Why a \(125\GeV\) Higgs boson isn't quite compatible with the Standard Model

David Gross made a similar point as the article below, but in a funny way, during a Lindau gathering of the Nobel prize winners.

Breaking Higgs news: A 7-minute video about the discovery of a new particle (so far) compatible with the SM Higgs that the tired but excited CMS boss Joe Incandela of Santa Barbara will be recording tomorrow (July 4th) was already leaked today, thanks to CERN's time machines. He says that they have just found enough data to be sure that it's almost certainly there and won't go away (the sentence implicitly means 5 sigma, I think).

The particle has a clear sharp peak in the diphoton channel, clear signal in the ZZ channel, inconclusive behavior in other channels, and extra tests are needed to find out whether it deviated from the Standard Model which it should. See the transcript.

Sarah Kavassalis and others were told by CERN that the European laboratory has "filmed all eventualities" in parallel universes in which there were different outcomes of the experiment. ;-) Sarah (and your humble correspondent) will only believe this explanation by CERN once they also show a video claiming that they had found Carmen Sandiego.

I was shocked by an article written for Wired whose basic point – affecting the formulation of pretty much every sentence in the text – is a downright lie. In its most concentrated form, the lie is contained in the following sentence:

Furthermore, all indications are that scientists will find that the Higgs weighs \(125\) gigaelectronvolts (\(\GeV\)) – or about 125 times more than a proton – which means that it sits exactly where the Standard Model expected it to be.

\(125\GeV\) – the expected Higgs mass plus minus one \(\GeV\) – is really 133 times the proton mass, not 125 times the proton mass, but that's just the smallest problem with the sentence above.

A remotely related poll: Imagine you're in charge of a regional science museum, let's call it Techmania ;-), and you may fight to get a LEP cavity. Would you struggle a lot? How much would you pay for it from your budget?

What's more important is what Adam Mann wrote that the value says about SUSY. In reality, \(125\GeV\) sits exactly where the Minimal Supersymmetric Standard Model allows the Higgs boson to sit but it sits outside the interval that allows the Standard Model to be a complete and consistent theory of all non-gravitational interactions in Nature.

The sentence above is exactly the opposite of the truth, it is a lie. It's partly due to Adam Mann's being a sucking journalist that may be blamed for the wrongness of the whole text; and it's partly due to his previous discussions with hardcore dishonest jerks such as one codenamed Lawrence Krauss that leads to the propagation of this kind of utter misinformation.




First, let me begin with a chart that has been posted a few times on this blog, e.g. in 2009:



Authors of the chart etc. are listed here...

The two-dimensional plane is labeled by the top-quark mass (x-axis) and the W-boson mass (y-axis). The red strip is allowed by the Standard Model (SM) assuming it is the complete theory of non-gravitational phenomena; the green strip is allowed by the Minimal Supersymmetric Standard Model (MSSM) assuming that it is the complete theory of non-gravitational phenomena. Note that the strips are comparably wide; comments about supersymmetric theories' being less predictive are just rubbish when it comes to actual observables such as the relationships between masses of known particles.




Now, the top-quark mass and the W-boson mass may be measured. The result is depicted by the blue color. The measured value tells you whether either of these theories have a chance to be a complete theory of non-gravitational forces. As you see, the blue disk falls squarely in the green, supersymmetric region. So the measured masses show that the Standard Model is nearly excluded by the mass data. Because the Standard Model red strip isn't too far from the blue disk (and the blue disk is just a depiction of a probability distribution that never strictly drops to zero), it's not quite excluded but this measurement increases the odds that supersymmetry is right – relatively to the odds that the Standard Model is right – by more than one order of magnitude. The measured data favor supersymmetry.

The precise values of the top-quark mass and the W-boson mass determine the Higgs mass according to the SM and the MSSM. And indeed, the Higgs mass was implicitly calculated when the graph above was drawn. How and why is the Higgs mass restricted in the SM and the MSSM? It has something to do with stability. Let me offer you the following two independent 1994 papers on the lower bound for the Higgs mass:

Improved Higgs Mass Stability Bound in the Standard Model and Implications for Supersymmetry (PDF) by J.A. Casas, J.R. Espinosa, M. Quiros

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Lower limit on the Higgs mass in the Standard Model : an update (PDF) by G. Altarelli, G. Isidori

When we consider the Higgs field "Mexican hat" potential energy\[

V = \frac{\lambda}{8} |h|^4 - \frac{m^2}{2}|h|^2,

\] it is not quite true that the coefficients \(\lambda,m^2\) are constant. In fact, they depend on a characteristic energy scale \(\Lambda\) we have to choose whenever we define a quantum field theory. Because effects at lower energies are similar to analogous effects at higher energies but they include extra corrections (the low-energy electron is a high-energy "core" electron dressed in the syrup of photons and electron-positron pairs, among other spices), the coupling constants "run" i.e. depend on \(\Lambda\).



Picture from Altarelli et al.

As we are increasing \(\Lambda\), the parameter \(m^2\) typically increases as well. It is dimensionful so there's no strict upper bound. However, what's more important is that the dimensionless parameter \(\lambda\) in front of the quartic term runs, too. As you go towards the right side of the picture above – energy scale about \(10^{18}\GeV\), approximately the reduced Planck scale where gravity is still very weak and a non-gravitational "theory of nearly everything" should still describe pretty much everything – we may observe \(\lambda\) to drop.

