<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:msup><mml:mi>π</mml:mi><mml:mo>+</mml:mo></mml:msup><mml:msup><mml:mi>π</mml:mi><mml:mo>−</mml:mo></mml:msup></mml:math> Coulomb interaction study and its use in data proc

In this work, the Coulomb effects (Coulomb correlations) in <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:msup><a:mi>π</a:mi><a:mo>+</a:mo></a:msup><a:msup><a:mi>π</a:mi><a:mo>−</a:mo></a:msup></a:math> pairs produced in <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:mrow><c:mi mathvariant="normal">p</c:mi><c:mo>+</c:mo><c:mi>Ni</c:mi></c:mrow></c:math> collisions at <f:math xmlns:f="http://www.w3.org/1998/Math/MathML" display="inline"><f:mrow><f:mn>24</f:mn><f:mtext> </f:mtext><f:mtext> </f:mtext><f:mi>GeV</f:mi><f:mo>/</f:mo><f:mi>c</f:mi></f:mrow></f:math>, are studied using experimental <h:math xmlns:h="http://www.w3.org/1998/Math/MathML" display="inline"><h:msup><h:mi>π</h:mi><h:mo>+</h:mo></h:msup><h:msup><h:mi>π</h:mi><h:mo>−</h:mo></h:msup></h:math> pair distributions in <j:math xmlns:j="http://www.w3.org/1998/Math/MathML" display="inline"><j:mi>Q</j:mi></j:math>, the relative momentum in the pair center-of-mass system (c.m.s.), and its projections <l:math xmlns:l="http://www.w3.org/1998/Math/MathML" display="inline"><l:msub><l:mi>Q</l:mi><l:mi>L</l:mi></l:msub></l:math> (longitudinal component) and <n:math xmlns:n="http://www.w3.org/1998/Math/MathML" display="inline"><n:msub><n:mi>Q</n:mi><n:mi>t</n:mi></n:msub></n:math> (transverse component) relative to the pair direction in the laboratory system (LS). The major part of the pion pairs (“Coulomb pairs”) is produced in the decay of <p:math xmlns:p="http://www.w3.org/1998/Math/MathML" display="inline"><p:mrow><p:mi>ρ</p:mi></p:mrow></p:math>, <r:math xmlns:r="http://www.w3.org/1998/Math/MathML" display="inline"><r:mrow><r:mi>ω</r:mi></r:mrow></r:math> and <t:math xmlns:t="http://www.w3.org/1998/Math/MathML" display="inline"><t:mi mathvariant="normal">Δ</t:mi></t:math> resonances and other short-lived sources. In these pairs, the significant Coulomb interaction occurs at small <w:math xmlns:w="http://www.w3.org/1998/Math/MathML" display="inline"><w:mi>Q</w:mi></w:math>, dominating the <y:math xmlns:y="http://www.w3.org/1998/Math/MathML" display="inline"><y:msup><y:mi>π</y:mi><y:mo>+</y:mo></y:msup><y:msup><y:mi>π</y:mi><y:mo>−</y:mo></y:msup></y:math> interaction in the final state. The minor part of the pairs (“non-Coulomb pairs”) is produced if one or both pions arose from long-lived sources like <ab:math xmlns:ab="http://www.w3.org/1998/Math/MathML" display="inline"><ab:mi>η</ab:mi><ab:mo>,</ab:mo><ab:msup><ab:mi>η</ab:mi><ab:mo>′</ab:mo></ab:msup></ab:math> or from different interactions. In this case, the final state interaction is practically absent. The <cb:math xmlns:cb="http://www.w3.org/1998/Math/MathML" display="inline"><cb:mi>Q</cb:mi></cb:math>, <eb:math xmlns:eb="http://www.w3.org/1998/Math/MathML" display="inline"><eb:msub><eb:mi>Q</eb:mi><eb:mi>L</eb:mi></eb:msub></eb:math>, and <gb:math xmlns:gb="http://www.w3.org/1998/Math/MathML" display="inline"><gb:msub><gb:mi>Q</gb:mi><gb:mi>t</gb:mi></gb:msub></gb:math> distributions of the Coulomb pairs in the c.m.s. have been simulated assuming they are described by the phase space modified by the known point-like Coulomb correlation function <ib:math xmlns:ib="http://www.w3.org/1998/Math/MathML" display="inline"><ib:msub><ib:mi>A</ib:mi><ib:mi>C</ib:mi></ib:msub><ib:mo stretchy="false">(</ib:mo><ib:mi>Q</ib:mi><ib:mo stretchy="false">)</ib:mo></ib:math>, corrected for small effects due to the nonpointlike pair production and the strong two-pion interaction. The same distributions of non-Coulomb pairs have been simulated according to the phase space, but without <mb:math xmlns:mb="http://www.w3.org/1998/Math/MathML" display="inline"><mb:msub><mb:mi>A</mb:mi><mb:mi>C</mb:mi></mb:msub><mb:mo stretchy="false">(</mb:mo><mb:mi>Q</mb:mi><mb:mo stretchy="false">)</mb:mo></mb:math>. In all <qb:math xmlns:qb="http://www.w3.org/1998/Math/MathML" display="inline"><qb:msub><qb:mi>Q</qb:mi><qb:mi>t</qb:mi></qb:msub></qb:math> intervals, the