ASYMPTOTIC FREEDOM

Asymptotic freedom is a characteristic property of quantum chromodynamics (QCD), the component of the Standard Model that describes the strong interactions. Asymptotic freedom ensures that when QCD is probed over short enough distances and times, it is well described by weakly interacting quarks and gluons. The idea that the strong interactions should somehow involve weakly interacting quarks may seem a bit paradoxical, but it is one of the triumphs of the Standard Model.

The strength of the color force is quantified by the QCD coupling s , the analog of the fine-structure constant of quantum electrodynamics (QED). There are a number of ways to conceptualize α_s, but perhaps the most intuitive is to define it in terms of the potential energy U_QCD between a quark and an anti-quark held apart by a distance r through the relation:

Here c is the speed of light, and ħ = h /2π, with h being Planck's constant. In Equation (1), α_s is itself a function of the distance. This dependence is a quantum mechanical effect called the running of the coupling. Asymptotic freedom is the property that α_s decreases as r decreases. It states that the color charge on a quark grows weaker when measured at shorter distances and stronger at longer distances.

In quantum field theory, the origin of Equation (1) is gluon exchange, shown in Figure 1, for a quark and antiquark separated by distance r . The quark spontaneously emits a gluon, which travels to the anti-quark (or vice versa). This violates energy conservation by the energy of the gluon Egluon for the period of time that it takes the gluon, which travels at the same speed as light, to arrive at the antiquark:

In quantum mechanics, such a process is allowed, so long as it satisfies the time-energy uncertainty relation

where Equation (2) has been used in the second step. Given this inequality, the potential energy U_QCD(r ) in Equation (1) can be thought of as the maximum energy of a gluon that the quark and antiquark can exchange at distance r times the probability that, if one were to look, such a gluon would be present. This probability is identified with α_s(r ). Such an interpretation is only approximate, however, because the direct exchange of a single gluon is only the first of an endless set of possibilities that generalize Figure 1.

The next most important scenarios are shown in Figure 2. In one, the gluon is emitted as above, but before it arrives at the antiquark, it splits for a while into a quark-antiquark pair, then reforms, and next is absorbed. Similarly, the gluon sometimes splits into two gluons for part of its journey.

The running of the coupling is due to the mixture of the processes in Figures 1 and 2. Suppose that the potential energy is measured at some fixed distance r₀, and α_s(r₀) is defined by Equation (1), α_s(r₀) = r₀U_QCD(r₀)/(ħc ). Then one changes the distance between the quark and antiquark to r₀ - Δr , with Δr being much less than r , and measures again. The change in αs is entirely due to Figure 2 and is proportional

FIGURE 1

to the fractional change in distance, Δr /r₀. The extra splitting in Figure 2 can be thought of as an extra emission and absorption so that the change in probability is also proportional to ∝_s²(_r0). In summary, the change in the coupling is of the form

Determining the constant b₀ requires careful calculation, but the answer is quite simple: b =(1/2π) (11 - 2n_f/3), where n_f is the number of quarks whose rest energies mc² are less than the energy of the gluon. The number "11" is the contribution of the graph with only gluons in Figure 2, and 2n_f/3 is the contribution of the graph with a quark pair. In the Standard Model, n_f is always less than 6, so that b₀ is always positive. This means that the coupling decreases as r decreases. This is asymptotic freedom.

How did this come about? When the gluon in Figure 2 spends time as a pair of quarks, the charges of the quark pair tend to screen the charge of the original quark. The larger r is, the longer the screening goes on, which tends to decrease the charge as r increases, just the opposite of asymptotic freedom. On the other hand, when the gluon spends part of its time as two gluons, the charges of the produced gluons tend to enhance, or antiscreen, the original quark charge. There is a close analogy between the screening of color and diamagnetism, in which a material establishes an internal

FIGURE 2

magnetic field that opposes an applied field. Correspondingly, the antiscreening of color may be compared to paramagnetism, in which a medium tends to enhance the applied field. In the case of QCD, the external field is the color field of a quark or gluon, and the medium is the QCD vacuum itself, capable of excitations to virtual states. The running of the coupling in QCD results from the competition of these two effects, and antiscreening wins out.

As long as α_s(r ) is much less than 1, the following explicit formula is equivalent to Equation (4):

where b₀ is the same constant as above, and Λ is an energy that must be determined from experiment. In practice, Λ turns out to be of order 2 × 10⁸ electron volts (200 MeV). One can easily verify that Equation (5) is an approximate solution to Equation (4). To do so, substitute r₀ - Δr for r in (5). Then denote x = r₀Δ/hc and x = δr₀Δ/hc , and use the relation

which is accurate when δx is much smaller than x .

αs is measured experimentally by accelerating quarks. This can be done by colliding them with leptons, or with other quarks. For example, in high-energy jet production, a quark whose energy is in the range of 10¹¹ electron volts radiates a gluon of comparable energy (E_gluon) about one-tenth of the time. This can be thought of as the likelihood that the quark was struck just at the time it had emitted such a virtual gluon, as in Figure 1. However, this probability is α_s(r ), r = ħc /E_gluon, which implies that α_s ≈ 0.1 for r = ħc /(10¹¹ eV) ≈ 2 × 10^-16cm. Given one such result, Equation (5) enables one to determine

FIGURE 3

Λ and then to extrapolate to any value of r, all the way up to r = (hc )/Λ ≈ 1 fm = 10^-13cm. At this distance scale, α_s in Equation (5) becomes undefined because the logarithm of 1 is zero. This means that the probability of finding a gluon at such an r is too large for this method of calculating α_s(r ) to be valid. Such distances are in the realm where the strong interactions become truly strong. It leaves, however, a wide range in which asymptotic freedom applies.

Experimental results on α_s are shown in Figure 3, compiled from a variety of sources, plotted against a scale of energy μ, and related to distances by μ = hc /r . The running of the coupling and its asymptotic freedom are more than evident; the change in α_s is greater than a factor of 2. Only because of asymptotic freedom is it possible to understand jet cross sections and the scaling phenomenon in electron-proton scattering through which quarks were first observed directly. The discovery of asymptotic freedom lifted the veil from the strong interactions making possible their consistent description in the Standard Model.