Rabindra Mohapatra is among the phenomenologists who seem to be convinced that the detailed properties of the neutrinos, together with some previously known characteristics of charged leptons and quarks, make the grand unification, especially one based on the \(SO(10)\) gauge group, much more likely than it was before.

*Grand unification in proper physics is even grander than the LHC alarmist Luis Sancho's unification theory sketched above. ;-)*

This is the main conclusion you may deduce from his intriguing talk at the Perimeter Institute:

Neutrino Mass and Flavor Grand Unification: video (63 minutes), PDF (42 MB)On Saturday night, we had a 20-year high school class reunion which was a source of nostalgia and interesting information. I wrote a skeleton for a blog entry on it but at the end, the text looked too subjective and emotional for a blog that tries to be objective and maintain a Sheldonite image so I haven't posted it yet and chances are that I will never post it.

First of all, I don't like to reveal personal things too much. Second of all, I don't really believe that people are sufficiently interested of those things and it is not my intent to transform TRF into another blog whose descriptions of what I just cooked

*asymptotically*converge to 100% of the content.

Minutes ago, I returned from a nice afternoon talk on investments combined with a one-hour lecture on champagnes that included champagne tasting – my approximate calculation shows that the price of the champagnes I tried exceeded the price of my lunch today by the factor of \(\pi\). This was an example of a less personal event but numbers still suggest that this is not what the readers are looking for here.

Still, I have some sleeping deficit after those reunions and watching of movies around the midnight etc. and TBBT begins in one hour so I am afraid that I will postpone the writing of a richer text on the \(SO(10)\) grand unification for tomorrow. It should appear on this very URL. It's already in my head but I don't see an efficient algorithm to reprint the content of my brain onto blogger.com.

**Why grand unification is sensible**

It's Tuesday so let us begin. The Standard Model of particle physics describes three non-gravitational interactions: electromagnetism, the weak nuclear force, and the strong nuclear force. Electromagnetism came from the unification of electricity and magnetism in the 19th century. The strong nuclear force holds quarks bound together inside the protons and neutrons and the residual force is still enough to keep protons and neutrons together in the nuclei. The weak nuclear force is only know from the beta decays.

The gravitational interaction is a bit different – and much weaker than the three forces above. It is linked to the metric tensor \(g_{\mu\nu}\) describing the spacetime geometry. The spin of the resulting particles is \(j=2\), higher than \(j=1\) that the photons, gluons, W-bosons, and Z-bosons carry, and I will not discuss gravity in this text.

The three non-gravitational forces are described by extremely similar mathematics – by Yang-Mills theory. It means that the physical configurations are declared to be equivalent to configurations that are transformed by a transformation that belongs to the gauge symmetry group. To make this symmetry group local, we need fields that transform as spacetime vectors, \(A_\mu(x,y,z,t)\), as well as adjoint representations of the gauge group.

The Standard Model's gauge group is \(SU(3) \times SU(2) \times U(1)_Y\). It is a direct product of three independent factors, three independent groups. Each of them has a different fate. The first factor mixes the three colors of quarks, jokingly called "red", "green", and "blue", into each other. Gluons are the gauge bosons that communicate the resulting strong nuclear force. And this force is confined.

The remaining two factors are responsible for the weak and electromagnetic interactions; the symmetry in these two factors is broken by the Higgs field which makes the weak force a short-range one while electromagnetism remains massless and a long-range, unconfined force (that was the third fate). In particular, \(SU(2)\) is the factor that mixes the weak doublets and its gauge bosons include the W-bosons, the more important messengers of the weak nuclear force. Finally, the remaining \(U(1)_Y\) factor is the generator of the hypercharge \(Y\) which is almost the same thing as the electric charge. Well, more precisely, the hypercharge is the average electric charge in a weak \(SU(2)\) multiplet; the full electric charge is \(Q=Y+T_3\) where \(T_3\) is the third generator of the \(SU(2)\) factor.

It means that the photons mediating electromagnetism are linear superposition of the gauge bosons for the hypercharge, the B-bosons, and the neutral \(W_0=W_3\) bosons from the \(SU(2)\) factor. Similarly, the weak nuclear force includes not only the "charged currents" coupled together by the charged W-bosons but also "neutral currents" coupled by the Z-bosons. The latter are mixtures of the B-bosons of \(U(1)_Y\) and the third generator of \(SU(2)\), much like the photons, although it's a different mixture (these two are orthogonal to each other under a proper inner product).

