小$x$的螺旋度：带回夸克产生的振荡
Helicity at Small $x$: Oscillations Generated by Bringing Back the Quarks
摘要：
我们构造了最近导出的大- $N_c \& N_f$小- $x$螺旋度演化方程的数值解，目的是建立夸克螺旋度分布的小- $x$渐近性，该渐近性超出先前在同一框架中探索的大- $N_c$极限。(Here $N_c$和$N_f$是夸克颜色和味道的数字。）大$N_c$螺旋度演化只涉及胶子，而大$N_c \& N_f$螺旋度演化也包括夸克的贡献.我们发现，加入夸克的演化使夸克的螺旋度分布作为$x$的函数振荡。我们的数值计算结果在大$N_c \& N_f$限制导致$x$依赖的味道单线态夸克螺旋度分布，这是很好的近似\begin{align} ΔΣ(x, Q^2)\bigg|_{\mbox{large-}N_c \& N_f} \sim \left( \frac{1}{x} \right)^{α_h^q} \, \cos \left[ ω_q \, \ln \left( \frac{1}{x} \right) + φ_q \right]. \end{align}的功率$α_h^q$表现出弱$N_f$依赖，并为所有的$N_f$值考虑，仍然非常接近$α_h^q (N_f=0) = (4/\sqrt{3}) \sqrt{α_s N_c/(2 π)}$较早获得的大$N_c$限制。新的振荡频率$ω_q$和相移$φ_q$更强烈地依赖于风味$N_f$的数量（其中$ω_q =0$处于纯胶大- $N_c$极限）。$ΔΣ$的典型振荡周期相当长，跨越许多快度单位。我们推测我们发现的振荡是否与从数据中提取的奇异夸克螺旋度分布中看到的$x$的符号变化有关。
英文摘要：
We construct a numerical solution of the recently-derived large-$N_c \& N_f$ small-$x$ helicity evolution equations with the aim to establish the small-$x$ asymptotics of the quark helicity distribution beyond the large-$N_c$ limit explored previously in the same framework. (Here $N_c$ and $N_f$ are the numbers of quark colors and flavors.) While the large-$N_c$ helicity evolution involves gluons only, the large-$N_c \& N_f$ evolution includes contributions from quarks as well. We find that adding quarks to the evolution makes quark helicity distribution oscillate as a function of $x$. Our numerical results in the large-$N_c \& N_f$ limit lead to the $x$-dependence of the flavor-singlet quark helicity distribution which is well-approximated by \begin{align} ΔΣ(x, Q^2)\bigg|_{\mbox{large-}N_c \& N_f} \sim \left( \frac{1}{x} \right)^{α_h^q} \, \cos \left[ ω_q \, \ln \left( \frac{1}{x} \right) + φ_q \right]. \end{align} The power $α_h^q$ exhibits a weak $N_f$-dependence, and, for all $N_f$ values considered, remains very close to $α_h^q (N_f=0) = (4/\sqrt{3}) \sqrt{α_s N_c/(2 π)}$ obtained earlier in the large-$N_c$ limit. The novel oscillation frequency $ω_q$ and phase shift $φ_q$ depend more strongly on the number of flavors $N_f$ (with $ω_q =0$ in the pure-glue large-$N_c$ limit). The typical period of oscillations for $ΔΣ$ is rather long, spanning many units of rapidity. We speculate whether the oscillations we find are related to the sign variation with $x$ seen in the strange quark helicity distribution extracted from the data.
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