Unity of All Elementary-Particle Forces Originally Published by:

1. Unity of All Elementary-Particle Forces • Originally Published by: • Howard Georgi and S. L. Glashow on 10 January 1974 • Presented by: Stephen Bello

2. What are the Three Elementary Particle Forces? Images courtesy of http://hyperphysics.phy-astr.gsu.edu/hbase/forces/funfor.html

3. The Marvels of SU(5)! Today I will work to show you all how the gauge group SU(5) literally runs the world. You will see that every single elementary particle force (Weak, Strong, and Electromagnetic) are merely different manifestations of the same fundamental interaction of one single coupling strength: The Fine Structure Constant! We are going to need to make an assumption as our starting point. Assume, for the moment, that the Weak and Electromagnetic forces are mediated by vector bosons in a gauge-invariant theory that includes spontaneous symmetry breaking. Weinberg Model – First proposed by Glashow as a model to describe the lepton interactions. The gauge group SU(2) x U(1) was improved by Weinberg and Salam to also include spontaneous symmetry breaking.

4. The Marvels of SU(5)! The Weinberg Model is not just limited to leptons, it can also be used to describe hadrons. It turns out that this is just one of an infinite amount of models that are compatible with the observed weak-interaction phenomenology. Now if we also assume that there are the minimal amount ofFermionfields (example being no unobserved leptons), the Weinberg Model becomes unique up to extensions of the gauge group. All observed leptons can be easily described by the following six left-handed Weyl Fields (as well as their charge conjugates): eL-, μL-, νL-, eL+, μL+, and νL+ If quarks do not mix with the gauge couplings then these six Weyl fields must transform as a representation of the gauge group. The thing is that it could be one of 23 subgroups of U(6) that contain a SU(2) x U(1) subgroup that this could be part of and still behave as they do in the Weinberg Model. We have our work cut out for us…

5. How can we include hadrons in this theory? • We can include hadrons in our theory by using what is known as the Glashow-Iliopoulos-Maiani (GIM) mechanism while also introducing a fourth quark p’ that carries charm. Decisions now need to be made on how to proceed: • Do quarks have fractional electric charge or integer charge? • Are we working with a single quartet of quarks or several quartets? • Iliopoulos and other suggested the most attractive alternative: three quartets of fractionally charged quarks. • This combination of the GIM mechanism with the notion of colored quarks keeps the quark model intact and gives the added bonus of cancelling Lepton and Hadron anomalies so that the theory of weak and electromagnetic interactions is renormalizable.

6. Our next step is to add Strong Interactions to our theory. Assume that strong interactions are mediated by an octet of neutral vector gauge gluons that are associated with local color SU(3) symmetry. Also assume there are no fundamental strongly interacting scalar-meson fields. These assumptions assure parity is conserved and the theory remains renormalizable. It turns out that the strongest binding forces are in color singlet states. This explains why observed hadrons lie in configurations of qqq and qqbar. Since strong interactions are non-Abelian they may be asymptotically free. With all this it makes sense for all three forces to spring from the gauge theory based on the group F = SU(3) x SU(2) x U(1) but there is a problem. This theory does not truly unify the weak and EM interactions. SU(2) x U(1) couplings describes two interactions with independent coupling constants. A true unification would only have one constant. What are we going to do???

7. Let’s try something else: • To make things work we have to assume the gauge group is larger than F. Suppose it takes the form of SU(3) x W where W contains SU(2) x U(1) but has its own unique gauge coupling constant. W also has to be rather simple. This form of the Weinberg model implies a relation between the coupling constants of the SU(2) and U(1) subgroups. • Since leptons are singlets under color SU(3), leptons and quarks must be in separate representations of W. If only the six observed leptons states are used, W must be one of the 23 subgroups of U(6). Out of these 23 groups the only ones that involve a single gauge coupling constant are SU(3), SU(3) x SU(3) and SU(6). • None of these can describe hadrons though since the generator does not admit fractional electric charges. Since it is also not traceless it can’t explain why the quark charge sum is not zero. • Therefore no gauge group of the form SU(3) x W can work.

8. We’ve failed once again but we now see that unifying weak and EM interactions without the strong force is impossible. The only remaining possibility to us is that the new gauge group G contains F as a subgroup but is itself simple. Leptons and quarks must lie together in the same representation of such a group with some of the gauge fields carrying lepton & quark numbers. The same coupling strength (fine-structure constant) characterizes all three kinds of interactions. This is explained by what is known as Infrared Slavery, which has to deal with color confinement. At that time no one had found an asymptotically free model where gauge symmetry is completely broken and all the vector mesons develop mass. Weinberg, Gross and Wilczek saw this and proposed something completely different – leave the gauge symmetry unbroken. While the Yang-Mills Langragian appears to describe massless vector bosons, the infrared divergences conspire to prevent their actual appearance in a physical state.

9. What is essential about a theory of strong interactions based on unbroken non-Abelian gauge symmetry is that the strength of the strong interactions no longer depends on a large coupling constant. Even though the gauge-coupling constant may be small the infrared divergences lead to interactions strong enough to keep quarks bound. So what we wanted was not asymptotic freedom but infrared slavery! Our new theory must involve a unifying gauge group G with a single coupling constant being the unit of electric charge and contains the subgroup F. To proceed our unifying group G must be of at least Rank 4. There are exactly nine rank-4 local groups that involve only one coupling strength: [SU(2)]4, [O(5)]2, [SU(3)]2, [G2]2, O(8), O(9), Sp(8), F4, and SU(5). Right away we can eliminate the first two since they do not contain SU(3). To go any further I will need to review quark and lepton behavior under F.

10. Our Weyl notation contains all fermion fields that are left-handed spinors. There are thirty such fields (the value of i says that there are three color values): • Four Leptons: (μ-, ν’, e-, ν)L • Two Antileptons: (μ+, e+)L • Twelve Quarks: (pi, pi’, λi, ni)L • Twelve Antiquarks: (pi, pi’, λi, ni)L • Within the SU(3) x SU(2) subgroup, leptons are in SU(3) singlets and SU(2) doublets, antileptons remain in singlets under both groups, quarks are in SU(3) triplets and SU(2) doublets and antiquarksare in SU(3) complex conjugate triplets but normal SU(2) singlets. • Skipping the entire heavy theoretical math, this representation is complex and not equivalent to its complex conjugate. The corresponding representation of G is the same. • Now then, of our nine candidates only [SU(3)]2 and SU(5) admit complex representations. [SU(3)]2 can be eliminated due to the weak and EM interactions so we are simply left with SU(5).

11. Our theory is entirely anomaly free! SU(5) is the only group of any rank with a 30-dimensional, anomaly free representation with the correct F content. Here’s some extra fun information… Two irreducible representations of Higgs mesons are needed to finish this up: First we need to find a multiplet that has a very large vacuum expectation value so that we can break the SU(5) symmetry back down to F. This is most simply done by using 24 real scalar-meson fields. It turns out that this process is the analog of the superstrong breaking process Weinberg discussed about SU(3) x SU(3). We also need Higgs mesons to give mass to the fermions and the weak-interaction intermediaries. Finally we have the superweak interactions and SU(3)-colored superheavy vector bosons. Besides mediating bizarre interactions they make the PROTON UNSTABLE!. Since the proton is very….very stable this vector boson must be very massive. The higgs mesons can also mediate proton decay and must also be massive.