On the positivity of MSbar parton distributions

This paper revisits and clarifies the theoretical argument that MS-bar parton distribution functions remain non-negative within the perturbative region, addressing recent findings of negativity at low scales and providing an estimate for the threshold scale above which positivity is guaranteed.

The Gist

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

Imagine a proton not as a solid ball, but as a bustling, chaotic city. Inside this city, there are tiny messengers called partons (quarks and gluons) zooming around. To understand how the city works, physicists try to count how many messengers are there and how much energy they carry. These counts are called Parton Distribution Functions (PDFs).

In the real world, you can't have "negative" messengers. You can't have -5 cars on a highway or -3 people in a room. If a calculation says there are negative messengers, something is wrong with the math or the map.

However, in the complex world of quantum physics, the way we draw these maps (called schemes) can sometimes make the numbers look negative, even if the physical reality is positive. This paper is about proving that, under the right conditions, our maps should always show positive numbers, and figuring out exactly when those conditions break down.

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The Problem: Two Different Maps for the Same City

Physicists use two main ways to draw these maps:

1. The "Physical" Map (The Reality Check): This map is drawn directly from what we can actually measure in experiments. Since you can't measure a negative number of particles, this map is always positive. It's like counting the actual cars on the road.
2. The "MS" Map (The Theoretical Standard): This is the most popular map used by theorists because it's mathematically convenient. It's like a highly stylized, abstract blueprint. The problem is that because it's so abstract, the math sometimes spits out negative numbers for the number of messengers.

The Question: If the "Physical" map is always positive, does the "MS" map have to be positive too? And if it does, when does it become positive?

The Analogy: The "Tax" and the "Refund"

To understand the math, imagine you are calculating your final bank balance.

• The Physical Map is your actual bank account. You deposit money (positive), and you withdraw money (negative). But the total number of coins you physically hold can never be negative.
• The MS Map is a theoretical accounting method where we apply a "tax" (subtraction) to simplify the math.

In the past, the authors of this paper argued: "If we start with the real coins (Physical Map) and apply our tax rules to get to the MS Map, the result should still be positive, as long as the tax isn't too huge."

However, a recent study (Ref. [2]) pointed out a flaw: "Wait a minute. If you look at the raw ingredients before we even start the accounting, they can actually be negative at very low energies. So, if the ingredients are negative, the final MS Map might be negative too, no matter how you do the math."

The Paper's Solution: Finding the "Safe Zone"

This paper revisits the argument to clarify exactly where the "Safety Zone" is. They break the problem down into two steps:

1. The Low-Energy Trap (The "Foggy Morning")

At very low energies (low "scale"), the quantum world is messy. It's like trying to drive in thick fog. The math shows that if you try to count partons at this level, the numbers can indeed turn negative.

• Why? Because at low energies, the "ingredients" (the raw parton counts) haven't settled down yet. They are fluctuating wildly.
• The Takeaway: You cannot trust the "positive" rule at very low energies. The math simply breaks down because the particles are interacting too strongly and chaotically.

2. The High-Energy Clearing (The "Sunny Day")

As you increase the energy (the "scale"), the fog lifts. The particles behave more predictably.

• The Discovery: The authors prove that once you cross a certain energy threshold, the "MS Map" must be positive if the "Physical Map" is positive.
• The Magic Number: They estimate this threshold to be around 5 GeV² (a specific unit of energy). Above this energy, the "tax" applied by the MS scheme is small enough that it doesn't flip the sign of the numbers. Below this, the "tax" is too aggressive, and the numbers can go negative.

The "Sudakov" Effect: Why the Math Gets Weird

The paper explains why the MS map tries to go negative. It's due to a phenomenon called Sudakov suppression.

• Analogy: Imagine a crowd of people trying to leave a stadium. If they all try to leave at once (low energy), they get jammed up, and the flow looks chaotic (negative numbers in the math).
• But if you give them a wide highway and plenty of time (high energy), they flow smoothly. The "MS" math essentially subtracts the "jammed" part of the crowd to make the math easier. If you do this subtraction at the wrong time (low energy), you subtract too much, resulting in a "negative crowd." At high energy, the subtraction is just right.

