<p>We calculate the electromagnetic multipole moments of the Σ-type strange hidden-charm pentaquarks <InlineEquation ID="IEq2"> <EquationSource Format="MATHML"><math display="inline"> <msubsup> <mi>P</mi> <mi mathvariant="italic">ψs</mi> <mi mathvariant="normal">Σ</mi> </msubsup> </math></EquationSource> <EquationSource Format="TEX">\( {P}_{\psi s}^{\Sigma} \)</EquationSource> </InlineEquation> — which form an isospin triplet with three distinct charge states (Σ<sup>+</sup>, Σ<sup>0</sup>, Σ<sup>−</sup>) offering independently measurable electromagnetic properties — using QCD light-cone sum rules, employing six interpolating currents for the spin-<InlineEquation ID="IEq3"> <EquationSource Format="MATHML"><math display="inline"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </math></EquationSource> <EquationSource Format="TEX">\( \frac{1}{2} \)</EquationSource> </InlineEquation> channel and seven for the spin-<InlineEquation ID="IEq4"> <EquationSource Format="MATHML"><math display="inline"> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> </math></EquationSource> <EquationSource Format="TEX">\( \frac{3}{2} \)</EquationSource> </InlineEquation> channel, each built from diquark-diquark-antiquark operators with scalar or axial-vector diquark content. We compute the magnetic dipole moment <i>μ</i> for all channels and, for spin-<InlineEquation ID="IEq5"> <EquationSource Format="MATHML"><math display="inline"> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> </math></EquationSource> <EquationSource Format="TEX">\( \frac{3}{2} \)</EquationSource> </InlineEquation>, the electric quadrupole <InlineEquation ID="IEq6"> <EquationSource Format="MATHML"><math display="inline"> <mi mathvariant="script">Q</mi> </math></EquationSource> <EquationSource Format="TEX">\( \mathcal{Q} \)</EquationSource> </InlineEquation> and magnetic octupole <InlineEquation ID="IEq7"> <EquationSource Format="MATHML"><math display="inline"> <mi mathvariant="script">O</mi> </math></EquationSource> <EquationSource Format="TEX">\( \mathcal{O} \)</EquationSource> </InlineEquation> moments, the latter two for the first time for this system, and provide the first systematic quark-flavor decomposition of all three multipole moments for <InlineEquation ID="IEq8"> <EquationSource Format="MATHML"><math display="inline"> <msubsup> <mi>P</mi> <mi mathvariant="italic">ψs</mi> <mi mathvariant="normal">Σ</mi> </msubsup> </math></EquationSource> <EquationSource Format="TEX">\( {P}_{\psi s}^{\Sigma} \)</EquationSource> </InlineEquation>. The results show a systematic dependence on the diquark spin. Scalar diquark currents yield charm-sector-dominated, flavor-insensitive moments (<i>μ</i> ∈ [−1.92, −1.21] <i>μ</i><sub><i>N</i></sub> for spin-<InlineEquation ID="IEq9"> <EquationSource Format="MATHML"><math display="inline"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </math></EquationSource> <EquationSource Format="TEX">\( \frac{1}{2} \)</EquationSource> </InlineEquation> and |<i>μ</i>| ≲ 1.2 <i>μ</i><sub><i>N</i></sub> for spin-<InlineEquation ID="IEq10"> <EquationSource Format="MATHML"><math display="inline"> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> </math></EquationSource> <EquationSource Format="TEX">\( \frac{3}{2} \)</EquationSource> </InlineEquation>), consistent with the heavy-quark spin symmetry limit. Axial-vector diquark currents produce larger, flavor-sensitive moments with sign reversals governed by the <i>e</i><sub><i>u</i></sub>/<i>e</i><sub><i>d</i></sub> = –2 charge ratio. For <InlineEquation ID="IEq11"> <EquationSource Format="MATHML"><math display="inline"> <mi mathvariant="script">Q</mi> </math></EquationSource> <EquationSource Format="TEX">\( \mathcal{Q} \)</EquationSource> </InlineEquation>, scalar-diquark currents give oblate deformations (<i>Q</i><sub>0</sub> ≈ –2.0 × 10<sup>−2</sup> fm<sup>2</sup>) dominated by the charm sector, while two-axial-vector-diquark currents predict prolate values up to <i>Q</i><sub>0</sub> = +8.0 × 10<sup>−2</sup> fm<sup>2</sup>, with a sign reversal for <InlineEquation ID="IEq12"> <EquationSource Format="MATHML"><math display="inline"> <mfenced close="]" open="["> <mi mathvariant="italic">su</mi> </mfenced> <mfenced close="]" open="["> <mi mathvariant="italic">uc</mi> </mfenced> <mover accent="true"> <mi>c</mi> <mo stretchy="true">¯</mo> </mover> </math></EquationSource> <EquationSource Format="TEX">\( \left[ su\right]\left[ uc\right]\overline{c} \)</EquationSource> </InlineEquation> in two currents. Currents with scalar antiquark coupling yield a topology-independent octupole value <InlineEquation ID="IEq13"> <EquationSource Format="MATHML"><math display="inline"> <mi mathvariant="script">O</mi> </math></EquationSource> <EquationSource Format="TEX">\( \mathcal{O} \)</EquationSource> </InlineEquation> ≈ –0.25 × 10<sup>−3</sup> fm<sup>3</sup>, which may serve as a lattice QCD benchmark. Comparison with constituent quark model predictions identifies four qualitative discriminants: |<i>μ</i>| ≳ 3 <i>μ</i><sub><i>N</i></sub> in the spin-<InlineEquation ID="IEq14"> <EquationSource Format="MATHML"><math display="inline"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </math></EquationSource> <EquationSource Format="TEX">\( \frac{1}{2} \)</EquationSource> </InlineEquation> sector; the sign of <i>μ</i> for the <InlineEquation ID="IEq15"> <EquationSource Format="MATHML"><math display="inline"> <mfenced close="]" open="["> <mi mathvariant="italic">su</mi> </mfenced> <mfenced close="]" open="["> <mi mathvariant="italic">uc</mi> </mfenced> <mover accent="true"> <mi>c</mi> <mo stretchy="true">¯</mo> </mover> </math></EquationSource> <EquationSource Format="TEX">\( \left[ su\right]\left[ uc\right]\overline{c} \)</EquationSource> </InlineEquation> state in the spin-<InlineEquation ID="IEq16"> <EquationSource Format="MATHML"><math display="inline"> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> </math></EquationSource> <EquationSource Format="TEX">\( \frac{3}{2} \)</EquationSource> </InlineEquation> sector; a non-zero <InlineEquation ID="IEq17"> <EquationSource Format="MATHML"><math display="inline"> <mi mathvariant="script">Q</mi> </math></EquationSource> <EquationSource Format="TEX">\( \mathcal{Q} \)</EquationSource> </InlineEquation>, which vanishes in the <i>S</i>-wave molecular approximation; and the sign correlation between <InlineEquation ID="IEq18"> <EquationSource Format="MATHML"><math display="inline"> <mi mathvariant="script">Q</mi> </math></EquationSource> <EquationSource Format="TEX">\( \mathcal{Q} \)</EquationSource> </InlineEquation> and <InlineEquation ID="IEq19"> <EquationSource Format="MATHML"><math display="inline"> <mi mathvariant="script">O</mi> </math></EquationSource> <EquationSource Format="TEX">\( \mathcal{O} \)</EquationSource> </InlineEquation>, which probes the 1/<i>m</i><sub><i>q</i></sub> mass weighting of the magnetization distribution.</p>

