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Quantum Chromodynamics PHY646 - Quantum Field Theory and the Standard Model Even Term 2020 Dr. Anosh Joseph, IISER Mohali LECTURE 42 Wednesday, April 1, 2020 (Note: This is an online lecture due to COVID-19 interruption.) Topic: Quantum Chromodynamics Quantum Chromodynamics The most natural candidate for a model of the strong interactions is the non-Abelian gauge theory with gauge group SU(3), coupled to fermions in the fundamental representation. This theory is known as Quantum Chromodynamics or QCD. Lagrangian of QCD The gauge invariant QCD Lagrangian is 1 L = − (F a )2 + (δ i@= + gA=aT a − mδ ) ; (1) QCD 4 µν i ij ij ij j where i, with the color index i = 1; 2; 3, is the quark field, in the fundamental representation of a a a abc b c a SU(3); Fµν = @µAν − @νAµ + gf AµAν is the gluon field strength tensor; Aµ with a = 1; 2; ··· ; 8 are the eight gluon fields; T a are the generators of SU(3); f abc are the structure constants of SU(3); p and the dimensionless quantity g = 4παs is the QCD coupling parameter. We have the gluon field 8 X λa A (x) = Aa (x)T a; with T a = : (2) µ µ 2 a=1 The matrices λa are called the Gell-Mann matrices. For six flavors of quarks, we can write the f quark field as a four-component Dirac spinor i , with the index f running over the quark flavors: a µa f = u; d; s; c; t; b. Gluons are massless: A term like mgAµA would break gauge invariance. PHY646 - Quantum Field Theory and the Standard Model Even Term 2020 Quarks Quarks are the fundamental objects taking part in the strong interactions. According to our current knowledge, like electrons, quarks are assumed to be simple structureless spin-1/2 particles. In a relativistic theory, similar to electrons, we can also describe the quark fields using Dirac spinors: α(x) with four components α = 1; 2; 3; 4. If quarks were completely decoupled from other particles or fields, they would obey the free Dirac equation (i@= − m) (x) = 0; (3) where m would be the mass of such a free quark. However, in the real world, free quarks have never been observed in laboratories. The reason is due to a property of strong interactions, associated with the low energy dynamics, called quark confinement. At present, we have discovered six flavors of quarks through high energy experiments: up (u), down (d), charm (c), strange (s), top (t), and bottom (b). These flavors organize themselves into three families or generations under the weak interactions. (Again, we do not know why quarks come in three generations. Different generations interact via the weak nuclear force. At least three generations are needed for the observed CP violation in nature and thus the abundance of matter over antimatter in our universe; the so-called matter-antimatter asymmetry.) Up and down quarks form the first family, charm and strange the second, and top and bottom the third. According to our current knowledge, these three families are basically repetitions of the same pattern (same quantum numbers). Quarks also come in fractional electric charges: the electric charges of the up, charm, and top quarks are all 2=3 of that of the proton, and the charges of the down, strange, and bottom are all −1=3. The quarks are distinguished by their masses and associated “flavor” quantum numbers. Quark Masses Since we are unable to observe free quarks, their masses cannot be measured directly, and so the very meaning of the mass of a quark requires some explanation. The mass of a quark is just a parameter in the Lagrangian of the theory, which describes the self interaction of the quark, and is not directly observable. Similar to the coupling parameter of a quantum field theory, the mass parameter is dependent on the momentum scale at which we probe the theory, and the renormalization scheme. A widely used convention is to express the quark masses in terms of the dimensional regularization and modified minimal subtraction MS scheme. According to the Standard Model, the quark masses are generated through a symmetry breaking phase transition of the electroweak interactions. (This transition is similar to that of a normal conductor to a superconductor in condensed matter physics, in which the effective mass of the photon is produced.) Our present knowledge of the quark masses is shown in Table. 1. 