**A good idea to get used to left-right-symmetric models**

When I listed some of the excesses seen at the LHC that the ongoing run will either confirm or disprove, the #1 bump I mentioned was the \(2\TeV\) bump of ATLAS that looks like a new \(W\)-like boson decaying to two normal electroweak bosons. The local significance was about 3.5 sigma and the global one was 2.5 sigma. Moreover, CMS saw similar (but weaker) effects at a nearby place.Sad news:Yoichiro Nambu died of heart attack on July 5th. This forefather of string theory and other things shared the 2008 Nobel prize in physics.

It seems increasingly clear that the high-energy phenomenological community actually agrees with my choice of the "most interesting bump of all". The Symmetry Magazine published by SLAC+Fermilab just printed a story

where two ladies describe the same bump. And make no mistake about it. Lots of phenomenologists also think that it is an extremely interesting bump because new papers appear on a daily basis.Something goes bump in the data

The CMS and ATLAS experiments at the LHC see something mysterious, but it’s too soon to pop the Champagne

For example, a new paper today suggests that this bump results from a new, heavy Higgs boson, either a charged one or a neutral one. And another hep-ph paper released today discusses this excess and another one which may be a hint of light supersymmetry and/or left-right-symmetric models. I am pretty sure that the number of experts who are excited by this excess is large.

Supersymmetry has been discussed many times but I think that I haven't ever written about the concept of the left-right-symmetric models. They contain an idea that is less revolutionary or far-reaching than supersymmetry, I think, but it's still remarkably cool and it is a potentially important step towards grand unification, too.

**Left-right-symmetric models: more natural than the Standard Model?**

In the Standard Model, the electromagnetic \(U(1)_{\rm em}\) group generated by the electric charge \(Q\) is embedded into a larger group, the electroweak group \(SU(2)_W\times U(1)_Y\), composed of the electroweak isospin and the hypercharge. The electric charge is written as a combination of two generators of the electroweak group\[

Q = \frac Y2 + I_{3L}

\] Quarks and leptons are described as Dirac, four-component spinors but to describe their electroweak interactions, these four-component spinors have to be divided to two two-component Weyl spinors. The left-handed and right-handed parts of the spinor (particle) have to be treated independently.

In the Standard Model, the left-handed components of the quarks and leptons (and, similarly, the right handed components of their antiparticles – produced by the Hermitian conjugate fields) transform nontrivially, as a doublet, under \(SU(2)_W\). That's why they interact with the \(W\)-bosons. The left-handed electron and the left-handed neutrino are combined into a doublet – which are therefore analogous particles (they behave exactly the same at energies much higher than the Higgs mass. However, these particles' right-handed spinorial parts (well, at least the right-handed electron: the right-handed neutrinos don't have to exist, as far as direct experimental evidence goes) have to be treated as singlets.

These doublets' and singlets' values of the hypercharge \(Y\) have to be adjusted in the right way for the value of \(Q\), given by the sum above, to be the same for the left-handed electron and the right-handed one – and similarly for the down-type and up-type quarks, too. So in effect, the Standard Model depends on the assignment of lots of independent values of \(Y\) to singlets and doublets that have the property that all (charged) quarks and leptons may be combined to full Dirac fermions. You always find pairs of Weyl spinors for which the values of \(Q\) match.

You might say that this structure of the Standard Model is contrived and the Standard Model doesn't explain why you may always construct the whole Dirac spinors with a uniform value of \(Q\). The separate treatment of the left-handed and right-handed parts of the fermions may look "contrived" to you by itself. Isn't there a more symmetric treatment that explains why the Dirac spinors exist (why the \(Q\) agrees for both parts) etc.?

Yes, there is an extension of the Standard Model that explains that, the left-right-symmetric models!

