Sign Reversal of Boer-Mulders Functions from Semi-inclusive Deep-Inelastic Scattering to the Drell-Yan Process

This paper reviews the theoretical and experimental status of the predicted sign reversal of Boer-Mulders functions between semi-inclusive deep-inelastic scattering and Drell-Yan processes, demonstrating that current data supports this reversal for proton valence quarks while highlighting future prospects for testing it in pions at the Electron-Ion Collider.

The Gist

Imagine the inside of a proton (the particle found in the center of every atom) not as a solid ball, but as a chaotic, swirling dance floor. Inside this dance floor, tiny particles called quarks are zooming around.

For a long time, scientists thought these quarks just moved straight forward. But we now know they also wiggle side-to-side. This side-to-side wobble is called transverse momentum.

This paper is about a specific, strange rule in the universe of quantum physics (Quantum Chromodynamics, or QCD) regarding how these quarks spin and wobble. Here is the story in simple terms:

1. The "Boer-Mulders" Dance Move

Scientists have named a specific pattern of movement the Boer-Mulders function.

• The Analogy: Imagine a quark is a dancer. Usually, we think of the dancer's spin (their "transverse spin") as being independent of their side-to-side wobble.
• The Twist: The Boer-Mulders function says there is a secret link. If a quark is spinning one way, it prefers to wobble in a specific direction. It's like a dancer who, whenever they spin clockwise, instinctively steps to the left.

2. The Two Different Dance Floors (SIDIS vs. Drell-Yan)

The universe has two main "dance floors" where we can watch these quarks perform:

1. SIDIS (The "Shooting" Floor): We shoot a beam of electrons at a proton target. The electron hits a quark, and the quark flies out. This is like a billiard ball hitting another ball.
2. Drell-Yan (The "Crash" Floor): We smash two protons (or a proton and a pion) together. The quarks inside crash and create a new pair of particles. This is like a head-on car crash.

3. The Big Prediction: The "Sign Reversal"

Here is the mind-bending part. A fundamental rule of QCD predicts that the "secret link" (the Boer-Mulders function) should flip its sign depending on which dance floor you are on.

• The Analogy: Imagine the dancer has a rule: "If I am on the Billiard table, I step Left when I spin Clockwise."
• The Prediction: The theory says, "If you move that same dancer to the Car Crash table, the rules of the universe change. Now, if they spin Clockwise, they must step Right."

This is called Sign Reversal. It's like a mirror image. If the effect is positive in one experiment, it must be negative in the other.

4. The Mystery of the "Pion"

The paper focuses on two types of dancers:

• Protons: We have known for a while that protons have this "wobble-spin" link.
• Pions: These are lighter, unstable particles. Scientists predicted that pions also have this link, and it should also flip signs between the two dance floors. But nobody had proven it yet because pions are hard to catch and study.

5. What Did This Paper Do?

The authors, Jen-Chieh Peng and his team, acted like detectives reviewing the evidence.

• Checking the Proton: They looked at all the data from past experiments (like HERMES, COMPASS, and Fermilab). They found that the data for protons matches the prediction perfectly. The "wobble-spin" link in the proton did flip signs when moving from the Billiard table (SIDIS) to the Car Crash table (Drell-Yan). This confirms the theory is working for protons.
• The New Clue: They pointed out a recent experiment (COMPASS) that measured the "wobble-spin" link in pions during a crash. The data suggests the pion link also flips signs, just like the theory says it should.
• The Future Plan: To be 100% sure about the pion, we need a new kind of experiment. The authors suggest using a future machine called the Electron-Ion Collider (EIC).
  • The Trick: Since we can't hold a pion in a target (they decay too fast), they propose using a "Sullivan Process." Imagine the proton is a cloud. Sometimes, the proton sheds a tiny piece of itself (a pion) and keeps going. We can shoot electrons at this "ghost pion" floating in the air to study it directly.

Why Does This Matter?

If this sign reversal didn't happen, it would mean our understanding of the fundamental forces holding the universe together (the Strong Force) is broken. It would be like discovering that gravity pushes you up instead of down.

In Summary:
This paper confirms that the universe has a consistent, albeit weird, rule: the way quarks wobble and spin is linked, and this link flips direction depending on whether the quarks are being shot out of a target or crashing into each other. The authors have gathered the evidence to say, "Yes, the rule holds for protons, and it likely holds for pions too," and they are calling for a new high-tech experiment to prove it once and for all.

Technical Summary

1. Problem Statement

The paper addresses a fundamental prediction of Quantum Chromodynamics (QCD) regarding Transverse Momentum Dependent (TMD) parton distribution functions. Specifically, it focuses on the Boer-Mulders (BM) functions, which describe the correlation between a quark's transverse momentum ($\vec{k}_T$) and its transverse spin within an unpolarized hadron.

• The Core Prediction: QCD predicts that T-odd (time-reversal odd) functions, such as the Sivers and Boer-Mulders functions, must undergo a sign reversal when extracted from Semi-Inclusive Deep-Inelastic Scattering (SIDIS) (a space-like process) compared to the Drell-Yan (DY) process (a time-like process). This reversal arises from the different gauge-link structures (initial-state vs. final-state interactions) required for gauge invariance in these two processes.
• The Gap: While this sign reversal has been extensively studied and partially verified for the Sivers function (correlation of $\vec{k}_T$ and nucleon spin), the status of the Boer-Mulders function remains less clear. Existing SIDIS data has not yet provided a definitive determination of the BM function's sign, and previous DY data (unpolarized) suffered from a two-fold ambiguity regarding the signs of the underlying parton distributions.

