Why can’t we just build an accelerator to test the predictions of string theory? Because the obstacles are much bigger than money or social commitment.

This section uses units where (Planck’s constant)/2p and the speed of light = 1. This choice of units is called natural units. With this choice, mass has units of inverse length, and vice versa. The conversion factor is 2×10^{-7} eV = 1/meter.

Contents

**Spontaneous symmetry breaking**

On the previous page we mentioned that spontaneous symmetry breaking was the phenomenon that allowed gauge bosons to acquire mass without spoiling the gauge invariance that protects quantum consistency of the theory. But this trick is not special to electroweak theory; spontaneous symmetry breaking is a powerful phenomenon that is tremendously important in understanding unified particle theories in general. So we will explain this phenomenon in more detail here.

The simplest example begins with a complex scalar field f(x) with the Lagrangian

The potential V(f) has a strange looking shape: the minimum is not at the center, but in a circle around the center, as shown below.

The scalar field f(x) can be written in terms of real and imaginary components, as below top, or expressed in terms of radial and angular degrees of freedom, shown on the bottom.

The minimum values of the potential lie along the circle where

The problem with describing f(x) in terms of f1 and f2 is that f1 and f2 don’t describe the normal modes of oscillation around the minimum of the potential. The normal modes for this potential are illustrated in the animation above by the two distinct motions of the yellow ball. One normal mode goes around and around the circle at the bottom of the potential. The other normal mode bobs up and down in the radial direction at a fixed value of angle, oscillating about the minimal value. Written in terms of the normal modes, the field becomes

The physical states in the theory are the massive field r(x) with mass r_{0}, and the massless field b(x). The radial oscillations are resisted by the curved sides of the scalar potential in the radial direction. That’s why the radial field is massive. But the minimum of the potential is flat in the angular direction. That’s why the angular mode is massless. This is called a **flat direction**. Flat directions in the surface that form the minimum of the scalar potential lead to massless scalars. This issue comes up again in string theory in not a good way.

The most crucial chapter in this story is what happens when this scalar field is coupled to a massless gauge boson A with a local **U(1)** gauge invariance. The Lagrangian is

The story for the scalar field is as before. The physical scalar fields that oscillate as normal modes about the potential minimum are the massless angular mode and the massive radial mode. But the plot thickens with the addition of the massless gauge boson. At the minimum of the scalar potential, the Lagrangian above remains invariant under the transformation

This transformation relates the normal modes of both the scalar and vector fields so that they can be written as

The most important thing to notice about the redefined fields above is that the angular oscillations b(x) of the scalar field end up as part of the the physical gauge boson Ã(x). This is the secret behind the power of spontaneous symmetry breaking. The massless normal mode of the scalar field winds up mixed into the definition of the physical gauge boson, because of gauge symmetry.

The oscillations of the scalar field around the flat angular direction of the scalar potential turn into longitudinal oscillations of the physical gauge field. A massless particle travels at the speed of light and cannot oscillate in the direction of motion. Therefore, the addition of a **longitudinal mode of oscillation** means the gauge field has become **massive**.

The gauge field has a mass, but the gauge invariance has not been spoiled in the process. The value of the scalar field at the potential minimum determines the mass of the gauge boson, and hence the range of the force carried by the gauge boson.

This whole coupled system is called the **Higgs mechanism**, and the massive scalar field that remains in the end is called a **Higgs field**.

This section uses units where (Planck’s constant)/2p and the speed of light = 1. This choice of units is called natural units. With this choice, mass has units of inverse length, and vice versa. The conversion factor is 2×10^{-7} eV = 1/meter.

**Electroweak unification**

The Higgs mechanism forms the basis of the experimentally well-tested theory of the weak and electromagnetic interactions that is referred to as **electroweak theory**. The initial gauge invariance in the theory is **SU(2)xU(1)**, with three massless gauge bosons from **SU(2)** and one from **U(1)**. In the end there has to be only one massless gauge boson — the photon that carries the electromagnetic force — and three massive gauge bosons mediating the short range weak nuclear force.

Therefore, three massless scalar normal modes (also known as **Goldstone bosons**) are needed to serve as longitudinal modes to turn the **four massless gauge bosons** into **one massless** gauge boson and **three massive** gauge bosons.

Remember that for a single complex scalar field, the massless mode, or **Goldstone boson**, comes from the **angular normal mode** that oscillates around the flat circle at the potential minimum.

