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Flavour decomposition of the nucleon tensor multipole moments

This paper computes the flavor-separated chiral-odd nucleon tensor form factors and their resulting monopole, dipole, and quadrupole moments using light-cone QCD sum rules, revealing new exact analytical relations between quark flavors and providing numerical predictions for the transverse spatial deformation of the nucleon's $u$ and $d$ quarks.

Original authors: U. Özdem

Published 2026-10-09

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: U. Özdem

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Inside every proton and neutron, the building blocks of atomic nuclei, a complex internal architecture is hidden. These particles are not solid spheres but dynamic clouds of smaller constituents called quarks, held together by the strong force. For decades, physicists have mapped how these quarks move and spin, creating a detailed picture of their behavior. However, one specific aspect of their motion has remained elusive: how they are polarized sideways, or transversely, relative to the spin of the particle they inhabit. While we know how quarks spin in the same direction as the proton, or in the opposite direction, understanding their sideways orientation is like trying to see a shadow cast by a light source that refuses to shine directly. This sideways polarization is crucial because it reveals how the internal spin of the quarks connects to their orbital motion, a relationship that shapes the very structure of matter. Furthermore, measuring this property precisely is essential for testing the limits of our current understanding of physics, as it could reveal subtle signs of new, undiscovered forces.

A recent study by U. Özdem at Istanbul Aydin University has taken a significant step forward in mapping this hidden landscape. Using a sophisticated theoretical framework known as light-cone QCD sum rules, the researcher calculated the internal "shape" of the nucleon's sideways polarization. Instead of treating the proton as a single, uniform object, the study separated the contributions of the two main types of quarks found inside: the up quark and the down quark. By doing this, the researcher was able to organize the complex data into three distinct geometric descriptions: a monopole, which represents the total amount of sideways spin; a dipole, which shows how the average position of these spinning quarks shifts; and a quadrupole, which describes how the distribution of quarks stretches or squashes when the proton is polarized. This quadrupole measurement is particularly novel, as it had never been successfully calculated using this specific method before.

The results paint a vivid picture of the internal world. The study found that the up and down quarks behave in strikingly different ways. When the proton is polarized, the distributions of these two quark types are pushed in opposite directions, shifting sideways by nearly equal amounts but in reverse. Specifically, the center of the up quark distribution moves about 0.37 femtometers in one direction, while the down quark moves the same distance in the opposite direction. A femtometer is a unit of length so small that it is difficult to visualize; to grasp its scale, one might imagine that if a proton were the size of a large stadium, a femtometer would be roughly the width of a single human hair. Beyond this shift, the study revealed that both quark distributions stretch out along the axis of polarization, but the down quark stretches much more dramatically—about five to six times more than the up quark. This indicates that the down quark is far more susceptible to deformation by the proton's spin than its up quark counterpart.

One of the most profound findings concerns the total amount of sideways spin carried by the proton. The study confirmed that the contributions from the up and down quarks largely cancel each other out when looking at the proton as a whole. This cancellation is so precise that the net sideways spin of the proton is extremely small, a result that had been theoretically predicted but never explicitly demonstrated with this level of detail in this framework. The research also uncovered a strict mathematical relationship between the different ways the down quark responds to polarization, suggesting that its behavior is governed by a single, independent function. This finding simplifies the theoretical description of the down quark's role within the proton.

The study also tested a long-standing theoretical prediction regarding the relationship between different aspects of the proton's shape. While the prediction held true in terms of the general direction and scale of the effects, the calculated values were about two-thirds of what the simple theory suggested. This discrepancy, while not a complete contradiction, points to the complexity of the proton's internal dynamics and suggests that the simple models used to describe it may need refinement. The researchers noted that their calculations are subject to certain uncertainties, particularly because the method relies on approximations that work best when certain complex interactions are minimized. Nevertheless, the results provide a robust new set of numbers that other scientists can use to refine their own models.

Perhaps the most significant outcome of this work is the successful calculation of the quadrupole moment, a measure of the proton's deformation that had been absent from previous calculations using this method. This new data fills a critical gap in our understanding of how the proton's shape changes under the influence of its own spin. By separating the contributions of the up and down quarks, the study has shown that the internal structure of the nucleon is not a uniform blend but a complex interplay of distinct components that move and deform in unique ways. These findings offer a clearer, more detailed map of the nucleon's interior, bringing physicists closer to a complete understanding of how the fundamental building blocks of matter are arranged and how they interact.

Technical Summary: Flavour Decomposition of the Nucleon Tensor Multipole Moments

Problem and Motivation
The internal structure of the nucleon is encoded in the matrix elements of local QCD operators. While vector and axial-vector currents are well-mapped, the tensor current, which probes the transverse polarization of quarks relative to the nucleon spin, remains the least constrained leading-twist structure. The tensor charge, the first moment of the transversity distribution $h_1$ , is chiral-odd and decouples from inclusive deep inelastic scattering, restricting its extraction to semi-inclusive processes and dihadron production. Theoretically, the tensor form factors $H_T$ , $E_T$ , and $\tilde{H}_T$ are not the most natural objects for physical interpretation. Instead, expanding the matrix element in irreducible tensors separates the physics into a monopole (tensor charge $g_T$ ), a dipole (anomalous tensor magnetic moment $\kappa_T$ ), and a quadrupole ( $Q_T$ ). The quadrupole moment is peculiar to the chiral-odd sector at leading twist, as chiral-even operators cannot form this structure. Previous light-cone QCD sum rule (LCSR) studies have addressed the isovector channel or derived sum rules for the isoscalar channel without evaluating them, and none have previously obtained the quadrupole moment within this framework. Furthermore, resolving the flavor contributions ( $u$ and $d$ quarks) is essential, as structural results often reside in specific flavor channels or their interplay.

