Mass Gap And Its Applications, The

Vakhtang Gogokhia, Gergely Gabor Barnaföldi

• Publisher: WSPC
• Year: 2013
• ISBN:
  9789814440721
• Topic: Physical Sciences
• Subtopic: Quantum Theory

Quantum Chromodynamics (QCD) is the most up-to-date theory of the strong interaction. Its predictions have been verified experimentally, and it is a cornerstone of the Standard Model of particle physics. However, standard perturbative procedures fail if applied to low-energy QCD. Even the discovery of the Higgs Boson will not solve the problem of masses originating from the non-perturbative behavior of QCD.

This book presents a new method, the introduction of the ‘mass gap’, first suggested by Arthur Jaffe and Edward Witten at the turn of the millennium. It attempts to show that, to explain the mass-spectrum of QCD, one needs the mass scale parameter (the mass gap) instead of other massive particles. The energy difference between the lowest order and the vacuum state in Yang–Mills quantum field theory, the mass gap is in principle responsible for the large-scale structure of the QCD ground state, and thus also for its non-perturbative phenomena at low energies. This book not only presents the mass gap, but also details the applications and outlook of the mass gap method. A detailed summary of references and problems are included as well.

This book is best for scientists and highly advanced students interested in non-perturbative effects and methods in QCD.

Contents:

• Theory of the Mass Gap:
  • Quantum Chromodynamics and the Mass Gap
  • Color Gauge Invariance and the Origin of the Mass Gap
  • Formal Exact Solutions for the Full Gluon Propagator at Non-zero Mass Gap
  • Renormalization of the Mass Gap
  • General Discussion
• Applications of the Mass Gap:
  • Vacuum Energy Density in the Quantum Yang–Mills Theory
  • The Non-perturbative Analytical Equation of State for the Gluon Matter I
  • The Non-perturbative Analytical Equation of State for the Gluon Matter II
  • The Non-perturbative Analytical Equation of State for SU (3) Gluon Plasma

Readership: Scientists and very advanced students of Quantum Chromodynamics (QCD).

Chapter 1

Quantum Chromodynamics and the Mass Gap

1.1 Quantum Chromodynamics

Quantum Chromodynamics (QCD) [1–6] is widely accepted as a realistic, dynamical quantum gauge theory of strong interactions not only at the fundamental (microscopic) quark–gluon level, but at the hadronic (macroscopic) level as well. This means that, in principle, it should describe the properties of the observed hadrons in terms of never experimentally seen quarks and gluons, i.e., to describe the hadronic world from first principles — the ultimate goal of any fundamental dynamical theory. However, this is a formidable task because of the color confinement phenomenon, whose dynamical mechanism is not understood yet, and therefore the confinement problem remains unsolved up to the present day. It prevents colored quarks and gluons from being experimentally detected as asymptotic states, which are colorless (color-singlet) by definition, i.e., color is permanently confined, being thus absolute [2]. At present, there are no doubts left that color confinement as well as other dynamical effects, such as spontaneous/ dynamical breakdown of chiral symmetry, bound-state problems, etc., are inaccessible to perturbative techniques, and thus they are very essential non-perturbative effects. In turn, this means that for their investigation, non-perturbative solutions, methods and approaches need to be found, developed and used. This is especially necessary taking into account that the above-mentioned non-perturbative effects are low-energy/momentum (large distances) phenomena and, as it is well known, the perturbative methods, in general, fail to investigate them.

The Lagrangian density, which describes the properties, symmetries and interactions between fundamental constituents (gluons and quarks) of QCD, can be given in the following simplified form based on,

where space-time indices are μ, ν = 0, 1, 2, 3, and the color indices run as follows: α, β = 1, 2, 3, while a = 1, 2…8. The number of different quarks, the so-called flavor number, is N_f and thus A = 1, 2…N_f . Let us also point out that a paper which provides an excellent guide with brief comments to the literature on QCD can be found in [7].

The gluon field strength tensor is given by

while the covariant derivative is defined as:

Note that both quantities depend on the same coupling constant g, i.e., the Lagrangian of QCD has a single universal dimensionless coupling constant for all types of the interactions between the fundamental constituents. It is worth reminding here that the QCD fine-structure coupling constant α_s = g²/4π (calculated at any scale) is much bigger numerically than its Quantum Electrodynamics (QED) counterpart. This restricts the application of the perturbation theory methods to QCD apart from in the limit of high energies due to asymptotic freedom phenomenon in this theory [8–10].

The λ^as generators are SU(3) matrices, which obey the commutator relation below,

with f^abc the structure constants of SU(3) color gauge group.

