Dimuon production in neutrino-nucleus collisions at next-to-next-to-leading order in perturbative QCD

This paper presents a self-contained next-to-next-to-leading-order (NNLO) perturbative QCD calculation of dimuon production in neutrino-nucleus deep-inelastic scattering based on the semi-inclusive DIS framework, demonstrating that these corrections significantly reduce scale uncertainties at large momentum fractions and alleviate the tension between dimuon and LHC data at small momentum fractions by introducing negative corrections.

The Gist

Here is an explanation of the paper, translated into everyday language with some creative analogies.

The Big Picture: Solving a "Strange" Mystery

Imagine the universe is built out of tiny Lego bricks called quarks. Most of the time, we see the common bricks: "up" and "down" quarks. But there are also "strange" bricks, which are heavier and rarer.

Scientists have been trying to build a perfect model of how these bricks are arranged inside a proton (the nucleus of an atom). This model is called a Parton Distribution Function (PDF). Think of the PDF as a recipe book that tells you exactly how much of each type of quark is in a proton at different speeds.

The Problem:
For a long time, two different groups of scientists have been arguing about the "strange" bricks:

1. The Neutrino Team: They shoot neutrinos (ghostly particles) at heavy metal targets. When a neutrino hits a "strange" quark, it creates a charm quark, which eventually decays into a pair of muons (a "dimuon"). Based on their data, they think there are very few strange bricks in the proton.
2. The LHC Team: They smash protons together at the Large Hadron Collider (LHC). Their data suggests there are plenty of strange bricks, and they aren't suppressed at all.

It's like two chefs looking at the same cake. One says, "There's barely any chocolate in here!" while the other says, "It's half chocolate!" They can't agree, and it's messing up the recipe book.

The Solution: A New, More Precise Calculator

The authors of this paper decided to build a super-precise calculator to settle the argument.

In physics, calculations are done in steps of accuracy:

• LO (Leading Order): The rough sketch. Good for a quick guess.
• NLO (Next-to-Leading Order): A detailed drawing. Better, but still has some fuzzy edges.
• NNLO (Next-to-Next-to-Leading Order): The high-definition, 4K photo. This is what this paper is about.

They took the "dimuon" experiment (the neutrino team's method) and recalculated it using this super-precise NNLO math.

The "Dimuon" Machine: A Factory Analogy

To understand how they measured the strange quarks, imagine a factory assembly line:

1. The Input: A neutrino hits a nucleus (the factory).
2. The Collision: It hits a "strange" quark inside the nucleus.
3. The Transformation: The strange quark turns into a "charm" quark (like a raw material turning into a semi-finished product).
4. The Decay: The charm quark is unstable. It quickly breaks apart (decays) into a muon and other stuff.
5. The Output: Because the original neutrino also turned into a muon, you end up with two muons flying out. This is the "dimuon."

The Old Way (The Flawed Approximation):
Previously, scientists calculated the collision (Step 3) very carefully, but then they just guessed the rest of the process (Steps 4 and 5) using a simple multiplier. It was like calculating the cost of the raw wood for a chair, but just guessing the cost of the glue and the labor. At high precision, this "guess" introduces errors.

The New Way (The SIDIS Approach):
The authors changed the whole approach. Instead of calculating the collision and then guessing the decay, they calculated the entire chain as one continuous process.

• They treated the decay of the charm quark not as a guess, but as a specific, measured event (using data from electron-positron collisions).
• They accounted for the "mass" of the heavy quarks (like the weight of the raw materials) right from the start.

This is like calculating the cost of the wood, the glue, the labor, and the shipping all in one go, rather than guessing the shipping cost later.

What Did They Find?

When they ran their super-precise NNLO calculator, two major things happened:

1. The "Scale" Problem Got Smaller
In physics, calculations often depend on arbitrary choices (like where you set the "zero" point). If your answer changes wildly depending on where you set that zero, your calculation is shaky.

• At low speeds (small $x$): The NNLO calculation was still a bit shaky, similar to the old NLO method.
• At high speeds (large $x$): The NNLO calculation became incredibly stable. The "fuzziness" (uncertainty) shrank significantly. This means we can trust the results much more when the particles are moving fast.