Methodology
The author computes the chiral-odd form factors of the nucleon using light-cone QCD sum rules, explicitly resolving $u$ and $d$ quark contributions. The calculation relies on the correlation function of the nucleon interpolating current $J_N$ and the tensor current $\bar{q} i\sigma_{\mu\nu} q$ .

Interpolating Current: The Ioffe current is used, fixing the mixing parameter to maximize the coupling to the nucleon ground state relative to the continuum. This choice also allows the current to be written with like-flavor pairs adjacent, making a specific selection rule manifest.
Lorentz Structure and Basis: A critical methodological step involves the canonical ordering of Dirac strings. The Dirac structures generated by the operator product are not independent until reduced to a canonical arrangement (e.g., $\gamma_\mu \gamma_\nu \not{q} \not{p}$ ). The author identifies eight mutually irreducible antisymmetric structures. Of these, only three are sufficient to extract the three form factors separately. Notably, the structure typically used to extract the tensor charge in earlier sum-rule work ( $\not{q}\sigma_{\mu\nu}$ ) is not among the eight independent structures after ordering.
Selection Rules: Due to the specific form of the Ioffe current and the properties of the charge conjugation matrix, a selection rule emerges: the $d$ -quark channel receives contributions only from the six vector ( $V$ ) distribution amplitudes, while the $u$ -quark channel is fed by scalar ( $S$ ), tensor ( $T$ ), and vector amplitudes (with $V$ amplitudes reaching the $u$ channel only via mass terms).
Numerical Analysis: The sum rules are evaluated for two sets of distribution amplitudes (Set I: estimated from QCD sum rules; Set II: asymptotic set). The results are extrapolated to $Q^2=0$ using a fit function with a free exponent. The working regions for the Borel mass ( $M^2$ ) and continuum threshold ( $s_0$ ) are fixed to ensure convergence and pole dominance.

Key Contributions and Results
The paper provides the first evaluation of the tensor quadrupole moment and the isoscalar channel within the LCSR framework. The results at $\mu^2 = 1 \text{ GeV}^2$ are:

Isovector Moments:
- Tensor charge: $g_T^{u-d} = 1.27(7)$ (Set I) and $1.08(6)$ (Set II).
- Dipole moment: $\kappa_T^{u-d} = 1.35(62)$ and $1.58(59)$ .
- Quadrupole moment: $Q_T^{u-d} = -4.40(83)$ and $-4.20(73)$ .
Isoscalar Moments:
- Tensor charge: $g_T^{u+d} = 0.39(7)$ and $0.38(6)$ .
- Dipole moment: $\kappa_T^{u+d} = 4.47(62)$ and $4.32(59)$ .
- Quadrupole moment: $Q_T^{u+d} = 2.41(53)$ and $1.56(30)$ .

Key Theoretical Findings

Vanishing Leading-Twist Isoscalar Charge: The isoscalar tensor charge receives no leading-twist contribution. The twist-three contributions from the $u$ and $d$ channels are equal in magnitude and opposite in sign, leading to a cancellation. Consequently, $g_T^{u+d}$ is built entirely from twist-four and higher terms, explaining its small value and higher relative uncertainty compared to the isovector charge.
Identity in the $d$ -Quark Channel: In the chiral limit, the $d$ -quark contributions to $E_T$ and $\tilde{H}_T$ are exactly equal and opposite ( $E_T^d = -\tilde{H}_T^d$ ). The breaking of this identity is proportional to the quark mass and the twist-six amplitude $V_6$ . This implies the $d$ -quark sector carries only a single independent function. Numerically, the breaking is a $4 \times 10^{-4}$ effect.
Large- $N_c$ Relation Test: The mean-field relation $2\tilde{H}_T^{u-d} = -E_T^{u-d}$ , derived under large- $N_c$ assumptions, is tested without assuming large $N_c$ . The ratio $2\tilde{H}_T^{u-d} / (-E_T^{u-d})$ is found to be $0.64(16)$ and $0.67(15)$ , consistent in sign and order of magnitude but deviating by approximately $2.2\sigma$ from unity. The deviation resides entirely in the $u$ -quark channel.

Impact-Parameter Interpretation
When mapped to the impact-parameter plane, the multipole moments reveal distinct flavor behaviors:

Displacement: The $u$ and $d$ quark distributions are displaced in opposite transverse directions by nearly equal amounts ( $\langle b_y \rangle \approx \pm 0.37$ fm). This opposite shift arises because the tensor charge $g_T^d$ is negative while $\kappa_T^d$ is positive, whereas both are positive for the $u$ quark.
Deformation: Both distributions are elongated across the polarization axis (negative quadrupole deformation). However, the deformation is five to six times stronger for the $d$ quark than for the $u$ quark. This hierarchy reflects the flavor asymmetry in the ratio $|\tilde{H}_T/E_T|$ , which is unity for the $d$ quark and an order of magnitude smaller for the $u$ quark.

Significance
The paper claims that resolving the flavor structure is not merely a refinement but the substance of the calculation, as it reveals identities and cancellations invisible in isospin-summed quantities. The methodological shift to a canonical Dirac string ordering provides a unique basis for extracting form factors, differing from previous sum-rule approaches. The results provide the first LCSR determination of the tensor quadrupole moment and the isoscalar tensor charge, offering a benchmark for lattice QCD and phenomenological models. The findings highlight that the isoscalar tensor charge is a higher-twist dominated quantity and that the $d$ -quark sector is governed by a single independent function in the chiral limit. The work also confirms that the mean-field relation for the isovector channel holds in sign and magnitude but exhibits a mild tension in value, residing entirely in the $u$ -quark contribution.

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