The moments of the single inclusive momentum distribution of hadrons in
QCD jets, are studied in the next-to-modified-leading-log approximation
(NMLLA) including next-to-leading-order (NLO) corrections to the
Article funded by SCOAP3
0$ such that, $k\cdot k_\perp=n\cdot k_\perp=0$. Then, the scalar product $k_1\cdot k_2$ reads: \begin{equation}\label{eq:k1k2} k_\perp^2=2z(1-z)k_1\cdot k_2. \end{equation} Writing now the 4-momenta $k=\left(E,\vec{k}\right)$, $k_1=\left(zE,\vec{k}_1\right)$, $k_2=\left((1-z)E,\vec{k}_2\right)$ one has, $\mid\!\vec{k}_1\!\mid=zE$, $\mid\!\vec{k}_2\!\mid=(1-z)E$ for on-shell and massless partons $k_i^2\approx0$. From energy-momentum conservation: \begin{equation}\label{eq:ksq} k^2=2k_1\cdot k_2=2z(1-z)E^2(1-\cos\theta) \end{equation} such that, replacing eq.~(\ref{eq:ksq}) in~(\ref{eq:k1k2}), one finally obtains: \begin{equation}\label{eq:kperp} k_\perp=2z(1-z)E\sin\frac{\theta}2. \end{equation} In the collinear limit, one is left with $k_\perp\approx z(1-z)Q$, where $Q=E\theta$ is the jet virtuality, or transverse momentum of the jet. \begin{figure} \centering \epsfig{file=kinematics_evoleqs.eps,width=9.5truecm} \vskip 0.15cm ]]>
\lqcd$. The numerical expression for the width (for $n_f=5$ quark flavours) reads: \begin{eqnarray} \sigma(Y)\!\!&=&\!\!0.36499\sqrt{(Y+\lambda)^{3/2}-\lambda^{3/2}} \left\{1-0.299739f_1(Y,\lambda)\frac1{\sqrt{Y+\lambda}}-\left[1.12479f_2(Y,\lambda)\right.\right.\cr &&+\left.\left. 0.0449219f_1^2(Y,\lambda) +\left(0.32239-0.246692\ln(Y+\lambda)\right)f_3(Y,\lambda)\right]\frac1{Y+\lambda}\right\}\,. \end{eqnarray} \paragraph{Skewness.} The NMLLA term of the third DG moment, $K_3$, turns out to vanish like the leading order one~\cite{Dokshitzer-ml-1991za}. According to the definition in eq.~(\ref{eq:moments}), the skewness $s=K_3\sigma^{-3}$ presents an extra subleading term which in this resummation scheme comes from the expansion of the second contribution to $\sigma^{-3}$, proportional to $1/\sqrt{(Y+\lambda)}$, as written in eq.~(\ref{eq:sigmaminus3}) of appendix~\ref{app:moments}, such that \begin{equation} \label{eq:skewness} s(Y,\lambda)=-\frac{a_1}{16} \frac{\left(\frac{144N_c}{\beta_0}\right)^{1/4}} {\sqrt{(Y+\lambda)^{3/2}-\lambda^{3/2}}}\left[1- \frac{\beta_0}{64}f_1(Y,\lambda)\sqrt{\frac{16N_c}{\beta_0(Y+\lambda)}}\right]. \end{equation} In~\cite{Fong-ml-1990nt}, only the first term of this expression was provided, the subleading contribution given here is thus new. This NMLLA+NLO$^{*}$ correction to eq.~(\ref{eq:skewness}) increases the skewness of the distribution, while for increasing $\lambda$ it should decrease again, thus revealing two competing effects. The net result is a displacement of the tails of the HBP distribution downwards to the left and upwards to the right from the peak position and depending on the sign given by both effects (figure~\ref{fig:DG}). The final numerical expression for the skewness (for $n_f=5$ quark flavours) reads: \begin{equation} s(Y)=-\frac{1.94704}{\sqrt{(Y+\lambda)^{3/2}-\lambda^{3/2}}} \left[1-0.299739f_1(Y,\lambda)\frac1{\sqrt{Y+\lambda}}\right]. \end{equation} \paragraph{Kurtosis.} The evolution of the kurtosis follows from the expressions for the fourth DG moment, given in eqs.~(\ref{eq:K4}) and~(\ref{eq:sigmaminus4}) of appendix~\ref{app:moments}. As shown in the same appendix, the proper Taylor expansion in powers of $(1/\sqrt{Y+\lambda})$ which keeps trace of higher-order corrections and leads to: \begin{eqnarray} k(Y,\lambda)\!