It shows the phase diagram for models of natural electroweak symmetry breaking. These models can be characterized by 2 quantum numbers:

**B**[Baroqueness], describing how complicated is the model relative to the standard model;**S**[Specialness], describing the fine-tuning needed to achieve electroweak symmetry breaking with the observed Higgs boson mass.

To allow for a fair comparison, in all models the cut-off scale is fixed to Λ=10 TeV. The standard model (SM) has, by definition,

*B=1*, while

*S≈(Λ/mZ)^2≈10^4.*The principle of naturalness postulates that

*S*should be much smaller,

*S ≲ 10*. This requires introducing new hypothetical particles and interactions, therefore inevitably increasing

*B*.

The most popular approach to reducing

*S*is by introducing supersymmetry. The minimal supersymmetric standard model (MSSM) does not make fine-tuning better than 10^3 in the bulk of its parameter space. To improve on that, one needs to introduce large A-terms (aMSSM), or R-parity breaking interactions (RPV), or an additional scalar (NMSSM). Another way to decrease

*S*is achieved in models the Higgs arises as a composite Goldstone boson of new strong interactions. Unfortunately, in all of those models,

*S*cannot be smaller than 10^2 due to phenomenological constraints from colliders. To suppress

*S*even further, one has to resort to the so-called neutral naturalness, where new particles beyond the standard model are not charged under the SU(3) color group. The twin Higgs - the simplest model of neutral naturalness - can achieve

*S*

*≈*

*10*at the cost of introducing a whole parallel mirror world.

The parametrization proposed here leads to a striking observation. While one can increase

*B*indefinitely (many examples have been proposed the literature), for a given

*S*there seems to be a minimum value of

*B*below which no models exist. In fact, the conjecture is that the product

*B*S*is bounded from below:

*BS ≳ 10^4.*

*minimum BS conjecture*is the existence of a very complicated (

*B=10^4*) yet to be discovered model with no fine-tuning at all. The take-home message is that one should always try to minimize

*BS*, even if for fundamental reasons it cannot be avoided completely ;)

## 32 comments:

How to quantify B?

[sigh] party crashers again... for any particular model you ask me and I will quantify :)

You should include on that graph a model that just has fire, air, water, and earth.

I love this plot. And I am morally certain that making it onto the BS plane is the high water mark of my career.

Jester, could you explain in a sentence to a physicist outside particle physics why Lambda is 10 TeV for the standard model? Where does that value come from?

It's arbitrary, im just assuming sm is valid up to 10 tev where it is replaced by an unspecified theory. there is nothing particular about that scale, but if i chose a higher one the plot would be less concise :)

Nathaniel, yes this is the naturalness' hall of fame :) Yuri, you dont solve the naturalness problem with just fire and water... Obviously you need sfire and swater.

Harry G Frankfurt needs a citation ?

Serious question: where does "cosmological relaxation of the weak scale" or the WGC plus a gauged U(1)B with g ~ 10^-17 fit on this plane? In those cases, S < 1 is forced by some UV physics by definition; but must you now take B ~ 1/small number?

(I don't know anything about twin Higgs, but) why did you give twin Higgs such a high B? It seems to me that if you're going to introduce new fields, then copying the known fields, and not cherry-picking them but just duplicating them all, is about the simplest thing you can do. It's fewer Ockham's Razor entities than other extensions, even if more fields.

It is rather surprising to see how many theories are minimizing BS. Based on my reading of the Second Law, a theory that minimizes BS could only be constructed by a Carnot Physicists. A Carnot Physicist would be reversible (which is contradicted by experimental observations that physicists only age in one direction in the absence of an external input of plastic surgery) and would work infinitely slowly during isothermal steps. Academic senates can indeed work* infinitely slowly, but not physicists who publish** at finite rates.

*To the extent that one can fairly call academic senate output "work".

**Though it's not clear if information is being transmitted in some publications.

I look forward to the next post in this series: "Piled Higher & Deeper: The Maximal BS Conjecture" :)

You're not tempted to claim that http://arxiv.org/abs/1504.07551 is a model that satisfies your "robust prediction" and fill in the green question mark region on your plot?

Clayton, Liam, yes I think the cosmological relaxation has the potential to fill that spot. This idea is too new to deserve a place in such a prestigious plot. But if it stands the test of time I will add it in version 2 :)

Ben, if twin Higgs was just the mirror symmetry then one would have B=10 as for the MSSM. However, you need more model building to get the right Higgs potential and keep the protection mechanism intact. Moreover, one needs to make sure that all this light stuff in the mirror sector does not affect cosmology.

