Accessing nucleon transversity with one-point energy correlators

This paper proposes using the one-point energy correlator (OPEC) in transversely polarized proton-proton collisions as a novel, infrared-and-collinear safe method to probe the nucleon's transversity distribution with a clean angular asymmetry over a wider kinematic range than traditional hadron transverse momentum measurements.

The Gist

Imagine the proton (the core of every atom) not as a solid marble, but as a bustling, chaotic city made of tiny, fast-moving particles called quarks and gluons. For decades, physicists have been trying to map this city. They know where the "traffic" (unpolarized quarks) is heavy and where the "police cars" (spinning quarks) are driving in a straight line. But there's a hidden layer of the city they can't see well: the sideways spin of these particles.

This sideways spin is called transversity. It's like trying to figure out if a spinning top is wobbling left or right while it's moving forward. It's incredibly hard to measure because the tools we usually use only see the top moving forward, not its wobble.

This paper proposes a brand-new, high-tech "drone" to map this wobble. Here is the breakdown in simple terms:

1. The Old Way: Catching a Single Fly

Traditionally, to see this sideways spin, physicists smash protons together and look for a specific particle (like a pion) flying out of the collision. They try to guess the spin based on which way that single "fly" (particle) veers off course.

• The Problem: It's like trying to understand the wind patterns of a hurricane by watching just one leaf blow by. You get a hint, but the leaf might have been pushed by other things (like the messy process of the particle breaking apart, called "fragmentation"). It's noisy and hard to trust.

2. The New Way: The "Energy Correlator" Drone

The authors propose using something called a One-Point Energy Correlator (OPEC).

• The Analogy: Instead of watching one leaf, imagine you have a super-sensitive camera that measures the total energy of everything flying out of the crash in a specific direction.
• How it works: When two protons smash, they create a "jet" (a spray of particles). The OPEC measures how the energy is distributed across the whole spray, not just one particle. It's like measuring the shape of the entire explosion cloud rather than just tracking one shrapnel piece.
• Why it's better: Because it sums up all the energy, it cancels out a lot of the "noise" and messy details. It's a cleaner, more direct way to see the underlying structure.

3. The "Wobble" Signal

The paper predicts that if you smash protons where one is spinning sideways (transversely polarized), this energy cloud won't be a perfect circle. It will have a specific, rhythmic wobble (a sine wave pattern) as you look around the circle.

• The Metaphor: Imagine spinning a pizza dough in the air. If you spin it perfectly, it's a flat circle. But if you give it a little sideways nudge while spinning, it wobbles. The OPEC measures that wobble. The size and shape of the wobble tell us exactly how strong the "sideways spin" (transversity) of the proton's ingredients is.

4. Why This Matters

• Mapping the Proton: This gives us a new, high-resolution map of the proton's 3D structure. We can finally see how the "sideways spin" is distributed.
• Testing the Rules of Physics: The paper suggests this method can test a fundamental rule of the universe called "universality." It asks: Does the way a quark spins and breaks apart look the same whether it happens in a particle collider on Earth or in a deep-space collision? By comparing this new method with old ones, we can prove if our understanding of the universe's rules is consistent.
• Looking for New Physics: If the wobble doesn't match our predictions, it might mean there are new, unknown forces at play (Beyond the Standard Model), which could explain mysteries like why the universe has more matter than antimatter.

Summary

Think of this paper as inventing a new type of MRI machine for subatomic particles.

• Old MRI: Blurry, only sees the surface, relies on guessing.
• New MRI (OPEC): Crystal clear, sees the deep internal structure, and ignores the static noise.

The authors are saying, "We have a new tool that can see the proton's sideways spin much better than before. We can use existing machines (like the RHIC collider) to test this immediately, and it will pave the way for even better maps at the future Electron-Ion Collider."

Technical Summary

1. Problem Statement

The nucleon's internal structure, specifically the transversity distribution function ($h_1^q$), remains one of the least constrained parton distribution functions (PDFs). Unlike the unpolarized ($f_1^q$) and helicity ($g_1^q$) distributions, transversity is chiral-odd, making it inaccessible in inclusive scattering processes.

• Current Limitations: Existing extractions rely on Semi-Inclusive Deep Inelastic Scattering (SIDIS) and proton-proton ($pp$) collisions via the Collins effect (azimuthal asymmetry in hadron production) or di-hadron production.
• Theoretical Bottleneck: These methods depend heavily on non-perturbative fragmentation functions (FFs). The extraction of transversity is entangled with the modeling of these FFs, introducing significant uncertainties. Furthermore, traditional measurements of hadron transverse momentum ($j_\perp$) are limited in angular resolution and kinematic range.
• Goal: The authors propose a novel, theoretically cleaner observable to probe transversity that reduces model dependence and accesses a broader kinematic range, specifically utilizing One-Point Energy Correlators (OPEC) within jet substructure.

2. Methodology

The paper introduces the One-Point Energy Correlator (OPEC) as a probe for transversity in transversely polarized proton-proton collisions ($p^\uparrow p \to J + X$).

