Quark and Gluon Helicity Evolution at Small $x$: Revised and Updated
Fuente:
arXiv
Saved in:
| Main Authors: | , , , |
|---|---|
| Format: | Preprint |
| Published: |
2022
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866909293120323584 |
|---|---|
| author | Cougoulic, Florian Kovchegov, Yuri V. Tarasov, Andrey Tawabutr, Yossathorn |
| author_facet | Cougoulic, Florian Kovchegov, Yuri V. Tarasov, Andrey Tawabutr, Yossathorn |
| contents | We revisit the problem of small Bjorken-$x$ evolution of the gluon and flavor-singlet quark helicity distributions in the shock wave ($s$-channel) formalism. Earlier works on the subject in the same framework resulted in an evolution equation for the gluon field-strength $F^{12}$ and quark "axial current" ${\barψ}γ^+γ^5ψ$ operators (sandwiched between the appropriate light-cone Wilson lines) in the double-logarithmic approximation (DLA: summing powers of $α_s\,\ln^2(1/x)$ with $α_s$ the strong coupling constant). In this work, we observe that an important mixing of the above operators with another gluon operator, $\overleftarrow{D}^i\,{D}^i$, also sandwiched between the light-cone Wilson lines (with the repeated index $i=1,2$ summed over), was missing in the previous works. This operator has the physical meaning of the sub-eikonal (covariant) phase: its contribution to helicity evolution is shown to be proportional to another sub-eikonal operator, $D^i-\overleftarrow{D}^i$, which is related to the Jaffe-Manohar polarized gluon distribution. In this work we include this operator into small-$x$ helicity evolution, and construct a novel evolution mixing all three operators ($D^i-\overleftarrow{D}^i$, $F^{12}$, and ${\barψ}γ^+γ^5ψ$), generalizing the previous results. We also construct closed DLA evolution equations in the large-$N_c$ and large-$N_c\& N_f$ limits, with $N_c$ and $N_f$ the numbers of quark colors and flavors, respectively. Solving the large-$N_c$ equations numerically we obtain the following small-$x$ asymptotics of the quark and gluon helicity distributions $ΔΣ$ and $ΔG$, along with the $g_1$ structure function, \[ΔΣ(x,Q^2)\simΔG(x,Q^2)\sim g_1(x,Q^2)\sim\left(\frac{1}{x}\right)^{3.66\,\sqrt{\frac{α_s\,N_c}{2π}}},\] in complete agreement with the earlier work by Bartels, Ermolaev and Ryskin. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2204_11898 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Quark and Gluon Helicity Evolution at Small $x$: Revised and Updated Cougoulic, Florian Kovchegov, Yuri V. Tarasov, Andrey Tawabutr, Yossathorn High Energy Physics - Phenomenology High Energy Physics - Experiment Nuclear Experiment Nuclear Theory We revisit the problem of small Bjorken-$x$ evolution of the gluon and flavor-singlet quark helicity distributions in the shock wave ($s$-channel) formalism. Earlier works on the subject in the same framework resulted in an evolution equation for the gluon field-strength $F^{12}$ and quark "axial current" ${\barψ}γ^+γ^5ψ$ operators (sandwiched between the appropriate light-cone Wilson lines) in the double-logarithmic approximation (DLA: summing powers of $α_s\,\ln^2(1/x)$ with $α_s$ the strong coupling constant). In this work, we observe that an important mixing of the above operators with another gluon operator, $\overleftarrow{D}^i\,{D}^i$, also sandwiched between the light-cone Wilson lines (with the repeated index $i=1,2$ summed over), was missing in the previous works. This operator has the physical meaning of the sub-eikonal (covariant) phase: its contribution to helicity evolution is shown to be proportional to another sub-eikonal operator, $D^i-\overleftarrow{D}^i$, which is related to the Jaffe-Manohar polarized gluon distribution. In this work we include this operator into small-$x$ helicity evolution, and construct a novel evolution mixing all three operators ($D^i-\overleftarrow{D}^i$, $F^{12}$, and ${\barψ}γ^+γ^5ψ$), generalizing the previous results. We also construct closed DLA evolution equations in the large-$N_c$ and large-$N_c\& N_f$ limits, with $N_c$ and $N_f$ the numbers of quark colors and flavors, respectively. Solving the large-$N_c$ equations numerically we obtain the following small-$x$ asymptotics of the quark and gluon helicity distributions $ΔΣ$ and $ΔG$, along with the $g_1$ structure function, \[ΔΣ(x,Q^2)\simΔG(x,Q^2)\sim g_1(x,Q^2)\sim\left(\frac{1}{x}\right)^{3.66\,\sqrt{\frac{α_s\,N_c}{2π}}},\] in complete agreement with the earlier work by Bartels, Ermolaev and Ryskin. |
| title | Quark and Gluon Helicity Evolution at Small $x$: Revised and Updated |
| topic | High Energy Physics - Phenomenology High Energy Physics - Experiment Nuclear Experiment Nuclear Theory |
| url | https://arxiv.org/abs/2204.11898 |