Emergence of kaonium as a sharp resonance in photon-photon to meson-meson cross sections

We calculate the binding energies of the hypothetical mesonic atom, <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:msup> <a:mi>K</a:mi> <a:mo>+</a:mo> </a:msup> <a:msup> <a:mi>K</a:mi> <a:mo>−</a:mo> </a:msup> </a:math> (kaonium), using the <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"> <c:msup> <c:mi>K</c:mi> <c:mo>+</c:mo> </c:msup> <c:msup> <c:mi>K</c:mi> <c:mo>−</c:mo> </c:msup> <c:mo stretchy="false">→</c:mo> <c:msup> <c:mi>K</c:mi> <c:mo>+</c:mo> </c:msup> <c:msup> <c:mi>K</c:mi> <c:mo>−</c:mo> </c:msup> </c:math> elastic scattering amplitude. Our findings are in line with previously reported results, which involve solving an eigenvalue equation of the Kudryavtsev-Popov type. Using chiral perturbation theory, we show that kaonium manifests itself as a sharp resonance around 992 MeV accompanying <f:math xmlns:f="http://www.w3.org/1998/Math/MathML" display="inline"> <f:msub> <f:mi>f</f:mi> <f:mn>0</f:mn> </f:msub> <f:mo stretchy="false">(</f:mo> <f:mn>980</f:mn> <f:mo stretchy="false">)</f:mo> </f:math> or <j:math xmlns:j="http://www.w3.org/1998/Math/MathML" display="inline"> <j:msub> <j:mi>a</j:mi> <j:mn>0</j:mn> </j:msub> <j:mo stretchy="false">(</j:mo> <j:mn>980</j:mn> <j:mo stretchy="false">)</j:mo> </j:math> in cross sections for processes <n:math xmlns:n="http://www.w3.org/1998/Math/MathML" display="inline"> <n:mi>γ</n:mi> <n:mi>γ</n:mi> <n:mo stretchy="false">→</n:mo> <n:msup> <n:mi>π</n:mi> <n:mn>0</n:mn> </n:msup> <n:msup> <n:mi>π</n:mi> <n:mn>0</n:mn> </n:msup> </n:math> or <q:math xmlns:q="http://www.w3.org/1998/Math/MathML" display="inline"> <q:mi>γ</q:mi> <q:mi>γ</q:mi> <q:mo stretchy="false">→</q:mo> <q:msup> <q:mi>π</q:mi> <q:mn>0</q:mn> </q:msup> <q:mi>η</q:mi> </q:math> . The latter process is particularly striking: the peak at the kaonium resonance energy is highly pronounced, with the ratio of the cross sections <t:math xmlns:t="http://www.w3.org/1998/Math/MathML" display="inline"> <t:mi>σ</t:mi> <t:mo stretchy="false">(</t:mo> <t:mi>γ</t:mi> <t:mi>γ</t:mi> <t:mo stretchy="false">→</t:mo> <t:msup> <t:mi>π</t:mi> <t:mn>0</t:mn> </t:msup> <t:mi>η</t:mi> <t:mo stretchy="false">)</t:mo> <t:mo>/</t:mo> <t:mi>σ</t:mi> <t:mo stretchy="false">(</t:mo> <t:mi>γ</t:mi> <t:mi>γ</t:mi> <t:mo stretchy="false">→</t:mo> <t:msup> <t:mi>π</t:mi> <t:mn>0</t:mn> </t:msup> <t:msup> <t:mi>π</t:mi> <t:mn>0</t:mn> </t:msup> <t:mo stretchy="false">)</t:mo> <t:mo>≈</t:mo> <t:mn>9</t:mn> </t:math> . Due to the short lifetime of kaonium ( <bb:math xmlns:bb="http://www.w3.org/1998/Math/MathML" display="inline"> <bb:mo>∼</bb:mo> <bb:msup> <bb:mn>10</bb:mn> <bb:mrow> <bb:mo>−</bb:mo> <bb:mn>18</bb:mn> </bb:mrow> </bb:msup> <bb:mtext> </bb:mtext> <bb:mtext> </bb:mtext> <bb:mi mathvariant="normal">s</bb:mi> </bb:math> ) and its small decay width ( <eb:math xmlns:eb="http://www.w3.org/1998/Math/MathML" display="inline"> <eb:mo>∼</eb:mo> <eb:mn>0.4</eb:mn> <eb:mtext> </eb:mtext> <eb:mtext> </eb:mtext> <eb:mi>keV</eb:mi> </eb:math> ), direct detection of this exotic atom poses a significant challenge and requires high experimental resolution. However, we show that once the formation of kaonium is considered in the cross section, a better fit to the available experimental data is obtained.