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Spontaneous chiral-symmetry breaking in four-dimensional QCD with $A_x=A_y=0$ projection in dimensional reduction gauge

This paper demonstrates that in four-dimensional QCD under the dimensional reduction gauge, spontaneous chiral-symmetry breaking is driven primarily by the temporal and longitudinal gauge components ( $A_t$ and $A_z$ ) while the transverse components ( $A_x$ and $A_y$ ) play no significant role, as evidenced by the preservation of the chiral condensate under $tz$-projection but its vanishing under $xy$-projection.

Original authors: Kei Tohme (Kyoto U.), Hideo Suganuma (Kyoto U.)

Published 2026-10-23

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CC BY 4.0

Original authors: Kei Tohme (Kyoto U.), Hideo Suganuma (Kyoto U.)

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

Deep within the fabric of our universe, a force known as the strong interaction binds the smallest building blocks of matter together. This force, described by a theory called quantum chromodynamics, is responsible for holding protons and neutrons together inside the atomic nucleus. While scientists can easily predict how these particles behave when they are smashed together at incredibly high speeds, understanding how they stick together at low energies remains one of the most difficult challenges in modern physics. The theory suggests that under normal conditions, two distinct but related phenomena occur: quarks are permanently trapped inside particles, never found alone, and a symmetry that should exist between different types of matter is spontaneously broken. This breaking of symmetry is what gives particles their mass and is essential for the existence of the world as we know it. To understand how these complex processes work, physicists often turn to powerful computer simulations that break space and time into a grid, allowing them to watch the behavior of these particles step by step.

A team of researchers at Kyoto University has recently used this approach to investigate the mysterious mechanism behind the breaking of chiral symmetry. They focused on a specific way of organizing the mathematical description of the strong force, known as the dimensional reduction gauge. In this framework, the four dimensions of space-time are treated in a special way that highlights a hidden simplicity: the theory behaves as if the particles are confined to move primarily on two-dimensional sheets stacked within the four-dimensional world. The researchers wanted to know which parts of this complex system are actually responsible for the breaking of symmetry. To find out, they performed a series of precise computer experiments where they systematically removed different components of the force field from their simulations. They created two distinct scenarios: one where they kept only the components acting along the time and one spatial direction, and another where they kept only the components acting along the other two spatial directions.

The results of these simulations were striking and revealed a clear division of labor within the strong force. When the researchers removed the components acting in the other two spatial directions, the breaking of symmetry vanished completely. The mathematical density of the low-energy states, which acts as a direct measure of this symmetry breaking, dropped to zero. This finding indicates that these specific components are the sole drivers of the phenomenon; without them, the symmetry remains unbroken. In contrast, when they removed the components acting in the time and one spatial direction, the symmetry breaking remained almost entirely intact. The measure of the effect was preserved at roughly eighty-five percent of its original strength, showing that the remaining components are sufficient to sustain the phenomenon. This suggests that the force fields acting along the time axis and one spatial axis carry the dominant role in generating the mass and structure of matter, while the fields in the other two directions play no significant part in this specific process.

This discovery offers a new perspective on how the universe might be structured at its most fundamental level. It implies that the complex, four-dimensional dance of particles can be understood through a much simpler lens, where the essential physics is concentrated on two-dimensional planes. The researchers propose a picture where the force-carrying particles are effectively confined to these two-dimensional sheets, while the matter particles themselves still move freely through the full four-dimensional space. This arrangement allows the system to maintain the necessary conditions for symmetry breaking without violating known physical laws that would otherwise prevent such a process in a strictly two-dimensional world. The study confirms that the essential features of how matter acquires its mass can be described using only a subset of the force's components, providing a clearer, more focused view of the mechanisms that hold the atomic nucleus together.

Technical Summary: Spontaneous Chiral-Symmetry Breaking in 4D QCD with Dimensional Reduction Gauge

Problem Statement
Quantum Chromodynamics (QCD) exhibits two fundamental non-perturbative phenomena: color confinement and spontaneous chiral-symmetry breaking (SCSB). While confinement is often associated with the formation of one-dimensional color-flux tubes, suggesting an effective dimensional reduction of spatial dimensions, the compatibility of this reduction with SCSB remains a theoretical question. The Coleman theorem prohibits the spontaneous breaking of global continuous symmetries in 1+1 dimensional systems due to infrared instabilities. This paper investigates whether SCSB in 4D QCD can be described within a framework where gluonic degrees of freedom undergo effective dimensional reduction, specifically utilizing the Dimensional Reduction (DR) gauge.

