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PHY646 - Theory and the

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

The most natural candidate for a model of the strong is the non-Abelian with gauge SU(3), coupled to 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 + ψ (δ i∂/ + gA/aT a − mδ )ψ , (1) QCD 4 µν i ij ij ij j where ψi, with the color index i = 1, 2, 3, is the field, in the fundamental representation of a a a abc b c a SU(3); Fµν = ∂µAν − ∂νAµ + gf AµAν is the 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 of SU(3); √ and the dimensionless quantity g = 4παs is the QCD parameter. We have the gluon field

8 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 , we can write the f quark field as a four-component Dirac ψi , with the index f running over the quark flavors: a µa f = u, d, s, c, t, b. are massless: A term like mgAµA would break gauge invariance. PHY646 - 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 , quarks are assumed to be simple structureless -1/2 particles. In a relativistic theory, similar to electrons, we can also describe the quark fields using Dirac :

ψα(x) with four components α = 1, 2, 3, 4. If quarks were completely decoupled from other particles or fields, they would obey the free

(i∂/ − m)ψ(x) = 0, (3) where m would be the 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), (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 . At least three generations are needed for the observed CP violation in nature and thus the abundance of over in our ; the so-called matter-antimatter .) 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 , and the charges of the down, strange, and bottom are all −1/3. The quarks are distinguished by their 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 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 scheme. A widely used convention is to express the quark masses in terms of the dimensional and modified minimal subtraction MS scheme. According to the Standard Model, the quark masses are generated through a breaking phase transition of the electroweak interactions. (This transition is similar to that of a normal conductor to a superconductor in , in which the effective mass of the is produced.) Our present knowledge of the quark masses is shown in Table. 1.

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Quark flavor Mass

+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 : 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 : the color charges. Though the color charges are the analogs of in QED, there are significant differences. In QED, we can consider the electric charge as a quantity. That is, the total charge of an electric system is simply the algebraic sum of individual charges. The is a quantum vector charge; a concept similar to the angular momentum in . 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 in a 3-dimensional complex , and thus a general color state of a quark would be a vector in this space. We have

  qr   ψ = qg . (4) qb

The color symmetry is unbroken; for instance, a red 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 , have : 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- 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) . 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 , 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) 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 , , 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 |rgi, |rbi, |gbi, |gri, |bri, |bgi, (|rri, −|ggi), (|rri + |ggi − 2|bbi), 2 6 for the color octet and r1 (|rri + |ggi + |bbi), 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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Since gluons interact with themselves it is possible to postulate the existence of composite particles made out of gluons. They are called . They are extremely difficult to identify in particle accelerators, since they mix with ordinary states. Prediction on the existence of glueballs is one of the most important predictions of the Standard Model that has not yet been confirmed experimentally. Strong interactions conserve the total number of each type of quarks. However, quarks can be transformed from one flavor to another through weak interactions. (This happens through the CKM matrix, which will see in a later lecture.)

Asymptotic Freedom

A remarkable property of QCD is asymptotic freedom: It states that the interaction strength αs = g2/4π between quarks becomes smaller as the distance between them gets shorter. In 2004, Gross, Politzer and Wilczek were awarded the for this observation. We can write the one-loop for QCD as

1 β(α) = − β α2 + ··· , (5) 2π 0 where 2 β = 11 − n . (6) 0 3 f 2 The second term in β0, − 3 nf is due to the quark-antiquark pair effect. It scales like the number of quark flavors nf , and is negative (as it would be in QED). However, the first term, 11, has the opposite sign, and comes from the non-linear gluon contribution (these are absent in QED). Thus, gluon self-coupling has an anti-screening effect. In QCD with 6 flavors of quarks, the anti-screening effect dominates the screening effect, making QCD asymptotically free. From the equation, we can show the following scale dependence for the QCD 2π αs(µ) = . (7) β0 ln(µ/ΛQCD) It goes to zero as the momentum scale µ goes to infinity. This strange behavior of the strong coupling has been verified in high-energy experiments to very high precision. The running coupling introduces a dimensional parameter, the integration constant ΛQCD in the above equation. It is an intrinsic scale of QCD; it sets the scale at which the coupling constant becomes large and the physics becomes nonperturbative. In the MS-scheme with 3 quark flavors we have

ΛQCD ∼ 250 MeV. (8)

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Figure 1: Summary of measurements of αs as a function of the energy scale Q. Figure taken from 2019 Review of Particle Physics: M. Tanabashi et al. (Particle Data Group), Phys. Rev. D 98, 030001 (2018).

Chiral Symmetry

For a moment let us assume that the quarks are massless. Then, the spin of the quark can either be in the direction of motion, in which case, we call it a right-handed quark, or in the opposite direction, in which case, we call it a left-handed quark. The handedness (or ) of the quark is independent of any Lorentz frame from which the observation is made since a massless quark travels at the speed of .

