Parton helicities at arbitrary x and Q2 in double-logarithmic approximation

This paper derives explicit expressions for parton helicities at arbitrary kinematics within the double-logarithmic approximation, advocates for $k_T$-factorization over collinear factorization when accounting for orbital angular momentum, and demonstrates that DGLAP asymptotics are less singular at small $x$ than Regge asymptotics.

The Gist

The Big Picture: The Spin Puzzle

Imagine a proton (a tiny particle inside an atom) as a spinning top. Physicists want to know exactly how this top spins. They know the top is made of smaller, invisible pieces called partons (quarks and gluons).

The paper is about calculating the "spin direction" (helicity) of these tiny pieces. The author, B.I. Ermolaev, is trying to write a universal instruction manual that tells us exactly how these pieces spin, no matter how fast they are moving or how hard we are hitting them.

The Two Maps: Collinear vs. KT Factorization

To navigate the world of spinning particles, physicists use "maps" called Factorization. The paper argues there are two main maps, and they are not interchangeable:

1. The "Highway" Map (Collinear Factorization): This map assumes all the tiny pieces are driving perfectly straight down a single-lane highway. They have no side-to-side movement.
   • The Paper's Claim: This map is great for straight roads, but it breaks down if you want to talk about the pieces' "side-to-side" movement (Orbital Angular Momentum). You can't describe a car drifting if your map says cars only drive straight.
2. The "Off-Road" Map (KT Factorization): This map allows the pieces to drift, weave, and move sideways. It accounts for the full 3D movement of the particles.
   • The Paper's Claim: If you want to understand the full spin of the proton, including the "drifting" (Orbital Angular Momentum), you must use this Off-Road map. Using the Highway map for this job is mathematically inconsistent.

The Weather Report: Small x and Large Q2

The paper focuses on two specific conditions, which the author calls "Small x" and "Large Q2."

• Small x: Imagine looking at the proton through a telescope that only sees the tiniest, fastest-moving fragments.
• Large Q2: This is like hitting the proton with a very powerful, high-energy hammer.

In this "stormy weather" (high energy, tiny fragments), the math gets messy. The author uses a special technique called Double-Logarithmic Approximation (DLA).

• Analogy: Think of DLA as a noise-canceling headset. In a chaotic storm, there are millions of tiny sounds (mathematical terms). DLA filters out the background noise and lets you hear only the loudest, most important signals (the "double-logarithms") so you can actually make sense of the data.

The Construction Site: Building the Formula

The author builds his solution in three stages, like constructing a building:

1. The Foundation (The "Off-Shell" Amplitudes): First, he calculates the behavior of the particles when they are "off-shell."
   • Analogy: Imagine a car that hasn't been built yet, or a ghost car that exists in a theoretical state. The author calculates how these "ghost cars" behave before they become real, solid particles. He uses a method called IREE (Infra-Red Evolution Equations), which is like a blueprint that shows how the car changes as you add more parts.
2. The Renovation (Interpolation): The initial blueprint only works for the "stormy weather" (small x, large Q2). But what if the weather is calm (medium x) or the hammer is weak (small Q2)?
   • Analogy: The author takes his storm-proof blueprint and blends it with a standard "sunny day" blueprint (called DGLAP). He creates a hybrid formula that works in any weather, from calm to stormy.
3. The Final Touch (Arbitrary x and Q2): Finally, he extends this hybrid formula to cover every possible speed and energy level, creating a single, universal equation for parton spin.

The Race: Who Wins the Spin?

The paper compares two different ways of predicting how fast the proton spins at high speeds:

• The Regge Runner (The Author's Method): This runner follows a specific path derived from the "ghost car" calculations. The author proves that this runner's speed increases in a very specific, predictable way (like a square root) as you zoom in on the tiny fragments.
• The DGLAP Runner (The Standard Method): This is the traditional runner used by most physicists.
  • The Paper's Claim: The author shows that the DGLAP runner is actually slower and less "singular" (less dramatic) than the Regge runner when looking at the tiniest fragments.
  • The "False Intercept" Warning: The author warns that sometimes people look at the DGLAP runner and pretend they see a "Regge-like" finish line. He calls this a "False Intercept." It's like looking at a blurry photo and thinking you see a finish line that isn't actually there. The math shows the DGLAP runner doesn't actually reach that specific finish line unless you force it with experimental data fitting.

