Society 2024