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 to the standard model subgroup at some 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 B-L. 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.
(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.)
- 1 Breaking SU(5)
- 2 Minimal supersymmetric SU(5)
- 3 Lee Smolin's view of SU(5)
- 4 Popular culture
- 5 References
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[why?]
Under the unbroken subgroup the adjoint 24 transforms as
- (dc and l)
- (q, uc and ec)
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. Actually though, it is possible for there to be only left-handed neutrinos without any right-handed neutrinos if we could somehow introduce a tiny Majorana[disambiguation needed] 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)
|This section requires expansion. (April 2008)|
The N=1 superspace extension of 3+1 Minkowski spacetime.
N=1 SUSY over 3+1 Minkowski spacetime without R-symmetry.
Gauge symmetry group
Global internal symmetry
Z2 (matter parity)
To prevent unwanted couplings in the supersymmetric version of the model, we assign a Z2 matter parity to the chiral superfields with the matter fields having odd parity and the Higgs having even parity. This is unnecessary in the nonsupersymmetric version, but then, we can't protect the electroweak Higgs from quadratic radiative mass corrections. See hierarchy problem. In the nonsupersymmetric version the action is invariant under a similar Z2 symmetry because the matter fields are all fermionic and thus must appear in the action in pairs, while the Higgs fields are bosonic.
Those associated with the SU(5) gauge symmetry
As complex representations:
|Φ||GUT Higgs field||1||24||+|
|Hu||electroweak Higgs field||1||5||+|
|Hd||electroweak Higgs field||1||+|
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 of Nc is not zero (i.e. that a sterile neutrino exists). The coupling Hu 10i 10j has coefficients which are symmetric in i and j. The coupling NciNcj has coefficients which are symmetric in i and j. Note that the number of sterile neutrino generations need not be three, unless the SU(5) is embedded in a higher unification scheme such as SO(10).
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 constraint . So, 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 into SU(5), and respectively (the stabilizer of the VEV).
In other words, there 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.
The gauge algebra 24 decomposes as . This 24 is a real representation, so the last two terms need explanation. Both and are 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. The Higgs mechanism causes one real HALF of the and of the adjoint Higgs to be absorbed. The other real half acquires a mass coming from the D-terms. And the other three components of the adjoint Higgs, , and acquire GUT scale masses coming from self pairings of the superpotential, aΦ2+b<Φ>Φ2.
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 representations and 10 remain chiral.
It's the Higgs fields 5H and which are interesting.
The two relevant superpotential terms here are and . Unless there happens to be some fine 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. See doublet-triplet splitting problem for more details.
Lee Smolin's view of SU(5)
In his book The Trouble with Physics, Smolin states:
After some twenty-five years, we are still waiting. No protons have decayed. We have been waiting long enough to know that SU(5) grand unification is wrong. It's a beautiful idea, but one that nature seems not to have adopted. Page 64.
Indeed, it would be hard to underestimate the implications of this negative result. SU(5) is the most elegant way imaginable of unifying quarks with leptons, and it leads to a codification of the properties of the standard model in simple terms. Even after twenty-five years, I still find it stunning that SU(5) doesn't work. Page 65.
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, fulfilling Allen's blackly humorous foreshadowing of a world whose news was so baleful that the mainstream media were systematically reporting its material demise.
- Physics sometimes takes G.U.T.s, New York Times, September 19, 1982