QCD sum rules: Borel parameter vs. Euclidean time

This paper introduces a modification of QCD sum rules using Euclidean time correlators in coordinate space instead of Borel transforms, demonstrating that while nucleon mass and residue can be roughly estimated, this approach suffers from significantly larger uncertainties and a lack of a stable working window compared to traditional Borel sum rules.

The Gist

Imagine you are trying to figure out the weight and size of a hidden object inside a sealed, foggy box. You can't see the object directly, but you can shake the box and listen to how the sound echoes. In the world of particle physics, this "box" is the vacuum of space, and the "object" is a proton (a type of nucleon).

This paper by Andrei Smilga is a comparison of two different ways to "listen" to these protons using a method called QCD Sum Rules. The goal is to calculate the mass and other properties of the proton using only the fundamental laws of physics, without needing to run a giant particle collider.

Here is the breakdown of the two methods compared in the paper, using simple analogies:

The Two Methods: The "Hot Water" vs. The "Foggy Window"

1. The Traditional Method: Borel Sum Rules (The Hot Water Tap)
Think of the standard method as a shower with a hot water tap.

• The Problem: You need the water to be the perfect temperature to wash effectively.
  • If the water is too cold (mathematically, the parameter $M^2$ is too small), the "power corrections" (which represent the messy, complex interactions of the vacuum) are huge and drown out the signal. It's like trying to wash with ice water; you can't get anything done.
  • If the water is too hot (the parameter $M^2$ is too large), the signal from the proton gets lost in the steam of "excited states" (heavier, unstable particles). It's like the water is boiling; you can't see the object you are washing.
• The Sweet Spot: The paper shows that there is a "lukewarm" zone where the water is just right. In this zone, the messy vacuum effects are small enough to ignore, but the excited states are suppressed enough that you can clearly hear the proton's "voice."
• The Result: Because this "lukewarm" zone exists, scientists can use this method to estimate the proton's mass and "residue" (a measure of how strongly the proton interacts with the current used to create it) with about 10–15% accuracy. The two different equations used to check the math agree with each other perfectly in this zone.

2. The New Method: Euclidean Time Sum Rules (Looking Through a Foggy Window)
The author proposes a new way: instead of the "shower tap," let's just look at the object through a window over time (Euclidean time, $\tau$).

• The Idea: This seems more natural. Time is a real thing we experience, whereas the "Borel parameter" is a mathematical trick invented to make the equations work.
• The Problem: When you try to use this method, the "fog" (the background noise from excited states) never clears up enough.
  • In the traditional method, the mathematical "weight" given to heavy particles drops off very quickly (like a steep cliff).
  • In this new method, the weight drops off much more slowly (like a gentle slope).
• The Result: Even when you wait a long time (large $\tau$), the "noise" from the excited states is still three times louder than the signal from the actual proton. Furthermore, the mathematical corrections start flipping signs and making the whole equation break down.
• The Verdict: While you can roughly guess the proton's mass if you force the numbers to work, there is no "sweet spot" where the math is reliable. The "window" is too foggy. The author concludes that while this method is theoretically beautiful and uses more natural concepts, it is not practical for getting accurate numbers.

The Bottom Line

The paper is essentially a "reality check" for a new idea.

• The Old Way (Borel): It feels a bit artificial (like a mathematical trick), but it works. It finds a "Goldilocks zone" where the answer is stable and reliable.
• The New Way (Euclidean Time): It feels more natural and physical, but it fails in practice. There is no "Goldilocks zone" for it; the background noise is always too loud, and the math becomes unstable.

Conclusion: The author argues that while the Euclidean time approach is an attractive alternative in theory, it cannot replace the traditional Borel sum rules for calculating the properties of protons because it lacks a stable range of values where the results are trustworthy.

Technical Summary

Technical Summary: QCD Sum Rules – Borel Parameter vs. Euclidean Time

Problem Statement
Quantum Chromodynamics (QCD) sum rules are a established method for determining hadron properties, such as mass and residues, by exploiting quark-hadron duality. This involves comparing a theoretical calculation of a current correlator (based on QCD diagrams and condensates) with a phenomenological representation (a sum over hadron states). Traditionally, this comparison is performed in momentum space using a Borel transform, which introduces an auxiliary parameter $M^2$ (the Borel parameter). The Borel transform serves to suppress contributions from excited states (continuum) and higher-order power corrections, creating a "fiducial interval" where the ground state dominates and the theoretical expansion is reliable.

The paper investigates a modification of this method: instead of transforming to momentum space, it considers the correlators directly in coordinate space as functions of Euclidean time ($\tau$). The central question is whether a "fiducial interval" of Euclidean time exists where power corrections and continuum contributions are simultaneously under control, allowing for a reliable extraction of nucleon properties, similar to the success of the Borel method.

