Unraveling pentaquarks with the Born-Oppenheimer effective theory

The hidden-charm pentaquark states <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:mrow> <a:msub> <a:mrow> <a:mi>P</a:mi> </a:mrow> <a:mrow> <a:mi>c</a:mi> <a:mover accent="true"> <a:mrow> <a:mi>c</a:mi> </a:mrow> <a:mrow> <a:mo stretchy="false">¯</a:mo> </a:mrow> </a:mover> </a:mrow> </a:msub> <a:msup> <a:mrow> <a:mo stretchy="false">(</a:mo> <a:mn>4312</a:mn> <a:mo stretchy="false">)</a:mo> </a:mrow> <a:mrow> <a:mo>+</a:mo> </a:mrow> </a:msup> </a:mrow> </a:math> , <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"> <g:mrow> <g:msub> <g:mrow> <g:mi>P</g:mi> </g:mrow> <g:mrow> <g:mi>c</g:mi> <g:mover accent="true"> <g:mrow> <g:mi>c</g:mi> </g:mrow> <g:mrow> <g:mo stretchy="false">¯</g:mo> </g:mrow> </g:mover> </g:mrow> </g:msub> <g:msup> <g:mrow> <g:mo stretchy="false">(</g:mo> <g:mn>4380</g:mn> <g:mo stretchy="false">)</g:mo> </g:mrow> <g:mrow> <g:mo>+</g:mo> </g:mrow> </g:msup> </g:mrow> </g:math> , <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"> <m:msub> <m:mi>P</m:mi> <m:mrow> <m:mi>c</m:mi> <m:mover accent="true"> <m:mi>c</m:mi> <m:mo stretchy="false">¯</m:mo> </m:mover> </m:mrow> </m:msub> <m:msup> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mn>4440</m:mn> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mo>+</m:mo> </m:msup> </m:math> , and <s:math xmlns:s="http://www.w3.org/1998/Math/MathML" display="inline"> <s:msub> <s:mi>P</s:mi> <s:mrow> <s:mi>c</s:mi> <s:mover accent="true"> <s:mi>c</s:mi> <s:mo stretchy="false">¯</s:mo> </s:mover> </s:mrow> </s:msub> <s:msup> <s:mrow> <s:mo stretchy="false">(</s:mo> <s:mn>4457</s:mn> <s:mo stretchy="false">)</s:mo> </s:mrow> <s:mo>+</s:mo> </s:msup> </s:math> , all with isospin <y:math xmlns:y="http://www.w3.org/1998/Math/MathML" display="inline"> <y:mi>I</y:mi> <y:mo>=</y:mo> <y:mn>1</y:mn> <y:mo>/</y:mo> <y:mn>2</y:mn> </y:math> , were discovered by the LHCb Collaboration in the decay process <ab:math xmlns:ab="http://www.w3.org/1998/Math/MathML" display="inline"> <ab:mrow> <ab:msubsup> <ab:mrow> <ab:mi mathvariant="normal">Λ</ab:mi> </ab:mrow> <ab:mrow> <ab:mi>b</ab:mi> </ab:mrow> <ab:mrow> <ab:mn>0</ab:mn> </ab:mrow> </ab:msubsup> <ab:mo stretchy="false">→</ab:mo> <ab:mi>J</ab:mi> <ab:mo>/</ab:mo> <ab:mi>ψ</ab:mi> <ab:mi>p</ab:mi> <ab:msup> <ab:mrow> <ab:mi>K</ab:mi> </ab:mrow> <ab:mrow> <ab:mo>−</ab:mo> </ab:mrow> </ab:msup> </ab:mrow> </ab:math> . Although their quantum numbers remain undetermined, these states have generated significant theoretical interest. We analyze their spectrum and decay patterns—including those of their spin partners—within the Born-Oppenheimer (BO) effective field theory (BOEFT), a framework grounded in QCD. At leading order in BOEFT, we identify these pentaquark states as bound states in BO potentials that exhibit at short distance a repulsive octet behavior and a nonperturbative shift due to the adjoint baryons masses, while asymptotically approaching the <eb:math xmlns:eb="http://www.w3.org/1998/Math/MathML" display="inline"> <eb:msub> <eb:mi mathvariant="normal">Σ</eb:mi> <eb:mi>c</eb:mi> </eb:msub> <eb:mover accent="true"> <eb:mi>D</eb:mi> <eb:mo stretchy="false">¯</eb:mo> </eb:mover> </eb:math> threshold. We further incorporate <jb:math xmlns:jb="http://www.w3.org/1998/Math/MathML" display="inline"> <jb:mi mathvariant="script">O</jb:mi> <jb:mo stretchy="false">(</jb:mo> <jb:mn>1</jb:mn> <jb:mo>/</jb:mo> <jb:msub> <jb:mi>m</jb:mi> <jb:mi>Q</jb:mi> </jb:msub> <jb:mo stretchy="false">)</jb:mo> </jb:math> spin-dependent corrections to compute pentaquark multiplet spin splittings. Based on the spectrum, semi-inclusive decay widths to <ob:math xmlns:ob="http://www.w3.org/1998/Math/MathML" display="inline"> <ob:mi>J</ob:mi> <ob:mo>/</ob:mo> <ob:mi>ψ</ob:mi> </ob:math> and <qb:math xmlns:qb="http://www.w3.org/1998/Math/MathML" display="inline"> <qb:msub> <qb:mi>η</qb:mi> <qb:mi>c</qb:mi> </qb:msub> </qb:math> , and the decay width ratios to <sb:math xmlns:sb="http://www.w3.org/1998/Math/MathML" display="inline"> <sb:msub> <sb:mi mathvariant="normal">Λ</sb:mi> <sb:mi>c</sb:mi> </sb:msub> <sb:mover accent="true"> <sb:mi>D</sb:mi> <sb:mo stretchy="false">¯</sb:mo> </sb:mover> </sb:math> and <xb:math xmlns:xb="http://www.w3.org/1998/Math/MathML" display="inline"> <xb:msub> <xb:mi mathvariant="normal">Λ</xb:mi> <xb:mi>c</xb:mi> </xb:msub> <xb:msup> <xb:mover accent="true"> <xb:mi>D</xb:mi> <xb:mo stretchy="false">¯</xb:mo> </xb:mover> <xb:mo>*</xb:mo> </xb:msup> </xb:math> , we provide the first theoretical predictions for the adjoint baryon masses, which can be confirmed by future lattice QCD studies. Moreover, our analysis supports the quantum number assignments: <cc:math xmlns:cc="http://www.w3.org/1998/Math/MathML" display="inline"> <cc:msup> <cc:mi>J</cc:mi> <cc:mi>P</cc:mi> </cc:msup> <cc:mo>=</cc:mo> <cc:mo stretchy="false">(</cc:mo> <cc:mn>1</cc:mn> <cc:mo>/<