The Conclusion

The paper concludes with three main takeaways:

1. We have a new universal map: We now have explicit formulas for parton spin that work at any speed or energy, whether you are using the "Highway" map or the "Off-Road" map.
2. Off-Road is mandatory for Spin: If you want to include the "drifting" (Orbital Angular Momentum) in your explanation of how the proton spins, you must use the KT (Off-Road) factorization. Using the Collinear (Highway) method for this is mathematically wrong.
3. The Standard Model needs a check: The traditional way of calculating these spins (DGLAP) doesn't naturally produce the same "Regge" behavior as the author's method. If you see that behavior in experiments, it might be coming from the data fitting (the "initial conditions") rather than the standard equations themselves.

In short, the author has built a more robust, flexible, and mathematically consistent tool for understanding the spin of the universe's smallest building blocks, specifically arguing that we need to stop treating them like cars on a straight highway when we are trying to understand their full spin.

Technical Summary

Technical Summary: Parton helicities at arbitrary $x$ and $Q^2$ in double-logarithmic approximation

Problem Statement
The description of spin-dependent hadronic processes at high energies relies heavily on parton helicities ($h_{q,g}$). While recent work by Kovchegov et al. established the KPSCTT evolution equations for calculating quark and gluon helicities with double-logarithmic (DL) accuracy, these results were primarily formulated as small-$x$ asymptotics. Furthermore, existing formulae for parton helicities derived in previous studies (Refs. [22, 23]) are compatible only with Collinear Factorization and are fundamentally incompatible with $k_T$-Factorization ($k_T$F). This limitation is critical because $k_T$F is necessary when accounting for parton Orbital Angular Momenta (OAM), which contribute to the nucleon spin. Additionally, a complete description of parton helicities requires explicit dependence on both the Bjorken variable $x$ and the momentum transfer scale $Q^2$, extending beyond the fixed-$Q^2$ approximations used in earlier RHIC-focused analyses. The paper addresses the need for explicit expressions for $h_{q,g}(x, Q^2)$ valid at arbitrary $x$ and $Q^2$ that are compatible with both Collinear and $k_T$ Factorization, and investigates the small-$x$ asymptotic behavior of these helicities compared to DGLAP predictions.

Methodology
The authors employ the Infra-Red Evolution Equation (IREE) approach, a method developed by Gribov and Lipatov, to calculate scattering amplitudes in the Double-Logarithmic Approximation (DLA). The core of the methodology involves:

1. IREE Construction: The authors construct IREEs for parton-parton amplitudes by factorizing the softest virtual partons (those with minimal transverse momentum $k_\perp$). This is done for three kinematic configurations:
   • Totally off-shell amplitudes ($A_{ij}$) where all external partons are virtual.
   • Partly off-shell amplitudes ($A'_{ij}$) where initial partons are on-shell but intermediate states are virtual.
   • On-shell amplitudes ($A''_{ij}$) where all external partons are on-shell.
2. Kinematic Regions: The calculation is divided into specific kinematic regions based on $x$ and $Q^2$:
   • Region $D_B$ (DL region): Small $x$ and large $Q^2$. Here, IREEs are solved explicitly. This region is further subdivided into Moderate-Virtual (MV) and Deeply-Virtual (DV) kinematics based on the relation between $Q^2$, $k_\perp^2$, and $x$.
   • Region $D_A$ (DGLAP region): Large $x$ and large $Q^2$.
   • Regions $D_C$ and $D_D$: Small $Q^2$ regions.
3. Color Octet Contributions: Unlike the IREEs for the spin-dependent structure function $g_1$, the IREEs for parton helicities involve $t$-channel color octet contributions from the outset. The authors derive equations for these amplitudes, noting that exact solutions for octet contributions are complex and currently unknown in full generality; thus, they are treated approximately using the Born approximation for the octet terms.
4. Interpolation and Extension:
   • To extend results from the DL region ($D_B$) to arbitrary $x$ (Region $D_A \oplus D_B$), the authors construct interpolation formulae by combining the DLA results with DGLAP coefficient functions and anomalous dimensions, subtracting the overlapping DL terms to avoid double-counting.
   • To extend to small $Q^2$ ($D_C \oplus D_D$), they apply a shift $Q^2 \to \bar{Q}^2 = Q^2 + \mu^2$, where $\mu$ is the IR cut-off, a technique proven in previous works to handle non-perturbative regions.
5. Asymptotic Analysis: The small-$x$ asymptotics are derived using the Saddle-Point method applied to the Mellin-transformed expressions. These are compared against the asymptotics generated by DGLAP evolution equations at Leading Order (LO), Next-to-Leading Order (NLO), and beyond.

