Starting from (6.51), show that if quarks had spin 0, F2(x)...

Starting from (6.51), show that if quarks had spin 0, $F_2(x)$ would still be given by $(9.13)$ but that $F_1(x)=0$ and hence $\sigma_T=0$. Thus, in contrast to (9.26), spin-0 quarks would yield $\sigma_T / \sigma_L=0$. We can understand this difference by glancing at Fig. 9.4, which shows the head-on collision between the quark and the virtual photon. By conservation of $J_z$ (with $z$ along p), we see that a spin- 0 quark cannot absorb a photon of helicity $\lambda= \pm 1$, so $\sigma_T=0$. Suppose now the quark has spin $\frac{1}{2}$. We recall that its helicity is conserved in a high-energy interaction (see Section 6.6). This can only be achieved by a $\lambda= \pm 1$ photon; hence, $\sigma_L \rightarrow 0$ in this case.

Starting from (6.51), show that if quarks had spin 0, $F_2(x)$ would still be given by $(9.13)$ but that $F_1(x)=0$ and hence $\sigma_T=0$.

Thus, in contrast to (9.26), spin-0 quarks would yield $\sigma_T / \sigma_L=0$. We can understand this difference by glancing at Fig. 9.4, which shows the head-on collision between the quark and the virtual photon. By conservation of $J_z$ (with $z$ along p), we see that a spin- 0 quark cannot absorb a photon of helicity $\lambda= \pm 1$, so $\sigma_T=0$. Suppose now the quark has spin $\frac{1}{2}$. We recall that its helicity is conserved in a high-energy interaction (see Section 6.6). This can only be achieved by a $\lambda= \pm 1$ photon; hence, $\sigma_L \rightarrow 0$ in this case.

Key Concepts

Deep Inelastic Scattering Structure Functions

Deep inelastic scattering (DIS) uses structure functions such as F1(x) and F2(x) to characterize the internal partonic structure of hadrons. These functions are extracted from measured cross sections and provide insights into the momentum distribution and dynamics of quarks inside hadrons, forming the basis for understanding phenomena in quantum chromodynamics.

Spin of Particles

The intrinsic spin of particles, like quarks, fundamentally affects their interaction with other particles. In DIS, whether quarks are spin-0 or spin-1/2 changes how they absorb virtual photons, as their spin determines the allowed transitions according to conservation laws, which in turn affects observable quantities like structure functions and cross sections.

Helicity Conservation

Helicity, defined as the projection of a particle's spin along its direction of motion, is conserved in high-energy interactions. In the context of DIS, helicity conservation for spin-1/2 quarks implies that only certain photon polarizations can interact with them, dictating the relative size of the longitudinal and transverse cross sections.

Photon Polarization in DIS

Virtual photons involved in DIS can have transverse or longitudinal polarizations, corresponding to different interaction mechanisms with quarks. The ability of a quark to absorb a transverse (?=±1) or longitudinal photon depends on its spin structure, influencing the measured cross sections and helping differentiate between theoretical models of quark behavior.

Angular Momentum Conservation

Angular momentum conservation, particularly the conservation of the z-component (Jz), is a key constraint in DIS processes. It restricts the interactions between quarks and virtual photons; for instance, a spin-0 quark cannot absorb a transversely polarized photon, leading to a vanishing transverse cross section, which is a critical aspect of how different spin scenarios alter scattering outcomes.

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