Weak bosons as partons below 10 TeV partonic center-of-momentum

This paper derives unrenormalized tree-level weak boson parton densities for leptons and hadrons within the Effective $W$ Approximation, introduces kinematical consistency conditions to resolve theoretical pathologies, and demonstrates that these conditions enable accurate approximations of multi-leg processes while suggesting the feasibility of testing the framework with same-sign $WW$ scattering data at the LHC.

The Gist

The Big Picture: Finding the "Ghost" Particles Inside a Proton

Imagine a proton (the tiny particle inside an atom's nucleus) not as a solid marble, but as a busy, chaotic cosmic food truck.

In the world of physics, we know this truck is mostly filled with "valence" ingredients (the main quarks that define what the truck is). But, because of the high energy inside, these main ingredients are constantly radiating energy, creating a "sea" of extra particles. Usually, we think of this sea as being made of gluons (the glue holding things together) and light quarks.

However, this paper asks a bold question: Can we also find "Weak Bosons" (particles like the W and Z that carry the weak nuclear force) floating around in this sea?

For a long time, physicists thought you needed "ultra-high" energies to see these weak bosons acting like partons (ingredients) inside a proton. This paper argues that the threshold is actually much lower than we thought—around 800 GeV (about 800 times the mass of a proton). If the energy is high enough, these weak bosons behave just like the other ingredients in the sea, and we can treat them as standard parts of the proton for our calculations.

The Problem: The "Recipe" Was Broken

Physicists have a standard recipe for calculating how these particles interact, called the Effective W Approximation (EWA). Think of this like a simplified recipe for baking a cake: "If you have flour and eggs, you can approximate the cake's weight by just weighing the flour."

For decades, this recipe worked well in some cases but failed in others. Sometimes, the math predicted that you could have negative amounts of ingredients (like -5 eggs), which is physically impossible. This happened because the recipe was being used in conditions where it didn't quite fit, specifically when the particles weren't moving perfectly straight or when the energy wasn't high enough.

The Solution: A New Set of "Safety Rules"

The authors of this paper went back to the kitchen and derived a more precise version of the recipe. They didn't just look at the main ingredients (Leading Power); they also looked at the tiny, messy details (Next-to-Leading Power) that usually get ignored.

They found that the "negative ingredient" problem happens when you try to use the recipe in two specific situations:

1. When the weak boson doesn't have enough energy (less than about 800 GeV).
2. When the particle is moving at a weird angle relative to the beam.

To fix this, they created a new set of Safety Rules (Kinematical Consistency Conditions).

• The Analogy: Imagine a rule that says, "You can only use this simplified cake recipe if the oven is hotter than 800 degrees and the batter is poured straight down."
• The Result: As long as these rules are followed, the simplified recipe (EWA) matches the complex, full calculation almost perfectly. The "negative eggs" disappear, and the math becomes reliable again.

The "Magic" Threshold

The paper identifies a specific "tipping point."

• Below the line: The weak bosons are just fleeting, messy fluctuations. You can't treat them as stable parts of the proton.
• Above the line (800 GeV): The weak bosons become "partons." They settle into the proton's "sea" and behave predictably, just like the quarks and gluons we are used to.

The authors show that once you cross this energy threshold, the complex, full math and the simplified "EWA" math agree with each other. This suggests that factorization (the ability to break a complex problem into smaller, manageable parts) actually works for weak bosons at these energies.

Why This Matters (According to the Paper)

The authors don't claim this will cure diseases or build new engines. Instead, they focus on how this helps physicists understand the universe's history and test their theories.

1. Testing the Theory: They suggest that the Large Hadron Collider (LHC) has enough data to actually test this. By looking at collisions where two protons smash together and produce two "same-sign" W bosons (a rare event), they estimate that with enough data, we could see about 30 to 300 events. This would be enough to prove that weak bosons really do act like partons inside the proton.
2. Understanding the Early Universe: The paper notes that understanding these "Weak Boson PDFs" (Parton Distribution Functions) is like having a laboratory probe for the universe's "Electroweak Epoch"—a time shortly after the Big Bang when the forces of nature were unified.
3. Better Simulations: By fixing the recipe, physicists can now simulate high-energy collisions more accurately without needing to do impossible amounts of complex math every time.

Summary

This paper is like finding the missing instruction manual for a complex machine. It tells us exactly when and how we can simplify our calculations of weak bosons. It says, "Don't try to use the shortcut if the energy is too low, but once you hit 800 GeV, the shortcut works perfectly, and the math stops breaking." This allows scientists to confidently study high-energy particle collisions and potentially see these elusive particles acting as building blocks of the proton right here at the LHC.

Technical Summary

Technical Summary: Weak Bosons as Partons Below 10 TeV Partonic Center-of-Momentum

Problem Statement
While the QCD-improved parton model successfully describes hadron structure via a "sea" of gluons and quarks, extending this framework to the electroweak (EW) sector remains theoretically incomplete. In the Standard Model, quarks and leptons carry weak charges, implying that $W$ and $Z$ bosons should emerge as perturbatively generated sea partons in ultra-high-energy collisions. However, the energy scale at which this "weak boson parton" description becomes a reliable approximation of full matrix elements has never been firmly established. Historically, predictions for weak boson parton distribution functions (PDFs) have varied significantly, and the Effective $W$ Approximation (EWA) has faced reported breakdowns and pathologies, particularly regarding gauge dependence and the violation of positive definiteness in PDFs. There is no all-orders factorization theorem for the EW sector, and it remains unclear if such a formulation is possible.

