The transverse polarization of $$\Lambda $$ hyperons in $$e^+e^-\rightarrow \Lambda ^\uparrow h X$$ processes within TMD factorization
Hui Li, Xiaoyu Wang, Yongliang Yang, Zhun Lu
Abstract
Abstract We investigate the transverse polarization of the $$\Lambda $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Λ</mml:mi> </mml:math> hyperon in the processes $$e^+e^-\rightarrow \Lambda ^\uparrow \pi ^\pm X$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mi>e</mml:mi> <mml:mo>+</mml:mo> </mml:msup> <mml:msup> <mml:mi>e</mml:mi> <mml:mo>-</mml:mo> </mml:msup> <mml:mo>→</mml:mo> <mml:msup> <mml:mi>Λ</mml:mi> <mml:mo>↑</mml:mo> </mml:msup> <mml:msup> <mml:mi>π</mml:mi> <mml:mo>±</mml:mo> </mml:msup> <mml:mi>X</mml:mi> </mml:mrow> </mml:math> and $$e^+e^-\rightarrow \Lambda ^\uparrow K^\pm X$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mi>e</mml:mi> <mml:mo>+</mml:mo> </mml:msup> <mml:msup> <mml:mi>e</mml:mi> <mml:mo>-</mml:mo> </mml:msup> <mml:mo>→</mml:mo> <mml:msup> <mml:mi>Λ</mml:mi> <mml:mo>↑</mml:mo> </mml:msup> <mml:msup> <mml:mi>K</mml:mi> <mml:mo>±</mml:mo> </mml:msup> <mml:mi>X</mml:mi> </mml:mrow> </mml:math> within the framework of the transverse momentum dependent (TMD) factorization. The transverse polarization is contributed by the convolution of the transversely polarizing fragmentation function (PFF) $$D_{1T}^\perp $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>D</mml:mi> <mml:mrow> <mml:mn>1</mml:mn> <mml:mi>T</mml:mi> </mml:mrow> <mml:mo>⊥</mml:mo> </mml:msubsup> </mml:math> of the lambda hyperon and the unpolarized fragmentation function $$D_1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>D</mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:math> of pion/kaon. We adopt the spectator diquark model result for $$D_{1T}^{\perp }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>D</mml:mi> <mml:mrow> <mml:mn>1</mml:mn> <mml:mi>T</mml:mi> </mml:mrow> <mml:mo>⊥</mml:mo> </mml:msubsup> </mml:math> to numerically estimate the transverse polarization in $$e^+e^-\rightarrow \Lambda ^\uparrow h X$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mi>e</mml:mi> <mml:mo>+</mml:mo> </mml:msup> <mml:msup> <mml:mi>e</mml:mi> <mml:mo>-</mml:mo> </mml:msup> <mml:mo>→</mml:mo> <mml:msup> <mml:mi>Λ</mml:mi> <mml:mo>↑</mml:mo> </mml:msup> <mml:mi>h</mml:mi> <mml:mi>X</mml:mi> </mml:mrow> </mml:math> process at the kinematical region of Belle Collaboration. To implement the TMD evolution formalism of the fragmentation functions, we apply two different parametrizations on the nonperturbative Sudakov form factors associated with the fragmentation functions of the $$\Lambda $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Λ</mml:mi> </mml:math> , pion and kaon. It is found that our prediction on the polarization in the $$\Lambda \pi ^+$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>Λ</mml:mi> <mml:msup> <mml:mi>π</mml:mi> <mml:mo>+</mml:mo> </mml:msup> </mml:mrow> </mml:math> production and $${\bar{\Lambda }} \pi ^-$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mover> <mml:mrow> <mml:mi>Λ</mml:mi> </mml:mrow> <mml:mrow> <mml:mo>¯</mml:mo> </mml:mrow> </mml:mover> <mml:msup> <mml:mi>π</mml:mi> <mml:mo>-</mml:mo> </mml:msup> </mml:mrow> </mml:math> is consistent with the recent Belle measurement in size and sign, while the model predictions on the polarizations in $$\Lambda \pi ^-$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>Λ</mml:mi> <mml:msup> <mml:mi>π</mml:mi> <mml:mo>-</mml:mo> </mml:msup> </mml:mrow> </mml:math> and $$\Lambda K^\pm $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>Λ</mml:mi> <mml:msup> <mml:mi>K</mml:mi> <mml:mo>±</mml:mo> </mml:msup> </mml:mrow> </mml:math> productions show strong disagreement with the Belle data. The reason for the discrepancies is discussed and possible approaches to improve the calculation in the future are also discussed.