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abstract: 'We present the results of a calculation of the intrinsic charm quark contribution to $J/\psi$ plus open charm associated production at large transverse momentum in hadron collisions at high energies. It is shown that the value of the cross section for such process and its energy dependence significantly determine by the choice of charm quark distribution function. We find that intrinsic charm contribution to inclusive $J/\psi$ hadroproduction is about 1% of the total contribution of all mechanisms.'
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2<span>t</span> 2<span>u</span> 2<span>s</span>

<span>**INTRINSIC CHARM CONTRIBUTION TO $J/\psi$ PLUS OPEN CHARM ASSOCIATED PRODUCTION AT HIGH ENERGIES**</span>

A.P.Martynenko and V.A.Saleev

Samara State University, Samara, 443011, Russia

The study of charmonium production at hadron colliders is presently an active field of theoretical @1 [@2; @3] and experimental researches @4 [@5; @6]. Previous calculations of $J/\psi$ production at large $p_\perp$ in $pp$ and $p\bar p$ collisions have included direct charmonium production via gluon-gluon fusion @1 [@7] and production via decay $B\to J/\psi X$ @2. Recently it was presented the preliminary results of a calculation of the gluon and charm quark fragmentation contribution to $J/\psi$ production at large $p_\perp$ @8. Here we examine the intrinsic charm contribution to $J/\psi$ hadroproduction. Based on perturbative QCD and nonrelativistic quark model, our calculation is the same in part the approach used in @9 for description of pion electroproduction on protons.

The calculation of the intrinsic charm quark contribution to associated $J/\psi$ plus open charm hadroproduction and photoproduction @10 is very important for study of the charm quark distribution function in a proton @11. The study of inclusive $J/\psi$ production is of particular importance because the decay $B\to J/\psi X$ plays a crucial role in the measurement of $b$ quark production. In order to understand this phenomenon, it is necessary to take into consideration the background due to direct production of $J/\psi$. We start from amplitudes, which correspond diagrams in Fig.1. As usual in potential model, the $J/\psi$ is represented as a nonrelativistic quark-antiquark bound system in singlet colour state with specified mass $M=2m$ ($m$ is charm quark mass) and spin-parity $J^p=1^-$. The amplitudes of the $J/\psi$ production have the following forms: $$M_1=g^3T^cT^aT^a\bar U(q')\hat\varepsilon_{g}\frac{\hat q'-\hat k+m}
 {(k-q')^2-m^2}\gamma^{\mu}\frac{\hat P}{(p-q)^2}
  \gamma_{\mu}U(q),$$ $$M_2=g^3T^aT^aT^c \bar U(q')\gamma^{\mu}\frac{\hat P}{(p+q')^2}
\gamma_{\mu}
\frac{\hat k+\hat q+m}{(k+q)^2-m^2}\hat\varepsilon_{g}U(q),$$ $$M_3=g^3T^aT^cT^a\bar U(q')\gamma^{\mu}\frac{\hat P}{(q'+p)^2}\hat
\varepsilon_g
\frac{\hat p-\hat k+m}{(p-k)^2-m^2}\gamma_{\mu}U(q),$$ $$M_4=g^3T^aT^cT^a\bar U(q')\gamma^{\mu}
\frac{\hat k-\hat p+m} {(p-k)^2-m^2}\varepsilon_{g}
\frac{\hat P}{(p-q)^2}\gamma_{\mu}U(q),$$ $$M_5=-ig^3f^{bac}T^bT^a\bar U(q')\gamma^{\rho}\frac{\hat P}{(p+q')^2}
 \frac{\gamma^{\sigma}}{(q-p)^2}U(q)C_{\mu\sigma\rho}\varepsilon^
  {\mu}_g,$$ where $$C_{\mu\sigma\rho}=(q+q')_{\mu}g_{\sigma\rho}-(p+k+q')_{\sigma}
  g_{\mu\rho}+(k-q+p)_{\rho}g_{\mu\sigma}.$$ In these formulas: $$\hat P=\frac{F_c}{\sqrt 2}A\hat\varepsilon_J(\hat p+m),$$ $A=\Psi(0)/\sqrt m$, $F_c=\delta^{kr}/\sqrt{3}$, k and r are colour indexes of charm quarks, $T^a=\lambda^a/2$. It is well known that $\Psi(0)$, which is equal to $J/\psi$ wave function at the origin, can be extracted in the lowest order of perturbative QCD from the leptonic decay width of the $J/\psi$: $$\Gamma_{ee}=4\pi e_q^2\alpha^2\frac{|\Psi(0)|^2}{m^2}.$$ We shall put in our calculation $\Gamma_{ee}=5.4$ KeV @12 and $m=
1.55$ GeV.

