Tuesday, January 13, 2015

A model that agrees with tau-mu Higgs decays and 2 other anomalies

...and its incomplete divine stringy incarnation...

I originally missed a hep-ph preprint almost a week ago,
Explaining \(h\to \mu^\pm \tau^\mp\), \(B\to K^*\mu^+\mu^-\), and \(B\to K\mu^+\mu^-/B\to Ke^+e^−\) in a two-Higgs-doublet model with gauged \(L_\mu−L_\tau\)
by Crivellin, D'Ambrosio, and Heeck, probably because it had such a repulsively boring title. By the way, do you agree with the hype saying that the new Mathjax 2.5 beta is loading 30-40 percent faster than Mathjax 2.4 that was used on this blog up to yesterday morning?

The title of the preprint is uninspiring even though it contains all the good stuff. Less is sometimes more. At any rate, CMS recently reported a 2.4-sigma excess in the search for the decays of the Higgs boson\[

h\to \mu^\pm \tau^\mp

\] which is flavor-violating. A muon plus an antitau; or an antimuon plus a tau. Bizarre. The 2.4-sigma excess corresponds to the claim that about 1% of the Higgs bosons decay in this weird way! Correct me if I am wrong but I think that this excess has only been discussed in the comment section of this blog but I was very excited about it in July.

Aside from this flavor-violating hint, the LHCb experiment has reported several anomalies and the two most famous ones may be explained by the model promoted by this paper. One of them was discussed on TRF repeatedly:

The \(B\)-mesons may decay to \(K\)-mesons plus a charged lepton pair, \(\ell^+\ell^-\), and the processes with \(\ell=e\) and \(\ell=\mu\) should be almost equally frequent according to the Standard Model but LHCb seems to see a difference between the electron-producing and muon-producing processes. The significance of the signal is 2.6 sigma.

The final, third deviation is seen by LHCb, too. The rate of the \(B\) decay to an off-shell \(K^*\) along with the muon-antimuon pair, \(\mu^+\mu^-\), seems to deviate from the Standard Model by 2-3 sigma, too.

Each of these three anomalies is significant approximately at the 2.5-sigma level and they seem to have something in common. The second generation – muons – is treated a bit differently. It doesn't seem to be just another copy of the first generation (or the third generation).

The model by the CERN-Naples-Brussels team claims to be compatible with all these three anomalies. Within this model, the three anomalies are no longer independent from each other – which may strengthen your belief that they are not just flukes that will go away.

If you were willing to oversimplify just a little bit, you could argue that these three anomalies are showing "almost the same thing" so you may add these excesses in the Pythagorean way. And \(\sqrt{3}\times 2.5 \approx 4.3\). With this optimistic interpretation, we may be approaching a 5-sigma excess. ;-)

These three physicists construct a model. It is a two-Higgs-doublet model (2HDM). The number of Higgs doublets is doubled relatively to the Standard Model – to yield the spectrum we know from minimal SUSY. But 2HDM is meant to be a more general model of the Higgs sector, a model ignoring the constraints on the parameters that are implied by supersymmetry. (But it is also a more special model because it ignores or decouples all the other superpartners.)

And there's one special new feature that they need before they explain the anomalies. Normally, the lepton number \(L\) – and especially the three generation-specific lepton numbers \(L_e,L_\mu,L_\tau\) – are (approximate?) global symmetries. But these three folks promote one particular combination, namely the difference \(L_\mu-L_\tau\), to a gauge symmetry – one that is spontaneously broken by a scalar field.

This gauging of the symmetry adds a new spin-one boson, \(Z'\), which has some mass, and right-handed neutrinos acquire some Majorana masses because of that, too. These new elementary particles and interactions also influence the processes such as the decays of the Higgs bosons and \(B\)-mesons – those we encountered in the anomalies.

