Nucleon matrix elements of axial anomaly, axial currents and pseudoscalar currents in the QCD sum rule

This paper utilizes QCD sum rules to analyze one-nucleon matrix elements of various axial and pseudoscalar currents, expressing their coupling constants in terms of quark and gluon operators and parton distribution moments while notably addressing the previously overlooked nucleon matrix element of the axial anomaly.

The Gist

Imagine the proton (a particle inside an atom) not as a solid marble, but as a bustling, chaotic city made of tiny, invisible workers called quarks and gluons. These workers are constantly moving, spinning, and interacting. Physicists want to understand exactly how much "spin" (a type of intrinsic rotation) each worker contributes to the total spin of the city.

This paper is like a detailed architectural survey of that city, trying to measure the specific contributions of different groups of workers using a mathematical toolkit called QCD Sum Rules.

Here is a breakdown of what the author, Janardan Prasad Singh, did, using simple analogies:

1. The Goal: Measuring the "Spin" of the City

In physics, there are different ways to measure how these particles spin.

• Axial Currents: Think of these as measuring the direction the workers are spinning (like a spinning top).
• Pseudoscalar Currents: Think of these as measuring the intensity or "push" of that spin.
• The Axial Anomaly: This is the paper's main star. Imagine a hidden, invisible force in the city that messes up the usual rules of how things spin. For a long time, physicists ignored this "ghost" force because it was hard to catch. This paper tries to measure exactly how strong this ghost force is inside the proton.

2. The Method: The "Echo" Technique

The author doesn't just look at the proton directly (which is impossible). Instead, he uses a clever trick involving echoes.

• The Setup: He imagines sending a signal (a mathematical "correlator") into the proton.
• The Phenomenological Side (The Real World): He looks at what happens when the proton interacts with its own "excited states" (like a proton getting a little bump and vibrating) or its "continuum" (a sea of other particles). It's like listening to the echo of a shout in a canyon to figure out the shape of the canyon walls.
• The Theoretical Side (The Math): He calculates what the echo should sound like based on the known laws of physics (Quantum Chromodynamics or QCD). This involves looking at the "moments" of the parton distribution functions.
  • Analogy: Imagine trying to guess the weight of a bag of flour by looking at how much it bounces. The "moments" are like measuring the bounce at different speeds to figure out the weight.

3. The Big Discovery: Catching the Ghost

The most significant part of this paper is that the author finally managed to calculate the nucleon matrix element of the axial anomaly.

• The Problem: Until now, this "ghost force" (the anomaly) was largely ignored in literature because it was too tricky to measure.
• The Result: The author found a way to express the strength of this anomaly in terms of the quarks and gluons inside the proton. He found that this anomaly is a real, measurable quantity (represented by a value called $\chi_g$), and it plays a crucial role in balancing the equations of the proton's spin.

4. Two Ways to Solve the Puzzle

The author didn't just find one answer; he found two different mathematical paths to calculate the "pseudoscalar coupling" (the intensity of the spin).

• Path A: A complex route involving many different variables (quark masses, gluon condensates).
• Path B: A surprisingly simple route that relies only on the "moments" (the bounce measurements mentioned earlier).
• The Surprise: Even though Path B was much simpler and ignored many complex factors, it gave almost the exact same numerical result as Path A. This suggests that the "bounce" of the particles is the most important factor, and the result is very robust.

5. Checking the Work

To make sure his numbers weren't just lucky guesses, the author checked them against:

• Internal Consistency: Do the different parts of his math agree with each other? (Yes, mostly).
• Other Experiments: Do his numbers match what other scientists have found using different methods (like Lattice QCD or previous sum rule studies)?
  • Result: His numbers for the "isovector" spin (the difference between up and down quarks) matched well with known data.
  • Nuance: For the "octet" spin (involving strange quarks), there was a slight mismatch, which the author explains is likely because the math gets messier when dealing with heavier particles (like the eta and eta-prime mesons) compared to the lighter ones.

Summary

In plain English, this paper is a rigorous attempt to map the invisible spin dynamics inside a proton. The author successfully:

1. Caught the "Ghost": Measured the elusive "axial anomaly" contribution, which had been ignored in many previous studies.
2. Simplified the Math: Showed that you can get accurate results using a simpler method that relies mostly on the "bounce" (moments) of the particles, without needing every single complex variable.
3. Validated the Model: Confirmed that his theoretical calculations align well with experimental data and other theoretical models, giving us a clearer picture of how the proton's spin is built from its tiny constituents.

The paper concludes that these new measurements of the anomaly and the spin couplings are now available for other physicists to use in understanding the fundamental building blocks of matter.

