Transparent nuclei and deuteron-gold collisions at relativistic energies

The current normalization of the cross section of inclusive high-$p_T$ particle production in deuteron-gold collisions measured at RHIC relies on Glauber model calculations for the inelastic $d$-Au cross section. These calculations should be corrected for diffraction. Moreover, they miss Gribov’s inelastic shadowing which makes nuclei more transparent (color transparency) and reduces the inelastic cross section. The magnitude of this effect rises with energy and one may anticipate it to affect dramatically the normalization of the RHIC data. We evaluate the inelastic shadowing corrections employing the light-cone dipole formalism which effectively sums up multiple interactions in all orders. We found a rather modest correction factor for the current normalization of data for high-$p_T$ hadron production in $d$-Au collisions. The results of experiments insensitive to diffraction (PHENIX, PHOBOS) should be renormalized by about $20\%$ down, while those which include diffraction (STAR), by only $10\%$. In spite of smallness of the correction it eliminates the Cronin enhancement in the PHENIX data for pions. The largest theoretical uncertainty comes from the part of inelastic shadowing which is related to diffractive gluon radiation or gluon shadowing. Our estimate is adjusted to data for the triple-Pomeron coupling and is small, however, other models do not have such a restriction and predict much stronger gluon shadowing. Thus, one arrives at quite diverse predictions for the correction factor which may be even as small as $K=0.65$. Therefore, one should admit that the current data for high-$p_T$ hadron production in $d$-Au collisions at RHIC cannot exclude in a model independent way a possibility of initial state suppression proposed by Kharzeev-Levin-McLerran. To settle this uncertainty one should directly measure the inelastic $d$-Au cross sections at RHIC. Also, collisions with a tagged spectator nucleon may serve as a sensitive probe for nuclear transparency and inelastic shadowing. We found an illuminating quantum-mechanical effect: the nucleus acts like a lens focusing spectators into a very narrow cone.

Figure 1

Data and calculations [10] for the total neutron-lead cross section as a function of energy. The dashed curve corresponds to the Glauber model, while the solid curve is corrected for Gribov’s inelastic shadowing.

Figure 2

The impact parameter distribution of inelastic deuteron-gold collisions (three upper curves) including diffractive excitations (STAR trigger). Impact parameter $\vec{b}$ corresponds to the center of gravity of the deuteron. The dashed curve corresponds to the Glauber approximation, Eq. (10). The thin solid curve includes inelastic shadowing related to excitation of the valence quark skeleton, Eq. (45). The thick solid curve is final, it includes gluon shadowing as well. The bottom solid thick curve shows the difference between the Glauber and final curves. The dotted curve shows the range of model uncertainty and corresponds to gluon shadowing with $R_G=03$ (see Sec. 6B). All curves are calculated with total cross section ${\tilde{\sigma }}_{tot}^{NN}=51\text{mb}$.

Figure 3

The impact parameter distribution of interacting protons in tagged deuteron-gold collisions with spectator neutrons. The calculation includes diffractive excitations (STAR trigger). Impact parameter $\vec{b}$ corresponds to the proton. The dashed curve represents the Glauber approximation, Eq. (10). The thin solid curve includes inelastic shadowing related to excitation of the valence quark skeleton, Eq. (45). The thick solid curve includes gluon shadowing as well. All curves are calculated with total cross section ${\tilde{\sigma }}_{tot}^{NN}=51\text{mb}$.

Figure 4

The impact parameter distribution of spectator neutrons in tagged deuteron-gold collisions with interacting protons. The calculation includes diffractive excitations (STAR trigger). Impact parameter $\vec{b}$ corresponds to the proton. The dashed curve represents the Glauber approximation, Eq. (10). The thin solid curve includes inelastic shadowing related to excitation of the valence quark skeleton, Eq. (45). The thick solid curve includes gluon shadowing as well. All curves are calculated with total cross section ${\tilde{\sigma }}_{tot}^{NN}=51\text{mb}$.

Figure 5

Transverse momentum distribution of spectator neutrons in the tagged reaction $d+\text{Au}\to n+X$ (solid curve), and in the projectile deuteron (dashed curve). The inelastic reaction $p+\text{Au}\to X$ is assumed to include diffraction (STAR experiment). The calculations are performed in the Glauber approximation, Eq. (21).

Figure 6

Same as in Fig. 5, but for a central $\left(b=0\right)$ proton-gold collision, accompanied by a spectator neutron.

Figure 7

Diagonal and off-diagonal diffractive multiple interactions of the projectile hadron in intermediate state.

Figure 8

Ratio of the gluonic inelastic shadowing correction (minimal bias) to the total nuclear cross section as function of center of mass energy $\sqrt{s}$.

Figure 9

Solid line is the correction factor, Eq. (81), for inelastic shadowing to the number of participants in $p$-Au collisions as a function of impact parameter. Dashed curve also includes a correction to ${\sigma }_{in}^{NN}$ (see text).

Figure 10

The Cronin ratio $R_{\text{Au}∕d}\left(p_T\right)$ for pions. Open circles show the results of PHENIX with normalization base upon Glauber model calculations of the inelastic $d$-Au cross section using ${\sigma }_{in}^{NN}=42\text{mb}$ [1]. Full points show the same data corrected for a proper value of inelastic $NN$ cross section and Gribov’s inelastic shadowing. The error bars include statistic and systematic uncertainties. The curve is the prediction from Ref. 4.

Figure 11

Cronin ratio $R_{pd}\left(p_T\right)$ calculated at $\sqrt{s}=200\text{GeV}$ using the formalism developed in Ref. 4.