B→K+ axionlike particles: Effective versus UV-complete models and enhanced two-loop contributions

An axionlike particle <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mi>a</a:mi></a:math> (ALP) can explain the excess of <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:mi>B</c:mi><c:mo stretchy="false">→</c:mo><c:mi>K</c:mi><c:mo>+</c:mo><c:mtext>invisible</c:mtext></c:math> events at Belle II. However, many analyses of ALP scenarios are oversimplified. We revisit the <f:math xmlns:f="http://www.w3.org/1998/Math/MathML" display="inline"><f:mi>B</f:mi><f:mo stretchy="false">→</f:mo><f:mi>K</f:mi><f:mi>a</f:mi></f:math> transition rate in a popular minimal and UV complete model with two Higgs doublets (2HDM) and a complex singlet [Dine-Fischler-Srednicki-Zhitnitsky (DFSZ) model]. To this end we compare our results with previous studies which derived the <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"><i:mover accent="true"><i:mi>b</i:mi><i:mo stretchy="false">¯</i:mo></i:mover><i:mi>s</i:mi><i:mi>a</i:mi></i:math> vertex from the <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"><m:mover accent="true"><m:mi>b</m:mi><m:mo stretchy="false">¯</m:mo></m:mover><m:mi>s</m:mi><m:mi>A</m:mi></m:math> vertex, where <q:math xmlns:q="http://www.w3.org/1998/Math/MathML" display="inline"><q:mi>A</q:mi></q:math> is the heavy pseudoscalar of the 2HDM, in terms of an <s:math xmlns:s="http://www.w3.org/1998/Math/MathML" display="inline"><s:mi>a</s:mi><s:mo>−</s:mo><s:mi>A</s:mi></s:math> mixing angle. We find this approach to work only at the leading one-loop order, while it fails at the two-loop level. Furthermore, while an approximate <u:math xmlns:u="http://www.w3.org/1998/Math/MathML" display="inline"><u:msub><u:mi>Z</u:mi><u:mn>2</u:mn></u:msub></u:math> symmetry suppresses the leading-order amplitude by a factor of <w:math xmlns:w="http://www.w3.org/1998/Math/MathML" display="inline"><w:mn>1</w:mn><w:mo>/</w:mo><w:mi>tan</w:mi><w:mi>β</w:mi></w:math>, which is the ratio of the two vacuum expectation values of the Higgs doublets, we find the two-loop contribution unsuppressed and phenomenologically relevant for <y:math xmlns:y="http://www.w3.org/1998/Math/MathML" display="inline"><y:mi>tan</y:mi><y:mi>β</y:mi><y:mo>≳</y:mo><y:mn>5</y:mn></y:math>. We determine the allowed parameter space and underline the importance of better searches for <ab:math xmlns:ab="http://www.w3.org/1998/Math/MathML" display="inline"><ab:mrow><ab:mi mathvariant="normal">ϒ</ab:mi><ab:mo stretchy="false">→</ab:mo><ab:mi>γ</ab:mi><ab:mo>+</ab:mo><ab:mi>invisible</ab:mi></ab:mrow></ab:math> and for a possible excess in <eb:math xmlns:eb="http://www.w3.org/1998/Math/MathML" display="inline"><eb:mi>B</eb:mi><eb:mo stretchy="false">→</eb:mo><eb:mi>K</eb:mi><eb:msup><eb:mi>μ</eb:mi><eb:mo>+</eb:mo></eb:msup><eb:msup><eb:mi>μ</eb:mi><eb:mo>−</eb:mo></eb:msup></eb:math>. We further study the low-energy axion effective theory which leads to a divergent and basis-dependent amplitude. As a conceptual result, we clarify the ambiguities and identify which low-energy framework is consistent with the DFSZ model.