Methodology
The author, Andrei Smilga, focuses on the nucleon channel using the standard Ioffe current:
$$\eta(x) = \epsilon_{ABC} [u^A(x) C \gamma_\mu u^B(x)] \gamma_5 \gamma^\mu d^C(x)$$
The analysis proceeds in two parallel tracks:

1. Theoretical Side (OPE): The Euclidean correlator $\Pi(\tau)$ is calculated using the Operator Product Expansion (OPE). This includes:

   • The perturbative leading-order term (Fig. 1a).
   • Non-perturbative power corrections arising from the quark condensate $\langle \bar{q}q \rangle$, the gluon condensate $\langle G^2 \rangle$, and mixed condensates like $\langle \bar{q} g \sigma G q \rangle$.
   • Higher-order corrections involving products of condensates and logarithmic terms.
     The resulting expression for $\Pi(\tau)$ is a series in powers of $\tau$ and condensates (Eq. 2.10).

2. Phenomenological Side: The correlator is represented as a sum over physical states. This includes the nucleon pole (mass $m$, residue $\lambda$) and a continuum of excited states modeled by a threshold $W$ (resonance + continuum model).

3. Comparison:

   • Borel Sum Rules (Standard): The coordinate space correlator is Fourier transformed to momentum space, and a Borel transform is applied. This yields sum rules where the weight function is $e^{-s/M^2}$.
   • Euclidean Time Sum Rules (Proposed): The theoretical OPE expression is compared directly with the phenomenological spectral integral in coordinate space. The weight functions here are modified Bessel functions $K_1(\sqrt{s}\tau)$ and $K_2(\sqrt{s}\tau)$, which decay as $e^{-\sqrt{s}\tau}$.

Key Contributions and Results

• Derivation of Euclidean Time Sum Rules: The paper derives explicit sum rules (Eq. 4.21) relating the OPE expansion in $\tau$ to the nucleon mass and residue, including the continuum contribution modeled by integrals involving Bessel functions ($Q_1, Q_2$).
• Analysis of the Fiducial Interval:
  • Borel Case: The analysis confirms that for Borel sum rules, a fiducial interval exists (around $M \approx 0.96$ GeV). In this region, the continuum contribution is roughly 36% and power corrections are roughly 38% of the total, allowing both to be controlled. This leads to a consistent extraction of the nucleon mass ($m \approx 940$ MeV) and residue with 10–15% accuracy.
  • Euclidean Time Case: The paper finds that no such fiducial interval exists for the Euclidean time sum rules in the nucleon channel.
    • To suppress power corrections (specifically the $\Sigma^2$ term), $\tau$ must be small. However, to suppress the continuum, $\tau$ must be large.
    • At the value of $\tau$ where power corrections become manageable ($\tau \approx 3.5$ GeV$^{-1}$), the continuum contribution is approximately 75% of the total, three times larger than the nucleon pole contribution.
    • Furthermore, at these large $\tau$ values, higher-order power corrections (e.g., $\sim m_0^2 \Sigma^2$) become dominant and negative, causing the theoretical side of the sum rule to flip sign, rendering the expansion unreliable.
• Asymptotic Behavior: The fundamental reason for the failure of the Euclidean time method is the difference in weight functions. The Borel weight $e^{-s/M^2}$ suppresses high-energy states ($s$) much more effectively than the Euclidean weight $e^{-\sqrt{s}\tau}$. Consequently, excited states are not sufficiently suppressed in the coordinate space approach.
• Rough Estimates: Despite the lack of a fiducial interval, the author notes that if one fixes the nucleon mass to an assumed value (e.g., 800 MeV), the extracted residue remains roughly constant in a limited $\tau$ range ($1.5 \text{--} 2.0$ GeV$^{-1}$) and yields a value ($\tilde{\lambda}^2 \sim 2$ GeV$^6$) comparable to Borel results. However, this region is dominated by the continuum, not the nucleon pole.

Significance and Claims
The paper concludes that while Euclidean time sum rules are theoretically attractive because Euclidean time $\tau$ is a more "natural and physical quantity" than the artificial Borel parameter $M$, they are not suitable for practical applications in the nucleon channel.

The primary claim is that the absence of a fiducial interval in coordinate space prevents a reliable separation of the ground state from the continuum and power corrections. The author suggests this limitation likely extends to other light hadron channels as well. The work serves as a cautionary comparison, reinforcing the necessity of the Borel transform for the stability and reliability of QCD sum rule analyses in this context.