What I find particularly attractive is that the gauging of \(L_\mu-L_\tau\) may support an old crazy \(E_8\) idea of mine. It is a well-known fact that the adjoint (in this case also fundamental) representation \({\bf 248}\) of the exceptional Lie group \(E_8\) decomposes under the maximal \(E_6\times SU(3)\) subgroup as\[

{\bf 248} = ({\bf 78},{\bf 1}) + ({\bf 1},{\bf 8}) + ({\bf 27},{\bf 3}) + ({\bf \bar{27}},{\bf \bar 3})

\] It is the direct sum of the adjoint representations of the subgroup's factors; and of the tensor product of the fundamental representations (plus the complex conjugate representation: note that \(E_6\) is the only simple exceptional Lie group that has complex representations).

If you use \(E_6\) or its subgroup as a grand unified group, the representation \({\bf 27}\) produces one generation of quarks and leptons. It works but what is very cool is that the decomposition of the representation of \(E_8\) seems to automatically produce three copies of the representation \({\bf 27}\).

It almost looks like if the \(E_8\) group were predicting three generations. The three generations may be complex-rotated by the \(SU(3)_g\) group, the centralizer of the grand unified group \(E_6\) within the \(E_8\) group. Isn't it cool? I added the \(g\) subscript for "generational".

A problem with this cute story is that the most natural stringy reincarnation of this \(E_8\) picture, the \(E_8\times E_8\) heterotic string theory (or its strongly coupled limit, the Hořava-Witten heterotic M-theory) doesn't normally support this way of counting the generations. Recall that in 1985, this became the first realistic embedding of the Standard Model (and SUSY and grand unification, not to mention gravity) within string theory. But the number of generations is usually written as \(N_g=|\chi|/2\), one-half of the Euler characteristic of the Calabi-Yau manifold. The latter constant may be anything. All traces of the special role of \(3\) are eliminated, and so on. A related defect is that the rest of the \(E_8\) group outside \(E_6\) is broken "by the compactification" which is a "stringy effect" so no four-dimensional effective field theory description ever sees the other \(E_8\) gauge bosons – except for the GUT \(E_6\) gauge bosons.

But from a different perspective, there could still be something special about the three generations – due to some effective, approximate, or local restoration of the whole \(E_8\) symmetry. The simplest heterotic compactifications identify the field strength in the \(SU(3)_E\) part of the gauge group – a subgroup of \(E_8\) – with the field strength in the gravitational \(SU(3)_{CY}\) holonomy – this \(SU(3)_{CY}\) is a subgroup of \(SO(6)\) rotating the six Calabi-Yau dimensions.

The grand unified group is only an \(E_6\) or smaller because it's the centralizer of \(SU(3)_g\) within \(E_8\). And I had to take the centralizer of \(SU(3)_g\) because that's the components of the field strength that break the gauge group in \(d=10\) spacetime dimensions. Perhaps, we should think that this field strength – or some of its components – are "small" in magnitude, so that one generator of this \(SU(3)_g\), and \(L_\mu-L_\tau\) is indeed one generator of \(SU(3)_g\) if \((e,\mu,\tau)\) are interpreted as the fundamental triplet of \(SU(3)_g\), is "much less broken" than others.

If the relevant component of the field strength may be considered "small" in this sense, it could be possible to organize the fermionic spectrum into the part of the \(E_8\) multiplet. And one should find some field-theoretical \(Z'\) boson responsible for the spontaneous breaking of this generator of the generational \(SU(3)_g\).

As you can see, if the heterotic models may be formulated in a slightly special, unorthodox, outside-the-box way (and yes, it's a somewhat big "if"), one may have a natural stringy model that achieves "more than grand" unification, explains why there are three generations of fermions, and accounts for three so far weak anomalies observed by CMS and LHCb (which will dramatically strengthen in a few months if they are real).

Hat tip: Tommaso Dorigo


  1. Great stuff.

    Unification is why physics is beautiful to me. You explained it perfectly by something you said elsewhere on this blog ('If someone is not impressed by the fact that a formula . . . can explain a large number of physical situations, including chemistry and animals, as well as the sunset, she can never understand why the physicists think that string theory is beautiful'). But what worries and annoys me is that we don't know where the buck of unification ultimately stops—and perhaps never will due to either the limits of consciousness or of science or of both.

