The origin of Bjorken-$x$ dependence in DIS: a case for a $z$-dependent weight functional in the CGC

This paper proposes and validates a modified Color Glass Condensate framework for Deep Inelastic Scattering that introduces a $z$-dependent weight functional to resolve the standard model's inconsistency regarding Bjorken-$x$ dependence, thereby enabling a $x_b$-varying cross section that aligns with experimental data and $k_t$-factorization.

The Gist

The Big Picture: A Mystery in Particle Physics

Imagine you are trying to take a high-resolution photo of a tiny, fast-moving object (a proton) using a powerful camera (a particle accelerator). In physics, this is called Deep Inelastic Scattering (DIS).

Physicists have a popular theory called the Color Glass Condensate (CGC) to explain how these photos look. It's like a rulebook for how the proton's internal "glue" (gluons) behaves when hit by a photon.

The Problem:
The author of this paper, Benjamin Guiot, found a glitch in the rulebook.
According to the current rules, if you take a photo of the proton at different energies (different "Bjorken-x" values), the theory predicts the picture should look exactly the same, regardless of how hard you hit it.

The Analogy:
Think of the proton as a busy city and the photon as a drone flying over it.

• The Current Theory: Says that no matter how high or low the drone flies (the energy), the "map" of the city it sees is generated by a single, static filter. The theory suggests that if you change the drone's altitude, the map shouldn't change at all.
• The Reality: We know from experiments that the map does change. When you fly lower (higher energy), you see more details, more traffic, and different structures. The current theory fails to explain why the view changes with the energy.

The "All-Order" Paradox

The paper points out a logical trap. The current theory works fine for simple, low-level calculations. But if you try to add up every single possible correction (going to "all-orders"), the math says the final result becomes completely independent of the energy.

The Analogy:
Imagine you are baking a cake.

• Step 1: You add flour (the basic theory).
• Step 2: You add sugar and eggs (corrections).
• The Glitch: The current recipe says that if you keep adding ingredients forever, the cake eventually stops tasting like a cake and becomes a flavorless block of dough that tastes the same no matter how much sugar you added.
• The Author's Point: "Wait a minute. If the cake tastes the same regardless of the sugar, then my recipe is broken. The sugar (energy) must change the flavor."

The Proposed Solution: A "Smart" Filter

The author suggests a simple fix. The current theory treats the "weight" of the proton's internal structure as a static filter. The author proposes making this filter dynamic.

The New Idea:
Instead of a static filter, the filter should change based on how much energy the specific part of the proton is carrying.

The Analogy:
Imagine the proton is a crowded concert hall.

• Old View: The security guard (the theory) checks everyone with the same clipboard, regardless of who they are.
• New View: The security guard has a smart clipboard. If a VIP (a high-energy quark) walks in, the guard checks a different list. If a regular fan (a low-energy quark) walks in, the guard checks a different list.
• The Result: The "view" of the crowd changes depending on who you are looking at. This matches reality: the structure of the proton does depend on the energy of the probe.

The author calls this a $z$-dependent weight functional. In plain English: "The way we calculate the proton's structure depends on the momentum fraction ($z$) of the particles involved."

Why This Matters: Proving the "Evolution" isn't the Whole Story

For years, physicists have believed that the reason the proton's structure changes with energy is due to a specific mathematical process called "small-$x$ evolution" (like the BK or JIMWLK equations). They thought, "The proton evolves over time, and that's why the photo changes."

The Author's Twist:
Guiot shows that you can get a perfect match with experimental data without using those complex evolution equations.

The Analogy:
Imagine two detectives trying to solve a crime (fitting the data).

• Detective A (Standard View): Uses a super-complex, high-tech algorithm (small-$x$ evolution) to predict the outcome. It works, but it's very complicated.
• Detective B (Guiot's View): Uses a simpler, smarter rule (the $z$-dependent filter).
• The Result: Both detectives get the exact same answer.
• The Conclusion: Since the simpler rule works just as well, we can't say for sure that the complex algorithm is the only reason the data looks the way it does. The "evolution" might not be the main driver; the "smart filter" (kinematics) might be doing the heavy lifting.

The Bottom Line

1. The Flaw: The standard theory has a logical hole where it predicts the proton looks the same at all energies if you calculate it perfectly.
2. The Fix: Make the theory aware that the "probe" (the photon) sees different things depending on its energy.
3. The Impact: This suggests that the famous "small-$x$ evolution equations" might not be the sole reason the proton changes shape. We might have been overcomplicating the explanation.

In short, the author is saying: "The current map is missing a variable. Once we add 'energy' back into the map-making process, the theory works better, and we realize we don't need the most complicated machinery to explain the data."

Technical Summary

1. Problem Statement

The paper addresses a fundamental theoretical ambiguity and potential inconsistency within the Color Glass Condensate (CGC) effective theory as applied to Deep Inelastic Scattering (DIS) in the eikonal approximation.

• The Core Issue: In the standard eikonal CGC formalism, the dependence on the Bjorken scaling variable ($x_b$) enters the cross-section solely through the rapidity cutoff $\Lambda = x_b$.
• The Inconsistency: The author argues that if the cross-section is calculated to all orders in the strong coupling constant ($\alpha_s$), and the only $x_b$ dependence is via the renormalization scale $\Lambda$, the resulting all-order cross-section becomes independent of $x_b$. This contradicts the physical reality that DIS cross-sections vary significantly with $x_b$.
• The Conflict: While standard phenomenological fits often assume that the $x_b$ dependence is driven entirely by small-$x$ evolution equations (like JIMWLK or BK) acting on the dipole amplitude, the author contends that this view is mathematically inconsistent with the all-order eikonal formulation. If the cross-section were truly independent of the renormalization scale at all orders, the observed $x_b$ variation cannot be explained solely by the evolution of the CGC correlators.

