Parton distributions with higher twist and jet power corrections

This paper presents a global determination of parton distribution functions that incorporates higher twist corrections for deep-inelastic scattering and linear power corrections for LHC jet data using a theory covariance formalism, demonstrating improved data description and perturbative convergence for key phenomenological observables like Higgs production and $\alpha_s$ determination.

The Gist

Imagine you are trying to bake the perfect cake, but you don't have the exact recipe. Instead, you have to figure out the recipe by tasting thousands of cakes made by different bakers in different kitchens. In the world of particle physics, the "cake" is the Standard Model (our best theory of how the universe works), the "bakers" are the giant particle colliders like the LHC (Large Hadron Collider), and the "ingredients" are Parton Distribution Functions (PDFs).

PDFs are essentially a map that tells us how the tiny particles inside a proton (called quarks and gluons) are arranged and how much energy they carry. If we want to predict what happens when two protons smash together at the LHC, we need this map to be incredibly accurate.

This paper, titled "Parton distributions with higher twist and jet power corrections," is about refining that map to account for some messy, real-world "noise" that was previously ignored or smoothed over.

Here is the breakdown using simple analogies:

1. The Problem: The "Perfect" Theory vs. The "Messy" Reality

For a long time, physicists have used a "perfect" mathematical theory (called Perturbative QCD) to predict how particles behave. It's like a GPS that assumes the roads are perfectly straight and there is no traffic.

• The Issue: In reality, protons are messy. When you smash them, the particles don't just behave according to the "perfect" math. There are extra effects that get smaller as the energy gets higher, but they are still there.
• The Analogy: Imagine you are driving a car. The "perfect theory" says you should arrive in exactly 10 minutes. But in reality, you have to deal with potholes (Higher Twist) and wind resistance (Power Corrections). If you ignore these, your GPS will be slightly off, and if you need to be precise to the second (which LHC scientists do), that error matters.

2. The Two Types of "Noise"

The authors focused on two specific types of messiness:

A. Higher Twist (The "Potholes" in Deep-Sea Diving)

• What it is: In deep-inelastic scattering (smashing electrons into protons), the math usually assumes the proton is a simple bag of particles. But sometimes, the particles inside interact in complex, "twisted" ways that the simple math misses.
• The Analogy: Think of a deep-sea diver looking at a school of fish. From far away, the school looks like a smooth, solid blob. But if you get close (low energy), you see the individual fish bumping into each other, swirling, and creating turbulence. These "swirls" are the Higher Twist corrections.
• The Fix: The authors created a new method to measure exactly how big these "swirls" are, rather than just cutting out the data where they happen.

B. Linear Power Corrections (The "Wind Resistance" for Jets)

• What it is: When protons collide, they shoot out sprays of particles called Jets. The theory predicts the energy of these jets perfectly, except for the fact that the particles have to turn into actual matter (hadronization) and interact with the surrounding "soup" of particles (the underlying event). This steals a tiny bit of energy from the jet.
• The Analogy: Imagine shooting a cannonball. The physics equations say it will travel exactly 100 meters. But in reality, the air resistance (wind) slows it down. If you are shooting a cannonball at 100 meters, the wind matters a lot. If you shoot it at 1,000 meters, the wind matters less.
• The Twist: In particle physics, this "wind resistance" (Power Correction) doesn't just fade away quickly; it fades away slowly. The authors found that even for very high-energy jets, this "wind" is still pushing them off course by a noticeable amount.

3. The Solution: The "Theory Covariance" Method

How did they fix this? They didn't just guess. They used a sophisticated statistical tool called the Theory Covariance Method.

• The Analogy: Imagine you are trying to tune a radio. Usually, you just turn the knob until the static is gone. But here, the "static" (the theoretical errors) is actually part of the signal.
• The authors treated these theoretical errors (the potholes and wind) as variables in their math. They said, "Let's assume there is a pothole here, and a wind gust there, and let the data tell us how big they are."
• They built a "Covariance Matrix," which is just a fancy way of saying: "We know that if the wind is strong here, it's likely strong there too." This allowed them to separate the "real" signal from the "noise" without throwing away any data.

