# Renormalization

# RENORMALIZATION

Renormalization is a technique required for extracting meaningful predictions from a quantum field theory. The description of particles and their interactions involves various physical parameters, such as the masses and charges of particles. These parameters are not known a priori and must be measured. Quantum effects can shift the values of these parameters and the measured physical value will include the quantum modifications. These notions form the basis of the renormalization program. When considered more carefully, these concepts can be used to show that the predictions of many theories are independent of any new physics that might appear at very high energies, such as the issue of whether or not the divergences found in perturbation theory exist. Renormalization theory also leads to the idea of a "running" charge, that is, one that depends on the energy scale at which it is measured.

## The Renormalization Program

Briefly stated, renormalization refers to the process by which the sum of all contributions to a physical parameter is adjusted so that the total is equal to the experimentally measured value of this parameter, and all predictions are expressed in terms of the experimental value.

First consider the concept of different contributions to a parameter. As examples, consider two specific parameters: the mass and the charge of an electron. Normally, one just thinks of these as two numbers. However, the values that one sees in nature reflect various physical effects. Since electric fields carry energy, some of the rest energy of the electron is carried by its electric field. If one were able to turn off the electromagnetic interaction, the mass of the electron would be different. One can distinguish the value of the electron mass with the electromagnetic interaction turned off (sometimes referred to as the bare mass) and the physical mass including electro-magnetic effects. While the bare mass is independent of the electric charge *e,* the energy in the electric field would be of order *e*^{2}. Similarly, the electron's charge can be modified by electromagnetic interactions. Quantum mechanics allows the temporary production of a virtual electron-positron pair in the electro-magnetic field of an electron. This virtual pair can partially screen the charge of the electron, an effect referred to as vacuum polarization. This is shown in the Feynman diagram of Figure 1. In this figure, a photon splits into an electron-positron pair in the neighborhood of a charged particle, and the external photon interacts with the virtual pair rather than the original charged particle. This modifies the interaction of the external photon with the charged particle. Again, one often refers to a bare charge, which can be denoted by *e*_{0}, plus quantum corrections. Although normally the electric field is linear in the electric charge *e*_{0}, the modification of the field due to vacuum polarization depends on this charge cubed. In perturbation theory, there would also be corrections at higher powers of the charge. The physical value is the sum of these various contributions.

Next consider the measurement process. To accomplish a measurement, the experimenter must set up some interaction with the particles. Typically, this involves scattering experiments at particular energies and angles. Since there are also quantum corrections to each process, one uses the theory to provide the relationship of the experimental measurements to the underlying parameters. Given a specific condition for the measurements, one measures the physical values for the parameters.

One would also need to describe other predictions of the theory in terms of the physically measured values of the parameters. This is the process of renormalization. The original parameters with

**FIGURE 1**

which one starts calculating would generally be analogous to the bare values. Using these original parameters, one calculates the quantum corrections to the parameters and also the quantum corrections to the other processes that one is predicting. One then expresses the new predictions in terms of the physical values of the parameters, referred to as the renormalized parameters. The bare parameters never appear in these results—the true predictions of the theory are expressed in terms of well-defined measured renormalized parameters.

## Importance of Renormalization

In the history of physics, the renormalization procedure was first important in the treatment of infinities that one finds in perturbation theory. For a class of theories that are called renormalizable, the predictions of the theory are all finite when expressed in terms of the measured values of a small number of parameters. This was a surprising and important result because at intermediate steps infinities appear in the calculation of the quantum corrections. By expressing the calculation in terms of the physically measured parameters, all the infinities disappear, and one is left with finite predictions. The theory unambiguously connects one physical process (the measurement of the parameters) to another (the process that is being predicted). However, note that the idea of renormalization is independent of the possible existence of infinities. If one had a fully finite theory, or a method of calculation that did not include infinities, one would still need to express the predictions in terms of measured parameters.

It is the success of renormalization that allows physicists to make quantum predictions in the face of incomplete knowledge. One is always in the situation where one does not know the ultimate theory at the highest energies. Physics is an experimental science, and only the nature of the fundamental theories up to a given energy has been explored. There could be new particles and new interactions that will only show up at higher energy, and this may be a problem for quantum predictions. In calculating quantum corrections, one is instructed to sum over all possible intermediate states and integrate over all energies. If the true physics at high energies are not yet known, how can such a calculation be conducted? The answer is definitely not that the effects of high energies are unimportant in general. In some of the intermediate steps there are contributions that are sensitive to the highest energies. However, it is a general result that the major effects of unknown new physics at very-high-energy scales can be absorbed (along with the infinities) into the values of the renormalized parameters. The relationships between physical processes are not influenced by the unknowns when expressed in terms of the measured parameters.