The group \(U(1)_Y\) is extended into a new \(SU(2)\) – namely \(SU(2)_R\) (right), while the original \(SU(2)_W\) (weak) is renamed as \(SU(2)_L\) (left) – and the hypercharge \(Y\) is rewritten as\[

Y = 2I_{3R} + (B-L)

\] You can see that the adjustable part of the hypercharge "became" the third component of the new, right-handed isospin \(SU(2)_R\) group. However, the original hypercharge wasn't always an integer, like \(2I_{3R}\) is, so there has to be some extra additive shift. If you combine this new "definition" of \(Y\) with the Standard Model formula for \(Q\), you get a pretty neat formula for the electric charge in the left-right-symmetric models:\[

Q = I_{3L} + I_{3R} + \frac{B-L} 2

\] The left-handed fermions are doublets under \(SU(2)_L\) but singlets under \(SU(2)_R\); it's reversed for the right-handed fermions. The list of possible values of the sum of the first two terms is therefore the same for the left-handed particles as it is for the right-handed ones. The parts of the doublet always differ by \(\Delta I_3=\pm 1\) i.e. by \(\Delta Q=1\) which is right. So the only other thing you have to verify is the additive shift. It is clearly the average value of the electric charge of the "Diracized doublets" because the average of \(I_{3L}+I_{3R}\) is zero – as for every non-Abelian group.

But that's the correct value. The electron-neutrino Diracized doublet has the average \(Q\) equal to \((-1+0)/2=-1/2\) which matches \((B-L)/2\) because \(B=0\) and \(L=1\). Similarly, the quark Diracized doublets have the average electric charge \((+2/3-1/3)/2=+1/6\) which agrees with \((B-L)/2\) because \(B=+1/3\) and \(L=0\) for quarks.

In some counting, we have simplified the assignment of charges and representations to the known quarks and leptons. Why? We have copied the tricky assignment of the \(SU(2)_W\) representation twice – both for the left-handed particles

*and*the right-handed ones. So the addition of the new \(SU(2)_R\) group hasn't added any "arbitrariness" at all. And concerning the \(U(1)\) charges, we have replaced the seemingly arbitrary assignment of many values of \(Y\) to the left-handed doublets and the right-handed singlets by the many fewer assignments of the \((B-L)\) charge generating the \(U(1)_{B-L}\) group. And only

*two values*of \(B-L\) had to be chosen – one for leptons and one for fermions!

So this left-right-symmetric models "explains" the representations and charges of all the quarks and leptons "more naturally" than the Standard Model. Moreover, the new \(U(1)_{B-L}\) factor of the gauge group that we had to add may naturally arise from grand unified theories (GUT). You just need a good model that also breaks the grand unified symmetry – and you always need some mechanism that breaks the new group \(SU(2)_R\) that the left-right-symmetric models added. That breaking has to occur at a higher energy scale than the known electroweak \(SU(2)_L=SU(2)_W\) symmetry breaking, perhaps those \(2\TeV\) or so. But it can be done. Viable models like that exist.

Just to be sure: We extended \(U(1)_Y\) into an \(SU(2)_R\) group – if we treat \((B-L)\) to be a "constant" because that group is probably broken at much higher energies than the LHC energies – which basically means that the gauge boson \(Z\) of \(U(1)_Y\) (well, it's mixed with \(I_{3L}\) and a photon, but let's identify the generator with the \(Z\)-boson) is extended to the list of three bosons, \(Z\) and \(W_R^\pm\), and the latter two might have the mass of \(2\TeV\) and they may try to emphasize their existence in the form of the ATLAS bump.

I do find it totally plausible that by the end of the year, the LHC will confirm the existence of these new particles and in 2016, people will already be saying that "it had to be obvious" that the new

*status quo*model, the left-right-symmetric model, is nicer and more natural and people including geniuses like Weinberg et al. had been stupid for almost 50 years when they avoided the self-evident left-right extensions of the Standard Model!

Stay tuned. ;-)

**Off-topic but physics:**Weyl points were finally detected – they were theorized in 1929. The lesson is the same as with other recent discoveries such as the pentaquarks. Sometimes it just takes time to see things experimentally.

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