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June 27, 2010

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 8E_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 GG (a finite-dimensional compact Lie group), and fermions, ψ\psi, transforming in a representation RR of GG. In our conventions, ψ\psi is a left-handed Weyl fermion, ie it transforms as the 22 of Spin(3,1) 0=SL(2,)Spin(3,1)_0 = SL(2,\mathbb{C}). Its Hermitian conjugate, ψ¯\overline{\psi}, is right-handed (transforms as the 2¯\overline{2}) and transforms in representation R¯\overline{R} of GG.

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

(1)m(ψ,ψ)+m¯(ψ¯,ψ¯)m (\psi, \psi) + \overline{m} (\overline{\psi},\overline{\psi})

where we’ve combined the symmetric bilinear form on RR with the skew-symmetric bilinear form on the 22, and used the fact that the fermions are anti-commuting fields.

If RR is pseudoreal, then it has a GG-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 RR is pseudoreal then R˜=R 2n\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 RR. 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)G=SU(2), and R˜\tilde{R} an odd number of copies of the defining representation.

Finally, if RR is complex, it does not admit a nondegenerate GG-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 m0m\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, GHG\to H, such that RR admits an HH-invariant symmetric bilinear form. Specifically, there’s some scalar field, Φ\Phi, transforming in some representation, SS, and a GG-invariant trilinear form, SSym 2(R)S\otimes Sym^2(R)\to \mathbb{C}. Using this trilinear form, we can write down a term in the Lagrangian

(2)λΦ(ψ,ψ)+h.c. \lambda \Phi(\psi,\psi) + h.c.

where λ\lambda is called a Yukawa coupling constant. When Φ\Phi has a vacuum expectation-value, Φ\langle\Phi\rangle (whose stabilizer subgroup is the aforementioned HGH\subset G), this acts like a mass term, with mλΦm\sim\lambda \langle\Phi\rangle. Note that Φ\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 Φ\Phi, the essential features are

  • Φ\langle\Phi\rangle is the order parameter for the gauge symmetry breaking. The masses of the broken gauge bosons are M W 2g 2Φ 2M_W^2 \sim g^2 \langle {\Vert\Phi\Vert}^2\rangle where gg is the gauge coupling. If we fix the scale of gauge symmetry breaking, that fixes the scale of Φ\langle\Phi\rangle. (In more complicated examples, where SS is a highly reducible representation, there are still some free parameters, but the “overall scale” of Φ\langle\Phi\rangle is fixed.)
  • λ\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)SL(2,)Spin(10)\cdot SL(2,\mathbb{C}) embeds in the quaternionic real form of E 8E_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)E_{8(-24)} under this embedding. In the notation of our paper, the “fermions” are: V 2,1=V 1,2=16+16¯ V_{2,1} =V_{1,2} = 16 + \overline{16} The rest are bosons2. As we see, the representation, R=V 2,1R= V_{2,1} is real. Explicitly, the mass term (1) looks like mψ 16¯ψ 16+h.c. m \psi_{\overline{16}}\psi_{16} + h.c. If present, there are no light fermions in the theory, at all. If we tune m0m\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)E_{8(-24)}, one might wonder how to interpret the paper “There is no ‘Theory of Everything’ inside E 8E_8”. In their work, Distler and Garibaldi prove that, using a direct decomposition of E 8E_8, when one embeds gravity and the Standard Model in E 8E_8, there are also mirror fermions. They then claim this prediction of mirror fermions (the existence of “non-chiral matter”) makes E 8E_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 8E_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-generation1 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 8E_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 λ\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 λ\lambda is not an innocent operation, devoid of side-effects. The point is that certain components of Φ\Phi are the longitudinal polarizations of the massive WW and ZZ bosons. Via (2), the anti-generations (and, of course, the ordinary generations) contribute loop corrections to the effective action for the WW and ZZ bosons. As you make λ\lambda larger, these corrections get larger.

These so-called “oblique” electroweak corrections3 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,US,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 SS depend only logarithmically on λ\lambda. Each additional (anti-)generation contributes ΔSN c6π(12Ylogm u 2m d 2) \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)(N_c,Y)=(3,1/6) and for leptons, (N c,Y)=(1,1/2)(N_c,Y)=(1,-1/2). The contributions to TT grow quadratically with λ\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, (ΔS,ΔT)=(2/3π,0)(\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 SS and TT 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 SS and TT. In order to stay within the 95% confidence limit (even with a single extra generation), one has to cleverly arrange their masses so that ΔSΔT\Delta S\sim \Delta T, where the uncertainties from the LEP data are the largest.

