Grand Unification

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According to grand unification, all nongravitational forces are manifestations of one single fundamental force. In everyday life the influence of two forces is readily apparent: gravity holds one in a chair and electromagnetism, with the help of quantum mechanics, prevents the chair from collapsing under one's weight. As matter is probed in increasingly smaller bits, the influence of gravity is surpassed by the much stronger electromagnetic force. In a hydrogen atom, for example, the electromagnetic force between the proton and its orbiting electron is tremendously stronger than the gravitational force. At even smaller distances, at or below the size of a proton, two new forces appear. One is the strong (or color) force, which binds quarks together in a proton. The other is the weak force that (among other phenomena) gives rise to particle decays, such as the decay of a neutron into a proton, electron, and antineutrino. It is the strong, weak, and electromagnetic forces that are united in a Grand Unified Theory (GUT).

The bits of matter that feel these forces are subatomic particles called quarks and leptons. Quarks are spin-½ fermions that interact through the strong, weak, and electromagnetic forces. The proton, for example, is made up of two up quarks and one down quark. Leptons are spin-½ fermions that do not interact through the strong force. The electron is a lepton that interacts through both the weak and electromagnetic forces, whereas neutrinos are leptons that experience only the weak force. The forces themselves arise through the virtual exchange of spin-1 gauge bosons: eight gluons for the strong force, the W+, W, and Z for the weak force, and the photon for electromagnetism.

The world of particle physics does not end with merely up quarks, down quarks, electrons, and electron neutrinos. These comprise only the first of three "generations"; see Table 1. These three generations of quarks and leptons have identical couplings to the strong, weak, and electromagnetic forces. The generations differ from one another only in their masses and lifetimes. The reason for three generations and the pattern of masses remains a mystery.

The most elegant proposal to unify both the force structure as well as the quark and lepton fermion matter fields is SU(5) grand unification. SU(5) takes its name from the mathematical notation for the special unitary group of symmetry transformations with a five-component fundamental representation. It is the smallest group that incorporates the known forces SU(3) (strong), SU(2) (weak), and U(1) (hypercharge) as part of its symmetry transformations. At long distances the electroweak (weak


Matter Content of the Standard Model
Firstup quark, down quark, electron, electron neutrinoμ, d , e , υe
Secondcharm quark, strang quark, muon, muon neutrinoC , S , μ, υμ
Thirdtop quark, bottom quark, tau, tau neutrinot , b , τ, τμ
credit: Courtesy of Vernon Barger.

and hypercharge) forces dissolve into the electromagnetic force. Just as there are three colors of each quark in the fundamental representation of the SU(3) strong force, there are five varieties of fermions that comprise one matter multiplet in the fundamental representation: where the three dR's correspond to the three colors of the right-handed down quark and (eLe) is the electron and electron neutrino that make up the two components of the SU(2) left-handed lepton doublet. Here n (n̄) denotes an n -dimensional (conjugate) representation of the grand unified group. It is literally true that 5 = 3 + 2 , which means that the five-dimensional representation of SU(5) incorporates both a three-dimensional (triplet) representation of SU(3) plus a two-dimensional (doublet) representation of SU(2).

This 5̄ comprises only part of one generation of matter. The other particles of a given generation are accommodated in the next larger representation of SU(5): called the antisymmetric tensor representation. Here uL and dL are the left-handed up and down quarks, u *R is the right-handed up quark, and e *R is the right-handed electron. Each generation of quarks and leptons is reduced to just a 10 + 5̄ combination of SU(5) representations. The fact that the Standard Model is chiral (only left-handed fields feel the weak force) is embedded in the simple fact that each generation is a 10 + 5̄ and not, say, a 5 + 5̄ or 10 + 1̄0̄ . One of the fascinating consequences of embedding matter into a GUT like SU(5) is the prediction of electric charge quantization. This follows from the embedding of quarks and leptons into SU(5) representations and the requirement that the strengths of the forces are equal at some high energy scale (see below). Specifically, in terms of the proton charge e , SU(5) predicts that the electron has charge −e , the up-type quarks have charge +2e /3, the down-type quarks have charge −e /3, and the neutrinos are neutral. This is a major piece of circumstantial evidence in favor of grand unification.