    Is it possible that the universe unascertainably unifies endlessly and thereby it's rather pointless to be excited about the (practically useless) unification that we have managed to surprisingly ascertain so far? Yes, and that sucks. In that case, spending one's life finding deeper and deeper unification becomes simply silly. That would be like spending one's life counting from 1 onwards: an endless and pointless task.

    And yet to assume that the universe does end in a grand synthesis that a sufficiently evolved consciousness can fully grasp seems to presuppose the importance of consciousness, which seems rather unlikely due to how careless and indifferent nature is with human life and consciousness.

    I emotionally want the unification to end in a final grand synthesis—with some kind of awesome explanation behind everything and such an explanation that a sufficiently evolved consciousness can grasp.

  2. Absolutely yes about the faster load of Mathjax. Also seems as there is an intermediate state, less to 2 seconds, during which you can already see the symbols rendered, but, say, not in their final state, with other font and/or highlighted. Seems a big improvement.

  3. Lubos, if we want to think about possible realizations of your idea, does the following make sense as a starting point: an E8 x E8 superfield (with susy broken), with SM fermions coming from one E8 and SM bosons coming from the other E8? And then you could try to obtain this as a heterotic EFT?

  4. Thanks for the happy feedback!

    It's really 2.5 beta but I have never experienced the tiniest problem with the betas - they are too conservative in releasing the new versions.

  5. LOL,nope, both E8 factors have bosons and both E8 factors have fermions. Spin is independent from "which E8".

  6. Yes, I know both E8 factors have fermions and bosons (I did say superfields)... I'm just saying, if we accept your idea regarding the origin of the SM fermions, where do the SM gauge bosons come from? And I'm thinking they'd come from the other E8 factor.

  7. Right you are, Shannon. I have had the honor of being invited into many Muslim homes and have found these homes to be considerably warmer and more welcoming than most Christian homes. Over the last sixty years I have known scores of muslims, all of whom were reasonable, trustworthy people, not really different from non-Muslims.
    The problem is Muslim fundamentalists but there are at least few million of them if you count all of the Wahabi followers. They are, of course, a real threat.

  8. Seconded - it's *much* better! I'm also seeing it go through the same funny "pre-rendered" state with the symbols rendered but not yet shown in the final font, so I'm pretty sure we're seeing the same thing. It's not really annoying and the whole experience is way smoother.

  9. I did address the point. Most Muslims do not interpret their religion in the way that you do and are just as peace loving as their Christian counterparts. You try to claim that the acts of a few idiots represents their entire culture but that is not the way I see it or the way that most people see it. Terrorists should be dealt with no matter what they believe. But they are responsible for their own actions and there is no such thing as collective guilt.

  10. Aren't you missing the point here? I have made it clear that all who initiate violence against other people should be held accountable. But I note that includes the individuals who decide to bomb civilians and the individuals who choose to obey their orders. A drone operator that kills innocent kids at a funeral or wedding is no better than the two idiots who shot up the magazine. Why aren't we hunting down the drone operators too?

  11. Ever since the unexpected success of electroweak unification, physicists have been trying to force fit spontaneous symmetry breaking and large broken symmetries everywhere. GUTs are just one glaring symptom of this disease. But does everything have to be explained by a symmetry? It's easy to postulate extra broken symmetries; any lack of observational evidence for them can always be explained away by pushing away the symmetry breaking to inaccessibly high energies.

  12. Rage is just a symptom of a loss of control and in the real world it is far better to maintain control and to be reasonable and logical. All we should do is to go after the individuals who initiate violence against other individuals and kill innocent people no matter what 'side' they are on. What those two idiots did is clearly wrong. But so is bombing civilians in Serbia, Syria, Pakistan, Libya, Iraq, or anywhere else. The cowards who operate those drones and shoot missiles into markets or wedding party need to be jailed too.

  13. "I did address the point."

    No, you bloody well did NOT.

    You're wriggling like a worm on a hook, or being extraordinarily obtuse.

    Here again is Philip Weisler's statement that you quoted and purported to address, but didn't:
    "What has never ceased to astonish me is the immense ethical perversity of Islam, its Prophet and its Idol Allah."