One can verify that the Lagrangian (1.1.1) is invariant under local gauge transformation of the form

Here the local SU(3) color gauge transformation

is a function of space-time dependent parameters Θ^a(x). Thus equation (1.1.1) is the minimal locally gauge invariant Lagrangian density implied by this SU(3) color symmetry.

The Lagrangian given by equation (1.1.1) is invariant under the larger chiral group

with

being the right and left hand-side components of the quark fields in the fundamental representation. U_B(1) describes the baryon number conservation and U_A(1) describes the axial-baryon number conservation, which is not wanted, since it is not observed. The chiral SU(N_f ) × SU(N_f ) flavor symmetry is broken in QCD by adding to equation (1.1.1) a quark mass term

where m₀ is the so-called current quark mass, depending on flavor A. Let us underline that only this massive term is compatible with the SU(3) color gauge symmetry of QCD. At the same time, the massive gluon term

A_μA_μ explicitly violates it. By adding this term to Eq. (1.1.1), it is not invariant under local gauge transformations given by equations (1.1.5). This causes one of the important challenges of QCD.

1.2 The Jaffe –Witten theorem on the Mass Gap

Let us now bring the reader’s attention to one of the important features of the Lagrangian of QCD briefly discussed above. It does not contain a mass scale parameter which could have a physical meaning. This is true even after the corresponding renormalization programme is performed. The current quarks are colored objects and that is why the hadron mass cannot directly depend on their masses: the color-singlet mass scale parameter is needed for this purpose. Precisely this important problem has been addressed at the beginning of this century by Arthur Jaffe and Edward Witten who have formulated one of the Millennium Prize Problems as follows [11]:

Yang – Mills existence and the Mass Gap: Prove that for any compact simple gauge group G, quantum Yang – Mills theory on

⁴ exists and has a mass gap ∆ > 0.

In the description of this theorem [11] they have explained why the mass gap is needed.

(i) It must have a ‘mass gap’. Every excitation of the vacuum has energy at least ∆ — to explain why the nuclear force is strong but short-range.

(ii) It must have ‘quark confinement’ — why the physical particles are SU(3)-invariant, i.e., color-independent.

(iii) It must have ‘chiral symmetry breaking’ — to account for the ‘current algebra’ theory of soft pions.

Summarizing, we need the mass gap which is responsible for the nonperturbative dynamics of QCD, and thus it determines the large-scale structure of the QCD ground state. Any hadron mass finally has to be expressed in terms of the renormalized mass gap itself, i.e., M_h = const_h × ∆, where h denotes any hadron, while const_h is the corresponding dimensionless constant. In other words, the hadron spectrum should depend on the mass gap. It is different from Λ_QCD, which is responsible for its non-trivial perturbative dynamics (scale violation, asymptotic freedom), and thus is due to the short-scale structure of QCD ground state. As we already know any mass term (for example, the gluon mass term), apart from the current quark masses, violates SU(3) color gauge invariance/symmetry of QCD. However, in the next chapter we will show that the common belief (which comes from the perturbation the...

Table of contents

• Cover
• Halftitle
• Title
• Copyright
• Dedication
• Preface
• Acknowledgment
• Contents
• Theory of the Mass Gap
• 1. Quantum Chromodynamics and the Mass Gap
• 2. Color Gauge Invariance and the Origin of the Mass Gap
• 2.A Appendix: Application for Abelian case
• 3. Formal Exact Solutions for the Full Gluon Propagator at Non-zero Mass Gap
• 3.A Appendix: The dimensional regularization method in the perturbation theory
• 3.B Appendix: The dimensional regularization method in the distribution theory
• 4. Renormalization of the Mass Gap
• 4.A Appendix: The Weierstrass – Sokhatsky – Casorati theorem
• 5. General Discussion
• Applications of the Mass Gap
• 6. Vacuum Energy Density in the Quantum Yang – Mills Theory
• 6.A Appendix: The general role of ghosts
• 7. The Non-perturbative Analytical Equation of State forthe Gluon Matter I
• 7.A Appendix: The summation of the thermal logarithms
• 8. The Non-perturbative Analytical Equation of State forthe Gluon Matter II
• 9. The Non-perturbative Analytical Equation of State for SU(3) Gluon Plasma
• 9.A Appendix: Analytical and numerical evaluation of the latent heat
• 9.B Appendix: The β-function for the confining effective charge at non-zero temperature
• 9.C Appendix: Least Mean Squares method and the definition of the average deviation
• 9.D Appendix: Restoration of the lattice pressure below 0.9Tc
• Bibliography
• Index