Key Contributions and Results

• Explicit Expressions for $h_{q,g}(x, Q^2)$: The paper derives explicit integral expressions for quark and gluon helicities valid at arbitrary $x$ and $Q^2$. These expressions are formulated to be compatible with both Collinear Factorization and $k_T$ Factorization.
  • In Collinear Factorization, the helicities are expressed as convolutions of perturbative components $M'_{ij}$ and initial parton distributions $\Phi_{q,g}$.
  • In $k_T$ Factorization, the expressions involve additional integrations over transverse momentum $k_\perp$ and utilize totally off-shell amplitudes $M_{ij}(x, Q^2, k_\perp^2)$.
• Necessity of $k_T$ Factorization for OAM: The authors argue that while Collinear Factorization is sufficient for describing parton spins, it is theoretically inconsistent for describing parton Orbital Angular Momenta (OAM). Since the $z$-component of OAM requires non-zero transverse momentum components of partons, which are integrated out in Collinear Factorization, $k_T$ Factorization is the only self-consistent framework for including OAM in the description of longitudinal nucleon spin.
• Small-$x$ Asymptotics:
  • Identity of Asymptotics: Despite $h_q$, $h_g$, and the structure function $g_1$ being represented by different formulae in DLA, the authors prove that their small-$x$ asymptotics are identical (up to pre-exponential factors). They all exhibit Regge-like behavior $x^{-\omega_0}$ with an intercept $\omega_0 \approx 0.86$ (at $\mu=1$ GeV) that is independent of $Q^2$.
  • Comparison with DGLAP: The paper demonstrates that DGLAP asymptotics (LO, NLO, NNLO) do not naturally acquire the Regge form. Instead, they follow a behavior like $\exp(c \xi^{1-1/2n} y^{1/2n})$. The authors show that the "Regge-like" behavior often observed in DGLAP fits arises from the input parton distribution functions (the "fits" themselves), not from the evolution kernels.
  • False Intercept: The authors critique the interpretation of DGLAP fits where a $Q^2$-dependent "intercept" is extracted. They argue this is a "false intercept" that contradicts both Regge theory and the DLA results, where the intercept is $Q^2$-independent.
• Color Octet Impact: The study confirms that while color octet contributions affect the precise value of the intercept, they do not alter the fundamental Regge nature of the asymptotics or the identity between the asymptotics of helicities and $g_1$.

Significance
The paper provides a unified theoretical framework for calculating parton helicities across the entire kinematic plane of $x$ and $Q^2$. Its primary significance lies in bridging the gap between the high-energy double-logarithmic regime and the standard DGLAP regime, providing interpolation formulae that respect the total resummation of DL terms.

Crucially, the work establishes a theoretical boundary condition for the study of the proton spin: it asserts that any attempt to include parton Orbital Angular Momenta in the description of the nucleon spin must utilize $k_T$ Factorization. The authors contend that combining OAM with Collinear Factorization is theoretically inconsistent. Finally, the paper clarifies the nature of small-$x$ asymptotics, distinguishing between the intrinsic Regge behavior predicted by DLA and the apparent Regge behavior in DGLAP which is driven by input parameterizations rather than the evolution dynamics.