2. The "Strange" Tension is Easing
This is the most exciting part.

• The old calculations said: "The strange quark count is very low."
• The LHC data said: "The strange quark count is high."
• The New NNLO Calculation: It found that at low speeds, the corrections actually lowered the predicted cross-section (the probability of the event happening).

The Analogy:
Imagine you are trying to estimate how many people are in a crowded room.

• Old Method: You count the people near the door and guess the rest. You think there are 500 people.
• LHC Method: They look at the whole room from a drone and see 800 people.
• New Method: You use a high-tech scanner. You realize that at the back of the room (low speed), your old guess was actually too high because you missed some shadows. Your new scanner says, "Actually, there are only 600 people."

While 600 isn't 800, it's closer than 500 was. The gap between the two groups is shrinking. The new math suggests that the "strange" quarks aren't as suppressed as the neutrino data originally suggested, bringing the two conflicting datasets closer to an agreement.

The "New Channels" Surprise

When you go from a 2D drawing to a 4K photo, you sometimes see details you missed before.

• In the old math, only certain types of quarks could start the reaction.
• In the new NNLO math, new pathways opened up. For example, "up" quarks (which are very common) could now participate in the reaction in a way they couldn't before.
• The Result: These new pathways were interesting, but they turned out to be tiny. They didn't change the final answer much. The main story was still about the "strange" quarks.

Conclusion: Why This Matters

This paper is a crucial step toward a unified "Recipe Book" for the universe.

1. Better Tools: They built a more accurate calculator (NNLO) that handles heavy particles correctly.
2. Less Uncertainty: They reduced the "fuzziness" in the results, especially at high energies.
3. Bridging the Gap: They showed that when you do the math perfectly, the disagreement between the Neutrino experiments and the LHC experiments gets smaller.

It doesn't solve the mystery completely yet (we still need better data on how particles break apart), but it proves that the "strange" quark isn't as rare as we thought, and it gives us a much clearer path to understanding the fundamental building blocks of matter.

In short: They took a blurry, conflicting photo of the subatomic world, sharpened the lens, and found that the picture is actually much more consistent than we thought.

Technical Summary

Here is a detailed technical summary of the paper "Dimuon production in neutrino-nucleus collisions at next-to-next-to-leading order in perturbative QCD" by Helenius, Paukkunen, and Yrjänheikki.

1. Problem Statement

The production of dimuons in charged-current neutrino-nucleus deep-inelastic scattering (DIS) is a critical observable for constraining the strange-quark parton distribution function (PDF) in global analyses. Historically, neutrino data (from CCFR, NuTeV, NOMAD) have indicated a suppressed strange sea (ratio $(s+\bar{s})/(\bar{u}+\bar{d}) \approx 0.5$ at $x \sim 0.02$), whereas recent LHC data (W/Z production) favor an unsuppressed strange sea. This creates a significant tension in global PDF fits.

Current theoretical descriptions of dimuon production often rely on a factorized approximation: relating the dimuon cross-section to inclusive charm production via an acceptance correction and a branching ratio. This approach faces several theoretical limitations at higher orders:

• Infrared Unsafety: At Next-to-Next-to-Leading Order (NNLO), inclusive charm production becomes infrared unsafe due to logarithmic contributions ($\log(Q^2/m_c^2)$) that are not cancelled without a requirement for a charm quark in the final state.
• Approximation Errors: The traditional factorization formula (Eq. 1.1) assumes the acceptance correction is independent of the perturbative order, which is inaccurate at NNLO.
• Missing Channels: Standard calculations often neglect new partonic channels that open up at NNLO, such as up-quark initiated processes.

2. Methodology

The authors present a self-contained NNLO perturbative QCD calculation based on the Semi-Inclusive DIS (SIDIS) framework, rather than the traditional inclusive charm approach.