68% and 95% confidence ellipses in the S-T plane.
The ellipses are the 68% and 95% CL constraints on the (S,T)(S,T) parameters obtained by the LEP Electroweak Working Group. The shift in (S,T)(S,T) resulting from increasing the Higgs mass is shown in red. The shifts in (S,T)(S,T) from a fourth generation with various different choices of mass parameters, cleverly chosen by Kribs et al, are shown in blue.

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 SS and/or TT, 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 ΔT\Delta T. Circa 2001, it was possible to concoct a 2-Higgs model4 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)Spin(11,3) “graviGUT”; the red and magenta ones are new to E 8E_8) : V 1,1=45+10+1+10 V_{1,1} = 45 + {\color{blue} 10} + {\color{red} 1} + {\color{magenta} 10} are one-forms. The 45 are gauge fields for SO(10)SO(10); the 1 is the gauge field for a noncompact (wrong-signature) abelian gauge theory. V 3,1=V 1,3=1 V_{3,1} = V_{1,3} = 1 are the spin connection. V 2,2=1+10+1 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.

Posted by distler at June 27, 2010 9:17 PM

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13 Comments & 0 Trackbacks

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.

Posted by: nc on June 28, 2010 4:41 AM | Permalink | Reply to this

Re: Crib Notes

It works fine using iceweasel on Debian Linux so yes, it’s probably a windows issue. :)

Posted by: George on June 28, 2010 7:05 AM | Permalink | Reply to this


It strikes me as odd that MathPlayer renders equation (1), but not (2), since they are fairly similiar.

You should file a bug report.

Posted by: Jacques Distler on June 28, 2010 10:35 AM | Permalink | PGP Sig | Reply to this

Re: MathPlayer

Correct: I can’t see either equation 1 or 2 - just empty spaces - but IE8 with the MathPlayer has no problems displaying all the equations embedded within the text itself.

Regaqrding Dr Lisi’s E8 idea, isn’t there is a deep issue with dogmatic attempts to represent the known particles in their conventional SM classifications. (Apologies - and please delete - if the following idea is too wacky.)

The beta decay of leptons (muon and tauon) is analyzed in a way that’s inconsistent with quark decay. E.g., muons are supposed to beta decay into electrons via the intermediary of a W- weak boson.

For consistency with this picture, downquarks and strange quarks would need to also beta decay into electrons via the intermediary of a W- weak boson.

But the mainstream interpretation lacks this consistency in interpretation, and instead insists on seeing quark decay as the decay of quarks directly into other quarks, not requiring the intermediary of the W- weak boson (indicating decay of quarks into leptons, complimenting Cabbibo’s 1964 discovery “universality” e.g. the similarity of CKM weak interaction strengths for quarks and leptons within the same generation). For an illustration that makes the problem clear, see the first diagram in the blog post URL linked to nc.

The solution to this discrepancy also gets rid of the alleged and unexplained excess of matter over antimatter. Quarks and leptons differ in terms of having colour charge and fractional electric charge, but these differences are superficial masking effects of vacuum polarization which changes the observable electric charge of a particle. They’re not fundamental and deep properties of nature, as today assumed in the construction of the SM.

Suppose that an upquark is really a disguised electron: it’s lost 2/3rds of its electric charge due to vacuum polarization screening, and that electromagnetic energy has been transformed into strong colour charge. This is because the polarization (pulling apart) of virtual quarks by strong electric fields, which gives them potential energy and thus increases their survival time over that predicted by the uncertainty principle. So some of electromagnetic field energy gets converted by this virtual fermion polarization mechanism into the energy of gluon fields (which automatically accompany the pair-production created virtual quark-antiquark pairs). Virtual fermions which have been pulled apart by strong electric fields, using energy from the electric field in the process, both screens the electric charge and contributes to the colour charge.

Thus the difference between the total electromagnetic field energy from the fractionally chargesd downquark, and the integer charge electron, is converted into the colour charge of the downquark. This idea actually predicts that the total energy of the short-range gluon field of a downquark is precisely equal to (1 - 1/3)/(1/3) or twice the total energy of its electromagnetic field.

