Physics Notes by Jakob Schwichtenberg

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unification_of_gauge_symmetries

Unification of Gauge Symmetries

also known as: Grand Unified Theories (GUTs)

Which Groups?

All simple Lie group were classified by Wilhelm Killing and Élie Cartan. This means concretely, that all groups which can be used in GUTs are known.

Important groups and possible breaking chains: <diagram>

 X2 X3 X2=$SO(n)$ X3=$SU(n)$ ! ! … ! ! ! A1 A2 ! A1=$E_8$ ! ! ! ! ! ! C1 ! !  - - v - - ' ! D1 D1=$SO(10)$ ! , - '  - - - - - v - - - - ' E1 E2 E1=Pati-Salam

</diagram>

Source: Grand Unified Models by D. V. Nanopoulos In Les Arcs 1980, Proceedings, Electroweak Interactions and Unified Theories, Vol. 2*, 427-468 and CERN Geneva - TH. 2896

Problems

It is not so obvious that GUTs are interesting things to study. Some years ago, in a panel discussion, Feynman asked me what I would think about $SU(5)$ if proton decay was not observed at the predicted level. In my youthful enthusiasm, I replied that I would believe that it is right anyway. It is too pretty to be wrong. I think that I still believe that. But what I didn't see at the time was that $SU(5)$ or closely related GUTs could be right but not very interesting. If proton decay is actually observed, they become extremely interesting. But until then, apart from a few numbers which express the relations between parameters in our low energy world which follow from unification, their only connection with reality is through cosmology. Cosmology is fun, but it seems unlikely to me that we will know enough about it to extract much quantitative information about physics at very short distances, at least not anytime soon.

Effective Quantum Field Theories by H. Georgi page 457 in P. Davis (ed.) The New Physics, Cambridge University Press

existing grand unified theories suffer from a number of theoretical problems, such as the lack of an explanation for the fermion spectrum and families, naturalness problems, and the absence of quantum gravity. As attractive as grand unified theories may be, they are clearly not the complete story.

Grand Unified Theories by P. Langacker

There exist no simple or nontrivial semi-simple groups which unify both the family structure and GUT.

Do quarks and leptons know a simple group? by Yasunari Tosa

The bad news is that those people who went off to detect proton decay never found it! It became clear in the mid-1980s that the proton lifetime was at least $10^{32}$ years or so, much larger than what the SU(5) theory most naturally predicts. Of course, if one is desperate to save a beautiful theory from an ugly fact, one can resort to desperate measures. For example, one can get the SU(5) model to predict very slow proton decay by making the grand unification mass scale large. Unfortunately, then the coupling constants of the strong and electroweak forces don't match at the grand unification mass scale. This became painfully clear as better measurements of the strong coupling constant came in.

In construction the standard electroweak $SU(2) \times U(1)$ theory one must introduce a complex doublet of Higgs fields carrying the weak color charges. However, on phenomenological grounds one must not introduce their counterparts carrying strong color charge. Indeed exchange of the strongly colored Higgs particles can destabilize protons, and it leads to catastrophic rates for proton decay unless the mass of these particles is extremely large. There is no compelling understanding of why the strong-color Higgs particles are so heavy compared to their weak-color counterparts; this is one aspect of the gauge hierachy problem. (Actually what is puzzling is not so much the heaviness of the strong-color Higgs particles but the lightness of the ordinary ones.) At present it seems wise to be pragmatic and simply accept nature's unequivocal indication that this is so.

From Savas Dimopoulos, Stuart Raby, Frank Wilczek, Unification of couplings, Physics Today, October 1991

After a few years working mostly on GUTs, I was beginning to get bored with it. However, I might have gone on with what by this time had become a fashionable trend if it had not been for the Solvay conference, in Austin in 1982. I was not prepared for this conference. All the conferences I had been to before, even small theory workshops, had been about physics as an experimental science. Solvay was somehow contaminated by the Ghost of Albert Einstein. People seemed to be there to get into the picture; and to talk philosophy. The sessions were full of pompous pronouncements about theories of everything. Solvay cured me. After that conference, I started working less on GUTs and more on technicolor, composite fermions, SU(2) x U(1) breaking and the like. Whether I would have been able to resist if proton decay had been discovered, I don't know. But when the failure of the IMB and Kamiokande experiments to see proton decay ruled out the simplest SU(5) and extinguished the hope that GUTs would actually yield a lot of interesting physics any time soon, I devoted myself entirely to lower energy pursuits. To this day, I do not know what to make of SU(5). It remains a beautiful little jewel and I love it. I still hope that someday we will see proton decay and have some direct evidence that the perfect fit of the quarks and leptons into SU(5) is more than just a mathematical accident.

Grand Unified Theories by Howard Georgi in History of Original Ideas and Basic Discoveries in Particle Physics

Possible Points of Confusion

“We will only be dealing with left-handed fermion fields. Right-handed fields will be represented by the left-handed part of the charge conjugate field. (For an example of this praxis see pp 158 and 159 in Particle physics and cosmology by Collins Martin Squires. If one wishes to put different fermions into the same representation of a grand unified theory group, they must all be in the same representation of the Lorentz group (such as all LH fields) to preserve Lorentz invariance. ” Global Structure of the Standard Model by Joseph Hucks

Books and Reviews

• Grand Unified Theories by Howard Georgi in History of Original Ideas and Basic Discoveries in Particle Physics