### Crib Notes

It is difficult to get a man to understand something when his livelihood depends on him not understanding it.

— Upton Sinclair

You probably don’t want to read this post. It has an intended audience of *one* — my erstwhile coauthor, Skip Garibaldi.

Skip and I wrote a paper, last year, which proved that Garrett Lisi’s “Theory of Everything” (or any $E_8$-based variant thereof) could not yield chiral fermions (much less 3 Standard Model generations worth of fermions). Anyone with training in high energy theory *instantly* apprehends the consequence that this “theory” cannot, therefore, have anything remotely to do with the real world. Unfortunately, if your PhD is in pure mathematics (or, apparently, in hydrodynamics), this may not be immediately obvious to you.

Skip has the unenviable task of lecturing on our paper at a workshop, next week, with Garrett in attendance. (Well, OK, the workshop is in lovely Banff Alberta, so perhaps some envy is warranted.) This post is designed to help him fill in the dots. It contains only material which — to someone schooled in high energy theory — is of an embarrassingly elementary nature.

You have been warned!

Consider a gauge theory, with gauge group $G$ (a finite-dimensional compact Lie group), and fermions, $\psi$, transforming in a representation $R$ of $G$. In our conventions, $\psi$ is a left-handed Weyl fermion, ie it transforms as the $2$ of $Spin(3,1)_0 = SL(2,\mathbb{C})$. Its Hermitian conjugate, $\overline{\psi}$, is right-handed (transforms as the $\overline{2}$) and transforms in representation $\overline{R}$ of $G$.

The (gauged) kinetic term for $\psi$ and $\overline{\psi}$ uses the nondegenerate, $G$-invariant, bilinear pairing between $R$ and $\overline{R}$, which always exists. But to write down a mass term for these fermions requires something more. If $R$ is *real*, then there’s a $G$-invariant symmetric bilinear form, $(\cdot,\cdot)$, on $R$, and we can write

where we’ve combined the symmetric bilinear form on $R$ with the skew-symmetric bilinear form on the $2$, and used the fact that the fermions are anti-commuting fields.

If $R$ is pseudoreal, then it has a $G$-invariant bilinear form on it which is skew-symmetric, rather than symmetric. So we can’t write down a mass term (1). On the other hand, if $R$ is pseudoreal then $\tilde{R}=R^{\oplus 2n}$ has a real structure. So we *can* write down a mass term for a theory with fermions transforming as an even number of copies of $R$. It is frequently the case that a gauge theory, with fermions transforming as an odd number of copies of a pseudoreal representation, suffers from a global gauge anomaly, which renders it inconsistent. The simplest example is $G=SU(2)$, and $\tilde{R}$ an odd number of copies of the defining representation.

Finally, if $R$ is complex, it does not admit a nondegenerate $G$-invariant bilinear form, and we cannot write down such a gauge-invariant mass term. Such gauge theories are called *chiral*.

In a nonchiral gauge theory, the mass term (1) is always allowed, and generically one does not expect *any* fermions to survive to low energies. One needs to fine-tune $m\to 0$, to keep the fermions light.

In a chiral gauge theory, no such fine-tuning is necessary. Gauge-invariance forbids a mass term of the form (1). To give masses to the fermions in a *chiral* gauge theory *requires* that the gauge symmetry be spontaneously broken, $G\to H$, such that $R$ admits an $H$-invariant symmetric bilinear form. Specifically, there’s some scalar field, $\Phi$, transforming in some representation, $S$, and a $G$-invariant trilinear form, $S\otimes Sym^2(R)\to \mathbb{C}$. Using this trilinear form, we can write down a term in the Lagrangian

where $\lambda$ is called a Yukawa coupling constant. When $\Phi$ has a vacuum expectation-value, $\langle\Phi\rangle$ (whose stabilizer subgroup is the aforementioned $H\subset G$), this acts like a mass term, with $m\sim\lambda \langle\Phi\rangle$. Note that $\Phi$ could be an elementary scalar field (as in the conventional Higgs mechanism), or it could be a composite field (as in extended technicolour theories).

Whatever the detailed nature of $\Phi$, the essential features are

- $\langle\Phi\rangle$ is the order parameter for the gauge symmetry breaking. The masses of the broken gauge bosons are $M_W^2 \sim g^2 \langle {\Vert\Phi\Vert}^2\rangle$ where $g$ is the gauge coupling. If we fix the scale of gauge symmetry breaking, that fixes the scale of $\langle\Phi\rangle$. (In more complicated examples, where $S$ is a highly reducible representation, there are still some free parameters, but the “overall scale” of $\langle\Phi\rangle$ is fixed.)
- $\lambda$
*is*a free parameter. In principle, it could be made large. But, as we’ll see, that’s not an innocent modification.