If we average and sum over spins and colours of initial and final particles, we obtain the expression for square of matrix element: $$|\bar M|^2=B_{gc}\sum_{i\le j=1}^{5}C_{ij}K_{ij}(\s2,\t2,\u2),$$ where $$B_{gc}=\frac{\pi^2\alpha_s^3\Gamma_{ee}}
    {16\alpha^2m},$$ $$\tilde s=\hat s/m^2,\qquad \tilde t=\hat t/m^2,\qquad \tilde  u=\hat
u/m^2,$$ $\hat s, \hat t, \hat u$ are usual Mandelstam variables and $\hat s+\hat t+\hat u=6m^2$. The explicit analytical formulae for functions $K_{ij}$ are: $$\begin{aligned}
      K_{11}&=&-4(2\s2\t2-2\s2+\t2^2\u2-4\t2^2-8\t2\u2+14
      \t2+7\u2-106)\nonumber\\
     &&/(\t2^4-4\t2^3+6\t2^2-4\t2+1)\end{aligned}$$ $$\begin{aligned}
      K_{12}&=&4(\s2^3-\s2^2\t2-6\s2^2-\s2\t2^2-2\s2\t2\u2+16
      \s2\t2-\s2\u2^2+8\s2\u2\nonumber\\
      && -28\s2+\t2^3-6\t2^2-\t2\u2
      ^2+8\t2\u2-28\t2+4\u2^2-126\u2+276)/\nonumber\\
      && (\s2^2\t2^2-2\s2^2\t2+\s2^2-2\s2\t2^2+4\s2\t2-
      2\s2+\t2^2-2\t2+1)\end{aligned}$$ $$\begin{aligned}
      K_{13}&=&8(\s2^2\t2+\s2^2+2\s2\t2^2+\s2\t2\u2-22\s2\t2+7
      \s2\u2+16\s2+\nonumber\\
&&  2\t2^2\u2-24\t2^2-11\t2\u2+217
 \t2-2\u2^2+41\u2-511)/(\s2\t2^2\u2-4\s2\nonumber\\
&& \t2^2-2\s2\t2\u2+8\s2\t2+\s2\u2-4\s2-\t2^2\u2+4
      \t2^2+2\t2\u2-8\t2-\u2+4)\end{aligned}$$ $$\begin{aligned}
      K_{14}&=&-8(\s2^3+\s2^2\t2-18\s2^2+\s2\t2^2+\s2\t2\u2-16
      \s2\t2-\s2\u2^2-3\s2\u2+\nonumber\\
       &&166\s2-11\t2^2-5\t2\u2+
 115\t2+13\u2^2-65\u2-335)/\nonumber\\
 &&(\t2^3\u2-4
      \t2^3-3\t2^2\u2+12\t2^2+3\t2\u2-12\t2-\u2+4)\end{aligned}$$ $$\begin{aligned}
      K_{22}&=&-4(\s2^2\u2-4\s2^2+2\s2\t2-8\s2\u2+14\s2-2
      \t2+7\u2-106)/\nonumber\\
      &&(\s2^4-4\s2^3+6\s2^2-4\s2+1)\end{aligned}$$ $$\begin{aligned}
      K_{23}&=&-4(\s2^2\t2-11\s2^2+\s2\t2^2+\s2\t2\u2-16\s2\t2-
      5\s2\u2+115\s2+\nonumber\\
      &&\t2^3-18\t2^2-\t2\u2^2-3\t2\u2+
      166\t2+13\u2^2-65\u2-335)/\nonumber\\
        &&(\s2^3\u2-4
      \s2^3-3\s2^2\u2+12\s2^2+3\s2\u2-12\s2-\u2+4)\end{aligned}$$ $$\begin{aligned}
      K_{24}&=&8(2\s2^2\t2+2\s2^2\u2-24\s2^2+\s2\t2^2+\s2\t2
      \u2-22\s2\t2-11\s2\u2+\nonumber\\
     &&  217\s2+\t2^2+7\t2\u2+16
      \t2-2\u2^2+41\u2-511)/(\s2^2\t2\u2-4\s2^2\t2\nonumber\\
     && -\s2^2\u2+4\s2^2-2\s2\t2\u2+8\s2\t2+2\s2
      \u2-8\s2+\t2\u2-4\t2-\u2+4)\end{aligned}$$ $$\begin{aligned}
      K_{33}&=&-16(2\s2^2+2\s2\t2+6\s2\u2-46\s2+\t2^2\u2-10
      \t2^2-\nonumber\\
       &&4\t2\u2+74\t2+2\u2^3-22\u2^2+53\u2-118)/
 (\s2^2\u2^2-8\s2^2\u2+\nonumber\\
   && 16\s2^2-2\s2\u2