experimental <sb:math xmlns:sb="http://www.w3.org/1998/Math/MathML" display="inline"><sb:msub><sb:mi>Q</sb:mi><sb:mi>L</sb:mi></sb:msub></sb:math> spectrum shows a peak around <ub:math xmlns:ub="http://www.w3.org/1998/Math/MathML" display="inline"><ub:msub><ub:mi>Q</ub:mi><ub:mi>L</ub:mi></ub:msub><ub:mo>=</ub:mo><ub:mn>0</ub:mn></ub:math> caused by the Coulomb final state interaction. The full width at half maximum increases with <wb:math xmlns:wb="http://www.w3.org/1998/Math/MathML" display="inline"><wb:msub><wb:mi>Q</wb:mi><wb:mi>t</wb:mi></wb:msub></wb:math> from <yb:math xmlns:yb="http://www.w3.org/1998/Math/MathML" display="inline"><yb:mrow><yb:mn>3</yb:mn><yb:mtext> </yb:mtext><yb:mtext> </yb:mtext><yb:mi>MeV</yb:mi><yb:mo>/</yb:mo><yb:mi>c</yb:mi></yb:mrow></yb:math> for <ac:math xmlns:ac="http://www.w3.org/1998/Math/MathML" display="inline"><ac:mn>0</ac:mn><ac:mo>&lt;</ac:mo><ac:msub><ac:mi>Q</ac:mi><ac:mi>t</ac:mi></ac:msub><ac:mo>&lt;</ac:mo><ac:mn>0.25</ac:mn><ac:mtext> </ac:mtext><ac:mtext> </ac:mtext><ac:mi>MeV</ac:mi><ac:mo>/</ac:mo><ac:mi>c</ac:mi></ac:math> to <cc:math xmlns:cc="http://www.w3.org/1998/Math/MathML" display="inline"><cc:mrow><cc:mn>11</cc:mn><cc:mtext> </cc:mtext><cc:mtext> </cc:mtext><cc:mi>MeV</cc:mi><cc:mo>/</cc:mo><cc:mi>c</cc:mi></cc:mrow></cc:math> for <ec:math xmlns:ec="http://www.w3.org/1998/Math/MathML" display="inline"><ec:mn>4.0</ec:mn><ec:mo>&lt;</ec:mo><ec:msub><ec:mi>Q</ec:mi><ec:mi>t</ec:mi></ec:msub><ec:mo>&lt;</ec:mo><ec:mn>5.0</ec:mn><ec:mtext> </ec:mtext><ec:mtext> </ec:mtext><ec:mi>MeV</ec:mi><ec:mo>/</ec:mo><ec:mi>c</ec:mi></ec:math>. The experimental <gc:math xmlns:gc="http://www.w3.org/1998/Math/MathML" display="inline"><gc:msub><gc:mi>Q</gc:mi><gc:mi>L</gc:mi></gc:msub></gc:math> distributions have been fitted with two free parameters: the fraction of Coulomb pairs and the normalization constant. The precision of the description of these distributions is better than 2% in <ic:math xmlns:ic="http://www.w3.org/1998/Math/MathML" display="inline"><ic:msub><ic:mi>Q</ic:mi><ic:mi>t</ic:mi></ic:msub></ic:math> intervals 2–3, 3–4, and <kc:math xmlns:kc="http://www.w3.org/1998/Math/MathML" display="inline"><kc:mrow><kc:mn>4</kc:mn><kc:mi>–</kc:mi><kc:mn>5</kc:mn><kc:mtext> </kc:mtext><kc:mtext> </kc:mtext><kc:mi>MeV</kc:mi><kc:mo stretchy="false">/</kc:mo><kc:mi>c</kc:mi></kc:mrow></kc:math> and better than 0.5% in the total <nc:math xmlns:nc="http://www.w3.org/1998/Math/MathML" display="inline"><nc:msub><nc:mi>Q</nc:mi><nc:mi>t</nc:mi></nc:msub></nc:math> interval <pc:math xmlns:pc="http://www.w3.org/1998/Math/MathML" display="inline"><pc:mrow><pc:mn>0</pc:mn><pc:mi>–</pc:mi><pc:mn>5</pc:mn><pc:mtext> </pc:mtext><pc:mtext> </pc:mtext><pc:mi>MeV</pc:mi><pc:mo stretchy="false">/</pc:mo><pc:mi>c</pc:mi></pc:mrow></pc:math>. It is shown that the number of Coulomb pairs in all <sc:math xmlns:sc="http://www.w3.org/1998/Math/MathML" display="inline"><sc:msub><sc:mi>Q</sc:mi><sc:mi>t</sc:mi></sc:msub></sc:math> intervals, including the small <uc:math xmlns:uc="http://www.w3.org/1998/Math/MathML" display="inline"><uc:msub><uc:mi>Q</uc:mi><uc:mi>t</uc:mi></uc:msub></uc:math> (small opening angles <wc:math xmlns:wc="http://www.w3.org/1998/Math/MathML" display="inline"><wc:mi>θ</wc:mi></wc:math> in the LS) is calculated with theoretical precision better than 2%. The comparison of the simulated and experimental numbers of Coulomb pairs at small <yc:math xmlns:yc="http://www.w3.org/1998/Math/MathML" display="inline"><yc:msub><yc:mi>Q</yc:mi><yc:mi>t</yc:mi></yc:msub></yc:math> allows us to check and correct the detection efficiency for the pairs with small <ad:math xmlns:ad="http://www.w3.org/1998/Math/MathML" display="inline"><ad:mi>θ</ad:mi></ad:math> (0.06 mrad and smaller). It is shown that Coulomb pairs can be used as a new physical tool to check and correct the quality of the simulated events. The special property of the Coulomb pairs is the possibility of checking and correcting the detection efficiency, especially for the pairs with small opening angles. Published by the American Physical Society 2024