With that as a preamble, let’s turn to the latest missive from our surfer-saviour. It’s largely devoted to the unremarkable fact that $Spin(10)\cdot SL(2,\mathbb{C})$ embeds in the quaternionic real form of $E_8$, as pointed out in Remark 8.2 of our paper. He spends the first twelve pages computing the decomposition of the adjoint of $E_{8(-24)}$ under this embedding. In the notation of our paper, the “fermions” are:
$V_{2,1} =V_{1,2} = 16 + \overline{16}$
The rest are bosons^{2}. As we see, the representation, $R= V_{2,1}$ is *real*. Explicitly, the mass term (1) looks like
$m \psi_{\overline{16}}\psi_{16} + h.c.$
If present, there are no light fermions in the theory, at all. If we tune $m\to 0$, the theory is still non-chiral, but “accidentally” has a massless generation and a massless anti-generation. These fermions are exactly massless, so long as the gauge symmetry is unbroken, but can be given masses, through terms of the form (2), once the gauge symmetry is broken.

At the end of these 12 pages, Lisi says:

Given this explicit embedding of gravity and the Standard Model inside $E_{8(-24)}$, one might wonder how to interpret the paper “There is no ‘Theory of Everything’ inside $E_8$”. In their work, Distler and Garibaldi prove that, using a direct decomposition of $E_8$, when one embeds gravity and the Standard Model in $E_8$, there are also mirror fermions. They then claim this prediction of mirror fermions (the existence of “non-chiral matter”) makes $E_8$ Theory unviable. However, since there is currently no good explanation for why any fermions have the masses they do, it is overly presumptuous to proclaim the failure of $E_8$ unification – since the detailed mechanism behind particle masses is unknown, and mirror fermions with large masses could exist in nature. Nevertheless, it was helpful of Distler and Garibaldi to emphasize the difficulty of describing the three generations of fermions, which remains an open problem.

In light of the devastating difficulties with his “theory”, “presumptuous” isn’t an adjective I would bandy about, if I were Garrett. More importantly, just because we don’t yet know the precise mechanism of electroweak symmetry breaking, does *not* mean that it’s open season to just make stuff up.

Still, let’s try to play along.

*Obviously*, one generation + one anti-generation^{1} cannot *possibly* describe the real world, where there are at least *three* generations. In that sense, Lisi’s theory is *trivially* “unviable.”

As Skip and I showed, if he switched from a noncompact real form of $E_8$ to the complex form, he could get 2 generations + 2 anti-generations. *Still* non-chiral, but perhaps closer to the goal. Let’s imagine, then, that Lisi *did* manage to find a nonchiral version of the Standard Model, with 3 generations and 3 anti-generations of Standard Model fermions. Could this be phenomenologically-viable? What are we to make of the statement “[M]irror fermions with large masses could exist in nature.” ?

What is meant here is that the analogue of (2), which gives masses to the anti-generations, could have a large $\lambda$ (much larger than the corresponding Yukawa couplings which give masses to the generations). If the anti-generations are heavy enough, they may have escaped direct detection, heretofore.

But, as I said, merely cranking up the value of $\lambda$ is not an innocent operation, devoid of side-effects. The point is that certain components of $\Phi$ *are* the longitudinal polarizations of the massive $W$ and $Z$ bosons. Via (2), the anti-generations (and, of course, the ordinary generations) contribute loop corrections to the effective action for the $W$ and $Z$ bosons. As you make $\lambda$ larger, these corrections get larger.

These so-called “oblique” electroweak corrections^{3} have been intensely studied, both theoretically and experimentally, because they are a sensitive probe of beyond-the-Standard Model physics (3 heavy anti-generations certainly counts as dramatic beyond-the-Standard Model physics!). The most relevant oblique corrections are usually parametrized by the Peskin-Takeuchi parameters, $S,T,U$.

As far as their contribution to “oblique” electroweak corrections, heavy anti-generations contribute just as extra heavy generations would. (The direct detection signals for an anti-generation would be even more dramatic than for an extra generation; they’d better be heavy enough to have evaded direct detection!) The results are well-known (see He et al for a review). The contributions to $S$ depend only logarithmically on $\lambda$. Each additional (anti-)generation contributes $\Delta S \approx \sum \frac{N_c}{6\pi}\left(1-2Y\log\tfrac{m_u^2}{m_d^2} \right)$ where, for quarks, $(N_c,Y)=(3,1/6)$ and for leptons, $(N_c,Y)=(1,-1/2)$. The contributions to $T$ grow quadratically with $\lambda$, and are proportional to the “isospin splitting” (the square of the mass difference of the up-type and down-type fermions). In the isospin-preserving limit, $(\Delta S, \Delta T)= (2/3\pi,0)$ for each (anti-)generation, independent of how heavy you try to make it.

