Slepton pair production at next-to-leading power

✨

This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Big Picture: Predicting the "Impossible" Collision

Imagine you are trying to predict how often two specific, heavy cars will crash into each other on a massive, crowded highway (the Large Hadron Collider). In physics, these "cars" are particles called sleptons (hypothetical partners to electrons).

The problem is that when these heavy particles are created, they are moving very slowly, almost like they are stuck in traffic right at the edge of the highway's exit ramp. In physics terms, this is called the "threshold."

When things happen right at this threshold, the math gets messy. It's like trying to count the number of cars in a traffic jam where the cars are constantly honking and swerving. The standard math tools (called "fixed-order calculations") start to break down because they miss a huge number of tiny, repetitive honks (mathematical "logarithms") that pile up and change the final count.

The Old Way: Ignoring the "Almost"

For a long time, physicists have been good at counting the main honks (the Leading Power effects). They built a very accurate map for the main traffic flow. However, they ignored the "almost" honks—the tiny, subtle swerves that happen just before the cars fully stop or just after they start moving.

The authors of this paper argue that ignoring these "almost" swerves is dangerous. They call these the Next-to-Leading Power (NLP) effects.

The Analogy:
Imagine you are baking a cake.

Leading Power (Old Method): You measure the flour, sugar, and eggs perfectly. You get a good cake.
Next-to-Leading Power (New Method): You realize that the way the flour settles in the bowl, or the tiny bit of air trapped in the sugar, actually changes how the cake rises. If you ignore these tiny details, your cake might look right, but your prediction of how tall it will be is slightly off.

What This Paper Did

The authors went back to the math and calculated these "tiny swerves" (NLP contributions) for the first time in the context of supersymmetric particles (sleptons).

They found the missing pieces: They calculated the mathematical terms that were previously ignored.
They checked the "Uncertainty Meter": In physics, every prediction comes with an error bar (a range of "maybe"). The authors found that the old methods were too confident. They thought the error was small, but when you add in these new "tiny swerves," the error bar actually gets bigger.
- Metaphor: It's like a weather forecaster saying, "There is a 99% chance of sun," but they forgot to account for a tiny cloud that might form. The new calculation says, "Actually, there's a 90% chance of sun, and a 10% chance of a surprise cloud." The new forecast is more honest about the uncertainty.
They looked at the future: They ran these calculations for a hypothetical future super-collider (FCC-hh) that would be much bigger than the current one. They found that for this future machine, getting these "tiny swerves" right is even more critical because the particles being hunted will be heavier and harder to find.

The Key Findings

The "Tiny" things are actually Big: The effects they calculated (NLP) are just as important as the next level of precision in the old method. You can't just ignore them.
Old predictions were too optimistic: The current best tools (like the "Resummino" software used by the LHC) underestimate how uncertain we really are when looking for heavy particles. They think they know the answer better than they actually do.
Stability: By including these new terms, the predictions become more stable. They don't wiggle as much when you tweak the input numbers slightly.

Why It Matters

If you are a detective looking for a criminal (a new particle) in a crowd, you need to know exactly how many people are in the crowd to spot the stranger. If your math says "100 people" but you are actually off by 10 because you ignored the "tiny swerves," you might miss the criminal or think you found one when you didn't.

This paper provides a better, more honest map of the "traffic jam" at the edge of the energy threshold. It tells physicists: "Don't trust the old maps too much; the uncertainty is larger than you thought, and here is the new math to fix it."

Summary in One Sentence

This paper fixes a blind spot in our mathematical models for creating heavy particles, showing that we have been underestimating our uncertainty and that including these "almost" effects is crucial for finding new physics in the future.

1. Problem Statement

The production of heavy particles at hadron colliders (like the LHC and potential future FCC-hh) is dominated by kinematic configurations near the production threshold ( $z = Q^2/\hat{s} \to 1$ ). In this region, perturbative QCD calculations suffer from large logarithmic terms of the form $\alpha_s^n \ln^m(1-z)/(1-z)$ , which spoil the convergence of fixed-order expansions.

Current State-of-the-Art: Existing calculations for slepton pair production (a key Supersymmetric search channel) typically include fixed-order contributions up to Next-to-Leading Order (NLO) and resummation of Leading Power (LP) logarithms up to Next-to-Next-to-Leading Logarithmic (NNLL) or even N $^3$ LL accuracy (using SCET or the Resummino code).
The Gap: These calculations systematically neglect Next-to-Leading Power (NLP) contributions, which are suppressed by a factor of $(1-z)$ relative to the leading terms. Recent studies in Standard Model processes (e.g., Drell-Yan) suggest that NLP logarithms can be numerically comparable to LP terms at lower logarithmic orders (e.g., NLP-LL $\approx$ LP-NLL).
The Issue: The authors argue that relying solely on increasing the logarithmic accuracy of LP terms (e.g., going to N $^3$ LL) may lead to an underestimation of theoretical uncertainties, particularly for heavy new particles where the scale dependence is critical.