      ^2+16\s2\u2-32\s2+\u2^2-8\u2+16)\end{aligned}$$ $$\begin{aligned}
      K_{34}&=&-16(2(\s2^2+\s2\t2-11\s2+\t2^2+2\t2\u2-19\t2+\u2
      ^3-9\u2^2+28\u2+11))/\nonumber\\
       &&(\s2\t2\u2^2-8\s2
      \t2\u2+16\s2\t2-\s2\u2^2+8\s2\u2-16\s2\nonumber\\
       &&-\t2\u2^2+8
      \t2\u2-16\t2+\u2^2-8\u2+16)\end{aligned}$$ $$\begin{aligned}
      K_{44}&=&-16(\s2^2\u2-10\s2^2+2\s2\t2-4\s2\u2+74\s2+2
      \t2^2+6\t2\u2-\nonumber\\
       &&46\t2+2\u2^3-22\u2^2+53\u2-118
      )/(\t2^2\u2^2-8\t2^2\u2+16\t2^2-2\t2\u2^2\nonumber\\
&& +16\t2\u2-32\t2+\u2^2-8\u2+16)\end{aligned}$$ $$\begin{aligned}
      K_{55}&=&4(\s2^3-\s2^2\t2+\s2^2\u2-14\s2^2-\s2\t2
     ^2+2\s2\t2\u2\nonumber\\
&& +20\s2\t2-\s2\u2^2 -8\s2\u2+40\s2+
      \t2^3+\t2^2\u2-14\t2^2\nonumber\\
&& -\t2\u2^2-8\t2\u2+40\t2+2
     \u2^3+4\u2^2-4\u2-168)/\nonumber\\
&&(\s2^2\t2^2-2\s2^2\t2+\s2^2-2\s2\t2^
     2+4\s2\t2-2\s2+\t2^2-2\t2+1)\end{aligned}$$ $$\begin{aligned}
      K_{51}&=&4(s2^3+\s2^2\t2-20\s2^2-\s2\t2^2+2\s2
      \t2\u2-12\s2\t2-\s2\u2^2\nonumber\\
&&       -4\s2\u2+168\s2-\t2^3-2 \t2^2\u2+24\t2^2+\t2\u2^2
          -12\t2\u2-88\t2+14\u2^2-232)\nonumber\\
&&      /(\s2\t2^3-3\s2\t2^2+3\s2\t2-\s2-\t2^3+
      3\t2^2-3\t2+1)\end{aligned}$$ $$\begin{aligned}
      K_{52}&=&-4(\s2^3+\s2^2\t2+2\s2^2\u2-24\s2^2-\s2
      \t2^2-2\s2\t2\u2+12\s2\t2\nonumber\\
&&       -\s2\u2^2+12\s2\u2+88
     \s2-\t2^3+20\t2^2+\t2\u2^2+4\t2\u2-168\t2-14\u2^2
      +232)\nonumber\\
&&      /(\s2^3\t2-\s2^3-3\s2^2\t2+3\s2^2+3
      \s2\t2-3\s2-\t2+1)\end{aligned}$$ $$\begin{aligned}
      K_{53}&=&8(4\s2^2-\s2\t2^2+22\s2\t2+10\s2\u2-
      117\s2+\t2^3+2\t2^2\u2\nonumber\\
&&      -26\t2^2-\t2\u2^2+2\t2\u2
      +69\t2+4\u2^3-29\u2^2+34\u2+80)\nonumber\\
&&  /(\s2^2\t2\u2-4\s2^2\t2-\s2^2\u2+4\s2^2 -2\s2\t2\u2\nonumber\\
&&   +8\s2\t2+2\s2\u2-8\s2+\t2\u2-4\t2-\u2+4)\end{aligned}$$ $$\begin{aligned}
       K_{54}&=&8(\s2^3-\s2^2\t2+2\s2^2\u2-26\s2^2+22
      \s2\t2-\s2\u2^2+2\s2\u2\nonumber\\
&&       +69\s2+4\t2^2+10\t2\u2-
      117\t2+4\u2^3-29\u2^2+34\u2+80)\nonumber\\
&&      /((\s2\t2^2\u2-4\s2\t2^2-2\s2\t2\u2+8\s2\t2 +\s2\u2\nonumber\\
&&      -4\s2-\t2^2\u2+4\t2^2+2\t2\u2-8\t2-\u2+4))\end{aligned}$$ The apropriate colour factors are: $$\begin{aligned}
&&C_{11}=C_{22}=C_{12}=C_{21}=64/9,\nonumber\\
&&C_{33}=C_{44}=C_{34}=C_{43}=1/9,\nonumber\\
&&C_{13}=C_{31}=C_{14}=C_{41}=C_{23}=C_{32}=C_{24}=C_{42}=-8/9,
  \nonumber\\
&&C_{15}=C_{51}=C_{25}=C_{52}=8,\nonumber\\
&&C_{35}=C_{53}=C_{45}=C_{54}=-1, \qquad  C_{55}=-9\end{aligned}$$

