# Georgi–Glashow model

# Georgi–Glashow model

In particle physics, the **Georgi–Glashow model** is a particular grand unification theory (GUT) proposed by Howard Georgi and Sheldon Glashow in 1974. In this model the standard model gauge groups SU(3) × SU(2) × U(1) are combined into a single simple gauge group—SU(5). The unified group SU(5) is then thought to be spontaneously broken into the standard model subgroup below some very high energy scale called the grand unification scale.

Since the Georgi–Glashow model combines leptons and quarks into single irreducible representations, there exist interactions which do not conserve baryon number, although they still conserve the quantum number B-L associated with the symmetry of the common representation. This yields a mechanism for proton decay, and the rate of proton decay can be predicted from the dynamics of the model. However, proton decay has not yet been observed experimentally, and the resulting lower limit on the lifetime of the proton contradicts the predictions of this model. However, the elegance of the model has led particle physicists to use it as the foundation for more complex models which yield longer proton lifetimes, particularly SO(10) in basic and SUSY variants.

(For a more elementary introduction to how the representation theory of Lie algebras are related to particle physics, see the article Particle physics and representation theory.)

This model suffers from the doublet-triplet splitting problem.

Details

Breaking SU(5)

SU(5) breaking occurs when a scalar field, analogous to the Higgs field, and transforming in the adjoint of SU(5) acquires a vacuum expectation value proportional to the weak hypercharge generator

When this occurs SU(5) is spontaneously broken to the subgroup of SU(5) commuting with the group generated by *Y*. This unbroken subgroup is just the standard model group:

Under the unbroken subgroup the adjoint **24** transforms as

giving the gauge bosons of the standard model plus the new X and Y bosons. See restricted representation.

`The standard modelquarksandleptonsfit neatly into representations of SU(5). Specifically, the left-handedfermionscombine into 3 generations of. Under the unbroken subgroup these transform as`

giving precisely the left-handed fermionic content of the standard model, where for every generation dc, uc, ec and νc stand for anti-down-type quark, anti-up-type quark, anti-down-type lepton and anti-up-type lepton, respectively, and q and l stand for quark and lepton. Note that fermions transforming as a **1** under SU(5) are now thought to be necessary because of the evidence for neutrino oscillations, unless a way is found to introduce a tiny Majorana coupling for the left-handed neutrinos.

Since the homotopy group

this model predicts 't Hooft–Polyakov monopoles.

These monopoles have quantized Y magnetic charges. Since the electromagnetic charge Q is a linear combination of some SU(2) generator with Y/2, these monopoles also have quantized magnetic charges, where by magnetic here, we mean electromagnetic magnetic charges.

Minimal supersymmetric SU(5)

Spacetime

The N=1 superspace extension of 3+1 Minkowski spacetime.

Spatial symmetry

N=1 SUSY over 3+1 Minkowski spacetime without R-symmetry.

Gauge symmetry group

SU(5)

Global internal symmetry

`(matter parity)`

Matter parity

`To prevent unwanted couplings in thesupersymmetricversion of the model, we assign amatter parityto the chiral superfields with the matter fields having odd parity and the Higgs having even parity. This is unnecessary in the non-supersymmetric version, but then, we can't protect the electroweak Higgs from quadratic radiative mass corrections. Seehierarchy problem. In the non-supersymmetric version the action is invariant under a similarsymmetry because the matter fields are allfermionicand thus must appear in the action in pairs, while the Higgs fields arebosonic.`

Vector superfields

Those associated with the SU(5) gauge symmetry

Chiral superfields

As complex representations:

label | description | multiplicity | SU(5) rep | rep |
---|---|---|---|---|

Φ | GUT Higgs field | 1 | 24 | |

H_{u} | electroweak Higgs field | 1 | 5 | |

H_{d} | electroweak Higgs field | 1 | ||

matter fields | 3 | |||

10 | matter fields | 3 | 10 | |

N^{c} | sterile neutrinos | ??? | 1 |

Superpotential

`A generic invariantrenormalizablesuperpotentialis a (complex)invariant cubic polynomial in the superfields. It is a linear combination of the following terms:`

The first column is an Abbreviation of the second column (neglecting proper normalization factors), where capital indices are SU(5) indices, and i and j are the generation indices.

`The last two rows presupposes the multiplicity ofis not zero (i.e. that asterile neutrinoexists). The couplinghas coefficients which are symmetric in`

*i*and*j*. The couplinghas coefficients which are symmetric in*i*and*j*. Note that the number ofsterile neutrinogenerationsneed not be three, unless the SU(5) is embedded in a higher unification scheme such asSO(10).Vacua

The vacua correspond to the mutual zeros of the F and D terms. Let's first look at the case where the VEVs of all the chiral fields are zero except for Φ.

