Proton mass decompositions in the NNLO QCD

This paper presents next-to-next-to-leading order (NNLO) QCD evaluations of proton and pion mass decompositions using gravitational form factors, comparing standard quark/gluon and trace-anomaly breakdowns with a novel twist-two/twist-four separation to demonstrate its advantages and reveal distinct parton-correlation behaviors between the two hadrons.

The Gist

Imagine the proton not as a solid, tiny marble, but as a bustling, chaotic city. Inside this city, there are two main groups of citizens: Quarks (the heavy-duty workers) and Gluons (the energetic messengers zipping around).

For decades, physicists have been trying to answer a simple but profound question: "Where does the proton's mass actually come from?"

We know the proton weighs about 1 GeV (in physics units). But if you add up the weights of the three quarks inside it, they only account for about 1% of that total. Where is the other 99%?

This paper, written by physicist Kazuhiro Tanaka, is like a high-tech audit of that city. It uses the most advanced mathematical tools available (called NNLO QCD, which is like upgrading from a ruler to a laser scanner) to break down the proton's mass into its exact ingredients.

Here is the breakdown of what the paper does, using simple analogies:

1. The Old Map vs. The New GPS

In the past, scientists had a map of the proton's mass that was a bit blurry. They knew the mass came from "Quarks" and "Gluons," but the way they split the bill was messy.

• The Problem: In the old method, the "Quark" pile and the "Gluon" pile were mixed up. It was like trying to separate a smoothie into "strawberry" and "banana" parts, but the blender had already pureed them so well that you couldn't tell which was which. Also, the results changed depending on how you measured them (a problem called "scale dependence").
• The New Method: Tanaka introduces a new way to look at the data. He separates the mass into two distinct categories based on how the particles move versus how they interact.

2. The Two Types of Mass Ingredients

The paper proposes a new "Four-Ingredient Recipe" for the proton's mass. Think of it like baking a cake:

• Ingredient A & B: The "Motion" (Twist-2)

  • Analogy: Imagine the quarks and gluons running a marathon inside the proton. Their sheer speed and momentum create energy. In physics, energy equals mass ($E=mc^2$).
  • The Result: This part represents the "kinetic energy" of the particles. The paper finds that for both protons and pions (a lighter cousin of the proton), this "running around" contributes about 37% of the total mass each. It's the universal "engine" of the hadron.

• Ingredient C & D: The "Correlation" (Twist-4)

  • Analogy: Now imagine the runners aren't just running; they are constantly bumping into each other, grabbing hands, and forming a tight knot. This is the "glue" holding them together. In the proton, this is the Trace Anomaly—a weird quantum effect where the interactions themselves generate mass.
  • The Result: This is where the proton and the pion differ wildly.
    • In the Proton: The "glue" (interactions) is very strong and complex.
    • In the Pion: The pion is a "Goldstone boson," which is a fancy way of saying it's a ripple in the quantum field. Its mass comes almost entirely from the explicit weight of the quarks themselves, not the "glue."
  • The Discovery: The new method shows that while the "running" part is similar for both, the "glue" part is what makes a proton heavy and a pion light.

3. Why the "Old Map" Was Confusing

The paper explains that previous calculations were like trying to weigh a suitcase while it was still on a moving train. The numbers kept shifting depending on how fast the train was going (the "renormalization scale").

• The Fix: Tanaka's new method is like putting the suitcase on a stationary scale. By strictly separating the "motion" (twist-2) from the "interaction" (twist-4) for both quarks and gluons, the numbers become stable and make physical sense. It finally clears up the confusion about where the mass comes from.

4. The "Heavy" Problem

There was also a worry about "heavy" quarks (like charm quarks) that might be hiding inside the proton. The paper shows that with this new method, we can prove that these heavy quarks don't mess up the calculation; they naturally "decouple" (disappear from the equation) in a way that makes sense, just like a heavy guest leaving a party doesn't change the weight of the empty room.

The Big Takeaway

This paper is a major upgrade in our understanding of the universe's building blocks.

• Before: We knew mass came from quarks and gluons, but the recipe was messy and changed depending on how you looked at it.
• Now: We have a precise, stable recipe.
  • ~74% of the proton's mass comes from the motion of quarks and gluons (the "engine").
  • ~26% comes from the interactions (the "glue" or trace anomaly).
  • The explicit weight of the quarks themselves is a tiny fraction (less than 1%).

In short: The proton is heavy not because its parts are heavy, but because they are moving incredibly fast and are tightly bound together by a powerful quantum "glue." This paper gives us the clearest picture yet of exactly how that glue works, distinguishing the proton from its lighter cousin, the pion, and paving the way for future experiments at the Electron-Ion Collider (EIC) to see this "glue" in action.

Technical Summary

1. Problem Statement

The origin of the proton mass is a fundamental question in Quantum Chromodynamics (QCD), with the upcoming Electron-Ion Collider (EIC) aiming to probe this structure. The proton mass is derived from the matrix elements of the QCD Energy-Momentum Tensor (EMT). However, decomposing this mass into contributions from quarks and gluons has historically been plagued by several theoretical challenges:

• Renormalization Ambiguities: The EMT consists of composite operators that undergo complex renormalization mixing. In standard $\overline{\text{MS}}$ schemes, "taking the trace" and "performing the finite-part operation" (UV pole subtraction) do not commute, leading to anomalies.
• Incomplete Separation: Previous decompositions (e.g., Ji's four-term decomposition) failed to strictly separate the traceless part (twist-two, associated with partonic motion) from the trace part (twist-four, associated with parton correlations and the trace anomaly). This resulted in "incomplete separation" where quark contributions mixed twist-two and twist-four effects.
• Scale Dependence: Previous evaluations showed strong dependence on the renormalization scale ($\mu$), making it difficult to define a universal picture of the mass structure.
• Heavy Quark Decoupling: The treatment of heavy quark contributions (like charm) in the mass decomposition was not manifestly consistent with decoupling theorems.

