Triply heavy tetraquarks <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mover accent="true"> <mml:mi>b</mml:mi> <mml:mo stretchy="false">¯</mml:mo> </mml:mover> <mml:mi>c</mml:mi> <mml:mover accent="true"> <mml:mi>q</

A systematic investigation of <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:mi>S</a:mi> </a:math> -wave triply heavy tetraquark systems with quark content <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"> <c:mover accent="true"> <c:mi>b</c:mi> <c:mo stretchy="false">¯</c:mo> </c:mover> <c:mi>c</c:mi> <c:mover accent="true"> <c:mi>q</c:mi> <c:mo stretchy="false">¯</c:mo> </c:mover> <c:mi>c</c:mi> </c:math> and <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"> <i:mover accent="true"> <i:mi>c</i:mi> <i:mo stretchy="false">¯</i:mo> </i:mover> <i:mi>b</i:mi> <i:mover accent="true"> <i:mi>q</i:mi> <i:mo stretchy="false">¯</i:mo> </i:mover> <i:mi>b</i:mi> </i:math> ( <o:math xmlns:o="http://www.w3.org/1998/Math/MathML" display="inline"> <o:mrow> <o:mi>q</o:mi> <o:mo>=</o:mo> <o:mi>u</o:mi> </o:mrow> </o:math> , <q:math xmlns:q="http://www.w3.org/1998/Math/MathML" display="inline"> <q:mrow> <q:mi>d</q:mi> </q:mrow> </q:math> , <s:math xmlns:s="http://www.w3.org/1998/Math/MathML" display="inline"> <s:mrow> <s:mi>s</s:mi> </s:mrow> </s:math> ), spin-parity quantum numbers <u:math xmlns:u="http://www.w3.org/1998/Math/MathML" display="inline"> <u:msup> <u:mi>J</u:mi> <u:mi>P</u:mi> </u:msup> <u:mo>=</u:mo> <u:msup> <u:mn>0</u:mn> <u:mo>+</u:mo> </u:msup> </u:math> , <w:math xmlns:w="http://www.w3.org/1998/Math/MathML" display="inline"> <w:msup> <w:mn>1</w:mn> <w:mo>+</w:mo> </w:msup> </w:math> , <y:math xmlns:y="http://www.w3.org/1998/Math/MathML" display="inline"> <y:msup> <y:mn>2</y:mn> <y:mo>+</y:mo> </y:msup> </y:math> , and isospin <ab:math xmlns:ab="http://www.w3.org/1998/Math/MathML" display="inline"> <ab:mi>I</ab:mi> <ab:mo>=</ab:mo> <ab:mn>0</ab:mn> <ab:mo>,</ab:mo> <ab:mfrac> <ab:mn>1</ab:mn> <ab:mn>2</ab:mn> </ab:mfrac> </ab:math> is carried out within the constituent quark model framework. The four-body bound and resonance states are determined by solving the Schrödinger equation employing the high-precision and efficient Gaussian expansion method in conjunction with the powerful complex scaling method. Besides, a comprehensive coupled-channel analysis of the <cb:math xmlns:cb="http://www.w3.org/1998/Math/MathML" display="inline"> <cb:mi>S</cb:mi> </cb:math> -wave tetraquark systems is performed, taking into account meson-meson, diquark-antidiquark, and K-type configurations, as well as all allowed color structures. Several narrow resonant states are identified in each <eb:math xmlns:eb="http://www.w3.org/1998/Math/MathML" display="inline"> <eb:mi>I</eb:mi> <eb:mo stretchy="false">(</eb:mo> <eb:msup> <eb:mi>J</eb:mi> <eb:mi>P</eb:mi> </eb:msup> <eb:mo stretchy="false">)</eb:mo> </eb:math> channel. In particular, resonances for the <ib:math xmlns:ib="http://www.w3.org/1998/Math/MathML" display="inline"> <ib:mover accent="true"> <ib:mi>b</ib:mi> <ib:mo stretchy="false">¯</ib:mo> </ib:mover> <ib:mi>c</ib:mi> <ib:mover accent="true"> <ib:mi>q</ib:mi> <ib:mo stretchy="false">¯</ib:mo> </ib:mover> <ib:mi>c</ib:mi> </ib:math> system are found in the mass range of 8.87–9.36 GeV, while those for the <ob:math xmlns:ob="http://www.w3.org/1998/Math/MathML" display="inline"> <ob:mover accent="true"> <ob:mi>c</ob:mi> <ob:mo stretchy="false">¯</ob:mo> </ob:mover> <ob:mi>b</ob:mi> <ob:mover accent="true"> <ob:mi>q</ob:mi> <ob:mo stretchy="false">¯</ob:mo> </ob:mover> <ob:mi>b</ob:mi> </ob:math> system lie between 12.13–12.45 GeV. Most of the predicted resonances are compact tetraquark states with sizes smaller than 1.0 fm, although four states exhibit more extended structures with sizes around 1.3 fm, suggesting a more loosely bound nature. Magnetic moments and dominant wave function components of these exotic states are also analyzed. The results indicate that K-type configurations play a major role in the structure of the observed resonances. Finally, possible strong decay channels (golden modes) for these states are theoretically proposed.