1 | \section {Kinky string excitation.} |
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2 | |
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3 | \hspace{1.0em}Having sampled the configuration of kinky strings |
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4 | we generate outgoing gluon--kink momenta. |
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5 | |
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6 | We assume that kinky strings are produced as result of $gg \rightarrow |
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7 | gg$ hard interactions. Our generation of the outgoing gluons (kinks) |
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8 | momenta is based on the two-jets inclusive production cross section: |
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9 | \begin{equation} |
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10 | \label{KSE1} \frac{d\sigma_{gg}}{dx_g^{+}dx_g^{-}d\cos{\theta}}= |
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11 | f(x_g^{+},Q^2)f(x_g^{-},Q^2)\frac{d\sigma_{gg}(\hat{s})}{d\cos{\theta}}, |
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12 | \end{equation} |
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13 | where we take $\hat{s}=Q^2=x_g^{+}x_g^{-}s$ and $s$ is the total |
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14 | colliding system center of mass energy squared, which calculated using |
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15 | $x_i$ and $q_{ti}$ and $m^2_i$ string end partons. The value |
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16 | of $s$ should be large enough to produce gluons with the transverse momentum |
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17 | cutoff $Q^2_{0} = 2 \ GeV^2$ choosen. The QCD gluon -- gluon |
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18 | interaction cross section |
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19 | \begin{equation} |
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20 | \label{KSE2} |
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21 | \frac{d\sigma_{gg}(\hat{s})}{d\cos{\theta}} = \frac{9\pi |
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22 | \alpha^2_s(Q^2)}{32s}\frac{(3+\cos^2{\theta})^3}{(1-\cos^2{\theta})^2} |
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23 | \end{equation} |
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24 | was calculated in the Born approximation \cite{CKR77}. The $\theta$ is |
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25 | the scattering angle in the center of mass of the parton-parton system |
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26 | and $-z_0 < \cos{\theta} < z_0$ with |
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27 | \begin{equation} |
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28 | \label{KSE3} z_0 = \sqrt{1 - \frac{4Q^2_0}{sx^{+}x^{-}}}. |
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29 | \end{equation} |
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30 | The QCD running coupling constant |
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31 | \begin{equation} |
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32 | \label{KSE4}\alpha_s(Q^2) = \frac{12 \pi}{25 \ln{(Q^2/\Lambda^2)}}, |
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33 | \end{equation} |
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34 | which is corresponding to four flavours and $\Lambda^2 = 0.01$ GeV$^2$ |
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35 | is taken. |
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36 | |
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37 | $f(x,Q^2)$ is the momentum fraction distribution of gluons in hadron. It |
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38 | is choosen from\cite{CKMT95}: |
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39 | \begin{equation} |
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40 | \label{KSE5} |
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41 | f(x,Q^2) = A(Q^2)\frac{x^{-\Delta(Q^2)}}{x}(1 - x)^{n(Q^2)+3} |
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42 | \end{equation} |
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43 | with parameters $\Delta(Q^2) = \Delta (1 + \frac{2Q^2}{Q^2 + 1.12})$ and |
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44 | $n(Q^2) = \frac{3}{2}(1+\frac{Q^2}{Q^2 + 3.55})$ and $A(Q^2)$ is |
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45 | calculated from energy-momentum conservation sum rule. At $Q^2_{0} = 2 |
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46 | \ GeV^2$ |
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47 | \begin{equation} |
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48 | \label{KSE6} |
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49 | f(x,Q^2_{0}) = 1.71\frac{x^{-0.18}}{x}(1 - x)^{5}. |
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50 | \end{equation} |
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51 | |
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52 | Thus the MC procedure to build the kinky strings can be outlined as |
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53 | follows: |
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54 | % |
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55 | \begin{itemize} |
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56 | |
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57 | \item Sample $x_i$, ${\bf q_{it}}$ and $m^2_i$, where $i = 1,2,...,2n$, |
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58 | for partons, which will be on the $2n$ string ends for both the soft |
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59 | and kinky strings. |
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60 | |
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61 | \item For each pair of kinky string calculate total center of mass energy |
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62 | $s$ and sample $x^{+}_{min} < x^{+}$ and $x^{-}_{min} < x^{-}$, where |
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63 | $x_{min} = 2Q_{0}/\sqrt{s}$, |
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64 | for ingoing gluon momenta |
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65 | using gluon distribution function defined by Eq.($\ref{KSE6}$) at |
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66 | $Q^2_0 = 2$ GeV$^2$. |
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67 | |
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68 | \item Sample the outgoing gluon center of mass scattering angle $\theta$ |
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69 | using Eq. ($\ref{KSE2}$). |
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70 | |
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71 | \item For each kinky string recalculate parton string end energies and |
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72 | momenta. |
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73 | |
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74 | \end{itemize} |
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75 | This procedure should be improved taking into account initial and final state |
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76 | gluon radiation. |
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