QCD analysis of the ATLAS and CMS $W^{\pm}$ and $Z$ cross-section measurements and implications for the strange sea density

This paper presents a next-to-next-to-leading order QCD analysis of combined ATLAS and CMS $W^{\pm}$ and $Z$ boson production data to assess potential tensions between the datasets and constrain the strange sea quark density.

The Gist

The Big Picture: The Proton as a Busy City

Imagine a proton (the tiny particle inside an atom) not as a solid ball, but as a bustling, chaotic city. Inside this city, there are different types of "citizens" zooming around:

• The "Valence" Citizens: These are the permanent residents (up and down quarks) that give the city its identity.
• The "Sea" Citizens: These are the tourists and temporary workers (quark-antiquark pairs) that pop in and out of existence. Most are "light" tourists (up and down), but there is a specific group called "Strange" tourists (strange quarks).

For a long time, physicists believed that the "Strange" tourists were rare. They thought that for every two "light" tourists, there was only one "strange" tourist. They assumed the strange population was suppressed (about 50% of the light population).

The Investigation: Two Detective Agencies

To figure out exactly how many "Strange" tourists are in the city, two major detective agencies—ATLAS and CMS—at the Large Hadron Collider (LHC) in Switzerland started counting. They looked at how protons collide to create heavy particles called W and Z bosons (think of these as the "traffic jams" or "accidents" that happen when citizens crash into each other).

• The Old Clues: Previously, scientists relied on data from a machine called HERA. It was like looking at the city from a high tower; it could see the general layout but couldn't clearly distinguish the "Strange" tourists from the "Light" ones.
• The New Clues: ATLAS and CMS have much sharper eyes. They can see the "Strange" tourists more clearly. However, there was a problem:
  • ATLAS said: "Hey, there are actually as many Strange tourists as Light tourists! The population is equal!"
  • CMS said: "Not quite. There are fewer Strange tourists than ATLAS thinks, though maybe not as few as the old theory suggested."

The Detective Work: Putting the Puzzle Together

The authors of this paper (Cooper-Sarkar and Wichmann) decided to act as the Chief Investigators. They took all the data from both agencies (ATLAS and CMS) and combined it with the old HERA data. Their goal was to answer two questions:

1. Are the two detective agencies (ATLAS and CMS) fighting, or do their stories actually match?
2. What is the true ratio of "Strange" to "Light" tourists?

They used a sophisticated mathematical tool (called NNLO QCD) which is like a super-advanced simulation engine. It runs millions of scenarios to see which version of the "city" fits the observed traffic accidents best.

The Findings: The "Strange" Truth

Here is what they discovered:

1. The Agencies Agree (Mostly)
When they put the data together, they found that ATLAS and CMS are not in a major fight. While their numbers were slightly different, the difference wasn't big enough to say one was wrong. It's like two weather stations reporting slightly different temperatures; when you average them out, you get a very accurate picture.

2. The "Strange" Tourists are Abundant
The most exciting result is about the "Strange" population.

• The Old Theory: Said the ratio was roughly 0.5 (half as many strange as light).
• The New Result: The data shows the ratio is 1.0 (or very close to it).
• The Analogy: Imagine a party where you expected only 50% as many people wearing red hats (Strange) as blue hats (Light). After counting everyone carefully, you realize there are actually equal numbers of red and blue hats. The "Strange" sea is unsuppressed.

3. Why the Difference?
The paper explains that the "Strange" population changes depending on how hard you look (the energy scale).

• At low energy (like looking at the city from a distance), the old data suggested suppression.
• But at the high energy of the LHC (looking with a microscope), the "Strange" tourists multiply because of the way energy turns into matter. The data confirms that at these high energies, the strange population has grown to match the light population.

The Conclusion

The paper concludes that the proton's "sea" is much more balanced than we thought. The "Strange" quarks are not the shy, hidden minority they were once thought to be; they are a major part of the crowd.

In a nutshell:
Physicists combined data from two giant particle colliders to solve a mystery about the makeup of the proton. They found that a specific type of particle, the "strange" quark, is just as common as the lighter ones at high energies. The two major experiments (ATLAS and CMS) agree on this new picture, confirming that our understanding of the proton's internal city needs an update: the "Strange" residents are everywhere!

Technical Summary

1. Problem Statement

The primary objective of this study is to resolve uncertainties regarding the strange quark sea density ($s$ and $\bar{s}$) within the proton, specifically in the low Bjorken-$x$ region ($x \sim 0.01$).

• Historical Context: Traditionally, strange sea constraints came from neutrino scattering experiments (NuTeV, CCFR, NOMAD) via dimuon production ($W^\pm s \to c$). These suggested a suppressed strange sea, often parameterized as $\bar{s}(x) = r_s \bar{d}(x)$ with $r_s \approx 0.5$. However, these measurements were at higher $x$ ($x > 0.1$) and suffered from uncertainties related to charm fragmentation and nuclear corrections.
• The Conflict: Recent high-precision measurements of inclusive $W$ and $Z$ boson production at the LHC by the ATLAS and CMS collaborations offered new constraints.
  • ATLAS data (specifically $W+c$ and inclusive $W/Z$) suggested a symmetric light quark sea at low $x$ (i.e., no suppression, $r_s \approx 1$).
  • CMS $W+c$ data suggested a somewhat smaller strangeness.
• The Question: Is there a genuine tension between the ATLAS and CMS datasets regarding the strange sea fraction, and what is the precise value of the strange sea density at LHC scales?

2. Methodology

The authors performed a global Parton Distribution Function (PDF) fit within the framework of perturbative Quantum Chromodynamics (pQCD) at Next-to-Next-to-Leading Order (NNLO).