The differential cross section for subprocess $g c\to J/\psi c$ can be written as follows: $$\frac{d\hat\sigma}{d\hat t}=\frac{|\bar M|^2}{16\pi(\hat s-m^2)^2}$$

In the general factorization approach of QCD the measurable cross section $\sigma(p p\to J/\psi c X)$ and the partonic cross section $\hat\sigma(g c\to J/\psi c)$ are connected by the following expression: $$\begin{aligned}
\frac{d\sigma}{d^2p_{\perp}dy}&=&\int dx_1\int dx_2 \left[G_p(x_1,Q^2)
 C_p(x_2,Q^2)+(1\Leftrightarrow 2)\right]\nonumber\\
 &&\frac{x_1x_2s}{\pi}
 \delta(\hat s+\hat t+\hat u-6m^2)\frac{d\hat\sigma}{d\hat t}.\end{aligned}$$

Here: $$\hat t=M_J^2+m^2-x_2(M_J^2-t),$$ $$\hat u=M_J^2-x_1(M_J^2-u),$$ $$\hat s=x_1x_2s+m^2,~~~~~ M_\perp=\sqrt{M_J^2+p_\perp^2},~~~~~~
   M_J=2m,$$ $$t=M_J^2-\sqrt{s}M_\perp \exp(y),$$ $$u=M_J^2-\sqrt{s}M_\perp \exp(-y),$$ where $y$ is $J/\psi$ rapidity in s.c.m., $p_\perp$ is $J/\psi$ transverse momentum, $G_p(x,Q^2)$ and $C_p(x,Q^2)$ are gluon and $c$-quark distribution function in a proton (antiproton) at the scale $Q^2=M_\perp^2$.

