Fully strange tetraquark resonant states as the cousins of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>X</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mn>6900</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:math>

Abstract

We conduct systematic calculations of the <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:mrow> <a:mi>S</a:mi> </a:mrow> </a:math> -wave fully strange systems with “normal” <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"> <c:mo stretchy="false">(</c:mo> <c:msup> <c:mi>J</c:mi> <c:mrow> <c:mi>P</c:mi> <c:mi>C</c:mi> </c:mrow> </c:msup> <c:mo>=</c:mo> <c:msup> <c:mn>0</c:mn> <c:mrow> <c:mo>+</c:mo> <c:mo>+</c:mo> </c:mrow> </c:msup> <c:mo>,</c:mo> <c:msup> <c:mn>1</c:mn> <c:mrow> <c:mo>+</c:mo> <c:mo>−</c:mo> </c:mrow> </c:msup> <c:mo>,</c:mo> <c:msup> <c:mn>2</c:mn> <c:mrow> <c:mo>+</c:mo> <c:mo>+</c:mo> </c:mrow> </c:msup> <c:mo stretchy="false">)</c:mo> </c:math> and “exotic” <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"> <g:mo stretchy="false">(</g:mo> <g:msup> <g:mi>J</g:mi> <g:mrow> <g:mi>P</g:mi> <g:mi>C</g:mi> </g:mrow> </g:msup> <g:mo>=</g:mo> <g:msup> <g:mn>0</g:mn> <g:mrow> <g:mo>+</g:mo> <g:mo>−</g:mo> </g:mrow> </g:msup> <g:mo>,</g:mo> <g:msup> <g:mn>1</g:mn> <g:mrow> <g:mo>+</g:mo> <g:mo>+</g:mo> </g:mrow> </g:msup> <g:mo>,</g:mo> <g:msup> <g:mn>2</g:mn> <g:mrow> <g:mo>+</g:mo> <g:mo>−</g:mo> </g:mrow> </g:msup> <g:mo stretchy="false">)</g:mo> </g:math> <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"> <k:mrow> <k:mi>C</k:mi> </k:mrow> </k:math> -parities, which are the strange analog of the fully charmed tetraquark state <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"> <m:mi>X</m:mi> <m:mo stretchy="false">(</m:mo> <m:mn>6900</m:mn> <m:mo stretchy="false">)</m:mo> </m:math> . Within a constituent quark potential model, we employ the Gaussian expansion method to solve the four-body Schrödinger equation and the complex scaling method to identify resonant states. We obtain a series of resonant states and zero-width states in the mass range of 2.7 to 3.3 GeV, with their widths ranging from less than 1 to about 50 MeV. Their rms radii strongly indicate that they are compact tetraquark states. Among these states, the <q:math xmlns:q="http://www.w3.org/1998/Math/MathML" display="inline"> <q:msub> <q:mi>T</q:mi> <q:mrow> <q:mn>4</q:mn> <q:mi>s</q:mi> <q:mo>,</q:mo> <q:msup> <q:mn>2</q:mn> <q:mrow> <q:mo>+</q:mo> <q:mo>+</q:mo> </q:mrow> </q:msup> </q:mrow> </q:msub> <q:mo stretchy="false">(</q:mo> <q:mn>2714</q:mn> <q:mo stretchy="false">)</q:mo> </q:math> may be the most likely one to be observed experimentally. We urge the experimental exploration of the <u:math xmlns:u="http://www.w3.org/1998/Math/MathML" display="inline"> <u:msup> <u:mn>2</u:mn> <u:mrow> <u:mo>+</u:mo> <u:mo>+</u:mo> </u:mrow> </u:msup> <u:mtext> </u:mtext> <u:mtext> </u:mtext> <u:mi>s</u:mi> <u:mi>s</u:mi> <u:mover accent="true"> <u:mi>s</u:mi> <u:mo stretchy="false">¯</u:mo> </u:mover> <u:mover accent="true"> <u:mi>s</u:mi> <u:mo stretchy="false">¯</u:mo> </u:mover> </u:math> state around 2.7 GeV in the <ab:math xmlns:ab="http://www.w3.org/1998/Math/MathML" display="inline"> <ab:mi>ϕ</ab:mi> <ab:mi>ϕ</ab:mi> </ab:math> channel. Since the lowest S-wave <cb:math xmlns:cb="http://www.w3.org/1998/Math/MathML" display="inline"> <cb:mi>s</cb:mi> <cb:mi>s</cb:mi> <cb:mover accent="true"> <cb:mi>s</cb:mi> <cb:mo stretchy="false">¯</cb:mo> </cb:mover> <cb:mover accent="true"> <cb:mi>s</cb:mi> <cb:mo stretchy="false">¯</cb:mo> </cb:mover> </cb:math> state is around 2.7 GeV, the compact wave <ib:math xmlns:ib="http://www.w3.org/1998/Math/MathML" display="inline"> <ib:mi>s</ib:mi> <ib:mi>s</ib:mi> <ib:mover accent="true"> <ib:mi>s</ib:mi> <ib:mo stretchy="false">¯</ib:mo> </ib:mover> <ib:mover accent="true"> <ib:mi>s</ib:mi> <ib:mo stretchy="false">¯</ib:mo> </ib:mover> </ib:math> states are expected to be heavier. Hence, <ob:math xmlns:ob="http://www.w3.org/1998/Math/MathML" display="inline"> <ob:mi>ϕ</ob:mi> <ob:mo stretchy="false">(</ob:mo> <ob:mn>2170</ob:mn> <ob:mo stretchy="false">)</ob:mo> </ob:math> and <sb:math xmlns:sb="http://www.w3.org/1998/Math/MathML" display="inline"> <sb:mi>X</sb:mi> <sb:mo stretchy="false">(</sb:mo> <sb:mn>2370</sb:mn> <sb:mo stretchy="false">)</sb:mo> </sb:math> are unlikely to be compact tetraquark states. Published by the American Physical Society 2024

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