The full SU(5) force comprises twenty-four gauge bosons. Only those associated with strong, weak, and electromagnetic forces have been experimentally observed. This means SU(5) cannot be an exact symmetry. Particle physicists are very much accustomed to "broken" symmetries in nature. For example, the weak force is spontaneously broken at a characteristic energy of about 100 GeV (which corresponds to a distance of about 10−19 meters), called the weak scale. Spontaneous symmetry breaking leaves the interactions unchanged but makes the force carriers massive. The result of SU(5) breaking is that twelve of its twenty-four force carriers acquire a very large mass; the remaining twelve stay massless, namely, the photon, the three weak gauge bosons, and the eight gluons. (The three weak gauge bosons acquire a mass of about 100 GeV after the electroweak symmetry is broken.) The twelve heavy gauge bosons, given the names X and Y , have masses comparable to the characteristic energy scale of the spontaneous SU(5) breaking.

If the Standard Model forces are embedded into an SU(5) GUT, then there is one parameter that characterizes the strength of the SU(5) force. At low energies, however, the strong, weak, and electromagnetic forces have quite different strengths. How is this reconciled? A consequence of the fully quantum mechanical nature of the particles and forces of the Standard Model is that the strengths of forces depend in a calculable way on the distance scale (or energy scale) at which one is probing them. The change in the strengths of the forces from experimentally measured energies (approximately 100 GeV) up to the Planck scale (2 × 1018 GeV) is shown in Figure 1. This graph assumes that there are no additional kinds of matter beyond the known quarks, leptons, and gauge bosons, except for the Higgs fields needed to break the weak SU(2) gauge symmetry. This absence of new matter (called a particle desert) is assumed to persist up to the unification scale.

Intriguingly, the gauge couplings nearly intersect around 2 × 1014 GeV. The largest uncertainty is associated with the strength of the strong force, illustrated


by the width of the shaded band in the figure. The following general picture emerges: starting from a very large energy scale, the grand unified symmetry breaks, leaving the strong, weak, and electromagnetic forces as the unbroken remnants. No relics of the unified theory are to be found because the unification scale is so high. The huge disparity between the unification scale and the weak scale allows the strengths of the forces to deviate quite significantly from their near unified value. The deviations are determined by the type and amount of matter at low energies, and remarkably the predicted deviations from a unified SU(5) lead to strong, weak, and electromagnetic force strengths not too far from their experimentally observed low-energy values.

Since both quarks and leptons are unified into GUT representations, there is no fundamental distinction between them. This means, for example, that quarks can transmute into leptons and vice versa. Such processes are mediated by the twelve heavy X and Y GUT gauge bosons. This is entirely analogous to the transmutation of an electron into an electron neutrino upon emitting a weak gauge


boson. The most striking consequence of the transmutation of quarks into leptons is that, in principle, the proton can decay!

One of the Feynman diagrams representing a proton decay process is shown in Figure 2. The general idea is that two quarks within the proton fuse into a virtual X or Y gauge boson that then promptly decomposes into an antiquark and a lepton. The third quark of the proton does not participate in the proton decay process, but it does combine with the antiquark emitted from X or Y decay to form a meson (a quark-pair bound state). In the SU(5) unified theory the dominant experimental signal for proton decay is a positron and a neutral pion. The lifetime of a proton is estimated to be

proton lifetime ≃ Thus, one would have to wait about 1020 times the age of the universe to see a single proton decay. Fortunately, this tiny decay rate can be overcome by looking at more than 1030 protons and simply waiting a few years.