A. The Observable: OPEC

The OPEC measures the angular distribution of energy flux inside a jet. It is defined by the energy flow operator $\mathcal{E}(\hat{n})$:
$$\mathcal{E}(\hat{n}) = \int_0^\infty dt \lim_{r\to\infty} r^2 n^i T_{0i}(t, r\hat{n})$$
The observable is the weighted sum of energy fractions ($z_h$) of hadrons inside a jet, projected onto a specific angular direction $\hat{n}$ relative to the jet axis:
$$\frac{d\Sigma}{d\theta_n d\phi_n d\eta dp_T} = \sum_{h \in J} \int_0^1 dz_h \int d^2\Omega_h \, \delta(\phi_n - \phi_h)\delta(\theta_n - \theta_h) z_h \frac{d\sigma}{dz_h d^2\Omega_h d\eta dp_T}$$

• Key Feature: Unlike traditional hadron-in-jet studies that measure specific hadron species, OPEC measures the total energy flow, making it Infrared and Collinear (IRC) safe.

B. Factorization and Formalism

The authors derive a factorization formula for the OPEC in the collinear limit, introducing a new class of Fragmenting Jet Functions (FJFs):

• Unpolarized FJF ($J_q$): Describes the energy flow from an unpolarized quark.
• Transversely Polarized FJF ($J_{1,\perp}^q$): Describes the energy flow from a transversely polarized quark. This function captures the azimuthal asymmetry induced by the Collins effect but in terms of energy flow rather than specific hadrons.

The cross-section is decomposed into unpolarized ($Z_{UU}$) and spin-dependent ($Z_{UT}$) structure functions. The spin-dependent term exhibits a characteristic $\sin(\phi_s - \phi_n)$ modulation, where $\phi_s$ is the proton spin angle and $\phi_n$ is the detector angle.

C. Matching to Collinear FFs

In the limit where the angle $\theta_n \gg \Lambda_{QCD}/Q$, the OPEC FJFs are matched onto standard collinear Fragmentation Functions (FFs) and the Collins function ($H_1^{\perp}$) via Operator Product Expansion (OPE):

• The unpolarized OPEC matches to the standard unpolarized FF ($D_{h/i}$).
• The polarized OPEC matches to the first moment of the Collins function ($\hat{H}^{(3)}_{h/i}$).
  This matching allows the use of existing global fits for Collins FFs while providing a new, cleaner observable.

D. Numerical Frameworks

The authors perform numerical evaluations using two distinct frameworks to estimate the Single-Spin Asymmetry (SSA):

1. TMD Evolution Framework: Utilizes full Transverse Momentum Dependent (TMD) evolution at Next-to-Leading Logarithmic (NLL) accuracy, incorporating QCD evolution and Sudakov factors.
2. JAM3D-22 Framework: Uses the JAM3D-22 global analysis, which employs a Gaussian model for transverse momentum dependence and DGLAP evolution, omitting full TMD resummation.

3. Key Contributions

• Novel Observable: First proposal of using OPEC to access nucleon transversity, establishing a direct link between energy correlators and spin physics.
• Reduced Model Dependence: By integrating over the energy fraction $z_h$, OPEC projects the full $z_h$-dependence onto a single moment. This reduces the dimensionality of the fit (from $(b, z_h)$ to just $b$) and minimizes reliance on specific hadronization models.
• Extended Kinematic Reach: OPEC allows for measurements at much smaller angular scales ($\theta_n \sim 10^{-3}$ to $10^{-4}$ rad) compared to traditional $j_\perp$ measurements, accessing a broader range of jet substructure.
• Universality Test: The formalism provides a rigorous method to test the universality of the Collins function by comparing OPEC measurements in $e^+e^-$ annihilation, SIDIS, and $pp$ collisions.

4. Results

The authors present phenomenological predictions for the SSA ($A_{UT}^{\sin(\phi_s-\phi_n)}$) in $p^\uparrow p$ collisions at RHIC energies ($\sqrt{s} = 200$ GeV and $510$ GeV).

• Magnitude of Asymmetry:
  • The TMD evolution framework predicts systematically smaller asymmetries compared to the JAM3D-22 approach.
  • The dominant source of uncertainty in both frameworks arises from the Collins FF parametrization, while the uncertainty from the transversity PDF is comparatively smaller.
• Angular Dependence:
  • The SSA exhibits a distinct peak position in $\theta_n$ that shifts with collision energy, reflecting changes in the fragmentation dynamics.
  • The results show partial overlap between the two frameworks, suggesting that current data may not yet fully resolve TMD evolution effects, but future high-precision data (e.g., at higher $p_T$) could distinguish them.
• Weighting Sensitivity:
  • The paper explores generalizing the OPEC by weighting with $z_h^n$. Increasing the power $n$ (probing higher Mellin moments of the Collins function) leads to a remarkable enhancement in the magnitude of the SSA.
• Scale Uncertainties:
  • Theoretical scale uncertainties in the NLL TMD framework are comparable to parametrization uncertainties. The JAM3D approach shows smaller scale sensitivity due to the lack of resummation.

5. Significance

• Complementary Channel: OPEC offers a complementary and systematically distinct channel to study nucleon 3D structure, independent of traditional hadron-in-jet measurements.
• Experimental Feasibility: The observable is well-suited for current experiments like STAR at RHIC (using existing data) and future high-precision measurements at the Electron-Ion Collider (EIC).
• Theoretical Robustness: As an IRC-safe observable, OPEC provides a cleaner theoretical input, reducing the "noise" from non-perturbative hadronization models.
• Fundamental Physics: This work establishes a vital link between the formal theory of energy flow (energy correlators) and the non-perturbative partonic structure of the nucleon, paving the way for precision tests of QCD factorization and the universality of spin-dependent fragmentation.

In summary, the paper proposes a powerful new tool to extract the elusive transversity distribution and test the universality of the Collins effect, leveraging the precision and theoretical cleanliness of jet substructure observables.