<p>Chiral symmetry restoration and deconfinement at larger numbers of light-quark flavors, <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(N_f\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>N</mi> <mi>f</mi> </msub> </math></EquationSource> </InlineEquation>, may impact the properties of light hadrons. In this work, we study several properties of the pion and kaon—such as their masses and other related quantities across a range of <InlineEquation ID="IEq6"> <EquationSource Format="TEX">\(N_f\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>N</mi> <mi>f</mi> </msub> </math></EquationSource> </InlineEquation>. We employ the symmetry-preserving, confining vector–vector flavor-dependent contact interaction (FCI) as an input to the Schwinger–Dyson equation (SDE) and the homogeneous Bethe–Salpeter equation (BSE). In the chiral limit <InlineEquation ID="IEq7"> <EquationSource Format="TEX">\((m_f = 0)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <msub> <mi>m</mi> <mi>f</mi> </msub> <mo>=</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, increasing the number of flavors <InlineEquation ID="IEq8"> <EquationSource Format="TEX">\(N_f\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>N</mi> <mi>f</mi> </msub> </math></EquationSource> </InlineEquation> leads to the restoration of chiral symmetry and deconfinement at a critical number of flavors, <InlineEquation ID="IEq9"> <EquationSource Format="TEX">\(N^{c}_{f} \approx 8\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mi>N</mi> <mi>f</mi> <mi>c</mi> </msubsup> <mo>≈</mo> <mn>8</mn> </mrow> </math></EquationSource> </InlineEquation>, where at above the dress quark mass <InlineEquation ID="IEq10"> <EquationSource Format="TEX">\(M_{0}\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>M</mi> <mn>0</mn> </msub> </math></EquationSource> </InlineEquation> vanish. For <InlineEquation ID="IEq11"> <EquationSource Format="TEX">\(N_f &lt; N_{f}^{c}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>N</mi> <mi>f</mi> </msub> <mo>&lt;</mo> <msubsup> <mi>N</mi> <mrow> <mi>f</mi> </mrow> <mi>c</mi> </msubsup> </mrow> </math></EquationSource> </InlineEquation>, the Nambu-Goldstone boson mass <InlineEquation ID="IEq12"> <EquationSource Format="TEX">\(m^{0}_{GB}\)</EquationSource> <EquationSource Format="MATHML"><math> <msubsup> <mi>m</mi> <mrow> <mi mathvariant="italic">GB</mi> </mrow> <mn>0</mn> </msubsup> </math></EquationSource> </InlineEquation> remains unchanged, signalling the chiral symmetry broken. When <InlineEquation ID="IEq13"> <EquationSource Format="TEX">\(N_f &gt; N_{f}^{c}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>N</mi> <mi>f</mi> </msub> <mo>&gt;</mo> <msubsup> <mi>N</mi> <mrow> <mi>f</mi> </mrow> <mi>c</mi> </msubsup> </mrow> </math></EquationSource> </InlineEquation>, chiral symmetry is fully restored and <InlineEquation ID="IEq14"> <EquationSource Format="TEX">\(m^{0}_{GB}\)</EquationSource> <EquationSource Format="MATHML"><math> <msubsup> <mi>m</mi> <mrow> <mi mathvariant="italic">GB</mi> </mrow> <mn>0</mn> </msubsup> </math></EquationSource> </InlineEquation> rises rapidly, indicating a transition from a bound state to a resonant state. The bound-state dissociation occurs at a critical flavor number <InlineEquation ID="IEq15"> <EquationSource Format="TEX">\(N^{d}_{f}\)</EquationSource> <EquationSource Format="MATHML"><math> <msubsup> <mi>N</mi> <mi>f</mi> <mi>d</mi> </msubsup> </math></EquationSource> </InlineEquation>, which coincides with <InlineEquation ID="IEq16"> <EquationSource Format="TEX">\(N_{f}^c\)</EquationSource> <EquationSource Format="MATHML"><math> <msubsup> <mi>N</mi> <mrow> <mi>f</mi> </mrow> <mi>c</mi> </msubsup> </math></EquationSource> </InlineEquation>, providing a clear indicator of deconfinement as quarks and antiquarks detach from their bound states. On the other hand, when bare quark mass is considered <InlineEquation ID="IEq17"> <EquationSource Format="TEX">\((m_f \ne 0)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <msub> <mi>m</mi> <mi>f</mi> </msub> <mo>≠</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, the chiral symmetry remains explicitly broken and is partially restored at and above <InlineEquation ID="IEq18"> <EquationSource Format="TEX">\(N^{c}_{f} \approx 8.2\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mi>N</mi> <mi>f</mi> <mi>c</mi> </msubsup> <mo>≈</mo> <mn>8.2</mn> </mrow> </math></EquationSource> </InlineEquation>. Pion and kaon masses <InlineEquation ID="IEq19"> <EquationSource Format="TEX">\(m_{(\pi , K)}\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>m</mi> <mrow> <mo stretchy="false">(</mo> <mi>π</mi> <mo>,</mo> <mi>K</mi> <mo stretchy="false">)</mo> </mrow> </msub> </math></EquationSource> </InlineEquation> rises rapidly and dissociate at <InlineEquation ID="IEq20"> <EquationSource Format="TEX">\(N^{d}_{f} =N^{d}_{(\pi ,K)f} \approx 8.2\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mi>N</mi> <mi>f</mi> <mi>d</mi> </msubsup> <mo>=</mo> <msubsup> <mi>N</mi> <mrow> <mo stretchy="false">(</mo> <mi>π</mi> <mo>,</mo> <mi>K</mi> <mo stretchy="false">)</mo> <mi>f</mi> </mrow> <mi>d</mi> </msubsup> <mo>≈</mo> <mn>8.2</mn> </mrow> </math></EquationSource> </InlineEquation>, which separates the bound states from their constituents, indicating a Mott-like dissociation of bound states into their constituents. We also verified the consistency of our findings across different flavors using the Gell-Mann-Oakes-Renner relation.</p>

