Nuclear shadowing

Nuclear shadowing is the suppression of nuclear structure functions per nucleon compared to free nucleons. It is observed at high energies where the particle probing the nucleon, typically a photon or a gluon, can fluctuate into hadronic or partonic states that live long enough to interact coherently with multiple nucleons, leading to a destructive interference that reduces cross sections.

The Gribov–Glauber shadowing mechanism

Shadowing was first observed in the 1960s,^[1] and by the late 1960s and early 1970s, nuclear shadowing had been observed in several reactions: photon–nucleus interactions (photoabsorption, photoproduction), hadron–nucleus interactions (e.g., π–A scattering, proton–A scattering), lepton pair (Drell–Yan) production.

These observations lead to a picture mostly for real or nearly real photons (i.e. their off-shell mass ${\displaystyle Q^{2}}$ is very low) that explained shadowing qualitatively and quantitatively.^[2] This picture, known as the Gribov–Glauber shadowing mechanism^{[3][4][5][6][7]} is based on two principal ingredients. (A) Vector meson dominance (VMD), i.e. the fluctuations of a photon into hadronic states and (B) coherent multiple scattering in the nucleus (Glauber theory).^[8]

In the VMD model, a high-energy photon spends part of its time as a vector meson (ρ, ω, φ), which then interacts hadronically with nucleons. The distance over which such fluctuation occurs (known as the coherence length) can be estimated with the uncertainty principle to be of the order of ${\displaystyle l_{c}\approx {\frac {1}{2m_{N}x}}}$, where ${\displaystyle m_{N}}$ is the meson mass and ${\displaystyle x}$ the Bjorken scaling variable.

Since ${\displaystyle l_{c}}$ is inversely proportional to ${\displaystyle x}$, ${\displaystyle l_{c}}$ may become larger than the nuclear radius at sufficiently small ${\displaystyle x}$. Then, the hadronic component of the photon will interact simultaneously with several nucleons of the nucleus. To picture this, it may help to think of ${\displaystyle l_{c}}$ as the size of wavefunction of the meson: large ${\displaystyle l_{c}}$ will spatially overlap over several nucleons, thereby interacting simultaneously with them. The photon can then interact with more than one nucleon coherently, which leads to destructive interference between the amplitudes for scattering off different nucleons. The net effect is that the total cross section per nucleon is reduced compared to the sum of incoherent scattering off free nucleons.

This model gave good quantitatively account of data in photoproduction of nuclei.

Shadowing with highly virtual photons

The Gribov–Glauber shadowing mechanism was developed mostly for real photons or low-${\displaystyle Q^{2}}$ photons (where ${\displaystyle Q^{2}}$ is the absolute value of off-shell mass mass of the photon, i.e. minus its squared four-momentum).^[9]

Before the physicists of CERN's EMC experiment did deeply inelastic scattering (DIS) off nuclei, it was not clear if shadowing would be seen in DIS. This was because DIS occurs at large ${\displaystyle Q^{2}}$ (i.e. very virtual photon with large ${\displaystyle Q^{2}}$). In fact, because the four-momenta characterizing nuclear binding and nuclear distances are much less (MeV scale) than that of DIS (GeV scale), most particle physicists expected nuclear shadowing to vanish away at large Q^2. In Feynman’s parton model language, at large ${\displaystyle Q^{2}}$, each parton is struck individually because the probe space-time resolution goes as ${\displaystyle 1/Q^{2}}$, and furthermore, each parton is almost not interacting with its neighbors because the strong force coupling ${\displaystyle \alpha _{s}}$ becomes very small at large ${\displaystyle Q^{2}}$.^[10] Thus, the multiple scattering/coherence picture was expected to be less relevant in the DIS.

It was therefore a surprise that shadowing was observed in the EMC DIS nuclear data at very low-${\displaystyle x}$.^[11] It was an even bigger surprise that another opposite nuclear effect, an enhancement of the structure function, (therefore called antishadowing) was also observed at more moderate ${\displaystyle x}$, as well as another enhancement at the largest ${\displaystyle x}$ due to nuclear Fermi motion.^[11] These effects form what is now known as the EMC effect.

The modern interpretation of nuclear shadowing in DIS at small ${\displaystyle x}$ is very similar to Gribov’s and Glauber’s ideas, but is usually stated in partonic language: the incoming virtual photon (with large coherence length ${\displaystyle l_{c}}$ at small ${\displaystyle x}$) can fluctuate into a quark–antiquark pair (or more complex hadronic states). These states interact coherently with several nucleons, which leads to destructive interference and suppression of the nuclear structure functions compared to the sum of free nucleons. In Quantum chromodynamics language: at small ${\displaystyle x}$, gluon densities become large, and the virtual photon can interact via partonic configurations that overlap spatially over several nucleons. This leads to shadowing of parton distributions: the nuclear parton distribution functions (PDFs) become suppressed compared to the sum of nucleon PDFs.