2. Methodology

The author employs a rigorous, model-independent framework based on Next-to-Next-to-Leading Order (NNLO) QCD calculations.

• Renormalization Scheme: The study utilizes $\overline{\text{MS}}$-like schemes where the EMT operators are renormalized by subtracting UV poles associated with a specific operator basis. This ensures the conservation of the EMT ($\partial_\alpha [T^{\alpha\beta}] = 0$) while correctly accounting for trace anomalies.
• Gravitational Form Factors (GFFs): The proton matrix elements of the EMT are parameterized by GFFs, $A_r(\Delta^2, \mu)$ and $\bar{C}_r(\Delta^2, \mu)$ (where $r=q, g$).
  • $A_r(0, \mu)$ relates to the momentum fraction carried by partons (twist-two).
  • $\bar{C}_r(0, \mu)$ relates to the trace anomaly and parton correlations (twist-four).
• NNLO Inputs: The paper utilizes recent NNLO evaluations of these forward GFFs (at zero momentum transfer) derived from:
  • Three-loop renormalization group (RG) equations.
  • Global QCD analyses of Parton Distribution Functions (PDFs) (e.g., CT18).
  • Empirical inputs for sigma terms ($\sigma_{\pi N}, \sigma_s$) and lattice QCD results.
• Twist Decomposition: A new decomposition strategy is proposed that strictly separates the traceless (twist-two) and trace (twist-four) parts for both the quark and gluon sectors individually, rather than just for the total EMT.

3. Key Contributions

1. NNLO Quantitative Evaluation: The paper provides the first NNLO quantitative evaluation of proton mass decompositions, reducing uncertainties to the few percent level (significantly better than current lattice QCD results).
2. Resolution of "Incomplete Separation": The author introduces a new four-term mass decomposition (Eq. 22) that strictly separates twist-two and twist-four contributions for both quarks and gluons.
   • Old Issue: In previous schemes (like Ji's), the quark mass term ($M_m$) and trace anomaly ($M_a$) were mixed with kinetic energy terms, obscuring the physical interpretation.
   • New Solution: The new formula separates the mass into:
     • $M_{q}^{(tw2)}$ and $M_{g}^{(tw2)}$: Representing partonic motions (kinetic energy).
     • $M_{q}^{(tw4)}$ and $M_{g}^{(tw4)}$: Representing parton correlations induced by non-perturbative interactions (trace anomaly).
3. Resolution of Scale Dependence: The new decomposition demonstrates that the twist-two terms ($3/4 M A_r$) exhibit a crossing behavior that stabilizes at high scales, while the twist-four terms are nearly RG-invariant. This resolves the issue of strong scale dependence found in previous decompositions.
4. Heavy Quark Decoupling: The new framework naturally incorporates the decoupling of heavy quarks (e.g., charm) via the heavy-quark expansion, resolving ambiguities regarding heavy-flavor contributions to the mass.
5. Pion vs. Proton Comparison: The methodology is extended to the pion, allowing for a direct comparison of hadron mass structures.

4. Key Results

• Proton Mass Decomposition (at $\mu = 2$ GeV):
  • Twist-two (Partonic Motion): Quarks and gluons contribute roughly equally to the partonic motion component ($\approx 37\%$ each of the total mass, summing to $\approx 74\%$).
  • Twist-four (Correlations): The remaining mass is generated by trace anomalies (parton correlations).
  • Scale Stability: Unlike previous models where quark and gluon contributions crossed significantly with scale changes, the new twist-two components show a stable, universal qualitative picture across the range $0.7 \text{ GeV} \lesssim \mu \lesssim 20 \text{ GeV}$.
• Pion Mass Decomposition:
  • The twist-two (motion) contributions for the pion are similar to the proton.
  • The twist-four (correlation) contributions are drastically different. The pion's mass is dominated by the quark mass term (chiral symmetry breaking), reflecting its nature as a Nambu-Goldstone boson, whereas the proton's mass is dominated by the trace anomaly (gluon dynamics).
• Physical Interpretation:
  • Twist-two terms correspond to the average momentum fractions of quarks and gluons (partonic motion).
  • Twist-four terms correspond to forces and correlations between partons (trace anomaly), which differ significantly between the proton and pion.

5. Significance

• Theoretical Clarity: This work resolves long-standing ambiguities in the definition of proton mass components by enforcing a mathematically strict separation of twist-two and twist-four operators within the $\overline{\text{MS}}$ scheme.
• Precision: By achieving NNLO accuracy, the results provide a high-precision benchmark for future experimental measurements at the EIC and for lattice QCD calculations.
• Physical Insight: The decomposition clarifies that while the mechanism of mass generation (partonic motion vs. correlations) is universal in form, the magnitude of these effects distinguishes different hadrons. The proton's mass is largely generated by the QCD trace anomaly (gluon correlations), while the pion's mass is heavily influenced by explicit chiral symmetry breaking (quark mass terms).
• Future Directions: The paper establishes a robust framework for exploring confinement and hadron structure, suggesting that the "hybrid" nature of mass (universal motion + specific correlations) is a key feature of QCD.

In summary, Tanaka's paper provides a definitive, high-precision, and physically transparent decomposition of the proton mass, correcting previous theoretical inconsistencies and offering a new lens through which to view the internal dynamics of hadrons.