The data for an (isoscalar) carbon target indicate that the...

The data for an (isoscalar) carbon target indicate that the ratio $$ \frac{\sigma\left(\pi^{+} \mathrm{C} \rightarrow \mu^{-} \mu^{+} \mathrm{X}\right)}{\sigma\left(\pi^{-} \mathrm{C} \rightarrow \mu^{-} \mu^{+} \mathrm{X}\right)} $$ is approximately unity for small values of $Q^2 / s$ but decreases toward $\frac{1}{4}$ as $Q^2 / s$ approaches 1 . Explain why this is in agreement with the Drell-Yan prediction. Note that $x y=Q^2 / s$.

The data for an (isoscalar) carbon target indicate that the ratio
$$
\frac{\sigma\left(\pi^{+} \mathrm{C} \rightarrow \mu^{-} \mu^{+} \mathrm{X}\right)}{\sigma\left(\pi^{-} \mathrm{C} \rightarrow \mu^{-} \mu^{+} \mathrm{X}\right)}
$$
is approximately unity for small values of $Q^2 / s$ but decreases toward $\frac{1}{4}$ as $Q^2 / s$ approaches 1 . Explain why this is in agreement with the Drell-Yan prediction. Note that $x y=Q^2 / s$.

Key Concepts

Drell-Yan Process

The Drell-Yan process is a mechanism in high-energy collisions in which a quark from one particle and an antiquark from another particle annihilate into a virtual photon, which then decays into a lepton pair. This framework is essential for understanding the dependence of the produced dilepton spectra on the momentum fractions of the participating partons, and it provides predictions for how experimental observables—such as cross section ratios—vary with kinematic variables.

Parton Distribution Functions

Parton distribution functions (PDFs) describe the probability of finding a quark or antiquark carrying a certain fraction of the parent hadron’s momentum. They are fundamental in predicting outcomes in high-energy processes like the Drell-Yan reaction, as the cross sections depend on the convolution of these PDFs for the two interacting hadrons. The differences in quark and antiquark PDFs between the projectiles (pions, in this case) are key to understanding the variation in cross section ratios.

Valence and Sea Quarks

In hadrons, valence quarks are the primary constituents that determine the hadron’s quantum numbers, while sea quarks arise from quantum fluctuations. At small values of the momentum fractions (corresponding to small Q²/s), the contributions from the sea quarks in the target and projectile tend to dominate and are relatively flavor symmetric, leading to comparable cross sections for different pion charges. As Q²/s increases, the reaction becomes sensitive to the valence quark content, and the inherent asymmetry in the valence quark distributions of ?? (u and anti-d) versus ?? (d and anti-u) leads to a reduced cross section ratio.

Kinematic Scaling Variables

The variable Q²/s, where Q² is the virtuality of the photon and s is the squared center-of-mass energy, plays a critical role in controlling the momentum fractions (x and y) of the interacting partons. Since x·y = Q²/s, small values of Q²/s imply that both partons carry only a small fraction of the hadron’s momentum (sea-dominated region), while values approaching one imply that the partons are taking nearly all the momentum of their parent hadrons (valence-dominated region). This transition is reflected in the changing cross section ratio observed in the experimental data.

Isoscalar Targets

An isoscalar target, such as carbon with an equal number of protons and neutrons, simplifies the analysis by averaging over the differences in up and down quark distributions in the nucleons. This symmetry ensures that any difference in the measured cross section ratio primarily arises from the structure of the projectile pion rather than complications from the target, allowing a cleaner test of the Drell-Yan predictions regarding quark and antiquark contributions.

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