4. The Results: A Sharper Map

After running their new calculations, they found:

1. The Map Changed (Slightly): The new PDFs (the ingredient map) shifted a little bit. The most noticeable change was in the gluon distribution (the glue holding the proton together). The "wind" from the jets made the map smoother in certain areas.
2. Better Predictions: When they used this new, "messy-aware" map to predict what happens at the LHC (like creating a Higgs boson), the predictions matched the experimental data better.
3. The "1% Goal": The LHC is now so precise that scientists want their theories to be accurate to within 1%. The authors showed that if you ignore these "potholes" and "wind," you miss that 1% target. By including them, they improved the accuracy.

5. Why This Matters

Think of the LHC as a giant microscope. For years, we've been looking through a lens that was slightly blurry. This paper is like cleaning the lens.

• Before: We knew the lens was blurry, so we just avoided looking at the blurry parts of the image (cutting out low-energy data).
• Now: We figured out exactly how the lens is blurry, so we can mathematically correct the image. This allows us to look at more of the picture and see details we missed before.

In a nutshell:
The authors took the "perfect" theory of particle physics, admitted that the real world is messy (with potholes and wind), measured exactly how messy it is, and updated the "recipe book" for the universe. This makes our predictions for future particle collisions more accurate, helping us find new physics or confirm what we already know with greater confidence.

Technical Summary

1. Problem Statement

Global Parton Distribution Function (PDF) fits aim to provide precise inputs for Standard Model predictions at the Large Hadron Collider (LHC). However, theoretical uncertainties remain a limiting factor, particularly those arising from non-perturbative effects suppressed by powers of the hard scale ($Q$ or $p_T$).

• Deep-Inelastic Scattering (DIS): Standard factorization assumes an Operator Product Expansion (OPE) where leading-twist (twist-2) terms dominate. Higher Twist (HT) corrections (twist-4 and above) are suppressed by $1/Q^2$ but become significant at low $Q^2$ and large Bjorken-$x$. Historically, global fits have avoided this region by imposing kinematic cuts ($Q^2 > 3.49 \text{ GeV}^2$), potentially discarding valuable data.
• Hadronic Processes (Jets): Unlike DIS, hadronic processes like inclusive jet and dijet production lack a strict OPE. They are subject to linear power corrections of order $\mathcal{O}(\Lambda/p_T)$ arising from hadronization, underlying events, and multi-parton interactions. These corrections fall off much more slowly than HT corrections and can be significant even at scales as high as 100 GeV.
• The Challenge: Distinguishing these power corrections from Missing Higher Order Uncertainties (MHOUs) is difficult because both manifest as deviations from the leading-order perturbative prediction. Previous global fits have largely ignored linear power corrections for jets, relying instead on kinematic cuts or Monte Carlo-based hadronization corrections.

2. Methodology

The authors employ the Theory Covariance Formalism (previously used for MHOUs and nuclear uncertainties) to simultaneously determine PDFs and the parameters governing power corrections.

• Theory Covariance Matrix: The total covariance matrix $C$ is constructed as the sum of the experimental covariance ($C_{exp}$) and a theory covariance matrix ($C_{th}$).
  $$C_{ij} = C_{exp, ij} + C_{th, ij}$$
  $C_{th}$ accounts for correlated shifts in theoretical predictions induced by nuisance parameters representing power corrections.
• Modeling Higher Twist (DIS):
  • HT corrections are modeled as multiplicative shifts to structure functions ($F_2$) and reduced cross-sections.
  • The shift is parametrized as $1 + H(x)/Q^2$, where $H(x)$ is a piecewise linear function of $x$ determined empirically.
  • Separate functions are used for proton ($H^p_2$), deuteron ($H^d_2$), and charged current ($H_{CC}$) data.
• Modeling Linear Power Corrections (Jets):
  • Corrections for single-inclusive jets and dijets are modeled as multiplicative shifts of the form $1 + H(y)/p_T$ (or $1 + H(\eta)/m_{jj}$).
  • Unlike DIS, these are linear in the hard scale ($1/p_T$) rather than quadratic.
  • The functions $H(y)$ and $H(\eta)$ are parametrized as piecewise linear functions of rapidity.
• Fitting Procedure:
  • The analysis uses the NNPDF4.0 framework and dataset.
  • To determine the corrections, kinematic cuts are lowered (e.g., $Q^2 > 2.5 \text{ GeV}^2$) to increase sensitivity to non-perturbative effects.
  • Nuisance parameters ($h_\alpha$) representing the magnitude of the corrections are treated as stochastic variables with Gaussian priors.
  • The fit determines the posterior distribution of these parameters, which are then used to correct the theoretical predictions in a final global fit using standard NNPDF4.0 cuts.