Why Should You Care? (The Real-World Impact)

Why do physicists care if a number is negative?

1. Detecting New Physics: If an experiment measures a result that forces the "MS Map" to show negative numbers, it's a huge red flag. It tells physicists: "Hey, our standard model (the leading-twist theory) isn't working here. There must be some hidden, complex interactions (higher-twist effects) that we aren't accounting for."
2. Better Predictions: Knowing exactly where the "Safety Zone" starts (around 5 GeV²) helps experimentalists know when they can trust their data and when they need to be careful. It prevents them from trying to use simple math to describe a chaotic, foggy situation.

The Bottom Line

This paper is like a manual for a GPS system.

• Old belief: "The GPS is always accurate."
• New warning: "The GPS gets glitchy and shows negative distances in heavy fog (low energy)."
• This paper's conclusion: "We have mapped out the fog. If you are driving above 5 GeV² (sunny weather), the GPS is trustworthy and will never show negative distances. If you are below that, you need to drive carefully and realize the map might be lying to you."

It confirms that while the universe is always "positive" in reality, our mathematical tools need a specific "speed limit" (energy scale) to work correctly.

Technical Summary

1. Problem Statement

The paper addresses the theoretical and phenomenological issue of the positivity of Parton Distribution Functions (PDFs) within the $\overline{\text{MS}}$ (Modified Minimal Subtraction) factorization scheme.

• Context: At Leading Order (LO), PDFs are proportional to physical cross-sections and are inherently non-negative. However, beyond LO, PDFs depend on the factorization scheme. In the widely used $\overline{\text{MS}}$ scheme, PDFs are not guaranteed to be positive.
• The Conflict: Previous work (Ref. [1] by the same authors) argued that $\overline{\text{MS}}$ PDFs are positive in the perturbative region. However, recent results (Ref. [2]) demonstrated that "bare" (collinear-unsubtracted) PDFs can become negative at sufficiently low scales ($Q^2$).
• The Question: Does the finding that bare PDFs turn negative at low scales invalidate the argument for $\overline{\text{MS}}$ positivity? If not, what is the precise domain of validity (i.e., the minimum scale $Q^2$) above which $\overline{\text{MS}}$ PDFs are guaranteed to be non-negative?

2. Methodology

The authors revisit their previous argument through two distinct but complementary approaches:

A. Theoretical Analysis of Bare vs. Renormalized PDFs (Section 2)

• Model Calculation: They perform a perturbative computation of the PDF for a quark inside a free, off-shell quark target (mass $M$) at Next-to-Leading Order (NLO).
• Factorization Steps:
  1. Bare PDF: Defined via UV renormalization but before collinear factorization. They show that for a low scale $\mu_r < |M|$, the logarithmic term $\ln(\mu_r^2/|M^2|)$ becomes negative and dominates, causing the PDF to turn negative.
  2. Renormalized PDF: After collinear subtraction (factorization), the scale dependence shifts to the factorization scale $\mu_f$. The PDF takes the form $f(x, \mu_f) \sim \delta(1-x) + \frac{\alpha_s}{2\pi} P_{qq}(x) \ln(\mu_f^2/\Lambda^2) + \text{finite terms}$.
• Conclusion of this step: Positivity is restored only when $\mu_f$ is sufficiently large (an UV scale) such that the logarithmic enhancement overcomes finite negative terms. The transition scale is determined by non-perturbative physics (the target mass scale $\Lambda_h$), making a purely perturbative determination of the absolute lower bound difficult.