Technical Summary

Technical Summary: Nucleon Matrix Elements of Axial Anomaly, Axial Currents, and Pseudoscalar Currents in the QCD Sum Rule

Problem Statement
The paper addresses the calculation of one-nucleon matrix elements for various currents, specifically focusing on the axial anomaly, axial isovector/isoscalar currents, and pseudoscalar currents. A primary motivation is the lack of investigation into the nucleon matrix element of the axial anomaly ($Q = \frac{\alpha_s}{\pi} G\tilde{G}$), also known as the topological charge density, outside of the chiral limit. The authors note that while the axial anomaly is crucial for understanding the singlet axial charge ($g_A^0$) and the proton spin structure, its direct matrix element has been largely ignored in current literature. Furthermore, neglecting the anomaly leads to significant violations of isospin and flavor $SU(3)$ symmetry in sum rules like Bjorken and Ellis-Jaffe, which are not supported by experimental data. The study aims to express the coupling constants of the nucleon to these currents in terms of quark and gluon matrix elements and moments of parton distribution functions (PDFs).

Methodology
The authors employ the QCD sum rule approach, specifically following a third approach distinct from external field methods or simple three-point functions. The methodology involves:

1. Correlator Construction: The authors analyze two-point correlation functions of nucleon interpolating fields in the presence of specific currents. These include pseudoscalar-pseudoscalar correlators, axial-anomaly-pseudoscalar correlators, and axial-axial current correlators (isovector, isoscalar, and strange quark sectors).
2. Phenomenological Side: The spectral functions are derived using the Lehmann representation. Crucially, the authors account for non-diagonal matrix elements between the nucleon ground state and excited states or continuum states. These contributions are parameterized using an exponential form involving the mass differences between the nucleon and excited states, a step the authors note was not taken in previous similar works [18, 19, 20].
3. Theoretical Side (OPE): The Operator Product Expansion (OPE) is performed for the correlators. The nucleon matrix elements of operators containing covariant derivatives are expressed in terms of the moments of parton distribution functions ($A_n^q$). Factorization approximations are used for higher-dimensional condensates.
4. Borel Transformation: A Borel transform is applied with respect to the energy variable ($\omega^2$) to suppress higher-state contributions and enhance the ground state signal.
5. Matching: The phenomenological and OPE sides are matched by equating coefficients of powers of the momentum transfer squared ($\vec{q}^2$). This allows for the extraction of coupling constants and form factors at zero momentum transfer.

Key Contributions and Results
The paper derives expressions for nucleon matrix elements and coupling constants, presenting the following specific results:

• Axial Anomaly and Pseudoscalar Couplings:

  • The nucleon matrix element of the axial anomaly, parameterized as $\chi_g$, is calculated. The result is $|\chi_g| = 1.014 \pm 0.102$.
  • The octet pseudoscalar coupling, $|\chi_8|$, is found to be $0.483 \pm 0.025$.
  • Two distinct expressions for pseudoscalar coupling constants are derived. One expression depends solely on the moments of parton distribution functions ($A_n^q$), rendering it independent of quark masses and other QCD parameters in the chiral limit. Both expressions yield approximately consistent numerical results.
  • The isovector pseudoscalar coupling $|\chi_P^-|$ is determined to be $0.484 \pm 0.026$.

• Axial Coupling Constants:

  • The isovector axial coupling $|g_A^3|$ is calculated as $1.216$.
  • The isoscalar (non-strange) axial coupling $|g_A^{u+d}|$ is found to be $0.409$.
  • The strange quark axial coupling $|g_A^s|$ is determined to be $0.054$. The authors note a limitation here: the calculation does not fully account for chiral anomaly and instanton contributions, which are significant for singlet-singlet correlations, as no suitable hybrid correlator exists for the strange sector alone.

• Consistency Checks:

  • The authors verify the consistency between the derived pseudoscalar couplings and the corresponding axial couplings via the Goldberger-Trieman relation (in the limit $q^2 \to 0$).
  • The singlet axial charge $g_A^0 = g_A^{u+d+s}$ is calculated as $0.261 \pm 0.251$. This is compared with the COMPASS experimental result ($0.32 \pm \dots$) and lattice QCD results, showing qualitative agreement within the large error margins.
  • The octet axial charge $g_A^8$ is discussed; the authors observe that the relation between axial and pseudoscalar couplings holds better for the isovector current than the octet current, attributing the discrepancy to the $q^2 \to 0$ limit and the mixing of the octet current with $\eta$ and $\eta'$ mesons.

Significance and Claims
The paper claims that the inclusion of the nucleon matrix element of the axial anomaly is indispensable for a consistent theoretical description of isosinglet and strange axial couplings, preventing large violations of isospin and flavor symmetry.

A significant technical contribution is the derivation of pseudoscalar coupling constants using an expression dependent only on the moments of parton distribution functions. The authors highlight that this independence from quark masses and other QCD parameters makes these specific results robust in the chiral limit.

Furthermore, the work emphasizes the importance of including non-diagonal matrix elements (excited states and continuum) in the phenomenological side of the sum rules, a refinement over previous studies in this specific approach. The authors conclude that their calculated values for the nucleon matrix elements of pseudoscalar currents, particularly the axial anomaly, are consistent with existing literature and experimental constraints, and believe these results will find application in hadron physics, filling a gap in the current literature regarding the axial anomaly's nucleon matrix element.