A circle is just a one-dimensional sphere, or a “one sphere”. In general, an **N-dimensional sphere** has **N angular directions**, and for oscillations about the sphere, there is **one radial direction**. We need a set of scalar fields that transform under the group **SU(2)** with a potential whose minimum has the geometry of a three sphere. This can be accomplished by using two complex scalar fields, transforming as a two-component object under transformations by the group **SU(2)**, so that f(x) is given by

The potential minimum is at

which is the equation of a three sphere in f-space.

The normal modes for this potential will consist of one radial mode and three angular modes, just enough to create one massive Higgs boson, and give mass to the three of the four massless gauge bosons in the **SU(2)xU(1)** theory. This leaves leaving one massless gauge boson for the remaining unbroken **U(1)** gauge invariance.

A complicating factor in electroweak theory is the presence of electroweak mixing. The four massless gauge bosons in the unbroken **SU(2)xU(1)** theory are the three **SU(2)** bosons, let’s called them **W**** ^{+}**,

**W**

**and**

^{–}**W**

**, and the massless**

^{0}**U(1)**gauge boson, let’s call it

**B**. The spontaneous symmetry breaking winds up mixing the

**W**

**and the**

^{0}**B**, into two different gauge bosons — the massless

**photon**that carries the electromagnetic force, and the massive

**Z**

**boson that carries the weak nuclear force. The mixing is described by the weak mixing angle q**

^{0}_{w}as shown below

The final physical states of this theory are the massless photon, and the massive neutral weak boson, the Z^{0}.

The distance scale of the electroweak mixing is roughly 100 GeV, or about 10^{-17} m. At scales smaller than that distance scale, or equivalently, at energy scales much above 100 GeV, the weak gauge bosons look massless and the full **SU(2)xU(1)** symmetry is restored. But at larger distance scales, or lower energy, only the **U(1)** symmetry of electromagnetism is apparent in the conservation laws and amplitudes.

The mathematical beauty and experimental success of this idea have led physicists to extend it to higher energies and possible higher symmetries, as will be described below.

**Running coupling constants**

In quantum field theory, when computing a particle scattering amplitude, one has to sum over all possible intermediate interactions, including those that happen at zero distance, or, expressed in terms of momentum space according to the de Broglie rule, at infinite momentum. These calculations lead to integrals of the form

which diverge at infinite momentum for n=0,1,2. The limit has to be approached through the use of a momentum cutoff of some kind. But the physical quantities must be independent of the cutoff, so that they remain finite as the cutoff is removed.

This procedure is called **renormalization**, and it cannot be done for any quantum field theory, just those theories whose divergences obey certain patterns that allow them to be added consistently to the definition of a finite number of physical quantities, namely the masses and coupling constants, or charges, in the theory.

The end result is that the masses and charges of elementary particles are dependent on the momentum scale at which they are measured. For example, the coupling strength of a renormalizable gauge theory has the mass dependence

where M and m are two mass scales at which the coupling strength is being measured and compared. The function f(n) depends on the number of degrees of freedom in the theory. For electromagnetism, f(n) = 1, but for QCD with six flavors of quarks, f(n) =-5.25.

Notice that this means electromagnetism gets stronger at higher energies, while the strong nuclear force gets weaker as the energy of the particle scattering increases. This is very important for understanding what physics might look like at higher energies than we can currently measure, see below.

Quantum field theories whose divergences can be hidden in a finite number of physical quantities are called **renormalizable quantum field theories**. Quantum field theories that are not renormalizable are regarded as being physically realizable theories. Note that **the list of unrenormalizable quantum field theories** includes **Einstein’s theory of gravity**, which is one reason why string theory became popular.

**Unification and group theory**

The success of spontaneous symmetry breaking in explaining electroweak physics led physicis to wonder whether the three particle theories of the **SU(3)xSU(2)xU(1)** model could be the spontaneously broken version of a higher unified theory at some higher energy scale, a single theory with only one gauge group and one coupling constant. This type of theory is called a **Grand Unified Theory**, or **GUT** for short.