2. Methodology

The author proposes a modification to the CGC formalism and validates it through theoretical analysis and phenomenological fitting.

A. Theoretical Modification

The paper proposes that the weight functional ($W[\rho]$), which averages over color charge densities, must depend explicitly on the light-cone momentum fraction $z$ (the fraction of the photon's momentum carried by the quark in the $q\bar{q}$ dipole), rather than just the rapidity cutoff $\Lambda$.

• Standard Formulation: The cross-section is written as an integral over transverse space where the dipole amplitude $N$ depends on $\Lambda$ (set to $x_b$).
  $$\sigma \sim \int d^2r |\psi|^2 N(r, \Lambda=x_b)$$
• Proposed Formulation: The weight functional depends on $z$, and the integration limits for $z$ are determined by $x_b$ via kinematical constraints.
  $$\sigma(x_b, Q^2) = \sigma_0 \sum_f \int_{z_{min}(x_b)}^{z_{max}(x_b)} dz \int d^2r |\psi(z, Q^2, r)|^2 N(r, z; \Lambda)$$
• Kinematical Constraints: The limits $z_{min}$ and $z_{max}$ are derived from the on-shell condition of the outgoing quarks and the requirement that the dipole is "fast" ($k^- \gg k^+$). This introduces an explicit $x_b$ dependence into the integrand, independent of the evolution of $N$.
• Analogy: This modification draws a parallel with collinear factorization, where the Bjorken-$x$ dependence enters through the convolution variable $\xi$ in the Parton Distribution Functions (PDFs), not just the renormalization scale.

B. Phenomenological Analysis

To test the viability of this modified framework, the author performs a fit to experimental DIS data (H1 and ZEUS collaborations) and compares it with existing literature:

1. Re-evaluation of Standard Fits: The author analyzes previous studies (e.g., AAMQS) that claim the BK/JIMWLK evolution explains the data. They argue that these fits rely heavily on free parameters in the initial conditions and running coupling schemes, making the "success" of the evolution equations ambiguous.
2. New Fit: The author fits the data using the modified equation (Eq. 18) with a $\Lambda$-independent dipole amplitude (i.e., no small-$x$ evolution equations are solved). The dipole amplitude is parametrized with a specific functional form depending on $z$ and $r$.

3. Key Contributions

1. Identification of the "All-Order" Inconsistency: The paper rigorously demonstrates that in the eikonal approximation, if $x_b$ dependence is restricted to the rapidity cutoff $\Lambda$, the all-order cross-section loses its $x_b$ dependence. This challenges the standard interpretation of how $x_b$ evolution works in CGC.
2. Proposal of a $z$-Dependent Weight Functional: The author introduces a minimal modification where the target structure (weight functional) depends on the probe's momentum fraction $z$. This restores the physical expectation that the observed structure depends on the probe energy.
3. Compatibility with $k_t$-Factorization: The modified formulation (Eq. 18) is shown to be compatible with $k_t$-factorization, whereas the standard eikonal formulation (Eq. 13) is not. This provides a stronger theoretical foundation for the modification.
4. Phenomenological Decoupling: The paper demonstrates that a high-quality description of DIS data can be achieved without invoking small-$x$ evolution equations (JIMWLK/BK), provided the $z$-dependence and kinematical limits are correctly implemented.

4. Results

• Theoretical Consistency: The modified equation resolves the paradox of the $x_b$-independent all-order cross-section by introducing explicit kinematical dependence on $x_b$ through the integration limits of $z$.
• Data Fitting:
  • The author fitted the H1/ZEUS data ($2 < Q^2 \leq 45$ GeV$^2$) using the modified model with a $\Lambda$-independent dipole.
  • Performance: The fit achieved a $\chi^2/\text{d.o.f.} \approx 1.82$, which is comparable to (and in some error-bar interpretations, better than) the results from standard CGC dipole models (like the AAMQS fit) that rely on running-coupling BK evolution.
  • Implication: Since the modified model fits the data well without solving the BK/JIMWLK equations, the success of previous fits does not constitute proof that the $x_b$ dependence is driven by small-$x$ evolution.
• Visual Comparison: The paper presents plots (Fig. 3) showing that the modified model describes the reduced cross-section across various $Q^2$ bins with similar precision to the standard CGC dipole model.

5. Significance

• Paradigm Shift: The paper challenges the prevailing assumption in high-energy QCD phenomenology that the variation of DIS cross-sections with $x_b$ is solely driven by the evolution of the target (JIMWLK/BK). It suggests that kinematical constraints and the explicit dependence of the target structure on the probe's energy ($z$) play a crucial, perhaps dominant, role.
• Clarification of CGC Limits: It highlights a subtle but critical limitation of the eikonal approximation when applied to all-order calculations, suggesting that sub-eikonal corrections or a redefinition of the weight functional are necessary for a consistent theory.
• Phenomenological Caution: The results serve as a warning against over-interpreting the success of phenomenological fits as evidence for specific dynamical mechanisms (like BK evolution) without ruling out alternative kinematical explanations.
• Future Directions: The work opens the door for developing CGC-based models that naturally incorporate $k_t$-factorization and explicit energy dependence, potentially leading to more robust predictions for the Electron-Ion Collider (EIC).

In summary, Guiot argues that the standard CGC description of DIS is theoretically incomplete regarding $x_b$ dependence and proposes a $z$-dependent weight functional as a natural, consistent, and phenomenologically viable solution that does not strictly require small-$x$ evolution equations to describe experimental data.