\!&=&\!\!-\frac{27}{5(Y\!+\!\lambda)}\sqrt{\frac{\beta_0(Y\!+\!\lambda)}{16N_c}} \frac{1\!-\!\left(\frac{\lambda}{Y\!+\!\lambda}\right)^{5/2}} {\left[1\!-\!\left(\frac{\lambda}{Y\!+\!\lambda}\right)^{3/2}\right]^2} \left\{1\!+\!\frac{\beta_0}{16}(f_1(Y,\lambda)\!-\!\frac53f_4(Y,\lambda))\sqrt{\frac{16N_c}{\beta_0(Y\!+\!\lambda)}}\right.\cr &&+\left.\left[\left(\frac{25}{24}f_5(Y,\lambda) -\frac94f_2(Y,\lambda)\right)a_2+\frac{a_1^2}{256N_c^2} \left(9f_2(Y,\lambda)-\frac{25}{2}f_5(Y,\lambda)\right)\right.\right.\cr &&+\left.\left.\!\frac{a_1\beta_0}{256N_c^2}(6f_2(Y,\lambda)\!-\!5f_5(Y,\lambda)) \!+\!\frac{\beta_0^2}{256N_c^2}\bigg(\!-\frac{3}4f_2(Y,\lambda)\!+\!\frac{54}8f_1^2(Y,\lambda) \!+\!\frac{275}{24}f_5(Y,\lambda)\right.\right.\cr &&-\left.\left.15f_1(Y,\lambda)f_4(Y,\lambda)\bigg)+\frac{5\beta_1}{96\beta_0} \left(\ln2(Y+\lambda)-\frac{2}{3}\right)f_6(Y,\lambda)\right.\right.\cr &&-\left.\left.\frac{\beta_1}{16\beta_0}(\ln2(Y+\lambda)-2)f_3(Y,\lambda)\right] \frac{16N_c}{\beta_0(Y+\lambda)}\right\}, \label{eq:kurtosis} \end{eqnarray} where the functions $f_i$ can be again found in appendix~\ref{app:moments}. The new NMLLA+NLO$^{*}$ correction for the kurtosis affects the distribution by making it smoother in the tails and wider in the hump region. The final numerical expression for the kurtosis (for $n_f=5$ quark flavours) reads: \begin{eqnarray} k(Y)\!\!&=&\!\!-\frac{2.15812}{\sqrt{Y+\lambda}}\frac{1-\left(\frac{\lambda}{Y+\lambda}\right)^{5/2}} {\left[1-\left(\frac{\lambda}{Y+\lambda}\right)^{3/2}\right]^2}\left\{1+\left[1.19896f_1(Y,\lambda)- 1.99826f_4(Y,\lambda)\right]\frac1{\sqrt{Y+\lambda}}\right.\cr &&+\left.\left[1.07813f_1^2(Y,\lambda)+4.49915f_2(Y,\lambda)+1.28956f_3(Y,\lambda) -2.39583f_1(Y,\lambda)f_4(Y,\lambda)\right.\right.\cr &&-\left.\left.3.76231f_5(Y,\lambda)+0.0217751f_6(Y,\lambda)\right.\right.\cr &&-\left.\left.(0.986767f_3(Y,\lambda) -0.822306f_6(Y,\lambda))\ln(Y+\lambda)\right]\frac1{Y+\lambda}\right\}. \end{eqnarray} \paragraph{Final DG expression.} The final expression of the DG parametrisation of the single inclusive distribution of soft hadrons inside gluon and quark jets, eq.~(\ref{eq:xDg}), can be obtained summing all its individually-derived NMLLA$+$NLO$^*$-resummed components: the mean multiplicity ${\cal N}(Y,\lambda)$ eq.~(\ref{eq:meanmultip}), the mean peak position $\bar\xi(Y,\lambda)$ eq.~(\ref{eq:sxipmlla}), the dispersion $\sigma(Y,\lambda)$ eq.~(\ref{eq:sigma}), the skewness eq.~$s(Y,\lambda)$~(\ref{eq:skewness}), and kurtosis $k(Y,\lambda)$ eq.~(\ref{eq:kurtosis}). In figure~\ref{fig:DGmllavsnmlla}, we display the resulting DG for two different values of the hadronisation parameter $\lambda=1.4$ ($Y=5.8$, $Q_0=1$\,GeV, $\lqcd=0.25$\,GeV) and $\lambda=2.0$ ($Y=5.2$, $Q_0=1$\,GeV, $\lqcd=0.25$\,GeV) for a jet of virtuality $Q=350$\,GeV and reconstructed jet energy $E=500$\,GeV inside a radius cone $\theta=0.7$. The distribution is compared to the corresponding MLLA predictions with the Fong-Webber results from~\cite{Fong-ml-1990nt} after setting to zero all terms proportional to $1/Y$ in the same expressions. \begin{figure} \centering \epsfig{file=nmlla_vs_mlla_lambda14.eps,width=0.49\textwidth} \epsfig{file=nmlla_vs_mlla_lambda2.eps, width=0.49\textwidth} ]]>