The quantum behavior of the known particle coupling constants supports the idea of Grand Unification. Because of renormalization, the electromagnetic coupling constant grows **larger** at high energies, whereas the coupling constants for the weak and strong nuclear interactions grow **smaller** at higher energies. At the mass scale

the three coupling constants become equal. Therefore, this ought to be the mass scale where the single gauge symmetry of a Grand Unified Theory would become spontaneously broken into the three distinct gauge symmetries of the **SU(3)xSU(2)xU(1)** model.

The single gauge group of a GUT has to be mathematically capable of containing the group product **SU(3)xSU(2)xU(1)** of the three gauge groups relevant to low energy particle physcs. The best candidate for such a theory is unitary group **SU(5)**, which would give 24 gauge bosons mediating the single unified force, but there are also other GUT models based on other groups, such as the orthogonal group **SO(10)**, which would give 45 gauge bosons and contain the SU(5) theory as a subgroup.

The problem with Grand Unification is that the unified gauge bosons allow quarks to couple to leptons in such a way that two quarks can be converted into an antiquark and an antilepton. For example, two up quarks would be allowed to turn into a positron and and a down antiquark.

A proton consists of two up quarks and and down quark. A neutral pion consists of a down quark and a down antiquark. Therefore the unified gauge boson in a GUT could mediate proton decay by the interaction

and other related decays.

The proton lifetime predicted in a GUT is about

whereas the current best measurement of the proton lifetime is

It’s important to note here that proton decay can happen through radiative corrections even in the Standard Model, so we don’t expect the proton lifetime to be infinite.

However, it seems that the proton doesn’t decay as quickly as predicted by a GUT. This situtation is improved when supersymmetry is added to the GUT, and this will be explained in next section.

**What about gravity?**

Einstein’s elegant and experimentally tested theory of **gravity** called **General Relativity** is not a normal gauge theory like electromagnetism. The symmetry is not a unitary group symmetry like **U(1)** or **SU(3)**, but instead a symmetry under **general coordinate transformations** in four spacetime dimensions. This does not lead to a renormalizable quantum field theory, and so gravity cannot be unified with the other three known physical forces in the context of a Grand(er) Unified Theory.

But string theory claims to be a unified theory encompassing all known forces including gravity. How can that be? The main symmetry apparent in string theory is conformal invariance, or superconformal invariance, on the string world sheet. This symmetry dicates the spectrum of allowed mass and spin states in the theory. The spin two graviton and the spin one gauge bosons exist within this framework naturally as part of the tensor structure of the quantized string spectrum.

This is another reason why physicists have become so impressed by **string theory**. There exists a completely novel way of putting gravity and the other known forces together in the context of a single symmetry, that is much more powerful than the ordinary quantum gauge theory of particles. But the question is — is this really the way that nature does it? The answer to that may take a long time to sort out.

**Symmetry breaking in string theory**

The two string theories that have shown the most promise for yielding a pattern of symmetry breaking that is like Grand Unification plus gravity are the heterotic superstring theories based on the groups SO(32) and E_{8}xE_{8}. However, these are supersymmetric theories in ten spacetime dimensions, so the symmetry breaking scheme also has to be involved with breaking the supersymmetry (because fermions and bosons don’t come in pairs in the real world) and dealing with the extra six space dimensions in some manner. So the possibilities, and the possible complications, are much wider in string theory than in ordinary quantum gauge field theories.

Forgetting these complications for a moment, focus on the group theory of the **E**_{8}**xE**** _{8}** model. The group

**E**

**is an**

_{8}**exceptional group**with interesting properties too complex to explain here. The common suppostion is that one of the

**E**

**groups remains unbroken, and decouples from physical observation as a kind of shadow matter. The other**

_{8}**E**

**has the right mathematical structure to break down to an**

_{8}**SU(5)**GUT via

**E**

_{8}**-> E**

_{6}**-> SO(10) -> SU(5)**.

The symmetry breaking scale would presumably start somewhere near the Planck scale

and end up at the GUT scale of about 10^{14} GeV. The spontaneous symmetry breaking mechanism would presumably be scalar field potentials of the form shown above, where a subset of the scalar fields with normal modes like the radial mode become massive, and the remaining massless scalar fields become longitudinal modes of massive gauge bosons to break the gauge symmetry down to the next level.

But — in string theory, at the level of perturbation theory where the physics is most understood — the scalar potentials seem to be **flat in all directions** and hence the scalar fields all remain massless. The solution to symmetry breaking in string theory has to be nonperturbative and is still regarded as an unsolved problem.