3. Key Contributions

1. First Empirical Determination of Jet Power Corrections: This is the first global PDF fit to empirically determine linear power corrections for single-inclusive jets and dijets within the PDF fitting framework.
2. Unified Treatment of Non-Perturbative Effects: The paper integrates HT corrections in DIS and linear power corrections in jets into a single global fit, accounting for their correlations via the theory covariance matrix.
3. Validation of Methodology: The approach validates the theory covariance formalism for extracting non-perturbative parameters alongside PDFs, showing consistency with previous results for HT and theoretical expectations for jet corrections.
4. Release of NNPDF4.0HT Sets: The authors provide new PDF sets (NNPDF4.0HT) available in LHAPDF format that include these corrections and their uncertainties.

4. Key Results

A. Determination of Corrections

• Higher Twist (DIS):
  • The posterior HT corrections are consistent with zero at low and intermediate $x$ but grow at large $x$ (peaking around $4 \text{ GeV}^2$ for protons).
  • The magnitude of the HT is significantly reduced when Missing Higher Order Uncertainties (MHOUs) are included, and further reduced at approximate N3LO (aN3LO). This suggests that much of the "HT" seen at NNLO is actually due to missing N3LO perturbative terms.
• Linear Power Corrections (Jets):
  • A significant positive power correction of approximately 10 GeV is determined for single-inclusive jets, largely independent of rapidity. This aligns with previous theoretical estimates (~7 GeV).
  • For dijets, the evidence is weaker, with a tendency toward negative corrections at central rapidity, though uncertainties remain large.
  • The corrections are significant up to $p_T \sim 100$ GeV, affecting up to 14% of the cross-section at the lowest $p_T$ bins.

B. Impact on PDFs and Luminosities

• PDFs: The impact on the extracted PDFs is generally modest (within $1\sigma$).
  • Gluon: The most notable change is a reduction in the gluon distribution at intermediate $x$ ($0.005 < x < 0.05$) in the NNLO fit, resolving tensions between jet data and top-quark pair data. This effect diminishes when MHOUs or aN3LO evolution are included.
  • Uncertainties: At low $x$, including HT increases PDF uncertainties due to the down-weighting of precise HERA data, but this effect is mitigated by including MHOUs.
• Luminosities: The gluon-gluon luminosity shows a $\sim 1\%$ drop in the intermediate mass region ($100-250$ GeV) in the NNLO fit, which is reduced to negligible levels in the aN3LO fit.

C. Impact on LHC Observables and $\alpha_s$

• Cross-Sections:
  • For Higgs production via gluon fusion, the inclusion of HT and PCs reduces the cross-section by $\sim 1\%$ at NNLO, but this effect drops to $\sim 0.5\%$ with MHOUs and becomes negligible at aN3LO.
  • The corrections improve the perturbative convergence, reducing the discrepancy between predictions using NNLO vs. aN3LO PDFs.
• Strong Coupling ($\alpha_s$):
  • The extraction of $\alpha_s(m_Z)$ is sensitive to these corrections.
  • At aN3LO, the HT correction leads to a substantial increase in $\alpha_s$ (more than $1\sigma$), while jet power corrections provide a smaller positive shift.
  • The final result at aN3LO is consistent with the NNLO value but with larger uncertainties.

5. Significance and Conclusions

• Precision Physics: The study demonstrates that to achieve the 1% precision goal for LHC observables, global PDF fits must account for power corrections, either by cutting low-scale data or by including them in the theoretical model.
• Perturbative Convergence: Including these corrections helps disentangle non-perturbative effects from missing higher-order perturbative terms, leading to better convergence of the perturbative series.
• Future Directions: The authors suggest that future fits should consider power corrections for Drell-Yan and top production. They also note that if reliable, consistent hadronization correction factors (from Monte Carlos) were applied to data before fitting, the empirical power corrections found here would likely be smaller.

In summary, this paper provides a rigorous framework for quantifying and incorporating non-perturbative power corrections into global PDF analyses, resulting in more robust PDF sets and improved theoretical predictions for LHC phenomenology.