**Are the forces unified?**

The formalisms describing these three forces are analogous – they're quantum field theories with gauge fields. However, there are three different factors in the gauge group that are independent. The groups are different i.e. not isomorphic and each of them has its separate coupling constant, a number encoding the strength of the interactions, too. We have also seen that the weak interactions and electromagnetism have to be described together, by the \(SU(2)\times U(1)_Y\) group. Both of them factors are responsible for both of the forces (electromagnetism, weak force) although the electric force is "mostly" coming from the \(U(1)\) factor (in some measure) and the most famous weak nuclear interactions – beta-decay mediated by "charged currents" – come purely from the \(SU(2)\) factor. Nevertheless, the electromagnetic force and the weak force have to be mixed and described by a single theory and they're "unified" into the electroweak force in this sense. However, the gauge group for the electroweak force still has two factors so it isn't "quite unified".

Can't Nature be controlled by a gauge group that has fewer pieces? In 1974, Shelly Glashow and Howard Georgi – whom I know rather well from Greater Boston – gave a nice answer to the question: Yes. Both of them were slapped by Mother Nature when the proton decay prediction by the simplest version of their grand unified theory (GUT) was falsified so they didn't dare to do the most assertive research of grand unification afterwards but many more courageous followers did continue. In particular, Shelly Glashow has worked on the neutrinos for years. I am sure that he must realize that neutrinos were less exciting and less profound that grand unification but a slapped Shelly simply had to be radically down-to-Earth. Paradoxically enough, the title of this blog entry indicates that the shallow waters of particle physics – neutrinos – could perhaps help us to glimpse something in the deep waters, the grand unification.

What was the gauge group they proposed? They proposed the simplest solution, \(SU(5)\). Its basic property is that it contains the Standard Model group as a subgroup: because \(2+3=5\), we have enough room for \(SU(2)\) and \(SU(3)\) inside an \(SU(5)\) and there's one "relative" \(U(1)\) over there, too. So some of the \(5^2-1=24\) gauge fields, namely exactly one-half of them, twelve, are re-used as the Standard Model gauge fields. The other gauge fields – with particles often called X or Y – are supposed to be massive. When employed as intermediate particles in propagators, they may induce proton decay so these new gauge bosons should better be very heavy.

The remaining fields of the Standard Model have to come from full \(SU(5)\) representations. For example, the electroweak Higgs doublets probably arise from a \({\bf 10}\). The bold face indicates it is a representation and the number says that it is a 10-dimensional one. You may get it as the antisymmetric product of two fundamental representations i.e. \({\bf 5}\)'s because \(5\times 4/2\times 1=10\).

What about the fermions i.e. leptons and quarks? How many 2-component spinors do we have in the theory? Well, there are three structurally identical generations so the number from one generation has to be tripled. What is the number per one generation? There are two 2-component spinors for the Dirac electron, one 2-component spinor for the Majorana neutrino, and twelve more 2-component spinors for the six Dirac spinors describing the six quark flavors. In total, there are fifteen two-component spinors. If you look at the \(SU(5)\) representations from which these may arise, you will find out that they beautifully combine into a \({\bf 5}\oplus \bar{\bf 10}\).

I don't want to go into details of the \(SU(5)\) model building but it is true that one may embed the Standard Model into the \(SU(5)\) group so that everything works. And in fact, when supersymmetry is added to the mix, the three coupling constants of the \(SU(3)\), \(SU(2)\), \(U(1)\) factors exactly (within the known error margins) unify at some energy scale – the GUT scale so that they may arise from a single coupling constant \(g\) of the \(SU(5)\) group. It's pretty much trivial for two straight lines in a plane to intersect – any pair is "automatic" – but the condition that the third line crosses the same intersection of the other two is one real nontrivial condition. And it works.

The minimal \(SU(5)\) model, in the sense of the minimization of the number of fields, is excluded because it predicts too speedy a proton decay. A fraction of the "less minimal" models is compatible with the known experimental data.

**Other groups**

\(SU(5)\) isn't the only group that may be used for grand unification. Can another group, e.g. \(E_8\), be used? The answer is No. Grand unified groups have to admit complex representations to describe the left-handed fermions. If a gauge group only admitted real representations, it would always predict left-handed and right-handed spinors that come together (i.e. have the same charges under all forces) because there's no way to correlate the handedness (i.e. \({\bf 2}\) vs \({\bar{\bf 2}}\) of \(SL(2,\CC)\)) with the gauge group representations if the latter are real.

Among the five exceptional Lie groups, \(G_2,F_4,E_6,E_7,E_8\), only \(E_6\) has complex representations. What is special about \(E_6\)? Its Dynkin diagram above is the only one that has a \(\ZZ_2\) symmetry: you may exchange the two long branches. Such a discrete symmetry actually induces an outer automorphism of the group which also acts on the representations, by exchanging them with their complex conjugate ones.