• Framework: The analysis utilized the xFitter framework.
• Input Data:
  • HERA: Combined inclusive $e^\pm p$ cross-section data (covering $Q^2$ from 0.045 to 50,000 GeV$^2$ and $x$ down to $6 \times 10^{-7}$).
  • ATLAS: Inclusive $W^\pm$ and $Z/\gamma^*$ differential cross-sections at 7 TeV (lepton pseudo-rapidity and boson rapidity).
  • CMS: Inclusive $W^\pm$ (7 TeV asymmetry and 8 TeV separate $W^+/W^-$) and $Z/\gamma^*$ (7 TeV double-differential) data.
  • Note: The 8 TeV CMS $Z$ data was excluded from the main fit due to unreasonably large $\chi^2$ values in preliminary tests, consistent with other PDF fitting groups.
• Theoretical Calculations:
  • DIS: NNLO light quark coefficient functions (QCDNUM) and General-Mass Variable Flavor Number Scheme (GM-VFNS) for heavy quarks.
  • Drell-Yan ($W/Z$): Calculations performed at NLO in QCD and LO in Electroweak (EW) using MCFM/APPLGRID, corrected to NNLO using K-factors derived from FEWZ3.1 and DYNNLO1.5.
• Parameterization:
  • Initial scale $Q_0^2 = 1.9$ GeV$^2$ (below charm threshold).
  • Quark distributions parameterized as $xq_i(x) = A_i x^{B_i} (1-x)^{C_i} P_i(x)$.
  • Strange Sea: Parameterized with $B_{\bar{s}} = B_{\bar{d}}$ but with free normalization ($A_s$) and shape ($C_s$) parameters. The assumption $x s = x \bar{s}$ was made by default.
• Statistical Treatment:
  • Minimization using MINUIT.
  • Correlated systematic uncertainties handled via nuisance parameters (ATLAS) and covariance matrices (CMS).
  • Uncertainties estimated via the Hessian method ($\Delta \chi^2 = 1$) and verified with the Monte Carlo replica method.

3. Key Contributions

1. Joint Analysis of ATLAS and CMS: This is one of the first rigorous NNLO QCD fits to simultaneously analyze the inclusive $W$ and $Z$ data from both major LHC experiments to test for internal tensions.
2. Resolution of $W+c$ Discrepancy: The authors investigated whether the disagreement between ATLAS and CMS $W+c$ data (regarding strangeness) persisted in the inclusive Drell-Yan data. They found that while CMS $W+c$ favored lower strangeness, the inclusive $W/Z$ data from both experiments are largely consistent when analyzed together, with ATLAS data dominating the constraint due to higher precision.
3. Quantification of Systematic Shifts: The study explicitly analyzed the shifts in nuisance parameters (systematic uncertainties) to demonstrate that the datasets are statistically consistent, despite the central values of the extracted strange PDFs differing slightly.

4. Results

• Consistency of Data:
  • There is no significant tension between HERA and LHC data.
  • There is no strong tension between ATLAS and CMS datasets. The $\chi^2$ values remain good when datasets are combined.
  • The 8 TeV CMS $Z$ data showed poor fit quality (high $\chi^2$) and was excluded from the nominal fit, but its inclusion did not significantly alter the strange sea results in the sensitive $x \sim 0.01$ region.
• Strange Sea Density:
  • The combined fit (referred to as CSKK) indicates that the strange sea is unsuppressed at low $x$.
  • The ratio of the strange sea to the light sea, $R_s = (s + \bar{s}) / (\bar{u} + \bar{d})$, is found to be consistent with unity ($R_s \approx 1$) at low $x$.
  • Quantitative Results:
    • At $x = 0.023, Q_0^2 = 1.9$ GeV$^2$: $R_s = 1.14 \pm 0.05 (\text{exp}) \pm 0.03 (\text{model}) +0.03_{-0.05} (\text{param}) +0.01_{-0.02} (\alpha_s)$.
    • At $x = 0.013, Q^2 = M_Z^2$: $R_s = 1.05 \pm 0.02 (\text{exp}) +0.02_{-0.01} (\text{model}) +0.02_{-0.01} (\text{param}) \pm 0.01 (\alpha_s)$.
• Dominance of ATLAS: The ATLAS data, having higher experimental precision (0.4–1.0% vs CMS ~1–3%), dominates the combined fit, driving the result toward unsuppressed strangeness. However, the CMS data is not in contradiction with this result.
• Uncertainties: The total uncertainty is dominated by parameterization uncertainties (specifically allowing the low-$x$ slope of $\bar{d}$ and $\bar{s}$ to vary independently) rather than experimental errors.

5. Significance

• Paradigm Shift: The results challenge the long-standing historical assumption of a suppressed strange sea ($r_s \approx 0.5$) at low $x$. The data strongly supports a symmetric light quark sea ($s \approx \bar{u} \approx \bar{d}$) at the LHC scale.
• Theoretical Validation: The study confirms that inclusive Drell-Yan processes at the LHC provide robust, high-precision constraints on the strange quark distribution, complementary to and more precise than previous neutrino scattering data in the low-$x$ regime.
• PDF Global Fits: The findings provide a critical benchmark for global PDF fitting groups (like NNPDF, CT, MSHT), suggesting that future fits should allow for an unsuppressed strange sea at low $x$ to accurately describe LHC data.
• Robustness: The result holds even when varying model parameters (heavy quark masses, scales) and parameterization forms, confirming the robustness of the "unsuppressed" conclusion.

In conclusion, the paper establishes that the strange quark sea in the proton is not suppressed relative to the light quark sea at low $x$ ($x \sim 0.01$) and high $Q^2$, with the combined ATLAS and CMS inclusive $W/Z$ data providing the most stringent evidence for this conclusion to date.