There is actually nothing special in the "running" at the point \(\lambda=0\). If \(\lambda\) decreases by some rate (slope) at tiny positive values, it will just drop to zero and then below zero, continuing by almost the same rate for quite a while. In other words, this paragraph wants to say that if there are no new particles or forces, nothing will prevent \(\lambda\) from going negative.

Usefully enough, the 1994 chart above already uses pretty much the same top-quark mass, around \(174\GeV\), which may be just \(1\) or \(2\GeV\) or so above the currently believed central value. You see that unless the Higgs mass is at least \(135\GeV\) or so, the coupling constant \(\lambda\) will go negative at some energy scale well beneath the Planck scale. For a \(125\GeV\) Higgs boson, the Standard Model will send its quartic Higgs coupling to "red numbers" at an intermediate energy scale such as \(10^{10}\GeV\), it's hard to say exactly, but it's surely below the Planck scale.

The other paper, one by Casas et al., allows slightly lighter Higgs mass than \(135\GeV\) but only by a few \(\GeV\)'s so one may still be pretty certain that \(125\GeV\) is too low. See the "Note added" right above the references at the end of the paper. There have been many other papers, including a relatively recent 2009 paper

The Probable Fate of the Standard Model by J. Ellis, J.R. Espinosa, G.F. Giudice, A. Hoecker, A. Riotto

which is an example of a paper that happened to lower the lower bound even more than Casas et al. but \(125\) and even \(126\GeV\) still looks sick although not by much.

Now, what would happen if \(\lambda\) were negative? Imagine that you describe the world of particle physics and you choose the characteristic energy scale of your theory to be \(10^{10}\GeV\) or whatever is needed for \(\lambda\) to go negative. Then the potential would have to look like\[

V = -0.01 |h|^4 - 0.1 |h|^2.

\] It's actually negatively definite! The potential is unbounded from below. So if this is the Higgs potential, the Higgs field will obviously try to roll to high values of \(h\). If you want some "good news" in the middle of the bad news about this catastrophe, the value of \(h\) will actually stabilize at some huge values, imagine \(h=500 m_W\), because there are actually some additional terms we have neglected – imagine \(|h|^6\) i.e. a non-renormalizable interaction with a tiny positive coefficient.

But if \(h=500 m_W\) is what the vacuum chooses, it's still bad. It means that small values of \(h\) – values around which we are actually expanding if we assume the ordinary "small vev" of the Higgs field in the electroweak theory – are insanely far from the vacuum that the theory would actually predict. Try to find any apologies you want; but you won't be able to show that a world with a negative quartic coupling at an energy scale is consistent with itself as well as with the known observations of the electroweak force. It's not consistent. It looks sick because it is sick.

Now, supersymmetry modifies the "slope" by which \(\lambda\) runs. It diminishes the running. The stop squarks and the higgsinos are the most important players that modify the running; see e.g. Phil Gibbs' blog for some additional comments on these issues.

At any rate, the Higgs boson at \(125\GeV\) is safely compatible with stability in the supersymmetric framework. For other reasons, SUSY actually excludes higher Higgs masses. The maximum light Higgs mass you may get in the MSSM happens to be the same \(135\GeV\) so for years, people have been saying that \(135\GeV\) is the sacred boundary between the regions favoring the Standard Model – masses above \(135\GeV\) – and those favoring the Minimal Supersymmetric Standard Model – for the Higgs masses below \(135\GeV\). There's some tiny overlap but it's so thin that I omitted it at the red-green-blue picture at the top.

Now imagine that we learn about a \(125\) or \(126\GeV\) Higgs boson tomorrow. Which way will it go, what do you think?

The precise numerical values above could be a bit different from all those papers. For example, Mikhail Shaposhnikov and Christof Wetterich could be right and \(126\GeV\) could be marginally safe. It could also be compatible with the measurements at the LHC. If that's so, the Higgs coupling could run to zero exactly at the doorway to the Planck scale physics and quantum gravity could save it at the very last moment. And this could even reinforce arguments for "asymptotic safety" even though we know it is wrong as it disagrees with some well-established properties of quantum gravity.

However, even if that happened, it would still be incorrect to say that the Higgs mass of \(125\GeV\) reinforces the Standard Model against supersymmetry – because it's very unlikely that coupling constants get "saved" in the last split second; asymptotic safety isn't quite "just the Standard Model", either; and all these wishful-thinking constructions could be realized in a supersymmetric framework and probably more naturally so. What a \(125\GeV\) Higgs mass does is exactly the opposite: it strengthens the case for supersymmetry. Adam Mann's article is a pile of lies.

Isn't he a relative of Michael Mann?



Meanwhile, the Syrian secret police invited the attractive 32-year-old Ms Sandra Bitarová, a Czech citizen working in Damascus who has a Czech mother and a Syrian father (and who seems to be an apolitical babe; her father is a moderate member of the opposition), for an interview. She hasn't returned and it's been nine days. Pictures and videos of Assad's thugs beheading children are everywhere but he's really made me even more upset today. Why do we have NATO if it can't eliminate all these Assads from the face of the Earth? They have no right to oxidize in this Solar System. All supporters of Assad, Iran and similar stuff in Syria should be neutralized and the territory should be reorganized as a protectorate of Israel. But our very societies are contaminated by the left-wing and similar people who never consider the right response to be sufficiently politically correct.