2. Methodology

The authors employ a comprehensive review and synthesis of theoretical models and experimental data to test the sign-reversal hypothesis for BM functions.

• Theoretical Framework:

  • The paper reviews theoretical predictions from various models (Bag model, Quark-spectator-diquark model, Large-$N_c$ model, Relativistic constituent quark model, and Lattice QCD). These models consistently predict that the valence $u$ and $d$ quark BM functions in nucleons are negative in SIDIS.
  • Similar predictions are made for pion valence quarks and nucleon sea quarks (via meson-cloud models), suggesting a universal negative sign in SIDIS.
  • The QCD prediction dictates that these signs should flip to positive in the DY process.

• Data Analysis:

  • SIDIS Extraction: The authors analyze unpolarized SIDIS data (from HERMES and COMPASS) focusing on the $\cos 2\phi$ azimuthal angular modulation of produced pions. This modulation is proportional to the product of the nucleon BM function ($h_1^\perp$) and the Collins fragmentation function ($H_1^\perp$). By fitting the data and assuming a functional relationship between BM and Sivers functions, they extract the signs of the valence quark BM functions.
  • Unpolarized DY Analysis: They examine the $\cos 2\phi$ modulation in unpolarized DY experiments (NA10, E615, and Fermilab $p+p$/$p+d$). The coefficient $\nu$ of the $\cos 2\phi$ term is analyzed to determine the convolution of projectile and target BM functions.
  • Polarized DY Analysis: Crucially, the authors incorporate recent results from the COMPASS experiment involving a transversely polarized proton target in a $\pi^-$-induced DY process. They analyze the azimuthal asymmetry amplitude $A_T^{\sin(2\phi - \phi_S)}$, which isolates the convolution of the pion BM function and the proton transversity distribution.

3. Key Contributions

• Resolution of Sign Ambiguity: The paper demonstrates that while unpolarized DY data alone cannot distinguish between the "sign reversal" scenario (all positive in DY) and the "no sign reversal" scenario (all negative in DY), the inclusion of polarized DY data from COMPASS resolves this ambiguity.
• Validation of Sign Reversal: By correcting for the coordinate system convention used by COMPASS (where the $z$-axis definition differs from the standard convention), the authors show that the COMPASS measurement of a negative asymmetry actually corresponds to a positive sign in the standard convention. This confirms the predicted sign reversal for the proton valence quark BM function.
• Proposal for Pion BM Function: The paper proposes a novel experimental strategy to test the sign reversal for pion BM functions. Since pions cannot be used as fixed targets, the authors suggest using the Sullivan process at a future Electron-Ion Collider (EIC). This involves tagging a neutron to isolate the virtual pion cloud and performing SIDIS on the virtual pion to determine its BM sign, which can then be compared to DY results.

4. Results

• SIDIS Status: Analysis of HERMES and COMPASS SIDIS data, combined with theoretical constraints on the Collins function signs, yields negative signs for the proton valence $u$ and $d$ quark BM functions ($h_1^{\perp u} < 0$, $h_1^{\perp d} < 0$). This aligns with theoretical expectations.
• DY Status:
  • Unpolarized DY data shows a positive $\nu$ coefficient, consistent with either all-positive or all-negative signs for the convoluted BM functions.
  • Polarized DY (COMPASS): The measurement of the asymmetry $A_T^{\sin(2\phi - \phi_S)}$ in $\pi^- p$ collisions yields a value consistent with a positive sign for the proton's $u$-quark BM function in the DY process (after correcting for coordinate conventions).
• Conclusion on Sign Reversal: The transition from SIDIS (negative) to DY (positive) for the proton valence quark BM function is consistent with the QCD prediction of sign reversal. The two-fold ambiguity in Table II is resolved in favor of solution (a) (Sign Reversal).
• Pion BM Function: Theoretical models predict the pion valence BM function is also negative in SIDIS. The COMPASS data suggests it becomes positive in DY, but a definitive extraction of the pion BM sign from SIDIS is required for a complete test.

5. Significance

• Fundamental QCD Test: This work provides strong experimental evidence supporting the QCD prediction that T-odd TMDs change sign between space-like and time-like processes. This is a critical test of the gauge-invariant definition of TMDs and the role of initial/final state interactions (gauge links).
• Beyond Sivers: It extends the verification of sign reversal beyond the well-studied Sivers function to the Boer-Mulders function, which is crucial for understanding the spin-momentum correlations in unpolarized hadrons.
• Future Roadmap: The paper outlines a clear path for future research, specifically highlighting the Electron-Ion Collider (EIC) and the Sullivan process as the necessary tools to fully map the flavor and hadron-type dependence (proton vs. pion) of the Boer-Mulders function and confirm the universality of the sign-reversal phenomenon across the meson sector.

In summary, the paper synthesizes existing experimental data to argue that the Boer-Mulders function exhibits the predicted QCD sign reversal from SIDIS to Drell-Yan, marking a significant step in validating the theoretical framework of TMD factorization.