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Home , Strong interaction

PHY401 - Nuclear and Particle

Monsoon Semester 2020 Dr. Anosh Joseph, IISER Mohali

LECTURE 30

Wednesday, November 4, 2020 (Note: This is an online lecture due to COVID-19 interruption.)

Contents

1 The Strong 1 1.1 Bilinear ...... 6 1.2 Quark- Plasma ...... 6 1.3 Lattice QCD ...... 7

1 The Strong Interaction

The strong is left-right symmetric, unbroken, and acts upon only one family of : the . In the last lecture we saw that the short-range nature of the can be argued to arise from spontaneous breaking of local weak-isospin . The short-range nature of the strong has a totally different origin. The dynamical theory of quarks and that describes color is known as Quan- tum Chromodynamics (QCD). It is a of the non-Abelian color symmetry group SU(3). This theory is very similar to (QED), which describes the electro- magnetic interactions of charges with . We saw that QED is a gauge theory of phase transformations corresponding to the commuting symmetry group U(1)Q. Being a gauge theory of color symmetry, QCD also contains massless gauge (gluons) that have properties similar to photons. There are essential differences between the two theories. They arise because of the different nature of the two symmetry groups. The , which is the carrier of the force between charged particles, is itself charge neutral. PHY401 - Nuclear and Monsoon Semester 2020

As a result, the photon does not interact with itself. The gluon, which is the mediator of color interactions, also carries . The color symmetry is unbroken. A red and a blue up quark are identical in every measurable way. A universal substitution of one for the other would leave every measurable quantity unchanged. The non-abelian nature of color symmetry to self-interactions of gluons. It also describes how color-neutral states can be formed. We can obtain a color-neutral system by combining a quark with an anti-color antiquark. This is very much like how electric charges add. We have red + red = color neutral. (1)

Three quarks with distinct colors can also yield a color neutral . Thus, there must be an alternative way of obtaining a color neutral combination from three colored quarks. This must be red + blue + green = color neutral, (2)

This is clearly different from the way electric charges add together. This difference between color charge and has important physical consequences. Consider a classical test particle carrying positive electric charge, polarizing a dielectric medium by creating pairs of oppositely charged particles (dipoles). Due to the nature of the Coulomb interaction, the negatively-charged parts of the dipoles are attracted towards the test particle, while the positively-charged parts are repelled. See Fig. 1.

+ + + efective

+ efective

e + e + +

Distance Momentum transfer

Figure 1: Due to the nature of the Coulomb interaction, the negatively-charged parts of the dipoles are attracted towards the test particle, while the positively-charged parts are repelled.

As a consequence, the charge of the test particle is shielded, and the effective charge seen at large distance is smaller than the true charge carried by the test particle. (Recall that the electric field in a dielectric medium is reduced relative to that in vacuum by the value of the dielectric constant of the medium.) In fact, the effective charge depends on the distance (or scale) at which we probe the test particle.

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The magnitude of the charge increases as we probe it at ever smaller distances, and only asymp- totically (at largest momentum transfers) do we obtain the true point charge of the test particle. Since the distance probed is inversely proportional to the momentum transfer, it is stated con- ventionally that the effective electric charge, or the strength of the electromagnetic interaction, increases with momentum transfer. As we have just argued, this is purely a consequence of the screening of electric charge in a dielectric medium. Because of the presence of quantum fluctuations, a similar effect arises for charged particles in vacuum. The impact of which is that the fine structure constant

e2 α = (3) ~c increases with momentum transfer. + − 2 This has been confirmed in high-energy e e scattering, where α(µ = mZ c ) is found to be 1/127.9 or 7% larger than at low energy. Here µ is an energy scale. Let us consider how a test particle carrying color charge polarizes the medium. It does that in two ways. First, just as in the case of QED, it can create pairs of particles with opposite color charge. But it can also create three particles of distinct color, while still maintaining overall color neutrality. Consequently, for the color force, the effect of color charge on a polarized medium is more complex. A detailed analysis of QCD reveals that the color charge of a test particle is, in fact, anti-screened. In other words, far away from the test particle, the magnitude of the effective color charge is larger than that carried by the test particle. As we probe deeper, the magnitude of this charge decreases. Thus, the qualitative dependence of the color charge on probing distance, or on the probing momentum transfer, is exactly opposite of that for electromagnetic interactions. See Fig. 2. This implies that the strength of the strong interactions decreases with increasing momentum transfer, and vanishes asymptotically. Conventionally, this is referred to as . It refers to the fact that, at infinite energies, quarks behave as essentially free particles, because the effective strength of the for interactions vanishes in this limit. (Asymptotic freedom of QCD was discovered independently by Politzer and by Gross and Wilczek in 1973. They were awarded the Nobel Prize in 2004.) This principle has the additional implication that, in very high energy collisions, consist

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Distance Momentum transfer

Figure 2: The qualitative dependence of the color charge on probing distance, or on the probing momentum transfer, is exactly opposite of that for electromagnetic interactions. of quarks that act as essentially free and independent particles. This limit of QCD for high-energy hadrons is known as the parton model, and this model agrees with many aspects of high-energy scattering. The very fact that the strength of coupling in QCD decreases at high energies is extremely important. It means that the effect of color interactions can be calculated perturbatively at small distances (or large momentum transfers). Consequently, the predictions of QCD are expected to be particularly accurate at large momen- tum scales, and can be checked in experiments at high energies. At present, all such predictions are in excellent agreement with data. At low energies, color interactions become stronger, thereby making perturbative calculations less reliable. But this property also points to the possibility that, as color couplings increase, quarks can form bound states (colorless hadrons). Quarks alone cannot account for the properties of hadrons. As inferred from high energy collisions, quarks carry only about one half of the momentum of the hadrons. The rest has to be attributed to the presence of other point-like constituents that appear to be electrically neutral, and have J = 1. There have been many attempts to understand the low-energy, nonperturbative, behavior of QCD. The present qualitative picture can be summarized best by a phenomenological linear potential between quarks and antiquarks of the form