Heroic experiments conducted in underground laboratories in the United States, Japan, and elsewhere have done just this. The general principle is to assemble a huge quantity of protons in the form of ultrapure water in a large underground tank that is lined with photomultiplier tubes. These detectors are installed 1 to 2 km deep underground to minimize the possibility of mistaking the collision of a high-energy cosmic ray for the decay of a proton. Radioactive impurities in the water may also lead to signals resembling proton decay, hence the use of ultrapure water. The positron from the decay of a proton is emitted at such high speed that it emits Cherenkov radiation, resulting in a characteristic cone of light that can be detected by the surrounding photomultiplier tubes.

Using this method, the tightest limits on proton decay have been set by the Super-Kamiokande experiment in Japan. They find

proton lifetime > 2.6 × 1033 years (4)

to a 95 percent confidence level. This lower limit is several orders of magnitude beyond the lifetime predicted by the SU(5) GUT! Combining this null result with the mismatch of the intersection of the gauge couplings (Figure 1) leads to the conclusion that embedding the Standard Model into SU(5) with no other matter or gauge fields is strongly disfavored by experiment.

Extrapolating the Standard Model to very high energies is also problematic for theoretical reasons. The basic difficulty is that the mass of the Higgs scalar particle needed to break the weak symmetry is extremely sensitive to and dependent on the GUT physics (this is often called the gauge hierarchy problem). The Higgs particles must also be embedded into a unified representation, which requires enlarging the number of scalar particles to include a color triplet that interacts through the strong force. These color triplet scalar particles can also lead to proton decay, and so they must have a large GUT scale mass. How this happens such that the uncolored Higgs scalars stay light remains a mystery.

The preferred solution to the gauge hierarchy problem is supersymmetry. Supersymmetry is a symmetry that relates fermions to bosons. If nature is supersymmetric, there is a supersymmetric particle ("superpartner") for every Standard Model particle, differing by ½ unit of spin. The addition of supersymmetry to the Standard Model removes the extreme sensitivity of the Higgs mass to GUT scale physics. However, like SU(5), supersymmetry cannot be an exact symmetry of nature since no superpartners


have been found. The result of breaking supersymmetry is that the masses of the supersymmetric particles are lifted above the masses of the Standard Model particles. To ensure the insensitivity to GUT scale physics is preserved, the superpartners cannot have masses too far above the weak scale.

The novel features of supersymmetry in the context of grand unification are threefold: First, the unification of the gauge couplings is much more accurate than in the Standard Model, as illustrated in Figure 3. Second, the unification scale is higher, near 2 × 1016 GeV. A direct consequence of the higher unification scale is that the rate of proton decay through X and Y GUT gauge boson exchange is about 1035 years and is thus not inconsistent with current experimental bounds. Finally, there are new contributions to proton decay that lead to completely different signals in the underground proton decay detectors. In fact, the dominant mode for proton decay in a world that is both supersymmetric as well as SU(5) grand unified is p → v̄μK+ (antimuon neutrino plus a charged kaon). Unification may also relate the masses of fermions within a generation.

Finally, for the sake of brevity, only the SU(5) unification proposal has been focused on, but there are certainly other possibilities. One particularly interesting alternative is unification into SO(10), the mathematical group of orthogonal matrices corresponding to rotations in a ten-dimensional space. Interestingly, one 16 representation of SO(10) includes an entire generation of matter fermions plus an additional particle that interacts through none of the Standard Model forces. This additional field can be naturally incorporated into an extension of the Standard Model that includes both left-handed and right-handed neutrinos. This extension is well motivated by the recent evidence for neutrino masses. SO(10) contains SU(5) as a subgroup along with an additional U(1) symmetry that may or may not survive to the weak scale.

Grand unification remains an extremely active area of frontier research in particle theory nearly thirty years after SU(5) grand unification was suggested. Some of the most recent ideas propose supersymmetric unification in extra physical or deconstructed dimensions.

See also:Family; Gauge Theory; Planck Scale; String Theory; Unified Theories


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Vernon Barger

Graham D. Kribs

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Grand Unification

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