    Now you SPECIFICALLY need to address "the immense ethical perversity of Islam, its Prophet and its Idol Allah". I'll make it really simple for you: the subject is the immense ethical perversity of these things and NOT what pisslam's card-carrying members had for lunch or anything else they did or didn't do, or do or don't do, or will or won't do.

    Look at it this way. A similar kind of claim could be made about communism or Naziism. What would you think of someone who responded to that claim by saying nothing whatsoever about the respective dogmas involved but rambled off on irrelevancies about the virtues the workers who lived under and accepted them?

    Got it now?

    "You try to claim that the acts of a few idiots represents their entire culture..."

    You fucking LIAR. I made no such claim and made no attempt to either.

    Their "entire culture", if one could dignify it with such a classification—which one cannot, does however entail such acts. And that's precisely why I agree with Philip's judgement on it, though he does rather understate its vileness.

  14. Hmm, not a provocation, but deserves a reply---the question isn't isomorphic. When someone threatens to exterminate you and your country, the average citizen is much more likely to believe that violence is justified
    than if someone draws a cartoon satirizing a non-existent being.

  15. The main difference is their IQs...

  16. You asked a question, "And rage against what?" I think I answered it. The rest of my post was in the nature of additional considerations. I don't find your arguments equating collateral damage to terrorism convincing, so those I did ignore in their entirety.

  17. We have a Nazi party but it stands on the boundary between politics and crime.

    We have laws banning the promotion of fascism because it's a path to crippling of millions of human lives. No known way how to use 2+2=5 to cripple human lives is known yet.

  18. Hi thanks for stating this explicitly.

    It's true within QFT. But I am not really proposing that the whole SU(3) family symmetry has to be unbroken at some scale in field theory. It's possible to make compactifications where E6 times U(1)_{L_mu - L_tau} is unbroken, and the rest is broken at any scale according to field theory.

  19. Not everything has to be explained by symmetry but it is a *possibility* that many more things are explained by symmetry and using a concept that has worked elsewhere again is an example of Occam's razor, I would say. At any rate, the claim that it is a "disease" is nothing else than irrational propaganda lacking any justification. It is an intriguing *possibility*, not a "disease".

    I personally think that some grand unification is more likely than not.

  20. Dear Mitchell, everything in the Standard Model has to be charged under the same E6 - or E8. There is one electric charge, not one separate electric charge for bosons and one for fermions, for example. What you write is a deep misunderstanding of elementary particle physics.

  21. Do you think such laws should be introduced also in the USA in order to prevent parties like American Nazi Party from crippling of millions of human lives? Or could be denial of holocaust refuted in the same way as 2+2=5?

  22. Gordon, don't be silly, to "exterminate" a population you need way more than just a suicide bombing. You need at least something like... cutting electricity and water supply... https://www.middleeastmonitor.com/news/middle-east/10131-israel-cuts-off-water-supply-to-45000-palestinians

  23. Or rather the use they make of their IQ ;-)

  24. It's Americans' business, and not mine, to decide about their laws.

    I am just aware of some facts - one of them is the European experience and the other one is that the American feeling that they are completely different is a form of stupid unjustified exceptionalism.

    Various freedom-suppressing ideologies have already suppressed freedom in the U.S. markedly - they could because it was totally kosher for them to do so.

  25. Some holocausts are denied as holocausts but rather defined as "ethnic cleansing", which anyone can deny.

  26. Ugh, I was thinking that the E8s act on each other somehow. But no, the first E8 is a singlet under the second E8, and vice versa... See, I was also thinking that there isn't "room" in a single E8 superfield for both the SM fermions and the SM gauge bosons, on the basis of some intuition about susy breaking... that heavy superpartners of the fermionic 27s would break too much of E8. That's why I assumed that the other E8 must be involved somehow. But it was just a guess, I should try to check. Anyway, thanks for your (im)patience.

  27. Dear Mitchell, right, the reps of one E8 are singlets under the other, and vice versa.

    Things charged under one E8 don't attract/repel things charged under the other one at all.

    Whether something is a boson or a fermion is completely unrelated to the E8 charges, one or another - spin is a different quantum number than E8 charges.

