[Paper Review] Structure of hybrid static potential flux tubes in SU(2) lattice Yang-Mills theory

Lasse Mueller, Marc Wagner|arXiv (Cornell University)|Mar 29, 2018

Quantum Chromodynamics and Particle Interactions4 references3 citations

TL;DR

This study investigates the structure of hybrid static potential flux tubes in SU(2) lattice Yang-Mills theory using Wilson loop correlation functions to compute chromoelectric and chromomagnetic field strength distributions. It finds that in the $Π_u$ sector, chromomagnetic flux dominates centrally between the quark-antiquark pair, while chromoelectric flux is suppressed, indicating a distinct gluonic excitation not present in the $Σ_g^+$ ground state.

ABSTRACT

We study the structure of the hybrid static potential flux tube in the $Π_u$ sector in SU(2) lattice Yang-Mills theory. To this end, we compute the squares of the chromoelectric and chromomagnetic field strengths in the presence of a static quark-antiquark pair. We show clear evidence that the gluon distribution is significantly different compared to that of the ordinary static potential with quantum numbers $Σ_g^+$.

Motivation & Objective

• To investigate the spatial structure of flux tubes in hybrid static potentials in SU(2) lattice Yang-Mills theory.
• To compare the chromoelectric and chromomagnetic field distributions in the $Π_u$ hybrid state with those in the $Σ_g^+$ ground state.
• To identify the role of gluonic excitations in generating non-trivial quantum numbers in hybrid mesons.
• To resolve discrepancies with a recent lattice study [6] by analyzing field strength distributions with improved gauge-invariant observables.

Proposed method

• Computes field strength expectation values $\braket{E_j^2}_{Q\bar{Q}} - \braket{E_j^2}_{\textrm{vac}}$ and $\braket{B_j^2}_{Q\bar{Q}} - \braket{B_j^2}_{\textrm{vac}}$ using Wilson loop correlation functions.
• Uses a Wilson loop with spatial structures designed to project onto the $\Pi_u$ sector, differing from standard rectangular loops used for $\Sigma_g^+$.
• Employs APE-smeared spatial links for Wilson loop operators while keeping plaquette links unsmeared for field strength measurements.
• Applies the relation $\braket{E_j^2}_{Q\bar{Q}} - \braket{E_j^2}_{\textrm{vac}} \propto \left( \frac{\braket{W \cdot P_{0j}(t/2,\mathbf{x})}}{\braket{W}} - \braket{P_{0j}} \right)$ to extract chromoelectric field contributions.
• Uses $\braket{B_j^2}_{Q\bar{Q}} - \braket{B_j^2}_{\textrm{vac}} \propto \left( \braket{P_{kl}} - \frac{\braket{W \cdot P_{kl}(t/2,\mathbf{x})}}{\braket{W}} \right)$ for chromomagnetic field strength.
• Performs simulations on an $18^4$ lattice with $\beta = 2.5$, corresponding to $a \approx 0.073\,\textrm{fm}$, and computes field profiles along the $Q\bar{Q}$ axis, mediator axes, and in the $x$-$z$ plane.

Experimental results

Research questions

• RQ1 How does the gluonic field distribution differ between the $\Sigma_g^+$ and $\Pi_u$ hybrid static potential sectors in SU(2) lattice Yang-Mills theory?
• RQ2 What is the spatial structure of chromoelectric and chromomagnetic flux tubes in the $\Pi_u$ hybrid state?
• RQ3 Why does the $\Pi_u$ hybrid state exhibit a localized peak in chromomagnetic field strength at the midpoint between the quark and antiquark?
• RQ4 How do the field strength distributions in this work compare quantitatively and qualitatively with those reported in a recent lattice study [6]?
• RQ5 What is the origin of the observed rotational non-invariance in the gluonic excitation pattern of the $\Pi_u$ state?

Key findings

• In the $\Pi_u$ hybrid state, the chromoelectric field strength $E_z^2$ is strongly suppressed at the midpoint between the quark and antiquark, while $E_z^2$ dominates near the static charges.
• Chromomagnetic field strengths $B_j^2$ are significantly enhanced at the midpoint between the quark and antiquark in the $\Pi_u$ state, forming a localized flux tube.
• The chromomagnetic flux is not rotationally symmetric: field strengths along the $x$ and $y$ axes differ, indicating a non-spherical gluonic excitation pattern.
• Chromoelectric field contributions are negative in the $\Sigma_g^+$ state at the midpoint, while they are positive near the quark and antiquark, but suppressed centrally.
• The field strength profiles in this work show a clear transition from chromoelectric dominance near the charges to chromomagnetic dominance at the center, which is not observed in the recent study [6].
• A qualitative discrepancy exists between this work and [6], where no such transition from chromoelectric to chromomagnetic dominance is found, despite similar methods and gauge groups.