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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 Nf and thus A = 1, 2…Nf . 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 αs = 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 λas generators are SU(3) matrices, which obey the commutator relation below,
with fabc 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.
UB(1) describes the baryon number conservation and
UA(1) describes the axial-baryon number conservation, which is not wanted, since it is not observed. The chiral
SU(
Nf ) ×
SU(
Nf ) 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
AμAμ 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., Mh = consth × ∆, where h denotes any hadron, while consth is the corresponding dimensionless constant. In other words, the hadron spectrum should depend on the mass gap. It is different from ΛQCD, 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...