For comparison we plot in Fig.2 the partonic cross sections for subprocesses $g c\to J/\psi c$ and $g g\to J/\psi g$ versus value of the scaled momentum transfer, $-\hat t/\hat s$, for typical value of the subenergy, $\hat s=50$ GeV$^2$. We observe from Fig.2 that cross section of c-quark–gluon scattering subprocess is larger than gluon–gluon cross section on factor 2-50 in the wide region of $-\hat t/\hat s$. The comparebly large cross section of $cg\to J/\psi c$ subprocess compensates in part the small value of the charm quark structure function in the proton.

Fig.3 shows the result of our calculation for $J/\psi$ with large transverce momentum ($p_T>5$ GeV/c) plus open charm associated production in $pp(p\bar p)$ interaction versus $\sqrt s$. We use in calculations "hard" (nonperturbative) scaling parameterization @13a and two set of "soft" (perturbative) parameterization from refs. @13(DO) and @13b(GRV) for $C_p(x,Q^2)$ structure functions and take the usual value of QCD $K-$factor: $K=2$. Note that the mean value of the proton momentum, which is carried by charm quarks, is approximately equal for all parameterization (0.3-0.5%), and does not contradict data from EMC collaboration on $F_2^c(x,Q^2)$ @14.

We observe from Fig.3 that the contribution for "hard" parameterization @13a dominants at small energy region $\sqrt s<70$ GeV, opposite at more high energies the contribution for "soft" parameterization @13 is large and it grows from .1 nbn at $\sqrt s=100$ GeV to 4 nbn at $\sqrt s=1$ TeV. Such a way we predict sufficiently large measurable cross section for $J/\psi$ plus open charm production at hadron colliders via partonic subprocess $g c\to J/\psi c$. It gives us the opportunity of experimental investigation of the x-dependence of charm quark distribution function in the proton, especcialy at small-x region.

Figs. 4 and 5 show our predictions for $J/\psi 's$ $p_T$ and $y$-spectra at the energy $\sqrt s=630$ GeV in $p\bar p$ interactions. Curver 1 in Figs.4 and 5 is the result of calculation for intrinsic charm contribution, curver 2 is the contribution for direct quarkonium production via gluon-gluon fusion and curver 3 is the contribution for $J/\psi$ from decay $B\to J/\psi X$. Curvers 2 and 3 are taken from ref. @2.

<span>***Acknowledgements.***</span> This research was supported by the Russian Foundation of Basic Research (Grant 93-02-3545) and by State Committee on High Education of Russian Federation (Grant 94-6.7-2015). Authors thank S. Baranov, N. Zotov and A. Likhoded for useful discussions.

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<span>**Figure captions.**</span>

1.  Diagrams used to describe the partonic subprocess $gc\to
     J/\psi c$.

2.  The partonic cross sections as function of $x=-\hat t/\hat s$ at $\hat s=50$ GeV$^2$. The curver 1 is $gc\to J/\psi c$ subprocess, the curver 2 is $gg\to J/\psi g$ subprocess.

3.  The total cross section for $J/\psi$ plus open charm associated production via intrinsic charm in $pp$ or $p\bar p$ collisions. The curvers 1-3 correspond to parameterizations charm quark structure functions from refs.@13a [@13; @13b]

4.  The $p_\perp$ distribution for $J/\psi$ production in $p\bar p$ collisions at $\sqrt s=630$ GeV and $|y|<2$. The data are from @5 . The curver 1 is intrinsic charm quark contribution. The curver 2 is the sum of the direct charmonium production subprocesses. The curver 3 corresponds to $J/\psi$'s of $b\bar b$ origin. The curver 4 is the sum of all contributions.

5.  The $y$ distribution for $J/\psi$ production in $p\bar p$ collisions at $\sqrt s=630$ GeV and $p_\perp>5$ GeV. The same as in Fig. 4.

{'source': 'arXiv_out/hep-ph9407376_extract_9407376'}