    In SUSY theories, for every choice of E8 x E8 charges, there exist both bosons and fermions with the same charges. Superpartners.

    In E8 x E8 model building, only one E8 is being used for the GUT/Standard Model. The other one is passive - at most, the gluinos in the shadow other E8 may be used to break SUSY by gluino condensates.

  28. Let me be clear. I do not see Islam as ethically perverse because I do not see the ancient writings to be reflective of the true beliefs and behaviors of most Muslims. I see the few terrorists as outliers,

    Note that it has been Christians who have been engaging in mass genocide over the past century. Hitler, Leopold, and Stalin were not representatives of Islam. I don't know about you but I don't consider those Christians as representatives of the rest.

  29. Argh, I can't read. It's right there in your post, that the 248 decomposes into E6 adjoint plus SU(3) adjoint plus three generations plus three anti-generations. So even if we remove from consideration the E8 gauge bosons that are superpartners of the SM fermions, as having been made massive by susy breaking, the E6 subgroup will remain untouched. So much for my "intuition".

    I found Stephen Adler on arxiv, discussing the real phenomenological prospects for supersymmetric E8 gauge theory:

  30. Apologies, Mitchell, I don't know what to do with verbal ejaculations such as "E6 adjoint plus SU(3) plus three generations" etc. You are adding apples with oranges - in fact, your "addition" is much worse than adding apples and oranges because apples and oranges are at least fruits.

    The subgroup of E8 is E6 *times* SU(3), and there's no way to "extend" the group so that it has three generations (the generations and the groups are completely different objects), so "and three generation" is a completely meaningless sequence of words.

    So even if we remove from consideration the E8 gauge bosons that are superpartners of the SM fermions

  31. "E6 adjoint plus..." refers to the gauginos, not the gauge bosons. The "plus" is a direct sum, sorry if that informality wasn't clear.

    And since the SM generations here are a subset of the gauginos, clearly they do have superpartners, that are some subset of the *E8* gauge bosons. In fact, it looks like the superpartners of the three generations and anti-generations are precisely the E8 gauge bosons that become heavy when E8 is broken to E6 x SU(3)g.

  32. It doesn't matter what version of MathJax you use on your blog. This site is so full of crap that takes forever to load, it doesn't matter.

  33. GUTs have more freely adjustable parameters than the Standard Model. Clearly, Occam's razor doesn't apply.

  34. Or more likely, you are not using the razor in a safe way. Better stop it before you cut your throat.

  35. SU(5) is a bit more complicated than the Standard Model. SO(10) is significantly more complicated. E6 and E8 are plain baroque. Just count the number of different Higgs representations needed, and the number of exotic fermions, not to mention all the fine-tuned relations needed between the couplings.

    TeV scale SUSY breaking requires additional ad hoc assumptions like R-parity and a suppression of flavor changing neutral current couplings.

  36. Could you please stop posting these 100% wrong crackpot-like delusions on my blog? Why don't you go somewhere where this junk will be appreciated? The Internet contains hundreds of such websites.

    SU(5) and SO(10) are simpler groups than SU(3) x SU(2) x U(1), that's also why they are technically known as "simple groups" in group theory (mathematics) and why the theories based on these simple gauge groups are known as "grand unification". If you don't understand these things, you are a moron and I would find it appropriate if not necessary for you to stop pretending that you are something else.

  37. Selection bias is at work here. Beyond MSSM model builders mostly choose to focus only on models respecting R-parity. So, a cursory look at popular models can easily lead one to think R-symmetry is fairly generic. It's not.

    There are many models where flavor changing neutral currents are naturally suppressed, but there are also many other models where they aren't.

    So, R-parity and FCNC suppression are indeed additional ad hoc assumptions. Ad hoc as in "after the fact".

    The strong anthropic principle can explain the hierarchy problem naturally without any need for TeV scale SUSY breaking.

    A simple Lie group is one whose Lie algebra has no proper ideal. That's not the same thing as "less complicated". The complexity of a GUT is judged not just by its gauge group, but also by its number of Higgs and fermion representations, and the baroque constraints on the permissible couplings.