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\(\pi \)- and K-Mesons Properties for Large \(N_f\)

  • Aftab Ahmad

摘要

Chiral symmetry restoration and deconfinement at larger numbers of light-quark flavors, \(N_f\) N f , may impact the properties of light hadrons. In this work, we study several properties of the pion and kaon—such as their masses and other related quantities across a range of \(N_f\) N f . We employ the symmetry-preserving, confining vector–vector flavor-dependent contact interaction (FCI) as an input to the Schwinger–Dyson equation (SDE) and the homogeneous Bethe–Salpeter equation (BSE). In the chiral limit \((m_f = 0)\) ( m f = 0 ) , increasing the number of flavors \(N_f\) N f leads to the restoration of chiral symmetry and deconfinement at a critical number of flavors, \(N^{c}_{f} \approx 8\) N f c 8 , where at above the dress quark mass \(M_{0}\) M 0 vanish. For \(N_f < N_{f}^{c}\) N f < N f c , the Nambu-Goldstone boson mass \(m^{0}_{GB}\) m GB 0 remains unchanged, signalling the chiral symmetry broken. When \(N_f > N_{f}^{c}\) N f > N f c , chiral symmetry is fully restored and \(m^{0}_{GB}\) m GB 0 rises rapidly, indicating a transition from a bound state to a resonant state. The bound-state dissociation occurs at a critical flavor number \(N^{d}_{f}\) N f d , which coincides with \(N_{f}^c\) N f c , providing a clear indicator of deconfinement as quarks and antiquarks detach from their bound states. On the other hand, when bare quark mass is considered \((m_f \ne 0)\) ( m f 0 ) , the chiral symmetry remains explicitly broken and is partially restored at and above \(N^{c}_{f} \approx 8.2\) N f c 8.2 . Pion and kaon masses \(m_{(\pi , K)}\) m ( π , K ) rises rapidly and dissociate at \(N^{d}_{f} =N^{d}_{(\pi ,K)f} \approx 8.2\) N f d = N ( π , K ) f d 8.2 , which separates the bound states from their constituents, indicating a Mott-like dissociation of bound states into their constituents. We also verified the consistency of our findings across different flavors using the Gell-Mann-Oakes-Renner relation.