• Theoretical Framework:

  • Process: The calculation treats the full chain: $\nu N \to \mu h X$ (semi-inclusive charm hadron production) followed by the decay $h \to \mu X$.
  • Factorization: The cross-section is expressed as a convolution of Nuclear PDFs ($f_i$), hard coefficient functions ($C_{ij}$), and Fragmentation Functions (FFs, $D_j$).
  • Scheme: The calculation utilizes a General-Mass Variable-Flavor-Number Scheme (GM-VFNS). Specifically, it employs the SACOT-$\chi$ scheme up to NLO and implements NNLO corrections using massless coefficient functions (from Ref. [71]) combined with a slow-rescaling (SR) or intermediate-mass (IM) approximation. This accounts for the dominant kinematic mass effects ($Q^2 \gtrsim 4 \text{ GeV}^2$) without requiring full massive NNLO coefficient functions.
  • Decay Treatment: The non-perturbative decay of the charm hadron is handled by parametrizing the differential decay width (fitted to $e^+e^-$ data), allowing for a natural evaluation of fiducial cross-sections without ad-hoc acceptance corrections.

• Inputs and Scales:

  • PDFs: The TUJU21 nuclear PDF set (for Iron, $A=56$) is used, evolved to NNLO.
  • FFs: Since no dedicated NNLO FFs exist, the authors use NLO parametrizations (kkks08, bkk05) and evolve them to NNLO using the EKO framework.
  • Scales: Renormalization ($\mu_{ren}$), factorization ($\mu_{fact}$), and fragmentation ($\mu_{frag}$) scales are set to $\mu_0^2 = Q^2 + m_c^2$. Scale uncertainties are estimated via a 17-point variation.

3. Key Contributions

• Resolution of Infrared Unsafety: By treating the process as SIDIS (requiring a specific final-state hadron), the calculation naturally resums mass logarithms into the scale dependence of the FFs, resolving the infrared unsafety inherent in inclusive charm calculations at NNLO.
• Elimination of Acceptance Corrections: The differential nature of the decay width allows the calculation to directly predict the cross-section within experimental kinematic cuts, removing the need for the approximate acceptance factor $A$ used in previous factorized approaches.
• New Partonic Channels: The study explicitly investigates new NNLO channels, including up-quark initiated processes ($u \to c$) which become relevant at large $x$ due to the valence up-quark distribution.

4. Results

• Perturbative Convergence:
  • The NNLO corrections generally reduce the cross-section at small $x$ and enhance it at large $x$.
  • The magnitude of NNLO corrections is systematically smaller than NLO corrections, indicating good perturbative convergence, particularly at large $x$ where scale uncertainties are significantly reduced.
  • At small $x$, scale uncertainties remain comparable to NLO, suggesting slower convergence in this region.
• Partonic Channels:
  • The dominant contributions remain the strange-initiated ($s \to c$) and gluon-initiated ($g \to c$) channels.
  • New NNLO channels (e.g., $u \to c$) are found to be subleading (orders of magnitude smaller than dominant channels) but non-negligible in the valence region ($x > 0.2$).
  • Gluon-induced NNLO corrections are typically negative, while quark-induced ones are positive, leading to partial cancellation in the total sum.
• Comparison with Data (NuTeV):
  • Using the TUJU21 PDFs (which assume unsuppressed strangeness, $\bar{s}=s=\bar{d}=\bar{u}$), the theoretical prediction overestimates the NuTeV dimuon data by 20–30%.
  • When using EPPS21 PDFs (which incorporate a compromise between neutrino and LHC data, implying suppressed strangeness), the agreement with NuTeV data improves significantly.
  • This confirms that the "tension" between neutrino and LHC data is sensitive to the perturbative order and the specific PDF assumptions regarding strange suppression.

5. Significance

• Resolving PDF Tensions: The results suggest that NNLO corrections tend to be negative at small $x$, which helps alleviate the tension between the suppressed strange sea preferred by dimuon data and the unsuppressed sea preferred by LHC data.
• Precision for Future Experiments: As new data from the High-Luminosity LHC (HL-LHC) and the Electron-Ion Collider (EIC) become available, NNLO precision for nuclear PDFs will be essential. This work provides a robust theoretical framework for interpreting future neutrino-nucleus data.
• Methodological Advancement: The SIDIS-based approach sets a new standard for calculating heavy-flavor production in DIS, offering a more rigorous treatment of heavy-quark mass effects and decay kinematics than previous factorized approximations.
• Future Outlook: The authors note that while the current calculation captures dominant effects, a full GM-VFNS calculation with massive NNLO coefficient functions and proper NNLO FF fits (with uncertainty analysis) would further reduce theoretical uncertainties. The code and interpolation tables for this calculation will be made publicly available to facilitate global PDF fits.