The Φ sector

`The F zeros corresponds to finding the stationary points of W subject to the traceless constraintSo,where λ is a Lagrange multiplier.`

Up to an SU(5) (unitary) transformation,

`The three cases are called case I, II and III and they break the gauge symmetry intoandrespectively (the stabilizer of the VEV).`

In other words, there are at least three different superselection sections, which is typical for supersymmetric theories.

Only case III makes any phenomenological sense and so, we will focus on this case from now onwards.

It can be verified that this solution together with zero VEVs for all the other chiral multiplets is a zero of the F-terms and D-terms. The matter parity remains unbroken (right up to the TeV scale).

Decomposition

The gauge algebra **24** decomposes as

`This`

**24**is a real representation, so the last two terms need explanation. Bothandare complex representations. However, the direct sum of both representation decomposes into two irreducible real representations and we only take half of the direct sum, i.e. one of the two real irreducible copies. The first three components are left unbroken. The adjoint Higgs also has a similar decomposition, except that it is complex. TheHiggs mechanismcauses one real HALF of theandof the adjoint Higgs to be absorbed. The other real half acquires a mass coming from theD-terms. And the other three components of the adjoint Higgs,andacquire GUT scale masses coming from self pairings of the superpotential,The sterile neutrinos, if any exists, would also acquire a GUT scale Majorana mass coming from the superpotential coupling νc2.

`Because of matter parity, the matter representationsand`

**10**remain chiral.`It is the Higgs fields 5Handwhich are interesting.`

`The two relevant superpotential terms here areand. Unless there happens to be somefine tuning, we would expect both the triplet terms and the doublet terms to pair up, leaving us with no light electroweak doublets. This is in complete disagreement with phenomenology. Seedoublet-triplet splitting problemfor more details.`

Fermion masses

Proton decay in SU(5)

Unification of the Standard Model via an SU(5) group has significant phenomenological implications. Most notable of these is proton decay, which is present in SU(5) with and without supersymmetry. This is allowed by the new vector bosons introduced from the adjoint representation of SU(5), which also contains the gauge bosons of the standard model forces. Since these new gauge bosons are in (3,2)−5/6 bifundamental representations, they violated baryon and lepton number. As a result, the new operators should cause protons to decay at a rate inversely proportional to their masses. This process is called dimension 6 proton decay and is an issue for the model, since the proton is experimentally determined to have a lifetime greater than the age of the universe. This means that an SU(5) model is severely constrained by this process.

As well as these new gauge bosons, in SU(5) models the Higgs field is usually embedded in a **5** representation of the GUT group. The caveat of this is that since the Higgs field is an SU(2) doublet, the remaining part, an SU(3) triplet, must be some new field - usually called D. This new scalar would be able to generate proton decay as well and, assuming the most basic Higgs vacuum alignment, would be massless, allowing the process at very high rates.

While not an issue in the Georgi–Glashow model, a supersymmeterised SU(5) model would have additional proton decay operators due to the superpartners of the standard model fermions. The lack of detection of proton decay (in any form) brings into question the veracity of SU(5) GUTs of all types, however, while the models are highly constrained by this result, they are not in general ruled out.

Mechanism

In the lowest-order Feynman diagram corresponding to the simplest source of proton decay in SU(5), a left-handed and a right-handed up quark annihilate, yielding an X+ boson, which decays to a right-handed positron and a left-handed anti-down quark:

- .

`This process conservesweak isospin,weak hypercharge, andcolor. Note that GUTs equate anti-color with having 2 colors,, and SU(5) defines left-handed normal leptons as "white" and right-handed antileptons as "black." Note also that the first vertex only involves fermions of the`

**10**representation, while the second only involves fermions in the**5̅**, demonstrating the preservation of SU(5) symmetry.Popular culture

When the filmmaker Sandy Bates (played by Woody Allen) in the 1980 Woody Allen film *Stardust Memories* launches a depressive soliloquy with the quote, "Did anybody read on the front page of *The Times* that matter is decaying?", this was almost certainly a reference to the Georgi–Glashow model, given the film's period, the importance of the Georgi–Glashow model at the time and the many contemporary layperson articles in circulation about some of the model's most striking consequences, particularly its mechanism for proton decay. An actual *New York Times* article appeared two years later,^{[1]} fulfilling Allen's blackly humorous foreshadowing of a world whose news was so baleful that the mainstream media were systematically reporting its material demise.

## References

*New York Times*. September 19, 1982.