Reply to "Comment on "Chiral symmetry restoration, the eigenvalue density of the Dirac operator, and the axial U(1) anomaly at finite temperature""

This paper refutes Matteo Giordano's comment by demonstrating that the proposed counterexamples violate the fundamental QCD assumption of analyticity in the squared quark mass at high temperatures and by identifying a technical error in the critique, thereby upholding the validity of the authors' original arguments regarding chiral symmetry restoration and the axial U(1) anomaly.

The Gist

Imagine a high-stakes debate between two groups of physicists trying to understand how the universe behaves when it gets extremely hot. One group (the authors of this paper, led by Sinya Aoki and Hidenori Fukaya) made a specific claim about how particles interact at these temperatures. The other group (represented by Matteo Giordano) wrote a "comment" trying to prove them wrong by offering a few counter-examples.

This paper is the authors' reply. Their main message is simple: "The examples you used to try to disprove us don't actually work because they break the fundamental rules of the game."

Here is a breakdown of their argument using everyday analogies:

1. The "Smoothness" Rule (The Core Disagreement)

The authors' original theory relies on a rule they call $m^2$-analyticity.

• The Analogy: Imagine you are baking a cake. The "quark mass" ($m$) is like the amount of sugar you add. The authors claim that if you are in the "hot phase" (like a fully baked cake), the taste of the cake changes smoothly as you tweak the sugar. If you plot the taste against the sugar amount, you get a nice, continuous curve with no sudden jumps or sharp corners.
• The Critic's Move: Giordano tried to show that this rule isn't true by inventing some weird, mathematical "cakes" where the taste suddenly jumps or behaves strangely when you change the sugar.
• The Rebuttal: The authors point out that Giordano's weird cakes are illegal. In the real world of high-temperature physics (QCD), nature doesn't allow those sudden jumps. Giordano's examples only work if you break the fundamental laws of the universe. Since his examples are "unrealistic," they cannot be used to disprove a theory about the real universe.

2. The "Well-Defined" Probability

The authors also discuss a mathematical tool they use called $P(m, A)$, which acts like a probability map for how particles behave.

• The Critic's Move: Giordano argued that this map might be "ill-defined" or broken in certain scenarios, suggesting a specific formula for it that looked messy.
• The Rebuttal: The authors explain that if you build this map using a standard, step-by-step method (like a computer simulation on a grid), it works perfectly fine. They argue that Giordano's messy formula is just another one of those "illegal" examples that breaks the smoothness rule mentioned above. It's like trying to use a map of a city that doesn't exist to prove your navigation skills are bad.

3. The "Average" Mistake

Finally, the authors found a specific mathematical error in Giordano's logic.

• The Analogy: Imagine you have a bag of marbles.
  • The Mistake: Giordano acted as if the "average marble" in the bag was the only marble that existed. He assumed that if the average weight is 5 grams, then every single marble weighs exactly 5 grams.
  • The Reality: In the real world, you have marbles of 4g, 6g, 3g, etc. The average is 5g, but the variance (the spread) is real and important.
• The Rebuttal: Giordano confused the "average value" with the "distribution of values." He used a formula that assumes there is no variation at all (a delta function), which is mathematically incorrect for this type of problem. Because of this basic error, the conclusions he drew from it are invalid.

The Conclusion

The authors wrap up by saying:

1. The counter-examples Giordano used are "unphysical" (they break the rules of the universe).
2. Giordano made a technical mistake by confusing an average with a specific value.
3. Therefore, Giordano's attempt to disprove the original theory fails. The original theory stands.

In short, the authors are saying, "You tried to knock down our house by throwing rocks at it, but you were throwing rocks made of glass that shattered before they hit the wall. Also, you miscalculated the trajectory of your throw. Our house is still standing."

Technical Summary

Technical Summary: Reply to "Comment on Chiral symmetry restoration, the eigenvalue density of the Dirac operator, and the axial U(1) anomaly at finite temperature"

Problem Statement
This paper serves as a formal response to a comment by Matteo Giordano [1] regarding the authors' previous work [2] on chiral symmetry restoration, the eigenvalue density of the Dirac operator, and the axial U(1) anomaly at finite temperature. Giordano's comment proposed counterexamples intended to refute the arguments presented in [2]. The central issue addressed here is whether the counterexamples provided by Giordano are valid within the framework of Quantum Chromodynamics (QCD) at high temperatures, specifically concerning the analyticity of gluonic observables with respect to the squared quark mass ($m^2$).