V (r) ∝ kr. (4)

This kind of picture works particularly well for describing the interactions of the heavier quarks. Intuitively, we can think of the qq¯ system as being connected through a string. As a bound qq¯ pair is forced to separate, the potential between the two constituents increases. At some separation length, it becomes energetically more favorable for the qq¯ pair to split into

4 / 8 PHY401 - Nuclear and Particle Physics Monsoon Semester 2020 two qq¯ pairs. See Fig. 3.

q q¯ q q¯ q q¯ q q¯

Figure 3: Pair production of quarks.

In other words, the strong color attraction increases with separation distance between the quarks. Therefore it prevents the possibility of observing an isolated quark. This effect, known as confinement, is consistent with observation. That is, all observed particles appear to be color neutral. There has never been any evidence for the production of an isolated quark or gluon with color charge. When additional quarks are produced in high-energy collisions, they are always found in states whose total color adds up to zero (i.e., color neutral). As these quarks leave the region of their production, they dress themselves (become converted) into hadrons. Their presence can be inferred from a of particles that is formed from their initial energy. Similarly, gluons emitted in hadronic interactions also become dressed into hadrons and produce jets of particles as they leave the point of collision. While we presently believe in the confinement of quarks and gluons, a detailed proof of this requirement within the context of QCD is still lacking. In the context of the , the strong nuclear force between hadrons can be thought of as a residual-color Van der Waals force. This is analogous to the Van der Waals force that describes the residual electromagnetic inter- actions of charge-neutral molecules. Just as the Van der Waals force reflects the presence of charged atomic constituents that can interact through the Coulomb force, the strong nuclear force reflects the presence of far more strongly interacting color objects that are present within hadrons. The Van der Waals force falls off far more rapidly with distance than the Coulomb force. This suggests that a similar effect could be expected for the case of color, which would explain the origin of the short-range nature of the strong force between hadrons, both within as well as outside of nuclei. See Fig. 4.

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u d u d d u

d u

u d d u d u

Figure 4: Interaction between two .

1.1 Quark Bilinear

A quark state can be represented as   qr   Ψ = qg . (5) qb From this we can construct a quark bilinear of the form

ΨΨ¯ . (6)

Under SU(3) transformation

~ ~ Ψ → e−igS θ·λΨ, (7) ~ ~ Ψ¯ → Ψ¯ eigS θ·λ. (8)

The group SU(N) has N 2 − 1 orthogonal generators. The group SU(3) will have eight.

They are known as Gell-Mann matrices, λi, i = 1, 2, ··· , 8. They satisfy the algebra X [λi, λj] = 2ifijkλk. (9) k

Here fijk are called the structure constants.

1.2 Quark-Gluon Plasma

QCD tells us that quarks are confined within hadrons. However, increasing the temperature of the hadronic system, and thereby the random thermal motion of its constituents, could eventually to a complete disintegration of the into free quarks and gluons. This leads to a new kind of known as the quark-gluon plasma.

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This phase is quite similar to the plasma state of charged particles that exists inside the and the , where and from ionized hydrogen move about freely. The best theoretical evidence that a transition between the confined and the deconfined phase of quarks takes place as the temperature increases, comes from extensive computer simulations based on QCD (Lattice QCD). This kind of a quark-gluon plasma phase of matter was likely to have existed right after the . The phase is characterized by a large number of rapidly moving charged quarks that scatter and therefore radiate photons. The high temperature (or high energy), would lead to the production of low-mass flavors (up and down). As temperature is increased, more exotic flavors, such as strange and , could also be produced. Experimental verification of such signals in high-energy interactions is an interesting area of research. It is being pursued in heavy-ion collisions at the Relativistic Heavy Ion Collider (RHIC) at the Brookhaven National Lab, and ALICE at the LHC. These collider experiments study interactions of large-A nuclei (lead, gold etc), each with energies of several hundred GeV per . The energy and matter densities in these experiments are expected to be large enough to observe the transformation of normal nuclei into free quark-gluon systems.

1.3 Lattice QCD

As the distance between the quarks increases, the interaction gets stronger, and many higher-order Feynman diagrams become important. In this strong interaction regime perturbation theory is no longer applicable and it has not yet been possible to evaluate the theory precisely. We therefore have to rely on approximate results obtained by numerical simulations of the theory on very large computers. The demonstration of confinement in QCD rests largely on such simulations. They are done using an approach called in which space (and sometimes time) is approximated by a finite lattice of discrete points. The exact theory can then in principle be recovered by letting the lattice spacing go to zero, and the number of lattice points become infinite. In practice, the number of lattice points that can be handled is limited by the computing power available, but nonetheless good results have been obtained for several static properties, for example the masses and decay constants of the lower lying hadrons.

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References

[1] Dave Goldberg, The Standard Model in a Nutshell, Princeton University Press (2017).

[2] A. Das and T. Ferbel, Introduction To Nuclear And Particle Physics, World Scientific (2003).

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