Alex, the BS plane is densely populated above the line, however I focused on the most popular frameworks that typically saturate the limit. Later I may add a few more points at the risk of pissing off certain people :)

Ben, I don't think so. First, no serious physicist would ever admit taking inspiration from philosophers ;) Second, as far as i can see, Frankfurt writes about BS as a means of communication, while this post is about minimizing BS - a completely different concept :)

Jester,

I continue to remain skeptical on all current attempts on explaining naturalness via BSM models.

There is virtually no compelling evidence pointing to either MSSM, compositness, the Randall-Sundrum scenario, large extra dimensions, twin Higgs, Quantum Gravity near the electroweak scale or cosmological relaxation for that matter (axions are still hypothetical, among other things).

Time will tell, of course, but your B-S plot clearly suggests that we're missing something fundamental on the Higgs sector and what lies beyond the electroweak scale.

Would it be a correct conclusion from the plot (as it's standing now) that naturalness can only be preserved by a significant dose of BS (and cannot be without, naturally)?

That's not exactly what I meant. It's not BS but B that grows when naturalness is improved, while BS may remain constant.

So you claim BS(SM)=BS(MSSM)? I don't buy that :-)

You can't fight the implacable math :)

Great plot! I absolutely enjoy you comparing these (and hopefully more) theories in such a way!

Keep it up!

On a more serious but still somewhat philosophical note, SM itself being quite "baroque" what could be the chances (in the visible horizon, quantitatively) that next BSM extension would produce something less, not more baroque? Rather than keep reworking naturalness schemes (as I imagine ancient astronomers were updating Aristotle epicycles after every new observation) wouldn't HEP community be better off launching a comprehensive (if abstract) catalog of possible next step GUT models - and then perhaps narrowing them down against SM phenomenology?

Existing Grand Unified Theories do not really improve on B: one explains the relative values of the gauge couplings, but at the cost of introducing a new sector to break the GUT symmetry. I'm not aware of any motivated extension of the SM that decreases B. I think chances of uncovering it without any experimental hint are slim.

BS conjecture is for stuck-on-Earth particle physicists with zeptometer limited perspectives ;-).

What about a B'S' one testable with zettascale accelerators where B'=1 for a minimal extension of the SM with three Majorana right handed neutrinos, a Z' boson and a singlet Higgs for spontaneous breaking of the U(1)(B-L) symmetry? For S' the cut-off could be at the inflation scale and mZ' would stand for mZ...

Well, breakdown of a nice and shiny theory to a very messy and baroque vacuum state as we know it is rather expected to be more baroque (and messy) than clean, nice and shiny (and by the second antropic principle giants forefathers would have discovered it long time back). Guess the question here is, could there be something special about this particularly baroque state (such as e.g. the shape and stability time of universe) - or it's only Almighty's lottery ticket (and then good luck to us finding the right path to GUT without new experimental data).

Hi Jester,

all fine tuning is on couplings of different kinds, which is the way theory works - including B and S in this post:)

Previous post supposes a theory where all parameters can be computed. So it looks like coupling is just parameterizing our ignorance. Right?

So the question may be to understand if the BS > 10^4 is a property of the theoretical framework. It is what the graph suggests. Wrong?

Jacques

I guess your proposal is not that original ;-) http://arxiv.org/abs/hep-th/0503249

No, my point is more general :)

Jester, I find this post disturbing ans a little disappointing. Your conjecture is about comparing theories; in the best case, only one is valid (and possibly zero). I do not understand the interest; it looks meta-physical to me: say the "physics" of "hypothetical physics". :-)

You also state that fine-tuning is unavoidable for "fundamental reasons". Which ones? How do you prove this statement?

Yes - disturbing

after a post which deems "logical" the iperbaroque idea of mirror world, where there is a copy of the SM, similar but not exactly equal to it, quite decoupled but not completely decoupled, with a host of crazy assumptions and crazy phenomenology hidden from the experiments just enough to be around the corner.. and all that just to save a number (fine tuning)...

... after that, you promote yourself to the role of judge of theories?

at best paradoxical I'd say :D

The path from classical QM to everyday chemistry is very baroque, highly phenomenological and rarely completely calculable. We can hardly expect less from transition from Planck scale to SM. The next step of the ladder up the energy scale must be the GUT. We know that it'll be bound to have at least (three) four sectors: electroweak - Higgs, QCD, neutrino and DM. How about putting aside doom and gloom and getting to work on a comprehensive catalog of GUT models, that together with new precision accelerator results and cosmology studies could eventually (the timeline can be years or decades, e.g. solving the Great Fermat theorem took couple of centuries) may give us a hint of the next stage of HEP?

What do you think about the hypothesis that SM can actually predict self-consistently the Higgs mass: http://arxiv.org/abs/1408.0827 ?

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