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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

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.

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snail feedback (25) :

reader Dilaton said...

Knowing that "everything is in the green zone" of Fig1 makes me feel very comfortable and relaxed ... ;-)

Nice article :-)

reader JollyJoker said...

That pic is a bit outdated, at least with regards to the top and W masses. I hope this pic shows ok. I put in a yellow not-so-exact ellipse with Paint using 172-175 for the top and 80.37-80.40 for the W. SUSY is still favored, but the SM doesn't look that bad.

reader Ravi said...

Hi Lubos,

I appreciate the time you take to make memos on various interesting topics and the depth to which you write in some of the memos. I think what you wrote is a fairly accepted result on the stability so may be I am missing something in my understanding.

However the question that comes to my mind is this -- since the std model is a renormalizable theory, the physics doesn't depend on the cut-off scale \Lambda (capital Lambda). The running couplings all give the same physics. If this is so then isn't the quartic coupling \lambda (small lambda) becoming negative at some scale Lambda just an artifact of perturbation theory?

--Ravi K

reader Luboš Motl said...

Dear Ravi, thanks for your interest.

Physics (at low energies) ultimately doesn't depend on Lambda. What is the status of the previous correct sentence? It is the criterion that tells you how to relate the values of coupling constants for different values of Lambda.

So it tells you that the low-energy couplings that give a light Higgs boson etc. are *equivalent* to a theory in which the potential is unbounded from below, at least in a huge vicinity of h=0. The latter is obviously inconsistent so it follows from the equivalence that the former is inconsistent, too. You can't revert the logic because the behavior of a theory at higher energies is more fundamental and the effective laws at low energies, for lower lambda, are derived.

More generally, almost all the statements referring to "artifacts of perturbation theory" are just wrong. They're denial of perturbation theory. Perturbation theory isn't something that may be denied. It's a damn accurate and systematic approximation of the full theory. If a theory is inconsistent even in the perturbative expansion, it is pretty much guaranteed to be inconsistent as an exact theory, too! The full consistency is generally more constraining condition, and not a less constraining condition, than perturbative consistency!

I've discussed similar issues e.g. here:


The context was very similar, after all.

reader Luboš Motl said...

Nope, JollyJoker! My today's chart is more current than yours. Note that your URL links to a picture that says 2008. My picture is redrawn from
http://www.pd.infn.it/~dorigo/sven_09_mwmt.jpg which is a 2009 picture by the same authors and the blue disk already sits purely in the SUSY strip over there.

I agree, the ellipse will become a line, well, finally a point. The line you're talking about is pretty much a line parallel to the strips going through the center of the blue ellipse. The line is within the MSSM strip, isn't it?

reader JollyJoker said...

But why does the top mass have an interval of 164 to 174? Is it some kind of "bare mass" that doesn't match what the PDG says?

As for the line; the earlier pic I linked has the SM area labeled as going from a Higgs mass of 114 GeV to 400 GeV. If we know the mass is 125, the SM area should become a line, probably outside the top-W mass ellipse.

reader Shannon said...

" 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" (at 126 GeV). Is it a way towards GUT ?

reader Luboš Motl said...

More or less yes, the top mass is calculated from the running mass. But you may be misinformed about what the PDG says. See top of the 2nd page here:

It does say that the mass in MS-bar from cross section measurements is 160+5-4 GeV. There are various masses over here. Of course, one wants to use pretty much high-energy values of all these parameters to link all the masses and couplings in the easiest way: the low-energy, directly measured values are "polluted" by all kinds of RG corrections.

When you say that "the line" should go outside the ellipse, then you're a victim of a circular reasoning with a wrong assumption. There are several such lines for SM and MSSM to say the least and the "main line" for a known Higgs mass is the MSSM line and it goes right through the center of the ellipse.

reader Luboš Motl said...

Dear Shannon, the authors of the "asymptotic safety" paper linked in that sentence of mine want to make the act of saving even more dramatic. I really mean that the saving would occur at the Planck scale which is (even) above the GUT scale.

Otherwise, conceptually, these proposals about the asymptotic safety and "saving near the Planck or GUT scale" have no known relationship with the grand unified theories. They seem to be completely different issues. One doesn't achieve any unification if he insists that the Higgs self-interaction "almost dies" but is saved by some high-energy physics before it does. The unification is about simplified groups and representations for matter. One doesn't seem to get any of that by setting the quartic coupling to zero.

reader Ravi said...


reader Cliff Harvey said...

Nice, I like that chart very much! This morning Im also very intrigued by these charts: http://arxiv.org/abs/1206.6888

Cheers Lubos. Maybe Ill see you around tonight/tomorrow when its time to watch the event.

reader Ravi said...

Thanks for your comments.

My response is: The theories at different \Lambda related by RGE give the same physics. So if we suspect that the theory is unstable then we should be able to see this also in the low energy theory itself -- without any reference to the high energy RGE related theory. You will pbly agree to this reasoning so far.

So this must occur at some high order in perturbation in the theory at low scale -- as we know its not unstable in lower orders.

Or else we revert this argument (and it can be reverted the way I have framed it now) -- the theory is stable and we should be able to see this also in the high energy RGE related theory at some higher order in perturbation in that theory. If \lambda (the quartic terms coupling constant) is small (say zero) in high energy theory at some scale, it may mean that other couplings like gauge, Yukawa etc will determine the stability.in higher orders in perturbation.