Methodology
The authors investigate the modeling of weak boson number densities for leptons and hadrons within the framework of the Effective $W$ Approximation (EWA). The study focuses on the process of a massless fermion radiating a quasi-collinear weak boson ($\mu^- \to \ell V_\lambda$).

• Theoretical Framework: Calculations are performed at tree level in the $R_\xi$ and axial gauges (lightcone, temporal, and spatial).
• Expansion Order: The authors derive unrenormalized PDFs at Next-to-Leading Power (NLP) in the collinear expansion, going beyond the standard Leading Log Approximation (LLA) and Leading Power (LP).
• Regularization: To handle ultraviolet divergences in the transverse momentum integral, a regulator $\mu_f$ with mass dimension one is introduced, analogous to previous EWA studies.
• Implementation: The derived NLP PDFs are implemented into the MadGraph5_aMC@NLO simulation framework to compute cross-sections for multi-leg processes ($e^+\mu^- \to \nu_e \nu_\mu hh$, $e^+\mu^- \to \nu_e \nu_\mu 3\gamma$, and $e^+\mu^+ \to \nu_e \nu_\mu W^+W^+$) at $\sqrt{s} = 5$ TeV. These results are compared against full 2$\to$n matrix element (ME) calculations.

Key Contributions

1. Derivation of NLP PDFs: The paper presents the first NLP-accurate calculation of helicity-polarized $W/Z$ PDFs for massless chiral leptons. These expressions (Eq. 6) include corrections of order $O(M_V^2/E_V^2)$ and $O(\mu_f^2/E_V^2)$.
2. Identification of Pathologies: The analysis reveals that NLP corrections exhibit specific pathologies:
   • Transverse PDFs ($\lambda = \pm 1$) can become negative when $\mu_f/E_V$ is large, violating positive definiteness.
   • Longitudinal PDFs ($\lambda = 0$) can become negative when $E_V$ is small relative to $M_V$.
   • These pathologies arise when kinematic variables are taken outside the strict collinear limit.
3. Kinematical Consistency Conditions: To suppress these pathologies, the authors derive a novel set of kinematical constraints (Eq. 10). For the EWA to be reliable, the weak boson energy $E_V$ must satisfy:
   • $E_V \gg M_V$ (specifically $E_V \gtrsim 10 M_V$).
   • $E_V \gg \mu_f$ (where $\mu_f$ acts as a threshold scale).
   • These conditions ensure that power corrections are negligible ($O(M_V^2/E_V^2) \sim 1\%$).
4. Gauge Robustness: The study demonstrates that while LP and LLA PDFs are robust across $R_\xi$ and axial gauges, NLP longitudinal PDFs exhibit gauge dependence in axial gauges. This dependence is attributed to the shuffling of terms between individual graphs, reinforcing the necessity of the derived kinematical cuts.

Results

• Agreement with Full Matrix Elements: When the kinematical consistency conditions are enforced (specifically $E_V > 800$ GeV for $W$ and $900$ GeV for $Z$), the shapes and normalizations of full matrix elements for multi-leg processes are well approximated by the EWA combined with fragmentation/bremsstrahlung contributions.
• Convergence: For processes like $W^+W^- \to hh$, NLP and LP predictions converge with the full ME for invariant masses $M(hh) \gtrsim 1.6$ TeV. Below this threshold, NLP predictions deviate significantly, often becoming negative or underestimating the cross-section.
• LLA vs. NLP: While LLA often recovers the full ME at large invariant masses, it does so with untrustworthy uncertainties because it neglects $O(M_V^2/\mu_f^2)$ terms. NLP provides more reliable uncertainty estimates but requires the kinematical cuts to avoid unphysical negative cross-sections.
• LHC Applicability: The authors estimate that at $\sqrt{s} = 13.6$ TeV, same-sign $WW$ scattering ($pp \to W^\pm W^\pm jj$) with $M(WW) > 1.6$ TeV could yield approximately 6 events with $450 \text{ fb}^{-1}$ of data (assuming standard selection efficiencies), suggesting the onset of tree-level factorization is testable at the LHC.

Significance and Claims
The paper claims a breakthrough in establishing the regime where weak bosons can be treated as partons. The primary significance lies in:

• Defining the Energy Scale: Firmly establishing that the "ultra-high energy" regime for weak boson parton descriptions begins around $800$ GeV (partonic center-of-momentum frame), corresponding to $E_V \approx 10 M_V$.
• Resolving Discrepancies: The derived kinematical conditions resolve historic discrepancies between EWA predictions and full matrix element computations by suppressing the pathologies that arise from inappropriate choices of $z$ and $\mu_f$.
• Onset of Factorization: The findings suggest the onset of tree-level factorization for the electroweak sector, where full matrix elements can be approximated by the convolution of weak boson PDFs and partonic cross-sections.
• Experimental Motivation: The work motivates the search for weak boson PDFs at the LHC, framing them as laboratory-based probes of the universe's electroweak epoch and essential for a complete understanding of EW symmetry breaking.

The authors conclude that while an all-orders factorization theorem is not yet proven, the derived NLP PDFs and kinematical constraints provide a practical and testable framework for simulating and understanding high-energy electroweak processes.