Within the framework of nonrelativistic QCD (NRQCD) factorization, and based on the Collins-Soper operator definition of fragmentation functions, we present a systematic calculation of the fragmentation functions for a heavy quark fragmenting into color-singlet P-wave quarkonium states. After reproducing and confirming the known leading-order results, we further compute the relativistic corrections up to order <InlineEquation ID="IEq1"> <EquationSource Format="MATHML"><math display="inline"> <mi mathvariant="script">O</mi> <mfenced close=")" open="("> <msup> <mi>v</mi> <mn>2</mn> </msup> </mfenced> </math></EquationSource> <EquationSource Format="TEX">$ \mathcal{O}\left({v}^2\right) $</EquationSource> </InlineEquation>. Our analysis applies both to quarkonium systems composed of heavy quarks with the same flavor and to Bc-type mesons formed by heavy quarks of different flavors. Numerical results show that, for all color-singlet P-wave channels, the <InlineEquation ID="IEq2"> <EquationSource Format="MATHML"><math display="inline"> <mi mathvariant="script">O</mi> <mfenced close=")" open="("> <msup> <mi>v</mi> <mn>2</mn> </msup> </mfenced> </math></EquationSource> <EquationSource Format="TEX">$ \mathcal{O}\left({v}^2\right) $</EquationSource> </InlineEquation> relativistic corrections give sizable negative contributions over most of the momentum-fraction z region. We further compute inclusive cross sections for P-wave quarkonium plus charmed hadrons in e+e− annihilation via the single photon process up to <InlineEquation ID="IEq3"> <EquationSource Format="MATHML"><math display="inline"> <mi mathvariant="script">O</mi> <mfenced close=")" open="("> <msup> <mi>v</mi> <mn>2</mn> </msup> </mfenced> </math></EquationSource> <EquationSource Format="TEX">$ \mathcal{O}\left({v}^2\right) $</EquationSource> </InlineEquation> by applying our obtained fragmentation functions, and the resulting predictions are consistent with the full fixed-order results in the high-energy region.

Order-v² relativistic corrections to heavy-quark fragmentation into P-wave quarkonium states

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2026年4月15日

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Sai Cui,
Sheng-Juan Jiang,
Guang-Zhi Xu,
Kui-Yong Liu

摘要

Within the framework of nonrelativistic QCD (NRQCD) factorization, and based on the Collins-Soper operator definition of fragmentation functions, we present a systematic calculation of the fragmentation functions for a heavy quark fragmenting into color-singlet P-wave quarkonium states. After reproducing and confirming the known leading-order results, we further compute the relativistic corrections up to order $O (v^{2})$ $ \mathcal{O}\left({v}^2\right) $ . Our analysis applies both to quarkonium systems composed of heavy quarks with the same flavor and to B_c-type mesons formed by heavy quarks of different flavors. Numerical results show that, for all color-singlet P-wave channels, the $O (v^{2})$ $ \mathcal{O}\left({v}^2\right) $ relativistic corrections give sizable negative contributions over most of the momentum-fraction z region. We further compute inclusive cross sections for P-wave quarkonium plus charmed hadrons in e⁺e⁻ annihilation via the single photon process up to $O (v^{2})$ $ \mathcal{O}\left({v}^2\right) $ by applying our obtained fragmentation functions, and the resulting predictions are consistent with the full fixed-order results in the high-energy region.

Order-v2 relativistic corrections to heavy-quark fragmentation into P-wave quarkonium states

摘要

Order-v² relativistic corrections to heavy-quark fragmentation into P-wave quarkonium states