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Distribution of Energy Momentum Tensor around Static Charges in Lattice Simulations and an Effective Model

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Abstract

Quantum Chromodynamics exhibits a large variety of unique phenomena. The precise description of them is crucial to understand the strong interaction in nature. The purpose of our studies is to thoroughly investigate the vacuum and the deconfined phase of the SU(3) Yang-Mills (YM) theory. To this effect, we focus on one of the most fundamental observables, the energy momentum tensor (EMT), which can characterize the local structure of the non-Abelian fields with gauge invariant and non-perturbative information. We study the EMT distribution around static charges both in the lattice simulations based on the SU(3) YM gauge theory and in the effective model with the Abelian-Higgs model.

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Notes

  1. 1.

    The \(\alpha \) particle is the nucleus of the helium atom.

  2. 2.

    It was, at the same time, confirmed that the mass of the \(\pi \) meson is approximately equal to \(140~\mathrm {MeV}\). In 1937, Anderson and Neddermeyer discovered the particle, whose mass is approximately equal to that of the meson which Yukawa had predicted. This particle was, however, found insensitive to the strong interaction. This particle is today known as the muon.

  3. 3.

    More than a hundred hadrons are often called ‘Hadron Zoo.’

  4. 4.

    Although Zweig named the particle ace, the name quark becomes nowadays popular.

  5. 5.

    Although the Standard Model is almost consistent with the experimental results so far, there exist some phenomena beyond the Standard Model, such as the fine-tuning problem on the Higgs particle, the strong CP problem, the neutrino oscillation, and so on. For the purpose of the exploration beyond the Standard Model, the grand unified theory and super-string theory are discussed from the theoretical aspects, while the experiments at extremely high energy are in progress.

  6. 6.

    Thus, the physical quantities, such as the scattering cross section \(\sigma \), do not depend on \(\mu \), which is summarized in the renormalization group equation as follows:

    $$\begin{aligned} \mu \frac{d}{d\mu } \sigma (p,\mu ,g(\mu )) = 0. \nonumber \end{aligned}$$

    Here, p denotes the momentum in the external legs. Note that the quark mass dependence is suppressed. The renormalization equation provides the relation between the bare and renormalized quantities.

  7. 7.

    See also Fig. 6.6 and refer to the detailed directions on the cylindrical coordinate in Chap. 6.

  8. 8.

    In the pure YM gauge theory, there do not appear any hadrons and the only particle which can be excited in the confined phase is the massive glueball, whose mass is estimated to be \(\gtrsim 1\) GeV. Since the glueball is more difficult to be excited than light hadrons in QCD, the transition temperature in the pure gauge theory is higher than that in QCD. Note that, in this sense, the transition temperature depends on the quark masses.

  9. 9.

    Even at the high temperature slightly above the transition temperature, the perturbative calculation is not necessarily applicable. This is because the strong coupling is not small enough for the perturbation at \(T=\mathcal {O}(100)\) MeV. In addition, to make matters essentially difficult, the magnetic effect breaks down the perturbation at higher order due to the infrared divergence no matter how high the temperature is. In these senses, the non-perturbative formalism is crucial.

  10. 10.

    For the detail of the Polyakov loop, see Chap. 3.

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  1. Department of Physics, Graduate School of Science, Osaka University, Toyonaka, Osaka, Japan

    Ryosuke Yanagihara

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Yanagihara, R. (2021). Introduction. In: Distribution of Energy Momentum Tensor around Static Charges in Lattice Simulations and an Effective Model. Springer Theses. Springer, Singapore. https://doi.org/10.1007/978-981-16-6234-8_1

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