The experimental bounds on $S$ and $T$ are very stringent, and rule out many conceivable scenarios for beyond-the-Standard Model physics. Nevertheless, theorists are clever, and there does exist a narrow window (see, for instance Kribs et al) where one can hide a *single* extra generation, in a fashion just-barely-compatible with the data.

Below is a plot (from Kribs et al), illustrating the point. Each additional (anti-)generation makes a non-negative contribution to both $S$ and $T$. In order to stay within the 95% confidence limit (even with a *single* extra generation), one has to cleverly arrange their masses so that $\Delta S\sim \Delta T$, where the uncertainties from the LEP data are the largest.

With *three* extra (anti-)generations, not even that trick would suffice. You would need some *additional* source of oblique corrections, which made sizeable *negative* contributions to $S$ and/or $T$, to cancel the contributions from the fermions. There are such sources, for instance a 2-Higgs model (in some regimes of its parameter space) can produce sizeable negative $\Delta T$. Circa 2001, it was possible to concoct a 2-Higgs model^{4} with as many as 3 (anti-)generations, compatible with the then-extant experimental bounds. Since then, the direct-search bounds from the Tevatron have tightened up, and I believe that most (if not all) of those models are ruled out. Perhaps yet-more-complicated Higgs sectors would suffice, but this is starting to get silly. While it once was reasonable to contemplate, the possibility that the world could be described by some *non-chiral* variant of the Standard Model is now exceedingly remote. Moreover, achieving such would require baroque ingredients that Lisi’s model does not possess.

Lisi’s “theory” doesn’t accommodate multiple Higgs doublets any more than it accommodates 3 generations. Which is to say that it is as dead-in-the-water as it ever was.

If you’ve read this far, I apologize for the time you wasted. But I did warn you …

^{1} Lisi prefers the more ambiguous phrase “mirror fermions.” But that phrase is used to mean many *different* things in the high energy theory literature. We’ll stick to “anti-generations” which is both more descriptive (an “anti-generation” transforms in the complex-conjugate representation of whatever representation a “generation” transforms in) and unambiguous.

^{2}
If you can’t bear to read Lisi’s 12 pages, the result is trivial, and easily summarized. In the notation of our paper, (the fields in black and blue are present in the Nesti-Percacci $Spin(11,3)$ “graviGUT”; the red and magenta ones are new to $E_8$) :
$V_{1,1} = 45 + {\color{blue} 10} + {\color{red} 1} + {\color{magenta} 10}$
are one-forms. The 45 are gauge fields for $SO(10)$; the 1 is the gauge field for a noncompact (wrong-signature) abelian gauge theory.
$V_{3,1} = V_{1,3} = 1$
are the spin connection.
$V_{2,2} = 1 + {\color{blue} 10} + {\color{red} 1}$
are would-be frame-fields (vector-valued 1-forms).

In Nesti and Percacci, the fields in blue are assumed to pair up to form a massive field. Presumably, the same could be true, in Lisi’s theory for the fields in red. (Though what dynamics could achieve this in Lisi’s case is even more murky than it was in Nesti and Percacci’s.) This would leave just a single massless frame field, which is what’s needed for gravity. More than one massless frame field would be a *disaster*.

Unfortunately, this still leaves Lisi with a massless 1-form with values in the 10 (the field in magenta). This is physically disastrous (it has unphysical, negative-norm, polarizations, but no gauge symmetry to remove them). But, since there were enough problems with the Nesti-Percacci proposal, we’ll pass over the *additional* problems of the bosonic sector of Lisi’s theory in silence.

^{3} Physically, the origin of the masses in (2) is electroweak symmetry-breaking. You don’t expect to be able to crank those masses up much higher than the electroweak symmetry-breaking scale, without paying a price. The first indications of that price are large oblique corrections.

^{4} Note that supersymmetry requires (at least) two Higgs doublets. So, in the context of supersymmetry, it’s natural to allow for the effects of additional Higgs doublets, in studying the constraints on extra generations.

## Re: Crib Notes

“Using this trilinear form, we can write down a term in the Lagrangian”

On my windows browser the equation supposed to follow the sentence above does not appear, despite stopping and refreshing the browser twice. Apologies if this displays fine on your Mac. Maybe the absence of this particular equation is just a windows issue with the math player add on.