Twist-2 relations for the twist-3 tensor-polarized distribution function $f_{LT}$ of a spin-1 hadron by the operator-product-expansion method

This paper employs the local operator-product-expansion method to independently derive twist-2 relations, specifically a Wandzura-Wilczek-like relation and a Burkhardt-Cottingham-like sum rule, for the twist-3 tensor-polarized distribution function $f_{LT}$ of spin-1 hadrons, thereby providing a robust theoretical foundation for upcoming electron-deuteron deep inelastic scattering experiments at JLab.

The Gist

Imagine the inside of a proton or a neutron (the building blocks of atoms) not as a solid ball, but as a bustling city filled with tiny, fast-moving particles called quarks. Physicists have spent decades studying how these quarks spin and move in a single proton (which has a "spin" of 1/2). They have a very good map for this city, known as the "twist-2" map, which describes the most basic, dominant behaviors.

However, there are other particles in the universe, like the deuteron (a nucleus made of one proton and one neutron), that are a bit more complex. They have a "spin" of 1. Think of a proton as a spinning top that wobbles in one direction, while the deuteron is like a spinning top that can also be squashed or stretched in different directions. This extra squashing ability is called "tensor polarization."

Because the deuteron is more complex, its internal map has extra layers of detail that the simple proton map misses. One of these extra layers is a specific measurement called $f_{LT}$.

The Problem: A Missing Piece of the Puzzle

Scientists are preparing to run a new experiment at a giant particle accelerator called JLab (Thomas Jefferson National Accelerator Facility). They plan to shoot electrons at these "squashable" deuterons to see how the quarks inside react.

The tricky part is that JLab will be looking at these interactions at relatively low energy levels. In the world of particle physics, low energy means the "extra layers" of the map (called twist-3 effects) become very important. If you try to read a book with a blurry lens (ignoring the twist-3 effects), you might miss the most interesting details.

To understand the JLab data, scientists need a reliable way to predict what the $f_{LT}$ map should look like based on the simpler, well-known $f_{1LL}$ map.

The Previous Attempt: A Rough Sketch

In a previous study, scientists used a "non-local" method to draw a connection between the simple map ($f_{1LL}$) and the complex map ($f_{LT}$). It was like connecting two cities with a straight line on a map, ignoring the winding roads in between. They found a relationship that looked very similar to a famous rule in physics called the Wandzura-Wilczek (WW) relation. They also found a "sum rule" (a conservation law) similar to the Burkhardt-Cottingham (BC) rule.

However, this previous method was a bit like a sketch. It worked, but it relied on a specific type of mathematical tool (non-local operators) that some physicists felt wasn't the most rigorous way to prove these rules.

The New Solution: The "Local" Blueprint

This paper presents a new, more rigorous way to draw that connection. The authors, S. Kumano and Kenshi Kuroki, used a method called Operator Product Expansion (OPE) with local operators.

The Analogy:
Imagine you are trying to understand the structure of a complex machine.

• The Old Way: You looked at the machine from a distance and guessed how the gears turned based on the overall motion. It gave a good guess, but it wasn't a proof.
• The New Way (This Paper): The authors took the machine apart, looked at the individual gears and springs right where they touch (the "local" parts), and mathematically proved exactly how they must fit together.

By using this "local" method, which respects the fundamental symmetries of the universe (like rotational symmetry), they derived the same relationships they found before:

1. The WW-like Relation: They proved that the complex $f_{LT}$ map is largely determined by the simple $f_{1LL}$ map, just like the original WW rule.
2. The BC-like Sum Rule: They proved a conservation rule that says if you add up all the values of a specific part of the map, the result should be zero.

Why This Matters

The authors didn't just find a new number; they confirmed the previous sketch with a solid, independent mathematical proof.

• Reliability: Now, when scientists at JLab get their data, they can use these relations as a "reliable first estimate." They know that if the data deviates from this rule, it's not because the math was wrong, but because there is something new and exciting happening (called "dynamical twist-3 terms") that they need to study.
• The Goal: The ultimate goal is to use these rules to interpret the upcoming JLab experiments on the tensor-polarized deuteron. This will help physicists understand the "squashable" nature of spin-1 particles, opening up a new field of spin physics that goes beyond just adding up protons and neutrons.

In short, this paper builds a sturdy bridge between what we already know about simple particles and what we are about to discover about complex ones, ensuring that when the new data arrives, we have a solid foundation to stand on.

Technical Summary

1. Problem Statement

• Context: Spin-1 hadrons (such as the deuteron) possess a richer spin structure than spin-1/2 nucleons due to the existence of tensor polarizations. While spin-1/2 structure functions ($g_1, g_2$) are well-studied, tensor-polarized parton distribution functions (PDFs) for spin-1 hadrons require further theoretical investigation.
• Specific Challenge: An upcoming experiment at the Thomas Jefferson National Accelerator Facility (JLab) aims to measure electron-deuteron deep inelastic scattering (DIS) with a tensor-polarized target. These measurements will occur in a relatively low-$Q^2$ region (few GeV$^2$), where twist-3 contributions are expected to be significant.
• Theoretical Gap: Previous work (Ref. [13]) derived a "Wandzura-Wilczek (WW)-like" relation and a "Burkhardt-Cottingham (BC)-like" sum rule for the twist-3 tensor-polarized PDF $f_{LT}$ in terms of the twist-2 PDF $f_{1LL}$. However, those derivations relied on nonlocal operators and light-cone correlation functions.
• Objective: To independently verify and establish these relations using the Operator Product Expansion (OPE) with local operators. This method is considered more formal and rigorously satisfies rotational invariance, providing a reliable foundation for interpreting future experimental data.