B. Transformation from Physical to $\overline{\text{MS}}$ Schemes (Section 3)

• Strategy: Instead of analyzing the bare PDF, they relate the $\overline{\text{MS}}$ PDF to a Physical Scheme PDF (e.g., the DIS scheme), where the PDF is defined to be proportional to a physical observable (structure function) and is therefore strictly positive by definition.
• Mathematical Framework:
  • They express the $\overline{\text{MS}}$ PDF as a convolution of the Physical PDF and the inverse of the perturbative coefficient function matrix: $f^{\text{MS}} = [C^{\text{MS}}]^{-1} \otimes f^{\text{Phys}}$.
  • Handling Divergences: The coefficient function $C^{\text{MS}}$ contains divergent terms (soft gluon emissions) proportional to $[\frac{\ln^k(1-x)}{1-x}]_+$. The authors perform an exact inversion of these divergent terms (resummation) rather than a perturbative expansion, showing that the inverse of the divergent part remains finite and positive.
  • Positivity Condition: They derive a condition for the positivity of $f^{\text{MS}}$ based on the magnitude of the finite terms in the coefficient function relative to the Physical PDF. Specifically, they require:
    $$\left| \frac{\alpha_s}{2\pi} C^{(1), \text{MS}}_F \otimes f^{\text{Phys}} \right| \leq |f^{\text{Phys}}|$$
• Estimation: They calculate the integrated coefficient functions (cumulants) $C_{ij}(x)$ for deep-inelastic scattering and Higgs production. They impose the positivity condition conservatively at the valence peak ($x \sim 0.3$).

3. Key Contributions

1. Clarification of the "Bare" vs. "Renormalized" Distinction: The paper explicitly demonstrates that while bare PDFs (and physical cross-sections in certain schemes) can be negative at low scales due to large negative logarithms, the fully renormalized $\overline{\text{MS}}$ PDF recovers positivity at high scales. This resolves the apparent contradiction with Ref. [2].
2. Exact Inversion of Divergent Terms: The authors provide a rigorous treatment of the $x \to 1$ region by exactly inverting the divergent soft-gluon contributions in the coefficient function matrix, proving that this specific transformation preserves positivity.
3. Quantitative Estimate of the Positivity Domain: By analyzing the finite terms of the NLO coefficient functions, they estimate the scale above which $\overline{\text{MS}}$ PDFs are guaranteed to be positive.
   • Result: Positivity holds for $Q^2 \gtrsim 5 \text{ GeV}^2$.
   • This aligns with practical implementations in global fits (e.g., NNPDF4.0 imposes positivity constraints at $Q^2 = 5 \text{ GeV}^2$).

4. Results

• Low Scale Behavior: At low scales ($Q^2 \lesssim \Lambda_h^2$), the perturbative expansion breaks down, and PDFs can become negative regardless of the scheme due to large higher-twist corrections and the dominance of negative logarithmic terms.
• High Scale Behavior: In the perturbative region ($Q^2 \gtrsim 5 \text{ GeV}^2$), the $\overline{\text{MS}}$ PDFs inherit the positivity of the physical observables. The transformation from the physical scheme to $\overline{\text{MS}}$ involves a finite renormalization that, for the standard QCD splitting functions, does not flip the sign of the PDF.
• Phenomenological Criterion: If a global fit yields a negative $\overline{\text{MS}}$ PDF in the perturbative region ($Q^2 > 5 \text{ GeV}^2$), it implies that the fitted function is not a true leading-twist PDF. Instead, it suggests the presence of significant higher-twist contributions (multiparton correlations) that are being absorbed into the PDF fit to compensate for missing physics in the leading-twist approximation.

5. Significance

• Theoretical Consistency: The paper solidifies the theoretical foundation of PDF positivity in the $\overline{\text{MS}}$ scheme, confirming that it is a robust property of the leading-twist perturbative QCD framework, provided the scale is high enough.
• Diagnostic Tool for Global Fits: The authors propose a powerful diagnostic criterion. If a PDF fit requires negative values to describe data in the perturbative region, it is a signal that the leading-twist approximation is insufficient (i.e., higher-twist effects are non-negligible). This helps distinguish between "bad fits" and "physics beyond leading twist."
• Scheme Independence of Physics: It reinforces the idea that while PDFs are scheme-dependent, the underlying physical observables are positive. The $\overline{\text{MS}}$ scheme is a valid tool for extracting these observables as long as the scale is chosen to ensure the perturbative expansion is well-behaved and positivity is preserved.

Limitations noted by authors:

• The analysis assumes massless quarks; the extension to heavy quarks (e.g., intrinsic charm) is left for future work.
• The precise lower bound ($Q^2 \approx 5 \text{ GeV}^2$) is a semi-quantitative estimate based on NLO cumulants; a precise non-perturbative determination of the transition scale remains an open challenge.