Upon using the projection , P± = (1 ± γ5)/2, we can project out the quark field,

ψ = ψL + ψR, into left and right-handed (chirality) quarks,

1 ψf = (1 − γ )ψf , (9) L 2 5 1 ψf = (1 + γ )ψf , (10) 2 5 where f labels different flavors. Plugging in the above decomposition into QCD Lagrangian without the mass term, the quark part splits into two terms

Lq = Lq(ψL) + Lq(ψR), (11) with each depending on either the left-handed, or the right-handed field, but not both. Since the up and down quarks are both massless, we can regard them as two independent states of the same object forming a two component spinor in space, in the same way as the 1 ± 2 projection states of the spin-1/2 object. The Lagrangian is symmetric under the independent

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rotations of ψL and ψR in the left and right isospaces. That is, under ! ! u0 u L,R = U L,R , (12) 0 L,R dL,R dL,R with UL,R denoting 2 × 2 unitary matrices, the quark part of the Lagrangian is invariant. Thus we can say that the QCD Lagrangian has a chiral symmetry U(2)L × U(2)R. We can decompose this into

U(2)L × U(2)R = SU(2)L × SU(2)R × U(1)L × U(1)R. (13)

The symmetry SU(2)L × SU(2)R has not been seen explicitly in particle spectrum or scattering amplitudes. It is a hidden symmetry, in the sense that it breaks spontaneously into the so-called isospin subgroup SU(2)L+R, which corresponds to rotations UL = UR. Adding a mass term explicitly breaks the chiral symmetry as

f f f f f f mf ψ ψ = mf ψLψR + mf ψRψL. (14)

However, the main origin of the chiral symmetry breaking may be described in terms of the f f condensate hψLψRi ( condensate of bilinear expressions involving the quarks in QCD vacuum) formed through nonperturbative action of QCD gluons. The spontaneous symmetry breaking due to the strong low-energy QCD dynamics rearranges the QCD vacuum such that the fermion condensate acquires a non-zero mass:

f f 3 hψLψRi ∝ ΛQCD 6= 0. (15)

Color Confinement

QCD undergoes color confinement in low energies: Any strongly interacting system at zero temper- ature and density must be a color singlet at distance scales larger than 1/ΛQCD. As a consequence, isolated free quarks cannot exist in nature (quark confinement). The color confinement of QCD is a theoretical conjecture consistent with experimental facts. We can study QCD in the regime of strong coupling, using an approximation scheme introduced by Wilson in which the continuum gauge theory is replaced by a discrete statistical mechanical system on a four-dimensional Euclidean lattice, the so-called Lattice QCD. Using this approxima- tion, Wilson showed that for sufficiently strong coupling, QCD exhibits confinement of color: The only finite-energy asymptotic states of the theory are those that are singlets of color SU(3). If one attempts to separate a color- into colored components - for example, to dissociate a meson into a quark and an antiquark - a tube of gauge field forms between the two sources, as shown in Fig. 2. In non-Abelian gauge theory with sufficiently strong coupling, this tube has fixed radius and energy density, so the energy cost of separating color sources grows proportionally to the separation. A force law of this type can consistently be weak at short distances and strong at long distances, accounting for the fact that isolated quarks are not observed.

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Figure 2: of gauge fields associated with the separation of color sources in a strongly- coupled gauge theory.

The short distance limit of QCD can be readily studied using technology. Here the asymptotic freedom makes the coupling weak, and there is a sensible diagrammatic perturbation theory that begins from the model of free quarks and gluons. QCD simulations on the lattice shows that the quark potential does increase linearly beyond the distance of a few fermi. In QED, the electric lines between positive and negative charges spread all over the space, generating a 1/r potential. In QCD, however, the vacuum acts like a dual superconductor which squeezes the color electric field to a minimal geometric configuration - a narrow tube. (In a normal superconductor, the magnetic flux is expelled from the interior of the metal, as shown in the Meissner effect.) It costs energy for the flux to spread out in space. The tube roughly has a constant cross section and with constant energy density. Because of this, the energy stored in the flux increases linearly with the length of the flux. The flux tube has been seen in numerical simulations. If one calculates the gluon energy in Lattice QCD, one can see the flux tube from the plot of energy density in space. See Fig. 3.

Figure 3: The meson and flux tubes from Lattice QCD simulation data. More details are provided in F. Bissey, F. G. Cao, A. R. Kitson, A. I. Signal, D. B. Leinweber, B. G. Lasscock and A. G. Williams, Gluon flux-tube distribution and linear confinement in , Phys. Rev. D 76, 114512 (2007).

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Hadrons and Nuclei

What we usually observe in experimental apparatus are and nuclei which are bound states of quarks and gluons. Under the postulate that all wavefunctions must be invariant under SU(3) symmetry transformations, the only allowed combinations are

i ijk i j k q¯ qi,  qiqjqk, and, ijkq¯ q¯ q¯ . (16)

That is, the assumption that physical hadrons are singlets under color implies that the only possible light hadrons are the , baryons, and antibaryons. The lowest mass baryons are (bound states of udd) and (bound states of uud), which are together called . Protons are stable whereas neutrons have a life time of about 10 minutes. There are attractive interactions between protons and neutrons, which are like the residual color interactions, just like the Van der Waals between neutral and molecules. The nuclear forces are responsible for binding the nucleons together to form atomic nuclei and the origin or atomic energy.

References

[1] M. E. Peskin and D. Schroeder, Introduction to Quantum Field Theory, Westview Press (1995).

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