2. Methodology

The authors employed the standard OPE framework used for spin-1/2 nucleons but adapted it for the tensor-polarized case of spin-1 hadrons.

• Correlation Functions: The study starts with the definition of the collinear correlation function $\Phi(x, P, T)$ involving a nonlocal operator $\bar{\psi}(0)\gamma^\sigma \psi(\xi)$. This is expanded in terms of tensor-polarized PDFs: $f_{1LL}$ (twist-2), $f_{LT}$ (twist-3), and $f_{3LL}$ (twist-4).
• Taylor Expansion & Local Operators:
  • The nonlocal quark field $\psi(\xi)$ is expanded into a Taylor series around $\xi=0$.
  • Derivatives are shifted to the matrix element via partial integration.
  • This transforms the nonlocal operator into a series of local operators $R^{\sigma\{\mu_1 \dots \mu_{n-1}\}}$ involving covariant derivatives $D_\mu$.
• Operator Decomposition:
  • The local operators are decomposed into symmetric ($R^{\{\sigma\mu_1\dots\}}$) and mixed-symmetric ($R^{[\sigma\{\mu_1\}\dots]}$) parts.
  • The symmetric part corresponds to twist-2, while the mixed-symmetric part corresponds to twist-3.
  • Trace terms (twist-4 and higher) are subtracted to ensure the operators are traceless.
• Matrix Element Parametrization:
  • The matrix elements of these local operators between spin-1 hadron states are expressed using available Lorentz vectors ($P^\mu$) and the tensor polarization $T^{\mu\nu}$.
  • The authors calculate the $n$-th moments of the PDFs by matching the OPE results with the moments derived from the correlation function.
• Moment Analysis:
  • By comparing the coefficients of the tensor polarization parameters ($S_{LL}$ and $S_{LT}^\mu$) in the OPE expansion, the authors derived integral relations between the moments of $f_{1LL}$ and $f_{LT}$.
  • They utilized the relation between moments and integrals (Eq. 26) to transform the moment-space relations back into $x$-space (momentum fraction space).

3. Key Contributions

• Independent Derivation: The paper provides the first derivation of the WW-like and BC-like relations for the tensor-polarized function $f_{LT}$ using local OPE, confirming previous results obtained via nonlocal operators.
• Formal Rigor: By using local operators, the derivation explicitly satisfies rotational invariance, addressing a theoretical concern regarding the previous nonlocal derivation.
• Separation of Terms: The method clearly separates the twist-3 function $f_{LT}$ into two distinct components:
  1. Kinematical/WW-like term: Determined entirely by the twist-2 function $f_{1LL}$.
  2. Dynamical term: A genuine twist-3 contribution arising from multi-parton correlations (represented by the coefficient $d_n$).

4. Key Results

The study successfully derived the following relations:

• The WW-like Relation:
  The twist-3 function $f_{LT}(x)$ is related to the twist-2 function $f_{1LL}(x)$ by:
  $$f_{LT}^{\pm \text{twist-2}}(x) = \frac{3}{2} \int_x^1 \frac{dy}{y} f_{1LL}^{\pm}(y)$$
  where the superscript $\pm$ denotes the combination of quark and antiquark distributions. This mirrors the classic Wandzura-Wilczek relation for $g_2$ in nucleons.

• The BC-like Sum Rule:
  By defining a modified structure function $f_{2LT}^{\pm}(x) = \frac{2}{3}f_{LT}^{\pm}(x) - f_{1LL}^{\pm}(x)$, the authors showed that the WW-like part satisfies:
  $$\int_0^1 dx \, f_{2LT}^{\pm \text{twist-2}}(x) = 0$$
  This is the tensor-polarized analog of the Burkhardt-Cottingham sum rule.

• Dynamical Twist-3 Term:
  The deviation from the WW-like relation is identified as the dynamical twist-3 term, proportional to the coefficient $d_n$ in the OPE, which represents genuine multi-parton interactions.

5. Significance

• Experimental Preparation: These relations provide a reliable first estimate for the tensor-polarized PDF $f_{LT}$ of the deuteron. Since the JLab experiment operates in a low-$Q^2$ regime where twist-3 effects are large, having a robust theoretical baseline (the WW-like part) is crucial for isolating the dynamical twist-3 contributions.
• New Physics Field: The work lays the groundwork for a new field of spin physics focusing on spin-1 hadrons. It highlights that tensor-polarized PDFs offer sensitivity to dynamical aspects beyond the simple sum of proton and neutron structure functions.
• Validation: The confirmation of the relations via two independent methods (nonlocal vs. local OPE) strengthens the theoretical framework necessary for analyzing future data from JLab, Fermilab, NICA, LHC, and Electron-Ion Colliders (EICs).

In conclusion, this paper solidifies the theoretical understanding of twist-3 tensor-polarized distributions, ensuring that upcoming experimental measurements at JLab can be interpreted with high confidence.