-- Ravi K

reader Luboš Motl said...

Good to see you here, Cliff! Yup, it's a very intriguing paper. I've just grown a bit tired by various aromas and things hidden in plain sights as they seem to be overfitting - what they're good at describing is manifestly some random noise, not signal. If this were a real 2-sigma signal, then the LHC would already have a 3-sigma signal over there by now which would be cool so I just decide to wait for that.

reader Alejandro Rivero said...

It seems that D0 has pushed a bit the mass of the top quark again. The 2012 value, 173.5 ± 0.6 ± 0.8, is very near of my vixra prediction 173.263947(6)... but, most interestingly, keeps the yukawa coupling again closer to unity than ever: 0.9965±0.0035±0.0046 A coincidence that will be left unexplained if the Higgs is confirmed to be a boring one, tomorrow.

reader Shannon said...

OK thanks Lubos. (the "saving" has some biblical résonance) ;-)

reader Luboš Motl said...

Dear Ravi, nope, you can't see the instability purely in the low-energy effective theory. The low-energy effective theory looks healthy but this health is an artifact of the low-energy approximations.

If you demand that the theory is valid, without new degrees of freedom, up to higher scales, you may prove that it's inconsistent. But this inconsistency only occurs if you require a stronger set of consistency conditions - consistency for arbitrary high-energy processes. If you only use the consistency criterion that the theory "looks OK when probed with low enough energy probes", then you don't see the problem.

The instability at a higher scale generally appears as a nonperturbative effect in a low-energy effective theory that looks stable. That would be true even for instabilities at the same scale. for example, the QED would be unstable if the fine-structure constant had the opposite sign: clumps of positively charged matter and negatively charged matter would appear everywhere in the vacuum. This sickness of the theory for alpha= – epsilon may be actually argued to be the reason why the perturbative expansion's radius of convergence is zero: it must break down already for alpha=-epsilon, even though the theory looks OK for positive alpha.

reader Ravi said...

Thanks Lubos for your thoughts. Yeah we need the stronger set of consistency conditions -- thats what I had also thought.

Generally I agree that there will be some such constraints from stability and RGE is a good way of getting at them...though I am still not convinced by the accuracy of just looking for \lambda to go negative at high scale....it is the right way to estimate but maybe we need to look for some non-pert./higher order in pert. effects even at high scale.

125 GeV seems very near the boundary for the standard model case and is intriguing....

Thanks for linking the CERN leaked video.

reader Luboš Motl said...

Ravi, when lambda is just crossing zero, it's near zero and the perturbative expansions are an extremely good guide. At even higher energy scales, when the quartic coupling gets safely negative, nothing can save the configurations with low vevs from the catastrophic instability. There's really no loophole here. Your vague suggestions that there is a loophole are nothing else than the denial of maths. Sorry.

reader Ravi said...

yeah but there are other couplings in the high energy theory that are not near zero -- it is their effect that I was talking about in higher loop order. But you may be right all these effects may be negligible and we only need to look at the sign of \lambda.

reader FX said...

Nice picture but remember that its a 1-sigma contour! Somehow misleading to give it much more than the importance it deserves. It may be the beginning of something but so far both MH_SM and MH_MSSM with heavy scalars sit well within the 2-sigma contour.

reader Physics Junkie said...

On Syria and Iran

I would be very surprised if the Syrian resistance was not getting training and/or money to buy arms by the CIA. It looks obvious to an educated American that this is the case, they are getting weapons from somewhere with Turkey as a middleman. As for Iran, there day is coming soon, if they continue with the Nuke rpgrogram. If the US does not do something, and I believe it will, then Israel definitely will. And I agree, the UN and NATO need some more balls in dealing with these penny ante dictators.

reader Old Wolf said...

Hi Lubos

It's often reported that the SM tells us everything about the Higgs boson except its mass. But you claim here that the SM does specify a Higgs mass of 135GeV or thereabouts. Can you explain the other reports?

Also, we already know the SM cannot be the whole theory of everything except gravity, as it doesn't explain neutrino masses. What effect do the neutrino masses have on plotting the green area in your graph?

reader Luboš Motl said...

Hi, I didn't write that 135 GeV or so is the SM-predicted Higgs mass. I wrote it's the lower bound. SM predicts the Higgs mass to be somewhere between 135 GeV and 800 GeV if it is supposed to be a complete theory of non-gravitational forces. I showed where the lower bound 135 GeV comes from. 800 GeV comes from the avoidance of the Landau pole, the point where lambda would diverge and the theory would also become inconsistent. This upper bound is less important today.

Every particle physicist agrees with the statement above. You must have just misunderstood them much like you misunderstood me.

Neutrino masses have no impact on these discussions at all - the're negligible and for the purpose of running above the electroweak scale and below the Planck scale, neutrinos are massless. In fact, by "Standard Model", I really meant the Standard Model with Majorana neutrino masses included.

reader Peppermint said...

Assad or Muslim Brotherhood. Just like in Egypt, it was Mubarak (who apparently was going to die anyway) or Muslim Brotherhood). I'm not sure why you want to help send "The Christians to Beirut, the Alawites to the grave"

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