Dispersive determination of fourth generation quark masses

We determine the masses of the sequential fourth generation quarks <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mrow><a:msup><a:mrow><a:mi>b</a:mi></a:mrow><a:mrow><a:mo>′</a:mo></a:mrow></a:msup></a:mrow></a:math> and <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:msup><c:mi>t</c:mi><c:mo>′</c:mo></c:msup></c:math> in the extension of the Standard Model by solving the dispersion relations associated with the mixing between the neutral states <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"><e:mi>Q</e:mi><e:mover accent="true"><e:mi>q</e:mi><e:mo stretchy="false">¯</e:mo></e:mover></e:math> and <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"><i:mover accent="true"><i:mi>Q</i:mi><i:mo stretchy="false">¯</i:mo></i:mover><i:mi>q</i:mi></i:math>, with <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"><m:mi>Q</m:mi></m:math> (<o:math xmlns:o="http://www.w3.org/1998/Math/MathML" display="inline"><o:mi>q</o:mi></o:math>) being a heavy (light) quark. The box diagrams responsible for the mixing, which provide the perturbative inputs to the dispersion relations, involve multiple intermediate channels, i.e., the <q:math xmlns:q="http://www.w3.org/1998/Math/MathML" display="inline"><q:mi>u</q:mi><q:mi>t</q:mi></q:math> and <s:math xmlns:s="http://www.w3.org/1998/Math/MathML" display="inline"><s:mi>c</s:mi><s:mi>t</s:mi></s:math> channels, <u:math xmlns:u="http://www.w3.org/1998/Math/MathML" display="inline"><u:mi>u</u:mi></u:math> (<w:math xmlns:w="http://www.w3.org/1998/Math/MathML" display="inline"><w:mi>c</w:mi></w:math>, <y:math xmlns:y="http://www.w3.org/1998/Math/MathML" display="inline"><y:mi>t</y:mi></y:math>) being an up (charm, top) quark, in the <ab:math xmlns:ab="http://www.w3.org/1998/Math/MathML" display="inline"><ab:msup><ab:mi>b</ab:mi><ab:mo>′</ab:mo></ab:msup></ab:math> case, and the <cb:math xmlns:cb="http://www.w3.org/1998/Math/MathML" display="inline"><cb:mi>d</cb:mi><cb:msup><cb:mi>b</cb:mi><cb:mo>′</cb:mo></cb:msup></cb:math>, <eb:math xmlns:eb="http://www.w3.org/1998/Math/MathML" display="inline"><eb:mi>s</eb:mi><eb:msup><eb:mi>b</eb:mi><eb:mo>′</eb:mo></eb:msup></eb:math>, and <gb:math xmlns:gb="http://www.w3.org/1998/Math/MathML" display="inline"><gb:mi>b</gb:mi><gb:msup><gb:mi>b</gb:mi><gb:mo>′</gb:mo></gb:msup></gb:math> ones, <ib:math xmlns:ib="http://www.w3.org/1998/Math/MathML" display="inline"><ib:mi>d</ib:mi></ib:math> (<kb:math xmlns:kb="http://www.w3.org/1998/Math/MathML" display="inline"><kb:mi>s</kb:mi></kb:math>, <mb:math xmlns:mb="http://www.w3.org/1998/Math/MathML" display="inline"><mb:mi>b</mb:mi></mb:math>) being a down (strange, bottom) quark, in the <ob:math xmlns:ob="http://www.w3.org/1998/Math/MathML" display="inline"><ob:msup><ob:mi>t</ob:mi><ob:mo>′</ob:mo></ob:msup></ob:math> case. The common solutions for the above channels lead to the masses <qb:math xmlns:qb="http://www.w3.org/1998/Math/MathML" display="inline"><qb:msub><qb:mi>m</qb:mi><qb:msup><qb:mi>b</qb:mi><qb:mo>′</qb:mo></qb:msup></qb:msub><qb:mo>=</qb:mo><qb:mo stretchy="false">(</qb:mo><qb:mn>2.7</qb:mn><qb:mo>±</qb:mo><qb:mn>0.1</qb:mn><qb:mo stretchy="false">)</qb:mo></qb:math> and <ub:math xmlns:ub="http://www.w3.org/1998/Math/MathML" display="inline"><ub:msub><ub:mi>m</ub:mi><ub:msup><ub:mi>t</ub:mi><ub:mo>′</ub:mo></ub:msup></ub:msub><ub:mo>≈</ub:mo><ub:mn>200</ub:mn><ub:mtext> </ub:mtext><ub:mtext> </ub:mtext><ub:mi>TeV</ub:mi></ub:math> unambiguously. We show that these superheavy quarks, forming bound states in a Yukawa potential, barely contribute to Higgs boson production via gluon fusion and decay to photon pairs and bypass current experimental constraints. The mass of the <wb:math xmlns:wb="http://www.w3.org/1998/Math/MathML" display="inline"><wb:msup><wb:mover accent="true"><wb:mi>b</wb:mi><wb:mo stretchy="false">¯</wb:mo></wb:mover><wb:mo>′</wb:mo></wb:msup><wb:msup><wb:mi>b</wb:mi><wb:mo>′</wb:mo></wb:msup></wb:math> ground state is estimated to be about 3.2 TeV. It is thus worthwhile to continue the search for <ac:math xmlns:ac="http://www.w3.org/1998/Math/MathML" display="inline"><ac:msup><ac:mi>b</ac:mi><ac:mo>′</ac:mo></ac:msup></ac:math> quarks or <cc:math xmlns:cc="http://www.w3.org/1998/Math/MathML" display="inline"><cc:msup><cc:mover accent="true"><cc:mi>b</cc:mi><cc:mo stretchy="false">¯</cc:mo></cc:mover><cc:mo>′</cc:mo></cc:msup><cc:msup><cc:mi>b</cc:mi><cc:mo>′</cc:mo></cc:msup></cc:math> resonances at the (high-luminosity) Large Hadron Collider.