<p>We consider QCD with <i>n</i><sub><i>f</i></sub> = 0 and <i>n</i><sub><i>f</i></sub> = 3 light quarks. We present the most up-to-date determinations for the normalizations of the leading renormalons of the pole mass, the singlet static potential, the octet static potential, and the gluelump energy. These read <InlineEquation ID="IEq1"> <EquationSource Format="MATHML"><math display="inline"> <msubsup> <mi>Z</mi> <mi>m</mi> <mover accent="true"> <mi>MS</mi> <mo stretchy="true">¯</mo> </mover> </msubsup> </math></EquationSource> <EquationSource Format="TEX">\( {Z}_m^{\overline{\textrm{MS}}} \)</EquationSource> </InlineEquation> = <InlineEquation ID="IEq2"> <EquationSource Format="MATHML"><math display="inline"> <mo>−</mo> <msubsup> <mi>Z</mi> <msub> <mi>V</mi> <mi>s</mi> </msub> <mover accent="true"> <mi>MS</mi> <mo stretchy="true">¯</mo> </mover> </msubsup> <mo>/</mo> <mn>2</mn> </math></EquationSource> <EquationSource Format="TEX">\( -{Z}_{V_s}^{\overline{\textrm{MS}}}/2 \)</EquationSource> </InlineEquation> = {0.604(17), 0.551(20)}, <InlineEquation ID="IEq3"> <EquationSource Format="MATHML"><math display="inline"> <msubsup> <mi>Z</mi> <msub> <mi>V</mi> <mi>o</mi> </msub> <mover accent="true"> <mi>MS</mi> <mo stretchy="true">¯</mo> </mover> </msubsup> </math></EquationSource> <EquationSource Format="TEX">\( {Z}_{V_o}^{\overline{\textrm{MS}}} \)</EquationSource> </InlineEquation> = {0.136(8), 0.121(13)}, and <InlineEquation ID="IEq4"> <EquationSource Format="MATHML"><math display="inline"> <msubsup> <mi>Z</mi> <mi>A</mi> <mover accent="true"> <mi>MS</mi> <mo stretchy="true">¯</mo> </mover> </msubsup> </math></EquationSource> <EquationSource Format="TEX">\( {Z}_A^{\overline{\textrm{MS}}} \)</EquationSource> </InlineEquation> = {–1.343(36), –1.224(43)}, for <i>n</i><sub><i>f</i></sub> = 0 and <i>n</i><sub><i>f</i></sub> = 3, respectively. For <i>n</i><sub><i>f</i></sub> = 0, we obtain two independent renormalization group invariant and renormalization scale independent determinations of the energy of the ground state gluelump in the principal value summation scheme: <InlineEquation ID="IEq5"> <EquationSource Format="MATHML"><math display="inline"> <msubsup> <mi mathvariant="normal">Λ</mi> <mi>B</mi> <mi>PV</mi> </msubsup> <mo>=</mo> <mn>2.47</mn> <mfenced close=")" open="("> <mn>9</mn> </mfenced> <msubsup> <mi>r</mi> <mn>0</mn> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msubsup> </math></EquationSource> <EquationSource Format="TEX">\( {\Lambda}_B^{\textrm{PV}}=2.47(9){r}_0^{-1} \)</EquationSource> </InlineEquation> and <InlineEquation ID="IEq6"> <EquationSource Format="MATHML"><math display="inline"> <msubsup> <mi mathvariant="normal">Λ</mi> <mi>B</mi> <mi>PV</mi> </msubsup> <mo>=</mo> <mn>2.38</mn> <mfenced close=")" open="("> <mn>11</mn> </mfenced> <msubsup> <mi>r</mi> <mn>0</mn> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msubsup> </math></EquationSource> <EquationSource Format="TEX">\( {\Lambda}_B^{\textrm{PV}}=2.38(11){r}_0^{-1} \)</EquationSource> </InlineEquation> where <InlineEquation ID="IEq7"> <EquationSource Format="MATHML"><math display="inline"> <msubsup> <mi>r</mi> <mn>0</mn> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msubsup> </math></EquationSource> <EquationSource Format="TEX">\( {r}_0^{-1} \)</EquationSource> </InlineEquation> ≈ 400 MeV. Averaging these results, we obtain <InlineEquation ID="IEq8"> <EquationSource Format="MATHML"><math display="inline"> <msubsup> <mi mathvariant="normal">Λ</mi> <mi>B</mi> <mi>PV</mi> </msubsup> <mo>=</mo> <mn>2.44</mn> <mfenced close=")" open="("> <mn>7</mn> </mfenced> <msubsup> <mi>r</mi> <mn>0</mn> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msubsup> </math></EquationSource> <EquationSource Format="TEX">\( {\Lambda}_B^{\textrm{PV}}=2.44(7){r}_0^{-1} \)</EquationSource> </InlineEquation>.</p>

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The mass of the lightest gluelump

  • Cesar Ayala,
  • Antonio Pineda

摘要

We consider QCD with nf = 0 and nf = 3 light quarks. We present the most up-to-date determinations for the normalizations of the leading renormalons of the pole mass, the singlet static potential, the octet static potential, and the gluelump energy. These read Z m MS ¯ \( {Z}_m^{\overline{\textrm{MS}}} \) = Z V s MS ¯ / 2 \( -{Z}_{V_s}^{\overline{\textrm{MS}}}/2 \) = {0.604(17), 0.551(20)}, Z V o MS ¯ \( {Z}_{V_o}^{\overline{\textrm{MS}}} \) = {0.136(8), 0.121(13)}, and Z A MS ¯ \( {Z}_A^{\overline{\textrm{MS}}} \) = {–1.343(36), –1.224(43)}, for nf = 0 and nf = 3, respectively. For nf = 0, we obtain two independent renormalization group invariant and renormalization scale independent determinations of the energy of the ground state gluelump in the principal value summation scheme: Λ B PV = 2.47 9 r 0 1 \( {\Lambda}_B^{\textrm{PV}}=2.47(9){r}_0^{-1} \) and Λ B PV = 2.38 11 r 0 1 \( {\Lambda}_B^{\textrm{PV}}=2.38(11){r}_0^{-1} \) where r 0 1 \( {r}_0^{-1} \) ≈ 400 MeV. Averaging these results, we obtain Λ B PV = 2.44 7 r 0 1 \( {\Lambda}_B^{\textrm{PV}}=2.44(7){r}_0^{-1} \) .