Deep Inelastic Scattering: Kinematics and Hadronic Structure

1. A New Kind of Microscope

When we scatter particles, the momentum transfer of the probe particle determines the resolution of our "measurement." Low-energy scattering can reveal properties like a particle's overall charge radius, but to see finer details, we need higher momentum transfer. Deep Inelastic Scattering uses high-energy leptons (like electrons) as probes to achieve very high resolution.

Let's start with a conceptual overview.

As the video explains, the key idea is to use the lepton to probe the proton's structure. The process is:
$e^{-}(l)+p(p)\to e^{-}(l^{\prime })+X(p_X)$

• Inelastic: The final hadronic state $X$ is not just a proton. The proton is broken apart, so the invariant mass of the final state, $M_X$, is greater than the proton mass, $M_p$.
• Deep: The collision involves a large momentum transfer, allowing us to resolve very small distance scales within the proton.

2. The Kinematic Variables of DIS

To describe this inclusive process, where we only measure the scattered electron, we need a set of Lorentz-invariant variables. Your background in special relativity and four-vectors will be central here.

Let's define the four-momenta as in the diagram above:

• $l$ and $l^{\prime }$ are the four-momenta of the incoming and outgoing electron.
• $p$ is the four-momentum of the target proton.
• $q=l-l^{\prime }$ is the four-momentum of the exchanged virtual photon.

From these, we construct the key variables that characterize the interaction.

Let's summarize the essential variables from the reading:

1. Virtuality or Momentum Transfer Squared, $Q^2$:
   $Q^2=-q^2=-(l-l^{\prime }{)}^2>0$
   Since the virtual photon is spacelike ($q^2<0$), $Q^2$ is positive. It represents the "virtuality" of the exchanged photon. In the context of scattering, $\sqrt{Q^2}$ is inversely proportional to the spatial resolution of the probe ($\lambda \sim \hslash /\sqrt{Q^2}$). "Deep" scattering means large $Q^2$.

2. Energy Transfer, $\nu$:
   $\nu =\frac{p\cdot q}{M_p}$
   In the proton's rest frame ($p=(M_p,\vec{0})$), $\nu =E_l-E_{l^{\prime }}$. This is simply the energy lost by the electron, which is transferred to the proton.

3. Invariant Mass of the Hadronic Final State, $W^2$:
   $W^2=p_X^2=(p+q{)}^2=M_p^2+2p\cdot q+q^2=M_p^2+2M_p\nu -Q^2$
   For elastic scattering ($ep\to ep$), $X$ is just a proton, so $W^2=M_p^2$. For inelastic scattering, the proton breaks up, so $W^2>M_p^2$.

4. Bjorken Scaling Variable, $x$:
   $x=\frac{Q^2}{2p\cdot q}=\frac{Q^2}{2M_p\nu }$
   This dimensionless variable ranges from 0 to 1. For elastic scattering, $Q^2=2M_p\nu$, which means $x=1$. For inelastic scattering, $W^2>M_p^2⟹Q^2<2M_p\nu$, so $x<1$. As we will see, $x$ has a profound physical interpretation as the fraction of the proton's momentum carried by the constituent that was "struck" by the photon.

5. Inelasticity, $y$:
   $y=\frac{p\cdot q}{p\cdot l}$
   In the lab frame, $y=\nu /E_l$, which is the fractional energy loss of the electron. It also ranges from 0 to 1.

For an inclusive measurement where only the final electron's energy and angle are detected, the kinematics are fully described by any two of these variables (e.g., $x$ and $Q^2$).

3. The DIS Cross-Section and Structure Functions

How do we describe the cross-section for this process? We don't know the detailed structure of the hadronic vertex ${\gamma }^{\ast }p\to X$. The approach, much like in our last lesson, is to factorize the process. The amplitude squared can be written as a contraction of a known leptonic tensor $L_{\mu \nu }$ and an unknown hadronic tensor $W^{\mu \nu }$.
$d\sigma \propto L_{\mu \nu }{W}^{\mu \nu }$
The hadronic tensor $W^{\mu \nu }$ parameterizes our ignorance of the proton's internal dynamics. However, we can write down its most general form consistent with Lorentz invariance and electromagnetic current conservation ($q_{\mu }{W}^{\mu \nu }=0$).