## Example

One of the most sensitively measured quantities in physics is the magnetic moment of the muon. This provides a fine example of the success of the renormalization program. The theoretical calculation has proceeded to many orders of perturbation theory. In each order there are divergences that arise in individual diagrams. However, when all diagrams are considered and the result expressed in the measured value of the fine structure constant, one obtains a finite result. It is traditional to express the result using the gyromagnetic ratio *gμ* that describes the fractional deviation of the result from the standard Dirac magnetic moment (which corresponds to *gμ* = 2). The theoretical result is where α is the fine-structure constant α =*e*^{2}/(4*πħc* ), and the uncertainties in the last digits are listed in parentheses. In a recent experiment at Brookhaven National Laboratory (BNL), the value of was measured. Although physicists are intrigued as to whether the modest 1.6 standard deviation difference (25 ± 16 × 10^{10}) might be a harbinger of some new physics, for the purposes of the discussion here, one may marvel that theory and experiment agree so well through many orders of perturbation theory.

## Running Charges

There are many different processes or experimental conditions that one could use for the measurement of a physical parameter, and each situation has its own set of quantum corrections. This leads to a potential ambiguity involving which corrections should be included in the definition of the renormalized parameter, and indeed different choices are allowed. The different choices are referred to as different renormalization schemes, and each scheme involves a precise definition of the basic renormalized parameter. It may also be observed that it is useful to change the definition of the basic coupling when working at different energies. For example, the quantum corrections that contribute to Coulomb scattering depend on the momentum transferred to the scat-tered electron. At large momentum transfer, the Fourier transform of the Coulomb potential becomes where *q = p - p* is the momentum transfer. This relation can be used to define the coupling at large values of momentum transfer to be This is referred to as the running coupling. Predictions can be expressed in terms of either the static coupling or the running coupling, and the experimenter could choose to measure either α or α(*q*^{2}) at a given value of *q*^{2}. However, the definition of the running coupling is found to be very useful. In the case where the logarithm factor is large, the use of the running coupling will absorb the large logarithms in *all* processes with an energy scale *E*^{2} ∼*q*^{2}. In a theory such as quantum electrodynamics (QED), one can express predictions in terms of the low-energy coupling α = 1/137. However, at higher energies one could also calculate using the running coupling in order to avoid large logarithms.

In quantum chromodynamics (QCD), there is a coupling constant that gives the strength of the interactions of quarks and qluons. When formulated as a running coupling constant, it is found to become weaker at high energies (a phenomenon referred to as asymptotic freedom) and grows strong at low energies. Therefore, one can treat the theory perturbatively at high energy where the coupling constant is weak enough. This has been verified experimentally. Since the predictions occur at energies that are large compared to the masses in the theory, the only relevant scale for the prediction is the energy at which the experiment is being conducted. Therefore, the predictions are expressed in terms of the running coupling at that energy scale. This notion can be made quite precise. The results of different experimental measurements at different energies are shown in Figure 2, where the phenomenon of the running coupling is clearly visible.

There are three separate charges in the Standard Model. Listed in order of decreasing size, one has the strong charge of QCD, then the charge associated with the weak gauge bosons, and finally the electric charge. When formulated as running charges,

**FIGURE 2**

the first two decrease at high energy, with the strong charge changing the most rapidly, while the electric charge grows. This raises the possibility that all three charges might meet at some very high energy. Associated with this possibility is the theory of forces referred to as the Grand Unified Theory (GUT). The idea is that at some very high energy there is only one interaction that encompasses the strong, weak, and electromagnetic forces, tied together in a simple gauge theory. At this scale, only a single coupling constant would exist. This unified theory could undergo spontaneous symmetry breaking, leaving the three separate interactions at lower energy. (This would be much like how the unified theory of the electromagnetic and weak interactions in the Standard Model breaks to leave only electromagnetism at low energies.) The three couplings would then evolve to their observed values at low energy. To determine if the couplings observed are compatible with this hypothesis, one would need to evaluate the running couplings at high energy and see if there is some energy at which they are all the same. The way that the couplings change with energy depends on the particles that are able to be excited at such an energy. If it is assumed that the only particles existing at any energy are those already discovered, then the running couplings come very close to each other, at an energy about thirteen orders of magnitude greater than that of present accelerators. Although it appears that the couplings do not exactly all coincide at a single point, it is possible that some new particles exist in those many orders of magnitude which slightly modify the running of the charges and which allow the couplings to be fully unified.

*See also:*Gauge Theory; Quantum Field Theory; Salam, Abdus; Standard Model

## Bibliography

Brown, L. M., ed. *Renormalization: From Lorentz to Landau (and beyond)* (Springer-Verlag, New York, c.1993).

Collins, J. C. *Renormalization: An Introduction to Renormalization, the Renormalization Group, and the Operator-product Expansion* (Cambridge University Press, Cambridge, UK, 1984).

Feynman, R. P. *QED: The Strange Theory of Light and Matter* (Princeton University Press, Princeton, NJ, 1985).

Kane, G. L. *Modern Elementary Particle Physics* (Addison Wesley, Redwood City, CA, 1987).

*John F. Donoghue*

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