Methodology
The authors employ 4D $SU(3)$ quenched lattice QCD Monte Carlo calculations at $\beta=6.0$ on a $16^4$ lattice. The study focuses on the DR gauge, defined by minimizing the functional $R_{DR} \equiv \int d^4s \, \text{Tr}[A_x^2(s) + A_y^2(s)]$ . In this gauge, the transverse gluon fields $A_x$ and $A_y$ are strongly suppressed, effectively confining gluon propagation to $t$ - $z$ planes.

To isolate the specific contributions of gauge field components to SCSB, the authors perform two distinct projections on the gauge configurations generated in the DR gauge:

$tz$-projection: Setting $A_x(s) = A_y(s) = 0$ while retaining $A_t$ and $A_z$ .
$xy$-projection: Setting $A_t(s) = A_z(s) = 0$ while retaining $A_x$ and $A_y$ .

The order parameter for SCSB, the chiral condensate $\langle \bar{q}q \rangle$ , is estimated via the Banks-Casher relation, which links the condensate to the spectral density of the Dirac operator $\rho(\lambda)$ at zero eigenvalue ( $\lambda=0$ ). The authors calculate the low-lying Dirac eigenvalues for three categories of configurations: original (non-projected), $tz$-projected, and $xy$-projected, using the Kogut-Susskind fermion formalism and the shift-inverse Lanczos method.

Key Results
The numerical analysis yields the following findings regarding the Dirac eigenvalue density $\rho(\lambda)$ and the resulting chiral condensate:

Original Configurations: The standard low-lying Dirac eigenvalue density $\rho(0)$ is observed, corresponding to a non-zero chiral condensate.
$tz$-projection: The chiral condensate is largely preserved. The estimated density $\rho_{tz}(0)$ is approximately $85\%$ of the original value ( $\rho_{tz}(0) \simeq 0.85 \rho(0)$ ). This indicates that the $A_t$ and $A_z$ components carry the dominant contribution to SCSB.
$xy$-projection: The chiral condensate vanishes completely ( $\rho_{xy}(0) = 0$ ). This demonstrates that the $A_x$ and $A_y$ components alone are insufficient to generate spontaneous chiral-symmetry breaking.

Key Contributions

Identification of Dominant Degrees of Freedom: The study provides numerical evidence that in the DR gauge, SCSB is driven almost exclusively by the temporal ( $A_t$ ) and longitudinal ( $A_z$ ) gluon components, while the transverse components ( $A_x, A_y$ ) play a negligible role.
Compatibility of Dimensional Reduction and SCSB: The results demonstrate that SCSB is compatible with the effective dimensional reduction of the gluonic sector. While gluons are confined to $t$ - $z$ planes (2D-like behavior), the quark sector retains its 4D structure. The quark propagator in the $tz$-projected theory retains 4D kinetic terms ( $\gamma_x \partial_x + \gamma_y \partial_y$ ), allowing for the existence of massless Nambu-Goldstone bosons (pions) in 4D space-time, thereby circumventing the constraints of the Coleman theorem which applies to strictly 2D systems.

Significance and Conclusion
The paper concludes that the essential features of both quark confinement and spontaneous chiral-symmetry breaking in 4D QCD can be described using only the $A_t$ and $A_z$ components within the DR gauge. This suggests a unified picture where the gluonic sector undergoes effective dimensional reduction (resembling an ensemble of 2D Yang-Mills systems on $t$ - $z$ planes), while the fermionic sector (quarks and pions) propagates through the full 4D space-time.

The authors propose a physical analogy where the DR-gauged QCD system resembles "dual graphene": in graphene, electrons (fermions) are confined to 2D planes while photons (bosons) propagate in 3D; in this QCD scenario, gluons (bosons) are confined to 2D $t$ - $z$ planes, while quarks and pions (fermions) propagate in 4D. This work reinforces the utility of the DR gauge in isolating the relevant degrees of freedom for non-perturbative QCD phenomena.

Technical Summary: Spontaneous Chiral-Symmetry Breaking in 4D QCD with Dimensional Reduction Gauge

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