2. Methodology

The authors perform a comprehensive calculation for inclusive slepton pair production ( $pp \to \tilde{\ell}\tilde{\ell}^* + X$ ) within the Minimal Supersymmetric Standard Model (MSSM).

Fixed-Order Baseline: They establish a baseline calculation up to NLO, including virtual corrections (gluon exchange and squark-gluino loops) and real emission diagrams ( $q\bar{q}$ and $qg$ channels). Renormalization is performed using on-shell schemes for masses/fields and $\overline{\text{MS}}$ for $\alpha_s$ and PDFs.
Threshold Resummation:
- Formalism: They utilize the factorization properties of QCD in Mellin space ( $N$ -space), where threshold logarithms $\ln(1-z)$ map to $\ln N$ .
- Leading Power (LP): They resum LP logarithms up to NLL accuracy.
- Next-to-Leading Power (NLP): They extend the resummation to include NLP logarithms at Leading Logarithmic (LL) accuracy. This involves:
  - $q\bar{q}$ Channel: Using generalized Wilson lines and "next-to-soft" gluon radiation formalisms to derive an exponentiated expression that retains full $z$ -dependence in the soft function and splitting kernels.
  - $qg$ Channel: Deriving the NLP resummed contribution for the quark-gluon initial state, which is non-zero at NLP but zero at LP. This involves specific spin-sum suppression factors for soft quark emission.
Matching: The resummed NLP contributions are matched to the fixed-order NLO result using additive matching. The "expanded" NLO terms (the Taylor expansion of the resummed result to $\mathcal{O}(\alpha_s)$ ) are subtracted to avoid double-counting.
Numerical Implementation:
- The inverse Mellin transform is performed numerically using the Minimal Prescription to avoid the Landau pole singularity.
- To ensure numerical stability, they employ differentiated Parton Distribution Functions (PDFs) and a specific integration contour deflection.
- Calculations are performed for $\sqrt{s} = 13.6$ TeV (LHC) and $\sqrt{s} = 85$ TeV (FCC-hh).

3. Key Contributions

First NLP Application to BSM: This is the first application of NLP threshold resummation techniques to Supersymmetric particle production (specifically slepton pairs).
Analytic NLP Expressions: The paper provides complete analytic expressions for the NLP resummed cross-sections for both $q\bar{q}$ and $qg$ initial states, including the matching to fixed-order results.
Benchmarking against State-of-the-Art: The authors rigorously compare their results with:
- The Resummino code (which includes LP NLL/NNLL and collinear-improved terms).
- SCET-based N $^3$ LL calculations (from Ref. [19]).
Uncertainty Analysis: They provide a detailed breakdown of scale uncertainties and PDF uncertainties, specifically analyzing how NLP terms affect the theoretical error bands compared to pure LP resummation.

4. Key Results

Magnitude of NLP Effects: The NLP leading-logarithmic contributions are found to be significant, often comparable in size to the LP next-to-leading logarithmic (NLL) terms.
Scale Uncertainty Reduction:
- Including NLP terms reduces the scale dependence of the cross-section, stabilizing the prediction as a function of the slepton mass.
- Crucial Finding: The scale uncertainty estimated by the current state-of-the-art LP-only calculations (e.g., Resummino NLO+NNLL) underestimates the true theoretical uncertainty for heavy sleptons. The shift induced by adding NLP terms is often larger than the scale uncertainty band predicted by the LP-only calculation.
Mass Dependence:
- For light sleptons (near threshold), the NLP effects are significant but the LP resummation dominates.
- For heavy sleptons (farther from threshold kinematically, but still in the high-mass tail), the NLP contributions remain relevant and are essential for a realistic error estimate.
FCC-hh Prospects: At $\sqrt{s} = 85$ TeV, the Missing Higher Order Uncertainty (MHOU) becomes the dominant source of error for masses just beyond the LHC reach. The inclusion of NLP terms is critical for precision measurements in this regime.
PDF Uncertainty: The inclusion of NLP terms (which introduces the $qg$ channel) has a negligible impact on the overall PDF uncertainty, as the change in the central cross-section value is small relative to the PDF errors.

5. Significance and Conclusion

Theoretical Robustness: The paper demonstrates that simply increasing the logarithmic order of Leading Power terms (e.g., going from NNLL to N $^3$ LL) is not always sufficient to capture all relevant physics or to correctly estimate theoretical uncertainties. NLP terms provide a necessary correction that competes with higher-order LP terms.
Impact on Searches: For LHC and future collider searches for SUSY, the current theoretical error bands (based on scale variation of LP-only calculations) may be overly optimistic. The authors argue that NLP corrections must be included to provide a reliable estimate of the Missing Higher Order Uncertainty (MHOU).
Future Tools: The authors intend to incorporate these results into a future public tool for BSM cross-section calculations, moving beyond the current standard of LP-only resummation.

In summary, this work establishes that Next-to-Leading Power resummation is not a sub-leading correction to be ignored in precision BSM phenomenology; it is a critical component for stabilizing predictions and accurately quantifying uncertainties for heavy particle production at current and future colliders.