However, in heterotic string theory, heterotic M-theory, or F-theory, one may start from an \(E_8\). However, it must be broken to a smaller group by "stringy effects". When a field theory analysis starts, you must already have a gauge group with complex representations. The most common grand unified groups (with complex representations) are \(E_6\), \(SO(10)\), and \(SU(5)\). One may also talk about the flipped \(SU(5)\) models whose gauge group is \(SU(5)\times U(1)\) i.e. a \(U(5)\) group that may be embedded into an \(SO(10)\) group. The flipped \(SU(5)\) models have a different definition of the hypercharge than the \(SU(5)\) models because the extra \(U(1)\) generator (which may be found in an \(SO(10)\)) is also mixed into the hypercharge in the flipped models while this wouldn't be possible for the simple \(SU(5)\) models.

Aside from the grand unified groups, one also discusses the Pati-Salam models based on \(SU(4)\times SU(2)\times SU(2)\) and trinification based on three equal factors, \(SU(3)\times SU(3)\times SU(3)\). They're not strictly speaking grand unified theories but they are expected to have similar properties such as a high scale and the unification of their coupling constants.

The fermions of one generation in the largest \(E_6\) case transform as the fundamental representation, \({\bf 27}\). For \(SO(10)\), they're a \({\bf 16}\); yes, it is a power of two because it is the spinor representation of the orthogonal group. I've already said that the \(SU(5)\) produces fermions in \({\bf 5}\oplus \bar{\bf 10}\) which is reducible. The larger gauge groups produce irreducible representations which you may interpret as a "unification of the fermions' representations". But it's not necessary for a "grand unification"; this concept only refers to the simplicity of the gauge groups.

It may be fun to see how all these grand unified groups are embedded into an \(E_8\) group that is naturally found in heterotic string theory, heterotic M-theory, and F-theory. It has various maximum subgroups (meaning that there is no other gauge group you may insert into the chain of embeddings):\[

\begin{array}{rl}

G\subset E_8& {\bf 248}\to \dots

\\

E_6\times SU(3)& ({\bf 78},{\bf 1}) \oplus ({\bf 1},{\bf 8}) \oplus ({\bf 27},{\bf 3})\oplus (\bar{\bf 27},\bar{\bf 3})

\\

SO(10)\times SO(6)& ({\bf 45},{\bf 1})\oplus ({\bf 1},{\bf 15})\oplus\dots

\\

&\dots\oplus ({\bf 10},{\bf 6})\oplus ({\bf 16},{\bf 4}) \oplus (\bar{\bf 16},\bar{\bf 4})

\\

SU(5)\times SU(5) & ({\bf 24},{\bf 1}) \oplus ({\bf 1},{\bf 24}) \oplus ({\bf 5},{\bf 10}) \oplus \dots\\

& \dots\oplus

({\bf 10},\bar{\bf 5}) \oplus (\bar{\bf 5},\bar {\bf 10}) \oplus (\bar{\bf 10},{\bf 5})

\end{array}

\] The embeddings above show the decomposition of the fundamental representation of \(E_8\), \({\bf 248}\), which happens to be the adjoint one at the same moment. The parentheses represent the tensor product of the respective representations, \(\oplus\) is the direct sum. Just interpret the parentheses as multiplication and \(\oplus\) as addition and you will see that the resulting dimension is 248 in all cases.

**Anomaly cancellation**

You may remember that one of the nontrivial properties of the Standard Model is that all the gauge anomalies cancel. This fact is only true because both leptons and quarks are included in the spectrum; if you omitted all the quarks or all the leptons, the theory would be anomalous. And gauge anomalies are inconsistencies so the theory would be inconsistent!

These gauge anomalies arise from ultraviolet-divergent triangle Feynman diagrams but they only depend on the spectrum of massless or light particles so these anomalies may be viewed as a long-distance effect, too.

In the Standard Model, there are many terms you have to check to verify that the theory is anomaly-free. You have to see that traces\[

{\rm Tr}_\text{matter spectrum} (ABC+ACB)

\] vanish for each triplet of generators \((A,B,C)\). The three generators may be from any of the three factors of the gauge group and you have to compute the trace as the sum over various leptons and quarks so it's a lot of work. For example, if you sum \(Q^3\) over all left-handed 2-component spinors in the Standard Model, the sum equals zero. But there are many conditions of this kind. In \(SO(10)\) grand unification, things are much simpler. There is only one representation for the chiral matter (fermions), the 16-dimensional one, and all the generators look like \[

\eq{

A,B,C&= \gamma_{ab}\\

\gamma_{ab} &= \frac 12 \zav{\gamma_a\gamma_b-\gamma_b\gamma_a}

}

\] if you appreciate how the \(SO(10)\) generators act on the spinor representation. And the trace is a simple trace over the representation. So all the anomalies are proportional to \[