2 / 9 PHY646 - Quantum Field Theory and the Standard Model Even Term 2020 Quark flavor Mass Charge +0:49 u 2:16−0:26 MeV 2=3 +0:48 d 4:67−0:17 MeV −1=3 +0:02 c 1:27−0:02 GeV 2=3 +11 s 93−5 MeV −1=3 +0:4 t 174:9−0:4 GeV 2=3 +0:03 b 4:18−0:02 GeV −1=3 Table 1: Quark masses in the MS renormalization scheme at a scale of µ = 2 GeV. The data are taken from 2019 Review of Particle Physics: M. Tanabashi et al. (Particle Data Group), Phys. Rev. D 98, 030001 (2018). Color Charge In addition to electric charges, quarks also carry the charges of the strong interaction: the color charges. Though the color charges are the analogs of electric charge in QED, there are significant differences. In QED, we can consider the electric charge as a scalar quantity. That is, the total charge of an electric system is simply the algebraic sum of individual charges. The color charge is a quantum vector charge; a concept similar to the angular momentum in quantum mechanics. For a given system, the total color charge must be obtained by combining the individual charges of the constituents according to group theoretic rules, analogues to those of combining angular momenta in quantum mechanics. Quarks have three basic color-charge states, which can be labelled as i = 1; 2; 3. Mimicking the three fundamental colors we can also assign color charges as red, green, and blue. The three color states form a basis in a 3-dimensional complex vector space, and thus a general color state of a quark would be a vector in this space. We have 0 1 qr B C = @qgA : (4) qb The color symmetry is unbroken; for instance, a red up quark and a blue up quark are identical in every measurable way, and a universal substitution for one for the other would leave every measurable quantity unchanged. The color state can be rotated by SU(3) matrices. We can label the color charges by the spaces of the SU(3) representations. The three-dimensional color space 3 / 9 PHY646 - Quantum Field Theory and the Standard Model Even Term 2020 forms a fundamental representation of SU(3). For the case of the quark it is 3, representing the fundamental representation of SU(3). The rules for adding together color charges follow those of adding representation spaces of the SU(3) group; and they are nothing but straightforward extensions of adding up angular momenta in quantum mechanics. Quarks, like the electron, have antiparticles: They are called antiquarks, q. The antiquarks have the same spin and mass as that of the quarks but with opposite electric charges. For example, the anti-top quark has an electric charge −2=3 of the proton charge. The color charge of an anti-quark is denoted by 3, which is a representation space of SU(3) where the vectors are transformed according to the complex conjugate SU(3) matrix. The colors of anti-quarks are anti-red, anti-green, and anti-blue. More generally, we can view the quark confinement as color confinement: strong interactions do not allow states other than color singlet, or color-neutral to appear in nature. Lattice QCD simu- lations provide strong evidence for color confinement. However, there exists no rigorous analytical proof of color confinement. Gluons Two or more quarks interact with each other through the intermediate agents called gluons. Like photons, gluons are massless, spin-1 particles with two polarization states (left-handed and right- handed). Since they are also represented by a four-component (non-Abelian) vector potential Aµ(x) we must impose conditions on Aµ to select only the physical degrees of freedom. There are eight types of gluons mediating the strong interactions. Quarks can interact with gluons in a similar way electrons interacting with photons. A new feature here is that the quark can change its color from i a to j by emitting or absorbing a gluon of color a, through a vertex involving the SU(3) generator Tij. Gluons are physical degrees of freedom, and therefore must carry energy and momentum themselves. The interaction terms tell us that there are three- and four-gluon interactions. These interactions are responsible for many of the unique and salient features of QCD, such as asymptotic freedom, chiral symmetry breaking, and color confinement. According to SU(3), the 3 × 3 color combinations form a singlet and an octet. The octet states form a basis from which all other color states can be constructed. The way in which these eight states are constructed from colors and anti-colors is a matter of convention. A possible choice is r1 r1 jrgi; jrbi; jgbi; jgri; jbri; jbgi; (jrri; −|ggi); (jrri + jggi − 2jbbi); 2 6 for the color octet and r1 (jrri + jggi + jbbi); 3 for the color singlet. We can readily see that the color singlet is invariant under a redefinition of the color names (rotation in color space). Therefore, it has no effect in color space and cannot be exchanged between color charges.
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