As the slides demonstrate, the hadronic tensor for a spin-averaged unpolarized proton, respecting all symmetries, can be parameterized by two scalar functions that depend on the available Lorentz invariants, $Q^2$ and $\nu$:

These functions, $W_1$ and $W_2$, are called the proton structure functions. They contain all the non-perturbative information about the proton's internal structure as probed by a virtual photon.

The double differential cross-section in terms of the scattered electron's final energy $E^{\prime }$ and solid angle $\Omega$ in the lab frame is then given by:

This expression is the inelastic equivalent of the Rosenbluth formula for elastic scattering. But where elastic scattering had form factors $G_E(Q^2),G_M(Q^2)$ which fall off rapidly with $Q^2$, inelastic scattering has structure functions $W_1,W_2$. Their behavior is dramatically different.

It is conventional to define dimensionless structure functions, $F_1$ and $F_2$, which in the high-energy limit are related to $W_1$ and $W_2$:

• $F_1(x,Q^2)=M_p{W}_1(\nu ,Q^2)$
• $F_2(x,Q^2)=\nu W_2(\nu ,Q^2)$

4. Bjorken Scaling: A Glimpse at Partons

The crucial experimental discovery was made at SLAC in the late 1960s. Experimenters found that in the "deep" inelastic limit (large $Q^2$ and large $\nu$), the structure functions $F_1$ and $F_2$ were approximately independent of $Q^2$! They depended only on the dimensionless ratio $x=Q^2/(2M_p\nu )$. This phenomenon is known as Bjorken scaling.

The observation that $F_1(x,Q^2)\approx F_1(x)$ and $F_2(x,Q^2)\approx F_2(x)$ is revolutionary. Why?

A $Q^2$ dependence in form factors signifies a spatially extended object. The lack of $Q^2$ dependence in the structure functions implies that the electron is scattering off point-like objects inside the proton. This is analogous to how Rutherford scattering at large angles revealed a point-like nucleus inside the atom.

This scaling behavior was the first strong experimental evidence for the existence of the particles Richard Feynman dubbed partons, which were later identified as quarks and gluons. In this "Parton Model" picture, the DIS process is viewed as an elastic scatter of the electron off a single, quasi-free parton within the proton. The structure functions $F_1(x)$ and $F_2(x)$ are no longer just arbitrary parameterizations; they take on a physical meaning, describing the momentum distribution of these partons within the proton. The variable $x$ is precisely the fraction of the proton's total momentum carried by the struck parton.

Conclusion

Key Takeaways:

• Deep Inelastic Scattering (DIS) is the process of scattering a high-energy lepton off a hadron, causing it to break apart, providing a high-resolution probe of the hadron's internal structure.
• The kinematics are described by Lorentz-invariant variables, most importantly the momentum transfer squared $Q^2$ (sets the resolution) and the Bjorken scaling variable $x$ (momentum fraction of the struck parton).
• The unknown dynamics of the proton are parameterized by hadronic structure functions ($F_1$ and $F_2$), which appear in the general formula for the DIS cross-section.
• The experimental observation of Bjorken scaling—the fact that at high $Q^2$, $F_1$ and $F_2$ depend only on $x$—provided the first compelling evidence that protons are composed of point-like constituents, later identified as quarks.

You have now learned the language used to describe the experiments that revolutionized our understanding of hadrons. We've moved from scattering off point-like leptons to scattering off composite protons and introduced the functions that encode the proton's complex inner world.

Preview of the Next Lesson:

Now that we've established that structure functions describe the point-like partons inside the proton, the next logical step is to build a quantitative model. In the next lesson, we will use the naive parton model to calculate the structure functions for electron-proton scattering, connecting these macroscopic observables directly to the charges and spins of the constituent quarks.