Hence in this model downquarks and electrons are the same thing, and are merely disguised or cloaked merely by the vacuum polarization phenomena accompanying confinement. This solves the beta decay interpretation anomaly above, and also explains the alleged problem of excess of matter over antimatter in the universe. Observe that the universe is 90% hydrogen, with one electron, one downquark, and two upquarks. If upquarks are disguised positrons and downquarks are disguised electrons, there is a perfect balance of matter and antimatter in the universe; it’s just hidden by vacuum polarization phenomena.

Posted by: nc on June 28, 2010 1:13 PM | Permalink | Reply to this

Re: MathPlayer

File a bug report about the MathPlayer issue.

As to your physics comments, they are, alas, completely wrong.

Rather than derailing the discussion, here, I would suggest you consult a basic textbook, like Griffiths to clear up your misconceptions.

Posted by: Jacques Distler on June 28, 2010 2:10 PM | Permalink | PGP Sig | Reply to this

Re: MathPlayer

Thanks for your technical analysis, Professor. I suggest you read Feynman’s book QED for the faults in the Standard Model I discussed (it’s at your level). Cheers.

Posted by: nc on June 29, 2010 11:29 AM | Permalink | Reply to this

Re: Crib Notes

Hi Jacques

Thanks for the interesting post, and the discussion of our fourth generation paper. I would agree with you that our findings are that a fourth chiral generation compatible with precision data requires some coincidences between parameters, a fifth generation is squeezed into a vanishingly small parameter space, and a sixth generation is pretty much completely out without something else to provide large contributions to the oblique parameters.

One small comment (which struck me because of your generous referencing) – my coauthor’s name is “Kribs” rather than “Kribbs”. Of course, your link to the arxiv removes the confusion, but I thought you might be interested to know.

Tim Tait

Posted by: Tim Tait on June 29, 2010 10:50 AM | Permalink | Reply to this

Re: Crib Notes

One small comment (which struck me because of your generous referencing) – my coauthor’s name is “Kribs” rather than “Kribbs”.

Fixed. Thanks.

We seem to have offices down the hall from each other, so I’ll see you at the Center.

Posted by: Jacques Distler on June 29, 2010 11:14 AM | Permalink | PGP Sig | Reply to this


Thanks for the post, Jacques! It’s luxurious to get such a detailed and custom-tailored explanation.

Posted by: Skip on July 2, 2010 6:03 AM | Permalink | Reply to this

Re: Crib Notes

What a bizarre conference. Garrett is not just in attendance, he seems to be the main star of the show with three 1 and 1/2 hour talks. Are there really so many gullible mathematicians? Are they amused by having 64x64 matrices written out explicitly? Do they like pretty colors? I don’t get it.

Posted by: Jeff Harvey on July 7, 2010 12:09 PM | Permalink | Reply to this

Re: Crib Notes

Are there really so many gullible mathematicians?

I don’t think it requires a large number of gullible mathematicians. It suffices that the 4 organizers be divided between

  • gullible
  • indifferent

Berkeley and MIT certainly have high energy theory groups, with people who talk to their colleagues in Math. One imagines that they could have straightened out two of the four organizers, had anyone bothered to pick up the phone.

I don’t get it.

What’s not to get?

Lisi was the only “physicist” invited to speak at the festschrift for Gregg Zuckerman — particularly embarrassing, given the influence of Gregg’s work in string theory.

Half of the inaugural episode of Morgan Freeman’s Through the Wormhole was devoted to Lisi.

The naïve observer(/”gullible” mathematician) might be forgiven for concluding that Garrett Lisi is a serious physicist, just like Christopher Monckton is a serious climate scientist.

Posted by: Jacques Distler on July 7, 2010 1:00 PM | Permalink | PGP Sig | Reply to this

Re: Crib Notes

I guess the E8 theory dog t-shirts also make it clear that this stuff should be taken seriously.

Posted by: Jeff Harvey on July 8, 2010 9:39 AM | Permalink | Reply to this

Re: Crib Notes

The wikipedia page on Lisi’s theory has been protected from editing for the multiple attempts to revert it to a previews version. If people here are interested they can go give their two cents in the talk page, here:

(don’t be supporters or detractors, just state things calmly and give your opinion supported with facts to help the dispute, it seems like the page is finally coming to a correct formulation and it would be good to have support by impartial physicists)


Posted by: Dan on July 2, 2011 4:46 PM | Permalink | Reply to this

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