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Deciphering the nature of \( {P}_{\psi s}^{\Sigma} \) pentaquarks in the light of their electromagnetic multipole moments

  • Ulaş Özdem

摘要

We calculate the electromagnetic multipole moments of the Σ-type strange hidden-charm pentaquarks P ψs Σ \( {P}_{\psi s}^{\Sigma} \) — which form an isospin triplet with three distinct charge states (Σ+, Σ0, Σ) offering independently measurable electromagnetic properties — using QCD light-cone sum rules, employing six interpolating currents for the spin- 1 2 \( \frac{1}{2} \) channel and seven for the spin- 3 2 \( \frac{3}{2} \) channel, each built from diquark-diquark-antiquark operators with scalar or axial-vector diquark content. We compute the magnetic dipole moment μ for all channels and, for spin- 3 2 \( \frac{3}{2} \) , the electric quadrupole Q \( \mathcal{Q} \) and magnetic octupole O \( \mathcal{O} \) moments, the latter two for the first time for this system, and provide the first systematic quark-flavor decomposition of all three multipole moments for P ψs Σ \( {P}_{\psi s}^{\Sigma} \) . The results show a systematic dependence on the diquark spin. Scalar diquark currents yield charm-sector-dominated, flavor-insensitive moments (μ ∈ [−1.92, −1.21] μN for spin- 1 2 \( \frac{1}{2} \) and |μ| ≲ 1.2 μN for spin- 3 2 \( \frac{3}{2} \) ), consistent with the heavy-quark spin symmetry limit. Axial-vector diquark currents produce larger, flavor-sensitive moments with sign reversals governed by the eu/ed = –2 charge ratio. For Q \( \mathcal{Q} \) , scalar-diquark currents give oblate deformations (Q0 ≈ –2.0 × 10−2 fm2) dominated by the charm sector, while two-axial-vector-diquark currents predict prolate values up to Q0 = +8.0 × 10−2 fm2, with a sign reversal for su uc c ¯ \( \left[ su\right]\left[ uc\right]\overline{c} \) in two currents. Currents with scalar antiquark coupling yield a topology-independent octupole value O \( \mathcal{O} \) ≈ –0.25 × 10−3 fm3, which may serve as a lattice QCD benchmark. Comparison with constituent quark model predictions identifies four qualitative discriminants: |μ| ≳ 3 μN in the spin- 1 2 \( \frac{1}{2} \) sector; the sign of μ for the su uc c ¯ \( \left[ su\right]\left[ uc\right]\overline{c} \) state in the spin- 3 2 \( \frac{3}{2} \) sector; a non-zero Q \( \mathcal{Q} \) , which vanishes in the S-wave molecular approximation; and the sign correlation between Q \( \mathcal{Q} \) and O \( \mathcal{O} \) , which probes the 1/mq mass weighting of the magnetization distribution.