{\rm Tr}_{\bf 16} (\gamma_{ab}\gamma_{cd}\gamma_{ef}+\gamma_{ab}\gamma_{ef}\gamma_{cd})

\] It's very easy to verify that all such traces vanish. For the individual gamma matrices to be nonzero, we have to have \(a\neq b\), \(c\neq d\), \(e\neq f\). On the other hand, each distinct \(\gamma_a\) matrix has to appear an even number of times for the trace to be proportional to the trace of the identity, and therefore nonzero. So up to the permutations and renaming of the indices, all the possible arrangements of the indices \(abcdef\) have to be equivalent to \(ab,cd,ef=12,23,31\) but in that case, the trace vanishes simply because \(\gamma_{23}\) and \(\gamma_{31}\) anticommute with each other, so the anticommutator vanishes. That's it. The anomalies are zero.

You may also easily truncate this proof and convert it to a proof for the Standard Model. The trace vanishes for all triplets of \(SO(10)\) generators so it will still vanish if you only consider triplets of generators that belong to a subgroup: it's a special example of the calculation above. Moreover, it doesn't matter whether we trace the products of triplets of Standard Model generators over the 16-dimensional (spinor) representation or the actual 15-dimensional representation of one generation in the Standard Model: they only differ by one component, the "right-handed neutrino", which is neutral under all Standard Model generators so it contributes zero to all such traces.

**Neutrino textures**

In his talk, Mohapatra considers the \(SO(10)\) grand unified theory as the most promising example to explain the neutrino masses, including the rather large \(\sin^2(2\theta_{13})\sim 0.09\) mixing angle that was observed a few weeks ago. (A large enough size of this element of the matrix was also predicted by F-theory models, see equation 59 here.) Note that leptons and quarks arise from a single representation so the patterns of the Yukawa couplings weighting and mixing various generations affect both quarks and leptons.

In particular, one must notice that the \(SO(10)\) grand unified theories have a generator equal to \(B-L\), the difference between the baryon and lepton number. Well, its eigenvalue is equal to \(B-L\) for all the fields we know so there's no reason we shouldn't call it \(B-L\). This \(U(1)\) symmetry – trivially embedded in the \(U(5)\) inside the \(SO(10)\) – has to be broken because the corresponding gauge boson is clearly not massless or light: we don't observe any new electric-like force that would have \(B-L\) instead of \(Q\) as its charge.

Mohapatra wants to break this segment of the symmetry by a special field. The \({\bf 10}\) Higgs field used already in \(SU(5)\) etc. isn't enough. One actually needs a \({\bf 126}\). How do you get this particular representation? Consider an antisymmetric tensor with 5 indices. It has \[

\frac{10\times 9 \times 8 \times 7 \times 6}{5 \times 4 \times 3\times 2\times 1} = 252

\] components. However, one may also define a Hodge-duality and decompose this 252-dimensional representation to a part that is self-dual and another part that is anti-self-dual. Each of them has 126 components. Mohapatra doesn't "critically" depend on supersymmetry in much of his analyses. However, it's still useful for him to parameterize the physical phenomena by supersymmetric parameters.

On page 27/51, he writes a superpotential\[

W = h\psi\psi H + f \psi\psi \Delta

\] Here, \(h,f\) are coupling constants that still depend on two generational indices and may be constrained by "inter-generational" symmetries. \(H\) is the ordinary grand unified Higgs in the 10-dimensional representation while \(\Delta\) is the 126-dimensional Higgs discussed a moment ago. The parameters \(h,f\) depend on a small number of real numbers, some of them may be removed by gauge transformations, some of them have to be zero for the Standard Model gauge group to remain unbroken, and so on. When you appreciate all such things, you will find out that there is just a dozen of parameters or so. Moreover, they may be determined from the known properties of charged leptons and quarks, and all other parameters except for one in the neutrino sector may be predictions! One also has to extrapolate the properties from the GUT scale where they're given by the simple classical formulae above to the much lower LHC scale.

Mohapatra has discussed the neutrino mass matrix that emerges from such a calculation. To say the least, many of its qualitative properties may agree very well with the experimental data. The order of magnitude is right because of the seesaw mechanism that is compatible with his mechanism; there is very limited hierarchy in the neutrino mass matrix and most of the angles are rather large, including \(\theta_{13}\). The conclusions are actually in a much better shape when you compare them with the observations than what my words suggest. But you will have to listen to the talk or read his PDF file to know details; my blog entry was a kind of background you may need before you do so.

I hope that many of you have already read the PDF file or watched the one-hour talk linked at the top (the latter is especially recommended if you enjoy the Bengali accent in English).

*The degree of pedagogical detail of this blog entry was truncated relatively to the previous plans because of the preliminary discovery of the 130 GeV dark matter particle.*

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