Methodology and Argumentation
The authors employ a theoretical analysis based on the properties of the QCD partition function and gluonic observables in the chirally symmetric phase. Their methodology involves:

1. Re-evaluating Counterexamples: The authors scrutinize the specific models (such as a Gaussian model) and density-of-states functions $\rho_O(x, m)$ proposed in Giordano's comment to determine if they satisfy the fundamental assumptions of high-temperature QCD.
2. Analyzing $m^2$-Analyticity: The authors rely on the assumption that every gluonic observable in the chirally symmetric phase is an analytic function of $m^2$. They test whether the proposed counterexamples preserve this analyticity or if they introduce non-analytic behavior that would imply infrared divergences or criticality inconsistent with the symmetric phase.
3. Examining Well-Definedness: The authors address Giordano's concerns regarding the well-definedness of the function $P(m, A)$ used in their derivation. They utilize lattice regularization (specifically the overlap Dirac operator on a finite lattice) to demonstrate that $P(m, A)$ is well-defined and that its infinite volume limit ($L \to \infty$) can be taken while maintaining $m > 1/L$.
4. Identifying Mathematical Errors: The authors perform a direct mathematical check on the density-of-states equality proposed in Giordano's comment, specifically testing the relation between the expectation value of a delta function and the delta function of the expectation value.

Key Contributions and Results
The paper presents three primary technical findings:

• Violation of $m^2$-Analyticity in Counterexamples: The authors demonstrate that the counterexamples proposed by Giordano [1] inherently violate the crucial assumption that gluonic observables are analytic functions of $m^2$ at high temperatures. Specifically, in the Gaussian model example provided by Giordano, expectation values $\langle O(A)^\alpha \rangle$ for non-integer $\alpha$ yield non-analytic powers of $m^2$ in the $m \to 0$ limit. The authors argue that such non-analyticity implies infrared divergence or criticality, which is not observed in the symmetric phase of QCD. They note that Giordano himself acknowledges that these examples break $m^2$-analyticity.
• Validation of $P(m, A)$: The authors refute the claim that $P(m, A)$ is ill-defined. By defining the expectation value via the path integral with a characteristic function for the support of the operator, they show that $P(m, A)$ is a well-defined functional on a finite lattice. They argue that the specific "ill-defined" example of $P(m, A)$ suggested by Giordano (involving a step function $\Theta(m^2 - |A|)$) also fails because it leads to non-analytic contributions in derivatives with respect to $m^2$, thereby contradicting the $m^2$-analyticity assumption of QCD in the symmetric phase.
• Correction of a Mathematical Error: The authors identify a fundamental error in Giordano's discussion regarding nonlocal quantities. Giordano assumed the equality $\langle \delta(x - O(A)) \rangle = \delta(x - \langle O(A) \rangle)$. The authors demonstrate this is incorrect, as the expectation value of a delta function is not equal to the delta function of the expectation value. Using a simple Gaussian model for topological charge density, they show that assuming this equality leads to the false conclusion that $\langle O_L(A)^2 \rangle = \langle O_L(A) \rangle^2$, whereas the correct relation involves independent cumulants (e.g., $\langle O_L(A)^2 \rangle = 3\langle O_L(A) \rangle^2$).

Significance and Conclusion
The authors conclude that the arguments presented in Giordano's comment [1] are not valid. The significance of this reply lies in the defense of the original arguments in [2] regarding chiral symmetry restoration and the axial U(1) anomaly. The authors assert that the counterexamples intended to refute their work fail because they do not respect the basic analytic properties of QCD at high temperatures. Furthermore, the technical errors identified in the comment regarding the definition of $P(m, A)$ and the properties of state densities undermine the critique's foundation. Consequently, the authors maintain that their original conclusions regarding the eigenvalue density of the Dirac operator and the U(1) anomaly remain robust.